Effective quantum theories with short- and long-range forces

Transcription

1 Effective quantum theories with short- and ong-range forces Dissertation zur Erangung des Doktorgrades Dr. rer. nat.) der Mathematisch-Naturwissenschaftichen Fakutät der Rheinischen Friedrich-Wihems-Universität Bonn vorgeegt von Sebastian König aus Northeim Bonn, August 2013

2 Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftichen Fakutät der Rheinischen Friedrich-Wihems-Universität Bonn 1. Gutachter: Prof. Dr. Hans-Werner Hammer 2. Gutachter: Prof. Dr. Uf-G. Meißner Tag der Promotion: 23. Oktober 2013 Erscheinungsjahr: 2013

3 Abstract At ow energies, nonreativistic quantum systems are essentiay governed by their wave functions at arge distances. For this reason, it is possibe to describe a wide range of phenomena with short- or even finite-range interactions. In this thesis, we discuss severa topics in connection with such an effective description and consider, in particuar, modifications introduced by the presence of additiona ong-range potentias. In the first part we derive genera resuts for the mass binding energy) shift of bound states with anguar momentum 1 in a periodic cubic box in two and three spatia dimensions. Our resuts have appications to attice simuations of hadronic moecues, hao nucei, and Feshbach moecues. The sign of the mass shift can be reated to the symmetry properties of the state under consideration. We verify our anaytica resuts with expicit numerica cacuations. Moreover, we discuss the case of twisted boundary conditions that arise when one considers moving bound states in finite boxes. The corresponding finite-voume shifts in the binding energies pay an important roe in the study of composite-partice scattering on the attice, where they give rise to topoogica correction factors. Whie the above resuts are derived under the assumption of a pure finite-range interaction and are sti true up to exponentiay sma correction in the short-range case in the second part we consider primariy systems of charged partices, where the Couomb force determines the ong-range part of the potentia. In quantum systems with short-range interactions, causaity imposes nontrivia constraints on ow-energy scattering parameters. We investigate these causaity constraints for systems where a ong-range Couomb potentia is present in addition to a short-range interaction. The main resut is an upper bound for the Couomb-modified effective range parameter. We discuss the impications of this bound to the effective fied theory EFT) for nucear hao systems. In particuar, we consider severa exampes of proton nuceus and nuceus nuceus scattering. For the bound-state regime, we find reations for the asymptotic normaization coefficients ANCs) of nucear hao states. Moreover, we aso consider the case of other singuar inverse-power-aw potentias and in particuar discuss the case of an asymptotic van der Waas tai, which pays an important roe in atomic physics. Finay, we consider the ow-energy proton deuteron system in pioness effective fied theory. Amending our previous work, we focus on the doubet-channe spin configuration and the 3 He bound state. In particuar, we study the situation at next-to-eading order in the EFT power counting and provide numerica evidence that a charge-dependent counterterm is necessary for correct renormaization of the theory at this order. We furthermore argue

4 that the previousy empoyed power counting for the incusion of Couomb contributions shoud be given up in favor of a scheme that is consistent throughout the bound-state and the scattering regime. In order to probe the importance of Couomb effects directy at the zero-energy threshod, we aso present a first cacuation of proton deuteron scattering engths in pioness effective fied theory. Parts of this thesis have been pubished in the foowing artices: S. König and H.-W. Hammer, Low-energy p-d scattering and He-3 in pioness EFT, Phys. Rev. C 83, ) S. König, D. Lee, and H.-W. Hammer, Voume Dependence of Bound States with Anguar Momentum, Phys. Rev. Lett. 107, ) S. Bour, S. König, D. Lee, H.-W. Hammer, and U.-G. Meißner, Topoogica phases for bound states moving in a finite voume, Phys. Rev. D 84, R) 2011) S. König, D. Lee, and H.-W. Hammer, Non-reativistic bound states in a finite voume, Annas Phys. 327, ) S. König, D. Lee, and H.-W. Hammer, Causaity constraints for charged partices, J. Phys. G: Nuc. Part. Phys. 40, ) S. König and H.-W. Hammer, The Low-Energy p d System in Pioness EFT, Few-Body Syst. 54, ) S. Ehatisari, S. König, D. Lee, and H.-W. Hammer, Causaity, universaity, and effective fied theory for van der Waas interactions, Phys. Rev. A 87, )

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7 Contents 1 Introduction 1 2 Genera concepts Finite-range interactions Non-oca interactions Soutions of the free equation Asymptotic form of the wave function The effective range expansion Bound states and asymptotic normaization constants Some forma scattering theory The Lippmann Schwinger equation The T-matrix Universaity Hierarchy of partia waves Universaity for arge scattering ength The theorist s point of view From potentia modes to a modern perspective Effective fied theory Historica overview Pioness effective fied theory Effective fied theory for nucear hao states Lattice cacuations Lattice QCD Nucear attice simuations Numerica aspects i

8 3 Finite-voume cacuations Introduction Bound states in a finite voume Infinite voume Finite voume S-wave resut Extension to higher partia waves Resuts Sign of the mass shift Trace formua Numerica tests Lattice discretization Methods Resuts Two-dimensiona systems Twisted boundary conditions Generaized derivation Topoogica voume factors Summary and outook The Couomb force Couomb wave functions The Gamow factor Anaytic wave functions Modified effective range expansion Bound-state regime Asymptotic wave function Bound-state condition and ANC The Couomb T-matrix Yukawa screening Expression in terms of hypergeometric functions Causaity bounds for charged partices Introduction ii

9 5.2 Setup and preiminaries Derivation of the causaity bound Wronskian identities Rewriting the wave functions The causaity-bound function Cacuating the Wronskians Constant terms in the causaity-bound function The causa range Practica considerations Exampes and resuts Proton proton scattering Proton deuteron scattering Proton heion scattering Apha apha scattering Numerica cacuations Reation for asymptotic normaization constants Derivation for the neutra system Derivation for charged partices Appication to the oxygen-16 system Other ong-range forces Singuar potentias Causaity bounds for van der Waas interactions Summary and outook The proton deuteron system revisited Introduction Formaism and buiding bocks Fu dibaryon propagators Couomb contributions in the proton proton system Power counting Integra equations Numerica impementation Scattering phase shifts iii

10 6.3.1 Quartet channe Doubet channe Trinuceon wave functions Homogeneous equation Normaization condition Heium-3 properties Energy shift in perturbation theory Nonperturbative cacuation Leading-order resuts The Couomb probem at next-to-eading order Scaing of the dibaryon propagators Utravioet behavior of the ampitude Consequences Nonperturbative cacuation Back to the scattering regime Proton deuteron scattering engths Numerica cacuation of the Gamow factor Bubbe diagram with fu off-she Couomb T-matrix Resuts Summary and outook Concuding remarks 153 A The Couomb wave functions of Boé and Gesztesy 157 B Expicit expressions for the causaity bound function 159 B.1 Repusive case, γ > B.2 Attractive case, γ < C Equivaence of ANC reations 161 D Bound states in nonreativistic effective fied theory 163 D.1 Simpified nuceon deuteron system D.2 Bethe Sapeter equation D.2.1 Momentum space iv

11 D.2.2 Bound state contribution D.2.3 Homogeneous equation D.2.4 Three-dimensiona reduction D.3 Operator formaism D.3.1 Lippmann Schwinger equation D.4 Normaization condition D.4.1 Genera derivation D.4.2 Expicit form in three dimensions D.4.3 Factorization of the T-matrix D.5 Some remarks v

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13 Chapter 1 Introduction A ot of interesting physics can be done with short or even finite-range interactions. In many situations physicists encounter universaity in the sense that observabes are independent, or at east approximatey so, of the short-range detais of a given system. The simpe yet very powerfu concept behind this observation is that ow-energy phenomena do not probe physics at short distances. This separation of scaes is aso the foundation of effective fied theory. However, such universa aspects are usuay just the starting point for a deeper understanding. In genera, universa behavior is not exacty reaized in nature and as soon as one wants to go beyond a certain basic eve of precision, it is necessary to take into account corrections. One such correction that pays indeed a very prominent roe at ow energies is the Couomb interaction. It is particuary important in nucear physics because most processes in that fied invove charged partices. The above two paragraphs aready mention the most important aspects of the topics to be discussed in this thesis. It summarizes the work done in a itte more than three years of doctora studies, divided into severa projects that wi be presented in separate chapters. Starting with a quick tour through quantum mechanics with short- and finiterange interactions, we wi discuss ow-energy universaity and its connection to effective fied theory EFT) in Chapter 2. With a focus on appications in nucear physics, we wi present there the genera concepts that wi be important throughout this work, both as theoretica foundations and to put the resuts to be derived in a broader context. In Chapter 3 we consider how cacuations in finite voumes with periodic boundary conditions, as they appear frequenty in numerica attice cacuations, affect the properties of nonreativistic bound states. When the atter arise from a finite-range interaction, it is possibe to derive an anaytic formua for their eading-order voume dependence. Generaizing a resut for S-waves that has been obtained by Lüscher in the 1980s [8], we derive the finite-voume mass shift for states with arbitrary orbita anguar momentum. For the rest of this work we wi then mainy be concerned with systems of charged partices, where the Couomb force pays an important roe, and particuary so at ow energies. Chapter 4 is dedicated to coecting and reviewing severa resuts concerning this topic. 1

14 2 Chapter 1. Introduction Subsequenty, in Chapter 5, we derive a generaization of the so-caed Wigner causaity bound to interactions with Couomb tais. Resuts of this kind are both interesting theoreticay and as a guide to improve the convergence of effective-fied-theory cacuations. We aso consider other ong-range forces and in particuar the van der Waas interaction, which pays an important roe in ow-energy atomic physics. Whie in the derivation of the causaity bounds the Couomb force enters mainy as a theoretica chaenge in a configuration-space treatment, we wi ook at the difficuties it creates from a more practica point of view in Chapter 6, where we consider the owenergy proton deuteron system in pioness effective fied theory. Performing cacuations in both the scattering and the bound-state regime, we obtain resuts for phase shifts, scattering engths, and the Heium-3 binding energy. In a cases, Couomb effects raise the interesting question of how to treat them consistenty in an EFT power counting scheme. At the same time, they aso constitute a chaenge for numerica cacuations in momentum space. Finay, we cose with some concuding remarks in Chapter 7.

15 Chapter 2 Genera concepts Overview In the first part of this chapter, we review severa we-known resuts from nonreativistic quantum mechanics. Focussing first on soutions of the radia Schrödinger equation and the effective range expansion in Section 2.1, we subsequenty give an overview of scattering theory from a more forma point of view in Section 2.2. Moving on, we then discuss owenergy universaity and its connection to effective fied theories Sections 2.3 and 2.4. Finay, in Section 2.5, we briefy discuss attice cacuations as a preude to the foowing Chapter Finite-range interactions Consider a two-partice system with reduced mass µ interacting via a sphericay-symmetric potentia V r), where r = r 1 r 2 is the reative distance of the two partices in their center-of-mass frame. It is a we-known fact in quantum mechanics that a soution ψr) of the time-independent Schrödinger equation 1,2 [ r 2µ + V r) E ] ψr) = 0, 2.1) describing a state with energy E and anguar momentum quantum numbers, m), can be separated as θ, φ) = u r) ψr) = ψ m r) = R r)y m r into a known anguar part, given by the spherica harmonics Y m radia wave function u r), which is a soution of [ d2 + 1) + + 2µ [ V r) E ] dr2 r 2 Y m θ, φ) 2.2) θ, φ), and a reduced) ] u r) = ) 1 r denotes the Lapacian. Throughout this work, we aways indicate the coordinate that a differentia operator acts on with an expicit subscript. 2 Here and in the foowing we work in natura units where = c = 1. 3

16 4 Chapter 2. Genera concepts This is known as the radia Schrödinger equation. In the foowing, we are primariy interested in interactions with a finite range R. For the potentia V r) this means that it fufis the condition V r) = 0 for r > R. 2.4) Moreover, throughout this work we empoy the foowing canonica cassification scheme for potentias: if for r the potentia fas off faster than any power aw, e.g., ike an exponentia, we ca it short-ranged. if, on the other hand, the potentia has a power-aw form r α with α 1 at arge r, we ca it a ong-range potentia. A very prominent member of this cass is the Couomb potentia V C r) 1/r, which wi be discussed in more detai in Chapter 4. We do not consider potentias that do not fa off at arge distances Non-oca interactions The interaction of a quantum system can be described in more genera terms by a rea symmetric operator V r, r ) = r ˆV r. 2.5) Assuming, as we wi do throughout this work whenever we consider such non-oca interactions, that V r, r ) aows an expansion in Legendre poynomias partia-wave decomposition), V r, r ) = = )V r, r )P cos θ), cos θ = r r rr, 2.6) a generaized form of the radia Schrödinger equation 2.3) can be written as p 2 u r) = d2 dr u + 1) r) + u 2 r 2 r) + 2µ 0 dr 4πr V r, r ) u r ), 2.7) where we have furthermore introduced the momentum scae p 2 = 2µE. From this one directy sees that the interaction no onger depends on the wave function at just a singe point but over some extended range and is thus smeared out or non-oca in that sense. For the specia case where we simpy get back Eq. 2.3). V r, r ) = V r) δ 3) r r ), 2.8) Note that the partia-wave decomposition 2.6) is in particuar possibe for non-oca potentias which are a function of the reative distance ony, V r, r ) = V r r ), 2.9)

17 2.1. Finite-range interactions 5 which is the straightforward generaization of spherica symmetry to the case of non-oca interactions. This condition, however, is ony a sufficient and not a necessary one, and we do not make such expicit assumptions on the interaction in the foowing. Due to the symmetry of V r, r ), which ensures that the fu Hamitonian is a symmetric operator, the generaized condition for the interaction to have a finite range now reads V r, r ) = 0 if r = r > R or r = r > R, 2.10) which impies the same for a partia-wave components V r, r ). The integra in Eq. 2.7) does then not extend a the way to infinity but is cut off at r = R. Since in this work we are not concerned with the exact form of the interaction, we wi from now on absorb the prefactor 4πr in Eq. 2.7) into the potentia and aso omit the subscript in writing down the radia Schrödinger equation Soutions of the free equation In the absence of any interaction in particuar, outside the range of a finite-range potentia), one is eft with the free radia Schrödinger equation, ] p 2 u r) = [ d2 + 1) + u dr2 r 2 r), 2.11) the soutions of which are we known. Commony see, e.g., Ref. [9]) the Riccati Besse and Riccati Neumann functions S pr) and C pr) are chosen as a base pair of inearyindependent soutions. 3 They are defined as πz S z) = 2 J +1/2z), 2.12a) C z) = 1) πz 2 J 1/2z), 2.12b) where J n z) denotes the ordinary Besse function. Any soution of Eq. 2.11) can be written as a inear combination of S pr) and C pr). Their asymptotic behavior for arge arguments is determined by S z) z sin z π/2), C z) z cos z π/2), 2.13) whereas cose to the origin in the imit z 0) S and C scae ike z +1 and z, respectivey. Due to this behavior S pr) and C pr) are aso caed the reguar and irreguar soutions of 2.11). In some situations it is more convenient to work instead with the Riccati Hanke functions Ĥ ± z) = C z) ± is z), 2.14) which have the asymptotic form e ±iz as z and thus, for z = pr, correspond to outgoing and incoming waves for z = pr, respectivey. 3 Ref. [9] actuay uses the notation S z) = ĵ z) and C z) = ˆn z). The convention we empoy here is the same as in Ref. [10] with L = and d = 3.

18 6 Chapter 2. Genera concepts Asymptotic form of the wave function Going back to Eq. 2.7), we now consider an arbitrary interaction that fufis the finiterange condition 2.10), and which we assume to be sufficienty we-behaved at the origin to aow for a soution that is reguar there i.e., u r) 0 as r 0 with finite derivative). We denote such a soution for a given possiby compex, to aso cover the bound-state regime momentum parameter p with an expicit superscript p). Outside the range of the interaction, it can be written in the form u p) r) = p [cot δ p) S pr) + C pr)] r > R), 2.15) where he have adopted the momentum-dependent normaization convention from Ref. [10]. In the foowing, whenever we write u p) r), it is aso impied that the wave function is normaized exacty as in Eq. 2.15). The scattering phase shift δ p) can be interpreted as the additiona phase compared to the reguar soution S pr) of the free equation) that the wave function picks up due to the interaction. It is reated to the partia-wave S-matrix s p) via s p) = e 2iδ p), 2.16) such that up to an overa change in the normaization, the asymptotic form of u p) r) can aso be written as u p) r) H pr) s p)h + pr) r > R) 2.17) in terms of Riccati Hanke functions. Aternativey, by factoring out an overa e iδ p) / sin δ p), one finds that u p) r) [ cot δ p) i ] H pr) [ cot δ p) + i ] H + pr) r > R). 2.18) The atter form can aso be derived directy by inverting Eq. 2.14) and inserting the resut into Eq. 2.15). For interactions with a short but not stricty finite range, e.g., a potentia with an exponentia or Gaussian tai, the reations above do not hod exacty, but are fufied asymptoticay as r The effective range expansion For systems with short- or finite-range interactions, the scattering phase shift δ p) can be expressed in terms of the we-known effective range expansion p 2+1 cot δ p) = 1 a r p 2 +, 2.19) where a and r are the scattering and effective range parameters, respectivey, and the eipses stand for higher-order shape parameters p 4, p 6,...) that we have not written out here. Eq. 2.19) states that the cotangent of the scattering phase shift, mutipied by appropriate powers of the momentum, is an anaytic function of p 2 and thus of the energy. Historicay, this expansion was first used to describe and interpret nuceon nuceon scattering at ow-energies. This is discussed, for exampe, in the review by Jackson and

19 2.1. Finite-range interactions 7 Batt [11]. As noted there and aready in an earier pubication by the same authors [12]), the S-wave = 0) version of Eq. 2.19) was first derived by Schwinger, but ony pubished in ecture notes. A derivation based on a direct anaysis of the radia wave functions was ater given by Bethe in Ref. [13]. An extension of Bethe s formaism to aso treat higher partia waves can be found in the textbook of Godberger and Watson [14]. For a modern derivation of Eq. 2.19) for arbitrary, based directy on the anaytic properties of the partia-wave S-matrix and scattering ampitude both of which wi be discussed in more detai in the foowing sections), see, for exampe, Ref. [9]. In genera, an expansion as in Eq. 2.19), which is essentiay a Tayor series of the efthand side around p 2 = 0, ony has a finite radius of convergence. For exampe, if the interaction is given by a short-range potentia of the Yukawa form exp µr)/r as it arises from the exchange of a partice with mass µ), p 2+1 cot δ p) has a cut starting at µ 2 /4 in the compex p 2 pane see, for exampe, Ref. [15]), which naturay imits the radius of convergence of the expansion. Within its region of anayticity, however, p 2+1 cot δ p) can aso be expanded around points other than p = 0. For exampe, in the 3 S 1 channe of neutron proton scattering it is customary to work with an effective range expansion around the deuteron poe [16]. In the presence of ong-range potentias p 2+1 cot δ p) is in genera no onger anaytic in p 2 and the ordinary effective range expansion as given above thus not vaid for systems where the interaction has a power-aw tai. It is, however, often possibe to modify the eft-hand side of Eq. 2.19) in such a way that anayticity is restored. Essentiay, this means that known non-anaytic terms due to the ong-range component of the potentia can be taken into account expicity. Most notaby, for the scattering of charged partices one has the so-caed Couomb-modified effective range expansion that wi be discussed in Chapter 4. Note furthermore that a genera modified effective range function for the case where the interaction can be written as the sum of a short-range and a ong-range potentia has been derived by van Haeringen and Kok in Ref. [17]. The methods just mentioned no onger work if the ong-range part of the potentia has a power-aw form that is more singuar than 1/r 2 at the origin. For such cases, an extension of a formaism caed quantum defect theory [18 20] has been deveoped by Gao [21]. This is particuary interesting for appications in atomic physics, where a ong-range van der Waas tai 1/r 6 ) arises due to the mutua poarization of the atoms, a case we wi come back to in Chapter Bound states and asymptotic normaization constants Going back to the case of pure finite-range interactions, we now consider bound states. Their wave functions correspond to soutions of Eq. 5.2) for negative energies, p 2 = 2µE < ) More precisey, in the compex p-pane, the bound states are ocated on the positive imaginary axis, whereas the negative imaginary axis is the ocation of virtua states [9]. We write p = iκ in the bound-state regime and ca κ > 0 the binding momentum.

20 8 Chapter 2. Genera concepts A bound state with anguar momentum corresponds to a simpe) poe in the partiawave S-matrix s p) at p = iκ. From Eq. 2.17) it can be seen that this means that there is no incoming component because s p), the ratio of the outgoing wave compared to the incoming one, becomes infinite. Aternativey one can choose a normaization with an overa coefficient s p) factored out in Eq. 2.17), eaving s p) 1 in front of the incoming component H pr) that has to vanish for p = iκ. From Eq. 2.18) one furthermore sees that cot δ p = iκ) = i 2.21) is the bound-state condition for the scattering phase shift. The same resut can be found more directy by writing the S-matrix as with the partia-wave scattering ampitude f p) = s p) = 1 + 2ikf p) 2.22) p 2 p 2+1 [cot δ p) i]. 2.23) Since the poe in s p) has to come from the second term in 2.22) we again obtain the condition stated in Eq. 2.21). On the bound-state wave function we impose the usua normaization condition From the above discussion it foows immediatey that 0 dr u r) 2 = ) u iκ) r) H + iκr) e κr 2.25) as r, which together with our reguarity assumption on the interaction ensures that the wave function is indeed normaizabe. Note, however, that the bound-state normaization condition 2.24) is not directy compatibe with the asymptoticay fixed form as given in Eq. 2.15). In order to discuss the precise asymptotic form of the bound-state wave functions we cosey foow Ref. [9] and define soutions χ ±,p r) of 2.7) normaized such that they exacty fufi the condition χ ± r,p r) H ± pr). 2.26) The bound-state soution normaized according to 2.24), which in the foowing we denote as u iκ) A, r), can then be written as u iκ) A, r) = i A κ χ +,iκ r), 2.27) where ) 1/2 A κ = dr χ +,iκ r) ) 0

21 2.2. Some forma scattering theory 9 is the so-caed asymptotic normaization constant/coefficient ANC). The factor i convenienty adjusts the phase of u iκ) r) such that it is a rea function. Outside the range of the interaction one then has the exact identity u iκ) A, r) = i A κ H + iκr) for r > R. 2.29) The wave functions u iκ) r) with the asymptotic behavior determined by Eq. 5.3), on the other hand, have the form u iκ) r) = i κ H + iκr) for r > R. 2.30) We use the different notations given in Eqs. 2.29) and 2.30) to indicate which convention is used. Asymptotic normaization constants are interesting quantities because for shaow bound states i.e., states with sma binding energy/momentum) they are cosey reated to scattering processes. For exampe, they are directy connected to zero-energy capture reactions [22], which pay an important roe in nucear astrophysics. In Chapter 5 Section 5.7) we wi discuss how the ANC can be expressed directy in terms of the parameters that appear in the effective range expansion 2.19). 2.2 Some forma scattering theory In the preceding sections, we have estabished that for finite-range interactions the asymptotic radia wave functions of two-partice systems have an expicity-known anaytic form. Since ow-energy physics is governed by arge-distance scaes, this fact wi be very usefu in the foowing chapters to derive reations based on just the properties of the wave-function tais. Since the focus so far was on consequences of the finite-range assumption, we have worked directy with radia Schrödinger equations and wave functions in configuration space, and ony introduced scattering concepts ike the phase shift and the S-matrix as they appear in that context. In the foowing, we discuss quantum-mechanica scattering theory from a more forma point of view, estabishing the connection with the reations given in Section 2.1 aong the way and focussing on resuts that wi be reevant ater in this work. Uness otherwise indicated with an expicit citation, what foows is mosty taken from Sakurai s textbook [23], but uses sighty different conventions in some paces. Unti further notice, we aso ift the finite-range assumption on the interaction The Lippmann Schwinger equation In its abstract operator form, the time-independent Schrödinger equation 2.1) reads Ĥ ψ = E ψ 2.31) with the Hamiton operator Ĥ = Ĥ0 + ˆV given by r Ĥ ψ = r 2µ ψr) + d 3 r V r, r )ψr ) 2.32)

22 10 Chapter 2. Genera concepts in configuration space, where we have aowed the interaction to be non-oca. What we have ignored so far is that the Schrödinger equation aone does not specify the boundary condition for the soutions. In the time-independent scattering formaism we are considering here, one is primariy interested in soutions that reduce to a pane-wave state p when the interaction is switched off, ˆV 0. This behavior can be enforced by making the ansatz ψ p +) = p + E p Ĥ0 + iε) 1 ˆV ψ +) p, E p = p2 2µ, 2.33) where the sma imaginary part iε has been introduced to make the otherwise singuar operator E Ĥ0 invertibe. More precisey, the zero modes of E Ĥ0 are given by the pane-wave states p, E p Ĥ0) p = 0 for a p. 2.34) Acting on both sides of Eq. 2.33) with E Ĥ0 + iε and sending ε 0 which in the foowing is aways impicity understood to be done at the end of a manipuations) gives back the Schrödinger equation in the form 3.3). Equation 2.33) is caed the Lippmann-Schwinger equation for the scattering states. It can be interpreted as an integra formuation of the Schrödinger equation with the boundary conditions determined by the inhomogeneous term p and the iε-prescription. We have chosen a positive sign for the imaginary part and indicated this by writing ψ p +) for the scattering state. This gives the physicay most reevant soution that corresponds to an incoming pane wave and an outgoing scattered wave when one goes over to a time-dependent framework. By choosing a configuration-space basis { r : r R 3 } we get the ordinary wave functions ψ +) p r) = r ψ +) p, 2.35) and the pane-wave states are just r p = expip r). In genera, the wave function ψ p +) r) is not characterized by a singe pair of anguar-momentum quantum numbers, m) but rather has an expansion in spherica harmonics or Legendre poynomias [24], ψ +) p r) = ) 1/2 2µ πp =0 m= ) 1/2 2µ 1 = πp 4π =0 Y m ˆr)Y m ˆp) u r) r 2 + 1)i P cos θ) u r) r 2.36), cos θ = ˆp ˆr, 2.37) with the unit vectors ˆp = p/p and ˆr = r/r. The wave functions u r) are then soutions of the radia Schrödinger equation 5.2), and the normaization in Eq. 2.37) is chosen such that the atter can be rewritten as an integra equation with the inhomogeneous term given by the Riccati Besse function S pr) The T-matrix Defining the Green s function operator the free resovent) Ĝ +) 0 E) = E Ĥ0 + iε) 1, 2.38)

23 2.2. Some forma scattering theory 11 we can write the Lippmann-Schwinger equation 2.33) as ψ p +) = p + Ĝ+) 0 E p ) ˆV ψ p +). 2.39) For the free Hamitonian defined impicity in Eq. 2.32), the Green s function in configuration space is given by the expression G +) 0 E; r, r ) = r Ĝ+) 0 E) r = µ e ik r r 2π r r, 2.40) where k is a momentum scae defined as k = 2µE + iε). It satisfies the equation 1 2µ r + k 2 ) G +) 0 E; r, r ) = δ 3) r r ). 2.41) Note that other conventions exist in the iterature where the factor 1/2µ) is absorbed into the definition of G +) 0. Our choice here has the advantage of staying cose to the operator notation, such that from Eq. 2.38) one can directy read off the momentumspace expression G +) 0 E; q, q ) = q Ĝ+) 0 E) q = 2π)3 δ 3) q q ) E q 2 /2µ) + iε. 2.42) Introducing now the operator ˆT, which in the foowing we simpy refer to as the T-matrix, via the impicit definition ˆV ψ p +) = ˆT p, 2.43) we obtain the forma soution ψ +) p = 1 + Ĝ+) 0 E p ) ˆT ) p 2.44) for the scattering state. The origina probem of finding the soution for the scattering wave function is thus modified to the question of determining the T-matrix. In order to proceed in this direction, one can act with the potentia operator ˆV on both sides of Eq. 2.44) from the eft and then use Eq. 2.43) once again to find ˆT p = ˆV + ˆV Ĝ +) 0 E p ) ˆT ) p. 2.45) Demanding further that this hods for a states { p : p R 3 } yieds an operator equation for ˆT. In fact, at this point one can furthermore ift the restriction that the Green s function operator is evauated at the on-she energy E = E p but rather aow this vaue to be an arbitrary compex parameter. The resuting equation reads ˆT E) = ˆV + ˆV Ĝ+) 0 E) ˆT E) 2.46) and is caed the Lippmann Schwinger equation for the T-matrix. The momentum-space representation T E; q, p) = q ˆT E) p 2.47)

24 12 Chapter 2. Genera concepts with no imposed connection between the variabes E, p, and q is commony referred to as the fu off-she T-matrix, whereas the quantity T q, p) = q ˆT E = E p ) p 2.48) is caed the haf off-she or simpy haf-she) T-matrix. It has the usefu property of being directy reated to the scattering wave function in momentum space via according to Eq. 2.44). ψ p +) q) = q ψ p +) = 2π) 3 δ 3) 2µ T q, p) q p) + p 2 q 2 + iε, 2.49) The scattering ampitude From the Green s function in configuration space, Eq. 2.40), one can deduce that at asymptoticay arge distances the scattering wave function ψ p +) r) can be written as the sum a pane wave given by the state p and interpreted as the originay incoming partice fux) and an outgoing spherica wave describing the effect of the scattering process, where p = pˆr and ψ p +) r) r e ip r + eipr r fp, p), 2.50) fp, p) = µ 2π p ˆV ψ +) p = µ 2π p ˆT p 2.51) is the scattering ampitude. From the definitions above it is cear that fp, p) ony depends on the magnitude p of p and the ange θ between p and r. It thus has an expansion in partia waves, fp, p) = 2 + 1)f p)p cos θ), 2.52) =0 and it is precisey the partia-wave ampitudes f p) appearing in Eq. 2.52) that were aready mentioned in Section Assuming that the potentia can be expanded in Legendre poynomias cf. Section 2.1.1), an anaogous expansion aso exists for the T- matrix, T E; q, p) = 2 + 1)T E; q, p)p cos θ), 2.53) =0 where θ is now the ange between the momentum vectors q and p. It then foows that the partia-wave scattering ampitude and thus aso the scattering phase shift δ p) is determined by the partia-wave T-matrix at the on-she point: f p) = e2iδ p) 1 2ip = µ 2π T E p ; p, p). 2.54)

25 2.3. Universaity Universaity We now turn back our attention to short- and finite-range interactions. As shown in Section 2.1, the asymptotic form of the radia wave function for a given two-partice scattering system is known anayticay and can be parametrized in terms of the scattering phase shift δ p). The reations for the wave functions in Eqs. 2.15) and 2.29) hod rigorousy for pure finite-range potentias and sti up to exponentiay sma as r ) corrections if the interaction is short-ranged. In the atter situation, the interaction range is imited by the typica fa-off scae of the potentia. In either case, the effective-range expansion 2.19) furthermore provides a method to express the phase shifts in terms of just a few ow-energy parameters. In many situations, aready the scattering ength and effective range parameter a and r are sufficient for a reasonaby accurate description of the experimenta data. The physics behind a this is that at sufficienty ow energies and thus sma scattering momenta p) the detais of the interaction cannot be resoved because the de Brogie waveength λ p 1/p corresponding to the incoming partice fux is too arge compared to the spatia extent of the scattering center. More quantitativey, the criterion for not resoving the detais of an interaction with range R is p R 1 pr ) This simpe principe is very powerfu because it means that provided the underying interaction has a finite or short) range ow-energy quantum scattering can be described in a universa way by the parameters appearing in the effective range expansion Hierarchy of partia waves In the forma discussion so far we have mosty considered some fixed but arbitrary anguar momentum. The reay important quantity, however, is the scattering ampitude fp, p) defined in Section since it is directy reated to the differentia cross section that is determined in experiments, dσ dω = fp, p) ) This means that a physicay significant information about the process is contained in the scattering ampitude. Combining Eqs. 2.52) and 2.54), we can express the scattering ampitude in terms of the phase shifts δ p) as fp, p) = = ) e2iδ p) 1 2ip P cos θ). 2.57) If a terms in this sum were equay important, the universa parametrization mentioned above woud not be very usefu because describing the physica system woud sti require an infinite number of parameters. Fortunatey, the phase shifts themseves have the ow-

26 14 Chapter 2. Genera concepts energy behavior 4 δ p) p 2+1, 2.58) which means that at ow energies ony a few partia waves are important. Usuay, the dominant term is the S-wave = 0), but this is not necessariy aways the case. 5 The physica origin behind the behavior 2.58) is of course the centrifuga barrier +1)/r 2 that the scattering partices cannot penetrate appreciaby at ow energies Universaity for arge scattering ength A particuary interesting situation occurs if the S-wave scattering ength of a system is unnaturay arge. By unnaturay arge we mean in this context that it is much arger than the range R of the potentia, a 0 R, because naïvey one woud expect that a ength scaes in the system are of the natura order of magnitude. Usuay, one does of course not know the exact range R of the underying interaction, nor is the notion of a strict finite-range interaction a very reaistic picture. In genera, if the interaction is assumed to be mediated by some kind of partice, the corresponding potentia has an exponentia Yukawa) tai, and the inverse mass of the exchange partice provides a good estimate for the interaction range. In ow-energy nucear physics, for exampe, the typica ength scae is set by the inverse pion mass, m 1 π 1.4 fm. Typicay, the S-wave effective range is found to be of the order of magnitude estimated for the underying interaction, such that, in the absence of more direct information, a 0 r 0 can be used as a criterion for asserting that the scattering ength is unnaturay arge. Since the tota cross section for two-partice scattering at zero energy is given by 6 σ tot 0) = 4πa 2 0, 2.59) a arge S-wave scattering ength means that the partices interact strongy. In fact, one can say that the ow-energy physics of such a system is competey governed by the arge S-wave scattering ength. For exampe, if the interaction supports a two-partice bound state simpy caed a dimer in the foowing) at p = iκ, the combination of Eqs. 2.21) and 2.19) tes us that κ 1 a 0 + r 0 2a 2 0 = 1 a Oa0 /r 0 ) ), 2.60) where the correction is negigibe if r 0 a 0. This means that at eading order the dimer binding energy is just E d = ) 2µa This can be seen, for exampe, from the effective range expansion 2.19). Since cotx) = 1 + Ox), one has p 2+1 /δ p) const. as p 0. 5 For exampe, in the scattering of two identica fermions the Paui principe ony aows odd, such that the eading ow-energy contribution is given by the P-wave = 1). 6 This foows from the discussion in the preceding sections by noting that im p 0 f 0 p) = a 0.

27 2.3. Universaity 15 Whether or not a bound state actuay exists can furthermore be reated to the sign of the scattering ength. In the conventions that we are using here, a arge positive scattering ength impies the existence of a shaow dimer state. 7 The Efimov effect A very intriguing phenomenon occurs in the three-body sector of a system of partices that have a arge two-body scattering ength and in that sense a strong pairwise interaction. It can be shown that the bound-state spectrum of such a system exhibits a tower of approximatey) geometricay-spaced three-body trimer states, i.e., a series of boundstate energies fufiing E n+1) trimer / En) trimer const ) In the imit where the magnitude of a 0 goes to infinity, 8 so does the number of trimer states, and the geometrica spacing becomes exact with a universa scaing factor determined ony by the mass ratio of the partices. This effect was first proposed by and named after) V. Efimov [25] and ater proven by Amado and Nobe [26, 27]. For a detaied discussion of quantum systems with a arge scattering ength and the Efimov effect, we refer here to the review by Braaten and Hammer [28]. To concude this subsection, we summarize that we speak of ow-energy universaity in the genera sense whenever a description with finite-range interaction is a good approximation to describe the physics of a given two-partice) system. In a manner of speaking, physics at ow energies is governed by the tais of the radia) wave functions, which have a universa anaytic form, and the essentia features of the system can be we described by just a few parameters, namey the scattering engths and effective ranges for a sma number of partia waves. Since the effective range expansion is vaid in the compex p- pane, this incudes the description of bound states with sma binding energies. Moreover, if the S-wave) scattering ength is unnaturay arge, this parameter governs the whoe system and one finds reations that are even more universa, ike the binding energies of shaow dimer states or the Efimov effect in the three-body sector. Finay, it is important to point out again that the above statements sti hod up to corrections that are often negigibe) in the more reaistic setup where the interaction has no stricty finite range but is short-ranged and fas off rapidy The theorist s point of view Having read the preceding sections, the caption above this sentence might seem confusing at first. By no means do we want to say that we now switch gears to ook at physics from the theoretica side, but rather that this is what we have been doing a aong so far. Here, we want to emphasize that finite-range potentias are not an experimenta concept but, since potentias are even not observabe quantities, reay just a convenient theoretica 7 For a derivation of this statement see, for exampe, Ref. [23]. 8 More precisey, the reevant imit is a 0 /R, which means that the effect can aso be found for zero-range interactions.

28 16 Chapter 2. Genera concepts too. The same shoud be said in fact, at east to a good extent, about partia-wave scattering phase shifts. What is directy accessibe in scattering experiments via count rates and their anguar distribution) are differentia) cross sections and thus, according to Eq. 2.56), the absoute vaue squared of the scattering ampitude. Phase shifts are then ony obtained from an inversion of Eq. 2.52), necessariy truncated at some and based on data in ony a imited energy and usuay anguar) regime. This procedure can be very deicate in practice and that the resut, due to the imited amount of data, is not truy unique. Potentias are sti a very usefu too for the theoretica description of physica processes, and scattering phase shifts provide a convenient interface to compare the resuts of cacuations with experiments, but it is important to keep in mind the imitations described above From potentia modes to a modern perspective Athough ow-energy universaity arises naturay in the the theory of finite- or shortrange interactions, it was first found as an experimenta phenomenon. It was observed 9 that ow-energy S-wave scattering phase shifts in the two-nuceon system coud be we described with just two parameters since the data points coud be fitted by a straight ine in a suitabe representation the eft-hand side of the effective range expansion 2.19) or its Couomb-modified anaog that wi be discussed in Chapter 4. Different forms of potentias coud thus be used equay we to mode the system as ong as they had two parameters that coud be adjusted to reproduce that ine. At the time, it was concuded that measurements at higher energies were needed to reay determine the shape of the nucear potentias. This ed, subsequenty to the construction of very sophisticated potentia modes, many of which, ike Nijmegen I,II [29], AV18 [30], and CD-Bonn [31], are sti we known and often used today a comparison of the different modes and an historica overview of their deveopment can be found, for exampe, in Ref. [32]). A of these potentias describe the nuceon nuceon phase shifts and deuteron properties) very we, but differ quite substantiay in their detais. Of course, this ambiguity simpy iustrates expicity the fact that potentias are not observabe. This, together with the difficuty of these approaches to consistenty describe and/or impement the physics of more than two partices see, for exampe, Ref. [33]), as we as the essentiay unsystematic way they are constructed, has ed to the deveopment of a more modern perspective. Rather than continuing the utimatey futie endeavor of trying to find the nucear potentia, one simpy constructs so-caed effective potentias as a systematic expansion, where new operators are added to describe physics at subsequenty higher energy scaes and/or of an increasing number of nuceons). This approach expicity incorporates the observed ow-energy universaity, or can, in fact, be characterized as being based on it. More generay, the underying concept is that of effective fied theory EFT), which we now turn to discuss in some more detai. 9 See, for exampe the review of proton proton scattering by Jackson and Batt [11] and origina references therein.

29 2.4. Effective fied theory Effective fied theory The main concepts of effective fied theories can be summarized as foows: First, choose degrees of freedom that are appropriate for the energy scae under consideration nuceons instead of quarks, for exampe). Then, construct a Lagrangian out of fieds corresponding to these degrees of freedom, incuding a terms aowed by genera principes ike unitarity, anayticity and whatever other symmetries reevant for the system at hand ike isospin, in the exampe above). Finay, since such a Lagrangian contains in principe infinitey many terms, a further key ingredient is an ordering scheme power counting ) that serves to estimate the reative importance of the individua terms. The connection to ow-energy universaity is that the ordering in the fina step is often but not necessariy aways or excusivey) based on the reative momenta in the system one wishes to describe. We wi iustrate this in the foowing by ooking at a few expicit exampes. First, it is important to point out that the coefficients of operators in a Lagrangian constructed in the way aid out above are not normay dictated by genera concepts. Instead, these so-caed ow-energy constants have to be determined by matching the resuts of cacuations to known observabes. 10 The predictive power of the theory then ies in the fact that usuay the same coefficients or combinations thereof) appear in different observabes. Having measured some of them, it is possibe to make predictions for others Historica overview The approach aid out above was pioneered by Weinberg in Ref. [34], where in the context of pion physics) he formuated the idea that a quantum fied theory has, utimatey, no other content than anayticity, unitarity, custer decomposition, and symmetry. Such phenomenoogica Lagrangians had been used before, but were usuay based on currentagebra concepts see, e.g., Ref. [35]). The big step that Weinberg took was to promote them to the starting point of cacuations, thereby abandoning their unsystematic heritage. A key roe in Weinberg s origina appication to hadron physics at ow energies is payed by the approximate chira SU2) L SU2) R symmetry of the strong interaction, the spontaneous breaking of which is responsibe for the pion mass being so sma. 11 This mass and the momenta of ow-energy pions are used as sma scaes in the power counting. 10 Aternativey, if an underying more fundamenta theory is known, they can sometimes aso be cacuated from that. 11 If there was no expicit chira-symmetry breaking, they woud be massess Godstone bosons.

30 18 Chapter 2. Genera concepts Weinberg s approach, which became famous as chira perturbation theory, was worked out further by Gasser and Leutwyer [36, 37] and has a vast number of appications and extensions today. In particuar, the formaism has been extended to incude aso nuceons and other baryons see, for exampe, Refs. [38 40]). This important step made it possibe to construct the effective nucear potentias mentioned at the end of Section In an idea that again goes back to Weinberg [41, 42], a ow-energy expansion of the effective interaction between nuceons is buit out of diagrams derived from the chira Lagrangian. This means that rather than doing pain perturbation theory by summing Feynman diagrams up to a given order, the power counting is appied to derive an effective potentia, which can subsequenty used in cacuations based on Schrödinger or Lippmann Schwinger) equations. Of course, with the above summary we have ony scratched the surface of a very compex fied. Much more detaied discussions of the points mentioned here and many more) can be found in the reviews by Epebaum et a. [43], or Macheidt and Entem [44], which we just refer to here for simpicity. Furthermore, it is important to point out that athough the origins of effective fied theory ie in hadron and nucear physics, it is by no means imited to those appications. In fact, it is used in many areas of modern theoretica physics. Since the key ingredient is merey a separation of scaes and not that the tota energy of the system one wishes to describe is a sma scae, there are even appications in high-energy physics, ike the soft-coinear effective theory started in Refs. [45, 46]. We do not make here the futie attempt to give a comprehensive ist of current EFT appications. Rather, we focus on two exampes that wi pay a roe ater in this work Pioness effective fied theory At very ow energies in nucear physics, even the pions can be integrated out, which then eaves ony nuceons as effective degrees of freedom. The interactions between them are simpe contact terms, corresponding to deta-peak potentias and derivatives thereof in configuration space, and since in the ow-energy regime a reative momenta are sma, a nonreativistic description is appropriate. The atter means that a partices ony propagate forward in time and that there is no pair creation. In fact, this kind of EFT can be thought of as a convenient reformuation of quantum mechanics. Naturay, this pioness effective fied theory is imited to energy momentum) scaes beow the pion mass, but nevertheess it is very interesting and exhibits a rich set of features. The reason for this is that the S-wave nuceon nuceon system is an important exampe for the case of unnaturay arge scattering engths. Both a d 5.42 fm in the 3 S 1 isospin 0) and a t fm in the 1 S 0 isospin 1) channe are significanty arger than the typica scae of about 1.4 fm set by the inverse pion mass cf. Section 2.3.2). The corresponding effective ranges, on the other hand, have the vaues 1.75 and 2.73 fm, respectivey, and are thus indeed of the expected natura order of magnitude The numbers quoted here for the scattering engths and effective ranges are quoted from Refs. [16] and [47].

31 2.4. Effective fied theory 19 The deuteron appears in this picture as a neary universa shaow dimer state corresponding to the arge positive scattering ength a d. From Eq. 2.61) one obtains E d 1.4 MeV as eading-order resut for its binding energy, which is not far from the experimenta vaue = MeV [48]. The agreement here is rather coarse because the effective range in this channe is ony about a factor of three smaer than the scattering ength, which makes the range corrections to the eading-order expression quite significant. Indeed, if Eq. 2.60) is used to cacuate the binding momentum incuding the effective-range term, the resut is κ 45.6 MeV, corresponding to a binding energy of about 2.21 MeV, much coser to the experimenta vaue. E exp d The presence of the shaow deuteron state has a significant impact on the construction of pioness effective fied theory. Since perturbation theory cannot produce bound states, the appicabiity of the approach woud naïvey be imited to momentum scaes Q beow the deuteron poe, Q < κ d 1/a d. If one aso takes into account the poe corresponding the virtua bound state in the 1 S 0 channe, where the scattering ength is arger yet, the range of appicabiity is even narrowed down to Q < 1/a t. In either case, this is much smaer than the natura breakdown scae set by the pion mass. The soution, introduced in Refs. [49 52], is to incude certain contributions up to a orders in the perturbative expansion and thus generate the shaow states corresponding to the arge scattering engths in a nonperturbative manner. The theory with this scheme appied and its owenergy constants fixed by matching two-body ampitudes to the effective range expansion is then vaid for ow-energy scaes of the order Q 1/a d,t and with the natura breakdown scae Λ m 1 π, corresponding to an EFT expansion parameter Q/Λ 1/3. 13 In Refs. [53 55], the formaism has been extended to the three-nuceon sector. The situation there is particuary interesting because the triton can be interpreted as an approximate Efimov state. Since the physics it describes are to a significant extent governed by the arge S-wave scattering engths, pioness effective fied theory has a ot in common with a simper EFT that describes identica bosons with a arge two-body scattering ength. The key features mentioned above nonperturbative resummation to reproduce shaow dimer bound states and the Efimov effect in the three-body sector) can be studied there without compications due to different spin and isospin channes. A comprehensive review of this EFT and a broader discussion of universaity in systems with arge scattering ength can be found in the review by Braaten and Hammer [28]. For appications to cod-atomic systems with arge scattering engths, see aso K. Hefrich s doctora thesis [56] and further references therein. In Chapter 6, we wi discuss pioness effective theory in more detai and use it to anayze the proton deuteron system Effective fied theory for nucear hao states Amost the same formaism as for the pioness EFT discussed above can aso be used to construct an effective fied theory that is usefu to cacuate properties of nucear hao systems. Such states, aso caed hao nucei, can be thought of as a tighty bound core 13 Note that if the size of corrections is estimated directy in terms of the effective range parameters, one gets the same resut: 1.75/

32 20 Chapter 2. Genera concepts nuceus with one or more weaky-bound vaence nuceons for reviews of such states see, for exampe, Refs. [57, 58]). The separation of scaes, which is the crucia ingredient for the construction of an EFT, is given for such systems by the sma separation energy of the vaence nuceons compared to the binding energy of the core. 14 In an effective two-body picture that negects the interna structure of the core which cannot be resoved at ow energies), a one-nuceon hao nuceus can be thought of as a shaow dimer state that occurs due to a arge scattering ength in the corresponding nuceon nuceus scattering system. This is exacty the same situation as with the deuteron in the few-nuceon sector, and it is thus natura to adopt the concepts of the pioness effective fied theory described in the previous section. The resuting effective Lagrangian contains, in addition to the nuceon terms, an additiona fied to describe the core as a whoe, interacting with the nuceons via contact terms and/or derivatives thereof. This so-caed hao EFT was first introduced in Refs. [60, 61] to study neutron apha and has since then been extended to describe a number of other phenomena ike apha apha scattering [62], bound singe-neutron hao states such as 11 Be [59] and 8 Li [63, 64], and various two-neutron hao systems [65 67]. Recenty, is has aso been used to cacuate charge form factors of two-neutron hao nucei [68]. 2.5 Lattice cacuations In the preceding sections we have repeatedy touched the subject of nucear physics, but amost excusivey discussed it directy from an effective point of view. It is thus due time to mention quantum chromodynamics QCD) as the widey accepted underying 15 theory of the strong interaction which, up to eectromagnetic and weak effects, utimatey governs the properties of nuceons and nucei. 16 It is defined by the Lagrangian L QCD = f ψ f id µ γ µ m f )ψ f 1 F a 4 µνf µν,a, D µ = µ + iga a µt a, F a µν = µ A a ν ν A a µ + igf abc A b µa c ν, a 2.63) which is an essentia component of the Standard Mode of partice physics. In Eq. 2.63), the sum runs over a quark favors f represented by the Dirac fermion fied ψ f and A a µx) are the guon gauge fieds. Furthermore, t a and f abc are the SU3) group generators and structure constants, and a, b, c = 1,..., 8 are the corresponding coor indices. In the imit of vanishing quark masses, m f = 0 for a f, L QCD exhibits an exact chira symmetry, which means that right and eft-handed quark fieds ψ f,r/l = 1 ± γ 5 )/2 ψ f decoupe. If one ony considers two ight quark favors f = u, d, which is usuay sufficient 14 Aternativey, as it is done for exampe in Ref. [59], one can compare the arge matter radius of the hao nuceus as a whoe to the sma radius of the core aone. 15 We write underying here to avoid the question of whether QCD can aso be regarded as a fundamenta theory. It is of course possibe, if not ikey, that QCD and the standard mode in genera are aso just effective theories of some sort, and the author takes the stance that it is not even ogicay possibe to assert any given theory as truy fundamenta. 16 Note that in nucear physics eectromagnetic effects are not a sma correction, but pay an important roe for the description of a nucei heavier than the deuteron.

33 2.5. Lattice cacuations 21 for nucear-physics appications, this is broken down spontaneousy to the approximate isospin symmetry exact in the imit m u = m d ) that is sti visibe in the hadronic spectrum. We assume a this to be we-known and wi not go into further detai here. The ony aspect reay important for the present discussion is that the running of the strong couping constant α s = g 2 /4π) that makes QCD perturbative at high energy scaes and eads to the famous asymptotic freedom at the same time renders the theory strongy-interacting and thus non-perturbative at ow energies. This is directy refected by the fact that quarks and guons are not physicay observabe degrees of freedom, but ony exist confined into hadrons. One way to circumvent the breakdown of perturbation theory are the effective fied theories introduced in Section 2.4. Yet, as powerfu as they are, it is nevertheess desirabe to aso deduce the properties of hadrons, their interactions, and utimatey those of nucei, directy from L QCD. A way to achieve this is to sove the theory numericay by putting it on a discretized space-time attice Lattice QCD The idea just mentioned was initiay conceived by Wison [69] in 1974 as an attempt to expain confinement. Athough the atter is sti an unsoved probem, with the rapid advance of computer power during the ast two decades, the attice formuation of QCD has evoved into a very successfu too. In the foowing few paragraphs, we wi briefy review its main concepts. Much more thorough discussions of the topic can be found in introductory attice QCD texts such as Refs. [70, 71]. As Davies puts it in Ref. [70], Lattice QCD is just QCD, no more and no ess. Athough in practice there are a number of technica issues in particuar with the impementation of fermions and chira symmetry this statement provides a good summary of the approach. Its starting point is the Feynman path-integra formuation of fied theory, which encodes a physica information in the partition function Z = DψD ψda a µ e is QCD[ψ, ψ,a a µ ]. 2.64) Here, S QCD = d 4 x L QCD 2.65) denotes the QCD action and we have omitted a favor indices f for simpicity. Physica observabes can be cacuated by considering vacuum expectation vaues of suitabe operators O given by O = 1 DψD Z ψda a µ O[ψ, ψ, A a µ] e is QCD[ψ, ψ,a a µ]. 2.66) Under a Wick rotation that yieds a description in Eucidean spacetime, the exponentia term in the above expression becomes e is QCD[ψ, ψ,a a µ] e SE QCD [ψ, ψ,a a µ ], 2.67)

34 22 Chapter 2. Genera concepts where the superscript E indicates the Eucidean action. With this, compex phase osciations in Eqs. 2.64) and 2.66) go over into exponentia suppression factors of terms away from the cassica i.e., minima) action. The resuting expressions are then numericay we-behaved and one can cacuate the functiona integras over the symboic measure DψD ψda a µ by samping a fieds on a discrete space-time mesh, giving an ordinary four-dimensiona integra at each point. 17 This integra is, of course, a tremendousy high-dimensiona one and cannot be performed with straightforward numerica quadrature rues. Instead, one has to resort to Monte Caro methods to make the cacuation feasibe at a. Even with that, the procedure in genera sti requires supercomputing power, and in practice one furthermore has to cope with a number of probems reated to the numerica treatment, some of which we wi come back to shorty in Section Sti, over the ast few years attice QCD has been very successfu in cacuating hadronic properties from first principes. With steady improvements in both agorithms and computationa faciities, hadron spectroscopy, as one of attice QCD s prime discipines, is moving towards precision cacuations [72, 73]. Beyond that, it is aso possibe to extract resonance properties [74] and pion scattering parameters [75 77], to name just a few exampes. There are furthermore promising efforts to cacuate nucear physics processes directy with attice QCD see, for exampe, Ref. [78] and further references therein). Despite the successes just mentioned, attice QCD is sti far from repacing effective fied theory in ow-energy hadron and nucear physics. Instead, the two methods are to a good extent compementary. For exampe, due to computationa imitations attice cacuations are usuay performed at unphysicay arge quark masses, and resuts from chira perturbation theory are then needed to perform extrapoations of the resuts back to the point of physica quark masses. On the other hand, attice QCD can be used to cacuate ow-energy constants of chira perturbation theory that woud otherwise have to be fixed from experiments [79]. In that sense, it is possibe to cose the gap between the EFT and the underying theory Nucear attice simuations One can go even further and adopt the attice approach directy to perform EFT cacuations. This rather new idea opens the door for efficient cacuations of few- and many-body systems, ranging from nucear physics to condensed matter and atomic physics overviews of computationa methods and appications can be found in Refs. [80, 81]). For nucear physics, the EFT approach is interesting because in attice QCD it is sti very chaenging to extract the properties of systems with more than two nuceons. The reason for this is that in such cacuations most of the computationa effort has to be spent for generating the correct degrees of freedom rather than the interactions between them. An overview of nucear attice simuations based on chira effective theory and appied to seected nucei with mass number up to A = 12 can be found in Refs. [82, 83]. Unike 17 In practice, the integra over the fermion fieds is typicay performed anayticay with the hep of Grassmann variabes, yieding a modified action for the guon fieds that invoves the determinant of a arge matrix.

35 2.5. Lattice cacuations 23 other nucear many-body approaches ike Green s Function Monte Caro GFMC) [84] or the No-Core She Mode NCSM) [85], the nucear attice approach is particuary suited to study nucei with a pronounced custer substructure, such as the famous Hoye state in carbon-12 [86, 87] Numerica aspects A methods mentioned in the preceding sections have in common that by putting the physica system on a discrete space-time mesh one necessariy introduces numerica approximations. Inherent to the approach is that the degrees of freedom no onger reside in a continuous space and time, but ony on fixed sites 18 which are separated by a attice spacing a we focus on the spatia separation here and for simpicity ignore that one can choose a different attice spacing in the time direction). Most prominenty, this quantity enters in the definition of derivatives as finite differences on the attice, which ony gives the continuum resut up to higher orders in a. In practice, a has thus to be kept sma to avoid arge discretization artifacts, and ideay cacuations have to be performed at a number of different attice spacings in order to extrapoate the resuts to their continuum vaues. Furthermore, since computing power is a finite resource, so is the space and time that can be simuated in a attice cacuation. Typicay, one chooses a cube of box ength L with periodic boundary conditions for the spatia simuation voume, which is used to sampe the path integra for a number of time steps L t. Naturay, the voume must not be too sma compared to the typica ength scae of the system one wants to simuate. Both, the need to increase the simuation voume and decrease the attice spacing drive up the computer time and memory) required for a given cacuation. However, whie the continuum extrapoation is something that simpy has to be done, the voume dependence of physica observabes can actuay be used as a too. This idea, pioneered by Lüscher in the 1980s [8, 88], wi pay an important roe in the foowing chapter. 18 Stricty speaking, gauge fieds ike the guons in attice QCD are actuay defined not on the attice sites but on the inks between them.

36 24 Chapter 2. Genera concepts

37 Chapter 3 Finite-voume cacuations Overview In this chapter we derive genera resuts for the mass binding energy) shift of bound states with anguar momentum 1 in a periodic finite voume. Most of the foowing content is the same as pubished in Ref. [4], some resuts of which were first summarized in a etter [2]. Section 3.7 summarizes resuts from Ref. [3] after giving a detaied derivation of the present author s main contribution to that pubication. 3.1 Introduction As aready mentioned at the end of the previous chapter, attice simuations are used in many areas of quantum physics, ranging from nucear and partice physics to atomic and condensed matter physics [78, 80, 81, 89]. In such cacuations, the system is soved numericay using a discrete space-time attice over a finite voume. In practice, this finite voume is usuay taken to be a cubic box with periodic boundary conditions. When simuating composite objects such as bound states, these boundaries of the box modify quantum wave functions, eading to finite-voume shifts in the binding energies of the states. A detaied knowedge of such effects is necessary in order to improve high-precision attice cacuations. In Ref. [8], Lüscher derived a formua for the finite-voume mass shift of S-wave bound states of two partices with reduced mass µ interacting via a potentia with finite range R. When such a state with energy E = E B is put in a periodic cubic box of ength L, its mass energy) in the rest frame 1 is shifted by an amount m B = 3 A κ 2 e κl µl + O e 2κL ), 3.1) where κ = 2µE B is the binding momentum and A κ is the asymptotic wave function normaization defined by ψ B r) = A κ e κr / 4πr) for r > R; cf. Section For 1 Lüscher uses the term mass shift because he was more interested in a reativistic setup. We adopt this convention here, but sometimes aso use the term binding energy shift synonymousy. 25

38 26 Chapter 3. Finite-voume cacuations potentias with exponentia fa-off, V r) exp r/r) for arge r, the formua is modified by exponentiay sma corrections provided that the binding momentum κ is smaer than 1/R. The generaization of Lüscher s formua 3.1) for the finite-voume mass shift to bound states in higher partia waves was briefy discussed in Ref. [2]. In this chapter we present the fu derivation of these resuts as it appeared in Ref. [4]. We give expicit resuts for the mass shift of states with anguar momentum up to = 3 and discuss how, in genera, the mass shift for a given state depends on its transformation properties with respect to the symmetry group of the cubic box. In addition to reducing finite-voume effects in precision attice cacuations, our resuts can aso be used as a diagnostic too to probe the anguar momentum and radia structure of the bound-state wave function. Furthermore, we discuss how the mass-shift formua can be generaized to two-dimensiona systems and different twisted ) boundary conditions in three dimensions). The atter resut is a key ingredient for studying bound states that are moving in a finite periodic voume, which have a topoogica phase correction to the energy [3, 90]. This factor contains information about the number and mass of the constituents of the bound states, and it must be incuded when determining scattering phase shifts for composite objects in a finite voume. For a discussion of how scattering phase shifts in S- and higher partia waves can be extracted from finite-voume energy eves, see Refs. [88, 91]. Our resuts are universa and can be appied to a wide range of systems. In partice physics, for exampe, there is some interest in hadronic moecues with anguar momentum [92 94]. In the case of S-waves, the deuteron and some exotic weaky-bound states such as the H-dibaryon were recenty studied in attice QCD [95]. Simiar investigations for exotic bound states with anguar momentum appear feasibe in the future. In atomic physics, severa experiments have investigated strongy-interacting P-wave Feshbach resonances in 6 Li and 40 K [96 98], which can be tuned to produce bound P-wave dimers. If such systems are simuated in a finite voume, our resuts can be used to describe the voume dependence of the dimer binding energies. Other systems that are reevant in this context are the hao nucei introduced in Section Among these weaky-bound nucei with moecuar character there are some systems with nonzero orbita anguar momentum. A we-known exampe of a P-wave hao state is the J P = 1/2 excited state in 11 Be. The eectromagnetic properties of the ow-ying states in this nuceus can be we described in a two-body hao picture of a 10 Be core and a neutron [59, 99]. In order to study such a system in, for exampe, a nucear attice simuation as discussed in Section 2.5.2, it is crucia to understand the voume dependence. A reated cass of systems is given by nucei with an α-custer structure such as 8 Be and excited states of 12 C [86, 100, 101]. Finay, we point out that the asymptotic normaization coefficient ANC) of the boundstate wave function appears in the mass-shift formua. Our resuts can hence be used to extract this quantity from attice cacuations at finite voumes. The chapter is organized as foows. Based argey on the prerequisites given in Chapter 2, we start with a genera discussion of the finite-voume mass shift in Section 3.2. Lüscher s resut for S-waves is recovered in Section 3.3, whie our extension to higher partia waves is given in Section 3.4. In particuar, we discuss the mass shift for the irreducibe repre-

39 3.2. Bound states in a finite voume 27 sentations of the cubic group, reate the sign of the shift to the eading parity, and derive a trace formua for the mutipet-averaged mass shift of states with arbitrary anguar momentum. In Section 3.5, we verify our resuts numericay for two mode systems. The case of two spatia dimensions is treated in Section 3.6, whie in Section 3.7 we discuss the mass shift for twisted boundary conditions and how it eads to topoogica phase factors in the finite-voume cacuations of composite-partice scattering. Finay, a brief summary and outook is given in Section Bound states in a finite voume As a starting point, we first review severa resuts from Ref. [8]. We cosey foow Lüscher s derivation, but consider a more genera system with arbitrary anguar momentum and non-oca interactions Infinite voume Before we discuss the finite voume, we briefy review the infinite-voume case. Our basic setup is the one discussed in Section 2.1, i.e., we consider a system of two spiness partices with reduced mass µ and zero tota momentum with a rotationay-symmetric finiterange interaction described by the symmetric operator V r, r ) in configuration space. We furthermore assume that the interaction is such that it supports a bound state ψ B with energy E = E B = κ 2 /2µ) and anguar-momentum quantum numbers, m). We consider the fu three-dimensiona wave function in this chapter to capture the anguar dependence of the state. Reca that the finite-range assumption on the potentia impies that V r, r ) = 0 if r > R or r > R. 3.2) The Schrödinger equation can be written as 1 2µ r ψ B r) + in configuration space and for a oca potentia, Ĥ ψ B = E B ψ B, 3.3) d 3 r V r, r ) ψ B r ) = E B ψ B r) 3.4) V r, r ) = V r) δ 3) r r ), 3.5) it reduces to the famiiar form [ 1 ] 2µ r + V r) ψ B r) = E B ψ B r). 3.6) According to Eqs. 2.2) and 2.29), the asymptotic form of the wave function ψ B r) is determined by the Riccati Hanke function H +, ψ B r) = i A κ Y m r/r) H+ iκr) r r > R), 3.7)

40 28 Chapter 3. Finite-voume cacuations regardess of the ocaity of the interaction, and where we have written the spherica harmonics as a function of the unit vector r/r instead of the anges θ and φ. For future reference, we give here the expicit expressions for H + z) for = 0, 1, 2: H 0 + z) = e iz, H 1 + z) = 1 + i ) e iz π/2), z H 2 + z) = 1 + 3i z 3 ) e iz π). z 2 3.8a) 3.8b) 3.8c) Finite voume We now consider what happens when the two-body system is put into a periodic cubic box with edge ength L R. For this probem it is convenient to define a periodic extension of the potentia V L r, r ) = n Z 3 V r + nl, r + nl). 3.9) We take ψ to be an exact periodic soution of the finite-voume Schrödinger equation, Ĥ L ψ = E B L) ψ, 3.10) with the finite-voume Hamitonian ĤL = Ĥ0 + ˆV L and the voume-dependent binding energy E B L). The periodic boundary conditions that we impose require that ψr + nl) = ψr) 3.11) for a integer vectors n Z 3. It is cear that E B L) approaches the infinite-voume eigenvaue E B and that ψ ψ B as L. We now derive a formua for the finite-voume mass energy) shift, m B E B ) E B L). 3.12) To proceed, we define a state ψ 0 by adding together periodic copies of the infinite-voume wave function in 3.4), r ψ 0 = ψ 0 r) = n ψ B r + nl). 3.13) This ceary satisfies the periodicity condition 3.11). Acting upon this state with the

41 3.2. Bound states in a finite voume 29 finite-voume Hamitonian, we get H L ψ 0 r) = H 0 ψ B r + n L) + d 3 r V r + nl, r + nl)ψ B r + n L) n n n = {H 0 ψ B r + n L) + d 3 r V r + n L, r + n L) ψ B r + n L) n + } d 3 r V r + nl, r + nl) ψ B r + n L) n n = E B ) ψ B r + n L) + d 3 r V r + nl, r + nl) ψ B r + n L). n n n n 3.14) The fina resut can be written as Ĥ L ψ 0 = E B ) ψ 0 + η, 3.15) where we have defined η as ηr) = n n n d 3 r V r + nl, r + nl) ψ B r + n L). 3.16) With the substitution r r nl for each term in the sum, this can be rewritten as ηr) = d 3 r V r + nl, r ) ψ B r + n n)l ). 3.17) n n n Due to the finite range of the potentia we ony get contributions from the domain r < R. We note that r + n n)l > R when n n and R L. Therefore, we can use the asymptotic form of the wave function and find that η = O e κl). This means that ψ 0 is an approximate soution of the finite-voume Schrödinger equation 3.10) for arge L. Motivated by this, we write the exact finite- voume soution ψ expicity as ψ = α ψ 0 + ψ with ψ = O e κl). 3.18) We take ψ to be unit-normaized per voume L 3. The same is true of ψ 0 up to corrections of order e κl. We wi choose α in Eq. 3.18) such that ψ 0 ψ = ) Consider now the matrix eement ψ ĤL ψ 0. Acting with ĤL on ψ 0, we get ψ ĤL ψ 0 = E B ) ψ ψ 0 + ψ η = E B ) ψ 0 ψ 0 α + ψ η 3.20) according to 3.15) and 3.18). On the other hand, acting with ĤL on ψ yieds ψ ĤL ψ 0 = E B L) ψ ψ 0 = E B L) ψ 0 ψ 0 α. 3.21)

42 30 Chapter 3. Finite-voume cacuations Combining these two resuts we find E B ) E B L) = m B = We first consider the numerator in this expression. Obviousy, ψ η α ψ 0 ψ ) ψ η = α ψ 0 η + ψ η = α ψ 0 η + O e 2κL). 3.23) Note that the factor of α here wi cance the α in the denominator of Eq. 3.22). We can now simpify further, starting with ψ 0 η = d 3 r d 3 r ψbr + n L) V r + nl, r + nl) ψ B r + n L). 3.24) n n n n For each n we can make the substitutions r r nl and r r nl. These eave the integras invariant, and we get ψ 0 η = d 3 r d 3 r ψb r + n n)l ) V r, r ) ψ B r + n n)l ). 3.25) n n n n Setting m = n n and m = n n yieds ψ 0 η = C d 3 r d 3 r ψbr + m L) V r, r ) ψ B r + ml), 3.26) m m 0 where C counts the number of repeated periodic copies. The fact that C diverges simpy refects the fact that we are working with periodic wave functions with normaization measured per voume L 3, and C wi cance in the fina resut. For the integra to be non-zero, both r and r have to be cose to 0 due to the finite range of the potentia. From the assumption L R it then foows that a terms with m 0 are suppressed by at east a factor of e 2κL, and we have ψ 0 η = C d 3 r d 3 r ψbr) V r, r ) ψ B r + ml) + O e 2κL). 3.27) m 0 The possibe nonvanishing vaues of m are 1, 2, 3,.... We therefore arrive at ψ 0 η = C d 3 r d 3 r ψbr) V r, r ) ψ B r + ml) + O e ) 2κL. 3.28) m =1 For the denominator in 3.22), an anaogous procedure yieds ψ 0 ψ 0 = C d 3 r ψbr) ψ B r + ml) = C [1 + O e κl)] 3.29) m with the same constant C as above. Combining 3.28) and 3.29), the constant cances and we get m B = d 3 r d 3 r ψbr) V r, r ) ψ B r + nl) + O e ) 2κL, 3.30) n =1

43 3.3. S-wave resut 31 where we have renamed m back to n. Eq. 3.30) is a genera resut vaid for any anguar momentum. The dependence of the mass shift on quantum numbers, m) wi emerge from the wave function ψ B and the resuting overap integras in 3.30). In the foowing, we expore this dependence in detai and denote the mass shift as m,m) B. 3.3 S-wave resut For = 0 the asymptotic wave function 3.7) is simpy given as 1 u 0 r) ψ B r) = ψ B r ) = 4π r with n =1 3.31a) u 0 r) = A κ H + 0 iκr) = A κ e κr for r > R. 3.31b) Due to the finite range R L of the potentia we ony have contributions with r +nl > R in Eq. 3.30). Hence we can insert the asymptotic form for ψ B r + nl) and get m 0,0) B = A κ ) d 3 r d 3 r ψb r V r, r +nl ) e κ r 4π r + nl + O e 2κL ). 3.32) We can furthermore use the Schrödinger equation 3.4) to eiminate the potentia. Doing this and then renaming r r, we get m 0,0) B = A {[ ] κ d 3 r r 4π 2µ E ) } B ψb e κ r+nl r r + nl + O e 2κL ) n =1 = A κ ) d 3 r ψb 1 [ r nl r κ 2] e κr + O e ) 3.33) 2κL. 4π 2µ r n =1 In the second ine we have shifted the integration variabe and used partia integration to et the Lapacian act on exp κr)/r. Finay, we use the fact that exp κr)/4πr) is a Green s function for the operator r κ 2, [ r κ 2] e κr 4πr = δ3) r), 3.34) cf. Eq.2.41). This aows us to triviay sove the integra and arrive at m 0,0) πaκ ) B = ψb nl + O e ) 2κL µ n =1 = 3 A κ 2 e κl µl + O e 2κL ). 3.35) ) In the ast step we have inserted the asymptotic form of the wave function for ψb nl = ψb L), and the sum yieds a factor of six. This is just Lüscher s resut 3.1) as given in the introduction.

44 32 Chapter 3. Finite-voume cacuations 3.4 Extension to higher partia waves We now discuss the generaization of the mass-shift formua to arbitrary anguar momentum. The genera form for the asymptotic wave function is ψ B,,m) r) = Y m θ, φ) i A κ H + iκr). 3.36) r Inserting this into Eq. 3.30) and performing steps anaogous to those for the S-wave case, we find m B = { 1 [ d 3 r r κ 2] } ψ 2µ Br nl) Y m θ, φ) i A κ H + iκr) + O e ) 2κL. r n =1 The crucia ingredient is now the reation Y m θ, φ) H+ iκr) r = i) R m 3.37) 1 ) [ ] e κr κ r, 3.38) r where R m are the soid harmonics defined via R mx, y, z) = Rm r) = r Y m θ, φ). The resut 3.38) foows from Lemma B.1 in Ref. [102], which proves that ) 1 R m r )fr) = R m d r) fr) 3.39) r dr for any smooth function fr). From this we obtain Eq. 3.38) by using the reation 2, ) 1 d h 1) 0 z) = 1) z h 1) z) 3.40) z dz and noting that e κr = H 0 + iκr) and H + z) = iz h1) z), where h 1) z) is a spherica Hanke function of the first kind. We can iustrate Eq. 3.38) with an exampe. For the case = 1 and m = 0 we have H 1 + iκr) ) e κr 3.41) κr r and Y1 0 θ, φ) cos θ. A straightforward cacuation using cos θ = z/r shows that indeed cos θ ) e κr = 1 [ ] e κr. 3.42) κr r κ z r Rewriting Eq. 3.37) with the hep of Eq. 3.38), we get m B = A { κ [ r d 3 r κ 2] } { ψ 2µ Br nl) R m 1 ) [ ]} e κr κ r + O e ) 2κL. r n =1 3.43) 2 The reation 3.40) is just a specia case of Eq ) in [103], which aso hods for other spherica Besse functions.

45 3.4. Extension to higher partia waves 33 We now integrate by parts and pass the Lapacian through the differentia operator R m r/κ). Since the operators both consist of partia derivatives, this is not a probem when the wave function is smooth. We assume that this is the case, with the possibe exception of a measure-zero region that can be omitted from the integra. The partia integrations give a factor 1). We can now proceed in exacty the same way as for S-waves. Performing one more integration by parts so that the Lapacian acts on exp κr)/r yieds a deta function times a factor of 4π. The fina resut is then m,m) B = 1) +1 2πA κ µ n =1 R m 1 ) κ r ψb,,m)r nl) + O e 2κL ). 3.44) r=0 For ψb,,m) we can insert the asymptotic form 3.36) since it is evauated in the asymptotic region Resuts For = 1, we find the same resut for a three P-wave states: m 1,0) B = m 1,±1) B = 3 A κ 2 e κl µl + O e 2κL ). 3.45) Compared to the S-wave case, the sign of the P-wave mass shift is reversed whie the magnitude is exacty the same. Quaitativey, this means that S-wave bound states are more deepy bound when put in a finite voume whie P-wave bound states are ess bound compared to the infinite-voume resut). This behavior wi be anayzed in more detai ater. We next discuss the resuts for = 2. From Eq. 3.44) we find where m 2,0) B m 2,±1) B m 2,±2) B = 15 A κ 2 e κl µl F κl) + O e ) 2κL, 3.46) = +15 A κ 2 e κl µl F κl) + O e ) 2κL, 3.47) = 15 A κ 2 e κl µl F κl) + O e ) 2κL, 3.48) F2 0 x) = x x x x4, 3.49) F2 1 x) = 2x + 9x x x 4, 3.50) F2 2 x) = x x x x ) We note that here the size and even the sign of the mass shift both depend on the quantum number m. To understand this effects we need to take into account that our cubic finite voume breaks the rotationa symmetry group down to a cubic subgroup.

46 34 Chapter 3. Finite-voume cacuations Representations of the cubic group The cubic symmetry group O is a finite subgroup of SO3) with 24 eements. There are five irreducibe representations of O, conventionay caed A 1, A 2, E, T 1, and T 2. Their dimensionaities are 1, 1, 2, 3, and 3, respectivey. Irreducibe representations D of the rotation group SO3) are reducibe with respect to O for > 1. For further detais about the decomposition see, for exampe, Ref. [104]. In our discussion we assume that the infinite-voume system has no partia wave mixing, such that orbita anguar momentum is a good quantum number. We aso assume that there are no accidenta degeneracies in the bound state spectrum, so we can use as a abe for the famiy of cubic representations spit apart at finite voume. Parity invariance remains unbroken by the cubic voume, and we have P = 1) just as in the infinitevoume case. For carity, however, we wi indicate parity expicity with ± superscripts in the foowing. With our assumptions, an S-wave state in infinite voume wi map onto an A + 1 state at finite voume. Aso a P-wave tripet wi simpy map onto the three eements of the T1 representation at finite voume. For = 2, however, the five D-wave states are spit into a T 2 + tripet and an E + doubet, D 2 = T + 2 E ) In the foowing we use the notation Γ, ; α, α = 1,..., dimγ) for the basis vectors of the irreducibe cubic representations. We can rewrite the finite voume mass shift in Eq. 3.30) as m Γ,,α) B Γ, ; α ˆV n =1 ˆT nl) Γ, ; α, 3.53) where ˆT x) is the transation operator for dispacement by a vector x. We can aso cacuate the matrix eements of m B in the, m) basis. In this case there wi be offdiagona matrix eements connecting, m) and, m ) when m and m are equivaent moduo 4. According to Ref. [104], the unitary transformation between the two basis sets for the five D-wave states is and So, for exampe, we have m T + 2,2;1) B = 1 2 T + 2, 2; 1 = 1 2 2, 1 + 2, 1 ), T + 2, 2; 2 = i 2 2, 1 2, 1 ), T + 2, 2; 3 = 1 2 2, 2 2, 2 ) 3.54a) 3.54b) 3.54c) E +, 2; 1 = 2, 0, 3.55a) E +, 2; 2 = 1 2 2, 2 + 2, 2 ). m 2, 1, 1) B + 2 m 2, 1,1) B + m 2,1,1) B = 15 A κ 2 e κl µl 2 κl + 9 κ 2 L κ 3 L κ 4 L 4 ) + O e 2κL ), ) 3.55b) 3.56)

47 3.4. Extension to higher partia waves 35 where we have defined m,m 1,m 2 ) B = 1) +1 2πA κ µ n =1 R m 1 as a straightforward generaization of Eq. 3.44). 1 ) κ r ψb,,m 2 )r nl) + O e r=0 2κL ) 3.57) As expected from cubic symmetry, the mass shift is the same for a three T 2 + states and aso within the E + doubet. To summarize our resuts, we write the mass shift for a state beonging to irreducibe representation Γ with anguar momentum as m,γ) B = α ) 1 κl Aκ 2 e κl µl. + O e 2κL ) 3.58) We ist the coefficients α 1 κl) for = 0,..., 3 in Tabe 3.1. Γ αx) 0 A T T x + 135x x x 4 2 E + 1 / x + 405x x x 4 ) 3 A 2 315x x x x x 6 3 T2 1 /2 105x + 945x x x x x 6 ) 3 T1 1/ x + 735x x x x x 6 ) Tabe 3.1: Coefficient αx) in the expression for the finite-voume mass shifts for = 0,..., 3. Γ indicates the corresponding representation of the cubic group Sign of the mass shift The sign of the finite-voume mass shift can be understood in terms of the parity of the wave function. In infinite voume the tai of each bound state wave function must vanish at infinity. In the finite voume, however, the bound state wave functions with even parity aong a given axis can remain nonzero everywhere. Ony the derivative needs to vanish, and the kinetic energy is owered by broadening the wave function profie. On the other hand, a wave function with odd parity aong a given axis must change sign across the boundary. In this case the wave function profie is compressed and the kinetic energy thus increased. We have iustrated both cases for a one-dimensiona square-we potentia in Fig In three dimensions, the situation is sighty more compicated, which can be seen from the fact that for = 2 the sign of the mass shift depend on the representation of the cubic group even though the parity is just 1) 2 = +1 for a states. In order to understand this we consider the basis poynomias for the cubic representations. These basis poynomias are obtained by decomposing the cubic basis vectors in terms of soid harmonics which

48 36 Chapter 3. Finite-voume cacuations ψ odd ψ even x Figure 3.1: Wave functions with even bottom) and odd parity top) for a onedimensiona square we potentia in a box with periodic boundary conditions. The dashed ines give the infinite voume soutions for comparison. are homogeneous poynomias in x, y and z. For = 0,..., 4 the basis poynomias are given expicity in [102]. For a given poynomia P x, y, z), we define its eading parity as p P = 1) dmax, 3.59) where d max = max{deg x P, deg y P, deg z P } 3.60) is the maximum degree of P with respect to any one of the three variabes. It is this eading parity that determines the asymptotic behavior of the mass shift as κl. More precisey, we have α ) 1 κl 1) d max+1 1 dmax κl) as κl 3.61) for the α 1 κl) in Eq. 3.58). It can easiy be checked that this reation hods for a resuts presented in Tabe 3.1. For = 2, for exampe, we have the basis poynomias P 2,T + 2 xy, yz, zx, P 2,E + x 2 y 2, y 2 z 2, 3.62a) 3.62b) and hence d max = 1 for the T + 2 representation and d max = 2 for the E + representation.

49 3.4. Extension to higher partia waves Trace formua The expressions for the finite-voume mass shift become simper when we sum over a m for a given. We can rewrite Eq. 3.44) as m,m) B = 1) +1 2πA κ R m 1 ) µ κ r ψb,,m)r) + O e 2κL ). 3.63) r=nl n =1 Inserting the asymptotic form of the wave function, ψ B,,m)r) and using Eq. 3.38) a second time yieds m,m) B = 1) +1 2π A κ 2 µ [ = Y m θ, φ) i A κ H + iκr) ], 3.64) r=nl r r=nl n =1 R m Now, from the we-known reation 1 ) κ r R m 1 κ r ) [ e κr r ] +O e 2κL ). r=nl 3.65) m= Y m θ, φ)y m θ, φ) = π 3.66) and R mr) = r Y m θ, φ) we get an anaogous expression for the soid harmonics, which then carries over to R m 1 ) κ r R m 1 ) κ r fr) = κ2 4π r) fr) 3.67) m= for any sufficienty smooth function fr). Finay, we have r ) e κr r 2 e κr = κ r r 0), 3.68) which foows from Eq. 3.34). Putting everything together, we arrive at m= m,m) B = 1) +1 2π A κ µ 4π n =1 1 κ 2 r) [ e κr r ] + O e 2κL ) r=nl = 1) ) 3 A κ 2 e κl µl + O e 2κL ), 3.69) where the sum just yieds a factor of six. Dividing by 2 + 1, we obtain the average mass shift for states with anguar momentum, m ) B = 1)+1 3 A κ 2 e κl µl + O e 2κL ). 3.70)

50 38 Chapter 3. Finite-voume cacuations Apart from the aternating sign, this average shift is independent of. Eq. 3.70) can be verified expicity for the the resuts presented in Section cf. Tabe 3.1). For = 2, for exampe, one has to average over the three-dimensiona representation T + 2 and the two-dimensiona representation E Numerica tests In order to verify our predictions numericay, we put the Schrödinger equation 3.10) on a discrete spatia attice such that the Hamitonian becomes an ordinary matrix. We then cacuate the corresponding energy eigenvaues and eigenvectors Lattice discretization We use a hat symbo to denote dimensioness attice units. For exampe, we have ˆL = L/a and ÊB = E B a, 3.71) where a denotes the attice spacing. The free attice Hamitonian is given by [ Ĥ 0 = ˆn 3 ˆµ a ˆn)aˆn) 1 a ˆn)aˆn + ê ) + a ˆn)aˆn ê ) )] 3.72) 2ˆµ =1,2,3 where a ˆn) and aˆn) are creation and annihiation operators for a attice site ˆn and ê is a unit vector in the -direction. The corresponding attice dispersion reation is with the attice function Q 2 ˆq) = 2 =1,2,3 ʈq) = Q2 ˆq) 2ˆµ 1 cos ˆq i ) = =1,2,3 ˆq ) [ 1 + Oˆq 2 ) ] 3.74) and the attice momenta ˆq = 2πˆn/ˆL. 3.75) The binding momentum for a bound state with energy ÊB is determined by ˆµÊB = 1 cos iˆκ) = 1 coshˆκ). 3.76) The attice Green s function for the Hamitonian 3.72) is Ĝˆn, Ê) = 1 L 3 ˆq e iˆq ˆn Q 2 ˆq) + 2ˆµÊ. 3.77) 3 Note that the mapping from the anguar momentum eigenstates to the cubic group states is a unitary transformation.

51 3.5. Numerica tests 39 We impose periodic boundary conditions by defining the distance ˆr to the origin as ˆr ˆn) = =1,2,3 ) } 2 min {ˆn ˆL 2, ˆn. 3.78) Methods We cacuate the mass shift using three different methods: 1. As a direct difference in energies, Eq. 3.12), where we use a very arge voume L ) to approximate the infinite-voume resut. 2. From the overap formua 3.30). 3. Using discretized versions of Eqs. 3.35) and 3.45), which we obtain by repacing exp κr)/r with the attice Green s function. More precisey, we write the asymptotic bound-state wave function 3.7) as ψ B r) = i A κ Y m r/r) H + iκr) eκr 4πG κ r) for r > R 3.79) and repace the continuum Green s function G κ r) = e κr 4πr 3.80) with the attice version Ĝˆκ ˆn) Ĝ Effectivey, this amounts to the repacement in the mass-shift formua. ) ˆn, ˆκ ) 2ˆµ e ˆκ ˆL/ˆL 4πĜˆκˆL, 0, 0) 3.82) The attice Green s function is aso used to cacuate the asymptotic normaization A κ from the attice data. This procedure has the advantage of avoiding arge discretization errors Resuts In the foowing we report physica quantities in units where the reduced mass µ is set to one.

52 40 Chapter 3. Finite-voume cacuations Gaussian potentia We first use a Gaussian potentia, V Gauss r) = V 0 exp r 2 /2R 2 ) ), 3.83) with R = 1 and V 0 = 6. This potentia does not have a finite range in a strict mathematica sense, but the range corrections can be entirey negected in comparison with other errors in our numerica cacuation. The smoothness of the Gaussian potentia heps to minimize attice discretization artifacts. In Fig. 3.2 we show the S- and P-wave mass shifts obtained with the three methods described in Section The resuts from the three different methods described above agree we for both S- and P-waves. In order to compare the dependence on the box size L with the predicted behavior we have potted ogl m B ) against L we use the absoute vaue of m B since the S-wave mass shift is negative). For both S- and P-waves, the expected inear dependence is ceary visibe. 0 V = V Gauss og L mb ) S-wave direct difference overap integra Green s function P-wave L Figure 3.2: S-wave and P-wave mass shifts ogl m B ) as functions of the box size L in attice units) for a Gaussian potentia. We show the resuts obtained from the direct difference Eq. 3.12) crosses), evauation of the overap integra Eq. 3.30) squares), and discretized versions of Eqs. 3.35), 3.45) circes). The dashed ines show inear fits to the overap integra resuts. When we perform a inear fit to the overap integra data dashed ines in Fig. 3.2) we obtain κ = ± 0.005, A κ = 11.5 ± 0.2 for the S-wave resuts and κ = ± 0.004, A κ = 7.0 ± 0.1 for the P-wave resuts. The vaues for the asymptotic normaization constants are in good agreement with the resuts A κ 11.5 S-wave) and A κ 7.2 P-wave) that are obtained directy from the L = 40 data. Inserting the corresponding energy eigenvaue into the attice dispersion reation 3.76), we find κ S-wave) and κ P-wave), again in very good agreement with the fit resuts. The remaining sma discrepancies can be attributed to the mixing with higher partia waves induced by the attice discretization and the fact that we have not performed a continuum extrapoation to vanishing attice spacing.

53 3.5. Numerica tests 41 Simpe step potentia For a simpe step potentia, V step r) = V 0 θr r), 3.84) which we use with R = 2 and V 0 = 3, the numerica cacuation becomes more difficut because the discontinuous shape introduces considerabe attice artifacts. Sti, we discuss it here due to its strict finite range and since we find that for a sma attice spacing of a = 0.2 the resuts are satisfactory. In Fig. 3.3 we show a pot anaogous to the one presented for the Gaussian potentia. Again, the resuts from the different methods agree we and the expected inear behavior is ceary visibe. Furthermore, the resuts from the three methods agree we with each other aready for smaer L compared to the resuts for the Gaussian potentia), as expected from the fact that the step potentia does not have a tai. 0 V = V step og L mb ) S-wave direct difference overap integra Green s function P-wave L Figure 3.3: S-wave and P-wave mass shifts ogl m B ) as functions of the box size L in attice units) for a simpe step potentia. The symbos are as in Fig From fitting to the overap integra data dashed ines in Fig. 3.3) we obtain κ = ± , A κ = ± 0.06 for the S-wave resuts and κ = ± , A κ = 12.48±0.05 for the P-wave resuts. From the L = 40 data we find κ , A κ 29.6 S-wave) and κ , A κ 12.8 P-wave). Given that we do not have error estimates for the L = 40 resuts, the overa agreement is quite good. Finay, we aso check our resut for the D-wave mass spittings, using again the step potentia with a = 0.2. In Fig. 3.4 we show the mass shift for the D-wave states in both the T 2 + and the E + representation. Due to the poynomia coefficients α 1 κl) see Eq. 3.58) and Tabe 3.1 one does not expect a inear dependence on L for ogl m B ). Hence, we simpy pot m B as a function of L directy and do not perform a fit. Nevertheess, we see that except for very sma L, where obviousy the condition L R is not satisfied) the agreement between the three methods to cacuate m B is very good and hence concude that our mass-shift formua indeed gives the right resut aso for = 2.

54 42 Chapter 3. Finite-voume cacuations mb V = V step, D-wave, T2 rep. direct difference overap integra Green s function L mb direct difference overap integra Green s function V = V step, D-wave, E rep L Figure 3.4: D-wave, mass shift m B for T 2 + rep. eft pane) and E + rep. right pane) as a function of the box size L in attice units) for a simpe step potentia. The symbos are as in Fig Two-dimensiona systems In this section we derive a formua for the finite-voume or rather finite-area) mass shift of bound states in two-dimensiona systems. The resuts can be used, for exampe, in attice simuations of cod atomic systems, which can be prepared experimentay to be effectivey two-dimensiona [105, 106]. We note that the S-wave case in two dimensions was previousy investigated in Ref. [107]. In two dimensions, the Schrödinger equation is 1 2µ 2D r ψ B r) + d 2 r V r, r ) ψ B r ) = E B ψ B r) 3.85) with [ 1 2D r ψ B r) = r r + 2 r r 2 2 θ 2 ] ψ B r) 3.86) in poar coordinates. States are described by a singe anguar momentum quantum number m = 0, ±1, ±2,..., and for the wave function we have the separation ψ B r) = u m r)y m θ) 3.87) with Y m θ) = eimθ 2π. 3.88) The two ineary independent soutions of the free radia equation d 2 dr + 1 ) d 2 r dr m2 r + 2 p2 u m r) = 0, 3.89) are just the Besse and Neumann functions J m pr) and N m pr). For a bound state, we have p 2 = κ 2 = 2µE B, and the wave function has the asymptotic form u m r) = A κ K m κr) for r > R, 3.90)

55 3.6. Two-dimensiona systems 43 where K m is the modified Besse function of the second kind. It is reated to the Hanke function of the first kind, H m 1) z) = J m z) + in m z), 3.91) via K m x) = π 2 im+1 H 1) m ix). 3.92) As in the three-dimensiona case, A κ is the asymptotic normaization constant. Inserting Eq. 3.92) into Eq. 3.90) yieds a form which is more simiar to the three-dimensiona expression. To render the anaogy to the cacuations in Section 3.4 as expicit as possibe, we wi use the Hanke function in the foowing intermediate steps and ony express the fina resuts in terms of the modified Besse function. Neary a of the three-dimensiona cacuation carries over if we just repace a exponentia terms with Hanke functions. The overap integra for the mass shift is now = ) ) d 2 r d 2 r ψb,mr) V r, r ) ψ B,m r + nl) + O 2κL. 3.93) m m) B n =1 From the asymptotic form of the Hanke function, H 1) m z) ih 1) m 2 m πz eiz 2 π π 4 ) as z, 3.94) it is cear that in principe we sti have an exponentia behavior. In deriving Eq. 3.93) we have used this to write ) ) O H m 1) 2 iκl O )) ) H m 1) 2iκL O H ) m 1) 2iκL. 3.95) In the foowing we wi simpy write the correction terms as O e 2κL ), as in the threedimensiona case. The two-dimensiona anaog of the reation 3.38) is Y m θ)h m 1) iκr) = i) m R m 1 ) κ 2D r H 1) 0 iκr), 3.96) where R m r, θ) = r m Y m θ). This foows from and 1 z R m 2D r )fr) = R m r) d dz 1 r d dr ) m fr) 3.97) ) m H 1) 0 z) = 1) m z m H 1) m z). 3.98) The derivation of Eq. 3.97) can be carried out in the same manner as the three-dimensiona proof of Lemma B.1 in [102], using the expansion of e ip r 2D vectors) in terms of Besse functions. As the fina ingredient we have [ 2D r κ 2] i 4 H1) 0 iκr) = δ 2) r). 3.99)

56 44 Chapter 3. Finite-voume cacuations Using a this in steps competey anaogous to those in three dimensions, we get m m) B = 1)m+1 πa κ R m 1 ) µ κ 2D r ψb,mr nl) + O e 2κL ) ) r=0 n =1 For m = 0 two-dimensiona S-waves), this directy yieds m 0) B = 2 A κ 2 µ K 0κL) + O e 2κL ) ) In fact, Eq ) can be simpified further. Inserting the asymptotic form for the wave function for ψb,m and using 3.96) a second time gives m m) B = 1)m+1 π A κ 2 µ n =1 From Eq. 3.88) and R m r) = r m Y m θ) it is cear that R m 1 ) κ 2D r Rm 1 ) [ κ 2D r i π ] 2 H1) 0 iκr) R m θ)r mθ) = r2 ) m which then yieds R m 1 ) κ 2D r Rm 1 ) κ 2D r fr) = 1 κ 1 2m 2π r=nl + O e 2κL ) ) 2π, 3.103) 2D r ) m fr) 3.104) for any sufficienty smooth fr). This is essentiay the same reation that we used to derive the trace formua in the three-dimensiona case, ony that here we do not have to sum over different m. Together with the two-dimensiona anaog of Eq. 3.68), we then get m m) B = 1)m+1 Aκ 2 2µ = 1) m+1 2 A κ 2 2D r ) m H 1) 0 iκr) = κ 2m H 1) 0 iκr) r 0), 3.105) n =1 1 ) [ 2D m κ 2m r i π ] 2 H1) 0 iκr) µ K 0κL) + O e + O e 2κL ) r=nl 3.106) 2κL ). As we sha see in the foowing, this is the fina resut for m = 0 and any odd m, whereas for even m 0 things become sighty more compicated. In genera, we have to take into account that the finite voume breaks the origina panar rotationa symmetry of the system down to the symmetry group of a square. We find

57 3.7. Twisted boundary conditions 45 that states with the same absoute vaue of m may mix to form good eigenstates in the finite voume. More precisey, we have the symmetric and antisymmetric combinations m, ± = 1 2 m ± m ) 3.107) for m 0. When we cacuate the mass shift for these states in the same way as described in Section 3.4.1), we get mixing terms of the form m m,mixed) B = 1) m+1 πa κ µ n =1 R m 1 ) κ 2D r Since the condition for the mixing of states is ψ B, mr) + O e 2κL ) ) r=nl 2m 0 mod 4, 3.109) they do not pay a roe for odd m in fact, they vanish in this case). For even m, however, we have to take them into account and find m m,±) B = 1 ) m m) B 2 ± 2 mm,mixed) B + m m) B 3.110) as our fina resut. As an iustration, we give the expicit resuts for m = 2: m 2,+) B = 4 A κ 2 [ ) κ µ 2 L 2 K0 κl) ) 24 κl κ 3 L 3 K1 κl) ] + O e ) 2κL, m 2, ) B = 16 A κ 2 [ 3 K κ µ 2 L 2 0 κl) ) 6 κl κ 3 L 3 K1 κl) ] + O e ) 2κL a) 3.111b) 3.7 Twisted boundary conditions In this section, we go back to the three-dimensiona case and discuss a generaization of the mass-shift formua obtained by changing the boundary condition imposed on the finite-voume wave function. Instead of the periodicity 3.11) we now require that ψr + nl) = ψr) e iθ n 3.112) for a n Z 3, where θ is an arbitrary vector of phases. It is cear that setting θ = 0 in this so-caed twisted boundary condition gives back Eq. 3.11). As we wi discuss in more detai shorty in Section 3.7.2, a boundary condition of this form arises when one considers a system of more than two partices, two of which form the bound state whose mass shift we are interested in. In such a setup one has to consider the fu two-partice wave function Ψr 1, r 2 ) = ΨR, r) = e ip R ψr), 3.113) where r = r 1 r 2 and R = η 1 r 1 + η 2 r )

58 46 Chapter 3. Finite-voume cacuations are the reative and center-of-mass coordinates, respectivey, and η 1,2 are the mass ratios η 1 = m 1 m 1 + m 2, η 2 = 1 η ) If one demands that Ψr 1, r 2 ) is periodic in both coordinates, e.g., Ψr 1 + nl, r 2 ) = e ip R e iη 1LP n ψr + nl) = Ψr 1, r 2 ), 3.116) one finds that the behavior of ψr + nl) has to cance the additiona phase, ψr + nl) = e iη 1LP n ψr) ) This is just Eq ) with θ = η 1 LP. boundary condition in the form More generay, one can of course write the Ψr 1 + n 1 L, r 2 + n 2 L) = e ip R e iη 1LP n 1 e iη 2LP n 2 ψr + n 1 n 2 )L) = Ψr 1, r 2 ), 3.118) but this again gives Eq ) since e iη 1LP n 1 e iη 2LP n 2 = e iη 1LP n 1 e i1 η 1)LP n 2 = e iη 1LP n 1 n 2 ), 3.119) where the ast equaity foows by noting that the tota momentum for the center-of-mass movement P is quantized in the finite voume, P = 2π L K, K Z ) Keeping in mind these considerations as a motivation, we now derive the the finite-voume mass shift for a the two-partice system with twisted boundary conditions without writing θ in terms of the momentum P, however, but rather keeping it as an arbitrary parameter Generaized derivation We carry out the derivation aong the ines aid out in Section 3.2 and start by making an ansatz ψ θ for the finite-voume wave function of the form r ψ θ = ψ θ r) = n ψ B r + nl) e iθ n ) Like the ψ 0 defined in Eq. 3.13), this has at east the correct boundary behavior: ψ θ r + nl) = ψ B r + n L + nl) e iθ n n = ψ B r + n L) e iθ n n) = ψ θ r) e iθ n. n =n +n 3.122)

59 3.7. Twisted boundary conditions 47 Acting on this state with the finite-voume Hamitonian, we get H L ψ θ r) = H 0 d 3 r V r + nl, r + nl)ψ B r + n L) e iθ n ψ B r + n L) e iθ n + n n,n = { [ ] H 0 ψ B r + n L) + d 3 r V r + n L, r + n L) ψ B r + n L) n + } d 3 r V r + nl, r + nl) ψ B r + n L) e iθ n n n e iθ n = E B ) n ψ B r + n L) e iθ n + n n n which anaogousy to Eq. 3.15) we write as with η θ r) = n n n d 3 r V r + nl, r + nl) ψ B r + n L) e iθ n, 3.123) Ĥ L ψ θ = E B ) ψ θ + η θ 3.124) d 3 r V r + nl, r + nl) ψ B r + n L) e iθ n ) We see that in the end the phase factor e iθ n is simpy carried through the whoe cacuation. Since furthermore it does not depend on the variabe r, we can now concude that a steps carried out in Section remain vaid in spite of its presence and directy arrive at the the integra formua for the finite-voume mass shift with twisted boundary conditions: m B θ) = d 3 r ψbr) V r) ψ B r + nl) e iθ n + O e ) 2κL ) n =1 For S-wave states, the fina resut then is m 0) B θ) = A κ 2 e κl µl n=ê x,ê y,ê z cosθ n) + O e 2κL ), 3.127) i.e., the phases from pairs of opposite directions combine to give cosine factors. Again, it is cear that setting θ = 0 gives back the od resut 3.35) for periodic boundary conditions. For P-wave states, there is now a dependence on the quantum number m: m 1,0) B θ) = A κ 2 e κl µl 1 + κl)cos θ x + cos θ y ) κl2 + κl)) cos θ z, κ 2 L a) m 1,±1) B θ) = A κ 2 e κl µl 1 + κl1 + κl))cos θ x + cos θ y ) 21 + κl) cos θ z. 2κ 2 L b)

60 48 Chapter 3. Finite-voume cacuations However, this goes away as it shoud if one sends θ 0. For m = 0, for exampe, one finds that 1 + κl)cos θ x + cos θ y ) κl2 + κl)) cos θ z = 21 + κl) κl2 + κl)) = κ 2 L 2, 3.129) canceing the additiona factor in the denominator and thus giving back the od resut. Improved voume dependence Aready in the two-body sector the resuts given above are actuay quite interesting. From Eq ), for exampe, one can see that for a bound state with equa mass constituents such that the mass ratios are η 1 = η 2 = 1/2), the eading finite-voume mass shift can be made to vanish if one chooses to perform the cacuation in a boosted frame with K = 1, 1, 1)/2 rather than in the rest frame of the bound state, which woud be the naïve choice. In Ref. [3] it was argued and demonstrated numericay that this procedure can be generaized to bound states of more than two partices if one assumes that there is no custer substructure. Furthermore, in Ref. [90] Davoudi and Savage discuss a genera method for reducing finite-voume corrections in two-body cacuations by choosing appropriate combinations of boosted frames Topoogica voume factors As mentioned at the outset, the mass shift for twisted boundary conditions is important for systems of more that two partices in a finite voume, two of which form a dimer bound state. In the foowing, we wi discuss how these shifts yied correction factors that have to be taken into account in finite-voume determinations of scattering phase shifts where one or more of the partices is composite. 4 To this end, consider the scattering at ow energies of two partice abeed A and B, both of which can be either eementary pointike) or composite bound states. For the sake of definiteness we take A to be a pointike atom and B to be a dimer bound state. In order to extract the S-wave scattering phase shift δ 0 p) from finite-voume cacuations, a popuar strategy is to appy Lüscher s formua p cot δ 0 p) = 1 ) 2 Lp πl Sη), η =, 3.130) 2π where p is the center-of-mass momentum of the A B system and Sη) = 4π Z 00 1, η) = im s 1 n Z 3 1 n 2 η) s 3.131) 4 The reations presented in the foowing, pubished in Ref. [3], were derived by D. Lee [108]. The present author s main contribution to Ref. [3] was the derivation of the twisted-boundary mass-shift formua.

61 3.7. Twisted boundary conditions 49 is the Lüscher Zeta-function see, e.g., Ref. [102]). 5 The crucia point here is that the momentum p is determined from energy eves of the A B system in the finite voume which, in turn, are functions of the box size L, i.e. p = p E AB L) ) ) If now at east one of the partices A and B is a composite bound state, there wi be a contribution to the voume dependence of the tota energy that is soey due to the mass shift of the bound state. In order to appy Lüscher s formua 3.130), this contribution has to be determined and subtracted. Turning back to our atom dimer exampe, we assume that the underying interactions between the atoms has a finite range R and impose periodic boundary conditions. We denote an A B scattering state with momentum p = p in the center-of-mass frame by Ψ p and write it as r Ψ p = c ei 2πk L r ) ) 2πk/L p 2 k Z 3 in configuration space, with some normaization constant c. In writing this expression, which corresponds to a pure S-wave, we have assumed that contributions from higher partia waves can be negected. It is then vaid as an approximation for a r outside shifted copies of the finite-range interaction if we negect exponentiay-suppressed contributions in the effective interaction between composite partices that are introduced by the bound-state wave functions. As discussed in Ref. [3], this is vaid because for the current derivation we are ony interested in the contributions due to the shifts in the binding energies. According to the formuas derived above, these are aso exponentiay sma, but their contributions to the scattering are suppressed further by inverse powers of L. Note that upon discretization the expression in Eq ) essentiay goes over into the attice Green s function defined in Section Let Ĥ denote the Hamitonian for the system we are considering. Acting with this on the state Ψ p then gives [ ] ) 2 L r Ĥ Ψ e i 2πk L r p 2 2µ AB + E 2πk/L A L) + E 2πk/L B L) p = c 3.134) 2π k 2 η k Z 3 in configuration space [108], where µ AB is the reduced mass of the A B system and with η as defined in Eq ). In this expression, E ±2πk/L A,B L) are the energy contributions at voume L) due to the binding of the states A and B, moving with momentum ±2πk/L. By definition, they are zero for point partices. In particuar, for our specific exampe here we have E 2πk/L A L) = 0 for a L, but we keep it in the equations in order to give a more genera resut. The tota energy of the A B system is then E AB p, L) = Ψ p Ĥ Ψ p Ψ p Ψ p = 1 N k Z 3 p 2 2µ AB + E 2πk/L A L) + E 2πk/L B L) 3.135) k 2 η) 2 5 As discussed in Ref. [102], Z 00 s, η) is initiay defined by the sum in Eq ) for Res) > 3/2 and then extended to the whoe compex pane by anaytic continuation; this is what we express with the imit in Eq ).

62 50 Chapter 3. Finite-voume cacuations with the normaization factor N = k k 2 η ) ) The voume dependence of E 2πk/L B L) is known from the derivation in Section According to Eq ), it is given as E B k, L) E 2πk/L B = m 0) B L) E 2πk/L B ) θ = 2πηB k ) = A κ 2 e κl µ B L i=x,y,z cos 2πη B k i ) + O e 2κL ), 3.137) where µ B and η B are the reduced mass and mass ratio of the constituents comprising the state B. If the state A is composite as we, one has an anaogous expression for the contribution E 2πk/L A L). The tota voume dependence of the energy E AB can then be written as E AB p, L) E AB p, ) = τ A η) E A 0, L) + τ B η) E B 0, L) 3.138) with the topoogica voume factors τ A,B η) = 1 N k i=1,2,3 cos 2πη A.Bk i ) 3 k 2 η) ) They are obtained by inserting the expressions for E A,B k, L) into the sum over a k in Eq ) and factoring out the voume dependence of the states A and B at rest. Eq ) is the desired contribution to the voume dependence of the energy eves E AB determined in numerica cacuations) that shoud be subtracted before appying Lüscher s formua 3.130) for an extraction of the scattering phase shift. Figure 3.5 demonstrates the importance of this procedure. For the atom dimer more generay caed fermion dimer here) system it shows attice resuts for the scattering ength and effective range cacuated from S-wave phase shifts that were extracted using Lüscher s finite-voume formua 3.130). The resuts shown in the pot were obtained in a cacuation by Bour et a., the fu detais of which can be found in Ref. [109]. They are shown here with the kind permission of the authors. Using dimensioness units obtained by rescaing a quantities with the dimer binding momentum κ D, both the scattering ength a FD and the effective range r FD are potted against different vaues of the attice spacing a att on the x-axis, such that a continuum extrapoation can be performed. The two data sets shown in the pots were obtained using two different attice Hamitonians that yied the same continuum imit. One ceary sees that the correct vaues indicated by the bue trianges in Fig. 3.5) are ony reached when the topoogica corrections factors are incuded in the cacuation. The effect is particuary prominent for the effective range, where otherwise the resut is competey off. It shoud be pointed out here that the effect of the topoogica voume contributions is so strong in the chosen exampe because the dimer was tuned to be very shaow and

63 3.8. Summary and outook 51 the voumes used in the numerica cacuation were not very arge. Together, these two factors enhance the reative importance of the binding-energy shifts. In a cacuation with deeper dimers and/or arger voumes, the effect can be much weaker or even negigibe. However, in genera it is aways there and shoud be taken into account, especiay when the exact situation is a priori unknown. κdafd κdrfd H 1, fu H 2, fu H 1, τη) = 1 H 2, τη) = 1 STM equation κ D a att Figure 3.5: Lattice resuts and continuum extrapoation with error estimates for the fermion dimer scattering ength top) and effective range parameter bottom). For comparison we show the continuum resuts obtained via the Skorniakov Ter-Martirosian equation. 3.8 Summary and outook In this chapter we have derived expicit formuae for the mass shift of P- and higher-wave bound states in a finite voume and discussed their decomposition into states transforming according to the representations of the cubic group. We have compared our numerica

64 52 Chapter 3. Finite-voume cacuations resuts for 2 with numerica cacuations of the finite-voume dependence for attice Gaussian and step potentias and found good agreement with the predictions. For 2, the mass shift of a given state, m) depends on the anguar momentum projection m due to the breaking of rotationa symmetry. Averaged over a m in a mutipet, however, the absoute vaue of the mass shift is even independent of. The mass shift for states in representations of the cubic group is the same for a states, and its sign can be understood from the eading parity of the representations. We have furthermore derived corresponding expressions for the finite-voume mass shift in two-dimensiona systems. With the known voume dependence, attice cacuations provide a method to extract asymptotic normaization coefficients of bound state wave functions, which are of interest, for exampe, in ow-energy astrophysica capture reactions. Using a reation that wi be discussed in more detai in Chapter 5, the asymptotic normaization and binding momentum of a shaow bound state can be used to extract the effective range from a simuation. Furthermore, we have shown how twisted boundary conditions arise if one studies dimer states in moving frames and how the mass shift for this case eads to topoogica corrections factors that have to be taken into account in finite-voume cacuations of compositepartice scattering. A precise knowedge of these corrections is particuary important for processes invoving shaow dimer states in voumes that are not very arge. Our work provides a genera framework for future attice studies of moecuar states with anguar momentum in systems with short-range interactions. Appications to nucear hao systems and moecuar states in atomic and hadronic physics appear promising. An important next step woud be to incude Couomb effects into the framework in order to investigate the voume dependence of bound states of charged partices, which are much easier to treat experimentay. In particuar, this extension of the formaism is important if one wants to describe proton-hao nucei. Another interesting direction woud be to anayze the voume dependence of resonances aong the ines of Refs. [104, 110].

65 Chapter 4 The Couomb force Overview The Couomb potentia, athough notoriousy difficut to hande due to its ong-range nature, is one of the most important and probaby most thoroughy investigated interactions in quantum mechanics. We review here resuts from the vast iterature on the subject with a focus on aspects that wi be important for the foowing two chapters of this thesis. The first part, where we introduce the Couomb wave functions and the modified effective range expansion, is based argey on the introductory section of Ref. [5], but provides some additiona detais. In the second part we discuss the fu off-she Couomb T-matrix. In particuar, we give an approximate expression for this function in the case of a Yukawa-screened Couomb potentia, originay derived by Gorshkov [111]. This section contains, at east to the author s best knowedge, some resuts that have not previousy been pubished. 4.1 Couomb wave functions For a pure Couomb interaction the radia Schrödinger equation for two partices carrying eectromagnetic charges Z 1 e and Z 2 e reads with the Couomb parameter p 2 w r) = d2 dr w + 1) r) + w 2 r 2 r) + γ r w r) 4.1) γ = 2µ αz 1 Z 2, 4.2) and the fine-structure constant α = e2 4π ) We use here the etter w to denote the wave functions in order to distinguish them from the u r) describing the asymptoticay non-interacting neutra system discussed in Chapter 2. 53

66 54 Chapter 4. The Couomb force The soutions of 5.29) are the so-caed Couomb wave functions, and their properties are we-known. We use here the conventions introduced by Yost, Breit and Wheeer in Ref. [112] and summarize some important reations in the foowing. For a more comprehensive discussion we refer to the review artice by Hu and Breit [113]. Expicity, we write the wave functions as [114, 115] F p) r) = 1 e i π 2 k Γ 1 + m k) 2 2 Γ2m + 1) π e i m) M k,m z) 4.4a) where and r) = Γ 1 + m k) 2 Γ π 1 + m k) e i m+k) W k,m z) + if p) r), 4.4b) 2 G p) ρ = pr, η = γ 2p, 4.5) z = 2iρ, k = iη, m = ) The functions M k,m and W k,m are Whittaker functions, which can be expressed in terms of hypergeometric functions as M k,m z) = e 1 2 z z 1 2 +m 1F m k, 1 + 2m; z), 4.7) W k,m z) = e 1 2 z z 1 2 +m U m k, 1 + 2m; z). 4.8) 1F 1 a, b; z) is Kummer s function of the first kind, and Ua, b; z) = 1F 1 a, b; z) = n=0 a n) z n b n) n! Γ1 b) Γ1 + a b) 1 F 1 a, b; z) + Due to their behavior in the imit ρ = pr 0, F p), a n) = aa + 1) a + n 1), 4.9) Γb 1) z 1 b 1F 1 a b + 1, 2 b; z). 4.10) Γa) r) C η, ρ +1, 4.11a) G p) ρ r) C η, 2 + 1), 4.11b) F p) and G p) are commony caed the reguar and irreguar Couomb wave functions, respectivey. The factor C η,, which in the foowing we refer to as the Gamow factor, 1 is given by C 2 η,0 = 2πη e 2πη 1, 4.12) 1 Note that there is no genera agreement about the name of C η, in the iterature. It is sometimes aso referred to as the Sommerfed factor or, perhaps introduced as a sort of compromise, Gamow Sommerfed factor.

67 4.1. Couomb wave functions 55 and C 2 η, = 2 2 [2 + 1)!] 2 s 2 + η 2 ) Cη, ) Sometimes it is convenient to write it in the more genera form [103] s=1 C η, = 2 e πη 2 [Γ iη)γ + 1 iη)] 1 2 Γ2 + 2). 4.14) For asymptoticay arge ρ = pr, on the other hand the Couomb wave functions behave as with the Couomb phase shift From Eqs. 4.11) and 4.15) it is cear that F p) F p) r) sinρ π/2 η og2ρ) + σ ), 4.15a) G p) r) cosρ π/2 η og2ρ) + σ ) 4.15b) σ = arg Γ iη). 4.16) r) and G p) r) are the direct anaogues of the Riccati Besse functions S pr) and C pr) that sove the free radia Schrödinger equation. The key differences to point out are the presence of the factors of C η, in 4.11) and of the additiona phases η og2pr) σ in 4.15). The atter in particuar the ogarithmicaydivergent term refect the inherent ong-range nature of the Couomb potentia: no matter how arge the separation r of two charged partices becomes, the phase of their reative-motion wave function determined soey by the reguar function F p) r) for a pure Couomb interaction) never goes to a constant, and the partices are thus never free [116]) The Gamow factor The factor C 2 η,0 has a direct physica interpretation. To see this, note first that the fu three-dimensiona continuum Couomb wave function with outgoing asymptotics for the spherica wave and normaized such that the incoming pane-wave component expip r) has unit ampitude) can be written in terms of a confuent hypergeometric function as [117] ψ p +) r) = e πη 2 Γ1 + iη) e ip r 1 F 1 iη, 1; ipr ip r). 4.17) Its partia-wave expansion is given in terms of the reguar Couomb wave functions as see Refs. [118, 119] and cf. aso Ref. [120]) ψ +) p r) = 2 + 1)i e iσ F p) r) P cos θ). 4.18) pr =0 It is cear that ony the S-wave term contributes in the imit r 0, such that by inserting the threshod behavior of F p) r) from Eq. 4.11a) we find that im r 0 ψ +) p r) = e iσ C η, )

68 56 Chapter 4. The Couomb force Aternativey, one can derive directy from Eq. 4.17) that ψ +) p 0) 2 = e πη Γ1 + iη)γ1 iη) = C 2 η,0, 4.20) where the ast equaity foows from Eq. 4.14). As pointed out in Ref. [117] for exampe) this means that the S-wave Gamow factor squared) is the probabiity of two charged partices to be found at zero separation Anaytic wave functions Of a possibe ineary independent pairs of soutions for the Schrödinger equation 5.29), the Couomb wave functions F p) r) and G p) r) are convenient to use because with their behavior at the origin and for asymptoticay arge distances, given in Eqs. 4.11) and 4.15), respectivey, they correspond most directy to the soutions of the free Schrödinger equation discussed in Chapter 2. In contrast to those functions, however, F p) r) and G p) r) cannot directy be used in order to obtain expressions that are anaytic in p 2. Rather, the fact that they are expressed in terms of the variabes ρ = pr and η 1/p yieds series expansions [112, 113] with a compicated entangement of terms in p and r. As first done by Lambert in Ref. [121], it is possibe to define Couomb wave functions that are directy anaytic in p 2. Lambert s resut was ater generaized by Boé and Gesztesy in Ref. [122]. Their pair of anaytic wave functions, which we denote by F n 0) p, r) and G 0) n p, r) and discuss in more detai in Appendix A, wi be very usefu in the foowing Chapter 5. Here, we note for competeness that Couomb wave functions anaytic in the energy are aso discussed in a more recent pubication by Seaton [123]. 4.2 Modified effective range expansion Anaogous to the discussion in Section we now consider the case where a finite-range interaction V r, r ) is present in addition to the Couomb potentia. We again assume that the interaction aows for a soution that is reguar at the origin, and the finite-range condition 5.1) impies that V r, r ) vanishes if r > R or r > R for some fixed but arbitrary range R. The radia Schrödinger equation now reads p 2 w r) = d2 dr w + 1) r) + w 2 r 2 r) + 2µ R 0 dr V r, r ) w r ) + γ r w r). 4.21) A genera soution of Eq. 4.21) for momentum p can be written as a inear combination of the reguar and irreguar Couomb wave functions F p) and G p) defined in the previous section, [ w p) r) cot δ ] p) F p) r) + G p) r) for r > R, 4.22) 2 In particuar, the wave function at the origin stays finite athough the Couomb potentia diverges for r 0. The atter is ony a mathematica pathoogy that is aways reguated by screening effects in a rea physica system.

69 4.2. Modified effective range expansion 57 with an arbitrary overa normaization, and where δ is the phase shift of the fu soution w p) compared to the reguar Couomb function F p) [13]. Since it can be reated to the partia-wave expansion of a function T SC defined by [124, 125] T = T SC + T C T SC = T T C, 4.23) where T is the fu T-matrix corresponding to the combined interaction ˆV + ˆV C and T C is the pure Couomb T-matrix see Section 4.4), δ is caed the Couomb-modified or Couomb-subtracted scattering phase shift. Instead of the ordinary effective range expansion 2.19) we now have the more compicated expression Cη, 2 p 2+1 cot δ p) + γ h p) = a C 2 rc p 2 +, 4.24) where h p) = p 2 C2 η, hη), 4.25) Cη,0 2 hη) = Re ψiη) og η, 4.26) and the digamma function ψz) = Γ z)/γz) is the ogarithmic derivative of the Gamma function. It means that in order to get an expression that is anaytic in p 2, one has to mutipy p 2+1 cot δ p) by the Gamow factor squared and add an additiona function that cances the remaining non-anaytic terms. Eq. 4.24) is caed the Couomb-modified effective range expansion. For = 0, it simpifies to Cη,0 2 p cot δ 0 p) + γ hη) = a C 0 2 rc 0 p ) A derivation of Eq. 4.27) for the case of proton proton scattering can be found, for exampe, in Ref. [13]. 3 See aso Ref. [11] for a detaied discussion. The anaytic properties of the = 0 modified effective range function are investigated in Ref. [126]. In Ref. [122], Boé and Gesztesy derived a very genera form of the Couomb-modified effective range expansion for an arbitrary number of spatia dimensions. Speciaizing their resut to the three-dimensiona case, a version of Eq. 4.24) can be written as Cη, 2 p cot 2+1 δ ) p) i + γ h p) = 1 a C rc p ) with 4 h p) = 2p)2 Γ iη) 2 ψiη) + 1 ) Γ2 + 2) 2 Γ1 + iη) 2 2iη ogiη). 4.29) 3 As a remark we note that on first sight the expansion given in Eq. 51) of Bethe s paper [13] seems to be different from the one given here in Eq. 4.27), which is the same as given in ater pubications referring to Bethe s resut. The η-dependent function on the eft hand side of Bethe s expansion appears to differ from our hη) by two times the Euer-Mascheroni constant γ E. This apparent confict can be resoved by noting that the gη) in Eq. 51) of Ref. [13] is not the function defined in Eq. 47a) of the same paper, but rather given by im η1 [gη) gη 1 )], where in this atter expression the g from Eq. 47a) is meant. The imiting process then yieds exacty the term 2γ E. 4 This definition essentiay comes from combining Eqs. 4.1) and 4.2) of Ref. [122], with the correction that the exponent in Eq. 4.2) shoud be 2 rather than 2.

70 58 Chapter 4. The Couomb force The atter function can be rewritten using C 2 η, = 2 2 Γ2 + 2) 2 Γ iη) 2 Γ1 + iη) 2 C 2 η,0, 4.30) with C 2 η,0 as defined in Eq. 4.12). The expressions given here reproduce Eqs. 4.24) and 4.13) when one expicity assumes that the momentum p is rea. In fact, one has to rewrite Eq. 4.29) in this manner in order to get an effective range function that is anaytic in p 2 around threshod. The form of the Couomb-modified effective range expansion for genera that we have given in Eqs. 4.24) and 4.28) is the same as in Ref. [122]. Note that sometimes another convention, differing from ours by an overa momentum-independent factor, is used in the iterature. The effective range expansion given in Refs. [124, 127, 128] can be written as ) 2 Γ2 + 2) [C 2η, p 2+1 cot 2 Γ + 1) δ ) p) i + γ h ] p) = ã C 2 rc p ) This expression has the advantage of having a more direct connection to the ordinary effective range expansion without Couomb effects. For = 0, both our choice and the form in Eq. 4.31) give the same expression. In this work we wi, because of its simper form, primariy use the convention of Eqs. 4.24) and 4.28). 4.3 Bound-state regime In order to discuss the bound-state regime for systems of charged partices we need to know the soutions of Eq. 5.29) with the appropriate exponentiay decaying) behavior. In other words, we need the Couomb anaogs of the Hanke functions H ± Asymptotic wave function Essentiay, this roe is payed by the Whittaker functions W iη,+ 1. From the definitions 4.4) one directy sees 2 that G p) r) if p) r) = e iσ e i π 2 iη) W iη,+ 1 2ipr) 4.32) where the prefactor is found by noting that with m and k as in Eq. 4.6) we have Γ 1 + m k) 2 Γ 1 + m k) = ei arg Γ+1 iη) = e iσ 4.33) 2 due to the property Γz) = Γz) of the Gamma function. Using this, one can aso write the Couomb phase shift in the form e iσ = 2 )1 Γ iη) 2, 4.34) Γ + 1 iη)

71 4.3. Bound-state regime 59 which is usefu for showing that furthermore G p) r) + if p) r) = e iσ e i π 2 +iη) W iη,+ 1 2ipr). 4.35) More precisey, this foows from the definitions 4.4) after a short cacuation invoving the reation [114] M k,m z) = e iπk Γ2m + 1) Γ 1 + m k)w k,m z) e iπm 1 2 +k) Γ2m + 1) Γ 1 + m + k)w k,mz), 4.36) 2 2 the second term of which cances the W k,m z) from G p) r) after simpifying the prefactors with the hep of 4.34). These and further reations for Couomb wave functions with compex arguments and/or parameters can aso be found in the artices by Humbet [129] and Dziecio et a. [130]. According to Hu and Breit [113], the asymptotic behavior of the Whittaker functions for arge z is W k,m z) e z/2 z k, W k,m z) e z/2 z) k, 4.37) such that the normaizabe bound-state soution is given by for bound-state momenta p = iκ, κ > 0. W iη,+ 1 2ipr) e κr as z 4.38) 2 r η Bound-state condition and ANC We now go back to the case where a finite-range interaction is present in addition to the Couomb tai and consider soutions w p) r) of Eq. 4.21) with the asymptotic from as given in Eq. 4.22). Inverting Eqs. 4.32) and 4.35) in order to express W ±iη,+ 1 in terms 2 of F p) and G p) and inserting the resut into Eq. 4.22) gives w p) r) [ cot δ p) i ] W iη, ipr) [ cot δ p) + i ] e 2iσ e iπ W iη,+ 1 2ipr) 4.39) 2 for r > R, in direct anaogy to Eq. 2.18) that describes the case without Couomb interaction see Section 2.1.3). Repeating the argument that for a bound state the component representing the incoming wave given by the W iη,+1/2 has to vanish, one finds that the condition for the existence of a bound state with binding momentum κ is cot δ p = iκ) = i. 4.40) In other words, one simpy has to repace the scattering phase shift δ p) in Eq. 2.21) with its Couomb-modified anaog δ p). Furthermore, we define a bound-state soution r) that behaves exacty ike the Whittaker function, w iκ) A, w iκ) A, r) = Ãκ W iη,+ 1 2κr) for r > R, 4.41) where Ãκ denotes the asymptotic normaization constant ANC) for a bound state of charged partices. 2

72 60 Chapter 4. The Couomb force 4.4 The Couomb T-matrix For the pure Couomb interaction it is possibe to write down a cosed expression for the fu off-she T-matrix see Section 2.2.2). If we write the Couomb interaction as an operator ˆV C with r ˆV C r = δ 3) r r )V C r) with V C r) = V C r) γ 2µr, 4.42) p ˆV C q = 2πγ µ 1 p q) 2 V Cp, q), 4.43) the Lippmann-Schwinger equation 2.46) for the Couomb T-matrix T C reads where the energy E is a free compex) parameter. ˆT C E) = ˆV C + ˆV C Ĝ +) 0 E) ˆT C E), 4.44) Sighty atering our notation for the rest of this chapter, we introduce the center-of-mass momentum scae k instead of denoting it by p as done so far) and write E = k2 2µ, η = γ 2k. 4.45) With this, a soution of Eq. 4.44) in momentum space can be written in the Hoster form 5 [ ] { ) } iη s + 1 ds T C k; p, q) = V C p, q) 1 2iη, 4.46) s 1 s 2 1 ɛ where 1 ɛ = p2 k 2 )q 2 k 2 ) k 2 p q) ) Aternativey, it can be recast in terms of hypergeometric functions as { [ T C k; p, q) = V C p, q) 1 1 2F 1 1, iη, 1 + iη; 1 ) + 1 with the new variabe defined via 2 F 1 1, iη, 1 + iη; ) ] }, 4.48) 2 = 1 + ɛ. 4.49) For future reference we note that this can be shown by using the integra representation [103] Γc) 1 2F 1 a, b; c; z) = t b 1 1 t) c b 1 1 tz) a dt 4.50) Γb)Γc b) 0 5 Up to a prefactor e 2πη 1) 1 in front of the integra, this is the form given in Eq. 90) of Ref. [131]. Note, however, that the additiona factor shoud actuay not be there, which can be seen by starting from Eq. 86) in the same reference.

73 4.4. The Couomb T-matrix 61 for the hypergeometric function to obtain first 2F 1 1, iη; 1 + iη; 1 ) Γ1 + iη) = + 1 Γiη) and subsequenty, using the transformation t = s + 1 s t iη 1 dt 1 t 1 +1 to arrive at 2F 1 1, iη; 1 + iη; 1 ) 2 F 1 1, iη; 1 + iη; + 1 ) t iη dt = iη, 4.51) 1 t 1 +1, dt = 2 ds s 1) ) = iη 1 ) iη s ds s 1 s ) 2 This gives back the integra form 4.46) of T C k; p, q) when inserted into Eq. 4.48) Yukawa screening In many situations, most notaby in numerica cacuations, it is necessary to suppress the ong range of the Couomb potentia by screening it a arge distances. A popuar choice to impement this is to repace the pain Couomb interaction with a Yukawa potentia, V C r) V C,λ r) = γ e λr, 4.54) 2µ r where the screening parameter λ can be interpreted as a photon mass. In momentum space, the Yukawa potentia is given by V C,λ p, q) = 2πγ µ 1 p q) 2 + λ ) In fact, the Couomb potentia in momentum space is usuay defined by taking the imit λ 0 in this expression because the Fourier transform of 1/r is not immediatey wedefined. From the above expression it is cear that the photon mass reguates the singuarity that otherwise occurs in forward direction q = p, i.e., for vanishing momentum transfer). Ref. [131] gives an expression for what we in the foowing ca the partiay screened Couomb T-matrix ˆT C,λ, originay derived by Gorshkov [111]. It is defined by the reation ˆT C,λ = ˆV C,λ + ˆV C,λ Ĝ +) 0 ˆT C, 4.56) where we have not written out the energy dependence of the functions for notationa convenience. Note that this is not a Lippmann Schwinger equation because the operator that appears on the right-hand side is the unscreened Couomb T-matrix ˆT C. Sti, ˆT C,λ is an interesting

74 62 Chapter 4. The Couomb force object to study because it can be written down as a cosed expression that converges to the unscreened Couomb T-matrix ˆT C in the imit λ 0. Due to this property it is usefu in numerica cacuations where the the Couomb interaction with its poe at vanishing momentum transfer that woud otherwise create probems has to be reguated. Ideay, one woud of course ike to use an expression for the exact Yukawa T-matrix in such an approach, but no cosed soution for that quantity is known so far. We thus propose here to use ˆT C,λ as a pragmatic aternative and wi discuss its appication to the ow-energy proton deuteron system in Chapter 6. Since it has the right behavior in the imit λ 0, we expect it to adequatey describe most of the nonperturbative Couomb effects. Unfortunatey, the expression given for T C,λ k; p, q) in Eqs. 246) and 247) of Ref. [131] is no fuy correct. 6 Since in the origina paper by Gorshkov [111] the imit λ 0 is taken without first giving the expicit form of the partiay screened T-matrix, we wi derive it here in the foowing. To this end we start from Eq. 244) of Ref. [131], which in our notation reads T C,λ k; p, q) = V C,λ p, q) iη 1 where Λ 0 x) is defined as the positive root of 0 { dx 1 } Λ 0 x) V dx 1 C,λ ikλ 0 x)xp, q) exp iη, 4.57) x x 1 Λ 0 x 1 ) Λ 2 0x) = [ 1 p/k) 2 x ] 1 x), 4.58) and V C,λ ikλ0 x) is just the Yukawa potentia 4.55) with the substitution λ λ ikλ 0 x). We now consider the integra in Eq. 4.57). With the substitution [111] x = s2 1 dx, s 2 p/k) 2 ds = 2s ) 1 p/k) 2 s2 p/k) 2) ) one finds that and since furthermore the integra in the exponent is just 1 x dx 1 x 1 Λ 0 x 1 ) = Λ 0 x) = s1 x), 4.60) 1 x = 1 p/k)2 s 2 p/k) 2, 4.61) For the potentia term under the integra we find s 2 ds 1 s + 1 ) s = og. 4.62) s 1 V C,λ ikλ0 x)xp, q) = 2πγ µ s 2 p/k) 2 ) s 2 1)q p) ] 4.63) [λ k 2 k 2 s 2 p 2 ) 2iλksk 2 p 2 ) k 2 q 2 )k 2 p 2 ) 2 6 This can be seen by a straightforward dimensiona anaysis of Eq. 247) in Ref. [131]. Furthermore, the prefactor in Eq. 246) is written in terms of the unscreened Couomb potentia, which is ceary not correct.

75 4.4. The Couomb T-matrix 63 after a engthy but straightforward cacuation. Adding in the denominator, we can rewrite this as 0 = λ 2 s 2 1) λ 2 s 2 + λ ) V C,λ ikλ0 x)xp, q) = 2πγ µ s 2 p/k) 2 ) s 2 1) [ q p) 2 + λ 2] 1 ] [k k 2 q 2 )k 2 p 2 ) + 1 [ ]. k 2 k 2 p 2 )λ 2 2iλks) 2 Finay, noting that the term in the numerator cances against the same factor in dx Λ 0 x) = 4.65) 2 ds s 2 p/k) 2, 4.66) and factoring out the Yukawa potentia, we arrive at { ) } iη s + 1 ds T C,λ k; p, q) = V C,λ p, q) 1 2iη s 1 s 2 1 ɛ λ + ζ λ s) ) with and ɛ λ = k2 p 2 )k 2 q 2 ) k 2 [q p) 2 + λ 2 ] ζ λ s) = k2 p 2 )λ 2 2iλks) k 2 [q p) 2 + λ 2 ] 4.68). 4.69) This expression is very simiar to the integra form of the unscreened Couomb T-matrix. The ony differences are given by the new term ζ λ s) in the denominator and the fact that a singuarities, both in the overa prefactor and under the integra, are now reguated by adding λ 2. In fact, one directy sees that in the imit λ 0, Eq. 4.67) converges to the unscreened expression given in Eq. 4.46) Expression in terms of hypergeometric functions Something that is not noted in Refs. [131] and [111] is that just ike the unscreened Couomb T-matrix T C,λ k; p, q) can aso be expressed in terms of hypergeometric functions. To obtain this expression we first note that the denominator in Eq. 4.67) can be written as s 2 1 ɛ λ + ζ λ s) = s d 1 D 2 ) d 2 D 2 )s 4.70) with and d 1 = k 2 q 2 λ 2, d 2 = 2iλk 4.71) D 2 = k 2 p 2 k 2 [q p) 2 + λ 2 ]. 4.72)

76 64 Chapter 4. The Couomb force After making the transformation we get s = t + 1 t 1, 1 dt ds = ) 1 t 1) 2 T C,λ k; p, q) = V C,λ p, q) { 1 4iη To proceed further, we use the indefinite integra 7 1 t iη dt }. 4.74) D 2 d 1 + d 2 ) t D 2 d 1 ) t D 2 d 1 d 2 ) { t ν dt = 1 2 ν t ν 2F x 2 t 2 1 ν, ν; 1 ν; X + + x 1 t + x 0 X 1 ν 2 t) ) X 3 + t) ν 2 F 1 ν, ν; 1 ν; X 2 t) ) } X3 t) ν, 4.75) where X 1 = x 2 1 4x 0 x 2, X ± 2 t) = x 1 X 1 x 1 + 2tx 2 X 1, X ± 3 t) = tx 2 x 1 + 2tx 2 X ) Evauating this at t = 1 is straightforward, but considering t requires a itte more care. From Eq. 4.76) one sees that X 2 ± t) goes to zero ike 1/t as t, such that the hypergeometric functions simpy yied one in this imit. Since the potentiay probematic because ν = iη) prefactor t ν is canceed by the numerator of X 3 ± t) ν with the remainder then going to zero as t, we can concude that there is actuay no contribution to the integra from the upper boundary in Eq. 4.74) and that its vaue is hence given by the right-hand side of Eq. 4.82) with t = 1. Before inserting this into Eq. 4.74), we subsequenty appy the identities [103] 2F 1 a, b; c; z) = 1 z) c a b 2F 1 c a, c b; c; z) 4.77) and to rewrite 2F 1 a, b; c; z) = 1 z) a 2F 1 a, c b; c; ) z z ) 2F 1 ν, ν; 1 ν; z) = 1 z) 1+ν 2F 1 1, 1; 1 ν; z) = 1 z) ν 2F 1 1, ν; 1 ν; This is usefu because from Eq 4.76) one finds that for z = X ± 2 t), ) z. 4.79) z 1 1 z) ν = 2 ν X ± 3 t) ν, 4.80) 7 This resut has been obtained with the hep of computer agebra software Wofram Mathematica).

77 4.4. The Couomb T-matrix 65 canceing the inverse factors of this in Eq. 4.82). Moreover, the arguments simpify to With this, we then have 1 z z 1 = 1 X 2 ± t) 2 X 3 ± t) = x 1 X ) 2t x 2 t ν dt = 1 {2F x 2 t 2 1 1, ν; 1 ν; x ) 1 + X 1 + x 1 t + x 0 νx 1 2x 2 2 F 1 1, ν; 1 ν; x ) } 1 X ) 2x 2 Finay, appying the above resut to Eq. 4.74), we can write the partiay-screened Couomb T-matrix as T C,λ k; p, q) { = V C p, q) with and 1 1 λ [2F 1 1, iη, 1 + iη; X λ 2 λ = 1 + k2 p 2 )k 2 q 2 λ 2 ) k 2 [q p) 2 + λ 2 ] ) ) ] } 2 F 1 1, iη, 1 + iη; X + λ, 4.83) λ 2 k 2 p 2 ) 2 k 2 [q p) 2 + λ 2 ] ) X ± λ = 2k2 [q p) 2 + λ 2 ] 1 ± λ ) + k 2 p 2 )k 2 q 2 λ 2 ) k 2 p 2 ) [k + iλ) 2 q 2 ]. 4.85) As it shoud, this reduces to the hypergeometric expression 4.48) for the unscreened Couomb T-matrix in the imit λ 0. It is directy cear from Eqs. 4.84) and 4.49) that im λ 0 2 λ = 2, 4.86) and a straightforward cacuation then furthermore shows that im λ 0 X ± λ = ± )

78 66 Chapter 4. The Couomb force

79 Chapter 5 Causaity bounds for charged partices Overview In this chapter, we derive a generaization of the so-caed Wigner causaity bound for a system of charged partices, where the Couomb force determines the ong-range interactions. The majority of the materia presented in the foowing sections has been pubished in Ref. [5]. The review of Couomb wave functions and the modified effective range expansion from that reference have aready been given in the preceding chapter; the remaining parts are incuded here in a sighty re-arranged and amended form. A part of Section 5.7 is based on resuts from Ref. [4] that have been omitted in Chapter 3 to put them in a more suitabe context here. Finay, the discussion of causaity bounds for van der Waas tais in Section 5.8 is summarized from Ref. [7], which was mainy worked out by S. Ehatisari. 5.1 Introduction The constraints of causaity for two-body scattering with finite-range interactions were first derived by Wigner [134]. The causaity bound can be understood as a ower bound on the time deay t between the incoming and outgoing wave packets. When t is negative, the outgoing wave packet departs earier than for the non-interacting system. However, the incoming wave must first reach the interaction region before the outgoing wave can eave. In ow-energy scattering this manifests itsef as an upper bound on the effective range parameter. In Ref. [135], Phiips and Cohen derived this bound for S-wave scattering with finite-range interactions. Some constraints on nuceon nuceon scattering and the chira two-pion exchange potentia were considered in Ref. [136], and reations between the scattering ength and effective range have been expored for oneboson exchange potentias [137] and van der Waas potentias [138]. In Refs. [10, 139] the causaity bounds for finite-range interactions were extended to an arbitrary number of space-time dimensions and arbitrary anguar momentum. The extension to systems with 67

80 68 Chapter 5. Causaity bounds for charged partices partia-wave mixing was first studied in Ref. [140]. Here, we consider the causaity constraints for the scattering of two charged partices with an arbitrary finite-range interaction. This anaysis, pubished in Ref. [4], is the first study of causaity bounds that takes into account the ong-range Couomb force. The resuts presented here are reevant to studies of ow-energy scattering of nucei and nuceons using effective fied theory EFT), in particuar for the appication of effective fied theory to the nucear hao systems discussed in Section There is an important connection between causaity bounds and the convergence of effective-fied-theory cacuations with increasing order [140]. For oca contact interactions, the range of the effective interaction is controed by the momentum cutoff scae of the effective theory. In effective theories with non-perturbative renormaization, which typicay occur in nucear physics, exact cutoff-independence can generay not be achieved. There is a natura vaue of the cutoff at which a higher-order corrections scae as expected from dimensiona anaysis. If the cutoff is taken arger, new physics intervenes, the corrections scae unnaturay, and unitarity vioations may occur. This is different from what one encounters in high-energy partice physics where the renormaization is typicay perturbative and cutoff momenta can be chosen arbitrariy arge. For cacuations using dimensiona reguarization, the renormaization scae pays a simiar roe in reguating utravioet physics. The term new physics, in the above context, refers to detais eft out integrated out) in the effective theory. In the case of hao EFT, these detais are the finite size of the core nuceus and its interna excitations as we as the exponentia tai of the pion-exchange interaction. Probems with convergence of the effective theory can occur if the cutoff scae is set higher than the scae of the new physics. It is desirabe to have a more quantitative measure of when probems may appear, and this is where the causaity bound provides a usefu diagnostic too. For each scattering channe we use the physica scattering parameters to compute a quantity caed the causa range, R c. It is the minimum range for finite-range interactions consistent with the requirements of causaity and unitarity. For any fixed cutoff scae, the causaity bound marks a branch cut of the effective theory when viewed as a function of physica scattering parameters [140]. The couping constants of the effective theory become compex when scattering parameters vioating the causaity bound are enforced. These branch cuts do not appear in perturbation theory; however, a nearby branch point can spoi the absoute convergence of the perturbative expansion. Our resuts can be viewed as a guide for improving the convergence of hao-eft cacuations. In particuar, if the cutoff momentum used in a cacuation is too high, then probems with convergence may appear in some observabes. Consequenty, the causa range can be used to estimate the natura utravioet cutoff Λ of the effective theory as R 1 c. The natura cutoff is optima in the sense that no known infrared physics is eft out of the theory and that a corrections invoving the utravioet cutoff scae naturay [141, 142]. Increasing the cutoff beyond the natura vaue wi not improve the accuracy of the cacuation. The causaity bounds aso have an impact in the regime of bound states. For two-body hao states or more generay whenever there is a shaow two-body bound state cose to threshod the same integra identity that yieds the causaity bound for the effective

81 5.2. Setup and preiminaries 69 range can be used to derive a reation between the asymptotic normaization constant ANC) of the bound-state wave function, the binding momentum, and the effective range for the scattering of the two hao constituents. This reation can be shown to be equivaent to a resut previousy derived by Sparenberg et a. [143]. Its significance ies in the fact that the ANC is an important input parameter for the cacuation of near-threshod radiative capture and photodissociation reactions. The causaity bounds aso constrain the range of mode potentias that are fitted to scattering data in order to extract ANCs. The organization of this chapter is as foows. After briefy reviewing in Section 5.2 the theoretica setup for two charged partices with additiona short-range interactions discussed in detai in Chapter 4), we derive the charged-partice causaity bounds for arbitrary vaues of the orbita anguar momentum in Section 5.3. This anaysis incudes both attractive and repusive Couomb forces. In Section 5.4 we define the causa range and then extract and discuss this quantity in Section 5.5 for severa nucear scattering processes incuding proton proton, proton deuteron, proton 3 He, proton apha, and apha apha scattering. Some numerica cacuations are given in Section 5.6. In Section 5.7, we eucidate the reation for asymptotic normaization constants mentioned above and extract, as an appication, the ANCs of the excited 2 + and 1 states in 16 O from α 12 C scattering data. Before briefy touching the subject of causaity bounds for other ong-range forces in particuar, for a van der Waas potentia) in Section 5.8, we then concude with a summary of the main resuts and provide an outook. 5.2 Setup and preiminaries We consider a two-partice system with reduced mass µ interacting via a finite-range potentia with range R. As aready done in the previous chapters, we write the interaction as a rea symmetric operator with kerne V r, r ) satisfying the finite-range condition, V r, r ) = 0 if r > R or r > R. 5.1) In particuar, we assume that the interaction is energy-independent. After giving a detaied forma derivation of the causaity bounds in the foowing sections, we wi come back to the question what the above assumptions mean for the appication to hao) EFT cacuations in Section 5.5. In the absence of Couomb interactions the system with fixed but arbitrary) anguar momentum is described by the radia Schrödinger equation, p 2 u p) r) = d2 dr 2 up) r) + + 1) u p) r 2 r) + 2µ R 0 dr V r, r ) u p) r ). 5.2) As done in Chapter 2, we adopt the conventions of Ref. [10] and choose the normaization of u p) that for r R we have u p) r) = p [cot δ p) S pr) + C pr)], 5.3) where S and C are the Riccati-Besse functions and δ p) is the scattering phase shift.

82 70 Chapter 5. Causaity bounds for charged partices If the partices carry eectromagnetic charges Z 1 e and Z 2 e, respectivey, there is a Couomb potentia in addition to the finite-range interaction. As done in Chapter 4, we write this as V C r) = γ 2µr = αz 1Z 2, 5.4) r such that in the radia Schrödinger equation we simpy get a term γ/r because the factor of 2µ in the denominator cances out. From Section 4.2 we quote it in the form p 2 w p) r) = d2 dr 2 wp) r) + + 1) w p) r 2 r) + 2µ R 0 dr V r, r ) w p) r ) + γ r wp) r), 5.5) where again we use the superscript p) to denote the soution for a given center-of-mass momentum p. We choose the normaization of w p) such that for r R we have 1 [ w p) r) = p C η, cot δ ] p) F p) r) + G p) r), 5.6) with the Couomb-subtracted phase shift δ and the reguar and irreguar Couomb wave functions F p) and G p) as defined in Chapter 4. The incusion of the Gamow factor in the normaization C η, wi be convenient ater, when we rewrite Eq. 5.6) in terms of a different pair of functions and reate it to the Couomb-modified effective range expansion 4.24), rearranged in the form with h p) as defined in Chapter 4. C 2 η, p 2+1 cot δ p) = γ h p) 1 a C rc p 2 +, 5.7) 5.3 Derivation of the causaity bound With the Couomb wave functions and Couomb-modified effective range expansion at our hands, we can now cosey foow the derivation presented in Ref. [10] for scattering in the absence of Couomb interactions Wronskian identities We consider soutions of the radia Schrödinger equation 4.21) for two different momenta p A and p B. Introducing the short-hand notation w A,B r) = w p A,B) r), 5.8) i.e., suppressing the anguar-momentum subscript for convenience, we get p 2 B p 2 A) r ɛ dr w A r )w B r ) = w B w A w A w B) r ɛ r R 2µ dr dr [w B r)v r, r )w A r ) w A r)v r, r )w B r )] 5.9) ɛ 0 1 Note that for = 0 our normaization is the same as chosen in Ref. [13], i.e., for r R our soution w p) 0 coincides with the function ϕ defined in Eq. 42) of that paper.

83 5.3. Derivation of the causaity bound 71 by subtracting w A times the equation for w B from that for w B mutipied by w A, as it is done in Ref. [10], and integrating from some sma radius ɛ to r. We assume that our interaction V r, r ) is such that it aone without the additiona Couomb potentia) permits a soution that is sufficienty reguar at the origin, i.e., u 0) = 0 and r u stays finite as r 0, where u is a soution of Eq. 5.2). As boundary condition for the soutions w A,B of the fu radia Schrödinger equation we can then demand as we that they vanish with finite derivative at the origin. If we ony had the Couomb potentia and no additiona interaction, this is fufied by the reguar Couomb function F p) r), cf. Eq. 4.11a). We can thus take the imit ɛ 0 in Eq. 5.9) and get the reation W [w B, w A ]r) = p 2 B p 2 A) where the Wronskian W [w B, w A ] is defined as r 0 dr w A r )w B r ), 5.10) W [w B, w A ]r) = w B r)w Ar) w A r)w Br). 5.11) Rewriting the wave functions Foowing further the derivation presented in Ref. [10], we re-express the soutions w p) r) in terms of functions fp, r) and gp, r) such that for r R, with fp, r) anaytic in p 2, and w p) r) = p 2+1 C 2 η, cot δ p) fp, r) + gp, r) 5.12) fp, r) = f 0 r) + f 2 r) p 2 + Op 4 ), 5.13) gp, r) = gp, r) + φp) fp, r). 5.14a) The gp, r) contains a term which is non-anaytic in p 2 and is proportiona to fp, r). The remainder gp, r), however, is anaytic in p 2, gp, r) = g 0 r) + g 2 r) p 2 + Op 4 ). 5.14b) Combining Eqs. 5.6) and 5.12), we find and fp, r) = 1 p +1 C η, F p) r) 5.15a) gp, r) = p C η, G p) r). 5.15b) These functions are directy reated to the anaytic Couomb wave functions of Boé and Gesztesy [122] mentioned in Section and discussed further in Appendix A. In fact, one simpy has that fp, r) is exacty the F n 0) p, r) defined in Eq. A.3a), whereas comparison of Eqs. 5.15b) and A.3b) shows that 0) gp, r) = G n p, r) + γ h ) p) ip 2+1 Cη, 2 F n 0) p, r) 5.16)

84 72 Chapter 5. Causaity bounds for charged partices with 0) G n p, r) as defined in Eq. A.4). This impies that and gp, r) = G 0) n p, r) 5.17) φp) = γ h p) ip 2+1 C 2 η, = γ h p), 5.18) where the ast step foows from the combination of Eqs. 4.29) and A.4) after a short cacuation. When we insert now the modified effective range expansion 4.24) into the asymptotic Couomb wave function 5.12), the non-anaytic term invoving h p) convenienty drops out and we are eft with w p) r) = 1 a C + 1 ) 2 rc p 2 + fp, r) + gp, r) for r R. 5.19) Thus, it is possibe to choose a normaization such that w p) r) is anaytic in p 2. Combining this with the expansions 5.13) and 5.14), we arrive at w p) r) = 1 a C f 0 r) + g 0 r) + p 2 [ 1 2 rc f 0 r) 1 a C The causaity-bound function ] f 2 r) + g 2 r) + Op 4 ). 5.20) From here we can proceed exacty as in Ref. [10]. For the Wronskian of two soutions w A and w B for r R we find { ) 2 W [w B, w A ]r) = p 2 B p A) 2 rc W [f 0, g 0 ]r) + W [f 2, f 0 ]r) 1 a C [W [f 2, g 0 ]r) W [g 2, f 0 ]r)] + W [g 2, g 0 ]r) a C } + Op 4 A,B). 5.21) Note that in the Op 4 A,B ) we have aso incuded terms of the form p2 A p2 B. We set p A = 0 in Eq. 5.10) and furthermore take the imit p B 0. Using the expansion 5.21), we get r [ 2 r C W [f 0, g 0 ]r) = b C r) 2 dr w 0) r )] 5.22) for r R, with w 0) b C r) = 2W [g 2, g 0 ]r) 2 a C r) = im p 0 w p) r) and the causaity-bound function 0 {W [f 2, g 0 ]r) + W [g 2, f 0 ]r)} + 2 a C )2 W [f 2, f 0 ]r). 5.23) Written as a function of ρ = p r, the Wronskian of the Couomb wave functions is 2 W [F p) 2 See, for exampe, Eq ) in Ref. [103]., G p) ]ρ) = W [G p), F p) ]ρ) = )

85 5.3. Derivation of the causaity bound 73 Since d/dr = p d/dρ and W [f, f] 0, we aso have W [f, g]r) = W [f, g]r) = ) Pugging in the expansions 5.13) and 5.14), we see that W [f 0, g 0 ]r) = 1 for the eading-order functions, and W [f 2, g 0 ]r) = W [g 2, f 0 ]r) for the terms at Op 2 ). Inserting these reations into Eq. 5.22), we get r [ 2 r C = b C r) 2 dr w 0) r )], 5.26) where b C r) has been simpified to b C r) = 2W [g 2, g 0 ]r) 4 a C 0 W [f 2, g 0 ]r) + 2 a C )2 W [f 2, f 0 ]r). 5.27) Since the integra in Eq. 5.26) is positive definite, the resuting causaity bound is r C b C r), r R. 5.28) Cacuating the Wronskians We now derive the expicit form of the function b C r). To do this, we need expressions for the Wronskians that appear in Eq. 5.27). We can obtain them by first noting that fp, r) and gp, r), being inear combinations of Couomb wave functions with p-dependent coefficients), are soutions of the Couomb Schrödinger equation, [ d2 + 1) + + γ ] dr2 r 2 r p2 xp, r) = 0, 5.29) which, of course, corresponds to setting V r, r ) = 0 in Eq. 4.21). foowing, x stands for either f or g. Inserting the expansion Here and in the xp, r) = x 0 r) + p 2 x 2 r) + Op 4 ) 5.30) into Eq. 5.29) and comparing orders in p 2, we find that [ d2 + 1) + + γ ] x dr2 r 2 0 r) = 0, 5.31) r i.e., x 0 is a soution of the zero-energy Couomb Schrödinger equation, and [ d2 + 1) + + γ ] x dr2 r 2 2 r) = x 0 r). 5.32) r From this we readiy obtain the differentia equations d dr W [f 2, f 0 ]r) = [f 0 r)] 2, d dr W [g 2, g 0 ]r) = [g 0 r)] 2, d dr W [f 2, g 0 ]r) = f 0 r)g 0 r) 5.33a) 5.33b) 5.33c)

86 74 Chapter 5. Causaity bounds for charged partices for the desired Wronskians. Put together, this yieds a simpe first-order differentia equation for b C r), d dr bc r) = 2 g 0 r) 1 2 f a C 0 r)). 5.34) From Eqs. A.7) and A.8) in Ref. [122] we have the expicit expressions f 0 r) = 2 + 1)! γ 2+1 r I2+1 2 γr), 5.35a) g 0 r) = 2 γ 2+1 r K2+1 2 γr) 2 + 1)! 5.35b) for γ > 0, where I α and K α are modified Besse functions, and f 0 r) = 2 + 1)! γ) 2+1 r J2+1 2 γr), 5.36a) g 0 r) = π γ) )! r N2+1 2 γr) 5.36b) for γ < 0, where J α and Y α are the ordinary Besse functions. 3,4 Using these expressions for f 0 and g 0 and Eq. 5.34) we can determine b C r) up to an integration constant. In order to fix this constant, we must work directy with the Wronskians in Eq. 5.23). Before we do that, however, we first discuss the genera form of b C r). We break apart the function as a sum of two functions, X r) and Y r), and a constant term Z, b C r) = X r) + Y r) + Z. 5.37) We take X r) to be a function consisting entirey of a sum of terms that have a poe at r = 0, ranging from order 1 to, X r) = X,m r m. 5.38) m=1 By furthermore requiring the function Y r) to vanish at r = 0, the decomposition in Eq. 5.37) is unique. Where exacty the contributions to the three terms in the decomposition originate from can be inferred from the behavior of fp, r) and gp, r) at the origin. From Eq. 4.11a) combined with Eq. 5.15) we find that fp, r) r +1 as r ) 3 From Eq ) and the remark above Eq ) in Ref. [103] it is cear that these f 0 and g 0 are indeed soutions of Eq. 5.31). 4 Boé and Gesztesy actuay give an expression for g 0 in the attractive case γ < 0) that invoves the Hanke function H α 2) times i instead of the Neumann function N α aso caed Besse function of the second kind and denoted then by Y α ). With that, however, g 0 woud not be rea, which it shoud be. Our g 0 as in Eq. 5.36b) is taken from the resuts of Lambert [121].

87 5.3. Derivation of the causaity bound 75 This impies that every term in the expansion of fp, r) is Or +1 ). Therefore, im r 0 W [f 2, f 0 ]r) = ) for a, which means that this Wronskian ony yieds contributions to the Y r). Furthermore, from Eq. 17) in Ref. [112] we know that the irreguar Couomb wave function has the asymptotic behavior G p) D η, ρ as ρ 0, 5.41) with D η, fufiing C η, D η, = Using Eq. 5.15b) then yieds gp, r) r as r ) We note that g 0 r) has exacty the same behavior near r = 0 and can thus show that g 2 r) is subeading as r 0, g 2 r) r +c for c > 0. From this we infer that im r 0 W [f 0, g 2 ]r) = ) for a, so aso from this Wronskian we ony get contributions to Y r). Both the singuar X r) and the constant Z, therefore, ony come from the Wronskian W [g 2, g 0 ]r). For = 0 the situation is sti simpe because the above anaysis aso tes us that im r 0 W [g 2, g 0 ]r) = 0 for = 0, 5.44) i.e., b C 0 r) is given entirey by Y 0 r). With the knowedge that it vanishes at the origin, it is actuay straightforward to give an expicit expression for b C 0 r) in terms of antiderivatives of the right hand side of Eq. 5.34), where one has to insert the f 0 r) and g 0 r) from Eqs. 5.35) and 5.36). The resut, obtained by integrating from 0 to r, is b C 0 r) = 2r3 3 ) ) ) a C F 2 2 ; 2, 4; 4γr 4r2 ) a C 1 1 π 0 G 2,1 1,3 4γr 2 0, 1, ) 5.45) πγr 2 G 3,1 2,4 4γr 1, 1 2 1, 0, 1, 2 for the repusive case, and ) b C 0 r) = 2r3 ) a C F 2 2 ; 2, 4; 4γr + 4 πr 2 a C 0 + 2π 2 [ γ 2 r F 2 ) 3 ; 2, 4; 4γr 2γr2 2 π G 3,1 3,5 ) 1 G 2,1 2,4 4γr 4γr 1, , 1, 2, 1 2 1, 1 2, 1 2 1, 0, 1, 2, 1 2 for an attractive Couomb interaction. In the above equations, p F q and G m,n p,q generaized) hypergeometric and Meijer G-functions, respectivey. ) )] 5.46) denote the For genera 1, W [g 2, g 0 ]r) is singuar at r = 0 and the anaysis becomes more compicated. For practica purposes one can simpy use power-series expansions for the Besse functions that appear in the expressions for the zero-energy functions and integrate these term by term unti a desired precision is reached. The ony additiona ingredients needed are the vaues for the constant terms Z because these are obviousy not generated by the integration.

88 76 Chapter 5. Causaity bounds for charged partices Constant terms in the causaity-bound function Obtaining these constants turns out to be the centra difficuty in the derivation of the causaity bound for the charged-partice system, which so far apart from having to cope with more compicated expressions very cosey foowed the path aid out for the neutra system in Ref. [10]. In order to determine the Z we consider the expicit form of gp, r). From the resuts of Boé and Gesztesy [122], we have 5 gp, r) = N p) γ og γ r) fp, r) { where a,n p) = and + γ Re N p) r +1 e ipr N p) = Γ2 + 2) Γn + 1)Γn ) 2ip)n b,n p) = } [a,n p) + b,n p)] r n n=0 2p)2 Γ2 + 2) 2 { } r 2 + Re e ipr d,n p) r n, 5.47) n=0 s 2 + η 2 ), 5.48) s=1 n s + iη) [ψn + 1) + ψn ) ], 5.49a) s=1 Γ2 + 2) Γn + 1)Γn ) 2ip)n d,n p) = 1 Γn + 1) 2ip)n n s + iη) s=1 n s=1 n+ j=1 1 j iη, 5.49b) ) s 1 iη. 5.50) s 2 1 With this resut and the appropriate expression for g 0 r) from Eq. 5.35) or 5.36) one can use the foowing procedure to cacuate the Z, i.e., the terms of order r 0 in the Wronskian W [g 2, g 0 ]r). 1. Note that W [ g, g 0 ]r) = p 2 W [g 2, g 0 ]r) + Op 4 ) 5.51) and cacuate this Wronskian using a truncated version of gp, r) as given in Eq. 5.47). Incuding terms up to the order in r is sufficient. 2. From the resut, extract the terms that are of the order r From that expression then extract the terms that are of the order p 2. They constitute the Or 0 ) contributions in a series expansion of W [g 2, g 0 ]r) which cannot be obtained from a term-by-term integration of g 0 r) 2. 5 One gets this form from Eq. A.5) in Appendix A by inserting n = 2 + 3, η = γ/2p) and assuming that the momentum p is rea and positive.

89 5.4. The causa range 77 With the hep of computer agebra software, this prescription is straightforward to impement. The resuts for = 0,..., 2 are shown in Tabe Z 0 γ 6 2γ ) E 79 γ γ ) E 360 Tabe 5.1: Constant term Z in Eq. 5.37) for = 0, 1, 2. γ E = is the Euer-Mascheroni constant. The vaues are the same for repusive γ > 0) and attractive γ < 0) Couomb potentias. At this point we remark that, in principe, it is aso possibe to use the wave functions defined by Seaton [123] to get expicit expressions for f 2 r) and g 2 r) in addition to the aready known zero-energy functions), and then simpy cacuate a the Wronskians directy. However, the anaytic irreguar Couomb function defined by Seaton is sighty different from our gp, r). More importanty, not a coefficients needed for the expansions are given expicity. Finay, writing everything in terms of the wave functions of Boé and Gesztesy paves the way for a generaization of the resuts presented here to an arbitrary number of spatia dimensions. Knowing now what the constant terms Z, it is possibe to write down expicit expressions for the causaity-bound functions b C r) aso for > 0. To do that we use that the antiderivative of the right-hand side of Eq. 5.34) can be expressed in terms of generaized) hypergeometric functions p F q and Meijer G-functions G m,n p,q. The ony additiona point to be taken into account is that the antiderivative of g 0 r) 2 in genera incudes a constant term that is different from the desired Z. Hence, one has to determine this term and add another constant such that their sum is exacty equa to the Z given in Tabe 5.1. We have carried out this procedure expicity for = 1 and = 2. Since the resuts are rather engthy, we give the compete expressions for b C 1 r) and b C 2 r) in Appendix B. 5.4 The causa range The causaity-bound equation 5.28) can be rewritten as b C r) r C 0 r R. 5.52) In cases where the detais of the interaction in particuar its range, assuming that a description with finite-range potentias is appicabe) is not known, one can use Eq. 5.52) to define the causa range R c of a scattering system as that vaue of r for which the bound is just satisfied, i.e., b C R c ) r C = ) We note from Eq. 5.34) that the derivative of b C r) is non-negative, d dr bc r) = 2 g 0 r) 1 2 f a C 0 r)) )

90 78 Chapter 5. Causaity bounds for charged partices Hence, b C is an increasing function of r and the causa range is defined uniquey. For the case that Eq. 5.53) does not have a soution i.e., if b C r) is positive aready for r = 0) we define the causa range to be zero. Note that the causa range is a function ony of the scattering ength and the effective range. It can thus be cacuated from observabes in a we-defined way. The importance of the causa range is given by the fact that it can be interpreted as the minimum range a potentia is aowed to have to be consistent with causaity. If, for a given system, the vaues in individua partia waves differ significanty, the maximum vaue shoud be taken as the causa range of the underying potentia. Aternativey, one can mode the interaction with an -dependent potentia. For effective fied theories with short-range interactions such as hao EFT, the causa range constrains the aowed vaues of the momentum-space cutoff or the attice spacing used in numerica cacuations Practica considerations At this point we reca that our derivation of the causaity bounds was based on the assumption that the concrete system under consideration is described by a finite-range though possiby non-oca) two-body interaction which is energy-independent. In EFT cacuations one frequenty obtains effective interactions that expicity depend on the energy. We note that this energy dependence can be traded for momentum dependence at any given order in the power counting EFT expansion) by using the equations of motion obtained from the effective Lagrangian. However, the energy dependence introduces another ength scae into the system, and so the conversion to momentum-dependent interactions coud produce an interaction range so arge that the causaity bounds may not be usefu in practice. There are aso other theoretica frameworks, e.g., Feshbach reaction theory, that expicity use energy-dependent interactions. Here again the energy dependence introduces a ength scae which acts as an interaction range. This can be seen from the time deay of the scattered wavepacket, which is proportiona to the derivative of the phase shift with respect to energy. By setting up a very strong energy dependence for the interactions it is possibe to produce a time deay which is arbitrariy arge and negative. This has the same effect as interactions at arbitrariy arge separations. Furthermore, the assumption of a strict finite range certainy is an ideaization that is ony appicabe to a varying degree of vaidity to concrete physica systems. For exampe, there can be exchange forces arising from the Paui principe. Consider, for exampe, nucear hao systems with a tighty bound core and a hao nuceon which is ony weaky bound to the core. The exchange of a nuceon from the core and the hao nuceon that is necessary to anti-symmetrize the system can ony give a sizabe contribution if there is spatia overap between the wave function of the core and the wave function of the hao nuceon. This yieds a short-range exponentia tai that, within the domain of vaidity of the effective theory, can be subsumed in the effective range parameters of the hao core interaction. The same anaysis woud appy to ow-energy nuceon nuceus scattering upon the core nuceus. Another more prominent effect is given by exponentia tais generated by simpe pion-exchange contributions; cf. Ref. [140] and the discussion in the foowing section.

91 5.5. Exampes and resuts Exampes and resuts We now cacuate expicit vaues for causa ranges in few-nuceon systems. In Fig. 5.1 we pot the eft-hand side of Eq. 5.52) as a function of r for the case of proton proton S-wave scattering. The causa range can then be read off as the point where the function becomes zero. Fig. 5.2 shows anaogous pots for α α S- and D-wave scattering. In this system, there are visibe error bands due to the arger uncertainties in the effective range parameters. 4 3 b C r) r C fm) r fm) Figure 5.1: Causa range pot for S-wave proton proton scattering. In Tabe 5.2 we give a summary of the causa ranges that one finds for various two-body systems of ight nucei where ow-energy scattering parameters and/or phase shifts are avaiabe from experiments. The resuts are briefy discussed in the foowing subsections Proton proton scattering For p p S-wave scattering one finds a causa range of about 1.38 fm. This vaue is very cose to the range estimate obtained by assuming that the typica ength scae of the N N interaction is set by the inverse pion mass, c/m π 1.4 fm. The vaue one finds in the 3 P 0 channe is somewhat arger R c 2.3 fm), whereas the 3 P 1 effective range parameters impose amost no constraint on the range of the nucear potentia in this channe. As we wi discuss in more detai beow, this suggests some significant differences in the radia dependence of the interactions for the 3 P 1 channe. For effective-fied-theory cacuations with purey oca interactions e.g., pioness effective fied theory), our resuts suggest to keep the cutoff momentum smaer than M π for the 1 S 0 and 3 P 0 channe. However, there is more freedom to take a higher cutoff for the 3 P 1 channe. In Ref. [140], causaity bounds were investigated for neutron proton scattering. The

92 80 Chapter 5. Causaity bounds for charged partices S-wave b C r) r C fm) D-wave r fm) Figure 5.2: Causa range pot for S-wave α α scattering. System Reference Channe a C / fm 2+1 r C / fm 2+1 R c / fm p p [144] 1 S ± ± ± 0.01 p p [145] 3 P ± ± ± 0.05 p p [145] 3 P ± ± p d [146] 2 S 1/ ± ± ± 0.15 p d [147] 2 S 1/ p d [146] 4 S 3/ ± ± p d [147] 4 S 3/ p 3 He [148] 1 S ± ± p 3 He [148] 3 S ± ± p α [149] S 1/ ± ± p α [149] P 1/ ± ± ± 0.07 p α [149] P 3/ ± ± α α [62] S 1.65 ± 0.17) ± α α [150] D 7.23 ± 0.61) ± 0.22) Tabe 5.2: Summary of causa-range resuts obtained from experimenta input for various few-nuceon systems.

93 5.5. Exampes and resuts 81 resuts R c = 1.27 fm for 1 S 0, R c = 3.07 fm for 3 P 0, and R c = 0.23 fm for 3 P 1 ) are quaitativey very simiar to what we find for the p p system. This woud indicate ony a moderate amount of isospin breaking. In the same pubication [140], the infuence of the shape of the potentia upon the neutron proton causa range was aso studied numericay. When the potentia is repusive at shorter distances ess than 1 fm) and attractive at arger distances greater than 1 fm), the causa range comes out on the arger side, about 2 fm or more. When the potentia is attractive at intermediate distances and repusive at arger distances, then the causa range is smaer, about 1 fm or ess. The pion tai determines the sign of the potentia at arger distances. For both the n p and the p p interaction, the one-pion exchange tai is repusive in the 3 P 1 -channe whie it is attractive in the 3 P 0 -channe. Note that causaity bounds in the presence of pion-exchange contributions were aso discussed by Phiips and Cohen in Ref. [135]. Ideay, one woud account for the one-pionexchange tai expicity in the cacuation of the causa range, as it was done in this work for the ong-range Couomb potentia. Without knowing anaytica soutions of the Schrödinger equation invoving a Yukawa-ike potentia pus a Couomb part, in the p p case), however, such a procedure can at best be impemented numericay. For an exampe, see Ref. [151] Proton deuteron scattering There are severa experimenta determinations of p d effective range parameters. In Tabe 5.2 we have incuded resuts from Arvieux [146] and Hutte et a. [147]. Whie for the quartet-channe there is a good agreement between the scattering engths and effective ranges and, of course, of the resuting causa ranges, which come out as fm), there is a arge discrepancy for the doubet-channe resuts. The difficuty of determining the proton deuteron doubet-channe scattering ength has previousy been discussed by Orov and Orevkov [152]. Comparing different modes, the authors concude that fm is currenty the best theoretica estimate for the doubet-channe p d scattering ength. From Tabe 3 in Ref. [152] one reads off that the corresponding vaue for the Couomb-modified effective range is as huge as fm. Inserting these numbers into the causa-range cacuation one gets a very arge vaue of R c 8.15 fm. As a consequence, no definite concusion can be reached with the currenty avaiabe data. Note that three-body forces have been found to be very important for theoretica cacuations of the p d and n d) threshod scattering parameters. Our anaysis here, however, is independent of the microscopic origin of the effective interaction between proton and deuteron. In a detaied picture, the force might arise from two-nuceon forces or threenuceon forces, but the resut is aways some effective two-body interaction between the proton and the deuteron. The causa range we cacuate is the minimum range that this effective interaction has to have in order to be abe to reproduce the experimentay determined scattering parameters.

94 82 Chapter 5. Causaity bounds for charged partices Proton heion scattering For the scattering of protons off a heium nuceus we were abe to find data for both p 3 He and p α scattering. In the first case, there was ony enough data avaiabe to cacuate the causa range for the S-wave channes. Since both the scattering engths and effective ranges are very simiar for the singet and the tripet channe, so are the resuting causa ranges, which come out as approximatey 1.3 fm. Incidentay, one finds amost the same vaue for the S-wave in p α scattering. For this system, it is interesting to compare to the neutron apha system, where there is no Couomb repusion in the scattering process. Resuts for the n α causa ranges can be read off from Fig. 5 in Ref. [10] obtained using effective range parameters from Ref. [149]). Even though from the pot one ony gets quite rough estimates, one ceary sees that the resuts for the S 1/2 and P 1/2 channes agree very we between p α and n α scattering, which as in the nuceon nuceon case discussed above coud be interpreted as ony a moderate amount of isospin breaking. However, the causa ranges for the P 3/2 channes are very different 0.5 fm for p α, 2 fm for n α). It is an interesting question if this discrepancy hints at an error in the extraction of the effective range parameters either for one of the systems or possiby both), or if there actuay is a physica reason behind the difference in the causa ranges Apha apha scattering For α α S-wave scattering we use the vaues given by Higa et a. see Ref. [62] and experimenta references therein), a C 0 = 1.65±0.17) 10 3 fm and r0 C = 1.084±0.011) fm to find a causa range of about 2.58 fm. For the = 2 channe, no effective range parameters coud be found in the iterature. We have thus used the phase-shift data coected in the review artice by Afza et a. [150] to perform the fit to the effective range expansion Eq. 4.24) with = 2) ourseves. By incuding the phase-shift data up to E ab 6.5 MeV we find a C 2 = 7.23 ± 0.61) 10 3 fm 5 and r C 2 = 1.31 ± 0.22) 10 3 fm 3. However, the fit is strongy dominated by the Op 4 ) shape parameter, so the actua uncertainties of a C 2 and r C 2 shoud probaby be somewhat arger. For the causa range in this channe we find a vaue of about 2 fm, which is just sighty smaer than the S-wave resut. 5.6 Numerica cacuations In order to check our reations and to get a better understanding of the vaues for the causa range, we now present some expicit numerica cacuations. By cutting off the singuar parts of the potentia i.e., the Couomb potentia and the anguar momentum term for 0) at very sma distances, it is a simpe task to numericay sove the radia Schrödinger equation 4.21) in configuration space. From the radia wave functions one can extract the Couomb-modified phase shifts by ooking for a zero

95 5.6. Numerica cacuations 83 at some arge i.e., much arger than the range R of the short-range potentia) distance, and then cacuating For the simpest case of a oca step potentia, w p) r 0 ) = 0, r 0 R, 5.55) cot δ p) = G p)r 0 ) F p)r 0 ). 5.56) V r, r ) = V step r) δr r ) V 0 θr r) δr r ), 5.57) one can of course aso obtain the phase shift directy by matching the wave functions at r = R. The effective range parameters are then obtained by repeating the cacuation for severa sma) momenta and fitting Eq. 4.24) to the resuts. In order to test Eq. 5.26) directy one needs the wave function to cacuate the integra r [ ] 2 r [ 2 dr w 0) r ) = imp 0 dr w p) r )]. 5.58) 0 0 Even if we do not actuay take the imit p 0 but rather just insert some sma p 0 = 0.1 in units of an arbitrary inverse ength scae), we find that the reation r C b C R) 2 R 0 dr [ w p 0) r )] ) is typicay fufied to better than one-percent accuracy for the simpe step potentia defined above. For iustration, in the foowing we choose units where the radia distance is measured in fm. The potentia range is set to 1 fm and its strength is measured in MeV. 6 Furthermore, the reduced mass and Couomb parameter are set to the vaues for the proton proton system, i.e., 2µ = m N 940 MeV and γ = γ p p fm 1. In Figs. 5.3 and 5.4 we show the resuts for = 0, 1, 2) for both repusive and attractive step potentias. Quite interestingy, the = 0 causa range for the repusive potentia stays at zero meaning that one coud reproduce the same vaues of the scattering ength and the effective range aso with a contact interaction) unti a potentia strength of about 100 MeV. For higher partia waves the causa range takes a nonzero vaue for much weaker potentias, but the rise is ess steep. In genera, it is remarkabe that the causa range is typicay consideraby smaer than the actua potentia range R = 1 fm). For attractive potentias the causa range grows much faster as the potentia strength now negative) increases. In contrast to what one might expect, no specia features are seen in the causa ranges as the potentia becomes strong enough to support a new bound state cose to threshod, i.e., when there is a poe in the scattering ength parameter. To concude this section, we show the genera dependence of the causa range on both the scattering ength and the effective range, which has the advantage of not depending on a 6 Note that with these conventions, the quantity that is used in the numerica cacuation is v 0 2µV 0 / c) 2, where c MeV fm is used for the unit conversion.

96 84 Chapter 5. Causaity bounds for charged partices = 0 = 1 = 2 Rc fm) V 0 MeV) Figure 5.3: Causa range for a repusive step potentia and γ = γ p p Rc fm) = 0 = 1 = V 0 MeV) ac fm 2+1 ) Figure 5.4: Causa range thick ines) for an attractive step potentia and γ = γ p p. The thin ines show the corresponding scattering engths.

97 5.7. Reation for asymptotic normaization constants 85 certain mode potentia. For iustration, we again measure distances in fm and set the Couomb parameter to the vaue of the proton proton system. In Figs. 5.5 and 5.6 we show the resuts for = 0 and = 1. For negative r C, the causa range stays essentiay zero. For positive effective ranges, it increases as the absoute vaue of a C becomes arger. If one graduay turns off the Couomb interaction by etting γ 0, the = 1 pot stays amost unchanged, whereas the = 0 resut remains quaitativey the same, but with a much steeper rise in the quadrant where a C > 0 and r C > 0. Figure 5.5: Causa range for γ = γ p p and = 0 in dependence of a C 0 measured in fm. and r C 0, both 5.7 Reation for asymptotic normaization constants The formaism that has been used so far in this chapter in order to derive the causaity bound for the effective range can be extended to the bound-state regime to derive a reation between the asymptotic normaization of the bound state wave function and the effective range of the corresponding two-partice scattering process. It is we-known [9] and was, as noted aready in Chapter 3, aso pointed out by Lüscher [8] that the asymptotic normaization constant ANC) of the bound-state wave function is reated to scattering parameters. More precisey, the residue of the bound-state poe in the anayticay-continued eastic scattering ampitude the f p) defined in Eqs. 2.22) and 2.23) in Chapter 2) is proportiona to A κ 2. In the imit of shaow bound states, κ 0, it is possibe to make a more direct connection to the effective range in the corresponding scattering channe. Utimatey, however, it can be viewed again as a manifestation of the anayticity of the scattering ampitude as a function defined on the compex energy pane. Before we discuss the reation for our system of charged partices, we first present its

98 86 Chapter 5. Causaity bounds for charged partices Figure 5.6: Causa range for γ = γ p p and = 1 in dependence of a C 1 in fm 3 and fm 1, respectivey. and r C 1, measured derivation for the simper case without Couomb interaction Derivation for the neutra system As a starting point we quote the neutra version of our master equation 5.26) from Ref. [10], R [ 2 r = b R) dr u 0) r)], 5.60) where r is the effective range as defined by Eq. 2.19) and 0 u 0) r) = im p 0 u p) r) 5.61) is the zero-energy imit of the scattering wave function for the uncharged system, normaized as in Eq. 5.3). The neutra version of the causaity-bound function, b R), can be given in cosed form for arbitrary anguar momentum. Speciaizing the genera formua given in Ref. [10] to the three-dimensiona case one finds ) ) Γ + 1 ) b R) = 2Γ 1 2 π = 2Γ 1 2 π ) ) Γ For = 0, this simpy gives R 2 4 ) 2 1 R a 2 2 ) 2+1 R + Oa 1 ). 2 2π Γ + 3 2) Γ + 5 2) 1 a 2 ) 2+3 R ) b 0 = 2R + Oa 1 ). 5.63)