Alpha particles in effective field theory

Transcription

1 Alpha particles in effective field theory. aniu itation: AIP onference Proceedings 625, ; View online: View Table of ontents: Published by the American Institute of Physics

2 Alpha Particles In Effective Field Theory. aniu Instituto de Física - Universidade de São Paulo EP idade Universitária, São Paulo - Brasil Abstract. Using an effective field theory for alpha α particles at non-relativistic energies, we calculate the strong scattering amplitude modified by oulomb corrections for a system of two αs. For the strong interaction, we consider a momentumdependent interaction which, in contrast to an energy dependent interaction alone [], could be more useful in extending the theory to systems with more than two α particles. We will present preliminary results of our EFT calculations for systems with two alpha particles. Keywords: Alpha Particles, Effective Field Theory PAS: v;2.60.Gx INTRODUTION An effective field theory EFT aims at describing a physical system including only the relevant degrees of freedom at energies much smaller than a given cutoff Λ. The natural low-energy length scale for nucleons systems is the range of the one-pion exchange potential: l π hc/m π.4 fm, where m π 40 MeV is the pion mass. The internal alphaparticle dynamics is characterized by an intrinsic momentum scale associated with binding mechanism. An estimation is that this scale is set by the pion mass []. At energies below the pion mass, the effective theory of two alpha-particle αα system is built in a first approximation by a scalar-isoscalar field Φ representing each α particle and short-range strong interactions encoded in a derivative expansion of local operators. At low-energy the oulomb repulsion between two alpha particles tends to keep them apart. For momenta smaller than around k = Z Z 2 α em µ 60 MeV, where α em is the fine-structure constant, Z i = 2, i=,2 and µ = m α /2 are the electromagnetic charge and the reduced mass of the αα system respectively, the oulomb interaction must be included by non-perturbative methods. In this work in progress, we briefly discuss the EFT of two αs in the presence of oulomb interactions and then present preliminary results for the strong scattering amplitude modified by oulomb corrections. EFT FOR NON-RELATIVISTI αα SYSTEM WITH OULOMB ORETIONS At low energies the short-range strong interactions can be represented by contact interactions. In the effective theory for non-relativistic nucleons [2], the effective Lagrangian is written as a series of local operators with increasing dimensions, where only the leading one is treated non-perturbatively. In a similar way we can write, for the αα system, the following strong-interaction effective Lagrangian: L = Φ t + 2 2µ Φ + 0 Φ Φ ] 2 [ΦΦ Φ Φ + h.c., where the operator = /2, µ is the reduced mass of the αα system and 0 and 2 are the leading and sub-leading coupling constants, respectively. However, for the αα system these couplings have different counting rules, in comparison with the EFT for two nucleons. In order to reproduce the narrow αα resonance at an energy of about 0. MeV, 2 dimension 8 should be included in the same footing as 0 dimension 6 []. ounting rules for couplings will be discussed later in this paper. The above contact interaction correspond to a singular delta function potential and derivatives thereof. In momentum space, this potential is: q ˆV S p = q 2 + p 2. 2 The electromagnetic interaction needs to be included in the effective Lagrangian. We use the two-potential formalism where the pure oulomb interaction is separated from the oulomb-modified [3, 4]. The scattering amplitude XXXVI Brazilian Workshop on Nuclear Physics AIP onf. Proc. 625, ; doi: 0.063/ AIP Publishing LL /$

3 between an incoming state Ψ p with momentum p and an outgoing state Ψ +, takes the standard form Sp,p = Ψ p Ψ + p = δp p 2πiδE E T p,p. 3 The T-matrix element can be written as the sum of two parts p T = T + T S, 4 where T is the pure-oulomb scattering amplitude and T S is the strong scattering amplitude modified by oulomb corrections T S p,p = χ p ˆV S Ψ + p. 5 The bra-state χ p represents the state of an incoming oulomb state. The outgoing state in 5 can be written in terms of the outgoing oulomb state χ p +, the oulomb propagator Ĝ + Ψ + p = n=0 Ĝ + As a result, we then have for the modified strong scattering amplitude T S p,p = n=0 χ p ˆ The incoming and outgoing oulomb states are given by the oulomb wave function [5] where and the strong interaction ˆV S as: Vˆ S n χ p +. 6 V S Ĝ + Vˆ S n χ p +. 7 χ ± p r = e 2 πη Γ ± iηm iη,,±ipr ip re ip r, 8 η p = Z Z 2 α em µ, 9 p Γz is the Gamma function, and M a, b, z is the Kummer function or well-known confluent hyper-geometric function of first kind F a,b,z. With the oulomb eigenstates 8 we can now find a useful expression for the oulomb propagator. Since the scattering states form a complete set in the repulsive case, the spectral representation of Ĝ + is Ĝ ± E = 2µ d 3 q χ q ± χ q ± 2π 3 2µE q 2 ± iε. 0 For S-wave interactions only, the modified scattering amplitude in the repulsive channel is parametrized in terms of the oulomb-corrected phase shift δ 0 T S p = 2π µ e 2iσ 0 kcotδ l i = 2π µ 2 ηe 2iσ 0 Kη 2k Hη, where the σ 0 is the pure oulomb phase shift, the function Hη is related to the digamma function ψ by Hη ψiη + lniη 2 2iη and η 2 is known as the Sommerfeld factor [6] [7] and represents the probability to find the two particles at zero separation η 2 = χ p ± 0 2 = 2πη e 2πη. 3 The Landau-Smorodinsky Kη function is known to be real-meromorphic function in a large domain of the p plane, for a large class of potentials. It reduces to the effective-range expansion EFE in the presence of oulomb interaction, The parameter a is known as the scattering length and r 0 is the effective range. Kη = a + 2 r 0 p

4 OULOMB-MODIFIED STRONG SATTERING AMPLITUDE The non-perturbative series 7 was calculated resulting: T S = 2 ηe 2iσ J/2 2 / [p 2 2k µ 2 2β2 ] + 2 /2 2 J 2 J 0, 5 Using the standard representation for the oulomb propagator, we can evaluate the divergent integral terms J, J 0 p and J 2 p. When 2 is switched off, we reproduce the same result as Kong and Ravndal [2] T S = 2 ηe 2iσ 0 / 0 J 0. 6 The integral term J 0 is ultraviolet and must be regularized. The renormalization can be carried out both in Power Divergences Subtraction PDS regularization scheme and using a simple momentum cutoff Λ [2]. Writing J 0 = J0 div + J f in, the finite part is just the H-function 2 0 and using PDS regularization the divergent part is: J div 0 p = µ 2π J f in 0 = µk Hη, 7 π {2k [ ε ln κ π 2k + 32 ] } γ + κ, 8 where ε = 3 D, with D the space dimension, γ = is the Euler s constant and κ the renormalization mass. According to the oulomb-modified ERE, there is no contribution to the effective range r 0 in contrast to the scattering length which is related to the ultraviolet divergent part by The renormalized value for 0 is defined by = 2π [ + κ 2k a µ 0 ε + ln κ π 2k + 32 γ ]. 9 a κ = 2π + κ. 20 µ 0 κ ombining equations 9 and 20 we see that a κ is related to the physical scattering length by a κ = [ κ π + 2k a ε + ln + 32 ] 2k γ. 2 When the 2 interaction is considered, in a non-perturbative way, we have to deal with more divergences from the loop integrations. In the expression 5 appears the integral term: J = 2µ regularized within the PDS scheme, with ζ the Riemann s zeta-function and d 3 q 2πηq 2π 3 e 2πη = 2πk2 µ 2k ζ 2 + κ, 22 2 J 2 p = p 2 J 2µ d 3 q 2π 3 2πηq e 2πη q2, 23 which needs to be regularized within the PDS scheme as well. The renormalization proceeds in a similar way as before. From oulomb-modified ERE we have contribution for both, the effective range and the scattering length. In order to absorb the divergences, will be natural to define a new scattering length a κ and a new effective range r 0 κ in terms of the renormalized coupling constants 0 κ and 2 κ. 79

5 Within the effective field theory for nucleons, in the channels with no additional oulomb interactions, it was found by Kaplan, Savage and Wise [8] that the couplings 2n κ scale as 2n κ 2π µλ n, 24 κn+ so that if κ p, 2 κ /p 2. This relation is valid for 2 perturbative only, while the non-perturbative case is being investigated. ONLUSION We present an EFT for two alpha particles that includes momentum-dependent strong interactions and that could be a useful input to systems with more than two alpha particles in multi-body calculations. However, power counting for coupling constants are being investigated. In the modified scattering amplitude, integrals are being regularized and the renormalization is underway. AKNOWLEDGMENTS This work was supported by the onselho Nacional de Desenvolvimento ientífico e Tecnológico NPq, National ouncil for Scientific and Technological Development, Brazil and by the omisión Nacional de Investigación ientífica y Tecnológica ONIYT, National ommission for Scientific and Technological Research, hile. REFERENES. R. Higa, H.-W. Hammer, and U. van Kolck, Nucl. Phys. A X. Kong, F. Ravndal, Nucl. Phys. A L.D. Landau, J. Smorodinski, J. Phys. Acad. Sci. USSR L.D. Landau, J. Smorodinski, J. Phys. Acad. Sci. USSR F. L. Yost, J. A. Wheeler, and G. Breit, Phys. Rev L.D. Landau and E.M. Lifschitz, Quantum Mechanics, Pergamon Press, London, A. Sommerfeld, Atombau and Spektrallinien, Vol. II, Vieweg, Braunschweig, D.B. Kaplan, M.J. Savage, M.B. Wise, Nucl. Phys. B