7. Renormalization and universality in pionless EFT

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1 Renoralization and universality in pionless EFT (last revised: October 6, 04) 7 7. Renoralization and universality in pionless EFT Recall the scales of nuclear forces fro Section 5: Pionless EFT is applicable to the region Q π. This EFT offers a systeatic expansion in Q/Λ breakdown, where the breakdown scale should be of order π. We ve already seen in Section 6 how we can describe the scattering aplitude (or on-shell T-atrix) in perturbation theory in kr (reproducing the effective range expansion) if the theory is natural. In practice this eans all of the ERE coefficients, and a 0 in particular, are of order R. We first consider an explicit exaple of atching for the perturbative case and then consider the unnatural case. a. Perturbative atching to a (toy) underlying theory This section is a slightly edited excerpt fro Perturbative Effective Field Theory at Finite Density by Furnstahl, Steele, and Tirfessa, nucl-th/ or Nucl. Phys. A 67, 396 (000) [4]. As a concrete exaple of perturbative atching an underlying theory to a low-energy EFT, we take the underlying potential to be a separable potential in the S 0 -channel that falls off at large oenta and depends on the relative oentu k k k k only, k ˆV true k = 4π α 3 M (k + )(k + ). () [Recall that Galilean boost invariance requires the interaction between two free nucleons of oentu k and k to be independent of their center-of-ass oentu P = k + k.] The ass corresponds to the range (and non-locality) of the potential; it plays the role of the underlying short-distance scale. We adopt an overall noralization of 4π/M, with M the nucleon ass. The diensionless coupling α provides a perturbative expansion paraeter. If α is O(), then the effective range paraeters are of natural size, which eans a constant of order one ties, in this case, a power of (details in the Appendix of [4]). An EFT can describe observables fro the interaction in Eq. () to any desired accuracy as long as the details of the underlying potential are not probed. For scattering observables, this restricts the external oentu k to be uch less than the characteristic ass. The effective potential can then be written as a oentu expansion ( k ˆV k + k ) ( k + k ) ( EFT k = C 0 + C + C C k k ) , () 4 which ust be regulated. Two possible regularization schees are diensional regularization with power divergence subtraction (DR/PDS) [8] and cutoff regularization (CR). In the latter case we use An extension to higher partial waves is straightforward and does not introduce new features into the analysis.

2 Renoralization and universality in pionless EFT (last revised: October 6, 04) 7 a gaussian separable cutoff in oentu space that strongly daps oenta above an arbitrary cutoff Λ c : ( k ˆV k EFT k = [C + k ) ( k + k ) ( 0 + C + C C k k ) ] e (k +k )/Λ c. (3) 4 For perturbative calculations, analytic expansions can be obtained for these schees in both freespace and in a unifor finite-density syste. The idea of a perturbative atching calculation is that we can work to n th order in the oentu expansion and deterine the corresponding coefficients C 0,..., C n order-by-order perturbatively in the coupling α by equating observables calculated fro the true and EFT hailtonians [0]. We define the series expansion coefficients C (s) n by C n = s= α s C (s) n. (4) It is not necessary that perturbation theory for the observables converges in the low-oentu liit, but for our exaple it does. We will carry out the atching for C 0 in soe detail so that the generalization to higher orders and the extension to finite density is clear. We choose the on-shell T -atrix as our atching observable, since it has a natural perturbative expansion: the Born series ˆT = ˆV + ˆV Ĝ0 ˆV +, with Ĝ0 = (E Ĥ0). We start our discussion with cutoff regularization because the physical content of the renoralization is anifest. At O(α), the on-shell T -atrix is just the on-shell potential ˆV, which for k, Λ c can be expanded (it is sufficient to take k = k for S-waves): k ˆV 4π α 3 k = M (k + ) = 4πα M + O(k ), true αc () /Λ 0 e k c = αc () 0 + O(k ), EFT (CR) where we have used the shorthand notation O(k ) to denote natural corrections with the proper diension ultiplied by either k / or k /Λ c as applicable. Matching to this order fixes (5) and is indicated scheatically in Fig. a. C () 0 = 4π M, (6) Matching at O(α ) is illustrated in Fig. b. There are two contributions on the EFT side: ˆV Ĝ0 ˆV evaluated using C () 0 fixed by Eq. (6), and ˆV evaluated with the O(α ) contribution to The ain assuption is that the short-distance physics is perturbative. The coefficients will have convergent expansions in α since, as we will see explicitly, higher-order constants incorporate high oentu physics only and so are not sensitive to possible infrared divergences (e.g., fro a long-range Coulob potential). This eans that perturbative atching is sufficient even if the constants will then be used in a nonperturbative calculation (see Ref. [0] for an exaple).

3 Renoralization and universality in pionless EFT (last revised: October 6, 04) 7 3 Figure : Perturbative atching in free-space to O(k ). reinder that the potential is separable. The double line for the potential is a C 0, naely, C () 0. The latter renoralizes C 0. To illustrate the physics of this renoralization, we consider explicitly the difference of ˆV Ĝ0 ˆV true and ˆV Ĝ0 ˆV EFT : ˆV Ĝ0 ˆV = ( ) { 4πα M e k /Λ c k + [ d 3 q (π) 3 q + M k q + iη ] q + k + [ d 3 ] } q /Λ M (π) 3 e q c /Λ k q + iη e q c e k /Λ c. (7) Because we are only working to O(k ) in the effective potential, we can expand all k dependence except for Ĝ0, since q can be saller than k: ( ) ˆV Ĝ0 ˆV 4πα dq q [ M = M 0 π k q + iη ( = 4πα M Λ c π 4 (q + ) e q /Λ c ] + O(k ) ) + O(k ). (8) The constant C () 0 is chosen to cancel the k independent part of ˆV Ĝ0 ˆV, so that the net result for ˆV + ˆV Ĝ0 ˆV is O(k ): C () 0 = 4π ( M ) Λ c. (9) π Choosing the cutoff Λ c keeps the constants natural and gives the axiu range of validity for the EFT []. As expected fro the uncertainty principle, the local interaction C () 0 is deterined entirely by high-oentu (q Λ c ) contributions in the loop integral [0]. This is because for q Λ c, the true and EFT tree-level results have already been atched in Eq. (5) to leading order in q and therefore cancel in the integrand of Eq. (8). Note that C () 0 reoves the Λ c dependence while correcting for the contributions in the loop integral fro the high-energy states. The ability

4 Renoralization and universality in pionless EFT (last revised: October 6, 04) 7 4 to absorb the high-oentu coponents of the interaction into the constants of the effective potential is a generic feature of an EFT approach and is essential for a systeatic prediction. If we add a long-range potential to the true theory, we ust also reproduce its long-wavelength effects in the EFT, so we will still have agreeent in the low-oentu region where q Λ c, and the analysis goes through unchanged. If we extend the effective potential to include two constants, the αc () q piece serves to ake the low-oentu part of the loop integrals agree to O(q 4 ), and the high-oentu part of ˆV Ĝ0 ˆV is absorbed into α (C () 0 + C () k ). As a result, the EFT reproduces the observables to O(k 4 ). The addition of C requires an additional renoralization of C 0 in cutoff regularization (see Appendix of [4]), which eans that all the constants change with each successive order in the oentu expansion. Having convinced ourselves of the physical origin of the constants, we ove to the sipler but physically less transparent diensional regularization. It has been shown that power counting with only short-range potentials is regularization schee independent for free-space observables [, ], as considered here. Coparing the expressions for the true and effective T -atrix to leading order in α again leads to k ˆV 4π α 3 k = M (k + ) = 4πα M + O(k ), true (0) αc () 0 + O(k ), EFT (PDS) which fixes the first ter in Eq. (4) to be just as with the cutoff regulator. C () 0 = 4π M, () At the next order in α, the difference between ˆV Ĝ0 ˆV true and ˆV Ĝ0 ˆV EFT becoes ( ) { ˆV Ĝ0 ˆV 4πα [ d 3 q M ] = M k + (π) 3 q + k q + iη q + k + [ (µ ) 4 D d D q (π) D ] } M k q, () + iη where the second integral is diensionally regularized with D = 4 + ɛ and in general is evaluated in the PDS schee as [8] ( µ ) 4 D d D q (π) D Mk i q j Mk(i+j) k q + iη = (µ + ik). (3) 4π Here µ is the DR renoralization scale. As before, the low-oentu parts of the integrands already agree to O(q ), so the constant difference is fro the high-oentu behavior.

5 Renoralization and universality in pionless EFT (last revised: October 6, 04) 7 5 The results for the first two ters in the Born series are [ 4π α 3 k ˆV + ˆV Ĝ0 ˆV M (k k = + ) + α 6 ( k (k + ) 4 ) 0 αc () 0 + α C () 0 α M ( C () 4π )] ik, true (µ + ik), EFT (PDS) (4) which requires C () 0 = M ( 4π C () 0 ) ( µ ) ( ) ( 4π = M µ ), (5) for a proper atch to O(α, k ). Note that for Λ c = πµ, the DR/PDS constants are equivalent to the cutoff results Eqs. (6) and (9). This agreeent does not persist at higher orders in oentu. Carrying the atching to one ore order in α suggests a geoetric series for the C (s) 0 in the DR/PDS schee, which indeed sus to the full nonperturbative solution (see Appendix of [4]): C 0 = 4πα [ ( + α M µ )]. (6) Extending the analysis to higher orders in oentu (or expanding the full results fro the Appendix of [4] in powers of α) gives C () = ( 4π M ), ( ) ( 4π 5 C() = M 8 µ ) 4, (7) ( ) C () 4π 3 4 = M 4, ( ) ( 4π 7 C() 4 = M 0 µ ) 0 4. (8) As additional constants are added to the effective potential in DR/PDS, the previously fixed constants are not odified. This is not the case in CR (see Appendix of [4]) due to ters proportional to positive powers of the cutoff. The constant C 4 requires additional input, because it does not contribute to the on-shell twobody T -atrix. It is tepting to atch the true and EFT T -atrices off shell to deterine C 4, but this is never necessary, as only observables are required to fix the EFT constants. In particular, the additional on-shell constraint of atching the three-body scattering aplitudes [, ] can be used to find C 4. Because we are in a regie where the coupling is perturbative, the Faddeev equations are siplified to a set of three-to-three tree-level aplitudes at second-order in the potential, O(α ). The result fro atching is: ( ) C () 4π 4 = M 4. (9) () The second-order constant C 4 as well as true three-body ters (needed to absorb the divergences in three-to-three loop-level aplitudes) do not enter nuclear atter calculations until higher order in the α expansion.

6 Renoralization and universality in pionless EFT (last revised: October 6, 04) 7 6 [Note fro Ref. [5], which is about field redefinitions:... we can arbitrarily trade-off the value of C 4 with the coefficient of a three-body contact ter. This eans that off-shell atrix eleents of the potential (e.g., 0 ˆV EFT k ) and subsequently the off-shell T-atrix can be changed continuously by varying α, without changing any observables. One choice is to eliinate two-body off-shell vertices such as C 4 entirely in favor of any-body vertices that do not vanish on shell. This is the for of any-body EFT used in Ref. [7], which corresponds to Georgi s on-shell effective field theory [6]. In different situations, different choices ay be ore efficient [3].] By construction, the EFT systeatically reduces the error order-by-order in the oentu expansion. We can see this explicitly by exaining the truncation error in ˆT to O(α ), which is just ˆV + ˆV Ĝ0 ˆV, between calculations using the true and EFT potentials. 3 For an effective potential only containing C 0, the error is O(k ) [ ) ( ˆV + ˆV Ĝ0 ˆV = 4π α ( + 5 Λ + α c Λ )]( k CR c πλc ) M ( α + 5 )( k ) α DR/PDS and adding C brings the error to O(k 4 ) { α (3 ˆV + ˆV Ĝ0 ˆV = + O(k 3 ), (0) 4π M Λ c ) + 4 Λ 4 + α [ c Λ c + 4Λ ( c + π Λ 54 c Λ 4 c )] } ( k 4 4Λ 4 c ) 4 CR ( 3α 7α )( k ) 4 DR/PDS + O(k 5 ). () Note that the truncation error with CR depends on Λ c while in the DR/PDS schee the truncation error is independent of µ. Extending the analysis to nonperturbative calculations and finite density will in general require nuerical solutions, and so a connection between the analytical results above and a graphical error analysis is iportant. The error plots introduced by Lepage [0] are useful in this regard. Keeping constants to O(k 0 ), O(k ), and O(k 4 ) in the effective potential Eq. () or Eq. (3) leads to successively better approxiations to ˆV + ˆV Ĝ0 ˆV, as seen by the analytic expressions for the error Eqs. (0) and () and shown graphically in Fig.. With each additional order, the slope of the error increases, reflecting the iproved truncation error. (If a long-range potential is added to both the true and effective potentials, the absolute error in the DR/PDS power-counting schee 3 For the present perturbative case, this is ore convenient than looking at the error in k cot δ, which is appropriate for a nonperturbative calculation. In the error plots, we consider only the real part of the difference in ˆV + ˆV Ĝ0 ˆV. The iaginary part follows fro unitarity, which the EFT reproduces order-by-order.

7 Renoralization and universality in pionless EFT (last revised: October 6, 04) 7 7 T(k) [GeV ] O(k ) O(k 4 ) O(k 6 ) CR c = T(k) [GeV ] O(k ) O(k 4 ) O(k 6 ) DR/PDS k/ k/ Figure : Re k ˆT k to O(α ) vs. k/ for α = / and with M = 940 MeV and = 600 MeV. Results for cutoff regularization (CR) with Λ c = are shown on the left and for diensional regularization (DR/PDS) on the right. Note that a s = /3 and r e = 9/. will decrease but the error is always O(k ) [, 9].) For k the error is doinated by the first unatched ter, so the lines are straight on a log-log plot. The EFT should break down when the external oentu probes the details of the underlying potential; graphically the intersection of the lines 4 indicates the approxiate breakdown scale Λ. In our exaple, we see that Λ, as expected for a natural theory [with α O()]; indeed, all the errors are of the sae order for k. This breakdown scale does not change as ore orders in α are included in the contact interactions, but the accuracy does iprove. b. Non-perturbative atching First we consider the leading order coefficient C 0 for the unnatural case of large scattering length. That eans we expand about the /a 0 = 0 liit (so instead of ka 0, we have ka 0 ). The plan is to atch the expansion of the Lippann-Schwinger (LS) equation for the T-atrix to the on-shell for for the T-atrix fro the effective range expansion: T = 4π a 0 r 0 + ik. () We can do non-perturbative atching because we can solve the Lippann-Schwinger (LS) 4 To deterine the intersection scale, one should extend the lines fro the straight regions (k ) rather than looking at the actual intersection.

8 Renoralization and universality in pionless EFT (last revised: October 6, 04) 7 8 P/ + k P/ + k = P/ k P/ k it (k, cos θ) ic 0 (Λ) ic 0 (Λ)I 0 (k, Λ)C 0 ic 0 I 0 C 0 I 0 C 0 Figure 3: Lippann-Schwinger equation for the T atrix. equation exactly for V = C 0 : where T = C 0 + C 0 I 0 (k, Λ c )C 0 + C 0 I 0 (k, Λ c )C 0 I 0 (k, Λ c )C 0 + (3) = C 0 [ + I 0 (k, Λ c )C 0 + (I 0 (k, Λ c )C 0 ) + ] (4) C 0 = C 0 I 0 (k, Λ c ) = C 0 I 0 (k, Λ c ) (5) I 0 (k, Λ c ) = 4π ( ik + ) π Λ c O(k /Λ c). (6) If we take Λ c large, then the O(k /Λ c) correction is sall. For other ways to regulate the integral with a cutoff, Λ c will appear in each case, but the nuerical coefficient is different. Matching to the scattering length only by equating () and (5): which yields or 4π a 0 + ik = C 0 I 0 (k, Λ c ) = C 0 + 4π (ik + π Λ c + O(k /Λ c)), (7) = 4π + a 0 C 0 π Λ c (8) C 0 (Λ c ) = 4π a 0 π Λ. (9) c This running coupling C 0 will give cutoff independent results at low k (up to O(k /Λ c) corrections). So at leading order, we have to resu all of the C 0 interactions. Beyond leading order, we will still have to resu the C 0 interactions whenever they occur, but then treat higher-order two-body interactions perturbatively. Three-body interactions are another story, however (stay tuned)! Let s discuss the behavior of C(Λ c ) in the strong and weak interaction liits (unnatural and natural).. For strong interactions, a 0 0 in which case which is always attractive to give a zero-energy bound state. C 0 (Λ c ) = π Λ c < 0, (30)

9 Renoralization and universality in pionless EFT (last revised: October 6, 04) 7 9. For weak interactions, we can choose Λ c a 0 for a large cutoff range and expand C 0 (Λ c ) = 4πa 0 which is what we found before doing perturbative atching. π Λ ca 4πa 0 ( + π a 0Λ c ), (3) In the weak interaction case, each successive ter in the LS equation is suppressed by an additional power of a 0 Λ c. But in the strong interaction case, each vertex C 0 contributes a /Λ c factor while each I 0 (k, Λ c ) contributes (ik + Λ c ) Λ c. The result is that each ter in the expansion is equally iportant, which is our signal that we need to su the. Soe further coents: Once again, the result that the potential C 0 (Λ c ) depends on the choice of Λ c eans that it is not unique and hence can not be easured directly in an experient. We say that the potential is both scale and schee dependent. The scale dependence is fro changing Λ c, while the schee dependence is fro choosing a sharp cutoff as a regulator (cf. choosing C 0 e (k +k ) n /Λ n c with the interediate integrations running over the full range fro 0 to ). We need to use a consistent schee for currents and any-body forces. In coordinate space, the potential with a finite cutoff Λ c is a seared out delta function, fine-tuned in the unitary liit to have a bound-state at E = 0. c. More on errors in pionless EFT There are two sources of errors in the leading-order pionless EFT:. Fro oitted ters, such as C, C, which scale as ( ) Q ( ) Q Λ breakdown Λ b ( ) Q (3). Fro regularization artifacts, naely the finite cutoff leads to a Q /Λ c error, which effectively induces an effective range r 0 of order /Λ c. (That is, a contribution of this for corresponds to an effective range ter in the ERE.) Thus the total error should scale as the axiu of these errors: ( ( ) Q ( ) ) Q error ax,,. (33) Λ c Λ b This iplies that as long as Λ c Λ b, the regularization does not lead to errors larger than those already present fro the EFT truncation. π

10 Renoralization and universality in pionless EFT (last revised: October 6, 04) 7 0 The systeatics of the errors in phase shifts can be anifested by aking error plots (of the type used in ordinary nuerical analysis) that plot the logarith of the (relative) error versus the logarith of the oentu scale. These are generally called Lepage plots after their introduction in Ref. [0]. (Exaples for perturbative atching are in Fig..) For Λ c Λ b, we expect the low oentu region (i.e., well below the breakdown scale Λ b ) to be doinated by the leading power of k/λ b, so we get straight lines with increasing slope as we go fro LO to NLO to N LO. The asyptotic intercept should roughly deterine Λ b. If we consider a fixed order but vary Λ c < Λ b, we expect lines with the sae (low-oentu) slope, but for lower Λ c they will have the slope inherited fro the regulator. This will iprove (the line oves down) until it stops iproving for Λ c Λ b. The non-perturbative solution is increasingly non-linear as Λ c increases further, which could lead to a deterioration in the error. So nothing is gained in the error by taking Λ c toward and uch can be lost by aking the nuerical solution ore difficult. (You add increasingly incorrect physics in the su over interediate states, the loops, which has to be canceled by the interaction. This requires larger basis sizes and increased fine tuning, both of which are usually negatives fro a coputational point of view unless there are other copensations.) d. Renoralization group equation The running of C 0 (Λ c ) with the cutoff can be expressed in the for of a differential equation, which is called a renoralization group (RG) equation. The key is to deand that the on-shell T-atrix, which is easurable, ust be independent of Λ c (for sall k, because leading order): dt dλ c = d dλ c C 0 (Λ I c) 0(k, Λ c ) = 0 d = dλ c C 0 d dλ c I 0 (k, Λ c ) dc 0 (Λ c ) C 0 (Λ c ) dλ c dc 0(Λ c ) dλ c = π ( + O(k /Λ c) ) = π (C 0(Λ c )). (34) We ve neglected the k /Λ c correction at low k. Can you solve this equation? What is the initial condition? How does this copare to the QCD running coupling α s (Q )? This is usually written as a function of Q, while the C 0 running is with respect to Λ c ; how do you account for this? Can you explain how the source of the running is the sae? Can you devise an analogous quantity to Λ QCD? e. Generalized potential What is the ost general pionless potential? In Section 6 we considered a spin-dependent ter in the low-energy Lagrangian without isospin (that is, just spin-up and spin-down ferions, such as neutrons) and stated (without proof) that the ter was redundant. Let s revisit such a ter here

11 Renoralization and universality in pionless EFT (last revised: October 6, 04) 7 but at the level of the potential. So consider V = C S + C T σ σ. (35) We will account for the antisyetrization necessary in atrix eleents of V by adding operators that exchange the coordinates: V antisy. = ( P )V (36) where the exchange operator P acts on the relative oenta and the spin: P = P k k P spin where P spin = + σ σ. (37) (Do you agree that the action of P on the oenta is to siply exchange the relative oenta?) At leading order there is no oentu dependence, so P k k is just the identity operator, leaving V antisy. = ( P spin )(C S + C T σ σ ) = (C S 3C T + (3C T C S )σ σ ) = { 0 S = (C S 3C T ) S = 0 (38) where we have used (σ σ ) = 3 σ σ. (39) Thus we coe to the sae conclusion as in Section 6, which is that there is only one linearly independent cobination of C S and C T, naely C S 3C T. We could, for exaple, choose C S = C 0, C T = 0 or C S = 0, C T = C 0 /3 and we would get exactly the sae results. [Question: What is σ σ in spin-singlet and spin-triplet states?] [Question: How would you build construction operators that project onto spin-singlet and spin-triplet parts of a wave function?] Next consider LO pionless EFT but with spin and isospin. There are now four possible operators to consider:, σ σ, τ τ, σ σ τ τ, (40) but only two different S-waves (singlet and triplet). As you have probably guessed, only two of these are independent and we can pick any cobination. The conventional choice is: V LO NN = C S + C T σ σ. (4) At NLO there are 4 possible operators but only 7 are linearly independent. The conventional choice is VNN NLO =C (k + k ) + C k k + C S (k + k )σ σ + C S k k σ σ + ic LS (σ + σ ) (k k ) + C T σ (k k)σ (k k) + C T σ (k + k)σ (k + k). (4)

12 Renoralization and universality in pionless EFT (last revised: October 6, 04) 7 In the second line, the first ter is a spin-orbit interaction and the second and third ters lead to tensor interactions. The individual ters are explicitly non-local, because they do not depend only on the oentu transfer q = k k. However, it turns out that one can pick a different linear cobination that are (alost) all functions of q only. This turns out to be advantageous for soe applications. f. References [] P. F. Bedaque, H. W. Haer, and U. van Kolck. Effective theory for neutron-deuteron scattering: Energy dependence. Phys. Rev. C, 58:64 644, 998. [] P. F. Bedaque, H. W. Haer, and U. van Kolck. The three-boson syste with short-range interactions. Nucl. Phys. A, 646: , 999. [3] J.-W. Chen, G. Rupak, and M. J. Savage. Nucleon-nucleon effective field theory without pions. Nucl. Phys. A, 653:386 4, 999. [4] R. Furnstahl, J. V. Steele, and N. Tirfessa. Perturbative effective field theory at finite density. Nucl. Phys. A, 67:396 45, 000. [5] R. J. Furnstahl, H. W. Haer, and N. Tirfessa. Field redefinitions at finite density. Nucl. Phys. A, 689: , 00. [6] H. Georgi. On-shell effective field theory. Nucl. Phys. B, 36: , 99. [7] H. W. Haer and R. J. Furnstahl. Effective field theory for dilute feri systes. Nucl. Phys. A, 678:77 94, 000. [8] D. B. Kaplan, M. J. Savage, and M. B. Wise. Two-nucleon systes fro effective field theory. Nucl. Phys. B, 534:39 355, 998. [9] D. B. Kaplan and J. V. Steele. The Long and short of nuclear effective field theory expansions. Phys. Rev. C, 60:06400, 999. [0] G. Lepage. How to Renoralize the Schrödinger Equation [] J. V. Steele and R. Furnstahl. Regularization ethods for nucleon-nucleon effective field theory. Nucl. Phys. A, 637:46 6, 998. [] U. van Kolck. Effective field theory of short range forces. Nucl. Phys. A, 645:73 30, 999.