On the axial-vector form factor of the nucleon and chiral symmetry
OOn the axial-vector form factor of the nucleonand chiral symmetry
Matthias F.M. Lutz,
1, 2
Ulrich Sauerwein,
1, 2, 3 and Rob G.E. Timmermans GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH,Planckstraße 1, 64291 Darmstadt, Germany Technische Universit¨at Darmstadt, D-64289 Darmstadt, Germany Van Swinderen Institute for Particle Physics and Gravity,University of Groningen, 9747 AG Groningen, The Netherlands (Dated: March 24, 2020)
Abstract
We consider the chiral Lagrangian with nucleon, isobar, and pion degrees of freedom. The baryonmasses and the axial-vector form factor of the nucleon are derived at the one-loop level. We explorethe impact of using on-shell baryon masses in the loop expressions. As compared to results fromconventional chiral perturbation theory we find significant differences. An application to QCDlattice data is presented. We perform a global fit to the available lattice data sets for the baryonmasses and the nucleon axial-vector form factor, and determine the low-energy constants relevantat N LO for the baryon masses and at N LO for the form factor. Partial finite-volume effectsare considered. We point out that the use of on-shell masses in the loops results in non-analyticbehavior of the baryon masses and the form factor as function of the pion mass, which becomesprominent for larger lattice volumes than presently used. a r X i v : . [ h e p - l a t ] M a r ONTENTS
I. Introduction 3II. The SU(2) chiral Lagrangians with baryon fields 4III. Quark-mass dependence of nucleon and isobar mass 6IV. Axial charge and radius of the nucleon 9V. Fit to QCD lattice data 15VI. Conclusion and outlook 28VII. Acknowledgments 28VIII. Appendix A 29IX. Appendix B 31X. Appendix C 33XI. Appendix D 34References 362 . INTRODUCTION
The axial-vector form factor of the nucleon, and in particular its value at zero momentumtransfer, the axial charge, is a quantity of fundamental interest in hadronic physics. It iscentral to, for instance, β decay and neutrino-nucleon scattering. Next to the nucleon andisobar masses, it is therefore an important testing ground for our understanding of non-perturbative QCD in the framework of chiral perturbation theory. In recent years, good-quality QCD lattice data have become available, not only for the nucleon and isobar masses,but also for the axial-vector form factor of the nucleon.Chiral perturbation theory is the tool of choice to study the pion and momentum de-pendence of hadronic quantities. Unfortunately, previous work within flavor-SU(3) chiralperturbation theory has shown serious convergence problems of the chiral expansion. Withinflavor-SU(2) chiral perturbation theory, several works have addressed the nucleon and isobarmasses and the axial-vector form factor, by using several expansion schemes. The axial-vector form factor has been calculated to one-loop level in heavy-baryon chiral perturbationtheory [1–3] and in relativistic (or covariant) baryon chiral perturbation theory [4–7] withdifferent renormalization schemes. The ∆(1232) isobar was included in Refs. [8–11].Motivated by the recent progress in lattice QCD [12–14], we develop here a novel schemebased, for convenience, on the relativistic flavor-SU(2) chiral Lagrangian and apply it to thenucleon and isobar masses and the axial-vector form factor of the nucleon. It allows us toexplore in an important test case the use of on-shell hadron masses in a chiral approach tosystems of pions, nucleons, and isobars. This has implications for an analysis of QCD latticedata of the nucleon axial charge and radius and the pion-nucleon and the pion-isobar sigmaterms.The common challenge faced by treatments of masses and form factors with chiral per-turbation theory is the occurrence of power-counting violating terms, with m π ∼ ∆ ∼ smallmomenta (∆ is the isobar-nucleon mass difference), in the chiral expansion [15]. It is notclear how to deal with such terms in other approaches to relativistic baryon chiral pertur-bation theory. In our scheme, we have to deal with power-counting violating terms in thepresence of on-shell masses. The solution lies in considering the chiral Ward identities an-alyzed in terms of the Passarino-Veltman reduction [16] scheme of the one-loop integrals.We show that we are able to renormalize the one-loop amplitudes in terms of subtracted3assarino-Veltman integrals, where we drop scalar integrals that involve only baryons, andwe need subtractions only in tadpole and bubble terms. An additional challenge occurs inthe chiral domain, where we need a dimensional counting with ∆ ∼ M ∼ Q . We find thata further class of power-counting violating terms arises, and, therefore, additional subtrac-tions are needed. In this way, we arrive at consistently renormalized amplitudes that can becompared to the QCD lattice data.We have organized our paper as follows. In Section II we define the flavor-SU(2) chiralLagrangians for the nucleon and isobar fields as the starting point of our development.In Section III we derive the nucleon and isobar masses at one-loop level. We discussthe Passarino-Veltman framework and our resulting power-counting and renormalizationscheme. In Section IV we extend our approach to the axial-vector form factor of the nu-cleon. Next, in Section V, we present the results of the fits of our expressions to the availableQCD lattice data for the nucleon and isobar masses and for the axial-vector form factor.We show that the use of on-shell masses in the one-loop expressions results in non-analyticbehavior of the masses [17, 18] and form factor as function of the pion mass and its depen-dence on the lattice size. We discuss the quality of the fit and the resulting values of theparameters. In Section VI we summarize our findings and conclude with an outlook. II. THE SU(2) CHIRAL LAGRANGIANS WITH BARYON FIELDS
We consider the flavor-SU(2) chiral Lagrangian density with nucleon and isobar degreesof freedom [19]. We focus on the strict isospin limit with degenerate up- and down-quarkmasses m u = m d ≡ m . The isospin-doublet nucleon field is N t = ( p, n ). The isospin-tripletpion fields (cid:126)π enter via the SU(2) matrix Φ = (cid:126)τ · (cid:126)π . For the nucleon the relevant terms are L N = ¯ N (cid:0) i /D − M (cid:1) N + 2 ζ N ¯ N χ + (cid:0) i /D − M (cid:1) N + 4 b χ ¯ N χ + N + 4 c χ ¯ N χ N + g A ¯ N γ µ γ i U µ N + 4 g χ ¯ N γ µ γ χ + i U µ N + g R / N γ µ γ [ D ν , F − µν ] N − g S ¯ N U µ U µ N − g T ¯ N i σ µν (cid:2) U µ , U ν (cid:3) N − g V (cid:16) ¯ N i γ µ { U µ , U ν } D ν N + h .c. (cid:17) , (1)4here the nucleon mass parameter is denoted by M , and U µ = u † (cid:16) ( ∂ µ e i Φ /f ) − (cid:8) i a µ , e i Φ /f (cid:9)(cid:17) u † , Γ µ = u † (cid:2) ∂ µ − i a µ (cid:3) u + u (cid:2) ∂ µ + i a µ (cid:3) u † , F ± µν = u † F Rµν u ± u F Lµν u † ,D µ N = ∂ µ N + Γ µ N , u = e i Φ / (2 f ) , χ + = 2 B m cos (cid:0) Φ /f (cid:1) ,F Rµν = ∂ µ a ν − ∂ ν a µ − i [ a µ , a ν ] , F Lµν = − ( ∂ µ a ν − ∂ ν a µ ) − i [ a µ , a ν ] . (2)In the presence of the isobar field ∆ µ additional terms are required, viz. L ∆ = − tr (cid:104) ¯∆ µ · (cid:0) ( i /D − ( M + ∆)) g µν − i ( γ µ D ν + γ ν D µ ) + γ µ ( i /D + ( M + ∆)) γ ν (cid:1) ∆ ν (cid:105) − ζ ∆ tr (cid:104) ¯∆ µ · (cid:0) i /D − ( M + ∆) (cid:1) ∆ µ χ + (cid:105) − d χ tr (cid:104) ¯∆ µ · ∆ µ χ + (cid:105) − e χ tr (cid:104) ¯∆ µ · ∆ µ χ (cid:105) + h A tr (cid:104)(cid:0) ¯∆ µ · γ γ ν ∆ µ (cid:1) i U ν (cid:105) − h χ tr (cid:104)(cid:0) ¯∆ µ · γ γ ν ∆ µ (cid:1) i χ + U ν (cid:105) + 4 h (1) S tr (cid:2) ¯∆ µ · ∆ µ U ν U ν (cid:3) + 4 h (2) S (cid:0) ¯∆ µ · U ν (cid:1) (cid:0) U ν · ∆ µ (cid:1) + 2 h (3) S tr (cid:2) ¯∆ µ · ∆ ν (cid:8) U µ , U ν (cid:9)(cid:3) + 2 h (4) S (cid:16)(cid:0) ¯∆ µ · U µ (cid:1) (cid:0) U ν · ∆ ν (cid:1) + (cid:0) ¯∆ µ · U ν (cid:1) (cid:0) U µ · ∆ ν (cid:1)(cid:17) + h T tr (cid:104) ¯∆ λ · iσ µν ∆ λ (cid:2) U µ , U ν (cid:3)(cid:105) + h (1) V tr (cid:2)(cid:0) ¯∆ λ · i γ µ D ν ∆ λ (cid:1) (cid:8) U µ , U ν (cid:9)(cid:3) + h.c.+ h (2) V (cid:0)(cid:0) ¯∆ λ · U µ (cid:1) i γ µ (cid:0) U ν · D ν ∆ λ (cid:1) + (cid:0) ¯∆ λ · U ν (cid:1) i γ µ (cid:0) U µ · D ν ∆ λ (cid:1)(cid:1) + h.c.+ f S (cid:16) ¯∆ µ · i U µ N + h.c. (cid:17) − f χ (cid:16) ¯∆ µ · i U µ χ + N + h.c. (cid:17) − f (1) A tr (cid:2)(cid:0) ¯∆ µ · γ ν γ N (cid:1) (cid:8) U µ , U ν (cid:9)(cid:3) + h.c. − f (2) A tr (cid:2)(cid:0) ¯∆ µ · γ ν γ N (cid:1) (cid:2) U µ , U ν (cid:3)(cid:3) + h.c. − f (3) A (cid:16)(cid:0) ¯∆ µ · U ν (cid:1) γ ν γ U µ N + (cid:0) ¯∆ µ · U µ (cid:1) γ ν γ U ν N (cid:17) + h.c. − f (4) A (cid:16)(cid:0) ¯∆ µ · U ν (cid:1) γ ν γ U µ N − (cid:0) ¯∆ µ · U µ (cid:1) γ ν γ U ν N (cid:17) + h.c. , (3)where the isobar mass parameter is denoted by M + ∆, and∆ µ = ∆ ++ µ , ∆ µ = ∆ + µ / √ , ∆ µ = ∆ µ / √ , ∆ µ = ∆ − µ , (Φ · ∆ µ ) a = (cid:15) kl Φ ln ∆ knaµ , ( ¯∆ µ · Φ) b = (cid:15) kl ¯∆ µknb Φ nl , ( ¯∆ µ · ∆ µ ) ab = ¯∆ µbcd ∆ acdµ , ( ¯ N · ∆ µ ) ab = (cid:15) k b ¯ N n ∆ knaµ , ( ¯∆ µ · N ) ab = (cid:15) k a ¯∆ µknb N n , ( D µ ∆ ν ) abc = ∂ µ ∆ abcν + Γ ad,µ ∆ dbcν + Γ bd,µ ∆ adcν + Γ cd,µ ∆ abdν . (4)The Lagrangian densities in Eqs. (1) and (3) contain a number of coupling constants or5low-energy constants” (LECs). At leading order in large- N c they satisfy [20, 21] h A = 9 g A − f S , g V = h (1) V + 43 h (2) V ,g S = h (1) S + 43 h (2) S + 49 h (4) S , h (3) S = − h (4) S ,f ( A )1 = 0 , f ( A )2 = 43 h T , f ( A )4 = 103 h T − g T ,b χ = d χ , c χ = e χ , ζ N = ζ ∆ . (5) III. QUARK-MASS DEPENDENCE OF NUCLEON AND ISOBAR MASS
Given the chiral Lagrangian densities in Eqs. (1) and (3) it is straightforward to deriveits implications for the nucleon and isobar mass at the one-loop level in dimensional regu-larization [22, 23]. There are various schemes how to deal with the power-counting violatingcontributions [24–26]. Here we follow a framework based on the Passarino-Veltman reduc-tion [16]. It was argued in Ref. [23] that power-counting violating terms come with scalarloop integrals only, which depend on the renormalization scale µ . In turn it suffices to setup a suitable subtraction scheme for the scalar tadpole and bubble loop integrals. Such aprogram was developed for the flavor-SU(3) baryon masses in Refs. [15, 23, 27]. Adaptedto our flavor-SU(2) case, the results are M N = M + ¯Σ N ( M N ) , M ∆ = M + ∆ + (cid:60) ¯Σ ∆ ( M ∆ + i (cid:15) ) , ¯Σ N = − b χ B m − c χ m π + 3 f (cid:16) − (¯ g S + M ¯ g V / m π + 2 b χ m π (cid:17) ¯ I π − ζ N m π ( M N − M ) + ¯Σ bubble N /Z N , ¯Σ ∆ = − d χ B m − e χ m π + 3 f (cid:16) − (¯ h S + ( M + ∆) ¯ h V / m π + 2 d χ m π (cid:17) ¯ I π − ζ ∆ m π ( M ∆ − ( M + ∆)) + ¯Σ bubble∆ /Z ∆ , (6)with m π the pion mass, Z N and Z ∆ the wave-function renormlization factors of the baryons,and ¯ g S = g S − (4 M + ∆ M − ∆ )18 M ( M + ∆) f S , ¯ g V = g V −
19 ( M + ∆) f S , ¯ h S = h S + 4 M + 5 ∆ M + 2 ∆
72 ( M + ∆) f S , h S = h (1) S + 23 h (2) S + 14 h (3) S + 16 h (4) S , ¯ h V = h V − M + ∆) (cid:16) h A + 136 f S (cid:17) , h V = h (1) V + 23 h (2) V . (7)6ll quantities in Eq. (6) are expressed in terms of renormalized scalar loop functions, atadpole ¯ I π , and a bubble function ¯ I πR ( M B ) with R, B ∈ { N, ∆ } . We identify the mass ofthe isobar M ∆ in a quasi-particle approach where its width is determined by (cid:61) ¯Σ ∆ ( M ∆ + i (cid:15) ).The loop functions take the form¯ I π = m π π log (cid:104) m π µ (cid:105) , ¯ I π | R = m π π log (cid:104) m π M R (cid:105) , ¯ I πR (cid:0) M B (cid:1) = 116 π (cid:34) γ RB − (cid:16)
12 + m π − M R M B (cid:17) log (cid:16) m π M R (cid:17) + p πR M B (cid:16) log (cid:104) − M B − p πR M B m π + M R (cid:105) − log (cid:104) − M B + 2 p πR M B m π + M R (cid:105)(cid:17)(cid:35) , (8)where p πR = M B − M R + m π M R − m π ) M B , (9) γ RB = − lim m → M R − M B M B log (cid:12)(cid:12)(cid:12) M R − M B M R (cid:12)(cid:12)(cid:12) . (10)The tadpole ¯ I π depends on the renormalization scale µ of dimensional regularization, whilethe renormalized bubble ¯ I πR ( M B ) in our scheme does not. The subtraction term γ RB il-lustrates the necessity of a renormalization scheme that leads to results that are in accor-dance with the expectation of dimensional power-counting rules. In the chiral domain with m π < M ∆ − M N ∼ Q one expects ¯ I πN ( M N ) ∼ Q , but ¯ I π ∆ ( M N ) ∼ Q , with Q denotingthe chiral small scale Q ∼ m π . Given the form of our renormalized bubble function such abehavior is ensured. Clearly, this is not the case when γ RB = 0.Matters turn considerably more complex once we start to leave the chiral domain andconsider m π ∼ M ∆ − M N , or even M ∆ − M N < m π < πf . In order to avoid a proliferationof counting schemes we follow here a pragmatic path, where we simply keep all model-independent parts of the one-loop expressions. The bubble-loop contribution to the nucleonmass is given by (see Eq. (31) of Ref. [15]) f ¯Σ bubble B = N = 34 g A (cid:40) M N − M B M B ¯ I π | N − ( M B + M N ) E N + M N p πN ¯ I πN (cid:41) , + f S (cid:40) ( M ∆ + M B ) M B M (cid:0) M − M B (cid:1) ¯ I π | ∆ − M B M ( E ∆ + M ∆ ) p π ∆ ¯ I π ∆ + 43 α ( B ) π ∆ (cid:41) , (11)7ith α ( B = N ) π ∆ = α ∆ π (cid:16) M ∆ − (cid:16) M (cid:17) M B (cid:17) (cid:16) ∆ ∂∂ ∆ + 1 (cid:17) γ + ∆ m π π α γ ,E = p π ∆ + M , γ = 2 M + ∆ M log (cid:104) ∆(2 M + ∆)( M + ∆) (cid:105) , (12) α = (2 M + ∆) M ( M + ∆) , γ = − M + 2 ∆ M + ∆ M (2 M + ∆) log (cid:104) ∆(2 M + ∆)( M + ∆) (cid:105) − M M + ∆ . Consider first the term proportional to ¯ I π ∆ . The pre-factor p π ∆ is model independent,since it is the relativistic phase-space factor describing the N → π ∆ decay process, whichis accessible for some unphysical parameter choices of the chiral Lagrangian. At M N >M ∆ + m π the scalar bubble loop function ¯ I π ∆ turns complex. The full pre-factor cannotchange upon the consideration of higher-loop effects. The only effect expected is that thebare coupling constant f S is replaced by its on-shell physical value, which then of course mayshow some quark-mass dependence. The role of the other term proportional to ¯ I π | ∆ is moresubtle. In fact, there is important cross talk to the ¯ I π ∆ term [15]. The reason is that a term ∼ ∆ m π log m π cannot be absorbed into any of the other non-bubble term contributions inEq. (6). Similar arguments can be put forward in favor of the model-independent nature ofthe terms proportional to ¯ I πN and ¯ I π | N .There is yet the subtraction term α ( N ) π ∆ to be discussed. The purpose of the latter istwofold. First, it ensures that the baryon wave-function renormalization factor becomesunity in the chiral limit, viz. Z B = (cid:0) − m π ζ B + ∂∂M B ¯Σ bubble B (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) m → → . (13)Second, it is required to protect the non-analytic g A m π contribution to the nucleon mass.A corresponding term h A m π for the isobar mass would generate a contribution proportionalto h A g A ∆ m π , which must be cancelled by higher-loop effects [15, 28, 29]. We run into thisissue only because we wish to keep the on-shell hadron masses inside the loop functions.We now turn to the bubble-loop contributions to the isobar mass, f ¯Σ bubble B =∆ = f S (cid:40) ( M N + M B ) M B (cid:0) M N − M B (cid:1) ¯ I π | N −
13 ( E N + M N ) p πN ¯ I πN + 23 α ( B ) πN (cid:41) + 512 h A (cid:40)(cid:16) − M + M B M B M ∆ + M + M B + 12 M M B M B M (cid:17)(cid:0) M − M B (cid:1) ¯ I π | ∆ − ( M B + M ∆ ) M E ∆ ( E ∆ − M ∆ ) + 5 M E ∆ + M ∆ p π ∆ ¯ I π ∆ (cid:41) , (14)8ith α ( B =∆) πN = β ∆ π (cid:16) M B − (cid:16) M (cid:17) M N (cid:17)(cid:16) ∆ ∂∂ ∆ + 1 (cid:17) δ + ∆ m π π β δ ,E N = p πN + M N , δ = − M (2 M + ∆)( M + ∆) log (cid:104) ∆(2 M + ∆) M (cid:105) , (15) β = (2 M + ∆) M ( M + ∆) , δ = M M + ∆ + M (2 M + 2 ∆ M + ∆ )(2 M + ∆)( M + ∆) log (cid:104) ∆ (2 M + ∆) M (cid:105) , where we again argue that the terms shown are model independent. The subtraction term α (∆) πN is instrumental to avoid the consideration of explicit contributions from two-loop effects.By construction the wave-function renormalization factor of the isobar approaches unity inthe chiral limit.It is worth pointing out another subtle issue. The expressions in Eqs. (11) and (14) showa non-trivial dependence on the ratio ∆ /M . While an expansion in powers of that ratiois formally convergent, it is advantageous not to expand, since the convergence pattern israther poor. Any significant result would require more terms than one would consider evenat the N LO level. Thus it is better to keep the unexpanded form.
IV. AXIAL CHARGE AND RADIUS OF THE NUCLEON
Consider the matrix element of the axial-vector current in the nucleon state [22], (cid:104) N (¯ p ) | A µi (0) | N ( p ) (cid:105) = ¯ u N (¯ p ) (cid:16) γ µ γ τ i G A ( q ) + γ q µ M N τ i G P ( q ) (cid:17) u N ( p ) , (16)with q = ¯ p − p . Given the chiral Lagrangian densities of Eqs. (1) and (3) it is straightforwardto derive its tree-level and one-loop contributions to the two form factors G A ( q ) and G T ( q ).In this work we focus on the axial term G A ( q ), which defines the axial charge G A (0) andradius (cid:104) r A (cid:105) ≡ G (cid:48) A (0) /G A (0) of the nucleon. In the chiral limit at m = 0 it holds that G A (0) → g A .In the Passarino-Veltman reduction scheme [16] we find for the one-loop contributions tothe axial-vector form factor G A ( q ) = g A Z N + 4 g χ m π + g R q + K π ¯ I π + (cid:88) R = N, ∆ K πR ¯ I πR ( M N )+ (cid:88) L,R = N, ∆ K LπR ¯ I LπR ( q ) + (cid:88) L,R = N, ∆ K (cid:48) LπR ∆ ¯ I LπR q , (17)9here the kinematical functions K π , K πR , K LπR , and K (cid:48) LπR depend on the pion mass m π , themomentum transfer t = q , and the baryon masses M N and M ∆ . While we derived explicitexpressions thereof, they are too lengthy to be shown here in full detail. In Appendix A andAppendix B we document a recursion scheme in terms of which they were derived. Belowwe will display the leading-order terms of their chiral decomposition.The scalar tadpole and bubble integrals ¯ I π and ¯ I πR ( M N ) we encountered already in theprevious section on the baryon-mass evaluations, where also the wave-function renormaliza-tion factor for the nucleon Z N was introduced. We recall that within our renormalizationscheme all tadpole terms ¯ I N or ¯ I ∆ that involve a heavy field must be dropped, because theyimply power-counting violating contributions to the form factors. As emphasized, such aprocedure complies with the chiral Ward identities of QCD, simply because manipulatingboth sides of a Ward identity in a well-defined manner does not spoil it [23].The evaluation of the axial-vector form factor involves another function of the Passarino-Veltman basis, the scalar triangle function I LπR ( t ). This function is strictly finite and for R = L = N it follows the dimensional counting rules with I NπN ( t ) ∼ Q . In the presence ofthe isobar degrees of freedom it still holds that I LπR ( t ) ∼ Q for L, R ∈ { N, ∆ } , however,only if we count ∆ ∼ Q ∼ m π . The trouble starts in the chiral domain where m π < ∆ ∼ Q .In this case dimensional counting rules require a different scaling behaviour of I LπR ( t ) for L (cid:54) = N or R (cid:54) = N . In order to restore the proper scaling in that domain we implement asubtraction term in the scalar triangle, viz. ¯ I LπR ( q ) = (cid:90) d l (2 π ) i (( l − ¯ p ) − M L )( l − m π )(( l − p ) − M R ) − γ LπR ,γ ∆ πN = γ Nπ ∆ = − π M log (cid:104) M ∆ + ∆ ( M + ∆) (cid:105) + 116 π (2 M ∆ + ∆ ) log (cid:104) ( M + ∆) M (cid:105) ,γ NπN = 0 , γ ∆ π ∆ = − π M log (cid:104) M ∆ + ∆ ( M + ∆) (cid:105) , (18)∆ ¯ I LπR = ¯ I LπR ( q ) − ¯ I LπR (0) − q γ (cid:48) LπR ,γ (cid:48) Nπ ∆ = γ (cid:48) ∆ πN = − π M ∆ (2 M + ∆) (cid:32) ∆ (2 M + ∆) log (cid:104) ∆ M (cid:105) − M + ∆) (4 M − M ∆ + 3 M ∆ + 4 M ∆ + ∆ ) log (cid:104) M + ∆ M (cid:105) +∆ (2 M + ∆) (cid:16) M (4 M + 2 M ∆ + ∆ ) + ∆ (2 M + ∆) log (cid:104) M + ∆ M (cid:105)(cid:17)(cid:33) , (cid:48) NπN = γ (cid:48) ∆ π ∆ = 0 , (19)with p = ¯ p = M N . This is analogous to the subtraction term γ ∆ N introduced in the scalarbubble function, which leads to ¯ I π ∆ ( M N ) ∼ Q . With Eq. (18) we obtain ¯ I NπN ( t ) ∼ Q ,¯ I Nπ ∆ ( t ) ∼ Q , and ¯ I ∆ π ∆ ( t ) ∼ Q in the chiral domain. It is crucial to consider thesesubtractions, since otherwise the explicit evaluation of a class of two-loop diagrams wouldbe needed [29]. In Appendix C we provide some more explicit expression for the trianglefunction that are instrumental in the computation of the axial charge and radius of thenucleon.We return to the kinematic functions K π , K πR , K LπR , and K (cid:48) LπR . One may be temptedto simply keep their form as they come out of our computation scheme in Appendix A andAppendix B. However, we argue that this would lead to inconsistencies. This is immediatelyclear by looking for instance into the function K π . The renormalization scale dependence in¯ I π can be balanced by the LEC g χ only if the suitably renormalized term K π is independenton the quark mass. Within our scheme ¯ I π is the exclusive source of such a dependence.Both our bubble and triangle functions do not depend on µ .We conclude that it is crucial to consider a chiral decomposition of such kinematic func-tions. How to do so in a scheme wherein one keeps on-shell masses M N and M ∆ needsdevelopment. For this purpose the functions are taken to depend on t , m π , δ , and M N , with m π ∼ Q , δ = M ∆ − M N (1 + ∆ /M ) ∼ Q ,t ∼ Q , M N ∼ Q . (20)This choice implies a well-defined chiral expansion of the form K π = − g A + g A / f + 4 g A f S f α + 20 h A f S f α − f S M N (5 f (3) A + f (4) A )27 f ∆ M α + O (cid:0) Q (cid:1) ,K πN = − g A + g A / / g A M N ( g S − g T ) f m π + 2 g A f S f (cid:110) −
56 ∆
M α M N −
524 ∆
M α t − α m π − α M N δ (cid:111) + O (cid:0) Q (cid:1) ,K π ∆ = 2 g A f S f (cid:110) −
56 ∆
M α M N −
524 ∆
M α t + 1918 α m π − α M N δ (cid:111) − h A f S f (cid:110)
149 ∆
M α M N −
118 ∆
M α t − α m π + 149 α M N δ (cid:111) − f S M N (cid:0) f (3) A + f (4) A (cid:1) f (cid:110) −
209 ∆ M α M N + 209 α m π −
409 ∆
M α M N δ (cid:111) + O (cid:0) Q (cid:1) , α a b Δ /M
71 31 34765525453336 β a b Δ /M FIG. 1. All coefficients α ab and β ab with a (cid:54) = 0 (cid:54) = b as function of the ratio ∆ /M . K NπN = O (cid:0) Q (cid:1) ,K Nπ ∆ = 2 g A f S M N f (cid:110)
56 ∆ M α M N + 524 ∆ M α t − α m π + 53 ∆ M α M N δ (cid:111) + O (cid:0) Q (cid:1) ,K (cid:48) Nπ ∆ = 2 g A f S M N f (cid:110)
23 ∆ M α M N −
23 ∆ M α m π + 43 ∆ M α M N δ (cid:111) + O (cid:0) Q (cid:1) ,K ∆ π ∆ = − h A f S M N f (cid:110) −
43 ∆ M α M N −
79 ∆ M α t + 43 α m π −
83 ∆
M α M N δ (cid:111) + O (cid:0) Q (cid:1) ,K ∆ πN = K Nπ ∆ , K (cid:48) ∆ πN = K (cid:48) Nπ ∆ , K (cid:48) NπN = K (cid:48) ∆ π ∆ = 0 . (21)The next higher-order contributions to the K ( n ) coefficients are given by K (2) π = − g A f M N β (cid:16) g S + 23 g T + 54 M N g V (cid:17) m π + 2 g A f S f M N (cid:16) β m π + 19 β M N δ (cid:17) − h A f S f M N (cid:16) β t + 133108 β m π + 1027 β M N δ (cid:17) − f S (cid:0) f (3) A + f (4) A (cid:1) f M N (cid:16) β m π + 109 β M N δ (cid:17) ,K (4) πN = − g A f M N (cid:16) g V M N t m π + 2 (cid:0) g S + g T (cid:1) m π (cid:17) − g A f M N t m π + 2 g A f S f M N (cid:110) −
596 ∆
M β t + 124 β t m π − β t M N δ − β m π + 5336 β m π M N δ −
98 ∆
M β M N δ (cid:111) ,K (4) π ∆ = 2 g A f S f M N (cid:110) −
596 ∆
M β t + 124 β t m π − β t M N δ β m π + 14 β m π M N δ − β M N δ (cid:111) − h A f S f M N (cid:110) −
172 ∆
M β t + 4136 β t m π − β t M N δ − β m π − β m π M N δ + 127 β M N δ (cid:111) − f S (cid:0) f (3) A + f (4) A (cid:1) f M N (cid:110) − β m π + 49 β m π M N δ − β M N δ (cid:111) ,K (4) NπN = − g A f m π ,K (4) Nπ ∆ = 2 g A f S f (cid:110)
596 ∆ M β t −
14 ∆
M β t m π + 512 ∆ M β t M N δ + 76 β m π − β m π M N δ + 56 β M N δ (cid:111) ,K (4) (cid:48) ∆ πN = 2 g A f S M N f (cid:110) −
43 ∆
M β m π M N δ + 23 β M N δ (cid:111) ,K (4)∆ π ∆ = − h A f S f (cid:110)
136 ∆ M β t − ∆ M β t m π −
149 ∆
M β t M N δ − β m π + 43 β m π M N δ − β M N δ (cid:111) ,K (4)∆ πN = K (4) Nπ ∆ , K (4) (cid:48) ∆ πN = K (4) (cid:48) Nπ ∆ , K (4) (cid:48) NπN = K (4) (cid:48) ∆ π ∆ = 0 , (22)where the dimensionless coefficients α ab and β ab depend on the ratio ∆ /M only. They arenormalized with α ab , β ab → →
0. A complete collection of the coefficients α ab can be found in Appendix D. In Figure 1 we plot the coefficients α ab and β ab as functionof the ratio ∆ /M .A few comments on Eq. (21) are in order. The expansion of Eq. (20) is rapidly convergingwith a suppression factor √ t/M N ∼ Q , m π /M N ∼ Q , or δ/M N ∼ Q . The neglected terms oforder Q and higher lead to corrections of less than 1% typically. In Table I we illustrate theimportance of the ∆ /M effect in the coefficients α ab at three typical values for ∆ /M = 0 . α ab →
1. Anyattempt to recover this effect in terms of only a few moments appears futile.Particular attention should be paid to the terms in Eq. (21) proportional to α n and α n .In the chiral domain they all violate the expectation from dimensional power-counting rules.We expect such terms to be cancelled by contributions from two-loop diagrams [15, 29]. Wetherefore impose the renormalization condition α n → α n →
0, and also β n → ab ∆ /M = 0 . /M = 0 . /M = 0 . α ab ∆ /M = 0 . /M = 0 . /M = 0 . α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α ab , introduced in Eq. (21), for typical numerical values ∆ /M = 0 . , . , . α ab are given in the Appendix in Eq. (38). include isobar contributions, for bubble and tadpole diagrams, but we disagree with themfor the triangle diagram with two internal nucleons. Our results agree with Eq. (21) inRef. [4] if we make the replacement 16 M N I ( P P ) πNN ( t ) → (16 M N − tM N ) I ( P P ) πNN ( t ) and changethe sign of the g R -term. In Ref. [10] results are given for the loop contributions to theaxial charge, Eq. (A4), and the axial radius, Eq. (A5). By using Eqs. (35) and (36) fromAppendix C we can rewrite triangle integrals in terms of bubble integrals. The full bubblecontributions, denoted by B ( m N , M π , m ) and B ( m , M π , m ), can then be reproduced.The contributions proportional to the tadpole integral A ( M π ) differ by a minus sign fromours. We disagree with the normalization of the axial radius in Eq. (A5) by a factor 36.14 c h i r a l c o rr e c t i o n s t o G A ( ) m π [GeV] δ = M Δ - M N (1 + Δ / M)finite box: ETMCfinite box: CLSfinite box: RQCDphysical valuechiral limit δ [ G e V ] m π [GeV] FIG. 2. A comparison between using chiral, physical, and on-shell baryon masses in the chiralcorrection terms to the axial charge. Since for the two cases with approximate baryon masses thechiral correction terms are quite large, their contributions are scaled down by a factor of 10. Astrong effect is seen starting at pion mass m π (cid:39) . V. FIT TO QCD LATTICE DATA
As a first application of our results we consider the set of QCD lattice data on thenucleon and isobar mass (when available) and the nucleon axial-vector form factor. Weuse the evolutionary fit algorithm GENEVA 1.9.0-GSI [30]. We take into account the mostrecent two-flavor ensembles of the ETMC [31, 32], which provides also data for the isobarmass, the CLS [12, 33], and the RQCD [34] QCD lattice collaborations.Our fit strategy is the following. We fit our expressions for the nucleon and isobar masses, M N and M ∆ in Eq. (6) (with finite-volume effects included following Ref. [27]), and for thenucleon axial-vector form factor, G A ( t ) in Eq. (17) (without explicit finite-volume effects),to the lattice data. As explained, we use on-shell baryon masses in the loops, in order tokeep the analytical structures of the integrals. We include the K factors of Eq. (21), but notEq. (22). This means that higher-order terms ∼ Q are not taken into account. We checkedthat these higher-order contributions are small for low-enough pion masses, m π < .
55 GeV.These convergence properties are dramatically improved by the use of on-shell masses asillustrated in Figure 2. There we evaluate the chiral correction terms of the axial charge15 roup scale this work lattice groupETMC a β =3 . [fm] 0 . +0 . − . ) 0 . a β =3 . [fm] 0 . +0 . − . ) 0 . a β =4 . [fm] 0 . +0 . − . ) 0 . a β =4 . [fm] 0 . +0 . − . ) 0 . a β =5 . [fm] 0 . +0 . − . ) 0.079 [12] a β =5 . [fm] 0 . +0 . − . ) 0.063 [12] a β =5 . [fm] 0 . +0 . − . ) 0.050 [12]RQCD a β =5 . [fm] 0 . +0 . − . ) 0.081 [34] a β =5 . [fm] 0 . +0 . − . ) 0.071 [34] a β =5 . [fm] 0 . +0 . − . ) 0.060 [34]TABLE II. Lattice scales as determined in our fit. on all considered lattice ensembles by means of Eqs. (17) and (21). The results dependon the LECs, the pion mass of the ensemble, and the baryon masses. In the Figure threedifferent choices for the baryon masses are illustrated. While within our on-shell scheme wefind reasonably-sized chiral correction terms up to pion masses of a 500 MeV, this is notthe case for the conventional case of using chiral-limit baryon masses in the loop functionsor physical values as was used in Ref. [10]. The source of the large differences is due todistinct values for δ in the coefficients K (see the right-hand panel of the Figure), but alsothe use of the finite-box baryon masses in the scalar loop integrals as they are predicted onthe various lattice ensembles by our global fit. The scatter in our results reflects the decisiverole of finite-volume effects on the baryon masses.It is important to have accurate values for the QCD lattice scales available on all con-sidered lattice ensembles. There are different scale-setting schemes used by the lattice col-laborations, which differ by the size of the discretization effects. We follow here the pathof ETMC [31], which suggests to set the scale by the requirement to recover the isospin-averaged mass of the nucleon at the physical point. With this construction discretizationeffects are minimized in the baryon masses. Since this relies on a particular chiral extrap-olation scheme, we consider the various lattice scales as free parameters in our global fit.Such a strategy was successfully used in various global fits to lattice data [15, 18, 27]. Our16 EC Fit result LEC Fit result LEC Fit result f [MeV] 87 . +0 . − . ) b ∗ χ [GeV − ] − . +0 . − . ) g S [GeV − ] 0 . +0 . − . ) M [MeV] 884 . +0 . − . ) d ∗ χ [GeV − ] − . +0 . − . ) g V [GeV − ] − . +0 . − . ) M + ∆ [MeV] 1187 . +0 . − . ) c χ [GeV − ] 2 . +0 . − . ) g T [GeV − ] 1 . +0 . − . ) g A . +0 . − . ) e χ [GeV − ] 1 . +0 . − . ) g R [GeV − ] 0 . +0 . − . ) f S . +0 . − . ) ¯ l . +0 . − . ) h S [GeV − ] − . +0 . − . ) h ∗ A − . +0 . − . ) g χ [GeV − ] − . +0 . − . ) h V [GeV − ] − . +0 . − . ) f (3) A + f (4) A / − ] − . +0 . − . )TABLE III. Low-energy constants as determined in our fit. The ∗ parameters are not fitted to thelattice data. While b χ and d χ are adjusted to the isospin-averaged masses of the nucleon and theisobar at the physical point, the value of h A = 9 g A − f S is implied by its large- N c sum rule. results for all lattice scales are shown in Table II. The scales given by the ETMC and CLScollaborations differ significantly from our values. There is, however, a clear trend that thesmaller the lattice scale, the closer our fitted scales get to the ones given by the latticecollaborations. The RQCD scales, on the other hand, can be reproduced quite accurately.Since the lattice set up of the CLS and RQCD groups coincide, one would expect identicallattice scales on the β = 5 . m π = 0.55 GeV for thenucleon and isobar masses and the nucleon axial-vector form factor. For the form factor weinclude the data points up to momentum transfer t = 0 .
36 GeV and for lattice sizes with m π L ≥ .
0, where we expect good convergence properties and (explicit) finite-volume effectsto be small. In the plots all data points are shown, but we identify and reject two outliers, viz. the masses connected to the highest pion masses of ETMC ( a = 0 . m π = 0 . a = 0 . m π = 0 .
436 GeV). For bothpoints there is no G A data to be fitted, since there is none available for ETMC ( a = 0 . m π L < . a = 0 . m π = 0 .
436 GeV).The fit minimizes the least-squares differences χ of our expressions with respect to thelattice data points. In this χ determination, all available lattice points that meet ourrequirements (see previous section) contribute with equal weight. The χ per lattice point17 EC Tree-level matching [18] LEC Tree-level matching [18] f S h A b χ − d χ − c χ e χ − g V h V g S − h S − reached is χ /N data = 1 .
04. With 99 used lattice data points and 26 degrees of freedom19-3 LECs and 10 lattice scales), we reach for the total χ per degree of freedom χ /N df = 103 . / (99 −
26) = 1 . , (23)which qualifies as a good description of the available lattice data. We give asymmetric errorbars. They are based on a one standard deviation ( σ ) change for the value of χ ( i.e. an increase by 1). We determined the region for the LECs meeting this range, from whichfollow the errors for the LECs.The LECs determined in the fit are collected in Table III. Of the large- N c relations in Eq.(5) we use for now only the relation that eliminates h A . In addition, we set ζ N = ζ ∆ = 0,because in the axial-form factor their effect can be renormalized into the value of g χ . Wenote that in our current work the impact of the large- N c relations is quite limited, becausethe data set that we consider does not constrain most of the isobar parameters h ( n ) S and h ( n ) V . Given our fit results, Eq. (5) can be used to derive an estimate of the LECs that ourdata set is not directly sensitive to. We find reasonable values for f , M , M + ∆, and g A [35]. It is noteworthy that f S = 1 .
94 takes a large value, leading to a negative value for h A ,which was also reported in Ref. [36] (denoted as g ). The middle column of Table III listschiral-symmetry-breaking LECs. It shows an expected value for ¯ l [35, 37]. The leading-order large- N c relations b χ = d χ and c χ = e χ are approximately fulfilled. The value of g χ ispoorly known in literature. The right column shows the chiral-symmetry-conserving LECsof higher order. We find values for g S and g V that disagree significantly from previous SU(2)works like Refs. [37, 38]. However, in most papers the constants g S and g V are determinedin a theory without isobars. How to translate this to our case with isobars is not clear (seepage 10 of Ref. [39]). The other LECs of this column have not been determined reliably in18he literature so far. In general, our determined LECs are in the expected range and theyappear to be reasonably small. The one-sigma error bars in the LECs are rather small ingeneral, with a notable exception of g V , for which its size does not seem to depend verystrongly on the baryon masses and form factor.It is illuminating to also compare our set of LEC with results from flavor-SU(3) analyses.Here we focus on the most recent work [18], which achieved a global fit to the baryon octetand decuplet masses as provided by various lattice groups. In Table IV we collect valuesthat are obtained by a tree-level matching of the flavor-SU(2) with the flavor-SU(3) chiralLagrangian. A comparison of Table IV with Table III reveals an interesting pattern. Whilethe LECs b χ and d χ at order Q are quite compatible, the higher-order terms c χ and e χ are quite distinct, which is not necessarily surprising or problematic. Most striking is theopposite sign in h A , which we interpret as a signal for the importance of strangeness loopeffects. This is in line with a striking prediction of the flavor-SU(3) analyses [15, 18], whichobtained a strangeness sigma term of the isobar σ s ∆ (cid:39) −
270 MeV, significantly larger inmagnitude than its corresponding value for the nucleon with σ sN (cid:39)
45 MeV (here we provideunpublished results from Ref. [18]).The masses of the nucleon and the isobar are reproduced excellently (see Figure 3). Thecolored points represent the lattice data of the different ensembles and the white points areour theory predictions for the corresponding lattice volume. Most theory error bars are toosmall to be shown. The collaborations CLS and RQCD do not give results for the isobarmass M ∆ . In these cases, the white points are to be seen as predictions.Figures 4-7 show the axial-vector form factor of the nucleon G A ( t ) for each lattice point.The order is given by increasing pion mass. The color of the points determines the cor-responding lattice group, whereas the used symbol indicates the size of the lattice scale a (diamond-circle-square, from largest to smallest). We present our theoretical results in theinfinite-volume limit (black line with gray error band) and extrapolated to the box size ofthe corresponding lattice point (orange lines). Finite-volume effects of the form factor orig-inate only from finite-volume effects of the masses, which enter the expressions for the loopcontributions. Explicit finite-volume effects, originating directly from the loop integrals ofthe form factors, are not taken into account. In order to visualize the lattice points whichdo not enter the fit due to the restrictions described above ( m π L < . t > .
36 GeV ),we used dashed orange lines as compared to the fitted data points which are indicated by19olid orange lines. For convenience we also show the isobar-nucleon mass gap in the infinitevolume and the pion mass in the top-right corner and the mass gap in the box and thelattice size m π L and L in the bottom of each plot. The plots of the data points of theCLS collaboration include the fitted “Two-State Method” (dark blue) and the light blue“Summation” method, which does not enter the fit. We find a very good description of thelattice points with the corresponding solid orange theory results.A large error band of the infinite volume prediction appears in the upper-right cornerof Figure 6 at pion mass m π = 0 .
385 GeV. This plot is particularly interesting, because M box∆ − M box N < m π < M ∆ − M N . This means that the isobar is unstable in our theory,but stable in the lattice simulation, implying a different analytical behavior. In order toscrutinize this in more detail, we provide Figures 8 and 9, which show the full range ofour results for the nucleon and isobar masses and the form factor at the physical point g A ≡ G A ( t = 0).In general, the region around m π = 375 MeV seems is very interesting, because, whenvarying M N , M ∆ and g A in terms of the pion mass, we find a clear jump. This has beenobserved before [17, 18] in SU(3) and is nicely displayed in Figures 8 and 9. The full treatedrange is shown in Figure 8, where we confront the lattice data with our theory in finite(orange, L ∈ [2 . , .
52] fm) and infinite (black) volume. We find a large difference betweenthe finite volume and the infinite volume, especially for the axial charge G A (0). We keep inmind that the lattice points are not expected to match with any of the given lines if theirvolume L (cid:54)∈ [2 . , .
52] fm.We see a clear jump around m π = 375 MeV in all three observables. Figure 9 allowsa closer look into that. We show our results in infinite volume (black with gray errorband), large lattice volume (red), and small lattice volume (orange). Additionally we showlattice results, which are unfortunately only available for small lattice sizes. We find arather smooth curve for small box size, but there are clear jumps for larger boxes, butespecially in the infinite volume. In the infinite-volume limit we determine the positionof the jump as m π = 373 . +17 . − . ) MeV and its height for the nucleon/isobar mass at60 . +65 . − . ) MeV / . +17 . − . ) MeV. This means that we report 3-4 σ evidence for theexistence of this jump. Our plot shows that lattice size L ≥ .
38 fm should be sensibleto this jump, especially when determining the axial charge G A (0). We therefore encouragethe lattice community to investigate this suggested behavior, which is a direct consequence20f our scheme, namely the use of on-shell masses in the loop contributions and the self-consistent determination of the baryon masses. Conventional chiral perturbation theoryapproaches, which use expanded masses in the loops, will not see this phenomenon, becauseapproximated solutions of Eq. (6) are always linear and do not allow for non-linear, self-consistent equations that result from the use of on-shell masses in the loops.The observables that follow from our fit are summarized in Table V. The experimentallywell-known parameter G A (0) = 1 . (cid:104) r A (cid:105) = 0 . +0 . − . )fm , which is pretty small, compared to (cid:104) r A (cid:105) = 0 . [10]and the experimental data, (cid:104) r A (cid:105) = 0 . [41] and (cid:104) r A (cid:105) = 0 . [42]. Theavailable lattice results are (cid:104) r A (cid:105) = 0 . [13], (cid:104) r A (cid:105) = 0 . +80 − )fm [12], and (cid:104) r A (cid:105) = 0 . [43].The sigma terms are defined by σ j = m ∂∂ m M j . (24)We determine σ N = 49 . +0 . − . ) MeV. A recent study of ETMC [44] suggests that σ N =41 . .
8) MeV. The RQCD collaboration [45] finds a lower value, σ N = 35(5) MeV. Wefind good agreement with the sigma terms from the extensive flavor-SU(3) mass fits [15, 18] σ N (cid:39) σ ∆ (cid:39) σ πN = 41(5)(4) MeV based on a flavor-SU(2) extrapolation of an older set of lattice datafor the nucleon mass [32, 48–50]. A comparison with the empirical value σ πN = 58(5) MeVfrom Ref. [51] is of limited use for us, because it is not clear how important strangenesseffects are. 21 m a ss a m π N Δ ETMCN Δ CLSN Δ RQCD
FIG. 3. Comparison between our (white) theory results for the nucleon and isobar masses and(colorful) results of different lattice collaborations. We give our fitted lattice scales as identification.The two outliers are the highest red point of ETMC and the highest green point of RQCD. A t [GeV ] MeV 292M Δ - M N = MeV 273M Δ box - M Nbox = fm4.48L = 3.47m π L = MeV 153m π = MeV 292M Δ - M N = MeV 282M Δ box - M Nbox = fm4.5L = 3.95m π L = MeV 173m π = MeV 298M Δ - M N = MeV 337M Δ box - M Nbox = fm2.98L = 3.74m π L = MeV 248m π = incompletefit in the boxpredictionMeV 298M Δ - M N = MeV 330M Δ box - M Nbox = fm3.36L = 4.22m π L = MeV 248m π = MeV 298M Δ - M N = MeV 341M Δ box - M Nbox = fm2.82L = 3.55m π L = MeV 249m π = MeV 298M Δ - M N = MeV 330M Δ box - M Nbox = fm3.37L = 4.25m π L = MeV 249m π = MeV 299M Δ - M N = MeV 321M Δ box - M Nbox = fm4.04L = 5.15m π L = MeV 252m π = MeV 302M Δ - M N = MeV 342M Δ box - M Nbox = fm2.83L = 3.82m π L = MeV 267m π = MeV 304M Δ - M N = MeV 345M Δ box - M Nbox = fm2.66L = 3.69m π L = MeV 274m π = FIG. 4. The axial-vector form factor G A ( t ) of the nucleon for different pion masses. The coloredlattice points are to be compared to our finite-box results (the straight orange lines). The dashedlines visualize the lattice points that are not fitted, as explained in the text. The black linesrepresent our results in the infinite-volume limit. A t [GeV ] MeV 306M Δ - M N = MeV 349M Δ box - M Nbox = fm2.35L = 3.32m π L = MeV 279m π = MeV 308M Δ - M N = MeV 340M Δ box - M Nbox = fm2.98L = 4.28m π L = MeV 284m π = MeV 309M Δ - M N = MeV 349M Δ box - M Nbox = fm2.23L = 3.27m π L = MeV 289m π = incompletefit in the boxpredictionMeV 310M Δ - M N = MeV 334M Δ box - M Nbox = fm3.37L = 4.97m π L = MeV 291m π = MeV 312M Δ - M N = MeV 343M Δ box - M Nbox = fm2.8L = 4.19m π L = MeV 295m π = MeV 312M Δ - M N = MeV 324M Δ box - M Nbox = fm4.48L = 6.71m π L = MeV 296m π = MeV 312M Δ - M N = MeV 345M Δ box - M Nbox = fm2.69L = 4.05m π L = MeV 297m π = MeV 314M Δ - M N = MeV 349M Δ box - M Nbox = fm2.24L = 3.42m π L = MeV 301m π = MeV 321M Δ - M N = MeV 348M Δ box - M Nbox = fm2.52L = 4.02m π L = MeV 315m π = FIG. 5. The axial-vector form factor G A ( t ) of the nucleon for different pion masses; see the captionof Figure 4. A t [GeV ] MeV 318M Δ - M N = MeV 338M Δ box - M Nbox = fm2.69L = 4.67m π L = MeV 342m π = MeV 313M Δ - M N = MeV 344M Δ box - M Nbox = fm2.23L = 4.04m π L = MeV 357m π = MeV 386M Δ - M N = MeV 339M Δ box - M Nbox = fm2.35L = 4.58m π L = MeV 385m π = incompletefit in the boxpredictionMeV 360M Δ - M N = MeV 337M Δ box - M Nbox = fm2.52L = 5.21m π L = MeV 408m π = MeV 358M Δ - M N = MeV 339M Δ box - M Nbox = fm2.25L = 4.67m π L = MeV 409m π = MeV 355M Δ - M N = MeV 339M Δ box - M Nbox = fm2.23L = 4.66m π L = MeV 411m π = MeV 338M Δ - M N = MeV 333M Δ box - M Nbox = fm2.24L = 4.9m π L = MeV 431m π = MeV 336M Δ - M N = MeV 366M Δ box - M Nbox = fm1.68L = 3.71m π L = MeV 436m π = MeV 334M Δ - M N = MeV 342M Δ box - M Nbox = fm1.88L = 4.18m π L = MeV 438m π = FIG. 6. The axial-vector form factor G A ( t ) of the nucleon for different pion masses; see the captionof Figure 4. A t [GeV ] MeV 330M Δ - M N = MeV 327M Δ box - M Nbox = fm2.35L = 5.28m π L = MeV 444m π = MeV 330M Δ - M N = MeV 327M Δ box - M Nbox = fm2.69L = 6.06m π L = MeV 444m π = MeV 330M Δ - M N = MeV 328M Δ box - M Nbox = fm2.23L = 5.04m π L = MeV 445m π = incompletefit in the boxpredictionMeV 330M Δ - M N = MeV 338M Δ box - M Nbox = fm1.88L = 4.24m π L = MeV 446m π = MeV 292M Δ - M N = MeV 297M Δ box - M Nbox = fm1.88L = 4.81m π L = MeV 503m π = MeV 246M Δ - M N = MeV 247M Δ box - M Nbox = fm2.35L = 6.48m π L = MeV 545m π = FIG. 7. The axial-vector form factor G A ( t ) of the nucleon for different pion masses; see the captionof Figure 4. M Δ [GeV]M N [GeV] m π [GeV] G A (0) m π [GeV] FIG. 8. Nucleon and isobar masses and the axial-vector form factor of the nucleon as function ofthe pion mass compared to the QCD lattice data. The orange lines are our results for the indicatedrange of lattice sizes, the black lines are our predictions in the infinite-volume limit. Δ [GeV]M N [GeV] m π [GeV] G A (0) m π [GeV] FIG. 9. Nucleon and isobar masses and the axial-vector form factor of the nucleon as functionof the pion mass compared to the QCD lattice data, zoomed in to the region where our resultsshow the non-analytic behavior. The orange lines are our results for the indicated range of latticesizes, the red lines are our results for the indicated larger lattice size, and the black lines are ourpredictions in the infinite-volume limit.Observable Fit results G A (0) 1 . +0 . − . ) (cid:104) r A (cid:105) [fm ] 0 . +0 . − . ) σ N [MeV] 49 . +0 . − . ) σ ∆ [MeV] 45 . +0 . − . )jump position [MeV] 373 . +17 . − . )jump height nucleon [MeV] 60 . +65 . − . )jump height isobar [MeV] 18 . +17 . − . ) χ /N data χ /N df I. CONCLUSION AND OUTLOOK
In this work we studied QCD with up and down quarks. From an effective field theorypoint of view baryonic systems are of particular interest, since here the intricate interplay ofthe prominent low-energy scales, the pion mass, m π , and the isobar-nucleon mass difference,∆, can be scrutinized in the absence of additional complications from the strange quark.We considered QCD lattice data with two dynamical quark fields on the nucleon mass,the isobar mass, as well as the axial-vector form factor of the nucleon. A global fit tosuch data was performed successfully, as application of the two-flavor chiral Lagrangian.Accurate results are obtained for pion masses up to 500 MeV with a χ /N df (cid:39) . m π = 373( +18 − ) MeV, whenevaluated in the infinite-box limit. It was illustrated that at box sizes of current QCD latticeensembles such a phase transition is not visible. However, it should be easily detectable ifensembles on 64 lattices in that pion-mass region are generated. We suggest to measurethe nucleon and the isobar finite-box masses on such ensembles, since both enter the chiraldynamics of that system decisively.Further work is required to consolidate our results. So far we only considered volumeeffects that arise from an evaluation of the finite-box baryon masses. Such effects are instru-mental to recover the axial-form factor results on the various QCD lattice ensembles, and itremains for us to implement explicit finite-volume effects in our form-factor computation. Ageneralization of our framework to baryon form factors in flavor-SU(3) appears promising. VII. ACKNOWLEDGMENTS
We thank Gunnar Bali, John Bulava, Daniel Mohler, and Thomas Wurm for helpfuldiscussions. M.F.M. Lutz thanks Denis Bertini for support on distributed computing issues28
III. APPENDIX A
In the course of evaluating the axial-vector form factor of the nucleon the following scalarone-loop integrals occur: A ,kf,i = (cid:90) d d l (2 π ) d µ − d ( l · ¯ p ) f ( l ) k ( l · p ) i ( l − p ) − M R ,A ,kf,i = (cid:90) d d l (2 π ) d µ − d ( l · ¯ p ) f ( l ) k ( l · p ) i l − m Q ,A ,kf,i = (cid:90) d d l (2 π ) d µ − d ( l · ¯ p ) f ( l ) k ( l · p ) i ( l − ¯ p ) − M L ,B ,kf,i = (cid:90) d d l (2 π ) d µ − d ( l · ¯ p ) f ( l ) k ( l · p ) i (( l − ¯ p ) − M L )( l − m Q ) ,B ,kf,i = (cid:90) d d l (2 π ) d µ − d ( l · ¯ p ) f ( l ) k ( l · p ) i (( l − ¯ p ) − M L )(( l − p ) − M R ) ,B ,kf,i = (cid:90) d d l (2 π ) d µ − d ( l · ¯ p ) f ( l ) k ( l · p ) i ( l − m Q )(( l − p ) − M R ) ,C kf,i = (cid:90) d d l (2 π ) d µ − d ( l · ¯ p ) f ( l ) k ( l · p ) i (( l − ¯ p ) − M L )( l − m Q )(( l − p ) − M R ) , (25)with the space-time dimension d and the renormalization scale µ of dimensional regulariza-tion. In this Appendix we present a convenient recursion scheme in terms of which all suchintegrals can be systematically expressed in the Passarino-Veltman basis, I R = (cid:90) d d l (2 π ) d i µ − d l − M R = i A , , , I Q = (cid:90) d d l (2 π ) d i µ − d l − m Q = i A , , ,I L = (cid:90) d d l (2 π ) d iµ − d l − M L = i A , , ,I LQ (cid:0) ¯ p (cid:1) = (cid:90) d d l (2 π ) d − i µ − d (( l − ¯ p ) − M L )( l − m Q ) = − i B , , ,I LR ( t ) = (cid:90) d d l (2 π ) d − i µ − d (( l − (¯ p − p )) − M L )( l − M R ) = − i B , , ,I QR (cid:0) p (cid:1) = (cid:90) d d l (2 π ) d − iµ − d ( l − m Q )(( l − p ) − M R ) = − i B , , ,I LQR (cid:0) ¯ p , p (cid:1) = (cid:90) d d l (2 π ) d i µ − d (( l − ¯ p ) − M L )( l − m Q )(( l − p ) − M R ) = i C , . (26)29ur scheme goes in three steps. First we consider the class of triangle integrals C kf,i . Giventhe relations C kf,i = v R C kf,i − − B ,kf,i − + 12 B ,kf,i − , C kf,i = m Q C k − f,i + B ,k − f,i ,C kf,i = v L C kf − ,i − B ,kf − ,i + 12 B ,kf − ,i , (27)with v R = p − M R + m Q , v L = ¯ p − M L + m Q , v C = q − M R + M L , (28)for any values of f, i and k the function C kf,i can be expressed in terms of C , and the setof scalar bubble functions B − ,kf,i .In the second step we use three sets of recurrence relations. The first two read B ,kf,i = v L B ,kf − ,i − A ,kf − ,i + 12 A ,kf − ,i , B ,kf,i = m Q B ,k − f,i + A ,k − f,i ,B , ,i = i (cid:88) k =0 / (cid:18) ik (cid:19) ( i − k − p · ¯ p ) k ( p ) ( i − k ) / (cid:16) a ( i ) L,k A , , + b ( i ) L,k A , , + c ( i ) L,k B , , (cid:17) ,B ,kf,i = v R B ,kf,i − − A ,kf,i − + 12 A ,kf,i − , B ,kf,i = m Q B ,k − f,i + A ,k − f,i ,B , f, = f (cid:88) k =0 / (cid:18) fk (cid:19) ( f − k − p · p ) k (¯ p ) ( f − k ) / (cid:16) a ( f ) R,k A , , + b ( f ) R,k A , , + c ( f ) R,k B , , (cid:17) , (29)where the factors a ( f ) R,k , b ( f ) R,k and c ( f ) R,k are derived in Appendix B. Note that the sum over k in(29) runs in steps of 2. The coefficients a ( f ) L,k , b ( f ) L,k and c ( f ) L,k are obtained by the replacement R → L in the coefficients a ( f ) R,k , b ( f ) R,k and c ( f ) R,k . It remains to provide a recursion relation forthe B ,kf,i bubbles. We find B ,kf,i = 12 B ,k +1 f − ,i + ¯ p − M L B ,kf − ,i − A ,kf − ,i ,B ,kf,i = 12 B ,k +1 f,i − + p − M R B ,kf,i − − A ,kf,i − ,B ,k , = ∞ (cid:88) v,w,z =0 K ( k ) vwz A ,vw + z, + k (cid:88) n =0 n (cid:88) y =0 / K ( k ) ny (cid:16) a ( n ) C,y A , , + b ( n ) C,y A , , + c ( n ) C,y B , , (cid:17) ,K ( k ) vwz = ∞ (cid:88) n,j =0 (cid:18) kn (cid:19) (cid:18) k − nj (cid:19) n − (cid:88) u =0 (cid:18) uv (cid:19) (cid:18) u − vw (cid:19) (cid:18) jz (cid:19) × ( − j − z + w (¯ p ) k − n − z + u − v − w ( M L ) n − − u j + w ,K ( k ) ny = 2 n (¯ p + M L ) k − n (cid:18) ny (cid:19) ( n − y − p · p − ¯ p ) y (¯ p ) ( n − y ) / , (30)30here we use the convention (cid:0) ab (cid:1) = 0 for b > a . The coefficients a ( f ) C,k , b ( f ) C,k and c ( f ) C,k follow from a ( f ) R,k , b ( f ) R,k and c ( f ) R,k by the replacements p → q and m Q → M L . By means of the recurrencerelations (29) and (30) any B − ,kf,i can be expressed in terms of B − , , and the set of tadpoleintegrals A − ,kf,i .In our final step we need to apply the following recurrence relations for the tadpoleintegrals. We find A ,kf,i = ∞ (cid:88) j,w,h =0 K ( f,k,i ) jwg h ( j + w + g ) R, A , , , A ,kf,i = ∞ (cid:88) j,w,h =0 ¯ K ( f,k,i ) jwg h ( j + g + w ) L, A , , ,A ,kf,i = m Q A ,k − f,i , A , f,i = K f,i h ( i + f ) Q, A , , ,K ( f,k,i ) jwg = (cid:18) fg (cid:19) (cid:18) ij (cid:19) k (cid:88) v =0 (cid:18) kv (cid:19) (cid:18) k − vw (cid:19) w ( p ) i − j + v ( p · ¯ p ) f − g ( M R ) k − v − w Min(j+w , g) (cid:88) u =0 / (cid:18) gu (cid:19) × (cid:18) j + wu (cid:19) u ! ( j + w − u − g − u − p ) ( j + w − u ) / (¯ p ) ( g − u ) / ( p · ¯ p ) u , ¯ K f,k,ijwg = (cid:18) fg (cid:19) (cid:18) ij (cid:19) k (cid:88) v =0 (cid:18) kv (cid:19) (cid:18) k − vw (cid:19) w (cid:0) ¯ p (cid:1) f − g + v ( p · ¯ p ) i − j (cid:0) M L (cid:1) k − v − w Min(j , g+w) (cid:88) u =0 / (cid:18) g + wu (cid:19) × (cid:18) ju (cid:19) u ! ( j − u − g + w − u − p ) ( j − u ) / (¯ p ) ( g + w − u ) / ( p · ¯ p ) u ,K f,i = Min( i,f ) (cid:88) u =0 / (cid:18) fu (cid:19)(cid:18) iu (cid:19) u ! ( i − u − f − u − p ) ( i − u ) / (¯ p ) ( f − u ) / ( p · ¯ p ) u ,h ( f ) Q,k = ( f − k − / (cid:89) i =0 m f − kQ ( d + 2 i ) , (31)which upon iteration leads to explicit results for the tadpole integrals A − ,kf,i in terms of A − , , . IX. APPENDIX B
Our results for a ( f ) R,k , b ( f ) R,k and c ( f ) R,k are determined by the following recursion relations: v R a ( f ) R,k = 12 h ( f +1) Q, δ k + v R h ( f ) Q, δ k + (2 k + 1) p a ( f +2) R,k + k ( k − a ( f +2) R,k − + p a ( f +2) R,k +2 ,m Q a ( f ) R,k = ( k + f + d ) a ( f +2) R,k + p a ( f +2) R,k +2 , R b ( f ) R,k = − v R h ( f ) R,k − p h ( f +1) R,k +1 − k h ( f +1) R,k − + (2 k + 1) p b ( f +2) R,k + k ( k − b ( f +2) R,k − + p b ( f +2) R,k +2 ,m Q b ( f ) R,k = − h ( f ) R,k + ( k + f + d ) b ( f +2) R,k + p b ( f +2) R,k +2 ,v R c ( f ) R,k = (2 k + 1) p c ( f +2) R,k + k ( k − c ( f +2) R,k − + p c ( f +2) R,k +2 ,m Q c ( f ) R,k = ( k + f + d ) c ( f +2) R,k + p c ( f +2) R,k +2 , (32)once supplemented by the start values for f = 0 and f = 1 a (0) R, = 0 , b (0) R, = 0 , c (0) R, = 1 ,a (1) R, = − p , b (1) R, = 12 p c (1) R, = v R p , (33)with v R = p − M R + m Q . We derived explicit expressions for f = 2 , , a (2) R, = v R p ( d − , a (2) R, = − d v R p ( d − ,b (2) R, = − p ( d − (cid:18) v R − p (cid:19) , b (2) R, = 1 p ( d − (cid:18) d p + 14 d v R − p (cid:19) ,c (2) R, = − p ( d − (cid:18) v R − p m Q (cid:19) , c (2) R, = 1 p ( d − (cid:18) d v R − p m Q (cid:19) ,a (3) R, = − d − p (cid:18) − v R p − m Q d + m Q (cid:19) , a (3) R, = − d − p (cid:18) ( d + 2) v R p + m Q d − m Q (cid:19) b (3) R, = − d − (cid:18) −
12 + v R p + v R p − m Q p + M R d p (cid:19) ,b (3) R, = − p ( d − (cid:18) p (cid:18) − d (cid:19) + 32 m Q − v R p (2 + d ) − v R d ) − M R d (2 + d ) (cid:19) ,c (3) R, = 1( d − p (cid:18) m Q v R − v R p (cid:19) , c (3) R, = 1( d − p (cid:18) − m Q v R d v R p + v R p (cid:19) ,a (4) R, = − v R d ( d − p (cid:16) d v R − d m Q p + 4 m Q p (cid:17) ,a (4) R, = − v R d ( d − p (cid:16) − d ( d + 2) v R − d m Q p − m Q p (cid:17) ,a (4) R, = − v R ( d + 2)16 d ( d − p (cid:16) d ( d + 4) v R − d m Q p + 16 m Q p (cid:17) ,b (4) R, = v R − p d ( d − p (cid:16) M R p + d (cid:0) ( m Q − M R ) − p ( m Q + 3 M R ) + p (cid:1)(cid:17) , (4) R, = (cid:16) p (cid:0) d ( m Q + 11 M R ) − d ( m Q + 3 M R ) − M R (cid:1) − p ( m Q − M R ) (cid:0) d ( m Q − M R ) − d ( m Q + 3 M R ) + 8 M R (cid:1) − d ( d + 2) ( m Q − M R ) + d ( d + 2) p (cid:17) / (cid:16) (16 d (cid:0) d − (cid:1) p (cid:17) ,b (4) R, = (cid:16) (11 d − d − p (cid:0) d ( m Q − M R ) − M R (cid:1) + (5 d + 6 d − p ( m Q − M R ) (cid:0) d ( m Q − M R ) − M R (cid:1) + d ( d + 6 d + 8)( m Q − M R ) + 3 d (5 d − d − p (cid:17) / (cid:0) d ( d − p (cid:1) ,c (4) R, = ( v R − m Q p )
16 ( d − p , c (4) R, = v R − m Q p −
16 ( d − p (cid:16) ( d + 2) v R − m Q p (cid:17) ,c (4) R, = (cid:16) ( d + 6 d + 8) ( m Q − M R ) + 6 p (cid:0) d ( m Q − M R ) − d (cid:0) m Q + 2 m Q M R − M R (cid:1) + 8 M R (cid:1) + ( d + 6 d + 8) p + 4 ( d + 2) p (cid:0) ( d − m Q − ( d + 4) M R (cid:1) + 4 ( d + 2) p ( m Q − M R ) (cid:0) ( d − m Q − ( d + 4) M R (cid:1)(cid:17) / (cid:0)
16 ( d − p (cid:1) . (34) X. APPENDIX C
We link the triangle integral and its derivative at t = 0 to renormalized bubble integrals,which implies¯ I LπR ( t = 0) = ¯ I πL ( M N ) − ¯ I πR ( M N ) M R − M L + log (cid:2) M R M L (cid:3) − γ LN + γ RN π ( M R − M L ) − γ LπR , ¯ I RπR ( t = 0) = − ∂ ¯ I πR ( M N ) ∂M R + 116 π M R − γ RπR = 2 ¯ I π | R + ( M N + m π − M R ) ( ¯ I πR − γ RN π )(( M N − M R ) − m π ) (( M N + M R ) − m π ) − γ RπR , (35)and d ¯ I LπR ( M N , M N , t = 0) dt = 196 π M N ( M L − M R ) (cid:110) (cid:0) M L + 4 M N + 4 m π + M R − M N m π − M N M R − M L M N + 2 M L M R (cid:1) log (cid:20) M R M L (cid:21) − m π (cid:16) M L log (cid:20) M R m π (cid:21) − M R log (cid:20) M L m π (cid:21) (cid:17)(cid:111) + 16 M N ( M L − M R ) (cid:110)(cid:0) M R − M L − M L M N + 2 M L m π + 2 M N M R − m π M R (cid:1) ( ¯ I LR − γ LR )+ (cid:0) − M L − M L M N − M L m π + 3 M L M R + 2 M N − M N m π − M N M R
33 2 m π − m π M R (cid:1) (cid:0) ¯ I πL − γ NL (cid:1) + (cid:0) M L M N + 3 M L m π − M L M R − M N + 4 M N m π + M N M R − m π + m π M R + M R (cid:1) ( (cid:0) ¯ I πR − γ NR (cid:1)(cid:111) . (36)The Feynman parameterization of triangle integral (18) reads¯ I LπR ( t ) = − (cid:90) (cid:90) − u dv du π ) Ω ( t ) − γ LπR , (37)Ω ( t ) = − m π + v ((1 − v ) M N − M R + m π ) + u ((1 − u ) M N − M L + m π ) + uv ( t − M N ) . XI. APPENDIX D
We table the factors α ab introduced in Eq. (21): α = 12 M + 26 M ∆ + 18 M ∆ + 6 M ∆ + ∆ M ( M + ∆) ,α = (2 M + ∆) (6 M + 13 M ∆ + 18 M ∆ + 12 M ∆ + 2 ∆ )24 M ( M + ∆) ,α = (2 M + ∆) (5 M + ∆)40 M ( M + ∆) ,α = α = (2 M + ∆) (5 M + 5 M ∆ + ∆ )20 M ( M + ∆) ,α = 20 M − M ∆ − M ∆ − M ( M + ∆) ,α = 20 M + 60 M ∆ + 87 M ∆ + 65 M ∆ + 23 M ∆ + 3 ∆ M ( M + ∆) ,α = (2 M + ∆) (15 M + 31 M ∆ + 19 M ∆ + 4 ∆ )60 M ( M + ∆) ,α = (2 M + ∆) (5 M + 5 M ∆ + ∆ )20 M ( M + ∆) ,α = 76 M + 170 M ∆ + 170 M ∆ + 73 M ∆ + 12 ∆ M ( M + ∆) ,α = 60 M + 308 M ∆ + 645 M ∆ + 707 M ∆ + 421 M ∆ + 129 M ∆ + 16 ∆ M ( M + ∆) ,α = (2 M + ∆) (42 M + 106 M ∆ + 129 M ∆ + 72 M ∆ + 11 ∆ )336 M ( M + ∆) ,α = (2 M + ∆) (6 M + 114 M ∆ + 163 M ∆ + 48 M ∆ + 5 ∆ )48 M ( M + ∆) ,α = (2 M + ∆) (4 M + 268 M ∆ + 538 M ∆ + 449 M ∆ + 188 M ∆ + 29 ∆ )8 M ( M + ∆) , = (42 M + 170 M ∆ + 440 M ∆ + 599 M ∆ + 436 M ∆ + 163 M ∆ + 22 ∆ )168 M ( M + ∆) / (2 M + ∆) ,α = (2 M + ∆) (5 M + ∆)80 M ( M + ∆) ,α = 0 ,α = (2 M + ∆) (20 M + 24 M ∆ + 19 M ∆ + 3 ∆ )80 M ( M + ∆) ,α = (2 M + ∆) (20 M + 36 M ∆ + 29 M ∆ + 5 ∆ )160 M ( M + ∆) ,α = α = (2 M + ∆) (5 M + 5 M ∆ + ∆ )40 M ( M + ∆) ,α = (2 M + ∆) (48 M + 88 M ∆ + 76 M ∆ + 28 M ∆ + 5 ∆ )96 M ( M + ∆) ,α = (2 M + ∆) (20 M + 55 M ∆ + 63 M ∆ + 31 M ∆ + 5 ∆ )80 M ( M + ∆) ,α = (2 M + ∆) ( M + M ∆ + ∆) M ( M + ∆) ,α = 0 ,α = (2 M + ∆) M ( M + ∆) ,α = (2 M + ∆) (4 M + 11 M ∆ + 19 M ∆ + 15 M ∆ + 5 ∆ )16 M ( M + ∆) ,α = (2 M + ∆) (6 M + 6 M ∆ + ∆ )96 M ( M + ∆) ,α = (2 M + ∆) (14 M + 10 M ∆ + 7 M ∆ + 8 M ∆ + ∆ )224 M ( M + ∆) ,α = (2 M + ∆) (12 M + 24 M ∆ + 25 M ∆ + 13 M ∆ + 2 ∆ )48 M ( M + ∆) ,α = (2 M + ∆) (12 M + 36 M ∆ + 43 M ∆ + 21 M ∆ + 3 ∆ )96 M ( M + ∆) . (38)35
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