A lattice study of ππ scattering at large N c

JHEP06(2022)049 Published for SISSA by Springer Received:February 15, 2022 Accepted:May 6, 2022 Published:June 9, 2022 A lattice study of ππ scattering at large Nc Jorge Baeza-Ballesteros,a,1Pilar Hernándezaand Fernando Romero-Lópeza,b aIFIC, CSIC-Universitat de València, 46980 Paterna, Spain bCTP, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. E-mail: [email protected],[email protected],[email protected] Abstract: We present the first lattice study of pion-pion scattering with varying number of colors, Nc . We use lattice simulations with four degenerate quark flavors, Nf = 4, and Nc = 3 − 6. We focus on two scattering channels that do not involve vacuum diagrams. These correspond to two irreducible representations of the SU(4) flavor group: the fully symmetric one, SS , and the fully antisymmetric one, AA . The former is a repulsive channel equivalent to the isospin-2 channel of SU(2). By contrast, the latter is attractive and only exists for Nf≥ 4. A representative state is |D+ sπ+i−|D+K+i/√2 . Using Lüscher’s formalism, we extract the near-threshold scattering amplitude and we match our results to Chiral Perturbation Theory (ChPT) at large Nc . For this, we compute the analytical U( Nf )ChPT prediction for two-pion scattering, and use the lattice results to constrain the Ncscaling of the relevant low-energy couplings. Keywords: Hadronic Spectroscopy, Structure and Interactions, Lattice QCD, 1/NExpansion, Chiral Lagrangian ArXiv ePrint: 2202.02291 1corresponding author Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP06(2022)049 JHEP06(2022)049 Contents 1 Introduction 1 2 Chiral Perturbation Theory predictions 3 2.1 SU(Nf) predictions 4 2.2 U(Nf) predictions 5 2.3 Large Ncdependence 6 2.4 Matching the U(Nf)and SU(Nf)theories 8 3 Finite-volume formalism 9 4 Lattice setup 10 4.1 Scale setting 11 4.2 Extraction of energy levels 13 5 Results for ππ scattering 15 5.1 Convergence of threshold expansion 15 5.2 Discretization effects 16 6 Fits to Chiral Perturbation Theory 18 6.1 Fitting procedure 18 6.2 Fits at fixed Nc21 6.3 Simultaneous chiral and Ncfits 23 6.4 Comparison to previous literature 23 6.5 A resonance pole in the AA channel? 25 7 Conclusions 27 A SU(Nf) and U(Nf) ChPT scattering amplitudes 28 B Wilson ChPT with a twisted mass 30 B.1 Lagrangian of the theory 31 B.2 Discretization effects on physical observables 32 1 Introduction The ’t Hooft limit of QCD [ 1 ] constitutes a significant simplification of the theory preserving all the interesting non-perturbative phenomena, such as asymptotic freedom, confinement and chiral symmetry breaking. In the limit of large number of colors, Nc , and a gauge coupling scaled as g2∝N−1 c , QCD is believed to reduce to a theory of free, infinitely narrow and colorless resonances [1–3] (i.e., glueballs and hadrons). – 1 – JHEP06(2022)049 Some controversy has arisen as regards the existence of exotic states, such as tetraquarks, in this limit. The old standard lore concluded that tetraquarks cannot exist at large Nc [ 3 ]. However this reasoning has been revised recently and a different conclusion has been reached [ 4 , 5 ]. The old argument simply implies that the mixing of any tetraquark with a two-meson state vanishes in this limit, but says nothing on whether tetraquarks actually exist. This is an interesting and timely question, in view of recent experimental discoveries of exotic states [6]. The description of the interactions and decays of these resonances requires non-vanishing N−1 c corrections. A first-principle method that can quantify such corrections is the lattice formulation, where the number of colors can be varied [ 7 , 8 ]. Indeed, various groups have successfully studied the Nc scaling of some observables, such as the hadron or glueball spectrum, as well as matrix elements [9–19] — see ref. [20] for a recent review. In this paper we present the first study of the Nc scaling of the two-pion scattering amplitude using lattice QCD. We have used dynamical simulations at different number of colors in the range Nc = 3 − 6with Nf = 4 degenerate quarks. This setup was used in ref. [ 18 ] to study the Nc scaling of the famous ∆ I = 1 / 2rule in the so-called GIM limit, where the charm is degenerate with the up quark. In this limit, penguin contractions vanish and therefore the enhancement of the isospin-0 over the isospin-2 channel depends solely on long-range QCD effects. The completion of that study requires the incorporation of final state interactions of the two pions, for which the study of ππ scattering is a first step. We study the scattering in the two simplest channels, which do not involve vacuum contractions. They correspond to two irreducible representations (irreps) of SU(4). First, we have a fully symmetric 84-dimensional irrep, which is equivalent to the isospin-2 channel for Nf = 2. Second, a 20-dimensional one which is antisymmetric for both quark and antiquark indices, contains states with four open flavors, and only exists for Nf≥ 4. Interestingly, as we will see, the antisymmetric channel turns out to be an attractive one, in contrast with the symmetric one which is repulsive. We measure two-pion energy levels in lattice simulations from which we can extract infinite-volume scattering properties, such as the scattering length, using the celebrated Lüscher’s method [ 21 , 22 ]. The same observables can also be predicted in Chiral Perturbation Theory (ChPT) [ 23 , 24 ]. These predictions can be found in the literature for Nc = 3 and any Nf [ 25 ]. However in the context of large Nc , it is necessary to include the η0 meson in the effective theory, since it becomes degenerate with the pions. The vacuum manifold is therefore not SU( Nf )but U( Nf )[ 26 – 28 ]. Somewhat surprisingly we have not found any computation of pion scattering in U( Nf )ChPT in the literature, so we present it here for the first time. The comparison of lattice results with ChPT predictions can be used to determine the relevant low-energy couplings (LECs) of the chiral Lagrangian, and study their scaling in Nc . These results can be compared with those from large Nc inspired phenomenological approaches such as Resonant Chiral Theory [ 29 , 30 ]. On the other hand, unitarized Chiral Theory has been used to predict the fate of resonances at large Nc from low-energy dynamics [ 31 , 32 ]. We will briefly comment on this approach for the attractive channel. Other approaches based on dispersion relations can also be found in the literature [ 33 , 34 ]. – 2 – JHEP06(2022)049 The paper is organized as follows. In section 2, we present the results for the scattering amplitudes in ChPT with and without the η0 . In section 3, we review the finite-volume formalism, and in section 4we present our lattice setup. The results for two-pion scattering are presented in section 5, while the fits to ChPT and a comparison with other approaches is included in section 6. We conclude in section 7. The paper contains two appendices: in appendix Awe quote the main results for the scattering amplitudes in SU( Nf ) and U( Nf ) ChPT, while in appendix Bwe discuss how ChPT can be extended to study discretization effects. 2 Chiral Perturbation Theory predictions Chiral Perturbation Theory (ChPT) describes QCD at energies below the QCD scale, Λ QCD , in terms of the lightest non-singlet multiplet of pseudoscalar mesons (an octet for Nf = 3). These are the pseudo-Nambu-Goldstone bosons (pNGB) arising from spontaneous chiral symmetry breaking, SU(Nf)L×SU(Nf)R→SU(Nf)V.(2.1) The theory contains a number of unknown couplings, the low-energy couplings (LECs), that need to be determined from experiment or from first principles, i.e., via matching to lattice QCD. Mesonic observables can be computed in terms of these parameters as a perturbative expansion according to the standard power counting [23,24], O(mq)∼ O(M2 π)∼ O(k2),(2.2) where mq is the quark mass, and Mπ and k2 are the meson mass and momenta, respectively. A subtle point of ChPT in the ’t Hooft limit is the treatment of the flavor-singlet pseudoscalar meson — the η0 . According to the Witten-Veneziano relation [ 35 , 36 ], the mass of this particle receives a large contribution from the U(1)Aanomaly:1 M2 η0−M2 π=M2 0≡2NfχYM F2 π ,(2.3) being χYM the topological susceptibility of the pure Yang-Mills theory, and Fπ the pion decay constant. 2 Since the χYM ∼N0 c and F2 π∼Nc , this contribution is suppressed as N−1 c . Thus, the large Ncpattern of chiral symmetry breaking becomes instead U(Nf)L×U(Nf)R→U(Nf)V,(2.4) and the η0is a pNGB that needs to be included in the effective theory at low energies. The required modification of ChPT, sometimes referred to as U( Nf ) ChPT, has been studied before [ 26 – 28 , 37 – 39 ]. It requires a modified power counting scheme that includes the scaling of N−1 c. A consistent one has been shown [27,28] to be: O(mq)∼ O(M2 π)∼ O(k2)∼ O(N−1 c).(2.5) 1We implicitly assume Nfdegenerate quarks. 2We use the normalization for Fπcorresponding to ∼93 MeV in QCD. – 3 – JHEP06(2022)049 While several quantities have been computed in U( Nf ) ChPT, meson-meson scattering has not been studied before in this context to the best of our knowledge. We therefore present here the first computation of scattering amplitudes of non-singlet mesons in U( Nf ) ChPT, which will be necessary for this work. In a general theory with Nf active flavors, there are seven different two-meson scattering channels 3 of non-singlet mesons [ 25 ]. They correspond to different irreducible representations (irreps) of SU(Nf). In this work, we focus on two of them: • The irrep of maximal dimensionality — 84 for Nf = 4 — that is symmetric in both quarks and antiquarks. It is equivalent to the isospin-2 channel of SU(2). We will refer to it as SS channel. A representative state of this channel is the well-known |π±π±i. • The irrep containing states with four distinct quark flavors that is antisymmetric in both quarks and antiquarks. This one only exists for Nf≥ 4, and is 20-dimensional for Nf = 4. We denote it as the AA channel. A representative state is ( |D+ sπ+i− |D+K+i)/√2. Throughout this work, we will generically refer to any of them as R. 2.1 SU(Nf) predictions At leading order (LO) in the chiral expansion, the pion-pion scattering amplitude depends only on the ratio Mπ/Fπ . At this order the result is found to be the same for both channels up to a sign MSS LO =−MAA LO =M2 π F2 π (2 −s),(2.6) where sis the usual Mandelstam variable normalized by the pion mass. At next-to-leading-order (NLO) and for Nf≥ 4a total of 13 additional LECs become relevant in the chiral Lagrangian, albeit only some linear combinations appear in the observables of interest. The scattering amplitudes for the two channels are known up to next-to-next-to-leading-order (NNLO) for Nf degenerate quark flavors [ 25 ]. For completeness we reproduce the s -wave projected NLO result in appendix A. From these amplitudes, one can extract the scattering lengths which are found to be:4 MπaSS 0=−M2 π 16πF2 π"1−16M2 π F2 π LSS +M2 π 8F2 ππ2N2 f −M2 π 8F2 ππ2Nf +M2 π 8F2 ππ2log M2 π µ2+M2 π 8F2 ππ2N2 f log M2 π µ2−M2 π 8F2 ππ2Nf log M2 π µ2#,(2.7) MπaAA 0=M2 π 16πF2 π"1−16M2 π F2 π LAA −M2 π 8F2 ππ2N2 f −M2 π 8F2 ππ2Nf−M2 π 8F2 ππ2log M2 π µ2−M2 π 8F2 ππ2N2 f log M2 π µ2−M2 π 8F2 ππ2Nf log M2 π µ2#,(2.8) 3These are reduced to three for Nf= 2 and six for Nf= 3. 4We use the convention in which kcot δ0= 1/a0, where a0is the s-wave phase shift. – 4 – JHEP06(2022)049 where LR≡LR ( µ )are the following linear combinations of LECs, defined at the renormalization scale µ, LSS =L0+ 2L1+ 2L2+L3−2L4−L5+ 2L6+L8,(2.9) LAA =L0−2L1−2L2+L3+ 2L4−L5−2L6+L8.(2.10) Eqs. (2.7) and (2.8) are valid for any number of colors, but the LECs, F2 π and M2 π have an implicit Ncscaling [40]: O(Nc) : F2 π, L0, L3, L5, L8,O(1) : Mπ, L1, L2, L4, L6.(2.11) The leading Nc scaling of LR is thus the same, but subleading effects differ. We therefore expect: LSS =NcL(0) +L(1) SS +O(N−1 c), LAA =NcL(0) +L(1) AA +O(N−1 c). (2.12) Another quantity of interest that may be extracted from the scattering amplitude is the effective range, r0. At LO both channels have M2 πaR 0rR 0=−3,(2.13) meaning r0is positive (negative) in the SS (AA) channel. 2.2 U(Nf) predictions We use the same notation for the LECs in the two chiral theories, even though they will take different values, as we will see when we discuss the matching between the two. At NLO in the U( Nf )power counting of eq. (2.5), the different observables are given by the tree-level contribution from the LO Lagrangian, and the NLO corrections from the LECs of order O(Nc). For instance, the SS scattering length is given by MπaSS 0=M2 π 16πF2 π"1−16M2 π F2 π NcL(0)#,(2.14) and similarly for the AA channel. This prediction is however not sufficient [ 17 ], and one needs to go to NNLO in the U( Nf )power counting. At this order, loop diagrams — and thus chiral logarithms — start to contribute, and further care has to be taken to include the η0 meson. Note that, we use the same notation for the LECs in the two chiral theories, even though they will take different values, as we will see when we discuss the matching between the two. The contribution of the η0 meson is encoded in a few additional diagrams with respect to the SU( Nf )result. These are summarized in figure 1, where solid (dashed) lines represent multiplet (singlet) mesons. The NNLO result is given by MU(Nf) NNLO =MSU(Nf) NLO +MK+ ∆Z2 U(Nf)MLO +Mloop η0,(2.15) where ∆ Z2 represents the additional mass renormalization due to η0 loops — diagram 1(a) — and Mloop η0 represents diagrams 1(b)–1(d), that involve η0 loops. Finally, MK includes products of O(Nc)LECs. – 5 – JHEP06(2022)049 (a) (b) (c) (d) Figure 1 . Additional one-loop Feynman diagrams required to study two-pion scattering for the two channels of interest in U( Nf ) ChPT. Solid lines depict non-singlet mesons, while dotted represent the η0. Our results for the scattering amplitudes at NNLO are presented in appendix A. From these we obtain the following scattering lengths: MπaSS 0=−M2 π 16πF2 π 1−16M2 π F2 π LSS +KSS M2 π F2 π!2 +M2 π 4F2 ππ2N2 f−M2 π 8F2 ππ2Nf +M2 π 8F2 ππ2log M2 π µ2−M2 π+M2 η0 M2 π−M2 η0 M2 π 8F2 ππ2N2 f log M2 π µ2+M2 η0 M2 π−M2 η0 M2 π 8F2 ππ2Nf log M2 π µ2 +M2 π+M2 η0 M2 π−M2 η0 M2 π 8F2 ππ2N2 f log M2 η0 µ2−M2 η0 M2 π−M2 η0 M2 π 8F2 ππ2Nf log M2 η0 µ2 ,(2.16) MπaAA 0=M2 π 16πF2 π 1−16M2 π F2 π LAA +KAA M2 π F2 π!2 −M2 π 4F2 ππ2N2 f−M2 π 8F2 ππ2Nf −M2 π 8F2 ππ2log M2 π µ2+M2 π+M2 η0 M2 π−M2 η0 M2 π 8F2 ππ2N2 f log M2 π µ2+M2 η0 M2 π−M2 η0 M2 π 8F2 ππ2Nf log M2 π µ2 −M2 π+M2 η0 M2 π−M2 η0 M2 π 8F2 ππ2N2 f log M2 η0 µ2−M2 η0 M2 π−M2 η0 M2 π 8F2 ππ2Nf log M2 η0 µ2 ,(2.17) where Mη0 is given by the Witten-Veneziano formula in eq. (2.3). KR are combinations of products of LECs and are O ( N2 c ). From the results of ref. [ 25 ], we find that the leading Nc dependence of these factors is also equal for both channels, i.e., KSSO(N2 c)=KAAO(N2 c)= 128(L8−2L5)2O(N2 c).(2.18) It is also possible to get the prediction for the effective range in the U( Nf ) theory as was done for SU( Nf ) ChPT. The LO result is the same as in eq. (2.13) while the NNLO result can be reconstructed from the scattering amplitudes in appendix A. 2.3 Large Ncdependence In the U( Nf )theory, the Nc scaling of the LECs in eq. (2.12) holds, with a common leading Nc dependence for both channels. It remains to be seen whether it is possible to derive a – 6 – JHEP06(2022)049 (a) Ddiagram, O(N2 c)(b) Cdiagram, O(N1 c) Figure 2 . Diagrammatic representation of the quark Wick contractions contributing to the SS and AA channels. Grey squares represent a pion operator and solid lines are quark lines. simple Nf dependence for the L(1) R terms in this theory, by means of a perturbative analysis of correlation functions at large Nc. The scattering of two pions in these channels at the quark level involves two topologies for the quark Wick contractions, as depicted in figure 2. We denote them as disconnected, D , and connected, C . Their contribution to the meson-meson correlator in the two channels is CSS(t) = 2(D−C), CAA(t) = 2(D+C).(2.19) There are two color loops in the D diagram, and just one in the C diagram. Thus, they are O(N2 c)and O(Nc), respectively. In terms of correlation functions, the scattering length is roughly given by: aR 0∝CR−C2 π C2 π ,(2.20) where Cπ∼ O ( Nc )is the single-pion correlator, which contains a single color loop. One can see that factorizable diagrams do not contribute to the scattering length. For instance, the Nc -leading diagrams contributing to D in figure 2(a) are planar contributions that involve gluons exchanged within each loop, and get exactly cancelled by C2 π. We can now analyze the leading non-factorizable contributions of D and C . The latter is non-factorizable and of O ( Nc )[see figure 2(b)] yielding the leading contribution to the scattering length. Exchanging planar gluons does not change the scaling, and including a quark loop contributes with an extra power of Nf/Nc [figure 3(b)]. Concerning the nonfactorizable contributions in D , they necessarily involve gluon exchange between the two loops, for example figure 3(a). This diagram is suppressed as 1 /N2 c relative to the leading scaling of D , and is thus O ( N0 c ). Finally, the scaling Cπ scales like C . All this can be summarized as follows: C=Nca+bNf Nc+O(N−1 c), D−C2 π=c+O(N−1 c), Cπ=Ncd+eNf Nc+O(N−1 c), (2.21) – 7 – JHEP06(2022)049 (a) O(N0 c)(b) O(NfN0 c) Figure 3 . Subleading diagrams in the large Nc scaling of the Wick contractions D and C in figure 2. Grey squares represent a pion operator, solid lines represent quark lines, and wavy lines are gluons. The scaling of each diagram is given in their respective subcaptions. where a−e are numerical constants, independent of Nc and Nf . Combining the scaling of the diagrams, with the linear combinations in eq. (2.19), one can derive MπaR 0≈ ± 1 Nc˜a+˜ bNf Nc∓˜c1 Nc+O(N−3 c),(2.22) where ˜a−˜c are linear combinations of a−e , and the upper (lower) signs correspond to the AA ( SS ) channel. large Nc arguments therefore explain why the repulsive nature of one channel implies the attractive nature of the other. By matching eq. (2.22) to the ChPT predictions in eqs. (2.16) and (2.17), we can derive the first coefficients of the large Ncscaling of the NLO LECs. We find LRU(Nf)=NcL(0) +NfL(1) c∓L(1) a+O(N−1 c),(2.23) where the correlated and anticorrelated terms, L(1) c and L(1) a , are now independent of Nf . Comparing to eqs. (2.9) and (2.10), one can see that L0+L3−L5+L8=NcL(0) +NfL(1) c+O(N−1 c), 2L1+ 2L2−2L4+ 2L6=L(1) a+O(N−1 c)(2.24) Finally, one may also check that the factors of 1 /Nf in eqs. (2.16) and (2.17) explicitly cancel at large Nc . This can be seen analytically assuming the Witten-Veneziano relation for the η0 mass [eq. (2.3)], and expanding in 1 /Nc . In conclusion, the U( Nf ) ChPT predictions are found to be consistent with the diagramatic 1 /Nc expansion in eq. (2.22), with the Mπ dependence hiding in the values of the coefficients ˜a−˜c. 2.4 Matching the U(Nf)and SU(Nf)theories The standard form of SU( Nf )ChPT can be recovered from the U( Nf ) one in the limit of large η0 mass, Mη0Mπ . The effect of integrating out this particle remains encoded in the SU( Nf ) LECs, and results in corrections that scale as 1 /Nf and 1 /N2 f . The matching – 8 – JHEP06(2022)049 012345 qNc 3 Mπ Fπ −0.6 −0.4 −0.2 0.0 Nc 3MπaSS 0 LO ChPT Nc= 3, Ens. “A” Nc= 3, Ens. “B” Nc= 3, Ens. “C” Nc= 4 Nc= 5 Nc= 6 (a) SS channel 012345 qNc 3 Mπ Fπ 0.0 0.5 1.0 Nc 3MπaAA 0 LO ChPT Nc= 3, Ens. “A” Nc= 3, Ens. “B” Nc= 3, Ens. “C” Nc= 4 Nc= 5 Nc= 6 (b) AA channel Figure 5 . Results for the s -wave scattering length obtained using the threshold expansion to O ( L−5 ), against the LO predictions from ChPT (black solid line). Both axis have been multiplied by a Nc -dependent factor to eliminate leading Nc dependencies. The physical point is indicated by a vertical dashed line. 5 Results for ππ scattering In this section, we use our results for ground state energy levels, together with the 1 /L expansion of the ground state — given in eq. (3.4) — to obtain the s -wave scattering length. Since this is a perturbative expansion valid when a0/L  1, we test whether this condition is satisfied a posteriori. We also study discretization effects in the scattering amplitude. We start by computing the scattering length to O ( L−5 ). The results are summarized in the third and fifth columns of table 4. As can be seen, the SS channel is repulsive, while the AA one is attractive. Qualitatively, the scattering lengths are of the same order for both channels, but with opposite signs. This is expected from ChPT and large Nc , as discussed in section 2. In figure 5we compare the scattering length against the LO prediction of ChPT. We observe good agreement for the SS channel, suggesting that higher order corrections are small in this channel. The situation is different for the AA channel, where deviations from the LO are sizeable. We now study a possible lack of convergence of the threshold expansion. 5.1 Convergence of threshold expansion As explained in section 3, eq. (3.4) is an expansion in powers of a0/L . The truncation of the expression to O ( L−5 )introduces some systematic error, that may not be neglected if a0/L ∼ 1. In order to study this effect, a comparison is done between the O ( L−5 )and the O ( L−6 )expansions. Since r0 enters at O ( L−6 ), we will estimate its effect based on its LO ChPT prediction given in eq. (2.13). Using the value of ∆ E for each ensemble, we collate the results for Mπa0at both orders of the threshold expansion. – 15 – JHEP06(2022)049 −0.6−0.4−0.2 0.0 MπaSS 0 0.00 0.05 0.10 0.15 0.20 ∆ESS/Mπ ∆E/Mπ O(L−5) O(L−6), M2 πr0a0=−3 (a) SS channel 0.0 0.5 1.0 1.5 MπaAA 0 −0.06 −0.04 −0.02 0.00 ∆EAA/Mπ ∆E/Mπ O(L−5) O(L−6), M2 πr0a0=−3 (b) AA channel Figure 6 . Comparison between the O ( L−5 )(red) and O ( L−6 )(green) threshold expansion given in eq. (3.4). The left panel corresponds to the 3A10 ensemble in the SS channel, whereas the right one is the 3A11 ensemble and AA channel. The gray horizontal band is the energy level extracted from the lattice, and the points are the determinations of the scattering length. In the case of the SS channel, we observe that for all the ensembles, both assumptions produce very similar results, with discrepancies smaller than 1 σ . On the left panel of figure 6 we show this comparison for the 3A10 ensemble. The gray horizontal band corresponds to the lattice ∆ ESS result, while the red and green ones are the O ( L−5 )and O ( L−6 )threshold expansions, respectively, whose width comes from the errors of MπL . The red and green points are the results for the scattering length, which are compatible. The case of the AA channel is more complicated. For the ensembles with smaller ξ and larger L , we find acceptable convergence, and compatible results for aAA 0 at both orders. However, as ξ is increased we arrive to a point at which convergence fails. An example of this is shown in the right panel of figure 6, with the same color code as before. We conclude that some of our ensembles lie in a regime in which the 1 /L expansion is no longer valid, and so any results for the scattering lengths may have uncontrolled systematic errors. In view of this caveat, we avoid the expansion and use instead the full Lüscher formalism. 5.2 Discretization effects Discretization uncertainties are always a concern in lattice calculations. Previous work in the isospin-2 channel with twisted-mass fermions has shown that discretization errors are not very significant for this channel [ 57 ]. However, hints from the two-baryon sector [ 58 ] indicate that this may not always be the same in all scattering observables. Indeed, as we will see, we find significant discretization effects in the AA channel. This is, to our knowledge, the first time that such large effects are seen for meson-meson scattering. We will assess the magnitude of the cutoff effects at Nc = 3, since various ensembles at three different lattice spacings are available. First, we compare the results using the pure Wilson and mixed-action setups, which should coincide in the continuum limit. In figure 7 we represent the ratio between the energy shifts determined for both regularizations. We – 16 – JHEP06(2022)049 0.000 0.002 0.004 0.006 a2/fm2 1.0 1.2 1.4 1.6 ∆EW SS/∆ETM SS (a) SS channel 0.000 0.002 0.004 0.006 a2/fm2 1.0 1.2 1.4 1.6 ∆EW AA/∆ETM AA (b) AA channel Figure 7 . Dependence on the lattice spacing of the quotient between the energy shifts computed using two different fermion regularizations: the Wilson setup and the mixed-action one. The comparison is done both for the SS (left) and AA (right) channels. observe that for the SS channel all results are close to unity and thus discretization effects are small. However, this is not the case for the AA channel. We observe large discretization effects, which nevertheless seem to decrease as we approach the continuum limit. Since only the AA energy shifts show relevant cutoff effects, we will focus on this channel henceforth. To understand the absolute impact of discretization effects in both regularizations we consider the continuum extrapolation of the scattering phase shift, kcot δAA 0 . This implies additional difficulties, since the scattering amplitude is a function of both ξ and the CM momentum k , which differ for each ensemble. To keep a line of constant physics, we first need to extrapolate/interpolate to a reference point ( k2 ref, ξref ) for each lattice spacing and fermion regularization. The procedure is as follows. We first extrapolate each ensemble to k2 ref = − 0 . 08 M2 π , using eq. (3.3) with a prior for the effective range, M2 πrAA 0aAA 0∈ [ − 5 ,− 1]. This choice is inspired by the LO prediction from ChPT, eq. (2.13). The width of the interval is chosen based on some recent results that show that this quantity may have sizeable higher order corrections [ 59 ]. An example of this extrapolation for one ensemble is shown in figure 8. Note that the errors introduced by the width of the prior, depicted by dotted lines, is negligible compared to the statistical one. Next, at fixed lattice spacing, we interpolate linearly to ξref = 0 . 14, as shown in figure 9. Finally, we use the results at the reference point to do a constrained linear continuum extrapolation using both regularizations, and taking into account correlations. The result is shown in figure 10. Note that we expect O ( a )improvement in both regularizations, since we are using a non-vanishing value of csw . Our results are consistent with a regularizationindependent continuum limit and O(a2)scaling, as expected. Henceforth, we stick to the mixed-action setup and explicitly include discretization effects to analytical expressions. To do so we have studied a generalization of ChPT that considers the effects of a Wilson term as well as a twisted mass [ 60 – 62 ]. We have not found – 17 – JHEP06(2022)049 −0.12 −0.10 −0.08 −0.06 −0.04 k2/M2 π 0.5 1.0 1.5 2.0 2.5 k MπcotδAA 0 Wilson Twisted-mass Figure 8 . Results for the s -wave phase shift of the AA channel for two regularizations, as indicated in the figure legend. We graphically show how one of the points — corresponding to the 3A40 ensemble — is extrapolated to k2 ref = − 0 . 08 M2 π , using the ERE and a prior for the effective range based on ChPT, M2 πaAA 0rAA 0∈ [ − 5 ,− 1], as explained in the main text. Dotted lines illustrate the error introduced by the width of this interval. in ChPT any hint pointing to larger discretization effects in the AA channel compared to the SS one. We have chosen to parameterize the dominant corrections as Mlatt AA =Mcont AA + 32π2a2ξW −→ k Mπ cot δlatt 0=k Mπ cot δcont 0 1−32π2a2ξW TLO 0,AA !,(5.1) where W a combination of new LECs that appear in Wilson ChPT, and is expected to scale as O ( N0 c ), TAA 0,LO is the LO s -wave scattering amplitude for the AA channel, as explained in appendix A, and we use “cont” (“latt”) to refere to continuum (lattice) quantities. More details about this result are explained in appendix B. From the continuum extrapolation, it is possible to determine a value W=−42(29) fm−2.(5.2) This result will be used as an additional input when matching the AA channel results to ChPT. 6 Fits to Chiral Perturbation Theory We now match the lattice results for ∆ ER to ChPT predictions in section 2and appendix A. This way we extract the relevant LECs and study their large Ncscaling. 6.1 Fitting procedure In order to match our lattice results to ChPT, we need the prediction of ∆ ER in this effective theory. The starting point is the result for the scattering amplitudes that can be – 18 – JHEP06(2022)049 0.100 0.125 0.150 ξ 0.5 1.0 1.5 2.0 k MπcotδAA 0 Wilson Twisted-mass (a) “A” ensembles (a= 0.075 fm) 0.08 0.10 0.12 0.14 0.16 ξ 1 2 3 k MπcotδAA 0 Wilson Twisted-mass (b) “B” ensembles (a= 0.065 fm) 0.12 0.14 0.16 0.18 ξ 0.5 1.0 1.5 k MπcotδAA 0 Wilson Twisted-mass (c) “C” ensembles (a= 0.059 fm) Figure 9 . Results for the linear interpolation of ( kMπ ) cot δ0 to a fixed value ξref = 0 . 14, depicted as empty pointd, for the pure Wilson (blue) and mixed-action (red) setups, and for the three lattice spacings. In the coarser case, only three points are used in the interpolation. found in ref. [ 25 ] for the SU( Nf ) theory, and in eqs. (A.10) and (A.11) for the U( Nf ) case. Following ref. [ 17 ], we choose a convenient renormalization scale, which is related to 4 πFπ but has no leading Ncdependence, µ2=3 Nc (4πFπ)2.(6.1) Also, in the U(Nf) theory, we take the splitting between M2 πand M2 η0to be M2 η0 (4πFπ)2=ξ+a0 N2 c ,(6.2) with a0∼ 6 . 5, as determined using eq. (2.3) and the topological susceptibility from ref. [ 63 ]. The results for the scattering amplitudes are then projected to an s -wave following eq. (A.1), and the corresponding phase shift is computed using eq. (A.2). Discretization – 19 – JHEP06(2022)049 0.000 0.002 0.004 0.006 a2/fm2 0.8 1.0 1.2 1.4 1.6 k MπcotδAA 0 Wilson Twisted-mass Figure 10 . Continuum extrapolation of ( k/Mπ ) cot δ0 . Data from the pure Wilson (blue) and mixed-action (red) setups is fitted simultaneously with a constrained common continuum limit, depicted by an empty diamond. effects are also included for the AA channel as explained in eq. (5.1). Then, the Lüscher equation [eq. (3.1)] is numerically solved to obtain the finite-volume energies. 6 The energy shifts are finally determined as a function of ξ,MπLand the LECs. Since we are working only for small values of the momentum, the LECs that appear in the partial amplitudes multiplied by a power of the momentum — L0 R and L00 R in eqs. (A.4) and (A.5) and K0 R in eqs. (A.10) and (A.11) — are set to zero. This is justified, as we do not have enough information to constrain them. Due to non-negligible correlations between ∆ ER and ξ , Ordinary Least Squares is not a good option for fitting. We instead use York Regression [ 66 ], defining the χ2 function as χ2= min δi [R|V−1R],(6.3) where R is the data vector and V is the corresponding covariance matrix. For a simple fit to a single channel, R is composed by a succession of tuples ( f ( xi + δi ) −yi, δi )corresponding to all the ensembles i that are fitted. Here, xi = ξi and yi = ∆ ER,i are our lattice results, and f the ChPT prediction of the energy shift, computed as explained above. In this case V is a block-diagonal matrix, due to the statistical independence of the different ensembles. By contrast, when we fit discretization effects in the AA channel, additional correlations must be included. This is due inclusion in R of the value of W from eq. (5.2), and so, V is no longer block-diagonal. Similarly when we consider a simultaneous fit to both channels, the tuples need to be enlarged, fSS(xi+δi)−yI=2 i, fAA(xi+δi)−yAA i, δi.(6.4) 6We use an efficient implementation for the generalized zeta function, introduced in refs. [64,65]. – 20 – JHEP06(2022)049 0.0 0.1 0.2 0.3 N−1 c −1.0 −0.5 0.0 0.5 1.0 LSU(4) R/Nc·103 SS AA (a) SU(4) ChPT 0.0 0.1 0.2 0.3 N−1 c −1.0 −0.5 0.0 0.5 1.0 LU(4) R/Nc·103 SS AA (b) U(4) ChPT Figure 11 . Results for the fits to ChPT at fixed Nc for the SS (blue squares, table 5) and AA (red circles, table 6) channels. The lines are the best fit to eq. (2.12), and empty points represent the large Nclimit. In the case of the U(4) theory, we have imposed a common limit for the fit. Note that, even though the energy shifts depend on both ξ and MπL , only uncertainties in the former are considered. The reason is that relative errors in the latter quantity are much smaller, so we expect them to have a insignificant effect. We have checked this explicitly for some cases and found this to be true. 6.2 Fits at fixed Nc We first work at fixed Nc and fit the energy shifts of both channels to SU(4) and U(4) ChPT, following the procedure described above. In all cases, we determine the values of LR in eqs. (A.4) and (A.5) from the fits. Additionally, we determine KR , as defined in eqs. (A.10) and (A.11), when we fit to the U(4) theory. Finally, for the AA -channel, we also fit the parameter W that accounts for discretization effects, as introduced in eq. (5.1). The results for the fits are shown in tables 5and 6for the SS and AA channels, respectively. In the former case, ChPT at the order considered cannot describe the behaviour of the lattice results for the heaviest masses. We have thus not included ensembles with ξ& 0 . 14 in the fit. In addition, as we are using the W determination from the continuum extrapolation as an additional input when fitting the AA channel results, the number of degrees of freedom (dof) is increased by one unit in this case. The Nc scaling of the fitted values of LSS and LAA is shown in figure 11. It is well described by the expected leading and subleading Nc dependence. However, only in the U(4) theory the leading term is found to be consistent in the two channels — see eq. (2.12). In figure 12 we also study the Nc scaling of KR . At the order we are working, only the leading Nc dependence should be included, which implies a constant value of the ratio shown in figure 12. However, the results found for both channels (solid lines) are not consistent, as would be expected from eq. (2.18). A common limit can be found from a constrained linear fit that includes both channels (dashed line), with different subleading Nc corrections in KR . – 21 – JHEP06(2022)049 Nc SU(4) ChPT U(4) ChPT LSS ·103χ2/dof LSS ·103KSS ·103χ2/dof 3−1.85(7) 1.8/4 −2.5(7) −0.4(0.6) 1.6/3 4−1.79(8) 3.0/3 −0.9(8) 1.3(1.0) 0.1/2 5−1.83(11) 2.3/3 −2.2(6) −0.3(0.9) 2.3/2 6−2.10(16) 3.8/3 −1.7(8) 1.2(1.7) 2.8/2 Table 5 . Results of fits of ∆ ESS to SU(4) and U(4) ChPT at fixed Nc . Nc = 3 ensembles with ξ&0.14 are not fitted. Nc SU(4) ChPT U(4) ChPT LAA ·103W/fm−2χ2/dof LAA ·103KAA ·103W/fm−2χ2/dof 3−2.4(5) −72(17) 23.4/8 1.7(1.3) 2.2(0.7) −39(23) 12.8/7 4−1.8(1.2) −45(30) 1.2/3 −1.1(2.7) 0.5(2.3) −41(32) 1.2/2 5−4.2(1.1) −75(22) 8.5/3 1.2(2.8) 5.6(2.7) −41(32) 3.9/2 6−5.4(1.5) −72(24) 5.4/3 0.1(3.2) 6.6(3.3) −39(33) 1.6/2 Table 6. Results of fits ∆EAA to SU(4) and U(4) ChPT at fixed Nc. 0.0 0.1 0.2 0.3 N−1 c −0.2 0.0 0.2 0.4 KU(4) R/N2 c·103 SS AA Figure 12 . Results for KR from the U(4) ChPT fits for the SS (blue squares, table 5) and AA (red circles, table 6) channels, together with two options for large Nc extrapolations, represented by empty points. Solid lines represent fits to contant values, which are not consistent as expected from eq. (2.18). Dashed lines correspond to a linear fit with constrained dominant term and different subleading dependence. – 22 – JHEP06(2022)049 Channel Fit L(0) ·103L(1) R·103K(0) R·105W/fm−2χ2/dof SS SU(4) −0.04(1.3) −1.70(18) − − 12.8/15 U(4) −0.01(7) −1.78(20) 1.2(2.5) −12.2/14 AA SU(4) −1.22(19) 0.8(4) − −94(15) 38.5/19 U(4) −0.1(4) 1.8(4) 21(5) −32(23) 22.5/18 Table 7 . Results of fits to the energy shift in both SS and AA channels to SU(4) and U(4) ChPT, as indicated in the first column. L(0) ·103L(1) SS ·103L(1) AA ·103K(0) SS ·105K(0) AA ·105W/fm−2χ2/dof −0.02(8) −1.79(19) 1.7(4) 0.8(2.2) 22(3) −31(9) 35.6/33 Table 8 . Results of a global fit to the energy shift in U(4) ChPT. The goodness of the fit is ilustrated in figure 13. 6.3 Simultaneous chiral and Ncfits Given the results above, we are confident to perform global fits including all values of Nc . We have done this both for the SU(4) and the U(4) theories, parametrizing LR with a leading and subleading Nc terms. Regarding KR in the U(4) theory, we have opted to keep two different leadingNc parameters, one for each channel, since we do not have enough information to constrain also subleading corrections, KR=K(0) RN2 c+O(Nc).(6.5) First, we have performed fits to single channels for the two theories. The results are shown in table 7. The best fit results of L(0) are different for both channels when using the SU(4) theory, but they are compatible in the U(4) case, in agreement with the results from the previous section and theoretical expectations. This led us to perform a simultaneous fit of both channels to the U(4) theory with a constrained common value of L(0) . The results are presented in table 8and the quality of the fits is shown in figure 13. 6.4 Comparison to previous literature From the global fit to U(4) ChPT, we observe that the leading Nc dependence of the LEC combination that enters ∆ ER in both channels, L(0) , is anomalously small. The dominant terms in the LECs for the two channels of interest are therefore the ones subleading in Nc . If we recall eq. (2.23) and combine the results for the two channels, we can further obtain the Ncand Nfscaling of the following linear combinations of LECs: L0+L3−L5+L8=−0.02(8)Nc−0.01(5)Nf+O(N−1 c), L1+L2−L4+L6=−0.88(10) + O(N−1 c).(6.6) The combination that is in principle leading, seems to be suppressed (both in the leading and subleading terms) with respect to the subleading one. – 23 – JHEP06(2022)049 40 50 60 70 (∆ESS/Mπ)(MπL)3ξ−1 Nc= 3 Nc= 4 Nc= 5 Nc= 6 3C10 3A10 3B10 3A11 3C20 3A20 3A30 3A40 3B20 4A10 4A20 4A30 4A40 5A10 5A20 5A30 5A40 6A10 6A20 6A30 6A40 Ensemble −50 −40 (∆EAA/Mπ)(MπL)3ξ−1 Figure 13 . Results for the simultaneous chiral and Nc fit of the energy shifts for both SS (top) and AA (bottom) channels. Lattice results are depicted as points, with a different marker for different Nc as explained in the plot legend. Empty points in the SS channel are not fitted. Horizontal lines correspond to the best fit to U(4) ChPT, summarized in table 8. The values are multiplied by (MπL)3ξ−1to eliminate leading chiral and Ncdependence. Using this scaling, we can compare our results to previous literature at Nc = 3, which exists for the SU( Nf ) theory and Nf< 4. Thus, only the SS channel is available for comparison. We begin from the U(4) results of eq. (6.6), for which the Nf scaling is known, and determine the value of LSS for the desired number of flavors. Then, we use the matching of eq. (2.25) to translate our result to the SU( Nf ) case. Finally, we change the renormalization scale to that used in the literature. 7 Our initial scale is µ = 1 . 40(12) GeV, which we have determined averaging eq. (6.1) over all ensembles. The error of this quantity is a systematic one coming from subleading Nceffects, while the statistical error is negligible. We first compare to Nf = 3 results from ChPT fits to experiment. Setting the renormalization scale at the mass of the ρresonance, Mρ= 0.77 MeV, we obtain LNc=3, Nf=3 SS (Mρ) = −1.14(22)stat(11)µ·10−3,(6.7) where the first error comes from the fit and the second corresponds to the renormalization scale. This result agrees with the values reconstructed from previous literature,8 LNc=3, Nf=3 I=2 =−0.9(1.5) ·10−3from ref. [68] (table 1, 2nd column),(6.8) LNc=3, Nf=3 I=2 =−0.7(3.2) ·10−3from ref. [24].(6.9) 7Note the running of the LECs differs between SU(Nf) [67] and U(Nf) ChPT [28]. 8 The error of these results has been estimated adding in quadrature the independent errors from each LEC. It is expected to be reduced when correlations are taken into account. – 24 – JHEP06(2022)049 B.1 Lagrangian of the theory The O(p4)Lagrangian of this theory reads L=F2 4Tr[∂µΣ∂µΣ†] + F2 4Tr[χΣ†+χ†Σ] + F2 4Tr[AΣ†+A†Σ] +L0Tr[∂µΣ∂νΣ∂µΣ†∂νΣ†] + L1Tr[∂µΣ∂µΣ]2 +L2Tr[∂µΣ∂νΣ†] Tr[∂µΣ∂νΣ†] + L3Tr[(∂µΣ∂µΣ†)2] +L4Tr[∂µΣ∂µΣ†] Tr[χΣ†+χ†Σ] + W4Tr[∂µΣ∂µΣ†] Tr[AΣ†+A†Σ] +L5Tr[∂µΣ∂µΣ†(χΣ†+χ†Σ)] + W5Tr[∂µΣ∂µΣ†(AΣ†+A†Σ)] +L6Tr[χΣ†+χ†Σ]2+W6Tr[χΣ†+χ†Σ] Tr[AΣ†+A†Σ] + ˜ W6Tr[AΣ†+A†Σ]2 +L7Tr[χΣ†−χ†Σ]2+W7Tr[χΣ†−χ†Σ] Tr[AΣ†−A†Σ] + ˜ W7Tr[AΣ†−A†Σ]2 +L8Tr[χΣ†χΣ†+χ†Σχ†Σ] + W8Tr[χΣ†AΣ†+χ†ΣA†Σ] +˜ W8Tr[AΣ†AΣ†+A†ΣA†Σ],(B.3) with Wi , ˜ Wi new LECs that need to be determined numerically. The ones which are relevant in this work have a defined large Ncscaling, O(Nc) : W5, W8,˜ W8,O(1) : W4, W6,˜ W6.(B.4) In order to work with this theory, one first redefines the spurion field, χ0=χ+A= 2B0(m1+iµT) + W0a=M2eiω0T(B.5) with Mthe bare pion mass and ω0the twist angle at LO, as well as the new LECs, W0 4,5=W4,5−L4,5, W0 6,7,8=W6,7,8−2L6,7,8, ˜ W0 6,7,8=˜ W6,7,8+L6,7,8. (B.6) One then expands around the physical vacuum, Σ = eiωT/2eiφ/F eiωT/2,(B.7) where F is the bare pion decay constant, φ is the usual pion matrix, and ω = ω0 + ε is the non-perturbative twist angle. At NLO, we can determine ε by requiring the one-point function to vanish, ε=−8ˆas F2(4W0 6+W0 8) + 2ˆac M2 π (4 ˜ W0 6+˜ W0 8),(B.8) where we have abbreviated s= sin ωand c= cos ω. – 31 – JHEP06(2022)049 B.2 Discretization effects on physical observables We are now in position to study discretization effect for some mesonic observables of interest. The lattice corrections to the continuum decay constant, Fcont π , and field renormalization, Zcont, are, respectively, Fπ=Fcont π+4ˆac Fπ (4W0 4+W0 5),(B.9) Z=Zcont −8ˆac F2 π (4W0 4+W0 5),(B.10) all of which vanish at maximal twist. The effects on the pion mass are a bit more complicated, since distinct mesons get different corrections to its continuum value, Mcont π . For charged mesons one finds M2 ±= (Mcont π)2+16ˆaM2 πc F2 π(4W0 6+W0 8)+ cˆa M2(4 ˜ W0 6+˜ W0 8)−8M2 πˆac F2 π (4W0 4+W0 5),(B.11) which also vanishes at maximal twist. However, neutral multiplet mesons get further O ( a2 ) corrections that do not cancel in such limit and even lead to a non-diagonal mass matrix for flavorless ones, M2 K0,D0=M2 ±−16 ˜ W0 8s2ˆa2 F2 π ,(B.12) M[π0, η0, ηc] = M2 K0,D013×3−32 ˜ W0 6s2ˆa2 3F2 π   3√3−√6 √3 1 −√2 −√6−√2 2   .(B.13) These corrections have been discussed in the literature to be as large as ∼30% [62]. Next, one can consider NLO corrections to the continuum scattering amplitudes in the SS and AA channels, Mcont SS and Mcont AA , respectively. These are generated by new four-point vertices proportional to the new LECs, and by corrections to the LO result originated from eqs. (B.9)–(B.11), MSS =Mcont SS +8ˆacM2 π F4 π−2W0 4s−W0 5s+ 8W0 6+ 4W0 8+4ˆac M2 π (2 ˜ W0 6+˜ W0 8),(B.14) MAA =Mcont AA +8ˆacM2 π F4 π−2W0 4s+W0 5s+ 8W0 6−4W0 8+4ˆac M2 π (2 ˜ W0 6−˜ W0 8),(B.15) which are consistent with ref. [ 62 ]. Note that, as initially expected, all these corrections vanish at maximal twist, meaning that the usage of a properly tuned twisted mass provides automatic O ( a )improvement for these observables. However, this also implies that in order to understand the observed discretization effects for this regularization, we need to go to higher order. For Wilson fermions, on the other hand, O ( a )improvement depends on the correct choice of csw, and can only be empirically tested. – 32 – JHEP06(2022)049 Figure 15 . Additional Feynman diagram required to study O ( a2 )discretization effects in the AA channel for twisted-mass fermions. At non-zero twist, further corrections arise from new three-leg vertices that lead to the diagram depicted in figure 15 also contributing to the scattering amplitudes for two channels. For ω6 = π/ 2, this originates O ( a4/M4 π )and O ( a3/M2 π )corrections compared to the leading result. However, both vanish at maximal twist, and only O(a2)effects remain Mω=π/2 SS =Mcont SS +32ˆa2M2 π F6 π 1 t−1"8W02 4+ 4W0 4W0 5+W02 5−16W0 4W0 6−4W0 5W0 6+ 8W02 6 −4W0 4W0 8−2W0 5W0 8+ 4W0 6W0 8+W02 8+t(−8W02 4−4W0 4W0 5−W02 5+ 8W0 4W0 6 + 2W0 5W0 6+ 2W0 4W0 8+W0 5W0 8) + t2 4(8W02 4+ 4W0 4W0 5+W02 5)#+ (t↔u), (B.16) Mω=π/2 AA =Mcont AA +32ˆa2M2 π F6 π 1 t−1"8W02 4+ 4W0 4W0 5−W02 5−16W0 4W0 6−4W0 5W0 6+ 8W02 6 −4W0 4W0 8+ 2W0 5W0 8+ 4W0 6W0 8−W02 8+t(−8W02 4−4W0 4W0 5+W02 5+ 8W0 4W0 6 + 2W0 5W0 6+ 2W0 4W0 8−W0 5W0 8) + t2 4(8W02 4+ 4W0 4W0 5−W02 5)#+ (t↔u), (B.17) where t and u are the usual Mandelstam variables normalized by the pion mass. If we focus on the corrections near threshold and consider only the leading Nc terms, we get the result explained in eq. (5.1). Two comments are in place. First, these leadingNc corrections should have the same size for both the SS and AA channels. 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