Light- and strange-quark mass dependence of the ρ(770) meson revisited
LLight- and strange-quark mass dependence of the ρ (770) meson revisited R. Molina
1, 2, ∗ and J. Ruiz de Elvira † Institute of Physics of the University of Sao Paulo,Rua do Mat˜ao, 1371 -Butant˜a, S˜ao Paulo -SP, 05508-090 Universidad Complutense de Madrid, Departamento de Fisica Teorica II,& IPARCOS, Plaza Ciencias, 1, 28040 Madrid, Spain Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics,University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland (Dated: June 2, 2020)Recent lattice data on ππ -scattering phase shifts in the vector-isovector channel, pseudoscalarmeson masses and decay constants for strange-quark masses smaller or equal to the physical valueallow us to study the strangeness dependence of these observables for the first time. We perform aglobal analysis on two kind of lattice trajectories depending on whether the sum of quark massesor the strange-quark mass is kept fixed to the physical point. The quark mass dependence of theseobservables is extracted from unitarized coupled-channel one-loop Chiral Perturbation Theory. Thisanalysis guides new predictions on the ρ (770) meson properties over trajectories where the strange-quark mass is lighter than the physical mass, as well as on the SU(3) symmetric line. As a result, thelight- and strange-quark mass dependence of the ρ (770) meson parameters are discussed and precisevalues of the Low Energy Constants present in unitarized one-loop Chiral Perturbation Theory aregiven. Finally, the current discrepancy between two- and three-flavor lattice results for the ρ (770)meson is studied. PACS numbers: 13.75.Lb 11.30.Rd, 12.38.Gc, 14.40.-n
I. INTRODUCTION
The ρ (770) meson is the lightest vector meson in thehadron spectrum and one of the most studied hadrons inthe literature. It is one of the best examples of a q ¯ q res-onance well described within the quark model. Its phaseshift fits well into a simple Breit-Wigner (BW) parame-terization up to small corrections [1, 2] and it is usuallyconsidered as the prototype of narrow resonance in thelight-quark sector. It also dominates the ππ scatteringamplitude in the I = J = 1 channel below 1 GeV, de-caying almost exclusively to two pions [3]. The ρ -mesonmass and width are well known from experiment; theParticle Data Group (PDG) quotes for their BW values M = 775 . . Fromthe theory side, the most precise determination of itspole parameters comes from the Roy-equation analysisof ππ scattering [4–8]. The contribution of the ρ (770)is also important for the hadronic total cross section σ ( e + e − → hadrons) [9–11], which explains applicationsthat go well beyond low-energy meson physics, rangingfrom the hadronic-vacuum polarization and the light-by-light contributions to the anomalous magnetic momentof the muon (see, for instance, [12–15]) to the electro-magnetic and tensor-nucleon form factors [16–19]. Fur- ∗ Electronic address: [email protected] † Electronic address: [email protected] Where I and J refer to isospin and angular momentum respec-tively. These values correspond to the average value for the chargedmeson seen in τ decays and e + e − collisions. thermore, it also plays a crucial role in the analysis ofheavy meson decays [20, 21] and in the restoration ofchiral symmetry at high temperatures [22–30].Although, the ρ -meson properties fit well within thenaive quark-model picture, its nature in terms of QCDdegrees of freedom is still under discussion [31]. Never-theless, at low energies QCD becomes non perturbative,what hinders the study of hadron composition in terms ofthe fundamental QCD degrees of freedom. LatticeQCDsimulations attempt to tackle this problem, however, sev-eral challenges are met when dealing with hadron scatter-ing processes [32, 33]. In this regard, the m q and 1 /N c ex-pansions [34–36] provide model independent predictionsto identify different kinds or hadrons. These parameterscan be used to study whether the response of resonanceproperties to a change on N c or m q compares well withthe behavior expected for different QCD configurations.For instance, by studying the N c dependence of the ρ -meson properties, it was found that it also has a smallnon- q ¯ q component [37–41]. In addition, the analysis ofthe quark-mass dependence of the ρ (770) parameters bymeans of the generalization of the Feynman-Hellmanntheorem for resonances suggests that it requires non-negligible corrections beyond the quark model [42]. Inthis way, the extraction of the light- and strange-quarkmass dependence of the ρ -meson parameters from Lat-ticeQCD simulations provides a powerful tool to confrontquark-model predictions.At low energies, Chiral Perturbation Theory m q and N c stand for the quark mass and number of colors, re-spectively. a r X i v : . [ h e p - l a t ] J un (ChPT) [43–45] is the Effective Field Theory (EFT)that controls the quark-mass dependence of hadronicobservables. ChPT encodes the interactions of thepseudo-Goldstone bosons of the spontaneous chiralsymmetry breaking, and hence, it is capable to describethe quark-mass dependence of the light-pseudoscalarmeson masses and decay constants at low energies,being this completely inherited from QCD [44, 45].Nevertheless, ChPT is constructed as an expansion inquark masses and momenta and hence it is only validbelow a certain scale. Therefore, ChPT does not providedirect information about resonance properties. Onthe contrary, unitarized Chiral Perturbation Theories(UChPTs) [46–52] are based on imposing exact unitaritywhile matching ChPT at low energies. Thus, the regionof validity of the chiral expansion is extended, allowingone to generate poles on unphysical Riemann sheetsin the complex-energy plane and to access resonanceproperties. In particular, the Inverse Amplitude Method(IAM) [47–50] generates the ρ (770) resonance from ππ scattering and provides a tool to study the light-and strange-quark mass dependence of the ρ -mesonproperties, while reproducing the chiral series at lowenergies. Thus, in this work we utilize the IAM toinvestigate the quark mass dependence of the ρ -mesonpole parameters, such as its mass, width and couplingsto the ππ and K ¯ K channels. The analysis of theseproperties requires the determination of the Low EnergyConstants (LECs) involved in the pseudoscalar mesonmasses, decay constants and meson-meson scattering. Achiral trajectory specifies the way in which the light- andstrange-quark masses vary. In UChPT, the behavior ofresonance properties over chiral trajectories is controlledby chiral symmetry and unitarity. It is desired todetermine LECs which provide a full description of theresonance properties on chiral trajectories where m s and/or m u,d vary. These kind of predictions of an EFTcan be tested by lattice QCD simulations.LatticeQCD (LQCD) is the only known tool to extractnon-perturbative information from QCD. It is the instru-ment to determine the low-energy parameters of the chi-ral Lagrangian that govern the quark mass dependenceof resonance properties, hence, rendering evidence of theEFT predictions. In recent simulations, lattice data on I = J = 1 ππ scattering have been extracted for severalpion masses for two light flavors ( N f = 2) [53–59] andincluding also the strange quark ( N f = 2 + 1) [60–67].See also [68, 69] for recent N f = 2 + 1 + 1 simulations.Surprisingly, results for the ρ mass in N f = 2 simulationsare at odds with experimental predictions. Namely, the N f = 2 simulation with the lightest pion mass m π (cid:39) ρ -mesonmass around 60 MeV below the physical value [58]. Other N f = 2 simulations also show disagreement with the clos-est pion-mass result for N f = 2 + 1. For example, the N f = 2 GWU simulation at m π (cid:39)
226 MeV [59] gives a ρ -meson mass around 45 MeV lighter than the N f = 2+1Hadron Spectrum (HadSpec) outcome of the simulation for m π (cid:39)
236 MeV [60]. It has been argued in recentanalyses [31, 59] with the UChPT model of [46], thatthis difference can be explained through the effect of K ¯ K loops in the ππ → K ¯ K → ππ reaction, where the kaonis off-shell. This effect has been shown to be consistentamong the N f = 2 simulations [31]. Moreover, while theerror ellipses of N f = 2 lattice data analyses do overlap,hence showing consistency among the simulations, thesame cannot be stated for the N f = 2 + 1 results, whereone finds inconsistencies among lattice simulations [31].The light-quark mass dependence on decay constantshas also been studied in LQCD simulations in [70–77].However, almost no attention has been paid in the past totheir strange–quark-mass dependence, nor of the ρ (770)phase shift. Most of these simulations have been per-formed with a strange-quark mass kept fixed at the phys-ical point, m s = m s . A reflection of this can be foundin the Flag Review [78]; the averaged LEC values fromdifferent fits to decay constants with ChPT do not repre-sent a global analysis of data and they do not track othertrajectories rather than those with roughly m s = m s . Inaddition, these LECs do not describe the meson-mesoninteraction at the energies where the ρ meson begins toresonate. The only exception up to recently was a sim-ulation of the pion decay constant done by MILC overthe m s = 0 . m s trajectory [72]. Thus, in spite of thegreat advance of lattice simulations, data out of the chi-ral trajectory m s = m s are still scarce, even though theresponse of hadron properties to different chiral trajecto-ries could elucidate their strangeness nature, in particu-lar, and dynamical nature, in general. A larger amount ofhighly precise data on a variety of chiral trajectories arenecessary to shed light on the composition of hadrons.Recently, the CLS Collaboration generated ensembleson different chiral trajectories with Tr M = C , (whereTr M = m u + m d + m s and C denotes a constant), inlarge volumes [70, 79]. Moreover, this constant variesa little with the inverse gauge coupling, β , of the sim-ulation, which characterizes the set of ensembles gener-ated. These trajectories are of particular interest sincethe hadron response along the trajectory will manifest asa consequence of both, variations in the light and strangequarks. These recent lattice simulations motivate thepresent analysis by investigating them in combinationwith the simulations over m s = m s trajectories. Thisprovides a good ground to study the strange-quark massdependence of decay constants and the ρ (770) phase shift,which we intend to do here. Of course, new LQCD sim-ulations over trajectories with larger variations on theseconstants or for different values of the strange-quark massin the m s = k trajectories would improve the analysispresented here.The study of hadron properties ( ρ meson) we con-duct here needs to emphasize the role of pseudoscalar The supper-script “0” stands for physical point from now on. decay constants, which are strongly connected to thecoupling of vector mesons to pions. This is supportedby the assumption of dominance of vector mesons in thepion-photon coupling, the so-called Vector Meson Dom-inance (VMD) [80], which connects the size of the piondecay constant ( f π ) and the ρ → ππ coupling ( g ρππ ) inthe EFT [81]. In this context, the large experimentallyobserved decay width of the ρ (770) meson is directlyconnected to its coupling to two pions, which explainswhy the ρ -meson phase shift and ρ -meson properties aretightly related to the size of f π . In this sense, thesetwo observables should always be determined togetherin lattice simulations. Beyond that, the quark mass de-pendence of pseudoscalar decay constants fixes the chiraltrajectories in the lattice and hence, they can be used toset the lattice scale by letting them go to the physicalpoint.The analysis we perform here will be useful to fur-ther check the KSFR relation [82], which under VMDstates that g ρππ = m ρ / √ f π in the SU(3) limit where m u = m d = m s . While, it is common that in previouslattice/experimental data analyses of ρ -meson phase shiftthe ρ -meson mass increases monotonically with m π , sothat the KSFR relation is fulfilled [59, 83, 84], this behav-ior was not observed in the recent data of [79]. Whetherthis is a consequence of the lightness of the strange-quarkmass used in these simulations or not will also be checkedin the present analysis.In conclusion, we analyze here the lattice data on ρ -meson phase shifts in N f = 2 + 1 of [60–62, 79], in com-bination with decay constant lattice data from [70–74].Moreover, the lattice data of [63, 64, 75, 76] are also con-sidered in separated analyses.Let us make some previous remarks. Experimentalphase-shift data on ππ → ππ scattering in the I = J =1 channel were successfully reproduced using the IAM[83, 85]. Here, the LECs are extracted by performinga fit to lattice phase-shift data instead, taking into ac-count the covariance matrix for energy levels, similarlyas in [84]. The main differences with the work of [84] are:1. A global fit to lattice data on two distinct chiral tra-jectories, Tr M = C and m s = k , is done instead ofconsidering trajectories only over m s = m s simula-tions. As mentioned previously, this includes datafrom [70, 79] for Tr M = C and from [60–62, 71–74]for m s = k .2. We perform a simultaneous fit of phase shift anddecay constant lattice data. Note that in [84], onlyphase-shift data were analyzed, while the quarkmass behavior of pseudoscalar decay constants wasfixed with the LECs obtained in the fit done in [83],which only included lattice data on m s = m s . Here, k = m s or 0 . m s , where only data on f π are included inthe latter [72].
3. The theoretical framework used here is the IAMin coupled channels [85] instead of the simplifiedUChPT model considered in [46]. This is, we in-clude here one-loop diagrams not just in the s chan-nel, but also in the t and u channels, hence, consis-tently with chiral symmetry at low energies.4. The systematic error in the lattice spacing is takeninto account by using the bootstrap method, as-suming that the lattice spacing is normally dis-tributed with the standard deviation associated tothe lattice error.Although in the present analysis we only include N f =2 + 1 lattice data, this work can be considered as com-plementary to the previous N f = 2 and N f = 2 + 1analyses done in [31, 59] and [84], respectively. If thestrange-quark mass has no effect on the ρ -meson prop-erties extracted from the lattice simulations, then, thedisagreement among N f = 2 and N f = 2 + 1 lattice re-sults will be due to the scale setting or other finite volumeeffects, such as the lattice spacing or the box size. Thus,we study in detail in which particular quark-mass regimethe ρ -meson properties in the simulation are sensitive toboth, the strange- and light u -, d -quark masses.This paper is organized as follows. In section II weexplain the formalism considered. In section III, we showthe results of a global analysis on decay constants overseveral chiral trajectories. Section IV provides the resultsof the combined fit both to phase shift and decay constantlattice data. In particular, we first analyze in IV A latticedata over m s = k trajectories, while the same analyses forthe Tr M = C data are shown in section IV B. Followingthis, we present our final results on a global fit on bothtrajectories in section IV C. Finally, the main conclusionsare presented in section V. II. THEORETICAL FRAMEWORKA. Chiral Perturbation Theory
At low energies QCD interactions become non-perturbative and EFTs provide the proper frameworkto perform systematic calculations. The basic premiseof EFTs is that the dynamics at low energies (or largedistances) do not depend on the details of the dynamicsat high energies (or short distances). As a result, low-energy hadron physics can be described using an effec-tive Lagrangian containing only a few degrees of freedom,hence, ignoring those present at higher energy scales.Chiral perturbation theory is the low-energy EFT ofQCD. It is built as the most general expansion in termsof derivatives and quark masses [44, 45] compatible withQCD symmetries, which relevant degrees of freedom atlow energies are the pseudo Nambu–Goldstone bosons(NGB) of the chiral symmetry spontaneous breakdown,i.e., pion, kaon and eta mesons.At leading order (LO) in this expansion, the chiral La-grangian reads L = f (cid:10) ∂ µ U ( x ) † ∂ µ U ( x ) + χ † U ( x ) + χU ( x ) † (cid:11) , (1)where f coincides with the pion decay constant inthe chiral limit and χ = 2 B M , with B a constantto be related with the quark condensate and M =diag ( m ud , m ud , m s ) is the three-flavor quark-mass ma-trix, where exact isospin symmetry m ud = m u + m d isassumed. The matrix U ( x ) = exp( i √ φ ( x ) f ) collects thecontribution of pions, kaons and etas, with φ ( x ) = π ( x ) √ + √ η ( x ) π + ( x ) K + ( x ) π − ( x ) − π ( x ) √ + √ η ( x ) K ( x ) K − ( x ) ¯ K ( x ) − √ η ( x ) . By expanding the LO chiral Lagrangian in powers of f , one can identify the mass field terms obtained withthe pseudo NGB fields, which yields a relation betweenmeson and quark masses M π =2 m ud B ,M K =( m ud + m s ) B ,M η = 23 ( m ud + 2 m s ) B . (2) The constant B is related with the quark condensatevalue in the chiral limit,Σ = −(cid:104) | ¯ qq | (cid:105) = B f , (3)with q ∈ { u, d, s } , leading to the well known Gell–Mann–Oakes-Renner formula 2 m ud Σ = M π f [86], i.e., eventhough both m ud and Σ are scale dependent quantities,and hence, they are not observables, their product is scaleindependent.At higher orders, all terms in the Lagrangian comemultiplied by LECs, which contain information abouthigher energy scales. In addition, they absorb the di-vergences which appear in the chiral expansion, so that,the theory is renormalizable order by order. Unfortu-nately, the LECs cannot be determined perturbativelyfrom QCD. While the LECs which multiply energy-dependent terms can be extracted quite well from disper-sion theory [87–90], Lattice QCD provides in principle amodel independent way to determine the values of LECswhich fix the quark mass dependence [78, 91].The NLO Lagrangian was first derived in [44] for twoflavors. The effect of the strange quark was studiedin [45]. Omitting field tensor and vacuum terms, theSU(3) NLO ChPT Lagrangian reads L = L (cid:10) D µ U † D µ U (cid:11) + L (cid:10) D µ U † D ν U (cid:11) (cid:10) D µ U † D ν U (cid:11) + L (cid:10) D µ U † D µ D ν U † D ν (cid:11) + L (cid:10) D µ U † D µ U (cid:11) (cid:10) χ † U + χU † (cid:11) + L (cid:10) D µ U † D µ U ( χ † U + U † χ ) (cid:11) + L (cid:10) χ † U + χU † (cid:11) + L (cid:10) χ † U − χU † (cid:11) + L (cid:10) χ † U χ † U + χU † χU † (cid:11) . (4)In Eq. (4), L , L and L multiply massless terms andhence they also contribute in the chiral limit. L and L accompany terms depending linearly on the quarkmasses and they contribute to the renormalization of theNGB wave functions and decay constants. Lastly, L , L and L come together with quadratic terms on thequark mass. These only contribute to the renormaliza-tion of the NGB masses and have a minor role in thedetermination of the ρ (770) meson properties.One-loop correction to the pion, kaon and eta NGBmasses read [45] m π = M π (cid:20) µ π − µ η M K f (2 L r − L r ) (5)+ 8 M π f (2 L r + 2 L r − L r − L r ) (cid:21) , m K = M K (cid:20) µ η M π f (2 L r − L r )+ 8 M K f (4 L r + 2 L r − L r − L r ) (cid:21) , (6) m η = M η (cid:34) µ K − µ η + 8 M η f (2 L r − L r )+ 8 f (2 M K + M π )(2 L r − L r ) (cid:21) + M π (cid:20) − µ π + 23 µ K + 13 µ η (cid:21) +1289 f ( M K − M π ) (3 L + L r ) , (7)with µ P = M P π f log M P µ , P = π, K, η . (8)The superscript r denotes renormalized LECs, whichcarry the dependence on the regularization scale µ [45].This scale dependence cancels exactly in the calculationof any observable. In the following, we will identify thephysical NGB masses with the one-loop ChPT predictionabove. Nevertheless, note that the quark mass depen-dence is always expressed in terms of the leading orderNGB masses.In addition, while at LO the NGB decay constant f is independent of the quark mass, one-loop corrections inthe pseudoscalar decay constants lead to f π = f (cid:20) − µ π − µ K + 4 M π f ( L r + L r ) + 8 M K f L r (cid:21) , (9) f K = f (cid:20) − µ π − µ K − µ η M π f L r + 4 M K f (2 L r + L r ) (cid:21) , (10) f η = f (cid:34) − µ K + 4 L r f (cid:0) M π + 2 M K (cid:1) + 4 M η f L r (cid:35) , (11)which are also identified with the physical quantities. B. Meson-meson scattering in ChPT
The scattering of NGB meson is computed in ChPT asan expansion in momenta and meson masses. Denotingas A I ( s, t, u ) the scattering amplitude of the NGB pro-cess a → b with defined isospin I , one has the genericform A I ( s, t, u ) = A I ( s, t, u ) + A I ( s, t, u ) + . . . , (12)where s , t and u are the usual Mandelstam variables and A Ik = O ( p k ), where p means either meson momenta ormasses. The LO amplitude A is obtained at tree levelfrom the L Lagrangian. The NLO contribution containsone-loop diagrams from L plus tree-level contributionfrom L involving LECs.The ππ → ππ scattering amplitude at one-loop orderin ChPT was computed first in [44] in a two-flavor for-malism and in [45] for three flavors. The πK → πK and πη → πη scattering amplitudes were evaluatedin [92, 93] and [94], respectively. The one-loop expres-sions for the SU(3) pseudo NGB reactions used here canbe found in [85]. The SU(2) and SU(3) two-loop ππ scat-tering amplitudes were obtained in [95, 96] and [97],respectively. The two-loop πK → πK amplitude was de-termined in [98]. Recently, first three-loop calculationshave been explored in [99].Using the normalization conventions given in [100,101], the s -channel partial-wave projection of the am- FIG. 1: Generic one-loop diagrams entering in meson-mesonscattering. Top diagrams correspond to tadpoles, while bot-tom diagrams represent loops in the s , t , and u channels. plitude is defined as t IJ ( s ) = 132 πN (cid:90) − dx P J ( x ) A I ( s, t ( s, x ) , u ( s, x )) , (13)where N is a normalization factor equal to 2 if all theparticles are identical and 1 otherwise. The Mandelstamvariables t ( s, x ) and u ( s, x ) are defined by the kinematicsof the corresponding a → b process and x = cos θ , being θ the scattering angle in the center-of-mass frame.Being an expansion in momenta and masses, it is clearthat ChPT cannot satisfy unitarity, which in the elasticcase implies the relationIm t IJ ( s ) = σ ( s ) | t IJ ( s ) | ⇒ | t IJ | < /σ ( s ) , (14)where σ ( s ) = 2 q ( s ) /s and q is the momentum in thecenter-of-mass frame. In the following, we only considerthe I = J = 1 channel and the superscript index IJ will be suppressed to ease the notation. Nevertheless,ChPT satisfies elastic unitarity perturbatively. For in-stance, defining as t ( s ) = t ( s ) + t ( s ) + · · · , (15)the chiral series of the I = J = 1 ππ partial-wave am-plitude, with t ( s ) and t ( s ) the tree-level and one-loopChPT partial-wave amplitudes, in the elastic case onefinds the relationsIm t ( s ) = 0 , Im t ( s ) = σ ( s ) | t ( s ) | , · · · , (16)which implies that the unitarity bound in Eq. (14) isincreasingly violated in ChPT at larger energy values.In practice, it implies that the chiral series is limitedto scattering momenta around 200 MeV above thresh-old. Furthermore, the ChPT series does not convergeequally well in all parts of the low-energy region. This isparticularly evident in the scalar-isoscalar channel wherestrong pion-pion rescattering effects slow the convergenceof the chiral series [102]. Finally, at increasingly largemomenta, several partial-waves become resonant. Res-onances are non-perturbative effects and, as such, theycannot be reproduced within the ChPT power expansion.Furthermore, they usually saturate the unitarity boundin Eq. (14), which implies that elastic unitarity can beviolated in the resonance region. C. Unitarity and analyticity
Below the first production threshold, located at s =16 m π , ππ scattering is purely elastic and, consequently,it can be described in terms of its phase shift. Above thisenergy, there are possible intermediate processes such as2 π → π n , with n = 2 , , . . . or ππ → ¯ KK, ηη , which,in principle, have to be taken into account. In our caseof interest, the P -wave ππ -scattering partial wave, inelas-ticities are completely negligible below the K ¯ K thresholdand very small below 1.4 GeV [6, 8, 103–107]. Thus, inthis work elastic scattering is assumed to occur below the K ¯ K threshold and above only the ππ and K ¯ K channelsare considered.The unitarity condition for the S -matrix, SS † = ,implies that, for two-coupled channels, it can be param-eterized in terms of only three independent parameters.It is customary to choose them as the ππ → ππ and K ¯ K → K ¯ K phase shifts, denoted as as δ and δ , re-spectively, and the inelasticity η . Thus, the S-matrix isexpressed as S = (cid:18) η e i δ i (cid:112) − η e i ( δ + δ ) i (cid:112) − η e i ( δ + δ ) η e i δ (cid:19) . (17)The T -matrix elements t ij of the scattering amplitudeare related to S -matrix elements as, S ij = δ ij + 2 i √ σ i σ j t ij (18)with σ i = (cid:26) (cid:112) − m i /s √ s > m i i, j = 1 ,
2. The relation between the S - and T -matrix, Eq. (18), allows one to derive the following uni-tarity condition for the T -matrix elementsIm t = σ | t | + σ | t | , Im t = σ t t ∗ + σ t t ∗ , Im t = σ | t | + σ | t | , (20)or Im T = T Σ T ∗ , (21)in matrix form, being T = (cid:18) t t t t (cid:19) , Σ = (cid:18) σ σ (cid:19) . (22) Eq. (21) implies the coupled-channel unitarity relationIm T − = − Σ (23)is fulfilled. The phase space definition, Eq. (19), ensuresthat in the elastic case, i.e., below the K ¯ K threshold,elastic unitarity is satisfied. In the one channel case, Eq.(23) simplifies toIm 1 /t ( s ) = − σ ( s ) . (24)The unitarity conditions in Eqs. (23) and (24) im-ply that the inverse of the imaginary part of an scat-tering amplitude in the physical region is completelyfixed by unitarity. The strong relation between unitar-ity and resonances has motivated the development ofseveral ChPT inspired methods based on imposing ex-act unitarity. Some of them are the so-called K -matrixmethod [108] and the chiral unitarity approach. The lat-ter was considered first in [46, 109] to describe ππ and K ¯ K scattering in the scalar-isoscalar channel, leading tofairly precise determinations of the f (500) and f (980)resonance properties. There are also more involved unita-rization methods. For example, the Bethe-Salpeter (BS)equations were solved for ππ scattering in Refs. [51, 52],both in the on-shell and off-shell schemes, while the N/Dmethod was employed in Ref. [110] providing also resultsfor the rest of lightest scalars, namely the κ (700) and a (980). However, none of them generates the ρ (770)pole in the ππ scattering P wave.The energy-dependence of an scattering amplitude isalso strongly constrained by analyticity. Analyticity isbased on the Mandelstam hypothesis [111], i.e., the as-sumption that an scattering amplitude is represented bya complex function that presents no further singularitiesthan those required by general principles such as uni-tarity and crossing symmetry. In this way, poles in thereal axis are associated with bound states (absent in low-energy meson-meson scattering) and production thresh-olds give rise to cuts. Cuts are a consequence of the uni-tarity condition given in Eq. (21), which, together withthe Schwartz-reflection principle, imply that an scatter-ing amplitude must have a cut where unitarity demandsits imaginary part to be non-zero. It occurs due to both,direct and crossed channels, leading to a right- (RHC)and left-hand cut (LHC), respectively.Once analyticity is established, Cauchy’s integral for-mula allows one to construct a representation that re-lates the amplitude at an arbitrary point in the complexplane to an integral over its imaginary part along theright- and left-hand cuts, the so called dispersion rela-tions. The convergence of the dispersive integral often re-quires subtractions, which introduce a certain number ofa priori undetermined constants. The Froissart–Martinbound [112, 113] guarantees that at most two subtrac-tions are needed to ensure the convergence at infinity, butone subtraction is enough for the ππ scattering amplitudein the vector-isovector channel. Thus, a once-subtracteddispersion relation for I = J = 1 ππ scattering reads t ( s ) = t (0) + sπ ∞ (cid:90) m π d s (cid:48) Im t ( s (cid:48) ) s (cid:48) ( s (cid:48) − s − i (cid:15) )+ sπ (cid:90) −∞ d s (cid:48) Im t ( s (cid:48) ) s (cid:48) ( s (cid:48) − s − i (cid:15) ) , (25)where the first and second integrals stand for the RHCand LHC contributions, respectively. The subtractionconstants involve the evaluation of the amplitude at s = 0, so that, they can be pinned down by matchingto ChPT in the regime where the chiral expansion is ex-pected to show better convergence properties. However,while the value of Im t ( s ) in the physical RHC is con-strained from unitarity, the LHC contribution is in prin-ciple unknown. On the one hand, most UChPT meth-ods differ in the way the LHC is treated. While the K -matrix and chiral unitarity approach models simplyneglect the LHC contribution, the BS and N/D methodsapproximate it with ChPT. On the other hand, Roy–Steiner equations [114, 115] solve this problem exactlyusing crossing symmetry. They provide a representa-tion involving only the physical region, but which, atthe same time, intertwines all partial-waves with differ-ent isospin and angular momentum. Although, Roy–Steiner-equation solutions allow for high-precision de-scriptions of different scattering processes at low ener-gies [4–6, 116, 117], and provide the proper frameworkto extract resonance pole parameters [7, 118–122], or toevaluate an scattering amplitude in an unphysical re-gion [123–125], their analysis requires experimental in-formation for the high-energy contribution and higherpartial waves. Thus, they are in principle inappropriatefor the analysis of lattice data at different quark masses.In this article, we follow the IAM, which will be outlinedin the next section II D. D. Elastic Inverse Amplitude Method
The Inverse Amplitude Method exploits the relationbetween a dispersion relation for the inverse of an scat-tering amplitude and the ChPT amplitude at a givenorder. At NLO in the chiral expansion, taking into ac-count that ChPT amplitudes grow as s when s → ∞ ,one needs three subtractions to ensure the convergenceat high energies. Thus, a thrice-subtracted dispersionrelation for a elastic ChPT ππ -scattering partial wave reads t ( s ) = t (0) + t (cid:48) (0) s,t ( s ) = t (0) + t (cid:48) (0) s + t (cid:48)(cid:48) (0) s + s π ∞ (cid:90) m π ds (cid:48) σ ( s ) t ( s ) s (cid:48) ( s (cid:48) − s − i(cid:15) )+ s π (cid:90) −∞ ds (cid:48) Im t ( s ) s (cid:48) ( s (cid:48) − s − i(cid:15) ) , (26)where we have used Eq. (16) to fix the absorptive part of t ( s ) in the physical region. Note that Eq. (26) is stronglyrelated to a thrice subtracted dispersion relation for thefunction g ( s ) = t ( s ) /t ( s ), g ( s ) = g (0) + g (cid:48) (0) s + g (cid:48)(cid:48) (0) s − s π ∞ (cid:90) m π ds (cid:48) σ ( s ) t ( s ) s (cid:48) ( s (cid:48) − s − i(cid:15) )+ s π (cid:90) −∞ ds (cid:48) Im g ( s ) s (cid:48) ( s (cid:48) − s − i(cid:15) ) , (27)so that in an elastic approximation the RHC contributioncoincides exactly with that of − t ( s ). The subtractionconstants require the evaluation of the scattering ampli-tude and its derivatives at s = 0, the kinematic regionwhere ChPT provides a reliable description. Thus, usingChPT at NLO one gets g (0) (cid:39) t (0) − t (0) ,g (cid:48) (0) (cid:39) t (cid:48) (0) − t (cid:48) (0) ,g (cid:48)(cid:48) (0) (cid:39) − t (cid:48)(cid:48) (0) . (28)Being the RHC exactly fixed from unitarity, and oncethe subtraction constants are estimated using ChPT, theonly remaining unknown information in Eq. (27) is theLHC. The left-hand cut might indeed play a relevant rolebelow threshold, but it is expected that its contributionshould be less important as one moves into the physicalregion. Thus, for a qualitative description it is sufficientto approximate the left-hand cut using ChPT. At NLO,one findsIm g ( s ) (cid:39) t ( s ) Im 1 t ( s ) + t ( s ) (cid:39) − Im t ( s ) . (29)Inserting Eqs. (28) and (29) in Eq. 27 one obtains t ( s ) IAM = t ( s ) t ( s ) − t ( s ) , (30)which stands for the well-known equation of the IAMmethod. The IAM was derived first in [47, 48] using onlyunitarity for ππ scattering. Its derivation from a disper-sion relation and application thereafter to πK scatteringwas investigated in [49, 50], whereas the remaining IAMmeson-meson scattering processes were studied in [85] toone loop. The two-loop version of the IAM was derivedin [126] and its generalization to include the effect ofAdler zeros was obtained in [127].The IAM provides a simple algebraic equation thatensures elastic unitarity while at low energies reproducesthe chiral expansion. This fact implies that the IAM canbe used to describe the resonance region below 1 GeV,i.e., well beyond the applicability range of ChPT. Fur-thermore, it is based on a dispersion relation, hence, itsuse in the complex plane is justified, providing a sim-ple tool to study resonance properties. The main dif-ference between the IAM and the on-shell BS or N/Dmethod, is that, in the IAM only the absorptive part ofthe left-hand contribution is expanded at low energies. Itimplies that the left-hand cut energy dependence is stillcontrolled by a dispersion relation instead of being fullygiven by ChPT. In addition, the IAM generates not onlyscalar but also vector resonances [46], without involvingnew additional parameters rather than the ChPT LECs.Hence, it reproduces at low energies the quark mass de-pendence predicted by ChPT.Nevertheless, it has also several caveats. While theRHC is solved exactly using elastic unitarity, the LHCis approximated using ChPT. The direct consequence ofthis fact is that the IAM breaks crossing symmetry. Be-sides, while the IAM provides higher order ChPT con-tributions needed to fulfill unitarity, some of the leadingorder logarithms from higher-order loop graphs appearwith the wrong coefficients [128].In addition, it is worth mentioning that the IAM de-scribes experimental data, including resonance pole pa-rameters, of meson-meson scattering in the region below1 GeV only within a 10%-15% accuracy [85, 129]. Thissmall difference highlights the relevance of the LHC inthe physical region below 1 GeV.Clearly, leaving the LECs as free parameters to be ad-justed to data instead of being fixed to the ChPT valuesimproves the description of the experimental data. In-deed, ππ and πK scattering experimental data were de-scribed in [83, 85] using the IAM with LEC values com-patible with pure ChPT determinations. Small LECschanges are indeed expected since the IAM includes con-tributions that go beyond the pure chiral expansion ata given order. However, it is important to remark thatwhile ChPT is a natural theory in the sense that its pre-dictions are linear in LECs changes, the IAM as well asother UChPT models are strongly dependent on preciseLECs determinations. Small changes on the LEC valuesmight produce large effects on the phase-shift and poleparameter predictions.Finally, let us remark that the dispersive derivationof the IAM only constrains its energy dependence, andhence, it is not clear whether it provides the correctquark-mass dependence. While the IAM reproduces theChPT series at low energies, thus, ensuring that it pro-vides the quark-mass dependence predicted from QCDclose to the chiral limit, it also introduces higher-ordercontributions that spoil the chiral series at higher ener-gies and for heavier quark masses. Thus, high quality lattice data for different light- and strange-quark massesare key to ensure that the chiral extrapolation performedwithin the IAM is well consistent with QCD. E. Coupled channel formalism
The generalization of the inverse amplitude method tocoupled channels should be in principle straightforwardif one assumes the factorization of the RHC and LHCcontribution for the different channels involved. In thiscase, we can define the matrix version of the function g ( s )in Eq. (27) as G ( s ) = T ( s ) T ( s ) − T ( s ), where T k standsfor the O ( p k ) I = J = 1 ChPT matrix (see Eq. (22)).Similarly as in Eq. (27), a thrice-subtracted dispersionrelation for G ( s ) reads G ( s ) = G (0) + G (cid:48) (0) s + G (cid:48)(cid:48) (0) s − s π ∞ (cid:90) s th ds (cid:48) T ( s )Σ( s ) T ( s ) s (cid:48) ( s (cid:48) − s − i(cid:15) )+ s π s L (cid:90) −∞ ds (cid:48) Im G ( s ) s (cid:48) ( s (cid:48) − s − i(cid:15) ) , (31)where s th and s L stand for the corresponding right- andleft-hand cut branching points, respectively. The numer-ator of the RHC contribution corresponds to the matrixversion of Eq. (16), i.e.,Im T ( s ) = T ( s )Σ( s ) T ( s ) , (32)and hence, the right-hand cut of G(s) coincides with thatof the matrix − T ( s ). The subtraction constants can beevaluated using ChPT. By means of expanding T − as T − (cid:39) ( T + T + · · · ) − (cid:39) T − (cid:0) − T T − + · · · (cid:1) , (33)one recovers the equivalent version of Eq. (28) in ma-trix form. However, the problem now is the evalua-tion of the left-hand cut. Although the RHC branch-ing point s th = 4 m π is common for all the elementsof the T-matrix, the LHCs of the various channels dodiffer. Namely, while the ππ scattering LHC starts at s = 0, the LHC for the K ¯ K → K ¯ K partial wave opensat s = 4 m K − m π . In this way, proceeding as we did forthe elastic IAM, i.e., taking the perturbative expansionin Eq. (33) for the absorptive part of G ( s ) along the LHC,one is indeed mixing the LHCs of all T-matrix elements.This translates into a violation of the factorization hy-pothesis, which produces spurious left-hand cuts break-ing unitarity [41, 85, 130–132]. As a summary, the anal-ogous of Eq. (30) cannot be derived in coupled-channelsusing a dispersion relation.Alternatively, one can still exploit unitarity in order toderive a coupled channel version of Eq. (30) valid in thereal axis. Taking into account Eq. (23), Im T − = − Σ,one can write T = (cid:2) Re T − − i Σ (cid:3) − . (34)Now, Re T − can be approximated once more withChPT. Using Eq. (33) one gets T (cid:39) T [ T − Re T − iT Σ T ] − T = T [ T − T ] − T (35)which provides the IAM coupled channel unitarizationformula. Note that to derive Eq. (35) we have usedEq. (32). Nevertheless, it is important to note thatEq. (35) is only justified in the real axis where the ChPTcoupled channel unitarity relation (32) is fulfilled.At this point, it is also important to discuss at whichenergy the couple-channel formalism should be taken intoaccount. Given the phase-space definition in Eqs. (19)and (22), the unitarity relation in Eq. (21) acquires di-mension two only when one crosses the K ¯ K productionthreshold. Thus, Eq. (35) should be used only above the K ¯ K threshold, i.e., when its dimension coincides withthe number of states accessible and the coupled-channelunitarity relation in Eq. (23) is fulfilled. Below this en-ergy one should consider the one-dimensional IAM equa-tion. Thus, this procedure yields a discontinuity at 4 m K ,instead of a single continuous function. Alternatively,one can include the K ¯ K channel for all energies. Thisprovides a continuous function but it again introducesspurious left-hand cuts, leading to a violation of unitar-ity. Nevertheless, these violations are in general small,around 2%-5% [85, 132]. In this paper we consider thesecond approach for Eq. (35), but in order to reduce theeffect of spurious cuts, we introduce an extra term in the χ of our fit to lattice data, which penalizes unitarityviolations of the S-matrix by some factor, as explainedin Sect. II H.Eq. (35) was used in [85] to study all possible am-plitudes for meson-meson scattering leading to a fairlygood description of all available experimental data be-low 1 . t - and u -channel loop functions (Fig. 1) in the ππ → ¯ KK and¯ KK → ¯ KK ChPT amplitudes that generate the spuri-ous cuts, the mass and width of the ρ -meson obtained inthe global Fit IV (see Sect. IV C) change less than 1 and6 MeV, respectively, i.e., within the uncertainties quoted.Nevertheless, the effect of the t and u channels in the ππ amplitude lead to a shift of 6 and 15 MeV for the massand width of the ρ -meson in Fit IV, respectively (withoutreadjusting the LECs).To conclude, Eq. (35) is the tool we use to analyzelattice scattering data in the ρ (770) channel. The explicitexpressions for the elements of the T and T for ππ → ππ , ππ → K ¯ K and K ¯ K → K ¯ K are given in the appendixof [85]. F. Resonances
Resonances are formally defined as poles lying on un-physical Riemann sheets. An unphysical Riemann sheetis reached when the physical right-hand cut is crossedcontinuously from the upper-half plane to the lower-halfplane above a given production threshold. In the elasticscattering case, there are only two Riemann sheets, thephysical and unphysical one, which are called, first andsecond sheet, respectively. These two Riemann sheetsmust coincide in the real axis, S I ( s + i(cid:15) ) = S II ( s − i(cid:15) ) . (36)In addition, the scattering amplitude on the firstRiemann-sheet satisfies the Schwartz reflection principle,i.e., S ( s + i(cid:15) ) = S ∗ ( s − i(cid:15) ), which together with unitarity, SS ∗ = , yields the relation S II ( s − i(cid:15) ) = S I ( s − i(cid:15) ) − . (37)The analytic continuation of Eq. (37) into the complexplane implies that a pole on the second Riemann sheetcorresponds to a zero in the physical one. By means ofEq. (18) one can translate this relation to the T -matrixelements, leading to t II ( s ) = t I ( s )1 + 2 i σ ( s ) t I ( s ) . (38)where σ ( s ) = (cid:112) − m /s , and its determination is cho-sen as σ ( s ∗ ) = − σ ( s ) ∗ , to ensure the Schwartz reflectionsymmetry.When further channels are opened, more unphysicalRiemann sheets can be defined by continuing the squaremomenta of the intermediate states over the differentthresholds. Thus, there are 2 n Riemann-sheets for agiven number n of opened channels. The generalizationof Eq. (38) in a coupled-channel formalism is straightfor-ward T ( n ) ( s ) = T ( s ) (cid:16) + 2 i Σ( s ) ( n ) T ( s ) (cid:17) − , (39)where Σ ( n ) is a diagonal matrix containing the phasespace factors of those channels that have been crossedcontinuously. In particular, for the ππ and K ¯ K I = J =1 coupled-channel case, we will have four different Rie-mann sheets defined asΣ II = (cid:18) σ π
00 0 (cid:19) , Σ III = (cid:18) σ π σ K (cid:19) , Σ IV = (cid:18) σ K (cid:19) , (40)where σ π = (cid:112) − m π /s and σ K = (cid:112) − m K are thephase space factors of the ππ an K ¯ K channels, respec-tively.Therefore, a pole in the T matrix corresponds to a zeroof the determinant of the matrix inside the brackets of0Eq. (39), which is denoted by √ s pole = E = ( M − i Γ / g i g j = − π lim s → s pole ( s − s pole ) t ij ( s )(2 J + 1) / (2 p ( s )) J , (41)where p ( s ) stands for the center-of-mass-system momen-tum of the corresponding process. G. Formalism in the finite volume
The L¨uscher’s approach [134, 135] allows one to relatethe measured discrete value of the energy in a finite vol-ume to the scattering phase shift at the same energy inthe continuum. The volume-dependence of the discretespectrum of the lattice QCD gives the energy dependenceof the scattering phase shift. This method, originally de-rived for a single scattering process was soon extendedto coupled channels for potential scattering [136], non-relativistic effective theories [137, 138] and to relativisticscattering [139–142]. Extensions of the L¨uscher formal-ism to three-particle systems under certain conditions arealso available [143–146].The L¨uscher’s approach is based on the analysis ofthe dominant power-law volume dependence that entersthrough the momentum sums in a BS equation, where allquantities are written in terms of non-perturbative cor-relation functions. In order to extract this dependenceone assumes that the BS kernel, which accounts for theLHC and subtraction constant contributions in Eq. (25)and only involves a exponentially suppressed dependenceon the volume [134], coincides for large volumes with itsinfinite-volume form. In this way, the difference betweenfinite- and infinite-volume integrals entering on the BSequations only depends on on-shell values of the two-particle integrand leading to the the quantization condi-tion det (cid:2) i T + F − (cid:3) = 0 , (42)where T is the scattering amplitude in the finite volumeand F is a matrix that contains sums of the generalizedZeta functions subduced into the relevant finite volumelittle groups [139, 140].L¨uscher’s method was subsequently rederived in [147,148] by discretizing the s -channel loop functions whichappear in the IAM coupled-channel equation of Eq. (35)and neglecting the t - and u -channel contributions. Theconsideration of relativistic effects in the L¨uscher formu-lation as in Ref. [149] can lead to a significant differencefor small volumen sizes in the phase shifts extracted. Thediscretization of the t and u channels has been discussedin [150, 151]. In the latter, the exponentially suppressed volume dependence of the LHC contribution was explic-itly taken into account, concluding that the LHC vol-ume dependence is numerically negligible for lattice sizes L > m − π while for lattice volumes m − π < L < m − π ,it only affects noticeably the first energy level. Further-more, note that neglecting the volume dependence of theLHC contribution in the finite volume is by no meansequivalent to ignoring the LHC in the continuum; lat-tice energy levels are non-perturbative quantities and, assuch, they include all physical effects, both from the RHCand LHC contributions. The same cannot be stated forthe dispersive formalism defined in Sect. II D and II Esince one explicitly factorizes the RHC and LHC contri-butions. However, to extract information from the en-ergy levels and connect them with the T-matrix in thecontinuum one does need a generalized L¨uscher methodincluding all physical effects, which might become partic-ularly difficult, for example, in the case of multi-channeland intermediate states of three or more particles.In principle, one could use the formalism in [150, 151]to evaluate the energy levels and fit them to the latticedata. Nevertheless, in order to avoid the discretizationof loops we follow here the method used in [84]. Namely,we fit the phase shift values extracted from the latticeusing L¨uscher’s method, while the eigenenergies are re-constructed by means of a Taylor expansion taking intoaccount the correlation between energy E n and phaseshift δ ( E n ), as well as the covariance matrix of eigenen-ergies provided by the lattice. This method is explainedin the subsection below. H. Fitting procedure
The low energy constants of SU(3) Chiral Perturba-tion Theory to one loop are extracted from fits to latticephase-shift data in the I = J = 1 channel together withpseudoscalar meson decay constants and masses. Thisincludes the N f = 2 + 1 phase-shift data of [60–64, 79]together with data from [70–76] for decay constants.We analyze lattice simulations on two different chiraltrajectories, where either the sum of the three-lightestquarks or the strange-quark mass is fixed to the physicalpoint, i.e., Tr M = C or m s = k , respectively. The corre-sponding tree-level pseudoscalar meson masses relationsare m K = − m π + C B , (43)for Tr M = C and m K = + 12 m π + k B , (44)for m s = k , with k = m s or 0 . m s .As a result from a combined analysis of data on thesetwo kind of trajectories, in Sect. IV C we also show pre-dictions for ρ -meson phase shifts, pseudoscalar meson de-cay constants and masses in other trajectories where the1strange-quark mass is fixed to values smaller than thephysical one, m s = k with k < m s , on the SU(3) sym-metric trajectory, m s = m ud , i.e., m k = m π , (45)and for trajectories where the light-quark mass is keptfixed at the physical point m ud = m ud , i.e., m π = m π ,m K = m K, phys + ( m s − m s ) B . (46)We employ one-loop ChPT for the analysis of pseu-doscalar meson masses and decay constants, see Sect. III,in combination with the coupled-channel IAM discussedin Sect. II E for the ρ -meson phase shifts. The fittingparameters are the LECs entering into our expressions,i.e., L i , with i = { , . . . , } , L = 2 L − L , and theparameters which fix the chiral trajectories in Eqs. (43)and (44), C B and k B . The chiral scale µ is fixed to770 MeV and the pion decay constant in the chiral limit f is set to 80 MeV. We fixed f because its inclusionas a new fitting parameter did not entail any substan-tial reduction of the χ . In the following we describe thecontributions to the χ .Meson-meson scattering in the lattice translates intodiscrete energies which are correlated. In order to takeinto account those the following function is minimized, χ E = ( (cid:126)E − (cid:126) E ) T C − ( (cid:126)E − (cid:126) E ) , (47)where (cid:126) E is the vector of eigenenergies measured on thelattice, C their covariance matrix and (cid:126)E the correspond-ing energies of the fit function.Nevertheless, we do not fit directly lattice energy levelsbut phase shifts extracted using the L¨uscher formula. Inorder to take into account the energy correlations, wefollow the method considered in [84]. This is, for eachenergy level , E i , a Taylor expansion of both, the phaseshift extracted from the lattice, δ L , and the one evaluatedin the IAM, δ IAM , is performed around the energy givenby the lattice simulation, E i . If one assumes that both δ L and δ IAM coincide exactly at E i , at leading order, onefinds E i = E i + δ L ( E i ) − δ IAM ( E i ) δ (cid:48) IAM ( E i ) − δ (cid:48) L ( E i ) , (48)which provides a direct way to evaluate χ E in Eq. (47)in terms of phase shift values.Regarding pseudoscalar meson masses and decay con-stants from the lattice, we fit the ratios, h = m K /m π , h = m π /f π , h = m K /f K and h = m K /f π , which are,in principle, more stable against possible discretizationeffects. Thus, we also minimize χ f = (cid:88) ij (cid:0) h pi,j − h li,j (cid:1) ∆ h l i,j , (49) where i denotes the different ratios, j = 1 , · · · , n are themeasurements, and n is the length of lattice data. Thesuperscripts l and p indicate values from lattice simula-tions and predicted by one-loop ChPT, respectively.Finally, as already discussed in section II E, thecoupled-channel version of the IAM generates unphysi-cal LHC contributions arising from the on-shell coupled-channel approximation considered. These contributionsproduce small violations of unitarity, which translate intoundesirable phase shift picks at low energies and in theresonance region, starting below s = 4 m K − m π (thisenergy corresponds to 880 MeV for the HadSpec lighterpion mass). These small picks are enhanced when thereare lattice data around that energy. To eliminate theseunphysical artifacts, a term that minimizes S -matrix uni-tarity violations at a degree controlled by a parameter λ is added to the χ , χ λ = λ (cid:88) ij (cid:90) | ( S S † ) ij − δ ij | ds . (50)In summary, the total χ -like minimization functionreads as χ = χ E + χ f + χ λ . (51)In Fig. 2 we show the value of χ and χ λ in Eq. (50)as a function of λ for the minimization of the HadronSpectrum Collaboration ρ -meson phase-shift data at m π = 236 MeV [60] together with decay constants fromMILC [72]. The LEC values obtained are given in Fig. 3.Clearly, for λ ∼
40 the LECs become stable while χ λ /λ gets significantly reduced. One could also choose a highervalue of λ , however, at the cost of increasing χ . Thus,we set the value of λ to 40 .There is an additional caveat that one should take intoaccount; ChPT is built as an expansion in meson massesand, as such, the chiral series is only expected to con-verge for light pions. In order to study the convergenceradius of ChPT we perform first individual fits of latticedata sets and discard pion mass results for which the fitdoes not pass the Pearson’s χ test at a 90% upper con-fidence limit. This restricts the lattice data sets to pionmasses below around 430 MeV. Results presented in thefollowing sections beyond that pion mass are merely qual-itative. III. CHPT: DECAY CONSTANT ANALYSIS
In this section, we attempt to perform a globalfit of pseudoscalar meson masses and decay constants { m π , m K , f π , f K } from [70–76]. These data are simu-lated on the chiral trajectories m s = m s [71, 73, 74, 76], m s = 0 . m s [72], m s (cid:39) (cid:8) . m s , m s (cid:9) [75] and Tr M = C [70]. The free parameters are the LECs L i , with i = { , . . . , } , which appear in Eqs. (5)-(11), as wellas the variables, CB and kB , which fix the chiral tra-jectories, Tr M = C and m s = k , respectively, accordingto Eqs. (43) and (44).2 FIG. 2: The minimized function χ in Eq. (51) (left) and variation of χ λ /λ with λ (right).FIG. 3: Low energy constants of the IAM for Had-Spec(236)+MILC data as a function of λ . A few aspects need to be considered before. First,the role of the renormalization scheme used in the lat-tice simulations to fix quark masses. Here, we do not ad-just quark masses values but pseudoscalar meson masses,which, in principle, should be independent of the renor-malization scheme. Still, we checked if the pseudoscalarmeson masses in the lattice data sets with different renor-malization schemes are compatible. For example, we no-tice that UKQCD Collaboration uses the MS scheme at3 GeV [71], while the MILC Collaboration uses the samescheme at 2 GeV [72–74]. When we compare both setsof data, we do not observe any substantial inconsistency,but instead, their values do agree quite well.Second, other important issue is the size of the pionmasses used in the simulations. We observe that ingeneral the JL/TWQCD [75] and PACS-CS Collabora-tions [76] have larger pion and kaon masses. For instance,the JL/TWQCD pion and kaon masses are larger than300 and 600 MeV, respectively. These values might betoo large for the perturbative ChPT expansion and in-deed we are not able to fit these data sets in combination with MILC and UKQCD data. Thus, in this fit we onlyinclude data from [70–74]. The JL/TWQCD and PACS-CS data are studied in separated analysis in the nextsection.Third, we should discuss possible finite volume andlattice spacing effects. In [70–74], the dependence of thedecay constant determinations with the lattice spacingwas studied carefully and the results were extrapolatedto the continuum. These extrapolated data are the inputof the fit we show here. Another difficulty that we find tostudy data from [76] (PACS-CS) is the following. In [76],the chiral trajectory is set in such a way that the physicalpoint of the strange quark is determined and later fixedonto the chiral trajectory of the simulation. Thus, the m K dependence on m π in principle should agree withthat from MILC [72–74] and UKQCD [71], since thesesimulations are also performed at the physical strange-quark mass. However, we found substantial discrepanciesin the behavior of the chiral trajectory in [76] with thosefrom MILC and UKQCD, which may be due to finitevolume effects. These discretization effects can be partlyabsorbed by the free parameters. The result from an-alyzing PACS-CS data, pseudoscalar meson masses anddecay constants [76] together with ρ -meson phase-shiftdata [63], is shown in the next section.Lastly, in [70] (CLS collaboration) two different scalesettings are provided, called here scale settings A and B . In the first one, scale setting A , the lattice spacingis determined by fixing the chiral extrapolations of f π and f K to the physical point. The second one, scalesetting B , uses the Wilson flow ( t ) to set the scale byassuming that, for all different ensembles, the data overthe Tr M = C trajectory intersects the m ud = m s sym-metric line at φ = 1 .
15, with φ = 8 t (cid:0) m K + m π (cid:1) .This method requires small corrections in the quarkmasses from the ones used in the simulations [70], whichtranslates into small shifts for the pseudoscalar massesand decay constants. Nevertheless, the CLS ρ -mesonphase-shift data in [79] for the scale setting B were not3shifted accordingly, and hence, these corrections couldlead to a conflict among the CLS decay constant andphase shift data. Then, we take here the no-shiftedvalues, first rows of Table II in [70]. For each ensem-ble β , these two scale settings lead to different latticespacing values a β . Namely, { a . , a . , a . , a . } = { . , . , . , . } for scale setting A and { a . , a . , a . , a . } = { . , . , . , . } for B [70]. Nevertheless, we find that scale setting B pro-duces systematically smaller values of f π than A for thesame pion masses. For instance, we see a difference ofaround 4 MeV in f π for pion masses of around 200 − m π imply varia-tions on f π of around 4 MeV in these data. Because ofthese discrepancies, we are only able to find an optimal χ when data with scale setting A are included. Noticethat this is the method that fixes the scale using the f π and f K physical quantities. In section IV B we analyzethe decay constant data in combination with ρ -mesonphase-shift data for both scale settings and discuss themain differences. Fit I LEC × L − . L . L . L . C ( k ) B × − (MeV )[ a B ] β =3 . b B ] β =3 . c B ] β =3 . k B ] m s However, we show in section IV B that phase-shift lattice datain this scale setting cannot be reproduced globally. The reasonis that the data for the ensembles N200 & N401 produce lower ρ -meson masses than the predictions in the IAM. This problemis tackled in section IV C. In conclusion, it is only possible to do a combined fitof data from [70] ( scale setting A ) and [71–74]. In Ta-bles I and II, the values of the fitting parameters ob-tained from this analysis are presented. This result iscalled Fit I. We notice that the LECs in this fit are notvery sensitive to small variations of the CB and kB pa-rameters, being thus quite stable. Furthermore, they arein line with the compilation of the FLAG Review [78],which only includes results for m s = k data. However,note that we are obtaining much smaller LEC errors com-pared to the FLAG average. Notice also that since thesedata include variations of the strange-quark mass, one isable to fix well the strange-quark mass dependence of thepseudoscalar decay constants for the pion masses studied.The various chiral trajectories studied are shown inFig. 4 (top-left panel), where one can see that the kaonmass squared data for the Tr M = C trajectory [70] differconsiderably from the m s = k ones [71–74]. In addition,two of the ensembles simulated in [70], the ones with β = 3 .
55 and 3 .
7, lead to very similar curves and henceto similar values of CB in Table II. Furthermore, theUKQCD [71], MILC [72, 73] and Laiho [74] lattice dataare in very good agreement. Indeed, ChPT is able to re-produce well the data on these two different trajectories.The ratios m π /f π , m K /f π and m K /f K are also de-picted in Fig. 4. For the ratio m π /f π , it is worth notingthat all data, independently of the chiral trajectory, liealmost on the same curve. This suggests that the ratio m π /f π is indeed quite independent on m s . We discussthis further in Sect. IV C. In fact, all lattice data for thisratio fall into the gray error band plotted, which is just anextrapolation of the percentage error of this ratio at thephysical point determined by MILC [73]. For this collab-oration only m π , m K and f π data are provided, whichare shown with dashed black lines. The m s = 0 . m s trajectory from [72] is denoted by a solid gray line. Thedata of Laiho [74] is represented by dashed-orange lines.UKQCD data are denoted by black squares, while CLSdata [70] are given by dark-green squares ( β = 3 . β = 3 .
55) and yellow pentagons ( β = 3 . f K . In addition, we includein Fig. 4 the chiral prediction for the SU(3) m s = m ud trajectory. Both, m s = 0 . m s and m s = ˆ m trajecto-ries, lead to a substantial reduction of the ratios m K /f π and m K /f K . For pion masses larger than 430 MeV, theChPT prediction begins to differ from the data, whichsuggests the breakdown of the chiral series. This fit passes the Pearson’s test. FIG. 4: Chiral trajectories (top-left) and the ratios m π /f π (top-right), m K /f π (bottom-left), and m K /f K (bottom right)obtained in Fit I over the chiral trajectories m s = k and Tr M = C. The light-brown and orange bands correspond to the errorsof the MILC and Laiho data, in black and orange dashed lines, respectively. IV. IAM: RHO PHASE SHIFTS ANALYSESA. Chiral trajectories m s = k In this section we analyze the ρ -meson phase-shift datafrom [60–64] and pseudoscalar meson masses and decayconstants from [71–75]. All these data are taken fromsimulations over chiral trajectories where the strange-quark mass is kept fixed to the physical value, m s = m s ,except for the JL/TWQCD, where k (cid:39) { . , } m s [75].In fact, the pion and kaon masses used in the simulationsof [75] are larger than in the other simulations. This sim-ulation is studied independently and discussed at the endon in this section.First of all, we perform individual fits to thepseudoscalar masses and decay constant ratios fromUKQCD [71], MILC [72, 73] and Laiho [74] together withthe ρ -meson phase shift data from the HadSpec Collab-oration [60, 61] corresponding to m π = { , } MeV.The LECs obtained in these fits are shown in the second,third and fourth columns of Table III, respectively. Al- though some small differences among the individual fitsare observed for L and L , they provide in general com-patible LEC values within uncertainties. Thus, we con-duct a simultaneous analysis of the UKQCD, MILC andLaiho decay constants and HadSpec phase shifts, whichis denoted as MUL+HS in the fifth column of Table III.As expected, the fit provides a good description of alldata with consistent LECs.Finally, we include the phase-shift results from [62](JB) at m π = 233 MeV. This is denoted as Fit II inthe sixth column of Table III. Notice that this fit encom-passes a large bunch of data on m s = k ( k = { , . } m s ).The LECs obtained in these fits are very similar to theprevious ones suggesting consistency among the differ-ent data sets. Results for ρ phase shifts together withthe fitted lattice data are plotted in Fig. 5. As shownin Fig. 5 (left, top), the extrapolation of Fit II resultsto the physical point (light-blue solid line) is very closeto experimental data, depicted as light-blue squares [103]and orange circles [106].Regarding decay constant ratios and pseudoscalar me-son masses, results from Fit II are very similar to those5obtained in Sect. III over m s = k trajectories, and areshown in Fig. 38 in the Appendix VI B.Unfortunately, we could not obtain additional consis-tency with the lattice data from Refs. [63, 64, 75]. Thus,in the following, we analyze the remaining lattice re-sults separately. The simulation of [64] (CA) for ρ -mesonphase-shift data does not include decay constant deter-minations, thus, we analyze this data with the UKQCDmeson and decay constant values. If other decay constantdata are used instead, as for example, those from MILC,the results are very similar. The resulting LECs, givenin the second column of Table IV, are, in general, com-patible with the values from Fit II, but we find slightlydifferent values for L and L , and larger discrepanciesfor L . These differences have a large impact on thephase-shift values. As shown in the right-top panel inFig. 5, the extrapolation to the physical point providesresults incompatible with experimental data.Concerning the JL/TWQCD collaboration decay con-stant data [75], we only find good partial fits if we in-clude the three and two lightest pion mass data pointsfor the trajectories m s = 1 . m s and m s = 2 m s , re-spectively. This can be due to the breakdown of theChPT expansion for such large m s values. Since in thesesimulations decay constant determinations are providedbut not ρ phase shifts, we analyze them together withthe Hadron Spectrum Collaboration (HS) ρ -meson phaseshift results at m π = { , } MeV. The only pur-pose of this fit is to show the qualitative behavior of thepseudoscalar meson mass and decay constant ratios overtrajectories with larger m s values than the physical one.The corresponding LECs obtained in the fit are given inthe third column in Table IV. A comparison with theresult from Fit II in Table III shows up sizable discrep-ancies between both fits, which might be due to incon-sistencies of the JL/TWQCD data with data included inFit II apart from the breaking of the chiral series. Thesephase shift results are also plotted in the top-left panelof Fig. 5 in dashed lines. Nevertheless, the extrapolationto the physical point of this fit turns out to be also veryclose to the experimental data.For the PACS-CS collaboration, both ρ -meson phaseshift [63] and decay constant [76] data are available andanalyzed together. The LECs are given in the fourthcolumn of Table IV, and also the L i ’s, i = 4 , −
8, dif-fer considerably from the Fit II values. As explained inSect. III these data have larger kaon masses for the sametrajectory m s = m s than data in Fit II. This can be dueto sizable finite volume effects in these simulations. Asa consequence, these data are in disagreement with thedata included in Fit II. In this case, the extrapolationto the physical point, depicted in the bottom-right panelof Fig. 5, fails substantially to describe the experimentaldata.Let us note that these ρ -meson phase-shift data wereanalyzed before in [84] using the UChPT model in [46].Even though this model neglects the LHC contribution,which now is taken into account, we obtain here similar results to the ones of [84].The chiral trajectories and decay constant ratios forthese fits are shown in Fig. 38 of the Appendix VI B.We find that the pseudoscalar meson mass data on m s = k trajectories fit very well into a linear formula m K = a m π + b with slope a = 0 .
5, depicted in dottedlines. This behavior is qualitatively similar to the leadingorder ChPT prediction. For the ratios of decay constantswe find similarities with the results of Fit I over the tra-jectory m s = m s . The ratios m K /f π and m K /f K inother m s = k trajectories as a function of the pion massare parallel to the ones over m s = m s and take highervalues. For the m π /f π ratio, only the JL/TWQCD andPACS-CS data are just a bit out of the error band.Finally, the ρ (770) pole position on the second Rie-mann sheet obtained in the different fits are given inTable V. While the values obtained in Fit II and in theJL/TWQCD & HS fits are compatible with the most pre-cise theoretical prediction [7], the results obtained forUKQCD & CA and PACS-CS provide smaller and largervalues, respectively. In order to write down results thatcan be compared to the BW values provided in latticearticles, we perform a refit of the IAM solution to theBW formula in Eq. (60). As we show in Fig. 39 in theAppendix VI B, the data is also well described by a Breit-Wigner (BW) parameterization. The BW mass, couplingand width, normalized to the pion mass, are shown inTable V, where we also provide the result for the extrap-olation to the physical point. B. Chiral trajectories Tr M = C In this section we show the outcome of the analy-sis of ρ -meson phase-shift [79] and decay constant [70]data of the CLS Collaboration over trajectories whereTr M = 2 m ud + m s = C . Thus, in these trajectories thekaon becomes lighter as the pion mass increases. Twodifferent scale setting methods were considered in [70].These two methods lead to differences of around 10 − m π , 20 −
45 MeV in m K , and 4 − f π and f K . These differences entail several difficulties. Asdiscussed in section III, we could only find an optimalsolution to the minimization problem of Fit I, that alsoincludes m s = k data, when scale setting A was taken forthe pseudoscalar meson masses and decay constants overthe Tr M = C trajectories. When the scale setting B wasconsidered instead, the global χ minimum was found tobe around twice larger than with the scale setting A . Onthe contrary, we observe that, when using scale setting A ,the dependence of ρ -phase shift data with the pion massof [79] cannot be described well within the IAM for allensembles. While the ensembles D101, J303 and D200are well described, the ensemble N200 (or N401) cannotbe reproduced. This is because the IAM predicts highervalues of the ρ meson mass for the pion mass used in thisensemble, see Fig. 40 (Fit IIIA) in the Appendix VI B.Interestingly, by using scale setting B , we find a solution6 FIG. 5: Lattice phase shift data analyzed and fit results obtained as explained in the text. For each fit we also plot theextrapolations to the physical point in comparison with the available experimental data. The pion mass value (in MeV) foreach simulation is given in parenthesis.LEC × MILC+HS UKQCD+HS Laiho+HS MUL+HS Fit II L . − . . . . L − . − . − . − . − . L . . . . . L . . . . . L . . . . . L . . . . . L − . − . − . − . − . m s = k described in the main text.Uncertainties are obtained from the minimization with the MINUIT program. describing all Tr M = C lattice data, i.e., pseudoscalarmeson mass and decay constant ratios and ρ -meson phaseshift (excluding m s = k data). These phase-shift resultsare plotted in Fig. 41 (Fit IIIB) in the Appendix VI B.Nevertheless, it is possible to perform fits of decay con-stant and ρ -meson phase shift data for ensembles withthe same gauge coupling β [79]. Namely, C101, D101 ( β = 3 . β = 3 . β = 3 .
55) andJ303 ( β = 3 . . LEC × UKQCD+CA JL/TWQCD+HS PACS-CS L . . . L − . − . − . L − . . . L . . . L . . . L . . . L − . − . − . m s = k fits as described in the main text. Uncertainties are ob-tained from the minimization with the MINUIT program. m π (MeV) (cid:101) E (cid:101) m BWρ g BW (cid:101) Γ BW Fit II 140 5 . − . i .
51 5 .
88 1 . . − . i .
34 5 .
92 0 . . − . i .
20 5 .
81 0 . . − . i .
33 5 .
75 0 . . − . i .
52 5 .
79 0 . . − . i .
62 5 .
66 1 . . − . i .
35 5 .
68 0 . . − . i .
19 5 .
59 0 . . − . i .
97 5 .
93 1 . . − . i .
86 5 .
98 0 . . − . i .
18 5 .
91 0 . E , and BWparameters m BWρ , g BW and Γ BW obtained from the refit ofthe IAM solution to the BW formula in Eq. (60) normalizedto the pion mass, i.e., (cid:101) E stands for E /m π . volume but N200 has a lattice spacing 1 .
13 times smallerthan N401. Finally, J303 has the biggest volume andsmallest lattice spacing. In this way, possible differencesbetween individual fits in these pairs might highlight fi-nite volume and lattice spacing effects. The resultingLECs are shown in Tables VI and VII for the A and The volumes of the C101 and D101 ensembles are L × T =48 ×
96 and 64 ×
128 respectively, both with a lattice spacing a = 0 .
086 fm ( scale setting B) . The lattice spacings for N401and N200 are a = 0 .
076 fm and 0 .
064 fm, respectively, and bothhave the same volume 48 × a = 0 .
05 fm and issimulated in a volume L × T = 64 × ×
128 and 0 .
064 fm, respectively. B , respectively. The ensembles C101 and D101 are fit-ted separately in order to study the finite volume effect.Overall, the values of the LECs L and L are approxi-mately stable, but we find large differences for the others.We also attempt to perform combined fits includingmost ensembles for different β in order to check whetherthese effects can be absorbed in the LECs. Since theD101 and N200 ensembles supersede the C101 and N401ones, accordingly, we only include the ensembles D101,N200, D200 and J303. These fits are denoted as Fit III Aand III B for the A and B scale settings , respectively, andthey also include the corresponding pseudoscalar mesonmass and decay constant ratio data. The LECs obtainedare given in the last columns of Tables VI and VII.In general, the LECs of Fit IIIA agree better with thoseobtained for the m s = k trajectories. The chiral trajec-tories and decay constant ratios of these fits are depictedin Fig. 38 (right) in the Appendix VI B, where we alsoshow the result of the m s = k fits for comparison (leftpanel). Results for the fits IIIA and B are plotted inlike blue-solid and green-double-dot-dash lines, respec-tively. The kaon mass dependence on the pion massfor the trajectories Tr M = C also fit well into straightlines, m K = a m π + b , but now with a slope close to a = − . .
5, as we found for the m s = k ones. In this way, the IAM is able to reproduce verywell the Tr M = C trajectories, which appear as threeclose decreasing curves intersecting the symmetric line, m s = m ud . At pion masses of 300 MeV, the kaon massis around 60 MeV lower than for the m s = k trajectory.The ratios of decay constants are also well reproduced.For the scale setting A , the ratio m π /f π agrees well withthe m s = k data, emphasizing that this ratio is almostindependent of m s . In the case of scale setting B , it fallsa bit out of the m s = m s error band, depicted in a light-brown color. Note that this behavior is different from the m s = 0 . m s trajectory (MILC, dotted-gray), which liesinside the error band and does not show any substantialdifference with the m s = m s curve. This suggests thatthere could be small dependencies with the strange-quarkmass. We comment more on this issue in the next section.Results for ρ -phase shifts are provided in Figs. 40and 41 of the Appendix VI D. As commented before, ex-cept for the ensemble N200 in scale setting A , all the otherphase shifts can be described qualitatively well in thesefits. In Figs. 6 and 7 we show the ρ -meson phase shiftsobtained for the different gauge coupling fits. The IAMallows one to describe the ρ -meson phase-shift data inTr M = C trajectories for every ensemble. Nevertheless,note that one can not observe a trend of the overall dataindicating that the ρ -meson mass increases monotoni-cally with the pion mass. For scale setting A , the N200and N401 ensembles give rise to a lighter ρ -meson mass Understood here as the energy for which the phase shift is equalto 90 ◦ . ρ -meson mass takes about the same value for the J303 andD101 ensembles, although the pion mass used in J303 isaround 20 MeV larger.At low energies, phase shifts decrease as the pion massgrows, as expected from the p -wave centrifugal barrierand the chiral expansion. For scale setting B one ob-serves that the trend of the ρ -meson mass dependenceon the pion mass is flatter. Noticeably, the ρ -meson be-comes lighter for pion masses around 300 MeV in bothscale settings. In both cases, systematic effects due to afinite volume and lattice spacing are reflected in around8 MeV difference in the ρ -meson mass between the C101and D101 and 14 MeV between the N200 and N401 en-sembles, respectively.The corresponding pole positions and couplings forboth scale setting are given in Table VIII. In addition,given the large discrepancies observed between the scalesettings we perform a new fit with their average for eachgauge coupling β , denoted as Fit C in Table VIII. Forcomparison, the result of the global fit including bothdata on m s = k and Tr M = C trajectories, discussed inthe next section (Fit IV) is also shown. The values arenormalized to the pion mass, so that the dependence ofthe ratio m ρ /m π with the pion mass and scale settingused is visible. Overall, we see that the results for differ-ent lattice spacings are quite similar. For the ensembleJ303 the dependence on the scale setting considered isnegligible, while for other ensembles it produces shiftsof less than 1% for the normalized ρ (770) mass and lessthan 5% for the couplings. Regarding finite volume andlattice effects, the systematic differences between C101and D101 are of around 1% in the normalized ρ mass,and 1.5% between the N200 and N401, while these are ofless than 2% in the couplings in both cases. Finally, thecomparison between the individual fit solutions obtainedusing A and B scale settings is given in Fig. 8, where itcan be seen that the differences produced in phase shiftsas a function of
E/m π are in general reasonably small,and negligible for the J303 ensemble. The largest differ-ence is coming from the size of the lattice spacing used inthe simulation, i.e., the difference observed between theN200 and N401 ensembles.Finally, we can compare the result of Fit C in Ta-ble VIII with the result of Fit IV which includes also m s = k data. There are small differences between thesetwo fits of less than 3% in the normalized ρ -meson massand less than 6% in the couplings. We discuss this furtherbelow. C. Global fit over Tr M = C and m s = k trajectories (Fit IV) In this section we perform a simultaneous analysis oflattice data over both m s = k and Tr M = c trajectories.This final study will be denoted as Fit IV and it analyzes lattice ρ -meson phase shift data in N f = 2 + 1 of [60–62, 79] in combination with pseudoscalar meson massesand decay constants from [70–74]. Thus, this analysistakes into account all data included in the fits II and IIIof Sects. IV A and IV B.As discussed in Sects. III and IV B, we were not ableto find a solution with the IAM describing data onTr M = C trajectories using neither of the scale settingsin [70, 79] in combination with data over m s = k . Hence,in order to attempt a global fit some remarks are nec-essary. First of all, the ensemble C101 of [79] will bediscarded since the simulation for the ensemble D101 isperformed in a larger volume. Note that even when theN401 ensemble has larger lattice spacing than the N200,the former has more data points and its uncertaintiesare smaller, therefore, we include both ensembles in thepresent analysis. Secondly, it is important to highlightthat, according to Tables V and VIII, the CLS result forthe ratio m ρ /m π of the D101 ensemble is very close to theone from the HadSpec (HS) collaboration at m π = 236MeV, Fit II; the difference is only of around 2%. Thisfact points out that the pion masses used in these sim-ulations should also be very similar. Nevertheless, onlythe average between the scale setting A and B has a sim-ilar pion mass ( m π = 233 MeV). This facts motivates usto consider that the average between both scale settingsprovides a reasonable estimate to be used in order to per-form a global fit of data. Hence, we perform a bootstrapof the lattice spacing for the Tr M = C ensembles assum-ing that for every β , it is normally distributed around theaverage of scale setting A and B and the standard devia-tion being half the difference between them. Not only thelattice spacings, a β ’s, but also decay constant ratios andenergy levels (normalized respect to the pion mass) foreach ensemble are generated from a normal distributionaccordingly to their lattice data errors. Regarding thelattice energy levels, the resampling is performed assum-ing a multivariate normal distribution with the originalcovariance matrix. Remarkably, following this strategy we could repro-duce decay constant and phase-shift data simultaneouslyon both trajectories. The resampling is performed 300times and the error is evaluated from that sample. Thisnumber of fits turns out to be enough, since the average,median and fit solution (taking the average of the latticespacing) are indeed very close to each other. Namely,they produce differences in the ρ -meson mass of less than1 − Phase-shift data are then obtained from a first order Taylor ex-pansion around the lattice data energies. LEC × β = 3 . β = 3 . β = 3 . β = 3 . L . . . − . − . . L − . − . − . − . − . − . L . − . − . − . − . − . L . . . . . . L . . . . . . L . . . . . . L − . − . . . − . − . M = C with the scale setting A .LEC × β = 3 . β = 3 . β = 3 . β = 3 . L − . − . . − . − . − . L − . − . − . − . − . − . L − . − . − . − . − . − . L . . . . . . L . . . . . . L . . . . . . L − . − . . . − . . M = C with the scale setting B . In Figs. 9, 10 and 11, the chiral trajectories and pseu-doscalar meson mass and decay constant ratios studiedare plotted. The lattice data fitted correspond to theextrapolation to the continuum limit with finite volumeeffects corrected. In more detail, for the m s = m s tra-jectory we include the UKQCD [71] (purple diamonds),MILC [72, 73] (black dashed curves with light-brown er-ror bands ) and Laiho [74] (orange dashed curves anderror bands) lattice data. For other m s = k trajecto-ries there is not much data except for the ratio m π /f π extracted by MILC [72] for m s = 0 . m s (gray dot-ted line). The Tr M = C data from the CLS Col-laboration are given for the different lattice gauge cou- The data was sent to us by C. Bernard and the error band isextrapolated from the physical point as suggested by him. plings β = 3 . .
46 (red circles), 3 . . A and B scale settings . Although in principle chiral trajecto-ries for the several gauge couplings β are different, inpractice, we obtain very similar curves when the errorin the lattice spacing is considered, which only start toseparate more clearly when these cross the symmetricline. This is, we get c ( β =3 . (cid:39) c ( β =3 . and only asmall difference for β = 3 .
4. In addition, we includein Figs. 9, 10 and 11 the IAM prediction for the tra-jectories m s = { , . , . , . , . , . , . } m s , whichare almost parallel to the m s = m s one. Furthermore,in order to highlight the relevance of the strange-quarkmass, we also include the prediction for the trajectories0 FIG. 6: Phase shift lattice data [79] and individual fits (depending of β ) obtained with the scale setting A . The values inbrackets stand for the pion and kaon masses (in MeV), respectively. m u = { , . } m u and m π = m π . These three trajec-tories start at a small value of m s ( m s B = 2 MeV ),then, they cross the symmetric line and end up at the m s = m s curve. All ratios m K /m π , m π /f π , m K /f π and m K /f K are reproduced well inside the 95 % CI till m π (cid:39)
400 MeV, when the ChPT predictions start to de-viate. Therefore, the predictions for pion masses between m π = 400 −
500 MeV are merely qualitative.From Fig. 10 one sees that the ratio m π /f π does notdepend much on m s . However, this does not necessarilymean that f π is independent of the strange-quark mass.In fact, both m π and f π depend on m s . This dependenceis shown explicitly in Figs. 12 and 13. In Fig. 12, thesquared leading order mass, M π , is depicted as a func-tion of the pion mass for different strange-quark masses.Indeed, one can see that m ud kept constant does not im-ply that the pion mass is constant as well. In fact, ouranalysis at one-loop level predicts that the pion massgrows with m s for a constant value of m ud ; while for m s = 0 one obtains m K (cid:39) / √ m π (see the red linein Fig. 9) and M π (cid:39) m π (notice the almost quadratic In this case, it is understood that m d = m u . behavior of the red curve in Fig. 12), consistently withthe leading order ChPT prediction, effects coming fromthe kaon and eta particles in f π become more relevantas m s increases. Although this effect is invisible for verylight pion masses and small for physical pion masses (fora constant value of m ud , the difference between the phys-ical pion mass and the m π value at m s = 0 in Fig. 12 isaround 14%.), it becomes larger for heavy pions. On thecontrary, in Fig. 13, where the dependence of f π on m π for different strange-quark masses is shown, one sees thatthis dependence is more noticeable for light pion massesand smaller for heavier pion masses, when the uncertain-ties for f π increase. At the physical point, one indeedfinds a difference of 5 MeV in f π between its value at m s = 0 and m s = m s . This is intrinsically connectedwith the contribution of the terms which involve kaonsand etas in Eqs. (5) and (9), m π = M π + ∆ K,η m π f π = F π + ∆ K,η f π , (52)1 FIG. 7: Phase shift lattice data [79] and individual fits (depending of β ) obtained with the scale setting B . The values inbrackets stand for the pion and kaon masses (in MeV), respectively.FIG. 8: Comparison between the individual fit solutions using A and B scale settings . FIG. 9: Chiral trajectories ( m K /m π ratio) considered in Fit IV in comparison with lattice data.FIG. 10: The ratio m π /f π obtained in Fit IV in comparison with the lattice data. FIG. 11: Decay constant ratios, m K /f π and m K /f K , obtained in Fit IV in comparison with the lattice data. with M π = M π (cid:104) µ π + M π f (2 L r + 2 L r − L r − L r ) (cid:105) , (53) F π = f (cid:104) − µ π + M π f ( L r + L r ) (cid:105) , (54)which are∆ K,η m π = M π (cid:20) − µ η M K f (2 L r − L r ) (cid:21) , (55)∆ K,η f π = f (cid:20) − µ K + 8 M K f L r (cid:21) . (56)These terms, which involve kaon and eta meson loops(tadpoles) and kaon mass contact terms, called t K,η fromnow on, contribute slightly for m s = 0, around 1 MeVin the f π value, but they account for about 6 − m s = m s . Later, we show that this relatively smallvariation in f π is translated into a visible difference inthe ρ mass.Beyond that, one might wonder what would happen ina world where kaons and etas are not present. In orderto answer that question, we could explicitly set to zerothe t K,η contribution in Eqs. (5) and (9), and define M π and F π in Eqs. (53) and (54) as the pion mass and de-cay constant in this world. In such a way, one obtains adependence of the new pion decay constant, F π , with thepion mass, as the orange-dashed line in Fig. 13. Effec-tively, this limit is equivalent to set the coupling of pionsto kaons and etas to zero, which can be achieved when,first, f K,η in Eqs. (10) and (11) are sent to infinity, and, Note that it can only achieved if one breaks the SU(3) symmetry
FIG. 12: The squared leading order mass, M π , of Eqs. (2)and (5), as a function of m π for different values of the strange-quark mass, m s . second, when m K,η are set to zero. This is discussed inSect. VI A. Furthermore, note that this is different fromthe so-called SU(2) formalism (corresponding to take the in the partially conserved axial current (PCAC), i.e., one hasto differentiate between the pion and the kaon and eta decayconstants in the chiral limit. See explanation in Sect. VI A. m π (MeV) (cid:101) E (cid:101) m BWρ g BW (cid:101) Γ BW D200 A 209 3 . − . i .
80 6 .
20 0 .
60B 200 3 . − . i .
79 6 .
10 0 .
57C 204 3 . − . i .
79 6 .
16 0 . . − . i .
77 6 .
00 0 . . − . i .
44 6 .
43 0 .
51B 223 3 . − . i .
44 6 .
15 0 .
46C 233 3 . − . i .
42 6 .
22 0 . . − . i .
42 6 .
38 0 .
49B 223 3 . − . i .
40 6 .
24 0 .
46C 233 3 . − . i .
41 6 .
21 0 . . − . i .
35 6 .
01 0 . . − . i .
11 6 .
30 0 .
37B 258 3 . − . i .
11 6 .
27 0 .
36C 263 3 . − . i .
10 6 .
31 0 . . − . i .
03 5 .
99 0 . . − . i .
78 6 .
18 0 .
24B 283 2 . − . i .
77 6 .
13 0 .
23C 290 2 . − . i .
78 6 .
17 0 . . − . i .
79 5 .
97 0 . . − . i .
74 6 .
07 0 .
21B 283 2 . − . i .
74 6 .
03 0 .
21C 290 2 . − . i .
76 6 .
10 0 . . − . i .
79 5 .
97 0 . ρ -meson pole positions and couplings obtainedfor the individual Tr M = C fits given in Tables VI and VIIand for each scale setting. The IAM pole position is denotedby E , while the BW parameters m BWρ , g BW and Γ BW areobtained by refitting the IAM solution to the Breit-Wignerformula. The fit C stands for the average of both scale set-tings, while IV denotes the global fit discussed in section IV C,included for comparison, which is obtained performing a re-sampling of the lattice spacing and lattice data. The quan-tities with tilde are normalized to the pion mass, i. e. , (cid:101) E stands for E /m π . limit m s → ∞ ), where the effect of pions interacting withkaons and etas is absorbed in the SU(2) LECs, the con-stant B and f , which are fixed, together with f π , toget observables in the physical world with more than twoflavors.When t K,η = 0, f π reduces its value around 6 − m s = 0 and m s = m ud as the pionmass decreases, reaching f in the chiral limit. These in- FIG. 13: The pion decay constant, f π , see Eq. (9), as a func-tion of m π for different strange-quark masses, m s . The con-tinuous and dot-dashed red lines represent the solution forthe pion decay constant from Fit IV at different values of m s and in the symmetric line, while the orange-dashed curve isthe solution when t K,η = 0. teracting terms involving kaons and etas have a similareffect to the one caused by a change in m s for light pionmasses. In fact, varying m s from zero rises the contribu-tion of these terms, increasing the value of f π . Later, weshow the contribution of these terms also in the ρ me-son mass (using the one-channel ππ IAM equation), butbefore, we continue discussing results from the analysiswith the ( ππ − K ¯ K ) coupled-channel IAM.The extrapolation to the physical point for the massand decay constant ratios is given in Table IX. The cen-tral value represents the median, while the first upperand down indices show the limits of the 68% CI. The up-per (down) limits of the 95% CI are obtained by summingthe absolute values of the first and second upper (down)indices. These extrapolated ratios are compatible withthe experimental values, which are inside our 68%CI.The values of the LECs and remaining fit parametersare given in Table X, where errors also represent 68% and95% CI. The quark condensate Σ can be estimated fora given strange-quark mass from Table X. For instance,taking m s = 95 MeV we obtain Σ / = 247 MeV, inclose agreement with the MILC result [152], 245(5)(4)(4)MeV. Finally, the correlation matrix of the parameters isgiven in Eq. (61) in the Appendix VI E.We can also compare the ratios obtained here with theones of the N f = 2 simulation of [59]. For the pion massesused in that simulation, m π = 225 and 315 MeV, theratios { m K /m π , m π /f π , m K /f K } are { . , . , . } m K /m π m π /f π m K /f π m K /f K . +0 . . − . . . +0 . . − . . . +0 . . − . . . +0 . . − . . Experiment [PDG]3 . . . . and { . , . , . } , for the light and heavy pionmass, respectively. Looking at Figs. 9, 10 and 11, wesee that the deviations from the mean values in Fit IVat the m s = m s trajectory are of less than { , , } for the light pion mass, and around { , , } forthe heavy pion mass. These relative differences are smalland the ratios in Table III of [59] are compatible with ourerror bands. This indicates that the setup of the simu-lation of [59] is in line with the result of this analysisfor the m s = m s chiral trajectory. Thus, possible devia-tions in the ρ -meson parameters with the ones obtainedhere might be caused by a different reason. Notice alsothat the method used to determine these ratios in [59] isto take m K /f K to the physical point in a strange-quarkquenched approximation. As a consequence, the valuesof f π obtained are consistent with the ones from Fit IVat m s = m s and its extrapolation to the physical value.However, since the real world has more than two flavors,that approach misses the m s dependence of f π as dis-cussed before.Phase shift lattice data and solutions from Fit IV at thecorresponding pion masses are depicted in Figs. 14, 15,and 16, where one can see that lattice data are very welldescribed also inside the 95 % CI. The only exceptionsare few data points from the CLS data for the N200 andN401 ensembles (right-bottom panel in Fig. 15), whichlie outside the error band and are also far from the bulkof data. Beyond that, most of data for these ensemblesare inside of the 95% CI error bands. This is, the N200and N401 ensembles are compatible within uncertainties,as concluded also in [79]. Note that the error bands arelarger for the Tr M = C data since they include the vari-ation of the lattice spacing a β between A and B scalesettings .The extrapolation to the physical point in comparison Where we have divided f K from [59] by a factor √ f K used here. LECs × L . +0 . . − . . L − . +0 . . − . . L − . +0 . . − . . L . +0 . . − . . L . +0 . . − . . L . +0 . . − . . L . +0 . . − . . c × − , k × − Tr M( β = 3 .
4) 268 +14(8) − Tr M( β = 3 .
55) 254 +11(7) − Tr M( β = 3 .
7) 257 +12(7) − m s B +14(10) − TABLE X: Values of the parameters obtained in Fit IV. Theerrors can be interpreted in terms of probability. The centralvalue represents the median, the first upper and down indicesgives the 68% CI, while the sum of the absolute values of thetwo upper (down) indices provides the upper (down) limits ofthe 95% CI. with experimental data is plotted in Fig. 17, where onecan see that it indeed provides an excellent description ofthe experiment. In Fig. 18, the CLS D101 ensemble andHS data for m π = 236 MeV are plotted together. Wecan see that indeed both results are compatible.The phase-shift solution for the pion mass used in theD101 ensemble (dark green) is shown in comparison withthe result from an individual fit of the C101 data (lightgreen) in Fig. 19. Note that even when the C101 ensem-ble was not included in the global Fit IV, both solutions,and the bulk of data itself, lie well inside the 95% CI.The difference between both fits for the energy at whichthe phase shift crosses 90 ◦ ( E ( δ = 90 o ) /m π ) is of around3%. This indicates that the deviations due to the volumesize are not large.To show the trend of the Tr M = C data, the meansolution of Fit IV together with the lattice data are rep-resented in Fig. 20. As can be seen, these data are now The average between the lattice spacings in scale settings A and B is taken in this individual fit of the C101 data in order tocompare with the solution from Fit IV. This correspond to using the averaged lattice spacing values asthe data plotted in the figure. FIG. 14: Result of Fit IV in comparison with the HS data at m π = 236 and 391 MeV [60, 61]FIG. 15: Result of Fit IV in comparison with the Tr M = C data of the CLS ensembles, D200, D101, J303, N200 and N401 [79]. well described. Phase shift data corresponding to higherpion masses fall more to the right and the ρ -meson massincreases monotonically with the pion mass. The depen-dence of the ρ -meson mass with the pion mass is de-picted in Figs. 21 and 22 for the m s = k and Tr M = C trajectories, respectively, where we also show the valuesof the ρ -meson mass given in the corresponding latticepapers. The m ρ /m π ratios are also represented in the Defined as the value of the energy for which δ = 90 o . For the Tr M = C trajectories, the error due to the use of scale right panels. For clearness, we present again the latticedata and the resulting curves separately in Fig. 23. Inboth trajectories m ρ increases with m π . Furthermore, forthe trajectories m s = m s and Tr M = Tr M , we find al-most identical results till pion masses of around 400 MeV,when these start to separate. The reason for this behav-ior is well understood. On one side, the ρ (770) mesonbecomes a bound state at pion masses around m π = 450MeV in the m s = m s trajectory (above this value, the settings A and B is also depicted. FIG. 16: Result of Fit IV in comparison with the JB data at m π = 233 MeV [62].FIG. 17: Extrapolation to the physical point of the Fit IVsolution in comparison with the experimental data. ππ threshold is plotted in Figs. 21 and 23 (left) instead).On the other side, it starts to decay into K ¯ K in theTr M = Tr M trajectory when the ρ -meson pole crossesthis threshold and the kaon gets lighter than the pion.Indeed, it becomes a pole in the IV Riemann sheet as de-fined in Eqs. (39) and (40). Conversely, other Tr M = C trajectories tend to be flatter than the m s = k ones. Thisis actually in line with the trend of lattice data.It is relevant to note that close to the physical point wedo not observe any relevant change in the ρ -meson mass.This suggests that the ρ -meson properties are quite sta-ble against small variations of the strangeness around thephysical point, however, its coupling to the K ¯ K channelis still large, around 60% of its coupling to ππ . . Nev- This can be seen in Figs. 31 and 32, as discussed later in this
FIG. 18: Phase shift lattice data corresponding to the D101ensemble in comparison with the HadSpec data for m π = 236MeV and the IAM solution for D101.FIG. 19: Comparison between the IAM solutions in Fit IVfor D101, with error bands, and the individual fit to the C101data using the averaged lattice spacing. ertheless, for m s values below 0 . m s , the ρ -meson massstarts decreasing considerably reaching a value inside theinterval [675 , m s = 0. This behavior is evenmore clear in the m u,π = c trajectories where the massof the u quark (or pion) is kept fixed and only m s varies.Since the ρ meson starts to decay into K ¯ K for lighter section. FIG. 20: Phase shift lattice data and global Fit IV solution (only the solution for the average lattice spacing is shown). strange quarks, the effect in the real part of the pole, seeFig. 26, becomes significant.This behavior of the ρ -meson mass is also visible inthe corresponding m ρ /m π plots (right panel of Figs. 21and 22), where the errors in the y-axis are reduced. Theseplots also show that the error due to the lattice spacing(or scaling setting used) is smaller than the reduction ofthe ρ -meson mass around the m s = 0 limit. The behaviorof the ρ -meson mass and width respect to the kaon massis depicted in Fig. 26 for the m u = c and m π = m π trajectories. When m K decreases, both mass and widthdecrease, as commented before. In Figs. 24 and 25, thecontinuation of the m ρ /m π ratios for smaller pion massesare also depicted. This difference with respect to thephysical point is more abrupt as the quark masses getsmaller. The symmetric line is also plotted in dot-dashedlines.In Figs. 27, 28, 29 and 30 we provide the real andimaginary parts of ρ -meson pole position in a 3D plotrespect both, the pion and kaon mass. To render somereferences, we also give in Table XI the pole positions at m s = { , . , } m s for pion masses near the chiral limit,physical point and when the ρ gets bound ( m π ∼ ρ -meson behavior in the m s = m s trajectory. See Figs. 27, 28. In this case, we ob-tain a ρ -meson pole position E = (735 − i
82) MeV nearthe chiral limit, while at the physical pion mass we get E = (747 − i
70) MeV, see Table XI, consistently withprevious analyses [42]. Nevertheless, as m π increases, the m s /m s m π Re E Γ0 ∼ +3(5) − +1(2) −
140 MeV 695 +3(4) − +1(3) −
450 MeV 867 +3(6) − +2(1) − . ∼ +5(9) − +3(5) −
140 MeV 744 +4(6) − +3(4) −
450 MeV 905 +4(6) − ∼ +7(10) − +5(6) −
140 MeV 747 +5(11) − +4(7) −
450 MeV 908 +3(7) − ρ (770) meson mass, Re E , and width, Γ = − E , extracted from the pole position for several strange-quark masses at the chiral limit, physical pion mass and m π (cid:39) m s = 0 , m s trajectories. The central value represents themedian, the first upper and down indices gives the 68% CI,while the sum of the absolute values of the two upper (down)indices provides the upper (down) limits of the 95% CI. FIG. 21: The ρ -meson mass (left) and normalized ρ mass (respect to the pion mass) (right) as a function of the pion mass forthe m s = k , m u = c , m π = m π , and m s = m u trajectories.FIG. 22: The ρ -meson mass (left) and normalized ρ mass (respect to the pion mass) (right) as a function of the pion mass forthe Tr M = C , m s = 0 and m s = m u trajectories. ρ -meson mass moves slower than the ππ threshold, sothat, eventually, the ρ (770) meson becomes a ππ boundstate with a mass around 908 MeV for a pion mass ofaround 450 MeV. Note that, in the case of the m s = m s trajectory, the ρ -meson pole is always below the K ¯ K threshold for the pion masses analyzed here. We do not start to appreciate significant changes in the behavior ofthe ρ -meson mass till the strange-quark mass is reducedin half its physical value. For instance, for m s = 0 . m s ,we still get a pole at 732 − i
81 MeV in the chiral limit,which transforms into a bound state with a mass of905 MeV also for pion masses of around 450 MeV. For0
FIG. 23: The ρ -meson mass as a function of the pion mass over the m s = m s (left) and Tr M = Tr M (right) trajectories incomparison with the lattice data.FIG. 24: m ρ /m π ratio as a function of m π for the m s = k and m s = m ud trajectories continued towards lighter pion masses. lighter strange-quark mass trajectories relevant changesare observed. Both, ρ -meson mass and width, decreaseconsistently, so that we obtain E = 678 − i
38 MeV for m s = 0 in the chiral limit. Furthermore, for m s ≤ . m s ,both the ππ and K ¯ K thresholds get closer to each otheras m π increases, in such a way the kaon becomes lighterat a given point. In this regime, the ρ meson becomes a pole in the fourth Riemann sheet when its mass getsbelow the ππ threshold. In this case, the ρ -meson decaysonly into K ¯ K and its width starts increasing again until The Riemann sheet that is reached when only the K ¯ K cut iscrossed continuously from the first Riemann sheet. FIG. 25: m ρ /m π ratio as a function of m π for the Tr M = C , m s = m ud and m s = 0 trajectories continued towards lighterpion massesFIG. 26: The real and imaginary parts of the ρ -meson pole position as a function of m K over the m u = c and m π = m π trajectories. it gets a maximum, after which the ρ eventually becomesa K ¯ K bound state as m π increases.This behavior is even more noticeable for the Tr M = C trajectories, depicted in Figs. 29 and 30. In this case, thestrange-quark mass decreases as m π grows reaching thesymmetric m s = m ud line for pion masses of around 450MeV. Once the symmetric line is crossed, the K ¯ K chan-nels opens below the two-pion threshold and the ρ (770)meson becomes again a pole on the fourth Riemann sheet.Nevertheless, the kaon mass in this trajectory decreasestill it ends up in the m s = 0 line (red-solid curve). Hence,the ρ -meson mass (width) starts decreasing (increasing)at a given point (when the kaon gets lighter than the pionafter crossing the symmetric line) till ending at the zerostrangeness line. This behavior suggests that strangeness plays an important role in the ρ (770) meson near theSU(3) flavor limit. This can also be inferred from theincrease of its coupling to K ¯ K , as discussed below. The pion mass dependence of the ρ -meson couplingsto the ππ and K ¯ K channels, g ππ and g K ¯ K as defined inEq. (41), are shown in Figs. 31 and 32 for the m s = k andTr M = C trajectories, respectively. On one hand, g ππ varies smoothly with m π before the the ρ -meson tran-sition into a bound state, decreasing as it approachesthe K ¯ K threshold and increasing with m s . Once the At the symmetric line, the coupling to K ¯ K grows 20% of itsvalue at m s = m s for physical pions. ρ meson becomes bound, its coupling rises sharply till m π (cid:39)
480 MeV. Overall, it takes values g ππ (cid:39) . − . g ππ behavior, g K ¯ K decreases sig-nificantly with the mass of the strange quark. In addi-tion, while the pion-mass dependence of g K ¯ K flattens forlighter strange quarks, it becomes larger as m s reachesthe physical value. All in all, it takes values within therange g K ¯ K (cid:39) . . m π ≤ m K one ob-serves the ratio g K ¯ K /g ππ ≤ / √
2. On the contrary, g K ¯ K /g ππ > / √ m π > m K . In the symmetricline we obtain exactly g K ¯ K /g ππ = 1 / √
2. This is not acoincidence. In the SU(3) limit, the decomposition of a I = 1, I = 0 state of the antisymmetric octet represen-tation into two-Goldstone–Boson states with well definedisospin reads | Y = 0 , I = 1 , I =0 (cid:105) A = 1 √ | K ¯ K (cid:105) − √ | ¯ KK (cid:105) + 2 √ | ππ (cid:105) , (57)where, Y stands for the hypercharge and I is the isospin.Thus, taking into account the kaon degeneracy due tostrangeness, the ρ -meson coupling to pions should be afactor √ √ g ππ f π /m ρ is depicted. Thisratio lies within the interval [0 . , . m s = { , m ud } curves, and wellapplicable also around the physical point, with deviationsfrom KSFR of less than 4%. The largest deviations (stillsmaller than 8%) are found for pion masses between 200and 300 MeV.To compare the LECs obtained in the different analy-ses done here with the Flag average [78], we depict themin Fig. 35, where we show the results from Fit I (pseu-doscalar meson mass and decay constant ratios), Fit II,(analysis of data over the m s = k trajectories), Fit III,(Tr M = C trajectories), and Fit IV (mean and standarddeviation as a result of the study including both m s = k and Tr M = C trajectories) together with the Flag av-erage (pink color). Indeed, we see that LECs from fitsI and IV are very close, being also consistent with the In this figure, m ρ means the real part of the pole position, Re E ,so that we are able to plot the ratio for a larger range of pionmasses and after the transition. Since pole positions are slightlylower than the energy corresponding to δ = 90 , this ratio givesvalues lower in around 1 . E ( δ = 90 ◦ ) is taken. FLAG average, which has larger errors. In general, fits I,II, IIIA and IV give closer results, while the LECs L , L and L , from the analyses of PACS-CS and JL/TWQCDdata, strongly disagree with other analyses. Notice theprecise values of the LECs provided by Fit IV.As we did before when discussing f π , we can study the ρ -meson properties in a world where there are no kaonsor etas by setting to zero their corresponding interactingterms, i.e., contact mass terms and diagrams involvingloops with kaons and etas in Fig. 1 and pions in theinitial and final state, (what we called t K,η ). In practice,this means that we solve the one-loop ππ IAM equation,see Eq. (30), taking the limits f K,η → ∞ , and m K,η → M π and F π in Eqs. (53) and (54). Theresult is shown in Fig. 36. Taking into account that wedid not include N f = 2 data in our analysis but this resultcomes out as a prediction from our SU(3) IAM analysis,the agreement with the N f = 2 data is astonishing.In fact, starting from a three-flavor formulation, the N f = 2 formalism should be in principle obtained whenone decouples the strange-quark contribution by sending m s to infinity [45]. In this case the effect of kaons andetas is encoded into the bare pion decay constant, f ,the constant B , and in terms proportional to ν K,η = (cid:0) log( ˆ m K,η /µ ) + 1 (cid:1) / π ( ˆ m K,η refer to the limit where m ud →
0, which can be absorbed in a redefinition of theLECs. Here, though, we are simply studying the worldwhere there are no kaons or etas relying on the fact thattheir interaction with pions comes from terms where theirmasses and decay constants appear explicitly.Lattice N f = 2 simulations typically use either f K [59,153] or the nucleon mass (via the QCD static poten-tial) [54–56, 58] to fix the scale, which in general requiresto perform a chiral extrapolation. Nevertheless, given theobserved dependence of m π and f π on the strange quark,and the fact that f π is correlated with f K and the nucleonmass, these methods seems to neglect this dependence.In fact, the agreement between N f = 2 lattice simula-tions and the IAM prediction without kaons and etason the ρ -meson properties suggests that N f = 2 latticesimulations leave out the contributions coming from thestrange quark, and hence, they describe a world wherethe strange-quark is missing.In Fig. 37 we compare this result with Fit IV overthe chiral trajectories m s = { m s , m ud , } and m u = m u .Moreover, we also show the result of solving the one-channel ( ππ ) IAM equation with m s = { , m s } , this is,keeping the t k,η terms in the ππ channel. Remarkably,the one-channel IAM result for t K,η = 0 (orange line), m s = 0 (dashed-red), and the two-coupled channel so-lution over m s = m ud (dotted-red), provide very closeresults for the ρ mass, which are also consistent withthe N f = 2 lattice data. This is explained because thecontribution of t K,η terms for light strange-quark massesis small. As explained before, these terms contribute inaround 1 − . f π value when m s = 0, and3 FIG. 27: The real part of the ρ -meson pole position, Re E , as a function of m π and m K for the m s = k trajectories, m u = c , m π = m π , and m s = m ud . around 6 − m s = m s at the physical pionmass, see Fig. 13. This reduction on the value of f π for smaller strange-quark masses reflects also in smaller ρ -meson masses. Notice that in the m s = m ud trajecto-ries, pions and kaons are acting effectively in the ρ -mesonmass as if only one flavor, the quark u , is present.It is also interesting to see what happens if one keepsthe t K,η terms in f π and in the ππ scattering amplitudewhen solving the one-channel ππ IAM equation in the m s = m s trajectory (dot-dashed blue). We see that thistrajectory is consistent with the coupled-channel IAMsolution for m s = m s , telling that the effect of the off-diagonal elements t in Eq. (20) is very small for physicalstrange-quark masses. Nevertheless, the coupled-channeleffect becomes appreciable for lighter m s , as as one cansee by comparing the difference between the dashed andcontinuous red lines. The small contribution of the off-diagonal elements at the physical point is in contradictionwith the results in [31, 59], where the absence of theseelements are found to be responsible for the dropping ofthe ρ mass in the N f = 2 case. However, this is naturalsince in these works the same value of f π was used inthe N f = 2 and N f = 2 + 1 predictions, and then, theeffect of the kaon and eta contributions were absorbed inthe off-diagonal elements. Indeed, we obtain here sim-ilar predictions for the ρ meson mass in N f = 2 and N f = 2 + 1 simulations over m s = m s than in [31, 59]. Nonetheless, we have gone through a deeper analysis inthis work; by studying the m s dependence of f π , we haveobtained that m s regulates the contribution of the kaonand eta interacting terms and that the effect of theseloops are absorbed in f π instead. Consistently with thepredictions done in the works of [31, 59], we also obtainthat the ρ mass is reduced when these terms are omit-ted. In Ref. [59], the pion decay constant was determinedand these values were used to make predictions for the ρ mass using the UChPT model in [46] with two andthree flavors. Nevertheless, the pseudoscalar meson de-cay constants in Ref. [59] were determined following themethod of Ref. [153] where the kaon is introduced later inthe quenched approximation and m K /f K is fixed to thephysical point. This leads to extrapolated values of f π in N f = 2 simulations consistent with the experimentalvalue, where more than two flavors do exist. However,by doing this, one is missing the m s dependence and theeffect of the kaon and eta loops in the pion decay con-stant, that we have shown here. These missing effectscan lead to discrepancies between observables in N f = 2,such as the values of the ρ mass and pion decay con-stant determinations. We have shown that a lower valueof the ρ mass than the physical one should be reflectedalso in lower values of f π . In summary, assuming thatthe pion decay constant is the same in the two and threeflavor simulations totally misses the effect of the strange4 FIG. 28: The real part of the ρ -meson pole position, Re E , as a function of m π and m K .FIG. 29: The width of the ρ -meson pole position, − E , as a function of m π and m K for the m s = k , m u = c , m π = m π and m s = m ud trajectories. FIG. 30: The width of the ρ -meson pole position, − E , as a function of m π and m K for T r M = K in comparison with the m s = 0 and m s = m ud trajectories.FIG. 31: The couplings of the ρ -meson pole to the ππ and K ¯ K channels, g ππ and g K ¯ K , for the m s = k , m u = c , m π = m π and m s = m u trajectories. quark and loops containing kaon and eta particles in pionobservables. V. CONCLUSIONS
For the first time, we have studied simultaneously boththe light- and strange-quark mass dependence of pseu-doscalar meson masses, decay constants and ρ -mesonproperties, such as its mass, width and couplings to thepion and kaon channels. Our analysis is based on recent6 FIG. 32: The couplings of the ρ -meson pole to the ππ and K ¯ K channels, g ππ and g K ¯ K , for the Tr M = C trajectories incomparison with the m s = 0 and m s = m u ones.FIG. 33: Ratio of the g K ¯ K and g ππ couplings for different trajectories. lattice data of these observables on the chiral trajecto-ries m s = k and Tr M = C . In the analysis we resam-ple pseudoscalar meson observables, energy levels (takinginto account covariance matrices), and lattice spacings,providing a satisfactory solution at the 95% confidencelevel. The IAM proves itself to be able to explain thepseudoscalar meson masses, decay constants and ρ -meson properties over different chiral trajectories. Therefore,the LECs obtained here are the most precise and the onlyones up to now that are able to describe the strangenessdependence of these observables. The chiral extrapola-tion of ρ -meson phase shift data is also in remarkableagreement with experiment.The dependence of the pion decay constant, f π , with7 FIG. 34: The quantity √ f π g ππ /m ρ as a function of m π for m s = k, m u = c, m π = m π , and m s = m u trajectories (left), andover Tr M = C (right). Deviations from one reflect violations of the KSFR relation.FIG. 35: Values of the LECs obtained in the several combined and global fits in comparison with the FLAG average (last datapoint in pink). FIG. 36: Result for the ρ meson mass when interacting termsinvolving kaons and etas, t K,η , are set to zero as explained inthe text, in comparison with N f = 2 lattice data.FIG. 37: Result for the ρ meson mass when interacting termsinvolving kaons and etas, t K,η , are set to zero as explained inthe text, in comparison with N f = 2 lattice data, and with theresult from SU(3) IAM in previous sections, m s = m u , m s , m s = ms . the strange-quark mass, m s , is also studied for the firsttime. We have shown that, although to assume that theratio m π /f π is independent of m s can be a good approx-imation, the variation of f π with m s is abrupt for lightpion masses. Furthermore, this dependence is acting as aregulator of the size of the contribution of loops and con-tact terms involving kaons and etas. For instance, theseterms contribute slightly to f π for m s = 0 but account for around 6 − m s = m s . This contribution to f π is sufficiently large, so that, their absence is able toexplain successfully the lower values of the ρ -meson massobtained in N f = 2 simulations. Even when we did notanalyze here N f = 2 lattice data, but only N f = 2 + 1simulations, the IAM has demonstrated to be able to de-scribe simultaneously both, the ρ -meson mass over chiraltrajectories in two and three flavor lattice simulations.Regarding this last aspect, the results obtained here areconsistent with the ones of Refs. [31, 59]. However, we ob-tain here that the ρ -meson mass reduction in two-flavorcalculations is due to the absence of the strange-quarkmass and the contribution containing strange particleson the pion decay constant. Fixing the pseudoscalar de-cay constants to the physical point in two-flavor latticesimulations misses this dependence.Some other interesting effects observed involve the K ¯ K channel. First, as m s decreases the ρ -meson mass re-duces; when m s approaches zero it drops around 70 MeVrespect to its values at the physical point. This is ef-fectively more visible in the m u = m u trajectory. Sec-ond, in the m s = m s trajectory, as m π increases andthe ρ -meson mass gets closer to the K ¯ K threshold, itscoupling to K ¯ K increases, becoming eventually a boundstate. Around m π = 450 MeV, it starts to decay into K ¯ K in the Tr M = Tr M trajectory. For other trajec-tories, these transitions occur at different pion masseswhen the kaon becomes lighter than the pion. Third,the coupling ratio g ππ /g K ¯ K = √ VI. APPENDIXA. Connecting ( ππ − K ¯ K ) coupled-channel IAMwith N f = 2 lattice simulations In this section we analyze kaon and eta contributionsinto the ρ -meson properties, as well as we discuss how it ispossible to disconnect their effect. In the IAM coupled-channel formalism, kaons and etas contribute to pion-pion scattering through terms of the kind:1. The ππ → K ¯ K and K ¯ K → K ¯ K scattering ampli-tudes, which are named t and t in the coupled-channel IAM formulation, see Sect. II E.2. Tadpoles and one-loop diagrams involving kaonsand etas in the ππ → ππ scattering amplitude, t ,see Fig. 1.93. Kaon and eta contact mass terms and tadpoles en-tering into the pion mass and decay constant, i.e.,Eqs. (5) and (9).Regarding (1), the amplitudes t and t are propor-tional to 1 /f K and 1 /f K , respectively, since they involvediagrams with two and four external kaon legs. It isclear that by sending f K → ∞ these contributions disap-pear. Note, though, that f K is related to f throughEq. (10). Thus, in practice, taking this limit entailsbreaking the SU(3) symmetry in PCAC [154–156], i.e.,to assume that the pion and kaon bare decay constants f ,K and f ,π do differ. For the same reason, the termsin (2), i.e., kaon and eta tadpoles and one-loop diagrams,are all proportional to 1 /f K,η and they also vanish when f K,η → ∞ .Finally, concerning (3), while kaon and eta tadpolesentering in the pion mass and decay constant, Eqs. (5)and (9), vanish when f K,η → ∞ , there are still kaonmass contact terms, which can be removed only whenone takes the limit m K →
0. Thus, taking the limits m K = 0, f K , f η → ∞ , one obtains the pion mass anddecay constant in a world where kaons are etas are bothdecoupled. Namely, M π = M π (cid:20) µ π + 8 M π f π (2 L r + 2 L r − L r − L r ) (cid:21) , (58)and F π = f π (cid:20) − µ π + 4 M π f π ( L r + L r ) (cid:21) . (59)The above relations are the same given in Eqs. (53)and (54), which are depicted in Figs. 12 and 13, anddiscussed in the paragraphs around these figures.Then, once all the contributions in (1)-(3), which arecalled t K,η in the text, are removed, one can solve theone-channel ππ IAM, Eq. (30), t ( s ) IAM = t ( s ) t ( s ) − t ( s ) , using as input the pion mass and decay constant givenin Eqs. (58), (59), to obtain pion-pion scattering in a world where kaons and etas are absent. This is plottedin the orange line in Fig. (37) referred to as t Kη = 0,and discussed in the Sect. II C. B. Chiral trajectories and decay constant ratiosobtained in fits II and III
In Fig. 38, the chiral trajectories and decay constantratios obtained in fits II and III are depicted.
C. Breit-Wigner reanalyses of IAM solutions
In Fig. 39 we show the refit of the IAM solution forthe m s = m s lattice data analyzed in section IV A. TheBreit-Wigner parameterization used in these fits for thephase shift istan δ ( E ) = E Γ( E ) m ρ − E with Γ( E ) = g ρππ p πE . (60) D. Global fits of decay constant and ρ -mesonphase-shift data of section IV B In Figs. 40 and 41 we provide the solutions of theTr M = C global fit to ρ -meson phase shift and decayconstant lattice data performed in Sect. IV B using sep-arately the scaling methods A or B. As it can be seen,the ensemble N200 (or N401) is not well reproduced withmethod A. E. Covariance matrix
In Eq. 61 we provide the correlation matrix of our fit-ting parameters.0 . − . − . − . − . − . − .
25 0 .
20 0 .
26 0 .
29 0 . − .
71 1 . . .
33 0 .
24 0 .
22 0 . − . − . − . − . − .
12 0 . . − .
091 0 .
73 0 . − .
53 0 . − . − . − . − .
34 0 . − .
091 1 . . − .
46 0 . − . − . − . − . − .
16 0 .
24 0 .
73 0 .
26 1 . . − . − . − . − . − . − .
27 0 .
22 0 . − .
46 0 .
32 1 . − .
76 0 .
54 0 .
36 0 .
29 0 . − .
25 0 . − .
53 0 . − . − .
75 1 . − . − . − . − . . − .
21 0 . − . − .
53 0 . − .
36 1 . .
95 0 .
93 0 . . − . − . − . − .
65 0 . − .
25 0 .
95 1 . .
96 0 . . − . − . − . − .
70 0 . − .
21 0 .
93 0 .
96 1 . . . − . − . − . − . . − .
41 0 .
94 0 .
95 0 .
95 1 . (61) Acknowledgments
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