Pseudoscalar meson and vector meson interactions and dynamicallygenerated axial-vector mesons

Yu Zhou, Xiu-Lei Ren, Hua-Xing Chen, and Li-Sheng Geng*

School of Physics and Nuclear Energy Engineering and International Research Center for Nucleiand Particles in the Cosmos, Beihang University, Beijing 100191, China

(Received 1 May 2014; published 15 July 2014)

The axial-vector mesons a1ð1260Þ, b1ð1235Þ, f1ð1285Þ, h1ð1170Þ, h1ð1380Þ, and K1ð1270Þ aredynamically generated in the unitized chiral perturbation theory. Such a picture has been tested extensivelyin the past few years. In this work, we calculate the interaction kernel up to Oðp2Þ and study the impact onthe dynamically generated axial-vector states. In anticipation of future lattice QCD simulations, wecalculate the scattering lengths and the pole positions as functions of the pion mass, with the light-quarkmass dependence of the kaon, the eta, and the vector mesons determined by the nf ¼ 2þ 1 lattice QCDsimulations of the PACS-CS Collaboration.

DOI: 10.1103/PhysRevD.90.014020 PACS numbers: 12.39.Fe, 12.38.Gc, 13.75.Lb

I. INTRODUCTION

Understanding the quark contents of hadrons has been atthe forefront of nonperturbative strong interaction physics.In the naive quark model, baryons are composed of threequarks or antiquarks, and mesons consist of a quark-antiquark pair. This picture works extremely well for mosthadrons discovered before 2000, except a few, such as thescalar nonet of the f0ð500Þ [1] and the Λð1405Þ [2]. On theother hand, in the past decade many newly discoveredresonances cannot be easily accommodated by the naivequark model. Some of them clearly contain more thanthe minimum number of valence quarks, such as theZcð3900Þ [3,4], while others may have components ofboth qq (qqq) and multiquark configurations. Although avast amount of experimental and theoretical studies havebeen performed to understand their nature, more studiesneed to be done.Even among the seemingly well-established and under-

stood hadrons, some of them may be more complicatedthan originally thought. One such example is the lowest-lying axial-vector mesons. It was shown in Refs. [5,6] thatthe a1ð1260Þ, b1ð1235Þ, f1ð1285Þ, h1ð1170Þ, h1ð1380Þ,and K1ð1270Þ can be built from the interactions betweenthe pseudoscalar octet of the pion and the vector nonetof the ρ within the so-called unitized chiral perturbationtheory (UChPT) approach (see Refs. [7–16] for some earlyreferences).1 There are some technical differences betweenthe two approaches. In Ref. [5], both the K1ð1270Þ andthe K1ð1400Þ are claimed to be dynamically generated,while in Ref. [6] the two poles found in the isospin 1=2and strangeness 1 channel are both claimed to be

the K1ð1270Þ—the so-called two-pole picture.2 Such adynamical picture has been put to the test in many differentscenarios, e.g., the two-pole structure of theK1ð1270Þ [20],their radiative decays [21–24], their large Nc behavior[25], the composite and elementary nature of the a1ð1260Þ[26], and its finite volume dependence [27]. All thesestudies suggest that the axial-vector mesons contain largepseudoscalar meson-vector meson components.It should be noted that the studies of Refs. [5,6] are both

based on the leading order (LO) chiral potentials. It is notclear whether the scenario of these states being dynamicallygenerated will change when higher order kernels areincluded. Given the fact that lately there have been attemptsto study these states on the lattice [28,29], such a study isurgently needed. Recently it has been shown in the DKsector that, although the inclusion of higher-order kernelsdoes not qualitatively change the conclusions, it does havean impact on the quantitative description of the latticechromodynamics (LQCD) data, more specifically, theirdependence on the light-quark masses (see, e.g., Ref. [30]).The main purpose of the present work is to include thecontributions of the next-to-leading order (NLO) chiralpotentials, and to study their impact on the LO results andon the light-quark mass dependence of the scatteringlengths and the pole positions.The paper is organized as follows. In Sec. II, we briefly

recall the basic framework of UChPT in studies of theinteractions between the pseudoscalar octet of the pion andthe vector nonet of the ρ, and introduce the NLO chiralpotentials. We study their impact on the prediction of thedynamically generated axial-vector mesons in Sec. III A. Inanticipation of future studies of these resonances on the

*lisheng.geng@buaa.edu.cn1See, e.g., Ref. [17] for a different perspective on the nature of

the axial-vector mesons in the extended linear sigma model.

2The two-pole scenario is also found for the Λð1405Þ [15,18],which seems to be a universal feature of all the studies based oncoupled channel chiral dynamics. See Ref. [19] for a recentreview.

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lattice, we calculate the scattering lengths of the pseudo-scalar-vector meson interactions and study their light-quarkmass dependence in Sec. III B. The pole positions and theirlight-quark mass dependence are studied in Sec. III C. Ashort summary is given in Sec. IV.

II. THEORETICAL FRAMEWORK

UChPT has become an invaluable tool to study inter-actions between hadrons and has played an important rolein helping understand the nature of the many newlydiscovered resonances. At the center of the UChPT isthe kernel provided by chiral dynamics and a unitarizationprocedure. Although chiral dynamics largely fixes the formof the kernel, unitarization techniques differ. Nevertheless,various unitarization techniques generally lead to similarresults and only fine details may differ. In both Refs. [5,6],the Bethe-Salpeter equation approach was adopted tounitarize the chiral kernel, which has the following sche-matic form:

T ¼ V þ VGT; ð1Þ

where V is the kernel potential, T the scattering T matrix,and G the one-loop scalar two-point function.

A. LO chiral potentials

The LO amplitudes of pseudoscalar meson-vector mesoninteractions are calculated with the following interactionLagrangian [5,6]:

LI ¼ −1

4Trfð∇μVν −∇νVμÞð∇μVν −∇νVμÞg; ð2Þ

where Tr means SU(3) trace and ∇μ is the covariantderivative defined as

∇μVν ¼ ∂μVν þ ½Γμ; Vν�; ð3Þ

where ½; � stands for the commutator and Γμ is the vectorcurrent

Γμ ¼1

2ðu†∂μuþ u∂μu†Þ ð4Þ

with

u2 ¼ U ¼ eiffiffi2

pf P: ð5Þ

In the previous equations f is the pion decay constant in thechiral limit and P and V are the SU(3) matrices containingthe octet of pseudoscalar and the nonet of vector mesons,respectively,

P ¼

0BB@

1ffiffi2

p π0 þ 1ffiffi6

p η8 πþ Kþ

π− − 1ffiffi2

p π0 þ 1ffiffi6

p η8 K0

K− K0 − 2ffiffi6

p η8

1CCA; ð6Þ

Vμ ¼

0B@

1ffiffi2

p ρ0 þ 1ffiffi2

p ω ρþ K�þ

ρ− − 1ffiffi2

p ρ0 þ 1ffiffi2

p ω K�0

K�− K�0 ϕ

1CA

μ

: ð7Þ

The two-vector–two-pseudoscalar amplitudes can beobtained by expanding the Lagrangian of Eq. (2) up totwo pseudoscalar meson fields:

LVVPP ¼ −1

4f2Trð½Vμ; ∂νVμ�½P; ∂νP�Þ; ð8Þ

which is the so-called Weinberg-Tomozawa interaction forthe VP → VP process. In the pseudoscalar octet we assumeη8 ≡ η. In the vector meson multiplet idealω1–ω8 mixing isassumed:

ϕ ¼ ω1=ffiffiffi3

p− ω8

ffiffiffiffiffiffiffiffi2=3

p; ω ¼ ω1

ffiffiffiffiffiffiffiffi2=3

pþ ω8=

ffiffiffi3

p:

ð9Þ

Throughout this work, the following phase convention isused: jπþi ¼ −j1þ 1i,jρþi ¼ −j1þ 1i, jK−i ¼ −j1=2 −1=2i and jK�−i ¼ −j1=2 − 1=2i, corresponding to jII3iisospin states.From the Lagrangian of Eq. (8) one obtains the S-wave

amplitude:

VijðsÞ ¼ −ϵ · ϵ0

8f2Cij

�3s − ðM2 þm2 þM02 þm02Þ

−1

sðM2 −m2ÞðM02 −m02Þ

�; ð10Þ

where ϵ (ϵ0) stands for the polarization four-vector of theincoming (outgoing) vector meson. The masses M (M0), m(m0) correspond to the initial (final) vector mesons andinitial (final) pseudoscalar mesons, respectively. The indi-ces i and j represent the initial and final VP states,respectively. The Cij coefficients for VP coupled channelsin the ðS; IÞ ¼ ð1; 1=2Þ, (0,0), and (0,1) sectors can befound in Ref. [6], where S denotes strangeness and Irepresents isospin.

B. NLO chiral potentials

The chiral Lagrangians relevant to the present study up tochiral order Oðp2Þ and with just one trace in flavor space(leading order in the large Nc expansion) has the followingform [31]:

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Lð2ÞVV ¼ 1

2δ1TrðVμVμuνuνÞ þ

1

2δ2TrðVμuνVμuνÞ

þ 1

2δ3TrðVμVνuμuνÞ þ

1

2δ4TrðVμVνuνuμÞ

þ 1

2δ5TrðVμuμVνuν þ VμuνVνuμÞ

þ 1

2κVTrðVμVμχþÞ; ð11Þ

where uμ ¼ ifu†∂μu − u∂μu†g and χ� ¼ u†χu† � uχ†uwith χ ¼ diagðm2

π; m2π; 2m2

K −m2πÞ.

In Eq. (11) the low-energy constant (LEC) κV is readilydetermined from the K� − ρ mass splitting:

κV ¼ M2K� −M2

ρ

2ðm2K −m2

πÞ: ð12Þ

The NLO chiral potentials, after projection into the Swave, are

VNLO ¼�δ1D

ð1Þij þ δ2D

ð2Þij

f2EPEP0 þ κV

2f2Dð3Þ

ij

�ϵ · ϵ0; ð13Þ

where EP and EP0 are the energy of the initial and final

pseudoscalar mesons, and Dð1;2;3Þij are the corresponding

Clebsch-Gordan coefficients, which are tabulated inTables IV–XII of the Appendix.

C. Unitarization

The unitarized amplitude obtained from the Bethe-Salpeter approach [6] is

T ¼ ½1þ VG�−1ð−VÞ~ϵ · ~ϵ0; ð14Þ

where ~ϵ (~ϵ0) is the spatial polarization vector of the initial

(final) vector meson and G ¼ Gð1þ 13

q2lM2

lÞ is a diagonal

matrix with the lth element, Gl being the two-meson loopfunction containing a vector and a pseudoscalar meson:

Glðffiffiffis

p Þ ¼ iZ

d4qð2πÞ4

1

ðP − qÞ2 −M2l þ iϵ

1

q2 −m2l þ iϵ

ð15Þ

with P the total incident momentum, which is ð ffiffiffis

p; 0; 0; 0Þ

in the center-of-mass frame. In the dimensional regulari-zation scheme the loop function reads

Glðffiffiffis

p Þ ¼ 1

16π2

�aðμÞ þ ln

M2l

μ2þm2

l −M2l þ s

2sln

m2l

M2l

þ qlffiffiffis

p ½lnðs − ðM2l −m2

l Þ þ 2qlffiffiffis

p Þ

þ lnðsþ ðM2l −m2

l Þ þ 2qlffiffiffis

p Þ− lnðs − ðM2

l −m2l Þ − 2ql

ffiffiffis

p Þ

− lnðsþ ðM2l −m2

l Þ − 2qlffiffiffis

p Þ − 2πi��;

ð16Þwhere μ is the dimensional regularization scale. Changesin the scale are reabsorbed in the subtraction constantaðμÞ, so the results remain scale independent. InEq. (16), ql denotes the three-momentum of the vectoror pseudoscalar meson in the center-of-mass frame and isgiven by

ql ¼1

2ffiffiffis

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½s − ðMl þmlÞ2�½s − ðMl −mlÞ2�

q; ð17Þ

where Ml and ml are the masses of the vector andpseudoscalar mesons, respectively.In Ref. [30], the power-counting issue related to the

regularization of the loop function Gl [Eq. (15)] wasdiscussed. It was found that the usual treatment, as donein Eq. (16), is reasonable by assuming that aðμÞ is in fact anOðpÞ quantity in order to ensure thatGl is ofOðpÞ. The on-shell approximation adopted in the present work is care-fully examined in Ref. [32], where special attention is givento its influence on the light-quark mass dependence of thescattering lengths and the pole positions. We show thatresults obtained from the on-shell approximation can betrusted, at least for quantities not far away from theirrespective threshold.

III. RESULTS AND DISCUSSION

In this section, we study the impact of the NLO chiralpotentials on the dynamically generated axial-vector mes-ons. In anticipation of future LQCD simulations, we furthercalculate the scattering lengths and the pole positions asfunctions of the light-quark masses. As in Ref. [6], we fixf ¼ 92 MeV, a ¼ −1.85, and μ ¼ 900 MeV. The isospinaveraged physical pseudoscalar and vector meson massesare taken from Ref. [33].

A. Impact of NLO potentials

At Oðp2Þ, there are six additional LECs. As mentionedearlier, the LEC κV can be fixed from the ρ-K� masssplitting. In addition, the contributions proportionalto the LECs δ3;4;5 are suppressed compared to thoseproportional to δ1 and δ2. Indeed, the δ3, δ4, and δ5 termsvanish in the MV → ∞ limit, where MV is a generic

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vector meson mass.3 Therefore, we only take into accountthe terms proportional to δ1 and δ2 and study their effectson the dynamically generated axial-vector mesons.Since our primary purpose is to check whether the

conclusions of Refs. [5,6] are stable against the inclusionof the NLO chiral potentials as kernels in the Bethe-Salpeter equation, we decided to vary the LECs δ1 and δ2between −1 and 1, assuming that they are of natural size,and then check whether the axial-vector mesons can still bedynamically generated. Indeed, we find that the dynamicalgeneration of the axial-vector mesons does not dependsensitively on the values of the LECs δ1 and δ2.In Fig. 1, we compare the experimental masses of the

axial-vector mesons with those obtained from the UChPTwith the LO kernel and the NLO kernel. The LECs δ1 andδ2 are fixed at 1.0 and 0.2, respectively. These values arechosen to yield an overall good agreement between theexperimental and theoretical masses. It is clear that κV termin the NLO kernel alone deteriorates the LO results. Theinclusion of the δ1 and δ2 terms can improve the

description, but not much. The overall agreement withthe experimental masses is similar to the LO results.In Tables I, II, and III, we tabulate the pole positions and

their respective couplings to each channel. The couplingsare obtained from the residues of the T-matrix elements atthe pole position s0, i.e.,

Tij ¼gigjs − s0

: ð18ÞIt can be seen that the couplings obtained with the LO andNLO kernels are similar to each other.

B. Scattering lengths of pseudoscalar mesonsand vector mesons

Scattering lengths provide vital information on the natureof hadron-hadron interactions at threshold and can show afirst hint on whether the interaction is strong enough toform shallow bound states and resonances. On the lattice,the scattering lengths can be easily calculated and theirlight-quark mass dependences prove to be of great value tounravel the nature of the underlying interactions. Recently,several studies have shown that by fitting to the LQCD dataon the scattering lengths of the pseudoscalar mesons offthe D mesons, the UChPT can dynamically generate theD�

s0ð2317Þ without a priori assumption of its existence[30,34]. Such studies demonstrate that much can beaccomplished by combining first-principles LQCD simu-lations with effective field theories, such as the UChPT.The scattering lengths of channel i with strangeness S

and isospin I are related to the diagonal T-matrix elementsTii in the following way:

aðS;IÞi ¼ −1

8πðMi þmiÞTðS;IÞii ðs ¼ ðMi þmiÞ2Þ; ð19Þ

where Mi and mi are the masses of the vector meson andthe pseudoscalar meson of channel i. Since our aim is tostudy the light-quark mass dependence of the scatteringlengths, we need to know the light-quark mass dependenceof the pseudoscalar mesons and the vector mesons. One canturn to LQCD simulations for the relevant information.Here we use the recent nf ¼ 2þ 1 results of the PACS-CSCollaboration performed with the nonperturbativelyOðaÞ-improved Wilson quark action and the Iwasaki gaugeaction [35]. From the simulated pion and kaon masses, oneobtains the following relation [36]:

m2K ¼ aþ bm2

π; ð20Þ

where a ¼ 0.291751 GeV2 and b ¼ 0.670652. In ourpresent work, we use the LO chiral perturbation theoryto relate the eta meson mass to those of the kaon and thepion, i.e.,

m2η ¼

4m2K −m2

π

3: ð21Þ

FIG. 1 (color online). The experimental axial-vector masses[33] in comparison with the corresponding UChPT results. TheLO, NLO-6th, and NLO numbers denote the masses obtainedfrom the LO kernel, the NLO kernel containing only the κV term,and the complete NLO kernel. The error bars denote theuncertainties of the experimental masses while the shaded bandsindicate the experimental widths. For the K1ð1270Þ, only thehigh-energy pole, coupling mainly to ρK, is shown.

3For a related discussion in the heavy-light meson sector, seeRef. [30].

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The vector meson masses up to Oðp2Þ can be calculatedfrom the following Lagrangian:

LVV ¼ 1

2M2

0TrðVμVμÞ þλm2

TrðVμVμχþÞ

þ λ04

TrðVμVμÞTrðχþÞ; ð22Þ

yielding

M2ρ ¼ M2

0 þ 2λmm2π þ λ0ð2m2

K þm2πÞ;

M2K� ¼ M2

0 þ 2λmm2K þ λ0ð2m2

K þm2πÞ;

M2ϕ ¼ M2

0 þ 4λmm2K − 2λmm2

π þ λ0ð2m2K þm2

πÞ;M2

ω ¼ M20 þ 2λmm2

π þ λ0ð2m2K þm2

πÞ: ð23Þ

Performing a least-squares fit to the PACS-CS data [35], thechiral limit vector meson massM0, and the LECs λm and λ0are determined to be

M0 ¼ 0.711964GeV; λm ¼ 0.48901; λ0 ¼ 0.126032;

ð24Þ

with a χ2=d:o:f: ¼ 0.723.4 We notice that the λm deter-mined from fitting the PACS-CS data is only slightlydifferent from the κV determined using the physicalρ-K� mass difference, which is 0.434.

TABLE II. Pole positions and couplings in the ðS; IÞ ¼ ð0; 0Þ channel.h1ð1170Þ h1ð1380Þ f1ð1285Þ

LO NLO LO NLO LO NLOffiffiffis

p918 − i17 925 − i29 1244 − i7 1257 − i0 1286 − i0 1289 − i0

1ffiffi2

p ðK�K þ K�KÞ � � � � � � � � � � � � 7219þ i0 7884þ i0

ϕη −46þ i13 69 − i102 −3309þ i47 −5963 − i38 � � � � � �ωη −24þ i28 711 − i427 3019 − i22 2642 − i47 � � � � � �ρπ 3452 − i1681 3576 − i1909 650 − i961 134 − i233 � � � � � �1ffiffi2

p ðK�K − K�KÞ −784þ i499 −1488þ i757 6137þ i183 6435þ i35 � � � � � �

TABLE III. Pole positions and couplings in the ðS; IÞ ¼ ð0; 1Þ channel.a1ð1260Þ b1ð1235Þ

LO NLO LO NLOffiffiffis

p1011 − i84 1013 − i106 1246 − i27 1293 − i22

1ffiffi2

p ðK�K þ K�KÞ � � � � � � 6159 − i75 5501 − i38

ϕπ � � � � � � 2085 − i384 2898 − i480ωπ � � � � � � −1867þ i299 −111 − i77ρη � � � � � � −3040þ i496 −2458þ i345ρπ 3794 − i2328 3931 − i2794 � � � � � �1ffiffi2

p ðK�K − K�KÞ −1875þ i1485 −2649þ 1928 � � � � � �

TABLE I. Pole positions and couplings in the ðS; IÞ ¼ ð1; 1=2Þ channel.K1ð1270Þ

LO NLO LO NLOffiffiffis

p1111 − i64 1119 − i68 1215 − i4 1216 − i5

ϕK 1584 − i873 2148 − i593 1095 − i401 1867 − i505ωK −1858þ i649 −2434þ i1063 −1031þ i375 −462þ i605ρK −1522þ i1155 −1121þ i1311 5268þ i298 5365þ i317K�η 27þ i156 −107þ i382 3454 − i93 4964 − i147K�π 4186 − i2098 4352 − i2075 339 − i984 391 − i1197

4Fitting the LQCD vector meson masses together with theirexperimental counterpart yields very similar results and has noappreciable effect on the results shown below.

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In Fig. 2, we show the scattering length of the dominantchannels (determined by the couplings tabulated inTables I, II, and III) in each isospin, strangeness, andG-parity sector as a function of the pion mass, with thestrangeness quark mass fixed at its physical value using theLO ChPT and the vector meson masses given by Eq. (23).The difference between the results obtained with the LOkernel and the NLO kernel indicates inherent theoreticaluncertainty. Interestingly, we notice that the aρπ showssome “threshold” effects in the h1 channel and in the a1channel. Such threshold effects can be easily understoodfrom Eq. (18) and Fig. 3. When one varies the light-quarkmasses, as routinely done in LQCD simulations, a boundstate for a certain channel can become a resonance for thatchannel; i.e., the trajectory of the threshold crosses the poletrajectory, as shown in Fig. 3. If this happens, one canimmediately see that for s approaching s0 from below,a → ∞, while just above s0, a → −∞. In performingLQCD simulations of scattering lengths, one needs to beaware of such a scenario.

In principle, one can compare directly the scatteringlengths at unphysical light-quark masses with thoseobtained from a LQCD simulation. For instance, inRef. [29], the scattering length of ρ and π mesons, aρπ ,is found to be 0.62(28) at a pion mass of 266 MeV. In ourcase, the aρπ is much larger at the order of 10, as can beseen from Fig. 2. Such a comparison needs to be takenwith caution, however. First, the LQCD simulations areperformed with nf ¼ 2 configurations. But more impor-tantly, the threshold effects can make such a comparisonaround the “threshold” region unreliable because theseeffects depend sensitively on the particular light-quarkmass dependence pattern of the masses of the buildingblocks, in the present work, those of the pseudoscalar andvector mesons. This may or may not be realized for aparticular LQCD simulation. That being said, for reso-nances consisting of large hadron-hadron components,one may need to study the related scattering lengthscarefully in order to not misinterpret the data if such ascenario occurs.

FIG. 2 (color online). Scattering lengths in the dominant channels of each isospin, strangeness, and G-parity ðS; IðGÞÞ sector.

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C. Light-quark mass dependence of the pole positions

Because LQCD simulations are generally performedwith larger than physical light-quark masses, the so-obtained resonance properties are not those of the physicalworld. It is of great importance to have a reliable approachto perform the necessary chiral extrapolations. In addition,the light-quark mass dependence of the pole positions canyield valuable information on the nature of the resonances,as has been argued in a number of previous studies [37,38].In Fig. 3, the real parts of the pole positions corresponding

to the two K1ð1270Þ states, a1ð1260Þ, b1ð1235Þ, f1ð1285Þ,h1ð1170Þ, and h1ð1380Þ are shown as functions of the pionmass. It is interesting to note that all the states exist in therange ofmπ to 700MeV, and thus LQCD simulations shouldhave no problem in identifying them even at unphysicallight-quark masses. In addition, the two K1ð1270Þ statespersist and remain well separated, thus allowing for thepossibility of being identified on the lattice. Recently, astrategy has been proposed to extract the two-pole structure

of theΛð1405Þ from LQCD simulations in a finite box [39].Such a strategy may also be employed to study the two-polestructure of the K1ð1270Þ on the lattice.In Fig. 4, the imaginary parts of the pole positions are

shown. It is clear that as the light-quark mass becomes large,these axial-vector mesons become bound, while as the pionmass decreases smaller than 2 or 3 times (depending on thechannel) of its physical value, they become a resonance.Therefore, to obtain reliable results, LQCD simulations atsmall light-quark masses may need to take into accountcoupled channel effects and final state interactions explicitly,among others. Such studies arevery demanding both numeri-cally and theoretically. Nevertheless, encouraging attemptshave been made recently.Finally, we should mention that the vector mesons have

finite widths, particularly those of the K� and the ρ. Wehave checked that including the widths as suggested inRef. [20] does not qualitatively change the obtained polepositions and couplings. On the other hand, we have no

FIG. 3 (color online). Real part of the pole positions, Reffiffiffiffiffis0

p, as a function of mπ . The dashed lines indicate the thresholds of the

corresponding coupled channels, respectively.

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information on how the widths change with varying light-quark masses, and therefore, we cannot incorporate thewidths into our studies of the scattering lengths and the polepositions without making further assumptions. We willleave such a study to a future work.

IV. SUMMARY

In this work, we have studied the effects of the next-to-leading order chiral potential on the dynamically generatedaxial-vector mesons. We found that the inclusion of thehigher-order kernel does not change the results obtainedwith the leading-order kernel in any significant way, andthus, it lends more confidence to the dynamical picture[5,6]. In anticipation of future LQCD simulations of theseresonances, utilizing the PACS-CS simulations of thepseudoscalar and vector meson masses, we have calculatedthe scattering lengths and the pole positions as functions ofthe pion mass. These results, when contrasted with futureLQCD simulations, may provide a clue to the true nature ofthe axial-vector mesons.

We have shown that when the masses of the pseudoscalarand the vector mesons vary with the light-quark masses,one may observe thresholdlike effects in the relatedscattering lengths. Future LQCD simulations may needto be carefully analyzed and should not be misinterpreted ifsuch a scenario is realized.

ACKNOWLEDGMENTS

We thank Dr. Eulogio Oset for a careful reading of thismanuscript and for enlightening discussions. X.-L.R.acknowledges support from the Innovation Foundationof Beihang University for Ph.D. graduates. This work issupported in part by the National Natural ScienceFoundation of China under Grants No. 11375024 andNo. 11205011, the Fundamental Research Funds for theCentral Universities, and the New Century ExcellentTalents in University Program of Ministry of Educationof China under Grant No. NCET-10-0029.

FIG. 4 (color online). Imaginary part of the pole positions, Imffiffiffiffiffis0

p, as a function of mπ .

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APPENDIX: Dij COEFFICIENTS IN THE NLO KERNEL

In this section, we tabulate the Clebsch-Gordan coefficients Dð1;2;3Þij appearing in the NLO chiral potentials [Eq. (13)] for

ðS; IÞ ¼ ð1; 1=2Þ, (0,0), and (0,1).

TABLE IV. Dð1Þij of Eq. (13) for ðS; IÞ ¼ ð1; 1

2Þ.

ϕK ωK ρK K�η K�π

ϕK 2 0 0 − 1ffiffi6

pffiffi32

q

ωK 0 1 −ffiffiffi3

p− 1

2ffiffi3

pffiffi3

p2

ρK 0 −ffiffiffi3

p1 1

212

K�η − 1ffiffi6

p − 1

2ffiffi3

p 12

53

1

K�πffiffi32

q ffiffi3

p2

12

1 1

TABLE V. Dð1Þij of Eq. (13) for ðS; IÞ ¼ ð0; 0Þ.

G 1ffiffi2

p ðK�K þ K�KÞ ϕη ωη ρπ 1ffiffi2

p ðK�K − K�KÞþ 1ffiffi

2p ðK�K þ K�KÞ 3 0 0 0 0

− ϕη 0 83

0 0 −ffiffi23

q− ωη 0 0 2

3−2 − 1ffiffi

3p

− ρπ 0 0 −2 2 −ffiffiffi3

p

− 1ffiffi2

p ðK�K − K�KÞ 0 −ffiffi23

q− 1ffiffi

3p −

ffiffiffi3

p3

TABLE VI. Dð1Þij of Eq. (13) for ðS; IÞ ¼ ð0; 1Þ.

G 1ffiffi2

p ðK�K þ K�KÞ ϕπ ωπ ρη ρπ 1ffiffi2

p ðK�K − K�KÞþ 1ffiffi

2p ðK�K þ K�KÞ 1 −

ffiffiffi2

p−1 1ffiffi

3p 0 0

þ ϕπ −ffiffiffi2

p0 0 0 0 0

þ ωπ −1 0 2 2ffiffi3

p 0 0

þ ρη 1ffiffi3

p 0 2ffiffi3

p 23

0 0

− ρπ 0 0 0 0 2 −ffiffiffi2

p

− 1ffiffi2

p ðK�K − K�KÞ 0 0 0 0 −ffiffiffi2

p1

TABLE VII. Dð2Þij of Eq. (13) for ðS; IÞ ¼ ð1; 1

2Þ.

ϕK ωK ρK K�η K�π

ϕK 0ffiffiffi2

p−

ffiffiffi6

p−2

ffiffi23

q0

ωKffiffiffi2

p0 0 1ffiffi

3p

ffiffiffi3

p

ρK −ffiffiffi6

p0 0 −1 −3

K�η −2ffiffi23

q1ffiffi3

p −1 − 43

−2K�π 0

ffiffiffi3

p−3 −2 0

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TABLE VIII. Dð2Þij of Eq. (13) for ðS; IÞ ¼ ð0; 0Þ.

G 1ffiffi2

p ðK�K þ K�KÞ ϕη ωη ρπ 1ffiffi2

p ðK�K − K�KÞþ 1ffiffi

2p ðK�K þ K�KÞ −6 0 0 0 0

− ϕη 0 83

0 0 −4ffiffi23

q− ωη 0 0 2

3−2 2ffiffi

3p

− ρπ 0 0 −2 6 −2ffiffiffi3

p

− 1ffiffi2

p ðK�K − K�KÞ 0 −4ffiffi23

q2ffiffi3

p −2ffiffiffi3

p6

TABLE IX. Dð2Þij of Eq. (13) for ðS; IÞ ¼ ð0; 1Þ.

G 1ffiffi2

p ðK�K þ K�KÞ ϕπ ωπ ρη ρπ 1ffiffi2

p ðK�K − K�KÞþ 1ffiffi

2p ðK�K þ K�KÞ 2 0 −2 − 2ffiffi

3p 0 0

þ ϕπ 0 0 0 0 0 0þ ωπ −2 0 2 2ffiffi

3p 0 0

þ ρη − 2ffiffi3

p 0 2ffiffi3

p 23

0 0

− ρπ 0 0 0 0 −4 2ffiffiffi2

p

− 1ffiffi2

p ðK�K − K�KÞ 0 0 0 0 2ffiffiffi2

p−2

TABLE X. Dð3Þij of Eq. (13) for ðS; IÞ ¼ ð1; 1

2Þ.

ϕK ωK ρK K�η K�π

ϕK −4m2K 0 0 1ffiffi

6p ð5m2

K − 3m2πÞ −

ffiffi32

qðm2

K þm2πÞ

ωK 0 −2m2K 2

ffiffiffi3

pm2

K1

2ffiffi3

p ð5m2K − 3m2

πÞ −ffiffi3

p2ðm2

K þm2πÞ

ρK 0 2ffiffiffi3

pm2

K −2m2K − 1

2ð5m2

K − 3m2πÞ − 1

2ðm2

K þm2πÞ

K�η 1ffiffi6

p ð5m2K − 3m2

πÞ 1

2ffiffi3

p ð5m2K − 3m2

πÞ − 12ð5m2

K − 3m2πÞ − 4

3ð8m2

K − 3m2πÞ −2m2

π

K�π −ffiffi32

qðm2

K þm2πÞ −

ffiffi3

p2ðm2

K þm2πÞ − 1

2ðm2

K þm2πÞ −2m2

π − 83m2

π

TABLE XI. Dð3Þij of Eq. (13) for ðS; IÞ ¼ ð0; 0Þ.

G 1ffiffi2

p ðK�K þ K�KÞ ϕη ωη ρπ 1ffiffi2

p ðK�K − K�KÞþ 1ffiffi

2p ðK�K þ K�KÞ −6m2

K 0 0 0 0

− ϕη 0 323ðm2

π − 2m2KÞ 0 0

ffiffi23

qð5m2

K − 3m2πÞ

− ωη 0 0 − 83m2

π 4m2π

1ffiffi3

p ð5m2K − 3m2

πÞ− ρπ 0 0 4m2

π − 163m2

π

ffiffiffi3

p ðm2K þm2

πÞ− 1ffiffi

2p ðK�K − K�KÞ 0

ffiffi23

qð5m2

K − 3m2πÞ 1ffiffi

3p ð5m2

K − 3m2πÞ

ffiffiffi3

p ðm2K þm2

πÞ −6m2K

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[1] R. L. Jaffe, Phys. Rev. D 15, 267 (1977).[2] R. H. Dalitz and S. F. Tuan, Phys. Rev. Lett. 2, 425 (1959).[3] M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett.

110, 252001 (2013).[4] Z. Q. Liu et al. (Belle Collaboration), Phys. Rev. Lett. 110,

252002 (2013).[5] M. F. M. Lutz and E. E. Kolomeitsev, Nucl. Phys.A730, 392

(2004).[6] L. Roca, E. Oset, and J. Singh, Phys. Rev. D 72, 014002

(2005).[7] N. Kaiser, P. B. Siegel, and W. Weise, Nucl. Phys. A594,

325 (1995).[8] A. Dobado and J. R. Pelaez, Phys. Rev. D 56, 3057 (1997).[9] J. A. Oller and E. Oset, Nucl. Phys. A620, 438 (1997);

A652, 407(E) (1999).[10] E. Oset and A. Ramos, Nucl. Phys. A635, 99 (1998).[11] B. Krippa, Phys. Rev. C 58, 1333 (1998).[12] J. A. Oller, E. Oset, and J. R. Pelaez, Phys. Rev. D 59,

074001 (1999); 60, 099906(E) (1999); 75, 099903(E)(2007).

[13] N. Kaiser, Eur. Phys. J. A 3, 307 (1998).[14] J. A. Oller and E. Oset, Phys. Rev. D 60, 074023 (1999).[15] J. A. Oller and U. G. Meissner, Phys. Lett. B 500, 263

(2001).[16] M. F. M. Lutz and E. E. Kolomeitsev, Nucl. Phys.A700, 193

(2002).[17] D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa, and D. H.

Rischke, Phys. Rev. D 87, 014011 (2013).[18] D. Jido, J. A. Oller, E. Oset, A. Ramos, and U. G. Meissner,

Nucl. Phys. A725, 181 (2003).[19] T. Hyodo and D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012).[20] L. S. Geng, E. Oset, L. Roca, and J. A. Oller, Phys. Rev. D

75, 014017 (2007).[21] L. Roca, A. Hosaka, and E. Oset, Phys. Lett. B 658, 17

(2007).

[22] M. F. M. Lutz and S. Leupold, Nucl. Phys. A813, 96(2008).

[23] H. Nagahiro, L. Roca, and E. Oset, Phys. Rev. D 77, 034017(2008).

[24] H. Nagahiro, L. Roca, A. Hosaka, and E. Oset, Phys. Rev. D79, 014015 (2009).

[25] L. S. Geng, E. Oset, J. R. Pelaez, and L. Roca, Eur. Phys. J.A 39, 81 (2009).

[26] H. Nagahiro, K. Nawa, S. Ozaki, D. Jido, and A. Hosaka,Phys. Rev. D 83, 111504 (2011).

[27] L. Roca and E. Oset, Phys. Rev. D 85, 054507 (2012).[28] S. Prelovsek, C. B. Lang, D. Mohler, and M. Vidmar, Proc.

Sci., LATTICE2011 (2011) 137.[29] C. B. Lang, L. Leskovec, D. Mohler, and S. Prelovsek,

J. High Energy Phys. 04 (2014) 162.[30] M. Altenbuchinger, L.-S. Geng, and W. Weise, Phys. Rev. D

89, 014026 (2014).[31] J. Prades, Z. Phys. C 63, 491 (1994); 11, 571(E)

(1999).[32] M. Altenbuchinger and L.-S. Geng, Phys. Rev. D 89,

054008 (2014).[33] J. Beringer et al. (Particle Data Group Collaboration), Phys.

Rev. D 86, 010001 (2012).[34] L. Liu, K. Orginos, F.-K. Guo, C. Hanhart, and U.-G.

Meissner, Phys. Rev. D 87, 014508 (2013).[35] S. Aoki et al. (PACS-CS Collaboration), Phys. Rev. D 79,

034503 (2009).[36] X.-L. Ren, L. S. Geng, J. Martin Camalich, J. Meng, and

H. Toki, J. High Energy Phys. 12 (2012) 073.[37] M. Cleven, F.-K. Guo, C. Hanhart, and U.-G. Meissner, Eur.

Phys. J. A 47, 19 (2011).[38] C. Hanhart, J. R. Pelaez, and G. Rios, Phys. Rev. Lett. 100,

152001 (2008).[39] A. Martinez Torres, M. Bayar, D. Jido, and E. Oset, Phys.

Rev. C 86, 055201 (2012).

TABLE XII. Dð3Þij of Eq. (13) for ðS; IÞ ¼ ð0; 1Þ.

G 1ffiffi2

p ðK�K þ K�KÞ ϕπ ωπ ρη ρπ 1ffiffi2

p ðK�K − K�KÞþ 1ffiffi

2p ðK�K þ K�KÞ −2m2

K

ffiffiffi2

p ðm2K þm2

πÞ m2K þm2

π − 1ffiffi3

p ð5m2K − 3m2

πÞ 0 0

þ ϕπffiffiffi2

p ðm2K þm2

πÞ 0 0 0 0 0

þ ωπ m2K þm2

π 0 −4m2π − 4ffiffi

3p m2

π 0 0

þ ρη − 1ffiffi3

p ð5m2K − 3m2

πÞ 0 − 4ffiffi3

p m2π − 8

3m2

π 0 0

− ρπ 0 0 0 0 −6m2π

ffiffiffi2

p ðm2K þm2

πÞ− 1ffiffi

2p ðK�K − K�KÞ 0 0 0 0

ffiffiffi2

p ðm2K þm2

πÞ −2m2K

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