PHYSICAL REVIEW D 102, 036005 (2020)

Chiral anomaly and the pion properties in the light-front quark model

Ho-Meoyng Choi Department of Physics, Teachers College, Kyungpook National University, Daegu 41566, Korea

Chueng-Ryong Ji Department of Physics, North Carolina State University, Raleigh, North Carolina 27695-8202, USA

(Received 16 June 2020; accepted 20 July 2020; published 6 August 2020)

We explore the link between the chiral symmetry of QCD and the numerical results of the light-front quark model, analyzing both the two-point and three-point functions of the pion. Including the axial-vector coupling as well as the pseudoscalar coupling in the light-front quark model, we discuss the implication of the chiral anomaly in describing the pion decay constant, the pion-photon transition form factor and the electromagnetic form factor of the pion. In constraining the model parameters, we find that the chiral 2 anomaly plays a critical role and the analysis of FπγðQ Þ in timelike region is important. Our results indicate that the constituent quark picture is effective for the low and high Q2 ranges implementing the quark mass evolution effect as Q2 grows.

DOI: 10.1103/PhysRevD.102.036005

I. INTRODUCTION the corresponding form factor not only in the spacelike region but also in the timelike region [12]. In particular, we Due to just a single hadron involvement, the meson- presented the new direct method to explore the timelike photon transition is well known to be the simplest exclusive region without resorting to mere analytic continuation from process in testing the quantum chromodynamics (QCD) spacelike to timelike region [12]. Our direct calculation in and understanding the structure of the meson [1]. As the timelike region showed the complete agreement with not pion is regarded as the lightest pseudo-Goldstone boson only the analytic continuation result from spacelike region arising from the spontaneous symmetry breaking of the but also the result from the dispersion relation (DR) chiral symmetry in QCD, it is particularly important to between the real and imaginary parts of the form factor. analyze the pion production process via the two-photon γγ → π This development added more predictability to the light- collision, , which involves only one transition form – F Q2 q2 −Q2 front quark models (LFQMs) [12 22] which have been factor (TFF) πγð Þ, where ¼ is the squared successful in describing hadron phenomenology based on momentum transfer of the virtual photon. Its complete the constant constituent quark and antiquark masses. More understanding requires a formulation capable of explaining specifically, our LFQM [14,15] built on the variational both the nonperturbative Adler-Bell-Jackiw (ABJ) anomaly principle to the QCD-motivated Hamiltonian provided a F 0 (or the chiral anomaly) [2], which determines πγð Þ when good description of the pion electromagnetic and transition 2 both photons are real (i.e., Q ¼ 0), and simultaneously form factors [12,14,17,22]. the perturbative QCD (pQCD) prediction that governs the We have further discussed the link between the chiral 2 2 behavior of FπγðQ Þ at large momentum transfer Q region. symmetry of QCD and the numerical results of the LFQM, 2 Since the publication of the BABAR data [3] for FπγðQ Þ in analyzing both the two-point and three-point functions of a 0 ≤ Q2 ≤ 40 GeV2 showing the violation of the scaling pseudoscalar meson from the perspective of the vacuum law predicted by pQCD [1], many theoretical efforts [4–11] fluctuation consistent with the chiral symmetry of QCD have been made to clarify this issue. [23]. This link is due to a pair creation of particles with zero In an effort to examine the issue of the scaling light-front (LF) longitudinal momenta from the vacuum, 2 2 behavior of Q FπγðQ Þ, we have attempted to analyze which captures the vacuum effect for the consistency with the chiral symmetry properties of the strong interactions. With this link, the zero-mode contribution [16,24–29] in the meson decay process could effectively accommodate Published by the American Physical Society under the terms of the effect of vacuum fluctuation consistent with the chiral the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to symmetry of the strong interactions. In this respect, the the author(s) and the published article’s title, journal citation, LFQM with effective degrees of freedom represented by the and DOI. Funded by SCOAP3. constituent quark and antiquark was linked to the QCD

2470-0010=2020=102(3)=036005(15) 036005-1 Published by the American Physical Society HO-MEOYNG CHOI and CHUENG-RYONG JI PHYS. REV. D 102, 036005 (2020) since the zero-mode link to the QCD vacuum could provide symmetry. In particular, the ABJ anomaly [2] is the key to the view of an effective zero-mode cloud around the understand the π0 → γγ decay rate resolving the issue with quark and antiquark inside the meson. While the constitu- the Sutherland-Veltman theorem [38]. As the Thompson ents are dressed by the zero-mode cloud, they are ex- low-energy limit works for the Compton scattering on any pected to satisfy the chiral symmetry consistent with the target, the Sutherland-Veltamn theorem reveals that the QCD. Our numerical results [30] were indeed consistent nonanomalous term must vanish in the case when both with this expectation and effectively indicated that the photons are on-mass-shell [39]. Only the chiral anomaly is constituent quark and antiquark in the standard LFQM capable of explaining the π0 decay to the two real photons. [12,14,15,17,18,22] could be considered indeed as the Thus, it appears important to analyze the contribution from dressed constituents including the zero-mode quantum the axial-vector coupling together with the contribution fluctuations from the QCD vacuum. from the pseudoscalar coupling to explore the correlation Moreover, the lattice QCD results [31] indicated that the between the nontrivial QCD vacuum effect and the con- mass difference between η0 and pseudoscalar octet mesons stituent quark mass as well as the parameter of the trial wave due to the complicated nontrivial vacuum effect increases function in the LFQM built on the variational principle. or decreases as the extrapolating quark mass decreases or In this work, we include the axial-vector coupling in increases; i.e., the effect of the topological charge con- addition to the pseudoscalar coupling in our LFQM for the tribution should be small as the quark mass increases. This pion to explore a well-defined chiral limit providing still correlation between the quark mass and the nontrivial QCD a good description of the pion electromagnetic and tran- vacuum effect further supported the development of our sition form factors [12,14,17]. To examine the relative LFQM [14] because the complicated non-trivial vacuum contribution between the pseudoscalar coupling and the effect in QCD could be traded off by rather large constitu- axial-vector coupling, we take the more general vertex ent quark masses. As a precursor of this development of Γπ ¼ðAπ þ Bπ=PÞγ5 which goes beyond the specific vertex LFQM, the constituent quark model in the light-front Γπ ¼ðMπ þ =PÞγ5 previously taken in the spin-averaged quantization approach appeared based on the spin-averaged mass scheme [32,33] for the pion spin-orbit structure with π ρ μ mass scheme [32,33] of taking the and meson masses the four momentum P . We then describe the pion proper- M 1 M 3 M ≈ 2 2 equal to the spin-averaged value av ¼ð4 π þ 4 ρÞExp ties such as fπ;FπγðQ Þ;FπðQ Þ depending on the varia- 612 MeV. In retrospect, such early development was an tion of the quark mass in a self-consistent manner within attempt to trade off the non-perturbative QCD effect with this model. the constituent quark mass averaged between the π and ρ The paper is organized as follows. In Sec. II,we mesons although the spin-averaged mass scheme itself was introduce the spin-orbit wave function of the pion obtained too naive to accommodate the complicate non-trivial from the operator Γπ ¼ðAπ þ Bπ=PÞγ5 and show the chiral vacuum effect. More sophisticated analysis was developed limit expression of the spin-orbit wave function. We also later to take into account the effect of the mass evolution compare it with our previous spin-orbit wave function F Q2 (from constituent to current quark mass) on πð Þ at low obtained from the operator Γπ ¼ Aπγ5. In Sec. III, we apply Q2 2 2 and intermediate [34]. We have then also discussed a our LFQM for the calculation of fπ;FπγðQ Þ and FπðQ Þ constraint of conformal symmetry in the analysis of the pion using both constituent quark mass and the chiral limit elastic form factor both in spacelike and timelike regions result. Especially, we explicitly obtain the analytic form of [35,36], confirming the anti–de Sitter space geometry/ 2 fπ and FπγðQ Þ in the exact chiral limit (Mπ;m→ 0). We conformal field theory (AdS=CFT) correspondence [37]. also show that our chiral limit result for twist-2 pion While the early LFQM approach of the spin-averaged distribution amplitude (DA), which encodes the nonper- mass scheme [32,33] included both the pseudoscalar and turbative information on the pion, is exactly the same as the axial-vector couplings for the pseudoscalar meson vertex, anti-de Sitter/conformal field theory (AdS=CFT) prediction only the specific vertex given by Γπ ¼ðMπ þ =PÞγ5 with μ of the asymptotic DA [37,40,41]. In Sec. IV, we discuss the four momentum P was taken for the coupling with the how to determine the model parameters and show the constituent quark and antiquark in the triangle loop F Q2 amplitude. Since then, the later development of most numerical results of the pion DA, the pion TFF πγð Þ standard LFQM [12–22] including ours [12,14,17] built both in spacelike and timelike regions covering the full on the variational principle used typically only the pseu- momentum transfer region, and the pion electromagnetic F Q2 doscalar vertex given by Γπ ¼ Aπγ5, where Aπ is a constant form factor πð Þ in the spacelike region. In this section, of proportionality with the mass dimension which gets we show the results by varying the quark mass in a self- absorbed into the normalization of the spin-orbit wave consistent way so that one can effectively see the evidence 2 2 function. However, the generalization of the vertex includ- of quark mass evolution effect as Q changes for FπγðQ Þ 2 ing the axial vector coupling deserves further consideration and FπðQ Þ. Summary and conclusions follow in Sec. V. to include the exact chiral limit ðMπ;m→ 0Þ phenomena, In the Appendix, we provide the derivation of our new spin- where m represents the uðdÞ quark mass respecting isospin orbit wave function.

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II. MODEL DESCRIPTION pion with the same constituent quark and antiquark masses mQ ¼ mQ¯ ≡ m, we take the harmonic oscillator (HO) wave The key approximation in the LFQM is the mock-hadron approximation [42] to saturate the Fock state expansion by function as our trial wave function the constituent quark and antiquark and treat that Fock state rffiffiffiffiffiffiffi 3=4 2 as a free state as far as the spin-orbit part is concerned. pffiffiffiffiffiffiffiffiffi 4π ∂k k⃗ z − 2 ϕR x; k⊥ P ¯ e 2β ; 3 The assignment of the quantum numbers such as angular ð Þ¼ QQ β3=2 ∂x ð Þ momentum, parity and charge conjugation to the LF wave function is given by the Melosh transformation [43]. ∂k =∂x M =4x 1 − x For example, the pion state jπi is represented by where z ¼ 0 ð Þ is the Jacobian of the π ¯ ¯ x; k → k⃗ k ;k jπi¼Ψ ¯ jQQi, where QðQÞ is the effective dressed variable transformation f ⊥g ¼ð ⊥ zÞ with QQ 2 2 2 M0 ¼ðk⊥ þ m Þ=xð1 − xÞ being the invariant mass quark (antiquark). That is, the pion state as a valence 2 2 ¯ μ þ − k⃗ k⃗ k2 k2 QQ bound state with momentum P ¼ðP ;P ; P⊥Þ is square. In particular, is given by ¼ ⊥ þ z where determined by the light-front wave function (LFWF) kz ¼ðx − 1=2ÞM0 and the normalization of ϕR is given by

π Z Z Ψ ¯ ≡ Ψπðxi; ki⊥; λiÞ¼ϕRðxi; ki⊥Þχðxi; ki⊥; λiÞ; ð1Þ 1 d2k QQ dx ⊥ ϕ x; k 2 P : 3 j Rð ⊥Þj ¼ QQ¯ ð4Þ 0 16π þ þ where xi ¼ ki =P , ki⊥, and λi are the Lorentz-invariant longitudinal-, transverse-momenta and the helicity of each The covariant form of the spin-orbit wave function for constituent quarkP (antiquark), respectively,P with the proper- PC −þ 2 x 1 2 k 0 ϕ the pion (J ¼ 0 ) is given by ties satisfying i¼1 i ¼ and i¼1 i⊥ ¼ . Here, R is the radial wave function which is taken as the trial wave χ χ x; k N u¯ k Γ υ k ; function following the variational principle and is the LF λ1λ2 ð ⊥Þ¼ λ1 ð 1Þ π λ2 ð 2Þ ð5Þ spin-orbit wave function which is obtained by the inter- action-independent Melosh transformation from the ordi- where nary equal-time static spin-orbit wave function assigned by the quantum numbers JPC. Γ A B =P γ ; The LFWF is normalized according to π ¼ð π þ π Þ 5 ð6Þ

Ψπ Ψπ P ; h QQ¯ j QQ¯ i¼ QQ¯ ð2Þ and N is the normalization constant satisfying the unitary condition hχλ λ jχλ λ i¼1. ¯ 1 2 1 2 where PQQ¯ is the probability of finding the QQ component Here, we set Aπ ¼ Mπ and Bπ being a free parameter. ϕ x; k χ in the LFWF. For the radial wave function Rð ⊥Þ of the Explicitly, we obtain the normalized form of λ1λ2 as

 L  1 −k M mM þ xð1 − xÞBπεB χ x; k ffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; λ1λ2 ð ⊥Þ¼p 2 2 2 R ð7Þ 2 M k⊥ þ½mM þ xð1 − xÞBπεB −mM − xð1 − xÞBπεB −k M

RðLÞ x y where M ¼ Mπ þ 2Bπm, k ¼ k ik , and εB ¼ The operator Γπ givenbyEq.(6) can be compared 2 2 Mπ − M0 corresponds to the binding energy. The detailed with those used in the previous LFQMs using two derivation of Eq. (7) is given in the Appendix. Furthermore, popular schemes, i.e., the spin-averaged meson mass in the chiral limit (i.e., Mπ;m→ 0), Eq. (7) reduces to scheme and the invariant meson mass scheme. For the   spin-averaged meson mass scheme used in [32,33], Γ M =P γ 1 01 π ¼ð av þ Þ 5 was taken, i.e., the spin-averaged χchiral χ ffiffiffi −B ; 1 3 λ1λ2 ¼ lim λ1λ2 ¼ p sgnð πÞ ð8Þ M M M Mπ ;m→0 2 −10 meson mass av ¼ð4 π þ 4 ρÞExp was used instead of the physical pion mass as mentioned earlier in Sec. I. For the invariant mass scheme used in [12–18],the where sgnð−BπÞ¼−sgnðBπÞ is the sign function of Bπ, meson mass was mocked by the invariant mass M0 and i.e., sgnðBπÞ¼1 for Bπ > 0, sgnðBπÞ¼−1 for Bπ < 0 Γ A γ and sgnð0Þ¼0, respectively. This reveals already the π ¼ π 5 was taken to yield the normalized spin-orbit B ≠ 0 χðM0Þ x; k N u¯ k γ υ k nontriviality of the axial-vector coupling, i.e., π ,in wave function λ1λ2 ð ⊥Þ¼ λ1 ð 1Þ 5 λ2 ð 2Þ as also the chiral limit. We shall illustrate the way of determining mentioned in Sec. I. Its explicit normalized form for the the value of Bπ phenomenologically in Sec. IV. pion is then given by

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 L  1 M 1 −k m ψ x; k ffiffiffi χ − χ ϕ x; k ; χð 0Þ ffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : πð ⊥Þ¼p ð ↑↓ ↓↑Þ Rð ⊥Þ ð12Þ λ1λ2 ¼ p 2 2 R ð9Þ 2 2 k⊥ þ m −m −k corresponds to the valence jQQ¯ i state having Jz ¼ Sz ¼ However, we note that the more general spin-orbit wave Lz ¼ 0 together with the helicity components of the spin- functiongivenbyEq.(7) yields indeed Eq. (9) regard- orbit wave function given in Eq. (7). The twist-2 pion DA less of the value of Bπ in the limit Mπ → M0 (or εB → 0) ϕπðxÞ results from the k⊥ integral of ψ πðx; k⊥Þ in the LF taken in the invariant meson mass scheme. This indicates gauge Aþ ¼ 0 [1] that the order of the two limits, i.e., the zero-binding Z M → M ε → 0 Q2 d2k limit ( π 0 or B ) vs the chiral limit ϕ x ⊥ ψ x; k ; πð Þ¼ 3 πð ⊥Þ ð13Þ (Mπ;m→ 0), do not commute in general regardless of 16π the value of Bπ. While the LFQM adopting Eq. (9) has proven to be very effective in predicting various physical and satisfies the normalization condition Z observables, its non-commutability with the chiral 1 fπ limit hinders the description of the chiral anomaly dxϕπðxÞ¼ pffiffiffiffiffiffiffiffi : ð14Þ 0 2 2Nc which determines Fπγð0Þ. Unlike Eq. (8),Eq.(9) yields ðM0Þ the ordinary helicity components behaving as χ ¼ In the chiral limit (i.e., Mπ;m→ 0), the spin-orbit part ↑↓ pffiffiffi ðM0Þ χ − χ = 2 −B χ → 0 in the chiral limit (i.e., Mπ;m→ 0). in Eq. (12) becomes ð ↑↓ ↓↑Þ ¼ sgnð πÞ [see ↓↑ −B 1 B < 0 In contrast to the previous works, we now take the more Eq. (8)]. By taking sgnð πÞ¼ (or π ), we then general spin-orbit structure of the pion given by Eq. (6) obtain the decay constant and the pion DA analytically as pffiffiffi   which leads to the spin-orbit wave function given by Eq. (7) pffiffiffiffiffiffiffiffiffi 3β 5 that is versatile enough to explore the chiral limit as well as fchiral P Γ ; π ¼ QQ¯ 3=4 1=4 4 ð15Þ the previous LFQM adopting the spin-averaged meson 2 π mass scheme or the invariant mass scheme. and pffiffiffi 2 2fchiral pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi III. APPLICATION: PION DECAY CONSTANT, chiral π ϕπ ðxÞ¼ pffiffiffi xð1 − xÞ; ð16Þ TRANSITION, AND ELASTIC FORM FACTORS 3π

The charged pion decay constant is given in terms of the respectively. We should note that this derivation of the matrix element of the weak current between a physical pion chiral limit result could not be made in the case of Γπ ¼ and the vacuum state Aπγ5 [see Eq. (9)] due to the lack of the axial-vector coupling. Our chiral limit result for twist-2 pion DA μ μ h0jq¯γ ð1 − γ5ÞqjπðPÞi ¼ ifπP : ð10Þ given by Eq. (16) is exactly the same as the AdS=CFT prediction of the asymptotic DA [37,40,41]. We also find chiral chiral The experimental value of the pion decay constant is the exactly the same ratio ϕπ ðxÞ=fπ even if we use fExp 130 2 1 7 ϕPL x; k ∝ π ¼ . ð . Þ MeV [44]. Using the plus component thepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi power-law type radial wave function, R ð ⊥Þ μ ⃗2 2 −2 ( ¼þ) of the current, we obtain the decay constant in ∂kz=∂xð1 þ k =β Þ instead of using the HO wave terms of the valence pion LFWF [1] function. This appears to indicate that the ratio chiral chiral Z Z ϕπ ðxÞ=fπ is model independent. pffiffiffiffiffiffiffiffi 1 d2k F f 2 2N dx ⊥ ψ x; k ; The neutral pion transition form factor (TFF) πγ for the π ¼ c 3 πð ⊥Þ ð11Þ 0 0 16π π → γγ transition is defined from the matrix element μ μ 0 of electromagnetic current Γ ¼hγðP − qÞjJemjπ ðPÞi as where Nc is the number of color and follows:

FIG. 1. One-loop Feynman diagrams that contribute to π0ðPÞ → γðqÞγðP0Þ. The single covariant Feynman diagram (a) is in principle the same as the sum of the two LF time-ordered diagrams (b) and (c), respectively.

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μ 0 2 2 μνρσ hγðP − qÞjJemjπ ðPÞi ¼ ie FπγðQ Þε Pνερqσ; ð17Þ model parameters as we shall discuss in Sec. IV. On the other hand, the decay width for π0 → γγ is obtained from where Pμ and qμ are the four-momenta of the incident pion the TFF at Q2 ¼ 0 via and virtual photon, respectively, and ερ is the transverse π 2 3 2 Γ 0 α M F 0 ; polarization four-vector of the final (on-shell) photon. π →γγ ¼ 4 em πj πγð Þj ð19Þ As we discussed in [12], this process is illustrated by the α Feynman diagram in Fig. 1(a), where the intermediate where em is the fine structure constant. The form factor F 0 quark and antiquark propagators of mass m ¼ mQ ¼ mQ¯ πγð Þ is also well described by the following ABJ carry the internal four-momenta p1 ¼ P−k, p2 ¼ P−q−k, anomaly (or the chiral anomaly) [2] and k, respectively. It is well known that the single 1 ABJ covariant Feynman diagram Fig. 1(a) is in general equal Fπγ ð0Þ¼ pffiffiffi ; ð20Þ 2 2π2f to the sum of the two LF time-ordered diagrams Figs. 1(b) π þ and 1(c) if the q ≠ 0 frame is taken. However, if the ABJ −1 Exp which results in Fπγ ð0Þ ≃ 0.276 GeV for fπ ≃ qþ 0 q ≠ 0 q2 qþq− − q2 ¼ frame (but ⊥ so that ¼ ⊥ ¼ Exp 2 2 130 MeV agreeing with the experimental data Fπγ ð0Þ¼ −q ¼ −Q ) is chosen, the LF diagram 1(c) does not ⊥ 0.272 3 GeV−1 within a few percent. contribute but only the diagram 1(b) gives exactly the same ð Þ From Eq. (18), we obtain the TFF in the exact chiral limit result as the covariant diagram 1(a). However, as we found (Mπ;m→ 0) and its analytic form is given by in [12], if one takes the qþ ¼ Pþ (or α ¼ qþ=Pþ ¼ 1)   2 qffiffiffiffi   frame but with q⊥ 0, Fig. 1(b) does not contribute Q 2 ¼ 5 2 Q 1 Q Γ e8β Γ − ; pffiffiffiffiffiffiffiffiffi 4 β3 4 8β2 but only Fig. 1(c) contributes to the total transition chiral 2 Fπγ ðQ Þ¼ PQQ¯ pffiffiffipffiffiffi : ð21Þ amplitude and shows exactly the same as the one obtained 4 3 4π from the qþ ¼ 0 frame. While the TFF obtained from 2 the qþ ¼ 0 frame is defined in the spacelike region In particular, the TFF at Q ¼ 0 is obtained as 2 2 2 (q ¼ −q⊥ ¼ −Q < 0), the TFF obtained from the qffiffiffiffi 3 qþ Pþ q 0 5 π 1 2 ¼ frame with ⊥ ¼ is directly defined in the pffiffiffiffiffiffiffiffiffi Γ 32½Γð4Þ PQQ¯ 2 − chiral ð4Þ timelike region (q ¼ qþq > 0). Thus, one can analyze Fπγ ð0Þ¼ PQQ¯ pffiffiffi ¼ pffiffiffi ; ð22Þ 2 3 2π 1=4β 2 2π2fchiral the TFF in the timelike region using the qþ ¼ Pþ but ð Þ π q⊥ ¼ 0 frame without resorting to the analytic continuation where we used Eq. (15) to obtain the second expression of from spacelike region to timelike region as was did in the Eq. (22). Equating Eq. (22) with Eq. (20), we find that qþ ¼ 0 frame. P ¯ < 0.1 in the chiral limit of our model is required to fit The explicit form of the pion TFF obtained from the QQ fExp ΓExp qþ ¼ Pþ frame is given by [12] both π and π0→γγ correctly. This indicates a significant ffiffiffiffiffiffiffiffi Z Z higher Fock-state contribution in the chiral limit. This point 2 2 p 1 2 eu − ed 2Nc dx 2 ψ πðx; k⊥Þ has been also discussed in the LF holographic QCD based Fπγ q pffiffiffi d k⊥ ; ð Þ¼ 3 2 2 on the AdS=CFT correspondence in which PQQ¯ ¼ 0.5 was 2 4π 0 ð1 − xÞ M0 − q estimated to describe simultaneously Γπ0→γγ and the pion ð18Þ TFF at the asymptotic limit. As we shall show in Sec. IV, the probability PQQ¯ increases as the quark mass increases where ψ π is the same as Eq. (12). The salient feature of indicating the saturation of the LF Fock-state expansion Eq. (18) is that the external virtual momentum is com- x; k with the lower Fock-state contribution as the so-called pletely decoupled from the internal momenta ( ⊥) “ ” q2 current quarks get amalgamated with themselves to form and facilitates the analysis of the timelike region ð ¼ the constituent quark degrees of freedom. In our numerical −Q2 > 0Þ due to the simple and clean pole structure, 2 2 −1 calculation of Sec. IV, we shall analyze the mass variation ðM0 − q Þ as shown in Eq. (18). The TFF in the spacelike Q2 2 2 effect as gets evolved and also compare with the result region can also be easily obtained by replacing q with −Q [45] obtained from the LF holographic QCD based on the 2 2 −1 in ðM0 − q Þ and was shown to be exactly the same as the AdS=CFT correspondence. qþ 0 2 result obtained from the ¼ frame [12]. We note that The electromagnetic form factor FπðQ Þ of a pion is F Q2 μ the leading order QCD result [1] for πγð Þ with defined by the matrix elements of the current operator Jem: ϕπðxÞ¼6xð1 − xÞ, so called Brodsky-Lepage (BL) limit 2 Q → ∞ 0 μ 0 μ 2 at the asymptoticffiffiffi region (i.e., ), is given by hP jJemjPi¼ðP þ P Þ FπðQ Þ; ð23Þ 2 2 p Q FπγðQ Þ¼ 2fπ ≃ 0.185 GeV. As one can see clearly from Eq. (18), our model satisfies the scaling behavior where q ¼ P − P0 is the four momentum transfer. 2 2 2 2 2 Q FπγðQ Þ → constant as Q → ∞. But how large Q Our calculation for FπðQ Þ is carried out using the should be to reach the scaling behavior is related with the standard LF frame (qþ ¼ 0). The charge form factor of the

036005-5 HO-MEOYNG CHOI and CHUENG-RYONG JI PHYS. REV. D 102, 036005 (2020) pion can then be expressed as the convolution of the initial and final state LF wave functions for the “þ” component of the μ current operator Jem as follows Z Z 1 d2k F Q2 dx ⊥ Ψ x; k0 Ψ x; k πð Þ¼ 16π3 πð ⊥Þ πð ⊥Þ Z0 Z 1 d2k M2k k0 mM x 1 − x B ε mM x 1 − x B ε0 ⊥ 0 ⊥ · ⊥ þ½ þ ð Þ π B½ þ ð Þ π B ¼ dx ϕRðx; k⊥ÞϕRðx; k⊥Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 16π3 2 2 2 2 02 0 2 0 M k⊥ þ½mM þ xð1 − xÞBπεB M k ⊥ þ½mM þ xð1 − xÞBπεB ð24Þ

0 0 chiral chiral where k⊥ ¼ k⊥ þð1 − xÞq⊥ and εB is the same as εB but the results of fπ and ϕπ ðxÞ in the chiral limit given by 0 with the replacement of k⊥ with k⊥. One can also easily Eqs. (15) and (16), the negativity of Bπ, i.e., Bπ < 0,is find that the spin-orbit term in Eq. (24) becomes 1 in the essential for the consistency in the chiral limit. Varying the chiral limit (i.e., Mπ;m→ 0). The charge radius of the pion value of PQQ¯ , i.e., 1 >PQQ¯ > 0, we have confirmed that 2 2 2 r −6dF Q =dQ 2 can be calculated by h πi¼ πð Þ jQ ¼0. the value of Bπ should be taken to be negative in order to make a link to the chiral limit. IV. NUMERICAL RESULTS Laying out all the formulas for our model description 2 2 and its application to fπ, FπγðQ Þ and FπðQ Þ in Secs. II and III, respectively, we have already discussed the critical role of chiral anomaly in constraining the model param- eters. In particular, we noticed not only the nontriviality of the axial-vector coupling, i.e., Bπ ≠ 0, in the chiral limit but also the negativity of the axial-vector coupling, i.e., Bπ < 0, to dictate the model independence of the ratio chiral chiral ϕπ ðxÞ=fπ and the consistency with the AdS=CFT prediction of asymptotic DA. Moreover, the probability of the lowest LF Fock state PQQ¯ should diminish in the chiral ΓExp limit to obtain both the chiral anomaly (i.e., π0→γγ) and Exp fπ correctly, indicating a significant higher Fock-state contribution in the chiral limit. In this section, we present our numerical results and discuss the consistency of the constraints on the model parameters with the chiral anomaly that we discussed in previous sections. To find the optimum model parameters ðm; β;BπÞ,we first take PQQ¯ ¼ 1 and fit the two experimental data, Exp (1) pion decay constant fπ ¼ 130.2ð1.7Þ MeV, and Exp −1 (2) Fπγ ð0Þ¼0.272ð3Þ GeV , simultaneously. For an illustration, we show in Fig. 2 the possible solution sets Th Exp for ðm; βÞ satisfying fπ ¼ fπ when Bπ ¼þ1 (upper panel) and −1 (lower panel) for given pion physical mass Mπ ¼ 135 MeV. As one can see from Fig. 2, while the negative sign of Bπ has the solution set (i.e., overlap line between the blue and peach colors) covering all the possible range of 0 ≤ ðm; βÞ ≤ 1 GeV, the positive sign of Bπ has the solution set covering severely restricted range with rather unusually large uðdÞ-quark mass (i.e., m ≥ 0.7 GeV). The restriction on the model parameters for the case of Bπ ¼þ1 appears in line with the unusually 1 3 large Mock meson mass M Mπ Mρ ≈612 MeV Th Exp av ¼ð4 þ 4 ÞExp FIG. 2. Possible solution sets for ðm; βÞ satisfying fπ ¼ fπ in the spin-averaged mass scheme [32,33] for the consis- obtained from the pion vertex Γπ ¼ðMπ þ Bπ=PÞγ5 with the tency with the experimental data. As already indicated in Bπ ¼ 1 (upper panel) and Bπ ¼ −1 (lower panel).

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In Fig. 3, we show the possible solution sets for the 3 model parameters depending on ðm; β;Bπ;PQQ¯ Þ for (a) m 255; 200; 150; 100; 50; 5 m = 255 MeV (reference point) given quark mass ¼ð Þ MeV m = 200 MeV and Mπ ¼ 135 MeV, i.e., (−Bπ vs PQQ¯ ) in Fig. 3(a) and 2.5 m = 150 MeV m = 100 MeV (β vs PQQ¯ ) in Fig. 3(b), which were obtained by fitting both m = 50 MeV Exp Exp −1 m = 5 MeV fπ ¼ 130.2ð1.7Þ MeV and Fπγ ð0Þ¼0.272ð3Þ GeV simultaneously. In our previous work [12] with Bπ ¼ 0, 2 i.e., Γπ ¼ Aπγ5, the quark mass m ¼ 220 MeV and the Gaussian parameter β ¼ 0.3659 GeV were taken from our

earlier LFQM [14] spectroscopic analysis of the ground π 1.5 state pseudoscalar and vector meson nonets based on the -B variational principle. In the scope of present work involving only the pion, however, we do not attempt a spectroscopic analysis but focus on the effect of nonzero axial vector 1 coupling (Bπ < 0) for the consistency with the chiral anomaly. For this purpose, we first set our reference P ¯ 1 B −0 25 parameter set with QQ ¼ and π ¼ . which is a 0.5 rather small axial vector coupling compared to the pseu- doscalar coupling and find the corresponding optimum Exp Exp values of m and β to fit both fπ and Fπγ ð0Þ. Then, by 0 reducing the quark mass m from this reference point and 0 0.2 0.4 0.6 0.8 1 Exp Exp P again fitting both fπ and Fπγ ð0Þ simultaneously, we QQ obtain the rest of parameter sets shown in Fig. 3. We mark 0.8 the reference parameter set by asterisk () in Fig. 3, i.e., (b) ðMπ;m;βÞ¼ð0.135;0.255;0.1980Þ GeV and ðBπ;PQQ¯ Þ¼ m = 255 MeV (reference point) ffiffiffi 0.7 p 2 m = 200 MeV ð−0.25; 1Þ, with which we get Fπγð0Þ¼PQQ¯ =ð2 2π fπÞ¼ m = 150 MeV 0 271 −1 f 130 4 m = 100 MeV . GeV and π ¼ . MeV close enough to m = 50 MeV Exp −1 Exp 0.6 Fπγ ð0Þ¼0.272ð3Þ GeV and fπ ¼ 130.2ð1.7Þ MeV m = 5 MeV for our purpose in this work. In comparison with the value β ¼ 0.3659 GeV in the absence of the axial vector 0.5 coupling Bπ ¼ 0 [12], the value β ¼ 0.1980 GeV in the reference parameter set is somewhat reduced with the B −0 25 0.4 contribution of axial vector coupling π ¼ . , while [GeV] the quark mass m ¼ 255 MeV still represents the ordinary β “ ” constituent quark picture in our reference point . 0.3 In reducing the quark mass m from this reference point Exp Exp to fit both fπ and Fπγ ð0Þ simultaneously, we ultimately 0.2 reached the parameter set ðMπ;mÞ¼ð135; 5Þ MeV reproducing the Gell-Mann-Oakes-Renner (GMOR) 2 2 relation [46], i.e., Mπfπ ¼ −2ðmq þ mq¯ Þhqq¯i, where 0.1 hqq¯i¼−ð250 MeVÞ3 with the “current” quark mass m ¼ mq ¼ mq¯ . For the fixed value of the pion mass, 0 i.e., Mπ ¼ 0.135 GeV, we distinguish the two different 0 0.2 0.4 0.6 0.8 1 P cases of the quark-antiquark bound state, i.e., Mπ < 2m vs QQ Mπ > 2m, and call them as the “constituent quark picture” −B P β vs the “current quark picture,” respectively. In Fig. 3, the FIG. 3. Possible solution sets for ( π vs QQ¯ ) (a) and ( vs PQQ¯ ) (b) for given quark mass m ¼ð255; 200; 150; 100; parameter sets corresponding to Mπ < 2m and Mπ > 2m Exp Exp cases are denoted by black and blue data, respectively. 50; 5Þ MeV and Mπ ¼ 135 MeV satisfying fπ and Fπγ ð0Þ, From the results shown in Fig. 3, we summarize simultaneously. We set the solution of m ¼ 255 MeV with P 1 our main findings for the model parameters as follows: QQ¯ ¼ as a reference point.

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Pmin Exp (1) The minimum probability QQ¯ exists for a given quark and PQQ¯ ¼ 0.078 by fitting both fπ [see Eq. (15)] and Exp Exp Exp mass satisfying both fπ and Fπγ ð0Þ simultaneously, e.g., Fπγ ð0Þ [see Eq. (22)] simultaneously. Pmin 0 45; 0 25 m 200; 5 B ; β QQ¯ ¼ð . . Þ for ¼ð Þ MeV etc. This result Table I shows our typical model parameters ð π Þ depending on the variation of ðMπ;mÞ and PQQ¯ used in the is in line with the trend that the probability PQQ¯ increases as ϕ x the quark mass increases indicating the saturation of the LF analysis of the twist-2 DA πð Þ, the transition form factor F Q2 F Q2 Fock-state expansion with the lower Fock-state contribu- πγð Þ, and the electromagnetic form factor πð Þ. Exp tion as the current quarks get amalgamated with themselves Among many possible solutions satisfying both fπ and Exp to form the constituent quark degrees of freedom. (2) For Fπγ ð0Þ as shown in Fig. 3, we select a few parameter sets M < 2m the quark masses satisfying π (i.e., constituent ðMπ;mÞ¼fð135; 255Þ; ð135; 150Þ; ð135; 50Þ; ð0; 0Þg MeV β P quark picture), the Guassian parameter gets larger as QQ¯ corresponding to the variation of the probability PQQ¯ ¼ decreases. This indicates that the spatial size of the lowest f1; 0.3; 0.15; 0.078g in order to estimate the mass variation 2 2 Fock state gets smaller as the higher Fock states contribute effect on both FπγðQ Þ and FπðQ Þ form factors. m more. For a given quark mass , the axial vector coupling Using these typical parameter sets in Table I, we first −Bπ gets also reduced as the higher Fock states contribute Rshow the normalized twist-2 pion DA ΦπðxÞ satisfying more, i.e., PQQ¯ decreases. For a fixed PQQ¯ ,however,we 1 0 dxΦπðxÞ¼1 and compare them with the asymptotic notice that −Bπ increases quite significantly as m decreases asy DA, Φπ ¼ 6xð1 − xÞ in Fig. 4. The twist-2 pion DA with β while values do not change much indicating only larger quark mass such as m ¼ 255 MeV is strongly marginal size reduction in the lowest Fock state with the suppressed in the vicinity of endpoints (x ¼ 0, 1) but the m reduction of mass . (3) For the quark masses satisfying DA shows broader shape than the asymptotic DA (double- M > 2m β π (i.e., current quark picture), values are in dot-dashed line) as the quark mass is getting smaller. general greater for the current quark mass than the Our chiral limit result (dot-dashed line) is exactly the same P constituent one for given QQ¯ indicating that the spatial as the AdS=CFT prediction of the asymptotic DA size of the lowest Fock state consisted of the current quark [37,40,41]. We obtain Φπð1=2Þ¼ð1.70; 1.34; 1.28Þ for is smaller than the one consisted of the constituent quark. chiral m ¼ð255; 150; 50Þ MeV and Φπ ð1=2Þ¼1.27 for As PQQ¯ decreases, however, β values get reduced down to ðMπ;mÞ¼ð0; 0Þ, which should be compared with those in the constituent quark picture indicating that the spatial size of the lowest Fock state consisted of the current quark gets larger as the higher Fock states 2 contribute more. The similar merge of the axial vector coupling Bπ between the current quark picture and the constituent picture appears as PQQ¯ decreases in the upper panel Fig. 3(a). It is indeed fascinating to observe the merge of the parameter sets between the current quark 1.5 picture and the constituent picture as PQQ¯ decreases both in Fig. 3(a) and Fig. 3(b). It seems to indicate a nontrivial dynamic saturation process of the LF Fock-state expan- sion occurring as the current quarks get amalgamated with (x) themselves to form the constituent quark degrees of π 1 Φ freedom according to these results. m = 255 MeV For the case of exact chiral limit (Mπ ¼ m ¼ 0), our m = 150 MeV B m = 50 MeV results for any physical observables are independent of π Chiral limit (M, m → 0) as far as it is negative nonzero value (Bπ < 0) and depend 6x(1-x) only on ðβ;PQQ¯ Þ, which were obtained as β ¼ 0.6685 GeV 0.5

TABLE I. Model parameters ðBπ; βÞ depending on the variation of ðMπ;mÞ and PQQ¯ . We denote ðMπ;m;β;fπÞ in unit of MeV.

Th Th −1 ðMπ;mÞ PQQ¯ Bπ β fπ Fπγ ð0Þ [GeV ] 0 0 0.2 0.4 0.6 0.8 1 (135, 255) 1 −0.25 198.0 130.4 0.271 x (135, 150) 0.3 −0.60 346.9 130.6 0.272 Φ x (135, 50) 0.15 −0.7 493.0 130.7 0.271 FIG. 4. The normalized pion DA πð Þ obtained from the sets M ;m 135; 255 ; 135; 150 ; 135; 50 ; 0; 0 (0, 0) 0.078 < 0 668.5 130.9 0.276 of ð π Þ¼fð Þ ð Þ ð Þ ð Þg MeV P 1; 0 3; 0 15; 0 078 Exp. [44] 130.2(1.7) 0.272(3) with QQ¯ ¼f . . . g compared with the asymptotic DA.

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asy Φπ ð1=2Þ¼1.5 as well as other theoretical predictions from the LF quark degrees of freedom at the reference point ΦSR 1=2 1 2 0 3 m m 255 P ¯ P 1 such as π ð Þ¼ . . obtained from QCD sum with ref ¼ ¼ MeV and QQ ¼ mref ¼ with RLðDBÞ rules [47], Φπ ð1=2Þ¼1.16ð1.29Þ from Dyson- another contribution from the LF quark degrees of freedom m P Schwinger equation approach using the rainbow-ladder with the reduced and m. For instance, the form factors ðm ;mÞ 2 ðm ;mÞ 2 truncation (RL) and the dynamical chiral-symmetry break- Fπγ ref ðQ Þ and Fπ ref ðQ Þ obtained from the mixing of LFQM m 255 ing improved (DB) kernels [48,49], and Φπ ð1=2Þ¼ the LF quark degrees of freedom with ref ¼ MeV P 1 1.25 from our LFQM using the spin structure given by and mref ¼ and the LF quark degrees of freedom with Eq. (9) and the linear confining potential model parameters m ¼ 150 MeV with Pm ¼ 0.3 in Table I are respectively [17]. One can also define the expectation value of the given by longitudinal momentum, so called ξ ¼ 2x − 1 moments as pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi m m 150 ˜ ð ref Þ ˜ ð ¼ Þ Z ðm ;m¼150Þ 2 1 − PmFπγ þ PmFπγ 1 Fπγ ref ðQ Þ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi ; n n ˜ ˜ hξ i¼ dxξ ΦπðxÞ: ð25Þ 1 − Pm þ Pm 0 ð26Þ The odd power of ξ-moments for the pion DA are zero due to the isospin symmetry, and the first nonzero moment and (n ¼ 2) is obtained as hξ2i¼ð0.155; 0.230; 0.247Þ for m ;m 150 m m 150 ð ref ¼ Þ 2 ˜ ð ref Þ ˜ ð ¼ Þ m ¼ð255; 150; 50Þ MeV with PQQ¯ ¼ð1; 0.3; 0.15Þ, and Fπ ðQ Þ¼ð1 − PmÞFπ þ PmFπ ; ð27Þ 0.250 in the exact chiral limit (Mπ;m→ 0), whereas 2 ˜ it is 0.20 for the asymptotic DA. Our result for hξ i gets where the renormalized probability is denoted as Pm ¼ larger as the quark mass and the probability are getting Pm=ðPm þ PmÞ¼0.3=1.3 ≈ 0.23. Since our prediction 2 ref ξ max 0 25 Th Exp smaller and reaches maximum value h i ¼ . in the of the TFF satisfies Fπγ 0 ≃ Fπγ 0 for any quark mass as 2 ð Þ ð Þ ξ m ;m chiral limit. Especially, our result for h i obtained from Fð ref Þ Q2 2m

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0.3 (a) (b)

(mref=255) Fπγ (m , m=150) 0.25 ref 0.25 Fπγ

(mref, m=50) Fπγ

(mref, m=0) Fπγ 0.2 (m=0) 0.2 Fπγ pQCD

] CELLO -1 CLEO

BaBar )[GeV] 2

) [GeV 0.15 Belle 0.15 2 (Q πγ (Q

F 2 (m =255)

2 ref πγ Q Fπγ F Q 2 (m , m=150) Q F ref 0.1 0.1 πγ 2 (mref, m=50) Q Fπγ

2 (mref, m=0) Q Fπγ 2 (m=0) 0.05 0.05 Q Fπγ CLEO Belle BaBar 0 0 0246 810 0 1020304050 2 2 2 2 Q [GeV ] Q [GeV ]

2 2 2 FIG. 5. Predictions of (a) FπγðQ Þ and (b) Q FπγðQ Þ. The experimental data are taken from [3,51–53].

ðm¼0Þ 2 Our result Fπγ ðQ Þ obtained from the exact chiral prediction [12] using the spin-orbit structure given by limit (double-dot-dashed line) shows a disagreement with Eq. (9) is very close to the pQCD result. 2 2 F Q2 Q2F Q2 the experimental data for low Q ð< 3 GeV Þ region. The From the results of πγð Þ and πγð Þ shown in 2 ðm¼0Þ 2 2 2 Fig. 5, we may summarize our findings as follows: (1) For Q Fπγ Q exceeds the pQCD result for Q > 10 GeV ð Þ Q2 0 ≤ Q2 ≤ 10 2 and shows a consistency with the data from BABAR [3] for low- and intermediate- region ð Þ GeV as the intermediate region of 4 ≤ Q2 ≤ 14 GeV2 although its shown in Fig 5(a), we find that the nonzero quark mass mild rising behavior is however not enough to fit the data results are in better agreement with the data than the result Q2 in the chiral limit. As the constituent quark mass decreases from BABAR for the higher region. m 255 from the reference point ref ¼ MeV, the reduction of The results of combining the two different quark mass P ðm ;m¼150Þ 2 ðm ;m¼50Þ 2 the probability QQ¯ is necessary to agree with the exper- degrees of freedom, i.e., Fπγ ref ðQ Þ, Fπγ ref ðQ Þ, m ;m 0 imental data. These results indicate that the constituent Fð ref ¼ Þ Q2 and πγ ð Þ, are not much different from quark picture (2m>Mπ) is very effective and important in ðmref Þ 2 2 2 Fπγ ðQ Þ for low- and intermediate-Q regions but shows describing FπγðQ Þ in the low energy regime but the quark 2 better agreement with the pQCD result in high Q region mass evolution seems inevitable as Q2 grows. (2) As the accounting 93% of the pQCD result. While we have noticed quark mass evolves from the constituent to current quark Q2F Q2 ¯ that πγð Þ obtained from the quark mass in the masses, the probability PQQ¯ finding the valence QQ region 0 ≤ m ≤ 150 MeV exceeds the pQCD result for component inside the pion also needs to be reduced 2 2 10 ≤ Q ≤ 20 GeV region by itself, it is interesting accordingly. This indicates that the higher Fock states to see that the results of combining the quark mass degrees contribute more as the quark mass decreases. m ;m 2 m Q2Fð ref Þ Q2 q > 0 of freedom with ref, i.e., πγ ð Þ, approach the We show in Fig. 6(a) the timelike ( )behaviorofthe norm 2 2 norm 2 2 asymptotic result only from below as shown in Fig. 5(b). normalized Fπγ ðq Þ¼Fπγðq Þ=Fπγð0Þ, i.e., jFπγ ðq Þj Effectively, our results obtained from the combination of as a function of q for small q (0 ≤ q ≤ 0.2 GeV) region the quark mass degrees of freedom show a consistency with compared with the experimental data for the Dalitz decay the data from Belle [53] rather than the BABAR data [3]. π0 → eþe−γ measured from A2 Collaboration [54].The Our results for Fπγð0Þ are comparable with the simple LF same line codes are used as in Fig. 5. As discussed in 2 holographic QCD model [45] with a twist-2 valence pion Sec. III, our result for Fπγðq Þ in timelike region is obtained Exp state in which it requires PQQ¯ ¼ 0.5 to reproduce Fπγ ð0Þ. from the direct timelike region ðqþ ¼ PþÞ calculation It may be also noteworthy that our previous LFQM without resorting to the analytic continuation from spacelike

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1.8 4 (a) (b)

norm (mref=255) Fπγ norm(m =255) Re[F ref ] 1.6 norm (m , m=150) πγ ref 3 Fπγ norm(mref=255) Im[Fπγ ] norm (mref, m=50) Fπγ norm(m =255) |F ref | norm (m , m=0) πγ 1.4 ref Fπγ norm(m=0) Re[F ] norm (m=0) 2 πγ F πγ norm(m=0) Im[F ] A2 πγ

2 1.2 norm(m=0) ) )| 2 |F |

2 πγ (q (q 1 πγ norm πγ norm F |F 1

0 0.8

-1 0.6

0.4 -2 0 0.05 0.1 0.15 0.2 -2 -1 0 1 2 3 4 5 2 2 q [GeV] q [GeV ]

norm 2 2 norm 2 2 FIG. 6. Predictions of normalized Fπγ ðq Þ¼Fπγðq Þ=Fπγð0Þ in timelike region: (a) jFπγ ðq Þj for small timelike region norm 2 2 2 ð0 ≤ q ≤ 0.2Þ GeV and (b) Fπγ ðq Þ for both spacelike and timelike regions (−2 ≤ q ≤ 5)GeV . The experimental data are taken from [54].

Q2 to the timelike q2 in contrast to the case of the qþ ¼ 0 modulus including both real and imaginary parts but frame calculation. Figure 6(b) exhibits the sample results reaches its maximum [see Eq. (21)]atq2 ¼ 0 and decreases norm 2 of Fπγ ðq Þ for both spacelike and timelike region just after that. As a result, the chiral limit prediction in the −2 ≤ q2 ≤ 5 2 m ( GeV )obtainedfrom ref (black thick lines) timelike region shows an apparent disagreement with the normðm ;m¼50Þ 2 2 and the exact chiral limit (blue thin lines), in which we experimental data. Likewise, jFπγ ref ðq Þj (dotted 2 F q normðm ;m¼0Þ 2 2 separate the real Re½ πγð Þ (dashed lines) and imaginary line) and Fπγ ref q (dot-dashed line) represent 2 2 j ð Þj Im½Fπγðq Þ (dotted lines) parts from the modulus jFπγðq Þj the modulus including both real and imaginary parts, but (solid lines). We should note that our direct results of the disagree with the data. From the analysis of pion TFF in 2 2 2 form factor Fπγðq Þ¼ReFπγðq ÞþiImFπγðq Þ are in both spacelike and timelike regions, we find that the complete agreement with those obtained from the dispersion constituent quark picture is definitely necessary to describe relations as we have explicitly shown in [12].Thisassures the low energy behavior correctly. While the form factors the validity of our numerical calculation both in the spacelike obtained from the mixing of the different quark mass and timelike regions. eigenstates are not much different from each other in the In our model calculation for the timelike region, spacelike region, their predictions for the timelike region q2 the imaginary part starts from the threshold, th ¼ are very different due to the resonance feature in the 2 2 2 ðmQ þ mQ¯ Þ ¼ 4m and the modulus of the TFF reaches timelike region. Therefore, the analysis of Fπγðq Þ in maximum near threshold and decreases after the threshold. timelike region plays a critical role in constraining theo- normðm Þ 2 2 Because of this, jFπγ ref ðq Þj (solid line) and retical models. normðm ;m¼150Þ 2 2 In association with the experimental data for the Dalitz jFπγ ref ðq Þj (dashed line) in Fig. 6(a) represent decay π0 → eþe−γ, due to the smallness of the lightest the result including only the real part since the thresholds þ − e e invariant mass mee ¼ q, the normalized TFF is q for those quark masses (i.e., m ¼ 255 and 150 MeV) th typically parametrized as [44,54] are greater than the maximum q value shown in Fig. 6(a).

Both show an excellent agreement with the experimental 2 normðm¼0Þ 2 2 q data. On the other hand, jFπγ ðq Þj obtained from Fnorm q2 1 a ; πγ ð Þ¼ þ π m2 ð28Þ the chiral limit (double-dot-dashed line) represents the π0

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1 where the parameter aπ corresponds to the slope of the TFF 2 norm 2 (a) at q ¼ 0. As shown in Fig. 6(a), our results for Fπγ ðq Þ m obtained from the two parameter sets, i.e., ref and the mixture of m and m ¼ 150 MeV, show a good agree- ref 0.8 ment with the A2 data. Our results for aπ obtained from normðm Þ 2 normðm ;m¼150Þ 2 Fπγ ref ðq Þ and Fπγ ref ðq Þ are obtained as aπ ¼ 0.038 and 0.043, respectively. On the other hand, in our previous LFQM [12] analysis using the spin-orbit 0.6 LFQM aπ 0 036 structure given by Eq. (9), we obtained ¼ . . ) m 2 Our result obtained from and the previous LFQM (Q

ref π result [17] are in closer good agreement with the current F a 0 032 0 004 world average π ¼ . . [44] and the two recent 0.4 NA7 Coll. (CERN SPS) 0 þ − experimental data extracted from the π → e e γ decay, Fπ Coll. (JLab) (m =255) a 0 030 0 010 a 0 0368 ref π ¼ . . from A2 [54] and π ¼ . Fπ 0 0057 (m , m=150) . from NA62 [55]. This again indicates that the F ref m ≥ 200 π constituent quark degrees of freedom (i.e., MeV) 0.2 (mref, m=50) Fπ rather than the current quark degrees of freedom is much (mref, m=0) F q2 Fπ better in describing πγð Þ for small timelike region. The (m=0) F timelike data going beyond the Dalitz decay can provide π 0 further constraints on theoretical understanding of the 0 0.2 0.4 0.6 0.8 1 2 2 effective quark degrees of freedom. Q [GeV ] Figure 7 shows the pion electromagnetic form factor, i.e., 2 2 2 2 1 FπðQ Þ in Fig. 7(a) for small Q ð0 ≤ Q ≤ 1 GeV ) Q2F Q2 Q2 (b) region and πð Þ in Fig. 7(b) for the larger 0.9 2 2 (0 ≤ Q ≤ 10 GeV ) region. We compare our results NA7 Coll. (CERN SPS) with the experimental data [56–60]. The results Fπ Coll. (JLab) 0.8 2 (m =255) ðm Þ 2 ðm ;m¼150Þ 2 Q F ref Fπ ref ðQ Þ (solid line), Fπ ref ðQ Þ (dashed line), π 2 (mref, m=150) ðm ;m¼50Þ 2 ðm ;m¼0Þ 2 Q Fπ Fπ ref ðQ Þ (dotted line), and Fπ ref ðQ Þ (dot- 0.7 2 (mref, m=50) dashed line) are in good agreement with the experimental Q Fπ 2 (m , m=0) 2 2 2 0.6 ref data [56–60] for small Q ð0 ≤ Q ≤ 1 GeV ) region as Q Fπ )

2 2 one can see from the plot of FπðQ Þ in Fig. 7(a), while the (Q m 0 π 0.5 ð ¼ Þ 2 F chiral limit result Fπ ðQ Þ (double-dot-dashed line) 2 Q severely deviates from the data as one may have expected 0.4 2 from the previous analysis of FπγðQ Þ. Our predictions of

2 1=2 ðmref Þ the pion charge radius rπ ≡ hrπi obtained from Fπ , 0.3 ðm ;m¼150Þ ðm ;m¼50Þ ðm ;m¼0Þ Fπ ref , Fπ ref , and Fπ ref are given by 0.2 rπ ¼ð0.683; 0.657; 0.677; 0.679Þ fm, respectively. Those four results show a good agreement with the most recent PDG 0.1 value quoted by Particle Data Group [44], rπ ¼ ð0.672 0.008Þ fm. These results may also be compared LFQM 0 with the result rπ ¼ 0.651 fm obtained from the 0246 810 2 2 spin-orbit structure given by Eq. (9) in our previous Q [GeV ] 2 2 LFQM analysis [17]. For the plots of Q FπðQ Þ 2 2 2 m ;m 150 FIG. 7. Predictions of (a) FπðQ Þ for small Q ð0 ≤ Q ≤ Q2 10 2 Q2Fð ref ¼ Þ Q2 2 2 2 2 up to ¼ GeV , the results of π ð Þ 1 GeV ) region and (b) Q FπðQ Þ for the larger Q m ;m 50 2 2 2 ð ref ¼ Þ 2 0 ≤ Q ≤ 10 (dashed line), Q Fπ ðQ Þ (dotted line), and ( GeV ) region. The same line codes are used as 2 ðm ;m¼0Þ 2 in Fig. 5 and the data are taken from [56–60]. Q Fπ ref ðQ Þ (dot-dashed line) appear reasonably consistent with the current available experimental data, 2 ðmref Þ 2 rates. The result of combining the constituent quark mass while the result Q Fπ ðQ Þ reaches its maximum around m ;m 150 Q2Fð ref ¼ Þ Q2 ≃ 1.2 GeV and drops rather steeply after passing the degrees of freedom π provides an improved m 2 2 2 ð ref Þ maximum value. Indeed, all the results for Q FπðQ Þ drop description over the result of Q Fπ without any mixing after reaching their maximum values but with different for the broader Q2 range, which again indicates the vitality

036005-12 CHIRAL ANOMALY AND THE PION PROPERTIES IN THE … PHYS. REV. D 102, 036005 (2020) of quark mass evolution as Q2 gets larger. We note however as the current quark degrees of freedom get amalgamated that the mixture of the constituent quark mass degrees of together to form the constituent quark degrees of freedom. freedom with the current quark mass degrees of freedom as We may discuss the amalgamation of the current quarks 2 ðm ;m¼50Þ 2 2 ðm ;m¼0Þ 2 shown in Q Fπ ref ðQ Þ and Q Fπ ref ðQ Þ forming the constituent quark degrees of freedom from the provides characteristically different scaling behaviors com- perspective of the vacuum fluctuation consistent with the pare to the typical high Q2 behavior exhibited in the chiral symmetry of QCD. While the constituent degrees of m freedom in our LFQM get dressed by the light-front zero- Q2Fð ref Þ Q2 constituent quark quark picture results π ð Þ and mode cloud, they satisfy the chiral symmetry consistent 2 ðm ;m¼150Þ 2 Q Fπ ref ðQ Þ. We anticipate that the 12 GeV with the QCD. The correlation between the quark mass and upgraded Jefferson Lab would provide much more detailed the nontrivial QCD vacuum effect is on par with the trade- and accurate data of the pion form factor for the larger Q2 off between the complicated nontrivial vacuum and the range. This would help us in coming up with the more effective constituent quark degrees of freedom. Our results realistic quark mass evolution analysis beyond this first indicate that the constituent quark picture (2m>Mπ)is 2 order approximation. very effective and important in describing both FπγðQ Þ 2 and FπðQ Þ in the low energy regime but the quark mass 2 V. SUMMARY AND CONCLUSIONS evolution seems inevitable as Q grows. More elaborate analysis including the quark mass evolution effect deserves As the chiral anomaly [2] is the key to understand the 0 further consideration. One may also explore the spectro- π → γγ decay rate resolving the issue with the Sutherland- scopic analysis including the pseudoscalar and vector Veltman theorem [38], we attempt to include the axial- meson nonets beyond the pion. vector coupling in addition to the pseudoscalar coupling in our LFQM for the pion to explore a well-defined chiral ACKNOWLEDGMENTS limit still providing a good description of the pion electro- magnetic and transition form factors [12,14,17].We H.-M. C. was supported by the National Research Foundation of Korea (NRF) (Grant No. NRF- thus took the spin-orbit vertex structure given by Γπ ¼ 2020R1F1A1067990). C.-R. J. was supported in part by ðAπ þ Bπ=PÞγ5 versatile enough to explore the chiral limit 2 the US Department of Energy (Grant No. DE-FG02- and described the pion properties such as fπ;FπγðQ Þ and 2 03ER41260). FπðQ Þ depending on the variation of the quark mass in a self-consistent manner. APPENDIX: SPIN-ORBIT WAVE We find that the chiral anomaly plays a critical role in FUNCTIONS χðx;k⊥Þ constraining the model parameters. The negativity of the axial-vector coupling, i.e., Bπ < 0, appears essential to The constituent quarks can be described by Dirac spinors chiral chiral u k v k dictate the model independence of the ratio ϕπ ðxÞ=fπ λð Þ and λð Þ satisfying the Dirac equation and the consistency with the AdS=CFT prediction of asymptotic DA. Our chiral limit result for twist-2 pion ð=k − mÞuλðkÞ¼0; ð=k þ mÞυλðkÞ¼0; ðA1Þ DA given by Eq. (16) is exactly the same as the AdS=CFT =k k γμ prediction of the asymptotic DA [37,40,41], indicating also where ¼ μ . It is instructive to use the appropriate basis a significant higher Fock-state contribution in the chiral of Dirac spinors [13]: F q2 limit. We also note that the analysis of πγð Þ in timelike 1 1 region plays a critical role in constraining theoretical model uλðkÞ¼pffiffiffiffiffiffi ð=k þ mÞuðλÞ; υλðkÞ¼pffiffiffiffiffiffi ð=k − mÞυðλÞ; parameters. While the form factors obtained from the kþ kþ mixing of the different quark mass eigenstates are not ðA2Þ much different from each other in the spacelike region, their 0 1 0 1 predictions for the timelike region are very different due to 1 0   B C   B C the resonance feature in the timelike region. 1 B 0 C 1 B 0 C While the small probability of the lowest Fock-state such u ¼ B C;u− ¼ B C; ðA3Þ 2 @ 0 A 2 @ 0 A as PQQ¯ < 0.1 in the chiral limit implies a significant higher Fock-state contribution, our numerical results in Sec. IV 0 1 indicate that P ¯ increases as the quark mass increases. It QQ υ λ u −λ γ is interesting to note that the merge of the parameter sets and ð Þ¼ ð Þ. In this basis the matrices are repre- between the current quark picture and the constituent sented by P     picture occurs both in Fig. 3(a) and Fig. 3(b) as QQ¯ 0 I 0 σi decreases. These results seem to reflect a nontrivial γ0 ; γi ; ¼ ¼ i ðA4Þ dynamic saturation process of the LF Fock-state expansion I 0 −σ 0

036005-13 HO-MEOYNG CHOI and CHUENG-RYONG JI PHYS. REV. D 102, 036005 (2020) where I is the 2 × 2 unit matrix and σi are Pauli matrices and defined as       01 0 −i 10 −kL 1 2 3 π 1 σ ¼ ; σ ¼ ; σ ¼ : ðA5Þ ð1Þ χ↑↑ ¼ N pffiffiffiffiffiffiffiffiffi ðMπ þ 2BπmÞ; 10 i 0 0 −1 x1x2 −1 0 3 5 0 1 2 3 2 χπ N ffiffiffiffiffiffiffiffiffi mM − B k2 − m2 − x x M2 ; Using the γ matrices γ ≡ γ γ and γ ≡ iγ γ γ γ , ð Þ ↓↑ ¼ p f π πð ⊥ 1 2 πÞg x1x2 1 þ − − þ and =k ¼ 2 ðk γ þ k γ Þ − γ⊥ · k⊥, the spinors uλðkÞ and π 1 2 2 2 υλðkÞ are obtained as 3 χ N ffiffiffiffiffiffiffiffiffi mM − B k − m − x x M ; ð Þ ↑↓ ¼ px x f π πð ⊥ 1 2 πÞg 0 1 0 1 1 2 m −kL −kR 4 χπ N ffiffiffiffiffiffiffiffiffi1 M 2B m ; B C B þ C ð Þ ↓↓ ¼ p ð π þ π Þ ðA9Þ 1 B 0 C 1 B k C x1x2 u↑ðkÞ¼pffiffiffiffiffiffiB C;u↓ðkÞ¼pffiffiffiffiffiffiB C; ðA6Þ kþ @kþ A kþ @ 0 A kR m where

0 L 1 0 1 −k −m     B C B C M2 k2 m2 1 kþ 1 0 þ þ i⊥ þ i B C B C P ¼ P ; ; 0⊥ ;ki ¼ xiP ; ; ki⊥ : υ↑ðkÞ¼pffiffiffiffiffiffiB C; υ↓ðkÞ¼pffiffiffiffiffiffiB C: ðA7Þ Pþ x Pþ kþ @ 0 A kþ @ kþ A i −m kR ðA10Þ

The normalization is u¯ λðkÞuλðkÞ¼−υ¯λðkÞυλðkÞ¼2m and kR and kL are defined as kR ≡ k1 þ ik2 and kL ≡ k1 − ik2, Thus, the normalized spin-orbit wave function for pion satisfying the unitary condition χλ λ χλ λ 1 is given by respectively. h 1 2 j 1 2 i¼ For a pion with four momentum P and mass Mπ, the general spin structure may be given as χλλ¯ ¼ χ x; k N u¯ k M B =P γ υ k λ1λ2 ð ⊥Þ λ1ð 1Þð π þ π Þ 5 λ2 ð 2Þ, which satisfies the nor- P †  L  malization λ χ χλ λ ¼ 1. −k M mM þ x1x2BπεB i λ1λ2 1 2 N ; ¼ R Then, the operator Γπ ¼ðMπ þ Bπ=PÞγ5 is given by −mM − x1x2BπεB −k M 0 − 1 ðA11Þ −Mπ 0 BπP 0 B þ C B 0 −Mπ 0 BπP C Γπ ¼ B C; ðA8Þ @ −B Pþ 0 M 0 A N ffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 M M 2B m π π where ¼ p p 2 2 2, ¼ π þ π 2 M k⊥þ½mMþx1x2Bπ εB 0 −B P− 0 M 2 2 π π and εB ¼ Mπ − M0.

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