Distribution Amplitudes of K ∗ and ϕ at Physical Pion Mass from Lattice QCD
Jun Hua, Min-Huan Chu, Peng Sun, Wei Wang, Ji Xu, Yi-Bo Yang, Jian-Hui Zhang, Qi-An Zhang
DDistribution Amplitudes of K ∗ and φ at Physical Pion Mass from Lattice QCD ( Lattice Parton Collaboration (LPC))
Jun Hua, Min-Huan Chu,
1, 2
Peng Sun, ∗ Wei Wang, † JiXu,
1, 4
Yi-Bo Yang,
5, 6, 7
Jian-Hui Zhang, and Qi-An Zhang INPAC, Shanghai Key Laboratory for Particle Physics and Cosmology,Key Laboratory for Particle Astrophysics and Cosmology (MOE),School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Shanghai Key Laboratory for Particle Physics and Cosmology,Key Laboratory for Particle Astrophysics and Cosmology (MOE),Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China Department of Physics and Institute of Theoretical Physics,Nanjing Normal University, Nanjing, Jiangsu, 210023, China School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China School of Fundamental Physics and Mathematical Sciences,Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China Center of Advanced Quantum Studies, Department of Physics,Beijing Normal University, Beijing 100875, China (Dated: November 20, 2020)We present the first lattice QCD calculation of the distribution amplitudes of longitudinally andtransversely polarized vector mesons K ∗ and φ using large momentum effective theory. We use theclover fermion action on three ensembles with 2+1+1 flavors of highly improved staggered quarks(HISQ) action, generated by MILC collaboration, at physical pion mass and { } fm lattice spacings, and choose three different hadron momenta P z = { . , . , . } GeV. Theresulting lattice matrix elements are nonperturbatively renormalized in a hybrid scheme proposedrecently. Also an extrapolation to the continuum and infinite momentum limit is carried out. Wefind that while the longitudinal distribution amplitudes tend to be close to the asymptotic form,the transverse ones deviate rather significantly from the asymptotic form. Our final results providecrucial ab initio theory inputs for analyzing pertinent exclusive processes.
Introduction. –Searching for new physics beyond thestandard model (SM) is a primary goal of particle physicsnowadays. A unique possibility of doing so is to investi-gate flavor-changing neutral current processes which arehighly suppressed in the SM. Some prominent examplesof such processes include B → K ∗ (cid:96) + (cid:96) − and B s → φ(cid:96) + (cid:96) − decays, which host a number of observables sensitive tonew physics, ranging from differential decay widths, var-ious asymmetries to full angular distributions. Recentexperimental analyses by the LHCb collaboration [1–3]have revealed notable tensions between the SM predic-tions of such processes and data, attracting quite con-siderable theoretical interests (see Ref. [4] and referencestherein). Various new physics interpretations have beenproposed to resolve such tensions, but to firmly establishtheir existence requires an accurate and reliable theoret-ical understanding of the dynamics of weak decays.The standard description of heavy meson weak decaysis based on the QCD factorization theorem, in which thedecay amplitudes are split into short-distance hard ker-nels and long-distance universal inputs. In the case of B → K ∗ (cid:96) + (cid:96) − and B s → φ(cid:96) + (cid:96) − decays, the universalinputs that enter include the light-cone distribution am-plitudes (LCDAs) of the vector mesons K ∗ , φ which, to the leading-twist accuracy, specify the longitudinal mo-mentum distribution amongst the valence quark and an-tiquark in the meson. While the hard scattering kernelis perturbatively calculable, the LCDAs can only be ex-tracted from nonperturbative methods or from fits to rel-evant data. To date most of the available analyses havemade use of estimates based on QCD sum rules [5], whichintroduce considerable systematic errors.Lattice QCD provides an ideal ab initio tool to accessnonperturbative quantities in strong interaction. Whileit has been used to calculate the lowest few Gegenbauermoments of light meson LCDAs, e.g., of the ρ mesonLCDA [6], a direct calculation of the entire distributionhas not been feasible until the proposal of large momen-tum effective theory (LaMET) [7, 8] recently. In LaMET,the calculation of LCDAs is realized by calculating on thelattice appropriately chosen equal-time correlations thatshare the same infrared (IR) behavior with the corre-sponding lightcone correlations defining the LCDAs, andthen convert them to the latter through a perturbativematching. Since LaMET was proposed, a lot of progresshas been achieved in calculating various parton distri-bution functions (for a comprehensive list see [9, 10])as well as distribution amplitudes for light pseudoscalar a r X i v : . [ h e p - l a t ] N ov TABLE I. Information on the simulation setup. The light andstrange quark mass of the clover action are tuned such that m π =140 MeV and m η s =670 MeV.Ensemble a (fm) L × T m π (MeV) m η s (MeV)a12m130 0.12 48 ×
64 140 670a09m130 0.09 64 ×
96 140 670a06m130 0.06 96 ×
192 140 670 mesons [11–13]. Other variants have also been exploredin Refs. [14–16]. For recent reviews on LaMET and itsapplications we refer the readers to Refs. [9, 10].In this Letter, we present the first lattice calculationof LCDAs for vector mesons K ∗ , φ in LaMET with theclover fermion action, on three ensembles with 2+1+1flavors of highly improved staggered quarks (HISQ) ac-tion [17], generated by MILC collaboration [19], at phys-ical pion mass and 0.06, 0.09 and 0.12 fm lattice spac-ings. The simulation setup is given in Table I. A hybridrenormalization scheme [23] is used to renormalize thebare quantities, after which an extrapolation is takento the continuum limit as well as to the infinite mo-mentum limit based on data at three hadron momenta, P z = { . , . , . } GeV. Our final results indicatethat, while the longitudinal LCDAs tend to be close tothe asymptotic form, the transverse ones deviate rathersignificantly from the asymptotic form.
LCDAs from LaMET. –The leading-twist LCDAs forlongitudinally and transversely polarized vector mesons,Φ
V,L and Φ
V,T , are defined as follows [24]: (cid:90) dξ − e ixp + ξ − (cid:104) | ¯ ψ (0) n/ + U (0 , ξ − ) ψ ( ξ − ) | V (cid:105) = f V n + · (cid:15) Φ V,L ( x ) , (1) (cid:90) dξ − e ixp + ξ − (cid:104) | ¯ ψ (0) σ + µ ⊥ U (0 , ξ − ) ψ ( ξ − ) | V (cid:105) = f TV [ (cid:15) + p µ ⊥ − (cid:15) µ ⊥ p + ]Φ V,T ( x ) , (2)where U (0 , ξ − ) = P exp (cid:2) ig s (cid:82) ξ − dsn + · A ( sn + ) (cid:3) is thegauge-link defined along the minus lightcone direction, (cid:15) is the polarization vector of the vector meson, and n + is the unit vector along the plus lightcone direction. f V and f TV are the decay constants defined by the local vectorand tensor current, respectively. Here for K ∗ , ψ denotesthe strange quark field and ψ the light u/d quark. Forthe φ meson, both ψ , are strange quark fields.According to LaMET, the above LCDAs can be ob-tained by first calculating the following bare equal-timecorrelations on the lattice (cid:104) | ¯ ψ (0) γ t U (0 , z ˆ z ) ψ ( z ˆ z ) | V (cid:105) = H V,L ( z ) (cid:15) t f V , (3) (cid:104) | ¯ ψ (0) σ νρ U (0 , z ˆ z ) ψ ( z ˆ z ) | V (cid:105) = H V,T ( z ) f TV [ (cid:15) ν p ρ − (cid:15) ρ p ν ] , where the Lorentz indices are chosen as { ν, ρ } = z, y ,and the gauge-link U (0 , z ˆ z ) is along the z direction. Thequantities H V, { L,T } ( z ) can be renormalized nonperturba-tively in an appropriate scheme [16, 23, 25–31]. Here we choose the hybrid scheme [23] proposed recently whichhas the advantage that the renormalization factor doesnot introduce extra nonperturbative effects at large z which distort the IR property of the bare correlations,and thus ensures the ensuing factorization to LCDA. Thisscheme works as follows: At | z | ≤ z S where z S is withinthe region where the leading-twist approximation is valid,we can choose the RI/MOM scheme [28] to avoid certaindiscretization effects (alternative choices include, e.g., theratio [16] scheme), while for | z | > z S one applies thegauge-link mass subtraction scheme [25–27] H RV ( z, a, P z ) = H V ( z, a, P z ) Z ( z, a ) θ ( z S − | z | )+ H V ( z, a, P z ) e − δm (˜ µ ) z Z hybrid ( z S , a ) θ ( | z | − z S ) , (4)where the superscript R denotes the renormalized quan-tity, ˜ µ denotes the intrinsic scale dependence of thegauge-link including both UV and IR. We have chosen Z ( z, a ) as the RI/MOM renormalization factor computedfrom Z ( z, a ) = 112 Tr (cid:104)(cid:10) S ( p ) (cid:11) − × (cid:10) S ( p | z ) (cid:11) γ z γ (cid:89) n U z ( n ˆ z ) S ( p | (cid:10) S ( p ) (cid:11) − γ z γ (cid:105) p − µ R,pz =0 , (5)and Z hybrid denotes the endpoint renormalization con-stant which can be determined by imposing a continuitycondition at z = z S , Z hybrid ( z S , a ) = e δm (˜ µ ) z S /Z ( z S , a ) . (6)The mass counterterm δm (˜ µ ) can be extracted from theRI/MOM renormalization factor [23]. More details canbe found in the supplemental material [22].By Fourier transforming H RV, { L,T } to momentum space,we then obtain the quasi-DAs˜Φ V, { L,T } ( y, P z ) = (cid:90) dze − iyP z z H RV, { L,T } ( z, P z ) , (7)where we assume the continuum limit has been taken. Itcan be factorized into the LCDAs through the factoriza-tion theorem [32]:˜Φ V, { L,T } ( y, P z , µ R )= (cid:90) dx C V, { L,T } ( x, y, P z , µ R , µ )Φ V, { L,T } ( x, µ ) , (8)where the matching kernel C V, { L,T } was derived first inthe transverse momentum cutoff scheme in Ref. [20] andthen in the RI/MOM scheme in Ref. [21]. The µ and µ R reflect the generic renormalization scale dependenceof LCDAs and quasi-DAs. Numerical setup. –On the lattice, one directly calcu-lates the two-point correlation function defined as: C m ( z, (cid:126)P , t ) = (cid:90) d ye i (cid:126)P · (cid:126)y × (cid:104) | ¯ ψ ( (cid:126)y, t )Γ U ( (cid:126)y, (cid:126)y + z ˆ z ) ψ ( (cid:126)y + z ˆ z, t ) × ¯ ψ (0 , ψ (0 , | (cid:105) , (9)where the longitudinal polarization case ( m = L ) hasΓ = γ t , and Γ = γ z , and the transverse polarizationcase ( m = T ) has Γ = σ zy , and Γ = γ x /γ y . Then thequasi-DAs can be extracted from the following parame-terization, C m ( z, (cid:126)P , t ) C m ( z = 0 , (cid:126)P , t ) = H bV,m ( z )(1 + c m ( z ) e − ∆ Et )(1 + c m (0) e − ∆ Et ) , (10)where c m ( z ) and ∆ E are free parameters accounting forthe excited state contaminations, and H bV,m ( z ) is the barematrix elements for quasi-DA. When t is large enough,the excited state contaminations parameterized by c m ( z )and ∆ E are suppressed exponentially, and the ratio de-fined in Eq. (10) approaches the ground state matrix el-ement H bV,m ( z ). Based on the comparison on the jointtwo-state fit and constant fit shown in the supplemen-tal material [22], we choose to use the constant fit inthe range of t ≥ .
54 fm to provide a conservative errorestimate in the following calculation.The numerical simulation is based on three ensembleswith 2+1+1 flavors of HISQ [17] at physical pion masswith 0.06, 0.09 and 0.12 fm lattice spacings. The mo-mentum smeared grid source [18] with the source posi-tions ( x + i x L/ , y + i y L/ , z + i z L/
2) are used in thecalculation, where ( x , y , z ) is a random position and i x,y,z = 0 /
1. It allows us to obtain the even momentain unit of 2 π/L with ∼ × × ×
2, 730 × × × × × × a =0.06, 0.09 and 0.12fm, respectively. We have also reversed the ˆ z direction inEq. (9) to double the statistics. Results. –After renormalization in the hybrid scheme,we perform a phase rotation e izP z / to the renormalizedcorrelation, so that the imaginary part directly reflectsthe flavor asymmetry. Taking the transversely-polarized K ∗ as an example, we show in Fig. 1 the real (upperpanel) and imaginary part (lower panel) of the renor-malized quasi-DA matrix elements e izP z / H K ∗ ,T ( z ) withthe momentum P z = 2 π/L ×
10 = 2 .
15 GeV. As shownin the upper panel, the matrix elements at different lat-tice spacings are consistent with each other and it meanslinear divergences arising from the gauge-link have beencanceled up to the current numerical uncertainty. In thelower panel, we find a positive imaginary part at all the lattice spacings, which corresponds to a non-zero asym-metry with the peak at x < /
2. This is consistent withexpectations that lighter quarks carry less momentum ofthe parent meson. = zP z Re[ e izP z H K * , T ( z )] a :0.06fm a :0.09fm a :0.12fm = zP z Im[ e izP z H K * , T ( z )] a :0.06fm a :0.09fm a :0.12fm FIG. 1. The two-point correlation function for thetransversely-polarized K ∗ in coordinate space. We make aphase rotation by multiplying a factor e izP z / with P z =2 . As one can see from Fig. 1, the uncertainty of latticedata grows rapidly with the spatial separation of the non-local operator. Thus, to have a reasonable control of un-certainties in the final result we need to truncate the cor-relation at certain point. The missing long-range infor-mation can be supplemented by a physics-based extrap-olation proposed in Ref. [23], which removes unphysicaloscillations in a naive truncated Fourier transform withthe price of altering the endpoint distribution (at x ∼ H V, { L,T } ( z, P z ) = (cid:104) c ( − iλ ) a + e iλ c ( iλ ) b (cid:105) e − λ/λ , (11)where the exponential term accounts for the finite cor-relation length for a hadron at finite momentum, andthe two algebraic terms account for a power law behav-ior of the momentum distribution at x close to 0 and 1,respectively. λ = zP z , and the parameters c , , a, b, λ are determined by a fitting to the lattice data in theregion where it exhibits an exponential decay behavior.To account for systematics from such an extrapolation,we have done two different extrapolations, one includingthe exponential term and the other not, and taken theirdifference as an estimate of systematics. The detailedcomparison of two extrapolations can be found in thesupplemental material [22]. x K * , T ( x ) K * , T ( x ) FIG. 2. Quasi-DA and LCDA extracted from it for thetransversely-polarized K ∗ using data at a = 0 .
09 fm, P z =2 .
15 GeV.
After renormalization and extrapolation, we canFourier transform to momentum space and apply the cor-responding matching. In Fig. 2, we show as an examplethe comparison of the quasi-DA and extracted LCDA forthe transversely polarized K ∗ . The results correspond tothe case with P z = 2 .
15 GeV, and a = 0 .
09 fm. Onenotices that there is a non-vanishing tail for quasi-DA(yellow curve) in the unphysical region ( x > x < ψ ( a ) = ψ ( a →
0) + c a + O ( a ) , (12)with O ( a ) correction being due to the mixed action ef-fect from the clover valence fermion on HISQ sea. Asan example, we show the extrapolated results for thetransversely polarized K ∗ in Fig. 3 for three different mo-menta, P z = { . , . , . } GeV. From this figure, onecan see that the asymmetry increases with P z slightly.Since the strange quark is heavier than the up/down x a P z = 1.29GeV a P z = 1.72GeV a P z = 2.15GeV FIG. 3. The continuum limit of the LCDA for thetransversely-polarized K ∗ , extrapolated from three differentlattice spacings. quark, a slight preference of x < / x > / P z extrapolation is es-sential to suppress the power corrections and reproducethis correct preference behavior. Such a behavior is alsoobserved in the study of Kaon LCDAs in Ref. [13].Our final results for LCDAs of the K ∗ and φ aregiven in Fig. 4 and 5, respectively, where the upper andlower panels correspond to the longitudinal and trans-verse polarization cases. In these figures, we have madea P z → ∞ extrapolation using the following simple form: ψ ( P z ) = ψ ( P z → ∞ ) + c P z + O (cid:16) P z (cid:17) . (13)It is worth emphasizing that the endpoint regions are dif-ficult to access in LaMET. The smallest x that one canreach can be roughly estimated from the largest attain-able λ (conjugate variable of x in the Fourier transform)as 1 /λ max . Accordingly, the largest x can be estimatedas 1 − /λ max since the momentum fraction carried bythe other quark, 1 − x , shall also be bounded from belowby the above estimate [9]. In the present calculation, wehave λ max ≈
14, thus we take a conservative estimate ofthe predictable region as [0 . , . x (1 − x ) and theexpanded parametrization using the Gegenbauer polyno-mials with the second moments from earlier QCD sumrule calculations [5]. Our results indicate that while thelongitudinal LCDAs tend to be close to the asymptoticform, the transverse LCDAs have relatively large devia-tions from the asymptotic form. Summary. –We have presented the first lattice QCDcalculation of LCDAs of longitudinally and transverselypolarized vector mesons K ∗ , φ using LaMET. We did not x K * , L ( x ) Sum ruleAsymptoticThis work x K * , T ( x ) Sum ruleAsymptoticThis work
FIG. 4. LCDAs for the longitudinally-polarized K ∗ (upperpanel) and transversely-polarized K ∗ (lower panel). The re-sults are extrapolated to the continuous limit ( a →
0) andthe infinite momentum limit ( P z → ∞ ). Regions with x < . , x > . consider the ρ meson due to its large width which willintroduce sizable systematic errors. The continuum andinfinite momentum limits are taken based on calculationsat physical light and strange quark mass with three lat-tice spacings and momenta. Our final results are thencompared to the asymptotic form and QCD sum rule re-sults. While the longitudinal LCDAs tend to be close tothe asymptotic form, the transverse ones have relativelylarge deviations from the asymptotic form. ACKNOWLEDGEMENT
We thank Xiangdong Ji, Liuming Liu, MaximilianSchlemmer, and Andreas Sch¨afer for valuable discus-sions. We thank the MILC collaboration for provid-ing us their HISQ gauge configurations. The LQCDcalculations were performed using the Chroma software x , L ( x ) Sum ruleAsymptoticThis work x , T ( x ) Sum ruleAsymptoticThis work
FIG. 5. Similar to Fig. 4, but for the φ vector meson. suite [33] and QUDA [34–36] through HIP programmingmodel [37]. The numerical calculation is supported byStrategic Priority Research Program of Chinese Academyof Sciences, Grant No. XDC01040100. The setup for nu-merical simulations was conducted on the π ∗ Corresponding author: [email protected] † Corresponding author: [email protected][1] R. Aaij et al. [LHCb], JHEP , 179 (2015)doi:10.1007/JHEP09(2015)179 [arXiv:1506.08777 [hep-ex]].[2] R. Aaij et al. [LHCb], JHEP , 055 (2017)doi:10.1007/JHEP08(2017)055 [arXiv:1705.05802 [hep-ex]].[3] R. Aaij et al. [LHCb], Phys. Rev. Lett. , no.1,011802 (2020) doi:10.1103/PhysRevLett.125.011802[arXiv:2003.04831 [hep-ex]].[4] A. Cerri, V. V. Gligorov, S. Malvezzi, J. Martin Ca-malich, J. Zupan, S. Akar, J. Alimena, B. C. Allanach,W. Altmannshofer and L. Anderlini, et al. CERN YellowRep. Monogr. , 867-1158 (2019) doi:10.23731/CYRM-2019-007.867 [arXiv:1812.07638 [hep-ph]].[5] P. Ball, V. M. Braun and A. Lenz, JHEP , 090 (2007)doi:10.1088/1126-6708/2007/08/090 [arXiv:0707.1201[hep-ph]].[6] V. M. Braun, P. C. Bruns, S. Collins, J. A. Gracey,M. Gruber, M. G¨ockeler, F. Hutzler, P. P´erez-Rubio,A. Sch¨afer and W. S¨oldner, et al. JHEP , 082 (2017)doi:10.1007/JHEP04(2017)082 [arXiv:1612.02955 [hep-lat]].[7] X. Ji, Phys. Rev. Lett. , 262002 (2013)doi:10.1103/PhysRevLett.110.262002 [arXiv:1305.1539[hep-ph]].[8] X. Ji, Sci. China Phys. Mech. Astron. , 1407-1412(2014) doi:10.1007/s11433-014-5492-3 [arXiv:1404.6680[hep-ph]].[9] X. Ji, Y. S. Liu, Y. Liu, J. H. Zhang and Y. Zhao,[arXiv:2004.03543 [hep-ph]].[10] K. Cichy and M. Constantinou, Adv. High EnergyPhys. , 3036904 (2019) doi:10.1155/2019/3036904[arXiv:1811.07248 [hep-lat]].[11] J. H. Zhang, J. W. Chen, X. Ji, L. Jin andH. W. Lin, Phys. Rev. D (2017) no.9, 094514doi:10.1103/PhysRevD.95.094514 [arXiv:1702.00008[hep-lat]].[12] J. H. Zhang et al. [LP3], Nucl. Phys. B ,429-446 (2019) doi:10.1016/j.nuclphysb.2018.12.020[arXiv:1712.10025 [hep-ph]].[13] R. Zhang, C. Honkala, H. W. Lin and J. W. Chen,[arXiv:2005.13955 [hep-lat]].[14] Y. Q. Ma and J. W. Qiu, Phys. Rev. D ,no.7, 074021 (2018) doi:10.1103/PhysRevD.98.074021[arXiv:1404.6860 [hep-ph]].[15] Y. Q. Ma and J. W. Qiu, Phys. Rev. Lett. , no.2,022003 (2018) doi:10.1103/PhysRevLett.120.022003[arXiv:1709.03018 [hep-ph]].[16] A. V. Radyushkin, Phys. Rev. D , no.3, 034025 (2017)doi:10.1103/PhysRevD.96.034025 [arXiv:1705.01488[hep-ph]].[17] E. Follana et al. [HPQCD and UKQCD], Phys. Rev.D , 054502 (2007) doi:10.1103/PhysRevD.75.054502[arXiv:hep-lat/0610092 [hep-lat]].[18] Y. B. Yang, A. Alexandru, T. Draper, M. Gong,K. F. Liu [ χ QCD] Phys. Rev. D , 034503(2016) doi:10.1103/PhysRevD.93.034503 [arXiv:hep-lat/1509.04616 [hep-lat]]. [19] A. Bazavov et al. [MILC], Phys. Rev. D ,no.5, 054505 (2013) doi:10.1103/PhysRevD.87.054505[arXiv:1212.4768 [hep-lat]].[20] J. Xu, Q. A. Zhang and S. Zhao, Phys. Rev. D ,no.11, 114026 (2018) doi:10.1103/PhysRevD.97.114026[arXiv:1804.01042 [hep-ph]].[21] Y. S. Liu, W. Wang, J. Xu, Q. A. Zhang, S. Zhaoand Y. Zhao, Phys. Rev. D , no.9, 094036 (2019)doi:10.1103/PhysRevD.99.094036 [arXiv:1810.10879[hep-ph]].[22] Supplementary material of this work.[23] X. Ji, Y. Liu, A. Sch¨afer, W. Wang, Y. B. Yang,J. H. Zhang and Y. Zhao, [arXiv:2008.03886 [hep-ph]].[24] A. Ali, V. M. Braun and H. Simma, Z. Phys. C (1994),437-454 doi:10.1007/BF01580324 [arXiv:hep-ph/9401277[hep-ph]].[25] T. Ishikawa, Y. Q. Ma, J. W. Qiu and S. Yoshida,[arXiv:1609.02018 [hep-lat]].[26] J. W. Chen, X. Ji and J. H. Zhang, Nucl. Phys.B (2017), 1-9 doi:10.1016/j.nuclphysb.2016.12.004[arXiv:1609.08102 [hep-ph]].[27] J. Green, K. Jansen and F. Steffens,Phys. Rev. Lett. (2018) no.2, 022004doi:10.1103/PhysRevLett.121.022004 [arXiv:1707.07152[hep-lat]].[28] I. W. Stewart and Y. Zhao, Phys. Rev. D (2018) no.5, 054512 doi:10.1103/PhysRevD.97.054512[arXiv:1709.04933 [hep-ph]].[29] C. Alexandrou, K. Cichy, M. Constantinou, K. Had-jiyiannakou, K. Jansen, H. Panagopoulos andF. Steffens, Nucl. Phys. B (2017), 394-415doi:10.1016/j.nuclphysb.2017.08.012 [arXiv:1706.00265[hep-lat]].[30] V. M. Braun, A. Vladimirov and J. H. Zhang,Phys. Rev. D (2019) no.1, 014013doi:10.1103/PhysRevD.99.014013 [arXiv:1810.00048[hep-ph]].[31] Z. Y. Li, Y. Q. Ma and J. W. Qiu, [arXiv:2006.12370[hep-ph]].[32] X. Ji, A. Sch¨afer, X. Xiong and J. H. Zhang, Phys. Rev.D (2015), 014039 doi:10.1103/PhysRevD.92.014039[arXiv:1506.00248 [hep-ph]].[33] R. G. Edwards et al. [SciDAC, LHPC and UKQCD],Nucl. Phys. B Proc. Suppl. , 832 (2005)doi:10.1016/j.nuclphysbps.2004.11.254 [arXiv:hep-lat/0409003 [hep-lat]].[34] M. A. Clark, R. Babich, K. Barros, R. C. Brower andC. Rebbi, Comput. Phys. Commun. , 1517-1528(2010) doi:10.1016/j.cpc.2010.05.002 [arXiv:0911.3191[hep-lat]].[35] R. Babich, M. A. Clark, B. Joo, G. Shi, R. C. Browerand S. Gottlieb, doi:10.1145/2063384.2063478[arXiv:1109.2935 [hep-lat]].[36] M. A. Clark, B. Jo´o, A. Strelchenko, M. Cheng, A. Gamb-hir and R. Brower, [arXiv:1612.07873 [hep-lat]].[37] Y. J. Bi, Y. Xiao, M. Gong, W. Y. Guo, P. Sun,S. Xu and Y. B. Yang, PoS LATTICE2019 , 286 (2020)doi:10.22323/1.363.0286 [arXiv:2001.05706 [hep-lat]].[38] T. Regge, Nuovo Cim. (1959), 951doi:10.1007/BF02728177 SUPPLEMENTAL MATERIALSDispersion Relation
Based on the two-point function at z = 0, we canextract the effective mass through a two-state fit. Wecombine the longitudinally-polarized and transversely-polarized data to give the dispersion relation for K ∗ and φ in Fig. 6. The parametrized form is given as: E = m + c ( P z ) + c ( P z ) a . (14) c ,K ∗ = − . ± .
024 and c ,φ = − . ± .
010 re-flects the discretization effects, while with 2 σ deviation, c ,K ∗ = 1 . ± . c ,φ = 1 . ± .
009 are consistentwith the speed of light. P z (GeV) E ( G e V ) a=0.12fma=0.09fma=0.06fm P z (GeV) E ( G e V ) a=0.12fma=0.09fma=0.06fm FIG. 6. Dispersion relation for the K ∗ (upper panel) and φ (lower panel) meson. Quasi-DA in coordinate space
We perform a phase rotation e izP z / for the two-point correlation function in Fig. 1. For comparison, weshow the two-point correlation function of transversely-polarized K ∗ without phase rotation in Fig. 7. The upper panel is the real part and the lower panel is the imaginarypart. = zP z Re[ H K * , T ( z )] a :0.06fm a :0.09fm a :0.12fm = zP z Im[ H K * , T ( z )] a :0.06fm a :0.09fm a :0.12fm FIG. 7. The quasi-DA for the transversely-polarized K ∗ incoordinate space with P z = 2 . GeV . Comparison of constant fit with two-state fit
As shown in Fig. 1, we perform a phase rotation e izP z / to the renormalized correlations, so that the imaginarypart directly reflects the flavor SU(3) asymmetry. Thereliability of the two-state fit in Eq. (10) largely dependson whether the energy of the excited state can be fittedwell or not. However, after the phase rotation, weights ofthe excited state energy in the real and imaginary partsmay be different, therefore the two-state fit including ex-cited energy is unstable.An alternative plan is to obtain the ground state ma-trix element H bV,m ( z ) in Eq. (10) by the constant fit(set∆ E = 0) to eliminate the influence of excited state en-ergy. When t is large enough, the excited states contam-ination parameterized by c m ( z ) and ∆ E are suppressedexponentially, and the ratio C m ( z, (cid:126)P , t ) /C m ( z = 0 , (cid:126)P , t )approaches the ground state matrix element H bV,m ( z ). InFig. 8 we show as an example the trend of the aboveratio over time at lattice spacing a = 0 .
06 fm. As canbe seen from the figure, the imaginary part decays to theground state more slowly than the real part. We make amore conservative choice for the region t ≥ a = 0 .
54 fmfor the constant fit.Fig. 9 shows the comparison of H bV,m ( z ) derived bytwo-state fit and constant fit. The number (2 ,
6) in thetwo-state fit implies that the fit range for the local corre-lation functions is t ≥ t ≥
6. The two-state fitted H bV,m ( z ) will be-come stable only when the starting point of the non-localcorrelation function fit range becomes very large. In thiscase, the two-state fitted ratios H bV,m ( z ) are consistentwith the constant fitted ones within statistical uncertain-ties. t C ( Z n , t ) / C ( , t ) Imag z=4Imag z=8Real z=4Real z=8
FIG. 8. Trends in the ratios C m ( z, (cid:126)P , t ) /C m ( z = 0 , (cid:126)P , t ) ofthe real and imaginary parts over time at lattice spacing a =0 .
06 fm. Take z = 4 and z = 8 as examples.FIG. 9. The upper half plane shows the comparison of two-state fitted ratio H bV,m ( z ) and constant fitted ones. The lowerhalf of the plane shows the χ of the two-state fit. Strategy of renormalization and matching
In this work, we have performed the renormaliza-tion and matching in the hybrid scheme [23]. Thisscheme takes advantages of both the RI/MOM or ra-tio type renormalization and the gauge-link mass sub-traction scheme in that the former cancels discretizationeffects at short distances whereas the latter avoids intro-ducing extra nonperturbative effects at large distances atthe renormalziation stage.The renormalization in the hybrid scheme has beengiven in Eq. (4), where the mass counterterm can bewritten as δm = m − /a + m , (15)with m ∼ Λ QCD , and can be extracted in practice fromthe asymptotic behavior of various hadronic matrix ele-ments. Here we choose to use the RI/MOM renormal-ization factor at large distance and perform a fit to thequasi-LF correlation from moderate to large z using thefollowing form: C = C e − (1+ c ln zzs ) (cid:104) ( m − a + m z ) z + c ln zz (cid:105) , (16)where we have included an extra logarithmic term in frontof the square bracket to roughly account for the depen-dence of strong coupling on the distance, z = 1 fm fordimensionless ratios, and z s = 0 . z(fm) fitted_0.12fmfitted_0.09fmfitted_0.06fmRI/MOM_0.12fmRI/MOM_0.09fmRI/MOM_0.06fm FIG. 10. Fit of RI/MOM renormalization factor with differentlattice spacings.
In addition to nonperturbative renormalization, wehave to face another issue when Fourier transforming therenormalized correlations to momentum space. That is:For finite hadron momentum, the available lattice data ofquasi-LF correlations always end up at finite λ L = z L P z while we need the correlations at all quasi-light-front dis-tances λ to reconstruct the momentum distribution. Here = zP z cx n (1 x )) n , n1=1.13, n2=0.32 L Lattice dataExponential L =7.1Polynomial L =12.5 FIG. 11. Comparison of different extrapolations and latticedata. we follow the extrapolation strategy proposed in Ref. [23]based on generic properties of coordinate space correla-tions. At finite momentum, the quasi-LF correlation in general has a finite correlation length and exhibits an ex-ponential decay (usually associated with a power decay aswell). When the momentum gets larger, the correlationlength (in λ space) also gets larger and the exponentialdecay behavior is much delayed in the quasi-light-frontcorrelation, thus we expect that the decay behavior fol-lows more like an algebraic law rather than an exponen-tial one as λ ∼ λ L , where the power law behavior is alsoconsistent with the asymptotic Regge behavior of mo-mentum distribution [38].In practice, we take the following two extrapolationforms [23], one is exponential and the other algebraic˜ H ( z, P z ) = (cid:20) c ( − iλ ) a + e iλ c ( iλ ) b (cid:21) e − λλ , ˜ H ( z, P z ) = c ( − iλ ) a + e iλ c ( iλ ) b , (17)and use their difference as an estimate of systematic er-rors from extrapolation. Fig. 11 shows the comparison oftwo extrapolation forms, which are consistent with eachother within errors. We have also tested that the sameis true when the fitting range is varied.As for the hybrid matching, it turns out to approachthe RI/MOM one when p zz