Charmonium contribution to $B \rightarrow K\ell^+\ell^-: testing the factorization approximation on the lattice
aa r X i v : . [ h e p - l a t ] J a n Charmonium contribution to B → K ℓ + ℓ − : testing thefactorization approximation on the lattice Katsumasa Nakayama ∗ a , b , Tsutomu Ishikawa b , c , and Shoji Hashimoto b , c (JLQCDcollaboration) a NIC, DESY Zeuthen, Platanenallee 6, 15738 Zeuthen, Germany b KEK Theory Center, High Energy Accelerator Research Organization (KEK), Tsukuba305-0801, Japan c School of High Energy Accelerator Science, The Graduate University for Advanced Studies(Sokendai),Tsukuba 305-0801, JapanE-mail: [email protected]
We report the current status of a study of charmonium contribution to B → K ℓ + ℓ − on the lat-tice. Our lattice calculation tests the factorization approximation for this contribution. In orderto control the problem of the artificial divergence, we focus on the low q region with a smallb-quark mass. We also take into account the renormalization constants of relevant four-quark op-erators calculated through the temporal moments. Results suggest a violation of the factorizationapproximation. The 37th Annual International Symposium on Lattice Field Theory - LATTICE201916-22 June, 2019Wuhan, China. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ → K ℓ + ℓ − Factorization
Katsumasa Nakayama
1. Introduction
The rare decay B → K ( ∗ ) ℓ + ℓ − has received much attention as a clean probe of new physicssince the Standard Model contribution is suppressed due to flavor-changing neutral-current. Siz-able difference from the Standard Model has been reported for the differential decay rate of B → K ( ∗ ) ℓ + ℓ − by LHCb [1, 2].In order to confirm this tension, we have to control the uncertainty due to non-perturbativecontributions. The experimental analysis of the B → K ( ∗ ) ℓ + ℓ − decays focused on the region whereinvariant mass squared of the finial lepton pair q is not close to the charmonium resonances.However, long-distance effects between the final state kaon and the virtual charmonium state couldbe significant even outside such resonance regions.So far, theoretical estimates have been attempted by using the perturbative calculation and ap-plying the factorization approximation, although the intermediate state can be more complex. Inthe factorization, we approximate the amplitude by a product of the B → K part and the charmo-nium resonance part. In other words, we ignore the interaction between the B → K form factorand the charmonium two-point function. The factorization approximation has been studied withexperimental results and models, but reliable prediction for the B → K decay remains to be difficult[3, 4, 5, 6].In this proceedings, we report the recent progress of the numerical lattice calculation to studythe factorization approximation for the B → K ℓ + ℓ − amplitude. We calculate the B → K ℓ + ℓ − decayamplitude with and without the factorization. We take account of the renormalization constant andprovide a test of the factorization approximation using an explicit lattice calculation. B → K ℓ + ℓ − amplitude and the artificial divergence In this section, we review the calculation of the decay amplitude with special emphasis onthe artificial divergence. Avoiding such divergence is essential for the lattice calculation, and theproblem is extensively studied for the calculation of K → π ℓ + ℓ − amplitude on the lattice [7, 8].We consider the B → K amplitude with the charmonium contribution, which occurs throughthe weak effective Hamiltonian H eff with the Fermi constant G F , CKM matrix V cb , V cs , and Wilsoncoefficient C i , H eff = G F √ V ∗ cs V cb ( C O c + C O c ) . (2.1)The operators O ci , which include cc are O c = ( s i γ µ P − c j )( c j γ µ P − b i ) , O c = ( s i γ µ P − c i )( c j γ µ P − b j ) . (2.2)where indeces i and j represent the color index, and the chiral projection operator is defined as P − ≡ − γ .We define the B → K ℓ + ℓ − decay amplitude for a four-momentum q ≡ k − p as A ( q ) = Z d x e iqx h K ( ppp ) | T (cid:2) J µ ( ) H eff ( x ) (cid:3) | B ( kkk ) i . (2.3)1 → K ℓ + ℓ − Factorization
Katsumasa Nakayama V ( x ) B K O i J/ψ l + l − t t H t J t K Figure 1:
Setup of the lattice calculation of the B → K ℓ + ℓ − amplitude through charmonium J / ψ resonances. In order to calculate the amplitude, we integrate over the position of the weak effective Hamil-tonian and define I µ as I µ = e − [ E K ( ppp ) − E B ( kkk )] t J Z t J + T b t J − T a d t H Z d xxx Z d yyy e − iqqq · xxx h K ( t K , ppp ) | T (cid:2) J µ ( t J , xxx ) H eff ( t H , yyy ) (cid:3) | B ( , kkk ) i . (2.4)The setup of the lattice calculation is shown in Figure 1. We introduce t H , t J , t K , T a , and T b toidentify the time for each states and operators.We can rewrite this quantity using the complete set of the intermediate states, which can be de-scribed by the spectral densities ρ ( E ) for the states with strangeness, and ρ ( E ) for those withoutstrangeness. Namely, I µ = − Z ∞ d E ρ ( E ) E h K ( ppp ) | J µ ( ) | E ( kkk ) ih E ( kkk ) | H eff ( ) | B ( kkk ) i E B ( kkk ) − E (cid:16) − e ( E B ( kkk ) − E ) T a (cid:17) + Z ∞ d E ρ ( E ) E h K ( ppp ) | H eff ( ) | E ( ppp ) ih E ( ppp ) | J µ ( ) | B ( kkk ) i E − E K ( ppp ) (cid:16) − e − ( E − E K ( ppp )) T b (cid:17) . (2.5)In this representation, the T a , b → ∞ limit of I µ can be identified as the amplitude, A ( q ) = − i lim T a , b → ∞ I µ ( T a , T b , kkk , ppp ) . (2.6)In order that the integral (2.5) stays finite, the energy of the intermediate state plays an essentialrole. Since E − E K ( ppp ) > − ( E − E K ( ppp )) T b can be ignored in the T b → ∞ limit. Onthe other hand, E B ( kkk ) − E < ( E B ( kkk ) − E ) T a may diverge in the limit of large T a . At the physical point of the quark masses,this artificial divergence can be hardly removed, since the number of such intermediate states islarge. In this study, we set the b-quark mass smaller than that of the physical value in order to avoidthis problem. Since the energy of the intermediate state E is bounded by the ground state energyof the K and J / ψ meson, we choose the b-quark mass to realize the condition, E B < E J / ψ + E K .With this unphysical b-quark mass, we can define the decay amplitude from the four-pointcorrelators. In this work, however, we test the factorization approximation as the first step beforeproceeding to the extraction of the decay amplitude.2 → K ℓ + ℓ − Factorization
Katsumasa Nakayama B K J/ψ
Figure 2:
Factorization of the four-point correlator B → K ℓ + ℓ − with charmonium J / ψ resonances. B K O i J/ψ
Figure 3:
A typical example of the non-factorizable contribution for B → K ℓ + ℓ − with the charmonium.Gluon exchanging between B (K) and the charmoinum can not be factorized.
3. Factorization and renormalization
In order to investigate the factorization, we define operators O ( ) and O ( ) as O ( ) = ( c i γ µ P − c i )( s j γ µ P − b j ) , O ( ) = ( c i [ T a ] i j γ µ P − c j )( s k [ T a ] kl γ µ P − b l ) . (3.1)The operator with the color octet contraction O ( ) includes the SU ( ) generators T a .Figure 2 illustrates the factorization of the B → K ℓ + ℓ − four-point correlator. The contributionof O ( ) is simply represented in the factorization approximation as, h KJ / ψ | O ( ) | B i ≃ h K | s i γ µ P − b i | B ih J / ψ | c γ µ c | i . (3.2)Figure 3 is a typical example of the non-factorizable contribution. As we can see in the defini-tion of O ( ) , the simple factorization is not allowed for this operator, because the factorized piece iscolor-octet, which vanishes when sandwiched by the physical states. Namely, factorization of the O ( ) is represented as h KJ / ψ | O ( ) | B i ≃ . (3.3)Since our lattice calculation is done in the O c and O c basis, we need to transform them tothe O ( ) and O ( ) basis. The Firtz transformation q γ µ P − q q γ µ P − q = q γ µ P − q q γ µ P − q can beused to obtain the relation between O c , O c and O ( ) , O ( ) : O c = O ( ) , O c = O ( ) + O ( ) . (3.4)3 → K ℓ + ℓ − Factorization
Katsumasa Nakayama β a − L × T ( × L s ) meas. am uds am c am b × ( × )
400 0.025 0.27287 0.66619
Table 1:
The parameters for our lattice calculation.
Here, we also consider the renormalization of the operator O c and O c . The renormalizedoperators h O c i R and h O c i R are written in terms of O c and O c with the renormalization constants Z and Z , h O c i R ≡ Z h O c i + Z h O c ih O c i R ≡ Z h O c i + Z h O c i . (3.5)The renormalization constant are determined through temporal moments of three-point correlators[9]. In order to test the factorization relation, we define the ratios R and R / on the lattice ofvolume V , R ≡ V h K | J ν O c | B i R h | J ν J µ | i R h K | s j γ µ P − b j | B i R , R / ≡ h K | J ν O c | B i R h K | J ν O c | B i R , (3.6)which become 1 or when the factorization approximation is valid, respectively.
4. Preliminaly results
Our lattice setup is summarized in Table 4. The lattice configurations are generated with N f = + a − = . ( ) GeV, and each quark mass is set to am uds = . am c = . am b = . m π = ( ) MeV and am B = . ( ) GeV.We insert two-different momenta p = ( − π L , , ) and p = ( − π L , − π L , ) for the final state ofcharmonium cc . For the initial B meson state, we input a momentum k = ( , , ) . The momenta p and p are smaller than the physical one, but we focus on these two inputs as a first step. Theenergy spectrum with these input values are calculated as E K ( p ) = ( ) MeV, E K ( p ) = ( ) MeV, E J / ψ ( p ) = . ( ) GeV, and E J / ψ ( p ) = . ( ) GeV. As we discussed inthe definition of the B → K decay amplitude, the spectrum satisfies the condition m B < E J / ψ + E K .Namely, our setup does not suffer from artificial divergence, as we mentioned previously. Thesource operators are set at t K =
42 for K meson source, t J =
27 for electromagnetic coupling J µ . The B-meson source is set at the t =
0. In this study, statistical uncertainty is estimatedwith 100 independent configurations with four different source points per configuration. Since thepropagating mesons are heavy, the auto-correlation is not significant.We use the renormalization constants determined through the moments of the correspondingthree-point correlators [9]. They are Z = . ( ) and Z = . ( ) .4 → K ℓ + ℓ − Factorization
Katsumasa Nakayama -1-0.5 0 0.5 1 1.5 2 0 5 10 15 20 25 30 35 40 t H R (1,1,0)R (1,0,0)1/3 Figure 4:
The ratio R / are shown for each input momenta. The electromagnetic current is set at t J =
27 asshown by the dashed line. t H R (1,1,0)R (1,0,0)1.0 Figure 5:
The ratio R are shown for each input momenta. The electromagnetic current is set at t J =
27 asshown by the dashed line.
Figure 4 shows the result for the ratio R / , which should be equal to 1 / /
3, and we do not see any signif-icant violation of the approximation. On the other hand, the relation R ≃ cc contribution to B → K ℓ + ℓ − .
5. Discussions
A quantitative estimate of the cc contribution to B → K ( ∗ ) ℓ + ℓ − remains a notoriously difficulttask, because of the non-perturbative dynamics of QCD. The first principle calculation of the latticeQCD can not be directly applied since there are many intermediate states that contribute to the realand imaginary parts of the amplitude. In this work, we simplify the problem by considering anunphysical setup with a smaller b quark mass, hoping that it captures the important part of thedynamics. We find a significant violation of the factorization ansatz, which may be used as inputsfor phenomenological models to study more realistic situations. We also note that a large violation5 → K ℓ + ℓ − Factorization
Katsumasa Nakayama of factorization was previously found for the K → ππ amplitude [11], which suggests the need forfully non-perturbative calculation for similar processes. Acknowledgements
The lattice QCD simulation has been performed on Blue Gene/Q supercomputer at the HighEnergy Accelerator Research Organization (KEK) under the Large Scale Simulation Program (Nos.15/16-09, 16/17-14). Oakforest-PACS at JCAHPC under the support of the HPCI System ResearchProjects. K. N. is supported by the Grant-in-Aid for JSPS (Japan Society for the Promotion ofScience) Research Fellow (No. 18J11457). This work is supported in part by the Grant-in-Aid ofthe Japanese Ministry of Education (No. 18H03710).
References [1] LHC B collaboration, R. Aaij et al., Observation of a resonance in B + → K + µ + µ − decays at lowrecoil , Phys. Rev. Lett. (2013) 112003 [ ].[2] LHC B collaboration, R. Aaij et al., Measurement of the phase difference between short- andlong-distance amplitudes in the B + → K + µ + µ − decay , Eur. Phys. J.
C77 (2017) 161[ ].[3] M. Neubert and B. Stech,
Nonleptonic weak decays of B mesons , Adv. Ser. Direct. High Energy Phys. (1998) 294 [ hep-ph/9705292 ].[4] M. Beneke, T. Feldmann and D. Seidel, Systematic approach to exclusive B → V l + l − , V γ decays , Nucl. Phys.
B612 (2001) 25 [ hep-ph/0106067 ].[5] J. Lyon and R. Zwicky,
Resonances gone topsy turvy - the charm of QCD or new physics inb → s ℓ + ℓ − ? , .[6] D. Du, A. X. El-Khadra, S. Gottlieb, A. S. Kronfeld, J. Laiho, E. Lunghi et al., Phenomenology ofsemileptonic B-meson decays with form factors from lattice QCD , Phys. Rev.
D93 (2016) 034005[ ].[7] RBC, UKQCD collaboration, N. H. Christ, X. Feng, A. Portelli and C. T. Sachrajda,
Prospects for alattice computation of rare kaon decay amplitudes: K → π ℓ + ℓ − decays , Phys. Rev.
D92 (2015)094512 [ ].[8] RBC, UKQCD collaboration, N. H. Christ, X. Feng, A. Portelli and C. T. Sachrajda,
Prospects for alattice computation of rare kaon decay amplitudes II K → πν ¯ ν decays , Phys. Rev.
D93 (2016)114517 [ ].[9] T. Ishikawa, K. Nakayama and S. Hashimoto,
Renormalization of bilinear and four-fermion operatorsthrough temporal moments , PoS LATTICE2019 (2019) .[10] R. C. Brower, H. Neff and K. Orginos,
The Möbius domain wall fermion algorithm , Comput. Phys.Commun. (2017) 1 [ ].[11] RBC, UKQCD collaboration, P. A. Boyle et al.,
Emerging understanding of the ∆ I = / Rule fromLattice QCD , Phys. Rev. Lett. (2013) 152001 [ ].].