PoS(LATTICE 2015)104 http://pos.sissa.it/ 2 with its experimental − g ected-type diagrams. Here, we ery method to calculate them by urbative QED method. the muon ribution must be calculated by lattice y, USA y, USA ive Commons onal License (CC BY-NC-ND 4.0). ∗ For reliable comparison of the standard model prediction to start with recalling the pointlattice QCD, that and must present one be concrete taken method care called nonpert of in ev QCD simulation. HLbL contribution has many types of disconn value, the hadronic light-by-light scattering (HLbL) cont Speaker. ∗ Copyright owned by the author(s) under the terms of the Creat c The 33rd International Symposium on Lattice14 Field -18 Theory July 2015 Kobe International Conference Center, Kobe, Japan Attribution-NonCommercial-NoDerivatives 4.0 Internati M. Hayakawa On calculating disconnected-type hadronic light-by-light scattering diagrams from lattice QCD Department of Physics, Nagoya University, Japan Nishina Center, RIKEN, Japan T. Blum Physics Department, Connecticut University, USA RIKEN-BNL Research Center, Brookhaven National Laborator N. H. Christ Physics Department, Columbia University, USA T. Izubuchi Physics Department, Brookhaven National Laboratory, USA RIKEN-BNL Research Center, Brookhaven National Laborator L. C. Jin Physics Department, Columbia University, USA C. Lehner Physics Department, Brookhaven National Laboratory, USA

PoS(LATTICE 2015)104 ) ) ) 1 ( 87 40 ( ( because O . ) local QED 1 11 ) , 249 116 − a E ( 3 . Actually, no O ( ) ) ) ) th ( 2, unless it can be µ a , will leave a contro- − exists. There is thus a HLbL HLbL ) ( ( Comparison of the dis- g ) − µ µ next exp ) ( a a µ average represents the inverse a exp HLbL ) on the muon side δ ( HLbL ( agrams according to the way ( µ µ tribution whose details are pre- contribution by the lattice sim- µ a a in Fig. 2 are those with three EM a Table 1: crepancy between theory and exper- iment with HLbL contribution.are given All in units of 10 ) 1 . The diagrams with photon, and one internal EM vertex ns out that there are seven types of r, focus on so-called connected-type account in every method to compute on, , to the muon . [3]. Mainz group has attempted to U M) vertices lie on a single quark loop. annot be captured by hadron models. ne concrete method with such a point E ) on to 3 gluon as dynamical variables, such as ( th ( with manageable theoretical uncertainty. µ ) a t scattering diagrams from lattice QCD 2 can resort HLbL 2 in Fig. 3 differ from those of type ( ) − µ 3 a g , E 1 ( , which is comparable in size with ) th + 5 permutations of QED vertices ( ( for a given QCD configuration µ , we observe the discrepancy µ ] ) a U th [ ( QCD D µ a in Tab. 1 was obtained as such [1]. In- ) HLbL ( requires purely theoretical consideration. Thus µ Connected-type HLbL diagrams. Each quark line under the QCD ) a Here we turn our attention to the disconnected-type HLbL con The lattice QCD simulation tempts us into classifying the di Recently, feasibility was demonstrated to compute the HLbL The hadronic light-by-light scattering (HLbL) contributi The reason is as follows. While the hadronic vacuum HLbL ( µ vertices on a quark loop, oneon of the which other couples loop. to The the diagrams external of type how four EM vertices aredisconnected-type distributed diagrams in over total. quark The loops. diagrams of type It tur vertices are not shown here. sented in Sec. 2. We firstit see the by point lattice that simulation must in be Sec.taken called 3. into into account In by Sec. remedying 4, the we one also proposed present in o Ref. [5]. 2. Classification of disconnected-type HLbL diagrams ulation [2], and more efficientcalculate method the HLbL is amplitude investigated [4]. in Alldiagram Ref of shown in those Fig. works, 1, howeve where all of four electromagnetic (E potential possibility of significance ofTherefore, QCD dynamics the that first-principle c calculationlattice with QCD quarks simulation, is and crucial to provide cluding it as the part of of the quark Dirac operator versial uncertainty in the standard model prediction On calculating disconnected-type hadronic light-by-ligh 1. Introduction M. Hayakawa Figure 1: far, it has been only estimatedseveral according hadrons to such the as models pionsvalue with as of dynamical variables. The polarization (HVP) contribution to the muon between the experiment and proof supporting the validity of the low energy approximati to the experiments to evaluatea the relevant QCD dynamics, calculated by means of lattice QCD simulation. PoS(LATTICE 2015)104 QCD con QCD QCD rk loops with two those having three quark loops our quark loops with just one EM QCD nal EM vertex. The diagrams of type local QED vertices are not shown. QCD QCD ) QCD con a ( symbol will be clarified in Sec. 3. O t scattering diagrams from lattice QCD -type diagrams -type diagrams ) ) 3 3 2 , , E E 1 2 ( ( QCD Figure 3: Figure 4: QCD QCD QCD con -type diagrams. The diagrams with ) 1 , are also the disconnected-type diagrams having just two qua E 1 3 ( in Fig. 4 Figure 2: ) 2 The meaning of the superscript “con” attached to the average 1 , E 2 On calculating disconnected-type hadronic light-by-ligh M. Hayakawa ( there is no internal EM vertex on the quark loop with the exter internal EM vertices on each.with at The least one diagrams EM in vertexvertex on Figs. on each. 5 each. The and diagram 6 in Figs. are 7 has f PoS(LATTICE 2015)104 (3.1) . ) QCD con ) 4 ( x ( ) con 4 QCD ( µ j ) ) 3 ( x ( ) 3 ctation value (VEV) of four ( ur EM currents µ j ) ) 2 ( con x QCD con QCD ( he field-theoretically connected dia- ) 2 ( QCD con µ j ted diagrams. Here, a field-theoretically ) ) 1 ( x QCD -type diagrams ( ) ) t scattering diagrams from lattice QCD -type diagrams -type diagrams E 1 ) ) 1 ( ) ) , 1 2 µ 4 j 4 , , 1 ( ] , 1 1 x q , , 1 , ( con ) q QCD , E E 4 , ( E 2 1 U µ ( ( [ 1 j ( ) ) QCD 3 S ( − x ( ) qe 3 ( Figure 6: Figure 5: µ j Figure 7: ) dqd ) QCD con 2 ( Z x ) ( x ) ( 2 dU ( µ µ j j Z ) ) 1 ( 1 QCD x ( Z ) 1 ( = µ j D Lattice QCD simulation may enable to compute the vacuum expe On calculating disconnected-type hadronic light-by-ligh M. Hayakawa hadronic EM currents 3. Disconnected component in the correlation function of fo This VEV, however, contains not onlygrams, the but contribution also from that t from the field-theoretically disconnec PoS(LATTICE 2015)104 . ) . dis QCD 0 3 1 2 3 2 3 i C HVP D QED S ) A 3 , − , in every α ′ C ( − h C O M 3 0 2 1 0 1 0 QCD + ( QCD C can be canceled i generated by dy- d side of Fig. 9 is . For that purpose, C A ) ) h ) A D M 1 directly because con- , Emergence of degener- ) ) , = K . renormalization of EM 2 1 1 m U ) ) ) , , , ( ) − 3 1 2 1 1 1 ( , say, denotes con QCD α , , , , , , i con QCD ia three virtual photons, its QCD C ( i E E E E E E

E 1 3 2 1 2 1 ) O A ( ( ( ( ( ( 4 y h A -type diagram. It is generated ins the unwanted contribution, ( h Table 2: acy. ) th the renormalized EM charge unwanted contribution ρ 2 j ate , lized by ) E x e than one nontrivial connected sub- ith the of a pair of 2 ( unting, we must thus explicitly sub- ( to get tice to compute the disconnected-type heoretically disconnected (connected) es λ ks and gluons. The best we can do is to j , we prepare two sets of , the quantity relevant to the HLbL is D

(4.1) in our classification scheme. to the dis QCD K con QCD , ) i of that contribution. If the connected com- 3 1 i as in Tab. 2 [5]. 1 } , A D A t scattering diagrams from lattice QCD h 1 and the one in the lower loop only with respect h , ) K 1 5 − , ) , which differs from the one in Eq. (3.1) by the sum E D 1 QED ( , component S con QCD ) are defined in Fig. 8, E − ) ′ and ) D C 4 ) QCD ( ( 1 QED) for light quark sys- S x M is denoted by , ( ) + 1 4 ( , A µ triplicate degeneracy and E j ) + ( connected ′ ′ 2 ( ) C C ( ) , 3 S ( , while the muon is charged with respect to both QEDs. ) M x ) 2 , ( − , ) QED) system. We multiply ′ C 3 1 ( in Eq. (4.1) is the one added here to subtract the unwanted C in Fig. 9. The individual terms involve µ , + S j ) QED E , D M HVP contribution, which is henceforth called D ) , 1 C ) ) ( K and its disconnected component 2 K 3 ( , QED) simulation. The quark in the upper loop on the right-han M disconnected ) x α − ) + ( ( 2 ( + ( ) QCD C and 2 , ( QCD ( O i E µ S D j HVP contribution. Note that the HVP contribution coming fro 2 ( A ) − S ) ) h 3 C , 1 ( D α x ( M ( ) ( The nonperturbative QED method we propose here to The term Basically, lattice QCD simulation does not allow us to evalu We discuss how subtraction of unwanted contributions is rea If the VEV of four EM currents in Eq. (3.1) couples to the muon v M 1 O ( { µ j 3 1 Practically, they may be two independent important samples charge due to QCD. Anycontains calculation such of a the contribution HVPtract implicitly. contribution such an wi To avoid double co graph as a graph. We shalldiagrams refer as to the the contribution of field-t method based on lattice QCD. Subtraction isdiagrams; required in prac On calculating disconnected-type hadronic light-by-ligh disconnected diagram is a Feynman diagram consisting of mor M. Hayakawa quantity in the bracket inof Eq. HLbL (4.1) diagrams because emerge individual with 8 typ namical (QCD and contribution contained in the other terms. To construct the averages with respect to (QCD compute full HLbL contribution is given by it may be helpful to observe the situation by focusing on a 4. Nonperturbative QED method for full HLbL contribution charged only with respect to the first in three ways as showni.e. in Fig. 10 ; each diagram in Fig. 10 conta of three terms each of which is a product of two currents, tem. Note thatinverse each of muon muon Dirac line operator in inerated the Figs. QED by 8 configuration such and gen- (QCD 9 denotes the where the terms ponent of the QCD average of to the second nectivity is the attribute of each Feynmancompute diagram with quar D indirectly. disconnected component gives rise to the HVP contribution w PoS(LATTICE 2015)104 tive (mid- . D M QED) n them is as + appear exactly QCD (left) and C M QED QED + + exactly coincides with that QED QED QED D + + + QCD QCD QED QED vertices. + K  ) ) le average of (QCD QCD QCD QCD . p), but the uncanceled one does 2 ( a D A ( QCD ology turns out to , ) 2 O K ( contains one HVP function entirely U  D , ms. If the full HVP function is denoted  absence of the last term in Eq. (4.1) can and ) with 1 S ( ′ D QCD A C , ) S or 1 ( S , ′ U ,  = D = C ′ = ′ t scattering diagrams from lattice QCD C 1 C C 1 D − S M −  S S i ) , S M 6 ) 2 1 ( 1 C ( − A A i e e ) ′ S 2 q q QED QED ( Q Q A i i , and + + e -diagrams generated by − − QED µ e e C ) ) ) Q + 3 2 1 i ( ( S QCD QCD − The terms -type is generated in three ways from U U e α , h  ) ) ( QCD 1 C D 2 D ( A , , verifying that subtraction is realized in the nonperturba e M µ E Q 2 i ( − QCD e h ↑ ↑ D Figure 9: The terms , but the other two survive. The essential difference betwee a set of the O C = = = ′ = C D C S D K M M M Figure 8: An identical diagram of The red propagators and vertices are generated by the ensemb . One can show that On calculating disconnected-type hadronic light-by-ligh M. Hayakawa follows. The HVP contribution canceled by in Fig. 12 with theQED same method. degeneracy Figure 10: dle, right). by that supplied from supplied from QED average (QCDnot. average The of same fully is red true quark foras the loo in other Fig. disconnected-type 11, diagra the unwanted contributionsbe that summarized survive in in the Fig. 12, where each diagram of identical top twice PoS(LATTICE 2015)104 l HLbL 2, we must − g , 303 (2009). 20 , no. 1, 012001 (2015). C and L.C.J are supported . [3] encourage to develop 114 QCD QCD QCD QCD -02-98CH10996(BNL). T.I is QCD in-Aid for Scientific Research ction of the muon QED vertices are not shown. Rev. Lett. alutsa, arXiv:1507.01577 [hep-lat]. ) thod, which is based on the dynamical 400261. , 439 (2014). nd C. Lehner, arXiv:1510.07100 [hep-lat]. a ( O t. High Energy Phys. tual photons. 2013 unwanted diagrams. + QCD ) 3 t scattering diagrams from lattice QCD α QCD QCD ( 7 O QCD QCD = Summary of , 1452 (2015). QCD QCD 347 QCD HVP contributions in every method for the computation of ful Figure 12: ) 3 QCD α ( Full HVP function. The diagrams with QCD , Science O et al. Figure 11: QED) simulation as done in Ref. [6], though the results in Ref + Here, we remarked that, to avoid double counting in the predi T.B is supported by U.S. DOE grant #DE-FG02-92ER41989. N.H. 5. Summary On calculating disconnected-type hadronic light-by-ligh M. Hayakawa explicitly subtract contribution. We presented an idea (4.1)(QCD for the concrete me Acknowledgments alternative methods without stochastic realization of vir by U.S. DOE grant#25610053. #de-sc0011941. T.I M.H and is C.L supported arealso by supported supported by Grants- by Grants-in-Aid U.S. for DOE Scientific Research Contract #26 #AC References [1] J. Prades, E. de Rafael and A. Vainshtein, Adv. Ser. Direc [2] T. Blum, S. Chowdhury, M. Hayakawa and T. Izubuchi, Phys. [3] T. Blum, N. H. Christ, M.[4] Hayakawa, T. Izubuchi, J. L. Green, C. O. Jin Gryniuk, a G. von[5] Hippel, H. T. B. Blum, Meyer M. and Hayakawa V. and Pasc T. Izubuchi, PoS LATTICE [6] S. Borsanyi