Four quark operators for kaon bag parameter with gradient flow
Asobu Suzuki, Yusuke Taniguchi, Hiroshi Suzuki, Kazuyuki Kanaya
UUTHEP-750, UTCCS-P-133, KYUSHU-HET-213
Four quark operators for kaon bag parameter with gradient flow
Asobu Suzuki, ∗ Yusuke Taniguchi, † Hiroshi Suzuki, ‡ and Kazuyuki Kanaya § Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan Center for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan Department of Physics, Kyushu University,744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan Tomonaga Center for the History of the Universe,University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan (Dated: September 1, 2020)
Abstract
To study the CP -violation using the K − K oscillation, we need the kaon bag parameter which representsQCD corrections in the leading Feynman diagrams. The lattice QCD provides us with the only wayto evaluate the kaon bag parameter directly from the first principles of QCD. However, a calculation ofrelevant four quark operators with theoretically sound Wilson-type lattice quarks had to carry a numericallybig burden of extra renormalizations and resolution of extra mixings due to the explicit chiral violation.Recently, the Small Flow- t ime eXpansion ( SF t X ) method was proposed as a general method based on thegradient flow to correctly calculate any renormalized observables on the lattice, irrespective of the explicitviolations of related symmetries on the lattice. To apply the SF t X method, we need matching coefficients,which relate finite operators at small flow times in the gradient flow scheme to renormalized observables inconventional renormalization schemes. In this paper, we calculate the matching coefficients for four quarkoperators and quark bilinear operators, relevant to the kaon bag parameter. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ h e p - l a t ] A ug . INTRODUCTION In the study of the CP -violation, the K − K oscillation plays an important role. Here, to extractCabibbo-Kobayashi-Maskawa matrix elements in the leading Feynman diagrams for the K − K oscillation, we need to know the kaon bag parameter which represents QCD corrections in thesediagrams. The lattice QCD provides us with the only way to evaluate the nonperturbative valueof the kaon bag parameter directly from the first principles of QCD [1]. However, a calculationof relevant four quark operators with theoretically sound Wilson-type lattice quarks had to carrya numerically big burden of extra renormalizations and resolution of extra mixings, requiredmainly due to the explicit violation of the chiral symmetry by the Wilson quarks at nonzero latticespacings [2–11].Recently, a series of new methods based on the gradient flow introduced various advances inlattice QCD [12–22]. Among them, we adopt the S mall F low- t ime e X pansion ( SF t X ) method,which is a general method to correctly calculate any renormalized observables on the lattice [18,19, 23–25]. The gradient flow is a modification of bare fields according to flow equations drivenby the gradient of an action. It is shown that the operators constructed by flowed fields (“flowedoperators”) are free from UV divergence and also from short-distance singularities at nonzero flowtime t > t X method is as follows: Because of the strict finiteness offlowed operators, we can safely evaluate their nonperturbative values by (i) constructing their latticeoperators directly from their continuum expressions, (ii) evaluating their values on the lattice, and(iii) taking the continuum limit. The finiteness of the target operators leads us automatically totheir correct values by just taking the continuum extrapolation—we do not need to introduce anyadditional corrections due to the lattice artifacts, even if the lattice model at finite lattice spacingsviolates some symmetries relevant to the original derivation of the operators.The method has been applied to calculate the energy-momentum tensor, which is the generatorof the continuous Poincaré transformation and thus is not straightforward to evaluate on discretelattices. From test studies around the deconfinement transition temperature in quenched QCD [26–28] and in 2 + t X method correctly reproduce previous results of theequation of state estimated by the conventional integral methods.Because the method is applicable also to observables related to the chiral symmetry, we mayapply the method to cope with the difficulties of Wilson-type quarks associated with their explicit2hiral violation. Theoretical basis to study fermion bilinear operators in the SF t X method is givenin [25]. The method was applied to compute the disconnected chiral susceptibility in 2 + t X method, the two definitions are shown to agreewell with each other even at a finite lattice spacing [30]. These suggest that the SF t X method ispowerful in calculating correctly renormalized observables.In this paper, we extend the SF t X method to the study of four fermi operators. As the first stepof the study, we concentrate on the issue of the kaon bag parameter, B K = (cid:104) K | O ∆ S = | K (cid:105) (cid:12)(cid:12) (cid:104) | s γ µ γ d | K (cid:105) (cid:12)(cid:12) , (I.1)where, with γ L µ : = γ µ ( − γ ) , O ∆ S = = ( s γ L µ d )( s γ L µ d ) (I.2)is the ∆ S = O ∆ S = = ZO ∆ S = + (cid:205) i Z i O i . Preciseevaluation of the renormalization and mixing coefficients is computationally demanding [34]. Ina real scalar field theory, the gradient flow was shown to avoid the issue of operator mixing [35].See also recent studies [36–39]. We thus expect that the SF t X method will drastically simplify thecalculation of four quark operators in QCD .The SF t X method [18, 19, 23–25] is based on the expansion of flowed operators at small t in terms of renormalized operators at t = /√ t , inasymptotically free theories such as QCD, we can calculate the matching coefficients at small t byperturbation theory. In this paper, we perform a one-loop calculation of the matching coefficientfor the ∆ S = See Refs. [40, 41] for a different approach using twisted mass Wilson-type quarks.
II. FORMULATIONA. Gradient flow
In this section, we introduce the gradient flow with the background field method [24], whichsimplifies perturbative calculations of renormalization factors. Our conventions for the gaugegroup factors and Casimirs are as follows: We normalize the gauge group generators byTr ( T a T b ) = − T δ ab , [ T a , T b ] = f abc T c , (II.1)where f abc is the structure constant. The anti-Hermitian matrices T a satisfy T a T a = − C F . (II.2)For the fundamental representation of SU( N ), T = /
2, dim ( R ) = N , and C F = ( N − )/ N .We first decompose the gauge field A µ and quark field ψ into background fields and quantumfields as A µ ( x ) = ˆ A µ ( x ) + a µ ( x ) , (II.3) ψ f ( x ) = ˆ ψ f ( x ) + p f ( x ) , (II.4) ψ f ( x ) = ˆ ψ f ( x ) + p f ( x ) , (II.5)where f =
1, 2, · · · , N f is for the flavor, ˆ A µ , ˆ ψ f , ˆ ψ f are background fields, and a µ , p f , p f are theirquantum fields, respectively.The flow equations we adopt are basically the simplest ones as proposed by Lüscher [14, 16].The gradient flow drives the fields, ˆ A µ , ˆ ψ f , ˆ ψ f , a µ , p f , and p f , into their flowed fields, ˆ B µ , ˆ χ f , ˆ χ f , b µ , k f , and k f , respectively [24]. Flow equations for the background fields are given by ∂ t ˆ B µ ( t , x ) = ˆ D ν ˆ G νµ ( t , x ) , ˆ B µ ( t = , x ) = ˆ A µ ( x ) , (II.6) ∂ t ˆ χ f ( t , x ) = ˆ D ˆ χ f ( t , x ) , ˆ χ f ( t = , x ) = ˆ ψ f ( x ) , (II.7) ∂ t ˆ χ f ( t , x ) = ˆ χ f ( t , x ) ˆ ←− D , ˆ χ f ( t = , x ) = ˆ ψ f ( x ) . (II.8)4n this paper, we set the gauge parameter α in Ref. [24] to unity, α =
1. Then, the flow equationsfor the quantum fields are given by ∂ t b µ ( t , x ) = ˆ D b µ ( t , x ) + [ ˆ G µν ( t , x ) , b ν ( t , x )] + ˆ R µ ( t , x ) , ˆ b µ ( t = , x ) = ˆ a µ ( x ) , (II.9) ∂ t k f ( t , x ) = (cid:8) D − ˆ D µ b µ ( t , x ) (cid:9) k f ( t , x ) + (cid:8) b µ ( t , x ) ˆ D µ + b ( t , x ) (cid:9) ˆ χ f ( t , x ) , k f ( t = , x ) = p f ( x ) , (II.10) ∂ t k f ( t , x ) = k f ( t , x ) (cid:110) ←− D + ˆ D µ b µ ( t , x ) (cid:111) + ˆ χ f ( t , x ) (cid:26) − ←− D µ b µ ( t , x ) + b ( t , x ) (cid:27) , k f ( t = , x ) = p f ( x ) , (II.11)where we define ˆ G µν ( t , x ) = ∂ t ˆ B ν ( t , x ) − ∂ t ˆ B µ ( t , x ) + [ ˆ B µ ( t , x ) , ˆ B ν ( t , x )] , (II.12)ˆ D µ = ∂ µ + [ ˆ B µ ( t , x ) , · ] , ( for gauge fields ) (II.13)ˆ D µ = ∂ µ + ˆ B µ ( t , x ) , ( for quark fields ) (II.14)ˆ R µ ( t , x ) = [ b ν ( t , x ) , ˆ D ν b µ ( t , x )] − [ b ν ( t , x ) , ˆ D µ b ν ( t , x )] + (cid:2) b ν ( t , x ) , (cid:2) b ν ( t , x ) , b µ ( t , x ) (cid:3) (cid:3) . (II.15)In this paper, we set the background gauge field to zero and the background quark fields toconstant. Then, the solution of the flow equations for the background fields is given byˆ B ( t , x ) = ˆ A ( x ) = , (II.16)ˆ χ f ( t , x ) = ˆ ψ f ( x ) = ( const. ) , (II.17)ˆ χ f ( t , x ) = ˆ ψ f ( x ) = ( const. ) . (II.18)Taking the solution of the background fields into account, the flow equations for the quantum fieldscan be simplified as ∂ t b µ ( t , x ) = ∂ b µ ( t , x ) + ˆ R µ ( t , x ) , ˆ b µ ( t = , x ) = ˆ a µ ( x ) , (II.19) ∂ t k f ( t , x ) = (cid:8) D − ∂ µ b µ ( t , x ) (cid:9) k f ( t , x ) + (cid:8) b µ ( t , x ) ∂ µ + b ( t , x ) (cid:9) ˆ ψ f ( t , x ) , k f ( t = , x ) = p f ( x ) , (II.20) ∂ t k f ( t , x ) = k f ( t , x ) (cid:110) ←− D + ∂ µ b µ ( t , x ) (cid:111) + ˆ ψ f ( t , x ) (cid:110) − ←− ∂ µ b µ ( t , x ) + b ( t , x ) (cid:111) , k f ( t = , x ) = p f ( x ) , (II.21)withˆ R a µ ( t , x ) = f abc b b ν ( t , x ) ∂ ν b c µ ( t , x ) − f abc b b ν ( t , x ) ∂ µ b c ν ( t , x ) + f abc f cde b b ν ( t , x ) b d ν ( t , x ) b e µ ( t , x ) . (II.22)5ormal solution of the flow equations for the quantum fields is given by b a µ ( t , x ) = e t ∂ a a µ ( x ) + ∫ t d s e ( t − s ) ∂ ˆ R a µ ( s , x ) , (II.23) k f ( t , x ) = e t ∂ p f ( x ) + ∫ t d s e ( t − s ) ∂ (cid:8) b µ ( s , x ) ∂ µ + b ( s , x ) (cid:9) (cid:110) e s ∂ ˆ ψ f ( x ) + k f ( s , x ) (cid:111) , (II.24) k f ( t , x ) = p f ( x ) e t ←− ∂ + ∫ t d s (cid:26) ˆ ψ f ( x ) e s ←− ∂ + k f ( s , x ) (cid:27) (cid:110) − ←− ∂ µ b µ ( s , x ) + b ( s , x ) (cid:111) e ( t − s )←− ∂ . (II.25)In one-loop calculations discussed in this paper, we can disregard the O( (cid:49) ) terms in thepropagators. Thus, the propagator between b ( t , (cid:96) ) and b ( s , (cid:96) ) can be simplified as G ab µν ( t , s ; (cid:96) ) ∼ e −( t + s ) (cid:96) G ab µν ( (cid:96) ) (one loop) , (II.26)where b a µ ( t , (cid:96) ) = ∫ d D x b a µ ( t , x ) e − i (cid:96) · x (II.27)is the quantum gauge field in the momentum space and G ab µν ( (cid:96) ) is the gluon propagator at t = (cid:96) , G ab µν ( (cid:96) ) = (cid:49) (cid:96) δ ab δ µν . (II.28)Similarly, the solution for quantum quark fields k f and k f can also be simplified as k f ( t , x ) ∼ e t ∂ p f ( x ) + ∫ t d s e ( t − s ) ∂ (cid:16) b ( s , x ) ˆ ψ f + b µ ( s , x ) ∂ µ e s ∂ p f ( x ) (cid:17) , (II.29) k f ( t , x ) ∼ p f ( x ) e t ←− ∂ + ∫ t d s (cid:16) ˆ ψ f b ( s , x ) − p f ( x ) e s ←− ∂ ←− ∂ µ b µ ( s , x ) (cid:17) e ( t − s )←− ∂ , (II.30)in one-loop calculations.Because quark masses and external momenta appear as tm and t p in the matching coefficients,their dependence appear in higher orders of the flow time t . Here, we set all quark masses and allexternal momenta of the four quark operators to zero for simplicity. B. Dimensional reduction scheme
In the calculation of four quark operators, we use the Fierz rearrangement to organize the spinorindices. Because the Fierz rearrangement is defined for 4 × D = − (cid:15) dimensional space-time, we thus impose that only the internalloop momenta are reduced to the D = − (cid:15) dimensional space-time, while the other Lorentzindices run in four dimensional Lorentz space-time. This procedure is called the dimensionalreduction scheme [42].We denote the gamma matrices in four dimension as γ µ , and the gamma matrices in D dimen-sional space-time as γ µ . Denoting the remaining part as ˜ γ µ , the four dimensional gamma matricesare decomposed as γ µ = γ µ + ˜ γ µ , (II.31) γ µ = γ µ ( ≤ µ ≤ D ) , ( D < µ ≤ ) , (II.32)˜ γ µ = ( ≤ µ ≤ D ) ,γ µ ( D < µ ≤ ) . (II.33)The anticommutation relation between γ µ and γ ν can be calculated as (cid:8) γ µ , γ ν (cid:9) = (cid:110) (cid:16) γ µ + ˜ γ µ (cid:17) , γ ν (cid:111) = δ µν , (II.34)where the δ µν means the Kronecker delta in D dimension. The other relations can be calculatedsimilarly, e.g., γ µ γ ν γ µ = − D γ ν + γ ν , (II.35) γ µ γ ν γ µ = − γ ν . (II.36)Finally, we define the γ matrix which anticommutes with all the gamma matrices in thisscheme: (cid:8) γ , γ µ (cid:9) = , (II.37) (cid:8) γ , γ µ (cid:9) = , (II.38) (cid:8) γ , ˜ γ µ (cid:9) = . (II.39)We construct four quark operators with γ µ and γ , but the internal quark propagators contain γ µ only. 7 IG. 1. One-loop Feynman diagrams for the quark field renormalization factor ϕ ( t ) . See Ref. [19] for theFeynman rule. C. Quark field renormalization
It is known that, with the simple gradient flow driven by the pure gauge action as we adopt, quarkfield renormalization is required to keep the flowed fields finite [16]. Here, to avoid complicationsdue to the matching between the lattice and dimensional regularization schemes, we adopt the8uark field renormalization proposed in Ref. [19], in which the renormalized quark fields at t > χ f ( t , x ) = (cid:118)(cid:117)(cid:116) − ( R )( π ) t (cid:68) χ f ( t , x ) γ µ ←→ D µ χ f ( t , x ) (cid:69) χ f ( t , x ) : = ϕ / ( t ) χ f ( t , x ) . (II.40)Note that the summation of the flavor index is not taken in this expression. Because we treat allquarks massless, the renormalization factor ϕ ( t ) is independent of f .In Ref [19], ϕ ( t ) has been calculated to the one-loop order of the perturbation theory with thedimensional regularization scheme. We revisit the calculation and compute ϕ ( t ) with the dimen-sional reduction scheme (DRED). Feynman diagrams relevant to the quark field renormalizationare listed in Fig. 1. See Ref. [19] for the Feynman rule we adopt. The diagrams mean D
02 : ∫ (cid:96), p (− ip µ ) e − tp S F ν ( p ) S F ρ ( p − (cid:96) ) S F σ ( p ) G ab αβ ( (cid:96) ) T a T b tr (cid:2) γ µ γ ν γ α γ ρ γ β γ σ (cid:3) , (II.41) D
03 : ∫ t d s ∫ (cid:96), p (− ip µ ) e −( t − s ) p e − s ( p − (cid:96) ) e − tp S F ν ( p ) S F σ ( p − (cid:96) )× (− i ( p − (cid:96) ) λ ) G ab ρλ ( s , (cid:96) ) T a T b tr (cid:2) γ µ γ ν γ ρ γ σ (cid:3) , (II.42) D
04 : ∫ t d s ∫ s d u ∫ (cid:96), p (− ip µ ) S F ν ( p ) e −( t − s ) p e −( s − u )( p − (cid:96) ) e − up e − tp × (− i ( p − (cid:96) ) ρ )(− ip λ ) G ab ρλ ( s , u ; (cid:96) ) T a T b tr (cid:2) γ µ γ ν (cid:3) , (II.43) D
05 : ∫ t d s ∫ t d u ∫ (cid:96), p (− ip µ ) S F ν ( p − (cid:96) ) e −( t − s ) p e −( t − u ) p e −( s + u )( p − (cid:96) ) × (− i ( p − (cid:96) ) ρ )( i ( p − (cid:96) ) σ ) G ab ρσ ( s , u ; (cid:96) ) T a T b tr (cid:2) γ µ γ ν (cid:3) , (II.44) D
06 : ∫ t d s ∫ (cid:96), p (− ip µ ) S F ν ( p ) e −( t − s ) p e − sp e − tp G ab ρρ ( s , s ; (cid:96) ) T a T b tr (cid:2) γ µ γ ν (cid:3) , (II.45) D
07 : ∫ (cid:96), p e − tp e − t (cid:96) e − t ( p + (cid:96) ) S F ρ ( p ) S F λ ( p + (cid:96) ) G ab µν ( t , (cid:96) ) T a T b tr (cid:2) γ µ γ ρ γ ν γ λ (cid:3) , (II.46) D
08 : ∫ t d s ∫ (cid:96), p e −( t − s ) p e −( t + s )( p + (cid:96) ) (− i ( p − (cid:96) ) ρ ) S F ν ( p + (cid:96) ) G ab µρ ( t , s ; (cid:96) ) T a T b tr (cid:2) γ µ γ ν (cid:3) , (II.47)where ∫ (cid:96), p = ∫ (cid:96) ∫ p with ∫ (cid:96) : = ∫ d D (cid:96) /( π ) D , and S F µ ( (cid:96) ) = − i (cid:96) µ (cid:96) , (II.48) G ab µν ( t , s ; (cid:96) ) = (cid:49) e −( t + s ) (cid:96) (cid:96) δ ab δ µν . (II.49)Carrying out the computations similar to those given in Ref. [19], we find that the diagrams9ontribute as D | DRED : − (cid:15) − ( π t ) − , (II.50) D | DRED : 2 1 (cid:15) + ( π t ) + + ( ) − ( ) , (II.51) D | DRED : −
20 log ( ) +
16 log ( ) , (II.52) D | DRED : 12 log ( ) − ( ) , (II.53) D | DRED : − (cid:15) − ( π t ) − , (II.54) D | DRED : 8 log ( ) − ( ) , (II.55) D | DRED : − ( ) , (II.56)in units of − ( R )( π ) t (cid:49) ( π ) C F . (II.57)Collecting these contributions, we obtain the quark field renormalization factor in the dimen-sional reduction scheme, ϕ ( t ) DRED = ( π t ) − (cid:15) (cid:26) + (cid:49) ( µ )( π ) C F (cid:18) (cid:15) + γ E + ( t µ ) + − log ( ) (cid:19) (cid:27) , (II.58)where we have replaced the bare gauge coupling (cid:49) by the dimensionless (cid:49) ( µ ) using the prescriptionof the MS scheme [43], (cid:49) = (cid:18) µ e γ E π (cid:19) (cid:15) (cid:49) ( µ ) (cid:2) + O( (cid:49) ( µ )) (cid:3) , (II.59)with γ E the Euler-Mascheroni constant. Since ϕ ( t ) is defined in terms of the expectation value ofbare fields in Eq. (II.40), it is independent of the renormalization scale µ . We may thus choose anyvalue for µ in the expressions above, provided that the perturbative expansions are well converged.Some conventional choices for µ are µ d = /√ t [14] and µ ≡ /√ e γ E t [32]. III. FOUR QUARK OPERATORS
In this study, we consider four quark operators of the form O ± = (cid:104) (cid:16) ψ γ L µ ψ (cid:17) (cid:16) ψ γ L µ ψ (cid:17) ± (cid:16) ψ γ L µ ψ (cid:17) (cid:16) ψ γ L µ ψ (cid:17) (cid:105) , (III.1)where the subscripts 1 , · · · , (cid:44)
2, 2 (cid:44)
3, 3 (cid:44)
4, and 4 (cid:44)
1, to avoid closed quark loops within the four quarkoperator. For the calculation of B K , the case 1 = (cid:44) = O ± asˆ O ± = (cid:104) (cid:16) ˆ ψ γ L µ ˆ ψ (cid:17) (cid:16) ˆ ψ γ L µ ˆ ψ (cid:17) ± (cid:16) ˆ ψ γ L µ ˆ ψ (cid:17) (cid:16) ˆ ψ γ L µ ˆ ψ (cid:17) (cid:105) . (III.2)We then denote flowed four quark operators as O ± ( t ) = (cid:104) (cid:16) χ γ L µ χ (cid:17) (cid:16) χ γ L µ χ (cid:17) ± (cid:16) χ γ L µ χ (cid:17) (cid:16) χ γ L µ χ (cid:17) (cid:105) , (III.3)and their renormalized ones in terms of the ringed quark fields as˚ O ± ( t ) = (cid:104) (cid:16) ˚ χ γ L µ ˚ χ (cid:17) (cid:16) ˚ χ γ L µ ˚ χ (cid:17) ± (cid:16) ˚ χ γ L µ ˚ χ (cid:17) (cid:16) ˚ χ γ L µ ˚ χ (cid:17) (cid:105) . (III.4)Since the tree-level contributions of the flowed and bare operators are the same, (cid:104) O ± ( t )(cid:105) | tree = ˆ O ± ( for t → ) , (III.5) (cid:104) O ± (cid:105) | tree = ˆ O ± , (III.6)we can write the small flow time expansion for O ± ( t ) as O ± ( t ) = ( + I GF ± ( t )) O ± + O( t ) , (III.7)where we put the vertex correction as I GF ± ( t ) . To compute the I GF ± ( t ) , it is convenient to considerone-particle irreducible vertex correction of O ± ( t )− O ± , because, with the background field method,the vertex correction is proportional to the background part of the operator: (cid:104) O ± ( t ) − O ± (cid:105) = I GF ± ( t ) (cid:104) O ± (cid:105) = I GF ± ( t ) Z − O ± ˆ O ± ∼ I GF ± ( t ) ˆ O ± (one loop) . (III.8)In the second line of Eq. (III.8), we used the fact that the vertex correction I GF ± ( t ) is O( (cid:49) ) . Fromthese relations, renormalized operators at small- t are then given by˚ O ± ( t ) = ( + I GF ± ( t )) (cid:16) ϕ DRED ( t ) (cid:17) O ± + O( t ) . (III.9)We now consider renormalized four quark operators in the MS scheme with the dimensionalreduction, O MS;DRED ± = Z MS;DRED O ± (cid:16) Z MS;DRED ψ (cid:17) O ± (III.10)11here Z MS;DRED O ± and Z MS;DRED ψ are renormalization factors for the four quark operator O ± and forthe quark field ψ ( x ) , respectively. Combining these relations, we obtain the one-loop expressionfor the matching coefficient Z GF → MS;DRED ( t ) to compute O MS;DRED ± from ˚ O ± ( t ) in the t → O MS;DRED ± = lim t → Z GF → MS;DRED O ± ( t ) ˚ O ± ( t ) , (III.11) Z GF → MS;DRED O ± ( t ) = Z MS;DRED O ± ( + I GF ± ( t )) (cid:169)(cid:173)(cid:171) Z MS;DRED ψ ϕ DRED ( t ) (cid:170)(cid:174)(cid:172) . (III.12)The renormalization factor ϕ DRED ( t ) is given by Eq. (II.58). We calculate I GF ± ( t ) , Z MS;DRED O ± , and Z MS;DRED ψ in the following subsections. A. Calculation of I GF ± ( t ) FIG. 2. One-loop 1PI diagrams for four quark operators with zero external momentum in the gradient flowscheme.
Setting all the external momenta to zero, we find that five diagrams shown in Fig. 2 contribute12o (cid:104) O ± ( t ) − O ± (cid:105) in the one-loop order. Concrete forms of the diagrams are given by ( a ) : ∫ t d s ∫ (cid:96) (cid:104) (cid:16) ˆ ψ γ L σ γ ρ V a µ ˆ ψ (cid:17) (cid:16) ˆ ψ γ L σ γ λ V b ν ˆ ψ (cid:17) ± { Fierz } (cid:105) (− (cid:96) ) e − s (cid:96) S F ρ ( (cid:96) ) S F λ (− (cid:96) ) G ab µν ( (cid:96) ) , (III.13) ( b ) : ∫ t d s ∫ (cid:96) (cid:104) (cid:16) ( ˆ ψ V a µ γ ρ γ L σ ˆ ψ (cid:17) (cid:16) ( ˆ ψ γ L σ γ λ V b ν ˆ ψ (cid:17) ± { Fierz } (cid:105) (− (cid:96) ) e − s (cid:96) S F ρ ( (cid:96) ) S F λ ( (cid:96) ) G ab µν ( (cid:96) ) , (III.14) ( c ) : ∫ t d s ∫ (cid:96) (cid:104) (cid:16) ( ˆ ψ V a µ γ ρ γ L σ γ λ V b ν ˆ ψ (cid:17) (cid:16) ( ˆ ψ γ L σ ˆ ψ (cid:17) ± { Fierz } (cid:105) (− (cid:96) ) e − s (cid:96) S F ρ ( (cid:96) ) S F λ ( (cid:96) ) G ab µν ( (cid:96) ) , (III.15) ( d ) : 2 ∫ t d s ∫ (cid:96) (cid:104) (cid:16) ( ˆ ψ γ L σ (− i ) (cid:96) µ T a γ ν V b ρ ˆ ψ (cid:17) (cid:16) ( ˆ ψ γ L σ ˆ ψ (cid:17) ± { Fierz } (cid:105) e − s (cid:96) S F ν ( (cid:96) ) G ab µρ ( s . (cid:96) ) , (III.16) ( e ) : ∫ t d s ∫ (cid:96) (cid:104) (cid:16) ( ˆ ψ γ L σ T a T b ˆ ψ (cid:17) (cid:16) ( ˆ ψ γ L σ ˆ ψ (cid:17) ± { Fierz } (cid:105) G ab µµ ( s , s ; (cid:96) ) , (III.17)where V a µ = γ µ T a (III.18)is the quark-gluon vertex, and the symbol { Fierz } means the Fierz partner of each original operator, i.e. , (cid:16) ˆ ψ V A ˆ ψ (cid:17) (cid:16) ˆ ψ V B ˆ ψ (cid:17) ± { Fierz } : = (cid:16) ˆ ψ V A ˆ ψ (cid:17) (cid:16) ˆ ψ V B ˆ ψ (cid:17) ± (cid:16) ˆ ψ V A ˆ ψ (cid:17) (cid:16) ˆ ψ V B ˆ ψ (cid:17) . (III.19)with V A and V B some combinations of γ µ , T a , etc.
1. Contribution of diagrams c , d and e We first evaluate the diagrams ( c ) , ( d ) and ( e ) of Fig. 2. The calculation is similar to thosefor fermion bilinear operators discussed in Ref. [25]. The main difference comes from O( (cid:15) ) termcalled the evanescent operator, which is a byproduct of the dimensional reduction scheme [44, 45].The spinor factor of the diagram ( c ) is calculated as (cid:16) γ µ γ ρ γ L σ γ ρ γ µ (cid:17) αβ (cid:16) γ L σ (cid:17) γδ = D (cid:16) γ L σ (cid:17) αβ (cid:16) γ L σ (cid:17) γδ − (cid:16) γ L σ (cid:17) αβ (cid:16) γ L σ (cid:17) γδ . (III.20)13he new Dirac structure ( γ L σ ) αβ ( γ L σ ) γδ must be removed appropriately to achieve the correctphysical operator. Then, we define the corresponding evanescent operator ˆ E [44, 45] byˆ E : = (cid:16) γ L σ (cid:17) αβ (cid:16) γ L σ (cid:17) γδ − D (cid:16) γ L σ (cid:17) αβ (cid:16) γ L σ (cid:17) γδ . (III.21) = (cid:16) ˜ γ L σ (cid:17) αβ (cid:16) ˜ γ L σ (cid:17) γδ − (cid:15) (cid:16) γ L σ (cid:17) αβ (cid:16) γ L σ (cid:17) γδ + O( (cid:15) ) . (III.22)Because the remnant gamma matrices ˜ γ µ live in the 2 (cid:15) dimensional space, we consider that the firstterm of Eq. (III.22) is O( (cid:15) ) and thus ˆ E itself is O( (cid:15) ) . For the diagram ( c ) of Fig. 2, we subtract D ˆ E from Eq. (III.20) to obtain (cid:16) γ µ γ ρ γ L σ γ ρ γ µ (cid:17) αβ (cid:16) γ L σ (cid:17) γδ = D (cid:16) γ L σ (cid:17) αβ (cid:16) γ L σ (cid:17) γδ . (III.23)Note that the definition of evanescent operator links to a finite renormalization (or subtraction) offour quark operators because of O( / (cid:15) ) UV divergences. Together with its Fierz partner, ˆ O ± isformed. The spinor factors for other diagrams can be calculated similarly.The integrations over the internal momentum can be evaluated by the formula, ∫ (cid:96) (cid:96) e − t (cid:96) = t − D / ( π ) D / Γ ( D / − ) Γ ( D / ) , (III.24)where Γ ( x ) is the gamma function. We find that Eqs. (III.15) ∼ (III.17) are evaluated as ( c ) : − (cid:49) ( π ) C F (cid:26) (cid:15) + log ( π t ) + (cid:27) ˆ O ± , (III.25) ( d ) : (cid:49) ( π ) C F (cid:26) (cid:15) + log ( π t ) + (cid:27) ˆ O ± , (III.26) ( e ) : − (cid:49) ( π ) C F (cid:26) (cid:15) + log ( π t ) + (cid:27) ˆ O ± . (III.27)
2. Contribution of diagrams a and b We now evaluate the diagrams ( a ) and ( b ) . The color indices in Eqs. (III.13) and (III.14) canbe handled using the relation T ai j T akl = − T (cid:18) δ il δ jk − ( R ) δ i j δ kl (cid:19) . (III.28)The complicated structure of the spinor indices in Eqs. (III.13) and (III.14) can be simplified usingthe Fierz rearrangement, ( Λ ) αβ ( Λ ) γδ = − (cid:213) Γ A (cid:16) Λ Γ A Λ (cid:17) αδ (cid:16) Γ A (cid:17) γβ (III.29) Γ A = (cid:8) , γ , γ µ , i γ µ γ , σ µν (cid:9) , (III.30)14here σ µν = i [ γ µ , γ ν ] . Using relations (II.35) and (II.36) of the dimensional reduction scheme,we find ( a ) : (cid:16) γ L σ γ ρ γ µ (cid:17) αβ (cid:16) γ L σ γ ρ γ µ (cid:17) γδ = D (cid:16) γ L σ (cid:17) αδ (cid:16) γ L σ (cid:17) γβ = D (cid:16) γ L σ (cid:17) αβ (cid:16) γ L σ (cid:17) γδ , (III.31) ( b ) : (cid:16) γ µ γ ρ γ L σ (cid:17) αβ (cid:16) γ L σ γ ρ γ µ (cid:17) γδ = D (cid:16) γ L σ (cid:17) αδ (cid:16) γ L σ (cid:17) γβ − (cid:16) γ L σ (cid:17) αδ (cid:16) γ L σ (cid:17) γβ = D (cid:16) γ L σ (cid:17) αβ (cid:16) γ L σ (cid:17) γδ − D ˆ E = D (cid:16) γ L σ (cid:17) αβ (cid:16) γ L σ (cid:17) γδ , (III.32)where the evanescent operator defined by Eq. (III.21) is removed to obtain the second lineof Eq. (III.32).Carrying out the integrations, we obtain the contributions of the diagrams ( a ) and ( b ) given by ( a ) : − (cid:49) ( π ) (cid:18) T dim ( R ) ∓ T (cid:19) (cid:26) (cid:15) + log ( π t ) + (cid:27) ˆ O ± , (III.33) ( b ) : (cid:49) ( π ) (cid:18) T dim ( R ) ∓ T (cid:19) (cid:26) (cid:15) + log ( π t ) + (cid:27) ˆ O ± . (III.34)
3. Result for I GF ± ( t ) We now combine the results of Eqs. (III.25), (III.26), (III.27), (III.33), and (III.34), taking intoaccount the fact that there exist two different diagrams for each of the types ( a ) , ( b ) , and ( c ) , whilefour diagrams for each of the types ( d ) and ( e ) . Note that, by removing the evanescent operatordefined by Eq. (III.21), the background fields are correctly combined to form the ˆ O ± . Our resultfor the coefficient I GF ± ( t ) in front of the ˆ O ± is given by I GF ± ( t ) = − (cid:49) ( µ )( π ) (cid:18) T dim ( R ) ∓ T + C F (cid:19) (cid:26) (cid:15) + γ E + log ( t µ ) + (cid:27) , (III.35)where we have replaced (cid:49) by (cid:49) ( µ ) using Eq. (II.59). From the definition of I GF ± given in Eq. (III.7), I GF ± is independent of the renormalization scale µ . We may choose any value for µ provided thatthe perturbative expansions are well converged. B. MS renormalization factors Z MS ; DRED ψ and Z MS ; DRED O ± The last pieces to be calculated are the renormalization factors for the quark field ψ ( x ) andthe four quark operator O ± in the MS scheme with the dimensional reduction. See Refs. [46, 47]for previous efforts to connect four quark operators in the MS scheme with those in the lattice15 IG. 3. One-loop 1PI diagrams for four quark operators with zero external momentum in the MS scheme. scheme. We again set the external momentum to zero, but introduce a gluon mass λ to regularizethe infrared divergences. Then, the gluon propagator G ab µν ( (cid:96) ) is given by G ab µν ( (cid:96) ; λ ) = (cid:49) (cid:96) + λ δ ab δ µν . (III.36)A convenient formula in these calculations is ∫ (cid:96) ( (cid:96) ) a ( (cid:96) + λ ) = ( π ) D / λ D − a − Γ ( D / − a ) Γ ( a + − D / ) Γ ( D / ) . (III.37)The renormalization factor Z MS;DRED ψ for the quark field is calculated via the self-energy.Denoting / p = p µ γ µ , we find (cid:10) ψ ( x ) ψ ( y ) (cid:11) = ∫ p i / p e ip ·( x − y ) − (cid:49) C F ∫ p , q i / p γ µ i / q γ µ i / p ( p − q ) + λ e ip ·( x − y ) , = ∫ p i / p (cid:34) − (cid:49) ( π ) C F (cid:26) (cid:15) − γ E + log (cid:18) πλ (cid:19) + (cid:27)(cid:35) e ip ·( x − y ) = ∫ p i / p (cid:20) − (cid:49) ( ˜ µ )( π ) C F (cid:26) (cid:15) + log (cid:18) ˜ µ λ (cid:19) + (cid:27)(cid:21) e ip ·( x − y ) (III.38)in the dimensional reduction scheme. In the last line of Eq. (III.38), we have replaced (cid:49) by (cid:49) ( ˜ µ ) using Eq. (II.59), where ˜ µ is the renormalization scale for the MS scheme. A conventional choicefor ˜ µ is 2 GeV. Thus the quark field renormalization factor reads Z MS;DRED ψ = + (cid:49) ( ˜ µ )( π ) C F (cid:15) (III.39)in the one-loop order of the dimensional reduction scheme.The renormalization factors Z MS;DRED O ± for the four quark operators O ± are evaluated consideringthe 1PI vertex corrections to O ± . Diagrams showing what we need to evaluate are given in Fig. 3.16he contributions from the three diagrams are given by ( a ) : ∫ (cid:96) (cid:104) (cid:16) ˆ ψ γ L σ γ ρ V a µ ˆ ψ (cid:17) (cid:16) ˆ ψ γ L σ γ λ V b ν ˆ ψ (cid:17) ± { Fierz } (cid:105) S F ρ ( (cid:96) ) S F λ (− (cid:96) ) G ab µν ( (cid:96) ; λ ) , (III.40) ( b ) : ∫ (cid:96) (cid:104) (cid:16) ˆ ψ V a µ γ ρ γ L σ ˆ ψ (cid:17) (cid:16) ˆ ψ γ L σ γ λ V b ν ˆ ψ (cid:17) ± { Fierz } (cid:105) S F ρ ( (cid:96) ) S F λ ( (cid:96) ) G ab µν ( (cid:96) ; λ ) , (III.41) ( c ) : ∫ (cid:96) (cid:104) (cid:16) ˆ ψ V a µ γ ρ γ L σ γ λ V b ν ˆ ψ (cid:17) (cid:16) ˆ ψ γ L σ ˆ ψ (cid:17) ± { Fierz } (cid:105) S F ρ ( (cid:96) ) S F λ ( (cid:96) ) G ab µν ( (cid:96) ; λ ) . (III.42)We again use the Fierz rearrangement adopting the same evanescent operator defined by Eq. (III.21).We obtain ( a ) : 4 (cid:49) ( π ) (cid:18) T dim ( R ) ∓ T (cid:19) (cid:26) (cid:15) − γ E + log (cid:18) πλ (cid:19) + (cid:27) ˆ O ± , (III.43) ( b ) : − (cid:49) ( π ) (cid:18) T dim ( R ) ∓ T (cid:19) (cid:26) (cid:15) − γ E + log (cid:18) πλ (cid:19) + (cid:27) ˆ O ± , (III.44) ( c ) : (cid:49) ( π ) C F (cid:26) (cid:15) − γ E + log (cid:18) πλ (cid:19) + (cid:27) ˆ O ± . (III.45)Collecting them, we obtain the one-loop 1PI vertex correction (cid:104) O ± (cid:105) = (cid:20) + (cid:49) ( ˜ µ )( π ) (cid:18) T dim ( R ) ∓ T + C F (cid:19) (cid:26) (cid:15) + log (cid:18) ˜ µ λ (cid:19) + (cid:27)(cid:21) ˆ O ± . (III.46)The MS renormalization factor in the dimensional reduction scheme is extracted as Z MS;DRED O ± = − (cid:49) ( ˜ µ )( π ) (cid:18) T dim ( R ) ∓ T + C F (cid:19) (cid:15) . (III.47) C. Matching coefficient for four quark operators O ± Combining the results of Eqs. (II.58), (III.35), (III.39), and (III.47) for ϕ DRED ( t ) , I GF ± ( t ) , Z MS;DRED ψ , and Z MS;DRED O ± , we find that the matching coefficient for O ± is given by Z GF → MS;DRED O ± ( t ) = Z MS;DRED O ± ( + I GF ± ( t )) (cid:169)(cid:173)(cid:171) Z MS;DRED ψ ϕ DRED ( t ) (cid:170)(cid:174)(cid:172) = + (cid:49) ( µ ) − (cid:49) ( ˜ µ )( π ) (cid:18) T dim ( R ) ∓ T (cid:19) (cid:15) (III.48) + (cid:49) ( µ )( π ) (cid:26) (cid:18) T dim ( R ) ∓ T (cid:19) (cid:16) log ( t µ ) + γ E + (cid:17) + C F log 432 (cid:27) . From Eq. (II.59), we have the tree-level running of the coupling constant, µ dd µ (cid:49) ( µ ) = − (cid:15) (cid:49) ( µ ) . (III.49)17ntegrating this equation, we obtain (cid:49) ( µ ) − (cid:49) ( ˜ µ ) = (cid:20) − (cid:18) ˜ µ µ (cid:19) − (cid:15) (cid:21) (cid:49) ( µ ) = (cid:15) log (cid:18) ˜ µ µ (cid:19) (cid:49) ( µ ) + O ( (cid:15) ) . (III.50)We thus find that the one-loop matching coefficient for O ± ( t ) is given by Z GF → MS;DRED O ± ( t ) = + (cid:49) ( µ )( π ) (cid:26) (cid:18) T dim ( R ) ∓ T (cid:19) (cid:16) log ( t ˜ µ ) + γ E + (cid:17) + C F log 432 (cid:27) , (III.51)where µ in the log of Eq. (III.48) is replaced by ˜ µ due to the contribution from (cid:49) ( µ ) − (cid:49) ( ˜ µ ) .With the matching coefficient Z GF → MS;DRED O ± ( t ) , we evaluate the MS renormalized four quarkoperators O MS;DRED ± in the dimensional reduction scheme from the corresponding lattice opera-tors ˚ O ± ( t ) at small flow time t , O MS;DRED ± = lim t → Z GF → MS;DRED O ± ( t ) ˚ O ± ( t ) . (III.52)Note that the 1 / (cid:15) UV divergences in Eqs. (II.58), (III.35), (III.39), and (III.47) cancel out witheach other in the combination of the matching coefficient Z GF → MS;DRED O ± ( t ) . This is expected fromthe finiteness of the matching coefficients in the SF t X method: Because both O MS;DRED ± and ˚ O ± ( t ) are finite in the matching relation (III.52), Z GF → MS;DRED O ± ( t ) should also be finite. This is explicitlyconfirmed by Eq. (III.51). IV. QUARK BILINEAR OPERATORS
To calculate the kaon bag parameter, we also need the matching coefficient of the quark bilinearoperator in the denominator of Eq. (I.1). In this study, we consider general bilinear operators ofthe form ψ Γ ψ (IV.1)with Γ = γ , γ µ , i γ µ γ , and σ µν . We assume that the flavors satisfy 1 (cid:44) t , the one-loop 1PI vertex corrections for the bilinear operatorsare given by (cid:10) χ ( t ) Γ χ ( t ) − ψ Γ ψ (cid:11) = I GF Γ ( t ) (cid:16) ˆ ψ Γ ˆ ψ (cid:17) (IV.2)18ith I GF Γ ( t ) = (− ) (cid:49) ( µ )( π ) C F (cid:8) (cid:15) + γ E + log ( t µ ) + (cid:9) , Γ = , γ , (− ) (cid:49) ( µ )( π ) C F (cid:8) (cid:15) + γ E + log ( t µ ) + (cid:9) , Γ = γ µ , i γ µ γ , (− ) (cid:49) ( µ )( π ) C F (cid:8) (cid:15) + γ E + log ( t µ ) + (cid:9) , Γ = σ µν , (IV.3)where we have replaced (cid:49) by (cid:49) ( µ ) using Eq. (II.59). The evanescent operators we adopt aredefined by ˆ E Γ = , Γ = , γ ,γ µ − D γ µ , Γ = γ µ , i γ µ γ − i D γ µ γ , Γ = i γ µ γ , ( D − ) σ µν , Γ = σ µν . (IV.4)Corresponding results at t = (cid:10) ψ Γ ψ (cid:11) = I MS;DRED Γ (cid:16) ˆ ψ Γ ˆ ψ (cid:17) , (IV.5)with I MS;DRED Γ = + (cid:49) ( ˜ µ )( π ) C F (cid:110) (cid:15) + log (cid:16) ˜ µ λ (cid:17) + (cid:111) , Γ = , γ , + (cid:49) ( ˜ µ )( π ) C F (cid:110) (cid:15) + log (cid:16) ˜ µ λ (cid:17) + (cid:111) , Γ = γ µ , i γ µ γ , , Γ = σ µν , (IV.6)where we have replaced (cid:49) by (cid:49) ( ˜ µ ) with setting the renormalization scale of the MS scheme to ˜ µ .From the results of I MS;DRED Γ , we obtain the MS renormalization factors, Z MS;DRED Γ = − (cid:49) ( ˜ µ )( π ) C F (cid:15) , Γ = , γ , − (cid:49) ( ˜ µ )( π ) C F (cid:15) , Γ = γ µ , i γ µ γ , , Γ = σ µν . (IV.7)Combining these results as well as that for ϕ DRED ( t ) given in Eq. (II.58), we obtain the matchingcoefficients for the quark bilinear operators ψ Γ ψ , Z GF → MS;DRED Γ ( t ) = Z MS;DRED Γ ( + I GF Γ ( t )) Z MS;DRED ψ ϕ DRED ( t ) = + (cid:49) ( µ )( π ) C F (cid:8) γ E + ( t ˜ µ ) + + log ( ) (cid:9) , Γ = , γ , + (cid:49) ( µ )( π ) C F { log ( )} , Γ = γ µ , i γ µ γ , + (cid:49) ( µ )( π ) C F (cid:8) − γ E − log ( t ˜ µ ) − + log ( ) (cid:9) , Γ = σ µν . (IV.8)We confirm that the 1 / (cid:15) divergences in I GF Γ ( t ) etc. cancel out with each other in the combinationof the matching coefficient Z GF → MS;DRED Γ ( t ) . 19 . SUMMARY AND OUTLOOK In this paper we computed the matching coefficient Z GF → MS;DRED O ± ( t ) for four quark operators O ± defined by Eq. (III.1), and Z GF → MS;DRED Γ ( t ) for quark bilinear operators ψ Γ ψ defined by Eq. (IV.1),adopting the dimensional reduction scheme. Our results for the one-loop matching coefficientsare given by Eqs. (III.51) and (IV.8), respectively. Combining these results, we also obtain thematching coefficient for the kaon bag parameter B K defined by Eq. (I.1), Z GF → MS;DRED B K ( t ) = Z GF → MS;DRED O + ( t ) (cid:16) Z GF → MS;DRED γ µ γ ( t ) (cid:17) = + (cid:49) ( µ )( π ) − N + N (cid:16) log ( t ˜ µ ) + γ E + (cid:17) , (V.1)where N = t X method, adopting nonperturbatively O( a ) -improved dynamical Wilson quarks [48].These matching coefficients are important in evaluating the MS renormalized operators in thedimensional reduction scheme at the renormalization scale ˜ µ , from corresponding lattice operatorsmeasured at small flow time t of the gradient flow. A conventional choice for ˜ µ is 2 GeV. On theother hand, we are free to choose the renormalization scale µ for the matching of MS and gradientflow schemes, provided that the perturbative expansions are well converged. Some conventionalchoices for µ are µ d = /√ t [14] and µ ≡ /√ e γ E t [32], which are natural scales for flowedoperators because the smearing range is ∼ √ t by the gradient flow. In practice, however, becausethe perturbative expansions are truncated, the quality of the results may be affected by the choiceof µ . Recently, we found that an optimal choice of µ can improve the reliability and applicability ofthe SF t X method [33]. Such improvement may be important in evaluating complicated operators,such as O ± for the kaon bag parameter. ACKNOWLEDGMENTS
We thank the members of the WHOT-QCD Collaboration for valuable discussions. Thiswork is in part supported by JSPS KAKENHI Grants No. JP16H03982, No. JP18K03607, No.20P19K03819, and No. JP20H01903. [1] P. Dimopoulos, arXiv:1101.3069; S. Aoki et al. , Eur. Phys. J. C , 113 (2020).[2] M.B. Gavela, L. Maiani, S. Petrarca, F. Rapuano et al. , Nucl. Phys. B , 677 (1988).[3] C. Bernard and A. Soni, Nucl. Phys. B, Proc. Suppl. , 495 (1990); , 391 (1995).[4] R. Gupta, D. Daniel, G.W. Kilcup, A. Patel, and S.R. Sharpe, Phys. Rev. D , 5113 (1993).[5] A. Donini, G. Martinelli, C.T. Sachrajda, M. Talevi, and A. Vladikas, Phys. Lett. B , 83 (1995).[6] M. Crisafulli, A. Donini, V. Lubicz, G. Martinelli, F. Rapuano, M. Talevi, C. Ungarelli, and A. Vladikas,Phys. Lett. B , 325 (1996).[7] A. Donini, V. Giménez, G. Martinelli, G.C. Rossi, M. Talevi, M. Testa, and A. Vladikas, Nucl. Phys.B, Proc. Suppl. , 883 (1997).[8] L. Conti, A. Donini, V. Gimenez, G. Martinelli, M. Talevi, and A. Vladikas, Nucl. Phys. B, Proc.Suppl. , 880 (1998); Phys. Lett. B , 273 (1998).[9] R. Gupta, T. Bhattacharya and S. Sharpe, Phys. Rev. D , 4036 (1997).[10] R. Gupta, Nucl.Phys. B (Proc.Suppl.) , 278 (1998).[11] S. Aoki, M. Fukugita, S. Hashimoto, N. Ishizuka, Y. Iwasaki, K. Kanaya, Y. Kuramashi, M. Okawa,A. Ukawa, and T. Yoshié (JLQCD Collaboration), Nucl. Phys. B, Proc.Suppl. A, 67 (1998); Phys.Rev. D , 034511 (1999).[12] R. Narayanan and H. Neuberger, J. High Energy Phys. 03 (2006) 064.[13] M. Lüscher, Commun. Math. Phys. , 899 (2010).[14] M. Lüscher, J. High Energy Phys. 08 (2010) 071; 03 (2014) 092(E).[15] M. Lüscher and P. Weisz, J. High Energy Phys. 02 (2011) 051.[16] M. Lüscher, J. High Energy Phys. 04 (2013) 123.[17] M. Lüscher, Proc. Sci., LATTICE2013 (2014) 016 [arXiv:1308.5598].[18] H. Suzuki, Prog. Theor. Exp. Phys. , 083B03 (2013); , 079202(E) (2015)].[19] H. Makino and H. Suzuki, Prog. Theor. Exp. Phys. , 063B02 (2014); , 079202(E) (2015).[20] A. Ramos, Proc. Sci., LATTICE2014 (2015) 017 [arXiv:1506.00118].[21] H. Suzuki, Proc. Sci., LATTICE2016 (2017) 002 [arXiv:1612.00210].[22] J. Artz, R.V. Harlander, F. Lange, T. Neumann, and M. Prausa, J. High Energy Phys. (2019) 121.[23] T. Endo, K. Hieda, D. Miura and H. Suzuki, Prog. Theor. Exp. Phys. , 053B03 (2015).
24] H. Suzuki, Prog. Theor. Exp. Phys. , 103B03 (2015).[25] K. Hieda and H. Suzuki, Mod. Phys. Lett. A , 1650214 (2016).[26] M. Asakawa, T. Hatsuda, E. Itou, M. Kitazawa, and H. Suzuki (FlowQCD Collaboration), Phys. Rev.D , 011501 (2014), ; , 059902(E) (2015).[27] M. Kitazawa, T. Iritani, M. Asakawa, T. Hatsuda, and H. Suzuki, Phys. Rev. D , 114512 (2016).[28] T. Iritani, M. Kitazawa, H. Suzuki, and H. Takaura, Prog. Theor. Exp. Phys. , 023B02 (2019).[29] Y. Taniguchi, S. Ejiri, R. Iwami, K. Kanaya, M. Kitazawa, H. Suzuki, T. Umeda and N. Wakabayashi,Phys. Rev. D , 014509 (2017).[30] Y. Taniguchi, K. Kanaya, H. Suzuki, and T. Umeda, Phys. Rev. D , 054502 (2017).[31] K. Kanaya , S. Ejiri, R. Iwami, M. Kitazawa, H. Suzuki, Y. Taniguchi, and T. Umeda, EPJ Web Conf. , 07023 (2018).[32] R.V. Harlander, Y. Kluth, and F. Lange, Eur. Phys. J. C 78, 944 (2018).[33] Y. Taniguchi, S. Ejiri, K. Kanaya, M. Kitazawa, H. Suzuki and T. Umeda, Phys. Rev. D , 014510(2020).[34] S. Aoki, M. Fukugita, S. Hashimoto, N. Ishizuka, Y. Iwasaki, K. Kanaya, Y. Kuramashi, M. Okawa,A. Ukawa, and T. Yoshié (JLQCD Collaboration), Nucl. Phys. B, Proc.Suppl. , 67 (1998) ; Phys.Rev. D , 034511 (1999).[35] C. Monahan and K. Orginos, Phys. Rev. D , 074513 (2015).[36] M. Rizik, C. Monahan, and A. Shindler, Proc. Sci., LATTICE2018 (2019) 215 [arXiv:1810.05637[hep-lat]].[37] J.G. Reyes, J. Dragos, J. Kim, A. Shindler, and T. Luu, Proc. Sci., LATTICE2018 (2019) 219[arXiv:1811.11798] .[38] J. Kim, J. Dragos, A. Shindler, T. Luu, and J. de Vries, Proc. Sci., LATTICE2018 (2019) 260[arXiv:1810.10301].[39] M.D. Rizik, C.J. Monahan and A. Shindler, [arXiv:2005.04199].[40] M. Constantinou, P. Dimopoulos, R. Frezzotti, K. Jansen, V. Gimenez, V. Lubicz, F. Mescia,H. Panagopoulos, M. Papinutto, G.C. Rossi, S. Simula, A. Skouroupathis, F. Stylianou, and A. Vladikas,Phys. Rev. D , 014505 (2011).[41] N. Carrasco, P. Dimopoulos, R. Frezzotti, V. Lubicz, G.C. Rossi, S. Simula, and C. Tarantino, Phys.Rev. D , 034516 (2015).[42] W. Siegel, Phys. Lett. B , 193 (1979).
43] W.A. Bardeen, A.J. Buras, D. W. Duke and T. Muta, Phys. Rev. D , 3998 (1978).[44] A.J. Buras and P. H. Weisz, Nucl. Phys. B333 , 66 (1990).[45] S. Herrlich and U Nierste, Nucl. Phys.
B455 , 39 (1995).[46] G. Martinelli, Phys. Lett. B , 395 (1984).[47] Y. Taniguchi, J. High Energy Phys. (2012) 143.[48] Y. Taniguchi,