Hadronic vacuum polarization using gradient flow
FFERMILAB-PUB-20-249-T, IIT-CAPP-20-01, TTK-20-20 — July 2020
Hadronic vacuum polarization using gradient flow
Robert V. Harlander , Fabian Lange , and Tobias Neumann TTK, RWTH Aachen University, 52056 Aachen, Germany Department of Physics, Illinois Institute of Technology, Chicago, Illinois 60616, USA Fermilab, PO Box 500, Batavia, Illinois 60510, USA
Abstract
The gradient-flow operator product expansion for
QCD current correlators includ-ing operators up to mass dimension four is calculated through
NNLO . This paves analternative way for efficient lattice evaluations of hadronic vacuum polarization func-tions. In addition, flow-time evolution equations for flowed composite operators arederived. Their explicit form for the non-trivial dimension-four operators of
QCD isgiven through order α s . Contents
B.1 Vector and axial-vector currents . . . . . . . . . . . . . . . . . . . . . . . . . 20B.2 Scalar and pseudo-scalar currents . . . . . . . . . . . . . . . . . . . . . . . . 21
C Renormalized mixing matrix 23D Ancillary File 24 a r X i v : . [ h e p - l a t ] J u l Introduction
The vacuum polarization functions (
VPFs ) for (axial-)vector and (pseudo-)scalar particlesare among the most important objects when studying
QCD . On the one hand, this isbecause their imaginary part is directly related to physical observables such as the decayrates of the Z - or the Higgs boson, or the hadronic R-ratio. Since the characteristicenergy scale of these quantities is far above the QCD scale, a perturbative evaluation ofthe polarization functions is sufficient in these cases to arrive at high-precision results (see,e.g., Ref. [1]).But
VPF s also contribute indirectly to physical observables such as anomalous magneticmoments [2,3], the definition of short-distance quark masses [4], or hadronic contributionsto the
QED coupling [5, 6]. These applications involve an integration of the
VPF s overthe non-perturbative regime, which is typically achieved with the help of experimentaldata and dispersion relations. Only very recently, first-principle lattice calculations havebecome competitive with these dispersive approaches. In the case of the hadronic vacuumpolarization contribution to the muon’s anomalous magnetic moment, the two approachesturn out to lead to incompatible results [7] . It would therefore be highly desirable tohave additional independent first-principle calculations of the VPF .About ten years ago, the gradient-flow formalism (
GFF ) was suggested as a mechanism toimprove the efficiency of lattice calculations [10–12] (see also Refs. [13, 14]). Since then,it has become a standard for the scale-setting procedure [15, 16]. However, also otherapplications of the
GFF have been studied, among them a new way to determine theenergy-momentum tensor on the lattice. The underlying idea in this case is the small-flow-time expansion of composite operators [11], leading to the flowed Operator ProductExpansion (
OPE ) (also named smeared
OPE in Ref. [17, 18]), where the regular operatorsare replaced by operators taken at finite flow time. Its main advantages with respectto ( w.r.t. ) the regular
OPE is the absence of operator mixing, and the improved efficiencyof the evaluation of operator matrix elements. The translation of the regular to the flowedoperators can be done perturbatively. For the energy-momentum tensor, it is availablethrough next-to-next-to-leading order (
NNLO ) [19–21]. Quite recently, the small-flow-timeexpansion was applied at next-to-leading order (
NLO ) to CP -violating operators [22], andto four-quark operators [23].In this paper, we present the flowed OPE for the time-ordered product of two currentsthrough
NNLO QCD . Taking the vacuum expectation value (
VEV ) leads to the
VPF . Thisshould thus allow for an alternative first-principle evaluation of
VPF s on the lattice. Inaddition, we derive a general logarithmic flow-time evolution equation for flowed operatorswhich resembles the renormalization group ( RG ) equation of regular operators.The remainder of this paper is organized as follows. Section 2 introduces the regular OPE of current correlators with operators up to mass dimension four. This includes therenormalization of these operators as well as an overview of the literature which providesthe corresponding perturbative Wilson coefficients. (The coefficients for the dimension-four operators for various currents are reproduced in Appendix B.) The transition to the The lattice calculation of the light-by-light contribution to ( g − µ is in agreement with other deter-minations though [8, 9]. OPE is presented in Sect. 3. Section 4 describes the calculation of the mixing matrixbetween regular and flowed operators in the small-flow-time limit. While a large partof this mixing matrix is already known [21] and recollected in Appendix C, the missingcomponents require higher order mass terms of the
VEV of the flowed dimension-fouroperators and their renormalization with the help of the vacuum-energy renormalizationconstant. These results complete the ingredients required for the flowed
OPE of the
VPF through
NNLO . In Sect. 5, we derive a logarithmic evolution equation from the genericflow-time dependence of the mixing matrix. Section 6 presents our conclusions and givesa short outlook on possible extensions of this work.
Our results are presented for a general non-Abelian gauge theory based on a simple com-pact Lie group with n f quark fields ψ , . . . , ψ n f in the fundamental representation, of whichthe first n h are degenerate with mass m , while the remaining n l are massless. The gen-erators T a of the fundamental representation are normalized as Tr( T a T b ) = − T R δ ab , andthe structure constants f abc are defined through the Lie algebra [ T a , T b ] = f abc T c . Thedimensions of the fundamental and the adoint representation are n c and n A , respectively,and their quadratic Casimir eigenvalues are denoted by C F and C A . For SU( N ), it is n c = N , n A = N − , C F = T R N − N , C A = N , (1)and
QCD is recovered for T R = 1 / N = 3, i.e. C F = 4 / C A = 3. For brevity,we often use “ QCD ” also to refer to the more general gauge theory in the following.
The role of the perturbative and the non-perturbative regime of
VPFs can be made mostexplicit through the
OPE (see, e.g., Ref. [24]): T ( Q ) ≡ (cid:90) d x e iQx (cid:104) T j ( x ) j (0) (cid:105) Q →∞ ∼ (cid:88) d,n C ( d ) , B n ( Q ) (cid:104) O ( d ) n ( x = 0) (cid:105) , (2)where j ( x ) generically stands for a scalar, pseudo-scalar, vector, axial-vector, or tensorcurrent. Fig. 1 shows sample Feynman diagrams which arise from the perturbative evalu-ation of the current correlator in Eq. (2). In the following, we only consider the so-callednon-singlet diagrams, where the currents are connected by a common quark line. Anexample for a singlet-diagram, on the other hand, is shown in Fig. 1 (e).The coefficients C ( d ) , B n on the right-hand side of Eq. (2) depend on the quantum numbersof the current and may thus carry Lorentz indices. Apart from the explicit results forspecific currents in Appendix B, we suppress these indices throughout the paper. Wefurthermore assume that, upon transition from the left- to the right-hand side, possibleglobal divergences are subtracted off of T ( Q ).3a) (b)(c) (d) (e)Figure 1: Sample diagrams contributing to the perturbative calculation of the VPF , i.e., the left-hand side of Eq. (2). The currents are symbolized by wavy lines,gluons by spirals. We consider the case where n l quarks are massless (thin straightlines), and n h quarks are degenerate with mass m (thick straight lines). (a) One-loop contribution for non-diagonal currents; (b-e) sample three-loop diagramsfor diagonal currents. In (d), the currents couple to massless quarks. (e) is a“singlet” diagram. All diagrams in this paper were produced with the help of FeynGame [25].Up to mass dimension two, the only operators of
QCD which contribute to physical matrixelements are proportional to unity, i.e., O (0)1 ≡ O (0) = , O (2)1 ≡ O (2) = m B , (3)where m B is the bare mass of the n h degenerate massive quarks. This means that C (0)1 ≡ C (0) ≡ C (0) , B , C (2)1 ≡ C (2) ≡ Z m C (2) , B (4)are ultra-violet ( UV )-finite, where Z m is the renormalization constant of the quark massdefined in Appendix A.At mass dimension four, we choose the following basis of operators (the space-time argu-ment is suppressed in most of what follows): O (4)1 ≡ O = 1 g B F aµν F aµν ,O (4)2 ≡ O = n f (cid:88) q =1 ¯ ψ q ←→ /D ψ q ,O (4)3 ≡ O = m B , (5)where F aµν = ∂ µ A aν − ∂ ν A aµ + f abc A bµ A cν , ←→ D µ = ∂ µ − ←− ∂ µ + 2 A aµ T a , (6)4ith the regular (as opposed to “flowed”, see Sect. 3) quark and gluon fields ψ q ( x ) and A aµ ( x ), respectively, and the bare coupling g B .We employ Euclidean space-time, but translation of the intermediate formulas and thefinal results to Minkowski space is possible without difficulty. Working in d = 4 − (cid:15) (7)space-time dimensions, the mass dimensions of O and O are actually equal to d , whilethat of O is equal to 4. Higher dimensional operators are neglected in the following.The set in Eq. (5) does not contain gauge dependent operators, or operators that vanishdue to the equations of motion when acting on physical states, since they are irrelevantfor the scope of this paper. In fact, in this respect the upper limit of the sum over q in O could be replaced by n h , because the terms with massless quarks vanish on-shell. Forthe same reason, one could use O (cid:48) ≡ − m B (cid:80) n h q =1 ¯ ψ q ψ q instead of O in the definitionof the operator basis (5). Other choices are possible as well, but the basis in Eq. (5) isparticularly suitable for our purposes, because it is most directly related to the operatorbasis used in Refs. [20, 21, 26].Matrix elements of the dimension-four operators are divergent in general. However, onemay define “renormalized operators” O R n as linear combinations among them, for whichphysical matrix elements become finite: O R n = (cid:88) k Z nk O k . (8)Analogously, one defines renormalized coefficient functions through the condition (cid:88) n C B n O n ! = (cid:88) n C n O R n ⇒ C n = (cid:88) m C B m ( Z − ) mn , (9)where C B n ≡ C (4) , B n , cf. Eqs. (2) and (5). It is well known that, since the operators ofEq. (5) are part of the QCD
Lagrangian, the renormalization matrix Z can be expressedin terms of the anomalous dimensions of QCD [27, 28]: Z = (cid:32) Z × (cid:126)Z (cid:126) T Z − m (cid:33) , where Z × = (cid:18) − (cid:15)/β (cid:15) − γ m /β (cid:15) (cid:19) ,(cid:126)Z = 4ˆ µ − (cid:15) Z − m a s ∂∂a s Z Z , ˆ µ ≡ µ √ π e γ E / . (10)The ’t Hooft mass µ ensures that each renormalized operator O R n in Eq. (8) has the samemass dimension as the corresponding bare one, and ˆ µ appears because we will adopt the MS scheme by default ( γ E = − Γ (cid:48) (1) = 0 . . . . ). We also introduced the quantity a s = α s /π = g / (4 π ) here, where g is the renormalized strong coupling in the MS scheme. Z is the MS renormalization constant for the vacuum energy [28]. It is given inAppendix A, together with the anomalous quark mass dimension γ m and the d -dimensionalbeta function β (cid:15) . 5 .2 Coefficient functions The
OPE form of the current correlators is usually inconvenient for their perturbativeevaluation. Instead, one rather evaluates the l.h.s. of Eq. (2) directly by calculating therelevant two-point functions to the appropriate order. The exact analytical result forgeneral quark masses is known at the two-loop level [29–32], while higher orders up to thefour-loop level have been reconstructed by combining various kinematical limits [33–43],or through integral reduction [44–46] and subsequent numerical evaluation of the resultingmaster integrals [47].Since the dimension-zero and -two operators in Eq. (2) are proportional to unity, thecoefficients C (0) and C (2) are immediately determined from the small-mass expansion ofthese perturbative results for the VPF . They are thus known up to the four-loop level atthe moment [48–52]. The Wilson coefficients C n of the dimension-four operators, on the other hand, require adedicated calculation which keeps track of the contributions from the individual operators.This has been done through O ( a s ) for C and C , and through O ( a s ) for C in Refs. [57–60].For the purpose of this paper, only the O ( a s ) results are required. For completeness, weinclude them in Appendix B. Having introduced the setup in the “regular” theory, we now translate this to the flowed
OPE for the current correlators.
We introduce the flowed operators as˜ O ( t, x ) = Z s g B G aµν ( t, x ) G aµν ( t, x ) = ˆ µ − (cid:15) g G aµν ( t, x ) G aµν ( t, x ) , ˜ O ( t, x ) = ˚ Z χ n f (cid:88) q =1 ¯ χ q ( t, x ) ←→ / D ( t, x ) χ q ( t, x ) , ˜ O ( t, x ) = m , (11)where ←→D µ = D µ − ←−D µ , D µ = ∂ µ + B aµ T a , ←−D µ = ←− ∂ µ − B aµ T a . (12) The imaginary parts of the
VPF s are known even at the five-loop level in the phenomenologically mostrelevant cases [53–56]. B aµ = B aµ ( x, t ) and χ q = χ q ( x, t ) obey the equations[10, 12] ∂ t B aµ = D abν G bνµ + κ D abµ ∂ ν B bν ,∂ t χ q = ∆ χ q − κ∂ µ B aµ T a χ q ,∂ t ¯ χ q = ¯ χ q ←− ∆ + κ ¯ χ q ∂ µ B aµ T a , (13)with the initial conditions B aµ ( t = 0 , x ) = A aµ ( x ) , χ q ( t = 0 , x ) = ψ q ( x ) , q ∈ { , . . . n f } . (14)Here we used the flowed covariant derivative in the adjoint representation, D abµ = δ ab ∂ µ − f abc B cµ , (15)and the flowed Laplace operators∆ = D µ D µ , ←− ∆ = ←−D µ ←−D µ , (16)where the flowed covariant derivatives in the fundamental representation are given inEq. (12).˚ Z χ is the non-minimal renormalization constant for the flowed quark fields χ q , defined bythe all-order condition [20] (cid:104) ˜ O ( t ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) m =0 ≡ − n c n f (4 πt ) , (17)where (cid:104)·(cid:105) denotes the VEV . It reads ˚ Z χ = ζ χ Z χ , (18)where Z χ is the MS part, Z χ = 1 + a s γ χ, (cid:15) + a s (cid:104) γ χ, (cid:15) (cid:16) γ χ, − β (cid:17) + γ χ, (cid:15) (cid:105) + O ( a s ) , (19)and ζ χ = 1 + a s (cid:18) γ χ, L µt − C F ln 3 − C F ln 2 (cid:19) + a s (cid:40) γ χ, (cid:16) β + γ χ, (cid:17) L µt + (cid:104) γ χ, − γ χ, (cid:16) β + γ χ, (cid:17) ln 3 − γ χ, (cid:16) β + γ χ, (cid:17) ln 2 (cid:105) L µt + c (2) χ (cid:41) + O ( a s ) . (20)The short-hand notation L µt = ln µ µ t , µ t = 1 √ te γ E (21)7eflects our choice of the “central” renormalization scale µ t [21].The minimal renormalization constant Z χ is known analytically through NNLO [12, 21], γ χ, = 32 C F ,γ χ, = C F (cid:20) C A (cid:18) − ln 2 (cid:19) − C F (cid:18)
316 + ln 2 (cid:19) − T R n f (cid:21) . (22)The finite coefficient in Eq. (20) has been obtained numerically in Ref. [61]: c (2) χ = C A C F c χ, A + C c χ, F + C F T R n f c χ, R , (23)with c χ, A = − . , c χ, F = 30 . ,c χ, R = − ζ (2) + 9449 ln 2 + 1603 ln − − (1 / − (1 /
3) + 1123 Li (3 /
4) = − . . . . , (24)with Riemann’s zeta function ζ ( s ) ≡ (cid:80) ∞ n =1 n − s and the di-logarithm Li ( z ) = (cid:80) ∞ k =1 z k /k .The strong coupling renormalization constant Z s in Eq. (11) ensures that matrix elementsof ˜ O ( t ) are finite [10, 11]. The reason for keeping track of the non-integer mass dimensionof ˜ O ( t ) is clarified later.Eq. (24) displays only the first six leading digits in numerical results. Results with higheraccuracy are provided in an ancillary file, which also includes the L µt terms, see Ap-pendix D. We expect that these floating point numbers can be considered equivalent totheir exact results for all practical purposes. This is why we often use the numerical valuesfor the coefficients in what follows, even if the exact result is available.Similar to the regular operators in Eq. (5), one could trade the flowed operator ˜ O ( t ) for˜ O (cid:48) ( t ) = − m ˚ Z χ (cid:80) n f q =1 ¯ χ q ( t ) χ q ( t ). However, in this case the final results to be derivedbelow would be different, because the equations of motion for the flowed operators relate˜ O (cid:48) ( t ) to both ˜ O ( t ) and ˜ O ( t ) (see Refs. [20, 21]). A transformation of the results in thispaper to ˜ O (cid:48) ( t ) is straightforward though. The small-flow-time expansion allows us to relate the
QCD operators and coefficients withthe flowed quantities as follows:˜ O n ( t ) = ζ (0) n ( t ) + ζ (2) , B n ( t ) m B + (cid:88) k ζ B nk ( t ) O k + . . . ≡ ζ (0) n ( t ) + ζ (2) n ( t ) m + (cid:88) k ζ nk ( t ) O R k + . . . , (25) The sign on the r.h.s. of equation (B.3) in Ref. [61] is incorrect. t →
0, and ζ (2) n ( t ) = ζ (2) , B n ( t ) Z m , ζ nk ( t ) = (cid:88) l ζ B nl ( t ) Z − lk (26)are the renormalized, finite mixing coefficients. Inversion of Eq. (25) gives O R n = (cid:88) k ζ − nk ( t ) ¯ O k ( t ) + . . . , ¯ O n ( t ) ≡ ˜ O n ( t ) − ζ (0) n ( t ) − ζ (2) n ( t ) m , (27)where ζ − nk is the nk -element of the inverse of the mixing matrix ζ . This lets one definethe “flowed OPE ” for the current correlator: T ( Q ) Q →∞ ∼ ˜ C (0) ( Q , t ) + ˜ C (2) ( Q , t ) m + (cid:88) n ˜ C n ( Q , t ) ˜ O n ( t ) + . . . (28)where the corresponding coefficient functions are related to the regular Wilson coefficientsthrough ˜ C n ( Q , t ) = (cid:88) k C k ( Q ) ζ − kn ( t ) , ˜ C (0 , ( Q , t ) = C (0 , ( Q ) − (cid:88) n ˜ C n ( t, Q ) ζ (0 , n ( t ) . (29)The regular QCD coefficients C (0) and C (2) are given by the first two terms in m /Q of the large- Q expansion of the VPF s. Through the required order, they can be foundin Ref. [49] for diagonal vector- and axial-vector currents, and in Ref. [50] for scalar-and pseudo-scalar currents, for example. The dimension-four coefficients can be found inRefs. [57, 60]. For convenience of the reader, they are also collected in Appendix B.
We now determine the mixing matrix ζ in a perturbative calculation through NNLO . Byusing the known results for the regular Wilson coefficients given in Appendix B, one candetermine the flowed coefficients to the same order. Together with an evaluation of theflowed operator matrix elements on the lattice, the
VPF s can be extracted and used in thedetermination of various physical quantities.The bare mixing matrix ζ B can be determined with the help of the method of projectors: ζ (0 , , B n ( t ) = P (0 , [ ˜ O n ( t )] , ζ B nk ( t ) = P (4) k [ ˜ O n ( t )] , (30)where the action of P ( d ) is to take suitable derivatives of a specific Green’s function of theoperator such that P ( n ) [ O ( m ) ] = δ nm , P ( n ) [ O k ] = P (4) k [ O ( n ) ] = 0 , P (4) k [ O l ] = δ kl , (31)9or n, m ∈ { , } and k, l ∈ { , , } . For details, see Refs. [21, 62, 63].Specifically, the projectors onto , m B , and O B are given by derivatives of vacuum matrixelements w.r.t. m B : ζ (0) n ( t ) = P (0) [ ˜ O n ( t )] ≡ (cid:104) ˜ O n ( t ) (cid:105) (cid:12)(cid:12)(cid:12) m B =0 ,ζ (2) n ( t ) = Z m P (2) [ ˜ O n ( t )] ≡ Z m ∂ ∂m B (cid:104) ˜ O n ( t ) (cid:105) (cid:12)(cid:12)(cid:12) m B =0 ,ζ B n ( t ) = P (4)3 [ ˜ O n ( t )] ≡ ∂ ∂m B (cid:104) ˜ O n ( t ) (cid:105) (cid:12)(cid:12)(cid:12) m B =0 . (32)A crucial point is that the derivatives and limits must be taken before loop integration. Asa consequence, even though physical matrix elements of ˜ O n ( t ) are finite, the projectionscan be divergent, and this is why we need to carefully account for a possible non-integermass dimension of these operators, see Eq. (11).We directly obtain ζ B = 14! ∂ ∂m B m = Z − m , ζ (0) , B = ζ (2) , B = 0 , ζ B = ζ B = 0 , (33)where the third set of equations follows from ˜ O ( t ) = m = O and the projector property P (4)1 [ O ] = P (4)2 [ O ] = 0, see Eq. (31).The bare and renormalized mixing matrices for the dimension-four operators thus takethe form ζ B = (cid:32) ζ B × (cid:126)ζ B (cid:126) T Z − m (cid:33) , ζ = (cid:32) ζ × (cid:126)ζ (cid:126) T (cid:33) , (34)where (cid:126) T = (0 , ζ B × = (cid:32) ζ B ζ B ζ B ζ B (cid:33) , ζ × = ζ B × Z − × ,(cid:126)ζ B = (cid:0) ζ B , ζ B (cid:1) T , (cid:126)ζ = ( (cid:126)ζ B − ζ × (cid:126)Z ) Z m . (35) ζ × can be obtained from the mixing matrix of the operators occuring in the energy-momentum tensor and is accordingly known through NNLO [21]. Explicit results aregiven in Appendix C.The coefficient ζ (0) n is simply the VEV of ˜ O n ( t ) for m = 0. For ˜ O ( t ) it has been calculatedthrough NNLO in Refs. [10, 61, 64]: ζ (0)1 ( t ) = (cid:104) ˜ O ( t ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) m =0 = 34 π t n A (cid:26) a s (cid:20) C A (cid:18)
139 + 116 ln 2 −
34 ln 3 (cid:19) − T R n f (cid:21) + ¯ a s (cid:20) . C − . C A T R n f + 0 . C F T R n f + 0 . T n (cid:21)(cid:27) + O (¯ a s ) , (36) The coefficient of the C term in Ref. [61] contains a typo: instead of 27 . . ζ ( m ) n ( m ∈ { , } ) and ζ n , for n = 1(a-d) and n = 2 (e-h). The notation is the same as in Fig. 1; in addition, whitecircles denote flow-vertices, and lines with arrows next to them denote flow-lines(see Ref. [61] for details). Diagrams (a) and (b) only contribute to ζ (0)1 .where ¯ a s = a s ( µ t ). Due to the RG invariance of a s ˜ O ( t ) [10, 11], the result for generalvalues of the ’t Hooft mass µ can be obtained by multiplying the result given in Eq. (36)by ¯ a s /a s , replacing¯ a s = a s (cid:20) a s β L µt + a s L µt (cid:0) β + β L µt (cid:1) (cid:21) + O ( a s ) , (37)and re-expanding in a s .For ˜ O , on the other hand, we have Eq. (17) to all orders in perturbation theory bydefinition, i.e. ζ (0)2 ( t ) = (cid:104) ˜ O ( t ) (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) m =0 ≡ − n c n f (4 πt ) . (38)Therefore, the only coefficients that are not yet known through NNLO are ζ (2)1 , ζ (2)2 , ζ and ζ . (39)According to Eq. (32), they require the calculation of derivatives w.r.t. m B in (cid:104) ˜ O ( t ) (cid:105) and (cid:104) ˜ O ( t ) (cid:105) up to three loops. We achieved this using the setup described in Ref. [61], whichemploys qgraf [65, 66] and q2e/exp [67, 68] for the generation and subsequent categoriza-tion of the Feynman diagrams, FORM [69, 70] for the manipulation of the ensuing algebraic11xpressions, the color package [71] of
FORM for the calculation of the gauge group factors,and
Kira ⊕ FireFly [72–76] for the Feynman integral reduction using integration-by-partsidentities and the Laporta algorithm [44–46] over finite fields [77–80]. For the evaluationof the master integrals, we adopt the method described in Ref. [64], which performs sectordecomposition [81] with the help of
FIESTA [82,83] in order to extract the UV poles, alongwith a fully symmetric integration rule of order 13 for the numerical evaluation of their co-efficients [84], implemented with high precision arithmetics by using the MPFR library [85].Some intermediate steps of the calculation are done within
Mathematica [86].Multiplying the result by Z m suffices to obtain the renormalized expressions for ζ (2) n , andwe find, setting µ = µ t , ζ (2)1 ( t ) = 3 n A n h π t T R ¯ a s (cid:20) a s (cid:18) . C A + 2 C F − T R n f (cid:19)(cid:21) + O ( a s ) , (40a) ζ (2)2 ( t ) = n c n h π t (cid:20) a s C F (cid:18) −
34 ln 3 (cid:19) + ¯ a s C F (2 . C A + 2 . C F − . T R n f ) (cid:21) + O ( a s ) , (40b)where again ¯ a s = a s ( µ t ). Since quark loops appear in (cid:104) ˜ O ( t ) (cid:105) only at the two-loop level, ζ (2)1 ( t ) starts at O ( a s ). In the case of ζ (2)2 ( t ), the result for general µ can be obtainedthrough multiplication by m ( µ t ) m ( µ ) = 1 + a s γ m, L µt + a s L µt (cid:20) γ m, + 12 (cid:0) β γ m, + γ m, (cid:1) L µt (cid:21) + O ( a s ) , (41)expressing ¯ a s by a s through Eq. (37), and re-expanding in a s . For ζ (2)1 ( t ) one needs tomultiply by Eq. (41), and in addition by ¯ a s /a s . ζ and ζ require the more sophisticated renormalization given in Eq. (35). It is importanthere to work consistently in d space-time dimensions. Since (cid:126)Z contains a 1 /(cid:15) pole alreadyat O ( a s ), we need to keep the O ( (cid:15) ) terms of ζ × at NLO , and the O ( (cid:15) ) terms at NNLO .They were not required in the calculation of Ref. [21], so we recalculated ζ × , keepingthese higher terms in (cid:15) . Using the identity n A T R = n c C F [71], our final result for (cid:126)ζ reads: ζ ( t ) = n h n c C F π a s (cid:26) − ζ (2) − L µt − L µt + a s (cid:20) − . C A − . C F − . T R n h + 19 . T R n l + L µt (cid:18) − . C A − . C F − . T R n h + 19 . T R n l (cid:19) + L µt (cid:18) − C A − C F + 263 T R n f (cid:19) + L µt (cid:18) − C A − C F + 83 T R n f (cid:19)(cid:21)(cid:27) + O ( a s ) , (42a) ζ ( t ) = n c n h π (cid:26) L µt + a s C F (cid:20) −
152 ln 3 −
94 Li (1 / (cid:18) − ln 2 −
34 ln 3 (cid:19) L µt + 32 L µt (cid:21) + a s C F (cid:20) − . C A + 6 . C F − . T R n h − . T R n l + L µt (cid:18) . C A + 4 . C F − . T R n h − . T R n l (cid:19) + L µt (cid:18) . C A + 4 . C F − . T R n f (cid:19) + L µt (cid:18) C A + 32 C F − T R n f (cid:19)(cid:21)(cid:27) + O ( a s ) . (42b)The logarithmic terms at O ( a ns ) are determined by the RG equation derived in Sect. 5.Nevertheless, for the convenience of the reader, we provide the result for general µ in thiscase.This, together with the results for ζ × obtained in Refs. [21] and explicitely given inEq. (77), completes the result for the small-flow-time coefficients of the OPE up to dimen-sion four of Eq. (28).
In the final section of this paper we derive a general flow-time evolution equation forflowed operators. It resembles the RG equation for regular operators but with a “flowedanomalous dimension matrix”. While studies of the relation between the RG and the flow-time evolution have also been performed elsewhere in the litarature (see, e.g., Refs. [26,87–90]), to our knowledge the treatment described here has not been discussed before.Let us return to the small-flow-time expansion of the operators ¯ O defined in Eqs. (25),(27), employing a matrix rather than component-wise notation for the sake of clarity:¯ O ( t ) = ζ B ( t ) O = ζ ( t ) O R . (43)Since we work in the small-flow-time limit, the dependence of ζ ( t ) on t can be only through L µt , defined in Eq. (21). Taking the logarithmic derivative w.r.t. t of Eq. (43), one thusobtains t∂ t ¯ O ( t ) = ( t∂ t ζ ( t )) O R . (44)Using Eq. (43) to eliminate the regular operators O R , we find the flow-equation for flowedcomposite operators: t∂ t ¯ O ( t ) = γ f ( t ) ¯ O ( t ) , where γ f ( t ) ≡ ( t∂ t ζ ( t )) ζ − ( t ) . (45)So far the discussion is general and holds for any flowed OPE . Specializing to our caseof the
QCD dimension-four operators, we can write the “flowed anomalous dimension”13atrix as γ f = (cid:18) γ f2 × (cid:126)γ f3 (cid:19) , γ f2 × ( t ) = ( t∂ t ζ × ( t )) ζ − × ( t ) ,(cid:126)γ f3 ( t ) = − γ f2 × ( t ) (cid:126)ζ ( t ) + t∂ t (cid:126)ζ ( t ) . (46)Through O ( a s ), the result can be directly evaluated from Eqs. (77) and (42). A consistencycheck is obtained by noting that ζ ( t ) depends on t only through L µt : t∂ t ζ ( t ) = µ ∂∂µ ζ ( t ) = µ dd µ ζ ( t ) − a s β ∂∂a s ζ ( t ) . (47)On the other hand, we know that a s ˜ O ( t ) and ˜ O ( t ) are RG invariant [10, 11, 20] andtherefore, with Eq. (25), m γ m = µ dd µ H − ( a s ) ˜ O ( t ) == µ dd µ H − ( a s ) (cid:16) ζ (0) ( t ) + ζ (2) ( t ) m + ζ ( t ) O R (cid:17) , (48)where H ( x ) = (cid:18) H × ( x ) (cid:126) (cid:126) T (cid:19) , with H × ( x ) = (cid:18) (4 π x ) −
00 1 (cid:19) . (49)Since operators of different mass dimensions do not mix under RG evolution and ζ (0 , ( t ) =0, we can drop the first two terms in the brackets on the r.h.s. of Eq. (48). We thus arriveat µ dd µ ζ ( t ) = γ m − β ζ ( t ) − ζ ( t ) γ O , (50)where γ O is the anomalous dimension of the operators O R , defined through µ dd µ O R = γ O ( a s ) O R . (51)It can be written as γ O = (cid:18) µ dd µ Z (cid:19) Z − = (cid:32) γ O × (cid:126)γ O (cid:126) T γ m (cid:33) , (52)with Z from Eq. (10). Using the expressions of Sect. 2.1, one derives [27, 28] γ O × = (cid:18) µ dd µ Z × (cid:19) Z − × = − a s ∂∂a s β − a s ∂∂a s γ m ,(cid:126)γ O = Z m (cid:18) µ dd µ (cid:126)Z − γ O × (cid:126)Z (cid:19) = a s ∂∂a s γ γ . (53) This can also be seen by noting that these two terms, multiplied by H − ( a s ), are a s (cid:104) ˜ O ( t ) (cid:105) and (cid:104) ˜ O ( t ) (cid:105) ,expanded through order m . QCD renormalization group functions β and γ m have been defined in Eqs. (58) and(60), respectively. Since they are of O ( a s ), the explicit µ -dependence of ζ × ( t ) can bederived through O ( a s ) from the results of Ref. [21]. Thus, for γ f2 × , Eq. (47) is not just aconsistency check, but a means to derive higher order terms. In our case, we can obtainthe result through O ( a s ): γ f11 = a s (cid:20) C + 18 C A T R n f + 78 C F T R n f (cid:21) + a s (cid:20) − C A T n − C F T n + C A C F T R n f (cid:18) − (cid:19) + C T R n f (cid:18) −
920 ln 3 (cid:19) + C (cid:18) − (cid:19) + C T R n f (cid:18) − (1 / − ζ (2) (cid:19) + (cid:18) C T R n f + 1164 C + 7748 C A C F T R n f − C A T n − C F T n (cid:19) L µt (cid:21) + O ( a s ) , (54a) γ f12 = − a s C F + a s (cid:40) − C A C F + 53 C F T R n f + C (cid:20) − −
32 ln 2 −
98 ln 3 (cid:21) + (cid:20) − C A C F + C F T R n f (cid:21) L µt (cid:41) + a s (cid:40) C C A (cid:20) − − (1 / − ζ (2) (cid:21) + C F T n (cid:20) − − ζ (2) (cid:21) + C (cid:20) − −
98 ln − −
94 ln 2 ln 3 − (1 /
4) + 3964 ζ (2) (cid:21) + C C F (cid:20) − − (1 /
4) + 118 ζ (2) (cid:21) + C T R n f (cid:20) − (1 / − ζ (2) − ζ (3) (cid:21) + C A C F T R n f (cid:20) − −
18 Li (1 /
4) + 78 ζ (2) + 92 ζ (3) (cid:21) + 332 C F c (2) χ + (cid:20) − C C F + 64748 C A C F T R n f − C F T n + C C A (cid:18) − −
338 ln 2 − (cid:19) + C T R n f (cid:18) (cid:19)(cid:21) L µt + (cid:20) − C C F + 114 C A C F T R n f − C F T n (cid:21) L µt (cid:41) + O ( a s ) , (54b) γ f21 = a s (cid:20) C T R n f (cid:18) − (cid:19) + C A T n (cid:18)
320 ln 2 + 980 ln 3 (cid:19) + C A C F T R n f (cid:18) − (1 / − ζ (2) (cid:19) + C F T n (cid:18) −
54 ln 2 + 3748 ln 3 −
16 Li (1 /
4) + 112 ζ (2) (cid:19)(cid:21) + O ( a s ) , (54c) γ f22 = a s (cid:34) C A C F (cid:18) − − (cid:19) + C F T R n f (cid:18) −
23 + 13 ln 2+ 14 ln 3 (cid:19)(cid:35) + a s (cid:40) C A C F T R n f (cid:20) − −
16 ln
2+ 1537160 ln 3 −
124 Li (1 / (cid:21) + C C F (cid:20) − −
2+ 10716 ln 3 + 118 Li (1 / − ζ (2) (cid:21) + C F T n (cid:20) − − −
16 Li (1 /
4) + 112 ζ (2) (cid:21) + C T R n f (cid:20) − − −
238 Li (1 /
4) + 548 ζ (2) (cid:21) + C C A (cid:20) − −
998 ln 3 −
118 ln 2 ln 3 −
3+ 18732 Li (1 /
4) + 143192 ζ (2) (cid:21) + (cid:18) C A − T R n f (cid:19) c (2) χ + (cid:20) C C F (cid:18) − − (cid:19) + C F T n (cid:18) −
29 ln 2 −
16 ln 3 (cid:19) + C A C F T R n f (cid:18) − (cid:19)(cid:21) L µt (cid:41) + O ( a s ) . (54d)We verified that this agrees through O ( a s ) with the result which is obtained by directlyinserting Eq. (77) into Eq. (46). Due to the factor of 1 /g in ˜ O (see Eq. (11)), γ f2 × is not RG invariant, while H − × γ f2 × H × is. It may be useful to note that, by subtracting the VEV s off of ˜ O and ˜ O , ˜ O , sub ( t, x ) = ˜ O ( t, x ) − (cid:104) ˜ O ( t, x ) (cid:105) , ˜ O , sub ( t, x ) = ˜ O ( t, x ) − (cid:104) ˜ O ( t, x ) (cid:105) , (55)the resulting operators do not mix with ˜ O under t -evolution. Rather, their logarithmic t -evolution is fully governed by γ f2 × and thus known through O ( a s ).Eq. (50) does not analogously allow one to derive the O ( a s ) terms of (cid:126)γ f3 , because it in-volves γ which, in contrast to β and γ m , starts at O ( a s ) rather than O ( a s ), see Eq. (63).Therefore, we can only give the result through O ( a s ) for (cid:126)γ f3 : γ f13 = 3 n c n h π a s C F (cid:26) a s (cid:20) . C A − . C F . T R n h + 1 . T R n l + L µt (cid:18) C A + 3 C F − T R n f (cid:19)(cid:21)(cid:27) , (56a) γ f23 = n c n h π (cid:26) a s C F (cid:20) − ln 2 −
34 ln 3 + 3 L µt (cid:21) + a s C F (cid:20) . C A + 4 . C F − . T R n h − . T R n l + L µt (cid:18) . C A + 8 . C F − . T R n f (cid:19) + L µt (cid:18) C A + 92 C F − T R n f (cid:19)(cid:21)(cid:27) . (56b)We checked, of course, that Eq. (50) is consistent with the results for ζ and ζ of Eq. (42). We presented the flowed
OPE for general current correlators and its matching to regular
QCD through
NNLO in the strong coupling α s and through mass dimension four by usingthe small-flow-time expansion. Our calculation is based on the renormalization procedurefor the regular QCD dimension-four operators worked out in Ref. [27,28], the mixing matrixbetween flowed and regular operators derived in Ref. [21], the method of projectors [62],and the tools and results for perturbative calculations in the
GFF presented in Ref. [61].Overall, our results allow to combine the known perturbative results for the regular
QCD current correlators from the literature to gradient-flow lattice calculations. This laysout the path for an alternative determination of hadronic contributions to observablessuch as the anomalous magnetic moment of the muon. In addition, we derived a generallogarithmic flow-time evolution equation for flowed operators and presented its explicitform for the dimension-four operators considered in this paper.Our methods are sufficiently general to be applied to similar problems at higher ordersin perturbation theory, such as CP violating operators [22] relevant for the electric dipolemoment of the neutron, or four-quark operators occuring in flavor physics [23]. Acknowledgments.
We would like to thank A. Hoang for confirmative comments onthe validity of the approach pursued in this paper, and Y. Kluth for helpful comments andclarifications.
RVH would like to thank K. Chetyrkin for past initiation and collaborationon the material of Sect. 2.This work was supported by
Deutsche Forschungsgemeinschaft (DFG) through project HA2990/9-1, and by the U.S. Department of Energy under award No.
DE-SC0008347 . Thisdocument was prepared using the resources of the Fermi National Accelerator Laboratory(Fermilab), a U.S. Department of Energy, Office of Science, HEP User Facility. Fermilab ismanaged by Fermi Research Alliance, LLC (FRA), acting under Contract No. DE-AC02-07CH11359. 17
Renormalization group functions
The d -dimensional beta function is defined as µ dd µ a s ( µ ) = a s ( µ ) β (cid:15) ( a s ( µ )) , (57)where a s ≡ α s /π ≡ g / (4 π ). The renormalized coupling g = g ( µ ) is related to the bareone through g B = ˆ µ (cid:15) Z / s g , where Z s is the MS renormalization constant. From this followthe relations β (cid:15) ( a s ) = − (cid:15) (cid:18) a s ∂∂a s ln Z s ( a s ) (cid:19) − = − (cid:15) + β ( a s ) ≡ − (cid:15) − (cid:88) n ≥ a n +1 s β n ,Z s ( a s ) = 1 − a s (cid:15) β + a s (cid:18) (cid:15) β − (cid:15) β (cid:19) + O ( a s ) . (58)Through Sect. 5, we only need the first two perturbative coefficients, while β is requiredin order to derive the O ( a s ) terms of γ f in Eq. (54): β = 14 (cid:18) C A − T R n f (cid:19) , β = 116 (cid:18) C − C F T R n f − C A T R n f (cid:19) ,β = 164 (cid:18) C + 2 C T R n f − C F C A T R n f − C T R n f + 449 C F T n + 15827 C A T n (cid:19) . (59)The anomalous dimension of the quark mass is defined through γ m ( a s ) = − a s β (cid:15) ( a s ) ∂∂a s ln Z m ( a s ) ≡ − (cid:88) n ≥ a n +1 s γ m,n , (60)with the first three perturbative coefficients given by γ m, = 34 C F , γ m, = 332 C + 9796 C A C F − C F T R n f ,γ m, = 164 (cid:20) C − C C A + 11413108 C F C + C T R n f ( −
46 + 48 ζ (3)) + C F C A T R n f (cid:18) − − ζ (3) (cid:19) − C F T n (cid:21) . (61)It determines the MS renormalized mass m through m B = Z m m , Z m = 1 − a s (cid:15) γ m, + a s (cid:20) γ m, (cid:15) ( γ m, + β ) − (cid:15) γ m, (cid:21) + O ( a s ) . (62)Similarly to β (cid:15) , the third coefficient γ m, is needed only in Sect. 5.The renormalization constant of the vacuum energy Z is related to the correspondinganomalous dimension γ through γ ( a s ) = [4 γ m ( a s ) − (cid:15) ] Z ( a s ) + β (cid:15) ( a s ) a s ∂∂a s Z ( a s ) ≡ − n c n h (4 π ) (cid:88) n ≥ a ns γ ,n , (63)18hich leads to Z ( a s ) = n c n h (4 π ) (cid:15) (cid:26) a s (cid:18) γ , − γ m, (cid:15) (cid:19) + a s (cid:104) (cid:15) (cid:0) β γ m, + 4 γ m, (cid:1) − (cid:15) ( β γ , + 4 γ , γ m, + 8 γ m, ) + 13 γ , (cid:105)(cid:27) + O ( a s ) . (64)The first three perturbative coefficients are given by [28, 58] γ , = 1 , γ , = C F ,γ , = − C (cid:18) − ζ (3) (cid:19) − C F C A (cid:18) − ζ (3) (cid:19) − C F T R (cid:18) n f + 3 n h (cid:19) . (65) B Perturbative coefficient functions
This appendix cites the results for the coefficient functions C n of the regular dimension-fouroperators appearing in the OPE of the current correlators defined in Eq. (2). We considerscalar, pseudo-scalar, vector- and axial-vector currents, both diagonal and non-diagonal,i.e. the currents assume the form j ( x ) = ¯ ψ k ( x )Γ ψ l ( x ) , Γ ∈ { , iγ , γ µ , γ µ γ } ,k, l ∈ N ∪ M , M = { , . . . , n h } , N = { n h + 1 , . . . , n f } . (66)This means that ψ k and ψ l can be either both massive with mass m ( k, l ∈ M ), or bothmassless ( k, l ∈ N ), or one of them is massless, the other massive (e.g. k ∈ M , l ∈ N ).While C is independent of k and l , the coefficient C of the quark operator takes the form C = C ,N + 1 n h ( δ kM + δ lM ) ( C ,M + C , nd ) , (67)where δ kM = 1 for k ∈ M , and 0 otherwise. Also the results for C depend on whether thequarks k and l are massive or not. This dependence will be indicated explicitely below,using the δ kM symbol defined above.For convenience, we introduce the short-hand notation l µQ ≡ ln Q µ , (68)and the dimensionless quantitiesˆ C = Q C , ˆ C = − Q C , ˆ C = Q C . (69)The extra factor ( −
2) between C and ˆ C arises from using O (cid:48) in Ref. [60] rather than O from Eq. (5). For the sake of brevity, we insert the SU(3) color factors. Higher orders have been computed in Ref. [91]. .1 Vector and axial-vector currents In this case, the current correlator can be decomposed into a transversal and a longitudinalpart, each associated with a set of coefficient functions: (cid:90) d xe iqx j v/aµ ( x ) j v/aν (0) Q →∞ −→ (cid:88) n (cid:16) ( g µν Q − Q µ Q ν ) C v/a, T n − Q µ Q ν C v/a, L n (cid:17) O R n . (70)The upper sign refers to the vector, the lower sign to the axial-vector case. The resultshave been taken from Ref. [60]:ˆ C v/a, T1 = 112 a s + 772 a s , (71a)ˆ C v/a, T2 ,N = a s (cid:18) − l µQ + 43 ζ (3) (cid:19) , (71b)ˆ C v/a, T2 ,M = − a s + a s (cid:20) −
296 + 16 n f + l µQ (cid:18) −
114 + 16 n f (cid:19)(cid:21) , (71c)ˆ C v/a, T2 , nd = ± (cid:18) a s + a s (cid:20) − n f + l µQ (cid:18) − n f (cid:19)(cid:21)(cid:19) , (71d)ˆ C v/a, L1 = 0 , ˆ C v/a, L2 ,N = 0 , ˆ C v/a, L2 ,M = 1 , ˆ C v/a, L2 , nd = ∓ , (71e)ˆ C v/a, T3 = 316 π (cid:26) δ kM δ lM (cid:34) a s (cid:18) − ζ (3) (cid:19) + a s (cid:20) − ζ (3) + 120 ζ (5) + n f (cid:18) − ζ (3) (cid:19) + l µQ (cid:18) − ζ (3) + n f (cid:18) − ζ (3) (cid:19) (cid:19)(cid:21)(cid:35) ± δ kM δ lM (cid:34) − l µQ + a s (cid:18) −
563 + 16 ζ (3) − l µQ − l µQ (cid:19) + a s (cid:20) − ζ (3) + 43 ζ (4) − ζ (5) + n f (cid:18) − ζ (3) (cid:19) + l µQ (cid:18) − ζ (3) + n f (cid:18) − ζ (3) (cid:19) (cid:19) + l µQ (cid:18) −
77 + 229 n f (cid:19) + l µQ (cid:18) −
18 + 49 n f (cid:19)(cid:21)(cid:35) + ( δ kM + δ lM ) (cid:34) − a s (cid:18) − − l µQ (cid:19) + a s (cid:20) − ζ (3) − ζ (5) + n f (cid:18) − ζ (3) (cid:19) + l µQ (cid:18) −
36 + 109 n f (cid:19) − l µQ (cid:21)(cid:35) n h a s (cid:34) −
809 + 163 ζ (3) + l µQ (cid:18) − ζ (3) (cid:19) − l µQ (cid:35) + n h a s (cid:34) ( δ kM + δ lM ) (cid:18) − ζ (3) (cid:19) ± δ kM δ lM (cid:18) −
649 + 16 l µQ + 323 ζ (3) (cid:19)(cid:35)(cid:27) , (71f)ˆ C v/a, L3 = 316 π (cid:26) δ kM δ lM (cid:34) a s (cid:18) l µQ (cid:19) + a s (cid:20) − ζ (3) − ζ (5) + n f (cid:18) − ζ (3) (cid:19) + l µQ (cid:18) − n f (cid:19) + l µQ (cid:18) − n f (cid:19)(cid:21)(cid:35) ± δ kM δ lM (cid:34) l µQ + a s (cid:18) − ζ (3) + 163 l µQ + 8 l µQ (cid:19) + a s (cid:20) − − ζ (3) − ζ (4) + 9409 ζ (5) + n f (cid:18) ζ (3) (cid:19) + l µQ (cid:18) − ζ (3) + n f (cid:18) − ζ (3) (cid:19) (cid:19) + l µQ (cid:18) − n f (cid:19) + l µQ (cid:18) − n f (cid:19)(cid:21)(cid:35) + ( δ kM + δ lM ) (cid:34) − − l µQ + a s (cid:18) − ζ (3) − l µQ − l µQ (cid:19) + a s (cid:20) − ζ (3) + 43 ζ (4) − ζ (5) + n f (cid:18) − ζ (3) (cid:19) + l µQ (cid:18) − n f (cid:18) − ζ (3) (cid:19) + 3323 ζ (3) (cid:19) + l µQ (cid:18) − n f (cid:19) + l µQ (cid:18) −
18 + 49 n f (cid:19)(cid:21)(cid:35) + a s n h (cid:34)(cid:18)
329 + 8 l µQ (cid:19) ( δ kM + δ lM ) ± (cid:18) − − l µQ (cid:19) δ kM δ lM (cid:35)(cid:27) . (71g) B.2 Scalar and pseudo-scalar currents
Also the results for the scalar and the pseudo-scalar currents (upper and lower signs,respectively) are taken from Ref. [60]:ˆ C s/p = 18 a s + a s (cid:20) l µQ (cid:21) , (72a)21 C s/p ,N = a s (cid:20) −
56 + 12 l µQ (cid:21) , (72b)ˆ C s/p ,M = 12 + a s (cid:20)
116 + l µQ (cid:21) + a s (cid:20) − ζ (3) − n f + l µQ (cid:18) − n f (cid:19) + l µQ (cid:18) − n f (cid:19) (cid:21) , (72c)ˆ C s/p , nd = ± (cid:18) a s (cid:20)
143 + 2 l µQ (cid:21) + a s (cid:20) − ζ (3) − n f + l µQ (cid:18) − n f (cid:19) + l µQ (cid:18) − n f (cid:19) (cid:21)(cid:19) , (72d)ˆ C s/p = 316 π (cid:26) δ kM δ lM (cid:34) a s (cid:20)
323 + 24 l µQ + 16 ζ (3) (cid:21) + a s (cid:20) ζ (3) − ζ (5) + n f (cid:18) − ζ (3) (cid:19) + l µQ (cid:18) ζ (3) + n f (cid:18) − − ζ (3) (cid:19)(cid:19) + l µQ (105 − n f ) (cid:21)(cid:35) ± δ kM δ lM (cid:34) − l µQ + a s (cid:20) −
16 + 16 ζ (3) − l µQ − l µQ (cid:21) + a s (cid:20) − ζ (3) + 43 ζ (4) + 509 ζ (5) + n f (cid:18) ζ (3) (cid:19) + l µQ (cid:18) − ζ (3) + n f (cid:18) − ζ (3) (cid:19)(cid:19) + l µQ (cid:18) −
222 + 529 n f (cid:19) + l µQ (cid:18) −
53 + 109 n f (cid:19) (cid:21)(cid:35) + ( δ kM + δ lM ) (cid:34) − l µQ + a s (cid:20) − ζ (3) − l µQ − l µQ (cid:21) + a s (cid:20) − ζ (3) + 23 ζ (4) + 1909 ζ (5) + n f (cid:18) ζ (3) (cid:19) + l µQ (cid:18) − n f (cid:19) + l µQ (cid:18) −
532 + 59 n f (cid:19) + l µQ (cid:18) − ζ (3) + n f (cid:18) − ζ (3) (cid:19)(cid:19) (cid:21)(cid:35) + n h a s (cid:20) − ζ (3) + 4 l µQ − l µQ (cid:21) + n h a s (cid:20) ( δ kM + δ lM ) (cid:18) − − ζ (3) + 4 l µQ (cid:19) ± δ kM δ lM (cid:18) − − ζ (3) + 16 l µQ (cid:19) (cid:21)(cid:27) . (72e)22 Renormalized mixing matrix
For the reader’s convenience, we provide the relation between the mixing matrix ζ usedin this paper and its definition in Ref. [21], referred to as “ HKL ” in what follows. This iseasily derived from the relation between the operators O n defined in Eq. (5) and the O n,µν of HKL : (cid:18) O O (cid:19) δ µν = H × ( a B s ) (cid:18) O ,µν O ,µν (cid:19) . (73)The mixing matrix between regular and flowed operators in HKL , restricted to the twooperators which are relevant for this paper, is defined through (cid:18) ˜ O ,µν ( t )˜ O ,µν ( t ) (cid:19) = ζ HKL × ( t ) (cid:18) O ,µν O ,µν (cid:19) , ζ HKL × = (cid:18) ζ HKL ζ HKL ζ HKL ζ HKL (cid:19) , (74)where the ζ HKL ij are the entries of the full 4 × HKL . Inserting Eq. (73)and (cid:18) ˜ O ( t )˜ O ( t ) (cid:19) δ µν = H × (ˆ µ (cid:15) a s ) χ ( t ) (cid:18) ˜ O ,µν ( t )˜ O ,µν ( t ) (cid:19) , with χ ( t ) = (cid:18) ζ χ ( t ) (cid:19) , (75)with ζ χ ( t ) from Eq. (20), gives the renormalized mixing matrix used in the current paperin terms of the bare mixing matrix of HKL : ζ × ( t ) = H × ( a s ˆ µ (cid:15) ) χ ( t ) ζ HKL × ( t ) H − × ( a B s ) Z − × ( a s ) . (76)Explicitely, one finds: ζ ( t ) = 1 + 78 a s C A + a s (cid:40) C (cid:20) − L µt (cid:21) + C A T R n f (cid:20) −
118 + 18 L µt (cid:21) + C F T R n f (cid:20)
316 + 14 L µt (cid:21)(cid:41) , (77a) ζ ( t ) = a s C F (cid:20) − − L µt (cid:21) + a s (cid:20) C (cid:18) − − L µt (cid:19) + C A C F (cid:18) − ζ (2) − (1 / − L µt − L µt (cid:19) + C F T R n f (cid:18)
158 + 12 ζ (2) + 53 L µt + 12 L µt (cid:19)(cid:21) , (77b) ζ ( t ) = 512 a s T R n f + a s (cid:20) C A T R n f (cid:18) − (cid:19) + C F T R n f (cid:18)
14 + 12 Li (1 / − ζ (2) + 103 ln 2 −
218 ln 3 (cid:19)(cid:21) , (77c) ζ ( t ) = 1 + a s C F (cid:20) − ln 2 −
34 ln 3 (cid:21) ζ (0)1 ζ (2)1 ζ ζ (0)2 ζ (2)2 ζ Eq. (36) (40) (42) (38) (40) (42) code zeta01anc zeta21anc zeta13anc zeta02anc zeta22anc zeta23anc γ f2 × (cid:126)γ f3 ζ × Eq. (54) (56) (77) code gamma22anc gamma3anc ZetaMatrix22anc + a s (cid:40) C A C F (cid:20) − −
558 ln 2 + 14 ln (cid:18) − − (cid:19) L µt + 34 Li (1 / − ζ (2) (cid:21) + C F T R n f (cid:20) − (cid:18) −
23 + 13 ln 2 + 14 ln 3 (cid:19) L µt + 14 Li (1 / − ζ (2) (cid:21) + C (cid:20) − (1 /
4) + 1332 ζ (2) (cid:21) + 116 c (2) χ (cid:41) , (77d)where c (2) χ is given in Eq. (23). The results including higher orders in (cid:15) are provided in theancillary file to this paper, see Appendix D. D Ancillary File
The main results of this paper are provided in computer readable format (e.g. with
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