Isospin-breaking corrections to the muon magnetic anomaly in Lattice QCD
Davide Giusti, Vittorio Lubicz, Guido Martinelli, Francesco Sanfilippo, Silvano Simula
IIsospin-breaking corrections to the muon magneticanomaly in Lattice QCD
D. Giusti ∗ ( a , b ) , V. Lubicz ( a , b ) , G. Martinelli ( c ) , F. Sanfilippo ( b ) , and S. Simula ( b ) ( a ) Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Rome, Italy.Email: [email protected] , [email protected] ( b ) Istituto Nazionale di Fisica Nucleare, Sezione di Roma Tre, Rome, Italy.Email: [email protected] , [email protected] ( c ) Dipartimento di Fisica, Università degli Studi di Roma "La Sapienza" and INFN, Sezione diRoma, Rome, Italy.Email: [email protected]
In this contribution we present a lattice calculation of the leading-order electromagnetic andstrong isospin-breaking (IB) corrections to the quark-connected hadronic-vacuum-polarization(HVP) contribution to the anomalous magnetic moment of the muon. The results are obtainedadopting the RM123 approach in the quenched-QED approximation and using the QCD gaugeconfigurations generated by the ETM Collaboration with N f = + + a (cid:39) . , . , .
089 fm), at several lattice volumes and withpion masses between (cid:39)
210 and (cid:39)
450 MeV. After the extrapolations to the physical pion massand to the continuum and infinite-volume limits the contributions of the light, strange and charmquarks are respectively equal to δ a HVP µ ( ud ) = . ( . ) · − , δ a HVP µ ( s ) = − . ( ) · − and δ a HVP µ ( c ) = . ( ) · − . At leading order in α em and ( m d − m u ) / Λ QCD we obtain δ a HVP µ ( udsc ) = . ( . ) · − , which is currently the most accurate determination of the IBcorrections to a HVP µ . The 9th International workshop on Chiral Dynamics17-21 September 2018Durham, NC, USA ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] S e p sospin-breaking corrections to a HVP µ D. Giusti
1. Introduction
The anomalous magnetic dipole moments of charged leptons a (cid:96) are defined as the deviationsof the spin gyromagnetic ratios g (cid:96) from the result predicted by the Dirac equation, a (cid:96) = ( g (cid:96) − ) / a µ is one of the most accurately determined dimensionless physicalquantity in Nature: it is currently known both experimentally [1] and from a SM theoretical cal-culation [2] to approximately 0 . σ ÷ σ level. Since this tensionmight be an exciting indication of New Physics beyond the SM, an intense research program iscurrently underway in order to achieve a significant reduction of the experimental and theoreticaluncertainties. New ( g − ) experiments at Fermilab (E989) [3] and J-PARC (E34) [4] aim at afourfold reduction of the experimental uncertainty such that a similar reduction in the theoreticaluncertainty is of timely interest. Hadronic loop contributions due to the HVP and hadronic light-by-light terms [5] give rise to the main theoretical uncertainty and, with a view to the plannedexperimental accuracy, they will soon become a major limitation of this SM test.Nowadays the theoretical predictions for the hadronic contribution a HVP µ are most accuratelydetermined using dispersion relations for relating the HVP function to the experimental crosssection data for e + e − annihilation into hadrons [6, 7]. However, since the pioneering works ofRefs. [8–10], lattice QCD calculations of a HVP µ (see Ref. [11] for a recent review) have been madean impressive progress providing a completely independent cross-check from first principles.With the increasing accuracy of lattice calculations, it becomes mandatory to include electro-magnetic (em) and strong IB corrections, which contribute to the HVP to O ( α em ) and O ( α em ( m d − m u ) / Λ QCD ) , respectively. Here we present the results of a lattice calculation of the IB corrections tothe HVP contribution due to light-, strange- and charm-quark (connected) intermediate states, ob-tained in Ref. [12] using the RM123 approach [13, 14], which consists in the expansion of the pathintegral in powers of the mass difference ( m d − m u ) and of the em coupling α em . The quenched-QED (qQED) approximation, which treats the dynamical quarks as electrically neutral particles,has been adopted and quark-disconnected contractions have not been included yet because of thelarge statistical fluctuations of the corresponding signals.
2. Isospin-breaking corrections in the RM123 approach
We have evaluated the HVP contribution a HVP µ to the muon ( g − ) by adopting the time-momentum representation [15], namely a HVP µ = α em (cid:90) ∞ dt K µ ( t ) V ( t ) , (2.1)where the kernel function K µ ( t ) is given by K µ ( t ) = m µ (cid:90) ∞ d ω √ + ω (cid:32) √ + ω − ω √ + ω + ω (cid:33) (cid:20) cos ( ω m µ t ) − ω + m µ t (cid:21) (2.2)1 sospin-breaking corrections to a HVP µ D. Giusti with m µ being the muon mass. In Eq. (2.1) the quantity V ( t ) is the vector current-current Euclideancorrelator defined as V ( t ) ≡ − ∑ i = , , (cid:90) d (cid:126) x (cid:104) J i ( (cid:126) x , t ) J i ( ) (cid:105) , (2.3)where J µ ( x ) ≡ ∑ f = u , d , s , c ,... J f µ ( x ) = ∑ f = u , d , s , c ,... q f ψ f ( x ) γ µ ψ f ( x ) (2.4)is the em current with q f being the electric charge of the quark with flavor f in units of the electroncharge e , while (cid:104) ... (cid:105) means the average of the T -product over gluon and quark fields.We consider only the quark-connected HVP contributions, thus neglecting off-diagonal flavorterms. In this case each quark flavor f contributes separately a HVP µ = ∑ f = u , d , s , c ,... [ a HVP µ ( f )] ( conn ) . (2.5)For sake of simplicity we drop the suffix ( conn ) , but it is understood that in the following we referalways to quark-connected contractions only.In the RM123 method of Refs. [13, 14] the vector correlator for the quark flavor f , V f ( t ) , isexpanded into a lowest-order contribution V ( ) f ( t ) , evaluated in isospin-symmetric QCD ( i.e. m u = m d and α em = δ V f ( t ) computed to leading order in the small parameters ( m d − m u ) / Λ QCD and α em : V f ( t ) = V ( ) f ( t ) + δ V f ( t ) + . . . , (2.6)where the ellipses stand for higher order terms in ( m d − m u ) / Λ QCD and α em .The separation between the isosymmetric QCD and the IB contributions, V ( ) f ( t ) and δ V f ( t ) ,is prescription dependent. As in Ref. [12], here we impose the matching condition in which therenormalized coupling and quark masses in the full theory, α s and m f , and in isosymmetric QCD, α ( ) s and m ( ) f , coincide in the MS scheme at a scale of 2 GeV. Such a prescription is known as theGasser-Rusetsky-Scimemi (GRS) one [16].The calculation of the IB correlator δ V f ( t ) requires the evaluation of the self-energy, exchange,tadpole, pseudoscalar and scalar insertion diagrams depicted in Fig. 1. More specifically, the IB where m ud ¼ ð m d þ m u Þ = is the bare isosymmetric light quark mass. In the case of the neutral pion we obtainThe sea quark propagators have been drawn in blue (and with a different line) and the isosymmetric vacuum polarizationdiagrams have not been displayed explicitly. By combining the previous expressions we find the elegant formulaAll the isosymmetric vacuum polarization diagrams cancelby taking the difference of ! M ! þ and ! M ! together withthe disconnected sea quark loop contributions explicitlyshown in Eqs. (64) and (65). Note, in particular, the can-cellation of the corrections/counterterms corresponding tothe variation of the symmetric up-down quark mass m ud % m ud and to the variation of the strong coupling constant g s % ð g s Þ . This is a general feature: at first order of theperturbative expansion in ^ " em and ^ m d % ^ m u , the isosym-metric corrections coming from the variation of the stonggauge coupling (the lattice spacing), of m ud and of theheavier quark masses do not contribute to observables that vanish in the isosymmetric theory, like the mass splitting M ! þ % M ! . Furthermore, as already stressed, the electriccharge does not need to be renormalized at this order and,for all these reasons, the expression for the pion masssplitting can be considered a ‘‘clean’’ theoretical prediction.On the other hand, the lattice calculation of the discon-nected diagram present in Eq. (66) is a highly nontrivialnumerical problem and we shall neglect this contributionin this paper. Relying on the same arguments that lead tothe derivation of the flavor SU ð Þ version of Dashen’stheorem [see Eq. (39)], it can be shown that the neutralpion mass has to vanish in the limit ^ m u ¼ ^ m d ¼ for G. M. DE DIVITIIS et al.
PHYSICAL REVIEW D where m ud ¼ ð m d þ m u Þ = is the bare isosymmetric light quark mass. In the case of the neutral pion we obtainThe sea quark propagators have been drawn in blue (and with a different line) and the isosymmetric vacuum polarizationdiagrams have not been displayed explicitly. By combining the previous expressions we find the elegant formulaAll the isosymmetric vacuum polarization diagrams cancelby taking the difference of ! M ! þ and ! M ! together withthe disconnected sea quark loop contributions explicitlyshown in Eqs. (64) and (65). Note, in particular, the can-cellation of the corrections/counterterms corresponding tothe variation of the symmetric up-down quark mass m ud % m ud and to the variation of the strong coupling constant g s % ð g s Þ . This is a general feature: at first order of theperturbative expansion in ^ " em and ^ m d % ^ m u , the isosym-metric corrections coming from the variation of the stonggauge coupling (the lattice spacing), of m ud and of theheavier quark masses do not contribute to observables that vanish in the isosymmetric theory, like the mass splitting M ! þ % M ! . Furthermore, as already stressed, the electriccharge does not need to be renormalized at this order and,for all these reasons, the expression for the pion masssplitting can be considered a ‘‘clean’’ theoretical prediction.On the other hand, the lattice calculation of the discon-nected diagram present in Eq. (66) is a highly nontrivialnumerical problem and we shall neglect this contributionin this paper. Relying on the same arguments that lead tothe derivation of the flavor SU ð Þ version of Dashen’stheorem [see Eq. (39)], it can be shown that the neutralpion mass has to vanish in the limit ^ m u ¼ ^ m d ¼ for G. M. DE DIVITIIS et al.
PHYSICAL REVIEW D where m ud ¼ ð m d þ m u Þ = is the bare isosymmetric light quark mass. In the case of the neutral pion we obtainThe sea quark propagators have been drawn in blue (and with a different line) and the isosymmetric vacuum polarizationdiagrams have not been displayed explicitly. By combining the previous expressions we find the elegant formulaAll the isosymmetric vacuum polarization diagrams cancelby taking the difference of ! M ! þ and ! M ! together withthe disconnected sea quark loop contributions explicitlyshown in Eqs. (64) and (65). Note, in particular, the can-cellation of the corrections/counterterms corresponding tothe variation of the symmetric up-down quark mass m ud % m ud and to the variation of the strong coupling constant g s % ð g s Þ . This is a general feature: at first order of theperturbative expansion in ^ " em and ^ m d % ^ m u , the isosym-metric corrections coming from the variation of the stonggauge coupling (the lattice spacing), of m ud and of theheavier quark masses do not contribute to observables that vanish in the isosymmetric theory, like the mass splitting M ! þ % M ! . Furthermore, as already stressed, the electriccharge does not need to be renormalized at this order and,for all these reasons, the expression for the pion masssplitting can be considered a ‘‘clean’’ theoretical prediction.On the other hand, the lattice calculation of the discon-nected diagram present in Eq. (66) is a highly nontrivialnumerical problem and we shall neglect this contributionin this paper. Relying on the same arguments that lead tothe derivation of the flavor SU ð Þ version of Dashen’stheorem [see Eq. (39)], it can be shown that the neutralpion mass has to vanish in the limit ^ m u ¼ ^ m d ¼ for G. M. DE DIVITIIS et al.
PHYSICAL REVIEW D where m ud ¼ ð m d þ m u Þ = is the bare isosymmetric light quark mass. In the case of the neutral pion we obtainThe sea quark propagators have been drawn in blue (and with a different line) and the isosymmetric vacuum polarizationdiagrams have not been displayed explicitly. By combining the previous expressions we find the elegant formulaAll the isosymmetric vacuum polarization diagrams cancelby taking the difference of ! M ! þ and ! M ! together withthe disconnected sea quark loop contributions explicitlyshown in Eqs. (64) and (65). Note, in particular, the can-cellation of the corrections/counterterms corresponding tothe variation of the symmetric up-down quark mass m ud % m ud and to the variation of the strong coupling constant g s % ð g s Þ . This is a general feature: at first order of theperturbative expansion in ^ " em and ^ m d % ^ m u , the isosym-metric corrections coming from the variation of the stonggauge coupling (the lattice spacing), of m ud and of theheavier quark masses do not contribute to observables that vanish in the isosymmetric theory, like the mass splitting M ! þ % M ! . Furthermore, as already stressed, the electriccharge does not need to be renormalized at this order and,for all these reasons, the expression for the pion masssplitting can be considered a ‘‘clean’’ theoretical prediction.On the other hand, the lattice calculation of the discon-nected diagram present in Eq. (66) is a highly nontrivialnumerical problem and we shall neglect this contributionin this paper. Relying on the same arguments that lead tothe derivation of the flavor SU ð Þ version of Dashen’stheorem [see Eq. (39)], it can be shown that the neutralpion mass has to vanish in the limit ^ m u ¼ ^ m d ¼ for G. M. DE DIVITIIS et al.
PHYSICAL REVIEW D where m ud ¼ ð m d þ m u Þ = is the bare isosymmetric light quark mass. In the case of the neutral pion we obtainThe sea quark propagators have been drawn in blue (and with a different line) and the isosymmetric vacuum polarizationdiagrams have not been displayed explicitly. By combining the previous expressions we find the elegant formulaAll the isosymmetric vacuum polarization diagrams cancelby taking the difference of ! M ! þ and ! M ! together withthe disconnected sea quark loop contributions explicitlyshown in Eqs. (64) and (65). Note, in particular, the can-cellation of the corrections/counterterms corresponding tothe variation of the symmetric up-down quark mass m ud % m ud and to the variation of the strong coupling constant g s % ð g s Þ . This is a general feature: at first order of theperturbative expansion in ^ " em and ^ m d % ^ m u , the isosym-metric corrections coming from the variation of the stonggauge coupling (the lattice spacing), of m ud and of theheavier quark masses do not contribute to observables that vanish in the isosymmetric theory, like the mass splitting M ! þ % M ! . Furthermore, as already stressed, the electriccharge does not need to be renormalized at this order and,for all these reasons, the expression for the pion masssplitting can be considered a ‘‘clean’’ theoretical prediction.On the other hand, the lattice calculation of the discon-nected diagram present in Eq. (66) is a highly nontrivialnumerical problem and we shall neglect this contributionin this paper. Relying on the same arguments that lead tothe derivation of the flavor SU ð Þ version of Dashen’stheorem [see Eq. (39)], it can be shown that the neutralpion mass has to vanish in the limit ^ m u ¼ ^ m d ¼ for G. M. DE DIVITIIS et al.
PHYSICAL REVIEW D (a) (b) (c) (d) (e)Figure 1:
Fermionic connected diagrams contributing to the IB corrections δ a HVP µ ( f ) : self-energy (a), exchange (b),tadpole (c), pseudoscalar (d) and scalar (e) insertions. Solid lines represent the propagators of the quark with flavor fin isosymmetric QCD. corrections δ V f ( t ) consists of two (prescription-dependent) contributions: the em, δ V QEDf ( t ) , andthe strong IB (SIB), δ V SIBf ( t ) , one. Diagrams (1a)-(1d) contribute to the em corrections only,while the diagram (1e) to both δ V QEDf ( t ) and δ V SIBf ( t ) . Tadpole insertions (1c) are a feature of2 sospin-breaking corrections to a HVP µ D. Giusti lattice discretization and play a crucial role in order to preserve gauge invariance to O ( α em ) inthe expansion of the quark action [14]. Since the lattice fermionic action used in this contributionincludes a Wilson term, the insertions of pseudoscalar densities (1d) account for regularization-specific IB effects associated with the tuning of the quark critical masses in the presence of eminteractions [14, 17]. In the numerical evaluation of the photon propagator the zero-mode has beenremoved according to the QED L prescription [18], i.e. the photon field A µ satisfies A µ ( k ,(cid:126) k = (cid:126) ) ≡ k .Within the qQED approximation and neglecting quark-disconnected diagrams, the QED cor-relator δ V QEDf ( t ) is proportional to α em q f . Instead, the SIB one δ V SIBf ( t ) is proportional to q f ( m ( ) f − m f ) . Since in the GRS prescription we require m ( ) f ( MS , ) = m f ( MS , ) for f = ( ud ) , s , c , the SIB correlator at the renormalization scale µ = m d = m u = m ud ). In that casethe correction [ δ V SIBud ( t )]( MS , ) is proportional to the light-quark mass difference, whosevalue, m d − m u = . ( ) MeV has been determined in Ref. [17] at the physical pion mass in theMS ( ) scheme by using as inputs the experimental charged- and neutral-kaon masses.The isosymmetric QCD gauge ensembles used in this contribution are the same adoptedin Ref. [12], i.e. those generated by the European (now Extended) Twisted Mass Collaboration(ETMC) with N f = + + O ( a ) -improvement [22]. We consider three values of the inverse bare lattice coupling β , corresponding to lattice spacings varying from 0 .
089 to 0 .
062 fm, pion masses in the range M π (cid:39) ÷
490 MeV and different lattice volumes. For earlier investigations of finite volume ef-fects (FVEs) the ETM Collaboration had produced three dedicated ensembles, A40.20, A40.24 andA40.32, which share the same quark mass (corresponding to M π (cid:39)
320 MeV) and lattice spacing ( a (cid:39) .
09 fm ) and differ only in the lattice size L ( L / a = ÷ ) . To improve such an investigationa further gauge ensemble, A40.40, has been generated at a larger value of the lattice size, L / a = J µ ( x ) = Z A q f ψ f (cid:48) ( x ) γ µ ψ f ( x ) , (2.7)where ψ f (cid:48) and ψ f represent two quarks with the same mass, charge and flavor, but regularized withopposite values of the Wilson r -parameter ( i.e. r f (cid:48) = − r f ). Being at maximal twist the current (2.7)renormalizes multiplicatively with the renormalization constant (RC) Z A of the axial current. Byconstruction the local current (2.7) does not generate quark-disconnected diagrams. As discussedin Ref. [23], the properties of the kernel function K µ ( t ) in Eq. (2.1), guarantee that the contactterms, generated in the HVP tensor by a local vector current, do not contribute to a HVP µ .Since we have adopted the renormalized vector current (2.7), the em correlator δ V QEDf ( t ) receives a contribution from the em corrections to the RC of the vector current of Eq. (2.7) as well,namely Z A = Z ( ) A (cid:16) + α em π Z A (cid:17) + O ( α mem α ns ) , ( m > , n ≥ ) (2.8)3 sospin-breaking corrections to a HVP µ D. Giusti where Z ( ) A is the RC of the axial current in pure QCD (determined in Ref. [24]), while the product Z ( ) A Z A encodes the corrections to first order in α em . The quantity Z A can be written as Z A = Z ( ) A · Z f actA , (2.9)where Z ( ) A = − . q f is the pure QED correction to leading order in α em given by [25, 26]and Z f actA takes into account QCD corrections of order O ( α ns ) with n ≥ Z A = Z ( ) A ( i.e. Z f actA =
1) adoptedin Ref. [23]. We make use of the non-perturbative determination performed in Ref. [27] within theRI (cid:48) -MOM scheme.Similarly, the em corrections to the mass RC Z m enter in δ V QEDf ( t ) . For our maximallytwisted-mass setup 1 / Z m = Z P and Z ( ) P is the RC of the pseudoscalar density evaluated in Ref. [24]in isosymmetric QCD, in the MS ( ) scheme. The pure QED contribution Z ( ) P = q f [ ( a µ ) − . ] to leading order in α em is given in the MS scheme at the renormalization scale µ by [25, 26]. The values adopted for the coefficients Z f actP and Z f actA are collected in Table V ofRef. [12].
3. Results
A convenient procedure [23,28] relies on splitting Eq. (2.1) into two contributions correspond-ing to 0 ≤ t ≤ T data and t > T data , respectively. In the first contribution the vector correlator isnumerically evaluated on the lattice, while for the second contribution an analytic representation isrequired. If T data is large enough that the ground-state contribution is dominant for t > T data andsmaller than T / δ a HVP µ ( f ) for the quarkflavor f can be written as δ a HVP µ ( f ) ≡ δ a HVP µ ( < ) + δ a HVP µ ( > ) (3.1)with δ a HVP µ ( < ) = α em T data ∑ t = K µ ( t ) δ V f ( t ) , (3.2) δ a HVP µ ( > ) = α em ∞ ∑ t = T data + a K µ ( t ) Z fV M fV e − M fV t (cid:34) δ Z fV Z fV − δ M fV M fV ( + M fV t ) (cid:35) , (3.3)where M fV is the ground-state mass of the lowest-order correlator V ( ) f ( t ) and Z fV is the squared ma-trix element of the vector current between the ground-state | V f (cid:105) and the vacuum: Z fV ≡ ( / ) ∑ i = x , y , z q f |(cid:104) | ψ f ( ) γ i ψ f ( ) | V f (cid:105)| . In Refs. [23, 28] the ground-state masses M fV and the matrix elements Z fV have been determined for f = ( ud ) , s , c using appropriate time intervals t min ≤ t ≤ t max for eachvalue of β and of the lattice volume for the ETMC gauge ensembles adopted in this contribution.The quantities δ M fV and δ Z fV in Eq. (3.3) can be extracted respectively from the “slope” andthe “intercept” of the ratio δ V f ( t ) / V ( ) f ( t ) at large time distances (see Refs. [12–14, 17, 23]). Wehave checked that the sum of the two terms in the r.h.s. of Eq. (3.1) is independent of the specificchoice of the value of T data within the statistical uncertainties [12, 23]. The time dependences of4 sospin-breaking corrections to a HVP µ D. Giusti the integrand functions K µ ( t ) δ V QEDf ( t ) for f = ( ud ) , s , c and K µ ( t ) δ V SIBud ( t ) are shown in Fig. 2in the cases of the ETMC gauge ensembles B .
32 and D .
48. After summation over the timedistance t , the SIB contribution dominates over the QED one. t / a -1200-800-40004008001200 K µ ( t ) δ V ud Q E D ( t ) * exchselfSZ A QEDT + PSB55.32M π ≈
375 MeVlight t / a K µ ( t ) δ V ud S I B ( t ) * SIBM π ≈
375 MeVB55.32 t / a -0.008-0.00400.0040.008 K µ ( t ) δ V s Q E D ( t ) * exchselfSZ A QEDT + PSD20.48M π ≈
260 MeVstrange t / a -0.04-0.03-0.02-0.0100.010.020.030.04 K µ ( t ) δ V c Q E D ( t ) * exchselfSZ A QEDT + PSD20.48M π ≈
260 MeVcharm
Figure 2:
Time dependence of the integrand functions K µ ( t ) δ V SIBud ( t ) (top-right panel) and K µ ( t ) δ V QEDf ( t ) for thelight- (top-left panel), strange- (bottom-left panel) and charm-quark (bottom-right panel) contributions to the IB correc-tions δ a HVP µ ( f ) [see Eq. (3.2)] in the cases of the ETMC gauge ensembles B . (M π (cid:39) MeV, a (cid:39) . fm) andD . (M π (cid:39) MeV, a (cid:39) . fm). In the panels the labels “self”, “exch”, “T+PS”, “S’, “ Z A ” indicate the QEDcontributions of the diagrams (1a), (1b), (1c)+(1d), (1e) and the one generated by the QED corrections to the RC of thelocal vector current. The accuracy of the lattice data can be improved by forming the ratio of the IB corrections δ a HVP µ ( f ) over the leading-order terms a HVP , ( ) µ ( f ) , which is shown in the case of the light-quarkcontribution in Fig. 3. The attractive feature of this ratio is to be less sensitive to some of thesystematics effects, in particular to the uncertainties of the scale setting.For the combined extrapolations to the physical pion mass and to the continuum and infinite-volume limits we have adopted the following fit ansatz: δ a HVP µ ( ud ) a HVP , ( ) µ ( ud ) = δ A (cid:96) (cid:104) + δ A (cid:96) m ud + δ A (cid:96) (cid:96) m ud ln ( m ud ) + δ A (cid:96) m ud + δ D (cid:96) a + δ (cid:96) FV E (cid:105) , (3.4)where the FVE term is estimated by using alternatively one of the fitting functions (see later on) δ (cid:96) FV E = δ F (cid:96) e − ML or δ (cid:96) FV E = δ (cid:98) F (cid:96) n M π f e − ML ( ML ) n ( n = , , , ) (3.5)5 sospin-breaking corrections to a HVP µ D. Giusti m ud (GeV) δ a µ HV P ( ud ) / a µ HV P , ( ) ( ud ) β =1.90, L=20 β =1.90, L=24 β =1.90, L=32 β =1.90, L=40 β =1.95, L=24 β =1.95, L=32 β =2.10, L=48physical point Figure 3:
Results for the ratio δ a HVP µ ( ud ) / a HVP , ( ) µ ( ud ) versus the renormalized average u / d mass m ud in the MS ( ) scheme. The empty markers correspond to the raw data, while the full ones represent the lattice datacorrected by the FVEs obtained in the fitting procedure (3.4) with δ A (cid:96) (cid:96) = and δ A (cid:96) (cid:54) = . The solid lines correspond tothe results of the combined fit (3.4) obtained in the infinite-volume limit at each value of the lattice spacing. The blackasterisk represents the value of the ratio δ a HVP µ ( ud ) / a HVP , ( ) µ ( ud ) extrapolated to the physical pion mass, correspondingto m physud ( MS , ) = . ( ) MeV and to the continuum and infinite-volume limits, while the red area indicates thecorresponding uncertainty as a function of m ud at the level of one standard deviation. Errors are statistical only. with B and f being the leading-order low-energy constants of Chiral Perturbation Theory (ChPT)and M ≡ B m ud . For the chiral extrapolation we consider either a quadratic ( δ A (cid:96) (cid:96) = δ A (cid:96) (cid:54) =
0) or a logarithmic ( δ A (cid:96) (cid:96) (cid:54) = δ A (cid:96) =
0) dependence. Half of the difference of the correspondingresults extrapolated to the physical pion mass is used to estimate the systematic uncertainty dueto the chiral extrapolation. Discretization effects play a minor role and, for our O ( a ) -improvedsimulation setup, they can be estimated by including ( δ D (cid:96) (cid:54) = ) or excluding ( δ D (cid:96) = ) the termproportional to a in Eq. (3.4). The free parameters to be determined by the fitting procedure are δ A (cid:96) , δ A (cid:96) , δ A (cid:96) (cid:96) (or δ A (cid:96) ), δ D (cid:96) and δ F (cid:96) (or δ (cid:98) F (cid:96) n ).Before discussing the result of the fitting procedure we focus more on the FVEs and commenton the choice of the fitting functions of Eqs. (3.5). For the separate QED and SIB contributionsthe FVEs differ qualitatively and quantitatively, as shown in Fig. 6 of Ref. [12]. In the case ofthe QED data a power-law behavior in terms of the inverse lattice size 1 / L is expected to start to O ( / L ) because of the overall neutrality of the system [12, 23, 29]. In the case of the SIB cor-relator, since a fixed value m d − m u = . ( ) MeV [17] is adopted for all gauge ensembles, anexponential dependence in terms of the quantity M π L is expected [30]. Since the SIB contribu-tion dominate over the QED one (see Fig. 2), the FVEs for the ratio δ a HVP µ ( ud ) / a HVP , ( ) µ ( ud ) areexpected to be mainly exponentially suppressed in M π L . We remind the reader that the lowest-order term a HVP , ( ) µ ( ud ) has non-negligible FVEs, which are exponentially suppressed in terms of M π L [28, 30, 31]. In Ref. [28] the FVEs on a HVP , ( ) µ ( ud ) have been evaluated by using the samelattice setup adopted here and developing an analytic representation of the vector correlator based Had we used in fitting our data (3.4) δ (cid:96) FVE = δ (cid:101) F (cid:96) / L we would have observed a change in the result (3.6) wellwithin the statistical uncertainty. sospin-breaking corrections to a HVP µ D. Giusti on quark-hadron duality [32] at small and intermediate time distances and on the two-pion con-tributions in a finite box [33] at larger time distances. After the extrapolation to the continuumlimit, the lattice estimates of FVEs turn out to be much larger than the corresponding predictionsof ChPT to NLO [34]. In Table 1 we collect the values of the ratio of the lattice FVEs, ∆ lat FV ( L ) ≡ a HVP , ( ) µ ( ud ; L → ∞ ) − a HVP , ( ) µ ( ud ; L ) , computed in Ref. [28] at the physical pion mass over the cor-responding NLO ChPT predictions, ∆ ChPT , NLO FV ( L ) . The NNLO ChPT corrections ∆ ChPT , NNLO FV ( L ) have been recently computed in Ref. [35] and the ratio ∆ ChPT , NNLO FV ( L ) / ∆ ChPT , NLO FV ( L ) for physicalpion masses and lattices of size L = ÷ (cid:39) . ( ) , which points in the samedirection as our lattice corrections ∆ lat FV ( L = ÷ ) / ∆ ChPT , NLO FV ( L = ÷ ) (cid:39) . ( ) (seeTable 1). M phys π L L ( fm ) ∆ lat FV ( L ) / ∆ ChPT , NLO FV ( L ) . . . ( ) . . . ( ) . . . ( ) . . . ( ) . . . ( ) . . . ( ) . . . ( ) Table 1:
Values of the ratio of the lattice FVEs, ∆ lat FV ( L ) ≡ a HVP , ( ) µ ( ud ; L → ∞ ) − a HVP , ( ) µ ( ud ; L ) , computed inRef. [28] at the physical pion mass M phys π (cid:39)
135 MeV over the NLO ChPT ones, ∆ ChPT , NLO FV ( L ) . At the physical pion mass and in the continuum and infinite-volume limits we have obtained [12] δ a HVP µ ( ud ) a HVP , ( ) µ ( ud ) = . ( ) stat + f it ( ) input ( ) chir ( ) FVE ( ) a [ ] , (3.6)where the errors come in the order from (statistics + fitting procedure), input parameters of theeight branches of the quark mass analysis of Ref. [24], chiral extrapolation, finite-volume anddiscretization effects. In Eq. (3.6) the uncertainty in the square brackets corresponds to the sum inquadrature of the statistical and systematic errors.Using the leading-order result a HVP , ( ) µ ( ud ) = . ( . ) · − from Ref. [28], our deter-mination of the leading-order IB corrections δ a HVP µ ( ud ) is δ a HVP µ ( ud ) = . ( . ) stat + f it ( . ) input ( . ) chir ( . ) FVE ( . ) a [ . ] · − , (3.7)which comes (within the GRS prescription) from the sum of the QED contribution (cid:2) δ a HVP µ ( ud ) (cid:3) ( QED ) = . ( . ) · − (3.8)and of the SIB one (cid:2) δ a HVP µ ( ud ) (cid:3) ( SIB ) = . ( . ) · − . (3.9)The above results show that the IB correction (3.7) is dominated by the strong SU ( ) -breakingterm, which corresponds roughly to ≈
85% of δ a HVP µ ( ud ) .7 sospin-breaking corrections to a HVP µ D. Giusti
Our determination (3.7), obtained with N f = + + δ a HVP µ ( ud ) = . ( . ) · − , obtained by the BMW Collaboration [36] using results of the dispersive analysisof e + e − data [37], and the lattice determination δ a HVP µ ( ud ) = . ( . ) · − , obtained by theRBC/UKQCD Collaboration [38] at N f = +
1, which includes also one disconnected QED dia-gram. Recently, adopting N f = + + + (cid:2) δ a HVP µ ( ud ) (cid:3) ( SIB ) = . ( . ) · − [39].Thanks to the recent non-perturbative evaluation of QCD+QED effects on the RCs of bilinearoperators performed in Ref. [27] we have updated the determinations of the strange δ a HVP µ ( s ) andcharm δ a HVP µ ( c ) contributions to the IB effects made in Ref. [23], obtaining a drastic improvementof the uncertainty by a factor of ≈ ≈ .
5, respectively. In Fig. 4 the updated results for theratios δ a HVP µ ( f ) / a HVP , ( ) µ ( f ) for f = s , c are shown. m ud (GeV) -0.000500.0005 δ a µ HV P ( s ) / a µ HV P , ( ) ( s ) β =1.90, L=20 β =1.90, L=24 β =1.90, L=32 β =1.95, L=24 β =1.95, L=32 β =2.10, L=48physical point m ud (GeV) -0.00200.0020.0040.006 δ a µ HV P ( c ) / a µ HV P , ( ) ( c ) β =1.90, L=20 β =1.90, L=24 β =1.90, L=32 β =1.95, L=24 β =1.95, L=32 β =2.10, L=48physical point Figure 4:
Results for the strange (left panel) and charm (right panel) contributions to δ a HVP µ / a HVP , ( ) µ versus therenormalized average u / d mass m ud . The solid lines correspond to the linear fit (3.10) including the discretization termin the infinite-volume limit. The black asterisks represent the results of the extrapolation to the physical pion mass andto the continuum and infinite-volume limits while the red area indicates the corresponding uncertainty as a function ofm ud at the level of one standard deviation. Errors are statistical only. By adopting the same fitting function (5.13) of Ref. [23], namely δ a HVP µ ( s , c ) a HVP , ( ) µ ( s , c ) = δ A s , c + δ A s , c m ud + δ D s , c a + δ F s , c L (3.10)and after the extrapolations to the physical pion mass and to the continuum and infinite-volumelimits we have found δ a HVP µ ( s ) a HVP , ( ) µ ( s ) = − . ( ) stat + f it ( ) input ( ) chir ( ) FVE ( ) a [ ] % , (3.11) δ a HVP µ ( c ) a HVP , ( ) µ ( c ) = . ( ) stat + f it ( ) input ( ) chir ( ) FVE ( ) a [ ] % (3.12)where the error budget has been obtained as in Ref. [23].8 sospin-breaking corrections to a HVP µ D. Giusti
Using the leading-order results a HVP , ( ) µ ( s ) = . ( . ) · − and a HVP , ( ) µ ( c ) = . ( . ) · − [23], our updated IB determinations are [12] δ a HVP µ ( s ) = − . ( ) stat + f it ( ) input ( ) chir ( ) FVE ( ) a [ ] · − , (3.13) δ a HVP µ ( c ) = . ( ) stat + f it ( ) input ( ) chir ( ) FVE ( ) a [ ] · − (3.14)to be compared with δ a HVP µ ( s ) = − . ( ) · − and δ a HVP µ ( c ) = − . ( ) · − givenin Ref. [23]. The updated results confirm that the em corrections δ a HVP µ ( s ) and δ a HVP µ ( c ) arenegligible with respect to the current uncertainties of the corresponding lowest-order terms. Re-cently [38] in the case of the strange contribution the RBC/UKQCD Collaboration has found theresult δ a HVP µ ( s ) = − . ( ) · − , which deviates from our finding (3.13) by ≈ δ a HVP µ within the qQED approximation, namely δ a HVP µ ( udsc ) | conn = . ( . ) · − .Recently, in Ref. [38] one QED disconnected diagram has been calculated in the case of the u - and d -quark contribution and found to be of the same order of the corresponding QED connected term.Thus, we estimate that the uncertainty related to the qQED approximation and to the neglect ofquark-disconnected diagrams is approximately equal to our QED contribution (3.8), obtaining δ a HVP µ ( udsc ) = . ( . ) ( . ) qQED + disc [ . ] · − , (3.15)which represents the most accurate determination of the IB contribution to a HVP µ to date.Using the recent ETMC determinations of the lowest-order contributions of light, strange andcharm quarks, a HVP , ( ) µ ( ud ) = . ( . ) · − , a HVP , ( ) µ ( s ) = . ( . ) · − and a HVP , ( ) µ ( c ) = . ( . ) · − [23, 28], and an estimate of the lowest-order quark-disconnected diagrams, a HVP , ( ) µ ( disc ) = − ( ) · − , obtained using the results of Refs. [36] and [38], our finding(3.15) for the IB corrections leads to an HVP contribution to the muon ( g −
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