Leading-order hadronic contributions to the electron and tau anomalous magnetic moments
Florian Burger, Karl Jansen, Marcus Petschlies, Grit Pientka
EEur. Phys. J. C manuscript No. (will be inserted by the editor)
Leading-order hadronic contributions to the electron and tauanomalous magnetic moments
Florian Burger a,1 , Karl Jansen b,2 , Marcus Petschlies c,3,4 , Grit Pientka d,1 Humboldt-Universit¨at zu Berlin, Institut f¨ur Physik, Newtonstr. 15, D-12489 Berlin, Germany NIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germany The Cyprus Institute, P.O.Box 27456, 1645 Nicosia, Cyprus Rheinische Friedrich-Wilhelms-Universit¨at Bonn, Institut f¨ur Strahlen- und Kernphysik, Nußallee 14-16, D-53115 Bonn,GermanyReceived: date / Accepted: date
Abstract
The leading hadronic contributions to theanomalous magnetic moments of the electron and the τ -lepton are determined by a four-flavour lattice QCDcomputation with twisted mass fermions. The resultspresented are based on the quark-connected contribu-tion to the hadronic vacuum polarization function. Thecontinuum limit is taken and systematic uncertaintiesare quantified. Full agreement with results obtained byphenomenological analyses is found. Keywords quantum chromodynamics · lattice QCD · leptons · g-2 · hadronic vacuum polarization PACS · The standard model of particle physics (SM) containsthree charged leptons l , mainly differing in mass, theelectron, the muon, and the τ -lepton with m e : m µ : m τ ≈ a l = ( g − l /
2, control their behaviour in anexternal magnetic field.Being the lepton with the smallest mass, the elec-tron is stable. This leads to the electron magnetic mo-ment being one of the most precisely determined quan-tities in nature. The difference between the latest ex-perimental [2] and SM values [3,4] is of O (cid:0) − (cid:1) orapproximately 1.3 standard deviations, c.f. [5] and ref-erences therein, a e-mail: fl[email protected] b e-mail: [email protected] c e-mail: [email protected] d e-mail: [email protected] a Exp e = 115 965 218 07 . . × a SM e = 115 965 218 17 . . × − a Exp e − a SM e = − . . × − . This constitutes one of the cornerstone results forquantum field theories to be recognised as the correctmechanism for describing particle interactions. The verygood agreement of the electron magnetic moment be-tween experiment and SM calculations is not matchedby the muon anomalous magnetic moment. In fact, herea two to four sigma discrepancy is observed, see e.g. [6].One reason for the observed discrepancy could be thatthe magnetic moment of the muon receives larger non -perturbative contributions than the one of the electron.On the other hand, it is supposed to be also more sensi-tive to beyond the SM physics, since for a large class oftheories new physics contributions are expected to beproportional to the squared lepton mass. Thus it is aprime candidate for detecting physics beyond the SM.Due to the large mass of the τ -lepton, it would be theoptimal lepton for finding new physics. However, be-cause its lifetime is very short ( O (cid:0) − (cid:1) s) there cur-rently only exist bounds on its anomalous magnetic mo-ment from indirect measurements [7].The QED [8,3] and the electroweak contributions [9,10] to the lepton anomalous magnetic moments havebeen computed in perturbation theory to impressivefive and two loops, respectively. The main uncertain-ties remaining in the theoretical determinations of theanomalous magnetic moments originate thus from theleading-order (LO) hadronic contributions. Since theyare particularly sensitive to those virtual photon mo-menta that are of O (cid:0) m l (cid:1) , these contributions are in-herently non-perturbative and not accessible to per-turbation theory. In order to have a prediction of the a r X i v : . [ h e p - l a t ] A ug anomalous magnetic moments from the SM alone, anon-perturbative method needs to be employed and theonly such approach we presently know, which eventu-ally allows us to control all systematic uncertainties, islattice QCD (LQCD) which we use here.As highlighted in [5], the uncertainty in the com-parison between the experimental and the SM value forthe electron anomalous magnetic moment is currentlydominated by the experimental uncertainty of its deter-mination and the value for α QED from atomic physicsexperiments with rubidium atoms which both are to bereduced in the future. Recently, the Harvard group hasannounced to be working on a more accurate determi-nation of the electron as well as the positron ( g −
2) [11].According to Ref. [5], uncertainties in the sub-10 − re-gion might be expected which would clearly provide theopportunity to also detect new physics contributions inthe anomalous magnetic moment of the electron andthus to cross-check the muon discrepancy. In this situa-tion it will again be of utmost importance to know thehadronic contributions as precisely as possible.Furthermore, even for the τ -lepton, Ref. [12] listsseveral proposals for the first actual measurement of itsanomalous magnetic moment, e.g. [13]. A first success-ful measurement in this direction has been reported in[14]. As we will show in the following, compared to thecase of the muon it will be much easier to obtain a valuefor the LO QCD contribution to a τ from LQCD withthe required precision to detect new physics and it willprobably not take very long before the QCD contribu-tion entering the official SM result will be provided byLQCD.As mentioned before, the hadronic LO contributionsto the anomalous magnetic moments of the three SMleptons, a hvp l , strongly depend on the values of theirmasses. Since the magnitude of the lepton masses spansfour orders of magnitude, the corresponding contribu-tions to the anomalous magnetic moments differ sub-stantially and probe very different energy regions, seealso the discussion of Fig. 1 in Sect. 3.In this article, we present the results of our four-flavour computations of the quark-connected, LO hadronicvacuum polarisation contributions to the electron and τ -lepton anomalous magnetic moments obtained fromthe (maximally) twisted mass formulation of LQCD.The muon case has already been covered in [15]. Theimportant feature of the present calculation is that weadopt exactly the same strategy as for the muon [15] in-cluding the same chiral and continuum extrapolations.Thus, the results presented here are not only interest-ing in themselves, but also serve as an important cross-check for our treatment of the hadronic vacuum polar-isation function. The consistent picture we obtain for all three standard model leptons then reassures the va-lidity of the analysis approach.Additionally to the systematic uncertainties investi-gated in our previous paper, we quantify the light-quarkdisconnected contributions on one of our N f = 2 + 1 + 1ensembles in order to arrive at a rough estimate of theirsystematic effect on our estimates for the hadronic LOanomalous magnetic moments. A full quantitative con-straint on the quark-disconnected contribution, how-ever, is beyond the scope of this work.Another very important feature is that incorporat-ing the complete first two generations of quarks en-ables us to directly and unambiguously compare ourresults with the values obtained from phenomenolog-ical analyses relying on experimental data and a dis-persion relation. We note that the contributions fromthird-generation quarks can be neglected, since they aresmaller than the current theoretical accuracy, as can beinferred e.g. from the data tables of Ref. [16]. Recently,the bottom quark contribution to a hvp µ has been explic-itly computed on the lattice [17] confirming it to be oneorder of magnitude smaller than the current uncertaintyof the phenomenological determinations of a hvp µ .Additionally to the N f = 2 + 1 + 1 flavour ensem-bles [18,19] at unphysically large pion masses studiedin [15], we computed the dominant light quark contribu-tions to the anomalous magnetic moments on a N f = 2flavour ensemble directly at the physical point [20,21].This allows us to test the chiral extrapolations per-formed when using the reparametrisation introducedin [22,23]. Since we currently only have one ensembleat the physical point with one lattice spacing and weneglect the small influence of the strange and charmsea quarks when comparing the results from N f = 2and N f = 2 + 1 + 1 ensembles, a final conclusion is pre-cluded at this point. The significance of this comparisonis based on the empirically observed weak dependenceon the lattice spacing of the light quark contribution aswell as the marginal sea quark effects from strange andcharm on the latter.The next section comprises a short repetition of themost important equations needed to follow the discus-sion of the results for the LO hadronic vacuum polarisa-tion contributions to the anomalous magnetic momentsof the electron in Sect. 3 and the τ -lepton in Sect. 4. InSect. 5 we summarise our results and draw our conclu-sions. a hvp l The LO hadronic contribution to the lepton anomalousmagnetic moments, a hvp l , can be directly computed in Euclidean space-time according to [24,25] a hvp l = α (cid:90) ∞ dQ Q w (cid:18) Q m l (cid:19) Π R ( Q ) , (1)where α is the fine structure constant, Q the Euclideanmomentum, m l the lepton mass, and Π R ( Q ) the renor-malised hadronic vacuum polarisation function, Π R ( Q ) = Π ( Q ) − Π (0) . It is obtained from the hadronic vacuum polarisationtensor Π µν ( Q ) = (cid:90) d x e iQ · ( x − y ) (cid:104) J µ ( x ) J ν ( y ) (cid:105) = (cid:0) Q µ Q ν − Q δ µν (cid:1) Π ( Q ) , (2)which is transverse because of the conservation of theelectromagnetic current J µ ( x ) = 23 ¯ u ( x ) γ µ u ( x ) −
13 ¯ d ( x ) γ µ d ( x )+ 23 ¯ c ( x ) γ µ c ( x ) −
13 ¯ s ( x ) γ µ s ( x ) . (3)Here u stands for the up quark, d for the down quark,c denotes the charm quark, and s the strange quark.Eq. (2) shows that Π µν ( Q ) results from the Fouriertransformation of the correlator of two such currents.Taking up and down quarks together, since they aremass-degenerate in our setup, we decompose the quark-connected part of the hadronic vacuum polarisationtensor according to Π µν ( Q ) = Π ud µν ( Q ) + Π s µν ( Q ) + Π c µν ( Q ) . (4)In our lattice calculation this decomposition intoflavour contributions is particularly straightforward, be-cause for all quark flavours we use the one-point-splitvector currents, which are conserved at non-zero latticespacing and thus do not require further multiplicativeor additive normalisation. From eq. (4), we can step-wise add the flavour contributions which will be donein the sections below.The standard integral definition in Eq. (1) results ina strong non-linear pion mass dependence, in particularfor the light quark contribution to a hvp l . This behaviouroriginates from the introduction of the lepton mass m l as an external scale, which is not related to the latticeparameters and in particular does not have an inherentvalue in lattice units. Employing Eq. (1) in the latticecalculation requires the input of the dimensionless com-bination a · m l and this renders the initially dimension-less quantity a hvp l effectively dependent on the latticescale setting. In view of this external scale problem, Refs. [22,23] proposed a modified definition of a newfamily of observables a hvp¯ l = α (cid:90) ∞ dQ Q w (cid:32) Q H H m l (cid:33) Π R ( Q ) . (5) H denotes some hadronic scale determined on the lat-tice at unphysically high pion masses, which fulfills theconstraint, that H ( m PS ) → H phys as m PS → m π .For each choice of H we thus obtain a correspondinglymodified, m PS dependent lepton mass on the lattice m ¯ l ( m PS ) = m l · H ( m PS ) /H phys .Our choice for the hadronic scale H is the mass m V of the lowest-lying state in the light vector meson chan-nel, i.e. the ρ -meson state. This choice uniquely fixes thepion mass dependence of the lepton mass m ¯ l ( m PS ) andis subsequently used for all single-flavour contributionsto the vacuum polarization function. H = H phys = 1 reproduces the standard definitionin Eq. (1). Up to lattice artefacts, the standard defini-tion is also recovered at the physical value of the pionmass when the ratio H/H phys becomes onelim m PS → m π a hvp¯ l ( m PS ) = a hvp l ( m π ) . Henceforth we always use the definition in Eq. (5) with H = m V and drop the bar on the label for the lepton.The weight function w is known from QED pertur-bation theory as w ( r ) = 64 r (1 + (cid:112) /r ) (cid:112) /r . (6)It has a pronounced peak at r peak = Q /m l = √ −
2. As an illustrating example the corresponding peaklocations for the electron, muon and tau are shown bythe labels on the upper x -axis in the upper plot of Fig.1 for ensemble D30.48 in Table 1 below.For a thorough description of the lattice calculationand a proof of automatic O ( a ) improvement of the vac-uum polarisation function we refer to [15] and [26], re-spectively. In order to discuss systematic uncertaintieslater on, we briefly summarise our method of fitting thehadronic vacuum polarisation function here.First, the lowest lying vector meson masses, m i , anddecay constants, f i , are determined from the time de-pendence of the two-point function of the light, strange,and charm point-split vector current, individually, atzero spatial momentum. Then Π ( Q ) determined in themomentum range between 0 and Q is split into alow-momentum part for 0 ≤ Q ≤ Q and a high-momentum one for Q < Q ≤ Q and is fitted separately for each flavour and each ensemble. The low-momentum fit function is given by Π low ( Q ) = M (cid:88) i =1 f i m i + Q + N − (cid:88) j =0 a j ( Q ) j , (7)and the high-momentum piece is parametrised as fol-lows Π high ( Q ) = log( Q ) B − (cid:88) k =0 b k ( Q ) k + C − (cid:88) l =0 c l ( Q ) l . (8)They are combined according to Π ( Q ) = (1 − Θ ( Q − Q )) Π low ( Q )+ Θ ( Q − Q ) Π high ( Q ) , (9)where Θ ( x ) is the Heaviside function.This defines our so-called MNBC fit function. Ourstandard fit for the light and strange quark contribu-tions is M1N2B4C1 which means M = 1, N = 2, B = 4, and C = 1 in Eqs. (7) and (8) above. Asvalue of Q in the Heaviside functions in Eq. (9)we have chosen 2 GeV . We have checked that varyingthe value of Q between 1 GeV and 3 GeV doesnot lead to observable differences as long as the tran-sition between the low- and the high-momentum partof the fit is smooth. For the upper integration limit weuse Q = 100 GeV , since the integrals are saturatedthere as can be seen in Fig. 1 below.For each ensemble and flavour we perform a fit for Π as given in Eq. (9). From this fit we obtain thecorresponding Π (0) and thus the subtracted polarisa-tion function. The latter is integrated using Eq. (5)and the contributions from individual quark flavoursare summed including the appropriate charge factors e + e , e , and e . This results in an estimate forthe hadronic leading-order lepton anomalous magneticmoment for each gauge field ensemble, which dependson the lattice spacing, the pion mass, and the latticesize, a hvp l = a hvp l (cid:0) a, m , L (cid:1) . As final step we per-form a combined extrapolation to the continuum andto physical quark masses. For this extrapolation somedependencies turn out to be negligible. The strange andcharm quark mass in the sea and valence quark actionhave been tuned for each ensemble, so we do not needto consider these dependencies explicitly in a hvp l . More-over, the detailed discussion of the lattice data in thefollowing sections will show that we only find significantlattice artefacts in the strange and charm quark contri-bution to a hvp l and we will use an appropriately adaptedfit ansatz. The dependence on the finite volume is dis-cussed in detail as part of the systematic error analysesin sections 3.3.1 and 4.3.1 and thus not part of the ex-trapolation described above. In [27,28] the usage of Pad´e approximants has beenadvocated for fits in the small momentum region. ThePad´e fit functions are formally identical to the M N series of fits. We analysed the Pad´e approximants forour data in [29]. We found agreement for the locationof the poles provided the same number of fit parame-ters were used in both cases (See also the more elabo-rate study for the case of the muon performed by theRBC-UKQCD collaboration in [30].). We can thus em-ploy the same procedure as for the muon and show thatit produces results compatible with phenomenologicaldeterminations for both the electron and the τ -leptonwithout any modification.Our analysis has been performed on the same setof gauge field configurations [18,19] as have been usedin our previous work [15]. A detailed list of the latticeparameters can be found in table 1 below. Ensemble a [fm] m P S [MeV] L [fm] m P S · L D15.48 0 .
061 227 2.9 3 . .
061 318 2.9 4 . .
061 387 1.9 3 . .
078 274 2.5 3 . .
078 319 2.5 4 . .
078 314 3.7 5 . .
078 393 2.5 5 . .
078 456 2.5 5 . .
078 491 1.9 4 . .
086 283 2.8 4 . .
086 323 2.8 4 . .
086 361 2.8 5 . .
091 135 4.4 3 . Table 1
Parameters of the N f = 2 + 1 + 1 flavour gaugefield configurations that have been analysed in this work. a denotes the lattice spacing (cf. [31]), m P S the value of thelight pseudoscalar meson mass (cf. [18]) and L the spatialextent of the lattices. The right-most column gives the valuefor m P S · L . The ensemble in the last line has N f = 2 andphysical pion mass. It is described in Refs. [20,21,32]. Theensemble name in the first column gives the bare quark massin lattice units as the first pair of digits times 10 − and thespatial lattice size L/a as the second pair of digits. g − The LO hadronic contribution to the electron anoma-lous magnetic moment a e is dominated by momentabelow 10 − GeV . To a good approximation it can evenbe determined from the slope of the vacuum polarisa-tion at zero momentum a e ∝ dΠ/dQ ( Q = 0). There-fore, we only use the low-momentum part, Π low ( Q ),of the hadronic vacuum polarisation function Eq. (7).The saturation of the integral for one of our ensembles, namely B55.32 featuring m PS ≈
390 MeV, a ≈ .
08 fmand L = 2 . R l ( Q ) = a hvp l ( Q ) a hvp l (100 GeV ) , (10)where a hvp l ( Q ) is the LO hadronic contribution tothe lepton anomalous magnetic moment integrated upto Q . This plot also implies that for the electron we Q ,e Q ,µ Q ,τ R l Q [GeV ] eµτ i n t e g r a nd , r e l a t i v ee rr o r Q /m eµτ Fig. 1
Upper plot: Comparison of the dependence on theupper integration bound in Eq. (5) of the four-flavour leptonanomalous magnetic moments. The blue curve represents theratio defined in Eq. (10) for the electron, the orange one forthe muon, and the dark red one for the tau. Q ,l denotesthe momentum value where the kernel function in Eq. (5)attains its maximum. Lower plot: Comparison of the depen-dence of the relative statistical uncertainties of the integrandsin Eq. (5) on the squared momenta scaled by the leptonmasses. The plots are based on data for the D30.48 ensemblefeaturing a = 0 .
061 fm , m PS = 318 MeV, and L = 2 . have to rely mostly on the extrapolation of our vacuumpolarisation data to the small momentum region. Al-though saturating well beyond momenta of O (cid:0) (cid:1) ,also for the τ , the renormalised vacuum polarisationfunction requires the subtraction of Π (0), which is de-termined from the same extrapolation to the small mo-mentum region. Despite the different masses of the leptons, the lowerplot of Fig. 1 shows that the relative statistical uncer-tainties are of the same orders of magnitude for all threeleptons and display a universal dependence on Q /m l .3.1 Contribution from up and down quarksThe light quark contribution is depicted in Fig. 2. Here,we compare a ud e with the result at the physical value ofthe pion mass obtained with the standard definitionEq. (1) on one ensemble [20,21] with only one latticespacing.For the extrapolation in the upper plot of Fig. 2 weinitially use a ud e from all our ensembles, i.e. all volumes,lattice spacings, and pion masses. As the figure shows,with the present accuracy of the data we do not resolveany statistically significant dependence of a ude on thelattice spacing or the lattice size. Using the modifieddefinition (Eq. 5) with H = m V the ansatz a ud e (cid:0) a, m , L (cid:1) = A + B m (11)is then already sufficient to describe our data. The re-sult of this extrapolation is shown as the light grey bandin Fig. 2. In principle, we are to add terms of higher or-der in m for the extrapolation formula in Eq. (11), a ud e (cid:0) a, m , L (cid:1) = A + B m + B m + . . . (12)to account for any non-linear dependence on m . Thedark grey band in the background in Fig. 2 shows theextrapolation with an additional term B m . The dif-ference between this extrapolation and the previous onelinear in the squared pion mass is insignificant. This in-significance of terms in m beyond the linear one whenusing the improved definition in Eq. (5) turns out to bea universal property of a f l for all leptons and all flavourcombinations f = ud, udsc considered here and below.In addition, we check explicitly for lattice artefacts inthe extrapolation by adding a term C a to Eq. (11)with the result shown in the lower plot of Fig. 2.We assume that lattice artefacts for the data at thephysical point are negligible as well, which gives meritto the observed compatibility of the physical point re-sult with the value determined by the linear extrapola-tion of the data described here.3.2 Adding the strange and the charm quarkcontributionsWhen incorporating the heavy, second-generation flavours,which are described by the Osterwalder-Seiler action [33, m π a ud e m (cid:2) GeV (cid:3) N f = 2 result a = 0 .
086 fm , L = 2 . a = 0 .
078 fm , L = 1 . a = 0 .
078 fm , L = 2 . a = 0 .
078 fm , L = 3 . a = 0 .
061 fm , L = 1 . a = 0 .
061 fm , L = 2 . m π a ud e m (cid:2) GeV (cid:3) a → N f = 2 result a = 0 .
061 fm , L = 1 . a = 0 .
061 fm , L = 2 . a = 0 .
078 fm , L = 1 . a = 0 .
078 fm , L = 2 . a = 0 .
078 fm , L = 3 . a = 0 .
086 fm , L = 2 . ude ( m PS , .
061 fm)a ude ( m PS , .
078 fm)a ude ( m PS , .
086 fm)
Fig. 2
Upper plot: light-quark contribution to a hvpe withfilled symbols representing points obtained with Eq. (5), opensymbols refer to those obtained with Eq. (1), i. e. H = 1. Inparticular, the two-flavour result at the physical point hasbeen computed with the standard definition. The light greyerrorband belongs to the linear fit, whereas the dark greyerrorband is attached to the quadratic fit. Lower plot: com-bined chiral and continuum extrapolation of the light-quarkcontribution to a hloe allowing for lattice artefacts.
34] and whose masses have been tuned to their physicalvalues as shown in [15], we take O ( a ) lattice artefactsinto account. The four-flavour result for a hvpe at thephysical point in the continuum limit is obtained fromsimultaneously extrapolating in the pion mass, m PS ,and to zero lattice spacing a using the ansatz a hvpe ( m PS , a ) = A + B m P S + C a . (13) A, B, C denote the free parameters of the fit. In thepresence of the strange and charm contributions to a hvpe ,the parameter C will also contain terms ∼ m c,R , m s,R from the renormalised charm and strange quark massand also receives contributions from lattice artefactspossibly present in the light quark contribution to a hvpe .The reason for omitting a linear term in a is that auto-matic O ( a ) improvement is retained for our definitionof the hadronic vacuum polarisation function at maxi- mal twist [26]. As we have discussed in [15], systematiceffects from varying the heavy valence and sea quarkmasses within the range given there have been foundto be negligible. This is partly due to the contributionof strange and charm quark current correlators to vac-uum polarisation being at least an order of magnitudesmaller than those from the light quarks.For a hvpe the corresponding fit is shown in Fig. 3.Again we use data for a hvpe for all lattice spacings, pionmasses, and lattice volumes in this extrapolation. Theansatz in Eq. (13) is sufficient for a good description ofall our data. This is shown by the three dashed lines,which evaluate the fit function for the individual lat-tice spacings. We have checked that amending the fitfunction by higher powers in m does not lead to sig-nificantly different results for the extrapolated value.Comparing the results for different lattice volumes forlattice spacing a = 0 .
078 fm in Fig. 3 suggests the ab-sence of observable finite volume effects. However, forthe compilation of our complete error budget we inves-tigate these effects in more detail below.Our result with only statistical uncertainty is a hvpe = 1 .
78 (06) · − . (14) a e ( m PS , .
086 fm)a e ( m PS , .
078 fm)a e ( m PS , .
061 fm) a = 0 .
086 fm, L = 2 . a = 0 .
078 fm, L = 3 . a = 0 .
078 fm, L = 2 . a = 0 .
078 fm, L = 1 . a = 0 .
061 fm, L = 2 . a = 0 .
061 fm, L = 1 . a → (cid:2) GeV (cid:3) a ud s ce m π Fig. 3
Chiral and continuum extrapolation of the N f =2+1+1 contribution to a hvpe . The inverted red triangle showsthe value extrapolated to the continuum and to the physicalvalue of the pion mass. It has been displaced to the left to fa-cilitate the comparison with the dispersive result in the blacksquare [35]. a hvpe given in Eq. (14). Wehave investigated finite size effects (FSE), the depen-dence of our chiral extrapolation on the incorporation of large pion masses, vector meson fit ranges, and thedependence of our results on different vacuum polarisa-tion fit functions. Moreover, for one ensemble the lightquark-disconnected contribution is quantified. As described in detail in Ref. [15], the N f = 2+1+1 en-sembles analysed in this work feature 3 . < m PS L < .
93, where L is the spatial extent of the lattice. Re-stricting our data to the condition m PS L > . m PS L > .
5, respectively, yields a hvpe ( m PS L > .
8) = 1 .
77 (07) · − , (15) a hvpe ( m PS L > .
5) = 1 .
83 (10) · − , (16)after combined chiral and continuum extrapolation. Thismatches the result given in Eq. (14) and thus indicatesthat FSE are negligible in our computation. This find-ing is supported by comparing the results of two en-sembles only differing in lattice size provided in Tab. 2.The numbers do not change when restricting the mo-menta of the larger ensemble to those of the smallerone. The FSE attributed to the lowest achievable mo-mentum being πL mixes with FSE entering the choice ofdifferent fit functions. We take a conservative approachand consider these effects separately. Ensemble (cid:0) La (cid:1) × Ta a hvpe , ud a hvpe B35.32 32 ×
64 1 . · − . · − B35.48 48 ×
96 1 . · − . · − Table 2
Comparison of light-quark contribution to a hvpe and total a hvpe from ensembles of different volumes. We have checked the validity of the chiral extrapola-tion by restricting the data, comprising pion masses be-tween 227 MeV and 491 MeV, to the condition m PS <
400 MeV. The value we obtain a hvpe = 1 .
78 (07) · − (17)only features a slightly larger uncertainty compared tothe result in Eq. (14). Thus, we do not assign a sys-tematic uncertainty to the usage of pion masses above400 MeV. Our standard computation involves the determinationof the masses and decay constants of the vector me-son ground states for the different flavours. Their val-ues depend on the choice of fit ranges. We have anal-ysed different fit ranges for the two-point functions ofthe light, strange, and charm vector currents and prop-agated the uncertainties to the values for a hvpe . Thisshowed that excited state contaminations are significantonly for m V and f V determined from the light vectorcurrent-current correlator. Variations of the standardfit ranges by 0 . a hvpe for the sγ µ s - and the J/ψ correlator. Furthermore,the heavy flavour contributions are approximately oneorder of magnitude smaller than the light quark contri-bution such that their systematic uncertainties wouldnot noticeably impact the overall uncertainty of a hvpe .In the upper plot of Fig. 4, the dependence of thelight quark contribution to the electron anomalous mag-netic moment on the fitrange for the ρ -correlator is plot-ted. The lower limit 0 . . . , . . , . ∆ V = 0 . · − . (18) The number of terms in the fit function Eq. (7) is givenby M and N. M1N2 is our standard choice. Repeatingthe whole analysis with different numbers of terms forthe light quark contribution leads to the results shownin the lower plot of Fig. 4. We observe that the chirallyand continuum extrapolated results of fit functions in-volving one and two poles are not compatible and thuswe assign a systematic error by taking half the differ-ence of the central values of the result of the M2N3 andthe M1N2 fit. This leads to a systematic uncertainty of ∆ ud MN = 0 . · − . (19)This results in the dominant systematic uncertainty ofthe determination of a hvpe . For the strange quark thesystematic uncertainty from different values of M andN is ∆ s MN = 0 . · − (20) Fig. 4
Dependence of a ude on the fitrange of the ρ -correlator(upper plot) and on values chosen for M, N in the vacuum po-larisation fit function (lower plot). The standard ρ -correlatorfit range is [0 . , . which we add to the light quark one. The differences ofresults from different fit functions for the charm quarkcontribution have turned out to be negligible such thatthe total systematic error originating from employingvarious numbers of terms in the fit function amounts to ∆ MN = 0 . · − . (21) Leaving out the quark-disconnected contributions is asystematic uncertainty we cannot completely quantify,yet. We have started investigating their magnitude onthe B55.32 ensemble mentioned already before. Usingthe local vector current we have detected a signal forthe light quark part of the vacuum polarisation functionwhen using 24 stochastic volume sources on 1548 con-figurations and 48 stochastic volume sources on 4996configurations. Employing the one-end trick [37], theisovector part Π µν ( x, y ) = (cid:104) J µ ( x ) J ν ( y ) (cid:105) (22)with J µ = χγ µ τ χ is significantly different from zero.However, this is a pure lattice artefact and will not -0.12-0.1-0.08-0.06-0.04-0.0200.02 0 0.2 0.4 0.6 0.8 1 1.2 Z V Π ( Q ) Q / GeV -0.00400.004 0 0.2 0.4 0.6 0.8 1connected, combined isospin componentsdisconnected, isospin 0 componentdisconnected, isospin 1 component Fig. 5
Comparison of the light quark contributions to theunsubtracted hadronic vacuum polarisation function fromquark-connected and disconnected diagrams of the local cur-rent correlator. Z V has been obtained from the ratio of theconnected part of the conserved and local current-current cor-relators. The values have been computed with the analyticalcontinuation method described in [36] without correcting forfinite-size effects. contribute in the continuum limit. On the other hand,the more interesting isoscalar part Π µν ( x, y ) = 19 (cid:104) J µ ( x ) J ν ( y ) (cid:105) (23)with J µ = χγ µ χ is compatible with zero. The con-nected and disconnected pieces of the polarisation func-tion for the light flavours are depicted in Fig. 5.A comparison of the values of a hvp l, ud for all three lep-tons on the B55.32 ensemble with and without incor-porating the disconnected contributions is presented inTab. 3. Here, we have combined the connected piecesobtained from the point-split current correlator withthe isoscalar part of the disconnected contributions ob-tained from the local current correlator using the renor-malisation constant Z V determined from the ratio ofthe connected pieces of the conserved and the localvector current two-point functions. Therefore and be-cause we only have results for one ensemble, the num-bers below can only give hints on the influence of thedisconnected pieces. We observe the tendency that forall three leptons a hvp l, ud decreases when incorporating thedisconnected contributions as has been predicted in [38].However, this is statistically not significant. Further-more, we find that the magnitude of the disconnectedcontributions is comparable to our current uncertainty.Hence, it will be mandatory to compute them whenaiming at more precise results. For the muon the valueshifts by ≈ −
5% given in [39] as well as more recenthigh-statistics evaluations in [40,41]. without disc with disc a hvpe , ud . · − . · − a hvp µ, ud . · − . · − a hvp τ, ud . · − . · − Table 3
Comparison of light-quark contributions to a hvp l with and without disconnected pieces in the low-momentumregion for the B55.32 ensemble. For all contributions the re-definition Eq. (5) and our standard analysis have been used. The disconnected heavy flavour contributions needto be considered as well. We plan to check their sizein future calculations. The pure charm quark contribu-tions have been computed in perturbation theory andshown to be suppressed by a factor (cid:16) q m c (cid:17) [42], where q is the relevant energy scale of the problem.3.4 Comparison with the phenomenological valueAdding the quantified systematic uncertainties in quadra-ture we obtain as final result a hvpe = 1 .
782 (64)(86) · − . (24)This can directly be compared with the phenomenolog-ical determination of [35] a hvpe = 1 .
866 (10) (05) · − . (25)They are fully compatible with each other although ourlattice result still is afflicted with larger errors. τ -lepton ( g − The large mass of the tau lepton, m τ ≈ . Q = 0 .
745 GeV . This is verydifferent from the peak position of the electron weightfunction. Hence, the saturation of a hvp τ requires datafrom a different part of the subtracted vacuum polari-sation function, in particular, also the high-momentumpiece of our fit function Eq. (8) is important here.4.1 Contribution from up an down quarksAs for the electron, we start off by showing the contribu-tion of the first-generation flavours to a hvp τ in the upperplot of Fig. 6. The data show a qualitatively similar be-haviour to those of the electron in Fig. 2. Their values differ, however, by six orders of magnitude. In partic-ular, by comparing the upper and lower plot of Fig. 6we find that no significant lattice artefacts are presentand that the data at unphysical pion masses obtainedwith Eq. (5), can be linearly extrapolated to the physi-cal point. This demonstrates again that the method ofincluding HH phys in the weight function is advantageousfor the chiral extrapolation. The value extrapolated inthis way and using all available lattice ensembles agreeswith our calculation directly at the physical pion massshown as the open square in Fig. 6. m π a ud e m (cid:2) GeV (cid:3) N f = 2 result a = 0 .
086 fm , L = 2 . a = 0 .
078 fm , L = 1 . a = 0 .
078 fm , L = 2 . a = 0 .
078 fm , L = 3 . a = 0 .
061 fm , L = 1 . a = 0 .
061 fm , L = 2 . m π a ud e m (cid:2) GeV (cid:3) a → N f = 2 result a = 0 .
061 fm , L = 1 . a = 0 .
061 fm , L = 2 . a = 0 .
078 fm , L = 1 . a = 0 .
078 fm , L = 2 . a = 0 .
078 fm , L = 3 . a = 0 .
086 fm , L = 2 . ude ( m PS , .
061 fm)a ude ( m PS , .
078 fm)a ude ( m PS , .
086 fm)
Fig. 6
Upper plot: light-quark contribution to a hvp τ withfilled symbols representing points obtained with Eq. (5), opensymbols refer to those obtained with Eq. (1), i. e. H = 1. Wenote that the two-flavour result at the physical point has beencomputed with the standard definition. The light grey error-band belongs to the linear fit (dotted black line), whereas thedark grey errorband is attached to the quadratic fit (solidgreen line). Lower plot: combined chiral and continuum ex-trapolation taking into account leading order lattice artefacts.0 a termin Eq. (13). As can be seen in Figs. 8 and 9, for the taulepton both, the strange and the charm contributiondo not show significant cut-off effects and hence, alsofor the total contribution a effects are small. We never-theless perform the continuum extrapolation in order touse exactly the same analysis strategy as for the otherleptons. a e ( m PS , .
086 fm)a e ( m PS , .
078 fm)a e ( m PS , .
061 fm) a = 0 .
086 fm, L = 2 . a = 0 .
078 fm, L = 3 . a = 0 .
078 fm, L = 2 . a = 0 .
078 fm, L = 1 . a = 0 .
061 fm, L = 2 . a = 0 .
061 fm, L = 1 . a → (cid:2) GeV (cid:3) a ud s c τ m π Fig. 7
Chiral and continuum extrapolation of the N f =2 + 1 + 1 contribution to a hvp τ . The inverted red triangleshows the value in the continuum limit at the physical valueof the pion mass. It has been displaced to the left to facilitatethe comparison with the dispersive result depicted as blacksquare [43]. Our four-flavour result with only statistical uncer-tainty reads a hvp τ = 3 .
41 (8) · − . (26)4.3 Systematic uncertaintiesWe have investigated the same systematic uncertaintiesfor our determination of a hvp τ as for the case of the elec-tron. The influence of the disconnected contributionshas already been discussed in the section of a hvpe . CL with linear fitdata at fixed m PS ≈
320 MeV a (cid:2) fm (cid:3) a s τ Fig. 8
Continuum limit of strange quark contribution to a hvp τ at approximately fixed pion mass. CL with linear fitdata at fixed m PS ≈
320 MeV a (cid:2) fm (cid:3) a c τ Fig. 9
Continuum limit of charm quark contribution to a hvp τ at approximately fixed pion mass. Restricting our data to the conditions m PS L > . m PS L > . a hvp τ ( m PS L > .
8) = 3 .
40 (09) · − , (27) a hvp τ ( m PS L > .
5) = 3 .
54 (13) · − . (28)This is compatible with the result in Eq. (26). Compar-ing again the two ensembles at m PS ≈
315 MeV whichonly differ in the extent of the lattices also indicatesnegligible finite size effects as shown in Tab. 4. Hence,we do not assign a FSE related systematic uncertainty.
Ensemble (cid:0) La (cid:1) × Ta a hvp τ, ud a hvp τ B35.32 32 ×
64 2 .
62 (06) · − .
40 (07) · − B35.48 48 ×
96 2 .
60 (06) · − .
41 (07) · − Table 4
Comparison of light-quark contribution to a hvp τ and total a hvp τ from ensembles of different volumes.1 Restricting the analysed ensembles to those featuringpion masses m PS <
400 MeV, we get a hvp τ = 3 .
45 (09) · − . (29)This is again compatible with the value given in Eq. (26).Hence, we do not assign a systematic uncertainty to thefact that ensembles with pion masses above 400 MeVhave been employed when extrapolating to the physicalvalue of the pion mass. The situation is similar to the case of the electron re-ported above. Only the excited state contamination inthe ρ -correlator has to be taken into account as sys-tematic uncertainty. In the upper plot of Fig. 10 thedependence of the light quark contribution, a ud τ , on thefit range chosen to extract the spectral information fromthe ρ -correlator is depicted. Fig. 10
Dependence of a ud τ on the fit range of the ρ -correlator (upper plot) and on the values chosen for M, N,B, and C in the vacuum polarisation fit function (lower plot).The standard ρ -correlator fit range is [0 . , . Taking half the difference of the central values ob-tained for [0 . , . . , . ∆ V = 0 . · − . (30) Due to the large Q we have to take the whole vac-uum polarisation function Eq. (9) into account, includ-ing in particular the high-momentum piece in Eq. (8).Thus, we have four different types of terms in the fitfunction that can have different numbers of summands.We only find observable differences in the light quarksector. But also here the results from different fits areall compatible as shown in the lower plot of Fig. 10.Conservatively, we take half the difference between theM2N3B4C1 and M1N2B4C1 fit and assign a systematicuncertainty of ∆ MNBC = 0 . · − (31)to our choice of the fit function.4.4 Comparison with the phenomenological valueIncluding the identified systematic uncertainties addedin quadrature, our final four-flavour result reads a hvp τ = 3 .
41 (8)(6) · − . (32)This agrees with the one obtained by a dispersive anal-ysis [43] a hvp τ = 3 .
38 (4) · − . (33)Compared to the electron, even better agreement be-tween the lattice and the phenomenological result isobserved for the τ -lepton. In this case, the uncertaintyof our twisted mass LQCD calculation is only abouttwice the phenomenological one. In this article we have presented the first four-flavourLQCD computation of the LO hadronic vacuum polar-isation contributions to the anomalous magnetic mo-ments of the electron and the τ -lepton. Our resultshave been obtained with N f = 2 + 1 + 1 twisted massfermions mostly at unphysically large pion masses but,at least for the light quark contribution, also directly atthe physical point. We find that for both, the electronand the tau lepton, the chirally extrapolated values forthe light quark contributions agree with the one at thephysical point. For our data at unphysically large values of the pionmass we have investigated the systematic uncertaintiesof the method used to obtain our final results. In par-ticular, we have addressed the effects of non-zero latticespacings, the finite volumes, the fit range for extractingthe vector meson mass, and using different fit functionsfor the vacuum polarisation function. As an additionaluncertainty we have investigated the disconnected con-tributions on one of our 4-flavor ensembles (B55.32) byusing the local vector current. This led to the first ob-servation of a signal for the disconnected diagrams dur-ing our calculations, which, however, is compatible withzero within our current errors and which we thereforehave neglected. This will no longer be justified once theuncertainties of the connected pieces are reduced and afull quantification of the quark-disconnected contribu-tion will become significant.Our final results are summarised in Tab. 5 belowand agree with the phenomenological determinations ofthe electron and tau lepton magnetic moments whichare also shown there. This universal agreement acrossall three leptons and thus distinct weightings of thesubtracted polarisation functions is elucidated by ourfindings in Ref. [44]. There it was shown, that the sub-tracted vacuum polarisation function itself calculatedwith the methods used in this work and described inmore detail in Ref. [15], is compatible with the phe-nomenological result for Π R ( Q ) in the range 0 ≤ Q ≤O (cid:0)
10 GeV (cid:1) . this work dispersive analyses a hvpe .
782 (64)(86) · − .
866 (10) (05) · − [35] a hvp µ .
78 (24)(16) · − .
91 (01) (05) · − [45] a hvp τ .
41 (8)(6) · − .
38 (4) · − [43] Table 5
Comparison of our first-principle values for a hvpe , a hvp µ , and a hvp τ with phenomenological results. As expected from the graph in the lower plot ofFig. 1 the relative statistical uncertainties in all threecases are similar. For the electron the systematic un-certainty already exceeds the statistical one.As in the case of the muon, also for the electronand tau lepton anomalous magnetic moments the errorsof our calculations are still larger than those from thedispersive analyses quoted above. However, it can beexpected that with future lattice QCD calculations atthe physical value of the pion mass, increased statisticsand an even better control over systematic uncertaintiesthe phenomenological error can be matched, if not evenbeaten, especially for the τ -lepton. Acknowledgements
We thank the European Twisted MassCollaboration (ETMC) for generating the gauge field ensem-bles used in this work. Special thanks goes to the authorsof [46] who generously granted us access to their data for thedisconnected contributions of the local vector current cor-relators. This work has been supported in part by the DFGCorroborative Research Center SFB/TR9. G.P. gratefully ac-knowledges the support of the German Academic NationalFoundation (Studienstiftung des deutschen Volkes e.V.) andof the DFG-funded Graduate School GK 1504. The numer-ical computations have been performed on the
SGI systemHLRN-II and the
Cray XC30 system HLRN-III at the HLRNSupercomputing Service Berlin-Hannover, FZJ/GCS, BG/P,and BG/Q at FZ-J¨ulich.
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