The electroweak contributions to (g-2)_μ after the Higgs boson mass measurement
aa r X i v : . [ h e p - ph ] S e p The electroweak contributions to ( g − µ after theHiggs boson mass measurement C. Gnendiger, D. St¨ockinger, H. St¨ockinger-Kim
Institut f¨ur Kern- und Teilchenphysik, TU Dresden, Dresden, Germany
Abstract
The Higgs boson mass used to be the only unknown input parameter ofthe electroweak contributions to ( g − µ in the Standard Model. It entersat the two-loop level in diagrams with e.g. top loops, W- or Z-exchange.We re-evaluate these contributions, providing analytic expressions and ex-act numerical results for the Higgs boson mass recently measured at theLHC. Our final result for the full Standard Model electroweak contribu-tions is (153 . ± . × − , where the remaining theory error comes fromunknown three-loop contributions and hadronic uncertainties. The anomalous magnetic moment a µ = ( g − µ / : a exp µ = (116 592 089 ± × − . (1)This measurement has already reached a sensitivity to details of the weak in-teractions, which contribute at the order 10 − . Future experiments planned atFermilab [2] and J-PARC [3] aim to further reduce the uncertainty by a factor 4.The Standard Model theory prediction has also been continuously improv-ing, see Refs. [4, 5] for recent reviews and references. The 5-loop QED con-tribution has been completely calculated [6]. The hadronic vacuum polar-ization contributions make use of the most recent experimental data on the( e + e − → hadrons) cross section [7–9], and an earlier discrepancy to analy-ses based on τ -decays has been resolved [9, 10]. The latest results of variousgroups for the hadronic light-by-light contributions agree within the quotederrors [4, 11], and new non-perturbative approaches promise further progress[12, 13]. The change in the number compared to Ref. [1] is due to a new PDG value for themagnetic moment ratio of the muon to proton, see e.g. Ref. [2] g − µ in the StandardModel. They include contributions from the Higgs boson and are the only oneswhich depend on the Higgs boson mass M H . This quantity used to be the onlyunknown input parameter of the Standard Model, resulting in the dominantremaining theory uncertainty of the electroweak contributions. As a reference,the seminal evaluation of Ref. [14] obtained the result a EW µ = (154 ± ± × − , (2)where the first error is due to hadronic uncertainties, but the second is due tothe unknown Higgs boson mass.Now, the Higgs boson mass has been measured at the LHC to be M H =125 . ± . . ) +0 . − . (syst . ) GeV by ATLAS [15] and M H = 125 . ± . . ) ± . . ) GeV by CMS [16]. In the following we take the average central valueand a conservative error band, covering the 2 σ range of both measurements: M H = 125 . ± . . (3)Given this progress on all fronts regarding ( g − µ and the Higgs bosonit is appropriate to update the prediction of the electroweak contributions to( g − µ .In the present paper we therefore re-evaluate the electroweak StandardModel contributions at the two-loop level, making use of the LHC result. Weprovide the full M H -dependent part in numerical and, where not readily avail-able, in analytical form. This allows us to obtain the exact ( g − µ predictionfor the measured value of M H , and to compare with previously published resultsand error estimates. We combine this with the most advanced computations ofall other electroweak contributions up to leading 3-loop order and provide thefinal result and a complete discussion of the remaining theory error.In the following our input parameters besides Eq. (3) are [17]: m µ = 105 . ± . , (4a) M Z = 91 . ± . , (4b) m t = 173 . ± . ± . G F = (1 .
166 378 7 ± .
000 000 6) × − GeV − , (5a) α = 1 / .
035 999 (5b)for the muon decay constant and the fine-structure constant. Given these pa-rameters the W-boson mass is predicted by the Standard Model theory [18].We obtain M W = 80 . ± .
013 GeV . (6) In Ref. [19] another top quark mass has been used: m t = 173 . ± . M W equals 80 . ± .
010 GeV. µH W (a) γ µ µH γ , Z (b) f γ µ µZ γ (c) f γ µ µµ µZ γf (d) γ Figure 1: Sample two-loop diagrams: Higgs-dependent bosonic (a) andfermionic (b) diagram, diagram with γγZ -fermion triangle (c) and γ – Z mixing(d).The Standard Model electroweak contributions are split up into one-loop,two-loop and higher orders as a EW µ = a EW(1) µ + a EW(2) µ ;bos + a EW(2) µ ;ferm + a EW( ≥ µ , (7)where the two-loop contributions are further split into bosonic and fermioniccontributions, as discussed below.The one-loop contribution is given by [4, 5] a EW(1) µ = G F √ m µ π (cid:20)
53 + 13 (1 − s W ) (cid:21) = (194 . ± . × − , (8)where s W = 1 − M W /M Z is the square of the weak mixing angle in the on-shell renormalization scheme. One-loop contributions suppressed by m µ /M Z or m µ /M H are smaller than 10 − and hence neglected here. The parametrizationin terms of G F already absorbs important higher-order contributions. The errorin Eq. (8) is due to the uncertainty of the input parameters, in particular ofthe W-boson mass.Before discussing higher-order contributions we briefly explain possibleparametrizations in terms of G F and α . The one-loop contribution in Eq. (8)has been parametrized in terms of G F . Generally, n -loop contributions areproportional to G F α ( n − , and it is possible to reparametrize α in terms ofother quantities. Possibilities are to replace α by a running α at the scaleof the muon mass or the Z-boson mass, or to replace α → α ( G F ), where α ( G F ) ≡ √ G F s W M W /π = α × (1 + ∆ r ). The quantity ∆ r summarizes radia-tive corrections to muon decay. Different choices amount to differences whichare formally of the order n + 1. We will always choose α in the Thomson limit,i.e. given by Eq. (5b).We now turn to the first set of contributions with noticeable dependenceon the Higgs boson mass: the bosonic two-loop contributions a EW(2) µ ;bos . Theyare defined by two-loop and associated counterterm diagrams without a closedfermion loop, see Fig. 1(a) for a sample diagram. They are conceptually In the literature sometimes the experimental value for M W instead of the theory value isused. If we would use the current value of M W = 80 . ± .
015 GeV [17] instead of Eq. (6),the result would be shifted to a EW(1) µ = (194 . ± . × − . - - - - - - - M H @ GeV D a Μ ; bo s E W H L @ - D only logexact 122 124 126 128 130 - - - - - - - - M H @ GeV D a Μ ; bo s E W H L @ - D Figure 2: Numerical result for a EW(2) µ ;bos as a function of the Higgs boson mass.The vertical band indicates the measured value of M H . The dashed line inthe left plot corresponds to the leading logarithmic approximation as definedin Ref. [21]. In the right plot the dotted, solid, dashed lines correspond to avariation of M W by ( − , , +15) MeV, respectively.straightforward but involve many diagrams. Their first full computation inRef. [20] was a milestone — the first full computation of a Standard Modelobservable at the two-loop level. Actually, Ref. [20] employed an approxima-tion assuming M H ≫ M W . Ref. [21] confirmed the result but provided the full M H -dependence; Ref. [22] then published the result in semianalytical form.Here we re-evaluate the bosonic two-loop contributions using theparametrization discussed above, in terms of G F α . Fig. 2 shows the resultfor a range of Higgs boson masses. The numerical result differs by around 3%from the one given in Ref. [21], where the G F α ( G F ) parametrization was cho-sen. The measured value of M H now fixes the value of these contributions andwe obtain a EW(2) µ ;bos = ( − . ± . × − . (9)Here the remaining parametric uncertainty results from the experimental un-certainties of the input parameters M H , and to a smaller extent of M W , seethe right plot in Fig. 2. The result lies within the intervals given in the originalRefs. [21,22] and the recent reviews [4,5], which all differ slightly because of thedifferent Higgs boson mass ranges and central values used for the evaluations.The fermionic two-loop contributions a EW(2) µ ;ferm are defined by Feynman dia-grams with a closed fermion loop. The Higgs boson enters through diagramsof the type of Fig. 1(b), where a fermion loop generates a Hγγ or HγZ inter-action. The fermionic contributions involve also light quark loops, e.g. in thediagrams of Fig. 1(c), for which perturbation theory is questionable. Hence wesplit up these contributions further, slightly extending the notation of Ref. [5]: a EW(2) µ ;ferm = a EW(2) µ ( e, µ, u, c, d, s ) + a EW(2) µ ( τ, t, b ) + a EW(2) µ ;f-rest,H + a EW(2) µ ;f-rest,no H . (10)4ere the first two terms on the r.h.s. denote contributions from the diagramsof Fig. 1(c) with a γγZ -subdiagram and the indicated fermions in the loop.The third term denotes the Higgs-dependent diagrams of Fig. 1(b); the fourthcollects all remaining fermionic contributions, e.g. from W-boson exchange orfrom diagram Fig. 1(d).We first focus on the Higgs-dependent part, for which we write a EW(2) µ ;f-rest,H = X f h a EW(2) µ ;f-rest,H γ ( f ) + a EW(2) µ ;f-rest,HZ ( f ) i , (11)where the two terms in the sum denote the Higgs-dependent diagrams ofFig. 1(b) with either a photon or a Z-boson in the outer loop and the sumextends over the Standard Model fermions; the relevant ones are f = t, b, c, τ .Contributions from the remaining Standard Model fermions are below 10 − and thus negligible.The first full computation of the fermionic contributions, including the Higgsdependence was carried out in Ref. [23]. There, the dependence on the Higgs bo-son mass is provided in three limiting cases, M H ≪ m t , M H = m t , M H ≫ m t .Furthermore, since s W ≈ /
4, terms suppressed by a factor (1 − s W ), in partic-ular the entire Higgs–Z diagrams of Fig. 1(b) were neglected. Diagrams similarto Fig. 1(b) have also been evaluated in the more complicated case of extendedmodels, e.g. in the Two-Higgs-doublet model and the supersymmetric StandardModel [24, 25].We computed the Higgs-dependent diagrams without approximations in twoways: with the technique developed for Ref. [21,26] using asymptotic expansionand integral reduction techniques, and with the method of Barr and Zee, wherethe inner loop is computed first and then inserted into the outer loop [27]. Theresult from this is a EW(2) µ ;f-rest,H γ ( f ) = G F √ m µ π απ N C Q f f Hγ ( x fH ) , (12) a EW(2) µ ;f-rest,HZ ( f ) = G F √ m µ π απ N C Q f I f − s W Q f c W s W (1 − s W ) f HZ ( x fH , x fZ ) , (13)with x fH = m f /M H and x fZ = m f /M Z . The loop functions can be written interms of one-dimensional integral representations or in terms of dilogarithms: f Hγ ( x ) = Z dw x w − w + 1 w − w + x log w (1 − w ) x (14)= x [ f H ( x ) − , (15) f HZ ( x, z ) = Z dw x z w − w + 1 w − w + z " log w (1 − w ) x w − w + x + log xz x − z (16)= x zx − z [ f H ( z ) − f H ( x )] . (17)5 æ æ
50 100 150 200 250 300 350 - - - - - M H @ GeV D a Μ ; f - r e s t , H E W H L @ - D æ approx .exact 122 124 126 128 130 - - - - - - - - M H @ GeV D a Μ ; f - r e s t , H E W H L @ - D Figure 3: Numerical result for a EW(2) µ ;f-rest,H as a function of the Higgs boson mass.The vertical band indicates the measured value of M H . The fat dots in the leftplot correspond to the approximations for M H = 60 GeV , m t ,
300 GeV givenin Ref. [23]. In the right plot the dotted, solid, dashed lines correspond to avariation of m t by ( − . , , +1 .
4) GeV, respectively.The dilogarithms are contained in the function f H ( x ), defined as f H ( x ) = 4 x − y (cid:20) Li (cid:18) − − y x (cid:19) − Li (cid:18) − y x (cid:19)(cid:21) − x, (18)with y = √ − x . Further, the weak isospin I f is defined as ± for up (down)fermions, and the electric charge Q f equals + , − , − N C is 1for leptons and 3 for quarks.Fig. 3(a) shows the numerical result as a function of the Higgs boson massand compares with the numerical values obtained in Ref. [23], using their ap-proximations. We find that the approximation for large M H is surprisinglypoor. As a check of this case, we have explicitly computed the higher orders inthe expansion in m t /M H and verified that the terms neglected in Ref. [23] areimportant.Inserting the measured value of the Higgs boson mass, and taking into ac-count all contributions including top, bottom, charm and τ loops and diagramswith Higgs and Z-boson exchange, we obtain a EW(2) µ ;f-rest,H = ( − . ± . × − , (19)where the indicated error arises essentially from the uncertainty of the inputparameters m t and M H . Again, the result is in agreement with the intervalsgiven in Refs. [4,5,23], which differ because of the different allowed Higgs bosonmass ranges.Eqs. (9), (12)–(19) and Figs. 2 and 3 constitute our main new results. Inthe following we briefly review the remaining electroweak contributions, withslight updates. In Ref. [28], Eq.(70), a similar function f S ( x ) is defined, where f S ( x ) = xf H ( x ) − x .Additionally, Eqs. (15), (17) are connected by f Hγ ( x ) = lim z →∞ f HZ ( x, z ). a EW(2) µ ;f-rest,no H are given by: a EW(2) µ ;f-rest,no H = − G F √ m µ π απ (cid:20) s W (cid:18) m t M W + log m t M W + 73 (cid:19) (cid:21) − G F √ m µ π απ (cid:20) c W s W m t M W (cid:0) − s W (cid:1) (cid:21) − G F √ m µ π απ (cid:20) (cid:18)
89 log M Z m µ + 49 log M Z m τ (cid:19) (cid:0) − s W (cid:1) + 43 × . (cid:0) − s W (cid:1) (cid:21) . (20)The first line has been computed in Ref. [23] and was re-written in this forme.g. in Ref. [4, 29]; the other two terms correspond to additional terms addedin Ref. [14], where however no explicit formula was provided. These terms aresuppressed by (1 − s W ) but enhanced by either m t /M W or by large logarithms.The factor m t /M W enters via the quantity ∆ ρ , which arises by applying therenormalization s W → s W + δs W in the (1 − s W ) -term of the one-loop re-sult (8). The other term originates from diagrams with γ – Z mixing as shownin Fig. 1(d) with light fermions running in the loop. It can be computed us-ing renormalization-group techniques [14, 30]. The number 6 .
88 in the last linehas been obtained in Ref. [14] as a nonperturbative replacement of the pertur-bative expression 2 / P q = u,d,s,c,b N c (cid:0) I q Q q − Q q s W (cid:1) log M Z /m q . Numerically,we obtain − . , − . , − .
29 in units of 10 − for the three contributions, intotal a EW(2) µ ;f-rest,no H = ( − . ± . × − . (21)The error due to the uncertainty of the input parameters is negligible; the givenerror is our estimate of the still neglected terms which are suppressed by a factor(1 − s W ) or M Z /m t and not enhanced by anything. The estimate is obtainedby comparison with the computed terms in the second and third line of Eq.(20) and the respective enhancement factors.For the third generation contributions to Fig. 1(c) perturbation theory canbe applied, and these contributions have been evaluated in Refs. [14, 23, 31].The result and the error estimate from Ref. [14], including subleading terms in m t /M Z , read a EW(2) µ ( τ, t, b ) = − (8 . ± . × − . (22)We have re-evaluated these contributions for various definitions of quark masseswhich differ by higher orders in the strong interaction, similarly to the errorestimation by Ref. [14]. The result is shown in Fig. 4, and it confirms that Eq.(22) is still compatible with present values of quark masses.The contribution of the first two generations to Fig. 1(c) has first been fullycomputed in Ref. [23], approximating the light quark contributions by a naiveperturbative calculation with constituent-like quark masses. The treatment ofthe light quark contributions has been successively improved in later references,7 .0 3.5 4.0 4.5 5.0 - - - - - - - m b @ GeV D a Μ E W H L H Τ , t , b L @ - D Figure 4: Numerical result for a EW(2) µ ( τ, t, b ) as a function of the bottom quarkmass, for various values of the top quark mass. The dash-dotted line corre-sponds to the MS mass m t = 160 GeV; the solid, dotted, dashed lines to thepole mass m t = 173 . ∓ . m b ( m b ) = 4 .
18 GeV and the 1S-mass4 .
65 GeV [17]; the MS mass at higher scales has smaller values. The horizontalgray band corresponds to the result (22).by taking into account non-perturbative information on the longitudinal [31,32],then on both the longitudinal and transverse parts of the γγZ three-pointfunction [14]. The final result of Ref. [14] is a EW(2) µ ( e, µ, u, c, d, s ) = − (6 . ± . ± . × − , (23)where the uncertainties for the 1st and 2nd generation have been given sepa-rately.Contributions from beyond the two-loop level have been considered inRefs. [14, 30]. There, the leading logarithms at the three-loop level have beenobtained from renormalization-group methods. It was found that these loga-rithms amount to 0 . × − , if the two-loop result is parametrized in terms of G F α ( m µ ), where α ( m µ ) is the running fine-structure constant at the scale ofthe muon mass. If the two-loop result is parametrized in terms of G F α , how-ever, the shift of the coupling accidentally cancels the three-loop logarithms.Hence, since this is the parametrization we have used, we take a EW( ≥ µ = (0 ± . × − , (24)where the error estimate is from Ref. [14]. It corresponds to estimating thenon-leading logarithmic three-loop contributions to be below a percent of thetwo-loop contributions.In summary, we have re-evaluated the electroweak contributions to a µ using the measured Higgs boson mass and employing consistently the G F α The result is taken from the erratum of Ref. [14]. It is perfectly compatible with the oneprovided in Ref. [4]. The result quoted in Ref. [5] was taken from the original Ref. [14]; itdiffers slightly but is also compatible within the errors. M H @ GeV D a Μ E W @ - D Figure 5: Numerical result for a EW µ as a function of the Higgs boson mass. Thevertical band indicates the measured value of M H . The dashed lines correspondto the uncertainty of the final result, quoted in Eq. (25).parametrization at the two-loop level. We provide exact numerical results forthe full bosonic and the Higgs-dependent fermionic two-loop contributions, forthe latter also analytical results. These results are supplemented by updatesof the most advanced available results on all other electroweak contributions.Our final result obtained from Eqs. (8), (9), (19), (21), (22), (23), (24) reads a EW µ = (153 . ± . × − (25)and is illustrated in Fig. 5. We assess the final theory error of these contribu-tions to be ± . × − . This is the same value as the one given in Ref. [14]for the overall hadronic uncertainty from the diagrams of Fig. 1(c), which isnow by far the dominant source of error of the electroweak contributions. Theerror from unknown three-loop contributions and neglected two-loop terms sup-pressed by M Z /m t and (1 − s W ) is significantly smaller and the error due tothe experimental uncertainty of the Higgs boson, W-boson, and top-quark massis well below 10 − and thus negligible.Our result is consistent with the previous evaluations of the elec-troweak contributions in Refs. [4, 5, 14], whose central values range between(153 . . . × − , but the large uncertainty due to the unknown Higgs bo-son mass has been reduced. In comparison, the recent 5-loop calculation [6] hasshifted the QED result by +0 . × − . We can now combine Eq. (25) and theresult of Ref. [6] with the hadronic contributions. We take the recent leadingorder evaluations of Refs. [7] and [8] and the higher order results of Refs. [8,11].The resulting difference between the experimental result Eq. (1) and the fullStandard Model prediction is: a exp µ − a SM µ = ( (287 ± × − [7] , (261 ± × − [8] . (26)The Standard Model theory error remains dominated by the non-electroweak hadronic contributions. The QED and electroweak contributions9an now be regarded as sufficiently accurate for the precision of next generationexperiments. Acknowledgements
Communications with A. Czarnecki, E. de Rafael and B. Lee Roberts are grate-fully acknowledged. This work has been supported by the German ResearchFoundation DFG through Grant No. STO876/1-1.
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