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The Muon G-2 Discrepancy: Errors Or New Physics?

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The Muon G-2 Discrepancy: Errors Or New Physics? The muon g-2 discrepancy: errors or new physics? † M. Passera∗, W. J. Marciano and A. Sirlin∗∗ ∗Istituto Nazionale Fisica Nucleare, Sezione di Padova, I-35131, Padova, Italy †Brookhaven National Laboratory, Upton, New York 11973, USA ∗∗Department of Physics, New York University, 10003 New York NY, USA Abstract. After a brief review of the muon g 2 status, we discuss hypothetical errors in the Standard Model prediction that could explain the present discrepancy with the− experimental value. None of them looks likely. In particular, an hypothetical + increase of the hadroproduction cross section in low-energy e e− collisions could bridge the muon g 2 discrepancy, but is shown to be unlikely in view of current experimental error estimates. If, nonetheless, this turns out to− be the explanation of the discrepancy, then the 95% CL upper bound on the Higgs boson mass is reduced to about 130 GeV which, in conjunction with the experimental 114.4 GeV 95% CL lower bound, leaves a narrow window for the mass of this fundamental particle. Keywords: Muon anomalous magnetic moment, Standard Model Higgs boson PACS: 13.40.Em, 14.60.Ef, 12.15.Lk, 14.80.Bn SM 11 INTRODUCTION aµ = 116591778(61) 10− . The difference with the EXP× 11 experimental value aµ = 116592080(63) 10− [1] The anomalousmagnetic momentof the muon, aµ ,isone EXP SM 11 × is ∆aµ = aµ aµ =+302(88) 10− , i.e., 3.4σ (all of the most interesting observables in particle physics. errors were added− in quadrature).× Similar discrepan- Indeed, as each sector of the Standard Model (SM) con- HLO cies are found employing the aµ values reported in tributes in a significant way to its theoretical prediction, Refs. [5, 8, 9]. For recent reviews of aµ see [7, 14, 15]. the precise aµ measurement by the E821 experiment at HLO The term aµ can alternatively be computed in- Brookhaven [1] allows us to test the entire SM and scru- corporating hadronic τ-decay data, related to those of tinize viable “new physics” appendagesto this theory [2]. + hadroproduction in e e− collisions via isospin symme- The SM prediction of the muon g 2 is conve- − try [16, 17]. Unfortunately there is a large difference be- niently split into QED, electroweak (EW) and hadronic + τ HLO SM tween the e e−- and -based determinations of aµ , (leading- and higher-order) contributions: aµ = even if isospin violation corrections are taken into ac- QED EW HLO HHO aµ +aµ +aµ +aµ . The QED prediction, computed count [18]. The τ-based value is significantly higher, up to four (and estimated at five) loops, currently stands leading to a small ( 1σ) ∆aµ difference. As the e+e QED − 11 ∼ HLO at aµ = 116584718.10(16) 10− [3], while the EW ef- data are more directly related to the aµ calculation than EW × 11 fects provide aµ = 154(2) 10− [4]. The latest calcu- the τ ones, all recent analyses do not include the latter. lations of the hadronic leading-ordercontribution,via× the Also, we note that recently studied additional isospin- + hadronic e e− annihilation data, are in good agreement: breaking corrections somewhat reduce the difference be- HLO 11 11 aµ = 6909(44) 10− [5], 6894(46) 10− [6, 7], tween these two sets of data (lowering the τ-based deter- 11× × 11 arXiv:0809.4062v1 [hep-ph] 24 Sep 2008 6921(56) 10− [8], and 6944(49) 10− [9]. The mination) [19, 20], and a new analysis of the pion form × × + higher-order hadronic term is further divided into factor claims that the τ and e e− data are consistent af- HHO HHO HHO two parts: aµ = aµ (vp) + aµ (lbl). The first ter isospin violation effects and vector meson mixings one, 98(1) 10 11[6], is the O(α3) contribution are considered [21]. − σ of diagrams− containing× hadronic vacuum polarization The 3.4 discrepancy between the theoretical predic- α3 tion and the experimental value of the muon g 2 can be insertions [10]. The second term, also of O( ), is the − hadronic light-by-light contribution; as it cannot be explainedin several ways. It could be due, at least in part, determined from data, its evaluation relies on specific to an error in the determination of the hadronic light-by- models. Recent determinations of this term vary between light contribution. However, if this were the only cause HHO 11 11 of the discrepancy, aµ (lbl) would have to move up by 80(40) 10− [11] and 136(25) 10− [12]. The most recent one,× 110(40) 10 11[13], lies× between them. If we many standard deviations to fix it – roughly eight, if we − HHO HLO× use the a lbl result of Ref. [13] (which includes all add this result to aµ , for example the value of Ref. [6] µ ( ) (which also provides the hadronic contribution to the ef- known uncertainties), and more than ten if the estimate fective fine-structure constant, later required for our dis- of Ref. [12] is employed instead. Although the errors as- HHO cussion), and the rest of the SM contributions, we obtain signed to aµ (lbl) are only educated guesses, this solu- su tion seems unlikely, at least as the dominant one. us define ai = ds fi(s)σ(s) (i = 1,2), where the up- 4mπ2 Another possibility is to explain the discrepancy ∆aµ R 2 per limit of integration is su < MZ , and the kernels are via the QED, EW and hadronic higher-order vacuum po- 3 2 2 2 f1(s)= K(s)/(4π ) and f2(s) = [M /(M s)]/(4απ ). larization contributions; this looks very improbable, as Z Z − The integrals ai with i = 1,2 provide the contributions one can immediately conclude inspecting their values (5) to aHLO and ∆α (M ), respectively, from 4m2 up to s and uncertainties reported above. If we assume that the µ had Z π u (see Eqs. (1,2)). An increase of the cross section σ(s) g 2 experiment E821 is correct, we are left with two − of the form ∆σ(s)= εσ(s) in the energy range √s options: possible contributions of physics beyond the δ δ ε ∈ [√s0 /2,√s0 + /2], where is a positive constant SM, or an erroneous determination of the leading-order − δ δ HLO and 2mπ + /2 < √s0 < √su /2, increases a1 by hadronic contribution aµ (or both). The first of these δ − ∆ δ ε ε √s0+ /2 σ 2 2 two explanations has been extensively discussed in the a1(√s0, , ) = δ 2t (t ) f (t )dt. If we as- √s0 /2 literature; following Ref. [22] we will study whether the sume that the muonRg −2 discrepancy is entirely due to σ − ∆ δ ε ∆ second one is realistic or not, and analyze its implications this increase in (s), so that a1(√s0, , )= aµ , the for the EW bounds on the mass of the Higgs boson. ∆α (5) corresponding increase in had(MZ) is δ √s0+ /2 g t2 σ t2 t dt √s δ /2 ( ) ( ) ∆a (√s ,δ)= ∆aµ R 0− . (3) ERRORS IN THE HADRONIC CROSS 2 0 s δ 2 √ 0+ / f (t2)σ(t2)t dt SECTION? √s δ /2 R 0− The shifts ∆a (√s ,δ) were studied in Ref. [22] for The hadronic leading-order contribution aHLO can be 2 0 µ several bin widths δ and central values √s . computed via the dispersion integral [23] 0 The present global fit of the LEP Electroweak ∞ Working Group (EWWG) leads to the Higgs bo- HLO 1 aµ = dsK(s)σ(s), (1) +34 3 son mass MH = 84 GeV and the 95% confi- 4π Z4m2 26 π − UB dence level (CL) upper bound MH 154 GeV [26]. + ≃ where σ(s) is the total cross section for e e− annihila- This result is based on the recent preliminary top tion into any hadronic state, with extraneous QED cor- quark mass Mt = 172.4(1.2) GeV [27] and the value ∆α (5) rections subtracted off, and s is the squared momentum had(MZ )= 0.02758(35) [28]. The LEP direct-search LB transfer. The well-known kernel function K(s) (see [24]) 95%CL lower bound is MH = 114.4 GeV [29]. Although UB is positive definite, decreases monotonically for increas- the global EW fit employs a large set of observables, MH 2 ing s and, for large s, behaves as mµ /(3s) to a good is strongly driven by the comparison of the theoretical approximation. About 90% of the total contribution to predictions of the W boson mass and the effective EW HLO 2θ lept aµ is accumulated at center-of-mass energies √s be- mixing angle sin eff with their precisely measured HLO low 1.8 GeV and roughly three-fourthsof aµ is covered values. Convenient formulae providing the MW and 2 lept (5) by the two-pion final state which is dominated by the sin θ SM predictions in terms of MH , Mt , ∆α (MZ ), ρ + eff had (770) resonance [17]. Exclusive low-energy e e− cross and αs(MZ ), the strong coupling constant at the scale sections were measured at colliders in Frascati, Novosi- MZ, are given in [30]. Combining these two predic- birsk, Orsay, and Stanford, while at higher energies the tions via a numerical χ2-analysis and using the present total cross section was determined inclusively. world-average values MW = 80.399(25) GeV [31], EXP Let’s now assume that the discrepancy ∆aµ = aµ 2θ lept sin eff =0.23153(16) [32], Mt =172.4(1.2) GeV [27], SM 11 − aµ =+302(88) 10− , is due to – and only to – hy- α (M ) = 0.118(2) [33], and the determination × s Z pothetical errors in σ(s), and let us increase this cross ∆α (5) +37 had(MZ )= 0.02758(35) [28], we get MH = 89 27 GeV section in order to raise aHLO, thus reducing ∆aµ .
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