The Fermi constant from muon decay versus electroweak fits and CKM unitarity
Andreas Crivellin, Martin Hoferichter, Claudio Andrea Manzari
CCERN-TH-2021-017, PSI-PR-21-02, ZU-TH 04/21
The Fermi constant from muon decay versus electroweak fits and CKM unitarity
Andreas Crivellin,
1, 2, 3
Martin Hoferichter, and Claudio Andrea Manzari
2, 3 CERN Theory Division, CH–1211 Geneva 23, Switzerland Physik-Institut, Universit¨at Z¨urich, Winterthurerstrasse 190, CH–8057 Z¨urich, Switzerland Paul Scherrer Institut, CH–5232 Villigen PSI, Switzerland Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics,University of Bern, Sidlerstrasse 5, CH–3012 Bern, Switzerland
The Fermi constant ( G F ) is extremely well measured through the muon lifetime, defining oneof the key fundamental parameters in the Standard Model (SM). However, to search for physicsbeyond the SM (BSM), it is the precision of the second-best independent determination of G F thatdefines the sensitivity. The best alternative extractions of G F proceed via the global electroweak(EW) fit or from superallowed β decays in combination with the Cabibbo angle measured in kaon, τ , or D decays. Both variants display some tension with G F from muon decay, albeit in oppositedirections, reflecting the known tensions within the EW fit and hints for the apparent violation ofCKM unitarity, respectively. We investigate how BSM physics could bring the three determinationsof G F into agreement using SM effective field theory and comment on future perspectives. I. INTRODUCTION
The numerical value of the Fermi constant G F is con-ventionally defined via the muon lifetime within the SM.Even though this measurement is extremely precise [1–3] G µF = 1 . × − GeV − , (1)at the level of 0 . G µF to another independentdetermination. This idea was first introduced by Mar-ciano in Ref. [4], concentrating on Z -pole observables andthe fine-structure constant α . In addition to a lot of newdata that has become available since 1999, another op-tion already mentioned in Ref. [4]—the determination of G F from β and kaon decays using CKM unitarity—hasbecome of particular interest due to recent hints for the(apparent) violation of first-row CKM unitarity. Thesedevelopments motivate a fresh look at the Fermi con-stant, in particular on its extraction from a global EWfit and via CKM unitarity, as to be discussed in the firstpart of this Letter.The comparison of the resulting values for G F showsthat with modern input these two extractions are close inprecision, yet still lagging behind muon decay by almostthree orders of magnitude. Since the different G F deter-minations turn out to display some disagreement beyondtheir quoted uncertainties, the second part of this Let-ter is devoted to a systematic analysis of possible BSMcontributions in SM effective field theory (SMEFT) [5, 6]to see which scenarios could account for these tensionswithout being excluded by other constraints. This isimportant to identify BSM scenarios that could be re-sponsible for the tensions, which will be scrutinized withforthcoming data in the next years. II. DETERMINATIONS OF G F Within the SM, the Fermi constant G F is defined by,and is most precisely determined from, the muon life-time [2] 1 τ µ = ( G µF ) m µ π (1 + ∆ q ) , (2)where ∆ q includes the phase space, QED, and hadronicradiative corrections. The resulting numerical value inEq. (1) is so precise that its error can be ignored in thefollowing. To address the question whether G µF subsumesBSM contributions, however, alternative independent de-terminations of G F are indispensable, and their precisionlimits the extent to which BSM contamination in G µF canbe detected.In Ref. [4], the two best independent determinationswere found as G Z(cid:96) + (cid:96) − F = 1 . (cid:0) +11 − (cid:1) × − GeV − ,G (3) F = 1 . (cid:0) +18 − (cid:1) × − GeV − , (3)where the first variant uses the width for Z → (cid:96) + (cid:96) − ( γ ),while the second employs α and sin θ W , together withthe appropriate radiative corrections. Since the presentuncertainty in Γ[ Z → (cid:96) + (cid:96) − ( γ )] = 83 . G Z(cid:96) + (cid:96) − F = 1 . × − GeV − (4)does not lead to a gain in precision, but the shift in thecentral value improves agreement with G µF . The secondvariant, G (3) F , is more interesting, as here the main lim-itation arose from the radiative corrections, which haveseen significant improvements regarding the input valuesfor the masses of the top quark, m t , the Higgs boson, M H , the strong coupling, α s , and the hadronic runningof α . In fact, with all EW parameters determined, it now a r X i v : . [ h e p - ph ] F e b M W [GeV] [7] 80 . W [GeV] [7] 2 . W → had) [7] 0 . W → lep) [7] 0 . θ eff(QFB) [7] 0 . θ eff(Tevatron) [27] 0 . θ eff(LHC) [28–31] 0 . Z [GeV] [9] 2 . σ h [nb] [9] 41 . P pol τ [9] 0 . A (cid:96) [9] 0 . R (cid:96) [9] 20 . A ,(cid:96) FB [9] 0 . R b [9] 0 . R c [9] 0 . A ,b FB [9] 0 . A ,c FB [9] 0 . A b [9] 0 . A c [9] 0 . α × [7] 7 . α had × [15, 16] 276 . . α s ( M Z ) [7, 32] 0 . M Z [GeV] [7, 33–36] 91 . M H [GeV] [7, 37–39] 125 . m t [GeV] [7, 40–43] 172 . makes sense to use the global EW fit, for which G µF is usu-ally a key input quantity, instead as a tool to determine G F in a completely independent way.The EW observables included in our fit ( W mass,sin θ W , and Z -pole observables [8, 9]) are given in Ta-ble I, with the other input parameters summarized inTable II. Here, the hadronic running ∆ α had is takenfrom e + e − data, which is insensitive to the changes in e + e − → hadrons cross sections [10–17] recently suggestedby lattice QCD [18], as long as these changes are concen-trated at low energies [19–22]. We perform the globalEW fit (without using experimental input for G F ) in aBayesian framework using the publicly available HEPfit package [23], whose Markov Chain Monte Carlo (MCMC)determination of posteriors is powered by the BayesianAnalysis Toolkit (
BAT ) [24]. As a result, we find G EW F (cid:12)(cid:12)(cid:12) full = 1 . × − GeV − , (5)a gain in precision over G (3) F in Eq. (3) by a factor 5.As depicted in Fig. 1, this value lies above G µF by 2 σ ,reflecting the known tensions within the EW fit [25, 26].For comparison, we also considered a closer analog of G (3) F , by only keeping sin θ W from Table I in the fit,which gives G EW F (cid:12)(cid:12)(cid:12) minimal = 1 . × − GeV − , (6)consistent with Eq. (5), but with a larger uncertainty.The pull of G EW F away from G µF is mainly driven by M W ,sin θ W from the hadron colliders, A (cid:96) , and A ,(cid:96) FB .As a third possibility, one can determine the Fermi con-stant from superallowed β decays, taking V us from kaon ◆◆ μ→ e νν CKM EW ( full ) EW ( minimal ) G F [ - / GeV ] FIG. 1: Values of G F from the different determinations. or τ decays and assuming CKM unitarity ( | V ub | is alsoneeded, but the impact of its uncertainty is negligible).This is particularly interesting given recent hints for theapparent violation of first-row CKM unitarity, known asthe Cabibbo angle anomaly (CAA). The significance ofthe tension crucially depends on the radiative correctionsapplied to β decays [44–51], but also on the treatment oftensions between K (cid:96) and K (cid:96) decays [52] and the con-straints from τ decays [53], see Ref. [54] for more details.In the end, quoting a significance around 3 σ should givea realistic representation of the current situation, andfor definiteness we will thus use the estimate of first-rowCKM unitarity from Ref. [7] (cid:12)(cid:12) V ud (cid:12)(cid:12) + (cid:12)(cid:12) V us (cid:12)(cid:12) + (cid:12)(cid:12) V ub (cid:12)(cid:12) = 0 . . (7)In addition, we remark that there is also a deficit in thefirst-column CKM unitarity relation [7] (cid:12)(cid:12) V ud (cid:12)(cid:12) + (cid:12)(cid:12) V cd (cid:12)(cid:12) + (cid:12)(cid:12) V td (cid:12)(cid:12) = 0 . , (8)less significant than Eq. (7), but suggesting that if thedeficits were due to BSM effects, they would likely berelated to β decays. For the numerical analysis, we willcontinue to use Eq. (7) given the higher precision. Thedeficit in Eq. (7) translates to G CKM F = 0 . × G µF = 1 . × − GeV − . (9)Comparing the three independent determinations of G F in Fig. 1, one finds the situation that G EW F lies above G µF by 2 σ , G CKM F below G µF by 3 σ , and the tension be-tween G EW F and G CKM F amounts to 3 . σ . To bring allthree determinations into agreement within 1 σ , an effectin at least two of the underlying processes is thus neces-sary. This leads us to study BSM contributions to1. µ → eνν transitions,2. Z → (cid:96)(cid:96), νν , α /α , M Z /M W ,3. superallowed β decays,where the second point gives the main observables in theEW fit, with α /α a proxy for the ratio of SU (2) L and U (1) Y couplings. We do not consider the possibility ofBSM effects in kaon, τ , or D decays, as this would re-quire a correlated effect with a relating symmetry. Fur-thermore, as shown in Ref. [54], the sensitivity to a BSMeffect in superallowed β decays is enhanced by a factor | V ud | / | V us | compared to kaon, τ , or D decays. Thiscan also be seen from Eq. (7) as | V ud | gives the dominantcontribution.BSM explanations of the discrepancies between thesedeterminations of G F have been studied in the literaturein the context of the CAA [54–64]. In this Letter, wewill analyze possible BSM effects in all three G F deter-minations using an EFT approach with gauge-invariantdimension 6 operators [5, 6]. III. SMEFT ANALYSIS
Dimension-6 operators that can explain the differencesamong the determinations of G F can be grouped into thefollowing classesA. four-fermion operators in µ → eνν ,B. four-fermion operators in u → deν ,C. modified W – u – d couplings,D. modified W – (cid:96) – ν couplings,E. other operators affecting the EW fit.Global fits to a similar set of effective operators havebeen considered in Refs. [65–69], here, we will concentratedirectly on the impact on G F determinations, followingthe conventions of Ref. [6]. A. Four-fermion operators in µ → eνν Not counting flavor indices, there are only two oper-ators that can generate a neutral current involving fourleptons: Q ijkl(cid:96)(cid:96) = ¯ (cid:96) i γ µ (cid:96) j ¯ (cid:96) k γ µ (cid:96) l , Q ijkl(cid:96)e = ¯ (cid:96) i γ µ (cid:96) j ¯ e k γ µ e l . (10)Not all flavor combinations are independent, e.g., Q ijkl(cid:96)(cid:96) = Q klij(cid:96)(cid:96) = Q ilkj(cid:96)(cid:96) = Q kjil(cid:96)(cid:96) due to Fierz identities and Q jilk(cid:96)(cid:96) ( e ) = Q ijkl ∗ (cid:96)(cid:96) ( e ) due to Hermiticity. Instead of summing over flavorindices, it is easiest to absorb these terms into a redef-inition of the operators whose latter two indices are 12,which contribute directly to µ → eνν . Therefore, wehave to consider 9 different flavor combinations for bothoperators:1. Q (cid:96)(cid:96) contributes to the SM amplitude (its coefficientis real by Fierz identities and Hermiticity). Therefore,it can give a constructive or destructive effect in the muon lifetime and does not affect the Michel param-eters [70–77]. In order to bring G CKM F and G µF intoagreement at 1 σ we need C (cid:96)(cid:96) ≈ − . × − G F . (11)This Wilson coefficient is constrained by LEP searchesfor e + e − → µ + µ − [8] − π (9 . < C (cid:96)(cid:96) < π (12 . , (12)a factor 8 weaker than preferred by the CAA, butwithin reach of future e + e − colliders.2. Even though Q (cid:96)e has a vectorial Dirac structure,it leads to a scalar amplitude after applying Fierzidentities. Its interference with the SM amplitudeis usually expressed in terms of the Michel parame-ter η = Re C (cid:96)e / (2 √ G F ), leading to a correction1 − ηm e /m µ . In the absence of right-handed neu-trinos the restricted analysis from Ref. [75] applies,constraining the shift in G µF to 0 . × − , well belowthe required effect to obtain 1 σ agreement with G CKM F or G EW F .3. The operators Q (cid:96)(cid:96) ( e ) could contribute to muon decayas long as the neutrino flavors are not detected. Toreconcile G CKM F and G µF within 1 σ we need | C (cid:96)(cid:96) | ≈ . G F or | C (cid:96)e | ≈ . G F . Both solutions are ex-cluded by muonium–anti-muonium oscillations ( M = µ + e − ) [78] P ( ¯ M – M ) < . × − /S B , (13)with correction factor S B = 0 .
35 ( C (cid:96)(cid:96) ) and S B =0 .
78 ( C (cid:96)e ) for the extrapolation to zero magneticfield. Comparing to the prediction for the rate [79–81] P ( ¯ M – M ) = 8( αµ µe ) τ µ G F π (cid:12)(cid:12)(cid:12) C (cid:96)(cid:96) ( e ) /G F (cid:12)(cid:12)(cid:12) = 3 . × − (cid:12)(cid:12)(cid:12) C (cid:96)(cid:96) ( e ) /G F (cid:12)(cid:12)(cid:12) , (14)with reduced mass µ µe = m µ m e / ( m µ + m e ), the limitsbecome | C (cid:96)(cid:96) ( e ) | < . . × − G F .4. For Q (cid:96)(cid:96) ( e ) again numerical values of | C (cid:96)(cid:96) ( e ) | ≈ . G F are preferred (as for all the remaining Wilson coeffi-cients in this list). Both operators give tree-level ef-fects in µ → e , e.g.,Br [ µ → e ] = m µ τ µ π (cid:12)(cid:12) C (cid:96)(cid:96) (cid:12)(cid:12) = 0 . (cid:12)(cid:12)(cid:12)(cid:12) C (cid:96)(cid:96) G F (cid:12)(cid:12)(cid:12)(cid:12) , (15)which exceeds the experimental limit on the branchingratio of 1 . × − [82] by many orders of magnitude(the result for C (cid:96)e is smaller by a factor 1 / Q (cid:96)(cid:96) ( e ) and Q (cid:96)(cid:96) ( e ) contribute at the one-loop level to µ → e conversion and µ → e and at thetwo-loop level to µ → eγ [83]. Here the current best bounds come from µ → e conversion. Using Table 3in Ref. [83] we have (cid:12)(cid:12) C (cid:96)(cid:96) (cid:12)(cid:12) < . × − G F , (cid:12)(cid:12) C (cid:96)(cid:96) (cid:12)(cid:12) < . × − G F , (16)excluding again a sizable BSM effect, and similarly for Q (cid:96)e and Q (cid:96)e .6. Q (cid:96)(cid:96) ( e ) , Q (cid:96)(cid:96) ( e ) , Q (cid:96)(cid:96) ( e ) , and Q (cid:96)(cid:96) ( e ) contribute to τ → µµe and τ → µee , respectively, which excludes a sizableeffect in analogy to µ → e above [53, 84, 85].Other four-quark operators can only contribute vialoop effects, which leads us to conclude that the onlyviable mechanism proceeds via a modification of the SMoperator. B. Four-fermion operators in d → ueν First of all, the operators Q (1)1111 (cid:96)equ and Q (3)1111 (cid:96)equ giverise to d → ueν scalar amplitudes. Such amplitudes leadto enhanced effects in π → µν/π → eν with respect to β decays and therefore can only have a negligible impact onthe latter once the stringent experimental bounds [7, 86]are taken into account. Furthermore, the tensor ampli-tude generated by Q (3) ijkl(cid:96)equ has a suppressed matrix ele-ment in β decays.Therefore, we are left with Q (3)1111 (cid:96)q , for which we onlyconsider the flavor combination that leads to interferencewith the SM. The CAA prefers C (3)1111 (cid:96)q ≈ . × − G F .Via SU (2) L invariance, this operator generates effects inneutral-current (NC) interactions L NC = C (3)1111 (cid:96)q (cid:0) ¯ dγ µ P L d − ¯ uγ µ P L u (cid:1) ¯ eγ µ P L e. (17)Note that since the SM amplitude for ¯ uu ( ¯ dd ) → e + e − ,at high energies, has negative (positive) sign, we haveconstructive interference in both amplitudes. Therefore,the latest nonresonant dilepton searches by ATLAS [87]lead to C (3)1111 (cid:96)q < ∼ . × − G F . (18)Hence, four-fermion operators affecting d → ueν transi-tions can bring G CKM F into 1 σ agreement with G µF , butare at the border of the LHC constraints. C. Modified W – u – d couplings There are only two operators modifying the W cou-plings to quarks Q (3) ijφq = φ † i ↔ D Iµ φ ¯ q i γ µ τ I q j ,Q ijφud = φ † i ↔ D µ φ ¯ u i γ µ d j . (19) First of all, Q ijφud generates right-handed W –quark cou-plings, which can only slightly alleviate the CAA, butnot solve it [88]. Q (3) ijφq generates modifications of theleft-handed W –quark couplings and data prefer C (3)11 φq ≈ . × − G F . (20)Due to SU (2) L invariance, in general effects in D – ¯ D and K – ¯ K mixing are generated. However, in case ofalignment with the down-sector, the effect in D – ¯ D issmaller than the experimental value and thus not con-straining as the SM prediction cannot be reliably calcu-lated. Furthermore, the effects in D – ¯ D and K – ¯ K mixing could be suppressed by assuming that C (3) ijφq re-spects a global U (2) symmetry. D. Modified W – (cid:96) – ν couplings Only the operator Q (3) ijφ(cid:96) = φ † i ↔ D Iµ φ ¯ (cid:96) i γ µ τ I (cid:96) j (21)generates modified W – (cid:96) – ν couplings at tree level. Inorder to avoid the stringent bounds from charged lep-ton flavor violation, the off-diagonal Wilson coefficients,in particular C (3)12 φ(cid:96) , must be very small. Since theyalso do not generate amplitudes interfering with the SMones, their effect can be neglected. While C (3)11 φ(cid:96) af-fects G µF and G CKM F in the same way, C (3)22 φ(cid:96) only en-ters in muon decay. Therefore, agreement between G µF and G CKM F can be achieved by C (3)11 φ(cid:96) < C (3)22 φ(cid:96) > | C (3)22 φ(cid:96) | < | C (3)11 φ(cid:96) | without violating lepton flavoruniversality tests such as π ( K ) → µν/π ( K ) → eν or τ → µνν/τ ( µ ) → eνν [54, 56, 89]. However, C (3) ijφ(cid:96) alsoaffects Z couplings to leptons and neutrinos, which enterthe global EW fit. E. Electroweak fit
The impact of modified gauge-boson–lepton couplingson the global EW fit, generated by Q (3) ijφ(cid:96) and Q (1) ijφ(cid:96) = φ † i ↔ D µ φ ¯ (cid:96) i γ µ (cid:96) j , (22)can be minimized by only affecting Zνν but not
Z(cid:96)(cid:96) , byimposing C (1) ijφ(cid:96) = − C (3) ijφ(cid:96) . In this way, in addition to theFermi constant, only the Z width to neutrino changes andthe fit improves significantly compared to the SM [56], seeFig. 2 for the preferred parameter space. One can evenfurther improve the fit by assuming C (1)11 φ(cid:96) = − C (3)11 φ(cid:96) , C (1)22 φ(cid:96) = − C (3)22 φ(cid:96) , which leads to a better descriptionof Z → µµ data. Furthermore, the part of the tensionbetween G EW F and G µF driven by the W mass can bealleviated by the operator Q φW B = φ † τ I φW Iµν B µν . FIG. 2: Example of the complementarity between the G F de-terminations from muon decay ( G µF ), CKM unitarity ( G CKM F ),and the global EW fit ( G EM F ) in case of C (3) iiφ(cid:96) = − C (1) iiφ(cid:96) , cor-responding to modifications of neutrino couplings to gaugebosons (the EW fit also includes τ → µνν/τ ( µ ) → eνν [7,53, 89]). Here, we show the preferred 1 σ regions obtained byrequiring that two or all three G F determinations agree. Thecontour lines show the value of the Fermi constant extractedfrom muon decay once BSM effects are taking into account. IV. CONCLUSIONS AND OUTLOOK
Even though the Fermi constant is determined ex-tremely precisely by the muon lifetime, Eq. (1), its con-straining power on BSM effects is limited by the preci-sion of the second-best determination. In this Letter wederived in a first step two alternative independent deter-minations, from the EW fit, Eq. (5), and superallowed β decays using CKM unitarity, Eq. (9). The latter de-termination is more precise than the one from the EWfit, even though the precision of G EW F increased by a fac-tor 5 compared to Ref. [4]. Furthermore, as shown inFig. 1, both determinations display a tension of 2 σ and3 σ compared to G µF , respectively.In a second step, we investigated how these hints ofBSM physics can be explained within the SMEFT frame-work. For BSM physics in G µF we were able to rule outall four-fermion operators, except for Q (cid:96)(cid:96) , which gen-erates a SM-like amplitude, and modified W – (cid:96) – ν cou-plings, from Q (3) ijφ(cid:96) . Therefore, both constructive and destructive interference is possible, which would bring G µF into agreement with G CKM F or G EW F , respectively, atthe expense of increasing the tension with the other de-termination. To achieve a better agreement among thethree different values of G F , also BSM effects in G CKM F and/or G EW F are necessary. In the case of G CKM F , onlya single four-fermion operator, Q (3)1111 (cid:96)q , and Q (3) ijφ(cid:96) re-main. Finally, modified gauge-boson–lepton couplings,via Q (3) ijφ(cid:96) and Q (1) ijφ(cid:96) , can not only change G CKM F and G µF , but also affect the EW fit via the Z -pole observ-ables, which can further improve the global agreementwith data, see Fig. 2. This figure also demonstrates theadvantage of interpreting the tensions in terms of G F ,defining a transparent benchmark for comparison bothin SMEFT and concrete BSM scenarios, and allows oneto constrain the amount of BSM contributions to muondecay.Our study highlights the importance of improving theprecision of the alternative independent determinationsof G CKM F and G EW F in order to confirm or refute BSMcontributions to the Fermi constant. Concerning G CKM F ,improvements in the determination of | V ud | should arisefrom advances in nuclear-structure [90, 91] and EWradiative corrections [92], while experimental develop-ments [93–99] could make the determination from neu-tron decay [100–102] competitive and, in combinationwith K (cid:96) decays, add another complementary constrainton | V ud | / | V us | via pion β decay [103, 104]. Further, im-proved measurements of | V cd | from D decays [105] couldbring the precision of the first-column CKM unitarity re-lation close to the first-row one, which, in turn, could becorroborated via improved | V us | determinations from K (cid:96) decays [106–108]. The precision of G EW F will profit in thenear future from LHC measurements of m t and M W , inthe mid-term future from the Belle-II EW precision pro-gram [109], and in the long-term from future e + e − collid-ers such as the FCC-ee [110], ILC [111], CEPC [112], orCLIC [113], which could achieve a precision at the levelof 10 − . Acknowledgments
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