CERN-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,4 and Claudio Andrea Manzari2, 3
arXiv:2102.02825v1 [hep-ph] 4 Feb 2021
2
1
CERN Theory Division, CH–1211 Geneva 23, Switzerland
Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH–8057 Zürich, Switzerland
3
Paul Scherrer Institut, CH–5232 Villigen PSI, Switzerland
4
Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics,
University of Bern, Sidlerstrasse 5, CH–3012 Bern, Switzerland
The Fermi constant (GF ) is extremely well measured through the muon lifetime, defining one
of the key fundamental parameters in the Standard Model (SM). However, to search for physics
beyond the SM (BSM), it is the precision of the second-best independent determination of GF that
defines the sensitivity. The best alternative extractions of GF 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 GF from muon decay, albeit in opposite
directions, reflecting the known tensions within the EW fit and hints for the apparent violation of
CKM unitarity, respectively. We investigate how BSM physics could bring the three determinations
of GF into agreement using SM effective field theory and comment on future perspectives.
I.
II.
INTRODUCTION
The numerical value of the Fermi constant GF is conventionally defined via the muon lifetime within the SM.
Even though this measurement is extremely precise [1–3]
GµF = 1.1663787(6) × 10−5 GeV−2 ,
DETERMINATIONS OF GF
Within the SM, the Fermi constant GF is defined by,
and is most precisely determined from, the muon lifetime [2]
(GµF )2 m5µ
1
(1 + ∆q),
=
τµ
192π 3
(1)
at the level of 0.5 ppm, its determination of the Fermi
constant is not necessarily free of BSM contributions. In
fact, one can only conclude the presence or absence of
BSM effects by comparing GµF to another independent
determination. This idea was first introduced by Marciano in Ref. [4], concentrating on Z-pole observables and
the fine-structure constant α. In addition to a lot of new
data that has become available since 1999, another option already mentioned in Ref. [4]—the determination of
GF from β and kaon decays using CKM unitarity—has
become of particular interest due to recent hints for the
(apparent) violation of first-row CKM unitarity. These
developments motivate a fresh look at the Fermi constant, in particular on its extraction from a global EW
fit and via CKM unitarity, as to be discussed in the first
part of this Letter.
The comparison of the resulting values for GF shows
that with modern input these two extractions are close in
precision, yet still lagging behind muon decay by almost
three orders of magnitude. Since the different GF determinations turn out to display some disagreement beyond
their quoted uncertainties, the second part of this Letter is devoted to a systematic analysis of possible BSM
contributions in SM effective field theory (SMEFT) [5, 6]
to see which scenarios could account for these tensions
without being excluded by other constraints. This is
important to identify BSM scenarios that could be responsible for the tensions, which will be scrutinized with
forthcoming data in the next years.
(2)
where ∆q includes the phase space, QED, and hadronic
radiative corrections. The resulting numerical value in
Eq. (1) is so precise that its error can be ignored in the
following. To address the question whether GµF subsumes
BSM contributions, however, alternative independent determinations of GF are indispensable, and their precision
limits the extent to which BSM contamination in GµF can
be detected.
In Ref. [4], the two best independent determinations
were found as
Z`+ `−
GF
= 1.1650(14) +11
× 10−5 GeV−2 ,
−6
(3)
GF = 1.1672(8) +18
× 10−5 GeV−2 ,
(3)
−7
where the first variant uses the width for Z → `+ `− (γ),
while the second employs α and sin2 θW , together with
the appropriate radiative corrections. Since the present
uncertainty in Γ[Z → `+ `− (γ)] = 83.984(86) MeV [7] is
only marginally improved compared to the one available
in 1999, the update
+ −
Z`
GF
`
= 1.1661(16) × 10−5 GeV−2
(4)
does not lead to a gain in precision, but the shift in the
central value improves agreement with GµF . The second
(3)
variant, GF , is more interesting, as here the main limitation arose from the radiative corrections, which have
seen significant improvements regarding the input values
for the masses of the top quark, mt , the Higgs boson,
MH , the strong coupling, αs , and the hadronic running
of α. In fact, with all EW parameters determined, it now
2
MW [GeV]
[7] 80.379(12) A
`
ΓW [GeV]
[7] 2.085(42)
R`0
BR(W → had)
[7] 0.6741(27)
A0,`
FB
BR(W → lep)
[7] 0.1086(9)
0
R
2
b
sin θeff(QFB)
[7] 0.2324(12)
Rc0
sin2 θeff(Tevatron)
[27] 0.23148(33)
0,b
sin2 θeff(LHC)
[28–31] 0.23129(33) AFB
0,c
ΓZ [GeV]
[9] 2.4952(23) AFB
0
A
b
σh [nb]
[9] 41.541(37)
Pτpol
[9] 0.1465(33) Ac
[9]
[9]
[9]
[9]
[9]
[9]
[9]
[9]
[9]
0.1513(21)
20.767(25)
0.0171(10)
0.21629(66)
0.1721(30)
0.0992(16)
0.0707(35)
0.923(20)
0.670(27)
TABLE I: EW observables included in our global fit together
with their current experimental values.
Parameter
Prior
α × 103 [7]
7.2973525664(17)
∆αhad × 104 [15, 16]
276.1(1.1)
αs (MZ ) [7, 32]
0.1179(10)
MZ [GeV] [7, 33–36]
91.1876(21)
MH [GeV] [7, 37–39]
125.10(14)
mt [GeV] [7, 40–43]
172.76(30)
makes sense to use the global EW fit, for which GµF is usually a key input quantity, instead as a tool to determine
GF in a completely independent way.
The EW observables included in our fit (W mass,
sin2 θW , and Z-pole observables [8, 9]) are given in Table I, with the other input parameters summarized in
Table II. Here, the hadronic running ∆αhad is taken
from e+ e− data, which is insensitive to the changes in
e+ e− → hadrons cross sections [10–17] recently suggested
by lattice QCD [18], as long as these changes are concentrated at low energies [19–22]. We perform the global
EW fit (without using experimental input for GF ) in a
Bayesian framework using the publicly available HEPfit
package [23], whose Markov Chain Monte Carlo (MCMC)
determination of posteriors is powered by the Bayesian
Analysis Toolkit (BAT) [24]. As a result, we find
= 1.16716(39) × 10−5 GeV−2 ,
(5)
full
(3)
a gain in precision over GF in Eq. (3) by a factor 5.
As depicted in Fig. 1, this value lies above GµF by 2σ,
reflecting the known tensions within the EW fit [25, 26].
For comparison, we also considered a closer analog of
(3)
GF , by only keeping sin2 θW from Table I in the fit,
which gives
GEW
F
= 1.16728(83) × 10−5 GeV−2 ,
◆
EW (full)
EW (minimal)
1.165
1.1655
1.166
1.1665 1.167
GF [10-5 /GeV2 ]
1.1675
1.168
FIG. 1: Values of GF from the different determinations.
TABLE II: Parameters of the EW fit together with their
(Gaussian) priors.
GEW
F
μ→eνν
CKM
(6)
minimal
consistent with Eq. (5), but with a larger uncertainty.
The pull of GEW
away from GµF is mainly driven by MW ,
F
2
sin θW from the hadron colliders, A` , and A0,`
FB .
As a third possibility, one can determine the Fermi constant from superallowed β decays, taking Vus from kaon
or τ decays and assuming CKM unitarity (|Vub | is also
needed, but the impact of its uncertainty is negligible).
This is particularly interesting given recent hints for the
apparent violation of first-row CKM unitarity, known as
the Cabibbo angle anomaly (CAA). The significance of
the tension crucially depends on the radiative corrections
applied to β decays [44–51], but also on the treatment of
tensions between K`2 and K`3 decays [52] and the constraints from τ decays [53], see Ref. [54] for more details.
In the end, quoting a significance around 3σ should give
a realistic representation of the current situation, and
for definiteness we will thus use the estimate of first-row
CKM unitarity from Ref. [7]
Vud
2
+ Vus
2
+ Vub
2
= 0.9985(5).
(7)
In addition, we remark that there is also a deficit in the
first-column CKM unitarity relation [7]
Vud
2
+ Vcd
2
+ Vtd
2
= 0.9970(18),
(8)
less significant than Eq. (7), but suggesting that if the
deficits were due to BSM effects, they would likely be
related to β decays. For the numerical analysis, we will
continue to use Eq. (7) given the higher precision. The
deficit in Eq. (7) translates to
GCKM
= 0.99925(25) × GµF = 1.16550(29) × 10−5 GeV−2 .
F
(9)
Comparing the three independent determinations of
GF in Fig. 1, one finds the situation that GEW
lies above
F
GµF by 2σ, GCKM
below GµF by 3σ, and the tension beF
tween GEW
and GCKM
amounts to 3.4σ. To bring all
F
F
three determinations into agreement within 1σ, an effect
in at least two of the underlying processes is thus necessary. This leads us to study BSM contributions to
1. µ → eνν transitions,
2. Z → ``, νν, α2 /α, MZ /MW ,
3. superallowed β decays,
3
where the second point gives the main observables in the
EW fit, with α2 /α a proxy for the ratio of SU (2)L and
U (1)Y couplings. We do not consider the possibility of
BSM effects in kaon, τ , or D decays, as this would require a correlated effect with a relating symmetry. Furthermore, as shown in Ref. [54], the sensitivity to a BSM
effect in superallowed β decays is enhanced by a factor
|Vud |2 /|Vus |2 compared to kaon, τ , or D decays. This
can also be seen from Eq. (7) as |Vud | gives the dominant
contribution.
BSM explanations of the discrepancies between these
determinations of GF have been studied in the literature
in the context of the CAA [54–64]. In this Letter, we
will analyze possible BSM effects in all three GF determinations using an EFT approach with gauge-invariant
dimension 6 operators [5, 6].
III.
SMEFT ANALYSIS
Dimension-6 operators that can explain the differences
among the determinations of GF can be grouped into the
following classes
A. four-fermion operators in µ → eνν,
C. modified W –u–d couplings,
D. modified W –`–ν couplings,
E. other operators affecting the EW fit.
Global fits to a similar set of effective operators have
been considered in Refs. [65–69], here, we will concentrate
directly on the impact on GF determinations, following
the conventions of Ref. [6].
Not counting flavor indices, there are only two operators that can generate a neutral current involving four
leptons:
µ
¯ µ
Qijkl
`e = `i γ `j ēk γ el .
(11)
This Wilson coefficient is constrained by LEP searches
for e+ e− → µ+ µ− [8]
−
4π
4π
1221
< C``
<
,
(9.8 TeV)2
(12.2 TeV)2
(12)
a factor 8 weaker than preferred by the CAA, but
within reach of future e+ e− colliders.
2. Even though Q2112
has a vectorial Dirac structure,
`e
it leads to a scalar amplitude after applying Fierz
identities. Its interference with the SM amplitude
is usually expressed √
in terms of the Michel parame2112
ter η = Re C`e
/(2 2GF ), leading to a correction
1 − 2ηme /mµ . In the absence of right-handed neutrinos the restricted analysis from Ref. [75] applies,
constraining the shift in GµF to 0.68 × 10−4 , well below
the required effect to obtain 1σ agreement with GCKM
F
or GEW
F .
P(M̄ –M ) < 8.3 × 10−11 /SB ,
(10)
Not all flavor combinations are independent, e.g., Qijkl
`` =
klij
ilkj
kjil
jilk
Q`` = Q`` = Q`` due to Fierz identities and Q``(e) =
Qijkl∗
``(e) due to Hermiticity. Instead of summing over flavor
indices, it is easiest to absorb these terms into a redefinition of the operators whose latter two indices are 12,
which contribute directly to µ → eνν. Therefore, we
have to consider 9 different flavor combinations for both
operators:
1. Q2112
contributes to the SM amplitude (its coefficient
``
is real by Fierz identities and Hermiticity). Therefore,
it can give a constructive or destructive effect in the
(13)
1212
with correction factor SB = 0.35 (C``
) and SB =
1212
0.78 (C`e ) for the extrapolation to zero magnetic
field. Comparing to the prediction for the rate [79–81]
P(M̄ –M ) =
Four-fermion operators in µ → eνν
¯ µ ¯ µ
Qijkl
`` = `i γ `j `k γ `l ,
2112
C``
≈ −1.4 × 10−3 GF .
3. The operators Q1212
``(e) could contribute to muon decay
as long as the neutrino flavors are not detected. To
1212
reconcile GCKM
and GµF within 1σ we need |C``
|≈
F
1212
0.045 GF or |C`e | ≈ 0.09 GF . Both solutions are excluded by muonium–anti-muonium oscillations (M =
µ+ e− ) [78]
B. four-fermion operators in u → deν,
A.
muon lifetime and does not affect the Michel parameters [70–77]. In order to bring GCKM
and GµF into
F
agreement at 1σ we need
8(αµµe )6 τµ2 G2F
1212
C``(e)
/GF
π2
1212
= 3.21 × 10−6 C``(e)
/GF
2
2
,
(14)
with reduced mass µµe = mµ me /(mµ +me ), the limits
1212
become |C``(e)
| < 8.6(5.8) × 10−3 GF .
1112
4. For Q1112
``(e) again numerical values of |C``(e) | ≈ 0.09 GF
are preferred (as for all the remaining Wilson coefficients in this list). Both operators give tree-level effects in µ → 3e, e.g.,
Br [µ → 3e] =
m5µ τµ 1112
C
768π 3 ``
2
2
= 0.25
1112
C``
,
GF
(15)
which exceeds the experimental limit on the branching
ratio of 1.0 × 10−12 [82] by many orders of magnitude
1112
(the result for C`e
is smaller by a factor 1/2).
3312
5. The operators Q2212
``(e) and Q``(e) contribute at the oneloop level to µ → e conversion and µ → 3e and at the
two-loop level to µ → eγ [83]. Here the current best
4
bounds come from µ → e conversion. Using Table 3
in Ref. [83] we have
3312
< 6.4 × 10−5 GF ,
C``
2212
< 2.8 × 10−5 GF ,
C``
First of all, Qij
φud generates right-handed W –quark couplings, which can only slightly alleviate the CAA, but
(3)ij
not solve it [88]. Qφq generates modifications of the
left-handed W –quark couplings and data prefer
(16)
excluding again a sizable BSM effect, and similarly for
Q3312
and Q2212
`e
`e .
3212
1312
3112
6. Q2312
``(e) , Q``(e) , Q``(e) , and Q``(e) contribute to τ → µµe
and τ → µee, respectively, which excludes a sizable
effect in analogy to µ → 3e above [53, 84, 85].
Other four-quark operators can only contribute via
loop effects, which leads us to conclude that the only
viable mechanism proceeds via a modification of the SM
operator.
(3)11
(20)
Due to SU (2)L invariance, in general effects in D0 –D̄0
and K 0 –K̄ 0 mixing are generated. However, in case of
alignment with the down-sector, the effect in D0 –D̄0 is
smaller than the experimental value and thus not constraining as the SM prediction cannot be reliably calculated. Furthermore, the effects in D0 –D̄0 and K 0 –K̄ 0
(3)ij
mixing could be suppressed by assuming that Cφq re2
spects a global U (2) symmetry.
D.
B.
≈ 0.7 × 10−3 GF .
Cφq
Modified W –`–ν couplings
Four-fermion operators in d → ueν
Only the operator
(1)1111
Q`equ
(3)1111
Q`equ
First of all, the operators
and
give
rise to d → ueν scalar amplitudes. Such amplitudes lead
to enhanced effects in π → µν/π → eν with respect to β
decays and therefore can only have a negligible impact on
the latter once the stringent experimental bounds [7, 86]
are taken into account. Furthermore, the tensor ampli(3)ijkl
tude generated by Q`equ has a suppressed matrix element in β decays.
(3)1111
Therefore, we are left with Q`q
, for which we only
consider the flavor combination that leads to interference
(3)1111
with the SM. The CAA prefers C`q
≈ 0.7 × 10−3 GF .
Via SU (2)L invariance, this operator generates effects in
neutral-current (NC) interactions
(3)1111
LNC = C`q
¯ µ PL d − ūγ µ PL u ēγµ PL e.
dγ
(17)
¯ → e+ e− ,
Note that since the SM amplitude for ūu(dd)
at high energies, has negative (positive) sign, we have
constructive interference in both amplitudes. Therefore,
the latest nonresonant dilepton searches by ATLAS [87]
lead to
(3)1111
C`q
< 0.8 × 10−3 GF .
∼
(18)
Hence, four-fermion operators affecting d → ueν transitions can bring GCKM
into 1σ agreement with GµF , but
F
are at the border of the LHC constraints.
C.
Modified W –u–d couplings
There are only two operators modifying the W couplings to quarks
(3)ij
Qφq
†
↔I
(3)ij
Qφ`
= φ iDµ φq̄i γ τ qj ,
↔
†
µ
Qij
φud = φ iD µ φūi γ dj .
(19)
(21)
generates modified W –`–ν couplings at tree level. In
order to avoid the stringent bounds from charged lepton flavor violation, the off-diagonal Wilson coefficients,
(3)12
in particular Cφ` , must be very small. Since they
also do not generate amplitudes interfering with the SM
(3)11
ones, their effect can be neglected. While Cφ`
af(3)22
in the same way, Cφ`
only enfects GµF and GCKM
F
ters in muon decay. Therefore, agreement between GµF
(3)11
(3)22
can be achieved by Cφ`
< 0, Cφ`
> 0,
and GCKM
F
(3)22
(3)11
and |Cφ` | < |Cφ` | without violating lepton flavor
universality tests such as π(K) → µν/π(K) → eν or
(3)ij
τ → µνν/τ (µ) → eνν [54, 56, 89]. However, Cφ` also
affects Z couplings to leptons and neutrinos, which enter
the global EW fit.
E.
Electroweak fit
The impact of modified gauge-boson–lepton couplings
(3)ij
on the global EW fit, generated by Qφ` and
(1)ij
Qφ`
↔
= φ† iDµ φ`¯i γ µ `j ,
(22)
can be minimized by only affecting Zνν but not Z``, by
(1)ij
(3)ij
imposing Cφ` = −Cφ` . In this way, in addition to the
Fermi constant, only the Z width to neutrino changes and
the fit improves significantly compared to the SM [56], see
Fig. 2 for the preferred parameter space. One can even
(1)11
(3)11
further improve the fit by assuming Cφ`
= −Cφ` ,
(1)22
µ I
↔I
= φ† iDµ φ`¯i γ µ τ I `j
(3)22
Cφ`
= −3Cφ` , which leads to a better description
of Z → µµ data. Furthermore, the part of the tension
between GEW
and GµF driven by the W mass can be
F
I
alleviated by the operator QφW B = φ† τ I φWµν
B µν .
5
FIG. 2: Example of the complementarity between the GF deCKM
terminations from muon decay (Gµ
),
F ), CKM unitarity (GF
(3)ii
(1)ii
EM
and the global EW fit (GF ) in case of Cφ` = −Cφ` , corresponding to modifications of neutrino couplings to gauge
bosons (the EW fit also includes τ → µνν/τ (µ) → eνν [7,
53, 89]). Here, we show the preferred 1σ regions obtained by
requiring that two or all three GF determinations agree. The
contour lines show the value of the Fermi constant extracted
from muon decay once BSM effects are taking into account.
IV.
CONCLUSIONS AND OUTLOOK
destructive interference is possible, which would bring
GµF into agreement with GCKM
or GEW
F
F , respectively, at
the expense of increasing the tension with the other determination. To achieve a better agreement among the
three different values of GF , also BSM effects in GCKM
F
and/or GEW
are necessary. In the case of GCKM
, only
F
F
(3)1111
(3)ij
a single four-fermion operator, Q`q
, and Qφ` remain. Finally, modified gauge-boson–lepton couplings,
(3)ij
(1)ij
via Qφ` and Qφ` , can not only change GCKM
and
F
µ
GF , but also affect the EW fit via the Z-pole observables, which can further improve the global agreement
with data, see Fig. 2. This figure also demonstrates the
advantage of interpreting the tensions in terms of GF ,
defining a transparent benchmark for comparison both
in SMEFT and concrete BSM scenarios, and allows one
to constrain the amount of BSM contributions to muon
decay.
Our study highlights the importance of improving the
precision of the alternative independent determinations
in order to confirm or refute BSM
and GEW
of GCKM
F
F
,
contributions to the Fermi constant. Concerning GCKM
F
improvements in the determination of |Vud | should arise
from advances in nuclear-structure [90, 91] and EW
radiative corrections [92], while experimental developments [93–99] could make the determination from neutron decay [100–102] competitive and, in combination
with K`3 decays, add another complementary constraint
on |Vud |/|Vus | via pion β decay [103, 104]. Further, improved measurements of |Vcd | from D decays [105] could
bring the precision of the first-column CKM unitarity relation close to the first-row one, which, in turn, could be
corroborated via improved |Vus | determinations from K`3
decays [106–108]. The precision of GEW
will profit in the
F
near future from LHC measurements of mt and MW , in
the mid-term future from the Belle-II EW precision program [109], and in the long-term from future e+ e− colliders such as the FCC-ee [110], ILC [111], CEPC [112], or
CLIC [113], which could achieve a precision at the level
of 10−5 .
Even though the Fermi constant is determined extremely precisely by the muon lifetime, Eq. (1), its constraining power on BSM effects is limited by the precision of the second-best determination. In this Letter we
derived in a first step two alternative independent determinations, from the EW fit, Eq. (5), and superallowed
β decays using CKM unitarity, Eq. (9). The latter determination is more precise than the one from the EW
increased by a facfit, even though the precision of GEW
F
tor 5 compared to Ref. [4]. Furthermore, as shown in
Fig. 1, both determinations display a tension of 2σ and
3σ compared to GµF , respectively.
In a second step, we investigated how these hints of
BSM physics can be explained within the SMEFT framework. For BSM physics in GµF we were able to rule out
all four-fermion operators, except for Q2112
`` , which generates a SM-like amplitude, and modified W –`–ν cou(3)ij
plings, from Qφ` . Therefore, both constructive and
We thank David Hertzog and Klaus Kirch for valuable
discussions, and the ATLAS collaboration, in particular
Noam Tal Hod, for clarifications concerning the analysis of Ref. [87]. Support by the Swiss National Science
Foundation, under Project Nos. PP00P21 76884 (A.C.,
C.A.M) and PCEFP2 181117 (M.H.) is gratefully acknowledged.
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