10. Electroweak Model and Constraints on New Physics

1

10. Electroweak Model and Constraints on New Physics
Revised April 2024 by J. Erler (KPH, JGU Mainz) and A. Freitas (Pittsburgh U.).

10.1
10.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Renormalization and radiative corrections . . . . . . . . . . . . . . . . . . . . . . .
10.2.1 The Fermi constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.2 The electromagnetic coupling . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.3 Quark masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.4 The weak mixing angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.5 Radiative corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Low energy electroweak observables . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.1 Neutrino scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.2 Parity violating lepton scattering . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.3 Atomic parity violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4 Precision flavor physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.1 The τ lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.2 The muon anomalous magnetic moment . . . . . . . . . . . . . . . . . . . . .
10.5 Physics of the massive electroweak bosons . . . . . . . . . . . . . . . . . . . . . . .
10.5.1 The Z boson mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.5.2 Electroweak physics off the Z pole . . . . . . . . . . . . . . . . . . . . . . . .
10.5.3 Z pole physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.5.4 W and Z decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6 Global fit results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.7 Constraints on new physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.8 Alternative scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
3
3
4
5
7
9
10
10
13
14
16
16
18
19
20
21
23
27
28
33
39

10.1 Introduction
The standard model of the electroweak interactions (SM) [1–4] is based on the gauge group
SU(2)×U(1), with gauge bosons Wµi , i = 1, 2, 3, and Bµ for the SU(2) and U(1) factors, respectively,
0
and the corresponding gauge coupling constantsof the ν gand g.u The left-handed fermion fieldsiith
0 ≡ P V d ,
0
and
under
SU(2),
whered
i fermion family transform as doublets Ψi = `−dij ij jiiand V is the Cabibbo-Kobayashi-Maskawa mixing [5, 6] matrix. The right-handed fields are SU(2)singlets. From Higgs and electroweak precision data it is known that there are precisely three
sequential fermion families. Constraints on V and tests of universality are discussed in Ref. [7] andin Section 12 on the “CKM Quark-Mixing Matrix” in this Review. The extension of the formalism toallow an analogous leptonic mixing matrix is discussed in Section 14 on “Neutrino Masses, Mixing,
and Oscillations” in this Review.
A complex scalar Higgs doublet, φ, is added to the model for mass generation through spontaneous symmetry breaking with potential1 given by,
λ2
V (φ) = µ φ φ + (φ† φ)2 ,
2
2 †

φ≡

φ+
φ0

!

1

.

(10.1)

There is no generally accepted convention to write the quartic term. Our numerical coefficient simplifies
Eq. (10.5a) below and the squared coupling preserves the relation between the number of external legs and thepower counting of couplings at a given loop order. This structure also naturally emerges from physics beyond theSM, such as Supersymmetry.
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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10. Electroweak Model and Constraints on New Physics

2√
For µ2 negative, φ develops a vacuum expectation value, v/ 2 = |µ|/λ, where v = 246.22 GeV,
breaking part of the electroweak (EW) gauge symmetry, after which only one neutral Higgs scalar,H, remains in the physical particle spectrum. In non-minimal models there are additional charged
and neutral scalar Higgs particles. Higgs boson physics is reviewed in Section 11 on the “Status ofHiggs Boson Physics” in this Review.
After symmetry breaking the Lagrangian for the fermion fields, ψi , is
LF =

X



ψ̄i

i

−e

X

mi H
i 6 ∂ − mi −v

Qi ψ̄i γ µ ψi Aµ −

i



g X
ψi − √
Ψ i γ µ (1 − γ 5 )(T + Wµ+ + T − Wµ− ) Ψi
2 2 i

X
g
i 5
ψ̄i γ µ (gVi − gAγ ) ψi Zµ .
2 cos θW i

(10.2)

Here θW ≡ tan−1 (g 0 /g) is the weak mixing angle and e = g sin θW is the positron electric charge.
Furthermore,Aµ ≡ Bµ cos θW + Wµ3 sin θW ,Wµ± ≡(10.3a)Wµ1 ∓ iWµ2√,2(10.3b)Zµ ≡ −Bµ sin θW + Wµ3 cos θW ,(10.3c)are the photon field (γ) and the charged (W ± ) and neutral (Z) weak boson fields, respectively.
The Yukawa coupling of H to ψi in the first term in LF , which is flavor diagonal in the minimalmodel, is gmi /2MW . From the bosonic interaction Lagrangian,1
g 2 + µ− g 2 + g 02
LHV = (v + H)2
W W +Zµ Z µ − V
2
2 µ
4






v+H
√
2



,

(10.4)

one obtains the EW boson masses (at tree level, i.e., to lowest order in perturbation theory)
MH = λ v ,(10.5a)

gv
ev
=
,
2
2 sin θWqvev
MZ = g 2 + g 02 =,2
2 sin θW cos θWMW =Mγ = 0 .

(10.5b)(10.5c)(10.5d)The second term in LF represents the charged-current weak interaction [8–10], where T + and−T are the weak isospin raising and lowering operators. For example, the coupling of a W to anelectron and a neutrino is
h
i
e
− √
Wµ− ē γ µ (1 − γ 5 )ν + Wµ+ ν̄ γ µ (1 − γ 5 )e .
2 2 sin θW(10.6)For momenta small compared to MW , √this term gives rise to the effective four-fermion interaction
2 . CP violation is incorporated intowith the Fermi constant given by GF / 2 = 1/2v 2 = g 2 /8MW
the EW model by a single observable phase in Vij .
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3

10. Electroweak Model and Constraints on New Physics

The third term in LF describes electromagnetic interactions (QED) [11, 12], and the last is the
weak neutral-current interaction [9, 10, 13]. The vector and axial-vector couplings are
gVi ≡ t3L(i) − 2Qi sin2 θW ,(10.7a)igA≡ t3L(i) ,(10.7b)where t3L (i) is the weak isospin of fermion i (+1/2 for ui and νi ; −1/2 for di and ei ) and Qi is the
charge of ψi in units of e.
The first term in Eq. (10.2) also gives rise to fermion masses, and in the presence of right-handed
neutrinos to Dirac neutrino masses. The possibility of Majorana masses is discussed in Section 14
on “Neutrino Mass, Mixing, and Oscillations” in this Review.

10.2 Renormalization and radiative corrections
In addition to the Higgs boson mass, MH , the fermion masses and mixings, and the strong
coupling constant, αs , the SM has three parameters. The set with the smallest experimental
errors contains the Z mass2 , the Fermi constant, and the fine structure constant. The latter two
will be discussed in more detail in the next two subsections. (The numerical values quoted in
Sections 10.2–10.4 correspond to the main fit result in Table 10.7.)
10.2.1 The Fermi constant
The Fermi constant,
GF = 1.1663788(6) × 10−5 GeV−2 ,(10.8)

is derived from the µ lifetime formula3 ,G2 m5µb µ)b 2 (mµ )b 3 (mµ )~α(mαα+ H2 (ρ)+ H3= F 3 F (ρ) 1 + H1 (ρ)
2
τµ
192π
π
π
π3
"

#

,

(10.9)

where ρ = m2e /m2µ , and whereF (ρ) = 1 − 8ρ + 8ρ3 − ρ4 − 12ρ2 ln ρ = 0.99981295 ,
25 π 2 −− 9 + 4π 2 + 12 ln ρ ρ + 16π 2 ρ3/2 + O(ρ2 ) = −1.80793 ,82
67 4 53 2
156815 518 2 895
H2 (ρ) =−π −ζ(3) +
π + π ln 2
5184
81
36
720
6
5 √
− (0.042 ± 0.002)had − π 2 ρ + O(ρ) = 6.64 ,41b µ )−1 = α−1 +α(mln ρ + O(α) = 135.901 .
3π

H1 (ρ) =(10.10a)(10.10b)(10.10c)(10.10d)

H1 and H2 capture the QED corrections within the Fermi model. The results for ρ = 0 have beenobtained in Refs. [16] and [17, 18] for H1 and H2 , respectively, where the term in parentheses isfrom the hadronic vacuum polarization [17]. The mass corrections to H1 have been known forsome time [19], while those to H2 are more recent [20]. Notice the term linear in me in H2 whose2
We emphasize that in the fits described in Sec. 10.6 and Sec. 10.7 the values of the SM parameters are affectedby all observables that depend on them. This is of no practical consequence for α and GF , however, since they arevery precisely known. Also note that other choices for the three SM input parameters (see e.g. Ref. [14]) will lead toequivalent fit results within theoretical uncertainties.
3
2
In the spirit of the Fermi theory, we incorporated the small propagator correction, 3/5 m2µ /MW, into ∆r (seebelow). This is also the convention adopted by the MuLan collaboration [15]. While this breaks with historicalconsistency, the numerical difference was negligible in the past.

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10. Electroweak Model and Constraints on New Physics

4

appearance was unforeseen and can be traced to the use of the muon pole mass in the prefactor [20].
The coefficient H3 = −15.3 ± 2.3 has been estimated in Refs. [21–23]. The remaining uncertainty
in GF is mostly experimental and has been reduced by an order of magnitude by the MuLan
collaboration [15] at the PSI.
10.2.2 The electromagnetic coupling
The fine structure constant, α, can be extracted from the anomalous magnetic moment of theelectron, ae = (1159652180.59 ± 0.13) × 10−12 [24]. Application of QED corrections up to five loop
order [25–27] allows to extract the value α−1 = 137.035999166(15). (Here the number in paratheses
denotes the uncertainty in the last digits shown.) Another approach combines measurements of
the Rydberg constant and atomic masses with interferometry of atomic recoil kinematics. Applied
to 87 Rb [28] and 133 Cs [29], this method implies the results α−1 = 137.035999206(11) and α−1 =137.035999046(27), respectively, which differ by 5.5 σ from each other, and when combined would
give α−1 = 137.035999183(10). Finally, combining the anomalous magnetic moment and atomic
interferometry methods, which unlike in the past are now in agreement, leads to the world average,
α−1 = 137.035999178(8) .

(10.11)This combination differs from the value in Section 1 (“Physical Constants”) in this Review, which
does not use the latest experimental inputs.
In most EW renormalization schemes it is convenient to define a running α dependent on
the energy scale of the process, with α−1 ≈ 137.036 appropriate at very low energy, i.e. close
to the Thomson limit. The OPAL [30] and L3 [31] collaborations at LEP could also observe
the running directly in small and large angle Bhabha scattering, respectively. For scales above
a few hundred MeV the low energy hadronic contribution to vacuum polarization introduces atheoretical uncertainty in α. In the modified minimal subtraction (MS) scheme4 [32] (used forb (4) (mτ )−1 = 133.450 ± 0.006 andthis Review), and with αs (MZ ) = 0.1187 ± 0.0017 we have αb (5) (MZ )−1 = 127.930 ± 0.008. The latter corresponds to a quark sector contribution (without theαtop) to the conventional (on-shell) QED coupling,αα(MZ ) =,(10.12)
1 − ∆α(MZ )(5)of ∆αhad (MZ ) = 0.02783 ± 0.00006. These values are updated with respect to Ref. [33] withthe uncertainty reduced (partly due to a more precise charm quark mass). Various evaluations(5)of ∆αhad (MZ ) are summarized in Table 10.1, where the relation5 between the MS and on-shelldefinitions (obtained using Refs. [37, 38]) is given by,"

α 100 1 7
M2
αs (MZ )b∆α(M− − ln 2Z +Z ) − ∆α(MZ ) =π 27
6 4 MWπ

605 44− ζ(3)
108
9



α2
+ 2s
π



976481 253
781
275
−
ζ(2) −
ζ(3) +
ζ(5)
23328
36
18
27

αs3
π3



+

1483517111 22781
397264931521255−
ζ(2) −
ζ(3) − ζ(2)2 +ζ(5)3359232
144
7776
81
7776

−

7315
5819
14675
ζ(7) +
ζ(2)ζ(3) +
ζ(3)2
324
54
162





= 0.007122(2)(5) ,(10.13)

4
In this Section we denote quantities defined in the MS scheme by a caret; the exception is the strong coupling
constant, αs , which will always correspond to the MS definition and where the caret will be dropped. Furthermore,(n)α(n) and αs denote the running couplings with n quark flavors.
5
In practice, α(MZ ) is directly evaluated in the MS scheme using the FORTRAN package GAPP [34], including
the QED contributions of both leptons and quarks. The leptonic three-loop [35] and four-loop [36] contributions in
the on-shell scheme have also been obtained.

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510. Electroweak Model and Constraints on New Physics

and where the first entry of the lowest order term is from fermions and the other two are from W ±loops, which are usually excluded from the on-shell definition. Fermion mass effects and correctionsof O(α2 ) contributing to Eq. (10.13) are small, partly cancel each other and are not included here.
The first error in Eq. (10.13) is parametric (from αs ) and the second is from the truncation of(5)
the perturbative expansion. The most recent results of ∆αhad (MZ ) [39–42] typically assume thevalidity of perturbative QCD (PQCD) at scales of ∼ 2 GeV and above and are in good agreementwith each other. In regions where PQCD is not trusted, one can use e+ e− → hadrons cross-section
data (for a list of references see, e.g., Ref. [41]) and isospin rotated information derived fromτ decay spectral functions [43], where the latter derive from OPAL [44], CLEO [45], ALEPH [46],and Belle [47]. Very recently new results appeared from the CMD-3 experiment [48] at the e+ e−
collider VEPP-2000. There is noticeable spread in the results, with lattice QCD [49] and CMD-3(KLOE) suggesting stronger (weaker) running. While VEPP-2000 and VEPP-4M scanned centerof-mass (CM) energies up to 2 GeV and between about 3 and 4 GeV, respectively, the BaBar
collaboration studied multi-hadron events radiatively returned from the Υ (4S), reconstructing theradiated photon and normalizing to µ± γ final states.
(5)
We include all of this information by using as actual input (fit constraint) instead of ∆αhad (MZ )the low energy contribution by the three light quarks,(3)∆αhad (2.0 GeV) = (60.30 ± 0.43) × 10−4 ,(10.14)

which we obtained relying on the analysis in Ref. [41], with τ decay data and the latest CMD-3
results added following Ref. [50]. Its correlation (86% [51]) with aµ discussed in Sec. 10.4.2, isbincluded in the fits. The non-linear αs dependence of α(MZ ) and the resulting correlation with
the input variable αs are fully taken into account by calculating the perturbative and heavy quarkbcontributions to α(MZ ) in each call of the fits according to Ref. [33]. Part of the uncertainty(5)−4(±0.37 × 10 ) in ∆αhad (MZ ) is from the combination of e+ e− annihilation data, τ decays into
two-pion final states, and constraints from lattice QCD (LQCD), but un-calculated higher order
perturbative QCD corrections (±0.21 × 10−4 ) and the MS quark mass values (see below) also
contribute.
10.2.3 Quark masses
Further free parameters entering into Eq. (10.2) are the quark and lepton masses, where mi is
the mass of the ith fermion ψi . For the light quarks, as described in Section 60 on “Quark Masses” inb u = 2.16+0.49
b d = 4.67+0.48
b s = 93.4+8.6
this Review, m−0.26 MeV, m−0.17 MeV, and m−3.4 MeV. These are running
MS masses evaluated at the scale µ = 2 GeV. For the charm mass we use the constraint [58],b c (mb c ) = 1274 ± 8 + 2616 [αs (MZ ) − 0.1182] MeV ,m(10.15)b c in each call of our fits to account for
which is based on QCD sum rules [59, 60], and recalculate m
its αs dependence. Similarly, for the bottom quark mass we use [61],b b (mb b ) = 4180 ± 8 − 108 [αs (MZ ) − 0.1182] MeV ,m(10.16)

with a theoretical correlation of about 60% arising from the PQCD truncation uncertainty which isb c (mb c ) and mb b (mb b ). To improve the precisions in mb c (mb c ) and mb b (mb b ) in the future it
similar for mwould help to remeasure the threshold regions of the heavy quarks, as well as the electronic decay
widths of the narrow cc̄ and bb̄ resonances. It would also be important to obtain data on the R-ratio
in e+ e− annihilation for center-of-mass energies & 11.2 GeV, as in this region QCD perturbation
theory cannot be sufficiently relied upon for b quarks [61].
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10. Electroweak Model and Constraints on New Physics(5)

Table 10.1: Evaluations of the on-shell ∆αhad (MZ ) by different groups (for a more complete list ofevaluations see the 2012 edition of this Review). For better comparison we adjusted central values
and errors to correspond to a common and fixed value of αs (MZ ) = 0.120, except for Ref. [49], forwhich αs is not an explicit input. References quoting results without the top quark decoupled areconverted to the five flavor definition. Ref. [52] uses ΛQCD = 380 ± 60 MeV; for the conversion weassumed αs (MZ ) = 0.118 ± 0.003.
Reference
Krasnikov, Rodenberg [53]Kühn & Steinhauser [54]Groote et al. [52]Martin et al. [55]de Troconiz, Yndurain [56]
Burkhardt, Pietrzyk [57]
Erler, Ferro-Hernández [39]Jegerlehner [40]Davier et al. [41]Keshavarzi et al. [42]Cè et al. [49]Result
0.02737 ± 0.00039
0.02778 ± 0.00016
0.02787 ± 0.00032
0.02741 ± 0.00019
0.02754 ± 0.00010
0.02750 ± 0.00033
0.02761 ± 0.00010
0.02755 ± 0.00013
0.02760 ± 0.00010
0.02761 ± 0.00011
0.02773 ± 0.00015

Comment√
PQCD for s > 2.3 GeV√
full O(αs2 ) for s > 1.8 GeV
use of QCD sum rulesincl. new BES data
PQCD for s > 2 GeV2√
PQCD for s > 12 GeV
conv. from MS scheme
Euclidean split technique√
PQCD for s = 1.8−3.7 & > 5 GeV√
PQCD for s > 11.2 GeV
LQCD for Euclidian Q2 < 5 GeV2

The top quark “pole” mass (the quotation marks are a reminder that the experiments do
not strictly measure the pole mass and that quarks do not form asymptotic states) has been
kinematically reconstructed by the Tevatron collaborations, CDF and DØ, in leptonic, hadronic,
and mixed channels with the result [62],mt = 174.30 ± 0.35 stat. ± 0.54 syst. GeV (Tevatron) .
(10.17)√
Likewise, using data from CM energies s = 7 and 8 TeV (Run 1), ATLAS and CMS (including
alternative technique measurements) at the LHC obtained [63],mt = 172.52 ± 0.14 stat. ± 0.30 syst. GeV (LHC Run 1) .
(10.18)√In addition, there are results derived from s = 13 TeV data (Run 2). The CMS collaboration
obtained mt = 171.77 ± 0.04 stat. ± 0.37 syst. GeV [64] in the lepton + jets channel, mt = 172.33 ±0.14 stat. ±0.69 syst. GeV [65] in the di-lepton channel, mt = 172.34±0.20 mostly stat. ±0.70 syst. GeV [66]in the all-jets channel, and mt = 172.13 ± 0.32 stat. ± 0.70 syst. GeV [67] in t-channel single top events(leptonic decays). ATLAS quotes mt = 174.41 ± 0.39 stat. ± 0.71 syst. GeV [68] from the lepton +jets channel and mt = 172.21 ± 0.20 stat. ± 0.78 syst. GeV [69] from the di-lepton channel. Assumingthat the correlation matrix for individual channels of the Run 1 combination [63] is approximately
applicable also at Run 2 (and that the same applies to the effective 9% correlation between ATLAS
and CMS which can also be extracted from Ref. [63]), we arrive at the result,
mt = 172.13 ± 0.06 stat. ± 0.30 syst. GeV (LHC Run 2) .

(10.19)To combine the two LHC runs, we use the largest correlation coefficient (0.51) between 7 TeV and
8 TeV results from the same experiment and channel in Ref. [63], and obtain,
mt = 172.30 ± 0.07 stat. ± 0.27 syst. GeV (LHC) .
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7

Table 10.2: Notations used to indicate the various sin2 θW schemes discussed in the text. Numerical values and the uncertainties induced by the imperfectly known SM parameters and unknown
higher orders [83] are also given for illustration.
SchemeOn-shellMS
MS (ND)MS
Effective angle

Notation
s2W
sbZ2
2
sbND
sb02
s̄2`

Value
0.223480.231290.231470.238730.23161

Uncertainty±0.00010±0.00004±0.00004±0.00005±0.00004

This treatment is highly conservative in view of the fact that for their lepton + jets channels novel
leptonic invariant mass and profile likelihood approaches were used by ATLAS and CMS, respectively, implying reduced correlations with the more traditional analyzes at Run 1. Note that thestatistical uncertainty of the combination (0.07 GeV) is larger than that of the CMS lepton+jets
measurement at 13 TeV (0.04 GeV) [64] because the latter has a relatively large systematic uncertainty and thus gets a small weight in the average. See Ref. [70] for a detailed discussion.
The Tevatron and LHC average values, (10.17) with (10.20), differ by 2.8 σ. In the inter-collider
combination we treat them as uncorrelated, yielding
mt = 172.61 ± 0.25 exp. GeV + ∆mMC ,(10.21)

where ∆mMC is defined to account for any difference between the top pole mass, mt , and themass parameter implemented in the Monte Carlo event generators employed by the experimentalgroups. ∆mMC is expected to be of order αs (Q0 )Q0 with a low scale Q0 ∼ O(1 GeV) [71], but
its value is unknown in hadron collider environments so that we will treat it as an uncertainty
instead6 , and choose for definiteness Q0 = Γt = 1.42 GeV to arrive at ∆mMC = 0 ± 0.52 GeV.
We further assume that an uncertainty [74] of ±0.32 GeV in the relation [75] between mt and theb t (mb t ), entering electroweak radiative correction libraries, including the renormalon
MS definition, mambiguity [76], is already included in ∆mMC , as mt merely serves as an intermediate bookkeeping
device in Ref. [71]. A promising future direction to arrive at a competitive independent constraint
on mt is to analyze differential top quark pair production cross-sections at next-to-next-to-leading
order (NNLO) [77, 78] as mt extraction based on them are easier to interpret, and experimentallythey have become much more precise recently [79–82]. The combination in Eq. (10.21) differs
slightly from the average, mt = 172.57 ± 0.29 exp. GeV, which appears in the Top Quark Listings inthis Review, and which is based exclusively on published results and assumes vanishing correlations.
For more details and references, see Section 61 on the “Top Quark” and the Quarks Listings in thisReview.
10.2.4 The weak mixing angleb
The observables sin2 θW and MW can be calculated from MZ , α(MZ ), and GF , when values
2
for mt and MH are given, or conversely, MH can be constrained by sin θW and MW . The valueof sin2 θW is extracted from neutral-current processes (see Sec. 10.3) and Z pole observables (seeSec. 10.5.4) and depends on the renormalization prescription. There are a number of popular
schemes [9] leading to values which differ by small factors depending on mt and MH . The notation
for these schemes is shown in Table 10.2.
6

However, see Ref. [72, 73] for proposed procedures to calibrate the Monte-Carlo mass parameter at hadroncolliders.

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10. Electroweak Model and Constraints on New Physics

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2 /M 2 to a definition(i) The on-shell scheme [84] promotes the tree-level formula sin2 θW = 1−MWZ
2of the renormalized sin θW to all orders in perturbation theory, i.e.,sin2 θW → s2W ≡ 1 −

2
MW
,
MZ2

(10.22a)A0
MW,(10.22b)
,
MZ =
1/2
cW
sW (1 − ∆r)√where cW ≡ cos θW , A0 = (πα/ 2GF )1/2 = 37.28038(1) GeV, and ∆r includes the radiative
corrections relating α, α(MZ ), GF , MW , and MZ . One finds ∆r ∼ ∆r0 − ρt tan−2 θW , whereb∆r0 = 1 − α/α(MZ ) = 0.06646(6) is due to the running of α, and
MW =

ρt =

3GF m2t
m2t√=
0.00934
×,(172.61 GeV)28 2π 2(10.23)represents the dominant (quadratic) mt dependence. There are additional contributions to∆r from bosonic loops, including those which depend logarithmically on MH and higher-ordercorrections7 . One has ∆r = 0.03685 ∓ 0.00020 ± 0.00006, where the first uncertainty is frommt (the ∓ sign indicates that increasing mt has the effect of decreasing ∆r) and the second
is from α(MZ ). Thus the value of s2W extracted from MZ includes an uncertainty (∓0.00007)
from the currently allowed range of mt . This scheme is simple conceptually. However, therelatively large (∼ 3%) correction from ρt causes large spurious contributions in higher orders.
s2W depends not only on the gauge couplings but also on the spontaneous-symmetry breaking,and it is awkward in the presence of any extension of the SM which perturbs the value ofMZ (or MW ). Other definitions are motivated by the tree-level coupling constant definition
θW = tan−1 (g 0 /g):(ii) In particular, the MS scheme introduces the quantity,
sin2 θbW (µ) ≡gb 02 (µ),
gb 2 (µ) + gb 02 (µ)(10.24)

where the couplings gb and gb0 are defined by modified minimal subtraction and the scale µ
is conveniently chosen to be MZ for many EW processes. The value of sbZ2 ≡ sin2 θbW (MZ )extracted from MZ is less sensitive than s2W to mt (by a factor of tan2 θW ), and is less sensitive
to most types of new physics. It is also very useful for comparing with the predictions of grandunification. There are actually several variant definitions of sin2 θbW (MZ ), differing according
to whether or how finite α ln(mt /MZ ) terms are decoupled (subtracted from the couplings).
One cannot entirely decouple the α ln(mt /MZ ) terms from all EW quantities because mt  mbbreaks SU(2) symmetry. The scheme that will be adopted here decouples the α ln(mt /MZ )terms from the γ–Z mixing [32, 85], essentially eliminating any ln(mt /MZ ) dependence in theformulae for asymmetries at the Z pole when written in terms of sbZ2 . (A similar definition isb The on-shell and MS definitions are related byused for α.)
sbZ2 = c (mt , MH )s2W = (1.0349 ± 0.0003)s2W .

(10.25)The quadratic mt dependence is given by c ∼ 1 + ρt / tan2 θW . The expressions for MW and
MZ in the MS scheme are
MW =

A0
,
sbZ (1 − ∆rbW )1/2

7

MZ =

MW
,
ρb 1/2 cbZ(10.26)All explicit numbers quoted here and below include the two- and three-loop corrections described near the endof Sec. 10.2.

31st May, 2024

10. Electroweak Model and Constraints on New Physics

9and one predicts ∆rbW = 0.06937 ± 0.00006. ∆rbW has no quadratic mt dependence, because
shifts in MW are absorbed into the observed GF , so that the error in ∆rbW is almost entirelybb ∼ 1 + ρt ,due to ∆r0 = 1 − α/α(MZ ). The quadratic mt dependence has been shifted into ρ
where including bosonic loops, ρb = 1.01016 ± 0.00009.
2 (used in the 1992 edition of this Review) does not decouple the(iii) A variant MS quantity sbND
α ln(mt /MZ ) terms [86]. It is related to sbZ2 by
sb 2Z =

2sbND

1 + απb d

,

d=

1
3



1
8
−
2
sb
3

 

1+

αs
π



ln

mt
15αs
−
MZ
8π



.

(10.27)

2 = −0.0002.
Thus, sbZ2 − sbND(iv) Some of the low-energy experiments discussed in the next section are sensitive to the weakmixing angle at almost vanishing momentum transfer [39, 87–89]. Thus, Table 10.2 also
includes sb02 ≡ sin2 θbW (0).
f(v) Yet another definition, the effective angle [90, 91], s̄2f ≡ sin θeff
, for the Z vector coupling tofermion f , is based on Z pole observables and described in Sec. 10.5.

10.2.5 Radiative corrections
Experiments are at such level of precision [83] that complete one-loop, dominant two-loop, andpartial three and four-loop radiative corrections must be applied. For neutral-current and Z pole
processes, these corrections are conveniently divided into two classes:
1. QED diagrams involving the emission of real photons or the exchange of virtual photons
in loops, but not including vacuum polarization diagrams. These graphs often yield finite
and gauge-invariant contributions to observable processes. However, they are dependent onenergies, experimental cuts, etc., and must be calculated individually for each experiment.
2. EW corrections, including γγ, γZ, ZZ, and W W vacuum polarization diagrams, as well asvertex corrections, box graphs, etc., involving virtual W and Z bosons. One-loop corrections [92] are included for all processes, and many two-loop corrections are also important.
In particular, two-loop corrections involving the top quark modify ρt in ρb, ∆r, and elsewhere
by


ρt
.
(10.28)ρt → ρt 1 + R(MH , mt )3
R(MH , mt ) can be described as an expansion in MZ2 /m2t , for which the leading m4t /MZ4 [93,94]
and next-to-leading m2t /MZ2 [95, 96] terms are known. The complete two-loop calculation of∆r (without further approximation) has been performed in Refs. [97–101]. More recently,
Ref. [102] obtained the MS quantities ∆rbW and ρb to two-loop accuracy, confirming the prediction of MW in the on-shell scheme from Refs. [99, 103] within about 4 MeV. Similarly,
the EW two-loop corrections for the relation between s̄2`,b and s2W are known [104–109], as
well as for the partial decay and total decay widths and the effective couplings of the Z boson [110–113]. For s̄s,c only two-loop corrections from diagrams with closed fermion loops areavailable [114], but given the experimental precision this is more than adequate.
The mixed QCD-EW contributions to gauge boson self-energies of order ααs m2t [115, 116],
ααs2 m2t [117, 118], and ααs3 m2t [119–121] increase the predicted value of mt by 6%. This is,however, almost entirely an artifact of using the pole mass definition for mt . The equivalentb t (mb t ) increase mt by less than 0.5%. The subcorrections when using the MS definition mleading ααs corrections [122–125] are also included. Further three-loop corrections of order
ααs2 [126, 127], α3 m6t , α2 αs m4t [128, 129], αf3 and αf2 αs [130, 131] are rather small (here αf
31st May, 2024

10. Electroweak Model and Constraints on New Physics

10

4 [132]
denotes electroweak corrections with a closed fermion loop). The same is true for α3 MH
corrections unless MH approaches 1 TeV. The theoretical uncertainty from unknown higherorder corrections [83] is estimated to amount to 4 MeV for the prediction of MW [103] and4.5 × 10−5 for s̄2` [114].

Throughout this Review we utilize EW radiative corrections from the program GAPP [34],which works entirely in the MS scheme, and which is independent of the ZFITTER [133] andGRIFFIN [134] packages.

10.3 Low energy electroweak observables
In the following we discuss EW precision observables obtained at low momentum transfers [135],
i.e., Q2  MZ2 . It is convenient to write the four-fermion interactions relevant to ν-hadron, ν-e,
as well as parity violating e-hadron and e-e neutral-current processes, in a form that is valid in anarbitrary gauge theory (assuming massless left-handed neutrinos). One has8 ,
GF
νe
νe 5
−L νe = √ ν̄γµ (1 − γ 5 )ν ē γ µ (gLV− gLAγ )e ,
2(10.29a)i
X h νqGF
νq−L νh = √ ν̄ γµ (1 − γ 5 )νgLL q̄ γ µ (1 − γ 5 )q + gLRq̄ γ µ (1 + γ 5 )q ,2
q(10.29b)

GF ee−L ee = − √ gAVē γµ γ 5 e ē γ µ e ,2(10.29c)i
GF X h eq
gAV ē γµ γ 5 e q̄ γ µ q + gVeqA ē γµ e q̄ γ µ γ 5 q ,−L eh = − √2 q(10.29d)where one must include the charged-current contribution for νe -e and ν̄e -e and the parity conserving
QED contribution for electron scattering. The SM tree level expressions for the four-Fermi couplings
are given in Table 10.3. Note that they differ from the respective products of the gauge couplingsin (10.7) in the radiative corrections and in the presence of possible physics beyond the SM.
10.3.1 Neutrino scattering
The cross-section in the laboratory system for νµ e → νµ e or ν̄µ e → ν̄µ e elastic scattering [9,136]
is (in this subsection we drop the redundant index L in the effective neutrino couplings),
dσν,ν̄
G2 me Eννe 2
νe 2
νe2 y me= F(gVνe ± gA
) + (gVνe ∓ gA
) (1 − y)2 − (gVνe2 − gA
)
dy
2π
Eν




,

(10.30)where the upper (lower) sign refers to νµ (ν̄µ ), and y ≡ Te /Eν (which runs from 0 to (1+me /2Eν )−1 )is the ratio of the kinetic energy of the recoil electron to the incident ν or ν̄ energy. For Eν  methis yields a total cross-section
G2 me Eν
1
νe 2
νe 2
σ= F
(gVνe ± gA
) + (gVνe ∓ gA
)
2π
3




.

(10.31)

The most accurate measurements of sin2 θW from ν-lepton scattering (see Sec. 10.6) are fromthe ratio R ≡ σνµ e /σ ν̄µ e , in which many of the systematic uncertainties cancel. The results are
sin2 θW = 0.211 ± 0.037 [137], sin2 θW = 0.195 ± 0.022 [138], and sin2 θW = 0.2324 ± 0.0083 [139],
where radiative corrections (other than mt effects) are small compared to the precision of present
8

We use here slightly different definitions (and to avoid confusion also a different notation) for the coefficients ofthese four-Fermi operators than we did in previous editions of this Review. The new couplings [13] are defined in thestatic limit, Q2 → 0, with specific radiative corrections included, while others (more experiment specific ones) are
assumed to be removed by the experimentalist. They are convenient in that their determinations from very different
types of processes can be straightforwardly combined.

31st May, 2024

11

10. Electroweak Model and Constraints on New Physics
Table 10.3: SM tree level expressions for the neutral-current parametersfor ν-hadron, ν-e, and e− scattering processes. To obtain the SM values
in the last column, the tree level expressions have to be multiplied by
the low-energy neutral-current ρ parameter, ρNC = 1.00060, and further
vertex and box corrections need to be added as detailed in Ref. [13]. The
dominant mt dependence is again given by ρNC ∼ 1 + ρt .
Quantity
νµ e
gLV
νµ e
gLA
ν u

SM tree level− 12 + 2 sb02
− 12

SM value−0.0395−0.5063

νµ d
gLL
νµ u
gLR
νµ d
gLR

1
2 b2
2 − 3 s0
− 12 + 31 sb02
− 23 sb02
1 b2
3 s0

−0.4288−0.1553

ee
gAV
eu
gAV
ed
gAV
gVeuA
gVedA

1
b2
2 − 2 s01
4 b2
− 2 + 3 s0
2 b2
1
2 − 3 s0
− 12 + 2 sb02
1b2
2 − 2 s00.0224−0.1886
0.3418−0.0349
0.0246µgLL

0.34570.0777

νe from theexperiments and have negligible effect. As shown in Fig. 10.1, one can determine gV,Aexperimental data as well. The cross-sections for νe -e and ν̄e -e may be obtained from Eq. (10.30)
νe by g νe + 1, where the 1 is due to the charged-current contribution.
by replacing gV,AV,AA precise determination of the on-shell s2W , which depends only very weakly on mt and MH , isobtained from deep inelastic scattering (DIS) of neutrinos [136,143] from (approximately) isoscalar
N C /σ CC of neutral-to-charged-current cross-sections has been measured
targets. The ratio Rν ≡ σνNνN
to 1% accuracy by CDHS [144] and CHARM [145] at CERN. CCFR [146] at Fermilab has obtained an even more precise result, so it is important to obtain theoretical expressions for Rν andN C /σ CC to comparable accuracy. Fortunately, many of the uncertainties from the strongRν̄ ≡ σν̄Nν̄Ninteractions and neutrino spectra cancel in the ratio. A large theoretical uncertainty is associatedwith the c-threshold, which mainly affects σ CC . Using the slow rescaling prescription [147, 148] thecentral value of sin2 θW from CCFR varies as 0.0111(mc /GeV−1.31), where mc is the effective massb c (mb c ), but their exact relation is unknown at higher
which is numerically close to the MS mass morders. For mc = 1.31 ± 0.24 GeV, which was determined from ν-induced di-muon production [149],
this contributes ±0.003 to the total error of ∆ sin2 θW = ±0.004 (the experimental uncertainty was
also ±0.003). This uncertainty largely cancels, however, in the Paschos-Wolfenstein ratio [150],R− =N C − σN C
σνN
ν̄N
.
CC − σ CCσνNν̄N(10.32)

It was measured by Fermilab’s NuTeV collaboration [151] for the first time, and required a highintensity and high-energy anti-neutrino beam.
31st May, 2024

10. Electroweak Model and Constraints on New Physics

12

1.0

--

0.7

(

)e

ee

0.6

--

0.0

-1.0
-1.0

0.1

-0.5

--

--

0.0

--

0.2

gVe

--

0.4

--

0.3

0.5

0.5

ee
--

-0.5

0.0

0.5

1.0

gAe
νe vs. g νe from neutrino-electron scattering and the SM predicFigure 10.1: Allowed contours in gAV
2tion as a function of sbZ . (The SM best fit value, sbZ2 = 0.23129, is also indicated.) The νe -e [140,141]and ν̄e -e [142] constraints are at 1 σ, while each of the four equivalent νµ (ν̄µ )-e [137–139] solutions(gV,A → −gV,A and gV,A → gA,V ) are at the 90% CL. The global best fit region (shaded) almost
exactly coincides with the corresponding νµ (ν̄µ )-e region. The solution near gA = 0 and gV = −0.5
is eliminated by e+ e− → `+ `− data under the weak additional assumption that the neutral current
is dominated by the exchange of a single Z boson.

A simple zeroth -order approximation is,2Rν = gL2 + gRr,Rν̄ = gL2 +2
R− = gL2 − gR
,

r≡

2
gR
,
r

CC
σν̄N
,
CC
σνN

(10.33a)(10.33b)where r is the ratio of ν̄ to ν charged-current cross-sections which can be measured directly9 , and15− sin2 θW + sin4 θW ,2
95
νµ u 2
νµ d 22gR≡ (gLR) + (gLR) ≈ sin4 θW .
9
ν u

ν d

µ
µ
gL2 ≡ (gLL
)2 + (gLL)2 ≈9(10.34a)(10.34b)

In the simple parton model, ignoring hadron energy cuts, r ≈ (1 + 3)/(3 + ), where  ∼ 0.125 is the ratio of
the fraction of the nucleon’s momentum carried by anti-quarks to that carried by quarks.

31st May, 2024

13

10. Electroweak Model and Constraints on New Physics

In practice, Eq. (10.33b) must be corrected for quark mixing, quark sea effects, c quark threshold
effects, non-isoscalarity, W –Z propagator differences, the finite muon mass, QED and EW radiative
corrections. Details of the neutrino spectra, experimental cuts, x and Q2 dependence of structure
functions, and longitudinal structure functions, enter only at the level of these corrections andtherefore lead to very small uncertainties. CCFR quotes s2W = 0.2236 ± 0.0041 for the reference
values (mt , MH ) = (175, 150) GeV with very little sensitivity to (mt , MH ).
The NuTeV collaboration found s2W = 0.2277±0.0016 (for the same reference values), which was
3.0 σ higher than the SM prediction [151]. However, since then several groups have raised concerns
2about the interpretation of the NuTeV result, which could affect the extracted gL,R(and thus s2W )including their uncertainties and correlation. These include the assumption of symmetric strange
and anti-strange sea quark distributions, the electron neutrino contamination from Ke3 decays,
isospin symmetry violation in the parton distribution functions and from QED splitting effects,
nuclear shadowing effects, and a more complete treatment of EW and QCD radiative corrections.
A more detailed discussion and a list of references can be found in the 2016 edition of this Review.
The precise impact of these effects would need to be evaluated carefully by the collaboration, but
in the absence of such an effort we do not include the νDIS constraints in our default set of fits.
Recently, the COHERENT collaboration was the first to observe the coherent elastic neutrino
nucleus scattering (CEνNS) process [152] on a target consisting mostly of 133 Cs and 127 I, and at
the opposite end of the kinematic scale where the momentum transfer is significantly smaller than
the inverse of the nuclear radius. Subsequently, COHERENT [153] observed CEνNS using a liquid
40 Ar detector, as well. The coherence enhances the process roughly proportional to the square of
the number of neutrons in the nuclei, but the process is difficult to observe as the experimental
signature is a mere keV scale nuclear recoil.
10.3.2 Parity violating lepton scattering
Reviews on weak polarized electron scattering may be found in Refs. [9, 154]. The SLAC
polarized electron-deuteron DIS (eDIS) experiment [155] measured the parity violating right-leftasymmetry,σR − σL
ARL ≡,(10.35)σR + σL
where σR,L is the cross-section for the deep-inelastic scattering of a right- or left-handed electron,eR,L N → eX. In the quark parton model,ARL
1 − (1 − y)2
=
a
+
a
,
1
2
Q2
1 + (1 − y)2(10.36)

where Q2 > 0 is the momentum transfer and y is the fractional energy transfer from the electron
to the hadrons. For the deuteron or other isoscalar targets, one has, neglecting the s quark andanti-quarks,
3GF
1 ed
eu
a1 = √
gAV
− gAV
2
5 2πα




3GF
3 5
≈ √
− + sb02
4 3
5 2πα


3GF
1
9GF
1
a2 = √
gVeuA − gVedA ≈ √sb02 −
2
4
5 2πα
5 2πα








,

(10.37a).

(10.37b)

The Jefferson Lab Hall A collaboration [156, 157] improved on the SLAC result by measuring ARL
at Q2 = 1.085 GeV2 and 1.901 GeV2 , and determined the weak mixing angle to 2% precision,
sbZ2 = 0.2299 ± 0.0043. In another polarized electron scattering experiment on deuterons, but inthe quasi-elastic kinematic regime, the SAMPLE experiment [158, 159] at MIT-Bates extracted
31st May, 2024

14

10. Electroweak Model and Constraints on New Physics

the combination gVeuA − gVedA at Q2 values of 0.038 GeV2 and 0.091 GeV2 . What was actually
determined were nucleon form factors from which the quoted results were obtained by the removalof a multi-quark radiative correction [160]. Other linear combinations of the effective couplings
have been determined in polarized lepton scattering at CERN in µ-12 C DIS [161] (the observable
was the double charge-helicity cross-section asymmetry), at Mainz in e-9 Be (quasi-elastic) [162],
and at Bates in e-12 C (elastic) [163]. More recent polarized electron scattering experiments, i.e.,
SAMPLE, the PVA4 experiment at Mainz, and the HAPPEX and GØ experiments at Jefferson
Lab, have focussed on the strange quark content of the nucleon [164].
ARL can also be measured in fixed target polarized Møller scattering, e− e− → e− e− , andreads [165],
ARL
1−y
G
ee
√ F
= −2 gAV.
(10.38)24Q
2πα 1 + y + (1 − y)4
It has been determined at low Q2 = 0.026 GeV2 in the SLAC E158 experiment [166], with theresult, ARL = (−1.31 ± 0.14 stat. ± 0.10 syst. ) × 10−7 . Expressed in terms of the weak mixing angle
in the MS scheme this yields sb 2 (161 MeV) = 0.2403 ± 0.0013, and as shown in Fig. 10.2 establishedthe scale dependence of the weak mixing angle at the level of 6.4 σ. One also extracts the modelee = 0.0190 ± 0.0027 [13]. One-loop radiative corrections andindependent effective coupling, gAV
implications are discussed in Ref. [87].
In a similar experiment and at about the same Q2 = 0.0248 GeV2 , the Qweak collaboration at
Jefferson Lab obtained ARL = (−2.265±0.073 stat. ±0.058 syst. )×10−7 [167,168] in elastic e− p → e− pscattering. To extract the physical quantity of interest, the weak charge of the proton [169], a large(≈ 30%) correction had to be applied to ARL arising from electromagnetic, strange, and axial form
factors. This was achieved by performing a global fit [170] including a large number of ARL data
points at larger Q2 , dominated by the HAPPEX result at Q2 = 0.109 GeV2 [171]. Finally, theeu + g ed = 0.0356 ± 0.0023, which translates into a weak mixing angle measurement
constraint, 2gAV
AV
2of sb (157 MeV) = 0.2382 ± 0.0011, could be deduced, after correcting for a relatively large anduncertain contribution from the γZ box diagram [172–175].
10.3.3 Atomic parity violation
There are precise measurements of atomic parity violation (APV) [9,176,177] in 133 Cs [178,179](at the 0.4% level [178]), 205 Tl [180,181], 208 Pb [182], and 209 Bi [183]. The EW physics is containedin the nuclear weak charges QW (Z, N ), where Z and N are the numbers of protons and neutrons
in the nucleus. In terms of the nucleon vector couplings,
1
ep
eu
ed
gAV
≡ 2gAV+ gAV≈ − + 2sb02 ,
2
1
en
eu
ed
gAV
≡ gAV
+ 2gAV≈+ ,2(10.39a)(10.39b)

one has,α1−,(10.40)2π
where the numerically small adjustments are discussed in Ref. [13] and include the result of theγZ-box correction from Ref. [184].
E.g., QW (133
78 Cs) is extracted by measuring experimentally the ratio of the parity violating
amplitude, EPNC , to the Stark vector transition polarizability, β, and by calculating theoretically
EPNC in terms of QW . One can then write,epen
QW (Z, N ) ≡ −2 [Z(gAV+ 0.00005) + N (gAV
+ 0.00006)]

QW (133
78 Cs) = N

Im EPNC
β




exp.

|e| aB QWIm EPNC N

31st May, 2024


th.

β
a3B



!
exp.+th.



a2B
|e|

!

,

(10.41)

10. Electroweak Model and Constraints on New Physics

15

0.245

RGE Running
Particle Threshold
Measurements

SLAC E158

sin2 ( )

0.240

APV

Qweak

0.235

eDIS
Tevatron

0.230

0.225

10 4

10 3

10 2

10 1

1

[GeV]

101

LEPI
SLC

102

LHC

103

104

Figure 10.2: Scale dependence of the weak mixing angle defined in the MS scheme [39,88] (for the
scale dependence in a mass-dependent renormalization scheme, see Ref. [87]). The minimum of thecurve corresponds to µ = MW , below which we switch to an effective theory with the W ± bosons
integrated out, and where the β-function for sb 2 (µ) changes sign. At MW and each fermion mass
there are also discontinuities arising from scheme dependent matching terms, which are necessary toensure that the various effective field theories within a given loop order describe the same physics.
However, in the MS scheme these are very small numerically and barely visible in the figure providedb q (mb q ). The width of the curve exceeds the theory uncertainty fromone decouples quarks at µ = m
strong interaction effects which at low energies is at the level of ±2 × 10−5 [39]. The Tevatron and
LHC measurements are strongly dominated by invariant masses of the final-state di-lepton pair ofO(MZ ) and can thus be considered as additional Z pole data points. For clarity we displayed theTevatron and LHC points horizontally to the left and right, respectively.
where aB is the Bohr radius. There are currently two semi-empirical approaches to β of similar precision. The ratio of the off-diagonal hyperfine amplitude to the vector polarizability was measured
directly by the Boulder group [185]. Combined with the hyperfine amplitude, computed precisely
in Ref. [186], one finds β = (26.957 ± 0.044 exp. ± 0.027 th. ) a3B . Alternatively, one can combine [187]the measurement of the ratio of scalar to vector transition polarizabilities [188] with the recent
calculation of the scalar polarizability [189] to obtain β = (26.887 ± 0.030 exp. ± 0.021 th. ) a3B , inagreement with earlier results [190, 191] based on this approach. The two determinations averageto β = (26.911 ± 0.025 exp. ± 0.017 th. ) a3B .
The uncertainties associated with the atomic wave function calculations are relatively small for
cesium [9, 192–194]. State-of-the-art many-body atomic structure computations of the parity nonconserving amplitude, Im EPNC = (0.8977 ± 0.0040) × 10−11 |e| aB QW /N [195–200], together withthe measurements [178,179] which can be combined to give Im EPNC /β = −1.5924±0.0055 mV/cm,
imply,QW (133(10.42)78 Cs) = −72.41 ± 0.26 exp. ± 0.33 th. ,31st May, 2024

16

10. Electroweak Model and Constraints on New Physics

ep
en = 36.25 ± 0.21. Within the SM this can also beor equivalently the constraint, 55gAV+ 78gAVtranslated into a determination of the weak mixing angle, sb 2 (2.4 MeV) = 0.2349±0.0018, where thescale setting follows the estimate in Ref. [201] for the typical momentum transfer for parity violation
experiments in Cs (the corresponding estimate for Tl amounts to 8 MeV). By comparing different
hyperfine transitions, the Boulder experiment in cesium also observed the parity violating weakcorrections to the nuclear electromagnetic vertex, called the nuclear anapole moment [202–204].
The theoretical atomic structure uncertainties are 3% for thallium [205] and even larger forthe other atoms. However, they mostly cancel if one takes ratios of parity violation in different
isotopes [206]. The first result of this type of experiment was announced recently by the Mainzgroup [207], who studied APV in 100 Yb, 102 Yb, 104 Yb, and 106 Yb, at the 0.5% level. The resulting
three ratios can be interpreted as a measurement of sb02 = 0.258 ± 0.052, and represent a very
complementary approach to search for BSM physics [208]. If the precision increases in the future,
one would ultimately face uncertainties from differences in the neutron charge radii [209,210]. Thesecan be constrained experimentally [211], e.g., by measuring ARL in heavier nuclei as done by the
PREX collaboration at Jefferson Lab on 208 Pb [212] and 48 Ca [213].

10.4 Precision flavor physics
In addition to cross-sections, asymmetries, parity violation, W , Z, Higgs and other collider
physics, there is a large number of experiments and observables testing the flavor structure of theSM. These are addressed elsewhere in this Review, and are generally not included in this Section.
However, we identify three precision observables with sensitivity to similar types of new physics as
the other processes discussed here. The branching fraction of the flavor changing transition b → sγ
is of comparatively low precision, but since it is a loop-level process (in the SM) its sensitivity to
new physics (and SM parameters, such as heavy quark masses) is enhanced. A discussion can befound in the 2010 edition of this Review.
The τ lepton lifetime and leptonic branching ratios are primarily sensitive to αs and not affected significantly by many types of new physics. However, having an independent and reliable
low energy measurement of αs in a global analysis allows the comparison with the Z lineshape determination of αs which shifts easily in the presence of new physics contributions. By far the mostprecise observable discussed here is the anomalous magnetic moment of the muon. Its combined experimental and theoretical uncertainty is smaller than typical electroweak scale contributions. The
electron magnetic moment is measured to even greater precision, and as discussed in Sec. 10.2.2
can be used to determine α. Its new physics sensitivity, however, is suppressed by an additional
factor of m2e /m2µ , unless there is a new light degree of freedom such as a dark Z [214] boson.
10.4.1 The τ lifetime
The extraction of αs from the τ lifetime ττ [215, 216] is standing out from other determinations
because of a variety of independent reasons:(i) The τ -scale is low, so that upon extrapolation to the Z scale (where it can be compared tothe theoretically clean Z lineshape determinations) the αs error shrinks by about an order ofmagnitude.
(ii) Yet, this scale is high enough that perturbation theory and the operator product expansion(OPE) can be applied.
(iii) These observables are fully inclusive and thus free of fragmentation and hadronization effects
that would have to be modeled or measured.
(iv) Duality violation (DV) effects are most problematic near the branch cut but there they aresuppressed by a double zero at s = m2τ .
(v) There are data [44,46,217] to constrain non-perturbative effects both within and breaking the
31st May, 2024

10. Electroweak Model and Constraints on New Physics

17

OPE.
(vi) A complete four-loop order QCD calculation is available [218–222] in the massless limit.
(vii) Large effects associated with the QCD β-function can be re-summed [223] in what has become
known as contour improved perturbation theory (CIPT).
However, while CIPT certainly shows faster convergence in the lower (calculable) orders, doubts
have been cast on the method by the observation of discrepancies in a specific model [224] and therealization that CIPT appears to be incompatible with the OPE [225, 226]. Moreover, it has beenfound that the construction of a renormalon-free scheme for the gluon condensate brings the CIPT
results in agreement with ordinary fixed order perturbation theory (FOPT), without significantly
modifying the latter [227, 228]. We therefore use the FOPT expressions [60, 222, 229],ττ = ~

1 − Bτs= 290.75 ± 0.36 fs ,Γτe + Γτµ + Γτud

(10.43)andG2 m5 |Vud |2Γτud = F τ 3S(mτ , MZ )64π"

3 m2τ − m2µ
1+
2
5 MW

(3)

!

×

b
α2
α3
α4 α
αs (mτ )+ 5.202 2s + 26.37 3s + 127.1 4s +
1+
π
π
π
π
π

85 π 2
−
24
2

!

#

+ δNP

, (10.44)

where Γτe and Γτµ can be taken from Eq. (10.9) with obvious replacements. The relative fractionof strangeness changing (∆S = −1) decays, Bτs = 0.0292 ± 0.0004, is based on experimental datab s (mτ ), is not well known and the QCD expansion
since the value for the strange quark mass, m2b s converges poorly and cannot be trusted. S(mτ , MZ ) = 1.01908 ± 0.0003 is a
proportional to mlogarithmically enhanced EW correction factor [230] with higher orders re-summed [231].
δNP collects non-perturbative and quark-mass suppressed contributions, including the dimension
four, six and eight terms in the OPE, as well as DV effects. We use the average δNP = 0.0141 ±0.0072 derived from the τ decay spectral functions provided by OPAL [44] and ALEPH [46, 217],
which give δNP = 0.000 ± 0.012 and δNP = 0.022 ± 0.009, respectively. These numbers are basedon the original analyses in Refs. [232, 233], but are modified to correspond to a strict FOPT
analysis as is appropriate for our purpose10 (for alternative analyses, see Section 9 on “Quantum
Chromodynamics” in this Review ).
The dominant uncertainty arises from the truncation of the FOPT series and is conservatively
taken as the αs4 term (this is re-calculated in each call of the fits, leading to an αs -dependent andthus asymmetric error) until a better understanding of the numerical differences between FOPT
and CIPT has been gained. Our perturbative error covers almost the entire range from using
CIPT to assuming that the nearly geometric series in Eq. (10.44) continues to higher orders. The
experimental uncertainty in Eq. (10.43) is from the combination of the two leptonic branching ratioswith the direct ττ . Included are also various smaller uncertainties (±0.16 fs) from other sources.
Based on the method of Refs. [60, 234], we obtain in total
αs(4) (mτ ) = 0.312+0.016−0.013 ,αs(5) (MZ ) = 0.1171+0.0018−0.0017 ,(10.45)which represents a 1.5% determination of αs (MZ ). For more details, see Refs. [232, 233] where theτ spectral functions themselves and an estimate of the unknown αs5 term were used as additional
inputs.
10

We are indebted to Diogo Boito, Maarten Golterman, Kim Maltman and Santiago Peris for privately communicating these results to us.

31st May, 2024

10. Electroweak Model and Constraints on New Physics

18

10.4.2 The muon anomalous magnetic moment
The world average of the muon anomalous magnetic moment11 ,gµ − 2aexp= (1165920.59 ± 0.22) × 10−9 ,(10.46)µ =2is the combination of the final result of the BNL E821 collaboration [235] and the recent resultof the Muon g − 2 collaboration at Fermilab [236]. The QED contribution has been calculated
to five loops [25, 27, 237–239] (fully analytic to three loops [240–244]). The estimated SM EW
contribution [245–250], aEW= (1.54 ± 0.01) × 10−9 , includes two-loop [251–255] and leading threeµ
loop [256, 257] corrections and is at the level of more than four times the current total uncertainty.
The limiting factor in the interpretation of the result are the uncertainties from hadronic effects. The most recent evaluations of the leading-order (two-loop) hadronic vacuum polarization
contribution using data from e+ e− → hadrons obtained ahad,VP(α2 ) = (68.81 ± 0.41) × 10−9 [258],µ
ahad,VP(α2 ) = (69.23±0.33)×10−9 [259], ahad,VP(α2 ) = (69.40±0.40)×10−9 [41], and ahad,VP(α2 ) =
µ
µ
µ
−9
+
−
(69.28 ± 0.24) × 10 [42]. Our analysis is based on the e e data compiled in Ref. [41], where byutilizing the assessment in Ref. [50], we add τ -decay information and the very recent CMD-3 result [48] to estimate the dominant pion form factor contribution. We complement the contributions√
up to s = 2 GeV, for which we find ahad,VP(α2 , 2 GeV) = (65.12 ± 0.30) × 10−9 , with analyticalµ
PQCD expressions for energies beyond 2 GeV and for the c and b quark contributions [244, 260].
(α2 ) from lattice QCD calculations [261]. We
By now there are also precise results for ahad,VPµ(α2 , 2 GeV) = (65.71 ± 0.55) × 10−9 , obtained from the result with theuse the constraint, ahad,VPµ
smallest quoted uncertainty (by the BMW collaboration [262]), ahad,VP(α2 ) = (70.75±0.55)×10−9 ,µby subtracting the perturbative QCD contribution including from charm and bottom quarks. It
is important to note that there is a strong correlation (estimated to 86% in Ref. [51]) between(3)
ahad,VP(α2 , 1.8 GeV) and ∆αhad (1.8 GeV) discussed in Sec. 10.2.2. As a result, using the latticeµby slightly increasing
calculation from Ref. [49] as a constraint for ∆αhad has an impact on ahad,VPµ
2
12
had,VP(α , 2 GeV) = (65.71 ± 0.55) ×
it . Incorporating this effect together with the BMW result, aµ−9
had,VP2
10 , and the data-driven value, aµ(α , 2 GeV) = (65.12 ± 0.30) × 10−9 , we obtainahad,VP(α2 , 2 GeV) = (65.44 ± 0.25) × 10−9 ,µ(10.47)

which we use as an input to our fits. The BMW result is roughly 2 σ higher than the value derived
from the KLOE data [263], is slightly lower than that from CMD-3 [48], and agrees well with τ
decays. In fact, the discrepancy between KLOE and CMD-3 cross-sections exceeds 5σ locally, andamounts to 3.3σ if one considers the entire overlap region of the two experiments [50]. A moredetailed analysis [264, 265] of the difference between BMW and the data-driven evaluation using
KLOE input suggests that it originates mostly from hadronic contributions below 2 GeV to thevacuum polarization. The BMW result for this hadronic window is supported by findings from otherlattice collaborations [266–272]. Possible explanations for this discrepancy include underestimatedsystematic uncertainties in the analysis of some of the available experimental data and higher-order
initial-state radiation effects [50].
Sub-leading hadronic vacuum polarization effects at three-loop [273] and four-loop order [274]contribute ahad,VP(α3 ) = (−0.983±0.004)×10−9 [42] and aµhad,VP (α4 ) = (0.124±0.001)×10−9 [274],µ
respectively. To account for the larger vacuum polarization effects suggested by lattice QCD, we
11

In what follows, we summarize the most important aspects of aµ and give some details on the evaluation in ourfits. For more details and references, see Section 56 on the “Muon Anomalous Magnetic Moment” in this Review.
There are some numerical differences, which are well understood and arise because internal consistency of the fitsrequires the calculation of all observables from analytical expressions and common inputs and fit parameters, so thatan independent evaluation is necessary for this Section.
(3)
12
Conversely, adding the BMW constraint for ahad,VP
has the effect of increasing ∆αhad (2 GeV).
µ31st May, 2024

19

10. Electroweak Model and Constraints on New Physics

increase both ahad,VP(α3 ) and ahad,VP(α4 ) by 1%. The correlations with the two-loop hadronicµµcontribution and with ∆α(MZ ) (see Sec. 10.2) were considered in Ref. [244].
The other hadronic uncertainty is induced by the three-loop light-by-light scattering amplitude,
where a number of independent model calculations yield results which are in reasonable agreement−9 [276],with each other, ahad,γ×γ(α3 ) = (1.36 ± 0.25) × 10−9 [275], ahad,γ×γ(α3 ) = 1.37+0.15µµ−0.27 × 10
aµhad,γ×γ (α3 ) = (1.05 ± 0.26) × 10−9 [277], and ahad,γ×γ(α3 ) = (1.03 ± 0.29) × 10−9 [258], but theµsign of this effect is opposite [278] to the one quoted in the 2002 edition of this Review. Thereis also an upper bound given by ahad,γ×γ(α3 ) < 1.59 × 10−9 [276] but this requires an ad hocµassumption, too. Efforts to improve the evaluation by using experimental input where available
yield the lower values, ahad,γ×γ(α3 ) = (0.87 ± 0.13) × 10−9 [279] and ahad,γ×γ(α3 ) = (0.92 ± 0.19) ×µµ−910 [261]. See also Ref. [280] for a recent discussion of the axial-vector contribution and shortdistance constraints. Lattice calculations have reached similar levels of precision, ahad,γ×γ(α3 ) =µ
(1.10 ± 0.15) × 10−9 [281,282] and ahad,γ×γ(α3 ) = (1.25 ± 0.15) × 10−9 [283], and are consistent withµthe model and data-driven calculations. For our fits we average the two lattice results (with thesystematic errors considered as fully correlated) and combine them with the data-driven result [261]to obtain,
ahad,γ×γ(α3 ) = (1.11 ± 0.10) × 10−9 ,(10.48)µwhich we have shifted by 1 × 10−11 to account for the perturbative charm quark treatment ofRef. [276]. Higher-order contributions with a hadronic light-by-light scattering subgraph have been
estimated in Ref. [284] with the result, ahad,γ×γ(α4 ) = (0.02 ± 0.01) × 10−9 [261].
µ
Altogether, the SM prediction isatheory= (1165919.46 ± 0.27) × 10−9 ,µ(10.49)where the error is from the hadronic uncertainties excluding parametric ones, such as from αs andthe heavy quark masses. We evaluate the correlation of the total (experimental plus theoretical)
uncertainty in aµ with ∆α(MZ ) to amount to 43%. The overall 3.2 σ discrepancy between theexperimental value (10.46) and the SM prediction,
theoryaexp= (1.13 ± 0.35) × 10−9 ,µ − aµ(10.50)could be due to fluctuations (aexp
is statistics dominated) or underestimates of the theoreticalµuncertainties. On the other hand, the deviation could also arise from physics beyond the SM, suchas supersymmetric models with large tan β and moderately light superparticle masses [285], or adark Z boson [214].

10.5 Physics of themassive electroweak bosons√

If the CM energy s is large compared to the fermion mass mf , the unpolarized Born crosssection for e+ e− → f f¯ [286] can be written as,idσπα2(s) h=
F1 (1 + cos2 θ) + 2F2 cos θ + B ,d cos θ2s

(10.51a)f2
e2
F1 = Q2e Q2f − 2χQe Qf ḡVe ḡVf cos δR + χ2 (ḡVe2 + ḡA)(ḡVf 2 + ḡA),(10.51b)e f
e f f
F2 = −2χQe Qf ḡAḡA cos δR + 4χ2 ḡVe ḡAḡV ḡA ,(10.51c)wheretan δR =MZΓ Z
2

MZ − s

2

,

GF
sM Z
χ= √
.
h
i
2 2πα(s) (M 2 − s)2 + M 2 Γ 2 1/2
Z

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Z

Z

(10.52)

20

10. Electroweak Model and Constraints on New Physicsf
B accounts for box graphs involving virtual Z and W bosons, and the ḡV,A
are defined in Eq. (10.53)
below. M Z and Γ Z correspond to mass and width definitions based on a Breit-Wigner shape withan energy-independent width (see Section 55 on the “Z Boson” in this Review). The differential
cross-section receives important corrections from QED effects in the initial and final states, andinterference between the two [287]. For q q̄ production, there are additional final-state QCD corrections, which are relatively large. Note also that the equations above are written in the CM frameof the incident e+ e− system, which may be boosted due to the initial-state QED radiation.
Some of the leading virtual EW corrections are captured by the running QED coupling α(s)and the Fermi constant GF . The remaining corrections to the Zf f¯ interactions are absorbed byreplacing the tree-level couplings in Eq. (10.7) with the s-dependent effective couplings [288],ḡVf =fḡA=√

ρf tf3L − 2Qf κf sin2 θW√

ρf tf3L .

,(10.53a)(10.53b)√In these equations, the effective couplings are to be taken at the scale s, but for notational
simplicity we do not show this explicitly. At tree-level, ρf = κf = 1, but inclusion of EW radiative
corrections leads to ρf 6= 1 and κf 6= 1, which depend on the fermion f and on the renormalization
scheme. In the on-shell scheme, the quadratic mt dependence is given by,ρf ∼ 1 + ρt ,

κf ∼ 1 +ρt,tan2 θW(10.54)b f ∼ 1, for f 6= b, andwhile in MS, ρbf ∼ κ4ρbb ∼ 1 − ρt ,32
b b ∼ 1 + ρt .
κ(10.55)3√b sbZ2 cbZ2 in the
In the MS scheme the normalization is changed according to GF MZ2 /2 2π → α/4
second Eq. (10.52).
As reviewed in Sec. 10.2.5, for the high precision Z pole observables discussed below, many
additional bosonic and fermionic loop effects, vertex corrections, and higher order contributions,b ` = 1.0014,
etc., must be included. For example, in the MS scheme one then has ρb` = 0.9977, κbb
ρb = 0.9867, and κb = 1.0068.
To connect to measured quantities, it is convenient to define an effective angleb f sbZ2 = κf s2W ,s̄2f ≡ sin2 θ̄W f ≡ κ(10.56)√f
in terms of which ḡVf and ḡA
are given by ρf times their tree-level formulae. One finds that theb f (f 6= b) are almost independent of mt and MH , and thus one can write,κs̄2` = sbZ2 + 0.00032 ,(10.57)

while the κf for the on-shell scheme are mt dependent.
10.5.1 The Z boson mass
The mass of the Z boson, MZ = 91.1876±0.0021 GeV, has been determined from the Z lineshapescan at LEP 1 [288]. Very recently the CDF collaboration at the Tevatron [289] determined MZ
from fits to the dimuon and dielectron mass distributions. The dimuon channel strongly dominates
so that our combination, MZ = 91.192 ± 0.007 GeV, is insensitive (at the level of the quoted digits)
31st May, 2024

21

10. Electroweak Model and Constraints on New Physics

to variations in the assumption concerning possible cross-channel correlations. Combined we findthe world average,
MZ = 91.1880 ± 0.0020 GeV .
(10.58)
These values correspond to a definition based on a Breit-Wigner shape with an energy-dependent
width, which differs from M Z in eq. (10.52)13 .
10.5.2 Electroweak physics off the Z pole
Experiments at PEP, PETRA and TRISTAN have measured the unpolarized forward-backward
asymmetry, AF B , and the total cross-section relative to pure QED, R, for e+ e− → `+ `− , ` = µ or√
τ at CM energies s < MZ . They are defined as
AF B ≡

σF − σB,
σF + σBR=

σ,Rini ⊗ σQED

(10.59)where σF (σB ) is the cross-section for `− to travel forward (backward) with respect to the e−
direction, σQED is the tree-level cross-section from s-channel photon exchange, and Rini ⊗ denotes
convolution with initial-state QED corrections. Neglecting box graph contributions, they are givenby,
AF B =

3 F2
,
4 F1

R = F1 .

(10.60)For the available data, it is sufficient to approximate the EW corrections through the leading
running α(s) and quadratic mt contributions [290], as described above. Reviews and formulae fore+ e− → hadrons may be found in [9, 291, 292].
LEP 2 [294] ran at several energies above the Z pole up to ∼ 209 GeV. Measurements weremade of a number of observables, including the total production cross-sections of f f¯ pairs forf = µ, τ , and q (hadrons), of four-fermion final states, of γγ, ZZ, W W and W W γ. The differential
cross-sections for all three lepton flavors, and the leptonic and hadronic W branching ratios werealso extracted.
Among the most important LEP 2 results were the measurements [294] of the W boson mass,
MW , which were dominated by kinematic reconstruction, but included the complementary albeit
statistics limited and thus much less precise determination from a W W threshold cross-section
measurement. The kinematic method was also employed at the Tevatron [301] and by ATLAS [302]and LHCb [296] at the LHC. A recent combination [298] of all available MW measurements, using
a careful calibration of simulation tools and PDFs, obtained the world average,
MW = 80.3946 ± 0.0115 GeV .

(10.61)

However, the χ2 probability of this combination is 0.5% or less, depending on the chosen PDF
set, which is mostly due to the W mass measurement by CDF from Run II at the Tevatron [289],
MW = 80.432 ± 0.016 GeV (adjusted to the common PDF set CT18 [303] in Ref. [298]). It differs
by almost 4 σ from the other measurements of MW , while the latter agree well among each other,with the average [298]MW = 80.3692 ± 0.0133 GeV ,(10.62)
without CDF II. Subsequently, the ATLAS result has been updated [295], resulting in a significant
downward shift of MW . It also included a measurement of the W boson width, ΓW , together witha strong 30% anticorrelation with MW . The new ATLAS ΓW value differs by about 2 σ from
13
Note that M Z is defined through the complex pole of the propagator, which ensures that it is gauge-invariant
and theoretically consistent, and which leads to a Breit-Wigner shape with a constant width. The two definitions
differ numerically, and this difference has to be accounted for in theoretical calculations.

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22

10. Electroweak Model and Constraints on New Physics

Table 10.4: Non-Z pole observables, compared with the SM best fit predictions. The SM prediction for ΓH is without electroweak corrections as in Ref. [293]. The first and second MW andΓW values are from LEP 2 [294] and ATLAS [295], respectively. The remaining MW values arefrom LHCb [296], DØ [297], and CDF [289], respectively, adjusted according to Ref. [298], andwhere the last one [in brackets] is omitted from the fits (see text and Sec. 10.8). The third ΓW is
the Tevatron combination [299], shifted to correspond to the current SM prediction of MW (whatis actually measured is a linear combination containing an MW term). The hadronic branchingratio for W decays combines LEP 2 [294] and CMS [300] and assumes lepton flavor universality.
νe are dominated by the CHARM II [139] results, g νe = −0.035 ± 0.017
The world averages for gV,AV
νe = −0.503 ± 0.017. The τ value is the τ lifetime world average computed by combining
and gAτthe direct measurements with values derived from the leptonic branching ratios [60]; in this case,the theory error is included in the SM prediction. In all other SM predictions, the uncertaintyb
is parametric from MZ , MH , mt , mb , mc , α(MZ ), and αs , and theoretical from unknown higher
orders [83], where correlations due to both types have been accounted for. The column denoted by
Pull gives the standard deviations.
Quantity
mt [GeV]MH [GeV]ΓH [MeV]
MW [GeV]ΓW [GeV]

B(W → hadrons)
gVνe
νe
gA
QW (e)QW (p)QW (Cs)QW (Tl)sbZ2 (eDIS)ττ [fs]1α2 (gµ − 2 − π )Value
172.61 ± 0.58
125.10 ± 0.09
3.5 ± 1.5
80.376 ± 0.033
80.355 ± 0.016
80.347 ± 0.033
80.372 ± 0.026[80.432 ± 0.016]
2.195 ± 0.083
2.198 ± 0.049
2.059 ± 0.049
0.6736 ± 0.0018−0.040 ± 0.015−0.507 ± 0.014−0.0403 ± 0.0053
0.0719 ± 0.0045−72.41 ± 0.42
−116.4 ± 3.6
0.2299 ± 0.0043
290.75 ± 0.36(4510.86 ± 0.35) × 10−9Standard Model
172.85 ± 0.55
125.10 ± 0.09
4.09 ± 0.05
80.356 ± 0.005

2.089 ± 0.001

0.6751 ± 0.0001−0.0395 ± 0.0001−0.5063−0.0469 ± 0.0002
0.0705 ± 0.0002−73.26 ± 0.01
−116.93 ± 0.01
0.23129 ± 0.00004
288.59 ± 2.31
(4509.73 ± 0.03) × 10−9

Pull
−0.4
0.0
−0.4
0.6
−0.1
−0.3
0.6
—
1.3
2.2
−0.6
−0.8
0.0
0.0
1.3
0.3
2.0
0.1
−0.3
0.9
3.2

the Tevatron width result [299] and the SM prediction. To include it in the global fit, insteadof employing (10.62) verbatim, we combine (i) the adjusted results for DØ and LHCb quoted
in Ref. [298], (ii) LEP [294], (iii) the new ATLAS results [295], and (iv) the Tevatron width
determination [299], keeping the approximate 49% PDF anticorrelation between ATLAS and LHCb.
Including further small correlations between DØ and the LHC experiments, this results in the
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10. Electroweak Model and Constraints on New Physics

23

updated overall averages (without MW from CDF as recommended by the authors of Ref. [298]14 ),
MW = 80.360 ± 0.012 GeV ,(10.63)

ΓW = 2.136 ± 0.032 GeV ,(10.64)

with a correlation coefficient of close to −0.3. Alternatively, fixing ΓW to the SM prediction gives
MW = 80.366 ± 0.012 GeV .

(10.65)For details and references, see Section 54 on the “Mass and Width of the W Boson” in this Review.
Strong constraints on anomalous triple and quartic gauge couplings have been obtained at
LEP 2, the Tevatron, and the LHC. These are described in detail in “Extraction of Triple Gauge
Couplings (TGCs)”, “Anomalous W/Z Quartic Couplings (QGCs)” notes in the W particle listingand in “Anomalous ZZγ, Zγγ, and ZZV Couplings” note in the Z particle listing.
After their discovery of the Higgs boson [304, 305], the LHC experiments are now performing
high precision measurements of its mass, MH . We average the results, MH = 125.11 ± 0.09 stat. ±0.06 syst. GeV from ATLAS [306], and MH = 125.08 ± 0.10 stat. ± 0.07 syst. GeV from CMS [307], bytreating the smaller systematic error as common among the two determinations, and arrive at,
MH = 125.10 ± 0.07 stat. ± 0.06 syst. GeV (LHC) .

(10.66)

There are also first measurements of the Higgs boson width, ΓH = 2.9+1.9−1.4 MeV from CMS [307]and ΓH = 4.5+3.3
MeV
from
ATLAS
[308].
We
performasimpleGaussian
average using the upper−2.5
error of CMS and the lower error of ATLAS. For further references and more details on Higgs boson
properties, see Section 11 on the “Status of Higgs Boson Physics” in this Review.
The principal non-Z pole observables discussed here and in Sections 10.2−10.4 are summarizedin Table 10.4.
10.5.3 Z pole physics√
High precision measurements of various Z pole ( s ≈ MZ ) observables [9, 318, 319] have been
performed at LEP 1 and SLC [288, 315, 316, 320, 321], as summarized in Table 10.5. These include
the Z mass and total width, ΓZ , and partial widths Γf f¯ for Z → f f¯, where f = e, µ, τ , light
hadrons, b, and c. It is convenient to use the variables MZ , ΓZ ,σhad ≡

12πΓe+ e− Γhad,
MZ2 ΓZ2

R` ≡

Γhad
,
Γ`+ `−

Rq ≡

Γqq̄
,
Γhad

(10.67)for ` = e, µ or τ , and q = b or c, where Γhad is the partial decay width into hadrons. Most of these are
weakly correlated experimentally. The three values for R` are consistent with lepton universality15(although Rτ is somewhat low compared to Re and Rµ ), but we use the general analysis in which
the three observables are treated as independent. Similar remarks apply to A0,`F B defined through0,τ
Eq. (10.68) with Pe = 0, where AF B is somewhat high. Initial-state radiation reduces the peakcross-section by more than 25%, where O(α3 ) QED effects induce a large anti-correlation (−30%)
between ΓZ and σhad . The anti-correlation between Rb and Rc amounts to −18% [288]. The R` areinsensitive to mt except for the Z → bb̄ vertex, final-state corrections, and the implicit dependence
through sin2 θW . Thus, they are especially useful for constraining αs .
Very important constraints follow from measurements of various Z pole asymmetries. These
include the forward-backward asymmetry, AF B , and the polarization or left-right asymmetry, ALR ,14

See Sec. 10.8 for the impact of including the CDF W mass result.
The ratio of branching fractions for Z bosons decaying into e+ e− relative to µ+ µ− final states has also beenmeasured by ATLAS [322], obtaining Re/µ = 1.0026 ± 0.0050 , in perfect agreement with lepton universality.
15

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10. Electroweak Model and Constraints on New Physics

24

Table 10.5: Principal Z pole observables and their SM predictions (cf. Table 10.4). The first
MZ is from LEP 1 [288] and the second from CDF [289]. The first s̄2` is the effective weak mixing
angle extracted from the hadronic charge asymmetry at LEP 1 [288], the second is the combined
value from the Tevatron [309], and the third is from the LHC [310–314]. The values of Ae are (i)from ALR for hadronic final states [315]; (ii) from ALR for leptonic final states and from polarized
Bhabha scattering [316]; and (iii) from the angular distribution of the τ polarization at LEP 1 [288].
The Aτ values are from SLD [316], the total τ polarization from LEP [288], and from CMS [317],
respectively. Note that the SM errors in ΓZ , the R` , and σhad are largely dominated by theuncertainty in αs .
Quantity
MZ [GeV]Value
91.1876 ± 0.0021
91.192 ± 0.007
2.4955 ± 0.0023
41.481 ± 0.033
20.804 ± 0.050
20.784 ± 0.034
20.764 ± 0.045
0.21629 ± 0.00066
0.1721 ± 0.0030

2.4940 ± 0.0009
41.481 ± 0.009
20.736 ± 0.010
20.736 ± 0.010
20.781 ± 0.010
0.21583 ± 0.00002
0.17221 ± 0.00003

Pull
−0.4
0.6
0.7
0.0
1.4
1.4
−0.4
0.7
0.0

AF B

(0,e)

0.0145 ± 0.0025

0.01606 ± 0.00006

−0.6

(0,µ)
AF B
(0,τ )
AF B
(0,b)
AF B
(0,c)
AF B
(0,s)
AF B
s̄2`

0.0169 ± 0.0013

0.6

0.0188 ± 0.0017

1.6

ΓZ [GeV]σhad [nb]
Re
Rµ
Rτ
Rb
Rc

Ae

Aµ
Aτ

Ab
Ac
As

Standard Model
91.1884 ± 0.0019

0.0996 ± 0.0016

0.1026 ± 0.0002−1.8

0.0707 ± 0.0035

0.0732 ± 0.0002−0.7

0.0976 ± 0.0114
0.2324 ± 0.0012
0.23148 ± 0.00033
0.23145 ± 0.00028
0.15138 ± 0.00216
0.1544 ± 0.0060
0.1498 ± 0.0049
0.142 ± 0.015
0.136 ± 0.015
0.1439 ± 0.0043
0.144 ± 0.015
0.923 ± 0.020
0.670 ± 0.027
0.895 ± 0.091

0.1027 ± 0.0002
0.23161 ± 0.00004

−0.4
0.7
−0.4
−0.6
2.3
1.3
0.7
−0.3
−0.7
−0.6
−0.2
−0.6
0.1
−0.4

0.1463 ± 0.0003

0.93470.6674 ± 0.0001
0.9356

defined analogously to Eq. (10.35). The latter was measured precisely by the SLD collaboration at
the SLC [315], and has the advantages of being very sensitive to s̄2` and that systematic uncertain31st May, 2024

10. Electroweak Model and Constraints on New Physics

25

ties largely cancel. After removing initial-state QED corrections and contributions from photon
exchange, γ–Z interference, as well as the EW boxes in Eq. (10.51a), one can use the effective
tree-level expressions,3
Ae + P e,
A F B = Af
4
1 + P e Ae

ALR = Ae Pe ,(10.68)where,Af ≡f
2ḡVf ḡA

1 − 4|Qf |s̄2fḡVf 2 + ḡA

1 − 4|Qf |s̄2f + 8Q2f s̄4f

=
f2

.

(10.69)Pe is the initial e− polarization, so that the second equality in Eq. (10.70) is reproduced for Pe = 1,
(0,f )and the Z pole forward-backward asymmetries at LEP 1 (Pe = 0) are given by AF B = 43 Ae Af(0,q)

for f = e, µ, τ , b, c, s [288], and q, and where AF B refers to the hadronic charge asymmetry.
(0,e)
Corrections for t-channel exchange and s/t-channel interference cause AF B to be strongly anticorrelated with Re (−37%). Recently, the mb -dependence [323] of the O(αs2 ) QCD correction [324],(0,b)
affecting the reference axis of the b quark asymmetry [325], increased the extracted16 AF B by
(0,b)
(0,c)
about 0.2 σ. The correlation between AF B and AF B amounts to 15%.
In addition, SLD extracted the final-state couplings Ab , Ac [288], As [320], Aτ , and Aµ [316],
from left-right forward-backward asymmetries, using
B
AFLR(f ) =

f
f
f
f
+ σRB
− σRF
− σLB
σLF
f
f
f
f
+ σRB
+ σRF
+ σLB
σLF

3
= Af ,
4(10.70)f
is the cross-section for a left-handed incident electron to produce a fermion
where, for example, σLF
f traveling in the forward hemisphere. Similarly, Aτ and Ae were measured at LEP 1 [288] (Aτalso very recently by CMS [317]) through the τ polarization, Pτ , as a function of the scattering
angle θ, which can be written as,Pτ = −Aτ (1 + cos2 θ) + 2Ae cos θ.
(1 + cos2 θ) + 2Aτ Ae cos θ(10.71)

The average polarization, hPτ i, obtained by integrating over cos θ in the numerator and denominatorof Eq. (10.71), yields hPτ i = −Aτ , and Ae can be extracted from the Pτ angular distribution. The
initial-state coupling, Ae , was also determined through the left-right charge asymmetry [321] andin polarized Bhabha scattering [316] at the SLC. Because ḡV` is very small, not only A0LR = Ae ,
(0,`)
(0,q)
AF B , and Pτ , but also AF B for q = b, c, and s, as well as the hadronic asymmetries are mainly
sensitive to s̄2` . The combination of all LEP and SLC asymmetries (but excluding other observables
such as R` ) yields,s̄2` = 0.23151 ± 0.00016 (LEP + SLC) .
(10.72)f
As an example of the precision of the Z pole observables, the values of ḡA
and ḡVf for f = e, µ, τ ,and `, extracted from the LEP and SLC lineshape and asymmetry data, are shown in Fig. 10.3. It
may be compared with Fig. 10.1 as the two sets of parameters coincide at the SM at tree-level.
As for hadron colliders, the forward-backward asymmetry, AF B , for e+ e− and µ+ µ− final states(with invariant masses restricted to or dominated by values around MZ ) in pp̄ collisions has beenmeasured by the CDF [326] and DØ [327] collaborations, and the values s̄2` = 0.23221 ± 0.00046
16

We are grateful to Werner Bernreuther and Long Chen for the re-calculation of their result employing the more
appropriate MS mixing angle, b
s2Z , instead of the on-shell quantity, s2W .
31st May, 2024

10. Electroweak Model and Constraints on New Physics

26

--

0.233

--

0.232

-0.032-0.034-0.036+ -+ --0.038

--

0.231

gfV

39.35% C.L.

e+ e-

39.35% C.L.

39.35% C.L.

+ -

l l

--

0.230

90% C.L.

-0.040-0.502-0.501-0.500-0.499-0.498gfA
f
and ḡVf for f = e, µ and
Figure 10.3: 1 σ (39.35% CL) contours of the effective couplings ḡAτ from LEP and SLC, compared to the SM expectation as a function of sbZ2 . (The SM best fit`
value sbZ2 = 0.23129 is also indicated.) Also shown is the 90% CL allowed region in ḡA,Vobtained
assuming lepton universality.

and s̄2` = 0.23095 ± 0.00040 were extracted, respectively. The combination of these measurements(which differ by more than 2 σ) yields [309],s̄2` = 0.23148 ± 0.00033 (Tevatron) .

(10.73)

By varying the invariant mass and the scattering angle (and assuming the electron couplings),
u,dinformation on the effective Z couplings to light quarks, ḡV,A, could also be obtained [328,329], butwith large uncertainties, mutual correlations, and not independently of s̄2` above. Similar analyses
have also been reported by the H1 [330] and ZEUS [331] collaborations at HERA and by the LEP
collaborations [288]. This kind of measurement is harder in the pp environment due to the difficulty
to assign the initial quark and antiquark in the underlying Drell-Yan process to the protons, thus
requiring excellent control of uncertainties from parton distribution functions. ATLAS obtaineds̄2` = 0.2308 ± 0.0012 using 7 TeV data [310] and a preliminary result s̄2` = 0.23140 ± 0.00036 at
8 TeV [311], while CMS measured s̄2` = 0.23101 ± 0.00053 (8 TeV) [312] and a preliminary results̄2` = 0.23157±0.00031 (13 TeV) [313], and LHCb reported s̄2` = 0.23142±0.00106 (from both 7 and31st May, 2024

10. Electroweak Model and Constraints on New Physics

27

Table 10.6: Results derived from Table 10.5 and the corresponding covariance matrices [288,332],
and the SM predictions for the partial and total Z decay widths [in MeV]. In the (second) thirdcolumn lepton universality is (not) assumed.
Quantity
Γe+ e−
Γµ+ µ−
Γτ + τ −
Γinv
Γuū
Γcc̄
Γdd¯, Γss̄
Γbb̄
Γhad
ΓZ

Value
83.87 ± 0.12
83.95 ± 0.18
84.03 ± 0.21
498.9 ± 2.5—
300.3 ± 5.3—
377.4 ± 1.3
1744.8 ± 2.6
2495.5 ± 2.3

Value (universal)
83.942 ± 0.085
83.941 ± 0.085
83.759 ± 0.085
500.5 ± 1.5—
300.0 ± 5.2—
377.0 ± 1.2
1743.2 ± 1.92495.5 ± 2.3

Standard Model
83.955 ± 0.009
83.955 ± 0.009
83.772 ± 0.009
501.435 ± 0.045
299.87 ± 0.20
299.81 ± 0.20
382.75 ± 0.14
375.73 ∓ 0.18
1740.88 ± 0.86
2494.00 ± 0.87

8 TeV data, but only analyzing µ+ µ− final state) [314]. Assuming that the smallest theoretical and
PDF uncertainty (±0.00024 from ATLAS [311]) is fully correlated among the five determinations,they combine tos̄2` = 0.23145 ± 0.00028 (LHC) .
(10.74)
Combining Eqs. (10.72), (10.73), and (10.74) gives,s̄2` = 0.23149 ± 0.00013 (collider average) .

(10.75)10.5.4 W and Z decays
The partial decay widths for gauge bosons to decay into massless fermions f1 f¯2 (the numericalvalues include the small EW radiative corrections and final-state mass effects) are given by,
Γ (W + → e+ νe ) =3MW= 226.29 ± 0.04 MeV ,12πv 2(10.76a)3MW
Γ (W + → ui d¯j ) =|Vij |2 RqV = (705.3 ± 0.4 MeV)|Vij |2 ,
2
12πv

i
MZ3 h f f 2
f f2
Γ (Z → f f¯) =
R
ḡ
+
R
ḡ
,
V V
A A
12πv 2(10.76b)(10.76c)where the result for the latter are shown in Table 10.6. Final-state QED and QCD corrections [333]to the vector and axial-vector form factors are given by,"

RfV,A = NC3α(s) NC2 − 1 αs (s)
Q2f
+
1+
4
π
2NC
π

!

#

+ ···

,

(10.77)

where NC = 3 (1) is the color factor for quarks (leptons) and the dots indicate finite fermion masseffects proportional to m2f /s which are different for RfV and RfA , as well as higher-order QCD corrections [334], which are known to O(αs4 ) [222]. For the Z boson, these include singlet contributions
starting from two-loop order which are large, strongly top quark mass dependent, family universal,
31st May, 2024

28

10. Electroweak Model and Constraints on New Physics

and flavor non-universal [335–339]. The O(α2 ) self-energy corrections from Ref. [340] are also taken
into account.
For the W decay into quarks, Eq. (10.76b), only the universal massless part (non-singlet andmq = 0) of the final-state QCD radiator function in RV from Eq. (10.77) is used, and the QED
3corrections are modified. Expressing the widths in terms of GF MW,Z
incorporates the largest
radiative corrections from the running QED coupling. EW corrections to the Z widths are then
2 . Hence, in the on-shell scheme the Z widths
taken into account through the effective couplings ḡ iV,A
are proportional to ρi ∼ 1 + ρt . There is additional (negative) quadratic mt dependence in the
Z → bb̄ vertex corrections [341, 342] which causes Γbb̄ to decrease with mt . The dominant effect is
to multiply Γbb̄ by the vertex correction 1 + δρbb̄ , where δρbb̄ ∼ 10−2 (− 21 m2t /MZ2 + 15 ). In practice,b b , as discussed in Sec. 10.5.
the corrections are included in ρbb and κ
Starting at O(ααs ), the factorized form indicated in Eq. (10.76) is violated and corrections need
to be included [343–345]. They add coherently, resulting in a sizable effect, and shift αs (MZ ) when
extracted from Z lineshape observables by about +0.0007. Similar non-factorizable corrections arealso known for mixed QED-EW corrections [111, 112, 114, 346].
For three fermion families the total widths of the Z [347–351] and W [352, 353] bosons are
predicted to be,ΓZ = 2.4940 ± 0.0009 GeV ,ΓW = 2.0892 ± 0.0008 GeV .

(10.78)The uncertainties in these predictions are almost entirely induced by the parametric error in
αs (MZ ) = 0.1187 ± 0.0017 from the global fit. These predictions can be compared with theexperimental results, ΓZ = 2.4955 ± 0.0023 GeV [288, 332] and ΓW = 2.137 ± 0.032 GeV in (10.78).
Note regarding the latter that in this Section we include the very recent result from ATLAS [295]and an updated value of the Tevatron result [299], which differ from the treatment in the Gauge& Higgs Bosons Particle Listings and Section 54 on the “Mass and Width of the W Boson” in thisReview. The hadronic branching ratio of the W boson, B(W → hadrons) has been measured byboth LEP 2 [294] and CMS [300]. The measurements of the total and partial widths are generally ingood agreement with the SM. The exceptions are ΓW , which is 1.5 σ larger than the SM prediction,and the branching ratio W → τ + ντ from LEP 2, which is 2.6 σ larger than the electron-muon
average [294]17 .
The invisible decay width, Γinv = ΓZ − Γe+ e− − Γµ+ µ− − Γτ + τ − − Γhad , can be used to determine
the number of neutrino flavors, Nν , much lighter than MZ /2. The hadronic peak cross-section, andtherefore the extracted Γhad , depends strongly on the knowledge of the LEP 1 luminosity derived
from small-angle Bhabha scattering. However, the prediction for the Bhabha cross-section wasrecently found to be overestimated, and consequently the luminosity underestimated [332]. The
updated analysis involved an improved Z lineshape fit, significantly reducing σhad , while slightly
increasing ΓZ , with the result, Nν = 2.9963±0.0074 [332]. In practice, we determine Nν by allowing
it as an additional fit parameter and obtain,Nν = 3.0025 ± 0.0061 ,(10.79)which is now in perfect agreement with the observed number of fermion generations and Nν = 3 (a1.3 σ deviation was observed in the 2018 edition of this Review before including the correction in
the luminosity determination).

10.6 Global fit results
In this section, we present the results of global fits, subject to the experimental data andtheoretical constraints discussed in Section 10.2−10.5. For earlier analyses, see Refs. [83, 288, 355–
17
W -boson branching ratio measurements from CMS and ATLAS are in good agreement with lepton universality
and slightly more precise than LEP-2 [300, 354].

31st May, 2024

29

10. Electroweak Model and Constraints on New Physics
Table 10.7: Principal electroweak SM fit result including mutual correlations.
MZ [GeV]b t (mb t ) [GeV]mb b (mb b ) [GeV]mbb c ) [GeV]mc (mαs (MZ )(3)∆αhad (2 GeV)

91.1884 ± 0.0019
163.18 ± 0.54
4.180 ± 0.008
1.274 ± 0.009
0.1187 ± 0.0017
0.00608 ± 0.00004

1.00
−0.08
0.00
0.00
0.02
0.02

−0.08
1.00
0.00
−0.12
−0.23
0.04

0.00
0.00
1.00
0.19
−0.02
0.01

0.00
−0.12
0.19
1.00
0.48
0.01

0.02
−0.23
−0.02
0.48
1.00
−0.04

0.02
0.04
0.01
0.01
−0.04
1.00

358] and previous editions of this Review. Recent global fits by other groups [359–361] find asimilar pattern of agreement as discussed below, with some discrepancies beyond the 2σ level fora few quantities entering into the fit. They differ, however, from the fit presented here by (i) not
including inputs from low-energy parity-violation data and the muon magnetic moment18 ; (ii) the
set of input data; (iii) the implementation of radiative corrections; and (iv) the fitting tools used.
For the results in this Review, the values for mt (see Sec. 10.2.3), MH [364, 365], ΓH [308,366], MW [295, 298], ΓW [294, 295, 299], B(W → hadrons) [294, 300], the weak charges of theelectron [166], the proton [167], cesium [178, 179] and thallium [180, 181], the weak mixing angle
extracted from eDIS [156], νµ (ν̄µ )-e scattering [137–139], the τ lifetime, and the µ anomalous
magnetic moment [367] are listed in Table 10.4. Likewise, Table 10.5 summarizes the principal
Z pole observables, where the LEP 1 averages of the ALEPH, DELPHI, L3, and OPAL results
include common systematic uncertainties and correlations [288, 332]. The heavy flavor results [288,323] of LEP 1 and SLD are based on common inputs, and are thus correlated, as well.
(3)
Also shown in both tables are the SM predictions for the values of MZ , αs (MZ ), ∆αhad andthe heavy quark masses shown in Table 10.7. The predictions result from a global least-square(χ2 ) fit to all data using the minimization package MINUIT [368] and the EW library GAPP [34].
In most cases, we treat all input errors (the uncertainties of the values) as Gaussian. The reason
is not that we assume that theoretical and systematic errors are intrinsically bell-shaped (which
they are not) but because in most cases the input errors are either dominated by the statisticalcomponents or they are combinations of many different (including statistical) error sources, whichshould yield approximately Gaussian combined errors by the large number theorem. An exception
is the theory dominated error on the τ lifetime, which we recalculate in each χ2 -function call since
it depends itself on αs . Sizes and shapes of the output errors (the uncertainties of the predictions
and the SM fit parameters) are fully determined by the fit, and 1 σ errors are defined to correspondto ∆χ2 = χ2 − χ2min = 1, and do not necessarily correspond to the 68.3% probability range or the
39.3% probability contour (for 2 parameters).
The agreement is generally very good. Despite the few discrepancies addressed in the following,
the global electroweak fit describes the data well, with a very good χ2 /d.o.f. = 49.5/47. The
probability of a larger χ2 is 37%, and only gµ − 2 is currently showing a larger (3.2 σ) conflict. In(0,b)
addition, A0LR (SLD) from hadronic final states, AF B (LEP 1), ΓW (ATLAS) and QW (Cs) deviate
at the 2 σ level. gL2 from NuTeV is nominally in conflict with the SM, as well, but the precise
status is unresolved (see Sec. 10.3.1). Also, there is currently no understanding as to why the MW
value reported by the CDF collaboration is significantly larger than the findings by other groups.
In this context, we refer to Sec. 10.8 for a discussion of fits involving alternative data inputs. We
also emphasize that there are a number of discrepancies among individual measurements of certain
18
Refs. [51, 362, 363] report on specialized fits to study the impact of a shift of the hadronic vacuum polarizationin the running of the electromagnetic coupling, where such shift is motivated by the apparent mismatch of the direct
measurement and the data-driven SM prediction of gµ − 2, but they do not include gµ − 2 itself in the fit.

31st May, 2024

10. Electroweak Model and Constraints on New Physics

30

500

MH [GeV]300
200

Z, had, Rl, Rq (1 )Z pole asymmetries (1 )MW (1 )
direct mt (1 )direct MH
all except direct MH (90%)

100
50
30
20
10

160

165

170

mt [GeV]

175

180

Figure 10.4: Fit result and one-standard-deviation (approximately 39.35% for the closed contours
and 68% for the others) uncertainties in MH as a function of mt for various inputs, and the 90% CL
region (∆χ2 = 4.605) allowed by all data. αs (MZ ) = 0.1187 is assumed except for the fits including
the Z lineshape. The width of the horizontal dashed band is not visible on the scale of the plot.
quantities, as discussed in previous sections, but that they are not reflected in the overall χ2 of the
fit as only the corresponding combinations are used as constraints.
(0,b)
Ab can be extracted from AF B when Ae = 0.1501 ± 0.0016 is taken from a fit to leptonic
asymmetries (using lepton universality). The result, Ab = 0.885 ± 0.017, is 2.9 σ below the SM
B (b) at SLD. Thus, it appears
prediction19 and also 1.4 σ below Ab = 0.923±0.020 obtained from AFLR
that at least some of the problem in Ab is due to a statistical fluctuation or other experimental(0,b)
effect in one of the asymmetries. Note, however, that the uncertainty in AF B is strongly statistics
dominated. The combined value, Ab = 0.901 ± 0.013 deviates by 2.6 σ.
The left-right asymmetry, A0LR = 0.15138 ± 0.00216 [315], from hadronic decays at SLD, differs
by 2.3 σ from the SM expectation of 0.1463 ± 0.0003. The combined value of A` = 0.1513 ± 0.0021
from SLD (using lepton-family universality and including correlations) is also 2.4 σ above the
SM prediction; but there is experimental agreement between this SLD value and the LEP 1 value,(0,`)
A` = 0.1481±0.0027, obtained from a fit to AF B , Ae (Pτ ), and Aτ (Pτ ), again assuming universality.
The observables in Table 10.4 and Table 10.5, as well as some other less precise observables,
are used in the global fits described below. In all fits, the errors include full statistical, systematic, and theoretical uncertainties. The correlations from the LEP 1 lineshape and τ polarization
measurements, the LEP/SLD heavy flavor observables, the SLD lepton asymmetries, and the ν-e
19

Alternatively, one can use A` = 0.1481 ± 0.0027, which is from LEP 1 alone and in excellent agreement with theSM, and obtain Ab = 0.897 ± 0.022 which is 1.7 σ low. This illustrates that some of the discrepancy is related to theone in ALR .
31st May, 2024

10. Electroweak Model and Constraints on New Physics

MW [GeV]31

80.45
direct ( W fixed) w/o CDF II (1 )80.44
direct ( W free) w/o CDF II (1 )CDF II (1 )80.43
indirect (1 )80.42
all data w/o CDF II (1 )
80.41
80.40
80.39
80.38
80.37
80.36
80.35
80.34
80.33
170 171 172 173 174 175 176 177 178 179 180 181 182
mt [GeV]Figure 10.5: One-standard-deviation (39.35%) regions in MW as a function of mt for the indirect
data from the fit (green), the direct data for mt and MW without the CDF II measurement (blueand black contour, respectively, for two differently treatments of the W -boson width), and the
combination thereof (red). For the combination, the 90% CL region (∆χ2 = 4.605) is also shown
as a red dashed contour. The grey region uses only the MW value from CDF II. Compared
to previous editions of this Review, this plot additionally includes theoretical uncertainties andsubleading parameter dependencies.
(5)

scattering observables, are included. The theoretical correlations between ∆αhad , sb20 , and gµ − 2,and between the MW and ΓW extractions from the LHC and the Tevatron, are also accounted for.
The electroweak data allow a simultaneous determination of MZ , mt , and αs (MZ ). The direct
measurements of MH at the LHC [364, 365] have reached a precision that the global fit resultfor MH coincides with the constraint in Eq. (10.66) with negligible correlations with the other fit(3)b c, mb b , and ∆αhad are also allowed to float in the fits, subject to the theoreticalparameters. m
constraints [33, 58] described in Sec. 10.2, and are correlated with αs , which in turn is determined
mainly through R` , ΓZ , σhad , and ττ , but ΓW and B(W → hadrons) also have an impact. The
global fit to all data, including the hadron collider mt average in Eq. (10.21), yields the results inTable 10.7, while those for the weak mixing angle in various schemes are summarized in Table 10.2.
Removing the kinematic constraint on MH from LHC gives the loop-level determination fromthe precision data,
MH = 97+18(10.80)−16 GeV ,which is 1.6 σ below the value in Eq. (10.66). The latter is just inside the 90% central confidence
range,
71 GeV < MH < 126 GeV .
(10.81)
It is instructive to study the effect of doubling the uncertainty in Eq. (10.14) on the loop-level
31st May, 2024

32

10. Electroweak Model and Constraints on New Physics

Table 10.8: Values of sbZ2 , s2W , αs , mt and MH for various data sets. In the fit to the LHC data, theαs constraint is from an NNLO analysis of the transverse momentum distribution of Z bosons [369].
For the Tevatron fit we use the αs result from the inclusive jet cross-section at DØ [370] and add
the MW result from CDF [289] as adjusted in Ref. [298].
data set
all data
all data except MH
all data except MZ
all data except MW , ΓW
all data except mt
MH,Z + ΓH,Z + mt
LHC
Tevatron
LEP 1 + LEP 2
LEP 1 + SLD
SLD + MZ + ΓZ + mt
(b,c)
AF B + MZ + ΓZ + mt
MW,Z + ΓW,Z + mt
low energy + MH,Z

sbZ20.23129(4)0.23118(8)0.23121(6)0.23130(4)0.23123(6)0.23134(8)0.23122(8)0.23085(13)0.23138(19)0.23116(17)0.23064(29)0.23190(30)0.23119(11)0.23173(94)s2W
0.22348(10)0.22327(18)0.22346(10)0.22353(11)0.22324(20)0.22361(16)0.22349(11)0.22254(28)0.22350(46)0.22340(58)0.22224(54)0.22492(70)0.22329(23)0.2252(35)αs (MZ )0.1187(17)0.1190(17)0.1187(17)0.1188(17)0.1191(17)0.1221(45)0.1183 (9)0.1159(45)0.1233(29)0.1221(27)0.1196(52)0.1279(49)0.1212(42)0.1172(18)mt [GeV]
172.9 ± 0.6
172.6 ± 0.6
172.6 ± 0.6
172.7 ± 0.6
175.2 ± 1.8
172.6 ± 0.6
172.3 ± 0.6
174.4 ± 0.8
177 ± 11169 ± 10172.6 ± 0.6
172.6 ± 0.6
172.6 ± 0.6
159 ± 29

MH [GeV]
125
18
97+
− 16
125
125
125
125
125
25
67+
− 20
+237
173− 96
87
71+
− 33
27
31+
− 22
306+164
−111
23
96+
− 20
125

determination of the Higgs boson mass. The result, MH = 94+19−16 GeV, reduces the small tension
compared to Eq. (10.80) only slightly (1.5 σ) and demonstrates that the uncertainty in ∆αhad is(3)
currently of only secondary importance. In fact, even removing ∆αhad (2 GeV) altogether as a fit
constraint still gives the bound MH < 135 GeV at the 95% CL. The hadronic contribution toα(MZ ) is correlated with gµ − 2 (see Sec. 10.4). The measurement of the latter is higher than the
SM prediction, and its inclusion in the fit favors a larger α(MZ ) and a lower MH from the precision
data (currently by 3.2 GeV).
Alternatively, one can carry out a fit without including the direct constraint from the hadroncolliders. One obtains mt = 175.2 ± 1.8 GeV, which is 1.4 σ higher than the collider average
mt = 172.61 ± 0.58 GeV, and a reflection of the low value in Eq. (10.80). (The indirect predictionb t (mb t ) = 165.4 ± 1.7 GeV, which is in the end converted to the poleis for the MS mass definition, mmass.) Finally, one can remove the explicit MW and ΓW constraints from the global fit and use
MH = 125.10 ± 0.09 GeV to obtain MW = 80.353 ± 0.006 GeV, in very good agreement with theexperimental measurements except the one from CDF II. The situation is summarized in Fig. 10.4
and in Fig. 10.5, showing the mt dependence of MH and MW , respectively.
The weak mixing angle can be determined from Z pole observables, MW , and a variety ofneutral-current processes spanning a very wide Q2 range. The results (for older low energy neutralcurrent data see Refs. [355–358], as well as earlier editions of this Review) shown in Table 10.8 arein reasonable agreement with each other, indicating the quantitative success of the SM. One of thelargest discrepancies is the value sbZ2 = 0.23064 ± 0.00028 from the SLD asymmetries (combinedwith MZ , ΓZ , and mt ), which is 2.3 σ below the value 0.23129 ± 0.00004 from the global fit to all
(0,b)
(0,c)
data. On the other hand, sbZ2 = 0.23176 ± 0.00027 from AF B and AF B is 1.7 σ high.
The extracted Z pole value of αs (MZ ) is based on a formula with negligible theoretical un31st May, 2024

10. Electroweak Model and Constraints on New Physics

33

Table 10.9: Values of model-independent neutral-current parameters, compared with the SM
predictions, where the uncertainties in the latter are . 0.0001, throughout.
Quantity
νe
gLV
νe
gLA

Experimental Value−0.040 ± 0.015−0.507 ± 0.014

Standard Model−0.0395−0.5063

eu + 2 g ed
gAVAVeu − g ed
2 gAV
AV
2 gVeuA − gVedA

0.4927 ± 0.0031−0.7165 ± 0.0068−0.13 ± 0.06

0.4950−0.7189−0.0944gVeeA

0.0190 ± 0.0027

0.0224

Correlation
−0.05
−0.88

0.20
−0.22

certainty if one assumes the exact validity of the SM. One should keep in mind, however, that
this value, αs (MZ ) = 0.1221 ± 0.0027, which increased after the updated analysis in Ref. [332],
is very sensitive to certain types of new physics such as non-universal vertex corrections. A fit
to the wider set of high-energy data, i.e., including W -decays but without ττ and gµ − 2, returns+0.0018
αs (MZ ) = 0.1211 ± 0.0025. In contrast, the value derived from τ decays, αs (MZ ) = 0.1171−0.0017,
is theory dominated but less sensitive to new physics. The agreement between these values is only
marginal, but the latter does agree well with the averages deduced from heavy quarkonia spectroscopy (0.1181 ± 0.0037), DIS and global PDF fits (0.1161 ± 0.0022), hadronic final states of e+ e−annihilations (0.1189 ± 0.0037), hadron colliders (0.1168 ± 0.0027), as well as lattice QCD simulations (0.1184 ± 0.0008). For more details, other determinations, and references, see Section 9 on
“Quantum Chromodynamics” in this Review. We also provide the values, computed with five-loop
beta functions and four-loop matching,αs(4) (mτ ) = 0.325 ± 0.014 ,(10.82)αs(5) (MW ) = 0.1210 ± 0.0017 ,(10.83)αs(5) (MH ) = 0.1132 ± 0.0015 ,(10.84)b t ) = 0.1090 ± 0.0014 ,αs(6) (m(10.85)to be used in precision calculations.
Using α(MZ ) and sbZ2 as inputs, one can predict αs (MZ ) assuming grand unification. One finds
αs (MZ ) = 0.13 ± 0.01 [371, 372] for the simplest theories based on the minimal supersymmetricextension of the SM, where the uncertainty is from the unknown particle thresholds. This is slightly
larger, but consistent with αs (MZ ) = 0.1187 ± 0.0017 from our fit and most other determinations,while minimal non-supersymmetric theories predict much lower and excluded values (see Section 93
on “Grand Unified Theories” in this Review).
Most of the parameters relevant to ν-hadron, ν-e, e-hadron, and e-e processes are determineduniquely and precisely from the data in “model-independent” fits, i.e., fits allowing for an arbitrary
EW gauge theory. The values for the parameters defined in Eq. (10.29) are given in Table 10.9
along with the predictions of the SM. The agreement is very good. (The ν-hadron results including
NuTeV [151] and other ν-DIS data can be found in the 2006 edition of this Review, and fits withmodified NuTeV constraints in the 2008 and 2010 editions.)

10.7 Constraints on new physics
The masses and decay properties of the electroweak bosons and low energy data can be used
to search for and set limits on deviations from the SM. We will mainly discuss the effects of exotic
31st May, 2024

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10. Electroweak Model and Constraints on New Physics

particles (with heavy masses Mnew  MZ in an expansion in MZ /Mnew ) on the gauge boson selfenergies. (Brief remarks are made on new physics which is not of this type.) Most of the effects
on precision measurements can be described by three gauge self-energy parameters S, T , and U .
We will define these, as well as the related parameters ρ0 and bi , to arise from new physics only.
In other words, they are equal to zero (ρ0 = 1) exactly in the SM, and do not include any (loopinduced) contributions that depend on mt or MH , which are treated separately. Our treatment
differs from most of the original papers.
The dominant effect of many extensions of the SM can be described by the ρ0 parameter,
ρ0 ≡

2
MW
,
2
MZ cbZ2 ρb(10.86)

which describes new sources of SU(2) breaking that cannot be accounted for by the SM Higgs
doublet or by mt effects. ρb is calculated as in Eq. (10.26) assuming the validity of the SM. In
the presence of ρ0 6= 1, Eq. (10.86) generalizes the second Eq. (10.26) while the first remains
unchanged. Provided that the new physics which yields ρ0 6= 1 is a small perturbation which
does not significantly affect other radiative corrections, ρ0 can be regarded as a phenomenologicalparameter which multiplies GF in Eqs. (10.29) and (10.52), as well as ΓZ in Eq. (10.76c). Fromthe global fit,
ρ0 = 1.00031 ± 0.00019 ,(10.87a)αs (MZ ) = 0.1189 ± 0.0017 ,(10.87b)where as before the uncertainty is from the experimental inputs and includes an estimate of theerror from unknown higher-order electroweak corrections. The result in Eq. (10.87a) is 1.6 σ above
the SM expectation, ρ0 = 1, and not unrelated to the small tension observed in the context with
Eq. (10.80). It can be used to constrain higher-dimensional Higgs representations to have vacuum
expectation values of less than a few percent of those of the doublets. Indeed, the relation between
MW and MZ is modified if there are Higgs multiplets with weak isospin > 1/2 and significant
vacuum expectation values. For a general (charge-conserving) Higgs structure,22i [ti (ti + 1) − t3i ]|vi |P 2,
2 i t3i |vi |2

P

ρ0 =

(10.88)where vi is the expectation value of the neutral component of a Higgs multiplet with weak isospin ti
and third component t3i . In order to calculate to higher orders in such theories one must define
a set of four fundamental renormalized parameters which one may conveniently choose to be α,
GF , MZ , and MW , since MW and MZ are directly measurable. Then sbZ2 and ρ0 can be considereddependent parameters.
Eq. (10.87a) can also be used to constrain other types of new physics. For example, nondegenerate multiplets of heavy fermions or scalars break the vector part of weak  SU(2) and leadto a decrease in the value of MZ /MW . Each non-degenerate SU(2) doublet ff12 yields a positive
contribution to ρ0 [373–375] of
NC GF
√
∆m2 ,
(10.89)
2
8 2π
where
∆m2 ≡ m21 + m22 −4m21 m22
m1
ln≥ (m1 − m2 )2 ,2
2
m1 − m2 m2

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(10.90)10. Electroweak Model and Constraints on New Physics

35

and NC = 1 (3) for color singlets (triplets). Eq. (10.87a) taken together with Eq. (10.89) implies
the following constraint on the mass splitting at the 90% CL,(2 GeV)2 <

X Ni

C

i

3

∆m2i < (44 GeV)2 ,(10.91)

where the sum runs over all new-physics doublets, for example fourth-family quarks or leptons,
or

 0
ν
`0−

 
t0
b0

, vector-like fermion doublets (which contribute to the sum in Eq. (10.91) with an extra

factor of 2), and scalar doublets such as b̃t̃ in Supersymmetry (in the absence of L–R mixing).
Non-degenerate multiplets usually imply ρ0 > 1. Similarly, heavy Z 0 bosons decrease theprediction for MZ due to mixing and generally lead to ρ0 > 1 [376]. On the other hand, extra
Higgs doublets participating in spontaneous symmetry breaking [377–379] or heavy lepton doublets
involving Majorana neutrinos [380], both of which have more complicated expressions, and the vi
of higher-dimensional Higgs representations can contribute to ρ0 with either sign.
A number of authors [381–383] have considered the general effects on neutral-current, Z and
W boson observables of various types of heavy (i.e., Mnew  MZ ) physics which contribute to
the W and Z self-energies but which do not have any direct coupling to the ordinary fermions (analternative formulation is given by Ref. [384]). In addition to non-degenerate multiplets, which
break the vector part of weak SU(2), these include heavy degenerate multiplets of chiral fermions
which break the axial generators.
Such effects can be described by just three parameters, S, T , and U [385], at the (EW) one-loop
level20 . T is proportional to the difference between the W and Z self-energies at Q2 = 0 (i.e., vector
SU(2)-breaking), while S (S + U ) is associated with the difference between the Z (W ) self-energy
2
at Q2 = MZ,W
and Q2 = 0 (axial SU(2)-breaking). Denoting the contributions of new physics tothe various self-energies by Πijnew , we havebα(MZ )T ≡

new (0)new (0)
ΠZZ
ΠW
W
−
,
2
MW
MZ2

(10.92a)new (M 2 )new (M 2 )new (M 2 ) − Π new (0)
Πγγ
b
α(M
ΠZZ
cbZ2 − sbZ2 ΠZγ
Z)
Z
Z
Z
ZZ
S
≡
−
−
,
cbZ sbZ
4 sbZ2 cbZ2
MZ2
MZ2
MZ2

(10.92b)new (M 2 )new (M 2 )new (M 2 ) − Π new (0)
Πγγ
b
α(M
ΠW
cbZ ΠZγ
Z)
Z
Z
W
W
WW
(S
+
U
)
≡
−
−
.
2
sbZ
4 sbZ2MW
MZ2MZ2(10.92c)b removed, so that they are expected to be
S, T , and U are defined with a factor proportional to αof order unity in the presence of new physics. In the MS scheme as defined in Ref. [85], the last
two terms in Eqs. (10.92b) and (10.92c) can be omitted, as was done in some earlier editions of thisReview. These parameters are related to other parameter sets, Si [85], bi [389], and hi [390], by
b1
T = hV =,(10.93a)bα(MZ)

S = hAZ = SZ = 4 sbZ2

b3
,
b
α(M
Z)

U = hAW − hAZ = SW − SZ = −4 sbZ2
20

(10.93b)
b2
.
b
α(M
Z)

(10.93c)Three additional parameters are needed if the new physics scale is comparable to MZ [386]. Further generalizations, including effects relevant to LEP 2 and Drell-Yan production at the LHC, are described in Refs. [387] and [388],
respectively.
31st May, 2024

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10. Electroweak Model and Constraints on New Physics
A heavy non-degenerate multiplet of fermions or scalars contributes positively to T as
ρ0 − 1 =1b− 1 ≈ α(MZ )T ,b
1 − α(MZ )T(10.94)

where ρ0 − 1 is given in Eq. (10.89). The effects of non-standard Higgs representations cannot
be separated from heavy non-degenerate multiplets unless the new physics has other consequences,such as vertex corrections. Most of the original papers defined T to include the effects of loops only.
However, we will redefine T to include all new sources of SU(2) breaking, including non-standard
Higgs, so that T and ρ0 are equivalent by Eq. (10.94).
A multiplet of heavy degenerate chiral fermions yields
S=

2
NC X  L
t3i − tR,3i
3π i

(10.95)rd
where tL,R
3i is the 3 component of weak isospin of the left-(right-)handed component of fermion i.
For example, a heavy degenerate ordinary or mirror family would contribute 2/3π to S. In models
with warped extra dimensions [391], sizeable corrections to the S parameter are generated throughmixing between the SM gauge bosons and their Kaluza-Klein (KK) excitations, and one finds−2[392], where MKK is the mass scale of the KK gauge bosons. Large positive values
S ≈ 30 v 2 MKKof S can also be generated in models with dynamical electroweak symmetry breaking, where the
Higgs boson is composite. In simple composite Higgs models, the dominant contribution stems
from heavy spin-1 resonances of the strong dynamics leading to S ≈ 4πv 2 (MV−2 + MA−2 ), where
MV,A are the masses of the lightest vector and axial-vector resonances, respectively [393].
Negative values of S are possible, for example, in composite Higgs models with larger gauge
group representations [394, 395], or from loops involving scalars or Majorana particles [396–398].
The simplest origin of S < 0 would probably be an additional heavy Z 0 boson [376]. Supersymmetricextensions of the SM [399, 400] generally give very small effects. For more details and references,see Refs. [401–410] and Sections 88,89 on “Supersymmetry” in this Review. Most simple types of
new physics yield U = 0, although there are counter-examples, such as the effects of anomalous
triple gauge vertices [389].
The SM expressions for observables are replaced by,
2
MZ2 = MZ0
2
2
MW
= MW
0

b
1 − α(MZ )T√,
2 S/2 2π
1 − GF MZ0

1
2 (S + U )/2
1 − G F MW
0

(10.96a)√2π,(10.96b)where MZ0 and MW 0 are the SM expressions (as functions of mt and MH ) in the MS scheme.
Furthermore,ΓZ =MZ3 βZ,b1 − α(MZ )T3
ΓW = MW
βW ,

Ai =

Ai0
,
b
1 − α(MZ )T(10.97a)(10.97b)(10.97c)3 and Γ3
where βZ,W are the SM expressions for the reduced widths ΓZ0 /MZ0
W 0 /MW 0 , MZ and MW
are the physical masses, and Ai (Ai0 ) is a neutral-current amplitude (in the SM).
31st May, 2024

10. Electroweak Model and Constraints on New Physics

37
1.0
0.8
0.6
0.4

T

0.2
0.0
Z, had, Rl, Rq
asymmetries
e & scattering
MW w/o CDF
APV
all (90% CL) U=0 w/o CDF
all (90% CL) U=0
SM prediction

0.2
0.4
0.6
0.8
1.0

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

S

Figure 10.6: 1 σ constraints (39.35% for the closed contours and 68% for the others) on S and T(for U = 0) from various inputs combined with MZ . S and T represent the contributions of new
physics only. Data sets not involving MW or ΓW are insensitive to U . With the exception of the
fit to all data, we fix αs = 0.1187. The yellow dot indicates the Standard Model values S = T = 0.
The fits to all data are at the 90% CL, with the open black ellipse corresponding to the alternative
scenario discussed in Sec. 10.8, i.e. including the MW result from CDF.
The data allows for a simultaneous determination of MH and mt (from the hadron colliders),
S (from MZ ), T (mainly from ΓZ ), U (from MW ), sbZ2 = 0.23113 ± 0.00013 (from the Z poleasymmetries), and αs (MZ ) = 0.1191 ± 0.0018 (mostly from R` , σhad , and ττ ), giving,
S = −0.04 ± 0.10 ,(10.98a)0.01 ± 0.12 ,(10.98b)

U = −0.01 ± 0.09 ,(10.98c)

T =where the correlations among the SM parameters are similar to those in Table 10.7, and where theuncertainties are from unknown higher orders in the SM predictions and the inputs. The parametersin Eq. (10.98), which by definition are due to new physics only, are in excellent agreement with
the SM values of zero. Fixing U = 0, which is motivated by the fact that U is suppressed by an
2 /M 2 compared to S and T [411], greatly improves the precision on S andadditional factor MnewZ
particularly T ,S = −0.05 ± 0.07 ,(10.99a)0.00 ± 0.06 .

(10.99b)

T =Using Eq. (10.94), the value of ρ0 corresponding to T in Eq. (10.98b) is 1.0001 ± 0.0009, while theone corresponding to Eq. (10.99b) is 1.0000 ± 0.0005. Thus, the multi-parameter fits are consistent
31st May, 2024

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10. Electroweak Model and Constraints on New Physics

with ρ0 = 1, in contrast to the fit with S = U = 0 in Eq. (10.87a). There is a strong correlation(93%) between the S and T parameters. The U parameter is −70% (−87%) anti-correlated with S(T ). The allowed regions in S−T (for U = 0) are shown in Fig. 10.6. From Eqs. (10.98) one obtains
S ≤ 0.11 and T ≤ 0.20 at 95% CL, where the former puts the constraint MKK & 4 TeV on themasses of KK gauge bosons in warped extra dimensions. In minimal composite Higgs models, thebound on S requires MV & 4.8 TeV, which is obtained from a one-sided 95% CL bound on S > 0
for T = 0 and using Weinberg sum rules [412]. However, this constraint can be relaxed, e.g., if thefermionic sector is also allowed to be partially composite [413, 414] and in soft-wall models [415].
The S parameter can also be used to constrain the number of fermion families, under theassumption that there are no new contributions to T or U and therefore that any new families
are degenerate; then an extra generation of SM fermions is excluded with almost 9 σ confidence,corresponding to NF = 2.75±0.14. This can be compared to the fit to the number of light neutrinos
given in Eq. (10.79), Nν = 3.0025 ± 0.0061, but the S parameter fits are valid even for a very heavy
fourth family neutrino. Allowing T to vary as well, the constraint on a fourth family is weaker [416].
However, a heavy fourth family would increase the Higgs production cross-section through gluon
fusion by a factor of about 9 [417], which is in considerable tension with the observed Higgs signal
at the LHC [418]. Combining the limits from electroweak precision data with the measured Higgs
production rate and limits from direct searches for heavy quarks [419], a fourth family of chiralfermions is now excluded by more than five standard deviations [420, 421]. Similar remarks apply
to a heavy mirror family [422] involving right-handed SU(2) doublets and left-handed singlets. In
contrast, new doublets that receive most of their mass from a different source than the Higgs vacuum
expectation value, such as vector-like fermion doublets or scalar doublets in Supersymmetry, givesmall or no contribution to S, T , U , and the Higgs production cross-section and are therefore
still allowed. Partial or complete vector-like fermion families are predicted in many Grand Unified
Theories [423] (see Section 93 on “Grand Unified Theories” in this Review), and many other models
including supersymmetric and superstring inspired ones [424–427].
As discussed in Sec. 10.6, there is a 3.6% deviation in the asymmetry parameter Ab . Assumingthat this is due to new physics affecting preferentially the third generation, we can perform a fit
allowing additional Z → bb̄ vertex corrections ρb and κb as in Eq. (10.53) (here defined to be due to
new physics only with the SM contributions removed), as well as S, T , U , and the SM parameters,with the result,ρb = 0.057 ± 0.020 ,(10.100a)κb = 0.183 ± 0.067 ,(10.100b)with an almost perfect correlation of 99% (because Rb is much better determined than Ab ). The
central values of the oblique parameters are consistent with their SM values of zero (note, however,that S = −0.10 ± 0.10 is slightly outside its 1σ range), and there is little change in the SMparameters, except that the value of αs (MZ ) is lower by 0.0009 compared to the SM fit. Given thatan O(20%) correction to κb would be necessary, it would be difficult to account for the deviationin Ab by new physics that enters only at the level of radiative corrections. Thus, if it is due
to new physics, it is most likely of tree-level type affecting preferentially the third generation.
Examples include the decay of a scalar neutrino resonance [428], mixing of the b quark with heavy
exotics [429], and a heavy Z 0 with family non-universal couplings [430, 431]. It is difficult, however,to simultaneously account for Rb without tuning, which has been measured on the Z peak andoff-peak [432] at LEP 1.
There is no simple parametrization to describe the effects of every type of new physics on
every possible observable. The S, T , and U formalism describes many types of heavy physics
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10. Electroweak Model and Constraints on New Physics

which affect only the gauge self-energies, and it can be applied to all precision observables. However, new physics which couples directly to ordinary fermions cannot be fully parametrized in thisframework. Examples include heavy Z 0 bosons [376], mixing with exotic fermions [9, 433, 434], leptoquark exchange [294, 435, 436], supersymmetric models, strong EW dynamics [413], Little Higgs
models [437, 438], and TeV-scale extra spatial dimensions [439–442] (for more details and references, see Section 85 on “Extra Dimensions” in this Review). These types of new physics can beparametrized in a model-independent way by using an effective field theory description [443–446].
Here the SM is extended by a set of higher-dimensional operators, denoted Oi ,L = LSM +X X Ci
d>4 i

Λd−4

Oi ,

(10.101)

where Λ is the characteristic scale of the new physics sector, which is assumed to satisfy Λ  v.
For EW precision observables, the leading new operators enter at dimension d = 6. Note that S
and T can be identified with two of these operators (or linear combinations thereof, depending on
the chosen operator basis), while U corresponds to a dimension-8 operator [411,447]. With current
data on MW and Z pole observables, Λ is constrained to be larger than O(TeV) if the Wilson
coefficients Ci are of order unity [448–454].
Limits on new four-Fermi operators and on leptoquarks using LEP 2 and lower energy data are
given in Refs. [294, 455–457], while constraints on various types of new physics are addressed inRefs. [9, 169, 319, 458, 459]. For a particularly well motivated and explored type of physics beyond
the SM, see Section 87 on “Z 0 -Boson Searches” in this Review.

10.8 Alternative scenarios(5)

For our evaluation of gµ −2 and ∆αhad (MZ ) we took the recent vacuum polarization constraints
from CMD-3 [48] and LQCD into account. Both have the effect to move the SM prediction of gµ −2
closer to the new measurement result in Ref. [236], while reducing the indirect determination of
MH and slightly increasing the small tension with the direct LHC measurements [306, 307]. The
discrepancies of CMD-3 and LQCD with some of the older data (especially KLOE) is presently(3)
not understood. Thus, it is illustrative to repeat the global fit by replacing ∆αhad (2 GeV) =2−9−4
had,VP(α ) = (65.44±0.25)×10 with the values that we used for the 2022
(60.30±0.43)×10 and aµ(3)

edition of this Review, ∆αhad (2 GeV) = (58.84±0.51)×10−4 and aµhad,VP (α2 ) = (64.49±0.33)×10−9 .
As a result, the discrepancy between the SM prediction and the measurement of aµ would become
5.1 σ, and Eq. (10.50) would change totheoryaexp= (2.08 ± 0.41) × 10−9 .
µ − aµ(10.102)

On the other hand, the result for the indirect determination of MH in Eq. (10.80) would now readMH = 103+18−17 GeV .

(10.103)

The global fit reported in Sec. 10.6 excluded the recent MW result by the CDF collaboration [289], because there is a roughly 4 σ discrepancy with other measurements whose origin iscurrently not understood (Ref. [289] also contains the first high-precision measurement of MZ at
a hadron collider shown in Table 10.5, in perfect agreement with LEP). Here we study the effectof including this measurement as an additional constraint, using MW = 80.432 ± 0.016 GeV (seeTable 10.4), which is the result of the adjustment to the common PDF set CT18 [303] in Ref. [298].
The quality of the global fit deteriorates sharply with a χ2 /d.o.f. = 70.1/48 and a probability ofa larger χ2 of only 2%. The increase in χ2 amounts to more than 20 for 1 additional degree of
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10. Electroweak Model and Constraints on New Physics

freedom. Removing MH from the LHC as a constraint gives a somewhat improved fit quality witha χ2 /d.o.f. = 60.1/47, but with
MH = 75+14(10.104)−12 GeV ,driven to very low and excluded values. More acceptable is the ρ0 fit as in Eq. (10.87a) with aχ2 /d.o.f. = 58.5/47 (12%), andρ0 = 1.00059 ± 0.00016 ,(10.105)
corresponding to T = 0.08 ± 0.02. Thus, if one views the discrepancies in the MW measurements
as statistical fluctuations, then ample parameter space would open up, allowing, e.g., additional
non-degenerate SU(2) doublets, as reviewed in Sec. 10.7. In the STU fit,
S = −0.04 ± 0.10 ,(10.106a)T =0.01 ± 0.12 ,(10.106b)U=

0.05 ± 0.09 ,(10.106c)

the effect is mostly moved to the U parameter when compared to Eqs. (10.98). S, T and U arehighly correlated and despite appearances the probability of all vanishing simultaneously is just
0.3%. Finally, Eq. (10.99) changes only moderately (see also the open black ellipse in Fig. 10.6),
S = −0.00 ± 0.07 ,(10.107a)

0.07 ± 0.05 .

(10.107b)

T =Of course, more drastic changes would be observed with the constraint, MW = 80.4335 ± 9.4 GeVfrom the original publication [289] with its much smaller quoted uncertainty. For related discussionsof this issue, see Refs. [361, 460–463].
Acknowledgments
It is a pleasure to thank Rodolfo Ferro-Hernández for discussions, for performing some of thecalculations and checks, and for updating the plots. A.F. is supported in part by the National
Science Foundation under grant no. PHY-2112829.
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31st May, 2024