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Updated status of the global electroweak fit and constraints on new physics

  • Regular Article - Theoretical Physics
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Abstract

We present an update of the Standard Model fit to electroweak precision data. We include newest experimental results on the top-quark mass, the W mass and width, and the Higgs-boson mass bounds from LEP, Tevatron and the LHC. We also include a new determination of the electromagnetic coupling strength at the Z pole. We find for the Higgs-boson mass \(91^{+30}_{-23}~\mbox{GeV}\) and \(120^{+12}_{-5}~\mbox{GeV}\) when not including and including the direct Higgs searches, respectively. From the latter fit we indirectly determine the W mass to be \((80.360^{+0.014}_{-0.013})~\mbox{GeV}\). We exploit the data to determine experimental constraints on the oblique vacuum polarisation parameters, and confront these with predictions from the Standard Model (SM) and selected SM extensions. By fitting the oblique parameters to the electroweak data we derive allowed regions in the BSM parameter spaces. We revisit and consistently update these constraints for a fourth fermion generation, two Higgs doublet, inert Higgs and littlest Higgs models, models with large, universal or warped extra dimensions and technicolour. In most of the models studied a heavy Higgs boson can be made compatible with the electroweak precision data.

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Notes

  1. Up to two-loop electroweak corrections are available in Refs. [21–32]. All known QCD corrections are given in Refs. [22, 26, 33].

  2. In the analysis and throughout this paper we use the \(\overline {\mathrm {MS}}\) renormalised masses of the c and b quarks, \(\overline {m}_{c}(\overline {m}_{c})\) and \(\overline {m}_{b}(\overline {m}_{b})\), at their proper scales. In the following they are denoted with \(\overline {m}_{c}\) and \(\overline {m}_{b}\), respectively, and their values are taken from [36].

  3. Using an external precision measurement of \(\alpha _{\scriptscriptstyle S}(M_{Z}^{2})\) in the fit has been studied in Ref. [2] and found to have a negligible impact on the M H result.

  4. We do not include the CDF and D0 measurements of the forward-backward charge asymmetry in \(p\overline{p}\to Z/\gamma^{\star}+X\to e^{+}e^{-}+X\) events, used to extract the \(\sin ^{2}\theta ^{e}_{\mathrm {eff}}\) values 0.2238±0.0040±0.0030 by CDF [40], and 0.2326±0.0018±0.0006 by D0 [41], as their impact so far is negligible compared to the precision of the combined Z pole data in the fit, \(\sin ^{2}\theta ^{\ell}_{\mathrm {eff}}=0.23143 \pm 0.00013\). Also due to lack of precision, we do not include results from atomic parity violation measurements, and from parity violation left–right asymmetry measurements using fixed target polarised Møller scattering at low Q 2 (see [2] for references).

    The NuTeV Collaboration measured ratios of neutral and charged current cross sections in neutrino–nucleon scattering at an average Q 2≃20 GeV2 using both muon neutrino and muon anti-neutrino beams [42]. The results derived for the effective weak couplings are not included in this analysis because of unclear theoretical uncertainties from QCD effects such as next-to-leading order corrections and nuclear effects of the bound nucleon parton distribution functions [43] (for reviews see, e.g., Refs. [44, 45]).

  5. In Ref. [53] the \(\overline {\mathrm {MS}}\) scheme is used to predict the QCD scaling function versus scale ratios (including the dependence on the top mass) that, convolved with the parton luminosity and multiplied by \((\alpha _{\scriptscriptstyle S}/\overline{m}_{t})^{2}\), determines the inclusive \(t\overline{t}\) production cross section. The experimental cross section measurement thus allows one to infer \(\overline{m}_{t}\) and hence the pole mass (being the renormalised quark mass in the on-shell renormalisation scheme) from the ratio of the corresponding renormalisation factors known to three loops [54]. The numerical analysis must account for the dependence of the experimental cross section value on the top mass used to determine the detector acceptance and reconstruction efficiencies.

  6. The quoted error on the extracted top mass does not include the ambiguity in the Monte Carlo top-mass interpretation.

  7. A mistake has been found in the published result of \(\Delta \alpha _{\mathrm {had}}^{(5)}(M_{Z}^{2})\) [58]. The corrected result used here is reported in Version 2 of the arXiv submission [1010.4180].

  8. In lack of published CLs+b values by ATLAS [63] we approximate a chi-squared behaviour of the \(\tilde{q}_{\mu}\) test statistics used [65] and compute \(\mathrm{CL}_{\mathrm{s}+\mathrm{b}} \simeq \mathrm{Prob}(\tilde{q}_{1},1)\), where the published p 0 values have been converted into \(\tilde{q}_{0}\), and \(\tilde{q}_{1} = \tilde{q}_{0} -2\mathrm{LLR}\) with the definition \(\mathrm{LLR}=-2\ln({\mathcal{L}}(1,\hat{\hat{\theta}})/{\mathcal{L}}(0,\hat{\theta}))\). A nearly identical result is found with CLs+b≃Prob(−2LLR+offset,1), where the offset of 1.1 has been added to ensure positive values over the Higgs mass range, and using the published LLR numbers only. These are the numbers used in the fit.

  9. The CLs+b obtained from the direct Higgs searches has been left unaltered during the Monte Carlo-based p-value evaluation of the complete fit. This is justified by the strong statistical significance of the LEP constraint, which drives the contribution of the direct Higgs searches to the \(\chi ^{2}_{\mathrm {min}}\).

  10. It is noticeable that the values of the four theoretical uncertainty parameters converge at the limits of their allowed intervals. This is explained by their uniform contribution to the χ 2 function but the necessarily non-uniform values of the global χ 2 function that depends on these theory parameters. The fit thus converges at the extrema of the allowed ranges.

  11. We have verified the chi-squared property of the test statistics by sampling pseudo MC experiments.

  12. Repeating the standard fit with all theory uncertainties fixed to zero gives \(\chi ^{2}_{\mathrm {min}}=17.2\) and \(M_{H}=94^{+30}_{-24}~\mathrm {GeV}\). A direct comparison of this result with Eq. (1) is not straightforward as the fit uses the additional nuisance parameters, when let free to vary, to improve the test statistics (recall the value of \(\chi ^{2}_{\mathrm {min}}=16.7\) for the standard fit result). The impact on the parameter errors would become noticeable once the input observables exhibit better compatibility (cf. discussion in Ref. [2]).

  13. The uncertainties in the free fit parameters that are correlated to M H (mainly \(\Delta \alpha _{\mathrm {had}}^{(5)}(M_{Z}^{2})\) and m t ) contribute to the errors shown in Fig. 3, and generate correlations between the four M H values found.

  14. Note that by fixing M H the number of degrees of freedom of the fit is increased compared to the standard fit resulting in a larger average \(\chi ^{2}_{\min }\) and thus in a larger p-value.

  15. Had we determined M H by confronting experimental and predicted oblique parameters, we would reproduce Fig. 4 up to deviations due to the higher-order and non-oblique corrections present in the standard electroweak fit.

  16. A detailed numerical SM4 analysis [80] taking into account low-energy FCNC processes in the quark sector, electroweak oblique corrections, and lepton decays (but not lepton mixing) concludes that small mixing between the quarks of the first three and those of the fourth family is favoured. The value of |V tb | is found in this analysis to exceed 0.93. The no-mixing assumption allows us to use the measured value of G F , extracted from the muon lifetime under the SM3 hypothesis, to its full precision [84].

  17. The functions in Eqs. (18)–(20) are defined as follows. F(x 1,x 2)=(x 1+x 2)/2−x 1 x 2/(x 1−x 2)⋅ln(x 1/x 2), G(x)=−4yarctan(1/y), \(y=\sqrt{4x-1}\), and \(f(x_{1},x_{2})=-2\sqrt{\Delta}[\arctan((x_{1}-x_{2}+1)/\sqrt{\Delta})-\arctan((x_{1}-x_{2}-1)/\sqrt{\Delta})]\) for Δ>0, f(x 1,x 2)=0 for Δ=0, and \(f(x_{1},x_{2})=\sqrt{-\Delta}\cdot\ln((X+\sqrt{-\Delta})/(X-\sqrt{-\Delta}))\) with X=x 1+x 2−1 for Δ<0, and where Δ=2(x 1+x 2)−(x 1−x 2)2−1.

  18. Ignoring flavour mixing between the fourth and the SM generations, it was found in Ref. [100] that absolute vacuum stability of the running Higgs self coupling approximately requires the mass hierarchy \(M_{H} \gtrsim m_{u_{4}}\). This strong lower bound on the Higgs-boson mass may possibly be weakened by looser stability requirements. For example, in SM3 the absolute stability lower bound on M H is significantly reduced by allowing the minimum potential to be metastable with finite probability not to have tunnelled into another, deeper minimum during the lifetime of the universe [101, 102]. In the following discussion we will ignore the stability bound on M H .

  19. The functions defined in Eqs. (21)–(23) are defined as follows. \(\mathcal{B}_{22}(q^{2},m_{1}^{2},m_{2}^{2}) = q^{2}/24\{2\ln q^{2} + \ln(x_{1}x_{2})+[(x_{1}-x_{2})^{3}-3(x_{1}^{2}-x_{2}^{2}) + 3(x_{1}-x_{2})]\ln(x_{1}/x_{2})- [2(x_{1}-x_{2})^{2}-8(x_{1}+x_{2})+10/3 ] - [(x_{1}-x_{2})^{2}-2(x_{1}+x_{2})+1]f(x_{1},x_{2})-6F(x_{1},x_{2})\} \overset{m_{1}=m_{2}}{\Rightarrow}q^{2}/24 [2\ln q^{2} + 2\ln x_{1} + ( 16x_{1} - 10/3 ) +(4x_{1}-1)G(x_{1}) ]\), where \(x_{i} \equiv m_{i}^{2}/q^{2}\), \(\mathcal{B}_{0}(q^{2},m_{1}^{2},m_{2}^{2}) = 1 + 1/2 [(x_{1} + x_{2})/(x_{1} - x_{2}) - (x_{1} - x_{2}) ] \ln(x_{1}/x_{2}) + 1/2f(x_{1}, x_{2}) \overset{m_{1} = m_{2}}{\Rightarrow}2 - 2y\arctan(1/y), \quad y = \sqrt{4x_{1} - 1}\), \(\overline{B}_{0}(m_{1}^{2},m_{2}^{2},m_{3}^{2}) = (m_{1}^{2}\ln m_{1}^{2} - m_{3}^{2}\ln m_{3}^{2})/(m_{1}^{2}-m_{3}^{2}) - (m_{1}^{2}\ln m_{1}^{2} - m_{2}^{2} \ln m_{2}^{2})/(m_{1}^{2}-m_{2}^{2})\) [89], see also Footnote 17 on page 14.

  20. It has been shown [143] that the T-odd partner of the hypercharge gauge boson (the heavy photon) can give rise to the observed relic density of the universe.

  21. The parameter s λ is defined by \(s_{\lambda} = \lambda_{2}/\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}\), where λ 1 and λ 2 are the Yukawa couplings of the new top states.

  22. A different result for the gauge sector contribution has been published in Ref. [145], where the logarithmic term is found to cancel. We thank Masaki Asano for pointing that out to us. The numerical effect of this correction is contained within the theoretical uncertainty of ±5 assigned to the δ c coefficient.

  23. The δ c parameter is treated as theory uncertainty varying in the range [−5,5] in the fit.

  24. If the UED is embedded into large extra dimensions of size eV−1 accessible to gravity only, the lightest KK state could decay via KK-number violating gravitational interaction into a photon and an eV-spaced graviton tower of mass equivalent between zero and R −1 [167]. Such a model provides a clear collider signature with two isolated photons and missing transverse energy in the final state, which has been searched for at ATLAS [168] and D0 [169].

  25. This effect is similar to the cancellation of the negative SM Higgs contribution to T with the positive contribution from the top sector in the littlest Higgs model (cf. Sect. 4.4).

  26. The lower mass limits are determined by where the calculation of the oblique parameters can be trusted.

  27. Owing to U(1) Q gauge symmetry, for the photon propagator the term q μ q ν has no physical effect. For the (massive) W and Z propagators the terms are also negligible, since, in the interaction with light fermions, they are suppressed by the fermion mass scale compared with the g μν parts. From now on we shall ignore the q μ q ν terms.

  28. Throughout this appendix the superscript (0) is used to label tree-level quantities.

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Acknowledgements

We are indebted to the LEP-Higgs and Tevatron-NPH working groups for providing the numerical CLs+b results of their direct Higgs-boson searches. It is a pleasure to thank Alexander Lenz for instructive discussions on extended fermion generations. We are grateful to Hong-Jian He und Shu-fang Su for their help on the implementation of the 2HDM oblique corrections. We thank Gian Giudice for helpful correspondence on large extra dimensions. We also thank Kenneth Lane for his helpful comments and suggestions on the technicolour studies presented here. This work is funded by the German Research Foundation (DFG) in the Collaborative Research Centre (SFB) 676 “Particles, Strings and the Early Universe” located in Hamburg. A.H. thanks the Aspen Center for Physics, which is supported by an NSF grant, for its hospitality during the finalisation of this paper.

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  1. CERN, Geneva, Switzerland

    M. Baak, A. Hoecker & M. Schott

  2. DESY, Hamburg and Zeuthen, Germany

    M. Goebel, D. Kennedy & K. Mönig

  3. Institut für Experimentalphysik, Universität Hamburg, Hamburg, Germany

    J. Haller & D. Kennedy

  4. Department of Physics and Astronomy, Michigan State University, East Lansing, USA

    J. Stelzer

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  1. M. Baak
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Appendix A: Oblique parameter formalism

Appendix A: Oblique parameter formalism

1.1 A.1 Absorption of radiative corrections

Oblique corrections can generally be absorbed into the fundamental constants occurring at the tree level of the SM. Kennedy and Lynn [203] have shown that this statement is general to all vacuum polarisation orders. In this appendix we illustrate the absorption process with some explicit examples.

The effects of oblique corrections on fermion scattering can be determined by examining how the gauge-boson vacuum polarisation functions

$$ \varPi_{ab}^{\mu\nu}(q)=\varPi_{ab} \bigl(q^{2}\bigr)g^{\mu\nu} + \bigl(q^{\mu}q^{\nu}\ \textrm{terms}\bigr), $$
(50)

with a,b=γ,W,Z, appear in the electroweak observables of interest.Footnote 27 The functions (50) have an SM and an unknown new physics component: \(\varPi_{ab}^{\mu\nu}(q^{2})=\varPi_{ab}^{SM}(q^{2}) + \delta\varPi_{ab}^{NP}(q^{2})\).

For the W and Z bosons one finds the following mass corrections to the tree-level quantities:Footnote 28

$$ \begin{array} {l} M_{W}^{2} \equiv M_{W}^{2}\bigl(M_{W}^{2}\bigr) = M_{W}^{(0)2} + \varPi_{WW}\bigl(M_{W}^{2} \bigr), \\[6pt] M_{Z}^{2} \equiv M_{Z}^{2} \bigl(M_{Z}^{2}\bigr) = M_{Z}^{(0)2} + \varPi_{ZZ}\bigl(M_{Z}^{2}\bigr), \end{array} $$
(51)

where the vacuum polarisation functions are evaluated at the poles of the propagators.

For the massless photon one has

$$ \varPi_{\gamma\gamma}(0)=\varPi_{\gamma Z}(0)=0. $$
(52)

The impact on the electromagnetic constant α is obtained by taking the leading-order photon propagator plus the first-order correction. Together these yield

$$ \frac{-ie^{2}}{q^{2}} \biggl(1+i\varPi_{\gamma\gamma}\bigl(q^{2}\bigr) \cdot\frac{-i}{q^{2}} \biggr). $$
(53)

The observed value of the electric charge is then found by taking the limit q 2→0 of this expression

$$ 4\pi\alpha_{*}(0) \equiv e_{*}^{2}(0)=\frac{g^{2}g^{\prime 2}}{g^{2}+g^{\prime 2}} \bigl(1+ \varPi_{\gamma\gamma}^{\prime}(0) \bigr), $$
(54)

where

$$ \varPi_{\gamma\gamma}^{\prime}(0)=\frac{d\varPi_{\gamma\gamma}}{dq^{2}}\bigg|_{q^{2}=0}. $$
(55)

The weak mixing angle s W appears in the interactions of Z bosons to fermions, and is shifted by the vacuum polarisation amplitude Π Zγ . The corrections change a Z into a photon that decays to two fermions, with coupling strength Qe, leading to the contribution

$$ i\varPi_{Z\gamma}\bigl(q^{2}\bigr)\frac{-i}{q^{2}}\cdot(ieQ). $$
(56)

Including this correction, the Z-fermion interaction takes the form

$$ i\sqrt{g^{2}+g^{\prime 2}} \bigl(T^{3}-s_{*}^{2}Q \bigr), $$
(57)

with

$$ s_{*}^{2}\bigl(M_{Z}^{2}\bigr) = s_{W}^{(0)\,2} - \frac{e}{\sqrt{g^{2}+g^{\prime 2}}} \frac{\varPi_{Z\gamma}(M_{Z}^{2})}{M_{Z}^{2}}, $$
(58)

as evaluated at \(q^{2}=M_{Z}^{2}\).

The Fermi constant, obtained from muon decays, as mediated by W propagator, receives the first-order correction from the W vacuum polarisation function

$$ \frac{-ig^{2}}{q^{2}-M_{W}^{2}} \biggl(1+i\varPi_{WW}\bigl(q^{2}\bigr) \frac{-i}{q^{2}-M_{W}^{2}} \biggr). $$
(59)

At q 2=0, the observed Fermi constant process shifts to

$$ \frac{G_{F*}}{\sqrt{2}}=\frac{1}{2v^{2}} \biggl(1- \frac{\varPi_{WW}(0)}{M_{W}^{2}} \biggr). $$
(60)

These examples illustrate that oblique corrections can be absorbed into the fundamental constants occurring of the SM. This conclusion is applied in the following section.

1.2 A.2 Introduction of the S, T, U parameters

In the SM with a single Higgs doublet the relationship between the neutral and charged weak couplings is fixed by the ratio of W and Z boson masses

$$ \rho = \frac{M_{W}^{2}}{M_{Z}^{2}\cos^{2}\theta _{\scriptscriptstyle W}}, $$
(61)

where ρ 0=1 at tree level. Generally one writes

$$ \rho = 1+\Delta\rho, $$
(62)

where Δρ captures the radiative corrections to the gauge-boson propagators and vertices. Inserting the first-order mass corrections of Eqs. (51) into Eq. (61) gives

$$ \Delta\rho = \frac{\varPi_{WW}(0)}{M_{W}^{2}}-\frac{\varPi_{ZZ}(0)}{M_{Z}^{2}}. $$
(63)

The tree-level vector and axial-vector couplings occurring in the Z boson to fermion-antifermion vertex \(i\overline{f}\gamma_{\mu}({g_{{\scriptscriptstyle V},f}^{(0)}}+{g_{{\scriptscriptstyle V},f}^{(0)}}\gamma_{5})f Z_{\mu}\) are given by

(64)
(65)

where Q f and \(I^{f}_{3}\) are respectively the charge and the third component of the weak isospin. In the (minimal) SM, containing only one Higgs doublet, the weak mixing angle is defined by

$$ \sin^{2}\theta _{\scriptscriptstyle W}= 1-\frac{M_{W}^2}{M_{Z}^2}. $$
(66)

Electroweak radiative corrections modify these relations, leading to the effective weak mixing angle and effective couplings

(67)
(68)
(69)

where the radiative corrections are absorbed in the form factors \(\kappa ^{f}_{Z}=1+\Delta \kappa ^{f}_{Z}\) and \(\rho ^{f}_{Z}=1+\Delta \rho ^{f}_{Z}\).

Electroweak unification leads to a relation between weak and electromagnetic couplings, which at tree level reads

$$ {G_{\scriptscriptstyle F}}= \frac{\pi \alpha}{\sqrt{2} (M_{W}^{(0)} )^2 (1-\frac{(M^{(0)}_{W})^2}{M_{Z}^2} )}. $$
(70)

The radiative corrections are parametrised by multiplying the r.h.s. of Eq. (70) with the form factor (1−Δr)−1. Using Eq. (66) and resolving for M W gives

$$ M_{W}^2 = \frac{M_{Z}^2}{2} \Biggl(1+\sqrt{1-\frac{\sqrt{8}\pi\alpha(1+\Delta r)}{{G_{\scriptscriptstyle F}}{M_{Z}^2}}}\,\Biggr). $$
(71)

An extra correction is required for the \(Z\rightarrow b\overline{b}\) decay vertex. The bottom quark is the only fermion that receives unsuppressed vertex corrections from the top quark. These corrections turn out to be significant—at the level of \(G_{F}m_{t}^{2}\)—and must be accounted for. The vector and axial couplings receive an extra contribution ε b

(72)

where ε b contains all top-quark induced vertex corrections.

The entire dependence of the electroweak theory on m t and M H , arising from one-loop diagrams and higher, only enters through the four parameters Δκ, Δρ, Δr, and ε b . The quantities Δκ, Δρ, and Δr W are mostly sensitive to the absolute mass splittings between different weak-isospin partners. In practise this means the mass differences between the top and bottom quarks, and the Z and W bosons. For example, the dominant contributions to Δρ are [203]

(73)

exhibiting a quadratic dependence on the top mass, and a logarithmic dependence on the Higgs mass. Since M H >M Z >M W , ρ H is negative.

Ignoring terms proportional to lnm t /M Z and vertex corrections, which do not contain sizable terms containing M H and m t , the parameters on one-loop level can can be written as [204]:

(74)

All quantities are dominated by terms of \(G_{F}m_{t}^{2}\). Considering this term only, Δk, Δρ, Δr W are related as follows:

$$ \Delta r_{W} = \frac{c^{2}-s^{2}}{s^{2}} \Delta k=-\frac{c^{2}}{s^{2}} \Delta \rho. $$
(75)

Restoring the \(\ln\frac{m_{t}}{m_{z}}\) terms, the ε 1,2,3 parameters defined in Eqs. (6)–(8) on page 10 are given by

(76)

The SM subtraction results in the parameter set \(\hat{\varepsilon}\). In terms of propagator functions one has [36]

(77)

Equivalently, contributions to \(\hat{\varepsilon}_{b}\) are the NP vertex correction to \(Z\rightarrow b\overline{b}\). The S, T, U parameters expressed in terms of the \(\hat{\varepsilon}\) parameters read

$$ S = \frac{4s^{2}\hat{\varepsilon}_{3}}{\alpha(M_{Z}^{2})}, \qquad T = \frac{\hat{\varepsilon}_{1}}{\alpha(M_{Z}^{2})},\qquad U = \frac{-4s^{2}\hat{\varepsilon}_{2}}{\alpha(M_{Z}^{2})}. $$
(78)

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The Gfitter Group., Baak, M., Goebel, M. et al. Updated status of the global electroweak fit and constraints on new physics. Eur. Phys. J. C 72, 2003 (2012). https://doi.org/10.1140/epjc/s10052-012-2003-4

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