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10. Electroweak Model and Constraints on New Physics

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10. Electroweak Model and Constraints on New Physics 1 10. Electroweak Model and Constraints on New Physics 10. Electroweak Model and Constraints on New Physics Revised March 2020 by J. Erler (IF-UNAM; U. of Mainz) and A. Freitas (Pittsburg U.). 10.1 Introduction ....................................... 1 10.2 Renormalization and radiative corrections....................... 3 10.2.1 The Fermi constant ................................ 3 10.2.2 The electromagnetic coupling........................... 3 10.2.3 Quark masses.................................... 5 10.2.4 The weak mixing angle .............................. 6 10.2.5 Radiative corrections................................ 8 10.3 Low energy electroweak observables .......................... 9 10.3.1 Neutrino scattering................................. 9 10.3.2 Parity violating lepton scattering......................... 12 10.3.3 Atomic parity violation .............................. 13 10.4 Precision flavor physics ................................. 15 10.4.1 The τ lifetime.................................... 15 10.4.2 The muon anomalous magnetic moment..................... 17 10.5 Physics of the massive electroweak bosons....................... 18 10.5.1 Electroweak physics off the Z pole........................ 19 10.5.2 Z pole physics ................................... 21 10.5.3 W and Z decays .................................. 25 10.6 Global fit results..................................... 26 10.7 Constraints on new physics............................... 30 10.1 Introduction The standard model of the electroweak interactions (SM) [1–4] is based on the gauge group i SU(2)×U(1), with gauge bosons Wµ, i = 1, 2, 3, and Bµ for the SU(2) and U(1) factors, respectively, and the corresponding gauge coupling constants g and g0. The left-handed fermion fields of the th νi ui 0 P i fermion family transform as doublets Ψi = − and d0 under SU(2), where d ≡ Vij dj, `i i i j and V is the Cabibbo-Kobayashi-Maskawa mixing [5,6] matrix1. 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. A complex scalar Higgs doublet, φ, is added to the model for mass generation through sponta- neous symmetry breaking with potential2 given by, ! λ2 φ+ V (φ) = µ2φ†φ + (φ†φ)2 , φ ≡ . (10.1) 2 φ0 √ 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 1Constraints on V and tests of universality are discussed in Ref. [7] and in the Section on the “CKM Quark-Mixing Matrix” in this Review. The extension of the formalism to allow an analogous leptonic mixing matrix is discussed in the Section on “Neutrino Masses, Mixing, and Oscillations” in this Review. 2There is no generally accepted convention to write the quartic term. Our numerical coefficient simplifies Eq. (10.4a) below and the squared coupling preserves the relation between the number of external legs and the power counting of couplings at a given loop order. This structure also naturally emerges from physics beyond the SM, such as Supersymmetry. P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020) 27th August, 2020 2:40pm 2 10. Electroweak Model and Constraints on New Physics and neutral scalar Higgs particles. Higgs boson physics is reviewed in the Section on the “Status of Higgs Boson Physics” in this Review. After symmetry breaking the Lagrangian for the fermion fields, ψi, is X ¯ miH g X µ 5 + + − − LF = ψi i6∂ − mi − ψi − √ Ψ i γ (1 − γ )(T W + T W ) Ψi v 2 2 µ µ i i (10.2) g − e X Q ψ¯ γµ ψ A − X ψ¯ γµ(gi − gi γ5) ψ Z . i i i µ 2 cos θ i V A i µ i W i −1 0 Here θW ≡ tan (g /g) is the weak mixing angle and e = g sin θW is the positron electric charge. Furthermore, 3 Aµ ≡ Bµ cos θW + Wµ sin θW , (10.3a) W 1 ∓ iW 2 W ± ≡ µ √ µ , (10.3b) µ 2 3 Zµ ≡ −Bµ sin θW + Wµ 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 minimal model, is gmi/2MW . The boson masses in the EW sector are given (at tree level, i.e., to lowest order in perturbation theory) by, MH = λ v , (10.4a) gv ev MW = = , (10.4b) 2 2 sin θW q 2 02 v ev MZ = g + g = , (10.4c) 2 2 sin θW cos θW Mγ = 0 . (10.4d) + 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 an electron and a neutrino is e h − µ 5 + µ 5 i − √ Wµ e¯ γ (1 − γ )ν + Wµ ν¯ γ (1 − γ )e . (10.5) 2 2 sin θW For momenta small compared to MW ,√ this term gives rise to the effective four-fermion interaction 2 2 2 with the Fermi constant given by GF / 2 = 1/2v = g /8MW . CP violation is incorporated into the EW model by a single observable phase in Vij. 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 i 2 gV ≡t3L(i) − 2Qi sin θW , (10.6a) i gA ≡t3L(i) , (10.6b) 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 the Section on “Neutrino Mass, Mixing, and Oscillations” in this Review. 27th August, 2020 2:40pm 3 10. Electroweak Model and Constraints on New Physics 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 mass3, the Fermi constant, and the fine structure constant, to be discussed in turn (the numerical values quoted in Sections 10.2−10.4 correspond to the main fit result in Table 10.7). The Z boson mass, MZ = 91.1876 ± 0.0021 GeV, has been determined from the Z lineshape scan at LEP 1 [14]. This value of MZ corresponds to a definition based on a Breit-Wigner shape with an energy-dependent width4 (see the Section on the “Z Boson” in this Review). 10.2.1 The Fermi constant −5 −2 5 The Fermi constant, GF = 1.1663787(6) × 10 GeV , is derived from the µ lifetime formula , 2 5 " 2 # ~ GF mµ αb(mµ) αb (mµ) = 3 F (ρ) 1 + H1(ρ) + H2(ρ) 2 , (10.7) τµ 192π π π 2 2 where ρ = me/mµ, and where F (ρ) = 1 − 8ρ + 8ρ3 − ρ4 − 12ρ2 ln ρ = 0.99981295 , (10.8a) 25 π2 H (ρ) = − − 9 + 4π2 + 12 ln ρ ρ + 16π2ρ3/2 + O(ρ2) 1 8 2 = −1.80793 , (10.8b) 156815 518 895 67 53 H (ρ) = − π2 − ζ(3) + π4 + π2 ln 2 2 5184 81 36 720 6 5 √ − (0.042 ± 0.002) − π2 ρ + O(ρ) = 6.64 , (10.8c) had 4 1 α(m )−1 = α−1 + ln ρ + O(α) = 135.901 . (10.8d) b µ 3π H1 and H2 capture the QED corrections within the Fermi model. The results for ρ = 0 have been obtained in Refs. [16] and [17, 18] for H1 and H2, respectively, where the term in parentheses is from the hadronic vacuum polarization [17]. The mass corrections to H1 have been known for some time [19], while those to H2 are more recent [20]. Notice the term linear in me whose appearance was unforeseen and can be traced to the use of the muon pole mass in the prefactor [20]. 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 e± anomalous magnetic moment [21], −12 −1 ae = (1159652180.91 ± 0.26) × 10 [22], which gives the value α = 137.035999150(33). Another approach combines measurements of the Rydberg constant and atomic masses with interferometry of atomic recoil kinematics. Applied to 87Rb [23] and 133Cs [24], this method implies the results α−1 = 137.035998997(85) and α−1 = 137.035999047(28), respectively, which can be combined to give α−1 = 137.035999042(26). Finally, combining the anomalous magnetic moment and atomic interferometry methods leads to the world average of α−1 = 137.035999084(21), but notice that 3We emphasize that in the fits described in Sec. 10.6 and Sec. 10.7 the values of the SM parameters are affected by all observables that depend on them. This is of no practical consequence for α and GF , however, since they are very precisely known. 4The theoretically consistent and gauge-invariant definition of the Z-boson mass through the complex pole of the propagator would instead lead to a Breit-Wigner with a constant width. The two definitions differ numerically, and this difference has to be accounted for in theoretical calculations. 5 2 2 In the spirit of the Fermi theory, we incorporated the small propagator correction, 3/5 mµ/MW , into ∆r (see below).
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