Leading fermionic three-loop corrections to electroweak precision observables

Published for SISSA by Springer Received: March 12, 2020 Revised: May 6, 2020 Accepted: July 9, 2020 Published: July 29, 2020 Lisong Chen and Ayres Freitas Pittsburgh Particle-physics Astro-physics & Cosmology Center (PITT-PACC), Department of Physics & Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A. E-mail: , Abstract: Future electron-position colliders, such as the CEPC, FCC-ee, and ILC have the capability to dramatically improve the experimental precision for W and Z-boson masses and couplings. This would enable indirect probes of physics beyond the Standard Model at multi-TeV scales. For this purpose, one must complement the experimental measurements with equally precise calculations for the theoretical predictions of these quantities within the Standard Model, including three-loop electroweak corrections. This article reports on the calculation of a subset of these corrections, stemming from diagrams with three closed fermion loops to the following quantities: the prediction of the W-boson mass from the Fermi constant, the effective weak mixing angle, and partial and total widths of the Z boson. The numerical size of these corrections is relatively modest, but non-negligible compared to the precision targets of future colliders. In passing, an error is identified in previous results for the two-loop corrections to the Z width, with a small yet non-zero numerical impact. Keywords: Quark Masses and SM Parameters, Scattering Amplitudes ArXiv ePrint: 2002.05845 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP07(2020)210 JHEP07(2020)210 Leading fermionic three-loop corrections to electroweak precision observables Contents 1 2 Renormalization 2 3 Definition of the observables 3.1 Fermi constant Gµ f 3.2 Effective weak mixing angle sin2 θeff 3.3 Partial width Γ[Z → f f¯] 3.4 Technical aspects of the calculation 6 6 6 7 8 4 Numerical results 9 5 Conclusions 11 1 Introduction Precision measurements of processes mediated by W and Z bosons are crucial testbeds for the Standard Model (SM) and physics beyond the SM. Some of the most important of these electroweak precision observables (EWPOs) are (a) muon decay, mediated by a virtual W boson, and (b) e+ e− → f f¯, which is primarily mediated by an s-channel Z boson √ for center-of-mass energies s ≈ MZ . Here f denotes any SM lepton or quark, except the top quark. These processes receive sizable radiative corrections within the SM, which are currently known at the full two-loop level [1–22] and leading partial three- and four-loop yt results in powers of the top Yukawa coupling, αt = 4π , which have been calculated at order O(αt αs2 ) [23–25] , O(αt2 αs ), O(αt3 ) [26, 27], O(αt αs3 ) [28–30]. Including these corrections, the estimated theory uncertainties from missing higher orders are safely below the current experimental precision for these processes, see refs. [31–33] for recent reviews. However, proposals for future high-luminosity e+ e− colliders, such as the CEPC [34], FCC-ee [35], and ILC/Giga-Z [36] would dramatically improve the experimental precision for the relevant EWPOs, thus requiring significant additional higher-order corrections to meet the physics goals [37]. In this article, we report on the leading fermionic three-loop corrections to the EWPOs. Here “leading fermionic” refers to diagrams with the maximal number (i.e. three) of closed fermion loops. Generally, contributions with closed fermion loops are numerically enhanced since they are enhanced by powers of mt and a large number of light fermion flavors. Technically, the leading fermionic corrections require only the computation of one-loop integrals, but care has to be taken in the derivation of the counterterms for the renormalization, as well as the description of e+ e− → f f¯ as a Laurent expansion about the complex Z pole [38–41]. –1– JHEP07(2020)210 1 Introduction 2 Renormalization The calculations presented in this article are based on the on-shell renormalization scheme. In this scheme, the renormalized electromagnetic coupling is defined through the electronphoton vertex at zero momentum transfer, while the renormalized squared masses are defined at the real part of the propagator poles. For particles with a non-negligible decay width, such as the W and Z bosons, the propagator pole is complex and can be written as 2 s0 ≡ M − iM Γ, (2.1) where M is the on-shell mass, while Γ is the particle’s decay width. This definition of the mass and width is rigorously gauge-invariant [38–41], but it differs from the mass and width commonly used in the literature. Denoting the latter by M and Γ, respectively, they are related according to p p M =M 1 + Γ2 /M 2 , Γ=Γ 1 + Γ2 /M 2 . (2.2) See e.g. refs. [31, 44] for a more detailed discussion. Including radiative corrections, the massive gauge boson two-point function becomes 2 D(p2 ) = p2 − s0 + Σ(p2 ) − δM , (2.3) where Σ(s) is the transverse part of the gauge boson self-energy, and δM 2 is the mass counterterm. To avoid notational clutter, we do not include a field or wavefunction renormalization for the gauge boson. Since unstable particles can only appear as internal particles in a physical process, any dependence on their field renormalization drops out in the computation of such process.1 In the on-shell scheme, s0 is required to be a pole of the propagator, D(s0 ) = 0. This leads to the conditions
2 2 δM = Re Σ M − iM Γ , (2.4) Γ= 1
1 2 Im Σ M − iM Γ . M (2.5) In our calculation, we have checked explicitly that any field renormalization counterterms cancel. –2– JHEP07(2020)210 Partial results for the leading fermionic three-loop corrections have been discussed in refs. [42, 43], but the proper treatment of the complex gauge boson pole was not addressed there. In section 2, the renormalization procedure and relevant counterterms are discussed in more detail. Section 3 describes the calculation of the leading fermion three-loop corrections to the following quantities: (a) the Fermi constant for muon decay, which can be used to f predict the W mass, (b) the effective weak mixing angle sin2 θeff , which describes the ratio of the vector and axial-vector couplings of the Zf f¯ vertex, and (c) the partial widths for Z → f f¯. Numerical results are presented in section 4, together with a discussion of their impact. By recursively inserting eq. (2.5) into (2.4) and expanding in orders of perturbation theory, the W -mass counterterm is given by 2 2 δM W(1) = Re ΣW(1) (M W ) , (2.6)  2 2 2  2  δM W(2) = Re ΣW(2) (M W ) + Im ΣW(1) (M W ) Im Σ0W(1) (M W ) , (2.7) Here and in the following the numbers in brackets denote the loop order. For the Z-mass counterterm, one needs to include γ–Z mixing effects. The Z and photon fields get renormalized according to √ 1 Z ZZ Zµ + δZ Zγ Aµ , 2 √ 1 γZ Aµ → δZ Zµ + Z γγ Aµ . 2 Zµ → (2.9) (2.10) As already mentioned above, in the following we will simply set Z ZZ , Z (...truncated)