Leading fermionic three-loop corrections to electroweak precision observables

Future electron-position colliders, such as CEPC and FCC-ee, 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.


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 − → ff , which is primarily mediated by an s-channel Z boson for center-of-mass energies √ s ≈ M Z . Here f denotes any SM lepton or quark, except the top quark. These processes receive sizeable radiative corrections within the SM, which are currently known at the full two-loop level [1][2][3][4][5][6][7][8][9][10] and partial three-and four-loop level [11][12][13]. Including these corrections, the estimated theory uncertainties from missing higher orders are safely below the current experimental precision for these processes, see Refs. [14][15][16] for recent reviews. However, proposals for future high-luminosity e + e − colliders, such as CEPC [17] and FCC-ee [18] would dramatically improve the experimental precision for the relevant EWPOs, thus requiring significant additional higher-order corrections to meet the physics goals [19]. 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 m t 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 − → ff as a Laurent expansion about the complex Z pole [20]. Partial results for the leading fermionic three-loop corrections have been discussed in Refs. [21,22], 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 predict the W mass, (b) the effective weak mixing angle sin 2 θ f eff , which describes the ratio of the vector and axial-vector couplings of the Zff vertex, and (c) the partial widths for Z → ff . Numerical results are presented in section 4, together with a discussion of their impact.

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 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 [20], 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 See e. g. Refs. [14,23] for a more detailed discussion. Including radiative corrections, the massive gauge boson two-point function becomes 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 * .
In the on-shell scheme, s 0 is required to be a pole of the propagator, D(s 0 ) = 0. This leads to the conditions By recursively inserting eq. (5) into (4) and expanding in orders of perturbation theory, the W -mass counterterm is given by 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 As already mentioned above, in the following we will simply set Z ZZ , Z γγ → 1 for the Z and photon field renormalization constants, since these drop out anyways for the physical processes discussed in this work. However, the mixing counterterms generate extra terms compared to eqs. (4) and (5): where Here Σ V 1 V 2 is the self-energy with an incoming gauge boson V 1 and outgoing gauge boson V 2 .
The mixing counterterms are fixed through the conditionŝ Using eqs. (11)- (16) and expanding in orders of perturbation theory yields The self-energies receive contributions from one-loop diagrams with counterterm insertions, see Fig. 1, with relevant counterterm Feynman rules shown in Fig. 2. It is worth noting that when inserting these into eq. (19), δZ γZ (2) drops out without needing to include an explicit expression for it. Figure 1: Diagrams with closed fermion loops contributing to self-energies at different orders. A box with number n indicates a counterterm of loop order n.
Other relevant counterterms are given by with L(m 2 ) ≡ ln m 2 4πµ 2 + γ E . The simple results in eqs. (20)-(23) are a consequence of restricting ourselves to only closed fermion loop. The weak mixing angle counterterm is obtained by demanding that the relation s 2 holds to all orders. Order-byorder expressions for δs W (n) can be obtained by plugging the previous expressions for the mass counterterms into (24), but we refrain from spelling them out here.
The symbol ∆α in eq. (21) stems from light-fermion loop contributions in the photon vacuum polarization, Π γγ lf (q 2 ) can be divided into a leptonic part, which is perturbatively calculable [38], and a hadronic part that becomes non-perturbative for small q 2 . Therefore the hadronic contribution is commonly extracted from data [39]. When using results from the literature for ∆α had , one must remember that these references use M Z rather M Z in eq. (25). Accordingly, one must also use M Z rather M Z in eq. (21). Here I 3 f = ± 1 2 for up/down-type fermions, and Q f is the fermion electric charge in units of e. δZ e and δs W are the change and weak mixing angle counterterms, respectively. Furthermore, c W +δc W = 1 − (s W + δs W ) 2 . Note that field or wavefunction renormalization counterterms have not been included, since they are either irrelevant for the processes considered here, as same as explained in the text, or do not receive any contributions from closed fermion loops.
3 Definition of the observables The Fermi constant can be determined with high precision from the muon decay lifetime [41]. In the SM it is defined through where ∆r includes the contribution from radiative corrections. With s 2 W = 1 − M 2 W /M 2 Z , eq. (26) can be used to compute a prediction for the W -boson mass in terms of G µ and other SM parameters. Since ∆r itself depends on M W and s W , this is usually performed in a recursive procedure.
The corrections to ∆r from diagrams with closed fermion loops can be written as It is permissible to set the momentum transfer in the W propagator and in Σ W to zero since m µ ≪ M W . Eq. (27) can be computed straightforwardly by expanding in orders of perturbation theory and using the derivations from section 2.

Effective weak mixing angle sin
The effective weak mixing angle is defined in terms of the effective vector and axial-vector couplings of the Z-boson to an ff pair, denoted v f and a f , respectively, When considering corrections with closed fermion loops, the effective couplings are obtained from the relations where In writing these equations, we have used that δZ Zγ = 0, see eq. (20), and set Z ZZ , Z γγ → 1 as discussed above. The second term in (30) originates from photon-Z mixing self-energies, as illustrated in Fig. 3. Figure 3: Decomposition of the effective Zff vertex and Z self-energy into one-particle irreducible building blocks, indicated by the hatched blobs.
Through the optical theorem, the imaginary part of Σ Z can be expressed in terms of the decay rate for Z → ff , where a f and v f are the effective axial-vector and vector couplings, defined in eqs. (29) and (30), respectively. Then one obtains the following expressions for the total and partial widths of the Z boson: and analogously for F f A . Here N f c = 3(1) for quarks (leptons), and the functions R V,A are included in general to account for final-state QCD and QED corrections. When considering corrections with only closed fermion loops, R V,A = 1.

Technical aspects of the calculation
Since the algebraic expressions for our results are rather lengthy, the calculation has been carried out with the help of computer algrebra tools, within the framework of Mathematica. FeynArts 3.3 [30] has been used for the generation of diagrams and amplitudes, and FeynCalc 9.2.0 [42] has been employed for some of the Dirac and tensor algebra. The masses and Yukawa couplings of all fermions except the top quark have been neglected. Furthermore, CKM mixing of the top quark with other quark generations has been ignored.
We have compared results for ∆r, sin 2 θ f eff and Γ f with two fermionic loops with Refs. [2,3], [6] and [9], respectively. Exact algebraic agreement was found, with one exception: The Re Σ ′ Z(2) term in the third line of (35), together with (13) and the γ-Z mixing counterterms, leads to The second term in eq. (36) was missed in Ref. [9]. The numerical impact of this term will be discussed in the following section.

Numerical results
Let us now discuss the numerical impact of the leading fermionic three-loop corrections to the observables introduced in the previous section. For concreteness, input parameters in Tab. 1 are used, but the results do not depend very strongly on the specific input values within experimentally allowed ranges.
With these inputs, the fermionic three-loop corrections to ∆r is found to be This can be translated into a shift, ∆M W(3) , of the predicted value of the W -boson mass in the SM, using eq. (26). Given that ∆r (3) , it is sufficient to expand eq. (26) up to linear order in ∆r (3) and ∆M W(3) , leading to For the effective weak mixing angle, the fermionic three-loop correction amounts to This result does not depend on the type of fermion f . If we assume that M W is predicted from G µ , we can take into account the leading effect of the shift ∆M W(3) from eq. (26) according to In a similar fashion, one obtains the corrections to the partial decay widths The results in eqs. (38), (41) and (43) are presented in terms of the gauge-invariant complexpole definitions of the gauge-boson masses and widths. However, the corresponding corrections to the conventional (unbarred) masses and widths are the same within the precision presented above, since the translation factor in eq. 2 is about 1.00035, i. e. very close to 1. When comparing the above results with the experimental determination of these quantities [16], which are dominated by measurements from LEP, SLD and LHC, one can see that the fermionic three-loop corrections are negligible compared to the experimental uncertainties. For example, the direct measurements of the W mass, effective weak mixing angle, and Z width are M W = 80.379 ± 0.012 GeV, sin 2 θ ℓ eff = 0.23152 ± 0.00016 and Γ Z,tot = 2.4952 ± 0.0023 GeV. These are at least one order of magnitude larger than the corrections in eqs. (38), (40) and (43).
However, future high-luminosity e + e − colliders, such as FCC-ee and CEPC, are anticipated to dramatically improve the experimental precision for these quantities [17,18]: CEPC: It is evident that the corrections computed in this paper are important for the physics program of these machines. Finally, we also wish to study in the impact of the error that was found in the previous calculation of the fermionic two-loop contribution to the partial decay widths, ∆Γ f, (2) , as discussed in section 3.4. Using the inputs from Tab. 1, the difference amounts to ∆Γ f, (2) this work − ∆Γ f, (2) Ref. [2,3] = It turns out that the numerical impact is very small, but for the sake of consistency it is important to identify and correct this error.

Conclusions
Electroweak precision measurements at future e + e − colliders, such as CEPC and FCC-ee, require the inclusion of three-loop electroweak corrections in theoretical calculations to match the experimental precision. As a first step, this article presents results for contributions with three closed fermion loops for some of the most important electroweak precision observables (EWPOs): (i) The prediction of the W mass from the muon decay rate; (ii) the ratio of vector and axial-vector Z-boson couplings; and (iii) Z-boson decay rates into different final states. Corrections with a fixed number of closed fermion loops form a UV-finite and gaugeinvariant subset, and they are enhanced by powers of m t and the large multiplicity of light fermion degrees of freedom. Care must be taken when deriving the counterterms for the W and Z boson masses. Since these particles have non-negligible decay widths, their propagator poles are complex. A consistent theoretical definition of the gauge boson masses is then given by the real part of the complex propagator poles. Additional complications arise from γ-Z mixing.
Given the large size of the final expression, the calculation has been performed with the help of computer-algebra tools. The numeric size of the leading fermionic three-loop corrections turns out to be relatively small for all considered EWPOs, but not negligible for the anticipated precision of CEPC and FCC-ee.
In the course of the calculation, an error was found in literature for the previously known results for the leading fermionic two-loop corrections to the Z-boson decay width. The numerical impact of this error is found to be very small.
Experience from electroweak two-loop calculations [2][3][4][5][6][7][8][9] shows that loop corrections with maximal number of closed fermion loops and next-to-maximal number of closed fermion loops can be of comparable size. Thus the availability of the new results in this paper, while important, does not significantly reduce the theoretical error estimates given in Refs. [5,10] for the relevant EWPOs.