DESY 19-103 KW 19-005 Electroweak pseudo-observables and Z-boson form factors at two-loop accuracy

# Desy 19-103 Kw 19-005 Electroweak pseudo-observables and Z-boson form factors at two-loop accuracy

Ievgen Dubovyk, Institute of Physics, University of Silesia, Katowice, Poland Faculty of Science, University of Hradec Králové, Czech RepublicDeutsches Elektronen–Synchrotron, DESY, 15738 Zeuthen, Germany    Ayres Freitas, Institute of Physics, University of Silesia, Katowice, Poland Faculty of Science, University of Hradec Králové, Czech RepublicDeutsches Elektronen–Synchrotron, DESY, 15738 Zeuthen, Germany    Janusz Gluza, Institute of Physics, University of Silesia, Katowice, Poland Faculty of Science, University of Hradec Králové, Czech RepublicDeutsches Elektronen–Synchrotron, DESY, 15738 Zeuthen, Germany    Tord Riemann, Institute of Physics, University of Silesia, Katowice, Poland Faculty of Science, University of Hradec Králové, Czech RepublicDeutsches Elektronen–Synchrotron, DESY, 15738 Zeuthen, Germany    Johann Usovitsch Institute of Physics, University of Silesia, Katowice, Poland Faculty of Science, University of Hradec Králové, Czech RepublicDeutsches Elektronen–Synchrotron, DESY, 15738 Zeuthen, Germany
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###### Abstract

We present Standard Model predictions for the complete set of phenomenologically relevant electroweak precision pseudo-observables related to the -boson: the leptonic and bottom-quark effective weak mixing angles , , the -boson partial decay widths , where indicates any charged lepton, neutrino and quark flavor (except for the top quark), as well as the total decay width , the branching ratios , , , and the hadronic cross section . The input parameters are the masses , and , and the couplings , . The scheme dependence due to the choice of or its alternative as a last input parameter is also discussed. Recent substantial technical progress in the calculation of Minkowskian massive higher-order Feynman integrals allows the calculation of the complete electroweak two-loop radiative corrections to all the observables mentioned. QCD contributions are included appropriately. Results are provided in terms of simple and convenient parameterization formulae whose coefficients have been determined from the full numerical multi-loop calculation. The size of the missing electroweak three-loop or QCD higher-order corrections is estimated. We briefly comment on the prospects for their calculation. Finally, direct predictions for the vector and axial-vector form-factors are given, including a discussion of separate order-by-order contributions.

II. Institut für Theoretische Physik, Universität Hamburg, 22761 Hamburg, GermanyPittsburgh Particle Physics, Astrophysics & Cosmology Center (PITT PACC), Department of Physics & Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USASchool of Mathematics, Trinity College Dublin, University of Dublin, Dublin 2, Ireland

## 1 Introduction

In 2018 we celebrated 50 years of the Standard Model of elementary particles. The basics of the model were formulated and experimentally validated in the 1960s/70s. The next decade brought an intensive development of the calculation of quantum field theoretical radiative corrections in that model and in its alternatives. An experimental highlight in this context was the -collider LEP, which enabled us to check the Standard Model at an accuracy of better than the per-cent level, which corresponds to effects from more than one electroweak and two QCD loop orders. This proved, for the first time in a systematic way, the Standard Model as a quantum field theory. LEP 1 was running, from Summer 1989 to 1995, at and around the -boson peak. The expectation for the experimental precision of and was 20 MeV in 1986 Blondel:1986kj () and reached finally 2 MeV Arduini:1996pp (). This precision tag was extremely important because is one of the Standard Model input parameters to the commonly used on-mass-shell renormalization scheme. Indeed, the experimental accuracy of triggered much of the precision loop calculations, including the prediction of the top quark and Higgs masses prior to their discoveries from loop corrections to LEP observables in the Standard Model, see Refs. Ellis:1986jba (); Altarelli:1989YR (); Bardin:1995-YR03 () (as well as Refs. Blondel:2018mad (); Blondel:2019vdq () for an overview of the current state of the art). Data from the peak and the resonance curve (the line shape) allow to measure a large variety of observables, such as , , cross-sections for different two-fermion final states and their ratios and angular asymmetries, together with radiation of (sufficiently soft) photons, gluons, etc. From the real observables, the so-called electroweak pseudo-observables (EWPOs) are extracted by means of a de-convolution of initial-state radiation and subtraction of backgrounds. The fine details of relating EWPOs to real cross-sections at LEP 1 precision are described in detail in Ref. Gluza:inYR2018 () and references quoted therein.

On occasion of the 50 anniversary of the Standard Model, many of the related crucial developments were remembered at the conference ”SM@50” smat50:2019 ().

This article is devoted to the state-of-the art calculation of the Standard Model (SM) corrections to the -vertex and their inclusion into the predictions for the various EWPOs. We will mostly focus on recent advances in the calculation of the electroweak two-loop terms. QCD contributions, which are known up to four-loop order, have also been taken into account in the results presented here, but we will refer to the literature for further details.

The resonance curve can be described theoretically by writing the S-matrix elements as a Laurent series in the center-of-mass energy squared (also called S-matrix ansatz). This Laurent series ansatz is worked out up to two loops Consoli:1986kr (); Sirlin:1991fd (); Willenbrock:1991hu (); Stuart:1991xk (); Sirlin:1991rt (); Stuart:1991cc (); Stuart:1992jf (); Veltman:1992tm (); Passera:1998uj (). The coefficients of the leading series term contain the vertex form factors. Their one-loop corrections were studied in the 1980s; first with massless fermions Bardin:1980fe (); Bardin:1981sv (); Marciano:1980pb (); Marciano:1983wwa (), and slightly later with the full mass dependence of the Standard Model Akhundov:1985fc (); Bernabeu:1987me (); Jegerlehner:1988ak (); Beenakker:1988pv (). After several papers on approximate/partial higher-order corrections, the complete two-loop weak corrections were determined in a series of papers from 2004 to 2018 Awramik:2004ge (); Awramik:2006ar (); Awramik:2006uz (); Hollik:2005va (); Hollik:2006ma (); Awramik:2008gi (); Freitas:2013dpa (); Freitas:2014owa (); Freitas:2014hra (); Dubovyk:2016aqv (); Dubovyk:2018rlg (). The correct formulation of the interplay of the 22 loop corrections with higher order real QED corrections in the S-matrix approach, also called un-folding of the effective 22 Born terms from the realistic 2n observables, is a topic on its own. It was first studied in Refs. Leike:1991pq (); Riemann:1992gv (); Kirsch:1994cf (); Riemann:2015wpn (), but its extension beyond two-loop level will require more work Gluza:inYR2018-C3 (); Gluza:inYR2018 (); Jadach:2019bye (). The numerically relevant two-loop and partial higher-order corrections were included in the final analysis of LEP 1 data ALEPH:2005ab ().111The EWPOs at LEP 1 were determined order by order without a Laurent expansion. This was based on the ZFITTER software Bardin:1999yd (); Arbuzov:2005ma (), for both the Standard Model loop calculation and the unfolding of cross-sections. The relevant higher-order corrections to the input mass Awramik:2003rn () and to the leptonic weak mixing angle Awramik:2004ge () are implemented in ZFITTER v.6.42. While ZFITTER v.6.44beta Akhundov:2013ons () also contains the QCD four-loop corrections of Baikov:2012er (), they are of no experimental relevance so far.

The theoretical advances described here go beyond the Standard Model theory used for physics at LEP 1 Bardin:1999yd (); Arbuzov:2005ma () but will be needed for the FCC-ee Tera-Z project Jadach:April2018 (); Blondel:2018mad (); Mangano:2018mur (); Blondel:2019vdq (); Dubovyk:inYR2018 () whose unique experimental precision calls for perturbative predictions at three electroweak loops together with corresponding QCD terms.

In this work, the following pseudo-EWPOs are discussed: The partial widths for -boson decay into final states; the total width ; the branching fractions ; the total hadronic -pole quark-pair production cross-section ; and the effective weak mixing angles , defined from the ratio of vector- and axial -boson couplings. The precise definition of all these pseudo-observables will be given below. Pseudo-observables differ from real observables by removing from the former the effects of initial-state and initial-final QED radiation, as well as non-resonant photon-exchange, box and -channel contributions ALEPH:2005ab (); Blondel:2018mad ().

The Standard Model predictions for -pole pseudo-observables can be constructed in terms of the following three theoretical building blocks Awramik:2006uz ():

 vf(s) ≡vZf(s)−vγf(s)ΣγZ(s)s+Σγγ(s), (1) af(s) ≡aZf(s)−aγf(s)ΣγZ(s)s+Σγγ(s), (2) ΣZ(s) ≡ΣZZ(s)−[ΣγZ(s)]2s+Σγγ(s), (3)

where and are the one-particle irreducible vector- and axial-vector vertex contributions, respectively, whereas and are their counterparts for the vertex. The denotes the one-particle irreducible self-energy. At tree level,

 vZf(0) =eI3f−2Qfs2W2sWcW, vγf(0) =eQf, (4) aZf(0) =eI3f2sWcW, aγf(0) =0. (5)

Here and are the weak isospin and electric charge (in units of the elementary charge ) of the fermion , respectively. and are the sine and cosine of the weak mixing angle, respectively, and the subscript is used to denote tree-level order.

For the theory calculations, these building blocks must be evaluated at the complex pole Willenbrock:1991hu (); Sirlin:1991fd (); Stuart:1991xk (); Veltman:1992tm (), , where and are the on-shell mass and width of the -boson, respectively. Note that and differ from the mass and width reported in publications of LEP, Tevatron and LHC experiments by a fixed factor Bardin:1988xt (); Leike:1991pq ():

 ¯¯¯¯¯¯MZ =MZ/√1+Γ2Z/M2Z, ¯¯¯¯ΓZ =ΓZ/√1+Γ2Z/M2Z. (6)

Similar expressions hold for and Bardin:1986fi (); Denner:1990tx ().

## 2 Z-boson decay width, branching ratios and cross-sections

The width of the boson, , is related to the imaginary part of the self-energy. Using the optical theorem, one can derive the following expression for   Freitas:2013dpa (); Freitas:2014hra ():

 ¯¯¯¯ΓZ =∑f¯¯¯¯Γf,   f=e,μ,τ,νe,νμ,ντ,u,c,s,d,b, (7) ¯¯¯¯Γf =Nfc¯¯¯¯¯¯MZ12π[RfVFfV+RfAFfA]s=¯¯¯¯¯M2Z. (8)

Here is the color factor and are radiator functions that capture final-state QCD and QED corrections, see section 7 in Ref. Chetyrkin:1994js (), whereas the remaining electroweak and mixed electroweak–QCD corrections are contained in the form factors . Up to two-loop accuracy, the form factors can be written as follows Freitas:2014hra ():

 FfV=|vf|21+Re{Σ′Z−i2¯¯¯¯¯¯MZ¯¯¯¯ΓZΣ′′Z}∣∣∣s=¯¯¯¯¯M2Z, (9) FfA=|af|21+Re{Σ′Z−i2¯¯¯¯¯¯MZ¯¯¯¯ΓZΣ′′Z}∣∣∣s=¯¯¯¯¯M2Z, (10)

where and are shorthand expressions for and , respectively.

In addition to the partial widths, certain branching ratios are of phenomenological importance:

Here . Further, the cross-section for at the peak can be expressed in terms of partial widths Freitas:2014hra (),

Here accounts for non-resonant photon-exchange, box and -channel contributions. Furthermore, occurs from higher-order terms of the Laurent expansion of the full amplitude around the complex pole . At two-loop order, can be written as , where subscripts indicate the loop order. In the limit (), it is given by

 δX(2) =−[αs2Wc2W(78−53s2W+209s4W)]2. (13)

It is important to note that eq. (12) assumes that initial-state photon radiation effects have been removed by means of a de-convolution procedure, see e.g. Ref. Schael:2013ita ().

Results for the partial and total widths, branching ratios and including the full two-loop corrections have first been published in Ref. Dubovyk:2018rlg (). They can be expressed in simple parameterization formulae, which are adequate for most phenomenological applications. Here, we present slightly more complicated formulae that cover an extended numerical range of input parameters:222This extended input parameter range is useful for determining indirect constraints on various SM parameters from electroweak precision observables (see e.g. section 10 in Ref. Patrignani:2016xqp ()), since these indirect bounds often extend over larger intervals than the corresponding direct measurements.

 25GeV

Here is the shift in the running electromagnetic coupling from to , defined by . It can be divided into a leptonic and a hadronic part, . has been computed to three-loop order Steinhauser:1998rq (), whereas contains non-perturbative hadronic contributions, which are commonly extracted from data Davier:2017zfy (); Jegerlehner:2017zsb (); Keshavarzi:2018mgv (). We neglect the light fermion masses , , everywhere besides in and (at leading power) in the radiator functions . The boson mass can be computed from the Fermi constant Awramik:2003rn () and thus is not listed as an independent input parameter. Both and , the electromagnetic fine structure constant in the Thomson limit, are known with very small uncertainties, and thus we use their central experimental values Patrignani:2016xqp () without any uncertainty interval.

The fitting formulae for the EWPOs have the form

 (15) LH=logMH125.7GeV,ΔH=(125.7GeVMH)4−1,Δt=(mt173.2GeV)2−1,

The coefficients and are obtained from fits to a grid of 8750 data points of the full computation. The latter includes

Numerical values for the coefficients are given in Tab. 1.

## 3 Asymmetries and effective weak mixing angles

The effective weak mixing angle for the vertex is defined, from the theory side, as

 sin2θfeff≡14|Qf|(1+Re{vf(¯¯¯¯¯¯M2Z)af(¯M2Z)}). (16)

Here and are the real parts of the complex pole of the and propagators, respectively. They are related to the masses commonly reported by experiments at LEP, Tevatron, LHC according to eq (6). Moreover, denotes the electric charge of the fermion .

The effective weak mixing angles can be extracted from a range of asymmetries Gluza:inYR2018 (), defined from effective Born two-particle cross-sections, including the left-right asymmetry

 ALR =σeL−σeRσeL+σeR=Ae+Anon−resLR (17) and the forward-backward asymmetry AfFB =σcosθ>0−σcosθ<0σcosθ>0+σcosθ<0=34AeAf+Af,non−resFB, (18) where Af ≡2Re{vf/af}1+(Re{vf/af})2=1−4|Qf|sin2θfeff1−4|Qf|sin2θfeff+8(|Qf|sin2θfeff)2. (19)

Here and are the cross-sections for for left- and right-handed polarized electron beams, respectively, whereas and denote the cross-section for restricted to the forward and backward hemisphere, respectively. Furthermore, accounts for the non-resonant photon-exchange, box and -channel contributions.

The most precisely measured effective weak mixing angles are the leptonic effective weak mixing angle (extracted from ) and the bottom-quark one, (extracted from ) Schael:2013ita ().

Standard Model predictions for including the full two-loop corrections have been presented originally in Ref. Awramik:2006ar (); Awramik:2006uz (); Hollik:2006ma (). We reproduced by an independent calculation the contribution of the bosonic electroweak two-loop corrections using the methods of Ref. Dubovyk:2018rlg (). The corrections can be expressed in terms of a weak form factor , where

 Δκf=(1−¯¯¯¯¯¯MW¯¯¯¯¯¯MZ)−1sin2θfeff−1. (20)

The comparison with Ref. Awramik:2006ar () is shown in Tab. 2, which demonstrates that the two calculations agree to an accuracy of , which implies an accuracy of better than for . The full two-loop corrections for have been presented first in Ref. Dubovyk:2016aqv ().

In the following, we present simple parameterization formulae for and , which cover the extended range of input parameters of eq. (14). The parameterization formula

 sin2θfeff=s0+d1LH+d2L2H+d3L4H+d4Δα+d5Δt+d6Δ2t+d7ΔtLH+d8Δαs+d9ΔαsΔt+d10ΔZ (21) with LH=logMH125.7GeV,Δt=(mt173.2GeV)2−1,

provides a good description of the full result in the parameter region (14). Values for the coefficients are obtained by fitting (21) to a grid of 8750 data points.

Table 3 shows the result of a fit to a calculation that includes all known corrections:

As indicated by the last column in the table, the largest deviation of the fit formulae from the full result is , while for most of the parameter region in (14) the agreement is better than . The careful reader may realize that the parameterization for in Table 3 deviates slightly from Eqs. (20,22) in Dubovyk:2016aqv (). The difference is due to the larger grid of data points used here. A fit formula is, obviously, not able to reproduce the data points in a grid perfectly. The fitting aims to find the best average agreement between the data points (which are generated with our full numerical calculation) and the fit formula. A larger grid therefore can lead to some shifts of the coefficients. As a consequence, the formula in Dubovyk:2016aqv () will probably be more accurate for input values within the ranges in Tab. 1 there. On the other hand, while the formula here may be a little less accurate within these ranges, it covers a much larger range of input values.

In Tab. 4 it is shown that the technical accuracy of our fit formulae is adequate for the expected experimental precision of several future colliders, although it will get modified by anticipated future three-loop electroweak corrections.

## 4 Vector and axial-vector Z-boson form factors FfV and FfA

The pseudo-observables discussed in the previous sections aim to be closely related to actual observables, such as cross-sections, branching ratios, or asymmetries. On the other hand, for some purposes it is also useful to have numerical results for the underlying vertex corrections themselves Freitas:2014owa (), for example: (i) Inclusion of selected corrections from Beyond Standard Model (BSM) physics, (ii) Estimations of magnitudes of selected single terms, (iii) Partial cross-checks with other calculations. For such purposes, the form factors and introduced in eq. (8) are needed explicitly.

Tables 5 and 6 show the numerical contributions of different orders of perturbation theory to and . In Tab. 5 these are computed using the following input values:

 MZ=91.1876GeV, ΓZ=2.4952GeV, ⇒¯¯¯¯¯¯MZ=91.1535GeV (22a) MW=80.385GeV, ΓW=2.085GeV, ⇒¯¯¯¯¯¯MW=80.358GeV (22b) MH=125.1GeV, mt=173.2GeV, m¯¯¯¯¯MSb=4.2GeV, Δα=0.059, αs=0.1184 (22c)

For Tab. 6, on the other hand, the Fermi constant is used as an input instead of (22b), and is computed from

 ¯¯¯¯¯¯M2W(1−¯¯¯¯¯¯M2W¯¯¯¯¯¯M2Z)=πα√2Gμ(1+Δr), (23)

where has been evaluated to the same orders as given in each column of the table. More details about the calculation of can be found in Ref. Awramik:2003rn (). As before, the dependence of the Standard Model prediction on various input parameters can be expressed in terms of the simple parameterization formula eq. (15).

Table 7 shows the numerical values for the coefficients obtained by fitting this formula to the currently most precise computation, including the same corrections as in section 2, except for the final-state QED and QCD radiation effects, i.e.

 FfV =|vf(0)|2+FfV(α)+FfV(ααs)+FfV(α2)+FfV(αtα2s)+FfV(α2tαs)+FfV(α3t)+FfV(αtα3s), FfA =|af(0)|2+FfA(α)+FfA(ααs)+FfA(α2)+FfA(αtα2s)+FfA(α2tαs)+FfA(α3t)+FfA(αtα3s). (24)

Note that (rather than ) has been used as one input in Tab. 7.

The form factor results presented here can be easily augmented to include the effects of some new physics model:

 FfV,SM+NP ≈FfV,SM+2Re{vf(0)vf,NP}, (25) FfA,SM+NP ≈FfA,SM+2Re{af(0)af,NP}. (26)

Here “SM” denotes the SM contributions discussed in the present paper, while “NP” stands for the new physics correction on top of the SM. Since the existing experimental constraints imply that any possible new physics effect is small, it is sufficient to use the tree-level couplings and in the interference terms and neglect the and terms.