Constraining SMEFT BSM scenarios with EWPO and $\Delta_{CKM}$

1Mathew Thomas Arun<EMAIL_ADDRESS>2Kuldeep
<EMAIL_ADDRESS>2Tripurari
<EMAIL_ADDRESS>
1School of Physics, Indian Institute of Science Education and Research
Thiruvananthapuram, Vithura, Kerala, 695551, India

2Department of Physics and Astrophysics, University of Delhi, Delhi 110007,
India

###### Abstract

Precision observables are well known for constraining most of the Beyond
Standard Model (BSM) scenarios tightly. We present here a simple and
comprehensive fitting framework for various BSM scenarios to these
observables. We start with the fit of $S$, $T$ and $V$ parameter and their
correlations using the Electroweak Precision Observables (EWPO) including the
recent $m_{W}$ measurement from CDF-II. Utilizing these observables, we also
fit various New Physics (NP) scenarios consisting of different subsets of
dimension-6 Standard Model Effective Field Theory (SMEFT) operators out of a
total of 10 appearing at tree level in EWPO. To further constrain these
scenarios, we augment these observables with $\Delta_{CKM}$ measurement using
1-loop matching of the Low Energy Effective Field Theory (LEFT) to SMEFT
operators at the Z-pole. We show that the inclusion of $\Delta_{CKM}$
constraint indeed results in stronger bounds on the SMEFT Wilson Coefficients.
We also constrain the UV parameters of BSM extensions like Vectorlike leptons
(VLL) and find out that such a minimal extension is in tension with the
forward-backward asymmetry in $b$-sector ($A_{b}^{FB}$) and the recent
measurement of $M_{W}$. In order to lift the two blind directions, which one
encounters while fitting all the 10 SMEFT WCs at tree-level, we also include
the LEP-II observables pertaining to the $WW$ production and present the
results for the fits with and without $\Delta_{CKM}$ constraint.

##  1 Introduction

The discovery of Higgs at the LHC in 2012 [1] completes the Standard Model of
particle physics (SM). On the other hand, the absence of any heavy resonances
in the Run 2 of the Large Hadron Collider (LHC) reveals that any physics
beyond the Standard Model (BSM), with SM like coupling, might be significantly
heavier than the electroweak scale. Interestingly, however, there has been a
few discrepancies in low energy experiments like ($g-2$) of the muon [2, 3, 4,
5, 6], flavor anomalies [7, 8, 9, 10], Cabibbo anomaly [11] and the recent CDF
anomaly in W boson mass [12]. Given the elusiveness of the New Physics at
collider experiments, there is a huge motivation to understand the nature of
various anomalies in a model-independent Effective Field Theory formalism.
Here, the New Physics effects are included by extending SM with higher
dimensional operators which are invariant under the SM gauge group [13, 14,
15, 16, 17, 18]. Being well measured at run-1 and run-2 of LEP [19, 20, 21,
22, 23], the precise measurements of electroweak observables strongly
constrain the scales of these SMEFT operators.

The recent measurement of $W$ boson mass [12], $M_{W}=80,433.5\pm 9.4$ MeV, at
CDF II has put particle physicists in a precarious position since it disagrees
with the SM prediction, $M_{W}=80,360\pm 6$ MeV, at $7\sigma$ level. The new
combined world average now lies at $M_{W}=80,413.3\pm 9$ MeV, which is about
$5.3\sigma$ away from the Standard Model value. Several studies have been
conducted to address the impact of this new measurement on the Global EW fit
in terms of the oblique parameters [24, 25, 26, 27] assuming the Standard
Model Effective Field Theory (SMEFT) framework [28, 29, 30, 31, 32, 33, 34,
35, 30]. Models such as Technicolor models, Extradimensions, Composite Higgs
or Little Higgs scenarios, SU(2) Triplet Scalar Extensions [36], Models with
VLQ [37], Two Higgs Doublet Models [24] among many others [38, 39, 40, 41] are
also proposed to address this, with Higgs and additional gauge bosons
($Z^{\prime}$) [25].

Moreover, there have been anomalies reported from low-energy measurements.
These are studied by Low-energy Effective Field Theory (LEFT) description of
Standard Model. In particular, in this paper, we have considered the beta
decay and other semileptonic processes that contribute to the $\Delta_{CKM}$
anomaly in the LEFT basis. Since $\Delta_{CKM}$ and electro-weak precision
tests crucially depend on the measured value of $G_{F}$, though at different
scales, these measurements are connected and have to be taken together while
constraining the New Physics. To that end, we match these LEFT operators with
SMEFT at 1-loop at $Z$ boson scale and obtain experimental constraints on the
Wilson Coefficients (WCs) of the dimension-6 SMEFT operators.

In this article, we augment the recent CDF-II $M_{W}$ mass measurement in the
electroweak fit with $\Delta_{CKM}$ constraint and explore various BSM
scenarios. Along with these two observables, $A_{b}^{FB}$ and $A_{l}^{SLD}$
play a significant role in these fits, since both have a long-standing
discrepancy of more than 2$\sigma$ with respect to the SM predictions. One of
New Physics parametrisation, which we use here, is the $S,~{}T$ and $V$ [42,
43, 44]. Inclusion of $V$ parameter along with $S$ and $T$ parameters
identifies the New Physics that interacts via the weak coupling constant
$G_{F}$. Moreover, this analysis is also crucial in understanding the Cabibbo
anomaly as it solely depends on the $V$ parameter. We then switch to SMEFT
paramatrisation where we consider various operator subsets affecting the
observables of our interest and and perform the fit to constrain their WCs.
Depending on the UV complete scenarios, there can be only one subset of all
the WCs that contribute significantly at low energies. One such fit looks at
the scenario where the operators common to the $M_{W}$ and $\Delta_{CKM}$ are
the ones carrying the imprint of the New Physics. Another New Physics scenario
considered here is the set of WCs which appear in the expression for $M_{W}$
at tree-level. We also perform a fit for the vector-like lepton models, being
a popular extension of SM because of its ability to address the discrepancy in
muon $(g-2)$ and provide mass to the SM neutrinos through see-saw mechanism,
which contains the WCs carrying imprints of these new leptons at electroweak
scale. It also enables us to constrain the UV parameters of the model.
Finally, we also perform an all-parameter fit which takes into account all the
10 WCs affecting the EWPO at tree-level. However, we encounter two blind
directions as the EWPO can only constrain 8 independent combinations of the
WCs. Addition of $\Delta_{CKM}$ also does not help in lifting the blind
directions. Rather, this forces us to use LEP-II data, where the presence of
W-W production channel breaks the two blind directions, enabling us to
constrain all the 10 WCs of our interest.

This paper is organized as follows: In the next section, we revisit the
contributions of dimension-6 operators to the shift in fermion couplings to
gauge-bosons at leading order. In section [3], we briefly discuss near Z pole
observables (LEP-I) and the observables arising due to $WW$ production (LEP-
II). We then discuss the contributions of SMEFT operators to muon and beta
decay obtained by matching with LEFT at one loop in section [4] and discuss
the fitting framework in section [5]. Further, we analyze various model
independent frameworks, using $S,~{}T,~{}V$ and the subsets of dimension-6
operators affecting the EWPO and $\Delta_{CKM}$. Finally, in section [6] we
summarize the results.

##  2 SMEFT contributions to Electwoweak Fit

The low-energy effects of the massive BSM particles can be approximated by
integrating them out to obtain higher-dimensional interactions between the SM
fields. In such an approach, the SM can be regarded as an effective theory
whose known renormalizable interactions are supplemented by higher-order terms
and scaled by inverse powers of the BSM mass scale. These higher dimensional
operators can then be written in the form

${\cal L}={\cal
L}_{SM}+\sum_{d=5}^{\infty}\sum_{i=1}^{n}{\mathcal{C}_{i}^{d}\over\Lambda^{d-4}}O_{i}^{d}\,,$
(1)

where $d$ represents the dimension of the operator and $i$ runs over all the
redundant set of operators at a particular dimension. The operators
$O_{i}^{d}$ are all $SU(3)\times SU(2)_{L}\times U(1)_{Y}$ invariant where all
of the effects of the BSM physics reside in the WCs $\mathcal{C}_{i}^{d}$. We
assume the WCs to be real and use the Warsaw basis [15] to paramatrise them.

The Electroweak Precesion Observables (EWPO) are nothing but the $Z$ and $W$
pole observables coming from LEP studies and there are 10 six dimensional
operators which contribute to these observables at tree level. We collect
these operators in Table 1, where $H$ is the $SU(2)_{L}$ Higgs doublet,
$\tau^{a}$ are the Pauli matrices,
$D_{\mu}=\partial_{\mu}+ig_{s}T^{A}G_{\mu}^{A}+ig_{2}{\tau^{a}\over
2}W_{\mu}^{a}+ig_{1}YB_{\mu}$, $q$ is the $SU(2)_{L}$ quark-doublet with
$q^{T}=(u_{L},d_{L})$, $l$ is the $SU(2)_{L}$ lepton-doublet with
$l^{T}=(\nu_{L},e_{L})$, $W_{\mu\nu}^{a}$ is the $SU(2)_{L}$ field strength
with
$W_{\mu\nu}^{a}=\partial_{\mu}W^{a}_{\nu}-\partial_{\nu}W^{a}_{\mu}-g_{2}\epsilon^{abc}W^{b}_{\mu}W^{c}_{\nu}$
and $B_{\mu\nu}$ is the $U(1)_{Y}$ field strength with
$B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}W_{\mu}$. The other
definitions include:
$\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,H}$}=iH^{\dagger}(D_{\mu}H)-i(D_{\mu}H)^{\dagger}H$,
and
$\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}^{\,a}\,H}$}=iH^{\dagger}\tau^{a}D_{\mu}H-i(D_{\mu}H)^{\dagger}\tau^{a}H$.

${\cal O}_{HD}$ | $\left(H^{\dagger}D^{\mu}H\right)^{*}\left(H^{\dagger}D_{\mu}H\right)$ | ${{\cal O}}_{HWB}$ | $(H^{\dagger}\tau^{a}H)\,W^{a}_{\mu\nu}B^{\mu\nu}$
---|---|---|---
${{\cal O}_{ll}}$ | $(\overline{l}\gamma_{\mu}l)(\overline{l}\gamma^{\mu}l)$ | ${{\cal O}}_{He}$ | $(\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,H}$})(\overline{e}\gamma^{\mu}e)$
${{\cal O}}_{Hu}$ | $(\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,H}$})(\overline{u}\gamma^{\mu}u)$ | ${{\cal O}}_{Hd}$ | $(\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,H}$})(\overline{d}\gamma^{\mu}d)$
${{\cal O}}_{Hq_{3}}$ | $(\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}^{\,a}\,H}$})(\overline{q}\tau^{a}\gamma^{\mu}q)$ | ${{\cal O}}_{Hq_{1}}$ | $(\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,H}$})(\overline{q}\tau^{a}\gamma^{\mu}q)$
${{\cal O}_{Hl_{3}}}$ | $(\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}^{\,a}\,H}$})(\overline{l}\tau^{a}\gamma^{\mu}l)$ | ${{\cal O}}_{Hl_{1}}$ | $(\mbox{${H^{\dagger}i\,\raisebox{5.69054pt}{\boldmath${}^{\leftrightarrow}$}D_{\mu}\,H}$})(\overline{l}\tau^{a}\gamma^{\mu}l)$
Table 1: Dimension-6 operators contributing to the $Z$ and $W$ pole
observables of this study at tree level.

In order to get the theoretical predictions from the Electroweak Precesion
Data pertaining the pole observables, we first fix our choice of input
parameters, namely, the fine structure constant $\hat{\alpha}_{e}$ from the
low energy limit of electron Compton scattering, the Fermi constant in muon
decays $\hat{G}_{F}$ and the measured $Z$ mass ($\hat{m}_{Z}$). At tree level,
one can then define the effective measured mixing angle

$\displaystyle
s_{W}^{2}=\frac{1}{2}-\frac{1}{2}\sqrt{1-\frac{4\,\pi\hat{\alpha}_{e}}{\sqrt{2}\,\hat{G}_{F}\,\hat{m}_{Z}^{2}}},$
(2)

The value of the $\rm SU(2)_{L}$ gauge coupling can be taken as:

$\displaystyle\hat{g}_{2}\,s_{W}=2\,\sqrt{\pi}\,\hat{\alpha}_{e}^{1/2}.$ (3)

The effective measured vacuum expectation value (vev) in the SM can be defined
as $\hat{v}^{2}=1/\sqrt{2}\,\hat{G}_{F}$.

Once we allow for these new SMEFT operators, the gauge sector gets modified
resulting in the requirement for barred fields and couplings in order to get
canonical kinetic terms. These transformations are [45]:

$\displaystyle{\overline{W}}_{\mu}^{a}$ $\displaystyle\equiv$
$\displaystyle(1-\mathcal{C}_{HW}v^{2}/\Lambda^{2})W_{\mu}^{a}$
$\displaystyle{\overline{B}}_{\mu}$ $\displaystyle\equiv$
$\displaystyle(1-\mathcal{C}_{HB}v^{2}/\Lambda^{2})B_{\mu}$
$\displaystyle{\overline{g}}_{2}$ $\displaystyle\equiv$
$\displaystyle(1+\mathcal{C}_{HW}v^{2}/\Lambda^{2})g_{2}$
$\displaystyle{\overline{g}}_{1}$ $\displaystyle\equiv$
$\displaystyle(1+\mathcal{C}_{HB}v^{2}/\Lambda^{2})g_{1}\,,$ (4)

such that ${\overline{W}}_{\mu}{\overline{g}}_{2}=W_{\mu}g_{2}$ and
${\overline{B}}_{\mu}{\overline{g}}_{1}=B_{\mu}g_{1}$. The masses of the W and
Z fields to ${\cal{O}}\biggl{(}{1\over\Lambda^{2}}\biggr{)}$ are [45],

$\displaystyle M_{W}^{2}$ $\displaystyle=$
$\displaystyle\frac{{\overline{g}}_{2}^{2}v^{2}}{4},$ $\displaystyle
M_{Z}^{2}$ $\displaystyle=$
$\displaystyle\frac{({\overline{g}}_{1}^{2}+{\overline{g}}_{2}^{2})v^{2}}{4}+\frac{v^{4}}{\Lambda^{2}}\left(\frac{1}{8}({\overline{g}}_{1}^{2}+{\overline{g}}_{2}^{2})\mathcal{C}_{HD}+\frac{1}{2}{\overline{g}}_{1}{\overline{g}}_{2}\mathcal{C}_{HWB}\right).$
(5)

The 4-fermion operators contributing to the decay of the $\mu$ changes the
relation between vev, $v$, and the Fermi constant $\hat{G}_{F}$,

$\displaystyle\hat{G}_{F}\equiv\frac{1}{\sqrt{2}v^{2}}-\frac{1}{\sqrt{2}\Lambda^{2}}\mathcal{C}_{ll}+{\sqrt{2}\over\Lambda^{2}}\mathcal{C}_{Hl_{3}}\,.$
(6)

The shift in $M_{Z}^{2}$ can then be written as:

$\displaystyle\delta
M_{Z}^{2}\equiv\frac{1}{2\,\sqrt{2}}\,\frac{\hat{m}_{Z}^{2}}{\hat{G}_{F}}C_{HD}+\frac{2^{1/4}\sqrt{\pi}\,\sqrt{\hat{\alpha_{e}}}\,\hat{m}_{Z}}{\hat{G}_{F}^{3/2}}C_{HWB}.$
(7)

where $\hat{m}_{Z}$ is the input parameter The kinetic mixing introduced by
the operator with Wilson coefficient $C_{HWB}$ leads to a redefinition of the
usual $s_{\theta}=\sin\theta$ mixing angle of the SM given by

$\displaystyle\delta
s_{W}^{2}=-\frac{v^{2}}{\Lambda^{2}}\frac{s_{W}c_{W}}{c_{W}^{2}-s_{W}^{2}}\left[2s_{W}c_{W}\left(\delta
v+\frac{1}{4}\mathcal{C}_{HD}\right)+\mathcal{C}_{HWB}\right]$ (8)

We relate the Lagrangian parameters $\overline{g}_{2},\overline{g}_{1}$ to the
input parameters at tree level via

$\displaystyle\overline{g}_{1}^{2}+\overline{g}_{2}^{2}=4\,\sqrt{2}\,\hat{G}_{F}\,\hat{m}_{Z}^{2}\left(1-\sqrt{2}\,\delta
G_{F}-\frac{\delta M_{Z}^{2}}{\hat{m}_{Z}^{2}}\right),$ (9)
$\displaystyle\overline{g}_{2}^{2}=\frac{4\,\pi\,\hat{\alpha_{e}}}{s^{2}_{W}}\left[1+\frac{\delta
s_{W}^{2}}{s^{2}_{W}}+\frac{c_{W}}{s_{W}}\frac{1}{\sqrt{2}\,\hat{G}_{F}}\,C_{HWB}\right].$
(10)

Expressing $\overline{M}_{W}^{2}$ in terms of the inputs parameters we get:

$\displaystyle\overline{M}_{W}^{2}=\hat{m}_{W}^{2}\left(1+\frac{\delta
s_{W}^{2}}{s_{W}^{2}}+\frac{c_{W}}{s_{W}\sqrt{2}\hat{G}_{F}}C_{HWB}+\sqrt{2}\delta
G_{F}\right)=\hat{m}_{W}^{2}-\delta M_{W}^{2},$ (11)

where $\delta M_{W}^{2}=-\hat{m}_{W}^{2}\left(\frac{\delta
s_{W}^{2}}{s_{W}^{2}}+\frac{c_{W}}{s_{W}\sqrt{2}\hat{G}_{F}}C_{HWB}+\sqrt{2}\delta
G_{F}\right)$ and $\hat{m}_{W}^{2}=c_{W}^{2}\hat{m}_{Z}^{2}$.

The effective axial and vector couplings of the SMEFT $Z$ boson are defined as
follows

$\displaystyle\mathcal{L}_{Z,eff}=g_{Z,eff}\,\left(J_{\mu}^{Z\ell}Z^{\mu}+J_{\mu}^{Z\nu}Z^{\mu}+J_{\mu}^{Zu}Z^{\mu}+J_{\mu}^{Zd}Z^{\mu}\right),$
(12)

where $g_{Z,eff}=-\,2\,2^{1/4}\,\sqrt{\hat{G}_{F}}\,\hat{m}_{Z}$,
$(J_{\mu}^{Zx})^{ij}=\overline{\psi}_{i}\,\gamma_{\mu}\left[(\overline{g}^{x}_{V})_{eff}^{ij}-(\overline{g}^{x}_{A})_{eff}^{ij}\,\gamma_{5}\right]\psi_{j}$
for $\psi=\\{u,d,\ell,\nu\\}$. In general, these currents are matrices in
flavour space. When we restrict our attention to the case of a minimal linear
MFV scenario $(J_{\mu}^{Zx})_{ij}\simeq(J_{\mu}^{Zx})\delta_{ij}$. In the
standard basis, the effective axial and vector couplings are modified from the
SM values by a shift defined as

$\displaystyle\delta(g^{x}_{V,A})_{ij}=(\overline{g}^{x}_{V,A})^{eff}_{ij}-(g^{x}_{V,A})^{SM}_{ij},$
(13)

The tree level couplings are usual SM relations,

$\displaystyle g_{R}^{Zf}$ $\displaystyle=$ $\displaystyle-
s_{W}^{2}Q_{f}\quad{\rm and}\quad g_{L}^{Zf}=T_{3}^{f}-s_{W}^{2}Q_{f}$ (14)

with $T_{3}^{f}=\pm\displaystyle\frac{1}{2}$. The full SMEFT contributions to
the effective couplings shown in Table 2.

$\delta(g^{\ell}_{V})$ | $\delta\overline{g}_{Z}\,(g^{\ell}_{V})^{SM}-\frac{1}{4\sqrt{2}\hat{G}_{F}}\left(C_{He}+C_{Hl_{1}}+C_{Hl_{3}}\right)-\delta s_{W}^{2},$
---|---
$\delta(g^{\ell}_{A})$ | $\delta\overline{g}_{Z}\,(g^{\ell}_{A})^{SM}_{pr}+\frac{1}{4\,\sqrt{2}\,\hat{G}_{F}}\left(C_{He}-C_{Hl_{1}}-C_{Hl_{3}}\right),$
$\delta(g^{\nu}_{V})$ | $\delta\overline{g}_{Z}\,(g^{\nu}_{V})^{SM}_{pr}-\frac{1}{4\,\sqrt{2}\,\hat{G}_{F}}\left(C_{Hl_{1}}-C_{Hl_{3}}\right)$
$\delta(g^{\nu}_{A})$ | $\delta\overline{g}_{Z}\,(g^{\nu}_{A})^{SM}_{pr}-\frac{1}{4\,\sqrt{2}\,\hat{G}_{F}}\left(C_{Hl_{1}}-C_{Hl_{3}}\right)$
$\delta(g^{u}_{V})$ | $\delta\overline{g}_{Z}\,(g^{u}_{V})^{SM}_{pr}+\frac{1}{4\,\sqrt{2}\,\hat{G}_{F}}\left(-C_{Hq_{1}}+\,C_{Hq_{3}}-C_{Hu}\right)+\frac{2}{3}\delta s_{W}^{2}$
$\delta(g^{u}_{A})$ | $\delta\overline{g}_{Z}\,(g^{u}_{A})^{SM}_{pr}-\frac{1}{4\,\sqrt{2}\,\hat{G}_{F}}\left(C_{Hq_{1}}-\,C_{Hq_{3}}-C_{Hu}\right)$
$\delta(g^{d}_{V})$ | $\delta\overline{g}_{Z}\,(g^{d}_{V})^{SM}_{pr}-\frac{1}{4\,\sqrt{2}\,\hat{G}_{F}}\left(C_{Hq_{1}}+\,C_{Hq_{3}}+C_{Hd}\right)-\frac{1}{3}\delta s_{W}^{2}$
$\delta(g^{d}_{A})$ | $\delta\overline{g}_{Z}\,(g^{d}_{A})^{SM}_{pr}+\frac{1}{4\,\sqrt{2}\,\hat{G}_{F}}\left(-C_{Hq_{1}}-\,C_{Hq_{3}}+C_{Hd}\right)$
$\delta(g^{W_{\pm},\ell}_{V})=\delta(g^{W_{\pm},\ell}_{A})$ | $\frac{1}{2\sqrt{2}\hat{G}_{F}}\left(C_{Hl_{3}}+\frac{1}{2}\frac{c_{W}}{s_{W}}\,C_{HWB}\right)+\frac{1}{4}\frac{\delta s_{W}^{2}}{s^{2}_{W}}$
$\delta(g^{W_{\pm},q}_{V})=\delta(g^{W_{\pm},q}_{A})$ | $\frac{1}{2\sqrt{2}\hat{G}_{F}}\left(C_{Hq_{3}}+\frac{1}{2}\frac{c_{W}}{s_{W}}\,C_{HWB}\right)+\frac{1}{4}\frac{\delta s_{W}^{2}}{s^{2}_{W}}$
Table 2: Anomalous fermion couplings at LO

For the charged currents, we define

$\displaystyle\mathcal{L}_{W,eff}=-\frac{\sqrt{2\,\pi\,\hat{\alpha_{e}}}}{s_{W}}\left[(J_{\mu}^{W_{\pm},\ell})_{ij}W_{\pm}^{\mu}+(J_{\mu}^{W_{\pm},q})_{ij}W_{\pm}^{\mu}\right],$
(15)

where in the SM one has

$\displaystyle(J_{\mu}^{W_{+},\ell})_{ij}$ $\displaystyle=$
$\displaystyle\,\overline{\nu}_{i}\,\gamma^{\mu}\,\left(\overline{g}^{W_{+},\ell}_{V}-\overline{g}^{W_{+},\ell}_{A}\gamma_{5}\right)\,\ell_{j},$
(16) $\displaystyle(J_{\mu}^{W_{-},\ell})_{ij}$ $\displaystyle=$
$\displaystyle\,\overline{\ell}_{i}\,\gamma^{\mu}\,\left(\overline{g}^{W_{-},\ell}_{V}-\overline{g}^{W_{-},\ell}_{A}\gamma_{5}\,\right)\nu_{j}.$
(17)

The contribution of these shifts to the Observables of our interest can then
be calculated [46, 47, 48, 48, 49, 50, 51].

##  3 $W$ and $Z$ pole observables of interest

This section dedicate to review the contributions from SMEFT to the
observables considered. Our list of EWPO includes [28, 51, 52, 19]:

$\displaystyle
M_{W},\Gamma_{W},\Gamma_{Z},\sigma_{h},R_{l},A_{l}^{FB},R_{b},R_{c},A_{b}^{FB},A_{c}^{FB},A_{b},A_{c},A_{l},A_{l}^{SLD},BR_{W\rightarrow\nu
l}\,.$ (18)

In the SMEFT, at tree level the decay width of $Z$ boson to fermions can be
given by:

$\displaystyle\overline{\Gamma}\left(Z\rightarrow f\overline{f}\right)$
$\displaystyle=$
$\displaystyle\frac{\,\sqrt{2}\,\hat{G}_{F}\hat{m}_{Z}^{3}\,N_{c}}{3\pi}\left(|\overline{g}^{f}_{V}|^{2}+|\overline{g}^{f}_{A}|^{2}\right),$
(19) $\displaystyle\overline{\Gamma}\left(Z\rightarrow{\rm Had}\right)$
$\displaystyle=$ $\displaystyle 2\,\overline{\Gamma}\left(Z\rightarrow
u\overline{u}\right)+3\,\overline{\Gamma}\left(Z\rightarrow
d\overline{d}\right).$ (20)

With our chosen normalization of
$\overline{g}_{V}^{x}=T_{3}/2-Q^{x}\,\overline{s}_{\theta}^{2},\overline{g}_{A}=T_{3}/2$
where $T_{3}=1/2$ for $u_{i},\nu_{i}$ and $T_{3}=-1/2$ for $d_{i},\ell_{i}$
and $Q^{x}=\\{-1,2/3,-1/3\\}$ for $x=\\{\ell,u,d\\}$. The modification of the
decay widths in the SMEFT compared to the situation in the SM introduces
corrections of the form [47, 46, 48, 53, 49]

$\displaystyle\delta\Gamma_{Z\rightarrow\ell\overline{\ell}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}\,\hat{G}_{F}\hat{m}_{Z}^{3}}{6\pi}\,\left[-\delta
g^{\ell}_{A}+\left(-1+4s_{W}^{2}\right)\delta g^{\ell}_{V}\right],$ (21)
$\displaystyle\delta\Gamma_{Z\rightarrow\nu\overline{\nu}}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}\,\hat{G}_{F}\hat{m}_{Z}^{3}}{6\pi}\,\left[\delta
g^{\nu}_{A}+\delta g^{\nu}_{V}\right],$ (22)
$\displaystyle\delta\Gamma_{Z\rightarrow Had}$ $\displaystyle=$ $\displaystyle
2\,\delta\Gamma_{Z\overline{u}u}+3\,\delta\Gamma_{Z\overline{d}d},$
$\displaystyle=$
$\displaystyle\frac{\,\sqrt{2}\,\hat{G}_{F}\hat{m}_{Z}^{3}}{\pi}\left[\delta
g^{u}_{A}-\frac{1}{3}\left(-3+8s_{W}^{2}\right)\delta
g^{u}_{V}-\frac{3}{2}\delta
g^{d}_{A}+\frac{1}{2}\left(-3+4s_{W}^{2}\right)\delta g^{d}_{V}\right],$
$\displaystyle\delta\Gamma_{Z}$ $\displaystyle=$ $\displaystyle
3\delta\Gamma_{Z\rightarrow\ell\overline{\ell}}+3\delta\Gamma_{Z\rightarrow\nu\overline{\nu}}+\delta\Gamma_{had},$
$\displaystyle=$
$\displaystyle\frac{\,\sqrt{2}\,\hat{G}_{F}\hat{m}_{Z}^{3}}{2\,\pi}\left[\delta
g^{\nu}_{A}+\delta g^{\nu}_{V}-\delta
g^{\ell}_{A}+\left(-1+4s_{W}^{2}\right)\delta g^{\ell}_{V},\right.$
$\displaystyle\hskip 62.59596pt\left.+2\delta
g^{u}_{A}-\frac{2}{3}\left(-3+8s_{W}^{2}\right)\delta g^{u}_{V}-3\delta
g^{d}_{A}+\left(-3+4s_{W}^{2}\right)\delta g^{d}_{V}\right],$ . (26)

So that: $\overline{\Gamma}\left(Z\rightarrow
f\overline{f}\right)=\Gamma_{Z\rightarrow
f\overline{f}}+\delta\Gamma_{Z\rightarrow f\overline{f}}$ for all $f$ and the
same kind of relation holds for $\overline{\Gamma}_{Z}$.

When considering partial widths extracted from LEP data in the SM at the $Z$
pole, $\sigma_{e^{+}e^{-}\rightarrow had}$ has the theoretical expression

$\displaystyle\overline{\sigma}_{h}^{0}=12\pi\,\frac{\overline{\Gamma}_{Z\rightarrow
e\overline{e}}\overline{\Gamma}_{Z\rightarrow
Had}}{|\overline{\omega}(M_{Z}^{2})|^{2}},$ (27)

with $\overline{\Gamma}_{Z\rightarrow e\overline{e}}$,
$\overline{\Gamma}_{Z\rightarrow Had}$ being the decay widths in SM. With the
choice $\overline{\omega}(M_{Z}^{2})=\overline{M}_{Z}\,\overline{\Gamma}_{Z}$,
and the partial width taking on SM values, this expression simplifies to the
well known SM result. The contribution from SMEFT is relegated in the appendix
Appendix.

The shift of the ratios of decay rates defined in the SM as
$R^{0}_{f}=\frac{\Gamma_{had}}{\Gamma_{Z\rightarrow\overline{f}f}}$ where $f$
can be a charged lepton $\ell$ or a neutrino follows from

$\displaystyle\delta R^{0}_{f}=\frac{1}{(\Gamma(Z\rightarrow
f\overline{f})^{2})_{SM}}\left[\delta\Gamma_{Z\rightarrow
Had}(\Gamma(Z\rightarrow f\overline{f}))_{SM}-\delta\Gamma_{Z\rightarrow
f\overline{f}}(\Gamma\left(Z\rightarrow{\rm Had})_{SM}\right)\right],$ (28)

and we can then write that $\overline{R}^{0}_{f}=R^{0}_{f}+\delta R^{0}_{f}$.
For an identified quark the inverse ratio is used. The forward backward
asymmetry for 2-2 scattering is defined as

$\displaystyle A_{FB}=\frac{\sigma_{F}-\sigma_{B}}{\sigma_{F}+\sigma_{B}}.$
(29)

If $\theta$ is the angle between incoming lepton $\ell$ and outgoing fermion
$f$ then, $\sigma_{F}$ is the cross-section defined in the region
$\theta\in\left[0,\pi/2\right]$ and $\sigma_{B}$ is the cross-section defined
in the region $\theta\in\left[\pi/2,\pi\right]$. In terms of the left-right
asymmetry ($A$) for the incoming lepton and outgoing fermion:

$\displaystyle\quad
A_{e}=2\frac{g^{\ell}_{V}g^{\ell}_{A}}{(g^{\ell}_{V})^{2}+(g^{\ell}_{A})^{2}},\quad
A_{f}=2\frac{g^{f}_{V}g^{f}_{A}}{(g^{f}_{V})^{2}+(g^{f}_{A})^{2}}.$ (30)

we can express the forward backward asymmetry as:

$\displaystyle\quad A_{FB}^{0,f}=\frac{3}{4}A_{e}A_{f},$ (31)

Once we include the SMEFT operators, the $Z$ couplings receive corrections
which in turn brings corrections to these observables. The expressions for
these corrections have been shown in Appendix Appendix in detail.

The partial $W$ width in SMEFT is given by [54, 49]:

$\displaystyle\overline{\Gamma}_{W\rightarrow\overline{f}_{i}f_{j}}=$
$\displaystyle\Gamma_{W\rightarrow\overline{f}_{i}f_{j}}^{SM}+\delta\Gamma_{W\rightarrow\overline{f}_{i}f_{j}},$
(32) $\displaystyle\Gamma_{W\rightarrow\overline{f}_{i}f_{j}}^{SM}=$
$\displaystyle\frac{N_{C}\,|V^{f}_{ij}|^{2}\sqrt{2}\hat{G}_{F}\,\hat{m}_{W}^{3}}{12\pi},$
(33) $\displaystyle\delta\Gamma_{W\rightarrow\overline{f}_{i}f_{j}}=$
$\displaystyle\frac{N_{C}\,|V^{f}_{ij}|^{2}\sqrt{2}\hat{G}_{F}\,\hat{m}_{W}^{3}}{12\pi}\left(4\delta
g_{V/A}^{W_{\pm},f}+\frac{1}{2}\frac{\delta
m_{W}^{2}}{\hat{m}_{W}^{2}}\right).$ (34)

Here, $N_{C}$ is 3 for quarks and 1 for leptons. $V^{f}_{ij}$ is CKM or PMNS
mixing matrix elememts. Summing over all the modes we can compute the
contribution to the total decay width from SMEFT.

###  3.1 LEP-II data

One issue with using just the LEP-I data for the EWPO is that they can only
constrain 8 linear combination of the WCs out of a total of 10 WCs which
affect the EWPO at tree-level, resulting in two blind directions. The origin
of this can be traced to the fact that there is a reparamatrisation invariance
in the LEP-I data where the fields and the couplings have the property:
$\mathcal{F},g\rightarrow\mathcal{F}^{\prime}(1+\epsilon)g^{\prime}(1-\epsilon)$
[55, 54, 56] for the $2\rightarrow 2$ scattering processes. In order to break
this invariance, one has to go to $2\rightarrow 4$ scattering processes, which
can be realised in LEP-II due to the possibility of having resonant production
of $W^{+}$ and $W^{-}$. Pair produced $W$’s can then further can decay
leptonically to ($\ell,\nu_{\ell}$) or hadronically to ($j,j$). In addition to
breaking the invariance, measurement of these four fermions final states also
provide additional observables for the electroweak precision study. Recently
the contributions from the SMEFT to these processes have been done assuming
both resonant and non-resonant precesses. Deatils of these observables and
their chi-squared analysis has also been studied in detail in [57].

##  4 Going Beyond EWPO: $\Delta_{CKM}$

The aim of this section is to understand the SMEFT operators that enter the
LEFT observables like muon decay and $\beta-$decay which are relevant in
electro-weak precision study. In [58], the authors studied the maximal
deviation of $\Delta_{CKM}$ allowed by electro-weak precision measurements.
Their study was specific to the combination of
$\mathcal{O}_{ll},\mathcal{O}_{lq_{3}},\mathcal{O}_{Hl_{3}}$ and
$\mathcal{O}_{Hq_{3}}$ SMEFT operators that contribute at tree-level to the
$\Delta_{CKM}$ constraint, in the limit of $U(3)^{5}$ invariance.

Since the SMEFT operators $\mathcal{O}_{HD}$ and $\mathcal{O}_{HWB}$ among
others, given in Table 1, are crucial to electro-weak study, here we ask the
question regarding how much they influence the $\Delta_{CKM}$ at 1-loop order.
Moreover, the reason to study purely bosonic operators are crucial. This is
not too difficult to envisage, given that LHC direct bounds on these operators
are not as bad as leptonic ones. Moreover, in such a scenario with New Physics
coupling only to bosons, it becomes important to understand these operators
and their contributions in low-energy observables. Hence, in this section, we
match the muon decay and $\beta-$decay to 1-loop order, thus including the
SMEFT operators given in Table 1 in the $\Delta_{CKM}$ constraint.

Assuming $U(3)^{5}$ flavour symmetry [58], the correction to CKM unitarity is
given by

$\Delta_{CKM}=|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}-1\ .$ (35)

The CKM matrix element $|V_{ud}|$ is precisely measured from the study of
super allowed $0^{+}\to 0^{+}$ beta decay. A crucial input for the correct
determination is the precision of $G_{F}$. To understand the New Physics
contribution to $G_{F}$, lets study the muon decay process.

###  4.1 Muon Decay

At low energies, the Lagrangian density that leads to the process $\mu\to
e\overline{\nu}_{e}\nu_{\mu}$ is given by

$\mathcal{L}_{\mu}=L^{V,LL}_{\nu
e}(\overline{\nu}_{L\mu}\gamma^{\mu}\nu_{Le})(\overline{e}_{L}\gamma_{\mu}\mu_{L})+L^{V,LR}_{\nu
e}(\overline{\nu}_{L\mu}\gamma^{\mu}\nu_{Le})(\overline{e}_{R}\gamma_{\mu}\mu_{R})\
,$ (36)

where, all the WCs are measured at the muon mass scale $\mu=m_{\mu}$.

At tree-level, the matching of the LEFT Lagrangian with the SMEFT one gives,

$\displaystyle L^{V,LL}_{\nu e\ tree}$ $\displaystyle=$
$\displaystyle\displaystyle-\frac{2}{v^{2}}+\Big{(}C_{ll(1,2,2,1)}+C_{ll(2,1,1,2)})-2(C_{Hl_{3}(2,2)}+C_{Hl_{3}(1,1)})\Big{)}U_{(1,1)}U^{\dagger}_{(2,2)}\
.$ (37)

At 1-loop level, the matching of LEFT Lagrangian with the SMEFT Lagrangian
given,

$\displaystyle L^{V,LL}_{\nu e\ 1-loop}$ $\displaystyle=$
$\displaystyle\displaystyle-1.48946\log\left(\mu_{W}^{2}\right)-6.8206\log\left(\mu_{W}^{2}\right)\Big{(}0.024(C_{Hl_{3}(1,1)}+C_{Hl_{3}(2,2)})$
(38) $\displaystyle+$ $\displaystyle\displaystyle
0.252418C_{Hq_{3}(1,1)}-0.0029(C_{ll(1,1,2,2)}+C_{ll(2,2,1,1)})$
$\displaystyle+$ $\displaystyle\displaystyle
0.001(C_{ll(1,2,2,1)}+C_{ll(2,1,1,2)})+0.463C_{HD}+0.0449C_{HWB}\overline{g}\Big{)}$
$\displaystyle-$ $\displaystyle\displaystyle
0.0025(C_{Hl_{1}(1,1)}+C_{Hl_{1}(2,2)})+0.0473(C_{Hl_{3}(1,1)}+C_{Hl_{3}(2,2)})+1.0119C_{Hq_{3}(1,1)}$
$\displaystyle-$ $\displaystyle\displaystyle
0.0206(C_{ll(1,1,2,2)}+C_{ll(2,2,1,1)})+0.0067(C_{ll(1,2,2,1)}+C_{ll(2,1,1,2)})$
$\displaystyle+$ $\displaystyle\displaystyle
2.0729C_{HD}+0.3125C_{HWB}\overline{g}$

Note that Standard Model only has the left handed currents, and hence the
$L^{V,LR}$ are generated by new physics beyond the cut-off scale $\Lambda$.
Thus all square terms of this WCs are $\frac{v^{4}}{\Lambda^{4}}$ suppressed
in comparison with its interference term with $L^{V,LL}$. Moreover, since the
electron in $L^{V,LL}$ and $L^{V,LR}$ have opposite hierarchies, the
interference term will be helicity suppressed. Moreover, since the tree and
1-loop matching of $L^{V,LR}_{\nu e}$ operator with SMEFT Lagrangian consists
of only $C_{le(2,1,1,2)}$ operator, electro-weak precision study will not play
any role in constraining them. Hence we assume that this SMEFT operator
vanishes and neglect the corresponding LEFT operator in further analysis.

This means that,

$L^{V,LL}_{\nu e}=L^{V,LL}_{\nu e\ tree}+L^{V,LL}_{\nu e\
1-loop}=-\frac{4G_{F}}{\sqrt{2}}\ ,$ (39)

evaluated at $\mu_{W}=M_{Z}$ and is fixed by muon decay measurement.

###  4.2 Beta Decay

The low-energy effective Lagrangian for the process $d\to
ue^{-}\overline{\nu}_{e}$ is given by,

$\displaystyle\mathcal{L}_{n}$ $\displaystyle=$ $\displaystyle\displaystyle
L^{V,LL}_{\nu
edu}(\overline{\nu}_{L}\gamma^{\mu}e_{L})(\overline{u}_{L}\gamma_{\mu}d_{L})+L^{V,LR}_{\nu
edu}(\overline{\nu}_{L}\gamma^{\mu}e_{L})(\overline{u}_{R}\gamma_{\mu}d_{R})+L^{S,RR}_{\nu
edu}(\overline{\nu}_{L}e_{R})(\overline{u}_{L}d_{R})$ (40) $\displaystyle+$
$\displaystyle\displaystyle L^{S,RL}_{\nu
edu}(\overline{\nu}_{L}e_{R})(\overline{u}_{L}\gamma_{\mu}d_{R})+L^{T,RR}_{\nu
edu}(\overline{\nu}_{L}\sigma^{\mu\nu}e_{R})(\overline{u}_{L}\sigma^{\mu\nu}d_{R})$

where $L^{V,LL}_{\nu edu},L^{V,LR}_{\nu edu},L^{S,RR}_{\nu edu},L^{S,RL}_{\nu
edu},L^{T,RR}_{\nu edu}$ are the LEFT WCs.

At tree level, the matching of the above Lagrangian with SMEFT Lagrangian
gives,

$\displaystyle L^{V,LL}_{\nu edu\ tree}$ $\displaystyle=$
$\displaystyle\displaystyle\Big{(}-\frac{2}{v^{2}}-2(C_{Hl_{3}(1,1)}+C_{Hq_{3}(1,1)})\Big{)}U^{\dagger}_{(1,1)}V^{\dagger}_{(1,1)}$
(41)

Matching this Lagrangian with the SMEFT Lagrangian, at 1-loop, level we get,

$\displaystyle L^{V,LL}_{\nu edu\ 1-loop}$ $\displaystyle=$
$\displaystyle\displaystyle
U^{\dagger}_{11}V^{\dagger}_{11}\Big{(}(0.0241\log\left(\mu_{W}^{2}\right)+0.0595)C_{Hl_{3}(1,1)}$
(42) $\displaystyle-$
$\displaystyle\displaystyle(0.0139\log\left(\mu_{W}^{2}\right)+0.0712)C_{Hq_{1}(1,1)}+(0.1007\log\left(\mu_{W}^{2}\right)+0.3931)C_{Hq_{3}(1,1)}$
$\displaystyle+$
$\displaystyle\displaystyle(0.1338\log\left(\mu_{W}^{2}\right)+0.592)C_{HD}-(0.0115\log\left(\mu_{W}^{2}\right)+0.0715)C_{HWB}\overline{g_{1}}$
$\displaystyle+$ $\displaystyle\displaystyle 0.0007C_{Hl_{1}(1,1)}\Big{)}$

The contribution from $L^{V,LR}_{\nu edu},L^{S,RR}_{\nu edu}$ and
$L^{T,RR}_{\nu edu}$ do not contain any SMEFT coefficients that contribute to
the EWPO. Then, up to 1-loop order,

$\displaystyle L^{V,LL}_{\nu edu}=L^{V,LL}_{\nu edu\ tree}+L^{V,LL}_{\nu edu\
1-loop}=\frac{4G_{F}}{\sqrt{2}}V_{ud}\ ,$ (43)

Using Eq.39, we get

$\displaystyle V_{ud}=\frac{L^{V,LL}_{\nu edu}}{L^{V,LL}_{\nu e}}$
$\displaystyle=$ $\displaystyle\displaystyle\frac{L^{V,LL}_{\nu edu\
tree}+L^{V,LL}_{\nu edu\ 1-loop}}{L^{V,LL}_{\nu e\ tree}+L^{V,LL}_{\nu e\
1-loop}}$ (44) $\displaystyle=$ $\displaystyle\displaystyle
V_{11}(1+\delta_{SMEFT\ tree}+\delta_{SMEFT\ 1-loop})\ .$
$\Delta_{CKM}=\delta_{SMEFT\ tree}+\delta_{SMEFT\ 1-loop}$ (45)

where,

$\displaystyle\delta_{SMEFT\ tree}$ $\displaystyle=$
$\displaystyle\displaystyle 2(C_{Hl_{3}(1,1)}+C_{Hq_{3}(1,1)})$
$\displaystyle+$
$\displaystyle(-2C_{Hl_{3}(2,2)}+C_{ll(1,2,2,1)}+C_{ll(2,1,1,2)}-2C_{Hl_{3}(1,1)})$
$\displaystyle\delta_{SMEFT\ 1-loop}$ $\displaystyle=$
$\displaystyle\displaystyle-0.0029C_{Hl_{1}(1,1)}-\displaystyle(0.0026\log\left(\mu_{W}^{2}\right)+0.0172)C_{Hl_{3}(1,1)}$
(46) $\displaystyle+$
$\displaystyle\displaystyle(0.0139\log\left(\mu_{W}^{2}\right)+0.0711)C_{Hq_{1}(1,1)}$
$\displaystyle-$
$\displaystyle\displaystyle(0.0255\log\left(\mu_{W}^{2}\right)+0.0917)C_{Hq_{3}(1,1)}$
$\displaystyle+$
$\displaystyle\displaystyle(0.004\log\left(\mu_{W}^{2}\right)+0.0252)C_{HD}$
$\displaystyle+$
$\displaystyle\displaystyle(0.0249\log\left(\mu_{W}^{2}\right)+0.1645)C_{HWB}\overline{g_{1}}$

Assuming flavor universal scenarios and so removing indices the contributions
to SMEFT can be written as:

$\displaystyle\delta_{SMEFT\ tree}$ $\displaystyle=$
$\displaystyle\displaystyle 2(C_{Hl_{3}}+C_{Hq_{3}})$ $\displaystyle+$
$\displaystyle(-2C_{Hl_{3}}+C_{ll}+C_{ll}-2C_{Hl_{3}})$ $\displaystyle=$
$\displaystyle(2C_{ll}-2C_{Hl_{3}}+2C_{Hq_{3}})$ $\displaystyle\delta_{SMEFT\
1-loop}$ $\displaystyle=$
$\displaystyle\displaystyle-0.0029C_{Hl_{1}}-\displaystyle(0.0026\log\left(\mu_{W}^{2}\right)+0.0172)C_{Hl_{3}}$
(47) $\displaystyle+$
$\displaystyle\displaystyle(0.0139\log\left(\mu_{W}^{2}\right)+0.0711)C_{Hq_{1}}$
$\displaystyle-$
$\displaystyle\displaystyle(0.0255\log\left(\mu_{W}^{2}\right)+0.0917)C_{Hq_{3}}$
$\displaystyle+$
$\displaystyle\displaystyle(0.004\log\left(\mu_{W}^{2}\right)+0.0252)C_{HD}$
$\displaystyle+$
$\displaystyle\displaystyle(0.0249\log\left(\mu_{W}^{2}\right)+0.1645)C_{HWB}\overline{g_{1}}$

Note that $V_{ud}=V_{11}$ in the limit where all the SMEFT coefficients vanish
and the tree-level result matches with [58].

##  5 Fitting procedure

The results of the fit involving EWPO has strong dependence on the input
scheme[49]. In our analyses, we choose the $(G_{F},M_{Z}\text{ and }\alpha)$
scheme and the input parameters we used are as follows:

$\displaystyle G_{F}$ $\displaystyle=$ $\displaystyle 1.1663787(6)\times
10^{-5}\,\mathrm{Ge\kern-1.00006ptV}^{-2}$ $\displaystyle M_{Z}$
$\displaystyle=$ $\displaystyle 91.1876\pm.0021\,\mathrm{Ge\kern-1.00006ptV}$
$\displaystyle{1\over\alpha_{e}}$ $\displaystyle=$
$\displaystyle{137.035999139(31)}$

Let the contribution of high dimensional operators, along with SM, to the
electro-weak observables be represented as,

$\displaystyle\mathcal{O}_{i}^{SMEFT}$ $\displaystyle=$
$\displaystyle\mathcal{O}_{i}^{SM}+\delta\mathcal{O}_{i}^{SMEFT}~{},$ (49)

where $\delta\mathcal{O}_{i}^{SMEFT}$’s are the functions of WCs $C_{i}$’s.

Measurement | Experiment | Precise Theory | Pull
---|---|---|---
$\Gamma_{Z}$(GeV) | $2.4955\pm 0.0023$ | $2.4945\pm 0.0006$ | 0.42
$\sigma_{h}$(nb) | $41.481\pm 0.033$ | $41.482\pm 0.008$ | -0.29
$R_{l}$ | $20.767\pm 0.025$ | $20.749\pm 0.009$ | 0.67
$R_{b}$ | $0.21629\pm 0.00066$ | $0.21582\pm 0.00002$ | 0.71
$R_{c}$ | $0.1721\pm 0.0030$ | $0.17221\pm 0.00003$ | -0.03
$R_{uc}$ | $0.166\pm 0.009$ | $0.172227\pm 0.000032$ | -0.69
$A_{l}$ | $0.1465\pm 0.0033$ | $0.1468\pm 0.0003$ | -0.09
$A_{l}(SLD)$ | $0.1513\pm 0.0021$ | $0.1468\pm 0.0003$ | 2.12
$A_{b}$ | $0.923\pm 0.020$ | $0.92699\pm 0.00006$ | -0.19
$A_{c}$ | $0.670\pm 0.027$ | $0.6677\pm 0.0001$ | 0.08
$A_{s}$ | $0.895\pm 0.020$ | $0.9356\pm 0.00004$ | -0.44
$A_{l,FB}$ | $0.0171\pm 0.0010$ | $0.01617\pm 0.00007$ | 0.92
$A_{b,FB}$ | $0.0996\pm 0.0016$ | $0.1029\pm 0.0002$ | -2.04
$A_{c,FB}$ | $0.0707\pm 0.0035$ | $0.0735\pm 0.0002$ | -0.79
$M_{W}$(GeV) | $80.4133\pm 0.008$ | $80.360\pm 0.006$ | 5.33
$\Gamma_{W}$(GeV) | $2.085\pm 0.042$ | $2.0904\pm 0.0003$ | -0.13
$BR_{W\rightarrow\nu l}$ | $0.1086\pm 0.0009$ | $0.108271\pm 0.000024$ | 0.36
$\Delta_{CKM}$ | $-0.0015\pm 0.0007$ | $0\pm 0$ | -2.14
Table 3:  Experimental and theoretical values and uncertainities of the
observables. Pulls of the measurement with best theory is also provided in the
last column.

For $O_{i}^{SM}$ in Eq. 49, we have collected the most precisely calculated
value and are given in Table 3. The form of the $\chi^{2}$ is given by,

$\displaystyle\chi^{2}$ $\displaystyle=$
$\displaystyle\Sigma_{i,j}(\mathcal{O}_{i}^{exp}-\mathcal{O}_{i}^{SMEFT})\sigma^{-2}_{ij}(\mathcal{O}_{j}^{exp}-\mathcal{O}_{j}^{SMEFT})\,.$
(50)

where the covariance matrix
$\sigma^{2}_{ij}=\Delta_{i}^{exp}\rho^{exp}_{ij}\Delta_{j}^{exp}+\Delta_{i}^{th}\rho^{th}_{ij}\Delta_{j}^{th}$.
Here $\rho^{exp}_{ij}$ is the experimental correlation matrix obtained from
[19] and $\rho^{th}_{ij}$ is identity. $\Delta_{i}^{exp}$ and
$\Delta_{i}^{th}$ are the experimental and theory errors of the $i^{\rm th}$
observable. Inclusion of theoretical uncertainity and the correlation among
the observables lead to better constraints over the parameters of interest.

In Table 3, we summarize the current status of the SM theory and the
experimental results, including $\Delta_{CKM}$. First we compute the
$\chi^{2}$ including the relevant observables from Table 3. The $\chi^{2}$
thus becomes a function of WCs which affects the observables that we are
interested in. The fit can be carried out by including all the relevant WCs or
by taking only a subset. The $\chi^{2}$ function is then minimized to get the
best-fit values of the WCs and the 1$\sigma$ ranges of the WCs are calculated
by marginalizing over the other parameters. This is done in the following way:
For the parameter $c_{i}$ of our interest, we choose a random value and then
define a partial $\chi^{2}$ as follows.

$\chi^{2}_{par}=\chi_{c_{i}=x_{0},c_{j}}^{2}=\chi^{2}(x_{0},c_{j}),$

where $x_{0}$ is the one value of random variable $c_{i}$ and $c_{j}$ are the
other parameters present in the $\chi^{2}$. We then define the following test-
statistic which is a difference between the minima of the partial $\chi^{2}$
and the minima of the total $\chi^{2}$

$min(\chi^{2}(c_{i}(i\neq j)=x_{0},c_{j}))-min(\chi^{2}(c_{i},c_{j}))$

For various random values of $c_{i}$, we get some result for this new test-
statistics, which can be used to obtain an interploating function. The
marginalized result is then obtained by demanding the value of the
interpolating function to be smaller than the significant region of $\chi^{2}$
distribution for a single variable.

The correlation between two WCs affecting the fit can also be calculated in a
similar way by treating the two WCs at hand as random variables and then
marginalizing over the others.

###  5.1 S, T, V fitting

The effects of new heavy particles(with mass $M_{new}$) on the gauge boson
self-energies can be described by three parameters $S$, T, and U. The U
parameter is not very sensitive to heavy New Physics because of the presence
of an extra factor of $M_{Z}/M_{new}$ and hence can be neglected. This S and T
parametrisation can be mapped to the SMEFT WCs $C_{HD}$ and $C_{HWB}$ and has
been studied in literature [59, 60, 61, 62]. Along with them, we append the V
parameter which parametrise the change in $G_{F}$. This parameter is crucial
if we are to make the quarks and leptons interact with BSM resonances. As
shown in Eq. 6, this shift in $G_{F}$ can be mapped to combination
$2C_{Hl_{3}}-C_{ll}$. The expressions for the Electroweak Observables in terms
of this S, T and V paramatrisation is given in Table 4. For the
$\Delta_{CKM}$, we infer the result by considering only the tree-level
matching, where the dependence on V comes from the muon decay parameters
sitting in the denominator of Eq 44. As can be seen from Table 4, this V
parameter is highly crucial for both $M_{W}$ as it has the highest sensitivity
to this parameter, which can be inferred from its largish coefficient of V
compared to other observables.

Table 4: Expressions for observables in terms of $\Delta S$, $\Delta T$ and
$\Delta V$.

In Figure 1, we have plotted the change in S, T and V parameters with EWPO
data and EWPO $+$ $\Delta_{CKM}$ data in a 2-D plane by marginalising over the
third parameter. Note that, inclusion of $\Delta_{CKM}$ constraints the
Peskin-Takeuchi parameters better in comparison with just the EWPO data. The
results of the fit are presented in Table 5.

Figure 1: S, T and V parameters with EWPO and EWPO $+$ $\Delta_{CKM}$ data WC |  B.F(EWPO) | Correlation |  B.F(EWPO+CKM) | Correlation
---|---|---|---|---
|  ($\chi^{2}_{\rm fit}$/$\chi^{2}_{\rm SM}=11.41/40.08$) |  ($\chi^{2}_{\rm fit}$/$\chi^{2}_{\rm SM}=11.68/44.67$)
$\Delta{S}$ | $-0.0032\pm 0.1077$ | $1.00$ |  |  | $-0.0292\pm 0.0965$ | $1.00$ |  |
$\Delta{T}$ | $0.1646\pm 0.0649$ | $0.79$ | $1.00$ |  | $0.1631\pm 0.06484$ | $0.86$ | $1.00$ |
$\Delta{V}$ | $-0.001\pm 0.0007$ | $-0.59$ | $-0.07$ | $1.00$ | $0.0013\pm 0.0005$ | $-0.44$ | $-0.04$ | $1.00$
Table 5: Global fit of (S,T,V) paramatrisation tree-level to EWPO and
$\Delta_{CKM}$. Table 6: Observables posteriors using STV fit.

The posterior values of the observables after the (S,T,V) can be read off from
Table 6. Even without including the $\Delta_{CKM}$ constraint, we can see that
this paramatrisation has very good agreement with the experimental results of
$M_{W}$ and $\Delta_{CKM}$ simultaneously. The discrepancy in $A_{l}^{SLD}$
and $A_{b}^{FB}$ also moves below 2$\sigma$, thereby making this a
paramatrisation worth considering to look for BSM physics. Inclusion of
$\Delta_{CKM}$ in the fit improves it’s agreement with the measured value,
while slightly deteriorating $A_{l}^{SLD}$ and $A_{b}^{FB}$.

###  5.2 Tree level study: $C_{Hl_{3}}$ and $C_{ll}$

As already seen from the expression of the recent anomalies in terms of SMEFT
WCs, all tree-level $C_{Hl_{3}}$ and $C_{ll}$ are the ones which are common
both in $m_{W}$ and $\Delta_{CKM}$. Incidentally, these WCs are also the ones
which affects the weak coupling constant $G_{F}$. However, this goes beyond
just $\delta G_{F}$ because of the additional dependences on $C_{Hl_{3}}$ in
the shift in couplings for the observables as shown in Table 2. Similar to the
(S,T,V) fit, we present two distinct cases, where first we have shown the fit
to the EWPO provided by LEP-I. We then augment the number of observables by
including $\Delta_{CKM}$ and then present the corresponding pulls of the
observables after each step.

Figure 2: Marginalised 2-D plot for WCs common to $W$-mass and $\Delta_{CKM}$ at tree-level. WC |  B.F(LEP-I) | Correlation |  B.F(LEP-I+CKM) | Correlation
---|---|---|---|---
| ($\chi^{2}_{\rm fit}$/$\chi^{2}_{\rm SM}=24.27/40.08$) | ($\chi^{2}_{\rm fit}$/$\chi^{2}_{\rm SM}=105.36/126.37$)
$C_{Hl_{3}}$ |  -$0.0138\pm 0.0065$ | $1.00$ |  |  -$0.0152\pm 0.0058$ | $1.00$ |
$C_{ll}$ |  -$0.0008\pm 0.0114$ | $0.86$ | $1.00$ |  -$0.0034\pm 0.0084$ | $0.88$ | $1.00$
Table 7: Global fit of the WCs common to $W$-mass and $\Delta_{CKM}$ at tree-
level. Table 8: Posterior results and Pull for WCs common to $M_{W}$ and
$\Delta_{CKM}$, once just with EWPO observables and once including the
$\Delta_{CKM}$ constraint.

As seen from Figure 2, the inclusion of $\Delta_{CKM}$ has a more constraining
effect on the two WCs. Table 7 presents the result of the fit and correlation
among the WCs. The posterior results for the observables are given in Table 8.
For the first case with only the EWPO observables in the fit, the discrepancy
in $M_{W}$ comes down to 2.85$\sigma$. The discrepancy in $A_{b}^{FB}$ however
deteriorates to -2.9$\sigma$. Discrepancy in $\Delta_{CKM}$ however comes down
to a mere 0.14$\sigma$. Inclusion of the $\Delta_{CKM}$ constraint in the fit
does not significant constrain it. Thus one can argue that any New Physics
effect coming through these two parameters (or $\delta G_{F}$) cannot fully
satisfy the EWPO data. This also strengthens the case for the (S,T,V)
paramatrisation, where presence of extra parameters S and T (or equivalently
$C_{HD}$ and $C_{HWB}$) makes helps us to explain all the observables within
2$\sigma$ discrepancy.

###  5.3 Tree level study: $C_{HD}$, $C_{HWB}$, $C_{Hl_{3}}$ and $C_{ll}$

With the result from CDF impacting the world-average of W-mass, the
discrepancy now stands at around 5.3 $\sigma$ from the SM value. Taking it as
a sign of BSM physics, we look at a fit with just the SMEFT operators
contributing to W-mass at tree-level. The shift in W mass in SMEFT can be
written as:

$\frac{\delta m_{W}^{2}}{m_{W}^{2}}=v^{2}\
\frac{s_{W}c_{W}}{s_{W}^{2}-c_{W}^{2}}\left[2\,C_{HWB}+\frac{c_{W}}{2s_{W}}\,C_{HD}+\frac{s_{W}}{c_{W}}\left(2\,C_{Hl}^{(3)}-C_{ll}\right)\right]\,,$
(51)

This can also be thought of as augmenting the previous fit by including two
more parameters $C_{HD}$ and $C_{HWB}$. The results of the fit and the 2-D
marginalised plots are shown in Table 9 and Figure 3. The plot here shows that
inclusion of $\Delta_{CKM}$ constraint has significant impact in the results
of the fit with $C_{HWB}$ and $C_{ll}$ having noticeable shifts in their best
fit points and their 2$\sigma$ reaches. The correlations among the WCs also
show significant changes, with $C_{ll}$-$C_{HWB}$ and $C_{ll}$-$C_{Hl_{3}}$
showing the highest change.

Figure 3: 2-D marginalised global fit of the four WCs affecting $M_{W}$ at tree level. WC |  B.F(EWPO) | Correlation |  B.F(EWPO+CKM) | Correlation
---|---|---|---|---
| ($\chi^{2}_{\rm fit}$/$\chi^{2}_{\rm SM}=11.07/40.08$) | ($\chi^{2}_{\rm fit}$/$\chi^{2}_{\rm SM}=14.96/44.67$)
$C_{HD}$ | $-0.0376\pm 0.0159$ | $1.00$ |  |  |  | $-0.0392\pm 0.0159$ | $1.00$ |  |  |
$C_{HWB}$ | $-0.00007\pm 0.0079$ | $-0.76$ | $1.00$ |  |  | $0.0085\pm 0.0066$ | $-0.87$ | $1.00$ |  |
$C_{Hl_{3}}$ | $-0.0042\pm 0.0072$ | $-0.21$ | $-0.04$ | $1.00$ |  | $-0.0048\pm 0.0072$ | $-0.21$ | $-0.01$ | $1.00$ |
$C_{ll}$ | $-0.0198\pm 0.0146$ | $-0.15$ | $0.51$ | $0.47$ | $1.00$ | $0.0028\pm 0.0091$ | $-0.19$ | $0.16$ | $0.81$ | $1.00$
Table 9: Global fit of the WCs affecting the W-boson mass at tree-level Table
10: Posterior results and Pull for WCs effecting $M_{W}$, once just with EWPO
observables and once including the $\Delta_{CKM}$ constraint.

The posterior values for the observables are given in Table 10. For the fit
with EWPO, the dicrepancy in $M_{W}$, as expected, is drastically reduced.
$A_{l}^{SLD}$ also comes down under 2$\sigma$. However $A_{b}^{FB}$ and
$\Delta_{CKM}$ shows the highest discrepancy with a deviation of -2.09$\sigma$
and -1.97$\sigma$ respectively. Inclusion of the $\Delta_{CKM}$ constraint
along with EWPO reduces it’s dicrepancy significantly to -0.59$\sigma$ at the
expense of worsening $A_{b}^{FB}$ to -2.24$\sigma$. The agreement of
$\Gamma_{Z}$ also suffers with it’s discrepancy increasing from -0.03$\sigma$
to -1.49$\sigma$. This study thus indicates that we need to go beyond these
four WCs in order to satisfy the precesion data.

###  5.4 VLL inspired study

Inspired by various see-saw like scenarios along with possibility to solve
other shortcomings of the SM like muon $(g-2)$, neutrino mass etc., we take a
closer look at various Vectorlike leptons multiplets by identifying the
leading dimension-6 operators which gets affected. The list of leptons along
with their corresponding yukawa coupling $\lambda_{l}$ (where $l$ represents
the corresponding vectorlike lepton) and the masses of the heavy states
$M_{l}$ are shown in Table 11. As seen from the table, the tree-level imprints
of the model parameters on the SMEFT WCs are restricted to $C_{He}$,
$C_{Hl}^{(1)}$ and $C_{Hl}^{(3)}$.

$\begin{array}[]{|cccccc|}\hline\cr\hbox{VLF}&(SU(2)_{L},Y)&Interaction&C_{He}&C_{Hl}^{(1)}&C_{Hl}^{(3)}\cr\hline\cr\hline\cr
N^{a}&(3,0)&N^{a}(H\epsilon\tau^{a}L)&0&+\lambda_{N}^{2}/4M_{N}^{2}&-\lambda_{N}^{2}/4M_{N}^{2}\cr\hline\cr
N^{\prime}&(1,0)&N^{\prime}LH&0&+3\lambda_{N}^{2}/4M_{N}^{2}&+\lambda_{N}^{2}/4M_{N}^{2}\cr\hline\cr
L^{\prime}&(2,-1/2)&EL^{\prime}H^{*}&-\lambda_{L}^{2}/2M_{L}^{2}&0&0\cr\hline\cr
L_{3/2}&(\overline{2},-3/2)&E(L_{3/2}\epsilon
H)&+\lambda_{L}^{2}/2M_{L}^{2}&0&0\cr\hline\cr
E^{\prime}&(1,1)&E^{\prime}LH^{*}&0&-\lambda_{E}^{2}/4M_{E}^{2}&-\lambda_{E}^{2}/4M_{E}^{2}\cr\hline\cr
E^{a}&(3,1)&E^{a}(H^{*}\tau^{a}L)&0&-3\lambda_{E}^{2}/4M_{E}^{2}&+\lambda_{E}^{2}/4M_{E}^{2}\cr\hline\cr\hline\cr\end{array}$
Table 11:  List of new leptons that can couple to the SM lepton doublet
$L=(\nu_{\mu},l_{L})$ or singlet $E=l_{R}$ (with the same gauge quantum
numbers as $L^{\prime}$ and $E^{\prime}$) and to the Higgs doublet
$H=(0,v+h/\sqrt{2})$ (an SU(2) doublet with $Y=1/2$).

Figure 4 and Table 12 represents the result of the fit once for the LEP
observables alone and second for the case including $\Delta_{CKM}$ alongwith
LEP observables. The posterior values of the observables for these two cases
can be read off from Table 13. We can see that even without including the
$\Delta_{CKM}$ constraint, we can satisfy both $M_{W}$ and $\Delta_{CKM}$ with
2$\sigma$. $A_{l}^{SLD}$ improves to 1.56 $\sigma$ while $A_{B}^{FB}$
deteriorates to -2.18$\sigma$. Inclusion of the $\Delta_{CKM}$ constraint
improves the posterior value of $\Delta_{CKM}$ only marginally, while pushing
the $M_{W}$ beyond 2$\sigma$.

Figure 4: 2-D Marginalised plot for the WCs impacting the VLLs WC | B.F(EWPO) | Correlation | B.F(EWPO+CKM) | Correlation
---|---|---|---|---
|  ($\chi^{2}_{\rm fit}$/$\chi^{2}_{\rm SM}=20.53/40.08$) |  ($\chi^{2}_{\rm fit}$/$\chi^{2}_{\rm SM}=21.32/44.67$)
$C_{He}$ | $-0.0148\pm 0.0075$ | $1.00$ |  |  | $-0.0125\pm 0.0073$ | $1.00$ |  |
$C_{Hl_{1}}$ | $-0.0037\pm 0.0043$ | $0.57$ | $1.00$ |  | $-0.0027\pm 0.0045$ | $0.56$ | $1.00$ |
$C_{Hl_{3}}$ | $-0.0184\pm 0.0039$ | $0.55$ | $0.21$ | $1.00$ | $-0.0158\pm 0.0027$ | $0.47$ | $0.17$ | $1.00$
Table 12: Global fit of the WCs affecting VLLs at tree-level to EWPO and
$\Delta_{CKM}$. Table 13: Posterior results and Pull for the VLL inspired
study, once just with EWPO observables and once including the $\Delta_{CKM}$
constraint.

We can put limits on the ratio of the coupling and mass for these scenarios.
For eg. if we only consider only $N^{\prime}$, we get a limit of
${v\lambda_{N}}/{M_{N}}<0.074$, $v$ is the vev. Similarly for $E^{\prime}$, we
get a limit of ${v\lambda_{E}}/{M_{E}}<0.05$. If we look at $L^{\prime}\oplus
E^{\prime}$, the bounds come out to be ${v\lambda_{L}}/{M_{L}}<0.045$ and
${v\lambda_{E}}/{M_{E}}<0.066$.

###  5.5 LEP-I 8 parameter fit

Though at leading order LEP-I observables (EWPO) depend on ten SMEFT
operators, only eight among them can be constrained using LEP-I electroweak
data. Some of the earlier literatures [28, 46] redefined the combinations of
the corresponding WCs by rotating away the two purely bosonic contributions
($C_{HD}$ and $C_{HWB}$) to get eight independent parameters which are finally
constrained. When analyzed in that fashion, we showed that our results are in
well agreement with those in the literature. We also found out that similar
constraints can also be achieved by setting those two purely bosonic WCs just
to zero. Phenomenologically, there can be frameworks where these bosonic
operators are highly suppressed and so this procedure can direct provide
limits for such kind of scenarios.

The results of this analyses is presented in the Table 14. The second column
of the table shows the best fit and deviation for the WCs given in the first
column. Deviations are computed using marginalization procedure discussed in
the section 5. We also tabulated the correlations between the coefficients in
the last column. As shifts in the gauge couplings due to SMEFT depends on
$\delta G_{f}$ which is a combination of $C_{Hl_{3}}$ and $C_{ll}$ these two
are expected to be highly correlated with a value of $0.94$. In a similar
manner, leptonic WCs have higher correlation among each other such as
correlation coefficients for $C_{Hl_{1}}$ and $C_{ll}$ is $0.81$ but negative
and for $C_{Hl_{1}}$ and $C_{Hl_{3}}$ it is $-0.67$. From the shifts in the
gauge coupling shown in Table 2, one can expect very small correlations among
quark WCs and leptonic one $\sim\mathcal{O}(0.01)$. The value of SMEFT
$\chi^{2}$ after the fit comes out to be $3.24$ compared the SM $\chi^{2}$ of
$40.08$, implying a very good quality of the fit. The 2-D marginalized
distributions of the WCs are shown in Figure 5 in purple colour.

Posterior values of the EWPO observables show very good agreement with the
experimental data with all of them within 1 $\sigma$ as shown in Table 16.
Notable improvements among them are $M_{W}$ at -0.2$\sigma$, $A_{b}^{FB}$ at
-0.19$\sigma$ and $A_{l}^{SLD}$ at 0.84$\sigma$. $\Delta_{CKM}$ however
becomes worse as the pull with the EWPO fitted parameters increases its
discrepancy to -2.7$\sigma$.

WC |  B.F(EWPO) | Correlation
---|---|---
| ($\chi^{2}_{\rm fit}$/$\chi^{2}_{\rm SM}=3.24/40.08$)
$C_{Hd}$ |  -0.4646 $\pm$ 0.1715 | 1.00 |  |  |  |  |  |  |
$C_{He}$ |  -0.0099$\pm$ 0.0085 | -0.36 | 1.00 |  |  |  |  |  |
$C_{Hl_{1}}$ |  -0.0031$\pm$ 0.011 | -0.12 | 0.42 | 1.00 |  |  |  |  |
$C_{Hl_{3}}$ |  -0.0398$\pm$0.0159 | -0.05 | 0.08 | -0.67 | 1.00 |  |  |  |
$C_{Hq_{1}}$ |  0.0101$\pm$ 0.0341 | 0.21 | -0.08 | 0.01 | -0.04 | 1.00 |  |  |
$C_{Hq_{3}}$ |  -0.0989$\pm$ 0.0310 | 0.59 | -0.17 | 0.03 | 0.06 | -0.42 | 1.00 |  |
$C_{Hu}$ |  0.1162$\pm$ 0.1195 | -0.14 | 0.10 | 0.07 | -0.03 | 0.44 | -0.76 | 1.00 |
$C_{ll}$ |  -0.0204$\pm$ 0.0284 | -0.06 | -0.10 | -0.81 | 0.94 | -0.04 | -0.03 | -0.01 | 1.00
Table 14: Global fit to EWPO for the case with 8 WCs. WC |  B.F(EWPO+$\Delta_{CKM}$) | Correlation
---|---|---
| ($\chi^{2}_{\rm fit}$/$\chi^{2}_{\rm SM}=10.88/44.67$)
$C_{Hd}$ |  -0.2125$\pm$ 0.1453 | 1.00 |  |  |  |  |  |  |
$C_{He}$ |  -0.0166$\pm$ 0.0083 | -0.24 | 1.00 |  |  |  |  |  |
$C_{Hl_{1}}$ |  -0.0136$\pm$ 0.0107 | 0.08 | 0.35 | 1.00 |  |  |  |  |
$C_{Hl_{3}}$ |  -0.0238$\pm$ 0.0148 | -0.31 | 0.19 | -0.63 | 1.00 |  |  |  |
$C_{Hq_{1}}$ |  -0.0291$\pm$ 0.0311 | 0.55 | -0.23 | -0.16 | 0.12 | 1.00 |  |  |
$C_{Hq_{3}}$ |  -0.0221$\pm$ 0.0139 | 0.30 | 0.18 | 0.78 | -0.64 | -0.13 | 1.00 |  |
$C_{Hu}$ |  -0.1206$\pm$ 0.0832 | 0.40 | -0.15 | -0.27 | 0.37 | 0.23 | -0.37 | 1.00 |
$C_{ll}$ |  0.0076$\pm$ 0.0265 | -0.32 | 0.01 | -0.79 | 0.93 | 0.13 | -0.83 | 0.38 | 1.00
Table 15: Global fit to EWPO and $\Delta_{CKM}$ for the case with 8 WCs. Table
16: Posterior Value and Pulls for the EWPO observables with 8 WCs, once
without $\Delta_{CKM}$ and once with $\Delta_{CKM}$

Inclusion of the CKM anomaly ($\Delta_{CKM}$), along with EWPO, shifts the
best-fit values of the WCs, constraining them better. The results of the fit
can be read off from Table 15 and the 2-D marginalized distributions of the
WCs are given in Figure 5 in green colour. $\Delta_{CKM}$ has the largest
dependence on $C_{Hl_{1}}$, $C_{ll}$, $C_{Hl_{3}}$ and $C_{Hq_{3}}$ therefore
there is significant change in the correlations involving these WCs.
$C_{Hq_{3}}$ and $C_{ll}$ becomes highly correlated with a value of $-0.83$.
As $\Delta_{CKM}$ depends only on left chiral operators, the correlation of
their WCs to that of the right chiral ones decrease. $\Delta_{CKM}$ also
introduce high correlation between the WCs corresponding to leptonic and quark
operators, e.g., the correlation factor for $C_{Hq_{3}}$ with $C_{Hl_{1}}$ and
$C_{Hl_{3}}$ now stands at 0.78$\sigma$ and -0.64$\sigma$ respectively.

On the other hand, with the inclusion of $\Delta_{CKM}$, observable reduces
its discrepancy to a mere -0.32$\sigma$. However, discrepancy in $A_{c}^{FB}$,
$A_{b}^{FB}$ and $A_{l}^{SLD}$ goes beyond 1$\sigma$. The other observables
also show good agreement with the experimental observations.

Figure 5: 2-D marginalised plot for the fit with 8 WCs

###  5.6 LEP-I+ LEP-II 10 parameter fit

As discussed in the previous subsection, EWPO can only constrain eight of 10
WCs with Z pole LEP-I data. This can be overcomed by including LEP-II data. In
second run of LEP, energy reached more than the need for production of pair of
on-shell $W$ bosons. This provides an opportunity to test four-fermionic final
states at the collider. Precise measurements of cross-sections and angular
distributions at various energy enhance the list of precision observables.
Ref. [55, 54, 57], explored this in the context of SMEFT and showed that it
lifts up the flat directions which were there in the case of LEP-I. Charged
current productions of a pair of $W$’s are sensitive to triple gauge coupling
(TGC) and that break the redundency and provide the possibility to constrain
all 10 WCs of EWPO. In addition to the 10 WC, introduction of LEP-II data
brings the anomalous triple gauge operator $\mathcal{O_{W}}$ in the set of
operators contributing at tree-level. As this is not part of EWPO or
$\Delta_{CKM}$ we set $C_{W}=0$ for our analyses.

Including the LEP-II data, increases the $\chi^{2}$ to 121 due to the increase
in number of observables. The results of the fit without including
$\Delta_{CKM}$ are shown in the Table 17. The minimum value of $\chi^{2}$ now
decreases to $83.8$. But, the best fit values of $C_{HD}$, $C_{HWB}$ and
$C_{He}$ are slightly on the larger side. The correlations among the WCs
change drastically due to the presence of the new LEP-II observables.
Correlation between $C_{Hl_{3}}$ and $C_{ll}$ become almost negligible while
many pairs of WCs become highly correlated. To show the strength of
correlation we also plot color density matrix plot in Figure 6(a). Color
coding decreases from blue to green showing decrease in strength of the
correlation. It can be easily observed that many of them are dark blue with
correlation $\sim 0.9$. The correlation of all WCs with Cll is small $\sim
0.3$ and almost no correlation with $C_{Hl_{3}}$ and $C_{Hq_{3}}$. The
correlation of $C_{Hl_{3}}$ and $C_{Hq_{3}}$ is almost 1 whereas both of them
are less correlated with others. The posterior results of the fit for the
precision observables are shown in Table 19. The discrepancy in all of these
observables are within $1\sigma$ with the exception of $\Delta_{CKM}$ where
the pull for is -2.7$\sigma$.

Incorporation of $\Delta_{CKM}$ as an observable in the fit makes it more
constraining as shown in the Table 18. The absolute best-fit values of the WCs
lie well within 1 with $C_{ll}$ becoming very small. There is no significant
change found in $1\sigma$ deviations of WCs and the correlations among the
pairs of WCs also do not show much change. The density plot for correlation
matrix assuming magnitude of elements is also shown in Figure 6(b).

Posteriors results and pull of the EWPO and $\Delta_{CKM}$ are collected in
the fourth column and fifth column respectively of Table 19. All the pulls are
again within $1\sigma$. As expected, pull for $\Delta_{CKM}$ improves to
-0.36$\sigma$. The other observables continue to remain well within 1$\sigma$
making the fit highly competitive with respect to the experimental data.

| Result | Correlation
---|---|---
| ($\chi^{2}_{\rm Fit}$/$\chi^{2}_{\rm SM}=83.82/121.79$)
$C_{Hd}$ |  -0.0841$\pm$ 0.3937 | 1.00 |  |  |  |  |  |  |  |  |
$C_{HD}$ |  -2.1541$\pm$ 2.2945 | -0.90 | 1.00 |  |  |  |  |  |  |  |
$C_{He}$ |  1.0669$\pm$ 1.1474 | 0.90 | -0.99 | 1.00 |  |  |  |  |  |  |
$C_{Hl_{1}}$ |  0.5383$\pm$ 0.5713 | 0.90 | -0.99 | 0.99 | 1.00 |  |  |  |  |  |
$C_{Hl_{3}}$ |  0.01423$\pm$ 0.4149 | 0.58 | -0.60 | 0.60 | 0.61 | 1.00 |  |  |  |  |
$C_{Hq_{1}}$ |  -0.1717$\pm$ 0.1912 | -0.87 | 0.98 | -0.98 | -0.98 | -0.60 | 1.00 |  |  |  |
$C_{Hq_{3}}$ |  -0.0346$\pm$ 0.41 | 0.57 | -0.58 | 0.58 | 0.58 | 0.99 | -0.58 | 1.00 |  |  |
$C_{Hu}$ |  -0.6179$\pm$ 0.7532 | -0.90 | 0.99 | -0.99 | -0.99 | -0.61 | 0.98 | -0.59 | 1.00 |  |
$C_{HWB}$ |  1.0071$\pm$ 0.9923 | 0.88 | -0.98 | 0.98 | 0.98 | 0.46 | -0.97 | 0.43 | -0.97 | 1.00 |
$C_{ll}$ |  -0.0295$\pm$ 0.0273 | 0.27 | -0.29 | 0.29 | 0.27 | 0.005 | -0.30 | -0.028 | -0.30 | 0.32 | 1.00
Table 17: Best fit with $1\sigma$ deviation of WCs after fitting using EWPO observables. Correlation matrix of coefficients are also shown in the third column. | Result | Correlation
---|---|---
| ($\chi^{2}_{\rm Fit}$/$\chi^{2}_{\rm SM}=90.93/126.37$)
$C_{Hd}$ |  -0.1266$\pm$ 0.3934 | 1.00 |  |  |  |  |  |  |  |  |
$C_{HD}$ |  -0.5455$\pm$ 2.2138 | -0.93 | 1.00 |  |  |  |  |  |  |  |
$C_{He}$ |  0.2564$\pm$ 1.1066 | 0.93 | -0.99 | 1.00 |  |  |  |  |  |  |
$C_{Hl_{1}}$ |  0.125$\pm$ 0.5498 | 0.93 | -0.99 | 0.99 | 1.00 |  |  |  |  |  |
$C_{Hl_{3}}$ |  -0.3029$\pm$ 0.3974 | 0.59 | -0.57 | 0.57 | 0.58 | 1.00 |  |  |  |  |
$C_{Hq_{1}}$ |  -0.07295$\pm$ 0.1876 | -0.88 | 0.98 | -0.99 | -0.99 | -0.58 | 1.00 |  |  |  |
$C_{Hq_{3}}$ |  -0.2994$\pm$ 0.403 | 0.57 | -0.55 | 0.55 | 0.56 | 0.99 | -0.56 | 1.00 |  |  |
$C_{Hu}$ |  -0.2947$\pm$ 0.7434 | -0.90 | 0.99 | -0.99 | -0.99 | -0.60 | 0.98 | -0.58 | 1.00 |  |
$C_{HWB}$ |  0.4054$\pm$ 0.9663 | 0.90 | -0.98 | 0.98 | 0.98 | 0.42 | -0.97 | 0.39 | -0.97 | 1.00 |
$C_{ll}$ |  -0.00004$\pm$ 0.0249 | 0.31 | -0.44 | 0.44 | 0.43 | 0.14 | -0.42 | 0.08 | -0.41 | 0.47 | 1.00
Table 18: Best fit with $1\sigma$ deviation of WCs after fitting using EWPO
observables and $\Delta_{CKM}$. Correlation matrix of coefficients are also
shown in the third column. Table 19: Observable’s posterior values for the
all-parameter fit with 10 WCs

(a) (b)

Figure 6: Density plot of correlation matrix: (a) LEP-I+LEP-II (b) LEP-I+LEP-
II+$\Delta_{CKM}$ fitting.

##  6 Summary and conclusions

Electro-weak precision measurements place one of the most stringent
constraints on all New Physics extensions of Standard Model. We explore
dimension-6 operator subsets of the SMEFT in the context of LEP-I $\&$ II data
along with Cabibbo anomaly hinting to need for Beyond Standard Model. In
addition to EWPO, we analyze the status of Cabbibo anomaly within SMEFT using
the most recent data. We compute the contributions to the beta decay and muon
decay using the SMEFT operators that enters in the LEFT at one loop matching.
We classify the precision study of SMEFT into various frameworks.

In the first study, we parameterize the impact of BSM heavy states in oblique
parameters $S$, $T$ and $V$ and compute the posterior fit of the EWPO and
$\Delta_{CKM}$. The results are impressive as pulls for almost all the
observables (except $A_{l}^{SLD}$ and $A_{b}^{FB}$) are within $1\sigma$.
Another thing to note is that even without including $\Delta_{CKM}$ into the
fit the pull for Cabbibo anomaly comes down to $\sim-0.52$. Including the
$\Delta_{CKM}$, pull improves to $\sim-0.27$. In the next study, we choose the
SMEFT operators $C_{Hl_{3}}$ and $C_{ll}$ as the ones which dictates the UV
BSM physics as both of them affect the $m_{W}$, $G_{F}$ and $\Delta_{CKM}$ at
tree level. We analyze this case for LEP-I, LEP-I+$\Delta_{CKM}$. Although
they can explain the Cabibo anomaly very well, they fare very badly with
$M_{W}$ and $A_{b}^{FB}$.

Since $M_{W}$ has the highest pull away from the SM among the observables of
our interest, we next considered the BSM case which consists of the four
operators that affect the $M_{W}$ at tree level ($C_{HD}$, $C_{HWB}$,
$C_{Hl_{3}}$ and $C_{ll}$). The minimum value of $\chi^{2}$ comes down to
$\sim~{}11$ from $\sim~{}40$ and the fit fares very well with $M_{W}$ with
discrepancy well within 1$\sigma$. Bosonic operators $C_{HD}$ and $C_{HWB}$
are highly correlated in this case. Inclusion of $\Delta_{CKM}$ shifts the
best fit towards slightly higher values while simultaneously shrinking the
allowed 2$\sigma$ ranges. Discrepancy in $A_{l}^{SLD}$ and $A_{b}^{FB}$
continues to remain towards the higher side.

We then choose the set of operators motivated from Vectorlike lepton (VLL)
model. Integrating out heavy degrees of freedom, at tree-level, generates the
operators $C_{He},C_{Hl_{1}}$ and $C_{Hl_{3}}$. The allowed range for these
WCs, after constraining using EWPO and $\Delta_{CKM}$, turn out to be
$\mathcal{O}(0.01)$. Inclusion of $\Delta_{CKM}$ shifts allowed ranges to
slightly larger values. We also explore a few minimal VLL frameworks and set
the constraints over ratios of VLF Yukawas and the scale of the model. The
posterior fit of the observables show that pulls for $\Gamma_{Z}$ becomes
worse with a discrepancy over $1.5\sigma$. New posterior value of $M_{W}$ is
improved comparison to SM, but pulls remain at $\sim 1.9\sigma$ and $\sim
2.4\sigma$ for the fits without and with $\Delta_{CKM}$ respectively. Thus we
can conclude that minimal VLL frameworks are severely constrained by the
precision measurements and slight tension exists with recent measurement of
the $M_{W}$ mass and $A_{b}^{FB}$.

Considering all the dimension-6 SMEFT operators, that appear in the EWPO at
the leading order, it can be seen that only eight of ten WCs, which appear at
tree-level, can be constrained by the LEP-I observables. We compute their best
fit, first with only EWPO and then including $\Delta_{CKM}$. In this case,
best fit $\chi^{2}$ relaxes to $\sim 3$ from a SM $\chi^{2}$ value of $40$
from EWPO. On the other hand, inclusion of $\Delta_{CKM}$ relaxes it to $\sim
10.8$ from a SM $\chi^{2}$ of $44$. We also analyzed 2D correlations of the
pairs of WCs with the highest correlation being of $C_{Hl_{3}}$ and $C_{ll}$.
We find that SMEFT at the leading order brings pull for all the Electroweak
observables within $1\sigma$. Recent $M_{W}$ measurement also fares
exceptionally well with the pull coming down to $\sim-0.03\sigma$. However,
the EWPO fit worsens the $\Delta_{CKM}$ compared to the SM as the discrepancy
increases to $-2.7\sigma$. When $\Delta_{CKM}$ constraint is also taken into
the account in the fit, discrepancies in $A_{l}^{SLD}$, $A_{b}^{FB}$ and
$A_{c}^{FB}$ go beyond 1$\sigma$ while improving the agreement with
$\Delta_{CKM}$ at $-0.3\sigma$. Overall, we can conclude that even after
including the $\Delta_{CKM}$ into the fit, the pulls for all the observables
are within $2\sigma$ and with most them within $1\sigma$.

Interestingly, the LEP-II data lifts the two blind directions which are
present for LEP-I. These new observables correspond to the pair production of
$W$’s, which subsequently leads to four fermion final states, resulting in a
number of angular observables at various energies. For the case without
including the $\Delta_{CKM}$ constraint, limits on the purely bosonic WCs
$C_{HD}$ and $C_{HWB}$ are rather weak and their best-fit comes out to be
slightly on the higher side. The correlations among the WCs are also generally
on the higher side. The pulls, however, on the posterior values of the
precesion observables are very small showing excellent agreement with the
experimental data. Like the previous 8 parameter fit, this also worsens the
$\Delta_{CKM}$ by increasing the pull to -2.7$\sigma$. The inclusion of
$\Delta_{CKM}$ constraint in the fit constraints the WCs further, including
that of $C_{HD}$ and $C_{HWB}$ along with simultaneously making their best-fit
values smaller. Another peculiar aspect of this fit is that it makes the best-
fit value of $C_{ll}$ very small. Correlations among the WCs still continues
to be on the higher side and the agreement of the posterior values of the
precesion observables including $\Delta_{CKM}$ with the experimental data
still remains excellent.

Our subsequent aim is to automatise this code with the current observables and
make it available for public use very soon. Finally, in order to realise a
true global fit, we would also be including the LHC observables in future.

## Acknowledgments

The authors would like to thank Dr.Nilanjana Kumar for valuable discussions
during the initial days of the work. M.T.A. acknowledges the financial support
of Department of Science and Technology, Government of India, India (DST)
through INSPIRE Faculty grant DST/INSPIRE/04/2019/002507. K.D. acknowledges
Council for Scientific and Industrial Research (CSIR), India for JRF/SRF
fellowship with the award letter no. 09/045(1654)/2019-EMR-I. T.S. would like
to acknowledge the support from the Dr. D.S. Kothari Postdoctoral fellowship
scheme no. F.4-2/2006 (BSR)/PH/20-21/0163.

## Appendix

The general correction to $\hat{\sigma}_{h}^{0}$ near the $Z$ pole
($s-M_{Z}^{2}\equiv\Delta$) in the SMEFT is

$\displaystyle\frac{\delta\sigma_{h}^{0}}{\sigma_{h}^{0}}$
$\displaystyle\simeq$
$\displaystyle\frac{\delta\Gamma_{Z\rightarrow\ell\overline{\ell}}}{\Gamma_{Z\rightarrow\ell\overline{\ell}}}+\frac{\delta\Gamma_{Z\rightarrow
Had}}{\Gamma_{Z\rightarrow
Had}}-\frac{\delta\omega(M_{Z}^{2})}{\omega(M_{Z}^{2})}-\frac{\delta\omega^{\star}(M_{Z}^{2})}{\omega^{\star}(M_{Z}^{2})}.$
(52)

where

$\displaystyle\overline{w}(s)$ $\displaystyle=$ $\displaystyle
s\frac{\overline{\Gamma}_{Z}}{\overline{M}_{Z}}\text{ we get : }\delta
w(s)=s\left(\frac{(\Gamma_{Z})_{SM}}{\hat{m}_{Z}}\right)\left(\frac{\delta\Gamma_{Z}}{(\Gamma_{Z})_{SM}}\right).$
(53) $\displaystyle\overline{w}(s)$ $\displaystyle=$
$\displaystyle\overline{\Gamma}_{Z}\overline{M}_{Z}\text{ we get : }\delta
w(s)=(\Gamma_{Z})_{SM}\hat{m}_{Z}\left(\frac{\delta\Gamma_{Z}}{(\Gamma_{Z})_{SM}}\right).$
(54)

but we note the following simplified expressions. In the SMEFT
$\overline{A}_{f}$ can be written as

$\displaystyle\overline{A}_{f}=\frac{2\overline{r}_{f}}{1+\overline{r}_{f}^{2}},$
(55)

where $\overline{r}_{f}=\frac{\overline{g}^{f}_{V}}{\overline{g}^{f}_{A}}$.
The redefinition of the $Z$ coupling then leads to a shift of
$\overline{A}_{f}$ such that
$\overline{A}_{f}=(A_{f})_{SM}\left(1+\frac{\delta
A_{f}}{(A_{f})_{SM}}\right)$ where

$\displaystyle\frac{\delta A_{f}}{(A_{f})_{SM}}=\delta
r_{f}\left(1-\frac{2(r_{f}^{2})_{SM}}{1+(r_{f}^{2})_{SM}}\right).$ (56)

Here $\delta r_{f}$ is defined by $r_{f}=(r_{f})_{SM}\left(1+\delta
r_{f}\right)$ with $\delta r_{f}=\delta g^{f}_{V}/G^{f}_{V}-\delta
g^{f}_{A}/G^{f}_{A}$. We again use : $(...)_{SM}$ for leading order SM
predictions and $G^{f}_{A,V}$ for leading order SM predictions for the
couplings. Then the corrections to $A_{FB}^{0,f}$ from the shifts in the
effective couplings are

$\displaystyle\delta A_{FB}^{0,f}=\frac{3}{4}\left[\delta
A_{\ell}\,(A_{f})_{SM}+(A_{\ell})_{SM}\,\delta A_{f}\right].$ (57)

The contribution to the total width of $W$ is
$\overline{\Gamma}_{W}=\Gamma_{W}^{SM}+\delta\Gamma_{W}$ where

$\displaystyle\Gamma^{SM}_{W}$
$\displaystyle=\frac{3\sqrt{2}\hat{G}_{F}\hat{m}_{W}^{3}}{4\pi},$
$\displaystyle\delta\Gamma_{W}$
$\displaystyle=\Gamma^{SM}_{W}\left(\frac{4}{3}\delta
g^{W_{\pm},\ell}+\frac{8}{3}\delta g^{W_{\pm},q}+\frac{\delta
m_{W}^{2}}{2\hat{m}_{W}^{2}}\right).$ (58)

Where $\hat{m}_{W}$ is the SM value.

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