# Anomalous gauge couplings vis-$\grave{a}$-vis $(g-2)_{\mu}$ and flavor
observables

Debajyoti Choudhury<EMAIL_ADDRESS>Kuldeep Deka
<EMAIL_ADDRESS>Suvam Maharana<EMAIL_ADDRESS>Lalit
Kumar Saini<EMAIL_ADDRESS>Department of Physics & Astrophysics,
University of Delhi

###### Abstract

We reassess non-standard triple gauge couplings in the light of the recent
$(g-2)_{\mu}$ measurement at FNAL, the new lattice theory result of
$(g-2)_{\mu}$ and the updated measurements of several $B$-decay modes. In the
framework of SMEFT, three bosonic dimension-6 operators are invoked to
parametrize physics beyond the Standard Model and their contributions to such
low-energy observables computed. Constraints on the corresponding Wilson
coefficients are then derived from fits to the current experimental bounds on
the observables and compared with the most stringent ones available from the
13 TeV LHC data in the $W^{+}W^{-}$ and $W^{\pm}Z$ production channels.

Muon magnetic moment, ATLAS

††preprint: APS/123-QED

##  I  Introduction

Despite its remarkable compatibility with most experimental observations in
elementary particle physics, the Standard Model (SM) is plagued by a number of
shortcomings warranting explorations in the vistas beyond. While some of the
experimental “discrepancies” such as the observation of neutrino masses and
mixings are relatively easy to address, others such as anomalies in
$B$-decays, the seeming deviations in the anomalous magnetic moments of the
muon and the electron or that in the forward-backward asymmetry in
$e^{+}e^{-}\to b\bar{b}$ at the $Z$-peak have not only been long-standing, but
also beg for more complicated solutions.

Solutions to individual issues are, of course, relatively easy to construct,
but a coherent explanation is much more difficult to achieve, some recent
examples being afforded by efforts that address $(g-2)_{\mu}$ in conjunction
with discrepancies such as those in low-energy flavour anomalies Du  _et al._
(2021); Ban  _et al._ (2021); Darmé  _et al._ (2021); Bhattacharya  _et al._
(2021), neutrino masses Nomura and Okada (2021); Zhang (2021); Borah  _et al._
(2022), dark matter Arcadi  _et al._ (2021); Bai and Berger (2021); Choudhury
_et al._ (2021a); Borah  _et al._ (2021) and others Abdughani  _et al._
(2021); Babu  _et al._ (2021); Keung  _et al._ (2021); Amaral  _et al._
(2021). The difficulty of the exercise can be gauged by the fact that
attempted explanations of even a single deviation are constrained by the need
to be in accordance with other observables so as to admit only a relatively
small parameter space Choudhury  _et al._ (2018); Altın  _et al._ (2018);
Choudhury and Sachdeva (2019); Altmannshofer  _et al._ (2020); Choudhury  _et
al._ (2021b); Cao  _et al._ (2021); Ke and Slavich (2021); Kim  _et al._
(2022).

The absence of any new resonances at the LHC strongly suggests that any new
physics (NP) explanation of the extant discrepancies would require the
operative scale $\Lambda$ to be at least a few TeVs or larger. Thankfully,
even such a little hierarchy between $\Lambda$ and the electroweak scale
validates the use of an effective field theory (EFT) to address these
discrepancies Falkowski  _et al._ (2017a); Choudhury  _et al._ (2020);
Buttazzo and Paradisi (2020); Aebischer  _et al._ (2021). While a given
ultraviolet-complete theory would yield, on the heavy fields being integrated
out, a very specific structure for the ensuing EFT (in other words, specific
relations between the Wilson coefficients), in the absence of such a
theoretically motivated completion, the coefficients are completely
independent. It has to be realized, though, that even apparently independent
gauge-invariant operators may be related through equations of motions and the
literature has seen several different choices for the truly independent
operators, viz. the Warsaw basis Buchmüller and Wyler (1986); Grzadkowski  _et
al._ (2010), the HISZ basis Hagiwara  _et al._ (1993), the SILH basis Giudice
_et al._ (2007) etc. The lack of discernible deviations from the SM
expectations have led to constraints on such SMEFT operators, whether of the
4-fermion form Falkowski  _et al._ (2017b) or otherwise Faham  _et al._
(2021); Aebischer  _et al._ (2021); Bellan  _et al._ (2021); Yin (2021);
Ethier  _et al._ (2021); Aad  _et al._ (2020). In this work, we carry on in
this spirit, sticking to the HISZ basis.

Of much recent interest has been the Fermi National Accelerator Laboratory
(FNAL) measurement Abi  _et al._ (2021) of the anomalous magnetic moment of
the muon, namely $a_{\mu}\equiv[(g-2)/2]_{\mu}$. The result is consistent with
the previous one from the Brookhaven National Laboratory (BNL) measurement,
and it is customary to combine the two so as to reduce the experimental
errors. As is well-known, $a_{\mu}$ receives many corrections within the SM,
of which a particularly difficult calculation is that for the contribution
from the hadronic vacuum polarization. The traditional method has been to use
dispersion relations alongwith experimental results and if this is adopted,
the experimental result would imply a discrepancy $\Delta a_{\mu}^{\mbox{\tiny
DISP}}=251(59)\times 10^{-11}$, a $4.2\sigma$ deviation from the the SM Aoyama
_et al._ (2020); Abi  _et al._ (2021); Davier  _et al._ (2017);
*Keshavarzi:2018mgv; *Colangelo:2018mtw; *Hoferichter:2019mqg;
*Davier:2019can; *Keshavarzi:2019abf; *Keshavarzi:2019abf; *Kurz:2014wya;
*Chakraborty:2017tqp; *Borsanyi:2017zdw; *Blum:2018mom; *Giusti:2019xct;
*Shintani:2019wai; *FermilabLattice:2019ugu; *Gerardin:2019rua;
*Aubin:2019usy; *Giusti:2019hkz; *Melnikov:2003xd; *Masjuan:2017tvw;
*Colangelo:2017fiz; *Hoferichter:2018kwz; *Gerardin:2019vio; *Bijnens:2019ghy;
*Colangelo:2019uex; *Pauk:2014rta; *Danilkin:2016hnh; *Jegerlehner:2017gek;
*Knecht:2018sci; *Eichmann:2019bqf; *Roig:2019reh; *Colangelo:2014qya;
*Blum:2019ugy; *Melnikov:2003xd; *Masjuan:2017tvw; *Colangelo:2017fiz;
*Hoferichter:2018kwz; *Gerardin:2019vio; *Bijnens:2019ghy; *Colangelo:2019uex;
*Pauk:2014rta; *Danilkin:2016hnh; *Jegerlehner:2017gek; *Knecht:2018sci;
*Eichmann:2019bqf; *Roig:2019reh; *Blum:2019ugy; *Aoyama:2012wk;
*Aoyama:2019ryr; *Czarnecki:2002nt; *Gnendiger:2013pva; *Davier:2017zfy;
*Keshavarzi:2018mgv; *Colangelo:2018mtw; *Hoferichter:2019mqg;
*Davier:2019can; *Keshavarzi:2019abf; *Kurz:2014wya; *Melnikov:2003xd;
*Masjuan:2017tvw; *Colangelo:2017fiz; *Hoferichter:2018kwz; *Gerardin:2019vio;
*Bijnens:2019ghy; *Colangelo:2019uex; *Pauk:2014rta; *Danilkin:2016hnh;
*Jegerlehner:2017gek; *Knecht:2018sci; *Eichmann:2019bqf; *Roig:2019reh;
*Blum:2019ugy; *Colangelo:2014qya; *Aoyama:2012wk; *Aoyama:2019ryr;
*Czarnecki:2002nt; *Gnendiger:2013pva; *Davier:2017zfy; *Keshavarzi:2018mgv;
*Colangelo:2018mtw; *Hoferichter:2019mqg; *Davier:2019can;
*Keshavarzi:2019abf; *Kurz:2014wya; *Melnikov:2003xd; *Masjuan:2017tvw;
*Colangelo:2017fiz; *Hoferichter:2018kwz; *Gerardin:2019vio; *Bijnens:2019ghy;
*Colangelo:2019uex; *Blum:2019ugy; *Colangelo:2014qya. On the other hand, the
adoption of the Lattice QCD results from the Budapest-Marseille-Wuppertal
(BMW) collaboration Borsanyi  _et al._ (2021) for the same significantly
reduces the deviation down to $\Delta a_{\mu}^{\mbox{\tiny BMW}}=107(69)\times
10^{-11}$ or an agreement at the $2\sigma$ level111 It should be recognized,
though, that while such a positive change in the hadronic vacuum polarization
(HVP) minimizes the apparent $(g-2)_{\mu}$ discrepancy, it could possibly
engender conflicts with the global EW fit prediction of the hadronic
contributions to the QED coupling $\Delta\alpha^{(5)}_{\mbox{had}}$ or
elsewhere in low-energy hadron phenomenology Crivellin  _et al._ (2020);
Keshavarzi  _et al._ (2020b); Colangelo  _et al._ (2021). (see ref Keshavarzi
_et al._ (2021); Gérardin (2021); Keshavarzi (2019) for reviews).

Also of interest are $B$ physics observables, where neutral current
$b\rightarrow sll$ transitions have been showing persistent discrepancy from
the SM values in recent years with the most recent result being of the $R_{K}$
anomaly and $BR(B_{s}\rightarrow\mu^{+}\mu^{-})$. In Moriond 2021, the LHCb
collaboration has reported the measured value of $R_{K}$ Aaij  _et al._
(2021a) to be $0.846^{+0.044}_{-0.041}$ in the 1.0 GeV $\leq$ $q^{2}$ $\leq$
6.0 GeV2 bin. With the central value remaining virtually unchanged from the
earlier result Aaij  _et al._ (2019), and the errors shrinking by almost 30%,
this has strengthened the deviation from the erstwhile $2.5\sigma$ level to
$3.1\sigma$. Most interestingly, the value of
$BR(B_{s}\rightarrow\mu^{+}\mu^{-})$ as reported by LHCb is
$3.09^{+0.483}_{-0.443}\times 10^{-9}$ Aaij  _et al._ (2021b) and compatible
with the SM prediction within $1\sigma$ whereas the previous world average
(ATLAS, CMS and LHCb) of $2.69^{+0.37}_{-0.35}\times 10^{-9}$ was below the SM
prediction by $2\sigma$.

In the present paper, we reexamine possible anomalous self-interactions of the
electroweak gauge bosons in the light of these experiments and other such.
Concentrating on three particular dimension-6 terms in the SMEFT Lagrangian
that lead to anomalous triple gauge boson couplings (TGCs), we evaluate the
corresponding one-loop contributions to both $(g-2)_{\mu}$ and $(g-2)_{e}$,
another observable that shows a discrepancy, albeit smaller as well as certain
electroweak precision measurements. While similar exercises have been
undertaken in the past Burgess  _et al._ (1994); Arbuzov and Zaitsev (2021),
our study differs in our inclusion of not only the direct constraints from CMS
and ATLAS Sirunyan  _et al._ (2019, 2020); Aaboud  _et al._ (2019), but also
the recent results from the LHCb experiment. Assuming that the aforementioned
operators are the leading ones, we show that radiative and rare $B$ and $K$
decays such as $B\to X_{s}\gamma$, $B_{s}\to\mu^{+}\mu^{-}$, $B\to
X_{s}\ell^{+}\ell^{-}$, $B\to K^{(\ast)}\mu^{+}\mu^{-}$,
$B_{s}\to\phi\mu^{+}\mu^{-}$, $K\to\pi\nu\bar{\nu}$ provide very important
constraints. Even though a comprehensive study with the preceding assumption
seems quite restrictive at the outset and would, presumably, have very little
to say about the mentioned anomalies in terms of a unified explanation, we
find that the ensuing results have interesting implications nonetheless,
especially with regards to constraining certain classes of new physics models
that could potentially serve to explain such discrepancies more efficiently.

This paper is organised as follows. In section II we will introduce the
effective Lagrangian for the Electro-Weak (EW) Gauge Bosons and relate the
couplings to the Wilson coefficients of the relevant dimension-6 SMEFT
operators. In Section III we discuss the anomalous contribution to
$(g-2)_{\mu}$, $b\rightarrow s\gamma$, $b\rightarrow s\mu^{+}\mu^{-}$ and
$Z\rightarrow b\bar{b}$ using the effective Lagrangian. In Section IV we will
present our results using the current experimental observations and also
discuss future projections in context of $(g-2)$ experiments. Finally we
conclude in Section V.

##  II  Effective Lagrangian for the Gauge Sector

With $\Lambda$ as the characteristic scale of the UV-complete theory, the EFT
operative between the electroweak scale and $\Lambda$ would be described by a
Lagrangian as an expansion in $\Lambda^{-1}$ with each term being invariant
under the full SM gauge group. As the only possible dimension-5 terms do not
respect global lepton number, we are eschewing this, and to the leading order,
the effective Lagrangian can be expressed as

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{SM}+\sum_{i}\,\frac{c_{i}}{\Lambda^{2}}\,\mathcal{O}_{i}$
(1)

where $\mathcal{L}_{SM}$ is the SM lagrangian and $\mathcal{O}_{i}$ is the set
of gauge-invariant dimension-6 SMEFT operators with corresponding Wilson
coefficients $c_{i}$. We make a further assumption that the superheavy fields
couple primarily to the bosonic sector and thus the leading corrections are
those that involve the latter rather than the SM fermions222This, of course,
is a strong assumption and we would return to this point later.. The
assumption of larger Wilson coefficients for the bosonic operators than those
for the 4-fermion operators (despite both sets nominally being of mass-
dimension six) may seem an ad hoc prescription. However, there exist a
plethora of scenarios wherein this could (and is indeed likely to) emerge
naturally. Perhaps the most famous of these are Randall-Sundrum-like scenarios
with bulk fermions and bosons. The localizations of the light fermions as
dictated by the warping, whether a single one Chang  _et al._ (2000); Csaki
_et al._ (2002); Agashe  _et al._ (2003) or a multiple and nested one Arun and
Choudhury (2015, 2016), ensures that the overlap integrals for the KK-gauge
bosons with the SM fermions are much smaller than those with the SM bosons.
This, immediately, leads to an hierarchy in the Wilson coefficients as
examined in this analysis333Such a hierarchy would obviously be manifested
when the bosonic operators can contribute to an observable at the Born level
itself. On the other hand, even when they can contribute only at the loop-
level, the hierarchy—itself depending on the features of the theory such as
the extent of warping or the profiles of the low energy fields—can be such
that these still overcome the pure-fermionic or mixed fermion-boson operators.
We shall implicitly assume this to be so.. Indeed, any model that would
include additional particles that couple preferentially to the electroweak
bosons rather than to the SM fermions could, in principle, result in larger
$c_{i}$ corresponding to bosonic operators rather than four-fermion operators
involving the SM fields alone. Examples are provided by numerous scenarios
that contain higher gauge-multiplet fermions such as those in a wide class of
scenarios seeking to explain, amongst others, neutrino masses and mixings Babu
_et al._ (2009); *Bonnet:2009ej; *Li:2009mw; *Picek:2009is; *Liao:2010ku;
*Delgado:2011iz; *Kumericki:2011hf; *Bonnet:2012kz; *Kumericki:2012bh;
*Ma:2013tda; *McDonald:2013hsa; *Ma:2014zda; *Yu:2015pwa; *Ko:2015uma;
*Cepedello:2017lyo; *Anamiati:2018cuq; *KumarAgarwalla:2018nrn;
*Agarwalla:2018xpc; *Arbelaez:2019cmj; *Kumar:2019tat; *Ashanujjaman:2020tuv.
In this sense, the analysis presented here serves to assess the viability of
such explicit models (many of which, potentially, lead to interesting, albeit
complicated signals at the LHC Kumar and Sahdev (2021); Ashanujjaman  _et al._
(2022)) when juxtaposed with the most recent limits on the low-energy
observables, to be described later.

The purely bosonic interactions are expressible in terms of 11
operatorsHagiwara  _et al._ (1993), of which two
($\mathcal{O}_{\Phi,2},\mathcal{O}_{\Phi,3}$) contribute, at the tree-level,
solely to the Higgs self-interaction. Containing terms proportional to the
gauge boson kinetic terms, $\mathcal{O}_{BB}$ and $\mathcal{O}_{WW}$ not only
lead to a finite renormalization of the $W$ and $B$ fields respectively, but
also contribute to the $H\rightarrow\gamma\gamma/Z\gamma$ decays, and, hence,
are constrained by these. Similarly, $\mathcal{O}_{\Phi,1}$ contributes to the
$Z$ boson mass but not to the $W$ mass and hence leads to deviations of the
$\rho$ parameter from 1. Finally, $\mathcal{O}_{DW}$ and $\mathcal{O}_{DB}$
lead to an anomalous running of the QED fine structure constant and of the
weak mixing angle.

Concentrating, for the sake of simplicity, on only those operators that would
leave the largest imprint on the observables of interest, we would examine
Hagiwara  _et al._ (1993)

$\begin{split}\mathcal{O}_{WWW}\ &=\
\mbox{Tr}\left[\hat{W}_{\mu}^{\nu}\hat{W}_{\nu}^{\rho}\hat{W}^{\mu}_{\rho}\right]\\\
\mathcal{O}_{W}\ &=\
\left(D_{\mu}\Phi\right)^{\dagger}\hat{W}^{\mu\nu}\left(D_{\nu}\Phi\right)\\\
\mathcal{O}_{B}\ &=\
\left(D_{\mu}\Phi\right)^{\dagger}\hat{B}^{\mu\nu}\left(D_{\nu}\Phi\right),\end{split}$
(2)

assuming the others to be absent (or, at the least, severely constrained as
discussed above). Here,
$D_{\mu}\Phi=\left(\partial_{\mu}+ig\frac{\sigma^{a}}{2}W_{\mu}^{a}+i\frac{g^{\prime}}{2}B_{\mu}\right)\Phi$,
$\hat{W}_{\mu\nu}=ig\frac{\sigma^{a}}{2}W_{\mu\nu}^{a}$ and
$\hat{B}_{\mu\nu}=i\frac{g^{\prime}}{2}B_{\mu\nu}$. Note that, while analogous
$CP$-odd operators (obtained by replacing a field-strength tensor by its dual)
would exist as well, these are irrelevant for our analysis as we deal with
only $CP$-even observables444 The leading contributions of the $CP$-odd
dimension-6 operators to the $CP$-even observables would appear only at second
order in the corresponding Wilson coefficients. In the ensuing analysis,
however, we limit contributions only upto a linear order in the WCs. Assuming
higher order terms would further require the inclusion of operators of similar
mass-dimensions (dimension-8 and higher) in the theory which is beyond the
scope our current study..

The operators in eqn.(2) give rise to, amongst other terms, anomalous triple
gauge boson couplings (TGCs). The latter are also often parametrized in terms
of a convenient phenomenological Lagrangian Hagiwara  _et al._ (1993);
Gounaris  _et al._ (1996) namely

$\begin{split}\mathcal{L}^{WWV}_{eff}=g_{WWV}\Big{\\{}&g_{1}^{V}\left(\tilde{W}^{-}_{\mu\nu}\tilde{W}^{+\nu}-\tilde{W}^{+}_{\mu\nu}\tilde{W}^{-\nu}\right)V^{\mu}\\\
&+\kappa_{V}\tilde{W}^{+}_{\mu}\tilde{W}^{-}_{\nu}\tilde{V}^{\mu\nu}\\\
&+\frac{\lambda_{V}}{m_{W}^{2}}\tilde{W}_{\mu}^{+\nu}\tilde{W}_{\nu}^{-\rho}{\tilde{V}_{\rho}}^{\,\,\mu}\Big{\\}},\end{split}$
(3)

with $V\equiv\gamma,Z$. Here, $g_{WW\gamma}=e$, $g_{WWZ}=e\cot\theta$ (with
$\theta$ being the Weinberg angle) and the field strengths correspond to only
the abelian part,
$\tilde{W}_{\mu\nu}=\partial_{\mu}W_{\nu}-\partial_{\nu}W_{\mu}$ and
$\tilde{V}_{\mu\nu}=\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}$. Within the
SM, we have $g_{1}^{V}=\kappa_{V}=1$, $\Delta g_{1}^{\gamma}$ = 0 and
$\lambda_{V}=0$. In other words, $\Delta\kappa_{V}\equiv\kappa_{V}-1$, $\Delta
g_{1}^{Z}\equiv g_{1}^{Z}-1$ and $\lambda_{V}$ suitably define the anomalous
couplings, and, post symmetry-breaking, can be related to the Wilson
coefficients $c_{W}$, $c_{B}$ and $c_{WWW}$ as follows:

$\begin{split}\Delta g_{1}^{Z}&=c_{W}\frac{m_{Z}^{2}}{2\Lambda^{2}}\\\
\Delta\kappa_{Z}&=\left[c_{W}-s^{2}_{\theta}\left(c_{W}+c_{B}\right)\right]\frac{m_{Z}^{2}}{2\Lambda^{2}}\\\
\Delta\kappa_{\gamma}&=\left(c_{W}+c_{B}\right)\frac{m_{W}^{2}}{2\Lambda^{2}}\\\
\lambda_{\gamma}&=\lambda_{Z}=\frac{3m_{W}^{2}g^{2}}{2\Lambda^{2}}c_{WWW}.\end{split}$
(4)

In the above and in the following sections we use the notation
$s_{\theta}=\sin{\theta}$ and $c_{\theta}=\cos{\theta}$.

Although the operators $\mathcal{O}_{DW}$ and $\mathcal{O}_{BW}$ too
contribute to the anomalous triple gauge couplings, their primary effect on
low-energy physics accrue through the modification of the gauge boson
propagators and, hence, these are not as visible in processes such as
$W^{+}W^{-}$ production in LEPDe Rújula  _et al._ (1992) and LHC experiments.

##  III Contributions to various observables

At this stage, the choice of the gauge is an important one. While full
$SU(2)\otimes U(1)$ gauge invariance is manifest in the formulation of
eqn.(2), it is not so for the case where eqn.(3) is supposed to encapsulate
the entire non-SM part of the effective Lagrangian. In view of this, and for
the sake of convenience, we adopt the unitary gauge for all our calculations.

###  III.1 $a_{\mu}$

Starting with the effective Lagrangian as described in the preceding section,
we may now compute the contribution to the magnetic moment of the muon. At the
one-loop order, the only contributing diagram is as shown in Fig. 1.

Figure 1:  Diagram contributing to $\Delta a_{\mu}^{\rm anom}$ at 1-loop
involving an aTGC vertex.

To this order, then, the only relevant part of the higher-dimensional terms is
that subsumed in eqn.(3), and more specifically

$\mathcal{L}_{\Delta
a_{\mu}}\supset\frac{g}{\sqrt{2}}\left(\bar{\mu}\gamma^{\mu}P_{L}\nu_{\mu}\right)W^{-}_{\mu}+\mbox{h.c.}+\mathcal{L}^{WW\gamma}_{\rm
eff},$ (5)

with $P_{L}$ being the left-handed projection operator. Since we are dealing
with a nonrenormalizable Lagrangian, we expect the integral to be a divergent
one (a simple power counting ensuring that it is only a logarithmic
divergence), necessitating the introduction of a cutoff. A natural choice for
the latter is $\Lambda$ itself555While this identification might seem an ad
hoc one, note that the cutoff has to be $\raisebox{-1.72218pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.72218pt}{$<$}\Lambda$, and given the
logarithmic nature of the dependence, a small variation would be numerically
insignificant., leading to

$\begin{split}\Delta
a_{\mu}^{anom.}=\frac{e^{2}}{48\pi^{2}s^{2}_{\theta}}&\frac{m_{\mu}^{2}}{m_{W}^{2}}\Bigg{\\{}\Delta\kappa_{\gamma}\left(\frac{1}{3}+\ln\left[\frac{\Lambda^{2}}{m_{W}^{2}}\right]\right)\\\
&+\lambda_{\gamma}\left(\frac{7}{6}-\ln\left[\frac{\Lambda^{2}}{m_{W}^{2}}\right]\right)\Bigg{\\}}.\end{split}$
(6)

where we have retained only terms at the leading order in
$m_{\mu}^{2}/m_{W}^{2}$ and neglected $m_{\nu}$ altogether. This expression
can, of course, be trivially translated in terms of $c_{WWW}$ and the
combination $(c_{W}+c_{B})$ using eqns.(4). While Fig.1 represents the only
one-loop contribution to $\Delta a_{\mu}$ emanating from the Lagrangian of
eqn.(3), when the SMEFT language of eqn.(2) is used instead, additional
contributions arise from diagrams with a Higgs in the loop. Contrary to naive
expectations, these are not small compared to that in eqn.(6). The analytic
expressions thereof are more unwieldy though, and, for the sake of breivity we
omit presesnting those. However, we do include such contributions in our
numerical analysis. Note that the four-point (or higher) vertices that eqn.(2)
engenders—and not included in eqn.(3)—contribute to $a_{\mu}$ only at the two-
loop or higher orders.

In the preceding and subsequent loop calculations we adopt dimensional
regularization alongwith the $\overline{\mbox{MS}}$ renormalization scheme.
While it is often argued that the simple pole at $d=4$ ($d$ being the number
of dimensions) could be straightforwardly exchanged for a logarithmic
dependence on the cutoff $\Lambda$, the said dependence is better understood
as a consequence of the renormalization group (RG) evolution of the effective
operators from the scale $\Lambda$ down to the EW scale Burgess and London
(1992). Note that we ignore the (subdominant) contributions of terms with
higher powers of the logarithm and the EW couplings which appear in
perturbative solutions of the RGEs.

It might seem that the contribution $\mathcal{O}_{WWW}$ is in disagreement
with the results of ref.Alonso  _et al._ (2014), which were expressed in terms
of the dipole operator. Of particular significance is the presence of the
$\ln(\Lambda^{2}/m_{W}^{2})$ dependence in our result vis á vis the apparent
absence of $c_{WWW}$ in the anomalous dimension (as computed in the Warsaw
basis) of $c_{e\gamma}$ (or, equivalently, $c_{eB}$ and $c_{eW}$). However, a
comparison between operators in different bases needs to be made with care,
especially since the dipole operators (sans the Higgs field) are admissible
only post electroweak symmetry breaking. A careful matching reintroduces this
effect, and, indeed our result for $\Delta a_{\mu}$ is, consistent with that
given in ref.Aebischer  _et al._ (2021) obtained from the one-loop matching
between LEFT and SMEFT operators at the electroweak scale.

While we have argued for the Wilson coefficient $c_{e\gamma}$ being, at best,
tiny at the scale $\Lambda$, it is worthwhile to examine whether it could be
amplified at the EW scale as a result of mixing, under RG evolution, with the
other operators in the theory. Note that the operators we use are a part of
the HISZ set which, in consisting of only eleven purely bosonic operators,
does not constitute a closed set under RG evolution. However, confined to this
subset, these modulo a difference in the normalizations, can be exactly mapped
onto operators in the SILH basis. Inspired by the latter, if we were to
augment the set by the inclusion of the dipole operator ($c_{e\gamma}$), its
evolution would depend mainly on $\mathcal{O}_{BW}$ ($\equiv
H^{\dagger}HW_{\mu\nu}B^{\mu\nu}$) apart from $\mathcal{O}_{B}$ and
$\mathcal{O}_{W}$. The dependence of $\Delta a_{\mu}$ on the last two
operators has already been calculated above to one-loop order and any residual
dependence would be further loop-suppressed and negligible. The Wilson
coefficient for $\mathcal{O}_{BW}$, on the other hand, is constrained by the
LEP experiments (from oblique corrections) to be tiny Hagiwara  _et al._
(1993). Consequently, any enhancement, on evolving down to the EW scale (or
even lower) in such an explicitly introduced $c_{e\gamma}/\Lambda^{2}$ is
tiny. For example, a value of starting with $c_{e\gamma}/\Lambda^{2}\sim{\cal
O}(10^{-3}~{}\mbox{TeV}^{-2})$ and $c_{BW}/\Lambda^{2}\sim{\cal
O}(0.1~{}\mbox{TeV}^{-2})$ at $\Lambda=1$ TeV, the RG evolution down to the EW
scale results in a change of less than $1\%$ in either. In other words, the
running effects can be neglected.

It is easy to see that an expression analogous to that in eqn.(6) would hold
for $\Delta a_{e}$ with $m_{\mu}$ replaced by $m_{e}$. Most importantly, the
sign of the new contributions would be identical in the two cases, which is in
disagreement with the experimental results.

###  III.2  Flavor Observables

The anomalous gauge couplings can also contribute to various loop-mediated
flavour changing neutral current hadronic decays. The said decays may occur
through a multitude of effective operators such as the electromagnetic dipole
or semi-leptonic vector and axial-vector ones, namely

$\begin{split}\mathcal{Q}_{7}&=\frac{e}{(4\pi)^{2}}m_{b}\left(\bar{s}_{L}\sigma_{\alpha\beta}b_{R}\right)F^{\alpha\beta}\\\
\mathcal{Q}_{9}&=\frac{e^{2}}{(4\pi)^{2}}\left(\bar{s}_{L}\gamma_{\alpha}b_{L})(\bar{l}\gamma^{\alpha}l\right)\\\
\mathcal{Q}_{10}&=\frac{e^{2}}{(4\pi)^{2}}\left(\bar{s}_{L}\gamma_{\alpha}b_{L})(\bar{l}\gamma^{\alpha}\gamma_{5}l\right)\\\
\end{split}$ (7)

where $L$ and $R$ denote the chirality of the fermionic fields,
$\sigma_{\alpha\beta}=i[\gamma_{\alpha},\gamma_{\beta}]/2$ and
$F^{\alpha\beta}$ is the electromagnetic field tensor. The $\Delta B,\Delta
S=1$ operator is traditionally written as

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\dfrac{4G_{F}}{\sqrt{2}}\Big{(}C_{7}Q_{7}+C_{9}Q_{9}+C_{10}Q_{10}\Big{)}+H.c$
(8)

with $C_{7}$, $C_{9}$ and $C_{10}$ being the corresponding Wilson coefficients
that factorise the short distance physics. We have omitted above the right-
handed analogues of $C_{9,10}$ as, to the leading order, the operators of
interest do not contribute to these. The anomalous contributions appear as a
result of the diagrams in Figure 2.

Figure 2: Diagram contributing to $b\rightarrow s\,\gamma(Z^{*})$ at 1-loop.

Although we use dimensional regularisation, it is instructive to trace the
divergences of the 1-loop contributions in generality. The photonic diagram
thus generated mirrors that for $\Delta a_{\mu}^{\rm anom}$ and is only
logarithmically divergent. For the $Z$ vertex, on the other hand, if a
momentum cutoff were used instead, the anomalous contribution from an
individual quark loop would be found to be quadratically divergent (in
dimensional regularisation, this is manifested as a pole at $d=2$). However,
thanks to the GIM mechanism, the quadratically divergent pieces (as with any
other term independent of the internal quark mass) cancel, leaving behind only
a logarithmic divergence. Of these, the top-quark contribution dominates
overwhelmingly Bobeth and Haisch (2015) and we can fairly approximate
($x\equiv m_{t}^{2}/m_{W}^{2}$)

$C_{i}\approx(C_{i})_{SM}+\left[V_{tb}V_{ts}^{*}\,\frac{m_{W}^{2}}{\Lambda^{2}}\,\right]\Delta
C_{i}$ (9)

with

$\begin{array}[]{rcl}\Delta
C_{7}&=&\displaystyle\frac{-(c_{B}+c_{W})}{8\left(x-1\right)^{2}}\left[2x+\frac{x^{3}-3x^{2}}{\left(x-1\right)}\,\ln
x\right]\\\\[10.76385pt] &+&\displaystyle\frac{3g^{2}\,c_{WWW}}{8}\left[\hskip
0.7113pt\frac{x^{2}+x}{\left(x-1\right)^{2}}-\frac{2x^{2}\,\ln
x}{\left(x-1\right)^{3}}\right]\ ,\\\\[15.0694pt] \Delta
C_{9}&=&\displaystyle\Bigg{\\{}\frac{-(c_{B}+c_{W})}{8}\,x\,+\frac{3c_{W}}{16}\,\frac{1-4s^{2}_{\theta}}{s^{2}_{\theta}}\,x\Bigg{\\}}\ln\left(\frac{\Lambda^{2}}{m_{W}^{2}}\right)\\\\[8.61108pt]
&+&\displaystyle\frac{3g^{2}c_{WWW}}{4\,\left(x-1\right)^{2}}\left[x-3x^{2}+\frac{2x^{3}\ln
x}{\left(x-1\right)}\right],\\\\[15.0694pt] \Delta
C_{10}&=&\displaystyle\dfrac{-3c_{W}}{16s^{2}_{\theta}}x\ln\left(\frac{\Lambda^{2}}{m_{W}^{2}}\right)\
.\end{array}$ (10)

The subdominant contributions from the up- and charm-loop can be easily read
off. For $b\rightarrow d$ and $s\rightarrow d$ transitions, analogous analyses
follow, but with the identity of the dominant loop changing.

###  III.3  $Z\to b\bar{b}$

The presence of the anomalous gauge couplings also leads to a modification in
the electroweak precision variables, whether these be the oblique corrections
or the fermion-gauge couplings. Of particular importance is the $Zb\bar{b}$
coupling and the $\rho$ (equivalently, $T$) parameter. The effective
$Zb\bar{b}$ vertex may be parametrized as

$\begin{split}\mathcal{L}_{Zb\bar{b}}=\dfrac{e}{s_{\theta}c_{\theta}}\left[\left(g_{L}^{b}+\delta
g_{L}^{b}\right)\bar{b}_{L}\hskip 0.7113pt\not{Z}\hskip
0.7113ptb_{L}+\left(g_{R}^{b}+\delta g_{R}^{b}\right)\hskip
1.42262pt\bar{b}_{R}\hskip 0.7113pt\not{Z}\hskip
0.7113ptb_{R}\right]\end{split}$ (11)

where $g_{L}=\left(-1/2+s_{\theta}^{2}/3\right)$ and
$g_{R}=\left(s_{\theta}^{2}/3\right)$ are the SM values of the couplings. The
one-loop contributions of the dimension-6 operators are encapsulated by the
two diagrams666There is, of course, a diagram analogous to Fig. 3$(a$), but
incorporating the $ZZH$ vertex instead of the $ZWW$ one. However, being
suppressed by an extra factor of $m_{b}^{2}/m_{W}^{2}$, the corresponding
contribution is negligible in size. of Fig. 3. It should be noted here that,
unlike in the previous two cases, an analysis with eqn.(3) is no longer
appropriate for it does not possess the full $SU(2)\otimes U(1)$ gauge
invariance. With a treatment of the oblique corrections being contingent on
this gauge invariance, the use of full eqn.(2) becomes mandatory.

(a)

(b)

Figure 3: Diagrams contributing to the $Z\rightarrow b\bar{b}$ decay
corresponding to (a) vertex correction and (b) wavefunction renormalisation of
$Z$-boson propagator.

With the most sensitive data on $\delta g_{L,R}^{b}$ being obtained at the
$Z$-pole, it is convenient to separate the corrections into two parts. One of
these originates from the wave-function renomalization of the $Z$ boson due to
a one-loop oblique correction $\Pi_{ZZ}$ (Fig. 3(b)) and is given by777It
should be remembered that, in the spirit of EFTs, only terms linear in the
Wilson coefficients should be retained.

$\begin{array}[]{rcl}(\delta
g_{L,R}^{b})_{ob}&=&\displaystyle\frac{-\alpha_{em}}{4\pi\Lambda^{2}}g_{L,R}{\cal
A}\\\\[8.61108pt] {\cal
A}&\equiv&\displaystyle\frac{gv\,m_{Z}}{4c_{\theta}}\Bigg{(}\frac{m_{H}^{2}}{2m_{Z}^{2}}+\frac{2}{3}\Bigg{)}\Bigg{(}\frac{c_{B}}{c_{\theta}^{2}}+\frac{c_{W}}{s_{\theta}^{2}}\Bigg{)}\log\frac{\Lambda^{2}}{m_{H}^{2}}\\\\[12.91663pt]
&+&\displaystyle\frac{m_{Z}^{2}}{2s_{\theta}^{2}}\,\log\frac{\Lambda^{2}}{m_{W}^{2}}\\\\[8.61108pt]
&\times&\displaystyle\Bigg{(}3(1-6c_{\theta}^{2})g^{2}\frac{m_{W}^{2}}{m_{Z}^{2}}c_{WWW}+\left[4c_{\theta}^{2}-\frac{5}{6}\right]c_{W}\\\\[8.61108pt]
&&\hskip
10.00002pt\displaystyle-\frac{1+4c_{\theta}^{2}-36c_{\theta}^{4}}{12c_{\theta}^{2}}\,\left[c_{\theta}^{2}{c_{W}}-s_{\theta}^{2}c_{B}\right]\Bigg{)}\end{array}$
(12)

The second contribution emanates from the direct one-loop correction to the
vertex (Fig 3(a)). Applicable only to the left-handed coupling888There is,
indeed, a vertex correction engendered by the anomalous $ZZH$ coupling that
contributes to $\delta g_{R}^{b}$. However, the contribution is suppressed by
the bottom mass., it is given by

$\begin{array}[]{rcl}(\delta
g_{L}^{b})_{v}&=&\displaystyle\left[\frac{\alpha_{em}\,\,|V_{tb}|^{2}}{16\,\pi\,s_{\theta}^{2}}\;\frac{m_{Z}^{2}}{2\Lambda^{2}}\right]\,{\cal
B}\,\log\frac{\Lambda^{2}}{m_{W}^{2}}\\\\[12.91663pt] {\cal
B}&\equiv&\displaystyle\left[c_{W}-s^{2}_{\theta}\left(c_{W}+c_{B}\right)\right]\left(-1+\frac{x}{2\,}-\frac{1}{6}\,\frac{m_{Z}^{2}}{m_{W}^{2}}\right)\\\\[12.91663pt]
&-&\displaystyle
c_{W}c^{2}_{\theta}\,\Bigg{(}-\frac{5m_{Z}^{2}}{3\,m_{W}^{2}}\,+\,3x\Bigg{)},\end{array}$
(13)

Tracing the divergences, we again note that, unlike in the preceding cases,
the GIM mechanism is not operative here, resulting in quadratic and higher
divergences as well. However, following the arguments in refs. Burgess and
London (1992, 1993), we presume that such loop contributions provide the
correct dependence on the new physics scale only upto terms scaling
logarithmically with $\Lambda$ and that higher divergences would be cancelled
by counter-terms in the EFT. The expressions in eq.(13) differ from those
given in ref. Bobeth and Haisch (2015) by a relative sign between the
$\Delta\kappa_{Z}$ and $\Delta g^{Z}_{1}$ contributions. The oblique
correction $\Pi_{ZZ}$ in eq. (12) matches with that calculated in ref Burgess
_et al._ (1994) for the $WW$-loop. Besides, we have also calculated the
$ZH$-loop contribution to the $Z$-propagator which hasn’t been included in
ref. Bobeth and Haisch (2015).

The logarithms in above equations are associated to the RG evolution between
$\Lambda$ and $m_{W}$. As argued in ref. Bobeth and Haisch (2015), since
$y_{t}^{2}/(4\pi)^{2}\,\ln(\Lambda^{2}/m_{W}^{2})\ll 1$ (with $y_{t}$ being
the top-quark Yukawa coupling) resumming the corresponding leading-logarithms
is not numerically important and retaining only the unresummed
$\ln(\Lambda^{2}/m_{W}^{2})$ contribution, as above, is a very safe
approximation.

##  IV Results

TGCs have been probed in many collider experiments over the years and
deviations from the SM values have been increasingly constrained with an
increase in the collision energy. The most recent studies by ATLAS and CMS
experiments at the LHC Sirunyan  _et al._ (2017); Aad  _et al._ (2016);
Khachatryan  _et al._ (2016); Sirunyan  _et al._ (2019, 2020); Aaboud  _et
al._ (2019) have primarily analysed multiple production channels such as
$W^{+}W^{-}$, $W^{\pm}\gamma$ and $W^{\pm}Z$ with the first-mentioned proving
to be the most restrictive. Each such underlying process can result in
different final states and several have been explored. Of the two most
sensitive studies, one Sirunyan  _et al._ (2020) considers the $W^{+}W^{-}$
channel with both $W$s decaying leptonically, leading to a
$l_{a}^{+}\nu_{a}l_{b}^{-}\bar{\nu}_{b}$ final state. The major reducible
backgrounds emanate from the Drell-Yan production of lepton pairs and
$t\bar{t}$ events in which both top quarks decay leptonically. The Drell-Yan
events can be suppressed by selecting the two leptons to be of different
flavor (one electron and one muon), wheras contributions from $t\bar{t}$
events can be reduced by rejecting events with b-tagged jets. The second
important analysis Sirunyan  _et al._ (2019) considers both $W^{+}W^{-}$ and
$W^{\pm}Z$ production where one $W$ decays leptonically and the other $W/Z$
decays hadronically resulting in a $l^{\pm}\nu q\bar{q^{\prime}}$ final state.
In this case, the two major backgrounds are $W+jets$ and $t\bar{t}$, and a
major portion can be reduced by using a combination of kinematical cuts and
jet substructure techniques. On the other hand, LEP experiments also posit
bounds on the same operators from $W^{+}W^{-}$ and single $W$ production
channels Schael  _et al._ (2013). Of the several studies, the most stringent
bounds on the anomalous TGCs, in terms of the Wilson coefficients
$c_{B}/\Lambda^{2}$, $c_{W}/\Lambda^{2}$ and $c_{WWW}/\Lambda^{2}$, have been
provided in Sirunyan  _et al._ (2019) for the LHC operating $\sqrt{s}=13$ TeV
and we use these for all subsequent comparisons. We note that the one-
parameter best fit limits on $c_{B}/\Lambda^{2}$ and $c_{W}/\Lambda^{2}$ from
Higgs production and decay measurements presented in ref. CMS (2020), although
stricter, are comparable with those from the TGC measurements. However, we do
not refer to these limits in our analysis explicitly as the ensuing sections
(IV.1–IV.3) deal only with two-dimensional parameter spaces in the WCs. In
what follows, we compare the current limits from the LHC against those
obtained from the aforementioned observables for a benchmark value of the
cutoff scale $\Lambda$.

A particular point needs to be noted at this juncture. An EFT such as that in
eqn.(2) should, ideally, be used in calculating cross sections only when
$\sqrt{\hat{s}}\lesssim\Lambda$, where, $\hat{s}$ denotes the partonic center-
of-mass energy. In the case of the LHC, this would nominally imply that
$\Lambda\raisebox{-1.72218pt}{\hbox
to0.0pt{$\sim$\hss}}\raisebox{1.72218pt}{$>$}{\cal O}(1~{}{\rm TeV})$—even
accounting for the nontrivial parton distribution functions—a constraint that
is often overlooked in interpreting results, generally presented in terms of
$c_{i}/\Lambda^{2}$.

In Table 1 we list the current limits for the aforementioned observables. For
the case of $(g-2)_{\mu}$ we quote two different values for the discrepancy
$\Delta a_{\mu}^{\rm anom.}$, namely, $\Delta a_{\mu}^{\mbox{\tiny DISP}}$
(WP20) and $\Delta a_{\mu}^{\mbox{\tiny BMW}}$ (BMW). The limits on $\Delta
C_{9}$ and $\Delta C_{10}$ are derived from single parameter fits to all $b\to
sl^{+}l^{-}$ branching ratio and angular observables (including that of
$\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$) excluding999The global fit that we
refer to, from (Hurth  _et al._ , 2022), does not include the
$B_{s,d}\to\mu^{+}\mu^{-}$ branching ratios. Including this, however, would
only have a marginal effect on the fit result and, hence, can be safely
ignored. the observables sensitive to lepton flavour universality (LFU) such
as the ratios $R_{K^{(*)}}$ Hurth  _et al._ (2022). Similar fits have also
been performed by other groups, see e.g.Carvunis  _et al._ (2021);
Altmannshofer and Stangl (2021); Algueró  _et al._ (2021). On the other hand,
the limits on $\Delta C_{7}$ are extracted from the inclusive B-meson
radiative decay $(\mathcal{B}_{s\gamma}:B/\bar{B}\to X_{s}\gamma)$ observable
$R_{X_{s}}\equiv\mathcal{B}_{s\gamma}/(\mathcal{B}_{s\gamma})_{SM}$
Altmannshofer and Straub (2015); Arbey  _et al._ (2018). It is important to
note that the exclusion of the observables such as $R_{K^{(*)}}$, is enforced
by the fact that whereas their current measurements hint towards a violation
of LFU, the operators under discussion are strictly flavour blind.

Current limits
---
Observable($\mathcal{F}$) |  $1\sigma$ limit
$\Delta a_{\mu}^{\mbox{\tiny DISP}}$(WP20) | $251\pm 59\times 10^{-11}$
$\Delta a_{\mu}^{\mbox{\tiny BMW}}$(BMW) | $107\pm 69\times 10^{-11}$
$\Delta C_{7}$ | $-0.03\pm 0.03$
$\Delta C_{9\mu\mu}$ | $-1.03\pm 0.13$
$\Delta C_{10\mu\mu}$ | $0.41\pm 0.23$
$\Delta C_{9ee}$ | $0.70\pm 0.60$
$\Delta C_{10ee}$ | $-0.50\pm 0.50$
$\delta g_{L}$ | $0.0016\pm 0.0015$
$\delta g_{R}$ | $0.019\pm 0.007$
Table 1:  Current experimental limits on various observables affected by
anomalous TGCsHurth  _et al._ (2022); Ciuchini  _et al._ (2013).

###  IV.1 $2\sigma$ bands for all observables for current limits

In our quest to study the low-energy constraints in their totality, we begin
by studing each in isolation. Assuming, for the purpose of easy visualization,
that only the Wilson coefficients $c_{B}$, $c_{W}$ are nonzero, we present, in
Fig. 4, the ensuing bounds in this plane emanating from the individual
observables in Table 1 with the assumption that the new physics
scale101010Note that, unlike the collider bounds (which have been derived by
neglecting all subleading dependence on $\Lambda$), the SMEFT constraints in
Fig. 4 have additional logarithmic dependence of $\Lambda$, and we would
return to this point later. $\Lambda\sim 2$ TeV.

(a)

(b)

Figure 4: The 2$\sigma$ ranges in the $c_{W}/\Lambda^{2}$–$c_{B}/\Lambda^{2}$
plane allowed by individual observables. $\Lambda=2$ TeV has been used along
with the lattice result Borsanyi  _et al._ (2021). The lower panel shows an
enlarged portion of the same. The black ellipse identifies the LHC limit.

Several features are worth noting:

  * •

The $\Delta a_{\mu}$-allowed band (purple), as calculated using the WP20
result Aoyama  _et al._ (2020) does not include the SM point, reflecting the
fact that the data does not agree with the SM value at the $2\sigma$ level.
For the lattice result (BMW) Borsanyi  _et al._ (2021), though, it is indeed
included. For brevity’s sake, here, and in subsequent discussions, we display
only the figures corresponding to the BMW result, while keeping under
consideration the WP20 alternative as well.

  * •

The corresponding band for $\Delta a_{e}$ sits on the opposite side of the
origin, owing to the sign of the discrepancy. However, with the anomalous
contribution being suppressed by $m_{e}^{2}$, the required sizes of the Wilson
coefficients are too large to be meaningful.

  * •

$\delta g_{L}$ has a relatively weaker dependence on $c_{B}$ than on $c_{W}$
(see eqns.(12&13)) leading to the slightly tilted band. $\delta g_{R}$, on the
other hand, receives a small correction only from the correction to the $Z$
self-energy, and the ensuing bounds are too weak to be relevant.

  * •

Since $\Delta C_{7}$, just like $\Delta a_{\mu}$, parametrizes the coupling of
a fermion current to the photon, both are understandably proportional (in the
absence of a nonzero $c_{WWW}$) to the combination $(c_{B}+c_{W})$ and the
ensuing bands are parallel to each other.

  * •

$\Delta C_{10}$, being dependent on $c_{W}$ alone, leads to a relatively
narrow vertical band in this plane. Most restrictive of all the observables,
the difference in its value as calculated from the electronic and muonic
channels exert opposite pulls leading to the two parallel bands. Although both
bands overlap with the collider limit, the one corresponding to the muonic
channel has a greater sensitivity to $c_{W}$ and, hence, its partial overlap
presents a comparatively stronger constraint on the allowed region, favouring
negative values for $c_{W}$.

This leads to an interesting possibility wherein $\Delta C_{10}$ is the
dominant flavour-blind Wilson coefficient parametrizing new physics effects in
FCNC $B$ decays. A sizable range of $c_{W}$ values compatible with the LHC
limits exists that could, then, ameliorate the discrepancies in the
aforementioned $B$ decay observables (excluding LFU ones). Not contributing to
$\Delta C_{10}$, a similar-sized $c_{B}$ would lead to only tiny changes in
the low-energy observables (see the zoomed-in view of Fig.4(b)) and would be
primarily constrained by collider experiments.

  * •

Similar to the preceding observable, the opposing experimental numbers for
$\Delta C_{9}$ from the two ($e$ and $\mu$) channels lead to two bands.

  * •

It is quite apparent that the flavor observables $\Delta C_{7,9,10}$ give some
of the strongest constraints. Indeed, with many of the bands intersecting each
other at different angles, it is expected that combining the individual data,
when independent, would lead to constraints much stronger than individual
ones. Whether such combined constraints agree with the LHC results is an
aspect which we address in the following subsection.

Before we attempt this, it behoves us to consider the limits afforded on
$c_{WWW}$ and we display these in Fig.5, once in the $c_{B}$–$c_{WWW}$ plane
(keeping $c_{W}=0$) and once in the $c_{W}$–$c_{WWW}$ plane (keeping
$c_{B}=0$). With their dependences on $c_{WWW}$ being different, the bands due
to $\Delta a_{\mu}$ and $\Delta C_{7}$ are no longer parallel. $\Delta C_{9}$
continues to be a restrictive force in the $c_{B}$–$c_{WWW}$ plane, while its
effect is reduced in the $c_{W}$–$c_{WWW}$ plane (owing to the smaller
dependence on $c_{B}$). For $\delta g_{L}$, it is, quite understandably, the
other way around.

(a)

(b)

Figure 5: The 2$\sigma$ bands as derived from the individual observables from
anomalous TGCs considering BMW result Aoyama  _et al._ (2020) as the theory
value in the $c_{WWW}/\Lambda^{2}$\- $c_{B}/\Lambda^{2}$ (upper panel) and
$c_{WWW}/\Lambda^{2}$\- $c_{W}/\Lambda^{2}$ (lower panel) planes along with
the 13 TeV experimental constraints.

It needs to be appreciated at this point that the nominal restrictions from
the low-energy observables are much weaker than those obtained at the LHC, as
exhibited by all the panels of Figs.4&5. This situation changes drastically
once we attempt to simultneously fit all the observables as we see next. Also
worth noting is the fact that much of the parameter space satisfying the
individual observables fall outside the domain wherein the EFT series
expansion can be safely deemed valid and, hence, caution must be exercised
while reading off such regions as constraints on the WCs.

###  IV.2 Fitting all observables

We now perform a combined fit of the non-standard parameters
$(c_{B},\,c_{W},\,c_{WWW})$ to the observables defined in sec.(III). To this
end, we define a $\chi^{2}$ as a function of the anomalous parameters, namely

$\chi^{2}(c_{B},c_{W},c_{WWW};\Lambda)=\sum_{i}\left(\frac{\mathcal{F}^{\rm
exp}_{i}-\mathcal{F}^{\rm th}_{i}}{\sigma_{i}}\right)^{2},$ (14)

where $i$ runs through all the observables, $\mathcal{F}^{\rm exp}_{i}$ are
the experimental values with standard deviations $\sigma_{i}$ and
$\mathcal{F}^{\rm th}_{i}$ denote the corresponding theoretical expectations
for a given set of values for the Wilson coefficients. The best-fit point
would then be given by the minimum of the $\chi^{2}$ and parameter points
leading to $\chi^{2}\leq\chi^{2}_{min}+\delta\chi^{2}$ being inseparable from
the best-fit point at a confidence level determined by $\delta\chi^{2}$. As
before, in order to render the constraints more tractable, we consider only
two nonzero Wilson coefficients at a time, holding $\Lambda$ to a fixed value.

(a)

(b)

Figure 6: 6(a) shows the result of the $\chi^{2}$ analysis in the BMW case for
$\Lambda=2$ TeV, and the collider result at $95\%$ confidence level, in the
$c_{W}/\Lambda^{2}$\- $c_{B}/\Lambda^{2}$ plane. 6(b) shows the same for
$\Lambda=10$ TeV.

The calculation of the $\chi^{2}$ entails inclusion of either only independent
measurements, or if more variables are to be considered, an appropriate
inclusion of the correlation matrix. For the sake of simplicity, we adopt the
former method despite the attendant loss of sensitivity. This allows us to use
the corresponding measurements (alongwith the errorbars) from Table 1 in a
straightforward manner. In particular, thanks to the flavour universal nature
of the Wilson coefficients in the theory, we make use of both the muonic
($\Delta C_{\mu\mu}$) as well as the electronic ($\Delta C_{ee}$) limits
simultaneously.

Calculation | Descriptor | $(c_{B},c_{W},c_{WWW})/\Lambda^{2}$ | $\chi^{2}$
---|---|---|---
|  | [TeV-2] |
WP20 | SM | (0,0,0) | 101.76
| 2-param B.F. | $(39.26,-1.64,0)$ | 25.76
| 3-param B.F. | $(38.48,-1.63,-2.97)$ | 25.71
BMW | SM | (0,0,0) | 86.121
| 2-param B.F. | $(35.05,-1.83,0)$ | 28.663
| 3-param B.F. | $(36.65,-1.85,6.41)$ | 28.446
Table 2: Results of the different $\chi^{2}$ analyses described in the text.

We, thus, have two inequivalent scenarios (depending on whether we choose the
WP20 or BMW calculation) that the theory is to be compared with. The resultant
$\chi^{2}_{min}$ values are tabulated in Table 2. Also tabulated, for
comparison, are the $\chi^{2}$ values for the SM point. The strikingly
different $\chi^{2}$ values for the latter under the two schemes is but a
consequence of the difference in the theoretical predictions for
$(g-2)_{\mu}$. Also, note that the rather large values for the SM point are
driven, to a large extent, by the flavour universal $B$-decay observables.

To interpret these results, it is instructive to consider a projection to a
two-parameter space, which we consider to be the
($c_{W}/\Lambda^{2},c_{B}/\Lambda^{2}$) plane with $c_{WWW}$ held zero. The
corresponding global fit, in the best case scenario (i.e., BMW), is
illustrated in Fig.6(a) wherein we show the best-fit with the attendant 95%
C.L. contour. We note that the $\chi^{2}$-fit not only prefers the best-fit
value to be far away (as determined on the scale of the LHC limits) from the
SM value, but even the two 95% ellipses (direct limits and low energy
preference) do not overlap with each other. That the separation is larger
along the $c_{B}$–axis can be understood qualitatively by comparing the
individual bands in Fig.4(a). On account of the relatively small uncertainty
associated with it, the muonic $\Delta C_{9}$ measurement exerts the strongest
pull on $c_{B}$, thereby pulling the best-fit point upward.

As can be gleaned from eq.10, and as has already been reflected in Fig.5, the
observables under consideration have only a very weak dependence on $c_{WWW}$.
Consequently, its contribution to the $\chi^{2}$ is relatively small, and, on
its inclusion in the minimization, the erstwhile minimum in the two-parameter
fit (with $c_{WWW}=0$) only expands into a very shallow basin. This is
reflected by Table 2, where the inclusion of $c_{WWW}$ doesn’t have any
significant impact on the $(c_{B},c_{W})$ coordinates of the best-fit point.
Similarly, the improvements in the attendant $\chi^{2}_{\rm min}$ values are
only marginal.

A caveat needs to be discussed at this point. Although we have been
characterizing our fits as functions of $c_{i}/\Lambda^{2}$, in actuality we
have held $\Lambda$ to a specific value (2 TeV) especially in the calculation
of the logarithms. Increasing it to higher values not only brings the best-fit
point closer to the SM, but also, understandably, shrinks the 95% C.L. ellipse
significantly (see Fig. 6(b)). The apparent tension with the LHC constraints
is maintained. This disagreement is expected to persist as, with additional
luminosity (LHC Run III, HL-LHC, etc.) or higher energies (a future collider
or an upgrade of the LHC), the collider limits are likely to shrink too (as it
already has for the 13 TeV run as compared to the 8 TeV oneSirunyan  _et al._
(2019)).

###  IV.3 Future Projections

The analysis in the preceding subsection establishes that an effort to explain
$(g-2)_{\mu}$ or several other discrepancies in the low-energy data in terms
of anomalous triple-gauge boson couplings (or, equivalently, the corresponding
bosonic operators in the SMEFT) would require couplings (Wilson coefficients)
that are too large given the LHC constraints. It is, then, of interest to
speculate whether near-future improvements in the low-energy data are likely
to lead to constraints stronger than those already imposed by the LHC. As a
particular example, we choose to do this for $(g-2)_{\mu}$. With the
experiment at FNAL projecting a reduction in the uncertainties by a factor of
about four Aoyama  _et al._ (2020), it is worth re-examining our analysis with
the following two assumptions as to the tentative outcomes by the end of the
FNAL experiment(s):

####  IV.3.1 Reduced uncertainties in $\Delta a_{\mu}$ with the same central
value

With no significant improvement in the theoretical calculations envisaged in
the near future, if the central value of the experimental measurement remains
unchanged, we would be likely to face $\Delta a_{\mu}^{\mbox{\tiny
DISP}}=(251\pm 15)\times 10^{-11}$ ($\Delta a_{\mu}^{\mbox{\tiny BMW}}=(107\pm
17.15)\times 10^{-11}$) pertaining to the WP20 (BMW) analyses. The
corresponding $\chi^{2}$-fit for the projected $\Delta a_{\mu}^{\mbox{\tiny
BMW}}$ (best case scenario) is shown in Fig.7(blue curve). Comparing with the
current limits presented in Fig.6(a) we note that while the best fit point has
shifted marginally towards the SM point along the $c_{B}/\Lambda^{2}$
direction, the corresponding $95\%$ C.L. contour has, expectedly, shrunk. The
overall tension would actually increase.

Figure 7: The $\chi^{2}$-fit and the collider result at $95\%$ confidence
level for $\Lambda=2$ TeV in the $c_{W}/\Lambda^{2}$\- $c_{B}/\Lambda^{2}$
plane: (a) (blue) assuming same deviations as of BMW result and corresponding
errors reduced by a factor of 4 and (b) (orange) assuming no deviations from
the SM result and errors reduced by a factor of 4.

####  IV.3.2 Reduced uncertainties in $\Delta a_{\mu}$ with a vanishing
central value

In the other hypothetical case of a vanishing discrepancy by the end of the
FNAL run, we would have $\Delta a_{\mu}=0\pm(17.15)\times 10^{-11}$, and
Fig.7(orange curve) shows the corresponding global fit. The best-fit point
would now shift by a significant amount towards the SM point with the contour
ellipse having shrunk as in the preceding case. Furthermore, the best-fit
ellipse now has a small overlap with the LHC contour. This fitting, thus,
gives a true measure of the pull exerted by the cohort of low-energy flavour-
universal observables (other than $\Delta a_{\mu}$).

Note that the preceding projections correspond specifically to $\Lambda=2$
TeV. For higher cutoff scales, the best fits would shift considerably towards
the SM point, as had been indicated in Fig. 6(b), and would offer a better
reconciliation with the LHC limits as well. However, any claim about this
indicating a resolution of the discrepancies through the WCs residing in the
overlap are fraught with danger. For one, were hypothetical resonances that
generate these WCs interacting only weakly with each other and with the SM
sector, one would expect
$\mathcal{O}(g^{2}_{SM}/16\pi^{2})\lesssim(|c_{B}|,|c_{W}|)\lesssim\mathcal{O}(1)$
Marzocca  _et al._ (2020); Giudice  _et al._ (2007), with $g_{SM}$ denoting a
typical SM coupling. However, much of the overlap region would most probably
correspond to $(|c_{B}|,|c_{W}|)>\mathcal{O}(1)$ indicating a strongly
interacting sector that calls into question the very method of calculating
quantum corrections that we have adopted. In particular, no definite
conclusion with regards to the matching of the SMEFT operators with a UV
theory can be drawn.

##  V  Summary and Outlook

Our study indicates that the limits on low-energy observables, taken
individually, lead to weak bounds on the bosonic SMEFT Wilson coefficients
when compared with the existing LHC limits, except for the bounds on
$c_{W}/\Lambda^{2}$ emanating from the limits on $\Delta C_{10}$ which are
comparable and consistent with the collider results. On the other hand, a
global fit in the $(c_{W}/\Lambda^{2},c_{B}/\Lambda^{2})$ plane, while
imposing significantly stronger constraints on the WCs, exhibits disagreement
with the LHC results. The naive expectation is that this disagreement would
reduce with a (speculative) lowering of the $a_{\mu}$ tension in the future;
on the other hand, improvements in the collider limits on the WCs are as
likely to maintain the disagreement. At this juncture a cautionary remark
ought to be underscored to aid our judgement. The LHC limits have used cross-
sections that also include terms quadratic in the TGCs, whereas we consider
contributions to the concerned observables only upto a linear order in the
same. Had we included quadratic contributions, we would have obtained stronger
bounds as well. Thus, in view of this caveat, it would be rash to rule out
completely the scenario wherein anomalous TGCs embody some of the most
dominant new physics effects addressing all of the aforementioned anomalies.

Notwithstanding the caveats, the fact that $\Delta C_{10}$ also favours
$c_{W}/\Lambda^{2}$ values that are very close to the origin indicates that
any explicit new physics model designed to explain the discrepancies (e.g.,
models which give rise to lepton flavour universality violating (LFUV) or a
combination of LFUV and LFU 4-fermion operators111111See London and Matias
(2021) for a review.) should either induce $\mathcal{O}_{W}$ with a suppressed
(or vanishing) Wilson coefficient or, otherwise, one must account for the WC
$c_{W}$ generated therein, in addition to other parameters, while performing a
fit to the concerned observables. This is a crucial outcome of our study, to
be regarded as a check on the existing models concerning low-energy
observables, as well as a note to the model builders developing new scenarios
along this line.

###### Acknowledgements.

We thank Martin Hoferichter and Diego Guadagnoli for insightful comments and
for bringing to our notice some important references. 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. S.M.
acknowledges research Grant No. CRG/2018/004889 of the SERB, India. L.K.S.
acknowledges the UGC SRF fellowship and research Grant No. CRG/2018/004889 of
the SERB, India, for partial financial support.

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