††institutetext: 1Department of Physics, University of Calcutta, 92 Acharya
Prafulla Chandra Road,
Kolkata 700009, India

# Exploring Higgs-Photon Production at the LHC

Tisa Biswas1, Anindya Datta1<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We have investigated the signal for physics beyond the Standard Model via
Higgs plus photon final state, hitherto unobserved at the LHC, in the
framework of Standard Model Effective Field Theory. Using the relevant
dimension-6 operators, we probe two distinct classes of interactions, based on
the Lorentz structure of Higgs couplings to gauge bosons and fermions. To
begin with, constraints on the Wilson coefficients of these operators have
been derived from existing experimental data. We then focus on the hadronic
decay of the Higgs boson to two bottom quarks in the boosted regime, leading
to a high $p_{T}$ fat-jet recoiling against a hard photon. Following a CMS Run
II search for a resonance decaying into a Higgs boson and a photon, a detailed
signal and background analysis for this channel has been done and limits on
the relevant Wilson coefficients have been obtained. By performing a cut-based
analysis, we identify some kinematic observables that distinguish between
signal and SM background. Minimum values of the Wilson coefficients that will
yield $3\sigma$ signal significance at the 14 TeV run of the LHC with
$3000\leavevmode\nobreak\ \rm fb^{-1}$ data, have been obtained. A
multivariate analysis using a boosted decision tree and exploiting the jet
substructure techniques further help to isolate the regions of phase space
where the contribution of SMEFT driven signal is significantly enhanced.
Finally, we briefly discuss the parameter dependencies and interpretations of
the allowed values of the coefficients on a particularly interesting UV
complete model namely, Minimal Supersymmetric Standard Model.

††preprint:

##  1 Introduction

With the Large Hadron Collider (LHC) in operation, our venture to shed more
light on the fundamental interactions of elementary particles has seen
impressive advances. After a few years that the first beam circulated at the
LHC, the discovery of the 125 GeV scalar boson Aad:2012tfa ;
Chatrchyan:2012xdj  has paved the way towards the Era of Higgs Physics and
intense efforts have been put in the decade thereafter, studying its
properties to a high degree of accuracy. So far, properties of this 125 GeV
scalar have been found to be in good agreement with those of the Standard
Model (SM) Higgs boson. However, the exact mechanism for electroweak symmetry
breaking remains to be verified. It is also possible that so far elusive
signal of new physics beyond the SM may arise from the interactions of the
Higgs boson itself. In this endeavour, precision measurement of Higgs
couplings are very important. The continuously increasing amount of data
recorded at the LHC opens the possibilities to explore properties of Higgs
boson in a multitude of ways. Improved analysis techniques, especially in the
context of High Luminosity-LHC (HL-LHC), will be extremely useful in the
pursuit of physics beyond the SM.

There are several approaches to search for the new particles and their
interactions. One way is to look for the signature of new particles that can
be produced at resonance at high energies. Yet, after years of intense
collective efforts searching for new particles, no clear evidence of such new
resonances have been found. Another fully consistent and theoretically-sound
framework is Effective Field Theory (EFT). This framework relies on the idea
that new physics affecting the LHC measurements is too heavy to be produced
on-shell. However, such heavy states leave their footprints via their
contributions when they act as propagators either at tree level or higher
order processes involving the SM particles leading to deviations in their
couplings. So, it is of utmost importance to measure the interactions
involving SM particles at LHC energies as an indirect probe for such new heavy
particles and their properties Weinberg:1978kz ; Buchmuller ; Leung:1984ni ;
Dawson:2018dcd . We will follow this second avenue and stick to a framework
where the SM is augmented by a set of higher dimensional operators respecting
all the symmetries of the SM, namely, the Standard Model Effective Field
Theory (SMEFT),

$\mathcal{L}_{\rm SMEFT}=\mathcal{L}_{\rm
SM}+\sum_{d=5}^{\infty}\sum_{i=1}^{n}{C_{i}^{(d)}\over\Lambda^{d-4}}\mathcal{O}_{i}^{d}$
(1)

Here, $\mathcal{O}_{i}^{d}$ are the mass dimension-$d$ operators, constructed
out of the SM fields only. Any non-SM physics effects are encapsulated in the
Wilson coefficients $C_{i}^{(d)}$. The energy scale $\Lambda$ can be connected
to the masses of the heavy particles which manifest themselves via virtual
effects. With these assumptions there are 59 linearly independent dimension-6
operators in a so-called Warsaw basis Grzadkowski:2010es  which we will use to
probe any new physics that they contain in their structure.

In this work, we will investigate the Higgs boson production in association
with a photon. The Higgs boson production in association with a vector boson,
$(W^{\pm},Z)$ has been extensively studied in the SM Gupta:2014rxa ;
Cohen:2016bsd . In the limit of vanishing quark masses, Higgs production in
association with a photon from quark$-$anti-quark annihilation, is absent at
the tree level. This process can also take place via internal particle loops,
rate for which is also not very promising. Therefore, any deviation from the
SM predictions and observation of $h\gamma$ signal at the LHC will point to
the presence of beyond-SM (BSM) physics. At hadron colliders, it is very
challenging to observe this particular Higgs signal due to the large
background and its small cross-section.

Thus, it is important to understand how the new physics can be manifested via
this channel and accordingly optimise the search strategy. In this work, we
will identify the SMEFT operators that give leading contributions to the
production of the Higgs and a photon. The Higgs decays into a pair of bottom
quarks, resulting in a final state consisting of a Higgs-like fat-jet and a
hard photon. The CMS collaboration has looked for this signal using
$35.9\leavevmode\nobreak\ \rm fb^{-1}$ CMS:2018cno  data at 13 TeV run in
search of a heavy resonance decaying into a Higgs and a photon. By comparing
our estimated rates, with those published in Ref. CMS:2018cno , we put model
independent constraints on the Wilson coefficients of the relevant dimension-6
operators. These Wilson coefficients of SMEFT operators relevant in Higgs
physics have been constrained through various channels, for example in Refs.
Sanz ; Cepeda:2019klc ; Biswas:2021qaf . We compare the bounds obtained from
our analysis with the existent limits on such couplings.

With these constraints on the operator coefficients, we study the impact of
possible BSM physics in this channel at the HL-LHC. Our study proposes more
than one kinematic variables which are very instrumental in separating the new
physics effects from the SM background.

In the following, we highlight the novel features of our study:

  * •

To the best of our knowledge, this is the first comprehensive study of
$h\gamma$ production in the framework of SMEFT at the LHC in the boosted
kinematic region of the phase space. The earlier investigations
Dobrescu:2017sue ; Khanpour:2017inb ; Aguilar-Saavedra:2020rgo  in this
channel, dealt with chosen few dimension-6 operators involving either only
bosonic fields or fermionic fields without contrasting the results with
available experimental data.

  * •

We focus in the high energy regime where the Higgs is boosted and will
manifest itself as a fat-jet containing two b-tagged subjets. We investigate
topologies of the final state with such a fat-jet in association with a hard
photon for possible signature of new interactions with Lorentz structures
different from the SM.

  * •

To better discriminate between signal and background, a multivariate analysis
is performed using Boosted Decision Tree (BDT) techniques with jet
substructure observables such as N-subjettiness, jet mass and energy
correlators as input variables. Going beyond the standard cut-based approach
and using the BDT technique shows a clear improvement in terms of signal
significance as explicitly demonstrated in the following.

  * •

We have segregated two types of interactions based on the operator structures.
We propose to correlate these distinct types of interactions in producing
$h\gamma$ by using the ratio of rates in the $\gamma\gamma$ and $Z\gamma$
decay modes of Higgs and ratio of rates of $Vh$ and $VBF$ Higgs production.
The effect of various dimension-6 operators considered in this paper, can also
be observed in the process $pp\to h\to b\bar{b}\gamma$. We briefly discuss the
effect of such operators in the final state kinematics for this process.

  * •

We briefly tried to discuss the possible sources of such dimension-6 operators
arising in a UV complete theory like MSSM.

The plan of the paper is the following. In section 2, we enlist the relevant
operators in the Warsaw basis that contribute to the $pp\to h\gamma$ process.
This has been followed by a discussion on various constraints on these
operators in the light of experimental data. In section 3, a detailed collider
analysis studying the kinematical features of the signal and background
processes is performed. Firstly, we present these features with a cut-based
analysis. Thereafter, we discuss the projected sensitivities to the EFT
couplings. The improvement over the cut-based results using boosted decision
trees has been explored. In section 4, we comment on the numerical results and
interpretations and discuss some parameter dependencies of the cross-section
in detail. Finally, we draw our conclusions in section 5.

##  2 Higher Dimensional Operators Parametrisation

We will follow the effective field theory approach to probe for any possible
deviations from the SM interactions in the Higgs sector. The Higgs observed at
the LHC is assumed to be a part of an $SU(2)_{L}$ doublet and that the
$SU(2)_{L}\times U(1)_{Y}$ symmetry is linearly realized in the effective
theory. We consider the following general set of dimension-6 gauge invariant
operators which give rise to such anomalous $hV\gamma$ interactions and the
$h\gamma ff$ contact interactions.

  * •

Operators containing the Higgs doublet $\varphi$ and its derivatives:

$\mathcal{O}_{H\Box}=(\varphi^{\dagger}\varphi)\raisebox{-1.42262pt}{$\Box$}(\varphi^{\dagger}\varphi);\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{O}_{HD}=\left(\varphi^{\dagger}D^{\mu}\varphi\right)^{\star}\left(\varphi^{\dagger}D_{\mu}\varphi\right)$
(2)

They lead to corrections to the SM Higgs couplings to other particles by
multiplicative factors without modifying the Lorentz structure of the
interactions. Here, $D_{\mu}$ has the usual meaning referring to the covariant
derivative and contain $SU(2)_{L}$, $U(1)_{Y}$ gauge couplings and bosons.

  * •

These set of operators can induce $h\gamma qq$ contact interactions. Due to
the gauge field strengths, there is a presence of explicit dependence on the
momentum of the gauge bosons in such operators. This will play a significant
role in kinematics in which we are interested in this analysis.

$\mathcal{O}_{uW}=(\bar{q}_{p}\sigma^{\mu\nu}u_{r})\tau^{I}\widetilde{\varphi}\,W_{\mu\nu}^{I};\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{O}_{dW}=(\bar{q}_{p}\sigma^{\mu\nu}d_{r})\tau^{I}\varphi\,W_{\mu\nu}^{I}$
$\mathcal{O}_{uB}=(\bar{q}_{p}\sigma^{\mu\nu}u_{r})\widetilde{\varphi}\,B_{\mu\nu};\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{O}_{dB}=(\bar{q}_{p}\sigma^{\mu\nu}d_{r})\varphi\,B_{\mu\nu}$ (3)

The field content consists of $q$, the quark doublet under $SU(2)_{L}$ and
$u$, $d$ are the $SU(2)_{L}$ singlet quarks, with $p$ and $r$ denoting the
generation indices and $SU(2)_{L}$ and $U(1)_{Y}$ gauge fields are denoted by
$W$ and $B$, respectively. Neglecting fermion masses, the magnetic dipole type
operators connect fermions of different helicities.

  * •

The anomalous $hV\gamma$ interactions arise from the following set of
dimension-6 operators Buchmuller ; Grzadkowski:2010es :

$\mathcal{O}_{HW}=\varphi^{\dagger}\varphi\,W^{I}_{\mu\nu}W^{I\mu\nu};\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{O}_{HB}=\varphi^{\dagger}\varphi\,B_{\mu\nu}B^{\mu\nu};\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{O}_{HWB}=\varphi^{\dagger}\tau^{I}\varphi\,W^{I}_{\mu\nu}B^{\mu\nu}$
(4)

These operators induce $hV\gamma$ couplings which are SM-like in Lorentz
structure, while modify the $hVV$ couplings by introducing new Lorentz
structure in the Lagrangian.

The couplings involving the Higgs boson and the photon are generated by the
combinations of the aforementioned operators. We note down the additional new
interactions in the effective Lagrangian:

$\displaystyle\mathcal{L}_{eff}$ $\displaystyle=$ $\displaystyle
g_{h\gamma\gamma}hA_{\mu\nu}A^{\mu\nu}+g_{hZ\gamma}hA_{\mu\nu}Z^{\mu\nu}$ (5)
$\displaystyle+\left(g_{uuh\gamma}(\overline{u}\sigma^{\mu\nu}P_{R}u)hA_{\mu\nu}+g_{ddh\gamma}(\overline{d}\sigma^{\mu\nu}P_{R}d)hA_{\mu\nu}+h.c.\right)\,,$

The effective couplings contributing to $h\gamma$ production in $pp$
collisions in Eqn. 5 are related to the coefficients of the operators
appearing in Eqns. 2-4 as follows:

| Warsaw Basis
---|---
$g_{h\gamma\gamma}$ | $\frac{2m_{W}}{\Lambda^{2}}\frac{1}{g}\left(s_{W}^{2}C_{HW}+c_{W}^{2}C_{HB}\right)$
$g_{hZ\gamma}$ | $\frac{4m_{W}}{\Lambda^{2}}\frac{1}{g}s_{W}c_{W}\left(C_{HW}-C_{HB}\right)+\frac{2m_{W}}{\Lambda^{2}}\frac{1}{g}(s_{W}^{2}-c_{W}^{2})C_{HWB}$
$g_{uuh\gamma}$ | $\frac{-1}{\sqrt{2}\Lambda^{2}}\left(c_{W}C_{uB}+s_{W}C_{uW}\right)$
$g_{ddh\gamma}$ | $\frac{-1}{\sqrt{2}\Lambda^{2}}\left(c_{W}C_{dB}-s_{W}C_{dW}\right)$
Table 1: Anomalous Higgs couplings in the Warsaw basis.

###  2.1 Model-independent constraints

After parametrising new physics in terms of dimension-6 operators, we now
locate the region of parameter space favoured by the existing experimental
data.

The operators $\mathcal{O}_{HW}$ and $\mathcal{O}_{HB}$ contribute to the so-
called Peskin-Takeuchi $S,T$ and $U$ parameters Peskin ; Altarelli  and are
therefore constrained by electroweak precision observables Hagiwara ; Garcia
(EWPO). The Wilson coefficients $C_{HWB}$ (that drives $Z-\gamma$ mixing) and
$C_{HD}$ are constrained from the EWPD. Following Ref. Garcia , the bounds at
$95\%$ C.L., taking one operator at a time, are given by
$-24\leavevmode\nobreak\ {\rm
TeV}^{-2}<\frac{C_{HW}}{\Lambda^{2}}<14\leavevmode\nobreak\ {\rm
TeV}^{-2}$,$-79\leavevmode\nobreak\ {\rm
TeV}^{-2}<\frac{C_{HB}}{\Lambda^{2}}<47\leavevmode\nobreak\ {\rm TeV}^{-2}$.
However, Higgs data puts much stronger bounds on these two operators compared
to the bounds derived from the oblique parameters. Couplings arising from the
bosonic operators affect the tree-level SM Higgs couplings, namely, $hWW$,
$hZZ$ and contribute in the loop-induced Higgs decays as well, namely,
$h\rightarrow\gamma\gamma$ and $h\rightarrow Z\gamma$. The dipole operators
are also subjected to the constraints imposed by the LHC data. Taking one
operator at a time, bounds on them are obtained by comparing the expected
cross-section with experimental data from ATLAS and CMS collaborations (Table
2). The results are summarised in Table 3. The limits on $C_{H\Box}$ and
$C_{HD}$ have been presented in Ref. Biswas:2021qaf  and will not be discussed
here anymore.

Channel | Experiment | Energy (GeV) | Luminosity
---|---|---|---
$h\rightarrow\gamma\gamma$ | CMS CMS:2018piu  | 13 TeV | 35.9 fb-1
$h\rightarrow\tau\bar{\tau}$ | CMS CMS:2017zyp  | 13 TeV | 35.9 fb-1
$h\rightarrow WW$ | CMS CMS:2018zzl  | 13 TeV | 35.9 fb-1
$h\rightarrow\gamma\gamma$ | ATLASATLAS:2018hxb  | 13 TeV | 36.1 fb-1
$h\rightarrow ZZ$ | ATLAS ATLAS:2017azn  | 13 TeV | 36.1 fb-1
$h\rightarrow b\bar{b}$ | ATLAS ATLAS:2018kot  | 13 TeV | 79.8 fb-1
Table 2: Listing of the Higgs signal strength measurements at LHC at 13 TeV
included in our study

The rate of the process $b\to s\gamma$ can be modified in presence of the
operator $\mathcal{O}_{bW}$. Stringent bounds on such an interaction has been
obtained in ParticleDataGroup:2020ssz ; Cirigliano:2016nyn . However, these
operators are flavour non-diagonal and we will not use such flavour non-
diagonal interactions further in our discussion. Also, constraints can be put
considering the contribution of dipole operators to the anomalous magnetic
moment of light-quarks. But for the light quarks, $(g-2)$ is hard to extract
in a model independent way, is therefore subjected to large uncertainties
Giudice:2012ms .

On introducing these higher dimensional operators, one may reasonably expect a
violation of unitarity in $2\to 2$ scattering processes
($V_{L}V_{L}\rightarrow V_{L}V_{L}\leavevmode\nobreak\
{\rm{and}}\leavevmode\nobreak\ f\bar{f}\to VV$, where $V=W,Z$), depending on
the values of the Wilson coefficients at high energies. Bounds on these
coefficients may be obtained demanding that there be no such violation of
unitarity below the scale of the effective theory. Unitarity constraints
stemming from the Higgs scattering processes on our dimension-6 operators of
interest, assuming an incoming parton inside a proton carries typically 1 TeV
of energy at the LHC, are summarised in Table 3.

Constraints | Unitarity | Higgs data ($95\%$ C.L.)
---|---|---
$\frac{C_{uW}}{\Lambda^{2}}$ | $[-7.0,7.0]$ | $[-0.417,0.417]$
$\frac{C_{uB}}{\Lambda^{2}}$ | $[-27.75,27.75]$ | $[-0.745,0.745]$
$\frac{C_{dW}}{\Lambda^{2}}$ | $[-7.0,7.0]$ | $[-0.483,0.483]$
$\frac{C_{dB}}{\Lambda^{2}}$ | $[-27.75,27.75]$ | $[-0.909,0.909]$
$\frac{C_{HB}}{\Lambda^{2}}$ | $[-150.75,150.75]$ | $[-0.062,0.342]$
$\frac{C_{HW}}{\Lambda^{2}}$ | $[-24.75,24.75]$ | $[-0.018,0.201]$
Table 3: Limits for the fermionic and bosonic operator coefficients in the
Warsaw basis. The bounds are given in $\rm TeV^{-2}$ units.

In the subsequent study, values of the effective coupling strengths consistent
with the electroweak data and the Higgs measurements have been used. Within
such constraints, we concentrate on the operators
$\\{\mathcal{O}_{uB},\mathcal{O}_{dB},\mathcal{O}_{uW},\mathcal{O}_{dW}\\}$
and $\\{\mathcal{O}_{HW},\mathcal{O}_{HB}\\}$ and examine their role in
modifying the $h\gamma$ production rates at the LHC.

###  2.2 $h\gamma$ production in the SMEFT

We implemented the effective Lagrangian in FeynRules Alloul:2013bka  to create
the UFO model file for the event generator Madgraph5_aMC@NLO Alwall:2014hca
which is used to generate all signal and background events. These events are
generated at the leading order (LO) followed by Pythia (v8) Sjostrand:2006za
for showering and hadronization. We use NNPDFNLO parton distribution functions
pdf  for event generation by setting factorisation and renormalisation scales
equal to the Higgs mass ($\mu_{R}=\mu_{F}=m_{H}$). The matching parameter,
QCUT is appropriately determined for different processes as discussed in Ref
MLMMerging . To incorporate detector effects, events are passed through
Delphes-v3.4.1 deFavereau:2013fsa . Jets are reconstructed by employing the
Cambridge-Achen(CA) Dokshitzer:1997in  algorithm with radius parameter
$R=0.8$, as implemented in the Fastjet-v3.3.2 Cacciari:2011ma .

In Fig. 1, we present a representative set of Feynman diagrams for $h\gamma$
production arising from the dimension-6 operator interactions and the SM
$h\gamma$ production for illustration. In this work, we will investigate the
effect of the contact interactions $qqh\gamma$ and $hV\gamma$ couplings in
producing the Higgs in association with a photon at the LHC and do not aim to
probe the deviations of the Higgs couplings to first and second generation SM
quarks via the higher dimensional operators (corrections to Yukawa
interactions) as shown in Fig. 1(k).

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

Figure 1:  Representative Feynman diagrams showing production of a Higgs boson
in association with a photon. Diagrams (a)-(i) correspond to the SM process,
while (j) involves the contact interaction denoted with a blob, between the
Higgs, gauge boson and fermions (k),(l) processes involve the anomalous
couplings of Higgs with gauge bosons and fermions respectively.

The variation of Higgs + photon production cross-section for different values
of effective couplings has been presented in Fig. 2 (left panel). One can see
that the effect of fermionic operators is more pronounced than the bosonic
operators. The enhancement of cross-section in case of dimension-6 operators
involving fermions can be accounted by the absence of an extra propagator
which is present in the SM-like $h\gamma$ production involving the bosonic-
type operators. Most of the high-energy contribution due to these fermionic-
type operators have an amplitude that is distinct from the SM contribution
because of a quadratic growth of energy.

Cross-sections for $pp\to h\gamma$ as function of $\sqrt{\hat{s}}$ have been
presented for different dimension-6 operators of our interest along with the
SM in Fig. 2 (right panel). A value of
${C_{i}\over\Lambda^{2}}=0.3\leavevmode\nobreak\ \rm TeV^{-2}$ is used in this
plot. The $h\gamma$ production rate, in presence of dimension-6 operators,
falls off at a rate slower than that in the SM. The slower rate of fall of the
cross-section with partonic centre-of-mass energy, can be accounted by
presence of (i) a term proportional to $C\over\Lambda^{2}$ and independent of
energy of scattering and (ii) a term proportional to
${C^{2}}\over{\Lambda^{4}}$ having a quadratic dependence on the energy of
collision, in the expression of the total cross-section. However, it must be
borne in mind that the relative rate of fall of the cross-section also depends
on the value of $C_{i}$. The aforementioned observation clearly tells that
these operators can be constrained from the considerations of unitarity.

(a) (b)

Figure 2: (a) The cross-section variation for $pp\to h\gamma$ generated by
different dimension-6 operators at the parton level (colored solid lines at 14
TeV and dashed lines at 13 TeV). (b) Variation of cross-sections with center
of mass energies $\sqrt{\hat{s}}$ for the various Higgs anomalous couplings.
Cross-section for SM background from $q\bar{q}$ annihilation shown as black
dashed line.

In this study, we consider the regime where the Higgs and the photon are
highly boosted. The total hadronic decay products of Higgs can be contained in
a large radius jet (fat-jet). We demand at least one Higgs jet in the signal
sample by reconstructing its invariant mass using the jet substructure
technique, along with a hard photon recoiling against this fat-jet in the
transverse plane. The radius of the fat-jet can be approximated by the
relation $R\sim\frac{2m_{h}}{p_{T_{h}}}$. The following SM processes can mimic
the $h\gamma$ signal.

  * •

Continuum $\gamma j$ turns out to be the most dominant background. This
background can be efficiently suppressed in the case of $h\gamma$ by applying
a Higgs tagger to the jet. The efficiency of b-tagging and the
misidentification rates of a $c$-jet or a light jet posing as a b-tagged jet
are taken to be dependent on the transverse momentum of jets as given in Ref.
CMS:2012feb .

  * •

The $Z/W\gamma+jets$ process has almost similar topology to the signal, but is
subdominant due to its low cross-section.

  * •

The production of $t\bar{t}\gamma$ followed by hadronic decays of the top
quarks will also contribute to the background. However, demanding a high
$p_{T}$ photon along with Higgs tagging would suppress this background.
Similarly, single top production $tj\gamma,tb\gamma$ also contribute in the
background.

  * •

The $pp\to h\gamma$ associated production in the SM has a nominal rate either
due to the fact that Higgs boson has negligibly small couplings with the
partons or the process is driven by higher order one loop contribution. The
gluon-initiated contribution $gg\to h\gamma$ is vanishing due to Furry’s
theorem Furry:1939 ; PeskinSchroeder .

  * •

Some other SM processes, such as QCD, $t\bar{t}$+jets, $W$+jets, $Z$+jets,
despite having no direct sources of hard photons, may also contribute to the
background owing to their large production cross-sections coupled with
mistagging of jets or leptons leading to fake photons. However, the cumulative
effect of hard transverse momentum of photon as well as the boosted sytem of
the final state requirement and Higgs tagging renders these contribution
negligible.

All the aforementioned background processes have been estimated at the LO and
multiplied by appropriate K-factors to obtain the NLO cross-sections following
Ref. CMS:2018cno . The cross-sections as used in present analysis are listed
in Table 4.

Process |  $\sigma$ (fb) at 13 TeV |  $\sigma$ (fb) at 14 TeV
---|---|---
$J\gamma$ Alwall:2014hca  | $b-tagged$ | $47.20\times 10^{3}$ | $54.19\times 10^{3}$
$untagged$ | $188.9\times 10^{3}$ | $212.92\times 10^{3}$
$V\gamma+jets$ Krause:2017nxq ; Lombardi:2020wju  | $Z\gamma+jets$ | $140.80$ | $158.56$
$W\gamma+jets$ | $541.89$ | $933.04$
$Top\gamma+jets$ Pagani:2021iwa  | $t\bar{t}\gamma+jets$ | $6.69$ | $8.22$
$tj\gamma+jets$ | $5.418$ | $6.978$
Electroweak Gabrielli:2016mdd  | $H\gamma$ | $0.01377$ | $0.01617$
Table 4: Cross-sections for the background processes considered in this
analysis at the 13 TeV and 14 TeV LHC. These processes except the electroweak
$h\gamma$ production in the SM have been estimated at the LO and multiplied by
appropriate K-factors to obtain the NLO cross-sections following the procedure
in Ref CMS:2018cno .

The CMS CMS:2018cno  and the ATLAS ATLAS:2018sxj  Collaborations have looked
the $h\gamma$ final state in search of a resonance decaying into Higgs and
photon. No significant excess over the SM expectation has been reported. The
CMS has collected 187 events with $\int\mathcal{L}dt=36\leavevmode\nobreak\
\rm fb^{-1}$ in the signal region $720\leavevmode\nobreak\ \rm
GeV<m_{J\gamma}<1.82\leavevmode\nobreak\ \rm TeV$. This corresponds to a
cross-section of 5.2 fb including acceptance and efficiency of selction cuts.
Our simulation of SM events yield 164 events with a scale uncertainty of 5%.
We have rescaled the LO cross-section 4.56 fb by the corresponding QCD-NLO
factor. With this SM background expectation, we compare Fig. 2 (left panel) of
Ref. CMS:2018cno  as shown in Fig. 3.

Figure 3:  The measurement of $J\gamma$ invariant mass distribution at 13 TeV
by the CMS experiment CMS:2018cno  compared to the SM prediction at the NLO
level. Also shown are the predictions corresponding to Wilson coefficients
$\frac{C_{uB}}{\Lambda^{2}}$ and $\frac{C_{HB}}{\Lambda^{2}}$ with values
allowed by the data.

We now consider the process involving contributions from dimension-6 operators
and perform a goodness-of-fit test with the six SMEFT coefficients on the
$J\gamma$ invariant mass distribution. We construct the $\chi^{2}$ function
for the aforementioned EFT couplings as

$\chi^{2}(C)=\sum_{i}^{bins}\frac{(N_{i}(C)^{theo}-N_{i}^{exp})^{2}}{(\sigma_{i}^{exp})^{2}}$
(6)

where $N_{i}^{exp}$ denotes the observed number of events, $N_{i}(C)^{theo}$
estimates the expected number of events, assumed to be different from the SM
due to the presence of non-zero EFT couplings. $\sigma_{i}^{exp}$ quantifies
the error including both statistical and systematic uncertainties added in
quadrature. We have assumed an overall systematic uncertainty which is $3\%$
of statistical uncertainty.

$\sigma_{i}^{exp}=\sqrt{N_{i}^{exp}+\Delta_{sys}N_{i}^{exp}}$

We minimise the $\chi^{2}$-function to derive the limits of the parameter
space allowed by the experimental data at $2\sigma$ level.

Wilson Coefficient | 95 % C.L. (TeV-2)
---|---
$\frac{C_{uW}}{\Lambda^{2}}$ | $[-0.39,0.39]$
$\frac{C_{uB}}{\Lambda^{2}}$ | $[-0.20,0.20]$
$\frac{C_{dW}}{\Lambda^{2}}$ | $[-0.37,0.37]$
$\frac{C_{dB}}{\Lambda^{2}}$ | $[-0.28,0.28]$
$\frac{C_{HW}}{\Lambda^{2}}$ | $[-0.85,0.79]$
$\frac{C_{HB}}{\Lambda^{2}}$ | $[-1.75,1.54]$
Table 5: 95 % C.L. bounds for the fermionic and bosonic operator coefficients
in the Warsaw basis. The bounds are given in $\rm TeV^{-2}$ units.

It is noteworthy that the limits on the dipole operators obtained here more
restrictive than our previous study on their effects on VBF Higgs production
Biswas:2021qaf . We also note that the Wilson coefficients are more
constrained from the higher energy bins. We will use these allowed values of
the effective couplings to investigate the prospect of finding such a signal
at 14 TeV run of the LHC with $\mathcal{L}=3000$ fb-1 integrated luminosity in
the following sections.

##  3 Collider Simulation

###  3.1 Cut-based Analysis

We generated samples for the signal that lead to the $pp\to
h\leavevmode\nobreak\ (\to\rm fat$-$\rm jet)\gamma$ final states where non-
zero EFT couplings in dimension-6 are present along with the SM. The cross-
sections are evaluated upto $\mathcal{O}\left(\frac{1}{\Lambda^{4}}\right)$ at
14 TeV $pp$ center-of-mass energy. We also generated the SM background events
for the processes listed in Table 4. We choose the following basic selection
criteria to select events in the signal and background for further analysis:

  * •

Photons are required to have $p_{T_{\gamma}}>150\leavevmode\nobreak\ \rm GeV$
and $|\eta_{\gamma}|<2.47$ excluding $1.37<|\eta_{\gamma}|<1.52$.

  * •

We veto the events if any lepton with $p_{T}>10\leavevmode\nobreak\ \rm GeV$
lies in the central psuedorapidity range $|\eta|<2.4$.

  * •

We only select those events with at least one fat-jet of radius 0.8 and with
$p_{T}>70$ GeV.

  * •

To suppress any theoretical uncertainties and veto additional jets in the
event sample, the fat-jet should satisfy the criteria of
$|\Delta\phi(J,\gamma)|>0.4$.

  * •

To obtain statistically significant events, we select events with jet mass to
be $m_{J}>50\leavevmode\nobreak\ \rm GeV$.

We now proceed to discuss the main kinematic characteristics of signal and the
SM background. To accentuate the distinguishability, we choose a few benchmark
points as listed in Table 6. Operators $\mathcal{O}_{fW}$ and
$\mathcal{O}_{fB}$ differ in the cross-section by a factor of $\rm
tan^{2}\theta_{W}$. BP3 closely mimics the SM interaction. Benchmark points
$\\{BP1,BP2,BP3\\}$ have been chosen (alongside the SM) with only one non-zero
EFT couplings at a time. The values of such couplings are allowed by the
available Higgs data from the LHC itself. We also choose benchmark couplings
with two couplings non-zero at a time so as to emulate a scenario where a
significant deviation from the SM is observed. These values are chosen based
on the fact that certain linear combinations of couplings are more sensitive
to the new physics interactions as expressed in Table 1.

Signal |  $\frac{C_{i}}{\Lambda^{2}}$ (TeV-2)
---|---
BP1 | $\frac{C_{uB}}{\Lambda^{2}}=0.1$
BP2 | $\frac{C_{dW}}{\Lambda^{2}}=0.2$
BP3 | $\frac{C_{HW}}{\Lambda^{2}}=0.2$
BP4 |  $\frac{C_{uW}}{\Lambda^{2}}=0.08$,$\frac{C_{uB}}{\Lambda^{2}}=0.08$
BP5 |  $\frac{C_{dW}}{\Lambda^{2}}=0.15$,$\frac{C_{dB}}{\Lambda^{2}}=-0.15$
BP6 |  $\frac{C_{HB}}{\Lambda^{2}}=0.3$,$\frac{C_{HW}}{\Lambda^{2}}=-0.3$
Table 6: Listing of benchmark points used in our study

Fig. 4 shows the normalised distributions of some of the kinematic variables
for the three BPs and the SM background processes. The variables $p_{T_{J}}$
and $E_{\gamma}$ (Fig. 4(a) and (c)) are quite efficient in distinguishing the
new interactions from most of the SM backgrounds. The availability of larger
parton center-of-mass energy in these interactions pushes the energy of photon
($E_{\gamma}$) and the transverse momentum of fat-jet ($p_{T_{J}}$) to higher
values. The photons in the signal events exhibit a hard $p_{T}$ and mainly
appear in the central region of the detector. Values of $E_{\gamma}$ and
$p_{T_{J}}$ extend to 1.8 TeV and 1.1 TeV. We see that significant enhancement
from the SM backgrounds can be seen due to the contribution of both the type
of dimension-6 operators. In presence, of dipole operators the cross-section
grows faster at higher energies compared to the bosonic operators or in the
case of SM. This can be attributed to the absence of an extra s-channel vector
boson propagator in the amplitudes involving dipole operators.

|
---|---
(a) |  (b)
|
(c) |  (d)
(a)  Figure 4:  Normalized distribution for signal as well as backgrounds of
(a) $p_{T_{J}}$, (b) $m_{J}$, (c) $E_{\gamma}$ and (d) $\Delta\eta_{J\gamma}$.
For the EFT predictions, we have chosen BP1 =
$\frac{C_{uB}}{\Lambda^{2}}=0.1\leavevmode\nobreak\ \rm{TeV^{-2}}$ (red), BP2
= $\frac{C_{dW}}{\Lambda^{2}}=0.2\leavevmode\nobreak\ \rm{TeV^{-2}}$ (blue)
and BP3 = $\frac{C_{HW}}{\Lambda^{2}}=0.2\leavevmode\nobreak\ \rm{TeV^{-2}}$
(green).

Instead of performing a standard resolved analysis, where two separate
b-tagged jets are demanded, we demand a fat-jet with a considerably large cone
radius. We employ the so-called BDRS approach Butterworth:2008iy  with minor
modifications to maximize the sensitivity. This technique helps in
discriminating boosted electroweak-scale resonances from large QCD
backgrounds. This approach recombines jets using the CA algorithm with a
significantly large cone radius to contain all the decay products of the
resonance (Higgs boson). The candidate jet j is broken into two subjets by
unwinding the last stage of jet clustering. The two subjets $j_{1}$ and
$j_{2}$ are labelled such that $m_{j_{1}}>m_{j_{2}}$. And a mass drop
condition is applied such that $m_{j_{1}}<\mu m_{J}$ with $\mu=0.66$ ($m_{J}$
being the mass of the fat-jet) and further a symmetry criterion
$\frac{min(p_{T,j}^{2},p_{T,j}^{2})}{m_{j}^{2}}\Delta R>0.09$ is applied
between the two sub-jets. If the condition is not satisfied, the softer
subjet, $j_{2}$ is discarded and only the subjet $j_{1}$ forms the part of the
remaining analysis and the process is reiterated. The algorithm is stopped
with the final jet $j$ obtained if the mass drop condition is obtained. This
criteria is fairly efficient in eliminating QCD jets, but still can vitiate
the jet reconstruction at LHC due the underlying events, which is also a
problem at the enhanced energies and luminosities of LHC. To remove the
remaining rare QCD events where a hard gluon is emitted and the underlying
events, we need to filter the Higgs neighbourhood. The constituents of $j_{1}$
and $j_{2}$ are again recombined using the CA algorithm with a cone radius
$R_{filt}=min(0.2,R_{b\bar{b}}/2)$111We find that in our study, choosing
$R_{filt}=min(0.2,R_{b\bar{b}}/2)$ facilitates in removing the backgrounds..
Only the hardest three filtered subjets are retained to reconstruct the
resonance. We show the distribution of this filtered jet mass in Fig. 4(b) for
the different signals and the dominant backgrounds. It is evident from these
distributions that the peak around $115-140\leavevmode\nobreak\ \rm GeV$
reflect the Higgs peak for the signal processes whereas for most of the
backgrounds, the peaks are below 50 GeV reflecting that the fat-jet mimicing
from either single prong hard QCD jet or a peak around 90 GeV reflects $Z$
boson or peak about 165-185 GeV from a top.

The most dominant backgrounds for this process produce events with larger
angular separations, $\Delta\eta$ between the photon and the fat-jet. For the
new physics signal, the separation of the final states is isotropic leading to
narrow separations in $\eta$ as shown in Fig. 4(d).

Having discussed the kinematic observables, we proceed to find a suitable cut-
flow, that will significantly discriminate between signal and background.

| Cross-sections at $\sqrt{s}=14$ TeV (fb)
---|---
Cut flow | EFT driven signals
BPs | $BP1$ | $BP2$ | $BP3$ | $BP4$ | $BP5$ | $BP6$
Preselection | $14.42$ | $10.956$ | $14.5268$ | $22.24$ | $31.19$ | $21.195$
Atleast 1 fat-jet with 2 B-meson tracks with $p_{T}>200\leavevmode\nobreak\ \rm GeV$ | $6.6475$ | $5.051$ | $6.1131$ | $10.2525$ | $14.3786$ | $8.987$
Atleast 1 isolated photon and lepton veto | $6.007$ | $4.5728$ | $5.4312$ | $9.2622$ | $13.0374$ | $7.98$
Photon $p_{T}>200$ GeV | $5.7103$ | $4.28293$ | $4.2829$ | $8.8007$ | $12.1695$ | $5.811$
Atleast 1 fat-jet with two B-meson tracks with $p_{T}>250$ GeV | $5.3027$ | $3.8976$ | $2.9003$ | $8.1638$ | $11.0354$ | $4.153$
2 mass drop subjets and $\geq 2$ filtered subjets | $2.357$ | $1.7931$ | $1.3862$ | $3.6205$ | $5.0887$ | $1.864$
2 b-tagged subjets | $1.9529$ | $0.7422$ | $0.5744$ | $1.4843$ | $2.1114$ | $1.599$
$110.0<m_{H}<140.0$ GeV | $0.9663$ | $0.7348$ | $0.5649$ | $1.4694$ | $2.0904$ | $1.543$
$\Delta R(\gamma,b_{i})>0.4,\not{E_{T}}<30\rm GeV,|\eta_{h}|<2.5$ | $0.3771$ | $0.2978$ | $0.2698$ | $0.5840$ | $0.8389$ | $0.784$
$\Delta R(\gamma,h)>2.4$ | $0.3695$ | $0.2895$ | $0.2649$ | $0.5665$ | $0.8175$ | $0.770$
$m_{b\bar{b}\gamma}>600\leavevmode\nobreak\ \rm GeV$ | $0.3654$ | $0.2822$ | $0.2548$ | $0.5545$ | $0.7981$ | $0.739$
$p_{T}^{\gamma}>300\leavevmode\nobreak\ \rm GeV$ | $0.3509$ | $0.2683$ | $0.2496$ | $0.5313$ | $0.7601$ | $0.659$

Table 7: The cut-flow table for Higgs photon production from the dimension-6
operators.

As evident from Table 7, cut efficiencies vary depending on the Wilson
coefficients associated with the operators with different Lorentz structures
involved.

| Cross-sections at $\sqrt{s}=14$ TeV (fb)
---|---
Cut flow | SM Backgrounds
| jets + $\gamma$ | $W/Z+\gamma$ | $t\bar{t}\gamma+tj\gamma$ | $SM(h+\gamma)$
Preselection | $297407.2$ | $1148.86$ | $18.9696$ | $0.4731$
Atleast 1 fat-jet with 2 B-meson tracks with $p_{T}>200\leavevmode\nobreak\ \rm GeV$ | $9932.125$ | $142.06$ | $16.483$ | $0.3763$
Atleast 1 isolated photon and lepton veto | $7836.135$ | $119.946$ | $13.643$ | $0.3224$
Photon $p_{T}>200$ GeV | $3550.661$ | $66.034$ | $8.005$ | $0.1857$
atleast 1 fat-jet with two B-meson tracks with $p_{T}>250$ GeV | $2068.434$ | $48.855$ | $6.286$ | $0.0987$
2 mass drop subjets and $\geq 2$ filtered subjets | $672.283$ | $23.737$ | $4.485$ | $0.0508$
2 b-tagged subjets | $115.134$ | $14.024$ | $0.824$ | $0.0263$
$110.0<m_{H}<140.0$ GeV | $101.294$ | $7.541$ | $0.734$ | $0.0237$
$\Delta R(\gamma,b_{i})>0.4,\not{E_{T}}<30\rm GeV,|\eta_{h}|<2.5$ | $38.532$ | $5.159$ | $0.279$ | $0.0115$
$\Delta R(\gamma,h)>2.4$ | $38.066$ | $5.114$ | $0.276$ | $0.0108$
$m_{b\bar{b}\gamma}>600\leavevmode\nobreak\ \rm GeV$ | $20.063$ | $3.289$ | $0.179$ | $0.00892$
$p_{T}^{\gamma}>300\leavevmode\nobreak\ \rm GeV$ | $13.456$ | $2.507$ | $0.147$ | $0.0060$

Table 8: The cut-flow table for jets + $\gamma$ production at
$\sqrt{14}\leavevmode\nobreak\ \rm TeV$.

It is worth pointing out here that with these choice of cuts, the SM
background is reduced by 4 orders of magnitude.

A comment is in order here. An immense progress has been made in identifying
the Higgs boson itself from its hadronic decay products despite the large QCD
background at the LHC when the Higgs is being produced with sufficiently high
$p_{T}$ or is in highly boosted regime Bhattacherjee:2012bu . Using the
properties of jet substructures also greatly helps in identifying signals of
Higgs boson via various final states such as $b\bar{b}b\bar{b}$
FerreiradeLima:2014qkf , $b\bar{b}\gamma\gamma$ Yao:2013ika  and
$b\bar{b}WW^{*}$ Shao:2013bz . Fig. 5 shows the Higgs tagging efficiency, as a
function of transverse momentum of Higgs boson for two cases. The so called
Higgs tagger algorithm described above is implemented in our analysis and its
efficiency i.e the ratio of the number actually tagged as Higgs jets to the
original number of $h$ produced is computed. In case one, the double b-tagged
case, both the harder sub-jets in the filtered fat-jet are required to be
b-tagged. In the second case, two-prong identifier is applied by requiring the
fat-jet to have N-subjettiness, $\tau_{21}<0.5$ (see Section 3.3). This is
also an effective way to identify such signal events with two-prong decays of
Higgs DiSipio:2019cfm . The Higgs tagging method by using 2-prong finder
tagger is more efficient in finding the Higgs events than the b-tagging based
technique. For instance, the method involving b-tagging, causes the event
number to significantly fall to as low as $30\%$ at
$p_{T}=600\leavevmode\nobreak\ \rm GeV$ of Higgs. The latter can, however, be
highly advantageous in removing events from the large QCD background untagged
jets.

---
(a)  Figure 5: Higgs tagging efficiency as a function of large radius jet
$p_{T}$.

It is to be noted that this figure is process independent and the tagging
process can also serve to give information about the other heavy SM particles,
distinct by their masses and the new particles that any UV theory predicts.

|
---|---
(a) |  (b)
|
(c) |  (d)
(a)  Figure 6: Two dimensional histograms showing the correlation of the
invariant mass of jet and photon, $m_{J\gamma}$ with the scattering angle of
photon in Higgs-photon rest frame, $cos\theta^{*}$. The $z$-axis indicates the
normalised frequency of events, in arbitrary units. The SM contributions in
the absence of dimension-6 interactions have been subtracted in (b), (c) and
(d). ${|C_{i}|\over{\Lambda^{2}}}=0.08\leavevmode\nobreak\ \rm TeV^{-2}$,
${|C_{i}|\over{\Lambda^{2}}}=0.3\leavevmode\nobreak\ \rm TeV^{-2}$ and
${|C_{i}|\over{\Lambda^{2}}}=0.15\leavevmode\nobreak\ \rm TeV^{-2}$ have been
assumed in (b), (c) and (d) respectively.333The new physics contributions
shown in this figure include two operator-coefficients taken non-zero at a
time. Fig. 3 (b) includes two Wilson coefficients with equal magnitude and
same sign for the couplings ($C_{uB},C_{uW}$) whereas (c) and (d) include the
two Wilson coefficients with equal magnitude but opposite signs of couplings
($C_{dB},C_{dW}$) and ($C_{HB},C_{HW}$) respectively.

As the process of our consideration is a hard $2\to 2$ process, its amplitude
can be completely specified by two variables, the energy variable and an
angle. We demonstrate the correlation between two such variables, the
invariant mass of jet and photon, $m_{J\gamma}$ and center-of-mass opening
angle, $cos\leavevmode\nobreak\ \theta^{*}$ in Fig. 3, where it is found that
the populated regions in the phase-space display a shift when the new physics
effects due to higher dimensional operators are included. The following
features emerge from Fig. 3:

  * •

For the background, $m_{J\gamma}$ distribution is smoothly and rapidly falling
while new physics contributions have rather slowly falling $m_{J\gamma}$
distributions. For the angular dependence analysis, it is to be noted that the
main contributions to photon+jet final state background come from a quark
exchange leading to production of photon and jet, t-channel gluon exchange
leading to QCD jet production along with a bremsrahlung radiation of a photon.
In the center-of-mass frame, the angular distribution resulting from such
background processes show a steeper increase as $|\rm cos\theta^{*}|\to 1$
than the new physics signal where either there is direct production of photon
and jet from quarks or dominated by an s-channel propagator (red colour
indicates larger larger number of events). A cut on the invariant mass of jet
and photon of 1 TeV will characteristically feature the new phase space
regions. Complementarily a cut $|\rm cos\leavevmode\nobreak\ \theta^{*}|\leq
0.8$ for instance would remove a large sample of background events that peak
in forward and backward directions.

  * •

This implies that the new vertices tend to push the Higgs-like jet and the
photon towards higher energy and the opening angle of the photon with respect
to the beam line tends to be larger compared to the SM background. Such a
correlation is most prominent for $C_{dB}$-$C_{dW}$ which have a very flat
angular distribution as we look into regions with high $m_{J\gamma}$. The
effects of other two operators, too, extend to $m_{J\gamma}$ values as high as
4 TeV.

###  3.2 Projected sensitivities on the EFT couplings

The statistical significance $(\mathcal{S})$ cowan  of the signal $(s)$ over
the total background $(b)$ has been calculated using

$\mathcal{S}=\sqrt{2\times\left[(s+b)ln(1+\frac{s}{b})-s\right]}$ (7)

We now focus on the sensitivity reach of the different benchmark points. We
show the significance of the signal for all the benchmark points in the fat-
jet$+\gamma$ channel in Table 9 with the possible choice of integrated
luminosity (${\mathcal{L}}=300,1000$ and $3000$ fb-1).

| Statistical significance | Required Luminosity
---|---|---
|  (${\mathcal{S}}$) |  (in $fb^{-1}$)
Signal |  ${\mathcal{L}}=300$ fb-1 |  ${\mathcal{L}}=1000$ fb-1 |  ${\mathcal{L}}=3000$ fb-1 | ${\mathcal{S}}=3{\sigma}$ | ${\mathcal{S}}=5{\sigma}$
BP1 | 0.456 | 1.517 | 4.543 | 1308.39 | 3634.43
BP2 | 0.364 | 1.21 | 3.62 | 2051.25 | 5697.92
BP3 | 0.324 | 1.078 | 2.96 | 3079.33 | 8553.71
BP4 | 0.647 | 2.156 | 6.447 | 649.62 | 1804.49
BP5 | 0.442 | 1.475 | 4.434 | 1373.72 | 3815.89
BP6 | 0.1935 | 0.645 | 1.933 | 7226.67 | 16074.1
Table 9: Statistical significance of the signal for different benchmark points
in the Higgs $+\gamma$ analysis at 14 TeV LHC. The significance is estimated
for three values of integrated luminosity (${\mathcal{L}}=300,1000$ and $3000$
fb-1). We also estimate the required integrated luminosity to achieve a
$3\sigma$ and $5\sigma$ excess for each benchmark point at LHC with
$\sqrt{s}=14$ TeV.

We find that BP4 is the most promising of all and should be observable in near
future at the LHC with luminosity as low as $700\leavevmode\nobreak\ \rm
fb^{-1}$. BP3 and BP6 are the scenarios involving bosonic operators which
would be difficult to observe in this final state. In fact, an integrated
luminosity of above $3000\leavevmode\nobreak\ \rm fb^{-1}$ would be required
to observe a notable excess for the $hV\gamma$ coupling given by the above
benchmark points. This is however understandable as the corresponding process
is SM-like process, suppressed by a photon or $Z$ propagator. The rest of the
benchmark points lead to $3\sigma$ and $5\sigma$ excess over the SM
backgrounds with relatively nominal to higher integrated luminosities as shown
in the last two columns of Table 9.

Using the distribution of $m_{J\gamma}$, we perform a binned analysis to
project the reach of LHC in probing the higher dimensional operators. In this
approach, we consider the range of $m_{J\gamma}$ divided into bins of 400 GeV
from $800$-$4000$ GeV. We study the required luminosity in each bin for
achieving $3\sigma$ significance as a function of the Wilson coefficients.
This is shown for the operators $\mathcal{O}_{uB}$ and $\mathcal{O}_{HW}$ in
Fig. 7. Significance increases monotonically for both the operators with
increasing bins of invariant mass of jet and photon. This is due to the
quadratic growth of energy of the amplitude-square compared to that of the SM.
With increasing energies, the photon and the Higgs get highly boosted allowing
for a regime where background events are depleted. However, in such a region,
the rate of cross-section of the EFT driven signal also decreases. It is the
higher invariant mass bins that can give best signal to background
sensitivity.

|
---|---
(a) |  (b)
(a)  Figure 7: Required integrated luminosity at 14 TeV for achieving a
3$\sigma$ significance in bins of $m_{J\gamma}$ as a function of Wilson
coefficients of operators (a)$\frac{C_{uB}}{\Lambda^{2}}$ and (b)
$\frac{C_{HW}}{\Lambda^{2}}$. The different colored bands signify bins of
$m_{J\gamma}$ each of width 500 GeV.

Results for sensitivities of different benchmark points have been presented in
Table 9. We now present the results for the sensitivity projections in four
cases assuming two non-zero Wilson coefficients at a time, Quark-specific
scenario ($C_{uB}$-$C_{uW}$ and $C_{dB}$-$C_{dW}$ planes), bosonic-specific
scenario in $C_{HW}$-$C_{HB}$ plane and quark-bosonic interaction in
$C_{uB}$-$C_{HB}$ plane (Fig. 8).

|
---|---
(a) |  (b)
|
(c) |  (d)
(a)  Figure 8: Projections for a 3$\sigma$ signal significance in blue shaded
regions in the parameter space in in the $C_{uB}-C_{uW}$ (top left) ,
$C_{dB}-C_{dW}$ (top right), $C_{HB}-C_{uB}$ (bottom left), and
$C_{HB}-C_{HW}$ (bottom right) planes with all other EFT coefficients assumed
to be zero with 3000 $fb^{-1}$ data at 14 TeV run of the LHC. The 2$\sigma$
level approximating the 95% confidence level exclusion bounds are shown in
magenta shaded regions.

In the limit of vanishing fermion masses, operators $\mathcal{O}_{uB}$ and
$\mathcal{O}_{uW}$ do not interfere with the SM amplitude, however they
interfere constructively among themselves which is clearly evident in Fig.
8(a). The blue shaded region marks the bounds that can be probed at $3\sigma$
significance level. Similar interference effect prevails in the case of non-
zero $C_{dB}$ and $C_{dW}$ (shown in Fig. 8(b)). These couplings are only
slightly correlated with each other due to a smaller interference effect in
the $h\gamma$ production. A more optimistic limit on the Wilson coefficients
of the operators with $u$-type quarks are obtained than that due to the
$d$-type fermionic operators. Operator $\mathcal{O}_{uB}$ neither interferes
with the SM amplitude and nor with the bosonic-type operator and thus, the
leading BSM effect arises at $\mathcal{O}\sim\frac{1}{\Lambda^{4}}$. This can
be seen from the $C_{uB}\to-C_{uB}$ symmetry in the Fig. 8(c) whereas the
$C_{HB}$ interferes with the SM. The yellow shaded region in Fig. 8(d) shows
the $2\sigma$ significance region in the $C_{HW}$-$C_{HB}$ plane. In this
case, there are two couplings contributory. One, the coupling proportional to
$g_{h\gamma\gamma}$ whose contribution to the Higgs-diphoton interaction
depends on $s_{W}^{2}C_{HW}+c_{W}^{2}C_{HB}$ combination. The other coupling
due to the presence of Z propagator $g_{hZ\gamma}$ receives contribution via
the combination $s_{W}c_{W}(C_{HW}-C_{HB})$, thus indicating a region in the
same place when the latter decreases the rate for positive $C_{HB}$. In all,
their total contribution requires for larger values of the Wilson coefficients
for probing the $h\gamma$ production at $3\sigma$ significance.

###  3.3 Multivariate Analysis

In the previous sub-section, we presented the sensitivity of the new
interactions at the end of a cut-based analysis. We now explore the
possibility of any improvement in our analysis using a multivariate analysis
(MVA). We use the adaptive Boosted Decision Tree (BDT) algorithm in the TMVA
framework Hocker:2007ht . The multivariate analysis complements the cut-based
analysis in the following way. A cut-based analysis can select out the phase
space by applying a series of grids in succession (bounded by either both
sided or one sided boundaries). However, many of the events may be
characterised by kinematic variables that may be correlated in a complicated
way. The decision trees proceed by recursively partitioning the phase space
into hyperspace regions with edges aligned along the axes of the phase space.
Each of these regions in the hyperspace is then classified for signal and
background segregation. A non-linear boundary in the multi-dimensional phase
space can, thus, help identifying the signal region more accurately.

We have used slightly looser cuts for the MVA compared to those used in the
previous section (in Table 8). In addition to the preselection cuts, a minimum
transverse momentum of $175$ GeV requirement has been applied on the leading
fat-jet and the photon. The reconstructed fat-jet mass is required to be at
least 60 GeV. We ensured that the cuts are very effective in reducing the
large background but not the signal. This is because the MVA does not perform
well if we use events just with the baseline cuts as the signal is very small
compared to the large background. Also, the strong cuts of Table 8 do not give
an improved sensitivity while performing MVA. Thus, the cuts in MVA are chosen
in such a way that they are neither the same nor too relaxed cuts compared to
the cut-based analysis.

We identify the various observables with maximal discerning potency which are
chosen to be the input variables to BDT. The normalised distributions of these
input variables are shown in Fig. 9a.

  * •

$p_{T_{\gamma}}$, the transverse momentum of photon.

  * •

$m_{J}$, the mass of leading jet.

  * •

The fat-jet appearing from the $h\to b\bar{b}$ potentially retains some
information of its two prong structure. This is revealed by the jet-shape
variable, N-subjettiness Thaler:2010tr ; Thaler:2011gf , defined as:

$\displaystyle\tau_{N}^{(\beta)}=\frac{1}{\mathcal{N}_{0}}\sum\limits_{i}p_{i,T}\,\min\left\\{\Delta
R_{i1}^{\beta},\Delta R_{i2}^{\beta},\cdots,\Delta R_{iN}^{\beta}\right\\}$
(8)

Here, $N$ is the axis of the subjet assumed within the fat-jet, $i$ runs over
the constituents of the jet, $p_{i,T}$ are their respective transverse
momenta, $\Delta R_{ij}=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$ is the
distance in $\eta-\phi$ plane, between candidate subjet $j$ and constituent
particle $i$ and $\mathcal{N}_{0}=\sum_{i}p_{i,T}R_{0}$ is the normalisation
factor. $R_{0}$ is the radius parameter of the fat-jet. In particular, ratio
$\tau_{N}/\tau_{N-1}$ discriminates between jets, predicably having $N$
internal energy clusters and those with ($N$-$1$) clusters. In our case, the
jet coming from the Higgs tends to have lower values for $\tau_{21}$ as
compared to QCD jet or top jets as shown in Fig 9a(c).

  * •

$cos\theta^{*}$, the scattering angle of photon in $h\gamma$ rest-frame.

  * •

A variable, $\psi=\rm log(1+\rm exp|\eta_{\gamma}-\eta_{J}|)$ renders the
dominant QCD background distributions flat, and the distribution we expect
from a new physics signal are peaked at low
$\rm{exp|\eta_{\gamma}-\eta_{J}|}$.

  * •

We introduce few other new observables namely $Q^{R}$, $Q^{T}$, $R_{2}$. They
are defined as follows.

$\displaystyle Q^{R}$ $\displaystyle=$
$\displaystyle\sqrt{(|\vec{p}_{J}|+|\vec{p}_{\gamma}|)^{2}-({p}_{Z_{J}}+{p}_{Z_{\gamma}})^{2}}$
(9) $\displaystyle Q^{T}$ $\displaystyle=$
$\displaystyle\sqrt{{p}_{T_{J}}{p}_{T_{\gamma}}(1-\rm
cos(\Delta\phi_{J\gamma}))/2.}$ (10) $\displaystyle R_{2}$ $\displaystyle=$
$\displaystyle\left(\frac{{Q^{T}}}{{Q^{R}}}\right)^{2}$ (11)

The $Q^{R}$ variable contains both longitudinal and transverse information of
the final-state particles while the $Q^{T}$ variable is obtained only from the
transverse momenta of the final particles. The ratio $R_{2}$ of $Q^{R}$ and
$Q^{T}$ encapsulates information about the flow of energy along the plane
perpendicular to the beam and separating the final states.

  * •

At the LHC, boosted objects such as the gauge bosons, the Higgs boson and the
top quark decaying into quarks can be studied by indentifying the jet
substructures. These substructures are characterised by energy correlation
functions Larkoski:2014gra , discriminating against the QCD jets, defined as:

$\displaystyle C_{2}^{\beta}=\frac{e_{3}^{\beta}}{(e_{2}^{\beta})^{2}},$ (12)
$\displaystyle e_{2}^{\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{p^{2}_{T,\mathbf{J}}}\sum_{i,j\in\mathbf{J},i<j}p_{T,i}p_{T,j}R^{\beta}_{ij},$
(13) $\displaystyle e_{3}^{\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{p^{3}_{T,\mathbf{J}}}\sum_{i,j,k\in\mathbf{J},i<j<k}p_{T,i}p_{T,j}p_{T,k}R^{\beta}_{ij}R^{\beta}_{jk}R^{\beta}_{ki},$
(14)

where, $e_{2}^{\beta}$ and $e_{3}^{\beta}$ are 2-point and 3-point energy
correlation functions respectively. The $\beta$ represents the angular
exponent. If the jet has a two-prong like substructure, $C_{2}$ is much less
than 1. The distribution for $C_{2}$, distinguishing the Higgs jet from the
backgrounds is shown in Fig 9a(g).

  * •

Angular distance between the jet and the photon, $\Delta R_{J\gamma}$.

  * •

Subjet variable $\Delta R_{b_{i},b_{j}}$ given by the angular separation
between two b-tagged subjets.

It is to be noted that although $m_{J\gamma}$ is one of the most important
distinguishing feature between signal and background, to minimize the bias we
do not use it as an input variable to the BDT.

|
---|---
(a) |  (b)
|
(c) |  (d)
|
(e) |  (f)
|
(g) |  (h)
(a)

Figure 9: Normalised distributions for the signal (blue) and the background
(red) of the input variables at the LHC (at $\sqrt{s}=14$ TeV) used in the
MVA. Signal distributions are obtained for BP4 and the background includes all
the dominant backgrounds discussed in Sec. 2

.

The linear correlation between two variables $x$ and $y$ is computed with the
correlation coefficient, $\rho$, defined by the following equation:

$\rho=\frac{cov(x,y)}{\sigma(x)\sigma(y)}=\frac{E(xy)-E(x)E(y)}{\sigma(x)\sigma(y)}$
(15)

where $E(x),E(y)$ and $E(xy)$ are the expectation value of the variables $x,y$
and $xy$ respectively. Here, $\sigma(x),\sigma(y)$ represent the standard
deviation of $x$ and $y$ respectively. The value of linear correlation
coefficient, $\rho$ lies between -1 and 1 and its sign indicates the direction
of linear relationship among the variables. The linear correlation coefficient
matrix for signal and background is illustrated in Fig. 10. Here, most of the
variables are independent of each other. The energy variables such as
$p_{T_{\gamma}}$ and $Q^{R}$ and angular variables $\psi$ and $\Delta
R_{J\gamma}$ show slightly strong linear correlation in case of signal events
while are less correlated in background events. The choice of kinematic
variables used, relies on factors that they should less correlation among
themselves and more discriminating power between the signal and the
background. The choice of kinematic variables used, relies on factors that
they should less correlation among themselves and more discriminating power
between the signal and the background. The analysis always has the scope of
improvement, by choosing a better set of variables. However, the variables
that we have used are good discriminators as demonstrated in the following.

|
---|---
(a) | (b)
(a)  Figure 10: The linear correlations coefficients in % between the input
variables that are used for the MVA for (a) signal (with benchmark point BP4)
and (b) background. Two variables are positively correlated and negatively-
correlated if the coefficients have positive and negative signs respectively.

Based on the separation between signal and background, we show the method
unspecific ranking (relative importance) for each observable in Fig. 11. The
separation power in terms of an observable $w$ is defined using the integral:

$<S^{2}>=\int\frac{(\hat{w}_{s}(w)-\hat{w}_{b}(w))^{2}}{\hat{w}_{s}(w)+\hat{w}_{b}(w)}dw$
(16)

where $\hat{w}_{s}$ and $\hat{w}_{b}$ are the probability distribution
functions for signal and background for a given observable $w$ respectively.
The limits of integration correspond to the allowed range of $w$. Here
$<S^{2}>$ quantifies the discrimination performance of an observable $w$. The
separation $<S^{2}>$ lies within the range $[0,1]$. If $<S^{2}>=0$ implies
$\hat{w}_{s}(w)=\hat{w}_{b}(w)$, which means identical signal and background
distributions for the observable $w$ and $<S^{2}>=1$ corresponds to
distributions with no overlap. Among the variables used, the five most
important variables to distinguish the signal from the backgrounds are :
$m_{J},p_{T_{\gamma}},Q^{R},\psi\leavevmode\nobreak\
\rm{and}\leavevmode\nobreak\ \tau_{21}$.

Figure 11: Kinematic variables used as input for our MVA and their relative
importance. We obtain these using numbers for the benchmark point BP4.

We then employ the BDT algorithm. The datasets consisting of the statistically
independent event samples for the signal and the background have been split
randomly in two equal parts. One part of the dataset is used to train the BDT
algorithm and the other part is used for validation for both the signal and
the background. The parameters used to train the BDT algorithm are listed in
Table 10.

NTrees | Number of trees in the forest | 400
---|---|---
MaxDepth | Max depth of the decision tree allowed | 2
MinNodeSize | Minimum % of training events required in a leaf node | $5.6\%$
BoostType | Boosting type for the trees in the forest | AdaBoost
AdaBoostBeta | Learning rate for AdaBoost algorithm | 0.5
nCuts | Number of grid points in variable | 20
| range used in finding optimal cut in node splitting |
Table 10: The list of BDT parameters (definition and values) used

For optimal performance of the BDT, we also need to minimise the possibility
of overtraining. The case of overtraining the signal/background usually arises
during the training of the algorithm due to insufficient statistics to train
or without the proper choices of the algorithm specific tuning parameters.
Here, the distinction between signal and background becomes extremely good for
the training sample while for the test sample, it fails to achieve the same
level of difference. We have explicitly checked that our algorithm is not
over-trained by ensuring that Kolmogorov-Smirnov probability is atleast of the
order $\sim 0.1$ during training and that the area under the Receiver Operator
Charateristic (ROC) curve (that estimates the degree of rejecting the
backgrounds with respect to the signal) remains almost same for training and
testing sample. We train the BDT for each benchmark point separately.
Kolmogorov-Smirnov probability for training and testing sample are shown in
Fig. 12, indicating that both signal and background samples are not
overtrained.

(a)

(b)

Figure 12: (a) The normalized BDT response distributions for the signal and
the background and (b) the cut efficiencies as functions of BDT cut values for
the benchmark point BP4, using integrated luminosity of $3000$ fb-1 at the 14
TeV LHC.

We apply an appropriate cut on BDT response and obtain the corresponding
number of signal $\mathcal{N}_{S}$ and background $\mathcal{N}_{B}$. Results
from BDT analysis considering one sample benchmark point (BP4) is shown in
Fig. 12.

Finally, we calculate the statistical significance using formula for
$\mathcal{S}$ defined in Eqn. 7. The cut value of BDT response is
$\mathrm{BDT}_{opt}$, where the maximum significance is achieved. This has
been presented in Fig. 12b for the sample benchmark point BP4. The results for
all benchmark points are displayed in Table 11. The best LHC sensitivity is
obtained for the BP4.

BPs | $\mathcal{N}_{S}^{bc}$ | $\mathrm{BDT}_{opt}$ | $\mathcal{N}_{S}(\epsilon_{S})$ | $\mathcal{N}_{B}(\epsilon_{B}\times 10^{-2})$ | $\mathcal{S}$ | $\mathcal{L}^{req}_{(5\sigma)}(fb^{-1})$
---|---|---|---|---|---|---
BP1 | 2669 | 0.2431 | 1311 (0.4913) | 2220(0.5533) | 25.59 | 114.45
BP2 | 2077 | 0.2554 | 585(0.2817) | 495(0.1234) | 22.69 | 145.59
BP3 | 533 | 0.3687 | 12(0.0225) | $1(1.705\times 10^{-4})$ | 6.53 | 1756.91
BP4 | 1913 | 0.2453 | 779 (0.4071) | 1104 (0.2753) | 21.28 | 165.65
BP5 | 4117 | 0.2370 | 1993(0.484) | 2436(0.6072) | 36.18 | 89.48
BP6 | 1467 | 0.2393 | 683(0.4659) | 5952(1.484) | 8.69 | 992.87
$\mathcal{N}_{\textrm{Bkg}}$ | 401224 | - | - | - | - |
Table 11: Number of signal events $\mathcal{N}_{S}^{bc}$ and background events
$\mathcal{N}_{\textrm{SM}}$ before imposing the optimum BDT criteria
($\mathrm{BDT}_{opt}$) for an integrated luminosity of $3000$ fb-1 at the 14
TeV LHC. $\mathcal{N}_{S}$ and $\mathcal{N}_{B}$ are the number of signal and
background events respectively after applying the $\mathrm{BDT}_{opt}$ cut.
$\epsilon_{S}$ and $\epsilon_{B}$ represent the signal acceptance and
background acceptance efficiency at the $\mathrm{BDT}_{opt}$ cut value.
Finally, $\mathcal{S}$ and $\mathcal{L}^{req}_{(5\sigma)}$ denote the
statistical significance for an integrated luminosity of $3000$ fb-1 and
required luminosity for a $5\sigma$ discovery in case of each BP,
respectively. Figure 13: Significance as a function of Wilson coefficient
$\frac{C_{i}}{\Lambda^{2}}$ (solid lines) at the 14 TeV LHC with 3000 fb-1
integrated luminosity. We also present the corresponding required luminosity
for $5\sigma$ discovery (dashed lines).

We also show the signal significance as a function of
$\frac{C_{i}}{\Lambda^{2}}$ at the 14 TeV LHC with 3000 fb-1 integrated
luminosity in Fig. 13. While presenting these results, we have assumed two
non-zero couplings of equal strength but with opposite signs at a time. The
blue curve represents ($\mathcal{C}_{HW},\mathcal{C}_{HB}$) combination while
the pink curve represents the ($\mathcal{C}_{dW},\mathcal{C}_{dB}$)
combination. For $\mathcal{C}_{HW}$-$\mathcal{C}_{HB}$ combination, the
interference with the SM explains the slightly asymmetric nature of
significance as a function of $\frac{C_{i}}{\Lambda^{2}}$. It is evident from
Fig. 13 that with the BDT analysis, for
${|C_{i}|\over\Lambda^{2}}=0.2\leavevmode\nobreak\ \rm TeV^{-2}$, an
integrated luminosity of $200\leavevmode\nobreak\ \rm fb^{-1}$ for the contact
$ff\gamma h$ coupling (${\cal O}_{dW,dB}$) and $2000\leavevmode\nobreak\ \rm
fb^{-1}$ for the effective $hV\gamma$ coupling (${\cal O}_{HW,HB}$) is needed
to achieve a 5$\sigma$ significance. We observe a clear improvement of the
statistical significance of the signal resulting from the BDT analysis over
the cut-based analysis (also see Tables 9 and 11).

###  3.4 Correlation of $h\gamma$ signal with other Higgs signal

We now turn our attention to any possible correlation between the Higgs-photon
production and other Higgs signals which are driven by the same dimension-6
operators in which we are interested. With this in mind, we define the
following quantities as ratio of event rates:

$R_{h\gamma}=\frac{\sigma_{pp\to h\gamma}(C_{i})\times BR_{h\to
b\bar{b}}}{\sigma_{pp\to h\gamma}\times BR_{h\to
b\bar{b}}};\leavevmode\nobreak\ R_{\gamma\gamma}=\frac{\sigma_{pp\to h}\times
BR_{h\to\gamma\gamma}(C_{i})}{\sigma_{pp\to h}\times
BR_{h\to\gamma\gamma}};\leavevmode\nobreak\ R_{Z\gamma}=\frac{\sigma_{pp\to
h}\times BR_{h\to Z\gamma}(C_{i})}{\sigma_{pp\to h}\times BR_{h\to Z\gamma}}$
(17) $R_{Vh}=\frac{\sigma_{pp\to hV}(C_{i})\times BR_{h\to
b\bar{b}}}{\sigma_{pp\to hV}\times BR_{h\to b\bar{b}}};\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ R_{VBF}=\frac{\sigma_{pp\to
hjj}(C_{i})\times BR_{h\to b\bar{b}}}{\sigma_{pp\to hjj}\times BR_{h\to
b\bar{b}}}$ (18)

Figure 14: Left: The correlations of $R_{h\gamma}$ with $R_{\gamma\gamma}$ and
$R_{Z\gamma}$ and Right: correlations of $R_{h\gamma}$ with $R_{VBF}$ and
$R_{Vh}$ at the LHC. The solid lines correspond to experimental values of
signal strengths measured and the dashed lines are $1\sigma$ uncertainties in
the measurements.

Note that, in the aforementioned ratios, any dependence on the PDF and scale
choice are absent. In Fig. 14 (left panel), we study the correlation between
$R_{h\gamma}$ , $R_{\gamma\gamma}$ and $R_{Z\gamma}$ evaluated at the 13 TeV
run of LHC. These ratios depend on two particular combinations of bosonic
Wilson coefficients $C_{HB}$, $C_{HW}$ and $C_{HWB}$ as listed in Table 1.
However, $C_{HWB}$ is constrained to be small from the measurement of
$sin^{2}\theta_{W}$. In the following analysis, we set $C_{HWB}=0$. The other
two couplings have been varied in the range allowed by $h\gamma$ measurement
and listed in Table 5. The $h\to\gamma\gamma$ rate with respect to the SM is
anticorrelated to $h\to Z\gamma$. But $pp\to h\gamma$ is positively correlated
to both the aforesaid processes. This is accounted by the fact that
$R_{\gamma\gamma}$ and $R_{Z\gamma}$ are proportional to
$s_{W}^{2}C_{HW}+c_{W}^{2}C_{HB}$ and $s_{W}c_{W}(C_{HW}-C_{HB})$ (see Table
1) respectively while both the above combinations of couplings contribute to
$h\gamma$ production. The $h\to\gamma\gamma$ signal strength measured by ATLAS
collaboration at 13 TeV is $1.18^{+0.17}_{-0.14}$ CMS:2018piu  while we have
only upper limits on $R_{Z\gamma}$ so far which is a factor of 3.6
ATLAS:2020qcv . We have shown the $1\sigma$ uncertainty in the signal strength
with the dashed lines. It is evident from the plot that the $R_{h\gamma}$ can
be as large as 3 in the region of parameter space allowed by both
$R_{Z\gamma}$ and $R_{\gamma\gamma}$. This is a consequence of more stringent
limits of $C_{HB}$ and $C_{HW}$ imposed from $R_{\gamma\gamma}$ and
$R_{Z\gamma}$ measurement at the LHC.

Now let us see how the fermionic operators could correlate some of the Higgs
observables. To illustrate this, we have shown the correlation of
$R_{h\gamma}$ with $R_{Vh}$ and $R_{VBF}$ in Fig. 14 (right panel). A positive
correlation between $R_{Vh}$ and $R_{VBF}$ is observed from this plot. One can
see that the rate of $h\gamma$ production is largely enhanced compared to $Vh$
production and $VBF$ Higgs production over the allowed range of couplings for
$g_{uuh\gamma}$ and $g_{ddh\gamma}$ (see Table 1). For instance, a particular
region occurs where the rate gets enhanced by 100-200 times the SM rate of
$h\gamma$ and yet in allowed band of signal strength $Vh\leavevmode\nobreak\
(1.1^{+0.16}_{-0.15})$ and $VBF\leavevmode\nobreak\ (1.21^{+0.24}_{-0.22})$
CMS:2021kom . This result clearly shows the efficacy of $h\gamma$ channel over
$VBF$ and $Vh$ channels atleast in case of fermionic operators.

At this end, we would like to comment that the $h\gamma$ production via non-
standard interactions followed by $h\to b\bar{b}$ decay via the SM would
produce $b\bar{b}\gamma$ final state, with $b\bar{b}$ mass peaked around higgs
mass $m_{h}$. This final state also arises when Higgs is being produced
resonantly via SM interactions and decaying to $b\bar{b}\gamma$ via the same
non-standard interaction. These two processes could easily be separated by
looking at the invariant mass of $b\bar{b}\gamma$ system. The ratio of the
above two rates defined below will depend on Wilson coefficients $C_{dB}$ and
$C_{dW}$ and can be used as a discriminator of new physics from the SM.

$r_{b}(C_{i})=\frac{\sigma_{h\gamma}(C_{i})\times BR_{h\to
b\bar{b}}}{\sigma_{ggF}\times BR_{h\to b\bar{b}\gamma}(C_{i})}$ (19)

This ratio crucially depends on the $p_{T}$ of the photon. In Fig. 15, we have
plotted the variation of the ratio $r_{b}$ with $\frac{C_{dW}}{\Lambda^{2}}$
for three different choices of minimum value of photon $p_{T}$. The values of
$r_{b}$ for vanishing $\frac{C_{dW}}{\Lambda^{2}}$ correspond to its SM
values.

In the case of $b\bar{b}\gamma$ coming from the decay of the Higgs, all the
final state particles tend to have a small transverse momentum and small
mutual angular separation. Consequently, the method of Higgs tagging will not
be effective in this scenario. Moreover, the photon in the background process
arising from final-state radiation, tends to be close to one of the jets and
have a small $p_{T}$. So, we would like to investigate the signal to
background ratio in case when the effective operator controls the Higgs decay
and produces the $b\bar{b}\gamma$ final state. We have specifically checked
that distributions of (i) $E_{\gamma}$ (ii) the larger
($\Delta\eta_{j\gamma}^{max}$) and the smaller ($\Delta\eta_{j\gamma}^{min}$)
one of the rapidity separations between the b-jets and photon, (iii) the
larger ($m_{j\gamma}^{max}$) and the smaller ($m_{j\gamma}^{min}$) one of the
invariant mass of the leading jets and photon, (iv) transverse momentum of the
three final states from $h\to b\bar{b}\gamma$, (v) angular separation between
the two jets and (vi) the larger $p_{Tj}^{max}$ and the smaller $p_{Tj}^{min}$
transverse momentum of the two leading jets, when given as input variables to
the BDT network serve as distinguishing features between signal and background
characteristics leading to a signal significance of 2.87 with
$C_{dW}=0.2\leavevmode\nobreak\ \rm{TeV^{-2}}$ at 3000 fb-1 14 TeV LHC. While
the inclusive $pp\to h(\to b\bar{b}\gamma)$ process alone may not be a
promising channel to probe these new contact interactions, however, the ratio
$r_{b}$ can serve as a better discriminator between the SM and the effective
interactions.

Figure 15: The variation of ratio $r_{b}$ with $\frac{C_{dW}}{\Lambda^{2}}$.

##  4 Validity of EFT and interpretation of the results

###  4.1 EFT Applicability

The SMEFT framework is a good approximation to describe the effects of new
physics at low energy but cannot be used above a certain energy scale. As the
SMEFT Lagrangian is expanded in power series of $\frac{\rm
energy^{2}}{\Lambda^{2}}$, the steadily growing cross-section with the
partonic center-of-mass energy $\sqrt{\hat{s}}$ indicates the breakdown of the
EFT expansion when $\sqrt{\hat{s}}$ is comparable with the masses of the heavy
particles which have been integrated out. We need to estimate the validity of
the EFT expansion which is particularly important for the collider bounds,
since a wide range of energies are naturally probed at the hadron collider
experiments. We thereby discuss the viable range of Wilson coefficients that
admit a valid EFT interpretation of cross-sections. While estimating the
cross-sections, we have kept the interference
$(\mathcal{O}(\frac{1}{\Lambda^{2}}))$ of dimension-6 operators with the SM
along with the square of the purely dimension-6 operator contributions
$(\mathcal{O}(\frac{1}{\Lambda^{4}}))$. We have checked that in case of the
bosonic operators interfering with the SM, the squared term is much smaller
than the interference terms for the range of center-of-mass energies probed.
So, the EFT series is converging in this case. The other set of operators,
namely, the dipole operators, involve fermions of different chirality and thus
do not interfere with the SM, for vanishing light-quark mass. The validity of
EFT expansion in this case is ensured provided that their contribution
dominates over the SM in the energy range
$\Lambda>E>\frac{g_{SM}}{g_{x}}\Lambda$, where $g_{x}$ is the dimensionless
coupling in the UV completion Contino:2016jqw . The dimension-8 operator
interfering with the SM is indeed suppressed by a factor of $g_{SM}^{2}$ with
respect to the dimension-6 squared term, although both contributing at
$\frac{energy^{2}}{\Lambda^{2}}$ and thus, dimension-8 effect can be
neglected.

The invariant mass of the jet-photon system ($m_{J\gamma}$) determines the
energy scale being probed in a collision. The validity of our analysis can be
ensured by restricting $m_{J\gamma}$ to be less than the cut-off scale
($\Lambda$). However, the cutoff scale for the EFT cannot be defined precisely
in a model independent approach, without knowing the details of the underlying
UV complete theory. To have some idea about the cut-off scale we use a method
following Ref. Busoni:2013lha ; Bhattacharya:2015vja . When $m_{J\gamma}$
remains smaller than $\Lambda$ for most of the collisions taking place, the
ratio $R_{m_{J\gamma}}$, defined in the following, would tend to 1.

$R_{m_{J\gamma}}\equiv\frac{\int^{m_{J\gamma}<{m^{max}_{J\gamma}}}\frac{d\sigma}{dm_{J\gamma}}\;dm_{J\gamma}}{\int\frac{d\sigma}{dm_{J\gamma}}\;dm_{J\gamma}}$
(20)

We illustrate this effect in Fig. 16 with different non-zero EFT couplings at
$\sqrt{s}=$14 TeV. Values of $R_{m_{J\gamma}}$ close to unity signifies that
transfer of energy in the process is much smaller than ${m^{max}_{J\gamma}}$,
the maximum $m_{J\gamma}$ value. Such values of $m_{J\gamma}^{max}$ could be
identified as the cut-off scale $\Lambda$ of the EFT. In fact, for both the
fermionic and bosonic operators, $R$ tends to unity as
$m^{max}_{J\gamma}>2.5\leavevmode\nobreak\ \rm TeV$. Minimum values of Wilson
coefficient for which $3\sigma$ significance sensitivity can be obtained, have
been assumed while estimating the above ratio. However, higher values of the
Wilson coefficients would not lead to better sensitivity on
$m^{max}_{J\gamma}$.

|
---|---
(a) |  (b)
(a)  Figure 16: The ratio $R_{m_{J\gamma}}$ variation as a function of maximum
$m_{J\gamma}$ in the $h\gamma$ production process with (a) the dipole type
operators and (b) the bosonic operators.

###  4.2 Reinterpretation in UV theory

It is of interest to assimilate how the SMEFT operators parametrise a scenario
with ultra-violet completion. In a top-down approach, the SMEFT provides an
interface between the different UV models and the _low energy_ theory. Thus,
SMEFT approach gives a good approximation for both the cases where new physics
has effects at tree level as well as loop level. However, the sensitivity to
the scale of new physics varies in each case. The Wilson coefficients of the
higher dimensional operators contributing to a process like $pp\rightarrow
h\gamma$ have particular relationships with the parameters of the UV
completion. Thus, unlike in the model-independent SMEFT case, where the
operator couplings are independent of each other, when translated into regions
of parameter space of specific models, can be related to each other. The
matching of dimension-6 operators to the UV models such as extensions in the
Higgs sector of the SM have been discussed in recent literature
Henning:2014wua ; Dawson:2020oco ; DasBakshi:2018vni . For example, one of
such popular models, the minimal supersymmetric standard Model (MSSM), that
predicts several new particles such as the additional Higgs bosons and the
superpartners of SM particles, can give additional contributions to the loop-
induced production of Higgs and photon. Figs. 17 and 18 illustrate few
representative one-loop processes where the effects can be parametrised by
dimension-6 operators.

|
---|---
(a) | (b)
(a)  Figure 17: Additional triangle-type diagrams contributing to the process $q\bar{q}\to h\gamma$ at one-loop level that can be parameterised with $\mathcal{O}_{HB}$ and $\mathcal{O}_{HW}$. |  |  |
---|---|---|---
(a) | (b) | (c) | (d)
|  |  |
(e) | (f) | (g) | (h)
Figure 18: Additional box-type and triangle-type diagrams contributing to the
process $q\bar{q}\to h\gamma$ at one-loop level that can be parameterised with
$\mathcal{O}_{fB}$ and $\mathcal{O}_{fW}$.

The triangle and box contributions in Fig. 18 can lead to the dipole-type
operators (see Eqn. 3) without a requiring a chirality flip of the SM fermion
lines. Fig. 17 lead to bosonic-type operators (see Eqn. 4) that can contribute
to the loop-induced $h\gamma$ production as well as decays of
$h\to\gamma\gamma$ and $h\to Z\gamma$. As an illustration, we show how the
limits of bosonic operators obtained in our analysis can provide interesting
constraints on masses and mixings of stop squarks in the framework of MSSM in
the limit when the electorweak gaugions are much heavier than the stop-quarks.
In presenting our results, we follow Henning:2014gca , assuming a common
diagonal mass term $m_{\tilde{t}}$ for right- and left- stop squarks and
$X_{t}=A_{t}-\mu\leavevmode\nobreak\ \rm cot\leavevmode\nobreak\ \beta$ that
governs the $\tilde{t}_{L}$-$\tilde{t}_{R}$ mixing and the splitting between
the two stop mass eigenstates. It has been already pointed out that $C_{HB}$
and $C_{HW}$ are severely constrained from $R_{\gamma\gamma}$. The
contribution of stop squarks to the $h\gamma\gamma$ vertex at one loop
Dermisek:2007fi , is given by:

$\displaystyle C^{MSSM}$
$\displaystyle=\frac{N^{\tilde{t}}}{D^{\tilde{t}}}\,,{\rm{here}}\leavevmode\nobreak\
N^{\tilde{t}}\leavevmode\nobreak\ {\rm{and}}\leavevmode\nobreak\
D^{\tilde{t}}\leavevmode\nobreak\ {\rm{are\leavevmode\nobreak\
given\leavevmode\nobreak\ by\leavevmode\nobreak\ the\leavevmode\nobreak\
following\leavevmode\nobreak\ equations:}}$ $\displaystyle N^{\tilde{t}}$
$\displaystyle=2{m_{t}^{4}}+2{m_{t}^{2}}m_{\tilde{t}}^{2}-{m_{t}^{2}}{X_{t}^{2}}+\frac{1}{8}{m_{Z}^{4}}{c_{2\beta}^{2}}+{m_{t}^{2}}{m_{Z}^{2}}{c_{2\beta}}+\frac{1}{2}{m_{Z}^{2}}c_{2\beta}m_{\tilde{t}}^{2}$
$\displaystyle-\frac{1}{144}{m_{Z}^{2}}c_{2\beta}(-3+8s_{W}^{2})c_{2\theta_{t}}(144m_{\tilde{t}}^{4}(1+r)^{2}-288m_{\tilde{t}}^{4}(1-r^{2})+144m_{\tilde{t}}^{2}(1-r)^{2}$
$\displaystyle+18m_{Z}^{4}-96m_{Z}^{4}s_{W}^{2}+128m_{Z}^{4}s_{W}^{4}+576m_{t}^{2}X_{t}^{2}+288m_{Z}^{2}c_{2\beta}m_{\tilde{t}}^{2}-768m_{Z}^{2}s_{W}^{2}c_{2\beta}m_{\tilde{t}}^{2}$
$\displaystyle+m_{Z}^{4}(18-96s_{W}^{2}+128s_{W}^{4})c_{4\beta})^{1/2}$
$\displaystyle D^{\tilde{t}}$
$\displaystyle={m_{t}^{4}}+m_{\tilde{t}}^{4}(1-r^{2})+2m_{t}^{2}m_{\tilde{t}}^{2}+\frac{4}{3}s_{W}^{2}m_{Z}^{2}m_{\tilde{t}}^{2}c_{2\beta}r+\frac{1}{2}m_{Z}^{2}m_{\tilde{t}}^{2}c_{2\beta}(1-r)+\frac{1}{2}m_{Z}^{2}m_{t}^{2}c_{2\beta}$
$\displaystyle+\frac{1}{3}m_{Z}^{4}c_{2\beta}s_{W}^{2}-\frac{4}{9}m_{Z}^{4}s_{W}^{4}c_{2\beta}^{2}-m_{t}^{2}X_{t}^{2}$
(21)

where $m_{t}$ denotes the top mass and $\tan\beta$ is equal to the ratio of
vacuum expectation values of the two Higgses: $\tan\beta=\frac{\langle
H_{u}\rangle}{\langle H_{d}\rangle}$ and
$m_{\tilde{t}}^{2}=\frac{m_{\tilde{t}L}^{2}+m_{\tilde{t}_{R}}^{2}}{2}$ and
$r=\frac{m_{\tilde{t}_{L}}^{2}-m_{\tilde{t}_{R}}^{2}}{m_{\tilde{t}_{L}}^{2}+m_{\tilde{t}_{R}}^{2}}$.
While deriving Eqn. 21, it has been assumed that the electroweak gauginos are
much heavier than the stop quarks.

Thus, the loop contribution arising from MSSM scales as inverse square of the
stop mass in the limit when stop is heavier than all other SM particles. This
is also evident from Fig. 19(a). For large stop masses, this $c_{MSSM}$ can be
identified with the following combination of SMEFT operators,
$C^{EFT}=c_{W}^{2}C_{HB}+s_{W}^{2}C_{HW}$ (see Table 1). Thus the upper limits
of $C_{HB}$ and $C_{HW}$ can be translated to stop parameters $m_{\tilde{t}}$
and $X_{t}$ through the same relation J_Ellis_Tevlong_Y .

We show (in right panel of Fig. 19,) the allowed (pink coloured) region in
$(\frac{X_{t}}{m_{\tilde{t}}},m_{\tilde{t}})$ plane from our analysis for
$\tan\beta=20$. However, the entire pink region is not allowed from the
consideration of Higgs mass444To have the lightest CP-even Higgs boson in the
MSSM to be identified with the 125 GeV scalar discovered at the LHC, one
requires large stop masses and large stop mixing.. One can easily see that the
Higgs mass allowed region (with $m_{h}\in(122,\,128)\leavevmode\nobreak\
\textnormal{GeV}$) runs almost parallel to $m_{\tilde{t}}$ axis implying a
loose constraint on $m_{\tilde{t}}$ resulting from our analysis. When the stop
mixing parameter $X_{t}=0$, the stop mass $m_{\tilde{t}}\sim\mathcal{O}(500)$
GeV. Limits on stop masses extend upto 1 TeV for large stop mixings. These
limits also depend on the values of lightest neutralino masses, vanishing
entirely for $m_{\tilde{\chi}_{1}^{0}}\leq 400\leavevmode\nobreak\ \rm GeV$
ATLAS:2021twp . Thus, these Wilson coefficient constraints, generated at one-
loop level provide complementary information to those from direct searches for
gluinos and squarks production. However, a global SMEFT fit of data resulting
from top production, Higgs production and decay along with di-boson production
at the LHC resulted into tighter constraint on stop parameters as discussed in
Ref. Ellisglobalfit2021 .

|
---|---
(a) |  (b)
(a)  Figure 19: (a) One loop-level MSSM contribution for the $h\gamma\gamma$
coupling, in the degenerate case for $\tan\beta=1$ and $\tan\beta=20$ as a
function of $m_{\tilde{t}}$ and (b) The allowed degenerate stop parameter
space : $(\frac{X_{t}}{m_{\tilde{t}}},m_{\tilde{t}})$ plane with
$\tan\beta=20$ obtained from the $C_{HW}\leavevmode\nobreak\
\rm{and}\leavevmode\nobreak\ C_{HB}$ projections. In addition, the green
shading indicates the region where $m_{h}\in(122,\,128)\leavevmode\nobreak\
\textnormal{GeV}$

Future lepton colliders also provide a viable prospect of precise and model
independent measurement of Higgs and photon production. Predictions for the
model independent parameters would not only allow to test the $h\gamma$
interactions but also will be helpful to distinguish the production of a
possible new scalar in association with a photon. A study including $h\gamma$
production at the lepton colliders and taking into account the additional
Higgs decay modes such as $h\to WW^{*},h\to\tau^{+}\tau^{-}$ will be reported
in a follow-up study. The production of Higgs and photon followed by the Higgs
decays into diphoton would lead to a distinctive $3\gamma$ peak, that would
provide more precise limits on the couplings and masses of the UV dynamics.

##  5 Conclusions

We have presented a study of the production of Higgs boson in association with
a high $p_{T}$ photon at the LHC, as the probe of new physics beyond the
Standard Model. The $pp\to h\gamma$ rate being highly suppressed in the
framework of SM, it can serve as a highly sensitive probe of new interactions.
Instead of confining ourselves to a particular BSM scenario, we stick to the
SMEFT framework in which operators involving SM fields of dimension-6 are
constructed keeping various local and global symmetries of the SM. The
coefficients of these operators encapsulate the effects of any high scale
dynamics that may originate from a particular BSM scenario or even simply from
the higher order processes in the SM. We have been particularly interested in
the boosted kinematic regime where a very high $p_{T}$ photon recoils against
a fat-jet originating from a pair of $b$-quarks resulting from the decay of
the Higgs boson. Dimension-6 SMEFT operators which contribute in the $h\gamma$
production rate arise in a couple of varieties, namely the bosonic-type
involving a Higgs boson and two gauge bosons and fermionic-type, involving two
fermions of opposite helicities along with a Higgs boson and a gauge boson.
These interactions have energy dependencies in the cross-sections that
essentially grow with the partonic center-of-mass energy.

To begin with, we have briefly reviewed the constraints on the dimension-6
operators of our interest, arising from the unitarity, electroweak precision
tests and the available Higgs data. Finally, we compare the mass distribution
of boosted $h\gamma$ system estimated in SMEFT framework with the same
measurement done at the Run II phase of 13 TeV LHC by the CMS collaboration.
This exercise helps us in validating our simulation as well as to constrain
the relevant Wilson coefficients from $h\gamma$ data in particular. We choose
six representative benchmark points from the allowed parameter space and
explore their detection possibility at the 14 TeV LHC with 3000 $fb^{-1}$
integrated luminosity. Different Lorentz structures appearing in the
dimension-6 SMEFT operators leave their signatures in the different kinematic
distributions which can be exploited to distinguish new physics signal from
the SM backgrounds. We have constructed new observables like
$\psi,Q^{R},Q^{T},R_{2}$ to enhance the characteristic features of the tensor
structures of $hV\gamma$ coupling and the $h\gamma ff$ contact interactions
and discriminate against the SM backgrounds. The fact that hadronic decay
products of the Higgs boson can form boosted fat-jet, opens up an intriguing
new possibility that can be beneficial, contrary to cleaner final states from
the $ZZ^{\ast}$ (or $WW^{\ast}$) decays of the Higgs. The reconstructed fat-
jet still carries the characteristics of the parent particle in its mass and
substructures, thus enabling one to identify the Higgs boson itself with
significant tagging efficiency. We have exploited such properties to counter
the large SM background coming from the QCD jets of light flavour. The
projected 3$\sigma$ upper limits derived from our cut-based analysis for
integrated luminosity of 3000 $\rm fb^{-1}$ seem to be more stringent than the
bounds coming from precision electroweak observables.

To explore any possible improvement of the sensitivity for the new
interactions at the LHC, we further undertook a multivariate analysis where a
total of 10 kinematic variables have been used as input to the boosted
decision tree network which provide the optimum separation between the signal
and the background. Unlike the rectangular cut-based analysis, the
multivariate analysis can use the potential of various kinematic variables at
one go. The above analysis clearly established that the LHC sensitivity can be
greatly improved for most of the benchmark points used by us. Benchmark points
involving the fermionic operators can be probed with more than 5$\sigma$
significance with 200 $\rm fb^{-1}$ data only while the more accumulated data
is need to probe bosonic operators at the same level.

Finally, we have briefly discussed the connection between the model-
independent coefficients of dimension-6 operators with an ultra-violet
complete model, namely the MSSM. After qualitatively discussing how the
bosonic and fermionic operators may arise from MSSM, constraints have been
derived on the MSSM parameters, namely stop masses and mixing,555We have
assumed that electroweak gauginos are heavy responsible for generating
$h\gamma\gamma$ coupling at one loop. As we are headed towards the new phase
of LHC run with the higher energy and luminosity, it is important to push for
the rare processes like $pp\to h\gamma$ production which are very sensitive to
the physics beyond the standard paradigm.

## Acknowledgements

We thank Satyaki Bhattacharya and Amit Chakraborty for useful discussions at
various stages of the work. TB acknowledges the support from Council of
Scientific and Industrial Research, Government of India. TB is thankful to
Somenath Pal for providing computational facilities at various stages of the
work.

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