††institutetext: PITT PACC, Department of Physics and Astronomy, University of
Pittsburgh, 3941 O’Hara St., Pittsburgh, PA 15260, USA

# Higgs to $b\bar{b}$ from Vector Boson Fusion for High-Scale Physics

Tao Han  Sze Ching Iris Leung  Matthew Low<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Vector boson fusion is arguably the most direct collider probe of electroweak
symmetry breaking. Typically, the signature includes two forward/backward jets
with low transverse momenta with a scale that is set by the mass of the vector
boson. For this reason, an upper cut is used when searching for vector boson
fusion processes in the Standard Model. Alternatively, the upper cut on the
forward jets can be removed and the high-momentum exchange region of vector
boson fusion can be studied. This phase space region has sensitivity to new
physics via higher dimensional operators and form factors. In this work, we
study the high-momentum region of the vector boson fusion channel where the
Higgs decays to $b\bar{b}$. We show that, depending on the form of new
physics, the limits on the new physics scale range from $0.5$ TeV to $1.8$
TeV.

##  1 Introduction

The milestone discovery of the Higgs boson $(h)$ by the ATLAS ATLAS:2012yve
and CMS CMS:2012qbp  collaborations at the CERN Large Hadron Collider (LHC) is
of great theoretical and experimental significance in understanding our
microscopic world. The Standard Model (SM) of the strong and electroweak
interactions, along with the Higgs mechanism, provides us with a consistent
theoretical framework valid up to high scales. On the other hand, we are still
lacking an understanding of if and how the Higgs mass, the only dimensionful
parameter in the theory, is stabilized against quantum corrections from a
higher new physics scale. This is the electroweak hierarchy problem
Weinberg:1975gm ; Gildener:1976ai ; tHooft:1980xss . It is thus of fundamental
importance to experimentally probe the interactions of the Higgs at higher
energy scales, which should hold the key to understanding the hierarchy
problem.

Some efforts have been made along these lines to probe the largest SM Higgs
coupling $ht\bar{t}$ at the LHC in high scales via the leading Higgs
production channel of gluon fusion $gg\to h\to ZZ$ Goncalves:2020vyn  and
associated production $pp\to t\bar{t}h$ MammenAbraham:2021ssc . The next
leading production mechanism of the Higgs boson is the process by which two
vector bosons collide to produce a Higgs boson, known as vector boson fusion
(VBF). With the Higgs coupling directly proportional to the gauge boson mass,
this process is one of the most direct probes of electroweak symmetry breaking
that we have. It is well-known that without a Higgs boson, the amplitude for
longitudinally-polarized gauge boson scattering exhibits quadratic growth with
energy, violating perturbative unitarity at a scale of $\Lambda\approx
2.2~{}{\rm TeV}$ Lee:1977eg ; Lee:1977yc . This is cornerstone of the no-lose
theorem of the LHC uncovering the first step of electroweak symmetry breaking
Glashow:1961tr ; Salam:1964ry ; Weinberg:1967tq .

The traditional VBF signal leverages the fact that the two outgoing quarks
exchange a relatively low amount of momentum with the vector bosons. These
quarks, often called tagging jets Kleiss:1987cj ; Barger:1988mr  tend to have
high pseudorapidity $(\eta)$ and low transverse momentum $(p_{T})$, dictated
by the (collinear) $W$ propagator (see Refs. Rauch:2016pai ; Green:2016trm ;
Anders:2018oin ; Bellan:2019xpr ; Baglio:2020bnc ; Covarelli:2021gyz ;
Chen:2016wkt  for reviews). Signal regions are then defined by a lower cut (to
pass trigger thresholds) and an upper cut (to maximize the VBF sensitivity) on
the tagging jet $p_{T}$. As expected, VBF processes have contributed to the
Higgs boson discovery ATLAS:2012qaq ; CMS:2012zhx . In the hope to probe Higgs
physics at a high scale, an alternative approach is to examine the Higgs boson
with a higher transverse momentum. This signal region, by contrast, would look
to isolate events with a large momentum exchange between the vector bosons and
the quarks. At a cost of lower signal event rate, we may have increased
sensitivity to physics beyond the Standard Model (BSM).

The idea of probing high energy tails of distributions at a hadron collider,
in the search for higher dimensional operators, was elucidated in Drell-Yan
Farina:2016rws  and subsequently applied to diboson production
Franceschini:2017xkh . Searches for higher dimensional operators in VBF have
been initiated in a few studies Banerjee:2013apa ; Gomez-Ambrosio:2018pnl ;
Biekotter:2020flu ; Araz:2020zyh ; Ethier:2021ydt ; Biswas:2021qaf ;
Hwang:2023wad . Ref Araz:2020zyh , in particular, applied the techniques from
Franceschini:2017xkh  to VBF production for the $h\to\gamma\gamma$ and
$h\to\tau^{+}\tau^{-}$ decay channels. They found with $3~{}{\rm ab}^{-1}$ a
limit of $\Lambda>3.9~{}{\rm TeV}$ (at 68% C.L.) could be set on the scale of
new physics.

In this work, we first compute the sensitivity to new physics, parametrized by
the leading dimension-6 operators and a non-local momentum-dependent form
factor, using the $b\bar{b}$ decay of the Higgs, including detector simulation
and backgrounds. A sample Feynman diagram for this process is shown in Fig. 1.
We find that the sensitivity is slightly worse than in the $h\to\gamma\gamma$
channel, primarily due to the fact that the Higgs mass resolution is reduced
and the backgrounds are larger. Combining multiple decay channels of the Higgs
would further improve the overall sensitivity.

Secondly, we point out that the high momentum Higgs region can be effectively
isolated by cutting on the tagging jets. In channels where the Higgs four-
momentum can be fully reconstructed, like in $h\to\gamma\gamma$, the $p_{T}$
of the Higgs is the best variable. However, in channels where that is not
possible, like in $h\to W(\ell\nu)W(\ell\nu)$, cutting on the tagging jets
only loses a little sensitivity.

The outline of the paper is as follows. In Sec. 2 we describe the
parametrizations we use for new physics. Section 3 studies the kinematics of
this topology and compares a few approximations. Next, we discuss the details
of our simulation in Sec. 4. The results are shown in Sec. 5 and we present
our conclusions in Sec. 6.

Figure 1: Representative Feynman diagrams of vector boson fusion in the
Standard Model (left), via a contact interaction $q\bar{q}\kern
1.00006ptV\kern-1.00006ptH$ from dimension-6 operators (center), and the QCD
background (right).

##  2 New Physics

In this section, we describe the parametrizations we use for exploring new
physics associated with Higgs production at high scales.

###  2.1 Higher-Dimensional Operators

A consistent and convenient scheme to parametrize new physics effects from a
higher scale is the effective field theory (EFT) approach with higher
dimensional operators. The leading operators are at dimension-6. We follow the
usual parametrization of

$\Delta\mathcal{L}_{d=6}=\sum_{i}\frac{c_{i}}{\Lambda^{2}}\mathcal{O}_{i}.$
(1)

The coefficients $c_{i}$ are dimensionless effective couplings and $\Lambda$
is the scale of new physics. From an EFT perspective the coefficients $c_{i}$
could naturally be of the size $g^{2}$ or $\mathcal{O}(1)$ (where $g$ is some
coupling), in weakly-coupled BSM theories Giudice:2007fh , or any values
$\lesssim(4\pi)^{2}$ Manohar:1983md ; Georgi:1984zwz  for a strongly-coupled
theory. They could have either sign. Although coefficients as large as
$c_{i}\sim(4\pi)^{2}$ are technically possible, it requires the unlikely
scenario of strong coupling between new physics and all particles of the
$q\bar{q}\kern 1.00006ptV\kern-1.00006ptH$ interaction. Even a coupling of
$c_{i}\sim 4\pi$ is not likely Liu:2016idz ; Franceschini:2017xkh .

In VBF, our interaction of interest is $q\bar{q}\kern
1.00006ptV\kern-1.00006ptH$, for which there are four leading operators
Franceschini:2017xkh . These are operators leading in the energy growth of the
SM-BSM interference term. In the Warsaw basis Grzadkowski:2010es , they are
written as $\mathcal{O}_{Q}^{(3)}$, $\mathcal{O}_{Q}$, $\mathcal{O}^{u}_{R}$,
and $\mathcal{O}^{d}_{R}$

$\displaystyle\mathcal{O}_{Q}^{(3)}$
$\displaystyle=(\bar{Q}\sigma^{a}\gamma^{\mu}Q)(iH^{\dagger}\sigma^{a}\overleftrightarrow{D_{\mu}}H),$
(2a) $\displaystyle\mathcal{O}_{Q}$
$\displaystyle=(\bar{Q}\gamma^{\mu}Q)(iH^{\dagger}\overleftrightarrow{D_{\mu}}H),$
(2b) $\displaystyle\mathcal{O}^{u}_{R}$
$\displaystyle=(\bar{u_{R}}\gamma^{\mu}u_{R})(iH^{\dagger}\overleftrightarrow{D_{\mu}}H),$
(2c) $\displaystyle\mathcal{O}^{d}_{R}$
$\displaystyle=(\bar{d_{R}}\gamma^{\mu}d_{R})(iH^{\dagger}\overleftrightarrow{D_{\mu}}H),$
(2d)

and have been labelled as the high energy primaries Franceschini:2017xkh . The
explicit contribution is depicted by the diagram in Fig. 1 (center).

In searches for BSM physics, the largest contribution usually comes from the
interference between the SM amplitude and the BSM amplitude, as the leading
power correction. The ideal situation for maximal sensitivity is for the BSM
amplitude to grow with energy while the SM amplitude remains constant. For
$q\bar{q}\kern 1.00006ptV\kern-1.00006ptH$ this is the case for the operators
in Eq. (2) Franceschini:2017xkh ; Araz:2020zyh  because they are sensitive to
the longitudinal polarization of the vector boson Azatov:2019xxn .

One subleading operator that can be considered is

$\mathcal{O}_{\phi W}=H^{\dagger}HW^{a}_{\mu\nu}W^{a,\mu\nu}.$ (3)

This operator is sensitive to the transverse polarizations of the vector boson
and has a smaller interference term with the SM. In the case of diboson, the
sensitivity to $c_{\phi W}$ is about an order of magnitude worse than to
$c^{(3)}_{Q}$ Bishara:2020vix . The coefficient $c_{\phi W}$ can technically
saturate perturbativity despite the fact that this does not occur in common
BSM theories. Its expected size is actually loop suppressed and is $c_{\phi
W}\sim g^{4}/(16\pi^{2})^{2}$ when the new physics in the loop is weakly-
coupled or $c_{\phi W}\sim g^{2}/(16\pi^{2})$ when the new physics in the loop
is strongly-coupled Giudice:2007fh ; Bishara:2020vix .

For VBF production, Ref. Araz:2020zyh  found that combining the
$h\to\gamma\gamma$ and $h\to\tau^{+}\tau^{-}$ decay channels resulted in a
sensitivity to $\Lambda\approx 3.9$ TeV for $3~{}{\rm ab}^{-1}$ at 68% C.L.
The same operators have been constrained in diboson searches in the channels:
$Wh$, $Zh$, $WZ$, and $WW$ Ellis:2014dva ; Franceschini:2017xkh ; Liu:2018pkg
; Banerjee:2018bio ; Grojean:2018dqj ; Azatov:2019xxn ; Freitas:2019hbk ;
Banerjee:2019pks ; Brehmer:2019gmn ; Chiu:2019ksm ; Banerjee:2019twi ;
Baglio:2020oqu ; Bishara:2020vix ; Bishara:2020pfx ; Huang:2020iya ;
Bishara:2022vsc . For the same amount of data, but at 95% C.L., the limits
extend to $\Lambda\approx 10~{}{\rm TeV}$ in the diboson channel
Franceschini:2017xkh . VBF and diboson production probe a different linear
combination of the operators in Eq. (2) so measuring both channels is
essential to measure the full operator dependence.

###  2.2 Form Factors

If the underlying mechanism for electroweak symmetry breaking is due to some
new strong dynamics, it is conceivable to consider that the Higgs boson may
not be fundamental, but a composite particle arising from strongly interacting
new dynamics at a scale $\Lambda$ Panico:2015jxa . In such scenarios, the
Higgs interaction may exhibit a momentum-dependent form-factor near or above
the new physics scale $\Lambda$, rather than a point-like interaction.

It is challenging to write a form-factor, in a general form, without prior
knowledge of the strong dynamics of the specific composite scenario. Inspired
by the nucleon form-factor Punjabi:2015bba , we adopt the following
phenomenological ansatz by adding a form factor to the $hW^{+}W^{-}$ and $hZZ$
vertices

$\displaystyle(gm_{W})hW^{+}W^{-}$
$\displaystyle\to\Gamma(Q^{2};n,\Lambda^{2})\;(gm_{W})hW^{+}W^{-},$ (4a)
$\displaystyle\left(\frac{gm_{Z}}{2c_{W}}\right)hZZ$
$\displaystyle\to\Gamma(Q^{2};n,\Lambda^{2})\;\left(\frac{gm_{Z}}{2c_{W}}\right)hZZ,$
(4b)

where $c_{W}$ is the cosine of the weak mixing angle and the form factor is of
the form Goncalves:2020vyn ; MammenAbraham:2021ssc

$\Gamma(Q^{2};n,\Lambda^{2})=\frac{1}{(1+Q^{2}/\Lambda^{2})^{n}}.$ (5)

This form factor approaches 1 at low energies as $Q^{2}/\Lambda^{2}\to 0$, and
suppresses this interaction as $Q^{2}/\Lambda^{2}$ grows at high energies. The
suppression depends on the value of $n$ chosen; the larger $n$ is, the faster
the suppression. The case of $n=2$ corresponds to a dipole form factor, as
often adopted for the nucleon form factor. Limits from a recent ATLAS search
can be applied to this form factor for the $ht\bar{t}$ vertex ATLAS:2023dnm .

In our study we use $Q=p_{T}^{h}$ since the transverse momentum of the Higgs
characterizes the momentum exchange. This also ensures that Eq. (5) always
results in a suppression, which is the expected behavior given considerations
to the hierarchy problem.

Another option, which we do not pursue in this study, is to use form factors
computed from integrating out heavy resonances Bittar:2022wgb . This would be
useful for a future study since these form factors could be matched directly
to known BSM theories.

##  3 Kinematics

It is important to explore the kinematics of the Higgs boson produced in
vector boson fusion in the hope to optimize the dynamical effects. The main
purpose is to identify to what extent the phase space with a boosted Higgs can
be isolated purely from the kinematics of the two tagging jets.111In the
$b\bar{b}$ channel that we study here, as well as in the $\gamma\gamma$ and
hadronic $\tau^{+}\tau^{-}$ channels Araz:2020zyh  the Higgs is fully
reconstructable so one can always directly use $p_{T}^{h}$. Using the tagging
jets would be useful in other channels like $W(\ell\nu)W(\ell\nu)$.

At leading order there are three objects in the final state: the Higgs $h$ and
two tagging jets $j_{1}$ and $j_{2}$ (where $p_{T}^{j_{1}}\geq p_{T}^{j_{2}}$
by convention). Balancing transverse momentum leads to the relationship

$(\tilde{p}_{T}^{h})^{2}=(p_{T}^{j_{1}})^{2}+(p_{T}^{j_{2}})^{2}+2p_{T}^{j_{1}}p_{T}^{j_{2}}\cos(\phi^{j_{1}}-\phi^{j_{2}}),$
(6)

where $\phi^{x}$ is the azimuthal angle of object $x$. The notation
$\tilde{p}_{T}^{h}$ is to indicate that this would be the transverse momentum
of the Higgs in the three-body limit.

We are interested in the kinematical region with a high $p_{T}$ Higgs. There
are two notable limits of Eq. (6). The first is when $p_{T}^{j_{2}}\ll
p_{T}^{j_{1}},p_{T}^{h}$. In this case the transverse momentum of the Higgs is
primarily balanced by the hardest jet and the three-body space approximately
collapses to a two-body space. Even at the high luminosity LHC, this kinematic
configuration is rather rare and out of reach. It could be relevant at a
future collider.

The second is when $\cos(\phi^{j_{1}}-\phi^{j_{2}})=\mathcal{O}(1)$. Then
$p_{T}^{h}\approx p_{T}^{j_{1}}+p_{T}^{j_{2}}$. In Fig. 2 we compare this
quantity with $p_{T}^{h}$ and with $\tilde{p}_{T}^{h}$ from Eq. (6). The
events are generated using Pythia 8.302 Sjostrand:2014zea  with $p_{T}^{h}\geq
150~{}{\rm GeV}$. The quantity $\tilde{p}_{T}^{h}$ is generally an
underestimate due to additional radiation that is ignored in the three-body
limit. Empirically, $p_{T}^{j_{1}}+p_{T}^{j_{2}}$ is a reasonable proxy for
$p_{T}^{h}$ at high values of $p_{T}^{h}$.

Figure 2: Distribution of Higgs $p_{T}$-like variables for operator
$\mathcal{O}_{Q}^{(3)}$ (left) and for the $n=2$ form factor (right) with
$\mathcal{L}=3~{}{\rm ab}^{-1}$.

A number of other Higgs $p_{T}$-like variables can be studied. These include
$p_{T}^{j_{1}}$, $p_{T}^{j_{2}}$, and the geometric mean
$\sqrt{p_{T}^{j_{1}}\;p_{T}^{j_{2}}}$ which was suggested in Refs.
Banerjee:2013apa ; Biswas:2021qaf . We find that $p_{T}^{h}$ gives the best
results, followed closely by the sum $p_{T}^{j_{1}}+p_{T}^{j_{2}}$ and the
quantity $\tilde{p}_{T}^{h}$. The other Higgs $p_{T}$-like variables are less
correlated with $p_{T}^{h}$.

##  4 Simulation

We generate events using MadGraph 5 v2.7.3 Alwall:2011uj . Both signal and
background are generated at leading order at a center-of-mass energy
$\sqrt{s}=13$ TeV with the parton distribution function NNPDF23_lo_as_0130_qed
Ball:2013hta  through LHAPDF Buckley:2014ana . Events are showered and
hadronized with Pythia 8.302, and detector simulation is performed by Delphes
3.4.1 deFavereau:2013fsa . Jets are reconstructed with the anti-$k_{T}$
algorithm with radius $R=0.5$ Cacciari:2008gp . For $b$-tagging we use the
truth flavor from Delphes and apply a flat $b$-tagging efficiency of $77\%$
for the leading $b$-jet and $85\%$ for the subleading $b$-jet.222This means
for events with the leading truth $b$-jet with $p_{T}>85~{}{\rm GeV}$ and the
subleading truth $b$-jet with $p_{T}>65~{}{\rm GeV}$, 65% of them pass the
$b$-jet cuts ($0.77\times 0.85=0.65$).

Our signal is the process $pp\to h(b\bar{b})jj$ with either the dimension-6
operators or the form factor turned on. Our calculation for this process
includes the contributions from the Standard Model, new physics, and the
interference between these two. We label the signal as BSM. We use dimension-6
operators from the SMEFTatNLO model Degrande:2020evl  via FeynRules
Alloul:2013bka .

There are two dominant backgrounds: the Standard Model VBF contribution and
the Standard Model QCD contribution. Standard Model VBF includes a resonant
Higgs and is generated from $pp\to h(b\bar{b})jj$ with all new physics
contributions set to zero. We label this background as VBF. The QCD background
is generated by $pp\to b\bar{b}jj$ (excluding on-shell Higgs contributions).
We label this background as QCD.

### Pre-Selection and Validation

We apply cuts to the detector-level events in two stages. The first is the
pre-selection which we use for validation with the forward event selection
from ATLAS ATLAS:2020bhl  and is listed in this section. The second is the
event selection which will be described in the following section. Cuts applied
at the generator level are listed in App. A.

We denote $b_{1}$ and $b_{2}$ as the leading and subleading $b$-tagged jets
which form the highest $p_{T}$ $b$-jet pair in the events. $j_{1}$ and $j_{2}$
refer to the tagging jets, where $j_{1}$ is the jet with highest $p_{T}$ and
$j_{2}$ is the sub-leading jet333Despite the fact that we are looking for a
high $p_{T}$ Higgs, in the VBF channel, the peak sensitivity comes from the
range $p_{T}^{h}\approx 300~{}{\rm GeV}$. Therefore, the analysis can safely
resolve the $b$-jets rather than using a boosted analysis looking for a Higgs
jet with substructure Butterworth:2008iy . This was also the case in Ref.
Araz:2020zyh  where they searched for resolved $\gamma\gamma$ and resolved
hadronic $\tau^{+}\tau^{-}$.

  * •

$p_{T}^{b_{1}}>85$ GeV and $|\eta^{b_{1}}|<2.5$. The operating point used for
$b$-tagging this $b$-jet has a 77% tagging efficiency.

  * •

$p_{T}^{b_{2}}>65$ GeV and $|\eta^{b_{2}}|<2.5$. The operating point used for
$b$-tagging this $b$-jet has a 85% tagging efficiency.

  * •

$p_{T}^{j_{1}}>60$ GeV and $3.2<|\eta^{j_{1}}|<4.5$.444Note that $j_{1}$ is
always the hardest jet. An event where, for example, $p_{T}^{j_{1}}=90~{}{\rm
GeV}$, $\eta^{j_{1}}=3.0$, $p_{T}^{j_{2}}=80~{}{\rm GeV}$, $\eta^{j_{2}}=-4.0$
does not pass the selection.

  * •

$p_{T}^{j_{2}}>30$ GeV and $|\eta^{j_{2}}|<4.5$.

  * •

$p_{T}^{b\bar{b}}>150$ GeV, where $p_{T}^{b\bar{b}}$ is the transverse
momentum of the $b\bar{b}$ system.

To validate our analysis, we compare our VBF and QCD event yields with the
ATLAS study ATLAS:2020bhl  with the above pre-selection applied. The numerical
results are shown in Table 1. Note that Ref. ATLAS:2020bhl  is searching for
the Standard Model Higgs signal so their signal corresponds to our VBF while
their (data-driven) background corresponds to our QCD.

| VBF | ATLAS signal | QCD | ATLAS data sidebands
---|---|---|---|---
after pre-selection | 755.8 | 930.9 | 1,683,155 | 2,584,704
Table 1: Event yields from our simulation for VBF and QCD and expected event
yields from Ref. ATLAS:2020bhl , at $\sqrt{s}=13$ TeV with an integrated
luminosity of $126~{}\text{fb}^{-1}$.

It is expected that our expected number of events is lower than those from
ATLAS because we only generate events at leading order. Higher-order
corrections can account for this difference. In VBF the $k$-factor at next-to-
leading order is known to be $\approx 1.1$ Greljo:2017spw ; Buckley:2021gfw .
The ratio between the ATLAS event yield and ours is about $1.2$. We apply an
effective $k$-factor of $1.2$ to our VBF and BSM samples. In our QCD sample we
use an effective $k$-factor of $1.5$ which makes our results consistent with
the ATLAS event yield.

### Event Selection

In addition to the above pre-selection, we require the events to satisfy
following selections which are based on the characteristics of our BSM signal,
inspired by Araz:2020zyh , but tailored to our final state.

Since the VBF signal features a wide rapidity gap between the two tagging
jets, events with large separation in pseudorapidity are selected for

$\Delta\eta^{j_{1},j_{2}}>3.$ (7)

The two tagging jets tend to be in the forward and backward region, so we
require them to be in the opposite directions relative to the beam axis

$\eta^{j_{1}}\cdot\eta^{j_{2}}<0.$ (8)

Due to this back-to-back nature, the two tagging jets also tend to have large
invariant mass. We select events with

$m^{j_{1},j_{2}}>600~{}{\rm GeV}.$ (9)

To limit additional emissions, we demand the minimum angular separation
between any $b$-jet and any tagging jet to be

$\Delta R_{\rm min}^{b_{i},j_{k}}>1.5,\quad\quad i,k=1,2.$ (10)

Furthermore, we make use of the azimuthal angular separation between the
$b$-jet pair and the tagging jet pair, and we require

$\Delta\Phi^{b_{1}b_{2},j_{1}j_{2}}>1.5,$ (11)

where $\Delta\Phi^{b_{1}b_{2},j_{1}j_{2}}$ is the difference in azimuthal
angle between the $b_{1}b_{2}$ system and the $j_{1}j_{2}$ system.

Since the QCD background tends to have jets with a relatively large azimuthal
angular separation, we select events such that

$\Delta\Phi^{j_{1},j_{2}}<2.$ (12)

In addition, because of the presence of the Higgs boson in the central region,
we require it to lie between the two tagging jets

$y_{\rm min}^{j_{1},j_{2}}<y^{h}<y_{\rm max}^{j_{1},j_{2}},$ (13)

where $y$ is the the rapidity of the jet or the reconstructed Higgs boson.

Finally, we cut on the invariant mass of the $b\bar{b}$ system. Due to the
mass resolution for $b\bar{b}$, we require a relatively wide mass window
around the Higgs boson mass

$115~{}{\rm GeV}<m^{b_{1}b_{2}}<135~{}{\rm GeV}.$ (14)

These cuts effectively distinguish VBF from QCD. Our goal, however, is to
identify the contribution of new physics to BSM from both QCD and from VBF.

To distinguish BSM from VBF, consider Fig. 3 which shows the distribution of
the Higgs $p_{T}$ (after both pre-selection and event selection) with a non-
zero value for $c_{Q}^{(3)}$ (left) and including the $n=2$ form factor
(right). We see that this is a powerful discriminant to separate our signal
from the backgrounds. Note that new physics from dimension-6 operators can
cause an increase or decrease in the number of expected events because the
leading effect is the interference between new physics and the SM.

Figure 3: Distribution of the Higgs $p_{T}$ in QCD, VBF, and BSM. On the left,
$c_{Q}^{(3)}/\Lambda^{2}$ is set to $-0.32~{}{\rm TeV}^{-2}$, while on the
right, a form factor with $n=2$ and a scale $\Lambda=1$ TeV is used. The
integrated luminosity is $\mathcal{L}=3~{}{\rm ab}^{-1}$.

##  5 Results

To derive the sensitivity to a new physics scale at the LHC, we adopt the
usual statistical significance,

$\frac{S}{\sqrt{B}}=\frac{|N_{\tt BSM}-N_{\tt VBF}|}{\sqrt{N_{\tt VBF}+N_{\tt
QCD}}},$ (15)

where $N_{i}=\mathcal{L}\cdot\sigma_{i}$ denotes the expected numbers of
events after event selections with luminosity $\mathcal{L}$ for sample
$i=\\{{\tt BSM},{\tt VBF},{\tt QCD}\\}$.

The contribution of new physics can result in either an enhancement or a
suppression relative to the SM VBF yield. In particular, the form factor
causes a suppression in the signal where the effect becomes more prominent
when $\Lambda$ is small. The effect of the dimension-6 operators depends on
the size of the Wilson coefficient and the sign of the interference with the
SM. In the case that $N_{\tt BSM}$ is less than $N_{\tt VBF}$, we take the
absolute value to ensure a positive significance.

In Fig. 4, we show how selecting different observables at different thresholds
affects the significance achieved at high luminosity LHC assuming
$\mathcal{L}=3~{}\text{ab}^{-1}$, for the case where
$c_{Q}^{(3)}/\Lambda^{2}=-0.32~{}{\rm TeV}^{-2}$ (left) and the $n=2$ form
factor with $\Lambda=1~{}{\rm TeV}$ (right).

Figure 4: Search significance as a function of cut on Higgs $p_{T}$-like
variables for coefficient $c_{Q}^{(3)}$ (left) and for the $n=2$ form factor
(right). Variables used are described in Sec. 3. The integrated luminosity is
$\mathcal{L}=3~{}{\rm ab}^{-1}$.

From Fig. 4 we find the transverse momentum of the Higgs boson to be the most
powerful observable to distinguish new physics from SM, as we discussed in
Sec. 4. The other variables we consider are $p_{T}^{j_{1}}+p_{T}^{j_{2}}$,
$\tilde{p}_{T}^{h}$ (see Eq. (6)), $p_{T}^{j_{1}}$,
$\sqrt{p_{T}^{j_{1}}p_{T}^{j_{2}}}$, and $p_{T}^{j_{2}}$. The tagging jets, in
particular the leading jet, can be useful to probe the high momentum-exchange
region of VBF. The subleading jet alone, however, does not correlate strongly
with the Higgs $p_{T}$ and is less useful.

Consequently, to compute the new physics reach, we use $p_{T}^{h}$ as the
final discriminant and select the threshold that optimizes the significance.
The optimal value does not differ much between operators and is around 250
GeV. We do not apply an upper cut on $p_{T}^{h}$ (as would be safer for EFT
validity), however, this has very little effect on the results since the
sensitivity is driven by the bins near $p_{T}^{h}\approx 300~{}{\rm GeV}$
Araz:2020zyh . We have explicitly verified that bins above 400 GeV only affect
our sensitivity at the percent level. The corresponding $1\sigma$ and
$2\sigma$ constraints on the dimension-6 operators are shown in Fig. 5. The
asymmetric limit between positive and negative coefficients is expected
because the leading contribution to the rate is the interference term between
the SM and new physics.

Figure 5: Limits on $c_{i}/\Lambda^{2}$ values of dimension-6 operators with
an integrated luminosity of $3~{}{\rm ab}^{-1}$.

The 95% C.L. limits are given numerically in Table 2. In the top row, we give
the limits on $c_{i}/\Lambda^{2}$ for each operator, one at a time. The VBF
search is most sensitive to the $\mathcal{O}_{Q}^{(3)}$ operator, as expected
Araz:2020zyh . Fixing $c_{i}=\pm 1$ allows us to translate the bound into a
limit on the scale $\Lambda$ which we show in the middle and bottom rows of
the table, respectively. Compared to the combination of the $\gamma\gamma$ and
$\tau^{+}\tau^{-}$ decay channels, we find the $b\bar{b}$ channel to be
roughly a factor of $2$ less sensitive Araz:2020zyh  (in comparing the
coefficients).

| $c_{Q}^{(3)}$ | $c_{Q}^{(1)}$ | $c_{u}$ | $c_{d}$ | $c_{\phi W}$
---|---|---|---|---|---
$c_{i}/\Lambda^{2}~{}[\text{TeV}^{-2}]$ | [$-0.32$, 0.40] | [$-3$, 1.72] | [$-1.5$, 3.1] | [$-5.24$, 4.20] | [$-0.82$, 0.57]
$\Lambda$ [TeV] ($c_{i}=+1$) | 1.6 | 0.76 | 0.57 | 0.49 | 1.3
$\Lambda$ [TeV] ($c_{i}=-1$) | 1.8 | 0.58 | 0.82 | 0.44 | 1.1
Table 2: 95% C.L. limits for dimension-6 operators with an integrated
luminosity of $3~{}{\rm ab}^{-1}$.

We can also compare our reach for $\mathcal{O}_{Q}^{(3)}$ with the reach in
the diboson channel. Ref. Franceschini:2017xkh  studied the channel
$q\bar{q}\to W^{*}\to W(\ell\nu)Z(\ell\ell)$ and found the reach to be roughly
$30$ times better.555Note that Ref. Franceschini:2017xkh  sets limits on
dimensionful coefficients $a_{i}$, which correspond to our parametrization via
$a_{i}=4c_{i}/\Lambda^{2}$. While diboson is an experimentally cleaner channel
than VBF, it is interesting to note that the rate for longitudinally-polarized
vector bosons is heavily suppressed relative to the rate for transversely-
polarized vector bosons Contino:2010mh . One consequence of this is that the
limits on $c_{\phi W}$ in VBF are only a factor of $2$ weaker than on
$c_{Q}^{(3)}$, while in diboson the limits are a factor of $24$ weaker
Bishara:2020vix .

Figure 6: Search significance as a function of the scale $\Lambda$ for the
form factor Eq. (5) for $n=2$ (blue) and for $n=3$ (orange). An integrated
luminosity of $3~{}{\rm ab}^{-1}$ is used.

Figure 6 shows the search significance as a function of $\Lambda$ for the form
factor, Eq. (5), with $n=2$ and $n=3$. Since $n=3$ has a steeper suppression
than $n=2$, the significance for fixed $\Lambda$ is larger. The 95% C.L.
limits are 0.8 TeV ($n=2$) and 1 TeV ($n=3$). There are no other searches for
this form factor of $VVh$ interactions, but for the analogous form factor
applied to the $t\bar{t}h$ interaction, the projected limits for $n=2$ are
$\Lambda=0.8~{}{\rm TeV},1.5~{}{\rm TeV},2.1~{}{\rm TeV}$, respectively for
the channels $h^{*}\to ZZ\to\ell\ell\ell\ell$, $h^{*}\to ZZ\to\ell\ell\nu\nu$,
and $t\bar{t}h$ Goncalves:2020vyn ; MammenAbraham:2021ssc . For $n=3$ they are
$\Lambda=1.1~{}{\rm TeV},2.1~{}{\rm TeV},2.7~{}{\rm TeV}$ in the same
channels.

Finally, we summarize all new physics results together in Fig. 7. The
strongest limits are set on the $c_{Q}^{(3)}$ coefficient, as the VBF rate
depends mostly strongly on its value Araz:2020zyh . Reasonable sensitivity is
still maintained if only one of the other coefficients $c_{Q}$, $c_{R}^{u}$,
$c_{R}^{d}$ is present. This is because in VBF there are always contributions
from both $Z$ and $W$ bosons.

Figure 7: Limits on the scale of new physics $\Lambda$ at 95% C.L. for
dimension-6 operators (assuming positive coefficients) and the form factor
with $n=2$ and $n=3$ with an integrated luminosity of $3~{}{\rm ab}^{-1}$.

##  6 Conclusions

In this paper we study the phase space region of VBF where the Higgs $p_{T}$
is large. In this region, there is sensitivity to new physics coming from high
scales, complementary to off-shell Higgs boson physics. We study the impact of
both dimension-6 operators and a nuclear-physics-inspired form factor.

We explore the $h(b\bar{b})jj$ channel where the Higgs decays on-shell to
$b\bar{b}$ at a high transverse momentum, in the hope to probe the physics at
high scales. The main backgrounds are the SM contribution to VBF and QCD where
the $b\bar{b}$ is from the splitting of a gluon. For dimension-6 operators,
the reach for new physics is up to 1.8 TeV at 95% C.L. This reach is a little
weaker than previous searches in the $\gamma\gamma$ and $\tau^{+}\tau^{-}$
Araz:2020zyh  where the decrease is due to the reduced Higgs mass resolution
and the much larger backgrounds in the $b\bar{b}$ channel.

Ultimately to maximize our reach for new physics the next step would be to
combine all existing channels. Additionally, it would be useful to include
additional decay channels of the Higgs such as $W(\ell\nu)W(\ell\nu)$. As we
have discussed, for such a channel with invisible particles, one can cut on
$p_{T}^{j_{1}}+p_{T}^{j_{2}}$, since $p_{T}^{h}$ would be unavailable, and
maintain sensitivity to the high momentum exchange region of phase space. With
enough data, it would also be interesting to do a simultaneous fit to VBF and
diboson data to fully characterize the full space of high energy primary
operators. Combining as many channels as possible is especially critical in
VBF due to the suppression in the distribution of longitudinal vectors
Contino:2010mh . The contribution of the $b\bar{b}$ channel is an
indispensable component.

###### Acknowledgements.

The authors would like to thank Joseph Boudreau for computing assistance and
Gauthier Durieux for consultation on the FeynRules model SMEFTatNLO. This work
was supported by the U.S. Department of Energy under grant No. DE-
FG02-95ER40896 and by the PITT PACC.

##  Appendix A Generator-Level Cuts

In each sample the final state at parton-level includes two $b$-quarks and two
light flavor quarks. The $b$-quarks are labelled as $b_{1}$ and $b_{2}$, in
descending order of $p_{T}$, while the light flavor quarks are labelled as
$j_{1}$ and $j_{2}$ following the same scheme.

  * •

$p_{T}^{j_{1}}>40$ GeV, $p_{T}^{j_{2}}>10$ GeV.

  * •

$p_{T}^{b_{1}}>65$ GeV, $p_{T}^{b_{2}}>45$ GeV.

  * •

$3<|\eta^{j_{1}}|<5$, $|\eta^{j_{2}}|<5$, $|\eta^{b_{1}}|<3$,
$|\eta^{b_{2}}|<3$

  * •

$|\eta^{j_{1}}-\eta^{j_{2}}|>2.5$.

  * •

$\Delta R^{j_{1},j_{2}}>0.4$, $\Delta R^{b_{1},b_{2}}>0.4$, $\Delta
R^{b,j}>0.4$, where $\Delta R^{b,j}$ refers to any pair of $b$-quark and light
flavor quark and $\Delta
R^{a,b}=\sqrt{(\eta^{a}-\eta^{b})^{2}+(\phi^{a}-\phi^{b})^{2}}$.

  * •

$p_{T}^{b\bar{b}}>120$ GeV, where $p_{T}^{b\bar{b}}$ is the transverse
momentum of the $b\bar{b}$ system.

  * •

$m^{jj}>550$ GeV where $m^{jj}$ is the invariant mass of the dijet system.

  * •

110 GeV $<m^{b\bar{b}}<140$ GeV, where $m^{b\bar{b}}$ is the invariant mass of
the $b\bar{b}$ system.

Already with these generator-level cuts, the impact of dimension-6 is
observable. Figure 8 shows the cross section of BSM where the coefficient of a
single dimension-6 operator coefficient is varied.

Figure 8: Cross section for BSM sample where the coefficient of a single
operator is varied.

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