aainstitutetext: Department of Physics, University of Siegen, 57068 Siegen,
Germanybbinstitutetext: Department of Physics and INFN, Sapienza University of
Rome, Rome I-00185, Italyccinstitutetext: School of Physics Sciences,
University of Chinese Academy of Sciences, Beijing 100039,
Chinaddinstitutetext: Center for future high energy physics, Institute of High
Energy Physics, Chinese Academy of Sciences, Beijing 100039,
Chinaeeinstitutetext: Centre for Cosmology, Particle Physics and Phenomenology
(CP3), Universit catholique de Louvain, Chemin du Cyclotron, 2, B-1348
Louvain-la-Neuve, Belgium

# Highly Boosted Higgs Bosons and Unitarity
in Vector-Boson Fusion at Future Hadron Colliders

Wolfgang Kilian b Sichun Sun c,d Qi-Shu Yan e Xiaoran Zhao d Zhijie Zhao
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the observability of new interactions which modify Higgs-pair
production via vector-boson fusion processes at the LHC and at future proton-
proton colliders. In an effective-Lagrangian approach, we explore in
particular the effect of the operator $h^{2}W_{\mu\nu}^{a}W^{a,\mu\nu}$, which
describes the interaction of the Higgs boson with transverse vector-boson
polarization modes. By tagging highly boosted Higgs bosons in the final state,
we determine projected bounds for the coefficient of this operator at the LHC
and at a future 27 TeV or 100 TeV collider. Taking into account unitarity
constraints, we estimate the new-physics discovery potential of Higgs pair
production in this channel.

††preprint: SI-HEP-2021-05, CP3-21-02

##  1 Introduction

In 2012, the Higgs boson was discovered at the LHC Aad:2012tfa ;
Chatrchyan:2012ufa . Current experimental data show that its properties agree
with the predictions of Standard Model (SM), but further measurements are
still necessary to examine the SM and search for new physics.

The SM predicts that the Higgs boson has three kinds of interaction at tree
level: (1) the Yukawa interaction with fermions; (2) the interaction with
massive vector bosons ($W^{\pm}$ and $Z$); (3) the cubic and quartic Higgs
self-interactions. One of the prime targets of the second run and future runs
of the LHC is to measure the Higgs self-couplings (3).

Higgs pair-production in vector-boson fusion (VBF) Jones:1979bq , $VV\to hh$,
is sensitive to the last two kinds of interaction. VBF-type processes are
possible at both electron-positron and hadron colliders. At a hadron collider,
the two incoming vector bosons $V=W^{\pm},Z$ are radiated from two initial
quarks. In the final state, in addition to two Higgs bosons, there are two
back-to-back hard jets which should be tagged in the forward and backward
regions of a detector, respectively. This allows us to use VBF cuts to reject
the QCD type backgrounds efficiently.

In this paper, we study double Higgs production in the VBF process in an
effective-field theory (EFT) approach. We follow the conventions given in our
previous paper Kilian:2018bhs , and use the following phenomenological
effective Lagrangian:

$\displaystyle\mathcal{L}_{EFT}=$
$\displaystyle\mathcal{L}_{\overline{SM}}+\mathcal{L}_{VVh}+\mathcal{L}_{Vh},$
(1) $\displaystyle\mathcal{L}_{VVh}=$
$\displaystyle-\left(g_{W,b1}\frac{h}{v}+g_{W,b2}\frac{h^{2}}{2v^{2}}+g_{W,b3}\frac{h^{3}}{6v^{3}}+\cdots\right)W^{+}_{\mu\nu}W^{-\,\mu\nu}$
$\displaystyle-\left(g_{A,b1}\frac{h}{2v}+g_{A,b2}\frac{h^{2}}{4v^{2}}+g_{A,b3}\frac{h^{3}}{12v^{3}}+\cdots\right)F_{\mu\nu}F^{\mu\nu}$
$\displaystyle-\left(g_{X,b1}\frac{h}{v}+g_{X,b2}\frac{h^{2}}{2v^{2}}+g_{X,b3}\frac{h^{3}}{6v^{3}}+\cdots\right)F_{\mu\nu}Z^{\mu\nu}$
$\displaystyle-\left(g_{Z,b1}\frac{h}{2v}+g_{Z,b2}\frac{h^{2}}{4v^{2}}++g_{Z,b3}\frac{h^{3}}{12v^{2}}+\cdots\right)Z_{\mu\nu}Z^{\mu\nu}$
(2) $\displaystyle\mathcal{L}_{VH}=$ $\displaystyle
g_{W,a1}\frac{2m_{W}^{2}}{v}hW^{+,\mu}W^{-}_{\mu}+g_{W,a2}\frac{m_{W}^{2}}{v^{2}}h^{2}W^{\mu}W_{\mu}+g_{W,a3}\frac{m_{W}^{2}}{3v^{3}}h^{3}W^{\mu}W_{\mu}$
$\displaystyle+g_{Z,a1}\frac{m_{Z}^{2}}{v}hZ^{\mu}Z_{\mu}+g_{Z,a2}\frac{m_{Z}^{2}}{2v^{2}}h^{2}Z^{\mu}Z_{\mu}+g_{Z,a3}\frac{m_{Z}^{2}}{6v^{3}}h^{3}Z^{\mu}Z_{\mu}+\cdots\,.$
(3)

Dots indicate higher-dimensional interactions which are not relevant for the
VBF Higgs production process that we consider. We only include the CP-
conserving interactions and omit any CP-violating operators. The relations
$g_{W,a1}=g_{W,a2}=g_{Z,a1}=g_{Z,a2}=1$ and
$g_{V,b1}=g_{V,b2}=g_{V,b3}=g_{W,a3}=g_{Z,a3}=0$ characterize the SM reference
values at the tree level, where the subscript letter $V$ denotes $W,A,X,Z$,
respectively. The corresponding terms have been removed from the SM
Lagrangian, as indicated by the overline notation $\overline{SM}$, such that
they are not double-counted.

Introducing gauge degrees of freedom in this phenomenological Lagrangian, the
vertices can be rewritten as gauge-invariant operators which are understood as
the low-energy effect of short-range structure or new heavy degrees of freedom
beyond the SM. Such effects would appear, for instance, in a composite Higgs
model – for concrete examples, cf. Qi:2019ocx ; Agrawal:2019bpm ; Xu:2019xuo ;
Li:2019ghf . In Sec.3, we relate the parameterization to the formulation in
terms of gauge-invariant operators, adopting a concrete basis and truncating
the expansion at dimension six, as commonly done in the literature.

The single Higgs couplings $g_{V,a1}$ can be determined to up to $5-10\%$ via
measuring the decay fractions $h\to WW^{*}$ and $h\to ZZ^{*}$ at the LHC.
Current LHC data exclude deviations from the SM prediction Aaboud:2017vzb ;
Sirunyan:2017exp ; Aaboud:2018jqu ; Aad:2019mbh  of more than about $15\%$.
(This limit, as well as the bounds discussed below, depends on a universal
assumption about the absence of undetected Higgs decays which we will adopt
for this paper.) For the couplings of type $g_{V,b1}$, the bounds are weaker.
For instance ($g_{V,b1}\in[0.8,4.5]$) was reported in Ref. Aaboud:2017vzb .
The measurement of double Higgs couplings hhVV ($g_{V,a2}$ and $g_{V,b2}$) is
challenging at the LHC. Recently, ATLAS reported a search for double Higgs
production in VBF Aad:2020kub , which excludes the ranges $g_{V,a2}<-0.56$ and
$g_{V,a2}>2.89$.

At the LHC, searches for a double Higgs final state focus on the gluon-gluon
fusion process $gg\to hh$. The Higgs decay channels $b\bar{b}b\bar{b}$
Aaboud:2018knk ; Sirunyan:2018tki , $b\bar{b}\gamma\gamma$ Aaboud:2018ftw ;
Sirunyan:2018iwt , $b\bar{b}\tau\tau$ Aaboud:2018sfw ; Sirunyan:2017djm  and
$b\bar{b}VV$ Aaboud:2018zhh ; Sirunyan:2017guj  have been investigated by both
ATLAS and CMS. In addition, result for the channels $WW\gamma\gamma$
Aaboud:2018ewm  and $WWWW$ Aaboud:2018ksn  were reported by ATLAS. A
combination of these searches can be found in Ref. Sirunyan:2018ayu ;
Aad:2019uzh . It is shown that the Higgs self-coupling $\lambda_{3}$ can be
constrained to $[-5,12]$ by data, while no constraint of $\kappa_{5}$ has been
reported.

As a complementary process, double Higgs production via VBF at hadron
colliders has been extensively studied in the literature Dolan:2013rja ; Liu-
Sheng:2014gxa ; Dolan:2015zja ; Bishara:2016kjn ; Arganda:2018ftn . The NLO
QCD and higher order correction for this process has been calculated in Ref.
Baglio:2012np ; Frederix:2014hta ; Dreyer:2018qbw ; Dreyer:2018rfu ;
Dreyer:2020urf ; Dreyer:2020xaj ; they find an enhancement of around $7\%$ as
it is natural for a pure electroweak process. For the high-luminosity LHC (HL-
LHC), assuming $\mathcal{L}=3$ ab-1 at 14 TeV, it is expected that the
couplings of type $g_{V,a2}$ can be constrained to $20\%$ Dolan:2015zja ,
while a precision of around $1\%$ should be achievable at a future 100 TeV
hadron collider Bishara:2016kjn . Furthermore, the hhVV coupling is also
accessible via $hVV$ or $hhV$ final states. Ref. Englert:2017gdy  argues that
a measurement of the $W^{\pm}W^{\pm}h$ final state can constrain the $hhWW$
coupling to $\mathcal{O}(100\%)$ at the HL-LHC, and to $20\%$ at a 100 TeV
collider, while the determination of this coupling from the $hhV$ final state
can only yield a weak bound Nordstrom:2018ceg .

Possible measurements of the $g_{V,a2}$ couplings, i.e., the Higgs interacting
with the logitudinal components of massive vector bosons, have thus been
covered in some detail in previous work. However, without further assumptions
it is not evident that couplings of the Higgs to transversal vector bosons
play a lesser role. In this work, we will aim at filling this gap and thus
perform a Monte-Carlo study of the sensitivity to couplings of type
$g_{V,b2}$, both for the LHC and for future high-energy hadron colliders, and
correlate this with the determination of $g_{V,a2}$.

The VBF process $VV\to hh$ receives a contribution from the Higgs cubic self-
coupling. It is well known that at hadron colliders, the Higgs self-coupling
is most accessible in the gluon-gluon fusion process Plehn:1996wb , whose
cross section is one order of magnitude larger than that of the VBF process.
This fact has received a lot of attention Baur:2002rb ; Li:2013flc ;
Cao:2015oaa ; Cao:2016zob ; Baur:2002qd ; Ren:2017jbg ; Baur:2003gp ;
Yao:2013ika ; Kling:2016lay ; Chang:2018uwu ; Kim:2018uty ; He:2015spf ;
Papaefstathiou:2012qe ; Baur:2003gpa ; Dolan:2012rv ; Barr:2013tda ;
deLima:2014dta ; Behr:2015oqq ; Barger:2013jfa ; Barr:2014sga ;
Papaefstathiou:2015iba ; Li:2015yia ; Zhao:2016tai ; Contino:2016spe ;
Goncalves:2018yva . A measurement of the VBF process will provide additional
precision for the Higgs self-coupling. However, for simplicity we will assume
here that the couplings which are accessible in gluon-gluon fusion are known,
and we fix those at their SM value. This allows us to focus on the couplings
which are specific to the VBF class of processes, and lets us more easily
estimate the sensitivity potential for those.

This paper is organized as follows. In Sec. 2, we briefly introduce the mass-
drop method used for tagging highly boosted Higgs bosons, and use it to
explore the SM case at the LHC and at a 100 TeV collider. In Sec. 3, we
investigate the potential for the discovery of new physics via multi-Higgs
production, in form of the interactions discussed above. In Sec. 4, we provide
projections for bounds on $g_{V,b2}$ and $g_{V,a2}$ for different collision
energies. We conclude this paper with a discussion of our findings in Sec. 5.

##  2  Higgs pair production in the Standard Model

We will focus on the signal process $pp\to hhjj\to 4b2j$ due to the reason
that the decay channel $h\to b\bar{b}$ has the largest branch ratio. The
partonic signal events are generated using WHIZARD Kilian:2007gr  with the
cuts listed in Table 1. We takes the parton distribution functions from
CTEQ6l1 Pumplin:2002vw .

In this work, we consider the main backgrounds $pp\to t\bar{t}\to 2b4j$,
$pp\to 2b4j$ (QCD) and $pp\to 4b2j$. The background events $pp\to t\bar{t}\to
2b4j$ are generated by WHIZARD. We have cross-checked the results with
Madgraph Alwall:2014hca . Pure-QCD partonic events of type $pp\to 2b4j$ and
$pp\to 4b2j$ are generated using ALPGEN Mangano:2002ea .

For all event samples, parton shower and hadronization are performed by Pythia
8 Sjostrand:2007gs . Jets are reconstructed by FastJet Cacciari:2011ma  using
the anti-$k_{t}$ algorithm Cacciari:2008gp  with a jet radius $R=0.4$ and
transverse momentum cut $P_{t}>20$ GeV. We do not account for detector effects
in detail but insert values for efficiency and mistagging rates where
appropriate.

###  2.1 Analysis method at the 14 TeV LHC

In Table 2 (first column) we list the expected number of signal and background
events in the SM for the LHC with $\sqrt{s}=14$ TeV and luminosity
$\mathcal{L}=3$ ab-1. To suppress the QCD background, we require four b-tagged
jets in the final state ($n_{b}=4$). We assume the b-tagging efficiency
$\epsilon_{b}=0.7$ and mistagging rate $\epsilon_{miss}=0.001$. The number of
events after applying the b-tagging requirement are listed in the 2nd column
of Table 2.

To identify two forward jets in the VBF process, we first select the jet with
highest energy and label it as $j_{1}$. If its energy satisfies
$E_{j_{1}}>500$ GeV, we scan over all other jets and determine the maximal
rapidity difference $\Delta\eta(j_{1},j)$ and invariant mass $m(j_{1},j)$ with
respect to the leading jet. If the conditions $\Delta\eta(j_{1},j)_{max}>3.6$
and $m(j_{1},j)_{max}>500$ GeV are met simultaneously, we identify the most
energetic j as $j_{2}$ and label corresponding pair of jets as the tagging
forward jets of a VBF process. Otherwise, the event is rejected. The number of
events after applying this VBF cut are listed in the 3rd column of Table 2.

With these b-jet and forward-jet tagging requirements, the backgrounds $pp\to
t\bar{t}$ and $pp\to 2b4j$ are greatly reduced. The dominant remaining
background originates from QCD processes $pp\to 4b2j$, where the cross section
after cuts is still five orders of magnitude larger than that of the signal.

To further suppress this huge background, we select those events with massive
jets formed by highly boosted Higgs bosons and adopt the mass drop method
Butterworth:2008iy .

Cuts |  $\sqrt{s}=14$ TeV |  $\sqrt{s}=27$ TeV |  $\sqrt{s}=100$ TeV
---|---|---|---
$P_{t}(j)$ |  $>20$ GeV |  $>20$ GeV |  $>30$ GeV
$\Delta R(j,j)$ | $>0.8$ | $>0.8$ | $>0.8$
$|\eta(j)|$ | $<5.0$ | $<5.0$ | $<8.0$
$\Delta\eta(j_{1},j_{2})$ | $>3.6$ | $>3.6$ | $>4.0$
$m(j_{1},j_{2})$ |  $>500$ GeV |  $>500$ GeV |  $>800$ GeV
Table 1: Acceptance cuts used for the calculation of VBF Higgs production in $pp$ collision (VBF cuts), for three different collider energies. The $j_{1}$ and $j_{2}$ are the tagged forward jets for VBF process.  Process | $\sigma\times\mathcal{L}$ | $n_{b}=4$ | VBF
---|---|---|---
SM signal | $993$ | $238$ | $171$
$pp\to 4b2j$ | $2.28\times 10^{8}$ | $5.47\times 10^{7}$ | $1.86\times 10^{7}$
$pp\to 2b4j$ (QCD) | $2.38\times 10^{10}$ | $1.14\times 10^{4}$ | $3.85\times 10^{4}$
$pp\to t\bar{t}\to 2b4j$ | $7.89\times 10^{8}$ | $387$ | $58$
Table 2: The cut efficiencies of b-tagging and VBF at 14 TeV LHC are
demonstrated. The total integrated luminosity is assumed to be
$\mathcal{L}=3000$ fb-1. The b-tagging efficiency is $\epsilon_{b}=0.7$, and
the mis-tagging rate of light quarks is $\epsilon_{miss}=0.001$.

####  2.1.1 The mass drop method for highly-boosted Higgs boson tagging

A significant fraction (of the order of $10\;\%$) of the VBF event sample in
the SM contains highly boosted Higgs bosons. A highly boosted Higgs boson has
a large transverse momentum ($P_{t}>200$ GeV) and can be detected in the
central region of the detector. The decay products of the Higgs boson
typically form a fat jet with a large jet mass, if a large cone parameter is
used for the analysis. A fraction of the QCD background events also contains
massive jets, but the fat jets originating from Higgs pairs can in principle
be distinguished by their characteristic jet sub-structure.

In recent years, various methods for jet substructure analysis have been
developed: (1) Jet-grooming methods aim at removing soft radiation which is
unlikely to originate from the hard process (see, e.g.,Butterworth:2008iy ).
(2) Radiation-constraint methods impose a cut on jet shape to separate the
signal from the background (see, e.g., Thaler:2010tr ). (3) Prong-finder
methods detect a massive boosted object as a fat jet with multiple hard cores
by exploiting the recombination history of the jet algorithm. As a particular
prong-finder algorithm, the mass-drop tagger method is particularly suited for
isolating boosted Higgs bosons, decaying to $b\bar{b}$, from the QCD
background Butterworth:2008iy . A detail review of jet substructure can be
found in Ref. Marzani:2019hun

In this work, we adopt the mass-drop tagger Butterworth:2008iy  as a means for
tagging highly boosted Higgs bosons in the final state of the VBF process. The
method consists of the following two steps: (1) We first identify jets by the
standard anti-$k_{t}$ algorithm with the cone parameter $R=0.4$. After
identifying the forward jets associated with the VBF process, we recluster the
remaining jets using the Cambridge/Aachen (CA) algorithm Dokshitzer:1997in ;
Wobisch:1998wt  with $R=1.2$. (2) If there are 2 to 4 CA jets in an event, and
its leading or subleading jet has a transverse momentum satisfying $P_{t}>200$
GeV and a jet mass $m_{j}>100$ GeV, we apply the following procedure to tag
candidates for highly boosted Higgs bosons.

  1. 1.

Undo the last step of clustering of jet $j$ to get two daughter jets $j_{1}$
and $j_{2}$ with $m_{j1}>m_{j2}$.

  2. 2.

If the conditions

$\displaystyle m_{j1}<\mu
m_{j}\qquad\text{and}\qquad\frac{\min(P_{t}(j_{1}),P_{t}(j_{2}))}{m^{2}_{j}}\Delta
R^{2}_{j1,j2}>y_{cut}$ (4)

are satified, we identify $j$ as a fat jet associated with a highly boosted
Higgs boson.

  3. 3.

Otherwise, we redefine $j$ as $j_{1}$ and repeat the above procedure.

There are two dimensionless parameters $\mu$ and $y_{cut}$ in this method, as
given in Eq. (4). In this work, we fix them as $\mu=0.67$ and $y_{cut}=0.09$.
After two subjets are found, we also apply the filtering method to remove soft
radiation which originates from the underlying event and contaminates CA jets
with a larger cone size.

After applying this tagging algorithm, both signal and background events fall
into three classes: 2-boosted Higgs (2BH), 1-boosted Higgs (1BH) and 0-boosted
Higgs (0BH) candidate events. The number of events for each class are listed
in Table 3.

We observe that the fraction of signal events in the 2BH category is still
small ($2-3\%$). Nevertheless, the corresponding background is two orders of
magnitude lower than for the 0BH category, where the fraction of signal events
is even less. The 1BH category falls in between. The tagger significantly
improves the chance for finding signal events, but by itself it is clearly not
sufficient for a measurement if the rate is SM-like.

(a) 2BH

(b) 1BH

(c) 0BH

Figure 1: Distributions of the reconstructed Higgs mass with (a) 2-boosted
Higgs events, (b) 1-boosted Higgs events, and (c) 0-boosted Higgs events.

In order to construct kinematic observables which improve the signal vs.
background discrimination, we can reconstruct the mass peaks of the Higgs
bosons for each event category. For each of the 2BH events, two jet masses
$m_{j}$ should peak around Higgs the boson mass $m_{h}$, as shown in Fig. (1).
For a 1BH event, the jet mass of the leading fat jet should peak around the
mass of the Higgs boson, while the mass of the second reconstructed Higgs
candidate should coincide with the invariant mass of two b jets. For a 0BH
event, two Higgs boson candidates are reconstructed by using the $\chi^{2}$
method. To this end, we define $\chi^{2}$ as follows:

$\displaystyle\chi^{2}(m)$ $\displaystyle=$
$\displaystyle\frac{|m(j_{1},j_{2})-m_{h}|^{2}}{\sigma^{2}_{j}}+\frac{|m(j_{3},j_{4})-m_{h}|^{2}}{\sigma^{2}_{j}}\,,$
(5)

We assume $m_{h}=125$ GeV, and $m(j_{1},j_{2})$ and $m(j_{3},j_{4})$ are the
invariant mass of two jet pairs in the final state, scanning over each
combination. $\sigma_{j}=10$ GeV is used to take into account the error in jet
energy resolution. The pairing which minimizes $\chi^{2}$ is selected, and the
corresponding invariant masses determined by the pairs of jets are taken as
the reconstructed masses of Higgs bosons.

We display the distributions of the reconstructed mass of the leading Higgs
boson in Fig. 1. The shapes of the 2BH and 1BH cases are almost identical. We
note that the Higgs peak in the 2BH case is narrower than in the 0BH case,
since wrong pairings are rejected more efficiently.

| 2-boosted Higgs | 1-boosted Higgs | 0-boosted Higgs
---|---|---|---
| (2BH) | (1BH) | (0BH)
SM Signal | $4$ | $21$ | $146$
$pp\to 4b2j$ | $1.17\times 10^{5}$ | $1.56\times 10^{6}$ | $1.69\times 10^{7}$
$pp\to 2b4j$ (QCD) | $28$ | $349$ | $3.81\times 10^{4}$
$pp\to t\bar{t}\to 2b4j$ | $3$ | $13$ | $42$
Table 3: The numbers of events in the 2BH case, 1BH case and 0BH case at 14
TeV LHC are tabulated.

####  2.1.2 Multivariate analysis

After applying the VBF cuts and boosted-Higgs tagging, in order to further
improve the ratio of signal over background, we employ the method of boosted
decision tree (BDT) which utilizes the correlation of observables in the
signal and can help to further suppress background. We select the following
observables as the input to our BDT analysis:

  * •

$P_{t}(h_{i})$: the transverse momenta of the two reconstructed Higgs bosons.

  * •

$m(h_{i})$: the invariant masses of the reconstructed Higgs bosons.

  * •

$P_{t}(j_{i})$: the transverse momenta of the two forward jets.

  * •

$E(j_{i})$: the energies of the forward jets.

  * •

$\eta(j_{i})$: the pseudo-rapidity of the forward jets.

  * •

$m(j,j)$: the invariant mass of the forward jets.

  * •

$\Delta\eta(j,j)$: the rapidity difference of the forward jets.

  * •

$P_{t}(j^{sub}_{i})$: the transverse momenta of the two subjets of each highly
boosted Higgs boson.

  * •

$m(h,h)$: the invariant mass of the two Higgs boson candidates.

  * •

$\chi^{2}_{min}$ (only for the 0BH case): the minimum value of $\chi^{2}$.

The results of the BDT response are presented in the Fig. 2. Obviously, signal
and background can separated best in the 2BH case. For the 1BH and 0BH cases,
extracting the signal is challenging even after exploiting the BDT method.

We can optimize the BDT cut to achieve the maximal significance, which is
defined as $S/\sqrt{S+B}$, where $S$ is the event number of the signal and $B$
is the event number of the total background. The efficiencies and
significances of the optimized BDT cut are listed the Table 4 for all three
event classes.

Comparing the results given in Table 3 and Table 4, we conclude that the BDT
reduces the background by one additional order of magnitude, improving on the
sequential cut method. Nevertheless, the final number of SM signal events is
still tiny compared to the background in all cases, and the significance can
only reach $0.02$ (2BH, 1BH) or $0.04$ (0BH). This is still far from the
requirement of a discovery at the LHC.

(a) 2-boosted Higgs

(b) 1-boosted Higgs

(c) 0-boosted Higgs

Figure 2: The response of the discriminants to the SM signal and background at 14 TeV LHC with (a) 2-boosted events, (b) 1-boosted events, and (c) 0-boosted events. | 2-boosted Higgs | 1-boosted Higgs | 0-boosted Higgs
---|---|---|---
| (2BH) | (1BH) | (0BH)
SM Signal with BDT cut | $3$ | $13$ | $90$
Background with BDT cut | $2.06\times 10^{4}$ | $3.05\times 10^{5}$ | $4.42\times 10^{6}$
$S/B$ | $1.40\times 10^{-4}$ | $4.30\times 10^{-5}$ | $2.04\times 10^{-5}$
$S/\sqrt{S+B}$ | $0.020$ | $0.024$ | $0.043$
Table 4: The significances of the BDT at 14 TeV LHC are demonstrated.

###  2.2 Analysis of $pp\to hhjj$ in SM at 100 TeV hadron collider

We now apply the methods as described above to the VBF process at a future 100
TeV hadron collider. We assume a high integrated luminosity of
$\mathcal{L}=30$ ab-1, and adapt all selection parameters to the different
environment as appropriate.

We require that the most energetic jet has an energy $E_{j}>800$ GeV. The VBF
cuts are adjusted as $\Delta\eta_{jj,max}>4.0$ and $m_{jj,max}>800$ GeV. Table
5 shows the number of events after imposing b-tagging and VBF cuts. Due to the
increased cross section at high energy and the high luminosity of the
collider, the number of signal events is increased by a factor 1000 compared
to the high-luminosity LHC. After applying b-tagging and VBF cuts, the total
number of the signal events is $8.96\times 10^{4}$. Of course, the number of
background events also increases and reaches $10^{9}$, so background reduction
is still essential.

The number of events for the 2BH, 1BH and 0BH cases are listed in Table 6. As
one would expect, more events with highly boosted Higgs bosons can be observed
at the 100 TeV collider than at the LHC. This is illustrated by the SM curves
of Fig. (4). To enable signal/background discrimination in these three cases,
we apply the BDT method as described above. The results are shown in Fig. 3
and Table 7. We can set the BDT cut in a region where the residual background
is effectively zero.

These results show that it is possible in principle to discover the SM signal
at a 100 TeV collider. The large rate of the VBF process at high collision
energy leaves enough signal events after all measures for background reduction
have been applied. We note, however, that pileup effects in a high luminosity
run might pose a challenge for analyses such as this one. For a final verdict
on the detectability of the SM signal, a more sophisticated full-simulation
study would be necessary which is beyond the scope of this paper.

Process | $\sigma\times\mathcal{L}$ | $n_{b}=4$ | VBF
---|---|---|---
SM signal | $4.28\times 10^{5}$ | $1.03\times 10^{5}$ | $8.96\times 10^{4}$
$pp\to 4b2j$ | $5.02\times 10^{10}$ | $1.21\times 10^{10}$ | $8.51\times 10^{9}$
$pp\to 2b4j$ | $5.04\times 10^{12}$ | $2.47\times 10^{6}$ | $1.83\times 10^{6}$
$pp\to t\bar{t}\to 2b4j$ | $3.93\times 10^{11}$ | $1.93\times 10^{5}$ | $6.20\times 10^{4}$
Table 5: The cut efficiencies of b-tagging and VBF at 100 TeV collider are demonstrated. Here, the total integrated luminosity is assumed to be $\mathcal{L}=30$ ab-1. b-tagging efficiency is $\epsilon_{b}=0.7$, and miss tagging rate is $\epsilon_{miss}=0.001$.  | 2-boosted Higgs | 1-boosted Higgs | 0-boosted Higgs
---|---|---|---
| (2BH) | (1BH) | (0BH)
SM Signal | $4265$ | $1.76\times 10^{4}$ | $6.77\times 10^{4}$
$pp\to 4b2j$ | $3.65\times 10^{8}$ | $2.01\times 10^{9}$ | $6.13\times 10^{9}$
$pp\to 2b4j$ | $8.35\times 10^{4}$ | $4.40\times 10^{5}$ | $1.31\times 10^{6}$
$pp\to t\bar{t}\to 2b4j$ | $8244$ | $2.20\times 10^{4}$ | $3.18\times 10^{4}$
Table 6: The numbers of events for 2BH, 1BH and 0BH cases at 100 TeV collider
are tabulated.

(a) 2BH

(b) 1BH

(c) 0BH

Figure 3: The response of the discriminants to the SM signal and background at 100 TeV collider with (a) 2BH case, (b) 1BH case, and (c) 0BH case. | 2-boosted Higgs | 1-boosted Higgs | 0-boosted Higgs
---|---|---|---
| (2BH) | (1BH) | (0BH)
SM Signal with BDT cut | $298$ | $12$ | $90$
Background with BDT cut | $0$ | $0$ | $78$
$S/B$ | - | - | $1.15$
$S/\sqrt{S+B}$ | $17.27$ | $3.43$ | $6.923$
Table 7: The significances of the BDT at 100 TeV collider are demonstrated.

##  3 Multi-Higgs production in VBF processes with dimension-6 operators

In this section, we extend our study of the $pp\to hhjj$ process to
contributions beyond the SM. Given the phenomenological Lagrangian 3, such
effects are parameterized in terms of the coefficients $g_{i}$, where we focus
in particular on the $g_{V,b2}$ coupling that describes a non-SM double-Higgs
interaction with the transverse polarization components of the vector bosons.

###  3.1 Effective-Theory description

If we introduce the gauge symmetry of the SM in the phenomenological
description, any anomalous effects can be re-expressed in terms of higher-
dimensional gauge-invariant operators. To avoid redundancy, it is convenient
to choose a particular operator basis such as the SILH basis Giudice:2007fh
that we adopt for this work. As usual, we truncate the power-series expansion
of the gauge-invariant effective theory at dimension six. This truncation
allows us to express the phenomenological couplings in 3 in terms of a small
number of SILH operator coefficients. Such a simplification allows for
relating quantitative results for the VBF Higgs-pair process to the analysis
of existing data, as well as to studies of different processes and
interations.

We list the relevant terms in the SILH basis,

$\displaystyle\begin{split}{\cal
L}_{\text{SILH}}\supset&\frac{ic_{W}g}{2m_{\rho}^{2}}\left(H^{\dagger}\sigma^{i}\overleftrightarrow{D^{\mu}}H\right)(D^{\nu}W_{\mu\nu})^{i}+\frac{ic_{B}g^{\prime}}{2m_{\rho}^{2}}\left(H^{\dagger}\overleftrightarrow{D^{\mu}}H\right)(\partial^{\nu}B_{\mu\nu})\\\
&+\frac{ic_{HW}g}{16\pi^{2}f^{2}}(D^{\mu}H)^{\dagger}\sigma^{i}(D^{\nu}H)W_{\mu\nu}^{i}+\frac{ic_{HB}g^{\prime}}{16\pi^{2}f^{2}}(D^{\mu}H)^{\dagger}(D^{\nu}H)B_{\mu\nu}\\\
&+\frac{c_{\gamma}{g^{\prime}}^{2}}{16\pi^{2}f^{2}}\frac{g^{2}}{g_{\rho}^{2}}H^{\dagger}HB_{\mu\nu}B^{\mu\nu}.\end{split}$
(6)

In the following, we make use of the result from Kilian:2017nio ;
Kilian:2018bhs  in Table 8 which relates the phenomenological coefficients
such as $g_{V,b}$ to the parameters of 6. In particular, the couplings
$g_{V,b}$ that we are interested in, only depend on $c_{HW},c_{HB}$ and
$c_{\gamma}$, but not on $c_{W}$ and $c_{B}$ which determine $g_{V,a}$. The
SILH parameterization has been evaluated as the low-energy limit of various
ultra-violet theories, such as models where the Higgs becomes a pseudo Nambu-
Goldstone boson theory or holographic completions with extra dimensions; cf.
Qi:2019ocx ; Agrawal:2019bpm ; Xu:2019xuo ; Li:2019ghf . We note the
correlations between $g_{V,a1}$ and $g_{V,a2}$, $g_{V,b1}$ and $g_{V,b2}$
which follow from the dimension-six truncation, and should be tested against
real data if actual deviations from the SM show up.

In the SILH effective Lagrangian, Higgs interactions include further
dimension-six operators such as

$\displaystyle\frac{c_{H}}{2f^{2}}\partial^{\mu}\left(H^{\dagger}H\right)\partial_{\mu}\left(H^{\dagger}H\right)+\frac{c_{T}}{2f^{2}}\left(H^{\dagger}{\overleftrightarrow{D^{\mu}}}H\right)\left(H^{\dagger}{\overleftrightarrow{D}}_{\mu}H\right).$
(7)

The coefficients of these operators are switched off in our analysis because
they are to be measured in different processes. In particular, the first one
entails a global shift to all Higgs interactions which is equivalent to a
modified Higgs total width, while the latter violates the custodial symmetry
of weak interactions and globally modifies $ZZ$ vs. $WW$ Higgs couplings. In
our current work we assume that no custodial-symmetry violation beyond the SM
is present, as detailed below.

In the third column of Table 8, we present some typical numerical values for
these parameters. Considering $g\sim 0.654,g^{\prime}\sim 0.350,v\sim
246\text{GeV},\text{tan}\theta=g^{\prime}/g=0.535$,
$\alpha=\frac{g^{2}v^{2}}{32\pi^{2}f^{2}}=8.2\times 10^{-5}$ with $f=1$ TeV
and $g_{\rho}\sim\frac{4\pi}{\sqrt{3}}$, $m_{\rho}=g_{\rho}f=7.3$ TeV, we can
simplify the expressions in this table with

$\displaystyle\zeta_{h}\sim 1,\quad\zeta_{A}\sim 1,\quad
y_{ZA}\sim\alpha\frac{g^{\prime}}{g}(c_{HW}-c_{HB})\quad$ (8)
$\displaystyle\zeta_{Z}\sim
1-\frac{1}{8}\alpha^{2}(\frac{g^{\prime}}{g})^{2}(c_{HW}-c_{HB})^{2},\quad\zeta_{AZ}\sim\alpha\frac{g^{\prime}}{4g}(c_{HW}-c_{HB}),\quad\zeta_{W}\sim
1-\frac{\alpha}{2}c_{HW}$ (9)

(Our notation slightly differs from the definitions for $\bar{c}_{i}$ used in
Ref. Ellis:2018gqa .) To simplify our discussion on the future collider
sensitivity on $g_{V,b2}$ and $g_{V,a2}$, we assume custodial-symmetry
relations, namely that $g_{W,b2}=g_{Z,b2}$ and $g_{X,b2}=g_{A,b2}=0$ and
$g_{W,a2}=g_{Z,a2}$. This roughly renders $c_{\gamma},c_{HB},c_{B}\sim 0$, so
we can perform a two-parameter study on $c_{W}$ and $c_{HW}$. The interactions
proportional to $g_{W,b2}$ and $g_{W,a2}$ ($g_{Z,b2}$ and $g_{Z,a2}$) account
for dominant contributions to the cross section of $pp\to hhjj$ process, up to
$70\%$ (30$\%$), respectively.

| SILH | numerics with assumptions below
---|---|---
$g_{W,b1}$ | $c_{HW}\frac{g^{2}v^{2}}{32\pi^{2}f^{2}}\zeta_{h}\zeta^{2}_{W}$ | $10^{-4}c_{HW}$
$g_{W,b2}$ | $g_{W,b1}\zeta_{h}$ | $10^{-4}c_{HW}$
$g_{A,b1}$ | $-c_{\gamma}\frac{g^{2}v^{2}}{8\pi^{2}f^{2}}\frac{{g^{\prime}}^{2}}{g^{2}_{\rho}}\cos^{2}{\theta}\zeta_{h}\zeta_{A}^{2}$ |  -$10^{-6}c_{\gamma}$
$g_{A,b2}$ | $g_{A,b1}\zeta_{h}$ |  -$10^{-6}c_{\gamma}$
$g_{X,b1}$ | $\frac{g{g^{\prime}}v^{2}}{64\pi^{2}f^{2}}\left[(c_{HW}-c_{HB})+8c_{\gamma}\frac{g^{2}}{g^{2}_{\rho}}\sin^{2}\theta\right]\zeta_{h}\zeta_{A}\zeta_{Z}$ |
$+c_{\gamma}\frac{g^{2}v^{2}}{4\pi^{2}f^{2}}\frac{{g^{\prime}}^{2}}{g^{2}_{\rho}}\cos^{2}{\theta}\zeta_{h}\zeta_{AZ}^{2}$ | $10^{-5}(c_{HW}-c_{HB})$
$g_{X,b2}$ | $g_{X,b1}\zeta_{h}$ | $10^{-5}(c_{HW}-c_{HB})$
$g_{Z,b1}$ | $\frac{g^{2}v^{2}}{32\pi^{2}f^{2}}(c_{HW}+c_{HB}\tan^{2}{\theta})\zeta_{h}\zeta^{2}_{Z}-c_{\gamma}\frac{g^{2}v^{2}}{8\pi^{2}f^{2}}\frac{{g^{\prime}}^{2}}{g^{2}_{\rho}}\cos^{2}{\theta}\zeta_{h}\zeta_{AZ}^{2}$ |
| $-\frac{g{g^{\prime}}v^{2}}{64\pi^{2}f^{2}}\left[(c_{HW}-c_{HB})+8c_{\gamma}\frac{g^{2}}{g^{2}_{\rho}}\sin^{2}\theta\right]\zeta_{h}\zeta_{AZ}\zeta_{Z}$ | $10^{-4}(c_{HW}+0.29c_{HB})$
$g_{Z,b2}$ | $g_{Z,b1}\zeta_{h}$ | $10^{-4}(c_{HW}+0.29c_{HB})$
$g_{W,a1}$ |  $\left[1-\left(c_{W}\frac{g^{2}v^{2}}{m^{2}_{\rho}}+c_{HW}\frac{g^{2}v^{2}}{16\pi^{2}f^{2}}\right)\right]\zeta_{h}\zeta^{2}_{W}$. |  $1-2\times 10^{-4}(3c_{W}+c_{HW}$ )
$g_{W,a2}$ | $\left[1-3\left(c_{W}\frac{g^{2}v^{2}}{m^{2}_{\rho}}+c_{HW}\frac{g^{2}v^{2}}{16\pi^{2}f^{2}}\right)\right]\zeta^{2}_{h}\zeta^{2}_{W}$ |  $1-6\times 10^{-4}(3c_{W}+c_{HW}$ )
$g_{Z,a1}$ | $\left[1-\left(c_{W}\frac{g^{2}v^{2}}{m^{2}_{\rho}}+c_{B}\frac{{g^{\prime}}^{2}v^{2}}{m^{2}_{\rho}}+c_{HW}\frac{g^{2}v^{2}}{16\pi^{2}f^{2}}+c_{HB}\frac{{g^{\prime}}^{2}v^{2}}{16\pi^{2}f^{2}}\right)\right]\zeta_{h}\zeta^{2}_{Z}$ | $1-2\times 10^{-4}[3(c_{W}+0.29c_{B})+c_{HW}+0.29c_{HB}]$
$g_{Z,a2}$ |  $\left[1-3\left(c_{W}\frac{g^{2}v^{2}}{m^{2}_{\rho}}+c_{B}\frac{{g^{\prime}}^{2}v^{2}}{m^{2}_{\rho}}+c_{HW}\frac{g^{2}v^{2}}{16\pi^{2}f^{2}}+c_{HB}\frac{{g^{\prime}}^{2}v^{2}}{16\pi^{2}f^{2}}\right)\right]\zeta^{2}_{h}\zeta^{2}_{Z}$. | $1-6\times 10^{-4}[3(c_{W}+0.29c_{B})+c_{HW}+0.29c_{HB}]$
Table 8:  Relations between the phenomenological Lagrangian parameters in
(1-3) (first column), the SILH effective Lagrangian 6 (second column), The
extra parameters $\zeta^{n}_{h}$, $\zeta^{n}_{W}$, $\zeta^{n}_{Z}$,
$\zeta^{n}_{A}$, $\zeta^{n}_{AZ}$ (defined in Kilian:2017nio ; Kilian:2018bhs
) are induced by the Higgs and gauge-boson wave-function normalization,
respectively.

###  3.2 Higgs-pair couplings to transverse vector polarizations in VBF

Introducing non-SM effects proportional to the $g_{V,b2}$ couplings, we
consider kinematical distributions and their discriminating power. In Fig. 4,
we display the distributions of the $P_{t}$ of the leading Higgs boson and the
invariant mass the Higgs-bosons pair, respectively. The subfigures (a) and (b)
show the distributions at the LHC with collision energy 14 TeV. The new-
physics effect is illustrated by the green and blue curves which correspond to
two different values of $g_{V,b2}$, namely $g_{V,b2}=0.09$ and
$g_{V,b2}=0.18$, repectively. We observe a huge enhancement at high $P_{t}$ in
the curves which include the new interaction of Higgs bosons with transversal
gauge bosons. For the selected parameter values, the fraction of events in the
region with $P_{t}>200$ GeV increases to $50\%$ and $70\%$, respectively,
while in the SM this fraction is just $18\%$.

The corresponding distributions at a 100 TeV hadron collider are shown in the
Fig. 4 (c) and (d), where we select $g_{V,b2}=0.018$ and $0.024$. In this
case, the fraction of events in the region with $P_{t}>200$ GeV increases to
$50\%$ and $60\%$, respectively, while in the SM, it is $25\%$.

Subfigures (b) and (d) contain the Higgs-pair invariant-mass distributions for
the LHC and for the 100 TeV collider, which are likewise enhanced in the high-
mass region if anomalous effects are included.

()

()

()

()

Figure 4: Distributions of (a) the $P_{t}$ of the leading Higgs and (b) the
invariant mass of the Higgs pair at the 14 TeV LHC, with $g_{V,b2}=0.09$ and
$g_{V,b2}=0.18$. The corresponding distributions for a 100 TeV collider are
shown in (c) and (d), with $g_{V,b2}=0.018$ and $g_{V,b2}=0.024$.

As a concrete numerical example for the analysis in presence of new-physics
effects, in Table 9 we demonstrate the cut flow for the value $g_{V,b2}=0.18$
at the 14 TeV LHC. For this parameter value, the total signal cross section
($\sigma\times\mathcal{L}$) is enhanced by a factor of 4 over the SM value. As
Figure (4a) demonstrates, most of enhancement occurs in the boosted region.
After b-tagging and VBF cuts have been applied, there are more than $600$
signal events left that can be observed. In Table. 10 we present the resulting
numbers for the three classes of events. In this scenario, the signal accounts
for $25\%$ and $35\%$ of the total events in the 2BH and 1BH classes,
respectively. Compared with the results given in Table 4, the increase in
signal events in the 2BH (1BH) case amounts to a factor 35 (10), respectively.

When the BDT method is applied, this result is further improved, as shown in
Fig. 5 and Table. 11. By comparing this with the results of Table 4, we
conclude that isolating the boosted-Higgs region significantly enhances the
discovery potential of this process, both for the 2BH and 1BH cases.

Process | $\sigma\times\mathcal{L}$ | $n_{b}=4$ | VBF
---|---|---|---
$g_{V,b2}=0.18$ signal | $4243$ | $1019$ | $617$
$pp\to 4b2j$ | $2.28\times 10^{8}$ | $5.47\times 10^{7}$ | $1.86\times 10^{7}$
$pp\to 2b4j$ | $2.38\times 10^{10}$ | $1.14\times 10^{4}$ | $3.85\times 10^{4}$
$pp\to t\bar{t}\to 2b4j$ | $7.89\times 10^{8}$ | $387$ | $58$
Table 9: The cut efficiencies of b-tagging and VBF at 14 TeV LHC are demonstrated. Here, the total integrated luminosity is assumed to be $\mathcal{L}=3$ ab-1. b-tagging efficiency is $\epsilon_{b}=0.7$, and mis-tagging rate is $\epsilon_{miss}=0.001$.  | 2-boosted Higgs | 1-boosted Higgs | 0-boosted Higgs
---|---|---|---
| (2BH) | (1BH) | (0BH)
$g_{V,b2}=0.18$ Signal | $153$ | $217$ | $247$
$pp\to 4b2j$ | $1.17\times 10^{5}$ | $1.56\times 10^{6}$ | $1.69\times 10^{7}$
$pp\to 2b4j$ | $28$ | $349$ | $3.81\times 10^{4}$
$pp\to t\bar{t}\to 2b4j$ | $3$ | $13$ | $42$
Table 10: The numbers of 2BH, 1BH and 0BH cases at 14 TeV LHC are tabulated.

By contrast, in the 0BH case the signal significance is too low for the chosen
collider parameters and benchmark values of $g_{V,b2}$.

(a) 2BH

(b) 1BH

(c) 0BH

Figure 5: The response of the discriminants to the $g_{V,b2}$ signal and background at 14 TeV LHC with (a) 2BH, (b) 1BH, and (c) 0BH. | 2-boosted Higgs | 1-boosted Higgs | 0-boosted Higgs
---|---|---|---
| (2BH) | (1BH) | (0BH)
$g_{V,b2}=0.18$ Signal with BDT cut | $54$ | $30$ | $10$
Background with BDT cut | $0$ | $0$ | $3430$
$S/B$ | - | - | $2.99\times 10^{-3}$
$S/\sqrt{S+B}$ | $7.348$ | $5.477$ | $0.175$
Table 11: The significances of the BDT at 14 TeV collider are demonstrated.

###  3.3 Unitarity limits and discovery reach

The interactions of Higgs bosons with transverse gauge bosons in the
Lagrangian (3) involve derivative couplings, and therefore are enhanced over
the SM interactions for high values of the four-momenta. This is clearly
visible in Fig. 4, where the contribution of the new interactions dominates
for sufficiently large values of the transverse momentum $P_{t}$ or the Higgs-
pair invariant mass $m(h,h)$.

If this behavior is naively extrapolated, the computed amplitudes will violate
unitarity constraints. In Ref. Kilian:2018bhs , we have derived a generic
unitarity bound for the $pp\to hhjj$ process, which relates the value of
$g_{V,b2}$ to a UV cutoff $\Lambda_{UV}$.

$\displaystyle\frac{\Lambda^{4}_{UV}}{2^{9}\pi^{2}v^{4}}|g_{V,b2}|^{2}\leq\frac{1}{4},$
(10)

For energy-momentum values beyond $\Lambda_{UV}$, the EFT expansion breaks
down. We expect higher-order contributions and, eventually, a new structure of
underlying interactions to dampen the rise of the amplitudes, e.g., a
resonance. Conversely, for the EFT description to remain useful, the value of
$g_{V,b2}$ must be such that the cutoff implied by (10) is outside the
accessible kinematic range, or we would have to apply a cutoff or form factor
on the calculated distributions.

In Fig. 6, we display the sensitivity on $\Lambda_{UV}$ by using the (a) 2BH
case and (b) 1BH case at the 14 TeV LHC, respectively. To this end, we
determine the maximally allowed $g_{V,b2}$ value as a function of
$\Lambda_{UV}$ using (10). The y-axis indicates the number of signal events
that result for corresponding values of the cutoff $\Lambda_{UV}$ and the
maximal $g_{V,b2}$, respectively. The dashed blue line, for each plot, marks
the $5\sigma$ discovery threshold as it follows from the signal and background
event rates derived above.

The crossing points of the signal curves and the discovery thresholds in Fig.
6 determine the sensitivity of the analysis to heavy new physics, assuming
that the bound (10) for $g_{V,b2}$ is saturated. We conclude that the effect
of a heavy resonance, for instance, can be accessible up to $4.4$ TeV in the
2BH case and up to $3.6$ TeV in the 1BH case. The cleaner environment of the
2BH case is clearly preferred. By contrast, the discovery reach of the 0BH
case is limited to about $2.4$ TeV.

()

()

()

()

Figure 6: The $5\sigma$ discovery constraints on new physics cutoff at 14 TeV
LHC are demonstrated. (a) is the constraint obtained from 2BH case, and (b) is
the constraint obtained from 1BH case. (c) and (d) are the corresponding
constraints on the $g_{V,b2}$ in the 2BH and 1BH case, respectively.

We can apply the analogous analysis to a 100 TeV hadron collider. Based on the
results obtained for the LHC with $\sqrt{s}=14$ TeV where the 2BH case has
been most useful, below we only present the results for this analysis which
requires two highly boosted Higgs bosons.

As shown above, at a 100 TeV machine the signal can be discovered already in
the SM case where no enhancement is present. Therefore, we define the
observability of new physics in terms of the deviation in rate,
$N_{2BH}-N^{SM}_{2BH}$, where $N^{SM}_{2BH}$ is the number of events in the
2BH category in the SM case.

By using the BDT analysis as before, we derive the $5\sigma$ excess bound for
new physics at the 100 TeV collider, as marked in Fig. 7 by the blue dashed
line. We read off this figure that this collider can yield a meaningful
constraint on $g_{V,b2}$ at a value which corresponds to a sensitivity in the
new-physics scale of $\Lambda_{UV}=27$ TeV.

()

()

Figure 7: The $5\sigma$ excess constraint obtained from 2BH events on new
physics cutoff at 100 TeV hadron collider is demonstrated in (a), while (b) is
the corresponding $g_{V,b2}$.

##  4 Two-parameter bounds

So far, we have considered only the dependence of the Higgs-pair VBF process
on the couplings $g_{V,b2}$ to the transverse polarization of vector bosons,
which dominate the distribution in the highly boosted region. In this section,
we complement the discussion by taking into account also the $g_{V,a2}$
couplings which describe the interaction of a Higgs pair with longitudinally
polarized vector bosons. This interaction exists in the SM but can receive a
correction if dimension-6 operators are included.

In order to express the cross sections of $pp\to hhjj$ in terms of the
parameters given in Eq. (1), we impose some simplifying assumptions on the
phenomenological information from expected data on single-Higgs processes. For
instance, the $WWh$ vertex should be strongly constrained by data from the
Higgs decay to $WW$ as well as from VBF single-Higgs production. As stated
before, we ignore the universal ambiguity in the Higgs-coupling determination
due to undetected Higgs decays at the LHC, which can be lifted by model-
independent measurements at Higgs factories such as the CEPC, the ILC, or the
CLIC collider. Accordingly, we fix $g_{W,a1}$ and $g_{W,b1}$, the single-Higgs
couplings to longitudinal and transverse $W$ polarizations, to their SM
values. As a further simplification, we assume the custodial-symmetry
relations $g_{W,ai}=g_{Z,ai}$ and $g_{W,bi}=g_{Z,bi}$ ($i=1,2$) whenever
contributions of Z bosons are considered.

Since the amplitudes are at most linear in the parameters, we can parameterize
the cross section of $pp\to hhjj$ in terms of $g_{V,a2}$ as given below

$\displaystyle\sigma(pp\to hhjj)$ $\displaystyle=$
$\displaystyle\sigma^{0}_{a_{2}}+\sigma^{1}_{a_{2}}g_{V,a2}+\sigma^{2}_{a_{2}}g_{V,a2}^{2}\,.$
(11)

We compute the coefficients $\sigma^{0}_{a_{2}}$, $\sigma^{1}_{a_{2}}$, and
$\sigma^{2}_{a_{2}}$ numerically using Monte-Carlo methods, evaluating the
total cross section for a sufficient number of different coupling values. The
results are given in Table 12.

|  $\sigma^{0}_{a_{2}}$ (fb) |  $\sigma^{1}_{a_{2}}$(fb) |  $\sigma^{2}_{a_{2}}$(fb)
---|---|---|---
14 TeV | $17.71$ | -$29.33$ | $12.68$
27 TeV | $88.3$ | \- $152.2$ | $68.1$
100 TeV | $1601.2$ | -$2963.8$ | $1401$
Table 12: Coefficients $\sigma^{0}_{a_{2}}$, $\sigma^{1}_{a_{2}}$, and
$\sigma^{2}_{a_{2}}$ in the expression (11) for $pp\to hhjj$ at three
different collider energies.

Analogously, we can parameterize the cross section as a function on $g_{V,b2}$
as follows

$\displaystyle\sigma(pp\to hhjj)$ $\displaystyle=$
$\displaystyle\sigma^{0}_{b_{2}}+\sigma^{1}_{b_{2}}g_{V,b2}+\sigma^{2}_{b_{2}}g_{V,b2}^{2}\,,$
(12)

The numerical results for $\sigma^{0}_{b_{2}}$, $\sigma^{1}_{b_{2}}$, and
$\sigma^{2}_{b_{2}}$ are given in Table 13.

When both $g_{V,b2}$ and $g_{V,a2}$ are turned on, we have to include a mixed
coefficient proportional to $g_{V,b2}g_{V,a2}$. The corresponding results can
be found in Kilian:2018bhs ; the values are $1.7$ fb for 14 TeV, $9.6$ fb for
27 TeV, and $95$ fb for 100 TeV, respectively.

|  $\sigma^{0}_{b_{2}}$(fb) |  $\sigma^{1}_{b_{2}}$(fb) |  $\sigma^{2}_{b_{2}}$(fb)
---|---|---|---
14 TeV | $1.06$ | $1.52$ | $106.8$
27 TeV | $4.2$ | $6.97$ | $1135.2$
100 TeV | $38.4$ | $83.08$ | $54070$
Table 13: Coefficients $\sigma^{0}_{b_{2}}$, $\sigma^{1}_{b_{2}}$, and
$\sigma^{2}_{b_{2}}$ in the expression (12) for $pp\to hhjj$ at three
different collider energies.

The numerical results given in Table 12 and Table 13 have been obtained with
Madgraph5 and Whizard by independent calculations, with very good numerical
agreement. It should be pointed out that the results in Table 12 clearly
reflect the strong gauge cancellation which occurs between individual terms of
(11) in the SM limit. Some of the coefficients $\sigma^{0}_{a_{2}}$,
$\sigma^{1}_{a_{2}}$, and $\sigma^{2}_{a_{2}}$ are one order of magnitude
larger than that of the cross section in the SM, as given as
$\sigma^{0}_{b_{2}}$ in Table 13. Using this parameterization and applying the
results of our study, we derive the parameter ranges tabulated in Table 14.
Outside the given limits, the deviation from the SM can be detectable as a
$5\sigma$ discovery.

| 14 TeV (3 ab-1) | 27 TeV (3 ab-1) | 100 TeV (30 ab-1)
---|---|---|---
$\delta g_{V,a2}$ | $(-0.31,0.39)$ | $(-0.11,0.13)$ | $(-0.013,0.047)$
$g_{V,b2}$ | $(-0.10,0.11)$ | $(-0.03,0.02)$ | $(-0.003,0.003)$
Table 14: $5\sigma$ discovery (excess) bounds of $g_{V,a2}$ and $g_{V,b2}$ at
a 14 TeV, 27 TeV and 100 TeV hadron collider, assuming the respective total
integrated luminosity in brackets.

We show the projected bounds on $g_{V,a2}$ and $g_{V,b2}$ in Fig. (8). We have
compared our results for $g_{V,a2}$ with those given in Ref. Bishara:2016kjn
and found good agreement. Regarding the unitarity bounds shown in the plots,
the bounds on $g_{V,b2}$ correspond to Eq. (10), while for $g_{V,a2}$ we make
use of the result Kilian:2018bhs

$\displaystyle\frac{\Lambda^{4}_{UV}}{2^{9}\pi^{2}v^{4}}|g_{V,a2}-g_{V,a1}^{2}|^{2}\leq\frac{1}{4}\,.$
(13)

(a) 14 TeV

(b) 27 TeV

(c) 100 TeV

(d) 14 TeV

(e) 27 TeV

(f) 100 TeV

Figure 8: Total cross section after VBF cuts for the process $pp\to hhjj$ as a
function of the $WWhh$ couplings $g_{a_{2}}$ (upper row) and $g_{b_{2}}$
(lower row), for three different collider energies. The vertical lines are
unitarity bounds, which are derived from Eq. (13) and Eq. 10).

It is interesting to explore whether it is possible to disentangle the effects
of the operator $h^{2}W^{a}_{\mu\nu}W^{a,\mu\nu}$ from those of the operator
$h^{2}W_{\mu}^{a}W^{a,\mu}$. As mentioned in our previous work Kilian:2018bhs
, the azimuthal angle between two forward jets can be a useful observable for
this purpose.

In Fig. (9), we display the azimuthal angle distributions for the 14 TeV, 27
TeV and 100 TeV cases. We take into account only events in category 2BH and
apply no further cuts beyond the VBF cuts. The relative azimuthal angle is
defined as $\Delta\phi=|\phi(j_{1})-\phi(j_{2})|$, where $\phi(j_{1})$ denotes
the azimuthal angle of the leading forward jet, and $\phi(j_{2})$ denotes that
of the second leading forward jet.

We note the main features of the distributions given in Fig. (9):

  * •

In the SM, the azimuthal-angle distribution of the SM is flat due to the
dominance of longitudinal polarized vector bosons in the process $pp\to hhjj$,
as shown by the red curve in each of plot. Similarly, the distribution is also
flat for the case where the term $h^{2}W_{\mu}^{a}W^{a,\mu}$ is turned on, as
shown by the green curve in each of plot where both the contrbutions of the SM
and new physics have been included. Although new physics represented by the
operator $h^{2}W_{\mu}^{a}W^{a,\mu}$ enhances the total cross section, the
azimuthal angle distribution remains similar to that of the SM.

  * •

If the operator $h^{2}W_{\mu\nu}^{a}W^{a,\mu\nu}$ is turned on, the Higgs pair
being coupled to transversely polarized vector bosons leads to distributions
that significantly differ from that of the SM. As the blue curves indicate,
this type of new interaction causes more events with back-to-back outgoing
jets.

  * •

At the 100 TeV collider, the difference in shape of the angular distribution
is much less pronounced. In fact, the chosen benchmark values of the operator
coefficients cause just a small disturbance of the SM signal in this plot,
which is still detectable due to the much larger event rate for the high
energy and high luminosity of the machine. To distinguish the two kinds of
contributions, a more detailed analysis of this and other distributions
becomes necessary.

(a) 14 TeV

(b) 27 TeV

(c) 100 TeV

Figure 9: The relative azimuthal angle distributions of two forward jets for
14 TeV, 27 TeV, and 100 TeV .

Figure 10: The fit results with inclusive cross section(green) and
differential cross section(blue) with three different assumptions on the
central values: $(\delta
g_{V,a2}^{\textrm{true}},g_{V,b2}^{\textrm{true}})=(0,0)$, $(0.07,0)$,
$(0,0.01)$ with $\sqrt{s}=100$ TeV and an integrated luminosity 30 ab-1.

To fully utilise the differential information, we perform a $\chi^{2}$-fit on
the distribution of $\Delta\phi(j,j)$, using the 2BH events after the BDT cuts
with the collision energy $\sqrt{s}=100$ TeV and integrated luminosity 30
ab-1. In Fig. 10, we show the $2\sigma$ allowed region obtained with
differential information (blue) or just from the total cross section (green).
For the leftmost plot, we assume the SM for the true values of the
coefficients. It is evident that the information from the differential
distribution significantly improves the precisions on both $\delta g_{V,a2}$
and $g_{V,b2}$. The middle plot assumes $(\delta g_{V,a2}^{\rm
true},g_{V,b2}^{\rm true})=(0.07,0)$ for the true value, i.e., only $g_{V,a2}$
receives a new-physics contribution. The inclusive cross section confines the
region allowed by a measurement to a ring, due to the quadratic dependence on
both $\delta g_{V,a2}$ and $g_{V,b2}$, while the differential distribution
singles out a small area around the point $(\delta
g_{V,a2},g_{V,b2})=(0.07,0)$. Analogously, for the right plot we assume the
true values $(\delta g_{V,a2}^{\rm true},g_{V,b2}^{\rm true})=(0,0.07)$.
Again, the differential distribution selects a small region, but in this case
there is a two-fold sign ambiguity left for $g_{V,b2}$. This reflects the fact
that the effects of $g_{V,b2}$ is dominated by the squared term, and thus the
sign of $g_{V,b2}$ cannot be determined.

Finally, using the relations given in Table 8, we can map this contour onto
the plane spanned by $c_{W}$ and $c_{HW}$. The assumption of a linearly
realized symmetry and truncation at the dimension-6 order enforces the
constraint $\delta g_{V,a2}\sim 6g_{V,b2}$, if both couplings depend only on a
single parameter $c_{HW}$. The two-parameter analysis that we describe in this
paper, would allow us to search for the relation of $c_{W}$ and $c_{HW}$,
which would point to the presence of contributions that do not follow the
simple power-counting assumption underlying the dimension-six truncation of
the SILH basis. Currently, the experimental data constraints on the parameter
set are rather weak, to the level of $c_{V}\sim O(10^{3})$ Ellis:2018gqa . In
this study, we conclude that we can reach a sensitivity of up to $c_{V}\sim
O(10)$ at future colliders.

##  5 Discussions and Conclusions

We have studied double-Higgs production in vector-boson at a proton-proton
collider, $pp\to hhjj\to 4b2j$. We compare the 14 TeV LHC with future high-
energy and high-luminosity colliders of 27 TeV and 100 TeV. The analysis of
this process benefits greatly from identifying highly-boosted Higgs bosons. To
this end, we use the mass-drop method to analyse the jet substructure, and we
optimize the significance by the boosted decision-tree method. Our results
show that the 2BH case where two highly-boosted Higgs are tagged can provide
the cleanest experimental environment to discover new physics in the the VBF
signal.

At the LHC, the number of 2BH events is too small to discover this channel if
the SM is valid without new contributions. Conversely, at a 100 TeV collider
with a high luminosity of the order of 30 ab-1, the number of 2BH events is
large enough to discover the SM signal.

To study the effects of new physics, we use a phenomenological effective
Lagrangian (3). We explore the effect of the interactions of type
$h^{2}V_{\mu}^{a}V^{a,\mu}$ and $h^{2}V_{\mu\nu}^{a}V^{a,\mu\nu}$ and extract
bounds for the associated coefficients $g_{V,a2}$ and $g_{V,b2}$. Our results
demonstrate that new-physics scales up to $4.4$ TeV at the 14 TeV LHC, and
$27$ TeV at a 100 TeV hadron collider, are within reach of discovery.

Fig. 11 collects the projected bounds on $g_{V,a2}$ and $g_{V,b2}$ that we
have determined in this work, cf. also Table 14. The bounds that can be
obtained at 27 TeV, are one order of magnitude stronger than for the 14 TeV
LHC. The 100 TeV machine can further constrain these parameters by another
order of magnitude. At both the 27 TeV and 100 TeV machines, this offers a
significant indirect sensitivity to new physics in this sector.

Figure 11: The summarized bounds of $g_{V,a2}$ and $g_{V,b2}$ at 14 TeV, 27
TeV and 100 TeV hadron collider, which is the same as Table 14.

Our study does not account for underlying-event or pile-up effects. These
produce additional soft radiation and may render the extraction of a hard-
process signal more difficult. In fact, the mass-drop method selects two-
pronged jets in the final state. This reduces the impact of QCD jets from
underlying events or pile-up, which are one-pronged. We also use filtering to
remove soft radiation after reconstructing two sub-jets in the final state,
which is helpful to reject jets from extra radiation produced by the parton
shower. Recent studies indicate that modern pile-up mitigation techniques
Cacciari:2014gra  can minimize the pile-up contamination efficiently for the
4b final state Behr:2015oqq . A detailed pile-up analysis can be done but is
beyond the scope of the current paper.

For a further improvement of our result, we may consider color-flow properties
Maltoni:2002mq  as a tool to further discriminate $h\to b\bar{b}$ decays from
b jets in the QCD background. Color-connection information can be quantified
by observables such as the pull vector Gallicchio:2010sw . In a recent study
of double Higgs production at LHC Kim:2019wns , it was argued that while the
color flow is very different between the double-Higgs signal and the QCD
background, this information may be diluted after applying kinematics cuts.
The authors of Kim:2019wns  proposed to use jet-image and a Deep Neutral
Network analysis methods to discriminate the signal from background, rather
than constructing the pull vector. We defer these refinements of our study to
future work.

In summary, the analysis of highly boosted Higgs-boson pairs in vector-boson
fusion is promising and can be used to significantly improve our knowledge
about Higgs-sector interactions. In particular, the method should become
important for a future high-energy proton collider where the sensitivity is
sufficient to extract a signal down to the SM rate.

###### Acknowledgements.

W.K. is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under grant 396021762 – TRR 257. S. Sun is supported by MIUR in
Italy under Contract(No. PRIN 2015P5SBHT) and ERC Ideas Advanced Grant (No.
267985) “DaMeSyFla"; Q.S. Yan is supported by the Natural Science Foundation
of China under the grant NO. 11475180 and NO. 11875260. X.R. Zhao is supported
by the “Fonds Spécial de Recherche” (FSR) of the UCLouvain.

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