aainstitutetext: École Polytechnique, Laboratoire Leprince-Ringuet, Av.
Chasles, 91120 Palaiseau, Francebbinstitutetext: INFN Sezione di Torino and
Università degli Studi di Torino, Via P. Giuria 1, 10125 Torino,
Italyccinstitutetext: ICTP South American Institute for Fundamental Research &
IFT-UNESP, R. Dr. Bento Teobaldo Ferraz 271, São Paulo, Brazil
ddinstitutetext:  Centro de Ciencias Naturais e Humanas, Universidade Federal
do ABC, Santo Andre, 09210-580 SP, Brazil eeinstitutetext: Departamento de
Fısica, Pontifıcia Universidade Catolica de Rio de Janeiro, Rio de Janeiro,
Brazilffinstitutetext: CERN, CH 1211, Geneva 23, Switzerlandgginstitutetext:
The University of Kansas, Department of Physics and Astronomy, 1251 Wescoe
Hall Dr , 66045 Lawrence, Kansas, US

# Searching for anomalous top quark interactions with proton tagging and
timing detectors at the LHC

Cristian Baldenegro b Andrea Bellora c,d Sylvain Fichet e Gero von Gersdorff f
Michael Pitt g Christophe Royon

###### Abstract

We study the LHC sensitivity to new broad neutral resonances produced in two-
photon fusion and decaying to a top quark pair, $\gamma\gamma\to t\bar{t}$.
This is probed in central exclusive $t\bar{t}$ production in proton-proton
collisions, $pp\to pt\bar{t}p$. We use the tagging of the intact protons by
PPS (CMS) and AFP (ATLAS) and consider the semi-leptonic $t\bar{t}$ channel.
The sensitivity is also mapped onto a set of dimension-8 $\gamma\gamma
t\bar{t}$ operators in the large mass limit. Using the kinematical
correlations between the intact protons and the reconstructed $t\bar{t}$
system, we obtain a sensitivity to the couplings of the dimension-8 operators
of $1.4\cdot 10^{-11}$ GeV-4 at 95% CL. The sensitivity to the anomalous
couplings is significantly improved down to about $7\cdot 10^{-12}$ GeV-4 if
the proton time-of-flight is known with a precision of 20 ps in future
measurements. The 95% CL sensitivity to broad neutral resonances reaches
masses of order $1500$ GeV when using timing information.

##  1 Introduction

Under the hypothesis that new physics with strong interactions exists beyond
the Standard Model (SM) of particle physics, it is possible that a new state
with broad width lies at the TeV scale. Searching for such broad resonances at
the LHC is somewhat challenging since the analyses cannot efficiently rely on
standard bump-searching methods. Instead, the search may substantially benefit
from observation channels featuring a highly reduced background. The
observation channels from central exclusive production (CEP) processes, in
which both protons remain intact in the final state, belong to this category
and therefore provide an environment favorable to the search for broad
resonances.

Among all processes, the CEP topology stands out because it can be efficiently
selected using the forward proton detectors built for this purpose in the CMS
and ATLAS experiments, namely the Precision Proton Spectrometer (PPS) PPStdr
and the ATLAS Forward Proton detector (AFP) AFPtdr , respectively. Forward
detectors measure the outgoing intact protons, leading to the reconstruction
of the entire final state kinematics of both the proton and central systems.
This set of techniques leads to drastic background reduction. A key example is
the $pp\to\gamma\gamma pp$ process for which the background for $300$ fb-1
drops below one expected event in the high mass region (see e.g.
Fichet:2014uka ). It was demonstrated in Baldenegro:2018hng  that this channel
is an efficient precision probe of broad neutral particles 111 Other studies
of new physics searches based on proton tagging at the LHC can be found in
usww ; usw ; Sahin:2009gq ; Atag:2010bh ; Gupta:2011be ; Epele:2012jn ;
Lebiedowicz:2013fta ; Fichet:2013ola ; Fichet:2013gsa ; Sun:2014qoa ;
Sun:2014qba ; Sun:2014ppa ; Sahin:2014dua ; Inan:2014mua ; Fichet:2014uka ;
Fichet:2015nia ; Cho:2015dha ; Fichet:2016clq ; Fichet:2015vvy ;
Fichet:2016pvq ; Baldenegro:2017aen ; Baldenegro:2017lzv . .

The SM central exclusive production of $t\bar{t}$ has been investigated in
recent phenomenological studies Luszczak:2018dfi ; Goncalves:2020saa ;
Martins:2022dfg . The process has not been observed experimentally, but a
recent search by the CMS Collaboration has set an upper bound on its cross
section of $\sigma=0.59$ pb at 95% CL at 13 TeV CMS-PAS-TOP-21-007 . It is
likely that the SM process could be observed at the high-luminosity LHC
CMS:2021ncv . Photoproduction of top quark pairs can be used to constrain the
electromagnetic dipole moments of the top quark, as discussed in Ref.
Koksal:2019gyo  for $\gamma$-gluon fusion processes, for example.

The goal of the present paper is to expand the prospects of CEP searches to
the case of neutral particles coupled to both photons and top quarks. Our
focus is on evaluating the sensitivity of the exclusive $pp\to t\bar{t}pp$
process to the presence of (possibly broad) neutral resonances. In the large
mass limit the photon-top quark interactions become local and can be described
using local dimension-8 effective operators. Thus, the sensitivity to these
dimension-8 operators is also studied throughout this paper as a natural
extension.

We emphasize that our focus on dimension-8 operators is motivated by the
neutral particle scenario. In principle, searching for the dimension-6 top
quark dipole operators in the CEP topology is also well-motivated. This
interesting case is kept for a separate investigation. In the case of neutral
particles, we will verify that the loop-generated contribution to the dipole
is negligible in the scope of our study, hence justifying our focus on the
dimension-8 operators.

Importantly, the sensitivity to the $pp\to t\bar{t}pp$ process is expected to
be substantially enhanced by the use of timing detectors embedded into the
forward detectors Cerny:2020rvp . These timing detectors provide an
independent reconstruction of the primary vertex using intact proton
information, improving further the selection of signal events. Our study
serves to illustrate the power of these timing detectors in the specific case
of broad neutral resonances.

The plan of the paper is as follows. In Section 2 the models for neutral
particles are introduced. We also work out the basis of effective operators
describing local $\gamma\gamma t\bar{t}$ interactions. Section 3 provides
basics on forward detectors and describes the simulation framework. Section 4
briefly describes the statistical framework. Section 5 describes the search
for $\gamma\gamma\to t\bar{t}$ in the semi-leptonic decay channel. Section 6
presents our results and a summary is given in Section 7.

##  2 Effective field theory and neutral particles

Here, we define extensions of the SM which induce anomalous $\gamma\gamma\to
t\bar{t}$ scattering. We start by deriving the relevant basis of higher
dimensional operators that contribute to such anomalous processes. The lowest
dimensional operators turn out to have dimension eight, and we identify three
CP-even and three CP-odd operators. Any new physics model contributing to
anomalous $t\bar{t}$ production via photon fusion can be matched to these
operators at low energy. We then introduce generic models of neutral
resonances with trilinear couplings to both photons and top quarks. They can
themselves be considered subsectors of well-motivated extensions of the SM,
such as composite Higgs models, extra dimensions, etc. Moreover, we explicitly
provide the matching of these generic models to the aforementioned dimension-8
effective theory. These (effective) models provide the basis for our
simulations in the subsequent sections.

###  2.1 Operator basis

The physics candidates probed by our experiment may have a mass scale
$\Lambda$ much higher than the other scales involved in the amplitudes. In
such a regime, the amplitudes can be expanded in inverse powers of $\Lambda$.
Whenever this expansion is possible, the observable effects of the high mass
physics can be described by a series of local operators encoded in a low-
energy effective Lagrangian, ${\cal L}={\cal L}_{\rm
SM}+\sum_{n,i}\frac{a_{i,n}}{\Lambda^{n}}{\cal O}_{i,n}$ .

Our interest is in the set of dimension-8 operator (i.e. $n=4$) involving two
top quarks and two photons. We allow for both CP-even and CP-odd bilinear
invariant in $\bar{t}t$ and photon field strength. At a given order in the
expansion, the equations of motion can be used to reduce the set of the
effective operators. We find an irreducible basis of six operators (omitting
the subscript $n$),

$\displaystyle\mathcal{O}_{1}$
$\displaystyle=m_{t}F^{\mu\nu}F_{\mu\nu}\bar{t}t$
$\displaystyle\mathcal{O}_{2}$
$\displaystyle=im_{t}F^{\mu\nu}\tilde{F}_{\mu\nu}\bar{t}\gamma_{5}t$
$\displaystyle\mathcal{O}_{3}$
$\displaystyle=m_{t}F^{\mu\nu}\tilde{F}_{\mu\nu}\bar{t}t$
$\displaystyle\mathcal{O}_{4}$
$\displaystyle=im_{t}F^{\mu\nu}F_{\mu\nu}\bar{t}\gamma_{5}t$
$\displaystyle\mathcal{O}_{5}$
$\displaystyle=iF^{\mu\rho}F^{\nu}_{\rho}\bar{t}\gamma_{\mu}D_{\nu}t$
$\displaystyle\mathcal{O}_{6}$
$\displaystyle=F^{\mu\rho}F^{\nu}_{\rho}\bar{t}\gamma_{5}\gamma_{\mu}D_{\nu}t$
(1)

with
$\tilde{F}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}$. The
${\cal O}_{1\ldots 4}$ operators involve two derivatives, the ${\cal O}_{5,6}$
involve three derivatives. The ${\cal O}_{1,3,5}$ operators involve CP-even
bilinear terms while ${\cal O}_{2,4,6}$ involve CP-odd bilinear ones.

Bilinear terms of the form $\bar{t}\Gamma\gamma_{\mu}\partial^{\mu}t$, where
$\Gamma$ is an arbitrary Lorentz structure, are reduced by using the top quark
equation of motion into other operators from the above basis plus operators
involving other fields, which are irrelevant for the process considered. One
can also check that
$F^{\mu\rho}\tilde{F}^{\nu}_{\rho}=\frac{1}{4}g_{\mu\nu}F\tilde{F}$ such that
operators of the form
$F^{\mu\rho}\tilde{F}^{\nu}_{\rho}\bar{t}\Gamma\gamma_{\mu}\partial_{\nu}t$
are reduced into $\mathcal{O}_{3,4}$ using the top quark equation of motion.

The effective Lagrangian description breaks down at momenta of order of the
$\Lambda$ mass scale. In case of strong coupling, this breakdown coincides
with the breakdown of the perturbative expansion in the effective couplings.
It also roughly coincides with violation of unitarity, signalling the
necessity of a UV completion.

We parametrize the coefficients of the basis of operators shown in Eq. (1) as

${\cal L}={\cal L}_{\rm SM}+\sum_{i=1}^{6}\zeta_{i}{\cal O}_{i}\,.$ (2)

###  2.2 Neutral particles

Neutral particles with non-renormalizable couplings to SM operators are
present in common extensions of the SM. Such theories often contain scalar,
pseudo-scalar and spin-2 particles, respectively denoted by $\varphi$,
$\tilde{\varphi}$ and $h^{\mu\nu}$. Examples include the Peccei-Quinn axion
Peccei:1977hh , the Kaluza-Klein (KK) graviton and the radion in warped extra
dimensions Randall:1999ee , the dilaton in theories of strongly coupled
electroweak breaking Goldberger:2007zk , Goldstone bosons of extended
composite Higgs models Kaplan:1983sm , mesons and glueballs of strongly-
coupled theories Hill:2002ap , extra scalars breaking the global symmetry of
composite Higgs models vonGersdorff:2015fta ; Fichet:2016xvs ; Fichet:2016xpw
, Higgs portal models Schabinger:2005ei , and many more.

Independently of the model they originate from, their leading couplings to
photons and quarks can be can be classified in terms of CP quantum numbers and
written as

${\cal L}^{\gamma\gamma}_{\rm eff}=\frac{1}{f^{\gamma\gamma}_{{\bf
0}^{+}}}\varphi(F_{\mu\nu})^{2}+\frac{1}{f^{\gamma\gamma}_{{\bf
0}^{-}}}aF_{\mu\nu}\tilde{F}_{\mu\nu}+\frac{1}{f^{\gamma\gamma}_{{\bf
2}}}h^{\mu\nu}\left(-F_{\mu\rho}F^{\rho}_{\,\,\nu}+\frac{1}{4}\eta_{\mu\nu}(F_{\rho\sigma})^{2}\right)\,,$
(3) ${\cal L}^{\bar{t}t}_{\rm eff}=\frac{m_{t}}{f^{\bar{t}t}_{{\bf
0}^{+}}}\varphi\bar{t}t+i\frac{m_{t}}{f^{\bar{t}t}_{{\bf
0}^{-}}}a\bar{t}\gamma_{5}t+i\frac{1}{f^{\bar{t}t}_{{\bf
2}}}h^{\mu\nu}\bar{t}\gamma_{\mu}D_{\nu}t\,$ (4)

where $\varphi$ is the CP-even scalar, $a$ the CP-odd scalar, and $h^{\mu\nu}$
the CP-even spin-2 field.

The couplings $f_{X}$ appearing in Eq.(3) e (4) are typically restricted by
unitarity to satisfy

$\frac{1}{f_{X}}\lesssim\frac{4\pi}{m}\,.$ (5)

Let us note that in some models that address both the hierarchy and flavor
problems of the SM (such as warped extra-dimensions with SM fermions in the
bulk), it is common that the heavy neutral resonances couple more strongly to
the top quark than to the lighter quarks.

The partial width of both the CP-even and CP-odd scalar into $\gamma\gamma$ is

$\Gamma_{\phi_{\pm}\to\gamma\gamma}=\frac{m^{3}}{4\pi
f^{2}_{\gamma\gamma}}\,.$ (6)

The partial width for the CP-even and CP-odd scalar into $t\bar{t}$ are
respectively

$\Gamma_{\phi_{+}\to\bar{t}t}=\frac{3}{8\pi}\frac{m_{t}^{2}}{(f^{+}_{t})^{2}}m\left(1-\frac{4m_{t}^{2}}{m^{2}}\right)^{3/2}$
(7)
$\Gamma_{\phi_{-}\to\bar{t}t}=\frac{3}{8\pi}\frac{m_{t}^{2}}{(f^{-}_{t})^{2}}m\left(1-\frac{4m_{t}^{2}}{m^{2}}\right)^{1/2}\,.$
(8)

### Matching to dimension-8 operators at tree-level

When any of these particles is heavy, it contributes to the effective EFT
operators classified in Eq. (1). The matching is as follows, depending on
which of the particles is integrated out

  * •

Spin-0, CP-even scalar $\phi$

$\zeta_{1}=\frac{1}{f^{\gamma\gamma}_{{\bf 0}^{+}}f^{\bar{t}t}_{{\bf
0}^{+}}m^{2}},\quad\quad\quad\quad\quad{\zeta_{i\neq 1}=0}$ (9)

  * •

Spin-0 CP-odd scalar $\tilde{\phi}$

$\zeta_{2}=\frac{1}{f^{\gamma\gamma}_{{\bf 0}^{-}}f^{\bar{t}t}_{{\bf
0}^{-}}m^{2}},\quad\quad\quad\quad\quad{\zeta_{i\neq 2}=0}$ (10)

  * •

Spin-2 field $h_{\mu\nu}$

$\zeta_{1}=\frac{1}{4f^{\gamma\gamma}_{{\bf 2}}f^{\bar{t}t}_{{\bf
2}}m^{2}},\quad\quad\quad\quad\quad\zeta_{5}=-\frac{1}{f^{\gamma\gamma}_{{\bf
2}}f^{\bar{t}t}_{{\bf 2}}m^{2}},\quad\quad\quad\quad\quad{\zeta_{i\neq
1,5}=0}$ (11)

This EFT regime typically applies in our context whenever $m\gtrsim 2.8$ TeV
as the experimental acceptance is very low beyond this scale and a heavier
resonance cannot be resolved, leaving only the imprint in the higher
dimensional operators.

### Estimate of dipole operator at loop-level

The above couplings of the top quark to the neutral resonance also generate a
CP-even dimension-6 dipole operator

$\zeta_{d}m_{t}F_{\mu\nu}\bar{t}\gamma^{\mu}\gamma^{\nu}\gamma^{5}t$ (12)

where a factor of $m_{t}$ arising from electroweak symmetry breaking is shown
explicitly and $\zeta_{d}$ has a mass dimension of $m^{-2}$. This operator
modifies the SM top quark coupling to the photon, and hence the
$\gamma\gamma\to t\bar{t}$ production amplitude via the standard $t$-channel
topology. However, the coefficient $\zeta_{d}$ is loop suppressed and goes as

$\zeta_{d}\sim\frac{e}{16\pi^{2}}\frac{m_{t}^{2}}{(f^{tt})^{2}m^{2}}$ (13)

and even in the strong-coupling scenario in Eq. (5),
$\zeta_{d}\to\frac{em_{t}^{2}}{m^{4}}$ suffers an additional strong
$(\frac{m_{t}}{m})^{2}$ suppression. The combined effect for the modification
of the SM coupling is

$\frac{\delta e}{e}\sim\frac{m_{t}^{3}E}{m^{4}}$ (14)

which is of the order of $\sim 10^{-6}$ for a 1 TeV resonance and a typical
energy of $E=$500 GeV. Therefore, such an effect is completely negligible.

##  3 Simulation framework

###  3.1 Signal modeling

In our setup the anomalous $pp\to t\bar{t}pp$ process (Fig. 1(c)) is probed in
the configuration of forward protons, i.e. in very peripheral proton-proton
collisions. In very peripheral collisions, the electromagnetic field generated
by the electrically charged protons can be treated as a source of quasi-real
photons. These quasi-real photons can then interact to produce the top quark
pair system. This photon emission off the proton is treated within the
equivalent photon approximation (EPA) epa1 ; epa2 . In contrast to the
standard parton distribution functions used in the quantum chromodynamics
(QCD) factorization theorem, the flux and energy spectra of the quasi-real
photons are under better theoretical control. This is because they are
parametrized with the elastic electric and magnetic form factors extracted
from precision electron-proton elastic scattering data. Due to additional soft
interactions between the protons, not all of them will remain intact after the
quasi-real photon exchange. This means that the cross section for photon-
induced processes with intact protons is reduced Khoze:2001xm ; Khoze:2002dc ;
Gotsman:2005rt ; Harland-Lang:2015cta ; Harland-Lang:2018iur . For our
projections, this effect is taken into account by supplementing the cross
section calculation with a survival probability factor that accounts for the
probability that the protons remain intact after the photon exchange. For our
calculations, we assume a survival probability of 90%.

The signal process originates from the squared anomalous amplitude plus the
interference with the SM exclusive production of $t\bar{t}$. The SM exclusive
production of $t\bar{t}$ mainly happens via photon-initiated interaction (Fig.
1(a)), while the amplitude for gluon initiated CEP (Fig. 1(b)), which can be
calculated with the perturbative QCD Durham model kmr , is at least two orders
of magnitude lower, thus negligible.

In the present study the SM cross section at $\sqrt{s}=14$ TeV, $\sim 0.35$
Goncalves:2020saa ; d_Enterria_2010 , is between one and two orders of
magnitude lower than its anomalous counterpart when considering coupling
values equal to the predicted sensitivity. This difference is further enhanced
when applying the selection criteria. The fact that the square SM amplitude is
smaller than the square BSM amplitude necessarily implies that the latter
dominates the signal while the interference is subleading (see Fichet:2016iuo
). Anticipating the final sensitivity, we can thus neglect beforehand the
interference between anomalous and SM amplitudes.

The anomalous central exclusive production of $t\bar{t}$ via $\gamma\gamma$
interaction (Fig. 1(c)) is generated with the Forward Physics Monte Carlo
(FPMC) event generator fpmc , which implements the EPA approach. The
scattering amplitudes were computed using MadGraph mg5 , and implemented in
FPMC.

(a)

(b)

(c)

Figure 1: Feynman diagram for exclusive production of $t\bar{t}$ via photon
fusion (a), color-singlet two-gluon fusion (b), and anomalous exclusive
production via a local effective operator from the basis (1) (c).

###  3.2 Background modeling

The present study focuses on the reconstruction of anomalous exclusive
$t\bar{t}$ production in the semi-leptonic decay channel. The dominant
background components to $t\bar{t}$ exclusive production, in high-pileup
conditions, are processes with final states similar to $t\bar{t}$ semi-
leptonic decays, in coincidence with intact outgoing protons coming from soft
diffractive pileup events. A next-to-leading order cross section of $903$ pb
for QCD $t\bar{t}$ production at $14$ TeV with resummation at next-to-leading
logarithmic accuracy is considered following Ref. ttxsec .

Soft diffractive reactions at $14$ TeV are expected to have a production cross
section of the order of $20$ mb. During the CERN LHC Run 2, average
simultaneous interactions (pileup) in the $30$-$50$ range were reached at the
ATLAS and CMS interaction points. This high number of interactions, combined
with the large soft single and central diffractive cross section yields a
significant amount of forward protons coming from pileup collisions. Even if
kinematic constraints between forward protons and the central system
measurement can be exploited to tame this background component, as shown in
the later sections, it still represents the main challenge when searching for
$t\bar{t}$ exclusive production.

To take into account this effect, a conservative average number of concurrent
interactions of $50$ is considered. In each $t\bar{t}$ event, the number of
pileup interactions is sampled from a Poisson distribution. For each pile up
vertex, 0, 1 (in one detector arm) or 2 protons (in both arms) are assigned.
The probability per-pileup vertex of having protons in the forward detectors
is calculated using the Pythia8 event generator pythia8 . Then, the $\xi$
distribution for these intact protons is sampled from a $1/\xi$ distribution,
which is an approximation to the expected $\xi$ distribution of soft
diffractive interactions. The proton fractional momentum loss is considered in
the range between $0.015$ and $0.2$ for our projections. A $2\%$ Gaussian
smearing is applied on $\xi$ to mimic detector resolution effects.

The main background component is given by QCD $t\bar{t}+jets$ production.
Simulated events for this process are generated with MadGraph and matched to
parton showers and hadronization with Pythia8. A secondary background
contribution is the QCD $WW+jets$ production in coincidence with pileup
protons from soft diffraction. In this case the cross section considered is
$131.3$ pb, following Ref. wwxsec .

Another background contribution might arise from single-diffractive $t\bar{t}$
production in coincidence with pileup protons. The process cross section is
expected to be $\mathcal{O}(1)$ pb. However, even though one of the intact
protons might be kinematically correlated to the central system, the second
pileup proton will not. Hence, it is effectively suppressed by exploiting this
kinematical correlation, making it negligible.

In double-pomeron exchange, not all of the energy of each exchanged pomeron
goes into the production of the $t\bar{t}$ system. This means that the
invariant mass of the $t\bar{t}$ system reconstructed with the intact protons
information heavily underestimates the true $t\bar{t}$ mass (see for example
the diffractive mass fraction reported by CDF in CDF:2000odl  or Figure 3 of
Ref. Martins:2022dfg ). This process is therefore suppressed in the selection
process.

CEP of $t\bar{t}$ by color-singlet two-gluon exchange, which is an irreducible
background component, is suppressed at high $t\bar{t}$ masses. This is because
the cross section needs to be supplemented with a Sudakov form factor that
accounts for the probability that there is no initial-state radiation that
could spoil the exclusivity of the process. The probability that there are no
gluon emissions decreases rapidly with the mass of the central system, since
there is more phase-space available for them. No such Sudakov form factor is
present in calculations with two-photon fusion processes. This has been shown
explicitly in Refs. Fichet:2014uka ; Harland-Lang:2018iur .

###  3.3 Detector simulation

The detector simulation is performed with the Delphes package, tuned with the
CMS datacard de2014delphes . Detector efficiencies and energy/momentum
smearings are parametrized as a function of the particle kinematics (pseudo-
rapidity, transverse momentum for tracking detectors, pseudo-rapidity and
energy for calorimeters) cms_electron ; cms_ecal . Jets are clustered with the
FastJet package Cacciari:2011ma , using the anti-$k_{t}$
algorithmCacciari:2008gp  with a distance parameter
$R=\sqrt{\Delta\phi^{2}+\Delta y^{2}}=0.5$, and transverse momentum $p_{T}>20$
GeV. $b$-tagging is included following Ref. CMS_BTagging  and takes into
account both the tagging efficiency and the misidentification rate. Leptons
are reconstructed within the pseudo-rapidity range $|\eta|<2.5$, and jets for
$|\eta|<5$. Missing transverse energy is computed in a particle-flow approach.

To account for the proton measurement resolution, a $2\%$ Gaussian smearing on
the proton fractional momentum loss ($\xi=(p_{i}-p_{f})/p_{i}$) is applied and
only protons with $0.015<\xi<0.20$ are considered, to match the typical
forward detectors acceptance.

The longitudinal position of primary vertices is randomly sampled from a
Gaussian distribution of $5$ cm RMS, according to the reported CMS
observations cms_vertex . Timing detector resolution effects are included by
applying Gaussian smearings on the proton time of arrival, assuming a $200$ m
distance of forward detectors from the interaction point. Timing resolutions
of $20$ and $60$ ps are considered, the former, being compatible with the
design scenario presented in PPStdr , and the latter accounting for
performance already achieved today.

##  4 Statistical framework

For our projections, we need to assume a set of observed data. As commonly
done, we assume that no statistical fluctuations are present in these pseudo-
data, which are usually dubbed “Asimov” data and that we denote here with a
prime. As mentioned before, the integrated luminosity is chosen to be $L=300$
fb-1, which is about the expected integrated luminosity to be taken by CT-PPS
and AFP in Run-3 (in Run-2, CMS and ATLAS collected about 100 of data with
forward detectors at 13 TeV).

The observed events follow a Poisson distribution and for this kind of
analysis we can safely neglect the systematic uncertainties. The statistics
for the events together with the prediction for the event rates sets the
likelihood function needed for our analysis,

${\cal L}(\sigma)={\rm Pr}(n^{\prime}|b+\sigma L)\,,\quad{\rm
Pr}(\hat{n}|n)=\frac{{n^{\hat{n}}}e^{-n}}{\hat{n}!}\,$ (15)

where $b$ is the expected number of events from the background. In our study,
depending on the cases, $b$ may be much larger than $1$ or $O(1)$. In order to
draw projected exclusion contours, we take the background-only hypothesis for
the Asimov data,

$n^{\prime}=b\,.$ (16)

Our method to obtain the projected exclusion contours is the following. We
define the posterior probability density for $\sigma$ as
$L(\sigma)\pi(\sigma)$ where the prior is $\pi(\sigma)=1$ if $\sigma>0$ and
$0$ otherwise. The highest posterior density region at $1-\alpha$ credibility
level is solved analytically and is simply given by

$1-\alpha=\frac{\int^{\sigma_{\alpha}}_{0}L(\sigma)\pi(\sigma)}{\int_{0}^{\infty}L(\sigma)\pi(\sigma)}=1-\frac{\Gamma(1+b,b+\sigma
L)}{\Gamma(1+b,b)}\,,$ (17)

where $\Gamma(x,y)$ is the incomplete Gamma function. The credibility regions
for different values of $\alpha$ amount respectively to exclusion at
$2\sigma$, $3\sigma$, $5\sigma$ in terms of standard statistical significance
222 The standard statistical significance is given by
$Z_{\alpha}=\Phi^{-1}(1-\alpha)=-\sqrt{2}{\rm Erf}^{-1}(2\alpha-1)$. We can
also notice that for $b\gg 1$, upon approximating $\Gamma(x,y)\sim
y^{x-1}e^{-y}$ and expanding in small $\frac{s}{b}$ we obtain
$\frac{s_{\alpha}}{\sqrt{b}}=\sqrt{-2\log\alpha}$, where
$s_{\alpha}=\sigma_{\alpha}L$. If $\alpha\ll 1$ we also have
$Z_{\alpha}\approx\sqrt{-2\log\alpha}$, therefore in this limit our formula
reproduces the standard relation
$Z_{\alpha}=\frac{s_{\alpha}}{\sqrt{b}}(1+O(\frac{s}{b}))$. . The projected
sensitivity to the signal that we quote is the value of $\sigma$ satisfying
Eq. (17) for a given statistical significance.

##  5 Anomalous $\gamma\gamma\rightarrow t\bar{t}$ in the semi-leptonic decay
channel

Since each of the top quarks decays to a $W$ boson and a $b$ quark, events can
be classified depending on the $W$ boson decays. Events in which both $W$
boson decays are leptonic are called fully-leptonic. When one $W$ boson decay
is hadronic and other is leptonic the event is called semi-leptonic. When two
hadronic $W$ boson decays take place, events are called fully-hadronic.

Fully-leptonic decays generally present lower backgrounds, but have a low
branching fraction of the order of $10\%$, and the presence of two neutrinos
in the final state lowers the precision of the $t\bar{t}$ system mass
measurement, effectively reducing the background rejection provided by the
measurement of the leading protons. Fully-hadronic decays on the other hand
have the highest branching fraction. They are however strongly impacted by QCD
jets backgrounds, so they represent a challenging environment for exclusive
$t\bar{t}$ measurements. For the present study, the event selection and
reconstruction was therefore tuned for semi-leptonic events, as they represent
a good compromise under all the above-mentioned aspects.

Throughout this section the signal is taken to be generated from the basis of
$t\bar{t}\gamma\gamma$ dimension-8 operators (see Eq. (1)). The analysis is
applied to signal events from neutral particles in the next section.

###  5.1 Event selection and reconstruction

Exclusive semi-leptonic $t\bar{t}$ events are selected in three steps: pre-
selection, central selection and matching and eventually timing. A pre-
selection is applied to select events with at least one lepton in the final
state, two $b$-tagged jets, two non $b$-tagged jets, and one tagged proton per
detector side. Furthermore, a missing energy transverse $E_{T}^{miss}>20$ GeV
is required.

EFT operators yield $t\bar{t}$ mass distributions that can be grouped into two
families: $\mathcal{O}_{5}$ and $\mathcal{O}_{6}$, which contain an additional
derivative, produce cross sections that grow faster with $m_{t\bar{t}}$,
compared to
$\mathcal{O}_{1}$,$\mathcal{O}_{2}$,$\mathcal{O}_{3}$,$\mathcal{O}_{4}$. The
two families are thus studied in parallel, and $\mathcal{O}_{1}$ and
$\mathcal{O}_{5}$ are shown as their representatives.

As illustrated in Fig. 2, anomalous $t\bar{t}$ events are expected to show
mass distributions peaked towards higher values with respect to SM $t\bar{t}$.
The two $b$-tagged jets and non $b$-tagged jets with the largest transverse
momenta are therefore chosen, as well as the largest $p_{T}$ lepton.

Figure 2: Mass of the $t\bar{t}$ system at generator level (stable-particle
level). All histograms are normalized to unity.

The mass of the central system can be expressed in relation to the proton
measurement with:

$m_{t\bar{t}}^{p}=\sqrt{s\xi_{+}\xi_{-}}$ (18)

with $\xi_{+}$ and $\xi_{-}$ being the fractional momentum losses of the
protons measured in the two proton detectors. The proton with the largest
$\xi$ is chosen to enhance the event selection efficiency of anomalous
$t\bar{t}$ exclusive events in case of multiple protons on each detector side.

After the pre-selection is performed, each top quark is reconstructed from its
decay product. In the following, $t_{h}$ refers to the top quark that decays
into hadrons, while $t_{\ell}$ refers to the one decaying into leptons.

To assign the $b$-tagged jets to their respective top (anti)quark, the mass of
the system formed by the two non $b$-tagged jets and a $b$-tagged jet is
computed. The $b$-jet assigned to $t_{h}$ is thus the one yielding a mass
measurement closest to the top quark mass. The other $b$-jet is consequently
assigned to $t_{\ell}$. $t_{\ell}$ is reconstructed together with the highest
$p_{T}$ lepton and missing transverse energy. In order to improve the
$t_{\ell}$ reconstruction, the longitudinal component of the neutrino four-
momentum ($p_{z}^{\nu}$) is estimated. The value of $p_{z}^{\nu}$ is computed
analytically based on the relation of the four-momentum of the lepton-neutrino
system and the $W$ boson mass,

$\sqrt{(E^{\ell}+E^{\nu})^{2}-(\mathbf{p}^{\ell}+\mathbf{p}^{\nu})^{2}}=m_{W}$
(19)

where the lepton kinematics are measured, $p_{T}^{\nu}=E_{T}^{miss}$, and the
neutrino mass is neglected. The solution that minimizes
$|p_{z}^{\nu}-p_{z}^{\ell}|$ is chosen, and in case of imaginary solutions
only the real part is considered.

With both $t_{h}$ and $t_{\ell}$ reconstructed, further selection is performed
based on their kinematics. As shown by Fig. 3, the top quark $p_{T}$ spectrum
in anomalous $t\bar{t}$ exclusive production is broader with respect to the
SM, and higher values are more frequent. A selection of $p_{T}>425$ GeV for
both $t_{h}$ and $t_{\ell}$ is therefore applied.

(a)

(b)

Figure 3: Transverse momentum spectrum, reconstructed from $t\bar{t}$ decay
products, for the top quark decaying hadronically (a) and leptonically (b).
Only events passing the pre-selection are shown. Histograms are normalized to
unity.

A similar approach is applied to $m_{t\bar{t}}$ (Fig. 4), the mass measured by
the central detector, selecting $m_{t\bar{t}}>960$ GeV. A mass spectrum
shifted towards higher values is indeed characteristic of anomalous exclusive
processes. With this mass selection, the SM CEP of $t\bar{t}$ is strongly
suppressed, making the interference with BSM production negligible.

Figure 4: Invariant mass spectrum of the $t\bar{t}$ system as reconstructed
from its decay products. Only events passing the pre-selection requirements
are shown. Histograms are normalized to unity. The blue histogram represents
the non-diffractive QCD $t\bar{t}$ component. The green and orange
distributions represent the prediction of anomalous $t\bar{t}$ production
assuming the coupling values of $\zeta_{1}=10^{-11}$ and $\zeta_{5}=10^{-11}$
GeV-4.

Since no further radiation is expected in the final state, the tops are
produced back-to-back in the transverse plane. Selecting events with
acoplanarity $|1-\frac{|\phi^{t_{h}}-\phi^{t_{\ell}}|}{\pi}|<0.09$, as Fig.
5(a) illustrates, thus favors anomalous production with respect to SM
$t\bar{t}+jets$ production. Further exploiting the fact that top quarks are
balanced in the transverse plane, we require the ratio of their transverse
momenta $p_{T}^{t_{h}}/p_{T}^{t_{\ell}}>0.88$ (see Fig. 5(b)). In exclusive
events, $t\bar{t}$ pairs are also produced in a narrower rapidity range (see
Fig. 6), therefore we require $|y_{t\bar{t}}|<0.72$.

(a)

(b)

Figure 5: Acoplanarity (a) and transverse momentum balance (b), reconstructed
from $t\bar{t}$ decay products. Only events passing the pre-selection are
shown. Histograms are normalized to unity. Figure 6: Rapidity distribution of
the $t\bar{t}$ system as reconstructed from its decay products. Only events
passing the pre-selection are shown. Histograms are normalized to unity.

Finally, a selection on the sum of the transverse momenta of the decay
products $H_{T}$ is performed. $H_{T}$ is computed as

$H_{T}=\sum_{i}p_{T}^{i},\mathrm{with}~{}i=\ell,\nu,j_{1},j_{2},j_{1}^{b},j_{2}^{b}$
(20)

where $j$ indicates the non $b$-tagged jets used for reconstruction and
$j^{b}$ indicates the $b$-tagged ones. As Fig. 7 shows, selecting high $H_{T}$
values above $1100$ GeV suppresses the SM $t\bar{t}$ background while
retaining anomalous exclusive $t\bar{t}$ events.

Figure 7: Sum of the transverse momenta of the decay products of the
$t\bar{t}$ system ($H_{T}$). Only events passing the pre-selection are shown.
Histograms are normalized to unity.

The final selection step is performed by comparing the mass and rapidity of
the final state reconstructed in the central detector from the decay products
with the measurements that can be obtained from proton tagging. As shown in
Eq. (18), the mass of the central system can be measured via the fractional
momentum losses of the protons. Similarly, the rapidity of the central system
can be computed from the proton kinematics:

$y_{t\bar{t}}^{p}=\frac{1}{2}\log(\frac{\xi_{+}}{\xi_{-}})$ (21)

where $\xi_{+}$ is the fractional momentum loss of the outgoing proton with
positive $p_{z}$, and $\xi_{-}$ the one of the proton with negative $p_{z}$.

In background events, pileup protons are uncorrelated with the processes
observed in the central system. Mass and rapidity measurements done with
outgoing protons are thus expected to be uncorrelated with the same
measurements given by the $t\bar{t}$ system reconstructed from its central
decay products. This is reflected in variables such as the mass match ratio
and $\Delta y_{t\bar{t}}$, defined as follows:

$\frac{\Delta
m_{t\bar{t}}}{m_{t\bar{t}}^{p}}=1-\frac{m_{t\bar{t}}^{c}}{m_{t\bar{t}}^{p}}$
(22) $\Delta y_{t\bar{t}}=y_{t\bar{t}}^{c}-y_{t\bar{t}}^{p}$ (23)

where the $c$ index indicates reconstructed quantities from decay products
measured in the central detector.

(a)

(b)

Figure 8: Mass match ratio (a) and rapidity difference (b) between the $t\bar{t}$ reconstructed from its decay products and from tagged protons. Events passing both pre-selection and central selection are shown. Histograms are normalized to unity. Pre-selection | Central selection | Proton matching | Timing
---|---|---|---
leptons $\geq 1$ | $p_{T}^{t_{h}},p_{T}^{t_{\ell}}>425$ |  |
$b$-jets $\geq 2$ |  $m_{t\bar{t}}>960$ GeV | $|\frac{\Delta m_{t\bar{t}}}{m_{t\bar{t}}^{p}}|<0.08$ | $|z^{p}-z^{c}|<1.5\sigma$
light jets $\geq 2$ |  $H_{T}>1100$ GeV |  |
$E_{\mathrm{T}}^{\mathrm{miss}}$ $\geq 20$ GeV | $|y_{t\bar{t}}|<0.72$ | $|\Delta y|<0.05$ | $|t_{+}+t_{-}-\frac{2\times 200\mathrm{m}}{c}|<1.5\sigma$
protons $\geq 1$ per side | $p_{T}^{t_{h}}/p_{T}^{t_{\ell}}>0.88$ |  |
| $|1-\frac{|\phi^{t_{h}}-\phi^{t_{\ell}}|}{\pi}|<0.09$ |  |
Table 1: Summary of the analysis sequential cuts applied at each selection
stage.

As Fig. 8 illustrates, signal events are expected to show a narrow
distribution around zero, while background events form a broader one. This is
due to the uncorrelated nature of the two pileup protons selected in
background events. Since pileup protons follow a $1/\xi$ momentum loss
distribution, the mass computed with a random combination peaks at lower
values with respect to $m_{t\bar{t}}$, so negative mass match ratios are
favored. Event candidates are classified as signal when $|\frac{\Delta
m_{t\bar{t}}}{m_{t\bar{t}}^{p}}|<0.08$ and $|\Delta y|<0.05$.

Finally, a selection based on the proton time of arrival is applied. The
longitudinal vertex position reconstructed by the central tracking detectors
($z^{c}$) is required to be compatible with the one reconstructed from timing
measurements ($z^{p}$), which is computed as

$z^{p}=\frac{c}{2}(t_{+}-t_{-})$ (24)

where $c$ denotes the speed of light and $t_{+}$ ($t_{-}$) indicates the time
of arrival of the proton with positive (negative) $p_{z}$. The distribution of
$z^{c}-z^{p}$ is Gaussian in our setup, and only events falling within
$1.5\sigma$ limits are accepted for maximum significance.

A similar selection is performed on the ($t_{+}+t_{-}$) variable, which is
expected to be compatible with twice the forward detectors distance from the
interaction point, divided by the speed of light, in case of protons
originating from the same vertex. Also in this case, a $1.5\sigma$
compatibility requirement is applied.

A summary of the cuts applied at each selection stage is given in Table 1.

##  6 Results

In this section we first display our results on the $t\bar{t}\gamma\gamma$
dimension-8 operator basis, turning on one operator at a time. We then apply
the analysis to models with broad neutral resonances.

###  6.1 Sensitivity to dimension-8 operators

Table 2 summarizes the expected events for $300$ fb-1 for both signal and
background, along the cut flow established in Table 1.

The non-diffractive $WW$ background is fully removed by the mass and rapidity
matching of the central event with the proton system. The non-diffractive
$t\bar{t}$ background gives $\sim 100$ events without the use of timing
detectors. This is in contrast with the case of CEP of electroweak gauge boson
pairs ($\gamma\gamma$, $\gamma Z$, $W^{+}W^{-}$, $ZZ$), where the non-
diffractive backgrounds can be suppressed to just a few event
countsggWW_ChaponKepkaRoyon ; Fichet:2014uka ; Baldenegro_ALP ;
Baldenegro:2017aen . The reason is that non-diffractive $t\bar{t}$ is
dominated by gluon-initiated processes with a partonic cross section that
scales with $\alpha_{\mathrm{s}}^{2}$, whereas non-diffractive electroweak
gauge boson pair production are due to quark-initiated processes with cross
sections proportional to $\alpha_{\mathrm{weak}}^{2}$ at leading order.
Furthermore, the $t\bar{t}$ decay is significantly more challenging to
reconstruct, which complicates the use of tighter event selection requirements
to suppress the non-diffractive background. Using the timing detectors at the
nominal resolutions of $60$ ps and $20$ ps, this background is reduced by a
factor of $\sim 7$ and $\sim 60$.

We see that, if the timing detectors performance matches expectations, this
channel may actually be a precision channel with almost no background, along
the same lines as the well-known CEP $\gamma\gamma\to\gamma\gamma$ channel
(see e.g. Fichet:2014uka ; Baldenegro_ALP ).

The statistical significance is computed using the expression presented in
Section 4, and a scan of the dimension-8 EFT operator couplings
($\zeta_{1\ldots 6}$) is performed to find the $95\%$ CL limits and the
$5\sigma$ sensitivities, presented in Table 3. The results are presented for
three possible scenarios, considering a $300$ fb-1 integrated luminosity:
without using information from timing detectors, and using them with nominal
resolutions of $60$ ps and $20$ ps.

| Signal | Background
---|---|---
Selection step | $\zeta_{1}=5\cdot 10^{-11}$ | $\zeta_{5}=5\cdot 10^{-11}$ | $t\bar{t}$ | $WW$ |  CEP $t\bar{t}$
| $\,\quad\mathrm{GeV^{-4}}$ | $\,\quad\mathrm{GeV^{-4}}$ | (non-diffractive) | (non-diffractive) | (diffractive)
Pre-selection | $2.6\cdot 10^{3}$ | $2.2\cdot 10^{3}$ | $5.0\cdot 10^{6}$ | $3.0\cdot 10^{3}$ | 13
Central selection | 341 | 487 | $5.5\cdot 10^{3}$ | 25 | 0
Proton matching | 246 | 355 | 95 | 0 | 0
Timing (60 ps) | 224 | 323 | 13.8 | 0 | 0
Timing (20 ps) | 224 | 323 | 1.7 | 0 | 0
Table 2: Expected events for $300$ fb-1 at each of the selection steps, for the representatives of the operator sets ${\cal O}_{1\ldots 4}$, ${\cal O}_{5\ldots 6}$ and for backgrounds. All other couplings are fixed to zero. Differences among samples of the same family are within Monte Carlo statistical fluctuations. Coupling $[\mathrm{10^{-11}\,GeV^{-4}}]$ | 95% CL | $5\sigma$ |  95% CL ($60$ ps) |  $5\sigma$ ($60$ ps) |  95% CL ($20$ ps) |  $5\sigma$ ($20$ ps)
---|---|---|---|---|---|---
$\zeta_{1}$ | 1.5 | 2.5 | 1.1 | 1.9 | 0.74 | 1.5
$\zeta_{2}$ | 1.4 | 2.4 | 1.0 | 1.7 | 0.70 | 1.4
$\zeta_{3}$ | 1.4 | 2.4 | 1.0 | 1.7 | 0.70 | 1.4
$\zeta_{4}$ | 1.5 | 2.5 | 1.0 | 1.8 | 0.73 | 1.4
$\zeta_{5}$ | 1.2 | 2.0 | 0.84 | 1.5 | 0.60 | 1.2
$\zeta_{6}$ | 1.3 | 2.2 | 0.92 | 1.6 | 0.66 | 1.3
Table 3:  $95$% CL and $5\sigma$ projected limits on each of the couplings,
setting the other ones to zero. Multiple timing detector performance scenarios
are considered: no timing information, timing detector resolution
$\sigma_{t}=60$ ps, and timing detector resolution $\sigma_{t}=20$ ps.

###  6.2 Sensitivity to neutral scalars

In this section we consider a model of neutral scalar coupling to photons and
top quarks (see section 2). In such a case the direct competition with other
channels should be taken into account. We have verified that for any value of
mass and coupling, a $5\sigma$ discovery of the neutral scalar would first
occur in the $\gamma\gamma\to\gamma\gamma$ channel according to the analysis
from Baldenegro:2018hng . As a result the question that should be raised is
the following: In the hypothesis of a discovery of the neutral scalar in the
$\gamma\gamma\to\gamma\gamma$ channel, how well could we measure the scalar’s
coupling to top quarks?

To answer this question, we work at the level of $95\%$ confidence level. We
assume two typical values for the coupling to the photons,

$f_{\gamma\gamma}=\frac{m}{4\pi}\,,\quad\quad\quad\quad\Gamma_{\gamma\gamma}=4\pi
m\quad\quad\quad\quad{\rm(Maximally~{}broad~{}width)}$ (25)
$f_{\gamma\gamma}=\frac{m}{\sqrt{4\pi}}\,,\quad\quad\quad\quad\Gamma_{\gamma\gamma}=m\quad\quad\quad\quad{\rm(Moderately~{}broad~{}width)}$
(26)

where we also indicate the value of the partial width into photons. For both
scenarios the neutral scalar can be considered as broad since $\Gamma\geq m$.
The case of maximally broad width also corresponds to the strongest coupling
allowed by EFT validity or unitarity.

Figure 9:  Projected sensitivity to the $pp\to t\bar{t}\,pp$ cross section at
$95\%$ CL as a function of the mass of the neutral scalar.

Figure 10:  Projected sensitivity at $95\%$ CL in the
$(m_{\phi}-f^{-1}_{\bar{t}t})$ plane. Regions above the curves are sensitive
to our $pp\to t\bar{t}\,pp$ search.

In Fig. 9 we present the projected sensitivity of our search expressed in the
$pp\to t\bar{t}pp$ cross section. The CP-even and CP-odd results are very
similar. We can see that in the mass range considered, the scenario with
moderately broad width has better sensitivity. This is because the width from
$f_{\gamma\gamma}=\frac{m}{4\pi}$ is so broad that it tends to suppress the
total cross section in this mass range.

Importantly, we can see that the use of the timing detectors is key in order
to increase the sensitivity of the search. Without timing detectors, the
projected sensitivity in the moderately broad case goes only up to $1200$ GeV
and is vanishing for the maximally broad case over the considered mass range.
In contrast, using the nominal performance for the timing detectors, it turns
out that the sensitivity in both cases reaches $\sim 1450-1500$ GeV, a
substantial improvement. In Fig. 10 we display the $95\%$ CL sensitivity to
the $f_{t\bar{t}}$ coupling, either with or without timing detectors. We can
again see that the timing detectors are key to expand the reach of the
measurement.

##  7 Summary and perspectives

In this work, we have studied the sensitivity of the central exclusive $pp\to
pp\,t\bar{t}$ process to anomalous interactions between top quarks and photons
at $\sqrt{s}=14$ TeV. The intact protons are detected with forward proton
detectors — the PPS for CMS and the AFP detector for ATLAS. We focused on the
semi-leptonic decay channel of the top quark pair. We considered signals from
broad neutral resonances and from $t\bar{t}\gamma\gamma$ dimension-8 local
operators. Our analysis does not rely on a bump search method nor on shape
analysis.

The main background for central exclusive production (CEP) processes consists
of quantum chromodynamics (QCD) processes paired with forward protons from
uncorrelated pileup interaction. CEP processes are efficiently selected by
using the mass and rapidity of the outgoing two-proton system provided by the
forward detectors. Furthermore, in the channel under consideration, it turns
out that the timing detectors complementing the forward detectors open the
possibility of significant, additional background reduction. For example, at
$L=300$ fb-1, while the background with no timing information is about $100$
events, the background gets reduced to $O(1)$ events when using the timing
detectors with nominal resolution of $20$ ps. This shows that the timing
detectors have the potential to make the CEP $\gamma\gamma\to t\bar{t}$
process a high precision channel, which can be used to search for physics
beyond the Standard Model.

The projected sensitivities obtained in terms of the dimension-8 operator
coefficients are of order $\sim 1.5\cdot 10^{-11}$ GeV-4 ($95\%$ CL), $\sim
2.5\cdot 10^{-11}$ GeV-4 ($5\sigma$) without timing information and are of
order $\sim 0.7\cdot 10^{-11}$ GeV-4 ($95\%$CL ), $\sim 1.5\cdot 10^{-11}$
GeV-4 ($5\sigma$) when using the timing detectors with nominal resolution of
$20$ ps.

We considered a scenario of broad neutral scalars with either moderately
($\Gamma\sim m$) or maximally ($\Gamma\sim 4\pi m$) broad width. We find that
the timing information is key in order to significantly increase the
sensitivity of the channel. For example for resolution of $20$ ps we find
$95\%$ CL sensitivity up to mass of $\sim 1500$ GeV, while the sensitivity is
significantly worse if no timing information is used.

An extension of the present work could be to investigate hadronic decays with
highly boosted topologies. The kinematics of the new physics processes
considered in this study are such that boosted hadronic decays of top quarks
can be reconstructed as large radii jets more frequently than in the SM.
However, to fully exploit this channel, a judicious choice on jet substructure
techniques needs to be taken to separate the boosted top quark topology from
the overwhelming background of light-quark and gluon jets. For example,
variables that are sensitive to the number of prongs in the large-radius jet,
such as $N$-subjettiness variables, and their ratios, could be used. Also
(un)groomed variables could be exploited to separate QCD top quarks, which
have contributions from underlying event activity and initial-state radiation
effects, from the photon-induced top quark production, which does not. We
leave this as possible future work.

The present study was only focused on dimension-8 operators, as motivated by
the neutral particle scenario, which generates suppressed dimension-6
operators which have been neglected here. Investigating the sensitivity of CEP
$t\bar{t}$ channels to dimension-6 photon-top quark operators (namely the top
quark anomalous dipoles) is an interesting direction, especially when the
timing information is taken into account. This exciting possibility is left
for further investigation.

###### Acknowledgements.

GG wishes to thank the Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq) for support under fellowship numbers 422227/2018-8 and
309448/2020-4 . CB and CR thank the Brazil-US Exchange Program of the American
Physical Society (APS) for the support provided in the initial stages of this
study.

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11institutetext: The Centre for Digital Music, Queen Mary University of London
11email: {keunwoo.choi, g.fazekas<EMAIL_ADDRESS>
# Text-based LSTM networks for
Automatic Music Composition

Keunwoo Choi  George Fazekas  Mark Sandler This paper has been supported by
EPSRC Grant EP/L019981/1, Fusing Audio and Semantic Technologies for
Intelligent Music Production and Consumption.

###### Abstract

In this paper, we introduce new methods and discuss results of text-based LSTM
(Long Short-Term Memory) networks for automatic music composition. The
proposed network is designed to learn relationships within text documents that
represent chord progressions and drum tracks in two case studies. In the
experiments, word-RNNs (Recurrent Neural Networks) show good results for both
cases, while character-based RNNs (char-RNNs) only succeed to learn chord
progressions. The proposed system can be used for fully automatic composition
or as semi-automatic systems that help humans to compose music by controlling
a diversity parameter of the model.

###### Keywords:

LSTM, RNN, automatic composition, chord progressions

##  1 Introduction

Music composition is considered creative, intuitive and therefore inherently
human. Nevertheless, it has a long history of mathematical approaches since
Hiller and Isaacson proposed to use Markov chains for automatic composition
[6]. The field of automatic composition includes a wide range of tasks such as
the composition of melody, chord, rhythm [10], and even lyrics [3], i.e. every
typical components of music, and has been subject to numerous research
studies. There are many applications for automatic composition too; automatic
background music generation, AI-assisted composition systems and improviser
software111http://jukedeck.com, http://arpegemusic.com, Band-in-a-Box, PG
Music Inc. for example.

Music can be represented as a sequence of events and thus it can be modelled
as conditional probabilities between musical events. For example, in harmonic
tracks, some chords are more likely to occur than others given the previous
chords, while the whole chord progressions often depend on the global key of
the music. In many automatic composition systems, these relationships are
simplified by assuming that the probability of the current state $p(n)$ only
depends on the probabilities of the states in the past $p(n-k)...p(n-1)$. A
sequence of musical events - notes, chords, rhythm patterns - is generated by
predicting the following event given a seed sequence.

Hidden-Markov models (HMMs) are one of the most popular methods to model and
predict sequences. HMMs are based on the assumption of $k=1$ (Markov
assumption) given the sequence of the hidden states which determine the
visible states. Choral harmonisation is generated after learning chorales by
Bach using a HMM in [1], where $229$ and $153$ chorales are used for training
and testing, respectively. In [14], chord progressions are generated to
accompany a melody to help non-musicians to create music using a HMM. The
training set of the HMM consists of $298$ lead sheets including pop, rock,
R&B, jazz, and country music. In the prediction, the system generates chords
using a $62$$\times$$62$ chord transition probability matrix. In practice,
HMMs had been the most suitable for time-series modelling given the data,
computing power, and feasible optimisation strategies. One of the drawbacks of
HMMs, however, is the inefficiency of 1-of-K scheme of its hidden states. The
memory of HMM is limited to $log_{2}(N)$ bits when there is $N$ hidden states,
which requires to learn $N^{2}$ parameters for the transition matrix.

Recurrent Neural Networks (RNNs) allow for incorporating long term dependency
in the model. Jordan net [8], a simple version of RNNs, is used in [12] to
generate chord sequences. In [13], melodies were generated by a system named
CONCERT, which is trained on sets of $10$ Bach pieces to generate melodies by
note-wise prediction. One ability CONCERT lacks is to learn the global
structure; this may be due to the difficulty of training an RNNs.
Theoretically, it can remember infinitely long sequences, although in practice
it is limited by the vanishing gradient problem [7]. During the training of
back-propagation through time, the gradient is extremely diminished by
multiplications of sigmoid operations.

LSTM (Long Short-Term Memory) units solved this vanishing gradient problem
[7]. LSTM allows the gradient to be flowed by a separate path with not
multiplication but addition operations. LSTM is adopted in [4] to learn
$12$-bar Blues chords progressions and melodies. [11] focuses on the
generation of percussive tracks using LSTM network. The network in [11]
directly analyses audio content of drum tracks and learns features using LSTM.

In this paper, we introduce applications of character- and word-based RNNs
with LSTM units for the automatic generation of jazz chord progressions and
rock music drum tracks. Our work is differentiated from previous works by two
aspects. First, the LSTM networks we use are designed to learn from text data
rather than representations of musical symbols or numeric values. Directly
using text data minimises the overall design procedures for the encoding-
decoding scheme and the network. Second, compared to the previous research
[1],[4],[14], the LSTM networks is trained using a large dataset, which
enables itself to learn more complex relationship between the chords in a
large set.

In the Section 2, we introduce character-based RNNs and the proposed
architecture. In Sections 3 and 4, two case studies on the applications of
RNNs to automatic composition are explained - for jazz chord progressions and
rock music drum tracks. We conclude the work in Section 5.

##  2 The architecture

###  2.1 Character-based RNNs

Char-RNNs are RNNs with character-based learning [15], which is different from
the conventional approach of word-based learning. When applied to the texts of
chords, a char-RNN predict a vector that corresponds to a character (e.g.
predict a based on C:m, and predict j based on C:ma), while a word-RNN
predicts a vector, which corresponds to a unique chord (e.g. C:maj based on
G:maj). Using char-RNNs in this work has two merits.

First, it is based on the minimal assumption - there is no constraint on the
form of the text representation of music. It is worth inspecting if RNNs can
learn musical information with such a weak assumption.

Second, fewer number of characters means fewer number of states, which results
in reducing the computational cost. From a linguistics point of view, sequence
learning methods such as HMMs and RNNs used to model each word(e.g. chord) as
a state as it is natural to find the relationships between words. One drawback
of word-based learning is the large number of states (or the size of
vocabulary); in natural language processing tasks, the vocabulary size easily
exceeds few thousands to even few millions. In the proposed method the size of
the chord vocabulary is $1$,$259$. With character-based prediction, this
decreases to $39$.

The price of small vocabulary size is a longer sequence; as we need to learn
character by character, the model should remember a longer sequence of states.
As mentioned above, the LSTM unit helps the RNNs to learn this long-term
dependency better. This trade-off does not necessarily benefit as in Section
4.

###  2.2 The Proposed Architecture

We use two LSTM layers, each of which consists of 512 hidden units. Dropout of
$0.2$ is added after every LSTM layers [16].

We use the Keras deep learning framework [2]. During the optimisation,
categorical cross-entropy is used as a loss function and optimisation is
performed by ADAM [9]. This optimiser shows an equivalent final performance to
Stochastic Gradient Descent with Nestrov momentum with faster convergence.

The prediction is stochastic. In each prediction for time index $n$, the
network outputs the probabilities of every states. To make the system tunable,
We employ a diversity parameter $\alpha$ in the prediction stage (see Eqn.
LABEL:eq:diversity), which suppresses ($\alpha<1$) or encourages ($\alpha>1$)
the diversity of prediction by re-weighting the probabilities. In detail, the
probabilities of $i$-th state, $p_{i}$, are re-weighted as
$\hat{p_{i}}=\exp{(\log(p_{i})/\alpha)}$. Then, one of the states is selected
by sampling a state according to the re-weighted probabilities.

As stated in Section 3, we perform experiments with char- and words-RNNs. We
keep the same size and number of layers for both networks, although they
result in different effective lengths; for example, manifold states are needed
to be predicted to complete a chord in char-RNNs while each state correspond
to a chord in word-RNNs.

The dataset, code and audio files are released on
web.222https://github.com/keunwoochoi/lstm_real_book
https://github.com/keunwoochoi/LSTMetallica
https://soundcloud.com/kchoi-research/sets/lstm-realbook-1-5
https://soundcloud.com/kchoi-research/sets/lstmetallica-drums

##  3 Case Study 1: Chord progressions

###  3.1 Representation

The goal of this experiment is to generate chord progressions by training an
LSTM network on jazz chord progressions. Here, we do not use any musical
interpretation of the chords such as binary vectors to represent pitch and
chords (as in [5]) but completely rely on their text representations. Table 1
shows an example of a chord progression and the corresponding texts. The left
is an example of a chord notation in The Realbook score, where the positions
of chords are loosely related to the timings of chord changes. The score on
the left is converted into the text on the right, which specifies every chord
for each quarter note.

F:9  MMM M MM $|$ D:min7 MM G:9 M$|$ C:maj MaM F:9 al $|$ C:maj MMMMl M $|$ |  F:9 F:9 F:9 F:9 D:min7 D:min7 G:9 G:9 C:maj C:maj F:9 F:9 C:maj C:maj C:maj C:maj
---|---
Table 1: An example of the text representations of chord progressions in score
(left) and the training data (right). A 4-bar chord progression is generally
written in the form on the left, where the positions of the chords loosely
indicate the chord change timings. On the right, the text show how the score
on the left is represented in the training data. Here, the chords for every
quarter notes are explicitly written and bar indicators are removed.

We used $2,486$ scores from The Realbooks and The Fakebooks as training data.
Every score file was parsed from band-in-a-box format to .xlab format. Then
they were transposed to the key of C while every blank quarter note was filled
with its preceding chord as in the Table 1. Finally, we put _START_ and _END_
flags (any distinctive words can be used as flags) at the beginning and the
end of each score.

Although the key was transposed to C, only $867$ (out of $2$,$846$) scores end
with C:maj ($30\%$), followed by $489$ G:7 (17%), $186$ C:maj6 ($7\%$), $52$
F:maj ($2\%$), and $1$,$252$ scores end with the others – $237$ chords
($46\%$). This is because the The Realbook chord progressions usually end with
chords for a turn-around to make the progressions natural to repeat the score.

There were $1,259$ unique chords in the training dataset. In other words, the
vocabulary size of word-RNN was $1,259$. However there were only $39$
characters in total, which significantly reduced the computation of char-RNN.
The total numbers of chords (words) and characters were $539,609$ and
$3,531,261$, respectively.

###  3.2 Results

We set the system to output a chord progression for every diversity parameter
$\alpha$ after every iteration. In this paper, we present four results from
each networks (char-RNNs and word-RNNs), part of which are reported in the
Table 2. For simplicity, we added bar symbols $|$ and removed repeating chords
in the same bar, e.g. $|$ C:7 C:7 C:7 C:7 $|$ reduced to $|$ C:7 $|$ and $|$
C:7 C:7 E:min E:min $|$ reduced to $|$ C:7 E:min $|$.

$i$ |  $\alpha$ |  Chord progressions
---|---|---
1 |  0.8 |  C:maj $|$ G:7 $|$ … $|$ G:7 $|$ C:maj
1 |  1.2 |  A#:maj $|$ A:7 $|$ A:7 D:min7 D:min7 D:min7 $|$ D:hdim C:hdim $|$C:hdim $|$ C:hdim G:9 G:9 D:min7 $|$ D:min7 D#:dim
23 |  0.8 |  C:7 F:maj$|$F:min$|$C:maj…C:maj$|$G:7$|$ C:maj
23 |  1.2 |  C:7/5 C:7 $|$ F:maj6 F#:dim $|$ C:6(9) $|$ C:6(9) $|$ C:6(9) C:6(9) C:6(9) C:maj $|$ E:7(b9) $|$ A:min(6,9) A#:min(6,9) $|$ A#:min(6,9) $|$… D:min $|$ G:9 C:maj $|$ … G:7 _END_ _START_ C:maj
(a)
1 |  0.5 |  C:maj $|$ G:7 $|$ … $|$ G:7 $|$ C:maj6
1 |  1.2 |  …C:maj _END_ … _START_ _START_ C#:maj A#:min A:sus4/5 C:maj/3 $|$ F:min7 A:min7 D:min7 D:min7… _START_
8 |  0.5 |  C:maj A:min $|$ D:min7 G:7(b9) $|$ C:maj $|$ A:min7 $|$ D:9 $|$ D:9 $|$ D:7 $|$ D:min7 $|$ G:7 $|$ C:maj $|$ C:7 $|$ F:maj $|$ F:min $|$ C:maj
8 |  1.2 |  C:Maj $|$ G:min7 $|$ F:maj $|$ D:min7 D:min7 D:min7/4 G:sus4(b7) $|$ G:min9 G:min9 G:min9 F#:(1,3,b5,b7,9,13) $|$ C:6(9) G:sus4(b7,9) … … C:min _END_ _START_ C:maj

(b)

Table 2: Chord progressions generated by char-RNN (a) and word-RNN (b). Bar
symbols ($|$) are inserted for readability and repeated chords in each bar are
omitted.

First, both char-RNN and word-RNN showed well-structured results. They learned
the local structures of chords and bars after sufficient number of iterations.
In the result, the majority of chords continued for multiples of four,
implying a single chord for within a bar. They also learned the local
relationships between flags and chord. After one iteration, the flags are not
placed properly as in the table 2 (a), where _END_ is not followed by _START_
but repeats itself. As training continues, the flags start to appear in a
sequence of _END_ _START_ C:maj as in the training texts. The last chords of
the score, i.e., the chord before _END_ are not always same as the first chord
(C), which is also natural as they vary in the training file.

Second, after sufficient training, both results showed chord progressions that
lie in Jazz grammar. Examples are II-V-I progressions (D:min7- G:9 - C:maj),
passing chords (A:dim - Ab:dim - G:min7), modal interchange chords (C:min6,
Db:maj ) and substitutions (B:7 as a tritone subdominant of F:7) in char-RNN;
modal interchanges (G:min7), circle of fifths (Eb:sus - Gb:maj6 - B:maj7), and
descending bass (C:maj6,9 - B:dim - A:min7 - Ab:7) in word-RNN. The authors
noticed a subtle difference between the results from the two approaches. The
results from word-RNN are more conventional progressions than those of char-
RNN. However, it cannot be the fundamental difference of the two approaches.
Instead, it may be caused by the difference of effective lengths between char-
and word-RNNs layers - they have the same length of state sequences, but it
results in a longer chord sequence in the word-RNN as mentioned in Section
2.2. In other words, the short memory of char-RNN may result in predictions
that seem to be less constrained and stereotyped.

##  4 Case 2. Drum Tracks

###  4.1 Representation

There are issues when applying LSTM networks to drum tracks including finding
a way to create and effective text representation. Both chord progressions and
drum tracks are sequences of simultaneous events (pitches and drum
components). However, drum tracks do not have a meaningful and compressive
representation such as chord and it necessitate an encoding strategy of the
track into text. We also need a finer time resolution as generally there are
more than four events in a bar.

To encode simultaneous events in a track into texts, we used a binary
representation of pitches, i.e., components of drums - kick, snare, hi-hats,
cymbals, and tom-toms. For example, 100000000 and 010000000 represent kick and
snare, respectively, and a simultaneous playing of kick and snare can be
represented by 110000000.

For efficient representation and learning, only nine components were allowed;
kick, snare, open hi-hats, closed hi-hats, three tom-toms, crash cymbal, and
ride cymbal.333Some of the components in the texts also represent other
similar components, e.g. a closed hi-hats in the texts can mean either closed
hi-hats or pedalled hi-hats in the original midi file. We limited the number
of events in a bar to $16$ by quantising the drum track by $16$th-note.

In the experiment, we first loaded $60$ midi files of drum tracks of Metallica
and quantised them. Then they were encoded into the above described binary
representation. We also added a flag _BAR_ as an annotation of the bar
segments in order to check if the networks learns the local structure.

There can be theoretically $2^{9}=512$ words, but there are supposedly much
fewer words because the combinations of drum components that are played
simultaneously are limited. The size of the word vocabulary in the training
file is $119$ and the file consists of $2$,$141$,$692$ words in total.

###  4.2 Results

Figure 1: A score of a generated drum track.

Char-RNNs turned out to fail to learn the drum tracks and output arbitrary
$0$’s and $1$’s without any structures (the results have no spaces or _BAR_
flags). The length of network may be too short to learn the long-term
relationship between characters. In char-RNNs, representing a single bar
requires $16$ events$\times$$10$ characters$=$$160$ time steps. Encoding music
sequences with only two characters - $0$ and $1$ (+space to for segmentation)
- is an extreme approach for char-RNNs. In this paper, we therefore only
report the result of word-RNNs.

Figure 1 shows one example of our results - a part of the generated track with
$\alpha=1.0$ after $25$ iterations.444The score uses the percussion clef where
$\times$ refers to hi-hats, notes on middle and bottom lines refers to snare
and kick, respectively. It consists of reasonable rock drum patterns -
$8$-beat hi-hats, combinations of kick and snare, and occasional crash cymbals
and tom-toms. Although there are occasional kick/snare/tom-toms notes on back
beats (of sixteen notes), hi-hats remain consistent, playing on $4$-beat and
$8$-beat pattern, which is very common for instance in drum tracks of
Metallica.555https://soundcloud.com/kchoi-research/00-24-100-bonus-for-score,
The score in the figure starts from 34-second.

Controlling $\alpha$ provides a way to tune the technical virtuosity of the
track. Since large $\alpha$ increases the probabilities of occasional events,
large $\alpha$ (=$1.5$) results in tracks with many fill-ins with tom-toms and
a crash cymbal. On the other hands, when $\alpha<1$, the track almost never
contains anything but kick, snare, and hi-hats. As a result, it is possible to
use a combination of small and large $\alpha$ in a drum track generator that
is guided by user, who specifies where to add fill-ins.

##  5 Conclusion

We introduced an algorithm of text-based LSTM networks for automatic
composition and reported results for generating chord progressions and rock
drum tracks. Word-RNNs showed good results in both cases while char-RNNs only
successfully learned chord progressions. The experiments show LSTM provides a
way to learn the sequence of musical events even when the data is given as
text. With the diversity parameter, the proposed algorithm can be used as a
tool that helps human composers. In the future, a more complex network with
the capability of learning interactions within music (instruments,
melody/lyrics) will be examined for a more complete automatic composition
algorithm.

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