11institutetext: Facultad de Ciencias Físico-Matemáticas,
Benemérita Universidad Autónoma de Puebla,
C.P. 72570, Puebla, Pue., Mexico 22institutetext: Ingeniería en Mecatrónica,
Universidad Politécnica de Puebla,
Tercer Carril del Ejido Serrano s/n, San Mateo Cuanalá, Juan C. Bonilla,
Puebla, Puebla, México

# Bounds on the absorptive parts of the chromomagnetic and chromoelectric
dipole moments of the top quark from LHC data

A. I. Hernández-Juárez<EMAIL_ADDRESS>A. Moyotl
<EMAIL_ADDRESS>G. Tavares-Velasco<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Received: date / Accepted: date)

###### Abstract

Bounds on the absorptive (imaginary) parts of the top quark chromomagnetic
$\hat{\mu}_{t}$ and chromoelectric $\hat{d}_{t}$ dipole moments are obtained
by reinterpreting the most recent LHC data in top quark pair production. It is
found that both limits are of the order of $10^{-1}-10^{-2}$, which are
consistent with the standard model prediction of ${\rm
Im}\big{[}\hat{\mu}_{t}\big{]}$. The effects of the absorptive parts of the
top quark dipole moments are also studied via some kinematic distributions of
$\overline{t}t$ production, though no significant deviation from the standard
model leading order contribution is observed. Our bounds can be useful to
constrain the parameter space of standard model extensions.

††offprints: A. I. Hernández-Juárez (corresponding author)

##  1 Introduction

Quite recently, the study of the chromomagnetic dipole moment (CMDM)
$\hat{\mu}_{t}$ of the top quark has become a topic of great interest both
theoretically and experimentally. On the theoretical side, a new evaluation of
the lowest order contributions to $\hat{\mu}_{t}$ within the framework of the
standard model (SM) was presented in Refs. Hernandez-Juarez:2020drn ;
Aranda:2020tox , which has settled some ambiguities found in previous
evaluations: in contrast to what was claimed before Martinez:2007qf , it has
become clear that the CMDM is infrared divergent, with the divergent part
arising from the non-abelian term of the gluon field tensor Hernandez-
Juarez:2020drn ; Aranda:2020tox . Therefore, the study of the static CMDM has
no sense in perturbative QCD. Nevertheless, the off-shell CMDM is finite and
gauge independent in the SM Hernandez-Juarez:2020drn , and therefore it can be
a valid observable quantity. In addition several non SM contributions to the
CMDM have been calculated up to the one-loop level in the framework of
extension theories such as two Higgs doublet models (THDM) Gaitan:2015aia ,
fourth-generation THDMs Hernandez-Juarez:2018uow , 331 models Hernandez-
Juarez:2020xon , models with an extra $Z$ gauge boson Aranda:2018zis , etc. As
far as the top quark chromoelectric dipole moment (CEDM) $\hat{d}_{t}$ is
concerned, it is induced up to the three-loop level in the SM Czarnecki:1997bu
and thus could give a clear signal of $CP$ violation. The top quark CEDM has
also been a topic of interest the literature as it can arise at one-loop level
in some beyond the SM (BSM) theories Hernandez-Juarez:2018uow ; Hernandez-
Juarez:2020xon , thereby opening the possibility of a considerable
enhancement. In general, both the off-shell CMDM and CEMD can have non-zero
imaginary parts, whose effects remain almost unexplored.

On the experimental side, the leading order corrections to the cross section
of top quark pair production induced by the top quark CMDM and CEDM have been
studied in Atwood:1994vm ; Simmons:1995hb ; Haberl:1995ek ; Cheung:1995nt ;
Choi:1997ie ; Hikasa:1998wx ; Czakon:2008ii ; Antipin:2008zx ; Gupta:2009eq ;
Zhang:2010dr ; Degrande:2010kt ; Baumgart:2012ay ; Englert:2012by ;
Hayreter:2013kba ; Bernreuther:2013aga ; Fabbrichesi:2014wva ; Aguilar-
Saavedra:2014iga ; Bernreuther:2015yna ; Cao:2015doa ; Barducci:2017ddn ;
Malekhosseini:2018fgp , and the next-to-leading order corrections have also
been calculated more recently Englert:2014oea ; Franzosi:2015osa ;
AguilarSaavedra:2018nen , whereas the effects of the dipole moments have been
analyzed in some processes Kane:1991bg ; Atwood:1992vj ; Grzadkowski:1997yi ;
Lampe:1997sj ; Yang:1997iv ; Tsuno:2005qb ; Hioki:2009hm ; Gupta:2009wu ;
Hioki:2010zu ; HIOKI:2011xx ; Kamenik:2011dk ; Hioki:2012vn ; Hioki:2013hva ;
MammenAbraham:2021ssc . The CMS collaboration has imposed the following
current bounds on the top quark CMDM and CEDM: $-0.014<\hat{\mu}_{t}<0.004$
and $-0.020<\hat{d}_{t}<0.012$ CMS:2018jcg , which where obtained via two
opposite sign leptons ($e^{+}e^{-}$, $e^{\pm}\mu^{\mp}$, $\mu^{+}\mu^{-}$) in
the final state. Furthermore, the CMS collaboration also set the limits
$\hat{\mu}_{t}=-0.024^{+0.013}_{-0.009}\text{(stat)}^{+0.016}_{-0.011}\text{(syst)}$
and $|\hat{d}_{t}|<0.03$ Sirunyan:2019eyu  obtained by the analysis of
lepton+jets events in the final state. These bounds were extracted from
experimental data by assuming that the top quark CMDM and CEDM are real
quantities.

An appropriate approach to study the anomalous couplings and their effects on
observable processes in a model-independent way is provided by the effective
Lagrangian approach, where a $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ gauge-
invariant effective Lagrangian is introduced to parametrize the effects of
physics BSM. Such an effective Lagrangian contains the SM Lagrangian plus a
tower of effective operators $\mathcal{O}^{n}$ ($n>4$) constructed out of the
SM fields

$\mathcal{L}^{\rm
Eff.}=\mathcal{L}^{\text{S}M}+\sum_{n>0}\frac{\alpha_{\mathcal{O}}}{\Lambda^{n+4}}\mathcal{O}^{n+4},$
(1)

where the coupling constants $\alpha_{\mathcal{O}}$ parametrizes our ignorance
of the new physics and $\Lambda$ is the new physics scale. In particular, non-
standard top-gluon interactions arise from a dimension-six operator
Arzt:1994gp ; Aguilar-Saavedra:2009ygx , which after electroweak symmetry
breaking gives rise to the the following Lagrangian

$\mathcal{L}=-g_{s}\bar{t}T^{a}\Big{[}\frac{\sigma^{\mu\nu}}{2m_{t}}\Big{(}\hat{\mu}_{t}+i\hat{d}_{t}\gamma^{5}\Big{)}G^{a}_{\mu\nu}\Big{]}t,$
(2)

where $T^{a}$ are the $SU(3)$ color generators, $G^{a}_{\mu}$ are the gluon
fields,
$G^{a}_{\mu\nu}=\partial_{\mu}G^{a}_{\nu}-\partial_{\nu}G^{a}_{\mu}-g_{s}f_{abc}G^{b}_{\mu}G^{c}_{\nu}$
is the gluon field tensor, whereas $\hat{\mu}_{t}$ and $\hat{d}_{t}$ are
constant coefficients that parametrizes the anomalous contributions to the
$ttg$ coupling arising from new physics. It must be noted however that the
above Lagrangian does not yield the most general $\overline{t}tg$ interaction,
which in fact can be written in terms of four independent form factors
Nowakowski:2004cv .

The above Lagrangian also describes the interaction between an off-shell gluon
and two on-shell quarks Davydychev:2000rt . Since the off-shell CMDM and CEDM
of the top quark are complex in general, they will be written as

$\displaystyle\hat{\mu}_{t}$ $\displaystyle={\rm
Re}\big{[}\hat{\mu}_{t}\big{]}+i{\rm Im}\big{[}\hat{\mu}_{t}\big{]},$ (3)
$\displaystyle\hat{d}_{t}$ $\displaystyle={\rm
Re}\big{[}\hat{d}_{t}\big{]}+i{\rm Im}\big{[}\hat{d}_{t}\big{]}.$ (4)

As far as the SM predictions are concerned, the real and imaginary parts of
the off-shell CMDM of the top quark are of the order of $10^{-2}-10^{-3}$
Hernandez-Juarez:2020drn , whereas the predictions for the off-shell CEDM are
not available yet. Nevertheless, in BSM theories both real and imaginary parts
of the off-shell top quark CEDM are of the order of $10^{-19}$ Hernandez-
Juarez:2020xon . On the other hand, the effects of the absorptive parts of the
CMDM and CEDM at LHC were first studied in Bernreuther:2013aga  but to our
knowledge, there is no update on such analysis, which we believe is in the
order given the current experimental bounds on these observables.

We would like to note that although the off-shell dipole form factors are
dependent on the gluon transfer momentum, we will follow the authors of Ref.
Bernreuther:2013aga  and consider in our analysis below that both the real and
absorptive parts of the dipole form factors are constant, which is valid as
long as unitarity is not spoiled. This approach has also been used for
instance to obtain bounds on the trilinear neutral gauge boson couplings
$Z\gamma V^{*}$ ($V=\gamma,Z$) Baur:1992cd , which due to Bose statistics and
angular momentum conservation are non-vanishing only for off-shell $V$. In
fact, over a large interval of $q^{2}$, the SM contribution to the top quark
CMDM shows little variation Hernandez-Juarez:2020drn , and the same is true
for both the CMDM and CEDM in some BSM theories (see for instance Hernandez-
Juarez:2020xon ; Aranda:2018zis ). Our main goal is to obtain bounds on ${\rm
Im}\left(\hat{\mu}_{t}\right)$ and ${\rm Im}\left(\hat{d}_{t}\right)$ using
the data for top quark pair production at the LHC run 2, which in turn can be
useful to constraint the parameter space of some BSM theories.

Our work is organized as follows. In Sec. 2 we discuss the framework for the
study of the CMDM and CEDM absorptive parts. Section 3 we present a novel
calculation of the parton cross-sections of $t\bar{t}$ production for complex
CMDM and CEDM, which to our knowledge has not been reported before. In Sec. 4
a numerical simulation is presented for top quark pair production at the LHC
via MadGraph5, where the effective Lagrangian of Eq. (2) was implemented with
the help of the FeynRules package. The results for the $t\overline{t}$ cross
section as a function of the real and imaginary parts of $\hat{\mu}_{t}$ and
$\hat{d}_{t}$ are then used to obtain bounds on their absorptive parts. The
possibility that kinematic distributions could be helpful to disentangle the
top quark CMDM and CEDM absorptive parts is examined in Section 5. Finally, in
Sec. 6 we present our conclusions.

##  2 Remarks on the absorptive parts of the CMDM in the SM

In the SM, the CMDM of quarks arises at the one-loop level through the Feynman
diagrams of Figs. 1 (QCD contribution) and 2 (electroweak contribution). The
off-shell CMDM $\hat{\mu}_{q}(q^{2})$ can develop an absorptive (imaginary)
part when the gluon transfer four-momentum $\hat{q}=\sqrt{q^{2}}$ crosses the
threshold $\hat{q}\geq 2m$, with $m$ the mass of the virtual particles
attached to the off-shell gluon. In such region these particles are allowed to
be pair produced, which is true for all energies of the external gluon in
Feynman diagram (b) of Fig. 1, whereas the threshold is $\hat{q}\geq 2m_{q}$
for Feynman diagrams 1(a), 2(a) and 2(c), and $\hat{q}\geq 2m_{q^{\prime}}$ in
Feynman diagram 2(b). The absorptive contributions to the CMDM can also be
extracted by the Cutkosky rules Cutkosky_1960 , which yield the same results
obtained via the usual techniques for Feynman diagram calculation Zhou:2004gm
. It is also important to emphasize that the the SM contribution to the off-
shell dipole moment $\hat{\mu}_{q}(q^{2})$ is finite and gauge independent for
arbitrary $q^{2}$.

Figure 1: QCD contributions to the CMDM of quarks in the SM.  Figure 2:
Electroweak contributions to the CMDM of quarks in the SM.

For the top quark CMDM, the contribution from the diagram (b) of Fig. 2
develops an absorptive part at $\hat{q}=2m_{b}$, whereas that from the diagram
(b) of Fig. 1 is complex for any $\hat{q}$ value. The remaining contributions
become complex at $\hat{q}=2m_{t}$. Moreover, the corresponding contributions
to the top quark CEDM would also become complex at the same energy thresholds.
It is thus interesting to obtain a bound on the absorptive part of the top
quark CMDM and CEDM consistent with the CMS limits. Such a bound can be
interpreted as a limit on the new physics effects inducing new contributions
to the CMDM and CEDM of the top quark. To assess the order of magnitude of the
absorptive part of the CMDM at the LHC energies, we have performed a numerical
evaluation of the analytical expressions of Ref. Hernandez-Juarez:2020drn  to
find the energy interval of the transfer momentum of the gluon where the real
part of the top quark CMDM predicted by the SM matches the CMS bounds
CMS:2018jcg ; Sirunyan:2019eyu . We obtain that the value ${\rm
Re}\big{[}\hat{\mu}_{t}\big{]}=-0.024$ reported in Ref Sirunyan:2019eyu
corresponds to the 57 GeV$\leqslant\hat{q}\leqslant$ 59 GeV interval, where
the respective absorptive part value is ${\rm
Im}\big{[}\hat{\mu}_{t}\big{]}\approx-0.034$. As far as the bound $-0.014<{\rm
Re}\big{[}\hat{\mu}_{t}\big{]}<0.004$ reported in Ref. CMS:2018jcg , it is
consistent with energies above $\hat{q}=85$ GeV, in this case, the absorptive
part can be one order of magnitude smaller than in the previous one: for
values around $\hat{q}=85$ GeV, ${\rm
Im}\big{[}\hat{\mu}_{t}\big{]}\approx-0.028$, whereas at higher energies the
corresponding value is of the order of $10^{-3}$ and remains almost constant
as the energy increases.

##  3 Contributions of CMDM and CEDM to $\overline{t}t$ production

Top pair production can receive contributions from the anomalous
$\overline{t}tg$ coupling Haberl:1995ek ; Cheung:1995nt  of Eq. (2) but also
from the non-SM $\overline{t}tgg$ vertex arising from the non-abelian part of
the gluon field strength tensor. The corresponding Feynman rules follow
straightforwardly and are shown in Fig. 3. As already mentioned, in a strict
sense, the top quark dipole form factors are functions of the gluon transfer
momentum $q^{2}$, with such functions being model dependent. Furthermore, when
working with off-shell green functions one must address the problem of gauge
dependence. Along this line, methods such as the pinch technique have been
used in the past to remove any gauge-dependent terms by considering additional
Feynman diagrams contributing to the physical process under study. However, as
argued in Ref. Hernandez-Juarez:2020drn  the off-shell CMDM of the top quark
is gauge-independent in the SM and the same is true for both the CMDM and CEDM
in electroweak extension models. Also, as shown in Refs. Hernandez-
Juarez:2020xon ; Aranda:2018zis  there is little dependence of the top quark
dipole form factors on $q^{2}$ over a large energy interval in the SM and some
BSM theories. We will thus assume that both the real and absorptive parts of
the CMDM and CEDM are constant and obtain bounds on the absorptive parts from
the data on top quark production at the LHC.

Figure 3: Feynman rules for the anomalous $\bar{t}tg$ and $\bar{t}tgg$
couplings arising from Lagrangian (2).

The most recent analyses on top quark production assume that both CMDM and
CEDM are purely real CMS:2018jcg ; Sirunyan:2019eyu . In this work, we are
interested in the study of the contributions of the absorptive parts of these
dipole moments. Therefore we consider that $\hat{\mu}_{t}$ and $\hat{d}_{t}$
are complex and calculate the following parton cross-sections:

$\displaystyle\hat{\sigma}_{q\overline{q}}\equiv\sigma(\overline{q}q\rightarrow\overline{t}t),$
$\displaystyle\hat{\sigma}_{gg}\equiv\sigma(gg\rightarrow\overline{t}t),$ (5)

which apart from the SM contribution receive a new one from the Feynman
diagrams of Fig. 4, where the large dot represents the anomalous CMDM and CEDM
contributions.

Figure 4: Feynman diagrams for the contribution to the parton cross sections
$\hat{\sigma}_{q\overline{q}}$ and $\hat{\sigma}_{gg}$ at the leading order.
Crossed diagrams are not shown. The large dot represents the anomalous
couplings induced by the CMDM and CEDM. The SM tree-level contribution is
obtained after replacing the anomalous $\bar{t}tg$ coupling by the SM one.

After some algebra we obtain the respective differential cross sections for
general complex CMDM and CEDM:

$\displaystyle\frac{d\hat{\sigma}_{q\bar{q}}}{d\hat{t}}=$
$\displaystyle\frac{\pi\alpha_{s}^{2}}{\hat{s}^{2}}\frac{8}{9}\Bigg{[}\frac{1}{2}-v+z+2{\rm
Re}\big{[}\hat{\mu}_{t}\big{]}+\Big{(}{\rm
Re}\big{[}\hat{\mu}_{t}\big{]}^{2}+{\rm
Im}\big{[}\hat{\mu}_{t}\big{]}^{2}-{\rm Re}\big{[}\hat{d}_{t}\big{]}^{2}-{\rm
Im}\big{[}\hat{d}_{t}\big{]}^{2}\Big{)}$ $\displaystyle+\Big{(}{\rm
Re}\big{[}\hat{\mu}_{t}\big{]}^{2}+{\rm
Im}\big{[}\hat{\mu}_{t}\big{]}^{2}+{\rm Re}\big{[}\hat{d}_{t}\big{]}^{2}+{\rm
Im}\big{[}\hat{d}_{t}\big{]}^{2}\Big{)}\frac{v}{z}\Bigg{]},$ (6)

and

$\displaystyle\frac{d\hat{\sigma}_{gg}}{d\hat{t}}=$
$\displaystyle\frac{\pi\alpha_{s}^{2}}{\hat{s}^{2}}\frac{1}{12}\Bigg{[}\Big{(}\frac{4}{v}-9\Big{)}\Big{(}\frac{1}{2}-v-2z\big{(}1-\frac{z}{v}\big{)}+2\
{\rm Re}\big{[}\hat{\mu}_{t}\big{]}\Big{)}+\frac{1}{8vz}\Big{[}v\Big{(}55\
{\rm Re}\big{[}\hat{d}_{t}\big{]}^{2}+{\rm
Re}\big{[}\hat{\mu}_{t}\big{]}^{2}(55-144z)\Big{)}$ $\displaystyle+z\Big{(}4\
{\rm Re}\big{[}\hat{d}_{t}\big{]}^{2}+70\ {\rm
Re}\big{[}\hat{\mu}_{t}\big{]}^{2}\Big{)}+\frac{1}{vz}\Big{[}-16v^{3}\Big{(}4\Big{(}{\rm
Re}\big{[}\hat{\mu}_{t}\big{]}^{2}{\rm Im}\big{[}\hat{d}_{t}\big{]}^{2}-4\ \
{\rm Re}\big{[}\hat{\mu}_{t}\big{]}\ {\rm Im}\big{[}\hat{\mu}_{t}\big{]}\ {\rm
Re}\big{[}\hat{d}_{t}\big{]}\ {\rm Im}\big{[}\hat{d}_{t}\big{]}$
$\displaystyle+{\rm Im}\big{[}\hat{\mu}_{t}\big{]}^{2}\ {\rm
Re}\big{[}\hat{d}_{t}\big{]}^{2}\Big{)}+9z\Big{(}{\rm
Im}\big{[}\hat{\mu}_{t}\big{]}^{2}+{\rm
Im}\big{[}\hat{d}_{t}\big{]}^{2}\Big{)}\Big{)}+v^{2}z\Big{(}-512\ {\rm
Re}\big{[}\hat{\mu}_{t}\big{]}\ {\rm Im}\big{[}\hat{\mu}_{t}\big{]}\ {\rm
Re}\big{[}\hat{d}_{t}\big{]}\ {\rm Im}\big{[}\hat{d}_{t}\big{]}$
$\displaystyle+{\rm Im}\big{[}\hat{d}_{t}\big{]}^{2}\Big{(}16\ {\rm
Re}\big{[}\hat{\mu}_{t}\big{]}\Big{(}15\ {\rm
Re}\big{[}\hat{\mu}_{t}\big{]}+7\Big{)}+288z+63\Big{)}+3\ {\rm
Im}\big{[}\hat{\mu}_{t}\big{]}^{2}\Big{(}80\ {\rm
Re}\big{[}\hat{d}_{t}\big{]}^{2}+48z+21\Big{)}\Big{)}$
$\displaystyle-2vz^{2}\Big{(}92\ {\rm Im}\big{[}\hat{\mu}_{t}\big{]}\ {\rm
Re}\big{[}\hat{d}_{t}\big{]}\ {\rm Im}\big{[}\hat{d}_{t}\big{]}+\Big{(}1-8\
{\rm Re}\big{[}\hat{d}_{t}\big{]}^{2}\Big{)}{\rm
Im}\big{[}\hat{\mu}_{t}\big{]}^{2}+2\ {\rm
Im}\big{[}\hat{d}_{t}\big{]}^{2}\Big{(}-{\rm
Re}\big{[}\hat{\mu}_{t}\big{]}\Big{(}4\ {\rm
Re}\big{[}\hat{\mu}_{t}\big{]}+41\Big{)}$
$\displaystyle+72z+17\Big{)}\Big{)}+128\ {\rm
Im}\big{[}\hat{d}_{t}\big{]}^{2}z^{3}\Big{]}+{\rm
Re}\big{[}\hat{\mu}_{t}\big{]}\left({\rm
Re}\big{[}\hat{\mu}_{t}\big{]}^{2}+{\rm
Im}\big{[}\hat{\mu}_{t}\big{]}^{2}+{\rm
Re}\big{[}\hat{d}_{t}\big{]}^{2}\right)\left(\frac{14}{z}-\frac{5}{2v}\right)$
$\displaystyle+\left(\Big{(}{\rm Re}\big{[}\hat{\mu}_{t}\big{]}^{2}+{\rm
Im}\big{[}\hat{\mu}_{t}\big{]}^{2}\Big{)}^{2}+2\Big{(}{\rm
Re}\big{[}\hat{\mu}_{t}\big{]}^{2}\ {\rm Re}\big{[}\hat{d}_{t}\big{]}^{2}+{\rm
Im}\big{[}\hat{\mu}_{t}\big{]}^{2}\ {\rm
Im}\big{[}\hat{d}_{t}\big{]}^{2}\Big{)}+\Big{(}{\rm
Re}\big{[}\hat{d}_{t}\big{]}^{2}+{\rm
Im}\big{[}\hat{d}_{t}\big{]}^{2}\Big{)}^{2}\right)$
$\displaystyle\times\left(-\frac{1}{z}+\frac{1}{v}+\frac{4v}{z^{2}}\right)\Bigg{]},$
(7)

where $\hat{s}$, $\hat{t}$ and $\hat{u}$ are the usual parton Mandelstam
variables and we introduced the definitions

$\displaystyle z$ $\displaystyle=\frac{m_{t}^{2}}{\hat{s}},$ (8)
$\displaystyle v$
$\displaystyle=\frac{1}{\hat{s}^{2}}(\hat{t}-m_{t}^{2})(\hat{u}-m_{t}^{2}).$
(9)

In the $t\overline{t}$ center of mass frame, the parameter $\hat{t}$, is
related to the angle $\hat{\theta}$ between the momentum of the outgoing top
quark and that of the incoming parton as

$m_{t}^{2}-\hat{t}=\frac{\hat{s}}{2}(1-\beta\cos{\hat{\theta}}),$ (10)

with $\beta=\sqrt{1-4z}$.

For $\hat{\mu}_{t}=\hat{d}_{t}=0$ the above cross sections reduce to the known
SM results Gluck:1977zm  as expected. We also have verified that in the
scenario with purely real CMDM and CEDM, Eq. (3) reproduces the result
reported in Ref. Haberl:1995ek ; Cheung:1995nt . Nevertheless, in the same
scenario we do not find agreement with our result for Eq. (3) and the one
previously reported Haberl:1995ek ; Cheung:1995nt , which apparently is
incomplete as there is no agreement in the coefficients of $\hat{\mu}_{t}^{2}$
and $\hat{d}^{2}_{t}$.

##  4 Bounds on absorptive parts of the CMDM and CEDM of the top quark

We now turn to constrain the absorptive parts ${\rm
Im}\big{[}\hat{\mu}_{t}\big{]}$ and ${\rm Im}\big{[}\hat{d}_{t}\big{]}$ via
the LHC data on top quark pair production CMS:2018jcg ; Aad:2020tmz ;
Sirunyan:2019eyu . We follow a similar approach to that discussed in
Hayreter:2013kba  and use a Monte Carlo simulation to obtain the theory
predictions for the leading order contribution to the $\sigma(pp\to
t\overline{t})$ cross section. In order to compute the corresponding
contributions from the top quark CMDM and CEDM, we use MadGraph5
Alwall:2014hca , where the anomalous interactions of Eq. (2) were implemented
with the help of FeynRules Alloul:2013bka .

We will consider the most recent LHC results for top quark pair production at
center-of-mass energy $\sqrt{s}=$13 TeV. Therefore we use the ATLAS cross
section in the lepton plus jets channel Aad:2020tmz

$\sigma_{\text{Exp}}(pp\to t\overline{t})=(830\pm 39)\quad\text{pb},$ (11)

whereas for the theoretical SM prediction we use Zyla:2020zbs ; Czakon:2008ii

$\sigma_{\text{Theo}}(pp\to t\overline{t})=(831.8\pm 43)\quad\text{pb},$ (12)

wherein both cases the errors have been added in quadrature. We will assume
that the small deviation in $\sigma_{\text{Exp}}(pp\to t\overline{t})$ from
the theoretical leading order SM prediction $\sigma_{\text{Theo}}(pp\to
t\overline{t})$ is due to the real and absorptive parts of the top quark CMDM
and CEDM. While 90% (85%) of $\sigma(pp\to t\overline{t})$ arises dominantly
from the partonic process $gg\rightarrow\overline{t}t$ at $\sqrt{s}=14$ TeV
($\sqrt{s}$=7 TeV), the contribution of $gg\overline{t}t$ vertex has not been
considered in the computation of (12) as it is a non-SM interaction. However,
as observed in Eqs. (3) and (3), the dipole moments induce a deviation in
$\sigma(pp\to t\overline{t})$. Thus, the top quark CMDM and CEDM may explain
slight deviations from the SM prediction to $\overline{t}t$ production.

The ratio between the measured and predicted cross sections is

$\mathcal{R}=\frac{\sigma_{\text{Exp}}(pp\to
t\overline{t})}{\sigma_{\text{Theo}}(pp\to t\overline{t})}=0.99\pm 0.069.$
(13)

Following Ref. Hayreter:2013kba , we will interpret the error of Eq. (13) as a
window to BSM effects in top quark pair production and use it to set
constraints on the absorptive parts of $\hat{\mu}_{t}$ and $\hat{d}_{t}$. As
already mentioned, in the analysis of the CMS collaboration the CMDM and CEDM
of the top quark were assumed to be real quantities. In Ref. CMS:2018jcg  all
the measurements are analyzed under a linearized approximation of the
anomalous couplings Bernreuther:2015yna , which parametrizes the non-SM
interactions and include the $gg\overline{t}t$ vertex, though the dipole
moments are not interpreted as complex quantities and thus the absorptive
parts are not considered in the differential cross section used to study BSM
effects. Also, in Ref. Sirunyan:2019eyu  the kinematic distribution used to
obtain constraints on $\hat{\mu}_{t}$ and $\hat{d}_{t}$ were defined via Eqs.
(3) and (3), though once again it was assumed that both the top quark CMDM and
CEDM are real quantities. Thus, the effects of the absorptive parts have never
been studied by the CMS collaboration, however, such consequences has a
negligible effect on their analysis as the kinematic distributions measured at
the LHC Bernreuther:2015yna ; Baumgart:2012ay  are strongly constrained
Fabbrichesi:2014wva ; Cao:2015doa . Below we will consider a similar approach
to that followed in Refs. Hayreter:2013kba ; Sirunyan:2019eyu  to obtain
bounds on the absorptive parts of $\hat{\mu}_{t}$ and $\hat{d}_{t}$ by
reinterpreting the LHC data. We will also show explicitly that the relevant
kinematic distributions of $\overline{t}t$ production are not sensitive to the
imaginary parts of the dipole moments, as already pointed out in Refs.
Bernreuther:2013aga ; Bernreuther:2015yna ; Baumgart:2012ay .

To study the absorptive part of the top quark CMDM and CEDM we proceed as
follows: We first set $\hat{\mu}_{t}=\hat{d}_{t}=0$ and obtain the SM cross
section $\sigma_{\text{SM}}(pp\to t\overline{t})$, afterwards we generate the
new physics contribution $\sigma_{\text{NP}}(pp\to t\overline{t})$ for non-
zero ${\rm Re}\big{[}\hat{\mu}_{t}\big{]}$, whereas all the remaining
parameters are set to zero. This procedure is repeated for each one of the
${\rm Im}\big{[}\hat{\mu}_{t}\big{]}$, ${\rm Re}\big{[}\hat{d}_{t}\big{]}$ and
${\rm Im}\big{[}\hat{d}_{t}\big{]}$ parameters. All our event samples for the
$pp\rightarrow\overline{t}t$ cross section are generated at $\sqrt{s}=$14 TeV.

We show in Fig. 5 the ratio $\sigma_{\text{NP}}/\sigma_{\text{SM}}$ as a
function of the real and absorptive parts of the CMDM (left plot) and CEDM
(right plot), where the MadGraph5 estimated error is included. We plot the
best fit curves.

Figure 5: Ratio
$\mathcal{R}=\sigma_{\text{NP}}(pp\to\bar{t}t)/\sigma_{\text{SM}}(pp\to\bar{t}t)$
as a function of the real (green lines) and imaginary parts (blue lines) of
the CMDM (left plot) and CEDM (right plot) of the top quark at $\sqrt{s}=14$
TeV. The bars represent the MadGraph5 estimated errors and the solid lines are
the best fit curves.

To fit the data of Fig. 5, we have not considered the interference terms of
the real and absorptive parts of $\hat{\mu}_{t}$ and $\hat{d}_{t}$. Such an
approach has been used in the past to study the implications of the top quark
CMDM and CEDM in $t\bar{t}$ production Hayreter:2013kba ;
Malekhosseini:2018fgp ; Englert:2014oea . Thus, from Eqs. (3) and (3) we
observe that the ratio $\mathcal{R}$ is a fourth-order polynomial of the real
and imaginary parts of $\hat{\mu}_{t}$ and $\hat{d}_{t}$, though there are
only even powers of the absorptive parts. The expression obtained from the fit
of Fig. 5 reads

$\displaystyle\mathcal{R}$ $\displaystyle\simeq 1+5.33\ {\rm
Re}\big{[}\hat{\mu}_{t}\big{]}+19.14\ {\rm
Re}\big{[}\hat{\mu}_{t}\big{]}^{2}+21.98\ {\rm
Re}\big{[}\hat{\mu}_{t}\big{]}^{3}+5.78\ {\rm
Re}\big{[}\hat{\mu}_{t}\big{]}^{4}$ $\displaystyle+12.35\ {\rm
Im}\big{[}\hat{\mu}_{t}\big{]}^{2}+4.38\ {\rm
Im}\big{[}\hat{\mu}_{t}\big{]}^{4}+13.79\ {\rm
Re}\big{[}\hat{d}_{t}\big{]}^{2}+5.58\ {\rm Re}\big{[}\hat{d}_{t}\big{]}^{4}$
$\displaystyle+13.78\ {\rm Im}\big{[}\hat{d}_{t}\big{]}^{2}+6.15\ {\rm
Im}\big{[}\hat{d}_{t}\big{]}^{4}.$ (14)

We observe that the contributions of ${\rm Im}\big{[}\hat{\mu}_{t}\big{]}$,
${\rm Re}\big{[}\hat{d}_{t}\big{]}$ and ${\rm Im}\big{[}\hat{d}_{t}\big{]}$
are of the same order and similar size, which is actually in accordance with
Eq. (3). Moreover, $\mathcal{R}$ shows a similar dependence on the absorptive
parts and ${\rm Re}\big{[}\hat{d}_{t}\big{]}$, which makes it possible to
obtain bounds on ${\rm Im}\big{[}\hat{\mu}_{t}\big{]}$ and ${\rm
Im}\big{[}\hat{d}_{t}\big{]}$ following Refs. Hayreter:2013kba ;
Sirunyan:2019eyu , where limits on the real part of $\hat{d}_{t}$ are obtained
using Eqs. (3), (3) and (4). We also note that the leading contribution to
$\mathcal{R}$ arises from the linear term of ${\rm
Re}\big{[}\hat{\mu}_{t}\big{]}$ as the other dipole terms contribute
quadratically at the lowest order and are thus more suppressed. The values
predicted for the off-shell top quark CMDM are of the order of
$10^{-2}-10^{-3}$ in the SM Hernandez-Juarez:2020drn , whereas the typical
values predicted for the CEDM in some BSM theories are of the order of
$10^{-19}-10^{-20}$ Hernandez-Juarez:2020xon . Thus. the effects of the top
quark dipole moments would hardly induce a significant deviation to the top
quark production cross section.

To analyze the effects of the absorptive parts of the top quark dipole moments
we will proceed as follows. We fix the corresponding real parts using the CMS
limits CMS:2018jcg ; Sirunyan:2019eyu , which allow us to constrain the
absorptive parts ${\rm Im}\big{[}\hat{\mu}_{t}\big{]}$ and ${\rm
Im}\big{[}\hat{d}_{t}\big{]}$ via Eqs. (4) and (13), with the error being
attributed to the anomalous $\bar{t}tg$ contributions. In other words, we fix
${\rm Re}\big{[}\hat{\mu}_{t}\big{]}$ and ${\rm Re}\big{[}\hat{d}_{t}\big{]}$
to their current constraints and find the allowed area of ${\rm
Im}\big{[}\hat{\mu}_{t}\big{]}$ and ${\rm Im}\big{[}\hat{d}_{t}\big{]}$
values. With this aim we assume the following three scenarios:

  * •

Scenario I: we use the lower bounds ${\rm
Re}\big{[}\hat{\mu}_{t}\big{]}=-0.014$ and ${\rm
Re}\big{[}\hat{d}_{t}\big{]}=-0.02$ reported in CMS:2018jcg .

  * •

Scenario II: we use the upper bounds ${\rm
Re}\big{[}\hat{\mu}_{t}\big{]}=0.004$ and ${\rm
Re}\big{[}\hat{d}_{t}\big{]}=0.012$ reported in CMS:2018jcg .

  * •

Scenario III: we use the value ${\rm Re}\big{[}\hat{\mu}_{t}\big{]}=-0.024$
and the upper bound ${\rm Re}\big{[}\hat{d}_{t}\big{]}=0.03$ reported in
Sirunyan:2019eyu .

We do not consider the scenario where ${\rm Re}\big{[}\hat{d}_{t}\big{]}$ is
set to its lower (negative) bound Sirunyan:2019eyu  as it yields similar
bounds to those obtained in scenario III as $\mathcal{R}$ is an even function
of ${\rm Re}\big{[}\hat{d}_{t}\big{]}$. A similar situation arises for other
possible scenarios, which yield bounds of a similar order of magnitude.

The allowed areas in the ${\rm Im}\big{[}\hat{\mu}_{t}\big{]}-{\rm
Im}\big{[}\hat{d}_{t}\big{]}$ plane at the 95% C.L. are the concentric
ellipses shown in Fig. 6 for the three scenarios discussed above. The
corresponding bounds are: $|{\rm Im}\big{[}\hat{\mu}_{t}\big{]}|\lesssim
0.127$ and $|{\rm Im}\big{[}\hat{d}_{t}\big{]}|\lesssim 0.12$ in scenario I
(blue solid lines); $|{\rm Im}\big{[}\hat{\mu}_{t}\big{]}|\lesssim 0.139$ and
$|{\rm Im}\big{[}\hat{d}_{t}\big{]}|\lesssim 0.133$ in scenario III (green
dot-dashed lines); and $|{\rm Im}\big{[}\hat{\mu}_{t}\big{]}|\lesssim 0.094$
and $|{\rm Im}\big{[}\hat{d}_{t}\big{]}|\lesssim 0.09$ in scenario II (orange
dashed line). The latter scenario yields the intersected area allowed by the
three scenarios, which means that the corresponding bounds are consistent with
both CMS limits. Note that in both cases the bounds on the absorptive parts of
$\hat{\mu}_{t}$ and $\hat{d}_{t}$ are quite similar, of the order of
$10^{-1}-10^{-2}$ at the 95% C.L.

For completeness, we revisit the case where the gluon transfer four-momentum
($\hat{q}=\sqrt{q^{2}}$) dependence is considered for ${\rm
Re}\big{[}\hat{\mu}_{t}\big{]}$, which is possible using the expressions for
$\hat{\mu}_{t}(q^{2})$ reported in Ref. Hernandez-Juarez:2020drn . The real
part of the CEDM is set as ${\rm Re}\big{[}\hat{d}_{t}\big{]}=0.01$ since
there are not any analytic results for $\hat{d}_{t}(q^{2})$. In Fig. 7, the
allowed areas in the ${\rm Im}\big{[}\hat{\mu}_{t}\big{]}-{\rm
Im}\big{[}\hat{d}_{t}\big{]}$ plane at the 95% C.L. for three different
energies of $\hat{q}$ are shown. We note that at low energies ($\hat{q}=30$
GeV) the bounds are slightly larger than scenario III but they are still of
the same order, whereas from $\hat{q}=m_{Z}$ the limits are similar to those
found in Fig. 6. Thus, the bounds obtained by taking the real parts as
constants are compatible with those where their dependence on $\hat{q}$ is
considered. Especially, for energies above the $Z$ boson mass both constraints
are almost identical.

It is worth comparing our limits with the theoretical predictions of the SM
and some BSM theories. In particular, for a transfer momentum in the interval
30 GeV $\leqslant\hat{q}\leqslant$ 1000 GeV, the SM prediction for the
absorptive part of $\hat{\mu}_{t}(q^{2})$ can be as large as $10^{-2}$
Hernandez-Juarez:2020drn , which is close to our bounds. On the other hand,
several BSM theories predict values for the absorptive part of
$\hat{d}_{t}(q^{2})$ of the order of $10^{-19}$, which is far away from our
bound.

We have also made the same analysis but including the interference terms of
Eq. (3). Nonetheless, the obtained fit is still consistent with Eq. (4) and
the bounds are similar to those of Fig. 6. Thus, the interference terms can be
neglected as their contribution is not relevant.

Figure 6: Allowed area at the 95% C.L. for the imaginary parts of the CMDM and
CEDM of the top quark in the three scenarios discussed in the text for the
corresponding real parts.  Figure 7: Allowed area at the 95% C.L. for the
imaginary parts of the CMDM and CEDM of the top quark at three different
energies of the gluon transfer momentum.

##  5 Kinematic distributions

The effects of ${\rm Re}\big{[}\hat{\mu}_{t}\big{]}$ and ${\rm
Re}\big{[}\hat{d}_{t}\big{]}$ on top quark pair production have been analyzed
in the past as some kinematic distributions can be sensitive to such
parameters Atwood:1994vm ; Cheung:1995nt ; Franzosi:2015osa ; Barducci:2017ddn
. On the other hand, to our knowledge the possible effects of the absorptive
parts ${\rm Im}\big{[}\hat{\mu}_{t}\big{]}$ and ${\rm
Im}\big{[}\hat{d}_{t}\big{]}$ have only been explored in Ref.
Bernreuther:2013aga  through the longitudinal $t$ and $\overline{t}$
polarizations. Therefore, for completeness we will examine the possibility
that the differential cross sections for top quark pair production could be
sensitive to the absorptive parts of the top quark CMDM and CEDM, such
implications are supposed to be unobservable Bernreuther:2015yna ;
Baumgart:2012ay , nonetheless it has been never shown explicitly. To this end,
we use the CMS constraints on the real parts of the top quark dipole form
factors (${\rm Re}\big{[}\hat{\mu}_{t}\big{]}=-0.014$ and ${\rm
Re}\big{[}\hat{d}_{t}\big{]}=0.01$) and analyze any possible deviation in the
kinematic distributions of top quark production when both dipole moments
develop an absorptive part. We consider the following three cases for ${\rm
Im}\big{[}\hat{\mu}_{t}\big{]}$ and ${\rm Im}\big{[}\hat{d}_{t}\big{]}$:

  1. i)

${\rm Im}\big{[}\hat{\mu}_{t}\big{]}={\rm Im}\big{[}\hat{d}_{t}\big{]}=0.01$.

  2. ii)

${\rm Im}\big{[}\hat{\mu}_{t}\big{]}={\rm Im}\big{[}\hat{d}_{t}\big{]}=0.05$.

  3. iii)

${\rm Im}\big{[}\hat{\mu}_{t}\big{]}=0.01$ and ${\rm
Im}\big{[}\hat{d}_{t}\big{]}=-0.01$.

We consider such values as they are consistent with the SM prediction for the
CMDM and the constraints of Sec. 4. Scenarios i) and ii) allow us to explore
the possibility that the kinematic distributions can be sensitive to small
changes in the absorptive terms, whereas scenario iii) allows us to test the
effect of a flip of sign. For the graphical analysis, we use MADANALYSIS 5
Conte:2012fm .

In Figs. 8 and 8 we show the kinematic distributions of the $t\overline{t}$
invariant mass and the top quark transverse momentum in the scenarios
discussed above. It is observed that there is no considerable distinction
between the kinematic distributions obtained in the general case with complex
top quark dipole form factors and those obtained in the scenario in which they
are purely real. This was also observed in the case where the contributions of
the real part of the top quark dipole moments are compared with the SM leading
order contribution Degrande:2010kt ; Barducci:2017ddn . A similar situation
occurs for the kinematic distribution of the rapidity $\eta$, which is shown
in Fig. 8.

Figure 8: Invariant mass (a), top quark transverse momentum (b) and rapidity
(c) kinematic distributions for top quark pair production at the LHC at
$\sqrt{s}=14$ TeV in the three scenarios discussed in the text for the
absorptive parts of $\hat{\mu}_{t}$ and $\hat{d}_{t}$.

We have also examined the sensitivity of the forward-backward (FB) asymmetry
to the CMDM and CEDM in top quark pair production at the LHC, which is
possible at the leading order in some models Dorsner:2009mq ; Degrande:2010kt
, whereas in the SM there is only a significant deviation up to next-to-
leading order Dorsner:2009mq . Unfortunately, Eqs. (3) and (3) cannot be
expressed as a linear combination of $\cos{\theta}$ via Eq. (10). Thus, a
deviation to the FB asymmetry at the leading order is not possible
Frampton:2009rk . However, other asymmetries could be sensitive to the CMDM
and CEDM of the top quark, as shown in Ref. Bernreuther:2013aga ;
Hayreter:2013kba . In summary, all the kinematic distributions studied here
show no significant deviation from leading order contribution to top quark
pair production arising from the real and absorptive parts of the top quark
dipole form factors.

##  6 Conclusions

The off-shell CMDM and CEDM of quarks have become a topic of interest recently
Hernandez-Juarez:2020drn ; Aranda:2020tox . However, the study of their
absorptive (imaginary) parts remains almost unexplored. In this work, we have
obtained bounds on the new physics contributions to the absorptive parts of
the off-shell top quark CMDM and CEDM via the experimental data of top quark
pair production at the LHC, which to our knowledge are the first limits of
this kind. We present explicit expressions for the corresponding differential
parton cross-sections considering complex CMDM and CEDM, which have also been
calculated for the first time. We point out that there is a disagreement
between our result for the $gg\to t\bar{t}$ differential cross section and the
expression previously reported in the scenario where only the real part of the
top quark dipole form factors are considered Haberl:1995ek ; Cheung:1995nt .
Our bounds for the absorptive parts were obtained using the most recent data
for the top quark CMDM and CEDM reported by the CMS collaboration CMS:2018jcg
; Sirunyan:2019eyu . It was found that the upper bound on the absorptive parts
of both dipole moments are of order $10^{-1}-10^{-2}$. In particular, values
of order $10^{-2}$ are consistent with all the CMS results. We also note that
our bound on ${\rm Im}\big{[}\hat{\mu}_{t}\big{]}$ is consistent with the SM
prediction for the absorptive part of $\hat{\mu}_{t}$, which is of order
$10^{-2}-10^{-3}$ Hernandez-Juarez:2020drn ; Aranda:2020tox . On the other
hand, in some BSM theories the absorptive part of the CEDM is of the order of
$10^{-19}$ Hernandez-Juarez:2020xon , which seems well beyond the experimental
reach. Our limits could be useful to constrain the parameter space of BSM
theories and are consistent with the case where the gluon transfer four-
momentum ($\hat{q}$) dependence is considered for the real part of the top
quark CMDM.

We also explored the possibility that several kinematic distributions for top
quark pair production at the LHC can be sensitive to the absorptive parts of
the CMDM and CEDM, but we find that there are no significant deviation from
the scenario where the CMDM and CEDM are purely real. In fact, even in the
case of real CMDM and CEDM, there is no significant deviation from the leading
order SM contribution as discussed previously Degrande:2010kt ;
Barducci:2017ddn .

##  7 acknowledgements

We acknowledge support from Consejo Nacional de Ciencia y Tecnología and
Sistema Nacional de Investigadores. Partial support from Vicerrectoría de
Investigación y Estudios de Posgrado de la Benémerita Universidad Autónoma de
Puebla is also acknowledged. We also thank to M. A. Arroyo-Ureña for
discussions to implement our model in MadGraph5.

## Data Availability Statement

The datasets generated during and/or analysed during the current study are
available from the corresponding author on reasonable request.

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