# Odderon as reggeized oddball spin-3 in $pp$ and $p\bar{p}$ elastic
scattering

Jingle B. Magallanes<EMAIL_ADDRESS>Department of Physics,
Mindanao State University - Iligan Institute of Technology, Iligan City, 9200,
Philippines  Prin Sawasdipol<EMAIL_ADDRESS>Khon Kaen Particle
Physics and Cosmology Theory Group (KKPaCT), Department of Physics, Faculty of
Science, Khon Kaen University,123 Mitraphap Rd., Khon Kaen, 40002, Thailand
Chakrit Pongkitivanichkul<EMAIL_ADDRESS>Khon Kaen Particle Physics and
Cosmology Theory Group (KKPaCT), Department of Physics, Faculty of Science,
Khon Kaen University,123 Mitraphap Rd., Khon Kaen, 40002, Thailand  Daris
Samart<EMAIL_ADDRESS>corresponding author Khon Kaen Particle Physics and
Cosmology Theory Group (KKPaCT), Department of Physics, Faculty of Science,
Khon Kaen University,123 Mitraphap Rd., Khon Kaen, 40002, Thailand

###### Abstract

In this work, we propose that the odderon is a reggeized odd-glueball tensor
spin-3. To demonstrate our proposal, we study the $pp$ and $p\bar{p}$ elastic
scattering by including contributions of the spin-3 odderon and spin-2 pomeron
exchange in the process. The phenomenological effective Lagrangian approach is
used to calculate the $pp$ and $p\bar{p}$ elastic scattering amplitudes at the
tree level. We fit the theoretical results with the various experimental data
of the $pp$ and $p\bar{p}$ scattering at the TeV scale to determine the model
parameters in the present work. By using the model parameters, as a result,
the Chew-Frautschi plot of the tensor odderon regge trajectory is evaluated.
In addition, the odderon spin-3 mass is predicted to be about 3 GeV which
agrees with a double pole regge model extracted from the data. In addition,
the $pp$ total cross-section of our model is compatible with the results from
TOTEM and its extrapolation from D0 collaboration. Moreover, the total cross-
section also satisfies the Friossart bound at the regge limit.

##  I Introduction

Quantum ChromoDynamics (QCD) is a modern theory of strong interaction based on
non-abalien color SU(3) quantum gauge field theory describing the interaction
of quarks and gluons. QCD is highly successful in explaining hadronic
structures and interactions at high energies (momentum exchange) where the
strong coupling is small and perturbative quantum field theory is applied.
However, at a low energy regime, QCD is a strongly coupled theory that we can
not use the standard perturbative theory. On the other hand, hadron-hadron
scattering in high center of mass energy ($\sqrt{s}$) but low momentum
exchange ($t$) known as soft-high energy regime or a Regge limit of
$s\to\infty$ and $s\ggg t$, the perturbative QCD is also inapplicable. Before
the birth of QCD, Regge theory is invented to describe the hadron-hadron
collisions by using analytical properties of the scattering amplitudes Eden et
al. (1966) including a consideration of the complex angular momentum Gribov
(2007). In the Regge theory, the amplitudes of the hadronic processes are
scaled as $s^{J}$ where $J$ is the spin of the exchange particles called
Reggeons with fixed relevant quantum numbers Collins (2009). One can write
down the spin as a linear function of $t$ as $J=\alpha(t)$ in the complex
angular momentum plane and the linear function
$\alpha(t)=\alpha_{0}+\alpha^{\prime}\,t$ is called a regge trajectory. In
addition, the poles (regge poles) correspond to the families of the exchange
particles with increasing spins along the trajectories. As a result, the
amplitudes of regge theory are represented in terms of a sum over all possible
exchange particles lying on the regge trajectory. The cross-sections of
various hadronic processes in the soft-high energy scattering limit are
successfully described by the regge theory Eden (1971); Collins (1971); Chiu
(1972); Irving and Worden (1977).

According to the experimental data of hadron-hadron scattering in the regge
limit, the total cross-sections slowly grew up with the increase of $s$
whereas the regge trajectories of all known mesons are not sufficient to
explain the experimental data. Then, a so-called pomeron was introduced to
address this problem Chew and Frautschi (1961); Gribov (1961). The pomeron is
a reggeon carrying all even charge transformations, vacuum quantum number,
with the intercept of regge trajectory $\alpha_{0}\approx 1$. This yields the
slow growth of the total cross-section at large $s$ Donnachie et al. (2004);
Forshaw and Ross (2022). There are various approaches trying to extract the
information of the pomeron trajectory. The typical values of the parameters of
the pomeron trajectory from Collins et al. (1974); Donnachie and Landshoff
(1984, 1992) are $\alpha_{0}\approx 1.06-1.08$ and $\alpha^{\prime}\approx
0.025$ GeV-2 Collins et al. (1974); Donnachie and Landshoff (1984, 1992). The
pomeron is generally considered as a bound state of the gluons (glueball). On
the other hand, the odd charge-conjugation counterpart of the pomeron called
odderon has been proposed by Ref.Lukaszuk and Nicolescu (1973). Similar to the
pomeron, the odderon is considered as the glueball with the odd number of
gluon compositions. The odderon might cause the different observables between
$pp$ and $p\bar{p}$ due to its charge-conjugation property which is compatible
with the experiment. However, the nature and properties of the pomeron and
odderon are still unclear so far. A number of approaches have been used to
calculate the properties (mass, spin, regge trajectory and etc.) of pomeron
and odderon as glueballs Akkelin and Martynov (1991); Kuraev et al. (1977);
Balitsky and Lipatov (1978); Kuraev et al. (1976); Lipatov (1990); Narison
(1984, 1998); Tang and Qiao (2016); Meyer and Teper (2005); Morningstar and
Peardon (1999); Ochs (2013); Cotanch et al. (2007); Kaidalov and Simonov
(2006); Kaidalov and Simonov (2000a, b); Llanes-Estrada et al. (2002); Brower
et al. (2007, 2009); Boschi-Filho and Braga (2004); Boschi-Filho et al.
(2006); Colangelo et al. (2007); Li and Huang (2013); Brower et al. (2015);
Ballon-Bayona et al. (2016); Folco Capossoli and Boschi-Filho (2016); Folco
Capossoli et al. (2016); Dymarsky and Melnikov (2022); Meyer (2004); Gregory
et al. (2012); Chen et al. (2006); Llanes-Estrada et al. (2006); Mathieu et
al. (2009). A study of pomeron and odderon exchanges in $pp$ and $p\bar{p}$
elastic scatterings in the soft-high energy regime has been extensively
investigated in various frameworks for instances, phenomenological approaches
Ewerz et al. (2014, 2016); Covolan et al. (1996); Block and Halzen (2012);
Szanyi et al. (2020); Jenkovszky (2020); Broniowski et al. (2018); Csorgo and
Szanyi (2021); Csörgő et al. (2021, 2019); Ster et al. (2015); Block and Cahn
(1985); Khoze et al. (2018), QCD inspired models Halzen et al. (1993);
Donnachie and Landshoff (1983); Ma et al. (2001); Hu et al. (2002); He et al.
(2003); Hu et al. (2008); Zhou et al. (2006); Lu et al. (2020); Bartels et al.
(2000), holographic QCD or AdS/CFT correspondence Domokos et al. (2009, 2010);
Avsar et al. (2010); Hu et al. (2018); Xie et al. (2019); Burikham and Samart
(2019); Liu et al. (2022a, b); Ballon-Bayona et al. (2017); Iatrakis et al.
(2016).

Recently, however, TOTEM and D0 collaborations have confirmed the existence of
the odderon by comparing the experimental data between $pp$ (extrapolated from
previous several data) and $p\bar{p}$ at 1.96 TeV Abazov et al. (2012, 2021).
This reveals the contributions of the odderon in $t$-channel elastic
scattering. After the TOTEM and D0 collaborations claimed the discovery of the
odderon, several works have done to investigate the properties and scattering
processes of the odderon Chen et al. (2021a, b); Zhang et al. (2022); Bence et
al. (2021); Capossoli et al. (2022); Lebiedowicz (2022); Baldenegro et al.
(2022); Cui et al. (2022); Lebiedowicz et al. (2022); Bonanno et al. (2022).

Inspired by the discovery of the odderon and pertinent literature on field
theoretical model in Ref.Ewerz et al. (2014), we propose the odderon as a
spin-3 tensor odd-glueball within the standard field theoretical framework and
study its consequences in $pp$ and $p\bar{p}$ elastic scatterings. It has been
demonstrated that the pomeron is likely to be the spin-2 tensor glueball
Domokos et al. (2009); Ewerz et al. (2014, 2016); Szanyi et al. (2020);
Jenkovszky (2020); Lu et al. (2020); Ma et al. (2001); Hu et al. (2018); Xie
et al. (2019); Burikham and Samart (2019); Liu et al. (2022a) instead of the
scalar one. In analogy to the tensor pomeron, we therefore consider the
odderon as the tensor particle, not the vector particle. The lowest tensor
odderon in its regge trajectory is spin-3 while the slope, intercept and mass
of the odderon are poorly understood. In this work, we will investigate the
elastic scattering of $pp$ and $p\bar{p}$ with the contributions of the
odderon spin-3. The effective Lagrangian of the spin-3 odderon with protons is
constructed by using a similar approach in Ref.Ewerz et al. (2014). The
contributions of the spin-2 pomeron exchange are also included in the
calculation where the effective Lagrangian of pomeron and protons is taken
from Ref.Ewerz et al. (2014). Then the differential cross-sections of the $pp$
and $p\bar{p}$ elastic scattering are calculated. After a careful statistical
analysis, we fit the parameters of our model with several relevant
experimental data at the TeV scale. All Feynman rules of our model such as
vertices, propagators and etc., can be computed directly from the effective
Lagrangians in the conventional method of perturbative QFT. The aim of the
present work is to make a clear and systematic calculation in order to obtain
the amplitudes. The analysis is made with intention for compatibility with
other field theoretical models.

The present work is organized as follows, in the section II, we set up our
model for $pp$ and $p\bar{p}$ scattering with pomeron spin-2 and odderon
spin-3 exchanges. The amplitudes are also computed as well. The observables
will be calculated and free parameters of our model will be fitted with the
relevant experimental data at the TeV scales in the section III. In section
IV, we close this work by giving discussions and conclusions.

##  II Formalisms: Model set up and scattering amplitudes

###  II.1 Effective Lagrangians of the $pp$ and $p\bar{p}$ scattering in
spin-2 pomeron and spin-3 odderon exchange picture

In this section, we will set up the effective Lagrangians of the $pp$ and
$p\bar{p}$ scattering. It is well-known that the pomeron exchange plays a
major role in elastic proton-proton scattering at high energy but low momentum
exchange regimes. This work we assume the pomeron as spin-2 tensor particle
and the Lagrangian is given by Ewerz et al. (2014, 2016)

$\displaystyle\mathscr{L}_{\mathbb{P}}=-\frac{i}{2}\,g_{\mathbb{P}}\,\mathbb{P}^{\mu\nu}\,\mathcal{G}_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}\bar{\psi}\gamma_{\mu^{\prime}}\,\overset{\leftrightarrow}{\partial}_{\nu^{\prime}}\psi\,,$
(1)

where $\mathbb{P}^{\mu\nu}$ is symmetric spin-2 tensor field, $\psi$ is Dirac
proton field,
$\bar{\psi}\,\overset{\leftrightarrow}{\partial}_{\mu}\,\psi\equiv(\partial_{\mu}\bar{\psi})\,\psi-\bar{\psi}\,\partial_{\mu}\psi$
and the coupling $g_{\mathbb{P}}$ with dimension [mass]-1 is considered to be
a free parameter in this work. The totally symmetric tensor
$\mathcal{G}_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}$ is defined by

$\displaystyle\mathcal{G}_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}=g_{\mu}^{\mu^{\prime}}g_{\nu}^{\nu^{\prime}}+g_{\nu}^{\mu^{\prime}}g_{\mu}^{\nu^{\prime}}\,.$
(2)

The corresponding vertex function of the pomeron-proton-proton coupling is
given by

$\displaystyle\Gamma_{\mu\nu}^{\mathbb{P}}(q,q^{\prime})=\frac{i}{2}\,g_{\mathbb{P}}\,\mathcal{G}_{\mu\nu}^{\mu^{\prime}\nu^{\prime}}\gamma_{\mu^{\prime}}\,\big{(}q^{\prime}_{\nu^{\prime}}+q_{\nu^{\prime}}\big{)},$
(3)

where $q$ and $q^{\prime}$ are incoming and outgoing proton/anti-proton
momenta respectively.

For the spin-3 odderon exchange interaction, the effective Lagrangian reads,

$\displaystyle\mathscr{L}_{\mathbb{O}}$ $\displaystyle=$
$\displaystyle\frac{i}{3!}\,g_{\mathbb{O}}\,\mathbb{O}^{\mu\nu\rho}\,\mathcal{G}_{\mu\nu\rho}^{\mu^{\prime}\nu^{\prime}\rho^{\prime}}\,\bar{\psi}\,\gamma_{\mu^{\prime}}\,\overset{\leftrightarrow}{\partial}_{\nu^{\prime}}\,\overset{\leftrightarrow}{\partial}_{\rho^{\prime}}\,\psi\,,$
(4)

where $\mathbb{O}^{\mu\nu\rho}$ is the spin-3 tensor field that totally
symmetric under interchanging of the Lorentz indices, i.e.,
$\mathbb{O}^{\mu\nu\rho}=\mathbb{O}^{\nu\rho\mu}=\mathbb{O}^{\rho\mu\nu}=\mathbb{O}^{\mu\rho\nu}=\mathbb{O}^{\nu\mu\rho}=\mathbb{O}^{\rho\nu\mu}$.
In order to obtain the effective Lagrangian in Eq. (4), in addition, we have
followed the construction of the higher spin field coupling to the nucleons in
Ref.Ewerz et al. (2014) by adding the twist-2 operator as shown in Appendix B
of Ewerz et al. (2014) and all detail discussions are given in Ewerz et al.
(2014). Moreover, the coupling $g_{\mathbb{O}}$ is the free parameter carrying
the dimension [mass]-2. While the totally symmetric tensor
$\mathcal{G}_{\mu\nu\rho}^{\mu^{\prime}\nu^{\prime}\rho^{\prime}}$ is used to
ensure that the lower indices $(\mu,\nu,\rho)$ of the vertex functions,
$\Gamma^{\mathbb{O}}_{\mu\nu\rho}$ is totally symmetric and it is defined by

$\displaystyle\mathcal{G}_{\mu\nu\rho}^{\mu^{\prime}\nu^{\prime}\rho^{\prime}}=\frac{1}{3!}\left(g_{\mu}^{\mu^{\prime}}\,g_{\nu}^{\nu^{\prime}}\,g_{\rho}^{\rho^{\prime}}+g_{\nu}^{\mu^{\prime}}\,g_{\rho}^{\nu^{\prime}}\,g_{\mu}^{\rho^{\prime}}+g_{\rho}^{\mu^{\prime}}\,g_{\mu}^{\nu^{\prime}}\,g_{\nu}^{\rho^{\prime}}+g_{\nu}^{\mu^{\prime}}\,g_{\mu}^{\nu^{\prime}}\,g_{\rho}^{\rho^{\prime}}+g_{\mu}^{\mu^{\prime}}\,g_{\rho}^{\nu^{\prime}}\,g_{\nu}^{\rho^{\prime}}+g_{\rho}^{\mu^{\prime}}\,g_{\nu}^{\nu^{\prime}}\,g_{\mu}^{\rho^{\prime}}\,.\right)$
(5)

According to the Lagrangian in Eqs.(4), we can write the Feynman rules for the
vertex function for $\mathscr{L}_{\mathbb{O}}$ as

$\displaystyle\Gamma^{\mathbb{O}}_{\mu\nu\rho}(q,q^{\prime})$ $\displaystyle=$
$\displaystyle\frac{1}{3!}\,g_{\mathbb{O}}\,\mathcal{G}_{\mu\nu\rho}^{\mu^{\prime}\nu^{\prime}\rho^{\prime}}\,\gamma_{\mu^{\prime}}\,(q^{\prime}_{\nu^{\prime}}+q_{\nu^{\prime}})\,(q^{\prime}_{\rho^{\prime}}+q_{\rho^{\prime}})\,,$
(6)

where $q$ and $q^{\prime}$ represent the incoming and outgoing momenta of the
proton/anti-proton of the vertex functions.

###  II.2 Scattering amplitudes of the $pp$ and $p\bar{p}$ elastic scattering

Next step, we will calculate the amplitudes for the elastic $pp$ and
$p\bar{p}$ scattering processes under the external momentum specifications as
$p(q_{1})\,p(q_{2})\to p(q_{3})\,p(q_{4})$ and
$\bar{p}(q_{1})\,p(q_{2})\to\bar{p}(q_{3})\,p(q_{4})$ for $pp$ and $p\bar{p}$
elastic scattering processes, respectively. In addition, we assume that the
elastic $pp$ and $p\bar{p}$ scatterings are mainly dominated by the pomeron
and odderon exchanges since other mesons and reggeons exchange contributions
are negligibly small in these processes at TeV scale.

By using the standard method in QFT Peskin and Schroeder (1995), the elastic
$pp$ scattering amplitude of the pomeron exchange is given by

$\displaystyle i\,\mathcal{M}_{\mathbb{P}}^{pp}$ $\displaystyle=$
$\displaystyle\langle
p(q_{3})\,p(q_{4})\,|:\mathcal{T}\,\exp{\left(i\int\mathscr{L}_{\mathbb{P}}(x)\,d^{4}x\right)}:|\,p(q_{1})\,p(q_{2})\rangle$
(7) $\displaystyle=$
$\displaystyle-\frac{i}{4}\,g_{\mathbb{P}}^{2}\,\mathcal{G}_{\mu_{1}\nu_{1}}^{\mu_{1}^{\prime}\nu_{1}^{\prime}}\,\bar{u}(q_{3})\,\gamma_{\mu_{1}^{\prime}}\,\big{(}q_{3,\nu_{1}^{\prime}}+q_{1,\nu_{1}^{\prime}}\big{)}\,u(q_{1})\,\Delta_{\mathbb{P}}^{\mu_{1}\nu_{1};\mu_{2}\nu_{2}}(s,t)$
$\displaystyle\qquad\times\,\mathcal{G}_{\mu_{2}\nu_{2}}^{\mu_{2}^{\prime}\nu_{2}^{\prime}}\,\bar{u}(q_{4})\,\gamma_{\mu_{2}^{\prime}}\,\big{(}q_{4,\nu_{2}^{\prime}}+q_{2,\nu_{2}^{\prime}}\big{)}\,u(q_{2})\,\mathcal{F}_{\mathbb{P}}(t)^{2}\,,$

and the elastic $p\bar{p}$ scattering amplitude of the pomeron exchange is
given by

$\displaystyle i\,\mathcal{M}_{\mathbb{P}}^{p\bar{p}}$ $\displaystyle=$
$\displaystyle\langle\bar{p}(q_{3})\,p(q_{4})\,|:\mathcal{T}\,\exp{\left(i\int\mathscr{L}_{\mathbb{P}}(x)\,d^{4}x\right)}:|\,\bar{p}(q_{1})\,p(q_{2})\rangle$
(8) $\displaystyle=$
$\displaystyle-\frac{i}{4}\,g_{\mathbb{P}}^{2}\,\mathcal{G}_{\mu_{1}\nu_{1}}^{\mu_{1}^{\prime}\nu_{1}^{\prime}}\,\bar{v}(q_{1})\,\gamma_{\mu_{1}^{\prime}}\,\big{(}q_{1,\nu_{1}^{\prime}}+q_{3,\nu_{1}^{\prime}}\big{)}\,v(q_{3})\,\Delta_{\mathbb{P}}^{\mu_{1}\nu_{1};\mu_{2}\nu_{2}}(s,t)$
$\displaystyle\qquad\times\,\mathcal{G}_{\mu_{2}\nu_{2}}^{\mu_{2}^{\prime}\nu_{2}^{\prime}}\,\bar{u}(q_{4})\,\gamma_{\mu_{2}^{\prime}}\,\big{(}q_{4,\nu_{2}^{\prime}}+q_{2,\nu_{2}^{\prime}}\big{)}\,u(q_{2})\,\mathcal{F}_{\mathbb{P}}(t)^{2}\,.$

For the propagator of the spin-2 pomeron
$\Delta_{\mathbb{P}}^{\mu\nu;\mu^{\prime}\nu^{\prime}}$, we employ from
Ref.Ewerz et al. (2014) and it takes the following form

$\displaystyle\Delta^{\mu\nu,\rho\sigma}_{\mathbb{P}}(s,t)$ $\displaystyle=$
$\displaystyle\frac{1}{4\,s}\Big{[}g^{\mu\rho}\,g^{\nu\sigma}+g^{\mu\sigma}\,g^{\nu\rho}-\frac{1}{2}\,g^{\mu\nu}\,g^{\rho\sigma}\Big{]}\,\left(-i\,\alpha_{\mathbb{P}}^{\prime}\,s\right)^{\alpha_{\mathbb{P}}(t)-1}\,,$
(9)

with the conventional linear pomeron trajectory Ewerz et al. (2014, 2016)

$\displaystyle\alpha_{\mathbb{P}}(t)$ $\displaystyle=$ $\displaystyle
1+\epsilon_{\mathbb{P}}+\alpha_{\mathbb{P}}^{\prime}\,t\,,$ (10)
$\displaystyle\epsilon_{\mathbb{P}}$ $\displaystyle=$ $\displaystyle
0.0808\,,\qquad{\rm and}\qquad\alpha_{\mathbb{P}}^{\prime}=0.25\;{\rm
GeV}^{-2},$ (11)

where the factor $1+\epsilon_{\mathbb{P}}$ and $\alpha_{\mathbb{P}}^{\prime}$
represent the vertical interception and slope of the pomeron trajectory,
respectively. This formulation of the spin-2 pomeron propagator is an
alternative approach to study soft high energy hadronic collisions and it can
reproduce related several experimental data. In addition, the propagator in
this approach is proposed by Heidelberg group Ewerz et al. (2014). Finally,
the pomeron-$pp/p\bar{p}$ coupling form factor reads,

$\displaystyle\mathcal{F}_{\mathbb{P}}(t)$ $\displaystyle=$
$\displaystyle\left(1-\frac{t}{4m_{p}^{2}}\frac{\mu_{p}}{\mu_{N}}\right)\left(1-\frac{t}{4m_{p}^{2}}\right)^{-1}\left(1-\frac{t}{m_{D}^{2}}\right)^{-2},$
$\displaystyle\mu_{N}$ $\displaystyle=$
$\displaystyle\frac{e}{2m_{p}}\,,\qquad\frac{\mu_{p}}{\mu_{N}}=2.7928\,,\qquad
m_{D}^{2}=0.71\;{\rm GeV}^{2}\,.$ (12)

This form factor is the standard Dirac form factor of proton and it is widely
used to study the $pp$ and $p\bar{p}$ scattering see more details discussions
and its consequences of the form factor in Eq.(II.2) in chapter two of
Ref.Close et al. (2009).

We turn to consider the elastic $pp$ and $p\bar{p}$ scattering amplitudes for
the spin-3 odderon exchange contribution. Having used the same manner, the
$pp$ scattering amplitude is calculated and one finds,

$\displaystyle i\mathcal{M}_{\mathbb{O}}^{pp}$ $\displaystyle=$
$\displaystyle\langle
p(q_{3})\,p(q_{4})\,|:\mathcal{T}\,\exp{\left(i\int\mathscr{L}_{\mathbb{O}}(x)\,d^{4}x\right)}:|\,p(q_{1})\,p(q_{2})\rangle$
(13) $\displaystyle=$
$\displaystyle\frac{i}{36}\,g_{\mathbb{O}}^{2}\,\mathcal{G}_{\mu\nu\rho}^{\mu_{1}\nu_{1}\rho_{1}}\,\bar{u}(q_{3})\,\gamma_{\mu_{1}}\,\big{(}q_{3,\nu_{1}}+q_{1,\nu_{1}}\big{)}\,\big{(}q_{3,\rho_{1}}+q_{1,\rho_{1}}\big{)}\,u(q_{1})\,\Delta_{\mathbb{O}}^{\mu\nu\rho;\mu^{\prime}\nu^{\prime}\rho^{\prime}}\big{(}s,t\big{)}\,$
$\displaystyle\quad\;\;\,\times\,\mathcal{G}_{\mu^{\prime}\nu^{\prime}\rho^{\prime}}^{\mu_{2}\nu_{2}\rho_{2}}\,\bar{u}(q_{4})\,\gamma_{\mu_{2}}\,\big{(}q_{4,\nu_{2}}+q_{2,\nu_{2}}\big{)}\,\big{(}q_{4,\rho_{2}}+q_{2,\rho_{2}}\big{)}\,u(q_{2})\,\mathcal{F}_{\mathbb{O}}(t)^{2}\,,$

while the amplitude of the $p\bar{p}$ scattering with the odderon exchange
reads

$\displaystyle i\mathcal{M}_{\mathbb{O}}^{p\bar{p}}$ $\displaystyle=$
$\displaystyle\langle\bar{p}(q_{3})\,p(q_{4})\,|:\mathcal{T}\,\exp{\left(i\int\mathscr{L}_{\mathbb{O}}(x)\,d^{4}x\right)}:|\,\bar{p}(q_{1})\,p(q_{2})\rangle$
(14) $\displaystyle=$
$\displaystyle\frac{i}{36}\,g_{\mathbb{O}}^{2}\,\mathcal{G}_{\mu\nu\rho}^{\mu_{1}\nu_{1}\rho_{1}}\,\bar{v}(q_{1})\,\gamma_{\mu_{1}}\,\big{(}q_{1,\nu_{1}}+q_{3,\nu_{1}}\big{)}\,\big{(}q_{1,\rho_{1}}+q_{3,\rho_{1}}\big{)}\,v(q_{3})\,\Delta_{\mathbb{O}}^{\mu\nu\rho;\mu^{\prime}\nu^{\prime}\rho^{\prime}}\big{(}s,t\big{)}$
$\displaystyle\quad\;\;\,\times\,\mathcal{G}_{\mu^{\prime}\nu^{\prime}\rho^{\prime}}^{\mu_{2}\nu_{2}\rho_{2}}\,\bar{u}(q_{4})\,\gamma_{\mu_{2}}\,\big{(}q_{4,\nu_{2}}+q_{2,\nu_{2}}\big{)}\,\big{(}q_{4,\rho_{2}}+q_{2,\rho_{2}}\big{)}\,u(q_{2})\,\mathcal{F}_{\mathbb{O}}(t)^{2}\,.$

In addition, we have modified the proton form-factor in Eq.(II.2) for the
odderon coupling to $pp$ by adding new three free parameters, $A$, $B$ and $C$
as,

$\displaystyle\mathcal{F}_{\mathbb{O}}(t)$ $\displaystyle=$
$\displaystyle\left(1-\frac{A\,t}{4m_{p}^{2}}\frac{\mu_{p}}{\mu_{N}}\right)\left(1-\frac{B\,t}{4m_{p}^{2}}\right)^{-1}\left(1-\frac{C\,t}{m_{D}^{2}}\right)^{-2}.$
(15)

The $\Delta_{\mathbb{O}}^{\mu_{1}\nu_{1}\rho_{1};\mu_{2}\nu_{2}\rho_{2}}(s,t)$
term is the propagator of the spin-3 particle in momentum space. By analogy to
the massless spin-2 tensor pomeron proposed by Ewerz et al. (2014), we propose
the propagator of the spin-3 tensor odderon. It reads,

$\displaystyle\Delta^{\mu\nu\lambda,\rho\sigma\tau}_{\mathbb{O}}(s,t)$
$\displaystyle=$
$\displaystyle\frac{M_{0}^{2}}{6\,s^{2}}\Big{[}\sum_{C}\,g^{\mu\rho}\left(g^{\nu\sigma}\,g^{\lambda\tau}+g^{\nu\tau}\,g^{\lambda\sigma}\right)-\frac{1}{2}\sum_{C}\,g^{\mu\nu}\,g^{\lambda\rho}\,g^{\sigma\tau}\Big{]}\,\left(-i\,\alpha_{\mathbb{O}}^{\prime}\,s\right)^{\alpha_{\mathbb{O}}(t)-1}\,,$
(16) $\displaystyle\alpha_{\mathbb{O}}(t)$ $\displaystyle=$ $\displaystyle
1+\epsilon_{\mathbb{O}}+\alpha_{\mathbb{O}}^{\prime}\,t\,,$ (17)

where $\sum_{C}$ stands for sum over all distinct combinations of the Lorentz
indices ($\mu\nu\lambda$) and ($\rho\sigma\tau$), for instance,

$\displaystyle\sum_{C}\,g_{\mu\nu}\,g_{\lambda\rho}\,g_{\sigma\tau}$
$\displaystyle=$ $\displaystyle
g_{\mu\nu}\,g_{\lambda\rho}\,g_{\sigma\tau}+g_{\mu\nu}\,g_{\lambda\sigma}\,g_{\rho\tau}+g_{\mu\nu}\,g_{\lambda\tau}\,g_{\rho\sigma}+g_{\mu\lambda}\,g_{\nu\rho}\,g_{\sigma\tau}+g_{\mu\lambda}\,g_{\nu\sigma}\,g_{\rho\tau}+g_{\mu\lambda}\,g_{\nu\tau}\,g_{\rho\sigma}$
(18) $\displaystyle+$ $\displaystyle
g_{\nu\lambda}\,g_{\mu\rho}\,g_{\sigma\tau}+g_{\nu\lambda}\,g_{\mu\sigma}\,g_{\rho\tau}+g_{\nu\lambda}\,g_{\mu\tau}\,g_{\rho\sigma}\,.$

The mass parameter $M_{0}$ is a free parameter in this work and it is
introduced in order to correct the mass dimension of the odderon propagator.
In addition, we consider the parameters $\epsilon_{\mathbb{O}}$ and
$\alpha_{\mathbb{O}}^{\prime}$ as free parameters. However, the condition
$\epsilon_{\mathbb{O}}<\epsilon_{\mathbb{P}}$ is imposed due to the
experimental data fact that the total cross-sections of both $pp$ and
$p\bar{p}$ are identical at very at high energies. In the other words, the
pomeron exchange contributions for elastic $pp$ and $p\bar{p}$ scattering at
the Regge limit always dominate over the odderon ones. Moreover, the tensor
structure of the spin-3 odderon propagator has been constructed in Ref.
Berends and van Reisen (1980) (see more detailed derivations and discussions).
We close this section by giving the definitions of the four-momentum
conservation, the on-shell mass of the particles, and the Mandelstam variables
as

$\displaystyle q_{1}+q_{2}=q_{3}+q_{4}\,,\quad
q_{1}^{2}=q_{2}^{2}=q_{3}^{2}=q_{4}^{2}=m_{p}^{2}\,,\quad
s+t+u=4\,m_{p}^{2}\,,$ $\displaystyle
s=(q_{1}+q_{2})^{2}=(q_{3}+q_{4})^{2}\,,\quad
t=(q_{3}-q_{1})^{2}=(q_{4}-q_{2})^{2}\,,\quad
u=(q_{4}-q_{1})^{2}=(q_{3}-q_{2})^{2}.$ (19)

In the next section, we will provide analytical expressions of the
differential cross-section for the $pp$ and $p\bar{p}$ elastic scattering with
pomeron and odderon exchanges and fit the model parameters with the
experimental data of the $pp$ and $p\bar{p}$ elastic scatterings at TeV scale.

##  III  Results and Discussions

###  III.1 Differential cross section formulae

In this subsection, we will provide the analytical formulae of the
differential cross section with respect to the $t$ variable and then the model
parameters in the present work will be determined by fitting with all
available experimental data of the $pp$ and $p\bar{p}$ elastic scatterings at
TeV level. The differential cross-section of the $pp$ and $p\bar{p}$
scattering is given by Ewerz et al. (2016),

$\displaystyle\frac{d\sigma^{pp/p\bar{p}}}{dt}=\frac{1}{16\pi
s\left(s-m_{p}^{2}\right)}\,\frac{1}{4}\,\sum_{\rm
spin}\,\big{|}\mathcal{M}^{pp/p\bar{p}}\big{|}^{2}.$ (20)

First of all, let us briefly discuss the definitions of the scattering
amplitudes of $pp$ and $p\bar{p}$ in the pomeron and odderon exchange picture.
Considering the amplitude $\mathcal{M}^{ab}(s,t)$ of an elastic scattering for
the $a+b\to a+b$ process in $s$ channel. While the corresponding elastic
scattering by crossing to the $u$-channel as $a+\bar{b}\to a+\bar{b}$ with the
amplitude $\mathcal{M}^{a\bar{b}}(u,t)$. According to the crossing symmetry of
the scattering amplitudes, they are symmetric under interchange between the
Mandelstam variables, $s$ and $u$ as

$\displaystyle\mathcal{M}^{a\bar{b}}(s,t,u)=\mathcal{M}^{ab}(u,t,s)\,.$ (21)

Moreover, the amplitude $\mathcal{M}_{\pm}$ are defined from
$\mathcal{M}^{ab/a\bar{b}}$ as

$\displaystyle\mathcal{M}_{\pm}(s,t)=\frac{1}{2}\left(\mathcal{M}^{ab}(s,t)\pm\mathcal{M}^{a\bar{b}}(s,t)\right)\,.$
(22)

Interchange $s\to u$, one finds that the amplitude $\mathcal{M}_{+}$ is
invariant under the crossing symmetry whereas the amplitude $\mathcal{M}_{-}$
changes the relative sign. We therefore call $\mathcal{M}_{+}$ and
$\mathcal{M}_{-}$ as even and odd under the crossing symmetry. As a result,
one observes that the $\mathcal{M}_{+}$ and $\mathcal{M}_{-}$ correspond to
the even and odd under charge conjugation, respectively. Since the interchange
$s\to u$ is equivalent to charge conjugation transformation ($\mathcal{C}$)
i.e., changing particle-particle scattering to particle-antiparticle
scattering. We therefore identify $\mathcal{M}_{+}$ and $\mathcal{M}_{-}$ as
pomeron ($\mathcal{M}_{\mathbb{P}}$ with $\mathcal{C}=+1$) and odderon
($\mathcal{M}_{\mathbb{O}}$ with $\mathcal{C}=-1$) exchange amplitudes,
respectively.

As discussed above, we can define the total amplitude of the elastic $pp$ and
$p\bar{p}$ processes with pomeron and odderon exchange diagrams at the tree
level as follow

$\displaystyle\mathcal{M}^{pp}$ $\displaystyle=$
$\displaystyle\mathcal{M}_{\mathbb{P}}^{pp}-\mathcal{M}_{\mathbb{O}}^{pp}\,,$
(23) $\displaystyle\mathcal{M}^{p\bar{p}}$ $\displaystyle=$
$\displaystyle\mathcal{M}_{\mathbb{P}}^{p\bar{p}}+\mathcal{M}_{\mathbb{O}}^{p\bar{p}}\,.$
(24)

The absolute square amplitudes of the $\mathcal{M}^{pp}$ and
$\mathcal{M}^{p\bar{p}}$ with an average sum over unpolarized initial spin
states of the incoming particles are given by

$\displaystyle\sum_{\rm spin}\,\big{|}\mathcal{M}^{pp}\big{|}^{2}$
$\displaystyle=$ $\displaystyle\sum_{\rm
spin}\left(\big{|}\mathcal{M}_{\mathbb{P}}^{pp}\big{|}^{2}-\big{|}\mathcal{M}_{\mathbb{P}}^{pp}\,\mathcal{M}_{\mathbb{O}}^{pp\,*}\big{|}-\big{|}\mathcal{M}_{\mathbb{P}}^{pp\,*}\,\mathcal{M}_{\mathbb{O}}^{pp}\big{|}+\big{|}\mathcal{M}_{\mathbb{O}}^{pp}\big{|}^{2}\right),$
(25) $\displaystyle\sum_{\rm spin}\,\big{|}\mathcal{M}^{p\bar{p}}\big{|}^{2}$
$\displaystyle=$ $\displaystyle\sum_{\rm
spin}\left(\big{|}\mathcal{M}_{\mathbb{P}}^{p\bar{p}}\big{|}^{2}+\big{|}\mathcal{M}_{\mathbb{P}}^{p\bar{p}\,*}\,\mathcal{M}_{\mathbb{O}}^{p\bar{p}}\big{|}+\big{|}\mathcal{M}_{\mathbb{P}}^{p\bar{p}}\,\mathcal{M}_{\mathbb{O}}^{p\bar{p}\,*}\big{|}+\big{|}\mathcal{M}_{\mathbb{O}}^{p\bar{p}}\big{|}^{2}\right).$
(26)

The explicit forms of the absolute square amplitudes with an average sum over
initial states can be calculated in the following forms,

$\displaystyle\sum_{\rm
spin}\,\big{|}\mathcal{M}_{\mathbb{P}}^{pp}\big{|}^{2}$ $\displaystyle=$
$\displaystyle{\rm
Tr}\,\Big{[}\big{(}q\\!\\!\\!/_{3}+m_{p}\big{)}\,\gamma_{\mu_{1}^{\prime}}\,\big{(}q_{3,\nu_{1}^{\prime}}+q_{1,\nu_{1}^{\prime}}\big{)}\,\big{(}q\\!\\!\\!/_{1}+m_{p}\big{)}\,\gamma_{\mu_{3}^{\prime}}\,\big{(}q_{3,\nu_{3}^{\prime}}+q_{1,\nu_{3}^{\prime}}\big{)}$
(27)
$\displaystyle\qquad\times\,\big{(}q\\!\\!\\!/_{4}+m_{p}\big{)}\,\gamma_{\mu_{2}^{\prime}}\,\big{(}q_{4,\nu_{2}^{\prime}}+q_{2,\nu_{2}^{\prime}}\big{)}\,\big{(}q\\!\\!\\!/_{2}+m_{p}\big{)}\,\gamma_{\mu_{4}^{\prime}}\,\big{(}q_{4,\nu_{4}^{\prime}}+q_{2,\nu_{4}^{\prime}}\big{)}\Big{]}$
$\displaystyle\,\times\,\Big{(}\frac{i}{4}\,g_{\mathbb{P}}^{2}\Big{)}^{2}\mathcal{G}_{\mu_{1}\nu_{1}}^{\mu_{1}^{\prime}\nu_{1}^{\prime}}\,\mathcal{G}_{\mu_{3}\nu_{3}}^{\mu_{3}^{\prime}\nu_{3}^{\prime}}\,\mathcal{G}_{\mu_{2}\nu_{2}}^{\mu_{2}^{\prime}\nu_{2}^{\prime}}\,\mathcal{G}_{\mu_{4}\nu_{4}}^{\mu_{4}^{\prime}\nu_{4}^{\prime}}\,\Delta_{\mathbb{P}}^{\mu_{1}\nu_{1};\mu_{2}\nu_{2}}(s,t)\,\Delta_{\mathbb{P}}^{\mu_{3}\nu_{3};\mu_{4}\nu_{4}}(s,t)\,\mathcal{F}_{\mathbb{P}}(t)^{4}$
$\displaystyle\approx$
$\displaystyle\frac{16\,g_{\mathbb{P}}^{4}\left[1-\frac{t}{4m_{p}^{2}}\frac{\mu_{p}}{\mu_{N}}\right]^{4}}{\left[1-\frac{t}{m_{p}^{2}}\right]^{4}\left[1-\frac{t}{m_{D}^{2}}\right]^{8}}\,s^{2}\left(\alpha^{\prime}_{\mathbb{P}}s\right)^{2\epsilon_{\mathbb{P}}+2\alpha^{\prime}_{\mathbb{P}}t},$
$\displaystyle\sum_{\rm
spin}\,\big{|}\mathcal{M}_{\mathbb{O}}^{pp}\big{|}^{2}$ $\displaystyle=$
$\displaystyle{\rm
Tr}\,\Big{[}\big{(}q\\!\\!\\!/_{3}+m_{p}\big{)}\,\gamma_{\mu_{1}}\,\big{(}q_{3,\nu_{1}}+q_{1,\nu_{1}}\big{)}\,\big{(}q_{3,\rho_{1}}+q_{1,\rho_{1}}\big{)}\,\big{(}q\\!\\!\\!/_{1}+m_{p}\big{)}\,\gamma_{\bar{\mu}_{3}}\,\big{(}q_{3,\bar{\nu}_{3}}+q_{1,\bar{\nu}_{3}}\big{)}\,\big{(}q_{3,\bar{\rho}_{3}}+q_{1,\bar{\rho}_{3}}\big{)}$
(28)
$\displaystyle\quad\;\;\,\times\,\big{(}q\\!\\!\\!/_{4}+m_{p}\big{)}\,\gamma_{\mu_{2}}\,\big{(}q_{4,\nu_{2}}+q_{2,\nu_{2}}\big{)}\,\big{(}q_{4,\rho_{2}}+q_{2,\rho_{2}}\big{)}\,\big{(}q\\!\\!\\!/_{2}+m_{p}\big{)}\,\gamma_{\bar{\mu}_{4}}\,\big{(}q_{4,\bar{\nu}_{4}}+q_{2,\bar{\nu}_{4}}\big{)}\,\big{(}q_{4,\bar{\rho}_{4}}+q_{2,\bar{\rho}_{4}}\big{)}\Big{]}$
$\displaystyle\,\times\,\Big{(}\frac{i}{36}\,g_{\mathbb{O}}^{2}\Big{)}^{2}\mathcal{G}_{\mu\nu\rho}^{\mu_{1}\nu_{1}\rho_{1}}\,\mathcal{G}_{\bar{\mu}\bar{\nu}\bar{\rho}}^{\bar{\mu}_{3}\bar{\nu}_{3}\bar{\rho}_{3}}\,\mathcal{G}_{\mu^{\prime}\nu^{\prime}\rho^{\prime}}^{\mu_{2}\nu_{2}\rho_{2}}\,\mathcal{G}_{\bar{\mu}^{\prime}\bar{\nu}^{\prime}\bar{\rho}^{\prime}}^{\bar{\mu}_{4}\bar{\nu}_{4}\bar{\rho}_{4}}\,\Delta_{\mathbb{O}}^{\mu\nu\rho;\mu^{\prime}\nu^{\prime}\rho^{\prime}}\big{(}s,t\big{)}\,\Delta_{\mathbb{O}}^{\bar{\mu}\bar{\nu}\bar{\rho};\bar{\mu}^{\prime}\bar{\nu}^{\prime}\bar{\rho}^{\prime}}\big{(}s,t\big{)}\,\mathcal{F}_{\mathbb{O}}(t)^{4}$
$\displaystyle\approx$
$\displaystyle\frac{4\,g_{\mathbb{O}}^{4}M_{0}^{4}\left[1-\frac{At}{4m_{p}^{2}}\frac{\mu_{p}}{\mu_{N}}\right]^{4}}{{729}\left[1-\frac{Bt}{m_{p}^{2}}\right]^{4}\left[1-\frac{Ct}{m_{D}^{2}}\right]^{8}}\,s^{2}\left(\alpha^{\prime}_{\mathbb{O}}s\right)^{2\epsilon_{\mathbb{O}}+2\alpha^{\prime}_{\mathbb{O}}t},$
$\displaystyle\sum_{\rm
spin}\,\big{|}\mathcal{M}_{\mathbb{P}}^{pp}\mathcal{M}_{\mathbb{O}}^{pp\,*}\big{|}$
$\displaystyle=$ $\displaystyle{\rm
Tr}\,\Big{[}\big{(}q\\!\\!\\!/_{3}+m_{p}\big{)}\,\gamma_{\mu_{1}^{\prime}}\,\big{(}q_{3,\nu_{1}^{\prime}}+q_{1,\nu_{1}^{\prime}}\big{)}\,\big{(}q\\!\\!\\!/_{1}+m_{p}\big{)}\,\gamma_{\bar{\mu}_{3}}\,\big{(}q_{3,\bar{\nu}_{3}}+q_{1,\bar{\nu}_{3}}\big{)}\,\big{(}q_{3,\bar{\rho}_{3}}+q_{1,\bar{\rho}_{3}}\big{)}$
(29)
$\displaystyle\qquad\times\,\big{(}q\\!\\!\\!/_{4}+m_{p}\big{)}\,\gamma_{\mu_{2}^{\prime}}\,\big{(}q_{4,\nu_{2}^{\prime}}+q_{2,\nu_{2}^{\prime}}\big{)}\,\big{(}q\\!\\!\\!/_{2}+m_{p}\big{)}\,\gamma_{\bar{\mu}_{4}}\,\big{(}q_{4,\bar{\nu}_{4}}+q_{2,\bar{\nu}_{4}}\big{)}\,\big{(}q_{4,\bar{\rho}_{4}}+q_{2,\bar{\rho}_{4}}\big{)}\Big{]}$
$\displaystyle\times\,\Big{(}\frac{i^{2}}{4\cdot
36}\,g_{\mathbb{P}}^{2}\,\,g_{\mathbb{O}}^{2}\Big{)}\,\mathcal{G}_{\mu_{1}\nu_{1}}^{\mu_{1}^{\prime}\nu_{1}^{\prime}}\,\mathcal{G}_{\bar{\mu}\bar{\nu}\bar{\rho}}^{\bar{\mu}_{3}\bar{\nu}_{3}\bar{\rho}_{3}}\,\mathcal{G}_{\mu_{2}\nu_{2}}^{\mu_{2}^{\prime}\nu_{2}^{\prime}}\,\mathcal{G}_{\bar{\mu}^{\prime}\bar{\nu}^{\prime}\bar{\rho}^{\prime}}^{\bar{\mu}_{4}\bar{\nu}_{4}\bar{\rho}_{4}}\,\Delta_{\mathbb{P}}^{\mu_{1}\nu_{1};\mu_{2}\nu_{2}}(s,t)\,\Delta_{\mathbb{O}}^{\bar{\mu}\bar{\nu}\bar{\rho};\bar{\mu}^{\prime}\bar{\nu}^{\prime}\bar{\rho}^{\prime}}\big{(}s,t\big{)}\,\mathcal{F}_{\mathbb{P}}(t)^{2}\mathcal{F}_{\mathbb{O}}(t)^{2}$
$\displaystyle\approx$
$\displaystyle\frac{8\,g_{\mathbb{P}}^{2}g_{\mathbb{O}}^{2}M_{0}^{2}\left[1-\frac{t}{4m_{p}^{2}}\frac{\mu_{p}}{\mu_{N}}\right]^{2}\left[1-\frac{At}{4m_{p}^{2}}\frac{\mu_{p}}{\mu_{N}}\right]^{2}}{27\left[1-\frac{t}{m_{p}^{2}}\right]^{2}\left[1-\frac{t}{m_{D}^{2}}\right]^{4}\left[1-\frac{Bt}{m_{p}^{2}}\right]^{2}\left[1-\frac{Ct}{m_{D}^{2}}\right]^{4}}\,s^{2}\left({-}i\,\alpha^{\prime}_{\mathbb{P}}s\right)^{\epsilon_{\mathbb{P}}+\alpha^{\prime}_{\mathbb{P}}t}\left(i\,\alpha^{\prime}_{\mathbb{O}}s\right)^{\epsilon_{\mathbb{O}}+\alpha^{\prime}_{\mathbb{O}}t},$
$\displaystyle\sum_{\rm
spin}\,\big{|}\mathcal{M}_{\mathbb{P}}^{p\bar{p}}\big{|}^{2}$ $\displaystyle=$
$\displaystyle{\rm
Tr}\,\Big{[}\big{(}q\\!\\!\\!/_{1}-m_{p}\big{)}\,\gamma_{\mu_{1}^{\prime}}\,\big{(}q_{3,\nu_{1}^{\prime}}+q_{1,\nu_{1}^{\prime}}\big{)}\,\big{(}q\\!\\!\\!/_{3}-m_{p}\big{)}\,\gamma_{\mu_{3}^{\prime}}\,\big{(}q_{3,\nu_{3}^{\prime}}+q_{1,\nu_{3}^{\prime}}\big{)}$
(30)
$\displaystyle\qquad\times\,\big{(}q\\!\\!\\!/_{4}+m_{p}\big{)}\,\gamma_{\mu_{2}^{\prime}}\,\big{(}q_{4,\nu_{2}^{\prime}}+q_{2,\nu_{2}^{\prime}}\big{)}\,\big{(}q\\!\\!\\!/_{2}+m_{p}\big{)}\,\gamma_{\mu_{4}^{\prime}}\,\big{(}q_{4,\nu_{4}^{\prime}}+q_{2,\nu_{4}^{\prime}}\big{)}\Big{]}$
$\displaystyle\,\times\,\Big{(}\frac{i}{4}\,g_{\mathbb{P}}^{2}\Big{)}^{2}\mathcal{G}_{\mu_{1}\nu_{1}}^{\mu_{1}^{\prime}\nu_{1}^{\prime}}\,\mathcal{G}_{\mu_{3}\nu_{3}}^{\mu_{3}^{\prime}\nu_{3}^{\prime}}\,\mathcal{G}_{\mu_{2}\nu_{2}}^{\mu_{2}^{\prime}\nu_{2}^{\prime}}\,\mathcal{G}_{\mu_{4}\nu_{4}}^{\mu_{4}^{\prime}\nu_{4}^{\prime}}\,\Delta_{\mathbb{P}}^{\mu_{1}\nu_{1};\mu_{2}\nu_{2}}(s,t)\,\Delta_{\mathbb{P}}^{\mu_{3}\nu_{3};\mu_{4}\nu_{4}}(s,t)\,\mathcal{F}_{\mathbb{P}}(t)^{4}$
$\displaystyle\approx$
$\displaystyle\frac{16\,g_{\mathbb{P}}^{4}\left[1-\frac{t}{4m_{p}^{2}}\frac{\mu_{p}}{\mu_{N}}\right]^{4}}{\left[1-\frac{t}{m_{p}^{2}}\right]^{4}\left[1-\frac{t}{m_{D}^{2}}\right]^{8}}\,s^{2}\left(\alpha^{\prime}_{\mathbb{P}}s\right)^{2\epsilon_{\mathbb{P}}+2\alpha^{\prime}_{\mathbb{P}}t},$
$\displaystyle\sum_{\rm
spin}\,\big{|}\mathcal{M}_{\mathbb{O}}^{p\bar{p}}\big{|}^{2}$ $\displaystyle=$
$\displaystyle{\rm
Tr}\,\Big{[}\big{(}q\\!\\!\\!/_{1}-m_{p}\big{)}\,\gamma_{\mu_{1}}\,\big{(}q_{3,\nu_{1}}+q_{1,\nu_{1}}\big{)}\,\big{(}q_{3,\rho_{1}}+q_{1,\rho_{1}}\big{)}\,\big{(}q\\!\\!\\!/_{3}-m_{p}\big{)}\,\gamma_{\bar{\mu}_{3}}\,\big{(}q_{3,\bar{\nu}_{3}}+q_{1,\bar{\nu}_{3}}\big{)}\,\big{(}q_{3,\bar{\rho}_{3}}+q_{1,\bar{\rho}_{3}}\big{)}$
(31)
$\displaystyle\quad\;\;\,\times\,\big{(}q\\!\\!\\!/_{4}+m_{p}\big{)}\,\gamma_{\mu_{2}}\,\big{(}q_{4,\nu_{2}}+q_{2,\nu_{2}}\big{)}\,\big{(}q_{4,\rho_{2}}+q_{2,\rho_{2}}\big{)}\,\big{(}q\\!\\!\\!/_{2}+m_{p}\big{)}\,\gamma_{\bar{\mu}_{4}}\,\big{(}q_{4,\bar{\nu}_{4}}+q_{2,\bar{\nu}_{4}}\big{)}\,\big{(}q_{4,\bar{\rho}_{4}}+q_{2,\bar{\rho}_{4}}\big{)}\Big{]}$
$\displaystyle\,\times\,\Big{(}\frac{i}{36}\,g_{\mathbb{O}}^{2}\Big{)}^{2}\mathcal{G}_{\mu\nu\rho}^{\mu_{1}\nu_{1}\rho_{1}}\,\mathcal{G}_{\bar{\mu}\bar{\nu}\bar{\rho}}^{\bar{\mu}_{3}\bar{\nu}_{3}\bar{\rho}_{3}}\,\mathcal{G}_{\mu^{\prime}\nu^{\prime}\rho^{\prime}}^{\mu_{2}\nu_{2}\rho_{2}}\,\mathcal{G}_{\bar{\mu}^{\prime}\bar{\nu}^{\prime}\bar{\rho}^{\prime}}^{\bar{\mu}_{4}\bar{\nu}_{4}\bar{\rho}_{4}}\,\Delta_{\mathbb{O}}^{\mu\nu\rho;\mu^{\prime}\nu^{\prime}\rho^{\prime}}\big{(}s,t\big{)}\,\Delta_{\mathbb{O}}^{\bar{\mu}\bar{\nu}\bar{\rho};\bar{\mu}^{\prime}\bar{\nu}^{\prime}\bar{\rho}^{\prime}}\big{(}s,t\big{)}\,\mathcal{F}_{\mathbb{O}}(t)^{4}$
$\displaystyle\approx$
$\displaystyle\frac{4\,g_{\mathbb{O}}^{4}M_{0}^{4}\left[1-\frac{At}{4m_{p}^{2}}\frac{\mu_{p}}{\mu_{N}}\right]^{4}}{{729}\left[1-\frac{Bt}{m_{p}^{2}}\right]^{4}\left[1-\frac{Ct}{m_{D}^{2}}\right]^{8}}\,s^{2}\left(\alpha^{\prime}_{\mathbb{O}}s\right)^{2\epsilon_{\mathbb{O}}+2\alpha^{\prime}_{\mathbb{O}}t},$
$\displaystyle\sum_{\rm
spin}\,\big{|}\mathcal{M}_{\mathbb{P}}^{p\bar{p}}\mathcal{M}_{\mathbb{O}}^{p\bar{p}\,*}\big{|}^{2}$
$\displaystyle=$ $\displaystyle{\rm
Tr}\,\Big{[}\big{(}q\\!\\!\\!/_{1}-m_{p}\big{)}\,\gamma_{\mu_{1}^{\prime}}\,\big{(}q_{3,\nu_{1}^{\prime}}+q_{1,\nu_{1}^{\prime}}\big{)}\,\big{(}q\\!\\!\\!/_{3}-m_{p}\big{)}\,\gamma_{\bar{\mu}_{3}}\,\big{(}q_{3,\bar{\nu}_{3}}+q_{1,\bar{\nu}_{3}}\big{)}\,\big{(}q_{3,\bar{\rho}_{3}}+q_{1,\bar{\rho}_{3}}\big{)}$
(32)
$\displaystyle\qquad\times\,\big{(}q\\!\\!\\!/_{4}+m_{p}\big{)}\,\gamma_{\mu_{2}^{\prime}}\,\big{(}q_{4,\nu_{2}^{\prime}}+q_{2,\nu_{2}^{\prime}}\big{)}\,\big{(}q\\!\\!\\!/_{2}+m_{p}\big{)}\,\gamma_{\bar{\mu}_{4}}\,\big{(}q_{4,\bar{\nu}_{4}}+q_{2,\bar{\nu}_{4}}\big{)}\,\big{(}q_{4,\bar{\rho}_{4}}+q_{2,\bar{\rho}_{4}}\big{)}\Big{]}$
$\displaystyle\times\,\Big{(}\frac{i^{2}}{4\cdot
36}\,g_{\mathbb{P}}^{2}\,\,g_{\mathbb{O}}^{2}\Big{)}\,\mathcal{G}_{\mu_{1}\nu_{1}}^{\mu_{1}^{\prime}\nu_{1}^{\prime}}\,\mathcal{G}_{\bar{\mu}\bar{\nu}\bar{\rho}}^{\bar{\mu}_{3}\bar{\nu}_{3}\bar{\rho}_{3}}\,\mathcal{G}_{\mu_{2}\nu_{2}}^{\mu_{2}^{\prime}\nu_{2}^{\prime}}\,\mathcal{G}_{\bar{\mu}^{\prime}\bar{\nu}^{\prime}\bar{\rho}^{\prime}}^{\bar{\mu}_{4}\bar{\nu}_{4}\bar{\rho}_{4}}\,\Delta_{\mathbb{P}}^{\mu_{1}\nu_{1};\mu_{2}\nu_{2}}(s,t)\,\Delta_{\mathbb{O}}^{\bar{\mu}\bar{\nu}\bar{\rho};\bar{\mu}^{\prime}\bar{\nu}^{\prime}\bar{\rho}^{\prime}}\big{(}s,t\big{)}\,\mathcal{F}_{\mathbb{P}}(t)^{2}\mathcal{F}_{\mathbb{O}}(t)^{2}$
$\displaystyle\approx$
$\displaystyle\frac{8\,g_{\mathbb{P}}^{2}g_{\mathbb{O}}^{2}M_{0}^{2}\left[1-\frac{t}{4m_{p}^{2}}\frac{\mu_{p}}{\mu_{N}}\right]^{2}\left[1-\frac{At}{4m_{p}^{2}}\frac{\mu_{p}}{\mu_{N}}\right]^{2}}{27\left[1-\frac{t}{m_{p}^{2}}\right]^{2}\left[1-\frac{t}{m_{D}^{2}}\right]^{4}\left[1-\frac{Bt}{m_{p}^{2}}\right]^{2}\left[1-\frac{Ct}{m_{D}^{2}}\right]^{4}}\,s^{2}\left({-}i\,\alpha^{\prime}_{\mathbb{P}}s\right)^{\epsilon_{\mathbb{P}}+\alpha^{\prime}_{\mathbb{P}}t}\left(i\,\alpha^{\prime}_{\mathbb{O}}s\right)^{\epsilon_{\mathbb{O}}+\alpha^{\prime}_{\mathbb{O}}t},$

where the regge limit, $s\ggg t,\,m_{p}^{2}$ has been applied for the
approximations to obtain the final results. Here we have used the following
normalizations and sum over spin of the spinors as

$\displaystyle\bar{u}_{r}(p,m_{p})\,u_{s}(p,m_{p})$ $\displaystyle=$
$\displaystyle
2\,m_{p}\,\delta_{rs}\,,\qquad\quad\sum_{r}u_{r}(p,m_{p})\,\bar{u}_{r}(p,m_{p})=p\\!\\!\\!/+m_{p}\,,$
$\displaystyle\bar{v}_{r}(p,m_{p})\,v_{s}(p,m_{p})$ $\displaystyle=$
$\displaystyle-2\,m_{p}\,\delta_{rs}\,,\qquad\,\,\sum_{r}v_{r}(p,m_{p})\,\bar{v}_{r}(p,m_{p})=p\\!\\!\\!/-m_{p}\,,$
(33)

where $r$ and $s$ are spin indices of the spinors. As results, we note that
$\sum_{\rm spin}\,\big{|}\mathcal{M}_{X}^{pp}\big{|}^{2}=\sum_{\rm
spin}\,\big{|}\mathcal{M}_{X}^{p\bar{p}}\big{|}^{2}$ with
$X=\mathbb{P},\mathbb{O}$.

Next, we will present the scalar amplitudes of the pomeron and odderon
exchanges as $\mathcal{A}_{\mathbb{P}}$ and $\mathcal{A}_{\mathbb{O}}$
respectively. Having use the results in Eqs.(27) and (28), they are written
by,

$\displaystyle\mathcal{A}_{\mathbb{P}}(s,t)$ $\displaystyle=$
$\displaystyle\frac{{4}\,g_{\mathbb{P}}^{2}\left[1-\frac{t}{4m_{p}^{2}}\frac{\mu_{p}}{\mu_{N}}\right]^{2}}{\left[1-\frac{t}{m_{p}^{2}}\right]^{2}\left[1-\frac{t}{m_{D}^{2}}\right]^{4}}\,s\left(-i\,\alpha^{\prime}_{\mathbb{P}}s\right)^{\epsilon_{\mathbb{P}}+\alpha^{\prime}_{\mathbb{P}}t},$
(34) $\displaystyle\mathcal{A}_{\mathbb{O}}(s,t)$ $\displaystyle=$
$\displaystyle\frac{2\,g_{\mathbb{O}}^{2}M_{0}^{2}\left[1-\frac{At}{4m_{p}^{2}}\frac{\mu_{p}}{\mu_{N}}\right]^{2}}{{27}\left[1-\frac{Bt}{m_{p}^{2}}\right]^{2}\left[1-\frac{Ct}{m_{D}^{2}}\right]^{4}}\,s\left(-i\,\alpha^{\prime}_{\mathbb{O}}s\right)^{\epsilon_{\mathbb{O}}+\alpha^{\prime}_{\mathbb{O}}t},$
(35)

where the amplitude of the $pp$ scattering is given by
$\mathcal{A}^{pp}(s,t)=\mathcal{A}_{\mathbb{P}}(s,t)-\mathcal{A}_{\mathbb{O}}(s,t)$
. According to the optical theorem, one can write the total cross-section
formula as,

$\displaystyle\sigma_{\rm tot}^{pp}$ $\displaystyle=$
$\displaystyle\frac{1}{s}\,{\rm Im}\,\mathcal{A}^{pp}(s,0)=\frac{1}{s}\,{\rm
Im}\,\Big{[}\mathcal{A}_{\mathbb{P}}(s,0)-\mathcal{A}_{\mathbb{O}}(s,0)\,\Big{]}={\rm
Im}\,\Big{[}{4}\,g_{\mathbb{P}}^{2}\,\left(-i\,\alpha^{\prime}_{\mathbb{P}}s\right)^{\epsilon_{\mathbb{P}}}-\frac{2}{{27}}\,g_{\mathbb{O}}^{2}M_{0}^{2}\left(-i\,\alpha^{\prime}_{\mathbb{O}}s\right)^{\epsilon_{\mathbb{O}}}\Big{]}\,.$
(36)

In the following subsection, we will use the total cross-section in Eq.(36) to
compare with the data from the TOTEM collaboration at the TeV scale after the
model parameters are chosen.

###  III.2 Parameters fitting and discussion

Figure 1: The best fit plots of the differential cross-section of the $pp$
(blue plots) and $p\bar{p}$ (green plots) vs the model results with the
parameters from the combined data set fit (see the last column in Table.1).

In this section, we perform curve fitting of the model parameters with
experimental data. As mentioned in the section II, we have eight free
parameters i.e., the typical mass scale in odderon propagator $M_{0}$,
pomeron-$pp$ coupling constant $g_{\mathbb{P}}$, odderon-$pp$ coupling
constant $g_{\mathbb{O}}$, $\epsilon_{\mathbb{O}}$,
$\alpha_{\mathbb{O}}^{\prime}$ and the modified odderon-$pp$ form-factor
parameters $A$, $B$ and $C$. The observed value for differential cross
sections of $pp$ and $p\bar{p}$ scattering, ${d\sigma}^{\rm obs}/{dt}$, come
from various experiments with the center of mass energy ranging from $1.8$ TeV
Avila et al. (1999) for $p\bar{p}$, $1.96$ TeV Abazov et al. (2021) for $pp$
and Abazov et al. (2012) for $p\bar{p}$, $2.76$ TeV Antchev et al. (2020) with
$13\sigma$ and $4.3\sigma$, $7$ TeV Antchev et al. (2013) and $13$ TeV Antchev
et al. (2019) for $pp$ scattering. Since we are interested in the small $-t$
limit, we only use the observed data with the linear relation between the
differential cross-section and the momentum exchange because the effect of the
pomeron and odderon is highly manifested in the linear regime of the
differential cross-section. We define $\chi^{2}$ function as

$\chi^{2}(\alpha_{i})=\sum_{j=1}^{N}\left(\frac{\frac{d\sigma}{dt}(\alpha_{i})^{\rm
model}_{j}-\frac{d\sigma}{dt}^{\rm obs}_{j}}{\frac{d\sigma}{dt}^{\rm
obs}_{j}}\right)^{2},$ (37)

where $\alpha_{i}$ are eight parameters
$(g_{\mathbb{P}},g_{\mathbb{O}},M_{0},\epsilon_{\mathbb{O}},\alpha_{\mathbb{O}}^{\prime},A,B,C)$
and $j=1,\ldots,N$ is the index of the data points associated with the
momentum exchange, $-t$. In order to obtain the best fit parameters, we
minimize $\chi^{2}$ functions using iminuitDembinski and et al. (2020); James
and Roos (1975). The results are shown in the table 1. Note that the errors
are calculated using the Hessian matrix where more details will be provided in
Appendix A.

The central values of $g_{\mathbb{O}}$, $M_{0}$,
$\alpha_{\mathbb{0}}^{\prime}$, $\epsilon_{\mathbb{O}}$, $B$ and $C$ are
consistent among various data sets, whereas the central values of
$g_{\mathbb{P}}$, $A$ vary rather widely. The odderon mass which can be
calculated from its regge trajectory at the pole, $t=m_{\mathbb{O}}^{2}$ with
$\alpha_{\mathbb{O}}\big{(}m_{\mathbb{O}}^{2}\big{)}=J=3$ as

$m_{\mathbb{O}}(J=3)=\sqrt{\frac{J-1-\epsilon_{\mathbb{O}}}{\alpha_{\mathbb{0}}^{\prime}}}\,\Bigg{|}_{J=3},$
(38)

are consistent due to its dependency on $\epsilon_{\mathbb{O}}$ and
$\alpha_{\mathbb{0}}^{\prime}$.

The quality of parameter fitting can be determined using the minimized
$\chi^{2}$ per degree of freedom. Among the available data sets, the best
parameters with sufficient statistics come from the $pp$-TOTEM 13 TeV data.
However, using this particular data alone leads to an overfitting problem,
i.e., these parameters lead to unsatisfying fits with $p\bar{p}$ data sets.
We, therefore, take a more global analysis using the combined $\chi^{2}$
function of all available data sets. We then use the parameter fitting from
the combined data set as the representation of our model. The model
differential cross sections comparing with experimental data are shown in Fig
1. We have provided the statistical error analysis in detail in Appendix A.

By using the Eq.(38) with the best fit parameters, $\epsilon_{\mathbb{O}}$ and
$\alpha_{\mathbb{0}}^{\prime}$ from the combined data set, one can determine
the masses of the odderons with $J=3,\,5,\,7$ as shown below,

$\displaystyle m_{\mathbb{O}}(J^{PC}=3^{--})$ $\displaystyle=$ $\displaystyle
2.884\pm 0.088\;{\rm GeV}\,,$ $\displaystyle m_{\mathbb{O}}(J^{PC}=5^{--})$
$\displaystyle=$ $\displaystyle 4.055\pm 0.125\;{\rm GeV}\,,$ $\displaystyle
m_{\mathbb{O}}(J^{PC}=7^{--})$ $\displaystyle=$ $\displaystyle 4.977\pm
0.154\;{\rm GeV}\,.$ (39)

We note that the lowest mass (pole position) of the tensor odderon with spin-3
is around 3 GeV. In addition, the Chew-Frautschi plot of the odderon regge
trajectory is depicted in Fig.2. The odderon mass results in the present work
are consistent with Ref.Szanyi et al. (2020). In that work, the authors
considered the odderons as the oddballs in the double pole regge model with
spin-3, 5, and 7. Then, the masses of the odderon are extracted from the
experimental data. The theoretical estimation such as SU(3) lattice QCD for
isotropic and anisotropic cases Meyer and Teper (2005); Chen et al. (2006);
Morningstar and Peardon (1999), Wilson loop approach Kaidalov and Simonov
(2000b), vacuum correlation method in QCD Kaidalov and Simonov (2000a);
Kaidalov and Simonov (2006), QCD sum rules Narison (1984); Chen et al. (2021a,
b), relativistic many body framework Llanes-Estrada et al. (2006) give the
ranges of odderon masses as $3.5-4.5$ GeV for spin-3, $5.0-5.5$ GeV for
spin-5, and $6.0-6.5$ GeV for spin-7. We note that the theoretical model
calculations of the odderon masses are heavier than the mass estimations from
the data in this work and Ref.Szanyi et al. (2020) by about 0.5 GeV. From the
results in Table 1, the odderon trajectory slope,
$\alpha_{\mathbb{O}}^{\prime}=0.24$ GeV-2, and the pomeron slope,
$\alpha_{\mathbb{P}}^{\prime}$ = 0.25 GeV-2, coming from Donnachie-Landschoff
parameterization Donnachie and Landshoff (1984, 1992) agree very well with
approximation
$\alpha_{\mathbb{O}}^{\prime}\approx\alpha_{\mathbb{P}}^{\prime}$.

On the other hand, we obtain a “decent” odderon,
$\epsilon_{\mathbb{O}}=0.0540\ll 1$. As a result, the best fit values of
$\alpha_{\mathbb{O}}^{\prime}$ and $\epsilon_{\mathbb{O}}$ in this work
correspond to the assumptions in Ref.Ewerz et al. (2014) that
$\alpha_{\mathbb{O}}^{\prime}\approx\alpha_{\mathbb{P}}^{\prime}$ and
$\epsilon_{\mathbb{O}}\leq\epsilon_{\mathbb{P}}\,(=0.0808)$ which use for
fixing the parameters $\alpha_{\mathbb{O}}^{\prime}$ and
$\epsilon_{\mathbb{O}}$ due to the lack of data used to constrain at that
moment.

We close this section by considering the total cross-section of the $pp$ in
our model. The relevant parameters from the combined data set are substituted
to the total cross-section formula in Eq.(36). Then, we plot the total cross
section as a function of c.m. energy ($\sqrt{s}$) as shown in Fig.3 and we
found that our model of the odderon as reggeized oddball spin-3 is compatible
with the TOTEM data for $pp$ scattering at the TeV regime. In particular, the
extrapolation of TOTEM data for the $pp$ total cross-section at 1.96 TeV is
also laid within the error band of our model. Using the best fit parameters
$\epsilon_{\mathbb{P}}=0.0808$ and $\epsilon_{\mathbb{O}}=0.0540$, in
addition, the total cross-section of the present work in Eq.(36) has been
checked numerically and it also corresponds to a Froissart bound, i.e.,
$\sigma_{\rm tot}^{pp}\leq(\ln s)^{2}$ at $s\to\infty$ limit.

Figure 2: The Chew-Frautschi plot of the odderon regge trajectory is depicted
by using the $\alpha_{\mathbb{O}}^{\prime}=0.24$ GeV-2 and
$\epsilon_{\mathbb{O}}=0.054$ from combined data set fitting in Table1. The
masses of the odderon with spin-3, 5, and 7 including the error of the fitting
parameter estimations with red dots are given by Eq.(39).  Figure 3: The total
cross-section of the $pp$ scattering with pomeron spin-2 and odderon spin-3
exchanges in Eq.36 is plotted by using the relevant fitted parameters from the
combined data set in Table.1. As a result, our model is compatible with the
TOTEM $pp$ results at the TeV region. Moreover, the model also agrees with the
extrapolation of the TOTEM data at 1.96 TeV for the $pp$ scattering. All data
points are laid on the model prediction band.

Parameters Data set |  $pp$-TOTEM $2.76\;\rm{TeV}$ (13$\sigma$) |  $pp$-TOTEM $2.76\;\rm{TeV}$ (4.3$\sigma$) |  $pp$-TOTEM $7\;\rm{TeV}$ |  $p\bar{p}$-Tevatron $1.96\;\rm{TeV}$
---|---|---|---|---
$g_{\mathbb{P}}$ |  $-6.120\pm 0.409$ |  $-5.941\pm 0.354$ |  $-3.524\pm 0.490$ |  $4.906\pm 1.735$
$g_{\mathbb{O}}$ |  $0.968\pm 0.034$ |  $0.867\pm 0.020$ |  $0.837\pm 0.019$ |  $0.855\pm 0.093$
$M_{0}$ |  $120.947\pm 4.223$ |  $108.376\pm 2.487$ |  $104.674\pm 2.402$ |  $106.848\pm 11.655$
$\alpha_{\mathbb{O}}^{\prime}$ |  $0.240\pm 0.009$ |  $0.161\pm 0.015$ |  $0.140\pm 0.022$ |  $0.234\pm 0.034$
$\epsilon_{\mathbb{O}}$ |  $0.078\pm 0.005$ |  $0.061\pm 0.003$ |  $0.055\pm 0.003$ |  $0.059\pm 0.016$
$A$ |  $0.343\pm 0.100$ |  $-0.942\pm 0.096$ |  $-1.669\pm 0.141$ |  $0.286\pm 0.339$
$B$ |  $1.354\pm 0.109$ |  $0.639\pm 0.107$ |  $0.564\pm 0.159$ |  $1.347\pm 0.369$
$C$ |  $1.133\pm 0.045$ |  $0.688\pm 0.046$ |  $0.673\pm 0.068$ |  $1.152\pm 0.153$
$m_{\mathbb{O}}$ |  $2.830\pm 0.144$ |  $3.470\pm 0.253$ |  $3.732\pm 0.391$ |  $2.880\pm 0.602$
d.o.f |  12 |  45 |  83 |  9
$\chi^{2}$ |  0.036 |  0.057 |  0.048 |  0.028

Parameters Data set |  $pp$-D0 1.96 TeV |  $pp$-TOTEM 13 TeV |  $p\bar{p}$-Tevatron 1.8 TeV |  $p\bar{p}$-D0 1.96 TeV | Combined data
---|---|---|---|---|---
$g_{\mathbb{P}}$ | $-8.156\pm 0.301$ | $-5.845\pm 0.163$ | $4.095\pm 0.836$ | $2.181\pm 4.080$ | $0.009\pm 2.828$
$g_{\mathbb{O}}$ | $0.951\pm 0.027$ | $0.879\pm 0.010$ | $0.762\pm 0.033$ | $0.804\pm 0.108$ | $0.803\pm 0.011$
$M_{0}$ | $118.823\pm 3.395$ | $109.819\pm 1.296$ | $95.275\pm 4.126$ | $100.534\pm 13.483$ | $100.411\pm 1.344$
$\alpha_{\mathbb{O}}^{\prime}$ | $0.235\pm 0.007$ | $0.160\pm 0.008$ | $0.102\pm 0.031$ | $0.221\pm 0.033$ | $0.240\pm 0.007$
$\epsilon_{\mathbb{O}}$ | $0.075\pm 0.004$ | $0.061\pm 0.001$ | $0.044\pm 0.007$ | $0.051\pm 0.020$ | $0.054\pm 0.002$
$A$ | $0.635\pm 0.081$ | $-1.351\pm 0.058$ | $-1.252\pm 0.143$ | $0.645\pm 0.385$ | $-0.472\pm 0.063$
$B$ | $1.175\pm 0.077$ | $0.588\pm 0.063$ | $0.593\pm 0.211$ | $1.177\pm 0.367$ | $0.685\pm 0.066$
$C$ | $1.059\pm 0.033$ | $0.712\pm 0.027$ | $0.638\pm 0.090$ | $1.076\pm 0.156$ | $0.718\pm 0.028$
$m_{\mathbb{O}}$ | $2.863\pm 0.122$ | $3.486\pm 0.123$ | $4.373\pm 1.007$ | $2.968\pm 0.790$ | $2.884\pm 0.088$
d.o.f | 4 | 150 | 49 | 4 | 356
$\chi^{2}$ | 0.038 | 0.008 | 0.075 | 0.008 | 7.160

Table 1: Summary of best fit parameters for data sets. Each column represents
the best fit values of 8 parameters for a data set.

##  IV Conclusions

In this work, we considered the odderon as the reggeized tensor oddball
spin-3. The existence of odderon can be observed in a study of the difference
between $pp$ and $p\bar{p}$ elastic scattering. We, therefore, investigate
$pp$ and $p\bar{p}$ scattering at the regge limit by including the pomeron and
odderon exchanges in the present work. The effective Lagrangians of the
processes are constructed and the standard perturbative QFT method is used to
calculate the relevant observables in this work. The pomeron and odderon are
identified as reggeized tensor glueballs and oddballs with spin-2 and -3,
respectively. We have employed the Donnachie-Landschoff parameterization for
the pomeron regge trajectory with the electromagnetic type of the pomeron-$pp$
form-factor. Furthermore, we also modified the electromagnetic type of the
odderon-$pp$ by introducing the additional 3 free parameters. There are 8 free
parameters of the model in this work
$(g_{\mathbb{P}},g_{\mathbb{O}},M_{0},\epsilon_{\mathbb{O}},\alpha_{\mathbb{O}}^{\prime},A,B,C)$.
Having performed a careful statistical analysis, all free parameters have been
fixed by fitting with all combined data of the $pp$ and $p\bar{p}$
differential cross-sections at the TeV regime see results in Table 1. After
fixing the free parameters in the present work, the masses of the odderon
spin-3 and their excited states for spin-5 and -7 are estimated from its regge
trajectory by using the best fit of the combined data set. Considering the
best fit results in Table 1, the oddereon regge trajectory parameters are
found to be $\alpha_{\mathbb{O}}^{\prime}=0.240\,\pm\,0.007$ and
$\epsilon_{\mathbb{O}}=0.054\,\pm\,0.002$. These results are compatible with
the assumptions in Ref.Ewerz et al. (2014) that used to estimate those two
parameters as
$\alpha_{\mathbb{O}}^{\prime}\approx\alpha_{\mathbb{O}}^{\prime}$ and
$\epsilon_{\mathbb{O}}\leq\epsilon_{\mathbb{P}}$. As a result, we found that
the tensor odderon masses are compatible with the phenomenological approach by
using the double pole regge model extracted from the experimental data Szanyi
et al. (2020). However, the odderon masses in this work are lighter than other
model theoretical calculations for all odderons along their trajectory by
about 0.5 GeV. Having used the best fit parameters, in addition, the total
cross-sections also agree with the TOTEM data in the TeV regime and its
extrapolation from D0 of the $pp$ scattering at 1.96 TeV. The odderon spin-3
contribution also provided the amplitude in Regge limit that satisfied the
Froissart bound. According to our findings in this work, we can conclude that
the odderon could be the tensor reggiezed oddball particles. Further studies
to confirm our conclusion are needed to investigate in other scattering
processes such as polarized proton-proton scattering, meson-baryon scattering,
photoproduction. We plan to do this in the forthcoming future work.

###### Acknowledgements.

DS is supported by the Fundamental Fund 2565 of Khon Kaen University and DS
has received funding support from the National Science, Research and
Innovation Fund. The Mindanao State University - Iligan Institute of
Technology is also acknowledged through its research and extension support
extended to J.B. Magallanes for his travel to Khon Kaen University, Thailand.
PS, CP, and DS are financially supported by the National Astronomical Research
Institute of Thailand (NARIT). CP and DS are supported by Thailand NSRF via
PMU-B [grant number B05F650021]. CP is also supported by Fundamental Fund 2565
of Khon Kaen University and Research Grant for New Scholar, Office of the
Permanent Secretary, Ministry of Higher Education, Science, Research and
Innovation under contract no. RGNS 64-043.

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##  Appendix A Error analysis

We provide the detail of the error estimation method in this section. The
errors for parameter fitting in this analysis are handling outside the iminuit
package due to an inaccuracy of the results. The errors shown in Table 1 are
therefore improved by following. Consider the Taylor expansion around the
minimum of the $\chi^{2}$ function,

$\chi^{2}(\alpha_{i})\approx\chi^{2}(\alpha_{i}^{\rm
min})+\frac{1}{2}\left(\alpha_{i}-\alpha_{i}^{\rm
min}\right)^{2}\frac{\partial\chi^{2}}{\partial\alpha_{i}}|_{\alpha_{i}=\alpha_{i}^{\rm
min}}+\mathcal{O}\left(\left(\alpha_{i}-\alpha_{i}^{\rm
min}\right)^{3}\right).$ (40)

We can approximate the error of parameter estimation, $\sigma_{i}$, using the
width of parabolic function defined as

$\frac{1}{\sigma^{2}_{i}}=\frac{1}{2}\frac{\partial\chi^{2}}{\partial\alpha_{i}}|_{\alpha_{i}=\alpha_{i}^{\rm
min}},$ (41)

which is also the diagonal components of the hessian matrix. The second
derivative of the $\chi^{2}$ function is obtained via the finite difference
method

$\frac{\partial\chi^{2}}{\partial\alpha_{i}}|_{\alpha_{i}=\alpha_{i}^{\rm
min}}\approx\frac{\chi^{2}(a_{i}^{\rm min}+\Delta a_{i})+\chi^{2}(a_{i}^{\rm
min}-\Delta a_{i})-2\chi^{2}(a_{i}^{\rm min})}{\Delta a_{i}^{2}},$ (42)

where $\Delta a_{i}$ is chosen to be sufficiently small comparing to the value
of the error. The validity of the approximation is then confirmed by the
comparison between the parabolic functions and the real $\chi^{2}$ function
shown in Fig.4. One can see that both functions agree very well within the
range of error estimations.

Figure 4: The comparison plots between the parabolic function using
$\chi^{2}=\chi^{\rm min}+\frac{\left(\alpha_{i}-\alpha_{i}^{\rm
min}\right)^{2}}{\sigma^{2}}$ (orange line) and the real $\chi^{2}$ function
(blue line). The combined data set is used to obtain the parameter fit in
these plots. Each panel represents the variation of $\chi^{2}$ function in
each particular direction of the parameter space.