# Odderon contribution in light of the LHC low-$t$ data

E. G. S. Luna<EMAIL_ADDRESS>Instituto de Física, Universidade Federal do
Rio Grande do Sul, Caixa Postal 15051, 91501-970, Porto Alegre, Rio Grande do
Sul, Brazil  M. G. Ryskin<EMAIL_ADDRESS>Petersburg Nuclear Physics
Institute, NRC Kurchatov Institute,
Gatchina, St. Petersburg, 188300, Russia  V. A. Khoze<EMAIL_ADDRESS>Institute for Particle Physics Phenomenology, University of Durham,
Durham, DH1 3LE, UK

###### Abstract

We perform the analysis of elastic scattering $pp$ and $\bar{p}p$ data at low
momentum transfer $|t|<0.1$ GeV2 within large collider energy interval
$\sqrt{s}=50$ GeV - 13 TeV in order to evaluate quantitatively the possible
Odderon contribution. We use the two-channel eikonal model, which naturally
accounts for the screening of the Odderon amplitude by $C$-even (Pomeron)
exchanges.

##  I Introduction

The TOTEM publication Tot-13  of the measurements of the total cross section
and the real part of the forward elastic $pp$ amplitude at 13 TeV, prompted a
renewal of interest in the potential existence of the high-energy $C$-odd
(Odderon) contribution. This is because the observed value of the ratio of
real to imaginary parts of the forward scattering amplitude, namely
$\rho=(0.09-0.10)\pm 0.01$, turned out to be noticeably smaller than the
predicted value ($\rho=0.13-0.14$ Compete ) based on dispersion relations for
the case of pure $C$-even interactions.

The new ATLAS/ALFA data recently confirmed this value of $\rho$ LHC1c .
However, the value of the total cross section at 13 TeV reported by the
ATLAS/ALFA team, $\sigma_{tot}=104.68\pm 1.09$ mb, is approximately 5% lower
than the average of values determined by TOTEM ($\sigma_{tot}=111.6\pm 3.4$
mb, $\sigma_{tot}=109.5\pm 3.4$ mb, and $\sigma_{tot}=110.3\pm 3.5$
mb)111Recall that the ATLAS/ALFA measurements at $\sqrt{s}=$ 7 and 8 TeV LHC1a
; LHC1b  are also systematically lower than the corresponding TOTEM results
LHC2 ; Sigt ., indicating that a smaller value of the real part of the
$C$-even amplitude should be expected from the dispersion relations.

The relatively small value of $\rho$ can be explained by the admixture of the
$C$-odd amplitude, which survives at high LHC energies. Such amplitude with
the intercept $\alpha_{Odd}$ close to 1 was predicted by the perturbative QCD
BKP ; BLV  (see also the reviews BE ) where at the lowest $\alpha_{s}$ order
it is provided by the three gluon exchange diagrams. Alternatively, it was
shown in DL ; BFLP  that if we do not assume a constant value for the real
part of the nuclear amplitude near $|t|=0$, the whole ensemble of the elastic
scattering low $|t|$ data (from which the total cross sections and the
$\rho$-parameter for the forward amplitude were extracted) can be
satisfactorily described without any Odderon contribution.222Note however that
the analysis of DL  did not account for the Bethe phase in Coulomb-nuclear
interference.

Another indication in favor of the Odderon was claimed in T-D0 , where the
$\bar{p}p$ differential cross section $d\sigma/dt$ at $\sqrt{s}=1.96$ TeV was
compared with the corresponding $pp$ cross section (measured at 2.76 TeV but
extrapolated to 1.96 TeV) in the diffractive dip region. A clear difference
was observed.

The problem, however, is that using this method, any inaccuracy in the
extrapolation from one energy to another is treated as the additional
(Odderon) contribution. In particular, we recall that using another
prescription for this (from 2.76 to 1.96 TeV) extrapolation, the authors of
CNP  obtained for the Odderon signal significance less than 0.5 $\sigma$ (see
Table 3 of CNP ) analyzing the same data points.

Next, it should be noted that at very low $t$ close to zero and in the
diffractive dip region, we deal with different $C$-odd contributions. To get a
well-pronounced dip-bump structure near the dip in $pp$ scattering at 2.76 TeV
and rather flat behavior of $\bar{p}p$ at 1.96 TeV, the real part of the
Odderon $pp$ amplitude should be positive (in agreement with perturbative QCD
expectation for the three gluon exchange case DL ). On the other hand, to
explain the low value of $\rho$ at $t=0$, we need a negative Odderon real
part. This negative real part could be induced by non-perturbative effects.

In the present paper, we describe the full ensemble of the $pp$ and $\bar{p}p$
elastic differential cross-section data at low $|t|<0.1$ GeV2 from
$\sqrt{s}=50$ GeV up to 13 TeV. We use a two-channel eikonal model, which now
includes the Odderon exchange amplitude. This allows us to quantitatively
evaluate the size of the Odderon contribution and the parameters (the slope
and the intercept) of the soft Pomeron trajectory.

In our analysis, the Odderon amplitude is not written separately but included
in the eikonal. This inclusion allows the eikonal model to immediately account
for the screening of the Odderon exchange by the Pomeron(s).

Moreover, we explore the tension between the $d\sigma/dt$ data measured by the
TOTEM and ATLAS/ALFA Collaborations since the discrepancies in the results
presented by the two experiments lead us to different scenarios for the
behavior of the forward scattering amplitude and, consequently, for the
parameters of both the Pomeron and Odderon.

The outline of this paper is as follows. In Sec. II we describe the model and
collect the formulae used in our data description. The results are presented
and discussed in Sec. III. In Sec. IV we draw our conclusions.

##  II Formalism

Our analysis is focused on differential cross-section data involving very
small values of $t$, which requires considering the Coulomb-nuclear
interference (CNI) region. The full scattering amplitude, including the
electromagnetic (Coulomb, $C$) and hadronic (Nuclear, $N$) interactions, can
be expressed as

$\displaystyle F^{C+N}=F^{N}+e^{i\alpha\phi(t)}F^{C},$ (1)

where the phase factor $\alpha\phi(t)$ describes the distortion of the pure
amplitudes $F^{C}$ and $F^{N}$ arising from the simultaneous presence of both
Coulombic and hadronic scattering. We adopt an expression for the phase
$\phi(t)$ derived from an eikonal approach Cahn01 , given by

$\displaystyle\phi(t)$ $\displaystyle=$
$\displaystyle\kappa\left[\gamma+\ln\left(\frac{B|t|}{2}\right)+\ln\left(1+\frac{8}{B\Lambda^{2}}\right)\right.$
(2)
$\displaystyle+\left.\frac{4|t|}{\Lambda^{2}}\ln\left(\frac{\Lambda^{2}}{4|t|}\right)-\frac{2|t|}{\Lambda^{2}}\right],$

where $\kappa$ flips sign when going from $pp$ ($\kappa=-1$) to $\bar{p}p$
($\kappa=+1$). In (2), $\Lambda^{2}$ is fixed at 0.71 GeV2 (as determined in
the dipole fit to the proton’s electromagnetic form factor), and $B$ is the
$t$ slope of elastic $d\sigma/dt\propto exp(Bt)$ cross-section. The $\alpha$
in (1) is the electromagnetic coupling, which is usually kept constant at the
value of the fine structure constant, $\alpha(0)=1/137.03599911$ pdg .
Nevertheless, in cases where the Coulomb peak is used in the normalization of
diffractive data, the contribution of vacuum polarization to the Coulomb force
can be significant Cahn01 ; yennie01 . This effect results in the replacement
of the fine structure constant with an electromagnetic coupling dependent on
$q^{2}$,

$\displaystyle\alpha(q^{2})=\frac{\alpha(0)}{1-\frac{\alpha(0)}{3\pi}\ln\left(\frac{q^{2}+m_{e}^{2}}{m_{e}^{2}}\right)},$
(3)

where $q^{2}=-t$ and $m_{e}=0.510998918$ MeV pdg  is the mass of the electron.
This corresponds to a one percent correction for $q^{2}=0.1$ GeV2.

The Coulomb amplitude can be expressed as

$\displaystyle F^{C}=\kappa s\frac{2\alpha}{|t|}G^{2}(t).$ (4)

Here, $G(t)$ is the electromagnetic form factor of the proton, described by
the dipole form

$\displaystyle G(t)=\left[\frac{\Lambda^{2}}{\Lambda^{2}+q^{2}}\right]^{2}.$
(5)

To account for the eikonalization, it is convenient to calculate the nuclear
amplitude in terms of opacities. The opacity function $\Omega_{i}(s,b)$ is
related to the bare nuclear amplitude $F^{N}_{i}(s,t)$ through the Fourier-
Bessel transform

$\displaystyle\Omega_{i}(s,b)=\frac{2}{s}\int^{\infty}_{0}q\,dq\,J_{0}(bq)\,F^{N}_{i}(s,t),$
(6)

where $i={\mathbb{P}},{\mathbb{O}}$ represent the Pomeron and Odderon
exchanges, respectively.

The single Pomeron contribution is given by

$\displaystyle
F^{N}_{\mathbb{P}}(s,t)=\beta_{\mathbb{P}}^{2}(t)\,\eta_{\mathbb{P}}(t)\left(\frac{s}{s_{0}}\right)^{\alpha_{\mathbb{P}}(t)},$
(7)

where $\eta_{\mathbb{P}}(t)=-e^{-i\frac{\pi}{2}\alpha_{\mathbb{P}}(t)}$ is the
even signature factor, $\beta_{\mathbb{P}}(t)$ is the elastic proton-Pomeron
vertex, and

$\displaystyle\alpha_{\mathbb{P}}(t)=1+\epsilon+\alpha^{\prime}_{\mathbb{P}}t+\frac{m_{\pi}^{2}\beta_{\pi}^{2}}{32\pi^{3}}\,h(\tau)$
(8)

is the Pomeron trajectory, where

$\displaystyle h(\tau)$ $\displaystyle=$
$\displaystyle-\frac{4}{\tau}\,F_{\pi}^{2}(t)\left[2\tau-(1+\tau)^{3/2}\ln\left(\frac{\sqrt{1+\tau}+1}{\sqrt{1+\tau}-1}\right)\right.$
(9) $\displaystyle\left.+\ln\left(\frac{m^{2}}{m_{\pi}^{2}}\right)\right],$

with $\tau=4m_{\pi}^{2}/|t|$, $F_{\pi}(t)=1/(1-t/m^{2}_{\rho})$,
$m_{\pi}=139.6$ MeV, $m=1$ GeV, $m_{\rho}=0.776$ GeV, and
$\beta_{\pi}/\beta_{I\\!\\!P}(0)=2/3$. The function $h$ accounts for the
inclusion of the pion loop into the Pomeron trajectory AG .

The Odderon contribution is given by

$\displaystyle
F^{N}_{\mathbb{O}}(s,t)=\beta_{\mathbb{O}}^{2}(t)\,\eta_{\mathbb{O}}(t)\left(\frac{s}{s_{0}}\right)^{\alpha_{\mathbb{O}}(t)},$
(10)

where $\eta_{\mathbb{O}}(t)=-ie^{-i\frac{\pi}{2}\alpha_{\mathbb{O}}(t)}$ is
the odd signature factor, $\beta_{\mathbb{O}}(t)$ is the elastic proton-
Odderon vertex, and $\alpha_{\mathbb{O}}(t)=1$ is the Odderon trajectory.

The $t$-dependence of the $\beta$ vertices is parameterized accounting for the
observed deviation from a pure exponential behavior of the low-$|t|$
$d\sigma/dt$ data at LHC energies, as identified by the TOTEM Collaboration
Tot-1 ; Tot-2 ; Tot-3 . To get a better fit in the small-$t$ region, the TOTEM
group has extended the pure exponential to a cumulant expansion,

$\displaystyle\frac{d\sigma}{dt}(t)=\left.\frac{d\sigma}{dt}\right|_{t=0}\exp\left(\sum_{n=1}^{N_{b}}b_{n}t^{n}\right),$
(11)

where the optimal fit was achieved for $N_{b}=3$, yielding $\chi^{2}/DoF=1.22$
and a corresponding $p$-value of 8.0%. The same was done by the ATLAS/ALFA
group

Based on this result, we have written the Pomeron- and Odderon-proton vertices
as

$\displaystyle\beta_{\mathbb{P}}(t)=\beta_{\mathbb{P}}(0)e^{\left(At+Bt^{2}+Ct^{3}\right)/2}$
(12)

and

$\displaystyle\beta_{\mathbb{O}}(t)=\beta_{\mathbb{O}}(0)e^{Dt/2},$ (13)

respectively.

In order to allow for the low mass diffractive dissociation, the Good-Walker
formalism GW  is used, which provides a convenient form to incorporate $p\to
N^{*}$ diffractive dissociation. This approach introduces diffractive
eigenstates $|\phi_{k}\rangle$ that diagonalize the interaction $T$ matrix. In
our two-channel model ($k=1,2$), the proton wave function is described by two
components with equal weights, namely
$|p\rangle=\sqrt{\frac{1}{2}}\left(|\phi_{1}\rangle+|\phi_{2}\rangle\right)$.
To minimize the number of free parameters, we take the same $t$ dependence for
both components. The Pomeron and Odderon couplings to the two diffractive
states $|\phi_{k}\rangle$ are

$\displaystyle\beta_{i,k}(t)=(1\pm\gamma)\beta_{i}(t),$ (14)

with $i={\mathbb{P}}$ or ${\mathbb{O}}$, and $\gamma=0.55$.333With
$\gamma=0.55$ at $\sqrt{s}=7$ TeV we get the low mass dissociation cross-
section $\sigma^{D}=4.7$ mb in agreement with the TOTEM result sigD
$\sigma^{D}=2.6\pm 2.2$ mb.

The eikonalized amplitude in $(s,t)$-space is then given by kmr2000 ; kmr2001

$\displaystyle{\cal A}(s,t)$ $\displaystyle=$ $\displaystyle
is\int^{\infty}_{0}b\,db\,J_{0}(bq)\left[1-\frac{1}{4}\,e^{i(1+\gamma)^{2}\Omega(s,b)/2}\right.$
$\displaystyle\left.-\frac{1}{2}\,e^{i(1-\gamma^{2})\Omega(s,b)/2}-\frac{1}{4}\,e^{i(1-\gamma)^{2}\Omega(s,b)/2}\right],$
(15)

where $\Omega(s,b)$ is the total opacity.

We consider two versions for the total opacity with different signs for the
Odderon contribution. In the first version, referred to as ‘Model I’, we have

$\displaystyle\Omega(s,b)=\Omega_{\mathbb{P}}(s,b)\mp\Omega_{\mathbb{O}}(s,b);$
(16)

in the second version, called ‘Model II’, we have

$\displaystyle\Omega(s,b)=\Omega_{\mathbb{P}}(s,b)\pm\Omega_{\mathbb{O}}(s,b);$
(17)

in both cases the upper sign is for $pp$ and the lower sign is for
$\bar{p}p$.444Recall that the sign of the Odderon exchange amplitude is not
known from the beginning. In our parameterization (10) we keep
$\beta_{\mathbb{O}}^{2}>0$ to be positive. That is, model II corresponds to
the case when the Odderon contribution to the real part of $pp$ amplitude is
negative. This is opposite to the sign expected in perturbative QCD for the
leading order $C$-odd three gluon exchange diagram DL ; fukugita1 . However,
at $t\to 0$ the situation is more complicated and may not be described by the
pure three gluon exchange. In particular, within the quark-diquark model of
the proton, the Odderon coupling $\beta^{2}_{\mathbb{O}}$ nullifies at $t=0$
(for the point-like diquark) and becomes positive at larger $|t|$ values. In
such a case at $t\to 0$ the major $C$-odd contribution comes from the Pomeron-
Odderon cut and the real part of $C$-odd $pp$ amplitude becomes negative.

The total cross section and the $\rho$ parameter are expressed in terms of the
nuclear eikonalized amplitude ${\cal A}(s,t)$:

$\displaystyle\sigma_{tot}(s)=\frac{4\pi}{s}\textnormal{Im}\,{\cal A}(s,t=0),$
(18) $\displaystyle\rho(s)=\frac{\textnormal{Re}\,{\cal
A}(s,t=0)}{\textnormal{Im}\,{\cal A}(s,t=0)}.$ (19)

Considering the eikonalized nuclear amplitude (15) in the presence of
electromagnetic and hadronic interactions, our full scattering amplitude will
finally be written as

$\displaystyle F^{C+N}(s,t)={\cal A}(s,t)+e^{i\alpha\phi(t)}F^{C}(s,t).$ (20)

Thus we can write the differential and the total elastic cross sections as

$\displaystyle\frac{d\sigma}{dt}(s,t)=\frac{\pi}{s^{2}}\,\left|{\cal
A}(s,t)+e^{i\alpha\phi}{\cal F}^{C}(s,t)\right|^{2},$ (21)
$\displaystyle\sigma_{el}(s)=\frac{\pi}{s^{2}}\,\int_{-\infty}^{0}dt\left|{\cal
A}(s,t)\right|^{2}.$ (22)

##  III Results

The LHC has provided highly precise measurements of diffractive processes,
allowing stringent constraints on the scattering amplitude behavior at high
energies. These measurements, particularly of the $\rho$ parameter and the
total and differential cross sections from ATLAS/ALFA and TOTEM experiments
play a crucial role in the accurate determination of the Pomeron and Odderon
parameters. However, while the measurements of the $\rho$ parameter at
$\sqrt{s}=13$ TeV by both Collaborations are consistent Tot-13 ; LHC1c , the
total and differential cross sections at $\sqrt{s}=7$, 8, and 13 TeV reveal
some tension between the TOTEM Tot-1 ; Tot-13 ; totem7a ; Sigt ; Sigt2  and
ATLAS/ALFA LHC1a ; LHC1b ; LHC1c  results. This data discrepancy suggests
different scenarios for the rise of the total cross section and, hence, the
parameters of the Pomeron and the Odderon luna1 ; petrov1 .

In order to systematically explore the tension between the TOTEM and
ATLAS/ALFA results, we perform global fits to the $pp$ and $\bar{p}p$
differential cross sections considering three distinct data sets: one
comprising solely the TOTEM measurements, another consisting solely of the
ATLAS/ALFA results, and the third one combining both. These data sets are
complemented by the $d\sigma/dt$ data spanning from CERN-ISR, $S\bar{p}pS$, to
Tevatron energies. Specifically, the three data ensembles can be defined as
follows:

Ensemble A:
$\left.\frac{d\sigma^{\bar{p}p,pp}}{dt}\right|_{\footnotesize\textnormal{CERN-
ISR}}$ \+ $\left.\frac{d\sigma^{\bar{p}p}}{dt}\right|_{\footnotesize
S\bar{p}pS}$ \+
$\left.\frac{d\sigma^{\bar{p}p}}{dt}\right|_{\footnotesize\textnormal{Tevatron}}$
\+
$\left.\frac{d\sigma^{pp}}{dt}\right|_{\footnotesize\textnormal{ATLAS/ALFA}}$

Ensemble T:
$\left.\frac{d\sigma^{\bar{p}p,pp}}{dt}\right|_{\footnotesize\textnormal{CERN-
ISR}}$ \+ $\left.\frac{d\sigma^{\bar{p}p}}{dt}\right|_{\footnotesize
S\bar{p}pS}$ \+
$\left.\frac{d\sigma^{\bar{p}p}}{dt}\right|_{\footnotesize\textnormal{Tevatron}}$
\+ $\left.\frac{d\sigma^{pp}}{dt}\right|_{\footnotesize\textnormal{TOTEM}}$

Ensemble A$\oplus$T:
$\left.\frac{d\sigma^{\bar{p}p,pp}}{dt}\right|_{\footnotesize\textnormal{CERN-
ISR}}$ \+ $\left.\frac{d\sigma^{\bar{p}p}}{dt}\right|_{\footnotesize
S\bar{p}pS}$ \+
$\left.\frac{d\sigma^{\bar{p}p}}{dt}\right|_{\footnotesize\textnormal{Tevatron}}$
\+
$\left.\frac{d\sigma^{pp}}{dt}\right|_{\footnotesize\textnormal{ATLAS/ALFA}}$
\+ $\left.\frac{d\sigma^{pp}}{dt}\right|_{\footnotesize\textnormal{TOTEM}}$

The $\left.d\sigma^{\bar{p}p,pp}/dt\right|_{\footnotesize\textnormal{CERN-
ISR}}$ data enclose measurements of the differential cross section for $pp$
scattering at $\sqrt{s}=$ 52.8 exp_amos1 , and 62.5 exp_amaldi1  GeV and
$\bar{p}p$ scattering at $\sqrt{s}=$ 52.6 exp_amos1 , 53 exp_breakstone1 , 62
exp_breakstone1 , and 62.3 GeV exp_amos1 . The
$\left.d\sigma^{\bar{p}p}/dt\right|_{\footnotesize S\bar{p}pS}$ data set
comprise differential cross section data for $\bar{p}p$ channel at
$\sqrt{s}=540$ exp_arnison1  and 546 GeV exp_bernard1 . The
$\left.d\sigma^{\bar{p}p}/dt\right|_{\footnotesize\textnormal{Tevatron}}$ data
set consist of $d\sigma/dt$ data for the $\bar{p}p$ channel at $\sqrt{s}=1800$
GeV Tev1 ; Tev2 . The
$\left.\frac{d\sigma^{pp}}{dt}\right|_{\footnotesize\textnormal{ATLAS/ALFA}}$
contains measurements of the differential cross section for $pp$ scattering at
$\sqrt{s}=$ 7 LHC1a , 8 LHC1b , and 13 TeV LHC1c  obtained using the ATLAS
Roman Pot system ALFA romanalfa . Finally, the
$\left.\frac{d\sigma^{pp}}{dt}\right|_{\footnotesize\textnormal{TOTEM}}$ data
set comprise differential cross section data for $pp$ channel at $\sqrt{s}=$ 7
totem7a , 8 Tot-1 , and 13 TeV Tot-13  measured by the TOTEM experiment.

Hence, only the elastic scattering data are included in the analysis since the
experimental $\sigma_{tot}$ and $\rho$ values were obtained from fitting the
same $d\sigma/dt$ points already included in our analysis. Of course, to be
sensitive to the $\rho$, we keep all the very small $|t|$ data points and
account for the CNI region. Moreover, to minimize the number of free
parameters, we start from the relatively large energy, namely $\sqrt{s}>50$
GeV, where the secondary Reggeon contribution can be neglected, and fix the
Odderon trajectory ($\alpha_{\mathbb{O}}(t)=1$).

The values of $B$ to be used in the Coulomb phase (equation (2)) are the ones
obtained by determining the differential cross sections at different center-
of-mass energies, as indicated in the original articles. Specifically, the
values are 12.86 GeV-2 for the $pp$ at $\sqrt{s}=$ 52.8 GeV, 13.21 GeV-2 for
the the $pp$ at $\sqrt{s}=$ 62.5 GeV, 13.36 GeV-2 for the $p\bar{p}$ at
$\sqrt{s}=$ 52.6 GeV, 11.5 GeV-2 for the $p\bar{p}$ at $\sqrt{s}=$ 53 GeV,
11.12 GeV-2 for the $p\bar{p}$ at $\sqrt{s}=$ 62 GeV, 13.47 GeV-2 for the
$p\bar{p}$ at $\sqrt{s}=$ 62.3 GeV, 17.1 GeV-2 for the $p\bar{p}$ at
$\sqrt{s}=$ 540 GeV, 15.3 GeV-2 for the $p\bar{p}$ at $\sqrt{s}=$ 546 GeV,
16.3 GeV-2 for the $p\bar{p}$ at $\sqrt{s}=$ 1800 GeV (E710), 16.98 GeV-2 for
the $p\bar{p}$ at $\sqrt{s}=$ 1800 GeV (CDF), 19.73 GeV-2 for the $pp$ at
$\sqrt{s}=$ 7 TeV (ATLAS/ALFA), 19.89 GeV-2 for the $pp$ at $\sqrt{s}=$ 7 TeV
(TOTEM), 19.74 GeV-2 for the $pp$ at $\sqrt{s}=$ 8 TeV (ATLAS/ALFA), 19.90
GeV-2 for the $pp$ at $\sqrt{s}=$ 8 TeV (TOTEM), 21.14 GeV-2 for the $pp$ at
$\sqrt{s}=$ 13 TeV (ATLAS/ALFA), and 20.40 GeV-2 for the $pp$ at $\sqrt{s}=$
13 TeV (TOTEM).

Once our ensembles are defined, we start carrying out a fit to the Ensemble
A$\oplus$T. In our analyses, we fit the CERN-ISR data with $|t|\leq 0.2$ GeV2
and to the $S\bar{p}pS$, Tevatron, and LHC data with $|t|\leq 0.1$ GeV2. We
use a $\chi^{2}$ fitting procedure, with the value of $\chi^{2}_{min}$
distributed as a $\chi^{2}$ distribution with $\nu$ degrees of freedom. We
adopt an interval $\chi^{2}-\chi^{2}_{min}$ corresponding to a 90% confidence
level (CL).

Since the absolute values of cross sections measured at the same energy by
different groups do not agree, we have introduced normalization factors
$N_{i}$ for high-energy $d\sigma/dt$ data, with $i=7[A]$, $8[A]$, and $13[A]$
for the ATLAS/ALFA data and $i=7[T]$, $8[T]$, and $13[T]$ for the TOTEM data.
Here, the numbers within the indices $i$ correspond to the values of
$\sqrt{s}$, namely 7, 8, and 13 TeV. Analogous normalization factors are
introduced for the Tevatron data with $i=1.8[E]$ and $i=1.8[C]$, i.e.
$N_{1.8[E]}$ for the E710 data and $N_{1.8[C]}$ for the CDF data. Despite
being the only data set measured at $\sqrt{s}=$ 546 GeV, we also included a
normalization factor for
$\left.d\sigma^{\bar{p}p}/dt\right|_{\sqrt{s}=546\,\textnormal{GeV}}$, namely
$N_{546}$. Furthermore, when dealing with the data sets incorporating
normalization factors $N_{i}$, we make use of the formula

$\chi^{2}=\sum_{ij}\frac{(N_{i}ds^{th}_{ij}-ds^{exp}_{ij})^{2}}{(\delta^{rem}_{ij})^{2}}+\sum_{i}\frac{(1-N_{i})^{2}}{\delta^{2}_{i}}$
(23)

to calculate the total $\chi^{2}$ value, where $i$, as already specified,
denotes the particular set of data while $j$ denotes the point $t_{j}$ in this
set of data; $ds^{th}_{ij}$ is the theoretically calculated $d\sigma/dt$ cross
section (21) while $ds^{exp}_{ij}$ is the value measured at the same $ij$
point experimentally; $\delta_{i}$ is the normalization uncertainty of the
given ($i$) set of data and $\delta^{rem}_{ij}$ is the remaining error at the
point $ij$ calculated as
$(\delta^{rem}_{ij})^{2}=\delta^{2}_{tot,ij}-(\delta_{i}\cdot
ds^{exp}_{ij})^{2}$. As a rule the value of $\delta^{rem}$ is dominantly the
statistical error.555A similar approach was used in Sel .

The values of the free parameters determined by the fit to the Ensemble
A$\oplus$T, as derived from Models I and II, are listed in Table 1. These
results were obtained by fixing in (13) the Odderon amplitude slope $D=A/2$.
The second and third columns exhibit the outcomes obtained by permitting the
normalization factors to fluctuate within the interval [0.85, 1.15]; the
results of these fits are shown in Figs. 1-6. We first observe that the
parameters related to the Pomeron are not sensitive to the chosen model (Model
I or II) since their values are compatible with each other considering the
associated uncertainties. The same is not observed concerning the Odderon
coupling $\beta_{\mathbb{O}}(0)$: its value is consistent with zero (error
significantly surpassing central value) in the case of Model I. Consequently,
it is clear that a positive Odderon contribution to the real part of $pp$
elastic amplitude is rejected by the data, resembling a scenario where the
Pomeron dominates the scattering amplitude. Therefore, we consider the Model I
as the model without the Odderon. Moreover, from the statistical standpoint,
the fit using Model II is appreciably better than that of Model I, as the
resultant $\chi^{2}/\nu$ values are 1.11 and 1.44, respectively. Hence the
inclusion of a negative Odderon contribution to the real part of $pp$ elastic
amplitude decreases the value of $\chi^{2}$ when compared with the value
obtained in the analysis using the model without the Odderon; specifically,
the decrease is around 25%.

The value of $N_{13[T]}$ each time is at the edge of the allowed interval,
indicating some problem with the TOTEM 13 TeV data. All other $N_{i}$ factors
are inside the interval. For a larger interval, we get a smaller $\epsilon$
(i.e., cross section grows slower with energy) and correspondingly a smaller
Odderon coupling $\beta_{\mathbb{O}}(0)$ since a smaller $\epsilon$ leads to a
smaller $\rho$ for a pure even contribution. With a larger allowed interval,
we get a smaller $\chi^{2}$ (at least up to [0.6-1.4] case); however, we would
not consider so large normalization factor as a realistic value. Therefore, we
chose the [0.85-1.15] interval as the main one, with the fourth column of
Table 1 (for the interval [0.80,1.20]) provided merely to illustrate the
trend.

The effect of incorporating the Odderon becomes notably significant when
analyzing specific subsets of data. In particular, it becomes more evident in
part of the data in the ISR region, notably in data with
$0.1\lesssim|t|\lesssim 0.2$ GeV2 at energies $\sqrt{s}=$ 53 and 62 GeV (Fig.
1). Although less pronounced, its influence is also visible in part of the
data in the $S\bar{p}pS$ region, namely the data with $0.05\lesssim|t|\lesssim
0.1$ GeV2 at energy $\sqrt{s}=$ 540 GeV (Fig. 2); in the data in the Tevatron
region, specifically in the data with $0.03\lesssim|t|\lesssim 0.1$ GeV2 at
energy $\sqrt{s}=$ 1.8 TeV (Fig. 2); and in the LHC data, particularly in the
CNI region at the energies $\sqrt{s}=$ 7, 8, and 13 TeV (Figs. 3 and 4).

The influence of the Odderon is also apparent in our predictions for the
behavior of the total cross section $\sigma_{tot}(s)$, the total elastic cross
section $\sigma_{el}(s)$ and the parameter $\rho(s)$. After introducing the
Odderon, we can see in Fig. 5 a slight difference between the $pp$ (solid
curve) and the $\bar{p}p$ (dashed curve) channels, both in the case of
$\sigma_{tot}$ and $\sigma_{el}$. For Model I, the $pp$ (dotted curve) and
$\bar{p}p$ (dashed-dotted curve) channels are indistinguishable since then the
scattering amplitude is dominated asymptotically only by the even terms, the
total cross-section difference behaves as
$\left|\Delta\sigma\right|=\left|\sigma_{tot}^{\bar{p}p}-\sigma_{tot}^{pp}\right|\to$
0 in the limit $s\to\infty$.

The influence of the Odderon also becomes particularly evident when examining
the Model II predictions for the $\rho$ parameter (Fig. 6): there is a clear
separation between the $pp$ and $\bar{p}p$ channels. As illustrated in Table
1, the model predictions for $\rho^{pp}$ and $\rho^{\bar{p}p}$ at
$\sqrt{s}=13$ TeV are, respectively, 0.111 and 0.119. Conversely, under the
Model I, the channels $pp$ and $\bar{p}p$ predictions are identical and equal
to 0.114. Table 2 shows the predictions for the high energy total cross
section and $\rho$ parameter, obtained using Model I and Model II. Note that
similar values of $\sigma_{tot}=107.6\pm 1.7$ mb and $\rho=0.11\pm 0.01$ at 13
TeV were derived in PT  fitting the low $|t|$ TOTEM 13 TeV data only but
introducing the normalization coefficient $\lambda$ (analogous to $1/N_{13,T}$
in our Table 1). In this case, the fit was normalized to the Coulomb
scattering.

Regarding the behavior of the Pomeron- and Odderon-proton vertices, note that
the parameters $A$ and $D$ are not independent, as they must satisfy the
unitarity constraint $D<A$. In this way, as previously indicated, we choose
$D=A/2$ in our analysis. We have studied other relations between $A$ and $D$,
specifically the cases $D=0.1A$, $D=0.3A$, $D=0.7A$, and $D=0.9A$. As
illustrated in Table 3, these choices do not affect the global fits and,
therefore, the model predictions. While the Pomeron ($C$-even amplitude)
parameters remain relatively stable with respect to variations in the Odderon
slope $D$, the value of the Odderon coupling $\beta_{\mathbb{O}}(0)$ decreases
monotonically as $D$ increases (see Table 3). This can be explained by the
fact that for a smaller $D$, the Odderon contribution is concentrated at a
smaller impact parameter $b$, where screening by the $C$-even amplitude is
stronger. Thus, we need larger $\beta_{\mathbb{O}}(0)$ to get the same final
result.

We investigated the effect of replacing, in Ensemble A$\oplus$T, the data set
of the differential cross section $\bar{p}p$ at $\sqrt{s}=546$ GeV
exp_bernard1  by the measurement of the differential $dN/dt$ distribution
performed by the UA4/2 Collaboration at $\sqrt{s}=$ 541 GeV UA42  (see Fig.
7). This study was to verify whether using the $dN/dt$ distribution, carried
out with a substantial reduction in statistical uncertainty and more rigorous
control of systematic effects than the previous measurement at $\sqrt{s}=$ 546
GeV, could affect our results and predictions for the LHC energy. Our analysis
revealed that using the distribution $dN/dt$ instead of $d\sigma/dt$ in
Ensemble A$\oplus$T does not affect the results already presented in this
paper.

We further conducted an analysis focusing only on the ATLAS/ALFA or TOTEM
data. As depicted in Table 3, the ATLAS/ALFA results show slightly better
agreement with their respective Ensemble. However, accounting for all
normalization factors $N_{i}$, the value $\chi^{2}/\nu=1.11$ remains highly
satisfactory for the analysis using the Ensemble A$\oplus$T.666A slightly
larger $\chi^{2}/\nu$ in the pure TOTEM case (Ensemble T) could be caused by
the fact that in TOTEM data of 13 TeV, some oscillations occur in the $t$
behavior of $d\sigma^{pp}/dt$ (see e.g. Sel ; Per ) that are not accounted for
within our parameterization of the high-energy elastic amplitude.

##  IV Conclusions

The differential proton-proton and proton-antiproton cross sections
$d\sigma/dt$ at low $|t|<0.1$ GeV2 and collider energies (from $\sqrt{s}>50$
GeV to 13 TeV) are successfully described ($\chi^{2}/\nu=1.11$) within the
two-channel eikonal model. To avoid the double counting we do not include in
the fit the $\sigma_{tot}$ and $\rho$ data (which were obtained from the
description of the same $d\sigma/dt$ data points) but rather extend our fit to
the very low $|t|$ values describing the Coulomb-nuclear interference region.
The model accounts for the screening of the Odderon contribution by the
Pomerons including the $C$-even (Pomeron) and $C$-odd (Odderon) multiple
exchanges. That is the opacity, $\Omega$, is written as the sum of the bare
Pomeron and the bare Odderon amplitudes.

To resolve the discrepancy between the TOTEM and ATLAS/ALFA (CDF and E710 in
the Tevatron case) data we introduce the normalization coefficients, $N_{i}$
writing the theoretical prediction as $d\sigma^{exp}/dt=N_{i}d\sigma^{Th}/dt$.
The global analysis chooses $N_{i}>1$ for TOTEM and $N_{i}>1$ for ATLAS/ALFA.
The deviation of the values of $N_{i}$ from 1 compared with the published
luminosity uncertainties is included in the total $\chi^{2}$.

We show that the presence of $C$-odd (Odderon) contribution essentially
improves the fit (see Table 1) however it does not noticeably change the
predicted value of $\rho^{pp}$ at 13 TeV. A larger Odderon amplitude is
constrained by the $S\bar{p}pS$ data.

For completeness, we present also the results obtained by fitting only the
TOTEM or only the ATLAS/ALFA data.

The main lessons about the Odderon coming from this study are:

  * •

The description using the Odderon improves the fit (the $\chi^{2}/\nu$ is the
lowest one).

  * •

The sign of the Odderon amplitude needed to describe the very low $|t|$ data
is opposite to that predicted by the perturbative QCD three-gluon exchange
contribution.

  * •

The quality of the description weakly depends on the Odderon $t$-slope, $D$,
(leading to practically the same values of $\sigma_{tot}$ and $\rho$ ).
However, for smaller $D$ we need larger coupling $\beta_{\mathbb{O}}$ to
compensate for a stronger absorption caused by the Pomeron screening at small
impact parameters $b$.

  * •

The Odderon-proton coupling, $\beta_{\mathbb{O}}$, is smaller than that for
the Pomeron, $\beta_{\mathbb{P}}$. For $D=A/2$ we get
$\beta_{\mathbb{O}}/\beta_{\mathbb{P}}=0.40$, however after accounting for
screening by the Pomeron the final $C$-odd contribution to $\rho$ at 13 TeV
becomes quite small, $\delta\rho=(\rho^{\bar{p}p}-\rho^{pp})/2\leq 0.004$ (see
Table 1) and it will be challenging to enlarge it. Otherwise, we will get too
large $\rho^{\bar{p}p}$ at $\sqrt{s}\sim 541$ GeV in disagreement with the
data UA42 .

## Acknowledgment

This research was partially supported by the Conselho Nacional de
Desenvolvimento Científico e Tecnológico under Grant No. 307189/2021-0.

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Table 1: Values of the parameters obtained in the global fits to Ensemble $A\oplus T$.  | Model I | Model II | Model II
---|---|---|---
$\beta_{\mathbb{P}}(0)$ | 2.247$\pm$0.013 | 2.259$\pm$0.016 | 2.307$\pm$0.022
$\epsilon$ | 0.1173$\pm$0.0021 | 0.1180$\pm$0.0020 | 0.1134$\pm$0.0019
$\alpha^{\prime}_{I\\!\\!P}$ (GeV-2) | 0.124$\pm$0.024 | 0.128$\pm$0.022 | 0.133$\pm$0.023
$A$ (GeV-2) | 5.01$\pm$0.20 | 4.78$\pm$0.21 | 4.72$\pm$0.21
$B$ (GeV-4) | 6.61$\pm$0.99 | 6.7$\pm$1.1 | 6.9$\pm$1.2
$C$ (GeV-6) | 20.4$\pm$5.7 | 17.7$\pm$4.0 | 17.0$\pm$4.2
$\beta_{\mathbb{O}}(0)$ | (0.15 $\times 10^{-4}$) $\pm$39 | 0.90$\pm$0.18 | 0.88$\pm$0.18
$N_{546}$ | 0.941 | 0.933 | 0.958
$N_{1.8[E]}$ | 0.923 | 0.912 | 0.944
$N_{1.8[C]}$ | 1.087 | 1.070 | 1.109
$N_{7[A]}$ | 1.015 | 1.015 | 1.056
$N_{8[A]}$ | 1.003 | 1.003 | 1.045
$N_{13[A]}$ | 1.009 | 1.009 | 1.052
$N_{7[T]}$ | 1.077 | 1.077 | 1.121
$N_{8[T]}$ | 1.121 | 1.121 | 1.167
$N_{13[T]}$ | 1.150 | 1.150 | 1.200
$\rho^{pp}(\sqrt{s}=13\ \textnormal{TeV})$ | 0.114 | 0.111 | 0.109
$\rho^{\bar{p}p}(\sqrt{s}=13\ \textnormal{TeV})$ | 0.114 | 0.119 | 0.116
Allowed $N_{i}$ interval | [0.85,1.15] | [0.85,1.15] | [0.80,1.20]
$\nu$ | 504 | 504 | 504
$\chi^{2}/\nu$ | 1.44 | 1.11 | 1.03
Table 2: Predictions for $\sigma_{tot}^{\bar{p}p,pp}$, $\sigma_{el}^{\bar{p}p,pp}$, and $\rho^{\bar{p}p,pp}$ using Models I and II. These results were derived for the scenario with $D=A/2$. | Model I | Model II
---|---|---
$\sqrt{s}$ (TeV) |  $\sigma^{pp}_{tot}$ $|$ $\sigma^{\bar{p}p}_{tot}$ (mb) |  $\sigma^{pp}_{el}$ $|$ $\sigma^{\bar{p}p}_{el}$ (mb) |  $\rho^{pp}$ $|$ $\rho^{\bar{p}p}$ |  $\sigma^{pp}_{tot}$ $|$ $\sigma^{\bar{p}p}_{tot}$ (mb) |  $\sigma^{pp}_{el}$ $|$ $\sigma^{\bar{p}p}_{el}$ (mb) |  $\rho^{pp}$ $|$ $\rho^{\bar{p}p}$
0.541 | 64.2 $|$ 64.2 | 13.2 $|$ 13.2 | 0.130 $|$ 0.130 | 63.8 $|$ 64.1 | 13.3 $|$ 13.5 | 0.117 $|$ 0.144
1.8 | 78.0 $|$ 78.0 | 17.6 $|$ 17.6 | 0.124 $|$ 0.124 | 77.6 $|$ 77.8 | 17.7 $|$ 17.9 | 0.116 $|$ 0.133
7 | 95.9 $|$ 95.9 | 23.9 $|$ 23.9 | 0.117 $|$ 0.117 | 95.7 $|$ 95.9 | 24.0 $|$ 24.2 | 0.113 $|$ 0.123
8 | 97.9 $|$ 97.9 | 24.5 $|$ 24.5 | 0.116 $|$ 0.116 | 97.6 $|$ 97.8 | 24.7 $|$ 24.8 | 0.113 $|$ 0.122
13 | 105.1 $|$ 105.1 | 27.2 $|$ 27.2 | 0.114 $|$ 0.114 | 104.9 $|$ 105.1 | 27.3 $|$ 27.4 | 0.111 $|$ 0.119
Table 3: Results using Model II |  | Ensemble $A$ |  |
---|---|---|---|---
$D$ (GeV-2) | $0.1A$ | $0.3A$ | $0.5A$ | $0.7A$ | $0.9A$
$\beta_{\mathbb{O}}(0)$ | 0.93$\pm$0.22 | 0.85$\pm$0.22 | 0.80$\pm$0.21 | 0.77$\pm$0.19 | 0.74$\pm$0.18
$\beta_{\mathbb{P}}(0)$ | 2.370$\pm$0.035 | 2.384$\pm$0.036 | 2.386$\pm$0.037 | 2.386$\pm$0.040 | 2.388$\pm$0.039
$\nu$ | 332 | 332 | 332 | 332 | 332
$\chi^{2}/\nu$ | 0.96 | 0.97 | 0.97 | 0.97 | 0.96
$\rho^{pp}(\sqrt{s}=13\ \textnormal{TeV})$ | 0.105 | 0.105 | 0.105 | 0.104 | 0.104
$\rho^{\bar{p}p}(\sqrt{s}=13\ \textnormal{TeV})$ | 0.113 | 0.112 | 0.113 | 0.114 | 0.114
$\sigma_{tot}^{pp}(\sqrt{s}=13\ \textnormal{TeV})$ (mb) | 98.0 | 98.0 | 98.0 | 98.0 | 98.0
$\sigma_{tot}^{\bar{p}p}(\sqrt{s}=13\ \textnormal{TeV})$ (mb) | 98.2 | 98.2 | 98.2 | 98.2 | 98.1
|  | Ensemble $T$ |  |
$D$ (GeV-2) | $0.1A$ | $0.3A$ | $0.5A$ | $0.7A$ | $0.9A$
$\beta_{\mathbb{O}}(0)$ | 1.09$\pm$0.22 | 0.96$\pm$0.18 | 0.90$\pm$0.16 | 0.86$\pm$0.15 | 0.83$\pm$0.14
$\beta_{\mathbb{P}}(0)$ | 2.236$\pm$0.022 | 2.258$\pm$0.016 | 2.260$\pm$0.016 | 2.260$\pm$0.017 | 2.259$\pm$0.018
$\nu$ | 418 | 418 | 418 | 418 | 418
$\chi^{2}/\nu$ | 1.28 | 1.30 | 1.29 | 1.28 | 1.27
$\rho^{pp}(\sqrt{s}=13\ \textnormal{TeV})$ | 0.112 | 0.112 | 0.111 | 0.111 | 0.110
$\rho^{\bar{p}p}(\sqrt{s}=13\ \textnormal{TeV})$ | 0.119 | 0.118 | 0.119 | 0.119 | 0.120
$\sigma_{tot}^{pp}(\sqrt{s}=13\ \textnormal{TeV})$ (mb) | 104.9 | 104.9 | 104.9 | 104.9 | 104.9
$\sigma_{tot}^{\bar{p}p}(\sqrt{s}=13\ \textnormal{TeV})$ (mb) | 105.1 | 105.1 | 105.1 | 105.1 | 105.1
|  | Ensemble $A\oplus T$ |  |
$D$ (GeV-2) | $0.1A$ | $0.3A$ | $0.5A$ | $0.7A$ | $0.9A$
$\beta_{\mathbb{O}}(0)$ | 1.09$\pm$0.24 | 0.95$\pm$0.19 | 0.90$\pm$0.18 | 0.86$\pm$0.17 | 0.83$\pm$0.16
$\beta_{\mathbb{P}}(0)$ | 2.235$\pm$0.023 | 2.257$\pm$0.016 | 2.259$\pm$0.016 | 2.258$\pm$0.016 | 2.258$\pm$0.017
$\nu$ | 504 | 504 | 504 | 504 | 504
$\chi^{2}/\nu$ | 1.11 | 1.12 | 1.11 | 1.10 | 1.09
$\rho^{pp}(\sqrt{s}=13\ \textnormal{TeV})$ | 0.112 | 0.112 | 0.111 | 0.111 | 0.110
$\rho^{\bar{p}p}(\sqrt{s}=13\ \textnormal{TeV})$ | 0.119 | 0.118 | 0.119 | 0.119 | 0.120
$\sigma_{tot}^{pp}(\sqrt{s}=13\ \textnormal{TeV})$ (mb) | 104.9 | 104.9 | 104.9 | 104.9 | 104.9
$\sigma_{tot}^{\bar{p}p}(\sqrt{s}=13\ \textnormal{TeV})$ (mb) | 105.1 | 105.1 | 105.1 | 105.1 | 105.1
Figure 1: Description of the $t$ dependence of the elastic $pp$\- and
$\bar{p}p$-cross sections measured at CERN-ISR exp_amos1 ; exp_amaldi1 ;
exp_breakstone1 . The dashed and solid curves depict the results obtained
using Models I and II, respectively. Figure 2: Description of the $t$
dependence of the elastic $pp$\- and $\bar{p}p$-cross sections measured at the
$S\bar{p}pS$ exp_arnison1 ; exp_bernard1 , the Tevatron Tev1 ; Tev2  and the
LHC colliders LHC1a ; LHC1b ; LHC1c ; Tot-13 ; Tot-1 ; LHC2 . The dashed and
solid curves depict the results obtained using Models I and II, respectively.
The lower curves describe the ATLAS/ALFA (E710) data while the upper curves
correspond to the TOTEM (CDF) data; in both cases, the normalization factors
$N_{i}$ are accounted for. Figure 3: The same as Fig. 2 but for the CNI region
where the Odderon contribution reveals itself. Figure 4: The same as Figs. 2
and 3 but in another scale to better see the quality of precise 13 TeV data
description. Figure 5: Description of the total and elastic $pp$ ($\bullet$,
$\blacktriangle$, $\blacksquare$) and $\bar{p}p$ ($\circ$) cross sections. The
data are from LHC1a ; Sigt ; LHC2 ; totem7a ; LHC1b . The dotted and dashed-
dotted curves represent the results for $pp$ and $\bar{p}p$ channels,
respectively, obtained from the global fit to Ensemble A $\oplus$ T using
Model I. These curves are indistinguishable. The solid and dashed curves
represent the results for $pp$ and $\bar{p}p$ channels, respectively, obtained
from the global fit to Ensemble A $\oplus$ T using Model II. Figure 6: $\rho$
parameter for $pp$ ($\blacktriangle$, $\blacksquare$) and $\bar{p}p$ ($\circ$)
elastic amplitude. The data are from pdg ; Tot-2 ; Tot-1 ; Tot-13 ; LHC1c .
The dotted and dashed-dotted curves represent the results for $\rho^{pp}$ and
$\rho^{\bar{p}p}$, respectively, obtained from the global fit to Ensemble A
$\oplus$ T using the Model I. These curves are indistinguishable. The solid
and dashed curves represent the results for $\rho^{pp}$ and $\rho^{\bar{p}p}$,
respectively, obtained from the global fit to Ensemble A $\oplus$ T using the
Model II. Figure 7: Description of the $\bar{p}p$ differential $dN/d|t|$
distribution in the $|t|$-range $|t|\leq 0.1$ GeV2. Figure 8: Description of
$\rho$ parameter for $pp$ elastic amplitude measured by TOTEM
($\blacktriangle$) and ATLAS/ALFA ($\blacksquare$) Collaborations. The data
are from Tot-2 ; Tot-1 ; Tot-13 ; LHC1c . The dashed (solid) curve represents
the predicted $\rho^{pp}$ from the global fit using Ensemble T (Ensemble A).