Eur. Phys. J. C (2019) 79:237
https://doi.org/10.1140/epjc/s10052-019-6768-6
Regular Article - Theoretical Physics
The Phillips–Barger model for the elastic cross section
and the Odderon
V. P. Gonçalves
a
,P.V.R.G.Silva
b
High and Medium Energy Group, Instituto de Física e Matemática, Universidade Federal de Pelotas, Caixa Postal 354, Pelotas,
RS CEP 96010-900, Brazil
Received: 24 December 2018 / Accepted: 11 March 2019 / Published online: 15 March 2019
© The Author(s) 2019
Abstract Inspired by the recent TOTEM data for the elas-
tic proton–proton ( pp) scattering at
√
s = 8 and 13 TeV,
we update previous studies of the differential cross sections
using the Phillips–Barger (PB) model, which parametrizes
the amplitude in terms of a small number of free parameters.
We demonstrate that this model is able to describe the recent
pp data on a statistically acceptable way. Additionally, we
perform separate fits of the pp data for each center-of-mass
energy and propose a parametrization for the energy depen-
dence of the parameters present in the PB model. As a con-
sequence, we are able to present the PB predictions for the
elastic proton–proton cross section at
√
s = 546 GeV and 1.8
TeV, which are compared with the existing antiproton–proton
( ¯pp) data. We show that the PB predictions, constrained by
the pp data, are not able to describe the ¯ppdata. In particular,
the PB model predicts a dip in the differential cross section
that is not present in the ¯pp data. Such result suggests the
contribution of the Odderon exchange at high energies.
The LHC measurements for the total, elastic and differential
cross sections at high energies of 2.76, 7, 8 and 13 TeV [1–11]
provide very interesting results, which challenge the descrip-
tion of the theory of strong interactions at high energies.
In particular, the high-precision measurement of the elas-
tic differential cross section data (dσ/dt) in the Coulomb–
Nuclear interference region in proton-proton ( pp) scattering,
performed by the TOTEM Collaboration [7], has allowed
the determination of the ρ parameter, which is defined as
the ratio between the real and imaginary part of the elas-
tic scattering amplitude in the forward direction. The value
obtained experimentally is lower than those predicted by
different Regge inspired models that take into account the
Reggeon and Pomeron exchange contributions for the scat-
tering amplitude, whose predictions for the total cross section
a
e-mail: barros@ufpel.edu.br
b
e-mail: precchia@ifi.unicamp.br
(σ
tot
) are compatible with the measurements [12]. Such result
has motivated an intense debate about the possible contribu-
tion of the Odderon [13–20]. The existence of an Odderon is a
natural prediction of the Quantum Chromodynamics (QCD),
has a C-odd parity and determines the hadronic cross section
difference between the direct and crossed channel processes
at very high energies (for a review see Ref. [21]). If the Odd-
eron exchange contributes for the hadronic scattering at high
energies we will have σ
pp
tot
(
√
s) = σ
¯pp
tot
(
√
s). Moreover, the
squared momentum transfer (t) dependence of dσ/dt for
|t| = 0 is predicted to be different for pp and ¯pp scattering
at a fixed center-of-mass energy (
√
s)[22,23]. Namely, a dip
is expected to be present in pp and absent in ¯pp collisions,
which is directly associated to the fact that the Odderon con-
tribution, being a C-odd term in the elastic scattering ampli-
tude, enters with an opposite sign in each case, resulting in
a different t -behaviour for dσ/dt. Consequently, a compar-
ison between the data for the pp and ¯pp distributions at the
same energy can be considered a direct probe of the Odderon.
Such comparison have only been performed using the data
at low energies (
√
s = 53 GeV), but due to the low statistics
of the ¯ppdata, the difference between the pp and ¯ppdata is
not very large and depends on only a few data points. There-
fore, the interpretation of the difference as a evidence of the
Odderon remains a subject of debate [21].
Another important aspect of the recent TOTEM data for
the transverse momentum distributions is its high precision
and the large range of t values spanned at high energies. The
description of the t-dependence of dσ/dt in the full kine-
matical range is still a theoretical challenge, since the model
should be able to describe the optical point, the forward peak,
the dip and the tail for a fixed energy, as well as to predict
the change of these structures with the energy. As the elas-
tic process is dominated by non-perturbative physics, such
process have been described using phenomenological mod-
els based on different assumptions. Our goal in this letter is
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