Pomeron and Reggeon contributions to elastic proton-proton and proton-antiproton scattering in holographic QCD Zhibo LiuWei Xieand Akira Watanabe

2025-05-02 0 0 895.32KB 22 页 10玖币

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Pomeron and Reggeon contributions to elastic proton-proton and

proton-antiproton scattering in holographic QCD

Zhibo Liu,∗Wei Xie,†and Akira Watanabe‡

College of Science, China Three Gorges University,

Yichang 443002, People’s Republic of China and

Center for Astronomy and Space Sciences, China Three Gorges University,

Yichang 443002, People’s Republic of China

(Dated: March 27, 2024)

Abstract

The total and diﬀerential cross sections of elastic proton-proton and proton-antiproton scatter-

ing are studied in a holographic QCD model, considering the Pomeron and Reggeon exchanges in

the Regge regime. In our model setup, the Pomeron and Reggeon exchanges are described by the

Reggeized spin-2 glueball and vector meson propagators, respectively. How those contributions

change with the energy is explicitly shown, focusing on the contribution ratios. The adjustable pa-

rameters included in the model are determined with the experimental data, and it is presented that

the resulting total and diﬀerential cross sections are consistent with the data in a wide kinematic

region.

∗202107020021014@ctgu.edu.cn

†xiewei@ctgu.edu.cn

‡watanabe@ctgu.edu.cn (Corresponding author)

arXiv:2210.11246v4 [hep-ph] 26 Mar 2024

I. INTRODUCTION

Quantum chromodynamics (QCD), being a remarkably successful theory of the strong

interaction, meets some practical diﬃculties in the soft kinematic region. One of those

diﬃculties is related to the hadron-hadron forward scattering processes [1, 2], with high

center-of-mass energy sand small momentum transfer t. Decades ago, the scaling laws for the

hadron-hadron cross sections at s→ ∞ and ﬁxed t/s were investigated [3–5], but the story is

totally diﬀerent for the forward scattering due to the nonperturbative nature of the involved

partonic dynamics. The forward scattering is associated with the hadron diﬀraction, in

which the initial and ﬁnal states in the scattering process have the same quantum numbers.

The diﬀractive processes are generally classiﬁed as double diﬀraction dissociation, single

diﬀraction dissociation and elastic scattering. The simplest process is elastic hadron-hadron

scattering, in which the imaginary part of the scattering amplitude is related to the total

cross section. Thus, the investigation of total cross section is embedded into the elastic

scattering amplitude. The total cross sections of various hadronic processes have been

measured experimentally [6] and analyzed phenomenologically [7–10]. These measurements

provide crucial information for the soft kinematic region of QCD, in which quarks and gluons

are strongly interacting and perturbative method is basically not applicable.

Historically the Regge theory provided a useful framework to analyze the total cross sec-

tions. Even today, the Regge theory, with combined Reggeon and Pomeron contributions,

unrelated to the underlying partonic dynamics, still provides successful descriptions of total

cross sections of hadronic scattering. The Regge theory is based on the analysis with the

complex angular momentum, which is extensively illustrated in the literature [11, 12]. The

total cross sections of proton-proton (pp) and proton-antiproton (p¯p) scattering have been

ﬁtted by the exchanges of the Reggeon trajectories and the soft Pomeron [13–16], with their

respective slope and intercept parameters. The nature of the soft Pomeron is highly nonper-

turbative, and practically it is almost impossible to be derived from QCD. Its properties can

be inferred from experimental data accumulated over several decades. The 2++ glueball is

considered to be the lightest state on the leading Pomeron trajectory, which has an intercept

of 1.08. The increasing behavior of the total cross sections with respect to the center-of-mass

energy √sis associated with the Pomeron exchange. On the other hand, the exchange of

the Reggeon trajectories accounts for the decreasing behavior.

Holographic QCD, a nonperturbative approach to QCD, has been developed based on

the anti-de Sitter/conformal ﬁeld theory (AdS/CFT) correspondence [17–19], which, also

called gauge-string duality, relates a strongly coupled gauge theory to a weakly coupled

string theory in the curved spacetime. Most works in holographic QCD assume that the

curved space string theory reduces to the supergravity theory in vanishing string length.

This approach has been employed to analyze the spectrum and structure of hadrons with

successful results [20–32]. Holographic QCD has also been used to study high energy scat-

tering processes [33–46]. In particular, hadronic scattering in the Regge regime, which is

characterized by the condition, s≫t, is interesting. Historically string theory originated in

the phenomenological description of hadronic scattering in the Regge regime, in which the

scattering amplitude is accounted by the exchange of the Regge trajectories of mesons.

Studies of high energy scattering in the Regge regime have gathered interests in the high

energy physics community so far, since this regime is essential to better understanding and

testing the gauge-string duality. The strict treatment of the string dual requires complicated

string calculations in the curved space. Approximations are certainly needed to make prac-

tical progress in this ﬁeld. A string theory inspired holographic QCD model for hadronic

scattering [47–49], which aims to describe the experimental data of scattering cross sections,

has been developed. In this model the string scattering amplitude in the weakly curved

background approximately takes the same structure as the ﬂat space amplitude, with the

Regge parameters (Regge slopes and intercepts) which diﬀer from those in the ﬂat space.

The amplitude in the Regge regime is dictated by the amplitude of exchange of lightest states

on the Regge trajectories of mesons and glueballs, which are determined by low energy ef-

fective couplings in the top-down holographic approach [49–51]. The top-down construction

contains relatively fewer parameters and unambiguously gives the hadron couplings via the

supergravity action. The couplings involving the proton, mesons and glueballs have been

derived explicitly in the literature [49, 50, 52–54]. Based on these couplings, hadronic scat-

tering amplitudes in the Regge regime can be obtained for exchanges of the lightest mesons

or glueballs. Then the single particle propagators are replaced with the Reggeized ones

which are obtained by comparing with the string scattering amplitudes [47, 49].

This model has been applied to the proton-proton scattering via the Pomeron ex-

change [47, 48], and to the central production of ηwith the double Pomeron and dou-

ble Reggeon exchange [49]. The propagator for proton-proton scattering via the Pomeron

exchange is based on a comparison of the Virasoro-Shapiro amplitude with the classical

ﬂat space bosonic closed string amplitude. The mesonic Regge trajectories have also been

incorporated into this model, whose propagators are constructed by using the bosonic open

string amplitude in ﬂat space. The parameters of these amplitudes are modiﬁed to account

for the physical meson and glueball trajectories.

In this work we study the elastic pp and p¯pscattering in the Regge regime, taking into

account both the Pomeron and Reggeon exchanges. This work is an extension of the pre-

ceding study [43], in which only the Pomeron exchange was considered. Since the Pomeron

exchange gives the dominant contribution to cross sections in the high energy region with

√s≳100 GeV, successful results were obtained in the previous work. However, the Reggeon

exchange contribution needs to also be considered to describe the data in the lower energy

region. In our model the Pomeron and Reggeon exchanges are described by the Reggeized

spin-2 glueball and vector meson propagators, respectively. The scattering amplitudes are

obtained by combining the proton-vector meson and proton-glueball couplings with those

propagators.

The model includes several parameters, but the ones related to the Pomeron exchange

have already been determined in the previous work. Hence, in this study we use the param-

eter values obtained in Ref. [43] for the Pomeron, and determine the adjustable parameters

associated with the Reggeon exchange by numerical ﬁts with experimental data of the total

cross sections for the pp and p¯pscattering. Utilizing the linear relation for the Reggeon

trajectory, only three parameters in total need to be determined in this study. With the

obtained parameter values, we explicitly show how the both contributions change with the

energy, focusing on the contribution ratios. It is presented that the resulting total cross

sections are in agreement with the data in a wide kinematic region. Then, we show that our

predictions for the diﬀerential cross section are also consistent with the data for both the pp

and p¯pscattering in the kinematic range of |t|<0.45 GeV2and 10 GeV <√s≤13 TeV.

Overall, the results presented in this paper show that the present model works well and the

future experimental data could better constrain the model parameters.

This paper is organized as follows. In Sec. II we introduce the holographic description of

the elastic pp and p¯pscattering in the Regge regime, taking into account the Pomeron and

Reggeon exchanges. We brieﬂy review the formalism developed in the preceding studies, and

present the expressions for the total and diﬀerential cross sections. The energy dependence

p1p3

p2p4

p1p3

p2p4

FIG. 1. The left and right Feynman diagrams represent the pp(p¯p) scattering with the Pomeron

and Reggeon exchanges in the t-channel, respectively. p1, p2and p3, p4are four-momenta of the

initial and ﬁnal states, respectively, and kis the momentum transfer.

of the both contributions is shown in detail, focusing on the contribution ratios. With the

obtained expressions, numerical ﬁts are performed, and the results are displayed in Sec. III.

Our conclusion with the implications of this work is given in Sec. IV.

II. MODEL SETUP

A. Holographic description of the pp and p¯pscattering

In this work we investigate the elastic pp and p¯pscattering in the Regge regime, con-

sidering contributions of the Pomeron and Reggeon exchanges which are described by the

Reggeized spin-2 glueball and vector meson. The Feynman diagrams, which describe those

contributions, are shown in Fig. 1. The scattering amplitudes are written as

App,p¯p

tot =App,p¯p

g+App,p¯p

v.(1)

Following the preceding study [55], the matrix element of the energy momentum tensor Tµν

between the initial and ﬁnal proton states is expressed as

⟨p′, s′|Tµν |p, s⟩= ¯u(p′, s′)A(t)γµPν+γνPµ

+B(t)i(Pµσνρ +Pνσµρ)kρ

4mp

+C(t)kµkν−ηµν k2

mpu(p, s),(2)

where P= (p1+p3)/2, k=p3−p1,t=−k2and mpis the proton mass. A(t), B(t) and C(t)

are the proton form factors. The contributions of B(t) and C(t) are negligible in the Regge

limit, and in this work, similar to our previous work [43], we use the gravitational form factor

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PomeronandReggeoncontributionstoelasticproton-protonandproton-antiprotonscatteringinholographicQCDZhiboLiu,∗WeiXie,†andAkiraWatanabe‡CollegeofScience,ChinaThreeGorgesUniversity,Yichang443002,People’sRepublicofChinaandCenterforAstronomyandSpaceSciences,ChinaThreeGorgesUniversity,Yichang443002,People’...

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Pomeron and Reggeon contributions to elastic proton-proton and proton-antiproton scattering in holographic QCD Zhibo LiuWei Xieand Akira Watanabe

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