Quark-diquark model for p([p bar])-p elastic scattering at high energies

Abstract

A model for elastic scattering of protons at high energies based on the quark-diquark representation of the proton is discussed. The predictions of the model are compared with experimental data for the differential elastic cross-sections from available databases.

Eur. Phys. J. C (2013) 73:2320DOI 10.1140/epjc/s10052-013-2320-2

Regular Article - Theoretical Physics

Quarkdiquark model for p( p)p elastic scattering at high

energies

V.M. Grichine1,a, N.I. Starkov1, N.P. Zotov2

1Lebedev Physical Institute, Moscow, Russia

2Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russia Received: 12 December 2012 / Revised: 15 January 2013 / Published online: 20 February 2013 Springer-Verlag Berlin Heidelberg and Societ Italiana di Fisica 2013

Abstract A model for elastic scattering of protons at high energies based on the quarkdiquark representation of the proton is discussed. The predictions of the model are compared with experimental data for the differential elastic cross-sections from available databases.

1 Introduction

Recently the TOTEM Collaboration reported the rst experimental data on the pp elastic cross-section at the total energy in the center of mass system s = 7 TeV (everywhere

the Planck constant, [planckover2pi1], and the speed of light, c, are assumed to be unit, [planckover2pi1] = c = 1) [1]. Therefore there is opportunity to

describe one in the more wide area of energy using the early data (see, for example, [2]). In general there are a number of models of the elastic pp scattering description [2, 3].

We discuss in this note the quarkdiquark (qQ-) model in which baryons are considered as bound states of quark and diquark (a quasi-particle state of two quarks). This model appeared at the end sixties [4] and was used for description of different problems: baryon spectroscopy [5, 6], multiparticle production [7, 8], deep-inelastic processes [9] and others. The qQ-model was proposed by V.A. Tsarev [10] in 1979 to describe the characteristics of protonproton elastic scattering and to explain the absence of second dip in the protonproton differential cross-section, del/dt, at the four-momentum transfer t 45 GeV2, which should ex

ist, if a proton would be compounded of three quarks.A. Bialas and A. Bzdak [11] reinvented in 2007 the approach of [10] in more simplied form without the real part of the scattering amplitude. They showed that the quark diquark model is capable to predict the correct position

a e-mail: [email protected]

of the protonproton differential elastic cross-section minimum at low energies, however, the absence of the scattering amplitude real part results in dramatic overestimation of the value of rst dip at t 1.3 GeV2. The model [11] was

applied in [12] to describe at the same level of accuracy the TOTEM data [1] for del/dt at s = 7 TeV.

We recall below the main features of the qQ-model [10] suitable for numerical calculations and provide comparison with the experimental data on the pp elastic differential cross-section in the region of high energies, s 546 GeV.

2 Quarkdiquark model for p( p)p elastic scattering

The protonproton differential elastic cross-section can be expressed in terms of the scattering amplitude F (s, t):

del

dt =

p2

[vextendsingle][vextendsingle]F (s, t)[vextendsingle][vextendsingle]2, (1)

where p is the proton momentum in the center of mass system.

The model [10] limits the consideration of the scattering amplitude by contributions from one- and two-pomeron exchanges between quarkquark (11), diquarkdiquark (22) and quarkdiquark (12). In this approximation F (s, t) can be expressed as

F (s, t) = F1(s, t) F2(s, t) F3(s, t), (2)

where F1(s, t) is the scattering amplitude with one-pomeron exchange, while F2(s, t) corresponds to two-pomeron exchanges between the proton constituents, quark and diquark, and F3(s, t) corresponds to two-pomeron exchanges between the quark (or diquark) of one proton and the quark and the diquark of another proton at the same time. The amplitude F1(s, t) reads

Page 2 of 4 Eur. Phys. J. C (2013) 73:2320

F1(s, t) =

iptot(s)

4

[bracketleftbig]B1 exp(A11t) + B2 exp(A22t)

+2[radicalbig]B1B2 exp(A12t)[bracketrightbig], (3)

where tot(s) is the total protonproton cross-section. The coefcients B1, and B2 parametrize the quarkquark, 11, and the diquarkdiquark, 22, cross-sections, respectively:

11 = B1tot(s), 22 = B2tot(s).

The model assumes the quarkdiquark cross-section, 12 = 1122.The coefcients Ajk, (j, k = 1, 2) are derived taking into

account the Gauss distribution of quark and diquark in proton together with the standard pomeron parametrization. They read (s0 = 1 GeV2)

Ajk =

r2j + r2k

with tot(s)) the s-dependence of the del/dt. The diquark diquark cross-section and the parameter B2 are derived from the optical theorem, which results in the following equation:

totb1B1B2 + tot

[radicalbig]B1B2(b2B1 + b3B2)

= B1 + B2 + 2

[radicalbig]B1B2 1, (7)

where

b1 =

14 Re

[bracketleftbigg] 1

A12 + 4/9

[bracketrightbigg]

,

b2 =

14 Re

[bracketleftbigg] 1

A11 + A22 4/9

[bracketrightbigg]

,

b3 =

14 Re

[bracketleftbigg] 1

A12 + A22 + 2/9

[bracketrightbigg]

.

Equation (7) is the third-order equation relative to B2. For 0 < B1 < 1, it has rational root 0 < B2 < 1 which is used in the model.

3 Comparison with experimental data

Figures 1, 2, and 3 show the antiprotonproton differential elastic cross-section versus |t| at s = 546 GeV, s =

1960 GeV, and the protonproton differential elastic cross section at s = 7 TeV, respectively. The curves are the pre

dictions of our model. We see that the proposed qQ-model describes reasonably the differential elastic cross sections of the antiprotonproton and protonproton scattering in a wide region of energy.

The results shown in Figs. 1, 2 and 3 correspond to the s-dependence of the model free parameters which are shown

16 +

[bracketleftbigg]

ln s

s0

i

2

[bracketrightbigg]

. (4)

Here r1, m1 are the quark radius and mass, and r2, m2 are the diquark radius and mass, respectively;

= 0.15 GeV2

is the pomeron trajectory slope, and = r2/4, where r is the

proton radius. It is assumed in the model that m1 = m/3 and

m2 = 2m/3, where m is the proton mass. The radii r1 and r2

were found by the tting of experimental data: r1 = 0.173r,

r2 = 0.316r.

The amplitudes F2(s, t) and F3(s, t) are

F2(s, t) =

ip 4

+

[bracketleftbigg][parenleftbigg]m mj

m

2 [parenleftbigg]m mk

m

2[bracketrightbigg]

B1B22tot(s) 8(A12 + 4/9)

exp

[bracketleftbigg] [parenleftbigg]A11A22 (4/9)2 2(A12 + 4/9)

t

[parenrightbigg]

[parenleftbigg]A12 4/9

2 t

[parenrightbigg][bracketrightbigg]

, (5)

+exp

and

F3(s, t) =

ip 4

B1B22tot(s) 4

[bracketleftbigg] B1

A11 + A12 4/9

exp

[parenleftbigg]A11A12 (2/9)2

A11 + A12 4/9

t

[parenrightbigg]

B2A12 + A22 + 2/9

exp

, (6)

respectively. The quarkquark cross-section, 11 and the proton radius r are the free parameters dening (together

+ [parenleftbigg] A12A22 (/9)2

A12 + A22 + 2/9

t

[parenrightbigg][bracketrightbigg]

Fig. 1 The antiprotonproton differential elastic cross-section versus

|t| at s = 546 GeV. The curve is the prediction of our model. The

open circles are the experimental data [13, 14]

Eur. Phys. J. C (2013) 73:2320 Page 3 of 4

Fig. 2 The antiprotonproton differential elastic cross-section versus

|t| at s = 1960 GeV. The curve is the prediction of our model. The

open circles are the experimental data [15]

Fig. 4 The dependence ln(r), r in GeV1, versus the ln(s/s0)

Fig. 5 The dependence ln(10311/tot) versus ln(s/s0)

data, while the value of dip is overestimated, though less than in the model [11]. The reason is that the model [10] has nonzero real part of the scattering amplitude coming from the pomeron parametrization. However, the value of the real part is not enough for correct description of the dip value.

The model tuning with more sophisticated form-factors, investigation of additional sources for the scattering amplitude real part, and broader comparison of the model with experimental data are current plans.

Acknowledgements The authors are thankful to S. Giani for fruitful discussions of the paper contents. N.Z. is grateful to the DESY Directorate for the support within the MoscowDESY project on Monte-Carlo implementation for HERALHC, he was also supported by FASI of Russian Federation (grant NS-3920.2012.2), the RFBR State contract 02.740.11.0244 and the Ministry of Education and Science of Russian Federation under agreement No. 8412. V.G and N.S. were partly supported by the CERNRAS Program of Fundamental Research at LHC.

Fig. 3 The protonproton differential elastic cross-section versus |t|

at s = 7 TeV. The curve is the prediction of our model. The open

and closed circles are the LHC TOTEM experimental data from [1]

in Figs. 4 and 5. The parametrization of the proton radius reads

ln(r GeV) = 1.72 + 0.004 ln

[radicalbigg] s s0 .

The model results in increase of the proton radius and the quarkquark cross-section with growth of s.

4 Discussion and summary

We have considered the qQ-model of the pp-elastic scattering at high energies. It was obtained reasonable description of the differential cross section of elastic pp scattering in a wide region of energies. The position of the del/dt-minimum is in the satisfactory agreement with experimental

Page 4 of 4 Eur. Phys. J. C (2013) 73:2320

References

1. G. Antchev et al. (TOTEM Collaboration), Europhys. Lett. 95, 41001 (2011); CERN-PH-EP-2012-239

2. N.P. Zotov, S.V. Rusakov, V.A. Tsarev, Fiz. Elem. Chastits At.

Yadra 11, 1160 (1980) (in Russian)

3. I.M. Dremin, arXiv:1206.5474 [hep-ph]4. D.B. Lichtenberg, I.J. Tassie, Phys. Rev. 155, 1601 (1967)5. P.J. De Souze, D.B. Lichtenberg, Phys. Rev. 161, 1513 (1967)6. D.B. Lichtenberg, I.J. Tassie, P.J. Keleman, Phys. Rev. 167, 1535 (1968)

7. A.B. Kaidalov, Z. Phys. C 12, 63 (1982)8. A.B. Kaidalov, Surv. High Energy Phys. 13, 265 (1999)9. B. Anderrsson et al., Phys. Rep. 97, 31 (1983)10. V.A. Tsarev, Sov. Phys., Lebedev Inst. Rep. 4, 12 (1979)11. A. Bialas, A. Bzdak, Acta Phys. Pol. 38, 159 (2007)12. F. Nemes, T. Csrg, arXiv:1204.5617 [hep-ph]13. M. Bozzo et al., Phys. Lett. B 147, 385 (1984)14. M. Bozzo et al., Phys. Lett. B 155, 197 (1985)15. D0 Collaboration, D0 Note 6056-CONF