Reggeometry of lepton- and hadron-induced reactions

Reggeometry of lepton- and hadron-induced reactions Roberto Fiore, László Jenkovszky, Adelmo Lavorini, and Andrii Salii Citation: AIP Conf. Proc. 1523, 83 (2013); doi: 10.1063/1.4802122 View online: http://dx.doi.org/10.1063/1.4802122 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1523&Issue=1 Published by the American Institute of Physics.

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Reggeometry of lepton- and hadron-induced reactions Roberto Fiore∗ , László Jenkovszky†,∗∗ , Adelmo Lavorini‡ and Andrii Salii† ∗

Dipartimento di Fisica, Università della Calabria Istituto Nazionale di Fisica Nucleare, Gruppo collegato di Cosenza I-87036 Arcavacata di Rende, Cosenza, Italy † Bogolyubov Institute for Theoretical Physics (BITP), Ukrainian National Academy of Sciences 14-b, Metrolohichna str., Kiev, 03680, Ukraine ∗∗ Wigner Research Centre for Physics, Hungarian Academy of Sciences 1525 Budapest, POB 49, Hungary ‡ Dipartimento di Fisica, Universitá della Calabria Istituto Nazionale di Fisica Nucleare, Gruppo collegato di Cosenza I-87036 Arcavacata di Rende, Cosenza, Italy Abstract. We append a simple Pomeron pole amplitude by t and Q2 , MV dependencies inspired by geometrical ideas. The experimentally transition from soft to hard dynamics is realized by the introduction of a two-component Pomeron with different Q2 and MV -dependent residues. A unified description of deeply virtual Compton scattering (DVCS), of exclusive electroproduction of all vector mesons as well as of elastic pp scattering is suggested. Keywords: Diffraction, vector mesons, Compton scattering, proton, Regge poles, Pomeron PACS: 11.55.-m, 11.55.Jy, 12.40.Nn

The forward slope of the differential cross sections for elastic scattering is known to be related to the masses/virtualities of the interacting particles. This phenomenon is evident e.g. from Fig. 1, where the forward slope B(Q˜ 2 ) = dtd ln ddtσ is plotted against the variable Q˜ 2 = Q2 + MV2 1 . The slope is proportional to the interaction radius R(Q˜ 2 ), which decreases with increasing of Q˜ 2 until it reaches saturation value (about 4.5 GeV−2 ), that correspond to the mass of the nucleon in the lower vertex (Fig. 2). In this geometrical picture, the largest slope (radius) is expected for real Compton scattering Q˜ 2 = 0, which may require a separate treatment. In the present paper we deal with exclusive electroproduction of real photons, called deeply virtual Compton scattering (DVCS), vector meson production (VMP) as well as elastic proton-proton scattering, using the above geometrical considerations and writing the scattering amplitude in 2 the form: A(s,t, MV2 ) ∼ eB(s,MV )t , where B(M 2 ) ∼ 1/ f (MV2 ). This approach was used in Ref. [1] for the simpler case of photoproduction, Q˜ 2 = 0, leaving outside DVCS, VMP and nucleon scattering, to be also treated below. While the geometrical considerations proved to be efficient for photoproduction [1], they are not sufficient in the case of electroproduction, Q2 = 0, since the relevant cross sections will increase with Q˜ 2 contradicting the experimental data. To remedy this deficiency, the rise must be compensated by multiplying the amplitude by a function that decreases with Q˜ 2 . Moreover, to cope with the observed trend of hardening the dynamics as Q˜ 2 increases, and following The use of the variable Q˜ 2 = (MV2 + Q2 ) implies symmetry between the mass MV2 and virtuality Q2 , which should imply equal slopes (radii) for e.g. J/ψ production near Q2 = 0 and ρ electroproduction near Q2 ≈ 9 GeV2 , which is not obvious.

1

Diffraction 2012 AIP Conf. Proc. 1523, 83-86 (2013); doi: 10.1063/1.4802122 © 2013 AIP Publishing LLC 978-0-7354-1146-3/$30.00

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Refs. [3, 4], we introduce two components for the diffractive (Pomeron) amplitude, soft As

FIGURE 1. Slope of the diffraction cone as function of Q˜ 2 = Q2 + M 2 . Compilation of the data for various VMP and DVCS processes measured by ZEUS [2].

and hard Ah , each one to be multiplied by a relevant Q˜ 2 -dependent factor Hi (Q˜ 2 ), i = s, h. These factors should be chosen in such a way as to provide for the increasing of the hard component’s weight with increasing of Q2 . To avoid conflict with unitarity, the rise with Q˜ 2 of the hard component must be finite, and terminates at some saturation scale Q˜ 2, whose value will be determined phenomenologically. Recently a model for exclusive production of vector particles at HERA was suggested and successfully fitted to the HERA data [1, 5, 6]. In that model, the interplay between t and Q˜ 2 is achieved by introducing a new variable z = t − Q˜ 2 . Good fits were obtained at those papers, however only at the cost of fitting each reaction separately. e-

e-

γ

e-

*

V

e-

γ (q ) 1

p

p

p

P(p ) 1

2

IP(p - p )

W2

p

γ ,V(q )

V1(Q 2 ,t)

1

V2 (t)

2

P(p ) 2

t (a)

(b)

(c)

FIGURE 2. Diagrams of DVCS (a) and VMP (b); (c) DVCS (VMP) amplitude in a Regge-factorized form.

2 The scattering amplitude contains two terms, a soft and hard one, with two different Q dependent factors: s αs (t) 2 as + bs t ˜ π A 2 s e−i 2 αs (t) e Q2 2m p (1) A(s,t, Q2 , Mv 2 ) = 2 ns s0s Q 1 + 2 Qs


A˜h + 1+

2

ah bh (t) α 2 + h π s 2 t h e Q2 2m p . e−i 2 αh (t) n +1 h 2 s0h Q Q 2 Q

2 Q h

Then elastic differential and integrated cross sections are: d σel = Hs2 e2{Ls (αs (t)−1)+gst} + Hh2 e2{Lh (αh (t)−1)+ght} d|t| π +2Hs Hh e{Ls (αs (t)−1)+Lh (αh (t)−1)+(gs +gh )t} cos (αs (t) − αh (t)) , 2

σel =

(2)

Bcos φ0 + L sin φ0 Hs2 e2{Ls (α0s −1)} Hh2 e2{Lh (α0h −1)} , (3) + + 2Hs Hh eLs (α0s −1)+Lh (α0h −1) 2(αs Ls + gs ) 2(αh Lh + gh ) B2 + L2

where: Hs =

As

2

1+ Q

Q2 ˜ Q2 h 2 nh +1 Q 1+ Q2 h

ns ,

Hh =

Q2

Ah

,

s as bs s Ls = ln s0s , gs = 2 , αs (t) = α0s + αst, + 2 Q2 2m p ah bh Lh = ln s s , gh = 2 + 2m αh (t) = α0h + αh t, 2 , 2

0h

Q

p

B = Ls αs + Lh αh + (gs + gh ), L = π2 (αs − αh ), φ0 = π2 (α0s − α0h ), αs (t) = 1.08 + 0.25t, αh (t) = 1.44 + 0.01t. In this contribution we present fits with one term alone. "σ*γ p->φp #z8 2005"|Q2 2.40 [GeV2] 140 130

"σ/dt*γ p->φp #h3 2009|W=75"|Q2 15.80 [GeV2]

3

10

120

10

104

dσ /dt *γ p->φp [nb⋅GeV -2]

σ *γ p->φp [nb]

σ*γ p->φp [nb]

"σ*γ p->φp #z8 2005"|W 75.00 [GeV] 102

102

110 10

100 90

1

1

80 10-1

70

0

5

10

15

20

25

30

35

40

40 Q2 [GeV 2]

50

60

70

80

90

100 W [GeV]

10-20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45 t [GeV2 ]

FIGURE 3. A sample of representative fits: differential and integrated cross sections for φ production. The data are form [8, 11]. TABLE 1.

Fitting results. 2 A Q s

pp ρ0 φ J/ψ ϒ (1S) γ

5.9±5.7 59±29 32±35 34±19 37±101 9.7±9.0

s

∗∗∗ 1.33 1.30 1.4 ±0.7 0.9 ±1.7 0.4±0.5

ns

α0s

αs

as

bs

χ˜ 2

0.00 1.35±0.05 1.32±0.10 1.39±0.13 1.53±0.55 0.94±0.24

1.05±0.14 1.15±0.06 1.14±0.12 1.21±0.05 1.29±0.26 1.19±0.09

0.28±0.47 0.15 0.15 0.09 0.0±0.6 0.0±0.3

2.9±2.8 -0.22 -0.8±1.6 1.90 1.90 1.9±4.6

0.00 1.69 2.5±2.7 1.03 1.03 1.7±2.3

1.52 6.56 3.81 4.50 1.28 1.75


The experimental data in our fitting procedure were taken from the ISR and SPS experiments (pp scattering, see Appendix in [7]), and from the ZEUS and H1 experiments [8–21] for DVCS and vector meson production. The preliminary results of our fits are displayed in Table 1 and in Fig. 3. The parameter ns 2 can not be was set 0 for pp scattering since it does not depend on Q2 , hence the parameter Q s determined here. An alternative, effective way of treating the transition from soft to hard physics is possible within one (first) term of Eq. (1), by using a Pomeron trajectory accounting for the transition from soft to hard dynamics i.e. α (t, Q2 ) = α0 (Q˜ 2 ) + α (Q˜ 2 )t. By using α0 (Q˜ 2 ) = 1 1 , with d = 2.16, f = 2.744 and α (Q˜ 2 ) = ln(1 + 1/(c + Q˜ 2 )) ln d+

f +Q˜ 2

)

with c = 8.17, DVCS was successfully fitted. The parameters c, b and f were chosen such as to match the "soft" and "hard" limits: α0 (Q˜ 2 → 0) = 1.08, α0 (Q˜ 2 → ∞) = 1.3, α (Q˜ 2 → 0) = 0.12 and α (Q˜ 2 → ∞) = 0.0. Notice that in this modification the parameters of the Pomeron trajectory assume slightly different values.

ACKNOWLEDGMENTS L.J. thanks the Organizers of the Diffraction 2012 conference at the Canarias for their hospitality and financial support.

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