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arXiv:1206.4441v1 [hep-ph] 20 Jun 2012

DESY 12–102

June 2012

ISSN 0418-9833

Production of one or two vector mesons in peripheral high-energy collisions of heavy

ions

A. I. Ahmadov∗

Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, 141980 Russia and

Institute of Physics, Azerbaijan National Academy of Science, Baku, Azerbaijan

B. A. Kniehl†and E. S. Scherbakova‡

II. Institut f¨ ur Theoretische Physik, Universit¨ at Hamburg,

Luruper Chaussee 149, 22761 Hamburg, Germany

E. A. Kuraev§

Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, 141980 Russia

(Dated: June 21, 2012)

We study the production of spin-one mesons in high-energy heavy-ion collisions with peripheral

kinematics in the framework of QED. The cross sections of the production of a single vector meson

and of two different ones are presented. The explicit dependence on the virtuality of the intermediate

vector meson is obtained within a quark model. The effect of reggeization of the intermediate vector

meson state in the case of the production of two vector mesons is taken into account.

PACS numbers: 12.20.Ds, 13.85.Dz, 25.75.-q, 25.75.Dw

I.INTRODUCTION

The CERN Large Hadron Collider (LHC) provides the opportunity to study experimentally the production of scalar,

pseudo-scalar, and vector mesons in peripheral collision of heavy ions. Peripheral kinematics implies the detection of

particles produced in directions close to the axis of the colliding beams [1]. The main feature of these processes is the

non-decreasing total and differential cross sections. The invariant mass of the created particles is assumed to be small

in the fragmentation and central regions in comparison with the total energy in the center of mass of the colliding

beams√s = 2E. The application of the known theoretical approaches, such as the Nambu-Iona-Lasinio model as well

as chiral perturbation theory, seems to be legitimate. We shall consider, in pure quantum electrodynamics (QED), the

processes of the production of a single vector meson and of two vector mesons separated by a rapidity gap. It is known

that the main contributions to the amplitudes of peripheral processes arise from the interaction mechanism of ions,

mediated by the exchange of spin-one particles, such as virtual photons, vector mesons, and gluons. For sufficiently

large electric charges of the ions, virtual-photon exchanges will eventually play the dominant role. Actually, the effect

of the replacement α → Zα for the case of charged heavy ions, e.g. for Pb-Pb collisions, exceeds the corresponding

QCD contribution for typical regions of momentum transfer, where αs∼ 0.1− 0.2, which seems to be essential in the

experimental set-up. We shall consider processes of the creation of one and two vector mesons, such as ω, J/ψ, ρ, or

ortho-positronium:

Y1(Z1,P1) + Y2(Z2,P2) → V (e,r) + Y1(Z1,P′

Y1(Z1,P1) + Y2(Z2,P2) → V (e1,r1) + V (e2,r2) + Y1(Z1,P′

iare the four-momenta of the incoming and scattered ions, and e,eiand r,riare the polarization four-vectors

and four-momenta and of the created vector mesons, which obey the transversality conditions e(r)r = ei(ri)ri= 0. To

1) + Y2(Z2,P′

2),

1) + Y2(Z2,P′

2). (1)

Here, Pi,P′

∗Electronic address: ahmadov@theor.jinr.ru

†Electronic address: kniehl@desy.de

‡Electronic address: scherbak@mail.desy.de

§Electronic address: kuraev@theor.jinr.ru

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describe the peripheral kinematics, it is convenient to introduce the light-cone four-momenta pias linear combinations

of the incoming-ion four-momenta Pi:

p1= P1− λP2,

where mi are the masses of the ions. In the calculation of the differential cross section, the effects of the off-mass-

shell-ness of the exchanged photons and vector meson must be taken into account. Our approach is based on taking

the constituent quarks to be QED fermions. The additional factors for the QED amplitudes associated with the color

and charge of the quarks will be discussed later.

The details of the wave function of the bound state were discussed in Ref. [2]. For our approach, only one structure

R is relevant. The virtual-photon polarization effects in the process γ⋆gg → ψ were considered in Ref. [3]. The analysis

of the inclusive annihilation of heavy quarkonium beyond the Born approximation of QCD was presented in Ref. [4].

In our work, we shall obtain the differential cross sections for the creation of one or two vector mesons considered as

bound states of the relevant quarks.

p2= P2− ηP1,p2

1= p2

2= 0,P2

1= m2

1,P2

2= m2

2,2P1P2≈ 2p1p2= s ≫ m2

i, (2)

II.MATRIX ELEMENTS OF THE 2 → 3 PROCESSES

To lowest order in perturbation theory, there are two sets of Feynman diagrams involving three virtual-photon

exchanges (see Fig. 1). The contribution of each of them to the total cross section has the form σ ∼ σ0(aL2+bL+c),

with L = ln(s/M2) being the “large” logarithm. The interference of the relevant amplitudes only contributes terms

devoid of the “large” logarithm. Below, we restrict ourselves to the consideration of just one of the amplitudes, which

corresponds to exchanges of one virtual photon with one ion Y1(Z1,P1) and of two virtual photons with the other ion

Y2(Z2,P2). Using the prescriptions proposed in Ref. [1] for evaluating the matrix element of the peripheral process of

single vector meson production, we obtain

MY1Y2→Y1Y2V=(4παZ1)(4παZ2)2

q2

1

?2

s

?3

sN1

?

d4q

(2π)4q2q2

3

s2N2sFCV, (3)

where the factor CVaccounts for the color and charge of the quarks and N1, N2, and F are given by the following

expressions:

N1 =

1

s¯ u(P′

1

2s2¯ u(P′

1

s2

1) ˆ p2u(P1),

?

MA

4TrOµνλ(ˆ p + M1)ˆ ep1µp2νp2λ.

N2 =

2)ˆ p1

ˆ p2− ˆ q + m2

(p2− q)2− m2

2

ˆ p1+ ˆ p1

ˆ p2− ˆ q3+ m2

(p2− q3)2− m2

2

ˆ p1

?

u(P2), (4)

F =

1

(5)

Here, M is the mass of the created vector meson, and q3 = q2− q.

specify below, measures the strength of the interaction of the vector meson with incoming photons. The quantity

¯ u(q)Oµνλv(q+) is the matrix element of the subprocess 3γ → q¯ q depicted in Fig. 1. So, we have

Oµνλp1µp2νp2λ = ˆ p1ˆ q−− ˆ q1+ m

D1

?1

+

D2D3[ˆ p2(ˆ q−− ˆ q + m)ˆ p1(−ˆ q++ ˆ q3+ m)ˆ p2+ ˆ p2(ˆ q−− ˆ q3+ m)ˆ p1(−ˆ q++ ˆ q2+ m)ˆ p2],

The coupling constant A, which we shall

?1

D3ˆ p2(−ˆ q++ ˆ q3+ m)ˆ p2+

1

D3ˆ p2(ˆ q−− ˆ q3+ m)ˆ p2

1

D2ˆ p2(−ˆ q++ ˆ q2+ m)ˆ p2

?−ˆ q++ ˆ q1+ m

?

+

D2ˆ p2(ˆ q−− ˆ q + m)ˆ p2+

1

D1

ˆ p1

(6)

where q± are the four-momenta of the created quarks and the denominators are given by the expressions D1 =

(q−−q1)2−m2+i0, D2= (−q−+q)2−m2, and D3= (−q++q3)2−m2. Following the rules for the construction of

the matrix element of the quark-antiquark bound state [5], we must put q+= q−= p/2 in this expression. Let us now

introduce the Sudakov parametrization of the loop momenta and the momenta of the quarks and virtual photons:

q1 = β1p1+ q1⊥,

q± = α±p2+ β±p1+ q±⊥,

q3= α3p2+ β3p1+ q3⊥,

d4q =s

q = αp2+ βp1+ q⊥,

2dαdβd2? q.(7)

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From four-momentum conservation and the on-mass-shell conditions of the quarks, we have

β+ = β−=1

2β1,α + α3= α2;

? q1+ ? q2 = ? q1+ ? q + ? q3= ? p,sα±=

1

2β1(? p2+ 4m2). (8)

The expressions for the denominators can now be rewritten as:

D1 = −? q2

D2 = −? q2+ ? p? q −1

3+ ? p? q3−1

1+ ? p? q1−1

2(? p2+ 4m2) = −1

2(? q2

1+ ? q2

2+ M2) = −1

2R,

2sβ1α + i0,

D3 = −? q2

2s(α − α2)β1+ i0. (9)

Performing the integration over the component β2of the loop momentum, we obtain for N2

s

?

dβN2=1

s¯ u(P′

2)ˆ p1u(P2)

∞

?

−∞

sdβ

?

1

sβ + a + i0+

1

−sβ + b + i0

?

= (−2πi)1

s¯ u(P′

2)ˆ u(P2) = −2πiN′

2. (10)

Summing the squared moduli of N1and N′

2over the spin states, we obtain

?

|N1|2= 2,

?

|N′

2|2= 2. (11)

In a similar way, the integral over the Sudakov variable α gives us

∞

?

−∞

sdα

?1

D2

+

1

D3

?

= (−2πi)2

β1. (12)

As a result for the contribution of these terms to the matrix element, we have

8(4παZ1)(4παZ2)2

? q2

1

N1N′

2

(2πi)2

4(2π)4

1

D1

M1A

2

?

d2? q

π? q2? q2

3

1

4TrQ, (13)

with the trace

1

4TrQ =

1

4Tr[ˆ p2(−1

1

4Tr[ˆ p2ˆ q1ˆ p1+ ˆ p1ˆ q1ˆ p2]Mˆ e = sM(? q? e).

2ˆ p + ˆ q1+ m)ˆ p1− ˆ p1(1

2ˆ p − ˆ q1+ m)ˆ p2](ˆ p + M)ˆ e

= (14)

The contribution of the last two terms may be found using the relation

∞

?

−∞

sdα

1

D2D3

= (−2πi)2

β1

1

D,

(15)

where D = R − 4? q? q3. The calculation of the relevant trace leads to

1

4Tr ˆ p2[(ˆ q−⊥− ˆ q2⊥+ m)(−ˆ q+⊥+ ˆ q3⊥− m) + (ˆ q−⊥− ˆ q3⊥+ m)(−ˆ q+⊥− ˆ q2⊥− m)](ˆ p + M)ˆ e

= −2m1

4Tr ˆ p2ˆ q1ˆ pˆ e = −sM

2β1(? q1? e).(16)

Here, the expression contained within the square brackets is 2m(ˆ q3+ ˆ q2− ˆ q+− ˆ q−) = −2mˆ q1, the replacement

(ˆ p + M) → ˆ p is due to the relation pe = 0, and p2p =1

1

D−1

2sβ1. Furthermore, we have

R=4? q(? q2− ? q)

RD

.(17)

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We thus obtain

?

dαF =(2πi)8? q? q3(? e? q1)

RD

MA

2

. (18)

It is worth noting that the right-hand side of this equation vanishes in the the limit where the transverse momenta

of each of the three virtual photons vanish, which is a consequence of gauge invariance.

The final result for matrix element corresponding to the Feynman diagram in Fig. 1a is

MY1Y2→Y1Y2V= M2s27π2Z1Z2

2α3

? q2

1

AN1N′

2(? q1? e)1

RJ(? q2

2,R)CV, (19)

with

J(? q2

2,R) =

?

d2? q

π? q2? q2

3

(? q? q3)

R − 4(? q? q3), (20)

where R = ? q2

As shown in Appendix A, the function J has the form

1+ ? q2

2+ M2and ? q3= ? q2− ? q.

J(? q2

2,R) =

1

2? q22− R

?

2ln

R

2? q2

2

− lnR − ? q22

? q22

?

=

1

? q2

1+ M2L(x1), (21)

where

L(x) =

1

x − 1ln(1 + x)2

4x

,x1,2=

? q2

2,1

? q2

1,2+ M2. (22)

III.DIFFERENTIAL CROSS SECTION OF SINGLE VECTOR MESON PRODUCTION

The Feynman diagrams contributing to the process of single vector meson production are shown in Fig. 1. The

phase space volume has the form

dΓ3=(2π)4

(2π)9

d3P′

2E′

1

1

d3P′

2E′

1

1

d3P′

2E′

1

1

δ4(P1+ P2− P′

1− P′

2− p). (23)

We introduce the factor d4q1δ4(p1− q1− p′

Sudakov parametrization of Eq. (7) [1]. Allowing for the vector meson to be unstable, the three-particle phase space

volume then becomes

1)d4q2δ4(p2− q2− p′

2) in terms of Sudakov variables and use the standard

dΓ3=

1

4s

(2π)4

(2π)9π2d2? q1d2? q2

π2

dβ1

β1

dp2R(p2),(24)

where p2= sα2β1− (? q1+ ? q2)2and we replace the delta function by a Breit-Wigner resonance,

δ(p2− M2) → R(p2) =1

π

ΓM

(p2− M2)2+ M2Γ2, (25)

where M ≈ 2mqand Γ are the mass and the total decay width of the vector meson resonance and mqis the mass of

the bound quarks. The quantity p2= (P1+ P2− P′

of single vector meson production.

With the matrix element of Eq. (19), we now have

1− P′

2)2may be associated with the missing mass in the process

MY1Y2→Y1Y2V

= s27π2M2α3Z1Z2N1N2

R

AZ, (26)

where

Z =

?

Z2(? q1? e)

1(? q2

? q2

1+ M2)L(x1) +

Z1(? q2? e)

2(? q2

? q2

2+ M2)L(x2)

?

CV.(27)

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The relevant differential cross section is

dσ(1)

dp2=26π(Z1Z2)2α6A2M4

R2

Z2d2q1d2q2

2

π2

dβ1

β1

R(p2). (28)

The interference term in Z2is canceled by the average over the azimuthal angle. For the case of extremely small

transverse momentum |? q1| of ion Y1, a modification of this formula is necessary, which consists in replacing

1

(? q2

β1

(? q2

1)2

dβ1

→

1 − β1

1+ m2

1β2

1)2

dβ1

β1

, (29)

and a similar replacement for small values of |? q2|. This leads to the so-called Weizs¨ acker-Wiliams enhancement factor

in the total cross section:

dσ(1)

dp2= σ0R(p2)[Z2

2(L2

1− 5L1) + Z2

1(L2

2− 5L2) + c(Z2

1+ Z2

2) + O(m2

1

s,m2

2

s)], (30)

where

σ0 =

32π(Z1Z2α3)2A2

M2

s

m2

1,2

?

0

(1 − ln2),

L1,2 = ln

,

c = 2

1

dx

x

?

−

5 + x

2(1 − x)3+5

2+

?1 + 2x

(1 − x)4− 1

?

ln1

x

?

≈ 10.4565. (31)

IV. DIFFERENTIAL CROSS SECTION OF TWO VECTOR MESON PRODUCTION

In the case of two vector mesons in the final state, we must consider the two mechanisms due to the sets of Feynman

diagrams indicated Figs. 2a and 2b.

The phase space volume of the four-particle final state is

dΓ4=

(2π)4

(2π)12π3dβ1

β1

dβ′

β′

d2? q1d2? q′d2? q2

π3

dr2

1dr2

2R(r2

1)R(r2

2)1

8s,

(32)

where r2

introduce another auxiliary four-vector: q′= α′p2+ β′p1+ q′

The matrix element of the process of two vector meson production mediated by a virtual vector meson (see Fig. 2a)

has the form

1= sα′β1− (? q1− ? q′)2and r2

2= sα2β′− (? q2+ ? q′)2. Also here, we assume that m2

⊥.

1,2/s ≪ β′≪ β1≪ 1. Let us

MY1Y2→Y1Y2V1V2

a

= s(Z1Z2α2)2N1N2A1A2(MV1MV2)2g2π229

? q2+ M2

V

T(? q? e1)(? q? e2)CV1

2CV2

2, (33)

where T is given by the expression

T =J(q2

1,R1)J(q2

R1R2

2,R2)

+J(q2

1,¯R1)J(q2

¯R1¯R2

2,¯R2)

,(34)

with R1= q2+ q2

the virtual vector meson, g is its coupling constant, and M1,2are the masses of the created vector mesons.

The cross section of the process of two vector meson production is

1+ M2

1, R2= q2+ q2

2+ M2

2,¯R1= q2+ q2

1+ M2

2, and¯R2= q2+ q2

2+ M2

2. Here, MV is the mass of

dσ(2)

dr2

a

1dr2

2

= s26α8(M1M2Z1Z2)4

(? q2+ M2

V)2

g4(A1A2)2R(r2

1)R(r2

2)|T|2(? q? e1)2(? q? e2)2(CV1

2CV2

2)2d2? q1d2? q2d2? q

π3

dβ1

β1

dβ

β.

(35)

After integration over d2? q, we obtain:

dσ(2)

dr2

a

1dr2

2

=8Z4

1Z4

2g4α8

π

(1 + 2cos2θ12)dβ1

β1

dβ

β(A1A2)2

?M2

V1M2

M5

V

V2

?2

R(r2

1)R(r2

2)(CV1

2CV2

2)2d? q2

1d? q2

M4

V

2

P(? q2

1

M2

V

,

? q2

M2

2

V

), (36)