Page 1

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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4

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)

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where cosθ12= cos?

? e1? e2and

P(? q2

1

M2

V

,

? q2

M2

2

V

) =

∞

?

0

d? q2

(? q2+ M2

V)2

(? q2)2

M2

V

|TM8

V|2

(37)

is evaluated in Appendix B yielding the numerical values presented in Table I.

For the case of two-gluon exchange (see Fig. 2b), the matrix element has the form

MY1Y2→Y1Y2V1V2

b

= is212π3(Z1Z2α2α2

s)(MV1MV2)2N1N2

? q2

21? q2

A1A2QCV1

1CV2

1, (38)

where

Q =I(R1,R2)

R1R2

(? e1? q1)(? e2? q2) +I(¯R1,¯R2)

¯R1¯R2

(? e1? q2)(? e2? q1). (39)

The quantity I(R1,R2) is given in the Appendix A. For the cross section, we may write

dσ(2)

dr2

b

1dr2

2

=212π(Z1Z2α2α2

s)2(MV1MV2)4

1? q2

(? q2

2)2

R(r2

1)R(r2

2)Q2dβ1

β1

dβ

β(A1A2)2d2? q1d2? q2d2? q

π3

(CV1

2CV2

2)2. (40)

Performing the integration over d2? q, we obtain

dσ(2)

dr2

b

1dr2

2

=210π(z1z2α2α2

x1x2MV1MV2

s)2dx1dx2

R(r2

1)R(r2

2)dβ1

β1

dβ

β(A1A2)2(CV1

2CV2

2)2

?

Φ +¯Φ + 2cos2θ12G

?

W,(41)

where the functions Φ,¯Φ, and G are evaluated in Appendix B yielding the numerical values presented in Table II,

W =d? q2

1

? q2

1

d? q2

? q2

2

2

(42)

is the Weizs¨ acker-Williams enhancement factor, and

x1,2=

? q2

1,2

MV1MV2

. (43)

In the region of small values of ? q2

We note that the interference term of the two amplitudes is absent, so that |M(2)

1,2, we must replace ? q2

1→ ? q2

1+ β2

1m2

1, ? q2

2→ ? q2

2+ α2

2m2

+ M(2)

1.

b

a |2= |M(2)

b|2+ |M(2)

a |2.

V.CONCLUSION

We studied the production of one or two vector mesons in peripheral heavy-ion collisions at high energies. In the

case of Zα > αs, a simplified version of a general theory [4] can be used to lowest order of QED and QCD that is

based on the subprocesses γ⋆γ⋆g → V and γ⋆gg → V . In our study of two vector meson production, a vector meson

can also appear virtually as an intermediate state. In this case, it is important to replace it by the relevant vector

reggeon state, with the Regge trajectory

αV((? q′)2) = αV(0) + α′

V(0)(? q′)2≈ αV(0) ≈ 1/2.(44)

This results in an additional factor

?p2

s0

?2(αV(0)−1)

∼s0

p2, (45)

where s0∼ 1 GeV and p2is the missing mass square, in the cross section dσ(2)

When constructing the invariant mass square of the decay products of one vector meson, p2= (P1+P2−P′

sβ1α2−(? q1+? q2)2, we take into account that the part sβ1α2is the combination (?Ei)2−(?piz)2, with Eiand pizbeing

from the transverse components −(?pi⊥)2. Here, it is understood that the z direction is taken along the beam axis

a .

1−P′

2)2=

the energies and z components of the three-momenta of the decay products, while the part −(? q1+? q2)2is the contribution

Page 7

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in the center-of-mass frame. Such is the case for two jet production with r2

and r2

The coupling constant A of the meson-photon interaction in the case of single vector meson production, appearing

in Eq. (5), is given by A = α3/2/(2√π) for ortho-positronium and by Ai= 2fVi/MVi, with fρ= fω= 0.21 GeV and

fψ= 0.38 GeV, for the ω, ρ, and J/ψ mesons, respectively (see Ref. [6] for details).

In the case of single vector meson production, the color and charge factors are CV= 3?

mesons, and CV=8

3for the J/ψ meson. In the case of two vector meson production through the mechanism shown

in Fig. 2a, we have CV1

2

mechanism shown in Fig. 2b, we have CV1

1

meson.

Note that the mechanism involving single γ∗exchange (see Fig. 2b) yields a L4enhancement, whereas double γ∗

exchange (see Fig. 2a) only produces a L2enhancement. For pp collisions at the LHC, the “large” logarithm is as

large as L = ln

m2 ≈ 7.

We do not consider gluon exchange between heavy ions and vector mesons to avoid channels with ion exitation.

1= (P1− P′

1− q′)2= sα′β1− (? q1− ? q′)2

2= (P2− P′

2+ q′)2= sα2β′− (? q2+ ? q′)2.

u,d

Q3

q=7

3for the ρ and ω

= 3?Q2

q=5

3for the ρ and ω mesons, and CV1

= 3?Qq = 1 for the ρ and ω mesons, and CV1

2

=4

3for the J/ψ meson. In the case of the

= 32

1

3= 2 for the J/ψ

s

Appendix A

In this section, we shall explain how to calculate the two-dimensional Euclidean integrals appearing in Eqs. (19)

and (39),

J(? q2

2,R) =

?

d2?k

π

?k(? q2−?k)

?k2(? q2−?k)2D,

2+M2, Di= Ri−4?k(? q −?k), and Ri= q2

?

I(R1,R2) = 4

?

d2?k

π

(?k(? q −?k))2

?k2(? q −?k)2D1D2

, (A1)

where D = R −4?k(? q −?k), R = ? q2

1+? q2

i+q2+M2

?

i. For the first one, we have

4J =

d2?k

π

R − D

?k2(? q2−?k)2D

= lim

λ→0

?

RJ1− J0

, (A2)

with

J1=

?

d2?k

π

1

((? q2−?k)2+ λ2)(?k + λ2)D,

J0=

?

d2?k

π

1

(k2+ λ2)((? q2−?k)2+ λ2). (A3)

Using Feynman’s trick of joining the denominators,

1

ab=

1

?

0

dx

1

(xa + ¯ xb)2,(A4)

with ¯ x = 1 − x, we obtain for J0:

J0=

1

?

0

dx

?

d2?k

π

1

[(?k − ? q22x)2+ ? q22x¯ x + λ2]2=

1

?

0

dx

? q2

2x¯ x + λ2=

2

? q2

2

ln? q2

2

λ2. (A5)

For J1, we have

4J1=

1

?

0

dx

1

?

0

ydy

[? q22x¯ xy2+¯ y

4(R − ¯ y? q2

2) + yλ2]2,(A6)

where ¯ y = 1 − y. Introducing the variable t = 1 − 2x, we may cast this into the form:

J1= 4

1

?

0

ydy

1

?

0

dt

(A − Bt2)2,(A7)

Page 8

8

where

A = ? q2

2y2+ ¯ y(R − ? q22¯ y) + 4yλ2,B = ? q2

2y2. (A8)

Performing the integration over t, we obtain

J1=

1

?

0

ydy

A3/2B1/2

?2(AB)1/2

A − B

+ lnA1/2+ B1/2

A1/2− B1/2

?

= (I1+ I2)

?1

? q2

?2

, (A9)

where

I1= 2

1

?

0

ydy

[¯ y(ρ − ¯ y) + 4yσ]T,

I2=

1

?

0

dy

T3/2lnT1/2+ y

T1/2− y,

(A10)

with

T = y2+ ¯ y(ρ − ¯ y),ρ =R

? q2,σ =λ2

? q2. (A11)

The first integral I1contains an infrared singularity. Introducing the small parameters σ and ǫ, with σ ≪ ǫ ≪ 1,

we rewrite it as

I1 = 2

1−ǫ

?

0

ydy

¯ y(ρ − ¯ y)[ρ − 1 + y(2 − ρ)]+ 2

ρlnρ

1

?

1−ǫ

dy

¯ yρ + 4σ

=

2

4σ−

2

ρ(ρ − 1)[(ρ − 1)ln(ρ − 1) + lnρ]. (A12)

For the second integral I2, which is infrared finite, the substitutions T = t2, y =t2−a

b, a = ρ−1, and a+b = 1 yield

I2 =

2

b

1

?

√a

?

√a

dt

t2ln

tb + t2− a

tb − (t2− a)=2b

1

?

√a

dt(1 − t)

t

?

b + 2t

tb + t2− a−

b − 2t

tb − t2+ a

?

= 4

1

dt

t2+ a

t(1 + t)(t2− a2)=2b[lna − 2ln2 +a + 1

a

ln(a + 1)].(A13)

The total answer for J1is

J1(ρ) = 2

?1

? q2

?2?1

ρln1

σ+

2

ρ(2 − ρ)(2lnρ − 2ln2 − ln(ρ − 1))

?

. (A14)

For the sum J = RJ1− J0, we obtain

J(R,? q2

2) =

1

2? q2

2− Rln

R2

4(R − ? q2

2)? q2

2

=

1

? q2

1+ M2

1

x − 1ln(x + 1)2

4x

,(A15)

where x = ? q2

For the integral I, we have

2/(? q2

1+ M2).

4I =

?

d2?k

π

1

?k2(? q −?k)2

(R1− D1)(R2− D2)

D1D2

= J0+

R2

2

R1− R2J1(R2) −

R2

1

R1− R2J1(R1),(A16)

where J1given in Eq. (A9) and R1,2= ? q2

1,2+ ? q2+ M2

1,2. The infrared singularity is canceled using

I =

1

R1− R2

?

R2

2? q2− R2ln

R2

2

4? q2(R2− ? q2)−

R1

2? q2− R1

ln

R2

1

4? q2(R1− ? q2)

?

.(A17)

Page 9

9

For the case of R1= R2= R = ? q2

1+ ? q2+ M2, we have

I(R,R) = −∂

∂R

?

d2k

π

?k(? q −?k)(R − D)

?k2(? q −?k)2D

= −4∂

∂RRJ(R,? q2) = 4

?

1

R − ? q2

1

−

2? q2

1− Rln

1

2? q2

R2

4? q2

1(R2− ? q2

1)

?

. (A18)

For the cross section of single vector meson production, integrated over ? q2

1and ? q2

2, we have

dσ(1)

dp2R(p2)

=

16π(Z1Z2α3)2A2

M2

∞

?

0

dx

(x + 1)2(x − 1)2ln2

?(x + 1)2

4x

??

Z2

1

1

?

m2

1

s

dβ1

β1

(1 − β1)

∞

?

0

dt · t

(1 + t)3(t + β2

1ρ2

1)

+ (β1→ α2,Z1↔ Z2,ρ1=m2

1

M2→ ρ2=m2

2

M2)

?

. (A19)

Using

∞

?

0

dx

(x2− 1)2ln2((x + 1)2

4x

) = 2(1 − ln2), (A20)

we obtain the expression given in Eq. (31).

Appendix B

In this section, we shall explain how to evaluate the integrals appearing in Eqs. (36) and (41) relevant for the

mechanisms based on vector meson (see Fig. 2a) and two-gluon (see Fig. 2b) exchange, respectively.

In the first case, we have

P(x1,x2;ρ1,ρ2) =

∞

?

0

dx · x2

(x + 1)2τ2, (B1)

with

τ =i(x1,r1)i(x2,r2)

r1r2

+i(x1, ¯ r1)i(x2, ¯ r2)

¯ r1¯ r2

,(B2)

where

i(x,r) =

1

2x − rln

r2

4x(r − x), (B3)

and r1= x + x1+ ρ1, r2 = x + x2+ ρ2, ¯ r1 = x + x1+ ρ2, ¯ r2= x + x2+ ρ1, ρ1= M2

Numerical values of P(x1,x2;1,1), appropriate for the important case ρ1= ρ2= 1, are listed in Table I.

On the other hand, we have

V1/M2

V, and ρ2= M2

V2/M2

V.

Φ(x1,x2;ρ1,ρ2) =

∞

?

0

∞

?

0

∞

?

0

dx

?j(r1,r2)

?j(¯ r1, ¯ r2)

r1r2

?2

?2

,

¯Φ(x1,x2;ρ1,ρ2) =

dx

¯ r1¯ r2

,

G(x1,x2;ρ1,ρ2) =

dxj(r1,r2)j(¯ r1, ¯ r2)

r1r2¯ r1, ¯ r2

,(B4)

where

j(r1,r2) =

1

r1− r2

?

r2

2x − r2ln

r2

2

4x(r2− x)−

r1

2x − r1

ln

r2

1

4x(r1− x)

?

.(B5)

Page 10

10

x1 \ x2

0.1

0.5

1

1.5

2

3

4

5

0.1

0.076

0.0047

0.00049

0.000082

0.00004 3.66 · 10−6

0.000046 2.51 · 10−6

0.000045 2.48 · 10−6

0.000038 2.23 · 10−6

0.511.52345

0.0047

0.00036

0.000049

0.00001

0.00049

0.000049

9.0049 · 10−62.34 · 10−67.85 · 10−72.52 · 10−72.136 · 10−72.03 · 10−7

2.34 · 10−6

7.39 · 10−72.78 · 10−76.46 · 10−83.357 · 10−82.85 · 10−8

7.85 · 10−7

2.78 · 10−71.29 · 10−74.59 · 10−8

2.52 · 10−7

6.46 · 10−84.59 · 10−84.02 · 10−8

2.14 · 10−7

3.36 · 10−82.52 · 10−83.21 · 10−8

2.029 · 10−7

2.85 · 10−81.76 · 10−82.45 · 10−8

0.000082

0.00001

0.00004

3.66 · 10−62.51 · 10−6

0.0000460.000045

2.48 · 10−6

0.000038

2.23 · 10−6

2.52 · 10−8

3.21 · 10−8

2.91 · 10−8

2.35 · 10−8

1.76 · 10−8

2.45 · 10−8

2.35 · 10−8

1.96 · 10−8

Table I: Values of the function P(x1,x2;1,1), defined in Eq. (B1), for different values of x1 and x2.

x1 \ x2

0.1

0.5

1

1.5

2

3

4

5

0.1

1.67

1.006

0.623

0.428

0.314

0.191

0.1297 0.08399 0.0559 0.04078 0.0314 0.02073

0.0940.0615 0.0414 0.0304 0.0235 0.0157 0.01142 0.0088

0.5

1.006

0.6142

0.386

0.268

0.198

0.123

11.5

0.428

0.268

0.173

0.1221 0.0918 0.0583 0.04078 0.0304

0.0918 0.0695 0.0446

0.0808 0.0583 0.0446

2345

0.623

0.386

0.246

0.173

0.129

0.314

0.198

0.129

0.191

0.123

0.0808

0.129

0.08399 0.0615

0.0559

0.094

0.0414

0.0314

0.02073 0.0157

0.149

0.0235

0.029

0.01142

Table II: Values of the function Φ(x1,x2;1,1) =¯Φ(x1,x2;1,1) = G(x1,x2;1,1), defined in Eq. (B4), for different values of x1

and x2.

In the case r1= r2, we have (see Eq. (A18))

j(r1,r1) = 4 −

8x

(x − 1)2ln

?(x + 1)2

4x

?

, x =

? q2

1

? q2+ M2.(B6)

For ρ1= ρ2, we have Φ(x1,x2;ρ1,ρ1) =¯Φ(x1,x2;ρ1,ρ1) = G(x1,x2;ρ1,ρ1). Numerical values for the important choice

ρ1= ρ2= 1 are listed in Table II.

Acknowledgments

We are grateful to V. Pozdnyakov for a discussion about the experimental situation at the LHC. We are grateful

toˇC. Burd´ ık for the taking part in initial stage of work. This work was supported in part by the German Federal

Ministry for Education and Research BMBF through Grant No. 05 HT6GUA and by the Helmholtz Association HGF

through Grant No. Ha 101. The work of E.A.K. was supported in part by Russian Foundation for Basic Research

RFBR through Grant No. 01201164165 and the Heisenberg-Landau Grant No. HLP-2012-11.

[1] A. B. Arbuzov, V. V. Bytev, E. A. Kuraev, E. Tomasi-Gustafsson, and Yu. M. Bystritskiy, Phys. Part. Nucl. 41, 593 (2010).

[2] I. F. Ginzburg, S. L. Panfil, and V. G. Serbo, Nucl. Phys. B 284, 685 (1987); I. F. Ginzburg, S. L. Panfil, and V. G. Serbo,

Nucl. Phys. B 296, 569 (1988).

[3] E. V. Kuraev, N. N. Nikolaev, and B. G. Zakharov, JETP Lett. 68, 696 (1998) [Pisma Zh. Eksp. Teor. Fiz. 68, 667 (1998)]

[hep-ph/9809539].

[4] G. T. Bodwin, E. Braaten, and G. P. Lepage, Phys. Rev. D 51, 1125 (1995) [Erratum-ibid. D 55, 5853 (1997)]

[hep-ph/9407339].

[5] G. S. Adkins and Y. Shiferaw, Phys. Rev. A 52, 2442 (1995).

[6] M. V. Terentev, Sov. J. Nucl. Phys. 24, 106 (1976) [Yad. Fiz. 24, 207 (1976)]; V. B. Berestetsky and M. V. Terentev, Sov.

J. Nucl. Phys. 25, 347 (1977) [Yad. Fiz. 25, 653 (1977)]. S. P. Klevansky, Rev. Mod. Phys. 64, 649 (1992).

Page 11

11

γ

γ

γ

p2,z2

p1,z1

V

γ

γ

γ

p2,z2

p1,z1

V

a.) b.)

V

=

+ ··· ∗

V

c.)

Figure 1: Feynman diagrams pertinent to single vector meson production in peripheral heavy-ion collisions.

γγ

V

γγ

2

1

γγ

V

γγ

1

2

a.)

γ

gg

γ

2

1

γ

gg

γ

1

2

b.)

Figure 2: Feynman diagrams pertinent to two vector meson production in peripheral heavy-ion collisions via intermediate a)

vector meson and b) two-gluon states.