Table Of ContentThe BFKL Pomeron in Deep Inelastic Diffractive Dissociation near t = 0
J. Bartels, H.Lotter and M.Wu¨sthoff
II. Institut fu¨r Theoretische Physik, Universita¨t Hamburg
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3 Abstract: The small-t behaviour of the deep inelastic diffractive dissociation cross section in
1
0
the triple Regge region is investigated, using the BFKL approximation in perturbative QCD. We
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/ show that the cross section is finite at t = 0, but the diffusion in lnkt2 leads to a large contribution
h
p
of small momenta at the triple Pomeron vertex. We study the dependence upon the total energy
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and the invariant mass. At t = 0, there is a decoupling of the three BFKL singularities which is a
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v consequence of the conservation of the conformal dimension. For large invariant masses, the four
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gluon state in the upper t-channel plays an important role and cannot be neglected.
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1 Introduction
ThestudyofperturbativeQCDinthetriple-Regge limithasrecently attracted someinterest [1,2].
If one interpretes the observed strong rise of F at small x as a signal for the BFKL-Pomeron [3],
2
it is natural to ask for corrections to this new piece of perturbative QCD, and an obvious place to
look for such terms is the triple Regge limit. There may also be some interest in this limit from
an experimental point of view: some of the observed “rapidity gap events“ [4, 5] may belong to a
kinematical region where perturbative QCD is applicable.
In [2] an attempt has been made to derive an analytic formula for the triple Regge inclusive
cross section which lies at the same level of accuracy as the BFKL Pomeron. The result was given
in a somewhat abstract form, and, so far, only a few rather general properties have been studied.
The general structure of the cross section formula is illustrated in Fig.1a: starting from the top,
the fermion box firstcouples to a BFKL-ladder. At the transition: two gluons four gluons a new
→
vertex function appears. Below this vertex the four-gluon state starts where the gluons interact
pairwise in all possible ways. Finally, the four-gluon state branches into the two BFKL ladders
at the bottom. In addition to this general structure, there is also a contribution where the upper
BFKL-ladder couples directly to the lower ones (Fig.1b).
Based upon the experience with BFKL Pomeron, we expect that the formalism developed in
[2] is suited to study the whole range of momentum transfer t of the diffractive dissociation cross
section (provided that √ tis still smaller than M, the missingmass of producedhadronicsystem).
−
Nevertheless, there are several reasons to believe that the point t = 0 plays a very special role and
perturbation theory may even not be applicable at all. Firstly, the early hard scattering approach
of ref. [6, 7] (and similar calculations later on) has shown that the large transverse momenta at the
twogluons fourgluons-vertex(fig. 1a)aresuppressedlikedk2/k4,i.e. smalltransversemomenta
→ t t
dominate. Without invoking further corrections, the cross section would diverge at k2 = 0: in the
t
framework of the GLR equation which leads to a saturation of the Pomeron it is the unitarity
corrections (screening) to the lower Pomerons which provide the necessary supression at small k2.
t
Since the saturation begins at a rather large momentm scale (2 4GeV2, depending upon the
−
2
ratios M2/s [7]), this mechanism tends to predict hard final states. Secondly, also within BFKL
physics the point t = 0 is exceptional: whithin the lower Pomerons the diffusion in lnk2 extends
t
into both the ultraviolet and the infrared regions, whereas for t = 0 the diffusion into the infrared
6
region where perturbation theory becomes unreliable is stopped by the momentum scale t. One
therefore expects, for the point t = 0, the “dangerous“ region of small k to play a much more
t
important role than for the case t = 0. This expectation has recently been confirmed by Mueller
6
[1], usingthelarge-N approximation. As a resultof this approximation, thefour gluon state above
c
the triple Pomeron vertex is absent, and one is lead directly to a study of the diagrams shown in
Fig.1b. The final result of this study is an explicit formula for the inclusive cross section, showing
the dependenceupon the energy variables s, M, and the momentum transfer t. The latter one is of
particular interest: for an intermediate t-region, the cross section goes as 1/√ t. This behaviour
−
hints at some sort of singular behaviour at t = 0, in agreement with what one might expect in the
diffusion picture. From this study, however, it is not clear what happens at t = 0, in particular,
whether the whole perturbative analysis breaks down or not.
The more general reason why, from the theoretical point of view, it is important to understand
the small-t behaviour of the diffractive dissociation cross section is the problem of unitarization. It
is well-known that the sum of the leading logarithms at sufficiently large energies (or sufficiently
smallx runsintoconflictwithunitarity, andtheway inwhichunitarity isrestoredisstill anopen
Bj
question. Diffractive dissociation is not contained in the leading logarithmic approximation and,
therefore, represents a (observable) correction which contributes to the unitarization procedure.
Within the GLR-equation, the finite limit of the cross section at t = 0 requires the complete
unitarization (saturation) of the lower Pomerons. Similarly, in [12] it was shown that, while
single unitarity corrections have large infrared contributions, their resummation leads to strong
cancellations. Inbothcases, oneexpectsthefinalstate toberather hard. Theprocessofdiffractive
dissociation may therefore become a very sensitive tool in exploring the unitarization mechanism.
For example, one might count the number of events in dependence on a lower cutoff on k of
t
the final state or, alternatively, measure the momentum transfer t. At which scale the saturation
3
occurs is still an open question, since the GLR-equation is only a crude estimation compared to
a complete procedure of unitarization. The HERA data will soon show, whether a value for k2 of
t
about 2 4GeV2 is justified.
−
In this paper, we are going to investigate the small t-region of the cross section formula of [2],
in particular the point t = 0. As one of the main results of this paper we will show that the BFKL
approach to the diffractive dissociation, in spite of the dk2/k4-behaviour at large k , is infrared
t t t
safe, and the limit t = 0 exists. At the same time, however, the BFKL diffusion has entered into
the infrared region and thus emphasizes the need to include unitarizing corrections to the leading
logarithmic approximation. We believe that this observation is important from the point of view
of theoretical consistency: it shows that, within the BFKL approach, a smooth transition from a
finite t down to t = 0 is possible without running into infrared singularities. Consequently, the
approximation usedin [2]represents awell-defined startingpointfor approachingtheunitarization
problem. Whether the (leading logarithmic) formula of [2] can already be used for deducing
experimental signatures remains less clear. It is encouraging that a rough estimate of the ratio of
diffractive events over all DIS-events gives a reasonable value. Furthermore, HERA-data, so far,
seem to support the dominance [9] of events with low k : this is in qualitative agreement with the
t
strong diffusion into the infrared region. Therefore, it seems worthwile to study the dynamics of
the unscreened BFKL-Pomeron in more detail, even before adressing the question of unitarization.
Certain characteristics may very well survive the unitarization procedure, and it is important to
check whether they may serve as signals to support or rule out the BFKL-dynamics.
Apart from the result that the BFKL cross section formula has a finite limit at t = 0 our
analysis contains a detailed saddle point analysis and investigates, as a function of t near t = 0,
the dependence upon s, M2, and Q2. For t of the order of Q2, our analysis confirms the 1/√ t
| | −
behaviourfoundin [12]inthelarge-N -approximation. Movingtowardssmallert-values,theshape
c
of the t-distribution changes, and the cross section reaches, at t = 0, a finite limit. At the same
time, the derivative with respect to t tends to infinity. For small t one observes a shrinkage, i.e. the
cusp becomes narrower as s (the typical width shrinkswith some inverse power ofs/M2). All
→ ∞
4
these changes as a function of t are accompanied by a very peculiar dependence upon s and M2.
In particular, at t = 0 one observes a decoupling of the lower BFKL Pomerons from the BFKL
singularity above. Much of this striking behaviour can be traced back to the conformal invariance
of the BFKL approximation: at t = 0 we find a conservation law of the conformal dimension of the
BFKL ladders above and below the triple Pomeron vertex.
Our paper will be organized as follows. We begin with the simplest case, the diffractive pro-
duction of a qq¯-pair in the triple Regge region near t = 0. This simple case already shows the main
result, namely theconservation law of conformal dimensions and its implication for thehigh energy
behavior. The advantage of first presenting this simpler case lies if the fact that we are able to
present an analytic expression for the M2-integrated cross section which can directly be used for
a comparison with observed event rates. The discussion will first be done in momentum space; in
the subsequent section we repeat the derivation in coordinate space where a more intuitive picture
has been developed [12, 13]. The generalization to the production of qq¯+gluons will be described
in Section 4; since the analysis presented in this part will be rather technical, we shall give a short
summary at the end of this section. In the final section we discuss a few implications of the results
of this paper.
2 Diffractive Production of qq¯-Pairs Near t = 0
WebeginwiththeM2-integrated crosssection fortheprocess(Fig.2a) γ∗+proton (qq¯)+proton,
→
where M is the invariant mass of the quark pair, t the square of the momentum transferred from
the proton to the quark pair, and 1/x = s/Q2 the total energy. We are interested in the limit of
B
small x and keep t as a small variable parameter.
B
Following the notation of [2] we use the integral representation
dσ α (2π)3 dω dω 1 ω1+ω2
= e2 em 1 2 F(ω ,ω ,t), (2.1)
dt f 8πQ4 2πi 2πi x 1 2
f Z Z (cid:18) B(cid:19)
X
where the partial wave consists of the three building blocks illustrated in Fig.2b : at the upper end
(1;++)
we have the four-gluon amplitude D , below the two BFKL Pomerons, and at the bottom we
(4;0)
use a form factor for the coupling of the BFKL ladders to the proton. The four-gluon amplitude
5
has been studied in [2]. It can be rewritten as a sum of two-gluon amplitudes:
√2
D(1;+,+)(k ,k ,k ,k ) = g2
(4,0) 1 2 3 4 3 ·
D (k ,k +k +k ) + D (k ,k +k +k ) (2.2)
(2;0) 1 2 3 4 (2;0) 2 1 3 4
n
+D (k ,k +k +k ) + D (k ,k +k +k )
(2;0) 3 1 2 4 (2;0) 4 1 2 3
D (k +k ,k +k ) D (k +k ,k +k ) D (k +k ,k +k ) .
(2;0) 1 2 3 4 (2;0) 1 3 2 4 (2;0) 1 4 2 3
− − −
o
For D (k, k) it is convenient to use a Mellin transform with respect to the variable k2/Q2:
(2;0)
−
−µ
dµ k2
D (k2) = D˜ (µ), (2.3)
(2;0) 2πi Q2! (2;0)
Z
where the µ-contour runs along the imaginary axis, intersecting the real axis within the interval
( 1,0) (in thefollowing we shalluse, as theintersection with therealaxis, thepoint 1/2; we shall
− −
then use the notation µ = 1/2 iν). The function D˜ (µ) has poles at positive and negative
(2;0)
− −
integers, and a detailed discussion is contained in [2]. In this paper we only need the behaviour
near µ = 1
−
D˜ e2fαs√84 1 (2.4)
(2;0) ≈ 2π 3(µ+1)2
f
X
and near µ = 1:
−2
e2α 9√2π2
D˜ f s (2.5)
(2;0)
≈ 16
f
X
Next we turn to the BFKL Pomeron. Since we want to study the t- dependence for t = 0 we
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need an expression for the BFKL-pomeron for non-zero momentum transfer. The BFKL-pomeron
is determined by a Bethe-Salpeter type of equation in two dimensional transverse space. Lipatov
[14] has shown that the configuration space representation of this equation is invariant under two
dimensionalconformaltransformations. Duetothissymmetryitcanbediagonalizedbyaconformal
partial wave expansion. Using orthonormality and completeness of the conformal partial waves
Lipatov found an analytic expression for the sum of the nonforward ladders. By a straightforward
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Fourier transfomation, this expression leads to the following momentum representation 1:
+∞ dν 1
Φ (k,k′,q) = E(ν)(k,q k)E(ν)∗(k′,q k′) (2.6)
ω
−∞ 2πω χ(0,ν) − −
Z −
where we have restricted ourselves to zero conformal spin. The eigenvalues χ(0,ν) of the BFKL-
kernel are given by:
g2N 1 1
c
χ(0,ν) = [2ψ(1) ψ( +iν) ψ( iν)] (2.7)
4π2 − 2 − 2 −
The conformal partial waves have the momentum representation
E(ν)(k,q−k) = 44−πiν ΓΓ(1(+22iνiν))ΓΓ(−(312+−iνiν))ΓΓ22((112 −+iiνν)) d2ρ1d2ρ2eikρ1+i(q−k)ρ2(ρρ221ρ22)21−iν
− 2 2 Z 1 2
Γ(1+2iν)Γ( 1 iν)Γ(3 iν)
= 2π −2 − 2 −
Γ( 2iν) Γ(1 +iν) Γ(1 +iν)
− 2 2
1
dx[x(1 x)]−12+iν[q2x(1 x)+(k xq)2]−32−iν
· − − −
Z0
3 1 (k xq)2
F ( +iν,iν ,1; − ) (2.8)
·2 1 2 − 2 q2x(1 x)+(k xq)2
− −
The normalization was chosen in such a way 2 that in the limit q =0 the expression (2.6) coincides
with the familiar BFKL-Pomeron in the forward direction:
Φω(k,k′,q = 0) = 2(2π)2 +∞ dν 1 (k2)−32−iν(k′2)−32+iν (2.9)
−∞ 2πω χ(0,ν)
Z −
In (2.8) one has to be careful in taking the limit q 0, namely making use of the well known
→
properties of the hypergeometric functions one finds
E(ν)(k,q 0) = 2π[(k2)−3/2−iν + C(ν)(k2)−3/2+iν(q2)−2iν] (2.10)
→
where C(ν) is analytic in the range 1/2 < Im(ν) < 1/2 and has the property C( ν) = 1/C(ν).
− −
Hence, E(ν) remains finite at q = 0 only if Im(ν) > 0. If Im(ν) < 0, E(ν) becomes infinte. For
E(ν)∗ the converse is true. So, one of the two factors, E(ν) or E(ν)∗, becomes infinte, no matter
what value of ν we choose. However, in (2.6) only the product of the two E(ν)-functions ap-
pears: E(ν)(k,q 0)E(ν)∗(k′,q 0) = (2π)2[(k2)−3/2−iν(k′2)−3/2+iν +(k2)−3/2+iν(k′2)−3/2−iν +
→ →
1Our normalization differs from Lipatov’s one by factors of 2π which are included in the integration measure in
momentum space
2It differs from Lipatov’s functions by some Γ factors.
7
C(ν)2(k2)−3/2−iν(k′2)−3/2−iν(q2)2iν+1/C(ν)2(k2)−3/2+iν(k′2)−3/2+iν(q2)−2iν. In the third and the
fourth term, using the variable µ = 1/2 iν, we have to shift the contour of integration to
− −
the left and to the right, respectively. Since C(ν) is analytic, both terms vanish. In the sec-
ond term we change from ν to ν. As a result we have dν E(ν)(k,q 0)E(ν)∗(k′,q
− → →
0) = 2(2π)2 dν (k2)−3/2−iν(k′2)−3/2+iν, i. e. we get theR right answer, if we simply write
E(ν)(k,q 0)R= √22π (k2)−3/2−iν. In the following, whenever we take the limit q 0, we shall
→ →
use this effective prescription.
We mention a few properties of (2.8). For q = 0, the limit k 0 is singular [15]. By explicit
6 →
calculation one finds:
1 12+iν Γ(1+2iν)Γ( 1 iν)Γ(1 iν) q k
E(ν)(k,q k) = 2πδ(2)(k) −2 − 2 − +O( · ) (2.11)
− q2 Γ( 2iν) Γ(3 +iν)Γ(1 +iν) k2
(cid:18) (cid:19) − 2 2
The delta-function term is dictated by the conformal invariance of the BFKL-kernel, and it does
not contribute if the BFKL-Pomeron is coupled to an external color singlet state which vanishes
as k 0 (or q k 0). A useful regularization of the k = 0 - limit is obtained if in (2.8) we
→ − →
introduce a nonzero conformal dimension λ for the (reggeized) gluon field [16] :
4−iν Γ(1+2iν)Γ( 1 iν)Γ2(1 iν)
E(ν,λ)(k,q k) = −2 − 2 −
− 2π Γ( 2iν) Γ(3 +iν) Γ2(1 +iν)
− 2 2
d2ρ d2ρ eikρ1+i(q−k)ρ2 (ρ212)21−iν−λ
·Z 1 2 (ρ21)12−iν(ρ22)21−iν
Γ(1+2iν)Γ( 1 iν)Γ(3 +iν λ)Γ(3 iν λ)
= 41−λπ −2 − 2 − 2 − −
Γ( 2iν) Γ(3 +iν) Γ2(1 +iν)
− 2 2
1
dxx−12+iν(1 x)−21+iν[q2x(1 x)+(k xq)2]−32+iν+λ
· − − −
Z0
3 1 (k xq)2
F ( +iν λ, +iν +λ,1; − ) (2.12)
·2 1 2 − −2 q2x(1 x)+(k xq)2
− −
For small k2, one obtains :
Γ(1+2iν) Γ( 1 iν)
E(ν,λ)(k,q k) = 41−λπ −2 − Γ(1+2iν)Γ(1 2iν)
− Γ( 2iν) Γ(3 +iν)Γ2(1 +iν) −
− 2 2
1 12+iν(k2)−1+λ i Γ2(1−λ) +c.c. +O(q·k)(2.13)
·(cid:18)q2(cid:19) "2ν Γ(32 +iν −λ)Γ(−12 −iν +λ) # k2
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which in the limit λ 0 leads us to (2.11), provided we identify :
→
1 1−λ
lim 21−2λ λ = δ(2)(k) (2.14)
λ→0 k2 ·
(cid:18) (cid:19)
Finally, wenotefrom(2.6)that,atfixedν,theBFKL-pomeronfactorizesinmomentumspace. This
property is not present in configuration space where the pomeron is a function of two anharmonic
ratios which link together the primed and unprimed coordinates [17]. As a by-product of this
calculation, we find the conformal partial waves in a mixed-representation which we give here for
later use :
1 Γ(1+2iν)Γ( 1 iν) 1
E(ν)(ρ ,q) = −2 − (q2)−iνρ
12 2 Γ( 2iν) Γ(3 +iν) Γ2(1 +iν) 12
− 2 2
1
dx[x(1 x)]−12 e−iq·ρ12(1−x)K−2iν(q ρ12 x(1 x)) (2.15)
− | || | −
Z0 q
Finally, forthe(nonperturbative)couplingoftheBFKL ladderstotheprotonweuse,as aguide
for the dependence upon a hadronic scale Q2, the following simple model (at the point q= 0):
0
k2
V(k2) = C (2.16)
k2+Q2
0
where Q2 denotes a hadronic scale of the order of 1GeV2. As to dependence upon t and ν, we shall
0
assume that C is a slowly varying function.
Returning to (2.1) and putting together all these ingredients, we arrive at the following expres-
sion for the partial wave F:
d2l d2m
(1;++)
F(ω ,ω ,t) = D (l,q l,m, m q)
1 2 (2π)3 (2π)3 (4;0) − − −
Z Z
d2l′
′ ′ ′ ′
Φ (l,q l,l,q l)V(l,q l)
· (2π)3 ω1 − − −
Z
d2m′
′ ′ ′ ′
Φ (m, q m,m, q m)V(m, q m) (2.17)
· (2π)3 ω2 − − − − − −
Z
where q2 = t denotes the square of the momentum transfer, and Φ and V are given in (2.6),
−
(2.16), resp..
We now turn to (2.17) and study its dependence upon t, near t = 0. We begin with the
integrations in l and m. For D we use (2.3), (2.3), and from the two lower Pomerons we have
(4;0)
9
the two E(ν)-functions. This defines the vertex g (q2):
µµ1µ2
d2ld2m
(l2)−µ+((l q)2)−µ+(m2)−µ+((m+q)2)−µ (q2)−µ ((l+m)2)−µ
(2π)6 − − −
Z h
(q2)µ1+µ2−µ
((l m q)2)−µ E(ν1)(l,l q)E(ν2)(m, m q) = g (q2) (2.18)
− − − − − − µ +µ µ µµ1µ2
1 2
i −
where we have used µ = 1 iν . In course of performing the integrals over l and m one finds
i −2 − i
that, out of the seven terms on the lhs, the result will come from the last two terms only; the other
serve as regulators in either the infrared or the ultraviolet region. Rather than presenting details
of the calculations, we shall limit ourselves to a brief description of the major steps. First, for the
E(ν) functions weuse therepresentation (2.8): thehypergeometric functions are written as a power
series in their arguments, and we consider term by term. It is convenient to perform the shifts
l l qz , m m+qz where z and z denote the x-parameters of the representation (2.8) for
1 2 1 2
→ − →
thetwoBFKL-Pomerons. Now itis notdifficulttoperformtheintegration over l: oneendsupwith
a string of terms consisting of one dimensional finite integrals involving hypergeometric functions
and powers. For the remaining integral over m one observes that convergence in the ultraviolet
region holds as long as the condition
µ > µ +µ (2.19)
1 2
is satisfied (µ = 1 iν ), i.e. the µ integration contour is to the right of µ +µ . In the infrared
i −2 − i 1 2
region, q2 serves as a regulator. Consequently, the lhs of expression (2.18), in the neighbourhood
of µ = µ +µ , behaves as
1 2
(q2)µ1+µ2−µ
g (q2), (2.20)
µ +µ µ · µµ1µ2
1 2
−
and the remaining vertex g (q2) has a finite limit as q2 0:
νν1ν2 →
2π2 Γ(1 µ) Γ(µ )Γ(µ )
1 2
g (0) = − (2.21)
µµ1µ2 (2π)6 Γ(µ) Γ(1 µ )Γ(1 µ )
1 2
− −
We mention that (2.18) can also be evaluated at q2 = 0 directly. The result is
(2π)δ(µ +µ µ)g (0) (2.22)
1 2− µµ1µ2
10
