We present the predictions of a model for proton-proton total cross-section
at LHC. It takes into account both hard partonic processes and soft gluon
emission effects to describe the proper high energy behavior and to respect the
Froissart bound.Comment: 7 pages, 3 figures, presented at MPI08, Perugia, October 27-31, 200

Achilli, A.

Godbole, R.

Grau, A.

Pancheri, G.

Srivastava, Y. N.

English

arXiv

We present the predictions of a model for proton-proton total cross-section at LHC. It takes into account both hard partonic processes and soft gluon emission effects to describe the proper high energy behavior and to respect the Froissart bound

Publications of the IAS Fellows

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QCD Mini-jet contribution to the total cross section
A. Achilli1†, R. Godbole2, A.Grau3, G. Pancheri4, Y.N. Srivastava1
1INFN and Physics Department, University of Perugia, I-06123 Perugia, Italy
2 Centre for High Energy Physics, Indian Institute of Science,Bangalore, 560012, India
3Departamento de Fı´sica Teo´rica y del Cosmos, Universidad de Granada, 18071 Granada, Spain
4INFN Frascati National Laboratories, I-00044 Frascati, Italy
Abstract
We present the predictions of a model for proton-proton total cross-
section at LHC. It takes into account both hard partonic processes and
soft gluon emission effects to describe the proper high energy behavior
and to respect the Froissart bound.
1 Introduction
A reliable prediction of the total proton-proton cross section is fundamental to know which will
be the underlying activity at the LHC and for new discoveries in physics from the LHC data. In
this article, we shall describe a model [1] [2] for the hadronic total cross section based on QCD
minijet formalism. The model includes a resummation of soft gluon radiation which is necessary
to tame the fast high-energy rise typical of a purely perturbative minijet model. It is called the BN
model from the Bloch and Nordsiek discussion of the infrared catastrophe in QED. In the first
section, results are presented concerning the behavior of the QCD minijet cross section. It will
then be explained how this term is included into an eikonal formalism where infrared soft gluon
emission effects are added. The last section is devoted to the link between the total cross-section
asymptotic high energy behavior predicted by our model and the model parameters. This relation
also shows that our prediction is in agreement with the limit imposed by the Froissart bound.
2 Mini-jet cross section
Hard processes involving high-energy partonic collisions drive the rise of the total cross section
[3]. These jet-producing collisions are typical perturbative processes and we can describe them
through the usual QCD expression:
σABjet (s, ptmin) =
∫ √s/2
ptmin
dpt
∫ 1
4p2t /s
dx1
∫ 1
4p2t/(x1s)
dx2 ×
∑
i,j,k,l
fi|A(x1, p
2
t )fj|B(x2, p
2
t )
dσˆklij (sˆ)
dpt
,
(1)
with A,B = p, p¯. This expression depends on the parameter ptmin which represents the min-
imum transverse momentum of the scattered partons for which one allows a perturbative QCD
treatment. Its value is usually around ≈ 1 − 2 GeV and it distinguishes hard processes (that
are processes for which a perturbative approach is used) from the soft ones that dominate at low
† speaker
energy, typically for
√
s ≤ 10 ÷ 20 GeV , i.e, well before the cross-section starts rising. The
Minijet expression also depends on the DGLAP evoluted Partonic Densities Functions fi|A for
which there exist in the literature different LO parameterizations(GRV, MRST, CTEQ [4]). We
obtain an asymptotic growth of σjet with energy as a power of s. As shown in figure 1, the value
of the exponent depends on the PDF used and one has
σGRVjet ≈ s0.4 σMRSTjet ≈ s0.3 σCTEQjet ≈ s0.3.
This result can be derived by considering the relevant contribution to the integral in (1) in the√
s >> ptmin limit. In this limit, the major contribution comes from the small fractions of
momentum carried by the colliding gluons with x1,2 << 1. In this limit we know that the relevant
PDF’s behave approximately like powers of the momentum fraction x−J with J ∼ 1.3 [5].
From the previous consideration and noting that dσˆ
kl
ij
(sˆ)
dpt
∝ 1
p3t
we obtain from (1) the following
asymptotic high-energy expression for σjet:
σjet ∝ 1
p2tmin
[
s
4p2tmin
]J−1
. (2)
The dominant term is just a power of s and the estimate obtained for the exponent ǫ = J−1 ∼ 0.3
is in agreement with our previous results. We now need to understand how to incorporate into
a model for the total cross section this very fast rise at very high energy, which is present in
the perturbative regime. Firstly it is important to note that σjet is an inclusive cross section and
therefore contains in itself a multiplicity factor, linked to the average number < n > of partonic
collisions that take place during the hadronic scattering. We can approximate the energy driving
term at high energy [6] < n > as
< n >≈ σjet · A, (3)
where A is a function representing the overlap between the two hadrons.
Now we can derive an expression for the total cross section as a function of< n >. Assum-
ing that the number of partonic collisions follows a Poisson distribution, since each interaction is
independent from the other, the probability of having k partonic collisions is:
P (k,< n >) =
< n >k e−<n>
k!
. (4)
The average number of partonic collisions should depend on the energy and on the impact pa-
rameter b relative to the hadronic process < n >≡< n(b, s) >. From the previous expression it
is possible to obtain the inelastic hadronic cross section:
σinelastic =
∫
d2b
∑
k=1
P (k,< n(b, s) >) =
∫
d2b
[
1− e−<n(b,s)>
]
, (5)
which is the usual eikonal expression if we consider the link between < n(b, s) > and the eikonal
χ(b, s):
< n(b, s) >= 2Imχ(b, s). (6)
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
1 10 10
2
10
3
10
4
√s ( GeV )
s
jet
(m
b)
s jet  for various LO densities
green band MRST
GRV
GRV98
CTEQ
ptmin=1.15 GeV
Fig. 1: minijet cross section for different input parton densities.
3 Eikonal model
The eikonal representation allows to implement multiple parton scattering and to restore a finite
size of the interaction. Neglecting the real part of the eikonal function, an acceptable approxima-
tion in the high energy limit, the expression for the total cross section is
σtot = 2
∫
d2b
[
1− e−n(b,s)/2
]
. (7)
The average number of partonic collisions receives contributions both from hard and soft physics
processes and we write it in the form
n(b, s) = nsoft(b, s) + nhard(b, s), (8)
where the soft term parameterizes the contribution of all the processes for which the partons
scatter with pt < ptmin. It is the only relevant term at low-energy and it establishes the overall
normalization, while the hard term is responsible for the high-energy rise. From (3), we approx-
imate this term with
nhard(b, s) = A(b, s)σjet(s), (9)
where the minijet cross section drives the rise due to the increase of the number of partonic
collisions with the energy and A(b, s) is the overlap function which depends on the (energy
dependent) spatial distribution of partons inside the colliding hadrons. In some older models
[6] a simpler factorized expression for n(b, s) was used, with the overlap function depending
only on b. However, when up-to-date realistic parton densities are used, such impact parameter
distributions, inspired by constant hadronic form factors, led to an excessive rise of σtot with the
energy. In our BN model we include an s-dependence in the overlap function that has to tame
the strong growth due to the fast asymptotic rise of σjet [2].
We identify soft gluon emissions from the colliding partons as the physical effect respon-
sible for the attenuation of the rise of the total cross section. These emissions influence matter
distribution inside of the hadrons, hence changing the overlap function. They break collinearity
between the colliding partons, diminishing the efficiency of the scattering process. The number
of soft emissions increases with the energy and this makes their contribution important, also at
very high energy. The calculation of this effect uses a semiclassical approach based on a Block-
Nordsieck inspired formalism [7], the basic assumption of this technique is that all emissions
are independent from each other, so the number of gluons emitted follows a Poisson distribu-
tion. Thereof one obtains a distribution of the colliding partons as function of the transverse
momentum of the soft gluons emitted in the collision, i.e.
d2P (K⊥) = d
2
K⊥
1
(2π)2
∫
d2b eiK⊥·b−h(b) , (10)
the factor h(b) is given by
h(b) =
∫
d3ng(k)[1− e−ik⊥·b] =
∫
d3k
2k0
∑
m,n=colors
|jµ,m(k)jµ,n(k)|[1− e−ik⊥·b],
where d3ng(k) is the distribution for single gluon emission in a scattering process and it is linked
to the QCD current jµ responsible for emission.
We have proposed to obtain the overlap function as the Fourier transform of the previous
expression of the soft gluon transverse momentum resummed distribution, namely to put
ABN (b, s) = N
∫
d2K⊥ e
−iK⊥·b
d2P (K⊥)
d2K⊥
=
e−h(b,qmax)∫
d2b e−h(b,qmax)
, (11)
with
h(b, qmax) =
16
3
∫ qmax
0
αs(k
2
t )
π
dkt
kt
log
2qmax
kt
[1− J0(ktb)], (12)
this integral is performed up to a maximum value which is linked to the maximum transverse
momentum allowed by the kinematics for a single gluon emitted, qmax [8]. In principle, this
parameter and the overlap function should be calculated for each partonic sub-process, but in
the partial factorization of Eq.(9) we use an average value of qmax obtained considering all the
sub-processes that can happen for a given energy of the main hadronic process [2]:
qmax(s) =
√
s
2
∑
i,j
∫ dx1
x1
∫ dx2
x2
∫ 1
zmin
dzfi(x1)fj(x2)
√
x1x2(1− z)
∑
i,j
∫ dx1
x1
∫ dx2
x2
∫ 1
zmin
dzfi(x1)fj(x2)
, (13)
with zmin = 4p2tmin/(sx1x2). Notice that consistency of the calculation requires that the PDF’s
used in Eq. (13) be the same as those used in σjet. In Fig.2 are presented our results for qmax as
function of
√
s using ptmin = 1.15 GeV.
0
0.5
1
1.5
2
2.5
3
1 10 10
2
10
3
10
4
 √(s) (GeV)
q m
a
x 
(G
eV
)
qmax FOR HARD PROCESSES
FOR VARIOUS LO DENSITIES
GRV
GRV98
CTEQ
MRST
Fig. 2: Results for qmax using different input parton densities with ptmin = 1.15 GeV.
The integral in (12) has another relevant feature, it extends down to zero momentum val-
ues, and to calculate it we have to take an expression of αs different from the perturbative QCD
expression which is singular and not integrable in (12). We use a phenomenological expres-
sion [9], which coincides with the usual QCD limit for large kt, and is singular but integrable for
kt → 0:
αs(k
2
t ) =
12π
33− 2Nf
p
ln[1 + p(ktΛ )
2p]
. (14)
This expression for αs is inspired by the Richardson expression for a linear confining potential
[10], and we find for the parameter p that
• p < 1 to have a convergent integral (unlike the case of the Richardson potential where
p = 1)
• p > 1/2 for the correct analyticity in the momentum transfer variable.
Fig.3 [1] shows our predictions, obtained for the total cross-section using a set of phenomeno-
logical values for ptmin and p, and varying the parton densities. We also make a comparison with
data and other current models.
40
60
80
100
120
140
10 10
2
10
3
10
4
√s ( GeV )
s
to
t(m
b) G.G.P.S. model, PRD 72, 076001 (2005)
 using GRV and MRST P.D.F.
 G.G.P.S. model, using GRV P.D.F.
s 0=48.0 mb p=0.75 ptmin=1.15
Cudell et. al. hep-ph/0612046a
Luna-Menon, hep-ph/0105076b
c Block-Halzen PRD 73 054022 (2006)
Donnachie-Landshoff, PRL B296 227 (1992)d
Donnachie-Landshoff, PLB 595 393 (2004)DLhp
b
a
c
d
DLhp
proton-antiproton
UA5
UA1
UA4
CDF
E710
E811
proton-proton
Fig. 3: Results from our total cross-section model (for different parton densities) compared with data [11] and with
other models [12].
4 Restoration of Froissart Bound
The Froissart Martin Bound [13] states that σtot cannot rise faster than a function which is pro-
portional to log2(s). In order to see that in our model this bound is respected, we approximate
our total cross section at very large energies as
σtot ≈ 2π
∫
db2
[
1− e−nhard(b,s)/2
]
, (15)
with nhard(b, s) ≈ σjet(s)Ahard(b, s). We then take for σjet the asymptotic high energy expres-
sion:
σjet = σ1
(
s
GeV 2
)ε
,
with σ1 =constant and ǫ ∼ 0.3− 0.4. Being Ahard(b, s) ∝ e−h(b,s), we can consider in (12) the
infrared limit kt → 0 where the integral receives the dominant contribution. In this limit we have
αs(k
2
t ) ≈
(
Λ
kt
)2p
,
apart from logarithmic terms. Then, with h(b, s) ∝ (bΛ¯)2p [2] (again apart from logarithmic
terms), we have
Ahard(b) ∝ e−(bΛ¯)2p ,
and from this expression
nhard = 2C(s)e
−(bΛ¯)2p ,
with C(s) = A0σ12
(
s
GeV 2
)ε
. The very high energy limit of Eq. (15) then gives
σtot ≈ 2π
∫ ∞
0
db2[1− e−C(s)e−(bΛ¯)
2p
]→
[
ε ln
(
s
GeV 2
)]1/p
. (16)
The asymptotic growth of σtot in our model depends on the parameter ǫ which fixes the asymp-
totic rise of the minijet cross section, and on p which modulates the infrared behavior of αs.
Notice that 1/2 < p < 1 and thus this approximated result links the restoration of the Froissart
bound in our model with the infrared behavior of αs. We can now understand why a knowledge
of the confining phase of the strong interaction is necessary if we want to restore the finite size
of the hadronic interaction.
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QCD Mini-jet contribution to the total cross section

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