A model for total cross-sections incorporating QCD jet cross-sections and
soft gluon resummation is described and compared with present data on $pp$ and
${\bar p}p$ cross-sections. Predictions for LHC are presented for different
parameter sets. It is shown that they differ according to the small x-behaviour
of available parton density functions.Comment: 10 pages, 3 figures, uses appolb.cls, LaTeX. Talk presented by G.
  Pancheri at the EURIDICE Final Meeting, August 24-27th, 2006, Kazimierz,
  Polan

Godbole, R. M.

Grau, A.

Pancheri, G.

Srivastava, Y. N.

Acta Physica Polonica B

English

arXiv

A model for total cross-sections incorporating QCD jet cross-sections

and soft gluon resummation is described and compared with present data on

pp and p¯p cross-sections. Predictions for LHC are presented for different

parameter sets. It is shown that they differ according to the small x behaviour

of available parton density functions

Publications of the IAS Fellows

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Total cross-section at LHC from minijets and soft gluon
resummation in the infrared region∗
G. Pancheri
INFN Frascati National Laboratories, P.O. Box 13, I00044 Frascati, Italy
and
R. Godbole
Centre for High Energy Physics, Indian Institute of Science
Bangalore, 560 012, India
and
A. Grau
Departamento de F´ısica Teo´rica y del Cosmos, Universidad de Granada, Spain
and
Y.N. Srivastava
Physics Department and INFN, University of Perugia, Perugia, Italy.
A model for total cross-sections incorporating QCD jet cross-sections
and soft gluon resummation is described and compared with present data on
pp and p¯p cross-sections. Predictions for LHC are presented for different
parameter sets. It is shown that they differ according to the small x-
behaviour of available parton density functions.
PACS numbers: 12.38.-t;12.40.Nn;13.60.Hb;13.85.Lg
hep-ph/0703174
IISc/CHEP/4/07
LNF - 07 / 6(P)
∗ Contribution to the Proceedings of EURIDICE Final Meeting, August 24-27th, 2006,
Kazimierz, Poland
(1)
2 kazimierz˙pancheri printed on February 2, 2008
1. Introduction
The upcoming measurements at LHC are renewing considerable interest
regarding predictions for total cross-sections. The model [1, 2] we shall
describe in the following, attempts to link the rate with which total cross-
sections rise, to the infrared behaviour of the strong coupling constant αs
and to QCD hard parton-parton scattering, using known phenomenological
entities such as the available QCD parton density functions (PDFs).
2. The model
The energy behaviour of the total cross-section exhibits the following
properties [3]
• an initial decrease
• a sharp change in curvature occurring somewhere between 20 and 50
GeV in the c.m. of the scattering hadrons
• a smooth rise which asymptotically follows a ln s or ln2 s type increase
in consonance with the Froissart bound [4, 5]
The model we use is based on
1. hard component of scattering responsible for the rise of the total cross-
section [6, 7]
2. soft gluon emission from scattering particles which softens the rise [1]
3. eikonal transformation which implies multiple scattering and requires
impact parameter distributions inside scattering particles and basic
scattering cross-sections [8]
According to our model, soft gluon emission is responsible for the initial
decrease in pp, as well as for the transformation of the sharp rise due to
the increase in gluon-gluon interactions into a smooth behavior. Thus soft
gluon emission plays a crucial role, with care taken to extend resummation
to the zero energy modes, in complete analogy for what is required by the
Bloch-Nordsieck theorem for QED[9]. The model can then be referred to as
the BN model, for reasons which will also be clearer in the following.
2.1. Details of the BN model
We use the following eikonal expression for the total inelastic cross-
section:
σinel =
∫
d2~b[1− e−n(b,s)] (1)
kazimierz˙pancheri printed on February 2, 2008 3
where n(b, s) corresponds to the average number of inelastic collisions at
any given value of the impact parameter b. Neglecting the real part of the
eikonal, we then calculate the total cross-section as
σtotal = 2
∫
d2~b[1− e−n(b,s)/2] (2)
In our model n(b, s) is split as
n(b, s) = nsoft(b, s) + nhard(b, s) (3)
where we postulate the following factorization
nsoft/hard(b, s) = A
soft/hard
BN (b, s)σsoft/hard(s) (4)
with
ABN (b, s) = N
∫
d2K⊥ e−iK⊥·b
d2P (K⊥)
d2K⊥
(5)
where N is a normalization factor such that
∫
d2~bA(b) = 1 and
d2P (K⊥)
d2K⊥
=
1
(2π)2
∫
d2~b eiK⊥·b−
∫ qmax
0
d3n¯(k)[1−e−ikt·b] (6)
is the transverse momentum distribution of initial state soft gluons emitted
in the parton-parton collisions and where, for simplicity, k⊥ · b = ~k⊥ · ~b.
In Equation (6) qmax is the maximum transverse momentum allowed by
kinematics to single soft gluon emission in a given hard collision, averaged
over the parton densities. According to the basic ansatz of the Eikonal
Minijet Model (EMM),
σhard ≡ σABjet (s) =
∫ √s/2
ptmin
dpt
∫ 1
4p2t /s
dx1
∫ 1
4p2t /(x1s)
dx2
∑
i,j,k,l
fi|A(x1)fj|B(x2)
dσˆklij (sˆ)
dpt
.
(7)
Here A and B denote particles (γ, p, . . .), i, j, k, l are parton types and
x1, x2 the fractions of the parent particle momentum carried by the parton.
sˆ = x1x2s and σˆ are hard parton scattering cross–sections. As discussed in
[1], kinematical considerations suggest [10]
qmax(s) =
√
s
2
∑
i,j
∫ dx1
x1
fi|A(x1)
∫ dx2
x2
fj|B(x2)
√
x1x2
∫ 1
zmin
dz(1− z)∑
i,j
∫ dx1
x1
fi|A(x1)
∫ dx2
x2
fj|B(x2)
∫ 1
zmin
(dz)
(8)
with zmin = 4p
2
tmin/(sx1x2) and fi/a the valence quark densities used in
the jet cross-section calculation. The steps we follow to compare the model
with data are then the following:
4 kazimierz˙pancheri printed on February 2, 2008
1. choose the parameters for the hard scattering part, namely
(i) parton densities (PDF), ptmin and ΛQCD for the chosen PDF set
in Equation (7)
(ii) model for αs in the infrared region and relevant parameters in
Equation (6)
2. calculate qmax(s, ptmin) for the given densities and ptmin using Equa-
tion (8)
3. calculate nhard(b, s) = A
hard
BN (b, s)σjet(s, ptmin)
4. choose the parameters for the low energy part, namely
(i) the constant low energy cross-section σ0
(ii) values for qsoftmax
5. calculate nsoft(b, s) = A
soft
BN (b, s)σ0(1 + ǫ
2√
s
) with ǫ = 0, 1 depending
upon the process being pp or pp¯
6. calculate n(b, s) and thus σtot
7. choose the parameter set which gives the best description of the total
cross-section up to the Tevatron data [11, 12, 13]
Notice that once a good set of parameters is found, one can use n(b, s)
with fitted parameters to calculate survival probabilities or diffractive Higgs
production.
2.2. Application to total cross-section data
We show in this section the application of the model to the total cross-
section for different PDFs. We find that our model is flexible enough to
be able to reproduce the present data for σtot using all presently available
PDFs. In particular, all GRV [14, 15, 16] and MRST [17] densities give a
good description using the singular αs model described in [1], while CTEQ
densities [18] give an acceptable fit up to Tevatron data, but then fail to
rise further.
We present these results by following the previously listed steps. We
start by choosing the parameters for the hard scattering, and, following our
previous results [2], we fix ptmin = 1.15 GeV in the jet cross-sections and
calculate qmax for different PDF sets. We show the result in Fig.(1). We
notice the following:
kazimierz˙pancheri printed on February 2, 2008 5
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
ptmin=1.15 GeV
 qmaxsoft
MRST
Fig. 1. Average value of the maximum transverse momentum allowed to single
gluon emission, according to the model in [2].
• GRV densities are of two types, GRV98[16] for which qmax keeps on
increasing logarithmically and the older ones [14] for which qmax slows
down past the TeV region, albeit still slowly increasing
• CTEQ densities give values for qmax which increase more rapidly than
ln s after typical Tevatron energies
• MRST densities indicate a behaviour opposite to CTEQ, since they
give qmax values decreasing after the TeV cross mark.
We now turn to the jet cross-sections and examine the growth with energy of
6 kazimierz˙pancheri printed on February 2, 2008
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. 2. Mini-jet QCD cross-sections for different PDFs as indicated in the figure
σjet for different PDFs. In Fig.(2) we plot these cross-sections for the same
set of densities used to calculate qmax and for ptmin = 1.15 GeV . From this
figure we notice that :
• the jet cross-sections for GRV densities increase faster than all the
others
• the jet cross-sections for CTEQ increase more or less similarly to those
for the MRST group
The implications are that σjet with GRV densities, which increase faster
than σjet with MRST, need more softening, σjet with CTEQ, which increase
less than with GRV, should not be smeared that much.
kazimierz˙pancheri printed on February 2, 2008 7
To proceed further, we now need to calculate the b−distribution and fix
the low-energy parameters, like σ0. The b − distribution requires to input
the behaviour of αs in the infrared region. We have shown in [2] the need to
use a singular but integrable expression for αs in order to reproduce both
the sudden rise and the subsequent softening of the total cross-section. Our
choice is an expression like
αs(kt) =
12π
(33− 2Nf )
p
ln[1 + p(ktΛ )
2p]
(9)
which depends on the singularity parameter p, in addition to the scale Λ.
In [2] we have chosen the value p = 0.75 and Λ = 100 MeV , other choices
are also possible [19]. Turning now to the low-energy part, nsoft(b, s), we
choose σ0 = 48 mb and use for A
soft
BN a set of qmax values which reproduce
the low energy behaviour, and which appear in Fig. (1).
3. Comparison with data and expectations at LHC densities
We can now input all the above in the eikonal representation for the
total cross-section and obtain the results shown in Fig.(3) for the singular
αs case and GRV, MRST and CTEQ densities. For the sake of clarity,
we only plot curves for pp scattering, referring the reader to [2, 19] for
the curves for pp¯ and for a different parameter set, or for predictions from
other models [5, 20, 21, 22, 23]. From the figure we see that the calculation
with CTEQ densities appears very unlikely. The effect is due to the fact
that the jet cross-sections in the CTEQ case do not rise as much as the
others while the softening effect is stronger, as it is driven by qmax, which is
strongly increasing for these densities. As a result, the cross-section starts
decreasing. Notice that while the behaviour of σjet is dominated by the
gluon densities, that of qmax is determined only by the valence quarks, as
we assume this to be the leading order effect.
The curves shown in Fig.(3) indicate that the coming measurement at√
s = 900 GeV will be very important in determining which of these curves
best describes the data. It can then be used to select the parameter set,
basically PDF’s and ptmin, for a prediction at the project LHC energy,√
s = 14 TeV . If the UA5 [24] value at
√
s = 900 GeV is confirmed with a
comparable error, then, for the set of parameters discussed in this note, at√
s = 14 TeV our model gives σGRV 98total = 90.2 mb, σ
GRV
total = 100.2 mb and
σMRST76total = 103.4 mb. As shown in [19], changing the parameter set, namely
σ0, ptmin or the singularity index p, give values in the range 88 ÷ 111 mb.
8 kazimierz˙pancheri printed on February 2, 2008
30
40
50
60
70
80
90
100
110
120
1 10 10
2
10
3
10
4
√s ( GeV )
s
to
t(m
b)
s tot  for various LO densities, ptmin=1.15 GeV
GRV
GRV98
CTEQ
MRST
proton-antiproton
proton-proton
Fig. 3. Data for total cross-sections for pp and pp¯ compared with model predictions
at LHC for pp scattering, using different PDFs.
4. Conclusions
We have presented a version of the eikonal minijet model which allows a
good description of total cross-sections at asymptotic energies and discussed
its connection with the small x-behaviour of various types of parton densi-
ties. This model is based on a softening of the mini-jet cross-sections due
to an s-dependent b-distribution in the proton, which we calculate using a
soft gluon resummation model down to zero energy of the soft gluons.
kazimierz˙pancheri printed on February 2, 2008 9
Acknowledgments
We acknowledge partial support from EU CT-2002-0311.
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Total cross-section at LHC from minijets and soft gluon resummation in the infrared region

A model for total cross-sections incorporating QCD jet cross-sections and soft  gluon resummation  is described and compared with present data on pp and pp cross-sections. Predictions for LHC are presented for different parameter sets. It is shown that they differ according to the small x-behaviour of available parton density functions

Pancheri, G

Godbole, RM

Grau, A

Srivastava, YN

Name not available

Total cross-section at LHC from minijets and soft  gluon resummation  in the infrared region

Open Access Repository of IISc Research Publications

Presented at The Final Euridice Meeting "Effective theories of colours and flavours: from Eurodaphne to Euridice", Kazimierz, Poland, 24-27 august, 2006.A model for total cross-sections incorporating QCD jet cross-sections and soft gluon resummation is described and compared with present data on pp and pp cross-sections. Predictions for LHC are presented for different parameter sets. It is shown that they differ according to the small x-behaviour of available parton density functions.We acknowledge partial support from EU CT-2002-0311

Repositorio Institucional Universidad de Granada

Vol. 38 (2007) ACTA PHYSICA POLONICA B No 9TOTAL CROSS-SECTION AT LHC FROM MINIJETSAND SOFT GLUON RESUMMATIONIN THE INFRARED REGION∗G. PancheriINFN Frascati National Laboratories, P.O. Box 13, I00044 Frascati, ItalyR.M. GodboleCentre for High Energy Physics, Indian Institute of ScienceBangalore, 560 012, IndiaA. GrauDepartamento de Física Teórica y del Cosmos, Universidad de Granada, SpainY.N. SrivastavaPhysics Department and INFN, University of Perugia, Perugia, Italy(Received June 28, 2007)A model for total cross-sections incorporating QCD jet cross-sectionsand soft gluon resummation is described and compared with present dataon pp and p̄p cross-sections. Predictions for LHC are presented for dif-ferent parameter sets. It is shown that they differ according to the smallx-behaviour of available parton density functions.PACS numbers: 12.38.–t, 12.40.Nn, 13.60.Hb, 13.85.Lg1. IntroductionThe upcoming measurements at LHC are renewing considerable inter-est regarding predictions for total cross-sections. The model [1, 2] we shalldescribe in the following, attempts to link the rate with which total cross-sections rise, to the infrared behaviour of the strong coupling constant αs∗ Presented at The Final EURIDICE Meeting “Effective Theories of Colours andFlavours: from EURODAPHNE to EURIDICE”, Kazimierz, Poland, 24–27 August,2006.(2979)2980 G. Pancheri et al.and to QCD hard parton–parton scattering, using known phenomenologicalentities such as the available QCD parton density functions (PDFs).2. The modelThe energy behaviour of the total cross-section exhibits the followingproperties [3]• an initial decrease;• a sharp change in curvature occurring somewhere between 20 and50 GeV in the c.m. of the scattering hadrons;• a smooth rise which asymptotically follows a ln s or ln2 s type increasein consonance with the Froissart bound [4, 5].The model we use is based on1. hard component of scattering responsible for the rise of the total cross-section [6, 7];2. soft gluon emission from scattering particles which softens the rise [1];3. eikonal transformation which implies multiple scattering and requiresimpact parameter distributions inside scattering particles and basicscattering cross-sections [8].According to our model, soft gluon emission is responsible for the initialdecrease in pp, as well as for the transformation of the sharp rise due tothe increase in gluon–gluon interactions into a smooth behaviour. Thus softgluon emission plays a crucial role, with care taken to extend resummationto the zero energy modes, in complete analogy for what is required by theBloch–Nordsieck theorem for QED [9]. The model can then be referred toas the BN model, for reasons which will also be clearer in the following.2.1. Details of the BN modelWe use the following eikonal expression for the total inelastic cross-section:σinel =∫d2~b[1 − e−n(b,s)], (1)where n(b, s) corresponds to the average number of inelastic collisions atany given value of the impact parameter b. Neglecting the real part of theeikonal, we then calculate the total cross-section asσtotal = 2∫d2~b[1 − e−n(b,s)/2]. (2)Total Cross-Section at LHC from Minijets and Soft Gluon . . . 2981In our model n(b, s) is split asn(b, s) = nsoft(b, s) + nhard(b, s) , (3)where we postulate the following factorisationnsoft/hard(b, s) = Asoft/hardBN (b, s)σsoft/hard(s) (4)withABN(b, s) = N∫d2K⊥ e−iK⊥·bd2P (K⊥)d2K⊥, (5)where N is a normalisation factor such that∫d2~bA(b) = 1 andd2P (K⊥)d2K⊥=1(2π)2∫d2~b expiK⊥ · b −qmax∫0d3n̄(k)[1 − e−ikt·b](6)is the transverse momentum distribution of initial state soft gluons emittedin the parton–parton collisions and where, for simplicity, k⊥ · b = ~k⊥ ·~b. Inequation (6) qmax is the maximum transverse momentum allowed by kine-matics to single soft gluon emission in a given hard collision, averaged overthe parton densities. According to the basic ansatz of the Eikonal MinijetModel (EMM),σhard ≡ σABjet (s)=√s/2∫ptmindpt1∫4p2t/sdx11∫4p2t/(x1s)dx2∑i,j,k,lfi|A(x1)fj|B(x2)dσ̂klij (ŝ)dpt.(7)Here A and B denote particles (γ, p, . . .), i, j, k, l are parton types andx1, x2 the fractions of the parent particle momentum carried by the parton.ŝ = x1x2s and σ̂ are hard parton scattering cross-sections. As discussedin [1], kinematical considerations suggest [10]qmax(s) =√s2∑i,j∫dx1x1fi|A(x1)∫dx2x2fj|B(x2)√x1x21∫zmindz(1 − z)∑i,j∫dx1x1fi|A(x1)∫dx2x2fj|B(x2)1∫zmin(dz)(8)with zmin = 4p2tmin/(sx1x2) and fi/a the valence quark densities used in thejet cross-section calculation. The steps we follow to compare the model withdata are then the following:2982 G. Pancheri et al.1. choose the parameters for the hard scattering part, namely(i) parton densities (PDF), ptmin and ΛQCD for the chosen PDF setin equation (7),(ii) model for αs in the infrared region and relevant parameters inequation (6),2. calculate qmax(s, ptmin) for the given densities and ptmin using equa-tion (8),3. calculate nhard(b, s) = AhardBN (b, s)σjet(s, ptmin),4. choose the parameters for the low energy part, namely(i) the constant low energy cross-section σ0,(ii) values for qsoftmax,5. calculate nsoft(b, s) = AsoftBN (b, s)σ0(1 + ǫ2√s) with ǫ = 0, 1 dependingupon the process being pp or pp̄,6. calculate n(b, s) and thus σtot,7. choose the parameter set which gives the best description of the totalcross-section up to the Tevatron data [11–13].Notice that once a good set of parameters is found, one can use n(b, s)with fitted parameters to calculate survival probabilities or diffractive Higgsproduction.2.2. Application to total cross-section dataWe show in this section the application of the model to the total cross-section for different PDFs. We find that our model is flexible enough tobe able to reproduce the present data for σtot using all presently availablePDFs. In particular, all GRV [14–16] and MRST [17] densities give a gooddescription using the singular αs model described in [1], while CTEQ den-sities [18] give an acceptable fit up to Tevatron data, but the cross-sectionfails to rise further.We present these results by following the previously listed steps. Westart by choosing the parameters for the hard scattering, and, following ourprevious results [2], we fix ptmin = 1.15 GeV in the jet cross-sections andcalculate qmax for different PDF sets. We show the result in Fig. 1.Total Cross-Section at LHC from Minijets and Soft Gluon . . . 298300.511.522.531 10 102103104√s ( GeV )q max(GeV)qmax for hard processes for various LO densitiesGRVGRV98CTEQptmin=1.15 GeV qmaxsoftMRSTFig. 1. Average value of the maximum transverse momentum allowed to singlegluon emission, according to the model in [2].We notice the following:• GRV densities are of two types, GRV98 [16] for which qmax keeps onincreasing logarithmically and the older ones [14] for which qmax slowsdown past the TeV region, albeit still slowly increasing.• CTEQ densities give values for qmax which increase more rapidly thanln s after typical Tevatron energies.• MRST densities indicate a behaviour opposite to CTEQ, since theygive qmax values decreasing after the TeV cross mark.We now turn to the jet cross-sections and examine the growth with energyof σjet for different PDFs. In Fig. 2 we plot these cross-sections for the sameset of densities used to calculate qmax and for ptmin = 1.15 GeV. From thisfigure we notice that :• the jet cross-sections for GRV densities increase faster than all theothers;2984 G. Pancheri et al.100020003000400050006000700080009000100001 10 102103104√s ( GeV )σ jet(mb)σjet  for various LO densitiesMRSTGRVGRV98CTEQptmin=1.15 GeVFig. 2. Mini-jet QCD cross-sections for different PDFs as indicated in the figure.• the jet cross-sections for CTEQ increase more or less similarly to thosefor the MRST group.The implications are that σjet with GRV densities, which increase faster thanσjet with MRST, need more softening, σjet with CTEQ, which increase lessthan with GRV, should not be smeared that much.To proceed further, we now need to calculate the b-distribution and fixthe low-energy parameters, like σ0. The b-distribution requires to input thebehaviour of αs in the infrared region. We have shown in [2] the need touse a singular but integrable expression for αs in order to reproduce boththe sudden rise and the subsequent softening of the total cross-section. Ourchoice is an expression likeαs(kt) =12π(33 − 2Nf )pln[1 + p(ktΛ)2p](9)which depends on the singularity parameter p, in addition to the scale Λ.In [2] we have chosen the value p = 0.75 and Λ = 100 MeV, other choices areTotal Cross-Section at LHC from Minijets and Soft Gluon . . . 2985also possible [19]. Turning now to the low-energy part, nsoft(b, s), we chooseσ0 = 48 mb and use for AsoftBN a set of qmax values which reproduce the lowenergy behaviour, and which appear in Fig. 1.3. Comparison with data and expectations at LHC densitiesWe can now input all the above in the eikonal representation for the totalcross-section and obtain the results shown in Fig. 3 for the singular αs caseand GRV, MRST and CTEQ densities. For the sake of clarity, we only plotcurves for pp scattering, referring the reader to [2,19] for the curves for pp̄ andfor a different parameter set, or for predictions from other models [5,20–23].From the figure we see that the calculation with CTEQ densities appearsvery unlikely. The effect is due to the fact that the jet cross-sections inthe CTEQ case do not rise as much as the others while the softening effectis stronger, as it is driven by qmax, which is strongly increasing for thesedensities. As a result, the cross-section starts decreasing. Notice that while304050607080901001101201 10 102103104√s ( GeV )σ tot(mb)σtot  for various LO densities, ptmin=1.15 GeVGRVGRV98CTEQMRSTproton-antiprotonproton-protonFig. 3. Data for total cross-sections for pp and pp̄ compared with model predictionsat LHC for pp scattering, using different PDFs.2986 G. Pancheri et al.the behaviour of σjet is dominated by the gluon densities, that of qmax isdetermined only by the valence quarks, as we assume this to be the leadingorder effect.The curves shown in Fig. 3 indicate that the coming measurement at√s = 900 GeV will be very important in determining which of these curvesbest describe the data. It can then be used to select the parameter set,basically PDF’s and ptmin, for a prediction at the project LHC energy,√s = 14 TeV. If the UA5 [24] value at√s = 900 GeV is confirmed with acomparable error, then, for the set of parameters discussed in this note, at√s = 14 TeV our model gives σGRV98total = 90.2 mb, σGRVtotal = 100.2 mb andσMRST76total = 103.4 mb. As shown in [19], changing the parameter set, namelyσ0, ptmin or the singularity index p, give values in the range 88 ÷ 111 mb.4. ConclusionsWe have presented a version of the eikonal minijet model which allows agood description of total cross-sections at asymptotic energies and discussedits connection with the small x-behaviour of various types of parton densities.This model is based on a softening of the mini-jet cross-sections due to ans-dependent b-distribution in the proton, which we calculate using a softgluon resummation model down to zero energy of the soft gluons.We acknowledge partial support from EU CT-2002-0311.REFERENCES[1] A. Corsetti, A. Grau, G. Pancheri, Y.N. Srivastava, Phys. Lett. B382, 282(1996) [hep-ph/9605314]; A. Grau, G. Pancheri, Y.N. Srivastava, Phys. Rev.D60, 114020 (1999) [hep-ph/9905228].[2] R.M. Godbole, A. Grau, G. Pancheri, Y.N. Srivastava, Phys. Rev. D72,076001 (2005) [hep-ph/0408355].[3] Particle Data Group, J. Phys. G: Nucl. Part. Phys. 33, 1 (2006).[4] M. Froissart, Phys. Rev. 123, 1053 (1961).[5] M.M. Block, F. Halzen, Phys. Rev. D72, 036006 (2005), Erratum Phys. 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Total cross-section at LHC from minijets and soft gluon resummation in the infrared region

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