Inclusion of down to zero-momentum gluons and their kt resummation is shown to quench the too fast rise of the mini jet cross section and thereby obtain realistic total cross-sections

Achilli, A.

Godbole, R.

Grau, A.

Pancheri, G.

Srivastava, Y. N.

English

Publications of the IAS Fellows

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09 Relevance of ultra-soft gluons and kt resummation for total
cross-sections ∗
A. Achilli and Y.N. Srivastava
Physics Department and INFN, University of Perugia, Perugia, Italy
R. Godbole
Centre for High Energy Physics, Indian Institute of Science, Bangalore, India
A. Grau
Departamento de F´ısica Teo´rica y del Cosmos, Universidad de Granada, Spain
G. Pancheri
INFN Frascati National Laboratories, Frascati, Italy
Inclusion of down to zero-momentum gluons and their kt resummation
is shown to quench the too fast rise of the mini jet cross section and thereby
obtain realistic total cross-sections.
PACS numbers: 13.60.Hb,13.85.Lg,12.40.Nn,12.38.Cy,11.80.Fv
1. Introduction
We present details of the very high energy behavior of hadron-hadron
and hadron-photon total cross sections in a model which incorporates soft
gluon kt-resummation in the infrared (IR) region. The effects of this resum-
mation are discussed to highlight the mechanisms responsible for the rate
with which the total cross section rises with energy.
The Froissart-Martin (FM) bound [1, 2] says that the hadronic cross-
sections cannot rise faster than ln2 s. The most common parametrizations of
σtotal for pp processes, based on unitarity and constraints from analyticity,
impose the FM bound. By contrast, in our QCD based model [3, 4, 5], we
find an interesting relationship between the IR behavior of αs and the rise
of σtotal.
We first present a short introduction to the soft gluon kt-resummation
technique, then the main features of our eikonal model based on QCD-
minijet cross section and soft gluon resummation, after which we compare
our results with data from pp and γp processes. Finally, we show how the
∗ Presented at FLAVIAnet Workshop, Kazimierz, 23-27 July, 2009
(1)
2 kazimierz09˙2 printed on October 22, 2009
kt-resummation affects the asymptotic rise which, while respecting the FM
bound, does not necessarily saturate it.
2. kt-resummation and the IR limit
To compute the total cross-section one has to integrate over large val-
ues of the impact parameters b, which gets linked to the IR regions of the
underlying partonic processes. Here we are considering (within a semiclas-
sical approach) soft gluon emissions from the colliding partons and their kt-
resummation, whose Fourier transforms we assume to describe the matter
distribution within the colliding hadrons. The expression for the transverse
momentum distribution resulting from soft gluon radiation emitted by the
colliding quarks is
d2P (K⊥) = d
2K⊥
1
(2pi)2
∫
d2b e−iK⊥·b−h(b,M) (1)
with
h(b,M) =
16
3
∫ M
0
αs(k
2
t )
pi
dkt
kt
ln
2M
kt
[1− J0(ktb)] (2)
where M gives the energy scale of the process, in our case it is the maximum
allowed transverse momentum for a single gluon emission [6]. In various
QCD problems [7, 8] one splits this expression in two terms
h(b,M) = c0(µ, b,M) + ∆h(b,M), (3)
with the first term parameterizing the infrared behavior and the second
term which is calculable in the perturbative regime defined as
∆h(b,M) =
16
3
∫ M
µ
αs(k
2
t )
pi
[1− Jo(bkt)]
dkt
kt
ln
2M
kt
. (4)
Since this integral only runs down to a scale µ > ΛQCD, one uses the
asymptotic freedom expression for αs and one assumes that Jo(bkt) oscillates
to zero. In the range 1/M < b < 1/Λ an effective h is obtained fixing the
scale µ = 1/b. Then one finds [9]
e−heff (b,M) = [
ln(1/b2Λ2)
ln(M2/Λ2)
](16/25) ln(M
2/Λ2) (5)
In our model, the integral (4) includes zero momentum values and we
investigate their contribution in the very large impact parameter region.
Since in the IR limit, the perturbative expression for αs is not valid, we
kazimierz09˙2 printed on October 22, 2009 3
employ a phenomenological expression[3], which is singular but integrable
in the IR limit of the soft gluon integral
αs(k
2
t ) =
12pi
(33 − 2Nf )
p
ln[1 + p(
k2t
Λ2 )
p]
(6)
with the parameter 1/2 < p < 1. For p = 1 it coincides with the Richardson
potential [10]. In the large-b region (b > 1NpΛ >
1
M ), we can obtain an
approximate analytic expression:
h(b,M,Λ) = 83pi
[
b¯
8(1−p)(b
2Λ2)p
[
2 ln(2Mb) + 11−p
]
+ b¯2p(b
2Λ2)p
[
2 ln(Mb)− 1p
]
+ b¯
2pN2pp
[
−2 ln MΛNp +
1
p
]
+ b¯ ln MΛ
[
ln
ln M
Λ
lnNp
− 1 + lnNp
ln M
Λ
]]
(7)
where Np = (1/p)
1/2p > 1 for p < 1 , and b¯ = 12pi/(33 − 2Nf ).
Our soft gluon resummation approach allows us to obtain a value for
the “intrinsic” transverse momentum [11], through
hintrinsic =
16
3
∫ Λ
0
αs(k
2
t )
pi
dkt
kt
ln
2M
kt
[1− J0(ktb)] ≈
b2
4
< k2t >int (8)
with
< k2t >int=
8
3pi
b¯
(1− p)
(
1
2(1 − p)
+ ln
2M
Λ
)
Λ2, (9)
which corresponds to an intrinsic transverse momentum of a few hundred
MeV for Λ in the 100 MeV range and M ≤ 1 GeV.
3. The Bloch-Nordsieck model for total cross-sections
In our model for the total hadronic cross-section we use the eikonal
formalism which implies multiple scattering and requires impact parameter
distributions of the scattering particles [3]. In the high energy limit, neglect-
ing the real part of the eikonal, the total cross-section can be approximated
as given by
σtotal = 2
∫
d2b[1− e−Imχ(b,s)]. (10)
The imaginary part of the eikonal is related to the average number of in-
elastic collisions n(b, s)
2Imχ(b, s) = n¯(b, s) (11)
4 kazimierz09˙2 printed on October 22, 2009
for which we use the expression
n¯(b, s) = nNP (b, s) + nhard(b, s) (12)
with the non perturbative (NP) term relevant only for the low-energy de-
scription of σtot and for its normalization, and the hard one responsible for
the high-energy rise. This term is given by
nhard(b, s) = A(b, s)σjet(s) (13)
where σjet is the mini-jet cross section describing high-energy partonic col-
lisions. It drives the rise of the total cross section and depends on the
parameter ptmin, the minimum transverse momentum of the scattered par-
tons, which separates hard parton-parton scattering from all other low-pt
processes and on currently used, DGLAP evoluted, parametrizations for the
Partonic Density Functions.
A(b,s) is the overlap function given by the Fourier transform of the
transverse momentum distribution resulting from initial state soft gluon
radiation, which breaks the collinearity of the colliding partons, making the
scattering process less efficient.
ABN (b, s) = N
∫
d2K⊥ e
−iK⊥·b
d2P (K⊥)
d2K⊥
=
e−h(b,qmax)∫
d2b e−h(b,qmax)
= A0(s)e
−h(b,qmax) (14)
The parameter qmax is linked to the maximum transverse momentum al-
lowed by the kinematics for single gluon emission [6]. It should be calculated
for every partonic subprocess but we make the simplifying assumption of
assigning it a value averaged over all the possible subprocesses [3] and use
it in Eq. (13) to calculate the rising part of σtotal.
Fig.1 shows our results for proton−proton ( band ) compared with other
models [3, 12]. In this case, our central prediction (black line in the band)
has been obtained with ptmin = 1.15 GeV , p = 0.75 and GRV densities.
Application of the model to γp [5] is shown on the left panel.
4. Asymptotic limit of the total cross-section
We can now estimate the very large s-limit when the minijet cross-
section rises asymptotically like a power sε (ε ≈ 0.3 [14]). We find
nhard(b, s) = ABN (b, s)σjet(s, ptmin) ∼ A0(s)e
−h(b,qmax)σ1(
s
s0
)ε (15)
kazimierz09˙2 printed on October 22, 2009 5
0.1
0.2
0.3
0.4
0.5
0.6
10 10 2 103 10 4
√s ( GeV )
s
to
tg
p (m
b)
Our model
Top: GRS, ptmin=1.2 GeV, p=0.75
Lower: CJKL, ptmin=1.8 GeV, p=0.8
Photoproduction data before HERA
ZEUS 96
H1 94
Vereshkov 03
(s pp from BN)/330
GRS ptmin=1.3 GeV, p=0.75
Aspen Model
FF model GRS ptmin=1.5 GeV
Block-Halzen log2(Eg /mp) fit
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. 1. γp total cross section (left) from [5] and pp total cross section (right) from
[4] are shown compared with data and with other phenomenological models [12, 13].
From it, we can deduce the very large b-limit to be
nhard(b, s) ∼ A0(s)σ1e
−(bΛ¯)2p(
s
s0
)ε (16)
with [9]
Λ¯ ≡ Λ¯(b, s) = Λ{
b¯
3pi(1 − p)
[ln(2qmax(s)b) +
1
1− p
]}1/2p (17)
where A0(s) ∝ Λ
2, and qmax a very slowly varying function of s. In this
limit, the total cross section becomes
σT (s) ≈ 2pi
∫
∞
0
db2[1− e−C(s)e
−(bΛ¯)2p
] (18)
with 2C(s) = A0(s)σ1(s/s0)
ε. We thus obtain
σT ≈ (constant)[ln
s
s0
]1/p. (19)
The constant in Eq.(19) depends on Λ and qmax [9] and is about one order
of magnitude smaller than the constant which appears in the Martin bound.
We thus find that the introduction of soft gluon resummation introduces a
new scale which imposes a more stringent limit on the high energy behavior
of σtotal.
6 kazimierz09˙2 printed on October 22, 2009
5. Conclusions
In an eikonal mini-jet model, soft gluon kt-resummation down to zero
gluon momenta does reduce the strong power-like rise of the minijet cross-
section. We find a link between the infrared region in soft kt-resummation
and the very high energy behaviour of total cross-sections. Introduction
of a dynamical scale related to the very soft gluon emission, reduces the
constant occuring in the Martin bound.
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Relevance of ultra-soft gluons and k<SUB>t</SUB> resummation for total cross-sections

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Relevance of ultra-soft gluons and k<SUB>t</SUB> resummation for total cross-sections

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