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9
Soft Gluon kt-Resummation and the Froissart bound
Agnes Graua, Rohini M. Godboleb, Giulia Pancheric, Yogendra N.Srivastavad
aDepartamento de Fisica Teorica y del Cosmos, Universidad de Granada, SpainbCentre for High Energy Physics, Indian Institute of Science, Bangalore, 560012,India
cINFN Frascati National Laboratories, P.O. Box 13, Frascati I00044, ItalydPhysics Department and INFN, University of Perugia, Perugia, Italy
Abstract
We study soft gluon kt-resummation and the relevance of InfraRed (IR) glu-ons for the energy dependence of total hadronic cross-sections. In our model,consistency with the Froissart bound is directly related to the ansatz thatthe IR behaviour of the QCD coupling constant follows an inverse power law.
Key words: Froissart bound, QCD, total cross-section, resummationPACS: 13.85.Lg, 12.38.Cy
1. Introduction
In this paper, we discuss soft gluon kt-resummation in the InfraRed (IR)region, with the aim to connect it to the energy dependence of the totalhadronic cross-section in the high energy region. We shall make use of amodel [1] for total cross-sections, which incorporates in an eikonal formu-lation such QCD inputs as mini-jets and soft gluon kt-resummation. Thismodel has been successfully applied both to proton and photon processes:our aim here is to describe its physical content, without explicit referenceto data fitting, and explain how the model incorporates a taming effect onthe rapidly rising QCD cross-sections, thus inducing a more temperate riseof the total cross-section.
Email addresses: [email protected] (Agnes Grau), [email protected] (RohiniM. Godbole), [email protected] (Giulia Pancheri),[email protected] (Yogendra N. Srivastava)
Preprint submitted to Physics Letters B October 21, 2009
The energy behaviour of the total hadronic cross-section has been thefocus of both theoretical and experimental enquiries for a long time. Fitsinspired by theoretical arguments have been the subject of many debates. Atraditional Regge-Pomeron type fit [2] such as
σtotal = Xs−η + Y sǫ, (1)
with η, ǫ > 0, presents the difficulty of not agreeing with the Froissart-Lukaszuk-Martin bound [3, 4, 5]
σtotal ≤π
m2π
ln2(s/s0). (2)
However, with√
s0 ≃ O(1 GeV ), the large constant factor at the r.h.s ofEq. (2) makes possible for Eq. (1) to be valid only in the present region, andthe observed behaviour to be not yet asymptotic. Still, phenomenologicalreasons run against the validity of the Regge-Pomeron type expression witha universal value for ǫ, since LEP data on photon-photon indicate that thepower, with which σγγ
total rises, differs from that in pp/pp [6].Currently, many models focus on QCD perturbative processes to drive
the rise with energy, with QCD inspired models on the one hand [7, 8] andapproaches based on Reggeon calculus [9, 10] on the other. However, itis still being debated how to implement this dynamics and simultaneoulydescribe both the early rise, which starts for
√s between 10 ÷ 20 GeV and
50 ÷ 60 GeV , and the subsequent levelling off at higher energies [11].In this paper we shall show how both of the above features are present
in our model for the total cross-section [1] and relate the rate, at which thecross-section asymptotically rises, to the IR limit of soft kt-resummation,thus linking directly the rise of the total cross-section to the IR region ofQCD. We start by recalling in Sect. 2 some features of resummation andpresent, in Sect. 3, our proposal for handling the ultra soft gluon emissionswhich affect scattering at the very large impact parameter values, relevantto total cross-section calculations. After a brief discussion of our model inSect. 4, we show in Sect. 5 the connection between the Froissart bound andour proposed eikonalized mini-jet model with ultra soft gluons. Finally, inSect. 6 we discuss both the various energy scales and the constants involvedin the model. We find that resummation of soft gluons emitted with verysmall transverse momentum kt introduces a new energy scale, which modifiesthe constant in front of the asymptotic limit derived in our model for the totalcross-section.
2
2. Resummation and the IR limit
The high energy behaviour of the total cross-section depends on the prop-erties of the scattering amplitude at large values of the impact parameter bin the plane perpendicular to the scattering. Recent attempts to study thislarge-b behaviour have focused on relating Yang-Mills theories to string the-ories through the AdS/CFT correspondance [12, 13, 14]. Our approach tothe large b-limit of the impact picture in the eikonal representation [15] isof a more phenomenological nature: within such a picture, we exploit softgluon kt-resummation in the IR region to describe matter distribution insidethe hadrons as they engage in hard scattering and their parton constituents“see” each other. We have already supplied phenomenological evidence forthe applicability of our model to high energy scattering [1]. Here we discusssome features of soft gluon kt-resummation which bear on the asymptoticform of the total cross-section.
We start by recalling some properties of soft photon resummation. InQED, the general expression for soft photon resummation in the energy-momentum variable Kµ can be obtained order by order in perturbation theory[16, 17] as
d4P (K) = d4K
∫
d4x
(2π)4eiK·x−h(x,E) (3)
where d4P (K) is the probability for an overall 4-momentum Kµ escapingdetection,
h(x, E) =
∫ E
0
d3n(k)[1 − e−ik·x] (4)
with d3n(k) being the single soft photon differential spectrum, and E themaximum energy allowed for single photon emission.
Eq. (3) leads to the well known, power-like, form of the energy distribution[18]. This is not possible for the momentum distribution, but it is also notnecessary, since the first order expression in αQED is adequate. On the otherhand, for strong interactions we require the resummed, transverse momentumdistribution, namely
d2P (K⊥) = d2K⊥1
(2π)2
∫
d2b e−iK⊥·b−h(b,E) (5)
with
h(b, E) =
∫
d3n(k)[1 − eik⊥·b] (6)
3
For large transverse momentum values, by neglecting the second term andusing a constant cut-off as lower limit of integration, the above expressioncoincides with the Sudakov form factor [19].
Eq. (5) has been applied in QCD [20, 21, 22, 23, 24] with
h(b, E) =16
3
∫ E αs(k2t )
π
dkt
ktln
2E
kt[1 − J0(ktb)] (7)
but its use is complicated by our ignorance of the IR behaviour of the theory.To overcome the difficulty arising from the IR region, the function h(b, E),which describes the relative transverse momentum distribution induced bysoft gluon emission from a pair of, initially collinear, colliding partons at LO,is split into
h(b, E) = c0(µ, b, E) + ∆h(b, E), (8)
where
∆h(b, E) =16
3
∫ E
µ
αs(k2t )
π[1 − Jo(bkt)]
dkt
ktln
2E
kt. (9)
Since the integral in ∆h(b, E) now extends down to a scale µ 6= 0, forµ > ΛQCD one can use the asymptotic freedom expression for αs(k
2t ). Fur-
thermore, having excluded the zero momentum region from the integration,Jo(bkt) is assumed to oscillate to zero and neglected. The integrand in Eq. (9)is now independent of b and the integral can be performed. In the range1/E < b < 1/Λ, the effective heff(b, E) is obtained by setting µ = 1/b [21].This choice of the scale introduces a cut-off in impact parameter space whichis stronger than any power, since the radiation function, for Nf = 4, is now[21]
e−heff (b,E) = [ln(1/b2Λ2)
ln(E2/Λ2)](16/25) ln(E2/Λ2) (10)
The remaining b-dependent term, namely exp[−c0(µ, b, E)], is dropped, a rea-sonable approximation if one assumes that there is no physical singularity inthe range of integration 0 ≤ kt ≤ 1/b. This contribution however reappearsas an energy independent smearing function which reproduces phenomeno-logically the effects of an intrinsic transverse momentum of partons. For mostapplications, this may be a good approximation. However, when the integra-tion in impact parameter space extends to very large-b values, as is the casefor the calculation of total cross-sections, the IR region may be importantand the possibility of a physical singularity for αs in the IR region becomes
4
relevant. It is this possibility, which we exploit in studying scattering in thevery large impact parameter region, b → ∞.
3. A proposal for the IR limit in the soft gluon integral
In this Section we discuss a phenomenological expression for the couplingof ultra soft gluons to the emitting quarks, and compare the resulting large-bbehaviour with the one discussed in the previous section.
Our choice for the IR behaviour of αs(Q2) used in obtaining a quantitative
description of the distribution in Eq. (6), is inspired by the Richardsonpotential for quarkonium bound states [25], as we have proposed in a numberof related applications [26]. Assume a confining potential (in momentumspace) given by the one gluon exchange term
V (Q) = K(αs(Q
2)
Q2), (11)
where K is a constant, calculable from the asymptotic form of αs(Q2). Let
us choose for Q2 << Λ2 the simple form
αs(Q2) =
B
(Q2/Λ2)p, (12)
(with B a constant), so that V (Q) for small Q goes as
V (Q) → Q−2(1+p). (13)
For the potential, in coordinate space, V (r) =∫
d3Q/(2π)3eiQ.rV (Q), Eq.(13)implies
V (r) → (1/r)3 · r(2+2p) ∼ C r(2p−1), (14)
for large r (C is another constant). A simple check is that for p equal tozero, the usual Coulomb potential is regained. Notice that for a potentialrising with r, one needs p > 1/2. Thus, for 1/2 < p < 1, this correspondsto a confining potential rising less than linearly with the interquark distancer, while a value of p = 1 coincides with the IR limit of the Richardson’spotential and is also found in a number of applications to potential estimatesof quarkonium properties [27].
5
Then, again following Richardson’s argument, we connect our IR limitfor αs(Q
2) to the asymptotic freedom region using the phenomenologicalexpression:
αs(k2t ) =
12π
(33 − 2Nf)
p
ln[1 + p(k2
t
Λ2 )p](15)
which coincides with the usual one-loop formula for values of kt >> Λ, whilegoing to a singular limit for small kt, and generalizes Richardson’s ansatzto values of p ≤ 1. The range p < 1 has an important advantage, i.e., itallows the integration in Eq.(6) to converge for all values of kt = |k⊥|. UsingEq. (15), one can study the behaviour of h(b, E) for the very large-b valueswhich enter the total cross-section calculation and recover the perturbativecalculation as well. The behaviour of h(b, E) in various regions in b-space wasdiscussed in [28], both for a singular and a frozen αs, namely one whose IRlimit is a constant. There we saw that, for the singular αs case, the followingis a good analytical approximation in the very large-b region:
h(b, M, ) =2cF
π
[
bb2Λ2p
2
∫ 1b
0
dk
k2p−1ln
2M
k+ 2bΛ2p
∫ NpΛ
1b
dk
k2p+1ln
M
k+ b
∫ M
NpΛ
dk
k
ln Mk
ln kΛ
]
=2cF
π
[
b
8(1 − p)(b2Λ2)p
[
2 ln(2Mb) +1
1 − p
]
+b
2p(b2Λ2)p
[
2 ln(Mb) − 1
p
]
+
b
2pN2pp
[
−2 lnM
ΛNp
+1
p
]
+ b lnM
Λ
[
lnln M
Λ
ln Np
− 1 +ln Np
ln MΛ
]
]
(16)
This approximation is valid in the region b > 1/(NpΛ) > 1/M , with Np =(1/p)1/2p, cF = 4/3 for emission from quark legs and b = 12π/(33 − 2Nf ).The upper limit of integration, M , indicates the maximum allowed transversemomentum, to be determined by the kinematics of single gluon emission as in[29]. The above expression exhibits the sharp cut-off at large-b values whichwe shall exploit to study the very large energy behaviour of our model. Onthe other hand, the possibility that αs becomes constant in the IR [21, 22, 23]in the same large b-limit leads to
h(b, M, Λ) = (constant) ln(2Mb) + double logs (17)
namely no sharp cut-off in impact parameter b, as expected.
6
4. The Bloch-Nordsieck(BN) model for the total cross-section
Our model for the total cross-section [1] is a modified mini-jet model, inwhich the rise with energy is driven by perturbative parton-parton scatter-ing [30], tempered by an energy dependent acollinearity effect. This effect isdue to kt-resummation of soft gluon emission from the initial state, hereafterreferred to as soft gluon kt-emission. The emphasis on resummation, first in-troduced for electron scattering by Bloch and Nordsieck [31], gives the modelits name. The model is built through the eikonal representation in impactparameter space, so as to satisfy unitarity, and allows to implement multipleparton scattering and to restore a finite size of the interaction through theimpact parameter distribution in the scattering hadrons. The details of themodel can be found in [1, 28, 32], here we shall recall some aspects relevantto its asymptotic energy behaviour.
In hadron-hadron scattering at a c.m. energy√
s, unitarity allows towrite a simple model for the total cross-section, namely
σtotal = 2
∫
d2b[1 − e−Imχ(b,s)cosℜeχ(b, s)] ≈ 2
∫
d2b[1 − e−n(b,s)/2] (18)
where the approximation on the r.h.s is obtained by neglecting the real partof the eikonal function (at the hadronic level, an acceptable approximationin the high energy limit) and 2Imχ(b, s) = n(b, s). The latter follows froma semiclassical argument relating σinelastic to a sum of a Poisson distributedindependent, single and multiple collisions.
We use perturbative QCD to calculate the cross-section in order to obtainthe average number of inelastic collisions. While implementing the QCDcalculation, albeit approximate, we distinguish between the average numberof collisions receiving contributions from hard physics processes and thosefrom non perturbative ones, and write n(b, s) in the form
n(b, s) = nNP (b, s) + nhard(b, s) (19)
where the non perturbative (NP) term parametrizes the contribution of allthose processes for which initial partons scatter with pt < ptmin, with ptmin
a suitable low energy cut-off for the QCD parton-parton cross-section. Weparametrize nNP (b, s), which establishes the overall normalization, and focusour attention on the hard term, which is responsible for the high-energy riseand which we expect to dominate in the extremely high energy limit. We
7
approximate this term as
nhard(b, s) = A(b, s)σjet(s). (20)
The QCD jet cross section drives the rise due to the increase with energy ofthe number of partonic collisions. It is calculated from the usual perturbativeQCD expression, with DGLAP evoluted parton densities and perturbativepartonic differential cross-sections. In Eq. (20), A(b, s) is the overlap func-tion which depends on the (energy dependent) spatial distribution of partonsinside the colliding hadrons, averaged over the densities [1, 32]. Before dis-cussing this function we shall examine the energy behaviour of the mini-jetcross-sections.
In the√
s >> ptmin limit, the major contribution to the mini-jet cross-sections comes from collisions of gluons carrying small momentum fractionsx1,2 << 1, a region where the relevant PDFs behave approximately likepowers of the momentum fraction x−J with J ∼ 1.3 [33]. This leads to theasymptotic high-energy expression for σjet
σjet ∝1
p2t min
[
s
4p2t min
]J−1
(21)
where the dominant term is a power of s. Fits to the mini-jet cross-sections,obtained with different PDF sets [34] confirm the value ε ≡ J − 1 ∼ 0.3.
Such energy behaviour as in Eq. (21) is at odds with the gentle riseof the total pp and pp cross-sections at very high energy, described ratheras ln s, ln2 s [11] or s0.08 power [2]. However, we shall see that a properimplementation of other QCD processes can modify this strong rise. To doso we now examine the energy behaviour of the next component of our BNmodel, the impact parameter distribution.
We have identified soft gluon kt-emission from the colliding partons asthe physical effect responsible for the attenuation of the rise of the totalcross section. These soft emissions break collinearity between the collidingpartons, diminishing the efficiency of the scattering process. Their numberincreases with energy and thus their contribution remains important, even atvery high energy, influencing matter distribution inside the hadrons, hencechanging the overlap function, which is proposed to be the Fourier transformof the previous expression for the soft gluon transverse momentum resummed
8
distribution, i.e,. we put
ABN (b, s) = N
∫
d2K⊥ e−iK⊥·bd2P (K⊥)
d2K⊥=
e−h(b,qmax)
∫
d2b e−h(b,qmax)= A0(s)e
−h(b,qmax)
(22)The integral in h(b, qmax) is performed up to a value qmax, which is linked tothe maximum transverse momentum allowed by kinematics of single gluonemission [29]. In principle, this parameter and the overlap function shouldbe calculated for each partonic sub-process, but in the partial factorization ofEq.(20) we use an average value of qmax obtained by considering all the sub-processes that can happen for a given energy of the main hadronic process[28].The energy parameter qmax is of the order of magnitude of ptmin. For presentlow−x behaviour of the PDFs, in the high energy limit, qmax is a slowly vary-ing function of s, starting as ln s, with a limiting behaviour which dependson the densities [35]. From Eqs. (16) and (21) one can estimate the verylarge s-limit
nhard(b, s) = ABN (b, s)σjet(s, ptmin) ∼ A0(s)e−h(b,qmax)σ1(
s
s0)ε (23)
and, from this, using the very large b-limit,
nhard(b, s) ∼ A0(s)σ1e−(bΛ)2p
(s
s0)ε (24)
with A0(s) ∝ Λ2 and with a logarithmic dependence on qmax, i.e. a veryslowly varying function of s. We also have
Λ ≡ Λ(b, s) = Λ{ cF b
4π(1 − p)[ln(2qmax(s)b) +
1
1 − p]}1/2p (25)
In the next section, we shall see how the two critical exponents of our model,namely the power ε with which the mini-jet cross-section increases with en-ergy and the parameter p dictating the IR behaviour of the QCD couplingconstant, combine to obtain a rise of the total cross-section in agreementwith the ln2 s limitation imposed by the Froissart bound.
5. Ultra soft gluons in the IR limit and the asymptotic limit ofthe total cross-section: the Froissart bound
In this section we consider the very large s-limit of the total cross-section,in the approximation that all constant (or decreasing ) terms in the eikonal
9
function can be neglected, thus studying only the QCD effects from mini-jetsand soft kt-resummation. We find a link between the infrared behaviour ofthe ultra soft gluons and the asymptotic Froissart-like behaviour of the totalcross-section and discuss it.
Let us consider the total cross-section in the eikonal representation atvery large asymptotic energies. At such large energies that nNP << nhard,the total cross-section in our model [1] reads
σT (s) ≈ 2π
∫ ∞
0
db2[1 − e−nhard(b,s)/2] (26)
We consider the asymptotic expression for σjet at high energies, which growslike a power of s, and ABN (b, s), which was obtained through soft gluonresummation, and which decreases in b-space at least like an exponential(1 < 2p < 2). In such large-b, large-s limit, we can write
nhard = 2C(s)e−(bΛ)2p
(27)
where 2C(s) = A0(s)σ1(s/s0)ε. The resulting expression for σT is
σT (s) ≈ 2π
∫ ∞
0
db2[1 − e−C(s)e−(bΛ)2p
] (28)
With the variable transformation u = (Λb)2p, and neglecting the logarithmicb-dependence in Λ by putting b = 1/Λ, Eq. (28) becomes
σT (s) ≈ 2π
p
1
Λ2
∫ ∞
0
duu1/p−1[1 − e−C(s)e−u
] (29)
Notice that, as s → ∞, C(s) also grows indefinitely as a power law. Thismeans that the quantity between square brackets I(u, s) = 1− e−C(s)e−u
hasthe limits I(u, s) → 1 at u = 0 and I(u, s) → 0 as u = ∞. Calling u0 thevalue at which I(u0, s) = 1/2 we then put I(u, s) ≈ 1 and integrate only upto u0. Thus
Λ2σT (s) ≈ (2π
p)
∫ u0
0
duu1−p
p = 2πu1/p0 (30)
and since, by construction
u0 = ln[C(s)
ln 2] ≈ ε ln s (31)
10
we finally obtain
σT ≈ 2π
Λ2[ε ln
s
s0
]1/p (32)
to leading terms in ln s. We therefore derive the asymptotic energy depen-dence
σT → [ε ln(s)](1/p) (33)
apart from a possible very slow s-dependence from Λ2. The same result isalso obtained using the saddle point method.
This indicates that the Froissart bound is saturated if p = 1/2. We shallnow show that, in our model, analyticity demands p > 1/2 and thus that,no matter how fast the mini-jet cross-section may grow with energy, theFroissart bound is always satisfied.
The requirement that p > 1/2 follows from analyticity arguments in thecomplex zs-plane, where zs = cos θs = 1 + 2t/(s − 4m2) → 1 + 2t/s is thecosine of the s-channel scattering angle for the equal mass case. Basically, thislimitation comes from asking that for large values of the impact parameterspace b, the eikonal be such to decrease at least like an exponential, i.e. tohave at least
Imχ(b, s) → e−b√
t0 (34)
where t0 > 0 is the boundary of the Lehmann ellipse on the real axis in thezs plane. To see the argument, consider the elastic scattering amplitude fora process a + b → a + b, normalized in such a way that
σtot = 2πImF (s, t = 0) (35)
with
F (s, t) = i
∫
d2b[1 − eiχ(b,s)]J0(b√−t) (36)
While in the s-channel physical region, and for equal mass particles, s ≥4m2 and t ≤ 0, for the Lehmann ellipse F (s, t) is analytic for t ≤ t0 ≤ µ2
where µ is some hadronic mass (e.g. twice the pion mass, as the smallestmass exchanged in the t-channel). The actual value is unimportant, what isneeded is t0 > 0. Then, for 0 < t ≤ t0 the argument of J0 becomes imaginaryso that
F (s, t > 0) = i
∫
d2b[1 − eiχ(b,s)]I0(b√
t) (37)
where I0 is the Bessel function of the second kind. The asymptotic behaviourof the Bessel functions is such that I0(y) for y real, grows exponentially as y
11
becomes large. The integral at the r.h.s. of the above equation has to existup to values of t = t0 and for fixed (0 ≤ t ≤ t0). Since, for large b-values,
I0(b√
t) ≃ eb√
t, (38)
for the integrand to be finite the imaginary part of the eikonal function,Imχ(b, s) must go to zero at least like an exponential, i.e. at least
Imχ(b, s) ⇒ e−b√
t0 (39)
Now, let us return to the expression for Imχ(b, s) from the BN model, whereImχ(b, s) ≃ e−h(b,s) up to exponential accuracy. Using the large-b behaviourof the function h(b, s), derived in previous papers and reproduced in theprevious section, we see that
Imχ(b, s) ≃ e−h(b,s) ≃ e−( b
b0)2p
(40)
whence follows that p > 1/2.
6. About scales and parameters in the BN model
The model we have described contains different scales. As mentionedearlier, in this paper we only study the rising part of the cross-section, whosebehaviour is proposed to come from processes for which the outgoing partonshave transverse momentum pt ≥ ptmin ≃ O(1 GeV ). In our model, ptmin isthe scale which separates perturbative scattering processes from everythingelse. Soft gluon emission introduces two more scales, namely Λ and qmax.The latter is the maximum transverse momentum in the integral for softgluon emission, it is of order ptmin and plays the role of the energy scale Ewhich appeared in QED radiative correction factors. Thus soft gluons satisfythe condition
kt ≤ qmax ≃ O(ptmin), (41)
since most of the parton-parton cross-section is peaked at pt = ptmin. Thenext scale Λ ≃ O(ΛQCD) separates the region of ultra soft gluons from therest. This region was originally neglected, on the basis that gluons with|k⊥| ≪ Λ would see the hadron as a point-like object [36] and such emissionswould have a small probability, because of colour screening. This argumentis appealing, and similar to the one mentioned in Sect. 2, but in our opin-ion, there is no compelling theoretical reason to assume that ultrasoft gluon
12
emission in high energy reactions has low probability. This argument couldbe applied to an isolated hadron, but not to high energy hadronic scatter-ing described through the scattering of partons, where soft gluon emission isstimulated by QCD interactions. It is through this interaction that we canexpect the transition between hadrons and quarks to arise. A singularity inthe infrared region would indeed provide a cut-off to separate quarks fromhadrons and lead to such transition. This is the rationale behind going intothe zero momentum region, and enter it with a singular confining couplingbetween ultra-soft gluons and the quark current. In our model, these ultrasoft gluons are important for the extremely large impact parameter values,which enter eikonal formulations of the total cross-section at very large en-ergy. For processes where such large-b values do not play a role, this regionmay be irrelevant, though.
The ultra soft gluon distribution which we have introduced depends on theparameter p which regulates the infrared region of the soft gluon integral ofEq. (7). In [1], we have used the value p = 3/4 but this is a phenomenologicalvalue which was obtained after performing various averages over the PDFs.A determination of the actual value of p is beyond the scope of this paper,except for the fact that p < 1 for the soft gluon integral to converge andp > 1/2 for analyticity of the scattering amplitude.
One can evaluate the coefficient of the (ln s)1/p term in Eq. (33) as givenby
C =2π
m2π
(mπ
Λ)2(
27(1 − p)2ε
4[1 + (1 − p) ln(2qmax/Λ)])1/p (42)
for Nf = 3, as appropriate for the total cross-section limit. In the approxi-mation in which the qmax term is neglected, Eq. (42) gives σT ≈ π/m2
π ln2 sfor p = 1/2, Λ = 100 MeV and ε = 0.3. However, qmax, which providesan extra dynamical scale, cannot, in general, be neglected: using Eq. (42)with qmax ≃ 1 GeV , the above equation gives C ≃ O(0.1)π/m2
π, namely aconstant which is one order of magnitude smaller than in the case of theactual Froissart bound [4].
We note that in the eikonal model as used here, one can derive the limitfor the inelastic cross-section [37] following the same steps as above obtaininga constant reduced by a factor 2, as in a black disk model.
13
Conclusions
Using an eikonal mini-jet model for the total cross-section we have shownhow soft gluon kt-resummation in the IR region can reduce the strong power-like rise due to the minijet cross-section. We have found that this model willalways satisfy the bound σtotal ≤ ln2 s provided that the infrared behaviourof αs reflect a rising one gluon exchange potential: for a potential risinglike r2p−1 the total cross-section is limited by an asymptotic behaviour ≃(ln s)1/p, with 1/2 < p < 1. This establishes the connection, in our BNmodel, between confinement and the satisfaction of limitations imposed bythe Froissart bound.
Acknowledgments
G.P. acknowledges enlightening discussions with E. Lomon, R. Ferrariand M. Procura, and thanks the MIT Center for Theoretical Physics forhospitality. This work was partially supported by the Department of Sci-ence and Technology, India, under the J.C. Bose fellowship, and by MEC(FPA2006-05294) and Junta de Andalucıa (FQM 101 and FQM 437).
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