[kutak, k.; surówka, p.] non-linear evolution of unintegrated gluon density at large values of...

7/24/2019 [Kutak, K.; Surwka, P.] Non-linear Evolution of Unintegrated Gluon Density at Large Values of Coupling Constant

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arXiv:1309

.3450v1

[hep-ph]

13Sep2013

arXiv:

Non-linear evolution of unintegrated gluon density atlarge values of coupling constant

Krzysztof Kutaka, Piotr Surowkab

aInstytut Fizyki Jadrowej im H. Niewodniczanskiego,Radzikowskiego 152, 31-342 Krakow, Poland

b Theoretische Natuurkunde, Vrije Universiteit Brussel,and International Solvay Institutes,

Pleinlaan 2, B-1050 Brussels, Belgium.

[email protected],

[email protected]

ABSTRACT

We propose an evolution equation for unintegrated gluon densities that is valid for thelarge values of QCD coupling constant s. Our approach is based on the linear resum-mation model introduced by Stasto. We generalize the model including non-linear termin the diffusive regime. The validity of the diffusive evolution at the strong coupling is

supported by the AdS/CFT consideration, as well as perturbative arguments. We solve theevolution equation numerically and extract saturation scale, which we compare with theweak coupling counterpart.
http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1http://arxiv.org/abs/1309.3450v1


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1 Introduction

Strongly coupled Quantum Field Theory like Quantum Chromodynamics is the basic frame-work which is used in the interpretation of hadronic observables data at the high-energyphysics experiments. Despite its correctness many open questions remain and the full the-oretical description is far from complete as neither perturbative methods, nor lattice gaugetheory, can provide full description of hadronic phenomena. One of the open problems isfor instance the gluon saturation [1] which is expected on theoretical grounds and thereis a growing evidence that it occurs[24]. Another open problem is the derivation of thedynamics of strongly coupled systems, such as quark-gluon plasma, directly from QCDlagrangian.

When energy is high enough quarks and gluons are elementary degrees of freedom inQCD. Therefore an essential ingredient to understand the collisions is the parton contentof the hadrons that are being collided. At present we do not have analytic methods to

derive parton distribution functions. We can either use perturbation theory for carefullychosen observables and resum infrared and collinear logarithms or use some simplifiedholographic model under analytic control.

A particularly interesting resummation approach, offering possible although not definiteinterpretation of low x data from the electron-proton collider HERA at DESY, was devel-oped by Balitsky, Fadin, Kuraev, and Lipatov (BFKL)[57]. The idea is that the scatteringprocess occurs through the exchange of the so-called Reggeized gluons. The two interactingReggeized gluons are known in the literature as the Pomeron. These are effective particlesemerging after resummation of (s ln

1x

)n. Such procedure gives an evolution equation forunintegrated gluon distribution functions schematically written as

f(x, k2

)ln x0/x

=K f(x, k2), (1.1)

where K is the evolution kernel and denotes convolution with respect to transversemomenta. The main prediction of the BFKL evolution is given by the hadronic cross-section of the form

sP, (1.2)where Pis known as the intercept. There regime of the applicability of the LO and NLOBFKL equation is limited to the weak coupling physics. Moreover, the BFKL leads topower-like growth of gluon density with energy. This is a consequence of the violation ofunitarity by the BFKL equation. The point at which the linear BFKL formalism has to

be corrected to include non-linear effects is known as saturation scale. Several approacheswere proposed to encapsulate parton saturation effects such as recombination or rescatter-ing [1, 817]. The common feature of these approaches is that the non-linearity takes intoaccount saturation effects.The BFKL equation, as observed in[18], can be useful in studies of infinite strong couplingeffects. It has been noted that with appropriate introduction of DGLAP anomalous di-mension into the BFKL framework, and after resummation of kinematical effects to infiniteorder one is able to extend formally the solution of the BFKL equation to large values of

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coupling constant. One can ask whether at the strong coupling the saturation effects willbe neccessary. In principle the strong coupling itself might suppress gluonic interaction. Inthis paper using the framework developed by Stasto, which allows to obtain gluon density

in the larg s regime, we conclude that it is not enough to use linear BFKL approach tosuppress gluons. As a way out we propose to use an appropriately resummed Balitsky-Kovchegov equation. The paper is organized as follows. In section2we review the basicsof the BFKL equation. In section 3 we solve the BFKL equation in the large values ofs and argue that the gluon distribution continues to evolve in a diffusive way. In section4 we further motivate the use of the diffusion approximation by holography. Finally insection5we show how the non-linear Balitsky-Kovchegov equation can be applied to studythe dynamics of unintegrated gluon densities at large values ofs. Moreover, we extracta saturation scale at strong coupling which is qualitatively similar to the result obtainedpreviously from gauge/gravity duality.

2 The BFKL equation in diffusion approximation

In this section we review some basics of the integral form of the BFKL equation at LOand its solution in a diffusion approximation [19, 20]. The diffusion form of the BFKLequation as turns out later is the form of BFKL at infinite strong coupling when kinematicalconstraint and DGLAP corrections are imposed. The forward BFKL equation written forthe unintegrated gluon density in the integral form reads:

f(x, k2) = f0(x, k2) + sk

2

1x/x0

dz

z

0

dl2

l2

f(x/z,l2) f(x/z,k2)

|l2 k2| +f(x/z,k2)

4l4 +k4

, (2.3)

wherexis a longitudinal momentum fraction carried by gluon kis its transversal momentumand we usek2 to indicate that there is no angle dependence the normalization of the gluon

is such that in the double logarithmic limit one has relationxg(x, Q2) =Q20

dk2f(x, k2)/k2.The LO BFKL equation due to conformal invariance can be solved by the Mellin transform.The Mellin transform w.r.t xand its inverse read:

f(, k2) =

10

dxx1f(x, k2), f(x, k2) = 1

2i

c+ici

dxf(, k2). (2.4)

Applying it to both sides of (2.3) and using 1

f(, k2) = 12i

c+i

ci

(k2)f(, )d (2.5)

one obtains

f(, ) = f0(, ) +s

()f(, ), (2.6)

1To simplify the notation we kept the same letter for Mellin transform w.r.t k2.

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where

() =

0

du

u

u 1|u 1| +

14u2 + 1

. (2.7)

The integral2.7after evaluation gives:() = 2(1) (1 ) (), (2.8)

where is a digamma function. The solution to the Eq. ( 2.3) can be written as:

f(x, k2) = 1

2i

d(k2)

1

2i

dx

f0(, )

s() . (2.9)

Taking for boundary condition

f0(x, k2) =f(x0, k

2), (2.10)

which corresponds in the Mellin to

f0(, ) =f(x0, )

, (2.11)

we arrive at the following expression:

f(x, k2) = 1

2i

d(k2)f(x0, )

x

x0

s(). (2.12)

In order to evaluate the integral above one needs to know the characteristic function alongthe imaginary axis in plane. The characteristic function is an analytic function given bythe above formula and its value along the imaginary axis can be easily obtained, the plot isshown on Fig. (1) where we plot the characteristic function along the real and imaginaryaxis for various values of the strong coupling constant. We also see that the larger value ofthe coupling constant the characteristic function diverges and from this one concludes thatthe LO BFKL equation can not be naively extended to the large strong coupling constantregime. Knowing that the characteristic function has a saddle point along the = 1/2 + icontour we can write the solution as:

f(x, k2) = 1

2

d(k2)1/2+if(x0, 1/2 +i)xs(1/2+i). (2.13)

The dimension full unintegrated gluon density reads:

F(x, k2) = F(x0, 1/2) 14 ln(x0/x)1/2

e ln(x0/x)1/2ln(k2/k20)e

ln(k2/k20)24 1/2 ln(x0/x) , (2.14)

whereF(x0, 1/2) = f(x0, 1/2)/k2 and the (1/2 +i) 122 with = s4 l n 2and = s(3). From this explicit form one may extract the coefficients of the diffusionequation 2

YF(Y, ) = 12

2F(Y, ) +1

2F(Y, ) + (+/8)F(Y, ). (2.15)

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0 0.5 1

5

10

15

20

s

3 2 1 1 2 3

6

4

2

2

4

6

Res12

Figure 1: Left: BFKL characteristic function multiplied bys along real axis. Right: BFKLcharacteristic function multiplied bys along imaginary axis. The evaluation is fors=0.2, 0.5,1, 3. Increasing values of y axis indicates the direction of growth ofs

In the above expression we used the following variables

Y = lnx0

x, (2.16)

= lnk2

k20. (2.17)

The variables (2.16) and (2.17) are convenient because they allow to write Eq. (2.15) in asimple diffusive form.

3 The BFKL equation with higher order corrections and the

gluon density in the whole range of coupling constantThe BFKL equation has been obtained at NLO accuracy in [21, 22] and recently solvedin [23]. However, it turns out that in order for the eigenvalue of the kernel to be stableone needs to perform resummations of corrections to infinite order [27]. One source ofsuch corrections is provided by the so-called kinematical constraint effects. The LO BFKLequation has been derived assuming strong ordering in energies of gluons emitted in thes-channel however the integral over the k is in principle unconstraint. The way to improvethese situation is to demand that the emitted gluons when z 1 are on shell. This putscertain restrictions on the k . The kinematical constraint refined equation reads:

f(x, k

2

) =f0(x, k

2

)+ sk

2

1x

dz

z

0

dl2

l2

f(x/z,l2)(l kz)(k/z l) f(x/z,k2)

|l2 k2| +f(x/z,k2)

4l4 +k4

,

(3.18)

and since it does not brake the conformal invariance the improved equation can be againsolved by the Mellin transform technique. Performing the transform with respect to x we

2For the details about the diffusion equation we refer the reader to the AppendixA.

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obtain3:

f(, k2) =f0(, k2) + s

k2

0

dl2

l2

f(, l2)(l kz)(k/z l) f(, k2)

|l2

k2

| +

f(, k2)4l4 +k4

.

(3.19)We have to combine contributions coming both from l2 > k2 and l2 < k2. Keeping thatinto account we perform the Mellin transform w.r.t k2 and obtain

f(, ) = f0(, ) +s

(, )f(, ), (3.20)

where

kc(, ) =

0

du

u

u+/2(1 u) +u/2(u 1) 1

|1 u| + 14u2 + 1

. (3.21)

We relegate the details of evaluation of the integrals to the appendix, the final result is:

kc(, ) = 2(1) (1 +/2) (+/2). (3.22)After rearranging (3.20) and using the inverse Mellin transforms w.r.t andwe obtain

f(x, k2) = 1

2i

d(k2)

1

2i

dx

f0(, )

skc(,, ) . (3.23)

The equations (3.23) define transcendental equation and its solution gives modified energydependence of the BFKL gluon density

= skc(, ) eff kc(, ). (3.24)Solving the Eq. (3.24) we see that the kinematical effects limits the growth of the eigenvalueto large values. We notice however that the eigenvalue along the imaginary axis is unlimitedfrom below see Fig. (2). Besides the kinematical constraint effects, as it has been suggestedin [18] in order to have more complete treatment of contribution of higher orders onemodifies last equation to the following one:

1

s=(0)()(, ). (3.25)

(0)

() =

1

+A() (3.26)

is the LO DGLAP anomalous dimension, where

A() = 1+ 1

+ 1

+ 2 1

+ 3 (2 +) +(1) +11

12 (3.27)

The equation (3.25) can be written as effective eigenvalue equation in a form

3For the technical details about the evaluation of such integrals we refer the reader to the Appendix B

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0. 0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

effk.c.,

4 2 0 2 4

3

2

1

0

1

2

Reeffk.c.12,

0. 0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

effk.c.,

4 2 0 2 4

1000

500

200

1

200

Reeffk.c.12,

Figure 2: Kinematical constraint effects. Upper right plot: function eff k.c.(, ) along realcontour for s = 0.2, 0.5, 1, 2. Upper left plot: function eff k.c. along imaginary contour for

s = 0.2, 0.5, 1, 2. Lower left plot: functioneff k.c.(, ) along real contour fors = 2, 10, 100.Lower right plot: functioneff k.c.(, ) along real contour fors = 2, 10, 100.

= ef f(, ), (3.28)

withef f(, ) = skc(, ) (1 +A) . (3.29)

The crucial behavior providing the graviton-like intercept is the vanishing of the eigenvaluefunction when 1. As a practical application of the result obtained in [18] we ask aquestion about the properties of the gluon density while evaluated at the increasing values

of strong coupling. To obtain gluon density we know from previous sections that we need toknow the eigenvalue function along the imaginary axis to perform inverse Mellin transform.Since the analytical solution is not possible we solve the equation above numerically alongthe imaginary axis

= Re (ef f(1/2 +i, )) (3.30)

We see (Fig. 5) that the additional contributions stabilize completely the eigenvalue andallow for investigations of the BFKL in the whole spectrum of coupling constant. Thisstems from the fact that there is no divergency after taking the linit S . Keeping

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0. 0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

eff,

4 2 0 2 4

1

0

1

2

Reeff12,

0. 0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

eff,

4 2 0 2 4

1

0

1

2

Reeff12,

Figure 3: Kinematical constraint effects and resummation effects. Upper right plot: functionef f(, ) along real contour fors = 0.2, 0.5, 1, 2. Upper left plot: function ef f(, ) along

imaginary contour fors = 0.2, 0.5, 1, 2. Lower left plot: functionef f(, ) along real contourfors = 2, 10, 100. Lower right plot: functionef f(, ) along real contour fors = 2, 10, 100.

this in mind we can ask a question what is the shape of the gluon density as we flow to thelarger values of the coupling constant. Naively one can think that if the coupling constantincreases, there the number of gluons vanishes or at least it is constant. Below we show thatthis is not the case, the gluon density grows, and we get infinitely many soft gluons. Thus inorder to achieve the stabilization one has to include some additional effects, most probablyof non-linear type. The solution for various values of s of the transcendental equationalong real and imaginary axis is presented on Fig. (4). In order to obtain the gluon densitywe interpolate the solution and integrate it numerically. The resulting gluon density fordifferent values of the coupling constant is shown on Fig. (4). The quite simple structureof the solution of eigenfunction equation for omega can be parameterized in polynomial of

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0 2 4 6 8 10k

2

1

3

5ln10 Fx,k

2

0 2 4 6 8 1 0k

2

1

3

5ln10 Fx,k

2

Figure 4: Gluon density obtained for various values of the coupling constant s =0.2, 0.5, 1,10, 104. The densities with smaller coupling constants are smaller. Upper leftx= 106, upper right: x= 104.

where An are fit parameters:

ef f(, 1/2 +i) =N

n=M

Ann (3.31)

Applying this prescription we obtain at infinite value of the strong coupling 4 the followinganalytical formula:

ef f(, 1/2 + i) = P10()(+ 0.683)(0.683 )(0.683)(0.683), (3.32)

where the tenth order polynomial P10() takes the form

P10() = 0.998873 2.013192 + 15.90084 154.0396 + 540.2088 657.20310. (3.33)The form of the eigenvalue function used in order to evaluate gluon density can be simplified.By inspection we see Fig. (5, right) that if for the evaluation of the gluon density we usesimplified fit (3.34) the resulting gluon almost does not change. The reason of this is thatthe net contribution to the integral from regions of the eigenvalue function where it isnegative and where the functions start to differ is negligible see Fig. (5, left).

ef f(, 1/2 +i) = 1.02795 2.046352 st 12

st2 (3.34)

The formula above can be used to obtain analytically a solution of the BFKL equationin strong coupling regime and to deduce a partial differential equation which it obeys:

Y(Y, ) =1

2st

2 (Y, ) +

1

2st(Y, ) + (st+

st/8)(Y, ) (3.35)

where the values are read off from formula (3.34) st = 4.08 st= 1.02

4It works also at finite values

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2 2

1

0

1

2

eff12i,

2 4 6 8 10k

2

0.5

1.0

1.5

2.0

2.5

3.0

3.5N ratio

Figure 5: Comparison of from formula (3.32) and (3.34). Right: ratio of gluon densities fordifferent values of x as obtained from (3.32) and (3.34. In order to visualize the tiny effect ofdependence on exact integration path we plot ratios and shift the appropriate curve by 2 and by 3respectively.

4 AdS/CFT and the PomeronThe main message that comes from our discussion so far is that the diffusive behavior isnot particular to weak coupling regime of the BFKL evolution. In fact it is the dominantcontribution in the strong coupling. This is not an artefact of the model we are considering.Similar observation was done in the context of gauge/gravity duality [2830] where ananalytic expression for the gluon distribution related function (or an evolution kernel) wasderived inN= 4 SYM theory

fPSBC(, s) 14DYe

j0Ye2

4DY , (4.36)

where

j0= 2 2

+O(1/) , D= 12

+O(1/) . (4.37)

and we introduced notation fPSBC to indicate that the function is not directly gluon dis-tribution function but related to it after rescaling by k1/2 [31]

fPSCB (Y, )

Y = D

2fPSCB(Y, )

2 +j0fPSCB(Y, ). (4.38)

We note that inN= 4 SYM has several remarkable properties, such as integrability of theevolution equation or the so-called maximal transcendentality property [32]. These prop-

erties allow to calculate the spectrum of anomalous dimensions and the Pomeron interceptto a high order in the inverse coupling expansion [3337]. Therefore the most activity inthe subject was devoted to the understanding of linear BFKL equation. At present thedetailed studies of unitarization is beyond reach as one has to resum multi-loop string am-plitudes to all orders. However, under some simplified assumption the saturation line hasbeen extracted from holographic considerations [38, 39]. In the next section we will showthat this result is consistent with the saturation line for the model we are considering.This is a striking feature of the strongly coupled physics, despite the differences between

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5 0 5 10 15 20

1

2

Y,

0 5 10 15 20

0.0002

0.0004

0.0006

0.0008

0.001

FY,

Figure 6: Left figure: solution of strongly coupled BK equation for Y=10, 20,30, 40. Right figure:the corresponding dipole gluon density.

QCD andN= 4 SYM they seem to share common properties such as validity of diffusiveapproximation and quantitatively similar saturation properties.

5 The BK equation in the limit of infinite coupling constant s

The linear BFKL evolution equation misses a very important aspect of the high-energy scat-tering, namely the saturation physics. As pointed out in the introduction several approacheswere constructed in order to include non-linear effects, like multiple scattering and gluonsaturation, responsible for the unitarization of the scattering amplitudes. A particularlyuseful and simple enough approach to unitarize the cross section is the Balitsky-Kovchegov

equation, it reads: (x, k2) = 0(x, k2) + 1(x, k

2), (5.39)

where

1(x, k2) = s

1x

dz

z

0

dl2

l2

l2(x/z,l2) k2(x/z,k2)

|l2 k2| +k2(x/z,k2)

4l4 +k4

s

R22(x/z,k2)

.

(5.40)We note that if one neglects the non-linear term one recovers the linear BFKL equation.An important feature of the BK equation as observed in [40] is that it lies within theuniversality class of the Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) equation:

tu(t, x) =2xu(t, x) +u(t, x)

u2(t, x). (5.41)

One can view this equation as a diffusion equation supplemented with a non-linear termthat encodes saturation. The question arises how to extend the BK equation to the wholestrong coupling regime. We do not have an answer yet and at this point we do not havea derivation of such equation from first principles. Nevertheless we can postulate suchextension based on our numerical analysis and phenomenological arguments

Y(Y, ) =1

2st

2 (Y, ) +

1

2st(Y, ) + (st+

st/8)(Y, ) NcsR2

2(Y, ), (5.42)

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where the values are read off from formula (3.34) st = 4.08 st= 1.02The coefficient in front of the non-linear term has to be consistent with the strong-

coupling. We take the t Hooft limit (Ncs ) and assume large target approximation

(R

2

), the ratio Ncs

R2 being fixed. The solution of the above equation is presented onFig. (6) and it shows that at some point where the shape of the curve flattens the numberof gluons saturates. We notice that the gluon density becomes constant in the saturatedregime therefore the derivatives vanish and we obtain from (5.42) the saturation scale:

sat =st+

st/8 (5.43)

0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8Y

Figure 7: Red line: saturation line obtained from solution of (5.42) using the definition (5.45).Blue line: saturation scale as follows from the weak coupling equations and models: Qs(Y) e0.29Y

The gluons, however, do not contribute much to the momentum distribution since themomentum gluon density or dipole gluon density which is calculated via:

FBK(Y, ) = Nc4

2(Y, ) (5.44)

drops off after the saturation scale has been reached and its maxima signalize emergenceof the saturation scale which can be defined as[41]:

FBK(Y, )|=lnQs2(Y) = 0 (5.45)using the formula we can calculate the saturation line that follow from our equation andwe obtain

Qs e0.85Y (5.46)The behavior of the gluon number density (Y, ) is to be contrasted with full form of

the BK in the weak coupling regime where rate of production of gluons slows down but stilldiverges logarithmically and can only approximately obeys FKKP equation.

The result for the saturation scale suggests that at the same values of ln( 1x

) saturationstrong correlations occur earlier than at weak coupling. This can be easily understood sincethe strong coupling tends to pack gluons closer and therefore the overlapping or screeningis larger. The result (5.46) is quite close to the one obtained in [38, 39] by very differentholographic approach, therefore justifying our derivation.

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6 Conclusions

Parton distribution functions are important objects in high-energy phenomenology as theyare necessary ingredients to determine physical observables. In this paper we proposedan evolution equation gluon density obeys when the coupling constant is very large. Westay within a QCD framework that, through certain resummations, allows to probe strongcoupling physics. Our proposal stays within the diffusive regime, in which we introducesaturation physics. Solving this equation we are able to extract saturation scale, whichagrees qualitatively with results from holography. For the future plans we postpone thestudy of the running coupling effect as well as possible extensions of the framework towardsthe whole range of strong coupling.

One very important aspect of high-energy physics is the generation of entropy after thecollision. For this phenomenon we lack theoretical tools that can handle the dynamics inQCD. Therefore it is often convenient to consider the same questions in the context of

strongly coupled plasma in theN = 4 supersymmetric gauge theory for which one canuse the AdS/CFT correspondence[42]. However, these methods will always be restrictedto some universal properties that QCD andN = 4 SYM share. In the context of QCDit has been suggested that the notion of a thermodynamical entropy is associated withthe production of gluons in the saturation regime of dense initial states in hadron-hadroncollisions [43]. Later a microscopic definition of entropy was given in [44] in which thenotion of gluon distribution function plays a crucial role. It will be interesting to employsimilar ideas to study entropy generation in the model presented in this paper.

Acknowledgments

We would like to thank Dimitri Colferai for useful correspondence and Anna Stasto forinteresting discussions. PS acknowledges the hospitality and partial support from the Insti-tute of Nuclear Physics, Polish Academy of Sciences where the work has been completed.The work of KK was supported by the NCBiR grant LIDER/02/35/L-2/10/NCBiR/2011.The work of PS was supported in part by the Belgian Federal Science Policy Office throughthe Interuniversity Attraction Pole IAP VI/11 and by FWO-Vlaanderen through projectG011410N.

A Diffusion equation

Following [45] we intend to solve an equation of the form

w(t, x)

t =a

2w(t, x)

x2 +b

w(t, x)

x +cw(t, x). (A.47)

The substitutionw(t, x) = exp(t +x)u(t, x), (A.48)

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with

= c b2

4a, and = b

2a, (A.49)

which leads to the homogenous heat equation for u(t, x)u(t, x)

t =a

2u(t, x)

x2 . (A.50)

As a next step we solve a Cauchy problem on the domain x with the initialcondition

u(0, x) =f(x). (A.51)

The solution can be written as

u(t, x) =

df()G(t,x,), (A.52)

where the Greens function is

G(t,x,) = 1

2

atexp

(x )

2

4at

. (A.53)

Iff(x) =(x) we get the solution as a Gaussian function

u(t, x) = 1

2

atexp

x

2

4at

. (A.54)

B Integrals

In this appendix we collect useful formulae needed in order to evaluate the function.

A1=

0

du

u

u+2

|1 u|(1 u). (B.55)

A2=

0

du

u

u2

|1 u|(u 1) = 10

dvv+

2

|1 v| . (B.56)

A3 =

1

0

du 1

u(1 u)=

1

0

du 1

1 u

1

0

du

u. (B.57)

TheA3 integral we split into

A3a= 10

du 1

1 u, (B.58)

A3b = 10

du

u, (B.59)

13


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A4 =

1

du 1

u(u 1)= 10

dv 1

1 v , (B.60)

A5 =

0

du

u

1

4u2 + 1, (B.61)

10

u+21 1

1 u du 10

1

u 1du = (1) (1 +

2)

10

1

u 1du. (B.62)

We combine the integrals in the following way:

A1+A4=(1) (+/2), (B.63)

A2+A3a= (1) (1 +/2). (B.64)Introducing regulators in the remaining integrals we get

A5+A3b=

0

duu

u 14u2 + 1

1

0

duu

u =21

12 2 2

1

. (B.65)

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[kutak, k.; surówka, p.] non-linear evolution of unintegrated gluon density at large values of...

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