# Fluctuations of the gluon distribution from the small-x effective action

###### Abstract

The computation of observables in high energy QCD involves an average over stochastic semi-classical small- gluon fields. The weight of various configurations is determined by the effective action. We introduce a method to study fluctuations of observables, functionals of the small- fields, which does not explicitly involve dipoles. We integrate out those fluctuations of the semi-classical gluon field under which a given observable is invariant. Thereby we obtain the effective potential for that observable describing its fluctuations about the average. We determine explicitly the effective potential for the covariant gauge gluon distribution both for the McLerran-Venugopalan (MV) model and for a (non-local) Gaussian approximation for the small- effective action. This provides insight into the correlation of fluctuations of the number of hard gluons versus their typical transverse momentum. We find that the spectral shape of the fluctuations of the gluon distribution is fundamentally different in the MV model, where there is a pile-up of gluons near the saturation scale, versus the solution of the small- JIMWLK renormalization group, which generates essentially scale invariant fluctuations above the absorptive boundary set by the saturation scale.

## I Introduction

High-energy scattering in QCD at fixed transverse momentum scales
probes strong color fields, i.e. the regime of high gluon
densities GLR (). In the high-energy limit physical observables,
such as the forward scattering amplitude of a dipole from a hadron or
nucleus, are typically expressed in terms of expectation values of
various Wilson line operators ; see, for example,
Ref. Weigert:2005us (). The expectation value
corresponds to a statistical average^{1}^{1}1Kovner describes this as
an average over the Hilbert space of the target, i.e. that the
weight which determines the probability
for a given configuration of is analogous to the modulus
squared of the wave function of the target Kovner:2005pe ().
over the distribution of “small-x gluon fields”. For example, if
quantum corrections are neglected this distribution is commonly
described by the McLerran-Venugopalan (MV) model MV ():

(1) | |||||

(2) |

Here, is the covariant gauge classical field (describing the
small-x gluon fields) sourced by the random valence charge density
which one averages over. corresponds
to the average color charge density squared per unit transverse area
and is the only parameter of the model; it is proportional to the
thickness of the nucleus . The expectation value of an
electric Wilson line , for example, is then computed
as^{2}^{2}2 is power divergent in the IR
and so requires a cutoff. We simply write the formal
Eq. (3) to illustrate the averaging procedure. The
dipole probe from Eq. (4) does not exhibit such a
power-law divergence in the IR.

(3) |

The forward scattering amplitude of a quark - antiquark dipole of size is given by

(4) |

We employ hermitian generators. The size where grows to order 1 defines the (inverse) saturation scale . In the MV model one finds that . For transverse momenta the Fourier transform of the forward scattering amplitude defines the dipole unintegrated gluon distribution

(5) |

Quantum corrections to the MV model modify the statistical weight . Ref. Iancu:2002aq () proposed a Gaussian “mean-field” approximation for at small light-cone momentum fractions (far from the valence sources) which reproduces the proper gluon distribution (or dipole scattering amplitude) both at small () as well as at high () transverse momentum:

(6) |

This non-local Gaussian can be rewritten in -space as^{3}^{3}3Note
that we define .

(7) | |||||

This action reproduces the correct dipole scattering amplitude and Weizsäcker-Williams gluon distribution in the short distance (high transverse momentum) limit, c.f. ref Iancu:2002aq (), with

(8) |

Here, is the BFKL anomalous dimension bfkl () (in the presence of a saturation boundary Mueller:2002zm ()). and are evaluated at the rapidity of interest (like in the MV model is again proportional to the thickness of the nucleus ). We will not spell out this dependence on explicitly since our focus here is not on the growth of with which is well known. For the present purposes the most important effect of the resummation of quantum fluctuations is that increases with transverse momentum when .

The paper is organized as follows. In Sec. II we present the basic idea for computing an effective potential for a given observable by introducing a constraint into the functional integral. In Sec. III, in order to illustrate the approach with a simple example we compute the effective potential for the number in the MV model on a single site. We then compute the effective potential for the covariant gauge gluon distribution function in Sec. IV. We proceed to calculate the fluctuations of the gluon multiplicity and of the average squared transverse momentum in Sec. V. In Sec. VI we present results of numerical Monte-Carlo simulations within the MV model and for the solution of the JIMWLK renormalization group equation. We end with a discussion and outlook in Sec. VII.

## Ii The basic idea: introducing the constraint effective potential

Expectation values such as those written in Eqs. (3,4) refer to a statistical average of an observable over all configurations from the ensemble . On the other hand, we may be interested in the value of an observable for a specific subset of configurations such as configurations with a high number of gluons or with a specific unintegrated gluon distribution. These represent more “global” measures averaging over all fluctuations of which do not affect, say, the unintegrated gluon distribution. In other words, our goal is to perform the integral over subject to the contraint that, for example, is fixed, thereby decomposing the space of all , or , into invariant subspaces (w.r.t. the given observable).

We illustrate the fluctuations of the gluon distribution originating from the fluctuations of the classical valence color charge density in Fig. 1. For simplicity we show a simple example corresponding to the fluctuations of the color charge representation of a system composed of a quark and an anti-quark. The MV model describes the fluctuations of a system of many valence charges in a high-dimensional representation about the most likely representation JeonVenugopalan (); Petreska_rho4 ().

The logarithm of the inverse of the partition function obtained after
integrating out the orthogonal fluctuations of then defines an
effective potential^{4}^{4}4More generally, this would give the
effective action for the field . for
:

(9) | |||||

(10) |

The stationary point of corresponds to the
extremal gluon distribution . In the limit of an infinite
number of degrees of freedom, i.e. the large- limit in our
case^{5}^{5}5To go beyond the large- limit one would have to
actually compute the integral over in Eq. (9)
which can be done by means of a Legendre transformation.,
of course is equal to the expectation value of . Away from the stationary solution, the potential
provides insight into the form of fluctuations
about the extremum. Specifically, we shall analyze the correlation
of fluctuations of the number of gluons (above the saturation
scale) and their typical transverse momentum.

Fluctuations of various observables induced by the fluctuations of have been analyzed before. For example, the multiplicity distribution of gluons with transverse momenta above Gelis:2009wh () and the fluctuations of the real and imaginary parts of spatial Wilson loops Lappi:2014opa () in the central region of a collision of two sheets of color charge have been analyzed. Angular harmonics of the dipole scattering amplitude for random individual configurations of the target have been shown in Ref. Dumitru:2014vka () (the ensemble average is, of course, isotropic). The evolution of the imaginary part of the dipole -matrix, i.e. of the odderon , has been discussed in Ref. Lappi:2016gqe (). Here, we describe how one can explicitly integrate out the fluctuations of or under which a given observable is invariant in order to derive an effective potential for the observable itself. We apply our method specifically to compute the effective potential for the covariant gauge gluon distribution from which we deduce the correlation of fluctuations of the multiplicity of hard gluons and of their typical transverse momentum. A somewhat similar procedure was previously used to compute the density matrix of the soft gluon fields and the associated entanglement entropy, see Ref. Kovner:2015hga ().

## Iii Warm-up: effective potential for

We begin with the effective potential for on a single site. The procedure is essentially identical to that used in sec. III of Ref. Dumitru:2004gd () to compute the effective potential for Polyakov loops in a single-site matrix model.

The partition function for a single site is

(11) |

denotes the random color charge density at the site and is an element of the algebra of the color group in the fundamental representation: . We shall keep only contributions to of order and drop terms of order 1.

The goal is to write (11) in the form

(12) |

where is the effective potential for . In other words, is the partition sum for scalars satisfying the constraint .

To compute we introduce a -function constraint in Eq. (11),

(13) | |||||

The integral for is easily computed in spherical coordinates,

(14) | |||||

In the last step we have dropped an irrelevant -independent normalization factor. This expression for then leads to

(15) |

Hence, the effective potential is given by

(16) |

The stationary point of this potential is

(17) |

Of course, this result can be obtained directly from the correlator which follows from the action in Eq. (11).

## Iv Effective potential for the gluon distribution

In this section we compute the effective potential for the gluon distribution obtained from the field in covariant gauge. We will also comment briefly on the potential for the gluon distribution obtained from the light-cone gauge field .

It is convenient to work directly in momentum space. The partition function of the Gaussian model is taken as

(18) |

The constraint is implicit. In general, may depend on the transverse momentum which would correspond to a non-local Gaussian action in coordinate space; this dependence is also left implicit from now on since it is not essential for the following steps.

Our goal is to obtain an expression of the form

(19) |

for

(20) |

where and . Since we can write the partition sum as

(21) | |||||

The first line of the equation above is obtained from the original partition sum (18) by inserting a -functional

(22) |

which fixes . To compute we again introduce the constraint field ,

(23) | |||||

Inserting this into Eq. (21) we obtain

(24) | |||||

In the last step we have taken the continuum limit, is the transverse area covered by the integration over the impact parameter in Eq. (20). The effective potential for the function is therefore

(25) |

and its stationary point corresponds to the average gluon distribution (at order )

(26) |

The contribution to at zeroth order in can be restored by comparing to obtained directly from the Gaussian two-point function:

(27) |

Hence, to restore the contribution to Eq. (25) should be modified to

(28) |

We shall mostly focus on Eq. (25) in what follows but refer to (28) when accuracy of the gluon distribution would be needed.

The two-point correlator of the color charge, averaged over all of its fluctuations, follows from the action and is written in Eq. (27). In order to explicitly split off the fluctuations of at fixed we can write this in the form

(29) |

Replacing the integration over by the extremal solution reproduces the correlator from Eq. (27). This last expression should be useful for future applications where one may want to explicitly isolate the fluctuations of the gluon distribution from more complicated expressions involving two-point functions of .

We briefly pause our derivation at this point to comment on the potential describing fluctuations of the Weizsäcker-Williams gluon distribution defined via the light-cone gauge field . Because of the non-linear dependence of on we are unable to compute the effective potential analytically except in the weak field regime where . Hence, in this regime both the diagonal as well as the off-diagonal components of the WW gluon distribution, and , respectively, are equal to . The effective potential for these distributions is therefore again given by Eq. (25) with the replacement in the first term of the integrand.

As a second aside we briefly illustrate the modifications due to adding a quartic color charge operator to the quadratic action. We choose a particularly simple form in order to be able to compute the effective potential exactly without having to resort to a perturbative expansion:

(30) |

This replaces eq. (21) by

(31) | |||||

(32) |

Hence, in this case

(33) |

In the MV model , where denotes the thickness of the nucleus, while the coupling for the quartic color charge density operator involves two additional powers of Petreska_rho4 (). Such a quartic in operator therefore represents a higher order correction in the high gluon density power counting scheme where , c.f. next subsection. Moreover, in fig. 5 below we shall show that the exact numerical solution of the LO small- evolution equation agrees rather well with the effective potential for the gluon distribution derived from a quadratic action. We will therefore neglect in what follows.

We now return to our discussion of the fluctuations of in the model with a quadratic action and write

(34) |

and expand to quadratic order in . This “one loop” approximation leads to

(35) | |||||

(36) |

However, it is clear from the form of that the quadratic approximation can not describe fluctuations far from the extremal solution . We therefore follow a different route. We introduce the fluctuation field through

(37) |

with as written in Eq. (26). A fluctuation from the extremal “path” has action

(38) | |||||

This is a Liouville action (without kinetic term and with negative
Ricci scalar) for the field in two dimensional
-space^{6}^{6}6In Ref. Iancu:2007st () Iancu and McLerran
proposed a Liouville action to describe the fluctuations of in
the transverse impact parameter plane (-space) due to stochastic
high-energy evolution; this is unrelated to our discussion of
fluctuations in the ensemble of gluon distributions which
occur even at fixed , as considered here, and exist even in the
absence of QCD evolution (MV model).. Indeed, the canonical
dimension of the fluctuation field as introduced in
Eq. (37) is zero. This will become important below to
understand the spectrum of fluctuations from small- evolution.

In the following section we use expression (38) to analyze the correlation of gluon number and transverse momentum fluctuations.

### iv.1 Parametric dependence on the number of colors and on the thickness of the target

In this subsection we discuss the parametric dependence of the fluctuations on and on the thickness of the target nucleus which is proportional to the third root of its atomic number, . In particular, we outline that the fluctuations of the gluon distribution considered here are of the same order in as the “extremal” (or average) gluon distribution , and of the same or lower order in . As explained by Kovchegov Kovchegov:1999ua (), quantum evolution at leading order applies when with . The latter condition implies that contributions which do not exhibit longitudinal coherence, i.e. those which are not proportional to the thickness of the nucleus, in this power counting scheme formally correspond to higher order corrections.

Recall from the previous section that the average gluon distribution . The action (25) evaluated at is (times a numerical factor equal to zero in dimensional regularization in dimensions). This corresponds to the action of classical gluon fields times a factor of from the coupling to the sources (see Fig. 1).

In order to be able to evolve initial fluctuations to small using leading order evolution these fluctuations must also be of order . This is satisfied since the effective action (38) for the fluctuation field does not involve the thickness explicitly. Indeed, the MV model MV () outlined in the Introduction describes precisely these longitudinally coherent valence color charge fluctuations. In other words, fluctuations corresponding to a penalty action which is independent of are of the same order in as the average gluon distribution and can be evolved to small . However, one can not study fluctuations with a suppression probability such that since that would correspond to and . Such fluctuations are of higher order in the coupling Kovchegov:1999ua ().

Power counting in proceeds along similar lines. is explicitly proportional to , so corresponds to fluctuations at the same order in as the average gluon distribution . These can be selected by an external “trigger” probability such that . However, it is allowed to select less suppressed fluctuations corresponding to provided such terms in the effective action are accounted for, c.f. Eq. (28).

## V Gluon multiplicity and transverse momentum fluctuations

In this section we analyze fluctuations of the semi-hard gluons above the saturation momentum and up to a maximum momentum scale . The number of such gluons for a given is given by

(39) |

The integral extends from up to . As already mentioned above in the linear regime approaches the Weizsäcker-Williams gluon distribution , so counts the number of Weizsäcker-Williams gluons from to . We focus first on the MV model with =const; analogous results for a -dependent shall be summarized at the end of this section.

The number of additional gluons due to the fluctuation about the extremal gluon distribution is given by

(40) |

This quantity does not depend on the UV cutoff because the fluctuation has finite support in order to have a finite action.

The average (squared) transverse momentum of gluons between and can be defined through (see analogous discussion in Ref. Baier:1996sk ())

(41) |

Here refers to an average over the transverse momentum distribution for a given gluon distribution but not to an average over all configurations of . Once again we subtract the value at the saddle point,

(42) |

We now proceed to discuss the effect of fluctuations, . Our strategy is to introduce a trial function for for which we then evaluate , , and the “penalty action” via Eq. (38). Consider the ansatz

(43) |

Thus, the fluctuation has support on the interval within the window , i.e. , with . Also, by dimensional analysis the multiplicative fluctuation can depend only on since is the only dimensionful scale in the MV action (2). We recall from our discussion in sec. IV.1 that, parametrically,

(44) |

so that for , is of the same order in and as the average gluon distribution .

For a fluctuation of the form (43) the excess gluon multiplicity is given by

(45) |

The excess (squared) transverse momentum of gluons with transverse momentum above the saturation scale is given by

(46) |

or

(47) |

We have simplified the expression by linearizing in the fluctuation amplitude. The factor of in this equation arises due to the fact that we only integrate over gluons with with . According to Eq. (47) the average squared transverse momentum due to the fluctuation is proportional to the excess number of gluons it contains. In the next section we shall confirm such a tight nearly linear correlation of and via Monte-Carlo simulations.

Finally, the penalty action for such a fluctuation is

(48) |

The goal now is to pay as low a price as possible while maximizing and . Fluctuations with , corresponding to increasing , come with a large penalty . In fact, even a flat with corresponds to while, at the same time, and increase only logarithmically with . Similarly, fluctuations with , which drop off very rapidly with , give small , but also a small multiplicity excess . Therefore, we expect that in the MV model the dominant “high multiplicity” fluctuations would have a high- tail corresponding to .

We now turn to a -dependent as written in Eq. (8). This corresponds to the non-local Gaussian approximation to the JIMWLK action at small proposed in Ref. Iancu:2002aq () which accounts for the small- anomalous dimension. Here, the gluon excess above is

(49) |

while the additional transverse momentum contributed by the fluctuation is