High energy factorization
in nucleusnucleus collisions
Abstract
We derive a high energy factorization theorem for inclusive gluon production in A+A collisions. Our factorized formula resums i) all order leading logarithms of the incoming partons momentum fractions, and ii) all contributions that are enhanced when the color charge densities in the two nuclei are of order of the inverse coupling– . The resummed inclusive gluon spectrum can be expressed as a convolution of gauge invariant distributions from each of the nuclei with the leading order gluon number operator. These distributions are shown to satisfy the JIMWLK equation describing the evolution of nuclear wavefunctions with rapidity. As a byproduct, we demonstrate that the JIMWLK Hamiltonian can be derived entirely in terms of retarded light cone Green’s functions without any ambiguities in their pole prescriptions. We comment on the implications of our results for understanding the Glasma produced at early times in A+A collisions at collider energies.

Theory Division, PHTH, Case C01600, CERN,
CH1211, Geneva 23, Switzerland

Institut de Physique Théorique (URA 2306 du CNRS)
CEA/DSM/Saclay, Bât. 774
91191, GifsurYvette Cedex, France

Physics Department, Brookhaven National Laboratory
Upton, NY11973, USA
Preprint IPhTT08/068, CERNPHTH2008074.
1 Introduction
Collinear factorization theorems [1] that isolate long distance nonperturbative parton distribution functions from perturbatively calculable short distance matrix elements are central to the predictive power and success of QCD. These theorems can be applied to compute inclusive crosssections of the form , where is a set of heavy particles or jets with invariant mass and corresponds to the sum over all possible states (including soft and collinear hadrons) that can accompany the object . This crosssection, for center of mass energy , can be expressed as [2, 3, 4, 5, 6]
(1) 
In this equation, are the nonperturbative “leading twist” parton distribution functions which gives the distribution of a parton in the hadron , as a function of the longitudinal momentum fraction evolved up to the factorization scale , while the hard scattering matrix element can be computed systematically in a perturbative expansion in powers of , where is the QCD coupling constant. Higher twist contributions to this formula are suppressed by powers of the hard scale . This factorization formula is valid in the Bjorken limit when (where MeV is the intrinsic QCD scale).
Our interest here is instead in a different regime of high energy scattering where, for fixed invariant mass , one takes and thus . We shall call this the Regge–Gribov limit of QCD. An important insight is that in this limit the field strengths squared can become very large () corresponding to the saturation of gluon densities [7, 8]. The onset of saturation is characterized by a saturation scale , which opens a kinematic window accessible at very high energies. The physics of the Regge–Gribov regime is quite different from that of the Bjorken limit discussed previously. The typical momenta of partons are and higher twist contributions are not suppressed. These considerations are especially relevant for the scattering of large nuclei because the large transverse density of partons in the nuclear wavefunctions (proportional to the nuclear radius ) provides a natural enhancement of the saturation scale, . Our goal is to derive a formula similar to eq. (1) for inclusive gluon production in the Regge–Gribov limit.
The dynamics of large parton phase space densities in the Regge–Gribov limit can be described in the Color Glass Condensate (CGC) effective field theory where small partons in hadrons and nuclei are described by a classical field, while the large partons act as color sources for the classical field [9, 10, 11]. The lack of dependence of physical observables on the (arbitrary) separation between large color sources and small dynamical fields is exploited to derive a renormalization group (RG) equation, known as the JIMWLK equation [12, 13, 14, 15, 16, 17, 18, 19]. This equation is a functional RG equation describing the change in the statistical distribution of color sources with rapidity (= ). It can be expressed as
(2) 
where is the JIMWLK Hamiltonian^{1}^{1}1The explicit form of this Hamiltonian will be given later in the text.. For a physical observable defined by an average over all the source configurations,
(3) 
one obtains
(4) 
We have used here eq. (2) and integrated by parts (using the hermiticity of ). The structure of is such that is an object distinct from , so that one obtains in principle an infinite hierarchy of evolution equations for operators expectation values [20]. In the large and large meanfield limit, this hierarchy simplifies greatly. When is the “dipole” operator, corresponding to the forward scattering amplitude in deep inelastic scattering, the resulting closed evolution equation is known as the BalitskyKovchegov (BK) equation [21, 22].
In refs. [23, 24, 25], we developed a formalism to compute observables related to multiparticle production in field theories with strong time dependent sources. This formalism is naturally applicable to the CGC description of high energy scattering^{2}^{2}2Although the color sources of each nucleus are independent of the corresponding lightcone time, their sum constitutes a timedependent current. albeit, for simplicity, we considered only a scalar field theory. (The corresponding QCD framework was briefly considered in ref. [26].) In these papers, the formalism for multiparticle production was developed for a fixed distribution of sources, with the assumption that the final results could be averaged over, as in eq. (3), with unspecified distributions of sources and (one for each of the projectiles). However, we did not discuss in these papers the validity of such a factorization formula.
In the formalism of refs. [23, 24, 25], one can formally arrange the perturbative expansion of an observable like the single inclusive gluon spectrum as
(5) 
where each term corresponds to a different loop order. Each of the coefficients is itself an infinite series of terms involving arbitrary orders in . We call “Leading Order” the contribution that comes from the first coefficient :
(6) 
In the case of the single gluon spectrum, the first term has been studied extensively. In [24] we developed tools to calculate the next term . Following this terminology, we denote
(7) 
However, this strict loop expansion ignores the fact that large logarithms of the momentum fractions can appear in the higher order coefficients when is very large. The term can have up to powers of such logarithms, and a more precise representation of these coefficients is
(8) 
The “Leading Log” terms are defined as those terms that have as many logarithms as their order in ,
(9) 
In this work, we will go significantly further than the Leading Order result, and resum the complete series of Leading Log terms. We will prove that, after averaging over the sources , all the Leading Log corrections are automatically resummed by the JIMWLK evolution of the distribution of sources, and that the event averaged Leading Log result is given by the factorized expression
(10) 
In this formula, is the rapidity at which the gluon is measured, and the subscripts indicate the amount of rapidity evolution^{3}^{3}3In terms of the center of mass energy of the collision (for a nucleonnucleon pair) and the longitudinal momentum components of the measured gluon, one has also – at leading log – and . of the source distributions of the two projectiles, starting in their respective fragmentation regions.
The expressions in eq. (10) are gauge invariant functionals describing the source distributions in each of the nuclei. In analogy to the parton distribution functions we introduced previously, they contain nonperturbative information on the distribution of sources at rapidities close to the beam rapidities. Just as the latter evolve in with the DGLAP [27, 28, 29] evolution equations, the former, as suggested by eq. 2, obey the JIMWLK evolution equation in rapidity which evolves them up to the rapidities and from the nuclei and respectively. As we will discuss in detail, the leading order inclusive gluon spectrum, for given sources , can be computed by solving the classical YangMills equations with simple retarded boundary conditions. Eq. (10) suggests that the result resumming all the leading logarithms of the collision energy can be obtained by averaging over this leading order result with the weight functionals evolved from the beam rapidity to the rapidity at which the gluon is produced.
In the Regge–Gribov limit, eq. (10) is the analog of the factorization formula eq. (1) proved in the Bjorken limit. While we will prove that eq. (10) holds for leading logarithmic contributions at all orders in perturbation theory, we have not attempted to show that it is valid for subleading logarithms. There is currently an intense activity in computing sub leading logarithmic contributions in the high parton density limit [30, 31, 32, 33, 34, 35, 36] so an extension of our results beyond leading logs is feasible in future. There is another aspect of A+A collisions that we have not discussed thus far. Our power counting does not account for the so called “secular divergences” [37, 38, 39]. These are contributions that diverge at least as powers of the time elapsed after the collision. Including these contributions will not alter our factorization theorem; it does affect how “observables” defined at finite times after the nuclear collisions are related to quantities measured in A+A experiments. We will address this issue briefly. A fuller treatment requires more work.
The paper is organized as follows. In section 2, we derive an important formula for the Next to Leading Order corrections to the inclusive gluon spectrum. This formula will play a crucial role later, in disentangling the initial state effects from the rest of the collision process. In section 3, we will derive the expressions stated in eqs. (2)–(4) for JIMWLK evolution of a single nucleus. Albeit the result is well known, our derivation is quite different from those existing in the literature [12, 13, 14, 15, 16, 17, 18, 19, 40, 41, 42, 43, 44]. We will obtain our result entirely in terms of retarded lightcone Green’s functions without any recourse to timeordered propagators. We will show that there are no ambiguities in specifying the pole prescriptions in this approach. More importantly, our derivation allows us to straightforwardly extend our treatment of the JIMWLK equation to the case of the collision of two nuclei. This is discussed separately in section 4 where we show explicitly that nonfactorizable terms are suppressed and our key result, stated in eq. (10), is obtained. In the following section, we will relate our work to previous work in this direction and briefly explore some of the connections between the different approaches. In section 6, we will discuss how one can relate our result for the Glasma produced at early times in heavy ion collisions [45, 25] and its subsequent evolution into the Quark Gluon Plasma. We conclude with a brief summary and discussion of open issues. There are three appendices dealing with properties of Green’s functions in light cone gauge relevant to the discussion in the main text of the paper.
2 NLO corrections to inclusive observables
Before studying the logarithmic divergences that arise in loop corrections to observables, let us derive a formula that expresses the 1loop corrections to inclusive observables in terms of the action of a certain operator acting on the same observable at leading order. As we shall see, this formula – albeit quite formal – can be used to separate the physics of the initial state from the collision itself.
We have in mind an operator made of elementary color fields, which probes multigluon correlations. To be specific, for a given source distribution, we shall consider the quantum expectation value
(11) 
in the limit where the time arguments of the two fields go to . We chose this particular operator because we wish to study the single gluon spectrum –the first moment of the multiplicity distribution– in the collision of two nuclei; it is obtained by Fourier transforming this bilinear combination of fields. Note that the two fields are not timeordered. The expectation value of such a product can be calculated in the SchwingerKeldysh formalism [46, 47, 48], by considering that lies on the branch of the contour and on the branch (A representation of the Schwinger–Keldysh contour is shown in fig. 1.)
This section is organized as follows. We first recall the expression of eq. (11) at leading order in terms of retarded solutions of the classical Yang–Mills equations. This result is well known and has been derived in a number of different ways. We will then discuss the nexttoleading order computation of this quantity in the CGC framework. There are two sorts of NLO corrections; these are the virtual corrections arising from oneloop corrections to the classical fields and the real corrections which are obtained by computing the propagator of a small fluctuation in lightcone gauge. We will show that can be expressed as a linear operator with real and virtual pieces acting on , plus an unimportant (as far as the resummation of logs of is concerned) additional term.
2.1 Leading order result
We showed in [23] that, at leading order, is the product of two classical solutions of the YangMills equations, with null retarded boundary conditions^{4}^{4}4The retarded nature of the boundary conditions is intimately related to the inclusiveness of the observable under consideration. For instance, if instead of the single inclusive gluon spectrum, one wanted to calculate at leading order the probability of producing a fixed number of gluons, one would have to solve the classical YangMills equations with boundary conditions both at and at (see [49]).,
(12) 
with
(13) 
Here, denotes the classical field, and is the color current corresponding to a fixed configuration of the color sources. The current is comprised of one or two sources depending on whether we consider only one nucleus or the collision of two nuclei – this distinction is not important in this section. It is important to note that this current, which has support only on the lightcone, must be covariantly conserved,
(14) 
This means that in general, there is a feedback of the gauge field on the current itself, unless one chooses a gauge condition such that the gauge field does not couple to the nonzero components of the current on the lightcone.
Although one can solve analytically the YangMills equations with these boundary conditions in the case of a single nucleus [12, 50], this is not possible in the case of two nuclei, and one must resort to numerical methods to obtain results in this case. Fortunately, as we shall see, the discussion of factorization in the case of two nuclei does not require that we know this solution analytically.
Because the solution of the YangMills equations we need is defined with retarded boundary conditions, its value at the points and (where the observable is measured) is fully determined if we know its value^{5}^{5}5Since the YangMills equations contain second derivatives with respect to time, one must also know the value of the first time derivative of the field on this initial surface. on an initial surface which is locally spacelike^{6}^{6}6This means that at every point , the vector normal to at the point ( for any displacement on around the point ) must be timelike. This condition prevents a signal emitted at the point , propagating at the speed of light, from encountering again the surface .– located below the points and , as illustrated in fig. 2.
Therefore, we will write
(15) 
which means that the observable is considered as a functional of the value of the color field on the initial surface . Note that we use the same symbol for the color field and for its initial value on , although mathematically these objects depend on a different number of variables and are therefore different functions.
2.2 Next to leading order corrections
A detailed discussion of the power counting for moments of the inclusive multiplicity distribution can be found in Ref. [23]. The leading order contributions to involves only tree diagrams, which explains why it can be obtained from classical solutions of the YangMills equations. As mentioned previously, this leading order contribution is of order but includes all orders in . In the rest of this section, we shall study the 1loop corrections to this quantity, that are of order in the coupling and to all orders in .
The framework to compute these 1loop corrections (hereafter called “NLO”) to quantities such as eq. (11) has been developed for a scalar theory in ref. [24]. Much of this analysis can be carried over to QCD. To avoid complications such as ghost loops, we shall work in a gauge such as the light cone gauge . Following the discussion for the scalar case, we obtain at NLO,
(16) 
In this equation, is the component of the small fluctuation SchwingerKeldysh propagator in the presence of the classical background field and the field is the one loop correction to the classical field. It is obtained by solving the small fluctuation equation of motion
(17) 
with null retarded boundary conditions :
(18) 
Here is the potential term in the YangMills Lagrangean^{7}^{7}7Unless one chooses a nonlinear gauge condition, is made of the usual 3gluon and 4gluon couplings., obtained by writing
(19) 
where is defined in eq. (136) of appendix A. We refer the reader to appendix A for more details. The source term in this small fluctuation equation includes the closed loop formed by the SchwingerKeldysh propagator to be defined shortly, the third derivative corresponds to the 3gluon vertex in the presence of a background field and is a symmetry factor.
Following [24], we can write the propagator in eq. (16) as a bilinear combination of small fluctuations of the gauge field whose initial conditions are plane waves,
(20) 
where
(21)  
The sum over is over the two physical polarizations for the initial plane wave and the index represents the initial color carried by the small fluctuation field. In eq. (20), our notation is such that the lower color index () represents the initial color of the fluctuation, while the upper color index ( or ) refer to its color after it has evolved on top of the classical background field^{8}^{8}8For future reference, note that quantities with only the lower color index are matrices in the adjoint representation of defined by
The propagator at equal points can be written in a similar fashion as^{9}^{9}9When the two endpoints are separated by a timelike interval, there can be an additional term contributing to this propagator – see [24] for more general formulas.
(23) 
We note that in a generic gauge, covariant current conservation may require the incoming field fluctuation to induce a color precession of the classical current . This modification of the current will in turn induce an additional contribution to the field fluctuation. Our strategy [51, 52, 53] to avoid this complication will be to perform all intermediate calculations in a gauge where this phenomenon does not happen. For instance, on the line where the color charges moving in the direction live, on should use a gauge in which . Indeed, because the color current only has a component, covariant conservation is trivial in this gauge. A gauge rotation of the final result is then performed to return to the lightcone gauge of interest. All effects due to current conservation are then taken care of by this final gauge transformation.
2.3 Rearrangement of the NLO corrections  I
In this subsection, we will express the small fluctuation propagator as the action of a differential operator on the classical fields and . This operator contains functional derivatives with respect to the initial value of the color field on . In the following subsection, we will repeat the exercise for the one loop correction to the classical field and write it in terms of a similar operator acting on the classical field . These identities, besides providing a transparent derivation of the JIMWLK equation for a single nucleus, will be especially powerful in our treatment of nucleusnucleus collisions.
Let us begin from the Green’s formula for the classical field ,
(24) 
where is the free retarded propagator (discussed in appendix A in the case of the lightcone gauge) and is the boundary term that contains the initial value of the classical field on . (Boundary terms for the classical and small fluctuation fields in lightcone gauge are discussed in detail in appendix B.) denotes the region of spacetime above the surface . Now, consider an operator (to be defined explicitly later) that acts on the initial value of the fields on the surface , and assume that this operator is linear, which implies
(25) 
Now apply this operator to both sides of eq. (24), we get
(26) 
By comparing this equation with the Green’s formula for a small fluctuation (see appendix B),
(27) 
we see that we can identify provided that we have
(28) 
Because is a linear functional of the initial value of the color fields on the surface , it is easy to see that the operator that fulfils this goal is
(29) 
where is the generator of translations of the initial fields^{10}^{10}10For now, it is sufficient to think of this operator as an operator which is linear in first derivatives with respect to the color field on . at the point . We denote by the measure on the surface (for instance, if is a surface defined by , this measure reads .) The detailed expression of this operator can be obtained by writing explicitly the Green’s formula for the retarded propagation of color fields above the surface , and it usually depends both on the choice of the surface and on the choice of the gauge condition. An explicit expression of this operator will be given in the next section when the initial surface is parallel to the lightcone () and when the fields are in the lightcone gauge . Therefore, we have established the following identity,
(30) 
Eq. (30) provides a formal expression of a fluctuation at point in terms of its value on some initial surface (in the right hand side of eq. (30), only the value of the fluctuation on appears). This formula is especially useful in situations where we can calculate analytically the initial value of the fluctuation on , but were we do not know analytically the classical background field above this surface.
The single nucleus case is a bit academic in this respect because one can analytically compute the background gauge field and the fluctuation at any point in spacetime. Rather, eq. (30) will prove especially powerful for nuclear collisions because in that case one does not have an analytic expression for the classical background field after the collision.
2.4 Rearrangement of the NLO corrections  II
The terms involving the 1loop correction can also be written in terms of the operator , but this is not as straightforward as for . The first step is to write down the formal Green’s function solution of eq. (17). It is convenient to write it as
where is identical to except that all occurrences of the bare propagator in the latter are replaced in the former by the dressed propagator in the background field . This dressed propagator, denoted , satisfies the equation
(33) 
plus a retarded boundary condition such that it vanishes if .
The second term on the right hand side of eq. (LABEL:eq:beta1+2) is the value would have if one turns off the source term (proportional to ) in the domain above the initial surface. It is therefore given by a formula identical to eq. (30),
(34) 
To calculate , let us first make explicit the interactions with the background field by writing it as
(35)  
This expression is obtained by substituting the expression for the dressed retarded propagator in terms of the free retarded propagator in the definition of .
Consider now the quantity
(36) 
We shall prove that and are identical. Using eq. (30), we can write
(37) 
Replace in this equation by the r.h.s of eq. (27). Because the boundary term does not depend on the initial value of the classical field , the action of on this term gives zero. We thus obtain
(38)  
which is identical to eq. (35). We therefore obtain . Combining the two contributions and , we finally arrive at the compact expression
(39) 
We can now use eqs. (31) and (39) to obtain a compact expression for NLO corrections to as
(40) 
where we recall that is the same observable at leading order, considered as a functional of the value of the gauge fields on the initial surface . The corrective term is defined by
(41) 
As we shall see later, this term does not contain any large logarithm. Only the terms in the first and second lines of eq. (40) will be important for our later discussion of factorization.
3 JIMWLK evolution for a single nucleus
Eq. (40) is central to our study of NLO corrections and of factorization. In the rest of this section, we will show how this formula is used to derive the JIMWLK evolution equation. In section 4, we will show that it can be generalized to the collision of two nuclei. A very convenient choice of initial surface in the derivation of the JIMWLK equation is the surface defined by . One should choose so that all the color sources of the nucleus are located in the strip . An illustration of the objects involved in eq. (40) and their localization in spacetime is provided in figure 3.
3.1 Gauge choice
We need first to choose the gauge in which to perform this calculation. Because the observable we wish to calculate and everything else in eq. (40) is expressed in terms of light cone gauge () quantities, we need to obtain and in this gauge as well. However, as previously mentioned, covariant current conservation is most easily preserved in a gauge where the field fluctuations have no component. This is because they do not induce a precession of the color current while crossing the light cone. We are therefore going to adopt the strategy advocated in refs. [51, 52, 17, 19], that consists in performing intermediate calculations in a gauge where and then gauge transforming the final result to gauge.
As discussed in detail in appendix B, if one uses the LC gauge and the surface as the initial surface, the linear differential operator that appears in the identity (30) should be defined as^{11}^{11}11We have omitted the color indices in this equation. should be understood as a matrix in the group, and as a column vector. is therefore a column vector whose components are .
(42)  
where is the adjoint color matrix^{12}^{12}12At first sight, does not play any role in the definition of – the necessity to introduce this matrix in the definition of is also explained in the appendix B. that will be defined in eq. (46) . Note that this operator in eq. (42) contains a term for each of the field components that must be specified on the initial surface to know completely the field above this surface. This operator can therefore be interpreted as the generator of translations of the initial condition for a classical solution of the YangMills equations. It is also important to note that the fluctuation field that multiplies this operator is evaluated just above the initial surface (at ). Therefore, because one does not require its entire history beyond this surface, it can in general be calculated analytically.
3.2 Classical field
Let us recall the structure of the classical background field itself. As is well known, the field in the Lorenz gauge () has no component, and therefore fulfills the condition. Its explicit expression in terms of the color source^{13}^{13}13The density of color sources is a gauge dependent quantity. When defined in the Lorenz gauge, we denote it with a tilde. in given by
(43) 
The gauge transformation that relates the classical background fields in the gauge and the corresponding fields in Lorenz gauge is^{14}^{14}14In this expression, is a matrix in the group , while is a matrix in the adjoint representation of the algebra . The product is a matrix in the algebra. Note that depending on the context we use the same symbol for an element of the algebra (i.e. a matrix), and for the vector column made of its components on the basis of the algebra. The relation between the two is of course .
(44) 
where the tilde denotes fields in the Lorenz gauge; those without a tilde are in light cone gauge. Using the light cone gauge condition , we get
(45) 
which admits the Wilson line
(46) 
as a solution. Note that because the color sources do not depend on , and depend only on and . The solution of the classical equations of motion in light cone gauge is then
(47) 
We should comment here on the residual gauge freedom of the classical solution. The most general solution of eq. (45) is
(48) 
where is an arbitrary independent gauge transformation. With this more general choice, one obtains
(49) 
The arbitrariness in the solution is because the condition does not fix completely the gauge and independent ’s span the residual gauge freedom. Requiring that the classical gauge field be of the form given in eq. (47) amounts to the choice . This choice is assumed in the rest of this paper.
3.3 Field fluctuations on the light cone
To readers familiar with the structure of the JIMWLK Hamiltonian, the structure of eq. (40) is already suggestive. In the rest of this section, we will show that the leading logarithmic contributions in this formula – terms that are linear in the rapidity differences between the projectile and target relative to the observed gluon – can be absorbed into a redefinition of the distribution of color sources of the nucleus. Our first task towards this conclusion is to compute the value of the field fluctuations and just above the light cone on the initial surface .
Let us consider a small fluctuation on top of the classical field . The relation between the two gauges must be modified,
(50) 
with
(51) 
where has components of order unity. Using this ansatz in eq. (50), and keeping in mind that while , we obtain the relation
(52) 
To determine , as previously, apply the gauge condition . This gives
(53) 
the solution of which can be written as
(54) 
In this equation is an arbitrary function that does not depend on , and is an “incomplete” Wilson line defined by
(55) 
The arbitrariness in the choice of the function again means that there is a residual gauge freedom after we have imposed .
A crucial point in our derivation is how the residual gauge freedom is fixed. We need small field fluctuations in order to represent the propagators as in eqs. (20) and (23) as bilinear forms in these fluctuations. These equations are valid only if the initial value of the fluctuations are plane waves with onshell momenta; one can check easily that this is true for the free propagators. Thus eq. (54) must give plane wave solutions for the field fluctuations in light cone gauge when . This is simply achieved by taking plane waves for the fluctuation in the original gauge and setting the function to zero^{15}^{15}15We note that it is also possible to choose ’s that are not plane waves and a nonzero to achieve our requirement that be a plane wave. This however makes the intermediate calculations more tedious.. Therefore, the requirement that eqs. (20) and (23) be valid leaves no residual gauge freedom.
We only need to know on our initial surface – at . Because the components of and of are all of order unity, it is legitimate to neglect the values of that are between and in the integration in eq. (54). For and , the incomplete Wilson line is equal to the complete Wilson line (which has the lower bound at ). We therefore obtain
(56) 
Note also that when , the Wilson line becomes independent of because all the color sources are in the strip . This explains why we only indicate in its list of arguments.
Once has been determined, the and components of the fluctuation in light cone gauge are determined from those in the gauge to be
(57) 
As we shall see shortly when we discuss the leading logarithmic divergences, the only quantity we need is^{16}^{16}16Note that , from the definition of the adjoint representation. With the notation where is a column vector, this quantity would also be denoted by .
(58)  
where we have used eq. (53) and the fact that in order to eliminate a few terms. Using the equation for , as well as the fact that is zero at , we get
(59) 
Let us now consider specifically the fluctuations . In the gauge , their expression below the light cone reads^{17}^{17}17Therefore, .
(60) 
with