The non–linear evolution of jet quenching
We construct a generalization of the JIMWLK Hamiltonian, going beyond the eikonal approximation, which governs the high-energy evolution of the scattering between a dilute projectile and a dense target with an arbitrary longitudinal extent (a nucleus, or a slice of quark–gluon plasma). Different physical regimes refer to the ratio between the longitudinal size of the target and the lifetime of the gluon fluctuations. When , meaning that the target can be effectively treated as a shockwave, we recover the JIMWLK Hamiltonian, as expected. When , meaning that the fluctuations live inside the target, the new Hamiltonian governs phenomena like the transverse momentum broadening and the radiative energy loss, which accompany the propagation of an energetic parton through a dense QCD medium. Using this Hamiltonian, we derive a non–linear equation for the dipole amplitude (a generalization of the BK equation), which describes the high–energy evolution of jet quenching. As compared to the original BK–JIMWLK evolution, the new evolution is remarkably different: the plasma saturation momentum evolves much faster with increasing energy (or decreasing Bjorken’s ) than the corresponding scale for a shockwave (nucleus). This widely opens the transverse phase-space for the evolution and implies the existence of large radiative corrections, enhanced by the double logarithm , with the temperature of the medium. This confirms and explains from a physical perspective a recent result by Liou, Mueller, and Wu (arXiv:1304.7677). The dominant, double–logarithmic, corrections are smooth enough to be absorbed into a renormalization of the jet quenching parameter . This renormalization is controlled by a linear equation supplemented with a saturation boundary, which emerges via controlled approximations from the generalized BK equation alluded to above.
The concept of jet quenching globally denotes the modifications in the properties of a ‘hard probe’ (an energetic parton, or the jet generated by its evolution) which occur when this ‘jet’ propagates through the dense QCD matter (‘quark–gluon plasma’) created in the intermediate stages of a ultrarelativistic nucleus–nucleus collision Mehtar-Tani:2013pia (); Majumder:2010qh (); d'Enterria:2009am (); CasalderreySolana:2007zz (); Kovner:2003zj (). This encompasses several related phenomena like the transverse momentum broadening, the (radiative) energy loss, or the jet fragmentation via medium–induced gluon branching, and also the associated observables, like the nuclear modification factor, or the di–jet asymmetry. A common denominator of these phenomena is that, within most of their theoretical descriptions to date, they depend upon the medium properties via a single parameter : a transport coefficient known as the ‘jet quenching parameter’ . This explains the importance of this quantity for both theory and phenomenology, and motivates the recent attempts to obtain better estimates for it from first principles, at least in special cases Arnold:2008vd (); CaronHuot:2008ni (); Majumder:2012sh (); Liou:2013qya (); Laine:2013lia (); Panero:2013pla ().
Roughly speaking, the jet quenching parameter measures the dispersion in transverse momentum111By ‘transverse’ we mean the two dimensional plane orthogonal to the parton direction of motion, conventionally chosen along . accumulated by a fast parton after crossing the medium over a distance : . At weak coupling, the dominant mechanism responsible for this dispersion is multiple scattering off the medium constituents. At leading order in , can be computed as the second moment of the ‘collision kernel’ (the differential cross–section for elastic scattering in the medium; see Sect. 4.1 for details). Beyond leading order, one needs a non–perturbative definition for the transverse momentum broadening. The one that we shall adopt here and which is often used in the literature involves the ‘color dipole’, a light–like Wilson loop in the color representation of the fast parton. Physically, this Wilson loop describes the –matrix for a small ‘color dipole’ (a quark–antiquark pair, or a set of two gluons, in a color singlet state) which propagates through the medium. Via unitarity, the Fourier transform of this –matrix determines the cross–section for transverse momentum broadening Mueller:2001fv (); Mueller:2012bn (). At tree–level, these definitions imply , with logarithmically dependent upon the medium size . This dependence enters via the resolution of the scattering process: the transverse momenta transferred by the medium can be as large as the ‘saturation momentum’ (see Sect. 4.1). Beyond leading order, it is a priori unclear whether the notion of ‘jet quenching parameter’ (as a quasi–local transport coefficient) is still useful, or even well–defined. Our criterion in that sense will be to check whether a formula like does still hold, with a reasonably slowly–varying function. But even when this appears to be the case, we shall see that the –dependence of is generally enhanced by the radiative corrections, due to the intrinsic non–locality of the quantum fluctuations.
So far, two different classes of next–to–leading order corrections, which correspond to very different kinematical regimes, have been computed at weak coupling CaronHuot:2008ni (); Liou:2013qya (). In Ref. CaronHuot:2008ni (), Caron-Huot considered a medium which is a weakly–coupled quark–gluon plasma (QGP) with temperature and computed the corrections of to the ‘collision kernel’, as generated by the soft, highly–populated (and hence semi–classical), thermal modes, with energies and momenta . (The corresponding leading–order value has been computed by Arnold and Xiao Arnold:2008vd ().) These corrections do not modify the logarithmic dependence of upon the medium size , which is rather introduced by the hardest collisions, with transferred momenta . (We implicitly assume that .)
By contrast, in Ref. Liou:2013qya (), Liou, Mueller, and Wu have studied the relatively hard and nearly on–shell gluon fluctuations, with large transverse momenta and even larger longitudinal momenta, (in the plasma rest frame). Such fluctuations, which are most naturally viewed as bremsstrahlung by the projectile, are not sensitive to the detailed properties of the medium. They depend upon the latter only via the tree–level value of the jet quenching parameter and via two basic scales — the longitudinal extent of the medium and the wavelength of its typical constituents (with for the QGP) — which constrain the phase–space for bremsstrahlung. Ref. Liou:2013qya () found large one–loop corrections222See also Ref. Wu:2011kc () for a similar but earlier observation, which has motivated the more elaborate analysis in Ref. Liou:2013qya (). to , of relative order , where the double logarithm comes from the phase–space: one logarithm is generated by integrating over the lifetime of the fluctuations, over the range , and the other one comes from the respective transverse momenta, within the interval (for a given value of ). The lower limit on refers to multiple scattering: the condition means that the relevant fluctuations are hard enough to suffer only one scattering during their lifetime.
Note that this ‘single scattering’ property refers to a color dipole, and not to a charged parton. That is, the large radiative corrections identified in Ref. Liou:2013qya () should not be viewed as a renormalization of the collision kernel above mentioned, which represents the differential cross–section for individual scatterings in the plasma, but rather as a change in the transport cross–section relevant for transverse momentum broadening, which is controlled by the relatively rare collisions involving a sufficiently large momentum transfer (which can be as large as ).
For what follows, it is important to notice that the medium size sets the upper limit on the lifetime of the fluctuations, and hence on their energy . Accordingly, when increasing , one opens the phase–space for fluctuations which are more and more energetic. Such fluctuations can then evolve towards lower energies, via soft gluon emissions. This evolution is represented by Feynman graphs of higher–loop order (gluon cascades which are strongly ordered in energy), which are enhanced by the phase–space: the powers of associated with soft gluon emissions can be accompanied by either double, or at least single, logarithms of , depending upon the kinematics of the emissions. Ref. Liou:2013qya () not only computed the first step in this evolution, for both the double–logarithmic and the single–logarithmic corrections, but also provided a simple recipe for resuming the corrections enhanced by double–logarithms to all orders.
Yet, already the one–loop calculation of the single–logarithmic corrections in Ref. Liou:2013qya () has met with serious difficulties, which reflect the lack of a systematic theoretical framework for addressing this complicated, non–linear, high-energy evolution. Namely, in order to compute the effects of order , one had to estimate the effects of multiple scattering beyond the eikonal approximation and also to heuristically include the ‘virtual’ corrections, which were truly missed by that analysis, but were essential for that purpose. Vice–versa, the only reason why the double–logarithmic corrections appear to be comparatively simple, is because they are neither sensitive to multiple scattering (except for the restriction on their phase–space), nor to the effects of the ‘virtual’ corrections (which are generally important to ensure probability conservation333The ‘virtual’ terms express the reduction in the probability that the evolving system remain in its original state, as it was prior to the evolution. They become negligible to double–logarithmic accuracy because, in that limit, the scattering of the original projectile is much weaker than that of the evolved system including additional gluons., but become irrelevant at double–logarithmic accuracy). Remarkably, it appears that the subset of radiative corrections which are enhanced by powers of forms an ‘island’ of effectively linear evolution, which besides being structurally simple, it also plays a major physical role, in that it gives the dominant contributions in the limit of a large medium .
By itself, the prominence of a double–logarithmic approximation in the context of pQCD evolution is not new — other familiar examples include the fragmentation of a virtual jet in the vacuum Dokshitzer:1991wu (), or the evolution of the parton distribution in the ‘double–leading–log–approximation’ (a common limit of the DGLAP and BFKL equations Kovchegov:2012mbw ()). What is a bit surprising though, is the importance of such an approximation in the context of a non–linear evolution. All the other examples listed above refer to linear processes. And in the only other example of a non–linear pQCD evolution at our disposal — the BK–JIMWLK evolution of the gluon distribution in a large nucleus (or of particle production in proton–nucleus collisions) Balitsky:1995ub (); Kovchegov:1999yj (); JalilianMarian:1997jx (); JalilianMarian:1997gr (); JalilianMarian:1997dw (); Kovner:2000pt (); Weigert:2000gi (); Iancu:2000hn (); Iancu:2001ad (); Iancu:2001md (); Ferreiro:2001qy () —, it is well known that the ‘double–logarithmic approximation’ (DLA) is not a good approximation at high energy.
For instance, the high–energy scattering between a small dipole and a dense nucleus is described by the non–linear BK equation or, at least (in the single scattering regime), by a linear approximation to it that can be described as ‘the BFKL equation supplemented with a saturation boundary’ Iancu:2002tr (); Mueller:2002zm (); Munier:2003vc (). The ‘saturation boundary’ expresses the reduction in the phase–space for linear evolution introduced by multiple scattering (or, equivalently, by gluon saturation in the nucleus). Both the saturation effects and the ‘virtual’ BFKL corrections are essential for that dynamics and together they lead to a dramatic change in the behavior of the scattering amplitude for small dipole sizes : they introduce a non–perturbative anomalous dimension (the amplitude behaves like with , instead of the tree–level result ). If a similar change was to occur in the context of transverse momentum broadening, it would have important consequences for the phenomenology : such corrections could not be simply absorbed into a redefinition of , unlike the comparatively smooth corrections introduced by the DLA (see below). We thus see that the prominence of the DLA in the problem of jet quenching is a remarkable simplification, which is unusual in the context of the non–linear evolution and is important for the phenomenology.
Such considerations invite us to a deeper understanding of the high–energy evolution of jet quenching from first principles. It is our main purpose in this paper to provide a general framework in that sense — that is, a theory for the non–linear evolution of jet quenching to leading order in perturbative QCD at high energy — and then use this framework to address some of the questions aforementioned. In particular, we shall try to clarify issues like the comparison with the BK–JIMWLK evolution, the origin and calculation of the virtual corrections (which is particularly tricky for an extended target), the physics of gluon evolution and saturation in the plasma, the emergence of the double–logarithmic approximation (including the precise phase–space), the possibility to include the radiative corrections into a renormalization of the jet quenching parameter, and the consequences of such a renormalization for the related problem of the radiative energy loss.
In developing the general formalism below, it will be convenient to assume that the projectile enters the medium from the outside and that it was on–shell prior to the collision. This guarantees that the quantum fluctuations which matter for the evolution of the –matrix are generated exclusively via interactions in the target444If the projectile is produced by a hard process occurring inside the medium or at some finite distance from it, then there is additional radiation, associated with the initial virtuality, that would mix with the evolution that we are here interested in (see e.g. the discussion in Kang:2013raa ()) By choosing an on–shell projectile, we avoid this mixing.. Then the main difference between the problem of jet quenching and the BK–JIMWLK formalism for collisions refers to the longitudinal extent of the target and, more precisely, to the ratio between and the lifetime of the typical gluon fluctuations. In collisions, the center–of–mass energy is so high that the nuclear target looks effectively like a shockwave (), due to Lorentz contraction. Then the multiple scattering can be treated in the strict eikonal approximation, which assumes that the transverse coordinates of the projectile partons are not affected by the interactions. By contrast, in the context of jet quenching, the energies are much lower and the fluctuations live fully inside the medium (), so the effects of the multiple scattering can accumulate during their whole lifetime. Then the strict eikonal approximation is not applicable anymore, although the individual scatterings are still soft: one cannot ignore the transverse motion of the fluctuations during their lifetime.
These considerations also show that the two problems aforementioned ( collisions and jet quenching) can be viewed as limiting situations of a common set–up: the high–energy scattering between a dilute projectile and a dense target with an arbitrary longitudinal extent. This is the first problem that we shall address and solve in this paper. Specifically, in Sect. 2 and Appendix A, we shall construct an effective Hamiltonian which, when acting on the –matrix of the projectile (a gauge–invariant product of Wilson lines), generates one additional soft gluon emission in the background of a strong color field representing the medium. (The medium correlations are reproduced by averaging over this background field, in the spirit of the color glass condensate Iancu:2002xk (); Gelis:2010nm ().) This Hamiltonian may be viewed as a generalization of the JIMWLK Hamiltonian beyond the strict eikonal approximation. It looks compact and simple, but it is less explicit than the JIMWLK Hamiltonian, in the sense that the integrals over the emission times cannot be performed in general (i.e. for an arbitrary target). Accordingly, the general Hamiltonian is non–local both in the transverse coordinates and in the light–cone (LC) times. The formal manipulations with this Hamiltonian are complicated by potential (infrared and ultraviolet) divergences which require prescriptions at the intermediate steps and cancel only in the final results. In Sect. 2.2, we demonstrate a general mechanism ensuring such cancellations — this involves a particular ‘sum–rule’ for the gluon propagator in the LC gauge, Eq. (13) — and clarify its connexion to probability conservation. In particular, we show that the ‘virtual’ corrections can be alternatively implemented as a local ‘counter–term’, which is particularly convenient when the target is an extended medium.
As a first test of the new Hamiltonian and of our ability to use it for explicit calculations, we consider in Sect. 3 the example of a shockwave target (). Then the integrals over the emission times can be explicitly performed (the adiabatic prescription for regulating the large time behavior turns out to be important for that purpose) and, as a result, we recover the JIMWLK Hamiltonian JalilianMarian:1997jx (); JalilianMarian:1997gr (); JalilianMarian:1997dw (); Kovner:2000pt (); Weigert:2000gi (); Iancu:2000hn (); Iancu:2001ad (); Iancu:2001md (); Ferreiro:2001qy (), as expected. We also show that the ‘counter–term’ alluded to above generates the ‘virtual’ piece in the BK equation, as it should.
Starting with Sect. 4, we turn to the case of an extended target (), as appropriate for the physics of jet quenching and related phenomena. The general equations generated by the evolution Hamiltonian in that case are extremely complicated (see the discussion in Sect. 4.2): they are non–local in LC time (because gluon emissions can occur anywhere inside the medium and they can have any lifetime ) and also functional (the transverse trajectories of the gluons are random, due to the quantum diffusion, and they are distributed according to a path–integral). An useful approximation is to assume that the medium correlations are Gaussian and local in LC time. (A similar mean field approximation proved to be successful in the case of the BK–JIMWLK equations Kovner:2001vi (); Iancu:2002xk (); Iancu:2002aq (); Blaizot:2004wv (); Kovchegov:2008mk (); Dominguez:2011wm (); Iancu:2011ns (); Iancu:2011nj (); Dumitru:2011vk ().) Under this assumption, the equation obeyed by the dipole –matrix takes the form shown in Eq. (4.2), which is recognized as a functional generalization of the BK equation. The solution to this equation resums all the corrections enhanced by at least one power of the large logarithm . It remains as an open question whether such a functional equation can be solved via numerical methods. Our main point though is that, for the present purposes — i.e. for a study of the leading–order evolution of the jet quenching in the limit —, one can drastically simplify this equation and even obtain analytic results.
This is based on the following two observations. First, in order to compute the transverse momentum broadening , one needs the dipole –matrix for dipole sizes in the vicinity of the plasma saturation line ; that is, when increasing the medium size , one must simultaneously decrease the typical dipole size, according to , with . Second, the dominant radiative corrections in the interesting regime at and are those which are enhanced by a double logarithm . They can be resumed to all orders by solving a simplified, linear, equation, namely Eq. (94), which emerges from the generalized BK equation aforementioned and is equivalent to the resummation proposed in Ref. Liou:2013qya (). This equation, that can be described as ‘the DLA equation supplemented with a saturation boundary’, is different from ‘the BFKL equation with a saturation boundary’ alluded to before — it is actually simpler and in particular it does not lead to a (non–perturbative) anomalous dimension. That is, the scattering amplitude obtained by solving this equation is still proportional to up to a slowly varying function, as it would be at tree–level555But a perturbative anomalous dimension, of , can be generated by the all–order resummation of the double–logarithmic corrections; this is the ‘saturation exponent’ to be discussed in Sect. 4.4. (see Sect. 4.3 for details). This in turn implies that the dominant corrections to can be absorbed into a renormalization of the jet quenching parameter, which thus becomes mildly non–local.
Given the central role played by the DLA, it is interesting to understand the emergence of this approximation on physical grounds. As we explain in Sect. 4.4, this is related to the specificity of the high–energy evolution of the gluon distribution in the medium, that we here address for the first time. Namely, we show that the non–linear effects in the generalized BK equation (4.2) can be also understood as gluon saturation in the medium, but with a saturation scale which increases very fast when decreasing (the longitudinal momentum fraction carried by the gluons) — much faster than the corresponding scale in a shockwave. Specifically, one has already at tree–level and this growth becomes even faster after including the effects of the small– evolution. The physical explanation is quite simple: the quantity is proportional with the longitudinal size of the region where the gluons can overlap with each other. For gluons inside the medium, this region is their wavelength ; hence, , as anticipated. In turn, this rapid growth of with widely opens the transverse phase–space and thus favors a double–logarithmic evolution : when increasing , one opens not only the longitudinal phase–space at , but also the transverse one at . The upper limit (the conventional ‘saturation momentum’ in the literature on jet quenching) is simply the largest value of , corresponding to . This situation should be contrasted to the more familiar case of a shockwave, where the variation of with is a parametrically small effect, of order (a pure effect of the evolution), so the transverse phase–space increases much slower than the longitudinal one in the approach towards saturation. Incidentally, this explains why, in that context, the DLA is generally not a good approximation Iancu:2002tr (); Mueller:2002zm (); Munier:2003vc ().
Such considerations will allow us to recover the double–logarithmic corrections of Ref. Liou:2013qya () from a more fundamental perspective and with a transparent physical interpretation. An additional clarification refers to the phase–space for the high–energy evolution: as we shall discuss in Sect. 4.3.3, the original argument in that sense in Ref. Liou:2013qya () must be supplemented with the kinematical constraint . This has consequences when the medium is a weakly–coupled QGP (more generally, whenever ): in that case, the validity of the high–energy approximations requires the stronger constraint (and not just ). In more suggestive terms, the necessary condition can be written as .
As a further application, we consider in Sect. 5 the evolution of the radiative energy loss, within the framework of the BDMPSZ mechanism for medium–induced gluon radiation Baier:1996kr (); Baier:1996sk (); Zakharov:1996fv (); Zakharov:1997uu (); Baier:1998yf (); Baier:1998kq (); Wiedemann:2000za (); Wiedemann:2000tf (); Arnold:2001ba (); Arnold:2001ms (); Arnold:2002ja (). Within the approximations of interest, this problem is closely related to that of the transverse momentum broadening and in Sect. 5 we shall merely emphasize the differences. Once again, the cross–section (and its evolution) can be related to the dipole –matrix, which obeys the equations established in Sect. 4. The new feature is that, now, the eikonal approximation fails not only for the soft gluon fluctuation responsible for the evolution, but also for its relatively hard parent gluon, which is responsible for the energy loss. Yet, this failure poses no difficulty for the calculation of the high–energy evolution, because of the strong separation in lifetime between the fluctuations and the radiation. In particular, to double–logarithmic accuracy, the evolution of the radiative energy loss is obtained by simply using the renormalized value of (the solution to Eq. (94)) within the respective formula at tree–level. Similar conclusions have been independently reached in Ref. Blaizot:2014 (), where Eq. (94) has been obtained via a different method (namely, via the direct calculation of the relevant Feynman graphs to DLA accuracy).
Finally, Sect. 6 summarizes our results and conclusions, together with some open problems.
2 The evolution Hamiltonian in the high–energy approximation
Throughout this paper, we shall consider the high–energy evolution of the scattering amplitude for the collision between a dilute projectile and a dense target. The projectile is a set of partons in an overall color singlet state (the prototype being a color dipole), while the target can be either a large nucleus, or the dense partonic medium created in the intermediate stage of an ultrarelativistic heavy ion collision. In both cases, the target is characterized by a dense gluon distribution, which for the present purposes will be described in the spirit of the CGC formalism, that is, as a random distribution of strong, classical, color fields. The interactions between the projectile and the target will be treated in a generalized eikonal approximation, which allows one to resum the multiple scattering between the partons in the projectile and the strong color fields in the target to all orders, via Wilson lines, while also keeping trace of the transverse motion of the partons.
One step in the high–energy evolution consists in the emission of a relatively soft gluon by one of the partons in the projectile and in the background of the target field. Such an emission modifies the partonic content of the projectile and hence the –matrix for the elastic scattering between the projectile and the target. In this section we shall present and motivate a rather compact expression for the Hamiltonian which ‘generates this evolution’, that is, which describes the change in the –matrix induced by one soft gluon emission. A more formal derivation of this Hamiltonian from the QCD path integral is given in Appendix A.
2.1 The evolution Hamiltonian
To be specific, let us assume that the projectile propagates in the positive direction and introduce light–cone (LC) vector notations: , with , , and . Each parton in the projectile has a color current oriented in the LC ‘plus’ direction, which couples to the component of the target color field. If the parton energy is sufficiently high (see below for the precise condition), its transverse coordinate is not affected by the interaction. Then the only effect of the latter is a rotation of the parton color state, as encoded in the Wilson line :
The ’s are the color group generators in the appropriate representation and P stands for path ordering w.r.t. (the LC ‘time’ for the projectile) : with increasing , matrices are ordered from right to left. The integral over formally extends along the whole real axis, but in practice it is limited to the support of the target field. The coordinate has been omitted in Eq. (1) since it is understood that for the ultrarelativistic projectile, by Lorentz contraction.
The elastic –matrix for a color–singlet projectile involves the trace of a product of such Wilson lines, one for each parton (quark, antiquark, or gluon) in the projectile. For more clarity, in what follows we shall keep the notations and for the color group generators and the Wilson lines in the adjoint representation, and use and respectively for quarks in the fundamental representation. As anticipated, most of the examples below will refer to a color dipole, for which the –matrix reads (in the fundamental representation, for definiteness)
where and are the transverse coordinates of the quark and the antiquark, respectively, and , etc. This dipole enters the calculation of a variety of physical processes, like the total cross-section for deep inelastic scattering, the cross–section for single inclusive hadron production in proton–nucleus () collisions, or the transverse momentum broadening of a ‘hard probe’ (here an energetic quark) propagating through the dense partonic medium (‘quark–gluon plasma’) created at the intermediate stages of a nucleus–nucleus () collision.
Below we shall refer to Eq. (1) as the strict eikonal approximation. For a quantum particle, this is correct only so long as the target is ‘sufficiently thin’ — namely, so long as the duration of the interaction (which is the same as the extent of the target in the direction) is small enough for the effects of the quantum diffusion to remain negligible. Indeed, a high energy particle with longitudinal momentum is similar to a non–relativistic quantum particle with mass equal to and living in two spatial dimensions, in that it undergoes a Brownian motion in the transverse plane: the dispersion in its transverse position grows with time according to . (This transverse dynamics is explicit in Eq. (9) below.) The dispersion thus accumulated during the interaction time can be neglected so long as it remains smaller than the respective quantum uncertainty (with the particle transverse momentum). This requires666In evaluating the coherence time one should use the maximal value of accumulated by the particle via rescattering in the target, that is, the saturation momentum to be later introduced. , a condition which is indeed satisfied when the target is a shockwave, but not also for the case of an extended medium. Hence, in the case of the medium, we shall need the generalization of Eq. (1) to an arbitrary trajectory in the transverse plane, where is the LC time. This reads
and is a functional of the trajectory. As compared to Eq. (1) we have also generalized the definition in Eq. (3) to trajectories which start at some generic (light–cone) time and end up at a later time .
We are now in a position to present the operator which generates the emission of a soft gluon by the dilute projectile in the presence of the strong color field of the target. This operator acts on gauge–invariant operators built with products of Wilson lines, like that in Eq. (2), and reads
in notations to be explained below.
The variable is the LC longitudinal momentum of the emitted gluon; by assumption this is much smaller than the respective momentum of the parent parton (to be below denoted as ), but much larger than any ‘plus’ component that can be transferred by the target in the collision process. Accordingly, the component is conserved by the interactions, which makes it useful to use the mixed Fourier representation , as we did above. The ‘strip integral’ in Eq. (4) runs over an interval in which is symmetric around :
Here is the typical ‘plus’ momentum of the emitters, which is the relevant ‘hard’ scale, whereas , with , is the smallest longitudinal fraction of the emitted, ‘soft’, gluon. In what follows, we shall be mostly concerned with situations where the above integral is logarithmic, ; in such a case, the evolution operator takes of the form , with playing the role of a Hamiltonian for the evolution with ‘time’ (the rapidity difference between the valence partons in the projectile and the softest evolution gluons).
Furthermore, denotes the functional derivative w.r.t. the component of the gauge field and plays the role of the color charge density operator. When acting on a Wilson line like that in Eq. (3), this operator generates the emission of a soft gluon from the parton represented by that Wilson line:
As visible on this equation, each functional derivative brings a factor of , so starts at order (but in general includes effects of higher order in , via the background field; see below). The operator is also the generator of the infinitesimal color rotations. Using (2.1), one can check the following equal–time commutation relation (with the structure constants for SU and )
which confirms that these operators obey the color group algebra, as they should.
The last ingredient in Eq. (4) is the background field propagator of the emitted gluon. This is a functional of the target field , via Wilson lines. Its construction is well documented in the literature and will be briefly discussed in Appendix B, where we show that
Here, is the ‘scalar’ propagator, defined as the solution to the following equation
with Feynman prescription for the pole at the mass–shell. This prescription ensures that modes with positive (negative) values of propagate forward (backward) in time (see e.g. Eq. (157)). For definiteness, we shall refer to the two pieces in the r.h.s. of Eq. (8) as the ‘radiative piece’ and respectively the ‘Coulomb piece’ of the gluon propagator.
Eq. (9) exhibits the eikonal coupling between the large component of the 4–momentum of the gluon and the conjugate component of the color field of the target, and also the transverse dynamics responsible for quantum diffusion. Given the formal analogy between this equation and the Schrödinger equation for a non–relativistic particle in two spatial dimensions, it is clear that its solution can be written as a path integral. Namely, for and hence , one has777The ‘reduced propagator’ is formally the same as the non–relativistic evolution operator.
where one integrates over paths with boundary conditions and . For (and hence ), the propagator can be computed by using the following symmetry property, which follows from Eq. (9) together with the Feynman prescription:
By exploiting the above properties, one can limit the time integrals in Eq. (4) to , while simultaneously restricting the integral to the positive side of the strip, , and multiplying the result by two. More precisely, we have here in mind the integral over the ‘radiation’ piece of the propagator (8), which is non–local in time. The local, Coulomb, piece must be treated separately.
Note finally that there is no ambiguity concerning the ordering of the various factors within the integrand of Eq. (4) : (i) the two charge operators act at different times, and , so they commute with each other; (ii) the radiation piece of the propagator involves the background field only at intermediate times , between and , so it commutes with any of the two functional derivatives; (iii) the Coulomb piece is local not only in time, but also in color.
The structure of the evolution Hamiltonian (4) looks both simple and intuitive: this operator does precisely what it is expected to do, namely, it generates the evolution of an –matrix like (2) via the emission and the reabsorption of a soft gluon by any of the color sources within the projectile. But this apparent simplicity hides several subtleties which show up when trying to use this Hamiltonian in practice. These subtleties will be discussed in the next subsection, where we shall derive an alternative form for the evolution Hamiltonian — more precisely, for its action on a generic operator — which is more convenient in practice, especially for an extended target.
2.2 Virtual corrections and probability conservation
The purpose of this subsection is to render the Hamiltonian (4) ‘less formal’. First, we shall argue that, in order to be well defined, this operator must be supplemented with an adiabatic prescription for switching off the interactions at large times. Second, we shall discuss a sum–rule for the free LC gauge propagator, which ensures probability conservation and also the cancellation of ultraviolet and infrared divergences between the ‘radiative’ piece and the ‘Coulomb’ piece of the Hamiltonian. Finally, we shall derive an alternative expression for the action of where this cancellation occurs locally in time and probability conservation becomes manifest.
Throughout this paper, we shall assume that the target is localized in , within the longitudinal888An interval is ‘longitudinal’ from the viewpoint of the target (a left mover), but ‘temporal’ from that of the projectile (a right mover). In what follows, we shall often mix the two viewpoints and the respective terminologies. The precise meaning should be clear from the context. strip at , so the collision has a finite duration . The scattering amplitude can only be affected by gluon emissions which occur sufficiently close to this interaction region, within a time interval . (We recall that the ‘coherence time’ is the typical lifetime of the fluctuation.) Vice–versa, virtual fluctuations in the wave function of the projectile which occur very far away from the interaction region, either in the remote past or the remote future, should have no influence on the evolution of the –matrix. As we shall see, this property is correctly encoded in the present formalism, but it involves delicate cancellations between various terms, which might be invalidated by careless manipulations at intermediate stages. It turns out that a proper way to deal with this problem is to adiabatically switch off the interactions at very large times Chen:1995pa (); Mueller:2012bn (). (Other, less smooth, prescriptions, like a sharp cutoff on , could induce spurious radiation and thus alter the Fock space of the projectile.) To that aim, we shall supplement each functional derivative within with an exponential attenuation factor,
where should be much smaller than . The physical predictions will not be sensitive to the precise value of because the limit of the final results, as obtained after performing the integrals over the emission times and , is indeed well defined.
With this adiabatic switch–off, the free LC gauge propagator , Eq. (156), obeys the following sum–rule, with paramount consequences for what follows:
This will be demonstrated in Appendix C, where we show that the l.h.s. of Eq. (13) is a quantity of and hence vanishes when . A simple way to understand this cancellation is to notice that the integral over isolates the Fourier component with , which vanishes because , cf. Eq. (156). But this property holds only for the complete propagator, , as obtained after adding its radiative and Coulomb pieces. In the presence of a background field, we have to distinguish between these two pieces, since they are differently dressed by the background, cf. Eq. (8). Taken separately, the radiative piece and the Coulomb piece generate contributions to the l.h.s. of Eq. (13), which however cancel, together with the finite terms of , in their sum (see Appendix C).
In view of the above, the sum–rule (13) is expected to be important for the limit of our formalism. In that limit, it ensures an important property, that we now explain. As previously mentioned, quantum fluctuations which are not measured by the collision should not matter for the evolution of the –matrix. Consider in particular the situation where, after acting with on some generic –matrix (to produce the fluctuation), one sets , so that there is no scattering. Without scattering, the evolution cannot be measured (the –matrix must be equal to one both before and after the evolution), hence the action of must vanish :
This is precisely ensured by the identity (13), as it can be easily seen: the action of the functional derivatives on becomes independent of time after we set (since all the Wilson lines are replaced by the unity matrix). Accordingly, the result of first acting with on any and then letting is indeed proportional to the integral in the l.h.s. of Eq. (13).
These properties, Eqs. (13) and (14), allows one to compute the action of on in an alternative way, where the Coulomb piece of the propagator is not explicitly present anymore and the cancellation of would–be divergent contributions occurs quasi–locally in time. Namely, Eq. (14) implies, with obvious notations,
Also, as we shall shortly demonstrate, the action of the Coulomb piece of the Hamiltonian on any observable amounts to
where the second equality follows after using Eq. (15). By using the above, one can write
or, less formally,
where is a Hamiltonian density built with the ‘radiation’ piece of the propagator alone:
In Eq. (19), the transverse derivatives act only on the ‘scalar’ propagator. In particular,
with the free propagator (157). Notice that the r.h.s. of Eq. (18) cannot be written as the action of a linear operator on . Hence, this equation does not provide an alternative expression for the Hamiltonian , but rather a new method for computing its action on a generic observable.
Using , one sees that the property (14) is now satisfied locally in time, that is, it is already verified by the integrand in Eq. (18). This allows for a natural probabilistic interpretation: the term describes the change in the –matrix associated with a real emission which occurrs during the time interval from to ; the virtual term represents the reduction in the probability that the projectile remain in its original state during that time interval. The local (in time) version of (14) is then the expression of probability conservation.
To better appreciate the advantages of Eq. (18) over the direct use of Eq. (4), let us consider the action of in more detail. (This will also allow us to verify the first equality in Eq. (16).) What we would like to show is that any operator is an eigenstate of , but with an ill–define eigenvalue, which suffers from both infrared (large time and small ) and ultraviolet (small , or high ) divergences. Chosing for definiteness (this brings no loss in generality), we can write (cf. Eq. (8))
Because of the ultra–local nature of the Coulomb propagator , the two functional derivatives must act on a same Wilson line within , either the quark one at or the antiquark one at . This feature, together with identities like
explains why the result is again proportional to . But for the very same reason, the proportionality coefficient exhibits several types of divergences, as anticipated: a large–time divergence as , a small– divergence when , and a transverse ‘tadpole’ . Being independent of , this coefficient is necessarily the same as the limit of , in agreement with Eq. (16). Clearly, a similar argument holds for any observable .
The above discussion shows that the action of the Coulomb piece of generates severe divergences. By virtue of Eq. (13), there divergences are guaranteed to cancel against similar ones generated by the radiation piece, but only after performing the two time integrations. This cancellation can be explicitly verified whenever one is able to perform the time integrations, as in the case of a shockwave target to be discussed in Sect. 3. But even in such a case, the calculation of the finite terms is quite subtle and relies in an essential way on the use of the adiabatic prescription (see e.g. Sect. 3.1). By contrast, the calculations based on Eq. (18) are more robust, because the potential divergences cancel between the ‘real’ and ‘virtual’ terms quasi–locally in time, so one is not sensitive to the regularization prescription used for the time integrations. This second method becomes particularly useful in those cases where one is not able to explicitly perform the time integrals, like that of an extended target to be discussed in Sect. 4.
3 A shockwave target: recovering the JIMWLK Hamiltonian
In this section, we shall specialize the general formalism developed so far to the case where the target is a ‘shockwave’. By this, we more precisely mean a target which looks localized in on the resolution scale set by the lifetime of the quantum fluctuations. For this case, we will be able to explicitly perform the time integrations which appear in Eq. (4) and thus recover the JIMWLK Hamiltonian JalilianMarian:1997jx (); JalilianMarian:1997gr (); JalilianMarian:1997dw (); Kovner:2000pt (); Weigert:2000gi (); Iancu:2000hn (); Iancu:2001ad (); Iancu:2001md (); Ferreiro:2001qy (), as expected. Besides giving us more confidence with the use of Eq. (4) in practice, the subsequent manipulations will also illustrate some of the subtleties discussed in Sect. 2.2, notably the role of the adiabatic prescription and the cancellation of the ill–defined contributions between the ‘radiation’ piece and the ‘Coulomb’ piece of .
More precisely, the physical problem that we here have in mind is ‘dense–dilute’ (e.g. proton–nucleus) scattering in the high–energy regime where the longitudinal extent of the dense target is much smaller than the coherence time of the typical gluons fluctuations associated with the evolution of the projectile : . This condition involves both the ‘energy’ (actually, LC longitudinal momentum) and the transverse momentum of the gluon fluctuations. In practice, is at least as large as the target saturation momentum , since this is the typical transverse momentum acquired by either the soft gluon, or its parent parton, via interactions with the target (see e.g. Mueller:2001fv (); Iancu:2002xk (); Gelis:2010nm ()). Hence, the ‘shockwave condition’ can be written as a lower limit on the gluon energy : , with
This limiting energy is an intrinsic scale of the target and grows with the target size like (since ). To have a significant phase–space for the high–energy evolution, the energy of the incoming projectile must be considerably larger than , namely such that with assumed to be small ().
3.1 Performing the time integrations
What is special about the shockwave (SW) target, is that the probability for a gluon to be emitted or absorbed inside the target is negligible999Strictly speaking, this statement is gauge–dependent, but it is indeed correct in the gauge that we currently use; see e.g. the discussion in Blaizot:2004wu ()., since suppressed by a factor . This physical statement is boost invariant, but the mathematics becomes simpler by working in the ‘target infinite momentum frame’, i.e. a frame in which the nucleus is ultrarelativistic and it looks like a ‘pancake’ (our intuitive representation of a SW). In such a frame, the target can be effectively treated as a –function at . This drastically simplifies the structure of the background field propagator and the action of the functional derivatives on the Wilson lines.
where is the free propagator (157), is the adjoint Wilson line introduced in Eq. (1), and . The physical interpretation of Eq. (3.1) is quite transparent: when and are both positive, or both negative, the gluon does not cross the SW, so it propagates freely; when and are on opposite sides of the SW, the gluon propagates freely from the initial point up to the SW, then it crosses the latter at some transverse position , thus accumulating a color precession represented by the Wilson line , then it moves freely again, up to the final point.
Furthermore, since gluons cannot be emitted or absorbed inside the SW, the action of the functional derivative on the Wilson lines is piecewise independent of time. Indeed for any negative value of the time argument, one has (compare to Eq. (2.1))
where we have used and for and a target field localized at . Similarly, for a positive value , one can write
The above equations have introduced the ‘right’ and ‘left’ functional derivatives, and respectively , which act on the Wilson lines as infinitesimal color rotations of the right, respectively on the left, and measure the color charge density in the projectile prior, respectively after, the collision. They are related by the condition , which expresses the color rotation acquired by a color current which crosses the shockwave.
The fact that the r.h.s.’s of Eqs. (25)–(26) are independent of time allows us to perform the time integrations directly at the level of the evolution Hamiltonian (4), that is, before acting with on the observable. To that aim, we need to distinguish three regions for the time integrations:
(i) and : the evolution gluon crosses the SW
where the adiabatic prescriptions are implicit (they will be exhibited when needed) and, cf. Eq. (3.1),
Due to the factorized structure of the background field propagator (28), the two time integrations are independent of each other. To be specific, consider the integral over . This involves
The final result is recognized as the Weizsäcker–Williams field created at by a point-like source at . Note that the complex exponential in the integral over has restricted the respective phase–space to an interval of order after the SW. A similar conclusion holds for the emission time , which is restricted to an interval before the SW. The respective integral yields
Importantly, the final results in Eqs. (3.1)–(30) are independent of . In both cases, this is due to a cancellation between the factor implicit in the free propagator and the phase–space factor produced by the time integral. As a consequence, the ensuing integral over in Eq. (27) is logarithmic : . Putting all together, one finds
with the following notations:
(ii) : the evolution gluon is emitted and reabsorbed prior to the SW
In this case, both functional derivatives within act as ‘right’ derivatives, cf. Eq. (25). Also, the gluon propagator reduces to the free propagator , as shown in Eq. (156). Consider first the ‘radiation’ piece of this propagator, which gives
The time integrations involve (with the shorthand notation )
The use of the adiabatic prescription has been essential in obtaining the above result, as we now explain. The time separation is restricted by the oscillatory phase to values of order , but the central value is only restricted by the adiabatic switch–off, so the corresponding integral yields an ‘infrared’ divergence proportional to . By itself, this divergence is pretty harmless, since ultimately cancelled by a similar contribution due to the Coulomb piece, as we shall see. What is more subtle though, is the obtention of the finite term accompanying the divergence (namely, the term in Eq. (3.1)) : this term is correctly computed when using the adiabatic prescription, as above, but it would be mistreated by other regularizations, like a sharp cutoff on