QCD-Gravity double-copy in the Regge regime: shock wave propagators

In two recent papers [1, 2], we explored universal features of multiparticle production in $2\rightarrow N$ gluon and graviton final states (with $N\gg 1$), respectively, in QCD and in Einstein gravity (GR). In the GR context, the program of such studies was first initiated by Lipatov [3, 4] and by Amati, Ciafaloni and Veneziano (ACV) [5, 6, 7] in the language of scattering amplitudes. Our work adopts a complementary semi-classical framework of gravitational shock wave scattering. This approach was previously developed successfully in QCD to describe multiparticle production in hadron-hadron and electron-hadron scattering in QCD at collider energies. In particular, since the rapid growth in gluon occupancies in high energy Regge asymptotics leads to the dominance of saturated states of maximal occupancy [8, 9] in multiparticle production, their properties could be computed systematically and efficiently employing many-body methods [10, 11]. This QCD framework, now called the Color Glass Condensate effective field theory (CGC EFT) [12, 13, 14], allows one to recover results previously obtained for $2\rightarrow N$ scattering by employing amplitude methods; a specific example is the celebrated BFKL equation of Lipatov and collaborators [15, 16]. More generally, the CGC EFT allows one to derive powerful nonlinear renormalization group (RG) equations-the BK/JIMWLK equations  [17, 18, 19, 20, 21, 22]- that allow for quantitative descriptions of a wide range of high occupancy final states in hadron and nuclear collisions. Such applications pose a formidable challenge for first principles Feynman amplitude computations.

In Regge asymptotics, the structure of $2\rightarrow N$ amplitudes, in both QCD and GR, can be constructed from the imaginary part of the $2\rightarrow 2$ amplitude $A(s,t)$. In the $t$-channel, this has a ladder structure. The ladder is ordered in relative rapidities (the logarithms of sub-energies of each cut rung) and is comprised of two building blocks making up, respectively, the individual rungs and legs of the ladder. The former are so-called Lipatov vertices anchoring each end of a given rung; these sum all possible $2\rightarrow 3$ amplitudes that are allowed in these asymptotics. The Lipatov vertices are nonlocal functions of the two momenta along the legs attached to a rung. Their product, from both ends of the rung, is a gauge invariant quantity.

The other building block is the reggeized propagator describing momentum transfer along the legs of the ladder. Reggeization in this context refers to the multi-loop iteration of the bare t-channel propagator $\frac{1}{t}\rightarrow\frac{1}{t}\,e^{\alpha_{R}(t)\,Y}$, where only contributions containing the leading double logarithmic contributions $\alpha_{R}(t)=\alpha\,\ln{t/\Lambda}$ and in the rapidity variable $Y=\ln{s/t}$ are retained. Here $\alpha$ refers to the coupling constant, $t$ and $s$ are, respectively, the squared momentum transfer and squared center-of-mass energy of a given sub-ladder, and $\Lambda$ is an infrared cutoff. The function $\alpha_{R}(t)$ satisfies a Regge theory interpretation as the poles $1/(j-\alpha_{R}(t))$ of the scattering amplitude in the $t-j$ plane, where $j$ denotes the spin of the $t$-channel exchange. Lipatov and ACV showed that these structures can be incorporated in a so-called reggeon field theory for both QCD and GR [23]; for a recent review of this eikonal approach, see [24].

In QCD, for weak coupling $\alpha\ll 1$, this BFKL construction resums leading logarithms $(\alpha Y)^{N}$ to all orders in perturbation theory. The intercept of this amplitude at $t=0$ corresponds to the BFKL pomeron, which describes the $2\rightarrow N$ cross-section $\sigma=2\,{\rm Im}A(s,t=0)\propto s^{\alpha_{R}(t=0)}$, with $\alpha_{R}(t=0)\sim 0.5$, for $\alpha=0.2$. The BFKL pomeron leads to a growth of the cross-section that is far more rapid than its nonperturbative counterpart; a smooth transition from soft to hard dynamics is clearly observed in deeply inelastic scattering (DIS) collider data [25].

An important point to note here is that the longitudinal momentum are soft in both regimes, corresponding to momentum fractions $x\ll 1$ of gluons in the $2\rightarrow N$ scattering. The soft to hard transition is in transverse momenta, where in the latter case, the gluons carry large transverse momenta $k_{\perp}\gg\Lambda_{\rm QCD}$. Specifically, the BFKL framework is valid for $|t|\gg\Lambda_{\rm QCD}^{2}$ with $|t|/s\rightarrow 0$. Systematic higher order corrections to this BFKL construction are therefore feasible. For an excellent recent summary of state-of-the art, we refer readers to [26].

Lipatov’s remarkable result in [3, 4] is that the gravitational Lipatov vertex satisfies a double-copy relation

where $C^{\mu}$ is the QCD Lipatov vertex [15] and $N^{\mu}$ is the QED bremsstrahlung factor. In [1], we showed that this result can be recovered from a semi-classical computation of radiation emitted in the scattering of gravitational shock waves in GR. In particular, we showed that the computation is exactly analogous to prior computations of gluon radiation in the scattering of gluon shock waves in the CGC EFT, where the amplitude for gluon radiation is proportional to $C^{\mu}$ [27, 28, 29]. Independently of the work of Lipatov, double-copy relations between Yang-Mills theory and gravity have a long history [30], an example of this being the BCJ color-kinematic double-copy [31, 32]; this powerful duality has recently been applied to compute high order corrections to gravitational radiation from the inspiral phase of black hole mergers [33, 34].

While there have been interesting attempts to understand the Lipatov double-copy within the framework of the BCJ color kinematic duality [35, 36, 37], we showed in [2] that the Lipatov double-copy can be simply understood as a classical double-copy [38, 39, 40, 41, 42, 43]. Building on prior work by Goldberger and Ridgway [39] computing gluon radiation from the scattering of classical color charges coupled to slowly varying Yang-Mills background fields [44], we observed that their classical double-copy replacement rules¹¹1As emphasized in [2], in order to recover the gravitational Lipatov vertex, it is very important to keep the sub-eikonal corrections to $C^{\mu}$ computed in the Yang-Mills + Wong framework. These sub-eikonal terms are crucial in correctly recovering the second term in Eq. (1) proportional to the bilinear of the QED bremsstrahlung factor. for gluon radiation from ultrarelativistic color charges $c^{a}$ with color index $a$,

and $g\rightarrow\kappa$ (with $g$ the QCD coupling, and $\kappa^{2}=8\pi G$, where $G$ is Newton’s constant), leads to Eq. (1). Interestingly, formally taking the soft limit of gluon radiation in the Yang-Mills+Wong computation, where both transverse and longitudinal momenta are small, and performing the corresponding classical double-copy, leads to Weinberg’s result [45] for soft graviton radiation [46].

Thus as emphasized by Ciafaloni et al. [47], there are two regimes of $2\rightarrow N$ multiparticle production in gravity. One is the Weinberg regime of soft gravitons, and the other is the Lipatov regime of semi-hard “wee” partons, in Feynman’s terminology of partons with momentum fractions $x\rightarrow 0$ but transverse momenta $\sim 1/R_{S}$, where $R_{S}$ is the Schwarzchild radius. As we discussed at length in [1], inelastic production of semi-hard gravitons becomes significant for impact parameters $b$ satisfying $\sqrt{|t|}\sim 1/b\leq 1/R_{S}$.

In QCD, the copious growth in gluon occupancy generates many-body screening and recombination effects that regulate the growth in the occupancy to saturate at a maximal occupancy $N\sim 1/\alpha$. The overoccupied gluon fields form a semi-classical lump characterized by an emergent close packing semi-hard scale $Q_{S}(x)$. It is their many-body dynamics that is captured by the BK/JIMWLK equations, with $Q_{S}(x)$ the scale determined by the nontrivial fixed point of their RG evolution with rapidity. The classical lump also saturates unitarity perturbatively at scales $1/Q_{S}\ll 1/\Lambda_{\rm QCD}$. In ultrarelativistic heavy-ion collisions, it is the formation, decay and rescattering of gluons forming these classical lumps that leads to the formation of the quark-gluon plasma [48].

The outstanding question of interest to us is whether an analogous picture emerges in $2\rightarrow N$ scattering in gravity for impact parameters $b\rightarrow R_{S}$. Indeed, it has been argued previously that black holes can be understood as overoccupied self-bound semi-classical graviton lumps, with the Schwarzchild radius representing the close packing scale $R_{S}\sim 1/Q_{S}$ [49, 50]. Simple power counting in this Black Hole Quantum Portait (BHQP) picture suggests that the probability to form such semi-classical lumps in $2\rightarrow N$ processes is exponentially suppressed unless the corresponding microstate entropy is $O(1/\alpha)$; this has been argued to saturate the Bekenstein bound in units of the Goldstone decay scale of these microstates [51]. As noted in [52], this reasoning is remarkably similar to the understanding of the emergence of semi-classical gluon lumps in the CGC EFT. It was argued further that the underlying physics of such semi-classical lumps that saturate unitarity is universal and independent of the underlying dynamics. Specifically, the CGC-BHQP correspondence suggests an analogous Goldstone picture of modes in the CGC EFT representing the overoccupied gluon microstates. Some of the consequences in the CGC EFT context have been noted separately [53, 54].

One way to confirm this conjecture of universality across widely separated energy scales is to develop a similar quantitative RG analysis for gravitational radiation in the Lipatov regime, as was previously developed for gluon radiation in the CGC EFT. In concrete terms, can we
I) derive the gravitational equivalent of the BFKL equation in the shock wave scattering formalism,
II) generalize this study to derive the equivalent of the BK/JIMWLK equations in gravity, and
III) observe the emergence of a nontrivial fixed points corresponding to black hole formation.
Questions I and II are of phenomenological interest in the computation of gravitational radiation from the merger phase of black hole scattering where velocities reach ultrarelativistic speeds and numerical relativity simulations become extremely challenging [55]. Developing quantitative methods to address the final instants of gravitational radiation [56, 57] is an important practical motivation for this work.

We observed earlier that the key ingredients in question I are the Lipatov vertices and the reggeized propagators. In [2, 1], we computed the former in the shock wave formalism. The focus of this work will be the computation of shock wave propagators; the mapping of these to the language of graviton propagators has been discussed previously [58, 59, 60]. In subsequent work, we will employ these propagators to develop the RG framework necessary to address questions I through III. Our discussion throughout will be grounded in the Schwinger-Keldysh formalism of cut and retarded propagators in quantum field theory rather than in in the language of Feynman amplitudes alone or in that of regggeon field theory, with the latter, as noted, employed in the Lipatov-ACV approach [23, 61, 62, 63, 64]. As observed in [65, 66], the so-called Abramovsky-Gribov-Kancheli (AGK) cutting rules [67] employed extensively by ACV can be understood in the Schwinger-Keldysh formalism (SK) as a specialized application of Cutkosky’s rules in the presence of time dependent sources. Further, as emphasized in an elegant discussion in [68], such SK methods are closely related to the generalized unitarity based approaches to scattering amplitude computations [69, 70].

The SK methods we will adopt are especially suited to the computation of gravitational radiation. In particular, we will be interested in the computation of the inclusive radiation spectrum as a function of impact parameter and rapidity. It was shown in [65, 66] that the computation of inclusive distributions can be formulated systematically in powers of the coupling $\alpha$ as an initial value problem where the ingredients are retarded and cut propagators. For gravitational (and gluon) radiation in strong backgrounds, the leading order spectrum is governed by the Lipatov vertex which, as we observed, can be computed from the equations of motion in classical shock wave scattering. To go beyond, and include the rapidity evolution of this spectrum, as well as rescattering contributions that become important as $b\rightarrow R_{S}$, we will need to compute next-to-leading order (NLO) corrections²²2As we will discuss in section IV, these NLO contributions in the shock wave formalism are one order beyond the 2-loop H-diagram of ACV in perturbation theory. in the coupling $\alpha$. This is where the computation of shock wave propagators is essential. As we will discuss, in order to obtain these propagators, one needs to solve the initial value problem of computing the solutions of the classical GR shock wave equations of motion as well as solutions of small fluctuation equations.

The paper is organized as follows. In section II, we will review the derivation of results in the CGC EFT for colored scalar and gauge field propagators in the background of a gluon shock wave. We shall focus our derivation on expressions for propagators in the eikonal limit. In section III, we will present results for the full off-shell propagator of scalar and the graviton fields at the leading eikonal order. We restrict our derivation here primarily to eikonal shock wave propagators. This limit will likely be sufficient as long as one is computing radiative corrections that are not too close to the event horizon. We will also comment on the sub-eikonal corrections which are substantially more involved; these will be important in addressing question III articulated above. In section IV, we will sketch the computation of the NLO corrections in GR following guidance from previous such computations in the QCD. Detailed computations carrying out the program outlined earlier will be undertaken in subsequent work. Appendix A provides a brief introduction to Schwinger-Keldysh propagators. The convolution properties of Green functions are outlined in appendix B.

In this section, we will first review the construction of eikonal shock wave propagators in QCD in high energy (Regge) asympotics. As noted in the introduction, these quark and gluon propagators are a key feature of the derivation of the BK/JIMWLK RG equations. Shock wave propagators in QCD were computed in several papers [71, 17, 72, 73, 74, 75, 76, 27, 28] using various techniques in both position and momentum space. In what follows, we shall review the derivation of a scalar propagator in the background of the CGC shock wave closely following the discussion in [27, 28]. The result for the gluon propagator analogously follows that we will present at the end of the section. This will be relevant for the classical double-copy map of this propagator to the corresponding propagator in gravity.

In Lorenz gauge $\partial_{\mu}A^{\mu}=0$, the shock wave background of overoccupied gluons appearing in the CGC effective field theory takes the form [1]:

Here $\rho({\boldsymbol{x}};y)$ denotes the color charge density generated by fast quarks and gluons carry large longitudinal momentum fractions $x\sim 1$ of the hadron momentum; these couple coherently to wee (small $x\ll 1$) gluons, corresponding to a rapidity scale $y$. This color charge density is classical due to the large occupancy, and parametrically of order $O(1/g)$, where $g$ is the QCD coupling. The background it couples to is a singular solution of the classical Yang-Mills (YM) equations in the presence of the static light cone current corresponding to the color charge density.

The solution of the YM equations also make clear that the field strengths of the shock wave vanish outside a narrow region in $x^{-}$ outside $x^{-}=0$. The corresponding pure gauge field configurations are not identical in the spacetime region separated by the shock wave since one cannot perform a continuous gauge transformation between the two sides of the shock wave. This is manifest in light cone gauge $A^{+}=0$, obtained by performing the gauge transformation $A_{\mu}\to UA_{\mu}U^{\dagger}+\frac{i}{g}U\partial_{\mu}U^{\dagger}$ on the solution in Eq. (3).

In light cone gauge, only transverse components of the gauge field are nonvanishing and are given by

This expression makes transparent that the vacua are not identical in the regions $x^{-}>0$ and $x^{-}<0$ and correspond to distinct pure gauge solutions that cannot be transformed into one another³³3This property of shock wave solutions in gauge theories (and gravity) has deep connections to soft theorems and memory effects [77], as we will discuss further later..

The direct computation of shock wave propagators for the light cone gauge $A^{+}=0$ (where $P^{+}$ corresponds to the direction of large momentum) is nontrivial, as shown in [73]. However they are considerably simpler in the “wrong” light cone gauge $A^{-}=0$ [71], which shares the same background field as the Lorenz gauge solution in Eq. (4). This background field also has a simple double-copy relation to gravitational shock waves [78]; we will therefore henceforth work in this gauge.

We will first compute the (retarded) propagator of an adjoint scalar field coupled minimally to this background. A necessary ingredient is the solution to the small fluctuation equations of the scalar field, given by⁴⁴4The scalar equations of motion are $\displaystyle D_{\mu}D^{\mu}\phi=0~{},\qquad D_{\mu}\phi=\partial_{\mu}\phi-ig[A_{\mu},\phi]~{}.$ (5)

where $\square=\partial_{\mu}\partial^{\mu}$ is the free Laplacian. Using $[A,B]_{a}=if^{abc}A_{b}B_{c}=-\left(T^{b}\right)_{ca}A_{b}B_{c}=[(A\cdot T)B]_{a}$, we can rewrite the commutator in the above equation,

Hence the Green function satisfies (the subscript $R$ stands for retarded)

supplemented by boundary data specified at a fixed $x^{-}$ slice in the region $x^{-}<0$. As shown in [27] (and reviewed in appendix B), $G_{R}(x,y)$ satisfies the recursive equation,

The surface $z^{-}=z^{-}_{0}$ specifies the spacetime slice where the initial data is defined.

Since the shock wave background in Eq. (3) is pure gauge everywhere, except at $x^{-}=0$, the propagators $G_{R}(x,y)$ in this background too are free propagators everywhere, except for paths corresponding to $x^{-}>0,y^{-}<0$ or its converse. For these configurations, the propagator encounters the shock wave background, and is therefore nontrivial. A straightforward way to calculate $G_{R}(x,y)$ for $x^{-}>0,y^{-}<0$ is to use Eq. (9) to write it as

and set $z_{0}^{-}=\delta$, $w^{-}_{0}=0$, where $\delta$ specifies the shock wave width in the $x^{-}$ direction, a parameter we will set to zero at the end of the calculation. This choice of $z^{-}_{0}$ and $w^{-}_{0}$ allows us to replace $G_{R}(x,z)$ and $G_{R}(w,y)$ by their corresponding free expressions $G_{R}^{0}(x,z)$, $G_{R}^{0}(w,y)$ in Eq. (10), that are given by

The intermediate steps in obtaining the last equality are provided in appendix B.1.

In order to compute $G_{R}(z,w)$ in Eq. (10), we begin with the definition of the retarded Green function,

where $\phi_{k}(x)$ is the eigenfunction of the small fluctuation equation in Eq. (7). In the absence of the shock wave background, the eigenfunctions are simply plane waves, and we recover Eq. (11).

To solve for $\phi_{k}(x)$ in the neighborhood of the shock wave (in the $x^{-}$ direction), we make the eikonal assumption that changes in the field in the transverse directions are negligible compared to those in the longitudinal direction. We can then approximate Eq. (7) to read

With the input from the scalar field solution in the region $x^{-}<0$, the solution for $x^{-}>0$ is

Here $c^{a}$ is a pointlike color charge and $U({\boldsymbol{x}})$ is a light-like Wilson line, path ordered in $x^{-}$:

Thus, in the eikonal approximation, the result of the scattering of the colored scalar with the shock wave is to simply rotate it in color space.

In summary, the solution of the small fluctuation scalar equation in the eikonal approximation is

Inserting this result into Eq. (12) gives the free propagator for $\theta(-x^{-})\theta(-y^{-})$ or the color rotated free propagator for $\theta(x^{-})\theta(y^{-})$. For the nontrivial component of the Green function in the r.h.s of Eq. (10), the term crossing the shock wave is given by

Note that in Eq. (10) we need to take the limit $\delta\to 0$. This gives

To obtain this result, we first performed the $k^{+}$ integral by closing the contour from below, and then took the limit $\delta\to 0^{+}$. We also used the identity

Plugging this result into Eq. (10), we obtain

which agrees⁵⁵5In [74], the authors are computing the Feynman propagator instead of the retarded propagator; this explains the $U^{\dagger}$ in their Eq. (9) instead of $U$ in our case. See [28] for a discussion retarded versus Feynman propagators in this context. with Eq. (9) of [74].

The Fourier transform

of the propagator then takes the compact form (derived in appendix B)

where $\tilde{G}_{R}^{0}(p)$ is the free propagator in momentum space,

and the effective vertex is

The dressed propagator can be represented pictorially as

The results derived above for the colored scalar propagator in the shock wave background can be straightforwardly extended to fermion and gluon propagators in the gluon shock wave background; these were first computed in [71, 17, 75] in position space and later given in momentum space [76]. Indeed, the gluon shock wave propagator in the lightcone gauge $A_{+}=0$ can be written similarly to Eq. (22) as

where the effective vertex is [76, 79, 80, 28]

Here $\tilde{G}^{0}_{R,\mu\nu}(p)=\Lambda_{\mu\nu}\tilde{G}^{0}_{R}(p)$ with $\Lambda_{\mu\nu}=\eta_{\mu\nu}-\frac{p^{\mu}n^{\nu}+p^{\nu}n^{\mu}}{p\cdot n}$. We emphasize that for phenomenological computations this light cone gauge $A_{+}=0$ should be distinguished from the light cone gauge $A_{-}=0$, where the parton interpretation of structure functions extracted from experiments (with hadron momenta $P^{+}$) is more transparent.

The result in Eq. (25) follows in a similar manner to the scalar case. One defines the spin-1 shock wave propagator in terms of the small fluctuation fields as

After gauge fixing to light cone gauge $a_{+}=0$, the solution to the small fluctuation field $a_{\mu,k}^{(\lambda)}(x)$ is readily determined [29]. Asymptotically, the solutions are plane waves $a_{\mu,k}^{(\lambda)}(x^{-}<0)=e^{-ikx}\,\epsilon_{\mu}^{(\lambda)}(k)$, where $\epsilon_{\mu}^{(\lambda)}(k)$ is the polarization tensor in light cone gauge. In arriving at the result for the gluon shock wave propagator, one employs the completeness relation

where $n$ is a null four vector such that $\epsilon\cdot n=0$ and $k\cdot n\neq 0$. Note that this completeness relation is identical to the one in the absence of shock wave background and will not hold for arbitrary background gauge fields. The reason we can use it here is because in the eikonal approximation the interactions of small fluctuation fields with the shock wave do not modify the polarization tensor at leading eikonal order. A similar reasoning will be used in the gravity discussion in the next section.

We will now follow the methods outlined in the previous section on QCD at high energies and apply these to construct shock wave propagators in gravity. After first reviewing the properties of gravitational shock waves, we will compute scalar propagators in shock wave spacetime and follow this by a computation of the graviton shock wave propagator. The latter computation will be performed in the eikonal regime where graviton polarizations are not affected by interactions with the shock wave. We will comment on sub-eikonal modifications to these results. We will also briefly comment on classical double-copy relations to the Yang-Mills results.