Classical Effective Field Theory for Weak Ultra Relativistic Scattering

Classical Effective Field Theory
Weak Ultra Relativistic Scattering

Barak Kol
Racah Institute of Physics, Hebrew University
Jerusalem 91904, Israel

Inspired by the problem of Planckian scattering we describe a classical effective field theory for weak ultra relativistic scattering in which field propagation is instantaneous and transverse and the particles’ equations of motion localize to the instant of passing. An analogy with the non-relativistic (post-Newtonian) approximation is stressed. The small parameter is identified and power counting rules are established. The theory is applied to reproduce the leading scattering angle for either a scalar interaction field or electro-magnetic or gravitational; to compute some subleading corrections, including the interaction duration; and to allow for non-zero masses. For the gravitational case we present an appropriate decomposition of the gravitational field onto the transverse plane together with its whole non-linear action.
On the way we touch upon the relation with the eikonal approximation, some evidence for censorship of quantum gravity, and an algebraic ring structure on 2d Minkowski spacetime.

1 Introduction

The effective field theory (EFT) approach to General Relativity (GR) GoldbergerRothstein1 () borrows ideas from effective quantum field theories and applies them to the two-body post-Newtonian dynamics in classical General Relativity.111See DamourFarese () for early precursors of the EFT approach to GR. In addition to contributing insight and a fresh perspective the approach allowed to push forward the state of the art PortoRothstein () by computing for the first time the next to leading spin1-spin2 interaction in the effective two-body action.222Even if imperfectly, since it was missing certain contributions found in SteinhoffHergtSchaefer () using Hamiltonian methods and also found later in Porto:2008tb () to arise from indirect contributions in the EFT method. See Levi:2008nh () for a derivation using Non-Relativistic Gravitational (NRG) fields.

Most of the ensuing work on the EFT approach centered on the post-Newtonian approximation – see for example NRGR (). However, the EFT approach has a very wide applicability range, namely whenever a field theory (not necessarily of gravity) contains two widely separated length (or time) scales CLEFT-caged (). Thus far some such applications appeared: a post-Newtonian theory for caged black holes (black holes localized within a compact dimension) caged (); black hole internal degrees of freedom at low frequencies dissip (); new observations on finite size corrections to the radiation reaction force in classical electrodynamics Galley:2010es (); some results on the gravitational two body problem in the extreme mass ratio limit, see GalleyHu1 () and continuations of it; and finally see Cannella:2009he () for some tests of GR.

In this work we shall present a new classical effective field theory, namely for weak ultra-relativistic scattering, in the context of Einstein’s gravity or more generally any other field theory.

We start in subsection 1.1 by reviewing past work on Planckian scattering including 'tHooft87 (); ACV () as a concrete challenge for quantum gravity to address. This part serves as motivation for our work but the ensuing parts are independent of it.

In section 2 we set up the effective field theory and determine its central Feynman rules. The main ingredient is to set the field propagator to be


This means that the field is instantaneous and transverse. The intuition is that while a particle at rest has a spherical configuration of field strength lines emanating from it, an ultra relativistic boost Lorentz contracts the longitudinal direction, thereby confining the field lines to a transverse pancake attached to the particle’s “nose” (see figure 1). Accordingly the particles interact only at the moment of passing when they each intersect with the other pancake, and the equations of motion degenerate to that instant.

We conclude this section by computing the leading momentum transfer during scattering for several cases of interaction fields: scalar, electromagnetic and gravitational, reproducing the literature value in the latter case.333The other cases may also exist in the literature. On the way we make use of the light-cone formalism where the dynamics of each particle is nearly Galilean, we define effective ultra-relativistic charges and we define a conserved global charge for the interaction fields. We observe that this theory is analogous to the non-relativistic EFT 444In the gravitational case the non-relativistic approximation is known as post-Newtonian, and the associated EFT GoldbergerRothstein1 () is known as Non-Relativistic GR (NRGR). only in one fewer dimension: the fields propagate instantaneously and the particle dynamics is nearly non-relativistic.

Figure 1: The geometry of the scattering process. Left: 3d geometry at the moment of passing, including equipotential lines. Note that the field lines are Lorentz compressed to the transverse plane. Right: top view. The initial condition parameters are the center of mass energy and the impact parameter . The right moving particle, namely the one moving in the positive direction, is denoted by R, while the other is denoted by L.

In section 3 we proceed to compute some higher order corrections, present even for linear fields, which are due to non-linearities in the solutions of the equations of motion, where in particular retardation effects must be considered. Setting-up a perturbative theory for ultra-relativistic particles and using the two-body effective action (where the field is integrated out) allows us to find the leading order longitudinal momentum exchange. It comes at second order in the small parameter, while the transverse momentum transfer is of first order, thereby confirming the assumed hierarchy of scales which defines the EFT.

In section 4 we choose a different time for initial conditions thereby changing our “regularization condition” and hence “resumming” and simplifying the perturbation series. This ingredient enables us to proceed to the third order and in particular to resolve the instantaneous interaction and to obtain its duration. This result demonstrates the utility of our formalism.

In section 5 we compute corrections due to non-zero masses, which are controlled by another small parameter. Finally in section 7 we summarize our results and discuss open questions. The appendices contain some conventions and calculations, as well as an algebraic curiosity.

In section 6 we describe the dimensional reduction of the gravitational field onto the transverse plane and its whole non-linear action. The analogue reduction for electromagnetic case is considerably simpler and is treated in section 3.

The EFT in this paper is related to the eikonal approximation in quantum field theory. Actually the latter is the semi-classical approximation corresponding to this EFT. Some of our results including an effective field theory for Planckian scattering appeared elsewhere including in Amati-Ciafaloni-Veneziano[93] ACV93 (); however, we have a different approach: our EFT is purely classical, we define the small parameter and establish power counting rules, we analyze fields other than gravity, and we compute various corrections including the interaction time.

We note that the ultra-relativistic limit is part of the post-Minkowski approximation (see postMink () and references therein) where one considers perturbations around a flat space-time due to bodies with arbitrary velocities, possibly relativistic but not ultra-relativistic. However, for our more restricted velocity range we are able to construct an EFT which allows to compute arbitrary higher order corrections, while in the post-Minkowski case so far only a first order correction was demonstrated to be possible.

1.1 Planckian scattering and classical dominance

In order to explore quantum gravity it is natural to consider a gedanken scattering experiment where two gravitationally interacting bodies are scattered (see figure 1), especially when the center of mass energy is Planckian or higher Dray-`tHooft (); 'tHooft87 (); ACV (); ACV93 (); ACV2008 (). For simplicity we may assume the particles to be light-like (massless) and point-like. Later we shall study corrections when the masses are non-zero but still much smaller than the energies. We shall also discuss finite-size corrections.

The initial conditions are specified by two parameters


the center of mass energy, and the impact parameter.

Figure 2: The parameter space for trans-Planckian scattering, demarcating the boundaries between three theoretical regions.

Let us discuss the physics as a function of this parameter space (see figure 2). For generality we consider a general space-time dimension . For we may compare with the Planck energy (where is Newton’s gravitational constant). For large we expect a semi-classical black hole to form, followed by a long period of Hawking evaporation, while for small tree-level quantum GR would be valid (loops are suppressed by ).555Actually in the quantum regime the impact parameter is not a good parameter due to uncertainty, and it should be replaced say by the Mandelstam variable .

For large on the other hand, we should compare with the would-be Schwarzschild radius which is the only other parameter with length dimension. Accordingly large means . In this regime the interaction is dominated by a single field exchange, so having no loops this is essentially classical scattering.

It is interesting to consider varying from large to small, while keeping fixed and large in Planck units. For large we have a small attractive deflection. As is reduced the deflection angle increases together with the deflection time. While (classical) gravitational radiation is weak for weak deflection, it too increases as is lowered. At some critical (or marginal) a black hole forms. By dimensional analysis must be order one in units of . We see no reason why the marginal black hole mass would not be zero. That means that at all the incoming energy is emitted in the form of gravitational radiation. From general considerations the scattering angle and time diverge in this limit666The generic dependence of these quantities in such a marginal event would be . The simplest example for that is a non-relativistic, non-quantum particle scattering in 1d off a potential maximum such that the particle’s energy is close to that of the peak.. Recently numerical GR simulations succeeded for the first time in simulating a black hole creation in this process ChoptuikPretorius () (and references therein). Incidentally we note that the simulations indicate that the marginal small black hole is very close to extremal spin.

For the black hole mass is non-zero, and the dependence on defines a certain characteristic exponent (and the same holds for its angular momentum). After the appearance of a black hole horizon, the black hole relaxes to its stationary form through the classical process of “ringdown” emission of gravitational waves. The ringdown radiation is a smooth continuation of the inspiral radiation.

As we know that the colliding light-like shock fronts do not interact until they reach . Being at sub-Planckian distance this collision is outside the validity of classical GR, yet we expect most of the energy to be converted to a black hole, and therefore the results of this quantum gravity process would remain hidden behind a horizon. At infinity one observes the Hawking radiation which is semi-classical apart for the last burst of one Planck mass which is an essential quantum gravity effect. Due to Hawking evaporation black hole formation does not necessitate a genuine phase transition (since for all eventually all energy returns to infinity) but rather it is probably a “cross-over”.

Altogether we observe that full quantum gravity is not among the theories tiling the parameter space of figure 2, being needed perhaps only at the boundary at and small . Hence it appears that the horizon conspires to hide such effects in a way that can be called censorship of quantum gravity.

In particular, the high energy limit is dominated by classical gravity. This phenomenon of “classical dominance” was stated by ’t Hooft in 'tHooft87 (). There the line of argument is somewhat different. The author analyzed the collision of two point-like and light-like gravitational shock waves. These shock waves are described by the Aichelburg-Sexl metric Aichelburg-Sexl () which was shown to be equivalent to a non-trivial gluing of a pair of flat half space-times – the one before the shock and the one after it Dray-`tHooft (). Considering the evolving wave function of a particle as it passes through the shock wave carried by another, the author computes the scattering amplitude, and finally concludes that it is dominated by classical physics.

Other interesting aspects of Planckian scattering are discussed in 'tHooft90 (); Verlinde91 ().

While the whole high energy range is covered by classical GR, there is a big difference between large and small : for large we have weak scattering and a perturbative descriptions is possible, while for small a black hole is created in what is surely a non-perturbative process. Accordingly we may hope to find a novel effective field theory for large , but not for the more interesting non-perturbative regime, which yields only to a full numerical simulation. Yet, a perturbative analytic theory does serve several purposes: it can test and validate the numerics, and it provides insight into the problem, especially into the dependence on parameters. It could even be used to provide indications for the non-perturbative regime via extrapolation. Finally and most importantly we wish to set-up a novel classical effective field theory. Hence from now on we shall study only the weak scattering limit.

2 Setting-up the effective field theory

Consider the two-body interaction of two ultra-relativistic bodies, see figure 1, as described in the previous subsection. We are interested in the perturbative limit of weak scattering, namely large impact parameter . Within this limit gravitation is not special and a scalar or vector field will also present similar EFT’s. Actually the scalar case suffices to derive the special form of the EFT propagator, and so we begin with it.

The total action for our two particles (denoted R,L for right and left moving) coupled to a scalar interaction field is


where the field action is 777Recall that with this normalization of the kinetic term the propagator in the 3d Euclidean configuration is .


The action for each particle is


where the particle kinetic action and the interaction action are

where are the mass and scalar charge of each body, is the particles trajectory as a function of its world-line parameterization and


where the dot denotes a derivative with respect to , and in this paper arrowed vectors always denote transverse space vectors


– see appendix A for a summary of our light cone conventions. Finally is an auxiliary field, the world-line metric, which guarantees reparameterization invariance by the transformation law under . Later we shall redefine the variable for convenience.

The field propagator - the main observation. Recall that when constructing an effective field theory for a non-relativistic approximation such as the Post-Newtonian approximation of gravity GoldbergerRothstein1 () one considers first field modes whose typical length scale is the inter-body separation. These modes are carried along with the particles and are called potential modes. Since by definition the velocities are low in this limit the potential modes satisfy , namely their temporal gradients are negligible with respect to spatial gradients. Accordingly one decomposes the quadratic bulk action as follows – the space-gradients constitute the unperturbed action and define the propagator, while the time-gradients are a perturbation and define a vertex. Apart from the potential modes, there are radiation modes which are generated at higher orders. The radiation modes have the same typical frequency as the potential modes, but being on-shell they have a much longer typical wavelength.

Similarly in the current ultra-relativistic case we shall find that the interaction field decomposes into potential and radiation modes, and that for potential modes certain components of , the wavenumber vector, dominate.

Consider computing the effective action for two bodies, in the straight line approximation, namely the one-field exchange diagram. It can be computed by evaluating (minus) the bulk kinetic term for the field. Substituting


into the action (3) where is the field sourced by particle , we have


When we compute (9) in space we essentially evaluate the integral


Since is independent of then and vice versa for body L (since ). The product integrand is now localized in both and in . Therefore, while has a finite scale


the longitudinal wavenumbers are negligibly small


More quantitatively we shall find later that


where , to be defined later, is schematically the product of effective charges


and is the classical and dimensionful analogue of the fine structure constant. Eq (13) defines the hierarchy of scales and the small parameter of the EFT. The scattering deflection angle will be seen to be of the same order as this small parameter.

The hierarchy of wavenumbers (12) is equivalent to a momentum transfer which is dominantly transverse since


Note that unlike quantum field theory the factor is essential on dimensional grounds.

Now we proceed to obtain the field propagator and its correction. We decompose the kinetic term as follows


Much like the non-relativistic case reviewed above, this decomposition implies the following Feynman rules for the perturbation theory: the first and dominant term is considered to belong to the unperturbed action and determines the field propagator


while the second term is subleading and determines a 2-vertex (correction to the propagator)


In 4d the transverse space is 2d and we find that in position space the propagator becomes


where is an arbitrary scale which does not affect the physics (see appendix B for details). From now on we shall set .

The transverse propagator (17) means that within the EFT the field lines spread instantaneously and only in the transverse space and thus they are confined to an infinitesimally thin (due to Lorentz contraction) “pancake” shock wave carried by the particle.

We recall that the transverse propagator applies only to the potential modes, and will not apply to radiation modes, which are needed only at higher orders than those calculated in this work.

The -particle vertex. The particle action (5) is singular in the massless (and hence ultra-relativistic) limit . To overcome that we rescale .

In the unperturbed light-like motion of particle R we have hence it is useful to parameterize the world-line by , namely we choose the gauge


Similarly we choose for particle L.

Finally, we perform another rescaling of , such that altogether we define


where is the initial light-cone momentum ( stands for “initial”), namely for particle R while for particle L . The factor in the definition is convenient as it sets the initial to unity


as we shall shortly see.

The action (5) becomes


where the effective ultra-relativistic scalar charge is defined by


Note that the limit is indeed non-singular now. The the 4-momentum is


Computing in with the gauge choice (20) we confirm that indeed .

We observe that the new form of action (23) is very similar to that of a non-relativistic particle: the action is quadratic in the transverse velocity and plays the role of the non-relativistic mass. This is a standard feature of the light cone or infinite momentum frame Dirac:1949cp (); Weinberg66 (); Susskind67 ().

The action (23) allows us to read the –particle vertex, namely


where the double lines represent the particle.

2.1 Leading scalar, electromagnetic and gravitational momentum transfer

Scalar case. At first order in the small parameter (13) one finds a transverse momentum transfer. In order to compute it we shall need the propagator (17), the -particle vertex (26) and the following equation of motion


derived from (23,25). We shall defer the full formulation of the equations of motion and the perturbation theory for the particle dynamics in the ultra relativistic limit, to the next section where we compute corrections.

The change in transverse momentum of particle L is opposite to that of particle R888As long as we can ignore momentum leakage to infinity through radiation. and is called simply “the momentum transfer”. The full expression for it is


where is the full retarded propagator sourced by at and without any approximations.

At leading order we may substitute the unperturbed straight line trajectory for both particles and the leading EFT propagator (17) and we have


This expression describes the momentum transfer in terms of a diagram which is actually the leading part of the two-body effective action.

This computation is quite straightforward in the EFT. Using the position space -propagator (19) we have


For light-like particles the integrations annihilate the functions and we have


We found that this result reproduces the computation in the full “microscopic” theory. Substituting back in (29) we finally obtain the leading momentum transfer


The weak scattering angle is can now be stated quantitatively


and it is of first order in our small parameter (13).

Electromagnetism. Now that we reproduced the leading momentum transfer for a scalar interaction we proceed to generalize it to the electromagnetic and gravitational interactions. In electromagnetism the total action is


where the field action is


and the particle action is


and where and are its mass and electric charge.

Since we assume that the longitudinal gradients are negligible compared with the transverse ones (12) it is appropriate to perform a dimensional reduction of the electro-magnetic field onto the transverse plane. Accordingly we decompose


where are a pair electro-static potentials which are scalars from the transverse perspective. Since on the unperturbed trajectory for particle R (and similarly for 2) we see that at leading order particle R couples only to , while particle L couples only to namely


This defines the electro-magnetic field-particle vertex, and in particular there is no need to define effective electric charges (unlike the effective scalar charges).

The propagator for in the limit (12) can be read from the kinetic term in (35)


Note that the sign which is opposite that in the scalar case signifies a repulsive interaction for identical particles, while in the scalar case it is attractive.

The longitudinal boost symmetry of the electro-magnetic action (35) is realized in the EFT as a global symmetry under which have charges . Accordingly we may represent the fields by the same oriented line and distinguish them by its orientation, which represents the flow of charge.

Having determined the Feynman rules we proceed to compute the leading scattering angle with the same method which we used in the scalar case (32). We find


See JKO () for a related study which is consistent with ours.

Gravitation. We shall now find the field propagator and field-particle vertex in order to compute the leading scattering angle for a gravitational interaction.

The gravitational field is


where is the flat space metric and are gravitational perturbations around it.

The gravitational action is


where the particle action is


where now


and so the field-particle coupling is encoded by the particle’s kinetic term.

Again we perform a dimensional reduction onto the transverse plane. We see that at leading order particle R couples only to , while particle L couples only to namely


This defines the gravitational field-particle vertex, where half the light cone momentum plays the role of an effective gravitational charge


The propagator for which are both scalars from the transverse perspective can be read from the Einstein-Hilbert kinetic term in (42)


(see for example subsection 6.1). The negative sign signifies an attractive interaction and there is an factor relative to the electromagnetic case. Here the longitudinal boost symmetry assigns to charges and again we may represent the fields by oriented lines.

Altogether the momentum transfer is


where we used


This result, eq (48) reproduces the value in the literature (see for example eq. (2.7) of ACV93 () as well as Dray-`tHooft ()).

Summary of leading momentum transfer. Summarizing the three cases above
(32,40,48), the leading momentum transfer is


where corresponds to the spin of the field and the fine structure constant is given by


3 Two body effective action

In this section we wish to consider higher order corrections while setting the masses to zero . Such corrections can arise at one of two steps: either in the two body effective action, or while solving the equations of motion which arise from it. As usual the two body effective action is defined by integrating out the interaction field, namely


where one must include all the connected diagrams without quantum loops, namely diagrams which consist of field propagators, field vertices and field-particle vertices, but no particle propagators, such that the diagram is connected and without loops when the source lines are ignored; and external field propagators end on both sources.

In general, and by analogy with the PN approximation we expect 5 possible diagrams (or effects) to contribute: retardation, vertex corrections, exchange of another field (such as other field polarizations), and finally bulk and world-line non-linearities.

Scalar two body potential. Let us compute the two body effective potential for the scalar interaction (3). Since the field is linear the sum over all diagrams (52) consists only of diagrams with one field exchange between the two sources and may include an arbitrary number of retardation vertices. Namely


where the retardation correction (sum over all retardation vertices) is given by the expansion




where the Taylor coefficient is given by the expansion


Combining all the ingredients we have in the scalar case




Electromagnetism. Since electromagnetism is a linear as well, we can express along the lines of the scalar case. First we must complete the discussion of the Maxwell action (35). After decomposing we need to add a gauge fixing term to the action for the transverse vector . As usual one could take


to render the (transverse) propagators non-singular. However one can still add a independent term to the gauge constraint . Just like in post-Coulomb approximation (non-relativistic electromagnetism) it is convenient to take the full -dimensional Lorentz gauge thereby avoiding a coupling between and . Accordingly we take


The total gauge fixed action for the electromagnetic field is

The coupling of particle R to the field is


and similarly for particle L.

The total action above determines the Feynman rules with which we can proceed to calculate the electromagnetic two body action. The sum over diagrams is similar to the scalar case, only now the particles can exchange either with either orientation or , which is the magnetic interaction. The factor is now absent, and there is an overall sign change, signifying as usual electromagnetic repulsion between like charges. Altogether the two body effective action is the same as in the scalar case (57) apart for substituting the following factor


Ultra Relativistic particle dynamics. In order to find higher order corrections for the scattering trajectories we should solve the equations of motion for the corrected two body effective action. But first we pause to conveniently formulate the equations of motion and the perturbation theory for Ultra Relativistic dynamics.

A point-like particle has 3 degrees of freedom. In the non-relativistic case they can be identified with , namely the spatial coordinates as a function of time. Looking at the action for particle R (23) we may now identify the 3 degrees of freedom as follows. account for 2 out of 3. and appear in the action only in a single -derivative term. Hence each requires only a single initial condition and should be counted as half a degree of freedom. Together they constitute the remaining degree of freedom.

We find it convenient to use a canonical (Hamiltonian) formalism with respect to the transverse coordinates, but not with respect to and .

3.1 Second order

We shall now use the formalism of the two body effective action to go beyond the first order in the small parameter. For concreteness, we specialize to the electromagnetic case and derive the equations of motion from the two body effective action (63). We denote the magnitude of the leading momentum transfer by


First we can recover the first order results including


where is the Heavyside step function.

We proceed to obtain the second order momentum transfer. By varying the action with respect to we find


from which we have


Varying the action with respect to we find that in order to saturate the factor we should extract an factor by expanding where is given by (66). Integrating over we obtain the total momentum change


The total transverse momentum change can be seen to vanish at this order.

Our results for the second order momentum change are related to the first order results in the following way. The change in (68) is required in order to preserve the mass-shell condition . The change in is such that in the center of mass frame where the change in energy (of each particle) vanishes as one could expect on general grounds. Actually (69) is the unique generalization outside the center of mass frame as it has the correct transformation law under a longitudinal boost.

4 Interaction duration at the third order

In this section we compute the interaction duration thereby resolving the instantaneous approximation. For that purpose we introduce an improved “renormalization condition”.

An improved renormalization condition. The perturbative computations become simpler if one takes the initial conditions not at early time but rather at the interaction time , since the latter enjoys a symmetry, namely time reversal, at least as long as radiation is ignored. For that purpose one pretends that the system’s state at is known and propagates both forward and backward in time, namely where is a provisional notation for the system’s state at time (actually the two relations are essentially the same due to the symmetry). Then one solves for and substitutes back into to obtain the required . By way of analogy we refer to this procedure as a change of renormalization condition.

We demonstrate this procedure at the leading order. Using early time initial conditions we already found the leading momentum transfer to be (50) where is the standard impact parameter at early time. Repeating the calculation with initial conditions at we obtain the same result only with where is the distance between the two bodies at . The scattering angle (in the center of mass frame) is given by


where (it is the same for both bodies), and is related to via


altogether yielding


This last expression sums beyond leading contributions to from expanding the left hand side, yet additional higher order contributions could arise on the right hand side as well.

Interaction duration. Repeating the perturbative procedure with initial conditions the first order results (66) change slightly into


where . The antisymmetry of the step function causes certain cancelations at higher orders which were not present in the original perturbation theory thereby fulfilling our expectation for simplifications. In particular it allows us to proceed to compute the transverse accelerations up to third order. The full computation is presented in appendix C. We obtain


We note that is proportional to . First, this means that the momentum transfer vanishes at this order. Moreover, since , the ratio of the corresponding prefactors has dimensions of time squared and can be taken as an estimate for the (square) of the interaction duration . More precisely, a function of the form


implies that is (the standard deviation of) its duration, and in particular it is finite, despite the use of delta functions. Following (74) we find


We pause to make several comments on this result. First, it is important that the relative sign between the terms in (74) is positive, allowing us to have a positive , while the opposite would mean that counter-intuitively does not have a constant sign. Secondly, we note that is the small parameter of the EFT (13) and hence the EFT’s basic tenet for a hierarchy of scales can now be expressed as , namely the longitudinal interaction length is much smaller than the transverse impact parameter. Thirdly, the constants in (76) have a simple physical interpretation: a deviation in direction through an angle over a duration (arc-length) implies that the radius of curvature is , namely the center of curvature for the motion of particle R resides, as expected, at particle L which exerts the force on it.

We note that the result (74) provides a strong test for our EFT: it contains integrals which diverge for (poles in dimensional regularization) yet the final answer is finite for all . In particular, the cancellation of the pole at relies on the sum of three different terms. Moreover, the sign and the precise numerical factor of the third order part match exactly with expectation, as explained.

5 Non-zero mass corrections

We now turn to compute non-zero mass corrections in the scalar case.999 At this order it would be essentially the same also for EM and gravitational fields. Here one only needs to make a relatively small correction in the derivation of the leading momentum transfer. When passing from (30) to (31) one must retain a Jacobian from the integration over functions as follows.


To evaluate the Jacobian we note that


Neglecting the transverse motion (more precisely ) we have


where initially (and approximately throughout the motion) , is the rapidity (), and similarly for particle L. From (78,79) we compute the Jacobian


where the relative rapidity is , and in the last equality we used (49).

This Jacobian corrects both the effective action diagram (31) and the momentum transfer (32) which becomes


This expression could be expanded as a power series in the following small parameter


6 The Fields of Ultra-Relativistic Gravitation

In this section we treat the case of the gravitational interaction in more detail. In the limit of weak ultra-relativistic scattering the components of the gravitational field split into several fields from the transverse perspective. In the literature (see for example ACV93 ()) the gravitational Einstein-Hilbert action is expanded perturbatively for weak gravitational fields. Here we shall define the field decomposition (84) and compute its whole non-linear action (85). Expanding it allows to readily read off all the gravitational bulk propagators and vertices for the perturbation theory. For example we reproduce a leading propagator term in (96).

In the related case of the post-Newtonian approximation it is natural and useful to decompose the gravitational field through a temporal Kaluza-Klein reduction into Non-Relativistic Gravitational (NRG) fields CLEFT-caged (); NRG (). These consist of the Newtonian potential, the gravito-magnetic 3-vector and a spatial metric. The full, non-linear gravitational action for these fields was determined in NRGaction ().

Here we go further by allowing dimensions larger than one (and arbitrary in principle) for both (the longitudinal) fiber and (the transverse) base. Computing the action by the standard method metric Christoffel symbols curvature tensor action would be a very complicated analytical task, perhaps hopelessly so. Here we simplify the computation to manageable form with no computerized computation by using a non-orthonormal frame within Cartan’s method, namely a hybrid method which incorporates both a non-trivial frame and a non-trivial metric as in NRGaction ().101010I was notified that an action whose mathematical form is essentially the same was given by Yoon Yoon (). While the mathematical context and tools there are essentially the same as here, namely a Kaluza-Klein reduction, the physical context and application is completely different and has no relation to ultra-relativistic gravitation. Instead Yoon () interprets 4d General Relativity as a 1+1 gauge theory. Technically, there the space is the base while here it is a fiber, and here spacetime dimension is arbitrary. I thank S. Carlip for alerting me to Yoon ().

Field decomposition and action. We work in the center of mass frame of a dimensional spacetime and we denote the longitudinal direction by , and the transverse directions by .

In the leading ultra-relativistic limit transverse gradients dominate over longitudinal ones


Therefore it is natural to perform a dimensional reduction à la Kaluza-Klein (KK) Kaluza-Klein () of the metric over the light-cone coordinates , a reduction which highlights the transformation properties (or tensor nature) with respect to gauge transformations which depend only on the transverse directions


In this expression and . Note that this dimensional reduction is more general and it applies to a reduction to a any base manifold parameterized by over a any fiber (not necessarily 2d) parameterized by the coordinates . We observe that from the transverse perspective the fields are a symmetric matrix of 3 scalars, a pair of transverse vectors, and the transverse metric. To put this dimensional reduction in the context of the literature we recall that the original and standard KK reduction Kaluza-Klein () is over spatial directions, the Non-Relativistic Gravitational reduction (NRG) CLEFT-caged (); NRG () is over the time direction, while here the reduction is over a Lorentzian fiber.

The action is simply


We proceed to define all the symbols and conventions. First


The extrinsic curvature of the fiber is 111111In general the extrinsic curvature evaluated on two vector fields which lie in a sub-manifold is defined by . This defines a symmetric tensor.


where in the last expression denotes the Lie derivative with respect to the longitudinal vector . 121212Recall that the Lie derivative by a vector field of a vector field is defined as