1 Introduction
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1cm

Paolo Lodone and Slava Rychkov

[0.7cm] Scuola Normale Superiore and INFN, Pisa, Italy

We investigate hard radiation emission in small-angle transplanckian scattering. We show how to reduce this problem to a quantum field theory computation in a classical background (gravitational shock wave). In momentum space, the formalism is similar to the flat-space light cone perturbation theory, with shock wave crossing vertices added. In the impact parameter representation, the radiating particle splits into a multi-particle virtual state, whose wavefunction is then multiplied by individual eikonal factors.

As a phenomenological application, we study QCD radiation in transplanckian collisions of TeV-scale gravity models. We derive the distribution of initial state radiation gluons, and find a suppression at large transverse momenta with respect to the standard QCD result. This is due to rescattering events, in which the quark and the emitted gluon scatter coherently. Interestingly, the suppression factor depends on the number of extra dimensions and provides a new experimental handle to measure this number. We evaluate the leading-log corrections to partonic cross-sections due to the initial state radiation, and prove that they can be absorbed into the hadronic PDF. The factorization scale should then be chosen in agreement with an earlier proposal of Emparan, Masip, and Rattazzi.

In the future, our methods can be applied to the gravitational radiation in transplanckian scattering, where they can go beyond the existing approaches limited to the soft radiation case.

September 2009

## 1 Introduction

Scattering at center-of-mass (CM) energies exceeding the quantum gravity scale (transplanckian scattering, or T-scattering, for short) is an exotic process of significant theoretical interest. In particular, it provides a laboratory to study the black hole information loss paradox. Microscopic black hole formation and its subsequent evaporation is expected for impact parameters of the order of the Schwarzschild radius of a black hole of mass [1],[2],[3],[4]. The detailed description of how this happens depends on the unknown underlying theory of quantum gravity and is at present out of reach. On the other hand, large impact parameters correspond to elastic small-angle scattering, whose amplitude can be predicted on the basis of General Relativity alone. It is given by eikonalized single-graviton exchange [1],[5],[6],[7]. Computing the corrections in to the elastic scattering, one hopes to learn about the strong inelastic dynamics at [8],[9],[10],[11],[12].

T-scattering is also interesting phenomenologically. If large extra dimension scenarios of TeV-scale gravity [13] are realized in Nature, this process could be observed at the LHC and other future colliders [14],[15], as well as in collisions of high-energy cosmic neutrinos with atmospheric nucleons [16],[17]. In these scenarios the total T-scattering cross section is finite, grows with energy, and is dominated by calculable small-angle scattering between partonic constituents [17],[15], see Fig. 1. The subleading black hole production cross section at present can only be estimated from geometrical arguments.

In spite of the small scattering angle, the typical momentum transfer in these scattering events is well above the QCD scale, and the typical impact parameter is much smaller than the proton size, which sets the typical distance between two uncorrelated partons inside the proton. It is unlikely that a multiple parton interaction, Fig. 2, will occur in the same T-scattering. Thus it is clear that the partonic picture should be applicable at leading order in the QCD coupling. In other words, we can compute the total cross section via a convolution of the partonic cross section and the parton distribution functions (PDF) .

Several interesting questions arise when one tries to think what happens beyond the leading order. For example, it’s not known how to treat events of the type shown in Fig. 3, where one of the colliding partons (say a quark) radiates a gluon just before the collision. Now the quark-gluon separation is not necessarily large, and the pair may scatter coherently. What is the correct description of such rescattering processes, and what is the resulting effect on the total T-scattering cross section?

A related question is: which factorization scale should we choose in the computation of the total cross section? As is well known, QCD-initiated processes have significant higher-order logarithmic corrections, associated with the collinear QCD radiation off initial partons. By choosing appropriately, these corrections can and should be reabsorbed into the PDF. As we will see below, the familiar choice is likely not the right one for the T-scattering. If so, we would like to see this explicitly.

The purpose of this paper is to answer the above-mentioned questions. We will be focusing on the QCD radiation since it is the dominant phenomenological effect due to the relative largeness of . However, our methods are equally applicable to the radiation of photons or any other spin 1 gauge bosons. We also hope that these methods may later prove useful in the more complicated problem of gravitational radiation emitted in T-scattering, and in particular to provide an alternative to the existing computations which are limited to the case when the emitted radiation is soft [8],[18].

The paper is organized as follows. In Section 2 we review the eikonalization of small-angle partonic scattering amplitude, following [17],[15]. To keep close contact with phenomenology, we work in the context of large extra dimension scenarios with the quantum gravity scale around a TeV. A key feature of the extra-dimensional situation with compactified dimensions is the appearance of a new length scale , which sets the range of a typical T-scattering interaction. In -dimensional Planck units, we have

 bc∼s1/n,RS∼s1/[2(n+1)],

so that in the deep transplanckian regime . We also present an alternative computation of the amplitude, based on generalizing to dimensions the early idea of ’t Hooft [1], who considered the small-angle T-scattering by solving the Klein-Gordon equation for one particle propagating in the classical gravitational field of the other particle.

In Section 3 we start discussing small-angle T-scattering with hadronic initial states, as in Fig. 1, in which QCD effects are expected to play a role. According to the existing proposal of Emparan, Masip and Rattazzi [17], the total cross section for these processes must be computed with the following prescription for the PDF scale :

 μF(q)=⎧⎪⎨⎪⎩qifqb−1c, (1.1)

where . For sufficiently high momentum transfers this deviates from the familiar prescription . An intuitive justification for this scale in terms of the typical impact parameter was given in [17], but we would like to check it via a direct computation.

The first step is to be able to evaluate the amplitude for small-angle T-scattering accompanied by collinear QCD radiation. In the resummation approach [17],[15], this computation seems prohibitively difficult even for one-gluon emission. Indeed, the eikonal amplitude for quark-quark T-scattering is a sum of an infinite number of crossed ladder gravition exchanges. The outgoing gluon may be attached anywhere on the quark lines, both external and internal. Moreover, in the splitting, the emitted near-collinear gluon is not necessarily soft, and thus may also exchange gravitons. The number of diagrams to resum skyrockets.

’t Hooft’s approach is a much better starting point. As we point out, it can be easily ‘upgraded’ to the case when radiation is present, provided that only one of the two colliding particles radiates. This covers completely lepton-quark scattering and is an important special case for quark-quark scattering. The idea is very simple. In the scattering, ’t Hooft treated one particle classically, the other one quantum mechanically. The only new twist is to allow the quantum particle to radiate. In other words, we should treat the non-radiating parton classically, while the radiating parton and the gluonic radiation field with which it interacts quantum mechanically.

This trick reduces the problem to a quantum field theory computation in the classical gravitational background produced by a relativistic point particle, the Aichelburg-Sexl (AS) shock wave [19]. In Section 4 we develop the necessary formalism. We first consider the simplest perturbative quantum field theory in the AS background: a scalar field with cubic self-interactions. We introduce a diagram technique for computing arbitrary transition amplitudes in this theory, which turns out to be closely related to the standard rules of light-cone perturbation theory in flat space. We then explain the changes necessary for the gluon field and for the scalar-gluon interactions, and compute the one-gluon emission amplitude as an example.

Notice that while fermionic matter fields can be considered analogously, we do not include them in this work in order to keep technical details to a minimum. Thus we stick to a toy model in which the partonic constituents of colliding hadron(s) are scalars.

Armed with the knowledge of gluon emission amplitudes, in Section 5 we attack the question of QCD corrections to T-scattering. For definiteness and simplicity, we consider the gravitational analogue of the DIS: a transplanckian electron-proton collision. The observable is the total cross section as a function of the Bjorken and the transverse momentum transfer , in the small-angle region . At leading order (LO) in the QCD coupling , the partons scatter elastically on the electron (no gluon emission). At next-to-leading order (NLO), we demonstrate the appearance of logarithmic corrections whose scale is precisely the from Eq. (1.1). We find that the cross section factorizes, in the sense that these logarithms appear multiplied by the DGLAP splitting functions, and can be reabsorbed into the PDFs. Finally, we are able to show that this factorization holds to all orders in , in the leading-logarithm approximation (LLA).

Our computation gives an explicit check for the validity of the partonic picture for the T-scattering. Moreover, it gives an interesting and unexpected explanation for why the PDF scale deviates from the usual . It turns out that rescatterings like in Fig. 3 suppress the initial state QCD radiation at transverse momenta . As a result the transverse momentum distribution of emitted gluons has the form:

 dNd2q=f(q)(dNd2q)% 0, (1.2)

where is the standard distribution without rescattering, and is a function interpolating between for and for . Logarithmic corrections to the cross section are obtained, as usual, by integrating Eq. (1.2) over the gluon phase space, and the scale of these logarithms is a geometric mean of and as in Eq. (1.1).

Notice that one could imagine other distributions giving rise to , for example the standard with a sharp cutoff at . In this sense Eq. (1.2) contains more information than the identification of the correct factorization scale. The predicted suppression of the initial state radiation is -dependent and could in principle be used to determine the number of extra dimensions.

A crucial insight into the physics of radiative processes is obtained by going into the impact parameter representation. In this picture, we find that the scattering is described via a multi-particle wavefunction of the virtual state (partonradiated quanta), which is multipled by individual eikonal factors when crossing the shock wave. This interpretation suggests a possible generalization of our formalism to the case when both colliding partons radiate, which we discuss in Section 6.

In conclusion, this work shows that factorization holds for QCD effects in T-scattering, and that the factorization scale has a nontrivial dependence on , in agreement with the earlier proposal of Ref. [17]. The novelty is that we arrive at these results by a concrete computation, and that we derive the modified distribution of the initial state radiation due to rescattering effects. The new distribution should be now incorporated into a ‘transplanckian parton shower algorithm’, to be used in Monte-Carlo simulations of T-scattering. We will come back to this issue in a future publication.

## 2 Review of the eikonal approach to T-scattering

In this section we review the basics of small angle T-scattering in the eikonal approximation. We will work within the large flat extra dimensions scenario of TeV-scale gravity [13], see [20] for the current experimental constraints.

Consider then two transplanckian massless Standard Model (SM) particles, thus confined to the SM 3-brane, which scatter due to the -dimensional gravitational field, , being the number of large extra dimensions, for phenomenological reasons. For now we ignore all interactions except for gravity. In particular, we suppose that the colliding particles are not charged, and thus cannot emit photons or gluons. We are interested in the scattering amplitude for small momentum transfer In this regime gravitational radiation is also suppressed (see [15]), and we have elastic scattering.

### 2.1 Resummation

The most direct way to compute the amplitude is by resumming the crossed ladder graviton exchange diagrams [17],[15], see Fig. 4. For small momentum transfer, exchanged gravitons are soft, and well-known simplifications occur in the vertices and the intermediate state propagators, allowing the resummation. The first term in the series, the one-graviton exchange, is given by

 ABorn(q)=−s2Mn+2D∫dnlq2+l2, (2.1)

where is the momentum transfer, which lies mostly in the direction transverse to the beam: . The -dimensional Planck scale TeV is normalized as in [15],[20]. The divergent integral over the extra dimensional momentum needs to be treated properly; see below. The second term in the series, the sum of two one-loop diagrams, turns out to be equal to a convolution of two Born amplitudes:

 A1-loop(q)=i4s∫d2k(2π)2ABorn(k)ABorn(q−k) ,

and this pattern continues to higher orders. As a result the series can be summed by going to the impact parameter representation. The amplitude acquires the eikonal form:

 Aeik(q)=ABorn+A1-loop+…=−2is∫d2be−iq.b(eiχ−1), (2.2)

with the eikonal phase given by the Fourier transform of the Born amplitude in the transverse plane:

To evaluate the eikonal phase, we need to regulate the divergent Born amplitude (2.1). In [15], dimensional regularization was used, and it was argued that since the subtracted divergent terms are local, they do not affect the small angle scattering amplitude1. The eikonal phase was found to be:

 χ(b)=(bc|b|)n,bc=1MD⎡⎣(4π)n2−1Γ(n/2)2⎤⎦1/n(sM2D)1/n. (2.3)

The corresponding amplitude is then given by:

 Aeik=4πsb2cFn(bc|q|) , Fn(y)=−i∫∞0dxxJ0(xy)[eix−n−1]. (2.4)

The functions are plotted in Fig. 5 (see also Fig. 2 of [15]). Their most salient features are as follows. At moderate , we have , 2 the integral (2.4) receiving contributions from . On the other hand, for the integral has a saddle point at , and the amplitude decays:

 |Fn(y)|≃n1n+1√n+1y−n+2n+1(y≫1).

The appearance of the scale is a peculiar feature of T-scattering for . Since the amplitude is the largest in the region , a typical scattering will have . Yet a classical particle trajectory for these is undefined, all impact parameters contributing to the scattering. On the other hand, for the scattering is dominated by a characteristic impact parameter , corresponding to the above saddle point. In this case the particle trajectory is well defined and the T-scattering is truly semiclassical, with many gravitons being exchanged.

### 2.2 ’t Hooft’s method

An alternative computation of the small angle T-scattering amplitude can be given using a method due to ’t Hooft [1], originally formulated in four dimensions. In this approach, particle scatters on the classical gravitational field created by particle . In other words, particle is treated as a classical point particle, while particle is treated quantum-mechanically.

Consider then the gravitational field of a relativistic classical point particle of energy propagating in the positive direction. This field is the -dimensional generalization of the AS [19] shock wave:

 ds2=−dx+dx−+Φ(x⊥)δ(x−)(dx−)2+dx2⊥. (2.5)

Here while denotes transverse directions. Einstein’s equations with the lightlike source

 T−−=EAδ(x−)δ(D−2)(x⊥)

reduce to one linear equation for the shock wave profile :

 −∂2⊥Φ=16πGDEAδ(D−2)(x⊥). (2.6)

The solution of this equation coincides with the eikonal phase (2.3) per unit of particle energy:

 Φ(x⊥)=E−1Bχ(x⊥). (2.7)

The right-moving particle is confined to the SM 3-brane, and its wavefunction solves the Klein-Gordon equation in the metric induced on the brane by the shock wave (2.5). At the wavefunction is a standard plane wave

 ϕ(x)=exp(ipB.x)=exp(−iEBx+).

The metric (2.5) has a strong discontinuity at . To solve the Klein-Gordon equation across the discontinuity, it is convenient to make a coordinate transformation [3],[23]

 x− =˜x−, x+ =˜x++θ(x−)Φ(˜xi)+x−θ(x−)(∂Φ(˜xi))24, (2.8) xi =˜xi+x−2θ(x−)∂iΦ(˜xi).

In the coordinates the metric is continuous across . When crossing the shock wave, the wavefunction remains continuous in these coordinates. This means that for small positive we have:

 ϕ(˜x)=exp(−iEB˜x+)=exp[−iEB(x+−Φ(x)]. (2.9)

The -dependent shift of the coordinate has a well-known classical origin: it is related to the time delay experienced by geodesics crossing the AS shock wave, see Fig. 6.3

We now see from (2.9) that the wavefunction immediately before and after the collision is related by a pure phase factor ), which via (2.7) is identical with the eikonal phase factor in (2.3). An alternative derivation, by directly solving the Klein-Gordon equation, is given in Appendix A.

Thus, ’t Hooft’s method is equivalent to the resummation. This is not surprising, because the external field approximation in quantum field theory resums precisely crossed ladder diagrams [24]. The AS shock wave is a solution to both linearized gravity and the full nonlinear Einstein’s equations. In retrospect, this explains why the diagrams in which gravitons emitted by particles and interact did not have to be taken into account in the resummation method. See [7] for a detailed discussion and comparison of the two methods in .

Still, an attentive reader will notice two small differences between the two results. First, Eq. (2.3) contains under the integral sign, while ’t Hooft’s method gives a pure phase. This is the usual difference between the S- and T-matrices, . Second, the amplitude (2.3) is relativistically normalized, while in the new derivation normalization needs yet to be determined.

Modulo the normalization issue (which will be resolved in Section 4.1 below), the power of ’t Hooft’s method relative to the resummation is quite evident. The eikonal phase is given a simple physical interpretation—it is related to the time delay experienced by geodesics upon crossing the shock wave. The exponential factor emerges as a whole rather than by summing infinitely many individually large terms.

## 3 T-scattering with hadrons: intuition and questions

If TeV-scale gravity is the way of Nature, then transplanckian collisions may be within the energy reach of the LHC. Moreover, transplanckian collisions may be constantly happening in the atmosphere, between the atmospheric nucleons and high-energy cosmic rays ( GeV for GeV of the order of the GZK cutoff). In case of cosmic ray neutrinos this signal could actually be observable.

Since protons are not elementary particles, the theory of small angle T-scattering from Section 2 should be applied instead to collisions between the partonic constituents. Notice that since we are dealing with CM energies well over a TeV, the typical momentum transfers will be hard compared to the QCD scale, even though the scattering angle has to be small for the eikonal approximation to be valid. Thus the collision resolves the internal structure of the proton(s), and the partonic picture is applicable [17].

Viewed another way, when two protons collide, there is a phase factor for each pair of partons moving in the opposite directions, see Fig. 2. This factor tends to zero rapidly at transverse separations , where GeV) for T-scattering at the LHC energies. Since partons are distributed in the disk of radius GeV, it is unlikely that more than one pair will undergo a hard collision.

We would like to briefly mention which observables one usually computes in phenomenological studies. In collisions one is mostly interested in the total interaction cross section as a function of the energy transfer to the proton [17],[22],[25]. We will discuss a similar observable in Section 5 below. On the other hand, in the collisions at the LHC one studies two jet final states of high invariant mass, produced at a small angle to the beam [15],[26], see Fig. 1. These jets originate from all possible parton pairs ) with the same partonic cross section, the eikonal amplitude being independent of the particle spin. For not much above a TeV, the dijet T-scattering signal turns out to be visible over the QCD background.

So far it may look that from the point of view of QCD, the T-scattering is just like any other hard process. Let us however discuss which parton distribution factorization scale one should use when evaluating the T-scattering cross sections—a necessary prerequisite for any practical computation.

For the usual hard processes, we are accustomed to the choice , but for the T-scattering this turns out to be more subtle. As we discussed in Section 2.1, T-scattering becomes semiclassical in the region of large momentum transfers . In this regime, the transverse distance characterizing the process is the typical impact parameter which is parametrically larger than . It is for this reason that Ref. [17] advocated a hybrid prescription: one should use for and switch to for see Eq. (1.1).4

For T-scattering at the LHC energies, the factorization scale will be hard with respect to the QCD scale as long as the momentum transfer is hard. This gives a self-consistency check on the proposed picture.5

The above is a summary of the current understanding of QCD effects in T-scattering. Clearly, it is based mostly on intuition. We would like to develop a systematic theory of these phenomena. In particular, such a theory should allow to check the factorization scale proposal by a concrete computation. We have to evaluate the leading log corrections to the T-scattering cross section due to the initial state radiation emission, and to show that they can be absorbed into a shift of the PDF factorization scale. Since is conjectured to have a nontrivial dependence on , some nontrivial physics is likely to come out.

Two equivalent methods were given in Section 2 to describe T-scattering without radiation. Which one shall we try to generalize to the case when radiation is present?

For the resummation method, generalization does not seem to be easy, not even for the one-gluon emission. Think about infinitely many crossed-ladder diagrams, infinitely many places to attach the gluon line, and the necessity to take into account the gravitational exchanges of the emitted gluon!

For ’t Hooft’s method, on the other hand, the situation looks hopeful: if only particle radiates, it is quite clear how to include its radiation. Namely, we should keep working in the classical gravitational background created by particle , but switch from relativistic quantum mechanics (wavefunctions, the Klein-Gordon equation) to quantum field theory (Green’s functions and interaction vertices). We will follow this path and will see that it allows relatively straightforward computations of the gluon emission amplitudes.

Physically, the assumption that particle does not QCD-radiate is realized if is a lepton. If both and are strongly interacting, one could first compute the radiation off (taking classical), then off (taking classical). Such an approximation of independent emission is valid for the dominant, collinear, radiation in the usual perturbative processes. For the T-scattering, we will be able to partially justify it below. But first we have to understand well the case of non-radiating .

## 4 Quantum field theory in the shock wave background

### 4.1 Scalar field

To compute the QCD radiation accompanying a transplanckian collision, we will replace particle with the classical background it generates, but will keep particle and the gluons as quantum fields. Thus we will be doing perturbative QFT computations in the shock wave background. We start with the simplest interacting QFT, the massless theory:

 L=√g(12gμν∂μϕ∂νϕ−λ3!ϕ3). (4.1)

We will describe how to compute transition amplitudes in this theory, and how these are related to the amplitudes in the full theory (i.e. before particle was replaced by a classical gravitational field).

The in (4.1) is the 4-dimensional metric obtained by restricting the -dimensional AS shock wave (2.5) to the SM brane on which both particles and the radiation propagate. We will continue using the coordinates as in (2.5), only restricting the number of components from to . Two features make this theory much simpler than it would be for generic curved backgrounds treated in [29]:

1. the metric is invariant under shifts. The conjugate momentum is conserved. This leads in particular to the absence of spontaneous particle creation.

2. the spacetime is flat except on the plane. The Feynman rules are simplified by using the flat-space coordinates.

We start by canonically quantizing the quadratic part of the lagrangian. The scalar field modes are found by solving the equations of motion (EOM) in the shock wave background with the plane wave conditions in the asymptotic past :6

 ϕinp−,p(x) =θ(−x−)ei[p].x+θ(x−)∫d2q(2π)2I(p−,q)ei[p+q].x, (4.2) I(p−,q) ≡∫d2xe−iq.xei12p−Φ(x).

The compact “vector in square brackets” notation denotes an on-shell 4-vector whose component is computed in terms of the known and , i.e. etc.

The function is identical to the eikonal amplitude (2.3), up to the normalization and the absence of under the integral sign (which means that it contains an extra -function piece).

The modes (4.2) solve the Klein-Gordon equation both for and for Across the shock wave, they satisfy the matching condition of Section 2.2:

 ϕ(x−=+ε,x+,x)=ϕ(x−=−ε,x+−Φ(x),x).

We proceed to quantize the field by expanding in oscillators:

 [ap−1p1,a†p−2p2]=(2π)3δ(p−1−p−2)δ(2)(p1−p2).

Such normalization of the creation/annihilation operators is standard for quantizing on the light cone; it differs from the usual one by a simple rescaling.

Equivalently, we can quantize using the outgoing modes, which reduce to plane waves for :

 ϕoutp−,p(x)=θ(x−)ei[p].x+θ(−x−)∫d2q(2π)2I(p−,q)ei[p−q].x. (4.3)

The in and out modes are related by a unitary Bogoliubov transformation, which acts only on the transverse momentum but not on . Thus there is no spontaneous particle creation in this background; the vacuum is unambiguously defined.

Let us now build a perturbation theory for transition amplitudes. The logic is simplest in the position space. Even though the metric is singular at   it is easy to see that: the metric determinant drops out of the interaction lagrangian. Thus the Feynman diagrams will be given by flat space integrals, with no singular contribution from the shock wave. For instance, the -channel diagram contributing to the transition amplitude will be given by:

 \raisebox−25.229457pt\includegraphics[natheight=1.399700in,natwidth=2.187900in,height=64.956276pt,width=100.722699pt]phi3.pdf=(−iλ)2∫d4xd4y[ϕout3(x)]∗[ϕout4(y)]∗G(x,y)ϕin1(x)ϕin2(y). (4.4)

The and enter as the in and out state wavefunctions. The propagator must be -ordered:

 G(x,y)=θ(x−−y−)⟨0|ϕ(x)ϕ(y)|0⟩+(x↔y).

In (4.4) we have to integrate in all possible orderings of and with respect to each other and to the shock wave sitting at . The propagator will take different forms depending on the ordering. For and on the same side of the shock, we get the flat space result:

 Gflat(x,y)=θ(x−>y−>0)∫p−>0dp−d2p2p−(2π)3ei[p].(x−y)+(x↔y).

On the other hand, across the shock wave we have

 Gcross(x,y)=θ(x−>0>y−)∫p−>0dp−d2p2p−(2π)3∫d2q(2π)2I(p−,q)ei[p+q].x−i[p].y+(x↔y).

The momentum-space Feynman rule can now be found by straightforward Fourier transformation; they are as follows. The in and out states are specified by the and of all incoming and outgoing particles. The transition amplitude in the external gravitational field of particle is then given by:

 out⟨f|i⟩in=2(2π)δ(p−f−p−i)M(i→f).

The is a function of the external momenta computed as a series in according to the following rules. To obtain the term:

• Draw the Feynman diagrams with vertices, considering all possible -orderings of these vertices with respect to each other and to the shock wave at .

• Consider all shock wave crossings as additional vertices, with entering transverse momenta representing momentum exchange with the shock wave.

• Assign , internal lines momenta by using their conservation in all vertices ( and shock wave crossings). Momentum flow is in the direction of increasing . The internal momenta are not conserved but are assigned by using the on-shell condition

• For each vertex multiply by

• For each shock wave crossing vertex multiply by , where is conserved in the crossing.

• For each internal line (i.e. a line connecting two vertices, or shock wave crossing) carrying momentum , multiply by

• The vertices and the shock wave at divide the axis into two unbounded and bounded intervals. For each bounded interval, we define an intermediate state, consisting of all the particles whose internal lines traverse this interval. For each intermediate state at negative , the amplitude is multiplied by

 i∑ip+−∑intermp++iε.

For each intermediate state at positive , it is multiplied by

 i∑fp+−∑intermp++iε.

The sums are over all particles in the initial (), intermediate, and final () state.

• Integrate over the momenta exchanged with the shock wave:

 ∫(2π)2δ(2)(∑qa+pi−pf)∏d2qa(2π)2.
• For loop diagrams, integrate over all undetermined momenta :

 ∫dk−d2k2(2π)3.

The reader will notice a striking similarity to the usual light-cone perturbation theory (PT) rules [30]. Notice in particular the light-cone energy denominators, and the factors, which eliminate some of the diagrams present in the time-ordered ‘old’ perturbation theory. New features in our case are the shock wave crossing vertices, and that there are two types of energy denominators, depending on the ordering with respect to the shock wave. We thus have ‘light-cone PT in presence of an instantaneous interaction’. Many years ago, Bjorken, Kogut and Soper [31] have developed light-cone PT in external electromagnetic field, and argued that at sufficiently high energies interaction with the external field can be represented as an instantaneous eikonal scattering.7 In our case, the eikonal factor has gravitational origin, but the formalism is the same. The formalism of [31] has found application in the dipole scattering approach to the DIS at small : an almost-real photon splits into two quarks which then undergo eikonal scattering in the gluon field of the proton [32]. The difference is that the gluon field of the proton is not really known, while in our case the eikonal phase can be computed exactly.

We will now demonstrate the rules by computing a couple of amplitudes. The elastic one-particle amplitude is given by just one diagram with a shock wave crossing vertex (denoted by a cross):

 M(p→p′)=p−I(p−,p−p′)(p2=p′2=0,p−=p′−) . (4.5)

As a more complicated example, let us compute one of the diagrams appearing in the computation of the amplitude :8

We have two vertices and three shock wave crossings. The dotted lines stress the ordering of the vertices. The are the internal line momenta, whose and components are fixed via momentum conservation, while the components are determined by being on shell. There are two intermediate states: one before the shock wave , and one after . The value of this diagram is thus:

 ∫(2π)2δ(2)(∑qa+p1+p2−p3−p4)3∏a=1d2qa(2π)2 ×(−iλ)2k−1I(k−1,q1)k−2I(k−2,q2)p−2I(p−2,q1) ×i(p+1+p+2)−(k+1+k+2+p+2)+iεi(p+3+p+4)−(p+3+k+3+k+4)+iε4∏i=1θ(k−i)k−i.

Finally, we have to discuss the relation between the transition amplitude and the full relativistic scattering amplitude , i.e. the one obtained when we reinstate particle as a quantum particle as opposed to replacing it with its classical field. We have:

 (4.6)

The incoming (outgoing) momenta of particle are assigned as follows:

In other words, particle absorbs the total transverse momentum exchanged with the shock wave. As long as the momentum transfer is small compared to , is almost on shell and the approximation is justified.

The relative factor in (4.6) is related to the normalization of the particle state, which is lost when we replace it with a classical field. This factor is thus process-independent. For instance one can extract it from the external field approximation in QED [24]. The extra can be traced back to the external field creation vertex, which carries a factor of .

This settles the question of relativistic normalization of the amplitudes computed via ’t Hooft’s method. We can now complete the comparison with the resummation method. Using Eqs. (4.5), (4.6) we have

 Mrel(A+B→A′+B′)=−2ip+Ap−BI(p−B,q), (4.7)

which agrees with the eikonal amplitude from Eq. (2.2) including the normalization, modulo the difference between the S- and T-matrices already discussed in Section 2.2.

### 4.2 Gauge field

In order to keep technical details to a minimum, we will not consider fermionic fields in the shock wave background. Instead, we will stick to a toy model in which charged matter (partonic constituents of the colliding hadrons) consists of massless scalars. This will be sufficient given our general goals. On the other hand, since the coupling constant of the lagrangian has dimension of mass, the cubic self-interaction is not a good model for the QCD radiation. We do have to introduce gauge fields. Thus we switch from (4.1) to a different microscopic lagrangian, describing the SU(3) Yang-Mills theory and a massless complex scalar in the fundamental representation, propagating in the shock wave background:

 L=√g(12gμν(Dμϕ)∗Dνϕ−12Tr[FμνFλσ]). (4.8)

Most of the formalism is carried over with trivial changes. Instead of repeating the whole discussion, we will introduce the necessary modifications on a concrete example.

Namely, let us consider the one-gluon emission: particle , while scattering in the gravitational field of particle , emits a gluon. The amplitude is given by the sum of the following two diagrams:

 (I)\includegraphics[natheight=2.171600in,natwidth=2.380900in,height=108.101466pt,width=118.327671pt]Mg+.pdf(II)\includegraphics[natheight=2.171600in,natwidth=2.987300in,height=105.246801pt,width=144.359325pt]Mg−.pdf (4.9)

The new objects are the gluon emission and the gluon shock wave crossing vertices.

To simplify the computations, we will impose the Lorenz and light-cone gauge conditions:

 DμAμ=0,A+=0.

The treatment in a general gauge and demonstration of gauge invariance is given in Appendix A.

Gluon emission in curved space is described by the cubic term in the lagrangian:

 igs∫d4x√ggμνϕ∗i↔∂μϕj(Ta)ijAaν. (4.10)

Here is the strong coupling constant, and the SU(3) generators are normalized by Tr. In the light-cone gauge the singular component drops out (see Appendix A for a more detailed discussion). The gluon emission vertex is thus the same as in flat space:

 gsTaij(p1+p2).ε⟶ gsTaij(p1+p2−p−1+p−2l−l).\upepsilon(ε+=0,l.ε=0), (4.11)

where we used the Lorenz gauge to eliminate the component.

The gluon shock wave crossing vertex contains the same factor as in the scalar case. A new feature is that the polarization component changes in the crossing according to:

 \raisebox−28.821276pt\includegraphics[natheight=1.567800in,natwidth=1.833700in,height=61.176555pt,width=71.40276pt]gauge−vertex.pdfε2−=ε1−−\upepsilon1.qp−1, \upepsilon2=\upepsilon1(ε+≡0), (4.12)

This rule is easy to guess from consistency with the imposed gauge; see Appendix A for an explicit derivation. Notice however that we don’t have to keep track of this change in if we use the simplified gluon emission vertex in (4.11).

We are now ready to evaluate the above two diagrams. Working for simplicity in the frame where , we get:

 M(I) =igsTaijI(p−,q)(2p′+l−2p′−+l−l−l).\upepsilonp′++l+−[p′+l]+, M(II) =igsTaij∫d2k(2π)2I(l−,k)I(p′−,q−k)(k−l−2p−−l−l−(l−k)).\upepsilon−[p−l+k]+−[l−k]++iε. (4.13)

Physical consequences of the derived expressions will be discussed below.

As a final comment, we note that lagrangian (4.8) contains also a cubic gluon self-interaction vertex, which could be discussed analogously to (4.11), as well as two quartic vertices ( and ). The quartic vertices do not contribute to the amplitudes in the collinearly enhanced region, and we will not need their precise expressions.

## 5 Initial state radiation in T-scattering

The dominant QCD radiation effects in the usual perturbative hard scattering processes are the collinear initial and final state radiation. We now proceed to see how these effects manifest themselves in the T-scattering. We will focus on the initial state radiation and its effect on the parton distribution scale. Final state radiation, which happens after the partons cross the shock wave, is expected to be as usual.

### 5.1 Observable

To discuss radiative corrections to the PDFs, we need to choose a process and an observable which can be defined and computed beyond LO. The simplest such process is the T-scattering analogue of the DIS. In other words, we will consider an electron-proton T-scattering +anything at a fixed momentum transfer. This is like in Fig. 1 with an electron instead of a neutrino.

The scattering is characterized by and , the energy transfer to the proton. These can be measured by observing the electron. As usual, we assume small angle scattering: . We will also assume that the relative electron energy loss is small, . Under these conditions, and also since the electron does not QCD-radiates, we can represent it by a classical relativistic point particle of fixed energy. This is our ’particle ’. Using the on shell condition , it is easy to show that the momentum transfer is mostly in the transverse plane, as expressed by the relation:

 q2=q2(1+2q+/p+A)≃q2.

Like in the DIS, we are interested in the differential cross section with respect to and the Bjorken :

 dσd2qdx,x=q2p−Bq+,0

As is customary, we will first analyze the partonic cross section between the electron and a quark (particle ). We will work in a toy model of scalar quarks. At LO (no gluon emission), the amplitude is (4.7) and the partonic cross section is given by

 d^σLOd2qdx=δ(x−1)14π2|I(p−B,q)|2.

### 5.2 One gluon emission in momentum space

Armed with the formalism from Section 4, we can easily write down the gluon emission amplitudes. At the NLO we have diagrams with real gluon emission, as in Eq. (4.9), as well as virtual corrections to the external legs and the vertices in the elastic amplitude. As usual, the latter diagrams do not have to be computed explicitly, since they only correct the coefficient of 9 We thus focus on the real emission.

The partonic cross section with one gluon emitted is given by a phase space integral (see Appendix B)

 d^σNLOd2qdx=116π2^s∫d2l2(2π)3∫10dzz(1−z)δ(x−q2/(p−Bq+))|Mrel|2, (5.1) q+=l2/l−+(q−l)2/p−B′.

Here is the relativistic scattering amplitude , related to the transition amplitude in the external field via Eq. (4.6). The amplitude is in turn the sum of the two diagrams (4.9), evaluated in Eq. (4.13).

We are using notation from (4.9) with , . The is the total + momentum of the quark-gluon system after the collision. The is the momentum fraction carried off by quark :

 p′−=zp−,l−=(1−z)p−.

Let us first analyze which region of the plane contributes to the integral (5.1). For which is there a saturating the -function? The relevant function (see Fig. 7)

 X(z)≡q2/(p−Bq+)≡q2(q−l)2/z+l2/(1−z),

has a maximum value

 max0

Thus, the integrand of (5.1) is nonzero for belonging to the ellipse:

 |q−l|+|l|<|q|/√x.

In other words, phase space limits the transverse momentum of the emitted gluon to be at most .

Let us now examine the amplitude, whose two parts are given in Eq. (4.13). By analogy with the usual DIS, we expect that part (I), corresponding to the gluon emission after the hard scattering, gives only a finite correction to the cross section, while part (II) contains a logarithmic IR divergence which has to be absorbed by redefining the PDFs. Let us see formally how this happens.

Notice that part (I) of the amplitude is non-singular in the plane. In particular, the intermediate state denominator is completely fixed at . Omitting the dependent factors, the amplitude is thus and its square is After integrating over the ellipse in the plane (area we get a finite contribution to the differential cross section of the relative order This is as expected.

Interesting physics is associated with part (II), whose expression can be simplified as follows:

 M(II) =−igsTaij2p−z\upepsilon.M, Mi =∫d2k(2π)2(k−l)i(k−l)2I(zp−,q−k)I((1−z)p−,k) (5.2) ≡−lil2˜I(zp−,q)+(q−l)i(q−l)2˜I((1−z)p−,q) (5.3)

Here we separated the regular part of the from the -function piece describing the propagation without scattering:

 I(p−,q)=(2π)2δ(2)(q)+˜I(p−,q),

where are the same functions as in Eq. (2.4). We omitted a total piece from (5.3).

The physical meaning of the decomposition (5.3) is as follows. In the first two terms, only one of the two splitting products of quark participates in the gravitational interaction, the other one passing the shock wave without scattering. The last term instead describes their coherent gravitational scattering, as in Fig. 3. We call it the rescattering term, since it corresponds to the situation when the emitted QCD radiation changes its direction in the field of the shock wave.

We now proceed to studying corrections to the cross section. Consider first the case In this case all the entering functions are The rescattering term can be estimated by integrating up to beyond which point the decrease faster than , and the integral converges. We get

 |Mresc|∼πb−2c(2π)2bc(2πb2c)2∼bc(2πb2c)<1|q|(2πb2c)(|q|≲b−1c). (5.4)

We see that rescattering is subleading to the first two terms in (5.3)

Concentrating on the first two terms, the dominant contribution to the cross section comes from the singularities at and . Squaring the amplitude and integrating we get:

 d^σNLOd2qdx≃14π2{∣∣˜I(xp−,q)∣∣2PQ→Q(x)+∣∣˜I(xp−,q)∣∣2PQ→g(x)}αs2πlogq2μ2IR(|q|≲b−1c), PQ→Q(x)=CF2x1−x,PQ→g(x)=PQ→Q(1−x),CF=4/3. (5.5)

Here we used that in the relevant regions of integration (see Fig. 7)

 x=X(z)≃z(|l|≪|q|),x=X(z)≃1−z(|q−l|≪|q|).

Eq. (5.5) has the standard factorized form expected from an NLO QCD correction to a hard scattering [27]. The IR divergent logarithm multiplies the quark-electron and gluon-electron LO cross sections, with the scalar quark DGLAP splitting functions and as coefficients10. As usual, we can absorb the IR divergence into the quark (first term) and gluon (second term) PDFs. If we fix the parton distribution scale at the upper cutoff, , then the whole logarithmic correction is absorbed.

It is of course not surprising that we managed to recover the standard factorization for  : rescattering was not important in this case, and without rescattering there is no difference between transplanckian and any other hard scattering.

Let us proceed to the case . The situation here is more complicated since the rescattering is no longer subleading. Consider for example the region . The rescattering integral is dominated by , where is maximal, and not, say, by the region of where is maximal. The reason is that decreases faster than for . We get an estimate:

 |Mresc|∼πb