Conformal spacelike-timelike correspondence in QCD

# Conformal spacelike-timelike correspondence in QCD

## Abstract

This paper is a study of a spacelike-timelike conformal correspondence in QCD. When the times at vertices are fixed in an gauge calculation the distribution of gluons in a highly virtual decay have an exact correspondence with the gluons in the lightcone wavefunction of a high energy dipole with the identification of angles in the timelike case and transverse coordinates in the lightcone wavefunction. Divergences show up when the time integrals are done. A procedure for dropping these divergences, analogous to the Gell-Mann Low procedure in QED, allows one to define a conformal QCD, at least through NLO. Possible uses of such a conformal QCD are discussed.

## 1 Introduction

Over the past fifteen years or so it has become increasingly clear that there are nontrivial relations between the distribution of particles in the decay of a highly timelike current and properties of high energy scattering processes. The first hint of such relations was the fact that the BMS equation1 (), an equation which was developed to describe nonglobal properties of jet decays2 (), is essentially identical to the BK equation3 (); 4 (), an equation describing high energy scattering. Shortly after the appearance of the BMS equation it was discovered5 (); 6 (); 7 () that in certain kinematic regions of a jet decay the number of produced heavy quarks, or minijets, is given by the BFKL equation8 (); 9 (), an equation long used to describe high energy hard scattering away from the unitarity limit.

The relationship between jet decays and high energy scattering became more interesting when Hofman and Maldacena10 () and Hatta11 () recognized that in the AdS/CFT correspondence the angular distribution of energy and charge in the decay of a highly virtual current is directly related to the transverse coordinate distribution of these same quantities in a high energy hadron. Hatta11 () then exhibited a stereographic projection relating the angular distribution of these quantities in jet decays to their transverse coordinate distributions in a high energy hadron, thus making the conformal relationship more explicit.

However, this is all a bit mysterious. Jet decays and the corresponding distribution of energy and particle densities are physical while the wave function of a high energy hadron is gauge and quantization dependent. To avoid this issue one could simply interpret the spacelike-timelike equivalence as one of evolution, In 11 () the equivalence of BMS evolution (timelike) to BK evolution (spacelike) was demonstrated while in 12 () the double logarithmic resummations necessary to tame the NLO kernels in BMS and BK evolution were shown to be related. However, the correspondence appears to be stronger than just an equivalence of evolutions.

In this paper we compare the distribution of particles in the decay of a timelike current into a quark-antiquark pair, along with an arbitrary number of gluons, with the distribution of partons(gluons) in the lightcone wavefunction of a high energy dipole13 () and find a one to one correspondence. More precisely, in the decay of a timelike current we suppose that the quark and antiquark, initially produced by the current, have longitudinal momenta much greater than that of the soft gluons subsequently emitted, and we fix to be the angle between the quark and the antiquark in a highly boosted frame where . (For simplicity we suppress additional quark-antiquark production.) On the hadron side we suppose an initial quark-antiquark dipole with transverse coordinate separation and we only consider gluons in the lightcone wavefunction whose longitudinal momenta are much less than that of the parent quark and antiquark dipole. Further, we suppose the quark and antiquark longitudinal momenta are identical in the decay and in the high energy dipole wavefunction. In the decay into soft gluons we do not suppose any strong ordering among the longitudinal momenta of the gluons, but later we shall only explicitly consider evolutions at NLO due to subtleties of coupling renormalization. The requirement that the gluon momenta be soft compared to the parent quark-antiquark pair is, we believe, an essential assumption. If a gluon had longitudinal momentum comparable to that of the quark-antiquark dipole(spacelike case), then that gluon emission would be sensitive to how the parent dipole was created and we believe that is beyond the correspondence we are considering.

At a given order of perturbation theory we observe a graph by graph equivalence for a timelike decay probability of and the square of the dipole wavefunction of when we identify with . The timelike and spacelike quantities are written as integrands over which integrations over the times at all the vertices present in a given graph are to be done. The integrands of the two processes, with the identification, are identical with no restriction on the gluon momenta except that the longitudinal momentum of every gluon must be small compared to the parent quark and antiquark momentum.

However, there are divergences when the time integrations are done. In some cases, when a time in the tinelike proess goes to infinity, corresponding to a time in the spacelike process going to zero, there are other graphs which cancel these divergences. These are “real-virtual” cancellations. (In the timelike case the cancellation will happen when one measures a jet rather than an individual particle, while in the spacelike case the cancellation will happen when the real and virtual configurations are not distinguished by a scattering.) These cancellations are always of collinear singularities in the timelike case and ultraviolet singularities in the spacelike case.

Other corresponding singularities do not cancel. They are ultraviolet singularities in both spacelike and timelike cases and represent the necessity of coupling renormalization in QCD. The introduction of the QCD -parameter breaks the conformal invariance and with it the spacelike-timelike correspondence. In section 4 we suggest a precise way of removing the coupling divergences, much like that originally done by Gell-Mann and Low14 () for QED, occurring only in self-energy graphs in our gauge dynamics. This removal does not introduce any new scale and leaves a “conformal QCD” and a correspondence between spacelike and timelike processes. Howevver, we have only been able to demonstrate this subtraction throughNLO in soft emissions.

One of the most ambitious, and interesting, programs using the spacelike-timelike correspondence has been that of Caron-Huot15 () who showed that in SYM the NLO kernel for BK evolution16 (); 17 (); 18 () could be obtained purely from the evaluation of decays. His procedure does not work when the -function is not zero. However, in this case one should be able to evaluate the timlike process with certain (see section 4) self-energy graphs in gauge removed, translate that to the contribution to the NLO BK kernel and then add the self-energy contributions back in with the appropriate renormalization in the spacelike process.

## 2 An example and its generalization

We start with a nontrivial example of a graph having a three gluon vertex as well as couplings to the parent quarks in which the conformal correspondence of the graph as part of the decay of a timelike photon to the graph as part of the lightcone wavefunction of a high energy dipole will be exhibited.The graphs are illustrated in Figure 1. We work in a frame where the timelike virtuality of the photon, , in Figure 0(a) obeys so that the angle between the quark and the antiquark is very small. For the lightcone wavefunction illustrated in Figure 0(b) the lines will be labelled by a transverse coordinate and a longitudinal momentum, although to begin we write the wavefunction only in terms of gluon momenta. The correspondence will relate the decay rate(), at given time values at the vertices and fixed on each of the lines to the square of the lightcone wavefunction(), also for fixed times at each of the vertices but with the corresponding lines labelled by . In the correspondence and are related by

 θ–i=√2k––iki+↔x––i. (1)

We begin by writing the graph, corresponding to a decay, of Figure 0(a) in detail. Then we shall write the corresponding graph for the square of the lightcone wavefunction, shown in Figure 0(b), and observe that they are the same. We always assume that the fermion lines, and , have a much larger longitudinal momentum than the gluon lines but there will be no assumed ordering as to the relative magnitude of and .

### 2.1 The decay graph of figure 0(a)

The decay rate of the virtual photon without radiative correction is . If is the rate with radiative corrections, then we are going to write an expression for as

 w=−ig4(2π)4N2c−14∫AL(θ–1,θ–2,k1+,k2+)⋅A∗R(θ–1,θ–2,k1+,k2+)dk1+2k1+dk2+2k2+k21+2d2θ1k22+2d2θ2 (2)

for the graph of figure 0(a). We shall then identify and with corresponding expressions for the graph of figure 0(b) with the time integrations fixed in each expression. Further write the part to the left of the cut, , as

 AL=aLbL, (3)

with a similar separation for , where includes exponential factor and time integrations while vertex factors are included in . Then

 aL=∫∞0dt1∫∞t1dt2eiΔE1t1+iΔE2t2 (4)

where

 ΔE1=(k––1+k––2)22(k1+k2)++p–2a2pa+−(p–a+k––1+k––2)22(pa+k1+k2)+ (5)

and

 ΔE2=k––212k1++k––222k2+−(k––1+k––2)22(k1+k2)+ (6)

It is straightforward to get

 ΔE1=(k1+k2)+4(θ–−θ–a)2 (7)
 ΔE2=k1+k2+4(k1+k2)+(θ–1−θ–2)2 (8)

where

 θ–=1(k1+k2)+[k1+θ–1+k2+θ–2]. (9)

Thus

 aL=∫∞0dt1∫∞t1dt2exp{i4[(k1+k2)+(θ–−θ–a)2t1+k1+k2+(k1+k2)+(θ–1−θ–2)2t2]} (10)

Similarly

 a∗R=∫∞0dt′1∫∞0dt′2exp{−i4[k1+(θ–1−θ–b)2t′1+k2+(θ–2−θ–a)2t′2]}. (11)

Now turn to the vertex factors, the term in (3). In the amplitude of the graph of 0(a) there is a vertex at and a three-gluon vertex at . Call where the vertex is given by

 P1L=¯u(pa)γ⋅ϵλ√2pa+u(pa+k1+k2)√2(pa+k1+k2)+≃ϵ–λ⋅(k––1+k––2)(k1+k2)+−p–a⋅ϵ–λpa+ (12)

or

 P1L=1√2[θ–−θ–a]⋅ϵ–λ (13)

In reaching (13) we have assumed that but we suppose that , and may all be of comparable magnitude.

The three gluon vertex, , is given by

 P2L=ϵλαϵλ1γϵλ2β[−gαγ(2k1+k2)β+gαβ(2k2+k1)γ−gαγ(k2−k1)α] (14)

or

 P2L=√2(θ–2−θ–1)⋅[k1+ϵ–λ2(ϵ–λ⋅ϵ–λ1)+k2+ϵ–λ1(ϵ–λ⋅ϵ–λ2)−k1+k2+(k1+k2)+ϵ–λ(ϵ–λ1⋅ϵ–λ2)] (15)

In (12)-(15) we imagine using real polarization vectors in order to avoid a proliferation of complex conjugate symbols. An abbreviated notation is being used where and . is obtained as

 bL=∑λP1LP2L=(θ−θa)i(θ2−θ1)j[k1+ϵλ1iϵλ2j+k2+ϵλ2iϵλ1j−δijk1+k2+(k1+k2)+ϵ–λ1⋅ϵ–λ2]. (16)

is easily found to be

 bR=12(θ–2−θ–a)⋅ϵ–λ2(θ–1−θ–b)⋅ϵ–λ1. (17)

Thus the integrand in (2) is given by

 ALA∗R= ∫∞0dt1∫∞t1dt2∫∞0dt′1∫∞0dt′2 ×exp{i4[(k1+k2)+(θ−θa)2t1−k1+(θ1−θb)2t′1−k2+(θ2−θa)2t′2+k1+k2+(k1+k2)+(θ1−θ2)2t2]} ×12∑λ1,λ2bL⋅bR. (18)

Equation (2.1) with and given by (16) and (17) respectively is a convenient form for the decay to compare to the high energy dipole wavefunction which we turn to next.

### 2.2 The high energy wave function graph of figure 0(b)

Our goal is to express the square of the high energy wavefunction contained in figure 0(b) in terms of an integration over coordinates and to identify the integrand with (2.1). We begin in momentum space and write the vertices as

 V1=eik––2t12k+ϵ–λ⋅k––k+e−ik––⋅x––a (19)
 V′∗1=e−i(k––′1)2t′12k1+ϵ–λ1⋅k––′1k1+eik––′1⋅x––b (20)
 V′∗2=e−i(k––′2)2t′22k2+ϵ–λ2⋅k––′2k2+eik––′2⋅x––a (21)
 V2=ei[k––212k1++k––222k2+−(k––1+k––2)22k+]t2ϵλαϵλ1γϵλ2β[−gαγ(2k1+k2)β+gαβ(2k2+k1)γ−gγβ(k2−k1)α] (22)

where , , but where, for the moment, we do not take and or and to be equal. Instead we put a coordinate on each line with phase factors which, after the coordinates are integrated, given transverse momentum conservation. Then in addition to the factors above we include the factors , , where

 L1=d2x1(2π)2ei(k––1−k––′1)⋅x––1 (23)
 L2=d2x2(2π)2ei(k––2−k––′2)⋅x––2 (24)
 L=d2x(2π)2ei(k––−k––1−k––2)⋅x––. (25)

Clearly the integrations over , and give transverse momentum conservation.

In analogy with the previous section, we group the factors together as

 ¯AL=∫0−∞dt1∫0t1dt2V1V2Ld2kd2k1d2k2. (26)

The various -integrals in (26) are easily done

 ~V1=∫d2kV1eik––⋅x––=−2πiϵ–λ⋅(x––−x––a)(k1+k2)+t21e−i(x––−x––a)2(k1+k2)+2t1 (27)
 ~V2=∫d2k1d2k2V2e−i(k––1+–k2)⋅x––+ik––1⋅x––1+ik––2⋅x––2. (28)

Using (22) one finds

 ~V2= −2i(x––2−x––1)t22⋅[k1+ϵ–λ2(ϵ–λ⋅ϵ–λ1)+k2+ϵ–λ1(ϵ–λ⋅ϵ–λ2)−ϵ–λ(ϵ–λ2⋅ϵ–λ1)k1+k2+(k1+k2)+] (29)

Using (27) and (2.2) in (26) along with , gives

 ¯AL= −∫∞0dτ1∫∞τ1dτ2ei(x––−x––a)2(k1+k2)+τ14+i(x––1−x––2)2k––1+k––2+4(k1+k2)+τ22π2d2x(k1+k2+) ×(x−xa)i(x2−x1)j[k1+ϵλ1iϵλ2j+k2+ϵλ1jϵλ2i−δijk1+k2+(k1+k2)+ϵ–λ1⋅ϵ–λ2] ×δ(x––−k1+(k1+k2)+x––1−k2+(k1+k2)+x––2). (30)

Similarly defined by

 ¯A∗Rd2x1d2x2=∫0−∞dt′1∫0−∞dt′2V′∗1V′∗2L1L2d2k′1d2k′2 (31)

is easily evaluated to be

 ¯A∗Rd2x1d2x2= −∫∞0dτ′1∫∞0dτ′2e−i(x––1−x––b)2k1+τ′14−i(x––2−x––a)2k1+τ′24d2x1d2x2k1+k2+4(2π)2 ×ϵ–λ1⋅(x––1−x––b)ϵ–λ2⋅(x––2−x––a). (32)

Multiplying (2.2) and (2.2) and doing the sum over , gives, in analogy with (2.1),

 ¯AL¯A∗R d2x1d2x2=∫∞0dτ1∫∞τ1dτ2∫∞0dτ′1∫∞0dτ′2 ×exp{i4[(k1+k2)+(x––−x––a)2τ1−k1+(x––1−x––b)2τ′1−k2+(x––2−x––a)2τ′2+k1+k2+(k1+k2)+(x––1−x––2)2τ2]} ×18∑λ1,λ2¯bL⋅¯bR(k1+k2+)2d2x1d2x2 (33)

where and are identical the and , in (16) and (17), with the replacement . To make the correspondence precise write in (2) as

 w=−ig4(N2c−1)4(2π)4∫∞0dt1∫∞t1dt2∫∞0dt′1∫∞0dt′2Idk1+2k1+dk2+2k2+k21+2d2θ1k22+2d2θ2 (34)

and write the amount of probability that graph 0(b) contributes to the square of the dipole wavefunction as

 ¯w=−ig4(N2c−1)4(2π)4∫∞0dτ1∫∞τ1dτ2∫∞0dτ′1∫∞0dτ′2¯Idk1+2k1+dk2+2k2+k21+2d2x1k22+2d2x2 (35)

then when the , variables of the are identified with the , variables of .

Although we are identifying variables with different dimension in the correspondence we note that both and are dimensionlesss so that one could always introduce a (fictitious) dimensional parameter to scale , and to dimensionless varaibles.

In dealing with the graphs of figure 1 we have separated the graphs into vertices and lines, as for example in (19)-(21) and (23)-(25). It should be clear that for any graph built out of three-gluon vertices and causal propagation the procedure we have used here will work and lead to a correspondence between the probability of a given configuration of gluons appearing in the decay of a timelike photon and the probability that the corresponding gluons appear in the square of the lightcone wavefunction. It is straightforward to see that the correspondence continues to be valid when four-gluon vertices and instantaneous propagation is included, but we omit the details here for simplicity.

Our result might seem to be too strong. After all, we expect the decay-wavefuntion correspondence to reflect conformal symmetry and it is known that running coupling corrections will break conformal symmetry. So how does the breaking of conformal symmetry come into our discussion? The correspondence identifying in (34) with in (35), once , variables in have been changed to , variables to get is for fixed times. We believe this correspondence to be exact. However, the integrations over and have divergences when two times approach each other. In some circumstances these divergences can be removed simply by considering a more appropriate “jet” variable. In other circumstances these divergences must be removed by renormalization. Renormalization requires introducing a scale which breaks the conformal symmetry and that breaking corresponds to the running of the coupling in QCD. The graphs we have considered in this section have no divergences when the time integrations are done and so the correspondence survives time integration. In the next section of this paper we consider graphs which include running coupling effects.

## 3 Graphs with running coupling divergences

We now turn to graphs having running coupling corrections, in particular the two graphs shown in figure 2. We begin with graph 1(b). For fixed , , , it is straightforward to write the graph as

 ¯w= −g4(N2c−1)2(2π)4∫dk1+2k1+dk2+2k2+d2kd2k′(2k+)2Vλ1(Vλ′4)∗Vλλ1λ22(Vλ′λ1λ23)∗ ×LL1L2(L′)∗2∏i=1d2kid2k′idt1dt2dt3dt4 (36)

where , and are as in (23)-(25) while is the same as (25) after the replacement . A sum over all ’s is understood in (3), while is as in (22) and is obtained from by the replacements . The limits on the -integrations will be given later.

Call

 Iλλ′=∫2∏i=1d2kid2k′ieik––1⋅(x––1−x––)+ik––2⋅(x––2−x––)−ik––′1⋅(x––1−x––′)−ik––′2⋅(x––2−x––′)∑λ1λ2Vλλ1λ22(Vλ′λ1λ23)∗. (37)

Then, using (28) and (2.2), it is straightforward to get

 Iλλ′= 4(2π)6(k1+k2+)2t22t23(k1+k2)2δ(x––−x––′)δ(x––−zx––1−(1−z)x––2)e−i(x––1−x––2)212k+z(1−z)(1t2−1t3) ×k2+{[z2+(1−z)2](x––1−x––2)2δλλ′+2[z(1−z)]2–ϵλ⋅(x––1−x––2)⋅ϵ–λ′⋅(x––1−x––2)} (38)

where . Write

 d2x1d2x2=d2x12d2~x (39)

with and . Using (27), and a similar expression for the Fourier transform of one finally gets

 ¯w= −g2(N2c−1)8(2π)4∫dt1dt2dt3dt4(t1t2t3t4)2dk+k+k4+e−i(x––−x––a)2k+2t1+i(x––−x––b)2k+2t4−ix––21212k+z(1−z)(1t2−1t3) ×[z(1−z)]2ϵ–λ⋅(x––−x––a)ϵ–λ′⋅(x––−x––b)d2xd2x12x212[(z1−z+1−zz)δλλ′+2z(1−z)ϵ–λ⋅x––12ϵ–λ′⋅x––12x212]. (40)

Now

 2d2x12ϵ–λ⋅x––12ϵ–λ′⋅x––12x212→d2x12δλλ′ (41)

since the rest of (3) depends only on but not on the orientation of . Now write