pQCD physics of multiparton interactions

# pQCD physics of multiparton interactions

B. Blok Department of Physics, Technion—Israel Institute of Technology, 32000 Haifa, Israel    Yu. Dokshitzer Laboratory of High Energy Theoretical Physics (LPTHE), University Paris 6, Paris, France 111On leave of absence: St. Petersburg Nuclear Physics Institute, Gatchina, Russia    L. Frankfurt School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978 Tel Aviv, Israel    M. Strikman Physics Department, Penn State University, University Park, PA, USA
###### Abstract

We study production of two pairs of jets in hadron–hadron collisions in view of extracting contribution of double hard interactions of three and four partons (, ). Such interactions, in spite of being power suppressed at the level of the total cross section, become comparable with the standard hard collisions of two partons, , in the back-to-back kinematics when the transverse momentum imbalances of two pairing jets are relatively small.

We express differential and total cross sections for two-dijet production in double parton collisions through the generalized two-parton distributions, GPDs BDFS1 (), that contain large-distance two-parton correlations of non-perturbative origin as well as small-distance correlations due to parton evolution. We find that these large- and small-distance correlations participate in different manner in 4-jet production, and treat them in the leading logarithmic approximation of pQCD that resums collinear logarithms in all orders.

A special emphasis is given to double hard interaction processes that occur as an interplay between large- and short-distance parton correlations and were not taken into consideration by approaches inspired by the parton model picture. We demonstrate that the mechanism, being of the same order in as the process, turns out to be geometrically enhanced compared to the latter and should contribute significantly to 4-jet production.

The framework developed here takes into systematic consideration perturbative evolution of GPDs. It can be used as a basis for future analysis of NLO corrections to multi-parton interactions (MPI) at LHC and Tevatron colliders, in particular for improving evaluation of QCD backgrounds to new physics searches.

## I Introduction

Understanding the rates and the structure of multi-jet production in hadron–hadron collisions is of primary importance for new physics searches.

Production of high transverse momentum jets is a hard process which implies a head-on collision of QCD partons — quarks and/or gluons — from the small-distance wave functions of initial hadrons. Cross section of a hard collision is small compared with the size of hadron, , with the scale related to transverse momenta of the produced jets, . Therefore, typically it is two partons that experience a hard collision in a given event. A large angle scattering of these two partons produces two (or more) final state partons that manifest themselves as hadron large transverse momentum jets. At the same time, one cannot exclude a possibility that more than one pair of partons happen to collide in a given event, giving rise to a multi-jet event. A possibility of a double hard collision becomes more important with increase of the energy of the collision where scattering off small partons which have much higher densities becomes possible.

In recent years multiparton collisions have attracted close attention. Following the pioneering work of Refs. TreleaniPaver (); mufti (), a large number of related theoretical papers appeared Treleani (); Wiedemann (); Frankfurt (); Frankfurt1 (); SST (), based on the parton model and geometrical picture in the impact parameter space. More recently, this topic has been intensively discussed in view of the LHC program Perugia (); Fano (). Monte Carlo event generators that produce multiple parton collisions are being developed Pythia (); Herwig (); Lund (); theoretical papers exploring properties of double parton distributions and discussing their QCD evolution have appeared BDFS1 (); Diehl (); DiehlSchafer ().

In our view, however, important elements of QCD that are necessary for theoretical understanding of the multiple hard interactions issue have not yet been properly taken into account by above-mentioned intuitive approaches.

The problem is, sort of, educational: both the probabilistic picture, the MC generator technology is based upon, and the familiar Feynman diagram technique, when used in the momentum space, prove to be inadequate for careful analysis and understanding of the physics of multiple collisions.

From experience gained by treating standard (single) hard processes, one became used to a motto that a large momentum transfer scale ensures the dominance of small distances, , in a process under consideration. With the multiple collisions under focus, however, one has to distinguish two space-time scales: that of localization of the parton participating in a hard interaction, , and that of transverse separation, , between the two hard collision vertices. The latter can be large, of the order of the hadron size, even for large .

In order to be able to trace the relative distance between the partons, one has to use the mixed longitudinal momentum–impact parameter representation which, in the momentum language, reduces to introduction of a mismatch between the transverse momentum of the parton in the amplitude and that of the same parton in the amplitude conjugated.

Another unusual feature of the multiple collision analysis that may look confusing at the first sight is the fact that — even at the tree level — the amplitude describing the double hard interaction process contains additional integrations over longitudinal momentum components; more precisely — over the difference of the (large) light-cone momentum components of the two partons originating from the same incident hadron (see Section IV.1).

In the previous short publication BDFS1 () we have considered production of two pairs of nearly back-to-back jets resulting from simultaneous hard collisions of two partons from the wave function of one incident hadron with two partons from the other hadron (“four-to-four” processes). As we have shown, this necessitates introduction of a new object — a generalized double parton distribution, GPD, that depends on a new transverse momentum parameter conjugate to the relative distance between the two partons in the hadron wave function. Generalized double parton distributions provide a natural framework for incorporating longitudinal and transverse correlations between partons in the hadron wave function at the scale, and for tracing the perturbative -evolution of the correlations.

The corresponding 4-jet cross section can be expressed in terms of GPD’s as follows

 dσ(x1,x2,x3,x4)d^t1d^t2=dσ13d^t1dσ24d^t2×∫d2→Δ(2π)2Da(x1,x2;→Δ)Db(x3,x4;−→Δ). (1)

The factor on the second line has dimension of inverse area:

 1πR2int=∫d2→Δ(2π)2D(x1,x2,→Δ)D(x3,x4,−→Δ)D(x1)D(x2)D(x3)D(x4), (2)

where are the corresponding one-parton distributions. The ratio of the product of two single-inclusive cross sections and the double-inclusive cross section () is often referred to in the literature as “an effective cross section” . We prefer, however, not to look at this quantity as a cross section, since it reflects transversal area of parton overlap as well as longitudinal correlations of the partons. At the same time, it has little to do with the measure of the strength of the interaction, which is what “cross section” represents.

In a two-parton collision, scattered partons form two nearly back-to-back jets, while additional jets (should there be any) tend to be softer and to align with the directions of initial and final partons, because of collinear enhancements due to radiative nature of secondary partons. Such will be typical characteristics of a 4-jet event, in particular. On the other hand, four jets produced as a result of a double hard collision of two parton pairs would, on the contrary, form two pairs of nearly back-to-back jets. This kinematical preference is in stark contrast with “hedgehog-like” configurations of four jets stemming from a single collision and can be used in order to single out double hard collisions experimentally.

Such experimental studies were recently carried out by the CDF and D0 collaborations who have studied production of three jets + photon Tevatron1 (); Tevatron2 (); Tevatron3 (); Perugia (). The analysis of the data performed in Frankfurt (); Frankfurt1 () using information about generalized parton distribution (GPDs) obtained from the study of hard exclusive processes at HERA has found that observed rates were a factor higher than the expected rates based on a naive model that neglected correlations between partons in the transverse plane.

The use of GPD allows one to incorporate such correlations and predict their evolution.

On the theory side, the back-to-back enhancement has been discussed, at tree level, in a number of studies of various channels (see, for example, discussion of the 2 jets+ in Berger () and references therein).

In the present paper we study perturbative radiative effects in the differential 4-jet distribution in the back-to-back kinematics and derive the expression for the corresponding cross section in the leading logarithmic collinear approximation. It takes into account QCD evolution of the generalized double parton distributions as well as effects due to multiple soft gluon radiation, and turns out to be a direct generalization of the known “DDT formula” for back-to-back production of two large transverse momentum particles in hadron collisions DDT ().

We also discuss and treat new specific correlations between transverse momenta of jets due to 3-parton interactions producing 4 jets,“three-to-four”. Such processes are induced by perturbative splitting of a parton from one of the hadrons, the offspring of which enter double hard collision with two partons from the wave function of the second hadron. The hard scale of this parton splitting is determined by transverse momentum imbalances of pairs of jets, , , and exhibits specific collinear enhancement in the kinematical region where two jet imbalances practically compensate one another, .

Consistently taking into account three-to-four parton process solves a longstanding problem of double counting in treating multi-parton interactions.

Discussion of the 2-parton distribution has a long history. It is commonly defined in the momentum space as a 2-particle inclusive quantity depending on two parton momenta, see Kirschner (); Snigirev (). Being related to (the imaginary part of) a certain forward scattering amplitude, it therefore disregards impact parameter space geometry of the interaction. Exploring properties of 2-parton distributions so defined, an approach to the study of the multiple jet production has been recently suggested in Ref. stirling (). The reason why this approach has faced difficulties, stirling1 (), and did not solve, in our view, the problem of systematic pQCD analysis of 4-jet production is clear: it did not incorporate effects due to variations of the transverse separation between the partons — information encoded by GPD’s but not by the 2-parton momentum distributions.

The GPD’s were recently used in Ref. Ryskin () for intuitive description of the total 4-jet production cross section. However, the differential distributions were not discussed in that paper, and not all relevant pQCD contributions were included, so that our results are different from the ones obtained in Ryskin ().

The paper is organized as follows.

In Section II we recall the main ingredients of the perturbative analysis based on selection of maximally collinear enhanced contributions in all orders. In Section III we present the evolution equation for generalized two-parton distributions. Section IV is devoted to the perturbative analysis of small-distance correlations between partons. The main result of the paper — the differential distribution of 4-jet production in the back-to-back kinematics — is formulated in Section V, and the total cross section of two-dijet production is described in Section VI. Conclusions and outlook are presented in Section VII.

## Ii Perturbative Analysis

### ii.1 Hard scales

The perturbative approach implies that all hardness (transverse momentum) scales that characterize the problem are comfortably larger than the intrinsic QCD scale : . The process under consideration may have up to five hard scales involved.

Indeed, in the leading order in , large transverse momentum partons are produced in pairs and have nearly opposite transverse momenta, setting the hard scale . Within the parton model framework (neglecting finite smearing due to intrinsic transverse momenta of incident partons), one has . Secondary QCD processes — evolution of initial parton distributions and accompanying soft gluon radiation — introduce transverse momentum imbalance: which constitutes another hard scale: . For production of four jets in the back-to-back kinematics, this gives four different hard scales. As we shall see below, in the 3-partons collisions producing four jets yet another scale enters the game: with — the total transverse momentum of the 4-jet ensemble.

In what follows we consider transverse momenta of all four jets to be of the same order, . This is not necessary but helps to avoid complications in the hierarchy of relevant scales.

Finally, let us mention that in what follows it will be tacitly implied that fixing these scales — from the largest one, , down to smaller ones, and , — is not compromised by uncertainties in determination of the transverse momenta of the jets.

### ii.2 Back-to-back kinematics

The basic 2-jet production cross section scales, asymptotically, as

 dσ(2→2)d^t∝α2sQ4. (3)

According to (1), production of four jets in simultaneous hard collisions of four partons yields

 (4a) with R2=1/⟨Δ2⟩ the characteristic distance between the two partons in the hadron wave function. At large Q2 this cross section is parametrically smaller than that for production of four well separated jets with transverse momenta j2i⊥∼Q2 in a 2-parton collision: dσ(2→4)d^t1d^t2∝α4sQ6 (4b)

(with transverse momenta of two out of four jets being integrated over).

Qualitatively, the production mechanism (4a) can be labelled a “higher twist effect”. Nevertheless, it may turn out to be essential — comparable with the “leading twist” , Eq. (4b) — if one looks at specific kinematics of the 4-jet ensemble.

Let be the direction of colliding hadron momenta. Imagine that we are triggering on two jets moving along the and axes in the transverse plane, and look for two accompanying jets inside some solid angles around the and directions. The production mechanism (4b) does not populate this region: the higher order QCD matrix element is enhanced when two final state partons become quasi-collinear but is perfectly smooth in the back-to-back kinematics. Therefore, its contribution will be suppressed,

 (ΔΩ4π)2vs.1R2Q2,

contrary to the production mechanism (4a) which is concentrated in this very kinematical region.

### ii.3 Collinear approximation

The differential 4-jet production cross section possesses two collinear enhancements. Depending on the kinematics of the jets, they are, symbolically,

 dσ(4→4) ∝ α2\rm\scriptsize sδ213δ224d2j3⊥d2j4⊥⋅dΣ, (5a) δ213≪Q2,δ224≪Q2; dσ(3→4) ∝ α2\rm\scriptsize sδ′2δ2d2j3⊥d2j4⊥⋅dΣ, (5b) δ′2≪δ2≪Q2,δ2=δ213≃δ224.

Here is the cross section integrated over the transverse momenta of the “backward” jets and . The integrated cross section contains the squared matrix element of the four-parton production and is of the order of , cf. Eq. (4a). At the Born level, the jets in pairs are exactly back-to-back, so that in (5a). To have a non-zero value of the transverse momentum imbalance, one has to have additional large transverse momentum parton(s) produced.

In the second important contribution to the cross section, Eq. (5b), one power of the coupling emerges from the splitting of a parton from one of the incident hadrons into two, and the second power is due to production of an additional final state parton with .

In both cases the smallness due to additional powers of the coupling is compensated by two broad (logarithmic) integrations over transverse momentum imbalances as indicated in (5).

### ii.4 Double Logarithmic parton form factors

In the leading order in , it suffices to have just one parton present with in order to assure . At the same time, inclusive production of accompanying partons with transverse momenta turns out to be suppressed in a broad interval , as long as one wants to preserve the collinear enhancement factor in the jet correlation (5a).

This dynamical “veto” has two consequences.

First of all, it results in reduction of the hardness scale of the parton distributions from the natural scale (scale of the parton distributions in the integrated cross section) down to the observation-induced scale .

Then, it introduces double logarithmic (DL) form factors of participating initial state partons, since the transverse momentum of the jet pair can be compensated not only by a hard (energetic) parton from inside initial parton distributions but also by a soft gluon whose radiation did not affect inclusive parton distributions due to real–virtual cancellation.

The presence of the DL form factors depending on the logarithm of a large ratio of scales, , is typical for the so-called “semi-inclusive” processes DDTsemi (); DDT ().

Production of massive lepton pairs in hadron collisions (the Drell–Yan process) is a classical example of a two-scale problem. Here enter form factors of colliding quarks that depend on the ratio of the invariant mass to the transverse momentum of the lepton pair, , in the dominant kinematical region :

 dσdq2dq2⊥=dσ% \scriptsize totdq2 ×∂∂q2⊥{Dqa(x1,q2⊥)D¯qb(x2,q2⊥)S2q(q2,q2⊥)}. (6)

is the double logarithmic QCD quark form factor.

Sudakov quark and gluon form factors can be expressed via the exponent of the total probability of the parton decay in the range of virtualities (transverse momenta) between the two hard scales:

 Sq(Q2,κ2) = exp{−∫Q2κ2dk2k2αs(k2)2π∫1−k/Q0dzPqq(z)}, (7a) Sg(Q2,κ2) = exp{−∫Q2κ2dk2k2αs(k2)2π∫1−k/Q0dz[zPgg(z)+nfPqg(z)]}. (7b)

Here are the non-regularized one-loop DGLAP splitting functions (without the “+” prescription):

 Pqq(z)=CF1+z21−z,Pgq(z)=Pqq(1−z),Pqg(z)=TR[z2+(1−z)2],Pgg(z)=CA1+z4+(1−z)4z(1−z); (8)

the upper limit of -integrals properly regularizes the soft gluon singularity, (in physical terms, it can be looked upon as a condition that the energy of a gluon should be larger than its transverse momentum, DDT ()).

The case of hadron interactions producing large transverse momentum partons (instead of colorless objects like a Drell–Yan pair or an intermediate boson) is more involved since here the transverse momentum imbalance may be compensated by QCD radiation from the final state partons too.

The azimuthal correlation between two nearly back-to-back large transverse momentum particles was considered in DDT (). An analog of the “DDT formula” has been derived in the collinear approximation, which expression contained the product of four form factors, two initial parton distributions and two fragmentation functions.

### ii.5 Single Logarithmic soft gluon effects

The case when jets are being reconstructed in the final state is more complicated to analyze as it yields an answer depending on the jet finding algorithm. The problem has been addressed by Banfi and Dasgupta in ABC () where a smart way of defining the final state jets was formulated that permitted to write down a resummed QCD formula for soft gluon effects in “2 partons 2 jets” cross sections.

Collinear logarithms due to hard splittings of the final state partons do not pose a problem: such secondary partons populate the jets. Partons that appear as separate out-of-jet radiation — and are relevant for transverse momentum imbalance compensation — have to be produced at sufficiently large angles with respect to the jet axis. This is the domain of large-angle gluon radiation. Production of soft gluons in-between jets is also logarithmically enhanced and induces single logarithmic (SL) corrections, , that may also be significant and should be resummed in all orders.

Contrary to collinear enhanced effects (that drive evolution of parton distributions and fragmentation functions and determine the Sudakov form factors), the large-angle gluon radiation cannot be attributed to one or another of the partons participating in the hard scattering. It is coherent and depends on the kinematics and color topology of the hard parton ensemble as a whole. As a result, resummation of these SL corrections becomes a matrix problem that involves tracing various color states of the parton system, see ABC () and references therein.

In the present paper we concentrate on resummation of collinear enhanced DL and SL terms and avoid complications due to soft SL corrections. This means ignoring color transfer effects in hard interactions. Thus, production of four jets with large transverse momenta and pair imbalances will be equivalent, in our treatment, to production of two colorless Drell–Yan pairs with invariant masses and transverse momenta and . Generalization of the results of ABC () to the case of double parton scattering seems straightforward and should be considered separately.

## Iii Generalized double parton distribution

### iii.1 Geometry of 2Gpd

The GPD in the expression for the multiparton production cross section has a meaning of a two body form factor when partons and receive transverse momenta and leaving the hadron intact. Nonrelativistic analogue of this form factor is familiar from the double scattering amplitude in the momentum space representation of the Glauber model, see e.g. LevinStrikman (). Recall that BDFS1 () the scale in GPD is conjugate to the relative transverse distance between the two partons in the GPD in the impact parameter representation considered in TreleaniPaver (); mufti (); Diehl ().

Two partons may originate from soft low-scale fluctuations inside the hadron; they can also emerge from a perturbative splitting of a common parent parton at relatively large momentum scales. It is clear that these two contributions to GPD will have essentially different dependence on the parameter .

The first contribution we will denote

 [2]Dbca(x1,x2;q21,q22;→Δ), (9)

with the subscript stressing that here the partons and emerge from the no-perturbative wave function of the hadron . It should decrease rapidly at scales larger than a natural scale of short-range parton correlation in a hadron (this scale may be slightly different for quarks and gluons and could in principle be significantly larger than the as there exists another non-perturbative scale of the chiral symmetry breaking which maybe as large as 700 MeV).

GPD should rapidly decrease for where is the transverse gluonic radius of the nucleon. In the mean field approximation when the correlations are neglected, it can be approximated by a factorized expression BDFS1 ()

 [2]Db,ca(x1,x2;q21,q22;→Δ) = Fg(Δ2;x1,q21)Fg(Δ2;x2,q22) (10a) × Gba(x1,q21)Gca(x2,q22), where G are the single-parton distributions and the two-gluon form factor F can be parametrized as Fg(Δ2)=(1+Δ2m2g(x))−2. (10b)

The parameter is of the order of 1 GeV for and gradually drops with decrease of and with increase of virtuality FSW ().

The second contribution we will denote

 [1]Db,ca(x1,x2;q21,q22;→Δ), (11)

where the subscript stands as a reminder of the fact that and originate from perturbative splitting of a single parton from the hadron wave function. This contribution is practically -independent and should decrease with much more slowly, due to logarithmic pQCD effects. (A steep power falloff starts only when exceeds the relevant hard scale, .)

### iii.2 On the geometrical enhancement of the interference effects due to pQCD correlations

By total cross section in the present context we mean the back-to-back 4-jet cross section integrated over pair jet imbalances in the dominant logarithmic region .

We start by noting that the product of two small-distance parton fluctuations, , does not contribute to the process we are interested in. Indeed, in this case the integral over in Eq. (1) formally diverges and yields a hard scale (instead of ) in the numerator. This means a significant contribution to the cross section but not the one we are looking for. Below in Section IV.2 we will explicitly verify that a double hard collision of two parton pairs each of which originates from perturbative splitting, lacks the back-to-back enhancement. In fact, the product corresponds to a one-loop correction to the “leading twist” perturbative production of four jets in a hard collision of two partons (“two-to-four”) whose distribution is smooth in the back-to-back region and as such gets subtracted as background.

Keeping this in mind, the back-to-back 4-jet production cross section is proportional to the inverse “interactions area” described by the expression

 1S=∫d2Δ(2π)2([2]Da(Δ)[2]Db(Δ)+[2]Da(Δ)[1]Db(Δ)+[1]Da(Δ)[2]Db(Δ)), (12)

where indices , mark two interacting nucleons. This expression is somewhat symbolic; a careful analysis of the “interaction area” will be carried out below in Sec. V (see Eq. (32)).

The first term in Eq. (12) we will refer to as a “four-to-four” process: two partons from the wave functions of the hadron interact with two partons from the hadron producing four jets. The second and the third terms in Eq. (12) describe hard collisions of one parton from one hadron with two partons from the second hadron. Until recently, these “three-to-four” processes were commonly ignored in the literature (see, however, Ryskin () ). At the same time, they turn out to be somewhat enhanced.

Indeed, the contribution due to four-to-four processes to the geometrical factor Eq. (12) is given by

 ∫d2Δ(2π)2F2g(Δ2)×F2g(Δ2)=m2g7π. (13a) Fast decrease of the product of two squared form factors leads to fast convergence of the integral whose median is positioned at a value as low as Δ2≈0.1m2g. The case of the three-to-four process is different. This process corresponds, as we explained above, to interaction of the offspring of the perturbative splitting of a parton from the wave function of one hadron, with two partons from the non-perturbative wave function of the second colliding hadron. On the side of [1]D, the parameter Δ enters “perturbative loop” due to parton splitting and as a result the dependence of [1]D on Δ turns out to be only logarithmic, that is, parametrically much slower than that of the non-perturbative form factor Fg(Δ2). Thus, the three-to-four contribution to the double interaction cross section reduces to ∫d2Δ(2π)2[1]D(x1,x2;Δ)F2g(Δ2)≃[1]D∣∣Δ=0∫d2Δ(2π)2F2g(Δ2), (13b) (where we have neglected the logarithmic Δ-dependence of [1]D). This corresponds to the fact that in the impact parameter space, the distance between partons coming from a perturbative splitting is much smaller than the hadron size, so that the answer is proportional to the density of non-perturbative two-parton correlation at small distances — “in the origin”: ∫d2Δ(2π)2F2g(Δ2)=m2g3π. (13c)

Comparison with the estimate (13a) shows that the contribution to the cross section of the “interference term” is enhanced, relative to the process, by the factor

 73×2∼5. (14)

(For the case of the Gaussian form factors this enhancement is 15% smaller — a factor of 4.) This estimate was obtained for the case when all four partons participating in the hard collisions are gluons. A detailed numerical study of the -dependence of the effective interaction area will be presented in the paper under preparation.

So we conclude that the three-to-four processes may provide a sizable contribution to the cross section even if they constitute a small correction to GPD.

### iii.3 Perturbative QCD effects in 2Gpd

Thus, we represent the generalized double parton distribution GPD as a sum of two terms:

 Db,ca(x1,x2;q21,q22;→Δ) = [2]Db,ca(x1,x2;q21,q22;→Δ) (15) + [1]Db,ca(x1,x2;q21,q22;→Δ).

The term describes the distribution of two partons from the non-perturbative wave function of the hadron that are independently evolved to large perturbative scales and according to the standard one-parton evolution equation. The perturbative evolution involves momentum scales much larger than the hadron wave function correlation scale . Therefore, the evolution practically does not affect the -dependence of the two-parton spectrum. This piece of the GPD acquires but a mild additional logarithmic dependence at the tail of the -distribution in addition to a non-perturbative power falloff (10).

The integral QCD evolution equation for reads

 [2]Db,ca(x1,x2;q21,q22;→Δ)=Sb(q21,Q2min)Sc(q22,Q2min)[2]Db,ca(x1,x2;Q20,Q20;→Δ)+∑b′∫q21Q2mindk2k2α\rm\scriptsize s(k2)2πSb(q21,k2)∫dzzPbb′(z)[2]Db′,ca(x1z,x2;k2,q22;→Δ)+∑c′∫q22Q2mindk2k2α\rm\scriptsize s(k2)2πSc(q22,k2)∫dzzPcc′(z)[2]Db,c′a(x1,x2z;q21,k2;→Δ). (16)

Here are the non-regularized one-loop DGLAP splitting functions (8) and — the double logarithmic Sudakov parton form factors defined in Eq. (7).

The lower limit of the perturbative evolution in Eq. (16),

 Q2min=max(Q20,Δ2)≃Q20+Δ2, (17)

is the only source of additional (logarithmic) -dependence. It emerges when exceeds — and substitutes — the starting non-perturbative scale of the perturbative evolution.

The second term in Eq. (15), , represents the small-distance correlation between the two partons that emerge from a perturbative splitting of a common parent parton taken from the hadron wave function. It can be expressed in terms of standard inclusive single-parton distributions as follows

 [1]Db,ca(x1,x2;q21,q22;→Δ)=∑a′,b′,c′∫min(q21,q22)Q2mindk2k2α\rm\scriptsize s(k2)2π∫dyy2Ga′a(y;k2,Q20)×∫dzz(1−z)Pb′[c′]a′(z)Gbb′(x1zy;q21,k2)Gcc′(x2(1−z)y;q22,k2). (18)

The -dependence of is very mild as it emerges solely from the lower limit of the logarithmic transverse momentum integration .

## Iv Analysis of perturbative two-parton correlations

### iv.1 3→4

Let us analyze the lowest order interaction amplitude shown in Fig. 1 that produces a double hard collision and involves parton splitting.

We express parton momenta in terms of the Sudakov decomposition using the light-like vectors , along the incident hadron momenta:

 k1 = x1pa+βpb+k⊥,k3≃(x3−β)pb; k2 = x2pa−βpb−k⊥,k4≃(x4+β)pb; →k⊥ = →δ12=−→δ34(δ′≡0);k0≃(x1+x2)pa.

Here , and are momenta of incoming (real) partons, and and — virtual ones. Light-cone fractions , i=1,..4, are determined by jet kinematics (invariant masses and rapidities of jet pair). The fraction that measures the difference in longitudinal momenta of the two partons coming from the hadron , is arbitrary. Fixed values of the parton momenta and correspond to the plane wave description of the scattering process in which the longitudinal distance between the two scatterings is arbitrary. This description does not correspond to the physical picture of the process we are interested in. In order to ensure than the partons and originate form the same hadron of finite size, we have to introduce an integration over in the amplitude, in the region .

The Feynman amplitude contains the product of two virtual propagators. The virtualities and the that enter the denominator of the amplitude in terms of the Sudakov variables read

 k21=x1βs−k2⊥,k22=−x2βs−k2⊥,

with and the square of the two-dimensional transverse momentum vector.

A singular contribution we are looking for originates from the region , so that precise form of the longitudinal smearing does not matter and the integral yields

 N∫dβ(x1βs−k2⊥+iϵ)(−x2βs−k2⊥+iϵ)=2πiN(x1+x2)1k2⊥.

The numerator of the amplitude is proportional to the first power of the transverse momentum . As a result, the squared amplitude (and thus the differential cross section) acquires the necessary factor that enhances the back-to-back jet production.

### iv.2 2→4

Now we should verify that the diagram of Fig. 2 where both incident partons split, and their offspring engage into double hard scattering, does not favor back-to-back jet kinematics. In other words, it does not lead to a small imbalance factor in the differential cross section.

Sudakov decomposition:

 k1=(x1−α)pa+βpb+k′⊥,k3=(x3−β)pb+αpa−k⊥;k2=(x2+α)pa−βpb−k′⊥,k4=(x4+β)pb−αpa+k⊥;→k′⊥−→k⊥=→δ12=−→δ34(δ′≡0);k0≃(x1+x2)pa,k5≃(x3+x4)pb.

This is a loop diagram and it contains explicit integration over the loop momentum:

 s∫dαdβ(2π)2i∫d2k⊥d2k′⊥(2π)2δ2(→k⊥+→δ−→k′⊥).

To get an enhanced contribution we have to have parton virtualities that enter the denominator of the Feynman amplitude to be relatively small, of the order of . This implies in the essential integration region. Adopting this approximation, we can simplify parton propagators and reduce the longitudinal momentum integrations to the product of two independent integrals:

 is∫dβ2πis(βx1s−k2⊥+iϵ)(βx2s+k2⊥−iϵ)×∫dα2πis(αx3s−k′2⊥+iϵ)(αx4s+k′2⊥−iϵ)=i(x1+x2)(x3+x4)s1k2⊥k′2⊥.

The remaining transverse momentum integration takes the form

 ∫d2k⊥(2π)2V→k⊥2(→k⊥+→δ)2 (20)

Due to gauge invariance the numerator of the diagram — the “vertex factor” — is linear in transverse momenta of the loop partons: . Therefore, the integral (20) produces no more than a logarithmic enhancement factor, , instead of the power back-to-back singularity we were looking for.

So, the diagram Fig. 2 with double parton splitting constitutes but a negligible loop correction to the usual “hedgehog” 4-jet kinematics typical for QCD processes. The fact that this loop diagram does not produce a pole singularity in could have been extracted, e.g., from numerical studies of double -boson production in two-parton collisions GloverBij () and, more generally, of multi-leg parton amplitudes NagySoper ().

The logarithmic character of this correction has been recently confirmed by the systematic study of “box integrals” in stirling1 ().

The presence of the double parton splitting contribution of Fig. 2 is being treated in the literature as a source of potential problem of double counting (see, e.g., NagySoper (); CaSaSa (); DiehlSchafer ()). The present paper solves this problem.

### iv.3 3→4 with additional parton emission

We have to return now to the process and examine the possibility of producing an additional parton, in collinear enhanced manner, in order to lift off the Born level kinematical constraint .

Consider the diagram of Fig. 3a. The momenta of quasi-real colliding partons are

 k0≃(x1+x2+α)pa;k3≃(x3−β)pb,k4≃(x4+β+δ′2αs)pb,

and the radiated on-mass-shell parton carries momentum

 ℓ−k2=αpa+δ′2αspb−δ′,δ′=δ13+δ24.

We have three virtual propagators subject to integration over :

 k1 = x1pa+βpb+δ13, (21a) k2 = x2pa−(β+δ′2αs)pb+δ24, (21b) ℓ = (x2+α)pa−βpb−δ13. (21c)

Closing the contour around the pole , we obtain and

 −ℓ2 = (x2+α)βs+δ213=δ213⋅x1+x2+αx1, (22a) −k22 = x2(β+δ′2αs)s+δ224=x1δ224+x2δ213x1+x2αδ′2. (22b)

Taken together with the residue of the -integration, , Eq. (22b) produces the universal factor present in all the amplitudes considered, (including the diagrams with parton emission off the external lines, see Fig. 4 below):

 P−1=x1δ224+x2δ213+x1x2αδ′2. (23)

We observe that both propagators (22) are enhanced in the back-to-back kinematics. The amplitude of Fig. 3a gives a double collinear enhanced contribution to the cross section in the region

 δ213≪δ224≃δ′2. (24a) The inequality (24a) corresponds to the following physical picture. An incident parton k0 splits into k1 and ℓ early, at time O(√s/δ213) corresponding to some comparatively low perturbative scale δ13. At this time scale the parton 1 collides with 3, while the parton ℓ keeps evolving and scatters off 4 with a much larger momentum transfer δ24. Evolution of the parton ℓ in between these two scales is the origin of probable (logarithmically enhanced) production of additional parton(s). Analogously, the diagram Fig. 3b with a parton produced off the virtual line 1 contributes in the complementary kinematical region δ224≪δ213≃δ′2. (24b)

Full perturbative analysis of the production of a parton from inside the “splitting fork”, Fig. 3, together with emission off the incoming line “0” (to be treated below in Sec. V.2) is sketched in the Appendix.

## V Differential distribution

### v.1 2+2→4

Now that we know the structure of the perturbative corrections to GPD, Eqs. (15)–(18), we are in a position to write down the generalization of the DDT formula (II.4) for (the first contribution to) the differential cross section of 4-jet production in nearly back-to-back kinematics. It reads

 π2dσ(4→4)d2δ13d2δ24 = dσ\scriptsize partd^t1d^t2⋅∂∂δ213∂∂δ224{[2]D1,2a(x1,x2;δ213,δ224)×[2]D3,4b(x3,x4;δ213,δ224) (25) × S1(Q2,δ213)S3(Q2,δ213)×S2(Q2,δ224)S4(Q2,δ224)}.

Here is the cross section of double hard parton scattering, and stand for Sudakov form factors of four participating partons. Sum over parton species and convolution over as in Eq. (1) is implied.

Taking derivative over the scale of the function depending on , produces the factor . Differentiating the Sudakov form factor of a given parton describes the situation when the jet imbalance is compensated by radiation of a soft gluon off this parton. Differentiation of the parton distribution corresponds to the situation when a hard parton takes the recoil.

### v.2 1+2→4

The differential transverse momentum imbalance distribution due to the cross-terms contains two pieces.

#### v.2.1 Two compensating partons

The first one has the same structure as Eq. (25):

 π2dσ(3→4)1d2δ13d2δ24 = dσ\scriptsize partd^t1d^t2⋅∂∂δ213∂∂δ224{[1]D1,2a(x1,x2;δ213,δ224)⋅[2]D3,4b(x3,x4;δ213,δ224) (26) × S1(Q2,δ213)S3(Q2,δ213)⋅S2(Q2,δ224)S4(Q2,δ224)}.

Sum over parton species and convolution over is implied as above in Eq. (25).

As above, taking the derivatives in Eq. (26) corresponds to fixing transverse momenta of two final state partons that compensate jet pair imbalances: and .

Consider now the correlation term of Eq. (26). If we apply the derivatives to the parton distributions in the integrand of Eq. (18) for the correlation term , this contribution will correspond to production of two momentum compensating quanta in the course of evolution of the system of two partons with account of small-distance perturbative correlation between them. In this case the scale of the core parton splitting stays smaller than the two external scales , , and is being integrated over.

#### v.2.2 One compensating parton

The correlation term (26) contains an additional option. Namely, instead of creating intermediate state partons that keep evolving up to external scales, the perturbative splitting may produce that very parton that gets engaged in the hard scattering. This possibility is also contained in Eq. (26): it corresponds to the differentiation of the upper limit of the virtuality integral in Eq. (18) over the smaller of the two imbalances, . One of the two parton distribution functions in the integrand then collapses to . Taking the second derivative over the larger imbalance of the second parton distribution produces the contribution described by the diagram of Fig. 3 that we have discussed above in Section IV.3.

### v.3 1+2→4, endpoint contribution

Finally, there is a possibility that both partons emerging from the perturbative splitting in experience hard collisions straight away, without any further evolution. Jet imbalances stemming from such an eventuality are no longer independent but, on the contrary, are strongly correlated. At the “Born level” one has

 dσd2δ13d2δ24∝α\rm% \scriptsize sδ2δ(→δ13+→δ24),δ2≡δ213=δ224.

With account of additional radiation, the delta-function gets replaced by the pole enhancement of the differential cross section in the imbalance of imbalances in a specific region of jet momenta when the pair imbalances are practically equal-and-opposite::

 dσd2δ13d2δ24∝α2\rm% \scriptsize sδ2δ′2,δ′2≪δ2≡δ213≃δ224.

Importantly, this contribution is also double-collinear enhanced and therefore has to be taken into full consideration. This eventuality is not incorporated in Eq. (26) and should be taken care of separately.

In this kinematical region the parton that carries the compensating transverse momentum can be produced off one of the external legs, as shown in Fig. 4.

The corresponding contributions to the differential 4-jet production cross section can be combined into the following relatively compact expression:

 π2dσ(3→4)2d2δ13d2δ24=dσ\scriptsize part% d^t1d^t2⋅α\rm\scriptsize s(δ2)2πδ2∑cP1,2c(x1x1+x2)S1(Q2,δ2)S2(Q2,δ2)×∂∂δ′2{S