# Diffractive Bremsstrahlung in Hadronic Collisions

Authors: Roman Pasechnik; Boris Kopeliovich; Irina Potashnikova
Journal: Advances in High Energy Physics
 (2015)

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## Abstract

Production of heavy
photons (Drell-Yan), gauge bosons, Higgs bosons, and
heavy flavors, which is treated within the QCD
parton model as a result of hard parton-parton
collision, can be considered a bremsstrahlung
process in the target rest frame. In this
review, we discuss the basic features of the
diffractive channels of these processes in the
framework of color dipole approach. The main
observation is a dramatic breakdown of
diffractive QCD factorisation due to the
interplay between soft and hard interactions,
which dominates these processes. This
observation is crucial for phenomenological
studies of diffractive reactions in high energy
hadronic collisions.


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## Body

## 1. Introduction

Diffractive production of particles in hadron-hadron scattering at high energies is one of the basic tools, experimental and theoretical, giving access to smallx and nonperturbative QCD physics. The characteristic feature of diffractive processes at high energies is the presence of a large rapidity gap between the remnants of the beam and target.The understanding of the mechanisms of inelastic diffraction came with the pioneering works of Glauber [1], Feinberg and Pomeranchuk [2], and Good and Walker [3]. Here diffraction is conventionally viewed as a shadow of inelastic processes. If the incoming plane wave contains components interacting differently with the target, the outgoing wave will have a different composition; that is, besides elastic scattering a newdiffractive state will be created resulting in a new combination of the Fock components (for a detailed review on QCD diffraction, see [4, 5]). Diffraction, which is usually a soft process, is difficult to predict from the first principles, because it involves poorly known nonperturbative effects. Therefore, diffractive reactions characterized by a hard scale deserve a special attention. It is tempting, on analogy to inclusive reactions, to expect that QCD factorization holds for such diffractive processes. Although factorization of short and long distances still holds in diffractive DIS, the fracture functions are not universal and cannot be used for other diffractive processes.Examples of breakdown of diffractive factorization are the processes of production of Drell-Yan dileptons [6, 7], gauge bosons [8], and heavy flavors [9]. Factorization turns out to be broken in all these channels in spite of presence of a hard scale given by the large masses of produced particles; it occurs due to the interplay of short- and long-range interactions.The main difficulty in formulation of a theoretical QCD-based framework for diffractive scattering is caused by the essential contamination of soft, nonperturbative interactions. For example, diffractive deep-inelastic scattering (DIS),γ∗p→q¯qp, although it is a higher twist process, is dominated by soft interactions [10]. Within the dipole approach [11] such a process looks like a linear combination of elastic scattering amplitudes for q¯q dipoles of different sizes. Although formally the process γ∗→q¯q is an off-diagonal diffraction, it does not vanish in the limit of unitarity saturation, the so-called black-disc limit. This happens because the initial and final q¯q distribution functions are not orthogonal. Similar features exhibit the contribution of higher Fock components of the photon, for example, the leading twist diffraction γ∗→q¯qg.Diffractive excitation of the beam hadron has been traditionally used as a way to measure the Pomeron-hadron total cross section [4]. This idea, extended to DIS, allows measuring the structure function of the Pomeron [12]. The next step, which might look natural, is to assume that QCD factorization holds for diffraction and to employ the extracted parton distributions in the Pomeron in order to predict the hard diffraction cross sections in hadronic collisions. However, such predictions for hard hadronic diffraction, for example, high-pT dijet production, failed by an order of magnitude [13, 14]. In this case the situation is different and more complicated; namely, factorization of small and large distances in hadronic diffraction is broken because of presence of spectator partons and due to large hadronic sizes.The cross section of diffractive production ofW boson in pp¯ collisions, measured by the CDF experiment [15, 16], was also found to be six times smaller than what was predicted relying on factorization and diffractive DIS data [17]. Besides, the phenomenological models based on diffractive factorization, which are widely discussed in the literature (see, e.g., [18, 19]), predict a significant increase of the ratio of the diffractive-to-inclusive gauge bosons production cross sections with energy. The diffractive QCD factorization in hadron collisions is, however, severely broken by the interplay of hard and soft dynamics, as was recently advocated in [7, 8], and this review is devoted to study of these important effects within the color dipole phenomenology.The processes under discussion—single diffractive Drell-Yan [6, 7], diffractive radiation of vector (Z,W±) bosons [8], diffractive heavy flavor production [9], and diffractive associated heavy flavor and Higgs boson production [20]—correspond to off-diagonal diffraction. While diagonal diffraction is enhanced by absorption effects (in fact it is a result of absorption), the off-diagonal diffractive processes are suppressed by absorption and even vanish in the limit of maximal absorption, that is, in the black-disc limit.The absorptive corrections, also known as the survival probability of rapidity gaps [21], are related to initial- and final-state interactions. Usually the survival probability is introduced into the diffractive cross section in a probabilistic way [22] and is estimated in simplified models such as eikonal, quasi-eikonal, and two-channel approximations.According to the Good-Walker basic mechanism of diffraction, the off-diagonal diffractive amplitude is a linear combination of diagonal (elastic) diffractive amplitudes of different Fock components in the projectile hadron. Thus, the absorptive corrections naturally emerge at the amplitude level as a result of mutual cancellations between different elastic amplitudes. Therefore, there is no need to introduce any additional multiplicative gap survival probability factors. Within the light-cone color dipole approach [11] a diffractive process is considered a result of elastic scattering of  q¯q dipoles of different sizes emerging in incident Fock states. The study of the diffractive Drell-Yan reaction performed in [6] has revealed importance of soft interactions with the partons spectators, which contribute on the same footing as hard perturbative ones and strongly violate QCD factorization.One of the advantages of the dipole description is the possibility to calculate directly (although in a process-dependent way) the full diffractive amplitude, which contains all the absorption corrections by employing the phenomenological universal dipole cross section (or dipole elastic amplitudes) fitted to DIS data. The gap survival amplitude can be explicitly singled out as a factor from the diffractive amplitude being a superposition of dipole scatterings at different transverse separations.Interestingly, besides interaction with the spectator projectile partons, there is another important source for diffractive factorization breaking. Even a single quark, having no spectator comovers, cannot radiate Abelian fields(γ,Z,W±,H) interacting diffractively with the target with zero-transverse-momentum transfer [23], that is, in forward direction scattering. This certainly contradicts the expectations based on diffractive factorization. In the case of a hadron beam the forward directions for the hadron and quark do not coincide, so a forward radiation is possible but is strongly suppressed (see below).Interaction with the spectator partons opens new possibilities for diffractive radiation in forward direction; namely, the transverse momenta transferred to different partons can compensate each other. It was found in [6–8] that this contribution dominates the forward diffractive Abelian radiation cross section. This mechanism leads to a dramatic violation of diffractive QCD factorization, which predicts diffraction to be a higher twist effect, while it turns out to be a leading twist effect due to the interplay between the soft and hard interactions. Although diffractive gluon radiation off a forward quark does not vanish due to possibility of glue-glue interaction, the diffractive factorization breaking in non-Abelian radiation is still important.In this review, we briefly discuss the corresponding effects whereas more details can be found in [6–9, 20].


## 2. Color Dipole Picture of Diffractive Excitation

Single diffractive scattering and production of a new (diffractive) state, that is, diffractive excitation, emerge as a consequence of quantum fluctuations in projectile hadron. The orthogonal hadron stateh can be excited due to interactions but can be decomposed over the orthogonal and complete set of eigenstates of interactions α as [11, 24, 25] (1)h=∑Cαhα,f^elα=fαα,where f^el is the elastic amplitude operator and fα is one of its eigenstates. The eigenamplitudes fα are the same for different types of hadrons. Hence, the elastic h→h and single diffractive h→h′ amplitudes can be conveniently written in terms of the elastic eigenamplitudes fα and coefficients Cαh; that is, (2)felhh=∑Cαh2fα,fsd⁡hh′=∑Cαh′∗Cαhfα,respectively, such that the forward single diffractive cross section (3)∑h′≠hdσsd⁡dtt=0=14π∑h′fsd⁡hh′2-felhh2=fα2-fα24πis given by the dispersion of the eigenvalues distribution.It was suggested in [11] that eigenstates of QCD interactions are color dipoles, such that any diffractive amplitude can be considered a superposition of universal elastic dipole amplitudes. Such dipoles experience only elastic scattering and are characterized only by transverse separation r→. The total hadron-proton cross section is then given by its eigenvalue, the universal dipole cross section (4)σr→≡∫d2b2Im⁡felb→,r→,as follows: (5)σtothp=∑Cαh2σα=∫d2rΨhr→2σr→≡σr→,where Ψh(r→) is the “hadron-to-dipole” transition wave function (incident parton momentum fractions are omitted). The dipole description of diffraction is based on the fact that dipoles of different transverse size r⊥ interact with different cross sections σ(r⊥), leading to the single inelastic diffractive scattering with a cross section, which in the forward limit is given by [11](6)σsd⁡dp⊥2p⊥=0=σ2r→-σr→216π,where p⊥ is the transverse momentum of the recoil proton, σ(r) is the universal dipole-proton cross section, and operation ⋯ means averaging over the dipole separation. For low and moderate energies, σ(r) also depends on Bjorken variable x whereas, in the high energy limit, the collision c.m. energy squared s is a more appropriate variable [23, 26]. The phenomenological dipole cross section fitted to data on inclusive DIS implicitly incorporates the effect of gluon bremsstrahlung. The latter is more important on a hard scale; this is why the small-distance dipole cross section rises faster with 1/x.Even for the simplest quark-antiquark dipole configuration, a theoretical prediction of the partial dipole amplitudefelqq¯(b→,r→) and the dipole cross section σqq¯(r→) from the first QCD principles is still a big challenge so these are rather fitted to data. The universality of the dipole scattering, however, enables us to fit known parameterizations to one set of known data (e.g., inclusive DIS) and use them for accurate predictions of other yet unknown observables (e.g., rapidity gap processes).Indeed, at small Bjorkenx in DIS the virtual photon exhibits partonic structure as shown in Figure 1. The leading order configuration, the qq¯ dipole, then elastically rescatters off the proton target p providing a phenomenological access to σqq¯(r→,x). When it comes to diffractive DIS schematically represented in Figure 2, the corresponding single diffractive cross section in the forward proton limit t→0 is given by the dipole cross section squared; that is, (7)16πdσsd⁡γ∗px,Q2dtt=0=∫d2r∫01dαΨγ∗r→,α,Q2σqq¯2r→,x,where α is the light-cone momentum fraction of the virtual photon carried by the quark. Here, the dipole size r→ is regulated by the photon light-cone wave function Ψγ∗ which can be found, for example, in [29]. The mean dipole size squared is inversely proportional to the quark energy squared: (8)r2~1ϵ2=1Q2α1-α+mq2.The dipole size is assumed to be preserved during scattering in the high energy limit.Figure 1
The DIS cross section via phenomenological Pomeron exchange (a) and a perturbative QCD ladder (b). At smallx the virtual photon fluctuates into qq¯ dipole and more complicated Fock states which then interact with the hadronic target.
(a)
(b)Figure 2
The diffractive DIS cross section via double ladder exchange.Hard and soft hadronic fluctuations have smallr2~1/Q2 (nearly symmetric α≫mq2/Q2 configuration) and large r2~1/mq2, mq~ΛQCD (aligned jet α~mq2/Q2 configuration) sizes, respectively. Remarkably enough, soft fluctuations play a dominant role in diffractive DIS in variance with inclusive DIS [10]. Although such soft fluctuations are very rare, their interactions with the target occur with a large cross section σ~1/mq2 which largely compensate their small ~mq2/Q2 weights. On the other hand, abundant hard fluctuations with nearly symmetric small-size dipoles r2~1/Q2 have vanishing (as 1/Q2) cross section. It turns out that, in inclusive DIS, both hard and soft contributions to the total cross section behave as 1/Q2 (semihard and semisoft), while in diffractive DIS the soft fluctuations ~1/mq2Q2 dominate over the hard ones ~1/Q4. This also explains why the ratio σsd⁡/σinc in DIS is nearly Q2 independent as well as a higher twist nature of the diffractive DIS.The main ingredient of the dipole approach is the phenomenological dipole cross section, which is parameterized in the saturated form [26],(9)σq¯qr,x=σ01-e-rp2/R02x,and fitted to DIS data. Here, x is the Bjorken variable, σ0=23.03mb, and R0(x)=0.4fm×(x/x0)0.144 and x0=0.003. In pp collisions x is identified with gluon x2=M2/x1s≪1, where M is the invariant mass of the produced system and s is the pp c.m. energy. This parametrization, though oversimplified (compare with [30]), is rather successful in description of experimental HERA data with a reasonable accuracy.In soft processes, however, the Bjorken variablex makes no sense, and gluon-target collision c.m. energy squared s^=x1s (s is the pp c.m. energy) is a more appropriate variable, while the saturated form (9) should be retained [23]. The corresponding parameterizations for σ0=σ0(s^) and R0=R0(s^) read as follows:(10)R0s^=0.88fms0s^0.14,σ0s^=σtotπps^1+3R02s^8rch2π,where the pion-proton total cross section is parametrized as [31] σtotπp(s^)=23.6(s^/s0)0.08 mb, s0=1000GeV2; the mean pion radius squared is [32] rch2π=0.44 fm2. An explicit analytic form of the x- and s^-dependent parameterizations for the elastic amplitude fel(b→,r→) accounting for information about the dipole orientation with respect to the color background field (i.e., the angular dependence between r→ and b→) can be found in [33–35].Ansatz (9) incorporates such important phenomenon as saturation at a soft scale since it levels off at r≫R0. Another important feature is vanishing of the cross section at small r→0 as σqq¯∝r2 [11]. This is a general property called color transparency which reflects the fact that a point-like colorless object does not interact with external color fields. Finally, the quadratic r-dependence is an immediate consequence of gauge invariance and nonabeliance of interactions in QCD.


## 3. Diffractive Abelian Radiation: Regge versus Dipole Approach

### 3.1. Diffractive Factorization

The cross section of the inclusive Drell-Yan (DY) is expressed via the dipole cross section in a way similar to DIS [36]: (11)dσDYqp⟶γ∗XdαdM2=∫d2rΨqγ∗r→,α2σqq¯αr→,where α is the light-cone momentum fraction carried by the heavy photon off the parent quark. QCD factorization relates inclusive DIS with DY, and similarity between these processes is the source of universality of the hadron PDFs.Now, consider the forward single diffractive Drell-Yan (DDY) and vector bosons productionG=Z, W± in pp collisions which is characterized by a relatively small momentum transfer between the colliding protons. In particular, one of the protons, for example, p1, radiates a hard virtual gauge G∗ boson with k2=M2≫mp2 and hadronizes into a hadronic system X both moving in forward direction and separated by a large rapidity gap from the second proton p2, which remains intact. In the DDY case,(12)p1+p2⟶X+gap⁡+p2,X≡γ∗l+l-+Y.Both the dilepton and X, the debris of p1, stay in the forward fragmentation region. In this case, the virtual photon is predominantly emitted by the valence quarks of proton p1.In some of the previous studies [18, 37] of the single diffractive Drell-Yan reaction the analysis was made within the phenomenological Pomeron-Pomeron and γ-Pomeron fusion mechanisms using the Ingelman-Schlein approach [12] based on diffractive factorization. In analogy to the proven collinear factorization [38] for inclusive processes, one assumes factorization of short and long distances in diffractive processes characterized by a hard scale. Besides one assumes that the soft part of the interaction is carried out by Pomeron exchange, which is universal for different diffractive processes; that is, Regge factorization is assumed as well. That could be true if the Pomeron were a true Regge pole, which is not supported by any known dynamical model. The above two assumptions lead to the following form of the diffractive DY cross section [37, 39] expressed in terms of the Pomeron PDFs Fq¯/P: (13)σsd⁡DY=GP/p⊗Fq¯/P⊗Fq/p⊗σ^qq¯⟶l¯l.The diffractive factorization leads to specific features of the differential DY cross sections similar to those in diffractive DIS process, for example, a slow increase of the diffractive-to-inclusive DY cross sections ratio with c.m.s. energy s, its practical independence on the hard scale, and the invariant mass of the lepton pair squared, M2 [18].However, presence of spectator partons in hadronic collisions leads to a dramatic breakdown of diffractive factorization of short and long distances. Contrary to inclusive processes, where spectator partons do not participate in the hard reactions in leading order, below we demonstrate that in diffraction the spectator partons do participate on a soft scale; that is, their contribution is enhanced byQ2/Λ2. In particular, the spectator partons generate large absorptive corrections, usually called rapidity gap survival probability, which cause a strong suppression of the diffractive cross section compared with (13).One can derive Regge behavior of the diffractive cross section of heavy photon production in terms of the usual light-cone variables:(14)x1=pγ+p1+;x2=pγ-p2-,so that x1x2=(M2+kT2)/s and x1-x2=xF, where M, kT, and xF are the invariant mass, transverse momentum, and Feynman variable of the heavy photon (dilepton).In the limit of smallx1→0 and large zp≡p4+/p2+→1 the diffractive DY cross section is given by the Mueller graph shown in Figure 4. In this case, the end-point behavior is dictated by the following general result:(15)dσdzpdx1dtt→0∝11-zp2αPt-1x1ε,where αP(t) is the Pomeron trajectory corresponding to the t-channel exchange and ε is equal to 1 or 1/2 for the Pomeron P or Reggeon R exchange corresponding to γ∗ emission from sea or valence quarks, respectively. Thus, the diffractive Abelian radiation process pp→(X→G∗+Y)p at large Feynman xF→1, or small (16)ξ=1-xF=MX2s≪1,is described by triple-Regge graphs in Figure 5 where we also explicitly included radiation of a virtual gauge boson G∗. The Feynman graphs, corresponding to the corresponding triple-Regge terms, are shown in Figure 5 (second and third rows). The (ba) and (ca) diagrams illustrate the 3-Pomeron term; that is, (17)dσdiffPPPdξdt∝ξ-αP0-2αP′t.It is worth to mention that the smallness of the triple-Pomeron coupling is related to the known shortness of gluon correlation length. The amplitude q+g→q+G is given by open circles as in Figure 3. So the 3-Pomeron term is interpreted as an excitation of the projectile proton due to the gluon radiation. The diffractive valence quark excitation is shown in Figure 5, graphs (bb) and (cb), and contributes to (18)dσdiffPPRdξdt∝ξαR0-αP0-2αP′t.Figure 3
Gauge boson radiation by a projectile quark in the target rest frame.Figure 4
The diffractive DY cross section summed over excitation channels at fixed effective massMX (a). The latter corresponds to the Mueller graph in Regge picture (b).
(a)
(b)Figure 5
The upper row contains the triple-Regge graphs forpp→(XG∗)+p. A few key examples of diagrams for diffractive excitation of a large invariant mass are given by 2nd and 3rd rows.


### 3.2. Diffractive Factorization Breaking in Forward Diffraction

As an alternative to the diffractive factorization based approach, the dipole description of the QCD diffraction was presented in [11] (see also [40]). The color dipole description of inclusive Drell-Yan process was first introduced in [41] (see also [36, 42]) and treats the production of a heavy virtual photon via bremsstrahlung mechanism rather than q¯q annihilation. The dipole approach applied to diffractive DY reaction in [6, 7] and later in diffractive vector boson production [8] has explicitly demonstrated the diffractive factorization breaking in diffractive Abelian radiation reactions.It is worth emphasizing that the quark radiating the gauge boson cannot be a spectator but must participate in the interaction. This is a straightforward consequence of the Good-Walker mechanism of diffraction [3]. According to this picture, diffraction vanishes if all Fock components of the hadron interact with the same elastic amplitudes. Then an unchanged Fock state composition emerges from the interaction; that is, the outgoing hadron is the same as the incoming one, so the interaction is elastic.For illustration, consider diffractive photon radiation off a quark [23]. The relevant contributions and kinematics of the process are schematically presented in Figure 6 where the Pomeron exchange is depicted as an effective two-gluon (BFKL) ladder. The corresponding framework has previously been used for diffractive gluon radiation and diffractive DIS processes in [23, 43, 44] and we adopt similar notations in what follows. Applying the generalized optical theorem in the high energy limit with a cut between the “screening” and “active” gluon as shown by dashed line in Figure 6, we get(19)
A
^
SD
=
i
2
∑
Y
8
∗
A
^
†
q
γ
p
⟶
q
Y
8
∗
A
^
q
p
⟶
q
Y
8
∗
+
A
^
†
q
γ
p
⟶
q
γ
Y
8
∗
A
^
q
p
⟶
q
γ
Y
8
∗
,with summation going through all octet-changed intermediate states {Y8∗}. In (19), the first and second terms correspond to the first and second diagrams in Figure 6. Then we switch to impact parameter representation: (20)A^b→,r→=12π4∫d2q→⊥d2κ→A^q→⊥,κ→e-iq→⊥·b→-iκ→·r→,κ→=αp→2-1-αp→,where q→⊥, p→2, and p→ are the transverse momenta of the Pomeron, final quark, and the radiated photon γ∗, α is the longitudinal momentum fraction of the photon taken off the parent quark p1, and κ is the relative transverse momentum between the final quark and γ∗. Thus, the amplitude of the “screening” gluon exchange summed over projectile valence quarks j=1,2,3 reads as follows: (21)A^qp⟶qY8∗=∑aτafγ^ab→1i,A^qγp⟶qγY8∗=∑aτafγ^ab→2i,A^qp⟶qγY8∗=A^qγp⟶qγY8∗=∑aτafγ^ab→1i-fγ^ab→2i·Ψq→qγr→,α,where b→1≡b→ and b2≡b→-αr→ are the impact parameter of the quark before and after photon radiation, r→ is the transverse separation between the quark and the radiated photon, α is the momentum fraction taken by the photon, Ψq→qγ is the distribution function for the qγ fluctuation of the quark, λa=2τa are the Gell-Mann matrices from a gluon coupling to the quark, and matrices γ^a are the operators in coordinate and color space for the target quarks:(22)γ^aR→=∑iτaiχR→-s→i,χs→≡1π∫d2qαsqeiq→·s→q2+Λ2,which depend on the effective gluon mass, Λ~100MeV, and on the transverse distance between ith valence quark in the target nucleon and its center of gravity, s→i.Figure 6
Schematic illustration of the typical contributions and kinematics of the diffractive Drell-Yan process in the quark-target collision.Combining these ingredients into the diffractive amplitude (19) one should average over color indices of the valence quarks and their relative coordinates in the target nucleon 3q1. The color averaging results in(23)
τ
a
j
·
τ
a
′
j
′
3
q
1
=
1
6
δ
a
a
′
:
j
=
j
′
-
1
12
δ
a
a
′
:
j
≠
j
′
.Finally, averaging over quark relative coordinates s→i leads to(24)iγ^ab→kγ^a′b→li3q1=34δaa′Sb→k,b→l,where S(b→k,b→l) is a scalar function, which can be expressed in terms of the quark-target scattering amplitude χ(r→) and the proton wave function [23]. Then, the total amplitude is(25)A^b→,r→∝Sb→,b→-Sb→-αr→,b→-αr→.After Fourier transform one notices that in the forward quark limit q⊥→0 the amplitude for single diffractive photon or any Abelian radiation vanishes, Aq→⊥,κq⊥→0=0, and(26)dσsd⁡DYdαdqT2qT=0=0,in accordance with the Landau-Pomeranchuk principle. Indeed, in both Fock components of the quark, q and qγ∗, only the quark interacts, so these components interact equally and thus no diffraction is possible. One immediately concludes that the diffractive factorization must be strongly broken.The functionS(b→k,b→l) above is directly related to qq¯ dipole cross section as (27)
σ
q
¯
q
r
→
1
-
r
→
2
≡
∫
d
2
b
S
b
→
+
r
→
1
,
b
→
+
r
→
1
+
S
b
→
+
r
→
2
,
b
→
+
r
→
2
-
2
S
b
→
+
r
→
1
,
b
→
+
r
→
2
.Thus, following analogical scheme one can obtain the diffractive amplitude of any diffractive process as a linear combination of the dipole cross sections for different dipole separations. As was anticipated, the diffractive amplitude represents the destructive interference effect from scattering of dipoles of slightly different sizes. Such an interference results in an interplay between hard and soft fluctuations in the diffractive pp amplitude, contributing to breakdown of diffractive factorization.When one considers diffractive DY off a finite-size object like a proton, in both Fock components,3q and 3qγ∗, only the quark hadron-scale dipoles interact. These dipoles are large due to soft intrinsic motion of quarks in the projectile proton wave function. The dipoles, however, have different sizes, since the recoil quark gets a shift in impact parameters. So the dipoles interact differently giving rise to forward diffraction. The contribution of a given projectile Fock state to the diffraction amplitude is given by the difference of elastic amplitudes for the Fock states including and excluding the gauge boson: (28)Im⁡fdiffn=Im⁡feln+G-Im⁡feln,where n is the total number of partons in the Fock state and fel(n+G) and fel(n) are the elastic scattering amplitudes for the whole n-parton ensemble, which either contains the gauge boson or does not, respectively. Although the gauge boson does not participate in the interaction, the impact parameter of the quark radiating the boson gets shifted, and this is the only reason why the difference (28) is not zero (see the next section). This also conveys that this quark must interact in order to retain the diffractive amplitude nonzero [6, 7]. For this reason in the graphs depicted in Figure 5 the quark radiating G always takes part in the interaction with the target.Notice that there is no one-to-one correspondence between diffraction in QCD and the triple-Regge phenomenology. In particular, there is no triple-Pomeron vertex localized in rapidity. The colorless “Pomeron” contains at least twot-channel gluons, which can couple to any pair of projectile partons. For instance, in diffractive gluon radiation, which is the lowest order term in the triple-Pomeron graph, one of the t-channel gluons can couple to the radiated gluon, while another one couples to another parton at any rapidity, for example, to a valence quark (see Figure 3 in [23]). Apparently, such a contribution cannot be associated literally with either of the Regge graphs in Figure 5. Nevertheless, this does not affect much xF and energy dependencies provided by the triple-Regge graphs, because the gluon has spin one.It is also worth mentioning that in Figure5 we presented only the lowest order graphs with two-gluon exchange. The spectator partons in a multiparton Fock component also can interact and contribute to the elastic amplitude of the whole parton ensemble. This gives rise to higher order terms, not shown explicitly in Figure 5. They contribute to the diffractive amplitude (28) as a factor, which we define as the gap survival amplitude [8].As was mentioned above the amplitude of diffractive gauge boson radiation by a quark-antiquark dipole does not vanish in forward direction, unlike the radiation by a single quark [6, 23]. This can be understood as follows. According to the general theory of diffraction [1–3, 5], the off-diagonal diffractive channels are possible only if different Fock components of the projectile (eigenstates of interaction) interact with different elastic amplitudes. Clearly, the two Fock states consisting of just a quark and of a quark plus a gauge boson interact equally, if their elastic amplitudes are integrated over impact parameter. Indeed, when a quark fluctuates into a state qG containing the gauge boson G, with the transverse quark-boson separation r→, the quark gets a transverse shift Δr→=αr→. The impact parameter integration gives the forward amplitude. Both Fock states q and qG interact with the target with the same total cross section; this is why a quark cannot radiate at zero-momentum transfer and hence G is not produced diffractively in the forward direction. This is the general and model independent statement. The details of this general consideration can be found in [23] (Appendices A 1 and A 4). The same result is obtained calculating Feynman graphs in Appendix B 4 of the same paper. Unimportance of radiation between two interactions was also demonstrated by Brodsky and Hoyer in [45].Note that in all these calculations one assumes that the coherence time of radiation considerably exceeds the time interval between the two interactions, which is fulfilled in our case, since we consider radiation at forward rapidities. The disappearance of both inelastic and diffractive forward Abelian radiation has a direct analogy in QED: if the electric charge gets no “kick,” that is, not accelerated, no photon is radiated, provided that the radiation time considerably exceeds the duration time of interaction. This is dictated by the renown Landau-Pomeranchuk principle [46]: radiation depends on the strength of the accumulated kick, rather than on its structure, if the time scale of the kick is shorter than the radiation time. It is worthy to notice that the non-Abelian QCD case is different: a quark can radiate gluons diffractively in the forward direction. This happens due to a possibility of interaction between the radiated gluon and the target. Such a process, in particular, becomes important in diffractive heavy flavor production [9].The situation changes if the gauge boson is radiated diffractively by a dipole as shown in Figure7. Then the quark dipoles with or without a gauge boson have different sizes and interact with the target differently. So the amplitude of the diffractive gauge boson radiation from qq¯ dipole is proportional to the difference between elastic amplitudes of the two Fock components, |qq¯〉 and |qq¯G〉 [6]; that is,(29)Mq¯qdiffb→,R→,r→,α∝Ψq→G∗qα,r→·2Im⁡felb→,R→-2Im⁡felb→,R→+αr→,where Ψq→G∗q is the light-cone (nonnormalised) wave function of q→G∗q fluctuation corresponding to bremsstrahlung of virtual gauge bosons G=γ,Z,W± of mass M [8], R→=r→1-r→2 is the transverse size of the qq¯ dipole, α is the momentum fraction of the gauge boson G taken off the parent quark q, and r~1/M is the hard scale.Figure 7
Leading order contribution to the diffractive Drell-Yan in the dipole-target collision.When applied to diffractivepp scattering the diffractive amplitude (29), thus, occurs to be sensitive to the large transverse separations between the projectile quarks in the incoming proton. Normally, transition to the hadron level is achieved by using the initial proton Ψi and remnant Ψf wave functions which encode information about distributions of constituents. The completeness relation to the wave function of the unobservable proton remnant is(30)∑finΨfinr→1,r→2,r→3;xq1,2,…,xg1,2,…·Ψfin∗r→′1,r→′2,r→′3;xq′1,2,…,xg′1,2,…=δr→1-r→′1·δr→2-r→′2δr→3-r→′3∏jδxq/gj-x′q/gj.Here, r→q/gi and xq/gi are the coordinates and fractional momenta of the valence/sea partons.Since gluons and sea quarks are mostly accumulated in a close vicinity of valence quarks (inside gluonic “spots”), to a reasonable accuracy the transverse positions of sea quarks and gluons can be identified with the coordinates of valence quarks. The valence part of the wave function is often taken to be a Gaussian distribution such that(31)Ψin2=3a2π2e-ar12+r22+r32Rx1,xq1,2,…,xg2,3,…·δr→1+r→2+r→3δ1-x1-∑jxq/gj,where all the partons not participating in the hard interaction are summed up, x1 is the photon fraction taken from the initial proton, a=rch2-1 is the inverse proton mean charge radius squared, and R is a collinear multiparton distribution in the proton. Once the latter is integrated over all the partons not participating in the hard interaction, one gets a conventional collinear PDF g(x1,μ2) for gluons and q(x1,μ2) for a given quark flavor q. Since the diffractive pp cross section appears as a sum of diffractive excitations of the proton constituents, valence/sea quarks and gluons are incorporated as (32)Ψinr→i,xi2∝13∑qqx+q¯x+8116gx,after integration over spectator impact parameters and momentum fractions with a proper color factor between quark and gluon PDFs. Note that only sea and valence quarks are excited by the photon radiation in the diffractive DY process which provide a direct access to the proton structure function in the soft limit of large x [36]:(33)∑qZq2qx+q¯x=1xF2x.For diffractive gluon radiation one should account for both quark and gluon excitations whose amplitudes, however, are calculated in different ways [23].Due to the internal transverse motion of the projectile valence quarks inside the incoming proton, which corresponds to finite large transverse separations between them, the forward photon radiation does not vanish [6, 8]. These large distances are controlled by a nonperturbative (hadron) scale R→, such that the diffractive amplitude has the Good-Walker structure:(34)Mq¯qdiff∝σR→-σR→-αr→∝r→·R→,while the single diffractive-to-inclusive cross sections ratio behaves as (35)σsd⁡DYσinclDY∝exp⁡-2R2/R02x2R02x2assuming the saturated GBW shape of the dipole cross section (9), where x2 is defined in (14). Thus, the soft part of the interaction is not enhanced in Drell-Yan diffraction which is semihard/semisoft like inclusive DIS. Linear dependence on the hard scale r~1/M≪R0(x2) means that even at a hard scale the Abelian radiation is sensitive to the hadron size due to a dramatic breakdown of diffractive factorization [37]. It was firstly found in [47, 48] that factorization for diffractive Drell-Yan reaction fails due to the presence of spectator partons in the Pomeron. In [6–8] it was demonstrated that factorization in diffractive Abelian radiation is thus even more broken due to presence of spectator partons in the colliding hadrons as reflected in (34).The effect of diffractive factorization breaking manifests itself in specific features of observables like a significant damping of the cross section at highs compared to the inclusive production case as illustrated in Figure 8. This is rather unusual, since a diffractive cross section, which is proportional to the dipole cross section squared, could be expected to rise with energy steeper than the total inclusive cross section, like it occurs in the diffractive DIS process. At the same time, the ratio of DDY to DY cross sections was found in [6, 7] to rise with the hard scale, and the photon virtuality M2 was also shown in Figure 8. This is also in variance with diffraction in DIS, which is associated with the soft interactions and where the diffractive factorization holds true [10].Figure 8
The single diffractive-to-inclusive DY cross sections ratio as a function of the photon virtualityM2 for x1=0.5 (solid lines) and x1=0.9 (dashed lines) and c.m.s. energies s=40GeV, 500GeV, and 14TeV (from top to bottom) [6].Such striking signatures of the diffractive factorization breaking are due to an interplay of soft and hard interactions in the corresponding diffractive amplitude. Namely, large and small size projectile fluctuations contribute to the diffractive Abelian radiation process on the same footing providing the leading twist nature of the process, whereas diffractive DIS dominated by soft fluctuations only is of higher twist [6, 7]. But this is not the only source of the factorization breaking; another important source is the absorptive (or unitarity) corrections.


### 3.3. Gap Survival Amplitude

In the limit of unitarity saturation (the so-called black-disk limit) the absorptive corrections can entirely terminate the large rapidity gap process. The situation close to this limit, in fact, happens in high energy (anti)proton-proton collisions such that unitarity is nearly saturated at small impact parameters [49, 50]. The unitarity corrections are typically parameterized by a suppression factor also known as the soft survival probability which significantly reduce the diffractive cross section. In hadronic collisions this probability is controlled by the soft spectator partons which are absent in the case of diffractive DIS causing the breakdown of diffractive factorization.It is well known that the absorptive corrections affect differently the diagonal and off-diagonal terms in the hadronic current [5, 51], in opposite directions, leading to an additional source of the QCD factorization breaking in processes with off-diagonal contributions only. Namely, the absorptive corrections enhance the diagonal terms at larger s, whereas they strongly suppress the off-diagonal ones. In the diffractive DY process a new state, the heavy lepton pair, is produced; hence, the whole process is of entirely off-diagonal nature, whereas the diffractive DIS process contains both diagonal and off-diagonal contributions [5].Amplitude (29) is the full expression, which includes by default the effect of absorption and does not need any extra survival probability factor [8]. This can be illustrated in a simple example of elastic dipole scattering off a potential. In this case, the dipole elastic amplitude has the eikonal form:(36)Im⁡felb→,r→1-r→2=1-exp⁡iχr→1-iχr→2,where(37)χb=-∫-∞∞dzVb→,zand V(b→,z) is the potential, which depends on the impact parameter and longitudinal coordinate, and is nearly imaginary at high energies. The difference between elastic amplitudes with a shifted quark position, which enters the diffractive amplitude, reads as follows:(38)Im⁡felb→,r→1-r→2+αr→-Im⁡felb→,r→1-r→2≃exp⁡iχr→1-iχr→2exp⁡iαr→·∇→χr→1.Here, the first factor exp⁡[iχ(r→1)-iχ(r→2)] is exactly the survival probability amplitude, which vanishes in the black disc limit, as it should do. This proves that the diffractive amplitude (29) includes the effect of all absorptive corrections (gap survival amplitude), provided that the dipole cross section is adjusted to the data. Note that usually the survival probability factor is introduced into the diffractive cross section probabilistically, while in (29) it is treated quantum-mechanically, at the amplitude level.The diffractive gluon radiation is known to be rather weak (the 3-Pomeron coupling is small). This phenomenological observation can be explained assuming that gluons in the proton are predominantly located inside small “gluonic spots” of sizer0~0.3fm around the valence quarks (see, e.g., [23, 52–54]). The smallness of gluonic dipole is an important nonperturbative phenomenon which may be connected, for example, to the small size of gluonic fluctuations in the instanton liquid model [53]. Therefore, a distance between a valence quark and a gluon in a vicinity of another quark can be safely approximated by the quark-quark separation.Besides the soft gluons in the proton light-cone wave function, virtual gauge boson production triggers intensive gluon radiation such that there are many more spectator gluons in a vicinity of the quark which radiates the gauge boson. The separations of such gluons from the parent quark are controlled by the QCD DGLAP dynamics. In practice, one may replace such a set of gluons by dipoles [55] whose transverse sizes rd vary between 1/MG and r0 scales [56]. Then the mean dipole size is regulated by a relation: (39)rd2=r02ln⁡r02MG2,leading to 〈rd2〉≈0.01fm2, which means that it is rather small and the corresponding dipole cross section σ≃C(x)rd2, where C(x)=σ0/R02(x) rises with energy, is suppressed. For x=MG2/s and naive GBW parameterization [26] we get σ≈0.9mb at the Tevatron energy. Each such small dipole brings up an extra suppression factor to the large rapidity gap survival amplitude given by (40)Sds=1-Im⁡fdb,rd.Here, the second term is small and thus is simplified to (for more details, see [57]) (41)Im⁡fdb,rd≈σd4πBde-b2/2Bd,where Bd is the standard dipole-nucleon elastic slope Bd≈6GeV-2 measured earlier in diffractive ρ electroproduction at HERA [58]. At the mean impact parameter given by 〈b2〉=2Bd and for the Tevatron energy s=2TeV we arrive at negligibly small value for the absorptive correction (41): Im⁡fd(0,rd)≈0.01.On the other hand, the overall number of such dipoles increases with hardness of the process, which can amplify the magnitude of the absorptive effect. Generalizing the gap survival amplitude tond projectile dipoles, we obtain (42)Sdnd=1-Im⁡fdb,rdnd. The DGLAP evolution formulated in impact parameter representation [56] enables estimating that the mean number of such dipoles can be estimated in the double-leading-log approximation: (43)nd=12β0ln⁡1αsMG2ln⁡1-xFss0. Here, the typical Bjorken x values of the radiated gluons are restricted by the diffractive mass as x>s0/MX2=s0/(1-xF)s. In typical kinematics at the Tevatron collider, the mean number of such dipoles is roughly nd≲6. The amplitude of survival of a large rapidity gap is controlled by the largest dipoles in the projectile hadron only, such that the first exponential factor in (38) provides a sufficiently good approximation to the gap survival amplitude. The absorptive corrections (42) to the gap survival amplitude are proven to be rather weak and do not exceed 5% (or 10% in the survival probability factor) which is small compared to an overall theoretical uncertainty. For the pioneering work on hard rescattering corrections to the gap survival factor see [59].The popular quasi-eikonal model for the so-called “enhanced” probabilityS^enh (see, e.g., [21, 59]), frequently used to describe the factorization breaking in diffractive processes, is not well justified in higher orders, whereas the color dipole approach, considered here, correctly includes all diffraction excitations to all orders [5]. Such effects are included into the phenomenological parameterizations for the partial elastic dipole amplitude fitted to data. This allows predicting the diffractive gauge bosons production cross sections in terms of a single parameterization for the universal dipole cross section (or, equivalently, the elastic dipole amplitude) known independently from the soft hadron scattering data.For more details on derivations of diffractive gauge boson production amplitudes and cross sections see [7, 8]. Now we turn to a discussion of numerical results for the most important observables.


## 3.1. Diffractive Factorization

The cross section of the inclusive Drell-Yan (DY) is expressed via the dipole cross section in a way similar to DIS [36]: (11)dσDYqp⟶γ∗XdαdM2=∫d2rΨqγ∗r→,α2σqq¯αr→,where α is the light-cone momentum fraction carried by the heavy photon off the parent quark. QCD factorization relates inclusive DIS with DY, and similarity between these processes is the source of universality of the hadron PDFs.Now, consider the forward single diffractive Drell-Yan (DDY) and vector bosons productionG=Z, W± in pp collisions which is characterized by a relatively small momentum transfer between the colliding protons. In particular, one of the protons, for example, p1, radiates a hard virtual gauge G∗ boson with k2=M2≫mp2 and hadronizes into a hadronic system X both moving in forward direction and separated by a large rapidity gap from the second proton p2, which remains intact. In the DDY case,(12)p1+p2⟶X+gap⁡+p2,X≡γ∗l+l-+Y.Both the dilepton and X, the debris of p1, stay in the forward fragmentation region. In this case, the virtual photon is predominantly emitted by the valence quarks of proton p1.In some of the previous studies [18, 37] of the single diffractive Drell-Yan reaction the analysis was made within the phenomenological Pomeron-Pomeron and γ-Pomeron fusion mechanisms using the Ingelman-Schlein approach [12] based on diffractive factorization. In analogy to the proven collinear factorization [38] for inclusive processes, one assumes factorization of short and long distances in diffractive processes characterized by a hard scale. Besides one assumes that the soft part of the interaction is carried out by Pomeron exchange, which is universal for different diffractive processes; that is, Regge factorization is assumed as well. That could be true if the Pomeron were a true Regge pole, which is not supported by any known dynamical model. The above two assumptions lead to the following form of the diffractive DY cross section [37, 39] expressed in terms of the Pomeron PDFs Fq¯/P: (13)σsd⁡DY=GP/p⊗Fq¯/P⊗Fq/p⊗σ^qq¯⟶l¯l.The diffractive factorization leads to specific features of the differential DY cross sections similar to those in diffractive DIS process, for example, a slow increase of the diffractive-to-inclusive DY cross sections ratio with c.m.s. energy s, its practical independence on the hard scale, and the invariant mass of the lepton pair squared, M2 [18].However, presence of spectator partons in hadronic collisions leads to a dramatic breakdown of diffractive factorization of short and long distances. Contrary to inclusive processes, where spectator partons do not participate in the hard reactions in leading order, below we demonstrate that in diffraction the spectator partons do participate on a soft scale; that is, their contribution is enhanced byQ2/Λ2. In particular, the spectator partons generate large absorptive corrections, usually called rapidity gap survival probability, which cause a strong suppression of the diffractive cross section compared with (13).One can derive Regge behavior of the diffractive cross section of heavy photon production in terms of the usual light-cone variables:(14)x1=pγ+p1+;x2=pγ-p2-,so that x1x2=(M2+kT2)/s and x1-x2=xF, where M, kT, and xF are the invariant mass, transverse momentum, and Feynman variable of the heavy photon (dilepton).In the limit of smallx1→0 and large zp≡p4+/p2+→1 the diffractive DY cross section is given by the Mueller graph shown in Figure 4. In this case, the end-point behavior is dictated by the following general result:(15)dσdzpdx1dtt→0∝11-zp2αPt-1x1ε,where αP(t) is the Pomeron trajectory corresponding to the t-channel exchange and ε is equal to 1 or 1/2 for the Pomeron P or Reggeon R exchange corresponding to γ∗ emission from sea or valence quarks, respectively. Thus, the diffractive Abelian radiation process pp→(X→G∗+Y)p at large Feynman xF→1, or small (16)ξ=1-xF=MX2s≪1,is described by triple-Regge graphs in Figure 5 where we also explicitly included radiation of a virtual gauge boson G∗. The Feynman graphs, corresponding to the corresponding triple-Regge terms, are shown in Figure 5 (second and third rows). The (ba) and (ca) diagrams illustrate the 3-Pomeron term; that is, (17)dσdiffPPPdξdt∝ξ-αP0-2αP′t.It is worth to mention that the smallness of the triple-Pomeron coupling is related to the known shortness of gluon correlation length. The amplitude q+g→q+G is given by open circles as in Figure 3. So the 3-Pomeron term is interpreted as an excitation of the projectile proton due to the gluon radiation. The diffractive valence quark excitation is shown in Figure 5, graphs (bb) and (cb), and contributes to (18)dσdiffPPRdξdt∝ξαR0-αP0-2αP′t.Figure 3
Gauge boson radiation by a projectile quark in the target rest frame.Figure 4
The diffractive DY cross section summed over excitation channels at fixed effective massMX (a). The latter corresponds to the Mueller graph in Regge picture (b).
(a)
(b)Figure 5
The upper row contains the triple-Regge graphs forpp→(XG∗)+p. A few key examples of diagrams for diffractive excitation of a large invariant mass are given by 2nd and 3rd rows.


## 3.2. Diffractive Factorization Breaking in Forward Diffraction

As an alternative to the diffractive factorization based approach, the dipole description of the QCD diffraction was presented in [11] (see also [40]). The color dipole description of inclusive Drell-Yan process was first introduced in [41] (see also [36, 42]) and treats the production of a heavy virtual photon via bremsstrahlung mechanism rather than q¯q annihilation. The dipole approach applied to diffractive DY reaction in [6, 7] and later in diffractive vector boson production [8] has explicitly demonstrated the diffractive factorization breaking in diffractive Abelian radiation reactions.It is worth emphasizing that the quark radiating the gauge boson cannot be a spectator but must participate in the interaction. This is a straightforward consequence of the Good-Walker mechanism of diffraction [3]. According to this picture, diffraction vanishes if all Fock components of the hadron interact with the same elastic amplitudes. Then an unchanged Fock state composition emerges from the interaction; that is, the outgoing hadron is the same as the incoming one, so the interaction is elastic.For illustration, consider diffractive photon radiation off a quark [23]. The relevant contributions and kinematics of the process are schematically presented in Figure 6 where the Pomeron exchange is depicted as an effective two-gluon (BFKL) ladder. The corresponding framework has previously been used for diffractive gluon radiation and diffractive DIS processes in [23, 43, 44] and we adopt similar notations in what follows. Applying the generalized optical theorem in the high energy limit with a cut between the “screening” and “active” gluon as shown by dashed line in Figure 6, we get(19)
A
^
SD
=
i
2
∑
Y
8
∗
A
^
†
q
γ
p
⟶
q
Y
8
∗
A
^
q
p
⟶
q
Y
8
∗
+
A
^
†
q
γ
p
⟶
q
γ
Y
8
∗
A
^
q
p
⟶
q
γ
Y
8
∗
,with summation going through all octet-changed intermediate states {Y8∗}. In (19), the first and second terms correspond to the first and second diagrams in Figure 6. Then we switch to impact parameter representation: (20)A^b→,r→=12π4∫d2q→⊥d2κ→A^q→⊥,κ→e-iq→⊥·b→-iκ→·r→,κ→=αp→2-1-αp→,where q→⊥, p→2, and p→ are the transverse momenta of the Pomeron, final quark, and the radiated photon γ∗, α is the longitudinal momentum fraction of the photon taken off the parent quark p1, and κ is the relative transverse momentum between the final quark and γ∗. Thus, the amplitude of the “screening” gluon exchange summed over projectile valence quarks j=1,2,3 reads as follows: (21)A^qp⟶qY8∗=∑aτafγ^ab→1i,A^qγp⟶qγY8∗=∑aτafγ^ab→2i,A^qp⟶qγY8∗=A^qγp⟶qγY8∗=∑aτafγ^ab→1i-fγ^ab→2i·Ψq→qγr→,α,where b→1≡b→ and b2≡b→-αr→ are the impact parameter of the quark before and after photon radiation, r→ is the transverse separation between the quark and the radiated photon, α is the momentum fraction taken by the photon, Ψq→qγ is the distribution function for the qγ fluctuation of the quark, λa=2τa are the Gell-Mann matrices from a gluon coupling to the quark, and matrices γ^a are the operators in coordinate and color space for the target quarks:(22)γ^aR→=∑iτaiχR→-s→i,χs→≡1π∫d2qαsqeiq→·s→q2+Λ2,which depend on the effective gluon mass, Λ~100MeV, and on the transverse distance between ith valence quark in the target nucleon and its center of gravity, s→i.Figure 6
Schematic illustration of the typical contributions and kinematics of the diffractive Drell-Yan process in the quark-target collision.Combining these ingredients into the diffractive amplitude (19) one should average over color indices of the valence quarks and their relative coordinates in the target nucleon 3q1. The color averaging results in(23)
τ
a
j
·
τ
a
′
j
′
3
q
1
=
1
6
δ
a
a
′
:
j
=
j
′
-
1
12
δ
a
a
′
:
j
≠
j
′
.Finally, averaging over quark relative coordinates s→i leads to(24)iγ^ab→kγ^a′b→li3q1=34δaa′Sb→k,b→l,where S(b→k,b→l) is a scalar function, which can be expressed in terms of the quark-target scattering amplitude χ(r→) and the proton wave function [23]. Then, the total amplitude is(25)A^b→,r→∝Sb→,b→-Sb→-αr→,b→-αr→.After Fourier transform one notices that in the forward quark limit q⊥→0 the amplitude for single diffractive photon or any Abelian radiation vanishes, Aq→⊥,κq⊥→0=0, and(26)dσsd⁡DYdαdqT2qT=0=0,in accordance with the Landau-Pomeranchuk principle. Indeed, in both Fock components of the quark, q and qγ∗, only the quark interacts, so these components interact equally and thus no diffraction is possible. One immediately concludes that the diffractive factorization must be strongly broken.The functionS(b→k,b→l) above is directly related to qq¯ dipole cross section as (27)
σ
q
¯
q
r
→
1
-
r
→
2
≡
∫
d
2
b
S
b
→
+
r
→
1
,
b
→
+
r
→
1
+
S
b
→
+
r
→
2
,
b
→
+
r
→
2
-
2
S
b
→
+
r
→
1
,
b
→
+
r
→
2
.Thus, following analogical scheme one can obtain the diffractive amplitude of any diffractive process as a linear combination of the dipole cross sections for different dipole separations. As was anticipated, the diffractive amplitude represents the destructive interference effect from scattering of dipoles of slightly different sizes. Such an interference results in an interplay between hard and soft fluctuations in the diffractive pp amplitude, contributing to breakdown of diffractive factorization.When one considers diffractive DY off a finite-size object like a proton, in both Fock components,3q and 3qγ∗, only the quark hadron-scale dipoles interact. These dipoles are large due to soft intrinsic motion of quarks in the projectile proton wave function. The dipoles, however, have different sizes, since the recoil quark gets a shift in impact parameters. So the dipoles interact differently giving rise to forward diffraction. The contribution of a given projectile Fock state to the diffraction amplitude is given by the difference of elastic amplitudes for the Fock states including and excluding the gauge boson: (28)Im⁡fdiffn=Im⁡feln+G-Im⁡feln,where n is the total number of partons in the Fock state and fel(n+G) and fel(n) are the elastic scattering amplitudes for the whole n-parton ensemble, which either contains the gauge boson or does not, respectively. Although the gauge boson does not participate in the interaction, the impact parameter of the quark radiating the boson gets shifted, and this is the only reason why the difference (28) is not zero (see the next section). This also conveys that this quark must interact in order to retain the diffractive amplitude nonzero [6, 7]. For this reason in the graphs depicted in Figure 5 the quark radiating G always takes part in the interaction with the target.Notice that there is no one-to-one correspondence between diffraction in QCD and the triple-Regge phenomenology. In particular, there is no triple-Pomeron vertex localized in rapidity. The colorless “Pomeron” contains at least twot-channel gluons, which can couple to any pair of projectile partons. For instance, in diffractive gluon radiation, which is the lowest order term in the triple-Pomeron graph, one of the t-channel gluons can couple to the radiated gluon, while another one couples to another parton at any rapidity, for example, to a valence quark (see Figure 3 in [23]). Apparently, such a contribution cannot be associated literally with either of the Regge graphs in Figure 5. Nevertheless, this does not affect much xF and energy dependencies provided by the triple-Regge graphs, because the gluon has spin one.It is also worth mentioning that in Figure5 we presented only the lowest order graphs with two-gluon exchange. The spectator partons in a multiparton Fock component also can interact and contribute to the elastic amplitude of the whole parton ensemble. This gives rise to higher order terms, not shown explicitly in Figure 5. They contribute to the diffractive amplitude (28) as a factor, which we define as the gap survival amplitude [8].As was mentioned above the amplitude of diffractive gauge boson radiation by a quark-antiquark dipole does not vanish in forward direction, unlike the radiation by a single quark [6, 23]. This can be understood as follows. According to the general theory of diffraction [1–3, 5], the off-diagonal diffractive channels are possible only if different Fock components of the projectile (eigenstates of interaction) interact with different elastic amplitudes. Clearly, the two Fock states consisting of just a quark and of a quark plus a gauge boson interact equally, if their elastic amplitudes are integrated over impact parameter. Indeed, when a quark fluctuates into a state qG containing the gauge boson G, with the transverse quark-boson separation r→, the quark gets a transverse shift Δr→=αr→. The impact parameter integration gives the forward amplitude. Both Fock states q and qG interact with the target with the same total cross section; this is why a quark cannot radiate at zero-momentum transfer and hence G is not produced diffractively in the forward direction. This is the general and model independent statement. The details of this general consideration can be found in [23] (Appendices A 1 and A 4). The same result is obtained calculating Feynman graphs in Appendix B 4 of the same paper. Unimportance of radiation between two interactions was also demonstrated by Brodsky and Hoyer in [45].Note that in all these calculations one assumes that the coherence time of radiation considerably exceeds the time interval between the two interactions, which is fulfilled in our case, since we consider radiation at forward rapidities. The disappearance of both inelastic and diffractive forward Abelian radiation has a direct analogy in QED: if the electric charge gets no “kick,” that is, not accelerated, no photon is radiated, provided that the radiation time considerably exceeds the duration time of interaction. This is dictated by the renown Landau-Pomeranchuk principle [46]: radiation depends on the strength of the accumulated kick, rather than on its structure, if the time scale of the kick is shorter than the radiation time. It is worthy to notice that the non-Abelian QCD case is different: a quark can radiate gluons diffractively in the forward direction. This happens due to a possibility of interaction between the radiated gluon and the target. Such a process, in particular, becomes important in diffractive heavy flavor production [9].The situation changes if the gauge boson is radiated diffractively by a dipole as shown in Figure7. Then the quark dipoles with or without a gauge boson have different sizes and interact with the target differently. So the amplitude of the diffractive gauge boson radiation from qq¯ dipole is proportional to the difference between elastic amplitudes of the two Fock components, |qq¯〉 and |qq¯G〉 [6]; that is,(29)Mq¯qdiffb→,R→,r→,α∝Ψq→G∗qα,r→·2Im⁡felb→,R→-2Im⁡felb→,R→+αr→,where Ψq→G∗q is the light-cone (nonnormalised) wave function of q→G∗q fluctuation corresponding to bremsstrahlung of virtual gauge bosons G=γ,Z,W± of mass M [8], R→=r→1-r→2 is the transverse size of the qq¯ dipole, α is the momentum fraction of the gauge boson G taken off the parent quark q, and r~1/M is the hard scale.Figure 7
Leading order contribution to the diffractive Drell-Yan in the dipole-target collision.When applied to diffractivepp scattering the diffractive amplitude (29), thus, occurs to be sensitive to the large transverse separations between the projectile quarks in the incoming proton. Normally, transition to the hadron level is achieved by using the initial proton Ψi and remnant Ψf wave functions which encode information about distributions of constituents. The completeness relation to the wave function of the unobservable proton remnant is(30)∑finΨfinr→1,r→2,r→3;xq1,2,…,xg1,2,…·Ψfin∗r→′1,r→′2,r→′3;xq′1,2,…,xg′1,2,…=δr→1-r→′1·δr→2-r→′2δr→3-r→′3∏jδxq/gj-x′q/gj.Here, r→q/gi and xq/gi are the coordinates and fractional momenta of the valence/sea partons.Since gluons and sea quarks are mostly accumulated in a close vicinity of valence quarks (inside gluonic “spots”), to a reasonable accuracy the transverse positions of sea quarks and gluons can be identified with the coordinates of valence quarks. The valence part of the wave function is often taken to be a Gaussian distribution such that(31)Ψin2=3a2π2e-ar12+r22+r32Rx1,xq1,2,…,xg2,3,…·δr→1+r→2+r→3δ1-x1-∑jxq/gj,where all the partons not participating in the hard interaction are summed up, x1 is the photon fraction taken from the initial proton, a=rch2-1 is the inverse proton mean charge radius squared, and R is a collinear multiparton distribution in the proton. Once the latter is integrated over all the partons not participating in the hard interaction, one gets a conventional collinear PDF g(x1,μ2) for gluons and q(x1,μ2) for a given quark flavor q. Since the diffractive pp cross section appears as a sum of diffractive excitations of the proton constituents, valence/sea quarks and gluons are incorporated as (32)Ψinr→i,xi2∝13∑qqx+q¯x+8116gx,after integration over spectator impact parameters and momentum fractions with a proper color factor between quark and gluon PDFs. Note that only sea and valence quarks are excited by the photon radiation in the diffractive DY process which provide a direct access to the proton structure function in the soft limit of large x [36]:(33)∑qZq2qx+q¯x=1xF2x.For diffractive gluon radiation one should account for both quark and gluon excitations whose amplitudes, however, are calculated in different ways [23].Due to the internal transverse motion of the projectile valence quarks inside the incoming proton, which corresponds to finite large transverse separations between them, the forward photon radiation does not vanish [6, 8]. These large distances are controlled by a nonperturbative (hadron) scale R→, such that the diffractive amplitude has the Good-Walker structure:(34)Mq¯qdiff∝σR→-σR→-αr→∝r→·R→,while the single diffractive-to-inclusive cross sections ratio behaves as (35)σsd⁡DYσinclDY∝exp⁡-2R2/R02x2R02x2assuming the saturated GBW shape of the dipole cross section (9), where x2 is defined in (14). Thus, the soft part of the interaction is not enhanced in Drell-Yan diffraction which is semihard/semisoft like inclusive DIS. Linear dependence on the hard scale r~1/M≪R0(x2) means that even at a hard scale the Abelian radiation is sensitive to the hadron size due to a dramatic breakdown of diffractive factorization [37]. It was firstly found in [47, 48] that factorization for diffractive Drell-Yan reaction fails due to the presence of spectator partons in the Pomeron. In [6–8] it was demonstrated that factorization in diffractive Abelian radiation is thus even more broken due to presence of spectator partons in the colliding hadrons as reflected in (34).The effect of diffractive factorization breaking manifests itself in specific features of observables like a significant damping of the cross section at highs compared to the inclusive production case as illustrated in Figure 8. This is rather unusual, since a diffractive cross section, which is proportional to the dipole cross section squared, could be expected to rise with energy steeper than the total inclusive cross section, like it occurs in the diffractive DIS process. At the same time, the ratio of DDY to DY cross sections was found in [6, 7] to rise with the hard scale, and the photon virtuality M2 was also shown in Figure 8. This is also in variance with diffraction in DIS, which is associated with the soft interactions and where the diffractive factorization holds true [10].Figure 8
The single diffractive-to-inclusive DY cross sections ratio as a function of the photon virtualityM2 for x1=0.5 (solid lines) and x1=0.9 (dashed lines) and c.m.s. energies s=40GeV, 500GeV, and 14TeV (from top to bottom) [6].Such striking signatures of the diffractive factorization breaking are due to an interplay of soft and hard interactions in the corresponding diffractive amplitude. Namely, large and small size projectile fluctuations contribute to the diffractive Abelian radiation process on the same footing providing the leading twist nature of the process, whereas diffractive DIS dominated by soft fluctuations only is of higher twist [6, 7]. But this is not the only source of the factorization breaking; another important source is the absorptive (or unitarity) corrections.


## 3.3. Gap Survival Amplitude

In the limit of unitarity saturation (the so-called black-disk limit) the absorptive corrections can entirely terminate the large rapidity gap process. The situation close to this limit, in fact, happens in high energy (anti)proton-proton collisions such that unitarity is nearly saturated at small impact parameters [49, 50]. The unitarity corrections are typically parameterized by a suppression factor also known as the soft survival probability which significantly reduce the diffractive cross section. In hadronic collisions this probability is controlled by the soft spectator partons which are absent in the case of diffractive DIS causing the breakdown of diffractive factorization.It is well known that the absorptive corrections affect differently the diagonal and off-diagonal terms in the hadronic current [5, 51], in opposite directions, leading to an additional source of the QCD factorization breaking in processes with off-diagonal contributions only. Namely, the absorptive corrections enhance the diagonal terms at larger s, whereas they strongly suppress the off-diagonal ones. In the diffractive DY process a new state, the heavy lepton pair, is produced; hence, the whole process is of entirely off-diagonal nature, whereas the diffractive DIS process contains both diagonal and off-diagonal contributions [5].Amplitude (29) is the full expression, which includes by default the effect of absorption and does not need any extra survival probability factor [8]. This can be illustrated in a simple example of elastic dipole scattering off a potential. In this case, the dipole elastic amplitude has the eikonal form:(36)Im⁡felb→,r→1-r→2=1-exp⁡iχr→1-iχr→2,where(37)χb=-∫-∞∞dzVb→,zand V(b→,z) is the potential, which depends on the impact parameter and longitudinal coordinate, and is nearly imaginary at high energies. The difference between elastic amplitudes with a shifted quark position, which enters the diffractive amplitude, reads as follows:(38)Im⁡felb→,r→1-r→2+αr→-Im⁡felb→,r→1-r→2≃exp⁡iχr→1-iχr→2exp⁡iαr→·∇→χr→1.Here, the first factor exp⁡[iχ(r→1)-iχ(r→2)] is exactly the survival probability amplitude, which vanishes in the black disc limit, as it should do. This proves that the diffractive amplitude (29) includes the effect of all absorptive corrections (gap survival amplitude), provided that the dipole cross section is adjusted to the data. Note that usually the survival probability factor is introduced into the diffractive cross section probabilistically, while in (29) it is treated quantum-mechanically, at the amplitude level.The diffractive gluon radiation is known to be rather weak (the 3-Pomeron coupling is small). This phenomenological observation can be explained assuming that gluons in the proton are predominantly located inside small “gluonic spots” of sizer0~0.3fm around the valence quarks (see, e.g., [23, 52–54]). The smallness of gluonic dipole is an important nonperturbative phenomenon which may be connected, for example, to the small size of gluonic fluctuations in the instanton liquid model [53]. Therefore, a distance between a valence quark and a gluon in a vicinity of another quark can be safely approximated by the quark-quark separation.Besides the soft gluons in the proton light-cone wave function, virtual gauge boson production triggers intensive gluon radiation such that there are many more spectator gluons in a vicinity of the quark which radiates the gauge boson. The separations of such gluons from the parent quark are controlled by the QCD DGLAP dynamics. In practice, one may replace such a set of gluons by dipoles [55] whose transverse sizes rd vary between 1/MG and r0 scales [56]. Then the mean dipole size is regulated by a relation: (39)rd2=r02ln⁡r02MG2,leading to 〈rd2〉≈0.01fm2, which means that it is rather small and the corresponding dipole cross section σ≃C(x)rd2, where C(x)=σ0/R02(x) rises with energy, is suppressed. For x=MG2/s and naive GBW parameterization [26] we get σ≈0.9mb at the Tevatron energy. Each such small dipole brings up an extra suppression factor to the large rapidity gap survival amplitude given by (40)Sds=1-Im⁡fdb,rd.Here, the second term is small and thus is simplified to (for more details, see [57]) (41)Im⁡fdb,rd≈σd4πBde-b2/2Bd,where Bd is the standard dipole-nucleon elastic slope Bd≈6GeV-2 measured earlier in diffractive ρ electroproduction at HERA [58]. At the mean impact parameter given by 〈b2〉=2Bd and for the Tevatron energy s=2TeV we arrive at negligibly small value for the absorptive correction (41): Im⁡fd(0,rd)≈0.01.On the other hand, the overall number of such dipoles increases with hardness of the process, which can amplify the magnitude of the absorptive effect. Generalizing the gap survival amplitude tond projectile dipoles, we obtain (42)Sdnd=1-Im⁡fdb,rdnd. The DGLAP evolution formulated in impact parameter representation [56] enables estimating that the mean number of such dipoles can be estimated in the double-leading-log approximation: (43)nd=12β0ln⁡1αsMG2ln⁡1-xFss0. Here, the typical Bjorken x values of the radiated gluons are restricted by the diffractive mass as x>s0/MX2=s0/(1-xF)s. In typical kinematics at the Tevatron collider, the mean number of such dipoles is roughly nd≲6. The amplitude of survival of a large rapidity gap is controlled by the largest dipoles in the projectile hadron only, such that the first exponential factor in (38) provides a sufficiently good approximation to the gap survival amplitude. The absorptive corrections (42) to the gap survival amplitude are proven to be rather weak and do not exceed 5% (or 10% in the survival probability factor) which is small compared to an overall theoretical uncertainty. For the pioneering work on hard rescattering corrections to the gap survival factor see [59].The popular quasi-eikonal model for the so-called “enhanced” probabilityS^enh (see, e.g., [21, 59]), frequently used to describe the factorization breaking in diffractive processes, is not well justified in higher orders, whereas the color dipole approach, considered here, correctly includes all diffraction excitations to all orders [5]. Such effects are included into the phenomenological parameterizations for the partial elastic dipole amplitude fitted to data. This allows predicting the diffractive gauge bosons production cross sections in terms of a single parameterization for the universal dipole cross section (or, equivalently, the elastic dipole amplitude) known independently from the soft hadron scattering data.For more details on derivations of diffractive gauge boson production amplitudes and cross sections see [7, 8]. Now we turn to a discussion of numerical results for the most important observables.


## 4. Single Diffractive Gauge Bosons Production

In [8] the dipole framework has been used in analysis of single diffractive gauge bosons production, and here we briefly overview these results. The corresponding cross sections for Z0,γ∗, and W± bosons production (s=14 TeV), differential in the lepton pair mass squared dσsd⁡/dM2, and its longitudinal momentum fraction, dσsd⁡/dx1, are shown in Figures 9(a) and 9(b), respectively. M2 distributions here are integrated over the forward rapidities (i.e., 0.3<x1<1) while the invariant mass distribution is integrated over 5<M2<105 GeV2.Figure 9
Diffractive gauge boson production cross section as function of invariant mass squaredM2 (a) and momentum fraction x1 (b) at the LHC.
(a)
(b)In the resonantsZ0 and W± region the M2 distributions exceed the corresponding DDY component; only for low masses γ∗ contribution becomes important. As for x1 distribution, the diffractive W+ and γ∗ components are comparable to each other, whereas W- and Z-boson production cross section are much lower. A precision measurement of differences in forward diffractive W+ and W- rates may enable further constraining of the quark content of the proton at large fractions xq≡x1/α.Another phenomenologically interesting observable is the lepton pair transverse momentumq⊥ distribution at the LHC shown in Figure 10(a) at the corresponding Z or W resonance. One of the important observables, sensitive to the difference between u- and d-quark PDFs at large x, W± charge asymmetry is defined as (44)AWx1=dσsd⁡W+/dx1-dσsd⁡W-/dx1dσsd⁡W+/dx1+dσsd⁡W-/dx1.And it is shown in Figure 10(b). The quantity is independent of both M2 and s.Figure 10
The lepton pair transverse momentumq⊥ distribution of the diffractive cross section at LHC s=14 TeV (a) and the charge asymmetry in SD W± cross sections at fixed M2=MW2 (b).
(a)
(b)Similarly to diffractive DY discussed above, an important feature of the SD-to-inclusive gauge boson radiation cross sections ratio is(45)RM2,x1=dσsd⁡/dx1dM2dσincl/dx1dM2,which is different from the one obtained in conventional diffractive QCD factorization-based approaches [18, 19] due to its unusual energy and scale dependence shown in Figure 11. In analogy to DDY case, this ratio behaves with respect to the energy and the hard scale in opposite way to what is expected from diffractive factorization. The ratio appears to be independent of the gauge boson properties and quark PDFs and is sensitive only to the structure of the universal elastic dipole amplitude. This means it can be efficiently used as a probe for the QCD diffractive mechanism. Therefore, measurements of the single diffractive gauge boson production cross section in the dilepton channel which enhanced compared to DDY around the Z0 and W± resonances would provide important information about the interplay between soft and hard interactions in QCD.Figure 11
The SD-to-inclusive ratio of the gauge bosons production cross sections as a function of the c.m. energys (a) and the lepton pair invariant mass M2 at the Tevatron (b).
(a)
(b)


## 5. Diffractive Non-Abelian Radiation

As we have seen in the discussion above, diffractive DY is one of the most important examples of leading twist processes, where simultaneously large and small size projectile fluctuations are at work. It turns out that the participation of soft spectator partons in the interaction with the gluonic ladder is crucial and results in a leading twist effect. What are other examples of the leading twist behavior in diffraction?


### 5.1. Leading Twist Diffractive Heavy Flavor Production

One might naively think that the Abelian (or DIS) mechanism of heavy flavor productionγ∗→QQ¯ is of the leading twist as well since it behaves as ~1/Q2. However, in the limit mQ2≫Q2 the corresponding cross section σsd⁡∝1/mQ4, that is, behaving as a higher twist process. One has to radiate at least one gluon off QQ¯ pair for this process to become the leading twist one, for example, σsd⁡(γ∗→QQ¯g)∝1/mQ2, since the mean transverse separation between G and small QQ¯ dipole is typically large although formally such a process is of the higher perturbative QCD order in αs.Consider now the non-Abelian mechanism for diffractive hadroproduction of heavy quarks viag∗→QQ¯ hard subprocess. Production of heavy quarks at large xF→1 is a longstanding controversial issue even in inclusive processes. On one hand, QCD factorization approach predicts vanishingly small yields of heavy flavor due to steeply falling gluon density as ~1-xF5 at large xF. On the other hand, the end-point behavior is controlled by the universal Regge asymptotics dσ/dxF(xF→1)∝(1-xF)1-2αR(t) in terms of the Regge trajectory of the t-channel exchange αR(t). Apparently, the Regge and QCd factorization approaches contradict each other. The same problem emerges in the DY process at large xF as is seen in data [60] which means that in the considered kinematics the conventional QCD factorization does not apply [61]. At the same time, the observation of an excess of diffractive production of heavy quarks at large xF→1 compared to conventional expectation may provide a good evidence for intrinsic heavy flavors if the latter is reliably known. Calculations assuming that diffractive factorization holds for hard diffraction [12, 39] may not be used for quantifying the effect from intrinsic heavy flavor. Instead, the dipole framework has been employed to this process for the first time in [9]. Here we briefly overview the basic theory aspects concerning primarily heavy quarks produced in the projectile fragmentation region (for inclusive QQ¯ production at mid rapidites in the dipole framework, see [62]).Typical contributions to the single diffractiveQQ¯ production rate are summarized in Figure 12. Figures 12(a) and 12(b) correspond to the leading order gluon splitting into QQ¯ contributions in the color field of the target proton (diffractive gluon excitation). The latter gluon as a component of the projectile proton wave function can be treated as real (via collinear gluon PDF) or virtual (via unintegrated gluon PDF). Due to hard scale mQ Figure 12(a) with Pomeron coupling to a small-size gQQ¯ system is of the higher twist due to color transparency and is therefore suppressed. Figure 12(b) involves two scales: the soft hadronic one ~ΛQCD associated with large transverse separations between a gluon and constituent valence quarks and the hard one ~mQ associated with small QQ¯ dipole. An interplay between these two scales similar to that in DDY emerges as the leading twist effect; thus, Figure 12(b) is important. Possible higher order terms with an extra gluon radiation contributing to the leading twist diffractive heavy flavor production were discussed in detail in [9].Figure 12
Leading order contributions to single diffractive heavy flavor production in gluon-proton (a, b) and quark-proton (c, d) scattering subprocesses inpp collisions. Diagrams (b, d) emerge due to the presence of soft spectator partons in the proton wave function (the screening gluon couples to every spectator parton besides the active one). Grey effective vertices account for all possible couplings of the incident partons.
(a)
(b)
(c)
(d)Figures12(c) and 12(d) account for QQ¯ production via diffractive quark excitation. Just as in leading twist diffraction in DIS γ∗→QQ¯g, these processes are associated with two characteristic transverse separations, a small one, ~1/mQ, between the Q¯ and Q, and a large one, either ~1/mq between q and QQ¯ (Figure 12(c)) or 1/ΛQCD between another constituent valence quark and QQ¯ (Figure 12(d)). While all the terms contributing to Figure 12(d) are of the leading twist (see [9]), only a special subset of Figure 12(c) are of the leading twist. Indeed, the hard subprocess q+g→(QQ¯)+q is characterized by five distinct topologies illustrated in Figure 13, and similar graphs are for gluon-proton scattering with subprocess g+g→(QQ¯)+g.Figure 13
Five different topologies contributing to inclusiveQQ¯ production in quark-proton scattering. These can be split into two gauge-invariant subsets of amplitudes as described in the text.These graphs can be grouped into two amplitudes attributed to bremsstrahlung (BR) and production (PR) mechanisms, which do or do not involve the projectile light quarks or gluons, respectively (for more details, see Figure 2 and Appendix A in [9]). The BR mechanism includes the same graphs as radiation of a gluon (see [43, 63]), that is, interaction with the source parton before and after radiation and interaction with the radiated gluon. The PR mechanism, responsible for the transition g→Q¯Q, includes the interactions with the gluon and the produced Q¯Q (also known as gluon-gluon fusion gg→QQ¯ mechanism). The total amplitude is (46)Mq,g=Mq,gBR+Mq,gPR,where subscripts q,g denote contributions with hard gluon radiation by the projectile valence or sea quarks and gluons, respectively. Such grouping is performed separately for transversely and longitudinally polarised gluons as described in [9]. One of the reasons for this grouping is that each of these two combinations is gauge invariant and can be expressed in terms of three-body dipole cross sections, σgq¯q and σgQ¯Q, respectively.Another physical reason for such a separation is different scale dependence of the BR and PR components. Introducing the transverse separationsr→,r→1, and r→2 within the Q¯Q, qQ¯, and qQ pairs, respectively, the three-body dipole cross sections can be expressed via two scales: the distance between the final light quark (or gluon) and the center of gravity of the QQ¯ pair, ρ→=r→-βr→1-(1-β)r→2 (β is the heavy quark momentum fraction taken from the parent gluon which takes fraction α of the parent parton) and the QQ¯ transverse separation, s→=r→1-r→2. The corresponding distribution amplitudes of QQ¯ production in diffractive quark/gluon scattering off proton(47)ABR∝ΦBRρ→,s→Σ1ρ→,s→,APR∝ΦPRρ→,s→Σ2ρ→,s→are given in terms of the effective dipole cross sections for colorless gq¯q and gQ¯Q systems and rather complicated wave functions Φ of subsequent gluon radiation and then its splitting into Q¯Q pair in both cases. In the case of bremsstrahlung, both mean separations are controlled by the hard scale such that(48)ABR~ρ2~s2~1mQ2,and, thus, the corresponding contribution is a higher twist effect and thus suppressed (note that in the case of forward Abelian radiation this contribution is equal to zero). On the contrary, in the production mechanism only the Q¯Q separation is small, 〈s2〉~1/mQ2; the second scale appears to be soft, 〈ρ2〉~1/mq2, leading to the leading twist behavior(49)APR~s→·ρ→in analogy to diffractive DY process. This is a rather nontrivial fact, since in the case of the DY reaction such a property is due to the Abelian nature of the radiated particle while here we consider a non-Abelian radiation. The bremsstrahlung-production interference terms are of the higher twist and thus are safely omitted.The situation with scale dependence in the case ofQ¯Q production in diffractive pp scattering is somewhat similar to diffractive quark-proton scattering discussed above but technically more involved due to extra terms (b) and (d) in Figure 12 and color averaging over the projectile proton wave function. Although bremsstrahlung terms from Figures 12(b) and 12(d) are formally of the leading twist due to interactions with distant spectator partons, numerically they are always tiny due to denominator suppression by large Q¯Q mass. Thus, the leading twist production terms from (b), (c), and (d) sets are relevant whereas the set (a) does not contain production terms and is a higher twist effect. Thus, like the case in diffractive Drell-Yan in the considered process the leading twist effect, at least, partly emerges due to intrinsic transverse motion of constituent quarks in the incoming proton. However, due to a non-Abelian nature of this process extra leading twist terms production from the “production” mechanism, which are independent of the structure of the hadronic wave function, become important. Diffractive production cross sections of charm, beauty, and top quark pairs, p+p→QQ¯X+p, as functions of c.m.s. pp energy are shown in Figure 14. The experimental data points available from E690 [27] and CDF [28] experiments have been compared with theoretical predictions evaluated with corresponding phase space constraints (for more details, see [9]).Figure 14
The total cross section of diffractivecc¯, bb¯, and tt¯ pairs production as function of energy in comparison with experimental data from E690 [27] and CDF [28] experiments (a) and the differential cross section as function of fraction x1 of the initial proton momentum carried by the charm quark (b) [9].
(a)
(b)


### 5.2. Single Diffractive Higgsstrahlung

Typically large Standard Model (SM) backgrounds and theoretical uncertainties due to higher order effects strongly limit the potential of inclusive Higgs boson production for spotting likely small but yet possible New Physics effects. Some of the SM extensions predict certain distortions in Higgs boson Yukawa couplings such that the precision multichannel measurements of the Higgs-heavy quarks couplings become a crucial test of the SM structure. As a very promising but challenging channel, the exclusive and diffractive Higgs production processes (involving rapidity gaps) offer new possibilities to constrain the backgrounds and open up more opportunities for New Physics searches (see, e.g., [64–71]).The QCD-initiated gluon-gluon fusiongg→H mechanism via a heavy quark loop is one of the dominant and most studied Higgs bosons production modes in inclusive pp scattering which has led to its discovery at the LHC (for more information on Higgs physics highlights, see, e.g., [72–77] and references therein). The same mechanism is expected to provide an important Higgs production mode in single diffractive pp scattering as well as in central exclusive Higgs boson production [64, 65, 68]. The forward inclusive and diffractive Higgsstrahlung off intrinsic heavy flavor at xF→1 has previously been studied in [78, 79], respectively.Very recently, a new single diffractive production mode of the Higgs boson in association with a heavy quark pairQ¯Q, namely, pp→X+QQ¯H+p, at large xF where conventional factorization-based approaches are expected to fail has been studied in [20]. The latter process is an important background for diffractive Higgs boson hadroproduction off intrinsic heavy flavor. Here, we provide a short overview of this process which is analogical to forward diffractive Q¯Q production discussed above.For a reasonably accurate estimate one retains only the dominant gluon-initiated leading twist terms illustrated in Figure15 where the “active” gluon is coupled to the hard QQ¯+H system, while the soft “screening” gluon couples to a spectator parton at a large impact distance. The latter is illustrated by tree-level diagrams with Higgs boson radiation off a heavy quark or Higgsstrahlung. In practice, however, one does not calculate the Feynman graphs explicitly in Figure 15. Instead one should adopt the generalized optical theorem within the Good-Walker approach to diffraction [3] such that a diffractive scattering amplitude turns out to be proportional to a difference between elastic scatterings of different Fock states [20]. The contributions where both “active” and “screening” gluons couple to partons at small relative distances are the higher twist ones and thus are strongly suppressed by extra powers of the hard scale (see, e.g., [9]). This becomes obvious in the color dipole framework due to color transparency [11] making the medium more transparent for smaller dipoles.Figure 15
Dominant gluon-initiated contributions to the single diffractiveQ¯Q+H production [20].The hard scales which control the diffractive Higgsstrahlung process are〈r2〉~1/mQ2 and 〈ρ2〉~1/τ2, where τ2=MH2+α3MQQ¯2 in terms of the Higgs boson mass, MH, and the QQ¯ pair invariant mass, MQQ¯. Another length scale here is the distance between ith and jth projectile partons, 〈rij2〉~〈R〉2, which is soft for light valence/sea quarks in the proton wave function. Before the hard gluon splits into Q¯QH system it undergoes multiple splittings g→gg populating the projectile fragmentation domain with gluon radiation with momenta below the hard scale of the process p⊥rad<MQ¯QH. The latter should be accounted for via a gluon PDF evolution.The SD-to-inclusive ratio of the cross sections for different c.m. energiess=0.5,7,14 TeV and for two distinct rapidities Y=0 and 3 as functions of Q¯QH invariant mass M is shown in Figure 16(a). The ratio is similar to that for heavy quark production [9] and thus in good agreement with experimental data from the Tevatron. Note that this ratio has falling energy-dependence and rising M-dependence, where M is the invariant mass of the produced Q¯QH system. This is similar to what was found for diffractive Drell-Yan process [7, 8] and has the same origin, namely, breakdown of QCD factorization and the saturated form of the dipole cross section.Figure 16
The SD-to-inclusive ratioR(M) as a function of Q¯QH invariant mass M for different c.m. energies s=0.5,2,14 TeV (a) and the differential cross section dσ/dx1 of the SD Higgsstrahlung off tt¯ and bb¯ pairs for s=14 TeV (b) [20].
(a)
(b)The differential cross sections of single diffractivebb¯ and tt¯ production in association with the Higgs boson are shown in Figure 16(b) as functions of x1 variable at the LHC energy s=14 TeV implied that the Higgs boson transverse momentum is large; that is, κ≳mH. In this case the asymptotic dipole formula based upon the collinear projectile gluon PDF (32) and the first (quadratic) term in the dipole cross section is a good approximation and reproduces the exact k⊥-factorization result for the inclusive Higgsstrahlung transverse momentum distribution in both the shape and normalisation (for more details, see [20]). The contribution of diffractive gluon excitations to the Higgsstrahlung dominates the total Higgsstrahlung cross section due to a large yield from central rapidities. Besides, the SD Higgsstrahlung off top quarks is larger than that from the bottom while shapes of x1 distributions are similar. Additional radiation of the Higgs boson enhances the contribution of heavy quarks and thus compensates the smallness of their diffractive production modes.Analogically to other diffractive bremsstrahlung processes discussed in previous sections, breakdown of QCD factorization leads to a flatter hard scale dependence of the cross section. This is a result of leading twist behaviour which has been discussed above and which has been confirmed by the comparison of data on diffractive production of charm and beauty [9].


## 5.1. Leading Twist Diffractive Heavy Flavor Production

One might naively think that the Abelian (or DIS) mechanism of heavy flavor productionγ∗→QQ¯ is of the leading twist as well since it behaves as ~1/Q2. However, in the limit mQ2≫Q2 the corresponding cross section σsd⁡∝1/mQ4, that is, behaving as a higher twist process. One has to radiate at least one gluon off QQ¯ pair for this process to become the leading twist one, for example, σsd⁡(γ∗→QQ¯g)∝1/mQ2, since the mean transverse separation between G and small QQ¯ dipole is typically large although formally such a process is of the higher perturbative QCD order in αs.Consider now the non-Abelian mechanism for diffractive hadroproduction of heavy quarks viag∗→QQ¯ hard subprocess. Production of heavy quarks at large xF→1 is a longstanding controversial issue even in inclusive processes. On one hand, QCD factorization approach predicts vanishingly small yields of heavy flavor due to steeply falling gluon density as ~1-xF5 at large xF. On the other hand, the end-point behavior is controlled by the universal Regge asymptotics dσ/dxF(xF→1)∝(1-xF)1-2αR(t) in terms of the Regge trajectory of the t-channel exchange αR(t). Apparently, the Regge and QCd factorization approaches contradict each other. The same problem emerges in the DY process at large xF as is seen in data [60] which means that in the considered kinematics the conventional QCD factorization does not apply [61]. At the same time, the observation of an excess of diffractive production of heavy quarks at large xF→1 compared to conventional expectation may provide a good evidence for intrinsic heavy flavors if the latter is reliably known. Calculations assuming that diffractive factorization holds for hard diffraction [12, 39] may not be used for quantifying the effect from intrinsic heavy flavor. Instead, the dipole framework has been employed to this process for the first time in [9]. Here we briefly overview the basic theory aspects concerning primarily heavy quarks produced in the projectile fragmentation region (for inclusive QQ¯ production at mid rapidites in the dipole framework, see [62]).Typical contributions to the single diffractiveQQ¯ production rate are summarized in Figure 12. Figures 12(a) and 12(b) correspond to the leading order gluon splitting into QQ¯ contributions in the color field of the target proton (diffractive gluon excitation). The latter gluon as a component of the projectile proton wave function can be treated as real (via collinear gluon PDF) or virtual (via unintegrated gluon PDF). Due to hard scale mQ Figure 12(a) with Pomeron coupling to a small-size gQQ¯ system is of the higher twist due to color transparency and is therefore suppressed. Figure 12(b) involves two scales: the soft hadronic one ~ΛQCD associated with large transverse separations between a gluon and constituent valence quarks and the hard one ~mQ associated with small QQ¯ dipole. An interplay between these two scales similar to that in DDY emerges as the leading twist effect; thus, Figure 12(b) is important. Possible higher order terms with an extra gluon radiation contributing to the leading twist diffractive heavy flavor production were discussed in detail in [9].Figure 12
Leading order contributions to single diffractive heavy flavor production in gluon-proton (a, b) and quark-proton (c, d) scattering subprocesses inpp collisions. Diagrams (b, d) emerge due to the presence of soft spectator partons in the proton wave function (the screening gluon couples to every spectator parton besides the active one). Grey effective vertices account for all possible couplings of the incident partons.
(a)
(b)
(c)
(d)Figures12(c) and 12(d) account for QQ¯ production via diffractive quark excitation. Just as in leading twist diffraction in DIS γ∗→QQ¯g, these processes are associated with two characteristic transverse separations, a small one, ~1/mQ, between the Q¯ and Q, and a large one, either ~1/mq between q and QQ¯ (Figure 12(c)) or 1/ΛQCD between another constituent valence quark and QQ¯ (Figure 12(d)). While all the terms contributing to Figure 12(d) are of the leading twist (see [9]), only a special subset of Figure 12(c) are of the leading twist. Indeed, the hard subprocess q+g→(QQ¯)+q is characterized by five distinct topologies illustrated in Figure 13, and similar graphs are for gluon-proton scattering with subprocess g+g→(QQ¯)+g.Figure 13
Five different topologies contributing to inclusiveQQ¯ production in quark-proton scattering. These can be split into two gauge-invariant subsets of amplitudes as described in the text.These graphs can be grouped into two amplitudes attributed to bremsstrahlung (BR) and production (PR) mechanisms, which do or do not involve the projectile light quarks or gluons, respectively (for more details, see Figure 2 and Appendix A in [9]). The BR mechanism includes the same graphs as radiation of a gluon (see [43, 63]), that is, interaction with the source parton before and after radiation and interaction with the radiated gluon. The PR mechanism, responsible for the transition g→Q¯Q, includes the interactions with the gluon and the produced Q¯Q (also known as gluon-gluon fusion gg→QQ¯ mechanism). The total amplitude is (46)Mq,g=Mq,gBR+Mq,gPR,where subscripts q,g denote contributions with hard gluon radiation by the projectile valence or sea quarks and gluons, respectively. Such grouping is performed separately for transversely and longitudinally polarised gluons as described in [9]. One of the reasons for this grouping is that each of these two combinations is gauge invariant and can be expressed in terms of three-body dipole cross sections, σgq¯q and σgQ¯Q, respectively.Another physical reason for such a separation is different scale dependence of the BR and PR components. Introducing the transverse separationsr→,r→1, and r→2 within the Q¯Q, qQ¯, and qQ pairs, respectively, the three-body dipole cross sections can be expressed via two scales: the distance between the final light quark (or gluon) and the center of gravity of the QQ¯ pair, ρ→=r→-βr→1-(1-β)r→2 (β is the heavy quark momentum fraction taken from the parent gluon which takes fraction α of the parent parton) and the QQ¯ transverse separation, s→=r→1-r→2. The corresponding distribution amplitudes of QQ¯ production in diffractive quark/gluon scattering off proton(47)ABR∝ΦBRρ→,s→Σ1ρ→,s→,APR∝ΦPRρ→,s→Σ2ρ→,s→are given in terms of the effective dipole cross sections for colorless gq¯q and gQ¯Q systems and rather complicated wave functions Φ of subsequent gluon radiation and then its splitting into Q¯Q pair in both cases. In the case of bremsstrahlung, both mean separations are controlled by the hard scale such that(48)ABR~ρ2~s2~1mQ2,and, thus, the corresponding contribution is a higher twist effect and thus suppressed (note that in the case of forward Abelian radiation this contribution is equal to zero). On the contrary, in the production mechanism only the Q¯Q separation is small, 〈s2〉~1/mQ2; the second scale appears to be soft, 〈ρ2〉~1/mq2, leading to the leading twist behavior(49)APR~s→·ρ→in analogy to diffractive DY process. This is a rather nontrivial fact, since in the case of the DY reaction such a property is due to the Abelian nature of the radiated particle while here we consider a non-Abelian radiation. The bremsstrahlung-production interference terms are of the higher twist and thus are safely omitted.The situation with scale dependence in the case ofQ¯Q production in diffractive pp scattering is somewhat similar to diffractive quark-proton scattering discussed above but technically more involved due to extra terms (b) and (d) in Figure 12 and color averaging over the projectile proton wave function. Although bremsstrahlung terms from Figures 12(b) and 12(d) are formally of the leading twist due to interactions with distant spectator partons, numerically they are always tiny due to denominator suppression by large Q¯Q mass. Thus, the leading twist production terms from (b), (c), and (d) sets are relevant whereas the set (a) does not contain production terms and is a higher twist effect. Thus, like the case in diffractive Drell-Yan in the considered process the leading twist effect, at least, partly emerges due to intrinsic transverse motion of constituent quarks in the incoming proton. However, due to a non-Abelian nature of this process extra leading twist terms production from the “production” mechanism, which are independent of the structure of the hadronic wave function, become important. Diffractive production cross sections of charm, beauty, and top quark pairs, p+p→QQ¯X+p, as functions of c.m.s. pp energy are shown in Figure 14. The experimental data points available from E690 [27] and CDF [28] experiments have been compared with theoretical predictions evaluated with corresponding phase space constraints (for more details, see [9]).Figure 14
The total cross section of diffractivecc¯, bb¯, and tt¯ pairs production as function of energy in comparison with experimental data from E690 [27] and CDF [28] experiments (a) and the differential cross section as function of fraction x1 of the initial proton momentum carried by the charm quark (b) [9].
(a)
(b)


## 5.2. Single Diffractive Higgsstrahlung

Typically large Standard Model (SM) backgrounds and theoretical uncertainties due to higher order effects strongly limit the potential of inclusive Higgs boson production for spotting likely small but yet possible New Physics effects. Some of the SM extensions predict certain distortions in Higgs boson Yukawa couplings such that the precision multichannel measurements of the Higgs-heavy quarks couplings become a crucial test of the SM structure. As a very promising but challenging channel, the exclusive and diffractive Higgs production processes (involving rapidity gaps) offer new possibilities to constrain the backgrounds and open up more opportunities for New Physics searches (see, e.g., [64–71]).The QCD-initiated gluon-gluon fusiongg→H mechanism via a heavy quark loop is one of the dominant and most studied Higgs bosons production modes in inclusive pp scattering which has led to its discovery at the LHC (for more information on Higgs physics highlights, see, e.g., [72–77] and references therein). The same mechanism is expected to provide an important Higgs production mode in single diffractive pp scattering as well as in central exclusive Higgs boson production [64, 65, 68]. The forward inclusive and diffractive Higgsstrahlung off intrinsic heavy flavor at xF→1 has previously been studied in [78, 79], respectively.Very recently, a new single diffractive production mode of the Higgs boson in association with a heavy quark pairQ¯Q, namely, pp→X+QQ¯H+p, at large xF where conventional factorization-based approaches are expected to fail has been studied in [20]. The latter process is an important background for diffractive Higgs boson hadroproduction off intrinsic heavy flavor. Here, we provide a short overview of this process which is analogical to forward diffractive Q¯Q production discussed above.For a reasonably accurate estimate one retains only the dominant gluon-initiated leading twist terms illustrated in Figure15 where the “active” gluon is coupled to the hard QQ¯+H system, while the soft “screening” gluon couples to a spectator parton at a large impact distance. The latter is illustrated by tree-level diagrams with Higgs boson radiation off a heavy quark or Higgsstrahlung. In practice, however, one does not calculate the Feynman graphs explicitly in Figure 15. Instead one should adopt the generalized optical theorem within the Good-Walker approach to diffraction [3] such that a diffractive scattering amplitude turns out to be proportional to a difference between elastic scatterings of different Fock states [20]. The contributions where both “active” and “screening” gluons couple to partons at small relative distances are the higher twist ones and thus are strongly suppressed by extra powers of the hard scale (see, e.g., [9]). This becomes obvious in the color dipole framework due to color transparency [11] making the medium more transparent for smaller dipoles.Figure 15
Dominant gluon-initiated contributions to the single diffractiveQ¯Q+H production [20].The hard scales which control the diffractive Higgsstrahlung process are〈r2〉~1/mQ2 and 〈ρ2〉~1/τ2, where τ2=MH2+α3MQQ¯2 in terms of the Higgs boson mass, MH, and the QQ¯ pair invariant mass, MQQ¯. Another length scale here is the distance between ith and jth projectile partons, 〈rij2〉~〈R〉2, which is soft for light valence/sea quarks in the proton wave function. Before the hard gluon splits into Q¯QH system it undergoes multiple splittings g→gg populating the projectile fragmentation domain with gluon radiation with momenta below the hard scale of the process p⊥rad<MQ¯QH. The latter should be accounted for via a gluon PDF evolution.The SD-to-inclusive ratio of the cross sections for different c.m. energiess=0.5,7,14 TeV and for two distinct rapidities Y=0 and 3 as functions of Q¯QH invariant mass M is shown in Figure 16(a). The ratio is similar to that for heavy quark production [9] and thus in good agreement with experimental data from the Tevatron. Note that this ratio has falling energy-dependence and rising M-dependence, where M is the invariant mass of the produced Q¯QH system. This is similar to what was found for diffractive Drell-Yan process [7, 8] and has the same origin, namely, breakdown of QCD factorization and the saturated form of the dipole cross section.Figure 16
The SD-to-inclusive ratioR(M) as a function of Q¯QH invariant mass M for different c.m. energies s=0.5,2,14 TeV (a) and the differential cross section dσ/dx1 of the SD Higgsstrahlung off tt¯ and bb¯ pairs for s=14 TeV (b) [20].
(a)
(b)The differential cross sections of single diffractivebb¯ and tt¯ production in association with the Higgs boson are shown in Figure 16(b) as functions of x1 variable at the LHC energy s=14 TeV implied that the Higgs boson transverse momentum is large; that is, κ≳mH. In this case the asymptotic dipole formula based upon the collinear projectile gluon PDF (32) and the first (quadratic) term in the dipole cross section is a good approximation and reproduces the exact k⊥-factorization result for the inclusive Higgsstrahlung transverse momentum distribution in both the shape and normalisation (for more details, see [20]). The contribution of diffractive gluon excitations to the Higgsstrahlung dominates the total Higgsstrahlung cross section due to a large yield from central rapidities. Besides, the SD Higgsstrahlung off top quarks is larger than that from the bottom while shapes of x1 distributions are similar. Additional radiation of the Higgs boson enhances the contribution of heavy quarks and thus compensates the smallness of their diffractive production modes.Analogically to other diffractive bremsstrahlung processes discussed in previous sections, breakdown of QCD factorization leads to a flatter hard scale dependence of the cross section. This is a result of leading twist behaviour which has been discussed above and which has been confirmed by the comparison of data on diffractive production of charm and beauty [9].


## 6. Summary

In this short review, we have discussed major properties and basic dynamics of single diffractive processes ofγ∗, Z0, and W± bosons production processes at the LHC, as well as leading twist heavy flavor hadroproduction at large Feynman xF and diffractive Higgsstrahlung off heavy quarks. We outlined the manifestations of diffractive factorization breaking in these single diffractive reactions within the framework of color dipole description, which is suitable for studies of the interplay between soft and hard fluctuations. The latter reliably determine diffractive hadroproduction in the projectile fragmentation region.The first, rather obvious source for violation of diffractive factorization, are related to the absorptive corrections (called sometimes survival probability of large rapidity gaps). The absorptive corrections affect differently the diagonal and off-diagonal diffractive amplitudes [5, 51], leading to a breakdown of diffractive QCD factorization in hard diffractive processes, like diffractive radiation of heavy Abelian particles and heavy flavors. The dipole approach enables accounting for the absorptive corrections automatically at the amplitude level.The second, more sophisticated reason for diffractive factorization breaking is specific for Abelian radiation; namely, a quark cannot radiate in the forward direction (zero-momentum transfer), where diffractive cross sections usually have a maximum. Forward diffraction becomes possible due to intrinsic transverse motion of quarks inside the proton, although the magnitude of the forward cross section remains very small [6, 7]. A much larger contribution to Abelian radiation in the forward direction in pp collisions comes from interaction with the spectator partons in the proton. Such an interplay of hard and soft dynamics is specific for the processes under consideration, which makes them different from diffractive DIS involving no comoving spectator partons.These mechanisms of diffractive factorization breaking lead to rather unusual features of the leading twist diffractive Abelian radiation with respect to its hard scale and energy-dependence.The outlined sources of factorization breaking are also presented in diffractive radiation of non-Abelian particles. Interactions of the radiated gluon make it possible to be radiated even at zero-momentum transfer. These processes have been quantitatively analysed in such important channels as diffractive heavy flavor production and Higgsstrahlung in the projectile fragmentation region. Further studies of these effects, both experimentally and theoretically, are of major importance for upcoming LHC measurements.



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