Diffractive Bremsstrahlung in Hadronic Collisions

Production of heavy photons (Drell-Yan), gauge bosons, Higgs bosons, heavy flavors, which is treated within the QCD parton model as a result of hard parton-parton collision, can be considered as 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.


I. INTRODUCTION
Diffractive production of particles in hadron-hadron scattering at high energies is one of the basic tools, experimental and theoretical, giving access to small-x 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], 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, i.e. besides elastic scattering a new diffractive state will be created resulting in a new combination of the Fock components (for a detailed review on QCD diffraction, see Ref. [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 characterised 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, non-perturbative interactions. For example, diffractive deep-inelastic scattering (DIS), γ * p →qqp, 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 forqq dipoles of different sizes. Although formally the process γ * →qq 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 finalqq distribution functions are not orthogonal. Similar features exhibit the contribution of higher Fock components of the photon, e.g. the leading twist diffraction γ * →qqg.
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 to measure 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, e.g. highp T dijet production, failed by an order of magnitude [13]. 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 of the W boson in pp collisions measured by the CDF experiment [14,15], was also found to be six times smaller than what was predicted relying on factorisation and diffractive DIS data [16]. Besides, the phenomenological models based on diffractive factorisation, which are widely discussed in the literature (see e.g. Refs. [17,18]), predict a significant increase of the ratio of the diffractive to inclusive gauge bosons production cross sections with energy. The diffractive QCD factorisation in hadron collisions is, however, severely broken by the interplay of hard and soft dynamics, as was recently advocated in Refs. [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 [19] -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, i.e. in the black-disc limit.
The absorptive corrections, also known as the survival probability of rapidity gaps [20], are related to initial-and final-state interactions. Usually the survival probability is introduced into the diffractive cross section in a probabilistic way [21] and is estimated in simplified models such as eikonal, quasi-eikonal, two-channel approximations, etc.
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 extra ad hoc gap survival probability factors. Within the light-cone color dipole approach [11] a diffractive process is considered as a result of elastic scattering ofqq dipoles of different sizes emerging in incident Fock states. The study of the diffractive Drell-Yan reaction performed in Ref. [6] has revealed importance of soft interactions with the partons spectators, which contributes 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.
Interesting, that besides interaction with the spectator projectile partons, there is another important source for diffractive factorization breaking. Even a single quark, having no spectator co-movers, cannot radiate Abelian fields (γ, Z, W ± , H) interacting diffractively with the target with zero transverse momentum transfer [22], i.e. in forward direction scattering. This is certainly contradicts the expectations based of 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 Refs. [6][7][8] that this contribution dominates the forward diffractive Abelian radiation cross section. This mechanism leads to a dramatic violation of diffractive QCD factorisation, 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 factorisation breaking in non-Abelian radiation is still important.

II. COLOR DIPOLE PICTURE OF DIFFRACTIVE EXCITATION
Single diffractive scattering and production of a new (diffractive) state, i.e. diffractive excitation, emerges as a consequence of quantum fluctuations in projectile hadron. The orthogonal hadron state |h can be excited due to interactions but can be decomposed over the orthogonal and complet set of eigenstates of interactions |α as [11,23,24] wheref el is the elastic amplitude operator and f α is one of its eignestates. The eigen amplitudes f α 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 eigen amplitudes f α and coefficients C h α , i.e.
respectively, such that the forward single diffractive cross section is given by the dispersion of the eigenvalues distribution. It was suggested in Ref. [11] that eigenstates of QCD interactions are color dipoles, such that any diffractive amplitude can be considered as a superposition of universal elastic dipole amplitudes. Such dipoles experience only elastic scattering and characterized only by transverse separation r. The total hadron-proton cross section is then given by its eigenvalue, the universal dipole cross section as follows where Ψ h ( 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], σ sd dp 2 where p ⊥ is the transverse momentum of the recoil proton, σ(r) is the universal dipoleproton 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 [22,25]. Even for the simplest quark-anitiquark dipole configuration, a theoretical prediction of the partial dipole amplitude f qq el ( b, r) and the dipole cross section σ qq ( 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 Bjorken x in DIS the virtual photon exhibits partonic structure as shown in Fig. 1. The leading order configuration, the qq dipole, then elastically rescatters off the proton target p providing a phenomenological access to σ qq ( r, x). When it comes to diffractive DIS schematically represented in Fig. 2, the corresponding single diffractive cross section in the forward proton limit t → 0 is given by the dipole cross section squared, i.e.
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 Ψ γ * which can be found e.g. in Ref. [26]. The mean dipole size squared is inversely proportional to the quark energy squared (2.8) The dipole size is assumed to be preserved during scattering in the high energy limit. Hard and soft hadronic fluctuations have small r 2 ∼ 1/Q 2 (nearly symmetric α ≫ m 2 q /Q 2 configuration) and large r 2 ∼ 1/m 2 q , m q ∼ Λ QCD (aligned jet α ∼ m 2 q /Q 2 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/m 2 q which largely compensate their small ∼ m 2 q /Q 2 weights. On the other hand, abundant hard fluctuations with nearly symmetric small-size dipoles r 2 ∼ 1/Q 2 have vanishing (as 1/Q 2 ) cross section. It turns out that in inclusive DIS, both hard and soft contributions to the total cross section behave as 1/Q 2 (semi-hard and semi-soft), while in diffractive DIS the soft fluctuations ∼ 1/m 2 q Q 2 dominate over the hard ones ∼ 1/Q 4 . This also explains why the ratio σ sd /σ inc in DIS is nearly Q 2 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 [25], and fitted to DIS data. Here, x is the Bjorken variable, σ 0 = 23.03 mb; R 0 (x) = 0.4 fm × (x/x 0 ) 0.144 and x 0 = 0.003. In pp collisions x is identified with gluon where M is the invariant mass of the produced system and s is the pp c.m. energy. This parametrization, although is oversimplified (compare with Ref. [27]), is rather successful in description of experimental HERA data with a reasonable accuracy. In soft processes, however, the Bjorken variable x makes no sense, and gluon-target collision c.m. energy squaredŝ = x 1 s (s is the pp c.m. energy) is a more appropriate variable, while the saturated form (2.9) should be retained [22]. The corresponding parameterisations for σ 0 = σ 0 (ŝ) and R 0 = R 0 (ŝ) read R 0 (ŝ) = 0.88 fm (s 0 /ŝ) 0.14 , σ 0 (ŝ) = σ πp tot (ŝ) 1 + where the pion-proton total cross section is parametrized as [28] σ πp tot (ŝ) = 23.6(ŝ/s 0 ) 0.08 mb, s 0 = 1000 GeV 2 , the mean pion radius squared is [29] r 2 ch π = 0.44 fm 2 . An explicit analytic form of the x-andŝ-dependent parameterisations for the elastic amplitude f el ( b, r) accounting for an information about the dipole orientation w.r.t. the color background field (i.e. the angular dependence between r and b) can be found in Refs. [30][31][32].
The ansatz (2.9) incorporate such important phenomenon as saturation at a soft scale since it levels off at r ≫ R 0 . Another important feature is vanishing of the cross section at small r → 0 as σ qq ∝ r 2 [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. The color transparency is the major effect governing diffraction. Finally, the quadratic r-dependence is an immediate consequence of gauge invariance and nonabeliance of interactions in QCD.

A. Diffractive factorisation
The cross section of the inclusive Drell-Yan (DY) is expressed via the dipole cross section in a way similar to DIS [33] dσ DY (qp → γ * X) where α is the light-cone momentum fraction carried by the heavy photon off the parent quark. QCD factorisation 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 production G = Z, W ± in pp collisions which is characterized by a relatively small momentum transfer between the colliding protons. In particular, one of the protons, e.g. p 1 , radiates a hard virtual gauge G * boson with k 2 = M 2 ≫ m 2 p and hadronizes into a hadronic system X both moving in forward direction and separated by a large rapidity gap from the second proton p 2 , which remains intact. In the DDY case, Both the di-lepton and X, the debris of p 1 , stay in the forward fragmentation region. In this case, the virtual photon is predominantly emitted by the valence quarks of the proton p 1 . In some of the previous studies [17,34] of the single diffractive Drell-Yan reaction the analysis was made within the phenomenological Pomeron-Pomeron and γ-Pomeron fusion mechanisms using the Ingelman-Shlein approach [12] based on diffractive factorization. According to this concept, similarly to usual (e.g. collinear) QCD factorisation, a hard scale can be used to probe partonic structure of the Pomeron. Once the parton densities in the Pomeron are known, one can predict the cross section of any hard diffractive hadronic reaction. So the single diffractive DY cross section has a similar to inclusive DY form in terms of diffractive PDFs Fq /IP . This relation, however, contradicts the basic principles of diffraction as will be discussed below. The diffractive factorisation leads to specific features of the differential DY cross sections similar to those in diffractive DIS process, e.g., 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, the invariant mass of the lepton pair squared, M 2 [17]. One can derive a Regge behavior of the diffractive cross section of heavy photon production in terms of the usual light-cone variables, where M, k T and x F are the invariant mass, transverse momentum and Feynman variable of the heavy photon (di-lepton).
In the limit of small x 1 → 0 and large z p ≡ p + 4 /p + 2 → 1 the diffractive DY cross section is given by the Mueller graph shown in Fig. 4. In this case, the end-point behavior is dictated by the following general result where α IP (t) is the Pomeron trajectory corresponding to the t-channel exchange, and ε is equal to 1 or 1/2 for the Pomeron IP or Reggeon IR 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 x F → 1 of the recoil proton, or small is described by triple Regge graphs IP IP IP and IP IP IR as represented in Fig. 5, (aa) and (ab) respectively, were we also explicitly included radiation of a virtual gauge boson G * . Examples of Feynman graphs corresponding to the above triple-Regge terms, are shown in the second and third rows in Fig. 5. The graphs (ba) and (ca) illustrate the triple-Pomeron term in the diffraction cross section, with the gauge boson radiated by either a sea, (ba), or a valence quark, (ca). The effective radiation amplitude q + g → q + G is depicted by open circles and is defined in Fig. 3. These Feynman graph interpret the triple-Pomeron term as a diffractive excitation of the incoming proton due to radiation of gluons with small fractional momentum. The proton can also dissociate via diffractive excitation of its valence quark skeleton, as is illustrated in Fig. 5 (bb) and (cb). The corresponding term in the diffraction cross section reads, Again, the gauge boson can be radiated either by a sea quark, (bb), or by the valence quark, (cb).

B. Diffractive factorisation breaking in forward diffraction
As an alternative to the diffractive factorization based approach, the dipole description of the QCD diffraction, was presented in Refs. [11] (see also Ref. [35]). The color dipole description of inclusive Drell-Yan process was first introduced in Ref. [36] (see also Refs. [33,37]) and treats the production of a heavy virtual photon via Bremsstrahlung mechanism rather thanqq annihilation. The dipole approach applied to diffractive DY reaction in Refs. [6,7] and later in diffractive vector boson production [8] has explictly demonstrated the diffractive factorisation 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, i.e. the outgoing hadron is the same as the incoming one, so the interactions is elastic.
For illustration, consider diffractive photon radiation off a quark [22]. The corresponding framework has previously been used for diffractive gluon radiation and diffractive DIS processes in Refs. [22,38,39] 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 we get,Â with summation going through all octet-changed intermediate states {Y * 8 }. Then we switch to impact parameter representation, (3.10) Thus, the amplitude of the "screening" gluon exchange summed over projectile valence quarks j = 1, 2, 3 readŝ where b 1 ≡ b and b 2 ≡ b − α r 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, Ψ q→qγ is the distribution function for the qγ fluctuation of the quark, λ a = 2τ a are the Gell-Mann matrices from a gluon coupling to the quark, and the matricesγ a are the operators in coordinate and color space for the target quarks,γ which depend on the effective gluon mass Λ ∼ 100 MeV, and on the transverse distance between i-th valence quark in the target nucleon and its center of gravity, s i . Combining these ingredients into the diffractive amplitude (3.9) one should average over color indices of the valence quarks and their relative coordinates in the target nucleon |3q 1 . The color averaging results in, Finally, averaging over quark relative coordinates s i leads to 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 [22]. Then, the total amplitude, After Fourier transform one notices that in the forward quark limit q ⊥ → 0 the amplitude for single diffractive photon or any Abelian radiation vanishes, A( q ⊥ , κ)| q ⊥ →0 = 0, and 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 factorisation must be strongly broken.
The function S( b k , b l ) above is directly related to the qq dipole cross section as, Thus, following anological 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 factorisation. 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, where n is the total number of partons in the Fock state; f el 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 Eq. (3.12) 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 Fig. 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 two t-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, e.g. to a valence quark (see Fig. 3 in [22]). Apparently, such a contribution cannot be associated literally with either of the Regge graphs in Fig. 5. Nevertheless, this does not affect much the x F -and energy dependencies provided by the triple-Regge graphs, because the gluon has spin one.
It is also worth mentioning that in Fig. 5 we presented only the lowest order graphs with two gluon exchange. The spectator partons in a multi-parton 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 Fig. 5. They contribute to the diffractive amplitude Eq. (3.12) 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 quarkantiquark dipole does not vanish in forward direction, unlike the radiation by a single quark [6,22]. This can be understood as follows. According to the general theory of diffraction [1][2][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 quarkboson separation r, the quark gets a transverse shift ∆ r = α r. 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 Ref. [22] (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 Stan Brodsky and Paul Hoyer in Ref. [40].
Note, in all these calculations one assumes that the coherence time of radiation considerably exceeds the time interval between the two interactions, what 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", i.e. 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 [41]: 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 worth 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 Fig. 6. 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 the qq dipole is proportional to the difference between elastic amplitudes of the two σ qq ( r 1 − r 2 ) Fock components, |qq and |qqG [6], i.e. (3.14) Here, r i q/g , x i q/g 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 where all the partons not participating in the hard interaction are summed up; x 1 is the photon fraction taken from the initial proton; a = r 2 ch −1 is the inverse proton mean charge radius squared; R is a collinear multi-parton 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(x 1 , µ 2 ) for gluons and q(x 1 , µ 2 ) 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 , (3.16) after intergation over spectator impact parameters and momentum fractions with a proper color factor between quark and gluon PDFs. Note, 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 For diffractive gluon radiation one should account for both quark and gluon excitations whose amplitudes, however, are calculated in different ways [22]. 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, while the single diffractive-to-inclusive cross sections ratio behaves as assuming the saturated GBW shape of the dipole cross section (2.9) where x 2 is defined in Eq. (3.4). Thus, the soft part of the interaction is not enhanced in Drell-Yan diffraction which is semi-hard/semi-soft like inclusive DIS. Linear dependence on the hard scale r ∼ 1/M ≪ R 0 (x 2 ) means that even at a hard scale the Abelian radiation is sensitive to the hadron size due to a dramatic breakdown of diffractive factorization [34]. It was firstly found in Refs. [42,43] that factorization for diffractive Drell-Yan reaction fails due to the presence of spectator partons in the Pomeron. In Refs. [6][7][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 Eq. (3.17). The effect of diffractive factorisation breaking manifests itself in specific features of observables like a significant damping of the cross section at high √ s compared to the inclusive production case as illustrated in Fig. 7. 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 the DDY to DY cross sections was found in Ref. [6,7] to rise with the hard scale, the photon virtuality M 2 also shown in Fig. 7. This is also in variance with diffraction in DIS, which is associated with the soft interactions and where the diffractive factorisation holds true [10]. Such striking signatures of the diffractive factorisation 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 the higher twist [6,7]. But this is not the only source of the factorisation breaking -another important source is the absorptive (or unitarity) corrections.

C. 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 [44]. 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 factorisation.
It is well-known that the absorptive corrections affect differently the diagonal and offdiagonal terms in the hadronic current [5,45], in opposite directions, leading to an additional source of the QCD factorisation 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]. The amplitude Eq. (3.13) 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, where and 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, 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 Eq. (3.13) includes the effect of all absorptive corrections (gap survival amplitude), provided that the dipole cross section is adjusted to the data. Note, usually the survival probability factor is introduced into the diffractive cross section probabilistically, while in Eq. (3.13) it is treated quantum-mechanically, at the amplitude level. Data on diffraction show that diffractive gluon radiation is quite weak (due well known smallness of the triple-Pomeron coupling), and this can be explained assuming that gluons in the proton are located within small "spots" around the valence quarks with radius r 0 ∼ 0.3 fm [22,[46][47][48]. Such smallness of gluonic dipoles is an important nonperturbative phenomenon which may be connected e.g. to the small size of gluonic fluctuations in the instanton liquid model [47]. Therefore, the large distance between one valence quark and a satellite-gluon of the other quark is approximately equal (with 10% accuracy) to the quarkquark separation. Since a valence quark together with co-moving gluons is a color triplet, in our calculations the interaction amplitude of such an effective ("constituent") quark with the target is a coherent sum of the quark-target and gluon-target interaction amplitudes.
In addition to the soft gluons, which are present in the proton light-cone wave function at a soft scale, production of a heavy gauge boson certainly lead to an additional intensive hard gluon radiation. In other words, there might be many more spectator gluons in the quark which radiates the gauge boson. The transverse separation of those gluons is controlled by the DGLAP evolution. One can replace a bunch of gluons by dipoles [49] which transverse size r d varies from 1/M G up to r 0 , and is distributed as dr d /r d [50]. Therefore the mean dipole size squared, is about r 2 d ≈ 0.01 fm 2 , i.e. quite small. The cross section of such a dipole on a proton is also small, σ d = C(x) r d 2 , where according to Eq. (2.9) factor C(x) = σ 0 /R 2 0 (x) rises with energy. Fixing x = M 2 G /s and using the parameters fitted in Ref. [25] to DIS data from HERA we get at the Tevatron collider energy σ d ≈ 0.9 mb.
Presence of each such a dipole in the projectile light-cone wave function brings an extra suppression factor to the survival amplitude of a large rapidity gap, (3.23) We aimed here at a demonstration that the second term in (3.23) is negligibly small, so we rely on its simplified form (see more involved calculations in Ref. [51]), where B d is the dipole-nucleons elastic slope, which was measured at B d ≈ 6 GeV −2 in diffractive electro-production of ρ mesons at HERA [52]. We evaluate the absorptive correction (3.24) at the mean impact parameter b 2 = 2B d and for the Tevatron energy √ s = 2 TeV arrive at the negligibly small value Im f d (0, r d ) ≈ 0.01. However, the number of such dipole rises with hardness of the process,and may substantially enhance the magnitude of the absorptive corrections. The gap survival amplitude for n d projectile dipoles reads, The mean number of dipoles can be estimated in in the double-leading-log approximation to the DGLAP evolution formulated in impact parameters [50], the mean number of such dipoles is given by Here the values of Bjorken x of the radiated gluons is restricted by the invariant mass of the diffractive excitation, x > s 0 /M 2 X = s 0 /(1 − x F )s. For the kinematics of experiments at the Tevatron collider (see next section), 1 − x F < 0.1, √ s = 2 TeV, the number of radiated dipoles is not large, n d 6. We conclude that the absorptive corrections Eq. (3.25) to the gap survival amplitude are rather weak, less than 5%, i.e. about 10% in the survival probability. This correction is certainly small compared to other theoretical uncertainties of our calculations. Notice that a similar correction due to radiation of soft gluons was found in Ref. [51] for the gap survival probability in leading neutron production in DIS. We conclude that 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 Eq. (3.21) provides a sufficiently good approximation to the gap survival amplitude.
The popular quasi-eikonal model for the so-called "enhanced" probabilityŜ enh (see e.g. Refs. [20,53]), frequently used to describe the factorisation 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 to predict 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 Refs. [7,8]. Now we turn to a discussion of numerical results for the most important observables.

IV. SINGLE DIFFRACTIVE GAUGE BOSONS PRODUCTION
In Ref. [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 Z 0 , γ * and W ± bosons production ( √ s = 14 TeV), differential in the di-lepton mass squared dσ sd /dM 2 and its longitudinal momentum fraction, dσ sd /dx 1 are shown in Fig. 8 at left and right panels, respectively. The M 2 distributions here are integrated over the forward rapidity interval 0.3 < x 1 < 1. The mass distribution is integrated over the invariant mass interval 5 < M 2 < 10 5 GeV 2 . In the resonant Z 0 and W ± region the M 2 distributions exceed the corresponding DDY component; only for low masses the γ * contribution becomes important. As for x 1 distribution, the diffractive W + and γ * components are comparable to each other, whereas the W − and Z-boson production cross section are much lower. The precise measurement of differences in forward diffractive W + and W − rates would potentially allow to constrain quark content of the proton at large x q ≡ x 1 /α.
Another phenomenologically interesting observable is the di-lepton transverse momentum q ⊥ distribution at the LHC shown in Fig. 9 (left panel) for the di-lepton invariant mass, fixed at a corresponding resonance value -the Z or W mass. One of the important observables, sensitive to the difference between u-and d-quark PDFs at large x, the W ± charge asymmetry A W is shown in Fig. 9 (right panel). The ratio is independent on both the hard scale M 2 and the c.m. energy √ s. Due to different x-shapes of valence u, d quark PDFs, at smaller x 1 0.9 the diffractive W + bosons' rate dominates over W − one. However, at large x 1 → 1 the W − boson cross section becomes increasingly important and strongly dominates over the W + one. Similarly to diffractive DY discussed above, an important feature of the diffractive-to- inclusive gauge boson radiation cross sections ratio which makes these predictions different from ones obtained in traditional diffractive QCD factorisation-based approaches (see e.g. Refs. [17,18]), is their unusual energy and scale dependence demonstrated in Fig. 10. This ratio is independent of the gauge boson type, polarisation, and quark PDFs. In this respect, it is the most convenient and model independent observable, which is sensitive only to the structure of the universal elastic dipole amplitude (or the dipole cross section), and can be used as an important probe for the QCD diffractive mechanism for forward diffractive reactions, essentially driven by the soft interaction dynamics. In analogy to DDY case, this ratio behaves w.r.t. the energy and the hard scale in opposite way to what is expected in the diffractive factorisation-based approaches. Therefore, measurements of the single diffractive gauge boson production cross section in the di-lepton channel which enhanced compared to DDY around the Z 0 and W ± resonances would provide important information about the interplay between soft and hard interactions in QCD.

V. 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?
A. Leading-twist diffractive heavy flavor production One might naively think that the Abelian (or DIS) mechanism of heavy flavor production γ * → QQ is of the leading twist as well since it behaves as ∼ 1/Q 2 . However, in the limit m 2 Q ≫ Q 2 the corresponding cross section σ sd ∝ 1/m 4 Q i.e. behaves as a higher twist process. One has to radiate at least one gluon off the QQ pair for this process to become the leading twist one, e.g. σ sd (γ * → QQg) ∝ 1/m 2 Q , 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 α s .
Consider now the non-Abelian mechanism for diffractive hadroproduction of heavy quarks via g * → QQ hard subprocess. Production of heavy quarks at large x F → 1 is a longstanding controversial issue even in inclusive processes. On one hand, QCD factorisation approach predicts vanishingly small yields of heavy flavor due to steeply falling gluon density as ∼ (1 − x F ) 5 at large x F . On the other hand, the end-point behavior is controlled by the universal Regge asymptotics dσ/dx F (x F → 1) ∝ (1 − x F ) 1−2α R (t) in terms of the Regge trajectory of the t-channel exchange α R (t). Apparently, the Regge and QCd factorisation approaches contradict each other. The same problem emerges in the DY process at large x F as is seen in data [54] which means that in the considering kinematics the conventional QCD factorisation does not apply [55]. At the same time, the observation of an excess of diffractive production of heavy quarks at large x F → 1 compared to conventional expectation may provide a good evidence for intrinsic heavy flavors if the latter is reliably known. Calculations assuming that diffractive factorisation holds for hard diffraction [12,56] 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 Ref. [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 Ref. [57]).
Typical contributions to the single diffractive QQ production rate are summarized in Fig. 11. Diagrams (a) and (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 m Q the diagram (a) with Pomeron coupling to a small-size gQQ system is of the higher twist due to color transparency and is therefore suppressed. Diagram (b) involves two scales -the soft hadronic one ∼ Λ QCD associated with large transverse separations between a gluon and constituent valence quarks, and the hard one ∼ m Q associated with small QQ dipole. An interplay between these two scales similar to that in DDY emerges as the leading twist effect; thus, diagram (b) is important. Possible higher order terms with an extra gluon radiation contributing to the leading twist diffractive heavy flavor production were disscussed in detail in Ref. [9]. Diagrams (c) and (d) account for QQ production via diffractive quark excitation. Just as in leading twist diffraction in DIS γ * → QQg, these processes are associated with two characteristic transverse separations, a small one, ∼ 1/m Q , between theQ and Q, and a large one, either ∼ 1/m q between q and QQ (diagram (c)) or 1/Λ QCD between another constituent valence quark and QQ (diagram (d)). While all the terms contributing to (d) are of the leading twist (see Ref. [9]), only a special subset of diagrams (c) are of the leading twist. Indeed, the hard subprocess q + g → (QQ) + q is characretized by five distinct topologies illustrated in Fig. 12, and similar graphs are for gluon-proton scattering with subprocess g + g → (QQ) + g.
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 Fig. 2 and Appendix A in Ref. [9]). The FIG. 12: Five different topologies contributing to inclusive QQ production in quark-proton scattering. These can be split into two gauge-invariant subsets of amplitudes as described in the text.
BR mechanism includes the same graphs as radiation of a gluon (see Refs. [38,58]), i.e. interaction with the source parton before and after radiation, and interaction with the radiated gluon. The PR mechanism, responsible for the transition g →QQ, includes the interactions with the gluon and the producedQQ (also known as gluon-gluon fusion gg → QQ mechanism). The total amplitude is 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 Ref. [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, σ gqq and σ gQQ respectively. Another physical reason for such a separation is different scale dependence of the BR and PR components. Introducing the transverse separations r, r 1 and r 2 within theQQ, 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 − β r 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 are given in terms of the effective dipole cross sections for a colorless gqq and gQQ systems, and rather complicated wave functions Φ of subsequent gluon radiation and then its splitting intoQQ pair in both cases. In the case of bremsstrahlung, both mean separations are controlled by the hard scale such that thus, the corresponding contribution is a higher twist effect and thus suppressed (note, in the case of forward Abelian radiation this contribution is equal to zero). On the contrary, in the production mechanism only theQQ separation is small, s 2 ∼ 1/m 2 Q , the second scale appears to be soft, ρ 2 ∼ 1/m 2 q , leading to the leading twist behavior 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. FIG. 13: The total cross section of diffractive cc, bb and tt pairs production as function of energy in comparison with experimental data from E690 [59] and CDF [60] experiments (left panel) and the differential cross section as function of fraction x 1 of the initial proton momentum carried by the charm quark (right panel) [9].
The situation with scale dependence in the case ofQQ 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 Fig. 11 and color averaging over the projectile proton wave function. Although bremsstrahlung terms from diagrams (b), (d) are formally of the leading twist due to interactions with distant spectator partons, numerically they are always tiny due to denominator suppression by a largeQQ 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 in diffractive Drell-Yan in the considering 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 → QQX + p, as functions of c.m.s. pp energy are shown in Fig. 13. The experimental data points available from E690 [59] and CDF [60] experiments have been compared with theoretical predictions evaluated with corresponding phase space constraints (for more details, see Ref. [9]).

B. 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 becomes a crutial 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. Refs. [61][62][63][64]).
The QCD-initiated gluon-gluon fusion gg → 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. Refs. [65][66][67][68] 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 [61]. The forward inclusive and diffractive Higgsstrahlung off intrinsic heavy flavor at x F → 1 has previously been studied in Refs. [69,70], respectively.
Very recently, a new single diffractive production mode of the Higgs boson in association with a heavy quark pairQQ, namely pp → X + QQH + p, at large x F where conventional factorisation-based approaches are expected to fail has been studied in Ref. [19]. 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 diffractiveQQ production discussed above. For a reasonably accurate estimate one retains only the dominant gluon-initiated leading twist terms illustrated in Fig. 14 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 are 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 Fig. 14. 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 [19]. 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. Refs. [9]). This becomes obvious in the colour dipole framework due to colour transparency [11] making the medium more transparent for smaller dipoles.
To a good approximation, it is instructive to study basic features of this process in the limit of small momentum fraction α 3 of the Higgs boson taken off the parent quark compared to the quark fraction α taken from the parent gluon, i.e. α 3 ≪ α. The hard scales which control the diffractive Higgsstrahlung process are, r 2 ∼ 1/m 2 Q and ρ 2 ∼ 1/τ 2 , where τ 2 = M 2 H + α 3 M 2 QQ in terms of the Higgs boson mass, M H , and the QQ pair invariant mass, M QQ . Another length scale here is the distance between ith and jth projectile partons, r 2 ij ∼ R 2 , is soft for light valence/sea quarks in the proton wave function. Before the hard gluon splits intoQQH 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 can be accounted for by the gluon PDF evolution (3.16). The differential cross sections of the single diffractive bb and tt production in association with the Higgs boson are shown in Fig. 15 as functions of Higgs boson transverse momentum k 3 (left panel) and relative transverse momentum between Q andQ quarks k 12 (right panel) at the LHC energy √ s = 14 TeV. Besides, the corresponding inclusive pp → X + (bbH) and pp → X + (ttH) + p cross sections are shown for comparison. The contribution of diffractive gluon excitations to the Higgsstrahlung dominate the total Higgsstrahlung cross section due a large yield from central rapidities.
Analogically to other diffractive bremsstrahlung processes discussed in previous sections, breakdown of QCD factorization leads to a flatter scale dependence of the cross section, 1/m 2 Q compareed to 1/m 4 Q dependence of diffractive DIS. This is a result of leading twist behaviour which have been discussed above and which has been confirmed by the comparison of data on diffractive production of charm and beauty [9]. Additional radiation of the Higgs boson enhances the contribution of heavy quarks and thus compensates the smallness of their diffractive production modes. Fig. 15 illustrates that single diffractive Higgsstrahlung is of the same order for top and beauty while the former dominates. Indeed, the contribution from top quarks into both SD and inclusive production channels turn out to be larger in magnitude and flatter than those of the bottom quarks (the slope for beauty is larger than for top). This is due to a relative hardening of the diffractive Higgs boson spectrum for heavier quarks similarly to that in the inclusive case. This could be anticipated, since the main fraction of the transferred momentum originates from the short distance interaction, which is the same in inclusive and diffractive processes. The charm contribution is suppressed by a few orders of magnitude, and thus is not shown. The SD-to-inclusive ratio of the cross sections for different c.m. energies √ s = 0.5, 7, 14 TeV and for two distinct rapidities Y = 0 and 5 as functions ofQQH invariant mass M are shown in Fig. 16. The diffraction-to-inclusive ratio is similar to that for heavy quark production [9] and thus in good agreement with experimental data from the Tevatron. Note, this ratio has falling energy-and rising M-dependence, where M is the invariant mass of the producedQQH 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.

VI. SUMMARY
In this short review, we have discussed major properties and basic dynamics of single diffractive processes of γ * , Z 0 and W ± bosons production processes at the LHC, as well as leading twist heavy flavor hadroproduction at large Feynman x F and diffractive Higgsstrahlung off heavy quarks. We outlined the manifestations of diffractive factorisation 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 factorisation, is 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 [5,45], leading to a breakdown of diffractive QCD factorisation in hard diffractive processes, like diffractive radiation of heavy Abelian particles and heavy flavors. The dipole approach enables to account for the absorptive corrections automatically at the amplitude level.
The second, more sophisticated reason for diffractive factorisation 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 co-moving spectator partons.
These mechanisms of diffractive factorisation breaking lead to rather unusual features of the leading-twist diffractive Abelian radiation w.r.t. its hard scale and energy dependence.
The outlined sources of factorisation breaking are also presented in diffractive radiation of non-Abelian particles. Interactions of the radiated gluon makes 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.