Minireview on diffraction

A short review is presented on diffractive excitation in high energy collisions. Figure 1. In optics a hole is equivalent to a black absorber.

Abstract.A short review is presented on diffractive excitation in high energy collisions.

Optical analogy and Good-Walker
In diffraction in optics, a hole is equivalent to a black absorber, with a forward peak with an angular width θ ∼ λ/(opening width), see fig. 1. Rescattering is described by a convolution in transverse momentum space, which corresponds to a product in transverse coordinate space.This implies that diffraction and rescattering is more easily described in impact parameter space.
The optical theorem says that Here the sum runs over all inelastic channels j.For a structureless projectile (e.g. a photon), diffraction corresponds to elastic scattering driven by absorption.If the absorption a e-mail: gosta.gustafson@thep.lu.se probability in Born approximation is given by 2F, then rescattering exponentiates in b-space, giving and the optical theorem in eq. ( 1) gives: 4)

Diffractive excitation
As an example we can look at a photon in an optically active medium.Here righthanded and lefthanded photons move with different velocities, meaning that they propagate as particles with different mass.Study a beam of righthanded photons hitting a polarized target, which absorbes photons polarized in the x-direction.The diffractively scattered beam is then a mixture of right-and lefthanded photons.If the righthanded photons have lower mass, this means that the diffractive beam contains also photons excited to a state with higher mass.

Good-Walker formalism
For a projectile with a substracture, the mass eigenstates can differ from the eigenstates of diffraction.Call the diffractive eigenstates Φ n , with elastic scattering amplitudes T n .The mass eigenstates Ψ k are linear combinations of the states Φ n : The elastic scattering amplitude is given by and the elastic cross section The amplitude for diffractive transition to the mass eigenstate Ψ k is given by  which gives a total diffractive cross section (including elastic scattering) Consequently the cross section for diffractive excitation is given by the fluctuations: 3 Soft diffraction

Reggeon theory
Pomeron exchange is described by a ladder exchanged between the projectile and the target, as illustrated in fig. 2. The elastic and total cross sections are given by dσ el /dt ∼ (g 2 • s α(t)−1 ) 2 = g 4 s 2(α(0)−1) e 2(ln s)α t (12) Note that α(0) > 1 implies that σ el > σ tot for large s, which means that multi-pomeron exchange must be important.Inelastic diffraction is described by the Mueller triple-Regge formalism, illustrated in fig. 3. The triple-pomeron contribution to the cross section is given by where g 3P denotes the triple-pomeron coupling.
The triple (and multiple) pomeron couplings give loops, as illustrated in fig.4, which leads to complicated resummation schemes.The diagrams can also contain multipomeron vertices, see fig. 5.
In particular three groups have studied these problems: Tel Aviv (GLM) [2], Durham (KMR) [3], and Ostapchenko [4] (based on work by Kaidalov and coworkers).In this approach low-mass diffraction is included with the Good-Walker formalism, approximated by one excited state N * , while high-mass diffraction is treated with the triple-regge formalism.At lower energies or excitations  also reggeons with α(0) ≈ 0.5 are included besides the pomeron.The regge intercepts and couplings are fitted to experimental data.We note here that these fits have been significantly modified after the presentation of the Totem data at 7 TeV, with σ tot = 98.6±2.2 mb and σ el = 25.4±1.1 mb [5].
• The Tel-Aviv group [2] has a single pomeron, with α P (0) = 1.23 and α ≈ 0. This implies that the pomeron propagator is approximately a delta-function, δ(b), with no diffusion in b-space.Only 3-pomeron vertices are included.
• The Durham group [3] has in its new version from 2014 a single "effective" pomeron with couplings dependent on k ⊥ .It interpolates between a "bare I P" with α P (0) ≈ 1.3 and α small, and a "soft I P" with α P (0) ≈ 0.08 and α = 0.25.Multi-pomeron couplings are large, with g n,m ∝ n m γ n+m .
• Ostapchenko [4] has a formalism with two pomerons, with α P (0) so f t = 1.14 and α P (0) hard = 1.31.The multipomeron couplings are fixed by the relation g n,m ∝ γ n+m , and thus not growing with n and m as fast as assumed by the Durham group.
The results obtained for the single diffractive cross section are presented in fig.6, reproduced from Cartiglia [6].The different experiments have different acceptance range in M X , and adjusting for this there is a general agreement between the results.We note in particular that Atlas and CMS have similar results, and also that these high energy results agree with tunes presented after the presentation of cross section data from Totem.
The models can also reproduce the dependence on the excitation mass M X .As an example fig.7 shows  X .Data from Atlas [7] and model calculations from ref. [3].
X from KMR compared with data from Atlas.

Mueller's dipole model
As mentioned above, unitarity constraints and saturation are much easier to account for in transverse coordinate space.Mueller's dipole model [8,9] is a formulation of LL BFKL evolution in impact parameter space.A color charge is always screened by an accompanying anticharge.A charge-anticharge pair can emit bremsstrahlung gluons in the same way as an electric dipole.The probability per unit rapidity for a dipole (r 0 , r 1 ) to emit a gluon in the point r 2 , is given by (cf fig.8) The important difference from electro-magnetism is that the emitted gluon carries away colour, which implies that the dipole splits in two dipoles.These dipoles can then emit further gluons in a cascade, producing a chain of dipoles as illustrated in fig.8.A dipole can radiate a gluon.The gluon carries away colour, which implies that the dipole is split in two dipoles, which in the large N c limit radiate further gluons independently.When two such chains, accelerated in opposite directions, meet, they can interact via gluon exchange.This implies exchange of colour, and thus a reconnection of the chains as shown in fig.9.The elastic scattering amplitude for gluon exchange is in the Born approximation given by BFKL evolution is a stochastic process, and many subcollisions may occur independently.Summing over all possible pairs gives the total Born amplitude The uniterized amplitude becoms and the cross sections The Lund cascade model DIPSY The DIPSY model [10][11][12] is a generalization of Mueller's cascade, which includes a set of corrections: • Important non-leading effects in BFKL evolution.
Most essential are those related to energy conservation and running α s .
• Saturation from pomeron loops in the evolution.Dipoles with identical colours form colour quadrupoles, which give pomeron loops in the evolution.These are not included in Mueller's model or in the BK equation.
• Confinement via a gluon mass satisfies t-channel unitarity.Figure 12.The virtual BFKL cascades are assumed to represent eigenstates for diffractive scattering.When they interact with a target, it can be absorbed in an inelastic event, give elastic scattering or diffractive excitation.
• The DIPSY MC gives also fluctuations and correlations.
• It can be applied to collisions between electrons, protons, and nuclei.
Some results for pp total and elastic cross sections are shown in figs.10, 11 [13].(Here the initial proton wave function is approximated by three dipoles in a triangle.)We note that there is no input structure functions in the model; the gluon distributions are generated within the model.

Good-Walker vs triple-regge
It is natural to assume that the diffractive eigenstates for a colliding proton are the BFKL cascades, which can come on shell through interaction with the target, as illustrated in fig.12, cf refs.[14][15][16][17].
These diffractive states have a continuous distribution up to high masses, with large fluctuations.As demonstrated in ref. [18], calculating diffractive excitation via the Good-Walker or the triple-regge formalism, is just different formulations of the same phenomenon.An essential feature of the BFKL cascade is its stochastic nature.If the probability for a dipole split is given by dP/dy ∼ λ, then the average number of dipoles, and the variance grow according to Thus the distribution satisfies approximate KNO scaling.For two colliding cascades, evolved to rapidities y 1 and y 2 respectively, we have s ≈ exp(y 1 + y 2 ) = exp(Y), with Y ≡ y 1 + y 2 .Assuming a dipole-dipole interaction probability 2 f , we get for the bare pomeron exchange (neglecting unitarization effects) This corresponds to a pomeron intercept α(0) = 1 + λ In the triple-regge formalism, the triple-pomeron diagram in fig.13 gives the following result for the integrated single diffractive cross section with M 2 X < M 2 max ≈ exp(y 1 ): In the Good-Walker formalism this cross section is determined by the fluctuations.For projectile excitation but intact target we obtain We see that we get exactly the same expression in the two approaches.Most essential for this result is the approximate KNO scaling.We also note that the DIPSY results, calculated with the Good-Walker formalism, have the expected triplepomeron form.The results for total, elastic, and singlediffractive cross sections, calculated for the bare pomeron (meaning the Born amplitude without saturation effects) are shown in fig.14.We see that the cross sections grow EPJ Web of Conferences 06002-p.4 as powers of s, in accordance with a regge fit with a single pomeron pole with parameters α(0) = 1.21, α = 0.2 GeV −2 (26) g pP (t) = (5.6GeV −1 ) e 1.9 t (27) g 3P (t) ≈ 1GeV −1 (dep.on def.) (28) We note also that when unitarization is omitted, the elastic cross section is larger than the total for √ s > 2 TeV.Fig. 15 shows preliminary results for dσ S D /d ln M X at 7 TeV, which should be compared with the LHC data in fig. 7. (Note that the diffractive mass grows in opposite directions in the two figures.)

Hard diffraction
Factorization and factorization breaking UA8 at the CERN S p pS collider (consisting of the UA2 central detector plus roman pots at 630 GeV) observed high p ⊥ jets in diffractive events [19].Jets have also been observed in gap events at HERA and the Tevatron.These events are often analyzed within the Ingelman-Schlein model [20], which assumes that the pomeron has a universal parton substructure f P q,g (z, Q 2 ).This implies that the diffractive cross section factorizes (cf fig.16):  Here x P is the energy fraction carried by the pomeron, and the sum runs over different parton species i.
A fit with DGLAP evolution to Hera DIS data for hard and soft diffraction by Zeus [21] is shown in fig.17, together with a comparison with data for the distribution in the observed parton fraction of the pomeron momentum, z obs P .We note in particular that the distributions are gluon dominated.The Ingelman-Schlein model is implemented in a number of MC generators, e.g.POMPYT, Pythia8, and POMWIG.
Factorization was proved by Collins for hard scattering in DIS [22].Results from the Tevatron showed, however, that factorization is strongly broken when comparing diffractive two-jet events in DIS and pp collisions [23].In pp scattering the gaps become frequently filled by soft interactions.Figure 18 shows the ratio between single diffractive and non-diffractive events (R = S D/ND) from CDF, which are a factor 0.1 -0.2 below the corresponding DIS data.Similarly fig.19 shows a fit to data for dN/d ξ ≈ dN/d(M 2 X /s), where the pomeron flux is renormalized in the MC with a gap survival probability ≈ 0.2 [24].

Similarities between diffractive and non-diffractive scattering
CDF has noted that there are strong similarities between jet production in single diffractive and nondiffractive events, hinting at a similar dynamical origine.Fig. 20 shows the distribution of two-jet events vs mean  E ⊥ , for diffractive and non-diffractive events, which indicates that the same hard subprocess is at work.There is a gap survival probability S 2 , but no extra suppression ∼ 1/Q 2 for diffractive events.This is consistent with Goulianos' empirical "renormalized pomeron" [26], and also with the assertion that hard diffraction is leading twist by Kopeliovich et al. [27].
The gap survival probability for multiple gaps is difficult to calculate.Some processes with multiple gaps are shown in fig.21.CDF has studied the ratios 2-gap/nogap (SDD/SD) and one-gap/no-gap (DD/tot).The results, reproduced in fig.22, show that multiple gaps are not multiply suppressed.This feature is also consistent with Goulianos' renormalized pomeron [26].

Central exclusive production
Many schemes are proposed for gap survival in central exclusive production (see e.g.refs.[31][32][33][34]).• Jet-gap-jet events in double diffraction, as a means to study BFKL evolution.
• Study γγ → γγ or γγ → W + W − , which can give information about possible anomalous weak couplings.

Soft diffraction
Several groups have presented analyses of elastic and diffractive cross sections within the Regge formalism.They can contain either one pomeron (GLM), a soft and a hard pomeron (Ostapchenko), or a pomeron interpolating between soft and hard (KMR).Unitarization is taken into account by summation of pomeron loop diagrams, where the results also depend on the assumptions made for multipomeron vertices, which vary between the groups.For lower masses, M X , contributions from low-lying reggeon trajectories are also important, and have to be included with extra parameters.The Regge-based formulations also generally include production of low mass N * resonances within the Good-Walker formalism, approximating it with two or three diffractive eigenstates.
High mass diffraction has, however, also been described using the Good-Walker formalism.The dynamics of BFKL evolution implies large fluctuations, where the gluon multiplicity satisfies approximate KNO scaling.This implies that Good-Walker reproduces the regge form for diffractive excitation.Thus the triple-pomeron and Good-Walker formalisms actually describe the same physics.The Good-Walker formalism can here have the advantage that the results do not depend on new tunable parameters.
(When comparing theory with data, we note that at high energies, low mass diffraction is very difficult to measure experimentally, and its behaviour is therefore less well-known.)

Hard diffraction
Hard diffraction is commonly analyzed by means of the Ingelman-Schlein formalism, assuming a factorized form with a partonic structure for the pomeron.Factorization is, however, strongly broken when comparing data for pp and γp scattering, due to soft exchange in pp re-actions.The survival probability in pp collisions is estimated to ∼ 0.1 − 0.2 at the Tevatron, and ∼ 0.03 at LHC.Data from the Tevatron indicate that the same hard subprocess is at work in diffractive and non-diffractive hard processes, and that multiple gaps are not multiply suppressed.These features are in agreementt with Goulianos' empirical renormalized pomeron.
The LHC detectors have larger acceptance in rapidity than the detectors at Hera or the Tevatron, which implies that we can look forward to many interesting analyses using roman pots at the LHC.

Figure 1 . 1 Content 1 .
Figure 1.In optics a hole is equivalent to a black absorber.

Figure 4 .
Figure 4. Examples of pomeron loop diagrams (copied from the Tel-Aviv group).

Figure 5 .
Figure 5. Multi-pomeron vertices.Left: A cut through an event with diffractive excitation.Right: A vertex with coupling between n and m pomerons.

Figure 8 .
Figure 8.A colour dipole cascade in transverse coordinate space.A dipole can radiate a gluon.The gluon carries away colour, which implies that the dipole is split in two dipoles, which in the large N c limit radiate further gluons independently.

Figure 9 .
Figure 9.In a collision between two dipole cascades, two dipoles can interact via gluon exchange.As the exchanged gluon carries colour, the two dipole chains become recoupled.

Figure 10 .
Figure 10.DIPSY results for total and elastic pp cross sections, compared to experimental data.

Figure 11 .
Figure 11.DIPSY results for the differential elastic pp cross section.

Figure 16 .
Figure 16.A diagram for hard diffraction in the Ingelman-Schlein model.

Figure 17 .
Figure 17.Pomeron pdf:s determined by Zeus, together with a comparison with data for the distribution in the observed parton fraction of the pomeron momentum, z obs P [21].

Figure 18 .
Figure 18.The ratio R = S D/ND.Data from CDF compared to expectation from NLO fit to HERA data, assuming factorization [25].

Figure 19 .
Figure 19.Data on dN/d ξ ≈ dN/d(M 2X /s) measured by CMS, compared with expectations with a pomeron flux rescaled by a factor 0.2[24].

Fig. 23
shows a diagram including eikonal and enhanced survival factors from the Durham group.Gap sur-