20. High Energy Soft QCD and Diffraction

1

20. High Energy Soft QCD and Diffraction
Revised August 2023 by V.A. Khoze (Durham U.), M.G. Ryskin (Petersburg Nuclear Phys. Inst.)
and M. Taševský (Prague, Inst. Phys.).

20.1 Introduction
Despite the enormous successes of Quantum Chromodynamics (QCD) (see Section 9 in [1]and [2]) there remain a number of deep questions to be answered in the domain of strong interaction
physics. These concern first of all small momentum transfer processes which are generically calledsoft interactions.
One of the most challenging problems is the high-energy behaviour of hadronic scattering pro√
cesses. At high collision energies, s, soft interactions play a dominant role. Unfortunately,
soft interactions cannot be described in terms of perturbative QCD. These are non-perturbative
phenomena related to confinement which are generally considered in the context of the analytic
S-matrix, based on first principles, such as analyticity, crossing symmetry and unitarity of partial
waves, see e.g. [3, 4]. At high energies the most self-consistent way to perform the calculations andto describe the data is the Regge approach (see for example [5–7]), which will be considered below.
As discussed in Section 20.5, this formalism could be smoothly matched with perturbative QCD
calculations at larger transverse momenta. Therefore, here we will concentrate on the propertiesof high energy soft interactions that can be expected from the extension of the perturbative QCD
domain.
The main aim of this review is to present the well-established theoretical framework, basedon Regge theory and QCD, used for describing high-energy collisions. A limited number of somenew experimental results, mainly from the LHC, are shown in order to demonstrate that the grossfeatures of the data are in agreement with this approach. We are not focussing on any particular
phenomenological or Monte Carlo model, which are covered in the dedicated reviews and books,
see e.g. Section 43 in [1], [2, 8–14] and Chapter 2 in [15].
Typically, in multiparticle production, the secondaries1 fill the whole available rapidity interval.2
However, there exists an important class of events in which a large interval of rapidity (typicallyat least 4 units) is devoid of any hadronic activity. Such an interval is called a Large Rapidity
Gap (LRG). The most frequent case with a LRG is elastic scattering. There are also events inwhich one of the incoming protons (or both) is transformed (dissociates) into a set of two or more√
final state particles with the mass M  s and proton quantum number. All these events haveproperties similar to those of the well-known from optics pattern of diffraction of a beam of lighton an obstacle. By analogy, in high-energy physics, the corresponding processes are usually called
diffractive. The classic example is the elastic scattering of hadrons on nuclei (see e.g. [16]), which
manifests an angular distribution with a series of minima and maxima, analogous to the diffractionof light on a black disk. At LHC energies diffractive processes constitute up to 40% of the total(pp) cross section, σtot . Therefore, we will pay special attention to the description of the elastic
scattering amplitude and proton diffractive dissociation. Diffraction dissociation can be considered
as a quantum mechanical process caused by the fact that different components of the incoming
hadron wave function have different probabilities for interaction with a target [17]. This featureallows us to probe the transverse size of the interaction region.
Note that besides being of a fundamental interest in their own right for understanding the highenergy behaviour of the QCD amplitude, there are several reasons why it is important to study
1

Here and in what follows, we call secondaries the new particles produced in the course of the interaction.
E+pz
For definition of particle rapidity (pseudorapidity), see Section 49.5.2 in [1]; y = 12 ln E−pz(η = − ln(tan(θ/2)); the correct variable is the rapidity y, however, experimentally it is simpler to use the pseudorapidity η which does not require identifying the particles, setting m = 0. For pT  m, η ' y.
2

S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
31st May, 2024 10:19am

20. High Energy Soft QCD and Diffraction

2

soft and diffractive processes. Firstly, soft interactions unavoidably give an underlying component
to rare ‘hard’ events, from which we hope to extract signals for New Physics. Secondly, we shouldbe able to estimate the probability that rapidity gaps, which occur in ‘hard’ diffractive events,
survive rescattering effects, that is, survive the population of the gaps by the secondary particles
from the underlying event. Thirdly, an understanding of diffractive processes is very important forevaluation of pile-up backgrounds in high-luminosity pp collisions, which have a direct impact onvarious experimental measurements. Pile-up corresponds to soft independent interactions in thesame bunch crossing whose number rises with increasing instantaneous luminosity. And, finally,studies of diffractive processes should help in the understanding of the structure of high-energy
cosmic ray cascades, which requires a very detailed knowledge of the spectra of particles carrying
a large fraction x of the incoming momentum in proton-air and nucleus-air interactions, see for
instance [18].
Experimentally, diffractive processes are selected using two distinct features:
1. large regions (typically at least ∆η > 4) in the detector are devoid of hadronic activity (LRG)
and/or
2. one or both incoming particles stay intact after collision and are registered by the dedicatedforward detectors placed a few hundred meters from the interaction point. The momentum
loss of the initial particle, ξ = 1 − x, is typically smaller than 0.15.
Thus, in the case of proton-proton collisions, diffractive events correspond to elastic pp → pp
scattering and to pp → p + X (Single Dissociation, SD) and pp → X + Y (Double Dissociation, DD)processes, where the + sign denotes a large rapidity gap. Note that strictly speaking in high energy
physics it is impossible to define (and select) rigorously purely diffractive events. We can always
have some admixture of events of different origin. As a rule we call ’diffractive’ the events with
sufficiently large gap (with say ∆y > 4, see above) and the vacuum quantum numbers transferred
across the gap. Typically at the LHC the integrated cross sections of diffractive dissociation, σSD ,σDD , are of the order of 5–10 mb depending on the gap size. Schematic diagrams of all discussed
processes are shown in Fig. 20.1.

20.2 Regge pole approach
In pre-QCD times, in order to describe the behaviour of scattering amplitudes at high en√ergy, s, and small momentum-transfer squared, −t, Regge theory was developed and successfully
applied in a wide range of energies. The Regge approach [5–7] is based on the singularities ofamplitudes in the complex angular momentum, j, plane.
p

p

p

p

p

p

p

p

(a)

Y

X
(c)

(b)

p

p

p
X
(d)

Figure 20.1: Schematic diagrams of soft pp processes. (a) non-diffractive processes, (b) elastic
scattering, (c) single dissociation and (d) double dissociation. The double line corresponds to the
Pomeron exchange.
For instance, the measured π − p → π 0 n amplitude behaves as
Tπp (s, t) ∝ sαρ (t) ,31st May, 2024

(20.1)

20. High Energy Soft QCD and Diffraction

3

where the process is described by the exchange of the ρ-trajectory, j = αρ (t) ' 0.5 + 0.9t (witht = (pπ− − pπ0 )2 in GeV2 ). This trajectory passes through the spin-1 ρ-meson resonance in the‘crossed’ t-channel π − π 0 → p̄n; that is, αρ (t = m2ρ ) = 1. The corresponding cross section decreases
with increasing s.

Figure 20.2: Illustration of the optical theorem for the total cross section and for high-massdiffractive dissociation in the absence of absorptive corrections.
On the other hand, high-energy total and elastic pp cross sections are observed to grow slowly
with energy (see e.g. Section 53 in [1]) and in terms of Regge theory are dominated by the exchange
of a trajectory with vacuum quantum numbers, σtot ∝ sj−1 . The simplest possibility is to assume
that the rightmost singularity in the j-plane, which drives the high-energy behaviour of the crosssection, is the leading (at t ≤ 0) Regge pole at j = α(t). Then the pp elastic amplitude reads
Tel (s, t) ∝ sαP (t) .

(20.2)The total cross section can then be conveniently expressed using the so called optical theorem whichstates thatsσtot = ImTel (s, t = 0),(20.3)
as illustrated in the upper part of Fig. 20.2, and thus
σtot ∝ sαP (0)−1 .

(20.4)The pole with the largest intercept, originally assumed to be αP (0) = 1 since high-energy total
cross sections were thought to have a constant asymptotic behaviour, is called the Pomeron3
Prior to the LHC, the energy behaviour of pp, pp̄, πp, Kp cross sections was satisfactorily reproduced by the sum of the Pomeron and secondary Reggeons (the poles at lower values of j,
typically with αρ (0) ' 0.5, see [22, 23] and Section 51 in [24]). However, above Tevatron energies
the secondary Reggeon contributions (which all have intercepts α(0) ' 0.5) are highly suppressed,
which enables us to study the properties of the Pomeron only.
A popular parameterization of the elastic pp-scattering amplitude by Donnachie-Landshoff (DL)is the Regge form [25]Tel (s, t) = ηP σ0 F12 (t)sαP (t) ,(20.5)
3

Pomeron pole was named after I. Y. Pomeranchuk. The history of the Pomeron is discussed in [19–21].

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20. High Energy Soft QCD and Diffraction

where σ0 = 21.7 mb [26] and ηP is the signature factor
ηP =1 + exp(−iπαP (t)),
sin(−παP (t))(20.6)

F1 is the Dirac electromagnetic form factor of the proton and the effective Pomeron trajectoryαP (t) = 1 + ∆ + α0 t ' 1 + 0.0808 + 0.25t,(20.7)

with t given in GeV2 . The intercept αP (0) just above 1 reproduces the observed slow growth of thetotal hadron-hadron cross sections at high energies.
However, this simple parameterization is becoming increasingly deficient at higher energies. Thisis because due to unitarity we have to take into account not only Regge poles, but also the cutsin the j-plane [27, 28], which correspond to the multiple exchange of Regge poles in the t-channel,
see for instance [29–31]. A powerful technique to evaluate Reggeon diagrams was developed by
Gribov [7,32] (Reggeon calculus or Reggeon Field Theory (RFT)), which allows us to calculate themulti-Pomeron contributions.

20.3 Theoretical description of high-energy diffraction
Diffractive processes (see e.g. reviews [33–37]) represent a rich testing ground for the dynamics
of soft interactions as well as Monte Carlo models for soft hadron-hadron physics (see for reviews
e.g. [8], Section 43 in [1] and Chapter 2 in [15]).
There is no universally agreed definition of diffractive processes. Theoretically, diffraction is theeffect caused by the absorption of the incoming plane-wave in some region of impact parameter,b. After a decomposition of the distorted plane-wave over the outgoing momentum, q, due toabsorption we arrive at some set of plane-waves with non-zero transverse momentum, qt 6= 0.
Experimentally, we call diffractive the events with large rapidity gaps (LRG) in the distribution of
the final state particles. However, this definition is appropriate only for the events with very large
gap sizes (∆η > 4 − 5); otherwise gaps can also be caused by fluctuations in the hadronization
process [38].
In the case of proton-proton collisions, diffraction corresponds to elastic pp → pp scattering andto the pp → p + X and pp → X1 + X2 processes where one or both protons are allowed todissociate into a system X with the quantum numbers of the proton. The p → X dissociation iscaused by the fact that the individual components of the incoming proton wave function interact
differently with the target (see Section 20.3.1).
Theoretically, high-energy diffraction may be studied from either the s-channel or the t-channel
viewpoint.
20.3.1 Diffraction from the s-channel viewpoint
Unitarity plays a central role in diffractive processes. To discuss unitarity effects it is convenient
to work in terms of impact parameter, b. The total cross section is closely related to the elastic
scattering amplitude and the scattering into inelastic final states via the s-channel unitarity of the
S-matrix (see Sections 50 and 53 in [1]), SS † = I, ordisc T ≡ T − T † = iT † T(20.8)

with S = I + iT . If we were to focus, for example, on the unitarity for elastic and quasielastic
processes, then disc T would simply denote a cut in s-channel between incoming and outgoingparticles as visualized by crosses in Fig. 20.3.
At high energies, the s-channel unitarity relation is diagonal in the b basis such that
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20. High Energy Soft QCD and Diffraction

5

1

1

1
x

1

1

x
x

1

1

x
x

1

x
P

=

P

P

P

+

2

P

+

P

2

P
x
x

x
2

2

2

x

P

x
2

1

2

2

2

x

2

(b)

(a)

Figure 20.3: Two-Pomeron exchange in the t channel expressed as a sum over all diffractive
intermediate states in the s-channel. The crosses indicate that the particles are on the mass shell.

2Im Tel (s, b) = |Tel (s, b)|2 + Ginel (s, b)(20.9)

withZ

σtot = 2
Z

σel =
Z

σinel =

d2 b Im Tel (s, b)(20.10)d2 b |Tel (s, b)|2

(20.11)hid2 b 2Im Tel (s, b) − |Tel (s, b)|2 .

(20.12)

The general solution of Eq. (20.9) is
Tel (b) = i(1 − e−Ω(b)/2 )(20.13)Ginel (s, b) = 1 − e−ReΩ(b) = 1 − Pnointer (s, b),(20.14)

andwhere Ginel is the sum over all inelastic intermediate states and Pnointer is a probability to haveno inelastic interactions. Ginel (s, b) describes the b-profile of inelastic particle collisions. It satisfies
the condition 0 ≤ Ginel ≤ 1 and determines how absorptive the interaction region is at a given
impact parameter (with Ginel = 1 for full absorption and Ginel = 0 for the complete dominance ofelastic scattering). As seen from Eq. (20.14), exp(−ReΩ(s, b)) is the probability that no inelastic
interactions occur at impact parameter b. Ω (ReΩ ≥ 0) is called the opacity (optical density) or
eikonal. The quantity
S 2 (b) ≡ e−ReΩ(b) = Pnointer (b)(20.15)is the so-called survival factor, which enables us to calculate the probability that the LRG survives
soft rescattering.
In terms of the opacity the elastic cross section takes the form
dσel
1
1
=
|Tel (s, t)|2 =
dt
16πs2
4π

Z

~

d2 b ei~qt ·b (1 − e−Ω(b)/2 )

2

Z

=π

2

bdb J0 (qt b)(1 − e−Ω(b)/2 ) ,(20.16)

where qt =p|t| and J0 is the zeroth-order Bessel function.
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20. High Energy Soft QCD and Diffraction

To describe the elastic scattering at one fixed energy we can always find an appropriate parameterization for the opacity Ω(b) and tune the parameters to reproduce the observed dσel /dt crosssection. Moreover, we can fix the form of the parameterization, but choose, at each particularenergy, the corresponding values of parameters; see, e.g. [39]. Alternatively, we may simply takethe Fourier-Bessel transform from the experimental data [33, 40, 41]Z

ImTel (b) =qt dqt4πsdσel 16πJ0 (qt b),
dt 1 + ρ2(20.17)

where the square root represents ImTel (qt ), with ρ ≡ ReTel /ImTel . In this way, we first determine
Tel from the data for dσel /dt, and then calculate Ω(b) using Eq. (20.13), assuming in accordancewith data that ρ is small (or ρ(t) = constant).
At high energies ρ2  1, which is usually well justified except in the diffractive dip region (seeSection 20.3.3.1 for discussion of the dip region).
The value of ρ can be derived via the dispersion relation, see [3]:1
1
ReTel (s) =
s
π

Z +∞
−∞

ds0
1
σtot (|s0 |) =
0
s −s
π

Z ∞
0

2sds0
σtot (s0 ) 02.
s − s2(20.18)Since we consider just the charge-parity C-even amplitude, here for negative s0 we put σpp̄ = σpp .
That is, for negative s0 , which corresponds to the interaction with an antiparticle, we use the sameσpp (|s0 |). The major contribution comes from s0 ' s. Thus, with a good accuracy we can evaluate
ρ(t = 0) as
π ∂ ln σtot (s)ρ '.
(20.19)2∂ ln s20.3.2 Diffractive dissociation
The elastic cross section probes the optical density of the proton. The well known example ofscattering on a black disk, with Ginel = 1 for b < R, gives σel = σinel = πR2 and σtot = 2πR2 . In
general, the absorption of the initial wave (due to inelastic channels) leads, via s-channel unitarity,
to elastic scattering.
Inelastic diffraction (i.e. proton dissociation) is a consequence of the internal structure ofhadrons. This can be conveniently described at high energies, where the lifetimes of each particular
Fock component of the incoming hadron/proton wave function (the hadronic fluctuations) arelarge, τ ∼ E/m2 , and during these time intervals the corresponding Fock states can be consideredas ‘frozen’. Each hadronic constituent can undergo a scattering with its own probability and thus
destroys coherence of the fluctuations4 . As a result, the outgoing superposition of states will bedifferent from the incident particle, and will most likely contain multiparticle states, so we will haveinelastic, as well as elastic scattering.
To calculate diffractive dissociation we can enlarge the set of intermediate states (p, Na∗ ), fromjust the single elastic channel, and introduce a multichannel eikonal. However, it is more convenient
to follow Good and Walker [45], and to introduce states φk diagonalising the T matrix (which e.g. in
the proton case describes different p → N ∗ , Na∗ → Nb∗ transitions). Such eigenstates only undergoelastic scattering. Since there are no off-diagonal transitions,hφi |T |φk i = 0

for i 6= k,(20.20)a state k cannot diffractively dissociate into a state j 6= k. Working in terms of the Good-Walker
eigenstates φi , we have a simple one-channel eikonal for each state. We denote the orthogonal
matrix which diagonalizes T by a, so that4
At high energies the configurations with different transverse separation, r, between the quarks (valence partons)
can serve as an example of such Fock states. An interaction with the QCD Pomeron does not change the value of r,
while the cross section σ ∝ αs2 · r2 (see Section 20.4.2 and [42–44]).

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20. High Energy Soft QCD and Diffraction

T = aF aT

hφi |F |φk i = Fk δik ,with(20.21)

where Fk is the probability amplitude of the hadronic process proceeding via the diffractive eigenstate φk .
Now consider the diffractive dissociation of an incoming state |hi. We can write
X

|hi =

ahk |φk i.

(20.22)k

The elastic scattering amplitude satisfies
hh|T |hi =X|ahk |2 Fk = hF i,(20.23)kwhere Fk ≡ hφk |F |φk i and where the brackets of hF i mean that we take the average of F over the
initial probability distribution of diffractive eigenstates. After the diffractive scattering describedby Tf h , the final state |f i will, in general, be a different superposition of eigenstates from that of|hi, which was shown in Eq. (20.22). Neglecting the real parts, for the cross sections at a given
impact parameter b, we haveXdσtot
= 2 Imhh|T |hi = 2|ahk |2 ImFk = 2hImF i
2
d b
k
2
X
dσel
2
2
=
|hh|T
|hi|
=
|a
|
F
= h|F |i2
hk
k
d2 b
k

(20.24)

Xdσel + SD X
2
=
|hφ
|T
|hi|
=
|ahk |2 |Fk2 | = h|F 2 |i.
k
d2 b
k
k

It follows that the cross section for the single diffractive dissociation of a proton,dσSD= h|F 2 |i − h|F |i2 ,(20.25)d2 b
is given by the statistical dispersion in the absorption probabilities of the diffractive eigenstates.
Here the average is taken over the components k of the incoming proton which dissociates. If theaverages are taken over the components of both of the incoming particles, then Eq. (20.25) is thesum of the cross sections for single and double dissociation, see Fig. 20.3.
Note that if all the components φk of the incoming proton |hi were absorbed equally, thenthe diffracted superposition would be proportional to the incident one and the probability of theinelastic diffraction would be zero. Thus if, at very high energies, the amplitudes Fk at small impact
parameters are equal to the black disk limit, Fk = i, then diffractive production will be equal tozero in this impact parameter domain, and so will only occur in the peripheral b region where theedge of the disk becomes not completely black. Hence the impact parameter structure of diffractive
dissociation and elastic scattering is drastically different in the presence of absorptive s-channel
unitarity effects (see the Ginel term in Eq. (20.9)). Under the assumption that amplitudes Fk athigh energies cannot exceed the black disk limit, Im Fk ≤ 1, equations 20.24 lead to the following
bounddσel + SD1 +SD2 +DD
1 dσtot≤.
(20.26)d2 b
2 d2 bknown as the Pumplin bound [46] 5 .
5Strictly speaking the proof of the Pumplin bound is justified only for low mass dissociation. When the masses√
M1,2 become so large (say, Mi2 > ss0 ) that the Good-Walker states |φi i, corresponding to two incoming protons
overlap, we may face double counting. Therefore, the high mass dissociation will be considered in the next Section,in terms of the multi-Pomeron diagram. Here and in what follows s0 is a constant which should be defined for aparticular theoretical model or fitted from experiment.

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20. High Energy Soft QCD and Diffraction

20.3.3 Diffraction from the t-channel viewpoint
The t-channel approach is based on the Regge model (see Section 20.2), where high-energy
diffractive processes are mediated by the exchange of a Pomeron (P). In the case of the elastic
pp-scattering amplitude in the eikonal model (see Eq. (20.13)), the opacity corresponding to theexchange of one Pomeron is
Z 2
d qt
~
Ω(s, b) =Ω(s, qt ) ei~qt ·b(20.27)4π 2with
 αP (t)−11sΩ(s, qt ) = Tel0 = −iηP (t)gN (t)gN (t),(20.28)ss0where Tel0 is the two-particle s-channel irreducible elastic amplitude, cf. Eq. (20.5), and gN (t) is the
proton-Pomeron coupling.
If we assume an exponential t-dependence of the coupling, gN (t) = gN (0) exp(B0 t), and neglect
the Pomeron phase, then the opacity issΩ(s, qt ) = gN (0)gN (0)s0αP (0)−1

eBt ,(20.29)with the t-slope given by
s
B =
.
s0
At high energies the opacity has a Gaussian form in the b-space:
2B0 + αP0 ln

g 2 (0)Ω(s, b) = N
4πB



s
s0





αP (0)−1(20.30)2e−b /4B .

(20.31)q

In terms of opacity the effective radius of interaction increases at high energies as αP0 ln(s/s0 ).
This means that with energy increasing the differential cross section becomes steeper (the so calledshrinkage of the diffractive peak).
If we were to take for the Pomeron the DL parametrisation [25,26], that is to keep just the first,
T (b) = Ω(b)/2, term in the elastic amplitude (Eq. (20.13)) then, at LHC energies, the Gaussian
would exceed the black disk limit at small b. However, the eikonal unitarization reduces the power
growth of the one-Pomeron exchange cross section. Thus, in Eq. (20.31) Ω(s, b) ∝ (s/s0 )αP −1 gives
an amplitude Im Tel (s, b) = 1 − e−Ω/2 < 1. Hence the total cross section is limited by the size ofthe effective interaction area σtot < 2πR2 , where the interaction radius R can be estimated from
Eq. (20.31) as the value of b where ReΩ(b) becomes ∼ 1.
For the parameterization of Eq. (20.31) the corresponding radius grows at very large energies
as"#
2 (0)  s αP (0)−1gN
22b = R = 4B ln' 4∆αP0 ln2 (s/s0 ) .
(20.32)4πB s0That is for ∆ = 0.1 and αP0 =0.25 GeV−2 we may expect that the cross section increases as
σtot = 2πR2 ' c · ln2 s ,(20.33)

with c = 8π∆αP0 = 0.24 mb. This value is close to that obtained by the COMPETE parameterization (c = 0.27 mb [22, 24]) but much smaller than the Froissart-Lukaszuk-Martin (FLM)
bound [47–49]. With cFLM = π/m2π ' 60 mb, see Section 20.7,πsσtot ≤ 2 ln2
.
mπ
s0


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

(20.34)

20. High Energy Soft QCD and Diffraction

9

The fact that c =(0.24–0.27) mb  cFLM = 60 mb demonstrates that even at the LHC we are very
far from true high-energy asymptotics 6 , and the observed growth of the cross section is driven by
the interactions at relatively large transverse momenta kt  mπ rather than the smallest hadronmass mπ in the denominator of Eq. (20.34).
20.3.3.1 The t-slope and dip in the elastic cross section
We first start with a relatively small one-Pomeron amplitude and consider the two-Pomeron
contribution corresponding to the Ω2 term inp the expansion of the eikonal 1 − exp(−Ω/2). In
this term the momentum transferred, qt = |t|, is divided between the two Pomerons so thateach Pomeron carries about a momentum qt /2. Correspondingly, the t dependence of the whole‘two-Pomeron’ amplitude will be exp(2B(t/4)) = exp(Bt/2) 7 .
Since the two-Pomeron contribution has an opposite sign in comparison with the one-Pomeron
exchange, their interference will result in the appearance of the first diffractive minimum whichmoves to smaller |t| with energy increasing. Such interference effects are largely responsible for thezero in the imaginary part of the amplitude (with the minimum filled by the real part).
It is worth mentioning that the one-channel eikonal discussed so far is a rather oversimplified
approximation. It provides some indications about the behaviour we may expect for the elastic crosssection, but clearly it does not give the whole story. Moreover, even within the framework of the onechannel eikonal, the expectation for the elastic slope t-dependence could be masked by other effects.
Firstly, there is no reason why the t-dependence of the proton-Pomeron coupling gN (t) has to be apure exponent. Next, there exists a two-pion singularity at t = 4m2π (close to the physical region)in the Pomeron trajectory which generates some curvature in the behaviour of dσel /dt [50–52].
So there may be some compensation between the effects caused by the eikonal (arising from theinterference between the different multi-Pomeron contributions), and the curvatures coming fromthe form of the proton-Pomeron coupling and the two-pion singularity of the Pomeron trajectory.
However, an exact compensation looks quite non-trivial and a pure exponential behaviour of dσel /dt
looks highly unlikely.
Indeed, the measurements by the TOTEM collaboration at 8 TeV [53] and at 13 TeV [54] clearly
demonstrate that the local slope of the elastic pp cross section,
B = d[ln(dσel /dt)]/dt,(20.35)2
at −t <
∼ 0.3 GeV varies with t.

20.3.3.2 High mass dissociation
Let us turn to inelastic diffractive processes that is, to single and double proton dissociations,
pp → X + p and pp → X1 + X2 , where the + sign denotes the presence of a LRG in the distributionof final state particles. For example, for the diffractive dissociation of a proton into a system ofmass M , the rapidity gap between the incoming proton and the remaining hadrons is
s
1
= ln
,
(20.36)2Mξwhere ξ = 1 − x and x is the initial momentum fraction (Feynman variable) carried by the outgoing√proton. The masses, M , of the diffractively excited states, produced in high s collisions, can belarge. To separate dissociation from the common inelastic process, usually the condition M 2  s
is imposed.
The simplest multi-Pomeron diagram used to describe the diffractive dissociation is the so-called
triple-Pomeron graph, shown at the end of Fig. 20.2.




 

∆y = ln6

As usual, we assume s0 = 1 GeV2 , but the qualitative conclusion does not depend on any realistic choice of s0 .
The two-Pomeron contribution has a factor of two smaller t-slope, and in terms of the impact parameter, theΩ2 (b) term is concentrated in the domain of a smaller radius. In such a simplified picture, the impact parameters
corresponding to an exchange of n Pomerons will rapidly decrease with n increasing.
7

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20. High Energy Soft QCD and Diffraction

10

In the Regge pole model, the cross section for the inclusive single diffractive (SD) dissociation
process [55–57] can be written in the form (see Fig. 20.2)
2 (t)
ξdσSD
M 2 dσSD
g3P (t)gN (0)gN==dtdξ
dtdM 2
16π 2



s
M2

2αP (t)−2M2s0!αP (0)−1(20.37),

where g3P (t) is the triple-Pomeron coupling. The value of the coupling g3P is usually obtained froma triple-Regge analysis of lower energy data (see e.g. [34]) .
In an analogous way the cross section for double dissociation reads
2 (t)g 2 (0)ξ1 ξ2 dσDD
M 2 M 2 dσDD
g3PN
= 1 22=dtdξ1 ξ2
16π 3
dtdM1 dM22



ss0
M12 M222αP (t)−2

M12 M22s20!αP (0)−1,(20.38)where t is the momentum squared transferred through the LRG. As discussed in Section 20.5, from

Figure 20.4: Pomeron exchange with schematic diagrams for the enhanced and semi-enhanced
exchanges.
a microscopic point of view the Pomeron exchange is described by a set of ladder-type diagrams(see [58–60]), which can lead to a rescattering of the intermediate partons (produced inside thisladder during the evolution), see Fig. 20.4. The left plot shows the Pomeron exchange complemented
with the rescattering of partons 1 and 2 and the scattering of a parton 3 on the target. In termsof multi-Pomeron exchanges this corresponds to the diagram on the right hand side, where the
Pomeron exchange is shown by the double line of a corresponding colour. The blue one is called“enhanced” (its contribution is integrated over the rapidities of both upper and lower vertices, i.e.
of partons 1 and 2). The loop formed by the Pomerons shown in red is called “semi-enhanced” (itis integrated over the rapidity of one intermediate parton).
While the rescattering of the incoming hadron (proton) is already embedded in the eikonal
formula (Eq. (20.13)), the rescattering of the intermediate partons in RFT is accounted for by then , which couple m to n Pomerons.
so-called enhanced diagrams 8 with multi-Pomeron vertices, gm
It is quite a challenging task to resum all the enhanced diagrams, however this was successfully
performed within the framework of the QGSJET Monte Carlo [61]. An elegant approach to sum
up all enhanced diagrams in the case when each extra effective Pomeron contribution is very largen vertices in n and m in the right half of
was proposed in [62], assuming the analyticity of the gm
the complex n- and m-planes. The resulting amplitude becomes a black disk.
The simplest triple-Pomeron vertex g21 = g3P produces the first multi-Pomeron graph considered
above (see the end of Fig. 20.2). However, numerically the multi-Pomeron vertices are relatively
8This contribution is enhanced due to the large parton multiplicity.

31st May, 2024

11

20. High Energy Soft QCD and Diffraction

small. Note also that the value of g3P , determined from the fit to experimental data (e.g. [63]), is
actually an effective vertex with coupling
geff = g3P hS 2 i,

(20.39)which already includes the survival factor S 2 (b), see Eq. (20.15).
Since the opacity Ω increases with energy, at large Ω the number of multiple interactions grows
as N ∝ Ω, leading to a smaller S 2 . An explicit analysis [64] accounting for the survival effects gives
a coupling g3P about a factor of 3 larger than geff , namely g3P ' 0.2gN .
Recall that the Pomeron exchange simultaneously describes both the elastic scattering amplitude, Tel , and the multiparticle production cross section, Ginel . The discontinuity (disc Tel ) of theladder diagram corresponds to the production of secondary particles, practically homogeneously
distributed over the whole available rapidity interval covered by the Pomeron, as illustrated by theright-hand diagram in Fig. 20.5.

Figure 20.5: Cut Pomeron contribution to the inelastic cross section.
For the one-Pomeron case this discontinuity is called the “cut Pomeron”. Correspondingly each
multi-Pomeron diagram describes a series of different processes. Cutting k Pomerons in the diagramwith n Pomerons we get the inelastic interaction with the multiplicity (density of secondaries) ktimes larger than that, N0 , produced by one cut Pomeron, dN/dy = k · N0 . The remaining n − k(elastic) Pomerons account for the absorptive corrections to the subprocess with k cut Pomerons.
Indeed, the contribution of the diagram with n Pomerons includes also the processes with larger,(k+i)·N0 multiplicities (cut Pomerons), where (i = 1, 2, ..., n−k). Absorptive corrections, describedby the remaining elastic Pomerons, play a role of the survival factor S 2 for the process with thefixed particle density k · N0 . They ensure probability conservation (the sum of the probabilities ofall possible different channels is equal to one) and restore unitarity. Note that the multi-Pomeron
diagrams represent all possible interactions between partons from the protons and partons from the
Pomerons. In the case of Monte Carlo generators, the non-enhanced multi-Pomeron contributions
are included in terms of the multiple parton interaction (MPI) option, see [13, 65] and Section 7.2
in [2]. However, as a rule, this option accounts mainly for the multiple interactions between thepartons from the protons (incoming hadrons). The energy-momentum sharing between the various
inelastic rescattering processes (including the cut and uncut Pomerons) was performed at theamplitude level within the EPOS Monte Carlo [66].
20.3.3.3 AGK cutting rules
The relation between the cross sections of subprocesses with a different number of cut Pomerons
within a given diagram with n Pomerons) is given by the AGK (Abramovsky-Gribov-Kancheli [67])cutting rules. These rules include also the cut between the Pomerons with k = 0 which corresponds
to the contribution of the particular diagram to the elastic cross section. By applying these rules,
31st May, 2024

20. High Energy Soft QCD and Diffraction

12

it is possible to show the self-consistency of the approach, which was lacking in the pure Regge-polemodel.
Consider a diagram where the elastic scattering amplitude is mediated by an exchange of n
Pomerons. The AGK cutting rules specify the coefficients ckn arising when k of these Pomerons are
cut. Recall that the Pomeron cut discontinuities give the corresponding inelastic contributions to
σtot . The terms with k = 0 correspond to the diffractive cutting of the diagram (that is, the cut is
between the Pomeron exchanges, and not through the Pomerons themselves), while the terms with(n)
k = 1, 2, ... describe the processes with k cut Pomerons. The coefficients ckn = σnk /|σtot | are 9
ck=0= (−1)n (2n−1 − 1),nck6n=0 = (−2)n−1(−1)k−1 n!
k!(n − k)!

(20.40)(n)where σtot denotes the contribution of the n-Pomeron diagram to the total cross section. Note the(n)alternating sign of σtot expressed as (−1)n−1 .

Figure 20.6: Two-Pomeron exchange diagram as a sum of different AGK cuts shown by thedashed lines.
For the two-Pomeron exchange, n = 2, the coefficients are +1, −4, or +2 according to whether
k = 0, 1 or 2 Pomerons are cut, respectively. As shown in Fig. 20.6, the amplitude of the twoPomeron exchange corresponds to a sum of three processes: i) inelastic interaction with particle
density twice that caused by one Pomeron (see Fig. 20.6(c)) which enters with the coefficient ‘2’, ii)
shadowing (absorptive) correction to the one-Pomeron exchange contribution, which corresponds
to events with a single Pomeron density (only one Pomeron is cut), see Fig. 20.6 (b), which enterswith a factor ‘-4’, and iii) diffractive elastic scattering or proton dissociation (when different components of the proton wave function correspond to different interaction cross sections), caused by
the distortion of the incoming plane wave, see Fig. 20.6(a).
Note that the inclusive cross section is not affected by the multi-Pomeron contribution: 2 ×(2) + 1 × (−4) = 0. This is a general property of the AGK rules valid for any number of Pomerons
n. Thus in order to calculate the inclusive single-particle cross section, it is sufficient to consider
just the one-Pomeron exchange diagram.
Let us emphasize that the AGK rules provide a framework to consistently work with multiPomeron diagrams, that is, with the Regge cuts, accounting for their contributions to different
processes (elastic scattering and diffractive dissociation, inelastic events with different densities,
dN/dy, of secondaries, etc.).
Measurements of diffractive dissociation cross sections have been made in a wide range of preLHC energies, see e.g. [68–73]. At the LHC, cross sections of events with a LRG were measured
by the ATLAS, CMS and ALICE collaborations at 7 and 8 TeV, see [74–77]. ATLAS [78] and9In their complete form the AGK cutting rules were implemented in the QGSJET Monte Carlo [61].

31st May, 2024

20. High Energy Soft QCD and Diffraction

13

CMS and TOTEM [79] presented first measurements of SD cross sections at 8 TeV with a tagged
forward proton. While ATLAS measured inclusive SD cross section, CMS and TOTEM studied
SD dijet production. Note that in [78] the measured slope B = 7.65 ± 0.34 GeV−2 of the inclusiveSD
SD cross section as well as the differential distributions ξdσdtdξ for 0.0001 ≤ ξ ≤ 0.025 are (within
the experimental uncertainties) in a good agreement with the theoretical expectations [29, 80].
Moreover a relatively small (in comparison with the dσel /dt) slope B indicates that the size of thetriple-Pomeron vertex is much smaller than the proton size.
20.3.4 Central Diffractive processes
Processes pp → p+X+p, where an object X, produced in the central rapidity region, is separated
from the outgoing protons by a LRG on each side, are called Central Exclusive Production (CEP).
They are described by the double Pomeron exchange (DPE) diagrams. When the mass of thecentral system, MX , is large and the interaction in the MX region can be described by Pomeron
exchange, the corresponding cross section reads
2 (t )
ξ1 ξ2 dσ CEP
g 2 (t1 )gN
2= Ndξ1 dt1 dξ2 dt2(16π 2 )21
ξ1

2αP (t1 )−2 1ξ22αP (t2 )−2

2
g3P
(0)

2
MX
s0

!αP (0)−1.

(20.41)

If the mass MX is not too large or for the cases (such as exclusive Higgs boson or dijet production) where the mass MX is comparable with the corresponding hard scale, the last factor
2 (0)(M 2 /s )αP (0)−1 should be replaced by the corresponding ‘Pomeron-Pomeron cross section’,g3P
X 0
see for instance [81, 82].
Note that equations (20.37), (20.38) and (20.41) are written in a simplified way without accounting for absorptive corrections. That is, the cross sections in equations (20.37) (20.38) and(20.41) should be multiplied by the gap survival factor S 2 (see Eq. (20.15)).
Since the QCD Pomeron is built mainly from gluons it is natural to search for glueballs indouble Pomeron exchange processes, and in particular, in CEP.
Resonance production in the Pomeron-Pomeron fusion was extensively studied at the CERN√
ISR at s from 22 GeV to 63 GeV (see for reviews [82–84]) and, after the ISR closure in 1983,
in fixed target experiments at the CERN SPS [85] and E690 at the Tevatron [86, 87]. Glueballs
were actively searched for and the properties of the f0 and f2 production studied in detail using
multiparticle spectrometers, such as the Omega facility at the CERN SPS experiments (WA76,
WA91 and WA102), see for a review [85].
An important property of CEP processes, which can be expected from matching with theperturbative QCD LO (leading order) calculation, is the Jz = 0 dominance. Perturbatively, for the
CEP of a heavy object, the leading contribution comes from a configuration with the projectionof this object spin onto the beam axis Jz = 0 [81]. Note that the CEP cross section is suppressedat large MX by a strong bremsstrahlung off the incoming gluons (from the Pomeron) which would
violate the ‘exclusivity’. The small probability of not having such radiation is described by the
Sudakov suppression factor, TSud , [88], see [81] for details.
20.3.5 Diffractive parton distributions
Selecting in Deep Inelastic Scattering (DIS) events with a LRG (see e.g. [89, 90]) or detecting
the leading proton (see Section V.C. in the review [91]) we can study the parton (quark and gluon)
distributions of the Pomeron 10 . In other words, such events can be treated as DIS on the Pomerontarget with the incoming Pomeron flux given byZ

fP (xP ) =10

g 2 (t) 2(1−αP (t))
dt N 2 xP,
16π

see also Section 18.5 in [1]31st May, 2024

(20.42)

20. High Energy Soft QCD and Diffraction

14

where the proton momentum fraction transferred through the Pomeron xP = ξ = M 2 /s.
These Pomeron PDFs were extracted from the HERA measurements of ep scattering with leading protons or a LRG and can be used to describe the inclusive production of high ET dijets oranother hard process based on the collinear factorization theorem in the same way as that in nondiffractive collisions (see [91]). The inclusive measurements of these PDFs are described in [92–94],
with the combined H1 and ZEUS data using tagged protons analyzed in [95]. The impact of diffractive jet measurements is addressed e.g. in [96] and the measured charm contribution is presented
in [97,98]. As far as the parton distributions are known, we can calculate the corresponding inelastic cross section of the Pomeron-proton interaction using one of the ‘general purpose’ Monte Carlogenerators (see e.g. [13]), multiply it by the Pomeron flux and compare the obtained result with theRegge formula in Eq. (20.37). This approach provides another way to evaluate the triple-Pomeronvertex g3P . The corresponding analysis was performed in [99] and leads to practically the same(within the error bars) value of g3P = 0.2gN (0).
It is worth mentioning that in DIS at large Q2 we are dealing with small-size objects and therescattering effects are small. Therefore, the survival factor S 2 ' 1 and does not affect the results.

20.4 Experimental data on diffraction at high energies
20.4.1 Total and elastic cross sections
The elastic scattering of protons is a process with a special and rather simple experimental
signature: the central detector is empty while the incoming protons after the collisions are detectedin the dedicated forward proton detectors (FPD) placed far from the interaction point (IP). Elastic
scattering data are taken in special runs in order to be able to reach different ranges of t-values
and thanks to the very large value of the cross section the data can be collected with a relatively
low instantaneous luminosity and hence a negligible pile-up.11
These special runs usually have very few proton bunches and differ in the t range covered, whichis governed roughly by the relation tmin ∝ d2 /β ∗ . Here d is the distance, expressed in multiples ofthe beam size at the detector, from the centre of the LHC beam and β ∗ is defined as the distancefrom the IP to the point where the transverse area of the beam is twice as wide as that at the IP(see Section 31 in [1]). Note that if we work at large β ∗ , the incoming protons have very small
angular divergence leading to small average transverse momentum, which allows us to measure verysmall |t| values. The lowest |t| values measured so far at the LHC are 4 × 10−4 GeV2 (ALFA) and6 × 10−4 GeV2 (TOTEM) reached with the 8 TeV LHC beam configured with β ∗ =1 km optics.
The largest t values of about 4 GeV2 were measured by TOTEM at 8 and 13 TeV with β ∗ = 90 m
thanks to special triggers. Other β ∗ values used in special runs are 3.5 m, 11 m and 2.5 km.
There are four ways to determine the σtot value:
1. Elastic and Inelastic. This method does not require the optical theorem and hence no extrapolation of dσel /dt to t = 0 and no ρ (defined below Eq. (20.17)) but rather the luminosityand measuring rates Nel (elastic) and Ninel (inelastic). The total cross section is then simply:
σtot =

1(Nel + Ninel ).
L(20.43)

Of course, both Nel and Ninel should be corrected for the detector acceptance and efficiency.
This is especially important for Ninel since the detectors never cover the whole rapidity region(i.e. the whole 4π).
2. Elastic only. This approach necessitates measuring dσel /dt and using the optical theorem
with a known value of ρ. As explained in Section 20.2, the optical theorem states that11

The pile-up is formed by additional pp collisions which typically produce low-pT particles. These may affect thesignal sample and worsen various reconstruction and identification efficiencies.

31st May, 2024

15

20. High Energy Soft QCD and Diffraction

Figure 20.7: Overview of elastic (σel ), inelastic (σinel ) and total (σtot ) cross section data for pp√
and pp̄ collisions as a function of s. The continuous black lines (lower for pp, upper for pp̄)
represent the best fits of the total cross section data by the COMPETE collaboration [22]. The
dashed line is a fit of the elastic cross section data. The dashed-dotted lines refer to the inelastic
cross section and are obtained from the difference between the continuous and dashed lines. Figure
from Ref. [100].
σtot ∝ Im[Tel (t → 0)], see Eq. (20.3). Since in practice it is not possible to measure down
to t = 0, we need to extrapolate. To minimize the model dependence when extrapolating,it is vital to measure down to as low |t| values as possible (i.e. high β ∗ ). This method
requires an independent luminosity measurement. Once the luminosity is known, dσel /dt canbe normalized and used to extract σtot using the formula:
2
σtot
=

16π dσel.
1 + ρ2 dt t→0(20.44)

3. Coulomb normalization. Similarly to the previous method, this approach relies on theelastic observables only and requires a measurement of the elastic cross section at very low
values of |t|, where it is sensitive to the theoretically well known Coulomb QED contribution
24παQED/t2 . The normalization of dσel /dt is then determined by fitting the experimental data
at very low |t| using a formula including the Coulomb amplitude and its interference with thestrongly interacting (the so-called nuclear) term. This method has been successfully used by
UA4/2 [101] and TOTEM [102].
4. Luminosity-independent. This method does not rely on the knowledge of luminosity but
rather on the knowledge of Nel and Ninel and on the optical theorem:
1 dNelel
combining equations (20.43) and (20.44) with dσ
dt = L dt we get
σtot =

16π dNel /dt t=0,
1 + ρ2 Nel + Ninel

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(20.45)

20. High Energy Soft QCD and Diffraction

16

where dNel /dt t=0 corresponds to the extrapolation to t = 0 of the nuclear term only. By independently and simultaneously measuring Nel and Ninel , and applying the optical theorem, we can also
determine the luminosity.
The TOTEM [100, 103–105] and ATLAS [106, 107] collaborations at CERN have covered an√energy range from s=2.76 TeV to 13 TeV. A compilation of high energy total pp and pp̄ crosssection measurements is shown in Fig. 20.7 (for discussion of the pre-LHC elastic scattering data
see review [108]).
Despite some tension between the Tevatron CDF [109] and E811 [110] data 12 and to a lesser
extent between the TOTEM [104, 105] and ATLAS [106, 107] measurements, the data clearly indicate that in the Tevatron – LHC energy interval the total cross section starts to grow faster thanthe power-law parametrization [26] describing the data below the Tevatron energy. In particular,√
while the DL fit [26] predicts σtot = 90.7 mb at s = 7 TeV, the TOTEM experiment observes
98.6±2.2 mb [104].

√
Figure 20.8: The diffractive slope B for pp and pp̄ elastic scattering as a function of s. The
experimental uncertainties represent the quadratic sum of statistical and systematic uncertainties.
√
The dashed line is a result of a linear fit to data at s < 3 TeV. The data points come from[103–107, 111–113]. Figure from Ref. [100].
A compilation of the high-energy data on the elastic slope is shown in Fig. 20.8. It is clearly√seen that in the TeV energy range the slope increases with s more rapidly than the logarithmic
behaviour expected in the case of one-Pomeron exchange, see Eq. (20.30). Such an acceleration of
the t-slope derivative, dB/d lns, is a clear manifestation of the increasing role of the multi-Pomeron
exchanges, where asymptotically the slope should rise as ln2 s, see [117]. Finally, Fig. 20.9 illustrates
the energy dependence of the differential elastic pp cross section. As expected (see Section 20.3.3.1),
the diffractive dip moves to smaller |t| with increasing energy.
12

The CDF 1.8 TeV point [109] is 2.8 σ higher than the corresponding E811 result [110].

31st May, 2024

20. High Energy Soft QCD and Diffraction(mb/GeV2 )106

dσ /dt17

103

elastic-scattering measurements by TOTEM√
s = 2.76 TeV (scaled 0.01×):

105

√

104

β∗ = 3.5 m [EPL 95 (2011) 41001]

√

102

√

β∗ = 90 m [EPL 101 (2013) 21002]
s = 8 TeV (scaled 10×):β∗ = 90 m [PRELIMINARY]

s = 13 TeVβ∗ = 90 m [NPB 899 (2015) 527]

101

√

√

100

β∗ = 90 m [EPJ C (2019) 79:861]β∗ = 2500 m [EPJ C (2019) 79:785]

PRE

LIM

10−2

√

10−3
10−4

√
0

β∗ = 1000 m [EPJ C (2016) 76:661]
s = 13 TeV (scaled 1000×):s = 8 TeV

10−1

10−5

β∗ = 11 m [CERN-EP-2018-341]
s = 7 TeV:

INA

RY

s = 7 TeV

s = 2.76 TeV

0.5

1

1.5

2

2.5

3

4
3.5
|t| (GeV2 )√Figure 20.9: The t-dependence of the pp elastic cross section for collision energies s = 2.76 TeV[103], 7 TeV [104, 114], 8 TeV [53, 115, 116] and 13 TeV [54, 102]. The experimental uncertainties
represent the quadratic sum of statistical and systematic uncertainties. Figure from Ref. [116].
20.4.2 Diffractive vector meson production
The exclusive production of vector mesons was studied in detail at HERA (see for a review[91]). It is well described within the ‘dipole model’ (see for review and references [120]), where
the incoming photon first fluctuates into a quark-antiquark, which then interacts with the target
proton and, finally, with the probability given by the overlap integral between the vector meson
wave function and the outgoing q q̄-pair, the vector meson is produced. The crucial quantity is thevalue of cross section, σ(q q̄ − p), of elastic scattering of the q q̄-pair on the proton. The energy
behaviour of σ(q q̄ − p) is driven by the intercept, αeff (0), of the effective Pomeron 13 (rightmost
singularity in the j-plane), while the value of the cross section depends on the quark separation, r,
in the transverse plane, σ(q q̄ − p) ∝ αs2 hr2 i [43, 44]. Thus different processes with the same hr2 i are
driven by the same σ(q q̄ − p) cross section.
In the DIS case this separation in turn is controlled by the photon virtuality, Q2 , and the quark
mass, mq : hr2 i ' 1/(z(1 − z)Q2 + m2q ) (z is the photon momentum fraction carried by the quark).
2
Indeed, the cross section of the ρ meson diffractive production in DIS at Q2 = MJ/ψis close (upto the difference in the quark electric charges) to that for the J/ψ photoproduction, see Fig. 20.10(Left).
The production cross section depends non-trivially on W , the energy of the γ ∗ p center ofmass system. It increases with W as W n , where n = 0.2 for ρ, ω and φ (light quark)-mesons
but n = 0.8 for J/ψ. Note that in the J/ψ case the energy dependence is close to that of the
BFKL (Balitsky-Fadin-Kuraev-Lipatov) Pomeron [59, 122, 123], that is, the singularity calculated
within the leading (and next-to-leading) approximation in perturbative QCD. But at lower scales
the absorptive (multi-Pomeron) corrections tame the growth which leads to smaller values of n
( [124–127]), see Fig. 20.10 (Right).
A similar situation reveals in the dependence of αeff on Q2 , as can be seen in Fig. 47 of [91].
At a large scale µ2 = (Q2 + MV2 )/4 the value of αeff ' 1.3 is close to the prediction for the QCD
13

Effective Pomeron means that this is not an original pole in the j-plane, but it includes the corrections (renormalizations) caused by the enhanced diagrams (see e.g. [121]).
31st May, 2024

20. High Energy Soft QCD and Diffraction

18

Figure 20.10: (Left) The ρ, ω, φ and J/ψ elastic production cross sections as a function of the scaleµ2 = (Q2 + MV2 )/4. For readability of the figure, the J/ψ cross sections are multiplied by a factor
2. Figure from Ref. [118]. (Right) Compilation of photoproduction cross section measurements asa function of the γp centre-of-mass energy, W . The total cross section and various vector meson
production cross sections are included, with the approximate power law dependences σ ∝ W δ
indicated for each process. Figure from Ref. [119].
Pomeron, while for a smaller scale, the absorptive corrections described by the multi-Pomeron
diagrams start to reduce the cross section, and αeff decreases.

20.5 Pomeron in QCD
All features described in the previous Sections were based on first principles, such as analyticity(based on causality), unitarity, crossing symmetry, etc. Since QCD theory satisfies all these principles it should reveal a corresponding “Regge” behaviour. Indeed, within perturbative QCD there
is a Pomeron: an even-signature singularity in the j-plane with vacuum quantum numbers. Whilein the old Regge theory the Regge trajectories and their couplings were phenomenological numbers
fitted from experiment, perturbative QCD allows one to calculate the positions of the singularities
and the corresponding couplings with O(αs ) and even with O(αs2 ) accuracy [59, 122, 123, 128–131].
In terms of Feynman diagrams, the QCD Pomeron may be viewed as a sum of multi-particle
ladders built by the exchange of two t-channel (reggeized 14 ) gluons, see the left-hand side ofFig. 20.5.
The sum of ladder diagrams of the type of Fig. 20.5 is the simplest multiparticle structure
which reproduces the power-like sα behaviour of the Pomeron pole. In other words it corresponds
to a sum of completely inelastic 2 → n processes, that is, to the last term Ginel = 1 − exp(−Ω)
in the unitarity equation (20.9). This set of diagrams was resummed in the limit of a small QCD
coupling, αs  1, but large energy, such that αs ln(s/s0 ) ∼ O(1) [59]. The summation results in
the rightmost singularity at j = 1 + ω0 > 1. After accounting for the next-to-leading logarithmic
14

That is, the virtual loop corrections to the one-gluon exchanges are included. These corrections are importantin order to provide infrared stability of the results.
31st May, 2024

19

20. High Energy Soft QCD and Diffraction(NLL) corrections, the position of the singularity (Pomeron intercept) corresponds to ω0 = 0.25–
0.3 depending only weakly on the scale [122, 123, 132–136], whose value is characterized by thetransverse momentum, kt , of gluons in the ladder.
It was demonstrated (see e.g. [137]) that the resummation of the (αs ln(1/x))n terms based on
the QCD Pomeron results essentially improves the description of low-x inclusive HERA data within
the framework of the NNLO DGLAP evolution.
At this stage the singularity is the cut in the j-plane. However we have to account for theboundary conditions at relatively small kt . Imposing a reasonable boundary, we arrive at a seriesof Regge poles in the interval from j = 1 to j = 1 + ω0 instead of the cut [134]. Note that thefirst (corresponding to the rightmost pole in the j-plane, i.e. to the pole with the largest Re j)eigenfunction consists of gluons with relatively small kt , while for the next poles the kt increases.
DIS inclusive γ ∗ p cross sections were fitted in [135] using the QCD based approach in which Pomeronis represented by series of Regge poles obtained within the perturbative QCD BFKL approach. It
was concluded that the first pole has a small coupling to the proton. It is possible that this smallvalue of the coupling to the proton is related to the fact that the enhanced multi-Pomeron diagrams(i.e. the rescattering of intermediate partons) were neglected in the fit. The main effect of thisenhanced contribution is the “renormalization” of the intercept which diminishes the effective valueof ω0 . Besides this, the enhanced diagrams provide a saturation by reducing the rise of the partondensities in the (b, kt , y)-space (see e.g. [138, 139]).
Note that perturbative QCD allows us to understand why the values of the phenomenologicalmulti-Pomeron vertices and the shift, ω0 , of the intercept, are small (due to αs  1 and somenumerical factors such as Nc and π). Indeed, at the lowest αs orders we get for the ω0 value andthe simplest multi-Pomeron vertices (see e.g. [59, 139, 140]):
ω0 ∝

Nc αs
,
π

g3P ∝

Nc αs22(Nc − 1)π 2

and

g22 ∝

Nc αs
,
(Nc2 − 1)2(20.46)

where g22 is the coupling corresponding to the transition of 2 into 2 Pomerons.
20.5.1 BFKL evolution in the ‘dipole’ representation
It was shown in [141–144] that the LO BFKL Pomeron equation [59] can be written in termsof the evolution of the dipole density, N (xd , yd ; y), in rapidity y (here xd and yd are the transverse
coordinates of two t-channel gluons which form the colour singlet dipole). Indeed, after the emissionof a new gluon at point zd , the initial colour dipole with coordinates (xd , yd ) turns into a pair ofdipoles (xd , zd ) and (zd , yd ). This can be considered as a development of a ‘dipole cascade’. Moreoverin this formalism it is easy to include the non-linear absorptive corrections (last term in the square
brackets in Eq. (20.47)), which accounts for the rescattering of the intermediate partons (gluons)on the target proton. The corresponding contribution is described by the so-called “fan” diagramsand these are the most important corrections to the linear DGLAP (Dokshitzer-Gribov-LipatovAltarelli-Parisi) evolution [145] in the case of DIS at not large scales but at very small momentum
fraction [139].
The resulting non-linear evolution (Balitsky-Kovchegov equation [146–148]) readsd(xd − yd )2
αs Nc
2N (xd , yd ; y) =
d
z
d
dy
2π 2
(xd − zd )2 (yd − zd )2(20.47)× [N (xd , zd ; y) + N (yd , zd ; y) − N (xd , yd ; y) − N (xd , zd ; y)N (yd , zd ; y)] .
Z

For a small density N the last term in the square brackets can be neglected, and the first threeterms in Eq. (20.47) reproduce the conventional BFKL equation in the coordinate representation.
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20. High Energy Soft QCD and Diffraction

However, for large N → 1 the right-hand side of Eq. (20.47) vanishes and we reach the saturation
N = 1. It is worth mentioning that, as shown in [149], in terms of ‘dipole’ formalism, with thetriple-Pomeron vertex generated by the ‘one dipole to two dipoles’ transition, it is possible to relate
the Good-Walker approach to high mass diffraction with the triple-Pomeron diagram.
20.5.2 Distribution of secondaries: theory versus experiment
As already discussed, in terms of Feynman diagrams the cut Pomeron can be viewed as aset of ladder diagrams corresponding to a sum of completely inelastic 2 → n processes, that is,to the last term Ginel = 1 − exp(−Ω) in the unitarity equation (20.9). Here n > 2 means the
production of additional (n − 2) gluons which, after hadronization, form minijets. 15 Therefore,
in the final state driven by one Pomeron, we expect to observe gluon minijets with a flat rapidity
distribution in the central (plateau) rapidity region. This would correspond to a flat pseudorapidity
distribution of produced particles if they were massless. A typical pseudorapidity distribution of
charged particles in inclusive events (up to |η| = 7) is shown in Fig. 20.11 (left) [150] (see also
Fig. 53.1 in [1]). The central part (|η| < 2.5) was measured by CMS, while the forward region wascovered by TOTEM. The dip observed atqη = 0 is explained by the presence of massive particles(the Jacobian J(pT , m, η) = pT /E → pT / p2T + m2 at η = 0). A photon energy spectrum is shownin Fig. 20.11 (right) [151], measured by LHCf inclusively and in events with a diffraction topology,
i.e. no charged particles with pT > 100 MeV and |η| <2.5 observed by ATLAS. As expected indiffractive events the energy flow decreases with Eγ more slowly than that in the inclusive case.

Figure 20.11: (Left) Charged-particle pseudorapidity distribution for inclusive events measured
by CMS and TOTEM [150]. The error bars represent the statistical and uncorrelated systematic
uncertainties between neighboring bins, while the shaded areas denote the combined statisticaland full systematic uncertainties. The coloured lines indicate model predictions. (Right) Photon
energy spectrum measured by LHCf at |η| >10.94. The filled circles show the inclusive photon
spectrum measured by LHCf [152] and filled squares the spectrum for Nch = 0 events where no
charged particles with pT > 100 MeV and |η| <2.5 are observed by ATLAS [153]. The colouredlines indicate model predictions. The error bars correspond to the statistical uncertainties and theshaded areas denote the combined statistical and systematic uncertainties. Figure from Ref. [151].
The energy dependence of the particle density dNch /dη at η = 0 is shown in Fig. 20.12 (left).
Neglecting absorptive corrections given by the enhanced diagrams (which mainly change (‘renor15
Minijets result from hadronization of partons emitted from the cut QCD Pomeron. Typically these are groupsof hadrons with comparatively low overall ET <∼ 5–10 GeV.

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20. High Energy Soft QCD and Diffraction

Figure 20.12: (Left) Energy dependence of the charged particle density dNch /dη at η ≈ 0 for pp
and pp̄ collisions. Shown are measurements performed with different Non-SD event selections from
UA1 [154], UA5 [155], CDF [156,157], ALICE [158] and CMS [159]. The dashed line is a power-law
fit to the data. Figure from Ref. [150]. (Right) Differential cross section of charged particles with√
|η| < 0.8 in inelastic pp collisions at s = 0.9, 2.76 and 7 TeV as a function of pT . Only statisticaluncertainties are shown. Figure from Ref. [160].
malize’) the effective Pomeron intercept αeff (0) = 1 + ∆ [121]), we conclude that according to
the AGK rules the plateau height dσ/dη ∝ s∆ is driven just by the one-Pomeron exchange witheffective ∆ ∼ 0.2 (see Section 20.3.3.3). That is, the density of secondaries observed in the inclusive
process increases with increasing energy faster than the total cross section, whose growth is tamedby the multi-Pomeron diagrams. Indeed, as is seen from Fig. 20.12 (left), in the interval of collider
energies dNch /dη = (1/σinel )dσ/dη ∝ s0.115 (i.e. dσ/dη ∝ s0.215 ), while σinel ∝ s0.1 .
Contrary to the ‘old’ Regge theory where it was assumed (based on the experimental data
existing in the 1950s and 1960s) that all transverse momenta are limited, in QCD the kt distributionsof jets (charged particles) have a long kt tail (dσ/dkt2 ∝ αs2 (kt2 )/kt4 at large kt and very large energys  kt2 ). An example of the pT distribution of charged secondaries is shown in Fig. 20.12 (right).
Note that the mean transverse momentum of secondaries, produced via jet fragmentation, slowly
increases with collision energy, see Fig. 20.13 (right). This is caused by the stronger absorption (at√larger s) of the gluons with a smaller kt (σ abs ∝ 1/kt2 ). The growth of hpT i with multiplicity (seeFig. 20.13 (left)) can be explained by the fact that events with larger Nch correspond to a smaller
impact parameter, b, where the absorption of a low kt component is stronger and, next, larger
multiplicity can be originated by the events with jets/minijets with higher pT . Since the mean pT√√of secondaries grows with s, the increase with s of transverse energy flow is a bit faster thanthat of particle density.
The model [162] based on a modification of the classic RFT allows one to trace the smoothtransition from the pure perturbative, large kt , region into the soft domain. A strong absorptionof the low kt partons plays a crucial role here since it produces an effective infrared cutoff, ksat ,and provides the possibility of extending the parton approach, used for ‘hard’ processes, to also
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20. High Energy Soft QCD and Diffraction

22
CMS

CMS

pp √
s = 13 TeV

1.6

1.4

Pythia8 CUETP8M1
EPOS LHC
Pythia6 Z2*

1.4

pp DS
π±±
K−
p,p

pp inel
π±±
K−
p,p

1.2

〈pT〉 [GeV/c]〈pT〉 [GeV/c]

1.2
1
0.8

1

0.8

0.6

0.6
0.4

±

0.4

π±
K−
p,p

0.2
0

50

100

150

200

0.2
0.5

1

2

5

10

s [TeV]√〈Ntracks〉

Figure 20.13: Average pT of pions, kaons and protons in the range |η| < 1.0 as a function of (left)
track multiplicity at |η| < 2.4 and (right) of center-of-mass energy where the curves show linear fitsusing lns. The error bars indicate the uncorrelated combined uncertainties, while the boxes show
the uncorrelated systematic uncertainties. Figures from Ref. [161].
describe high-energy soft and semihard interactions. This approach combines a description of soft
physics and diffraction with jet physics in a coherent self-consistent way.
Another way is to include the soft and hard components independently [37,66,163,164]. In thisapproach the soft part is described in terms of RFT with the phenomenological “soft” Pomeronpole while the hard part is calculated in terms of the parton model for minijet production with theenergy dependent cutoff kt > k0 (s). A combined description of soft and hard processes in hadronic
collisions is reached within the QGSJET Monte Carlo model (e.g. [61]) in the framework of theso-called “semi-hard Pomeron” approach (see e.g. [165]).
In [166] a model was constructed, which incorporated the attractive features of the two successful
theoretical approaches to high energy QCD: BFKL Pomeron calculus [59, 60] and the Colour Glass
Condensate/saturation [167].
20.5.2.1 Correlations
All LHC experiments routinely measure tracks with pT > pmin , where pmin can vary in different
studies. Typically, pmin = 200 MeV, where tracking reconstruction efficiencies are larger than 70%.
In order to identify particle species, each experiment has sophisticated identification procedures
usually based on the ionization energy loss, dE/dx, or other techniques, with different regions ofapplicability for different particle species. Thanks to usually relatively large cross sections of soft
QCD processes, most of the results below come from event samples with very low or negligible
pile-up.
Following the notation in [168], symmetrized inclusive particle number densities for q points at
y1 , ..., yq (where yi represents the 4-momentum of the ith particle), ρq (y1 , ..., yq ), are related to theinclusive differential cross section by
1σinel

dσ = ρ1 (y)dy,

1
σinel

d2 σ = ρ2 (y1 , y2 )dy1 dy2

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etc.

(20.48)

20. High Energy Soft QCD and Diffraction

23
By integrating we getZρ1 (y)dy = hni,

Z Z

ρ2 (y1 , y2 )dy1 dy2 = hn(n − 1)i etc.,(20.49)where the angular brackets denote averaging over the event sample and n is the particle multiplicity.
Since the inclusive q-particle densities in general contain trivial contributions from lower-orderdensities, it is convenient to consider quantities Cq which vanish when one of their arguments
becomes statistically independent of (uncorrelated with) the others. These quantities Cq , called
correlation functions (or cumulant functions), are defined as:C2 (1, 2) = ρ2 (1, 2) − ρ1 (1)ρ1 (2), C3 (1, 2, 3) = ρ3 (1, 2, 3) −X

ρ1 (1)ρ2 (2, 3) + 2ρ1 (1)ρ1 (2)ρ1 (3),(3)

C4 (1, 2, 3, 4) = ρ4 (1, 2, 3, 4) −X

ρ1 (1)ρ3 (1, 2, 3) −(4)X

ρ2 (1, 2)ρ2 (3, 4) + 2(3)Xρ1 (1)ρ1 (2)ρ2 (3, 4)(6)− 6ρ1 (1)ρ1 (2)ρ1 (3)ρ1 (4).
(20.50)The 2D two-particle correlation function is defined asC(∆η, ∆φ) =ρ2 (∆η, ∆φ).
ρ1 (ηa , φa )ρ1 (ηb , φb )(20.51)The distribution ρ2 (∆η, ∆φ) is usually interpreted as a conditional probability to observe a particle
a at the phase-space point (ηa , φa ) if a particle b at (ηb , φb ) is observed as well, and ∆η = ηa − ηband ∆φ = φa − φb . The distributions ρ1 (ηa , φa ) and ρ1 (ηb , φb ) are probabilities to observe a single
particle at (ηa , φa ) and (ηb , φb ), respectively. The denominator of Eq. (20.51) is constructed as a
product of two single-particle distributions using an event mixing technique, where each particle
in the pair comes from a different event. Experimentally, each reconstructed track is weighted by
the inverse of an efficiency factor which accounts for the detector acceptance, the reconstruction
and particle identification efficiencies, the contamination by secondary particles and the fraction of
misreconstructed tracks.
An example of two-particle correlation functions measured in pp collisions at 7 TeV is shown in
Fig. 20.14 for identical-particle pairs (right panel) and for particle–anti-particle pairs (left panel)[169].
We observe two distinct features which can be explained by short-range (in rapidity) correlations: 1) a near-side peak at ∆φ ≈ 0 and 2) an away-side peak or rather a ridge at ∆φ ≈ π. The
near-side peak is considered to be caused by at least three effects:• fragmentation of partons scattered at a hard scale. These relatively high pT partons produce
showers which after the hadronization form the mini-jets which create a broad structure
extending over at least one unit in ∆η and ∆φ.
• resonance decays. The decay of resonances contributes to the near-side peak at ∆η ∼ 0 and
extended in ∆φ [170–172], depending on the released kinetic energy of the given resonance.
This effect is mostly visible for unlike-sign particle pairs.
• femtoscopic correlations. The term “femtoscopic” refers to a length scale of the order of 10−15
m. These correlations are present at low relative momenta of the particles in a pair (representing a very small phase-space corner, so they are practically invisible in terms of (∆η, ∆φ))and give rise to an enhancement of the correlation function (due to Bose-Einstein quantum
statistics for identical bosons) or its suppression (due to Fermi-Dirac quantum statistics for
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20. High Energy Soft QCD and Diffraction

24

ALICE pp s = 7 TeV(e) π+πC (∆ϕ, ∆η)C (∆ϕ, ∆η)(a) π+π+ + π-π1.2
1
0.8
1
0

-

d)
ϕ (ra

-

(f) K+KC (∆ϕ, ∆η)C (∆ϕ, ∆η)(b) K+K+ + K K

∆

1.2
1
0.8
1
0

∆

d)
ϕ (ra

1.2
1
0.8
1
0

−1

0

−1

0

2.5
2
1.5
1
1
4
2

∆

d)
ϕ (ra

0

2.5
2
1.5
1
1
4

∆η

∆η

2

rad)

∆ϕ (

2

∆

d)
ϕ (ra

(h) ΛΛ
C (∆ϕ, ∆η)(d) ΛΛ + ΛΛC (∆ϕ, ∆η)

0

-

0

4

1.2
1
0.8
1

∆η

4
2
0

rad)

∆ϕ (

2.5
2
1.5
1
1
0

4

∆η

0
−1

−1

∆

d)
ϕ (ra

(g) ppC (∆ϕ, ∆η)C (∆ϕ, ∆η)(c) pp + pp

−1

0

∆η

∆η

2

0

2

−1

0

4
−1

4

∆η

∆η

0

1
0

4
2

−1

2.5
2
1.5
1

2

)

rad
∆ϕ (

Figure 20.14: Two-particle correlation functions for identical-particle pairs: π + π + + π − π − ,K + K + +K − K − , pp+ p̄p̄, and ΛΛ+ Λ̄Λ̄ (left panel) and particle–anti-particle pairs: π + π − , K + K − ,pp̄ and ΛΛ̄ (right panel). Figure from Ref. [169].

identical fermions). Besides this, at low relative momenta there are correlations caused by
Coulomb and/or other final state interactions. The shape of all these effects in (∆η, ∆φ) spacedepend strongly on the mass of the particle type as well as on the size of the particle-emitting
system. The latter is traditionally measured in Bose-Einstein correlation (BEC) analyses andis not part of this review.
The away-side peak originates from energy-momentum conservation which manifests itself by
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20. High Energy Soft QCD and Diffraction

25

the quark and the anti-quark going back-to-back in φ. In this case the rapidity width of the awayside peak is much larger than the near-side peak since in the original matrix element the quark andthe antiquark can be separated by some ∆η interval.
As discussed in Sections 20.3.3.2 and 20.3.3.3, there may be several cut Pomerons in the same
event, each giving rise to particle sets which are, in general, independent of each other (except forsmall Bose-Einstein correlations). This leads to long-range (in rapidity) correlations. Since thedensity of secondaries, dN/dy, is proportional to the number of cut Pomerons, k, the probability toobserve at least one particle is proportional to hki, while the probability to observe simultaneously
two particles separated by some (rather large) rapidity interval is proportional to hk 2 i. Thus thelong-range correlations are pedicted to be C2 = hk 2 i/hki2 − 1 > 0 which depends weakly on theseparation ∆η between the two particles [173, 174]. In the case of the pure eikonal approach,neglecting the enhanced diagrams and the conservation law effects in the proton fragmentation
region, we expect that these long-range correlations,C2 (∆y) =σinel d2 σ/dy1 dy2− 1 ∼ const,
dσ/dy1 dσ/dy2(20.52)

do not depend on the rapidity separation, ∆y = |y1 − y2 |, between the two particles. The contribution of the processes with more cut Pomerons also results in a much wider multiplicity distribution
and in a larger density of soft particles coming from the ‘underlying event’.
CMS

0.01

ATLAS
EvSel_Mref
0.3 < pT < 3 GeV|η| < 2.5

p+p s = 5.02 TeV
p+p s = 13 TeV
p+Pb sNN = 5.02 TeV
Pb+Pb sNN = 2.76 TeV

pp

0.10

s = 13 TeVvsub
2 {2, |Δη|>2}
v2{4}
v2{6}

v2

c2{2,|∆η|>2}

0.015

0.05

0.005

0
0

0.3 < p < 3.0 GeV/c
T|η| < 2.4

100

200

300
400
〈Nch(p > 0.4 GeV)〉
T

0

50

100

Noffline
trk

150

Figure 20.15: (Left) The two-particle cumulant, c2 {2, |∆η| > 2}, as a function of hNch (pT >√√0.4 GeV)i for pp collisions at s = 5.02 and 13 TeV, pP b collisions at sN N = 5.02 TeV and low√multiplicity P bP b collisions at sN N = 2.76 TeV. The data are constructed from particles with
0.3 < pT < 3.0 GeV. Figure from Ref. [175]. (Right) The v2 {2, |∆η| > 2}, v2 {4} and v2 {6} values
as a function of number of charged particles, averaged over 0.3 < pT < 3.0 GeV and |η| < 2.4, in√
pp collisions at s = 13 TeV. Figure from Ref. [176]. The error bars correspond to the statisticaluncertainties, while the shaded areas denote the systematic uncertainties.
20.5.2.2 Color reconnection
In this context, we have to mention also the so-called ’colour reconnection’ phenomenon. Thisis a pure ‘soft QCD’ effect. The point is that after a number of coloured secondary partons areproduced, there are different possibilities to form the colour flow between these partons and togroup the partons into colourless clusters. In the process of reconnection, one rearranges the colour
flow in such a way as to minimize the size of the clusters. This is especially important when
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20. High Energy Soft QCD and Diffraction

26

dealing with MPI contributions. The reconnection between the different cut Pomerons diminishes
the final multiplicity and can change the form of the Nch distributions (see e.g. Section 43.3.2 of [1]and [2, 13, 177, 178]).
20.5.2.3 Double parton scattering
The probability of MPI depends on the spatial distribution of partons in the incoming protons.
The effects of MPI are suppressed if the density of partons is low and the partons from the incoming
beam particles are separated from each other by a large interval in transverse coordinate space ~xt .
Events in which two hard subprocesses, caused by interactions of two different parton pairs (say,(a1 b1 ) and (a2 b2 )), take place simultaneously, are called Double Parton Scattering (DPS). The
DPS cross section is driven by the ‘double parton distributions’, D(ya1 , ya2 , ...), where ya1 and ya2
are momentum fractions carried by the partons from the proton a and the dots denote all othercoordinates. As a rule, experiments study DPS processes at relatively small momentum fractions
yi . Here, correlations due to momentum conservation (like ya1 + ya2 < 1) are not so important,
and with a reasonable accuracy we can assume a factorization
D(ya1 , ya2 , ...) ∝ F (ya1 ) · F (ya2 ) ,(20.53)where F (yai ) are the single parton distributions. In such a case the DPS cross section takes theform
σa b σa b
σ DPS = c · 1 1 2 2 ,(20.54)
σeff
where σa1 b1 and σa2 b2 are cross sections for the two independent hard processes, while σeff characterizes the mean area occupied by the partons a1 and b1 ; the constant factor c = 1/2 if both hard
processes (a1 b1 ) and (a2 b2 ) are identical, otherwise c = 1. Thus the DPS cross section is sensitive
to the spatial separations between partons in the proton (see Section 7.2.3 in [2] and [179, 180] formore explanations and reviews).
One problem is that within this approach we assume that the partons a1 and a2 are produced by
two independent parton showers (and similarly for the other incoming proton). On the other hand,
there is a probability that from the beginning we start with the evolution of a single shower whichfurther splits into two different branches. In this case the separation between theptwo partons (twoshower branches) becomes very small – of the order of the inverse scale (∼ 1/ q 2 ) at which the
splitting occurs. The exact value of this ‘splitting’ scale q 2 depends on the particular kinematicsof the DPS process. So, different experiments (with different kinematical conditions) can givesomewhat different values of σeff . In general, the value of σeff depends on the following features:a) on the measured process since the spatial (bt ) distributions of different
p incoming partons (lightquarks, heavy quarks, gluons) can be different; b) on the splitting scale, q 2 , of one parton cascade
into two branches. The typically high value of the splitting scale then explains the fact that theexperimentally measured values of σeff ∼ 7–25 mb (see Fig. 4 of [181]) are smaller than σtot ormostly even lower than the proton area πRp2 ∼ 22–24 mb (see e.g. [182]); c) on the pT balance,−→
kT , in the individual hard process (e.g. for two dijet productions kT = |−p→
T 1 + pT 2 | where pT 1 andpT 2 are jet pT ’s of the first hard process (similarly for the second
phard process). A small value of
kT indicates that there were no splittings or the splitting scale q 2 was small and, therefore, we
expect larger σeff ; d) on the contribution of single parton scatterings misidentified as DPS. For alower scale of the hard process this contribution is larger (see [183] for more detailed discussion).
20.5.2.4 Final state interactions
The formalism of the RFT does not include ‘final state interactions’ 16 . Therefore, besides thecorrelations considered in the previous Section 20.5.2.1 we have to expect the correlation caused by
16

In general, final state interactions can be included into the detailed structure of the multi-Pomeron vertices.
However these vertices are phenomenological objects which are not well known experimentally.
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〈 Nch/ δη δφ 〉2.5

pT > 0.5 GeV, |η| < 2.5〈 Σ pT/ δη δφ 〉 [GeV]

20. High Energy Soft QCD and Diffraction

27

ATLAS
s = 13 TeV, 1.6 nb-1

2

1.5

8
7
6
5

pT > 0.5 GeV, |η| < 2.5

ATLAS
s = 13 TeV, 1.6 nb-1

Towards region
Transverse region
Away region

4

1

3

0

2

Towards region
Transverse region
Away region

0.5

5

10

15

20

25

1

30

0

5

10

plead [GeV]
T

15

20

25

30

plead [GeV]T

Figure 20.16: Mean charged particle density (left) and the sum of transverse momenta of secondaries (right) in events with the ‘leading’ high pT particle as a function of pleadin the transverse,T
towards and away azimuthal regions. Secondaries with pT > 0.5 GeV and |η| < 2.5 are registered.
The error bars (mostly hidden by the data markers) represent combined statistical and systematic
uncertainties. Figures from [184].
partons and hadrons rescattering in the final state. These effects are not crucial at lower energies,
but become more important at high LHC energies, in particular in heavy-ion collisions where theparticle density is large. For example, the final state interactions (FSI) lead to the formation of thecollective flow of secondaries (see e.g. [185] for a review), especially in high-multiplicity events. To
study the collective flow experimentally, one has to subtract correlations coming from few-particle
sources such as resonance decays, mini-jets, multi-jets and BEC (so called “non-flow”). The nonflow can efficiently be suppressed using the sub-event method, that is by studying the azimuthal
correlations between particles separated in η [186], or subtracted using the multi-particle correlation(or cumulant) techniques.
The cumulant method is based on calculating 2k-particle azimuthal correlations, corrn {2k},and cumulants cn {2k} (where k = 1, 2, ...), for nth Fourier harmonics. The corrn {2k} are definedas [187, 188]:
hhcorrn {2}ii = hhein(φ1 −φ2 ) ii, hhcorrn {4}ii = hhein(φ1 +φ2 −φ3 −φ4 ) ii,
hhcorrn {6}ii = hhein(φ1 +φ2 +φ3 −φ4 −φ5 −φ6 ii
and similarly for higher numbers of correlated particles. The double-brackets hhii denote averaging
first over particles in an event and then over events within a given event class. For every event,the average is taken over all possible combinations of azimuthal angles φl (l = 1, ..., 2k) of the
2k particles. The cumulants are then obtained from multi-particle azimuthal correlations after
subtracting correlations between 2(k − 1) particles according to the following formulae [187, 188]:cn {2} = hhcorrn {2}ii, cn {4} = hhcorrn {4}ii − 2hhcorrn {2}ii2 ,cn {6} = hhcorrn {6}ii − 9hhcorrn {2}ii × hhcorrn {4}ii + 12hhcorrn {2}ii3 .
The cumulants for higher particle multiplicities are calculated in [187, 188]. The cumulants thenserve to estimate the Fourier harmonics vn as follows [187]:pppvn {2} = cn {2}, vn {4} = 4 −cn {4}, vn {6} = 6 cn {6}/4.
Some of the long-range correlation (|∆η| > 2) results obtained on a sample of charged particles
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28

20. High Energy Soft QCD and Diffraction

with 0.3 < pT < 3.0 GeV and |η| < 2.4 are summarized in Fig. 20.15. The left plot shows the
cumulant c2 measured for pp, pP b and P bP b collisions [175], while the right plot shows the ellipticalharmonics v2 measured for pp collisions [176], both as functions of multiplicities of charged particles.
The two-particle correlations are observed to be strongest and rising with Nch for P bP b collisions,and weakest and rather flat for pp collisions. The elliptical-flow harmonics for 4- and 6-particle
correlations show again a rather flat multiplicity dependence (at least for large multiplicities).
Within experimental uncertainties, the values of v2 {2}, v2 {4} and v2 {6} measured in pp collisions
at 13 TeV are consistent with each other. The similarity between v2 {4} and v2 {6} suggests thatsome collective effects are occurring in pp collisions at high multiplicity and the observations are
similar to those in P bP b collisions, where the v2 {4} values were measured to be close to v2 {6} but
they are both lower than v2 {2} (not shown here)).
Another example of long-range correlations is the so-called “ridge effect”. Here not only the‘back-to-back’ jet correlations are registered, but also an excess of particles going in the same (inthe azimuthal plane) direction as the leading (relatively high pT ) hadron. Moreover, this excess isseen at the rapidities separated from the leading hadron by a rather large interval (see e.g. [189]for a review).
It is popular to describe such FSI effects within the hydrodynamic model [190], which operates
with collective (thermodynamic) variables. In terms of microscopic interactions, the collective flow
can be caused by the geometry of a particular collision (the absorption is smaller for the secondaries
flying in the direction orthogonal to the impact parameter vector ~b [191, 192]), or by the colour
reconnection at the hadronization stage [193], or accounting for the rescattering of secondaries
directly, as was done, for example, in the AMPT model [194].
20.5.3 The underlying event
Except for the exclusive case, any ‘hard’ subprocess is accompanied by soft secondaries coming from initial state radiation (ISR), final state radiation (FSR) and multiple parton interaction(MPI), see Subsection 7.2.2 in [2]. These extra particles distort the signal we are looking for.
In particular, they affect the isolation criteria applied to photons and charged leptons and thevertex reconstruction efficiency. In general, also the effects of colour reconnection (discussed in
Section 20.5.2.2) contribute to the underlying event.
The usual procedure of estimating the amount of underlying event (UE) is to spatially divide
tracks in each event according to their azimuthal angle into the Toward region (where the highest
pT jet points), the Away region (opposite to the Toward region) and to two Transverse regions. The
standard observables are the average track multiplicity per unit area and the average scalar sumof track pT per unit area. Figure 20.16 shows the particle density and the sum of pT for the UE in
ATLAS events containing at least one charged particle with pT > 0.5 GeV and |η| < 2.5 [184].
Note that by construction the largest values of hpT i are observed in the ‘Toward’ region, whilein the ‘Away’ region we observe a slightly larger density than in the Toward region. These areresults of the ‘leading’ and ‘backward’ jet fragmentation. In the transverse region, mostly filledby particles from the UE, the particle density and sum of pT per unit (∆η, ∆φ) area practicallydo not depend on the plead
since these secondaries come from the other cut Pomeron(s), that is,Tfrom other ‘multiple interactions’. For low plead< 2 GeV the distributions in all three regions areTclose to each other. These events actually do not contain a ‘hard’ subprocess. Moreover, for avery small plead→ 1 GeV we start to select soft events with abnormally low pT < plead
particles.
T
T
Since only particles with pT > 0.5 GeV are registered, the signal drops fast for plead→
1
GeV. AsT√P
a function of collision energy s, the energy flow in the transverse region increases as pT ∼ s0.2
(as follows from Fig. 7 (right) in [184]) due to the larger number of MPI collisions and larger hpT i
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20. High Energy Soft QCD and Diffraction29

in each collision 17 . As follows from this and other UE-dedicated LHC studies [195], from thecomparisons of the data to the models with and without MPI, the necessity of MPI is convincingly
demonstrated.

20.6 The Odderon

Figure 20.17: Comparison of the t-dependence of the elastic cross sections from pp and pp̄
collisions. (Left) Data from the ISR energy of 53 GeV are shown by closed triangles [196] forpp collisions and by open circles [197] and open squares [198] for pp̄ collisions. Only t-dependent
uncertainties are shown and the systematic scale uncertainty is estimated to be ±30%. Figure from
Ref. [197]. (Right) Data from the D0 experiment at 1.96 TeV [199] are compared with data fromthe TOTEM experiment [103]. The green dashed line indicates the normalization uncertainty of
the D0 measurement. Figure from Ref. [103].

ρ

0.25

PDG:

0.2

pp

0.15

pp

0.1

TOTEM:−0.05

13 TeV

7 TeV
8 TeV

1.8 TeV

2.76 TeV

0

indirect
0.9 TeV

0.546 TeV

0.05

via CNICOMPETE(pre-LHC model RRPnf L2u ):

−0.1

pp

−0.15

pp

−0.2
101

102

103

104

105
√
s [GeV]

Figure 20.18: The dependence of the ρ parameter on the collision energy. The pp (blue) and pp̄
(green) data are taken from [1]. The TOTEM measurements are marked in red. The two points
at 13 TeV correspond to two fit cases, discussed in [102], using the same data. The lines represent
fits to the data using the COMPETE parameterization [22]. Figure from Ref. [102].
17

Recall the stronger absorption of the low kt partons.

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20. High Energy Soft QCD and Diffraction

30

Apart from the even-signature singularity (Pomeron), in QCD with Nc = 3 there exists its
counterpart, the odd-signature singularity placed at j ' 1 and formed by three t-channel reggeized
gluons connected in colour space by the symmetric dabc tensor of the colour SU (3) group [200,201].
This object is called the Odderon. The Odderon exchange amplitude has opposite sign for pp andpp̄ scatterings. Its intercept is predicted to be very close to j = 1 [202–204], while according toperturbative estimates the coupling to the nucleon is rather small [205, 206]. The corresponding
amplitude is mainly real and is about 100 times smaller than the imaginary part of the Pomeron
exchange amplitude. Calculating the elastic amplitude via the eikonal formula (20.13) we have toreplace the opacity Ω(b) by the sum Ω = Ωeven + Ωodd , where Ωeven is mainly real and Ωodd isimaginary. Note that at t = 0 this QCD Odderon does not couple to mesons, and the t-slope of theOdderon amplitude is expected to be smaller than that for the Pomeron; instead of the singularity
at t = 4m2π in the Pomeron case, the nearest singularity in the Odderon channel is at t = 9m2π , seefor instance [204]. Thus, in the impact parameter b space the QCD Odderon occupies an area of asmaller radius, see e.g. [207].
Experimentally an indication in favour of a manifestation of the high energy C-odd amplitude
was observed by comparing the elastic pp and pp̄ cross sections in the dip region (where the contribution from the C-even amplitude has a minimum) at the CERN-ISR [197], see Fig. 20.17 (left)
18 .
To get a better understanding of the Odderon effects it would be very instructive to have thedσel /dt data for both pp and pp̄ reactions at the same but higher energy ∼ 1 TeV (ideally in thesame apparatus) and, in the ideal case, to study the energy dependence.
√
At the moment we can only compare the pp cross section measured by TOTEM at s = 2.76 TeV[103] with the dσ/dt values measured by the D0 collaboration at 1.96 TeV in pp̄ collisions [199],
see Fig. 20.17 (right).
The situation looks quite intriguing, but needs further investigation. Note that in the TeV energy
range the ω, ρ and ωP , ρP exchange contributions, which may be responsible for the difference
between the pp and pp̄ cross sections in the dip region at the ISR energies, are practically negligible.
Another way to search for the Odderon is to measure the real part of the elastic pp scattering
amplitude via the interference with the pure QED one-photon exchange. Since the one-photon
exchange amplitude contribution is sizeable only at very small |t|, this way we can study the
Odderon at or near to t = 0. Indeed, the value of the ratio ρ ≡ ReTel /ImTel , obtained by TOTEM
at 13 TeV (ρ = 0.10 ± 0.01 [102]), turns out to be smaller than that expected for the pure evensignature amplitude, see Fig. 20.18.
Based on dispersion relations and assuming the C-even contribution only, from the known total
cross sections we would rather expect ρ ' 0.13–0.14. The difference could indicate that the rise of
the total cross section at energies above those of the LHC slows down (see the dispersion relation,
Eq. (20.19)) or this could be attributed to an Odderon contribution (see e.g. [210]). However, theOdderon exchange amplitude extracted in this analysis has opposite sign to that for the lowestαs -order QCD Odderon, see e.g. [205, 206, 211, 212]. Besides this, the Odderon contribution to ρ,√√
obtained in [210], grows with s (for s > 0.5 TeV), while in QCD we expect that the Odderon
contribution to ρ decreases with energy, since the QCD Odderon intercept is smaller than that of
the QCD Pomeron. Note also that recently the ATLAS Collaboration has published new results
on elastic pp scattering at 13 TeV using the ALFA proton spectrometer [213]. While the measured
value of ρ = 0.098 ± 0.011 appears to be in a good agreement, a smaller total cross-section (with asmaller error bar) σtot = 104.7 ± 1.1 mb, is obtained when both are compared with TOTEM results
at 13 TeV [102].
18

Note that a qualitatively similar behaviour to the pp̄ ISR data, namely a filling in of the dip in the t-distribution,√
was observed by the UA4 collaboration at the CERN Spp̄S collider at s=546 and 630 GeV(see [208, 209] and inparticular Fig. 2 in [209]) and by the D0 collaboration at the Tevatron at 1.96 TeV [199].
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20. High Energy Soft QCD and Diffraction

31

The values of σtot measured by ATLAS at 7, 8, and 13 TeV are found systematically lower
than the TOTEM numbers. We note that if we used these ALFA values in dispersion relations, wewould obtain a smaller ρ ratio for the C-even amplitude as well. Indeed, as shown in Figure 3 ofRef. [214], a good description of both, the σtot and ρ measurements, is achieved based on a global fit
to the data performed using a damped ln2 s amplitude and without an assumption of the Odderon.
A good description of the ATLAS-ALFA data (χ2 /n.d.f. = 0.85, see table 11 of Ref. [213]) of bothσtot and ρ without an assumption of the Odderon is also obtained in the KMR model [215].
It is worth mentioning also that the Odderon contribution is strongly screened by the multiPomeron diagrams, which facilitate the falling-off of ρ with energy increasing, see [216,217]. On theother hand, analyzing the whole ensemble of high energy elastic pp (pp̄) low |t| data, a reasonable
description can be obtained using the even-signature amplitude only, that is, without the Odderon.
In particular, the RR(PL2)qc model/version of the COMPETE parameterization is consistent with
the TOTEM 13 TeV data on σtot and ρ within 1σ 19 . Another example is the recent analysis
in [212] of the low |t| < 0.1 GeV2 elastic data. Fitting all the low-t pp and pp̄ data in the range√of s between 13 GeV and 13 TeV without Odderon, Donnachie and Landshoff [212] succeeded todescribe the TOTEM cross section with less than 1σ deviation in each dσel /dt point (see Fig. 8of [212]). Note that in this analysis, they get a larger value of ρ close to 0.14 at 13 TeV.
It was proposed also to search for the Odderon in exclusive C-even meson (π 0 , η, f2 , ηc , ...)
photoproduction (see e.g. [218, 219]). However the expected cross sections are small (e.g. for ηc )and in each channel there is a large background caused either by Pomeron-Pomeron fusion (suchas CEP of the f2 meson production in pp or pP b collisions) or due to the vector meson radiative
decay (such as ω → π 0 γ for the case of pion) [220]. Up to now, no definitive Odderon signal in theC-even meson production has been observed. At the moment there exist only upper limits on the√photoproduction cross sections obtained in the measurements at HERA at s ' 200 GeV [221–223].
To conclude, let us emphasize that the existence of the C-odd singularity with intercept
αodd (0) ' 1 is a firm prediction of QCD. At least in the high kt region there is a well established
C-odd three-gluon contribution to the scattering amplitude. However the expected coupling ofsuch an Odderon singularity is numerically very small. Therefore it is quite challenging to observe
its manifestation experimentally. Currently it seems to be a bit premature to draw any definite
conclusion about an experimental observation of the Odderon signal.

20.7 Asymptotics
The high-energy behaviour of total hadronic cross sections has been one of the oldest problems ofstrong interactions over many decades, beginning from Heisenberg [224]. The most important boundobtained based on general analytical properties of scattering amplitudes is the FLM bound [47–49].
It states that the growth of the total hadronic cross section with energy does not exceed ln2 s, see
Eq. (20.34).
Recall that we neglected the photon contribution as well as the whole electro-weak sector, andthat the parameter in Eq. (20.34) s0 is an a priori unknown scale. However, if we were to assume
a reasonable hadronic scale, s0 ' 1 GeV2 , we would find that Eq. (20.34) implies an unrealisticallyhigh upper bound in comparison with the cross sections observed at present collider energies.
Nevertheless there is a common trend in the literature (see for instance, reviews [225, 226] andreferences therein) to fit phenomenologically the total cross section with ln2 s, keeping in mind thesaturation of the FLM bound. Such an asymptotic behaviour is assumed also by the COMPETE
collaboration [22], which achieved a comprehensive description of all soft pre-LHC data measured√
at s ≥ 4 GeV as well as total pp cross sections from the LHC available in the first half of 2015(see Section 51 in [24]).
19

We thank Jean-Rene Cudell for clarifying this issue.

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32

20. High Energy Soft QCD and Diffraction

It is interesting that the Froissart-type ln2 s asymptotics of the pp total cross section are also
supported by numerical results in lattice QCD [227]. Such a behaviour is also observed in theapproach [228] based on Colour Glass Condensate saturation.
Finally, it is worth mentioning that the possibility that asymptotically the Pomeron intercept
becomes smaller than 1, αP (0) < 1, and at very high energies the total cross section starts todecrease with energy, though highly unlikely, is not yet completely rejected. For instance, such abehaviour is expected in a theory with only the triple-Pomeron coupling, g3P , and which neglects
n , such as the 2 → 2 Pomeron coupling [229, 230].
the more complicated multi-Pomeron vertices gm
It was also argued that in the case of an increasing (with energy) cross section the only regime
consistent asymptotically with both the s- and the t-channel unitarities is that of a black disc whose
radius increases as R = c · lns [231] (i.e. R ∝ (ln s)γ , with γ = 1 exactly).

20.8 Acknowledgements
It is a pleasure to thank Michael Albrow, Robert Cahn, Robert Ciesielski, Per Grafstrom,Frank Krauss, Paul Newman, Sergey Ostapchenko and Edward Sarkisyan-Grinbaum for discussions,suggestions and comments on this review. Special thanks to Graeme Watt who read the article
through and helped to improve the presentation. MT is supported by MEYS of the Czech Republic
within project LTT17018. VAK thanks the Institute of Physics of the Czech Academy of Sciencesin Prague for hospitality.
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