Diffraction

Introduction

Diffraction is not well understood, and several alternative approaches have been proposed. Here we follow a fairly conventional Pomeron-based one, in the Ingelman-Schlein spirit [Ing85], but integrated to make full use of the standard PYTHIA machinery for multiparton interactions, parton showers and hadronization [Nav10,Cor10a]. This is the approach pioneered in the PomPyt program by Ingelman and collaborators [Ing97].

For ease of use (and of modelling), the Pomeron-specific parts of the generation are subdivided into three sets of parameters that are rather independent of each other:
(i) the total, elastic and diffractive cross sections are parametrized as functions of the CM energy, or can be set by the user to the desired values, see the Total Cross Sections page;
(ii) once it has been decided to have a diffractive process, a Pomeron flux parametrization is used to pick the mass of the diffractive system(s) and the t of the exchanged Pomeron, see below;
(iii) a diffractive system of a given mass is classified either as low-mass unresolved, which gives a simple low-pT string topology, or as high-mass resolved, for which the full machinery of multiparton interactions and parton showers are applied, making use of Pomeron PDFs.
The parameters related to multiparton interactions, parton showers and hadronization are kept the same as for normal nondiffractive events, with only one exception. This may be questioned, especially for the multiparton interactions, but we do not believe that there are currently enough good diffractive data that would allow detailed separate tunes.

The above subdivision may not represent the way "physics comes about". For instance, the total diffractive cross section can be viewed as a convolution of a Pomeron flux with a Pomeron-proton total cross section. Since neither of the two is known from first principles there will be a significant amount of ambiguity in the flux factor. The picture is further complicated by the fact that the possibility of simultaneous further multiparton interactions ("cut Pomerons") will screen the rate of diffractive systems. In the end, our set of parameters refers to the effective description that emerges out of these effects, rather than to the underlying "bare" parameters.

In the event record the diffractive system in the case of an excited proton is denoted p_diffr, code 9902210, whereas a central diffractive system is denoted rho_diffr, code 9900110. Apart from representing the correct charge and baryon numbers, no deeper meaning should be attributed to the names.

Pomeron flux

As already mentioned above, the total diffractive cross section is fixed by a default energy-dependent parametrization or by the user, see the Total Cross Sections page. Therefore we do not attribute any significance to the absolute normalization of the Pomeron flux. The choice of Pomeron flux model still will decide on the mass spectrum of diffractive states and the t spectrum of the Pomeron exchange.

mode  Diffraction:PomFlux   (default = 1; minimum = 1; maximum = 5)
Parametrization of the Pomeron flux f_Pom/p( x_Pom, t).
option 1 : Schuler and Sjöstrand [Sch94]: based on a critical Pomeron, giving a mass spectrum roughly like dm^2/m^2; a mass-dependent exponential t slope that reduces the rate of low-mass states; partly compensated by a very-low-mass (resonance region) enhancement. Is currently the only one that contains a separate t spectrum for double diffraction (along with MBR) and separate parameters for pion beams.
option 2 : Bruni and Ingelman [Bru93]: also a critical Pomeron giving close to dm^2/m^2, with a t distribution the sum of two exponentials. The original model only covers single diffraction, but is here expanded by analogy to double and central diffraction.
option 3 : a conventional Pomeron description, in the RapGap manual [Jun95] attributed to Berger et al. and Streng [Ber87a], but there (and here) with values updated to a supercritical Pomeron with epsilon > 0 (see below), which gives a stronger peaking towards low-mass diffractive states, and with a mass-dependent (the alpha' below) exponential t slope. The original model only covers single diffraction, but is here expanded by analogy to double and central diffraction.
option 4 : a conventional Pomeron description, attributed to Donnachie and Landshoff [Don84], again with supercritical Pomeron, with the same two parameters as option 3 above, but this time with a power-law t distribution. The original model only covers single diffraction, but is here expanded by analogy to double and central diffraction.
option 5 : the MBR (Minimum Bias Rockefeller) simulation of (anti)proton-proton interactions [Cie12]. The event generation follows a renormalized-Regge-theory model, successfully tested using CDF data. The simulation includes single and double diffraction, as well as the central diffractive (double-Pomeron exchange) process (106). Only p p, pbar p and p pbar beam combinations are allowed for this option. Several parameters of this model are listed below.

In options 3 and 4 above, the Pomeron Regge trajectory is parametrized as
alpha(t) = 1 + epsilon + alpha' t
The epsilon and alpha' parameters can be set separately:

parm  Diffraction:PomFluxEpsilon   (default = 0.085; minimum = 0.02; maximum = 0.15)
The Pomeron trajectory intercept epsilon above. For technical reasons epsilon > 0 is necessary in the current implementation.

parm  Diffraction:PomFluxAlphaPrime   (default = 0.25; minimum = 0.1; maximum = 0.4)
The Pomeron trajectory slope alpha' above.

When option 5 is selected, the following parameters of the MBR model [Cie12] are used:

parm  Diffraction:MBRepsilon   (default = 0.104; minimum = 0.02; maximum = 0.15)

parm  Diffraction:MBRalpha   (default = 0.25; minimum = 0.1; maximum = 0.4)
the parameters of the Pomeron trajectory.

parm  Diffraction:MBRbeta0   (default = 6.566; minimum = 0.0; maximum = 10.0)

parm  Diffraction:MBRsigma0   (default = 2.82; minimum = 0.0; maximum = 5.0)
the Pomeron-proton coupling, and the total Pomeron-proton cross section.

parm  Diffraction:MBRm2Min   (default = 1.5; minimum = 0.0; maximum = 3.0)
the lowest value of the mass squared of the dissociated system.

parm  Diffraction:MBRdyminSDflux   (default = 2.3; minimum = 0.0; maximum = 5.0)

parm  Diffraction:MBRdyminDDflux   (default = 2.3; minimum = 0.0; maximum = 5.0)

parm  Diffraction:MBRdyminCDflux   (default = 2.3; minimum = 0.0; maximum = 5.0)
the minimum width of the rapidity gap used in the calculation of Ngap(s) (flux renormalization).

parm  Diffraction:MBRdyminSD   (default = 2.0; minimum = 0.0; maximum = 5.0)

parm  Diffraction:MBRdyminDD   (default = 2.0; minimum = 0.0; maximum = 5.0)

parm  Diffraction:MBRdyminCD   (default = 2.0; minimum = 0.0; maximum = 5.0)
the minimum width of the rapidity gap used in the calculation of cross sections, i.e. the parameter dy_S, which suppresses the cross section at low dy (non-diffractive region).

parm  Diffraction:MBRdyminSigSD   (default = 0.5; minimum = 0.001; maximum = 5.0)

parm  Diffraction:MBRdyminSigDD   (default = 0.5; minimum = 0.001; maximum = 5.0)

parm  Diffraction:MBRdyminSigCD   (default = 0.5; minimum = 0.001; maximum = 5.0)
the parameter sigma_S, used for the cross section suppression at low dy (non-diffractive region).

Separation into low and high masses

Preferably one would want to have a perturbative picture of the dynamics of Pomeron-proton collisions, like multiparton interactions provide for proton-proton ones. However, while PYTHIA by default will only allow collisions with a CM energy above 10 GeV, the mass spectrum of diffractive systems will stretch to down to the order of 1.2 GeV. It would not be feasible to attempt a perturbative description there. Therefore we do offer a simpler low-mass description, with only longitudinally stretched strings, with a gradual switch-over to the perturbative picture for higher masses. The probability for the latter picture is parametrized as
P_pert = P_max ( 1 - exp( (m_diffr - m_min) / m_width ) )
which vanishes for the diffractive system mass m_diffr < m_min, and is 1 - 1/e = 0.632 for m_diffr = m_min + m_width, assuming P_max = 1.

parm  Diffraction:mMinPert   (default = 10.; minimum = 5.)
The abovementioned threshold mass m_min for phasing in a perturbative treatment. If you put this parameter to be bigger than the CM energy then there will be no perturbative description at all, but only the older low-pt description.

parm  Diffraction:mWidthPert   (default = 10.; minimum = 0.)
The abovementioned threshold width m_width.

parm  Diffraction:probMaxPert   (default = 1.; minimum = 0.; maximum = 1.)
The abovementioned maximum probability P_max.. Would normally be assumed to be unity, but a somewhat lower value could be used to represent a small nonperturbative component also at high diffractive masses.

Low-mass diffraction

When an incoming hadron beam is diffractively excited, it is modeled as if either a valence quark or a gluon is kicked out from the hadron. In the former case this produces a simple string to the leftover remnant, in the latter it gives a hairpin arrangement where a string is stretched from one quark in the remnant, via the gluon, back to the rest of the remnant. The latter ought to dominate at higher mass of the diffractive system. Therefore an approximate behaviour like
P_q / P_g = N / m^p
is assumed.

parm  Diffraction:pickQuarkNorm   (default = 5.0; minimum = 0.)
The abovementioned normalization N for the relative quark rate in diffractive systems.

parm  Diffraction:pickQuarkPower   (default = 1.0)
The abovementioned mass-dependence power p for the relative quark rate in diffractive systems.

When a gluon is kicked out from the hadron, the longitudinal momentum sharing between the the two remnant partons is determined by the same parameters as above. It is plausible that the primordial kT may be lower than in perturbative processes, however:

parm  Diffraction:primKTwidth   (default = 0.5; minimum = 0.)
The width of Gaussian distributions in p_x and p_y separately that is assigned as a primordial kT to the two beam remnants when a gluon is kicked out of a diffractive system.

parm  Diffraction:largeMassSuppress   (default = 2.; minimum = 0.)
The choice of longitudinal and transverse structure of a diffractive beam remnant for a kicked-out gluon implies a remnant mass m_rem distribution (i.e. quark plus diquark invariant mass for a baryon beam) that knows no bounds. A suppression like (1 - m_rem^2 / m_diff^2)^p is therefore introduced, where p is the diffLargeMassSuppress parameter.

High-mass diffraction

The perturbative description need to use parton densities of the Pomeron. The options are described in the page on PDF Selection. The standard perturbative multiparton interactions framework then provides cross sections for parton-parton interactions. In order to turn these cross section into probabilities one also needs an ansatz for the Pomeron-proton total cross section. In the literature one often finds low numbers for this, of the order of 2 mb. These, if taken at face value, would give way too much activity per event. There are ways to tame this, e.g. by a larger pT0 than in the normal pp framework. Actually, there are many reasons to use a completely different set of parameters for MPI in diffraction than in pp collisions, especially with respect to the impact-parameter picture, see below. A lower number in some frameworks could alternatively be regarded as a consequence of screening, with a larger "bare" number.

For now, however, an attempt at the most general solution would carry too far, and instead we patch up the problem by using a larger Pomeron-proton total cross section, such that average activity makes more sense. This should be viewed as the main tunable parameter in the description of high-mass diffraction. It is to be fitted to diffractive event-shape data such as the average charged multiplicity. It would be very closely tied to the choice of Pomeron PDF; we remind that some of these add up to less than unit momentum sum in the Pomeron, a choice that also affect the value one ends up with. Furthermore, like with hadronic cross sections, it is quite plausible that the Pomeron-proton cross section increases with energy, so we have allowed for a power-like dependence on the diffractive mass.

parm  Diffraction:sigmaRefPomP   (default = 10.; minimum = 2.; maximum = 40.)
The assumed Pomeron-proton effective cross section, as used for multiparton interactions in diffractive systems. If this cross section is made to depend on the mass of the diffractive system then the above value refers to the cross section at the reference scale, and
sigma_PomP(m) = sigma_PomP(m_ref) * (m / m_ref)^p
where m is the mass of the diffractive system, m_ref is the reference mass scale Diffraction:mRefPomP below and p is the mass-dependence power Diffraction:mPowPomP. Note that a larger cross section value gives less MPI activity per event. There is no point in making the cross section too big, however, since then pT0 will be adjusted downwards to ensure that the integrated perturbative cross section stays above this assumed total cross section. (The requirement of at least one perturbative interaction per event.)

parm  Diffraction:mRefPomP   (default = 100.0; minimum = 1.)
The mRef reference mass scale introduced above.

parm  Diffraction:mPowPomP   (default = 0.0; minimum = 0.0; maximum = 0.5)
The p mass rescaling pace introduced above.

Also note that, even for a fixed CM energy of events, the diffractive subsystem will range from the abovementioned threshold mass m_min to the full CM energy, with a variation of parameters such as pT0 along this mass range. Therefore multiparton interactions are initialized for a few different diffractive masses, currently five, and all relevant parameters are interpolated between them to obtain the behaviour at a specific diffractive mass. Furthermore, A B → X B and A B → A X are initialized separately, to allow for different beams or PDF's on the two sides. These two aspects mean that initialization of MPI is appreciably slower when perturbative high-mass diffraction is allowed.

Diffraction tends to be peripheral, i.e. occur at intermediate impact parameter for the two protons. That aspect is implicit in the selection of diffractive cross section. For the simulation of the Pomeron-proton subcollision it is the impact-parameter distribution of that particular subsystem that should rather be modeled. That is, it also involves the transverse coordinate space of a Pomeron wavefunction. The outcome of the convolution therefore could be a different shape than for nondiffractive events. For simplicity we allow the same kind of options as for nondiffractive events, except that the bProfile = 4 option for now is not implemented.

mode  Diffraction:bProfile   (default = 1; minimum = 0; maximum = 3)
Choice of impact parameter profile for the incoming hadron beams.
option 0 : no impact parameter dependence at all.
option 1 : a simple Gaussian matter distribution; no free parameters.
option 2 : a double Gaussian matter distribution, with the two free parameters coreRadius and coreFraction.
option 3 : an overlap function, i.e. the convolution of the matter distributions of the two incoming hadrons, of the form exp(- b^expPow), where expPow is a free parameter.

parm  Diffraction:coreRadius   (default = 0.4; minimum = 0.1; maximum = 1.)
When assuming a double Gaussian matter profile, bProfile = 2, the inner core is assumed to have a radius that is a factor coreRadius smaller than the rest.

parm  Diffraction:coreFraction   (default = 0.5; minimum = 0.; maximum = 1.)
When assuming a double Gaussian matter profile, bProfile = 2, the inner core is assumed to have a fraction coreFraction of the matter content of the hadron.

parm  Diffraction:expPow   (default = 1.; minimum = 0.4; maximum = 10.)
When bProfile = 3 it gives the power of the assumed overlap shape exp(- b^expPow). Default corresponds to a simple exponential drop, which is not too dissimilar from the overlap obtained with the standard double Gaussian parameters. For expPow = 2 we reduce to the simple Gaussian, bProfile = 1, and for expPow → infinity to no impact parameter dependence at all, bProfile = 0. For small expPow the program becomes slow and unstable, so the min limit must be respected.