## Total Cross Sections

The `SigmaTotal` class returns the total, elastic, diffractive and nondiffractive cross sections in hadronic collisions. By implication it also has to provide differential elastic and diffractive cross sections, since many models start out from the differential expressions and then integrate to obtain more inclusive rates. In principle it would have been possible to decouple the overall normalization from the differential shape, however.

The current page describes the options available for integrated and differential cross sections alike. The number of options is especially large for diffraction, reflecting the lack of a well-understood theory. Conversely, the wide spectrum of options should allow for detailed comparisons that eventually will improve our understanding. The Diffraction page contains those further parameters needed to describe the hadronization of a diffractive system, or at least those that set diffraction apart from the nondiffractive topologies. There are borderline cases, that could have been described in either place, such as the ones related to the pomeron-proton cross section, which mainly are relevant for the description of MPIs in diffractive systems, and therefore have been put on the Diffraction page. That page also contains the "hard diffraction" framework, i.e. the modelling of diffractive events that contain a hard process.

Several different parametrization options are available for p p and pbar p collisions, of special interest for hadron colliders, while the selection for other processes is considerably more limited. As a simple generalization, a neutron is assumed to have the same hadronic cross section as a proton.

Historically most of the parametrizations used are from [Sch94, Sch97] which borrows some of the total cross sections from [Don92]. A few parameters allow some possibility to vary the basic setup. The allowed combinations of incoming particles are p + p, pbar + p, pi+ + p, pi- + p, pi0/rho0 + p, phi + p, J/psi + p, rho + rho, rho + phi, rho + J/psi, phi + phi, phi + J/psi, J/psi + J/psi, Pomeron + p, gamma + gamma and gamma + p. The strong emphasis on vector mesons is related to the description of gamma + p and gamma + gamma interactions in a Vector Dominance Model framework (which is not explicitly used in the current implementation of photoproduction, but is retained for potential future applications).

The other options available for total, elastic and diffractive cross sections are:

- A do-it-yourself selection of the main parameters.
- The MBR (Minimum Bias Rockefeller) model [Cie12], which is mainly intended for diffractive physics, but also parametrizes the total and elastic cross sections.
- The ABMST model [App16], which is based on a quite sophisticated Pomeron-inspired framework, and addresses total, elastic and single diffractive cross sections. The tuning to single diffractive data has mainly been performed at lower energies, so we also include variants that (hopefully) improves agreement with LHC data, and also introduce simple extensions to double and central diffraction.
- The RPP 2016 parametrization [Pat16], which is also Pomeron-inspired. It does not address diffractive cross sections.

The elastic cross section is differential in the squared momentum transfer t. The single diffractive additionally is differential in the mass of the diffractive system, or in xi = x_Pom, where M^2_diff = xi * s. For double diffraction the two masses can accordingly be related to xi_1 and xi_2 values. For central diffraction M^2_diff = xi_1 * xi_2 * s, and additionally the cross section is differential in t_1 and t_2.

### Master switches

The total and elastic cross sections are intimately connected via the optical theorem. Therefore the two should be calculated within a common setup. The diffractive cross sections are not as easily related, and can therefore be chosen separately, hence the two switches below. This allows different combinations to be tried out.

`mode` SigmaTotal:mode (`default = 1`; `minimum = 0`; `maximum = 4`)
Choice of parametrization of the total and elastic cross sections.
`option` 0 : Make your own choices (the "own model"), set as fixed values.
`option` 1 : The DL model for total cross sections, extended to more processes and to elastic cross sections according to SaS ("SaS/DL").
`option` 2 : The MBR model for p p and p pbar, else as option 1.
`option` 3 : The ABMST parametrizations for p p and p pbar, else as option 1.
`option` 4 : The RPP2016 parametrizations for p p and p pbar, else as option 1.

`mode` SigmaDiffractive:mode (`default = 1`; `minimum = 0`; `maximum = 3`)
Choice of parametrization of diffractive cross sections: single, double and central ditto. Note that there is no option 4.
`option` 0 : Make your own choices, set as fixed values.
`option` 1 : The SaS parametrizations, available for a larger set of incoming hadron combinations.
`option` 2 : The MBR model for p p and p pbar, else as option 1.
`option` 3 : The ABMST parametrizations for p p and p pbar, else as option 1.

Note that the total cross section subtracted by the elastic and various diffractive ones gives the inelastic nondiffractive cross section, which therefore is not set separately. However, since the nondiffractive inelastic cross section is what makes up the minimum-bias event class, and plays a major role in the description of multiparton interactions, it is important that a consistent set is used.

In the following subsections all the parameters available for the various values of the master switches are described. A final subsection covers the possibility to include Coulomb corrections in elastic scattering, and is relevant for all scenarios.

### Set your own cross sections

The following four parameters can be set for the `SigmaTotal:mode = 0` option. The default values are in the right ballpark for LHC physics, but precise numbers depend on the energy used.

`parm` SigmaTotal:sigmaTot (`default = 100.`; `minimum = 0.`)
The assumed total cross section in mb.

`parm` SigmaTotal:sigmaEl (`default = 25.`; `minimum = 0.`)
The assumed elastic cross section in mb.

`parm` SigmaElastic:bSlope (`default = 18.`; `minimum = 0.`)
The assumed slope b of the strong-interaction term exp(bt), in units of GeV^-2.

`parm` SigmaElastic:rho (`default = 0.13`; `minimum = -1.`; `maximum = 1.`)
The assumed ratio of the real to the imaginary parts of the nuclear scattering amplitude. This value is also used in the SaS/DL option.

The following four parameters can be set for the `SigmaDiffractive:mode = 0` option. Again the default values are in the right ballpark for LHC physics, but with a considerable measure of uncertainty.

`parm` SigmaTotal:sigmaXB (`default = 8.`; `minimum = 0.`)
Single Diffractive cross section A + B → X + B in mb.

`parm` SigmaTotal:sigmaAX (`default = 8.`; `minimum = 0.`)
Single Diffractive cross section A + B → A + X in mb.

`parm` SigmaTotal:sigmaXX (`default = 4.`; `minimum = 0.`)
Double Diffractive cross section A + B → X_1 + X_2 in mb.

`parm` SigmaTotal:sigmaAXB (`default = 1.`; `minimum = 0.`)
Central Diffractive cross section A + B → A + X + B in mb.

The key parameter to set the differential shape of single diffraction is the `SigmaDiffractive:PomFlux` switch below. Seven different options are included, that provide the differential shape in diffractive mass and t of the scattered proton, based on the assumed Pomeron flux parametrizations. Only the SaS option contains a (published) extension to double diffraction, but the other alternatives have been extended in a minimal manner consistent with Pomeron phenomenology. These basic shapes can be further modified by the other settings below.

`mode` SigmaDiffractive:PomFlux (`default = 1`; `minimum = 1`; `maximum = 8`)
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.
`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.
`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.
`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.
`option` 5 : the MBR simulation of (anti)proton-proton interactions [Cie12]. The mass distribution follows a renormalized-Regge-theory model, successfully tested using CDF data.
`option` 6 : The H1 Fit A parametrisation of the Pomeron flux [H1P06,H1P06a]. The flux factors are motivated by Regge theory, assuming a Regge trajectory as in options 3 and 4. The flux has been normalised to 1 at x_Pomeron = 0.003 and slope parameter and Pomeron intercept has been fitted to H1 data.
`option` 7 : The H1 Fit B parametrisation of the Pomeron flux [H1P06,H1P06a].
`option` 8 : The same functional form as with the H1 Fit A and B above, f_Pom(x_Pom) = exp(B0 t) / x_Pom^(2 \alpha(t) - 1), but with user-supplied values for parameters alpha', epsilon and B0 described below.

In options 3, 4, 6, 7 and 8 above, the Pomeron Regge trajectory is parametrized as
alpha(t) = 1 + epsilon + alpha' t
The epsilon and alpha' parameters can be set separately in options 3 and 4, and additionally alpha' is set in option 1, while values are fixed in options 6 and 7 as these are linked to specific Pomeron PDF fits. The option 8 applies the same form as 6 and 7 but provides user a freedom to change the values of above parameters along with the slope parameter B0 to modify the shape of Pomeron flux.

`parm` SigmaDiffractive:PomFluxEpsilon (`default = 0.085`; `minimum = 0.02`; `maximum = 0.15`)
The Pomeron trajectory intercept epsilon above for the 3, 4 and 8 flux options. For technical reasons epsilon > 0 is necessary in the current implementation.

`parm` SigmaDiffractive:PomFluxAlphaPrime (`default = 0.25`; `minimum = 0.05`; `maximum = 0.4`)
The Pomeron trajectory slope alpha' above for the 1, 3, 4 and 8 flux options.

`parm` SigmaDiffractive:PomFluxB0 (`default = 5.5`; `minimum = 0.5`; `maximum = 500.`)
The B0 parameter for the H1-like Pomeron flux parametrization applied with option 8 above.

The options above might give vanishing (or even negative) b slope values, and also do not enforce the presence of a rapidity gap. Furthermore the lowest allowed central diffractive mass is not well-defined; it would not be meaningful to go all the way down to the pi pi kinematical limit, since exclusive states are not modelled. Therefore the following parameters have been introduced to address such issues.

`parm` SigmaDiffractive:OwnbMinDD (`default = 1.`; `minimum = 0.5`; `maximum = 5.`)
In the options with a simple exp(b * t) falloff for the t spectrum, ensure that b is at least this large. (Recall that the b formula typically contains one term for each incoming hadron that does not break up, and for double diffraction such terms are absent. This leaves only the pomeron propagator part, which often vanishes in the limit of vanishing rapidity gap.)

`flag` SigmaDiffractive:OwndampenGap (`default = off`)
Switch on damping of small rapidity gaps in single, double and central diffraction. The reason for this option is that the separation between diffraction and nondiffraction is blurred for events with small gaps. Therefore a damping factor for small gaps is imposed with this option, of the form
1 / (1 + exp( -p * (y - y_gap))) = 1 / (1 + exp(p * y_gap) * (exp(-y))^p),
where y is the rapidity gap(s) in the current event, and p and y_gap are two parameters. Thus the damping kicks in for y < y_gap, and the transition region from small to large damping is of order 1/p in y. The exp(-y) values are xi for SD, xi_1 * xi_2 * s / m_p^2 for DD, and xi_1 and xi_2 for CD. The two parameters of the damping are described below.
Note: if the integrated diffractive cross sections are kept fixed, switching on this option will increase the rate of diffractive events with large rapidity gaps, so do consistent changes.

`parm` SigmaDiffractive:Ownygap (`default = 2.`; `minimum = 0.1`)
Assume a damping of small rapidity gaps, as described above, to set in around the value y_gap given by this parameter.

`parm` SigmaDiffractive:Ownypow (`default = 5.`; `minimum = 0.5`)
Assume a damping of small rapidity gaps, as described above, to set in over a rapidity region of width 1/p, with p given by this parameter.

`parm` SigmaDiffractive:OwnmMinCD (`default = 1.`; `minimum = 0.5`)
The smallest allowed central diffractive mass, with a sharp cut at this value.

### Modify the SaS/DL cross sections

The default description of total, elastic and diffractive interactions was parameterized and fit in [Sch94, Sch97]. There is no freedom for total and elastic cross sections, except that the rho parameter is not modelled but taken from the `SigmaElastic:rho` parameter above.

The following three parameters allow for some modification of the mass distribution of the diffractive system, relative to the default setup. The parametrized cross sections explicitly depend on them, so that integrated diffractive cross section are changed acordingly.

`parm` SigmaDiffractive:mMin (`default = 0.28`; `minimum = 0.0`)
Lowest mass of a single or double diffractive system is set to be mHadron + mMin.

`parm` SigmaDiffractive:lowMEnhance (`default = 2.0`; `minimum = 0.0`)
Normalization factor for the contribution of low-mass resonances to the diffractive cross section (cRes in eq. (22) of [Sch94]).

`parm` SigmaDiffractive:mResMax (`default = 1.062`; `minimum = 0.0`)
The contribution of low-mass resonances is dampened at around the scale mHadron + mResMax (the sum is Mres in eq. (22) of [Sch94]). To make sense, we should have mResMax > mMin.

Central diffraction (CD) was not part of the framework in [Sch94]. It has now been added for p p or pbar p, but only for multiparticle states, i.e. excluding the low-mass resonance region below roughly 1 GeV, as well as other exclusive states. It uses the same proton-Pomeron vertex as in single diffraction, twice, to describe x_Pomeron and t spectra. This fixes the energy dependence, which has been integrated and parametrized. The absolute normalization has been left open, however. Furthermore, since CD has not been included in previous tunes to data, a special flag is available to reproduce the old behaviour (with due complications when one does not want to do this).

`parm` SigmaDiffractive:mMinCD (`default = 1.`; `minimum = 0.5`)
The smallest allowed central diffractive mass, with a sharp cut at this value.

`parm` SigmaTotal:sigmaAXB2TeV (`default = 1.5`; `minimum = 0.`)
The CD cross section for p p and pbar p collisions, normalized to its value at 2 TeV CM energy, expressed in mb. The energy dependence is then parametrized, and behaves roughly like ln^1.5(s).

`flag` SigmaTotal:zeroAXB (`default = on`)
several existing tunes do not include CD. An inclusion of a nonvanishing CD cross section directly affects the nondiffractive phenomenology, even if not dramatically, and so this flag is used to forcibly set the CD cross section to vanish in such tunes. You can switch CD back on after the selection of a tune, if you so wish, by resetting `SigmaTotal:zeroAXB = off`.

LHC data have suggested that diffractive cross sections rise slower than predicted in the original studies. A likely reason is that unitarization effects may dampen the rise of diffractive cross sections relative to the default parametrizations. The settings here allows one way to introduce a dampening, which is used in some of the existing tunes.

`flag` SigmaDiffractive:dampen (`default = on`)
Allow a user to dampen diffractive cross sections; on/off = true/false.

When `SigmaDiffractive:dampen = on`, the three diffractive cross sections are damped so that they never can exceed the respective values below. Specifically, if the standard parametrization gives the cross section sigma_old(s) and a fixed sigma_max is set, the actual cross section becomes
sigma_new(s) = sigma_old(s) * sigma_max / (sigma_old(s) + sigma_max).
This reduces to sigma_old(s) at low energies and to sigma_max at high ones. Note that the asymptotic value is approached quite slowly, however.

`parm` SigmaDiffractive:maxXB (`default = 65.`; `minimum = 0.`)
The above sigma_max for A + B → X + B in mb.

`parm` SigmaDiffractive:maxAX (`default = 65.`; `minimum = 0.`)
The above sigma_max for A + B → A + X in mb.

`parm` SigmaDiffractive:maxXX (`default = 65.`; `minimum = 0.`)
The above sigma_max for A + B → X_1 + X_2 in mb.

`parm` SigmaDiffractive:maxAXB (`default = 3.`; `minimum = 0.`)
The above sigma_max for A + B → A + X + B in mb.

As above, a reduced diffractive cross section automatically translates into an increased nondiffractive one, such that the total (and elastic) cross section remains fixed.

`parm` SigmaDiffractive:SaSepsilon (`default = 0.0`; `minimum = -0.2`; `maximum = 0.2`)
The SaS ansatz starts out from a dM^2/M^2 shape of diffractive spectra, a shape that then is modified by t-spectra integration and small-mass enhancement. For exploratory purposes it is possible to modify the base ansatz to be dM^2/M^(2 * (1 + epsilon)). In principle the integrated diffractive cross sections ought to be recalculated accordingly, but for simplicity they are not modified.

### Modify the MBR cross sections

The MBR differential cross section also comes with a selection of parameters that can be changed from their default values, to modify diffractive event rates and shapes, while the total and elastic cross sections remain unaffected. These parameters are described in the following.

`parm` SigmaDiffractive:MBRepsilon (`default = 0.104`; `minimum = 0.02`; `maximum = 0.15`)
`parm` SigmaDiffractive:MBRalpha (`default = 0.25`; `minimum = 0.1`; `maximum = 0.4`)
the parameters of the Pomeron trajectory.

`parm` SigmaDiffractive:MBRbeta0 (`default = 6.566`; `minimum = 0.0`; `maximum = 10.0`)
`parm` SigmaDiffractive:MBRsigma0 (`default = 2.82`; `minimum = 0.0`; `maximum = 5.0`)
the Pomeron-proton coupling, and the total Pomeron-proton cross section.

`parm` SigmaDiffractive:MBRm2Min (`default = 1.5`; `minimum = 0.0`; `maximum = 3.0`)
the lowest value of the mass squared of the dissociated system, including central diffraction.

`parm` SigmaDiffractive:MBRdyminSDflux (`default = 2.3`; `minimum = 0.0`; `maximum = 5.0`)
`parm` SigmaDiffractive:MBRdyminDDflux (`default = 2.3`; `minimum = 0.0`; `maximum = 5.0`)
`parm` SigmaDiffractive: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` SigmaDiffractive:MBRdyminSD (`default = 2.0`; `minimum = 0.0`; `maximum = 5.0`)
`parm` SigmaDiffractive:MBRdyminDD (`default = 2.0`; `minimum = 0.0`; `maximum = 5.0`)
`parm` SigmaDiffractive: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). The cross section is damped smoothly, such that it is suppressed by a factor of a half at around this scale.

`parm` SigmaDiffractive:MBRdyminSigSD (`default = 0.5`; `minimum = 0.001`; `maximum = 5.0`)
`parm` SigmaDiffractive:MBRdyminSigDD (`default = 0.5`; `minimum = 0.001`; `maximum = 5.0`)
`parm` SigmaDiffractive: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). The smaller this value, the more narrow the rapidity region over which the suppression sets in.

### Modify the ABMST cross sections

The ABMST model provides a detailed description of the total, elastic and single diffractive cross sections. The former two components are accepted as is, while we have allowed alternative shapes for single diffraction, notably to enforce a rapidity gap. The ABMST model does not address double and central diffraction, so we have extended it on our own, as described below.

`mode` SigmaDiffractive:ABMSTmodeSD (`default = 1`; `minimum = 0`; `maximum = 3`)
Setup of single diffraction in the ABMST scenario.
`option` 0 : Keep the pure ABMST ansatz, which notably vanishes above |t| = 4 GeV^2, and has a constant term up to that scale.
`option` 1 : Use a slightly modified ansatz without an upper |t| cut, but instead an exponential fall-off that gives the same integrated diffractive rate and average |t| value. In addition the low-mass background term is modified as a combination of a linear and a quadratic term, instead of a qudratic only.
`option` 2 : Option 0, with a scaling factor of k * (s / m_p^2)^q, where k is `SigmaDiffractive:multSD` and q is `SigmaDiffractive:powSD`
`option` 3 : Option 1, with a scaling factor of k * (s / m_p^2)^q, where k is `SigmaDiffractive:multSD` and q is `SigmaDiffractive:powSD`
Note: also the `SigmaDiffractive:ABMSTdampenGap` and `SigmaDiffractive:ABMSTuseBMin` flags below very much affect the behaviour; you have to switch them off and use option 0 above to recover the pure ABMST model.

`parm` SigmaDiffractive:ABMSTmultSD (`default = 1.`; `minimum = 0.01`)
possibility to rescale the double diffractive cross section by a factor k as described above.

`parm` SigmaDiffractive:ABMSTpowSD (`default = 0.0`; `minimum = 0.0`; `maximum = 0.25`)
possibility to rescale the double diffractive cross section by a factor (s / m_p^2)^q, as described above, with q set here.

`mode` SigmaDiffractive:ABMSTmodeDD (`default = 1`; `minimum = 0`; `maximum = 1`)
Setup of double diffraction in the ABMST scenario. Note that ABMST does not provide any answer here, so the single-diffractive framework is extended by a simple factorized ansatz
dsigma_DD( xi_1, xi_2, t) / (dxi_1 dxi_2 dt) = dsigma_SD (xi_1, t) / (dxi_1 dt) * dsigma_SD (xi_2, t) / (dxi_2 dt) / (dsigma_El( t) / dt) .
The above ansatz is marred by the dip in dsigma_El / dt by destructive interference, however, so in this extension we only allow for Pomerons in the elastic cross section, which is intended to represent the bulk of the cross section. As such, the equation gives a parameter-free prediction for the double diffractive cross section. For flexibility we introduce a (default) option where the absolute normalization can be modified, while retaining the shape of the ansatz.
`option` 0 : Describe the double diffractive cross section by the simple factorized ansatz introduced above, within the allowed phase-space limits. Note that the single diffractive cross section is affected by the choice made for `SigmaDiffractive:ABMSTmodeSD`.
`option` 1 : The double diffractive cross section can be rescaled by a factor k * (s / m_p^2)^q, where k is `SigmaDiffractive:multDD` and q is `SigmaDiffractive:powDD`.
Note: also the `SigmaDiffractive:ABMSTdampenGap` and `SigmaDiffractive:ABMSTuseBMin` flags below very much affect the behaviour.

`parm` SigmaDiffractive:ABMSTmultDD (`default = 1.`; `minimum = 0.01`)
possibility to rescale the double diffractive cross section by a factor k as described above.

`parm` SigmaDiffractive:ABMSTpowDD (`default = 0.1`; `minimum = 0.0`; `maximum = 0.25`)
possibility to rescale the double diffractive cross section by a factor (s / m_p^2)^q, as described above, with q set here.

`mode` SigmaDiffractive:ABMSTmodeCD (`default = 0`; `minimum = 0`; `maximum = 1`)
Setup of central diffraction in the ABMST scenario. Note that ABMST does not provide any answer here, so the single-diffractive framework is extended by a simple factorized ansatz
dsigma_CD( xi_1, xi_2, t_1, t_2) / (dxi_1 dxi_2 dt_1 dt2_) = dsigma_SD (xi_1, t_1) / (dxi_1 dt_1) * dsigma_SD (xi_2, t_2) / (dxi_2 dt_2) / sigma_total(s) ,
and again a variant is introduced below.
`option` 0 : Describe the central diffractive cross section by the simple factorized ansatz introduced above, within the allowed phase-space limits. Also here, we only allow for Pomerons in the total cross section. Note that the single diffractive cross section is affected by the choice made for `SigmaDiffractive:ABMSTmodeSD`.
`option` 1 : In addition to option 0, the central diffractive cross section can be rescaled by a factor k * (s / m_p^2)^q, where k is `SigmaDiffractive:multCD` and q is `SigmaDiffractive:powCD`.
Note: also the `SigmaDiffractive:ABMSTdampenGap` and `SigmaDiffractive:ABMSTuseBMin` flags below very much affect the behaviour.

`parm` SigmaDiffractive:ABMSTmultCD (`default = 1.`; `minimum = 0.01`)
possibility to rescale the central diffractive cross section by a factor k as described above.

`parm` SigmaDiffractive:ABMSTpowCD (`default = 0.1`; `minimum = 0.0`; `maximum = 0.25`)
possibility to rescale the central diffractive cross section by a factor (s / m_p^2)^q, as described above, with q set here.

`parm` SigmaDiffractive:ABMSTmMinCD (`default = 1.`; `minimum = 0.5`)
The smallest allowed central diffractive mass, with a sharp cut at this value.

`flag` SigmaDiffractive:ABMSTdampenGap (`default = on`)
Switch on damping of small rapidity gaps in single, double and central diffraction. The reason for this option, on by default, is that the the ABMST SD ansats contains terms that peak near xi = 1. This leads to very large integrated SD cross sections at higher energies, such that the diffractive cross section is larger than the nondiffractive one. It then becomes a challenge e.g. how to implement and interpret PDFs, which by definition are inclusive, but would have to be split consistently between the different contributions. (For the hard-jet subsample it can be done e.g. as in [Ras16], but it would be more complicated for softer jets in the MPI context.) Furthermore the separation between diffraction and nondiffraction is blurred for events with small gaps. Therefore a damping factor for small gaps is imposed with this option, of the form
1 / (1 + exp( -p * (y - y_gap))) = 1 / (1 + exp(p * y_gap) * (exp(-y))^p),
where y is the rapidity gap(s) in the current event, and p and y_gap are two parameters. Thus the damping kicks in for y < y_gap, and the transition region from small to large damping is of order 1/p in y. The exp(-y) values are xi for SD, xi_1 * xi_2 * s / m_p^2 for DD, and xi_1 and xi_2 for CD. The two parameters of the damping are described below.

`parm` SigmaDiffractive:ABMSTygap (`default = 2.`; `minimum = 0.1`)
Assume a damping of small rapidity gaps in the ABMST model, as described above, to set in around the value y_gap given by this parameter.

`parm` SigmaDiffractive:ABMSTypow (`default = 5.`; `minimum = 0.5`)
Assume a damping of small rapidity gaps in the ABMST model, as described above, to set in over a rapidity region of width 1/p, with p given by this parameter.

`flag` SigmaDiffractive:ABMSTuseBMin (`default = on`)
The slope b of an approximate exp(b * t) fall-off is xi-dependent in the ABMST model for single diffraction. In particular it can become close to zero for large xi, which means that the t-integrated cross section becomes very large. While the general trend is reasonable, the behaviour in the xi → 1 limit is questionable. Therefore it makes sense to impose some minimal b slope. For double diffraction such issues become even more pressing, since the division by the elastic cross section could even lead to a negative b slope, which would not be physical. The central diffractive cross section is more well-behaved, but for consistency it is meaningful to ensure a minimal fall-off also here. Therefore, when this flag is on, a minimal fall-off exp(b_min * t) is assumed for each of the three components, with the respective b_min value stored in the three parameters below. The fall-off is defined relative to the value at t = 0, a point that is outside the physical region, but the parametrization of the diffractive cross sections can still be used there meaningfully. Only positive b_min values are acted on, so the SD/DD/CD components can be switched off individually even when this flag is on.

`parm` SigmaDiffractive:ABMSTbMinSD (`default = 2.`)
Assume a minimal fall-off exp(b_min * t) in the ABMST model for single diffraction, as described above.

`parm` SigmaDiffractive:ABMSTbMinDD (`default = 2.`)
Assume a minimal fall-off exp(b_min * t) in the extension of the ABMST model to double diffraction, as described above.

`parm` SigmaDiffractive:ABMSTbMinCD (`default = 2.`)
Assume a minimal fall-off exp(b_min * (t_1 + t_2)) in the extension of the ABMST model to central diffraction, as described above.

### Modify the RPP cross sections

The RPP approach only addresses total and (differential) elastic cross sections, and there are no free parameters that can be changed.

### Coulomb corrections to elastic scattering

By default there is no Coulomb-term contribution to the elastic (or total) cross section, which of course becomes infinite if this contribution is included in the collision between charged particles, owing to the 1/t^2 singularity of t-channel photon exchange. You can switch on Coulomb corrections below, however, including interference with the conventional strong-interaction term. The own, SaS/DL and MBR models share a common machinery to evaluate the interference [Ber87], while ABMST and RPP use a slighly different expression for the (poorly known) difference in phases between the hadronic and the electromagnetic amplitudes.

`flag` SigmaElastic:Coulomb (`default = off`)
Include Coulomb corrections to the elastic and total cross sections.

`parm` SigmaElastic:tAbsMin (`default = 5e-5`; `minimum = 1e-10`; `maximum = 1e-3`)
since the Coulomb contribution is infinite a lower limit on |t| must be set to regularize the divergence, in units of GeV^2. This means that the elastic and total cross sections are reduced by the amount of the ordinary cross section in the cut-out region, but increased by the Coulomb contribution itself and the interference term (of either sign). This variable has no effect if Coulomb corrections are not switched on or not relevant (e.g. for neutral particles), i.e. then t = 0 sets the limit.

`parm` SigmaElastic:lambda (`default = 0.71`; `minimum = 0.1`; `maximum = 2.`)
the main parameter of the electric form factor G(t) = lambda^2 / (lambda + |t|)^2, in units of GeV^2, as used in the own, SaS/DL and MBR models.

`parm` SigmaElastic:phaseConst (`default = 0.577`)
The Coulomb term is taken to contain a phase factor exp(+- i alpha phi(t)), with + for p p and - for pbar p, where phi(t) = - phaseConst - ln(-B t/2). This constant is model dependent [Cah82]. This expression is used in the own, SaS/DL and MBR models, where the hadronic cross section is modelled as a simple exp(B t).

### Low-energy and combined cross sections

The above descriptions are mainly aimed for high-energy applications. The SaS/DL and especially the ABMST ones still are intended to be used down to reasonably low energies, say around or a bit below 10 GeV CM energy, also for diffraction.

In hadronic rescattering typical energy scales are much lower, and extend all the way down to the kinematical threshold. For the studies in [Sjo20] it was therefore necessary to implement separate low-energy cross sections. This was done using data and/or models from various sources. The Additive Quark Model was applied to extend the expressions to unconstrained cross sections. These cross sections are encoded in the `SigmaLowEnergy` class, which exists separately from the other options on this page.

There are other applications where it is necessary to have access to cross sections at all energy scales. One example is a high-energy particle cascading in a medium, giving rise to more and more particles of lower and lower energy. A special `SigmaCombined` class has therefore been created, that contains one `SigmaLowEnergy` object and one `SigmaSaSDL` object. The latter class has been extended to cover a wide range of incoming particles; see [Sjo21] for details. For now, the medium is assumed to consist of a mix of protons and neutrons, thereby somewhat limiting the list of required hadron combinations. Output from the `Pythia::getSigmaTotal` and `Pythia::getSigmaPartial` methods provides user-access to these cross sections, see Program Flow. This output can be based purely on the assumed low-energy or high-energy behaviour, but the default is a mix of the two. This is done by a linear transition specified by the following two parameters.

`parm` SigmaCombined:eMinHigh (`default = 6.`; `minimum = 5.`; `maximum = 20.`)
Energy below which the low-energy cross sections are used exclusively. The number actually applies for collisions of hadrons with up to the proton mass; to allow for heavier hadrons with masses m_A and m_B the threshold is at eMinHigh + max(0., m_A - m_p) + max(0., m_B - m_p).

`parm` SigmaCombined:deltaEHigh (`default = 8.`; `minimum = 0.`; `maximum = 20.`)
If the energy is above eMinHigh + deltaEHigh the high-energy cross sections are used exclusively, while in between the two cross sections are mixed, with a fraction `(e - eMinHigh) / deltaEHigh` taken by the high-energy expressions. This applies for pp collisions; otherwise there is an offset for eMinHigh as already explained.

It is worth noting that the transition is far from perfect, and typically worse for some partial cross sections than for the total ones. In some cases the disagreement can be less than it seems, with pp/pbarp as the prime example. The low-energy description includes an explicit nucleon excitation term, which is absent in the high-energy formulae. There, instead, the enhanced low-mass spectrum in diffraction fills a similar function, but with a different classification.