Total Cross Sections
- Master switches
- Set your own cross sections
- Modify the SaS/DL cross sections
- Modify the MBR cross sections
- Modify the ABMST cross sections
- Modify the RPP cross sections
- Coulomb corrections to elastic scattering
- Low-energy and combined 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.