Spacelike Showers
The PYTHIA algorithm for spacelike initial-state showers is
based on the recent article [Sjo05], where a
transverse-momentum-ordered backwards evolution scheme is introduced.
This algorithm is a further development of the virtuality-ordered one
presented in [Sj085], with matching to first-order matrix
element for Z^0, W^+- and Higgs (in the
m_t -> infinity limit) production as introduced in
[Miu99].
The normal user is not expected to call SpaceShower
directly, but only have it called from Pythia
,
via PartonLevel
. Some of the parameters below,
in particular SpaceShower:alphaSvalue
,
would be of interest for a tuning exercise, however.
Main variables
The maximum pT to be allowed in the shower evolution is
related to the nature of the hard process itself. It involves a
delicate balance between not doublecounting and not leaving any
gaps in the coverage. The best procedure may depend on information
only the user has: how the events were generated and mixed (e.g. with
Les Houches Accord external input), and how they are intended to be
used. Therefore a few options are available, with a sensible default
behaviour.
mode
SpaceShower:pTmaxMatch
(default = 0
; minimum = 0
; maximum = 2
)
Way in which the maximum shower evolution scale is set to match the
scale of the hard process itself.
option
0 : (i) if the final state of the hard process
(not counting subsequent resonance decays) contains at least one quark
(u, d, s, c ,b), gluon or photon then pT_max
is chosen to be the factorization scale for internal processes
and the scale
value for Les Houches input;
(ii) if not, emissions are allowed to go all the way up to
the kinematical limit.
The reasoning is that in the former set of processes the ISR
emission of yet another quark, gluon or photon could lead to
doublecounting, while no such danger exists in the latter case.
option
1 : always use the factorization scale for an internal
process and the scale
value for Les Houches input,
i.e. the lower value. This should avoid doublecounting, but
may leave out some emissions that ought to have been simulated.
option
2 : always allow emissions up to the kinematical limit.
This will simulate all possible event topologies, but may lead to
doublecounting.
Note 1: These options only apply to the hard interaction.
Emissions off subsequent multiple interactions are always constrainted
to be below the factorization scale of the process itself.
Note 2: Some processes contain matrix-element matching
to the first emission; this is the case notably for single
gamma^*/Z^0, W^+- and H^0 production. Then default
and option 2 give the correct result, while option 1 should never
be used.
parm
SpaceShower:pTmaxFudge
(default = 1.0
; minimum = 0.5
; maximum = 2.0
)
In cases where the above pTmaxMatch
rules would imply
that pT_max = pT_factorization, pTmaxFudge
introduced a multiplicative factor f such that instead
pT_max = f * pT_factorization. Only applies to the hardest
interaction in an event. It is strongly suggested that f = 1,
but variations around this default can be useful to test this assumption.
mode
SpaceShower:pTdampMatch
(default = 0
; minimum = 0
; maximum = 2
)
These options only take effect when a process is allowed to radiate up
to the kinematical limit by the above pTmaxMatch
choice,
and no matrix-element corrections are available. Then, in many processes,
the fall-off in pT will be too slow by one factor of pT^2.
That is, while showers have an approximate dpT^2/pT^2 shape, often
it should become more like dpT^2/pT^4 at pT values above
the scale of the hard process. Whether this actually is the case
depends on the particular process studied, e.g. if t-channel
gluon exchange is likely to dominate. If so, the options below could
provide a reasonable high-pT behaviour without requiring
higher-order calculations.
option
0 : emissions go up to the kinematical limit,
with no special dampening.
option
1 : emissions go up to the kinematical limit,
but dampened by a factor k^2 Q^2_fac/(pT^2 + k^2 Q^2_fac),
where Q_fac is the factorization scale and k is a
multiplicative fudge factor stored in pTdampFudge
below.
option
2 : emissions go up to the kinematical limit,
but dampened by a factor k^2 Q^2_ren/(pT^2 + k^2 Q^2_ren),
where Q_ren is the renormalization scale and k is a
multiplicative fudge factor stored in pTdampFudge
below.
Note: These options only apply to the hard interaction.
Emissions off subsequent multiple interactions are always constrainted
to be below the factorization scale of the process itself.
parm
SpaceShower:pTdampFudge
(default = 1.0
; minimum = 0.25
; maximum = 4.0
)
In cases 1 and 2 above, where a dampening is imposed at around the
factorization or renormalization scale, respectively, this allows the
pT scale of dampening of radiation by a half to be shifted
by this factor relative to the default Q_fac or Q_ren.
This number ought to be in the neighbourhood of unity, but variations
away from this value could do better in some processes.
The amount of QCD radiation in the shower is determined by
parm
SpaceShower:alphaSvalue
(default = 0.127
; minimum = 0.06
; maximum = 0.25
)
The alpha_strong value at scale M_Z^2
.
Default value is picked equal to the one used in CTEQ 5L.
The actual value is then regulated by the running to the scale
pT^2, at which it is evaluated
mode
SpaceShower:alphaSorder
(default = 1
; minimum = 0
; maximum = 2
)
Order at which alpha_strong runs,
option
0 : zeroth order, i.e. alpha_strong is kept
fixed.
option
1 : first order, which is the normal value.
option
2 : second order. Since other parts of the code do
not go to second order there is no strong reason to use this option,
but there is also nothing wrong with it.
QED radiation is regulated by the alpha_electromagnetic
value at the pT^2 scale of a branching.
mode
SpaceShower:alphaEMorder
(default = 1
; minimum = -1
; maximum = 1
)
The running of alpha_em.
option
1 : first-order running, constrained to agree with
StandardModel:alphaEMmZ
at the Z^0 mass.
option
0 : zeroth order, i.e. alpha_em is kept
fixed at its value at vanishing momentum transfer.
option
-1 : zeroth order, i.e. alpha_em is kept
fixed, but at StandardModel:alphaEMmZ
, i.e. its value
at the Z^0 mass.
There are two complementary ways of regularizing the small-pT
divergence, a sharp cutoff and a smooth dampening. These can be
combined as desired but it makes sense to coordinate with how the
same issue is handled in multiple interactions.
flag
SpaceShower:samePTasMI
(default = on
)
Regularize the pT -> 0 divergence using the same sharp cutoff
and smooth dampening parameters as used to describe multiple interactions.
That is, the MultipleInteractions:pT0Ref
,
MultipleInteractions:ecmRef
,
MultipleInteractions:ecmPow
and
MultipleInteractions:pTmin
parameters are used to regularize
all ISR QCD radiation, rather than the corresponding parameters below.
This is a sensible physics ansatz, based on the assumption that colour
screening effects influence both MI and ISR in the same way. Photon
radiation is regularized separately in either case.
Warning: if a large pT0
is picked for multiple
interactions, such that the integrated interaction cross section is
below the nondiffractive inelastic one, this pT0
will
automatically be scaled down to cope. Information on such a rescaling
does NOT propagate to SpaceShower
, however.
The actual pT0
parameter used at a given cm energy scale,
ecmNow, is obtained as
pT0 = pT0(ecmNow) = pT0Ref * (ecmNow / ecmRef)^ecmPow
where pT0Ref, ecmRef and ecmPow are the
three parameters below.
parm
SpaceShower:pT0Ref
(default = 2.2
; minimum = 0.5
; maximum = 10.0
)
Regularization of the divergence of the QCD emission probability for
pT -> 0 is obtained by a factor pT^2 / (pT0^2 + pT^2),
and by using an alpha_s(pT0^2 + pT^2). An energy dependence
of the pT0 choice is introduced by the next two parameters,
so that pT0Ref is the pT0 value for the reference
cm energy, pT0Ref = pT0(ecmRef).
parm
SpaceShower:ecmRef
(default = 1800.0
; minimum = 1.
)
The ecmRef reference energy scale introduced above.
parm
SpaceShower:ecmPow
(default = 0.16
; minimum = 0.
; maximum = 0.5
)
The ecmPow energy rescaling pace introduced above.
parm
SpaceShower:pTmin
(default = 0.2
; minimum = 0.1
; maximum = 10.0
)
Lower cutoff in pT, below which no further ISR branchings
are allowed. Normally the pT0 above would be used to
provide the main regularization of the branching rate for
pT -> 0, in which case pTmin is used mainly for
technical reasons. It is possible, however, to set pT0Ref = 0
and use pTmin to provide a step-function regularization,
or to combine them in intermediate approaches. Currently pTmin
is taken to be energy-independent.
parm
SpaceShower:pTminChgQ
(default = 0.5
; minimum = 0.01
)
Parton shower cut-off pT for photon coupling to a coloured
particle.
parm
SpaceShower:pTminChgL
(default = 0.0005
; minimum = 0.0001
)
Parton shower cut-off mass for pure QED branchings.
Assumed smaller than (or equal to) pTminChgQ.
flag
SpaceShower:rapidityOrder
(default = off
)
Force emissions, after the first, to be ordered in rapidity,
i.e. in terms of decreasing angles in a backwards-evolution sense.
Could be used to probe sensitivity to unordered emissions.
Only affects QCD emissions.
Further variables
These should normally not be touched. Their only function is for
cross-checks.
There are three flags you can use to switch on or off selected
branchings in the shower:
flag
SpaceShower:QCDshower
(default = on
)
Allow a QCD shower; on/off = true/false.
flag
SpaceShower:QEDshowerByQ
(default = on
)
Allow quarks to radiate photons; on/off = true/false.
flag
SpaceShower:QEDshowerByL
(default = on
)
Allow leptons to radiate photons; on/off = true/false.
There are three further possibilities to simplify the shower:
flag
SpaceShower:MEcorrections
(default = on
)
Use of matrix element corrections; on/off = true/false.
flag
SpaceShower:phiPolAsym
(default = on
)
Azimuthal asymmetry induced by gluon polarization; on/off = true/false.
Not yet implemented.
mode
SpaceShower:nQuarkIn
(default = 5
; minimum = 0
; maximum = 5
)
Number of allowed quark flavours in g -> q qbar branchings,
when kinematically allowed, and thereby also in incoming beams.
Changing it to 4 would forbid g -> b bbar, etc.
Technical notes
Almost everything is equivalent to the algorithm in [1]. Minor changes
are as follows.
-
It is now possible to have a second-order running alpha_s,
in addition to fixed or first-order running.
-
The description of heavy flavour production in the threshold region
has been modified, so as to be more forgiving about mismatches
between the c/b masses used in Pythia relative to those
used in a respective PDF parametrization. The basic idea is that,
in the threshold region of a heavy quark Q, Q = c/b,
the effect of subsequent Q -> Q g branchings is negligible.
If so, then
f_Q(x, pT2) = integral_mQ2^pT2 dpT'2/pT'2 * alpha_s(pT'2)/2pi
* integral P(z) g(x', pT'2) delta(x - z x')
so use this to select the pT2 of the g -> Q Qbar
branching. In the old formalism the same kind of behaviour should
be obtained, but by a cancellation of a 1/f_Q that diverges
at the theshold and a Sudakov that vanishes.
The strategy therefore is that, once pT2 < f * mQ2, with
f a parameter of the order of 2, a pT2 is chosen
like dpT2/pT2 between mQ2 and f * mQ2, a
nd a z flat in the allowed range. Thereafter acceptance
is based on the product of three factors, representing the running
of alpha_strong, the splitting kernel (including the mass term)
and the gluon density weight. At failure, a new pT2 is chosen
in the same range, i.e. is not required to be lower since no Sudakov
is involved.