<chapter name="Spacelike Showers">
<h2>Spacelike Showers</h2>
The PYTHIA algorithm for spacelike initial-state showers is
based on the article <ref>Sjo05</ref>, where a
transverse-momentum-ordered backwards evolution scheme is introduced,
with the extension to fully interleaved evolution covered in
<ref>Cor10a</ref>.
This algorithm is a further development of the virtuality-ordered one
presented in <ref>Sj085</ref>, with matching to first-order matrix
element for <ei>Z^0</ei>, <ei>W^+-</ei> and Higgs (in the
<ei>m_t → infinity</ei> limit) production as introduced in
<ref>Miu99</ref>.
<p/>
The normal user is not expected to call <code>SpaceShower</code>
directly, but only have it called from <code>Pythia</code>,
via <code>PartonLevel</code>. Some of the parameters below,
in particular <code>SpaceShower:alphaSvalue</code>,
would be of interest for a tuning exercise, however.
<h3>Main variables</h3>
The maximum <ei>pT</ei> to be allowed in the shower evolution is
related to the nature of the hard process itself. It involves a
delicate balance between not double-counting 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.
<modepick name="SpaceShower:pTmaxMatch" default="0" min="0" max="2">
Way in which the maximum shower evolution scale is set to match the
scale of the hard process itself.
<option value="0"><b>(i)</b> if the final state of the hard process
(not counting subsequent resonance decays) contains at least one quark
(<ei>u, d, s, c ,b</ei>), gluon or photon then <ei>pT_max</ei>
is chosen to be the factorization scale for internal processes
and the <code>scale</code> value for Les Houches input;
<b>(ii)</b> 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
double-counting, while no such danger exists in the latter case.
</option>
<option value="1">always use the factorization scale for an internal
process and the <code>scale</code> value for Les Houches input,
i.e. the lower value. This should avoid double-counting, but
may leave out some emissions that ought to have been simulated.
(Also known as wimpy showers.)
</option>
<option value="2">always allow emissions up to the kinematical limit.
This will simulate all possible event topologies, but may lead to
double-counting.
(Also known as power showers.)
</option>
<note>Note 1:</note> Some processes contain matrix-element matching
to the first emission; this is the case notably for single
<ei>gamma^*/Z^0, W^+-</ei> and <ei>H^0</ei> production. Then default
and option 2 give the correct result, while option 1 should never
be used.
<note>Note 2:</note> as enumerated in the text, these options take effect
both for internal and external processes. Whether a particular option
makes sense depends on the context. For instance, if events for the same
basic process to different orders are to be matched, then option 1 would
be a reasonable first guess. Note, however, that a program like the
POWHEG BOX uses a <ei>pT</ei> definition for ISR and FSR that does not
quite agree with the PYTHIA evolution scale, and thus there will be some
amount of mismatch. In more sophisticated descriptions, therefore,
option 2 could be combined with <code>UserHooks</code> vetoes on emissions
that would lead to double-counting, using more flexible phase space
boundaries. Further details are found in the
<aloc href="MatchingAndMerging">Matching and Merging</aloc> description,
with an example in <code>examples/main31</code>.
Option 0, finally, may be most realistic when only Born-level
processes are involved, possibly in combination with a nonzero
<code>SpaceShower:pTdampMatch</code>. The rules used for avoiding
double-counting are not foolproof, however. As an example, for the
<ei>t</ei>-channel process <ei>gamma gamma → e^+ e^-</ei> its <ei>pT</ei>
scale is the plausible upper shower limit, with only dampened emissions
above it. But the initial state is not checked and, had only incoming
quarks and gluons been taken into account, only the <ei>s</ei>-channel
process <ei>q qbar → gamma^*/Z^0 → e^+ e^-</ei> would have
been possible, where indeed the whole phase space should be populated.
So this is erroneously used, giving too much emissions.
<note>Note 3:</note> These options only apply to the hard interaction.
If a "second hard" process is present, the two are analyzed and
set separately for the default 0 option, while both are affected
the same way for non-default options 1 and 2.
Emissions off subsequent multiparton interactions are always constrained
to be below the factorization scale of each process itself.
</modepick>
<parm name="SpaceShower:pTmaxFudge" default="1.0" min="0.25" max="2.0">
In cases where the above <code>pTmaxMatch</code> rules would imply
that <ei>pT_max = pT_factorization</ei>, <code>pTmaxFudge</code>
introduces a multiplicative factor <ei>f</ei> such that instead
<ei>pT_max = f * pT_factorization</ei>. Only applies to the hardest
interaction in an event, and a "second hard" if there is such a one,
cf. below. It is strongly suggested that <ei>f = 1</ei>, but variations
around this default can be useful to test this assumption.
</parm>
<parm name="SpaceShower:pTmaxFudgeMPI" default="1.0" min="0.25" max="2.0">
A multiplicative factor <ei>f</ei> such that
<ei>pT_max = f * pT_factorization</ei>, as above, but here for the
non-hardest interactions (when multiparton interactions are allowed).
</parm>
<modepick name="SpaceShower:pTdampMatch" default="0" min="0" max="2">
These options only take effect when a process is allowed to radiate up
to the kinematical limit by the above <code>pTmaxMatch</code> choice,
and no matrix-element corrections are available. Then, in many processes,
the fall-off in <ei>pT</ei> will be too slow by one factor of <ei>pT^2</ei>.
That is, while showers have an approximate <ei>dpT^2/pT^2</ei> shape, often
it should become more like <ei>dpT^2/pT^4</ei> at <ei>pT</ei> values above
the scale of the hard process. Whether this actually is the case
depends on the particular process studied, e.g. if <ei>t</ei>-channel
gluon exchange is likely to dominate. If so, the options below could
provide a reasonable high-<ei>pT</ei> behaviour without requiring
higher-order calculations.
<option value="0">emissions go up to the kinematical limit,
with no special dampening.
</option>
<option value="1">emissions go up to the kinematical limit,
but dampened by a factor <ei>k^2 Q^2_fac/(pT^2 + k^2 Q^2_fac)</ei>,
where <ei>Q_fac</ei> is the factorization scale and <ei>k</ei> is a
multiplicative fudge factor stored in <code>pTdampFudge</code> below.
</option>
<option value="2">emissions go up to the kinematical limit,
but dampened by a factor <ei>k^2 Q^2_ren/(pT^2 + k^2 Q^2_ren)</ei>,
where <ei>Q_ren</ei> is the renormalization scale and <ei>k</ei> is a
multiplicative fudge factor stored in <code>pTdampFudge</code> below.
</option>
<note>Note:</note> These options only apply to the hard interaction.
Specifically, a "second hard" interaction would not be affected.
Emissions off subsequent multiparton interactions are always constrained
to be below the factorization scale of the process itself.
</modepick>
<parm name="SpaceShower:pTdampFudge" default="1.0" min="0.25" max="4.0">
In cases 1 and 2 above, where a dampening is imposed at around the
factorization or renormalization scale, respectively, this allows the
<ei>pT</ei> scale of dampening of radiation by a half to be shifted
by this factor relative to the default <ei>Q_fac</ei> or <ei>Q_ren</ei>.
This number ought to be in the neighbourhood of unity, but variations
away from this value could do better in some processes.
</parm>
<p/>
The amount of QCD radiation in the shower is determined by
<parm name="SpaceShower:alphaSvalue" default="0.137" min="0.06" max="0.25">
The <ei>alpha_strong</ei> value at scale <code>M_Z^2</code>.
Default value is picked equal to the one used in CTEQ 5L.
</parm>
<p/>
The actual value is then regulated by the running to the scale
<ei>pT^2</ei>, at which it is evaluated
<modepick name="SpaceShower:alphaSorder" default="1" min="0" max="2">
Order at which <ei>alpha_strong</ei> runs,
<option value="0">zeroth order, i.e. <ei>alpha_strong</ei> is kept
fixed.</option>
<option value="1">first order, which is the normal value.</option>
<option value="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.</option>
</modepick>
<p/>
The CMW rescaling of <ei>Lambda_QCD</ei> (see the section on
<aloc href="StandardModelParameters">StandardModelParameters</aloc>)
can be applied to the <ei>alpha_strong</ei> values used for spacelike showers.
Note that tunes using this option need lower values of
<ei>alpha_strong(m_Z^2)</ei> than tunes that do not.
<flag name="SpaceShower:alphaSuseCMW" default="false">
<option value="false">Do not apply the CMW rescaling. </option>
<option value="true">Apply the CMW rescaling, increasing
<ei>Lambda_QCD</ei> for spacelike showers by a factor roughly 1.6.
</option>
</flag>
<p/>
QED radiation is regulated by the <ei>alpha_electromagnetic</ei>
value at the <ei>pT^2</ei> scale of a branching.
<modepick name="SpaceShower:alphaEMorder" default="1" min="-1" max="1">
The running of <ei>alpha_em</ei>.
<option value="1">first-order running, constrained to agree with
<code>StandardModel:alphaEMmZ</code> at the <ei>Z^0</ei> mass.
</option>
<option value="0">zeroth order, i.e. <ei>alpha_em</ei> is kept
fixed at its value at vanishing momentum transfer.</option>
<option value="-1">zeroth order, i.e. <ei>alpha_em</ei> is kept
fixed, but at <code>StandardModel:alphaEMmZ</code>, i.e. its value
at the <ei>Z^0</ei> mass.
</option>
</modepick>
<p/>
The natural scale for couplings and PDFs is <ei>pT^2</ei>. To explore
uncertainties it is possibly to vary around this value, however, in
analogy with what can be done for
<aloc href="CouplingsAndScales">hard processes</aloc>.
<parm name="SpaceShower:renormMultFac" default="1." min="0.1" max="10.">
The default <ei>pT^2</ei> renormalization scale is multiplied by
this prefactor. For QCD this is equivalent to a change of
<ei>Lambda^2</ei> in the opposite direction, i.e. to a change of
<ei>alpha_strong(M_Z^2)</ei> (except that flavour thresholds
remain at fixed scales). Below, when <ei>pT^2 + pT_0^2</ei> is used
as scale, it is this whole expression that is multiplied by the prefactor.
</parm>
<parm name="SpaceShower:factorMultFac" default="1." min="0.1" max="10.">
The default <ei>pT^2</ei> factorization scale is multiplied by
this prefactor.
</parm>
<p/>
There are two complementary ways of regularizing the small-<ei>pT</ei>
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 multiparton interactions.
<flag name="SpaceShower:samePTasMPI" default="off">
Regularize the <ei>pT → 0</ei> divergence using the same sharp cutoff
and smooth dampening parameters as used to describe multiparton interactions.
That is, the <code>MultipartonInteractions:pT0Ref</code>,
<code>MultipartonInteractions:ecmRef</code>,
<code>MultipartonInteractions:ecmPow</code> and
<code>MultipartonInteractions:pTmin</code> 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 MPI and ISR in the same way. Photon
radiation is regularized separately in either case.
<note>Warning:</note> if a large <code>pT0</code> is picked for multiparton
interactions, such that the integrated interaction cross section is
below the nondiffractive inelastic one, this <code>pT0</code> will
automatically be scaled down to cope. Information on such a rescaling
does NOT propagate to <code>SpaceShower</code>, however.
</flag>
<p/>
The actual <code>pT0</code> parameter used at a given CM energy scale,
<ei>ecmNow</ei>, is obtained as
<eq>
pT0 = pT0(ecmNow) = pT0Ref * (ecmNow / ecmRef)^ecmPow
</eq>
where <ei>pT0Ref</ei>, <ei>ecmRef</ei> and <ei>ecmPow</ei> are the
three parameters below.
<parm name="SpaceShower:pT0Ref" default="2.0"
min="0.5" max="10.0">
Regularization of the divergence of the QCD emission probability for
<ei>pT → 0</ei> is obtained by a factor <ei>pT^2 / (pT0^2 + pT^2)</ei>,
and by using an <ei>alpha_s(pT0^2 + pT^2)</ei>. An energy dependence
of the <ei>pT0</ei> choice is introduced by the next two parameters,
so that <ei>pT0Ref</ei> is the <ei>pT0</ei> value for the reference
cm energy, <ei>pT0Ref = pT0(ecmRef)</ei>.
</parm>
<parm name="SpaceShower:ecmRef" default="1800.0" min="1.">
The <ei>ecmRef</ei> reference energy scale introduced above.
</parm>
<parm name="SpaceShower:ecmPow" default="0.0" min="0." max="0.5">
The <ei>ecmPow</ei> energy rescaling pace introduced above.
</parm>
<parm name="SpaceShower:pTmin" default="0.2"
min="0.1" max="10.0">
Lower cutoff in <ei>pT</ei>, below which no further ISR branchings
are allowed. Normally the <ei>pT0</ei> above would be used to
provide the main regularization of the branching rate for
<ei>pT → 0</ei>, in which case <ei>pTmin</ei> is used mainly for
technical reasons. It is possible, however, to set <ei>pT0Ref = 0</ei>
and use <ei>pTmin</ei> to provide a step-function regularization,
or to combine them in intermediate approaches. Currently <ei>pTmin</ei>
is taken to be energy-independent.
</parm>
<parm name="SpaceShower:pTminChgQ" default="0.5" min="0.01">
Parton shower cut-off <ei>pT</ei> for photon coupling to a coloured
particle.
</parm>
<parm name="SpaceShower:pTminChgL" default="0.0005" min="0.0001">
Parton shower cut-off mass for pure QED branchings.
Assumed smaller than (or equal to) <ei>pTminChgQ</ei>.
</parm>
<flag name="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.
</flag>
<h3>Weak showers</h3>
The emission of weak gauge bosons is an integrated part of the initial-
and final-state radiation, see <aloc href="WeakShowers">Weak Showers</aloc>.
The following settings are those specifically related to the initial-state
weak radiation, while common settings are found in the
<aloc href="WeakShowers">Weak Showers</aloc> description.
<flag name="SpaceShower:weakShower" default="off">
Allow a weak shower, yes or no.
</flag>
<modeopen name="SpaceShower:weakShowerMode" default="0" min="0" max="2">
Determine which branchings are allowed.
<option value="0"> both <ei>W^+-</ei> and <ei>Z^0</ei> branchings.
</option>
<option value="1"> only <ei>W^+-</ei> branchings. </option>
<option value="2"> only <ei>Z^0</ei> branchings. </option>
</modeopen>
<parm name ="SpaceShower:pTminWeak" default="1.0" min="0.1" max="2.0">
Parton shower cut-off <ei>pT</ei> for weak branchings.
<h3>Further variables</h3>
These should normally not be touched. Their only function is for
cross-checks.
<p/>
There are three flags you can use to switch on or off selected
branchings in the shower:
<flag name="SpaceShower:QCDshower" default="on">
Allow a QCD shower; on/off = true/false.
</flag>
<flag name="SpaceShower:QEDshowerByQ" default="on">
Allow quarks to radiate photons; on/off = true/false.
</flag>
<flag name="SpaceShower:QEDshowerByL" default="on">
Allow leptons to radiate photons; on/off = true/false.
</flag>
<p/>
There are some further possibilities to modify the shower:
<flag name="SpaceShower:MEcorrections" default="on">
Use of matrix element corrections; on/off = true/false.
</flag>
<flag name="SpaceShower:MEafterFirst" default="on">
Use of matrix element corrections also after the first emission,
for dipole ends of the same system that did not yet radiate.
Only has a meaning if <code>MEcorrections</code> above is
switched on.
</flag>
<flag name="SpaceShower:phiPolAsym" default="on">
Azimuthal asymmetry induced by gluon polarization; on/off = true/false.
</flag>
<flag name="SpaceShower:phiIntAsym" default="on">
Azimuthal asymmetry induced by interference; on/off = true/false.
</flag>
<parm name="SpaceShower:strengthIntAsym" default="0.7"
min="0." max="0.9">
Size of asymmetry induced by interference. Natural value of order 0.5;
expression would blow up for a value of 1.
</parm>
<modeopen name="SpaceShower:nQuarkIn" default="5" min="0" max="5">
Number of allowed quark flavours in <ei>g → q qbar</ei> branchings,
when kinematically allowed, and thereby also in incoming beams.
Changing it to 4 would forbid <ei>g → b bbar</ei>, etc.
</modeopen>
<flag name="SpaceShower:useFixedFacScale" default="off">
Allow the possibility to use a fixed factorization scale, set by
the <code>parm</code> below. This option is unphysical and only
intended for toy-model and debug studies.
</flag>
<parm name="SpaceShower:fixedFacScale" default="100." min="1.">
The fixed factorization scale, in GeV, that would be used in the
evaluation of parton densities if the <code>flag</code> above is on.
</parm>
<h3>Technical notes</h3>
Almost everything is equivalent to the algorithm in [1]. Minor changes
are as follows.
<ul>
<li>
It is now possible to have a second-order running <ei>alpha_s</ei>,
in addition to fixed or first-order running.
</li>
<li>
The description of heavy flavour production in the threshold region
has been modified, so as to be more forgiving about mismatches
between the <ei>c/b</ei> 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 <ei>Q</ei>, <ei>Q = c/b</ei>,
the effect of subsequent <ei>Q → Q g</ei> branchings is negligible.
If so, then
<eq>
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')
</eq>
so use this to select the <ei>pT2</ei> of the <ei>g → Q Qbar</ei>
branching. In the old formalism the same kind of behaviour should
be obtained, but by a cancellation of a <ei>1/f_Q</ei> that diverges
at the threshold and a Sudakov that vanishes.
<br/>
The strategy therefore is that, once <ei>pT2 < f * mQ2</ei>, with
<ei>f</ei> a parameter of the order of 2, a <ei>pT2</ei> is chosen
like <ei>dpT2/pT2</ei> between <ei>mQ2</ei> and <ei>f * mQ2</ei>, a
nd a <ei>z</ei> flat in the allowed range. Thereafter acceptance
is based on the product of three factors, representing the running
of <ei>alpha_strong</ei>, the splitting kernel (including the mass term)
and the gluon density weight. At failure, a new <ei>pT2</ei> is chosen
in the same range, i.e. is not required to be lower since no Sudakov
is involved.
</li>
<li>
The QED algorithm now allows for hadron beams with non-zero photon
content. The backwards-evolution of a photon in a hadron is identical
to that of a gluon, with <ei>CF → eq^2</ei> and <ei>CA → 0</ei>.
Note that this will only work in conjunction with parton distributions
that explicitly include photons as part of the hadron structure, such
as the NNPDF2.3 QCD+QED sets. The possibility of a fermion
backwards-evolving to a photon has not yet been included, nor has
photon backwards-evolution in lepton beams.
</li>
</ul>
</chapter>
<!-- Copyright (C) 2014 Torbjorn Sjostrand -->
Marcin Chrzaszcz