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<title>Couplings and Scales</title>
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<h2>Couplings and Scales</h2>
Here is collected some possibilities to modify the scale choices
of couplings and parton densities for all internally implemented
hard processes. This is based on them all being derived from the
<code>SigmaProcess</code> base class. The matrix-element coding is
also used by the multiparton-interactions machinery, but there with a
separate choice of <i>alpha_strong(M_Z^2)</i> value and running,
and separate PDF scale choices. Also, in <i>2 → 2</i> and
<i>2 → 3</i> processes where resonances are produced, their
couplings and thereby their Breit-Wigner shapes are always evaluated
with the resonance mass as scale, irrespective of the choices below.
<h3>Couplings and K factor</h3>
The size of QCD cross sections is mainly determined by
<p/><code>parm </code><strong> SigmaProcess:alphaSvalue </strong>
(<code>default = <strong>0.1265</strong></code>; <code>minimum = 0.06</code>; <code>maximum = 0.25</code>)<br/>
The <i>alpha_strong</i> value at scale <i>M_Z^2</i>.
<p/>
The actual value is then regulated by the running to the <i>Q^2</i>
renormalization scale, at which <i>alpha_strong</i> is evaluated
<p/><code>mode </code><strong> SigmaProcess:alphaSorder </strong>
(<code>default = <strong>1</strong></code>; <code>minimum = 0</code>; <code>maximum = 2</code>)<br/>
Order at which <i>alpha_strong</i> runs,
<br/><code>option </code><strong> 0</strong> : zeroth order, i.e. <i>alpha_strong</i> is kept
fixed.
<br/><code>option </code><strong> 1</strong> : first order, which is the normal value.
<br/><code>option </code><strong> 2</strong> : 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.
<p/>
QED interactions are regulated by the <i>alpha_electromagnetic</i>
value at the <i>Q^2</i> renormalization scale of an interaction.
<p/><code>mode </code><strong> SigmaProcess:alphaEMorder </strong>
(<code>default = <strong>1</strong></code>; <code>minimum = -1</code>; <code>maximum = 1</code>)<br/>
The running of <i>alpha_em</i> used in hard processes.
<br/><code>option </code><strong> 1</strong> : first-order running, constrained to agree with
<code>StandardModel:alphaEMmZ</code> at the <i>Z^0</i> mass.
<br/><code>option </code><strong> 0</strong> : zeroth order, i.e. <i>alpha_em</i> is kept
fixed at its value at vanishing momentum transfer.
<br/><code>option </code><strong> -1</strong> : zeroth order, i.e. <i>alpha_em</i> is kept
fixed, but at <code>StandardModel:alphaEMmZ</code>, i.e. its value
at the <i>Z^0</i> mass.
<p/>
In addition there is the possibility of a global rescaling of
cross sections (which could not easily be accommodated by a
changed <i>alpha_strong</i>, since <i>alpha_strong</i> runs)
<p/><code>parm </code><strong> SigmaProcess:Kfactor </strong>
(<code>default = <strong>1.0</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 4.0</code>)<br/>
Multiply almost all cross sections by this common fix factor. Excluded
are only unresolved processes, where cross sections are better
<a href="TotalCrossSections.html" target="page">set directly</a>, and
multiparton interactions, which have a separate <i>K</i> factor
<a href="MultipartonInteractions.html" target="page">of their own</a>.
This degree of freedom is primarily intended for hadron colliders, and
should not normally be used for <i>e^+e^-</i> annihilation processes.
<h3>Renormalization scales</h3>
The <i>Q^2</i> renormalization scale can be chosen among a few different
alternatives, separately for <i>2 → 1</i>, <i>2 → 2</i> and two
different kinds of <i>2 → 3</i> processes. In addition a common
multiplicative factor may be imposed.
<p/><code>mode </code><strong> SigmaProcess:renormScale1 </strong>
(<code>default = <strong>1</strong></code>; <code>minimum = 1</code>; <code>maximum = 2</code>)<br/>
The <i>Q^2</i> renormalization scale for <i>2 → 1</i> processes.
The same options also apply for those <i>2 → 2</i> and
<i>2 → 3</i> processes that have been specially marked as
proceeding only through an <i>s</i>-channel resonance, by the
<code>isSChannel()</code> virtual method of <code>SigmaProcess</code>.
<br/><code>option </code><strong> 1</strong> : the squared invariant mass, i.e. <i>sHat</i>.
<br/><code>option </code><strong> 2</strong> : fix scale set in <code>SigmaProcess:renormFixScale</code>
below.
<p/><code>mode </code><strong> SigmaProcess:renormScale2 </strong>
(<code>default = <strong>2</strong></code>; <code>minimum = 1</code>; <code>maximum = 5</code>)<br/>
The <i>Q^2</i> renormalization scale for <i>2 → 2</i> processes.
<br/><code>option </code><strong> 1</strong> : the smaller of the squared transverse masses of the two
outgoing particles, i.e. <i>min(mT_3^2, mT_4^2) =
pT^2 + min(m_3^2, m_4^2)</i>.
<br/><code>option </code><strong> 2</strong> : the geometric mean of the squared transverse masses of
the two outgoing particles, i.e. <i>mT_3 * mT_4 =
sqrt((pT^2 + m_3^2) * (pT^2 + m_4^2))</i>.
<br/><code>option </code><strong> 3</strong> : the arithmetic mean of the squared transverse masses of
the two outgoing particles, i.e. <i>(mT_3^2 + mT_4^2) / 2 =
pT^2 + 0.5 * (m_3^2 + m_4^2)</i>. Useful for comparisons
with PYTHIA 6, where this is the default.
<br/><code>option </code><strong> 4</strong> : squared invariant mass of the system,
i.e. <i>sHat</i>. Useful for processes dominated by
<i>s</i>-channel exchange.
<br/><code>option </code><strong> 5</strong> : fix scale set in <code>SigmaProcess:renormFixScale</code>
below.
<p/><code>mode </code><strong> SigmaProcess:renormScale3 </strong>
(<code>default = <strong>3</strong></code>; <code>minimum = 1</code>; <code>maximum = 6</code>)<br/>
The <i>Q^2</i> renormalization scale for "normal" <i>2 → 3</i>
processes, i.e excepting the vector-boson-fusion processes below.
Here it is assumed that particle masses in the final state either match
or are heavier than that of any <i>t</i>-channel propagator particle.
(Currently only <i>g g / q qbar → H^0 Q Qbar</i> processes are
implemented, where the "match" criterion holds.)
<br/><code>option </code><strong> 1</strong> : the smaller of the squared transverse masses of the three
outgoing particles, i.e. min(mT_3^2, mT_4^2, mT_5^2).
<br/><code>option </code><strong> 2</strong> : the geometric mean of the two smallest squared transverse
masses of the three outgoing particles, i.e.
<i>sqrt( mT_3^2 * mT_4^2 * mT_5^2 / max(mT_3^2, mT_4^2, mT_5^2) )</i>.
<br/><code>option </code><strong> 3</strong> : the geometric mean of the squared transverse masses of the
three outgoing particles, i.e. <i>(mT_3^2 * mT_4^2 * mT_5^2)^(1/3)</i>.
<br/><code>option </code><strong> 4</strong> : the arithmetic mean of the squared transverse masses of
the three outgoing particles, i.e. <i>(mT_3^2 + mT_4^2 + mT_5^2)/3</i>.
<br/><code>option </code><strong> 5</strong> : squared invariant mass of the system,
i.e. <i>sHat</i>.
<br/><code>option </code><strong> 6</strong> : fix scale set in <code>SigmaProcess:renormFixScale</code>
below.
<p/><code>mode </code><strong> SigmaProcess:renormScale3VV </strong>
(<code>default = <strong>3</strong></code>; <code>minimum = 1</code>; <code>maximum = 6</code>)<br/>
The <i>Q^2</i> renormalization scale for <i>2 → 3</i>
vector-boson-fusion processes, i.e. <i>f_1 f_2 → H^0 f_3 f_4</i>
with <i>Z^0</i> or <i>W^+-</i> <i>t</i>-channel propagators.
Here the transverse masses of the outgoing fermions do not reflect the
virtualities of the exchanged bosons. A better estimate is obtained
by replacing the final-state fermion masses by the vector-boson ones
in the definition of transverse masses. We denote these combinations
<i>mT_Vi^2 = m_V^2 + pT_i^2</i>.
<br/><code>option </code><strong> 1</strong> : the squared mass <i>m_V^2</i> of the exchanged
vector boson.
<br/><code>option </code><strong> 2</strong> : the geometric mean of the two propagator virtuality
estimates, i.e. <i>sqrt(mT_V3^2 * mT_V4^2)</i>.
<br/><code>option </code><strong> 3</strong> : the geometric mean of the three relevant squared
transverse masses, i.e. <i>(mT_V3^2 * mT_V4^2 * mT_H^2)^(1/3)</i>.
<br/><code>option </code><strong> 4</strong> : the arithmetic mean of the three relevant squared
transverse masses, i.e. <i>(mT_V3^2 + mT_V4^2 + mT_H^2)/3</i>.
<br/><code>option </code><strong> 5</strong> : squared invariant mass of the system,
i.e. <i>sHat</i>.
<br/><code>option </code><strong> 6</strong> : fix scale set in <code>SigmaProcess:renormFixScale</code>
below.
<p/><code>parm </code><strong> SigmaProcess:renormMultFac </strong>
(<code>default = <strong>1.</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 10.</code>)<br/>
The <i>Q^2</i> renormalization scale for <i>2 → 1</i>,
<i>2 → 2</i> and <i>2 → 3</i> processes is multiplied by
this factor relative to the scale described above (except for the options
with a fix scale). Should be use sparingly for <i>2 → 1</i> processes.
<p/><code>parm </code><strong> SigmaProcess:renormFixScale </strong>
(<code>default = <strong>10000.</strong></code>; <code>minimum = 1.</code>)<br/>
A fix <i>Q^2</i> value used as renormalization scale for
<i>2 → 1</i>, <i>2 → 2</i> and <i>2 → 3</i> processes
in some of the options above.
<h3>Factorization scales</h3>
Corresponding options exist for the <i>Q^2</i> factorization scale
used as argument in PDF's. Again there is a choice of form for
<i>2 → 1</i>, <i>2 → 2</i> and <i>2 → 3</i> processes
separately. For simplicity we have let the numbering of options agree,
for each event class separately, between normalization and factorization
scales, and the description has therefore been slightly shortened. The
default values are <b>not</b> necessarily the same, however.
<p/><code>mode </code><strong> SigmaProcess:factorScale1 </strong>
(<code>default = <strong>1</strong></code>; <code>minimum = 1</code>; <code>maximum = 2</code>)<br/>
The <i>Q^2</i> factorization scale for <i>2 → 1</i> processes.
The same options also apply for those <i>2 → 2</i> and
<i>2 → 3</i> processes that have been specially marked as
proceeding only through an <i>s</i>-channel resonance.
<br/><code>option </code><strong> 1</strong> : the squared invariant mass, i.e. <i>sHat</i>.
<br/><code>option </code><strong> 2</strong> : fix scale set in <code>SigmaProcess:factorFixScale</code>
below.
<p/><code>mode </code><strong> SigmaProcess:factorScale2 </strong>
(<code>default = <strong>1</strong></code>; <code>minimum = 1</code>; <code>maximum = 5</code>)<br/>
The <i>Q^2</i> factorization scale for <i>2 → 2</i> processes.
<br/><code>option </code><strong> 1</strong> : the smaller of the squared transverse masses of the two
outgoing particles.
<br/><code>option </code><strong> 2</strong> : the geometric mean of the squared transverse masses of
the two outgoing particles.
<br/><code>option </code><strong> 3</strong> : the arithmetic mean of the squared transverse masses of
the two outgoing particles. Useful for comparisons with PYTHIA 6, where
this is the default.
<br/><code>option </code><strong> 4</strong> : squared invariant mass of the system,
i.e. <i>sHat</i>. Useful for processes dominated by
<i>s</i>-channel exchange.
<br/><code>option </code><strong> 5</strong> : fix scale set in <code>SigmaProcess:factorFixScale</code>
below.
<p/><code>mode </code><strong> SigmaProcess:factorScale3 </strong>
(<code>default = <strong>2</strong></code>; <code>minimum = 1</code>; <code>maximum = 6</code>)<br/>
The <i>Q^2</i> factorization scale for "normal" <i>2 → 3</i>
processes, i.e excepting the vector-boson-fusion processes below.
<br/><code>option </code><strong> 1</strong> : the smaller of the squared transverse masses of the three
outgoing particles.
<br/><code>option </code><strong> 2</strong> : the geometric mean of the two smallest squared transverse
masses of the three outgoing particles.
<br/><code>option </code><strong> 3</strong> : the geometric mean of the squared transverse masses of the
three outgoing particles.
<br/><code>option </code><strong> 4</strong> : the arithmetic mean of the squared transverse masses of
the three outgoing particles.
<br/><code>option </code><strong> 5</strong> : squared invariant mass of the system,
i.e. <i>sHat</i>.
<br/><code>option </code><strong> 6</strong> : fix scale set in <code>SigmaProcess:factorFixScale</code>
below.
<p/><code>mode </code><strong> SigmaProcess:factorScale3VV </strong>
(<code>default = <strong>2</strong></code>; <code>minimum = 1</code>; <code>maximum = 6</code>)<br/>
The <i>Q^2</i> factorization scale for <i>2 → 3</i>
vector-boson-fusion processes, i.e. <i>f_1 f_2 → H^0 f_3 f_4</i>
with <i>Z^0</i> or <i>W^+-</i> <i>t</i>-channel propagators.
Here we again introduce the combinations <i>mT_Vi^2 = m_V^2 + pT_i^2</i>
as replacements for the normal squared transverse masses of the two
outgoing quarks.
<br/><code>option </code><strong> 1</strong> : the squared mass <i>m_V^2</i> of the exchanged
vector boson.
<br/><code>option </code><strong> 2</strong> : the geometric mean of the two propagator virtuality
estimates.
<br/><code>option </code><strong> 3</strong> : the geometric mean of the three relevant squared
transverse masses.
<br/><code>option </code><strong> 4</strong> : the arithmetic mean of the three relevant squared
transverse masses.
<br/><code>option </code><strong> 5</strong> : squared invariant mass of the system,
i.e. <i>sHat</i>.
<br/><code>option </code><strong> 6</strong> : fix scale set in <code>SigmaProcess:factorFixScale</code>
below.
<p/><code>parm </code><strong> SigmaProcess:factorMultFac </strong>
(<code>default = <strong>1.</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 10.</code>)<br/>
The <i>Q^2</i> factorization scale for <i>2 → 1</i>,
<i>2 → 2</i> and <i>2 → 3</i> processes is multiplied by
this factor relative to the scale described above (except for the options
with a fix scale). Should be use sparingly for <i>2 → 1</i> processes.
<p/><code>parm </code><strong> SigmaProcess:factorFixScale </strong>
(<code>default = <strong>10000.</strong></code>; <code>minimum = 1.</code>)<br/>
A fix <i>Q^2</i> value used as factorization scale for <i>2 → 1</i>,
<i>2 → 2</i> and <i>2 → 3</i> processes in some of the options
above.
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