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-<head>
-<title>Couplings and Scales</title>
-<link rel="stylesheet" type="text/css" href="pythia.css"/>
-<link rel="shortcut icon" href="pythia32.gif"/>
-</head>
-<body>
-
-<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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