<chapter name="Couplings and Scales"> 
 
<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 <ei>alpha_strong(M_Z^2)</ei> value and running, 
and separate PDF scale choices. Also, in <ei>2 &rarr; 2</ei> and 
<ei>2 &rarr; 3</ei> 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. 
 
<p/> 
We stress that couplings and scales are set separately from the 
values on this page for 
<aloc href="MultipartonInteractions">multiparton interactions</aloc>, 
<aloc href="TimelikeShowers">timelike showers</aloc>, and 
<aloc href="SpacelikeShowers">spacelike showers</aloc>. 
This allows a bigger flexibility, but also requires a bit more work 
e.g. if you insist on using the same <ei>alpha_s</ei> everywhere. 
 
<h3>Couplings and K factor</h3> 
 
The size of QCD cross sections is mainly determined by 
<parm name="SigmaProcess:alphaSvalue" default="0.13" 
min="0.06" max="0.25"> 
The <ei>alpha_strong</ei> value at scale <ei>M_Z^2</ei>. 
</parm> 
 
<p/> 
The actual value is then regulated by the running to the <ei>Q^2</ei> 
renormalization scale, at which <ei>alpha_strong</ei> is evaluated 
<modepick name="SigmaProcess: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/> 
QED interactions are regulated by the <ei>alpha_electromagnetic</ei> 
value at the <ei>Q^2</ei> renormalization scale of an interaction. 
<modepick name="SigmaProcess:alphaEMorder" default="1" min="-1" max="1"> 
The running of <ei>alpha_em</ei> used in hard processes. 
<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/> 
In addition there is the possibility of a global rescaling of 
cross sections (which could not easily be accommodated by a 
changed <ei>alpha_strong</ei>, since <ei>alpha_strong</ei> runs) 
<parm name="SigmaProcess:Kfactor" default="1.0" min="0.5" max="4.0"> 
Multiply almost all cross sections by this common fix factor. Excluded 
are only unresolved processes, where cross sections are better 
<aloc href="TotalCrossSections">set directly</aloc>, and 
multiparton interactions, which have a separate <ei>K</ei> factor 
<aloc href="MultipartonInteractions">of their own</aloc>. 
This degree of freedom is primarily intended for hadron colliders, and 
should not normally be used for <ei>e^+e^-</ei> annihilation processes. 
</parm> 
 
<h3>Renormalization scales</h3> 
 
The <ei>Q^2</ei> renormalization scale can be chosen among a few different 
alternatives, separately for <ei>2 &rarr; 1</ei>, <ei>2 &rarr; 2</ei> and two 
different kinds of <ei>2 &rarr; 3</ei> processes. In addition a common 
multiplicative factor may be imposed. 
 
<modepick name="SigmaProcess:renormScale1" default="1" min="1" max="2"> 
The <ei>Q^2</ei> renormalization scale for <ei>2 &rarr; 1</ei> processes. 
The same options also apply for those <ei>2 &rarr; 2</ei> and 
<ei>2 &rarr; 3</ei> processes that have been specially marked as 
proceeding only through an <ei>s</ei>-channel resonance, by the 
<code>isSChannel()</code> virtual method of <code>SigmaProcess</code>. 
<option value="1">the squared invariant mass, i.e. <ei>sHat</ei>. 
</option> 
<option value="2">fix scale set in <code>SigmaProcess:renormFixScale</code> 
below. 
</option> 
</modepick> 
 
<modepick name="SigmaProcess:renormScale2" default="2" min="1" max="6"> 
The <ei>Q^2</ei> renormalization scale for <ei>2 &rarr; 2</ei> processes. 
<option value="1">the smaller of the squared transverse masses of the two 
outgoing particles, i.e. <ei>min(mT_3^2, mT_4^2) = 
pT^2 + min(m_3^2, m_4^2)</ei>. 
</option> 
<option value="2">the geometric mean of the squared transverse masses of 
the two outgoing particles, i.e. <ei>mT_3 * mT_4 = 
sqrt((pT^2 + m_3^2) * (pT^2 + m_4^2))</ei>. 
</option> 
<option value="3">the arithmetic mean of the squared transverse masses of 
the two outgoing particles, i.e. <ei>(mT_3^2 + mT_4^2) / 2 = 
pT^2 + 0.5 * (m_3^2 + m_4^2)</ei>. Useful for comparisons 
with PYTHIA 6, where this is the default. 
</option> 
<option value="4">squared invariant mass of the system, 
i.e. <ei>sHat</ei>. Useful for processes dominated by 
<ei>s</ei>-channel exchange. 
</option> 
<option value="5">fix scale set in <code>SigmaProcess:renormFixScale</code> 
below. 
</option> 
<option value="6">Use squared invariant momentum transfer <ei>-tHat</ei>. 
This is a common choice for lepton-hadron scattering processes. In that 
case <ei>-tHat=Q^2</ei>. 
</option> 
</modepick> 
 
<modepick name="SigmaProcess:renormScale3" default="3" min="1" max="6"> 
The <ei>Q^2</ei> renormalization scale for "normal" <ei>2 &rarr; 3</ei> 
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 <ei>t</ei>-channel propagator particle. 
(Currently only <ei>g g / q qbar &rarr; H^0 Q Qbar</ei> processes are 
implemented, where the "match" criterion holds.) 
<option value="1">the smaller of the squared transverse masses of the three 
outgoing particles, i.e. min(mT_3^2, mT_4^2, mT_5^2). 
</option> 
<option value="2">the geometric mean of the two smallest squared transverse 
masses of the three outgoing particles, i.e. 
<ei>sqrt( mT_3^2 * mT_4^2 * mT_5^2 / max(mT_3^2, mT_4^2, mT_5^2) )</ei>. 
</option> 
<option value="3">the geometric mean of the squared transverse masses of the 
three outgoing particles, i.e. <ei>(mT_3^2 * mT_4^2 * mT_5^2)^(1/3)</ei>. 
</option> 
<option value="4">the arithmetic mean of the squared transverse masses of 
the three outgoing particles, i.e. <ei>(mT_3^2 + mT_4^2 + mT_5^2)/3</ei>. 
</option> 
<option value="5">squared invariant mass of the system, 
i.e. <ei>sHat</ei>. 
</option> 
<option value="6">fix scale set in <code>SigmaProcess:renormFixScale</code> 
below. 
</option> 
</modepick> 
 
<modepick name="SigmaProcess:renormScale3VV" default="3" min="1" max="6"> 
The <ei>Q^2</ei> renormalization scale for <ei>2 &rarr; 3</ei> 
vector-boson-fusion processes, i.e. <ei>f_1 f_2 &rarr; H^0 f_3 f_4</ei> 
with <ei>Z^0</ei> or <ei>W^+-</ei>  <ei>t</ei>-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 
<ei>mT_Vi^2 = m_V^2 + pT_i^2</ei>. 
<option value="1">the squared mass <ei>m_V^2</ei> of the exchanged 
vector boson. 
</option> 
<option value="2">the geometric mean of the two propagator virtuality 
estimates, i.e. <ei>sqrt(mT_V3^2 * mT_V4^2)</ei>. 
</option> 
<option value="3">the geometric mean of the three relevant squared 
transverse masses, i.e. <ei>(mT_V3^2 * mT_V4^2 * mT_H^2)^(1/3)</ei>. 
</option> 
<option value="4">the arithmetic mean of the three relevant squared 
transverse masses, i.e. <ei>(mT_V3^2 + mT_V4^2 + mT_H^2)/3</ei>. 
</option> 
<option value="5">squared invariant mass of the system, 
i.e. <ei>sHat</ei>. 
</option> 
<option value="6">fix scale set in <code>SigmaProcess:renormFixScale</code> 
below. 
</option> 
</modepick> 
 
<parm name="SigmaProcess:renormMultFac" default="1." min="0.1" max="10."> 
The <ei>Q^2</ei> renormalization scale for <ei>2 &rarr; 1</ei>, 
<ei>2 &rarr; 2</ei> and <ei>2 &rarr; 3</ei> 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 <ei>2 &rarr; 1</ei> processes. 
</parm> 
 
<parm name="SigmaProcess:renormFixScale" default="10000." min="1."> 
A fix <ei>Q^2</ei> value used as renormalization scale for 
<ei>2 &rarr; 1</ei>, <ei>2 &rarr; 2</ei> and <ei>2 &rarr; 3</ei> processes 
in some of the options above. 
</parm> 
 
<h3>Factorization scales</h3> 
 
Corresponding options exist for the <ei>Q^2</ei> factorization scale 
used as argument in PDF's. Again there is a choice of form for 
<ei>2 &rarr; 1</ei>, <ei>2 &rarr; 2</ei> and <ei>2 &rarr; 3</ei> 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. 
 
<modepick name="SigmaProcess:factorScale1" default="1" min="1" max="2"> 
The <ei>Q^2</ei> factorization scale for <ei>2 &rarr; 1</ei> processes. 
The same options also apply for those <ei>2 &rarr; 2</ei> and 
<ei>2 &rarr; 3</ei> processes that have been specially marked as 
proceeding only through an <ei>s</ei>-channel resonance. 
<option value="1">the squared invariant mass, i.e. <ei>sHat</ei>. 
</option> 
<option value="2">fix scale set in <code>SigmaProcess:factorFixScale</code> 
below. 
</option> 
</modepick> 
 
<modepick name="SigmaProcess:factorScale2" default="1" min="1" max="6"> 
The <ei>Q^2</ei> factorization scale for <ei>2 &rarr; 2</ei> processes. 
<option value="1">the smaller of the squared transverse masses of the two 
outgoing particles. 
</option> 
<option value="2">the geometric mean of the squared transverse masses of 
the two outgoing particles. 
</option> 
<option value="3">the arithmetic mean of the squared transverse masses of 
the two outgoing particles. Useful for comparisons with PYTHIA 6, where 
this is the default. 
</option> 
<option value="4">squared invariant mass of the system, 
i.e. <ei>sHat</ei>. Useful for processes dominated by 
<ei>s</ei>-channel exchange. 
</option> 
<option value="5">fix scale set in <code>SigmaProcess:factorFixScale</code> 
below. 
</option> 
<option value="6">Use squared invariant momentum transfer <ei>-tHat</ei>. 
This is a common choice for lepton-hadron scattering processes. In that 
case <ei>-tHat=Q^2</ei>. 
</option> 
</modepick> 
 
<modepick name="SigmaProcess:factorScale3" default="2" min="1" max="6"> 
The <ei>Q^2</ei> factorization scale for "normal" <ei>2 &rarr; 3</ei> 
processes, i.e excepting the vector-boson-fusion processes below. 
<option value="1">the smaller of the squared transverse masses of the three 
outgoing particles. 
</option> 
<option value="2">the geometric mean of the two smallest squared transverse 
masses of the three outgoing particles. 
</option> 
<option value="3">the geometric mean of the squared transverse masses of the 
three outgoing particles. 
</option> 
<option value="4">the arithmetic mean of the squared transverse masses of 
the three outgoing particles. 
</option> 
<option value="5">squared invariant mass of the system, 
i.e. <ei>sHat</ei>. 
</option> 
<option value="6">fix scale set in <code>SigmaProcess:factorFixScale</code> 
below. 
</option> 
</modepick> 
 
<modepick name="SigmaProcess:factorScale3VV" default="2" min="1" max="6"> 
The <ei>Q^2</ei> factorization scale for <ei>2 &rarr; 3</ei> 
vector-boson-fusion processes, i.e. <ei>f_1 f_2 &rarr; H^0 f_3 f_4</ei> 
with <ei>Z^0</ei> or <ei>W^+-</ei>  <ei>t</ei>-channel propagators. 
Here we again introduce the combinations <ei>mT_Vi^2 = m_V^2 + pT_i^2</ei> 
as replacements for the normal squared transverse masses of the two 
outgoing quarks. 
<option value="1">the squared mass <ei>m_V^2</ei> of the exchanged 
vector boson. 
</option> 
<option value="2">the geometric mean of the two propagator virtuality 
estimates. 
</option> 
<option value="3">the geometric mean of the three relevant squared 
transverse masses. 
</option> 
<option value="4">the arithmetic mean of the three relevant squared 
transverse masses. 
</option> 
<option value="5">squared invariant mass of the system, 
i.e. <ei>sHat</ei>. 
</option> 
<option value="6">fix scale set in <code>SigmaProcess:factorFixScale</code> 
below. 
</option> 
</modepick> 
 
<parm name="SigmaProcess:factorMultFac" default="1." min="0.1" max="10."> 
The <ei>Q^2</ei> factorization scale for <ei>2 &rarr; 1</ei>, 
<ei>2 &rarr; 2</ei> and <ei>2 &rarr; 3</ei> 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 <ei>2 &rarr; 1</ei> processes. 
</parm> 
 
<parm name="SigmaProcess:factorFixScale" default="10000." min="1."> 
A fix <ei>Q^2</ei> value used as factorization scale for <ei>2 &rarr; 1</ei>, 
<ei>2 &rarr; 2</ei> and <ei>2 &rarr; 3</ei> processes in some of the options 
above. 
</parm> 
 
</chapter> 
 
<!-- Copyright (C) 2020 Torbjorn Sjostrand --> 
