<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 → 2</ei> and <ei>2 → 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. <h3>Couplings and K factor</h3> The size of QCD cross sections is mainly determined by <parm name="SigmaProcess:alphaSvalue" default="0.1265" 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 → 1</ei>, <ei>2 → 2</ei> and two different kinds of <ei>2 → 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 → 1</ei> processes. The same options also apply for those <ei>2 → 2</ei> and <ei>2 → 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="5"> The <ei>Q^2</ei> renormalization scale for <ei>2 → 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> </modepick> <modepick name="SigmaProcess:renormScale3" default="3" min="1" max="6"> The <ei>Q^2</ei> renormalization scale for "normal" <ei>2 → 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 → 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 → 3</ei> vector-boson-fusion processes, i.e. <ei>f_1 f_2 → 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 → 1</ei>, <ei>2 → 2</ei> and <ei>2 → 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 → 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 → 1</ei>, <ei>2 → 2</ei> and <ei>2 → 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 → 1</ei>, <ei>2 → 2</ei> and <ei>2 → 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 → 1</ei> processes. The same options also apply for those <ei>2 → 2</ei> and <ei>2 → 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="5"> The <ei>Q^2</ei> factorization scale for <ei>2 → 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> </modepick> <modepick name="SigmaProcess:factorScale3" default="2" min="1" max="6"> The <ei>Q^2</ei> factorization scale for "normal" <ei>2 → 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 → 3</ei> vector-boson-fusion processes, i.e. <ei>f_1 f_2 → 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 → 1</ei>, <ei>2 → 2</ei> and <ei>2 → 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 → 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 → 1</ei>, <ei>2 → 2</ei> and <ei>2 → 3</ei> processes in some of the options above. </parm> </chapter> <!-- Copyright (C) 2014 Torbjorn Sjostrand -->