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1<html>
2<head>
3<title>Couplings and Scales</title>
4<link rel="stylesheet" type="text/css" href="pythia.css"/>
5<link rel="shortcut icon" href="pythia32.gif"/>
6</head>
7<body>
8
9<h2>Couplings and Scales</h2>
10
11Here is collected some possibilities to modify the scale choices
12of couplings and parton densities for all internally implemented
13hard processes. This is based on them all being derived from the
14<code>SigmaProcess</code> base class. The matrix-element coding is
15also used by the multiparton-interactions machinery, but there with a
16separate choice of <i>alpha_strong(M_Z^2)</i> value and running,
17and separate PDF scale choices. Also, in <i>2 -> 2</i> and
18<i>2 -> 3</i> processes where resonances are produced, their
19couplings and thereby their Breit-Wigner shapes are always evaluated
20with the resonance mass as scale, irrespective of the choices below.
21
22<h3>Couplings and K factor</h3>
23
24The size of QCD cross sections is mainly determined by
25<p/><code>parm&nbsp; </code><strong> SigmaProcess:alphaSvalue &nbsp;</strong> 
26 (<code>default = <strong>0.1265</strong></code>; <code>minimum = 0.06</code>; <code>maximum = 0.25</code>)<br/>
27The <i>alpha_strong</i> value at scale <i>M_Z^2</i>.
28 
29
30<p/>
31The actual value is then regulated by the running to the <i>Q^2</i> 
32renormalization scale, at which <i>alpha_strong</i> is evaluated
33<p/><code>mode&nbsp; </code><strong> SigmaProcess:alphaSorder &nbsp;</strong> 
34 (<code>default = <strong>1</strong></code>; <code>minimum = 0</code>; <code>maximum = 2</code>)<br/>
35Order at which <i>alpha_strong</i> runs,
36<br/><code>option </code><strong> 0</strong> : zeroth order, i.e. <i>alpha_strong</i> is kept
37fixed. 
38<br/><code>option </code><strong> 1</strong> : first order, which is the normal value. 
39<br/><code>option </code><strong> 2</strong> : second order. Since other parts of the code do
40not go to second order there is no strong reason to use this option,
41but there is also nothing wrong with it. 
42 
43
44<p/>
45QED interactions are regulated by the <i>alpha_electromagnetic</i>
46value at the <i>Q^2</i> renormalization scale of an interaction.
47<p/><code>mode&nbsp; </code><strong> SigmaProcess:alphaEMorder &nbsp;</strong> 
48 (<code>default = <strong>1</strong></code>; <code>minimum = -1</code>; <code>maximum = 1</code>)<br/>
49The running of <i>alpha_em</i> used in hard processes.
50<br/><code>option </code><strong> 1</strong> : first-order running, constrained to agree with
51<code>StandardModel:alphaEMmZ</code> at the <i>Z^0</i> mass.
52 
53<br/><code>option </code><strong> 0</strong> : zeroth order, i.e. <i>alpha_em</i> is kept
54fixed at its value at vanishing momentum transfer. 
55<br/><code>option </code><strong> -1</strong> : zeroth order, i.e. <i>alpha_em</i> is kept
56fixed, but at <code>StandardModel:alphaEMmZ</code>, i.e. its value
57at the <i>Z^0</i> mass.
58   
59 
60
61<p/>
62In addition there is the possibility of a global rescaling of
63cross sections (which could not easily be accommodated by a
64changed <i>alpha_strong</i>, since <i>alpha_strong</i> runs)
65<p/><code>parm&nbsp; </code><strong> SigmaProcess:Kfactor &nbsp;</strong> 
66 (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 4.0</code>)<br/>
67Multiply almost all cross sections by this common fix factor. Excluded
68are only unresolved processes, where cross sections are better
69<a href="TotalCrossSections.html" target="page">set directly</a>, and
70multiparton interactions, which have a separate <i>K</i> factor
71<a href="MultipartonInteractions.html" target="page">of their own</a>. 
72This degree of freedom is primarily intended for hadron colliders, and
73should not normally be used for <i>e^+e^-</i> annihilation processes.
74 
75
76<h3>Renormalization scales</h3>
77
78The <i>Q^2</i> renormalization scale can be chosen among a few different
79alternatives, separately for <i>2 -> 1</i>, <i>2 -> 2</i> and two
80different kinds of <i>2 -> 3</i> processes. In addition a common
81multiplicative factor may be imposed.
82 
83<p/><code>mode&nbsp; </code><strong> SigmaProcess:renormScale1 &nbsp;</strong> 
84 (<code>default = <strong>1</strong></code>; <code>minimum = 1</code>; <code>maximum = 2</code>)<br/>
85The <i>Q^2</i> renormalization scale for <i>2 -> 1</i> processes.
86The same options also apply for those <i>2 -> 2</i> and <i>2 -> 3</i>
87processes that have been specially marked as proceeding only through
88an <i>s</i>-channel resonance, by the <code>isSChannel()</code> virtual
89method of <code>SigmaProcess</code>.
90<br/><code>option </code><strong> 1</strong> : the squared invariant mass, i.e. <i>sHat</i>.
91 
92<br/><code>option </code><strong> 2</strong> : fix scale set in <code>SigmaProcess:renormFixScale</code> 
93below.
94 
95 
96 
97<p/><code>mode&nbsp; </code><strong> SigmaProcess:renormScale2 &nbsp;</strong> 
98 (<code>default = <strong>2</strong></code>; <code>minimum = 1</code>; <code>maximum = 5</code>)<br/>
99The <i>Q^2</i> renormalization scale for <i>2 -> 2</i> processes.
100<br/><code>option </code><strong> 1</strong> : the smaller of the squared transverse masses of the two
101outgoing particles, i.e. <i>min(mT_3^2, mT_4^2) =
102pT^2 + min(m_3^2, m_4^2)</i>.
103 
104<br/><code>option </code><strong> 2</strong> : the geometric mean of the squared transverse masses of
105the two outgoing particles, i.e. <i>mT_3 * mT_4 =
106sqrt((pT^2 + m_3^2) * (pT^2 + m_4^2))</i>.
107 
108<br/><code>option </code><strong> 3</strong> : the arithmetic mean of the squared transverse masses of
109the two outgoing particles, i.e. <i>(mT_3^2 + mT_4^2) / 2 =
110pT^2 + 0.5 * (m_3^2 + m_4^2)</i>. Useful for comparisons
111with PYTHIA 6, where this is the default.
112 
113<br/><code>option </code><strong> 4</strong> : squared invariant mass of the system,
114i.e. <i>sHat</i>. Useful for processes dominated by
115<i>s</i>-channel exchange.
116 
117<br/><code>option </code><strong> 5</strong> : fix scale set in <code>SigmaProcess:renormFixScale</code> 
118below.
119 
120 
121 
122<p/><code>mode&nbsp; </code><strong> SigmaProcess:renormScale3 &nbsp;</strong> 
123 (<code>default = <strong>3</strong></code>; <code>minimum = 1</code>; <code>maximum = 6</code>)<br/>
124The <i>Q^2</i> renormalization scale for "normal" <i>2 -> 3</i> 
125processes, i.e excepting the vector-boson-fusion processes below.
126Here it is assumed that particle masses in the final state either match
127or are heavier than that of any <i>t</i>-channel propagator particle.
128(Currently only <i>g g / q qbar -> H^0 Q Qbar</i> processes are
129implemented, where the "match" criterion holds.)
130<br/><code>option </code><strong> 1</strong> : the smaller of the squared transverse masses of the three
131outgoing particles, i.e. min(mT_3^2, mT_4^2, mT_5^2).
132 
133<br/><code>option </code><strong> 2</strong> : the geometric mean of the two smallest squared transverse
134masses of the three outgoing particles, i.e.
135<i>sqrt( mT_3^2 * mT_4^2 * mT_5^2 / max(mT_3^2, mT_4^2, mT_5^2) )</i>.
136 
137<br/><code>option </code><strong> 3</strong> : the geometric mean of the squared transverse masses of the
138three outgoing particles, i.e. <i>(mT_3^2 * mT_4^2 * mT_5^2)^(1/3)</i>.
139 
140<br/><code>option </code><strong> 4</strong> : the arithmetic mean of the squared transverse masses of
141the three outgoing particles, i.e. <i>(mT_3^2 + mT_4^2 + mT_5^2)/3</i>.
142 
143<br/><code>option </code><strong> 5</strong> : squared invariant mass of the system,
144i.e. <i>sHat</i>.
145 
146<br/><code>option </code><strong> 6</strong> : fix scale set in <code>SigmaProcess:renormFixScale</code> 
147below.
148 
149   
150 
151<p/><code>mode&nbsp; </code><strong> SigmaProcess:renormScale3VV &nbsp;</strong> 
152 (<code>default = <strong>3</strong></code>; <code>minimum = 1</code>; <code>maximum = 6</code>)<br/>
153The <i>Q^2</i> renormalization scale for <i>2 -> 3</i> 
154vector-boson-fusion processes, i.e. <i>f_1 f_2 -> H^0 f_3 f_4</i>
155with <i>Z^0</i> or <i>W^+-</i>  <i>t</i>-channel propagators.
156Here the transverse masses of the outgoing fermions do not reflect the
157virtualities of the exchanged bosons. A better estimate is obtained
158by replacing the final-state fermion masses by the vector-boson ones
159in the definition of transverse masses. We denote these combinations
160<i>mT_Vi^2 = m_V^2 + pT_i^2</i>.
161<br/><code>option </code><strong> 1</strong> : the squared mass <i>m_V^2</i> of the exchanged
162vector boson.
163 
164<br/><code>option </code><strong> 2</strong> : the geometric mean of the two propagator virtuality
165estimates, i.e. <i>sqrt(mT_V3^2 * mT_V4^2)</i>.
166 
167<br/><code>option </code><strong> 3</strong> : the geometric mean of the three relevant squared
168transverse masses, i.e. <i>(mT_V3^2 * mT_V4^2 * mT_H^2)^(1/3)</i>.
169 
170<br/><code>option </code><strong> 4</strong> : the arithmetic mean of the three relevant squared
171transverse masses, i.e. <i>(mT_V3^2 + mT_V4^2 + mT_H^2)/3</i>.
172 
173<br/><code>option </code><strong> 5</strong> : squared invariant mass of the system,
174i.e. <i>sHat</i>.
175 
176<br/><code>option </code><strong> 6</strong> : fix scale set in <code>SigmaProcess:renormFixScale</code> 
177below.
178 
179 
180
181<p/><code>parm&nbsp; </code><strong> SigmaProcess:renormMultFac &nbsp;</strong> 
182 (<code>default = <strong>1.</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 10.</code>)<br/>
183The <i>Q^2</i> renormalization scale for <i>2 -> 1</i>,
184<i>2 -> 2</i> and <i>2 -> 3</i> processes is multiplied by
185this factor relative to the scale described above (except for the options
186with a fix scale). Should be use sparingly for <i>2 -> 1</i> processes.
187 
188
189<p/><code>parm&nbsp; </code><strong> SigmaProcess:renormFixScale &nbsp;</strong> 
190 (<code>default = <strong>10000.</strong></code>; <code>minimum = 1.</code>)<br/>
191A fix <i>Q^2</i> value used as renormalization scale for <i>2 -> 1</i>,
192<i>2 -> 2</i> and <i>2 -> 3</i> processes in some of the options above.
193 
194
195<h3>Factorization scales</h3>
196
197Corresponding options exist for the <i>Q^2</i> factorization scale
198used as argument in PDF's. Again there is a choice of form for 
199<i>2 -> 1</i>, <i>2 -> 2</i> and <i>2 -> 3</i> processes separately.
200For simplicity we have let the numbering of options agree, for each event
201class separately, between normalization and factorization scales, and the
202description has therefore been slightly shortened. The default values are
203<b>not</b> necessarily the same, however.
204 
205<p/><code>mode&nbsp; </code><strong> SigmaProcess:factorScale1 &nbsp;</strong> 
206 (<code>default = <strong>1</strong></code>; <code>minimum = 1</code>; <code>maximum = 2</code>)<br/>
207The <i>Q^2</i> factorization scale for <i>2 -> 1</i> processes.
208The same options also apply for those <i>2 -> 2</i> and <i>2 -> 3</i>
209processes that have been specially marked as proceeding only through
210an <i>s</i>-channel resonance.
211<br/><code>option </code><strong> 1</strong> : the squared invariant mass, i.e. <i>sHat</i>.
212 
213<br/><code>option </code><strong> 2</strong> : fix scale set in <code>SigmaProcess:factorFixScale</code> 
214below.
215 
216 
217
218<p/><code>mode&nbsp; </code><strong> SigmaProcess:factorScale2 &nbsp;</strong> 
219 (<code>default = <strong>1</strong></code>; <code>minimum = 1</code>; <code>maximum = 5</code>)<br/>
220The <i>Q^2</i> factorization scale for <i>2 -> 2</i> processes.
221<br/><code>option </code><strong> 1</strong> : the smaller of the squared transverse masses of the two
222outgoing particles.
223 
224<br/><code>option </code><strong> 2</strong> : the geometric mean of the squared transverse masses of
225the two outgoing particles.
226 
227<br/><code>option </code><strong> 3</strong> : the arithmetic mean of the squared transverse masses of
228the two outgoing particles. Useful for comparisons with PYTHIA 6, where
229this is the default.
230 
231<br/><code>option </code><strong> 4</strong> : squared invariant mass of the system,
232i.e. <i>sHat</i>. Useful for processes dominated by
233<i>s</i>-channel exchange.
234 
235<br/><code>option </code><strong> 5</strong> : fix scale set in <code>SigmaProcess:factorFixScale</code> 
236below.
237 
238 
239 
240<p/><code>mode&nbsp; </code><strong> SigmaProcess:factorScale3 &nbsp;</strong> 
241 (<code>default = <strong>2</strong></code>; <code>minimum = 1</code>; <code>maximum = 6</code>)<br/>
242The <i>Q^2</i> factorization scale for "normal" <i>2 -> 3</i> 
243processes, i.e excepting the vector-boson-fusion processes below.
244<br/><code>option </code><strong> 1</strong> : the smaller of the squared transverse masses of the three
245outgoing particles.
246 
247<br/><code>option </code><strong> 2</strong> : the geometric mean of the two smallest squared transverse
248masses of the three outgoing particles.
249 
250<br/><code>option </code><strong> 3</strong> : the geometric mean of the squared transverse masses of the
251three outgoing particles.
252 
253<br/><code>option </code><strong> 4</strong> : the arithmetic mean of the squared transverse masses of
254the three outgoing particles.
255 
256<br/><code>option </code><strong> 5</strong> : squared invariant mass of the system,
257i.e. <i>sHat</i>.
258 
259<br/><code>option </code><strong> 6</strong> : fix scale set in <code>SigmaProcess:factorFixScale</code> 
260below.
261 
262   
263 
264<p/><code>mode&nbsp; </code><strong> SigmaProcess:factorScale3VV &nbsp;</strong> 
265 (<code>default = <strong>2</strong></code>; <code>minimum = 1</code>; <code>maximum = 6</code>)<br/>
266The <i>Q^2</i> factorization scale for <i>2 -> 3</i> 
267vector-boson-fusion processes, i.e. <i>f_1 f_2 -> H^0 f_3 f_4</i>
268with <i>Z^0</i> or <i>W^+-</i>  <i>t</i>-channel propagators.
269Here we again introduce the combinations <i>mT_Vi^2 = m_V^2 + pT_i^2</i>
270as replacements for the normal squared transverse masses of the two
271outgoing quarks.
272<br/><code>option </code><strong> 1</strong> : the squared mass <i>m_V^2</i> of the exchanged
273vector boson.
274 
275<br/><code>option </code><strong> 2</strong> : the geometric mean of the two propagator virtuality
276estimates.
277 
278<br/><code>option </code><strong> 3</strong> : the geometric mean of the three relevant squared
279transverse masses.
280 
281<br/><code>option </code><strong> 4</strong> : the arithmetic mean of the three relevant squared
282transverse masses.
283 
284<br/><code>option </code><strong> 5</strong> : squared invariant mass of the system,
285i.e. <i>sHat</i>.
286 
287<br/><code>option </code><strong> 6</strong> : fix scale set in <code>SigmaProcess:factorFixScale</code> 
288below.
289 
290 
291
292<p/><code>parm&nbsp; </code><strong> SigmaProcess:factorMultFac &nbsp;</strong> 
293 (<code>default = <strong>1.</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 10.</code>)<br/>
294The <i>Q^2</i> factorization scale for <i>2 -> 1</i>,
295<i>2 -> 2</i> and <i>2 -> 3</i> processes is multiplied by
296this factor relative to the scale described above (except for the options
297with a fix scale). Should be use sparingly for <i>2 -> 1</i> processes.
298 
299
300<p/><code>parm&nbsp; </code><strong> SigmaProcess:factorFixScale &nbsp;</strong> 
301 (<code>default = <strong>10000.</strong></code>; <code>minimum = 1.</code>)<br/>
302A fix <i>Q^2</i> value used as factorization scale for <i>2 -> 1</i>,
303<i>2 -> 2</i> and <i>2 -> 3</i> processes in some of the options above.
304 
305
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307</html>
308
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