Version 4 (modified by md987, 8 years ago) (diff)

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## Matrix Element Method

The Matrix Element Method consist in minimizing a likelihood.

The likelihood for N events is defined as

The best estimate of the parameter is obtained through a maximisation of the likelihood. It is common practice to minimize with respect to ,

In general, the probability that an event is accepted depends on the characteristics of the measured event, and not on the process that produced it. The measured probability density can be related to the produced probability density :

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The command:

'/usr/bin/pdflatex -interaction=nonstopmode f04536e1f28716cfeaabaf7f87c44cc561dca9e3.tex'
failed with the following output:
"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./f04536e1f28716cfeaabaf7f87c44cc561dca9e3.tex\nLaTeX2e <2011/06/27>\nBabel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo\naded.\n(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo))\n(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty\n(/usr/share/texlive/texmf-dist/tex/latex/base/utf8.def\n(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.dfu)\n(/usr/share/texlive/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/usr/share/texlive/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/usr/share/texlive/texmf-dist/tex/latex/cmap/cmap.sty) (./type1ec.sty\n(/usr/share/texlive/texmf-dist/tex/latex/base/t1cmr.fd))\n(/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty\n(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap>>)\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the ?' option.\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty)\n(/usr/share/texlive/texmf-local/texmf-compat/tex/latex/preview/preview.sty\n(/usr/share/texlive/texmf-local/texmf-compat/tex/latex/preview/prtightpage.def)\n) (./f04536e1f28716cfeaabaf7f87c44cc561dca9e3.aux)\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n<<ot1.cmap>> (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Missing } inserted.\n<inserted text> \n                }\nl.14 $\\bar{P}(x,\\alpha){{{Acc(x) P(x,\\alpha)$\n                                             \n! Missing } inserted.\n<inserted text> \n                }\nl.14 $\\bar{P}(x,\\alpha){{{Acc(x) P(x,\\alpha)$\n                                             \n! Missing } inserted.\n<inserted text> \n                }\nl.14 $\\bar{P}(x,\\alpha){{{Acc(x) P(x,\\alpha)$\n                                             \nPreview: Tightpage -32891 -32891 32891 32891\n[1{/usr/share/texlive/texmf/fonts/map/pdftex/updmap/pdftex.map}]\n(./f04536e1f28716cfeaabaf7f87c44cc561dca9e3.aux) )\n(see the transcript file for additional information)</usr/share/texlive/texmf-d\nist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/share/texlive/texmf-dist/fo\nnts/type1/public/amsfonts/cm/cmr10.pfb>\nOutput written on f04536e1f28716cfeaabaf7f87c44cc561dca9e3.pdf (1 page, 18546 b\nytes).\nTranscript written on f04536e1f28716cfeaabaf7f87c44cc561dca9e3.log.\n"

where is the detector acceptance, which depends only on . So the quantity that we have to minimize is
TracMath macro processor has detected an error. Please fix the problem before continuing.

The command:

'/usr/bin/pdflatex -interaction=nonstopmode 56c05a207a38a4f6eb352518e3d09bd39caca1f2.tex'
failed with the following output:
"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./56c05a207a38a4f6eb352518e3d09bd39caca1f2.tex\nLaTeX2e <2011/06/27>\nBabel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo\naded.\n(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo))\n(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty\n(/usr/share/texlive/texmf-dist/tex/latex/base/utf8.def\n(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.dfu)\n(/usr/share/texlive/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/usr/share/texlive/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/usr/share/texlive/texmf-dist/tex/latex/cmap/cmap.sty) (./type1ec.sty\n(/usr/share/texlive/texmf-dist/tex/latex/base/t1cmr.fd))\n(/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty\n(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap>>)\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the ?' option.\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty)\n(/usr/share/texlive/texmf-local/texmf-compat/tex/latex/preview/preview.sty\n(/usr/share/texlive/texmf-local/texmf-compat/tex/latex/preview/prtightpage.def)\n) (./56c05a207a38a4f6eb352518e3d09bd39caca1f2.aux)\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n<<ot1.cmap>> (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Extra }, or forgotten $.\nl.14$-ln (\\tilde{L})}\n                      }}-\\sum_{i=1}^{N} ln(P(x_i;\\alpha)) + N \\int Acc(x) P(...\n\n! Extra }, or forgotten $.\nl.14$-ln (\\tilde{L})}}\n                       }-\\sum_{i=1}^{N} ln(P(x_i;\\alpha)) + N \\int Acc(x) P(...\n\n! Extra }, or forgotten $.\nl.14$-ln (\\tilde{L})}}}\n                        -\\sum_{i=1}^{N} ln(P(x_i;\\alpha)) + N \\int Acc(x) P(...\n\nPreview: Tightpage -32891 -32891 32891 32891\n[1{/usr/share/texlive/texmf/fonts/map/pdftex/updmap/pdftex.map}]\n(./56c05a207a38a4f6eb352518e3d09bd39caca1f2.aux) )\n(see the transcript file for additional information)</usr/share/texlive/texmf-d\nist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/share/texlive/texmf-dist/fo\nnts/type1/public/amsfonts/cm/cmmi10.pfb></usr/share/texlive/texmf-dist/fonts/ty\npe1/public/amsfonts/cm/cmmi7.pfb></usr/share/texlive/texmf-dist/fonts/type1/pub\nlic/amsfonts/cm/cmr10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/ams\nfonts/cm/cmr7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm\n/cmsy10.pfb>\nOutput written on 56c05a207a38a4f6eb352518e3d09bd39caca1f2.pdf (1 page, 51545 b\nytes).\nTranscript written on 56c05a207a38a4f6eb352518e3d09bd39caca1f2.log.\n"

where the term has been omitted since it does not depend on .

## Definition of the Weight

The Matrix Element Method associates a weight to each experimental event where

1. is the set of information describing the events in the detector (momenta,tag,...)
2. describe a theoretical hyppothesis
3. is the cross section of this theoretical hyppothesis
4. is the aplitude linked to this theoretical framework
5. is the parton distribution function associate to the initial parton
6. is the TransferFunction

## Computation of those elements

The MadWeight has created a series of tool to compute the transfer function, the weight, the cross-section, the likelihood,... Some of these tools have their own specific page/

-- Main.OlivierMattelaer - 22 May 2009