introduction to state of the art calculations for lhcuwer/lectures/... · introduction to state of...
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Introduction to state of the art calculations for LHC
Peter Uwer*)
Universität Karlsruhe
*) Heisenberg Fellow of the Deutsche Forschungsgemeinschaft
Heidelberg, 01/23/2008
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2 Contents
1. Introduction - Setting the scene
2. Current state of the art
3. Example – Born approximation
4. Example – NLO approximation
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3 What we will see at the LHC…
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4… and how we understand it
[Frank Krauss]
● Hard scattering● Parton shower● Hadronization● Underlying event
Aspects: Tools:
● Perturbation theory● MC-Tools like Herwig
and Phythia
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5 The perturbative part
Parton-parton scattering
● The matrix elements describing the transition ij X are calculable in perturbation theory
How do we calculate the corresponding hadronic cross sections
?
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6 Simplified picture of the hadronic cross section
QCD improved parton model
p p
Parton distribution functions (PDF)(non-perturbativ
experiment, lattice)
Partonic cross section
Specific process, i.e.Parton ≈ constituent
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7 Partonic cross section
∫∫ ∫+
+
x*
2Re*
x2Re
2
∫2
∫+*
x2Re +2
∫+
+ …
Leading-order, Born approximation
Next-to-leading order(NLO)
Next-to-next-to-leading order (NNLO)
n-legs
(n+1)-legs, real corrections
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8 Pictorial representation of amplitudes
∫
= + …
= + …
= + …
Complex functionsof the kinematics
Phase spaceintegral
Born approximation
2-loop approximation
1-loop approximation
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9 Current state of the art
● Leading-order:
2 8 + n processes calculable in automated way
Drawback: matrix element evaluation and phase space evaluation might be slow
● Next-to-leading:
2 3 processes feasible with current technology, no true 2 4 process @ NLO currently available for LHC
● Next-to-next-to-leading order:
21 processes can be done, do we need NNLO for 22?
Note: many phase space points needed for good accuracy (high dim. phase space integrals)
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10 Les Houches wishlist
High demand for one-loop calculations for the LHC
[Heinrich 07]
NLO
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11
pp t t + 1 Jet@ NLO
A concrete example:
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12 Motivation: Topquark as background for Higgs search
Higgs search at LHC
[Atlas]
“Weak Boson Fusion” (WBF)
Background processes:
Precise predictions for pp t t + 1-Jet are important
[Alves, Eboli, Plehn, Rainwater ’04]
HW
W
W
W
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13 Side remark: New physics search at the LHC
LHC-Physics = Standardmodell + X
X = LHC-Physics – StandardmodellExperiment Theory prediction
new physics
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14 Scattering amplitudes for ij t t + 1Jet
complexe function of momenta and polarisation
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15Methods to calculate scattering amplitudes (LO)
1. Analytically by hand on a piece of paper
2. Analytically using computer algebra
3. purely numerical
Lets take a closer look to see how it works by hand and whywe don’t want to do it that way
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16 A simple example how to do it by hand
Color is not observerd average over incoming color, sum over outgoing
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17A simple example how to do it by hand (cont’d)
If spin is not observed: average over incoming sum over outgoing
Use:
Calculating the traces gives:
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18A simple example how to do it by hand (cont’d)
Last step to obtain total cross section: phase space integral
The differential (partonic) cross section becomes:
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19
What are the problems when going to more
complicated processes
?
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20 A simple example how to do it by hand
Color is not observerd average over incoming color, sum over outgoing
more diagrams, longer expressions
many terms i.e. ~16x16 = 256
more structures
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21A simple example how to do it by hand (cont’d)
If spin is not observed: average over incoming sum over outgoing
Use:
Calculating the trace gives:
more complicated traces more γ matrices
function of many variables
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22A simple example how to do it by hand (cont’d)
Last step to obtain cross section: phase space integral
The differential (partonic) cross section becomes:more particles phase space more complicated
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23 One solution: Use computer algebra
Generate diagrams
(Topologies)[QGRAF, Feynarts]
Algebraic expressions( Maple, Mathematica, Form)
Feynman rules
Analytic expressions for amplitudesfor specific helicty configurations
Evaluate amplitude numericallyas complex number using C/C++ or Fortran,
calculate the square numerically
explicit representationof the spinors and ε’s
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24 Another approach: Completely numerical approach
Two common approaches for amplitude calculations:
1. Feynman diagram based i.e. Madgraph,…
2. Use recurrence relation i.e. Alpgen,…
In 1. for every diagram a code is generated to evaluate it numerically
In 2. amplitudes are calculated from simplerobjects via recurrence relation
some progress recently from string inspired methods
[Long, Stelzer ’94]
[Mangano et al]
Want to use it as a black box don’t care what is inside!
We care about speed and numerical accuracy!
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25 Example Madgraph
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26 Example Madgraph – Output
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27 Example Madgraph – Output
Input: QCD coupling+ masses and widths
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28 Example Madgraph – Output
Postscript figure also producedby Madgraph
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29 What about phase space integration ?
● High dimensional for multiparton processes (i.e. 5 for 23)
● Want to include arbitrary cuts / observables
Do integration numerically using Monte Carlo techniques
Basic idea:
Computer Code (F77) i.e. Vegas by Lepage
call vegas(ndim, fxn, avg, sd, chi2)
integrates fxn over [0,1]ndim
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30 Missing piece: mapping [0,1]n dLIPS
dLIPS = lorentz invariant phase space measure
RAMBO by Ellis, Kleiss, Stirling
● Flat mapping:
SUBROUTINE RAMBO(N,ET,XM,P,WT)
● Sequential splitting
disadvantage: flat and [0,1]4n dLIPS
[0,1]3n-4 dLIPS [Book: Byckling,Kajantie p. 273]
● Multi channel algorithms Adopt MC to structure of the integrandby using different mappings in parallel
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31 Last missing piece: Parton distribution functions
Remember:
2 additional integration over x1,x2, no problem in MC approach
How to evaluate the PDF’s ?
use LHAPDF, MRST/MSTW or CTEQ code
Subroutine SetCtq6 (Iset) Function Ctq6Pdf (Iparton, X, Q)
Cteq6Pdf-2007.fCTEQ:
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32 Topquark pair production + 1 Jet (Born)
Large scale dependence(~100%)
we need NLOLHC
Perturbation theory:
Born one-loop corrections
but
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33 One-loop diagrams
~350 diagrams
Computer-Algebra numerical methods+
…
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34 Diagram generation with QGRAF
Model file (output) style file
QGRAF
File with all Feynman diagrams
Process info:qgraf.dat
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35Diagram generation with QGRAF: Input
Model file
style file
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36Diagram generation with QGRAF: Output LO
Repetition of input
Output
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37Diagram generation with QGRAF: Output NLO
“snail”
a1 could be suppressed by option nosnail
polarisation vectors
propagators,vertices
dummy index i49
No tadpols
No correctionsOn external lines
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38Diagram generation with QGRAF: Output NLO
pentagondiagram
5 propagators containingthe loop momenta
pentagon diagrams are the most complicated once
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39 More on pentagon diagrams
complicated complex function of 5 variables, i.e.
loop momenta appearsin numerator tensor integrale
loop integration needs to be done in d dimensions toregulate UV and IR singularities
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40 How to calculate the loop diagrams ?
many diagrams many topologies
we cannot calculate every tensor integral analytically by hand
Solution:
Tensor integrals can be expressed in termsof a small set of scalar “master integrals”
many different tensor integrals
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41 Tensor reduction à la Passarino & Veltman
Passarino-Veltman
Contract with p
Scalar integrals:
Terms in red add up to zero
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42 Passarino-Veltman reduction (cont’d)
problematic for
General problem:
Numerical stable and efficient calculationof tensor integrals
Analytically the limit “0/0” can be taken, numerically it mightresult in severe instabilities
Basic version of Passarino-Veltman implemented in LoopTools
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43 Improvement of Passarino-Veltman
● Derive special reduction formulae for problematic phase space regions
● Special reductions for 5- and 6-point tensor integrals
[Denner, Dittmaier and others]
Remark about scalar integrals:
● Only 1-,2-,3-,and 4-point scalar integrals needed, higher point integrals can be reduced
● Evaluation of scalar integrals can be assumed as solved
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44 Alternative reduction procedure – first step
From Schwinger or Feynman parametrizationof tensor integrals:
[Davydychev]
Reduction of tensor integrals to scalar integrals with raised powers of the propagators and in higher dimensions!
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45 Alternative reduction procedure – second step
Integration-by-parts (IBP)
Linear relation between different scalar integrals with raised powers of the propagators
Problematic phase points can be studied systematically
[Chetyrkin, Kataev, Tkachov]
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46 General feature of the reduction
apart from the presence of Ii calculation is similar to leading-order calculation
Same techniques:
helicity basis, numerical evalualtion of spinor products,numerical evaluation of amplitude
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47 For pp t t + 1Jet we used:
F77/C++ library to calculate tensor integrals
1.) Impoved Passarino-Veltman reduction, Feynarts, F772.) 2-loop inspired techniques (IBP), QGRAF, C++
Methods completely general, also applicable to other processes
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48 Real corrections
Note: Virtual corrections contain UV and IR singularities
UV singularities are cancelled via the renormalization procedure
IR singularities are cancelled by real corrections
*x2Re +
2
∫(n+1)-legs, real corrections
∫divergent divergent
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49 Real corrections (cont’d)
● In the real corrections the singularity is produced by the phase space integration over soft and collinear regions
● When we use dimensional regularization for the virtual corrections the same has to be done for the real corrections
● d dimensional integration of the phase space integrals in general not feasible
Solution:
Subtraction Method
[Catani,Seymour,…]
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50 Real corrections: Dipole subtraction method
Add and subtract a counterterm which is easy enough to be integrated analytically:
Construction of subtraction for real corrections more involved,Fortunately a general solution exists:
Dipole subtraction formalism
Can be done numerically
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51 Dipole subtraction method (2)
How it works in practise:
Requirements:
in all single-unresolved regions
Due to universality of soft and collinear factorization,general algorithms to construct subtractions exist
[Frixione,Kunszt,Signer ´95, Catani,Seymour ´96, Nason,Oleari 98, Phaf, Weinzierl, Catani,Dittmaier,Seymour, Trocsanyi ´02]
Recently: NNLO algorithm [Daleo, Gehrmann, Gehrmann-de Ridder, Glover, Heinrich, Maitre]
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52 Dipole subtraction method (3)
Universal structure:
Generic form of individual dipol:Leading-order amplitudes
Vector in color space
Color charge operators,induce color correlation
Spin dependent part,induces spin correlation
universal
Example ggttgg: 6 different colorstructures in LO,36 (singular) dipoles
! !Color charge operators,induce color correlation
Spin dependent part,induces spin correlation
Color charge operators,induce color correlation
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53 Example
For ggttgg the LO amplitude ggttg is required:
Six component vector in color space
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54 Dipole subtraction method — implementation
LO – amplitude, with colour information,
i.e. correlations
List of dipoles we want to calculate
0
1234
5
reduced kinematics,“tilde momenta”
Dipole di
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55 Topquarkpaar + 1-Jet-Production (NLO)
[Dittmaier, Uwer, Weinzierl, Phys. Rev. Lett. 98:262002, ‘07]
● scale dependence is improved● tools are completely general: arbitrary infrared save
observables are calculable (work in progress)
Tevtron LHC
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56 Differential distributions
[Dittmaier, Uwer, Weinzierl 07]
currently studied at the Tevatron
Pseudo rapidity
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57