Matching of Fixed Order and Resummed Calculation in pQCD

1
Matching of Fixed Order and Resummed Calculation
in pQCD

• Doktorandenseminar
• ETHZ/UZH
• Gionata Luisoni
• 13.09.2007

2
Outline

• Motivation The example of 3-jet observable in
  ee- annihilation.
• Two complementary ways to compute cross
  sections
• Fixed order calculations,
• Resummed calculations.
• Matching
• R-matching scheme,
• Log(R)-matching scheme,
• Modifications of the matching schemes.
• Conclusions and Outlook.

3
Motivation The example of 3-Jet Observables in
ee- Annihilation

• Phenomenologically very important
• Three-jet events at PETRA first evidence for
  gluon
• Determination of the QCD coupling constant
• Measured very precisely at LEP
• Error in the determination of ?S comes mainly
  from theoretical uncertainty.
• NLO calculation Ellis, Ross, Terrano, Kunszt,
  Nason, Giele, Glover, Catani, Seymour
• NLL resummation Catani, Turnock, Webber,
  Trentadue

Up to now
4
3-Jet Observable in ee- Annihilation

• First NNLO results are becoming available

Gehrmann-De Ridder, Gehrmann, Glover,
Heinrich arXiv0707.1285v1 hep-ph
arXiv0709.1608v1hep-ph
5
Two ways for computing cross sections

• Fixed order calculations

• Resummed calculations

Why?
To understand this we consider first fixed order
calculations.
6
Fixed Order Calculation

• Fundamental assumption in pQCD the strong
  coupling ?S is small enough to permit to compute
  things as a perturbation expansion in it

• Cross sections computed by making an expansion in
  the strong coupling

7
Fixed Order Calculation

• Each coefficient function contains powers of

• Cross section computed by making an expansion in
  the strong coupling

8
Fixed Order Calculation

• Cross section computed by making an expansion in
  the strong coupling

Contribution becomes smaller

• If L is not large sum line by line!

9
Fixed Order Calculation

• But
• In region where y ! 0
• For IR-safe observables the limit is finite,
  nevertheless the coefficient functions becomes
  large, spoiling the convergence of the series
  expansion.

Need to find another way to compute things in
this region
10
Example Thrust

• Fixed order coefficients
• LO known analytically
• NLO NNLO known numerically

11
Fixed Order Calculation

• Consider cumulative cross section
• then
• and

12
Resummed Calculation

• In region where y ! 0 main contribution comes
  from highest powers of logarithms.
• Need to find a way to consider the different
  contributions in order of numerical magnitude.

13
Resummed Calculation

• Resum highest powers of the logarithms to all
  orders in perturbation theory

Leading logarithms LL
Next-to-Leading logarithms NLL
14
Resummed Calculation

• For suitable observables the sum over LL and NLL,
  leads to exponentiation

Logarithmic part
Remainder function as
15
Resummed Calculation

• For suitable observables the sum over LL and NLL,
  leads to exponentiation
• The functions in the exponent can be expanded

16
Two complementary ways for computing cross
sections
Need to match them!

• Fixed order calculations

• Resummed calculations

17
Matching

• Different possibilities
• The naïve way
• R-Matching Scheme
• Log(R)-Matching Scheme

18
Matching the naïve way

• Choose function
• then
• Uncertainty due to the choice of function makes
  this procedure unreliable.

19
Matching R- Matching Scheme

• In region where L is small, reexpand resummed
  result
• The coefficient (Ri(y)di(y)) have to be equal to
  the fixed order expansion

20
Matching R- Matching Scheme

• At NNLO/NNLL, matched result can then be written
  as
• For y ! ymax fixed order result is dominant.
• For y ! 0 resummed result is dominant.

• Disadvantage
• Need to know the coefficients G31, C2 and C3
  which cannot be determined analytically.

as
21
Matching Log(R)-Matching Scheme

• Consider again
• Taking the logarithm on both sides and expanding

22

• Comparison of the matching schemes at NLO/NLL

Example Thrust
Log(R)-matching
R-matching
Resummed result
23
Modifications of the Matching Schemes

• Modifications in region
• As in fixed order results
• Renormalization scale dependence
• Up to now we have not considered the dependence
  on the renormalization scheme
• Dependence on the resummed logarithm
• Choice of the logarithms to be resummed is
  arbitrary.

R.W.L.Jones, M.Ford, G.P.Salam, H.Stenzel,
D.Wicke
24

• Comparison of the matching schemes at NLO/NLL

Example Thrust
Modified Log(R)-matching
Modified R-matching
Unmodified Log(R)-matching
25
Conclusions and Outlook

• Fixed order calculations

Resummed calculations

• Matching two schemes
• R-matching scheme
• Log(R)-matching scheme

• Precise determination of NNLO resummed
  coefficient,
• Matching of NNLO and NLL calculation,
• NNLO/NLL matching for improving the determination
  of ?S (in collaboration with G. Dissertori , H.
  Stenzel and ALEPH).

26
Backup Slides
27

• NLO

Thrust
28

• NNLO

Thrust
29
Thrust

• Exponentiated functions
• where

30
Thrust

• Expanding them in the coefficients one
  finds

31
Matching R- Matching Scheme

• This leads to the following equality between
  fixed order and resummed calculation
• Can find unknown coefficient by fitting numerical
  result

32

• Modification in region
• Cross section has to vanish in the multijet
  region in perturbative calculations only a
  finite number (very few) of final state particles
  is considered.
• This means putting the following conditions
• Log(R)-matching scheme
• R-matching scheme

33

• Renormalization scale dependence
• In pQCD coupling runs
• Cross section can not depend on renormalization
  scale ? up to higher orders in perturbation
  theory.
• Thus, every second or higher order coefficient
  acquires an explicit dependence on ?.

34

• Renormalization scale dependence
• Fixed order coefficient functions
• Resummation coefficient

35

• Renormalization scale dependence
• Exponentiated functions

36

• Dependence on the resummed logarithm
• It is not clear whether powers of ?Slog(1/y) or
  powers of ?Slog(2/y) have to be resummed.
• Can express arbitrariness by introducing a
  constant xL which rescales the resummed logs
• This leads to further redefinitions of the
  resummation coefficients Gij and Ci, and the
  exponentiated functions gi(?SL).

37

• Dependence on the resummed logarithm
• Resummation coefficients Gij
• Exponentiated functions

38

• Dependence on the resummed logarithm
• Resummation coefficients Ci

39

• Renormalization scale dependence

Example Thrust
Modified Log(R)-matching
NLO fixed order
Comparison between Log(R)-matching scheme
prediction for ?0.211 GeV and ?QMZ0 and NLO
fixed order prediction for ?0.100 GeV and
?20.0012 Q2 (QMZ0).
40

• Resummed logarithms dependence

Example Thrust
Modified Log(R)-matching xL1
Modified Log(R)-matching xL2
Thrust distribution in the Log(R)-matching
scheme. The solid curve shows the prediction for
xL2, the dotted curve the prediction for xL1.
41

• Modification in region
• For Log(R)-matching scheme constraints are
  fulfilled by substituting
• For R-matching scheme a further substitution is
  needed