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Robert Harlander:
Research Interests
/ Home
/ TTP Karlsruhe
/ CERN Theory
/ HET
Brookhaven
Quark mass effects in higher order QCD  Asymptotic Expansions
Asymptotic expansions
A central part of my Ph.D. thesis was concerned with the precise
determination of the influence of finite quark masses to the
hadronic cross section at electronpositron colliders. A
powerful tool for this kind of calculations is provided by the method of
asymptotic expansion, an effective reformulation of the
operator product expansion (for a review see
[9]).
Previously, the application of this
procedure was restricted to lower orders of perturbation theory because
it requires the evaluation of the huge number of diagrams generated by
this approach. Therefore we automated the method which allowed to
compute the third order QCD corrections to the hadronic R ratio as an
expansion in the quark mass
[8],
[7]
(for a review on the automatic computation of Feynman diagrams, see
[6]).
Subdiagrams of the nonplanar threeloop diagram that
contribute to the large momentum procedure (taken from [7]).
Top quark production at a Linear Collider
It turned out that using the same strategy one could also examine
top quark pair production at a linear collider. If the
energy is slightly above the threshold region (click here for a treatment of the threshold region
itself), the top quark mass still influences the cross section
significantly. A detailed investigation of the corresponding production
cross section was performed in
[5].
Higgs boson decay
Also the decay of (heavy) scalar or pseudoscalar Higgs
bosons into top
quarks could be investigated. Although the current limits on a
Standard Model Higgs mass rule out this decay, the existence of
other, heavier scalars or pseudoscalar particles as predicted, for
example, in supersymmetric theories is well possible.
In a certain mass range, top quark mass effects may have a
significant influence on the decay rate. This dependence was studied in
[4]
with the help of asymptotic
expansions and the tools developed at the first stage of my thesis.
Padé approximations (for singlet diagrams)
(For a review on Padé approximations see
here.)
The results obtained through asymptotic expansions can be combined so
that the corresponding cross sections or decay rates will be valid over
the full energy range. This strategy was already known for nonsinglet
contributions, but in order to correctly account for axialvector
couplings one needs its generalization to singlet
diagrams.
"Singlet" or "double triangle" diagrams.
Furthermore, the original approach was tailored to include only the
leading mass corrections, and it was important to respect the previously
mentioned higher order mass terms. Both the extension to singlet
diagrams and the inclusion of higher mass terms was performed in
[3],
so that now, for example, the hadronic R ratio is known
completely up to order \alpha_s^2, including its axial vector part.
Quartic mass terms at alpha_s^3
All the discussion above is concerned with mass corrections at order
alpha_s^2. In the massless case, most of the quantities above are known
up to order alpha_s^3. To obtain an estimate on the mass effects at this
order, the above strategy of asymptotic expansion leads to fourloop
diagrams that can not be solved in general. However, using
renormalization group methods, one can derive the quadratic mass terms
at order alpha_s^3 from the ones at alpha_s^2.
If one aims for the quartic mass terms, one may follow a similar strategy.
Operator product expansion in combination with renormalization group methods
allows to reduce the fourloop diagrams to threeloop ones.
The method is briefly described in
[2],
while the results are presented in
[1].
Literature:
[10] 
R.V. Harlander, M. Steinhauser  
rhad: a program for the evaluation of the hadronic Rratio in the
perturbative regime of QCD
,  
Comp. Phys. Comm. 153 (2003) 244274.
[hepph/0212294]
[Journal
Version]  
(the program is available from
http://www.rhad.de/)
   
 [9] 
R. Harlander,  
Asymptotic Expansions  Methods and Applications,  
Acta Phys. Pol. B30 (1999) 34433462.  
(Pedagogical overview,
relation among different approaches.)
   
 [8] 
K.G. Chetyrkin, R. Harlander, J.H. Kühn, M. Steinhauser  
Mass Corrections to the Vector Current Correlator,  
Nucl. Phys. B503 (1997) 339353.  
(Expansion of
R_{had} in terms of m^{2}/s.)
   
 [7] 
K.G. Chetyrkin, R. Harlander, J.H. Kühn, M. Steinhauser,  
Automatic Computation Of ThreeLoop TwoPoint Functions In Large
Momentum Expansion,  
Nucl. Instrum. Meth. A389 (1997) 354356.  
(Describes the technical realization
of the method used in [8].)    
 [6] 
R. Harlander, M. Steinhauser,  
Automatic Computation of Feynman Diagrams,  
Prog. Part. Nucl. Phys. 43 (1999) 167228.  
(General overview of methods and realizations
for
calculating Feynman diagrams on a computer.)    
 [5] 
R. Harlander, M. Steinhauser,  
O(\alpha_s^2) Corrections to Top Quark Production at e^+e^
Colliders,  
Eur. Phys. J. C2 (1998) 151158.  
(Expansion of the total cross section in terms of
m_{top} /sqrt(s).)    
 [4] 
R. Harlander, M. Steinhauser,  
Higgs Decay to Top Quarks at O(\alpha_s^2),,  
Phys. Rev. D56 (1997) 39803990.  
(Expansion of the decay rate for scalar and
pseudoscalar Higgs in terms of
m_{top} /M_{Higgs}.)    
 [3] 
K.G. Chetyrkin, R. Harlander, M. Steinhauser,  
Singlet Polarization Functions at O(\alpha_s^2),,  
Phys. Rev. D58 (1998) 014012.  
(Combines asymptotic expansion and Padé
approximation.)    
 [2] 
K.G. Chetyrkin, R.V. Harlander, J.H. Kühn,  
Quark mass effects to sigma(e+ e > hadrons) at O(alpha_s^3)
,  
(Proceedings of EPSHEP 99, Finland)  
hepph/9910345  
(Describes the techniques used in
[1].)
   
 [1] 
K.G. Chetyrkin, R.V. Harlander, J.H. Kühn,  
Quartic mass corrections to R_{had} at O(\alpha_s^3),
,
  Nucl. Phys. B586 (2000) 56.
 
(Uses OPE and RG techniques.)
 

Robert Harlander:
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last modified: Jan 5, 2004, by RH
