Analytical calculation of quantum electrodynamics S

Nuclear Instruments and Methods in Physics Research A 502 (2003) 605–606
Analytical calculation of quantum electrodynamics S-matrix
V.V. Andreev1
Physics Department, Gomel State University, Gomel, Belarus
Abstract
The program is presented for analytical calculations of a Quantum Electrodynamics S-matrix as a sum of products of
propagators and normal-ordered products of ﬁeld operators.
r 2003 Elsevier Science B.V. All rights reserved.
PACS: 4.70.Hp; 14.80.Bn; 13.85.Qk
Keywords: Wick; S-matrix; Normal product; Chronological product; Spinor ﬁeld; Vector ﬁeld; Hamiltonian
1. Introduction
2. Method of calculations
As well-known, all observable quantities in high
energy physics are calculated in terms of the
quantum ﬁeld theory. Most of the programs for
analytical calculations in high-energy physics
make use of the Feynman rules, which are set to
be the input objects [1–4].
In this work, we propose that Wick’s theorem
should be used in the analytical calculations. The
input object in this approach is the interaction
Lagrangian described by particle ﬁelds. We
propose the computer program of Wick’s theorem,
which is called ‘‘WickT’’(Wick’s Theorem).
The program ‘‘WickT’’ permits us to convert the
time-ordered products of ﬁeld operators into a
sum of products of propagators and normalordered products of ﬁeld operators.
The interaction term in the Langrangian of
QED for coupling of a fermion (electron and
muon) f with electric charge Qf ; described by the
Dirac spinor ﬁeld cf ðxÞ; to the photon ﬁeld Aa ðxÞ
is well known (see, Ref. [5]):
X
Ql c% l ðxÞga cl ðxÞAa ðxÞ:
ð1Þ
LI ðxÞ ¼
1
E-mail address: [email protected] (V.V. Andreev).
Participation in ACAT’2002 is supported by HP-grant.
l¼e;m
In the quantum ﬁeld theory the S-matrix can be
presented as
X ðiÞn Z
dx1 ?dxn S* ðnÞ ðx1 ; y; xn Þ
ð2Þ
S¼
n!
n
where
S* ðnÞ ðx1 ; y; xn Þ ¼ TðHI ðx1 Þ?HI ðxn ÞÞ
ð3Þ
with interaction Hamiltonian HI ðxÞ ¼ LI ðxÞ
and Wick chronological T-product.
Our main purpose is to calculate the
* 1 ; x2 ; y; xn Þ-matrix operator. Wick’s theorem
Sðx
permits us to convert the time-ordered products
of ﬁeld operators into a sum of products of
0168-9002/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0168-9002(03)00517-5
606
V.V. Andreev / Nuclear Instruments and Methods in Physics Research A 502 (2003) 605–606
propagators and normal-ordered products of ﬁeld
operators. A particular Feynman graph is, then,
neither more nor less than the symbolical representation of a particular operator in the series of
the Wick decomposition.
For the T-product of any ﬁeld operators we
have
TðUVWR?XY Þ
¼ NðUVWR?XY Þ þ NðU a V a WR?XY Þ
þ NðU b VW b R?XY Þ
þ ? þ NðU a V a W b R?X b Y Þ þ ? :
ð4Þ
The normal product with the internal contractions of operators is determined as
NðU a V a W b R?X b YZÞ
¼ dDðU; V ÞDðW ; X ÞNðRyYZÞ
ð5Þ
where d ¼ 1 ðd ¼ 1Þ in the case of even (odd)
number permutations of the Fermi operators.
The ﬁrst line of Eq. (5) contains a normal
product of operators. The second line contains
the sum of all possible normal products with one
contraction inside them. The third line contains
the sum of all possible normal products with two
contractions, and so on.
The program ‘‘WickT’’ permits us to calculate a
S-matrix of QED (3) in the given order n of the
perturbation theory.
3. Program summary
Title of the program: WickT.
Computer: any computer with the environment
‘‘MATHEMATICA’’ [6] version 4.0.
3.1. The syntax for WickT
The main input object of the program ‘‘WickT’’
(Wick’s Theorem) is the interaction Lagrangian of
quantum electrodynamics (QED) LI ðxÞ:
Input object:
X
% x1 ; c½l; x2 ;g½a; B0 ½A; x;ag:
Ql H½fc½l;
ð6Þ
l¼e;m
Main command:
WickTheorem½QED; n;
ð7Þ
where n is the calculation order.
Output object: Sum of products of propagators
DðxÞ and normal-ordered products of ﬁeld operators N: For example,
% x1 ;c½e; x1 ;g½a1 ;
ðN½fc½e;
% x2 ;c½e; x2 ;g½a2 gþ
c½e;
% x1 ;c½e; x2 ; g½a2 g
N½fc½e;
% x2 S½D½c½e; x1; g½a1 ;c½e;
% x1 ;g½a1;c½e; x2 g
N½fc½e;
% x1 S½D½c½e; x2; g½a2 ;c½e;
% x2 S½D½c½e; x1; g½a1 ;c½e;
% x1 Þ
D½c½e; x2; g½a2 ;c½e;
ðN½fB0 ½A; x1 ;a1 ; B0 ½A; x2 ;a2 gþ
S½D½B0 ½A; x1 ;a1 ; B0 ½A; x2 ;a2 Þ:
ð8Þ
This program is part of the project ‘‘WickVikCalc’’ of automatic analytical calculations of the
scattering amplitude (the work is in progress).
References
[1] A.E. Pukhov, et al., Preprint INP MSU 98-41/542, 1998;
hep-ph/9908288.
[2] KEK Report 92-19, February 1993.
.
[3] R. Mertig, M. Bohm,
A. Denner, Comput. Phys. Commun.
64 (1991) 345.
[4] M. Moretti, T. Ohl, J. Reuter, hep-ph/0102195.
[5] S.M. Bilenky, Introduction to the Physics of Electroweak
Interaction, Pergamon, London, 1982.
[6] S. Wolfram, The Mathematica Book, 4th Edition, AddisonWesley, Reading, MA, 1999.

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