One Particle Off Shell Wave Functions

2	One Particle Off Shell Wave Functions

Ward Identities

One Particle Off-Shell Wave Functions (1POWs) are obtained from connected Greensfunctions by applying the LSZ reduction formula to all but one external line while the remaining line is kept off the mass shell

W(x; p₁,...,p_n; q₁,...,q_m) =áf(q₁),...,f(q_m);out|F(x) |f(p₁),...,f(p_n);inñ . (1)

Depending on the context, the off shell line will either be understood as amputated or not. For example, áf(q₁),f(q₂);out|F(x)|f(p₁);inñ in unflavored scalar f³-theory is given at tree level by

The number of distinct momenta that can be formed from n external momenta is P(n)=2^n-1-1. Therefore, the number of tree 1POWs grows exponentially with the number of external particles and not with a factorial, as the number of Feynman diagrams, e. g. F(n)=(2n-5)!!=(2n-5)·...5·3·1 in unflavored f³-theory.

At tree-level, the set of all 1POWs for a given set of external momenta can be constructed recursively

where the sum extends over all partitions of the set of n momenta. This recursion will terminate at the external wave functions.

For all quantum field theories, there are---well defined, but not unique---sets of Keystones K [1] such that the sum of tree Feynman diagrams for a given process can be expressed as a sparse sum of products of 1POWs without double counting. In a theory with only cubic couplings this is expressed as

T =

F(n)

i=1

D_i =

P(n)

k,l,m=1

K_{f_kf_lf_m}³(p_k,p_l,p_m) W_{f_k}(p_k)W_{f_l}(p_l)W_{f_m}(p_m) , (2)

with obvious generalizations. The non-trivial problem is to avoide the double counting of diagrams like

where the circle denotes the keystone. The problem has been solved explicitely for general theories with vertices of arbitrary degrees [1]. The solution is inspired by arguments [4] based on the equations of motion (EOM) of the theory in the presence of sources. The iterative solution of the EOM leads to the construcion of the 1POWs and the constraints imposed on the 1POWs by the EOM suggest the correct set [4] of partitions {(p_k,p_l,p_m)} in equation (2).

The maximally symmetric solution selects among equivalent diagrams the keystone closest to the center of a diagram. This corresponds to the numerical expressions of [4]. The absence of double counting can be demonstrated by counting the number F(d_max,n) of unflavored Feynman tree diagrams with n external legs and vertices of maximum degree d_max in to different ways: once directly and then as a sum over keystones. The number F^~(d_max,N_d,n) of unflavored Feynman tree diagrams for one keystone N_d,n={n₁,n₂,...,n_d}, with n = n₁ + n₂ + ··· + n_d, is given by the product of the number of subtrees and symmetry factors

(d_max,N_d,n) =

|S(N_d,n)|s(n_d,n)

i=1

F(d_max,n_i+1)

n_i!

(3)

where |S(N)| is the size of the symmetric group of N, s(n,2n) = 2 and s(n,m) = 1 otherwise. Indeed, it can be verified that the sum over all keystones reproduces the number

F(d_max,n) =

d_max

d=3

N = {n₁,n₂,...,n_d}n₁ + n₂ + ··· + n_d = n1 £ n₁ £ n₂ £ ··· £ n_d £ ë n/2 û

(d_max,N) (4)

of all unflavored Feynman tree diagrams.

A second consistent prescription for the construction of keystones is maximally asymmetric and selects the keystone adjacent to a chosen external line. This prescription reproduces the approach in [5] where the tree-level Schwinger-Dyson equations are used as a special case of the EOM.

Recursive algorithms for gauge theory amplitudes have been pioneered in [6]. The use of 1POWs as basic building blocks for the calculation of scattering amplitudes in tree approximation has been advocated in [7] and a heuristic procedure, without reference to keystones, for minimizing the number of arithmetical operations has been suggested. This approach is used by MADGRAPH [8] for fully automated calculations. The heuristic optimizations are quite efficient for 2®4 processes, but the number of operations remains bounded from below by the number of Feynman diagrams.

2.1

Ward Identities

A particularly convenient property of the 1POWs in gauge theories is that, even for vector particles, the 1POWs are `almost' physical objects and satisfy simple Ward Identities

¶ x_µ

áout|A_µ(x)|inñ_amp. = 0 (5)

for unbroken gauge theories and

¶ x_µ

áout|W_µ(x)|inñ_amp. = - m_W áout|f_W(x)|inñ_amp. (6)

for spontaneously broken gauge theories in R_x-gauge for all physical external states |inñ and |outñ. Thus the identities (2.1) can serve as powerful numerical checks both for the consistency of a set of Feynman rules and for the numerical stability of the generated code. The code for matrix elements can optionally be instrumented by O'Mega with numerical checks of these Ward identities for intermediate lines.