ELSEVIER

11 May 1995

Physics Letters B 350 (1995) 225-233

PHYSICS LETTERS B

A new method for helicity calculations *

Alessandro Ballestrero ‘, Ezio Maina 2 Diparfimento di Fisica Teorica, Vniversitci di Torino, Turin, Italy,

INFN, Sezione di Torino, Ku Giuria I, 10125 Turin, Italy

Received 21 November 1994 Editor: R. Gatto

Abstract

We propose a new helicity formalism based on the insertion in spinor lines of a complete set of states built up with appropriate unphysical spinors, which are eigenstates of d, where p is the momentum flowing in a fermion propagator. The corresponding eigenvalues can be real (p* > 0) or imaginary (p* < 0). The method is developed both for massless and massive fermions for which it turns out to be particularly fast. All relevant formulae are given.

1. Introduction

In high energy collisions often many particles or partons widely separated in phase space are produced.

The calculation of cross sections for these processes is made difficult by the large number of Feynman di-

agrams which appear in the perturbative expansion. This is due both to the complexity of non-abelian the- ories and to simple combinatorics, which generates more and more diagrams when the number of external

particles grows. If one computes unpolarized cross sections for a

process described by many Feynman diagrams with the textbook method of considering the amplitude modulus squared and taking the traces, one can end up

with a prohibitive number of traces to evaluate. The calculation becomes simpler if one uses the so-called helicity amplitudes techniques. In such an approach,

*Work supported in part by the Minister0 dell’Universit& e della Ricerca Scientifica.

I E-mail: balles&ero@to.infn.it. * E-mail: maina@to.infn.it.

given an assigned helicity to external particles, one computes the contribution of every single diagram k

as a complex number Uk, sums over all k’s and takes the modulus squared. To obtain unpolarized cross

sections one simply sums the modulus squared for the various external helicities.

The use of helicity amplitude techniques in high energy physics dates back to Refs. [ 1,2]. Many dif- ferent approaches have been developed [ 3-121, and even a brief survey of the vast literature on this sub-

ject goes far beyond our scope. Two among the most popular schemes are those of Refs. [ 7,101. They can be used both with massless and massive particles. We have employed them in the past for several calcula- tions. One of them is based [lo] on the choice of a specific representation for the Dirac matrices. It is then

possible to obtain explicit expressions for the spinors, for the matrices at the vertices and for the fermion

propagators. The complex number corresponding to a given diagram is obtained multiplying these matrices. In the other one [7,12] all spinors for any physical momentum are defined in terms of a basic spinor for

0370-2693/95/$09.50 @I 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(95)00351-7

226 A. Ballestrero, E. Maina / Physics Letters B 350 (1995) 225-233

an auxiliary lightlike momentum. Decomposing the internal momenta in terms of the external ones, and using the fact that CA uti = p -I- m, Cn UB = p - m, all spinor lines are reduced to an algebraic combina- tion of spinors products J(pl ) u(p2). In order to use this method the polarization vectors of spin- 1 particles have to be expressed as nyfiu currents.

In the following we describe an approach to helicity amplitudes which is based on the insertion in spinor lines of a complete set of states built up with unphys- ical spinors. By this we mean that for every internal spinor line (propagator) carrying momentum p in a Feynman diagram, we make use of the eigenstates of p with eigenvalues equal to the two determinations of

#. When p2 < 0, as often is the case, the eigen- values are imaginary. This generalization of the usual u and u type spinors allows us to treat in a unified way the case of spacelike and timelike momenta. This new formalism, which we have tested in several phys- ical computations [ 131, combines the best features of Refs. [ 7, lo], is highly flexible, has a modular structure and in our experience is faster than previous methods.

2. Spinors

The method we will describe makes use of general- ized spinors u(p) which coincide with the usual ones when p2 > 0, but are defined also for spacelike mo- menta. We discuss such a generalization in this sec- tion.

Spinors may be defined as eigenstates of p. This is equivalent to saying that they must satisfy the Dirac equations

plu(p) = +mu(p>, P~(P> = -mu(p). (I)

Since p is not hermitian, m need not to be real, but it is constrained to satisfy m2 = p* as p2 = p2. When p2 5 0, the two eigenvalues are imaginary and one can choose, for instance, to associate u(p) with the eigenvalue m whose imaginary part is positive. The remaining degeneracy is normally eliminated consid- ering eigenstates of .>15&

Y5$t4P*+) = U(P, +), Y5rddP, -) = -U(P, -),

T%(P, +> = dP7 +) 9 f%J(P7 -> = -U(P* -),

(2)

where the polarization vector s has to satisfy the two conditions s 9 p = 0, s2 = - 1. The first one implies that y$ commutes withp, the second that ( y5#)* = 1 and hence that its two eigenvalues are f 1.

One can easily construct an example of spinors satisfying Eqs. (I), (2) for any value of p2 with a straightforward generalization of the method of Ref. [ 71. In such a method, one first defines spinors w( ko, A) for an auxiliary massless vector ko satisfying

1+ AY5 w(ko,A)hqko,A) = -

2 to (3)

and with their relative phase fixed by

w(ko, A> = A$lW(kO> -A), (4)

with kl a second auxiliary vector such that kf = -1, ko. kl = 0. Spinors for a four-momentum p, with m2 = p2 are then obtained as

P+m 4~9 A) = dm Nko, -A),

P-m U(P+ A) = dm w(ko, -A).

One can readily check that u and u of Eq. (5) satisfy Eqs. ( l), (2) also when p2 5 0 and m is imaginary.

Some care must be taken in defining in the general case the conjugate spinors ii(p, A), C(p, A). In fact the usual quantities U = u+y’, V = v+p do not satisfy normal Dirac equations

ii(p)+ = +mfi(p), fi(p)$ = -ma(p), (6)

when m is imaginary. Taking the hermitian conjugate of Eqs. ( 1 ), one readily verifies on the contrary that u = u+yo, C = u+yo do.

We define ii(p, A), g(p, A) as the spinors satisfying both Eqs. (6) and

C(P, +>r’# = i(p9 +I, E(P, -Jr”, = -ii(p, -),

D(P, +,r’b = fi(P, +>, 3(P, -)r’C = -O(P, -).

(7)

With this definition the usual orthogonality relations among spinors are satisfied. As a consequence, choos- ing the normalization

A. Ballestrero, E. Maina / Physics Letters B 350 (1995) 225-233 227

ii(p, A)u(p, A) = 2m iqp, A)u(p, A) = -2m,

(8)

completeness relation

u(p, A)ii(p, - v(p, A) 2m A

is for any of p’.

The explicit relation between U, u and ii, fi is

fi(p, A) = u+(p, A)y”, tXp, A) = u+(p, WY’,

(10)

as usual when p* > 0. For p* < 0 one has

ii(p, A) = u+(p, A)y’, fi(p, A) = u+(p, A)y’,

(11)

if the components of the polarization vector s are real, and

ii(p, A> = u+(p, -A)y’, ii(p, A) = u+(p, -A)y”,

112)

if they become purely imaginary for imaginary m. This is precisely the case when one uses the spinors of Ref. [ 71: the polarization vector is sP = pP/m - (m/p . ko)c and for a spacelike momentum p it is equal to the imaginary unit times a timelike vector. It is interesting to notice that for kg = cu(p’ - IpI) /p* and k. = ap( IpI - p”)/lp(p2, with (Y an arbitrary factor, one recovers the usual helicity polarization vector s =

(I~12~~0~)lml~l. With the previous definitions, the spinors conjugate

to u and u defined in Rq. (5) are given by the simple formulae

#+m 4~3 A) = *(ko, -A) $?,

C(P, A) = Mko, -A) d& (13)

in all cases. At this point one may notice that there is an ambiguity in the definitions (5) and ( 13)) since we have not specified which of the two determinations

I I I I I I I

I I

PI P2 P3 Pit Pn+1

Fig. 1. Fennion line with n insertions.

m = im) since only m* appears at the end of the computation of a complete Feynman diagram. The same applies to the determination of dm.

3. Outline of the method. T functions

The spinor part of every massive fermion line (Fig. 1) with n insertions has a generic expression of the following type,

T(“) =~‘(PlA)Xl(#* + p*)x*C#, + p3) ._.

x t+, + ru*)XnU(Pn+19 An+1 19 (14)

where A1 and A,,+1 are the polarizations of the external fermions, p1 and pn+l their momenta. ~2,. . . ,p,, and

, p,, are the four-momenta and masses appear- rg f h’ the fermion propagators. U(p, A) (u( p, A) ) stands for either u(p, A) (ii(p,A)) or u(p, h)

(iJ(p, A)). The x’s are

(15)

when the insertion corresponds to a scalar (or pseu- doscalar), or

=Ili [cr; (F) +cli (?)I, (16)

when it corresponds to a vector particle whose “po- larization” is 7. Of course 17 can be the polarization vector of the external particle or the vector resulting from a complete subdiagram which is connected in the ith position to the fermion line.

Let us start considering the case in which there are only two insertions:

(17)

228 A. Balfestrero, E. Maina / Physics Letters B 350 (1995) 225-233

Here and in the following we indicate with ci both the couplings cr, and c~,. Obviously the vectors 77 only

appear as an argument for vector insertions.

Even if rnz f pz does not correspond to the mass of any physical particle, one can insert in Eq. (17), just

before (i2 + ~2)) a complete set of states in the form

1=x 4P2, A)iGP2, N&dPZv Nfi(P2* A) ( 18)

A

and make use of the Dirac equations to get

7-‘2’ = ; c [RPl9 4 )XIdP29 A2)

A?

x fi(p2, A2)x2u(p3, A3) x (1 + p2/m2)

+n(pl, Al ),k’lu(p2, A2) x fi(p2, h2),k’2U(p3,A3)

x (1 -p2/m2)1. (19)

This example can be generalized to any number of insertions and shows that the factors (#i + ,!.Q) can

be easily eliminated, reducing all fermion lines essen-

tially to products of T functions:

TA,A,(PI;~,c;P~) =~~P,~~I)xWP~&> (20)

defined for any value of pf and pi.

It is convenient to use the spinors u(p, A) and u(p, A) defined in Eqs. (5), (13). With this choice, to which we will adhere from now on, the T functions (20) have a simple dependence on ml and m2 and as a consequence the rules for constructing spinor lines

out of them are simple. Every T function has in fact an expression of the following kind,

TA,/\? (PI; 7. c; ~2)

= ds &&-%Tw~(PI; rl, c; ~2)

= AA,A~(PI;~LC;P~) + MlB~jnz(pl;v,~;~2)

+ M2CA,Az(p1;771;1,C;p2)

+ MIM~DA,A~(PI; 7, C;p2), (21)

where

Mi = +mi if Ll(pi, Ai) = u(pi, Ai),

= -m; if U(p,, Ai) = U(pi+ Ai). (22)

The functions A, B, C, D turn out to be independent of ml and m2 and of the u or u nature of u(p,, A1 )

and U(pz, AZ). This implies that the four T func-

tions U(PI, 4 ),w(P~, Ad, U(PI, AI )xu(P~, Ad,

~(PI, AI ),v(P~, Ad and ~PI, AI )xu(P;?, A2) can be computed simultaneously, speeding up the calculation considerably.

The functions A, B, C, D corresponding to a scalar and a vector insertion can be easily evaluated: using Eqs. (3),(5),(13),thespinorproductscanbewritten

as

u(p2, A2)u(p1, AI > = 1

4vGGXF-G

x ($2 + M2){(1+ AlA2) - (AI + A2)y5

+b[(Al - A21 - (1 - AIA~Y~I}~~O($, + MI).

(23)

Multiplying to the right by AS or xv of Eqs. ( 15),

( 16) , taking the trace and with the help of Eq. (2 1) , one gets

A:-=~GYP, kl.p2-ko.p2k1.pl

- ie (~O~h,Pl,P2)1,

AS - -+ - G[ -ko . p1 kl . p2 + ko . p2 kl . p1

- i~(ko9h,pl,p2)l,

$+=crko.p2, Bf_=clkoq2,

Cf+ = CI ko . PI, C!_ =c, ko.pl,

A~+=c,[-ko.rlpl.p2+ko.p] 71.~2

+ko.p2rl.p] +i~(ko,rl,pI,p2)1,

A!_ =cl[-ko.rlpl .p2+ko.p] v’p2

+ko.pzrl.p~ --i~(ko,rl,pl,p2)1,

BY- =cr[ko.rlkl.p2-ko.p2kl.rl

- i~(kotkl,rl,p2)lT

BV+=c,[-ko.77kl.p2+ko.p2k].rl

-i~(ko,k1,v,n)l.

C,V_=c,[-ko.r)kl.p]+ko.p] kl.7

+ ic (ko, kl, rl,pl) I,

C’, =cJ[ko.q kl .PI - ko.pI kl ‘q

+ie(ko,kl,qp1)1,

DL = cl ko .rl, D”__ = c,. k. v. (24)

A. Balkstrero, E. Maina /Physics Letters B 350 (19%) 225-233 229

All functions A, B, C, D for a single insertion not reported in the preceding list are identically zero.

The function E is defined to be the determinant:

1 p” q” r” so 1

E (P, 4, rr s) = det p1 41 g s1

p2 q2 r2 s2 . (25)

p3 q’ r3 s3

4. Spinor lines

We will show in this section how to compute recur- sively the functions

(P2 . ko) (P3 * ko) * *. h * ko) (26)

from which the T(“) themselves can then be immedi- ately obtained at the end of the computation, dividing by the appropriate factors.

Let us denote withf, A, B, C, D the 2 x 2 matrices whose elements are TA,A,, AA,A~, BA,A,, CA,A~, DA,A~.

With this notation, making use of Eqs. (21) and (20)) Eq. (19) reads

?2’(1,2,3) = &{[A(1,2)+M,B(1,2)

+mzC(1,2) + M1mzD(1,2)1

x (1 + /-52/~2) x [A(% 3) + mzB(2,3)

+ M&(2,3) + mzMsD(2,3)]

+ lA(1,2) + M1B(1,2)

-mzC(1,2) - M1mzD(1,2)1

X (1 - ~2/m2) X [A(2,3) - mzB(2,3)

+ MsC(2,3) - &&D(2,3)]}, (27)

where we have used the shorthands ( 1,2) and (l,2,3) for(pl;rll,cl;P;?) and (~~;71~,~~;~2;77~.c~; p3 ) , respectively.

Elementary algebra shows that T(2) has again the same dependence on the external (possibly unphysi- cal) masses as in (21):

?2’(1,2,3> =Ac2’(1,2,3) +M,Bc2)(1,2,3)

+&C(2)(l,2,3)+M,MsD(2)(l,2,3), (28)

A’2’(1,2,3) =A(1,2)[A(2,3)+~2B(2,3)]

+C(l,2)[~2A(2,3) +p@(2,3)i,

B’2’U,2,3) = B(1,2)[A(2,3) +,~2B(2,3)]

+D(l,2)l~A(2,3) +p;B(2,3)1,

C’2’(1,2,3) =A(1,2)[C(2,3) +,~2D(2,3)]

+C(l,2)[~2C(2,3) +p22D(2,3)],

Dc2’(1,2,3) = B(1,2)[C(2,3) +/&D(2,3)]

+D(l,2)[~2C(2,3) +&D(2,3)]. (29)

This implies that A (2), Bc2), Cc2), Dc2) can be rein- serted in an equation like Fq. (27) to give the y func- tion T(3) corresponding to a fermion line with three in- sertions, and so on. So one can generahze Eqs. (27)- (29) by induction: every 8:(‘) turns out to be of the form

f(i) = A(0 + M1B(i) + Mi+lCU) + MIMi+,D(i)

(30)

A(‘fj) = A(‘) [ A(j) +,&j)]

+ Cci) rpiA(j) + p?@] 1

B(‘+_d = B(i) [A(j) + @(j) ]

+ DC’) [ piA + p?B”’ ] I

C(i+j) = A(‘) [C(j) + ,@(j) ]

+ c(i) [pic(j) + &$j) ] ,

D(‘+j) = B(‘) [C(j) + @(j)]

+ D(i) [pic(i) +pfD”‘]) (31)

whichbuildupT(‘+j)(l,...,i+j).InEq. (31) ,LQ and pi are the mass and momentum of the propagator which connects the left (i) insertions with the right (j) ones.

It should by now be evident that the evaluation of any spinor line can be performed computing the A, B, C, D matrices relative to every single insertion and combining them with the help of Eq. (3 1) until one gets to the final T(“). It is important to point out that the unphysical masses mi (i = 2, . . . , n) do not appear in

230 A. Ballestrero, E. Maina / Physics Letters B 350 (1995) 225-233

Eqs. (29) -( 3 1) : only their squares p? do. Therefore, as anticipated after Eq. ( 13), the identical final result

is obtained using any of the two determinations of mi =

fi. The same conclusion can be drawn for Jx from Eq. (26) and the fact that the expressions (24) are independent of square roots.

5. 7 matrices

It is convenient to cast the previous formulae in a

matrix notation. We drop the superscripts (n) when

not necessary. Every piece of a spinor line as well as every com-

plete spinor line (for all fermion polarizations !) with IZ insertions is completely known when we know the matrix

(32)

The law of composition of two pieces of spinor

line, connected by a fermion propagator with four- momentum p and mass p, whose matrices are

r,=(;: ;;), 7*=(;; Z)> (33)

is simply (cf. Eq. (31))

We will sometimes indicate the above composition as

follows,

7=7-l 072.

If we call ri the matrix

(35)

(36)

corresponding to the propagator of four-momentum pi and ri the matrix associated with the ith insertion of Fig. 1, the T matrix of the whole spinor line can then be computed as follows,

Explicitly written as 4 x 4 matrices, r and ri are

A++ A+- C++ C+-

7= A-+ A__ C_+ C-w_

B++ B,- D++ D+- B-+ B__ D-+ D__

I1 O Pi O\

(38)

From the expressions of A, B, C, D given in Eq. (24)) one can see that the r’s of a single vector or scalar insertion have the particular form

(39)

When the insertion corresponds to a W boson, only

A!_, C!,, By-3 D;+ are different from zero. For practical computations we have implemented a set of routines which automatically write lines of Fortran code both for the expressions of the A, B, C, D func- tions and for their combination to form whole spinor lines. These routines of course avoid unnecessary and time consuming multiplications by the zeroes of the r

matrices.

6. Massless spinor lines

When one has to deal with massless spinor lines, all the formulae given in Eq. (24) remain valid. But in this case the fact that all pi’s as well as I)Z~ and m,+l are zero leads to significant simplifications. For example the rule (3 1) for combining pieces of spinor line is now

A(‘+j) = A( + pfC(i)B(j),

B(i+j) = B(‘)Afj) + p?D(‘)B(j),

cCi+j) = A(i)c(/) + pfC(i)D(j),

A. Ballestrero, E. Maina /Physics Letters B 350 (1995) 225-233 231

D(i+i) = B(i)c(I) +pf~(i)~W (40)

and the ri matrices (37)) (38) become diagonal. Moreover, in order to compute the whole spinor

line with n insertions it is not necessary to know the

whole r(“) matrix. It is clear from Eq. (30) that only A(“) is needed. Therefore, if one for instance multi-

plies recursively the r matrices of the single insertions starting from the left, at every single step one needs

to compute only A and C. In fact, from Eq. (40) with i=n-l,j= 1, one sees that B(“-‘) and D(“-‘) are not needed to compute A”. With i = n - 2, j = 1 one verifies that Bcne2) and Dcne2) are not needed

to compute A(“-‘) and C(‘+‘), and so on. Had one

started from the right, only A and B would have had to be calculated for every product.

In most theories, like, e.g., the standard model, one has to consider only vector and axial-vector couplings to massless spinor lines. This means that one has to compute only 7” matrices (39) for every insertion.

Combining two matrices of this type, one still gets a matrix whose only elements different from zero are on the two diagonals and therefore the r matrix of any piece of massless spinor line is “cross-diagonal” for

these theories.

7. Combining spinor lines. A simple example

As a simple example of how to use in practice the above method, in this section we consider the process e +e- --+ tfg. For a more complicated application, we

refer the reader to Ref. [ 131, where the calculation of the reaction e+e- + W+W-bh is presented in detail

together with its phenomenological relevance for top and Higgs physics.

In the following, whenever in the argument of a r matrix or of one of its elements there will be an index

/* in place of a “polarization vector” 7, we imply that the components $ have been taken to be $’ = g”‘.

In fact every r matrix satisfies the relation

=rlir*7(P1;7)1,...;cL,...,p,) (41)

and of course similar relations can be written for its A, B, C, D components. In case of just one insertion we have for example

t A e+ 9

7 e-

i

Fig. 2. Diagrams for the process e+e- --+ ffg.

4+h.ww2) =q.a A;+(p,;p,w;?).

From Eq. (24) it is immediately seen that

(42)

=c,(-pl.~:!k~+ko.pl~~+ko.p2~~

- icpvpr k,” pf ~20) , (43)

with ~0123 = 1.

The process efe- ---f ttg is described by the dia-

grams in Fig. 2. In this case all insertions on spinor lines correspond to vector particles. We will indicate

the four-momenta of the particles with their names, so

that s = (e+ + e-)2. We will also denote with cf” the couplings of a generic fermion f to a vector particle V.

One has to choose ko and kl and the form of the polarization vectors ~8 for the gluon in terms of his

momentum. One can for instance use ko = ( 1 , 1 , 0,O) ,

k, = (O,O, l,O) and the real polarization vectors proposed in Ref. [lo] or normal helicity eigen- vectors. One then starts computing with Eqs. (24) Ajj(e+;p,C’Y;e-) and Ajj(e+;p,c”;e-), with j (= +, -) the polarization of the electron. With them one constructs the “polarization” vectors

Y ?lj/l =

Ajj(e’; /L, PY; e-)

s

z rtjltL =

Ajj(e+;p,P’;e-)

s-rn2,i-irzmz . (44)

Using again Eqs. (24), it is easy to compute the tau matrices relative to the insertions in the upper (u) and

232 A. Ballestrero, E. Maina /Physics Letters B 350 (1995) 225-233

lower (d) part of the line:

r”,[j] =7(t;7$,cry;-i-g),

$[j] =7(t;qf,P;-i-g),

Ti[i] =7(t;&c’g;t+g),

+j1 =7(t+g;77jY,c’YA

&[jl =7(t+g;~;‘CfZ;o,

7i[i] =7(--f-g;$,c@;i)

and then the sum of ry and rz matrices

(45)

+[jl =~~[jl +$[jl,

7tz[jl =$Ul +4[jl. (46)

The final r matrices are obtained just composing with thelaw (29), (34) ~!$~[j] withri[i] and $[i] with

r;z [jl :

~l[i,jl =~5)z1jl~7~[il,

72[i,j] =~i[i] l ~t~[j]. (47)

From r[ i, j] the polarized amplitude is then obtained

with the help of Eq. (30). Indicating with 1 the polar- ization of the top and with m that of the antitop, one

has

Fl [i,j,I,ml = Alf,lL jl + mJ~f,[i,jl

- mKlf,[i,jl - m:Dlfm[i,jl,

%[i.j,E,ml = A2fm[i, jl + mJh,[i, jl

- d2f,[i,jl - m;l&fm[Ljl

and the amplitude

(48)

Ampi i, j, 1, m] =

( Tl[i,j,l,ml ’ (-i-g) .ko ((-f-g)2-mmf+ir,mt)

%[i, j,Z,ml

+ (t+g).ko ((t+g)2-mf+irtm,) ’ > (49)

8. Comments and conclusions

The method we have demonstrated has been tested in several computations of physical amplitudes, both

unpolarized and partially or fully polarized. Our re- sults have always been in perfect agreement with those obtained in other formalisms.

Even if we use for external fermions the same

spinors as in Ref. [7], we do not also have to use

the polarizations they suggest for vector particles. In effect, especially for massive vector particles, we often use the real polarization vectors suggested in Ref. [lo]. If one defines ke = (1,0,0,-l), ki =

(0, 1 , 0, 0). the massless spinors of Ref. [ 71 coincide with those of Ref. [ lo]. This implies that with this

choice of ku and kl and the polarization for vector particles of Ref. [ lo] our results must agree, in the

limit of spinor masses going to zero, with those ob- tained with the method of Ref. [lo] for every single diagram and every polarization of external particles.

This has been checked for several examples and it may be used as a valid test of the correctness of the results.

We believe that our method has some advantages with respect to those in Refs. [7,10]. In Ref. [lo] it is proposed, using the particular form of the chiral

representation for y matrices, to decompose the cal- culation of every spinor line in a sum of terms of the

type x+41,. . . , &,y, where & = a0 f u - a and up are either momenta of the propagators or “polariza- tion” vectors. When all up’s are real (and hence only when all “polarization” vectors correspond to exter- nal particles and not to other pieces of diagram in- serted on the fermion line), the &‘s are written as

(~0 T bl>x+(a),d(a) + (a0 f bl>x-(a>xt_(a>, where xk (a) are two-component spinor eigenstates of the helicity u - ala/. In this way the computation of spinor lines is reduced to spinorial products [ 15 1.

This expansion of the #h’s already used eigenstates depending on the four-momenta of the propagators, as we do. There are however important differences: the

decomposition of the C’s is performed also for vectors corresponding to external particles helicities, while we decompose only the propagators. We have the pos- sibility of using complex polarization vectors, which have been instrumental in obtaining extremely simple expressions for massless helicity amplitudes. Flexibil- ity in the choice of polarization vectors is a desirable feature also when massive fermions are involved. In Ref. [lo] the complete set is constructed with two- component spinors which are eigenstates of the he-

A. Ballesnero, E. Maina /Physics Letfers B 350 (1995) 225-233 233

licity, while our complete sets are formed with four- component spinor eigenstates of spy@. It is just the fact that we employ four-component spinors, that we do not separate the various terms of the spinor lines, and that we use eigenstates of 4 x 4 matrices a,yp when a corresponds to the four-momenta of the spinor propagators, that allows us to treat the massive case in a compact and economical way. In such a fashion we can easily combine different diagrams with common intermediate fermion legs and treat the insertion of parts of the diagram on a spinor line as simply as the insertion of an external particle with complex polar- ization vector. A similar modularity can be achieved in the formalism of Ref. [lo] when one multiplies spinors and matrices, without using spinor products. Compared with this latter approach, our method has only one T matrix to compute for each insertion, in- stead of one matrix for each vertex and one for each propagator. Moreover, we can freely choose the aux- iliary vectors ko and kl which can be useful both in simplifying the expressions generated in intermediate stages and as a test of the final result.

In comparison with the method of Refs. [7,12], our formalism is much more compact. It avoids the proliferation of terms generated by the expansion of the momenta flowing in fermion propagators in terms of the external momenta. Even with respect to the more efficient method suggested in Ref. [ 81 for treat- ing internal propagators, we have a smaller number of terms. The relationship between the matrix elements of u-type and v-type spinors (2 1) , (22), (30) leads to simpler expressions than the introduction of additional auxiliary vectors, especially when long fermion lines are present and the insertions do not all correspond to external particles. As a consequence, it is much easier in our formalism to keep track of partial results and to set up recursive schemes of evaluation which compute and store for later use subdiagrams of increasing size and complexity. Our method is also more flexible in the choice of the polarization vectors for external vec- tor particles and one can avoid the extra integration needed to obtain the correct sum over polarization

of the formalism of Refs. [7,12]. It allows one to directly compute cross sections with polarized W’s and Z’s for any desired polarization.

In conclusion, we have shown that it is possible to insert in spinor lines completeness relations in order to diagonalize the operators $ of the propagators of mo- mentum p. This is particularly convenient when one uses the spinors [ 71. We have presented the formal- ism necessary to perform actual calculations, both in the massless and in the massive case.

References

[ 11 M. Jacob and J.C. Wick, Ann. Phys. 7 (1959) 404. [2] J.D. Bjorken and M.C. Chen, Phys. Rev. 154 (1966) 1335;

0. Reading-Henry, Phys. Rev. 154 (1967) 1534. [3] P. De Causmaecker, R. Gastmans, W. Troosts and T.T. Wu,

Phys. Len. B 105 (1981) 215; Nucl. Phys. B 206 (1982) 53; EA. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans, W. Troosts and T.T. Wu, Nucl. Phys. B 206 (1982) 61; B 239 ( 1984) 382; B 239 (1984) 395; B 264 ( 1986) 243; B 264 (1986) 265; for a review, see R. Gastmans and T.T. Wu, The ubiquitous photon (Addison-Wesley, Reading, Ma, 1990).

[4] M. Caffo and E. Remiddi, Helv. Phys. Acta 55 (1982) 339. [5] G.R. Farrar and E Neri, Phys. L&t. B 130 (1983) 109. [6] G. Passarino, Phys. Rev. D 28 (1983) 2867; Nucl. Phys.

B 237 (1984) 249. [7] EA. Berends, PH. Daverveldt and R. Kleiss, Nucl. Phys.

B 253 (1985) 441; R. Kleiss and W.J. Stirling, Nucl. Phys. B 262 (1985) 235.

[8] R. Kleiss and W.J. Stirling, Phys. L&t. B 179 (1986) 159. [9] J. Gunion and Z. Kunszt, Phys. Lett. B 161 (1985) 333.

[ lo] K. Hagiwara and D. Zeppenfeld, Nucl. Phys. B 274 (1986)

[ 111 i Xu, Da-Hua Zhang and L. Chang Nucl. Phys. B 291 (1987) 392.

[ 121 C. Mana and M. Martinez, Nucl. Phys. B 287 (1987) 601. [ 131 A. Ballestnxo, E. Maina and S. More& Phys. Lett. B 333

(1994) 434; A. Ballestrero, E. Maina and S. Morel& Phys. Lett. B 335 (1994) 460.

[ 141 A. Ballestrero, E. Maina and S. Momtti, Nucl. Phys. B 415 (1994) 265.

[IS] R. Kleiss, Nucl. Phys. B 241 (1984) 61.