HELICITY WAVE-FUNCTIONS FOR MASSLESS AND MASSIVE SPIN-2 PARTICLES

PHYSICAL REVIE%D VOLUME 44,NUMBER 12 15DECEMBER 1991

Helicity ^mave functions

for massless and massive spin-2 particles

D.

Spehler

Instituto de Fisica Teorica, Universidade Estadual Paulista, Rua Parnplona 1$$, 01/05 ^Sao Paulo, Brazil and Universite Louis Pasteur, Institut Universitaire de Technologie, 8rue St. Paul, 67800Strasbourg, France

S.F.

Novaes

Instituto de Fisica Teorica, Universidade Estadual Paulista, Rua Parnplona 1$$, 01/0$ Sao Paulo, Brazil (Received 24 3une 1991)

Starting from general properties of a spin-2 field, we construct helicity wave functions in the framework ofthe Weyl —^van der Waerden spinor formalism. We discuss here the cases ofmassless and massive spin-2 particles.

A very eKcient technique for evaluating multiparticle amplitudes is achieved _by means

of

the helicity method.

It

turns out to be much more simple to calculate helicity matrix elements rather than evaluating the unpolarized amplitudes by summing the squared invariant amplitude over all possible spin states

[1].

This method was improved [2, 3]with the use of the Weyl —^van der Waerden

SL(2, C)

spinors conjointly with the spinor calculus [4]. This ^way

of

dealing with ^he- licity amplitudes is capable of eliminating the lengthy p—matrices algebra by unifying the descriptions of Dirac spinors and Minkowski four-vectors. With this treatment it was feasible to obtain exact expressions for amplitudes containing a large number of external legs

[5].

In the present work, we apply the Weyl—_{van der Waer-} den spinor method to spin-2 bosons. Making use ofgen- eral properties ofspin-2 helicity states, we construct he- licity wave functions in spinorial form for massless and massive particles. Our results have a smooth interface

to

the method ofWeyl —van der Waerden spinors applied for spin- ^—

'

_{and spin-1 particles}

[3] and also for spin--

2 2

fermions

6].

A spin-2 field is described by a symmetric tensor

4""

which satisfies the Klein-Gordon equation. The tensor C'"" _{is traceless}

(4"„=

0) ^{and has null} four-divergence

(B&4"" = 0).

^{The solution of the} Klein-Gordon equation for the spin-2 field

of

momentum p, ⁱⁿ the absence

of

a source, is given by the plane-wave expansion

The following normalization condition for the tensor

E"

can be adopted:

(p9)

— =

e _p&rIa

=

_{p Va (pV)}

=

e pj, iI

=p

^ilo'

where c is the metric spinor

Any four-vector

p" = (p,

^p') can be expressed as

+ltt ^~+Pab~_ba^.

where

P

_b ^——o.

"

_{p&. Therefore,} the Lorentz invariant p~p~ can be written as

p"p„= ~Pi„P= 2(P, P)

^{. .}

(6)

[8"'

_(k,A)]^t

_F„„(k,

A')

=

_6),g

In order

to

apply the Weyl—_van der Waerden method

to

spin-2 particles, we briefly summarize some results on spinor formalism. We use the conventions of Wess and Bagger [7],^with

diag(g„„) = _(1, —

_1,

—

_1,

—

_{1), vector in-}

dices are denoted by Greek letters

(p,

^{v, . ..}

=

_{0, 1,}2,3) and spinor ones are denoted by Latin letters

(a,

^{li,. . .}

=

1,

2).

We define the inner product between two spinors as

4&""(z)

=

d

p) [F"'(p,

A)

exp(+ip„z") +

^H.

^c. ],

_case,^We_we^startrequire the invariance^{by analyzing massless}under^spin-2the gauge transfor-^{particles. In this} mation

where

6"'(p,

A) is the polarization tensor corresponding

to

states

of

definite helicities _{(A) which} should satisfy

8""(p,

A)

= 8'"(p,

A),

P„E""(p,

A)

= P„Z""(p,

A)

=

_0,

(1a) (1b)

where A" are four arbitrary functions. This gauge trans- formation is able to decouple the vector and scalar parts

of

the tensor field [8].^We end up with only two physical helicity states,

i. e.

, A

= _+2,

which satisfy

Z"„(p,

A)

=

0 .

(1c) [&"'(p, +2)]l = 8""(p, _~2)

.

3990

1991

The American Physical Society

BRIEF

REPORTS 3991 Using

Eq. (5)

^{for each} vectorial index of

E"",

we can

write the helicity tensor as

~""(», +2) = (2~"'Eb')( , '~"-"Fd _),

_and the helicity tensor can be written as

F""(p —

2)

=

(-'o.

"

_G

₎₍ 'o"-"Hd, ),

(9) g

^pe_(p

+

2)

—

_&gab ^vcd—

_g

(1oa)

where

E, I", G,

^{and H are} arbitrary spinors to be de- termined. Equat,ions

(la)

^—

(lc), (2),

^and

(8)

impose some constraints on these spinors, For instance, the conditions

(la)

^—

(lc)

^require

Eab

lcd

@cdgab

epe2

—

_ege1

—

_Re2e1

e1e2 el^e1 ~ele1

—

_ie1e2

_ie]e1 —

_e1e1

e2e2

—

_e2e1

—

_ie2e1

—

ere2_e1e2

—

'Ee]e₂

Using the gauge freedom Eq.

(7)

ⁱⁿ the momentum space, i.

e.

, 6 "~

~ 8"" —

_i(p"A~

+ p"A")

we can choose A" _such that

P' _E,

_bI",d ^——

_P

_G

_-II, „=

^0,

(lob)

,e1e2 g e1e2

Z

2

'

₂

with this choice we have

E. r" ⁼ _G. ^{H" =}

O,

(loc)

whereas the normalization and Hermiticity conditions de- ITland

(lod)

(Eab)$(Fcd)f

^GbaHdc ₍loe)

P

_b

_=u'Sb

We search for an explicit form of

E, ^I",

^{C~, and H writing}

E,

b

= _e, e', _F,

b

= f, ^f',

Gb — —

_g g'-,

Hb —

_h

h',

where g, g', ..._,h, h' are arbitrary spinors. The relation

(loe)

constraints e

=

g', _g

=

e',

f ⁼

^{h', and h}

^{= f'}

The relation

(lob),

^t,aking into account Eq.

(11),

^yields

four possibilities: ^e

=

_{p and}

f ⁼

^{p; e'}

⁼

^{p and}

^{f' =}

^p;

e

=

_p and

f' =

_{p; or e'}

=

_p and

f ⁼

^p. Nevertheless,

Eq. (10a)

makes only the first, two possibilities acceptable with the additional constraint: ^e'

= f'

in the former case and e

= f

^{in the later one. In order}

^to

^{guarantee that}

F"'(p, +2)

corresponds to a positive-helicity state, the only possible choice is e'

= f'

_{=. p and} e

= f, ^i.e.

^,

8""(p, +2) = 2o" 'o"'"p,

_ebp,^e._d,

8""(p, —

2)

= 2cr"'a"'"e,

_pbe,pd .

We can verify that

8"'(p, +2)

corresponds indeed to a massless spin-2 particle ^with positive helicity

[9].

^Inorder

to

do this, let us assume that the particle is traveling in the z direction,

i.e.

,

p" = (E,

^0,0,

E),

^where

E

^{is its}

energy. In this case the momentum spinor is

We make use

of

the fact

that,

for

a

massless particle, the spinor

P

bassociated with the particle momentum ac- quires asimple form. Since _p

=

_z

_(P, P) =

^0,

^P

bshould bewritten asthe product

of

two momentum spinors:

i.e.

_,

(0

^{0 0} 0~

O~ io

8"'(p, +2) = — _eie,

&0 o o

0)

We can see that

if

we perform a rotation about the z direction,

i.e.

,

8' "" _{= R" (0, 0)}

R"&(0,

0)$

^{P we obtain '}

f' "' ₌ exp(+2ig)fi'"

which shows that

F""(p, +2)

^is a helicity

+2

^wave function.

From

Eq. (10d)

we obtain the normalization condition

~(pe)~

=

4, and consequently (pe)

= _~2exp(iu),

where

u

is an arbitrary phase.

If

we define the spinor

r

by

e

= _exp(iu)~2

(pp) ^'

we can write the final result for the helicity states

of

a massless spin-2 particle as

E""(p, ₊₂₎ = -oi""cr"'d exp(2i~).

(p~) (p~) ^'

(15) F""(p,—

2)

= ^— ai'"o '"exp( — _2iu)

(p&)t

4»')'

We see that

r

is a gauge spinor. For instance, if we make two different choices

of

^t,he spinor r

(r

^and

r')

we

obtain

~bade(~ )

—

~bade(P)

—

2(PdcAha

+

^PbaAdc) with

2i(t'i.

₎

(p

'}'(p )'

x ((p~')rbp-

+ , ^'

(&'&)_pbbs

')—

If

we take into account the results

of

Ref, [3] for helicity

of

a massless spin-I particle

[s" (+1)]

^{in spinorial}

forni, we notice that the helicity states Eq.

(15)

^{can be}

written as

6""(p, +2) = z"

_(p,

+I)z"

_(p,

+I).

Equation

(15),

^{with w}

=

_{0, agrees} with Ref. [10]^where the graviton helicity states are considered.

We can write the sum over the helicities as

3992

BRIEF

REPORTS

4

.

^Pb

^.

^Pg

^{. PI gPb+}

^(P

^),

pp

which is equivalent to the vectorial form

%=+2

)

&

(~~a

^p~P

+ ^~1

^P

^~~a

^p~~

^paP) (16)

where

p

+p

= —

_g

+

p ^0p

We can see that the sum over the polarizations depends on the gauge spinor ^r and that only the physical states

[11]

^contribute

to Eq. (16).

Let us now treat the case of a massive spin-2 particle.

As in the massless case, we give aspinorial description of the polarization vectors, by writing the following decom- positions for all the physical helicity

states:

Notice that the pairs

of

spinors (A,

B) (I, J)

^{play sim-}

ilar roles. Therefore, ⁱⁿ what follows, we will concentrate

just

on the first pair.

We can take advantage

of

the simple form

of

alightlike vector in spinorial notation,

Eq. (11),

by writing the spin- 2 massive boson momentum k as asum

of

two momenta p and q, with p

=

_q

=

0,

i.e.

,

PP

—

_pP

+

^qP

⁽¹⁸⁾

with

2(I' I') ⁼

l(pq)l'

=m'

and therefore,

(pq)

=

_p q

=

m

exp(iu)

.

We can expand the pair (A,

B)

^{in terms}

of

the momen- tum spinors p and q:

with

kl"k„=

^m2

=

2P4q„. In spinorial form, Eq.

(18)

becomes

I&

b

— P

_b

+

Q

b:

^PaPb

⁺

^qaqb

8""(k

₎

+2) = _— ^{— 1-} "A ^1- ""B—-

2 ^bi 2 di)

8""(k,+1) ⁼ 1- o""Cb, 1- ^— o"'"Dg,

_,

^—

^.

8""(k,

₀₎

= 1-

^cr~abEb

^— 1-- o"4Fg;—

_,

(17)

(nip +ao2qa)(ospb

+

o4q, )

B

b

—

_(bypa

+

b2qa)(bspb

+

bqqb) .

(2o)

1 ^. 1

F"'(k ^— ₁₎ = ^—

^o.

"'

_Gb

cr"~Hg—

2 2

F""(k +2) = ^1-

^oPabI6Q

^{— 1-}

^o.v—cd

J

2 ^dc

If

we substitute this expansion in Eqs.

(la)

^—

(lc)

^{we ob-}

tain the following constraints: a3 ^——a2, aq

— — _—

_a~,

b3

—

_b2, bq

— — —

_{bq, and aq 62}

— —

a2bq . Therefore, A _b

and

B,

& become

2 2

A,

b

— —

_ma,

l

cb(k, 0—

₎

₊ c,

b(k,

+I) ^— —

_q ^E b(k,

— _I)

I

=

p B

b )

(21)

where e (k,A) are the helicity spinors ofa massive spin-1 particle

of

^{momentum k} [6],

i.e.

,

s b(k,

+1) =

_q,

_p,

,

b-„(k,

0)

= — _(p,

_p,

^—

_q,q,

_),

s _b(k,

—

₁₎

=

_p~qg

m (22)

We obtain similar results for the remaining pairs of spinors of

Eq. (17). ^If

^{we take into account} the normalization condition

Eq. (2),

^{we obtain} the normalized helicity wave function for a massive spin-2 particle ⁱⁿ the Weyl—_{van der} Waerden notation, up to a phase factor

t"'(k, +2) = -o""o-"'" _p,

_qbp,qg, . . m

~" _(k, +1) = -~""~-"

_[(O'Pb

^—

_q-qb)P'qd+ P'qb(p'P~

—

_q'q~)],

4 m

1'g'd2

8"'(k, ₀₎ = ^— o"' o"'"

_{[(p pb}

—

q, qb)(p;pa

—

. _q&qz.)

—

_{p q q Pa}

—

_{qapbP qz]}

m~ 6

8""(k, ^— 1) = ^— -o"

rr

',

1 ^[(popb

^—

^{qoqb)q'p~+ qoPb(p'p~}

^—q'q~)l.

8""(k, ^—

2)

= -o""o ^'"

2qapbqcp~

4 ^rn2

(23)

BRIEF

REPORTS

where

(24)

k k )C"

= ^— g"

m2

We can verify that the sum over the polarization states, in vectorial form, is given by

Our results,

Eqs. (15)

^and

(23),

can be very useful in the evaluation

of

invariant amplitudes involving massless and massive spin-2 particles. In the first case, we can take advantage

of

the freedom in the choice

of

the gauge spinor r toeliminate some Feynman diagrams contribut- ing

to

agiven process. In the massive case, we can make convenient choices

of

the momenta p and q, defined in

Eq. (18), ^to

reduce the number

of

contributions.

This work was partially supported by Conselho Na- cional de Desenvolvimento Cientifxco e Tecnalagico, CNPq

(Brazil).

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