CAN WE MEASURE THE R-PHI(2) COUPLING PARAMETER AT TREE-LEVEL SCATTERING

PHYSICAL REVIEW

0

VOLUME 51,NUMBER 2 15JANUARY 1995

Can

we

measure

the RP2

coupling

parameter

at

tree-level scattering'?

A.

J.

Accioly,

D.

Spehler, 2

S.

F.

Novaes,

S.

F.

Kwok, and H. Mukai

Instituto de Eisica Teorica, Universidade Estadual Paulista, Rua Pamplona 2$$, OI$05-900 Sao Paulo, Sao Paulo, Brazil Universite Louis Pasteur, Institut Interuniversitaire de Technologie, 8rue StPaul, 67800Strasbourg, France

(Received 12September 1994)

The lowest order invariant amplitudes for the 2 + 2processes concerning the action S[g,P]

=

J

d

xg

9[2—

R/e

+

—

_(g""8„$8

_P+

_ARP

)] are computed. It is found that these results do not depend on the value ofA; therefore, it is impossible to measure the RgP coupling parameter at tree-level scattering in this case. It is also shown that the theory described by the above action is equivalent at the classical level to Einstein's theory with a massless minimally coupled scalar 6eld provided that 1

+

(Am _{P /4)}

)

0. Some consequences for cosmology and black holes are discussed. PACS number(s): 04.60.

—

m, 04.20.Cv, 04.50.

+h

It

has been suggested by various authors that the ac-tion for gravity should contain, in addition

to

the Ein-stein action, certain nonminimal functionals

of

the scalar field. There are many reasons for these suggestions: the necessity

to

soften the divergences

of

the stress tensor

_[1],

the possibility

of

a

theoretical explanation

of

Mach s prin-ciple [2],the incorporation

of

the spontaneous symmetry-breaking mechanism into gravity

_[3,

the construction

of

nonsingular models for the _{Universe [4],}the need

to

main-tain renormalizability in quantum field theory in curved space [5],and so on [6].

The candidates for such nonminimally coupled actions contain, in general, a term

of

the form ABgP, which in-cidentally is the only possible local term involving

a

di-mensionless coupling between the scalar field

_P

and the curvature scalar

B

[5]. Accordingly, here we shall con-centrate our attention ontheories described by the action functional

is no a priori reason why it could not have any other real value. So, which value

of

A should we use in our ordinary calculations? Toanswer this question

a

scheme must be devised which, at least in principle, allows us

to

gain

a

feel for this parameter A. Since the 2

~

2 pro-cesses concerning action

₍₁₎

for which A will very likely enter the invariant amplitudes are only the scalar-scalar and graviton-scalar scattering, we shall calculate in the following the lowest order invariant amplitudes for these processes.

Of

course, there isno need

to

consider the an-nihilation

of

scalars into gravitons since, as is well known, the invariant amplitude for this process can be promptly inferred &om the one concerning the graviton-scalar scat-tering.

The usual procedure in quantum gravity for obtaining the vertex functions in the absence

of

fermions isto

start

with the action functional describing the matter fields and to write the metric tensor

_g„„(x)

as

g„„(x)

=

g„„+

rh„(x),

(2)

with K2

=

_32aG

_{in natural units. The Ricci tensor} is de-fined by

R„„=

—8

I'„„+,

and the metric convention is

g„„=

diag(l,

—

1,

—

1,

—1).

Generally the coupling parameter A in

Eq. (1)

is cho-sen

to

be zero (minimal coupling) or

—

1/6 (conformal coupling) for the sake

of

simplicity. Nevertheless, there

I

where

_rl„„

isthe fiat space metric and

_h„„(x)

isthe gravi-ton field. Consequently, the Feynman rules for scalar-graviton interactions are obtained &om the action for

a

gravitational nonminimally coupled scalar field:

Ss

—

—

d4x

—

_g—

g"

„—

_P

m'

'+AH

',

expanding around

fat

space using

Eq.

(2).

This leads

to

Sg

—— _d

_z

—

₆

g""8„—

_m

—

2h"

„+

_{2A Clh}

—

_„h

"

4

[(6

—

2hphP)(rl""B„QB„Q

—

m _{P )}

+

4(2h"

6

"

—

h,

h"")8„$0

_g

+A/ (16k

prl""B„Oph

„+

80

h pO„hp

—

8h pB Bph

—

8h p H h p

—

88

hrl"

_B„h„

hq"

8

h~s„h

—

_p+4g"

_„s„h

_~sph

_„+4hclh

—

_4hs

_s„h

_"+2n

~s

_hsph)]I,

where the action forthe &eescalar field has been omitted. From the previous expression the Feynman rules for the elementary vertices may readily be deduced. These are shown in

Fig.

1.

Let us now analyze the gravitational scattering of identical massive scalars with arbitrary BP2 coupling. The Feynman diagrams for the process

_{P(pi)P(p2) -+}

(j5(p3)P(p4) in lowest order are displayed in

Fig.

2, and

932

BRIEF

REPORTS 51

(p,v)

—'"_[—_g„„(k2~_k3+m2+_2Akl2)

+k2pk3v +k3gak2v +2~klpklv]

The evaluation

of

the invariant amplitude for scalar-graviton scattering israther involved. The corresponding Feynman diagrams are shown in

Fig.

3.

We made use of the formalism

of

Refs. [8,9]

to

calculate the helicity amplitudes in this case, obtaining

k

',

k4

'4 ((k + ')(m~i~.m»~ +m~i~~»~. )

—_~»~.[k3~ik4~+k4~ik3~+2(kl~k2~i +kl~ik2~)]

A

gvi~[3yi 4pg+ 4gsi

3'

+2(lyse 2pi+ 1pi 2')]

—g„,„,[k3»k4~+k4»k3~ +2(kl~k2» +kl»k2~)]

IviW[k3vik4pa +k4vik3pg +2(klpak2vi +klvik2pg)]

+A[2(kl+k2)+ kl'_{k2](9pl+2gvl~} +_9pl~gvlpQ))

FIG.

1.

Feynman rules for scalar-graviton interactions with arbitrary RP coupling. The wavy lines represent gravitons, and the dashed lines stand for scalars. In the calculation concerning the quadrilinear vertex. we have made use ofthe fact that

8

h~"

=

0 and

6

=

0, since we are only interested in physical gravitons.

the corresponding invariant amplitude is given by ~4

t2+

u'

s'+

u'

s'+

t2

1~44-+441 ₆₄

+

_t

+

4(l

1 1

—

12m'I'

_{+ +}

I

+

8m'

s

t

_u)

2

—

8Am'(5+

6A)

where s,

t,

and u are the usual Mandelstam variables.

It

is clear that in the massless limit, which is the case for the action

_(1),

the invariant amplitude becomes totally independent

of

the parameter A. Equation

₍₃₎

with A

=

0 agrees with the result of Peet [7].

K4 _su

—

m4 4

16

t (s

—

m )2(u

—

m2)

l~ds~es(+

)I

=

l~es~es( +)

I

x4ms t2

16

(s

—

m2)'(u

—

m')'

'

where the

+

signs refer

to

the initial and final graviton helicities. We can see that even for massive scalars the invariant amplitude does not depend on A. Our result agrees with the one given by Berends and Gastmans [10], who have computed the elastic scattering

of

gravitons on massive scalars for gravitational minimally coupled scalars (A

=

0).

Thus, we come

to

the unexpected conclusion that the tree-level invariant amplitudes for the 2

~

2 processes concerning action

(1) (P

—

P and _P

—

_g scattering) are completely independent of the RP2 coupling parameter, which directly implies that this parameter cannot be measured

at

tree-level scattering in this case.

The preceding semiclassical result strongly suggests that there ought

to

be transformations which eliminate the nonminimal coupling from action

(1) at

no expense. Accordingly, we introduce the following reparametriza-tion in the aforementioned action:

where 1

+

""

P2 is supposed

to

be positive. Then we

PI _P3 Pl P3 pl

P2 4~ _p4 P2 p4

pl P3.

.

''

r r

X

r

P2 p4 P2 p4 P2 p4

BRIEF

REPORTS ₉₃₃

have

S[g,

g]

=

d

xg

g

—

2R

+

F—

(g)g""8„$8„$,

(4)

where

1+

"4

₍₁₊

6A)g ( )

(

where

_P(P)

is a new scalar field, we easily obtain the equation

which can be trivially integrated with the help of the boundary _{condition P(P)]@} o

—

—

0.

Hence, Einstein's

the-ory with

a

nonininimally coupled scalar field _P and Ein-stein's theory with a minimally coupled scalar field _P,

s

fg41

=

f

~'*,

4

g„,

+

g"

s„—

es

which are usually considered astotally di6'erent, are actu-ally one and the same theory provided that I+AK2$2/4 &

0.

What about the physical significance

of

this con-straint? To answer this question we write the field equa-tions obtained by variation

of

the action

(1)

in the sug-gestive form

where

G"

=

'K[A(g""H

V'"V")Q-+

_,

'g""~-4~

4

—

-~"4~"4],

HP

—

ARP

=

0,

(6-)

(6b)

K

g+

A]c

4

The nonlinearity with respect

to

the scalar field _Pmay, by means

of

afurther field redefinition, be removed from

(4).

Indeed, from the differential relation

g"

_{(~F8„Q)~F0„Q}

=

g"

0„$0

P,

is an efFective Einstein constant which will be positive

if

and only

if

1+

Av2$2/4

)

0.

Consequently, the adop-tion

of

the above-mentioned constraint leads

to

Einstein's standard formula t ~

=

'RTI"",

where

T~"

is given by

the right-hand side

of Eq.

(6a) and

'8

&

0. It

is worth mentioning, in passing, that the problem concerning the classical equivalence ofARP theories has been discussed by some authors in the last few years, although in a dif-ferent context &om the one in hand

[11].

Todetermine the cosmological effect

of

the ARP2 term for 'R

)

0,we may study the behavior

_{of (6),}

with A

=

_0, on

a

Friedmann-Robertson-Walker

(FRW)

metric. Us-ing dynamical system techniques, for instance,

it

is easy

to

show that the resulting solutions reach a singularity in

a

Gnite proper time, as is expected &om the singular-ity theorems

of

Hawking and Penrose, since the ordinary scalar Geld obeys the basic assumption

of

these theorems. Bekenstein's bouncing universe _[4],which was examined by Deng and Mannheim [4],is no exception

to

this rule: in fact, the value of the efFective Einstein constant for this Einstein-conformal scalar solution (A

=

—

—

)

is neg-ative

('8

(

0).

Thus, all

FRW

solutions concerning

Eq.

(6) are singular for 'R

)

_0.

Since there are no

static

black holes in the kame-work

of

Einstein-ordinary scalar field theory

[12],

we can promptly infer that ARP black holes have no hair pro-vided that 'R _&

_0.

_It

_{is straightforward}

_to

_show

_that

'R

(

_{0 for Bekenstein's black hole with}

_a

_{scalar charge} [4,

13].

Incidentally, this black hole was proved

to

be un-stable under monopole perturbations

_[14].

We believe that the semiclassical result presented in this

Brief

Report could be generalized in order

to

take into account quantum corrections.

Finally, we would like

to

mention the fact

that a

cos-mological constant,

a

scalar self-interaction, oreven other matter would reveal the RP2 coupling.

A.

J.

A. and

S.

F.

N. gratefully acknowledge financial support &om Conselho Nacional de Desenvolvimento Cientifico e Tecnologico

(CNPq).

S.

F.

K.

and H.M. are very indebted

to

Fundagao de Amparo

a

Pesquisa do

Es-tado de Sao Paulo

(FAPESP)

for financial support.

D.

S.

is grateful

to FAPESP

and CNPq for financial support.

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