PHYSICAL REVIEW D VOLUME 47, NUMBER 6 15 MARCH 1993

G ravitational laser backseat tering

S. F. Novaes
Instituto de Fisica Teorica, Universidade Estadual Paulista, Bua Pamplona 1/5, 01/05 90-0 Sao Paulo, Brazil

D. Spehler
Instituto de Fisica Teorica, Universidade Estadual Paulista, Rua Pamplona 1/5, 01/05 900-Sao Paulo Brazilau or
and Universite Louis Pasteur, Institut Universitaire de Technologie, 8 rue St. Paul, 67800 Strasbourg, Prance

(Received 19 October 1992)

A possible way of producing gravitons in the laboratory is investigated. We evaluate the cross
section for electron + photon —+ electron + graviton in the framework of linearized gravitation, and
analyze this reaction considering the photon coming either from a laser beam or from a Compton
backscattering process.
PACS number(s): 04.30.+x, 04.80.+z, 12.25.+e

The attempts to detect gravitational radiation usually
rely on the observation of astrophysical objects. These
sources, such as supernovae explosions or coalescing bina-
ries, have in general very uncertain frequency, strength,
and spectrum of gravitational waves. A considerable im-
provement in the search of the gravitational waves would
occur if we succeed in producing them in the laboratory.
In this case, the energy and propagation direction could,
at least in principle, be controlled.

In this paper, a mechanism of graviton production in a
linear electron-positron collider is analyzed. Inspired by
the Compton laser backscattering process [1], we suggest
that the collision of laser photons of few electron volts,
at small angle, with an energetic electron beam is able to
generate a very collimated graviton beam.

We evaluate, in a first step, the cross section for the
process electron + photon —+ electron + graviton, and es-

where 6„ is the graviton field and y. =—+32mG = 8.211x
10 GeV i in natural units. The vierbein is expanded

K
(2)

The above expressions allow us to write the relevant
part of the action as

timate the total radiated power assuming that the pho-
tons come from a laser beam. The scenario where the
photons are harder, as a result of a Compton backscat-
tering process, is examined in a second step.

The coupling of the matter fields with gravity can be
obtained using the weak field approximation [2]. The
metric is assumed to be close to the Minkowski one (ri„):

4 K 2

0(V &p + V—p~—) 0+
4 (&I 0V + cl Ay) 0 li."+

I 2VW"clI 4 — &pA'"0 —mA-'
I

h

——(q" q'~h —4h." q ~) F'„,F p
——(6,"h —h,,~) i (A„p'+ A'p„) g), (3)

where we omitted the free electron and photon Lagrangians, and the usual @ED fermion-photon coupling, and
neglected the O(K ) terms.

In order to evaluate the cross section for the process electron + photon ~ electron + graviton, we used the Weyl-
van der Waerden spinor technique to describe the graviton helicity wave function [3] and the graviton couplings with
bosons and fermions [2]. The cross section of the reaction e (p) + p(q) —+ e (p') +9(k) is given by

GO

dt
6 K 1
16~ (s —mz)z

(us —m4) m2 m2 ~ m2 m~ 1 s —m2
z+ i + + —— +t s —m u —m s —m u —m 4 u —m

u —m'l
s —m2) (4)

47 2432 1993 The American Physical Society



47 GRAVITATIONAL LASER BACKSCATTERING 2433

This expression can be written in a more compact form
if we define new variables:

S —Al 2EQp (1+Pcosn),

(5)
ko 1 + cos(8 —n)y=

s —m~ E 1+Pcosn

where m, E, and P = (1 —m /Ez) / are respectively
the electron mass, energy, and velocity in the laboratory
frame. The laser, of energy qo, is supposed to make an
angle a with the electron beam. The scattered graviton
has energy ko.

1+P cosn
1 —P cos 8 + (qo/E) [1 + cos(8 —n)]

'

where 8 is the angle between the graviton and the in-
coming electron. Note that, for n 0, the variable y
represents the fraction of the electron energy carried by
the graviton in the forward direction (8 = 0).

Equation (4) expressed in terms of x and y becomes

do. e'K' (1 x+1)
dy 64~ ky

-10p6 1 I I I IIIII

1O5

104

ef

a
O'e
b'c

103

102

1O'

1OO
1O

—6 10 1O-4
e (rad)

10
I I I I II

1p 2

The radiation power emitted in solid angle of semiangle
(8o) due to the graviton emission is

FIG. 1. The angular distribution der/cos8 in arbitrary
units versus the angle in radians for the process e + pg —+

e +g, where pg comes from a laser beam. We assume qo ——1
eV and plot for diferent electron beam energies: E=50 GeV
(dashed), 250 GeV (solid), 500 GeV (dotted), and 1000 GeV
(dot-dashed) .

with

1 4y 4y~

1 —y x(1 —y) x2 (1 —y) 2

Pg(8o) = Z,~,
d~

d8 sin8 (8) ko(8).dcos8

In Fig. 2 we show the spectral distribution of the emit-
ted radiation

Equation (7) shows that the cross section for large val-
ues of y is suppressed by an overall factor (1/y —1/ym~„)
which makes the distribution very sharp for small values
of y. This is a major difference from the Compton cross
section e+ pg ~ e+ pb, where pg(b) is the laser (backscat-
tered) photon. In the latter case, the y distribution as-
sumes its maximum value for y = y ~„= x/(x + 1),
giving rise to a hard photon spectrum.

Let us start by analyzing our result when the initial
photon originates from a laser beam. We consider four
different designs of e+e linear colliders [4]: SLAC Linear
Collider (SLC) (E = 50 GeV, 8 = 5 x 10~s cm ~ s r),
Palmer-G (E = 250 GeV, 8 = 5.85 x 10ss cm ~ s r),
Palmer-K (E = 500 GeV, 8 = 11.1 x10ss cm 2 s r), and
Serpukhov's VLEPP (E = 1000 GeV, 8 = 10ss cm
s—r)

Assuming qo = 1 eV, and n = 0, we compare the be-
havior of the graviton angular distribution do/d cos 8 for
different electron beam energies. The angular distribu-
tion is peaked for very small angles, close to the electron
direction [Fig. (1)]. By changing the value of n we are
of course able to shift this peak away from the electron
beam.

Let us suppose that the laser has flash energy of 2.5 J,
the same repetition rate as the electron pulse frequency,
and a rms radius of 20p,m. In this case, the electron laser
luminosity is Z,~, = qZ, 8 being the collider luminos-
ity, and rl = A, N~, /A~, N, . Here, A, (~, ) is the electron
(laser) beam cross section and N, (~,) is the number of
electrons (photons) in the bunch (flash).

104 I I I I I I I II I I I I I 1 I 1

102

4$

CI

C4'a

101

1001O-9
I I l I I I I I l

10—8
I ~ t ~ I ~ I I I

10 7

ko (GeV)

I ~ ~ I ~ I ~ I

FIG. 2. Spectral distribution of the graviton radiation
power de/dko in arbitrary units as a function of kp, for
E = 50 GeV.

ko (PE —qo x+11 ( ko —qo l
(X

!/F/
x,

dko PE —qo ( ko —qo x ) ( 'PE —qo)

We note that this distribution is maximum for ko

ko
'" = qo, and is almost independent of the electron

beam energy.
Another way of obtaining the reaction e+ p —+ e+g is

taking advantage of the energetic photons produced by
the Compton backscattered process. The energy spec-
trum of backscattered photons is [1,5]



2434 S. F. NOVAES AND D. SPEHLER 47

t' 4 8) 1 8
C(x) =

~
1 ————,[ln(1+x)+x x') 2 x 2(1+x)'

(10)

Defining the ratio of electron-photon invariant mass
squared (s = 4Eq~) to invariant mass squared of the
collider,

S gp
7

1 dOc 1
X~/, (x, z):—— ' =, , F(x, z),

o~ dz Q(Q)

where a., is the Compton cross section, and z = q~/E is
the fraction of the initial electron carried by the photon.
The function F(x, z) was defined in Eq. (8), with x given
by Eq. (5), and

Collider

SLC

Ps (eV/s)

1.17 x 10 '

ao

10'
20'
30'

Pb (eV/s)

1.69 x 10
4.48 x 10
750 x 10

Palmer-G 2.32 x10 " 10'
20'
30'

5.02 x 10
1.47 x 10 '
2.66 x 10

Palmer-K 6.70 x 10
10'
20
30'

1.51 x 10-"
4.57 x 10
8.47 x 10

TABLE I. Total graviton radiation power emitted in a
solid angle of semiangle 80. Pg refers to the direct laser photon
process and Pb to the backscattered photon one. We consider
in both cases a laser photon with energy of 1 eV.

and assuming a conversion coefficient (the average num-
ber of converted photons per electron) ( = 0.65 [6], the
ebb luminosity can be written in terms of the machine
luminosity as 2,» ——(Z. The luminosity distribution of
the backscattered photons is

VLEPP 1.53 x 10
10'
20'
30'

2.26 x 10
6 ~ 94x 10
1.32 x 10

dZ = ( l: P, /~(x, ~).

In this case, the radiation power, emitted in solid angle
of semiangle (8p), is

&b(eP) =
&IsLx dg ~O

d~ de sine„(e, r) kp(e, ~),
d7 dcosH

where 7,„=x/(x+ 1), and o. is the elementary cross
section for the process e+ p ~ e+ g evaluated at s = s.

In Table I we present the total radiation power for
difFerent values of Hp, and for a laser energy (qp) of 1
eV. Since x increases when we increase either the laser
energy or the electron beam energy, the diferent choices
of colliders considered here give a reasonable idea of the
laser backscattering capabilities to produce gravitons in
the laboratory.

As pointed out before, gravitons can be produced via
e + pg ~ e + g, where pg is a laser photon, and via
e + e(+ps) —+ e + pb —+ e + g, where pb is the laser
backscattered photon. In the former case, the radiation
power Pg is almost independent of the particular choice
of 80, since the angular distribution of the graviton peaks
at very small angles relative to the electron beam direc-
tion. On the other hand, in the latter case, the backscat-

tered photon is very hard, giving rise to a Hatter angu-
lar distribution and making the dependence of Pb on Hp

more evident. The total forward emitted power becomes,
therefore, small for hard photons.

In a recent paper, Chen [7] has examined the pro-
duction of gravitons due to the interaction between the
beams near the interaction point in an e+e collider
(gravitational beamstrahlung). His results for the to-
tal radiation power, at the SLC, is roughly one order
of magnitude below the one for the gravitational laser
backscattering process. For the next generation of linear
e+e colliders, the gravitational beamstrahlung yields a
larger radiation power. Nevertheless, the coherent con-
tribution is again smaller than the one coming from the
laser backseat tering.

We should stress that, in spite of the small emitted ra-
diation power, the gravitational laser backscattering pro-
cess yields a very collimated beam of gravitational waves
with a defined energy spectrum (see Fig. 2). These points
could be of great help for the complex issue of gravita-
tional wave detection [8].

We are very grateful to R. Aldrovandi, O. J. P. Eboli,
and C. O. Escobar for useful discussions. This work was
partially supported by Conselho Nacional de Desenvolvi-
mento Cientifico e Tecnologico, CNPq (Brazil).

[1] F. R. Arutyunian and V. A. Tumanian, Phys. Lett. 4, 176
(1963); R. H. Milburn, Phys. Rev. Lett. 10, 75 (1963); see
also C. Akerlof, University of Michigan Report No. UMHE
81-59, 1981 (unpublished).

[2] R. Aldrovandi, S. F. Novaes, and D. Spehler, Report No.
IFT-P.027/92 (unpublished).

[3] D. Spehler and S. F. Novaes, Phys. Rev. D 44, 3990
(1991).

[4] Particle Data Group, K. Hikasa et al. , Phys. Rev. D 45,
Sl (1992); R. B. Palmer, Annu. Rev. Nucl. Part. Sci. 40,

529 (1990).
[5] I. F. Ginzburg, G. L. Kotkin, V. G. Serbo, and V. I.

Telnov, Nucl. Instrum. Methods 205, 47 (1983); 219, 5
(1984).

[6] V. I. Telnov, Nucl. Instrum. Methods A 294, 72 (1990).
[7] P. Chen, Mod. Phys. Lett. A 6, 1069 (1991).
[8] See, for instance, The Detection of Gravitational Waves,

edited by D. G. Blair (Cambridge University Press, New
York, 1991).