Straight-line path approximation for studying Planckian-energy
scattering in quantum gravity (*)
NGUYENSUANHAN(1) (**) and EAPPONNA(2)
(1) International Centre for Theoretical Physics - Trieste, Italy
(2) Physics Faculty, Phnom Penh University - Phnom Penh, Cambogia (ricevuto il 27 Settembre 1996; approvato il 7 Maggio 1997)
Summary. — A method is proposed for constructing a scattering amplitude in
quantum gravity by means of functional integration. The straight-line path approximation is used to calculate the functional integrals required. The closed analytic, relativistically invariant expressions are obtained for the two-particle elastic-scattering amplitudes. In the limit of high energies s KQ and for given momentum transfers this expression for scattering amplitude takes a Glauber representation with an eikonal function depending on the energy. The connection between this representation for the potential scattering amplitude and the eikonal approximation in quantum field theory is discussed.
PACS 11.80 – Relativistic scattering theory. PACS 04.60 – Quantum gravity.
1. – Introduction
In the series of papers [1-7] a method has been developed for constructing the scattering amplitude in quantum field theory by means of a functional integral. Such a representation of scattering amplitudes as a sum over all possible trajectories of colliding nucleons has proved helpful to investigate the interaction of elementary particles in the infrared region [ 6 , 7 ] and at high energies [1-5]. The main advantage of this approach over the others is the possibility of performing calculations in a compact form.
In the present paper we attempt to extend this approach to the Planckian-energy gravitational scattering which has received considerable attention in recent years because of its relation to fundamental problems as strong gravitational forces near black holes, string modification of theory of gravity and some other effects of quantum gravity [8-20].
(*) The authors of this paper have agreed to not receive the proofs for correction.
(**) Permanent address: Department of Theoretical Physics, Vietnam National University, P.O. Box 600, BoHo, Hanoi 10000, Vietnam.
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The paper is organized as follows. In the second section we present a method to find the exact closed expression for the single-particle Green’s function in an external gravitational field gmn(x) in the form of the functional integrals. The Weierstrass
transformation [21] was applied to second-order differential equations in which the differential operator can be written as a product of two lower-order operators. As a result, the transition Tj-exponent to ordinary operator expression after
“disentangling” the differentiation operators by the terminology of Feynman [22] can be done without any series expansion with respect to a field. Considering the linear approximation for gravitational field we obtained the two-particle Green’s function necessary to construct the scattering amplitudes. In sect. 3 by transition to the mass shell of the external two-particle Green’s function we obtain a closed representations for the two-particle scattering amplitude expressed in the form of the functional integrals. For estimating the functional integrals we used the straight line path approximation [ 1 , 2 ] based on the idea of rectilinear paths of the interacting particles at asymptotically high energies and small momentum transfers. Section 4 is devoted to investigating the asymptotic behavior of this scattering amplitude in the Planckian-energy limit. Finally, we discuss the obtained results and possible generalizations of the approach.
2. – The scalar particle Green’s function in a gravitational field
Let us consider the model of interaction of a scalar “nucleons” field W(x) with a gravitational field gmn(x), where the interaction Lagrangian is of the form
L(x) 4 k2g 2 [g mn(x) ¯ mW(x) ¯nW(x) 2m2W2(x) ] 1Lgrav(x) , (2.1) where g 4det gmn(x) 4k2g gmn(x) .
Variating the Lagrangian (2.1) leads to the following equation for W(x):
[gAmn(x) i¯mi¯n2k2g m22 ¯mgAmn(x) ¯n] W(x) 40 , gAmn(x) 4k2g gmn(x) .
(2.2)
Equation (2.2) is conveniently investigated in the harmonic coordinates defined by the condition [23]
¯mgAmn(x) 40 .
(2.3)
The harmonic gauge (2.3) being the analogy of the Lorentz gauge in the electrodynamics has led to eliminate the nonphysical component of the tensor field. Taking eq. (2.3) into account, eq. (2.2) yields:
[gAmn(x) i¯
mi¯n2k2g m2] W(x) 40 .
For the single-particle Green’s function in the gravitational field we have the following equation:
[gAmn(x) i¯
mi¯n2k2g m2] G(x , yNgmn) 40 .
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If one uses the representation of the inverse operator
[
gAmn(x) i¯mi¯n2k2gm2
]
21proposed by Feynman [22], Fock [24] and as an exponent form, this equation can be written in an operator form
(2.5) G(x , yNgmn ) 4 4 i
0 Q dt expy
2im2 0 tk
2g(x , j) dj 1 i 0 t gAmn(x , j) i¯ m(j) i¯n(j) djz
d4(x 2y) ,the exponent in expression (2.5), which contains the non-commuting operators ¯m(x , j ),
gAmn(x , j ) and g(x , j ) is considered as T
j-exponent, where the ordering subscript j has
the meaning of the proper time divided by mass m. All operators in (2.5) are assumed to be commuting functions that depend on the parameters j . The coefficient of the exponent in eq. (2.5) is quadratic in the differential operator ¯m. However, the
transition from Tj-exponent to an ordinary operator expression (“disentangling” the
differentiation operators in the argument of the exponential function by the terminology of Feynman [22]) cannot be performed without the series expansion with respect to an external field. But one can lower the power of the operator ¯m in eq. (2.5)
by using the following formal transformation [21]: (2.6) exp
y
i 0 Q gAmn(x , j) i¯m(j) i¯n(j)z
4 4 Cnd4nn»
h expy
i 0 Q [gAmn(x , j) ]21i¯ m(j) i¯n(j) 22i 0 t dj nm(j) ¯ m(j)z
.The functional integral in the right-hand side of eq. (2.6) is taken in the space of 4-dimensional function nn(j ). The constant Cnis defined by the condition
Cn
d4nnexpy
2i0 Q
[gAmn(x , j) ]21n
m(j) nn(j)
z
4 1 ,from which it follows
Cn4
y
d4nnexpy
2i 0 Q [gAmn(x , j) ]21n m(j) nn(j)zz
21 4(
det [gAmn(x , j) ])
21 /2. After substituting (2.6) into (2.5), the operator expk
2 is
0t
dj nm(j ) ¯
m(j )
l
can be“disentangled” and we can find a solution to eq. (2.4) in the form of the functional integral (2.7) G(x , yNgmn ) 4i
0 Q dt e2im2t Q Q Cnd4n expy
2i 0 t dj [gAmn(x , j) ]21n m(j) nn(j) 2im2 0 t [k
2g(xj) 21] djz
Q Q d4u
x 2y22 0 t n(h) dhv
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eq. (2.7) is the exact closed expression for the scalar particle Green’s function in an arbitrary external gravitational field in the form of functional integrals [25, 26].
In the following we consider the gravitational field in the linear approximation,
i.e. put gmn
4 hmn1 khmn, where hmn is the Minkowski metric tensor with diagonal
( 1 , 21, 21, 21). Rewrite eq. (2.7) in the variables hmn(x) after dropping out the term
with an exponent power higher than first hmn(1), we have a Green’s function for
single-particle Klein-Gordon equation in a linearized gravitational field:
G(x , yNhmn) 4i
0 Q dt e2im2t[d4 n]t0Q (2.8) Q expy
ik 0 t nm(j) nn(j) hmnu
x 22 j t n(h) dhv
djz
d4u
x 2y22 0 t n(h) dhv
, where [d4n]tt214 d4 n exp [2is
tt21n 2 m(h)»
hd4h]s
d4 n exp [2is
tt21n 2 m(h)»
hd4h] , and [d4n]tt21 is a volume element of the functional space of the four-dimensional
functions n(h) defined in the interval t1G h G t2. The expression for the Fourier transform of the Green’s function (2.8) takes the form
(2.9) G(p , qNhmn ) 4
d4x d4 y G(x , yNhmn ) 4 4 i 0 Q dt ei(p22 m2) t dx ei(p 2q) x[d4n 0]t0expy
ik 0 t Jmnhmnz
,here we use the notation
Jih 4hmn(z) Jmn(z); i 41, 2, and Jmn(z) is the current of the“nucleon” defined by Jmn(z) 4
0 t1 dj(
nm(j) nn(j))
du
z 2xi1 2 pij 12 0 j ni(h) dhv
. (1) The Lagrangian (2.1) in the linear approximation to hmn(x) has the form L(x) 4L0 , W(x) 1 L0 , grav(x) 1Lint(x), where
L0(x) 4 1 2[¯ mW(x) ¯ mW(x) 2m2W2(x) ] , Lint(x) 42k 2h mn(x) T mn(x) , Tmn(x) 4¯mW(x) ¯nW(x) 2 1 2hmn[¯ sW(x) ¯ sW(x) 2m2W2(x) ]; Tmn(x) is the energy momentum tensor of the scalar field.
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Following the authors [13, 27, 28] we shall determine the two-particle Green’s function G
(
p1, p2Nq1, q2)
(2): (2.10) G(p1, p2; q1, q2) 4 4g
expy
i 2 d4 kDabgd(k) d 2 dhab(k) dhgd (2k)z
G(p1, q1Nh) G(p2, q2Nh) S0(h)h
Nh 4 0 . Here Dabgd(k) is the propagator of the free graviton field, and S0(h) is the average of the S-matrix over the fluctuations of the “nucleon” vacuum in the presence of the external field hmn. We shall henceforth disregard the contributions of the vacuum loops
and put S0(h) 41.
Substituting (2.9) into (2.10) and performing variational derivatives, we find for the two-particle Green’s function the following expression:
(2.11) G(p1, p2; q1, q2) 4 4 i2
»
i 41, 2u
0 Q dsie i(p2 i2 m2) si[d4n i] si 0d4xie i(pi2 qi) xiv
expk
2ik 2 2 D(J11 J2)2l
, where we have introduced the abbreviated notationJiDJk4
dz1dz2Jimn(z1) Dmnas(z12 z2) Jkas(z2); i , k 41, 2 . Expanding expression (2.11) with respect to the coupling constant k2and taking the functional integrals with respect to ni(h), we obtain the well-known series ofperturbation theory for the two-particle Green’s function.
The term in the exponent (2.11) can be rewritten in the following form: 2k 2 2
D(J11 J2)2 4 2ik2DJ1J22 ik2 2 DJ2 12 ik2 2 DJ2 2 , (2.12)The first term on the right-hand side of (2.12) corresponds to the one-graviton exchange between the two nucleons and the remainder lead to radiative corrections to the lines of the nucleons of the field W(x).
3. – Construction of the two-particle scattering amplitude
The two-nucleon scattering amplitude is expressed in terms of the two-particle Green’s function (2.11) by equation
( 2 p)4d4(p11 p22 q12 q2) iT(p1, p2; q1, q2) 4 (3.1) 4 lim p2 i, qi2K m2 i 41, 2
»
(qi22 m2)(pi22 m2) G(p1, p2Nq1, q2) .(2) The identity of the particles is allowed for by symmetrizing expression (2.10) with respect to the momenta of the initial or final particles.
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The transition to the mass shell pi2; qi2K m2; i 41, 2 calls for separating from
formula (3.1) the pole terms ( pi22 m2)21 and (qi22 m2)21, which cancel the factors
( pi22 m2) and (qi22 m2). In perturbation theory this compensation is obvious, since the
expression for the amplitude is made up of free propagation functions, but if the Green’s function is sought by means of methods other than perturbation theory, the separation of the pole terms entails certain difficulties. We shall be interested in the structure of scattering amplitude as a whole, therefore the development of a correct procedure for going to the mass shell in the general case is very important. Many approximate methods that are reasonable from the physical point of view when used before the transition on the mass shell, shift the positions of the poles of the Green’s function and render the procedure of finding the scattering amplitude mathematically incorrect. In present paper we shall use a method for separating the poles of the Green’s functions that generalizes the method introduced in [7] to find the scattering amplitude in a model of scalar nucleons interacting with a scalar meson field in which the contributions of closed nucleon loops are ignored.
Substituting (2.11) into (3.1) we get (3.2) ( 2 p)4d4(p 11 p22 q12 q2) iT(p1, p2; q1, q2) 4 4 lim pi, qiK m2
u
i 41, 2»
(q2 i 2 m2)(pi22 m2)d4xiei(pi2 qi) xi 0 Q dsi 0 si djei(pi22 m2) si[d4n i]s0iv
Q Q k2DJ 1J2 0 1 dl expk
2ik2lDJ 1J22 i k2 2 DJ2 il
,In deriving (3.2) we employed the operator of subtracting unity in the formula (2.11) from the exponential function containing the D-function in its argument in accordance with the formula
e2ik2
DJ1J2 2 1 4 2ik2 0 1 dl DJ1J2e2il DJ1J2.This corresponds to eliminating from the Green’s function the terms describing the propagation of two noninteracting particles. Taking into account the identity
»
k 41, 2 0 Q dsk 0 sk djkR»
k 41, 2 0 Q djk jk Q dskRand making a change of the ordinary and the functional variables
.
`
/
`
´
siK si1 ji, i 41, 2 , xiK xi2 2 0 ji [p 1n(h) ]idh , ni(h) Kni(h 2j)i2 (p 2 q)iu(h 2si) , (3.4)465
we transform eq. (3.2) into
(3.5) ( 2 p)4d4(p11 p22 q12 q2) T(p1, p2; q1, q2) 4 4 lim pi, qiK m2 i 41, 2
»
u
(pi22 m2)(qi22 m2)d4xiei(pi2 qi) xi 0 Q djiei(p 2 i2 m2) ji 0 Q dsiei(q 2 i2 m2) siv
Q Q[d4n 1]s2j1 1[d 4n 2]s2j2 2(k 2)[n a(j1) nb(j1)[ Dabgd(xi2 x2)[ng(j2) nd(j2) ] Q Q 0 1 expk
2ik2lDJ 1J2l
. In the following we consider the forward scattering, and the radiative corrections to the lines of the field W(x) in eq. (3.5) will be omitted. We now note that the integrals with respect to si and ji give pole factors ( pi22 m2)21 and (qi22 m2)21; i 41, 2.Therefore, in eq. (3.3) we can go over to the mass shell with respect to the external lines of the nucleon using the relation [29]
lim a , e K0ia
0 Q ds eias 2ef (s) 4f(Q) ,which holds for any finite function f (s). By means of the substitutions x14 ( y 1 x) O2 and x24 ( y 2 x) O2 in eq. (3.5) and performing the integration with respect to dy we can separate out the d-function of the conservation of the four-momentum d4( p
11 p22
q12 q2). As a result, the scattering amplitude takes the form (3.6) T(p1, p2; q1, q2) 4 (k2)
»
i 41, 2 [d4n i]Q2Qd 4x ei(p12 q1) xQ Q[p11 q11 2 n( 0 ) ]a[p11 q11 2 n( 0 ) ]bDabgd(x)[p21 q21 2 n( 0 ) ]g[p21 q21 2 n( 0 ) ]dQ Q 0 1 exp [2ik2lDJ 1J2] , where (3.7) Jimn(k)44 2Q Qdj [piu(j)1qiu(2j)1n(j)]m[piu(j)1qiu(2j)1n(j)]nQ
Qexp
y
2 iky
pijiu(j) 1qijiu(2j)10
j
ni(h) dh
zz
.The exp [2ik2l
s
DJ1J2] describes virtual-graviton exchange among the scattering “nucleons”. The integration with respect to dl ensures subtraction of the contribution of the freely propagating “nucleons” from the matrix element. The functional with respect to [d4n
i]; (i 41, 2) corresponds to summation over all possible trajectories of
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Let us consider briefly the physical meaning of the functional variables n1(h) and
n2(h) which were introduced formally in formula (2.7) in the derivation of the solution for the Green’s function (2.7). In high-energy forward scattering and small momentum transfers it is natural to assume that for each fixed value of the relative coordinate of two-particles the main contribution comes from all the possible nucleon trajectories, which are specified by the directions of nucleon momenta before and after the scattering. This means that the functional variables ni(h), which cause the nucleon
trajectories to deviate from straight lines, can be omitted
(
ni(h) 40)
from the nucleoncurrents (3.7) in the considered kinematic region. In the language of Feynman diagrams, this corresponds to linearization of the nucleon propagators with respect to the momenta of the virtual gravitons, i.e. to the substitution
1 (p 1
!
iki) 2 2 m2 K 1 2 p!
iki , (3.8)where p is the momentum of one of the scattered nucleons and ki are the momenta of
virtual gravitons. More satisfactory from the point of view of convergence of the Feynman integrals is to approximate the propagator approximation in which ki2 is
conserved 1 (p 1
!
iki)22 m2 K 1 2 p!
iki1!
iki2 . (3.9)In this approximation, the products pki are assumed to be effectively more
important in the high-energy region than kikj(i c j). In the functional approach this is
obtained by performing the approximate calculation for the continual integrals:
[d4n] exp
[
F [n]]
B exp[
F [n]–]
, (3.10)where F[n]–4
s
[d4n] F[n]. In research on high-energy process, such an approximationis called the straight-path approximation.
Therefore, the scattering amplitude (3.6) in this approximation takes the form (3.11) T(p1, p2; q1, q2) 4
d4x ei(p12 q1) x[k 1p11 q1]m[k 1p11 q1]nQ Q Dmnrs (x)[2k1p21 q2]r[2k1p21 q2]sQ Q 0 1 expy
2 ilk 2 ( 2 p)4 d 4kD mnrs(k) eikxJ mn 1 (2k) J rs 2 (k) –z
, where Jmn i (k) 4[d4ni]Q2QJ mn i (k) 4y
( 2 pi1 k)n( 2 pi1 k)n 2 pik 2k22 ie 2 ( 2 qi2 k) m( 2 q i2 k)n 2 p1k 2k22 iez
, (3.12)467 Jmn 1 (2k) J2rs(k) 4
[d4n1]Q2QJ mn 1 (2k) J2mn(k) 4 (3.13) 4y
( 2 p11 k) m( 2 p 11 k)n 2 p1k 1k21 ie 2 ( 2 q12 k) m( 2 q 12 k)n 2 p1k 2k22 iez
Q Qy
( 2 p22 k) r( 2 p 22 k)s 2 p2k 2k22 ie 2 ( 2 q21 k) r( 2 q 21 k)s 2 p2k 1k21 iez
. This approximation corresponds to the following interaction picture: the high-energy particles are scattered by successive independent exchange of virtual quanta, and the individual exchanges do not influence one another, so that the correlation termkikjdrops out of the propagator.
It is quite correct to regard, as is frequently done, the approximations (3.8) and (3.9) as a soft-quantum approximation. For example, the important role in the summation of the ladder diagrams of this approximation is played not by the smallness of the momenta ki, but by the absence of mutual correlation between individual exchanges. If
the exchange particles have additional degrees of freedom (e.g. charge), then the simple assumption that their momenta are small is still insufficient for such a summation, since the charge degrees of freedom remain interrelated in this case.
4. – Asymptotic behavior of the scattering amplitude at high energies
Let us consider the asymptotic behavior of the elastic forward amplitude of two scalar particles (3.10)-(3.12) in the high energies s KQ and for given momentum transfers. We perform the following calculation in the centre-of-mass system of the colliding particles, p14 2 p24 p , and we direct the z-axis along the momentum p1:
. / ´ p14 (p10, 0 , 0 , p 4pz); p24 (p20, 0 , 0 , 2p) , s 4 (p101 p20)24 4 p02; p104 p204 p0; t 4 (p12 q1)24 (p22 q2)24 2q2». (4.1)
In the high-energy limit
s c MPL2 ct , (4.2)
where MPL is the Planck mass, we find the following Glauber-type representation for the scattering amplitude [30] (3):
T(s , t) 422is
d2x »e2iq» x»g
expk
ik 2s 2 p K0(mNx»)l
2 1h
, (4.3)where K0
(
mNx»)
is the MacDonald function of zeroth order and can be determined by the formula K0(
mNx»)
4 1 O2 pd2k »(e
2ik»x») O(k2
»1 m
2) and m is a graviton mass which serves as an infrared cut-off, k2
4 16 pG . Note that in contradistinction to the scalar electrodynamics case (Lint4 2 gW * i ¯
D
sWAs1 g2AsAsW * W), the eikonal in eq.
(4.3) contains in the exponent an additional factor s which reflects that the graviton has
(3) Allowance for the identity of nucleons leads to terms that vanish in the limit s KQ and for t fixed when expression (4.3) is symmetrized.
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spin two. Performing the integral in eq. (4.3), one gets
T(s , t) 4 8 pGs 2 2t G( 1 2iGs) G( 1 1iGs)
g
4 m2 2th
2iGs . (4.4)Eikonal representation of the amplitude of high-energy scattering (4.3) agrees with the result found by the “shock-wave method” proposed by ’t Hooft [8], and by the method of effective topological theory in the Planckian limit proposed by Verlinde and Verlinde [12] and by the summing Feynman diagrams in the eikonal approximation [13].
To conclude the section we shall show how one can use the eikonal expression (A.4) to obtain a Glauber representation for the scattering amplitude of two scalar nucleons that exchange virtual gravitons. We apply the operation of functional averaging of the product of the classical amplitudes (A.4)
(4.5) ( 2 p)4d4(p 11 p22 q12 q2) iT(p1, p2; q1, q2) 4
g
expy
i 2 d4k Dabgd(k) d 2 dhab(k) dhgd (2k)z
F(p1, q1Nh) F(p2, q2Nh)h
h 40 , here F(
p , qNh)
is the amplitude for the scattering of the scalar nucleon on the tensor potential hmn, and defined by (A.4).After the substitution of (A.4) into (4.5), the variational differentiation can be readily performed to obtain
(4.6) ( 2 p)4 d4(p11 p22 q12 q2) iT(p1, p2; q1, q2) 4
»
i 41, 2 d4 xiexp [i(pi2 qi) xi] Q Qu
¯ 2 ¯a1¯a2 expy
2ik2 a1 Q dj1 a2 Q dj22 p m 12 p n 1Dmnsr(x12x222 p1j112 p2j2) 2 p s 22 p r 2z
v
a14a240 . In eq. (4.6), we consider forward scattering, and we have therefore omitted the radiative corrections to scattering particles. Now, making the change of variables [5, 31] x11 x24 y ; x12 x24 x , j1K j12 px02 p0xz 4 pp0 , j2K j21 px01 p0xz 4 pp0 , (4.7)and integrating over dy , dx0, dxz, we obtain for the scattering amplitude
(4.8) T(s , t) 42is
d2 x»e2iq »x»Q Qg
expk
ik2 2Q Q ds1 2Q Q ds22 p m 12 p1nDmnsr(x»2 p×1s11 p×2s2) 2 p2s2 p2rl
2 1h
, where p× im4 pimONpN , si4 2 NpN ji, i 41, 2.In the region of high energies (4.2), we obtain the same Glauber representation (4.3) for the scattering amplitude
T(s , t) 422is
d2x »e 2iq»x»g
expk
ik 2s 2 p K0( mNx»)l
2 1h
, (4.9)469
The connection between this method of approximation and perturbation theory and also the validity of the formula (4.9) in different models are discussed [27, 28, 32]. Here we shall content ourselves with the following remarks. The expression F
(
p , qNh)
in (A.4) is valid if the functions hmn satisfy conditions ofsmoothness. The substitution of each of the two smooth functions h1mn, h2sr by the singular Green’s function in (4.5) seems to be in direct contradiction to these conditions. However, the neighbourhood of the singular point
(
the zero in the integral (4.9))
can be neglected because of the rapidly oscillating factor ex0(x»),whereas in the remaining region of values of the variable x», the phase x0(x») and the effective potential satisfy the conditions of the eikonal approximations as s KQ,
i.e. the operation (4.5) is quite correct. The question of the smoothness of the
potential is also considered in [30, 33].
5. – Conclusions
In this paper, we have proposed a method to find a closed analytic relativistically invariant crossing symmetry expression for two high-energy particles within the framework of the functional integration method. In the limit of high energies s KQ and for a given momentum transfers t this expression for scattering amplitude takes a Glauber representation of the type (4.3) with an eikonal function depending on the energy. Our results and comparison with other approaches to this problem show that in our approach the Glauber representation of the scattering amplitude is constructed by an eikonal expansion of the exact two-particle Green’s function on the mass shell, in [13] a similar result is obtained by the averaging of two classical amplitudes found in the eikonal approximation. The equivalence of this more accurate method to the approximation described in the present paper is connected with the interesting problem of the commutativity of the operators of eikonal approximation and second quantization. If the latter do commute, the Glauber representation for amplitude (4.3) in quantum theory is a consequence of the eikonal approximation in quantum mechanics. Finally, we would like to emphasize that our method of writing the formal exact solutions of the equation for the particle Green’s function in an external gravitational field and deriving the quantum quantities from them can be used not only to find the asymptotic behavior of the elastic-scattering amplitude at the high energies, but also to investigate other problems as the infrared singularities, radiative corrections, the multiple production of “soft” gravitons, retardation effects ... to high-energy scattering. However, in the latter cases one must make further study and find different methods of approximation for the calculation of the functional integrals.
* * *
We are grateful to Profs. B. M. BARBASHOV, G. VENEZIANO, A. I. ANDREEV, A. V. EFREMOV, M. K. VOLKOV, V. V. NESTERENKO, V. N. PERVUSHIN for useful discussions. NSH is also indebted to Prof. S. RANDJBAR-DAEMI for support during his stay at ICTP in Trieste and warm hospitality. This work was supported in part by the Vietnam National Research Programme in Natural Science, the International Centre for Theoretical Physics, Trieste, UNESCO and the International Atomic Energy Agency.
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AP P E N D I X
In this appendix we consider the scattering of the scalar nucleon in the external potential hmn(x). Using expression (2.9) we find the scattering amplitude F
(
p , qNhmn)
( pand q are particle momenta before and after the scattering, respectively) by the following formula: F(p , qNhmn) 4 lim p2, q2 K m2 (p2 2 m2)(q22 m2) G(p , qNhmn) . (A.1)
Subtracting from G
(
p , qNhmn)
the free Green’s function G0( p , q) 4 (2p)4d4( p 2
q)( p2
2 q2), which does not contribute to the scattering amplitude (A.1). As a result, we obtain F(p , qNhmn ) 4 lim p2, q2 K m2 (p2 2 m2)(q22 m2) i
0 Q dt ei(p2 2 m2) td4x ei(p 2q) xQ (A.2) Q[d4n]t 0 0 t dj M(xNj) 0 1 dl eil 0 t M(xNj). with M(xNj) 4 [n(j)1p]m[n(j) 1p]nhmnu
x 1 0 j [n(h) 1p] dhv
.Using (A.2) and making a similar change of the ordinary and functional variables (3.4) similar to the one in sect. 3, then passing to the mass shell, we obtain the following expression for the scattering amplitude:
F(p , qNhmn) 4
dx ei(p 2q) x[d4n]Q 2QM(xN0) 0 1 dl expk
il 2Q Q dj M(xNj)l
, (A.3) whereM(xNj) 4 [pu(j)1qu(2j)1n(j) ]m[pu(j) 1qu(2j)1n(j) ]nQ
Q hmn
u
x 12pju(j)12qju(2j)12
0
j
n(h) dh
v
. To calculate the functional integrals we use the straight path approximation, i.e. we assume that in the high-energy particle scattering on the smooth potential hmn, onemay neglect the dependence of the functional variables nm(h) [33]. As a result, we
have F(p , qNhmn ) 4
dx ei(p 2q) xM(xN0) 0 1 expk
il 2Q Q dj M(xNj)l
4d4x ei(p 2q) xQ (A.4) Qu
d daexpy
ik a Q[pu(j) 1qu(2j) ]m[pu(j) 1qu(2j) ]nhmn
(
pju(j) 1qju(2j) dj]z
v
a 40471
To simplify, we consider the case when the potential does not depend on time:
hmn(x) 4hmn
(r , t) 4hmn(r). Therefore, for the scattering amplitude we obtain the
following closed expression (4):
F(p , q) 42(2p)2d(p
02 q0) f (p , q) , (A.5)
f (p , q) 4 1
4 p
dr ei(p 2q) rM 8(r8 N0)dl exp [ilj(r) ] ,(A.6) where
.
`
/
`
´
M 8(rN0) 4 1 4[p 1q]m[p 1q]nh mn(r) x0(r) 4 1 2 NpN 2Q Qds
(
[pu(s) 1qu(2s) ]m[pu(s) 1qu(2s) ]nhmn(
r 1p×u(s) s1q×u(2s) s)
)
,(A.7)
with p× 4NpNONpN. In the case of weak gravitational field, the tensor hmn has the
form [34] h00 (r) 42f(r) , hab (r) 42dabf(r) , h0 a(r) 4ha0(r) 40 (a, b41, 2, 3) . (A.8) As a result, we have f (p , q) 4 1 2 p
dr ei(p 2q) r(p2 01 p2) f(r) 0 1 dl exp [ilx0(r) ] , (A.9) where x0(r) 4 1 NpN 2Q Q ds(p2 01 p2) f(r 1p×s) . (A.10)We direct the z-axis along the momentum p, but we place the x-axis into the plane defined by vectors p and q. In this coordinate system, for the phase (A.10) in the presence of small angles u b 1 we obtain the following expression:
(A.11) x0(p0, r) 4 2 p2 0 NpzN
u
0 Q ds f(x , y , z 1s)1 2Q 0 f(x 1s sin u, y, z1s cos u)v
B 4 2 p 2 0 NpzN 2Q Q ds f(x , y , z 1s) 4 2 p 2 0 NpzN 2Q Q dz f(r) .(4) The amplitude f ( p , q) is normalized by the relations
s 4 4 p NpN
Im f (p , q); ds
dV 4 Nf (p , q) N 2.
472
At small angles the momentum transfer is nearly perpendicular to the z-axis, therefore in eq. (A.6) we can put exp [i(p 2q) r] 4exp[iq»x»]. Integrating over dz and dl and taking account of (A.11), we find the Glauber representation for the scattering amplitude [30] f (p , q) 42iNpzN 2 p
d2x »e iq»x»(
exp [ix 0(p0, x) ] 21)
, (A.12)where x»4 (x , y , 0 ) and the eikonal phase x0( p0, x) is determined by eq. (A.11). Let us consider the Newton potential as a limit of the Yukawa potential when m K0, f(r) 4 (2kMe2mr
Or)m K04 (2 kM) Or , where k is the gravitational constant, k 4 6 Q 10239mn,
mnis the mass nucleon, M is the mass creating the potential. In this case the eikonal
phase (A.11) depends on the energy and takes the following form:
x0(p0, r) 42 2 kp2 0M NpzN
2Q Q dze 2mkx2»1 z 2k
x2»1 z 2 4 2bK0(mNx») , (A.13)where b 4 (kMp02) O
(
2 pNpzN)
. For the scattering amplitude, we have the followingexpression: f (p , q) 42iNpzN 2 p
d2 x»e2iq »x»(
exp [2ibK0(mNx») ] 21)
. (A.14)Evaluating the integral in (A.14), and preserving only terms which do not disappear when m K0, we get f (p , q) 42kMp 2 0 pt G( 1 1ib)
G( 1 2ib)exp
y
2ib lng
kt mC
h
z
,(A.15)
where t 42q2
»; C is the Euler constant. The phase diverges when m K0 in (A.15) as it is well known, by the long-range of action of the Newton potential [35]. From comparing (A.14) with the result of [31], devoted to the scattering of the scalar particle by the vector potential, one may conclude that (agravOsvec) B (k2M2Oe2) p02. The poles of the amplitude determined by (A.14) give the discrete energy levels of particles in the Newton potential En4 2 k2m2 M(m 1M) 8 p2 1 n2 , n 41, 2, 3, R . (A.16)
If in eq. (A.16) we put m 4MBmnucleon, we obtain the energy of the ground state which is equal to E14 9 .4 Q 10270eV .
R E F E R E N C E S
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