Nội dung

VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 1 (2017) 41-47 The Leading Eikonal of the Scattering Amplitude of Particles in Gravitational Field at High Energy Using the Partial Wave Method Nguyen Suan Han1,*, Nguyen Nhu Xuan2 1 Department of Theoretical Physics, Hanoi National University, Vietnam 2 Department of Physics, Le Qui Don University, Hanoi, Vietnam Received 12 December 2016 Revised 16 January 2017; Accepted 20 March 2017 Abstract : The scattering amplitudes for two spineless particles colliding at Planckian center-ofmass energies are considered by the partial wave method in quantum gravity. In the framework of the partial method, a scheme for finding the leading eikonal scattering amplitudes is developed and constructed. The connection between the solutions obtained by partial wave method, quasipotential and functional approaches is also discussed. Keywords: Eikonal scattering, partial wave method, gravitational field. 1. Introduction In recent years there have been important advances in our understanding of Planck scattering in quantum field theory and string theory ( M Pl  G 1/ 2 is called Planck mass at the energy scale about 1019 GeV ) [1-6]. This understanding give us a scientific basis to investigate the singularity, the formation of black holes and the loss of information near black holes as well as the modification of the string theory in quantum gravity. The research results have confirmed that [1-6] if gravitational interaction is considred, Planck scattering amplitude of two particles at high energy, s  M Pl (s is the square of the total energy of two particles in the center of mass system) and small fixed momentum transfer t (t is the square of momentum transfer) has the form Glauber with the scattering phase depends on energy in the limit  t / s    [1]. The calculation of high-level correction terms to the eikonal leading term of scattering amplitude has been studied by many authors, but this problem remains an issues. By using the integral method and quasi – potential equation we have obtained the analytical expression for the eikonal leading term in this problem [7]. To confirm this result, we revisit this problem using a new approach that is the partial wave method [8]. All results which have been obtained are compared. _______  Corresponding author. Tel.: 84-913059195 Email: lienbat76@yahoo.com 41 N.S. Han, N.N. Xuan / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 1 (2017) 41-47 42 The paper is organized as follow. In section 2, we introduce briefly the method to find scattering amplitude and scattering phase by using the partial wave method. Section 3 is devoted to compute the leading term and correction terms of scattering amplitude at high energy and small momentum tranfer. Finally, in section 4, we discuss and compare the results that we obtained in previous sections and draw conclude. 2. Scattering of uncharged particles in the gravitational field The covariant Klein-Gordon equation for the massless test particle with no electric charge and moving in the gravitational field is 1   (  g g    ) =0 , g (2.1) where g  det g  ( x )   g g  . The solution of classical Schwarzschild background field of slow target particle that is obtained by the Einstein equation has the form 2GM  ds 2    1  r  1 2GM   2  2 2 2 2 2  dt   1   dr  r ( d  sin  d ) , r    (2.2) where M is the mass of the target particle. In the center of mass frame of the particles, M << s . Main diagonal terms of the Schwarschild metric is determined by the expression g   2GM  0 1   r   1  2GM    0 1    r     0 0   0 0  0 0 r2 0      0   0  r 2 sin 2   0 (2.3) with 1 2GM  2GM     rr 2  2  g  r 2 sin , g tt    1   , g  1   , g  r ,g  ( r sin ) r  r    Using expression (2.3), we rewrite equation (2.1) in the form 1 2GM  GM   r 2 sin  1    tt  2r sin  1  r  r   2GM   r 2 sin  1  r  The wave function form    r    (2.4)   rr  cos   sin     0  of the test particle that obtain from eq. (2.1) or (2.4) is assumed to have the  ( r ,t )   ( r )eiEt  f(r ) .Ylm (  , )eiEt r (2.5) N.S. Han, N.N. Xuan / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 1 (2017) 41-47 43 where E is energy of the test particle as measured by an asymptotic observer From the Eqs. (2.5)-(2.6), we have  tt   tt ( r ,t )   E 2  r  f(r ) Ylm (  , )eiEt r r r f ( r )  f ( r )   f (r ) f (r ) Ylm (  , ).eiEt   r  2  Ylm (  , ).eiEt ; 2 r r   r   f ( r ) 2 r f ( r ) 2 f ( r )  iEt  rr   rr    Ylm (  , ).e 2 3 r r r   (2.6) f(r )  Ylm (  , ).eiEt ; r f(r ) f(r )    Ylm (  , ).eiEt ;    Ylm (  , ).eiEt r r Substituting Eq.(2.5) into Eq. (2.1) and using Eq. (2.6), one obtains the equation for wave function of the radial coordinate   1 2  2 2GM  2GM  2    2 r 1  E   r ( 1  )   L    ( r )  0 , r  r  r  r      (2.7) 2 2 1  1   (sin )   here L    ; và L Ylm (  , )  l( l  1)Ylm (  , ) .   sin2   sin  From eq. (2.7), linearizing the Schwarzschild metric, substituting s  2ME ( s is the Mandelstam variable in quantum relativistic mechanics) and retaining terms up to order  2GM / r  , the radial equation of the l th partial wave is (for large l) 2  d 2 f ( r )  l( l  1)  G 2 s 2 2GsE    E2  f ( r )  0 . 2 2 dr r r   (2.8) Thus, we can find the solution of this equation without adding a further approximation while keeping in mind that at Planck scale (about 10-33 cm) very small impact parameter scattering cannot be probed The radial equation (2.8) will be solved by using hypergeometric functions with its asymptotic form. It is By setting pl ( s ) pl ( s )  1  l( l  1)  G 2 s 2 (2.9) The eq.(2.8) is rewritten in form d 2 f ( r )  2 pl ( s )  pl ( s )  1 2GsE   E    f(r )0. d 2r r2 r   (2.10) Equation (2.10) is the hypergeometric equation. At far enough distance, l , its solution has form [9] 44 N.S. Han, N.N. Xuan / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 1 (2017) 41-47 p ( s ).  i  fl ( r )  sin  Er  l  ( i ) pl ( s )1 ( e 2il  1 )eiEr 2  2  p ( s ).   pl ( s ) 1 i l  sin  Er  l e sin  l .eiEr , Er  pl ( s )   ( i ) 2   (2.11) the phase shift of the partial wave is  2GsE f l ( r ). f0l ( r )dr r 0 E.sin  l    p ( s ). 2GsE    sin  Er  l  r   2 0  p ( s ).   pl ( s )1 i l e sin  l .eiEr  .sin  Er  l   ( i ) 2      dr  (2.12) In the first order approximation, it has the form  1    l  2Gs  .sin 2  Er  r 0 pl ( s ). 2   pl ( s )  1  iGs    dr    pl ( s )  1  iGs   (2.13) If the particle is free motion, eq. (2.11) become to p ( s ).   f0l ( r )  sin  Er  l , Er  pl ( s ) . 2   Wave function is expressed in terms of a partial wave expansion  f (r )  ( r )  ( r, , )  ( 2l  1).i pl ( s ) l P(cos ) l Er l 0 (2.14) (2.15) in view of the spherical symmetry eiEr eiEr (2.16)  f0l ( r )  f (  ). r r The scattering amplitude in the gravitational field is found in terms of a partial wave expansion  ( r )   0 ( r )  f (  ). f ( )   i  )   2l  1 e2il  1 P(cos l 2E l 0 , (2.17) where, the phase shift of the partial wave, characterized by a fixed angular momentum quantum number l ≫ 1, is determined by eq. (2.16) and pl ( s ) is determined by eq. (2.9). In the centre of mass frame (cms) of the particles, because in ref.[10]), the scattering amplitude is s  2E is the total energy (see eq. (9) 1  ) (2.18)  2l  1 e2il  1 P(cos  l i s l 0 It is not difficult to show from eq. (2.13) that, for fixed l, the phase shift has singularities at center mass energies f ( )  pl ( s )  1  iGs   N with N  0 (2.19) i l( l  1)  N( N  1) 2N   1 (2.20) or Gs  N.S. Han, N.N. Xuan / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 1 (2017) 41-47 45 for any non-negative integer N. Although still located on the imaginary axis of the complex splane, clearly the locations of these poles are quite distinct from those seen in the eikonal limit [2], viz., at Gs  iN . Here, we only consider the first order correction term for the leading term (leading eikonal) of the scattering amplitude in the limit l   . In case of large and fixed l, the explanation of the existence above poles outside the eikonal limit according to string theory was disscussed in [8]. The formula above also permits us to extract the leading order corrections to the eikonal limit l   , by using the asymptotic expansion of the argument of the gamma function in increasing inverse powers of l. We obtain 1   Gs   1  l  Gs log l     O 3  . (2.21) 2 2l  2l  l  The first term in eq. (2.21) obviously corresponds to the eikonal result, and the sub-leading corrections have been anticipated from reggeized string exchange diagrams [3]. The leading correction 2   above to the eikonal phase shift behaves as  Gs  / l 2 log s . 2 By using quantum mechanics, we will not receive correction terms which are logarithmic functions [8]. Therefore, we need to use formalism of quantum field theory to achieve this aim. 3. The correction terms of leading term for scattering amplitude For the scattering of particles at high energy and small momentum transfer, we can convert the sum into an integral in l in the expression of the scattering amplitude (2.17): f ( s,t )  f (  )   i s   0 dl( 2l  1)P(cos  ) e 2il  1 l (3.1) ( 2l  1 ) 2l  1 s  so that dl  Edb  db , here b is called impact parameter and the 2E 2 s Legendre polynomial convert to the Bessel function of zezoth order Set b    k  high P(cos  )    J 0  ( 2l  1)sin  . l   small 2  The expression for the scattering amplitude is found f ( s,t )   i s 2   0   bdb.J 0  ( 2l  1)sin  e 2il  1 2  (3.2) (3.3)       When small angle  then sin    , ( 2l  1)sin    ( 2l  1)  Eb , Eq. (3.3) becomes to 2 2 2 2 f ( s,t )   i s 2   0 bdbJ 0 ( Eb ) e 2il  1  1 Note that  bdbJ 0 ( Eb ) e2il  1  0 2 obtained in general form (3.4)   0 d 2beiEb e2il  1 , so the scattering amplitude is 46 N.S. Han, N.N. Xuan / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 1 (2017) 41-47 i s  2 iEb 2il d be e  1 4 0 Substituting the shift phase in Eq.(2.21) into Eq(3.5) we obtain f ( s,t )   f ( s,t )   i s 4   0  2iGs log l  2l1  ...  i s   d 2beiEb e  1   4   Expanding at large l, l f ( s,t )   i s 4   0 0 iGs   d 2beiEb ( l )2iGs .e l  1    iGs ( iGs )2  d 2beiEb .( l )2iGs 1    ... . 2 l l     0 (3.6) 1 , we have In Eq.(3.7), if l is large, we substitute l  bE  i s f ( s,t )   4   (3.5) 2 d be iEb b s  .   2  2iGs (3.7) b s into Eq.(3.7) and obtain finally expression 2  2iGs  2iGs 2   1    ... b s b s    (3.8) The factor precedes the parenthesess in Eq.(3.8) is the leading eikonal term, the other terms in the parenthesess are the correction terms of scattering amplitude. Phases of them increase in proportion to the square root of energy s . In our recent paper [7, 11], we obtained exact expressions for the scattering amplitude of two particles in quantum gravity by using functional integration method. In cms system at Planck energy and small momentum transfer, we used eikonal approximation to calculate the integrals and obtained the leading term and the first order correction term for the leading term. (1/ 2)b  2 v1   d 0 p1 q2 q1 p2 (1/ 2)b Fig. 1. The accuracy of the eikonal approximation ki k j  0, i  j  at high energy and small momentum transfer was confirmed in the frame of perturbation theory. 2  t , the phase function of the scattering amplitude has the In the high-energy limit s  M PL eikonal form and increases with energy [7]. N.S. Han, N.N. Xuan / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 1 (2017) 41-47 47 If mass of the changed particle is small, we will obtain the following expression for the scattering amplitude in quantum gravity in the zeroth order of mass   f ( s,t )  2is  d be 0 2 i b   |b|    2  i 2 s 2    2 s    2 s 2  1      ...   2   2   (3.9) with   0.5772... is the Euler Mascheroni constant. The factor precedes the parenthesess in Eq.(3.9) is the leading eikonal term, the other terms in the parenthesess are the correction terms. Phases of them are proportional to s. 4. Conclusion From eq.(3.8) and eq.(3.9) give us that the dependence of the correction terms on energy are qualitatively not the same. To explane this problem is that eq.(3.8) was obtained basing on regular perturbation theory. It has not been proven in quantum gravity because of alternating signs of correction terms [4]. Eq. (3.9) was found in the frame of functional integration method in quantum gravity, it is not related to perturbation theory. Expression (3.9) is the same as the expression obtained by using quasi-potential equation [11, 12]. Note that eikonal representation for scattering amplitude of particles at high energy in quantum field theory was first found by using quasi-potential equation [11]. Acknowledgments This work was completed with financial support from the National Foundation for Science and Technology Development of Vietnam and the Hanoi University of Science. References [1] G.'t Hoof , Phys. Lett. B198 (1987) 61; Nucl.Phys. B304 (1988) 867. [2] D.Amati, M.Ciafaloni and G.Veneziano, Phys. Lett. B 197 (1987) 81; IJMPA, A3 (1988) 1615; Nucl. Phys. B347 (1990) 550; Phys. Lett. B289 (1992) 87; Nuc. Phys. B403 (1003) 707. [3] H.Verlinde and E.Verlinde, Nucl. Phys. B371 (1992) 246. [4] D.Kabat and M. Ortiz, Nucl. Phys. B388 (1992) 570. [5] Nguyen Suan Han and Eap Ponna; Nuo Cim A, N110A (1997) 459. [6] Nguyen Suan Han, Euro. Phys. J. C, vol.16, N3 (2000) 547. [7] Nguyen Suan Han, Nguyen Nhu Xuan, e-print arXiv: gr-qc/ 0203054, 15 Mar 2002, 16p. Euro. Phys, J. C, vol.24, N1 (2002) 643. [8] S.Das and P. Majumdar, Phys. Rev. Lett. 72 (1994) 2524, Phys.Rev. D15(1995); IMSc /94-95/ 61; hepth/9504060 11 Aprel 1995. [9] A. S. Davydov (1963), Quantum Mechanics, Fizmatgiz. [10] I. Lukaˇcevi´c (2013), Quantum Mechanics Lecture 8: Relativistic Quantum Mechanics, UJJS, Dept. of Physics, Osijek. [11] Nguyen Suan Han, Le Hai Yen and Nguyen Nhu Xuan, e-Print: arXiv:1201.0322 [hep-th] IJMP A, vol.27,N1 (2012) 1250004 (19 pages). [12] V.R.Garsevanishvili, V. A. Matveev, L.A. Slepchenko and A.N. Tavkhelidze, Phys. Lett. 29 B (1969) 191.