Regge poles and perturbation theory

7.A

I

Nuclear Physics 4 9 ( 1 9 6 3 ) 1 7 0 - - 1 7 6 ;

(~)

North-Holland Publishing Co., Amsterdam

I

Not to be reproduced by photoprint or microfilm without written permission from the publisher

R E G G E P O L E S AND P E R T U R B A T I O N T H E O R Y A. A. LOGUNOV, NGUYEN VAN-HIEU, A. N. TAVKHEL1DZE and O.A. KHRUSTALEV Joint Institute for Nuclear Research, Laboratory of Theoretical Physics, Dubna, USSR Received 8 April 1963 Abstract: The contributions of the cuts in the/-plane to the scattering amplitude are investigated. A

way of separating the pole terms is discussed. These terms should be understood as the main ones in the expansion in powers of e~m(m~-E2)-~. 1. Introduction

In our previous papers 1-4) we suggested a method for calculating Regge trajectories by combining the ideas of Regge with perturbation theory. As has been shown in ref. 3), the extraction of the expression for a Regge trajectory from the perturbation series is considerably embarrassed by the presence of the cuts in the/-plane. We give here some examples illustrating the contribution of the cuts and a method for selecting the terms which determine the Regge trajectory. It is pointed out that these terms are the main ones in an expansion in powers of e2m (m2-- E2) -~ . 2. Scattering o f Charged Particles in the C o u l o m b Field

Consider first the scattering of charged scalar particles in the Coulomb field. The interaction Lagrangian is of the form L = i e(Av+

AeXt

v ) ~ * r v * - e 2 ( A ~ + A ~ext) ( A ~ + A ext v )cb * *,

(1)

where F v ~--- _ _ _

Ox~

0x~

A,. is the quantized electromagnetic field, and Avext is the static Coulomb field A ex' = 0,

e

A~/~t = -- - .

(2)

r

The sign of the charge was chosen so that the particle would be attracted to the centre. It was stated in ref. 5) that the scattering amplitudes reads M = exp [~B(m, 2, t)]Mr,

(3)

where m and 2 are the particle and photon masses, respectively, and tis the m o m e n t u m 170

REGGE

POLES

171

transfer squared. Up to the second order in ct, we have Mr = M ( ' ) + M~2),

(4)

where M °) is the scattering amplitude in the first order, and M(~z) is related to the scattering amplitude in the second order M (2) by M~2) = M(2) _ ~ B M (1).

(5)

C a l c u l a t i n g M ~2) according to the diagrams in fig. 1 and using for B the expression from ref. 5), we get M~2) = 4root2

E _ 1 in t t

]

I

I

I

i

l

x

m

1

2rc2~ 2

(6)

E

/ \

/ /

\

\

× o

b

c

Fig. 1, Scattering of charged scalar particle in Coulomb field. Diagrams for second order amplitude.

Among the terms in (6) which give the contributions of the type (1/t)ln(t/m2), we wrote explicitly the first term only. The other terms are independent of E and in calculating the energy levels, [1 - (E2/m2)]-4 ~ 1, they give but small corrections. It should be noted that M (2) contains a term of the form (1/t)ln2(t/m2). However, this term vanishes when M~(2) is calculated with the aid of (5). As was shown in ref. 6) the presence of the term quadratic in the field leads to the appearance of a cut in the /-plane extending from - c t - ½ to ct-½. The second term in (6), corresponding to the diagram lc, contains a non-integral power of t, i.e. this is a contribution of the cut in the lowest order. Thus, the contribution of the Regge poles to the scattering amplitudes in the second order is 4rt~(

ctE

[M,]p = -~- 1 + x / m ~ E 2

In t )

~

.

(7)

Obviously the application of the renormalization group method gives the Regge trajectory ctE l(E) -- - 1 + x/m 2 - E 2 " (8) To elucidate the character of the contribution from the cut we consider the diagrams drawn in figs. 2 and 3. Those in fig. 2 fail to give the contribution, while the sum of the matrix elements of the diagram lc and of the diagrams in fig. 3 is found to be

1 {2~z~2+l[2g2~212+~[(2~2~2)a~]2+...}_

__

x/-t

oo

1 . = ~ (1~ ) . 8x/-t

2n

(9)

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A.A. LOGUNOV et al.

This sum is the total contribution of the cut if only the potential interaction is taken into account (the interaction with the quantized electromagnetic field is neglected). /\

I I I I

/ \ / /

\

/

\

~

A I 1 I

/

I-I 1 I

\

/

\

/

\

Fig. 2. Scattering of charged scalar particle in Coulomb field. Diagrams giving no contribution from cut. A

A

/\

~

/\

/

\

;~/

"\

[

/

\

/

\

A

/\ f

--A

/\ \

~\ //

/

\

\\

\ \

//

\\

Fig. 3. Scattering of charged scalar particle in Coulomb field. Diagrams giving contribution from cut.

The scattering of a spinor particle in the Coutomb field was treated in ref. 3). Here also only the potential part of the interaction is considered, whilst the interaction with the quantized electromagnetic field is neglected. In the second order consideration of the latter interaction leads to the appearance of diagrams like lb. Its matrix element contains terms of the form lnZ(t/m 2) or (1/t)lnZ(t/m2). However, these terms vanish when calculating 31(,2) in (3). Thus, 311(,2) looks like the amplitude T in ref. 3). The terms of the form ln(t/m z) or (1/t)ln(t/m z) corresponding to the diagram lb are now independent of E and are extremely small if compared with those of the form m

t In m2 #m 2 _E 2

m

or

1

t

# m 2 _ E 2 t In m 2 .

3. Boson-Fermion Scattering Now let us turn to the electromagnetic scattering of a fermion with mass m and charge + e on a boson with mass M and charge - e. The boson momenta in the initial and final states are denoted by ql and q2, respectively. The scattering amplitude is

Minv=eXp[~B(m'M'2's't'u)]{ A°+i(t1+~22

B°) "

(10)

Calculating Mi. v in the first and second orders and making use of the expression from ref. 5) for ~B, we get

Ao

= 4~2m

(s-2m2)(s+MZ-m2)f(s)

_1 In __t t m2 '

s

Bo =

-8rcc~l +4c~ 2 t

I_s+M2+2m2+m2

(11)

-m211t f(s)

lnmz

s

t

REGGEPOLES

173

where

f(m f(s) =

dsr

(13)

u+,.)2 (s,_s)x/Es,_(M +m)2][s,_(M_m)2]

In the given order the expression Ao and B 0 do not contain the terms of the form

(1/t)ln2(t/m2). As has been emphasized in ref. 4), the application of the renormalization group method to the invariant amplitudes A 0 and B 0 yields no information on the Regge poles as long as these amplitudes are not connected with physical states. To determine the Regge trajectory one has to consider the amplitudes fa and f2 in the centre-ofmass system connected with the total scattering amplitude by

M~m=eXp[~B]( fl+ia'[q2Aql]

fz •

(14)

The relationships between f l , f2 and A o, B o are given in many papers (see, e.g. ref. 7)). Using these, we have from (11) and (12)

[(~/s- m) 2 -- M2](~/s+ m) [

f' = - ~ [s-(M + m)2][s-(M-m) 2]

I + r/(s)In~]

-½~[(ff-s-m)Z-ME](~/s+m)Is -t [l+r/(s) In ~t ]

(15)

[(~/-s+ m)Z- M2](x/s-m ) I_ Fl+~(s)In tl s

t

m

m ~]

'

f,-zf 2 = -½~[(x/-s+m)2-M2](x/s-m)l [l+~(s)In t] -t ~'

(16)

where s is the square of the total energy in the centre-of-mass system,

{

¢(s) = 2n s - M 2 - 2 m Z - m 2 ~ { ~l(s) = ~ s - M Z - 2 m 2 - m

M2 _ m 2

s

m

+ #~_--~

(s-m2)(s+M2-m)

f(s),

M2-m2 m (s-m2)(s+M2-m2)}f(s) 2 --s + x/s+m s " "

(17)

(18)

Like in the extreme case M ~ oo treated in ref. 3), the first term in f l is the contribution of the cut from - ~ - 1 to e - 1. According to the results of this paper the contribution of the cut from - c t - 1 to c t - 1 is equal to

[ f , - z f 2 ] o = -~, [(~/;-mY'M?][(~/;+"O] - 1 + ~ ( s ) 2 s1t [

In ~ ] .

(19)

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A.A. LOGLINOV et al.

Subtracting (19) from (16) we obtain the following expression for the contribution of the Regge poles in f l - z f 2 : [ f , -- zf2]p = - ~

s-MZ--m21{l+ ~/s

t

~--m ( s + M 2 - m 2 ) f ( s ) I n

~2}

n #s

(20)

The renormalization group method gives the Regge trajectory in the form

l+(s) = - 1 + ~- m_ (s + M 2 - mZ)f(s). n ~/s

(21)

For ( M - m ) z < s < ( M + m ) z, we have

f(s) -

2

arctg s - (M

~j[(M+m)Z-s][s-(M-m)

2]

(22)

- m) 2

(M+m)2-s

In the extreme case the expressions (15), (16) and (19)-(21) go over into (.3), (4) and (9)-(11), respectively, of ref. 3). 4. Boson-Boson and Fermion-Fermion Scattering

In all the previous examples either the contributions of the cuts have an "incorrect" asymptotic behaviour in t, or it was known beforehand that they were the contributions of the cut. Consider now the electromagnetic scattering of scalar particles with masses m and M. The interaction Lagrangian in this case is

L = ieA~(4~*F~cb- ~o*F~q~)+ e2A~ Av(~b*q~ + 9*9).

(23)

Here 4~, ~* are the field operators with mass m, and q~, q~* are the fields with mass M, the charges of the particles being equal to + e and - e , respectively. The "main" diagrams for determining the Regge trajectory are those drawn in fig. 4, where Pl, ql are the initial momenta, and P2, q2 are the final ones, p2 = _rn 2,

q2 = _ M 2. I J *

I

J

___l

Fig. 4. E l e c t r o m a g n e t i c scattering of scalar particles.

The scattering amplitudes in the first and second orders in

e2

at t ~ oo read

T2 = e 2 (q' +q2)v(P, +P2)v = _ e 2 s - u ,

e4

r 4 = 16--;c2 4(p,q,)

[

t

t

(Pl +Pz)v(q, + q 2 ) v - ( P , + P 2 - q , - q 2 ) v ( P , - - q t ) ~

-

(24)

tl

~

Io

e4 [ 2 ( M 2 + m 2 - s ) ] 2 I o , 16n 2

(25)

REGGE POLES

175

where

t = -(Pl-P2)

s = -(pl+qa)

z,

io =

z,

u = -(pl-q2)

z,

(26)

In tf(s)" t

Here we are faced with the following situation: the amplitude in the lowest order has not the expected asymptotic behaviour, but in the next order it has. Let us suppose that with the aid of T 2 and T4 we constructed somehow the amplitude of the Regge type. Then we have to get the values of the energy of bound states up to e 2. However, in this approximation one should deal with the Coulomb interaction only. We pass to the centre-of-mass system and single out in (24) the products of the fourth components of the vectors Pl, q~ . . . . : (M2-m2)

2

s = ( E I + E 2 ) 2,

(ql+q2)4(Pl +P2)4 = S

(27)

where E1 and E 2 are the energies of the particles in the centre-of-mass system. This corresponds to the breaking up of s - u into

u

(M2- m2)2 and s S

(MS- m:)2 S

The first expression vanishes at threshold, the second one is finite there. Analogously, having distinguished in T4 the Coulomb term and having found the Regge trajectory corresponding to these terms, we can get the correct value for the Coulomb levels of a two-particle system. A similar situation arises in calculating the energy levels of positonium. It has already been pointed out in ref. 4) that none of the invariant amplitudes of the negaton-positon scattering in the fourth order has the expected asymptotic behaviour. Having separated the singlet part of the scattering amplitude, we got the amplitude with an asymptotic behaviour appropriate to obtain the energy levels. The triplet part of the scattering amplitude contains the terms which approach constants as t ~ oo. By performing the phase shift analysis, one can see that the appearance of these terms is due to the cuts. I f we remember that on the basis of the diagrams calculated in ref. 4) it is possible to get the energy levels up to e 2, while the spin effects contribute to the energy terms of order of e 4, we may hope to obtain the terms with a correct asymptotic behaviour. We followed this procedure, and as a result we got an expression coinciding with the singlet scattering amplitude, which is consistent with the spin independence of the energy levels in the e 2 approximation. Thus, in the case under consideration the terms determining the energy levels may not be the main ones at t --* oo. The asymptotic behaviour we are expecting to get is spoiled by the terms due to the cuts.

176

A.A. LOGUNOV et al.

The separation of the terms caused by the cuts becomes still much more difficult, for they do not only spoil the asymptotic behaviour, but also give terms "quite decent in their appearance" and with a correct asymptotic behaviour in t. The help comes from ref. a), where it was pointed out that to get the energy levels it is necessary to expand in powers of e 2 [ 1 - (E2/m2)] -½ rather than in powers of e z. Calculating the perturbation theory diagrams we get, as a rule,

e"qg(s, t)f(s, t), where f--* ~ as s approaches the threshold. Dividing ~(s, t) into the sum
The construction of the Regge trajectory with this term yields correct results. The authors express their deep gratitude to N. N. Bogolyubov, B. A. Arbuzov, G. Domokos, M. A. Markov, Ja. A. Smorodinsky, L. D. Soloviev, R. N. Faustov and A. T. Filippov for interest in the work and discussions. References 1) B. A. Arbuzov, A. A. Logunov, A. N. Tavkhelidze and R. N. Faustov, Phys. Lett. 2 (1962) 150 2) B. A. Arbuzov, A. A. Logunov, A. N. Tavkhelidze, R. N. Faustov and A. T. Filippov, Phys. Lett. 2 (1962) 305 3) B. A. Arbuzov, B. M. Barbashev, A. A. Logunov, Nguyen van Hieu, A. N. Tavkhelidze, R. N. Faustov and A. T. Filippov, Dubna, preprint, E-1095; Phys. Lett 4 (1963) 272 4) A. A. Logunov, Nguyen van-Hieu, A. N. Tavkhelidze and O. A. Khrustalev, Nuclear Physics, 44 (1963) 275 5) D. R. Yennie, S. C. Frautschi and H. Suura, Ann. Phys. 13 (1961) 379 6) R. Oehme, Nuovo Cim. 26 (1962) 183; V. Singh, Phys. Rev. 127 (1962) 631 7) G. Chew, M. Goldberger, F. Low and Y. Narnbu, Phys. Rev. 106 (1957) 1337