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Dispersion relations approach for the gluonic pomeron
Physics Letters B 273 Not-th-Holland
( 199 I ) I33- I36
Dispersion
PHYSICS
relations approach
for the gluonic pomeron
LETTERS
6
*
A.P. Contogouris
and F. Lebessis
Rcceibed
2 September
199
I
A nonperturbative approach for the gluon~c (Ltpatov ) pomeron 1s presented. The approach is based on dispersion theory (I%‘/ D method ). The input (Born term) IS the O(cu,) gauge invariant set of Feynman graphs forgluon-gluon scattering. The resulting amplitude. analytically Lipatov pomeron
continued
in angular
momentum.
Since some time the Lipatov pomeron has occupied a prominent position in the study of the small-s behaviour ofthe structure function F(x, Q’) [l-3]. This is a two-gluon object leading to a fixed branch point in complex angular momentum /with intercept a!,, = 1+N,(g’/n’)ln
2,
leads to a lixed branch
to that of the
principle that leads to such branch points in complex /. Consider the integral equation %
J-t-) =.f;,(a) +i
s K( \O
.Y.
s’ ),f( .Y’ )ds’ ,
(2)
and suppose that the kernel is “marginally” singular, i.e. its norm diverges logarithmically: examples of such kernels are K(s, s’ ) = I/ (x+x ) and In (s’ /s) / (s’ -.Y) [ 61. Then as a function of the parameter 3.. the resolvent has a branch point [ 61. If J. depends analytically on 1. the solution has a branch point in 1. Following well known procedures of the theory of complex angular momentum we consider gluongluon elastic scattering in the t-channel (fig. la); its singularities in complex I determine the asymptotic behaviour of the amplitude for s-co. f=fixed. Let 141 =CM momentum of the gluon and 8=CM scattering angle. With )q) ’ 3 Y and cos 0~ z: f II
* Also supported by the Natural Sciences and Engineering Research Council of Canada and by the Quebec Department of Education.
03.50 0 1991 Elsevtcr Sctencc Publishers
similar
(1)
where g= QCD coupling constant and lVc= number of colors. The techniques of Lipatov go beyond the Bjorken limit Q’-co, s=fixed: however. they are perturbative in nature. and it is questionable whether they are appropriate in the Regge limit Q’=fixed, In I/s+cx, [4]. The present paper is an effort to work directly in the Regge limit using nonpm.fLwhutive techniques. The approach. based on dispersion relations (DR). in particular the N/D method. has been extensively applied to strong interactions [ 51. Our result is also a fixed branch point in complex I, with similar intercept. We begin by briefly explaining the mathematical
0370-2693/91/S
point with an intercept
(y, +/‘1)z=4Y. (pi
-p2)2=
-2u(
s~(p,--p,)2=-2u(l-~).
1+:-)
(3)
We consider the nonflip helicity amplitude, to be denoted by I;.( 1,s), and its partial wave expansion:
B.V. ,AII rights reserved.
133
Volume 273, number 1,2
Fit, s) =
PHYSICS LETTERS B
~ (2•+ 1 ) F A t ) P / ( z ) ,
(4)
/
The Born term ( i n p u t ) of our approach is defined by the four O(g 2) graphs of fig. la which form a gauge invariant set. The term that determines the leading s asymptotic behaviour is
B(t,s)=-8g2N~V(s l l j + u l u S ) ,
(5)
We have left out terms contributing only to the partial w a v e / = 0 (graph ( d ) and part of graphs ( a ), ( b ) ) or to / = 1 (graph ( c ) ) and thus cannot be analytically continued in l. In the d e n o m i n a t o r s of (5) we have assigned a mass It to the exchanged gluon to permit a separation of the left-hand cut in t of F/(t) from the right-hand cut, necessary for writing dispersion relations; later we set/x = 0. The projection of ( 5 ) onto the hh partial wave gives
B/(v)=4geN,[l+(-l)qQ/
( '-) l+~v
12 December 1991
.... For these values o f / , along - c o < v < - II~2
AB/(v)=4g-~c~zPt
B~(v) has
1+
a discontinuity
0(- v-I/,2) .
The discontinuity Ak)(v) of the partial wave Fl(v) (the "force" of our interaction) will be defined in terms of (7). We remark that in this way the dem a n d s of the P h r a g m e n - L i n d e l 6 f theorem [7 ] are well satisfied: Partial wave unitarity requires (for 0
&( v) <~1/p ,
(8)
w h e r e p = 1/32zt a phase space factor. Then the theorem [ 7 ] d e m a n d s that for p ~ - oo: 0 < zSFt( v ) ~< l/p; for g2 not too large this is certainly true. On the other hand, for complex Z B/(p) has an additional discontinuity along - ],u 2 < v < 0. This extra piece is removed by considering the function fl/(v) = v - / B / ( v ) .
(6)
'
where Q/(~) = Legendre function of second kind. This has the correct signature, i.e. Bj(v) vaO for I=0, 2, 4, -L
(9)
Similar properties characterize the amplitude F~(v), which near the physical threshold (v = 0) behaves like v/ and for complex / develops an additional kinematic branch point [8]. Now we proceed with the N/D decomposition. As in (9), we factor out this branch point by carrying out the decomposition as follows:
v - % ( v ) =Nff v)/O/(v) , P,I a)
T P.
(a/
(b)
(d)
(7)
(10)
then N/(v) has the left-hand cut - ao < v < - ¼~z2 and D~(v) the right-hand cut 0 < v < o o due to unitarity. For N/(v ) we write a once-subtracted DR with subtraction point at v = vo:
&(v)=Xj(v0) --jr2~4
+
V-Vo [ - 7~
+ ....
B) Fig. 1. (a) The gauge invariant set o f O ( g -2 ) graphs defining our Born term. (b) Graphical representation depicting the content of our solution. (See also footnote 2. ) 134
J _ -/j
dr' Afll(v')Dt(v') /2'--
P
P'--
(11)
P0
where Aft/(v) the discontinuity along the left-hand cut to be defined in terms of the discontinuity of the Born term. For D/(v) we also write a once-subtracted DR; and as customary (with no loss of generality) take D/(v0) = 1, thus
dv' D / ( v ) = l - p -v-vo ~ - i --P' -- P 0
v'W/(v') P ' - - PO
(12)
273, n u m b e r
Volume
1,2
In the last expression we have adopted elastic unitarity throughout the range 0 < v < oo. The approach can be easily extended to incorporate any inelasticity provided that for v-~oo it tends to a constant ~ Eqs. ( 11 ) and ( 12 ) lead to an integral equation for D/(v). It is convenient to set v = - c o , v ' = - c o ' , vo= -COo and define
H~( co, ~o' ) -
(13)
d~ ~
1
h'~((J, oj')]_= ~
(~+co)(~+co') 0
Then we obtain the integral equation
~'({o) = Wo(co) g-N~.
+ ~-
Ii2/4
~o') & 1 dco' Hz({o, (0,1
qJ(co')
(14) where ~'o(co)-= 1 / ( o 9 o - o o ) - N / (
-o)o)H/(oo, COo)and
we use
Afll( co) = 4gZNcTr(o-/Pl(1- ~o)) ,
'
valid for - 1 < R e / < 1. Then we obtain W(oo) = ~'o(OJ)
03'~--Of
+ 87rsinJr~
1
doo' ( o ' - c o o ) , / g ( o o ' ) .
(15)
0
P - - Po "
Furthermore we introduce
CO'/--OJ
1
sin 7r/ o o ' - c o
gZNc f
De(v)
~,(o~) -
12 December 1991
PHYSICS LETTERS B
This is a marginally singular integral equation of the type solved and studied in ref. [6]. Its singular character can be traced to the fact that for [ vl --,oo the discontinuity zXB/(v) of the Born term tends to a constant. This results from the nature of the three-gluon vertices of the Q C D graphs of fig. la; or, stated differently, from the fact that the coupling g is dimensionless (see also ref. [10] ). Note that the Lipatov kernel is also marginally singular [ 1,2 ]. The integral equation (15) can be diagnonalized by a Mellin transform. Its general resolvent is
R(~o, (o':,~)
,f
- ilroo'
dr'
[sin 7 r ( 2 v + / - ½) - 2 ] -~
o ' - - ic~c
this coincides with the discontinuity of (9) for the interesting values l = 0 , 2, 4 ..... As in Lipatov et al. [ 1,2 ] we treat the coupling g as constant. A graphical representation of the content o f our solution is depicted in fig. lb ~2 Subsequently we set/~ = 0 and use the relation [ 9 ] =J Let q(v) be the inelasticity and let f o r v ~ o o : q ( u ) ~ M ( > 0 ) . We write q(u) = q ~ + [ q ( u ) - tl~]- The p a r t / / ~ leads to an integral equation identical to (14) with only g2N~ replaced by q**~2 ¥,. For the remaining part [ t/(v) - ~1~], using the resolvent of (14) (see below), a Fredholm integral equation can be set up giving the complete solution. For details see ref. [ 6 ]. ,2 Typically a partial wave DR is of the form
I
(
~k)( v' )
J
F/(v) = 7r
d u ' - -p ' _ _ p
left-hand cut
+ -
'
7(
f
dr' lmk?(v')) p'--
p
ilghl h a n d c u t
AFI(p)=ABI(p)
With the left-hand cut integral=B~(u), and with elastic unitarity the content of this DR is depicted in fig. lb.
(16) where 2-= cos (Jr(l+ 1 ) ) + ~g~Nc.The limits of the integration path lie in 0 < ( 7 < 1 - R e / ; thus the Mellin transform of ( l 5 ) exists for Re l < 1. The general resolvent ( 16 ) has fixed branch points at [6]
,Z:l, - 1 .
(17)
In fact, a resolvent with a branch point only at 2 = 1 is R(o), co'; 2)
(_oooo'_)~/~(oo_~)-//2co'+~sin(~qoln~o),
-- ().2-- l )'/2
09'--
(18) where qo= ( I / ~ ) l n [ - 2 + ( 2 2 - 1)~/2]. The solution of ( 15 ) corresponding to this resolvent can be determined explicitly [ 10]. We stress that (16) and (18) are strong interaction solutions, obtained by nonperturbative methods. 135
Volume 273, number 1,2
PHYSICS LETTERS B
12 December 1991
T h e b r a n c h p o i n t Z = 1, in the l i m i t o f w e a k coupling, leads to a b r a n c h p o i n t in l at
p o i n t in / at l = - 1 +g2Nc/4~z. T h i s implies a b r a n c h p o i n t i n j = / + a l +cr_~ at
I = - 1 + (g/~r) ( ~Nc) L/2
j = 1 + N~.g2/4~r.
(19)
A c c o r d i n g to a well k n o w n t h e o r e m [ 11,2 ], w h e n the external particles h a v e spin crj, or> a singularity in / o f the partial w a v e a m p l i t u d e c o r r e s p o n d s to a singularity in the total angular m o m e n t u m j = l + ~ + or> H e r e this i m p l i e s a b r a n c h p o i n t at j = 1 + (g/Tr)(~N,.),/2.
(20)
T h e result is s i m i l a r to eq. ( 1 ) and, as far as we can see, has s i m i l a r c o n s e q u e n c e s . An intercept e v e n closer in f o r m to ( 1 ) can be obt a i n e d by an a p p r o x i m a t i o n to the kernel o f eq. ( 1 4 ) that retains its m a r g i n a l l y singular character. We set c o = x 2, ca' = x '2 so that for Re l > 0 :
(24)
It is o f interest to note that for N , . = 3 and oLd-g2~ 4 ~ ~ 0 . 2 [2,3,12] the intercepts ( 2 0 ) , ( 2 4 ) a n d ( 1 ) are all close to each other. We are m u c h i n d e b t e d to L.N. L p a t o v and A.H. M u e l l e r for useful discussions and s t i m u l a t i n g c o m ments, and to S. P a p a d o p o u l o s and F.V. T k a c h o v , as well as to V. A n i s o v i c h , V. F a d i n , L. J e n k o v s k y and o t h e r p a r t i c i p a n t s o f the I n t e r n a t i o n a l C o n f e r e n c e on Elastic and d i f f r a c t i v e scattering (4th Blois W o r k s h o p ) , for interesting discussions.
References
eY
-
o)
(o' / -
x'
+
x
x'
F o r x ' - . c o and x fixed, neglecting in the p a r e n t h e s i s terms of O(x/x') or higher, and setting / t = 0 we obtain q/(x)-~q/o(x)+ '~i
dx' x,+.~-~,(x'),
(21)
0
where 2"- - ~g2N~,sin(tc(l+ 1 ) ). As m e n t i o n e d , this is also m a r g i n a l l y singular, o f a type solved and studied in detail in ref. [6]. T h e r e s o l v e n t kernel is [6]
R(-¢, x ' ; ,~)
,f
w0 + ix
-
2~zix'
dw
1 sin 7 r w - £ '
(22)
~ 0 - - i,:<
with 0 < Wo < 1. T h i s has again fixed b r a n c h p o i n t s at 2-'= - 1 and + 1. A r e s o l v e n t with a b r a n c h p o i n t only at 2 = - 1 is
R(x,x':2) 2 (xx')'/2 ( ~') = = ( 1 _ 2 e ) ~ / 2 x2_x,2 sin ( i + 0 o ) l n ,
(23)
w h e r e qo--- ( 1 / ~ ) l n [ - 2 + (2 -~- 1 )~/-~]. T h i s b r a n c h point, in the l i m i t o f w e a k coupling, leads to a b r a n c h
136
[ 1] v. Fadin, E. Kuraev and L. Lipatov, Phys. Lelt. B 60 (1975) 50; Zh. Eksp. Teor. Fiz. 72 (1977) 373. [2] L. Grivov, E. Levin and M. Ryskin, Phys. Rep. 100 (1982) 1: E. Levin and M. Ryskin, Phys, Rep. 189 (1990) 267. [3] A,H. Mueller, Nucl. Phys. Proc. Intern. Europhysics Confi on High Eenergy Physics ( Madrid, September 1989 ), (Proc. Suppl.) 16 (1990) 150. [4 ] J. Barrels, Particle World 2 ( 1991 ) 46. [5] See e.g.P.D.B. Collins and F.J. Squires, Regge poles in particle physics (Springer, Berlin, 1968 ). [6] D. Atkinson and A.P. Contogouris, Nuovo Cimento 39 (1965) 1082, 1102. [ 7 ] E.C. Titchmarseh, The theory of functions, 2nd Ed. (Oxford U.P.. Oxford), Section 5.6: A. Logunov, N. van Hieu, I. Todorov and O. Khrustalev. Phys. Len. 7 (1963) 69. [8] V. Gribov, Zh. Eksp. Teor. Fiz. 42 (1962) 1260; A. Barut and D. Zwanziger, Phys. Rcv. 127 (1962) 974; K. Bardaci, Phys. Rev. 127 (1962) 1832. [9] A. Erdelyi et al., cds., Tables of integral transforms, Vol. 2 (New York, 1954). [ 10] A.P. Comogouris, Nuovo Cimento 44 (1966) 927. [ 11 ] Ya. Azimov, Phys. Len. 3 (1963) 195. [ 12 ] L. Jenkovszky, F. Paeeanoni and E. Predazzi in: Proc. Intern. Conf. on Elastic and diffractive scattering (La Biodola, Italy, May 199l ), to appear; A.P. Contogouris and F. Lebessis, Proc. Interm. Conf. on Elastic and diffractive scattering (La Biodola, Italy, May 1990 ). to appear.
( 199 I ) I33- I36
Dispersion
PHYSICS
relations approach
for the gluonic pomeron
LETTERS
6
*
A.P. Contogouris
and F. Lebessis
Rcceibed
2 September
199
I
A nonperturbative approach for the gluon~c (Ltpatov ) pomeron 1s presented. The approach is based on dispersion theory (I%‘/ D method ). The input (Born term) IS the O(cu,) gauge invariant set of Feynman graphs forgluon-gluon scattering. The resulting amplitude. analytically Lipatov pomeron
continued
in angular
momentum.
Since some time the Lipatov pomeron has occupied a prominent position in the study of the small-s behaviour ofthe structure function F(x, Q’) [l-3]. This is a two-gluon object leading to a fixed branch point in complex angular momentum /with intercept a!,, = 1+N,(g’/n’)ln
2,
leads to a lixed branch
to that of the
principle that leads to such branch points in complex /. Consider the integral equation %
J-t-) =.f;,(a) +i
s K( \O
.Y.
s’ ),f( .Y’ )ds’ ,
(2)
and suppose that the kernel is “marginally” singular, i.e. its norm diverges logarithmically: examples of such kernels are K(s, s’ ) = I/ (x+x ) and In (s’ /s) / (s’ -.Y) [ 61. Then as a function of the parameter 3.. the resolvent has a branch point [ 61. If J. depends analytically on 1. the solution has a branch point in 1. Following well known procedures of the theory of complex angular momentum we consider gluongluon elastic scattering in the t-channel (fig. la); its singularities in complex I determine the asymptotic behaviour of the amplitude for s-co. f=fixed. Let 141 =CM momentum of the gluon and 8=CM scattering angle. With )q) ’ 3 Y and cos 0~ z: f II
* Also supported by the Natural Sciences and Engineering Research Council of Canada and by the Quebec Department of Education.
03.50 0 1991 Elsevtcr Sctencc Publishers
similar
(1)
where g= QCD coupling constant and lVc= number of colors. The techniques of Lipatov go beyond the Bjorken limit Q’-co, s=fixed: however. they are perturbative in nature. and it is questionable whether they are appropriate in the Regge limit Q’=fixed, In I/s+cx, [4]. The present paper is an effort to work directly in the Regge limit using nonpm.fLwhutive techniques. The approach. based on dispersion relations (DR). in particular the N/D method. has been extensively applied to strong interactions [ 51. Our result is also a fixed branch point in complex I, with similar intercept. We begin by briefly explaining the mathematical
0370-2693/91/S
point with an intercept
(y, +/‘1)z=4Y. (pi
-p2)2=
-2u(
s~(p,--p,)2=-2u(l-~).
1+:-)
(3)
We consider the nonflip helicity amplitude, to be denoted by I;.( 1,s), and its partial wave expansion:
B.V. ,AII rights reserved.
133
Volume 273, number 1,2
Fit, s) =
PHYSICS LETTERS B
~ (2•+ 1 ) F A t ) P / ( z ) ,
(4)
/
The Born term ( i n p u t ) of our approach is defined by the four O(g 2) graphs of fig. la which form a gauge invariant set. The term that determines the leading s asymptotic behaviour is
B(t,s)=-8g2N~V(s l l j + u l u S ) ,
(5)
We have left out terms contributing only to the partial w a v e / = 0 (graph ( d ) and part of graphs ( a ), ( b ) ) or to / = 1 (graph ( c ) ) and thus cannot be analytically continued in l. In the d e n o m i n a t o r s of (5) we have assigned a mass It to the exchanged gluon to permit a separation of the left-hand cut in t of F/(t) from the right-hand cut, necessary for writing dispersion relations; later we set/x = 0. The projection of ( 5 ) onto the hh partial wave gives
B/(v)=4geN,[l+(-l)qQ/
( '-) l+~v
12 December 1991
.... For these values o f / , along - c o < v < - II~2
AB/(v)=4g-~c~zPt
B~(v) has
1+
a discontinuity
0(- v-I/,2) .
The discontinuity Ak)(v) of the partial wave Fl(v) (the "force" of our interaction) will be defined in terms of (7). We remark that in this way the dem a n d s of the P h r a g m e n - L i n d e l 6 f theorem [7 ] are well satisfied: Partial wave unitarity requires (for 0
&( v) <~1/p ,
(8)
w h e r e p = 1/32zt a phase space factor. Then the theorem [ 7 ] d e m a n d s that for p ~ - oo: 0 < zSFt( v ) ~< l/p; for g2 not too large this is certainly true. On the other hand, for complex Z B/(p) has an additional discontinuity along - ],u 2 < v < 0. This extra piece is removed by considering the function fl/(v) = v - / B / ( v ) .
(6)
'
where Q/(~) = Legendre function of second kind. This has the correct signature, i.e. Bj(v) vaO for I=0, 2, 4, -L
(9)
Similar properties characterize the amplitude F~(v), which near the physical threshold (v = 0) behaves like v/ and for complex / develops an additional kinematic branch point [8]. Now we proceed with the N/D decomposition. As in (9), we factor out this branch point by carrying out the decomposition as follows:
v - % ( v ) =Nff v)/O/(v) , P,I a)
T P.
(a/
(b)
(d)
(7)
(10)
then N/(v) has the left-hand cut - ao < v < - ¼~z2 and D~(v) the right-hand cut 0 < v < o o due to unitarity. For N/(v ) we write a once-subtracted DR with subtraction point at v = vo:
&(v)=Xj(v0) --jr2~4
+
V-Vo [ - 7~
+ ....
B) Fig. 1. (a) The gauge invariant set o f O ( g -2 ) graphs defining our Born term. (b) Graphical representation depicting the content of our solution. (See also footnote 2. ) 134
J _ -/j
dr' Afll(v')Dt(v') /2'--
P
P'--
(11)
P0
where Aft/(v) the discontinuity along the left-hand cut to be defined in terms of the discontinuity of the Born term. For D/(v) we also write a once-subtracted DR; and as customary (with no loss of generality) take D/(v0) = 1, thus
dv' D / ( v ) = l - p -v-vo ~ - i --P' -- P 0
v'W/(v') P ' - - PO
(12)
273, n u m b e r
Volume
1,2
In the last expression we have adopted elastic unitarity throughout the range 0 < v < oo. The approach can be easily extended to incorporate any inelasticity provided that for v-~oo it tends to a constant ~ Eqs. ( 11 ) and ( 12 ) lead to an integral equation for D/(v). It is convenient to set v = - c o , v ' = - c o ' , vo= -COo and define
H~( co, ~o' ) -
(13)
d~ ~
1
h'~((J, oj')]_= ~
(~+co)(~+co') 0
Then we obtain the integral equation
~'({o) = Wo(co) g-N~.
+ ~-
Ii2/4
~o') & 1 dco' Hz({o, (0,1
qJ(co')
(14) where ~'o(co)-= 1 / ( o 9 o - o o ) - N / (
-o)o)H/(oo, COo)and
we use
Afll( co) = 4gZNcTr(o-/Pl(1- ~o)) ,
'
valid for - 1 < R e / < 1. Then we obtain W(oo) = ~'o(OJ)
03'~--Of
+ 87rsinJr~
1
doo' ( o ' - c o o ) , / g ( o o ' ) .
(15)
0
P - - Po "
Furthermore we introduce
CO'/--OJ
1
sin 7r/ o o ' - c o
gZNc f
De(v)
~,(o~) -
12 December 1991
PHYSICS LETTERS B
This is a marginally singular integral equation of the type solved and studied in ref. [6]. Its singular character can be traced to the fact that for [ vl --,oo the discontinuity zXB/(v) of the Born term tends to a constant. This results from the nature of the three-gluon vertices of the Q C D graphs of fig. la; or, stated differently, from the fact that the coupling g is dimensionless (see also ref. [10] ). Note that the Lipatov kernel is also marginally singular [ 1,2 ]. The integral equation (15) can be diagnonalized by a Mellin transform. Its general resolvent is
R(~o, (o':,~)
,f
- ilroo'
dr'
[sin 7 r ( 2 v + / - ½) - 2 ] -~
o ' - - ic~c
this coincides with the discontinuity of (9) for the interesting values l = 0 , 2, 4 ..... As in Lipatov et al. [ 1,2 ] we treat the coupling g as constant. A graphical representation of the content o f our solution is depicted in fig. lb ~2 Subsequently we set/~ = 0 and use the relation [ 9 ] =J Let q(v) be the inelasticity and let f o r v ~ o o : q ( u ) ~ M ( > 0 ) . We write q(u) = q ~ + [ q ( u ) - tl~]- The p a r t / / ~ leads to an integral equation identical to (14) with only g2N~ replaced by q**~2 ¥,. For the remaining part [ t/(v) - ~1~], using the resolvent of (14) (see below), a Fredholm integral equation can be set up giving the complete solution. For details see ref. [ 6 ]. ,2 Typically a partial wave DR is of the form
I
(
~k)( v' )
J
F/(v) = 7r
d u ' - -p ' _ _ p
left-hand cut
+ -
'
7(
f
dr' lmk?(v')) p'--
p
ilghl h a n d c u t
AFI(p)=ABI(p)
With the left-hand cut integral=B~(u), and with elastic unitarity the content of this DR is depicted in fig. lb.
(16) where 2-= cos (Jr(l+ 1 ) ) + ~g~Nc.The limits of the integration path lie in 0 < ( 7 < 1 - R e / ; thus the Mellin transform of ( l 5 ) exists for Re l < 1. The general resolvent ( 16 ) has fixed branch points at [6]
,Z:l, - 1 .
(17)
In fact, a resolvent with a branch point only at 2 = 1 is R(o), co'; 2)
(_oooo'_)~/~(oo_~)-//2co'+~sin(~qoln~o),
-- ().2-- l )'/2
09'--
(18) where qo= ( I / ~ ) l n [ - 2 + ( 2 2 - 1)~/2]. The solution of ( 15 ) corresponding to this resolvent can be determined explicitly [ 10]. We stress that (16) and (18) are strong interaction solutions, obtained by nonperturbative methods. 135
Volume 273, number 1,2
PHYSICS LETTERS B
12 December 1991
T h e b r a n c h p o i n t Z = 1, in the l i m i t o f w e a k coupling, leads to a b r a n c h p o i n t in l at
p o i n t in / at l = - 1 +g2Nc/4~z. T h i s implies a b r a n c h p o i n t i n j = / + a l +cr_~ at
I = - 1 + (g/~r) ( ~Nc) L/2
j = 1 + N~.g2/4~r.
(19)
A c c o r d i n g to a well k n o w n t h e o r e m [ 11,2 ], w h e n the external particles h a v e spin crj, or> a singularity in / o f the partial w a v e a m p l i t u d e c o r r e s p o n d s to a singularity in the total angular m o m e n t u m j = l + ~ + or> H e r e this i m p l i e s a b r a n c h p o i n t at j = 1 + (g/Tr)(~N,.),/2.
(20)
T h e result is s i m i l a r to eq. ( 1 ) and, as far as we can see, has s i m i l a r c o n s e q u e n c e s . An intercept e v e n closer in f o r m to ( 1 ) can be obt a i n e d by an a p p r o x i m a t i o n to the kernel o f eq. ( 1 4 ) that retains its m a r g i n a l l y singular character. We set c o = x 2, ca' = x '2 so that for Re l > 0 :
(24)
It is o f interest to note that for N , . = 3 and oLd-g2~ 4 ~ ~ 0 . 2 [2,3,12] the intercepts ( 2 0 ) , ( 2 4 ) a n d ( 1 ) are all close to each other. We are m u c h i n d e b t e d to L.N. L p a t o v and A.H. M u e l l e r for useful discussions and s t i m u l a t i n g c o m ments, and to S. P a p a d o p o u l o s and F.V. T k a c h o v , as well as to V. A n i s o v i c h , V. F a d i n , L. J e n k o v s k y and o t h e r p a r t i c i p a n t s o f the I n t e r n a t i o n a l C o n f e r e n c e on Elastic and d i f f r a c t i v e scattering (4th Blois W o r k s h o p ) , for interesting discussions.
References
eY
-
o)
(o' / -
x'
+
x
x'
F o r x ' - . c o and x fixed, neglecting in the p a r e n t h e s i s terms of O(x/x') or higher, and setting / t = 0 we obtain q/(x)-~q/o(x)+ '~i
dx' x,+.~-~,(x'),
(21)
0
where 2"- - ~g2N~,sin(tc(l+ 1 ) ). As m e n t i o n e d , this is also m a r g i n a l l y singular, o f a type solved and studied in detail in ref. [6]. T h e r e s o l v e n t kernel is [6]
R(-¢, x ' ; ,~)
,f
w0 + ix
-
2~zix'
dw
1 sin 7 r w - £ '
(22)
~ 0 - - i,:<
with 0 < Wo < 1. T h i s has again fixed b r a n c h p o i n t s at 2-'= - 1 and + 1. A r e s o l v e n t with a b r a n c h p o i n t only at 2 = - 1 is
R(x,x':2) 2 (xx')'/2 ( ~') = = ( 1 _ 2 e ) ~ / 2 x2_x,2 sin ( i + 0 o ) l n ,
(23)
w h e r e qo--- ( 1 / ~ ) l n [ - 2 + (2 -~- 1 )~/-~]. T h i s b r a n c h point, in the l i m i t o f w e a k coupling, leads to a b r a n c h
136
[ 1] v. Fadin, E. Kuraev and L. Lipatov, Phys. Lelt. B 60 (1975) 50; Zh. Eksp. Teor. Fiz. 72 (1977) 373. [2] L. Grivov, E. Levin and M. Ryskin, Phys. Rep. 100 (1982) 1: E. Levin and M. Ryskin, Phys, Rep. 189 (1990) 267. [3] A,H. Mueller, Nucl. Phys. Proc. Intern. Europhysics Confi on High Eenergy Physics ( Madrid, September 1989 ), (Proc. Suppl.) 16 (1990) 150. [4 ] J. Barrels, Particle World 2 ( 1991 ) 46. [5] See e.g.P.D.B. Collins and F.J. Squires, Regge poles in particle physics (Springer, Berlin, 1968 ). [6] D. Atkinson and A.P. Contogouris, Nuovo Cimento 39 (1965) 1082, 1102. [ 7 ] E.C. Titchmarseh, The theory of functions, 2nd Ed. (Oxford U.P.. Oxford), Section 5.6: A. Logunov, N. van Hieu, I. Todorov and O. Khrustalev. Phys. Len. 7 (1963) 69. [8] V. Gribov, Zh. Eksp. Teor. Fiz. 42 (1962) 1260; A. Barut and D. Zwanziger, Phys. Rcv. 127 (1962) 974; K. Bardaci, Phys. Rev. 127 (1962) 1832. [9] A. Erdelyi et al., cds., Tables of integral transforms, Vol. 2 (New York, 1954). [ 10] A.P. Comogouris, Nuovo Cimento 44 (1966) 927. [ 11 ] Ya. Azimov, Phys. Len. 3 (1963) 195. [ 12 ] L. Jenkovszky, F. Paeeanoni and E. Predazzi in: Proc. Intern. Conf. on Elastic and diffractive scattering (La Biodola, Italy, May 199l ), to appear; A.P. Contogouris and F. Lebessis, Proc. Interm. Conf. on Elastic and diffractive scattering (La Biodola, Italy, May 1990 ). to appear.