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Regge behaviour in an asymptotically free field theory
Volume 53B, number 4
PHYSICS LETTERS
23 December 1974
R E G G E B E H A V I O U R IN AN A S Y M P T O T I C A L L Y F R E E F I E L D T H E O R Y ~ J.L. CARDY *
Department of Physics, University of California, Santa Barbara, California 93106, USA Received 28 October 1974 We consider the high energy behaviour at f'Lxedmomentum transfer of ~oa theory in six dimensional space-time, as the simplest example of an exactly renormalisableasymptotically free theory. We find that the damping in transverse momentum of the full theory is sufficiently strong to give rise to moving Regge pole singularities, in contrast to a fixed-point theory which would naturally lead to a fixed square-root branch point. In studying the high energy behaviour of field theories at fLxed momentum transfer, it becomes clear that the large transverse momentum behaviour is crucial in determining the character of the relevant angular momentum plane singularities. In superrenormalisable theories, for example ~03 theory in four dimensions, the effective cutoff in k± is relatively strong, and these singularities are moving Regge poles and cuts [1 ]. The integrals over k± which remain after scaling out the Regge behaviour are convergent. However, in exactly renormalisable theories, for example quantum electrodynamics [2-4] and ~04 theory in four dimensions [5], these integrals do not converge, and give rise to extra powers of Ins m the asymptotic behaviour of a given diagram, which generally sum to fixed Regge singularities. Such considerations are, however, usually applied to the bare theory, and it becomes an interesting question as to whether, in an exactly renormalisable asymptotically free theory, the damping of the full vertices at large transverse momentum is sufficient to restore moving Regge pole behaviour in the full theory. In this paper we consider ~03 theory in six dimensions, which is the simples example of such a theory, albeit unphysical, and answer this question in the affirmative. This is an attractive idea if it extends to non-Abelian gauge theories, since moving Regge singularities would appear to be observed in nature. On the other hand, a theory with an ultraviolet stable fixed point not at the origin naturally leads to fixed Regge cuts of the square-root type already discovered in quantum electrodynamics [2-4]. We proceed to give an outline of our arguments; the details of the calculations will be given elsewhere. Considering firstly the bare theory, one can show that, in any given order in the coupling constant, the leading diagrams which are one-particle irreducible in the t-channel are, as usual, the simple ladder diagrams. In D < 6 dimensions the leading behaviour of the n-rung ladder diagram is (1)
L(nD) = G 2n s-1 [ l n ( - s ) ] n - I K(t) n-1
(1)
(n- 1)!
where G is essentially the coupling constant and d(D- 2)k.L
K(t) f =d(k2 + m 2) ((q± - k±)2 + m 2)
(t
~ q~ ~ ~
(2)
Summing over n we obtain a moving Regge pole. When D = 6 the expression (2) for K(t) diverges and (1) becomes L(n6) = G 2n s - l ( l n $)2n- 2/(2n _ 2)!,
(3)
independent of t. Summing over n gives the behaviour Supported by the National Science Foundation. * Permanent address: Daresbury Nuclear Physics Laboratory, Warrington, Lancashire,WA4 4AD. 355
Volume 53B, number 4
PHYSICS LETTERS
23 December 1974
L(n6) = ½{G2s -l+G2 + G2s-I-G2} ,
(4)
characteristic of two fixed poles. Such a behaviour violates elastic t-channel unitarity [6], which is an input of the ladder model. Thus there must be some subtle cancellation with nonleading terms. However, at this level one must also consider vertex and propagator insertions, since when D = 6 these are down by only powers of Ins. For example, the O(g 4) corrections to the single-rung ladder both behave like s -1 (In s). This leads us to consider the full ladder diagrams which satisfy the Bethe-Salpeter equation depected in fig. 1. (We could take into account all two-particle irreducible diagrams in the kernel. However it will emerge that only the large momentum behaviour of the kernel is relevant to our result. In that region the asymptotic freedom of the theory guarantees that the ladder diagrams dominate.) The Bethe -Salpeter equation is explicitly
(5)
F(p,p", q) = r(p,p", q) - i f d 6 p ' X(p,p',q) F(p',p",q) , where
K(p, p', q) = F (3) [(p - p ' ) 2 , (p + ½q)2, (p' + ½q)2 ] F(3) [(p = p ' ) 2 , (p _ ½q)2, (p' _ ½q)2 ] X p(E)(p _ p ' ) - I (r(2)(p + ½q) p(2)(p _ ½q) p(E)(p' + ½q) p(E)(p, _ ½ q ) ) - l / 2 ,
(6) F (n) are the truncated Green functions. Going into the/-channel centre of mass frame so that q = (V~, 0) we define the partial wave projection of K: It
Kl(Po,p, po,p , t) - (]+ 147r2 ) ( ] + 2 ) ~2~'2
f(sm0) K(p,p ,q) C?/2 ( c o s 0 ) d 0
(7) 0 where ~ = Ip I, etc. The Gegenbauer function C/3/2(cos 0) appears since we are in five spatial dimensions [7]. This projection partially diagonalises eq. (5), and after continuiing t below threshold and performing a Wick rot tation in the Po integration contour we find --
t
--P
6 ( Po,P, Po,P ,t) = K+j" f dPodP ' -' - "
"
3
t
Po,P, , t ) 6 ( Po,P ,Po,P , t). . . .Po,P . .
.
.
.
.
.
.
.
(8)
It is convenient to write K(p,p', q) as a dispersion integral over (p _ p ' ) 2
K(p,p',q)= f
dm2Tp.((P++'---~~:q)--~'(P~'+~q)2'm~2)-
(9)
m o ( P o - P o ) 2 +~2 +~ 2 -- 2p--fi' cos0 +m 2 ' so that the large m 2 behavior o f p reflects the large ( p - p ' ) 2 K/cc (] + 1 ) - l ( f f ~ ' )
fdm2p((p + ½q)2, (p' _+½q)2, m 2 ) ,
behaviour of K. We then find, using (7)
(lO)
where we have restricted ourselves to the vicinity o f ] = - 1 and explicitly exhibited the pole in K1. (It is possible to work without this restriction, and indeed strictly speaking, necessary, since we are not working in a weak coupling limit.) The condition that moving Regge poles arise is that eq. (8) be of the Fredholm type, that is f d Po d~ dPo, d~ , [K/(p o, p, - Po, t P- ,, t)] 2 < oo.
(11)
For then Kj has a discrete spectrum of eigenvalues [8], and if we denotes these by (] + 1 ) - l p n ( t ) and the corresponding eigenfunctions by In), the solution of (8) is 356
Volume 53B, number 4
PHYSICS LETTERS
p_½q ~
23 December 1974
p._½q
Fig. 1.
/~=~
(/+ l)-l#n(t) n 1-(j+l)-tpn(t)
In)(nl,
(12)
which exhibits moving poles at j = - 1 + Pn(t). The Fredholm condition (11) translates into the condition on p f d Po d~ dPo' d ? ' dm 2 dm'2 if2 if'2 p (Po, P, - Po, ' P- ' , m 2 ) p (Po, P, - Po, ' P- ', m '2~) < o .
(13)
To establish (13) we need to know the behaviour o f p for large values of its arguments, which is furnished bythe renormalisation group analysis. According to this the behaviour of the Green functions at large momenta is determined by the functions ~(g) and 7(g), which in this theory have the perturbation expansion [9] ~(g)-
3 ~ 4(41r) g3 + o ( g S ) ,
1
2
7(g) = 6(4rt) 3 g + O(g 4) .
(14, 15)
As a result the behaviour of r (2) and r(3) when their momenta become large O(X) is r (2) = O(X2(ln h ) - 2 a ) ,
r (3) = O((ln X ) - 1 / 2 - 3 = ) ,
(16)
where t~ = 1/18. This gives for p the asymptotic behaviour P = O(X-6(ln ?~)-1).
(17)
If we now scale all the integration variables in (13) by ?, we obtain an integral of the form oo
f ~.(lnd• )k)2 '
(18)
which converges. To complete the proof of (13) we need to investigate the convergence when some of the variables do not tend to infinity as rapidly as the rest. This corresponds to the case of exceptional momenta. To apply the renormalisation group to this case one must prove that the massless theory does not contain infrared divergences as some of the external masses tend to zero [10]. This appears to be the case in our model, since even two-particle phase space, which gives rise to trouble in four dimensions [10], is suppressed in six dimensions. With this assumption one can then examine each case in turn and confirm that (13) is indeed valid. We should emphasize that while a knowledge of the kernel at all values of the momenta is necessary to calculate the trajectory function, to prove the existence of a Regge pole we need only the behaviour at large momenta. However, because (18) converges very slowly, we can see that the approach of the trajectory 0~(t) to its asymptote at - 1 as t ~ -oo is also very slow. This is related to the fact that in the fixed angle limit, the usual power law behaviour is broken by logarithms. We now suppose that the ultraviolet behaviour of the theory is controlled by a simple zero of/3(g) which is not at the origin. In this case one finds p = O(~.-6),
(19)
so that the norm of the kernel is logarithmically divergent, and the equation is no longer of Fredholm type. Such 357
Volume 53B, number 4
PHYSICS LETTERS
23 December 1974
kernels have already been discussed in the literature [ 2 - 4 ] and we shall give the details elsewhere. In this case Kj has a continuous spectrum, and the equation analogous to (12) is ~o
#d/a
j,
Ira) (#1
(20)
The leading singularity is now a cut at/c = - 1 + go, and its nature is given by the behaviour of the normalised eigenfunctions I/a) as/a ~ / a o. Both these are determined by the behaviour of K. at large values of its arguments, so are independent o f t. We find that, as in the case of quantum electrodynamics ~, I#) (/~1 = O((#o - / a ) -1/2)
as
/~#o,
(21)
so that F / h a s a fixed cut o f the type ( [ - j c ) -1/2. We may hope that our result extends to the discussion of the vacuum singularity in gauge theories. The main difference from our simple model is in the question o f exceptional momenta, which may give trouble in four dimensions. This aside, we would expect such theories, if asymptotically free, to give rise to moving Regge poles above j = 1, in contrast to quantum electrodynamics, which has a fixed cut above j = 1. The author would like to thank R.L. Sugar for several useful discussions. There is an error in ref. [21 at this point. When corrected, it agrees with refs. [3,4].
References [1] R.J. Eden, P.V. Landshoff, D.I. Olive, J.C. Polkinghorne, The analytic S-matrix, Ch. III (Cambridge Univ. Press, 1966), and references cited therein. [2] H. Cheng and T.T. Wu, Phys. Rev. D1 (1970) 2775. [3] G.V. Frolov, V.N. Gribov, L.N. Lipatov, Phys. Letters 31B (1970) 34. [4] J.B. Kogut, Phys. Rev. D4 (1971) 3101. [5] J.D. Bjorken and T.T. Wu, Phys. Rev. 130 (1963) 2566. [6] V.N. Gribov, Nuclear Phys. 22 (1961) 249. [7] A. Erdelyi et al., Higher transcendental functions, Vol. lI, Ch. XI (McGraw-Hill, 1953). [8] R. Courant and D. Hilbert, Methods of mathematical physics, Vol. I, Ch. III (Intersclence, 1953) [9] A.J. Macfarlane and G. Woo, Nuclear Phys. B77 (1974) 91. [10] M. Creutz and L-L. Wang, preprint BNL-19078 (1974).
358
PHYSICS LETTERS
23 December 1974
R E G G E B E H A V I O U R IN AN A S Y M P T O T I C A L L Y F R E E F I E L D T H E O R Y ~ J.L. CARDY *
Department of Physics, University of California, Santa Barbara, California 93106, USA Received 28 October 1974 We consider the high energy behaviour at f'Lxedmomentum transfer of ~oa theory in six dimensional space-time, as the simplest example of an exactly renormalisableasymptotically free theory. We find that the damping in transverse momentum of the full theory is sufficiently strong to give rise to moving Regge pole singularities, in contrast to a fixed-point theory which would naturally lead to a fixed square-root branch point. In studying the high energy behaviour of field theories at fLxed momentum transfer, it becomes clear that the large transverse momentum behaviour is crucial in determining the character of the relevant angular momentum plane singularities. In superrenormalisable theories, for example ~03 theory in four dimensions, the effective cutoff in k± is relatively strong, and these singularities are moving Regge poles and cuts [1 ]. The integrals over k± which remain after scaling out the Regge behaviour are convergent. However, in exactly renormalisable theories, for example quantum electrodynamics [2-4] and ~04 theory in four dimensions [5], these integrals do not converge, and give rise to extra powers of Ins m the asymptotic behaviour of a given diagram, which generally sum to fixed Regge singularities. Such considerations are, however, usually applied to the bare theory, and it becomes an interesting question as to whether, in an exactly renormalisable asymptotically free theory, the damping of the full vertices at large transverse momentum is sufficient to restore moving Regge pole behaviour in the full theory. In this paper we consider ~03 theory in six dimensions, which is the simples example of such a theory, albeit unphysical, and answer this question in the affirmative. This is an attractive idea if it extends to non-Abelian gauge theories, since moving Regge singularities would appear to be observed in nature. On the other hand, a theory with an ultraviolet stable fixed point not at the origin naturally leads to fixed Regge cuts of the square-root type already discovered in quantum electrodynamics [2-4]. We proceed to give an outline of our arguments; the details of the calculations will be given elsewhere. Considering firstly the bare theory, one can show that, in any given order in the coupling constant, the leading diagrams which are one-particle irreducible in the t-channel are, as usual, the simple ladder diagrams. In D < 6 dimensions the leading behaviour of the n-rung ladder diagram is (1)
L(nD) = G 2n s-1 [ l n ( - s ) ] n - I K(t) n-1
(1)
(n- 1)!
where G is essentially the coupling constant and d(D- 2)k.L
K(t) f =d(k2 + m 2) ((q± - k±)2 + m 2)
(t
~ q~ ~ ~
(2)
Summing over n we obtain a moving Regge pole. When D = 6 the expression (2) for K(t) diverges and (1) becomes L(n6) = G 2n s - l ( l n $)2n- 2/(2n _ 2)!,
(3)
independent of t. Summing over n gives the behaviour Supported by the National Science Foundation. * Permanent address: Daresbury Nuclear Physics Laboratory, Warrington, Lancashire,WA4 4AD. 355
Volume 53B, number 4
PHYSICS LETTERS
23 December 1974
L(n6) = ½{G2s -l+G2 + G2s-I-G2} ,
(4)
characteristic of two fixed poles. Such a behaviour violates elastic t-channel unitarity [6], which is an input of the ladder model. Thus there must be some subtle cancellation with nonleading terms. However, at this level one must also consider vertex and propagator insertions, since when D = 6 these are down by only powers of Ins. For example, the O(g 4) corrections to the single-rung ladder both behave like s -1 (In s). This leads us to consider the full ladder diagrams which satisfy the Bethe-Salpeter equation depected in fig. 1. (We could take into account all two-particle irreducible diagrams in the kernel. However it will emerge that only the large momentum behaviour of the kernel is relevant to our result. In that region the asymptotic freedom of the theory guarantees that the ladder diagrams dominate.) The Bethe -Salpeter equation is explicitly
(5)
F(p,p", q) = r(p,p", q) - i f d 6 p ' X(p,p',q) F(p',p",q) , where
K(p, p', q) = F (3) [(p - p ' ) 2 , (p + ½q)2, (p' + ½q)2 ] F(3) [(p = p ' ) 2 , (p _ ½q)2, (p' _ ½q)2 ] X p(E)(p _ p ' ) - I (r(2)(p + ½q) p(2)(p _ ½q) p(E)(p' + ½q) p(E)(p, _ ½ q ) ) - l / 2 ,
(6) F (n) are the truncated Green functions. Going into the/-channel centre of mass frame so that q = (V~, 0) we define the partial wave projection of K: It
Kl(Po,p, po,p , t) - (]+ 147r2 ) ( ] + 2 ) ~2~'2
f(sm0) K(p,p ,q) C?/2 ( c o s 0 ) d 0
(7) 0 where ~ = Ip I, etc. The Gegenbauer function C/3/2(cos 0) appears since we are in five spatial dimensions [7]. This projection partially diagonalises eq. (5), and after continuiing t below threshold and performing a Wick rot tation in the Po integration contour we find --
t
--P
6 ( Po,P, Po,P ,t) = K+j" f dPodP ' -' - "
"
3
t
Po,P, , t ) 6 ( Po,P ,Po,P , t). . . .Po,P . .
.
.
.
.
.
.
.
(8)
It is convenient to write K(p,p', q) as a dispersion integral over (p _ p ' ) 2
K(p,p',q)= f
dm2Tp.((P++'---~~:q)--~'(P~'+~q)2'm~2)-
(9)
m o ( P o - P o ) 2 +~2 +~ 2 -- 2p--fi' cos0 +m 2 ' so that the large m 2 behavior o f p reflects the large ( p - p ' ) 2 K/cc (] + 1 ) - l ( f f ~ ' )
fdm2p((p + ½q)2, (p' _+½q)2, m 2 ) ,
behaviour of K. We then find, using (7)
(lO)
where we have restricted ourselves to the vicinity o f ] = - 1 and explicitly exhibited the pole in K1. (It is possible to work without this restriction, and indeed strictly speaking, necessary, since we are not working in a weak coupling limit.) The condition that moving Regge poles arise is that eq. (8) be of the Fredholm type, that is f d Po d~ dPo, d~ , [K/(p o, p, - Po, t P- ,, t)] 2 < oo.
(11)
For then Kj has a discrete spectrum of eigenvalues [8], and if we denotes these by (] + 1 ) - l p n ( t ) and the corresponding eigenfunctions by In), the solution of (8) is 356
Volume 53B, number 4
PHYSICS LETTERS
p_½q ~
23 December 1974
p._½q
Fig. 1.
/~=~
(/+ l)-l#n(t) n 1-(j+l)-tpn(t)
In)(nl,
(12)
which exhibits moving poles at j = - 1 + Pn(t). The Fredholm condition (11) translates into the condition on p f d Po d~ dPo' d ? ' dm 2 dm'2 if2 if'2 p (Po, P, - Po, ' P- ' , m 2 ) p (Po, P, - Po, ' P- ', m '2~) < o .
(13)
To establish (13) we need to know the behaviour o f p for large values of its arguments, which is furnished bythe renormalisation group analysis. According to this the behaviour of the Green functions at large momenta is determined by the functions ~(g) and 7(g), which in this theory have the perturbation expansion [9] ~(g)-
3 ~ 4(41r) g3 + o ( g S ) ,
1
2
7(g) = 6(4rt) 3 g + O(g 4) .
(14, 15)
As a result the behaviour of r (2) and r(3) when their momenta become large O(X) is r (2) = O(X2(ln h ) - 2 a ) ,
r (3) = O((ln X ) - 1 / 2 - 3 = ) ,
(16)
where t~ = 1/18. This gives for p the asymptotic behaviour P = O(X-6(ln ?~)-1).
(17)
If we now scale all the integration variables in (13) by ?, we obtain an integral of the form oo
f ~.(lnd• )k)2 '
(18)
which converges. To complete the proof of (13) we need to investigate the convergence when some of the variables do not tend to infinity as rapidly as the rest. This corresponds to the case of exceptional momenta. To apply the renormalisation group to this case one must prove that the massless theory does not contain infrared divergences as some of the external masses tend to zero [10]. This appears to be the case in our model, since even two-particle phase space, which gives rise to trouble in four dimensions [10], is suppressed in six dimensions. With this assumption one can then examine each case in turn and confirm that (13) is indeed valid. We should emphasize that while a knowledge of the kernel at all values of the momenta is necessary to calculate the trajectory function, to prove the existence of a Regge pole we need only the behaviour at large momenta. However, because (18) converges very slowly, we can see that the approach of the trajectory 0~(t) to its asymptote at - 1 as t ~ -oo is also very slow. This is related to the fact that in the fixed angle limit, the usual power law behaviour is broken by logarithms. We now suppose that the ultraviolet behaviour of the theory is controlled by a simple zero of/3(g) which is not at the origin. In this case one finds p = O(~.-6),
(19)
so that the norm of the kernel is logarithmically divergent, and the equation is no longer of Fredholm type. Such 357
Volume 53B, number 4
PHYSICS LETTERS
23 December 1974
kernels have already been discussed in the literature [ 2 - 4 ] and we shall give the details elsewhere. In this case Kj has a continuous spectrum, and the equation analogous to (12) is ~o
#d/a
j,
Ira) (#1
(20)
The leading singularity is now a cut at/c = - 1 + go, and its nature is given by the behaviour of the normalised eigenfunctions I/a) as/a ~ / a o. Both these are determined by the behaviour of K. at large values of its arguments, so are independent o f t. We find that, as in the case of quantum electrodynamics ~, I#) (/~1 = O((#o - / a ) -1/2)
as
/~#o,
(21)
so that F / h a s a fixed cut o f the type ( [ - j c ) -1/2. We may hope that our result extends to the discussion of the vacuum singularity in gauge theories. The main difference from our simple model is in the question o f exceptional momenta, which may give trouble in four dimensions. This aside, we would expect such theories, if asymptotically free, to give rise to moving Regge poles above j = 1, in contrast to quantum electrodynamics, which has a fixed cut above j = 1. The author would like to thank R.L. Sugar for several useful discussions. There is an error in ref. [21 at this point. When corrected, it agrees with refs. [3,4].
References [1] R.J. Eden, P.V. Landshoff, D.I. Olive, J.C. Polkinghorne, The analytic S-matrix, Ch. III (Cambridge Univ. Press, 1966), and references cited therein. [2] H. Cheng and T.T. Wu, Phys. Rev. D1 (1970) 2775. [3] G.V. Frolov, V.N. Gribov, L.N. Lipatov, Phys. Letters 31B (1970) 34. [4] J.B. Kogut, Phys. Rev. D4 (1971) 3101. [5] J.D. Bjorken and T.T. Wu, Phys. Rev. 130 (1963) 2566. [6] V.N. Gribov, Nuclear Phys. 22 (1961) 249. [7] A. Erdelyi et al., Higher transcendental functions, Vol. lI, Ch. XI (McGraw-Hill, 1953). [8] R. Courant and D. Hilbert, Methods of mathematical physics, Vol. I, Ch. III (Intersclence, 1953) [9] A.J. Macfarlane and G. Woo, Nuclear Phys. B77 (1974) 91. [10] M. Creutz and L-L. Wang, preprint BNL-19078 (1974).
358