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Rise and fall of Regge trajectories in potential scattering
Volume
PHYSICS
28B, number 9
RISE
AND
FALL
OF REGGE
LETTERS
TRAJECTORIES
3 March
IN POTENTIAL
1969
SCATTERING
H. H. ALY * International
Centre
for
Theoretical
Physics,
Trieste,
Italy
and Department
of Physics,
P. NARAYANASWAMY American University of Beirut, Received
Beirut,
Lebanon
17 January 1969
An outline is given of a model in potential scattering which exhibits indefinitely rising Regge trajectories. The model has built into it several features which one would require in a relativistic theory based on rising trajectories.
Extensive interest has recently been shown in dynamical schemes based on indefinitely rising Regge trajectories. There is also considerable experimental evidence for the asymptotic rise of trajectories [l] and relativistic models have been constructed which exhibit rising meson Regge trajectories, especially for the p-trajectory [2]. It has been commonly assumed that potential theory cannot give rise to rising trajectories and that the trajectories would turn over at sufficiently high energy. This led some authors to conclude that a similar feature would also prevail in relativistic theories. Specifically, we refer to the conclusion of Khuri [3] that Re (Y(S)cannot increase indefinitely without violating some basic assumptions of Regge pole theory and dispersion relations. The analyticity properties of the residue function for a Yukawa potential was the basis of such claims. On the other hand, a number of models have been suggested [2,4] which exhibit rising trajectories. In these models, the analyticity properties of a(s) and /3(s) have been exploited and detailed approximation schemes developed to calculate Re (Y(S)for s - 03. Mandelstam [4] has pointed out several attractive features of such models, viz., (1) it is possible to combine the narrow resonance width approximation with Regge asymptotic behaviour and (2) Levinson’s theorem is not satisfied as is indeed evident from the decrease of form factors. It is however still believed [4] that the trajectory will always turn over in potential theory. * Present address: American University of Beirut, Beitur, Lebanon.
It is therefore of great interest to look at potential theory more carefully with a view to show that the feature of indefinitely rising trajectories can also be built into potential theory. In this note we consider scattering off singular potentials and demonstrate the monotonic increase of Re (Y(S)as s increases to to. We also show that all the requirements (except crossing symmetry, of course) discussed by Mandelstam [4] are met with in non-relativistic potential scattering. In this context, we mean a very general class of potentials such that r2V(+‘) -) m as Y + 0 and for simplicity shall confine ourselves to the potential g2y-4, since this is an exactly solvable model for all values of k and 1. We consider the radial SchrGdinger equation for scattering off the potential V(T) = gh-4:
V(r) + p
-
$3
_cJ@(y) =O.
(1)
The form of the S-matrix for this potential has been given by various authors [5]:
s(Q)
=
exp( ink) cos 7$ + i[exp( - @) + sinan@]+
(2)
where and the function 9 is found by using the WKH method: cos n@ = 1 - 2A(O) sin2 &A
(3) 603
Volume 28B, number 9
PHYSICS LETTERS
~ ( k , k ) = y ( k 2 - 2 i k g ) F ( ~ , ½;2; k2 - 2 i k g ~ + k2 +2ikg/ 2(k2+2ikg) ½
(4) 1
+ O[(k2 + 2ikg)-~]. In eq. (3), A(0) is a c e r t a i n d e t e r m i n a n t function of g, m o m e n t u m R and k = l + ½. We m a y add h e r e that t h i s a p p r o x i m a t i o n i s good if Ik2 + 2ikgl>> >> 1. The Regge p o l e s and t h e i r r e s i d u e s can b e l o c a t e d by the p o s i t i o n s of the p o l e s of the S - m a t r i x , eq. (2), given by
exp(,2~) =- 1 .
(5)
I m a ( s ) = s a(O)+½ fl(s) .
(12)
Eq. (12) is a consequence of u n i t a r i t y (13)
2i
Re ¢ ( k , k ) = 0
for the s c a t t e r i n g a m p l i t u d e
and the r e s i d u e functions for each Regge t r a j e c t o r y can be r e a d i l y c a l c u l a t e d [6] f r o m eq. (4), by w r i t i n g
(k-km)s(k,k)
lira
w h e r e s o i s the t h r e s h o l d , a i s a constant and the d i s p e r s i o n r e l a t i o n h a s been s u b t r a c t e d n t i m e s (n >/ 1). The s u b t r a c t i o n p o l y n o m i a l has been r e p l a c e d by the leading t e r m a s we a r e i n t e r e s t e d in the a s y m p t o t i c r e g i o n , s -* oo. Eq. (11) is a conse.quence of r e a l a n a l y t i c i t y for a(s) and fl(s)/q 2a(s), which fact is e s t a b l i s h e d if t r a j e c t o r i e s do not c r o s s [7]. Our second a s s u m p t i o n i s the u n i t a r i t y ~ [8], condition in the a p p r o x i m a t e f o r m which r e l a t e s [9] Iron(s) to the r e s i d u e 9(s)
1__ [A(llk ) -A*(l*,k)] = q2l+lA(l,k)A*(l*,k)
T h e Regge p o l e s lie along a line
[3m(S)=
3 March 1969
k --, k m
(7)
exp (2UXrn) cos nflkrn
¢'(xm)
A(l,k) = ~(s)/ {l - a ( s ) }
(14)
valid n e a r the pole at 1 = a(s). Re a(0) i s taken to be l a r g e r than -~.~ The a p p r o x i m a t i o n is valid if Im a(s) i s s m a l l , a s h a s been d i s c u s s e d by Cheng and Sharp [10]. We can show that t h e s e two a s s u m p t i o n s i m m e d i a t e l y l e a d to r i s i n g t r a j e c t o r i e s for the l e a d ing t r a j e c t o r y . Using the r e s i d u e function given by eq. (9), the d i s p e r s i o n r e l a t i o n t a k e s the f o r m
w h e r e S = k 2. In the l i m i t of l a r g e m, i . e . , for
the l e a d i n g trajectory, we have [6] ~m(s)
=
[21n(4.k2~
- ins]
Re a(s) = a s n-1 -1
= - t/ins
.
(9)
In g e n e r a l , f o r s c a t t e r i n g off a p o t e n t i a l r -n, the r e s i d u e function i s given by
~m(s) =
(10)
If the t r a j e c t o r y function a(s) is a s s u m e d to obey r e a l a n a l y t i c i t y , the following d i s p e r s i o n r e l a t i o n holds:
n-1
sn ~ + -- P J
Ima(s')ds'
so ( s ' ) n ( s '
604
.
(15)
Re a(s) =as n-1 +S n-1 P / 7;
(s')a(O)+½-n d s ' (16) In s '
= [exp{a(0) +{ - n } - 1 ] s n-1 I n s .
(17)
(n - 2 ) 2nln I 2Xm ] L(ig) 2/n k 1 - 2 / n
R e d ( s ) = as
~ (s')a(O)+½-nds' So ( s ' - s ) l n s '
F o r l a r g e S, we obtain
We a r e i n t e r e s t e d in the a s y m p t o t i c b e h a v i o u r f o r S ~ 0%
fire(s)
sn
- -~- P
(8)
.
-
s)
(11)
F o r n = 1, the f a c t o r in the exponent n e v e r v a n i s h e s s i n c e Re a(0) > - ½ and Re ~(s) r i s e s l o g a r i t h m i c a l l y if a(s) s a t i s f i e s the once s u b t r a c t e d d i s p e r s i o n r e l a t i o n . F o r twice s u b t r a c t e d d i s p e r s i o n r e l a t i o n , Re a(s) r i s e s like S In S, p r o v i d e d Re a(0) ¢ ½. In g e n e r a l , indefinitely r i s ing t r a j e c t o r i e s r e s u l t f r o m s u b t r a c t e d d i s p e r s i o n r e l a t i o n s for the t r a j e c t o r y function. It should be e m p h a s i z e d h e r e that in the model The S-matrix for singular potentials of the form r - 4 has been shown to satisfy elastic unitarity [8]
Volume
28B, number
9
PHYSICS
LETTERS
considered here, we do not need to impose the vanishing of Im a(s) as s - m, a condition necessary in Mandelstam’s scheme [4]. On the other hand, the 1nS or S ln S behaviour of Re CT(S)comes out as a consequence of real analyticity of a(s). This is a very general feature and is true for all potentials which are singular like Y-~ (n > 3). Mandelstam has already pointed out [4] another feature of rising trajectory models, viz., that the phase shifts never fall. This is exactly what happens in the scattering by singular potentials where the phase shift is asymptotically [8]: 6(S) = ax - ,,a
.
(18)
This behaviour of the phase shift renders Levinson’s theorem invalid, which is what we expect. We would like to mention in passing the Pad6 approximant calculation by Masson [ll] who wrote down the first three leading terms of the expansion for large S for Z(Z + 1) = o(Y) [a(R) + l] and obtained the trajectories
Q)(s) =-a(,l)(s)_
and al is a constant. where n = 0,1,X.. jectory has the asymptotic behaviour
Ima This reinforces
- - (n+l)
rise of Re o(s) based on analytic solution. In conclusion, we want to emphasize the fact that the fall of the residue function S(S) - s-l in Mandelstam’s model is to be compared with the behaviour /3(s) - [Ins]-1 in the singular potential model, while the phase shift in the potential model rises as s - 00, a feature assumed by Mandelstam for his dynamical scheme. Finally we make no assumption in the potential model with regard to the vanishing of Im a(s) as s + 00 in order to establish the rising behaviour of Re a(s), which is to be contrasted with the case in Mandelstam’s scheme. One of us (H.H.A.) would like to thank Prof. A. Salam and Prof. P. Budini, International Centre for Theoretical Physics, where this work was started. Thanks are also due to Dr. J. G. Taylor for his very useful discussion.
1. S. W. Karmanyos
2. 3. 4. 5.
The tra-
RecY(s) - s +
(20) .
our conclusion
10.
regarding
1969
References
(19)
1
3 March
the
11.
et al., Phys. Rev. Letters 16 (1966 709; See ref. 2 for a complete list. G. Epstein and P. Kaus, Phvs. Rev 166 (1968) 1633. N. N: Khuri, Phys. Rev: Letters 18 (1968) 1094. S. Mandelstam, Phys. Rev. 166 (1968) 1539. H. H. Aly and H. J. W. MUller, Journ.‘Math. Phys. ‘7 (1966) 1. N. Dombey and R. H. Jones, Journ. Math. Phys. 9 (1968) 986. H. Cheng, Phys. Rev. 130 (1963) 1283. H. H. Aly and H. J. W. Miller, Journ. Math. Phys. 8 (1967) 367. E. J. Squires, Complex angular momenta and particle physics (W. A. Benjamin Inc., New York 1963) p.34. H. Cheng and D. Sharp, Ann. Phys. (N.Y.) 22 (1963) 481; Phys. Rev. 132 (1963) 1854. D. Masson, Nuovo Cimento 35 (1965) 125.
*****
605
PHYSICS
28B, number 9
RISE
AND
FALL
OF REGGE
LETTERS
TRAJECTORIES
3 March
IN POTENTIAL
1969
SCATTERING
H. H. ALY * International
Centre
for
Theoretical
Physics,
Trieste,
Italy
and Department
of Physics,
P. NARAYANASWAMY American University of Beirut, Received
Beirut,
Lebanon
17 January 1969
An outline is given of a model in potential scattering which exhibits indefinitely rising Regge trajectories. The model has built into it several features which one would require in a relativistic theory based on rising trajectories.
Extensive interest has recently been shown in dynamical schemes based on indefinitely rising Regge trajectories. There is also considerable experimental evidence for the asymptotic rise of trajectories [l] and relativistic models have been constructed which exhibit rising meson Regge trajectories, especially for the p-trajectory [2]. It has been commonly assumed that potential theory cannot give rise to rising trajectories and that the trajectories would turn over at sufficiently high energy. This led some authors to conclude that a similar feature would also prevail in relativistic theories. Specifically, we refer to the conclusion of Khuri [3] that Re (Y(S)cannot increase indefinitely without violating some basic assumptions of Regge pole theory and dispersion relations. The analyticity properties of the residue function for a Yukawa potential was the basis of such claims. On the other hand, a number of models have been suggested [2,4] which exhibit rising trajectories. In these models, the analyticity properties of a(s) and /3(s) have been exploited and detailed approximation schemes developed to calculate Re (Y(S)for s - 03. Mandelstam [4] has pointed out several attractive features of such models, viz., (1) it is possible to combine the narrow resonance width approximation with Regge asymptotic behaviour and (2) Levinson’s theorem is not satisfied as is indeed evident from the decrease of form factors. It is however still believed [4] that the trajectory will always turn over in potential theory. * Present address: American University of Beirut, Beitur, Lebanon.
It is therefore of great interest to look at potential theory more carefully with a view to show that the feature of indefinitely rising trajectories can also be built into potential theory. In this note we consider scattering off singular potentials and demonstrate the monotonic increase of Re (Y(S)as s increases to to. We also show that all the requirements (except crossing symmetry, of course) discussed by Mandelstam [4] are met with in non-relativistic potential scattering. In this context, we mean a very general class of potentials such that r2V(+‘) -) m as Y + 0 and for simplicity shall confine ourselves to the potential g2y-4, since this is an exactly solvable model for all values of k and 1. We consider the radial SchrGdinger equation for scattering off the potential V(T) = gh-4:
V(r) + p
-
$3
_cJ@(y) =O.
(1)
The form of the S-matrix for this potential has been given by various authors [5]:
s(Q)
=
exp( ink) cos 7$ + i[exp( - @) + sinan@]+
(2)
where and the function 9 is found by using the WKH method: cos n@ = 1 - 2A(O) sin2 &A
(3) 603
Volume 28B, number 9
PHYSICS LETTERS
~ ( k , k ) = y ( k 2 - 2 i k g ) F ( ~ , ½;2; k2 - 2 i k g ~ + k2 +2ikg/ 2(k2+2ikg) ½
(4) 1
+ O[(k2 + 2ikg)-~]. In eq. (3), A(0) is a c e r t a i n d e t e r m i n a n t function of g, m o m e n t u m R and k = l + ½. We m a y add h e r e that t h i s a p p r o x i m a t i o n i s good if Ik2 + 2ikgl>> >> 1. The Regge p o l e s and t h e i r r e s i d u e s can b e l o c a t e d by the p o s i t i o n s of the p o l e s of the S - m a t r i x , eq. (2), given by
exp(,2~) =- 1 .
(5)
I m a ( s ) = s a(O)+½ fl(s) .
(12)
Eq. (12) is a consequence of u n i t a r i t y (13)
2i
Re ¢ ( k , k ) = 0
for the s c a t t e r i n g a m p l i t u d e
and the r e s i d u e functions for each Regge t r a j e c t o r y can be r e a d i l y c a l c u l a t e d [6] f r o m eq. (4), by w r i t i n g
(k-km)s(k,k)
lira
w h e r e s o i s the t h r e s h o l d , a i s a constant and the d i s p e r s i o n r e l a t i o n h a s been s u b t r a c t e d n t i m e s (n >/ 1). The s u b t r a c t i o n p o l y n o m i a l has been r e p l a c e d by the leading t e r m a s we a r e i n t e r e s t e d in the a s y m p t o t i c r e g i o n , s -* oo. Eq. (11) is a conse.quence of r e a l a n a l y t i c i t y for a(s) and fl(s)/q 2a(s), which fact is e s t a b l i s h e d if t r a j e c t o r i e s do not c r o s s [7]. Our second a s s u m p t i o n i s the u n i t a r i t y ~ [8], condition in the a p p r o x i m a t e f o r m which r e l a t e s [9] Iron(s) to the r e s i d u e 9(s)
1__ [A(llk ) -A*(l*,k)] = q2l+lA(l,k)A*(l*,k)
T h e Regge p o l e s lie along a line
[3m(S)=
3 March 1969
k --, k m
(7)
exp (2UXrn) cos nflkrn
¢'(xm)
A(l,k) = ~(s)/ {l - a ( s ) }
(14)
valid n e a r the pole at 1 = a(s). Re a(0) i s taken to be l a r g e r than -~.~ The a p p r o x i m a t i o n is valid if Im a(s) i s s m a l l , a s h a s been d i s c u s s e d by Cheng and Sharp [10]. We can show that t h e s e two a s s u m p t i o n s i m m e d i a t e l y l e a d to r i s i n g t r a j e c t o r i e s for the l e a d ing t r a j e c t o r y . Using the r e s i d u e function given by eq. (9), the d i s p e r s i o n r e l a t i o n t a k e s the f o r m
w h e r e S = k 2. In the l i m i t of l a r g e m, i . e . , for
the l e a d i n g trajectory, we have [6] ~m(s)
=
[21n(4.k2~
- ins]
Re a(s) = a s n-1 -1
= - t/ins
.
(9)
In g e n e r a l , f o r s c a t t e r i n g off a p o t e n t i a l r -n, the r e s i d u e function i s given by
~m(s) =
(10)
If the t r a j e c t o r y function a(s) is a s s u m e d to obey r e a l a n a l y t i c i t y , the following d i s p e r s i o n r e l a t i o n holds:
n-1
sn ~ + -- P J
Ima(s')ds'
so ( s ' ) n ( s '
604
.
(15)
Re a(s) =as n-1 +S n-1 P / 7;
(s')a(O)+½-n d s ' (16) In s '
= [exp{a(0) +{ - n } - 1 ] s n-1 I n s .
(17)
(n - 2 ) 2nln I 2Xm ] L(ig) 2/n k 1 - 2 / n
R e d ( s ) = as
~ (s')a(O)+½-nds' So ( s ' - s ) l n s '
F o r l a r g e S, we obtain
We a r e i n t e r e s t e d in the a s y m p t o t i c b e h a v i o u r f o r S ~ 0%
fire(s)
sn
- -~- P
(8)
.
-
s)
(11)
F o r n = 1, the f a c t o r in the exponent n e v e r v a n i s h e s s i n c e Re a(0) > - ½ and Re ~(s) r i s e s l o g a r i t h m i c a l l y if a(s) s a t i s f i e s the once s u b t r a c t e d d i s p e r s i o n r e l a t i o n . F o r twice s u b t r a c t e d d i s p e r s i o n r e l a t i o n , Re a(s) r i s e s like S In S, p r o v i d e d Re a(0) ¢ ½. In g e n e r a l , indefinitely r i s ing t r a j e c t o r i e s r e s u l t f r o m s u b t r a c t e d d i s p e r s i o n r e l a t i o n s for the t r a j e c t o r y function. It should be e m p h a s i z e d h e r e that in the model The S-matrix for singular potentials of the form r - 4 has been shown to satisfy elastic unitarity [8]
Volume
28B, number
9
PHYSICS
LETTERS
considered here, we do not need to impose the vanishing of Im a(s) as s - m, a condition necessary in Mandelstam’s scheme [4]. On the other hand, the 1nS or S ln S behaviour of Re CT(S)comes out as a consequence of real analyticity of a(s). This is a very general feature and is true for all potentials which are singular like Y-~ (n > 3). Mandelstam has already pointed out [4] another feature of rising trajectory models, viz., that the phase shifts never fall. This is exactly what happens in the scattering by singular potentials where the phase shift is asymptotically [8]: 6(S) = ax - ,,a
.
(18)
This behaviour of the phase shift renders Levinson’s theorem invalid, which is what we expect. We would like to mention in passing the Pad6 approximant calculation by Masson [ll] who wrote down the first three leading terms of the expansion for large S for Z(Z + 1) = o(Y) [a(R) + l] and obtained the trajectories
Q)(s) =-a(,l)(s)_
and al is a constant. where n = 0,1,X.. jectory has the asymptotic behaviour
Ima This reinforces
- - (n+l)
rise of Re o(s) based on analytic solution. In conclusion, we want to emphasize the fact that the fall of the residue function S(S) - s-l in Mandelstam’s model is to be compared with the behaviour /3(s) - [Ins]-1 in the singular potential model, while the phase shift in the potential model rises as s - 00, a feature assumed by Mandelstam for his dynamical scheme. Finally we make no assumption in the potential model with regard to the vanishing of Im a(s) as s + 00 in order to establish the rising behaviour of Re a(s), which is to be contrasted with the case in Mandelstam’s scheme. One of us (H.H.A.) would like to thank Prof. A. Salam and Prof. P. Budini, International Centre for Theoretical Physics, where this work was started. Thanks are also due to Dr. J. G. Taylor for his very useful discussion.
1. S. W. Karmanyos
2. 3. 4. 5.
The tra-
RecY(s) - s +
(20) .
our conclusion
10.
regarding
1969
References
(19)
1
3 March
the
11.
et al., Phys. Rev. Letters 16 (1966 709; See ref. 2 for a complete list. G. Epstein and P. Kaus, Phvs. Rev 166 (1968) 1633. N. N: Khuri, Phys. Rev: Letters 18 (1968) 1094. S. Mandelstam, Phys. Rev. 166 (1968) 1539. H. H. Aly and H. J. W. MUller, Journ.‘Math. Phys. ‘7 (1966) 1. N. Dombey and R. H. Jones, Journ. Math. Phys. 9 (1968) 986. H. Cheng, Phys. Rev. 130 (1963) 1283. H. H. Aly and H. J. W. Miller, Journ. Math. Phys. 8 (1967) 367. E. J. Squires, Complex angular momenta and particle physics (W. A. Benjamin Inc., New York 1963) p.34. H. Cheng and D. Sharp, Ann. Phys. (N.Y.) 22 (1963) 481; Phys. Rev. 132 (1963) 1854. D. Masson, Nuovo Cimento 35 (1965) 125.
*****
605