Regge trajectories with left-hand branch lines

Volume

30B. number 6

REGGE

PHYSICS

TRAJECTORIES

LETTERS

WITH

ICTP,

LEFT-HAND

R. OEHME * Miramare, Trieste,

Received

10 November

22 September

BRANCH

1969

LINES

Italy 1969

The conditions are explored under which Regge trajectories can have left-hand branch lines without two or more crossing pole surfaces appearing in the physical sheet of t.he continued partial-wave amplitude,

It is well known that Regge pole surfaces a(s) and their residua /3(s) do not have the left-hand unitarity branch points which are present in the continued partial-wave amplitude F(s, X) [l]. Except for the physical thresholds (and the superimposed winding points), any further singularities must be such that they do not induce fixed singular points of F(s,X) in the complex s-plane. A familiar example are possible branch points due to the coincidence of different Regge trajectories at a point [2,3] It is the purpose of this note to discuss the conditions under which an individual Regge trajectory can have branch points other than those associated with physical thresholds. In particular, we are concerned with the role of fixed and moving branch points of F(s,)c) in generating left-hand cuts in Regge pole surfaces. This problem is of importance for high-energy reactions; for example, it has been pointed out several years ago [4] that a cut in a trajectory a(s) for s C 0 could make it possible to have Re c!(s) = con&.

for

with (Y(S) being a non-constant

s-‘o analytic function $.

This is of obvious interest for models of elastic diffraction scattering and for other high-energy properties of amplitudes [3]. Our conclusions are the following. Left-hand cuts in individual Regge trajectories cannot simply occur because a pole and a branch point collide at a given point (s,h) = (so,ao). If we do not want two or more crossing Regge trajectories, or a forbidden fixed branch point in the s-plane of the amplitude F(s, A), we must use possible branch cuts of F(s, A) in the A-plane in order to remove secondary branches of the pole surface a!(s) to unphysical sheets. There are models with fixed cuts in the X-plane which naturally perform this task for a closely correlated trajectory (Y(S). We also present mathematical examples with moving branch points instead of fixed ones, and we consider the problem of using the branch point trajectories actually generated by the poles. Regge trajectories are pole surfaces h = CY(s) of the continued signatured partial-wave amplitude F(s, X), or of its “asymptotic part” D(s, X), which is given by an expression like

D(S, A) = ; j-i a

* Permanent

address: Enrico Fermi Institute. University of Chicago. Chicago. Illinois. USA. $ Some time ago, we have shown [5] that those fixed singularities in the complex h-plane for which the amplitude is unbounded are generally not compatible with the continued unitarity condition unless there are very specific shielding cuts [6]. We note that this result is different from that of Gribov [ 71 who shows that a high-energy beyjour like [(s, t) - C(s)t is not allowed for 4~ s s 16~ . For example. an asymptotic behaviour like F(s, t) - id(s)t + i 1 for real s C 0 and + p (s) P(@ with Re@(s RecY(s) > 1 for s > 4/.l 2) could be excluded by our theorem. but not by Gribov’s argument.

414

dv+2Q, (1+z)

2q2

Ab, s) ,

0)

where A(v, s) is the sum or difference of the absorptive parts in the t- and u-channels. The lower limit a2 of the integral can be made as large as we please in order to remove left-hand branch points towards -Q, [8]. If D-l(s,X) is holomorphic in the neighbourhood of a point (s o, ao) belonging to a Regge pole surface h = (Y(S) given by D-l(s,

o!(s)) = 0 )

then, near s = so, this surface the form

(2) is in general

of

Volume

30B, number 6

(Y(s)= “0

PHYSICS

+ Jgo aj (s -so) j/n

)

where n is some integer. This expression is a direct consequence of the preparation theorem of Weierstrass [Q]. For n a 2, we have two or more Regge trajectories which all cross at the point (s o, 01~). They correspond to the n branches al(s), 02(s), * . . , an(s) of the multivalued function (3), and they must all be pole surfaces of F(s,X) in order to ensure that this function has no fixed branch point at s = so *. For later discussions, it may be instructive to consider the case n = 2. Here we may have near (so, oo)

F(S,A) =

Pi(S) A - q(s)

&3(s) + A - cY2(s) + * * * ’

(4)

with ‘Y12(S)=aO*al(s-sO)~+...

(5)

,

and

P1,2(s) = P(so) f bl(s -

~0)~ +... ,

(6)

so that F(s, A) is independent of the sign of the root. In physical applications, we may not want to have crossing Regge pole surfaces. On the other hand, we know from perturbation theory that there are branch point trajectories Q~(s) which have common points with related pole trajectories o(s) [lo]. If we have o(so) = oc(so) = oo, the question arises whether there can be a branch point in o!(s) at s = so so that only one branch of the function (Y(S) is a pole surface of F(s, X). However, it is easy to see that the pole trajectory and the branch-point trajectory cannot collaborate in a simple manner in order to eliminate the branch point at s = s from the continued partial-wave amplitude F(s, A9 . For example, we may have near the point (so, oo) F(s,A)

p(s)

=+y(s)lg(x-CYc(s)) x - (Y(s)

+ . . . , 0)

and we find that, even with (Y(S) = al(s), oc(s) = = o2(s) as given by eq. (5), we cannot arrange a cancellation. The function F(s, X) inherits the branch point at s = so. A compensation is only possible between singular surfaces of the same character, i.e. two pole trajectories, two logarithmic or two square root branch-point surfaces, or others. * In the following we consider explicitly only the trajectory functions. Similar results may be derived for residuum functions.

LETTERS

10 November

1969

We conclude that the simple fact that a branch and a pole trajectory collide at a point s = so does not make it possible to have a branch point at s = so in any one of these trajectories $. The question remains as to whether there could be more sophisticated ways in which a branch point of F(s, A) in the X-plane may be helpful in allowing a branch point in a single pole trajectory a(s) which does not appear in F(s, X). One could think of using such a cut of F(s, X) in order to move one of the branches of the double-valued function o(s) into a secondary sheet of the Xplane. Some time ago, we have shown that this is possible with fixed branch cuts of F(s, A) in the complex X-plane [12]. We must discuss this case briefly before we consider moving branch points. Fixed branch points of F(s,X) in the X-plane are known to occur in potential models with y-2 singularities [12,13] and in related graph summations in perturbation theory [14]. The physical interpretation of these branch points as transition points in the balance of centrifugal and other forces is well known. As we have pointed out repeatedly [12], it is an open question whether corresponding branch points are present in general dispersion theory, mainly because particle production is possible. Irrespective of these questions, we are interested here in the mechanism by which these X-cuts allow non-threshold branch points of Regge trajectories. Suppose the pole trajectory o!(s) has a square root branch line for - 2y C s 5 so. Then we could have Re (Y(S) = ffo = const. and Im ar(s*iO) Z 0 for real s along the cut. As s varies from so to - 2y, \Im a(s) 1 varies from zero to y > 0 and back to zero. We suppose now that the amplitude F(s, X) has fixed branch points at A = f iy + (Yo which are connected by a cut along the line ReX = oo. We then can have a situation where the first branch of the trajectory o(s) is a pole surface of F(s,X) in the physical sheet of the X-plane with respect to the fixed cut ReX = oo, /ImX / C y, whereas the second branch of Q!(s) is restricted to the unphysical sheet. Hence this second branch does not generate poles in the physical partial-wave amplitudes F(s, X = I). In the example given here, the Regge pole function (Y(s) traces the lips of the X-cut of F(s, A) as s varies along both sides of the branch line - 2y C s s so of (Y(S). The relevant point is that, as soon as we cross the latter cut into a second sheet of the s-plane, the pole (Y(S) of F(s, A) moves into another sheet of $ Our conclusion

here is different

from that reached

in ref. 11.

415

PHYSICS

Volume 30B, number 6

the A-plane. We note that here F(s, A) has branch points at A = ffo f iy, whereas, for s = so, the pole trajectory (Y(S)is at A = oo. Hence pole and branch points do not collide at s = so. The position A = a0 f iy may be reached by (Y(S) for some point s f i0 along the cut - 2y G s C so. From the example considered above, we see that a left-hand branch line in a single Regge trajectory becomes possible if we allow a closely related fixed cut of F(s, A) in the A-plane. We may now ask if we can use the moving branch points A = oo(s) in a similar way in order to remove the second branch of a double-valued pole trajectory from the physical sheet of F(s, A). Normally, with the branch point at A = IY~(s), we would draw a cut running to the left along the real A-axis if oc(s) is real for real s s 0. But here we would like to use this branch line in order to make it possible for the point A = a(s) to move into a secondary sheet if the variable s crosses the cut -sL C s s 0 in o(s) which implies that the “physical” branch al(s) goes over into the “unphysical” branch @2(s). Since (Y(S +iO) is complex along the cut - SL C s s 0, we appear to need at least complex branch points (Ye. This can be achieved by assuming that the surface (Ye has a square-root branch line similar to that of CY (s). Since oc(s) is generated by Regge poles, it is plausible that this surface could also have left-hand branch lines. But with (Y,(S) having two branches ‘all and (Yap, we presumably should assume that they are both present in the physical sheet of F(s, A) because, otherwise, new branch lines in the A-plane would be required in order to hide @,-.2(s), and then there would be no end to the problem. For example, both branch points could appear in F(s, A) in the form * [(A-c$l(s))

(A -@c2(s)1’

,

(8)

which is symmetric under the exchange ‘~~1 - cy,2 so that the branch points of (Y,(S) are not introduced into F(s, A). With branch point surfaces and pole surfaces both having left-hand cuts, the general analytic structure of F(s,A) becomes somewhat more involved. For brevity, we prefer simple mathematical example. plitude F(s,A) contains a term

[c;(s) - (A -@,(s))~ where (Ye,

416

for example,

ref. 6.

10

November 1969

we have branch point surfaces ‘yc1,2(s) = a0(s) * (-n(s))+

(10)

and Regge pole surfaces a1,2(s) = DO(S) f (t2(s) -77(s))+ ,

(11)

and we can write the term (9) in the form

-it(s) + [(A-@,I(s)

(A-ac2(s,]f}

(A-~~(s))@-c~~(s))

(12)

For special choices of t(s) and v(s), we may arrange the branch lines in eq. (12) so that only the branch A = al(s) is a pole surface of F(s,A) in

the “physical” sheet of the A-plane with respect to the branch points ocl(s) and oc2(s). Let us consider the particular choice [(s)=a+bs,

q(s) = (c+bs)2

- s(sT - s) , (13)

where we have the pole trajectories “1,2(s)

= a!0 f [s(sT -s)]’

(14)

and the branch point surfaces oc1,2(s)

= “0 f [s(sT -s) - (a+ be)‘+

.

(15)

For real values of s near s = 0, the branch points of (Y. are complex. We connect them by a cut via the point A = a0 and define the physical sheet with the positive (negative) root in eq. (12) for real A > a0 (A< (Ye). Then only the trajectory (Ye is a pole surface of F(s, A) in this physical sheet. We have not yet explored the problem of constructing model amplitudes with realistic moving cuts as generated by Regge poles with left-hand branch lines. We must remember, in this connection, that there appear to be actually infinite sequences of such moving branch points and that their accumulation may well give rise to fixed branch points of F(s, A) in the A-plane [15]. We hope to return to this problem elsewhere. It is a pleasure to thank Professors H. Epstein, V. Glaser and 0. Steinmann for very helpful discussions. Thanks are also due to Professors Abdus Salam and P. Budini as well as the International Atomic Energy Agency for their very kind hospitality at the International Centre for Theoretical Physics, Trieste.

of the form

+tl(s))+-’

,

t(s) and n(s) are polynomials.

* Concerning the character see,

to give here a Suppose the am-

LETTERS

(9) Then

of moving branch points

References Phys. Letters 2 (1962) 1. R. Oehme and G. Tiktopoulos, 86; V. N. Gribov and I. Ya. Pomeranchuk, Nucl. Phys. 38 (1962) 516; A. 0. Barut and D. E. Zwanziger, Phys. Rev. 127 (1962) 974.

Volume

30B, number

6

PHYSICS

2. Hung Cheng, Phys. Rev. 130 (1963) 1283. 3. R. Oehme, Strong interactions and high-energy physics, ed. R. G. Moorhouse (Oliver and Boyd, Edinburgh and London, 1964) pp. 129-222; also in: Dispersion relations and their connections with causality, ed. E. P. Wigner (Academic Prass, New York, 1964) pp. 167-256. 4. P. G. 0. Freund and R.Oehme, Phys. Rev. Letters 10 (1963) 450; ref. 3, p. 201. 5. R.Oehme, Phys. Rev. Letters 9 (1962) 358. 6. R.Oehme, Phys. Rev. Letters 18 (1967) 1222. 7. V.N.Gribov, Nucl. Phys. 22 (1961) 249. 8. Ref. 3, p. 146. 9. H. Behnke and P. Thullen, Theorie der Funktionen mehrerer komplexer Ver)dnderlichen (Springer Verlag, Berlin, 1934).

LETTERS

10 November

1969

10. D. Amati, S. Fubini and A. Stangellini, Nuovo Cimento 26 (1962) 896; S. Mandelstam, Nuovo Cimento 30 (1963) 1127. 11. J. S. Bali and F. Zachariasen, Phys. Rev. Letters 25 (1969) 346. 12. R.Oehme, Nuovo Cimento 25 (1962) 183; ref. 3, p. 163. 13. V.Singh, Phys. Rev. 127 (1962) 632; G. S. Guralnik and C. R. Hagen, Phys. Rev. 130 (1963) 1259. 14. J. D. Bjorken and T. T. Wu, Phps. Rev. 130 (1963) 2566; R.F.Sawyer, Phys. Rev. 131 (1963) 1384. 15. V. N. Gribov, Proc. 1967 Intern. Conf. on Particles and fields, ed. C. R. Hagen et al. (Interscience Publishers, New York, 1968) p. 621.

*****

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