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Asymptotic estimate of forward dispersion relations in the presence of Regge cuts
Volume 27B. number
9
ASYMPTOTIC
PHYSICS
LETTERS
30 September
ESTIMATE OF FORWARD DISPERSION IN THE PRESENCE OF REGGE CUTS
1968
RELATIONS
F. J. YNDURAIN CERN,
Geneva,
Received
Switzerland
26 July 1968
The asymptotic behaviour of forward dispersion relations is studied. An explicit estimate of the ratio Re T/Im T is deduced when Regge cuts are present. Its predictions are checked against experiment, with which they are compatible in the present energy range.
Recent measurements of the ratio (Y(E) = Re T(E, O)/Im T(E, 0) of real to imaginary part of forward scattering amplitudes [l] seem to indicate that the predicted limit o(E) -+ 0 is approached quite slowly. Whereas the appropriate tool for computing (Y is forward dispersion relations [2], a straightforward calculation is prevented (in the asymptotic region) by the arbitrariness in the extrapolation of total cross-sections for energies beyond the accessible region [2,3] (in fact, recent fits [4] seem to indicate that a pure Regge-pole expansion is questionable, and that one should include Regge-cut effects). This makes it desirable to work in a round-about fashion, using only assumptions as general as possible. As a matter of fact, the difficulty here (as compared with the simpler case of standard “asymptotic theorems” [5,6]) is due to the fact that lim
E-+m
1ogIm T(E)/log
E = 1 = integer
(1)
and that, precisely, one would like to detect derivations of Im T(E) from the pure behaviour Im T(E) = = (constant) X E, and distinguish whether these deviations are due to other Regge poles or to the presence of cuts. In spite of this, we want to show that, under what essentially amounts to assumptions of smoothness, one can still obtain general results, of which special cases are the current ones [e.g. 7,5]. In particular, we shall present a general analysis * and an explicit estimate of the asymptotic value of
(Y(E), the last particularly suited to the situation where Regge cuts aye present. Its predictions aye checked against experiment, showing that the existence of Regge cuts seems to be compatible with experimental results. We shall consistently use the notation Z”*(E) = D+(E) + iA*( where the (+/-) refer to the direct/ crossed channel, for the forward scattering amplitudes. The variables E (lab. energy) and k = iE2 - m2 (lab. momentum) will be used interchangeably, as E/k - 1 very quickly. Apart from the validity i) Total cross-sections
of forward dispersion relations ** we shall assume the following do not oscillate at infinity (in the sense of ref. 6), i.e., : lim
A(uE)/A(E)
exists for u near 1 .
(2)
E+m Then, also from ref. 6, we know that eq. (1) holds; the explicit value of the limit is taken from experiment. ii) The Pomeranchuk theorem [6,10] is satisfied in its strong version. By this we mean that the integral in the expression
* This result represents a refinement of the analysis of Lehmann [8]. ** More generally, we can take the results proved from field theory by Bros hold with trivial modifications.
et al. [9] and our analysis
will still
581
Volume 27B, number
9
PHYSICS
I = J= dE
A+@) -A_(E)
LETTERS
+ 4n2 T+(m) - T_(m) 2m
k2
0
30 September
+p
1968
8n2f 2
(3)
rFL&
exists in the Lebesgue sense. If Z is different from zero, these assumptions are all we need. If, however, (and this seems to be the actual situation), Z = 0, extra hypotheses are, of course, necessary. As a natural choice we have taken iii) the difference o(-) = IJ+ - o_ between total cross-sections is also non-oscillating at infinity (this assumption can be dropped to get some results. Cf. Section 4 below). iv) The present behaviour of u continues smoothly beyond the region of accessible energies. More precisely, we can state 6iv) as follows: as proved in ref. 11, from (i) it can be shown that, if we define (4)
then v(k) - 0 as k -m. [Eq. (4) will be recognized as a finite energy sum rule.] Plots of q(k) show [4] that this limit is attained slowly and without oscillation. Therefore, we formulate (iv) as requiring that 17be non-oscillating, namely, that the analogue of eq. (2) holds: lim k-m
exists
&k)/n(k)
(5)
.
Then the limit $irnWlog q(k)/log k also exists. Again we take its value from experiment, and, since numerical computations [4] seem to give it vanishing, we assume it to be zero *. Two more preliminary results will be introduced. First, as shown in ref. 6 (Theorem II. l), from (i), (iv), (iii), it follows, respectively, lim a,(uk)/a,(k) k-m
lim q+(uk)/q*(k)
JZ-‘m
= 1 = 1
lim c(-)(uk)/o(-)(k) k-+m Second,
solving
eq. (4) (cf. also ref. 11, appendix),
= us ,
p<0.
03
we obtain
o,(k)
= [I +~+(k)]o:(W
o:(k)
= (constant) x exp
k 1 dk’g,(k’)/k’ 1
1
I
.
Clearly, co/o -+ 1, as k -t a. Non-vanishing I. From (i), (ii) we can, as in ref. 6, deduce that D(E)/A(E) = IE/BA(E). Then, since D*(E) = D(G), A,(E) = rtA(+E), the result ol+(E)/a!_(E) - - 1 follows trivially. It is perhaps worth while to remark that this result is still true in some cases where (Y - 0 (as if, e.g., the Froissart bound is saturated and u - log2 k) or where (Y - m [as if o - l/(log k)‘, 6 > 0, as some recent fits seem to indicate [4]. Vanishing 1. In this case the foregoing analysis breaks down and a more refined estimate is necessary. For this, write the usual twice subtracted dispersion relation in the form **
* The analysis presented here breaks down if a non-zero limit is obtained. This will be discussed in detail below. ** The principal value i to be understood in all singular integrals. We are consistently omitting terms due to the poles of Z’(E)/(E2 -m 8 ); their inclusion is straightforward.
582
Volume 27B, or still,
number
with A(-)
9 q
PHYSICS
LETTERS
30 Seutember
1968
A+ - A_,
(9b) We shall study separately
the contributions
1p_A(-)(E’)
R=&JwdE
from
r and R . As to the second,
1 1 \m dEt E’ A(-)(E’) E 71’ kf2 E’ +E
kT2
=
/m dk, o(-)(a~)
1 -nE-
E’+E
where we have sed th act that Z = 0 to eliminate one term, and the than e of variables E’ - k’ and the substitution At-7 = ktcf-f in the other term. Multiplying and dividing by u (_g,(k), making the change of variables k’ = uk, and taking the limit * (6), we get 1
o(-)(k) R=--G--I
m
tip deru+l=-l
0
As to r,
making obvious
changes
of variables
dk’ ~ a&‘) kf2 _ k2
u as in eq. (‘Z), r = rl
Multiplying
(10)
’
we get
r =;Jm and, decomposing
o(-)(k) E
smnp
and dividing
+ r.
17(k’b’(k’) = 5 Jrn dk’ ~~ k12_k2
rl
ro(resp:
lYl) by o’(k)
(resp:
v(k)o’(k)),
rl
= 77(k) uo$k)
r
(11)
’
and letting k’ = uk,
2 .irn du V(uk)O”(uk)/V(kbo(k) 71 22 - 1 0
.
(12)
Using eq. (‘7), we see that, to first order in q(k),
a’(uk)/o’(k)
= exp
r](k) [ 1
du’
n(U’@)/n(JZ) u,
where we have used eq. (6) to take the limit u(u’k)/q(k).
v(k) 1’ du’/u’ 1
1 = ZL’(~)
Therefore,
(13)
Hence,
substituting
this and using again eq. (6), we get ~(Uk)oO(Uk)/n(k)o% 2
- 1
+- 4,Tdu&l=0. x
We can neglect the contribution of rl as compared to that of ro. Theorem: Under assumptions (i) to (iv), we have
0 Collecting
* We interchange limits and integrals freely, since, for non-oscillating functions, ficient to justify the use of Lebesque theorem [8, appendices A and C].
our results,
we get, finally,
one can obtain majorizations
suf-
583
Volume
27B, number
9
PHYSICS
LETTERS
1 - cos nTJ*(k) a*(k)
=
sin w*(k)
1
’ sinnp
30 September
1968
o(-)(k) u,(k)
I
I
I
I
I
I
I
6
a
10
12
i4
16
la
(14)
’
I 20
I
22
GeVlc
Fig. 1. Comparison of Regge cut predictions with experiment for np scattering: plot of (Y = CY++ CY versus momentum. Isolated points: data from ref. 1. Solid line: Regge cut prediction. Broken line: pure Regge pole prediction of refs. 1 and 5. From the preceding analysis, one can at once decide that Z = 0, in general. In fact, otherwise one would have (Y+/a! _ x - 1, which is far from the present values (- + 1 for nN, - 0 for NN scattering). Moreover, when the value of Z can be computed, it is found to be zero (Goldberger’s sum rule for nN scattering). Of course, the same conclusion may be obtained invoking the experimental values of da ch .x/dt/ t=O. Anyway, we have to apply the preceding analysis. Bbfore proceeding with the actual computations, some comments are due to the verification of hypotheses (i) to (iv). Clearly, all of them are satisfied if a generalized Regge-like behaviour (with cuts) of
the form
a(E) = cj
CjEQj(logE)
“j . . . holds [5,6,11].
More generally,
the situation is as follows:
as-
sumptions (i), (ii) are classical, and the same is true of (iii) (the last can also be dropped if we are interested only in the values of CY= u + + (Y_), and we shall not consider them any further. Assumption (iv) seems to be more difficult to justify on theoretical grounds, although it is apparently well satisfied experimentally [4]. Whereas the fact that q(k) is non-oscillating is not likely to rise objections, the special value of the limit is (5) [or, equivalently, (6)] has some implications. In fact, if a pure Regge pole behaviour holds for U, then q(k) - k-4 and q(uk)/q(k) -u-p, logq(k)/logk - -p. The analysis presented here breaks down, and one has to apply different techniques. As the results one obtains in this case are discussed in several other places [6,7,12], we shall not repeat them here. It should be emphasized again, however, that present experimental analysis [4] seems to favour assumption (iv), indicating that some deviation off a pure Regge pole behaviour might be present. In this spirit, testing of eq. (14) amounts to obtain a dispersion-theoretic confirmation of the existence (or non-existence) of Regge cuts in forward amplitudes. We have checked the predictions of eq. (14) for np scattering. In doing so, only the values of (Y+ + ff_ have been considered, as, from eq. (14), it is obvious that 01+ - a_ is given here essentially as in the usual case (i.e., by means of the charge-exchange amplitude) and therefore nothing new may occu We use eq. (14) as it is, rather than the expression we would have obtained making the analysis for A ?’‘) or he determination of TJ T(+) directly (instead of analyzing A,, T, sep rately) fo the following reason: is a very delicate process. If we work with A ?+), then 17G) is obtained through CJ~+)= a + u Therefore to the errors resulting from the (necessary) interpolation one has to perform for :om&ing the errors due to the fact that a integral in eq. (4), one would, if working with u(+) add (unnecessary) measured at exactly the same energies). It turns further interpolation is needed (since a+, u_ are& out that these errors contribute to s sizable deviation (-20%) off the more exact value (14). The results of our analysis are presented in fig. 1, for pn scattering. The experimental points are from ref. 1. We see that the upper part of the predicted curve is compatible with the experimental numbers, within the errors (“theoretical” errors can be estimated to be * - 10%). However, present experimental data are not sufficient to decide clearly against or for the existence of a cut. If a free hand extrapolation of the theoretical curve is made, we see that deviations from the pure Regge pole result * See footnote
584
on following
page.
Volume
27B, number
9
PHYSICS
LETTERS
30 September
1968
of refs. 1,3 will become unambiguously relevant from - 26 GeV on (the two curves seem to cross at that energy). However, we believe that the present results are very interesting, as they show that replacement of the P, P’ poles by a cut improves the agreement. What is perhaps more relevant is that we have shown practically that dispersion relations, by being sensitive to values of ototal above the accessible region, provide a “crystal ball” where effects of Regge cuts can manifest themselves before what one would think to be a high enough energy. We are grateful to Professor A. Martin for most valuable discussions. Thanks are also due to Dr. J. A. Rubio for interesting discussions and especially for help with the numerical computations.
* Sources of these errors
are: 1. the arbitrariness in the lower limit of the integral in eq. (4); a charge in c from 800 MeV to 200 MeV gives an error of - 5%; we have taken c = 400 MeV; 2. errors in the total cross-sections, again contributing to eq. (4); 3. errors in the determination of the integral in eq. (4); changing from a direct (linear) interpolation to a parabolic one improves the results by a - 7%; To the over-all effect of these errors, a purely theoretical (and unknown) one is added, due to the fact that eq. (14) is only true asymptotically.
References 1. K.J.Foley et al.. Phys. Rev. Letters 19 (1967) 193. 2. For a review, see S. J. Lindenbaum, Forward dispersion relations. Conference on liN scattering, University California. Irvine, December 1967. therein. 3. K. J.Foley et al., Phys. Rev. Letters 19 (1967) 330, and references 4. J.A.Rubio and F.J.Yndurain. to be published. 5. L.Van Hove, Phys. Letters 5 (1963) 252; 7 (1963) 76; A.A.Logunov. Nguyen Van Hieu and I.T.Todorov. Ann. Phys. (N.Y.) 31 (1965) 203. 6. J. L.Gervais and F. J.Yndurain, Phys. Rev. 167 (1968) 1289. 7. R. J. Eden, High energy collisions of elementary particles (Cambridge University Press, Cambridge, 1967). 8. H. Lehmann, Nuclear Phys. 29 (1962) 300. 9. J.Bros, H.Epstein and V.Glaser. Comm. Math. Phys. 1 (1965) 240. 10. I. Ya. Pomeranchuk. Soviet Phys. JETP 3 (1956) 306; D.Amati, M.Fierz and V.Glaser, Phys. Rev. Letters 4 (1960) 89; A. Martin, Nuovo Cimento 39 (1965) 704. 11. J.L.Gervais and F.J.Yndurain. Phys. Rev. 169 (1968) 1187. 12. A. Bialas and E. Bialas, Nuovo Cimento 37 (1965) 1686.
of
*****
585
9
ASYMPTOTIC
PHYSICS
LETTERS
30 September
ESTIMATE OF FORWARD DISPERSION IN THE PRESENCE OF REGGE CUTS
1968
RELATIONS
F. J. YNDURAIN CERN,
Geneva,
Received
Switzerland
26 July 1968
The asymptotic behaviour of forward dispersion relations is studied. An explicit estimate of the ratio Re T/Im T is deduced when Regge cuts are present. Its predictions are checked against experiment, with which they are compatible in the present energy range.
Recent measurements of the ratio (Y(E) = Re T(E, O)/Im T(E, 0) of real to imaginary part of forward scattering amplitudes [l] seem to indicate that the predicted limit o(E) -+ 0 is approached quite slowly. Whereas the appropriate tool for computing (Y is forward dispersion relations [2], a straightforward calculation is prevented (in the asymptotic region) by the arbitrariness in the extrapolation of total cross-sections for energies beyond the accessible region [2,3] (in fact, recent fits [4] seem to indicate that a pure Regge-pole expansion is questionable, and that one should include Regge-cut effects). This makes it desirable to work in a round-about fashion, using only assumptions as general as possible. As a matter of fact, the difficulty here (as compared with the simpler case of standard “asymptotic theorems” [5,6]) is due to the fact that lim
E-+m
1ogIm T(E)/log
E = 1 = integer
(1)
and that, precisely, one would like to detect derivations of Im T(E) from the pure behaviour Im T(E) = = (constant) X E, and distinguish whether these deviations are due to other Regge poles or to the presence of cuts. In spite of this, we want to show that, under what essentially amounts to assumptions of smoothness, one can still obtain general results, of which special cases are the current ones [e.g. 7,5]. In particular, we shall present a general analysis * and an explicit estimate of the asymptotic value of
(Y(E), the last particularly suited to the situation where Regge cuts aye present. Its predictions aye checked against experiment, showing that the existence of Regge cuts seems to be compatible with experimental results. We shall consistently use the notation Z”*(E) = D+(E) + iA*( where the (+/-) refer to the direct/ crossed channel, for the forward scattering amplitudes. The variables E (lab. energy) and k = iE2 - m2 (lab. momentum) will be used interchangeably, as E/k - 1 very quickly. Apart from the validity i) Total cross-sections
of forward dispersion relations ** we shall assume the following do not oscillate at infinity (in the sense of ref. 6), i.e., : lim
A(uE)/A(E)
exists for u near 1 .
(2)
E+m Then, also from ref. 6, we know that eq. (1) holds; the explicit value of the limit is taken from experiment. ii) The Pomeranchuk theorem [6,10] is satisfied in its strong version. By this we mean that the integral in the expression
* This result represents a refinement of the analysis of Lehmann [8]. ** More generally, we can take the results proved from field theory by Bros hold with trivial modifications.
et al. [9] and our analysis
will still
581
Volume 27B, number
9
PHYSICS
I = J= dE
A+@) -A_(E)
LETTERS
+ 4n2 T+(m) - T_(m) 2m
k2
0
30 September
+p
1968
8n2f 2
(3)
rFL&
exists in the Lebesgue sense. If Z is different from zero, these assumptions are all we need. If, however, (and this seems to be the actual situation), Z = 0, extra hypotheses are, of course, necessary. As a natural choice we have taken iii) the difference o(-) = IJ+ - o_ between total cross-sections is also non-oscillating at infinity (this assumption can be dropped to get some results. Cf. Section 4 below). iv) The present behaviour of u continues smoothly beyond the region of accessible energies. More precisely, we can state 6iv) as follows: as proved in ref. 11, from (i) it can be shown that, if we define (4)
then v(k) - 0 as k -m. [Eq. (4) will be recognized as a finite energy sum rule.] Plots of q(k) show [4] that this limit is attained slowly and without oscillation. Therefore, we formulate (iv) as requiring that 17be non-oscillating, namely, that the analogue of eq. (2) holds: lim k-m
exists
&k)/n(k)
(5)
.
Then the limit $irnWlog q(k)/log k also exists. Again we take its value from experiment, and, since numerical computations [4] seem to give it vanishing, we assume it to be zero *. Two more preliminary results will be introduced. First, as shown in ref. 6 (Theorem II. l), from (i), (iv), (iii), it follows, respectively, lim a,(uk)/a,(k) k-m
lim q+(uk)/q*(k)
JZ-‘m
= 1 = 1
lim c(-)(uk)/o(-)(k) k-+m Second,
solving
eq. (4) (cf. also ref. 11, appendix),
= us ,
p<0.
03
we obtain
o,(k)
= [I +~+(k)]o:(W
o:(k)
= (constant) x exp
k 1 dk’g,(k’)/k’ 1
1
I
.
Clearly, co/o -+ 1, as k -t a. Non-vanishing I. From (i), (ii) we can, as in ref. 6, deduce that D(E)/A(E) = IE/BA(E). Then, since D*(E) = D(G), A,(E) = rtA(+E), the result ol+(E)/a!_(E) - - 1 follows trivially. It is perhaps worth while to remark that this result is still true in some cases where (Y - 0 (as if, e.g., the Froissart bound is saturated and u - log2 k) or where (Y - m [as if o - l/(log k)‘, 6 > 0, as some recent fits seem to indicate [4]. Vanishing 1. In this case the foregoing analysis breaks down and a more refined estimate is necessary. For this, write the usual twice subtracted dispersion relation in the form **
* The analysis presented here breaks down if a non-zero limit is obtained. This will be discussed in detail below. ** The principal value i to be understood in all singular integrals. We are consistently omitting terms due to the poles of Z’(E)/(E2 -m 8 ); their inclusion is straightforward.
582
Volume 27B, or still,
number
with A(-)
9 q
PHYSICS
LETTERS
30 Seutember
1968
A+ - A_,
(9b) We shall study separately
the contributions
1p_A(-)(E’)
R=&JwdE
from
r and R . As to the second,
1 1 \m dEt E’ A(-)(E’) E 71’ kf2 E’ +E
kT2
=
/m dk, o(-)(a~)
1 -nE-
E’+E
where we have sed th act that Z = 0 to eliminate one term, and the than e of variables E’ - k’ and the substitution At-7 = ktcf-f in the other term. Multiplying and dividing by u (_g,(k), making the change of variables k’ = uk, and taking the limit * (6), we get 1
o(-)(k) R=--G--I
m
tip deru+l=-l
0
As to r,
making obvious
changes
of variables
dk’ ~ a&‘) kf2 _ k2
u as in eq. (‘Z), r = rl
Multiplying
(10)
’
we get
r =;Jm and, decomposing
o(-)(k) E
smnp
and dividing
+ r.
17(k’b’(k’) = 5 Jrn dk’ ~~ k12_k2
rl
ro(resp:
lYl) by o’(k)
(resp:
v(k)o’(k)),
rl
= 77(k) uo$k)
r
(11)
’
and letting k’ = uk,
2 .irn du V(uk)O”(uk)/V(kbo(k) 71 22 - 1 0
.
(12)
Using eq. (‘7), we see that, to first order in q(k),
a’(uk)/o’(k)
= exp
r](k) [ 1
du’
n(U’@)/n(JZ) u,
where we have used eq. (6) to take the limit u(u’k)/q(k).
v(k) 1’ du’/u’ 1
1 = ZL’(~)
Therefore,
(13)
Hence,
substituting
this and using again eq. (6), we get ~(Uk)oO(Uk)/n(k)o% 2
- 1
+- 4,Tdu&l=0. x
We can neglect the contribution of rl as compared to that of ro. Theorem: Under assumptions (i) to (iv), we have
0 Collecting
* We interchange limits and integrals freely, since, for non-oscillating functions, ficient to justify the use of Lebesque theorem [8, appendices A and C].
our results,
we get, finally,
one can obtain majorizations
suf-
583
Volume
27B, number
9
PHYSICS
LETTERS
1 - cos nTJ*(k) a*(k)
=
sin w*(k)
1
’ sinnp
30 September
1968
o(-)(k) u,(k)
I
I
I
I
I
I
I
6
a
10
12
i4
16
la
(14)
’
I 20
I
22
GeVlc
Fig. 1. Comparison of Regge cut predictions with experiment for np scattering: plot of (Y = CY++ CY versus momentum. Isolated points: data from ref. 1. Solid line: Regge cut prediction. Broken line: pure Regge pole prediction of refs. 1 and 5. From the preceding analysis, one can at once decide that Z = 0, in general. In fact, otherwise one would have (Y+/a! _ x - 1, which is far from the present values (- + 1 for nN, - 0 for NN scattering). Moreover, when the value of Z can be computed, it is found to be zero (Goldberger’s sum rule for nN scattering). Of course, the same conclusion may be obtained invoking the experimental values of da ch .x/dt/ t=O. Anyway, we have to apply the preceding analysis. Bbfore proceeding with the actual computations, some comments are due to the verification of hypotheses (i) to (iv). Clearly, all of them are satisfied if a generalized Regge-like behaviour (with cuts) of
the form
a(E) = cj
CjEQj(logE)
“j . . . holds [5,6,11].
More generally,
the situation is as follows:
as-
sumptions (i), (ii) are classical, and the same is true of (iii) (the last can also be dropped if we are interested only in the values of CY= u + + (Y_), and we shall not consider them any further. Assumption (iv) seems to be more difficult to justify on theoretical grounds, although it is apparently well satisfied experimentally [4]. Whereas the fact that q(k) is non-oscillating is not likely to rise objections, the special value of the limit is (5) [or, equivalently, (6)] has some implications. In fact, if a pure Regge pole behaviour holds for U, then q(k) - k-4 and q(uk)/q(k) -u-p, logq(k)/logk - -p. The analysis presented here breaks down, and one has to apply different techniques. As the results one obtains in this case are discussed in several other places [6,7,12], we shall not repeat them here. It should be emphasized again, however, that present experimental analysis [4] seems to favour assumption (iv), indicating that some deviation off a pure Regge pole behaviour might be present. In this spirit, testing of eq. (14) amounts to obtain a dispersion-theoretic confirmation of the existence (or non-existence) of Regge cuts in forward amplitudes. We have checked the predictions of eq. (14) for np scattering. In doing so, only the values of (Y+ + ff_ have been considered, as, from eq. (14), it is obvious that 01+ - a_ is given here essentially as in the usual case (i.e., by means of the charge-exchange amplitude) and therefore nothing new may occu We use eq. (14) as it is, rather than the expression we would have obtained making the analysis for A ?’‘) or he determination of TJ T(+) directly (instead of analyzing A,, T, sep rately) fo the following reason: is a very delicate process. If we work with A ?+), then 17G) is obtained through CJ~+)= a + u Therefore to the errors resulting from the (necessary) interpolation one has to perform for :om&ing the errors due to the fact that a integral in eq. (4), one would, if working with u(+) add (unnecessary) measured at exactly the same energies). It turns further interpolation is needed (since a+, u_ are& out that these errors contribute to s sizable deviation (-20%) off the more exact value (14). The results of our analysis are presented in fig. 1, for pn scattering. The experimental points are from ref. 1. We see that the upper part of the predicted curve is compatible with the experimental numbers, within the errors (“theoretical” errors can be estimated to be * - 10%). However, present experimental data are not sufficient to decide clearly against or for the existence of a cut. If a free hand extrapolation of the theoretical curve is made, we see that deviations from the pure Regge pole result * See footnote
584
on following
page.
Volume
27B, number
9
PHYSICS
LETTERS
30 September
1968
of refs. 1,3 will become unambiguously relevant from - 26 GeV on (the two curves seem to cross at that energy). However, we believe that the present results are very interesting, as they show that replacement of the P, P’ poles by a cut improves the agreement. What is perhaps more relevant is that we have shown practically that dispersion relations, by being sensitive to values of ototal above the accessible region, provide a “crystal ball” where effects of Regge cuts can manifest themselves before what one would think to be a high enough energy. We are grateful to Professor A. Martin for most valuable discussions. Thanks are also due to Dr. J. A. Rubio for interesting discussions and especially for help with the numerical computations.
* Sources of these errors
are: 1. the arbitrariness in the lower limit of the integral in eq. (4); a charge in c from 800 MeV to 200 MeV gives an error of - 5%; we have taken c = 400 MeV; 2. errors in the total cross-sections, again contributing to eq. (4); 3. errors in the determination of the integral in eq. (4); changing from a direct (linear) interpolation to a parabolic one improves the results by a - 7%; To the over-all effect of these errors, a purely theoretical (and unknown) one is added, due to the fact that eq. (14) is only true asymptotically.
References 1. K.J.Foley et al.. Phys. Rev. Letters 19 (1967) 193. 2. For a review, see S. J. Lindenbaum, Forward dispersion relations. Conference on liN scattering, University California. Irvine, December 1967. therein. 3. K. J.Foley et al., Phys. Rev. Letters 19 (1967) 330, and references 4. J.A.Rubio and F.J.Yndurain. to be published. 5. L.Van Hove, Phys. Letters 5 (1963) 252; 7 (1963) 76; A.A.Logunov. Nguyen Van Hieu and I.T.Todorov. Ann. Phys. (N.Y.) 31 (1965) 203. 6. J. L.Gervais and F. J.Yndurain, Phys. Rev. 167 (1968) 1289. 7. R. J. Eden, High energy collisions of elementary particles (Cambridge University Press, Cambridge, 1967). 8. H. Lehmann, Nuclear Phys. 29 (1962) 300. 9. J.Bros, H.Epstein and V.Glaser. Comm. Math. Phys. 1 (1965) 240. 10. I. Ya. Pomeranchuk. Soviet Phys. JETP 3 (1956) 306; D.Amati, M.Fierz and V.Glaser, Phys. Rev. Letters 4 (1960) 89; A. Martin, Nuovo Cimento 39 (1965) 704. 11. J.L.Gervais and F.J.Yndurain. Phys. Rev. 169 (1968) 1187. 12. A. Bialas and E. Bialas, Nuovo Cimento 37 (1965) 1686.
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