REAL PARTS OF FORWARD M ESON-NUCLEON SCATTERING AMPLITUDES

5. General Formalism

We use dispersion relations to analyze ^a forward scattering amplitude, whose discontinuity is determined by total cross sections

+ -

+

of two channels related by crossing (e.g., n: p and n: p or K p and K p). We refer the reader to Ref. 30 for the conventional formu-

lation of dispersion relations and previous calculations. One usually separates the symmetric _amplitude ^. _A ⁽⁺⁾

=

2(A(n: ^{1 -}p)

+

A(n: p)

+ J

from the antisyrru~etric one A ⁽

-

⁾

= 2

1 [A(n: p) -- A(n: p) + ] , and writes the dispersion relations

A(+) (V)

A ( -) (V) =

2 2

A ( +) (µ) + _ _ _ _...__..__ _ _ _ _ _ +

M~ - (~)Jt2 - ( ~)2J

00

+

_2n:2 k2

JdV'

^{V' cr(+) (V')}

µ

2f2 v

·+

v2 _

(L) 2M

²

00

V

r

^dV'

2n:2

J

µ

k' cr(-)(V') v•² -

v2 -

^iE:

(1)

(2)

(+) 1 -

+ ]

M is the nucleon mass andµ the meson mass. a -

=

2[crT(n: p)± crT(n: p) V and k are the meson's laboratory energy and momentum, respectively.

f2

specifies the strength of the Born term, and is equal to 0.082.

A(+)(V

=

µ) is the only subtraction constant. It is known to be zero within experimental errors, in agreement with Adler's PCAC self-

. d. . 31)

consistency con ition . In writing (2), one obviously makes the

assumption that 0(-) goes asymptotically to zero. This is the point which we now want to change. Follo:wing the approach of Ref. 19)

- +

we assume that both 0T(n p) and 0T(n p) remain constant from about 30 BeV on. This then implies that they have different values) and 0(-) is a non-zero constant. We want to see what the predictions of these assumptions for the real part are. Having to introduce a

subtraction into (2)) we therefore replace it by

A ( - ) (V) 2£2 v v

f

^{dV' k'} <J(-)(V')

=

+

v2 -

(~) 2

^{2n 2} µ ^{V'2 - v2 - l. E .}

00 ( 3)

. 2

r

^{c/-){V'2 v}

' Vk dV'

-:- - 2

.J

^{k r (V' 2 2 + c}

²

2n

-

^v

-

^{iE) M}

K

Note that) instead of performing a subtraction on the entire integral) we divide it into two parts. One is written in an un-

subtracted form) and the other in a subtracted one. ·This is done for practical purposes. It avoids stressing the low-energy input and thus increasing the errors in the calculation. The number c depends on the choice of K· Equation (3) also demonstrates the fact that the real part at low energy is not necessarily affected by the new assumptions on the high-energy behavior. We are actually able to reproduce at low energies (say) below 4 BeV) the same results pre- viously obtained by the use of (2) with any reasonably decreasing fit to 0(-).

To i llustrate the changes brought about by the assumptions on the behavior of the total cross section) let us discuss a rnathe-

matical example that is very similar to the actual situation in K p. ± Let us denote the two reactions in question by A and B (analogous

+ -

to K p and K p, respectively). Assume first that (case I):

Im A

1 = a ^V, Im B

1 = a V

+

b .{v, O<Y<oo. (4) It is then easy to find that

Re ^A = - b .[v

I ' Re B

1

=

0 . (5)

This is the expected result for K-p + if one uses a Regge repre-

sentation with a regular Pomeron and two pairs of exchange degenerate trajectories with intercepts at 1/2. If we now make the analogous assumption to that of Ref. 19, we have (case II):

L

^{v + b .[v} 0 < V < A

Im A

11 ^a

v,

Im B

11 ⁼ _b

+ - ) v _{A< Y <oo}

. [A

It is then readily established that

2b

.{v

arctan

fi + - -

^bV log

I

^Y

+

A

1 - ^c

1' 1t

.JA

^{M2 v J}

+ ^£_ ₂ ^v _.

M

(6)

(7)

It is now interesting to note that although Eqs. (5) and (7) are very different from one another, it is still possible to find a value of c that will show a similar behavior for low ^V, Thus i t is possible that even though Im B

1

i

Im B

11 for V > A,one still finds that. the

real parts of the various amplitudes can roughly agree for V

<

/1. To illustrate this point numerically, we choose a

=

b

=

3.6, A 22. (These values are close to those indicated by experiment if

-1

V is measured in BeV and the amplitudes in BeV .) We find such an agreement between I and II for c

=

1.1. We present in Fig. 3 the results for a(A)

=

Re A/Im A and a(B)

=

Re B/Im B, since this is the customary way in which the data are given in rcN experiments • •k

Note that after the value V

==

100 the logarithmic part in Re A 11 and Re B

11 is taking over. Nevertheless it does not reach a sizable amount even at high V values. To quote a number -- at V = 10 we find ⁶ a(B11)

= -

0.49 and a(A

11)

=

0.59. We will find a similar behavior in the next section when discussing the nN problem.

6. Real Parts of re ^± p Amplitudes

In Ref. 19, the rc±p total cross sections were fitted to a form (8)

An ionization point was then assumed to appear at V

=

30 BeV,

resulting in the flattening off of the cross sections at that point.

This meant that 2cr(-)

=

cr(rc-p) - cr(rc+p) > 1 mb even at high energies.

(-)

In Ref. 19, a was assumed to remain a constant for V ~ 30 BeV. Any breaking of the Pomeranchuk theorem results in a logarithmic rise of the real part of the amplitude, notably of A(-)(V)ll,l9) . Hence, a±(V)

=

Re A±(V)/Im A±(V) does not tend to 0 as V ~ oo. Once the logarithmic behavior begins to dominate,

a

rises in absolute value, with

a,

and

a

taking opposite signs. The strength of the

T -

*

It is unfortunate that the letter

a

is used for this ratio. It is not to be confused with a Regge trajectory.

logarithmic term is proportional to the value of cr(-).

The dispersion integrals were evaluated on ^a computer. In order to do the principal part integration, it is necessary to have a smooth fit to the data points, since the integral is sensitive to discontinuities near V'

=

V

*

For v < 4 BeV we used the fit of Ref.

30. The data between 4 and 30 Bev33134

) can be fitted in a variety of ways. We first fitted each cross section separately to a form

n-1

"' = a ^..L. b V

v± ± ' ± • ⁽⁹⁾

In such fits, a - a+ was invariably greater than 1 mb, and the choice of n was a matter of taste. We then tried a fit satisfying the Pomeranchuk theorem,

a + b vn-l ± c ^_µi-1 v • (10)

This was done in order to be able to compare the premise of a cutoff with the assumption that the Serpukhov data might be wrong, and that

the Pomeranchuk theorem might be right after all.

*

^{The principal} part integral is performed as ro ^~ 11 ows 32) :

B B B

P

J

_A ^{dx x - y f{x2 =}

I

_A ^{dx f{x2 x -- y f {y2 + f(y)}

^pf

_A ^{x dx - y}

B

=

J (

dx f(x2 - f(y2 + f(y) logjB - Yj

x - y A - y

A

The first integral one has a regular integrand. If A~ y ~Bone

substitutes f' (y) at x

=

y. This is easy if f(x) is given in functional form. If only discrete values are known, it is advisable to have them equally spaced in x, so that one can use Simpson's formula. To that approximation

f I (X )

n x - x

n+l n-1

The data of Citron et al.33

) do not seem to fit snoothly to those of Foley et al.34

) . We had to settle for a slightly low value of n. We chose

n

=

0.25, m = 0.6

a

=

22.5 b

=

18.9 c = - 2 .45

where V is measured in BeV and 0 in mb. Applying to fit (10) a cutoff at 30 BeV, we got for V above cutoff

1. 3 mb .

This n~~ber is consistent with the result of Ref. 14. In doing the (

-

)

same with fit_ (9), we got 20 above cutoff to depend on the fit. o(rt p) is, of course, determined by the Serpukhov data, but there is a slight freedom of play in 0(rt+p). First one has to choose the

cutoff point. We assumed it to be the same as in rt p (30 BeV). Since this is 8 BeV higher than the last data point, the extrapolation

depends on the fit. It was pointed out above that the strength of the logarithmic term in the real part of the amplitude is proportional to 0(-). If we constrained fit (9) to satisfy 20(-)

=

1.3 mb, our numerical dispersion calculations with it gave the same results for the real parts as fit (10). We adopted the latter for the purpose of testing the sensitivity of the calculation to the possible breaking of the Pomeranchuk theorem. We called case I that which assumes

(10) to be good for all

v.

In case II we applied the cutoff, ~o that for V ~ 30 BeV both cross sections were constant. The two cases are illustrated in Fig. 4. Note that if further structure appears in 0T at much higher energies, it may have negligible effects on our

calculation.

The calculated ratios a±(V)

=

Re A±(V)/Im A±(V) for the n p ± amplitudes are plotted in Fig. 5} together with the data35

). In case I there is no free parameter in the dispersion relations (1) and (2). In case II there is the arbitrariness of c in (3)J which can be chosen to best fit the data. (We used K

=

4 BeV.)

If one assUi~es exact charge independence} one can evaluate the forward CEX differential cross section. The predictions are plotted together with the data 36) in Fig. 6 and Fig. 7. We note that in case I the prediction seems to be too high by about 30 percent at} say}

20 BeV. If we attribute the discrepancy to I-spin violation of the electromagnetic amplitude} we find it to be 20 percent of the total A(-) amplitude. With 2cr(-) (V = 20) ,..,,. 1.5 mb} we would thus have

2cr (-) < 0 3 b

EM ""' • m • Since we do not expect the electromagnetic effects

to vary strongly with energy} we may conclude that the ansatz of the Pomeranchuk theorem is good only up to

2cr(~)(oo) <

_,..., 0.3 mb.

In case II we can adjust c so as to get a very good fit to the CEX data (c

=

^0.35). Alternatively} we can fix c to fit the

a

_±

data. Choosing here c

=

0.35} we find a good fit to a+ but a poor one to

a .

This is an improvement over case I. A change to c

=

^0.25

results in an equivalent over-all fit to a±} with a poorer fit to a+

and a better one to

a .

Note that such a change contributes oppo- sitely to a+ and a . Checking the CEX prediction with c = 0.25, we find it too low by about 40 percent. This corresponds to

2~ (-) < 0 5 b

vEM rJ • m •

Note that the small deviations that we found are a feature of our calculated real parts. Point by point) the experimental a±(V))

within their errors, are consistent with the CEX data without any I-spin violation . This was already pointed out by Foley et al. ³⁵⁾ .

. Although we can fit the data with no I-spin breaking, we cannot rule

out ^2a _EM ^(-) _,.., < ⁰ _' ^{5 b}

m ' However, this is still too small to account for the expected constant difference between crT(rt-p) and crT(rt+p).

We have to cortclude, then) that this difference is a genuine strong interaction effect.

The main difference between the two dispersion calculations I and II sets in around 100 BeV. At that point) the logarithmic part of Re A(-) in case II begins to dominate. Instead of going to zero,

a

(V) becomes positive and increases, while

a

(V) turns over and +

becomes more negative. The CEX forward cross section begins to rise again. On an absolute scale, both effects are small. We should be able to see the CEX forward cross section flattening, but for the real part to dominate the amplitude we will need fantastically high

energies. By that time, a new physics may very well set in. It was pointed out in Ref. 19, as well as in Ref. 17, that if Re A/Im A grows logarithmically, then one has to have the forward elastic peak

2 .

shrink like log s to avoid a conflict with unitarity. Strictly speaking) such a conflict would arise only at such large values of V that the whole problem looks rather academic. Nevertheless, tµe same conclusion about the shrinkage arises of course from the assumption that crel does not rise with energy, which might very well be the case.

Finally) a word about errors and low-energy behavior. The cross sections are accurate to about 1 percent. This leads to errors of approximately ± 0.003 in

a

(V). A change in o(-) above cutoff

±

causes a bigger correction. Varying the high-energy cross sections above 30 BeV does not change the low-energy (V S. 4 BeV) dispersion calculations. There) our results agree with those of Ref. 30.

7. Real Parts of K~p Amplitudes

We calculate9 the real parts of K-p forward + scattering amplitudes

-L

in the same way as for "~p. The data between threshold and V; 3.3 Bev37

) were slightly smoothed. Above that. point) the following fit was made:

cr(K

+

p) ; a

(11)

a ; 17.2 b ; 17.4

The dispersion relations were evaluated for cases I and II as in "P' with the cutoff in case II taken at 20 BeV. The errors involved here are much bigger than in "P· The uncertainties in the subthreshold singularities do not allow ^a good determination of the real parts at low energies. In particular) the Y (1405) is i: an S-wave) and thus is not quenched kinematically. We estimate its effect to be six times

*

as big as the Born term in rtN. This would be approximately 5 - 10 percent of the real part at V; 5 Bev. An additional unknown is the

*

We approximated the resonance by a pole. The coupling ^. g ² /4rt ; 0.32 is estimated from the Dalitz-Tuan mode1 38).

subtraction term of the symmetric amplitude, A(+)(V

=

µ). However, their combined effect remains constant, while the imaginary part grows like v, so that their contribution to o;(K-p) should fall like +

l/V. In case II there is the further difficulty of evaluating the

b t . . h . . 1 . t d A ( - ) Th

su raction constant c in t e antisymmetric amp i u e • e CEX reactions are not related by a simple I-spin rotation. Nor has a direct experimental determination of a±(V) by Coulomb interference been done. The only existing test is the forward elastic dif-

ferential cross section. This is a measurement of 1 +

a

²

.

I f a ^{is small, its} becomes difficult.

K+p data ³⁹⁾ , which The error in o; was

d (j

dt t = 0

determination Fortunately there suggests jo:(K+p)j

2

GT 2

(1

+

o; ) 16rc

exists relatively ,..,, 0.55 ± 0 .15 for evaluated by allowing the dcr/dt data

accurate v ^~ 7

-

to vary (12)

15

wi thin their error bars, and evaluat ing the variation in Cl through (12). If we allow a further variation of one standard deviation on dcr/dt, we can set a lower limit on

a

of,...., 0.25. The K ^-p data ⁴⁰⁾ is

Bev.

consistent wi th jo:(K-p) j

=

^{O, but} an upper limit of "' 0.3 has to be allowed within error bars. An additional standard deviation in-

creases this limit to,.,,, 0.5. The calculated values of o;(K±p), together with the experimental limits are plotted in Figure 8.

Case I seems to disagree with the data. In case II we can· ex- plain the discrepancy by means of the subtraction term. To fit

a(K

+

p), we can choose either one of two values, depending on the sign

of

a,

which cannot be determined by this method. We find for

K

=

3.3,

a(K

+

p)

>

O,

c

= -

1.6 is ruled·out because it gives O(K-p),...,, 0.65. Hence we conclude that O(K+p) < 0 and a(K-p)

>

0. The data points for

a(K+p) were plotted under this assumption in Fig. 8. The errors are clearly very large and allow us safely to ignore the subthreshold singularities!

The general features of np dispersion relations appear also in Kp. The logarithmic behavior is magnified biecause 2a ^(-) -.; 4 mb. However, at present energies the bulk of the real part seems to come from the subtraction term; and not from the logarithmic one. In fact, these appear to have opposite signs. Thus we expect la!

actually to fall until very high energies, when a changes signs and ja

I

begins to grow again. As in np, the real part does not dominate until extremely high energie~.

The difference between the pion and the kaon amplitudes lies in the energy range below the cutoff point. The usual Regge pie- ture -- which assumes the Pomeranchuk theorem to hold -- is compa- table with experiment for the pions, but appears not to be so for the kaons. In the latter case, the existence of an additional real term seems to be implied by the data.

APPENDIX

In this appendix we list Cutkosky's results 26) for SU(3) projection

operators, in 8 _... x _,.., 8 ~ _,,,..,, 8 x 8 and relate his couplings to those of Rosner 27) • . We use the normalization

(1)

= -

4 ₃ ^{5 Q} + 2d Q A

a::I-' 0:1-'Y Y (2)

Repeated indices are to be sunnned from 1 to 8. In this normalization

f f = 35 (3)

anm ~nm o;~

d d =

2

5 (4)

a.nm ~nm 3 o;(j

d f = 0 (5)

anm (jnm .

We use the shorthand notation fo;(3.fy 5 for fo;~mfy

5

^m. These scalar products form our couplings. They correspond to intermediate octets

in all three channels. Projection operators for other repre- sentations in a given channel are linear combinations of these couplings. Our couplings satisfy the following useful identities

fo;(j. fy5

+

^f

av.

f5(3

+

_fo;5·f~y

=

0 (6)

fo;~'dy5

+

fo;y•d5(3

+

_fo;5·d~y

=

^{0 (7)}

do;~'dy5

+

^d o;y'd5~

+

_do;5·d~y

=

₃ ¹ _(5a!35y5

+

5ciy55~

+

5a::55~y) . ⁽⁸⁾

fa~·dy

5 =

do;

5

^{·f~Y - do;y·f}

0

^{~ (9)}

1 3

5a!35yo 2 (fa

5

^{·f~Y - fo;y·f}

0

^{~) +}

2

^(do;y·d

0

^{~ + da::}

0

^{·d~Y) . (10)}

The projection operators for SU(3) representations in the s channel for an elastic process are

8 3 , d

~s (ll)

A yo' C43

5 °et13"

yo

8 1

"'a

(12) A _{yo' a:~}

3

fa:~·fyo

Ay5,l aj3

= ^l s

⁰ a:~ ⁰ yo (13)

Qyo,a:~ ⁼

.[s

^{-i (f} a:~ ^·d yo ^{- d} a:l3 ^{• f} yo ⁾ . ⁽¹⁴⁾

27 1 8

coayo

5

^{~ oa:oopy) ... s} Al, ⁽¹⁵⁾

A-;5 , ap

2 + A

yo,a13 yo ,a:l3

10 1 _{(o 0 - 0 0 )} ¹ ""a ⁸

J"s

(16)

I\~

=

^{- A} _{Qop ,ay}

• yo,a:~ 4 o;y op

ao

f3y 2 y5,Ctf3 3

10 1 ⁸ +

J" s

oa:0opy) 1 "'a (17)

f....,..., 'yo ,a:l3 =

4

(oayoof3 -2 yo A ,o:~ 3 Qo~,ay

Qyo,a:13 is the projection operator for an intermediate octet with f coupling in the initial state and d in the final or vice- versa. For an elastic process the two are related by time reversal

invariance? and only the combination Qyo,a:13 is allowed.

An alternate coupling scheme of Rosner27) uses the idea that each meson is built of a quark and an antiquark. His couplings are of the type (f...J-~f.../1.

0

⁾ where ( ) stands for 1/4 Tr. This means physically that the quark of particle

o:

annihilates the antiquark of particle 5, the qu·ark of 5 , the antiquark of y, and so on. A different coupling scheme has to be used in meson-baryon or in baryon-baryon scattering, since baryons are composed of three quarks. The connection between the Rosner scheme and ours is straightforward with the use of the

following identities.

fa~·fy5 = - t ^{\ [Aa,A~ ] [Ay,A5])}

ida~.

^{fy5 =}

t ( ^(Aa,A~}

^[Ay'A5])

ifa~·dy5 = t ([Aa,A~](Ay'A5})

da~ · dy5 ⁼ t ((Aa,A~}{Ay'A5})

^-

~ (Ac}-~)(AyA5)

In particular

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

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C.Y. Chien et al., Physics Letters 28B, 615 (1969). We use the