unitarized finite energy sum rules and their application to π n-charge exchange

Nuclear Physics B41 (1972) 557-586. North-Holland Publishing Company

U N I T A R I Z E D F I N I T E E N E R G Y S U M R U L E S A N D

T H E I R A P P L I C A T I O N T O r r N - C H A R G E E X C H A N G E

F. SCHREMPP I1. Institu t fu'r Theoretisehe Physik der Universitiit Hamburg, Germany

and Deutsches Elektronen-Synehrotron DES Y, Hamburg,

Received 21 December 1971

Abstract: The problem of 'unitarizing' finite energy sum rules (FESR) by the inclusion of Regge cuts is investigated. Instead of considering Regge cuts in FESR for the full scattering amplitude Tfi, new FESR are proposed for that part of Tfi, which has just pure Regge pole asymptotic behavior if the Regge cuts are included in Tfi. The Regge cuts in the high-energy region are evaluated according to the 'weak-cut' version of the Reggeized K-matrix model. The new FESR then look like the conventional ones, however, with Tfi replaced by the corresponding two-particle K-matrix element. With these K-matrix FESR the serious difficulties arising in conventional approaches to the problem disappear. With the new K-matrix FESR a detailed analysis of nN-charge exchange is performed. The K-matrix FESR turn out to be satisfied with only the o pole on the high energy side much better and more locally than the corresponding conventional FESR. The p residues in the K-matrix are predicted to choose 'nonsense' at c~ 0 = 0. If reduced o residues are defined as in the Regge limit of a nN-Veneziano formula, they are found to be practically constant between t = m 2 and t ~ 0.4 (GeV/c) 2. The values agree well with those known at t = m 2 and with results p from high-energy fits.

I. I N T R O D U C T I O N

Analy t ic i ty and the assumption o f a pure Regge pole high-energy behaviour have

led to the wel l -known fini te energy sum rules ( F E S R ) for a two-body scattering ampli tude Tfi(fu, t):

N

1 f dvvnlmTf i (v , t ) ~ ~ ~/(t)N~/(t)_ £ - Y - (1) S ~ + l o / ~ ( t , . n . 1

D

The impor tance o f these F E S R originates mainly in the fol lowing propert ies:

(a) Since in the low-energy region Im Tfi (v, t) is given to good approx imat ion by a sum o f resonances, the F E S R (1) imply the no t ion o f (global) duali ty [1 ], i.e. the resonances in the direct channel 'bui ld tip' the Regge poles in the crossed channel.

(b) The F E S R (1) are l inear in the matr ix e lements and, therefore , the di f ferent react ion channels may be t reated independent ly . Because o f this fact, the F E S R (1)

558 F. Schrempp, ~rN charge exchange sum rules

provide a simple linear bootstrap scheme, based on analyticity and crossing. (c) The FESR (1) permit to determine the Regge-pole trajectories a/(t) and the

residues ~j(t) as functions of t from the low-energy data. The main advantage compared to corresponding determinations from the high-energy data consists in the fact that one is always concerned with individual scattering amplitudes and moreover, that no parametrizations of the Regge-pole functions are required.

What will, however, happen to these important results, if we go beyond the pure Regge-pole model in the high-energy region?

In fact, it is now widely accepted that Regge behaviour and the bilinear character of unitarity implies the existence of Regge cuts. The importance of Regge cuts is, moreover, strongly supported by the high-energy data. Thus, in order to incorporate unitarity besides analyticity and crossing in the FESR scheme, suitable Regge-cut terms should be added on the right-hand side (r.h.s.) of the FESR (1). Since from general considerations only some qualitative properties of Regge cuts are known, for their concrete evaluation one has to rely on dynamical models [2-6], which yield the Regge cuts by an iteration of the Regge poles involved (see for instance eq. (3)). With a relatively small number of free parameters, these Regge cut models give surprisingly good fits to the high-energy data [3-8]. However, if one tries to 'unitarize' the FESR (I) by adding such Regge cuts on their high-energy sides, the nice properties (a)-(c) cannot be carried over because of the following reasons:

(i) As to the notion of duality, one is now confronted with an ambiguity, since it cannot be read off the FESR, which part of the low-energy contributions in the direct channel corresponds in a dual sense to the Regge poles and which part to the Regge cuts.

(ii) On the one hand - according to the Regge-cut models considered - the scattering amplitude is unitarized by the inclusion of Regge cuts [12, 13] (c.f. sect. 2). This advantage, however, goes at the expense of the fact that the linear character of the FESR gets lost by the appearance of coupled channels.

(iii) As another consequence, the Regge-pole functions ~/(t) and ai(t) result as so- lutions of complicated, coupled non-linear integral equations [15]. Therefore, a determination of these functions from FESR becomes impracticable, unless very spe- cial parametrizations and crude approximations are used. I f Regge cuts are included, it is, however, extremely important to make use of the additional information provided by the low-energy data. For the Regge pole contributions often can hardly be isolated from the Regge-cut 'background' by fits to the high-energy data alone. This becomes obvious e.g. by the fact that the pomeron as well as the pion appears in the literature with flat [8, 9] and with 'normally' steep [10, 11 ] trajectories, although the respective Regge-cut models are very similar.

It is the purpose of this paper to show up a possible way out of these difficulties (i)-(iii): We propose new FESR, not for the full scattering amplitude Tfi, but rather for that part of it, which has - after including Regge cuts - just Regge pole high-energy behaviour. Although Tfi is 'unitarized', the new FESR contain only Regge poles on their high-energy sides.

F. Schrempp, nN charge exchange sum rules 559

The plan of the paper is as follows: In sect. 2 we select a suitable Regge-cut model for the high-energy region, namely the Reggeized K-matrix model by Lovelace [5]. In sect. 3 we then propose our new FESR, which look like the FESR (1), however, with Tfi (u, t) replaced by the two-particle K-matrix element Kfi (v, t). Sect. 4 contains theoretical arguments in favour of the new FESR. In sect. 5 we perform a detailed analysis of nN-charge exchange by means of the K-matrix FESR. On the one hand the validity of the new FESR is verified directly from the data and on the othex hand a number of interesting predictions is obtained which are compared with the corresponding high-energy results.

2. CHOICE OF A REGGE-CUT MODEL

The dynamical Regge-cut models, which have been proposed for the high-energy region, are all based on the idea suggested by the structure of unitarity that besides single scattering, effected by Regge-pole exchange, one has also to take into account multiple-scattering type corrections arising from repeated exchange of Regge poles. Among these Regge-cut models the most popular ones are the Reggeized K-matrix model [5, 14] or the Reggeized absorption model [3, 4] and the Reggeized eikonal model [2] or the hybrid model [6]. Both, the first group and the second group of Regge-cut models, can be found in the literature in a 'weak-cut' or in a 'strong-cut' version, according as multiple scattering is thought to proceed via elastic intermediate states only or also via two-body states produced by diffractive dissociation from the initial or final states. For a detailed discussion of the different Regge-cut models the reader is referred to the existing reviews [16]. All the models mentioned agree to second order in the single scattering term and, therefore yield similar fits at very high energies and small momentum transfers for a specific choice of intermediate states.

Since we intend to study FESR and duality questions in the presence of Regge cuts, we have to select a multiple-scattering formalism which moreover allows a simple description of the resonance region in terms of a suitably parametrized single-scattering term. As a typical high-energy approximation, the eikonal picture is not well suited for that purpose. This can be seen from the fact that a resonance like pole ansatz for the single scattering (eikonal) produces an essential singularity in the full scattering amplitude. On the contrary, the K-matrix formalism was originally applied just in the low-energy region, since here it provides a simple unitary representa- tion of the scattering amplitude. Furthermore, referring to some basic results of the multiperipheral bootstrap model, Dash, Fulco and Pignotti [12] argued that at high energies the identification of the K-matrix with Regge-pole exchange leads to a T- matrix which satisfies approximately the fu l l s-channel unitarity equations. Thus , at present, the K-matrix formalism seems to provide the most suitable basis for a uni- tarization of FESR. At high energies the Regge cuts are, therefore, evaluated according to the Reggeized K-matrix model. Finally, we have to decide between the 'weak-

560 F. Schrempp. nN charge exchange sum rules

cut' and the 'strong-cut' version. From a theoretical point of view it is difficult to favour one of the two versions as long as it is not known, which intermediate states in the s-channel unitarity relation are already taken into account by a (complex) Regge pole exchanged in the t-channel. Notice, however, that the attractive concept of exchange degeneracy for the Regge poles cannot be realized in the 'strong-cut ' models ++ . Both versions yield fits to the high-energy data of similar quality. However, in the 'strong-cut' case a quantitative evaluation of all the contributions from non- elastic intermediate states is impracticable. Usually, these contributions are accounted for in an empirical way by introducing an additional free parameter X > 1 multiply- ing the 'weak' cuts. With this situation in mind, we appeal to simplicity arguments and choose the 'weak-cut ' version of the Reggeized K-matrix model: Starting from the following set of coupled two-body reactions,

-+ described by a T-matrix T = Ti Tif , . . .+

i ___>

we define the two-body K-matrix

K = ( Kii Kif 1

~Kfi K f f /

f o r all energies by +

T = K + iKp@T ,

where the elements of the matrix l:0 ) p = pf

(2)

are the appropriate two-particle phase-space factors. At high energies we assume the K-matrix elements Knm, (n, m = i, O, to be given

by the conventional (complex) Regge-pole exchange amplitudes Rnm. The multiple scattering structure of eq. (2) becomes obvious, if we consider its iterative solution:

T = K + F , [ I ~ K , K @ I ~ K , . . . ] " (3)

+ The symbol @denotes an ordinary multiplication in the angular momentum basis or - equiva- lently a convolution in momentum space.

++ See Fox, ref. [16].

F. Schrempp, , N charge exchange sum rules 561

at higher energies, when K _'z R, the second term in eq. (3) yields the Regge cuts.

3. FESR FOR THE K-MATRIX

In the introduction we emphasized the serious difficulties which arise if the FESR (1) are 'unitarized' by adding the Regge-cut t e r m s F[R@R ' R@R@R . . . . . ] to the Regge poles on the r.h.s, of the FESR.

We propose the following way out of these difficulties: Instead of modifying the high-energy side of the FESR (1) by Regge-cut contribu-

tions, one might also try to adjust its low-energy side, leaving the high-energy side un- changed. Thus, let us consider on the left hand side (1.h.s.) of the FESR (1)- instead of Im Tfi - the quantity

Im(Tfi - Ffi IK@K ' K@K@K . . . . . ] ) = ImKfi.

Im Kfi is just that part of Im Tfi which has a Regge pole high-energy behaviour, if the full scattering amplitude is 'unitarized' by the inclusion of Regge cuts as specified in sect. 2.

Thus, the central hypothesis of this paper consists in the statement that not the full scattering amplitude Tfi but rather the corresponding K-matrix element Kfi satisfies the usual FESR, i.e.

1 N ~j(t) Np (') f dv v n lmKfi (v , t ) ~- ~ (4)

N n . l j a/( t ) + n + 1 " 0

Kfi(u, t) can be calculated- as well as Tfi(v, t) - from the experimentally known (complex) phase shifts.

Before we give arguments for the validity of these K-matrix FESR (4) let us point out, why all the previous difficulties (c.f. subsects. 1(i) - 1 (iii) are removed, if the K-matrix FESR (4) actually hold:

(a) The notion of duality gets again a unique meaning by the FESR (4). The pre- cise duality content becomes evident in sect. 4, where realistic low-energy models for Im Kfi are discussed.

(b) Although the full scattering amplitude Tfi is 'unitarized' by the inclusion of Regge cuts, the K-matrix FESR (4) do not explicitely involve coupled channels. Thus the FESR (4) may serve as a basis for a 'unitarized' but nevertheless linear bootstrap.

(c) Since })n their high-energy sides, the K-matrix FESR (4) are identical with the usual FESR (1), the Regge-pole functions fl/(t) and ai( t ) can again be determined as functions of t from the low-energy data, although the full amplitude contains all the Regge cuts.

562 ~: Sehrempp, nN charge exchange sum rules

4. ARGUMENTS FOR THE VALIDITY OF THE K-MATRIX FESR

First of all, let us demonstrate that our FESR hypothesis (4) is at least not unrea- sonable: In fact, it has been shown by Gervais and Yndurain [ 17] that any function being asymptotically dominated by Regge poles for v -+ ~, satisfies the usual FESR for moments

n > - Min(~i(t)) - 1 g - -

in the sense of an asymptotic equality +. Thus, on the basis of the Reggeized K-matrix model in the high-energy region, at least the relative error of the K-matrix FESR (4) vanishes fo rN --> o~. In case of FESR for the full scattering amplitude one has not only a Regge-type high-energy behaviour, but also analyticity in the cut v plane. Given a certain high-energy model, analyticity leads not only to a decreasing relative error of the FESR for N -+ o~ but moreover to a decreasing absolute error. The attempt to prove a corresponding statement on the absolute error for the K-matrix FESR (4) is complicated by the fact that Kfi(v, t ) , f o r f i xed t, does not have in general the same simple analytic structure [18] in v as Tfi(v, t). Apart from a left hand cut and a right hand cut along the real v axis, Kfi(v, t) has also complex branch points in v clustering around the negative v axis. On the other hand, if instead of t, cos0 s is held fixed and IcOS0s[ ~< 1, the K-matrix has been shown (in the equal mass, spin zero case) to have no complex s branch points [18, 19]. Consequently it satisfies the same type of fixed cos0 s dispersion relations as Tfi(s, cOS0s) does, the only difference being that the r.h. and l.h. branch points on the real s axis are located differently. Therefore, for fixed

cOS0s, K-matrix FESR can be derived just as the FESR for Tfi, but unfortunately for fixed cOS0s, there is no definite idea on the high-energy behaviour of Kfi(s , cOS0s).

However, for elastic and charge exchange reactions, cos0 s = 1 corresponds to (fixed) t = 0 and thus, at least for t = 0, the K-matrix FESR can be shown to hold in the same sense as FESR for the full amplitude ++. From analyticity considerations alone, one is always left with FESR in which the integration extends also over negative v, i.e. over an unphysical region [ 1 ]. In case of the full scattering amplitude one can often make use of crossing symmetry in order to relate this unphysical piece to the physical region. The K-matrix lacks, however, in general s - u (~ v ~ - v ) crossing symmetry in the non-asymptotic region. Actually this does not cause difficulties, if the K-matrix FESR hold more or less locally +++. Then, the same arguments may be

applied as used in the case of the conventional FESR (1) in order to obtain exchange degeneracy [31]: If a FESR holds sufficiently local, it should hold separately

• f ( x ) + J~x) is asymptotically equal to g(x) for x -~ ~ ~ h m g ~ = 1.

+ +

Note, that the same result holds also for t # 0 if the total contribution of the complex v singularities to Re Kti(u, t) vanishes faster than 1/v for v -~ oo.

+++ The phrase of a 'local validity of FESR' is always to be understood in the sense that the extrapolated Regge term should be equal to a quasilocal average of the physical amplitude considered•

F. Schrempp, ~N charge exchange sum rules 563

[23] for the positive and negative regions of p, since the imaginary part of the amplitude has to vanish identically between the 1.h. cut and the r.h. cut. Therefore, if the local-average hypothesis is actually valid for the K-matrix FESR, they should not only hold for t = 0 in the form (4) but also for t 4* 0, since the complex u-branch points of Kfi(u, t) always remain in the left half plane.

Thus, we have to look for arguments in favour of a local validity of the K-matrix FESR in order to justify the integration range (4).

Analyticity provides only a statement on the validity of FESR f o r N -+ ~. The validity of the conventional FESR (1) for rather small integration intervals, which is actually needed for practical applications and which just led to nontrivial results, was established in a purely empirical way. With this in mind, we now give theoretical arguments, which depend on certain realistic low-energy models for Kfi(u,t), but which have the advantage to support directly the validity of our K-matrix FESR (4) in the sense of a more or less local average. The question, whether the new K-matrix FESR (4) are also valid independent of low-energy models for Kfi(u,t ) will be post- poned to sect. 5, where the K-matrix FESR are directly confronted with the data of 7rN scattering.

Now, let us give three arguments in favour of a local-average validity of the K-matrix FESR.

4. 1. The Veneziano )brmula Lovelace identified the K-matrix with the Veneziano formula [20] and that in the

low-energy region [21 ] as well as in the high-energy region [5] +. Apart from a tole- rably small violation of crossing symmetry the full amplitude then has the following properties:

(i) in the low-energy region elastic unitarity is explicitly satisfied and accord- ingly, the energy poles of the Veneziano formula show up as correct resonance terms in the full amplitude.

(ii) in the high-energy region, besides the Regge poles, there are also Regge cuts having the correct properties.

(iii) with relatively few free parameters this model leads to a surprisingly good agreement with the data in the low-energy region as well as in the high-energy region. On the other hand, the Veneziano amplitude fulfils automatically the usual FESR and that in a local way. Consequently, to the extend that the K-matrix is given by the Veneziano formula, our FESR (4) are satisfied locally.

4 .2 . Breit- Wigner resonance approximation Let us start from the full scattering amplitude and assume it to be given in the non-

asymptotic energy region by a sum of Breit-Wigner type resonances. This model is somewhat more general than a Veneziano ansatz for the K-matrix element. The K- matrix element KJfi(s), (i 4: f), with angular momentum j is calculated on this basis -from the following (2 X 2)-submatrix of the full T matrix:

+ At higher energies actually the Regge limit of the Veneziano formula was taken.

564 F. Schrempp, 7rN charge exchange sum rules

, m , n = i , f , (5) T~n(s) - Pmn(s) M 2 - s - i M r "

where Pmn = P'~-~mm " P@~nn is the relevant phase-space factor, M the mass, F the total width of the resonance with spin j and Xm, x n is its branching ratio into channel m, n. We find from eq. (3):

• _ l M F V ~ f ~ (6) K/fi(s)

pfi(s) M 2 - s - i M P ( 1 - x i - x f ) "

Obviously, K{-I(s ) has - jus t as T~i(s ) - Breit-Wigner like resonance form. The difference is that a resonance appears in K{i(s ) with a width reduced by a factor of (1 - x i - xf). It is intuitively clear that the local validity of FESR near a resonance requires the integration interval in the low-energy integral to be at least equal to the mean extension of the resonance peak considered. Since the width of a resonance appears in general to be considerably reduced in Kfi(v, t), we are led to the conjecture that the K-matrix FESR can be satisfied in a much more local way than the conventional FESR.

I f x i + xf = 1, the resonance pole in KJr,(s) is even shifted to the real s axis and, therefore, contributes only with a 6 function + to lm Kfi(p, t). Thus, in all cases, where - along the integration interval considered - I m Kfi(u, t) is approximately zero apart from 6 functions resulting from resonances with x i + x f = 1, we have the following general result from eq. (6): The K-matrix FESR (4) are identical with the conventional FESR evaluated in the narrow resonance approximat ion (NRA) and that independent of the total resonance widths F i. It follows that in this case the relations between resonance parameters, obtained from the conventional FESR (1) in the NRA, should be essentially exact - and not approximations to order O(F) - if our K-matrix FESR (4) hold. Such relations have actually turned out to be surprisingly accurate or consistent over a fairly large t range. As examples let us recall FESR analyses o f n n - ~ 7rn by Schmid [22] and of ~rrr -+ rrco and rrTr -* rrA 2 by Ademollo et al. [23]. In all these reactions the FESR evaluated in the NRA are well satisfied with the p and f resonances only. Both decay purely into 2~, whence x~O, f + xf , f -~ 1 is fulfilled. Moreover, this

resonance saturation turned out to be possible locally. The same is true for the K-matrix FESR.

Furthermore, from these considerations in the resonance approximation for Tfi @, t), we may read off the new duality prescription in the presence of Regge cuts: The Regge poles in the t channel are 'built up' in the average by 'narrow' resonances in the s channel.

+ Or more precisely: its residue has to be added to the l.h. side of the FESR (4).


4.3. Exchange degeneracy and ghost-killing mechanism Further support for the validity of the K-matrix FESR (4) for t v a 0 comes from

the following consideration: The conventional FESR (1) and the existence of exotic channels lead to exchange degeneracy (EXD) of the full scattering amplitudes. In a pure fi Regge-pole picture of nN charge exchange, EXD together with factorization of the fi residues require [24] the vanishing of both the helicity nonflip and flip amplitudes of ~rN-charge exchange at ap = 0 (i.e. the t) has to choose 'nonsense'at ao = 0). This prediction is, however, inconsistent, since at c~ o = 0 the differential cross section of 7rN-charge exchange does not vanish experimentally. On the other hand, if there are Regge cuts besides a choosing 'nonsense' fi pole the fits to the differential cross section do not vanish at ap = 0 in agreement with the data. In the presence of Regge cuts the fi may, however, also choose 'sense' at ap = 0 without being inconsistent with the arguments given above, since the full scattering amplitudes do not factorize anymore. Thus, the nice predictive power of EXD seems to be lost, if Regge cuts are included.

If we start, however, from our K-matrix FESR (4), EXD is found for the Regge poles in the K-matrix elements. With the plausible assumption that the Regge poles in the K-matrix factorize, we then immediately predict the choosing 'nonsense'mechanism for the p-Regge pole in the 7rN-K matrix. This mechanism is actually preferred by the high-energy fits of appropriate Regge-cut models to the differential charge exchange cross section. This consistent result obviously supports the validity of the K- matrix FESR (4) away from t = 0. Let us remark that in subsect. 5.5 we confirm the choosing 'nonsense' mechanism for the p - independent of EXD and factorization arguments - by evaluating our K-matrix FESR directly from the experimental nN-phase shifts.

5. ANALYSIS OF 7rN CHARGE EXCHANGE

5.1. K-matrix amplitudes For FESR analyses of ~N scattering it is convenient to use the standard t channel

helicity amplitudes A'(v, t) and vB(v, t) [25]. The even-signature (isospin zero) exchanges contribute to

A'(+)(v, t) = A'ron(V, t) = ½ (A'Tr_p(V, t) + A'lr+p(V, t)), (7)

the odd-signature (isospin one) exchanges contribute to

A'(-)(P, t) = - I~A' (v, t) = 1 (A 'Tr_p(V, t) - A 'rr+p(V, t)), (8) 7r - Pcex.

and similarly for vB(±)(v, t). The expansion of A'(v, t) and vB(v, t) into partial wave amplitudes fl+(s) of orbital angular momentum l and total angular momentum J = l ±1 is well known [26]. We define

566 F. Schrempp, lrN charge exchange sum rules

l ( e2 i al+(s) ) f! + (s) - 2 t o ( s ) - - 1 , (9)

where p(s) = 2q/x/s, q is the modulus of the c.m. momentum and 61+ - (s) are the (complex) phase shifts. Hence elastic unitarity reads for definite s-channel isospin Is:

I m f / (s) = p(s) fl+-(s) (10)

Next we consider the (2 X 2)K-matr ix defined by: ), + l/j/~+- ) (11)

By eq. (11) and eq. (9) together with the isospin decomposition of the)')+- the relation of the K-matrix elements kl+ - corresponding to definite charge configurations to the experimentally known phase shifts 6~S+(s) is established. Completely analoguous to A'(+-)(v, t) and vB(+-)(v, t), (eqs. (7, 8))~we then define K-matrix amphtudes/1 ~(+) (v, t) and VBk(+-)(v, t ) jus t by replacingfl+(s ) by kl+-(s ) in the relevant partial-wave expansions. The K-matrix amplitudes A'k (-) and vBk( ) we assume to be dominated for v ~ oo by t channel Regge poles w i t h l p G = 1-+(o = signature, G = G-parity) and the amplitudes A~. (+) and vBk (+) by Regge poles w i t h I t °G = 0 ++. In the following sections we discuss in detail the analysis of the K-matrix amplitudes A~ (-) and VBk (-) with our new K-matrix FESR. The results of a corresponding analysis o fA~ ~+) and uB/¢ (+) will be reported elsewhere [27].

5.2. Continuous moment sum rules For the concrete analysis of 7rN-charge exchange we prefer to use continous mo-

ment sum rules (CMSR) instead of the FESR (4) for the K-matrix amplitudes. With CMSR one can make much better use of the range of moments for which the sum

• , ( - ) ( ) . . rules hold best. Parallel to the analysis o f A k and uB~ with K-matrix CMSR, we perform an analysis of the full amplitudesA ' ( -) and uB (-) with the conventional CMSR [28] under completely equal conditions. The p meson is the only known particle which can be exchanged in the t channel. Therfore, we try to satisfy the K-matrix CMSR with only the p-Regge pole on their high-energy sides. Previous CMSR analyses [28] of the full 7rN amplitudes A ' ( -) and vB (-) have shown that on the basis of the pure Regge-pole model in addition to the p-Regge pole a further hypothetical p' is required. Nevertheless, in our CMSR analysis of the full amplitudes, performed for comparison with the K-matrix results, we only take into account the p. In this way the differences become most clear. If F(-)(u, t) stands for any of the four amplitudes A~ (-), UBk (-), A '(-) and vB (-) the p contribution is conveniently written as:

F. Sehrempp, ~rN charge exchange sum rules 567

where g =

F(-) (v , t ) v ~ F p ( v , t ) = f # ( , ) ( v )

1 GeV and

l ~ ( t )

7~(t)

~p(t)

%(0

i fF( - ) (v , t) =

V2o -- V Vz(ap(t) - l) (12)

A ' ( )- - k tv, t)

t) A ( - ) ( v , t)

vB( - ) (p, t)

(13)

The 1/cos (1Air a , ( t ) ) factor of the usual Regge-pole parametrization is absorbed in the coefficient function f , ( t ) . The characteristic signature factor is contained in 1(v2 _ v2)(~ o 1) This term is defined to be real and positive for real v with - v o < < v < v o. It is convenient to identify v o with the beginning of the r.h. cut o f F ( ) ( v , t ) . For the K-matrix it would be the first inelastic threshold. However, in this case we actually choose a slightly higher value of v o for reasons to be discussed in the next section. For the four amplitudes considered we use the following CMSR [28] :

of dv Im F ( - ) (v , t ) - - o

% ( 0 - a '

where pole terms are to be included on the 1.h.s. The 1.h.s. of the CMSR (14) is evaluated numerically from the CERN-phase shifts [29]. Since it is not known, for which value of N the CMSR (14) hold best, the following averaging procedure was consequently applied: The whole analysis was performed for ten values of N in the interval

1.2 GeV ~< N ~< Nmmx(t), (15)

where Nmax(t ) corresponds to the maximal energy available: X/Sma x = 2.19 GeV. Each quantity determined from the CMSR (14), which should be theoretically independent of N, is always given as the arithmetic mean value of the values resulting for each N. The standard deviation of this mean value then represents a suitable quantity to characterize the N dependence of the single values. Later on, the standart devia- tions will appear as error bars in the diagrams. Note, however, that they have nothing to do with experimental errors!


5.3. Pole contributions In this section we discuss the contributions from poles in the K-matrix elements

A'k(-)(u, t) and UBk( )(u, t) to the 1.h.s. of the K-matrix CMSR (14). Such poles may arise in two different ways:

(i) a certain K-matrix partial wave amplitude

f21s, l+_ (s)

k2I,, l+(s) = 1 + i p (s)f21s, le(S ) (16)

may have a pole if the denominator in eq. (16) vanishes. poles m A k and UBk (-) arising (ii) even if all k21s, l+_(s) are finite, there may be " ' (-)

from a (local) divergence of the partial-wave sum. Case (i). Let us assume that all poles ofk2ls, l+_(s) at complex s values occur on

some unphysical sheets. Then, we may confine to the discussion of poles on the real s axis of k21 I+_(S), where we can take advantage of the information provided by the experiment~ phase-shift data. In the integration range o f the K-matrix CMSR (14) we have to distinguish between two cases: A pole at s = Sp may occur in the region

= (~ (a) Sel (/14+/2) 2 < Sp < Sma x u = N )

or in (b) Sel > Sp > M 2 + ~t 2 - ½t (~ p = 0), (17)

where M and/~ denote the nucleon and pion mass respectively. As discussed already in subsect. 4.2, in region (a) there are poles corresponding to purely elastic resonances. In 7rN scattering the only candidate is the 2x(1236), leading to a pole in k3,t+(s ) at s = m 2. A Laurent expansion of

1 k3,1+(s ) = p ~ ctg (½rr - 63,1+(s)) (18)

around ½rr leads to the residue

lim (m I - s) k3,1+(s) = I (19) s_+ m 2" .o(m I) d(5 3,1+(s)

2 s = m A

The Breit-Wigner approximation gives

, 2 d _ 1 3'l+(mA) = ' ~ ~ 3, I+(s) 2 m a r &

s = m A

With max = 1235.8 MeV and Fax = 125 MeV [29] one obtains

, 2 83,1+(max ) = 6.45 GeV -2. t

We prefer, however, to determine 83,1+ directly from a least-square fit of N ~i (u) (m 2

(53,1+(s) = ½rr + ~ 3 ~3 ' l+V"ax" ( s - m 2 ) v u = 1 V!

(20)

(21)

(22)


to the values of the CERN-phase shifts [29] around s = m 2. The degree of the poly- nomial (22) was fixed by imposing minimal X2in/(degrees of freedom). With a poly- nomial of 4th degree we obtain

' 2 ~i3,1+(mA) = (7.45 +0.3) GeV -2. (23)

In connection with the evaluation of the A(1236) contribution the following should be added: generally, with the parameter ~'o we may fix the point, where Re F (-) starts to contribute in the CMSR (14). Since the A(1236) pole is located on the u axis slightly beyond the one-pion production threshold Vinel , it is very convenient to choose v o > u a 2> Pinel. Then, one has to inte}rate only over a 5-function contribution from the A(1236) in ImA)c ( ) and lm UBk ~-). Actually, in the K-matrix CMSR we choose for v o a value corresponding to xf i = 1250 MeV, i.e. slightly beyond the A pole. By varying u o over a certain range we made sure, of course, that the results do not depend appreciably on the specific choice of u o.

In region (b), a reliable determination of the locations and residues of possible poles meets considerable difficulties at present, since, here the partial wave ampli- tudesf21s, l._(s) are not known. They could, however, be calculated from the data in the physical region and a specific high-energy model by means of the partial-wave relations of Baacke and Steiner [30]. Unfortunately, such a calculation has not yet been performed separately for f3j+ and fl,l+ because of the existing ambiguities in the high-energy region. In order to get at least a rough idea about the pole structure in region (b) we used effective range expansions of the form

l = q 2 / + l q - (2 l + 1) 0 (s) kzls, l+(S) ctg62is, l+(S)

(24) = G an(S -- Sp)n

17

with coefficients a n determined near threshold by a fit to the phase shifts of Roper et al. [32] and of the CERN group [29]. The extrapolation of eq. (24) into the region (b), in fact, gives some indications for poles with the exception of the P11 wave, these indications depend, however, too strongly on the phase-shift solution used in order to be taken seriously. For the Pl l K-matrix element, we find a pole located at

Sp "" 0.95 GeV 2. (25)

It corresponds to the direct nucleon pole infl, l_(S), as may be verified by calculating kl,l_(S ) - according to eq. (16)- on the plausible assumption that near s = m~, f l , l _ (S ) is dominated by the nucleon-pole term. Then, we find Sp = 0.94 GeV 2, which is close to the value (25) found from the effective range extrapolation. The pole residue is, however, estimated to be relatively small. Therefore, we prefer to neglect this pole term at all. For t < - 0.08 GeV/e 2 it is, moreover, outside of the integration range.

Case (ii). Here, we only have the crossed nucleon pole, which appears in A~ (-) and in VBk(-) at the same position and with the same residue as in the corresponding full amplitudes A '( ) and uB (-). For the nucleon coupling constant, we take the usual value [1]


/-2 = 0.081 _+0.002. (26)

Thus, altogether in the K-matrix CMSR (14) we take into account the A(1236) pole and the crossed nucleon pole, whose separate contributions are displayed in figs. 1 a, b.

t,o-

:=..

- 20-

-20-

-l,o-

N

ofdV 3m F (-) (v,t)

i i

F(-)= A'~-}

N = 1.7 6eV

.....T_ j ~- /

/

. . . . CROSSED- CHANNEL-NUCLEON FROM CERN PHASE SHIFTS

d.2 o'.~ o'.6 oi~, 1;o -t [ (GeV/c)2]

~.2 12

i i i i i

- = -

-20-} - - -z- ~ -

I - . . . . CROSSED-CHANNEL-NUCLEON ~ - .40 i FROM CERN PHASE SHIFTS ~ I ~ .

b

o o.2 o2 0'.6 o:8 1'.o -t[(GeVlc) 2]

Fig. 1 (a) Various(co, n t r ibut ions to the low-energy integral of the K-matrix FESR for the K-matrix ampli tude A' k )(u, t). (b) Same as in fig. 1 (a) for the K-matrix amplitude UBk(-)(u, t).

5.4. Test o f the K-matrix CMSR In this section, we are concerned with the basic question, whether in fact the new

K-matrix CMSR hold independently of certain K-matrix models in the low-energy region; i.e. we check in detail whether the low-energy side of the K-matrix CMSR, evaluated from the data, is consistent with a Regge pole high-energy side. In particular, we shall examine the attractive conjecture that one needs to consider only known Regge poles - i.e. in our case only the p trajectory - in the K-matrix elements at high energy in order to get a good fit to the low-energy data. On the basis of the Reggeized K-matrix model the fairly strong p' contribution required in the full amplitudes [28] might turn out to be a Regge-cut effect (e.g. a p_fO cut) and then, it would not appear in the K-matrix CMSR.

The test of the new K-matrix CMSR is carried out using two different methods. Method A. Here, we proceed as usual in CMSR analyses by least square fitting the

r.h.s, of the relevant CMSR to its 1.h.s. taken as a function of the continuous parameter 8. The fits are always carried out for the ten different values of the upper integration limit N (eq. (14)) specified in (15).

The CMSR are applied over the following range of t:

0 ~> t ~> - 0.5 (GeV/c) 2, in steps of 0.1 (GeV/c) 2

and-0 .5 (GeV/c) 2 ~> t ~> -1 .1 (GeV/c) 2, in steps of 0.2 (GeV/e) 2. (27)


We try the same range of 6 as in ref. [28]:

0 ~ > 6 ~ > - 5 .

For the highest possible upper integration limit, Nm~v(t), the result of the fits is pre- ,~-~ ,( ) • sented in figs. 2 (a), (b) and (c) for the amplitudesAk andA . Figs. 3 (a), (b) and (c) show correspondingly the CMSR for the amplitudes uBk ( ) and uB (-). The fits for the other N values, which are not shown, are of similar quality as the presented ones, exept the three lowest N values yielding somewhat worse fits.

Figs. 2 (a), (b), (c) and figs. 3 (a), (b), (c) show that in the case of the K-matrix CMSR the fits are altogether surprisingly good. In particular they are essentially better on the average than the corresponding fits for the conventional CMSR.

If compared in more detail, the following points are worth emphasizing: (a) The fits to the conventional CMSR completely fail in the range

0 ~ > 6 / > - 2 (29)

already for t ~< - 0.1 (GeV/c) 2 in case of A '(-) and for t ~< - 0.3 (GeV/c) 2 in case of vB (-). This fact was already pointed out in ref. [33]. It was essentially this deviation of the low-energy points from the typical regularly oszillating 6 dependence of a single Regge-pole term which led in previous CMSR analyses to the introduction of a hypothetical p'-Regge pole [28]. But even with several Regge poles it is not possible to obtain a good fit in the 8 range considered because of the following reason. For 6 > - 1 the low-energy contributions to the CMSR integral are strongly enhanced because of the factor (Uo 2 - v2) - '/2(1 + 8). From the known threshold behaviour of the full amplitudes it follows that the 1.h.s. of the conventional CMSR has a pole at 8 = + 1. Any finite number of Regge poles, however, gives a finite contribution to the r.h.s, of the CMSR at 6 = + 1. Since the pole occuring in the 1.h.s. of the CMSR at 6 = + 1 has still a strong influence on the 6 region (29) there must be a discrepancy in the conventional CMSR! This effect becomes much stronger with increasing [t[, since then the threshold region becomes unphysical ([cOS0sl > 1) and is strongly weighted in the partial-wave expansion. Since the discrepancy arises from low-energy contributions it points to an inconsistency of the conventional CMSR with elastic uni- rarity, which plays an outstanding role in this region. In fact, if Regge cuts are added on the r.h.s, of the conventional CMSR good fits over the 6 region (29) can be obtained [33].

In contrast to the conventional CMSR with a pure Regge pole r.h.s., our K-matrix CMSR are just based on high-energy amplitudes which explicitely satisfy the unitarity relation in the elastic region. Therefore, it is not surprising that the K-matrix CMSR hold very well in the 6 region (29), although only the p-Regge pole appears on their r.h.s. This good agreement leads us to conclude that it is more consistent with unitarity, to assume a pure Regge pole high-energy behaviour for the K-matrix elements rather than for the full amplitudes. Moreover, this consideration clearly shows, how the low-energy constraint of elastic unitarity is transmitted via FESR to the high-energy region where it requires the presence of Regge cuts.

60

30 0

-15

-30 60

/,S

30

15 0

-15

-10 20

I0 0

-I0

-2{I

-30

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ydv ]

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bGeV

Z]

K-M

ATR

IX C

MSR

: F

{-)

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) -.a

K{

v, ~

)

I0 0

-I0

-20 30

t =-

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20

0 -10

-20 20

10

2O

3O

CONV

ENTI

ONAL

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: M

=

^,(

-)

F Iv

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t =-0.I

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IX C

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AL CM

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IX

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: C

ON

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: F

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t =-l

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Fig.

2 (

a) T

est

of t

he K

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rix

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for

A'k

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u,t )

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on 0

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For

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pari

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onal

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9]).

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• oo

,~

0 t -60 -B0

O -1

-2

6-

3 -k

-5

cb l

I o

-1

~5

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Fig.

3 (

a) S

ame

as i

n fi

g. 2

(a)f

or t

he K

-mat

rix

ampl

itud

e VB

k(-)(

v, t)

and

the

full

am

plit

ude

uB(-

)(~,

, t)

. (b

) S

ame

as i

n fi

g. 2

(b)

for

uBk(

-)(u,

t)

and

uB(-)

(v,

t). (

c) S

ame

as i

n fi

g 2(

c) f

or V

Bk(-)

(u,

t) an

d vB

(-)(u

, t).

574 F. Schrempp, rrN charge exchange sum rules

(b) For t < - 0.5 (GeV/c) 2 the discrepancy in the small 1(sf region 0 > (5 > - 1, may also be explained by the (theoretical) divergence of the partial-wave expansion. The divergence sets in in the low-energy region at smaller Itl values than in the high- energy region. So, for fixed t, the discrepancy will show up first in the small 16 I-range, since here the low-energy contributions dominate the CMSR integrals. Therefore, for t < - 0.5 (GeV/c) 2 all fits were confined to the range - 1/> (5/> - 5.

(c) For the t values - 0.3 (GeV/c) 2 ~> t >~ - 0.5 (GeV/c) 2 the fits are not very good both for A '(-) and for A ~(-) (fig. 2 (b)). Especially, in case of the K-matrix element A~c(-) this can be explained by the fact that the p Regge pole part is very small over the whole (5 range because it changes sign slightly above Itl = 0.5 (GeV/c) 2. Therefore, for such t values even a weak background can show up relatively strong.

(d) For - 0.7 (GeV/c) 2 ~> t >~ - 1.1 (GeV/e) 2 the fits to the K-matrix CMSR are considerably better than to the conventional CMSR over the whole range - 2 ~> (5>.-- - 5 (figs. 2 (c), 3 (c)). This fact is in accordance with the well-known result that the importance of Regge cuts in the full amplitudes increases with Jtl.

Method B. Let us denote the left-hand side of the CMSR (14) by SF((5, t). Instead of considering the different CMSR separately, we form the following ratios:

Qk ((5' t) = S A ,k ((5 , t)/SvBk(6, t) = ~ ( t ) / ~ / ~ t )

and (30)

Q (6, t) = S A,(6, t)/SuB(~, t) = 13o(t)/3,o(t ).

With regard to subsect. 5.5. it is more convenient to use instead of Q k and Q the quantities

t t 1 - - - 1

4M 2 4M 2 Vk(6, t) = and V((5, t) =

Qk((5' t) - t Q(6, t) t 4M 2 4M 2

(31)

The following properties of V k and V are very important: If, for fixed t, the CMSR (14) were fulfilled in an ideal manner, i.e. with only the

o-Regge pole on the high-energy side, because of eq. (30) the quantities Vk(6 , t) and V(6, t) would be (i) independent of (5 and (ii) independent of the upper integration limit N, for sufficiently high N. By checking (i) together with (ii) we therefore are provided with a sensible test on the presence of additional contributions besides the p-Regge pole term.

MoreoveK from the N dependence of V k and V respectively we may infer to which extent the CMSR hold locally.

The results for Vk((5, t) and V((5, t) are shown in fig. 6 and fig. 7. Note, that the error bars only characterize the N dependence of the points according to the averaging procedure described in subsect 5.2.

F. Schrempp, 7rN charge exchange sum rules 575

J I J J I I I i i I

] rnA'K (-)

. . . . . . lmA ' I - ) t=O0

~ o .

70-

o- VN VA ; r . . . . . . . . . . . . ~ . . . . . .

-2Q-

-~0 ' 0'.~ 0'~ 01~ 016 1'.0 llz 1'~ 1'8 1'6 L0

v 16eV]

I I I I I I I I I

6. fl I ~t ~ t :-0t, ~ I

.=t,- r , / \

l /.-. \ . . . . l m A " - '

O - r l " 1 j - I --w . . . . . . . . ~ - . . . . r . . . . . . .

" ~ i ,, i ~ I I I

o~ o~ 0'6 o18 1'o 11: 11~ 116 i'~ 2'o v [GeV]

Fig. 4 (a) Im A'k(- ) (p , t) and Im A '(-)(u, t) at t = 0, calculated f rom the CERN-phase shifts (ref. [29] ) as f u n c t i o n s o f u. (b) S a m e as in fig. 4 (a) for t = - 0 . 4 (GeV/c ) 2.

The following important features should be noticed: For all t values considered the importance ofl additional contributions besides the

o-Regge pole is much smaller in Vk(6 , t) than in V(6, t) because of the following reasons: (a) Vk(6, t) is much weaker dependent on 6 than V(6, t). Considered more closely, within the error bars, Vk(6, t) is almost consistent with a constant over the whole 6 range (fig. 6). On the contrary, V(6, t) shows a comparatively strong dependence on 6, especially at t around - 0.2 (GeV/c) 2 (fig. 7). (b) The error bars are considerably smaller for Vk(6, t) than for V(6, t), especially in the range + - 3 ~> 6 ~> - 5 . This gives good support to the conjecture that the K-matrix CMSR are valid more locally than the CMSR for the full scattering amplitudes with p-Regge pole high-energy behaviour. This result fits well into the picture we obtained from model consideration~ in subsect. 4.2.

+ The points , w h i c h are close to the zeros o f SA, k SoB. and S A ,, SoB respect ively, o f course,

' K have big error bars and should not be taken seriously.

576 F. Schrempp, ~rN charge exchange sum rules

I I I I f I I I I

Ot, ot, O otf' -

"2°ILJ v~ -&Olq ]m{vBK (')} -r~o I ..... ]rn (vB {')) - ~ f l l L ,

' ' ' i • ' ' '. ' o o2 o~ 06 o.B Io 12 I.~ I~ 18 2o v [ 6eV]

B 0 I 1 I i I L I I I I I

t :0.~ ;~° ~I!t /~

]m {ve~ ~) ~o-30. i t',,. / \ . . . . . 3rn{vBt-')

1 0 - "

lil O.

v -10.

-20 -

- 3 0 o 0.2 o'.4 d.~ o:~ 11o 112 1'~ 1'6 1'.~ 2'o v [GeV]

Fig. 5 (a) Same as in fig. 4 (a) for Im uBk(-)(u,t) and Im vB(-)(u, t). (b) Same as in fig. 5 (a) for t = 0.4 (GeVjc) 2.

5.5. Residues and trajectory o f the p-Regge pole In this subsection we shall go beyond a pure consistency test of the K-matrix

CMSR by making quantitative predictions for the p trajectory and especially for the p-residue functions. The p trajectory ako(t) and the coefficient functions/3~(t) and 7~(t) in Ajc(-) and UBk (-) (eq. (12)) are determined from the K-matrix CMSR by a least-square fit as described in subsect. 5.4. An analoguous determination of the p trajectory ap(t) and of the coefficient functions ~p(t) and 7p(t) from the conventional CMSR is carried out again for comparison. First the results so obtained for the p trajectory shall be compared with a corresponding trajectory determined from high-energy fits. With the assumption of a linear p trajectory, fits of the Reggeized absorption model [3] and of the Reggeized eikonal model [8] to the high-energy data of 7rN-charge exchange both lead to an intercept of the p-Regge pole in the single-scattering term of

c~(t = O) -~ 0.50. (32)


Vt~(6,t): r~/r~" (K-MATRIX CMSR)

o. -I-'?-'5.__ -2 -3 6 -4 -S

2o[• t: -O3

o+,,.~ ' -i'6i-~ -~

,0 t 1o] t :-0,2 t

-2 -3 5 -4 -s 20

Uol t:-O.3 ' [ r

-2 -3 8 -4 -5 2O

-2 -3 0 -4 -5

2°1 t:-0.0

'°t t t = - 0 . 7

1

-2 -3 6 -& -5

-2 -3 O -4 :5 20

- 5

Fig. 6 Test of the K-matrix CMSR prediction: Vk(6, t) = const. (cf. eqs. (30), (31) and (40) for

definition of Vk), The solid dots represent the K-matrix CMSR results calculated from the CERN- phase shifts (ref. [29]). (The error bars characterize the dependence on the upper integration li-

mit N(cf. sect. 5.2.). The solid line shows Vk(a, t) from eq. (42).

It should be mentioned that this value also results from the Veneziano formula [34] which just represents a good model for the K-matrix elements (see subsect. 4.1.). With eq. (32) and the condition

c~(m 2 = 0.585 GeV 2) = 1, (33)

we obtain the following p trajectory in the K-matrix +

a~o(t) = 0.50 + 0.86 t. (34)

+ The p-Regge pole parameters determined in the high-energy region from the Reggeized absorption model or from the Reggeized eikonal model should agree with those of the Reggeized K- matrix model at least in the small Itl region.

5 7 8 F. Schrempp, nN charge exchange sum rules

V (6,t) = O.p. rM/T v (CONVENTIONAL CMSR)

3O

q 0 I , , , , , ,

-2 -3 5 -l, -5 30 I t I ,04 .-o, ,,1

I I

O~ ....... ~ ......

4 -3 6 -~ -5

o !2 _ ~_ . . . .

-70 [ , , , ~ , m -2 -3 6 "~ -5

o f ~ _ L -10 l ,

-2 -3 6 ~ -5

, {I 2o-I t = - o & I ' " m ' t o T ~

'°-1 ~il~; m '

.,o;. ]1.. i.,i-. -2 -] 6 -L -5

30t 20 t = -0 .5 ~ ~ • , i

10

°T -lo , [ , t

- 2 -3 6 -L -5

11; - t=-O.£ T T : i

::::_! -2 -3 6 -~ -5

15 !

10_ I t=-11 6

5

-2 -3 5 -~ -5

Fig. 7. T e s t o f t h e c o n v e n t i o n a l - C M S R p r e d i c t i o n : V(6 , t ) = c o n s t (cf . eqs . ( 3 0 ) , ( 3 1 ) a n d

( 4 0 ) f o r t h e d e f i n i t i o n o f V as we l l as fig. 6 c a p t i o n ) .

The wrong-signature point aok = 0 thus corresponds to a t value of

t o = - 0.59 (GeV/c) 2. (35)

In case of a pure p-Regge pole model for the full scattering amplitudes the high- energy data prefer the following p trajectory [35]:

ap(t ) = 0.57 + 0.96 t (36)

Here, we again have ap = 0 for t ~ t o . The results for the p trajectory from the fits to the low-energy sides of the K- matrix CMSR and the conventional CMSR respectively are illustrated together with the corresponding p trajectories from high-energy fits, eq. (34) and eq. (36), in'fig. 10c and fig. 11 c. Compared to the sensitivity of this quantity the agreement is surprisingly good for small [tl and again for sufficiently large Itl. In particular at t = 0 we obtain from the CMSR fits


c~k(0) = 0.50 -+ 0.21, ctp(0 ) = 0.59 -+0.14. (37)

Note, that it is due to the averaging procedure described in sect. 5.2, that a determination of this sensitive quantity is possible at all from CMSR. The values around c~ok = 0 and C~p = 0 get obviously falsified by background effects, to which the p trajectories are most sensible in this region, since the Regge pole amplitudes are very small here.

Next, let us discuss the results for the coefficient func t ions fp ( t ) in eq. (12). The coefficient functions/3pk(t) and/3p (t) as well as 7ok(t) and 7p (t) are confronted to each other in fig. 8 (a) and fig. 8 (b) and in fig. 9 (a) and in fig. 9 (b) respectively +.

%?

I(-~4 ~,TRIX ~a

//6

0 i i ~t -2

f t l l -I12 -0'3 -O.G 0

t [ (GeV/c ] 2]

, I i I I i L

~ULL AMPLI[UDE

e

. . . . . . . . . ~7~ -2

' -1J2 ' -0'.B ' "0:/* • t [(G eV/c)2]

lO

8

6

2

0

-?

-6

-B

-10

Fig. 8 (a) a-residue function t3k(t) of the Reggeized K-matrix amplitude A~ ( )(u, t) from K-matrix CMSR. The error bars characterize the dependence on the upper integration limit N in the CMSR. (b) o-residue function ~p(t) of the full P Regge pole amplitude A'(-)(u, t)from conven-

tional CMSR.

The following points are important: (i) The ghost-killing mechanism at a o = 0. An interpolation of the points shown in

fig. 8 (a) and fig. 9 (a) yields a zero of #pk(t) and 7ok(t) at approximately the same t value:

t o ~ - 0.6 (GeV/c) 2.

This point t o agrees remarkably well with the zero o f c ~ in eq. (35). From the simul- taneous zero of/31 and 7g at a~ = 0 we conclude that on the basis of Reggeized K- matrix model the p residues choose 'nonsense ' for a k -~ O. This prediction of the new K-matrix CMSR agrees both with the theoretical expectations (cf. subsect. 4.3.) and with the high-energy fits of the relevant Regge-cut models [8, 27].

+ For I tl ~> 0.7 (GeV/c) 2 we use in the factor (N 2 - uz)~(ap_4, 1), on the r.h.s, of the CMSR (14), the

trajectories (34) and (36) respectively, since the corresponding P trajectories determined from the CMSR fits are too inaccurate here.

580 F. Sehrempp, 7rN charge exchange sum rules

~ ' Bo FULL AMPLITUDE 60

. . . . . . . lo . . . . . . . . , - : ' - - - o

~ i _ I , 0 -I -&O 50 , , , , , , -60

-1.2 -o.~ -o.~ o -1.2 -o.B -o.~ t [(GeV/c) 2 ] t [(GeV/c) 2 ] b

Fig. 9 (a) p-residue function ~,k(t) of the Reggeized K-matrix amplitude uBk(-)(u, t) from K-matrix CMSR (cf. fig. 8 (a) caption). (b) p-residue function 3,p(t) of the full p Regge pole amplitude

vB(-)(p, t) from conventional CMSR.

In case of the pure p Regge pole model an interpolation of the points shown in fig. 8 (b) and fig. 9 (b) yields a zero of 7p(t) at ap = 0, but no zero of ~p(t) at this point. Therefore, it follows in accordance with previous analyses [1,28] that the p residues of the pure Regge-pole model prefer the 'choosing-sense' mechanism at c~p = 0. This result agrees with the high-energy fits to the pure p Regge pole model [35, 3], but disagrees with the theoretical expectations due to exchange degeneracy [24] and factorization (c.f. subsect. 4.3.).

So these results from our new K-matrix CMSR again confirm the coniecture that it is much more consistent to identify the K-matrix elements A~ (-) and UBk (-) rather than the full scattering amplitudes A ' ( - ) and uB (-) in the high-energy region with a factorizing p Regge pole term.

(ii) The 'cross-over' effect. In the pure p Regge pole model the zero in ~p(t) at t c -~ - 0.18 (GeV/c) 2, (fig. 8 (b)), is welcome as an explanation of the cross-over of the elastic ?r+-p differential cross sections [1, 35]. According to Reggeized multiple- scattering models this 'cross-over' effect may be explained (at least qualitatively) by a destructive interference of the p-pole contribution with Regge-cut terms without having a zeroin/3o~ at the 'cross-over' point t c. In fact, the low-energy results from the K-matrix CMSR do not exhibit such a zero (fig. 8 (a)).

(iii) Dependence on N. The figs. 8 (a), (b) and 9 (a), (b) clearly show that in particular for t < - 0.7 (GeV/c) 2 the error bars of/3p(t) and 3,p(t) are substantially larger than those of/3k(t) and 3,k(t).Again, this fact reflects the importance of Regge cuts in the full scattering amplitudes at higher values of Itl.

(iv) Reduced P residues. In the following let us consider so-called 'reduced' p residues, from which all kinematical t singularities and the correct C~p ~ 0 behaviour are extracted. Furthermore, they have to satisfy automatically the kinematical constraint holding at t = 4M 2. Such reduced p residues are most probably suited to play the role of genuine coupling constants over an extended t range. From high-energy fits

F. Sehrempp, ~rN charge exchange sum rules 581

[3, 27] there is already some support that the reduced p residues in the K-matrix elements in fact have this property in contrast to the corresponding residues of the pure Regge pole model. In the Reggeized K-matrix model the Regge cuts appearing in the full amplitudes are thought to produce a formfactor effect in such a way that the reduced p residues in the K-matrix do not require any structure functions. Thus. the data could be predicted over a fairly large t range in terms of the residue values known at t = m 2.

In the following we shall determine the reduced p residues in the K-matrix independently and directly as functions of t by means of the new K-matrix CMSR. Again for comparison we calculate the corresponding reduced p residues of the pure Regge- pole model using the conventional CMSR.

First, suitable reduced p residues have to be defined. With regard to the fact that the Veneziano formula should be a good approximation for the K-matrix elements, it seems most natural to us to take just the P residues of a zrN-Veneziano formula [36] in the Regge limit as reduced residues. Since the Veneziano residues appear as coefficients of the invariant amplitudes A(v, t) and B(v, t) all kinematical requirements are automatically fulfilled. Moreover, the Veneziano residues are defined with respect to the 'choosing nonsense' mechanism in agreement with our results in subsect. 5.5.. Finally, they are defined with respect to an energy-scaling parameter s o = 1/a'o, which is required in all FESR sdaemes working in the narrow resonance approximation [23] (c.f. subsect. 4.2). In the Veneziano formula, these residues have to be constants from the requirement of crossing symmetry and Regge pole asymptotic behaviour in all three channels. Thus, we define as reduced p residues in the K-matrix

r~r( t ) - M (2 ,okMv ) 2vff~

--o~ k

_o¢ k r~l(t) - M (2a,okMv) o F(ak) cos (½~ak)Tk ,

2X/~F (38)

where ~ = 1 GeV. Since in the pure p-Regge pole model for the full scattering amplitudes the 'choosing sense' mechanism is preferred, in this case the quantities

7v(t) = ap(t)rv(t ) and rM(t), (39)

if any, should be slowly varying. At this point, the meaning of the quantities V k and V, defined in eq. (31), becomes obvious. For we obtain from eq. (31) and eqs. (38),

(39) k r M r M and V = ap •v V k - k (40)

r V

582 F. Schrempp, nN charge exchange sum rules

The indices 'V' and 'M' of the reduced p residues shall indicate their connection with the vector and Sachs-magnetic couplings of the p to nucleons. At the p pole we have v

v (mp ) k 2 w 2~ Tr3/2 gpzr. gpNN r = ,v(mo) = 1/2 4~r

V + T r kM(mp )2 = rM(m; ) = 7r3/2 -gP"~4~GM(m2) = 1/2 ~.3/2 gp.Tr4zrgoNN 1 gpNN

V T goNN

gpN N V k = V t = l + - - t = m 2 =m 2 V

P P gpN N (41)

The vector and tensor couplingsgVN T as well asgpnTr are defined as in ref. [37]. For the reduced p residues we take the values of ref. [38, 35] at the p pole:

k 2 v(mp) r = rv(mT) = 6.1

and (42) k 2 M(mp)

) . r = rM(mD) =31 .4

This corresponds to

V gpTv~ gpNN

477 - 2 . 1 9 . ( 4 3 )

which agrees with H6hlers value obtained with the Cini-Fubini approximation [37], and to

T gpNN

1 + G p (,~2~ V M~rrtp ) _ gpNN

m 2 T

4M2 V gpNN

= 3.1, (44)

which agrees with the result of ref. [39]. For the calculation of the reduced P residues according to eqs. (38) and (39) we

use the coefficient functions/3pk(t), 7pk(t)and 13p(t), 7p(t) determined previtmsly from the respective CMSR and the p trajectories as given in eq. (34) and eq. (36). We confine to the region [tl ~< 0.4 (GeV/c) 2 since around a0 k = 0 and ~o = 0 the values of the reduced p residues strongly depend on the p trajectory as well as on small background effects. The results obtained for rkv(t), r~(t) and Fv(t), rM(t ) are shown in figs. i0(a), (b) and figs 1 l(a), (b) together with the corresponding high-energy results from a fit to the Reggeized K-matrix model [27] and to the pure p-Regge pole model [35]. The values at t = m 2 from eq. (42) are also shown.

Let us point out the following important results: (a) The agreement between all ' low-energy' and 'high-energy' results is excellent


6D-

~0 -

-

~20-

0 -O.(+ -0.4

REGGEIZEO K-MATRIX MODEL

K-MATRIX CMSR

-0.2 0.0 0.& mp

12 I ,

10-

2 -

o - 0 .6

t [(6eVlc) 2 ]

I I J

--HIGH-ENERGY FIT ;~ K-MATRIX CMSR

i J i i i , i i i i i I i

-0.1, -0.2 0.2 0./, m 'p2

t[(GeVlc) 2] I l l l l

1.0- ~;

- 0 . 5 - . ~

" ~ I "

~ 8 6 t

K-MA'~R~X CMSR

-],0. , ,

t[( GeV/d 2 ]

Fig. ]O (a) 'Reduced' p res idue r~ ( t ) o f the Reggeized K-mat r i x (eq. (38)). The solid dots are the predictions from K-matrix CMSR. The error bars characterize the dependence on the upper integration limit N. The solid line represents the result of a high-enerzv fit to the Reggeized K-matrix model (ref. [27]), • denotes rk(m2,) from eq. (42). (b) 'Reduced' p residue rk( t ) of the Reggeized K-matrix; denotes 4 ( m ~ ) fro"~n e~l. (42). (c) p trajectory of the Reggeized K-*matrix model. Solid

dots, from K-matrix CMSR, solid line from high-energy fit (eq. (34)).

over the t range cons idered .

At this p o i n t it b e c o m e s ev iden t t h a t the averaging m e t h o d (c.f. subsec t . 5.2.) ,

w h i c h was c o n s e q u e n t l y appl ied in th is analysis , pe rmi t s to o b t a i n p r ed i c t i ons o f con-

s iderable accuracy also w i t h o u t t ak ing in add i t i on the h igh-energy data + .

(b ) The reduced K-matr ix residues, r k and r k , come ou t to be practically independent o f t and agree wi th the values at t = m 2. This fac t represen ts not only a nice P

+ In this connection it should be noted that - in contrast to the results of ref. [ 1 ] - the residues "Fv(t) and rM(t) of the pure O Regge pole model determined in this paper from the conventional CMSR do not violate the bounds given by Hohler et el. [35].


PURE p REGGE'POLE MOOEL i ~ i i h I i i 30 i ~ II

PROPOSAL OF HOHLER et al. zE FRDM HIBH-ENERBY FIT

E CONVENIIONAL CMSR

0 i , , , , , , , , i il a

-06 -0k -02 0.2 0~ m t [(GeV/c) 2 ]

2 . . . . . t . . . . . !'

I f f . / / l P~OPOS~L OF HOHLERelat

" ~ ~ I I FROM HIBH-ENERBY FIT

7 _ _-i- -06 -0~, -0.2 0 0.2 O J, m~

t [(GeV/c} 2 ]

I i i i i i i i i

0.5" a p (t} = 0.57.0961

[ i -

t [( 6eVlc )2 ] P

Fig. l 1 (a) 'Reduced' p residue rM(t) of the pure p Regge pole model (eq. (39)). The solid dots are the predictions from conventiona/CMSR. The error bars characterize the dependence on the upper integration limit N. The solid line illustrates the result of a high-energy fit to the pure p Regge-pole model (ref. [35]). (b) 'Reduced' p residue 7v(t) of the pure o Regge-pole model (eq. (39)). (c) p trajectory of the pure p Regge-pole model. Solid dots from conventional CMSR,

solid line from high-energy fit (ref. [35]).

success o f the new K-mat r ix FESR, but also an a posteriori jus t i f ica t ion o f the 'weak- cut' version of the Reggeized K-mat r ix model , since the residues ex t rapola te so well to t = m 2. The inf luence o f the Regge cuts on the t dependence o f the reduced P residues gets obvious f rom a compar ison of r k and r~l wi th the pure Regge-pole quant i t ies 'F v and r M, which shows a very s trong dependence on t be tween t ~< 0 and t = m 2,

Finally, f rom fig 6 it can be read o f f that to good approx imat ion

Vk(6, t) ~- Vk(6, rn2).

Note , that this result is independen t o f the p t ra jectory used.

F. Schrempp, nN charge exchange sum rules

SUMMARY AND CONCLUSION

585

It was the purpose of this investigation to study the problem of 'unitarizing' finite energy sum rules (FESR) by the inclusion of Regge cuts. To this end we proposed new FESR, not for the full two-body scattering amplitude Tfi, but rather for that part of it, which has just Regge pole asymptotic behaviour, if Tfi is 'unitarized' by Regge-cut terms. The Regge cuts in the high-energy region were evaluated according to the 'weak- cut' version of the Reggeized K-matrix model [5]. The new FESR then look like the conventional ones, however, with Tfi replaced by the corresponding two-particle K-matrix element. With these K-matrix FESR all the serious difficulties of conventional approaches to the problem disappear. If, for instance, we follow Lovelace [21] and identify the K-matrix elements with the Veneziano formula, our K-matrix FESR are fulfilled even locally. With the K-matrix FESR (actually we used continuous moment sum rules (CMSR)) a detailed analysis of nN-charge exchange was performed. The low-energy integrals over the nN K-matrix elements were evaluated directly from the CERN- phase shifts.

Let us sum up our main results: (i) The K-matrix CMSR are satisfied with a pure P Regge pole high-energy side

much better and more locally than the corresponding conventional CMSR. In particular, the K-matrix CMSR do not require the inclusion of a hypothetical p ' Regge pole in contrast to the conventional CMSR case.

(ii) The K-matrix CMSR predict the 'choosing-nonsense' mechanism for the P residues. This prediction agrees with theoretical expectations and with high-energy fits of corresponding Regge-cut models.

(iii) The p residue in the helicity nonflip K-matrix element does not vanish at the point where the elastic 7r+p and rr-p cross sections cross over. This result agrees with the idea that the 'cross-over' effect results from an interference of the p pole with the Regge-cut terms.

(iv) Apart from a certain region around a o = 0 the p trajectory found from the K- matrix CMSR is consistent with a o = 0.5 + 0.86 t from corresponding high-energy fits.

(v) If reduced/9 residues are defined in the same way as, e.g. the p residues appearing in the Regge limit of a 7rN Veneziano formula, they are predicted from the K-matrix CMSR to be practically constant between t = m 2 and t ~ - 0 . 4 (GeV/c) 2. The values obtained are in excellent agreement with those from corresponding high-energy fits. This nice result shows that the Regge cuts obtained from the 'weak-cut ' version of the Reggeized K-matrix model just produce the correct formfactor effect, which turns the reduced P residues into genuine coupling constants.

After having determined also the isospin-zero exchanges of 7rN scattering from the new K-matrix CMSR [27] it will be interesting to compare the predictions of the Reggeized K-matrix model for the different nN channels with the high-energy data.

I wish to thank Professor G. Kramer for many illuminating discussions.


R E F E R E N C E S

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[221 C. Schmid, Phys. Rev. Letters 20 (1968) 628. [231 M. Ademollo, H.R. Rubinstein, G. Veneziano and M.A. Virasoro, Phys. Rev. 176 (1968) 1904. [24] J. Finkelstein, Phys. Rev. Letters 22 (1969) 362. [25] V. Singh, Phys. Rev. 129 (1963) 1889. [26] S. Gasiorowicz, Elementary particle physics (John Wiley & Sons, New York). [27] B. Schrempp-Otto and F. Schrempp, to be published. [28] V. Barger and R.J.N. Phillips, Phys. Rev. 187 (1969) 2210. [29] A. Donnachie, R.G. Kirsopp and C. Lovelace, Phys. Letters 26B (1968) 161. [30] J. Baake and F. Steiner, Fortsch. Phys. 18 (1970) 67. [31] C. Schmid, Proc. Roy. Soc. A318 (1970) 257. [32] L.D. Rooper, R.M. Wright and B.T. Feld, Phys. Rev. 138B (1965) 190. 133] C. Ferro Fontan, R. Odorico and L. Masperi, Nuovo Cimento 58A (1968) 534. [34] C. Lovelace, Phys. Letters 28B (1968) 264. [35] G. H~3hler, J. Baacke and G. Eisenbeiss, Phys. Letters 22 (1966) 203. [36] K. Igi, Phys. Letters 28B (1968) 330. [ 37] Springer Tracts in Modern Physics, vol 55 (Ed. G. ffohler, Springer-Verlag). [38] B.R. Desai, Phys. Rev. 142 (1966) 1255. [391 E.B. ttughes, T.A. Griffy, M.R. Yearian, R. Hofstadter, Phys. Rev. 139 (1965) 458.

unitarized finite energy sum rules and their application to π n-charge exchange

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