MESON-MESON THEORY

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MESON-MESON THEORY

J. Basdevant

To cite this version:

J. Basdevant. MESON-MESON THEORY. Journal de Physique Colloques, 1973, 34 (C1), pp.C1-220-

C1-222. �10.1051/jphyscol:1973123�. �jpa-00215203�

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J. L. BASDEVANT

MESON-MESON THEORY J.L. BASDEVANT

Laboratoire de Physique Thgorique et Hautes Energies, Universit6 Paris VI

1.- FIELD THEORETICAL MODELS.

-

Two papers have been presented, both are continuations of quite sys- tematic programs. Ecker and Honerkamp [I] compute m amplitudes from an SU(3)xSU(3) Lagrangian following a method initiated by Lehmann 121. They obtain acceptable phases below 700 MeV and show that baryon loops are not important. Iagolnitzer et al. [3] compute all meson-meson channels with the second order Pad6 approximation of a Yang-Mills Lagrangian with p, K

m

and cp exchange. They obtain a nice explanation of the

Kk

threshold effect in nn. Such approaches m y gain great interest in the future, with the present development of field-theoretical models.

2.- TEK SCATTERING.

-

The phenomenological approach to 7% -t 6, rtn -t

~

^andⁿⁿ^-tⁿⁿamplitudes through the combined mechanism of dispersion relations, crossing and coupled channel unitarity is being pursued. There are considerable technical difficulties, and the results are somewhat preliminary. Ader et al. [4] re- port the large multiplicity of amplitudes which con- tain the vector resonances (p,

e).

Johannesson and Petersen [5] concentrate on the I = P = 0 m +

Kk

amplitude and show that the sign of its imaginary part can be determined below

Kk

threshold provided it behaves near t = O as predicted by soft pion theory. The result of their analysis is that the relative sign of e resonance couplings to nrr and

Kk

channels is positive, in agreement with the Lovelace-Veneziano model.

3.- nn DISPERSION RELATION ANALYSIS.

-

The situa- tion of m theory has considerably changed since the Amsterdam Conference [6], In fact, a new theoretical technique, the Roy equatjons 171, allows to impose the requirements of crossing symmetry and analyticity (fixed t dispersion relations) directly on partial wave amplitudes in the physical region. This has been used extensively by Pennington and Protopopescu 181, Bonnier and Gauron [Y], and Basdevant, Froggatt and Petersen [LO, 111, to construct phenomenological m

amplitudes consistent with the data and with funda- mental principles. The aim is to smooth experimental data, to gather new information and, possibly, to determine what further measurements are required to settle remaining ambiguities.

3.1.- PRESENT METHOD.

-

We denote the nn phase shifts as 6 I (s). The investigation is made for low

I

energies only : < 1.1 GeV (no discussion of the

m

Mnn

-

s ,

E ' , p' etc... is made). At present, in order to

construct a unique low energy nn amplitude one needs to incorporate in the Roy equations the following information :

a) existence and parameters of the p meson M

r

P P' absence of an I = 2 I = 0 resonance

b) semiquantitative information on intermediate and high energies ( S

m ,

E ' , fo, p-f Regge parameters, pomeron)

,

c) value of the I = O S-wave scattering length a0 d) set of values of ijOin some low-energy interval, typically 500-1100 MeV, including weak inelas- ticity below

Kk

threshold.

The remaining parameters of low-energy nrr amplitudes are then predicted.

3.2.- S-WAVES. - Given a set of data points for

6

: in the energy interval 500-1100 MeV, the scattering length a: cannot be obtained by extrapolating down to threshold. Figure 1 shows theoretical curves for 6: based on fits to the prefered solution of Estabrooks, Martin et al. [12]. A whole range of values for a: is allowed, and on the basis of such considerations only it cannot be restricted much more than - 0.05 5 :a 5 0.7 mi1

.

However, once a: is fixed, the I = 2 phase is determined within small uncertainties, and could be used as a counter check if the errors were small enough.

There are several experimental indications [13]

that a: may be noticeably larger than Weinberg's prediction (a:

-

0.16) and reach a:

-

^0.6

.

^How-

ever, Nielsen and Oakes 1141 have analysed A1

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973123

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MESON-MESON THEORY Cl-221

Fig. 1

-

Typical s-wave phases resulting from fit on EM [12]

a) 6;

,

^{data from}^{P EM,}

+

Grayer et al. [20]

1) dotted curve a8 = 0.17 ag =-0.06 2) solid curve " = 0.31 " =

-

^0.03

3 dashed curve

"

= 0.59 I' =+0.02

1

b) 60 data from Colton et al.[21] and

+

^Cohen

et a1. [22].

d-waves near threshold where the effect of the m scattering length in the long range contribution can be extracted, and conclude that the value is a0

-

0.1k0.2 in agreement with Weinberg, and that a

: z 0.6 is excluded. One must also keep in mind the remarkable success of Weinberg's assignments in ex-

+ -

plaing the rr-p + rr rr n cross section near threshold 1151, although Batusov et al. [16] have reported that the matrix element of this reaction cannot be used to determine rrrr scattering lengths.

A high statistics K: experiment is quite desirable for getting better insight in this question of scattering lengths.

3.3.- HIGHER PARTIAL WAVES.

-

^P 2 waves, in par- ticular d and f-waves, appear to be much more accu- rately defined by the dispersion relation analysis than experimentally [lo] below 1 GeV. Therefore one should probably

use

the theoretical higher waves in phase shift analysis rather than try to determine them with the higher moments.

3.4.- PROBLEMS IN A HIGH STATISTICS PHASE SHIFT ANALYSIS. - Owing to the smallness of error bars in

the EM 1121 phase shift analysis, one can detect a conflict between their results and the crossing

+

dispersion relation constraints below 650 MeV.

3 1

Figure 2 shows a plot of the quantity 2q

/fl

cotg &il as computed theoretically (including uncertainties)

and as a result of the EM analysis. One concludes that biases seem to be present in the data or in the analysis so that the result for 61 below 650 1 MeV cannot be correct (neglecting here the possibility that our understanding of crossing and analyticity could be inadequate). Since the subsequent EM-analysis relies on this 61, one expects corresponding 1 biases in the results for 6.:

3 1

Fig. 2

-

Plot of (2q

/fi)

cot 61

.

E~rperimental points EM [12] g solution 2 (preferred) @ solution 1. Theore- tical curves span the allowed possibilities.

However, when trying to reconstruct 6' from a

"corrected" 61 and the experimental 1 (Y;) moment, one runs into another difficulty. In figure 3 we repre- sent the quantity

0 1

Z. (G) -

⁴

^fi

^sin

⁴

12"" :6 f00(6~ -

4 +

^sin

6

^{'0s (6,}²

-

¹

]

Mxn ( M ~ v )

Fig.3

-

^{Plot of}

^Z.(Y) -

defined in text Solid lines span theoretical possibilities, experimental points from EM [12].

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C1-222 J. L. BASDE VANT

-

which i s t h e r e s t r i c t i o n t o s and p waves o f t h e unnormalized (Y:) moment

-

as computed from t h e EM p h a s e s ( i t i s t h e same f o r b o t h s o l u t i o n s ) and a s a r e s u l t o f t h e t h e o r e t i c a l a n a l y s i s . Again t h e r e i s a d i s a g r e e m e n t o f s e v e r a l s t a n d a r d d e v i a t i o n s below 650 MeV where a bump a p p e a r s which c a n n o t be accoun- t e d f o r by t h e d i s p e r s i o n r e l a t i o n s .

We i n s i s t t h a t t h i s c o n f l i c t between t h e d a t a and t h e t h e o r e t i c a l a n a l y s i s c a n o n l y b e d e t e c t e d owing t o t h e h i g h a c c u r a c y o f t h e experiment. However, t h e obvious c o n c l u s i o n s a r e t h a t (a) t h e phenomenologi- c a l a n a l y s i s s h o u l d be based d i r e c t l y o n e x p e r i m e n t a l q u a n t i t i e s (moments) r a t h e r t h a n o n phase s h i f t s , and (b) i t seems a n o b l i g a t i o n , w i t h h i g h a c c u r a c i e s , t o p e r f o r m any p h a s e s h i f t h a n a l y s i s t o g e t h e r w i t h a d i s p e r s i o n r e l a t i o n a n a l y s i s .

3.5.

-

ZERO OF THE HIGH ENERGY It = 1 m AMPLITUDE.

-

S e v e r a l a u t h o r s [17,18,19] have n o t i c e d t h a t by wor- k i n g w i t h t h e u n s u b t r a c t e d d i s p e r s i o n r e l a t i o n f o r t h e I t = l rn a m p l i t u d e

one c a n o b t a i n sum r u l e s which r e l a t e t h e t-depen- dence o f t h i s a m p l i t u d e a t h i g h e n e r g i e s t o t o t a l c r o s s s e c t i o n s and low e n e r g y a m p l i t u d e s . The p h y s i - c a l s t a n d p o i n t o f Tryon [17] i s somewhat d i f f e r e n t from t h e two o t h e r p a p e r s r18,19]. However, t h e s e l a t t e r a u t h o r s a g r e e on t h e f a c t t h a t t h e low energy pheno- menology seems t o imply a z e r o o f Im F ( s , t ) n e a r t

--

^{0.2 GeV}2 a t h i g h e n e r g i e s i n s t e a d o f t

--

^0.6

GeV 2 a s p r e d i c t e d by s i m p l e exchange degeneracy. The p o s i t i o n o f t h i s z e r o i s o f c o u r s e r e m i n i s c e n t o f

t h e c r o s s o v e r e f f e c t i n nN, b u t what makes t h e r e s u l t i n t e r e s t i n g i s t h a t i t i s o b t a i n e d e n t i r e l y on t h e b a s i s o f low e n e r g y c o n s i d e r a t i o n s (and t h e convergence o f t h e u n s u b t r a c t e d D.R.) i n a way v e r y d i f f e r e n t from u s u a l FESR arguments.

4.- CONCLUSION.

-

I t i s v e r y c o m f o r t i n g t h a t b o t h t h e o r y and experiment have made s u f f i c i e n t p r o g r e s s i n m s c a t t e r i n g t o e n a b l e u s t o e x p l o i t p r a c t i c a l l y t h e f a c t t h a t nn -, rcrr i s h i g h l y c o n s t r a i n e d by c r o s - s i n g and a n a l y t i c i t y . Many c o r r e l a t i o n s can be e l i c i - t e d , some i n c o n s i s t e n c i e s c a n b e p o i n t e d o u t . T h i s r e p r e s e n t s c e r t a i n l y a c o n s i d e r a b l e improvement o v e r p a s t y e a r s .

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$&

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,

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[6] SCHMID (C.), Rapporteur t a l k a t t h e Amsterdam Conference 1971

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,

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,

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,

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a

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