Resonances due to annihilation in the NN system

Volume 768, number 4

RESONANCES

PHYSICS LETTERS

DUE TO ANNIHILATION

A.M. BADALYAN, M.I. POLIKARPOV Institute

for Theoretical

and Experimental

19 June 1978

IN THE NN SYSTEM

and Yu.A. SIMONOV

Physics, Moscow,

USSR

Received 6 March 1978

We calculate the parameters of the s-wave NN resonances taking into account boson exchanges and annihilation explicitly in the many-channel N/D method. Two main results are: (1) the width of the usual barionium (quasinuclear) states is sensitive to an effective pionic space and might be as small as - 10 MeV. (2) a new kind of resonances appear near the NN threshold which do not exist without annihilation.

A physical idea of the “quasinuclear” or “barionium” states in Nrrj system, suggested by Shapiro et al. [l] , is supported by many calculations [l-6] which treated NN forces as due to the boson exchanges. The most controversial problem here is how to treat the annihilation of NN into mesons. A perturbation theory in the annihilation radius [7] was used in [ 1, 31 to predict the widths of the quasinuclear states, the total widths being of order of 100 MeV. Later on it was argued [8,9] that the actual widths and the energy shifts should be even much smaller, than in the perturbation method. Another treatment of the annihilation was done in the framework of an optical potential model [ 51 and in a boundary condition model [6] with the general result: the annihilation makes the quasinuclear states so wide that they do not appear as bumps in the cross-section and even the Argand loops disappear. However in the optical potential approach it is not clear whether a local, energy-independent and purely imaginary potential could well describe an annihilation contribution (see also discussion in [S]). At the same time in the perturbation approach an independent check of the quantitative accuracy of the perturbation formulas should be made in the realistic situation where the annihilation cross-section CT,, is large. In the present paper we describe the results of a different approach - the many-channel N/D formalism [lo]. This method takes into account rigorously the unitarity, analytic properties of all processes including annihilation and is appropriate to describe relativistic and large exchange processes and therefore is free of the 388

drawbacks of the methods discussed above. We consider a N?? system with a given isospin I, spin S and the zero orbital momentum L = 0 as a channel 1. For simplicity here we disregard a D-wave admixture to the triplet states. Our motivation for treating only the S-state of NR is two-fold. First, these states show up in the efee experiments and are possibly already seen [ 1 I] . Second, the role of the annihilation should be the most striking in the S-states of NN. A dynamical input consists of the OBE-diagrams with the rr, q, p, w, o o, al -meson exchanges taken into account with the coupling constants from the nonstatic Bryan-Scott potential [ 121. A two-pion exchange contribution is also taken into account and a formfactor cut-off of A = 1 GeV and 1.5 GeV as in [ 12, 131 was used. The consistency of our procedure was checked by calculating a single NN pole (0.287 MeV virtual state instead of 0.067 MeV experimental). In the absence of annihilation one-channel results are listed in table 1, first column. These figures coincide with the results for 1So-states in [ 141 and qualitatively close to the first calculations on NN in [ 151 and to the value in [l, 31 where a different potential [ 131 was used. We have taken the annihilation to be due to a general type diagram of fig. 1 with any number of pions in the final state and with an exchanged mass in the t-channel larger or equal to the nucleon mass M. We have disregarded in left-hand side discontinuities of the pion-pion forces, because the most striking effect of the pionic channel is the occurrence of purely pionic resonances which do not require any left-hand side dis-

Volume 76B, number 4

PItYSICS LETTERS

19 June 1978

Table 1 The position of the quasinuclear level E 1 - iF 1/2 in MeV for the different quantum states given respectively the threshold x~th = 2M. The value reran (0) = 26 rob, r a = 1/Mr. Quantum states

JP= O-,I G = 0+ JP= 0+,IG = 1JP= 1-,I G = 0-, JP= 1-,I G = 1+

The value of the parameters Cran= 0

A =0

A = 3.24 × 10.3 MeV-1

A = 2.38 X 10.2 MeV-1

-108

-141,-i80

-152 i34

-178-i9.4

-126

-260-i107

-291-i61

-376-i14

-95

-255-i158 a)

-258-i87 a)

-185-i14

-68

-190-i156

-188 i81

-253-i19

a) The permitable value of OOanat E = 0 is smaller than 15 rob.

d = ~ ' . b -1

(1)

and the positions of the resonances in NN system are given by zeros of the determinant: det Fig. 1. Annihilationprocess contributing to left-hand side discontinuities. continuities and can be accounted for in our method by a proper redefinition of an effective pionic phase space (see later). As a result all pionic channels in the many-channel N/D equations can be combined to one channel with an effective phase space x(E), which is a weighted average of all individual channel phase-space factors. Denoting this channel as 2, we have to solve the two-channel N/D problem. The left-hand side discontinuities of the annihilation diagrams of fig. 1 enter as a dynamical input of the annihilation amplitude C12, they are nonzero on the cut [_0% Ea ] in the energy plane, where E a ~ M for an effective mass of pions in fig. 1 much smaller than the nucleon mass, but E a ~ -3/(1 - moZ/2M2) 2 ~ ~ 0.44 M if the pions effectively combine into two p-mesons. Since we are going to treat the annihilation discontinuity phenomenologically (also using the fact that effective E on the cut are to left o f E a and therefore far from the effective OBE region E~ ~ -m2/4M), we replace the annihilation cut contribution by a pole at a position E 0 (varied around E 0 = - M ' ) and with a residue *7, which should be adjusted to reproduce the experimental annihilation and elastic cross-sections. With the usual matrix notations for the two-channel amplitude C' =- Cij (i, ] = 1, 2) we have

D(E) = D l l ( E ) + ~(E)D12(E ) .

(2)

Here both D 11 and D12 are found from the nonsingular integral equations with OBE dynamic input; constant X is proportional to r/2 and r(E) is a dispersion integral over the effective pionic phase space; a subtraction point is convenient to choose at E = E 0:

r(E)

E o - E I'~

~2

x(E') dE'

(3)

eJth (E' - ~ ) q ; ' - E0) 2

where E is the c.m. kinetic energy in the NN system. Unknown parameters here are X and x(E), some information of them can be taken from the annihilation and elastic cross sections: 4X x(E) reran(E) : M 3 / 2 ( E - E0) 2 Idet DI 2 '

(4)

here v is the c.m. velocity; function x(E) is normalized as x ( 0 ) = 1 and

_(Eth - E ~n x(E)-\ Et h i ,

E<0;

x(E)=I+AE,

E>0.

Eth is an effective threshold, in most calculations we have taken n = 3, Eth corresponds to 4n threshold, parameterA was varied between zero and several units 389

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PHYSICS LETTERS

Table 2 The position of the annihilation level E2 - iFa/ Quantum state

___--___

in MeV. The value uoa,(E = 0) = 26 mb. .-____

The values of the parameters ____

JP=()-,IG=f)+

r, = l/M

r, = l/M

A=0

A = 3.24

JP=O-,IG=O+ JP=O-,ZG=

19June1978

l-

26-i14 -28-i35

jP=l-,~Gzl+ -.____

ra = 1.75/M X

lob3 MeV-’

A = 5.58

29-i0.2

55-i76

JP=~-,IG=o-

-

g-i29

X

10e3 MeV-’

-1.2 +i8.5

5.6-il.2

-8.5 +i18

3-i27

4.1 +il.l

8.5-i3.2

-13 +i9.5

Table 3 The scattering length in fm for the different choices of the parameters. -____ The value of parameters

Quantum states lso,zG=o+

lSe,ZG=

1-

3

s,,zG=o-

%/

= 1+

__. Ya = l/M A = 3.54 X 10e3 MeV-’

-0.194-i0.99

ra = 1.3/M A = 5.6 X 10W3MeV-r

-1.66-i

of MeV-1 . The elastic cross-section expression: 47W&) ael =

0.34-i0.96

1.44

-1.22-i0.77

is given by the

+ x r(Q &,@)I2 (5) *

Ml det &!?)I2

im

Imb

t

Fig. 2. The movement of the quasinuclear (q) and the annihilation (a) levels in E-plane as a function ucran(0) is shown. The figures on the curves are the values of uoan(0) in mb. For the solid curves the parameters are A = 0, ra = l/M; for the dashed curve A = 3.24 X low3 MeV-‘, ra = l/M.

390

1.64-i0.95 -0.4-i3.1

0.64-i0.98 -1.54-i0.51

The results of our calculations are presented in tables l-3 and figs. 2,3. The entries in tables 1 and 2 are the resonance positions E - ir/2 in MeV. In (4) the parameter X was defined through the condition vu,@ = 0) x 26 mb (which is close to the experiment al value for E k. 20 MeV, see fig. 3) or through the condition: uo,,(E = 3 1 MeV) is maximal (curve T2 in fig. 3). For all states a maximum value of uo,(E = 0) is equal or larger 26 mb with an exception of the state with S = 1, I= 0 where the maximum uo,,(E = 0) is smaller (see remarks to the tables 1 and 2). Our calculations show that: 1) Due to annihilation the quasinuclear resonances shift downwards (more attraction) for all chosen values of parameters A and two values of radius of annihilation: ra = l/lEol. Note, that the shift of the level is larger: AE - -100 MeV, if uu,(E = 0) 2 26 mb. For vu,(O) < 10 mb AE is much smaller: AL?- -(lo-20.) MeV. In our case the sign of the energy shift is in contrast with the results of [S] , and coincide with the prediction of [9] in the sign but not in the value of the shift. The widths of levels depend strongly on the shape of the effective pionic phase space i.e. on the parameter A, and vary from I’ - 1SO-300 MeV for A = 0 to

Volume 76B, number 4

PHYSICS LETTERS

ff~Oll (rob> 20.

20 t~ {2

"'...

8

0

Ec.rn (m v) Fig. 3. The function rOan(E) for different choices of parameters. The dotted and dash-dotted curves T1 and T2 correspond to 3St-state (/= l) forA = 3.24 X 10-3 MeV-z andA = 3.24 X 10-2 MeV-z. The dashed curve S corresponds to 1So (I = 0)state for A = 3.24 X 10 -3 MeV. The solid curve UL shows the unitary limit of reran(E) in S-wave. The experimental points are from refs. [21]. I" ~ 20--40 MeV f o r A = 2.4 × 10 - 2 MeV, at the same time the shifts increase with A can be easily understood in the langauge of the optical potential model. Note, that X r(E) in (3) corresponds in some sense to the annihilation part of the NN effective optical potential V = V 1 + iV 2; parameter A being (in some units) roughly the ratio of the real to imaginary part of X r ( E = 0), i.e. V1/V 2. F o r A = 0 the situation is more similar to that of a purely imaginary annihilation potential (however strongly energy dependent in our case) and we get very large widths (compare [5]). But from the realistic annihilation diagrams we can see that Re X r ( E ) ~ Im Xr(E) i.e. V 1 ~ V2 and in this case the characteristic value of parameter A is A ~ 4 × 10 - 3 MeV -1. F o r A > 10 - 2 MeV -1 the real part X r ( E ) i s much larger than the imaginary one, i.e. IVII >> V2 and then it can be shown [16] that the width of a resonance goes to zero as V 2 / V 1, when V 2 / V 1 -+ O. 2) Another kind of resonances appears in present calculations with the following main features: a) they lie near the NTNthreshold at a distance of ~< 20 MeV; b) they do not exist without annihilation: when vaan ~ 0 (or X ~ 0) they move far away in the c o r n -

19 June 1978

plex plane (see fig. 2). For that reason we call them "the annihilation resonances". In some respect they resemble the "accidental resonances" found in [17] in a somewhat different situation; c) with growing roan the annihilation resonances approach the NN threshold, becoming more narrow. The fig. 2 illustrates this situation; d) if only annihilation is present, but OBE-interaction is switched off, they become very wide: P ~ 500 MeV; e) the annihilation resonances lie on the second sheet respectively to Eth and on the first sheet respectively to NN-threshold (when P > 0) and on the adjoint second sheet respectively to NI~ threshold (when P < 0), i.e. in k-plane pole is at k 0 = -/30 + ia 0,/3 > 0, k =~ Therefore the contribution of a annihilation resonance to the cross sections is not of the Breit-Wigner type but v% n =

4~T/3 M ( ( k + 13)2 + ~2} '

_ °el

4rr (6) (k + 13)2 + 52 "

With constant e and/3, eqs. (6) are zero range formulas given in [ 18], correspondingly the scattering length is a = [~(0) + i13(0)] - 1 ; for ~ = ~(k 2) and 13= 13(k2) expressions (6) generalize the two-channel unitary formulas investigated in [ 19]. f) A sharp enhancement at E = 0 takes place in the annihilation and elastic cross-sections in our exact calculations, see fig. 3 and table 3. In fig. 3 we also show the experimental points for roan for p~-annihilation and the unitary S-wave limit v • 7r/k2. The approximate formulas (6) reproduce our exact calculations rather well for E ~< 15 MeV. For Oel(P~) and ech(P ~ ~ nff) at E = 0 we get ~ 3 0 0 mb and 35 mb respectively with A -= 3.24 × 10 . 3 MeV. From analysis fig. 3 it is clear that the P, D and possibly higher waves are necessary to explain the constant value roan over the tange 22 E ~ 150 MeV. Besides the lowest waves should be rather close to the unitary limits to combine to the large experimental value roan ~ 26 mb. (Note, that the unitary limits at E = 31 MeV are 7.5 mb and 22.5 mb for S- and P wave respectively.) This argument justifies our choice of X which gives either maximal roan at E = 31 MeV or the value of V%n(0 ) ~ 26 rob. A check of stability of our results was made varying otherwise fixed parameters: (1) cut off parameter A from 1 GeV to 1.5 GeV; (2) the coupling constants in OBE input from 1.5 to 2/3 of its original value; 391

Volume 76B, number 4

PHYSICS LETTERS

(3) Eth from 37r to 00-threshold; (4) E 0 from ( - 2 3 4 ) to ( - 0 . 5 7 M ) ; (5) an effective phase space x(E) dependence for E < 0 from n = 1 to n = 3. Only the first variation was important producing 4 5 0 % change in the position of the deep resonance (more attraction); all other variations give ~<20% change in the cross-section and the position of the quasinuclear level. For any choice of parameters the annihilation resonances lie near the NN threshold. In this paper we have not made an optimal choice of all parameters emphasizing the qualitative effect of the existence of the annihilation resonances. An accurate description o f the experimental situation requires a selfconsistent treatment of all partial waves which is now in progress. Finally we would like to remark that the annihilation resonances might be quite common in the several channel reactions like K - p -+ Nn, An; K - d -+ Aprrnear NN threshold and also may show up in KK, 7r~ system (6(970)) and KK, rrrrsystem (S*(993)) [ 1 8 20]. In the last case the phenomenological analysis [20] predicts a pole with the same position as was discussed in the item (e) above.

[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

The authors are grateful to I.S. Shapiro for some important remarks and suggestions: to K.G. Boreskov and A.B. Kaidalov for many useful discussions, to L.N. Bogdanova, O.D. Dalkarov, A.E. Kudryavtsev, and V. Markushin for helpful comments. The authors are indebted to V.I. Lisin for important suggestions concerning method of computations.

References [1] I.S. Shapiro, Soviet Phys. Usp. 16 (1973) 173;

392

[17] [18] [19] [20] [21]

19 June 1978

L.N. Bogdanova, O.D. Dalkarov and I.S. Shapiro, Ann. Phys. 84 (1974) 261. J.M. Richard et al., Phys. Lett. 64B (1976) 121. C.B. Dover and M. Goldhaber, Phys. Rev. D15 (1977) 1997. J. Tjon, preprint LA-UR-77-1845, Los Alamos, 1977. F. Myhrer and A. Gersten, Nuovo Cimento 37A (1977) 21; F. Myhrer and A.W. Thomas, Phys. Lett. 64B (1976) 59. O.D. Dalkarov and F. Myhrer, CERN preprint TH-2280 (1977). A.M. Badalyan and Yu.A. Simonov, Yad. Fiz. 11 (1970) 1112. I.S. Shapiro, preprint ITEP-88 (1977). B.O. Kerbikov et al. JETP Letters 26 (1977) 505. G.F. Chew and Mandelstam, Phys. Rev. 119 (1960) 467; J.D. Bjorken, Phys. Rev. Lett. 4 (1960) 473. E. Lohrmann, A summary talk at 1977 Intern. Syrup. on Lepton and photon interactions, DESY, Hamburg. R.A. Bryan and B.L. Scott, Phys. Rev. 177 (1968) 1435. R.A. Bryan and R.J.N. Phillips, Nucl. Phys. B5 (1968) 201. A.M. Badalyan and I.M. Narodetsky, Yad. Fiz. 26 (1977) 971. J.S. Ball, A. Scotti and D.Y. Wong, Phys. Rev. 142 (1966) 1000. A.M. Badalyan and Yu.A. Simonov, Sov. J. Part. Nucl. 6 (1975) 119. R. Dashen, J.B. Healy and I. Musinich, Phys. Rev. D14 (1976) 2773. R. Dalitz and S.F. Tuan, Ann. Phys. 3 (1960) 307; J.D. Jackson et al., Nuovo Cimento 9 (1958) 834. M. Flatt6, Phys. Lett. 63B (1976) 224,228. B. Hyams et al., Nucl. Phys. 64B (1973) 162; D. Morgan, Phys. Lett. 51B (1974) 71. J.E, Ernstrom et al., A compilation - lqN and Iqd interactions, LBL-58 (1972).