Zeitschrift ftir Physik 200, 332--342 (1967)

Two Channel N]D Calculation of the Pn Partial Wave of n N Scattering at Low Energies I. BENDER*, D. HEISS and E. TR,~NKLE Institut fiir Theoretische Physik der Freien Universit~it Berlin Received December 21, 1966 The pll-phase shift of nN scattering is calculated by a two-channel matrix N]D method using the reaction aN-+tyN as second channel. The calculated amplitude has the two main features of the Pal partial wave at low energies: the nucleon pole and the zero of the phase near the inelastic threshold. I . Introduction

It is an experimental fact that the phase shift of the Pl 1-partial wave in pion-nucleon scattering is negative in the elastic region, whereas it is positive in the inelastic region 1. The question whether the change of sign appears below or above the inelastic threshold, is not yet experimentally settled, t h o u g h experiment seems rather to suggest that it appears below. I n earlier attempts to calculate ~N-scattering by solving partial-wave dispersion relations with the help of the N/D method, inelastic effects were ignored or - if they were n o t - only treated in a rather cursory m a n n e r 2. N o w the p l l - w a v e in teN-scattering is already relatively inelastic at low energies 3. Therefore it is n o t surprising that the "elastically" calculated p~l-phase disagrees with the experimental data, even at low energies. In particular, no change of sign is f o u n d in the calculated p l ~-phase shift. In a recent paper COULTER and SHAW* calculated the p~ ~-phase by solving the single-channel N/D equations with inelastic unitarity. Taking the inelasticity 17 (the m o d u l u s of the S-matrix element) f r o m the phaseshift analysis, the authors got results which were in clear disagreement with experiment, b o t h for the phase shift and for the residue of the * Present address: Institut fiir Hochenergiephysik, Heidelberg.

1 ROPER,L. D., and R. M. W~GHT: Phys. Rev. 138 B, 921 (1965). 2 ABERS,E., and C. ZE~CH: Phys. Rev. 131, 2305 (1963). -- BALL,J. S., and P. WONG: Phys. Rev. 133 B, 179 (1964). -- NARAYANASWAMuP., and LALITKUMAR PANDE: Phys. Rev. 136 B, 1760 (1964). -- BENDER,I., and D. HEIss: Z. Physik 193, 479 (1966). 3 Au~v~, p., A. DONNACHIE,A. T. LEA, and C. LOVELACE:Phys. Letters 12, 76 (1964). 4 COULTER,P. W., and G. L. SHAW:Phys. Rev. 141, 1419 (1966).

Two Channel N/D Calculation of thepl I Partial Wave of nN Scattering

333

nucleon pole whose position was fitted by a cut-off parameter. After including the nucleon pole in the direct channel in the left hand singularities - which means the nucleon is not a dynamical bound state - the calculated phase shift is in good agreement with experiment. The fact that COULTER and SHAW could not calculate the nucleon bound state and the zero of the pll-phase simultaneously may be understood by a reasoning similar to that of ROTHLEITNERand STECH5 : ROTHLEITNERand STECrI were able to show that, in a Chew-Mandelstam type one-channel N/D method, it is impossible to calculate, without introducing parameters, in the same time a bound state and a zero of the argument of the scattering amplitude under consideration. Now, COULTER and SHAW used the Frye-Warnock method 6 in calculating simultaneously the p l 1- and s t ~-phase of teN-scattering. Here, the situation is more complicated. But one can say that it is impossible, in a one-channel N/D method of the Frye-Warnock type, to calculate, without introducing parameters, a bound state and s~l- and pll-phases in agreement with experiment, which seems to indicate that both phases are positive at high energies. In this paper, we calculate the Pl 1-Pbase of TeN-scattering in a simple two channel model using the matrix N/D method ~. As second channel we chose the aN-system, a being the twopion state with isospin 0, spin 0, p a r i t y + , and mass about 400 MeV 8, whose existence seems to be asserted by recent experimental analysis. We decided for the aN-system since the aN-threshold is the lowest one above the teN-threshold and, therefore, the process rcN-~aN should play an important part in describing the inelasticity in ~N-scattering at low energies. Solving the integral equations not by matrix inversion, but by an approximation procedure proposed by PAGELS9, w e found a zero of the pll-phase. Of course, its position depends on the numerical value of the aNN-coupling constant G~. The zero appears slightly below the inelastic threshold for G~/]/-4~= 3.3 and is shifted to the left with increasing G~. In the approximation method applied, formaUy no cut-off parameter is needed. But without cut-off the force, produced by the possible exchange of the N*, turns out to be too strong. To diminish its influence we introduce a parameter, which is adjusted in such a way that the calculated nucleon mass agrees with the experimental mass. If we take the experimental values for the input parameters the residue of the 5 ROTHLEITNER, J., and B. STECH:Z. Physik 180, 375 (1964). 6 FRYE, G., and R. L. WARNOCK:Phys. Rev. 130, 478 (1963).

7 B~ORrO~N,J. D., and M. NAUENBERG:Phys. Rev. 121, 1250 (1961). 8 HAMILTON, J. : Internationale Universitfitswochen far Kernphysik, Schladming 1966 (reprint). 9 PAGELS, H. R." Phys. Rev. 140 ]3, 1599 (1965).

334

I. BENDER, D. HEms, and E. TR.~NKLE:

nucleon pole, which is induced by a zero of the determinant of the D-matrix, leads to a n N-coupling constant ~ 21. The calculated scattering length is in fair agreement with the experimental value: a~P= -0.105 m~"1,

a~1~= - 0.13 m7 ~.

In our model, a cusp occurs at the inelastic threshold, because the a N-system is in a s~-state if the initial n N-system is in apcstate. Another feature of the model is the fact that the p~-phase passes through 7~/2 immediately after passing through zero, if the zero does not occur in the immediate neighborhood of the inelastic threshold. II. Matrix

N/D Method

In order to introduce the notation, we briefly cite the formulas to be used in a multichannel N/D calculation. We define the scattering amplitude Tjk by

Sjk=C~jkq-2iVPj TjkV~, where P1 is a kinematic factor multiplied by the step function 0 ( s - s sj being the threshold of the j-th channel From unitarity we get

j)

Im Tjk= Tyzpi T~, where we have used time reversal invariance Ti k= Nil (D- 1)1k, we get the formulas 7

Nik--'-~ ~LIm Til(s')Dt~(S')s'-s

Tjk=Tkj.

With the ansatz

ds',

(la)

D,k=fi, k+--~ ! Ki(s,s") Im T~t(s")DtkCS")ds '',

(lb)

with

K,(s,s,,)=I ~

pi(s')ds'

rc R ( s ' - s ) s ' ( s ' - s " )

(lc)

"

In order to compute the matrices N and D from a given input Im Ti, on the left hand cut, which will be specified in the next chapter, we follow the idea of PAGEL89. Making an expansion in partial fractions of the denominator of the integrand (1 c) we obtain the expression:

s

s

tt

K, (s, s") = 7S_sTr I,(s) + ~

I, (s"),

where

rr '" ]

f ,P'(S')

n ~ s 2(s'-s)

ds'.

Two Channel N / D Calculation of the Pit Partial Wave of r N Scattering

335

PAGELS' procedure is now to approximate It(s) by a simple pole function as long as s lies on the left band cut, i.e. we put it(s)=

At

(if s is on the left hand cut),

a--s

where a and A~ have to be determined by a best fit. With this approximation, one obtains the formulas 2 Ats Dtk(S)=fiik--S It(s)N,g(S)+-aZ--S-_s(SN~k(s)-aN~k(a)),

(2a)

a

N,k(S)=Bik(S)+~(aBt,(a)--sB, z(s))A, Nlk(a ).

(2b)

We used the definition

Btk(S)=l l

ImTik(s')

L

ds'.

st--s

This approximation procedure does not depend on the point sl where Dtk is normalised to 5tk. We put sl equal to zero in the above formulas. Like the exact solution, this approximation does not have the correct threshold behavior, if one starts with an input which already has the desired behavior at threshold, as has the Born amplitude. But the correct threshold behavior can be enforced, at least in one fixed partial wave, by an appropriate choice of the kinematic factor p~. This will be discussed in detail in chapter IV.

HI. Input Forces a) To calculate the left hand cut in the process account the following exchange diagrams:

~ N ~ nN we take

into

9

z

'

/

'

/2

Denoting the projection of the corresponding Born amplitudes on the partial wave with I= 89 J = 8 9 l = 1, with h n~ , h~. N* and h ~ respectively, we have (in calculating the following Born amplitudes, Pt is chosen to be k(~ with k (1) the momentum in the 7cN-c.m.s., k (2) that in the Na-c.m.s.) :

s

h,~--

4~ 4k 2

(W-m)(E+m)Q 1

1-s-m2-2#2] 2k 2 -] +

+(w+m)(e-m)Qo(1- s-m2-2/ 2k 2

]J'

336 n* h~-

I. BENDER,D. HElSS,and E. TR~,NKLE:

FM {

(2(m2+#2)-s-M2)(E+m) •

3k,~- ~ Q1 1 +

2k 2

[ W--M-2m (

x -

+Q0

E*+m

(1+

3 1+

2k'2

2(m2+#=)-s-M a ~ 2 kz

] ( E - m) x

-t

) ~

]

+ (3)

\

h~=f[~ -~ Q1

1+-~

(E+m)(W-m)+

Here W = l / s is the total c.m. energy, E the nucleon energy in the c.m.s., k the c.m. momentum; m is the nucleon mass, M the N * mass, rnp the mass of the p meson (k*=k(M), E*=E(M)). g=/4n is the nN-coupling constant, f ~ the product of the pN/V-coupling constant and the p n n coupling constant* and F the half width of the N * resonance. Ql is the Legendre function of the second kind, defined by

Ql(z)- 2 dl z-z' dz'. b) In the u-channel of the inelastic process zcN~ aN, the least massive particle which can be exchanged is the nucleon. The nucleon exchange produces, apart from a long cut along the negative real s-axis, a cut on the positive real s-axis from s ~ 4 4 g2 to s~55 gz (for m2=49 gz and rn~ = 7.5 gz). The t-channel admits the one-pion exchange. This amplitude produces, apart from a cut along the negative real s-axis, a complex cut with branch points at s = (52.7__+i. 35.7) g2 which crosses the physical cut between the elastic and the inelastic threshold. Of course the approximation described in chapter II fails for such an input force. However, assuming that the coupling of the rr to the 7fro-system is of the same order of magnitude as the coupling of the a to the NN-system (an assumption which is supported by recent calculations of FURLAN and ROS~TTI1~ we may conclude - since the above mentioned complex branch points are three times as far away from the nN-threshold as the * We took into account only the coupling without derivative from the two possible couplings of the p to the N_N-system. 10 Ftn~LAN,G., and C. RosErrI: Phys. Rev. Letters 23, 499 (1966).

Two Channel

N/D Calculation

of the Pll Partial Wave of ~zN Scattering

337

nearest branch point of the N-exchange cut - that the n-exchange is less important than the N-exchange. In fact we found that the P~Iprojection of the one-pion exchange is at most one third of the P l : projection of the nucleon-exchange amplitude. Hence we only consider the diagram: x

/r

Because of the different intrinsic parity of a and 7~, the invariant decomposition of the T-matrix element is in this case:

if: Tu i =u:(A i y~ + B i y5 ~ Q) ul [ui, us are the nucleon spinors, Q=89 with q,, q: the n- and a-four momenta resp.], which in the c.m.s, can be brought into the form:

a Pi

a Pf

Zi) I

with

f l =]/(E: + m) (Ei- m) (A - ] / s B) f2 = - l / ( E : - m) (E, + m) (A + ]/s B) . Here p, and p: are the initial and final momenta of the nucleon, El and E: the corresponding energies. The connection of the partial-wave amplitudes w i t h f 1 and f2 is: 1

7Z

f~_+= ~ - (A__I .[ P~-+1 +A P3 d cos 0. Hence we obtain the result:

g.G~ 1 (l/(Ei_m)(Ey+m)(]/s+m).Qo(a) + 4n 4p p'

+ V(e~+ m) (es- m) (V~- m) Ql(a)

(4)

with

p=lp~l,p'=lp:l

and

a=(s--2 Vs(E~+ E:)+ 2E~E:+rnZ)/2p p '

G~ is defined by: Gtr

= (-~--~)a ~(P~) u(p).

(5)

338

I. BENDER,D. HEISS,and E. TRb.NKLE"

c) In the u-channel of the elastic process account only the nucleon exchange:

aN~Na,

we take into

The invariant decomposition and the partial-wave expansion of the scattering amplitude are just the same as for elastic teN-scattering. The NNa-vertex is defined by (5). Hence we obtain: N G~ 1 [L ]/s-3m h~-4n 2k 2 (E+m) 2 + ( E - m ) ],,/s+m32

Qo

( 1+

mZ+2mZ-s~ 2k 2

Q1 (1-t m2+22kzmzc-s)] :

]+ (6)

In principle, the a-exchange would be possible in the t-channel in both processes, the elastic nN-scattering (a) and o-N-scattering (c). In the elastic nN-channel, the a-exchange would have an effect similar to the p-exchange, which is the least important of the one-particle exchanges considered. Whether the a-exchange is of importance in the elastic aNchannel, depends on the unknown aaa-coupling constant. Because of the relatively indirect influence of the aN-channel on the teN-channel a variation of the forces in the former would not change the nN-amplitude too much. Since, finally, the a-exchange, if included in our calculation, would increase the number of free parameters, we decided to drop it. IV. Method of Calculation

Our aim is to calculate, by use of (2a, b) the Pl 1-partial wave of Tll, which will be called f l ~ henceforth. We approximate the left hand cut by the exchange diagrams specified in the last chapter, i.e. the matrix Bik in the formulas (2a, b) is given by Ba l = h ~ + B1 2 = B2 1 = h ~

(7)

B2 2 = h~r First we have to take care of the correct threshold behavior. In the nN-channel, we deal with a p-wave so that there the phase must have a k3-behavior at threshold and, therefore, the amplitude fl~ should have a k2-behavior. In the aN-channel, we are concerned with an s-wave, that is, the amplitude f2 2 should tend to a nonvanishing value at thresh-

Two Channel N/D Calculation of the Pll Partial Wave of nN Scattering 339 old. Now, the amplitudes fl k calculated by use of formulas (2a, b) do not vanish at threshold, which would be correct only if we were concerned with s-waves in all channels. In order to guarantee the correct threshold behavior for f l l and f12=f21, we perform the N/D calculation for di "fik" dk instead offi k with d 1 = 1~(E-m) ~, d z = 1, i.e. we have to replace the kinematic factor p 1 by k (~). (E-rn)/]/-s, whereas P2 remains unaltered. Apart from producing the correct threshold behavior, this special choice of the p~ has the effect of removing some unpleasant kinematic singularities occuring in B12. In order to calculate the input forces we take into account only those particles, whose exchange leads to singularities near the unitary cut. But all these one-particle exchange amplitudes considered have, in addition, cuts on the whole negative s-axis, where the Born amplitude surely fails to be a good approximation. In fact, the increase of the N*-exchange amplitude for s -~ oo contradicts unitarity. That is to say that at least the contribution from the N*-exchange is surely too large. We hope that for our purpose - the calculation of the phase shift in the low-energy region - the consideration of only the short cut on the positive s-axis represents a reliable estimation of the force produced by N*-exchange. If we evaluate the integral 1 r Imh~ - -

J

-

-

ds'

over this short cut, the resulting function is roughly one third of the full exchange amplitude h~n~ * in the low-energy region. To facilitate the calculation we replace this integral by c~.h ~N*, where a is considered as a parameter, which is adjusted in such a way that the calculated nucleon mass agrees with the input mass for G~ = 0. If a turns out to be of the order 0.3, this would confirm the assumption that the contribution to the forces of the N*-exchange, is reliably estimated by taking into account only the short N*-cut. In the case of N-exchange, the integral 1 ~

7~

Im h ~ d s' s' --S

extended over the short N-cut, differs only by about 5 ~ to 10 ~ from the full N-exchange amplitude in the low energy region. The quantities A1, A2, and a occuring in (2a, b) are calculated by a least squares fit, i.e. by minimizing the following expressions:

9 \7c(,,,+1), 23

Z. Physik. Bd. 200

s'2.V~(s'_s3

a-si!

340

I. BENDER,

D. HEISS, and E. TRANKLE:

and

X il

:,(s')as'

A2t

de--s,:

where s~ are ten equidistant points on the real s-axis chosen in the interval s = 0 and the 7rN- and aN-threshold, respectively. The resulting

s I

0

(mrc + raN) 2

s

S

(mr~+rnN') 2

(m6+mN)2

~ s

b Fig. 1. a Single-pole fit to ll(s), b Slngle-pole fit to I2(s) for m~r=370 MeV

fits are shown in Fig. 1 a, b. Of course, we have to calculate the least squares fit with the subsidiary condition that fit and exact function agree at the corresponding threshold. V. Numerical Results We put in the experimental values for the various parameters, as far as they are known: m2 =49 i.t2

nucleon mass,

g2 _ 15 4~r

~zN-coupling constant,

M2=81

].t 2

F=0.87g m 2 = 301x2 f p ~ "fp Nfi ~, 2 4re

mass of the 33-resonance, width of the 33-resonance, p-mass, fp ~ ~ from the p-width, fp N~Tas predicted by unitary symmetry.

The results depend only weakly on the mass of the a-meson. The curves shown in the Fig. 1, 2 and 3 are calculated with m2=7.5 I~2 ( ~ m , ~ 370 MeV). To study the dependence of the zero of the phase on the aN-coupling constant G,, we first put G~=0. Then the parameter ct, adjusted in such a way that the output nucleon mass agreed with the

Two Channel N/D Calculation of the Pll Partial Wave of ~N Scattering

341

input mass, turned out to be 0.4, so that the assumption, made above that N*-exchange is reasonably treated, if one takes into account only the short N*-cut - is on the whole consistent. The calculated nNcoupling constant has the value 22, the scattering length is ~ - 0 . 2 g-1. The phase is negative at threshold, of course, and remains negative. Turning on the aN-coupling there appears, as expected, a cusp in the phase at the inelastic threshold, brought about by the positive parity of the o-, which requires that the aN-system should be in an s-state, if the initial zcN-system is in a p~-state. For increasing G,, the position of the nucleon pole moves very slightly to the left, the zcN-coupling constant becomes smaller, i.e. ameliorates, as does the scattering length. (For ~G, / ~ - 3 , moZut=48.5 g2, g2/4rc=21, a l t = - 0 . 1 5 negative.

].t-l). For small G~ the phase remains still

2

=5/A 2;:z/l 64 (mr: +mN )2

3 ~ 100 (Ill d + tllN)2

Fig. 2. Calculated &11-phase for m~ =370 M e V and

Ga/]/~=3,

4 and 5

1

GE

94

I

(m6+mN)

2

7r sfm~J2 )m"

Fig. 3. Calculated Jnelasticity t / f o r m~ = 3 7 0 MeW and 23*

G~/V~=3, 4

and 5

342

I. BENDER et al.: Two Channel

N[D Calculation of the Pal Partial Wave

The cusp becomes more pronounced with increasing G~. For G~ - 3 . 3 the phase at inelastic threshold has a positive sign and a pointed peak. There is now a zero below as well as above the inelastic threshold. (The values of G~/47c, used by other authors la, range from 4 to 13.) If G~ increases further, the zero below threshold moves to the left. After passing through zero the phase rises very rapidly and passes through n/2, if the zero does not lie in the immediate neighborhood of the inelastic threshold. In this case the zero above threshold disappears. The calculated phase and the inelasticity ~/are shown in Fig. 2 and 3.

VI. Summary and Conclusion In this model we cannot, of course, expect a quantitative agreement of the calculated quantities with the experimental data. The value of the calculated phase at its minimum turns out to be ~ - 11~ whereas the experimental phase is ~ - 4 ~ at its minimum. Furthermore, if the zero occurs only slightly below the inelastic threshold, we get a very pronounced cusp, and if it occurs farther away from threshold, the phase passes subsequently through ~/2, while now the cusp is not so strong. Experimentally the very subtle question, whether there is a cusp or not, is not yet settled. In any case, the fact that the phase passes through n]2 below the inelastic threshold, is in disagreement with the experimental data. We cannot relate it with the l l-pion nucleon resonance, which occurs at a much higher energy (1500 MeV), far above inelastic threshold. However the calculated amplitude possesses the two main features of the Pl 1-wave at low energies: the nucleon bound state and the zero of the phase. As referred to in the introduction it is not possible to get these two phenomena in a one-channel calculation free of CDD-poles 12 Furthermore we remark that the zero of the phase in our model, if there is any, has to occur below the inelastic threshold. One could, certainly, improve the agreement of calculation and experiment by including further channels or by treating the remaining inelasticity with an absorption matrix. The fact, for instance, that the phase passes through n/2 immediately after passing through zero, is presumably a feature of the two channel calculation and would not occur in a more-channel calculation. We thank Professor F. PENZLIN for a discussion. ix Dosc~, H. G., and V. F. MIJLLER: 7N. C. 39, 886 (1965). 12 ATKINSON,D., and M. B. HALPERN: Possible one channel CDD-poles in n N and zcK scattering (to be published).