Inelastic effect of the ωπ0 channel on the pion form factor

PHYSICS LETTERS

Volume 67B, number 2

28 March 1977

INELASTIC EFFECT OF THE cm0 CHANNEL ON THE PION FORM FACTOR B. COSTA de BEAUREGARD,

T.N. PHAM, B. PIRE and Tran N. TRUONG

Centre de Physique ThPorique de YEcole Polytechnique,

Route de Saclay, 91128 Palaiseau Cedex, France

Received 4 January 1977 It is shown that the enhancement in the pion form factor in the l- 1.4 GeV region is due mostly to a fast energy dependence of the ratio R = o (e+e- -+ wm)/u(e+e-+ m). This result is obtained in a model independent way. Recent experiments at Orsay [l] and Novosibirsk [2] with the e+e- colliding beam show the following interesting features in the 1- 2 GeV: (i) The measured pion form factor above 1 GeV is enhanced with respect to that obtained from the simple vector meson dominance approximation. (ii) The 7i+n-n0rro cross section increases rapidly with energy but does not show any structure which can be due to a possible 4rr resonance. The shape of the cross section is consistent with a simple calculation based on the production of wn” via p dominance. (iii) The cross section in the above energy range is one order of magnitude smaller for x+rr+n-nthan for n+n-7&u. The purpose of this,note is to show that the enhancement in the pion form factor reflects the inelastic effect coming from the wrr” channel, but is not due to the existence of a new resonance ~‘(1200) as it is usually assumed [31. We shall neglect the rr+rr+7~-7~- and KK channels and study the influence of the n+rr-rrOnO channel in the form wrr”. Let us denote as F, and F2 respectively the pion and the mm0 form factor. The unitarity conditions for F, and F2 are as follows:

ImF, = TrI plFl

+ T,*,p2F2,

T&p2F2,

ImF2 = T&plFl+

Ua,b)

where T11 =fiJpl

> T,,=fd~z

3 T,,=f221~2 7

and the phase space factors p1 and p2 are given by PI=

Y:l&

P2 = 2q;ti

2

with

41=&G$/2,

q2=Js-(m,-mm,)2~s-(m,+m,)2/2~.

i

Eqs. (1 a) and (1 b) together with the dispersion relations for F, and F2 give us a system of coupled integral equations for Fl and F2. Setting u1 = Tf2p2F2 and u 2 = Tf2p1 F, the solutions of ei. (1) are [4]

lmal(s’)ds’ 1, fTIFc(s’)s’(s’-

(24

s - ie)

or 2Re(olei”r)ds’ s2 (l+g)F:l(s’)e-‘Sls’(s’-

s - ie)

1’

(2b) 213

Volume

67B, number

2

PHYSICS

LETTERS

28 March

1977

IFnl 4-

z-

I-

.6-

.4-

.2-

I I

Fig. 2. IF?,(s)1 in the 1 - 1.4 GeV region. Experimental points are from ref. [l] and [2]. The dashed line is the Breit-Wigner curve. Curve (a) is the result of the first step of the iteration orocedure described in the text, curve (b) is the result of the self consistency method, and cdrve (c) the result of the third method. In each case, eq. (10) or (12) is used withN= 1.25 GeV and I? = 100 MeV;n i= l/2 and l/3 for curve (a) and (b)

.S

n2

s2

Fig. 1. Behaviour

of a(s)

respectively.

as defined in the text.

and F2(s)=F;2(s)

lJ [

1 j2~(l;;);,;,(n;e~V;

@2(S’)dS’

sl Fs,2(s’)(s’-s-ifz)

where q1 is the phase offi

+is2

oz(SF]

)

(2c)

:

and 6’s are the phase shifts and

and similar expressions for Fi 1 and F: 2 ; the thresholds Si are given by sl=4rn;,

s2 = (m,

+ m71)2 .

We have written an unsubtracted dispersion relation for F2(s) as the situation is somewhat similar to the calculation of the magnetic form factor of the nucleon. In general, we can also subtract F2 at s = 0, the subtraction constant is then related to the o + 7r”y amplitude. 1. LXTOform factor. Let us first discuss the wr” form factor F2(s). We want to show that the p-dominance value for F2 would lead to difficulty of understanding F, (s). The first integral in the r.h.s. of eq. (2~) receives contribution from s1 to s2 due to the 27r intermediate state in the unitarity condition. Below the ano threshold, T12 has the phase of 6 1, hence ff2 = (Const)pl

lF’o0’I2 ,

in this region. Using a S-function 214

(3) approximation

for IFi1 I2 in the first integral, we get a pole term which should

28 March 1977

PHYSICS LETTERS

Volume 67B, number 2

be identified with the p propagator in the usual vector meson dominance approximation for IFtlI 2 in the dispersion integral yields

field theory language. A more refined

F2 = (Const) Fil (s) Fi2(s) ,

(4)

where the real constant is directly related to the usualfyp ,gwpn, etc ... . To get (4), we have assumed that 02 due to the 2n intermediate state dominates the dispersion integral and that it converges. One should note that (4) has the term F:‘(s) which corresponds to the field theory language of taking into account of a finite width of the “p”. Hence we identify eq. (4) with the p dominance model result. Eq. (4) indicates that F2 has the phase of 6 L+ 62, the inelastic spectral function u1 = Tr2p2F2 is then purely real. In this case Fl(s) = Fp (s) as given by eq. (2a). As it has been shown previously [5], using the CERN-Munich data [6] on nrr phase, agreement with experimental results is qualitative, i.e. it is not possible to explain the enhancement in the pion form factor in the 1 GeV region. In fact we get a result somewhat smaller than that given by the Breit-Wigner from [S, 71. The dispersion integral in eq. (2a) contributes and could provide an enhancement of FL if F2 does not have the phase of 6 L+ 6,. This is the case if the p-dominance result is not valid for F,, i.e. the second term on the r.h.s. of eq. (2~) contributes. Any attempt to calculate the contribution from this term to F, would depend not only on a detailed knowledge of 6, which is not available, but also on the approximation scheme involved. This leads US to consider the following possible scheme which requires experimental data on 1F21 but not on iS2, and does not need any analytic property of F2. 2. Pion form factor. Let us rewrite eq. (2b) as FL(s) = F; l (s>g1 6) >

(9

where 81 (s) is real for s < ~2. We define the phase of g1 as (Y(S)with CY(S = s2) = 0. Because the absorptive part of F(s) is real, using eq. (1) we can show [4] 21m(aleia1) RegL(s) =

(1 -$Fir

2Re (oL eisr)

e-i61 ’

ImgL(s) =

Using eqs. (5) and (6) it is straightforward

(1 t ~)F~ie+i

’

(da, b)

to calculate the phase (Y(S)of g1 (s) (7)

where R=

IF2(s)12 ~20)

o(e+e- + 071) u(e+e-+

by unitarity

n+n-)

=jFlol2p!(s)’

(8)

R is restricted as

g(s) is then obtained by a dispersion integral g(s) = exp is

sI(sI :!I)

ie) ds’ ,

(9).

$2

where we have assumed that g(s) does not have any zero. It should be noted that the sign of a(s) is not determined by this method. We choose the positive sign in order to get the correct enhancement factor for F1. The consequence of this condition on Fz(s) will be discussed elsewhere. 21.5

Volume

67B, number

2

PHYSICS

LETTERS

28 March 1977

Combining eqs. (5), (7) and (9) we get a fairly complicated integral equation. It is not difficult however to extract some physical consequences from this expression. We can proceed by two possible methods to solve this integral equation. We can first approximate F,(s) N Fil (s) and use this and the experimental value of IF2] to get the first approximation on R as given by eq. (8) and hence on (Y(S)by eq. (7). Fl(s) can then be calculated by eqs. (5) and (9) which in turn can be used to get a second approximation for R, this procedure being iterated until a converging result is obtained. An alternative method consists in using the experimental value of R (s) to calculate F1 (s) and check for self consistency i.e. the experimental result of F, (s). The important feature of eq. (7) is that R is a fast increasing function due to the threshold effect of the c3rr” channel. At threshold R is zero but it increases rapidly. At fi = 1.3 GeV, R E 10. Suppose 77= 0.9 from 1 .l GeV to 1.3 GeV as given (but with large error) by the CERN-Munich phase shift analysis [6], we get a(‘& = 1.3 GeV) E 90” by the first step of the iterative method and a(& = 1.3 GeV) = 50” by the second method. Such a rapid increase of the phase of a(s) from its zero value at threshold would result in an enhancement in Fl(s) as is observed experimentally. To be more quantitative let us assume two possibilities that o(s) behaves as in fig. 1, with (u(m) = nrr where n is of order unity. We get

g(s)=

(10)

where N is approximately the value of Gwhere (II(S)= n7r/2 and r is proportional to the derivative of (Yat this point. The expression given above is similar to a Breit-Wigner from. It does not correspond to a resonance however. Its effect is due entirely to a rapid increase of the ratio R. Its effect on IF,1 at the p mass is small and of the order of 15%. ’ It is also possible to use a third method which will give a sufficiently accurate solution when R S (1 -n)/(l t n), which is the case considered here but is less valid near the wrr” threshold. Using eq. (7) we get:

(11) where R. is defined persion relation for first approximation is that no interaction as above we get the

&(S)= 1+

as in eq. (8) with F, replaced by F”ot. We use eq. (11) to write down an once subtracted disgI (s) with the condition g1 (0) = 1 and we get a non linear integral equation for g1 (s) [8]. A can be obtained by putting IgI( = 1 in the r.h.s. of eq. (11). The advantage of this method scheme is needed for values of 4 above 1 .l GeV. Using the same numerical approximation following expression of gl(s):

Img~ttJJ+r>2) 71

In

A42

+ i ImgT(s) ,

M2 - s - iMI(s)

(12)

with gy defined by eq. (11) with 1gI l2 replaced by unity on the r.h.s. Again we used the positive sign in eq. (11) which corresponds to the same sign for Q in eq. (7). As can be seen from eq. (12) a fast increase of R gives an enhancement in F, . We show in fig. 2 the resulting pion form factor in the 1 - 1.4 GeV region. It should be emphasized that the parameters used in eqs. (10) and (12) are completely determined (with large errors) by the pion-pion interaction parameters 6, q and the 071 form factor IF2 I. There is no flexibility in fixing the absolute scale of the enhancement factor. [l] [2] [3] [4] [5] [6] [7] [8]

216

G. Cosme et al., Phys. Lett. 63B (1976) 349; and L.A.L. preprint (July 1976). Preliminary results presented at the 18th Intern. Conf. on High energy physics, Tbilisi, 1976. F.M. Renard, Nucl. Phys. 882 (1974) 1. T.N. Pham and Tran N. Truong, Ecole Polytechnique Preprint A 248.1176 (1976). T.N. Pham and Tran N. Truong, Phys. Rev. D14 (1976) 185. B. Hyams et al., Nucl. Phys. B64 (1973) 134. G. Bonneau and F. Martin, Nuovo Cimento 13A (1972) 413. This type of integral equation was also given by N.M. Budnev, V.M. Budnev and V.V. Serebryakov,

Phys. Lett. 64B (1976)

307.