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The influence of inelastic πω channel on pion form factor at s<(mω + mπ)2
Volume 64B, number 3
PHYSICS LETTERS
27 September 1976
THE INFLUENCE OF INELASTIC no CHANNEL P I O N F O R M F A C T O R A T s < ( m o + m n)2
ON
N.M. BUDNEV, V.M. BUDNEV and V.V. SEREBRYAKOV Department of Theoretical Physics, lnstitute for Mathematics, Novosibirsk-90, USSR 630090
Received 9 August 1976 It is shown that Fn(s) can be calculated in a model independent way if one knows the phase ~ l and the inelasticity rt of the p-wave 7r~r scattering and also F n and the form factor of the 7" ~ rr°to transition for s > (mto + rnTr)2 . The correction on FTr(S)for s < (mw +mn) 2 due to the *rw state with a strong 0'(1250) allows to explain the discrepancy be-
tween 0-dominance predictions and the experimental data.
During the last years considerable efforts have been invested in the study of pion form factor F~(s) at 4 GeV 2 < s < 9 GeV 2 [ 1 - 9 ] . These data were analyzed in a number of theoretical papers [ 1 0 - 1 6 ] . In this paper we consider the influence of inelastic unitarity taking into account 7rlr and lr~o intermediate states on F,r at s < (mto +m~) 2. The unitarity relations for F,r and the form factor of 3' -+ 7r°co transition Fro can be written:
F~o is connected with the cross section o(e+e---~r°co) by
_ °e*e-"~'~°w
212p 3
=F,r/e2i6l-1 2i
.FPLS(1-r12)] 112 +Ft°L
2p3
j
i(61 +62)
e
/i
(1)
=F,[_p3(1-n2)11/2 ImFt° ~rL J 8 .3 s Pro
e2i82_1
e i(~l+~2)+F*r/
Im N(s)
l l/f
F2_p3sT1/2 Re D(s)l~22(s)l
t'l+n
L-L-Tj
Pw =
+m) 2][s-(m 4s
(4)
la l(s)i
Prr
(5)
ll~r p3 ll/2 ReN(s)igZl(S)l Im D(s) = 1 / ~
L2-~sd
1~22(s)1
,
Using analaytical properties of N(s)(F~(s)) and eq.(5), one can write the following dispersion relation for N(s): oo
Re N(s)= 1 +L
/[s-(m
F w = a2(s)D(s ) .
Here ~l(S) and ~2(s) are the usual Omnes functions for 7rTrand 7rco scatterings, respectively. It is easy to see that at s > (into + r n ) 2 the unitarity relations (1) are satisfied if
2i
These relations are correct if we assume that isovector channel with JP = 1- is saturated by 7rTrand 7re states only. This assumption means that in the unitarity condition for F~ and Fw as well as for the partial amplitudes •'f jr,= 1 l i ' ~ - - - 1 " 7 r / r ~ dfJp =1- 7r~'.--c.,/r o,) ~ dFJP= 1- ~" (,o---~-~ o ~ it is sufficient to take into account mr and zr~o intermediate states only. We consider the experimental data to confirm our main assumption. In eq. (1) fi 1,62 are the phases of mr and 7rco scattering respectively; ~7 is the common inelasticity, and
(3)
Using eq. (1), it was shown (in paper [17]) that one can evaluate F,~ and F~o if 61,62 and ~7 are known. Indeed let us represent F~ and F~o by
F = al(S)N(s); im F
3
\ w/s] I F I 2 "
ds'
f st (m~o+rn,r)2 -
_3 rs,a-~ I/2 s'[(1-Lr/2a)v~o_______t ,.1
2s'p3r(s')
J
(6)
2 Re O(s')1~2(s') 1
-m)2l (2)
× (1 +r/(s'))l~l(S')l 307
Volume 64B, number 3
PHYSICS LETTERS
In the same manner it is also possible to write the dispersion relation for Re D(s). Combining the latter with eq. (6), the Fredholm integral equation for Re D(s) was obtained in [17]. Solving this equation for Re D and using eq. (6) one can determine Re N(s) in terms o f S 1 , 6 2 and r/. At the present time p-wave rr6o phase shift data are absent. Nevertheless, using eq. (5) it is easy to get 2 Re
D(s)l~22(s)l
27 September 1976
If one chooses F t o ( 0 ) = m p -1;
so = l G e v
~, ,.s.,[2
7
Pro
2s 3 J
m2
~ReD(s)=F
(0)
p m2_ s o
o
s -s m2 Re D(s) = Fw(O ) o_ p s o r n 2p - - s (9)
P
,m , P
,1 This conclusion does not depend upon the sign of Fro(0),
since the sign of Fro(0) coincides with the one of gpton/gpmr and the sign of root ~ is the same as the one of gp to,gpmr. Therefore, for simplicity we consider Fro(0) > 0 below. 308
A
for our estimations. Let us discuss the parametrization of ~ 1(s) before comparing our prediction on F~r with the data. The P partial wave phase of rrTr scattering is well known from the experiments [18, 19] hence one can obtain a good approximation of ~l(S). In the elastic unitarity region, 2Pzr d P =1-
mp Pp (s). A(s)
' (ll)
Im A _ mpPp(s) ReA ReA(s)
and the function ~21 (s) is expressed in terms of A(s) in the usual way ~ l ( S ) = A(o)/A(s).
(12)
A good approximation of tan 8, is obtained by the following choice of P(s) and Re A(s): mp p3(s) R 2pTr(mp) 2 2 +1 3 2 Plr(mp) R2p2,(s) + 1
[h(s)-h(m2p)](, 13)
where
ds'mpPp(s') a ,--~S.. , - - 7 2 ~ " 4m~ (S - s)(s --4mlr)
S - 4m 2. ~
P rap,
zsp¢ o
Re A(s)=(m2o-s)(1-h'(m2p))-
m 2,
2 _s_iP
nl/2
- (1 - r / ) . _ ~ 3 I F i 2 |
tan 8 1 -
m2
1 gpco~ P .(8) 2 gp~rlr m 2 - s
When using approximation (8) for Fto(s ) it is impossible to describe the data [6, 8, 9, 22] on FTr and F ~ at V t s ~ 1+1.4 GeV. In the previous paper [21] we have shown that these data may be described within experimental errors if the phase 6 2 is assumed to go through rr/2 at X/s-" 1250 MeV, i.e. the p'(1250)-meson exists. In this case
Y~2(s)=
p3,
iF ol
"
Eqs. (3, 4; 5, 6) allow to compute F , in a modelindependent way in terms of 61, r/(which are known from the experiments [18, 19] and of the data on lEvi 2 and l e v i 2 at s > ( m ~ + m~) 2 too. However, it is necessary to determine the sign of Re D to compute the integral (6). If one makes use of 0-dominance [20] for F ~ , then ~22(s) = 1 and Re D is negative ,1 in the domain of integration in eq. (6), i.e.
F
(10)
then the representation (9) approximates the data satisfactorily [21 ] and corresponds to the positive sign of the root in eq, (7). When computing the integral in eq. (6) we have used the approximation (9) of
1 -r/p.
-f7
rap, = 1 2 5 0 M e V ;
I"o, = 200 MeV
l+r/ (7)
2',
h(s) - - 7r
When fitting the data [18, 19] it is often assumed Re A = m 2 - s. This approximation is a good one at s = m 2. However, terms h(s), h(m 2) and h'(m 2) must be tak"en into account when ~21 (s~ is evaluate~ beyond
Volume 64B, number 3
PHYSICS LETTERS
27 September 1976
the p-meson region. Including these terms one can obtain 10% deviation o f A ( o ) from m 2, it is the so-called finite-with correction. In addition it is often assumed that p3(s)
I'(s) = rp
IF~I
X 0rsa~ //71 //
I
//
2 2 + 1 R 2 p~r(mp)
\
2 2 n2p2r(s ) + p~r(mp) 1
,// Let us notice that the factor m / v ~ in P(s) is necessary for a correct description o f the threshold behaviour of tan 61 . So eqs. (12), (13) explicitly exhibit the analytical properties o f ~1 (s) and approximate tan 61, in the proper way. Therefore in the following calculations of F~r(s) we shall use (12) and (13) where the parameters m o and Pp are chosen according to the rrTr phase shift analysis [18, 19]. We shall take R 2 = 9 GeV - 2 as for the most resonances %2 (see [23]). When computing the integral (6) a rather large ambiguity arises due to the factor 1 - r / 2 . The point is that we have to use an inelasticity connected with the transition rm -+ 7rco only. We have considered that %/1 - rl2 = ap 3 / m 3 at s < 1.04 C,eV 2 and %/1 - r/2 = 0.45 at s > 1.0t°4 ~ V . This approximation fits an average ( - - -) solution o f phase shift analysis [18]. The calculations have been made b y computer. Let us cover the consequences o f the account o f the 7roJ state in unitarity relations. i) The p-meson range. In this range there is 7 - 1 0 % correction to F,r due to the influence o f the Iro~ state. At s ~ m 2p F~r is equal to N(m2)A(o)/Ft, m p or to A(o)/mpPp if we take or do not take into consideration the ~r~ state respectively. Notice, since F~r(0 ) = 1 that one cannot consider A(o) as a free parameter. Therefore, if one does not take the 7rw state into account the effective p-meson width appears to have a value 7 - 1 0 % smaller than the one which was obtained from the phase shift analysis. The importance of t a k ing into account the inelastic part o f Frr(s ) for removing the discrepancy in the Fp obtained from e+e - data and 7rrr phase shift was pointed out by Ross [14]. Besides, one must keep in mind that the behaviour of ~1 (s) in the vicinity o f m 2 depends considerably
i 0.55
i 0.6
.
f
i
0.7
0.75
\\
i
>
~ deV
Fig. 1. The pion form factor FTr by taking into account the 7fro-state for different S21(s): (- - -) according to GounarisSakurai [24] with Fp = 150 MeV; ( - - ) the same with Fp = 163 MeV; (-. -. -) according to approximation (13) with Fp = 170 MeV. The parameters of p-to mixing are taken from [7]. on the value of R 2. In the paper [7] the G o u n a r i s Sakurai formula [24] was used to determine the O" meson parameters (in this case R 2 = 0). On the contrary phase shift analysis yields a considerable value o f the order o f a few GeV 2 f o r R 2. Thus the phase o f the form factor defined according to G o u n a r i s -
O.t~8
This work
0.47
,{
Lang, Sbefaaescu
9.46,
•
Kiehlmaa, Schmidb
~0.53
Bebek
0.55 ~
I x
14
Berezhnev 75
0.56
I
0.61 ~
I
~2 Let us notice that the parametrization (1 l) (13) continued in the threshold region with R 2 = 9 GeV -2 yields a large scattering length a[ = 0.067 m~ra (PCAC value a~ = 0.035 m~ra) which well agrees with CERN-Munich group data [lgal.
i 0.65
-
o'.5
o~6
I
o:7
Berezbnev 73 Adylov
o'.s
'
Fig. 2. The experimental data and theoretical estimates of (r2~>. Our estimate of the radius is (r2~> = 0 . 4 3 5 - 0 . 4 8 fm 2. 309
Volume 64B, number 3
PHYSICS LETTERS
I F~ Ilsl o.7
FTr(s)go for s < 0. Let us notice that for this range o f
X
c~
•
?0 r ~ e i l
71
0.6
o.5
27 September 1976
!
o.3
0.2
the energy dependence o f the function ~ 1(s) = F P(s) in parametrisation ( 1 2 - 1 3 ) does practically not differ from the pole approximation and the G o u n a r i s - S a k u rai formula. Therefore, the decrease of tF~r(S)lS is due to the inelastic contribution wholly. For comparison FTr(s) in p-dominance approximation is shown too in fig. 3. If one uses p-dominance approximation (8) for Fro instead of eq. (9), then N(s) > 1 and the deviation o f the theoretical estimation from the data increases. Thus one can conclude that the p-dominance breaking for F~(s) by means of adding a strong p'-meson [21] yields a concerted description of all known data on the pion form factor in the s < 2 GeV 2.
0.1
-1
-2
-3
$ ~eV2
Fig. 3. The pion form factor FTr(s)in the space-like region. (- - -) is the p-dominance sm~/(m~ - s) (in this region Fir does not depend on the approximation of f4 (s)). Hatched domain represents our prediction for Fn(s) taking into account the 7rto state. Sakurai formula and the phase of ~rn-scattering obtained in [19] differ from each other. But it is hardly possible to draw a conclusion on unitarity violation for Fn because of the poor statistics for F~r. The comparison o f the estimations ofFer and the data is shown in fig. 1. The parameters o f p - t o mixing were taken from the paper [7]. ii) The pion radius (r2). A simple vector-meson dominance leads to (r2DM) 7 0 . 3 9 fm 2. Experimental data on (r 2) yield the value which is 1 . 2 - 1 . 5 times larger in comparison with (r2DM). The account o f ~rw state, (i.e. N(s) function) in the vicinity of s ~- 0, yields a correction of the order o f 0.04 fm 2 for (r2). It should be mentioned that, if one uses a simple p-dominance approximation for Re D(s) then A 2 ( ~ ) < 0. Another contribution leading to an increase o f the radius is due to the parametrisation o f ~2i(s) being used (eqs. (12, 13). I f R 2 = 9 GeV - 2 then A l ( r 2 ) = 0.04 fm 2. As a result our estimation o f (r 2) is (r~) = 0 . 4 5 - 0 . 4 8 fm 2. iii) Space-like s. It is easy to see that at s < 0 Re N < 1 i.e. rrw state contribution leads to a decrease o f
310
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
G.T. Adylov et al., Phys. Lett. 51B (1974) 402. S.F. Berezhnev et al., Soviet J. Nucl. Phys. 18 (1974) 53. C.J. Bebek et al., Phys. Rev. 13D (1976) 25. C.N. Brown et al., Phys. Rev. D8 (1973) 92. C.J. Bebek et al., Phys. Rev. D9 (1974) 1229. V.E. Balakin et al., Phys. Lett. 41B (1972) 205. D. Benaksas et al., Phys. Lett. 39B (1972) 289. G. Barbiellini et al., Nuovo Cim. Lett. 6 (1973) 557. D. Bollini et al., Nuovo Cim. Lett. 14 (1975) 418. M. Gourden, Phys. Rep. 14C (1974) 29. N.D. Kiehlmann and W. Schmidt, Nucl. Phys. B94 (1975) 67. C.B. Lang and I.-S. Stefanescu, Phys. Lett. 58B (1975) 450. F.M. Renard, Phys. Lett. 47B (1973) 361. M. Ross, Nucl. Phys. B100 (1975) 205. S. Dudnicka and V.A. Mescheriakov, Nucl. Phys. B83 (1974) 311. A. Bramon, Lett. Nuovo Cim. 8 (1973) 659. V.M. Budnev and V.V. Serebryakov, Yad. Fiz 23 (1976) 1306. B. Hyams et al., Nucl. Phys. B100 (1975) 205; W. Manner, Contribution the 17th Intern. Conf. on High energy physics, London, July 1974. S.D. Protopopeseu et al., Phys. Rev. D7 (1973) 1279. F.M. Renard, Nuovo Cim. 54A (1969) 979. N.M. Budnev, B.M. Budnev and V.V. Serebryakov, Nucl. Phys., in print. M. Conversi et al., Phys. Lett. 52B (1974) 493. Data Particle Group, Phys. Lett. 50B (1974) 1. G.S. Gounaris and J.J. Sakurai, Phys. Rev. Lett. 21 (1968) 244.
PHYSICS LETTERS
27 September 1976
THE INFLUENCE OF INELASTIC no CHANNEL P I O N F O R M F A C T O R A T s < ( m o + m n)2
ON
N.M. BUDNEV, V.M. BUDNEV and V.V. SEREBRYAKOV Department of Theoretical Physics, lnstitute for Mathematics, Novosibirsk-90, USSR 630090
Received 9 August 1976 It is shown that Fn(s) can be calculated in a model independent way if one knows the phase ~ l and the inelasticity rt of the p-wave 7r~r scattering and also F n and the form factor of the 7" ~ rr°to transition for s > (mto + rnTr)2 . The correction on FTr(S)for s < (mw +mn) 2 due to the *rw state with a strong 0'(1250) allows to explain the discrepancy be-
tween 0-dominance predictions and the experimental data.
During the last years considerable efforts have been invested in the study of pion form factor F~(s) at 4 GeV 2 < s < 9 GeV 2 [ 1 - 9 ] . These data were analyzed in a number of theoretical papers [ 1 0 - 1 6 ] . In this paper we consider the influence of inelastic unitarity taking into account 7rlr and lr~o intermediate states on F,r at s < (mto +m~) 2. The unitarity relations for F,r and the form factor of 3' -+ 7r°co transition Fro can be written:
F~o is connected with the cross section o(e+e---~r°co) by
_ °e*e-"~'~°w
212p 3
=F,r/e2i6l-1 2i
.FPLS(1-r12)] 112 +Ft°L
2p3
j
i(61 +62)
e
/i
(1)
=F,[_p3(1-n2)11/2 ImFt° ~rL J 8 .3 s Pro
e2i82_1
e i(~l+~2)+F*r/
Im N(s)
l l/f
F2_p3sT1/2 Re D(s)l~22(s)l
t'l+n
L-L-Tj
Pw =
+m) 2][s-(m 4s
(4)
la l(s)i
Prr
(5)
ll~r p3 ll/2 ReN(s)igZl(S)l Im D(s) = 1 / ~
L2-~sd
1~22(s)1
,
Using analaytical properties of N(s)(F~(s)) and eq.(5), one can write the following dispersion relation for N(s): oo
Re N(s)= 1 +L
/[s-(m
F w = a2(s)D(s ) .
Here ~l(S) and ~2(s) are the usual Omnes functions for 7rTrand 7rco scatterings, respectively. It is easy to see that at s > (into + r n ) 2 the unitarity relations (1) are satisfied if
2i
These relations are correct if we assume that isovector channel with JP = 1- is saturated by 7rTrand 7re states only. This assumption means that in the unitarity condition for F~ and Fw as well as for the partial amplitudes •'f jr,= 1 l i ' ~ - - - 1 " 7 r / r ~ dfJp =1- 7r~'.--c.,/r o,) ~ dFJP= 1- ~" (,o---~-~ o ~ it is sufficient to take into account mr and zr~o intermediate states only. We consider the experimental data to confirm our main assumption. In eq. (1) fi 1,62 are the phases of mr and 7rco scattering respectively; ~7 is the common inelasticity, and
(3)
Using eq. (1), it was shown (in paper [17]) that one can evaluate F,~ and F~o if 61,62 and ~7 are known. Indeed let us represent F~ and F~o by
F = al(S)N(s); im F
3
\ w/s] I F I 2 "
ds'
f st (m~o+rn,r)2 -
_3 rs,a-~ I/2 s'[(1-Lr/2a)v~o_______t ,.1
2s'p3r(s')
J
(6)
2 Re O(s')1~2(s') 1
-m)2l (2)
× (1 +r/(s'))l~l(S')l 307
Volume 64B, number 3
PHYSICS LETTERS
In the same manner it is also possible to write the dispersion relation for Re D(s). Combining the latter with eq. (6), the Fredholm integral equation for Re D(s) was obtained in [17]. Solving this equation for Re D and using eq. (6) one can determine Re N(s) in terms o f S 1 , 6 2 and r/. At the present time p-wave rr6o phase shift data are absent. Nevertheless, using eq. (5) it is easy to get 2 Re
D(s)l~22(s)l
27 September 1976
If one chooses F t o ( 0 ) = m p -1;
so = l G e v
~, ,.s.,[2
7
Pro
2s 3 J
m2
~ReD(s)=F
(0)
p m2_ s o
o
s -s m2 Re D(s) = Fw(O ) o_ p s o r n 2p - - s (9)
P
,m , P
,1 This conclusion does not depend upon the sign of Fro(0),
since the sign of Fro(0) coincides with the one of gpton/gpmr and the sign of root ~ is the same as the one of gp to,gpmr. Therefore, for simplicity we consider Fro(0) > 0 below. 308
A
for our estimations. Let us discuss the parametrization of ~ 1(s) before comparing our prediction on F~r with the data. The P partial wave phase of rrTr scattering is well known from the experiments [18, 19] hence one can obtain a good approximation of ~l(S). In the elastic unitarity region, 2Pzr d P =1-
mp Pp (s). A(s)
' (ll)
Im A _ mpPp(s) ReA ReA(s)
and the function ~21 (s) is expressed in terms of A(s) in the usual way ~ l ( S ) = A(o)/A(s).
(12)
A good approximation of tan 8, is obtained by the following choice of P(s) and Re A(s): mp p3(s) R 2pTr(mp) 2 2 +1 3 2 Plr(mp) R2p2,(s) + 1
[h(s)-h(m2p)](, 13)
where
ds'mpPp(s') a ,--~S.. , - - 7 2 ~ " 4m~ (S - s)(s --4mlr)
S - 4m 2. ~
P rap,
zsp¢ o
Re A(s)=(m2o-s)(1-h'(m2p))-
m 2,
2 _s_iP
nl/2
- (1 - r / ) . _ ~ 3 I F i 2 |
tan 8 1 -
m2
1 gpco~ P .(8) 2 gp~rlr m 2 - s
When using approximation (8) for Fto(s ) it is impossible to describe the data [6, 8, 9, 22] on FTr and F ~ at V t s ~ 1+1.4 GeV. In the previous paper [21] we have shown that these data may be described within experimental errors if the phase 6 2 is assumed to go through rr/2 at X/s-" 1250 MeV, i.e. the p'(1250)-meson exists. In this case
Y~2(s)=
p3,
iF ol
"
Eqs. (3, 4; 5, 6) allow to compute F , in a modelindependent way in terms of 61, r/(which are known from the experiments [18, 19] and of the data on lEvi 2 and l e v i 2 at s > ( m ~ + m~) 2 too. However, it is necessary to determine the sign of Re D to compute the integral (6). If one makes use of 0-dominance [20] for F ~ , then ~22(s) = 1 and Re D is negative ,1 in the domain of integration in eq. (6), i.e.
F
(10)
then the representation (9) approximates the data satisfactorily [21 ] and corresponds to the positive sign of the root in eq, (7). When computing the integral in eq. (6) we have used the approximation (9) of
1 -r/p.
-f7
rap, = 1 2 5 0 M e V ;
I"o, = 200 MeV
l+r/ (7)
2',
h(s) - - 7r
When fitting the data [18, 19] it is often assumed Re A = m 2 - s. This approximation is a good one at s = m 2. However, terms h(s), h(m 2) and h'(m 2) must be tak"en into account when ~21 (s~ is evaluate~ beyond
Volume 64B, number 3
PHYSICS LETTERS
27 September 1976
the p-meson region. Including these terms one can obtain 10% deviation o f A ( o ) from m 2, it is the so-called finite-with correction. In addition it is often assumed that p3(s)
I'(s) = rp
IF~I
X 0rsa~ //71 //
I
//
2 2 + 1 R 2 p~r(mp)
\
2 2 n2p2r(s ) + p~r(mp) 1
,// Let us notice that the factor m / v ~ in P(s) is necessary for a correct description o f the threshold behaviour of tan 61 . So eqs. (12), (13) explicitly exhibit the analytical properties o f ~1 (s) and approximate tan 61, in the proper way. Therefore in the following calculations of F~r(s) we shall use (12) and (13) where the parameters m o and Pp are chosen according to the rrTr phase shift analysis [18, 19]. We shall take R 2 = 9 GeV - 2 as for the most resonances %2 (see [23]). When computing the integral (6) a rather large ambiguity arises due to the factor 1 - r / 2 . The point is that we have to use an inelasticity connected with the transition rm -+ 7rco only. We have considered that %/1 - rl2 = ap 3 / m 3 at s < 1.04 C,eV 2 and %/1 - r/2 = 0.45 at s > 1.0t°4 ~ V . This approximation fits an average ( - - -) solution o f phase shift analysis [18]. The calculations have been made b y computer. Let us cover the consequences o f the account o f the 7roJ state in unitarity relations. i) The p-meson range. In this range there is 7 - 1 0 % correction to F,r due to the influence o f the Iro~ state. At s ~ m 2p F~r is equal to N(m2)A(o)/Ft, m p or to A(o)/mpPp if we take or do not take into consideration the ~r~ state respectively. Notice, since F~r(0 ) = 1 that one cannot consider A(o) as a free parameter. Therefore, if one does not take the 7rw state into account the effective p-meson width appears to have a value 7 - 1 0 % smaller than the one which was obtained from the phase shift analysis. The importance of t a k ing into account the inelastic part o f Frr(s ) for removing the discrepancy in the Fp obtained from e+e - data and 7rrr phase shift was pointed out by Ross [14]. Besides, one must keep in mind that the behaviour of ~1 (s) in the vicinity o f m 2 depends considerably
i 0.55
i 0.6
.
f
i
0.7
0.75
\\
i
>
~ deV
Fig. 1. The pion form factor FTr by taking into account the 7fro-state for different S21(s): (- - -) according to GounarisSakurai [24] with Fp = 150 MeV; ( - - ) the same with Fp = 163 MeV; (-. -. -) according to approximation (13) with Fp = 170 MeV. The parameters of p-to mixing are taken from [7]. on the value of R 2. In the paper [7] the G o u n a r i s Sakurai formula [24] was used to determine the O" meson parameters (in this case R 2 = 0). On the contrary phase shift analysis yields a considerable value o f the order o f a few GeV 2 f o r R 2. Thus the phase o f the form factor defined according to G o u n a r i s -
O.t~8
This work
0.47
,{
Lang, Sbefaaescu
9.46,
•
Kiehlmaa, Schmidb
~0.53
Bebek
0.55 ~
I x
14
Berezhnev 75
0.56
I
0.61 ~
I
~2 Let us notice that the parametrization (1 l) (13) continued in the threshold region with R 2 = 9 GeV -2 yields a large scattering length a[ = 0.067 m~ra (PCAC value a~ = 0.035 m~ra) which well agrees with CERN-Munich group data [lgal.
i 0.65
-
o'.5
o~6
I
o:7
Berezbnev 73 Adylov
o'.s
'
Fig. 2. The experimental data and theoretical estimates of (r2~>. Our estimate of the radius is (r2~> = 0 . 4 3 5 - 0 . 4 8 fm 2. 309
Volume 64B, number 3
PHYSICS LETTERS
I F~ Ilsl o.7
FTr(s)go for s < 0. Let us notice that for this range o f
X
c~
•
?0 r ~ e i l
71
0.6
o.5
27 September 1976
!
o.3
0.2
the energy dependence o f the function ~ 1(s) = F P(s) in parametrisation ( 1 2 - 1 3 ) does practically not differ from the pole approximation and the G o u n a r i s - S a k u rai formula. Therefore, the decrease of tF~r(S)lS is due to the inelastic contribution wholly. For comparison FTr(s) in p-dominance approximation is shown too in fig. 3. If one uses p-dominance approximation (8) for Fro instead of eq. (9), then N(s) > 1 and the deviation o f the theoretical estimation from the data increases. Thus one can conclude that the p-dominance breaking for F~(s) by means of adding a strong p'-meson [21] yields a concerted description of all known data on the pion form factor in the s < 2 GeV 2.
0.1
-1
-2
-3
$ ~eV2
Fig. 3. The pion form factor FTr(s)in the space-like region. (- - -) is the p-dominance sm~/(m~ - s) (in this region Fir does not depend on the approximation of f4 (s)). Hatched domain represents our prediction for Fn(s) taking into account the 7rto state. Sakurai formula and the phase of ~rn-scattering obtained in [19] differ from each other. But it is hardly possible to draw a conclusion on unitarity violation for Fn because of the poor statistics for F~r. The comparison o f the estimations ofFer and the data is shown in fig. 1. The parameters o f p - t o mixing were taken from the paper [7]. ii) The pion radius (r2). A simple vector-meson dominance leads to (r2DM) 7 0 . 3 9 fm 2. Experimental data on (r 2) yield the value which is 1 . 2 - 1 . 5 times larger in comparison with (r2DM). The account o f ~rw state, (i.e. N(s) function) in the vicinity of s ~- 0, yields a correction of the order o f 0.04 fm 2 for (r2). It should be mentioned that, if one uses a simple p-dominance approximation for Re D(s) then A 2 ( ~ ) < 0. Another contribution leading to an increase o f the radius is due to the parametrisation o f ~2i(s) being used (eqs. (12, 13). I f R 2 = 9 GeV - 2 then A l ( r 2 ) = 0.04 fm 2. As a result our estimation o f (r 2) is (r~) = 0 . 4 5 - 0 . 4 8 fm 2. iii) Space-like s. It is easy to see that at s < 0 Re N < 1 i.e. rrw state contribution leads to a decrease o f
310
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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