- Email: [email protected]
Status of the Cabibbio - Radicati sum rule and a new test of vector meson dominance
Volume 41B, number 3
PHYSICS LETTERS
20ctober!1972
S T A T U S O F T H E C A B I B B I O - R A D I C A T I SUM R U L E A N D A NEW T E S T O F V E C T O R M E S O N D O M I N A N C E G.J.
GOUNARIS
CERN, Geneva, Switzerland and Nuclear Research Centre "Democritus" Athens, Greece Received 11 July 1972 Usingfinite energy sum rules the high energy contribution to the Cabibbo-Radicati sum rule is calculated and shown to be of the proper mag~etude and sign, so that this sum rule is exactly satisfied. In addition the coupling of the p Regge trajectory to two charged photons is calculated and found to be in agreement with the predictions of vector meson dominance and universality for the Reggeized p vertex, within the expected accuracy.
It is well known that the Cabibbo-Radicati [I ] (CR) sum rule states that * d F V ( q 2)
/dV ~2
2 - - -0,1 q+~ 21r2a
f
doo a(co) = 0 ¢o '
(1)
mlr+m~/2MN where a(¢o) = aT(7+p) -- aT('),-p) = 2 OT(TVp -->I=~) - OT(TVp +I=3)
,
(2)
and ¢o is the laboratory energy of the photon. When confronted against the experiment this sum rule seems to be rather well satisfied [2, 3]. Stated in more detail the first two terms are respectively -1720/xb and 1520/Jb while the integral gives about - 5 0 / J b ifa cutoffat co = 1.1 GeV is used. We note that this later result comes about after large cancellations between large positive and negative contributions. We conclude therefore that a reasonably small high energy contribution of about 250 lib will explain the remaining discrepancy between the left and right-hand side in 6q. (1). The purpose of the present work is twofold. First we would like to use finite energy sum rules (FESR) to calculate the high energy contribution to CR and see whether it has the sign and magnitude expected to ex• F]/(q2)is the isovector Dixac form factor of the nucleon norrealized to F y ( 0 ) = 1 ;/~V = 3.7.
plain the remaining discrepancy mentioned above. Second we would like to calculate the coupling of a Reggeized p to two "charged" photons. The reason is the following. We know [4,5] that the couplings of the Pomeranchukon singularity and the A 2 trajectory to a photon and a vector meson obey the vector meson dominance (VMD) predictions to within 22%. We want therefore to see whether the coupling of the Reggeized p to two "charged" photons obeys also the VMD predictions to the same accuracy. Below we first give our formalism and then state and discuss our results. Let us call e2F(7-p; ¢o) and e2F(7*p; 6o) the for, ward spin non-flip invariant amplitudes for the processes 7 - P ~ 7 - P and 7*P ~ 7*P, at a photon laboratory energy w. We define F(-)(¢o) = ½{F(7-p; co) - F(7+p; ¢o)} ,
(3)
and demand that F(-)(~o) is an analytic function in the 6o plane with branch points at +w o=+(m
+m~2MN)
Moreover, we demand that it satisfies the crossing relation
and the optical theorem a(~o) = (4,ra/COMN) Im F(-)(w)
,
(4)
where o(¢o) is defined by eq. (2). We expect that for high energies we can always parametrize this amplitude as
329
Volume 41B, number 3
PHYSICS LETTERS
2 October 1972
0(60)= {2O:r(TVp-+ (nN)l=}) - o~TVp -+ (rrN)i={} + 0'(60)
/37r
F(-)(60) ~" F~-)(60) - 2 sin (trap(0))
= %N(60) + 0'(60),
×{
_
,
(5)
where/3 is a constant and the crossing symmetry of the amplitude has been taken into account. Here ap(0) is the intercept of an effective trajectory which describes the contribution of the P trajectory as well as the contributions of associated cuts and lower lying trajectories. Inspired by the data on charge exchange rrN scattering *, we will assume ,vp(0) ~ 0.5. Using eqs. (3)(5) and the lowest moment FERS I2
f
Im
- Fr(-)(60))d60
0 ,
(6)
too
we find a2
/3 _ otp(0) + 1 f 60o(60) d60/~2ap(0)+l MN
(7)
2rr2a ¢O0
where ~ is the cut-off in the FESR. We now express the high energy contribution to the integral appearing in CR in terms of/3/MN. To do this we assume that there exists an energy £Z' such that for 60 > gY we:are allowed to use in 'the afore-mentioned high energy contribution the Regge expression for o(60). From (4) and (5) we find this Regge expression to be
Or(60) = 2rrZa~/MN) (.O~O(0)-1 ,
where OTrN(60) describes the contribution of the lrN channel while 0'(60) comes from all the other channels. For OrrN(CO)we note that Walker's [6] phenomenologieal analysis of the pion photoproduction in terms of the Born term, a few resonances and a background, gives a reasonable description of the data up to energies [6, 7] of 1.1 GeV. We assume that by including into his amplitudes [8] Pl1(1750) and F37(1940 ) we get a good description for OwN(g.O) for 60 ~ 1.73 GeV. We remark that Walker's background is presumably coming mainly from weak s channel resonances which he ignores, as well as from the tails of the u channel resonances. At least this is what is suggested by applying the two component duality hypothesis to the process 7N -+ rrN. We are then led to the assumption that the true background in the process 7N -* aN is coming mainly from the pion pole in the Born term **. Now we turn to o'(w) for which we know how to calculate the contributions of the various resonances but we do not know how to estimate the background. Ignoring therefore hereafter the background for 0'(60), we have that the resonance contribution is o'(60) = ~
2oi(7Vp'+ R 1 ~ rrN)( I't(R1)" - 1 ) \ r ( R 1 + wN)
RI
(g)
and then
(10)
-
(,
rt(R 3) aT(TVp-+R3-* ~N) \ r ( R 3 + ~N) - 1 /]
~
' (11)
R3 1
where the sums over R 1 and R 3 run over all contributing resonances of isospin ~ and ~ respectively *** Here Pt(R ) and P(R -+ rrN) stand for the total and the partial widths to the nN channel, of resonance R. Adding OrrN(60) and o'(60) we have our result for o(60). This result is presumably valid for 60 ~< 1.73 GeV, and it is shown by the continuous line in rigA.
f - 0(60) d60 =
2rr2~ J
60
to O
1 r 0(60) 27r2a J
60
oa O
1 I 2z~2a
1 27r2c~
Or(60) • d (.a,) 60 ~t
°(60) d60 60
/3 (g2')aP(0)-i MN ~p(0)- 1
(9)
60 0
To proceed further we need to know o(60). In order to obtain it we use the following procedure. We split o(60) into two parts * Notice that for ~rN charge exchange scattering the same kind of Regge singularities contribute.
330
This is essentially assumed in the analysis of ref. [8]. We use here the resonances Pn(1470), Sn(1520), Dt3(1520), Dis(1670), Fts(1690), Pn(1750), S3t(1630), F37(1940). The characteristics of these resonances are taken from Rankin's article mentioned in ref. [8]. We use Rankin's values instead of Walker's to make sure that w e also take into account the contributions of the weaker resonances which Walker absorbs into his background. Notice also that our results do not depend on the presence or not of an isotensor coupling in P33(1236).
Volume 41B, number 3
PHYSICS LETTERS
expected from the duality hypothesis t . Using now in (9) fZ' = 12 we calculate the integral appearing in CR. The result is presented in the first column of table 1, where the contributions o f the other terms are also given. As we see from table 1, the high energy contribution to CR is exactly what is needed to make this sum rule satisfied. We emphasize that our F E S R method determines both the sign and magnitude o f the high energy contribution to CR. We also remark that, since the integral in (1) emphasizes the low energy region, it should not depend critically on the detailed form of o(w) for high ¢o. This makes us expect that CR should still be satisfied even if ~2' is considerably smaller than ~2. Table 1 shows that this is indeed the case. This means that so far as the Cabibbo-Radicati sum rule is concerned Or(W) "averages" a(w) even for laboratory energies o f the photon as low as 0.85 GeV. We now come to the second aim of the present work, namely to provide still another check of VMD. As we see from fig. 2 and eq. (12), VMD tells us that
3011
200
El ,oo |,
t
°
ii t
o.
-1~1 -
ff b
2 October 1972
.200_
([3/MN)3'-P-*'~-P = ( 1/ f2) (filMN)p_p~ p-p Assuming that at t = 0 the main contribution comes from the p trajectory i t , and assuming Reggeized p universality we have that
-300 -
-400 -
MN
f2
MN lrN--lrN
/9
0.1
I
0.3
I
0.5
I
0.7 0.9 (~ (C..,eV)
D
Fig. 1. Input values for tr(to) = 2OT(,),Vp ~I=~) - oT(-rVp --yI=~) as a function of to (continuous line). Broken line gives the Regge asymptotic expression for o(to) when extrapolated to low energies. Assuming now that F E S R is satisfied when* ~2 = 1.73 (i.e., right after the F37(1940 ) resonance) we fred from (7)
~/MN = + 0.423 (GeV) -1-~o(0) .
f24 P
I
(12)
Substituting (12) in (8), we obtain the Regge expression or(co) for 0(¢o) which is the broken line in fig. 1. As it is shown there, at(to ) "averages" tr(¢o), for low ¢o, as * As it is seen from eq. (6), the lowest moment FESR stresses equally the low and the high energy contributions. Thilk~uggests that we should keep ~ as high as possible to make sure that this sum rule is satisfied.
where 7p(t=0) is the coupling of p trajectory in ~rN scattering defined by Barger and Phillips [9]. From eq. (12) and their result for 7(t=0) we find t t t
VMD
VMD
e2
1 r
Fig. 2. Diagrams for the coupling of O trajectory to "charged" photons according to VMD plus universality. ~"Actually the average would have been more convincing if too(w) had been plotted. t t This is reasonable since cuts are usually assumed to be important at higher t values. Moreover at least in the parametrization of Barger and Phillips [9] for ~rN scattering, the p' coupling vanishes at t = 0. t t t In our calculation we used c~p(0) = 0.5. Had we used ap(0) = = 0.55 as they suggest, we would have found f~p/4 = 1.65. 331
Volume 41B, number 3
PHYSICS LETTERS
2 October 1972
Table 1 Contributions to the Cabibbo-Radicati sum rule *
fZ' dFV(q2) q 2=0+ ( t~V ~2+
2 T dq
\~N/
1 I
21r2~-~
dto [2OT(TVp...I=~_)_ OT(,),Vp~I=3)] -~
too 1 ~ dw [20T(~V p - l = ~ ) - trT(~Vp -- /--{) ] = 0
+ 2.2--S
-S
(GeV)
~
1.73 1.50 1.30 1.10 0.85
Iq2=0
Sum of the terms in left-hand side
~o
~ub)
0tb)
(jttb)
(#b)
of the sum rule 0~b)
-1720 -1720 -1720 -1720 -1720
1520 1520 1520 1520 1520
- 48 - 40 - 37 - 54 -112
250 270 290 310 350
2 30 53 56 38
oo
* For the computation of the integral ! , . . . , a Regge parametrization is used (see the text).
f'a/4rr = 1.8 O
(13)
The new Orsay data suggest [10]
f2/4~r = 2.54 -+ 0.23
9
while the old Orsay data [11] were giving [10] .f2/4~r = 2.10 -+ 0.11 We see therefore that, at worse, we have a 30% violation of the VMD plus universality assumption for the coupling of a Reggeized p to two charged photons. Actually, part of this violation may well be due to our assumption that a particular Regge trajectory dominates. Nevertheless we remark that a similar phenomenon is also observed for the couplings of the Pomeranchukon and A 2 singularities to a phonon andavector meson. Indeed the sum rule [5] ~
V16rr(2/f, )2 ~dt0 (Tp--> Vp))(1 +*72) = 1(14)
P,~,~P ,.,,yp ~
V
is violated by 22% when the new Orsay data are used [4]. Above do0/dt(Tp -+ Vp) referes to the vector meson photoproduction cross-section when extrapolated to 332
t = 0 and r/v stands for the ratio of the real to the imaginary part of the forward vector meson production amplitude. In conclusion we summarize our results. We have shown in the present note that: 1) FESR for the forward amplitude of charged photon scattering indicates that the high energy contribution to the Cabibbo-Radicati sum rule is of correct sign and magnitude to make the sum rule exactly satisfied; 2) so far as this sum rule is concerned, the high energy expression for o ( ~ ) averages the low energy data up to an energy as low as 0.85 GeV; 3) comparing our FESR result for charge photon Compton scattering with the one for ~rN scattering, we conclude that the VMD assumption for the coupling of a Reggeized p to two isovector photons is violated by almost 30% and thid is in agreement with what we would have expected in analogy to the situation in photoproduction of vector mesons [4, 5]. I would like to thank Dr. W. Wada for informing me about his work on the high energy contributions to the
Volume 41B, number 3
PHYSICS LETTERS
Cabibbo-Radicati sum rule. This information prompted the present investigation. I would also like to thank Dr. C. Ferro-Fontan for many helpful discussions as well as for supplyinga program to calculate the amplitudes for pion photoproduction. Finally, I am grateful to the CERN Theoretical Study Division for the hospi, tality extended to me while this workwas done.
References [1] N. Cabibbo and L. Radicati, Phys. Letters 19 (1966) 697; J.D. Bjorken, Phys. Rev. 148 (1966) 1467; R.F. Dashen and M. Gell-Mann, Proc. Third Coral Gables Conf. on S._vmmetryprinciples at high energy, eds B. Kursunoglu, A. Pedmutter and I. Sakmar (W.H. Freeman and Co., San Francisco, 1966). [2] F.J. Gilman and H.J. Schnitzer, Phys. Rev. 150 (1966) 1362; S.L. Adler and F.J. Gilman, Phys. Rev. 156 (1967) 1596; F.J. Gilman, Phys. Rev. 167 (1968) 1365; G. Aubrecht and W. Wada, to be published.
2 October 1972
[3] G.C. Fox and D.Z. Freedman, Phys. Rev. 182 (1969) 1628. [4] J.J, Sakurai and D. Schfldknecht, Phys. Letters 40B (1972) 121. [5] L. Stodolsky, Phys. Rev. Letters 18 (1967) 135;, P.G.O. Freund, Nuovo Cimento 44A (1966) 411 ; H. Joos, Phys. Letters 24B (1967) 103; K. Kajantie and J.S. Trefil, Phys. Letters 24B (1967) 106. [6] R.L. Walker, Phys. Rev. 182 (1969) 1729. [7] P. Noelle, W. Pfeil and D. Schwela, Nucl. Phys. B26 (1971) 461. [8] R.C.E. Devenish, D.H. Lyth and W.A. Rankin, Lancaster and Dazesbury Preprints (June 1971). See also: A.W. Rankin, Proc. Daresbury Study Week End (June 1971). For the couplings of the Pn(1750) and F37(1240) we take the values quoted in these proceedings. [9] V. Barger and R.J.N. Phillips, Phys. Rev. 187 (1969) 2210. [10] D. Benaksas et al., Phys. Letters 39B (1972) 289. [11] J.E. Augustin et al., Phys. Letters 28B (1969) 508; Nuovo Cimento Letters 2 (1969) 214.
333
PHYSICS LETTERS
20ctober!1972
S T A T U S O F T H E C A B I B B I O - R A D I C A T I SUM R U L E A N D A NEW T E S T O F V E C T O R M E S O N D O M I N A N C E G.J.
GOUNARIS
CERN, Geneva, Switzerland and Nuclear Research Centre "Democritus" Athens, Greece Received 11 July 1972 Usingfinite energy sum rules the high energy contribution to the Cabibbo-Radicati sum rule is calculated and shown to be of the proper mag~etude and sign, so that this sum rule is exactly satisfied. In addition the coupling of the p Regge trajectory to two charged photons is calculated and found to be in agreement with the predictions of vector meson dominance and universality for the Reggeized p vertex, within the expected accuracy.
It is well known that the Cabibbo-Radicati [I ] (CR) sum rule states that * d F V ( q 2)
/dV ~2
2 - - -0,1 q+~ 21r2a
f
doo a(co) = 0 ¢o '
(1)
mlr+m~/2MN where a(¢o) = aT(7+p) -- aT('),-p) = 2 OT(TVp -->I=~) - OT(TVp +I=3)
,
(2)
and ¢o is the laboratory energy of the photon. When confronted against the experiment this sum rule seems to be rather well satisfied [2, 3]. Stated in more detail the first two terms are respectively -1720/xb and 1520/Jb while the integral gives about - 5 0 / J b ifa cutoffat co = 1.1 GeV is used. We note that this later result comes about after large cancellations between large positive and negative contributions. We conclude therefore that a reasonably small high energy contribution of about 250 lib will explain the remaining discrepancy between the left and right-hand side in 6q. (1). The purpose of the present work is twofold. First we would like to use finite energy sum rules (FESR) to calculate the high energy contribution to CR and see whether it has the sign and magnitude expected to ex• F]/(q2)is the isovector Dixac form factor of the nucleon norrealized to F y ( 0 ) = 1 ;/~V = 3.7.
plain the remaining discrepancy mentioned above. Second we would like to calculate the coupling of a Reggeized p to two "charged" photons. The reason is the following. We know [4,5] that the couplings of the Pomeranchukon singularity and the A 2 trajectory to a photon and a vector meson obey the vector meson dominance (VMD) predictions to within 22%. We want therefore to see whether the coupling of the Reggeized p to two "charged" photons obeys also the VMD predictions to the same accuracy. Below we first give our formalism and then state and discuss our results. Let us call e2F(7-p; ¢o) and e2F(7*p; 6o) the for, ward spin non-flip invariant amplitudes for the processes 7 - P ~ 7 - P and 7*P ~ 7*P, at a photon laboratory energy w. We define F(-)(¢o) = ½{F(7-p; co) - F(7+p; ¢o)} ,
(3)
and demand that F(-)(~o) is an analytic function in the 6o plane with branch points at +w o=+(m
+m~2MN)
Moreover, we demand that it satisfies the crossing relation
and the optical theorem a(~o) = (4,ra/COMN) Im F(-)(w)
,
(4)
where o(¢o) is defined by eq. (2). We expect that for high energies we can always parametrize this amplitude as
329
Volume 41B, number 3
PHYSICS LETTERS
2 October 1972
0(60)= {2O:r(TVp-+ (nN)l=}) - o~TVp -+ (rrN)i={} + 0'(60)
/37r
F(-)(60) ~" F~-)(60) - 2 sin (trap(0))
= %N(60) + 0'(60),
×{
_
,
(5)
where/3 is a constant and the crossing symmetry of the amplitude has been taken into account. Here ap(0) is the intercept of an effective trajectory which describes the contribution of the P trajectory as well as the contributions of associated cuts and lower lying trajectories. Inspired by the data on charge exchange rrN scattering *, we will assume ,vp(0) ~ 0.5. Using eqs. (3)(5) and the lowest moment FERS I2
f
Im
- Fr(-)(60))d60
0 ,
(6)
too
we find a2
/3 _ otp(0) + 1 f 60o(60) d60/~2ap(0)+l MN
(7)
2rr2a ¢O0
where ~ is the cut-off in the FESR. We now express the high energy contribution to the integral appearing in CR in terms of/3/MN. To do this we assume that there exists an energy £Z' such that for 60 > gY we:are allowed to use in 'the afore-mentioned high energy contribution the Regge expression for o(60). From (4) and (5) we find this Regge expression to be
Or(60) = 2rrZa~/MN) (.O~O(0)-1 ,
where OTrN(60) describes the contribution of the lrN channel while 0'(60) comes from all the other channels. For OrrN(CO)we note that Walker's [6] phenomenologieal analysis of the pion photoproduction in terms of the Born term, a few resonances and a background, gives a reasonable description of the data up to energies [6, 7] of 1.1 GeV. We assume that by including into his amplitudes [8] Pl1(1750) and F37(1940 ) we get a good description for OwN(g.O) for 60 ~ 1.73 GeV. We remark that Walker's background is presumably coming mainly from weak s channel resonances which he ignores, as well as from the tails of the u channel resonances. At least this is what is suggested by applying the two component duality hypothesis to the process 7N -+ rrN. We are then led to the assumption that the true background in the process 7N -* aN is coming mainly from the pion pole in the Born term **. Now we turn to o'(w) for which we know how to calculate the contributions of the various resonances but we do not know how to estimate the background. Ignoring therefore hereafter the background for 0'(60), we have that the resonance contribution is o'(60) = ~
2oi(7Vp'+ R 1 ~ rrN)( I't(R1)" - 1 ) \ r ( R 1 + wN)
RI
(g)
and then
(10)
-
(,
rt(R 3) aT(TVp-+R3-* ~N) \ r ( R 3 + ~N) - 1 /]
~
' (11)
R3 1
where the sums over R 1 and R 3 run over all contributing resonances of isospin ~ and ~ respectively *** Here Pt(R ) and P(R -+ rrN) stand for the total and the partial widths to the nN channel, of resonance R. Adding OrrN(60) and o'(60) we have our result for o(60). This result is presumably valid for 60 ~< 1.73 GeV, and it is shown by the continuous line in rigA.
f - 0(60) d60 =
2rr2~ J
60
to O
1 r 0(60) 27r2a J
60
oa O
1 I 2z~2a
1 27r2c~
Or(60) • d (.a,) 60 ~t
°(60) d60 60
/3 (g2')aP(0)-i MN ~p(0)- 1
(9)
60 0
To proceed further we need to know o(60). In order to obtain it we use the following procedure. We split o(60) into two parts * Notice that for ~rN charge exchange scattering the same kind of Regge singularities contribute.
330
This is essentially assumed in the analysis of ref. [8]. We use here the resonances Pn(1470), Sn(1520), Dt3(1520), Dis(1670), Fts(1690), Pn(1750), S3t(1630), F37(1940). The characteristics of these resonances are taken from Rankin's article mentioned in ref. [8]. We use Rankin's values instead of Walker's to make sure that w e also take into account the contributions of the weaker resonances which Walker absorbs into his background. Notice also that our results do not depend on the presence or not of an isotensor coupling in P33(1236).
Volume 41B, number 3
PHYSICS LETTERS
expected from the duality hypothesis t . Using now in (9) fZ' = 12 we calculate the integral appearing in CR. The result is presented in the first column of table 1, where the contributions o f the other terms are also given. As we see from table 1, the high energy contribution to CR is exactly what is needed to make this sum rule satisfied. We emphasize that our F E S R method determines both the sign and magnitude o f the high energy contribution to CR. We also remark that, since the integral in (1) emphasizes the low energy region, it should not depend critically on the detailed form of o(w) for high ¢o. This makes us expect that CR should still be satisfied even if ~2' is considerably smaller than ~2. Table 1 shows that this is indeed the case. This means that so far as the Cabibbo-Radicati sum rule is concerned Or(W) "averages" a(w) even for laboratory energies o f the photon as low as 0.85 GeV. We now come to the second aim of the present work, namely to provide still another check of VMD. As we see from fig. 2 and eq. (12), VMD tells us that
3011
200
El ,oo |,
t
°
ii t
o.
-1~1 -
ff b
2 October 1972
.200_
([3/MN)3'-P-*'~-P = ( 1/ f2) (filMN)p_p~ p-p Assuming that at t = 0 the main contribution comes from the p trajectory i t , and assuming Reggeized p universality we have that
-300 -
-400 -
MN
f2
MN lrN--lrN
/9
0.1
I
0.3
I
0.5
I
0.7 0.9 (~ (C..,eV)
D
Fig. 1. Input values for tr(to) = 2OT(,),Vp ~I=~) - oT(-rVp --yI=~) as a function of to (continuous line). Broken line gives the Regge asymptotic expression for o(to) when extrapolated to low energies. Assuming now that F E S R is satisfied when* ~2 = 1.73 (i.e., right after the F37(1940 ) resonance) we fred from (7)
~/MN = + 0.423 (GeV) -1-~o(0) .
f24 P
I
(12)
Substituting (12) in (8), we obtain the Regge expression or(co) for 0(¢o) which is the broken line in fig. 1. As it is shown there, at(to ) "averages" tr(¢o), for low ¢o, as * As it is seen from eq. (6), the lowest moment FESR stresses equally the low and the high energy contributions. Thilk~uggests that we should keep ~ as high as possible to make sure that this sum rule is satisfied.
where 7p(t=0) is the coupling of p trajectory in ~rN scattering defined by Barger and Phillips [9]. From eq. (12) and their result for 7(t=0) we find t t t
VMD
VMD
e2
1 r
Fig. 2. Diagrams for the coupling of O trajectory to "charged" photons according to VMD plus universality. ~"Actually the average would have been more convincing if too(w) had been plotted. t t This is reasonable since cuts are usually assumed to be important at higher t values. Moreover at least in the parametrization of Barger and Phillips [9] for ~rN scattering, the p' coupling vanishes at t = 0. t t t In our calculation we used c~p(0) = 0.5. Had we used ap(0) = = 0.55 as they suggest, we would have found f~p/4 = 1.65. 331
Volume 41B, number 3
PHYSICS LETTERS
2 October 1972
Table 1 Contributions to the Cabibbo-Radicati sum rule *
fZ' dFV(q2) q 2=0+ ( t~V ~2+
2 T dq
\~N/
1 I
21r2~-~
dto [2OT(TVp...I=~_)_ OT(,),Vp~I=3)] -~
too 1 ~ dw [20T(~V p - l = ~ ) - trT(~Vp -- /--{) ] = 0
+ 2.2--S
-S
(GeV)
~
1.73 1.50 1.30 1.10 0.85
Iq2=0
Sum of the terms in left-hand side
~o
~ub)
0tb)
(jttb)
(#b)
of the sum rule 0~b)
-1720 -1720 -1720 -1720 -1720
1520 1520 1520 1520 1520
- 48 - 40 - 37 - 54 -112
250 270 290 310 350
2 30 53 56 38
oo
* For the computation of the integral ! , . . . , a Regge parametrization is used (see the text).
f'a/4rr = 1.8 O
(13)
The new Orsay data suggest [10]
f2/4~r = 2.54 -+ 0.23
9
while the old Orsay data [11] were giving [10] .f2/4~r = 2.10 -+ 0.11 We see therefore that, at worse, we have a 30% violation of the VMD plus universality assumption for the coupling of a Reggeized p to two charged photons. Actually, part of this violation may well be due to our assumption that a particular Regge trajectory dominates. Nevertheless we remark that a similar phenomenon is also observed for the couplings of the Pomeranchukon and A 2 singularities to a phonon andavector meson. Indeed the sum rule [5] ~
V16rr(2/f, )2 ~dt0 (Tp--> Vp))(1 +*72) = 1(14)
P,~,~P ,.,,yp ~
V
is violated by 22% when the new Orsay data are used [4]. Above do0/dt(Tp -+ Vp) referes to the vector meson photoproduction cross-section when extrapolated to 332
t = 0 and r/v stands for the ratio of the real to the imaginary part of the forward vector meson production amplitude. In conclusion we summarize our results. We have shown in the present note that: 1) FESR for the forward amplitude of charged photon scattering indicates that the high energy contribution to the Cabibbo-Radicati sum rule is of correct sign and magnitude to make the sum rule exactly satisfied; 2) so far as this sum rule is concerned, the high energy expression for o ( ~ ) averages the low energy data up to an energy as low as 0.85 GeV; 3) comparing our FESR result for charge photon Compton scattering with the one for ~rN scattering, we conclude that the VMD assumption for the coupling of a Reggeized p to two isovector photons is violated by almost 30% and thid is in agreement with what we would have expected in analogy to the situation in photoproduction of vector mesons [4, 5]. I would like to thank Dr. W. Wada for informing me about his work on the high energy contributions to the
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Cabibbo-Radicati sum rule. This information prompted the present investigation. I would also like to thank Dr. C. Ferro-Fontan for many helpful discussions as well as for supplyinga program to calculate the amplitudes for pion photoproduction. Finally, I am grateful to the CERN Theoretical Study Division for the hospi, tality extended to me while this workwas done.
References [1] N. Cabibbo and L. Radicati, Phys. Letters 19 (1966) 697; J.D. Bjorken, Phys. Rev. 148 (1966) 1467; R.F. Dashen and M. Gell-Mann, Proc. Third Coral Gables Conf. on S._vmmetryprinciples at high energy, eds B. Kursunoglu, A. Pedmutter and I. Sakmar (W.H. Freeman and Co., San Francisco, 1966). [2] F.J. Gilman and H.J. Schnitzer, Phys. Rev. 150 (1966) 1362; S.L. Adler and F.J. Gilman, Phys. Rev. 156 (1967) 1596; F.J. Gilman, Phys. Rev. 167 (1968) 1365; G. Aubrecht and W. Wada, to be published.
2 October 1972
[3] G.C. Fox and D.Z. Freedman, Phys. Rev. 182 (1969) 1628. [4] J.J, Sakurai and D. Schfldknecht, Phys. Letters 40B (1972) 121. [5] L. Stodolsky, Phys. Rev. Letters 18 (1967) 135;, P.G.O. Freund, Nuovo Cimento 44A (1966) 411 ; H. Joos, Phys. Letters 24B (1967) 103; K. Kajantie and J.S. Trefil, Phys. Letters 24B (1967) 106. [6] R.L. Walker, Phys. Rev. 182 (1969) 1729. [7] P. Noelle, W. Pfeil and D. Schwela, Nucl. Phys. B26 (1971) 461. [8] R.C.E. Devenish, D.H. Lyth and W.A. Rankin, Lancaster and Dazesbury Preprints (June 1971). See also: A.W. Rankin, Proc. Daresbury Study Week End (June 1971). For the couplings of the Pn(1750) and F37(1240) we take the values quoted in these proceedings. [9] V. Barger and R.J.N. Phillips, Phys. Rev. 187 (1969) 2210. [10] D. Benaksas et al., Phys. Letters 39B (1972) 289. [11] J.E. Augustin et al., Phys. Letters 28B (1969) 508; Nuovo Cimento Letters 2 (1969) 214.
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