- Email: [email protected]
Electroproduction of hyperons
.__ f!ii
2s
3 April 1997
-83
EISEYlJZR
PHYSICS
LETTERS B
Physics Letters B 397 (1997) 311-316
Electroproduction
of hyperons
D. Drechsela, M.M. Giannini b BInstitut fiir Kernphysik, Johannes Gutenberg-Universitiit, D-55099 Mainz, Germany b Dipartimento di Fisica and Istituto Nazionale di Fisica Nucleare, Universitci di Geneva. Italy
Received 23 January 1997 Editor: R. Gatto
Abstract We present some estimates for the reaction ep -+ e’C+. At small scattering angles, the differential cross section is completely determined by the experimental data for the radiative decay H+ 4 py. For kinematical values typical of the proposed parity experiment at MAMI, we find a suppression factor of about 5 . lo-l5 as compared to elastic electron scattering. On the basis of conservative models for the transition form factors, this ratio might increase by about a factor of 20 at backward angles. Even higher values could be expected if the presently unknown convection currents should turn out to be large. Due to the parity violation involved in this process, we also expect a large asymmetry for polarized electrons. @ 1997 Published by Elsevier Science B.V.
1. Introduction The weak radiative decay Z+ -+ py has been studied experimentally and theoretically for nearly fouty years (see, e.g., Refs. [l-3] for reviews). The first events were seen in bubble chambers [4]. The low branching ratio r, /r M 10m3 was to be expected, but the observation of an angular asymmetry ay M -1 came as a big surprise. Due to the production mechanism, the hyperons are largely polarized, and the asymmetry appears as function of the angle between the hyperon’s polarization vector and the recoil momentum of the proton. The observation of such a large asymmetry was puzzling, indeed, because ay was expected to vanish in W(3) for CP invariant and lefthanded weak currents [ 51. Over the years the experimental data have considerably improved, the most recent information coming from experiment E761 at the FNAL high-energy hyperon beam. The new values I’,,/T = (1.19 f 0.08) . 1O-3 and +, = -0.72 f 0.10
are obtained with the statistics of some 30.000 events [ 61. The Particle Data Group presently lists Tr/T = ( 1.23 & 0.05) . 10e3 and c+, = -0.76 f 0.08 [ 71, and we will use these values for our numerical calculations. Despite of considerable efforts, a complete theoretical understanding of the weak radiative decays is still lacking. In fact, these decays have been called “the last low q* frontier of weak interaction physics” [ 21. Since the s-quark in the Xf has to change into a d-quark, the reaction requires the presence of the flavour-changing W+ gauge boson as a catalyzer in the intermediate state. A variety of diagrams can contribute to the process, e.g. one-, two- and three-quark as well as “penguin” diagrams [ 31, and the present consensus among theorists is that none of these diagrams can be neglected. It also seems that baryon intermediate states (i.e. multi-quark configurations) play an important role such that a complete calculation would have to be performed in the framework
0370.2693/97/$17.00 0 1997 Published by Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00172-X
312
D. Drechsel, M.M. Giannini/Physics
of chiral perturbation theory. In this context it is interesting to note that the relevance of chiral symmetry for these reactions was realized many years ago [ 81. In any case, weak radiative decays will continue to be of interest, because they combine elements of the strong, weak and electromagnetic interactions in a rather unique way. Therefore it has been recently suggested to study the inverse reaction ep --) e’Z,+ [ 91. In view of the huge counting rates necessary to determine the strangeness content of the nucleon, it had been suspected that such parity-violating experiments might measure the electroproduction of hyperons as a by-product. We therefore have calculated this reaction on the basis of the available data for the radiative decay. Unfortunately, forward electroproduction is even more suppressed than previously expected, because the leading contribution is due to a magnetization current, and hence vanishes at zero momentum transfer. The situation improves with increasing scattering angle, but at the expense of decreasing cross sections for both the elastic scattering and, most probably, the excitation process. While drafting this paper, we became aware of a similar calculation by Jin and Jaffe [ lo]. Our final results agree with that reference, though we have used a somewhat different notation. In addition, we evaluated the asymmetry for polarized electrons, which turns out to be quite large at energies of the order of 1 GeV. We have also modeled the dependence of the reaction on momentum transfer by introducing form factors similar to elastic scattering. Since the convection current is not determined by the radiative decay, there is some freedom which, in an optimistic scenario, could increase the counting rates considerably. This contribution is organized as follows. In Section 2 we discuss the kinematics, the transition currents and the form factors involved. Section 3 contains the calculation for the cross sections. Finally, we present our numerical results in Section 4, together with some brief discussion.
Letters B 397 (1997) 311-316
q’ =
(k -
sin*f ,
k’)* = -Q* = -4E’E
(1)
where cos 8 = k’_ k/ 1k’ 1 - 1 k 1defines the scattering angle, and E’ and E are the electron energies in the lab frame. The virtual photon is absorbed by a proton with momentum pp, leading to a positively charged sigma with momentum pz = pp + q. Furthermore we define P = (& -t-pP)/2, P = P - (P . q)q/q* and K = (k + k’) /2. Since the proton and the sigma are on shell, p: = m* and pz = M*, the energy of the outgoing electron is
E-(M2-m2)/(2m) 1 + 2(E/m) sin*(0/2>
E,_
The transition spin and parity tures yfi, Pp, qp, $ ( yPcyp - yI’yP). shell, the Gordon
(2)
’
current between two particles with l/2+ has the 5 Lorentz struccrpyqv, and uflvPv, where aF, = Since both particles are on the mass identity provides the relations (3)
2P, = (M + ml y, - ic,,q”, qp = -2ia,,Pv
+ (M - m) yp .
From the radiative decay of the Z we know, however, that there is a strong contribution of the hadronic background with spin and parity l/2-. Therefore, we have to add an axial current with the 5 Lorentz structures y5yP, etc. The corresponding Gordon identities (3) appear with an additional ys in front, and the terms with M change sign. The available information from radiative decay determines the structures up,,qy, and for reasons of symmetry we express the current in terms of the six structures
Jp = (Al + +
(Cl
y5A2)P,
(h
+ ‘Y5B2kfi
(4)
+ Y5c2)~/wqy.
Current conservation q.
+
requires
J = (A, +ysAdP.q+
(& +Wdq*
= 0,
(5)
leading to our final result, 2. Kinematics and transition currents The four-momenta of the incoming and outgoing electrons are denoted by k = (E, k) and k’ = (E’, k') , respectively. In the relativistic limit of m, < E, the 4-momentum transfer is
Jp = UMQ*> +75&Q*))& + (GdQ2)
+ rs&(Q*))
s
.
Obviously, both the first term (a convection current) and the second term (a spin current) are independently
D. Drechsel, M.M. Giannini/Physics
gauge invariant. We stress that Eq. (6) describes a purely electromagnetic interaction, i.e. it represents the vector current seen by the virtual photon. The appearance of axial structures in the current is due to the fact that this is a flavour changing process, which needs the exchange of a W* in addition to the photon. As a consequence the transition current is suppressed by a factor of about lo-l3 compared to the expectation value of the current in the nucleon. The weak neutral current, i.e. the exchange of a P, is not expected to change the flavour either. Again it will take the exchange of a 2” and a Wf loop for the reaction under consideration, thus reducing the transition probability by many more orders of magnitude in comparison with photoexcitation. Finally, we point out that the form factors in Eq. (6) are complex functions of Q* only, because both the initial and the final states are on the mass shell. In order to get information on the multipole structure of the transition current, we expand the current operator JP = (p, j) in inverse powers of the masses p=morM, (M+m)j=
(Ti:)*
M2-mm2 2mM (nxa)G~
+(M-m)&~+i (M-m)3 mMQ*
(G~--ff$Fc)n
2M-mq’
(Coulomb monopole form factor Fc) and a moving magnetic moment (magnetic dipole GM). The second term is a spin current (electric dipole GM), followed by a magnetization current (magnetic dipole GM). The last term, which is strongly suppressed by the mass factors, is the current due to the motion of a pseudoscalar structure (anapole moment PC). The current simplifies considerably in the lab frame, where ?I = -mq/( M - m),
CM+ m)jlab= +(M-m)&o+il(axq)GM M-m
--
4M2
Mfm
4mM-Fc(a.q)q, Q*
(10)
where all vectors have to be expressed in the lab frame. It is again obvious that both FC (Q*) and Fc (Q*) have to vanish with Q* in the real photon limit. Since 1q I+ M -m = A in that limit, the leading terms in Eq. ( 10) are the electric and magnetic dipoles, both linear in A. The first term is suppressed by an additional factor A, and the anapole moment (PC) appears with a factor As.
3. Cross sections
Fc(a.n)n,
In the notation of Bjorken and Drell [ 111, the invariant matrix element for the weak decay Z+ -+ p +y is
where ,=p_l”+rn
313
Letters B 397 (1997) 311-316
(8)
The leading terms of the charge operator may be constructed from Eq. (7) via gauge invariance, (9) because 40 = M - m + 0( ,L-’ ) . Since the transition current has to remain finite in the real photon limit, Q* -+ 0, the form factors FC (Q*) and & (Q*) have to vanish in that limit as -Q2(1-*)/6. The squares of the rms radii, (r2), could differ for the two form factors, but are expected to scale as (m + M) -*. The first structure on the rhs of Eq. (7) is a convection current due to a moving transition charge
M = iii,,Jzux . Ed,
(11)
where Ed is the polarization vector of the photon and JP the current operator of Eq. (6). The differential decay rate for a polarized Zf is dw z
=
Tr -&l
+cu,cose,)
,
(12)
with t9, the angle between the polarization vector Z of the sigma and the momentum of the outgoing proton, cos 0, = (pP . Z) / 1 pP /. The asymmetry is given by the interference of the vector with the axial vector current, ~ =
’
2
Re(GXO)~dO)) I GM(O) I* + I GM(O)I* ’
(13)
314
D. Drechsel, M.M. Giannini/Physics
and the photodecay I
width by
Letters B 397 (1997) 311-316
with the current operator of Eq. (6) yields the structure functions
_(M-@(Mim) Y
4mMWl(Q*)
2M3 x (I GM(O) I2 + (~~(0)
I*) 7
(14) +
with a = e2/4n M l/137. Since the decay photon is real, all form factors have been evaluated at Q2 = 0. Furthermore, the contributions of the convection (Fc ) and induced pseudovector (Fc) currents vanish in that limit. The total decay width of the Z is obtained from the lifetime CT = 2.4. 1013 fm [7] as Iror = /?c/cr = 8.2. 10w6eV The branching ratio is Ir/Iror = 1.23. 10w3, i.e. I,, = 1.01 + 10-8eV. Together with the measured value for the asymmetry, c+ = -0.76, we find
1GM(O) I2 + 1 c&(O) I*=s” + g* = 1.39. 10-13, 2Re (GL(O)GM(O))
= 2ggcos6
= -1.05.
10-13. (15)
In this equation we have written the complex form factors as absolute value and phase, GM (0) = geiY etc. The experimental finding is that GM and GM have the same magnitude and essentially opposite phase. More precisely, if the ratio x = g/g = 1, the phase difference M 140°. The kinematical limit of S = 180” S=T-y is reached for x = 0.46, and for the inverse value x = 2.17. The cross section for the reaction e + p -+ e’ + Z+ can be expressed in terms of 3 structure functions Wj. These structure functions are defined by 1
W PCLU = 2
EF’)IPUf))*
=itr ( =
+
(~~‘)I,U~))
C( sfs h++ 2M
dp+mr PL2m
Wt+ -g/U + z!& 4* ) (
4&p,, P”qp (M+m)* W3’
I2
I &I
4mMW(Q*)
4P&
+Q”)
M+m12+Q2)
= +Q2 (I GM I2+ 1GMI*)
+ (CM- mj2+ Q') 1PCI2 -2Q2Re
(G&Fc + G&PC)
4mMWs(Q*)
= -(M*
,
- m*)Re
(G&X&).
&l/2, f
P+me
--YF
VP = &
e
zrn,
yv
((4*&L, - qpqv) + 4&K”
f2&p,,q”
K?) .
(18)
The differential cross section is obtained by evaluating the usual phase space factors, following the rules of Bjorken and Drell [ 111, and multiplying with the contraction of the tensors W,, and $$ The first two terms of the tensors in Eqs. ( 16) and ( 18) are symmetrical and the last one is antisymmetrical under the exchange of p and v. The final result is
8m *(M+m)*
(16)
with FP = yc,I’~ ya, the symmetrical tensor gP, (gaa = 1) and the antisymmetrical Levi-Civita tensor 1). All three contributions to the E&U (E0123 = hadronic tensor are individually gauge-invariant. A straightforward calculation of the traces of Eq. (16)
(17)
In particular we note that the last contribution in Eq. (16), which is antisymmetrical under the exchange of (u and v, is due to parity violations in this weak process. In a similar way we evaluate the leptonic tensor for relativistic electrons with helicity
4m2 (M+m)2 W2
,
+ (CM+ ml*+ Q">) 1FC I2
> (A4 + m)*
=I GM I* ((M--m)*
(E’ + E) tan2 t W3 ,
w2 (19)
where
The two signs in Eq. ( 19) refer to the helicity of the incoming electrons, i.e. the +( -) sign has to be used
D. Drechsel, M.M. Giannini/Physics
for electrons 3-momentum.
with spin parallel
(antiparallel)
to the
4. Results and conclusions At cross cess. order peron Rf
forward scattering angles we may express the sections by the values known from the decay proUsing Eq. ( 19) and neglecting terms of higher in 8, we obtain for the ratio between weak hyproduction and elastic scattering
=
WV dv(ep
j
+ e/x+) = R1 + R2 & R3 -+ e’p’)
r,k13 sin2 812 + ((M2 - m2)2 + 8m2E’E - m2)3
am2(M2
q=2m(M2 - m2)(E’ +
E)a,} ,
R* (E = 850 MeV) 1.9) . 10-14sin2:
(0 530’))
R*(E=4GeV) = (O.O+ 1.8hO.l).
10-t2sin2
i
(0 5 10”). (22)
In particular, we expect an asymmetry due to strong admixtures of negative parity in the transition. At energies in the GeV region, electrons with positive helicity are more likely to produce a Z+, the asymmetry being of the order of 30%. In order to model the transition at larger scattering angles and momentum transfers, we assume the following behaviour of the form factors:
GdQ2>
= geiYhp(~),
Fc ( Q2) = feiP
&&tip(T).
315
Table 1 The contributions of the 3 structure functions to the ratio of hyperon electroproduction and elastic scattering, together with the corresponding ratios for electrons with positive and negative helicity (see l3.q. (21) for definition). The values without brackets are obtained with f = g, j; = 2, the values with brackets are for f = -log, f= -102. All ratios R are. given in units of lo-l5 E[GeV] 0.85 0.85 0.85 0.85 4.0 4.0 4.0 4.0 4.0
0[deg] 3.5 90 90 150 15 35 90 90 1.50
RI 1 25 (25) 133 1 6 66 (66) 154
R+
R-
R2
R3
5 16
2 19
8 60
4
(18) 72 1 5 23 (23) 45
(186) 211 19 39 108 (3408) 202
( 14;: 67 16 28 62 (3363) 112
(143) 6 17 27 19 (3319) 3
(21)
whereE’=E-(M2-m2)/(2m) inthatlimit.The3 terms on the rhs of Eq. (2 1) correspond to the leading contributions of the response functions WI, W2, and Ws, in that order. It is obvious that W2 dominates at the forward angles, while WI and Ws are suppressed by factors involving the mass difference M - m. Specifically we obtain the estimates
= (0.5 + 6.04
Letters B 397 (1997) 311-316
(23)
where T = Q2/(m + M)2 and FhP = (1-t 4.96~)-~, corresponding to the familiar dipole form factor for
elastic scattering (M = m). We note that this ansatz corresponds to a transition radius of about 0.7 fm, somewhat smaller than the proton radius because of the heavier s-quark involved. The condition FC (0) = 0 has been implemented by multiplying the dipole form factor with an additional factor of 7. This leads to a similar shape of the form factor as in the case of the electric neutron form factor (apart from the suppression factor f). The axial vector form factors are defined as in E&s. (23), with GM -+ GM, g -+ g, y + 7 etc. Furthermore, we assume g/g = 1.25, correspondingto6=7--_=141’,f=g,j:=~,~= y, and 6 = 7. Our result at 850 MeV and 35O is somewhat larger than the value of Ref. [lo], however we find complete agreement with the analytical results of that reference for forward scattering. In the same kinematics, our small angle expansion (21) agrees with the model calculation of Table 1 within about 0.5. 10-15. At the higher electron energy of 4 GeV, this expansion is only valid for 8 5 10”. While R2 turns out to be relatively constant as function of 19, RI and R3 increase quite strongly with the scattering angle. This is essentially due to the dependence of these ratios on tan2 8/2 (see Eq. ( 19) ). For 850 MeV and 150°, e. g., we predict ratios of 2. lo-l3 and 0.7. lo-l3 for electrons with positive and negative helicity, respectively, and an asymmetry of about 50%. Clearly the strong relative increase of the signal at backward angles is at the expense of smaller cross sections for both elastic
316
D. Drechsel, M.M. Giannini/Physics
and inelastic scattering, and even at those angles the signal to noise ratio is still rather tiny. Some results obtained from this model are shown in Table 1 for typical kinematical situations. We have listed the ratios RI, Rz, R3, R+ and R-, as defined by Eq. (21). The very strong asymmetry for the scattering of polarized electrons turns out to be a unique feature of hyperon electroproduction. We add that our estimates are obtained by rather conservative assumptions on the form factors Fc and PC. As may be seen from Eq. ( 17), only the structure function W2 depends on these quantities anyway. Moreover, the contribution of these convection currents to W2 is suppressed by an additional factor Q2. Altogether, the cross sections of Table 1 will change by less than lo%, even at backward angles, if FC = FC = 0. Since nothing is known about these form factors, one could also take a more optimistic view and assume that they might be considerably larger, e.g., f = -log, j: = -log. This would lead to an increase of R+ by a factor 3 for 850 MeV and 90”, while the same ratio at 4 GeV would be about 3 +10-12. Finally we have studied the dependence of the reaction on the form factors. With our model parameters, but with the most radical and certainly unrealistic assumption to replace F+ by unity for the inelastic events, the ratio R+ at 850 MeV increases by factors of 2 and 10 at B = 35” and 150”, respectively. The corresponding factors at 4 GeV would be 3 1 lo2 and 3 . 103, respectively. We conclude that the electroproduction of hyperons is strongly suppressed both by the small width of radiative hyperon decay and a factor of r relative to elastic scattering. The ratio of hyperon production and
Letters B 397 (1997) 311-316
elastic scattering is of the order of lo-l5 at forward and lo-l3 at backward angles. These values would increase in scenarios with large convective transition currents. The reaction is expected to show a strong asymmetry for polarized electrons, which might reach 50% or more. This distinct signature should help to identify the reaction if the severe problem of the huge suppression factors can ever be solved.
Acknowledgments We would like to thank Dietrich von Harrach for suggesting this exercise to us, and we are indebted to both him and Frank Maas for useful conversations. We also gratefully acknowledge the support of the Deutsche Forschungsgemeinschaft (SFB 201) and of the Istituto Nazionale di Fisica Nucleare.
References [l] R.E. Behrends, Phys. Rev. 111 (1958)
1691. B.L. Roberts, Nucl. Phys. A 479 (1988) 75~. B. Bassalleck, Nucl. Phys. A 585 (1995) 25%. L.K. Gershwin et al., Phys. Rev. 188 (1969) 2077. Y. Ham, Phys. Rev. Lett. 12 (1964) 378. M. Foucher et al., Phys. Rev. Lett. 68 (1992) 3004. R.M. Barnett et al. (Particle Data Group), Phys. Rev. D 54 (1996) 1. [8] B.R. Holstein, Nuovo Cimento A 2 (1971) 561. [ 91 D. von Harrach and FE. Maas (private communications). [lo] X. Jin and R.L. Jaffe, hep-th/9612316. [ 111 J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New-York, 1964). [2] [3] [4] [5] [6] [7]
2s
3 April 1997
-83
EISEYlJZR
PHYSICS
LETTERS B
Physics Letters B 397 (1997) 311-316
Electroproduction
of hyperons
D. Drechsela, M.M. Giannini b BInstitut fiir Kernphysik, Johannes Gutenberg-Universitiit, D-55099 Mainz, Germany b Dipartimento di Fisica and Istituto Nazionale di Fisica Nucleare, Universitci di Geneva. Italy
Received 23 January 1997 Editor: R. Gatto
Abstract We present some estimates for the reaction ep -+ e’C+. At small scattering angles, the differential cross section is completely determined by the experimental data for the radiative decay H+ 4 py. For kinematical values typical of the proposed parity experiment at MAMI, we find a suppression factor of about 5 . lo-l5 as compared to elastic electron scattering. On the basis of conservative models for the transition form factors, this ratio might increase by about a factor of 20 at backward angles. Even higher values could be expected if the presently unknown convection currents should turn out to be large. Due to the parity violation involved in this process, we also expect a large asymmetry for polarized electrons. @ 1997 Published by Elsevier Science B.V.
1. Introduction The weak radiative decay Z+ -+ py has been studied experimentally and theoretically for nearly fouty years (see, e.g., Refs. [l-3] for reviews). The first events were seen in bubble chambers [4]. The low branching ratio r, /r M 10m3 was to be expected, but the observation of an angular asymmetry ay M -1 came as a big surprise. Due to the production mechanism, the hyperons are largely polarized, and the asymmetry appears as function of the angle between the hyperon’s polarization vector and the recoil momentum of the proton. The observation of such a large asymmetry was puzzling, indeed, because ay was expected to vanish in W(3) for CP invariant and lefthanded weak currents [ 51. Over the years the experimental data have considerably improved, the most recent information coming from experiment E761 at the FNAL high-energy hyperon beam. The new values I’,,/T = (1.19 f 0.08) . 1O-3 and +, = -0.72 f 0.10
are obtained with the statistics of some 30.000 events [ 61. The Particle Data Group presently lists Tr/T = ( 1.23 & 0.05) . 10e3 and c+, = -0.76 f 0.08 [ 71, and we will use these values for our numerical calculations. Despite of considerable efforts, a complete theoretical understanding of the weak radiative decays is still lacking. In fact, these decays have been called “the last low q* frontier of weak interaction physics” [ 21. Since the s-quark in the Xf has to change into a d-quark, the reaction requires the presence of the flavour-changing W+ gauge boson as a catalyzer in the intermediate state. A variety of diagrams can contribute to the process, e.g. one-, two- and three-quark as well as “penguin” diagrams [ 31, and the present consensus among theorists is that none of these diagrams can be neglected. It also seems that baryon intermediate states (i.e. multi-quark configurations) play an important role such that a complete calculation would have to be performed in the framework
0370.2693/97/$17.00 0 1997 Published by Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00172-X
312
D. Drechsel, M.M. Giannini/Physics
of chiral perturbation theory. In this context it is interesting to note that the relevance of chiral symmetry for these reactions was realized many years ago [ 81. In any case, weak radiative decays will continue to be of interest, because they combine elements of the strong, weak and electromagnetic interactions in a rather unique way. Therefore it has been recently suggested to study the inverse reaction ep --) e’Z,+ [ 91. In view of the huge counting rates necessary to determine the strangeness content of the nucleon, it had been suspected that such parity-violating experiments might measure the electroproduction of hyperons as a by-product. We therefore have calculated this reaction on the basis of the available data for the radiative decay. Unfortunately, forward electroproduction is even more suppressed than previously expected, because the leading contribution is due to a magnetization current, and hence vanishes at zero momentum transfer. The situation improves with increasing scattering angle, but at the expense of decreasing cross sections for both the elastic scattering and, most probably, the excitation process. While drafting this paper, we became aware of a similar calculation by Jin and Jaffe [ lo]. Our final results agree with that reference, though we have used a somewhat different notation. In addition, we evaluated the asymmetry for polarized electrons, which turns out to be quite large at energies of the order of 1 GeV. We have also modeled the dependence of the reaction on momentum transfer by introducing form factors similar to elastic scattering. Since the convection current is not determined by the radiative decay, there is some freedom which, in an optimistic scenario, could increase the counting rates considerably. This contribution is organized as follows. In Section 2 we discuss the kinematics, the transition currents and the form factors involved. Section 3 contains the calculation for the cross sections. Finally, we present our numerical results in Section 4, together with some brief discussion.
Letters B 397 (1997) 311-316
q’ =
(k -
sin*f ,
k’)* = -Q* = -4E’E
(1)
where cos 8 = k’_ k/ 1k’ 1 - 1 k 1defines the scattering angle, and E’ and E are the electron energies in the lab frame. The virtual photon is absorbed by a proton with momentum pp, leading to a positively charged sigma with momentum pz = pp + q. Furthermore we define P = (& -t-pP)/2, P = P - (P . q)q/q* and K = (k + k’) /2. Since the proton and the sigma are on shell, p: = m* and pz = M*, the energy of the outgoing electron is
E-(M2-m2)/(2m) 1 + 2(E/m) sin*(0/2>
E,_
The transition spin and parity tures yfi, Pp, qp, $ ( yPcyp - yI’yP). shell, the Gordon
(2)
’
current between two particles with l/2+ has the 5 Lorentz struccrpyqv, and uflvPv, where aF, = Since both particles are on the mass identity provides the relations (3)
2P, = (M + ml y, - ic,,q”, qp = -2ia,,Pv
+ (M - m) yp .
From the radiative decay of the Z we know, however, that there is a strong contribution of the hadronic background with spin and parity l/2-. Therefore, we have to add an axial current with the 5 Lorentz structures y5yP, etc. The corresponding Gordon identities (3) appear with an additional ys in front, and the terms with M change sign. The available information from radiative decay determines the structures up,,qy, and for reasons of symmetry we express the current in terms of the six structures
Jp = (Al + +
(Cl
y5A2)P,
(h
+ ‘Y5B2kfi
(4)
+ Y5c2)~/wqy.
Current conservation q.
+
requires
J = (A, +ysAdP.q+
(& +Wdq*
= 0,
(5)
leading to our final result, 2. Kinematics and transition currents The four-momenta of the incoming and outgoing electrons are denoted by k = (E, k) and k’ = (E’, k') , respectively. In the relativistic limit of m, < E, the 4-momentum transfer is
Jp = UMQ*> +75&Q*))& + (GdQ2)
+ rs&(Q*))
s
.
Obviously, both the first term (a convection current) and the second term (a spin current) are independently
D. Drechsel, M.M. Giannini/Physics
gauge invariant. We stress that Eq. (6) describes a purely electromagnetic interaction, i.e. it represents the vector current seen by the virtual photon. The appearance of axial structures in the current is due to the fact that this is a flavour changing process, which needs the exchange of a W* in addition to the photon. As a consequence the transition current is suppressed by a factor of about lo-l3 compared to the expectation value of the current in the nucleon. The weak neutral current, i.e. the exchange of a P, is not expected to change the flavour either. Again it will take the exchange of a 2” and a Wf loop for the reaction under consideration, thus reducing the transition probability by many more orders of magnitude in comparison with photoexcitation. Finally, we point out that the form factors in Eq. (6) are complex functions of Q* only, because both the initial and the final states are on the mass shell. In order to get information on the multipole structure of the transition current, we expand the current operator JP = (p, j) in inverse powers of the masses p=morM, (M+m)j=
(Ti:)*
M2-mm2 2mM (nxa)G~
+(M-m)&~+i (M-m)3 mMQ*
(G~--ff$Fc)n
2M-mq’
(Coulomb monopole form factor Fc) and a moving magnetic moment (magnetic dipole GM). The second term is a spin current (electric dipole GM), followed by a magnetization current (magnetic dipole GM). The last term, which is strongly suppressed by the mass factors, is the current due to the motion of a pseudoscalar structure (anapole moment PC). The current simplifies considerably in the lab frame, where ?I = -mq/( M - m),
CM+ m)jlab= +(M-m)&o+il(axq)GM M-m
--
4M2
Mfm
4mM-Fc(a.q)q, Q*
(10)
where all vectors have to be expressed in the lab frame. It is again obvious that both FC (Q*) and Fc (Q*) have to vanish with Q* in the real photon limit. Since 1q I+ M -m = A in that limit, the leading terms in Eq. ( 10) are the electric and magnetic dipoles, both linear in A. The first term is suppressed by an additional factor A, and the anapole moment (PC) appears with a factor As.
3. Cross sections
Fc(a.n)n,
In the notation of Bjorken and Drell [ 111, the invariant matrix element for the weak decay Z+ -+ p +y is
where ,=p_l”+rn
313
Letters B 397 (1997) 311-316
(8)
The leading terms of the charge operator may be constructed from Eq. (7) via gauge invariance, (9) because 40 = M - m + 0( ,L-’ ) . Since the transition current has to remain finite in the real photon limit, Q* -+ 0, the form factors FC (Q*) and & (Q*) have to vanish in that limit as -Q2(1-*)/6. The squares of the rms radii, (r2), could differ for the two form factors, but are expected to scale as (m + M) -*. The first structure on the rhs of Eq. (7) is a convection current due to a moving transition charge
M = iii,,Jzux . Ed,
(11)
where Ed is the polarization vector of the photon and JP the current operator of Eq. (6). The differential decay rate for a polarized Zf is dw z
=
Tr -&l
+cu,cose,)
,
(12)
with t9, the angle between the polarization vector Z of the sigma and the momentum of the outgoing proton, cos 0, = (pP . Z) / 1 pP /. The asymmetry is given by the interference of the vector with the axial vector current, ~ =
’
2
Re(GXO)~dO)) I GM(O) I* + I GM(O)I* ’
(13)
314
D. Drechsel, M.M. Giannini/Physics
and the photodecay I
width by
Letters B 397 (1997) 311-316
with the current operator of Eq. (6) yields the structure functions
_(M-@(Mim) Y
4mMWl(Q*)
2M3 x (I GM(O) I2 + (~~(0)
I*) 7
(14) +
with a = e2/4n M l/137. Since the decay photon is real, all form factors have been evaluated at Q2 = 0. Furthermore, the contributions of the convection (Fc ) and induced pseudovector (Fc) currents vanish in that limit. The total decay width of the Z is obtained from the lifetime CT = 2.4. 1013 fm [7] as Iror = /?c/cr = 8.2. 10w6eV The branching ratio is Ir/Iror = 1.23. 10w3, i.e. I,, = 1.01 + 10-8eV. Together with the measured value for the asymmetry, c+ = -0.76, we find
1GM(O) I2 + 1 c&(O) I*=s” + g* = 1.39. 10-13, 2Re (GL(O)GM(O))
= 2ggcos6
= -1.05.
10-13. (15)
In this equation we have written the complex form factors as absolute value and phase, GM (0) = geiY etc. The experimental finding is that GM and GM have the same magnitude and essentially opposite phase. More precisely, if the ratio x = g/g = 1, the phase difference M 140°. The kinematical limit of S = 180” S=T-y is reached for x = 0.46, and for the inverse value x = 2.17. The cross section for the reaction e + p -+ e’ + Z+ can be expressed in terms of 3 structure functions Wj. These structure functions are defined by 1
W PCLU = 2
EF’)IPUf))*
=itr ( =
+
(~~‘)I,U~))
C( sfs h++ 2M
dp+mr PL2m
Wt+ -g/U + z!& 4* ) (
4&p,, P”qp (M+m)* W3’
I2
I &I
4mMW(Q*)
4P&
+Q”)
M+m12+Q2)
= +Q2 (I GM I2+ 1GMI*)
+ (CM- mj2+ Q') 1PCI2 -2Q2Re
(G&Fc + G&PC)
4mMWs(Q*)
= -(M*
,
- m*)Re
(G&X&).
&l/2, f
P+me
--YF
VP = &
e
zrn,
yv
((4*&L, - qpqv) + 4&K”
f2&p,,q”
K?) .
(18)
The differential cross section is obtained by evaluating the usual phase space factors, following the rules of Bjorken and Drell [ 111, and multiplying with the contraction of the tensors W,, and $$ The first two terms of the tensors in Eqs. ( 16) and ( 18) are symmetrical and the last one is antisymmetrical under the exchange of p and v. The final result is
8m *(M+m)*
(16)
with FP = yc,I’~ ya, the symmetrical tensor gP, (gaa = 1) and the antisymmetrical Levi-Civita tensor 1). All three contributions to the E&U (E0123 = hadronic tensor are individually gauge-invariant. A straightforward calculation of the traces of Eq. (16)
(17)
In particular we note that the last contribution in Eq. (16), which is antisymmetrical under the exchange of (u and v, is due to parity violations in this weak process. In a similar way we evaluate the leptonic tensor for relativistic electrons with helicity
4m2 (M+m)2 W2
,
+ (CM+ ml*+ Q">) 1FC I2
> (A4 + m)*
=I GM I* ((M--m)*
(E’ + E) tan2 t W3 ,
w2 (19)
where
The two signs in Eq. ( 19) refer to the helicity of the incoming electrons, i.e. the +( -) sign has to be used
D. Drechsel, M.M. Giannini/Physics
for electrons 3-momentum.
with spin parallel
(antiparallel)
to the
4. Results and conclusions At cross cess. order peron Rf
forward scattering angles we may express the sections by the values known from the decay proUsing Eq. ( 19) and neglecting terms of higher in 8, we obtain for the ratio between weak hyproduction and elastic scattering
=
WV dv(ep
j
+ e/x+) = R1 + R2 & R3 -+ e’p’)
r,k13 sin2 812 + ((M2 - m2)2 + 8m2E’E - m2)3
am2(M2
q=2m(M2 - m2)(E’ +
E)a,} ,
R* (E = 850 MeV) 1.9) . 10-14sin2:
(0 530’))
R*(E=4GeV) = (O.O+ 1.8hO.l).
10-t2sin2
i
(0 5 10”). (22)
In particular, we expect an asymmetry due to strong admixtures of negative parity in the transition. At energies in the GeV region, electrons with positive helicity are more likely to produce a Z+, the asymmetry being of the order of 30%. In order to model the transition at larger scattering angles and momentum transfers, we assume the following behaviour of the form factors:
GdQ2>
= geiYhp(~),
Fc ( Q2) = feiP
&&tip(T).
315
Table 1 The contributions of the 3 structure functions to the ratio of hyperon electroproduction and elastic scattering, together with the corresponding ratios for electrons with positive and negative helicity (see l3.q. (21) for definition). The values without brackets are obtained with f = g, j; = 2, the values with brackets are for f = -log, f= -102. All ratios R are. given in units of lo-l5 E[GeV] 0.85 0.85 0.85 0.85 4.0 4.0 4.0 4.0 4.0
0[deg] 3.5 90 90 150 15 35 90 90 1.50
RI 1 25 (25) 133 1 6 66 (66) 154
R+
R-
R2
R3
5 16
2 19
8 60
4
(18) 72 1 5 23 (23) 45
(186) 211 19 39 108 (3408) 202
( 14;: 67 16 28 62 (3363) 112
(143) 6 17 27 19 (3319) 3
(21)
whereE’=E-(M2-m2)/(2m) inthatlimit.The3 terms on the rhs of Eq. (2 1) correspond to the leading contributions of the response functions WI, W2, and Ws, in that order. It is obvious that W2 dominates at the forward angles, while WI and Ws are suppressed by factors involving the mass difference M - m. Specifically we obtain the estimates
= (0.5 + 6.04
Letters B 397 (1997) 311-316
(23)
where T = Q2/(m + M)2 and FhP = (1-t 4.96~)-~, corresponding to the familiar dipole form factor for
elastic scattering (M = m). We note that this ansatz corresponds to a transition radius of about 0.7 fm, somewhat smaller than the proton radius because of the heavier s-quark involved. The condition FC (0) = 0 has been implemented by multiplying the dipole form factor with an additional factor of 7. This leads to a similar shape of the form factor as in the case of the electric neutron form factor (apart from the suppression factor f). The axial vector form factors are defined as in E&s. (23), with GM -+ GM, g -+ g, y + 7 etc. Furthermore, we assume g/g = 1.25, correspondingto6=7--_=141’,f=g,j:=~,~= y, and 6 = 7. Our result at 850 MeV and 35O is somewhat larger than the value of Ref. [lo], however we find complete agreement with the analytical results of that reference for forward scattering. In the same kinematics, our small angle expansion (21) agrees with the model calculation of Table 1 within about 0.5. 10-15. At the higher electron energy of 4 GeV, this expansion is only valid for 8 5 10”. While R2 turns out to be relatively constant as function of 19, RI and R3 increase quite strongly with the scattering angle. This is essentially due to the dependence of these ratios on tan2 8/2 (see Eq. ( 19) ). For 850 MeV and 150°, e. g., we predict ratios of 2. lo-l3 and 0.7. lo-l3 for electrons with positive and negative helicity, respectively, and an asymmetry of about 50%. Clearly the strong relative increase of the signal at backward angles is at the expense of smaller cross sections for both elastic
316
D. Drechsel, M.M. Giannini/Physics
and inelastic scattering, and even at those angles the signal to noise ratio is still rather tiny. Some results obtained from this model are shown in Table 1 for typical kinematical situations. We have listed the ratios RI, Rz, R3, R+ and R-, as defined by Eq. (21). The very strong asymmetry for the scattering of polarized electrons turns out to be a unique feature of hyperon electroproduction. We add that our estimates are obtained by rather conservative assumptions on the form factors Fc and PC. As may be seen from Eq. ( 17), only the structure function W2 depends on these quantities anyway. Moreover, the contribution of these convection currents to W2 is suppressed by an additional factor Q2. Altogether, the cross sections of Table 1 will change by less than lo%, even at backward angles, if FC = FC = 0. Since nothing is known about these form factors, one could also take a more optimistic view and assume that they might be considerably larger, e.g., f = -log, j: = -log. This would lead to an increase of R+ by a factor 3 for 850 MeV and 90”, while the same ratio at 4 GeV would be about 3 +10-12. Finally we have studied the dependence of the reaction on the form factors. With our model parameters, but with the most radical and certainly unrealistic assumption to replace F+ by unity for the inelastic events, the ratio R+ at 850 MeV increases by factors of 2 and 10 at B = 35” and 150”, respectively. The corresponding factors at 4 GeV would be 3 1 lo2 and 3 . 103, respectively. We conclude that the electroproduction of hyperons is strongly suppressed both by the small width of radiative hyperon decay and a factor of r relative to elastic scattering. The ratio of hyperon production and
Letters B 397 (1997) 311-316
elastic scattering is of the order of lo-l5 at forward and lo-l3 at backward angles. These values would increase in scenarios with large convective transition currents. The reaction is expected to show a strong asymmetry for polarized electrons, which might reach 50% or more. This distinct signature should help to identify the reaction if the severe problem of the huge suppression factors can ever be solved.
Acknowledgments We would like to thank Dietrich von Harrach for suggesting this exercise to us, and we are indebted to both him and Frank Maas for useful conversations. We also gratefully acknowledge the support of the Deutsche Forschungsgemeinschaft (SFB 201) and of the Istituto Nazionale di Fisica Nucleare.
References [l] R.E. Behrends, Phys. Rev. 111 (1958)
1691. B.L. Roberts, Nucl. Phys. A 479 (1988) 75~. B. Bassalleck, Nucl. Phys. A 585 (1995) 25%. L.K. Gershwin et al., Phys. Rev. 188 (1969) 2077. Y. Ham, Phys. Rev. Lett. 12 (1964) 378. M. Foucher et al., Phys. Rev. Lett. 68 (1992) 3004. R.M. Barnett et al. (Particle Data Group), Phys. Rev. D 54 (1996) 1. [8] B.R. Holstein, Nuovo Cimento A 2 (1971) 561. [ 91 D. von Harrach and FE. Maas (private communications). [lo] X. Jin and R.L. Jaffe, hep-th/9612316. [ 111 J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New-York, 1964). [2] [3] [4] [5] [6] [7]