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Electromagnetic form factors of electrodisintegration of light nuclei near threshold
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A 635 (1998) 387-407
Electromagnetic form factors of electrodisintegration of light nuclei near threshold C. Yalqin a, M.E Rekalo a'l, H. Koru b'2 a Physics Department, Middle East Technical University, Ankara, Turkey b Gazi University, Physics Department, Ankara, Turkey
Received 6 June 1997; revised 12 March 1998; accepted 21 March 1998
Abstract The general structure of electromagnetic currents for electrodisintegration of nuclei near threshold, e - + A --* e - + A1 + A2 (A, At and A2 are arbitrary nuclei or hadrons with spins 0, l / 2 and 1), are established. Considering nuclei as elementary particles with the definite values of spin and space parity and using the conservation of hadronic electromagnetic current and P-invariance of hadron electromagnetic interaction, the spin structure of the amplitudes of the processes T* + A ---* A1 + A2 is derived in terms of threshold electromagnetic form factors. The present formalism is valid for any nuclear structure and any reaction mechanism. The structure functions, which characterize the scattering electrons on the polarized target, are calculated as quadratic combinations of the form factors. These structure functions are used for the analysis of the problem of complete experiment for the full determination of all threshold form factors. P-odd part of electromagnetic current, induced by P-odd nuclear forces, is parametrized also in terms of the corresponding form factors. The processes with the following combinations of spins of hadrons e - + 1 --~ e - + 0 + 0 , e - + 1 --~ e - + 1 + 0 , e - + 1/2 --* e - + 1 / 2 + 1 and e - + 1/2 --* e - + 1/2 + 1/2, and with different combinations of space parities of nuclei A, Ai and A2, are examined within this approach. @ 1998 Published by Elsevier Science B.V. PACS: 11.30.Er; 24.70.+s; 25.20.Lj; 25.30.Dh Keywords: Electromagnetic form factors; Polarization effects; Nuclear processes; P-invariance; Structure functions; Electrodisintegration
I Address: National Scientific Center, Kharkov Institute of Physics and Technology, Kharkov, Ukraine. 2 E-mail [email protected]. 0375-9474/98/$19.00 (~) 1998 Published by Elsevier Science B.V. All rights reserved. PH S 0 3 7 5 - 9 4 7 4 ( 9 8 ) 0 0 1 9 3 - 6
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1. Introduction The electromagnetic and weak hadronic currents, which determine the processes of electron and neutrino scattering by hadrons ( e - + A ~ e - + A ( At), u + A ---* u + A (A t), v~ + A ~ /z- + A" where A, At and A" are hadronic states, nucleon resonances and nuclei in ground or excited states, respectively), are parametrized in terms of corresponding form factors (FF). The calculation of differential cross-sections of these processes and the analysis of polarization effects are given in terms of these elastic (or quasielastic) FF's. The symmetry properties of fundamental interactions can be formulated also in terms of the electromagnetic and weak FF's. But electromagnetic current for the more complicated processes, for example, such as e - + A ~ e - + ~ ° + A t , which is equivalent to the process y* ÷ A ~ 7r° + At (y* is the virtual photon) is determined instead of FF's, which depend on the momentum transfer square only, by some set of amplitudes, which depend generally on three invariant kinematical variables. In this paper, we present an analysis of e - + A ~ e - + Ai + A2 types of processes (A, Al and A2 are hadrons and nuclei) in special kinematical conditions, namely at the reaction threshold. More exactly, we define the threshold as the minimal value of the effective mass of produced AI + A2-system, but with arbitrary value of momentum transfer square x 2. In such kinematical conditions the corresponding electromagnetic current is also characterized by the electromagnetic FF's [ 1], which are so fundamental as the electromagnetic FF's of one-particles transitions y* + A ~ A t. Similar to the case of elastic (or quasielastic) scattering, e - + A -+ e - + A ( A t ) , the number of independent electromagnetic FF's for e - + A ---+e - + A1 + A2 (at threshold) is determined by the values of spin and spin parity of A j, A2 and A. The spin structure of electromagnetic current for the processes e - + A --~ e - +Al +A2, which we consider in the framework of one-photon mechanism, can be established in the general form without any assumptions about the structure of hadrons in y* + A ~ AI + A2 and about details of the reaction mechanism. Only the general properties of hadron electromagnetic interaction, such as the P-invariance and the conservation of electromagnetic current (gauge invariance), are used. Detailed properties of hadrons or nuclei are not essential for such an analysis, only the spins and space parities of hadrons A, Al and A2 are important here. Therefore, all results which can be obtained in such a way must be valid and for that case, when A, A1 and A2 are arbitrary nuclei. The exact parametrization of electromagnetic current for threshold electrodisintegration of nuclei (or hadron electroproduction) provides the separation of the kinematical and dynamical aspect of description of electromagnetic processes. This allows one to take into account in the full volume the effects of relativistic kinematics and the conservation of electromagnetic hadron current. This leads to an effective formalism for theoretical analysis of the sensitivity of differential cross-section and different polarization observables, for threshold processes to the behavior of the corresponding wave functions at small distances and to the effects of mesonic exchange currents and isobar configurations in light nuclei [2-8]. In Section 2 we introduce the structure functions (SF) formalism for description of
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the threshold differential cross-section for any processes e - + A ---+ e - + A~ + A2. This formalism reveals the dependence of differential cross-sections on the polarization of target nuclei A with spin J. In Section 3 we parametrize threshold electromagnetic currents for 3'* + A ---, Al + A2, for relevant combinations of the spins of A, Al and A2, for s ~< 1. Details of the calculation of the corresponding SF's in terms of the threshold electromagnetic FF's are given. Methods of realization of complete experiment for different processes e - + A ---* e - + AI ÷ A2 are discussed in Section 4. In Section 5 we study the P-odd effects in e - + A ~ e - + Al ÷ A2, which are induced by the P-odd nuclear forces. We demonstrate how to parametrize the P-odd part of electromagnetic current in terms of the definite number of P-odd electromagnetic FF's. Section 6 is devoted to the calculation of P-odd structure functions in terms of P-odd and P-even threshold electromagnetic FF's.
2. T h e structure f u n c t i o n s for the processes e - + A ---, e - + A1 + A2
As it is known [9], the differential cross-section of any process of the type e - + A e - + A1 + A2 can be written in the framework of the one-photon mechanism as, 0'2 E2 Iql 1 1 dE2 doe dO - 647r 3 El M W 1 - • ( - K 2) d3o "
x
{
Hxx + Hyy + •cos2~o(Hxx - Hyy) - 2e-~oHzz
+•sin2~o(H~y + Hyx) - a
lx/f-Z~-e[x/1 + e(H~y - Hyx)
+ ¢ /2 6 - -( -- K ~ 022) ( c o s ~ o ( H y z - H z y ) -
sin~p(Hxz - H z x ) ) ] } ,
( 1)
1
K2
0e
K0
T"
• -1 = 1 - 2"~gs tan 2
Here, E1 (E2) is the energy of the initial (final) electron in laboratory system (LS), ds2e is the solid angle element for the scattered electron and 0e is the electron scattering angle in LS, dg2 is the solid angle element of one produced hadron in CMS of y* ÷ A A1 ÷ A2 reaction. W is the invariant mass of system A1 + A2, M is the mass of A, g/ is 3-momentum of A1 in CMS, ,~ = ±1, for two possible relative orientations of spin and 3-momentum of initial electron, K2 is square momentum transfer and ~o is azimuthal angle of final electron. The definition of the tensor Hij is given as
Hij = JiJj,
(2)
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where J is the 3-vector of electromagnetic current for the y* ÷ A ---, Al + A2 in the CMS, line in the definition of Hij denotes the summing over polarizations of AI and A2 particles. In Eq. (1) we use the coordinate system with z-axis along the 3-momentum t¢ of the virtual photon, the plane xz coincides with the plane defined by K and q. But at the threshold of (q ~ 0) such a coordinate system definition is not appropriate and the components of Hii depend only on one variable, namely K2. Therefore, for any unpolarized target, it is possible to integrate Eq. ( 1 ) over dO, so the differential cross-section d2~r/dezdOe (at threshold) will contain three contributions, namely, due to Hxx + H~,y, Hzz and ,~(Hx~, - H~.:,) only. But for the polarized target, it is possible to introduce the coordinate system with xz-plane defined by K and the spin vector of target (or direction of corresponding magnetic field). Eq. (1) is valid within the one photon mechanism with the assumption of the conservation of hadronic electromagnetic current for the y* + A -~ AI + A2 process. The other details, such as reaction mechanism for the 3/* + A ---, Al + A2 and the structure of particles A, Al and A2, are not relevant in deriving Eq. (1). Using P-invariance of hadronic electromagnetic interaction, it is possible to establish the dependence of tensor Hij on the polarization properties of target. For example, for the spin-one target (deuteron or 6Li) Hii can be written [10] as follows: (from now we will consider all processes e - + A ---+e - + Al + A2 as being near threshold):
Hi.j = ( 6ij - kik.j ) w j (K 2) + kif~iW2( K 2) + ieijeStw3( K2)
÷if:.ijgK,S " KW4(K2) ÷ ('iabSakbk.i ÷ 6jabSaKbKiOW5(K2) ÷( SabKaKb) [W6(K2) ( C~ij-- ~,f(j ) ÷ fqiK,jW7(K2)[ ÷ SijW8(K2) ÷(SiakakjWSjai'
(3)
where wl - w m are real structure functions, which depend near threshold on K2, only; is unit vector along the momentum of virtual photon, S describes the vector polarization of target, the tensor S,b stands for the quadrupole(or tensor) polarization. In the CMS of y* + A --~ A1 + A2, we have the following properties for Sab; M2
Sab = Sba,
Sxx + Syy + Szz ~ - = 0,
E =
W2 + M 2 _ t¢2 2W '
(4)
where E is the energy of A. Of course, the momentum transfer square K 2 in the threshold region can take any value in space-like region. Eq. (3) is valid also for a target with spin 1/2, but in this case only five SF's arise, Wl(K 2) -- WS(K2) [11]. If the spin of the target is greater than one, then Eq. (3) describes the corresponding contributions into the differential cross-section, induced by the vector and tensor polarizations of the target.
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The SF's ws(K 2) and wl0(x 2) characterize the possible T-odd effects in y* + A A} + A2. Note that T-odd effects in y* + A --~ A1 + A2 are defined by the interference of longitudinal and transversal components of hadron electromagnetic current (for the vector and for tensor polarization of target) independently from the mechanism of Tinvariance violation. For vector polarization, this has been formulated by Christ and Lee [11]. The SF's formalism allows the study of the polarization phenomena for any spins J of target A in cA-collisions [ 12]. In the framework of this formalism the polarization states of a target with spin J are characterized by a set of 2J real tensors T,~...~,,,, n = 1. . . . . 2J, symmetrical in any pair of indexes with zero trace, T~aa~...a, = 0. Each such tensor has ( 2 J + 1) independent components [ 13]. Tensors Ta,...a, with odd number of indexes do not change their sign in space reflections (pseudotensors). To discriminate the tensors with even and odd numbers of indexes, we denote them by Q~...,,, and P~,...,,,, respectively. Taking into account the properties of these P and Q, the general form of the hadronic tensors H U can be written as
a~l +) -- [(~ij -- K.iKj) x' (K2) ÷ KiKjX2(K2)] OaK.a 4- OijX3(K 2) ÷(QiKj ÷ OjKi)X4( K2) ÷ i ( Q i k j - O_.jki)xs( K2), O_.i = Qi(,2...~,, ka2 . . . ka,,
(5)
Oij = Qija~ ...a,,kay.., ka,,,
H!"t.l -) = ieijgkeff~gmkmYl (K2) + i~.ijgPgmY2(K 2) + i (eiabkbPja -- 6jabkbPia) y3(K2) 4- (~:iobkbPja ÷ ~:jobKbPia) Y4(K2) ÷ (KiEjabPaKb -]- Kj'iabPaKb) YS(X2),
(6)
P,j=P,jo3 . o, ...ko,,, where the indexes ( n + ) and ( n - ) indicate the polarization tensors with even and odd numbers of indexes. The contribution of each "irreducible" polarization in the cross-section of an inclusive cA-interaction is characterized by five real SF's. The number of SF's is independent from n (multipolarity of corresponding target polarization). The single exception is the case of vector polarization, n -- 1, where only three SF's appear. In T-invariant hadron electrodynamics, the number of SF's decreases, because the tensors H}ff+) and H I T - ) are symmetric and antisymmetric, respectively. From a set of 10J ( J >/ 1) independent SF's, which characterize the scattering electrons on the polarized target with spin J, only 3J - 1 SF's (if 2J is even number) or 3J - 1/2 SF's (if 2J is odd number) describe the effects of T-invariance violation. T-odd contributions in H}jn+) are defined necessarily by the interference of longitudinal and transversal components of hadron electromagnetic current. It is important to note that the T-odd contributions in H~j"-) can be defined also by interference of the transversal components of current, but only for J ~> 3/2. The T-odd effects at the scattering unpolarized electrons can be observed only at such target polarization, when the components of odd tensors P~,...o,, are non-zero. For the polarized electrons scattering, T-odd effects can appear in the case of polarized target
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with non-zero components of even tensor Qa,...a,,. In T-invariant hadron electrodynamics, non-zero asymmetries of the unpolarized electron scattering are induced by even polarization tensors only; in the general case, we have 4 J such asymmetries for the even value of 2J, and ( 4 J - 2) asymmetries for the odd values of 2J. Asymmetries of longitudinally polarized electron scattering are induced by the odd polarization tensors: for the even (odd) values of 2 J there are 3 J - 1 ( 3 J + 1/2) such asymmetries.
3. Parametrization of the threshold electromagnetic currents in terms of form factors
The aim of this section is to study a set of different processes e - + A ~ e - q - A l +A2, where the spins of A, A1 or A2 are equal or less, than unity. In the case of light nuclei, the one-photon approximation is good enough, therefore, the matrix element of any process e - + A ---, e - + A~ + A2 is determined by the hadronic electromagnetic current e2 (7)
A/I( eA ~ e A t A 2 ) = -~-ff( K2)yuU( Kl ) J u ,
where El and r2 are 4-momenta of initial and final electrons. Due to the conservation of hadronic and leptonic electromagnetic currents the amplitude (7) can be rewritten in terms of space components of these currents: e2 (8)
A / t ( e A --~ eA1A2) = ---K2 e • J ,
where e is 3-vector of virtual photon polarization, t¢./~ e=g-K
K2 ,
(9)
g~=~(K2)yuU(KI).
In this section we present the parametrization of the hadronic current J in terms of the threshold electromagnetic FF's. This parametrization can be exploited in the calculation of corresponding SF's in terms of threshold FF's. e- +1 +~e-
+0 ++0 +
and
e-+l---+e-+0
+ + 0 ~.
We employ through the notation given by S e = 1+ or 0 + for the spin S and spaceparity P of particles A, Al and A2. Taking into account the P-invariance of electromagnetic interaction of hadrons, we can write the following expression for the threshold amplitude of T* + A ~ AI q- A2 for the above mentioned set of spins and P-parities: Fth
= i f ( K Z ) e × f¢" U = e . J ,
(10)
where U is 3-vector of polarization of A-particles (with spin I). Therefore, the form factor f ( K 2) characterizes the threshold absorption of magnetic dipole virtual photons iny*+A~Al+A2.
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The structure functions Wi(K2) can be written as follows: 3Wl (K 2 ) = 2Wa(K 2 ) = --W6(K 2 ) = --W7(K 2) = --W8(K 2 ) = W9(K 2) = [f(K 2) 12, W2(K 2) =W3(K 2) = W5(K 2) = W10(K 2) = 0.
(11)
In y* + A ~ Al + A2 reaction only the transversal photon absorption takes place, therefore for which the relations: W2(K 2) = WS(K 2) = WI0(K 2) = WT(K2) + Ws(K 2) + 2W9(K 2) = 0
(12)
can be written directly. As a result of the presence of a single FF in Eq. (10), all asymmetries in e- + A --. e- + A1 + A2, which are due to collisions of polarized electrons with polarized target, do not depend on the FF f ( K 2) and therefore can be predicted exactly. This information about asymmetries can be exploited for the measurement of polarization of high energy electron beams (after scattering from target with vector polarization) and for the determination of tensor polarization of targets (at the scattering unpolarized electrons). e-+l
+~e-
+0 ±+0 T
and
e-+l----,e-+O
± + 0 ±.
The threshold amplitude of any such process is determined by two independent electromagnetic FF's: Fth = e . Ugl (K 2) + e . k U . ~g2(K2).
(13)
The form factor gl characterizes the absorption of transversal electric dipole (El) virtual photons in y * + A ---, A1 + A2, and the FF's combination gl(K 2) + g2(K2) characterizes the absorption of longitudinal E1 virtual photons. The SF's Wi(K2) can be expressed in terms of these FF's as, 3WI =W7 = W8 = Igl (K2)I 2,
E2 w2 = ~--M-2Ig] (K2) + g2(~2)12, w3 = 2-~-Re gl (K2) (g] (K2) + g2(K2) )*, w4-
E-2~/] - M g]( K2) 12- ~ER e g ] ( K 2)(g](K 2) +g2(K2)) *,
W5= ~Mlmg] (K2)g~(K2), W6 = 0 ,
w9 = Regl (K2)g~(K2), WlO= Imgl (K2)g~(K2).
(14)
In T-invariant hadron electrodynamics, both FF's gl (K2) and g2(K2) are complex functions of K2, but their relative phase must be equal 0 or 7. Therefore, ws(K2) = Wl0(K2) = 0 for any model of FF's g](K 2) and g2(K2).
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Reaction e - + 6Li ---+ e - + 7r° + 6Li* (0+)is one example of the above mentioned set of spins and parities of participating hadrons. e-+l+---+e-+l++0
q:
and
e-+l---+e-+l++0
+.
The neutral pion electroproduction of deuteron, e - ÷ d --+ e - + d + rr °, is in this class of process. The threshold amplitude of any such processes can be parametrized in the following form: Fth =gl (K2) e ' U1 × U~ + g2(K 2) (e × k . U j & . U~. + e × k - U ~ k - U I ) +g3(tc2)e • ~ k . UI )< U~,
(15)
where U1 and U2 are 3-vectors of polarizations of initial and final vector particles. The form factor gl(K 2) defines the absorption of transversal E1 virtual photons in 3'* + A + AI + A2, g2(KZ)-magnetic quadrupole (M2), and the sum gl(K 2) + g3(t<2) describes the absorption of longitudinal E1 virtual photons. Namely, only three such multipole transitions are allowed in y* ÷ A -+ A1 + A2 with the above mentioned set of the spins and P-parities. The expressions for the SF's in terms of these FF's are given as, w, - ~ Ig, - g2l 2 + ~-2lg' + g2l 2 , w2 = 2(Igl + g312), E w3 = - ~ - ~ R e (gj + g2) (gl + g3)*, E w4 = Igl -- g2] 2 ÷ ~ - ~ R e (gl ÷ g2) (gl ÷ g3)*, E I m ( g l + g2)(gl + g3)*, 2M w6 = 4Re gl g~, ws=-
w7 -- - [ g l - g2l 2 - [gl + gs[ 2, w8 = - I g ~
- g212 + 2Re (gl + g2) (gl + g3)*,
w9 -- IgJ - g21e - Re (gl + g2) (gl + g3)*, WlO = - I m (gl + g 2 ) ( g l + g3)*,
(16)
The SF's wl, w2 and w7 are determined only by combinations of FF's squares; [g~ +g212 and [gl +g312. Any experimental information about processes e - + d --+ e - + d + pO, where p0 = rr or r/-meson, is not available now. But near reaction threshold the experimental study of these processes is interesting, because it provides a possibility to test the predictions of low energy theorems (LET) for photo-and electro-production of the pions on nucleons, T + N ~ N + r r and e - + N ---+ e - + N + r r [14]. One example of such LET is the well known Kroll-Ruderman theorem [ 15 ] about threshold values of electric dipole amplitude
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EO+ for different y + N --~ N + 7r processes. Current algebra techniques advances the generalization of these predictions for the pion electroproduction [ 16]. After a long history with consistent predictions of experiment, it was found that the experimental value of E o + ( y p ~ p~r°) = ( - 0 . 5 d z O . 3 ) e / m ~ r 10 -3 [17] deviates essentially from the LET predictions E0+ (3"p ~ p¢r°) = - 2 . 4 e / m ~ r 10 -3. This result has been supported by another experiment [18], which has also demonstrated the strong energy dependence of the amplitude E0+ for 3"p --~ p~r ° (near threshold). All attempts the revision and improvement of classical LET were unsuccessful over a similar history [ 14]. New experiments [ 19,20] with tagged photons for 3' + p --* p + ~r° show a strong deviation from the predictions of LET. This discrepancy can be explained in the framework of chiral versions of non-perturbative QCD by some special contribution with non-analytical logarithmic singularity, which were not considered in previous studies of LET [21]. All these results show that the physics of threshold hadron electrodynamics is worth of renewal of interest. The next step in testing all these predictions is the study of 7r°-production in 3' + n ~ n + ~.0 and e - + n ~ e - + n + 7r° processes. And the best way to clarify this problem is to measure the coherent processes on the deuteron target, y + d --+ d + ¢r° and e - + d ~ e - + d + 7r°, which is expected to be sensitive to small neutron amplitudes. The ~7-production on deuteron, 3' + d ~ d + r/, remains also as an important problem of hadron electrodynamics. The contradiction between the large values of cross-section tbr reaction y + d --+ d + r/ near threshold [22] and the isovector nature of transition y + N ---* Sll (1535), which follows from the results of multipole analysis for 3' + N --~ N + ¢r processes [23-26] and predictions of quark models [27-31 ], is notified after the recent experiment [32], which demonstrated the large contribution of inelastic deuteron disintegration process, namely, 3" + d ~ n + p + r/. Electroproduction of ~/-mesons on deuteron, e - + d ~ e - + d + r / i n the near threshold region is another open problem. The formalism of the threshold FF's seems as the most suitable approach for phenomenological description of such processes. e-+l+-+e-+0±+l
i
and
e-+l---~e-+0±+l
i
Electrodisintegration of 6Li, e - + 6Li --+ e - + d + 4He, is one example of such a reaction. The threshold amplitude of any process 3'* + A --~ A1 + A2 with such set of spins and parities can be written in the following form: Fth = g l e . U l k
. U~ + g2e . U~2~C . U l + g3e . k U I . U~2 + g4e . k~c . U l i c . U~.
(17)
The absorption of MI virtual photons is characterized by the combination gl - g 2 and the absorption of electric quadrupole (E2) virtual photons with transverse polarizationby the sum of gl + g2. The form factors g3 defines the absorption of monopole (E0) virtual photons, and sum gl + g2 + g3 + g4 defines the absorption of longitudinal E2 photons.
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After summing over the polarizations of vector particles in the final state of y* + A Aj + A2, the following expression for the SF's wi(K 2) is obtained
w~ = ½(Ig~[ + Ig=12), E2 1 w2 = g(21g312 + ~--~lglr2), E w3 = ~--~Re (glg~ - g2g3), * W4 = l l g 112 _
E Re (glgL, -- g2g~), 2M
E
w5 = - ~---~Im (gjg~ - g2g~), W6 = Jg212,
w7 = dg212 + Ig412 + 2Re (g2 + g3)g~, w8 = Igl 12, W9 = R e
(gl(g2 ÷g3 ÷g4)*
÷g2g~),
wlo = Im (glg[ + g2g~),
(18)
where gL = gl + g2 + g3 ÷ g4. e - + ~ l+ - - ~ e - + ~ I + + 1 ~
and
e - + ~ 1 - - - - ~ e - + ~1 ± + i T"
The electrodisintegration of three nucleon nuclei, 3He and 3H, namely
e-+3He---~e-+d+p,
e-+3H--*e-+d+n
(19)
belongs to this class of processes. In general case, the amplitude of such processes have very complicated spin structure and is described in terms of eighteen independent amplitudes, which depend on the three different kinematical variables. But the situation is simplified essentially at the threshold of reaction, when the d + N system is produced in the S-state: Fth = .)(~ [ifl (t<2)e • U* x k + f2(K2)O ". k e " U* + f 3 ( t ¢ 2 ) o ". eU. f¢ + f 4 ( K 2 ) e • k o ' . U* + f s ( K 2 ) o -. k e . KU*. &IX1,
(2O)
where XJ,2 are the two component spinors of initial and final particles with spin 1/2 (in y* + A --, AI + A2 transitions). The allowed threshold multipole transitions are described by the following five independent combinations of FF's fi(K2): MI ---~ S = gJ
"
M1 ---+ S = .3 2 ." E2T --, S = !2 •I
EOL-+ S=~ :
(fl
+ f2 - fs),
(2f~ - f2 + f 3 ) , (f2 + f3),
(f2+f3
+f4+.fs),
C. Yal(inet al./NuclearPhysicsA 635 (1998)387-407 E2L --+ S= 3_. 2
(f2 + f3 + fs)
397 (21)
where S is total spin of A1 + A2-system; T and L denote the transverse and longitudinal polarizations of virtual photons. Target spin is equal to 1/2; therefore, only the following five SF's are non-zero: w](K 2) = [fll 2 + [f2l 2 + If3l 2, W2(K2) = If2 + f3 + f4 + fsl 2 + 21f412,
W3(K2) = - I f 3 l 2 - 2Re [(fl - f 2 ) ~ + f 3 ( f 2 + f4 + f s ) * ] , W4(K 2) =2Re [(fl - f2)f~4 + f 3 ( f 2 + f4 + fs)*] , w5(K 2 ) = 2 I m [ ( f l - f2)f~4+ f 3 ( f 2 + f 4 + f 5 ) * e
-
+l+--÷e-+~
14-
+$
14-
and
1•
(22) 14-
e-+l---+e-+$
lq: +$
The threshold amplitude of the process 3/* + A --+ AI + A2 is defined in this case by a set of following five electromagnetic FF's: Fth = 2"I [ig] (K2)e • U × ~' + g2(K 2) ( o ' . k e . U - o'- e k . U) +g3(K 2) (O'. K'e" U + o'. e~'. U - 2~r. k e - k U . ~) +g4(K2)o "" ~'e. k U . k + g5(K2)o ".
Ue.
k] o'2,~,
(23)
where Xl and 2"2 are the two component spinors of the produced particles A] and A2. The spin structures in Eq. (23) are chosen in this form, so that each FF describes the definite threshold transition in reaction y* + A ~ A1 + A2: MI --+ S = 0 (g]), M1 ---+ S = 1 (g2) E2T --+ S = 1 (g3), E2L ---+S = 1 (g4) and E0L --+ S = 1 (g5). Summing over the polarizations of A1 and A2, one can obtain the following expressions for the SF's in terms of the threshold electromagnetic FF's:
+ [g2 + g312 + --llTl-~lg2- g312
w1('¢2)=~
Igl
w2('~z) = g
Igs[2 + ~ l g 4 +g512
4(
w3(K2) = ~ R e
)
,
,
(g2 + g3) (g4 + g5)* + (g2 --
g3)g~5 ,
W4(K2) =lg|12 + Ig2+g312-- -~-Re [(g2+g3)(g4+g5)*+ (g2--g3)g;], Ws(K 2) = - ~ I m
(ga
+g3)(g4 +g5)* +
(g2 --
g3)g~5 ,
M2 [2, W6(K2) = _2~__5 [g] [2 + 21g2 + g3 wT(K2) = --21gil 2 + 2[g2 + g3l2 + 21g412 + 21g5[2
1-
398
C. Yal¢in et al./Nuclear Physics A 635 (1998) 387-407
+2Re [- (g2 + g3)g*4+ g4g~- 2g3g~1, ws(K 2) = - 2 ] g l 12 + 21g2 + g3l 2, W9( K2) ----21gl[ 2 -- 21g2 + g3[ 2 ÷ 2Re [(g2 +
WlO(K2)=2Im[(g2+g3)g*4 +
2gzg~] •
g3)g~+ 2g3g~], (24)
In this context, we now point out, that Eqs. (23) and (24) describe the threshold deuteron electrodisintegration, e - + d --~ e - + n + p. Interest in this kinematical region for this process is due to a number of reasons. These are: (i) A relatively simple spin structure of y* ÷ d -~ n + p amplitude. In the general case, at least 18 independent amplitudes are necessary for the complete description of matrix element of e - + d ---, e - + n + p (in the one-photon approximation). (ii) The square of momentum transfer K2 is related directly to the argument of the deuteron wave function. Therefore, the differential cross-section for d(e,e')pn near threshold measures the deuteron wave function. (iii) The threshold amplitudes of y* + d --~ n + p process depend only on one variable K2, while beyond the threshold, the same amplitudes depend on three different kinematical variables. (iv) A variety of possible polarization effects even near threshold of the y * + d ~ n + p reaction. (v) The relation between the isotopic spin selection rules and the spin structure of the threshold amplitude. (vi) The possibility of using the e - + d ~ e - + n + p process near threshold to test the symmetry properties of the hadron electromagnetic interactions (checking the P-and T-invariances). (vii) The threshold amplitudes are sensitive to such contributions arises from meson exchange currents, isobar configurations in deuteron and quark degrees of freedom. (viii) The threshold amplitudes of y* + d --, n + p process are real function of K2-for any reaction mechanism- due to the unitarity condition. (ix) Very short "distance" between elastic ed-scattering, e - + d ~ e - + d, and the threshold electrodisintegration, e - ÷ d --, e - + n + p , must result in some similarity between these processes. (x) The more simple experiment for the inclusive process d(e, e')np near threshold is also as informative as the more complicated experiment for the exclusive reaction
d(e, elp)n. The electrodisintegration of 6Li ( e - + 6Li ---+ e - + 3He ÷ 3H), is very similar to electrodisintegration of deuteron ( e - + d ~ e - + n + p ) in spin structure of amplitudes.
C. Yal(in et al./Nuclear Physics A 635 (1998) 387-407
399
4. Discussion of complete experiment Using results of the above analysis, it is possible to find the most optimal strategy of complete experiment for all the aforementioned processes, which allows to reconstruct the complete spin structure of the threshold amplitudes, i.e. for the determination of all the threshold electromagnetic FF's. For this discussion it is necessary to discriminate between two different possibilities. The simplest one corresponds to electro-production (or - disintegration) processes with a single value of the total angular momentum J of the produced A1 + A2-system, when all the threshold FF's are real function of K2 (with the relative phase 0 or ~')-following the theorem of Christ and Lee [11]. This case is realized when A1 (or A2) has spin equal to zero:
e- +N---~e- +N+~, e - + d ---* e - + d + 7r(r/), e - + 6Li ---+e - + d + 4He. Second possibility corresponds to the processes e - + A ---, e - + A1 + A2, when Al + A2-system has different values of J. Thus, the corresponding form factors can have in the general case non-zero relative phase which can be determined in special polarization experiments. In discussion of complete experiment it is necessary to keep in mind, that at the threshold of any process e - + A ~ e - + A1 + A2 not only the experiments with polarized target A is possible, but for a large enough IK21 it is possible to measure also the polarization of produced particles - due to the larger its 3-momentum - in comparison with the photo-production processes y + A ~ A~ + A2. Let us illustrate this with the example processes e - + N ~ e - + N + T r and e - + d -+ e - + n + p , when the threshold kinetic energy Tp of produced proton is controlled by the value x 2 only: K2
Tp-
4M'
e-+d---~e-+n+p,
where M is the deuteron mass (neglecting the deuteron binding energy in comparison with the nucleon mass), and l K 2 ÷ /~2
Tp
-
K2
- ~--2 m + lx 2m'
if
Iz/m<
e-+p---~e-+p+rr,
where m ( / . ) is the proton (pion) mass. Thus, examination of the rr°-production threshold in g-- + p --+ e - + ff + rr° with its accompanying new and interesting dynamical information, might best be studied in the TJNAF [33] experiment with its polarized e - (and unpolarized p) beam and proton polarization analyzer. In any case, the polarization measurements are unnecessary to realize the complete experiment for the process class e - + A --+ e - + Al + A2 which is characterized by one or two electromagnetic FF's:
C Yalgin et al./Nuclear Physics A 635 (1998) 387-407
400
e - q, 1+ - - ~ e - q , O + q . O ~
and
e- q,l- ~e-q.O
+ q . O m,
e - q.1 + - - - + e - + O + q . O m
and
e- q,1- ~e-q.O
± q , O ±.
In all these reactions the complete experiment is equivalent to measuring only the differential cross-section (with unpolarized particles). But, the Rosenbluth separation of the different contributions to the inclusive cross-section:
-
62o -
dE2 df2e
~
O-T q - 6OrL,
where O-T and OrL are the cross-sections for y* + A --~ A~ + A2 with absorption of virtual photons y" with transversal and longitudinal polarizations, respectively, must be done. Note, that these measurements cannot determine the sign of electromagnetic threshold FF's. So, in these cases, it is possible to use the standard procedure, which is so effective for the determination of nucleon electromagnetic FF's. Namely, starting from the precisely known values of the static electromagnetic characteristics of the nucleons (elastic charges and magnetic moments, corresponding to FF's at K 2 = 0 ) , it is possible to extrapolate the sign step by step for any value of momentum transfer square, K2. It is also possible to exploit the same method for any process e - q, A ---, e - q, A I q" A2, using the sign of corresponding multipole amplitudes for y q, A ---, A1 q, A2, i.e. at K2 = 0, which can be determined directly or with the help of a more resolved theoretical model. Note also, that if one form factor (longitudinal or transversal) is small, its determination with respect to a Rosenbluth fit may not be reasonable accuracy (for example, in the case of the electric FF of the neutron). In such a situation, the special polarization experiments are necessary, where the result is determined by the interference of longitudinal and transversal FF's. For the process e - q, A --~ e - q, Aj + A2 the most suitable experiment is the measurement of asymmetry of scattering longitudinally polarized electrons by a vector polarized target, when the corresponding SF w3 is determined by the product of transversal (gl) and the longitudinal (gl q" g2) electromagnetic FF's (c.f. Eq. (14)). But if the number of electromagnetic FF's is larger than 2, the polarization phenomena must be studied for the full reconstruction of the spin structure of the threshold electromagnetic current. For the reactions e - + 1+ --, e - q, 1± q.0 m and e - q, I - ---, e - q, 1+ q-0 +, the complete experiment for the determination of 3 independent FF's (Eq. (15)) must include the following measurements: differential cross-section with unpolarized particles (Rosenbluth separation of o-x and O-L); tensor analyzing power in scattering unpolarized electrons by tensorially polarized target. So in this case we have a situation, which is very similar to the elastic ed-scattering [34], where, for the separation of electric and quadrupole electromagnetic FF's the measurement of tensor polarization of scattered deuteron (or the tensor analyzing power) -
-
C. Yalgin et al./Nuclear Physics A 635 (1998) 387-407
401
is necessary. In this case, it is necessary to measure only 7"20 from three possible tensor analyzing powers; namely, T20, T21 and T22. Another situation appears for the process e - + 1+ ~ e - + 0 + + 1+ and e - + 1e - + 0 ~- + l T with four independent electromagnetic FF's (Eq. (17)). It can be see from Eq. (18), that the measurement of SF's Wl, w2, w6 and w7 will allow to determine moduli of all these FF's. More exactly, for this case it is necessary to know the relative phase of g4 and (g2 +g3 ). So, again it is necessary to "organize" the unpolarized electron scattering by the target with tensor polarization. But it is important to stress that, for all these reactions, the scattering unpolarized electrons only is necessary. However, the polarized electron beam is evidently necessary to realize the complete experiment for p r o c e s s e s e - + ~ l+ ~ e - + ~ 1+ + 1± a n d e - + ½- ~ e - + 7 1± + l t : . I t can be realized from Eq. (22), that the full reconstruction of five electromagnetic FF's is not possible without measuring polarization of produced nuclei. The processes e - + 1+ --~ e - + ½+ + ½~: and e - + 1- --+ e - + 7 + + 1:F (with five electromagnetic FF's) represent the most complicated case (from considered above). Eq. (24) for SF's wi reveals that SF's w l, w3 + w4 and w6 will allow to determine moduli of all three transversal FF's, namely, Ig~12 and [g2 + g312. Adding to these measurements the SF's w2 and w3, it is possible to determine the longitudinal FF's g4, gs. Moreover, the measurement of SF's w7 and w8 will be useful for decreasing the number of possible solutions for FF's. In conclusion, let us stress that all SF's Wl - WlO, which determine the scattering polarized electrons by polarized target, are insensitive to the relative phase of singlet form factor gl and triplet FF's g2 - gs. To determine this phase the measurement of polarization of one produced fermion is necessary. In threshold conditions, non-zero polarization is possible only in the case of scattering longitudinally polarized electrons.
5. P - o d d effects in processes e - + A --~ e - + A1 + A2
In this paper only the one photon mechanism, which arises as a result of P-odd effects due to P-invariance violation in 3/* + A ~ A1 + A2 processes [35-37], has been considered. Another possible source of P-odd effects in 7" + A ---, Al + A2 is the Z-boson exchange between weak neutral currents of electron and hadrons [38,39]. At relatively small values of K2, the one-photon mechanism of P-odd effects is more dominant, whereas the mechanism with the weak neutral currents is dominant at the large values of the momentum transfer. All P-odd asymmetries in e - + A ~ e - + Ai + A2 can be extracted from the general formula, given by Eq. (1), if the following changes are introduced,
Hij --+ Hij + fiIij,
ffIij = Ji]'} + ]iJ~,
(25)
where J is the P-odd part of the electromagnetic current for the process T* + A --+ Ai +A2.
C. Yalqin et al./Nuclear Physics A 635 (1998) 387-407
402
For the target with vector and tensor polarizations, the most general formula for the threshold tensor/4ii is as follows:
ffIi.j=i~ugKglTVI(K 2) ÷ S. t¢
f(~ij -
÷ ( SiKj ÷ SjKi)17V4( K2 ) ÷
KiKj)ITF2(K2) + KiKjlTv3(K2)]
i( Sikj - Siki)~vs( K2)
+ iEijgKg( SabKaKb)W6( K2)
+i(eiabSackbkckj - ".iabSacKb~
+(eiabSiako + ej~bSiakb)~V9(K 2) + (eiabki + e i~bki)Sackbkc#,O(K2),
(26)
where ~i(K 2) are real P-odd SF's. So, the P-odd effects near threshold of reaction e- + A ---+ e- + A] + A2 can arise at the scattering of unpolarized electrons by polarized target with vector polarization; in the case of scattering longitudinally polarized electrons, the P-odd effects arise for the unpolarized target or target with tensor polarization. SF's g'5 (K2) characterizes the P-and T-odd effects (simultaneously) in the scattering longitudinally polarized electrons by target with vector polarization, and ~9(K 2) and ~'lo(K2) characterize the P-and T-odd polarization effects at the scattering unpolarized electrons at the target with tensor polarization.
6. Calculation of P - o d d structure functions
In this section we present the parametrization of the P-odd part of electromagnetic current and the calculation of P-odd SF's ~i(K 2) in terms of corresponding FF's. e- + 1 + - - - , e - + O + + 0 :t: and
e-+l--+e-
+ 0 + + 1 ~:.
The P-odd amplitude is defined in terms of two electromagnetic FF's as Pth = e. Up] (1(2) + e. kU-1~'/02(K2),
(27)
which describe two possible P-odd transitions with absorption of E1T(jO1 ) and E1L(ffl +
P2). Using the P-even amplitude Eq. (10), for the P-odd SF's ~'i(K 2) the following expressions are found I~ 1 (K 2) = ½Replf*, l~2(K 2) = -Reffl f*,
~,3(,¢2) = - E R e (p~ + P2) f*, M
~'4(K2) = ~ E R e (Pl +/~2) f * ,
2M
C. Yalginet aL/Nuclear PhysicsA 635 (1998) 387-407 ~:5 (t<2) = ----~Im (/~1 + 2
403
P2)ff,
1~6(K 2) ----0,
u:7 (K2) = -Re/~2f ~, #8(K 2) = -Replf* ~9( K2) = -Imp1 i f , v~lO(K2) = - I m p z F .
(28)
The description of P-odd effects for e- + 1+ ---, e- + 0 :F + 0 ± and e- + 1- --~ e- + 0 ± + 0 ± processes is obtained by interchanging P-odd and P-even amplitudes in Eq. (28).
e-+l+~e-+l++0
~: and e - + l - - - + e - + l ± + 0
±.
The P-odd threshold amplitude for these processes is defined by four corresponding electromagnetic FF's: i/Wth = e. U I k - U ~ I + e. U~2 k - Ulg'2 + e. kU1 - U~3 + e. I~UI. I~U~' i~g4.
(29)
These form factors describe the following P-odd multipole transitions: M1 (~l -g'2), E2T(~I + g'2), EOL(~3) and E2L(~l + g2 -'F-g4). Using P-even amplitude for such processes, Eq. (15), and the definition of tensor /7/ij, after summing over the polarizations of vector particles in the final state, we obtain: ~l(K2)_
2 E2 3 ~-~Re~2(gl + g2)*,
~2(K 2) = Re~l (gl - g2)*, 1,~'3( K 2 ) = E R e [(gl + g2 + g3 + g4)(gl - g2)* - g2(gl + g3)* + gs(gl + gz)* --
2 ~M@ 3 ( g l
+
g2)*] ,
#4(K 2) = ~--~Re (~1 + ~2 + g3 -1- g4) ( - - g l -[- g2) * -[- g 2 ( g l ~- g3) * -- g 3 ( g l + g2)* , #5(K 2) = ~-~Im (~l + g2 + ~3 + ,~4)( - g l + g2)* - ~2(gl + g3)* + g 3 ( g l + g2)* , #6(K 2) = - 2 R e g'2(gl + g2)*,
WT(t<2)=Re [ ( g 2 - I - g , 3 + g 4 ) ( g l - g 2 ) * - g g ( g l + g 2 ) * ] , ~8(K 2) =Re~l (gl - g2)*, #9(K 2) = Im~l (gl -- g2)*,
7Vlo(K2) =Im [(~,2 + ~3 + ~4)(gl -- g2)* -- ~2(gl + g3)* -- ~3(gl + g2)*].
(30)
404
C. Yalfin et al./Nuclear Physics A 635 (1998) 387-407
After interchanging the P-odd and P-even amplitudes, these formulas describe the SF's ff~i( K2) for processes e - + 1+ --~ e - + 14- + 04- and e - + 1- ~ e - + 17: + 0 +. e
-
+~
! +
- - - ~ e - + i14- + 14-
and
e-+~
1 -
- - - ~ e - + ~1 + + lq: .
The threshold P-odd amplitude can be written in the following form: Fth = X~ [fie.
U* + if20".e × U* + if30".e x &ft. U*
+f4e. iek. U* +ifso'. k × U*e. k]X1.
(31)
P-odd form factors fi are determined five threshold multipole transitions with Pparity non-conservation: E1T ~ S = ½, EIT ~ S = 3; M2 ---, S = 3, E l L ~ S = ½ and E l L --~ S = 3, where S is total spin of particles A1 and A2. After summing over polarizations of particles Aj and A2, and by making use of the formulas for Fth, Eq. (20), and /~'th, Eq. (31), we obtain the following expressions for the P-odd SF's ~1 - ~5:
w,(K2) = 2Re [flf~t + f 2 ( f 2 - f 3 ) * - fgf~31, wz(K2) = 2Re [flf~2 + f2(f, + f3)* + faf~31, w3(K 2) = 2Re If1 (f2 +
fs)* + (f2 + fs) f~ + f2(f3 + fs)*
+f3 (f2 4- f3 + f4 4- f5)* -I- f4(f2 "Jr-f4 4- fs)* -W4(K2) = Re
fs(f2
+
2f4)* / ,
[f~(f3 + f4)* - (f2 + fs) f~l - f z ( f 3 + fs)* + f3 + f4 + fs)* + f4f~3 + fsf~2 -- 2f2f~4/ ,
ws(K 2) = Im [ ( f , -- f 2 ) ~ -- ( f , + J % ) ~ + (f2 +
f s ) ( f , -- f2)*
-- (f2 4- f3) (/2 + f3 4- f4 4- fs)*].
(32)
These formulas describe the P-odd effects at the threshold of electrodisintegration of polarized nuclei 3He, e - + 3He ~ e - 4- p + d. e
--
4 - 1 + - - - ~ e - + g1 -/- + g14-
and
e-+l---+e-+g
14- + gI T
P-odd electromagnetic current for these processes is characterized by five FF's:
Pth = 22 [~ze . U + i~,2o-. e x U + i~3o" . e x icU . i¢ L
C. Yalfinet aL/Nuclear PhysicsA 635 (1998)387-407 q-ig40" • k × U e . k q- ~5e. ~U- KI Or2"~"
405 (33)
These form factors describe five independent P-odd multipole transitions namely: E 1T --~ S = 0(~l), E1T --~ S = 1 (~2), M2 --* S = 1(~3), E l L ~ S = 0(~1 + g s ) and E l L ---+ S = |(g2 q- g4). The SF's w/(K 2) can be written in the following form: I~I(K2) = 2Re glg~ q-g2(g2 -~-g3)* -I- ~ ( g 2
-~g3)(g2 - g3)* ,
@2(K2) = Re lglg*l -- g2(g2 q- g3)*] , #3(K 2) = ~ R e
(gl + gs)g~ + ~2(g2 - g3 + g4 -t- 2g5)*
+g3g5 ~ * + g4(g2
- g3)*]
d
- 2Re (g2 q- g4)gs, ~ *
w4(K2) = - 2 M Re (gl + gs)g~ + g2(g2 - g3 + g4 + 2g5)* +g3g5 ~ * + g4(g2
-- g3)*]
..I
~5(K 2) = --~-E Im2M [(gl + gs)g~ + g2(-g2 +g3 -l-g4 + 2g5)*
]
- * +g395 + g4(--g2 + g3)* ,
#6 ( K2) = 2Re g3 ( -g2 + g3)*, #7(K2)=Re
g2(--2gz+g4)*--g3gs+g4(--g2+g3)*
gsg] ,
Ws(K2)=Re[glg*l+g2(g2+g3)* 1,
1 ~lo(K2)=Im[~2(293--g4)*+~,3g*5+~4(--g2q-g3)*+~Sg*l].
(34)
All these expressions, which we have presented above, can be used in the analysis of P-odd polarization effects at the threshold of deuteron electrodisintegration..
7.
Conclusion
The analysis, presented in this report, is valid for a wide class of hadronic and nuclear processes, such as, for example:
406
C. Yalgin et aL/Nuclear Physics A 635 (1998) 387-407
e - q. N--~ e - q. N + 7"r, e- +N--+ e- +N-t-o-,
e - + 3He ---+ e e - + 3He --+ e e q.N---~e-q,A+K, e-q.3He---~e e - + N --~ e - + N q, p(oJ, q~), e - + 4 H e --+ e e - q-d--+ e - q . d + 7 " r °, e- +6Li--~ ee - q. d ---~ e - q, n q,p, e - q,6Li --~ e -
+p+d, q, 3He q, 7r°, +Tr++3H. + 4He q, 7r°, q,3Heq.3H, q, d q, 4He,
It is demonstrated that the electromagnetic FF's formalism is effective enough for the parametrization of electromagnetic current for 3'*+ A --+ A 1q"A2 processes near threshold. The explicit forms of the structure functions can be taken as the kinematical base of analysis of different polarization effects in such processes. The accurate measurements of the threshold electromagnetic FF's will reveal the dynamics of 9'* q" A --~ A1 q, A2 processes. The parametrization of electromagnetic currents for the processes, considered in this report, in terms of threshold electromagnetic FF's, and the calculation of the structure functions, which parametrize the polarization effects in electron-hadron scattering processes, is the first necessary step in the analysis of electrodisintegration (and electroproduction) processes near threshold. Within this approach, we establish the spin structure of amplitudes for y* + A ~ A l + A2 in most general form, which is consistent with the symmetry properties of hadron electromagnetic interactions. The structure of interacting nuclei and the mechanism of reaction y* q. A ---, Al q, A2 contribute to the explicit form of corresponding FF's only. More exactly, the nuclei can be considered in such formalism as elementary particles with the definite values of spin and space parity. But we use essentially the conservation of the hadronic electromagnetic current and the P-and T-invariances of the electromagnetic interaction of hadrons. The relativistic kinematics and the spin effects for all considered above processes y*q,A --~ A1 q,A2 are taken into account exactly. The dynamics of processes y" q, A --+ Al q. A2, in conjunction with nuclear structure, such as, exact form of electromagnetic interaction, meson exchange currents, quark phenomenology etc., is reflected in the threshold electromagnetic FF's. Summarizing all this we must repeat once more, that like this formalism for the description of phenomenology of the threshold electrodisintegration (or electroproduction) processes in terms of corresponding FF's is, evidently, the necessary first step in the kinematical analysis of inelastic processes, similarly to the well-known procedure, which is used usually for the description of elastic electron-hadron scattering. In both case we have a limited sets of electromagnetic form factors, depending on the momentum transfer square u2, only. Therefore, all experimental information about these processes must be formulated in terms of the threshold electromagnetic FF's, as well as a theoretical predictions, which are used a definite models for the discussed processes. Various processes of neutrino-hadron interactions, such as u~ + A --~/z + Al q. A2 and u# + A --, u u q, Ai + A2, due to charged or neutral weak currents, can be described also in terms of weak FF's for the threshold transitions W* q, A ---+ Ai q, A2 and Z* q. A ---, Ai q, A2, where W* and Z* are virtual W and Z bosons.
C. Yale'in et al./Nuclear Physics A 635 (1998) 387-407
407
Acknowledgements W e t h a n k J. A r v i e u x , J.M. L a g e t , M. G a r c o n a n d E. T o m a s i - G u s t a f s s o n for useful d i s c u s s i o n s o f t h r e s h o l d p h y s i c s in C E B A F c o n d i t i o n s . O n e o f the a u t h o r s ( M . R . ) also t h a n k s T U B I T A K - N A T O - C P G r a n t , w h i c h m a d e this visit to M i d d l e E a s t T e c h n i c a l University (Ankara, Turkey) possible.
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Nuclear Physics A 635 (1998) 387-407
Electromagnetic form factors of electrodisintegration of light nuclei near threshold C. Yalqin a, M.E Rekalo a'l, H. Koru b'2 a Physics Department, Middle East Technical University, Ankara, Turkey b Gazi University, Physics Department, Ankara, Turkey
Received 6 June 1997; revised 12 March 1998; accepted 21 March 1998
Abstract The general structure of electromagnetic currents for electrodisintegration of nuclei near threshold, e - + A --* e - + A1 + A2 (A, At and A2 are arbitrary nuclei or hadrons with spins 0, l / 2 and 1), are established. Considering nuclei as elementary particles with the definite values of spin and space parity and using the conservation of hadronic electromagnetic current and P-invariance of hadron electromagnetic interaction, the spin structure of the amplitudes of the processes T* + A ---* A1 + A2 is derived in terms of threshold electromagnetic form factors. The present formalism is valid for any nuclear structure and any reaction mechanism. The structure functions, which characterize the scattering electrons on the polarized target, are calculated as quadratic combinations of the form factors. These structure functions are used for the analysis of the problem of complete experiment for the full determination of all threshold form factors. P-odd part of electromagnetic current, induced by P-odd nuclear forces, is parametrized also in terms of the corresponding form factors. The processes with the following combinations of spins of hadrons e - + 1 --~ e - + 0 + 0 , e - + 1 --~ e - + 1 + 0 , e - + 1/2 --* e - + 1 / 2 + 1 and e - + 1/2 --* e - + 1/2 + 1/2, and with different combinations of space parities of nuclei A, Ai and A2, are examined within this approach. @ 1998 Published by Elsevier Science B.V. PACS: 11.30.Er; 24.70.+s; 25.20.Lj; 25.30.Dh Keywords: Electromagnetic form factors; Polarization effects; Nuclear processes; P-invariance; Structure functions; Electrodisintegration
I Address: National Scientific Center, Kharkov Institute of Physics and Technology, Kharkov, Ukraine. 2 E-mail [email protected]. 0375-9474/98/$19.00 (~) 1998 Published by Elsevier Science B.V. All rights reserved. PH S 0 3 7 5 - 9 4 7 4 ( 9 8 ) 0 0 1 9 3 - 6
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1. Introduction The electromagnetic and weak hadronic currents, which determine the processes of electron and neutrino scattering by hadrons ( e - + A ~ e - + A ( At), u + A ---* u + A (A t), v~ + A ~ /z- + A" where A, At and A" are hadronic states, nucleon resonances and nuclei in ground or excited states, respectively), are parametrized in terms of corresponding form factors (FF). The calculation of differential cross-sections of these processes and the analysis of polarization effects are given in terms of these elastic (or quasielastic) FF's. The symmetry properties of fundamental interactions can be formulated also in terms of the electromagnetic and weak FF's. But electromagnetic current for the more complicated processes, for example, such as e - + A ~ e - + ~ ° + A t , which is equivalent to the process y* ÷ A ~ 7r° + At (y* is the virtual photon) is determined instead of FF's, which depend on the momentum transfer square only, by some set of amplitudes, which depend generally on three invariant kinematical variables. In this paper, we present an analysis of e - + A ~ e - + Ai + A2 types of processes (A, Al and A2 are hadrons and nuclei) in special kinematical conditions, namely at the reaction threshold. More exactly, we define the threshold as the minimal value of the effective mass of produced AI + A2-system, but with arbitrary value of momentum transfer square x 2. In such kinematical conditions the corresponding electromagnetic current is also characterized by the electromagnetic FF's [ 1], which are so fundamental as the electromagnetic FF's of one-particles transitions y* + A ~ A t. Similar to the case of elastic (or quasielastic) scattering, e - + A -+ e - + A ( A t ) , the number of independent electromagnetic FF's for e - + A ---+e - + A1 + A2 (at threshold) is determined by the values of spin and spin parity of A j, A2 and A. The spin structure of electromagnetic current for the processes e - + A --~ e - +Al +A2, which we consider in the framework of one-photon mechanism, can be established in the general form without any assumptions about the structure of hadrons in y* + A ~ AI + A2 and about details of the reaction mechanism. Only the general properties of hadron electromagnetic interaction, such as the P-invariance and the conservation of electromagnetic current (gauge invariance), are used. Detailed properties of hadrons or nuclei are not essential for such an analysis, only the spins and space parities of hadrons A, Al and A2 are important here. Therefore, all results which can be obtained in such a way must be valid and for that case, when A, A1 and A2 are arbitrary nuclei. The exact parametrization of electromagnetic current for threshold electrodisintegration of nuclei (or hadron electroproduction) provides the separation of the kinematical and dynamical aspect of description of electromagnetic processes. This allows one to take into account in the full volume the effects of relativistic kinematics and the conservation of electromagnetic hadron current. This leads to an effective formalism for theoretical analysis of the sensitivity of differential cross-section and different polarization observables, for threshold processes to the behavior of the corresponding wave functions at small distances and to the effects of mesonic exchange currents and isobar configurations in light nuclei [2-8]. In Section 2 we introduce the structure functions (SF) formalism for description of
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the threshold differential cross-section for any processes e - + A ---+ e - + A~ + A2. This formalism reveals the dependence of differential cross-sections on the polarization of target nuclei A with spin J. In Section 3 we parametrize threshold electromagnetic currents for 3'* + A ---, Al + A2, for relevant combinations of the spins of A, Al and A2, for s ~< 1. Details of the calculation of the corresponding SF's in terms of the threshold electromagnetic FF's are given. Methods of realization of complete experiment for different processes e - + A ---* e - + AI ÷ A2 are discussed in Section 4. In Section 5 we study the P-odd effects in e - + A ~ e - + Al ÷ A2, which are induced by the P-odd nuclear forces. We demonstrate how to parametrize the P-odd part of electromagnetic current in terms of the definite number of P-odd electromagnetic FF's. Section 6 is devoted to the calculation of P-odd structure functions in terms of P-odd and P-even threshold electromagnetic FF's.
2. T h e structure f u n c t i o n s for the processes e - + A ---, e - + A1 + A2
As it is known [9], the differential cross-section of any process of the type e - + A e - + A1 + A2 can be written in the framework of the one-photon mechanism as, 0'2 E2 Iql 1 1 dE2 doe dO - 647r 3 El M W 1 - • ( - K 2) d3o "
x
{
Hxx + Hyy + •cos2~o(Hxx - Hyy) - 2e-~oHzz
+•sin2~o(H~y + Hyx) - a
lx/f-Z~-e[x/1 + e(H~y - Hyx)
+ ¢ /2 6 - -( -- K ~ 022) ( c o s ~ o ( H y z - H z y ) -
sin~p(Hxz - H z x ) ) ] } ,
( 1)
1
K2
0e
K0
T"
• -1 = 1 - 2"~gs tan 2
Here, E1 (E2) is the energy of the initial (final) electron in laboratory system (LS), ds2e is the solid angle element for the scattered electron and 0e is the electron scattering angle in LS, dg2 is the solid angle element of one produced hadron in CMS of y* ÷ A A1 ÷ A2 reaction. W is the invariant mass of system A1 + A2, M is the mass of A, g/ is 3-momentum of A1 in CMS, ,~ = ±1, for two possible relative orientations of spin and 3-momentum of initial electron, K2 is square momentum transfer and ~o is azimuthal angle of final electron. The definition of the tensor Hij is given as
Hij = JiJj,
(2)
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where J is the 3-vector of electromagnetic current for the y* ÷ A ---, Al + A2 in the CMS, line in the definition of Hij denotes the summing over polarizations of AI and A2 particles. In Eq. (1) we use the coordinate system with z-axis along the 3-momentum t¢ of the virtual photon, the plane xz coincides with the plane defined by K and q. But at the threshold of (q ~ 0) such a coordinate system definition is not appropriate and the components of Hii depend only on one variable, namely K2. Therefore, for any unpolarized target, it is possible to integrate Eq. ( 1 ) over dO, so the differential cross-section d2~r/dezdOe (at threshold) will contain three contributions, namely, due to Hxx + H~,y, Hzz and ,~(Hx~, - H~.:,) only. But for the polarized target, it is possible to introduce the coordinate system with xz-plane defined by K and the spin vector of target (or direction of corresponding magnetic field). Eq. (1) is valid within the one photon mechanism with the assumption of the conservation of hadronic electromagnetic current for the y* + A -~ AI + A2 process. The other details, such as reaction mechanism for the 3/* + A ---, Al + A2 and the structure of particles A, Al and A2, are not relevant in deriving Eq. (1). Using P-invariance of hadronic electromagnetic interaction, it is possible to establish the dependence of tensor Hij on the polarization properties of target. For example, for the spin-one target (deuteron or 6Li) Hii can be written [10] as follows: (from now we will consider all processes e - + A ---+e - + Al + A2 as being near threshold):
Hi.j = ( 6ij - kik.j ) w j (K 2) + kif~iW2( K 2) + ieijeStw3( K2)
÷if:.ijgK,S " KW4(K2) ÷ ('iabSakbk.i ÷ 6jabSaKbKiOW5(K2) ÷( SabKaKb) [W6(K2) ( C~ij-- ~,f(j ) ÷ fqiK,jW7(K2)[ ÷ SijW8(K2) ÷(SiakakjWSjai'
(3)
where wl - w m are real structure functions, which depend near threshold on K2, only; is unit vector along the momentum of virtual photon, S describes the vector polarization of target, the tensor S,b stands for the quadrupole(or tensor) polarization. In the CMS of y* + A --~ A1 + A2, we have the following properties for Sab; M2
Sab = Sba,
Sxx + Syy + Szz ~ - = 0,
E =
W2 + M 2 _ t¢2 2W '
(4)
where E is the energy of A. Of course, the momentum transfer square K 2 in the threshold region can take any value in space-like region. Eq. (3) is valid also for a target with spin 1/2, but in this case only five SF's arise, Wl(K 2) -- WS(K2) [11]. If the spin of the target is greater than one, then Eq. (3) describes the corresponding contributions into the differential cross-section, induced by the vector and tensor polarizations of the target.
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The SF's ws(K 2) and wl0(x 2) characterize the possible T-odd effects in y* + A A} + A2. Note that T-odd effects in y* + A --~ A1 + A2 are defined by the interference of longitudinal and transversal components of hadron electromagnetic current (for the vector and for tensor polarization of target) independently from the mechanism of Tinvariance violation. For vector polarization, this has been formulated by Christ and Lee [11]. The SF's formalism allows the study of the polarization phenomena for any spins J of target A in cA-collisions [ 12]. In the framework of this formalism the polarization states of a target with spin J are characterized by a set of 2J real tensors T,~...~,,,, n = 1. . . . . 2J, symmetrical in any pair of indexes with zero trace, T~aa~...a, = 0. Each such tensor has ( 2 J + 1) independent components [ 13]. Tensors Ta,...a, with odd number of indexes do not change their sign in space reflections (pseudotensors). To discriminate the tensors with even and odd numbers of indexes, we denote them by Q~...,,, and P~,...,,,, respectively. Taking into account the properties of these P and Q, the general form of the hadronic tensors H U can be written as
a~l +) -- [(~ij -- K.iKj) x' (K2) ÷ KiKjX2(K2)] OaK.a 4- OijX3(K 2) ÷(QiKj ÷ OjKi)X4( K2) ÷ i ( Q i k j - O_.jki)xs( K2), O_.i = Qi(,2...~,, ka2 . . . ka,,
(5)
Oij = Qija~ ...a,,kay.., ka,,,
H!"t.l -) = ieijgkeff~gmkmYl (K2) + i~.ijgPgmY2(K 2) + i (eiabkbPja -- 6jabkbPia) y3(K2) 4- (~:iobkbPja ÷ ~:jobKbPia) Y4(K2) ÷ (KiEjabPaKb -]- Kj'iabPaKb) YS(X2),
(6)
P,j=P,jo3 . o, ...ko,,, where the indexes ( n + ) and ( n - ) indicate the polarization tensors with even and odd numbers of indexes. The contribution of each "irreducible" polarization in the cross-section of an inclusive cA-interaction is characterized by five real SF's. The number of SF's is independent from n (multipolarity of corresponding target polarization). The single exception is the case of vector polarization, n -- 1, where only three SF's appear. In T-invariant hadron electrodynamics, the number of SF's decreases, because the tensors H}ff+) and H I T - ) are symmetric and antisymmetric, respectively. From a set of 10J ( J >/ 1) independent SF's, which characterize the scattering electrons on the polarized target with spin J, only 3J - 1 SF's (if 2J is even number) or 3J - 1/2 SF's (if 2J is odd number) describe the effects of T-invariance violation. T-odd contributions in H}jn+) are defined necessarily by the interference of longitudinal and transversal components of hadron electromagnetic current. It is important to note that the T-odd contributions in H~j"-) can be defined also by interference of the transversal components of current, but only for J ~> 3/2. The T-odd effects at the scattering unpolarized electrons can be observed only at such target polarization, when the components of odd tensors P~,...o,, are non-zero. For the polarized electrons scattering, T-odd effects can appear in the case of polarized target
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with non-zero components of even tensor Qa,...a,,. In T-invariant hadron electrodynamics, non-zero asymmetries of the unpolarized electron scattering are induced by even polarization tensors only; in the general case, we have 4 J such asymmetries for the even value of 2J, and ( 4 J - 2) asymmetries for the odd values of 2J. Asymmetries of longitudinally polarized electron scattering are induced by the odd polarization tensors: for the even (odd) values of 2 J there are 3 J - 1 ( 3 J + 1/2) such asymmetries.
3. Parametrization of the threshold electromagnetic currents in terms of form factors
The aim of this section is to study a set of different processes e - + A ~ e - q - A l +A2, where the spins of A, A1 or A2 are equal or less, than unity. In the case of light nuclei, the one-photon approximation is good enough, therefore, the matrix element of any process e - + A ---, e - + A~ + A2 is determined by the hadronic electromagnetic current e2 (7)
A/I( eA ~ e A t A 2 ) = -~-ff( K2)yuU( Kl ) J u ,
where El and r2 are 4-momenta of initial and final electrons. Due to the conservation of hadronic and leptonic electromagnetic currents the amplitude (7) can be rewritten in terms of space components of these currents: e2 (8)
A / t ( e A --~ eA1A2) = ---K2 e • J ,
where e is 3-vector of virtual photon polarization, t¢./~ e=g-K
K2 ,
(9)
g~=~(K2)yuU(KI).
In this section we present the parametrization of the hadronic current J in terms of the threshold electromagnetic FF's. This parametrization can be exploited in the calculation of corresponding SF's in terms of threshold FF's. e- +1 +~e-
+0 ++0 +
and
e-+l---+e-+0
+ + 0 ~.
We employ through the notation given by S e = 1+ or 0 + for the spin S and spaceparity P of particles A, Al and A2. Taking into account the P-invariance of electromagnetic interaction of hadrons, we can write the following expression for the threshold amplitude of T* + A ~ AI q- A2 for the above mentioned set of spins and P-parities: Fth
= i f ( K Z ) e × f¢" U = e . J ,
(10)
where U is 3-vector of polarization of A-particles (with spin I). Therefore, the form factor f ( K 2) characterizes the threshold absorption of magnetic dipole virtual photons iny*+A~Al+A2.
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The structure functions Wi(K2) can be written as follows: 3Wl (K 2 ) = 2Wa(K 2 ) = --W6(K 2 ) = --W7(K 2) = --W8(K 2 ) = W9(K 2) = [f(K 2) 12, W2(K 2) =W3(K 2) = W5(K 2) = W10(K 2) = 0.
(11)
In y* + A ~ Al + A2 reaction only the transversal photon absorption takes place, therefore for which the relations: W2(K 2) = WS(K 2) = WI0(K 2) = WT(K2) + Ws(K 2) + 2W9(K 2) = 0
(12)
can be written directly. As a result of the presence of a single FF in Eq. (10), all asymmetries in e- + A --. e- + A1 + A2, which are due to collisions of polarized electrons with polarized target, do not depend on the FF f ( K 2) and therefore can be predicted exactly. This information about asymmetries can be exploited for the measurement of polarization of high energy electron beams (after scattering from target with vector polarization) and for the determination of tensor polarization of targets (at the scattering unpolarized electrons). e-+l
+~e-
+0 ±+0 T
and
e-+l----,e-+O
± + 0 ±.
The threshold amplitude of any such process is determined by two independent electromagnetic FF's: Fth = e . Ugl (K 2) + e . k U . ~g2(K2).
(13)
The form factor gl characterizes the absorption of transversal electric dipole (El) virtual photons in y * + A ---, A1 + A2, and the FF's combination gl(K 2) + g2(K2) characterizes the absorption of longitudinal E1 virtual photons. The SF's Wi(K2) can be expressed in terms of these FF's as, 3WI =W7 = W8 = Igl (K2)I 2,
E2 w2 = ~--M-2Ig] (K2) + g2(~2)12, w3 = 2-~-Re gl (K2) (g] (K2) + g2(K2) )*, w4-
E-2~/] - M g]( K2) 12- ~ER e g ] ( K 2)(g](K 2) +g2(K2)) *,
W5= ~Mlmg] (K2)g~(K2), W6 = 0 ,
w9 = Regl (K2)g~(K2), WlO= Imgl (K2)g~(K2).
(14)
In T-invariant hadron electrodynamics, both FF's gl (K2) and g2(K2) are complex functions of K2, but their relative phase must be equal 0 or 7. Therefore, ws(K2) = Wl0(K2) = 0 for any model of FF's g](K 2) and g2(K2).
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Reaction e - + 6Li ---+ e - + 7r° + 6Li* (0+)is one example of the above mentioned set of spins and parities of participating hadrons. e-+l+---+e-+l++0
q:
and
e-+l---+e-+l++0
+.
The neutral pion electroproduction of deuteron, e - ÷ d --+ e - + d + rr °, is in this class of process. The threshold amplitude of any such processes can be parametrized in the following form: Fth =gl (K2) e ' U1 × U~ + g2(K 2) (e × k . U j & . U~. + e × k - U ~ k - U I ) +g3(tc2)e • ~ k . UI )< U~,
(15)
where U1 and U2 are 3-vectors of polarizations of initial and final vector particles. The form factor gl(K 2) defines the absorption of transversal E1 virtual photons in 3'* + A + AI + A2, g2(KZ)-magnetic quadrupole (M2), and the sum gl(K 2) + g3(t<2) describes the absorption of longitudinal E1 virtual photons. Namely, only three such multipole transitions are allowed in y* ÷ A -+ A1 + A2 with the above mentioned set of the spins and P-parities. The expressions for the SF's in terms of these FF's are given as, w, - ~ Ig, - g2l 2 + ~-2lg' + g2l 2 , w2 = 2(Igl + g312), E w3 = - ~ - ~ R e (gj + g2) (gl + g3)*, E w4 = Igl -- g2] 2 ÷ ~ - ~ R e (gl ÷ g2) (gl ÷ g3)*, E I m ( g l + g2)(gl + g3)*, 2M w6 = 4Re gl g~, ws=-
w7 -- - [ g l - g2l 2 - [gl + gs[ 2, w8 = - I g ~
- g212 + 2Re (gl + g2) (gl + g3)*,
w9 -- IgJ - g21e - Re (gl + g2) (gl + g3)*, WlO = - I m (gl + g 2 ) ( g l + g3)*,
(16)
The SF's wl, w2 and w7 are determined only by combinations of FF's squares; [g~ +g212 and [gl +g312. Any experimental information about processes e - + d --+ e - + d + pO, where p0 = rr or r/-meson, is not available now. But near reaction threshold the experimental study of these processes is interesting, because it provides a possibility to test the predictions of low energy theorems (LET) for photo-and electro-production of the pions on nucleons, T + N ~ N + r r and e - + N ---+ e - + N + r r [14]. One example of such LET is the well known Kroll-Ruderman theorem [ 15 ] about threshold values of electric dipole amplitude
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EO+ for different y + N --~ N + 7r processes. Current algebra techniques advances the generalization of these predictions for the pion electroproduction [ 16]. After a long history with consistent predictions of experiment, it was found that the experimental value of E o + ( y p ~ p~r°) = ( - 0 . 5 d z O . 3 ) e / m ~ r 10 -3 [17] deviates essentially from the LET predictions E0+ (3"p ~ p¢r°) = - 2 . 4 e / m ~ r 10 -3. This result has been supported by another experiment [18], which has also demonstrated the strong energy dependence of the amplitude E0+ for 3"p --~ p~r ° (near threshold). All attempts the revision and improvement of classical LET were unsuccessful over a similar history [ 14]. New experiments [ 19,20] with tagged photons for 3' + p --* p + ~r° show a strong deviation from the predictions of LET. This discrepancy can be explained in the framework of chiral versions of non-perturbative QCD by some special contribution with non-analytical logarithmic singularity, which were not considered in previous studies of LET [21]. All these results show that the physics of threshold hadron electrodynamics is worth of renewal of interest. The next step in testing all these predictions is the study of 7r°-production in 3' + n ~ n + ~.0 and e - + n ~ e - + n + 7r° processes. And the best way to clarify this problem is to measure the coherent processes on the deuteron target, y + d --+ d + ¢r° and e - + d ~ e - + d + 7r°, which is expected to be sensitive to small neutron amplitudes. The ~7-production on deuteron, 3' + d ~ d + r/, remains also as an important problem of hadron electrodynamics. The contradiction between the large values of cross-section tbr reaction y + d --+ d + r/ near threshold [22] and the isovector nature of transition y + N ---* Sll (1535), which follows from the results of multipole analysis for 3' + N --~ N + ¢r processes [23-26] and predictions of quark models [27-31 ], is notified after the recent experiment [32], which demonstrated the large contribution of inelastic deuteron disintegration process, namely, 3" + d ~ n + p + r/. Electroproduction of ~/-mesons on deuteron, e - + d ~ e - + d + r / i n the near threshold region is another open problem. The formalism of the threshold FF's seems as the most suitable approach for phenomenological description of such processes. e-+l+-+e-+0±+l
i
and
e-+l---~e-+0±+l
i
Electrodisintegration of 6Li, e - + 6Li --+ e - + d + 4He, is one example of such a reaction. The threshold amplitude of any process 3'* + A --~ A1 + A2 with such set of spins and parities can be written in the following form: Fth = g l e . U l k
. U~ + g2e . U~2~C . U l + g3e . k U I . U~2 + g4e . k~c . U l i c . U~.
(17)
The absorption of MI virtual photons is characterized by the combination gl - g 2 and the absorption of electric quadrupole (E2) virtual photons with transverse polarizationby the sum of gl + g2. The form factors g3 defines the absorption of monopole (E0) virtual photons, and sum gl + g2 + g3 + g4 defines the absorption of longitudinal E2 photons.
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After summing over the polarizations of vector particles in the final state of y* + A Aj + A2, the following expression for the SF's wi(K 2) is obtained
w~ = ½(Ig~[ + Ig=12), E2 1 w2 = g(21g312 + ~--~lglr2), E w3 = ~--~Re (glg~ - g2g3), * W4 = l l g 112 _
E Re (glgL, -- g2g~), 2M
E
w5 = - ~---~Im (gjg~ - g2g~), W6 = Jg212,
w7 = dg212 + Ig412 + 2Re (g2 + g3)g~, w8 = Igl 12, W9 = R e
(gl(g2 ÷g3 ÷g4)*
÷g2g~),
wlo = Im (glg[ + g2g~),
(18)
where gL = gl + g2 + g3 ÷ g4. e - + ~ l+ - - ~ e - + ~ I + + 1 ~
and
e - + ~ 1 - - - - ~ e - + ~1 ± + i T"
The electrodisintegration of three nucleon nuclei, 3He and 3H, namely
e-+3He---~e-+d+p,
e-+3H--*e-+d+n
(19)
belongs to this class of processes. In general case, the amplitude of such processes have very complicated spin structure and is described in terms of eighteen independent amplitudes, which depend on the three different kinematical variables. But the situation is simplified essentially at the threshold of reaction, when the d + N system is produced in the S-state: Fth = .)(~ [ifl (t<2)e • U* x k + f2(K2)O ". k e " U* + f 3 ( t ¢ 2 ) o ". eU. f¢ + f 4 ( K 2 ) e • k o ' . U* + f s ( K 2 ) o -. k e . KU*. &IX1,
(2O)
where XJ,2 are the two component spinors of initial and final particles with spin 1/2 (in y* + A --, AI + A2 transitions). The allowed threshold multipole transitions are described by the following five independent combinations of FF's fi(K2): MI ---~ S = gJ
"
M1 ---+ S = .3 2 ." E2T --, S = !2 •I
EOL-+ S=~ :
(fl
+ f2 - fs),
(2f~ - f2 + f 3 ) , (f2 + f3),
(f2+f3
+f4+.fs),
C. Yal(inet al./NuclearPhysicsA 635 (1998)387-407 E2L --+ S= 3_. 2
(f2 + f3 + fs)
397 (21)
where S is total spin of A1 + A2-system; T and L denote the transverse and longitudinal polarizations of virtual photons. Target spin is equal to 1/2; therefore, only the following five SF's are non-zero: w](K 2) = [fll 2 + [f2l 2 + If3l 2, W2(K2) = If2 + f3 + f4 + fsl 2 + 21f412,
W3(K2) = - I f 3 l 2 - 2Re [(fl - f 2 ) ~ + f 3 ( f 2 + f4 + f s ) * ] , W4(K 2) =2Re [(fl - f2)f~4 + f 3 ( f 2 + f4 + fs)*] , w5(K 2 ) = 2 I m [ ( f l - f2)f~4+ f 3 ( f 2 + f 4 + f 5 ) * e
-
+l+--÷e-+~
14-
+$
14-
and
1•
(22) 14-
e-+l---+e-+$
lq: +$
The threshold amplitude of the process 3/* + A --+ AI + A2 is defined in this case by a set of following five electromagnetic FF's: Fth = 2"I [ig] (K2)e • U × ~' + g2(K 2) ( o ' . k e . U - o'- e k . U) +g3(K 2) (O'. K'e" U + o'. e~'. U - 2~r. k e - k U . ~) +g4(K2)o "" ~'e. k U . k + g5(K2)o ".
Ue.
k] o'2,~,
(23)
where Xl and 2"2 are the two component spinors of the produced particles A] and A2. The spin structures in Eq. (23) are chosen in this form, so that each FF describes the definite threshold transition in reaction y* + A ~ A1 + A2: MI --+ S = 0 (g]), M1 ---+ S = 1 (g2) E2T --+ S = 1 (g3), E2L ---+S = 1 (g4) and E0L --+ S = 1 (g5). Summing over the polarizations of A1 and A2, one can obtain the following expressions for the SF's in terms of the threshold electromagnetic FF's:
+ [g2 + g312 + --llTl-~lg2- g312
w1('¢2)=~
Igl
w2('~z) = g
Igs[2 + ~ l g 4 +g512
4(
w3(K2) = ~ R e
)
,
,
(g2 + g3) (g4 + g5)* + (g2 --
g3)g~5 ,
W4(K2) =lg|12 + Ig2+g312-- -~-Re [(g2+g3)(g4+g5)*+ (g2--g3)g;], Ws(K 2) = - ~ I m
(ga
+g3)(g4 +g5)* +
(g2 --
g3)g~5 ,
M2 [2, W6(K2) = _2~__5 [g] [2 + 21g2 + g3 wT(K2) = --21gil 2 + 2[g2 + g3l2 + 21g412 + 21g5[2
1-
398
C. Yal¢in et al./Nuclear Physics A 635 (1998) 387-407
+2Re [- (g2 + g3)g*4+ g4g~- 2g3g~1, ws(K 2) = - 2 ] g l 12 + 21g2 + g3l 2, W9( K2) ----21gl[ 2 -- 21g2 + g3[ 2 ÷ 2Re [(g2 +
WlO(K2)=2Im[(g2+g3)g*4 +
2gzg~] •
g3)g~+ 2g3g~], (24)
In this context, we now point out, that Eqs. (23) and (24) describe the threshold deuteron electrodisintegration, e - + d --~ e - + n + p. Interest in this kinematical region for this process is due to a number of reasons. These are: (i) A relatively simple spin structure of y* ÷ d -~ n + p amplitude. In the general case, at least 18 independent amplitudes are necessary for the complete description of matrix element of e - + d ---, e - + n + p (in the one-photon approximation). (ii) The square of momentum transfer K2 is related directly to the argument of the deuteron wave function. Therefore, the differential cross-section for d(e,e')pn near threshold measures the deuteron wave function. (iii) The threshold amplitudes of y* + d --~ n + p process depend only on one variable K2, while beyond the threshold, the same amplitudes depend on three different kinematical variables. (iv) A variety of possible polarization effects even near threshold of the y * + d ~ n + p reaction. (v) The relation between the isotopic spin selection rules and the spin structure of the threshold amplitude. (vi) The possibility of using the e - + d ~ e - + n + p process near threshold to test the symmetry properties of the hadron electromagnetic interactions (checking the P-and T-invariances). (vii) The threshold amplitudes are sensitive to such contributions arises from meson exchange currents, isobar configurations in deuteron and quark degrees of freedom. (viii) The threshold amplitudes of y* + d --, n + p process are real function of K2-for any reaction mechanism- due to the unitarity condition. (ix) Very short "distance" between elastic ed-scattering, e - + d ~ e - + d, and the threshold electrodisintegration, e - ÷ d --, e - + n + p , must result in some similarity between these processes. (x) The more simple experiment for the inclusive process d(e, e')np near threshold is also as informative as the more complicated experiment for the exclusive reaction
d(e, elp)n. The electrodisintegration of 6Li ( e - + 6Li ---+ e - + 3He ÷ 3H), is very similar to electrodisintegration of deuteron ( e - + d ~ e - + n + p ) in spin structure of amplitudes.
C. Yal(in et al./Nuclear Physics A 635 (1998) 387-407
399
4. Discussion of complete experiment Using results of the above analysis, it is possible to find the most optimal strategy of complete experiment for all the aforementioned processes, which allows to reconstruct the complete spin structure of the threshold amplitudes, i.e. for the determination of all the threshold electromagnetic FF's. For this discussion it is necessary to discriminate between two different possibilities. The simplest one corresponds to electro-production (or - disintegration) processes with a single value of the total angular momentum J of the produced A1 + A2-system, when all the threshold FF's are real function of K2 (with the relative phase 0 or ~')-following the theorem of Christ and Lee [11]. This case is realized when A1 (or A2) has spin equal to zero:
e- +N---~e- +N+~, e - + d ---* e - + d + 7r(r/), e - + 6Li ---+e - + d + 4He. Second possibility corresponds to the processes e - + A ---, e - + A1 + A2, when Al + A2-system has different values of J. Thus, the corresponding form factors can have in the general case non-zero relative phase which can be determined in special polarization experiments. In discussion of complete experiment it is necessary to keep in mind, that at the threshold of any process e - + A ~ e - + A1 + A2 not only the experiments with polarized target A is possible, but for a large enough IK21 it is possible to measure also the polarization of produced particles - due to the larger its 3-momentum - in comparison with the photo-production processes y + A ~ A~ + A2. Let us illustrate this with the example processes e - + N ~ e - + N + T r and e - + d -+ e - + n + p , when the threshold kinetic energy Tp of produced proton is controlled by the value x 2 only: K2
Tp-
4M'
e-+d---~e-+n+p,
where M is the deuteron mass (neglecting the deuteron binding energy in comparison with the nucleon mass), and l K 2 ÷ /~2
Tp
-
K2
- ~--2 m + lx 2m'
if
Iz/m<
e-+p---~e-+p+rr,
where m ( / . ) is the proton (pion) mass. Thus, examination of the rr°-production threshold in g-- + p --+ e - + ff + rr° with its accompanying new and interesting dynamical information, might best be studied in the TJNAF [33] experiment with its polarized e - (and unpolarized p) beam and proton polarization analyzer. In any case, the polarization measurements are unnecessary to realize the complete experiment for the process class e - + A --+ e - + Al + A2 which is characterized by one or two electromagnetic FF's:
C Yalgin et al./Nuclear Physics A 635 (1998) 387-407
400
e - q, 1+ - - ~ e - q , O + q . O ~
and
e- q,l- ~e-q.O
+ q . O m,
e - q.1 + - - - + e - + O + q . O m
and
e- q,1- ~e-q.O
± q , O ±.
In all these reactions the complete experiment is equivalent to measuring only the differential cross-section (with unpolarized particles). But, the Rosenbluth separation of the different contributions to the inclusive cross-section:
-
62o -
dE2 df2e
~
O-T q - 6OrL,
where O-T and OrL are the cross-sections for y* + A --~ A~ + A2 with absorption of virtual photons y" with transversal and longitudinal polarizations, respectively, must be done. Note, that these measurements cannot determine the sign of electromagnetic threshold FF's. So, in these cases, it is possible to use the standard procedure, which is so effective for the determination of nucleon electromagnetic FF's. Namely, starting from the precisely known values of the static electromagnetic characteristics of the nucleons (elastic charges and magnetic moments, corresponding to FF's at K 2 = 0 ) , it is possible to extrapolate the sign step by step for any value of momentum transfer square, K2. It is also possible to exploit the same method for any process e - q, A ---, e - q, A I q" A2, using the sign of corresponding multipole amplitudes for y q, A ---, A1 q, A2, i.e. at K2 = 0, which can be determined directly or with the help of a more resolved theoretical model. Note also, that if one form factor (longitudinal or transversal) is small, its determination with respect to a Rosenbluth fit may not be reasonable accuracy (for example, in the case of the electric FF of the neutron). In such a situation, the special polarization experiments are necessary, where the result is determined by the interference of longitudinal and transversal FF's. For the process e - q, A --~ e - q, Aj + A2 the most suitable experiment is the measurement of asymmetry of scattering longitudinally polarized electrons by a vector polarized target, when the corresponding SF w3 is determined by the product of transversal (gl) and the longitudinal (gl q" g2) electromagnetic FF's (c.f. Eq. (14)). But if the number of electromagnetic FF's is larger than 2, the polarization phenomena must be studied for the full reconstruction of the spin structure of the threshold electromagnetic current. For the reactions e - + 1+ --, e - q, 1± q.0 m and e - q, I - ---, e - q, 1+ q-0 +, the complete experiment for the determination of 3 independent FF's (Eq. (15)) must include the following measurements: differential cross-section with unpolarized particles (Rosenbluth separation of o-x and O-L); tensor analyzing power in scattering unpolarized electrons by tensorially polarized target. So in this case we have a situation, which is very similar to the elastic ed-scattering [34], where, for the separation of electric and quadrupole electromagnetic FF's the measurement of tensor polarization of scattered deuteron (or the tensor analyzing power) -
-
C. Yalgin et al./Nuclear Physics A 635 (1998) 387-407
401
is necessary. In this case, it is necessary to measure only 7"20 from three possible tensor analyzing powers; namely, T20, T21 and T22. Another situation appears for the process e - + 1+ ~ e - + 0 + + 1+ and e - + 1e - + 0 ~- + l T with four independent electromagnetic FF's (Eq. (17)). It can be see from Eq. (18), that the measurement of SF's Wl, w2, w6 and w7 will allow to determine moduli of all these FF's. More exactly, for this case it is necessary to know the relative phase of g4 and (g2 +g3 ). So, again it is necessary to "organize" the unpolarized electron scattering by the target with tensor polarization. But it is important to stress that, for all these reactions, the scattering unpolarized electrons only is necessary. However, the polarized electron beam is evidently necessary to realize the complete experiment for p r o c e s s e s e - + ~ l+ ~ e - + ~ 1+ + 1± a n d e - + ½- ~ e - + 7 1± + l t : . I t can be realized from Eq. (22), that the full reconstruction of five electromagnetic FF's is not possible without measuring polarization of produced nuclei. The processes e - + 1+ --~ e - + ½+ + ½~: and e - + 1- --+ e - + 7 + + 1:F (with five electromagnetic FF's) represent the most complicated case (from considered above). Eq. (24) for SF's wi reveals that SF's w l, w3 + w4 and w6 will allow to determine moduli of all three transversal FF's, namely, Ig~12 and [g2 + g312. Adding to these measurements the SF's w2 and w3, it is possible to determine the longitudinal FF's g4, gs. Moreover, the measurement of SF's w7 and w8 will be useful for decreasing the number of possible solutions for FF's. In conclusion, let us stress that all SF's Wl - WlO, which determine the scattering polarized electrons by polarized target, are insensitive to the relative phase of singlet form factor gl and triplet FF's g2 - gs. To determine this phase the measurement of polarization of one produced fermion is necessary. In threshold conditions, non-zero polarization is possible only in the case of scattering longitudinally polarized electrons.
5. P - o d d effects in processes e - + A --~ e - + A1 + A2
In this paper only the one photon mechanism, which arises as a result of P-odd effects due to P-invariance violation in 3/* + A ~ A1 + A2 processes [35-37], has been considered. Another possible source of P-odd effects in 7" + A ---, Al + A2 is the Z-boson exchange between weak neutral currents of electron and hadrons [38,39]. At relatively small values of K2, the one-photon mechanism of P-odd effects is more dominant, whereas the mechanism with the weak neutral currents is dominant at the large values of the momentum transfer. All P-odd asymmetries in e - + A ~ e - + Ai + A2 can be extracted from the general formula, given by Eq. (1), if the following changes are introduced,
Hij --+ Hij + fiIij,
ffIij = Ji]'} + ]iJ~,
(25)
where J is the P-odd part of the electromagnetic current for the process T* + A --+ Ai +A2.
C. Yalqin et al./Nuclear Physics A 635 (1998) 387-407
402
For the target with vector and tensor polarizations, the most general formula for the threshold tensor/4ii is as follows:
ffIi.j=i~ugKglTVI(K 2) ÷ S. t¢
f(~ij -
÷ ( SiKj ÷ SjKi)17V4( K2 ) ÷
KiKj)ITF2(K2) + KiKjlTv3(K2)]
i( Sikj - Siki)~vs( K2)
+ iEijgKg( SabKaKb)W6( K2)
+i(eiabSackbkckj - ".iabSacKb~
+(eiabSiako + ej~bSiakb)~V9(K 2) + (eiabki + e i~bki)Sackbkc#,O(K2),
(26)
where ~i(K 2) are real P-odd SF's. So, the P-odd effects near threshold of reaction e- + A ---+ e- + A] + A2 can arise at the scattering of unpolarized electrons by polarized target with vector polarization; in the case of scattering longitudinally polarized electrons, the P-odd effects arise for the unpolarized target or target with tensor polarization. SF's g'5 (K2) characterizes the P-and T-odd effects (simultaneously) in the scattering longitudinally polarized electrons by target with vector polarization, and ~9(K 2) and ~'lo(K2) characterize the P-and T-odd polarization effects at the scattering unpolarized electrons at the target with tensor polarization.
6. Calculation of P - o d d structure functions
In this section we present the parametrization of the P-odd part of electromagnetic current and the calculation of P-odd SF's ~i(K 2) in terms of corresponding FF's. e- + 1 + - - - , e - + O + + 0 :t: and
e-+l--+e-
+ 0 + + 1 ~:.
The P-odd amplitude is defined in terms of two electromagnetic FF's as Pth = e. Up] (1(2) + e. kU-1~'/02(K2),
(27)
which describe two possible P-odd transitions with absorption of E1T(jO1 ) and E1L(ffl +
P2). Using the P-even amplitude Eq. (10), for the P-odd SF's ~'i(K 2) the following expressions are found I~ 1 (K 2) = ½Replf*, l~2(K 2) = -Reffl f*,
~,3(,¢2) = - E R e (p~ + P2) f*, M
~'4(K2) = ~ E R e (Pl +/~2) f * ,
2M
C. Yalginet aL/Nuclear PhysicsA 635 (1998) 387-407 ~:5 (t<2) = ----~Im (/~1 + 2
403
P2)ff,
1~6(K 2) ----0,
u:7 (K2) = -Re/~2f ~, #8(K 2) = -Replf* ~9( K2) = -Imp1 i f , v~lO(K2) = - I m p z F .
(28)
The description of P-odd effects for e- + 1+ ---, e- + 0 :F + 0 ± and e- + 1- --~ e- + 0 ± + 0 ± processes is obtained by interchanging P-odd and P-even amplitudes in Eq. (28).
e-+l+~e-+l++0
~: and e - + l - - - + e - + l ± + 0
±.
The P-odd threshold amplitude for these processes is defined by four corresponding electromagnetic FF's: i/Wth = e. U I k - U ~ I + e. U~2 k - Ulg'2 + e. kU1 - U~3 + e. I~UI. I~U~' i~g4.
(29)
These form factors describe the following P-odd multipole transitions: M1 (~l -g'2), E2T(~I + g'2), EOL(~3) and E2L(~l + g2 -'F-g4). Using P-even amplitude for such processes, Eq. (15), and the definition of tensor /7/ij, after summing over the polarizations of vector particles in the final state, we obtain: ~l(K2)_
2 E2 3 ~-~Re~2(gl + g2)*,
~2(K 2) = Re~l (gl - g2)*, 1,~'3( K 2 ) = E R e [(gl + g2 + g3 + g4)(gl - g2)* - g2(gl + g3)* + gs(gl + gz)* --
2 ~M@ 3 ( g l
+
g2)*] ,
#4(K 2) = ~--~Re (~1 + ~2 + g3 -1- g4) ( - - g l -[- g2) * -[- g 2 ( g l ~- g3) * -- g 3 ( g l + g2)* , #5(K 2) = ~-~Im (~l + g2 + ~3 + ,~4)( - g l + g2)* - ~2(gl + g3)* + g 3 ( g l + g2)* , #6(K 2) = - 2 R e g'2(gl + g2)*,
WT(t<2)=Re [ ( g 2 - I - g , 3 + g 4 ) ( g l - g 2 ) * - g g ( g l + g 2 ) * ] , ~8(K 2) =Re~l (gl - g2)*, #9(K 2) = Im~l (gl -- g2)*,
7Vlo(K2) =Im [(~,2 + ~3 + ~4)(gl -- g2)* -- ~2(gl + g3)* -- ~3(gl + g2)*].
(30)
404
C. Yalfin et al./Nuclear Physics A 635 (1998) 387-407
After interchanging the P-odd and P-even amplitudes, these formulas describe the SF's ff~i( K2) for processes e - + 1+ --~ e - + 14- + 04- and e - + 1- ~ e - + 17: + 0 +. e
-
+~
! +
- - - ~ e - + i14- + 14-
and
e-+~
1 -
- - - ~ e - + ~1 + + lq: .
The threshold P-odd amplitude can be written in the following form: Fth = X~ [fie.
U* + if20".e × U* + if30".e x &ft. U*
+f4e. iek. U* +ifso'. k × U*e. k]X1.
(31)
P-odd form factors fi are determined five threshold multipole transitions with Pparity non-conservation: E1T ~ S = ½, EIT ~ S = 3; M2 ---, S = 3, E l L ~ S = ½ and E l L --~ S = 3, where S is total spin of particles A1 and A2. After summing over polarizations of particles Aj and A2, and by making use of the formulas for Fth, Eq. (20), and /~'th, Eq. (31), we obtain the following expressions for the P-odd SF's ~1 - ~5:
w,(K2) = 2Re [flf~t + f 2 ( f 2 - f 3 ) * - fgf~31, wz(K2) = 2Re [flf~2 + f2(f, + f3)* + faf~31, w3(K 2) = 2Re If1 (f2 +
fs)* + (f2 + fs) f~ + f2(f3 + fs)*
+f3 (f2 4- f3 + f4 4- f5)* -I- f4(f2 "Jr-f4 4- fs)* -W4(K2) = Re
fs(f2
+
2f4)* / ,
[f~(f3 + f4)* - (f2 + fs) f~l - f z ( f 3 + fs)* + f3 + f4 + fs)* + f4f~3 + fsf~2 -- 2f2f~4/ ,
ws(K 2) = Im [ ( f , -- f 2 ) ~ -- ( f , + J % ) ~ + (f2 +
f s ) ( f , -- f2)*
-- (f2 4- f3) (/2 + f3 4- f4 4- fs)*].
(32)
These formulas describe the P-odd effects at the threshold of electrodisintegration of polarized nuclei 3He, e - + 3He ~ e - 4- p + d. e
--
4 - 1 + - - - ~ e - + g1 -/- + g14-
and
e-+l---+e-+g
14- + gI T
P-odd electromagnetic current for these processes is characterized by five FF's:
Pth = 22 [~ze . U + i~,2o-. e x U + i~3o" . e x icU . i¢ L
C. Yalfinet aL/Nuclear PhysicsA 635 (1998)387-407 q-ig40" • k × U e . k q- ~5e. ~U- KI Or2"~"
405 (33)
These form factors describe five independent P-odd multipole transitions namely: E 1T --~ S = 0(~l), E1T --~ S = 1 (~2), M2 --* S = 1(~3), E l L ~ S = 0(~1 + g s ) and E l L ---+ S = |(g2 q- g4). The SF's w/(K 2) can be written in the following form: I~I(K2) = 2Re glg~ q-g2(g2 -~-g3)* -I- ~ ( g 2
-~g3)(g2 - g3)* ,
@2(K2) = Re lglg*l -- g2(g2 q- g3)*] , #3(K 2) = ~ R e
(gl + gs)g~ + ~2(g2 - g3 + g4 -t- 2g5)*
+g3g5 ~ * + g4(g2
- g3)*]
d
- 2Re (g2 q- g4)gs, ~ *
w4(K2) = - 2 M Re (gl + gs)g~ + g2(g2 - g3 + g4 + 2g5)* +g3g5 ~ * + g4(g2
-- g3)*]
..I
~5(K 2) = --~-E Im2M [(gl + gs)g~ + g2(-g2 +g3 -l-g4 + 2g5)*
]
- * +g395 + g4(--g2 + g3)* ,
#6 ( K2) = 2Re g3 ( -g2 + g3)*, #7(K2)=Re
g2(--2gz+g4)*--g3gs+g4(--g2+g3)*
gsg] ,
Ws(K2)=Re[glg*l+g2(g2+g3)* 1,
1 ~lo(K2)=Im[~2(293--g4)*+~,3g*5+~4(--g2q-g3)*+~Sg*l].
(34)
All these expressions, which we have presented above, can be used in the analysis of P-odd polarization effects at the threshold of deuteron electrodisintegration..
7.
Conclusion
The analysis, presented in this report, is valid for a wide class of hadronic and nuclear processes, such as, for example:
406
C. Yalgin et aL/Nuclear Physics A 635 (1998) 387-407
e - q. N--~ e - q. N + 7"r, e- +N--+ e- +N-t-o-,
e - + 3He ---+ e e - + 3He --+ e e q.N---~e-q,A+K, e-q.3He---~e e - + N --~ e - + N q, p(oJ, q~), e - + 4 H e --+ e e - q-d--+ e - q . d + 7 " r °, e- +6Li--~ ee - q. d ---~ e - q, n q,p, e - q,6Li --~ e -
+p+d, q, 3He q, 7r°, +Tr++3H. + 4He q, 7r°, q,3Heq.3H, q, d q, 4He,
It is demonstrated that the electromagnetic FF's formalism is effective enough for the parametrization of electromagnetic current for 3'*+ A --+ A 1q"A2 processes near threshold. The explicit forms of the structure functions can be taken as the kinematical base of analysis of different polarization effects in such processes. The accurate measurements of the threshold electromagnetic FF's will reveal the dynamics of 9'* q" A --~ A1 q, A2 processes. The parametrization of electromagnetic currents for the processes, considered in this report, in terms of threshold electromagnetic FF's, and the calculation of the structure functions, which parametrize the polarization effects in electron-hadron scattering processes, is the first necessary step in the analysis of electrodisintegration (and electroproduction) processes near threshold. Within this approach, we establish the spin structure of amplitudes for y* + A ~ A l + A2 in most general form, which is consistent with the symmetry properties of hadron electromagnetic interactions. The structure of interacting nuclei and the mechanism of reaction y* q. A ---, Al q, A2 contribute to the explicit form of corresponding FF's only. More exactly, the nuclei can be considered in such formalism as elementary particles with the definite values of spin and space parity. But we use essentially the conservation of the hadronic electromagnetic current and the P-and T-invariances of the electromagnetic interaction of hadrons. The relativistic kinematics and the spin effects for all considered above processes y*q,A --~ A1 q,A2 are taken into account exactly. The dynamics of processes y" q, A --+ Al q. A2, in conjunction with nuclear structure, such as, exact form of electromagnetic interaction, meson exchange currents, quark phenomenology etc., is reflected in the threshold electromagnetic FF's. Summarizing all this we must repeat once more, that like this formalism for the description of phenomenology of the threshold electrodisintegration (or electroproduction) processes in terms of corresponding FF's is, evidently, the necessary first step in the kinematical analysis of inelastic processes, similarly to the well-known procedure, which is used usually for the description of elastic electron-hadron scattering. In both case we have a limited sets of electromagnetic form factors, depending on the momentum transfer square u2, only. Therefore, all experimental information about these processes must be formulated in terms of the threshold electromagnetic FF's, as well as a theoretical predictions, which are used a definite models for the discussed processes. Various processes of neutrino-hadron interactions, such as u~ + A --~/z + Al q. A2 and u# + A --, u u q, Ai + A2, due to charged or neutral weak currents, can be described also in terms of weak FF's for the threshold transitions W* q, A ---+ Ai q, A2 and Z* q. A ---, Ai q, A2, where W* and Z* are virtual W and Z bosons.
C. Yale'in et al./Nuclear Physics A 635 (1998) 387-407
407
Acknowledgements W e t h a n k J. A r v i e u x , J.M. L a g e t , M. G a r c o n a n d E. T o m a s i - G u s t a f s s o n for useful d i s c u s s i o n s o f t h r e s h o l d p h y s i c s in C E B A F c o n d i t i o n s . O n e o f the a u t h o r s ( M . R . ) also t h a n k s T U B I T A K - N A T O - C P G r a n t , w h i c h m a d e this visit to M i d d l e E a s t T e c h n i c a l University (Ankara, Turkey) possible.
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