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Neutrino electron scattering in theories with neutral currents
Nuclear Physics B70 (1974) 61 69. North-Holland Publishing Company
NEUTRINO ELECTRON SCATTERING IN THEORIES WITH N E U T R A L CURRENTS * L.M. S E H G A L
111. Physikalisches Institut, Technisehe Hochschule Aachen, Aachen, W-Germany Received 7 October 1973
Abstract: Elastic neutrino electron scattering is considered in a general class of theories containing both charged and neutral currents. Assuming ~-e symmetry, the following results are proved. (i) x/O(vee) + x/o(vwe ) >- x/'2, (ii) xfo (~ee) + x/a (rue)/> x / ~ , (iii) Offee) - ague ) = ~ [O(Vee) - o(vue)] , (iv) ] [a (vee) - ~ a (~ee)] ~ +- [o (rue) - I a (~.e] ~ I = 4 , (where cross sections are in units of G2mEv/n). Constraints are derived on the energy spectrum of the recoil electron. The Weinberg-Salam theory and the V-A theory are considered as special cases.
1. Introduction There is n o w significant experimental evidence for the existence o f neutral currents in neutrino-induced reactions [1 ]. Both at CERN-Gargamelle [2] and at N.A.L. [3] a signal has been obtained for the semileptonic reactions vg(ffu) + nucleon
vu(vu) + hadrons. A candidate has also been found for the purely leptonic process v u + e -+T~ + e - in the Gargamelle e x p e r i m e n t [4]. These observations suggest that neutral currents play an essential role in weak interactions and compel a reexamination o f ideas based on the traditional V-A t h e o r y [5]. This paper studies the implications o f neutral currents for elastic n e u t r i n o electron scattering. We consider a broad class o f theories containing b o t h charged and neutral currents, which is a generalisation o f the intermediate vector boson model, and which is sufficiently general to a c c o m m o d a t e models that arise in spontaneously b r o k e n gauge theories. The Weinberg-Salam theory [6] and the V-A t h e o r y appear as special cases. Relations are derived among the four distinct processes o f interest: Vee -+ Vee , Fee ~ Ve e, rue ~ v u e , ~ e - ~ u e ; constraints are obtained on the total cross sections * Work supported by the German Bundesministerium ftir Forschung und Technologie.
L.M. Sehgal, Neutrino electron scattering
62
and the energy (or angle) distribution of the recoil electron. (A few of the results obtained here were reported earlier [7].)
2. A general Lagrangian for neutrino electron scattering We assume that weak teptonic processes are mediated by (i) an arbitrary number of charged spin-one bosons, (ii) an arbitrary number of neutral spin-one bosons, (iii) an arbitrary number of neutral spin-zero bosons. We require that the Lagrangian satisfy (i) the two-component neutrino hypothesis, (ii) separate conservation of muon and electron number, (iii)/a-e universality. The general Lagrangian consistent with these assumptions is
Z? = ~ g i + ~f/ ]
[e 7a(1 +3,5) .ve + (e ~ ) ]
w(i)+a h.c.
[VeTa(1 +')'5)Ve+eT~(CJV +C]A 75) e + ( e ~ t ) ]
(1) Z~')+ .~ (scalars).
In eq. (1), W (i) and Z(/) denote respectively the sets of charged and neutral spin-one intermediate bosons, and gi, fl" CIV' CIA are arbitrary real constants../3(scalars) denotes terms containing neutral spin-zero fields coupled to neutral lepton currents. We note that the above Lagrangian encompasses models of the type arising in spontaneously broken gauge theories, which contain in general neutral spin-zero particles (the Higgs fields) in addition to charged and neutral spin-one bosons (the gauge fields). Our objective is to derive the consequences of the interaction (1) for the processes Vee -+ Vee,~ee -+ fie e, rue -+ rue , Fue -+F,e in the lowest order of the coupling constants. To this end, it is crucial to observe that the expression for .~ (scalars) cannot contain any terms involving neutral neutrino currents, since a neutral spin-zero field cannot couple to (or decay into) a ~ pair. Thus, the existence of spin-zero neutral bosons (such as the Higgs particles of gauge theories) has no influence on lowest order neutrino electron scattering, and the term/2 (scalars) can simply be disregarded in the present discussion. We will be interested in the domain of energies for which
m 41, E 1)
~ ~ 1, M2
(2)
where rn is the electron mass, E v the lab energy of the incident neutrino, s (~ 2mEv) the centre-of-mass energy squared and M the mass of any of the spin-one intermediate bosons. The condition m/E v ,~ 1 is fulfilled at all accelerator energies (but is somewhat inaccurate for reactor experiments). The condition s/M 2 ~ 1 is also expected to be well satisfied even at the high energies of N.A.L. In these circumstances predictions for neutrino-electron scattering may be derived from an effective Lagrangian equivalent to (1),
63
L.M. Sehgal, Neutrino electron scattering
"/2eft-
G ~ [(e3'C~(1 +3'5)Ve)(VeYc~(1 +75) e) + (re 7a(1 +TS) re) (g 7c~(Dv +DA 3'5) e)
(3)
+ ( 7 3'~(1 +3'5) vu) (V 7a(Dv+DA3'5 ) e)] , where 2 v
%' (4)
M(i) n a,(J) being the masses of W(i) and Z q) respectively, and G the Fermi conw ~a"u~v*Z
stant. Eq. O) may be rewritten in a more convenient form [8] by applying a Fierz transformation to the first term, which yields "/2eft-
G ~ , , X/~ [(Ve3'a(1 +3'5) Ve)(e3' (Dv+DA3'5) e)
(5) +(vta 7"(1 +3'5) vg) (e 3'a(D V +D A 3"5) e)] , where D V= 1 +O V ,
DA
1 +D A
(6)
In the Weinberg-Salam theory, the effective Lagrangian for neutrino-electron scattering has the form (5), with [9] D V = - ½ + 2 s i n 20 , (7a) Dh = - ½ ,
(Weinberg-Salam),
0 being the Weinberg angle. The V-A theory, on the other hand, is defined by O V = D A = 0 ,(V-A) .
(7b)
3. Differential and total cross sections. General constraints
We consider here constraints on the differential and total cross sections of neutrino electron scattering arising from the vector (spin one) nature of the couplings'in eq. (5).
L.M. Sehgal, Neutrino electron scattering
64
The differential cross sections for the four types of reaction have the general form (neglecting terms of order m/Ev)
2do(r) ~ - 2G2m ~ IA (r) +B(r) ( k1 -~E- )
2] '
(8)
A (r), B (r) >~ O, where r stands for Vee,~ee , rue o r ~ u e , and E is the (lab.) recoil energy of the electron. We may equivalently write the cross section in terms of the centre-of-mass scattering angle (the angle of the outgoing electron) and the result is d cos 0 -
zr
(r)
+B(r)
.
(9)
The coefficients A (r) and B(r) are determined by the constants D V and D A appearing in ddeffand are listed in table 1, where, for comparison, we also indicate the values in the Weinberg-Salam and V-A theories. The total cross sections are given by G2mE
v [A(r)+~B(r)].
o(r) = 2
(10)
zr The following features emerge. (i) The differential cross section (8) is a quadratic function of E. (ii) Neutrino and antineutrino reactions are related by
A(vee)=B(~ee),
A(~ee)=B(Pee),
A(v e)=B(Y#e),
A(v e)=B(p e).
(11) Table 1 Values of the coefficients A and B occurring in the differential cross section for neutrino electron scattering (eq. (8)):
The constants DV,, DA, D~V,D'A are those appearing in the effective Lagrangian (eq. (5)); D ~ = 1 +Dv, D A = 1 +D A Predictions General
Weinberg-Salam
V-A
A
A B
Reaction A
B
Vee ~ Vee
(} (D~+D~)) 2 (~(D~d-D~)) 2
~ee~ee
2 (~-(D~v-D~)) 2 ({(D~+D~)) 2
v
e~v
e
~ e-+vge
(~(Dv+DA))2
(~.(Dv_DA))2
(~(D 1 V _DA))2 (~(Dv+DA)) 2
B
(~-+ sin20) 2 sin40 (~--sm l - 2 O)2 sin40
sin40 1_ • 2 0) 2 (2+sin
sin40 (~-sin20) 2
1 0 0 1 0 0 0 0
L.M. Sehgal, Neutrino electron scattering
65
(iii) The ratio of neutrino and antineutrino cross sections is bounded by ~<
o(k-ee )
FL O(Uee) '
o (ff u e) O(Uu e)
]<~ 3
(12)
These results are analogous to those obtained in the classic work of Lee and Yang [10], which analysed the consequences of the two-component neutrino hypothesis in V-A theory. In the following section, we derive a number of relations between e-type and/atype neutrino scattering that depend specifically on the assumption of/a-e symmetry.
4. Constraints arising from g-e symmetry
4.1. Constraints on electron energy spectrum From the expressions forA(uee ) and A(uue ) given in table 1, and the relation D V = 1 + D v , D~ = 1 + D A (eq. (6)), we deduce t
x/A-(uee) + vIA (uue) = 1 , or x/a (Uee) - X / ~ u e )
(13a)
= 1 ,
(13b)
or x/A-(pue ) - x/A((uee ) = 1 ,
(13c)
(where the positive square root is understood). This constraint can be written compactly as I x/-A(uee ) -+x/A(uue)l = 1 ,
(141)
and is shown geometrically in fig. 1. The allowed values ofvCA(uue) and x/~(Uee) must lie along the line segments PR, RS or PQ (corresponding to the conditions (13a), (13b) or (13c)). The prediction of the Weinberg-Salam model is shown by the dotted regions of the lines PR (for sin 2 0 < ½) and RS (for sin 2 0 > ½). The V-A prediction is the unique point R. Because of the relation between the A and B coefficients given in (1 1), a constraint similar to (14) exists for the antineutrino reactions, I X/~(Vee ) -+ x / ~ u e ) l
= 1 .
(151)
Also, by virtue ofD~r - D ~ = D V - DA, we note from table 1 the following further relations:
A(vee ) = A ( ~ e ) ,
B(vee ) = B ( v e) .
(16)
Constraints (14), (15) and (16) provide connections between the energy (or angle) distribution in e-type and/a-type neutrino scattering, and depend critically on the assumption of/~-e symmetry.
66
L.M. Sehgal, Neutrino electron scattering
Q
2-
3
P
S
1
l 2
1
3_ 2
2
Fig. 1. Geometrical relation between ~/A~ee ) and A ~ u e ) (eq. (14)). The allowed region consists of the line segments PR, RS and PQ. The Weinberg-Salam prediction is the dotted region of PR (RS) for sin2 0 < 1 (> {). The V-A prediction is the point R. The constraint (15) is obtained by replacing A (Vee) ~ B (~ee), A (vt~e) ~ B (~ue). 4.2. Constraints on total cross sections A n u m b e r of relations can now be derived among the total cross sections of the four reactions. We observe first that eq. (14) implies X/~-(Vee ) + X / ~ u e
) ~> 1
(17)
At the same time, we have from eq. (10) (expressing cross sections in units of G 2mEv/n ) X/O(Vee ) ~> x / 2 A ( v e e ) ,
VcO-(vue)/> ~ u e )
.
(18)
We thus obtain the following remarkable b o u n d on neutrino cross sections: X/b-(Vee) + ~ / o ( v u e ) >~ X / 2 .
(19)
A similar b o u n d is obtained for the antineutrino cross section, b y making use of eq. (15).
"v/oCvee) + X/o(-flue) ~> X/~5
•
(20)
L.M. Sehgal, Neutrino electron scattering
[ol :l-
in units of
61
G2mEv
6 S
/,
2
I
l
k
Fig. 2. Geometrical relation between [o(v~e) - ~ a(vue)] ~ and [a(vee) - ~-O(vee)] ~ (eq. 22)). The allowed region consists of the line segments PR, RS and PQ. The Weinberg-Salam prediction is the dotted region of PR (RS) for sin2 0 < -~(> ~-),while the V-A prediction is the point R. The inequalities in (19) and (20) become equalities in the V-A theory as well as in the Weinberg-Salam model for sin 2 0 = 0. Finally, we note that the cross sections o f the four different reactions are determined by only two unknown parameters D V and D A. Elimination of these gives the following two relations. a(Vee ) -
o(vue ) = 3 [O(~ee ) - o(~ue)]
,
I [O(Vee) - } O(~ee)l ~ -+ [o(v e ) - ~ a(~ue)] ~1 = ~-.
(21) (22)
The last relation is illustrated in fig. 2, where (as in fig. 1) the allowed region consists of the line segments PQ, PR and RS, and the predictions of the V-A and WeinbergSalam theories are as indicated.
5. Comments The discovery of neutral currents in semi-leptonic neutrino reactions suggests strongly the occurrence of such currents in purely leptonic processes as well. Neutrinoelectron scattering, in particular, should exhibit significant deviations from the ex-
L.M. Sehgal, Neutrino electron scattering
68
pectations based on the V-A theory. The precise manner in which neutral currents will be incorporated into the new theory is not known, but such currents do find a natural place (and, indeed, were anticipated) in gauge models unifying weak and electromagnetic interactions. The analysis of this paper has been made in the framework of a Lagrangian that is general enough to embrace most such models, and includes the Weinberg-Salam and V-A theories as special cases. Thus the constraints obtained above are expected to have rather general validity. As an example of the potential usefulness of these constraints, consider the bound in eq. (20). If the cross section forSee in the reactor experiment [11] is found to be significantly lower than ~ (the V-A value), eq. (20) will predict a lower bound on the neutral current reaction ~ue - + u , e . With the advent of high energy neutrino beams, measurements of neutrino electron scattering have entered the domain of feasibility, and one can hope that the relations derived in this paper will find useful application. The author is indebted to Professor L. Wolfenstein for some valuable comments.
Appendix. Electron mass corrections
If the mass of the electron is not neglected, the recoil electron spectrum for any of the four reactions is
do(r) 2G2rn [A dE -
n
(r)+B(r)
(
!
)
Eft7 2 _ C ( r )
mE]
~v
,
(A.1)
where E is the kinetic energy of the electron in the lab. frame. This should be compared with eq. (8) obtained by neglecting O(m/Ev). The coefficients A (r) and B (r) are as given in table 1, while the coefficients C(r) are
C(uee) = C(Vee)
----14 (D V,2_ D ~ ) (A.2)
C(vue ) = C(~ue ) = ~; (D2v - D2 ) . A consequence of the electron mass correction is that the total cross section is not strictly proportional to E v. In practice, the mass term is likely to be important only in the reactor experiment [11 ] measuring~ e + e - -+ue + e - , where E v is of the order of a few MeV. A separate determination of A, B and C may be possible for this reaction, in which case it would be interesting to check the relation C(uee)2 = A(>ee) B(Tee) .
(A.3)
The cross section relations (19), (20), (21) and (22) were obtained in the approximation of neglecting O(m/Ev) and are therefore valid for accelerator energies. If information from the reactor experiment is to be used, relations involving O(Vee) should be suitably corrected. For instance, eq. (20) should be replaced by
L.M. Sehgal, Neutrino electron scattering
X/-~(~ee ) + X/~ue
) 1> ~
.
69
(A.4)
Relations a m o n g the coefficients A and B are, o f course, unchanged.
References [ 1 ] G. Myatt, invited talk at the Int. Symposium on electron-photon-neutrino interactions at high energies, Bonn, August 1973; P. Musset, rapporteur talk at the II. Aix-en-Provence Int. Conf. on elementary particles, September 1973. [2] F.J. Hasert et al., (Gargamelle collaboration), Phys. Letters 46B (1973) 138. [3] A. Benvenuti et al., to be published. [4] F.J. Hasert et al., (Gargamelle collaboration), Phys. Letters 46B (1973) 121. [5] R.P. Feynman and M. Gell-Mann, Phys. Rev. 109 (1958) 193; R.E. Marshak and E.C.G. Sudarshan, Phys. Rev. 109 (1958) 1860. [6] S. Weinberg, Phys. Rev. Letters 19 (1967), 1264; A. Salam, Elementary particle theory, ed. N. Svartholm (Almquist and Forlag A.B., Stockholm 1969) p. 367. [7] L.M. Sehgal, Phys. Letters (to be published). [8] H. Terazawa, Phys. Rev. D8 (1973) 1817. [9] G. 't Hooft, Phys. Letters 37B (1971) 195; H.H. Chen and B.W. Lee, Phys. Rev. D5 (1972) 1874; D.H. Perkins, in Proc. of the 16th Int. Conf. on high energy physics at Chicago, vol. 4 (1972) 149. [10] T.D. Lee and C.N. Yang, Phys. Rev. 126 (1962) 2239. [11] H.S. Gurr, F. Reines and H.W. Sobel, Phys. Rev. Letters 28 (1972) 1406.
NEUTRINO ELECTRON SCATTERING IN THEORIES WITH N E U T R A L CURRENTS * L.M. S E H G A L
111. Physikalisches Institut, Technisehe Hochschule Aachen, Aachen, W-Germany Received 7 October 1973
Abstract: Elastic neutrino electron scattering is considered in a general class of theories containing both charged and neutral currents. Assuming ~-e symmetry, the following results are proved. (i) x/O(vee) + x/o(vwe ) >- x/'2, (ii) xfo (~ee) + x/a (rue)/> x / ~ , (iii) Offee) - ague ) = ~ [O(Vee) - o(vue)] , (iv) ] [a (vee) - ~ a (~ee)] ~ +- [o (rue) - I a (~.e] ~ I = 4 , (where cross sections are in units of G2mEv/n). Constraints are derived on the energy spectrum of the recoil electron. The Weinberg-Salam theory and the V-A theory are considered as special cases.
1. Introduction There is n o w significant experimental evidence for the existence o f neutral currents in neutrino-induced reactions [1 ]. Both at CERN-Gargamelle [2] and at N.A.L. [3] a signal has been obtained for the semileptonic reactions vg(ffu) + nucleon
vu(vu) + hadrons. A candidate has also been found for the purely leptonic process v u + e -+T~ + e - in the Gargamelle e x p e r i m e n t [4]. These observations suggest that neutral currents play an essential role in weak interactions and compel a reexamination o f ideas based on the traditional V-A t h e o r y [5]. This paper studies the implications o f neutral currents for elastic n e u t r i n o electron scattering. We consider a broad class o f theories containing b o t h charged and neutral currents, which is a generalisation o f the intermediate vector boson model, and which is sufficiently general to a c c o m m o d a t e models that arise in spontaneously b r o k e n gauge theories. The Weinberg-Salam theory [6] and the V-A t h e o r y appear as special cases. Relations are derived among the four distinct processes o f interest: Vee -+ Vee , Fee ~ Ve e, rue ~ v u e , ~ e - ~ u e ; constraints are obtained on the total cross sections * Work supported by the German Bundesministerium ftir Forschung und Technologie.
L.M. Sehgal, Neutrino electron scattering
62
and the energy (or angle) distribution of the recoil electron. (A few of the results obtained here were reported earlier [7].)
2. A general Lagrangian for neutrino electron scattering We assume that weak teptonic processes are mediated by (i) an arbitrary number of charged spin-one bosons, (ii) an arbitrary number of neutral spin-one bosons, (iii) an arbitrary number of neutral spin-zero bosons. We require that the Lagrangian satisfy (i) the two-component neutrino hypothesis, (ii) separate conservation of muon and electron number, (iii)/a-e universality. The general Lagrangian consistent with these assumptions is
Z? = ~ g i + ~f/ ]
[e 7a(1 +3,5) .ve + (e ~ ) ]
w(i)+a h.c.
[VeTa(1 +')'5)Ve+eT~(CJV +C]A 75) e + ( e ~ t ) ]
(1) Z~')+ .~ (scalars).
In eq. (1), W (i) and Z(/) denote respectively the sets of charged and neutral spin-one intermediate bosons, and gi, fl" CIV' CIA are arbitrary real constants../3(scalars) denotes terms containing neutral spin-zero fields coupled to neutral lepton currents. We note that the above Lagrangian encompasses models of the type arising in spontaneously broken gauge theories, which contain in general neutral spin-zero particles (the Higgs fields) in addition to charged and neutral spin-one bosons (the gauge fields). Our objective is to derive the consequences of the interaction (1) for the processes Vee -+ Vee,~ee -+ fie e, rue -+ rue , Fue -+F,e in the lowest order of the coupling constants. To this end, it is crucial to observe that the expression for .~ (scalars) cannot contain any terms involving neutral neutrino currents, since a neutral spin-zero field cannot couple to (or decay into) a ~ pair. Thus, the existence of spin-zero neutral bosons (such as the Higgs particles of gauge theories) has no influence on lowest order neutrino electron scattering, and the term/2 (scalars) can simply be disregarded in the present discussion. We will be interested in the domain of energies for which
m 41, E 1)
~ ~ 1, M2
(2)
where rn is the electron mass, E v the lab energy of the incident neutrino, s (~ 2mEv) the centre-of-mass energy squared and M the mass of any of the spin-one intermediate bosons. The condition m/E v ,~ 1 is fulfilled at all accelerator energies (but is somewhat inaccurate for reactor experiments). The condition s/M 2 ~ 1 is also expected to be well satisfied even at the high energies of N.A.L. In these circumstances predictions for neutrino-electron scattering may be derived from an effective Lagrangian equivalent to (1),
63
L.M. Sehgal, Neutrino electron scattering
"/2eft-
G ~ [(e3'C~(1 +3'5)Ve)(VeYc~(1 +75) e) + (re 7a(1 +TS) re) (g 7c~(Dv +DA 3'5) e)
(3)
+ ( 7 3'~(1 +3'5) vu) (V 7a(Dv+DA3'5 ) e)] , where 2 v
%' (4)
M(i) n a,(J) being the masses of W(i) and Z q) respectively, and G the Fermi conw ~a"u~v*Z
stant. Eq. O) may be rewritten in a more convenient form [8] by applying a Fierz transformation to the first term, which yields "/2eft-
G ~ , , X/~ [(Ve3'a(1 +3'5) Ve)(e3' (Dv+DA3'5) e)
(5) +(vta 7"(1 +3'5) vg) (e 3'a(D V +D A 3"5) e)] , where D V= 1 +O V ,
DA
1 +D A
(6)
In the Weinberg-Salam theory, the effective Lagrangian for neutrino-electron scattering has the form (5), with [9] D V = - ½ + 2 s i n 20 , (7a) Dh = - ½ ,
(Weinberg-Salam),
0 being the Weinberg angle. The V-A theory, on the other hand, is defined by O V = D A = 0 ,(V-A) .
(7b)
3. Differential and total cross sections. General constraints
We consider here constraints on the differential and total cross sections of neutrino electron scattering arising from the vector (spin one) nature of the couplings'in eq. (5).
L.M. Sehgal, Neutrino electron scattering
64
The differential cross sections for the four types of reaction have the general form (neglecting terms of order m/Ev)
2do(r) ~ - 2G2m ~ IA (r) +B(r) ( k1 -~E- )
2] '
(8)
A (r), B (r) >~ O, where r stands for Vee,~ee , rue o r ~ u e , and E is the (lab.) recoil energy of the electron. We may equivalently write the cross section in terms of the centre-of-mass scattering angle (the angle of the outgoing electron) and the result is d cos 0 -
zr
(r)
+B(r)
.
(9)
The coefficients A (r) and B(r) are determined by the constants D V and D A appearing in ddeffand are listed in table 1, where, for comparison, we also indicate the values in the Weinberg-Salam and V-A theories. The total cross sections are given by G2mE
v [A(r)+~B(r)].
o(r) = 2
(10)
zr The following features emerge. (i) The differential cross section (8) is a quadratic function of E. (ii) Neutrino and antineutrino reactions are related by
A(vee)=B(~ee),
A(~ee)=B(Pee),
A(v e)=B(Y#e),
A(v e)=B(p e).
(11) Table 1 Values of the coefficients A and B occurring in the differential cross section for neutrino electron scattering (eq. (8)):
The constants DV,, DA, D~V,D'A are those appearing in the effective Lagrangian (eq. (5)); D ~ = 1 +Dv, D A = 1 +D A Predictions General
Weinberg-Salam
V-A
A
A B
Reaction A
B
Vee ~ Vee
(} (D~+D~)) 2 (~(D~d-D~)) 2
~ee~ee
2 (~-(D~v-D~)) 2 ({(D~+D~)) 2
v
e~v
e
~ e-+vge
(~(Dv+DA))2
(~.(Dv_DA))2
(~(D 1 V _DA))2 (~(Dv+DA)) 2
B
(~-+ sin20) 2 sin40 (~--sm l - 2 O)2 sin40
sin40 1_ • 2 0) 2 (2+sin
sin40 (~-sin20) 2
1 0 0 1 0 0 0 0
L.M. Sehgal, Neutrino electron scattering
65
(iii) The ratio of neutrino and antineutrino cross sections is bounded by ~<
o(k-ee )
FL O(Uee) '
o (ff u e) O(Uu e)
]<~ 3
(12)
These results are analogous to those obtained in the classic work of Lee and Yang [10], which analysed the consequences of the two-component neutrino hypothesis in V-A theory. In the following section, we derive a number of relations between e-type and/atype neutrino scattering that depend specifically on the assumption of/a-e symmetry.
4. Constraints arising from g-e symmetry
4.1. Constraints on electron energy spectrum From the expressions forA(uee ) and A(uue ) given in table 1, and the relation D V = 1 + D v , D~ = 1 + D A (eq. (6)), we deduce t
x/A-(uee) + vIA (uue) = 1 , or x/a (Uee) - X / ~ u e )
(13a)
= 1 ,
(13b)
or x/A-(pue ) - x/A((uee ) = 1 ,
(13c)
(where the positive square root is understood). This constraint can be written compactly as I x/-A(uee ) -+x/A(uue)l = 1 ,
(141)
and is shown geometrically in fig. 1. The allowed values ofvCA(uue) and x/~(Uee) must lie along the line segments PR, RS or PQ (corresponding to the conditions (13a), (13b) or (13c)). The prediction of the Weinberg-Salam model is shown by the dotted regions of the lines PR (for sin 2 0 < ½) and RS (for sin 2 0 > ½). The V-A prediction is the unique point R. Because of the relation between the A and B coefficients given in (1 1), a constraint similar to (14) exists for the antineutrino reactions, I X/~(Vee ) -+ x / ~ u e ) l
= 1 .
(151)
Also, by virtue ofD~r - D ~ = D V - DA, we note from table 1 the following further relations:
A(vee ) = A ( ~ e ) ,
B(vee ) = B ( v e) .
(16)
Constraints (14), (15) and (16) provide connections between the energy (or angle) distribution in e-type and/a-type neutrino scattering, and depend critically on the assumption of/~-e symmetry.
66
L.M. Sehgal, Neutrino electron scattering
Q
2-
3
P
S
1
l 2
1
3_ 2
2
Fig. 1. Geometrical relation between ~/A~ee ) and A ~ u e ) (eq. (14)). The allowed region consists of the line segments PR, RS and PQ. The Weinberg-Salam prediction is the dotted region of PR (RS) for sin2 0 < 1 (> {). The V-A prediction is the point R. The constraint (15) is obtained by replacing A (Vee) ~ B (~ee), A (vt~e) ~ B (~ue). 4.2. Constraints on total cross sections A n u m b e r of relations can now be derived among the total cross sections of the four reactions. We observe first that eq. (14) implies X/~-(Vee ) + X / ~ u e
) ~> 1
(17)
At the same time, we have from eq. (10) (expressing cross sections in units of G 2mEv/n ) X/O(Vee ) ~> x / 2 A ( v e e ) ,
VcO-(vue)/> ~ u e )
.
(18)
We thus obtain the following remarkable b o u n d on neutrino cross sections: X/b-(Vee) + ~ / o ( v u e ) >~ X / 2 .
(19)
A similar b o u n d is obtained for the antineutrino cross section, b y making use of eq. (15).
"v/oCvee) + X/o(-flue) ~> X/~5
•
(20)
L.M. Sehgal, Neutrino electron scattering
[ol :l-
in units of
61
G2mEv
6 S
/,
2
I
l
k
Fig. 2. Geometrical relation between [o(v~e) - ~ a(vue)] ~ and [a(vee) - ~-O(vee)] ~ (eq. 22)). The allowed region consists of the line segments PR, RS and PQ. The Weinberg-Salam prediction is the dotted region of PR (RS) for sin2 0 < -~(> ~-),while the V-A prediction is the point R. The inequalities in (19) and (20) become equalities in the V-A theory as well as in the Weinberg-Salam model for sin 2 0 = 0. Finally, we note that the cross sections o f the four different reactions are determined by only two unknown parameters D V and D A. Elimination of these gives the following two relations. a(Vee ) -
o(vue ) = 3 [O(~ee ) - o(~ue)]
,
I [O(Vee) - } O(~ee)l ~ -+ [o(v e ) - ~ a(~ue)] ~1 = ~-.
(21) (22)
The last relation is illustrated in fig. 2, where (as in fig. 1) the allowed region consists of the line segments PQ, PR and RS, and the predictions of the V-A and WeinbergSalam theories are as indicated.
5. Comments The discovery of neutral currents in semi-leptonic neutrino reactions suggests strongly the occurrence of such currents in purely leptonic processes as well. Neutrinoelectron scattering, in particular, should exhibit significant deviations from the ex-
L.M. Sehgal, Neutrino electron scattering
68
pectations based on the V-A theory. The precise manner in which neutral currents will be incorporated into the new theory is not known, but such currents do find a natural place (and, indeed, were anticipated) in gauge models unifying weak and electromagnetic interactions. The analysis of this paper has been made in the framework of a Lagrangian that is general enough to embrace most such models, and includes the Weinberg-Salam and V-A theories as special cases. Thus the constraints obtained above are expected to have rather general validity. As an example of the potential usefulness of these constraints, consider the bound in eq. (20). If the cross section forSee in the reactor experiment [11] is found to be significantly lower than ~ (the V-A value), eq. (20) will predict a lower bound on the neutral current reaction ~ue - + u , e . With the advent of high energy neutrino beams, measurements of neutrino electron scattering have entered the domain of feasibility, and one can hope that the relations derived in this paper will find useful application. The author is indebted to Professor L. Wolfenstein for some valuable comments.
Appendix. Electron mass corrections
If the mass of the electron is not neglected, the recoil electron spectrum for any of the four reactions is
do(r) 2G2rn [A dE -
n
(r)+B(r)
(
!
)
Eft7 2 _ C ( r )
mE]
~v
,
(A.1)
where E is the kinetic energy of the electron in the lab. frame. This should be compared with eq. (8) obtained by neglecting O(m/Ev). The coefficients A (r) and B (r) are as given in table 1, while the coefficients C(r) are
C(uee) = C(Vee)
----14 (D V,2_ D ~ ) (A.2)
C(vue ) = C(~ue ) = ~; (D2v - D2 ) . A consequence of the electron mass correction is that the total cross section is not strictly proportional to E v. In practice, the mass term is likely to be important only in the reactor experiment [11 ] measuring~ e + e - -+ue + e - , where E v is of the order of a few MeV. A separate determination of A, B and C may be possible for this reaction, in which case it would be interesting to check the relation C(uee)2 = A(>ee) B(Tee) .
(A.3)
The cross section relations (19), (20), (21) and (22) were obtained in the approximation of neglecting O(m/Ev) and are therefore valid for accelerator energies. If information from the reactor experiment is to be used, relations involving O(Vee) should be suitably corrected. For instance, eq. (20) should be replaced by
L.M. Sehgal, Neutrino electron scattering
X/-~(~ee ) + X/~ue
) 1> ~
.
69
(A.4)
Relations a m o n g the coefficients A and B are, o f course, unchanged.
References [ 1 ] G. Myatt, invited talk at the Int. Symposium on electron-photon-neutrino interactions at high energies, Bonn, August 1973; P. Musset, rapporteur talk at the II. Aix-en-Provence Int. Conf. on elementary particles, September 1973. [2] F.J. Hasert et al., (Gargamelle collaboration), Phys. Letters 46B (1973) 138. [3] A. Benvenuti et al., to be published. [4] F.J. Hasert et al., (Gargamelle collaboration), Phys. Letters 46B (1973) 121. [5] R.P. Feynman and M. Gell-Mann, Phys. Rev. 109 (1958) 193; R.E. Marshak and E.C.G. Sudarshan, Phys. Rev. 109 (1958) 1860. [6] S. Weinberg, Phys. Rev. Letters 19 (1967), 1264; A. Salam, Elementary particle theory, ed. N. Svartholm (Almquist and Forlag A.B., Stockholm 1969) p. 367. [7] L.M. Sehgal, Phys. Letters (to be published). [8] H. Terazawa, Phys. Rev. D8 (1973) 1817. [9] G. 't Hooft, Phys. Letters 37B (1971) 195; H.H. Chen and B.W. Lee, Phys. Rev. D5 (1972) 1874; D.H. Perkins, in Proc. of the 16th Int. Conf. on high energy physics at Chicago, vol. 4 (1972) 149. [10] T.D. Lee and C.N. Yang, Phys. Rev. 126 (1962) 2239. [11] H.S. Gurr, F. Reines and H.W. Sobel, Phys. Rev. Letters 28 (1972) 1406.