SOVIET P H Y S I C S JOURNAL 75

INELASTIC SCATTERING OF ANTINEUTRINOS BY ELECTRONS

D. K. Ershov

Izvestiya VUZ. Fizika, Vol. 10, No. 7, pp. 118-120, 1967

The theory of the universal weak interaction [~] allows the fol- lowing lepton-lepton processes:

1. a~e+7+v , 2. e+v~e+v , 3. e+v--+e+v, 4. e+v~,~+v.

Only the first of these is known at present, but experimental confirmation of the others would be of great importance to testing the general picture of the weak interaction. Processes 1 -3 have been con- sidered theoretical ly [1-5] . Here we consider process 4 with allowance for polarization of the electron in the Fermi universal interaction scheme and also on the assumption that there exists an intermediate boson W.

1. The matr ix element for process 4 in the Fermi universal interaction scheme may [2] be put in the following form (a Firtz trans- formation has been applied in the expression for M):

G M = - - ~ - Us7= (1 + 7~) Ue-ffCr~ (1 - - %) U~, (1)

0-5 2 in which G is the constant of weak interaction (1 / m~ ~, mp being the proton mass). The cross section for the process in tile laboratory coordinate system ia

( i 4G'~m ~ 1 ~ ( I + Z . ) 2 ,o+ - - m U

o = 3~-~ 1+2~o 4 (1 + 2~o ) + ~ ( o + l + o X ) X

X (1 + 2~)s . (2 )

The cross section for e + v --~ I~ + v is

2~0 + 1 - - - - 2O~m ~ m~

_ (1 + X) o. r~ 1+2o ,

(a)

In (2) and (8), g is the mass of a g meson; m is electron mass; X is the projection of the target electron spin onto the direction of the mo-

mentum of the incident neutrino (antineutrino); and w is the energy of the latter in units of electron mass in the laboratory coordinate

system. dSp2daq~ ~, -- , To derive (2) we must find P2=q2~ 2E~2m 2 ~ I.Patql--Pa--q=),

in which Pi, as, Pz, and q2 are the 4 -momenta respectively of the electron, the incident antineutrino, the g meson, and the scattered antineutrino, while Eg and wz are the energies of formation of the meson and the antinentrino. It may be shown that

~ P2eq2~ dap2daq~ B* 2Et,2% (Pz + q, ~ P2 - - q2) =

= 31 L [ ~ (q~ 4q2-- ~2)~ -[ q~%2 q'~ (q2 +q41"r -- 2~4 X

(' dSpflaq2 ] ~ . X J 2Ep.2%] o (el + q*--P~.--q~), (4)

in which q = Pl + ql. The latter integral is readily calculated in any coordinate system. If g = m, formulas (2) and (3) coincide with those derived in [3].

2. If there exists a charged intermediate boson W, process 4 is described by a Feynman diagram (Fig. 1). The matrix element cor- responding to this diagram is

I

w-Ir162 I I

4/~q, , ( q%)

M = 4r.f2 q.~ ~-7-~-M-: 5=3 - - ~ U~'[= (I + 7s) UvUF f~ (1 + rs) Ue. (5)

f f O Here 4= M~ = ~ - , and M is the mass of the intermediate bosom It

is readily seen that

q ; ~ - m'l~ * U~7~ (1 4- 75) U~FI~ (I + %) Ue N M ~ (6)

Current es t imates give ~ - ~ N 10 -4 i . e . , much less than one, which

is negligible, so the expression for the cross section in the laboratory coordinate system is

[ / ~(1 + k ) 2~ -{- 1 - - 4O~m ~ 1

[ _~ ]2 1-i-2~o 4(1+2~o) 3= 1-- (1+2,0)

(1 + 2oJ) 2o~ -F 1 + m / m~ .,.o (,o + 1 + ,,,?,) . (7)

+ (1 + 20,),

For ~o of 10 MeV or less, the cross sections of (3) and (7) agree closely, but a difference appears at higher energies. If a neutral inter- media te boson W ~ exists as well as a charged one W • a different re- sult is obtained [5]. If the masses of W + and W ~ are equal, the con- tributions from diagrams corresponding to exchange between the neutral and charged bosons cancel out.

REFERENCES

1. R. Feynman and M. Gell-Marm, Phys. Rev., 109, 193, t958. 2. L. B. Okun, The Weak Interaction of Elementary Particles

[in Russian], Fizmatgiz, 1968. 3. S. S. Gershtein and V. N. Folomeshkin, ZhETF, 46, 818,

1964. 4. B. K. Kerimov and Yu. L Romanov, ZhETF, 46, 1912,

1964. 5. A. A. Bogush, A. I. Bolson, and L S. Satsunkevich,

Yadernaya Fizika, 1, 288, 1965.

19 December 1966 Marx Pedagogic Institute, smolensk