NEUTRINO TRANSITIONS IN AN EXTERNAL FIELD

V. V. Skobelev UDC 539.12

In the article the transition v + v is considered in a field of plane-wave e type, induced by the effective values of the anomalous magnetic and electric dipole moments of massive neutrinos. The possible contribution of the given mechanism to the background of quantum mechanical oscillatory phenomena is discussed.

Since the time of the original paper of Pontecorvo [I] with the subsequent development of the ideas discussed by the author [2], the problem of the mixing of massive neutrinos and its consequences has comprised an object of intensive theoretical and experimental in- vestigations [3, 4]. Of greatest interest is the possibility of oscillations of neutrinos participating in the weak interactions. In the case of the noted mixing, for example, of electron and muon neutrinos the prospect is opened of an adequate solution to the question of the balance of solar neutrinos. In the theoretical plan the detection of mixing would widen the existing analogies between the neutrino and quark generations, since for quarks the corresponding effect, as is kno~, takes place. In the standard model, a nondiagonal term in the Lagrangian can be induced by the interaction of a neutrino with a classical scalar Higgs field

where the indices e and ~ pertain to the electron and muon neutrinos, and the parameter meD has the dimension of mass (for simplicity, let us limit ourselves to the mixing of two generations). The parameterization of the functions ~e,B in tezm~s of the mixing angle leads to the dependence

W~=~lcosO+~2sinO, (2)

~.=--~lsinO+~2cosO,

where here and further the indices 1 and 2 pertain to the states of a neutrino with speci- fied masses m I and m2, where the mass parameters of the electron and muon neutrinos

m~ -~- ml cos~O + m2sin2O,

m~--- m l sin2,0+ m2cos20,

1 m ~ ~ (m2--ml)sin(20). (3)

The traditional approach to considering the transitions Ve +-~ ~D is, essentially, pure- ly quantum mechanical and is based on an analysis of the temporal evolution of the state vectors of the neutrino, taking into account the representations (2), (3), leading to an oscillatory temporal dependence of the probability, explicitly not connected with the elec- troweak characteristics of the neutrino. But in astrophysical situations the propagation of neutrinos occurs in the presence of external electromagnetic fields of a different type.

I! t! But then the process ~e +-+ ~D actually is not a two-particle one, but is connected with the capture or emission of photons as a consequence of the interaction of the vacuum of charged particles of the standard (or other) model with the external field. For sufficient- ly small momenta of the external lines, the electromagnetic interaction of the Dirac neu- trinos can be accounted for in the Farri picture by the introduction of anomalous magnetic

Moscow State Correspondence Pedagogical Institute. Uchebnykh Zavedenii, Fizika, No. I, pp. 31-35, January, August 3, 1992.

Translated from Izvestiya Vysshikh 1993. Original article submitted

1064-8887/93/3601-0025512.50 © 1993 Plenum Publishing Corporation 25

(A>~I) p and electric dipole (EDM) e moments of the neutrino (in the presence of preserva- tion of CP-invariance the EDM equals zero). In this interpretation, expression (I), in the usual manner, can be considered as the interaction Langrangian, computing the probability Wf caused by the external field of the transitions ~e +-+ vp per unit of time. It is evi- dent that this makes sense if the time of the "field" transition ~e ~-+ vD Wf -~ is less than the period of quantum mechanical oscillations.

In this paper, with the use of the invariant solutions found earlier of the generalized Dirac equation in the field of a plane wave [5, 6], the corresponding probabilities are computed in the field of a linearly polarized wave and in a constant crossed field, approxi- mating in the ultrarelativistic case constant fields of other configurations. Numerical estimates, characterizing the role of the considered mechanism in the transitions ~e ~-+ vp are also presented.

The invariant solution of the generalized Dirac equation for a neutral particle, taking into account the ~-~I and the EDM in the field of a plane wave with a factorized dependence on the phase A s = a~f(kx), has the form

W'= [cosz+ sinz(TN+--i77~N )] u ( p ) e_i(p.,, (4) (2po)~1 ~

where

z = [-- A 2 (p2 + s , ) l , ,2 ' AAA AAA

N+_ = ~cap +_ ptca 2 (gp) ( - a~)'l ~ '

= ~ / ( ~ + ~-D I,~, ~ = ,/(~,~ + ~)'/~.

Here p= = m =, u(p) is the Dirac spinor with the normalization u-u = 2m. properties

A A

N + = ~ N+, N ~ = - - I , N + p N + = - T - p , 1 A A

N + N = N N+ - - [p, ~c]. 2 (,cp)

N+_pN_ T = ~ -- , VcP

Considering the

(5)

(6)

it is easy to verify that the current (~¥~) is preserved, and the function (4) is normalized by one particle per unit of volume. Let us remark in passing that the solution presented in [6] for a circularly polarized plane monochromatic wave

A = alsin (nx) +a2cos (nx), a~ = a L ( a l ~ ) = (a~u) = ( a l a ~ ) = 0, ( 7 )

as is apparent from formula (4a) of this paper, indeed holds in the more general case with the replacement in (7) and in the solution of the phase (kx) by an arbitrary function f(kx).*

In the case of a monochromatic wave with linear polarization

As=a=sin (~x) (8)

in the computation of the matrix elements (for definiteness let us consider the transition v e + vp) in correspondence with formulas (I), (2), (4) it is necessary to use the expansions

cosz, cosza/= 1 ~ (j~_, + j.(+)) e_is(~.L (9a) sin z, sin za/ 2 even

*My attention was directed to this circumstance by A. S. Vshivtsev.

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cos zt sin z~] = i X (~+, ~ j~_)) e_,,,K~) ' sinzlcosz2] 2 od~ (9b)

where in (ga) t he sum over s runs over even v a l u e s , and in (gb) - over odd va lues and phys i - c a l l y s i g n i f i e s t he number of photons cap tu red from the wave (or given back) . The n o t a t i o n J s ( i ) i s app l i ed he re fo r t he Besse l f u n c t i o n s of va r ious combinat ions of arguments

~±) ~ 4 ( x , ± x2), (lO) x,,2 = [ - a ~ (~,2 + *~,2)1 ~/2,

and ~. and ¢. are the effective ~ and EDM of neutrinos with the specified masses. Using 3 J

the relations written out, we obtain for the value caused by the external field of the full transition probability v e + vB per unit of time

Wf=W12sin4O+W~lcos40, (II)

where

m~ y d3p] 1 ^ = - - , l l - ~ ( p 2 - ~ s K - - p J l ) Sp <(p] + m,) X

WI~ (P2) 2po2 2poI

× { ~ [(J~+) + J(~-)) + (J(~+) - J~-)) (1')(2)] (,~2 + m2) [(4(+)+ J(Z )) -4- 8 7 7 ~ / l

+ (L (+) - - J~-)) (2) (1')] + ~ [(J~+~ - - J~-)) (2) - - (J(~+) + J~(-)) (1')1 >< odd

X (p% + m2) [-- (Y(+) -- Jl -)) (2) + (i~+) + J(~-)) ( l ' ) ] /> (12)

and, for brevity, for example, it is denoted that

(1') ------ ~ N ~ )' ;Y~ ~ro)'

that is the last expression, pertains to neutrinos of sort 1 in a finite state with momentum PI', etc. The expression W21 is obtained from formula (12) by a transposition of the signs 1+-+2.

After uncomplicated computations we obtain

m' ( m 2 + m l ) 2~ 1 x~ ~ ( s - 60)) A (s), ( 13 )

2po~ 2 (tcp~) S ~ - - C O

A (~) = (JW~+ J(;)~) - ( : W ' - J(~-~') (~$~ + ~,*~), (13a)

slO) = .,,~ - r a g 2 (rip2) (13b)

If the values of the parameters a re such that I s2 (° ) [ >> i, then the "unnecessary" 6-function is removed by the approximative change from sunmmtion to integration over s with the analytic continuation of the Bessel functions to nonintegral values of the index. But in the general case, it is necessary to consider the nonmonochromaticity of the neutrino beam. If one denotes the energy distribution_ function F(P0), then the process Ve +-+ ~-~ is

characterized by a number of transitions Wf per unit of volume per unit of time. Here, in (II) it is necessary to let

w . (pO ~ ~z,2 = S w,2 (p2) F(p00 dp0~ =

m~.(m~ 4- m,) 2 ,~ A (s) F(pos) 7-, [ p0~cos~ ] (14)

4 no ~=-~o spo~ 1 (Pe~ - - m ~ ) 1~2

and W2x(pl) * ~'21 wi th t he e v i d e n t r e d e s i g n a t i o n s , where t h e va lue P0s i s e s t a b l i s h e d by equa t ing the argument of t he d - f u n c t i o n in (13) to zero , and ~ i s the angle between the momentum of t he n e u t r i n o and the wave v e c t o r .

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The probability value in a constant crossed field A~ = oa(kx), as previously, is deter- mined by (II), where now*

W~(p2 ) = ~ mg~(m, +m~) ~ ×

4 P0~ (15)

× ~ (1 - p~,~ - p~,~) ~ [2 ( p ~ (~)) - (m~ - m~)], L~=~I

(p) = { [ _ a z ( ~ + ~)]l/~ + p i _ a ' ( ~ + ~)]l;2} K, (15a )

and t h e t r a n s i t i o n t o t h e r e a l c a s e o f nonmonoch roma t i c beams i s done in t h e way d e s c r i b e d a b o v e .

Passing to a discussion of the results obtained, let us note at first that the expres- sions (13) and (15) contain the Abe4 and EDM in a symmetric manner, so that in possible ex- periments recording the transitions ~e +-+ ~, induced by the external field, it is impos- sible to uniquely establish violation of CP-invariance.

To obtain specific estimates let us consider an ultrarelativistic neutrino IP I >>mL=, the momentum of which is parallel to the wave vector. Since here

~0 ~m~ ~o \m~

then for not too large masses for attainable laser frequencies (k 0

..[s~,=(°)[ ~ I can be satisfied and we arrive at the result

(16)

- I0 -6 m e ) the condition

mS, (ml + m2) 2 A ( s ~ 0 ,

4 ~0ml

_ e~ " A (s~)) . W 2 ~ m ' (m, + m~ ~ (17)

4 ~0m~

By t h e same o p t i m i s t i c e s t i m a t e s t h e A~I o f t h e n e u t r i n o can be o f t h e o r d e r I0 - z ° ~e, and t h e maximal s t r e n g t h o f t h e f i e l d a t t h e f o c i o f t h e l a s e r s o n l y 10 9 G, and t h e v a l u e o f t h e a rgumen t o f t h e B e s s e l f u n c t i o n s has o r d e r 10 -9 Due t o t h e a s y m p t o t i c p r o p e r t i e s o f t h e l a t t e r and u n d e r r e a s o n a b l e a s s u m p t i o n s c o n c e r n i n g t h e m a s s e s (1 eV 5 m l , 2 , [mz - m2[ ~ 10 eV) and e n e r g y ( - 1 MeV) o f t h e n e u t r i n o d e t e r m i n e d by ( 1 7 ) , t h e mean f r e e p a t h f o r an o p t i m a l v a l u e o f t h e m i x i n g a n g l e p r o v e s t o be s e v e r a l o r d e r s g r e a t e r t h a n t h e d i s - t a n c e from the Earth to the Sun. Thus, the experimental observation of the effect, ap- parently, is not possible, until one can speak of its role in the problem of solar neu- trinos. Probably, the situation is otherwise in the presence of superstrong magnetic fields of collapsed astrophysical objects (up to I0 ~7 G), however, the estimates produced on the basis of expression (15) are too uncertain, in view of the absence of reliable in- formation concerning the form of the energy distribution function of the neutrino.

1. 2. 3. 4. 5. 6. 7.

LITERATb~E CITED

B. Pontecorvo, Zh. Eksp. Teor. Fiz., 53, 1717 (1967). S. M. Bilenky and B. Pontecorvo, Phys. Rep., 41, 225 (1978). P. Astier et al., Nucl. Phys., B335, 517 (1990). V. K. Barger et al., Phys. Rev., D22, 1636 (1980). V. V. Skobolev, Yad. Fiz., 54, 162 (1991). V. V. Skobolev, Zh. ~ksp. Teor. Fiz., I00, 75 (1991). V. V. Skobolev, Zh. ~ksp. Teor. Fiz., I01, 1724 (1992).

*See also the note to formula (7) in [7].

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