Stimulated conversion of neutrinos: a new method to search for radiative decay of neutrinos

Physics Letters B 289 (1992) 194-198 North-Holland

PHYSICS LETTERS B

Stimulated conversion of neutrinos: a new method to search for radiative decay of neutrinos S. Matsuki Institutefor Chemical Research, Kyoto University, Uji, Kyoto 61 l, Japan

and K. Yamamoto Department of Nuclear Engineering, Kyoto University, Kyoto 606, Japan

Received 6 April 1992

A new method is proposed to search for radiative decay of neutrinos: relativistic neutrinos from the Sun or reactors are converted into neutrinos of different flavors by stimulated emission/absorption of intense radiofrequency/microwave photons in a resonant high-Q cavity. A much more stringent laboratory limit on the radiative lifetime of neutrinos could be obtained from this method. Possible experimental schemes for realizing this method are discussed.

The quest for massive neutrinos is one o f the most i m p o r t a n t issues in current particle physics and cosmology. M a n y elaborate experiments have been dev o t e d to observe any effects o f massive neutrinos from, for example, the spectrum shape near the endpoint energy in beta decay, neutrinoless double-beta decay and neutrino oscillations [ 1 ]. A m o n g such effects o f massive neutrinos, we consider in this lette their possible radiative decay: v~v'+7.

(1)

Some a t t e m p t s have been m a d e so far in terrestrial experiments to observe this process [ 2 ]. They, however, just p r o v i d e b r o a d limits on the mass and decay rate o f neutrinos, while m u c h m o r e stringent b o u n d s m a y be o b t a i n e d from astrophysical a n d cosmological considerations [ 3 ]. It is thus desired to catalyze artificially the radiative decay o f neutrinos. We here propose a novel m e t h o d to i m p r o v e substantially the m e a s u r e m e n t o f radiative decay o f neutrinos: In a resonant high-Q cavity, i n c o m i n g relativistic neutrinos are c o n v e r t e d into neutrinos o f different flavors by s t i m u l a t e d e m i s s i o n / a b s o r p t i o n o f intense r a d i o f r e q u e n c y / m i c r o w a v e photons. The 194

n u m b e r o f photons stored in the cavity with a resonant energy q and quality factor Q a m o u n t s typically to N v ~ 4 X l026 ( Q / 1 0 9 ) ( P / 1 0 2 W ) (q/10_6eV) 2 ,

(2)

when a power P is applied. It is not difficult now to have a superconducting cavity with such a quality factor and an input power. The single-mode cavity with photons o f energy 10 -6 eV has a resonant frequency of 240 M H z which can span about 1 m E area, covering thus typical detector dimensions for neutrinos from a reactor or the Sun. Due to this huge photon number, the radiative conversion o f neutrinos would be enormously enhanced. This would thus enable us to observe artificially forced radiative decay of neutrinos in laboratory experiments, which is free from the assumptions m a d e in considering the celestial processes. Specifically, this m e t h o d is much more effectively applicable to the case o f smaller mass difference o f neutrinos c o m p a r e d to their masses, to which no previous terrestrial experiments could be reached yet. The conversion rate in the cavity is estimated as

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follows (with the unit of c=h= 1 ). The transition amplitude for the decay ( 1 ) compatible with gauge invariance is given generally by T~=½i f d4x17 ' (X)CTxp(fl'k'd75)P(x)F~P(x)

,

(3)

where/~ and d are the magnetic and electric dipole moments between the neutrinos, and F ~p is the photon field strength tensor. This may be calculated under the situation expected in the actual experiments where the neutrinos are ultra-relativistic, while the photon energy in the cavity is much lower:

E~-E'>>m,m',q, where E, E' and m,

(4)


(5)

m ' are the energies and masses of the neutrinos, respectively. (E' =E+ q due to energy conservation, as seen below. ) It may also be expected that IAm2l

where AmZ=---m2--m '2, and Eq~l e V 2 typically. Then, the wavefunctions of the neutrinos are given approximately by

u(x)~_exp(-ik'x)( ~°,) sq)~ '

leading contribution to the conversion rate happens to be suppressed by the factor (AmE/Eq) 2 for tam21
s' ~0,. '

"2

=\-q-~/

×cos(q~x)

[exp(iqt)+exp(-iqt)]

cos(qyy),

(7)

under the normalization ½fvd3x(E2-t-B2)=qN~, where V=LxLyLz, qx=yrL21, qy=yrL~l, and q = 2 2 1/2 (qx+qy) . The relevant boundary conditions for the electromagnetic fields are satisfied on the surfaces x = + ½L~ and y = _+½Lyof the cavity. By substituting these wavefunctions (6) and (7) into eq. (3), we find that the transition amplitude has nonzero contributions only for ss' = - 1; the helicity is flipped through the conversion under the condition (4). The conversion rate per unit time is then evaluated with the usual procedure (see for example the calculations in ref. [4] for the Coulomb scattering):

F~-NvK( I/tl2+ ×(

v,(x)~exp(-ik"x)(~°s" ) x/2ff

3 September 1992

~=+1

Idl 2)

f d3k' 4uVd(E'+°~q-E) G(Z~)) (2rr) 3 E2qI7

(6)

where kU= (E, k), k'u= (E', k' ), and s, s' = _+ 1 represent the helicities with the eigenvectors ~0~= (~) and ~0 ~= (o) of the Pauli matrix a~= (6 _o). Here the momentum transfer k' - k of the neutrinos is of order q through the radiative conversion in the cavity. (This will be seen below more explicitly in calculating the transition rate. ) Hence with a suitable choice of the spatial axes we may take k-~k ' -~ (0, 0, E) to give a good approximation under the condition (4), a.k/ (E + m) ~-a.k' / ( E' + m' ) ~-~, for the operators acting on (0s and ~,, in the lower parts of the four-component spinors in eq. (6). The volume factor may be given by ff~ [./~I - t with the incident neutrino flux J, ( I& I -~ 1 for the velocity). Consider typically the Ho~ mode in the cavity of rectangular form with the side scales L~~L (i=x, y, z), where the electric field is taken for definiteness parallel to the incident neutrinos along the z-axis. ( I f the electric field is perpendicular to the neutrinos, the

(8) G(Ak)

=-E2 ,=~x.yV-' f d3xexp(iAk.x)iq,(x) 2 , V

(9) where Ak--k' /~i(x), and

-k, B~(x) =-[exp(iqt) + e x p ( - i q t ) ] ><

Lx/2

fd3x= V

f --Lx/2

LI2

dx

Lz/2

dy f dz. --Ly/2

(10)

--Lz/2

The parameter x is determined depending on the polarization of the incident neutrino flux; x = l , I ¢ t - d l 2 / ( I/zl2+ Id12), and I/z+dl2/( 1#12+ Idl 2) for unpolarized, left-handed and right-handed fluxes, respectively ( x - 1 unless/z~ _+d accidentally). It is here convenient to make integrations by parts for eq. (9) by noting the relations such as Bx=OyAz and 195

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cos ( + qy' ½Ly) = 0 (the boundary conditions on the surfaces y = +_ ~Ly). Then, we obtain G(Ak) = E 2 ( Ak~ + Ak~)l T(Ak) 12 ,

E'=E-q k / ~ ~

( 11 )

T(Ak) - V-~ .f d3x exp(iAk.x)cos(qxx) c o s ( q y y ) .

k

V

A k ( Q )+,

/~A~/2-E

(12) The delta-functions 6(E' + a q - E ) representing energy conservation appear in eq. (8) from the time integral for eq. (3) which is taken over a large enough interval; a = + 1 ( - 1 ) corresponds to the photon emission v-~v' + " 7 " (absorption v + " 7 " ~ v ' ) in the coherent electromagnetic fields. (The dispersion around the resonant energy, 8q~q/Q<>q,

(13)

where q ~ q~= ~zL7 In performing the integration over the m o m e n t u m k' in eq. (8), we note, in addition to the above condition (13), that due to energy conservation, the neutrino m o m e n t u m transfer is restricted as a function of the direction of the final neutrino, O = (p, O) =k'/Ik'l (see fig. 1 for the relevant kinematical relations):

Ik'l=x/(E-aq)2-m'2~Ak=Ak(O)~.

(14)

By first performing the integration over Ik ' l in eq. (8) ( d 3 k ' = Ik' 12dlk'l dO) to eliminate the deltafunctions, we obtain for the conversion rate 196

Fig. 1. Kinematical relations in the case of photon emission (a = + 1) in the present scheme of stimulated conversion of neutrinos. See text for the meaning of the variables. The condition for the energy conservation E' =E-q is represented by a spherical surface, the radius of which is determined by [ k L ' -~ ]kl -q+Am2/2E from eq. (14). It should here be understood that Ik' [ and [k I are much larger than the scale q relevant to this figure. Similar relations may be given in the case of photon absorption (a= - 1).

F"~-2N~K( I#] 2+ idl2)q3ViJvli,

(15)

with I=½

E

f dO Ak~(O)~ + Ak~(O)~ £~ qZ

o~=_+1

× I T ( A k ( O ) ~ ) L2 ,

(16)

where IJvl -~ ~ - 1 , and ~--- (q/E) 2. Then, according to the condition (13), the integration over dO in eq. (16) acquires dominant contributions for p ~ IM~I / [kl ~ q/E<< I (see fig. I ). It is convenient for the numerical estimate of I to set p - ~(q/E) so as to give a good approximation d O / ~ - ~ d ~ d 0 and A k ( O ) , / q ~ (~cos0, ~sin0, -a+Am2/2Eq) under the conditions (4) and ( 13 ). A preliminary calculation gives 12__0.2 for lain21 <
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volume. Then, since the condition on the momentum transfer is relaxed as IA k l ~ q , the energy-conservation conditions for emission and absorption E' =E_+ q are both satisfied on the mass-shell of the neutrinos for the typical cases with IAm2[ < E q ~ 1 e V 2.

The number of converted neutrinos within a measurement time t is given by ANv=F't, while the total number of incident neutrinos amounts to Nv~L21Jvl.t ( L 2 ~ V/L.., Vq/~r), where the front area of the neutrino detector ~ L 2 is taken to be of the same order as that of the cavity. Then, the ratio of the converted neutrinos is estimated from the photon number given by eq. (2), which may be compared with the radiative decay rate in the vacuum, z~-I = ( [ / t 1 2 + i d [ 2 ) ( A m 2 ) 3 87tm 3 .

(17)

The result becomes R-----

N~

_~ 10-t (m/eV)3(Q/IO9)(P/IO2W)(I/0.2) (Arn 2/eV 2) 3

X (T~/s)-j ,

(18)

where x ~ 1 is omitted for definiteness. A schematic diagram for the proposed experiment is shown in fig. 2. Due to the helicity flip through the radiative conversion, the so-called active neutrinos change to sterile ones, and vice versa. By detecting

RF Cavity Sun O

Active (Sterile)~

Sterile (Actiy:~)

Neutrinos~ ~ Active (Sterile)

Active (Sterile) Neutrino detector

Fig. 2. Schematic diagram for the proposed experiment to search for radiative decay of neutrinos by the stimulated conversion of neutrino flavors with intense radiofrequency/microwave photons in a resonant high-Q cavity. Neutrinos from reactors can be also used for the experiment.

3 September 1992

the neutrinos passing through the cavity, appearance as well as disappearance experiments would be possible if the incoming neutrino flux contained the sterile as well as the active neutrinos. Half of the neutrinos coming from the Sun or a reactor pass through a cavity which contains intense rf/microwave photons. Then, the neutrino energy spectra with the cavity passage would turn out different compared to that without the cavity passage. By virtue of the quantum uncertainty for the momentum transfer in the finite scale of cavity as seen so far, any neutrino in a wide energy range can be converted so that the resultant neutrino energy spectra would be enhanced (appearance) or reduced (disappearance) uniformly over the entire neutrino energies. Recent observations of solar neutrinos in the 37C1 radiochemical experiment [ 6 ] in addition to the Kamiokande-H water Cherenkov detector [7] are claimed to be well explained by the neutrino oscillations in matter or in vacuum with a quite small A m 2 ~< 10 - 6 e V 2 for the relevant neutrinos [8]. In the case of A m 2 << m 2, usual methods to search for the radiative decay with the observation of gamma rays from the decaying neutrinos are not applicable, since the resultant gamma-ray energies, given by q
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3 September 1992

p e r i m e n t s for r a t h e r light n e u t r i n o s w i t h m < 1 eV. 3o

10E--

F

""-~r., b ~

.~ ~q /

T h e a u t h o r s w o u l d like to t h a n k A. M a s a i k e a n d M. F u k u g i t a for critical c o m m e n t s a n d discussions. T h i s r e s e a r c h was s u p p o r t e d in part by the M a t s u o F o u n d a t i o n a n d by a G r a n t - i n - A i d for the " S c i e n t i f i c R e s e a r c h o n P r i o r i t y Areas; E l e m e n t a r y P a r t i c l e Picture o f the U n i v e r s e " by the M i n i s t r y o f E d u c a t i o n , Science a n d Culture, J a p a n .

References

16 ° 10~

I

I

I

I

I

10-3

10-2

10'

1(~

10'

Neutrino Mass m

(eV)

Fig. 3. The regions of the radiative lifetime of neutrinos attainable with the present scheme of stimulated conversion of neutrino flavors as functions of neutrino mass m and mass-squared difference Am 2. The values of Q, P and I are taken to be 109, 102 W and 0.2, respectively. Also shown are the lower limits on the radiative lifetime of neutrinos obtained from the previous laboratory experiment (Gamma-rays from reactor), from the search for gamma-rays in the SN 1987A supernova (SN1987.4 ), and from the astrophysical arguments (Astrophysical limit). [ 9 ], a n d f r o m the l a b o r a t o r y search for g a m m a - r a y s from decaying reactor-neutrinos [2]. These limits w e r e o b t a i n e d by a s s u m i n g t h a t the m a s s o f the p r o d u c e d n e u t r i n o is negligibly small, rn' <
198

[1 ] See the relevant papers in: Proc. 14th Intern. Conf. on Neutrino physics and astrophysics, Neutrino 90, eds. J. Panman and K. Winter, Nucl. Phys. B (Proc. Suppl.) 19 ( 1991 ), and references therein. [2 ] L. Oberauer, F. yon Feilitzsch and R.U M6ssbauer, Phys. Lett. B 198 (1987) 113, and references therein. [3] Recent reviews include M. Roos, in: Proc. Workshop on Neutrino physics (Heidelberg, October 1987), eds. H.V. Klapdor and B. Povh (Springer, Berlin, 1988); F. von Feilitzsch, in: Neutrinos, ed. H.V. Klapdor (Springer, Berlin, 1989) p. 22; M. Fukugita, Nucl. Phys. B (Proc. Suppl.) 13 (1990) 401. [4]J.D. Bjorken and S.D. Drell, Relativistic quantum mechanics, (McGraw-Hill, New York, 1964)p. 100. [ 5 ] K. Yamamoto and S. Matsuki, to be published. [ 6 ] R. Davis Jr., in: Proc. 13th Intern. Conf. on Neutrino physics and astrophysics, Neutrino 88, eds. J. Scheps et al. (World Scientific, Singapore, 1989 ) p. 518. [7] K.S. Hirata et al., Phys. Rev. Lett. 63 (1989) 16. [8] S.P. Rosen and J.M. Gelb, Phys. Rev. D 39 (1989) 3190; J.N. Bahcall and H.A. Bethe, Phys. Rev. Lett. 65 (1990) 2233; Phys. Rev. D 44 ( 1991 ) 2962; S. Pakvasa and J. Pantaleone, Phys. Rev. Lett. 65 (1990) 2479; V. Barger, R.J.N. Phillips and K. Whisnant, Phys. Rev. Lett. 65 (1990) 3084; A.J. Baltz and J. Weneser, Phys. Rev. Lett. 66 ( 1991 ) 520; A. Acker, S. Pakvasa and J. Pantaleone, Phys. Rev. D 43 (1991) 1754. [9] F. von Feilitzsch and U Oberauer, Phys. Lett. B 200 (1988) 580; see also G. Raffelt, Phys. Rev. D 31 ( 1985 ) 3002; E.W. Kolb and M.S. Turner, Phys. Rev. Lett. 62 (1989) 509; M.S.A. Bershady, M.T. Ressel and M.S. Turner, Phys. Rev. Lett. 66 (1991) 1398.