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ELSEVIER

26 January 1995

Physics Letters B 344 ( 1995) 252-258

PHYSICS LETTERS B

Transition radiation of ultrarelativistic neutral particles *

W. Grimus, H. Neufeld Institutfir Theoretische Physik, Univemitiit Wien, Boltzmanngasse 5. A-1090 Wien, Austria

Received 12 October 1994

Editor: R. Gatto

Abstract

We perform a quantum theoretical calculation of transition radiation by neutral particles with spin l/2 equipped with magnetic moments and/or electric dipole moments. The limit of vanishing masses is treated exactly for arbitrary refraction index. Finally, we apply our result to the solar neutrino flux.

Neutrinos seem to be likely candidates for carrying features of physics beyond the Standard Model. Apart from masses and mixings also magnetic moments (MM) and electric dipole moments (EDM) are signs of new physics

and are of relevance in terrestrial experiments, the solar neutrino problem [ 11, astrophysics and cosmology. Elastic z+e _ scattering restricts MMs to p:c + 2.1 p.‘y, < 1.16.10-1R& [2] and 1 py71 <5.4.10P7p., [3] where pB is

the Bohr magneton. Limits on py7 from e +e -+ vVy are an order of magnitude weaker [4]. Note that in the above

inequalities p2 can be replaced by p*+d* since MMs p and EDMs d become indistinguishable when neutrino masses can be neglected. Astrophysical and cosmological bounds are considerably tighter but subject to additional

assumptions (see e.g. [ 51) .

In this paper we explore the possibility of using electromagnetic interactions of neutrinos with polarizable media

to get information on neutrino MMs and EDMs. The case of an infinitely extended homogeneous medium where Cherenkov radiation is emitted has already been discussed in a previous paper [ 61. Here we concentrate so to speak on the opposite situation, namely a sharp boundary between two different homogeneous media where transition radiation [ 71 is emitted when a particle crosses such a boundary. In contrast to Cherenkov radiation the particle can radiate at any non-zero velocity in this case. Classical discussions of transition radiation are e.g. found in [ 8,9] whereas the quantum theoretical approach has been used in [ 10,111. In this paper we will perform a calculation for general neutral spin l/2 particles with MMs and EDMs. As pointed out in [ lo], the quantum theoretical treatment of transition radiation of the electron changes the classical result very little but, on the other hand, is essential in our case as will become clear later. Our aim is not only to estimate the transition radiation caused by the solar neutrino flux but also to exhibit the general methods of our calculation and successive approximations. We think therefore that the present work can also be used as a basis for considering transition radiation in the case of e.g. non-relativistic particles, large diffractive indices etc. where the usual computations are not applicable.

The plane of the paper is as follows. To envisage a realistic situation we will assume a slab of a homogeneous dielectric medium in vacuum (thus we have two boundaries) and write down the corresponding plane wave solutions

” Supported in part by Jubillumsfonds der &terreichischen Nationalbank, Project No. 5051

0370-2693/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(94)01490-6

W. Grimus, H. Neufeld / Physics Letters B 344 (1995) 252-258 253

of Maxwell’s equations. These solutions will enable us to quantize the electromagnetic field in the presence of the medium. In the discussion of the probability for the emission of a photon when the particle crosses the slab we will

specialize to neutral particles by choosing the appropriate form factors of the electromagnetic current. The main

result of this paper will be obtained by deriving an exact but simple formula for the above probability in the

ultrarelativistic limit of zero particle masses and the case of the incident particle momentum being orthogonal to the

surface of the dielectric layer. Finally we will evaluate this formula at photon energies above hw,, where wP is the

plasma frequency. Finally neutrinos will come into the game by using the solar neutrino flux in the numerical

example and taking into account the experimental restrictions on p2 + d2. As already mentioned we envisage a situation where there is a layer of a medium with dielectric constant E = II’ # 1

(n is the refraction index) occupying the space defined by 1 z 1 <a/2. Thus the surfaces of the layer are orthogonal

to the z-axis. For simplicity we assume vacuum outside the layer and that the permeability of the medium is 1. For

the description we need the following wave vectors:

k,. = Sk , k,,, = Sk,, , with S=diag(l, 1, -1) . (1)

k denotes the wave vector of incident and transmitted waves whereas k,, pertains to the refracted solution inside the medium. In addition, to fulfill Maxwell’s equations there are reflected waves with k, in that part of the space from

where the wave is coming and k,,,. in the medium. The angles of incidence CY and of refraction p are related by

Snell’s law,

sin a=n sin p (2)

and therefore k,,_ = knLr,?. Since cos (Y > 0 the plus and minus signs in k, refer to waves incident from the spaces

z < - al2 and z > a/2, respectively. It is easy to show that, given a solution E, B of Maxwell’s equation for the above situation, the field

E’(x, t) +SE(Sx, t) , B’(x, t) = -SB(Sx, r) (3)

form another solution because the medium and its position in space is symmetric with respect to z+ -z. Conse-

quently, plane waves with k, < 0 can be obtained from solutions with k, > 0 by applying (3). Therefore we will

confine ourselves to k, > 0 for the time being. As a polarization basis for the electric field of the incident wave we will choose

and we will search for solutions E, B with time dependence given by e -io’. In such a situation the magnetic field is simply obtained by

B= -I-curlE. iw

(5)

It will turn out later that in the calculation of the transition radiation it is sufficient to know the electric field. Thus we content ourselves with describing Ej (j = I, II) in detail:

254 W. Grinus, H. Neufeld / Physics Letters B 344 (I 995) 252-258

-a/2<z<a/2,

-a/2<z<a/2, z>a/2.

The additional polarization vectors in En are defined by

eIIr = - Lk,.XeI= -Se,,, 0

eIInlr = - L k,,, X e, = - SeIIn, . nw

Continuity of E,, and CE, at the surfaces z = f. a/2 determines the coefficients in (6) where

E(Z) = -t

E, IzI <a/2 1, Izl >a/2

describes the variation of the dielectricity constant in space. With the definitions

K= eilkilolZ , K,, =eilkr*lzlnl2, P=k,,z/kz =n2p,

and

(6)

(7)

(8)

(9)

(10)

one can write down the list of coefficients:

a~=(1-p2)K*2(K~2-K~,)l.Y, af=4pKs2/M,

aI, =2( 1 +p)K*KzfN, a’,,,.= -2(1-p)K*K,,,IN;

a t1 = (1 -fi2)K**(K~* -Ki,)/.%, a:‘=4/TK*‘/2,

ai =2(1 +p)K*K~lnd?“, a:,.= -2( 1 -/T)K*K,,,/d. (11)

It can be checked that (6) and (5) are normalized like the vacuum solutions to

t J d”x(E(z)Ej(k, n)*.Ejt(k’+ X) +Bj(k, X)*.Bjr(k’, x)) = (2T)38jj,6(k-k’) ( 12)

and that the corresponding integrals without complex conjugation give zero. This allows for the straightforward

quantization of

J T (aj(k)Ej(k, X) +ajt(k)EJk, x)*> (13)

by

W. Grimes, H. Neujeld / Physics Letters B 344 (I 995) 252-258 255

[‘j(k), aJt(k’>l =ajj,S(k-k’) 7 [Uj(k), aj,(k’)] =O (14)

and consequently

H,= f j- d3x :(F(z)E*(x) +B*(x>):= ,zlI j- d3kwa;(k)uj(k) (15)

Now we turn to the calculation of the probability of

vi(pi, si) + r&+, sf> + y(k, j) o’=I, II) (16)

in the presence of the dielectric layer. Here vi, vfstand for any neutral fermion with magnetic and electric (transition)

moments p, d, respectively. The relation [ 111

i(Ei-Ef) I

d4XAp,(x)(Pf, sflJ”(X> Ipi> s;)= I

d4x E(x) . (pf, sflJ(x) Ipi, si) > (17)

where JP is the electromagnetic current and A, the vector potential, greatly simplifies the calculation. In our case

we have

(18)

with 4 =pi -pf and obvious abbreviations ui, z+ Assuming an initially polarized fermion but summing over the final

polarization sr we obtain

d6W= C C I&{ E”+iy,d)cT,,~~*qYu;/W(2.(2~)3S(pil -p; -k’) sf j=I,II

.6(p;-pj-k2)6(E;-Ef-w). mi. ~ p; (2:3E, d3pf’W2 d3k

(2rr)32w ’

with

and cFj(k, qz> = dz e - ‘qZZEj( k, x) *o=xl=x2=~

(19)

(20)

for the probability of the process ( 16). The Z-functions reflect the symmetries of the problems leading to energy

conservation and momentum conservation in the xy-plane which allows to express pf and q as functions of pi and k:

pf=p$Yj~ q=[p:‘.l with P=[(Ei-~)2-m~-(p,!-k’)2-(pj!-k2)2]1’2. (21)

77 = f 1 corresponds to forward and backward scattering of the fermion, respectively. In the computation of ~5’~ Cj = I, II) it turns out that uj. and a{ can suitably be expressed by a/, and a;,,. such that

these latter coefficients appear only in the expressions

sj_ E:{, sin( b> (k,, -a) ) sj+ =L&. sin( fa> (k,, + qz)

km, - qz km, + qz ’ (22)

Finally one arrives at

256 W. Grirnus, H. Neujkld / Physics Letters B 344 (1995) 252-258

fF’I(k, qJ =2(S’_ +S:)(l -n2) x k,2 -4; er’

w sin28+k,,,kZlw) +SF(q,w sin2&-k,,gkZ/w)]e~

+ [S!(qz -w2-q,k,,,z) +S’:(qf -w2+qzkn,&] sin &,} . (23)

The two unit vectors in B,, are defined as

cos c#J

e+=sin4, e,=O. 1 I 0

(1 0 1 (24)

Instead of the angle of incidence (Y we now use 8= Q(k, e,) for convenience to subsume forward (k, > 0) and

backward (k, < 0) moving incident photons. Thus

k,=wcos 8, k,,,, = (T( f3) wJn2-sin2e , cos 8

with a(e) = ~ ]cos 81

(25)

and 0 < 0 < r (0 # n-12). Note that the coefficients ai,, , a/n,,. do not change under i3 + r - 8.

Up to now no approximations have been made. In the following, however, we will evaluate (19) in the ultrare- lativistic limit. To this end we take advantage of the Gordon decomposition and obtain

4m,mfnf ~uf(~++iy5d)cT,,~‘*q”uj~“-t(~2+d2)(q~-k~)~(p~+p;)~~~2 (26) SJ

for rn;, rnp 0. Let us stress three consequences of this limit:

(i) The form of the dipole moment interaction (18) requires a spin flip of the fermion. Consequently, the

classical treatment of transition radiation is not applicable in this case ‘. (ii) The transition radiation is independent of the polarization si of the incident fermion.

(iii) Forp, = E,e, the polarization I of the photon does not contribute and the probability ( 19) is independent of

the angle 4. Adopting (iii) in addition to mi, mf+ 0 we thus obtain for the probability ( 19)

d2W= C sin3yf;‘dw; ($+d*) IS”(q2-w-k,,)+Su,(q,-w+k,,)12 (27) r)=+1

with

E=E,, qz=E-VP, P=(E=-2Ew+w2cos20)“*.

This is the main result of the present work.

(28)

We will now apply (27) to the case where the photon energy w is much larger than the plasma frequency w,, of the medium *. In this energy range, the dielectric constant is given by E = a * = 1 - IS; / w2. The differential production rate is dominated by the kinematical region where k, = qz, corresponding to small values fo the angle 8. This is a consequence of aw >> 1 for realistic situations (for example aw = lo6 for a = 10e3 cm and w = 20 keV) . For the same reason also qz = w + w*8’/2( E - w) holds for all values of o except for small and negligible range close to

’ In the classical approach, one calculates the radiation emitted by a magnetic (or electric) dipole moving with constant velocity and pointing

towards a fixed direction in space. Therefore, in the above limit, the classical calculation gives a zero result [ 121.

’ Polypropylene with op = 20 eV may serve as a typical example.

W. Gritnus, H. NeufeldIPhysics Letters B 344 (1995) 252-258 251

E. Furthermore, contributions from backscattering of the photon or the fermion in (27) are also completely negligible

and ]a::] =l,a& 2: 0. Thus to a very good approximation, the final result is then given by

This leads to the photon spectrum

(29)

(30)

where

m

(31)

We consider the situation where a K 4w/ wz. (For detectors of comparable size, the opposite case of large a

leads to a much smaller event rate.) In this case F(u) can be approximated by F( 0) = n-/2, and the total emission probability in the photon energy range w,, < w < E is given by

(32)

For our numerical example we take w,, = 20 keV and wP = 20 eV. With these values, formula (32) is valid for

a K 4.10 -3 cm. The total number of events for a flux of incoming particles Z, a detector cross section A and N

foils during a time interval T is given by

#(events) = WANT. (33)

As we are interested in the transition radiation produced by solar neutrinos, we normalize our result to I= 6.10” cm-2s-1, the solar neutrino flux expected from the standard solar model [ 131. The quantity $ + d2 in (32) has to be understood as some sort of “effective magnetic moment’ ’ [ 61,

(34)

wherep, is the probability to find a neutrino with flavour i in the solar neutrino flux. We normalize ~_~e~r to 10P9pD.

The remaining quantities are normalized to u =4. lop4 cm, A = 100 m2, N= lo6 and T= 1 yr = 3. lo7 s. Then we

obtain

(35)

Unfortunately, this event rate is extremely small leaving little hope to detect a neutrino MM by the mechanism

of transition radiation. However, one remaining possibility would be the case of a largep, and a p77 near its present upper bound.

In concluding this paper we want to mention a few points. In the numerical calculation for a stack of N foils we have simply multiplied the probability for one foil by N. This is certainly not correct if the distance between the foils becomes comparable to their thickness and thus interference effects become important. However, if the situation considered here is similar to the transition radiation of electrons in this respect then we cannot expect a substantial increase of the radiation yield [ 93 and Hq. (35) still holds as an order of magnitude estimate. Furthermore, we have not taken into account total reflexion but for w ZP wP its effect will be negligible. On the other hand, for large II - 1 one could have such geometries of the dielectric medium where total reflexion is important. Finally, we want to

258 W. Grimus, H. Neujeld / P hysics Letter~s B 344 (1995) 252-258

stress once more that the methods used in our calculation are quite general and could thus be carried over to cases where one is not satisfied with the usual approximations made in the field of transition radiation.

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