Fast secondary electrons and neutrino mass measurements

Volume 262, number 1

PHYSICS LETTERS B

13 June 1991

Fast secondary electrons and neutrino mass measurements E.G. D r u k a r e v Leningrad Nuclear Physics Institute, Gatchina, SU-188 350 Leningrad, USSR Received 24 January 1991; revised manuscript received 10 April 1991

The influence of fast secondary electrons, caused mainly by the final state interactions, on the spectrum of the l~-decayof tritium is considered. The possible applications to neutrino mass measurements are discussed.

1. Introduction

A

F(A, M ~ ) = . I f ( e ) (A--e) [ ( d - e ) 2 - M 2 v I l/2de, The 13-decay of tritium is a well-known tool for the measurement o f the neutrino mass My [ 1 ]. The information about the value of My is obtained from the analyses o f the spectrum o f the [3 near the end-point Eo = 18.6 keV. For the fixed value o f the ~ energy E the dependence of the spectrum on the difference A = E o - E is contained in the factor (M~+Ev)p~ o f the equation

0

(4) e is the energy of the secondary electron, while the distribution function f ( e ) is normalized by the condition oo

f

f(e) de= 1 .

(5)

0

d W = O(E) (M~ + Ev)p,,dE ,

(1)

Ev (pv) is the kinetic energy ( m o m e n t u m ) o f the neutrino, 0 ( E ) is the Fermi function multiplied by certain kinematic factors and by the matrix element of the decay of the nucleus. Thus the subject of the analyses is the function

dW - -

-F(A, M2) ,

~(E)dE

(2)

F(d, M 2) = ( d - t) 2 - ( M 2 - 2or2)/2

(3)

t=(e), a2= ( e 2 ) - (

Eqs. ( 1 ) and (3) are true if the possibility o f the transitions in the electronic shell of the decaying atom is neglected. If one takes into account the possible ionization (and excitation) of the bound electrons the energy ,4 = E o - E should be shared between the neutrino and the secondary electrons. Thus Elsevier Science Publishers B.V. (North-Holland)

(6)

with

while

F( ,4 ) = A ( "42- M2v) 1/2

If We include the interaction with the bound electrons in the shake-off (s.o.) approximation [2] we obtain f ( e ) ~ e - 4 for e >> J ( J is the binding energy o f the electron). Thus the integral o f eq. (4) is saturated by e ~ J and for A >> J we can replace the upper limit by infinity. U n d e r this condition it was obtained in ref. [ 3 ] that

(e) )2 ,

(7)

(e') = f def(e) e~ .

(8)

while 0o

0

105

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One of the approaches to the problem [3] is the determination of the value of M 2 by fitting the spectrum by eq. (6). If the interaction between the ~ and the electronic shell is neglected, i.e. f ( e ) = ~ ( e ) the right-hand side ofeq. ( 1 ) is given by the function Fo. For the latter we obtain

Fo(A', mE) =F(A, M~)

(9)

with

A ' = A - t (E'o=Eo-t) ,

(9')

rn2=M2v-2tr 2 .

(9")

Thus neglecting the transitions in the electronic shell leads to the imitation of the value m 2 = - 2 a 2. Now we shall see how accounting of the final state interaction (f.s.i.) of the fl with the electronic shell does change the situation.

If the Coulomb parameter of the fl, ~= (mot2/ 2E) 1/2, is small enough, the f.s.i, is taken into account in the lowest order of perturbation theory by the equation [4] ITI 2= I To 12+2to Re T1 + IIm T112 (10)

Here T, is the amplitude with n final state interactions. Since To ~ 1, Im Tl ~ ~, Re Tl ~ ~2, Re T2 ~ ~z the three last terms give a correction of the order ~2 to the s.o. term I To l 2. In ref. [ 5 ] the f.s.i.in 13-decay of tritium was considered but the last term of eq. (10) was overlooked. However it is as important as the others. In the calculation of the total probability it cancels the third one. The calculation of decay probabilities to various states of He 3+ based on eq. (10) was carried out in ref [6]. The influence of the f.s.i, in the transitions to the lowest lying excited states on the value of the neutrino mass is negligible. (The conclusion reached in ref. [ 5] is the same though the numerical results are different. ) However the role of the f.s.i, increases if we consider the transition to the continuum with energy e >> J. The f.s.i, terms i n f ( e ) are proportional to (J/e) 2 at e>>J. Thus in the integrals 106

d

z.

=

(8')

f f ( e ) e" de 0

the large values e >> J of the energies of the secondary electrons become important. On the other hand the f.s.i, corrections contain the small factor ~2= rna 2/2Eo--7 × 10-4 (~ is the Coulomb parameter of the 13). Thus one can carry out the expansion in powers of e in eq. (4). Hence, eq. (6) is still true with the upper limit of integration in eq. (8) being replaced by A. For the moments t, (A) one can obtain [6] A zl = q + 7 1 n ~ ,

1"2- 12 + ~)A,

(11)

if A >> J. Here the t, are the s.o. values of z,, B is of the order of the binding energy J while y=~2x,

2. Accounting of the final state interactions

+2ToRe T2.

13 June 1991

(12)

with x depending on the structure of the decaying system (atom or molecule). For an atom [ 5 ] 1 Ka=~mm(WI ~ r u 2 l ~ ) ,

(13)

g/is the atomic wave function. For the molecules we attribute the electrons which are involved into the chemical binding to the separate atoms [ 7 ] and present X=Xa+Xl with

xl =

~a o 1 R/2 2 m '

(14)

Ri is the distance between the decaying nucleus and the ith nucleus of the molecule, the a o are the occupancies of the initial electronic states. The values of Ri are assumed to be much larger then the radii of the atoms. Eq. (14) is not true for molecular tritium. Using eqs. (6) and ( 11 ) we obtain

F(A,M~)=

(

A-t,-yln~

5

-M~/2,

m v2 = - - 2 a o2 + M v2 + 4 t l y l n ~z~ + O ( y 2) , with a 2 being the s.o. value o f e 2.

(15) (16)

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PHYSICS LETTERS B

Before starting the analyses o f eqs. ( 15 ), (16) we present the values o f the parameters.

i 3 June 1991

If I am21 << M2~ M~ =M~e

1 8m 2

(24)

2 M~e

3. The values of the parameters

Here we present the values of tl, Y and B for two cases. The first one is atomic tritium for which the nonrelativistic Coulomb calculations are true. The second one is a tritium nucleus implanted into a valine molecule (CsHI INO2) used in the ITEP experiment [ 8 ]. For atomic tritium tl = J = 13.6 e V ,

(17)

8m2=4t~yln-~ + O ( y 2) .

while [ 6 ] B = 0 . 7 8 J = 10.5 e V ,

y = 2 . 0 × 10-2 e V .

Thus at large A 0 - 1.5 keV, we obtain that the imitated values o f the mass m~ are respectively

tl = 1 9 e V .

(19)

Using the values o f a o [ eq. (14) ] calculated in ref. [ 10 ] we obtain at ,4 ~<300 eV, when only the external bound electrons are involved, B,,~ 10 e V .

(20)

For the larger values of,4 the interaction with the Kelectrons o f C, N and O nuclei should be taken into account, giving y = 5 . 8 X 10-2 e V ,

(25)

(18)

For valine [ 9 ]

y = 4 . 7 X 10-2 e V ,

Undoubtly the analysis of a real experiment is much more complicated than the following one. The calculations o f the shift of m2 are given in order to illustrate the size o f the effect. For a better quantitative feeling o f the effects in valine we consider the spectrum at large '4 where the logarithmic approximation has chances. Assuming that In A does not vary much while '4 is close to a certain value ,4o we calculate using eqs.

B = 2 0 eV

(21)

8 m 2 = 5 . 4 e V 2,

8 m 2 ~ 1 9 e V 2.

(26)

while the accuracy o f the result for valine is about 25% in this case. N o w we come to the analysis o f the I~-spectrum. Considering the Kurie plot

K(,4, M 2) = [F(A, M 2) l '/2 =A-t~

M 2_2a 2 4,4

,4 yln~

(27)

at ,4>~500 eV. Note that the values of B are rather order of magnitude estimates in this case. Thus the equations for valine are really true in the logarithmic approximation.

we see that there is an additional nonlinear contribution o f the last term. The deviations from the linear law are characterized by the finite value o f the second derivative

4. The imitated value of m

d2K d~2 --

If all the effects but that o f the f.s.i, are taken into account the value o f M2e extracted from the experimental data is MZe = M 2 + 8m 2 ,

(22)

8m 2 is the value imitated by the f.s.i. If the real value My = 0 we obtain the imitated neutrino mass

Mv=(Sm2) w2

(23)

M 2 _ 2tro2 + 2,43 ,42 •

(28)

Thus the f.s.i, imitates the value 8m 2 = - 2y'4.

(29)

which is negative due to the different dependence on ,4 o f the two last terms of eq. (27). Note that 8m 2 does not depend on the value o f B. Thus eq. (29) is free from the uncertainties o f the logarithmic approximation which was used for the calculations in 107

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valine. Since the nonlinearity of the Kurie plot is visible at A ~<100 eV we obtain the imitated value ~m 2 that is of the order of several eV 2, e.g., ~m 2 = - 4 eV 2 and ~rn 2 = _ 9 eV 2 at A= 100 eV for atomic tritium and valine, respectively. If the experimental spectrum is approximated by eq. (6) with the value of M 2 being extracted by the minimization of the discrepancy in a certain interval of the values of/1 we see that the f.s.i, imitates the value 8mv=~K 1 2~4wfrln_~_"

(30)

Here Klabels the N points of the spectrum. Assuming Ax-- Kd with d being the distance between the experimental points, we obtain for N>> 1 8m2 = 2~,d~v(ln ~-~g-- 1 ) + O ( N - ' ) .

13 June 1991

limiting value. But the role of the effect increases rapidly if the sensitivity is improved. Assuming all the rest of the uncertainties to be removed, we obtain that the values of the neutrino mass that are of the order of 2-3 eV for atomic tritium and of the order of about 4-5 eV in the case ofvaline can be imitated if the true value of the neutrino mass is zero. The calculations of the effect in the important case of molecular tritium is a separate and more complicated problem. At any rate the results obtained for atomic tritium can be treated as the lower limit for the molecular one. The real experiment is undoubtedly much more complicated. The aim of this paper is not the analysis of the experimental data but just a note on the possible role of the fast secondary electrons.

(31)

Acknowledgement Here As= Nd is the largest value ofdK in the interval. If d = 100 eV ~m 2 = 7 . 2 e V 2, 5 m ~ = 1 7 e V 2

(32)

for atomic tritium and for valine. Note that in the experiments with molecules all the effects connected with the secondary electrons may become larger if the solid angle of the observation of the 13 is limited while the molecules have a certain orientation-see the Conclusion of ref. [ 11 ]. In the limiting case the enlargement is given by the factor 1.9 [ l l l .

5. Summary Accounting of the final state interactions leads to the influence of the fast secondary electrons (e >> J) on the spectrum of the ~. Until the sensitivity of the experiment to the neutrino mass is of the order of 10 eV (e.g., the case of valine) the f.s.i, causes a small contribution to the total uncertainty in M 2. It produces a correction of the order of I eV or less to the

108

I am indebted to D. Decman and V.A. Lubimov for useful discussions.

References [ 1 ] K.E. Bergkvist, Nucl. Phys. B 39 (1972) 317. [2] A. Migdal, J. Phys. (USSR) 4 ( 1941 ) 449. [ 3 ] V.A. Lubimov, XXII Intern. Conf. on High energy physics, (Leipzig), Vol. II (1984) p. 108. [4] E.G. Drukarev and M.I. Strikman, Zh. Eksp. Teor. Fiz. 91 (1986) 1160. [5] R.D. Williams and S.E. Koonin, Phys. Rev. C 27 (1983) 1815; J. Arafune and T. Watanabe, Phys. Rev. C 34 (1986) 336. [6] E.G. Drukarev, Sov. J. Nucl. Phys. 50 (1989) 876. [ 7 ] J.N. Murrell, F.A. Kettle and J.M. Tedder, The chemical bond (Wiley, New York, 1978) Ch. 9.3. [ 8 ] V.A. Lubimov et al., Phys. Lett. B 94 (1980) 266. [ 9 ] I.G. Kaplan, V.N. Smutnyi and G.V. Smelov, Zh. Eksp. Teor. Fiz. 84 (1983) 833. [ 10 ] E. Clementi, F. Gavallone and R. Scordamaglia, J. Amer. Chem. Soc. 99 (1977) 5531. [ 11 ] E.G. Drukarev and L.L. Frankfurt, Sov. J. Nucl. Phys. 43 (1986) 33.