Sum rules for two-particle operators and double beta decay

Volume 212, number 3

PHYSICS LETTERS B

29 September 1988

S U M RULES FOR TWO-PARTICLE OPERATORS AND DOUBLE BETA DECAY P. V O G E L Physics Department, California Institute of Technology, Pasadena, CA 91125, USA

M. E R I C S O N CERN, CH- 1211 Geneva 23. Switzerland and Institut de Physique Nucl~aire, F-69622 Villeurbanne, France

and J.D. V E R G A D O S Physics Department, University ofIoannina, GR 45332 loannina, Greece

Received 11 March 1988; revised manuscript received 11 July 1988 Sum rules for the double Gamow-Teller and Fermi operators are derived. They are exact when additional symmetries, isospin and SU (4) invariance, hold. Moreover, they represent a useful approximation independent of further assumptions for the cases of practical importance, nuclei with N - Z>> 1. The 2v mode of double beta decay exhausts only ~ 10 4 of the GT sum rule. Examples of the double strength distribution calculated within QRPA are shown and the importance of experimental determination of the double strength is stressed. Several processes which can be used for that purpose are identified and briefly discussed.

It has been p o i n t e d out [ 1-5 ] that the m a t r i x elements for the 2v 1313decay and to some extent also for the 0v 1313decay are suppressed when evaluated in the Q R P A with a consistent treatment o f the particle-hole a n d p a r t i c l e - p a r t i c l e c o m p o n e n t s o f the n e u t r o n proton interaction. To illustrate the difficulties in evaluating such small nuclear matrix elements we consider here a m o r e general situation, n a m e l y the p r o b l e m o f how large the total double G a m o w - T e l l e r and F e r m i strengths are, a n d how they are distributed a m o n g the states in the final nuclei. Let us recall how a similar p r o b l e m can be formulated for the usual single-particle G a m o w - T e l l e r a n d F e r m i operators Y_+ = ~tr, t + a n d T_+ = Z t ( . The difference o f the [3- a n d [3÷ strengths is then ( s u m m i n g over the angular m o m e n t u m projections in the G T case) S~r-S~-=(0I[Y

,Y+]I0)=(0]-6T=[0)

=3(N-Z), S~.--St

(la)

+ = <01 I T _ , T+ l [ 0 > = < 0 I - 2 T - I 0 >

= (N-Z).

(lb)

0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )

Eq. ( l a ) represents the famous G a m o w - T e l l e r sum rule, which is i n d e p e n d e n t o f the structure o f the ground state I 0 ) and is obeyed as long as nuclei are assumed to be m a d e o f nucleons only. In nuclei with a large neutron excess, N - Z > > 1, the 13+ transitions are Pauli blocked and the sum rules ( 1 ) provide an estimate o f the total 13- strength. I f one assumes isospin conservation the F e r m i 13+ strength vanishes entirely (for T = - T ~ > 0) and the F e r m i 13- strength is concentrated in the isobar analogue state. Similarly, i f one assumes the validity o f the Wigner SU ( 4 ) supermultiplet symmetry, the Gam o w - T e l l e r 13÷ strength vanishes and the [3- strength is concentrated in a single state with S = 1, Tf= T - 1. While the validity o f the isospin conservation is known to be very good, the S U ( 4 ) is not expected to be a good s y m m e t r y in nuclei. Nevertheless, experiments show that the "giant G T " state d o m i n a t e s the [3- strength distribution a n d exhausts most (typically ~ 60%) o f the sum rule for G a m o w - T e l l e r transitions [6]. At the same t i m e both 13+ a n d 13transitions to low lying states are suppressed by a factor o f ~ 100 when expressed in units o f the sum rule 259

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( 1a). Thus, it appears that the SU (4) selection rules offer useful guidance, even though the giant GT state does not actually exhaust the sum rule, it is quite broad, and a substantial part ( ~ 2 5 % ) of the observed strength is spread in the energy interval below it. Sum rules for the double operators can be again obtained from the corresponding commutators. The commutators of the double operators are three body operators, and their expectation values are no longer exactly conserved quantities. For the Fermi operator we obtain

[TZ_,T2+]=4T=-4T_T+~-4TzT+T_.

(2)

Taking the expectation value of (2) in a state of isospin T and ~ = ½( Z - N), we find for T = - Tz <0l [T2-, T2+] 10>

= 2 ( N - Z ) [2(T 2 - T 2 ) + 2 T - 1] =2(N-Z)(N-Z-1).

(3)

Eq. (3) is the sum rule for the double Fermi operator. If isospin is exactly conserved, the strength is concentrated in the double analogue state, and for N - Z>~ 0 the T 2_ strength vanishes. The double Gamow-Teller operator can be a scalar, vector, or second rank tensor in ordinary space. For application to double beta decay, as we are interested in the 0 ÷ 4 0 + transitions, we shall consider the scalar operator. Defining X= Zai, the commutator equals

[Y_ "Y_, Y+ "Y+ ] =-2

~ (Y~_T~Y"++TzY"_Y~+

=6(N-Z)(N-Z+I) +4(N-Z) +2<01X'(Y_×Y+)-(Y+×Y_)'XI0>.

(5)

For Pauli blocked nuclei the second line of (5) vanishes. It also vanishes if one assumes the validity of SU (4) symmetry and considers the simplest SU (4) representation for the even-even nucleus. Thus, for the double Fermi and Gamow-Teller operators we cannot obtain exact sum rules, equivalent to (1), without additional assumptions about nuclear symmetries, but we can obtain approximate sum rules. The distribution of the double Gamow-Teller strength can be evaluated if one assumes the validity of SU(4). (For notation and techniques needed to evaluate matrix elements in SU (4) see ref. [ 7 ]. ) For even-even nuclei N-Z>_- 1, the ground state belongs to the supermultiplet [y, y, 0] and has S = 0 , T=y= ½( N - Z). The ground state is the only state of the nucleus (N, Z) belonging to that supermultiplet. In the odd-odd nucleus ( N - 1, Z + 1 ) there are two states belonging to the supermultiplet [y, y, 0], the isobar analogue state with T=y, S = 0 and the GT state with T = y - 1, S = 1. There are no states belongingto [y, y, 0] in the odd-odd nucleus ( N + 1, Z - 1 ). In the even-even nucleus (N-2, Z + 2 ) there are three states belonging to [y, y, 0]; the double analogue T=y, S = 0 , and two double GT states with T = y - 2 , S = 0 and T=y-2, S = 2 . The double GT operator Y+- Y+ populates both S = 0 states in ( N - 2, Z + 2) with strengths

= 12y

+i ~ ~"a~(Y"_X~Y~+ + X ~ Y "_ Y~+ ,xpy (4)

Taking the expectationvalue of (4) in the state with Tz= ½( Z - N ) we find

260

(01 [Y_ "Y_, Y+ "Y+ ]10)

I [2

+ Y~_ Y~_ T~ + Y~_ T~ Y~_)

+ Y~+ Y~_Zr+ YP+XYY~_ ) ,

29 September 1988

4(y2-1) , 2y- 1

3 I < S = 0 , T=yl Y+'Y+ 10> [2= 12Y2y_ 1 "

(6) (7)

There is no double [3÷ strength in SU(4) for nuclei with N - Z > 0. In order to establish a link to the double beta decay let us recall that the rate of the 2v decay is proportional to the square of the amplitude [ 8-10 ]

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2~ ( 0 ~ IY+ [ l ~ ) ( l m + IY+ 10i+ ) M ~ T = ~'m Em-(Mi+Mr)/2 ( 0~- [ Y+" Y+ I0,+ > zXE '

(8)

where we obtained the last equation in the closure approximation. In this case the scalar double Gamow-Teller operator determines the rate of the 2v [313 decay. For the 0v mode of [313decay the rate formula contains terms proportional to the neutrino mass square, ( m y ) 2, and terms proportional to the strength of weak interaction which couples to the right-handed lepton current [8-1 0 ]. Here, for simplicity, we consider only the ( m y ) 2 part. In that case the nuclear structure part of the process is determined by the matrix elements M c0 vv = (0~-IR ~, ~r~.~rj t~-tf ~(r~j) [0i+ ) , i,j

rij

M°V= (0~ IR ~ t+ t+ q~(r~j) 10+ ) , ij

ri)

(9)

where R=roA 1/3 is the nuclear radius and the summation is over all nucleon pairs. The factor ~(r)/r in (9) comes from the neutrino propagator, q~(r) ~ 1. For qualitative estimates of these matrix elements we replace the propagator by a constant and again characterize the nuclear structure part of the problem by the ground-to-ground state matrix elements of the double Gamow-Teller and Fermi operators. Thus, the matrix elements which characterize both modes of 131~decay are, at least crudely, the quantities which determine the extreme low energy tail of the strengths that obey the sum rules (3) and (5). These sum rules, therefore, offer a convenient unit by which one can judge the degree of suppression of the matrix elements obtained by detailed calculation. The shell model calculations of ref. [8 ] correspond to 0.0010.005 of the sum rule for the heavier 1313decay candidates, and to 0.0004 of the sum rule for 48Ca. From the experimental I]13lifetimes we find that the fraction of the sum rule (5) is ~ 0.0005 for 8ZSe [ 11 ], and ~0.00002 for ~3°Te [12]. (In order to extract the closure matrix elements from experimental lifetimes one has to estimate the corresponding average energy denominators.) The calculations based on

29 September 1988

QRPA [ 1-5] usually give matrix elements smaller than the shell model values [ 8 ], and for the cases of interest the ground to ground state double G a m o w Teller strength represents about 10 -4 of the sum rule (5). It is, therefore, not surprising that the calculation of double beta decay lifetimes is so difficult; we are dealing with very hindered processes. The importance of the sum rules (3) and (5) for 1313decay goes beyond the convenient unit. In many cases, e.g., in the shell model calculations of [ 8 ], only the 0 ÷ states of the initial and final nuclei are considered. Evaluation of the double strength allow one to judge how serious the limitations caused by the necessary truncation of the single-particle space are. Thus, even though the "real" 1313decay amplitudes (8) and (9) are only very approximately related to the matrix elements of the double Gamow-Teller and Fermi operators, it is useful to evaluate the double strengths because their values and distributions may reveal the limitations of the underlying nuclear structure methods. QRPA obeys automatically ( 1 ) for single-particle operators. For the two-particle operators which populate two-phonon states of QRPA the sum rules (3) and (5) are obeyed in the case of Pauli blocked nuclei with N-Z>> 1. In fig. 1 we show an example of the QRPA calculation of the double strengths for the nucleus 76Gewhich is Pauli blocked; the programs and parameters of ref. [3] were used. The double [3÷ strengths are not displayed, as they are negligibly 250

I

i

I

76Ge double/3- strength 200

A

o~ 150

II II I I

~ loo cO

50

00

5

10

15

20

25

Energy (MeV) Fig. 1. Double [3- strengths for 76Ge versus the excitation energy of the final nucleus 768e. The discrete lines obtained in QRPA have been replaced by a continuous distribution by assigning a width to each of them. The GT strength is denoted by the full line, the Fermi strength by the dashed line.

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small. One can see that the double [3- Fermi strength is concentrated in the double analogue state; the small additional peaks reflect isospin non-conservation introduced by the approximate nature of the QRPA. The double ~- GT strength shows some concentration with the average energy of ~ 18 MeV. As expected the average energy equals approximately twice the energy of the single giant GT state. The spreading of the double [~- GT strength is, however, considerable. In order to illustrate the limitations of the validity of this approach and of the QRPA treatment we show in fig. 2 the double strengths of the nucleus 54Fe which is not Pauli blocked as the single-particle transition nfT/2~vfs/2 is allowed. Even in this case, however, the double [~+ Fermi strength is negligible. The double [3- Fermi strength, shown by the dot-and-dashed line, should be concentrated in the double analogue state of 54Fe, i.e., in the ground state of 54Ni. Again, the QRPA leads to some spreading. The double [3- GT strength is concentrated in two peaks which correspond approximately to the vfv/2~nfT/2 and vf7/2 --'~f5/2 transitions. The distance between the double GT and the double IA resonances is much larger here than in 76Ge, in agreement with the general trend for the single GT versus IA resonance [ 13 ]. The double [~+ GT strength of 54Fe, also shown in fig. 2, is sizeable. As expected in this case the simple expression 6 ( N - Z ) ( N - Z + 1 ) for the double GT sum rule is not applicable.

60 50 40 ~

30 i13~; '~ (~Ox)

~ 2o ~'~

~ 13GT

13+ p+ k 0~""

5

_a._ "t,,. 10 15 Energy (MeV)

I 20

25

Fig. 2. Double strength for 54Fe. The double GT 13 strength is shown by the full line, the double [3+ GT strength by the dashed line, and the double 13- Fermi strength by the dot-dashed line. (The Fermi strength was multiplied by the factor 10 so that the same scale can be used. )

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29 September 1988

We shall discuss now possible ways to determine experimentally the double charge changing strength. Since the realization that study of the (p, n) and (n, p) reactions furnishes information on the single charge changing Gamow-Teller strength, a whole field of nuclear study has been opened, complementing the traditional studies of the low energy tails of these strengths in ~- and [~+ decays. In analogy, a possible way to study the double charge-changing strength is through reactions based on strong interaction in which the projectile loses two units of charge and the target gains them, or vice versa. What is the motivation for such program? The relation to double beta decay has been stressed already. Moreover, it would be possible to see whether the ideas of SU (4) symmetry restoration (see, e.g., ref. [ 14], and references therein) is a viable one. An example of a projectile which can lose two units of charge is the pion through the double charge-exchange reaction rt + ~ r t - . On the basis of PCAC the dominant GT part of the double beta decay is proportional to the pion charge exchange amplitude taken for soft pions. However, the fact that the soft pions have a vanishing energy, instead of an energy ~ rn~ as is the case of physical pions, makes a fundamental difference. For physical pions the intermediate states with one pion are favored. The two body operator, responsible for the transition from the initial to the final nucleus involves then pion propagator and therefore cannot have the simple T+ T+ or Y+-Y+ form. On the contrary, in [313decay it is more advantageous to have no pions in the intermediate state. In order to retain the link between double charge exchange and double beta decay the charge exchange process should be soft. Double charge exchange of a hadronic probe induced by two successive spin-isospin interactions is an example. Since a single charge exchange, such as (p, n), has proven so successful in study of the giant GT states, one can hope that the double charge exchange of a hadronic probe will allow one to study the double GT states. A possible candidate is the (p, A - ) reaction. The momentum transferred to the nuclear system, q=Po--PA should be as small as possible. Thus the A- should be emitted in the forward direction, and the proton energy should be large when compared to the proton rest mass. (Note, however, that the smallness of Iql does not guarantee that there is no momentum transfer at

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each successive step. ) Unfortunately, observation o f the (p, A - ) reaction requires the detection o f two particles, n - a n d n, a n d represents a f o r m i d a b l e task. A n o t h e r probe would be a reaction involving the target (Z~, At) a n d the projectile ( Z v, Ap),

29 September 1988

The authors would like to thank the Theory Division o f C E R N for hospitality and one o f them, P.V., the U S D e p a r t m e n t o f Energy G r a n t No. DE-F60388ER40397 for support. Advice o f Professor K.T. Hecht on the SU ( 4 ) s y m m e t r y is appreciated.

(Zp, Ap) + (Zt, A,) ~ (Zp - 2 , Ap) + (Zt + 2, A , ) ,

(lO) References in which the transferred nucleon p a i r is d o m i n a n t l y in a relative s-state a n d which is d o m i n a n t l y a onestep process. R e a c t i o n (10) could allow d e t e r m i n a tion o f the double ~ - strength o f the target, p r o v i d e d the projectile is left in a well defined state. This is difficult as the projectile would have to undergo a " d o u b l e [3+'' transition a n d we argued that such transitions are usually suppressed in stable nuclei. A way a r o u n d it is to use a radioactive beam, e.g., ~40 or ~SNe, since in e v e n - e v e n projectile with Z = ½A+ 1 both the initial a n d final nuclei belong to the same SU ( 4 ) multiplet. Beams o f these a n d similar radioactive nuclei are available at low intensities, for example, at G A N I L [ 15 ]. While e x p e r i m e n t a l d e t e r m i n a t i o n o f the double strength is u n d o u b t e d l y not i m m i n e n t , it is worthwhile to keep this possibility in m i n d a n d hope that the ingenuity o f our e x p e r i m e n t a l colleagues will o v e r c o m e the f o r m i d a b l e difficulties associated with such a project.

[ 1] P. Vogel and M.R. Zirnbauer, Phys. Rev. Len. 57 (1986) 3148. [ 2 ] O. Civitarese, A. Faessler and T. Tomoda, Phys. Lett. B 194 (1987) 11. [3] J. Engel, P. Vogel and M.R. Zirnbauer, Phys. Rev. C (February 1988). [4] T. Tomoda and A. Faessler, Phys. Lett. B 199 (1987) 475. [ 5 ] W.M. Alberico et al., Torino preprint ( 1988 ). [6] C. Gaarde et al., Nucl. Phys. A 369 ( 1981 ) 258. [7] K.T. Hecht and S.C. Pang, J. Math. Phys. 8 (1969) 1571. [8] W.C. Haxton and G.J. Stephenson Jr., Prog. Part. Nucl. Phys. 12 (1984) 409. [9] M. Doi, T. Kotani and E. Takasugi, Prog. Theor. Phys. Suppl. 83 (1985) 1. [ 10 ] J.D. Vergados, Phys. Rep. 133 ( 1986 ) 1. [ 11 ] S.R. Ellion, A.A. Hahn and M.K. Moe, Phys. Rev. Lett. 59 (1987) 2020. [ 12] T. Kirsten et al., in: Nuclear beta decay and neutrinos, eds. T. Kotani et al. (World Scientific, Singapore, 1986) p. 81. [13] C. Gaarde, Nucl. Phys. A 396 (1983) 127c. [ 14] Yu.V. Gaponov, Sov. J. Nucl. Phys. 40 (1984) 55. [ 15 ] P.G. Hansen, Nature 328 (1987) 476.

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