11 September 1997

Physics Letters B 408 (1997) 439-444

PHYSICS LElTERS 6

Search for non-paulian transitions in 23Na and 1271 R. Bernabeia, P Belli a, F. Montecchiaa, M. De Sanctis b, W. Di Nicolantoniob,

A. Incicchitti b, D. Prosperi b, C. Bacci ‘, C.J. Dai d, L.K. Ding d, H.H. Kuang d, J.M. Mad a Dip. di Fisica, Universita’ di Roma “Tor Vergata” and INFN, sez. Roma2, I-00133 Rome, Italy b Dip. di Fisica, Universita’ di Roma “La Sapienza” and INFN, sez. Roma, I-00185 Rome, Italy

c Dip. di Fisica, Vniversita’ di Roma Ill and INFN, sez. Roma, 1-00185 Rome, Italy d IHEP: Chinese Academy, I?O. Box 91W3, Beijing 100039. China

Received 27 March 1997 Editor: K. Winter

Abstract

A new search of non-paulian nuclear processes, i.e. processes normally forbidden by the Pauli exclusion principle, was carried out by determining an improved upper limit for the spontaneous emission rate of protons in =Na and t2’I. A large set-up of highly radiopure NaI(TI) crystals was employed; it was placed deeply underground at the Gran Sasso National Laboratory of I.N.F.N. The upper limit on the unit time probability of non-paulian emission of protons with energy Et, > 10 MeV is A < 4.6 . 10-33s-1. The corresponding limit on the relative strength (S2) for the searched non-paulian transitions has been estimated to be S2 s 1 .O . lo-“. @ 1997 Elsevier Science B.V.

1. Introduction

The Pauli exclusion principle (PEP) plays a crucial role in our description of the structure and properties

of fermion aggregates as atoms and nuclei. Despite of

its well known success, the exact validity of PEP is

still an open question. In fact, the principles of quan- tum field theory allow to adopt various kinds of statis- tics more complex than the standard Fermi and Bose ones. In this context, let us remember that Mohapatra [ l] examined the possibility to adopt for the particle creation (destruction) operators a “q-commutation al- gebra” such as

UiUT - qUj+Ui = 6ij , -l<q<l.

This situation has inspired many experimental tests of the PEP validity. A list of these efforts, regard-

ing both atomic and nuclear physics, can be found

in the extensive reviews of Refs. [ 2-51. In particu- lar, the possibility of nuclear decay processes asso- ciated to non-paulian transitions, i.e. transitions nor- mally forbidden by PEP, has been examined by Ejiri and Toki [ 41; these authors showed that PEP viola- tions could give rise to long life nucleon emission by atomic nuclei usually considered to be stable. In the above context a relevant role is played by those nucleon-nucleon interaction processes in which one of the interacting particles falls from higher energy shell to a lower energy state, normally occupied, while the other one acquires enough energy to escape from the nucleus.

In Ref. [4], the authors searched for the non- paulian emission of single protons with energy Ep 2 18 MeV in 23Na and ‘27I obtaining for the PEP va- lidity a very stringent limit. Aim of the present work is to perform a non-paulian proton search analogous

0370-2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SO370-2693(97)00842-3

440 R. Bernabei et al. / Physics Letters B 408 (19971439-444

to that of Ref. [ 41, exploring a larger proton energy range (EP > 10 MeV).

Our results have been obtained with a large mass set-up, assembled by highly radiopure NaI(Tl) crys- tals placed deeply underground in the Gran Sasso Na- tional Laboratory of I.N.F.N. The crystals act both as probed proton sources and detectors. The apparatus was mainly realized for a particle Dark Matter search (DAMA experiment).

2. Experimental results

The results presented in the following were obtained by using four 7.05 kg (stored deeply underground since about 3 years) and nine 9.70 kg NaI( Tl) crystals (stored deeply underground since about 1 year) built by Crismatec company in the framework of I.N.F.N. contracts. The main characteristics of these detectors have been described elsewhere [6]. Each detector was viewed by two low background EMI9265B53/FL photomultipliers working in coincidence. The shield was made by 15 cm of low-radioactivity lead and 10 cm of highly radiopure copper; furthermore, the lead was surrounded by 1.5 mm Cd foils and about 10 cm of polyethylene. The different shield materials were stored underground since different time periods, vary- ing from several years to several months. A Lecroy transient digitizer, TRS828D, allowed us to record the pulse shapes in a 3 125 ns time window. In the course of the experiment, by summing the data collected in different runs, it was accumulated a statistics corre- sponding to 6.13 . lo7 kg . s, that is N. t = 2.46. 103* nuclei . s for both 23Na and 1271 nuclides.

The total single hit spectrum, i.e. the spectrum ob- tained considering the case in which only a single crys- tal fires is shown in Fig. 1. No event is observed in the energy region ( 10-36) MeV. Taking into account this result, we can immediately calculate an upper limit on rates (probabilities for unit time) of non-paulian processes leading to emission of protons in the energy range 10 MeV < E,, 6 36 MeV, we obtain:

A = A(23Na) + A(1271) < l.l4(eNt)-’

= 4.6. 1O-33 s-l (1)

with 68% confidence level (C.L.) . In the above ex- pression, 1.14 is the appropriate statistical factor for

40-

I

i

I

0 “““‘nh”I”I”,‘l”” 20 40 60 8C 100

Energy(MeV)

Fig. 1. Single hit energy spectrum for a statistics of 6.13. 107kg. s collected deeply underground in the Gran Sasso National Labora- tory of 1.N.F.N. The considered energy bin is 0.5 MeV.

zero observed events and E is the proton detection effi- ciency assumed to be 100 %. The corresponding limit on the widths is:

r = I’(23Na) + I’( 1271) = TLA < 3.0 a 1O-54 MeV.

(2)

3. The theoretical framework

Let us now define, according to Ref. [ 41, a conven- tional parameter a2 allowing to quantify possible PEP violations. The width Ti for a single nucleon transi- tion to the i-th occupied state can be conveniently ex- pressed in the form Ti = SfFi, where ri is the corre- sponding width calculated as if the final state would be empty. More exactly, 8; represents the mixing proba- bility of non-fermion statistics allowing the transition into the occupied state i.

Similarly, in the case of our interest, we can write the nuclear width r previously defined in the form:

r = s2F, (3)

Where S* is a suitable mean value of S?. In Ref. [4] it was estimated the limit S* < 1.4 - 10-53. Aim of the present work is to determine an improved upper

441 R. Bemabei et al. / Physics Letters B 408 (1997) 439-444

(a) (b)

F5g. 2. Schematic representation of the nucleon-nucleon interactiorl processes leading to non-paulian proton emission in the nuclear potential well. Graphs a) and b) represents the “direct” and “exchange” contributions, respectively. The dashed line schematizes a standard boson exchange. We have qdir = kl - k{ = k; - k2 and qexc = kl - ki = k{ - k2.

limit on S2 by using the experimental results of Fqs.

(l), (2). In order to theoretically estimate i: it is convenient

to adopt the standard impulse approximation. First let us define the partial contribution Fnln2 to f due to the interaction of a pair of nucleons nt 112 (np or pp) . In the following we will refer to the calculation scheme shown in Fig. 2. We take into account the interaction of a pair of nucleons nr, n2 with initial momenta kt and kz, and final momenta k’, and kk. In the final state a proton, of momentum ki, acquires enough energy to be emitted, while the second nucleon, of momen- tum ki , lowers its energy. To calculate f,,,, it is also needed to consider the momentum distribution func- tions, fi (ki), of the nucleons in the bound state. We have examined two different possibilities: a) a Fermi momentum distribution with k~ = 255 MeV/c; b) “re- alistic” functions taking into account correlation ef- fects, of the kind discussed in Ref. [ 71. Owing to the fact that they are quite similar for all finite nuclei with A > 12, we adopted for both 23Na and 127I the distri- bution functions calculated for 56Fe.

Furthermore, we assume, as usual, the nucleons to be subjected to an energy-dependent square well po- tential V(r,E) = Vo(r,E) + iWo(r,E). For r < R, R being the nuclear radius, the real part of V( r, E) has beenputintheformVa(r,E) =h(E) =-A+B.E. This kind of approach, that leads to the introduction of

an effective nucleon mass m* = rn,, . ( 1 - B) and an ef- fective potential Vo* = -A.(l-B)-‘,issupportedby the whole systematics of nuclear processes, in particu- lar by the inelastic electron-nucleus processes. More- over, to take into account that the elastic flux of out- going protons can be attenuated by collisions with the nuclear medium, we have also introduced, as in the standard optical model, an imaginary square well po- tential; for r < R and E > 0 we have adopted an ex- pressionofthekindWa(r,E)=Wa(E)=-(C-E+ D . E2). This scheme brings to neglect secondary en- ergy loss processes degrading higher energy protons into lower energy ones. Fortunately, their omission makes our estimate of If a pessimistic one.

The nucleon-nucleon interaction giving rise to the Pauli violating transitions has been put in a simple phe- nomenological form that reasonably reproduces the effective range and scattering length parameters [ 81. The adopted expression is:

(4)

where Pi\‘) and P$’ are the usual projection operators on singlet and triplet spin states, respectively, while I$:“’ and V$” are the corresponding two-body poten-

tials; furthermore, P.!” = -P!.“’ . P.‘.” is the standard ?I 11 v space exchange operator. Moreover, in the momentum

442 R. Bemabei et al./Physics Letters B 408 (1997) 439-444

representation, r/;l”’ and &jt’ take the forms:

$“(q*) = -@“(,, 77 31*,$ , (5)

The q values occurring in the “direct” and “exchange” matrix elements shown in Fig. 2 have been labelled as

%ir md qexcp respectively (see caption of Fig. 2). Ac- cording to Ref. [ 81 the parameters of the potentials have the values: e”’ = 29.05 MeV, Vd” = 66.92 MeV LY, = 1.137. lo4 MeV*, LY~ = 1.616. IO4 MeV*, u = 0.93. By means of a standard calculation the squared spin averaged matrix-elements of the interaction po- tential describing the processes of Fig. 2 can be put in the form:

Gp(qdiry !&xc)

Tpp ( qdir * 4exc ) = gu - u* [;(~y(q:xcH2

+~(~~‘)(q~i~))* - ~~“(q~~,)~~“(q~ir)]

+ ‘(v.!“‘(q2 >>* + ‘($“‘(q&))* 4 ‘J exe 4 *J

+ I@)(q2 )v.!“‘(q,2i,). 2 ‘J W.c ,J (7)

Moreover, the f( AX) width for a specific (AX) nuclide is given by:

i;(AX> = r,,JNZ + rppz - ‘Z(Z-I), (8)

where p,,, and fPP are the contributions due to single n-p and p-p pairs. For each rnlnz contribution we have, in the Born approximation (ti, c = 1) :

(2lr-2 cl,n, = -

VN s d3k,d3k2d3k’d3k’ I 2

xTnln,(qdirvQexc) .fl(kl) ..f2(k2) ..g~(k:)gc(‘$$)

xS3(k;+k;-k,-kz)2ms(k;*+k;-k:-k;),

(9)

Table 1 Values of Fo, F and (gw) calculated as described in the text. The assumed parameters are: A = 45 MeV, B = 0.2, C = 0.51 and

D = -5.5. 10m4 MeV-‘. Case a): Fermi momentum distributions with kF = 255 MeV/c; case b) fi(kt) momentum distributions given in Ref. [7] for %Fe

Case AX Eth r0 (SW) F Upper limit ( MeV) ( MeV) (MeV) for ?i2

23Na 10 ‘2’1 10

23Na 18 “‘I 18

*‘Na IO ‘2’1 10

23Na 18 ‘2’1 18

3.90 0.42 1.65 4.8 .10-5s 16.0 0.29 4.64

0.60 0.32 0.19 5.0 .10-s 1.65 0.25 0.41

14.2 0.32 4.59 1.9 .10-55 76.6 0.14 11.1

8.22 0.23 1.90 5.6 .10-55 44.6 0.08 3.49

being VN the nuclear volume. Furthermore, the factor g,(,$) = e-2R.1mK, with K = [2m,,[Ek; - Vf(&)

- NO (I$; ) ] ] !I, roughly represents the escape prob- ability of the excited proton; gc( ki) is the proton tun- neling probability through the Coulomb barrier calcu- lated in the W.K.B. approximation. Finally, the inte- grations present in Ifnlnz have been performed within thelimits (i= 1,2):

0 < ki < 500 MeV/c, kk 2 khr, (10)

being kmr the experimental momentum threshold. The width r = F(23Na) + F(*271) has been nu-

merically calculated by a Monte Carlo procedure for different choices of the parameters A, B, C, D and dif- ferent forms of the distribution functions fi ( ki) . The assumed nuclear radius was R = roA’j3 with r-0 = 1.45 fm. The parameters C and D have been fixed at the values C = 0.51 and D = -5.5 . 10m4 MeV-*, while the other ones have been varied around A = 45 MeV and B = 0.2. Any reasonable variation of A and B does not produce changes in f ( AX) higher than about a factor two. Similar results can be obtained by adopt- ing for xj the standard one-pion interaction as in Ref. [4]. In any case the most relevant contributions to i; ( AX) come from the direct np processes.

Typical results are shown in Table 1 for energy thresholds, ,?&,, of 10 and 18 MeV. In this threshold en- ergy range there is no appreciable reduction of F ( AX) due to the Coulomb barrier, both for 23Na and ‘271. In

R. Bernabei et al. / Physics Letters B 40R (1997) 439-444 443

Table 1, in addition to f( AX), are reported the fol- lowing quantities: Fu ( AX>, the width calculated for &(ki) =gw(k:) = Land (g,,(Q) = F(AX)/F~(AX), a parameter allowing to quantify the mean effect of the imaginary potential. Furthermore, case a) refers to a Fermi momentum distribution with kF = 255 MeV/c, while in case b) it was adopted the distribu- tion function given in Ref. [7] for saFe. No signifi- cant variations have been obtained by adopting other distribution functions, as those of 12C or nuclear mat- ter. As can be immediately checked, the use of a more realistic description for the momentum distributions produces two important effects: increases F( AX) of a large factor (varying from about 2.5 at &, = 10 MeV to 9 at E& = 18 MeV and strongly reduces the thresh- old dependence. These results are essentially due to the presence of high momentum components (up to ki N 2kF) in the momentum distribution functions of Ref. [7].

Finally, the whole behaviour of i!( 23Na) and F( 1271) versus the energy threshold Eh is shown in Figs. 3a and 3b; the flat trend of the curves at low E* values is due to the presence of the Coulomb barrier which prevents low energy protons to escape from the nucleus.

4. Conclusions

The upper limits obtained for a2 in both cases a) and b) are shown in the last column of the Table 1. For our threshold of 10 MeV we have: a2 < 4.8 . 10vs5 (case a), and S* < l.9~10-55 (case b) . Moreover, let us stress that these estimates are strongly model de- pendent: in addition to the uncertainties deriving from the choice of the fi( ki) functions, one must take into account those deriving from the adopted calculation scheme and from the relevant physical parameters. A cautious conclusion could be to consider S2 6 10ms4, a limit higher than that obtained in case b) by a factor nearly five. Let us also observe that the uncertainties due to our poor knowledge of the momentum distri- bution functions sensibly increase when rising the en- ergy threshold.

Concluding, this work sets a new limit, S* 5 1.0 . 10ms4, on the relative strength of non-paulian transi- tion in nuclei; our estimate substantially confirms that of Ref. [ 41; it is lower of a factor about ten and refers

t

Proton energy threshold (MeV)

0 ” 0 5 ’ ” :; -+ :o

Proton energy threshold (MeV)

Fig. 3. Typical behaviour of ?(23Na) and f( 12’1) versus the proton energy threshold. The cases a) and b) are defined in the text (see also Table 1)

to a more extended energy region. This result could not be reached without the very high radiopurity of the whole set-up used here. The study of different non- paulian processes in nuclear and atomic systems were not able to give so stringent limits (see the discussion in Refs. [2-51).

In addition, by assuming I to have the same thresh- old dependence of f, we can roughly estimate a lower limit of the mean life for non-paulian proton emission; in the framework of case b), we obtain r 2 7. 1O24 years and r 1271, respectively.

2 9. 1O24 years for 23Na and

Finally, let us note that our result could be used also to give limits on other exotic decay modes followed by an energy release in the range ( 10-36) MeV.

444 R. Bernabei et al. /Physics Letters B 408 (I 997) 439-444

Acknowledgements References

It is a pleasure to thank Prof. S. d’Angelo for many useful discussions on related topics and for his contin- uous and stimulating support to our activities. We also thank Mr. A. Bussolotti and G. Ranelli for qualified technical help and the LNGS staff for support.

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