Neutrino Mass and Forbidden Beta Decays

Transcription

1 NUCLEAR THEORY Vol ) ds. M. Gaidarov N. Minkov Hron Prss Sofia Nutrino Mass and Forbiddn Bta Dcays R. Dvornický 1 D. Štfánik F. Šimkovic 3 1 Dzhlpov Laboratory of Nuclar Problms JINR Dubna Russia Comnius Univrsity Mlynská dolina F Bratislava Slovakia 3 Bogoliubov Laboratory of Thortical Physics JINR Dubna Russia Abstract. A possibility to xploit th first- scond- and third- uniqu and th first non-uniqu forbiddn bta dcays to stimat th nutrino mass is discussd. Our findings show that th corrsponding Kuri function in th vicinity of th ndpoint is within a good accuracy linar in th limit of masslss nutrinos lik th Kuri function of th suprallowd bta dcay of tritium. 1 Introduction Th absolut nutrino mass scal is important for th physics byond th Standard Modl. Various xprimnts with nutrino oscillations provd that nutrinos ar massiv particls. Th absolut valu of th nutrino masss cannot b dtrmind by th oscillation xprimnts bing kind of th intrfrnc xprimnts. A prcis and modl indpndnt way to obtain th dirct information on th nutrino mass is th kinmatical analysis of th ndpoint of th lctron nrgy spctrum in singl β dcays such as th tritium β dcay. Th first masurmnt was prformd by Hanna and Pontcorvo in 1949 [1] and a limit of 1 kv was obtaind. Currntly from th Mainz and Troitsk xprimnts for th uppr bound on th ffctiv nutrino mass m β w hav m β <. V []. Th KA- TRIN xprimnt [3] aims at raching a snsitivity of 0. V. Anothr promising way to dtrmin th absolut nutrino mass scal is to xploit th first uniqu forbiddn β dcay of 187 R du to its low transition nrgy of.47 kv [4]. Th aim of this contribution is to study th thortical lctron nrgy spctrum in th first scond and third uniqu and th first non-uniqu forbiddn bta dcays. Th attntion is paid to th Kuri function nar th ndpoint as a function of th nutrino mass m β. Forbiddn Bta Dcays Th width of th first non-uniqu J π = 0 ) forbiddn β dcay for th spin of th initial nuclus J i = 0 taks th sam form dγ = de π G F Vud p E p E U j θe 0 E m j )B A G A E ) 1) j 141

2 R. Dvornický D. Štfánik F. Šimkovic as th diffrntial dcay rats of th first J π = ) scond J π = 3 + ) and third J π = 4 ) uniqu forbiddn β dcays. Hr G F is th Frmi constant and V ud is th lmnt of th Cabibbo-Kobayashi-Maskawa mixing matrix. p E and E 0 ar th momntum nrgy and th maximal ndpoint nrgy in th cas of zro nutrino mass) of th lctron rspctivly. Th nutrino nrgy and momntum rspctivly ar E = E 0 E and p = E 0 E ) m j. U j and m j ar th lmnt of nutrino mixing matrix and th nutrino mass rspctivly. θx) is a thta stp) function. W strss that thr is only on nuclar matrix lmnt B A which ntrs th dcay rat givn in Eq. 1. For a givn particular forbiddn β transition i.. for th first non-uniqu A = 0 ) and th first A = ) scond A = 3 + ) and third A = 4 ) uniqu forbiddn β dcay th nuclar matrix lmnt is givn as B 0 = ga 0π f B = g A J i + 1 J π f f B 3 + = g A J i + 1 J π f f B 4 = g A J i + 1 J π f f r n R τ n + {σ 1 n) Y 1 n)} 0 0 πi r n R τ + n {σ 1 n) Y 1 n)} J πi i r n R τ + n {σ 1 n) Y n)} 3 J πi i r 3 n R 3 τ + n {σ 1 n) Y 3 n)} 4 J πi i. ) Hr g A dnots an axial-vctor coupling constant. Th functions G A E ) dpnd solly on th lctron nrgy. G 0 E ) = 1 9 F p1/ E R)p R) + F sp1/ E R) p p E R +F s1/ E R)p R) ) G E ) = 1 Fp3/ 9 E R)p R) + F E s1/ R)p R) ) G 3 +E ) = F d 5/ E R)p R) F p 3/ E R)p p ) R F s 1/ E R)p R) 4 ) 3) 14 G 4 E ) = Ff7/ E R)p R) 6 + 7F d5/ E R)p 4 p R 6 +7F p3/ E R)p p 4 R 6 + F s1/ E R)p R) 6).

3 Nutrino Mass and Forbiddn Bta Dcays Th Frmi functions associatd with th mission of s 1/ p 1/ p 3/ d 5/ and f 7/ -stat lctron rspctivly ar dfind as F s1/ E R) = g 1E R) + f +1E R) F p1/ E R) = g +1E R) + f 1E R) p R) /3 F p3/ E R) = g E R) + f +E R) p R) /3 F d5/ E R) = g 3E R) + f +3E R) p R) 4 /15 F f7/ E R) = g 4E R) + f +4E R) p R) 6 /105. 4) Th β transitions with th J π = 0 ar non-uniqu first forbiddn. Thrfor th dcay rat consists of thr contributions s Eq. 3) associatd with mission of th lctron in s 1/ -stat p 1/ -stat and th intrfrnc btwn s 1/ -stat and p 1/ -stat rspctivly. Th corrsponding Frmi function for th intrfrnc trm is dfind as F sp1/ E R) = g +1E R)f +1 E R) g 1 E R)f 1 E R). 5) p R/3 Important ingrdints in th Frmi functions ar th radial lctron wav functions g k E R) and f k E R) which satisfy th radial Dirac quations. Th input is th potntial of Coulomb fild of daughtr nuclus distortd with scrning potntial of lctrons of daughtr atom. Th lctron radial wav functions ar valuatd by mans of th subroutin packag RADIAL [5]. Hr w Tabl 1. Nucli which bta dcays ar classifid as first scond and third uniqu and first non-uniqu ar listd. Both transitions to th ground stat with spin and parity J π and th first xcitd stat J π 1 of final nuclus ar considrd. W not that Q = E 0 m ParntJ π i i ) DaughtrJ π f f ) J π Q-valu kv T 1/ yrs 10 B0 + ) 10 B3 + ) K4 ) 40 Ca0 + ) S7/ + ) 79 Br3/ ) Sr0 + ) 90 Y ) Zr5/ + ) Nb1/ 1 ) Pd5/ + ) Ag1/ ) La5 + ) C + 1 ) Ba0 + ) La0 1 ) C0 + ) Pr0 ) R5/ + ) 187 Os1/ )

4 R. Dvornický D. Štfánik F. Šimkovic hav mad a rasonabl assumption that in β dcay of nucli participat mostly nuclons clos to th Frmi lvl i.. th radial lctron wav functions ar valuatd at th nuclar surfac R = 1.A 1/3 fm. In Tabl 1 w prsnt nucli which undrgo β dcays that ar classifid as th first scond and third uniqu and th first non-uniqu forbiddn. Our attntion has bn focusd primarily on th dcays with small Q valu and liftim largr than 10 days. 3 Kuri Functions Th currnt uppr limit on nutrino mass holds in th dgnrat nutrino mass rgion i.. m 1 m m 3 m β = 3 j=1 U j m j. Th Kuri function for th allowd transitions taks th following form KE m β ) E 0 E ) 4 1 m β E 0 E ). 6) W dfin th Kuri functions for th first non-uniqu and th first scond and third uniqu forbiddn β dcays rspctivly in such a way K 0 E m β ) = p E Fp1/ E R)p R) /3 ) K E m β ) = K 3 +E m β ) = K 4 E m β ) = 144 π B 0 E 0 E ) 4 1 p E Fp3/ E R)p R) /3 ) π B E 0 E ) 4 1 p E Fd5/ E R)p R) 4 /15 ) π B 3 +E 0 E ) 4 1 p E Ff7/ E R)p R) 6 /105 ) π B 4 E 0 E ) 4 1 m β E 0 E ) S0 E ) m β E 0 E ) S E ) m β E 0 E ) S3 +E ) m β E 0 E ) S4 E ) 7)

5 Prparation Nutrino of Paprs Mass for and Hron Forbiddn Prss Scinc Bta Dcays Sris Books that thy rsmbl th Kuri function for th allowd β dcays. Th shap factors S A E ) ar givn as S 0 E ) = 1 + F sp 1/ E R)p + F s E 1/ R)p ) F p1/ E R)p E F p1/ E R)p S 0 E ) = 1 + F sp 1/ E R)p + F s E ) 1/ R)p S E ) = 1 + F F p1/ E s 1/ E R)p ) R)p E F p1/ E R)p F p3/ E R)p S E ) = 1 + F ) s 1/ E R)p S 3 +E ) = F F p 3/ E p3/ E R)p R)p 3F d5/ E R)p + F s 1/ E R)p 4 ) F d5/ E R)p 4 S 3 +E ) = F p 3/ E R)p S 4 E ) = F d 5/ E 3F R)p F f7/ E R)p + 7 F p 3/ E R)p 4 F f7/ E R)p 4 + F s 1/ E R)p 6 d5/ E R)p + F ) s 1/ E R)p 4 F d5/ E R)p 4 ) F f7/ E R)p 6. S 4 E ) = F d 5/ E R)p F f7/ E R)p 8) W prformd numrical analysis of th shap factors S A E ) for th particular nucli listd in Tabl +7 F p 3/ E R)p 4 F 1. Thy ar plottd in Figur 1 against th lctron f7/ E nrgy clos to th ndpoint xcpt 187 R)p 4 + F ) s 1/ E R)p 6 F f7/ E R)p 6. 8) R du to an xtrmly low Q valu. Our findings show that th shap factors can b approximatd as S A E ) 1 clos to th Wndpoint prformd numrical all nucli givn analysis in Tabl of th1. shap Thnfactors th Kuri S A E functions ) for th forpar- ticular non-uniqu nucli listd andin th Tabl first 1. scond Thy ar andplottd third uniqu in Figur forbiddn 1 against βth dcays lctron can th first b nrgy approximatd clos to th with ndpoint th Kuri xcpt function 187 Rfor duth toallowd an xtrmly transitions low Q valu. Our findings show that th shap factors can b approximatd as S A E ) 1 clos to th ndpoint for K A E m β ) all nucli givn π B in Tabl 1. Thn th Kuri m AE 0 E ) 1 4 βfunctions E 0 E ). for th first non-uniqu and th first scond and third uniqu forbiddn β dcays can 9) S A E ) B 40 K 79 S 90 Sr 93 Zr 107 Pd 138 La 140 Ba 144 C E -m -Q kv Figur 1. Th shap factors as a function of th lctron nrgy. 1455

6 R. Dvornický D. Štfánik F. Šimkovic Th goal is that th Kuri plot is linar only in th cas of masslss nutrinos. Howvr th linarity is lost for th non-zro nutrino mass. 4 Conclusions Th kinmatical masurmnt of th nutrino mass has bn prformd in th laboratory by taking th advantag of th suprallowd β dcay of tritium. W found that this goal can b achivd also with th forbiddn β dcays. It is shown that th Kuri function for th first non-uniqu and th first scond and third uniqu forbiddn β dcays clos to th ndpoint coincids up to a factor to th Kuri function of th suprallowd β dcay of tritium having th sam dpndnc on th ffctiv nutrino mass m β. Rfrncs [1] G. Hanna B. Pontcorvo Phys. Rv ) 983. [] E.W. Ottn C. Winhimr Rpt. Prog. Phys ) [3] C. Winhimr Nucl. Phys. Proc. Suppl ) 5. [4] R. Dvornický K. Muto F. Šimkovic and A. Fasslr Phys. Rv. C ) [5] D. Štfánik R. Dvornický F. Šimkovic and P. Vogl Phys. Rv. C 9 015) [6] R. Dvornický F. Šimkovic Acta Physica Polonica B Proc. Supp )