Solution totheoretical Question 3. Neutrino Mass and Neutron Decay

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1 art A Soluto tothortal Qusto 3 Nutro ass ad Nutro Day a Lt,,,, ad, b th rgy-otu -tors of th ltro, th roto, ad th at-utro, rstly, th rst fra of th utro. Not that,,,,, ar all uts of ass. Th roto ad th at-utro ν ay b osdrd as forg a syst of total rst ass total otu. Thus, w ha ν, total rgy, ad,, A Not that th agtud of th tor s dotd as. Th sa oto also als to all othr tors. S rgy ad otu ar osrd th utro day, w ha A A3 Wh suard, th last uato lads to th followg ualty A Fro. A ad th thrd ualty of. A, w obta 7 A5 Wth ts sod ad thrd trs od to th othr sd of th ualty,. A5 ay b ddd by. A to g A6 As a syst of ould lar uatos, s. A ad A6 ay b sold to g A7 A8 Usg. A8, th last ualty. A ay b rwrtt as. A8 shows that a axu of rst ass A9 orrsods to a u of. Now th s th total rgy of th roto ad at-utro ar thr tr of ass or otu fra so that t ahs th u

2 8 A wh th roto ad th at-utro ar both at rst th tr of ass fra. H, fro s. A8 ad A, th axu rgy of th ltro s [ ] V.9 V.9569 ax A Wh. A holds, th roto ad th at-utro o wth th sa loty of th tr of ass ad w ha ax ax ax A whr th last ualty follows fro. A3. By s. A7 ad A9, th last xrsso. A ay b usd to obta th sd of th at-utro wh ax. Thus, wth, w ha A [Altrat Soluto] Assu that, th rst fra of th utro, th ltro os out wth otu ad rgy, th roto wth ad, ad th at-utro wth ad. Wth th agtud of tor α dotd by th sybol α, w ha,, A Cosrato of rgy ad otu th utro day lads to A 3A Wh suard, th last two uatos lad to A 5A Subtratg. 5A fro. A ad akg us of. A th gs 6A or, ualtly,

3 If s th agl btw lads to th rlato 7A ad, w ha os so that. 7A 8A Not that th ualty. 8A holds oly f,.., th rgy of th ltro taks o ts axu alu oly wh th at-utro ad th roto o th sa drto. Lt th sds of th roto ad th at-utro th rst fra of th utro b β ad β, rstly. W th ha β ad β. As show Fg. A, w trodu th agl φ φ < / for th atutro by taφ, sφ, β / sφ 9A φ Fgur A Slarly, for th roto, w wrt, wth φ < /, taφ, sφ, β / sφ A. 8A ay th b xrssd as sφ sφ A osφ osφ Th fator arthss at th d of th last uato ay b xrssd as sφ sφ osφ osφ sφ sφ osφ osφ osφ osφ os φ φ osφ osφ A ad larly assus ts u ossbl alu of wh φ φ,.., wh th at-utro ad th roto o wth th sa loty so that β β. Thus, t follows fro. A that th axu alu of s ax [ ] ad th axu rgy of th ltro s ax ax 3A.9569 V.9 V A 9

4 Wh th at-utro ad th roto o wth th sa loty, w ha, fro s. 9A, A, A,3A, ad A, th rsult β β 5A Substtutg th rsult of. 3A to th last uato, th sd of th at-utro wh th ltro attas ts axu alu ax s, wth, g by β ax ax ax art B Lght Ltato 6A b fr to Fg. B. frato of lght at th shral surfa obys Sll s law ad lads to s s B Ngltg trs of th ordr / 3 or hghr s futos,. B bos B For th tragl FAC Fg. B, w ha z β B3 t t t F Lt f b th fruy of th dt lght. If s th ubr of hotos dt o th la surfa r ut ara r ut t, th th total ubr of hotos dt o th la surfa r ut t s. Th total owr of hotos dt o th la surfa s, wth h bg lak s ostat. H, B Th ubr of hotos dt o a aular dsk of r radus r ad outr radus r dr o th la surfa r ut t s rdr, whr r ta. Thrfor, t β A C Fg. B rdr d B5 Th z-oot of th otu arrd away r ut t by ths hotos wh 3

5 3 rfratd at th shral surfa s o z d d rdr df β β 3 os B6 so that th z-oot of th total otu arrd away r ut t s 3 z d F B7 whr ta. Thrfor, by th rsult of. B5, w ha F z B8 Th for of otal ltato s ual to th su of th z-oots of th fors xrtd by th dt ad rfratd lghts o th glass hshr ad s g by F z B9 uatg ths to th wght g of th glass hshr, w obta th u lasr owr rurd to ltat th hshr as g B

6 Total Sors art A. ts. Sub Sors a. arkg Sh Thortal Qusto 3 Nutro ass ad Nutro Day arkg Sh for Aswrs to th robl Th axu rgy of th ltro ad th orrsodg sd of th at-utro..5 us rgy-otu osrato ad a ort t to uatos..5 obta a xrsso for that allows th dtrato of ts axu alu..5. for oludg that roto ad at-utro ust o wth th sa loty wh s axu.. for th sa drto.6 for stablshg th u alu of to b or a oluso ualt to t..5. for xrsso ad alu of ax..5 for oludg β /..5. for xrsso ad alu of /. art B. ts b. Lght Ltato Lasr owr dd to bala th wght of th glass hshr..3 for law of rfrato s st..3 for akg th lar aroxato t.. for rlato btw agls of dato ad d..3 for hoto rgy ε hν..3 for hoto otu ε /..3 for otu of dt hotos r ut t /..6 for otu of hotos rfratd r ut t as a futo of th agl of d.. for total otu of hotos rfratd r ut t [-- / ]/.. for for of ltato su of fors xrtd by dt ad rfratd hotos.. for for of ltato - /..3 for th dd lasr owr g /-. 3