Majorana Neutrino Oscillations in Vacuum

Transcription

1 Journl of odrn Pysis tt://dx.doi.org/0.46/jm Publisd Onlin August 0 (tt:// jorn Nutrino Osilltions in Vuum Yubr Frny Prz Crlos Jos Quimby Esul d Físi Univrsidd Pdgógi y nológi d Colombi unj Colombi Drtmnto d Fsi Univrsidd Nionl d Colombi Ciudd Univrsitri Bogotá D.C. Colombi Emil: yubr.rz@ut.du.o jquimby@unl.du.o ivd y 0; rvisd Jun 9 0; td July 0 ABSAC In t ontxt of ty I ssw snrio wi lds to gt ligt lft-ndd nd vy rigt-ndd jorn nutrinos w obtin xrssions for t trnsition robbility dnsitis btwn two flvor nutrinos in t ss of lftndd nd rigt-ndd nutrinos. W obtin ts xrssions in t ontxt of n ro dvlod in t nonil formlism of Quntum Fild ory for nutrinos wi r onsidrd s surositions of mss-ignstt ln wvs wit sifi momnt. xrssions obtind for t lft-ndd nutrino s ftr t ultr-rltivisti limit is tking ld to t stndrd robbility dnsitis wi dsrib ligt nutrino osilltions. For t rigt-ndd nutrino s t xrssions dsribing vy nutrino osilltions in t non-rltivisti limit r diffrnt rst to t ons of t stndrd nutrino osilltions. Howvr t rigt-ndd nutrino osilltions r nomnologilly rstritd s is sown wn t rogtion of vy nutrinos is onsidrd s surositions of mss-ignstt wv kts. Kywords: jorn Frmions; jorn Nutrino Osilltions; rnsition Probbility; Non-ltivisti nd Ultr-ltivisti Aroximtions. Introdution Nutrino ysis is vry tiv r of rsr wi involvs som of t most intriguing roblms in rtil ysis. ntur of nutrinos nd t origin of t smll mss of nutrinos r two xmls of ts kinds of roblms. Sin nutrinos r ltrilly nutrl t ntur of ts lmntry rtils n b jorn or Dir frmions. first ossibility i.. nutrinos bing jorn frmions ws introdud by Etor jorn [] wn suggstd tt mssiv nutrl frmions wit sifi momnt v ssoitd only two liity stts imlying tt nutrinos nd nti-nutrinos r t sm rtils. sond ossibility imlis tt Dir nutrinos r dsribd by four-omonnt sinoril filds wi r diffrnt from sinoril filds dsribing ntinutrinos. In tis work w will onsidr nutrinos s jorn frmions wi is fvord by simliity bus ty v only two dgrs of frdom []. Dirt nd indirt xrimntl vidns sow tt nutrinos r mssiv frmions wit msss smllr tn V [4]. most td wy to gnrt nutrino msss is by mn of t ssw mnism [5]. ss for nutrinos is nssry ingrdint to undrstnd t osilltions btwn nutrino flvor stts wi v bn obsrvd xrimntlly [4]. Nutrino osilltions r origintd by t intrfrn btwn mss stts wos mixing gnrts flvor stts. is nomnon mns tt nutrino rtd in wk intrtion ross wit sifi flvor n b dttd wit diffrnt flvor. Nutrino osilltions wr first dsribd by Pontorvo [6] s n xtnsion for t ltoni stor of t strng osilltions obsrvd in t nutrl Kon systm. Nutrino osilltions n b dsribd in ontxt of Quntum nis [7-] s n lition of t two lvl systm []. Dsrition of nutrino osilltions in t ontxt of Quntum Fild ory (QF) is vry wll studid toi [-0]. In t litrtur it is ossibl to find two kinds of QF modls dsribing nutrino osilltions: intrmdit modls nd xtrnl modls [7]. In t frmwork of intrmdit modls Sssroli dvlod modl bsd in n intrting grngin dnsity wi inluds t ouling btwn two flvor filds [-]. is modl ws frmd by But s ybrid modl owing to it gos lf--wy to QF [7]. Sssroli modl ws first dvlod for ould systm of two Dir qutions [ ] nd tn it ws xtndd for ould systm of two jorn ons []. robbility mlitud of trnsition btwn two nutrino flvor stts for ts two systms [-] ws obtind strting from flvor stts wi r usd on t stndrd trtmnt of nutrino osilltions. stndrd dfinition of flvor stts n origint som ossibl limittions in t dsrition of nutrino

2 804 osilltions s ws obsrvd by Giunti t l. [4]. Sifilly in rfrn [4] it ws sown tt flvor stts n dfin n roximt Fok s of wk stts in t following two ss: ) In t xtrmly rltivisti limit i.. if nutrino momntum is mu lrgr tn t mximum mss ignvlu of nutrino mss stt; ) for lmost dgnrtd nutrino mss ignvlus i.. if t diffrns btwn nutrino mss ignvlus r mu smllr tn t nutrino momntum. first s lds to t stndrd dfinition of flvor stts. sond s s ssoitd rl mixing mtrix wi is rstritd to sifi intrtion ross. Additionlly ts utors v roosd tt osilltions n b dsribd roritly for ultr-rltivisti nd non-rltivisti nutrinos by dfining rorit flvor stts wi r surositions of mss stts wigtd by tir trnsitions mlituds in t ross undr onsidrtion [4]. By onsidring t limittions mntiond in t lst rgr it ws st down by But in [7] tt Sssroli ybrid modl n only b lid onsistntly if lton flvor wv funtions r onsidrd s obsrvbl nd t ultr-rltivisti limit is tkn into ount. On t otr nd t Sssroli modl dsribing jorn nutrino osilltions [] ws dvlod witout onsidring t four-momntum onsrvtion for nutrinos wi imlis t xistn of sifi momntum for vry nutrino mss stt. min gol of tis work is to study nutrino osilltions in vuum btwn two flvor stts onsidring nutrinos s jorn frmions nd to obtin t robbility dnsitis of trnsition for lft-ndd nutrinos (ultr-rltivisti limit) nd for rigt-ndd nutrinos (non-rltivisti limit). is work is dvlod in t ontxt of ty I ssw snrio wi lds to gt ligt lft-ndd nd vy rigt-ndd jorn nutrinos. In tis ontxt w rform n xtnsion of t modl dvlod by Sssroli in wi t jorn nutrino osilltions r obtind for t s of flvor stts onstrutd s surositions of mss stts [ ]. Our xtnsion onsists in onsidring nutrino mss stts s ln wvs wit sifi momnt. modl tt w onsidr in tis work wi is dvlod in t nonil formlism of Quntum Fild ory s t dvntg tt in t sm tortil trtmnt it is ossibl to study nutrino osilltions for ligt nutrinos nd for vy nutrinos. o do tis w first rform t nonil quntiztion rodur for jorn nutrino filds of dfinit msss nd tn w writ t nutrino flvor stts s surositions of mss stts using quntum fild ortors. Nxt w lult t robbility mlitud of trnsition btwn two diffrnt nutrino flvor stts for t ligt nd vy nutrino ss nd w stblis normliztion nd boundry onditions for t robbility dnsity. s robbility dnsitis for t lft-ndd nutrino s ftr t ultr-rltivisti limit is tking ld to t stndrd robbility dnsitis wi dsrib ligt nutrino osilltions. For t rigtndd nutrino s t xrssions dsribing vy nutrino osilltions in t non-rltivisti limit r diffrnt rst to t ons of t stndrd nutrino osilltions. Howvr t rigt-ndd nutrino osilltions r nomnologilly rstritd s is sown wn t rogtion of vy nutrinos is onsidrd s surositions of mss-ignstt wv kts [5]. osilltions do not tk l in tis s bus t orn is not rsrvd: in otr words t osilltion lngt is omrbl or lrgr tn t orn lngt of t nutrino systm [5]. ontnt of tis work s bn orgnizd s follows: In Stion ftr stblising t jorn ondition w obtin nd solv t two-omonnt jorn qution for fr frmion; in Stion w onsidr ty I ssw snrio wi lds to gt ligt lft-ndd nutrinos nd vy rigt-ndd nutrinos; in Stion 4 w obtin t jorn nutrino filds wit dfinit msss tn w rry out t nonil quntiztion rodur of ts jorn nutrino filds nd w obtin rltion btwn nutrino flvor stts nd nutrino mss stts using ortor filds; in Stion 5 w dtrmin t robbility dnsity of trnsition btwn two lft-ndd nutrino flvor stts dditionlly w stblis normliztion nd boundry onditions nd tn w obtin lft-ndd nutrino osilltions for ultr-rltivisti ligt nutrinos; in Stion 6 w study t rigt-ndd nutrino osilltions for non-rltivisti vy nutrinos; finlly in Stion 7 w rsnt som onlusions.. wo-comonnt jorn Eqution In 97 Ettor jorn roosd symmtri tory for ltron nd ositron troug gnrliztion of vritionl rinil for filds wi oby Frmi-Dir sttistis []. Wn tis tory is lid to nutrl frmion wi s sifi momntum tn tr xist only two liity stts. jorn tory imlis tt it dos not xist ntirtils ssoitd to ts frmions i.. jorn frmions r tir own ntirtils. For onvnin w study t qution of motion for nutrl frmions but using t two-omonnt tory dvlod by Cs in [6]. In ontrst wit Dir frmion jorn frmion n only b dsribd by two-omonnt sinor. o sow it w onsidr fr rltivisti frmioni fild of mss m dsribd by t Dir qution i m 0 wr Dir mtrixs oby t ntionmuttion rltions g nd mtri tnsor stisfis g g dig. Using

3 805 t irlity mtrix givn by 5 i 0 t lftnd rigt-ndd irl rojtions of t frmion fild ψ 5 r rstivly. If w writ t Dir mtrixs rojtd on t irl subs s 5 w obtin tt t ould qutions for t irl omonnts of t frmioni fild r givn by i m () i m (). W introdu t rg onjugtion ortion tt will llow us to dsrib jorn frmions. rgd onjugtd fild (or onjugtd fild) is dfind s ˆ wr t rg onjugtion ortor stisfis t ˆ rortis ˆ ˆ ˆ ˆ ˆ []. Using ts rortis w find tt t onjugtd fild obys t Dir qution i m 0. As dsribs frmion wit sifi rg its onjugtd fild rrsnts frmion wit n oosit rg nd wit t sm mss i.. dsribs t ntifrmion of. Dir qution for sould b rojtd on t irl subs nd for tis rson it 5 is nssry to rmmbr tt ˆ 5 ˆ []. So t ould qutions for t irl omonnts of t onjugtd fild r () i m (4) i m. (5) W obsrv tt t irl omonnts of t frmioni fild undr rg onjugtion nd t irl omonnts of t onjugtd fild r rltd by sowing ow t rg-onjugtion ortion ngs t irlity of filds. W dfin t jorn ondition by tking t frmioni fild s roortionl to t onjugtd fild wr t roortionl onstnt is omlx s ftor of t form wi lys n imortnt rol on i litions of jorn tory. Equlity (6) imlis tt jorn frmions r tir own ntirtils. Now t irl omonnts of t jorn fild stisfy ˆ ˆ. (7) (6) So w n writ Equtions () nd () in t form ˆ i m (8) ˆ i m. (9) If w ly t jorn ondition (7) into t Equtions (4) nd (5) w obtin Equtions (8) nd (9). Additionlly w n obsrv tt Equtions (8) nd (9) r rltd to tmslvs by mns of omlx onjugtion. In tis wy w v gon from four ould qutions dsribing frmion nd its ntifrmion to two dould qutions dsribing lft-ndd irl fild nd rigt-ndd irl fild. Du to t ft tt t rigt-ndd irl fild n b onstrutd from t lft-ndd irl fild [6] s it is sown in (7) now w v only n indndnt fild givn by. For t lst ft w will b bl to dsrib jorn frmion by mns of fild wi now s two omonnts. o vrify tis sntn w rwrit Eqution (8) s ˆ 0 i m. (0) If w dfin ˆ 0 () nd if w tk tn Eqution (0) n b writtn s i m () wi is known s t jorn qution. is qution in wi rtil is indistinguisbl from its ntirtil s two omonnts bus t mtrixs r rojtd on t irl subss of two omonnts. mtrixs r lld jorn mtrixs nd ts sould not b onfusd wit t Dir mtrixs writtn in jorn rrsnttion. Now w r intrstd in knowing t kind of rltions tt jorn mtrixs oby. So w first ly dfinition () into Eqution (9) nd w obtin 0 i m wit ˆ. n w ly i into () nd w v m 0 or its quivlnt m 0 wr w v usd. Aordingly jorn mtrixs sould stisfy rltions g nd tn t fild is stisfying t Klin-Gordon qution givn by m 0. In tis work w v tkn rtiulr rrsnttion of mtrixs wi s rmittd us to writ t twoomonnt jorn qution in t form givn by (). Now w n onsidr mtrix A wi stisfis t following rltions [6] i i A A A A A A i wr rrsnts Puli mtrixs in givn rrsnttion. W tk i wr σ () bing t unit mtrix nd σ Puli mtrixs. Sin stisfis rortis () w

4 806 Wit t uros of studying t nonil quntiztion for t jorn fild w will obtin t fr-rtil solution of Eqution (4). On t outst w onsidr bisinors wi oby t following rltions σ (5) i (6) wr ts bi-sinors orrsond to liity ignstts. If w tk t momntum in sril oordints sin os sinsin os tn t liity ortor s t form i σ os sin. i sin os (7) W oos t following rrsnttion for ts bisinors i os sin. (8) i sin os W n rov tt t following solution stisfis t two-omonnt jorn Eqution (4) E E ix x ix E E (9) wit x x Et x. W obsrv tt jorn fild n b writtn s surosition of ositiv nd ngtiv nrgy stts. grngin dnsity wi dsribs fr twoomonnt jorn fild is givn by m i i i (0) wr t two-omonnt jorn fild nd its onjugtd fild bv s Grssmnn vribls. It is vry sy to rov tt t two-omonnt jorn v tkn A. So t Eqution () n b writtn s Eqution (4) n b obtind from t grngin dn- sity (0) using t Eulr-grng qution. Additionlly i im 0. (4) w n obtin t following nrgy-momntum tnsor is qution is t wll known two-omonnt jorn qution [78] wi will b solvd in nxt from (0) substion. m i g i i i. Cnonil Quntiztion for jorn Fild Following t stndrd nonil quntiztion rodur w now onsidr t jorn fild nd its onjugtd fild s ortors wi stisfy t usul nonil ntionmuttion rltions givn by ˆ ˆ ˆ ˆ r t r' t r t r' t ˆ ˆ r t r' t r r' 0 wr. Using t Hisnbrg qution for t jorn fild i ˆ ˆ ˆ t r t ' t H r w n obtin its orrsonding jorn Eqution (4). By mns of t nrgy-momntum tnsor it is ossibl to rov tt t Hmiltonin ortor n b writtn s ˆ ˆ ˆ m ˆ H ˆ ˆ ˆ x i i i σ. xnsion in Fourir sris for t jorn fild ortor is [4-6] (s Eqution ()). wr w v usd t fr-rtil solution (9) nd ortors ˆ ˆ wi stisfy t ntionmuttion rltions ˆ ˆ ' '. ˆ ˆ ' ˆ ˆ ' 0 n w n idntify ˆ s t nniiltion ortor nd ˆ s t rtion ortor of jo- rn frmion wit momntum nd liity.. sss for jorn Nutrino Filds most td wy to gnrt nutrino msss is troug t ssw mnism. In tis stion w onsidr ty I ssw snrio wi lds to gt ligt lft-ndd nutrinos nd vy rigt-ndd nutrinos. For t s of two nutrino gnrtions Dir-jorn mss trm is givn by [9] D ˆ D Y N N H.. () wr H.. rrsnts t rmiti onjugt trm is t vtor of flvor nutrino filds writtn s N ˆ d ix ix x E ˆ E ˆ () π E

5 807 N ˆ () wr rrsnts doublt of lft-ndd nutrino filds tiv undr t wk intrtion nd rrsnts doublt of rigt-ndd jorn nutrino filds non tiv (stril) undr t wk intrtion. s doublts r givn by ;. (4) D In t Dir-jorn trm () is 4 4 non-digonl mtrix of t form ' D D (5) ' D wr nd D r mtrixs. vtor of nutrino filds wit dfinit msss n n b writtn by mn of unitry mtrix U s follows N U n (6) wr n s t form n n. (7) n 4 unitry mtrix U is osn in su wy tt D t non-digonl mtrix n b digonlizd troug t similrity trnsformtion U D U (8) wr is digonl mtrix wi is dfind by m b b wr b 4. msss of t nutrino filds of dfinit msss r m wit 4. ssw snrio is stblisd imosing t following onditions into t mtrix (5): ' ' D 0 D kj tus t mtrix kj is digonlizd s D U U l 0 0 (9) wr l is t ligt nutrino mss mtrix nd is t vy nutrino mss mtrix. If t unitry mtrix U is xnding onsidring trms until of t ordr ' D t ligt nd vy nutrino mss mtrixs n b writtn s l m 0 m 0 ;. (0) 0 m 0 m4 Dir-jorn mss trm () n b writtn in trms of t nutrino filds of dfinit msss s D ˆ ˆ Y nln n n H.. () wr t mtrixs l nd r givn by (0) nd t doublts nd n r writtn s n n ; n. 4 () nutrino filds of dfinit msss nd v ssoit rstivly t ligt msss nd m m f v m m f v nd t nutrino filds of dfinit msss nd 4 v ssoit rstivly t vy msss m f v nd 4 44v v f m f wr b r Yukw oulings m is t ltron mss nd m is t muon mss. As it will b sown in t nxt stion strting from t Dir-jorn mss trm D ˆ ˆ Y H.. () wr nd r t flvor doublts of non-dfinit msss givn by (4) wil nd r non-digonl mtrixs it will b ossibl to obtin t Dir-jorn mss trm () ftr t digonliztion of t mtrixs nd. 4. ss nd Flvor Nutrino Stts In t nxt w suos tt t jorn filds nd dsrib t tiv ligt lft-ndd nutrinos tt r rodud nd dttd in t lbortory wil t jorn filds nd dsrib t stril vy rigt-ndd nutrinos wi tr xist in ty I ssw snrio. In t Stion () w v rsntd lgrngin dnsity (0) wi dsribs fr jorn frmion. is lgrngin dnsity n b xtndd to dsrib systm of two flvor lft-ndd nutrinos nd two flvor rigtndd nutrinos wit non-dfinit msss. Using t Dir-jorn mss trm givn by () t lgrngin dnsity dsribing tis systm is givn by i i i (4) i H.. wr t non-digonl mss mtrixs nd r writtn s i i m m (5) i i m m

6 808 i i 4 m m (6) i4 i4 m m W obsrv tt t form of t mtrixs nd is t sm. In t nxt w will rstrit to t lftndd jorn nutrinos but t rsults r dirtly xt ndd to t rigt-ndd jorn nutrinos. From t Eulr-grng qutions w obtin tt t ould qution of motion for t flvor lft-ndd nutrino filds nd r i i i im im (7) i i im i im (8) rstivly. W obsrv tt flvor nutrino filds r ould by mns of t rmtr m. Wit t uros of douling t qutions of motion for t flvor lft-ndd nutrino filds now w onsidr t most gnrl unitry mtrix U givn by U i i i i (9) i wr t ss nd i r s onsqun of t jorn ondition (7). dfinit-mss nutrino fild doublt n givn by () is rltd to t flvor lft-ndd nutrino doublt givn by (4) by mn of U n. (40) Witout lost of gnrlity w n ng t ss of t flvor lft-ndd nutrino filds by mns of i i x nd x. us i tr is just s x tt n not b limi- ntd. So t mtrix U n b rwrittn s U i i. (4) Now t digonliztion of t mss mtrix (5) givn by is vlid for wit U U dig m m (4) D m m m m m 4 m (4) nd tus t nutrino filds wit dfinit msss nd v rstivly t following msss m m m (44) i m m m. ( 45) W obsrv tt in t xrssion for m rs t ftor x i wi suggst tt tis mss ould b omlx. Howvr t digonliztion givn by (4) is not omltly rigt bus is symmtri mtrix. So from (4) t digonliztion sould b of t form D U U (46) wr w v onsidrd tt tis mtrix is rmiti i... So t vlus m nd m r t qudrti roots of t ignvlus of. is lst rsult imlis tt ts ignvlus n b multilid by omlx s. xrssion (40) givs t mixing of t flvor nutrino filds in trms of t nutrino filds wit dfinit msss. nutrino filds wit dfinit msss nd oby jorn fild qutions of t form i im (47) i im. (48) i wit t uros of liminting t s from t lst qution of motion w n mk t following i s trnsformtion x. Now t unitry mtrix n b writtn s U. i i (49) W obsrv tt t s x i ws limintd from t lst qution of motion but not from t unitrin mtrix U l. So it rovs tt tis s is ysil nd sould b involvd in som rosss. is s ould ly n imortnt rol in t s of doublt bt dy ross. Following similr rodur for t rigt-ndd jorn nutrinos w find tt t dfinit-mss nutrino fild doublt n givn by () is rltd to t flvor rigt-ndd nutrino doublt givn by (4) by mn of wr t unitry mtrix U n (50) U is givn by

7 809 wr U i. i is givn by m m m wit m m 4 m. (5) (5) Nxt w onsidr t nonil quntiztion of t nutrino filds wit dfinit mss by stting t ntionmuttion rltions givn by ˆ ˆ r b r' b r r ' ˆ ˆ r br ' 0 nd ˆ ˆ b r r ' 0 wr b 4 rrsnt nutrino mss stts. E on of t dfinit-mss nutrino fild ortors ˆ x obys jorn qution. It is ossibl to x on of ts fild ortors on ln-wv nd bsis st s ws sown in () (s Eqution (5)). wr E m is t nrgy of t nutrino fild wit dfinit mss wi is tggd by 4. flvor nutrino fild ortors tggd by ˆ r dfind s surosition of t dfinit-mss nutrino fild ortors ˆ givn by (5) troug t xrssion ˆ x U ˆ x (54) wr U is t unitrin mtrix dfind by (49) for lft-ndd nutrinos nd by (5) for rigt-ndd nutrinos mnwil nutrino flvor stts r dfind in trms of t nutrino mss stts s U. (55) us w v found rltion btwn nutrino flvor stts nd nutrino mss stts using ortor filds. As flvor stts r ysil stts sin ty ould b dttd in intrtion rosss flvor stts r non-sttionry. So tir tmorl volution givs t robbility of trnsition btwn tm. rfor tis robbility dsribs jorn nutrino osilltions studid s follows. 5. ft-hndd Nutrino Osilltions Now w will fouss our intrst in t dsrition of lft- ndd nutrino osilltions in vuum from inmtil oint of viw. For tis rson w will not onsidr in dtil t wk intrtion rosss involvd in t rtion nd dttion of lft-ndd nutrinos. Howvr ts rosss r mnifstd wn boundry onditions r imosd in t robbility mlitud of trnsition btwn two nutrino flvor stts. W suos tt nutrino wit sifi flvor is rtd in oint of s-tim x0 t0 r0 s rsult of rtin wk intrtion ross. W will dtrmin t robbility mlitud to find out t nutrino wit notr flvor in diffrnt oint of s-tim x t r. W ssum tt nutrinos r rtd undr t sm rodution ross wit diffrnt vlus of nrgy nd momntum. s dynmil quntitis r rltd mong tmslvs undr t sifi rodution ross. initil lft-ndd nutrino flvor stt in t rodution tim ( t 0 ) orrsonds to t following surosition of nutrino mss stts t 0 A B. (56) wr A B. E of ts nutrino mss stts s ssoitd sifi four-momntum. W ssum tt in t rodution oint it ws rtd lft-ndd ltroni nutrino wit mssiv fild ving four-momntum givn by E wit. initil lft-ndd ltroni nutrino stt stisfying t ondition A B is writtn s i t0 (57) wr t sum ovr liitis is tkn ovr t nutrino mss stts. is sum ovr liitis must b onsidrd to dsrib roritly t initil lft-ndd nutrino flvor stt bus t liity is rorty wi is not dirtly msurd in t xrimnts. mnnr s t ltroni lft-ndd nutrino stt s bn built in t rodution oint is in grmnt wit t xrimntl ft tt lft-ndd nutrinos r ultr-rltivisti. nutrino mss stts involv in t surosition givn by (57) r obtind from t vuum stt s ˆ 0 i x 0 wr w v inludd ts ftor x i x 0. is s ftor givs us informtion bout t fours tim wr t lft-ndd nutrino ws rtd. robbility mlituds for trnsitions to ltroni nd muoni lft-ndd nutrinos r rstivly givn by d i x i ˆ ˆ x E ˆ x E (5) π E

8 80 E E i X ix ˆ X 0 0 (58) x t π E E E E 0 ˆ X x t (59) i X ix 0 π E E wr w v usd som xnsions ovr t jorn filds nd w v tkn X x x0 wi orrsonds to four-vt or ssoitd to t distn nd tim of nutrino rogtion. robbility dnsitis rstivly r X π 4 X X E E EE sin X π EE E E X EE EE sin X E E E E (60) (6) ount tt sinors wr w v tkn into nd r t sm bus vtors nd r o-linr. robbility dnsitis (60) nd (6) tt w v found rsnt sr ious roblm. If w fix X 0 into (60) nd (6) w find tt X 0 π 4 E E EE (6) π X 0 (6) E E EE nd w obsrv tt t robbility dnsity (6) n b diffrnt from zro i.. it n xist muoni nutrino in t rodution oint wi disgrs wit t initil onditions. origin of tis roblm is rltd to t wk stt dfinition (55) tt w v usd bfor. As it ws rviously mntiond into t introdution t flvor dfinition (55) is not omltly onsistnt nd it is nssry to dfin rorit flvor stts [4]. Ultr-ltivisti imit: ft-hndd Nutrino Osilltions is roblm n b solvd by tking n roximtion in t robbility dnsitis (60) nd (6) bsd on t ft tt lft-ndd nutrinos r ultr-rltivisti rtils bus tir msss r vry smll. Hr w onsidr nrgy nd momntum diffrnt for vry mss stt. In gnrl w n writ 4 E m m (64) 4 E E m m (65) wr t rmtrs nd r dtrmind in t rodution ross nd E is t nrgy for t s in wi nutrinos wr msslss. For instn for t ion dy ross w v E m mπ m π m m π 4mπ (66) (67) wr m is t muon mss nd m π is t ion mss. Bus for t ultr-rltivisti limit m 0 w n roximt t xrssions (64) nd (65) to m E (68) E

9 8 m E E. (69) E Now it is ossibl to rov tt t rigt sid of t rltion E E m m EE 8E (70) n b roximtd to t unit bus m E 0 wr m m m. On t otr nd nutrino rogtion tim is not msurd in nutrino xri- mnts [97]. In tis kind of xrimnts is msurd t distn btwn t nutrino sour nd t d- btwn ttor. By tis rson it n b ossibl to find n- tt stbliss rltion lytil xrssion nd t rogtion distn. In our ro using ln wvs for t ultr-rltivisti limit w n writ. (7) is rltion imlis tt t rogtion distn nd t rogtion tim for nutrinos r roximtly qul bus in t ultr-rltivisti limit nutrino mss stt s mss too smll nd its vloity of rogtion v k is roximtly qul to sd vloity i.. vk. Howvr most ris rltion btwn nd must b dsribd by n xrssion tt sould inlud xliitly t vloitis of t two nutrino mss stts involvd in su wy tt tis xrssion for t ultr-rltivisti limit sould ld to (7). So for t ultr-rltivisti limit t robbility dnsitis (60) nd (6) n b writtn s m sin (7) π 4E m sin π 4E (7) wr w v usd nd m m m. Undr tis roximtion it is lr tt ts robbility dnsity dos not dnd from t rodution ross du to tt tr is no dndn from. us ts robbility dnsitis stisfy t boundry onditions tt w v imosd. In t nxt w will rov tt t robbility dnsitis (7) nd (7) v t form of t stndrd robbility dnsitis for nutrino osilltions. In t ontxt of t stndrd formlism of nutrino osilltions (ssuming CP onsrvtion) for t two gnrtion s onsidring r t rrsnttion of t unitry mtrix U tt rs into t xrssion (40) is givn b y [0] os sin U (74) sin os wr is t mixing ngl. If w omr t unitry mtrix givn by (49) wit t on givn by (74) w obsrv tt os nd tn it is vry sy to obtin tt sin. (75) Substituting (75) into (7) nd (7) w obtin t xrssions sin sin π π m 4E m 4E sin sin (76) (77) wi r t stndrd robbility dnsitis for lftndd nutrino osilltions in t two flvor s [0]. 6. igt-hndd Nutrino Osilltions initil rigt-ndd nutrino flvor stt in t rodution tim ( t 0 ) orrsonds to t following surosition of nutrino mss stts t C D. (78) 0 4 wr C D. E of ts nutrino mss stts s ssoitd sifi four-momntum. W ssum tt in t rodution oint it ws rtd rigt-ndd ltroni nutrino wit mssiv fild ving four-momntum givn by E wit 4. initil rigt-ndd ltroni nutrino stt stisfying t ondition C D is writtn s i t0 4 (79) wr t sum ovr liitis is tkn ovr t nutrino mss stts. robbility mlituds for trnsitions to ltroni nd muoni rigt-ndd nutrinos r rstivly givn by X 0 ˆ x t 0 E E 4 i X 4 i4 X 4 π E E4 (80)

10 8 X 0 ˆ x t π robbility dnsitis X 0 E E 4 4 i X i X 4 4. E E4 X X rstivly r 4 π EE 4 4 E E4 E E4 X EE sin (8) (8) X π EE wr w v tkn into oun nor nd 4 r t sm bus vtors nd r o-linr. robbility dnsitis (8) nd (8) 4 t tt si s 4 4 E E4 4 E EE 4 EE 4 4 E 4 sin X (8) tt w v found rsnt srious roblm. If w fix X 0 into (8) nd (8) w find tt 4 X 0 E E4 4 (84) π X 0 4 E E 4 (85) π EE 4 nd w obsrv tt t robbility dnsity (85) n b diffrnt from zro. Non-ltivisti imit: igt-hndd Nutrino Osilltions is roblm n b solvd by tking n roximtion in t robbility dnsitis (8) nd (8) bsd on t ft tt rigt-ndd nutrinos r non-rltivisti rtils bus tir msss r vry lrg. By tis rson w tk t non-rltivisti roximtion i.. m. So w v E m. (86) m rfor w suos tt vy rigt-ndd jorn nutrinos oby simly t rltivisti disrsion rltio n. So w obtin t following roximtion E E 4 4 EE 4 vv 4 v v4v v4 8 wr t non-rltivisti vloity of t nutrino is i vi 0 mnwil t s is roximtd to m i E E m (88) wit m4 m m4. So t robbility dnsitis of trnsition r givn by m4 sin π m4 sin π (86) (87) wr is givn by (5). lst robbility dnsitis stisfy t normliztion nd boundry onditions. Unlikly to t s of lft-ndd nutrino osilltions dsribd by (60) nd (6) t rgumnt of t riodi funtion for t rigt-ndd nutrino osilltions dnds on t linr mss diffrn m4 nd t rogtion tim. dsrition of vy rigt-ndd nutrino osilltions tt w rsnt r ould b of

11 8 intrst in osmologil roblms []. As it s bn roosd in t litrtur vy-vy nutrino osilltions ould b rsonsibl for t bryon symmtry of t univrs troug ltognsis mnism [-4]. But it sould b notd tt if t rogtion of vy rigt-ndd nutrinos is onsidrd s surositions of mss-ignstt wv kts [5] tn t osilltions do not tk l bus t orn is not rsrvd: in otr words t osilltion lngt is omrbl or lrgr tn t orn lngt of t rigt-ndd nutrino systm [5]. 7. Conlusions In tis work w v studid nutrino osilltions in vuum btwn two flvor stts onsidring nutrinos s jorn frmions. W v rformd tis study for t s of flvor stts onstrutd s surositions of mss stts xtnding t Sssroli modl wi dsribs jorn nutrino osilltions by onsidring nutrino mss stts s ln wvs wit sifi momnt. In t ontxt of ty I ssw snrio wi lds to gt ligt lft-ndd nd vy rigt-ndd jorn nutrinos t min ontribution of tis work s bn to obtin in sm formlism t robbility dnsitis wi dsrib t osilltions for ligt lftndd nutrinos (ultrrltivisti limit) nd for vy rigt-ndd nutrinos (non-rltivisti limit). In tis work w v rformd t nonil quntiztion rodur for jorn nutrino filds of dfinit msss nd tn w v writtn t nutrino flvor stts s surositions of mss stts using quntum fild ortrnsition btwn two diffrnt nutrino flvor stts for t ligt nd vy nutrino ss nd w v stb- lisd normliztion nd boundry onditions for t tors. W v lultd t robbility mlitud of robbility dnsity. Aftr t ultr-rltivisti limit ws tkn in t robbility dnsitis for t lft-ndd nutrino s ld to t stndrd robbility dnsitis wi dsrib ligt nutrino osilltions. For t rigt-ndd nutrino s t xrssions dsribing vy nutrino osilltions in t non-rltivisti limit wr diffrnt rst to t ons of t stndrd nutrino osilltions. Howvr t rigt-ndd nutrino osilltions r nomnologilly rstritd s is sown wn t ro- of mss-ignstt wv kts [5]. gtion of vy nutrinos is onsidrd s surositions is work stblis frmwork to study jorn nutrino osilltions for t s wr mss stts r dsribd by Gussin wv kts s will b rsntd in fortoming work [5]. wv kt trtmnt is nssry owing to t nutrinos r rodud in wk intrtion rosss witout sifi momnt. Additionlly t ln wv trtmnt n not dsrib rodution nd dttion lolizd rosss s our in nutrino osilltions. 8. Aknowldgmnts C. J. Quimby tnks DIB by t finnil suort rivd troug t rsr rojt Proidds ltromgnétis y d osilión d nutrinos d jorn y d Dir. C. J. Quimby tnks lso Virrtorí d Invstigions of Univrsidd Nionl d Colombi by t finnil suort rivd troug t rsr grnt orí d Cmos Cuántios lid sistms d l Físi d Prtíuls d l Físi d l tri Condnsd y l dsriión d roidds dl grfno. EFEENCES [] E. jorn ori simmtri dll lttron dl ositron Nuovo Cimnto Vol [] P. B. Pl nd. N. otr siv Nutrinos in Pysis nd Astroysis World Sintifi Publising Singor 004. [] C. Giunti nd C. Kim Fundmntl of Nutrinos in Pysis nd Astroysis Oxford Univrsity Prss Nw York 007. [4] K. Nkmur t l. Prtil Dt Grou t viw of Prtil Pysis Journl of Pysis G Vol Artil ID: [5] A. Bottino C. W. Kim H. Nisiur nd W. K. Sz odl for ton ixing Angls nd jorn Nutrino sss Pysil viw D Vol. 4 No doi:0.0/pysvd.4.86 [6] B. Pontorvo Nutrino Exrimnts nd t Problm of Consrvtion of toni Crg Sovit Pysis JEP Vol [7] S.. Bilnky nd B. Pontorvo ton ixing nd Nutrino Osilltions Pysis orts Vol. 4 No doi:0.06/070-57(78) [8] B. Kysr On t Quntum nis of Nutrino Osilltion Pysil viw D Vol. 4 No doi:0.0/pysvd.4.0 [9] C. Giunti C. W. Kim nd U. W. Wn Do Nutrinos lly Osillt? Quntum nis of Nutrino Osilltions Pysil viw D Vol. 44 No doi:0.0/pysvd [0] J. i Quntum nis of Nutrino Osilltions Pysil viw D Vol. 48 No doi:0.0/pysvd []. Zrl From Kons to Nutrinos: Quntum nis of Prtil Osilltions At Pysi Poloni B Vol [] E. Sssrolli Nutrino Osilltions: A ltivisti Exml of wo-vl Systm Amrin Journl of Pysis Vol. 67 No doi:0.9/.940 []. Blson nd G. Vitillo Quntum Fild ory of Frmion ixing Annls of Pysis Vol. 44 No doi:0.006/y.995.5

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