Propagation of Light in a Hot and Dense Medium

Transcription

1 Propagation of Light in a Hot and Dns Mdiu Saina S. Masood Dpartnt of Physics Univrsity of Houston Clar La Houston TX 7758 Photons as quanta of lctroagntic filds dtrin th lctroagntic proprtis of an xtrly hot and dns diu. Considring th proprtis of a photon in th intracting diu of chargd particls w xplicitly calculat th lctroagntic proprtis such as th lctric prittivity agntic prability rfractiv indx and th propagation spd of lctroagntic signals in xtrly hot and dns bacground in cosos. Photons acquir dynaically gnratd ass in a diu. Th scrning ass of photon Dby shilding lngth and th plasa frquncy ar functions of statistical paratrs of th diu and ar studid in dtail. W study th proprtis of th propagating particls in astrophysical systs of distinct statistical conditions. Th odifications in th diu proprtis lad to th quation of stat of th syst. W ainly calculat all ths paratrs for th xtrly high tpratur conditions of th arly univrs.. INTRODUCTION W r-invstigat th bhavior of th first gnration of lptons and corrsponding lctroagntic fild in xtrly hot and dns diu of lctrons blow th lctrowa tpratur. This syst is coposd of diffrnt typs of particls but th significant contribution cos fro th light particls which hav asss lowr than th tpratur in th arly univrs. Ovrall bhavior of particls in a diu is a nt rsult of intraction of propagating particls with th diu. Th bacground corrctions bco uch or significant at high tpraturs and dnsitis whr th propagating particls can odify th proprtis of th diu [-5]. W considr th light ass particls in a hat bath of lctrons and photons at tpratur blow th dcoupling tpratur. Thus th highr gnrations of lptons ar not xpctd to affct or b affctd significantly by tpratur which ar uch blow thir asss. Radiativ bacground corrctions du to th havy intrdiat vctor bosons of lctrowa intractions ar also supprssd bcaus of thir havy ass. Thrfor w study th bacground contribution of radiation whil it is intracting with attr and tpratur of hot lctrons is blow MV th dcoupling tpratur. Th syst is considrd to b in thral quilibriu in spcifid rgions of th stllar bodis. In this papr w study a pur gas of lctrons and photons that can b convrtd in to lctroagntic plasa of photons and lctrons at high tpraturs which ar sufficintly sallr than th W and Z asss. Thrfor th bacground contribution is coing fro th lpton and photon propagators and th nutrinos ar not yt dcoupld bcaus lctrons hav lowr than nutrino dcoupling nrgy. Howvr lctroagntic proprtis of th propagating nutrinos if thy hav ass will b odifid at high tpraturs du to th lctron inducd Elctronic addrss: asood@uhcl.du

2 agntic ont. Th hatbath with a high concntration of lctrons and photons can still b considrd as a rlativistic plasa of lctrons and photons for µ <<. Photon acquirs a tpratur dpndnt scrning ass and Dby scrning lngth can b calculatd fro th longitudinal coponnt of th vacuu polarization tnsor. Elctroagntic proprtis of th diu ar odifid. Massiv photon can thn b tratd as a Plason. Th Plason at highr nrgy can dcay in to a nutrino-antinutrino pair which can coupl with th Plason through lctrons as a highr ordr ffct. Th first ordr radiativ corrctions to lctroagntic vrtx ar shown in Figur. Ths diagras contribut to th agntic ont of lctron Figur a and nutrino Figur b andc. Elctron inducs a nonzro agntic ont to nutrino du to th intraction of lctron with nutrino in - ν - ν in th inial standard odls with vry tiny ass of γ γ γ Z γ ν ν ν ν W a b c Figur : First ordr radiativ corrctions to plasons in th lctrowa odl rsults in to lctrons or nutrinos or th intraction of lptons with th agntic fild. Radiativ corrctions to lctroagntic vrtx a th tadpol diagra b corrspond to nutral currnt and givs nonzro contribution for asytric cobination of lctrons-positron bacgroud only. Th bubbl diagra c givs a ajor contribution to agntic ont of nutrino. nutrino. Thral bacground corrctions du to th scond or highr gnration of particls will always b supprssd vn if thos particls ar injctd in th diu fro outsid.. Calculational Sch Th proprtis of lctrons as lctroagntically intracting particls in an xtrly hot and dns syst ar studid using th rnoralization sch of quantu lctrodynaics QED in statistical dia for diffrnt rangs of tpratur and chical potntial [7-3]. Tpraturs and chical potntial ar in th rang whr w dal with th ral particls only and th ral-ti foralis is usd. This foralis is valid in a hat bath of ral particls blow th dcoupling tpraturs. All of th Fynan ruls of QED rain unchangd. Th statistical ffcts ar includd through th statistical distribution functions. W incorporat th Bos- Einstin distribution function for asslss vctor bosons. Fri-Dirac distribution function is givn by [6] p p n F p p p W considr in Eq. a closd syst with qual and opposit chical potntial to th corrsponding antifrions in a CP sytric bacground. First tr in parnthsis corrsponds

3 to particl distribution whras th scond tr corrsponds to antiparticls distribution in hot and dns diu. It is convnint to xpand th distribution functions of particl and antiparticl in powrs of β for a constant chical potntial whr is th ass of th corrsponding particls and β=/t. All th statistical paratrs μ T and B ar xprssd in units of th lctron ass. In a hat bath of lctrons at vry high tpraturs proprtis of lctrons chang corrsponding to tpratur and dnsity of th syst. Th physically asurabl valus of lctron ass charg and wavfunction of lctrons in a diu ar calculatd as rnoralization constants [5] of QED in a hot and dns hat bath for diffrnt rangs of tpratur and chical potntial. Without gtting in to dtails of calculations w us th physically asurabl paratrs of th propagating particl with th rnoralization constants of QED to dtrin th lctroagntic proprtis of th diu as a rlativistic plasa. QED rnoralization constants cobining with th bar paratrs giv th physically asurabl quantitis of th syst. Ths rnoralizd finit quantitis corrspond to th physically asurabl valus of th paratrs such as lctron ass R [78] charg [9-] wavfunction [] and th agntic onts [3-]. W can thn rplac th rst ass of lctron by th physically asurabl rnoralizd ass as: R = + a R Phys whras th corrsponding rlations btwn rnoralizd wavfunction of lctron with that of th corrsponding vacuu valu is givn as R 3a Z ψ R = ψ Phys 3b such that th probability of finding particls in crtain stats bcos a function of th statistical paratrs of th diu. Th lctroagntic filds is xprssd as b A R A a Z 3 Aµ R =Aµ Phys b and th physical ass Eq.a wavfunction Eq.3a and th lctroagntic filds Eq.a giv th physically asurabl valus of th corrsponding paratrs.th QED Lagrangian of such a syst can thn b writtn as L Z F F iz D Z 3 R R R R 3 R R R Z Z A 5 In this sch of calculations th rnoralization constants of QED ar considrd to b th ffctiv paratrs of th thory. Th rnoralization constants of QED giv th physical ass and th charg of lctrons and th corrsponding wavfunction at finit tpratur and dnsity. Th vacuu polarization tnsor Πµν for such a syst can b writtn by rplacing th photon and lctron propagator in vacuu by th on in th diu in ral-ti foralis such that:

4 With [6] whras π μν K μ = i d p π Τr {γ μp + K + γ ν p + } [ p + K + Γ Fp + K μ ] [ p + Γ Fp μ] 6 Γ F p μ = πiδp [θp n F p μ + θ p n F p μ] 7 K = ω ω = K α u α. K = is in vacuu du to th transvrs natur of light which assurs th absnc of longitudinal coponnt as wll as th asslssnss of photon. u α is th -vlocity of th hatbath. Th vacuu part π T= μν K and th diu contribution π β μν K μ to Eq. 6 can b writtn as 8 with π μν K = π T= μν K+π β μν K μ 9 π β μν K μ = π d p Τr π {γ μp + K + γ μ p + } [ δ[p+k ] {n p F p + K μ + n F p + K μ} + δ[p ] p+k {n Fp μ + n F p μ}] Th polarization tnsor π β μν K μ can gnrally b writtn in trs of th longitudinal and transvrs coponnts π L ω and π T ω rspctivly such that it satisfis th rlation π μν K μ = P μν π T K μ+ Q μν π L K μ Whras th polarization tnsor corrsponding to th transvrsly polarizd wav is P μν =g μν + K μk ν a and th polarization tnsor corrsponding to th longitudinally polarizd wav only possibl in a statistical diu is whras and Q μν = K u μ + ωk μ u ν + ωk ν g μν =g μν u μ u ν K μ=k μ ωu μ b c d Such that thy satisfy th conditions: with P ν μ P α μ = P α μ Q α μ = Q ν μ Q α ν K μ P ν μ = K μ Q ν μ = 3 = L

5 in vacuu. In th absnc of th longitudinal coponnt of photon in vacuu Eq. rducs to π μν = P μν 5 Showing that all th light is transvrsally polarizd and K and K. Transvrsality of photon Eq. is th proprty associatd with th asslssnss of photon. Whn th photon acquirs a plasa scrning ass at nonzro tpratur it givs a nonzro contribution to th longitudinal coponnt of th polarization tnsor which shows th trapping of longitudinal coponnt of lctroagntic wavs in a diu. Nonzro paralll coponnt in thr dinsional spac can still aintain th circular polarization in an isotropic diu and th possibility of trapping of light in a diu affcts th transvrs propagation in anisotropic diu in xtr statistical conditions. Du to th ass of th photon light slows down in th transvrs dirction by losing so nrgy to longitudinal dirction. Tpratur and dnsity corrctions to QED paratrs in a hot and dns diu ar rviwd in th nxt sction. Th agntic fild is not includd xplicitly. Howvr in th prsnc of chargd lptons at high tpraturs significantly larg agntic fild is xpctd. In a closd syst with isotropic attr distribution has a constant agntic fild at a constant tpratur. Thrfor at a givn tpratur agntic fild ffct is incorporatd through th potntial nrgy contribution to th chargd lptons and th nrgy of particls will b odifid in th prsnc of th agntic fild as E = p = p + ±µn+l+b Whr l corrspond to th Landau lvl and B is th constant agntic fild. B can b rplacd by th ti varying agntic fild to incorporat th chang in agntic nrgy with ti. W postpon th dtaild study of th ffct of diffrnt typ of agntic filds for now. 3. QED Paratrs in a Mdiu Thral corrctions to QED paratrs can b writtn as a function of tpratur T and chical potntial µ in th for of Masood s abc.functions xprssd as a i µ [5-] and rfrrd to as Masood s functions hraftr T 6 T 3 c µ a µ b µ. 3 6 First tr in bract is a asur of thral corrctions du to th incras in intic nrgy du to th coupling of particls with radiation whras a b and c functions ar volvd fro th intgration of Fri-Dirac distribution and vanishs at low tpratur whr th prsnc of hot frions is ngligibl in th syst. a ln 7a

6 n b Ei n 7b n c. n n 7c n n Whr +-µ corrspond to th chical potntial of frion antifrion in th diu. Th wavfunction rnoralization constant of QED can b writtn as [5] Z Z T T v ln { ve v 6 d n B 5 b µ a µ c µ } 8 and th charg rnoralization constant is calculatd [5] as : a c Z 3. b 3 9 Th photon in th diu dvlop a plasa scrning ass which can b obtaind fro th longitudinal and transvrs coponnt of th vacuu polarization tnsor L and T whr K = ω -. In this sch of calculations longitudinal and transvrs coponnts ΠL and ΠT rspctivly of vacuu polarization tnsor Πµν play a crucial rol in th calculation of th lctroagntic proprtis of a diu. Th lctroagntic proprtis such as lctric prittivity εk and agntic prability µk. Othr rlatd proprtis of th diu including rfractiv indx propagation spd and th agntic ont of diffrnt particls in th diu ar studid by using ths basic proprtis of th diu. Elctric prittivity K and th agntic prability K can b xprssd [9] in trs of L and L K a K K T L K K T such that: b Such that L [ ln 37 7 b a c ] a

7 ln ]. } [{ b b c a T Whras at xtrly high tpraturs at 3 li T K L L T>> a KL corrspond to Dby lngth. If w st with thn 6 li T T T T>> b Whras in th rlativistic plasa it dpnds on tpratur quadratically. Now th lctric prittivity K and agntic prability K of such a diu can b calculatd fro th longitudinal and transvrs coponnts [9] as: and 3b ln }]. [ b c a K K Eq. and Eq. 3 show th dpndnc of th longitudinal and transvrs coponnts of th polarization as wll as th lctric prittivity and th agntic prability of th diu as a function of ω and corrsponding to th propagating lctroagntic wavs. Dpndnc of K and K on tpratur inducs tpratur dpndnc to th propagation vlocity and th rfractiv indx of th diu. Ths quantitis also varis with th nrgy and th ontu of photon. This is a distinct fatur of th xtrly hot diu of th arly univrs that th rfractiv indx will dpnd on th wav proprtis as wll as th tpratur of th diu. Th spd of propagation of lctroagntic wavs in such a diu can b xprssd as 3a 7 37 ln } { b c a K K

8 v prop K K. and th rfractiv indx of th diu cos out to b c K K n = = v K K 5 and th Dby shilding lngth cos out to b D T 3 In th arly univrs for an xtrly larg ontu of photon. It is nown that th Dby lngth in classical plasa dpnds on tpratur as Showing th diffrnc btwn classical and rlativistic plasa.. Propagation of light in th Early Univrs It is wll-nown fro thral history of th univrs that tpraturs of th univrs wr xtrly highr than th chical potntial and th valid liit of tpratur is T >> µ. In this sction w valuat all of Masood s functions for xtrly high tpraturs. Doinant thral contribution cos fro th intraction with th radiation at th quilibriu tpratur T and frions at th sa tpratur. Th valus of a i µ for xtrly high tpratur only c µ tr contributs giving c And in th xtrly high tpratur liit T>>>>µ with ignorabl dnsity th longitudinal and transvrs coponnts of th vacuu polarization tnsor can b writtn as T K ln 3K T K 3K ln 6a 6b For larg valus of for th rlativistic systs w can writ K and as K

9 T K 7a 3K K T K 6 K 7b Eqns. 6 and 7 can also b valuatd for two xtr conditions basd on th proprtis of light as i For larg nrgy of photons ω π L = ω T 3 π T = ω T 6 8 ε = + ω T 3 μ ω T 3 K 9 v prop T. 3 K 3 T and th rfractiv indx of th diu cos out to b c K 3 T n = = v T 3 ii For larg ontu of photons ω π L = T 3 π T = T 6 3 ε = T 6 μ T 6K 33 Thral corrction to th spd of th propagation of light in such a diu is T v prop. 3a 6 T And th rfractiv indx of this diu is givn by 6 T n. 3b T

10 Eqn 3 puts so natural liit on th valus of. = T 6 is not physically allowd as it will giv infinit spd and zro rfractiv indx. Ths quations cannot b tru for T= as wll. Th valus of lctric prittivity and agntic prability dpnd on th valus of th plason frquncy and th wav nubr at a givn tpratur and v prop and th rfractiv indx n also bco a function of tpratur and th nrgy and ontu of photon.. It is also to b noticd that th thral contributions in Eqs. 5 start at T >.5 MV. So th lctroagntic proprtis of th diu cannot s any thral ffct aftr th nuclosynthsis [3 ] in th arly univrs. This liit blow th nutrino dcoupling tpratur indicating that th nuclosynthsis startd right aftr th tpratur of th univrs droppd blow th nutrino dcoupling tpratur and th nutrino captur contributd to bta procsss in th production of lctrons. All th calculations in this papr ar rlvant for th tpraturs blow th nutrino dcoupling tpraturs and highr than th lctron ass. Existnc of hot lctrons in a diu at such tpraturs insurs a significant ffct on physically asurabl valus of lctron ass charg and concntration of lctrons in a diu which lads to th chang in th lctroagntic proprtis in trs of agntic onts of lptons lctric prittivity and agntic prability of th diu. Th lctroagntic proprtis of a diu thn ffct diffrntly on th particls that propagat through this diu. 5. Magntic Mont of chargd particls Rlativistically oving chargd particls hav an associatd lctric fild and thir rlativistic otion at high tpraturs crat wa non-ngligibl currnts in xtr situations in a diu. Whn chargd particls ar acclratd in a diu and a continuous chang in nrgy occurs through acclration of particls along with th chang in tpratur of th syst an associatd agntic fild is gnratd. In such an lctroagntic syst thr ar localizd lctroagntic filds that ar associatd with th distribution of chargd particls in th diu. Magntic ont is associatd with charg and ass of lptons. Lightr particls hav larg agntic ont ffct and thral contribution is highr for th lightr particls as tpratur is always copard to th ass of th particls for thral ffcts. Rnoralization sch givs a chang in ass in Eq.. Th charg of lctron is not affctd until th tpratur is xtrly high as indicatd in Eq. 5. Magntic ont is siply calculatd fro th chang in ass in thral bacground givn as aµ = α π δ µb 35 3 δ = δ vacuu + δ T 36 µb is th unit of agntic ont calld Bohr Magnton. So th statistical corrctions to th agntic ont of lctron is dirctly proportional to th statistical slfass corrctions givn in Eq. 6. Gnralization of rsults

11 A straightforward gnralization of all th abov rsults can b don by valuating Masood s functions ai such that a a a b a3 c and so on. All th Masood s statistical functions ai corrspond to lctron bacground contributions in th abov quations and always corrspond to frion bacground contribution for T > µ. Evaluating th abov quations w just considr th lctron bacground as is tan as th lctron ass. Thus ths functions corrspond to th lctron bacground for tpraturs highr than th lctron ass. Positrons contribution in th sa diu can b xprssd by rplacing µ with - µ. Evrything ls rains unchangd. Nt contributions fro th CP sytric bacground can b obtaind by taing an avrag of particl and antiparticl bacground contributions and a nt bacground contributions can b obtaind by rplacing ai functions by th corrsponding diffrnc functions giving [5] a avg [ a a ] 37a a nt [ a a ] 37b such that in th liit T chical potntial contribution is ignorabl and hnc th thral ffcts doinat such that a nt 38a aavg ai 38b So th avrag bacground contribution can b obtaind by th corrsponding functions to Eqns. 7 a avg n 39a b avg c avg n cosh n Ei n 39b n n cosh n 39c n n It can b asily sn that at xtrly high tpraturs th frion contribution fro th diu is controlld by c as it can b asily shown that c. 6. Rsults and Discussions It can b sn fro Eqs. 6-9 that th bacground contribution to lctron ass wavfunction and charg cos dirctly fro th hot bacground as a function of tpratur which brings in n

12 T/ as th doinant contribution. Th photon bacground contribution is always proportional to T/. Thral contribution to lctron ass wavfunction and charg of lctron at high tpratur is plottd in figur showing a coparison of thral contribution to all th thr rnoralization paratrs of QED. Thy hav a siilar dpndnc on tpratur with diffrnt cofficints. Thral contribution to th lctron slfass is a coupl of ordrs of agnitud gratr than th lctron charg and th coupling constant of QED. It can b sn fro Eqns. 6-9 that th wavfunction corrction is uch sallr than th lctron ass or vn th charg of lctron. It can b asily sn that th ass contribution doinats ovr th charg contribution bcaus lctron ass has radiativ corrctions by its intraction with th radiation photon in th bacground. Howvr charg dos not s th photon and thral contribution is only du to th frion bacground. Thral corrctions to th lctron wavfunction ar vry sall vn at T >. Howvr th charg of lctron is not affctd by th bacground at low tpraturs T< bcaus asslss photons do not intract aong thslvs. At T> photon acquirs dynaically gnratd ass which incrass with tpratur in th prsnc of high concntration of lctrons in a diu. Th QED paratrs indicating proprtis of lctrons at high tpratur ar plottd in Figur..6E-8.E-8.E-8 E-8 8E-9 6E-9 E-9 E-9 -E-9 b.wavfunction a.slfass c.charg 5 5 Figur: Proprtis of lctrons as a function of tpratur in units of. Thral corrctions to a.lctron ass; b.wavfunction; and c.charg. Slfass corrctions to lctron affct th agntic ont of lctron which is also proportional to th T /. Figur 3 givs th agntic ont of lctron as a function of tpratur as th lctron ass ps on incrasing with tpratur. Howvr th prsnc of chargd frions in th bacground affcts th lctroagntic proprtis of lctron also. Figur 3 plots th thral contributions to th agntic ont as copard to th thral contribution to th ass of lctron.

13 Elctrons proprtis 3 5 T/ Figur 3: A coparison of tpratur dpndnt slfass and th corrsponding agntic ont of lctron shows that th bhavior of agntic dipol ont at finit tpratur s xactly siilar to th ass of lctron at high tpraturs i.; T> Magntic ont is a for factor and is a proprty of ass and charg whras nutrino is a asslss nutral particl. So th nonzro agntic ont can only b obtaind through th wa intraction of nutrino with th corrsponding chargd lpton as shown in Figur. In th standard odl asslss and nutral nutrino cannot intract with th agntic fild and xhibits zro agntic ont. It is a highr ordr procss if nutrino has so ass. In this way th agntic ont of nutrino will dpnd on th xtnsion of th standard odl and will b a odl dpndnt quantity. W just considr th inially xtndd standard odl with th nutrino ass as MV just to copar thral ass of lctron and th agntic dipol ont of nutrino as a function of tpratur. This coparison is don at xtrly high tpraturs. Inducd agntic ont of nutrinos is plottd in Figur whr th agntic ont of nutrino is plottd as a function of tpratur for th uppr liit of th ass of lctron typ nutrino around V. In this first ordr corrction th inial standard odl obying th consrvation of th individual lpton nubr is considrd. Tpratur corrction is supprssd for nutrino bcaus th tpraturs is copard to th ass of W instad of lctron as W boson is th loop partnr of lctron in th bubbl diagra. Thrfor th bacground contribution to th agntic ont of nutrino is inducd as alost - tis th corrsponding contributions to lctron ass which is xactly of th ordr of /M and is of th ordr of th squar of th ratios btwn th ass of lctron and th ass of W boson. Figur shows that this inducd agntic ont of nutrino in th inial standard odl is ngligibly sall bing th highr ordr ffct.

14 6E-9 Chang in Magntic Mont of Nutrino 5E-9 E-9 3E-9 E-9 E T/ Figur : Th ratio of thral lctrons bacground corrctions to th agntic ont of lctronic typ nutrino with th corrsponding vacuu valu of th agntic ont is plottd as a function of tpratur blow th nutrino dcoupling tpratur to xtract th pur bacground ffct. An xplicit coparison of all ths valus is givn in Tabl. It shows th valus of all of th QED paratrs such as lctron proprtis and th agntic ont of lctron for a givn valu of tpratur including th inducd agntic ont of nutrino. Howvr it is clar fro th last colun of th tabl that th ratio of th agntic ont of lctron typ nutrino in lpton nubr consrving inially xtndd standard odl with that of lctron is constant which is actually proportional to th ratio btwn thir asss. T/ δ/ Δ/ Elctron Dipol ont Nutrino Dipol ont Ratio of Magntic Monts E E-.36E E E-.36E E E-9.36E E E-9.36E E E-8 Tabl : Elctron slfass charg agntic ont and nutrino agntic ont hav bn valuatd for th sa valus of tpraturs in th units of lctron ass and blow th coupling tpraturs. Chical potntial is ignord at this point to p this coparison sipl. Magntic ont is a proprty of ass. Thral corrctions to th agntic dipol ont of nutrino ar actually du to its swallowd ass in thral bacground. This xpctd bhavior is donstratd in a plot of lctron ass and its corrsponding odification in th dipol ont as a function of tpratur. Figur 3 shows a clar donstration of this bhavior of lctron. It is also intrsting to not th vacuu polarization contribution du to th prsnc of frions in th bacground for T >. Nutrino agntic ont can b inducd th sa way vn if

15 th nutrinos ar not dcoupld blow MV as thy acquir th inducd agntic dipol ont by lctrons which can occur vn if nutrino concntration is lowr but thr ar nough lctrons in th diu.. Proprtis of photons at high tpratur Photon as a quanta of nrgy acquirs nonzro ass in a diu with an abundanc of lctrons with xtrly high nrgy at high tpratur. Elctroagntic intraction of photon with lctrons in a diu givs a dynaically gnratd ass to photon which can b tratd as th scrning ass and th Dby shilding turns th diu into an lctron-photon rlativistic plasa undr suitabl conditions. Sinc th photon is asslss at lowr tpratur and dnsity and th phas transition in a diu occurs at tpraturs gratr than MV whr th frion bacground starts to contribut to th slf-nrgy of photon which lads to Dby shilding du to th dynaically gnratd ass of photons. Th bhavior of dynaically gnratd scrning ass and Dby shilding lngth as a function of tpratur is givn in Figur 5. Dby scrning lngth dcrass with th incras in tpratur and with th incras in scrning ass of photon. At th tpratur around 3- Mv th Dby lngth dcrass copard to th scrning ass itslf. Dby lngth corrspond to th potntial nrgy and dcras in th potntial nrgy is xpctd with tpratur Scrning ass as a function of tpratur 6 8 T/ Figur 5: Plot of plasa scrning ass solid lin and th corrsponding Dby shilding lngth bron lin as a function of tpratur in units of lctron ass. Nubr of particls in th Dby sphr can b calculatd fro Dby shilding lngth λd if th nubr dnsity rains unchangd. ND=/3 π n λ 3 D Thus th dcras in λd with tpratur is associatd with th nubr dnsity of th univrs. Th coposition of th univrs changs with tpratur and so dos th ND du to th chang in ass as wll as th othr paratrs of th thory such as λd and th propagation vlocitis.

16 W considr th sall longitudinal coponnt and th agnitud of th wav-vctor is uch largr than th frquncy ω aing it a vry high nrgy wav. Howvr th vlocity of wavs in a diu grows quicly with tpratur whras th rfractiv indx dcrass rlativly slowly with tpratur in such a diu vlocity Rfractiv Indx 3 Figur 6: Plot of propagation vlocity and th rfractiv indx in an xtrly hot diu. Th abov rsults indicat that QED can b usd as th only thory blow th nutrino dcoupling tpratur and that is MV. As soon as th tpratur of th syst crosss this liit concntration of nutrinos bco significant and th wa intractions co in to play so of th nrgy is usd in wa intraction and is not ignorabl with th ris of tpratur. It can b clarly sn that th thral corrctions lad to quadratic incras in th asurabl valus of physical paratrs of th thory [Figs. -]. Howvr th intraction basd bul proprtis such as th propagation spd agntic ont and th scrning ass has diffrnt bhavior. This diffrnc in bhavior balancs so of th ffcts and p th possibility of xistnc of physical systs at xtr tpraturs intic nrgis and larg dnsitis or chical potntial potntial nrgis and agntic fild hlp to dvlop an quilibriu and aintain th syst. Proprtis of nutrinos ar rlatd to th assiv nutrinos which opns up a whol list of possibl xtnsions of th standard odl to accoodat assiv nutrinos. Thral ffcts on th proprtis of nutrinos ar highly odl dpndnt [-3] and can only b calculatd individually for vry odl. Evn th Dirac and Majorana ass will contribut diffrntly to th agntic ont. It can b clarly sn that all th abov quations of sction rproduc th alrady xisting rsults in th tranvrsality liit whr ω w gt propagation vlocity lctric prittivity and agntic prability ar all qual to unity. REFERENCES AND FOOTNOTES. P. Landsan and Ch. G. Wrt Phys.Rp and rfrncs thrin.. L.Dolan and R.Jaciw Phys. Rv. D A.Wldon Phys.Rv.D

17 . R.Kobs and G.W.Snoff Nucl. Phys. B ; ibid B Saina Masood Phys.Rs.Int 893 arxiv:7. 6. E.J.Lvinson and D.H.Boal Phys.Rv. D K.Ahd and Saina Sal Masood Phys. Rv D K.Ahd and Saina Sal Masood Phys. Rv D K.Ahd and Saina Sal Ann.Phys Saina S.MasoodPhys.Rv.D Saina S.Masood Phys. Rv.D and rfrncs thrin.. Mahnaz Hasb and Saina Masood. Phys.Ltt.B7:66-73 and rfrncs thrin. 3. Saina Masood JMP 5 96 arxiv:5.39. Saina Sal Masood Phys. Rv. D Saina S.Masood Phys.Rv.D and rfrncs thrin. 6. D.Notzold and G. Rafflt Nucl. Phys. B C. S. Li and W. J. Marciano Phys. Rv. D Saina S.Masood Astroparticl Physics Saina Masood arxiv:56.8 [hp-phys.]. R.Barbiri and P.Mohapatra Phys.Ltt. B K.S.Babu and V.S.Mathur Phys.Ltt. B Pal and Mohapatra Massiv Nutrinos in Physics and Astrophysics World Scintific publication B. C. Canas t. al; Phys.Ltt. B753 96