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TOAPPEARIN “MAGNETICFIELDS IN THEUNIVERSE 4 (2013)” RevMexAA(SC) NEUTRINOOSCILLATION FROMHIDDEN GRBJETS NissimFraija1andEnriqueMorenoMeńdez2 RESUMEN Los Cola´psares son probablemente la fuente de los destellos de rayos gama de larga duracioń (lGRBs por sus siglas en ingle´s). Se han observado lGRBs con duracioń de miles y hasta decenas de miles de segundos, lo que hace poco 4 1 probablequesusjetsseanproducidospormotorescentralesaccionadosporneutrinos.Ene´stecontexto,elmecanismode 0 Blandford-Znajekesunbuencandidatoparaexplicarlaproducciońdejetsalolargodelejederotaciońsinlanecesidadde 2 grandestazasdeaccrecioń.Estosmotorescentralesrequierencamposmagne´ticosde1012G<B<1015Gatravezando n la parte interna del dicso de accrecioń y el agujero negro de Kerr. Obtenemos la auto-energ´ıa y el potencial efectivo a a orden O(1/M4 ) para una bola de fuego hecha de electrones, protones, neutrones y sus antipart´ıculas respectivas y J W con campo magne´tico de´bil y fuerte . Consideramos que la energ´ıa de los neutrinos producidos durante el colapso 8 gravitacional,as´ıcomoenfusiońdeobjetoscompactosoenlaboladefuegoens´ı,porprocesosdeaniquilaciońdepares, 1 decaimientobeta yporprocesosbremsstrahlungnucleońicosesde entre1 y 100MeV. Muchosde dichosneutrinosse ] propaganatrave´sdelaboladefuegoypuedenoscilarresonantemente. Usandolamezcladedosneutrinosestudiamos E laoscilaciońdeneutrinos. H ABSTRACT . h p CollapsarsarethelikelyprogenitorsofLongGRBs(lGRBs). lGRBshavebeenobservedtolastforthousandstotensof - thousandsofseconds,thusmakingunlikelytheneutrino-drivenengineasthemainmechanismfordrivingthejets. Inthis o context,theBlandford-Znajekmechanismseemslikelytoexplaintheproductionofrotational-axisdirectedjetswithout r t the need for large accretion rates. These engines, require magnetic fields between 1012G < B < 1015 G threading s a the innerdisk, Kerr-BH region to exist. We derive the neutrino self-energyand the effective potentialup to O(1/M4 ) W [ in a weakly andhighly magnetizedGRB fireballflow which is made up of electrons, protons, neutronsand their anti- 1 particles. We considerneutrinoenergiesof1-100MeVwhichareproducedduringstellarcollapse,mergereventsorin v the fireball itself by electron-positronannihilation, inverse beta decay and nucleonic bremsstrahlungprocesses. Many 5 oftheseneutrinospropagatethroughthefireballandmayoscillateresonantly. Usingtwo-neutrinomixingwestudythe 1 possibilityoftheseoscillations. 6 4 KeyWords:binaries:general—blackholephysics—gamma-raybursts:general—stars:evolution—stars:magneticfield—stars: 1. massive 0 Collapsars (Woosley 1993; MacFadyen&Woosley solve this issue, one of the proposed mechanisms is to 4 1 1999)aregenerallyacceptedastheprogenitorsofLong use a binary to transfer angular momentum back to the : Gamma-Ray Bursts (lGRBs). This is supported by starlateinitsevolution(see,e.g.Paczynski1998). v the prediction that lGRBs are accompanied by an en- Leeetal. (2002) studied several Galactic black-hole bi- i X ergetic supernova (SN). There is strong evidence that naries(BHBs)andreproducedthenatalspinoftheBHs. r six lGRBs are spectroscopically associated with SNe Using these results, Brownetal. (2007), Brownetal. a (GRB980425/SN 1998bw, GRB 030329/SN 2003dh, (2008) and MorenoMeńdezetal. (2011) estimated the GRB 031203/SN 2003lw, GRB 060218/SN 2006aj and rotational energies of the BHs at the time they were GRB 100316D/SN 2010bh, see Hjorth&Bloom 2011; born and estimated whether a Blandford-Znajek (BZ; and GRB120422/SN 2012bz, see Malesanietal. 2012). Blandford&Znajek1977)centralenginemayhavetrig- Thereis also evidenceof such a relation in anothertwo gered a lGRB in any of these BHBs. In Brownetal. dozenevents. (2008) it was found that, most likely, LMC X 3 is a − OneofthemajorissueswiththeCollapsarmodel,is likely lGRB relic, unlike most Galactic BHBs where theissueofproducingamassivestellarcorewithenough too much rotational energy may have destroyed the BZ angular momentum (see, e.g., Hegeretal. 2005). To engines too soon. In MorenoMeńdezetal. (2008) and MorenoMeńdez(2011)itwasshownthatthelargespins 1LucBinette-Fundacioń UNAMFellow. Instituto deAstronom´ıa, observed in some Galactic high-mass X-ray binaries UNAM,CU,Me´xico([email protected]). 2Instituto de Astronom´ıa, UNAM, CU, Me´xico (en- (HMXBs) cannotbenatalbuttheyratherhaveto beac- [email protected]). 1 2 N.FRAIJA,&E.MORENOMEŃDEZ quiredpost-BH formation. Thus, these HMXBs are not the medium, as for instance, chemical potential, parti- goodcandidatesforrelicsoflGRBs. cledensity,temperature,magneticfield,etc. Solvingthis Woosley&Heger(2012)havesuggestedthatthisbinary equationandusingDirac’salgebrawecanwritethedis- channelmaynotproducelGRBsastheinternalmagnetic persionrelationV =k k asafunctionofLorentz eff 0 −| | fields and/or dynamos may extract the angular momen- scalars: tum off the stellar core. MorenoMeńdez (2014) shows thatthisdiffucultymaybeovercomeiftheinternalfield Veff =b c cosφ a⊥ k sin2φ, (2) − − | | islowenoughandamagnetar-likefieldcanbeproduced where φ is the angle between the neutrino momentum duringcorecollapse. and the magnetic field vector. The Lorentz scalars a, b Now,LMXBs(low-massXBs),whichhavetoomuchro- and c are functions of the neutrino energy, momentum tationalenergymayinitiallytriggeraninternaljetwhich andthemagneticfield. Theycanbecalculatedfromthe quickly dies after the BZ engine is destroyed as a SN neutrino self-energy due to charge-current and neutral- is launched. Also, many succesful lGRBs may not be currentinteractionwiththebackgroundparticles. pointed in our direction. Similarly, were some stars to develop a rapidly rotating core without losing their en- 1.1. One-loopneutrinoself-energy velope, an internal jet could be launched which would be quenched before reaching the stellar surface. All of The total one-loopneutrino self-energyin a magne- theseshouldproduceastrongthermalneutrinosignalat tizedmediumisgivenby thecentralengine.Theseneutrinosshouldbeobservable Σ(k)=Σ (k)+Σ (k)+Σ (k) (3) withinaseveral-Mpcrange. W Z t Thepropertiesofneutrinosgetmodifiedwhentheyprop- where Σ (k) is the W-exchange, Σ (k) is the Z- agate in a strongly magnetized medium. Depending on W Z exchange and Σ (k) represents the tadpole. The W- theflavoroftheneutrinos,theyinteractwithdifferentef- t exchangediagramtotheone-loopself-energyis fective potentials because electron neutrinos (ν ) inter- e actwithelectronsviaboth,neutralandchargedcurrents (CC), whereas muon and tau neutrinos (νµ and ντ) in- d4p ig iΣ (k) = R − γ iS (p) teract with electrons only via the neutral current (NC). − W (2π)4 √2 µ ℓ This induces a coherent effect in which maximal con- (cid:20)Zig (cid:18) (cid:19) version of νe into νµ (ντ) takes place even for a small, −√2 γνiWµν(q) L (4) intrinsic,mixingangle. Theresonantconversionofneu- (cid:18) (cid:19) (cid:21) trinofromoneflavortoanotherduetothemediumeffect, whereg2 = 4√2G M2 istheweakcouplingconstant, F W iswellknownastheMikheyev-Smirnov-Wolfensteinef- Wµν depictstheW-bosonpropagatorwhich,intheeB fect. In this work, we study the propagation and reso- M2 limitandinunitarygauge,isgivenby ≪ W nant oscillation of thermal neutrinos in both, a weakly- andhighly-magnetizedfireball. Wetakeintoaccountthe gµν q2 qµqν 3ie Wµν(q)= 1+ + Fµν, (5) two-neutrinomixing(solar,atmosphericandaccelerator M2 M2 − M4 2M4 parameters). Finally, we discuss ourresultsin the GRB W(cid:18) W(cid:19) W W framework. hereM istheW-bosonmass,gµν isthemetrictensor, W Fµν is the electromagnetic field tensor and e being the 1. NEUTRINOEFFECTIVEPOTENTIAL magnitude of the electron charge. S (p) stands for the ℓ Weusethefinite-temperature,field-theoryformalism charged lepton propagator which can be separated into to study the effect of a heat bath on the propagation of two charged propagators; one in presence of a uniform elementary particles (Sahuetal. 2009a; Garc´ıa&Sahu backgroundmagneticfield(S0(p))andanotherinamag- ℓ 2007; Nieves 1990). The effect of the magnetic field netizedmedium(Sβ(p)). Itcanbewrittenas, ℓ is taken into account through Schwinger’s proper-time method (Schwinger 1951). The effective potential of a S (p)=S0(p)+Sβ(p). (6) ℓ ℓ ℓ particleiscalculatedfromtherealpartofitsself-energy diagram.Theneutrinofieldequationofmotioninamag- Assuming that the z-axis points in the direction of the netizedmediumis, magnetic field B, we can express the charged lepton propagator in presence of a uniform background mag- [/k Σ(k)]ΨL =0, (1) neticfieldas, − wheretheneutrinoself-energyoperatorΣ(k)isaLorentz ∞ iS0(p)= eφ(p,s)G(p,s)ds, (7) scalarwhichdependsonthecharacterizedparametersof ℓ Z0 REVMEXAA(SC)DEMODOCUMENT 3 wherethefunctionsφ(p,s)andG(p,s)aregiveby, andneutrinoswiththeZ boson. Theirformsareasfol- lows, G(p,s) = sec2z[A/+iB/γ 5 + mℓ(cos2z iΣ3sinzcosz) −21 +2sin2θW e − 1 ν tanz C = 2 , (13) φ(p,s) = is(p20−m2ℓ)−is[p23+ z (cid:3)p2⊥], (8) V  12 −2sin2θW p 1 n hanerdepm2 ℓ=ispt2h+epm2aasrseotfhethperocjheacrtgioendsleopfttohne,mp2kom=enpt20u−mopn23 and  −2 ⊥ 1 2 themagneticfielddirectionandz = eBs, Additionally, 1 ν,p thecovariantvectorsaregivenasfollows, CA =( −212 e,n . (14) A =p sin2z(p u u p b b ), µ µ− · µ− · µ By evaluating each contribution of one-loop neutrino energy, the effective potential in the weak Bµ =sinzcosz(p u bµ p b uµ), (Sahuetal. 2009b) (B m2/e) and strong (B · − · m2/e) (Fraija 2014b; F≪raija&eMorenoMendez 201≫4; and e Fraija&MorenoMeńdez 2014) -field limit is given re- Σ3 =γ5/b/u. spectivelyby Tishgeivcehnarbgye,dleptonpropagatorina magnetizedmedium Veff,w = √2GπF2m3eBBc ∞ (−1)lsinhαK1(σ) 1+CVe (cid:20)l=0 (cid:26) Sℓβ(p)=iηF(p·u)Z−∞∞eφ(p,s)G(p,s)ds, (9) +mm12W2e (cid:18)232E+ν22+mEXν22eB+ BBcc(cid:19)os−φ(cid:18)1−4CmA2ee+Eνmm2W∞2e ( 1)l wpahretirceleηsFi(npt·hue)mceodnituaminswthhiecdhiastrreibguivtieonnbfyu:nctionsofthe ×(cid:18)2 − m2e Bc(cid:19)(cid:19) (cid:27)− m2W me Xl=0 − 3 K1(σ) K1(σ) coshα K0(σ)+ cosφ (15) × 4 σ − σ θ(p u) θ( p u) (cid:26) (cid:27)(cid:21) η (p u)= · + − · , (10) F · eβ(p·u−µℓ)+1 e−β(p·u−µℓ)+1 and wtuhreeraenβdtahnedcµheℓmariceatlhpeoitnevnetirasleooffththeecmhaerdgieudmletpetmonp.era- Veff,s = √2GFmπ23e(cid:20)l∞=0(−1)lsinh α(cid:20)(cid:18)1+ 23MmW22e − MeBW2 (cid:19) TheZ-exchangediagramtotheone-loopself-energy 2 XB B m2 eB is (cid:18)σK2(σ)− BcK1(∞σ)(cid:19)(cid:21)− Bc(cid:18)1+ 2MeW2 − MW2 (cid:19) −iΣZ(k) = R(cid:20)Z (2dπ4p)4 (cid:18)√2−coisgθW(cid:19)γµiSνℓ(p) ×K1(σ)− 2mMeW2Eν Xl=0(−1)lcosh α(cid:20)(cid:18)σ82 − 52BBc(cid:19) ig B 16 K1(σ) (cid:18)√2−cosθW(cid:19)γνiZµν(q)(cid:21)L, (11) ×K0(σ)+(cid:18)2−4Bc + σ2(cid:19) σ (cid:21)(cid:21), (16) where, θ istheWeinbergangle,Zµν(q)istheZ-bosonpropa- W gatorinvaccum,S istheneutrinopropagatorinather- νl α=βµ(l+1) and σ =βme(l+1). (17) malbathofneutrinos. TheTadpolediagramtotheone-loopself-energyis 2. TWO-NEUTRINOMIXING Here we consider the neutrino oscillation process g 2 d4p iΣt(k) = R 2cosθ γµiZµν(0) (2π)4 νe ↔ νµ,τ. The evolution equationfor the propagation (cid:20)(cid:18) W(cid:19) Z ofneutrinosintheabovemediumisgivenby Tr[γ (C +C γ )iS (p)] L, (12) ν V A 5 ℓ (cid:21) i ν˙e = Veff −∆cos2θ ∆2 sin2θ νe , (18) wherethequantitiesC andC arethevectorandaxial- (cid:18)ν˙µ(cid:19) (cid:18) ∆2 sin2θ 0 (cid:19)(cid:18)νµ(cid:19) V A vector coupling constants which come in the neutral- where ∆ = δm2/2E , V is the potential difference ν eff currentinteractionofelectrons,protons(p),neutrons(n) betweenV andV , E istheneutrinoenergyandθ νe νµ,τ ν 4 N.FRAIJA,&E.MORENOMEŃDEZ istheneutrinomixingangle. Theconversionprobability 50 fortheaboveprocessatagiventimetisgivenby ∆2sin22θ ωt 40 P (t)= sin2 , (19) νe→νµ(ντ) ω2 2 (cid:18) (cid:19) 30 with T me 20 ω = (V ∆cos2θ)2+∆2sin22θ. (20) eff − q Thepotentialfortheaboveoscillationprocessisgivenby 10 eq. (2). The oscillation length for the neutrino is given by 0 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 L = Lv , (21) p osc cos22θ(1 Veff )2+sin22θ 50 − ∆cos2θ q andtheresonanceconditioncanbewrittenas 40 δm2 V cos2θ =0. (22) 30 eff − 2E ν T me Following Fraija (2014a) we apply the neutrinooscilla- 20 tionparametersintheresonancecondition(eq. 22). 10 3. RESULTSANDCONCLUSIONS Wehaveplottedtheresonanceconditionforneutrino 0 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 oscillationinweaklyandstronglymagnetizedcollapsars p at the base of the jet when the temperature is in the 50 range of 1- 25 MeV. We have used effective potential V uptoorderM4 inthestrong-andweak-fieldlimit, eff W thebestparametersforthetwo-solar(Aharmim&etal. 40 2011), atmospheric (Abe&etal. 2011) and accelerator (Athanassopoulos&etal.1996,1998)neutrinosandthe 30 neutrinoenergies(E =1,5,20,50and100MeV).The ν T me analysisofresonanceconditionshowsthat,thetempera- 20 tureasafunctionofchemicalpotentialisdegeneratefor theweaklimitandnotinthestronglimit. Inbothcases, thechemicalpotentialisbiggestwhenacceleratorparam- 10 eters are used (19 eV - 50 keV and 5 keV - 0.15 MeV, respectively)andsmallestforsolarparameters(0.08eV 0 -5eVand0.25-50eV,respectively). -5 -4 -3 -2 -1 0 Thebaryonload,resonancelengthandleptonicsymme- p try(twoandthreeflavors)willbeestimatedandstudied inFraija&MorenoMeńdezinprogress. Fig.1. Plotoftemperature(T/me)asafunctionofchemical potential (µ = me10p) for which the resonance condition is satisfied. Wehave used the best parameters of the two-flavor NF gratefully acknowledges a Luc Binette- solar(top),atmospheric(middle)andaccelerator(bottom)neu- Fundacioń UNAM Posdoctoral Fellowship. EMM trino oscillation, B=0.1 B and taken five different neutrino c was supported by a CONACyT fellowship and projects energies: Eν =1 MeV (red dashed line), Eν =5 MeV (blue CB-2007/83254 and CB-2008/101958. This research dot-dashedline),Eν =20MeV(magentadottedline),Eν =50 has made use of NASAs Astrophysics Data System as MeV(blackthin-solidline)andEν =100MeV(brownthick- wellasarXiv. solidline). 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Wehave used the best parameters of thetwo-flavor solar(top),atmospheric(middle)andaccelerator(bottom)neutrinooscillation,B=10B andtakenfivedifferentneutrinoen- c ergies: Eν =1MeV(reddashedline),Eν =5MeV(bluedot- dashed line), Eν =20 MeV (magenta dotted line), Eν =50 MeV(blackthin-solidline)andEν =100MeV(brownthick- solidline).