We point out that the Pauli blocking of neutrinos by cosmological relic
neutrinos can be a significant effect. For zero-energy neutrinos, the standard
parameters for the neutrino background temperature and density give a
suppression of approximately 1/2. We show the effect this has on three-body
beta decays. The size of the effect is of the same order as the recently
suggested neutrino capture on beta-decaying nuclei.Comment: 4 pages, 2 figures, contribution to procedings of PANIC 200

McElrath, Bob

English

arXiv

We point out that the Pauli blocking of neutrinos by cosmological relic neutrinos can be a significant effect. For zero-energy neutrinos, the standard parameters for the neutrino background temperature and density give a suppression of approximately 1/2. We show the effect this has on three-body beta decays. The size of the effect is of the same order as the recently suggested neutrino capture on beta-decaying nuclei

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arXiv:0901.3491v1  [hep-ph]  22 Jan 2009 Laboratorytestsforthecosmicneutrinobackgroundusingbeta-decayingnucleiBob M cElrathCER N ,G eneva 23,CH -1211 SwitzerlandA bstractW e pointoutthatthe Pauliblocking ofneutrinosby cosm ologicalrelic neutrinoscan be a sig-nicanteect.Forzero-energy neutrinos,the standard param etersfortheneutrino backgroundtem perature and density give a suppression ofapproxim ately 1=2.W e show the eectthishason three-body beta decays.The size ofthe eectisofthe sam e orderasthe recently suggestedneutrino capture on beta-decaying nuclei.K ey words:PACS:1. IntroductionThe lastrem aining rem nantofthe big bang,which is com posed ofa known particleis the Cosm ic Neutrino Background (CNB).It decouples from therm alequilibrium atT  2 M eV.It gives inform ation about the universe at a tim e signi cantly before thedecoupling ofphotons at T  1 eV.The processes responsible for their creation anddecoupling arewellunderstood nuclearphysics.Today these neutrinosare expected to be extrem ely cold (1:952K = 1:68 10 4 eV).Assuch,they areextrem ely di cultto detectdueto thefactthatweak interaction crosssections scale as (G F E )2.They have a density of = 3=22n = 56=cm3 per speciesofneutrino and anti-neutrino,corresponding to a lum inosity L = 1:7  1013=cm 2s ifthe neutrinoswere m assless.[1,2]These num bersrely on a speci c cosm ologicalm odelwhich could be substantially m odi ed,if neutrinos cluster gravitationally,or if theyhavenontrivialdynam icsafterfreeze-out.[3]Ithasalso recently been shown thattheseneutrinos are a quantum liquid,and their  uctuations have the quantum num bers ofa graviton,opening the prospect that m easurem ents ofrelic neutrinos could then becom pared with gravitationalconstants.[4]Preprintsubm itted to Elsevier 13 June 2013Nucleithatundergo-decay area preciseand speci claboratory to look fortheCNB.There existsa vastarray ofnucleithatcan em itorabsorb neutrinosata wide rangeofenergies.A signalseen using -decaying nucleiconstitutes a speci c test because theyarealready known to em itorabsorb lepton num berin speci cways.Allotherproposalscould not in principle tellifthe e ect was due to an object with lepton num ber.[1,2]High-energyneutrinos(e.g.Z-burst)can beabsorbed by thingswhich donotcarryleptonnum ber,and anom alousforcescould have a variety ofsourcesthathave nothing to dowith lepton num ber(forinstance,aDarkM atterwind).Finallythee ectsofneutrinosontheCM B cannotbedisentangled from otherrelativisticspeciesthatarenotbeferm ionicordo notcarry lepton num ber.Ratesusing decaying nucleiarein principlem uch higherthan otherproposalssuch ascoherentscattering,becausetheenergy fortheobservation iscom ing from nuclearm assdi erences,and notfrom theneutrinosthem selves.Theenergy Q in nucleartransitionsisO (M eV).G iving theCNB neutrinosthisenergy in a coherentexperim entwould requirem ovingwith avelocitycorrespondingtoaboostfactor = Q =T ’ 1010.Forcom parison, atthe LHC with protonsisabout15000,or5500 with Lead.Therearetwo waysto seean e ectofthe neutrino background using a beta-decayingnucleus:add a neutrino to it or rem ove a neutrino from it.Both were suggested byW einberg in 1962.[5]Adding a neutrino to the background is suppressed for m om enta which are alreadyoccupied by the CNB therm aldistribution,due to the factthatneutrinosare ferm ionsand theirchem icalpotentialand averageenergy aresim ilar.Thisisan O (1)e ect,ifonecan createneutrinoshaving the correctenergy.Rem oving a neutrino from the CNB using nucleiisknown asneutrino capture (C).Captureofreactorneutrinosistheoriginalm odeused to discovertheneutrino.Recentlythere hasbeen a surge ofinterestin thism ode fordetecting the CNB using -decayingnuclei,which can havezero threshold.[6,7,8]2. PauliB locking by T he C osm ic N eutrino B ackgroundThe CNB is a therm aldistribution in a particularfram e u which we assum e to becoincidentwith thedipolefrom theCosm icM icrowaveBackground,which pointsin thedirection (264:85 0:10);(48:25 0:04) in galactic coordinates,with velocity 368 2km /s.Itstherm aldistribution isthenFi(~p)=he(pu   i)=kT + 1i 1(1)foreach speciesofneutrinoand anti-neutrinoi,havingm assm iand chem icalpotentialiand four-m om entum p in thecosm icrestfram eu = (1;~0)and thisreducesto theusualnon-relativisticFerm i-Diracdistribution.Therelativisticchem icalpotentialistheFerm ienergy atzero tem perature,i = E F =pm 2 + p2F,and the nom inalFerm im om entumpredicted by the standard cosm ologicalm odelispF =3p32 =3p32 = 3:6 10 5eV,where  is the num ber density per  avor.W e willrefer to this as the \standard"chem icalpotential.A processwhich em its neutrinoshas a suppression [1  Fi(~p)]due to Pauliblockingfrom thistherm aldistribution.Thisisindependentofwhethertheneutrinosaredescribed20 0.002 0.004 0.006 0.008 0.01Neutrino momentum (eV c)0.50.60.70.80.91Suppression 1-Fmν = 0mν = 0 (inverted hierarchy)mν = 0.01 eVmν = 0.01 eV (inverted hierarchy)mν = 0.1 eVmν = 0.1 eV (inverted hierarchy)Fig.1.Sum of suppression factor 1   Fi(p) forthree m ass eigenstates vs. neutrino m om entumforseveralvaluesofthe neutrino m ass,assum inga standard chem icalpotential.0 0.005 0.01 0.015pν (eV)05e-171e-161.5e-162e-16dN/dp νΣ mν=0mν=0.1 eV (normal hierarchy)mν=0.1 eV (inverted hierarchy)Fig.2.The dierentialeventrateasa function ofneutrino m om entum for several choices of neu-trino m ass and norm al/inverted hierarchy, as-sum ing a standard chem ical potential and thekinem atics ofTritium .as a localized classicalgas having sm alluncertainty  x  n 1=3 or a quantum liquid x  n 1=3 fornum berdensity n.Fora beta decay thisisd  = 2XiZjM ij22i[1  Fi(~p)]dP S (2)where dP S is the di erentialphase space,i is the eigenvectorcom ponent ofneutrinom asseigenstateiin theelectron neutrinodirection,M isthem atrix elem entoftheem is-sion ofan electron-typeneutrino with m assm i,and onesum soverthem asseigenstatessince nalstateem itted particlesm ustbein a m asseigenstate.Thissuppression factorisexperim entally indistinguishablefrom 1 exceptin a region in which theem itted neutrinohasthe sam e energy asthe CNB.W e plotthe suppression factor,sum m ed over avorsin Fig.1The M atrix Elem entforthe beta decay isjM j2 =G 2F1283M 2I(gV + gA )2(pJ  pe)(pI  p)+ (gV   gA )2(pI  pe)(pJ  p) (3)+ (g2V   g2A )(pI  pJ)(pe  p)wheregV and gA arethevectorand axialvectorweakchargesoftheatom .ForI= neutron,gV = gA =   1=2.The m atrix elem ent also reaches a m inim um jM j2 = 0 atp= 0,sothe event rate at p = 0 is zero.Because ofthis,the m inim um in Fig.1 is deceptive.The di erentialeventrate including the m atrix elem entisplotted in Fig.2 fortritium ,assum ing a standard chem icalpotential.To placea neutrino into thebackground with signi cantsuppression,weneed thattheinvariantpu=kT < E F =kT.Since ourvelocity with respectto the cosm icrestfram eissm all( = vc= 1:23 10 3 ),one can ignore ourvelocity and pretend we can do theexperim entin the cosm icrestfram e. 11 O ne m ust use Special relativity and not G alilean relativity here. The atom s and electron m ay benon-relativistic,but the neutrino isrelativistic.3In a norm al decay an atom I decaysto atom J by em itting an electron orpositronand an anti-neutrino orneutrino.Ifonecan precisely m easurethem om enta ofI,J,ande ,onecan solvefortheneutrino m om enta.Thisrequiresm om entum resolution on eachoforderp<p2m kT.IftheinitialstateI isatrest,thiscorrespondsto a tem peratureT =23m M IT ’ 1:40 10 9m eVam uM IK :Stated another way,the de Broglie wavelength ofthe neutrinos is 1.2 m m .Since theuncertainty on m om entum scaleswith m om entum ,the initialstate m usthave a sim ilardeBrogliewavelength.M odern atom icBose-Einstein Condensateand DegenerateFerm iG asexperim entsusing laserand evaporativecooling routinely reach 10 9 K today.An-otherprom ising technology to getto theseprecisionsin the initialstateis\CrystallizedBeam s".[9]The nalstateofthedecayistheatom J alm ostexactlyback-to-backwith theelectronorpositron.Sim ilarly these nalstateparticlesm ustbe m easured with a precisionpp’s2m kTQ (Q + 2m e)’ 1:33 10 7rm eVs18keVQ:wherein the lastterm weassum eQ < 2m e.O ne m ightwonderifthe e ectshown here can im pactexperim entssuch asK ATRINwhich attem pt to m easure the neutrino m ass using the highest energy electrons in abeta decay.K ATRIN attem pts to m easure m ass due to the change in slope and ratesuppression near the endpoint,and they do not have the resolution to see the actualendpointitself.Theirresolutionisapproxim ately E e  1eV.Thee ectshereonlye ectthehighest(electron)energy bin,and reducethenum berofeventsthereby O (10 18 N )where N isthe totaldecaysthey see.Existing experim entssim ply do nothave the rateforthise ectto be a concern.3. A cknow ledgem entsW e thank Patrick Huberand M atsLandroosforfruitfuldiscussions.References[1] A .R ingwald,arX iv:hep-ph/0505024.[2] G .B.G elm ini,Phys.Scripta T 121 (2005) 131 [arX iv:hep-ph/0412305].[3] A .R ingwald and Y .Y .Y .W ong,JCA P 0412 (2004) 005 [arX iv:hep-ph/0408241].[4] B.M cElrath,arX iv:0812.2696 [gr-qc].[5] S.W einberg,Phys.R ev.128 (1962) 1457.[6] A .G .Cocco,G .M angano and M .M essina,JCA P 0706 (2007) 015 [arX iv:hep-ph/0703075].[7] R .Lazauskas,P.Vogeland C.Volpe,J.Phys.G 35 (2008) 025001 [arX iv:0710.5312 [astro-ph]].[8] M .Blennow,arX iv:0803.3762 [astro-ph].[9] H .D anared,A .K allberg,K .-G .R ensfelt,and A .Sim onsson,Phys.R ev.Lett.88,174801 (2002).4

Laboratory tests for the cosmic neutrino background using beta-decaying nuclei

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