Absolute values of three neutrino masses from atmospheric mixing and an ansatz for the mixi

a rXiv:h ep-ph/968354v117Aug1996HU-TFT-96-34hep-ph/9608354Cosmological abundances of right-handed neutrinos Kari Enqvist 1,Petteri Ker ¨a nen 2,Jukka Maalampi 3and Hannes Uibo 4Department of Physics,P.O.Box 9,FIN-00014University of Helsinki,Finland 17August 1996Abstract We study the equilibration of the right-helicity states of light Dirac neutrinos in the early universe by solving the momentum dependent Boltzmann equa-tions numerically.We show that the main effect is due to electroweak gauge boson poles,which enhance thermalization rates by some three orders of mag-nitude.The right-helicity states of tau neutrinos will be brought in equilibriumindependently of their initial distribution at a temperature above the poles if m ντ>∼10keV.1IntroductionPrimordial nucleosynthesis is a remarkable probe of neutrino properties[1].Althoughrecently the increasing accuracy of the cosmological data,such as observations relatedto the primordial abundances of helium,deuterium and the other light elements,hasemphasized systematic errors inherent to primordial nucleosynthesis analysis,thereremains a great potential for constraining neutrino physics via cosmological observa-tions.To some extent primordial nucleosynthesis could be sensitive even to the Diracvs.Majorana nature of neutrinos,because in the Dirac case the small relic abundanceof the inert right-handed component of Dirac neutrino would also contribute at nucle-osynthesis.Of course,presently one cannot hope to differentiate between the Diracand Majorana nature of neutrinos on cosmological grounds,but in principle this isan interesting problem.In practise,because of the smallness of the electron neutrinomass,from this point of view only the right-handed components ofνµandντcan haveinteresting relic abundances.Naively,the cosmological density of light right-handed neutrinosνR(or rather theright-helicity states of light neutrinos,ν+)is expected to be very small.IfνR’s ever werein equilibrium,they decoupled very early because of their low capability of interacting.Assuming this took place well above the electroweak phase transition temperatureT EW,at the onset of primordial nucleosynthesis the contribution of the right-handedcomponent of a Dirac neutrino is given byρR≃[g∗(1MeV)/g∗(T EW)]4/3ρL≃0.044ρL, where g∗(T)is the effective number of degrees of freedom in thermal equilibrium inthe temperature T andρL is the equilibrium energy density of left-handed neutrino.This assumes that at high temperature we may consider the gas of quarks,leptons,and gauge bosons nearly ideal,which may not be true.A recent lattice simulation[2]of the QCD energy density above the critical temperature has revealed that the actualenergy density is some15%smaller than expected,which might signify the existenceof a condensate at high T.One also assumes that there is no significant entropyproduction either at the electroweak or QCD phase ttice simulationsseem to indicate that this is true for QCD,and the latent heat in the electroweakphase transition is also known to be small[3].The right-helicity states of Dirac neutrinos are not completely inert in the StandardModel[5]but can be produced(and destroyed)in spin-flip transitions induced bythe Dirac mass[4,6]or the neutrino magnetic moment.If the neutrino mass islarge enough,ν+would be produced in collisions below the QCD phase transitiontemperature,resulting effectively in an additional neutrino species at nucleosynthesis.An important source ofν+’s are also the non-equilibrium neutrino scatterings and<∼130keV decays of pions,as was pointed out in[6].This gives rise to the bound mνµ<∼150keV[8],using T QCD=100MeV and assuming that nucleosynthesis and mντallows less than0.3extra neutrino families.This is very restrictive bound for tauneutrinos since,basing on primordial nucleosynthesis arguments,there seems to beno window of opportunity for a sufficiently stable(τν>∼102sec)tau neutrino in the MeV region because of the production of non-equilibrium electron neutrinos inντ¯ντannihilations[9].The actual cosmological density of the right-helicity neutrinos depends not onlyon the production rate near the QCD phase transition,but also on whether the right-helicity neutrinos had a chance to equilibrate at some point during the course of theevolution of the universe.In this paper we wish to point out that at T<∼100GeV there is an enhancement of theν+production rate due to the electroweak gauge boson poles.1As a consequence there is a temperature range in which the production rate can exceed the expansion rate of the universe and the right-helicity neutrinos may be brought into thermal equilibrium with other light particles.This will take place if the neutrino mass is sufficiently large.We show that for the tau neutrino the mass limit is about10keV,and of the same order of magnitude for the muon neutrino.At the temperature T=1MeV the energy density of right-helicity tau neutrinos with a mass of10keV is found to be about6%of the energy density of an ordinary left-helicity neutrinos,i.e.their contribution to the effective number of neutrino species is ∆Nν≃0.06.The contribution of right-helicity neutrinos with a smaller mass depends, apart from the mass,on their initial energy density above the electroweak scale.The paper is organized as follows.In Section2we will list all tree-level spin-flipreactions where right-helicity neutrinos are produced and discuss their relative im-portance.We will also demonstrate the W-pole effect by considering the reactionsu1The question of the cosmic abundance of the right-handed component of Dirac neutrinos waspreviously studied in[10].There the pole effect was not considered.neutrino mass squared.Another source for the spin-flip is neutrino magnetic moment[7],which appears in the Standard Modelfirst at one-loop level and is,therefore,toosmall to be of any significance for our considerations.We shall consider light neutrinos with mass less than0.2MeV at temperaturesbetween1MeV and100GeV.Since each right-helicity neutrino introduces in thematrix element a factor mν/|pν|,we may safely neglect processes in which more than one right-helicity neutrino is involved.All relevant processes are listed(up to crossing)in Table1.There are68purely fermionic2→2processes in which a right-helicity muon ortau neutrino can be produced.In addition,a right-helicity tau neutrino can also beproduced in11lepton and quark three-body decays,and the muon neutrino in anotherset of11three-body decays.Since we are especially interested in interactions occuringaround the poles of weak gauge bosons,we have to consider also processes involvingW±,Z and H.There are16such processes.Finally,there are3two-body decaysof W±,Z and H bosons which are capable producing right-helicity muon and tauneutrinos.However,as we will show below,the processes involving gauge or Higgsbosons can be neglected in comparision with the purely fermionic processes.In what follows,for definiteness,we will consider only processes including right-helicity tau neutrinos.Since we are mainly interested on processes at temperaturesabove the muon and tau lepton masses,we expect that the results below are roughlyvalid also for muon neutrinos.2.2Production ratesLet us consider a2→2scattering a+b→ν++d where one of thefinal state particlesis a right-helicity neutrino.To estimate the relative importance of various processeswe approximate the thermally averaged production rate per oneν+by1Γ+=ΠL(ω,k=0),which we approximate by M2i(T)≃M2i+0.1T2(i=W,Z).(This modification is necessary in t-channel propagators only,since s-channel propagators are not infrared sensitive.)In this approximation external particles or interaction vertices do not receive thermal corrections.Thus the dispersion relations in the thermal distributions in Eq.(1)remain unchanged.Admittedly,this is a simplistic approach, but for our purposes,and for the desired accuracies,this should be sufficient.In fact, in the region of interest the effects due to thermal masses turn out to be very small.In addition to thermal corrections,we must account for the imaginary parts of the gauge boson propagators,or the widths.This is particularly important for the s-channel.Thermal corrections will generate additional imaginary parts both in the s-channel and t-channel,but at T<∼100GeV they may safely be neglected.A technical detail worth pointing out is that for afixed helicity,the spin-flip matrix elements are not Lorentz-invariant since the direction of the spin picks out a preferred frame of reference,as was emphasized in[6].Indeed,a Lorentz boost changes the helicity of the particle,so that sometimes afixed-helicity reaction forbidden in the CM may actually take place in another frame.This means that it is not sufficient to compute matrix elements just in e.g.the CM-frame,but instead one should a use a general frame.The main purpose of this paper is to show that interactions at the weak boson pole may bring the right-helicity neutrinos to thermal equilibrium.To demonstrate this effect,let us consider the the t-channel reactionτ−u→ν+d and its crossed s-channel reaction u√M2)¯uνγµ(1−γ5)uτ¯u dγν(1−γ5)u u,(2)Wwhere q=pτ−pν,V ud is the appropriate CKM matrix element,and as in s-channel, the propagator gives rise to a term of the formM2WR W(q2)=M2W m2u(pν·p d)−m2d(pν·p u) (Kν·pτ)− (pτ·pν)−m2τ m2u(Kν·p d)−m2d(Kν·p u)−m2ν m2u(pτ·p d)−m2d(pτ·p u) ,T qq=16m ,E|p|.(6)In the ultra-relativistic limit Kλcan be approximated asKλ≃2pλfor h=−1,(7)Kλ≃m2production rateΓ+of the s-channel process is seen to exceed in a certain temperature interval the expansion rate of the universe,given by the Hubble parameterH≡˙R3M2P l1/2.(10)Hence,with this reaction alone a20keV right-helicity tau(and muon)neutrino wouldbe brought into thermal equilibrium while universe cools through this stage.Thecomplete analysis described in the next section,which is based on solving the Boltz-mann equation with all the relevant processes included,confirms this expectation.Thepole effects are indeed important for an estimate of the relic density of right-helicityneutrinos.In addition to fermionic2→2scattering processes,at high T a potential sourceforν+’s are2→2processes that involve the gauge or the Higgs bosons in thefinalor initial state(we dub such processes“bosonic”).Let us consider as an examplethe processτ−γ→ν+W−,where the photon couples either to the charged lepton(Compton scattering)or to the W-boson in a three-boson vertex.For comparision,the thermally averaged rate of this process is also displayed in Fig. 1.One can seethat it can be neglected even at temperatures around the pole,because there theproduction rate of the s-channel purely fermionic process is about three orders ofmagnitude higher.We have not considered all possible bosonic processes.It is however very plausiblethat generically among the2→2scattering processes only the purely fermionic pro-cesses are important,and among these,s-channel dominates over t-channel because ofthe pole in the s-channel.In what follows,we will always disregard bosonic processes.The importance of decays is less straightforward to discern.Numerical inspectionreveals that the two-body decays listed in the Table1can be neglected:their contribu-tion to the total rate is∼O(10−11)at T>∼10GeV,at lower temperatures their contri-bution vanishes exponentially.Three-body decays are more important.Wefind thatwhile the total contribution from three-body decays is negligible at higher momenta,|p+|/T>∼3,it is as large as few per cents for|p+|/T∼3at T<∼1GeV,increasing up to∼30%for very small momenta.For such small momenta alsoν+production by t→bτ+ν+,which has its maximal contribution(∼10%)around T∼30GeV,is important.3Evolution of the right-helicity neutrino density 3.1The Boltzmann equationOur goal is to estimate the relic abundance of the right-helicity neutrinosν+as a func-tion of time.Of particular interest is the contribution of the right-helicity neutrinosto the effective number of neutrinos,∆Nν,at the onset of primeval nucleosynthesis.Tofind∆Nνone has to determinefirst the evolution of the phase space distributionof the right-helicity neutrinos f+(|p+|,t)2.Then using a primordial nucleosynthesis code one computes the increase in the primordial4He abundance∆Y p resulting fromthe non-zero energy densities ofν+and240T n↔p,(11) where the subscript n↔p refers to the freeze-out of the reactions which transmute protons and neutrons into each other.The justification of this approximation comes from the fact that Y p is predominantly determined by the density of neutrons just after the n↔p freeze-out.Let us briefly describe the method we have applied for solving the evolution of theright-helicity neutrino density.The evolution of the distribution f+(|p+|,t)is governed by the Boltzmann equation∂∂|p+|f+= ∂f+∂R−|p+|∂H ∂f+R0,(16)where R0is arbitrary butfixed.Then Eq.(14)may be written asR ∂H ∂˜f+∂R coll=(C2→2+C1→3)(1−f+)−(C′2→2+C′1→3)f+,(19)where the coefficients C I represent production and C′I destruction ofν+,with I=2→2,1→3.The explicit expressions for the quantities C2→2and C1→3areC2→2(|p+|,R)= scatt12E+ dΠf dΠg dΠh(2π)4δ(4)(p f−p g−p+−p h)×S|M f↔g+h|2f FD f(1−f FD g)(1−f FD h),(20) where,as before,dΠi≡d3p i/((2π)32E i)and S is a symmetry factor taking into account identical particles in the initial and/orfinal states.Note that these expressions do not include the factors g i representing the number of spin degrees of freedom.According to our convention these factors are included already into the matrix elements squared since we sum over polarization states of all particles except the right-helicity neutrino under consideration.We have also used the well justified assumption that charged leptons,quarks and left-helicity neutrinos have thermal FD distributions.The expressions for C′I can be obtained from(20)by the replacementf FDi↔(1−f FD i).(21) Using(20)and(21)one easilyfinds that the primed and unprimed coefficients are related to each other byC′I=e E+/T C I,(22) where we have assumed that all the particles except the right-helicity neutrino are at a common temperature T and the chemical potentials of all the particles can be neglected.With Eq.(22)we are able to rewrite(19)in the more compact form,(23)∂f+f FD+=[exp(E+/T)+1]−1.where f FD+Let us introduce the total production rates per unit volume:Γ2→2(R)= scatt dΠa dΠb dΠ+dΠd(2π)4δ(4)(p a+p b−p+−p d)×S|M ab↔+d|2f FD a f FD b(1−f FD d),Γ1→3(R)= dec dΠf dΠg dΠ+dΠh(2π)4δ(4)(p f−p g−p+−p h)×S|M f↔g+h|2f FD f(1−f FD g)(1−f FD h).(24) Note that the expressions given above do not include the blocking factors(1−f+). Also,the dependence ofΓI’s on R arise through the distribution functions of ambientparing Eqs.(20)and(24)one easily sees thatΓI(R)= d3p+d(|p+|3/(6π2))ΓI(R).(26) From this expression we see that the quantities C I,which are of the dimension of mass,are the production rates ofν+’s per unit volume per unit interval of|p+|3/(6π2) around|p+|.Eq.(26)suggests also a general method for calculating C I’s.Onefirst generates by a Monte Carlo(MC)method a sample of unweighted events,together with their common weight,such that the total integral over the initial andfinal state phase space yieldsΓI.Having this sample,the evaluation of the derivative in(26)essentially reduces to building the histogram the number of unweighted events vs.|p+|3/(6π2). Although this method is very general(applicable both to scatterings and decays)and with a clear physical meaning,it yields a good accuracy only if one generates the initial sample of weighted events such that there are more weighted events in the regions of the phase space where the integrand inΓI is larger.This requires some knowledge of the behaviour of the integrand and an ability to build a MC event generator with the required distribution of events in the phase space.In the case of2→2scatterings there exists also another method of calculation of C2→2.This method follows directly from(20)and is based on the T-invariance of interactions allowing to rewrite C2→2in a form whereν+is an initial state particle. After this transformation the calculation of the corresponding integral proceeds in a usual way.In order to check our results we have determined C2→2with both methods.3.3ResultsThe total rates C(|p+|,R)=C2→2(|p+|,R)+C1→3(|p+|,R)for the right-helicity tau neutrino production for a number offixed momenta are shown in Fig.2.One can clearly see the effect of the s-channel pole,indicating that the rate is dominated by fermionic s-channel processes.Small momentum states pass through the pole at high temperatures,large at low temperatures.Another feature is the increase of the total rate for very small momentumν+’s at0.2GeV<∼T<∼2GeV.This enhancement is due to both scatterings(u s→ν+τ+)and the decays of the tau lepton.Since at these temperatures the thermally averaged center of mass energy of colliding particles is of the same order of magnitude as the mass of the tau lepton,the neutrinos(andτ’s) are preferentially produced in the small-momentum states,for which the probability of the creation of a right-helicity neutrino(spin-flip)is larger.For similar reasons thecontribution ofτ-decays to the total rate can be as large as∼30%.For right-helicity muon neutrinos one expects the analogous mechanisms to be effective at temperatures below100MeV.Because different momentum states have different interaction rates,this causes a distortion of the momentum distribution ofν+’s relative to the equilibrium distribu-tion.This effect is demonstrated in Fig.3.Here we have parametrised the distribution function ofν+forfixed R,f+(|p+|),introducing a momentum-dependent effective tem-perature T eff(|p+|)through f+(|p+|)=[exp(E+/T eff(|p+|))+1]−1.In Fig.3the ratioT eff/T for a tau neutrino of mass10keV is shown at T=30,3and0.3GeV for twoextreme initial conditions at a high temperature T=100GeV:(i)f+=f−=f FD+, plete equilibrium,and(ii)f+=0.One sees that only the lowest momentum right-helicity tau neutrinos of this mass will come into full equilibrium at the pole: T effdepends on the initial condition even at T=0.3GeV for|p+|/T>∼1.The rise of T effat the small-momentum end of the spectrum is because these states,interact-ing more strongly,are kept in good thermal contact with ambient matter even when higher momentum states are(already)decoupled.This kind of the distortion of the distribution function becomes more prominent as the mass of the neutrino increases. As mentioned before,the regime of free expansion reveals itself in the shape-preserving evolution of the distribution function.Another remarkable feature is the rise of T effat higher momenta clearly seen for the curve corresponding to T=30GeV and the zero initial condition f+=0.This rise is there despite the fact the total production rate C(|p+|,R)is a monotonically(at least for higher momenta)dereasing function of|p+|.The explanation can be found from Eq.(23):if f+/f FD+is much smaller than unity, the destruction ofν+’s can be neglected.The regime of well-out-of-equilibrium pro-duction ofν+is effectively maintained only while both the interaction rate C(|p+|,R)and the ratio f+/f FD+are sufficiently small.These conditions are more easily fulfilled by the higher momentum neutrinos at the beginning of their evolution from the zero initial condition.This out-of-equilibrium production results in the distribution func-tion decreasing with|p+|more slowly than the equilibrium distribution function,or, equivalently,an T effrising with|p+|.In contrast with the above discussed rise of T effat lower momenta,the later feature becomes more apparent as the mass of the neutrino decreses.Wefind that spectral distortion changes the energy desity ofν+’s (which were in equilibrium at temperatures above the weak interaction pole)at∼1 MeV typically by only a few percent.The major difference between the actual energy density and the naive estimateρR≃0.044ρL is due to the relatively late decoupling caused by the pole.Once the initial condition isfixed,we may follow the evolution of the energy density ofν+.In Fig.4we show the evolution of the right-helicity neutrino energy density formντ=1,6and20keV for two different initial conditions.If mντ=1keV,spin-flipinteractions are too weak to affect the evolution,and the resulting relic abundance at nucleosynthesis is,to a high accuracy,what one would naively expect.However,if =20keV,spin-flip interactions at the pole are strong enough to equilibrateν+ mντeven if initially f+=0.The right-helicity neutrino energy density at about nucleosynthesis time in units of left-helicity neutrino energy density is shown in Fig.5.We see that almost full>∼10keV. equilibration ofν+is obtained at some temperature below100GeV if mντWe have assumed here that there is no entropy production at QCD phase transition so that the only effect is the dilution of the right-helicity neutrino densities by the appropriate ratios of the effective degrees of freedom before and after the phase tran-sition.We have computed∆Nνassuming also that below T QCD all interactions can<∼30 be ignored.While this is a very good approximation for smaller masses(mντkeV),it has been found[6,8]that for higher masses out-of-equilibrium scatterings and the decayπ0→ν+References[1]For a recent review,see Subir Sarkar,Oxford preprint OUT-95-16P,hep-ph/9602260.[2]G.Boyd et al.,Phys.Rev.Lett.75(1995)4169.[3]K.Enqvist,J.Ignatius,K.Kajantie and K.Rummukainen,Phys.Rev.D45(1992)3415.[4]G.M.Fuller and R.A.Malaney Phys.Rev.D43(1991)3136;K.Enqvist and H.Uibo,Phys.Lett.B301(1993)376.[5]K.J.F.Gaemers,R.Gandhi and ttimer,Phys.Rev.D40(1989)309.[6]A.D.Dolgov,K.Kainulainen and I.Z.Rothstein,Phys.Rev.51(1995)4129.[7]K.Fujikawa and R.Schrock,Phys.Rev.Lett.45(1980)963.[8]B.D.Fields,K.Kainulainen and K.A.Olive,CERN-TH/95-335,UMN-TH-1417/95,hep-ph/9512321(1995).[9]A.D.Dolgov,S.Pastor and J.W.F.Valle,preprint FTUV/98-07,IFIC/96-08,hep-ph/9602233(1996).[10]S.L.Shapiro,S.A.Teukolsky and I.Wasserman,Phys.Rev.Lett.45(1980)689.[11]D.Buskulic et al.(ALEPH collaboration),Phys.Lett.B349(1995)585;K.As-samagan et al.,Phys.Lett.B335(1994)231.Table1:Processes,up to crossings,which produce a right-helicity tau neutrinoν+.fermionic2→23−body decaysντντ→ν+νττ−→ν+l−j W+→τ+ν+(l−jνe,µ−ντνj→ν+νj,(j=e,µ)τ−→ν+d n Z→ν+u m=d u,s c)ντH→ν+H H→ν+ντf ch→ν+f ch,(f ch=τ−)u mτ+ν+,u m=u)τ−H→ν+W−τ−γ→ν+W−Figure captionsFigure1.Thermally averaged production rates per one right-helicity tau neutrinoν+ of mass20keV for the s-channel process u¯d→ν+τ+,the t-channel processτ−u→ν+d and the bosonic processτ−γ→ν+W−.The Hubble expansion rate is also shown for comparision.Figure 2.The total production rates(dashed curves)C(|p+|,R)of right-helicity tau neutrinos as functions of the temperature T.From top to bottom,the rates are given for˜p+/100GeV=0.33,1,2,4,6,8and10.For comparison the evolution of the Hubble parameter H is also given(solid curve).Figure3.The ratio of the momentum-dependent effective temperature T effof the right-helicity tau neutrinos of mass10keV to the plasma temperature T at T= 30GeV,3GeV and0.3GeV.In each case the solid curve corresponds to the equilibrium initial condition f+=f FD,and the dashed curve to the zero initial condition f+=0+at T=100GeV.Figure4.The evolution ofρ+/ρFD+of tau neutrinos as a function of the temperature =20keV(solid curve),6keV(dot-dashed curve)and1keV(dashed curve). T for mντand f+=0at The evolution is given for two different initial conditions:f+=f FD+T=100GeV.Figure5.The energy density of right-helicity tau neutrinosν+in units of the effective number of two-component neutrino species,∆Nν,as a function of the neutrino mass.∆Nνis calculated at T=3MeV(solid curves),1MeV(dashed curves)and0.3MeV<20keV the two sets of curves differ by initial conditions (dot-dashed curves).For mντat T=100GeV.T (GeV)Figure 110-2310-2210-2110-2010-1910-1810-1710-1610-1510-1410-13R a t e (G e V )0110100T (GeV)Figure 210-3010-2510-2010-1510-10H , C (p ,R ) (G e V )+Hubble parameterproduction rates0.0 2.0 4.06.08.010.0p /T +0.60.81.0T /T eff3 GeV30 GeV0.3 GeVFigure 30110100T (GeV)Figure 40.00.20.40.60.81.0ρ /ρFD ++20 keV6 keV 1 keV1101001000m (keV)ντFigure 50.000.020.040.060.08N ∆νT=3 MeV T=1 MeV T=0.3 MeV。

BSHM Bulletin, 2014Did Weierstrass's differential calculus have a limit-avoiding character? His definition of a limit in£ —5 styleMICHIYO NAKANENihon University Research Institute of Science & Technology, JapanIn the 1820s, Cauchy founded his calculus on his original limit concept and developed his the-ory by using £—S inequalities, but he did not apply these inequalities consistently to all parts of his theory. In contrast, Weierstrass consistently developed his 1861 lectures on differential calculus in terms of epsilonics. His lectures were not based on Cauchy's limit and are distin-guished by their limit-avoiding character. Dugac's partial publication of the 1861 lectures makes these differences clear. But in the unpublishedportions of the lectures, Weierstrass actu-ally defined his limit in terms of £—Sinequalities.Weierstrass's limit was a prototype of the modern limit but did not serve as a foundation of his calculus theory. For this reason, he did not provide the basic structure for the modern e d style analysis. Thus it was Dini's 1878 text-book that introduced the definition of a limit in terms of £—B inequalities.IntroductionAugustin Louis Cauchy and Karl Weierstrass were two of the most important mathematicians associated with the formalization of analysis on the basis of the e d doctrine. In the 1820s, Cauchy was the first to give comprehensive statements of mathematical analysis that were based from the outset on a reasonably clear definition of the limit concept (Edwards 1979, 310). He introduced various definitions and theories that involved his limit concept. His expressions were mainly verbal, but they could be understood in terms of inequalities: given an e, find n or d (Grabiner 1981, 7). As we show later, Cauchy actually paraphrased his limit concept in terms of e, d, and n0 inequalities, in his more complicated proofs. But it was Weierstrass's 1861 lectures which used the technique in all proofs and also in his defi-nition (Lutzen€ 2003, 185-186).Weierstrass's adoption of full epsilonic arguments, however, did not mean that he attained a prototype of the modern theory. Modern analysis theory is founded on limits defined in terms of e d inequalities. His lectures were not founded on Cauchy's limit or his own original definition of limit (Dugac 1973). Therefore, in order to clarify the formation of the modern theory, it will be necessary to identify where the e d definition of limit was introduced and used as a foundation.We do not find the word ‘limit' in the published part of the 1861 lectures. Accordingly, Grattan-Guinness (1986, 228) characterizes Weierstrass's analysis as limit- avoid-ing. However, Weierstrass actually defined his limit in terms of epsilonics in the unpublished portion of his lectures. His theory involved his limit concept, although the concept did not function as the foundation of his theory. Based on this discovery, this paper re-examines the formation of e d calculus theory, noting mathematicians' treat-ments of their limits. We restrict our attention to the process of defining continuity and derivatives. Nonetheless, this focus provides sufficient information for our purposes.First, we confirm that epsilonics arguments cannot represent Cauchy's limit, though they can describe relationships that involved his limit concept. Next, we examine how Weierstrass constructed a novel analysis theory which was not based2013 British Society for the History of Mathematics52 BSHM Bulletinon Cauchy's limits but could have involved Cauchy's results. Then we confirm Weierstrass's definition of limit. Finally, we note that Dini organized his analysis textbook in 1878 based on analysis performed in the e d style.Cauchy's limit and epsilonic arguments Cauchy's series of textbooks on calculus, Cours d'analyse (1821), Resume des lecons, donnees a l'Ecole royale polytechnique sur le calcul infinitesimal tome premier (1823), and Lecons, sur le calcul differentiel (1829), are often considered as the main referen-ces for modern analysis theory, the rigour of which is rooted more in the nineteenth than the twentiethcentury.At the beginning of his Cours d'analyse, Cauchy defined the limit concept as fol-lows: ‘When the successively attributed values of the same variable indefinitely approach a fixed value, so that finally they differ from it by as little as desired, the last is called the limit of all the others' (1821, 19; English translation from Grabiner 1981, 80). Starting from this concept, Cauchy developed a theory of continuous func-tions, infinite series, derivatives, and integrals, constructing an analysis based on lim-its (Grabiner 1981, 77).When discussing the evolution of the limit concept, Grabiner writes: ‘This con-cept, translated into thealgebra of inequalities, was exactly what Cauchy needed for his calculus' (1981, 80). From the present-day point of view, Cauchy described rather than defined his kinetic concept of limits. According to his‘definition'—which has the quality of a translation or description—he could develop any aspect of the theory by reducing it to the algebra of inequalities.Next, Cauchy introduced infinitely small quantities into his theory. ’When the successive absolute values of a variable decrease indefinitely, in such a way as to become less than any given quantity, that variable becomes what is called an infinitesimal. Such a variable has zero for its limit' (1821, 19; English translation from Birkhoff and Merzbach 1973, 2). That is to say, in Cauchy's framework 'the limit of variable x is c' is intuitively understood as 'x indefinitely approaches c', and is represented as ’jx cj is as little as desired' or 'jx cj is infinitesimal'. Cauchy's idea of defining infinitesimals as variables of a special kind was original, because Leibniz and Euler, for example, had treated them as constants (Boyer 1989, 575; Lutzen€ 2003, 164).In Cours d'analyse Cauchy at first gave a verbal definition of a continuous func-tion. Then, he rewrote it in terms of infinitesimals:[In other words,] the function f dxD will remain continuous relative to x in a given interval if (in this interval) an infinitesimal increment in the variable always produces an infinitesimal increment in the function itself. (1821, 43; English translation from Birkhoff and Merzbach 1973, 2).He introduced the infinitesimal-involving definition and adopted a modified version of it in Resume (1823, 19-20) and Lecons, (1829, 278).Following Cauchy's definition of infinitesimals, a continuous function can be defined as a function f dxD in which ‘the variable f dx » aD f dxD is an infinitely small quantity (as previously defined) whenever the variable a is, that is, that f dx » aD f dxD approaches to zero as a does', as noted by Edwards (1979, 311). Thus, the definition can be translated into the language of e d inequalities from a modern viewpoint. Cauchy's infinitesimals are variables, and we can also take such an interpretation.Volume 29 (2014) 53Cauchy himself translated his limit concept in terms of e d inequalities. He changed ‘If the difference f dx » 1D f dxD converges towards a certain limit k, for increasing values of x, (. . .)' to ‘First suppose that the quantity k has a finite value, and denote by e a number as small as we wish. . . . we can give the numberh a value large enough that, when x is equal to or greater than h, the difference in question is always contained between the limits k e; k » e' (1821, 54; English translation from Bradley and Sandifer 2009, 35).In Resume , Cauchy gave a definition of a derivative: ‘if f dxD is continuous, then its derivative is the limit of the difference quotient,△y = f(x + i) - f(x) 国ias i tends to 0' (1823, 22-23). He also translated the concept of derivative as follows: ‘Designate by d and e two very small numbers; the first being chosen insuch a way that, for numerical values of i less than d, [. . .], the ratio f dx » iD f 6xD=i always remains greater than f 'dx D e and less than f 'dxD » e' (1823, 44-45; English transla-tion from Grabiner 1981, 115).These examples show that Cauchy noted that relationships involving limits or infinitesimals could be rewritten in term of inequalities. Cauchy's arguments about infinite series in Cours d'analyse, which dealt with the relationship between increasing numbers and infinitesimals, had such a character. Laugwitz (1987, 264; 1999, 58) and Lutzen€ (2003, 167) have noted Cauchy's strict use of the e N characterization of convergence in several of his proofs. Borovick and Katz (2012) indicate that there is room to question whether or not our representation using e d inequalities conveys messages different from Cauchy's original intention. But this paper accepts the inter-pretations of Edwards, Laugwitz, and Lutzen€.Cauchy's lectures mainly discussed properties of series and functions in the limit process, which were represented as relationships between his limits or his infinitesimals, or between increasing numbers and infinitesimals. His contemporaries presum-ably recognized the possibility of developing analysis theory in terms of only e, d, and n0 inequalities. With a few notable exceptions, all of Cauchys lectures could be rewrit-ten in terms of e d inequalities. Cauchy's limits and his infinitesimals were not func-tional relationships,1 so they were not representable in terms of e d inequalities.Cauchy's limit concept was the foundation of his theory. Thus, Weierstrass's full epsilonic analysis theory has a different foundation from that of Cauchy.Weierstrass’s 1861 lecturesWeierstrass's consistent use of e d argumentsWeierstrass delivered his lectures ‘On the differential calculus' at the Gewerbe Insti- tut Berlin2 in the summer semester of 1861. Notes of these lectures were taken by lEdwards (1979, 310), Laugwitz (1987, 260-261, 271-272), and Fisher (1978, 16318) point out that Cauchy's infinitesimals equate to a dependent variable function or adhP that approaches zero as h ! 0. Cauchy adopted the latter infinitesimals, which can be written in terms of e d arguments, when he intro-duced a concept of degree of infinitesimals (1823, 250; 1829, 325). Every infinitesimal of Cauchys is a vari-able in the parts that the present paper discusses.2A forerunner of the Technische Universit€at Berlin.54 BSHM BulletinHerman Amandus Schwarz, and some of them have been published in the original German by Dugac (1973). Noting the new aspects related to foundational concepts in analysis, full e d definitions of limit and continuous function, a new definition of derivative, and a modern definition of infinitesimals, Dugac considered that the nov-elty of Weierstrass's lectures was incontestable (1978, 372, 1976, 6-7).3After beginning his lectures by defining a variable magnitude, Weierstrass gave the definition of a function using the notion of correspondence. This brought him to the following important definition, which did not directly appear in Cauchy's theory:(D1) If it is now possible to determine for h a bound d such that for all values of h which in their absolute value are smaller than d, f dx » hD f dxD becomes smaller than any magnitude e, however small, then one says that infinitely small changes of the argument correspond to infinitely small changes of the function. (Dugac 1973, 119; English translation from Calinger 1995, 607)That is, Weierstrass defined not infinitely small changes of variables but ‘infinitely small changes of the arguments correspond(ing) to infinitely small changes of function' that were presented in terms of e d inequalities. He founded his theory on this correspondence.Using this concept, he defined a continuous function as follows: (D2) If now a function is such that to infinitely small changes of the argument there correspond infinitely small changes of the function, one then says that it is a continuous function of the argument, or that it changes continuously with this argument. (Dugac 1973, 119-120; English translation from Calinger 1995, 607)So we see that in accordance with his definition of correspondence, Weierstrass actually defined a continuous function on an interval in terms of epsilonics. Since (D2) is derived by merely changing Cauchy's term ‘produce' to ‘correspond', it seems that Weierstrass took the idea of this definition from Cauchy. However, Weierstrass's definition was given in terms of epsilonics, while Cauchy's definition can only be interpreted in these terms. Furthermore, Weierstrass achieved it without Cauchy's limit.Luzten€ (2003, 186) indicates that Weierstrass still used the concept of ‘infinitely small' in his lectures. Until giving his definition of derivative, Weierstrass actually continued to use the term'infinitesimally small' and often wrote of 'a function which becomes infinitely small with h'. But several instances of 'infinitesimally small' appeared in forms of the relationships involving them. Definition (D1) gives the rela-tionship in terms of e d inequalities. We may therefore assume that Weierstrass's lectures consistently used e d inequalities, even though his definitions were not directly written in terms of these inequalities.Weierstrass inserted sentences confirming that the relationships involving the term 'infinitely small' were defined in terms of e d inequalities as follows:(D3) If h denotes a magnitude which can assume infinitely small values, 'dhP is an arbitrary function of h with the property that for an infinitely small value of h it3The present paper also quotes Kurt Bing's translation included in Calinger's Classics of mathematics. Volume 29 (2014) 55also becomes infinitely small (that is, that always, as soon as a definite arbitrary small magnitude e is chosen, a magnitude d can be determined such that for all values of h whose absolute value is smaller than d, 'dhD becomes smaller than e). (Dugac 1973, 120; English translation from Calinger, 1995, 607)As Dugac (1973, 65) indicates, some modern textbooks describe 'dhP as infinitely small or infinitesimal. Weierstrass argued that the whole change of function can in general be decomposed asDf dxD % f dx » hD f dxD % p:h » hdhP; d 1Dwhere the factor p is independent of h and dh D is a magnitude that becomes infinitely small with h.4 However, he overlooked that such decomposition is not possible for all functions and inserted the term 'in general'. He rewrote h as dx. One can make the difference between Df dxD and p:dx smaller than any magnitude with decreasing dx. Hence Weierstrass defined 'differential' as the change which a function undergoes when its argument changes by an infinitesimally small magnitude and denoted it as df dxD. Then, df dxD % p:dx. Weierstrass pointed out that the differential coefficient p is a function of x derived from f dxD and called it a derivative (Dugac 1973, 120-121; English translation from Calinger 1995, 607-608). In accordance with Weierstrass's definitions (D1) and (D3), he largely defined a derivative in terms of epsilonics.Weierstrass did not adopt the term 'infinitely small' but directly used e d inequalities when he discussed properties of infinite series involving uniform conver-gence (Dugac 1973, 122-124). It may be inferred from the published portion of his notes that Cauchy's limit has no place in Weierstrass's lectures. Grattan- Guinness's (1986, 228) description of the limit-avoiding character of his analysis represents this situation well.However, Weierstrass thought that his theory included most of the content of Cauchy's theory. Cauchy first gave the definition of limits of variables and infinitesimals. Then, he demonstrated notions and theorems that were written in terms of the relationships involving infinitesimals. From Weierstrass's viewpoint, they were writ-ten in terms of e d inequalities. Analytical theory mainly examines properties of functions and series, which were described in the relationships involving Cauchy s limits and infinitesimals. Weierstrass recognized this fact and had the idea of consis-tently developing his theory in terms of inequalities. Hence Weierstrass at first defined the relationships among infinitesimals in terms of e d inequalities. In accor-dance with this definition, Weierstrass rewrote Cauchy's results and naturally imported them into his own theory. This is a process that may be described as follows: ‘Weierstrass completed the transformation away from the use of terms such as “infinitely small”, (Katz 1998, 728).Weierstrass's definition of limitDugac (1978, 370-372; 1976, 6-7) read (D1) as the first definition of limit with the help of e d. But (D1)does not involve an endpoint that variables or functions4Dugac (1973, 65) indicated that dhP corresponds to the modern notion of od1D. In addition, hdhP corre-sponds to the function that was introduced as 'dhP in the former quotation from Weierstrass's sentences.。

重要哲学术语英汉对照——转载自《当代英美哲学概论》a priori瞐 posteriori distinction 先验-后验的区分abstract ideas 抽象理念abstract objects 抽象客体ad hominem argument 谬误论证alienation/estrangement 异化，疏离altruism 利他主义analysis 分析analytic瞫ynthetic distinction 分析-综合的区分aporia 困惑argument from design 来自设计的论证artificial intelligence (AI) 人工智能association of ideas 理念的联想autonomy 自律axioms 公理Categorical Imperative 绝对命令categories 范畴Category mistake 范畴错误causal theory of reference 指称的因果论causation 因果关系certainty 确定性chaos theory 混沌理论class 总纲、类clearness and distinctness 清楚与明晰cogito ergo sum 我思故我在concept 概念consciousness 意识consent 同意consequentialism 效果论conservative 保守的consistency 一致性，相容性constructivism 建构主义contents of consciousness 意识的内容contingent瞡ecessary distinction 偶然-必然的区分 continuum 连续体continuum hypothesis 连续性假说contradiction 矛盾（律）conventionalism 约定论counterfactual conditional 反事实的条件句criterion 准则，标准critique 批判，批评Dasein 此在，定在deconstruction 解构主义defeasible 可以废除的definite description 限定摹状词deontology 义务论dialectic 辩证法didactic 说教的dualism 二元论egoism 自我主义、利己主义eliminative materialism 消除性的唯物主义 empiricism 经验主义Enlightenment 启蒙运动（思想）entailment 蕴含essence 本质ethical intuition 伦理直观ethical naturalism 伦理的自然主义eudaimonia 幸福主义event 事件、事变evolutionary epistemology 进化认识论expert system 专门体系explanation 解释fallibilism 谬误论family resemblance 家族相似fictional entities 虚构的实体first philosophy 第一哲学form of life 生活形式formal 形式的foundationalism 基础主义free will and determinism 自由意志和决定论 function 函项（功能）function explanation 功能解释good 善happiness 幸福hedonism 享乐主义hermeneutics 解释学（诠释学，释义学）historicism 历史论（历史主义）holism 整体论iconographic 绘画idealism 理念论ideas 理念identity 同一性illocutionary act 以言行事的行为imagination 想象力immaterical substance 非物质实体immutable 不变的、永恒的individualism 个人主义（个体主义）induction 归纳inference 推断infinite regress 无限回归intensionality 内涵性intentionality 意向性irreducible 不可还原的Leibniz餾 Law 莱布尼茨法则logical atomism 逻辑原子主义logical positivism 逻辑实证主义logomachy 玩弄词藻的争论material biconditional 物质的双向制约materialism 唯物论（唯物主义）maxim 箴言，格言method 方法methodologica 方法论的model 样式modern 现代的modus ponens and modus tollens 肯定前件和否定后件 natural selection 自然选择necessary 必然的neutral monism 中立一无论nominalism 唯名论non睧uclidean geometry 非欧几里德几何non瞞onotonic logics 非单一逻辑Ockham餜azor 奥卡姆剃刀omnipotence and omniscience 全能和全知ontology 本体论（存有学）operator 算符(或算子)paradox 悖论perception 知觉phenomenology 现象学picture theory of meaning 意义的图像说pluralism 多元论polis 城邦possible world 可能世界postmodernism 后现代主义prescriptive statement 规定性陈述presupposition 预设primary and secondary qualities 第一性的质和第二性的质 principle of non瞔ontradiction 不矛盾律proposition 命题quantifier 量词quantum mechanics 量子力学rational numbers 有理数real number 实数realism 实在论reason 理性，理智recursive function 循环函数reflective equilibrium 反思的均衡relativity (theory of) 相对（论）rights 权利rigid designator严格的指称词Rorschach test 相对性（相对论）rule 规则rule utilitarianism 功利主义规则Russell餾 paradox 罗素悖论sanctions 制发scope 范围，限界semantics 语义学sense data 感觉材料，感觉资料set 集solipsism 唯我论social contract 社会契约subjective瞣bjective distinction 主客区分 sublation 扬弃substance 实体，本体sui generis 特殊的，独特性supervenience 偶然性syllogism 三段论things瞚n瞭hemselves 物自体thought 思想thought experiment 思想实验three瞯alued logic 三值逻辑transcendental 先验的truth 真理truth function 真值函项understanding 理解universals 共相，一般，普遍verfication principle 证实原则versimilitude 逼真性vicious regress 恶性回归Vienna Circle 维也纳学派virtue 美德注释计量经济学中英对照词汇(continuous)2007年8月23日，22:02:47 | mindreader计量经济学中英对照词汇(continuous)K-Means Cluster逐步聚类分析K means method, 逐步聚类法Kaplan-Meier, 评估事件的时间长度Kaplan-Merier chart, Kaplan-Merier图Kendall's rank correlation, Kendall等级相关Kinetic, 动力学Kolmogorov-Smirnove test, 柯尔莫哥洛夫-斯米尔诺夫检验Kruskal and Wallis test, Kruskal及Wallis检验/多样本的秩和检验/H检验Kurtosis, 峰度Lack of fit, 失拟Ladder of powers, 幂阶梯Lag, 滞后Large sample, 大样本Large sample test, 大样本检验Latin square, 拉丁方Latin square design, 拉丁方设计Leakage, 泄漏Least favorable configuration, 最不利构形Least favorable distribution, 最不利分布Least significant difference, 最小显著差法Least square method, 最小二乘法Least Squared Criterion，最小二乘方准则Least-absolute-residuals estimates, 最小绝对残差估计Least-absolute-residuals fit, 最小绝对残差拟合Least-absolute-residuals line, 最小绝对残差线Legend, 图例L-estimator, L估计量L-estimator of location, 位置L估计量L-estimator of scale, 尺度L估计量Level, 水平Leveage Correction，杠杆率校正Life expectance, 预期期望寿命Life table, 寿命表Life table method, 生命表法Light-tailed distribution, 轻尾分布Likelihood function, 似然函数Likelihood ratio, 似然比line graph, 线图Linear correlation, 直线相关Linear equation, 线性方程Linear programming, 线性规划Linear regression, 直线回归Linear Regression, 线性回归Linear trend, 线性趋势Loading, 载荷Location and scale equivariance, 位置尺度同变性Location equivariance, 位置同变性Location invariance, 位置不变性Location scale family, 位置尺度族Log rank test, 时序检验Logarithmic curve, 对数曲线Logarithmic normal distribution, 对数正态分布Logarithmic scale, 对数尺度Logarithmic transformation, 对数变换Logic check, 逻辑检查Logistic distribution, 逻辑斯特分布Logit transformation, Logit转换LOGLINEAR, 多维列联表通用模型Lognormal distribution, 对数正态分布Lost function, 损失函数Low correlation, 低度相关Lower limit, 下限Lowest-attained variance, 最小可达方差LSD, 最小显著差法的简称Lurking variable, 潜在变量Main effect, 主效应Major heading, 主辞标目Marginal density function, 边缘密度函数Marginal probability, 边缘概率Marginal probability distribution, 边缘概率分布Matched data, 配对资料Matched distribution, 匹配过分布Matching of distribution, 分布的匹配Matching of transformation, 变换的匹配Mathematical expectation, 数学期望Mathematical model, 数学模型Maximum L-estimator, 极大极小L 估计量Maximum likelihood method, 最大似然法Mean, 均数Mean squares between groups, 组间均方Mean squares within group, 组内均方Means (Compare means), 均值-均值比较Median, 中位数Median effective dose, 半数效量Median lethal dose, 半数致死量Median polish, 中位数平滑Median test, 中位数检验Minimal sufficient statistic, 最小充分统计量Minimum distance estimation, 最小距离估计Minimum effective dose, 最小有效量Minimum lethal dose, 最小致死量Minimum variance estimator, 最小方差估计量MINITAB, 统计软件包Minor heading, 宾词标目Missing data, 缺失值Model specification, 模型的确定Modeling Statistics , 模型统计Models for outliers, 离群值模型Modifying the model, 模型的修正Modulus of continuity, 连续性模Morbidity, 发病率Most favorable configuration, 最有利构形MSC（多元散射校正）Multidimensional Scaling (ASCAL), 多维尺度/多维标度Multinomial Logistic Regression , 多项逻辑斯蒂回归Multiple comparison, 多重比较Multiple correlation , 复相关Multiple covariance, 多元协方差Multiple linear regression, 多元线性回归Multiple response , 多重选项Multiple solutions, 多解Multiplication theorem, 乘法定理Multiresponse, 多元响应Multi-stage sampling, 多阶段抽样Multivariate T distribution, 多元T分布Mutual exclusive, 互不相容Mutual independence, 互相独立Natural boundary, 自然边界Natural dead, 自然死亡Natural zero, 自然零Negative correlation, 负相关Negative linear correlation, 负线性相关Negatively skewed, 负偏Newman-Keuls method, q检验NK method, q检验No statistical significance, 无统计意义Nominal variable, 名义变量Nonconstancy of variability, 变异的非定常性Nonlinear regression, 非线性相关Nonparametric statistics, 非参数统计Nonparametric test, 非参数检验Nonparametric tests, 非参数检验Normal deviate, 正态离差Normal distribution, 正态分布Normal equation, 正规方程组Normal P-P, 正态概率分布图Normal Q-Q, 正态概率单位分布图Normal ranges, 正常范围Normal value, 正常值Normalization 归一化Nuisance parameter, 多余参数/讨厌参数Null hypothesis, 无效假设Numerical variable, 数值变量Objective function, 目标函数Observation unit, 观察单位Observed value, 观察值One sided test, 单侧检验One-way analysis of variance, 单因素方差分析Oneway ANOVA , 单因素方差分析Open sequential trial, 开放型序贯设计Optrim, 优切尾Optrim efficiency, 优切尾效率Order statistics, 顺序统计量Ordered categories, 有序分类Ordinal logistic regression , 序数逻辑斯蒂回归Ordinal variable, 有序变量Orthogonal basis, 正交基Orthogonal design, 正交试验设计Orthogonality conditions, 正交条件ORTHOPLAN, 正交设计Outlier cutoffs, 离群值截断点Outliers, 极端值OVERALS , 多组变量的非线性正规相关Overshoot, 迭代过度Paired design, 配对设计Paired sample, 配对样本Pairwise slopes, 成对斜率Parabola, 抛物线Parallel tests, 平行试验Parameter, 参数Parametric statistics, 参数统计Parametric test, 参数检验Pareto, 直条构成线图（又称佩尔托图）Partial correlation, 偏相关Partial regression, 偏回归Partial sorting, 偏排序Partials residuals, 偏残差Pattern, 模式PCA（主成分分析）Pearson curves, 皮尔逊曲线Peeling, 退层Percent bar graph, 百分条形图Percentage, 百分比Percentile, 百分位数Percentile curves, 百分位曲线Periodicity, 周期性Permutation, 排列P-estimator, P估计量Pie graph, 构成图,饼图Pitman estimator, 皮特曼估计量Pivot, 枢轴量Planar, 平坦Planar assumption, 平面的假设PLANCARDS, 生成试验的计划卡PLS（偏最小二乘法）Point estimation, 点估计Poisson distribution, 泊松分布Polishing, 平滑Polled standard deviation, 合并标准差Polled variance, 合并方差Polygon, 多边图Polynomial, 多项式Polynomial curve, 多项式曲线Population, 总体Population attributable risk, 人群归因危险度Positive correlation, 正相关Positively skewed, 正偏Posterior distribution, 后验分布Power of a test, 检验效能Precision, 精密度Predicted value, 预测值Preliminary analysis, 预备性分析Principal axis factoring,主轴因子法Principal component analysis, 主成分分析Prior distribution, 先验分布Prior probability, 先验概率Probabilistic model, 概率模型probability, 概率Probability density, 概率密度Product moment, 乘积矩/协方差Pro, 截面迹图Proportion, 比/构成比Proportion allocation in stratified random sampling, 按比例分层随机抽样Proportionate, 成比例Proportionate sub-class numbers, 成比例次级组含量Prospective study, 前瞻性调查Proximities, 亲近性Pseudo F test, 近似F检验Pseudo model, 近似模型Pseudosigma, 伪标准差Purposive sampling, 有目的抽样QR decomposition, QR分解Quadratic approximation, 二次近似Qualitative classification, 属性分类Qualitative method, 定性方法Quantile-quantile plot, 分位数-分位数图/Q-Q图Quantitative analysis, 定量分析Quartile, 四分位数Quick Cluster, 快速聚类Radix sort, 基数排序Random allocation, 随机化分组Random blocks design, 随机区组设计Random event, 随机事件Randomization, 随机化Range, 极差/全距Rank correlation, 等级相关Rank sum test, 秩和检验Rank test, 秩检验Ranked data, 等级资料Rate, 比率Ratio, 比例Raw data, 原始资料Raw residual, 原始残差Rayleigh's test, 雷氏检验Rayleigh's Z, 雷氏Z值Reciprocal, 倒数Reciprocal transformation, 倒数变换Recording, 记录Redescending estimators, 回降估计量Reducing dimensions, 降维Re-expression, 重新表达Reference set, 标准组Region of acceptance, 接受域Regression coefficient, 回归系数Regression sum of square, 回归平方和Rejection point, 拒绝点Relative dispersion, 相对离散度Relative number, 相对数Reliability, 可靠性Reparametrization, 重新设置参数Replication, 重复Report Summaries, 报告摘要Residual sum of square, 剩余平方和residual variance (剩余方差)Resistance, 耐抗性Resistant line, 耐抗线Resistant technique, 耐抗技术R-estimator of location, 位置R估计量R-estimator of scale, 尺度R估计量Retrospective study, 回顾性调查Ridge trace, 岭迹Ridit analysis, Ridit分析Rotation, 旋转Rounding, 舍入Row, 行Row effects, 行效应Row factor, 行因素RXC table, RXC表Sample, 样本Sample regression coefficient, 样本回归系数Sample size, 样本量Sample standard deviation, 样本标准差Sampling error, 抽样误差SAS(Statistical analysis system , SAS统计软件包Scale, 尺度/量表Scatter diagram, 散点图Schematic plot, 示意图/简图Score test, 计分检验Screening, 筛检SEASON, 季节分析Second derivative, 二阶导数Second principal component, 第二主成分SEM (Structural equation modeling), 结构化方程模型Semi-logarithmic graph, 半对数图Semi-logarithmic paper, 半对数格纸Sensitivity curve, 敏感度曲线Sequential analysis, 贯序分析Sequence, 普通序列图Sequential data set, 顺序数据集Sequential design, 贯序设计Sequential method, 贯序法Sequential test, 贯序检验法Serial tests, 系列试验Short-cut method, 简捷法Sigmoid curve, S形曲线Sign function, 正负号函数Sign test, 符号检验Signed rank, 符号秩Significant Level, 显著水平Significance test, 显著性检验Significant figure, 有效数字Simple cluster sampling, 简单整群抽样Simple correlation, 简单相关Simple random sampling, 简单随机抽样Simple regression, 简单回归simple table, 简单表Sine estimator, 正弦估计量Single-valued estimate, 单值估计Singular matrix, 奇异矩阵Skewed distribution, 偏斜分布Skewness, 偏度Slash distribution, 斜线分布Slope, 斜率Smirnov test, 斯米尔诺夫检验Source of variation, 变异来源Spearman rank correlation, 斯皮尔曼等级相关Specific factor, 特殊因子Specific factor variance, 特殊因子方差Spectra , 频谱Spherical distribution, 球型正态分布Spread, 展布SPSS(Statistical package for the social science), SPSS统计软件包Spurious correlation, 假性相关Square root transformation, 平方根变换Stabilizing variance, 稳定方差Standard deviation, 标准差Standard error, 标准误Standard error of difference, 差别的标准误Standard error of estimate, 标准估计误差Standard error of rate, 率的标准误Standard normal distribution, 标准正态分布Standardization, 标准化Starting value, 起始值Statistic, 统计量Statistical control, 统计控制Statistical graph, 统计图Statistical inference, 统计推断Statistical table, 统计表Steepest descent, 最速下降法Stem and leaf display, 茎叶图Step factor, 步长因子Stepwise regression, 逐步回归Storage, 存Strata, 层（复数）Stratified sampling, 分层抽样Stratified sampling, 分层抽样Strength, 强度Stringency, 严密性Structural relationship, 结构关系Studentized residual, 学生化残差/t化残差Sub-class numbers, 次级组含量Subdividing, 分割Sufficient statistic, 充分统计量Sum of products, 积和Sum of squares, 离差平方和Sum of squares about regression, 回归平方和Sum of squares between groups, 组间平方和Sum of squares of partial regression, 偏回归平方和Sure event, 必然事件Survey, 调查Survival, 生存分析Survival rate, 生存率Suspended root gram, 悬吊根图Symmetry, 对称Systematic error, 系统误差Systematic sampling, 系统抽样Tags, 标签Tail area, 尾部面积Tail length, 尾长Tail weight, 尾重Tangent line, 切线Target distribution, 目标分布Taylor series, 泰勒级数Test(检验)Test of linearity, 线性检验Tendency of dispersion, 离散趋势Testing of hypotheses, 假设检验Theoretical frequency, 理论频数Time series, 时间序列Tolerance interval, 容忍区间Tolerance lower limit, 容忍下限Tolerance upper limit, 容忍上限Torsion, 扰率Total sum of square, 总平方和Total variation, 总变异Transformation, 转换Treatment, 处理Trend, 趋势Trend of percentage, 百分比趋势Trial, 试验Trial and error method, 试错法Tuning constant, 细调常数Two sided test, 双向检验Two-stage least squares, 二阶最小平方Two-stage sampling, 二阶段抽样Two-tailed test, 双侧检验Two-way analysis of variance, 双因素方差分析Two-way table, 双向表Type I error, 一类错误/α错误Type II error, 二类错误/β错误UMVU, 方差一致最小无偏估计简称Unbiased estimate, 无偏估计Unconstrained nonlinear regression , 无约束非线性回归Unequal subclass number, 不等次级组含量Ungrouped data, 不分组资料Uniform coordinate, 均匀坐标Uniform distribution, 均匀分布Uniformly minimum variance unbiased estimate, 方差一致最小无偏估计Unit, 单元Unordered categories, 无序分类Unweighted least squares, 未加权最小平方法Upper limit, 上限Upward rank, 升秩Vague concept, 模糊概念Validity, 有效性VARCOMP (Variance component estimation), 方差元素估计Variability, 变异性Variable, 变量Variance, 方差Variation, 变异Varimax orthogonal rotation, 方差最大正交旋转Volume of distribution, 容积W test, W检验Weibull distribution, 威布尔分布Weight, 权数Weighted Chi-square test, 加权卡方检验/Cochran检验Weighted linear regression method, 加权直线回归Weighted mean, 加权平均数Weighted mean square, 加权平均方差Weighted sum of square, 加权平方和Weighting coefficient, 权重系数Weighting method, 加权法W-estimation, W估计量W-estimation of location, 位置W估计量Width, 宽度Wilcoxon paired test, 威斯康星配对法/配对符号秩和检验Wild point, 野点/狂点Wild value, 野值/狂值Winsorized mean, 缩尾均值Withdraw, 失访Youden's index, 尤登指数Z test, Z检验Zero correlation, 零相关Z-transformation, Z变换注释。

a r X i v :h e p -p h /0010077v 1 9 O c t 2000Mass Spectrum and the Nature of Neutrinos.M.Czakon,J.Gluza,J.Studnik and M.Zra l ek Department of Field Theory and Particle Physics,Institute of Physics,University of Silesia,Uniwersytecka 4,PL-40-007Katowice,Poland Taking as input the best fit solar neutrino anomaly description,MSW LMA,and the tritium beta decay results we estimate the allowed range of neutrino masses independently of their nature.Adding the present bound on the effective neutrino mass coming from neutrinoless double beta decay,we narrow this range for Majorana neutrinos.We complete the discussion by considering future perspectives on determining the neutrino masses,when the oscillation data will be improved and the next experiments on (ββ)0νand 3H decay give new bounds or obtain concrete life-times or distortions in the energy distribution.We know much more about neutrino masses than yet a few years ago.The observed anoma-lies in atmospheric,solar and possibly the LSND neutrino experiments,which we believe are explained by neutrino oscillations,supplied with the tritium beta decay data give hints on neutrino masses independently of whether they are Dirac or Majorana particles.Additional constraints on Majorana neutrino masses come from the fact that no neutrinoless double beta decay has been observed to this day.In this work we present an up to date analysis and future perspectives of finding the neutrino mass spectrum without any constraints from theoretical models.We consider only the three neutrino case (i.e.without considering the LSND anomaly),and the latest best fit solar neutrino problem solution,the MSW LMA 1.The oscillation param-eters inferred from atmospheric and solar data are given in Table 1.The four neutrino case and other currently acceptable solutions of the solar anomaly are considered elsewhere 3.As there are definitely two scales of δm 2,δm 2atm ≫δm 2sol ,two possible neutrino mass spectra must be con-sidered.The first,known as normal mass hierarchy (A 3)where δm 2sol =δm 221≪δm 232≈δm 2atm and the second,inverse mass hierarchy spectrum (A inv 3)with δm 2sol =δm 221≪δm 2atm ≈−δm 231.Both schemes are not distinguishable by present experiments.There is hope that future neutrino factories will do that 4.Two elements of the first row of the mixing matrix |U e 1|and |U e 2|can be expressed by theTable1:The allowed range(95%of CL)and the bestfit values of sin22θandδm2for the atmospheric neutrino oscillation and the bestfit MSW LMA solution of the solar neutrino problem.Allowed range Bestfitδm2[eV2]sin22θsolar(1.5−6)×10−30.84−1Solar neutrinos(MSW LMA)18×10−50.66third element|U e3|and the sin22θsolar|U e1|2=(1−|U e3|2)11−sin22θsolar),(1)and|U e2|2=(1−|U e3|2)11−sin22θsolar).(2)The value of the third element|U e3|is notfixed yet and only different bounds exist for it.We will take the bound directly inferred from the CHOOZ and SK experiments5|U e3|2<0.04(with95%of CL).(3) Since in both schemes there is(mν)2max=(mν)2min+δm2solar+δm2atm,(4) the oscillation experiments alone give(mν)max≥δm2solar+δm2atm.(6) Translating the above into numbers(again at95%CL)2we end up with(mν)max≥0.04eV,|m i−m j|<0.08eV.(7) The next important data comes from the tritium beta decay experiments.The following bound has been lately obtained63i=1|U ei|2m2i 1/2≡mβ<κ′=2.2eV(8) this obviously leads only to the double inequality(mν)min≤mβ≤(mν)max.(9) Therefore0≤(mν)min≤2.2eV.(10) (mν)max remains unfortunately unlimited from above.Supplying the tritium decay with oscil-lations wefind that7m2β=(mν)2min+Ωscheme,(11) and(mν)2max=m2β+Λscheme,(12)whereΩandΛare scheme dependent.For example,in the A3schemeΩ(A3)=(1−|U e1|2)δm2solar+|U e3|2δm2atm,(13) andΛ(A3)=|U e1|2δm2solar+(1−|U e3|2)δm2atm.(14) This provides limits for both(mν)min and(mν)max0≤(mν)min≤+δm2atm≤(mν)max≤δm2solarFigure2:of(mν)min in the case of the A3shaded and hashed regions represent the are taken into account.The present and band is an example of a0.05)eV.The observed10(19) There are future plans to go down to| mν |≃0.02eV or even to| mν |≃0.006eV11.Do we have a chance offinding the Majorana mass spectrum if a value of| mν |is found within such a small range12?This answer as we will see is not very promising.We shall neglect the difficulties connected with the determination of| mν |from the half life time of germanium13.As the phases of U ei remain unknown,we are not in position to predict the value of| mν |.However, the lower| mν |min and upper| mν |max ranges as function of(mν)min can be inferred14.They are shown in Fig.2for the A3scheme and for the MSW LMA solar neutrino problem solution. The shaded and hashed regions give the uncertainties connected with the allowed ranges of the input parameters(sin22θsolar,δm2atm(Table1)and|U e3|2(Eq.3).Future better knowledge of these parameters will reduce the uncertainty regions shown in Fig.2,but the min-max range caused by the unknown CP phases will remain.The present experimental bound on| mν |(Eq.19)gives the following limit on the possible (mν)min for Majorana neutrinos(mν)min<0.86eV.(20) This bound strongly depends on the unknown oscillation parameters,most notably on sin22θsolar. In Fig.3we plot this dependence for two different sets ofδm2atm and|U e3|2values.The limit given in Eq.20is valid for sin22θsolar=0.92,|U e3|2=0.04andδm2atm=6×10−3eV2.If in future,the(ββ)0νexperiments observe no decay,and a new bound is only found,the next better limit that can be derived from Fig.3(with the present oscillation results),is(mν)min<0.092eV GENIUS I,(21) and(mν)min<0.037eV GENIUS II.(22)Figure3:ofδm2atm and|U e3|2. In the we can try to predict the value of| mν | and on the of| mν |values is given of values allowed by oscillations(mν)min(ββ)0νmin ≤(mν)min≤(mν)max(ββ)0νmin.(23)With the present day uncertainties on the oscillation parameters,the range of possible values determined by Eq.23is not satisfactorily small.For example,with| mν |≃0.05eV(mν)min∈(0.03−0.6)eV.(24)For smaller values of| mν |we can only say that(mν)min<0.2eV.A better knowledge of the oscillation parameters changes the situation slightly.For example,if the oscillation parameters are known with negligible error bars for| mν |≃0.05eV,then the range Eq.24changes to(mν)min∈(0.04−0.1)eV.(25)The ignorance of the CP breaking phases in the mixing matrix is fully responsible for this smearing.The bounds on the effective neutrino mass| mν |in the inverse hierarchy mass scheme A inv3 and the MSW LMA solution of the solar neutrino problem are depicted in Fig.4.We see that the present bound on| mν |(Eq.19),gives a similar limit on the possible range of(mν)min of Majorana neutrino masses(mν)min<0.86eV.(26) Thefirst stage of GENIUS can yield(mν)min<0.077eV,(27)while the second would exclude the A inv3scheme.In conclusion,the present data allow for the following statementsFigure4:of(mν)min in the case ofthe A inv3and hashed regions represent the taken into account.The•we•the but the latter depends strongly on the oscillation parameters.•the oscillation and tritium beta decay experiments are able to determine the spectrum ofneutrino masses for values of mβwhich differ in the A3(mβ≥0.04eV)and the A inv3 (mβ≥0.2eV)schemes.•the oscillation and(ββ)0νexperiments are able tofind the range of possible(mν)min values.However,this range is not small even with oscillation parameters of negligible error bars. 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