On Neutrino Masses and Mixings from Extra Dimensions
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First District Association, based in Litchfield, Minnesota, U.S., produces a wide variety of cheeses. Quality is of the utmost importance for First District and, when the time came to renovate and remodel their testing labs, they chose the leader in quality instrumentation – METTLER TOLEDO.Upholding High StandardsFirst District Association is an independent dairy cooperative that maximizes returns for its pro-ducers and employees through innovation and providing progressive quality products to a glob-al market. They play a prominent role in the dairy industry and have a very distinct and proud history which has influenced the birth and formation of modern dairy cooperatives.First District produces a wide variety of cheese for use in all types of applications. They pro-duce 500 lb barrels and 40 lb blocks as well as whey protein concentrate (WPC) and lactose. Cheese produced by First District is used in processed cheese products, shredded cheese, and First District AssociationMoisture testing Ash testingQuality Cheese Guaranteed by Top-Quality Instruments Laboratory Case StudyFood & BeverageXP BalanceMettler-Toledo AGLaboratory DivisionIm LangacherCH-8606 Greifensee, Switzerland Subject to technical changes.© Mettler-Toledo AG 07/11Printed in SwitzerlandGlobal MarCom Switzerland cheese sauces and powders. The new testing labs required updated equipment to help speed up produc-tion. First District was also interested in a way to move from manual result transcription to an electronic format. Reliable Moisture & Ash Control First District prides itself on product quality. Each batch of WPC and lactose is lab-tested and must fall into a specific range of moisture and ash in order to meet the es-tablished standard. The moisture testing process begins by taking a small sample from each batch and weighing it. The sample is then dried out in an oven and weighed again. Determining the weight dif-ference in the sample before and after the drying process provides a simple calculation for the amount of moisture present. The ash testingprocess measures the amount of ash remaining in the sample after heating it in a muffle furnace. The remaining ash is weighed to ensure it falls within a standard range. First District chose the Excellence Plus (XP) series of analytical balances for both of these tests. They chose the XP for its speed and for the flex-ibility of the grid-style weighing pan for easy cleaning.Correct pH Level Ensures Quality First District also tests the pH levels of each batch of cheese. They grind the cheese samples and pack them into sample cups. During pH testing, a combination electrode is inserted into the cheese. The S40 pH meter then automatically reads the data and prepares it for output. Determin-ing the correct pH level ensures the quality of flavor, body and texture of each batch of cheese. To complete the package, First Dis-trict purchased METTLER TOLEDO’s LabX Direct pH and Balance soft-ware. This software suite will be connected through a LIMS system as the remaining lab renovations are completed. LabX software eliminates the need to manually note each quality test and the connection to the LIMS system will send the results directly into a host computer which reduces the likelihood of transcrip-tion errors. First District now has an up-to-date solution for their quality assurance testing. Production is moving faster and the new equipment ensures the quality and taste of each batch of cheese produced. The addition of the LIMS system and software will further enhance and streamline the lab completing the entire testing package. /xp-analytical /pHlab/LabX Office building of First District Association.S40 pH Meter.。
a r X i v :0803.2889v 2 [h e p -p h ] 14 J u l 2008Mapping Out SU (5)GUTs with Non-Abelian Discrete Flavor SymmetriesFlorian Plentinger ∗and Gerhart Seidl †Institut f¨u r Physik und Astrophysik,Universit¨a t W¨u rzburg,Am Hubland,D 97074W¨u rzburg,Germany(Dated:December 25,2013)We construct a class of supersymmetric SU (5)GUT models that produce nearly tribimaximal lepton mixing,the observed quark mixing matrix,and the quark and lepton masses,from discrete non-Abelian flavor symmetries.The SU (5)GUTs are formulated on five-dimensional throats in the flat limit and the neutrino masses become small due to the type-I seesaw mechanism.The discrete non-Abelian flavor symmetries are given by semi-direct products of cyclic groups that are broken at the infrared branes at the tip of the throats.As a result,we obtain SU (5)GUTs that provide a combined description of non-Abelian flavor symmetries and quark-lepton complementarity.PACS numbers:12.15.Ff,11.30.Hv,12.10.Dm,One possibility to explore the physics of grand unified theories (GUTs)[1,2]at low energies is to analyze the neutrino sector.This is due to the explanation of small neutrino masses via the seesaw mechanism [3,4],which is naturally incorporated in GUTs.In fact,from the perspective of quark-lepton unification,it is interesting to study in GUTs the drastic differences between the masses and mixings of quarks and leptons as revealed by current neutrino oscillation data.In recent years,there have been many attempts to re-produce a tribimaximal mixing form [5]for the leptonic Pontecorvo-Maki-Nakagawa-Sakata (PMNS)[6]mixing matrix U PMNS using non-Abelian discrete flavor symme-tries such as the tetrahedral [7]and double (or binary)tetrahedral [8]groupA 4≃Z 3⋉(Z 2×Z 2)and T ′≃Z 2⋉Q,(1)where Q is the quaternion group of order eight,or [9]∆(27)≃Z 3⋉(Z 3×Z 3),(2)which is a subgroup of SU (3)(for reviews see, e.g.,Ref.[10]).Existing models,however,have generally dif-ficulties to predict also the observed fermion mass hierar-chies as well as the Cabibbo-Kobayashi-Maskawa (CKM)quark mixing matrix V CKM [11],which applies especially to GUTs (for very recent examples,see Ref.[12]).An-other approach,on the other hand,is offered by the idea of quark-lepton complementarity (QLC),where the so-lar neutrino angle is a combination of maximal mixing and the Cabibbo angle θC [13].Subsequently,this has,in an interpretation of QLC [14,15],led to a machine-aided survey of several thousand lepton flavor models for nearly tribimaximal lepton mixing [16].Here,we investigate the embedding of the models found in Ref.[16]into five-dimensional (5D)supersym-metric (SUSY)SU (5)GUTs.The hierarchical pattern of quark and lepton masses,V CKM ,and nearly tribi-maximal lepton mixing,arise from the local breaking of non-Abelian discrete flavor symmetries in the extra-dimensional geometry.This has the advantage that theFIG.1:SUSY SU (5)GUT on two 5D intervals or throats.The zero modes of the matter fields 10i ,5H,24H ,and the gauge supermul-tiplet,propagate freely in the two throats.scalar sector of these models is extremely simple without the need for a vacuum alignment mechanism,while of-fering an intuitive geometrical interpretation of the non-Abelian flavor symmetries.As a consequence,we obtain,for the first time,a realization of non-Abelian flavor sym-metries and QLC in SU (5)GUTs.We will describe our models by considering a specific minimal realization as an example.The main features of this example model,however,should be viewed as generic and representative for a large class of possible realiza-tions.Our model is given by a SUSY SU (5)GUT in 5D flat space,which is defined on two 5D intervals that have been glued together at a common endpoint.The geom-etry and the location of the 5D hypermultiplets in the model is depicted in FIG.1.The two intervals consti-tute a simple example for a two-throat setup in the flat limit (see,e.g.,Refs.[17,18]),where the two 5D inter-vals,or throats,have the lengths πR 1and πR 2,and the coordinates y 1∈[0,πR 1]and y 2∈[0,πR 2].The point at y 1=y 2=0is called ultraviolet (UV)brane,whereas the two endpoints at y 1=πR 1and y 2=πR 2will be referred to as infrared (IR)branes.The throats are supposed to be GUT-scale sized,i.e.1/R 1,2 M GUT ≃1016GeV,and the SU (5)gauge supermultiplet and the Higgs hy-permultiplets 5H and2neously broken to G SM by a 24H bulk Higgs hypermulti-plet propagating in the two throats that acquires a vac-uum expectation value pointing in the hypercharge direc-tion 24H ∝diag(−12,13,15i ,where i =1,2,3is the generation index.Toobtainsmall neutrino masses via the type-I seesaw mechanism [3],we introduce three right-handed SU (5)singlet neutrino superfields 1i .The 5D Lagrangian for the Yukawa couplings of the zero mode fermions then readsL 5D =d 2θ δ(y 1−πR 1) ˜Y uij,R 110i 10j 5H +˜Y d ij,R 110i 5H +˜Y νij,R 15j5i 1j 5H +M R ˜Y R ij,R 21i 1j+h.c. ,(3)where ˜Y x ij,R 1and ˜Y x ij,R 2(x =u,d,ν,R )are Yukawa cou-pling matrices (with mass dimension −1/2)and M R ≃1014GeV is the B −L breaking scale.In the four-dimensional (4D)low energy effective theory,L 5D gives rise to the 4D Yukawa couplingsL 4D =d 2θ Y u ij 10i 10j 5H +Y dij10i 5H +Y νij5i ∼(q i 1,q i 2,...,q i m ),(5)1i ∼(r i 1,r i 2,...,r im ),where the j th entry in each row vector denotes the Z n jcharge of the representation.In the 5D theory,we sup-pose that the group G A is spontaneously broken by singly charged flavon fields located at the IR branes.The Yukawa coupling matrices of quarks and leptons are then generated by the Froggatt-Nielsen mechanism [21].Applying a straightforward generalization of the flavor group space scan in Ref.[16]to the SU (5)×G A represen-tations in Eq.(5),we find a large number of about 4×102flavor models that produce the hierarchies of quark and lepton masses and yield the CKM and PMNS mixing angles in perfect agreement with current data.A distri-bution of these models as a function of the group G A for increasing group order is shown in FIG.2.The selection criteria for the flavor models are as follows:First,all models have to be consistent with the quark and charged3 lepton mass ratiosm u:m c:m t=ǫ6:ǫ4:1,m d:m s:m b=ǫ4:ǫ2:1,(6)m e:mµ:mτ=ǫ4:ǫ2:1,and a normal hierarchical neutrino mass spectrumm1:m2:m3=ǫ2:ǫ:1,(7)whereǫ≃θC≃0.2is of the order of the Cabibbo angle.Second,each model has to reproduce the CKM anglesV us∼ǫ,V cb∼ǫ2,V ub∼ǫ3,(8)as well as nearly tribimaximal lepton mixing at3σCLwith an extremely small reactor angle 1◦.In perform-ing the group space scan,we have restricted ourselves togroups G A with orders roughly up to 102and FIG.2shows only groups admitting more than three valid mod-els.In FIG.2,we can observe the general trend thatwith increasing group order the number of valid modelsper group generally increases too.This rough observa-tion,however,is modified by a large“periodic”fluctu-ation of the number of models,which possibly singlesout certain groups G A as particularly interesting.Thehighly populated groups would deserve further system-atic investigation,which is,however,beyond the scopeof this paper.From this large set of models,let us choose the groupG A=Z3×Z8×Z9and,in the notation of Eq.(5),thecharge assignment101∼(1,1,6),102∼(0,3,1),103∼(0,0,0),52∼(0,7,0),52↔4FIG.3:Effect of the non-Abelian flavor symmetry on θ23for a 10%variation of all Yukawa couplings.Shown is θ23as a function of ǫfor the flavor group G A (left)and G A ⋉G B (right).The right plot illustrates the exact prediction of the zeroth order term π/4in the expansion θ23=π/4+ǫ/√2and the relation θ13≃ǫ2.The important point is that in the expression for θ23,the leading order term π/4is exactly predicted by thenon-Abelian flavor symmetry G F =G A ⋉G B (see FIG.3),while θ13≃θ2C is extremely small due to a suppression by the square of the Cabibbo angle.We thus predict a devi-ation ∼ǫ/√2,which is the well-known QLC relation for the solar angle.There have been attempts in the literature to reproduce QLC in quark-lepton unified models [26],however,the model presented here is the first realization of QLC in an SU (5)GUT.Although our analysis has been carried out for the CP conserving case,a simple numerical study shows that CP violating phases (cf.Ref.[27])relevant for neutri-noless double beta decay and leptogenesis can be easily included as well.Concerning proton decay,note that since SU (5)is bro-ken by a bulk Higgs field,the broken gauge boson masses are ≃M GUT .Therefore,all fermion zero modes can be localized at the IR branes of the throats without intro-ducing rapid proton decay through d =6operators.To achieve doublet-triplet splitting and suppress d =5pro-ton decay,we may then,e.g.,resort to suitable extensions of the Higgs sector [28].Moreover,although the flavor symmetry G F is global,quantum gravity effects might require G F to be gauged [29].Anomalies can then be canceled by Chern-Simons terms in the 5D bulk.We emphasize that the above discussion is focussed on a specific minimal example realization of the model.Many SU (5)GUTs with non-Abelian flavor symmetries,however,can be constructed along the same lines by varying the flavor charge assignment,choosing different groups G F ,or by modifying the throat geometry.A de-tailed analysis of these models and variations thereof will be presented in a future publication [30].To summarize,we have discussed the construction of 5D SUSY SU (5)GUTs that yield nearly tribimaximal lepton mixing,as well as the observed CKM mixing matrix,together with the hierarchy of quark and lepton masses.Small neutrino masses are generated only by the type-I seesaw mechanism.The fermion masses and mixings arise from the local breaking of non-Abelian flavor symmetries at the IR branes of a flat multi-throat geometry.For an example realization,we have shown that the non-Abelian flavor symmetries can exactly predict the leading order term π/4in the sum rule for the atmospheric mixing angle,while strongly suppress-ing the reactor angle.This makes this class of models testable in future neutrino oscillation experiments.In addition,we arrive,for the first time,at a combined description of QLC and non-Abelian flavor symmetries in SU (5)GUTs.One main advantage of our setup with throats is that the necessary symmetry breaking can be realized with a very simple Higgs sector and that it can be applied to and generalized for a large class of unified models.We would like to thank T.Ohl for useful comments.The research of F.P.is supported by Research Train-ing Group 1147“Theoretical Astrophysics and Particle Physics ”of Deutsche Forschungsgemeinschaft.G.S.is supported by the Federal Ministry of Education and Re-search (BMBF)under contract number 05HT6WWA.∗********************************.de †**************************.de[1]H.Georgi and S.L.Glashow,Phys.Rev.Lett.32,438(1974);H.Georgi,in Proceedings of Coral Gables 1975,Theories and Experiments in High Energy Physics ,New York,1975.[2]J.C.Pati and A.Salam,Phys.Rev.D 10,275(1974)[Erratum-ibid.D 11,703(1975)].[3]P.Minkowski,Phys.Lett.B 67,421(1977);T.Yanagida,in Proceedings of the Workshop on the Unified Theory and Baryon Number in the Universe ,KEK,Tsukuba,1979;M.Gell-Mann,P.Ramond and R.Slansky,in Pro-ceedings of the Workshop on Supergravity ,Stony Brook,5New York,1979;S.L.Glashow,in Proceedings of the 1979Cargese Summer Institute on Quarks and Leptons, New York,1980.[4]M.Magg and C.Wetterich,Phys.Lett.B94,61(1980);R.N.Mohapatra and G.Senjanovi´c,Phys.Rev.Lett.44, 912(1980);Phys.Rev.D23,165(1981);J.Schechter and J.W. F.Valle,Phys.Rev.D22,2227(1980);zarides,Q.Shafiand C.Wetterich,Nucl.Phys.B181,287(1981).[5]P.F.Harrison,D.H.Perkins and W.G.Scott,Phys.Lett.B458,79(1999);P.F.Harrison,D.H.Perkins and W.G.Scott,Phys.Lett.B530,167(2002).[6]B.Pontecorvo,Sov.Phys.JETP6,429(1957);Z.Maki,M.Nakagawa and S.Sakata,Prog.Theor.Phys.28,870 (1962).[7]E.Ma and G.Rajasekaran,Phys.Rev.D64,113012(2001);K.S.Babu,E.Ma and J.W.F.Valle,Phys.Lett.B552,207(2003);M.Hirsch et al.,Phys.Rev.D 69,093006(2004).[8]P.H.Frampton and T.W.Kephart,Int.J.Mod.Phys.A10,4689(1995); A.Aranda, C. D.Carone and R.F.Lebed,Phys.Rev.D62,016009(2000);P.D.Carr and P.H.Frampton,arXiv:hep-ph/0701034;A.Aranda, Phys.Rev.D76,111301(2007).[9]I.de Medeiros Varzielas,S.F.King and G.G.Ross,Phys.Lett.B648,201(2007);C.Luhn,S.Nasri and P.Ramond,J.Math.Phys.48,073501(2007);Phys.Lett.B652,27(2007).[10]E.Ma,arXiv:0705.0327[hep-ph];G.Altarelli,arXiv:0705.0860[hep-ph].[11]N.Cabibbo,Phys.Rev.Lett.10,531(1963);M.Kobayashi and T.Maskawa,Prog.Theor.Phys.49, 652(1973).[12]M.-C.Chen and K.T.Mahanthappa,Phys.Lett.B652,34(2007);W.Grimus and H.Kuhbock,Phys.Rev.D77, 055008(2008);F.Bazzocchi et al.,arXiv:0802.1693[hep-ph];G.Altarelli,F.Feruglio and C.Hagedorn,J.High Energy Phys.0803,052(2008).[13]A.Y.Smirnov,arXiv:hep-ph/0402264;M.Raidal,Phys.Rev.Lett.93,161801(2004);H.Minakata andA.Y.Smirnov,Phys.Rev.D70,073009(2004).[14]F.Plentinger,G.Seidl and W.Winter,Nucl.Phys.B791,60(2008).[15]F.Plentinger,G.Seidl and W.Winter,Phys.Rev.D76,113003(2007).[16]F.Plentinger,G.Seidl and W.Winter,J.High EnergyPhys.0804,077(2008).[17]G.Cacciapaglia,C.Csaki,C.Grojean and J.Terning,Phys.Rev.D74,045019(2006).[18]K.Agashe,A.Falkowski,I.Low and G.Servant,J.HighEnergy Phys.0804,027(2008);C.D.Carone,J.Erlich and M.Sher,arXiv:0802.3702[hep-ph].[19]Y.Kawamura,Prog.Theor.Phys.105,999(2001);G.Altarelli and F.Feruglio,Phys.Lett.B511,257(2001);A.B.Kobakhidze,Phys.Lett.B514,131(2001);A.Hebecker and J.March-Russell,Nucl.Phys.B613,3(2001);L.J.Hall and Y.Nomura,Phys.Rev.D66, 075004(2002).[20]D.E.Kaplan and T.M.P.Tait,J.High Energy Phys.0111,051(2001).[21]C.D.Froggatt and H.B.Nielsen,Nucl.Phys.B147,277(1979).[22]Y.Nomura,Phys.Rev.D65,085036(2002).[23]H.Georgi and C.Jarlskog,Phys.Lett.B86,297(1979).[24]H.Arason et al.,Phys.Rev.Lett.67,2933(1991);H.Arason et al.,Phys.Rev.D47,232(1993).[25]D.S.Ayres et al.[NOνA Collaboration],arXiv:hep-ex/0503053;Y.Hayato et al.,Letter of Intent.[26]S.Antusch,S.F.King and R.N.Mohapatra,Phys.Lett.B618,150(2005).[27]W.Winter,Phys.Lett.B659,275(2008).[28]K.S.Babu and S.M.Barr,Phys.Rev.D48,5354(1993);K.Kurosawa,N.Maru and T.Yanagida,Phys.Lett.B 512,203(2001).[29]L.M.Krauss and F.Wilczek,Phys.Rev.Lett.62,1221(1989).[30]F.Plentinger and G.Seidl,in preparation.。
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a r X i v :a s t r o -p h /0208563v 1 30 A u g 2002Proceedings of the 270.WE-Heraeus Seminar on:“Neutron Stars,Pulsars and Supernova Remnants”Physikzentrum Bad Honnef,Germany,Jan.21-25,2002,eds.W.Becker,H.Lesch &J.Tr¨u mper,MPE Report 278,pp.300-302Neutron Stars,Pulsars and Supernova Remnants:concluding remarksF.Pacini 1,21Arcetri Astrophysical Observatory,L.go E.Fermi,5,I-50125Firenze,Italy2Dept.of Astronomy and Space Science,University of Florence,L.go E.Fermi,2,I-50125Firenze,Italy1.IntroductionMore than 30years have elapsed since the discovery of pul-sars (Hewish et al.1968)and the realization that they are connected with rotating magnetized neutron stars (Gold 1968;Pacini 1967,1968).It became soon clear that these objects are responsible for the production of the relativis-tic wind observed in some Supernovae remnants such as the Crab Nebula.For many years,the study of pulsars has been car-ried out mostly in the radio band.However,many recent results have come from observations at much higher fre-quencies (optical,X-rays,gamma rays).These observa-tions have been decisive in order to establish a realistic demography and have brought a better understanding of the relationship between neutron stars and SN remnants.The Proceedings of this Conference cover many aspects of this relationship (see also previous Conference Proceed-ings such as Bandiera et al.1998;Slane and Gaensler,2002).Because of this reason,my summary will not re-view all the very interesting results which have been pre-sented here and I shall address briefly just a few issues.The choice of these issues is largely personal:other col-leagues may have made a different selection.2.Demography of Neutron Stars:the role of the magnetic field For a long time it has been believed that only Crab-like remnants (plerions)contain a neutron star and that the typical field strength of neutron stars is 1012Gauss.The basis of this belief was the lack of pulsars associated with shell-type remnants or other manifestations of a relativis-tic wind.The justification given is that some SN explo-sions may blow apart the entire star.Alternatively,the central object may become a black hole.However,the number of shell remnants greatly exceeds that of pleri-ons:it becomes then difficult to invoke the formation of black holes,an event much more rare than the formation of neutron stars.The suggestion that shell remnants such as Cas A could be associated with neutron stars which have rapidly lost their initial rotational energy because of an ultra-strong magnetic field B ∼1014−1015Gauss (Cavaliere &Pacini,1970)did receive little attention.The observa-tional situation has now changed:a compact thermal X-ray source has been discovered close to the center of Cas A (Tananbaum,1999)and it could be the predicted ob-ject.Similar sources have been found in association with other remnants and are likely to be neutron stars.We have also heard during this Conference that some shell-type remnants (including Cas A)show evidence for a weak non-thermal X-ray emission superimposed on the thermal one:this may indicate the presence of a residual relativis-tic wind produced in the center.Another important result has been the discovery of neutron stars with ultra-strong magnetic fields,up to 1014−1015G.In this case the total magnetic energy could be larger than the rotational en-ergy (”magnetars”).This possibility had been suggested long time ago (Woltjer,1968).It should be noticed,how-ever,that the slowing down rate determines the strength of the field at the speed of light cylinder and that the usually quoted surface fields assume a dipolar geometry corresponding to a braking index n =3.Unfortunately the value of n has been measured only in a few cases and it ranges between 1.4−2.8(Lyne et al.,1996).The present evidence indicates that neutron stars man-ifest themselves in different ways:–Classical radio pulsars (with or without emission at higher frequencies)where the rotation is the energy source.–Compact X-ray sources where the energy is supplied by accretion (products of the evolution in binary sys-tems).–Compact X-ray sources due to the residual thermal emission from a hot surface.–Anomalous X-ray pulsars (AXP)with long periods and ultra strong fields (up to 1015Gauss).The power emit-ted by AXPs exceeds the energy loss inferred from the slowing down rate.It is possible that AXPs are asso-ciated with magnetized white dwarfs,rotating close to the shortest possible period (5−10s)or,alternatively,they could be neutron stars whose magnetic energy is dissipated by flares.–Soft gamma-ray repeaters.2 F.Pacini:Neutron Stars,Pulsars and Supernova Remnants:concluding remarks In addition it is possible that some of the unidentifiedgamma ray sources are related to neutron stars.Thepresent picture solves some previous inconsistencies.Forinstance,the estimate for the rate of core-collapse Super-novae(roughly one every30-50years)was about a factorof two larger than the birth-rate of radio pulsars,suggest-ing already that a large fraction of neutron stars does not appear as radio pulsars.The observational evidence supports the notion of a large spread in the magnetic strength of neutron stars and the hypothesis that this spread is an important factor in determining the morphology of Supernova remnants.A very strongfield would lead to the release of the bulk of the rotational energy during a short initial period(say, days up to a few years):at later times the remnant would appear as a shell-type.A more moderatefield(say1012 Gauss or so)would entail a long lasting energy loss and produce a plerion.3.Where are the pulses emitted?Despite the great wealth of data available,there is no gen-eral consensus about the radiation mechanism for pulsars. The location of the region where the pulses are emitted is also controversial:it could be located close to the stellar surface or,alternatively,in the proximity of the speed of light cylinder.The radio emission is certainly due to a coherent pro-cess because of the very high brightness temperatures(T b up to and above1030K have been observed).A possible model invokes the motion of bunches of charges sliding along the curvedfield lines with a relativistic Lorentz fac-torγsuch that the critical frequencyνc∼c2π:Ψ∼10−2;B⊥∼104G;γ∼102−103.The model leads to the expectation of a very fast de-crease of the synchrotron intensity with period because of the combination of two factors:a)the reduced particles flux when the period increases;b)the reduced efficiency of synchrotron losses(which scale∝B2∝R−6L∝P−6)at the speed of light cylinder(Pacini,1971;Pacini&Salvati 1983,1987).The predictionfits the observed secular de-crease of the optical emission from the Crab Nebula and the magnitude of the Vela pulsar.A recent re-examination of all available optical data confirms that this model can account for the luminosity of the known optical pulsars (Shearer and Golden,2001).If so,the optical radiation supports strongly the notion that the emitting region is located close to the speed of light cylinder.4.A speculation:can the thermal radiation fromyoung neutron stars quench the relativisticwind?Myfinal remarks concern the possible effect of the ther-mal radiation coming from the neutron star surface upon the acceleration of particles.This problem has been inves-tigated for the near magnetosphere(Supper&Truemper, 2000)and it has been found that the Inverse Compton Scattering(ICS)against the thermal photons is impor-tant only in marginal cases.However,if we assume that the acceleration of the relativistic wind and the radiation of pulses occur close to the speed of light cylinder,the sit-uation becomes different and the ICS can dominate over synchrotron losses for a variety of parameters.The basic reason is that the importance of ICS at the speed of light distance R L scales like the energy density of the thermal photons uγ∝R−2L∝P−2;on the other hand, the synchrotron losses are proportional to the magnetic energy density in the same region u B∝R−6L∝P−6.Numerically,onefinds that ICS losses dominate over synchrotron losses ifT6>0.4B1/2121012G; P s is the pulsar period in seconds).The corresponding upper limit for the energy of the electrons,assuming that the acceleration takes place for a length of order of the speed of light distance and that the gains are equal to the losses is given by:E max≃1.2×103T6−4P s GeV.F.Pacini:Neutron Stars,Pulsars and Supernova Remnants:concluding remarks3Provided that the particles are accelerated and radi-ate in proximity of the speed of light cylinder distance,weconclude that the thermal photons can limit the acceler-ation of particles,especially in the case of young and hotneutron stars.It becomes tempting to speculate that thismay postpone the beginning of the pulsar activity untilthe temperature of the star is sufficiently low.The mainmanifestation of neutron stars in this phase would be aflux of high energy photons in the gamma-ray band,dueto the interaction of the quenched wind with the thermalphotons from the stellar surface.This model and its ob-servational consequences are currently under investigation(Amato,Blasi,Pacini,work in progress).ReferencesAloisio,R.,&Blasi,P.2002,Astrop.Phys.,Bandiera,R.,et al.1998,Proc.Workshop”The Relationshipbetween Neutron Stars and Supernova Remnants”,Mem.Societ Astronomica Italiana,vol.69,n.4Cavaliere,A.,&Pacini,F.1970,ApJ,159,170Gold,T.1968,Nature,217,731Hewish A.,et al.1968,Nature217,709Lyne,G.,et al.1996,Nature,381,497Pacini,F.1967,Nature,216,567Pacini,F.1968,Nature,219,145Pacini,F.1971,ApJ,163,L17Pacini,F.,&Salvati,M.1983,ApJ,274,369Pacini F.,&Salvati,M.1987,ApJ.,321,447Shearer,A.,and Golden,A.2001,ApJ,547,967Slane,P.,Gaensler,B.2002,Proc.Workshop”Neutron Starsin Supernova Remnants”ASP Conference Proceedings(inpress)Supper,R.,&Trumper,J.2000,A&A,357,301Tananbaum,B,et al.1999,IAU Circular7246Thompson,C.,Duncan,R.C.1996,ApJ,473,322Woltjer,L.1968,ApJ,152,179。
a rXiv:h ep-ph/9291v33Jan21An Explanation on Negative Mass-Square of Neutrinos Tsao Chang Center for Space Plasma and Aeronomy Research University of Alabama in Huntsville Huntsville,AL 35899Email:changt@ Guangjiong Ni Department of Physics,Fudan University Shanghai,200433,China Abstract:It has been known for many years that the measured mass-square of neutrino is probably negative.For solving this puzzle,we have further investigated the hypothesis that neutrinos are superlumi-nal fermions.A new Dirac-type equation is proposed and a tachyonic quantum theory is briefly discussed.This equation is equivalent to two Weyl equations coupled together via nonzero mass while respecting the maximum parity violation,and it reduces to one Weyl equation when the neutrino mass becomes zero.PACS number:14.60.Lm,14.60.Pq,14.60.St11.IntroductionThe square of the neutrino mass is measured in tritium beta decay experiments byfitting the shape of the beta spectrum near endpoint. In many experiments,it has been found to be negative.Most recent data are listed in”Review of Particle Physics,2000”[1]and references therein.The weighted average from two experiments reported in1999 [2-3]ism2(νe)=−2.5±3.3eV2(1) However,other nine measurements from different experiments in1991-1995are not used for averages.For instance,a value of m2(νe)=-130±20eV2with95%confidence level was measured in LLNL in1995[4].Furthermore,the pion decay experiment also obtained a negative value forµ-neutrinos[5].m2(νµ)=−0.016±0.023MeV2(2) The negative value of the neutrino mass-square simply means:E2/c2−p2=m2(νe)c2<0(3) The right-hand side in Eq.(3)can be rewritten as(-m2s c2),then m s has a positive value.Eq.(1)and(2)suggests that neutrinos might be particles faster than light,no matter how small the m s is.This possibility is further investigated in this paper.Based on special relativity and known as re-interpretation rule,su-perluminal particles were proposed by Bilaniuk et al in the Sixties[6-8]. The sign of4-D world line element,ds2,is associated with three classes of particles.For simplicity,let dy=dz=0,then>0ClassI(subluminal particles)ds2=c2dt2−dx2=0ClassII(photon)(4)<0ClassIII(superluminal particles) For Class III particles,i.e.superluminal particle,the relation of mo-mentum and energy is shown in Eq.(3).The negative value on the right-hand side of Eq.(3)for superluminal particles means that p2 is greater than(E/c)2.The velocity of a superluminal particle,u s,is2greater than the speed of light.The momentum and energy in terms of u s are as follows:p=m s u su2s/c2−1,E=m s c2u2s/c2−1(5)where the subscript s means superluminal particle,i.e.tachyon.From Eq.(5),it is easily seen that when u s is increased,both of p and E would be decreased.This property is opposite to Class I particle.Any physical reference system is built by Class I particles(atoms, molecules etc.),which requires that any reference frame must move slower than light.On the other hand,once a superluminal particle is created in an interaction,its speed is always greater than the speed of light.Neutrino is the most possible candidate for a superluminal particle because it has left-handed spin in any reference frame.On the other hand,anti-neutrino always has right-handed spin.Thefirst step in this direction is usually to introduce an imagi-nary mass,but these efforts could not reach a point for constructing a consistent quantum theory.Some early investigations of a Dirac-type equation for tachyonic fermions can be found in Ref.[9].An alterna-tive approach was investigated by Chodos et al.[10].They examined the possibility that neutrinos might be tachyonic fermions.A form of the lagrangian density for tachyonic neutrinos was proposed.Al-though they did not obtain a satisfatory quantum theory for tachyonic fermions,they suggested that more theoretical work would be needed to determine physically acceptable tachyonic theory.2.A new Dirac-type equationIn this paper,we will start with a different approach to derive a new Dirac-type equation for tachyonic neutrinos.In order to avoid introducing imaginary mass,Eq.(3)can be rewritten asE=(c2p2−m2s c4)1/2(6) where m s is called proper mass,for instance,m s(νe)=1.6eV from Eq.(1).We follow Dirac’s search[11],Hamiltonian must befirst order in momentum operatorˆp:ˆE=−c( α·ˆp)+βsm s c2(7)3with(ˆE=i¯h∂/∂t,ˆp=−i¯h∇),where α=(α1,α2,α3)andβs are4×4 matrix,which are defined asαi= 0σiσi0 ,βs= 0I−I0 (8)whereσi is2×2Pauli matrix,I is2×2unit matrix.Notice thatβs is a new matrix,which is different from the one in the traditional Dirac equation.the relation between the matrixβs and the traditional matrix βis as follows:βs=βγ5(9) whereβ= I00−I ,γ5= 0110 (9a) When we take square for both sides in Eq.(7),and consider the following relations:αiαj+αjαi=2δijαiβs+βsαi=0β2s=−1(10) the relation in Eq.(3)or Eq.(6)is reproduced.Since Eq.(6)is related to Eq.(5),this meansβs is a right choice to describe neutrinos as superluminal particles.Denote the wave function asΨ= ϕ( x,t)χ( x,t) withϕ= ϕ1ϕ2,χ= χ1χ2(11)From Eq.(7),the complete form of the new Dirac-type equation be-comesˆEΨ=−c( α·ˆp)Ψ+βsm s c2Ψ(7a) Sinceβ2s=−1in the Eq.(7a),the new Dirac-type equation is different from the traditional Dirac equation in any covariant representation in terms of theγmatrices.We now study the spin-1/2property of neutrino as a tachyonic Fermion.Eq.(7a),can be rewritten as a pair of two-component equa-tions:i¯h∂ϕ∂χi¯h+∇· j=0(14)∂tand we haveρ=ϕ†χ+χ†ϕ, j=−c(ϕ† σϕ+χ† σχ)(15) whereρand j are the probability density and current;ϕ†andχ†are the Hermitian adjoint ofϕandχrespectively.Eq.(15)can be rewritten asρ=Ψ†γ5Ψ, j=c(Ψ†γ5 αΨ)(15a) It is easy to see that the probability densityρis positive definite when the components inϕandχare positive.Considering a plane wave along the z axis for a left-handed particle ( σ· p)/p=−1,the equations(12)yields the following solution:cp−m s c2χ=√√In terms of Eq.(17),the equation(12)can be rewritten in the Weyl representation:∂ξi¯h=−ic¯h σ·∇η+m s c2ξ(19)∂tIn the above equations,bothξandηare coupled via nonzero m s.In order to compare Eq.(19)with the well known two-component Weyl equation,we take a limit m s=0,then thefirst equation in Eq.(19)reduces to∂ξνmass-square is negative.Therefore,more accurate tritium beta decay experiments are needed to further determine the neutrino mass-square.According to special relativity[16],if there is a superluminal par-ticle,it might travel backward in time.However,a re-interpretation rule has been introduced since the Sixties[6-8].Another approach is to introduce a kinematic time under a non-standard form of the Lorentz transformation[17-20].Therefore,special relativity can be extended to space-like region,and superluminal particles are allowed without causality violation.We wish to thank S.Y.Zhu and Y.Takahashi for helpful discussions References[1]”Review of Particle Physics”,Euro.Phys.Journ.C15(2000)350.[2]Ch.Weinhermer et al.,Phys.Lett.B460(1999)219.[3]V.M.Lobashev et al.,Phys.Lett.B460(1999)227.[4]W.Stoefflet al.,Phys.Rev.Lett.,75(1995)3237.[5]K.Assamagan et al.,Phys.Rev.D53(1996)6065.[6]O.M.P.Bilaniuk et al,Am.J.Phys.,30(1962)718.[7]E.Recami et al,Tachyons,Monopoles and Related Topics,North-Holland,(1978),and references therein.[8]G.Feinberg,Phys.Rev.159(1967)1089.[9]See e.g.E.C.G.Sudarshan:in Proceedings of the VIII Nobel Sym-posium,ed.by N.Swartholm(J.Wiley,New York,1970),P.335;J.Bandukwala and D.Shay,Phys.Rev.D9(1974)889;D.Shay, Lett.Nuovo Cim.19(1977)333[10]A.Chodos et al.,Phys.Lett.B150(1985)431.[11]P.A.M.Dirac,Proc.R.Soc.Ser,A117;610,118(1928)351.7[12]G-j Ni and Chen,On the essence of special relativity,Fudan Uni-versity,(Natural Science),35(1996)325.[13]G-j.Ni et al,Chin.Phys.Lett.,17(2000)393.[14]T.D.Lee and C.N.Yang,Phys.Rev.104(1956)254;Phys.Rev.105(1957)1671.[15]C.S.Wu et al.,Phys.Rev.105(1957)1413.[16]A.Einstein,H.A.Lorentz,H.Minkowski,and H.Weyl,The Prin-ciple of Relativity(collected papers),Dover,New York(1952).[17]R.Tangherlini,Nuov.Cim Suppl.,20(1961)1.[18]T.Chang,J.Phys.A12(1979)L203;”Does a free tachyon exist?”,Proceedings of the Sir A.Eddington Centenary Symposium, Vol.3,Gravitational Radiation and Relativity”,p.431(1986). [19]J.Rembielinski,Phys.Lett.,A78(1980)33;Int.J.Mod.Phys.,A12(1997)1677.[20]T.Chang and D.G.Torr,Found.Phys.Lett.,1(1988)343.8。