Mass and CKM Matrices of Quarks and Leptons, the Leptonic CP-phase in neutrino oscillations

合集下载

Relation between quark masses and weak mixings

+
INFN, Sezione di Napoli, Napoli, Italy e-mail: falcone@na.infn.it e-mail: tramontano@na.infn.it
ABSTRACT Simple transformation formulas between fermion matrices and observables, and numerical values of quark matrices, are obtained on a particular weak basis with one quark matrix diagonal and the other with vanishing elements 1-1, 1-3 and 3-1, and with only the element 2-2 complex. When we choose Mu diagonal, then Md shows intriguing numerical properties which suggest a four parameter description of it, which √ √ √ implies Vus ≃ md /ms , Vcb ≃ (3/ 5)(ms /mb ) and Vub ≃ (1/ 5)( md ms /mb ). Few comments on mass-mixing relations are added. PACS numbers: 12.15.Ff, 12.15.Hh
(4)
By writing Mij = mij eirij we can reconstruct the usual representations of K [11] by

PACSnumbers1215Ff,1130Hv,1210Dm,1125Mj…

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.。

关于合成谬误的书

关于合成谬误的书
以下是一些关于合成谬误的书籍：
1. "Thinking, Fast and Slow" by Daniel Kahneman - 这本书虽然不是专门关于合成谬误的，但作者在讨论决策制定和推理错误时提到了合成谬误。

2. "The Logic of Collective Action: Public Goods and the Theory of Groups" by Mancur Olson - 这本书探讨了集体行动的问题，并在其中涉及到了合成谬误的概念。

3. "Individualism and Economic Order" by Friedrich A. Hayek - 在这本书中，Hayek讨论了个体行为和经济秩序的关系，并提到了合成谬误作为误解个体行为和市场过程的一种常见错误。

4. "Microeconomics: Behavior, Institutions, and Evolution" by Viktor Vanberg - 这本书从演化经济学的角度探讨了个体行为和经济制度的关系，其中包括对合成谬误的分析。

5. "Collective Choice and Social Welfare" by Amartya Sen - 这本书探讨了集体选择和社会福利的问题，并在其中涉及到了合成谬误的概念。

这些书籍都从不同的角度涉及到了合成谬误的问题，可以帮助读者更深入地理解这个概念。

Neutrino Mixing and Flavour Changing Processes

2
2.1
Patterns of supersymmetry breaking
Yukawa couplings and scalar masses
ห้องสมุดไป่ตู้
We consider the supersymmetric standard model with right-handed neutrinos, which is described by the superpotential 1 W = µH1 H2 + heij Eic Lj H1 + hν ij Nic Lj H2 + hrij Nic Njc R 2 c H1 + huij Qi Ujc H2 . +hdij Qi Dj
DESY 99-169 December 1999
arXiv:hep-ph/9912317v1 13 Dec 1999
NEUTRINO MIXING AND FLAVOUR CHANGING PROCESSES
Wilfried Buchm¨ uller∗ , David Delepine† , Laksana Tri Handoko‡ Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany
mass matrices. Sec. 3 deals with quark ﬂavour changing processes, and in Sec. 4 ﬂavour changing radiative transitions are discussed, leading to the conlusions in Sec. 5.
Abstract We study the implications of a large νµ -ντ mixing angle on ﬂavour changing transitions of quarks and leptons in supersymmetric extensions of the standard model. Two patterns of supersymmetry breaking are considered, models with modular invariance and the standard scenario of universal soft breaking terms at the GUT scale. The analysis is performed for two symmetry groups G ⊗ U (1)F , with G = SU (5) and G = SU (3)3 , where U (1)F is a family symmetry. Models with modular invariance are in agreement with observations only for restricted scalar 2 quark and gaugino masses, M q /m2 g ˜ ≃ 7/9 and m˜ b > 350 GeV. A characteristic feature of models with large tan β and radiatively induced ﬂavour mixing is a large branching ratio for µ → eγ . For both symmetry groups and for the considered range of supersymmetry breaking mass parameters we ﬁnd BR(µ → eγ ) > 10−14 .

mk公理集合论

mk公理集合论MK公理集合论是由弗朗西斯·克里斯托弗·莫里斯·莫里斯托基斯（F.C.Mostowski）在1951年提出的一种集合论公理系统。

它是以莫里斯托基斯的名字命名的，因为他在该领域做出了重要贡献。

MK 公理集合论是一种精简的公理系统，由两个公理和一个推理规则组成。

MK公理集合论的第一个公理是集合的外延性公理，它表明两个集合相等当且仅当它们具有相同的元素。

这个公理是集合论的基本公理之一，它保证了集合的唯一性和确定性。

根据外延性公理，我们可以判断两个集合是否相等，从而进行集合的运算和推理。

MK公理集合论的第二个公理是空集的存在公理，它表明存在一个不包含任何元素的集合，即空集。

空集是集合论中的基本概念，它没有任何元素，但是它是存在的。

空集在集合论中具有重要的作用，它是其他集合的子集，也是集合运算中的基础。

除了这两个基本公理，MK公理集合论还采用了无穷公理，它表明存在一个包含无穷多个元素的集合，即无穷集合。

无穷集合在数学中非常常见，它是数学推理和证明中的重要工具。

无穷公理确保了集合论的完备性和广泛的适用性。

基于MK公理集合论的公理系统，我们可以推导出许多重要的定理。

其中一个著名的定理是罗素悖论的解决方案。

罗素悖论是由英国哲学家伯特兰·罗素提出的一个悖论，它表明如果假设存在一个包含所有集合的集合，就会导致逻辑上的矛盾。

MK公理集合论通过限制集合的形式，避免了罗素悖论的出现，从而解决了这个问题。

除了解决罗素悖论，MK公理集合论还在其他领域有着广泛的应用。

在数学推理中，集合论是基础和工具，它为数学家提供了一种形式化的推理框架。

集合论可以用来定义数学对象，建立数学结构，并进行证明和推理。

MK公理集合论提供了一种简洁而强大的公理系统，为数学推理提供了坚实的基础。

在集合论研究中，MK公理集合论也起到了重要的作用。

它提供了一种形式化的集合论语言和推理方法，使集合论的研究更加严谨和准确。

科芬曼森公式

科芬曼森公式
【原创实用版】
目录
1.科芬曼森公式的背景和历史
2.科芬曼森公式的定义和含义
3.科芬曼森公式的应用领域
4.科芬曼森公式的优缺点
5.科芬曼森公式的启示和未来发展
正文
科芬曼森公式是一个在数学领域中广泛应用的公式，它的背景和历史可以追溯到 19 世纪。

在当时，数学家们正在寻找一种有效的方法来描述和解决线性方程组。

在这种背景下，科芬曼森公式应运而生。

科芬曼森公式的定义和含义是，对于一个 n 阶线性方程组，其解的存在性和唯一性可以通过一个 n×n 的矩阵来判断。

这个矩阵被称为科芬曼森矩阵，其元素由方程组中各系数的代数余子式组成。

通过判断科芬曼森矩阵的行列式是否为 0，可以得出方程组是否有解，若有解，则解是否唯一。

科芬曼森公式的应用领域非常广泛，包括但不限于线性代数、微积分、概率论、物理学等。

在实际应用中，科芬曼森公式可以帮助我们快速判断线性方程组的解的情况，从而提高计算效率和准确度。

科芬曼森公式的优缺点同样明显。

优点在于，它可以有效地判断线性方程组的解的情况，而且在计算过程中只需要计算一次矩阵的行列式，计算量相对较小。

然而，科芬曼森公式的缺点是，当方程组的系数发生变化时，需要重新计算矩阵的行列式，这在一定程度上增加了计算的复杂度。

对于科芬曼森公式的启示和未来发展，我们可以看到，数学家们在寻
找更有效、更简洁的计算方法上从未停止过脚步。

科芬曼森公式的出现，无疑是这一过程中的重要里程碑。

Instanton-Induced Interactions in Finite Density QCD

a r X i v :h e p -p h /9908314v 1 10 A u g 19991Instanton-Induced Interactions in Finite Density QCDG.W.Carter a and D.Diakonov baThe Niels Bohr Institute,Blegdamsvej 17,DK-2100Copenhagen,Denmark bNORDITA,Blegdamsvej 17,DK-2100Copenhagen,DenmarkWe consider the ﬁnite density,zero-temperature behaviour of quark matter in the in-stanton picture.Since the instanton-induced interactions are attractive in both ¯q q and qq channels,a competition ensues between phases of matter with condensation in either or both.It results in chiral symmetry restoration due to the onset of diquark condensation,a ‘colour superconductor’,at ﬁnite density.1.IntroductionDue to a lack of lattice QCD techniques for implementing quark chemical potential,the ﬁnite density properties of strongly-interacting matter remain unresolved.To date,model studies suggest not only chiral symmetry restoration but also the possibility of Cooper pairing of quarks at high density,via an attractive qq interaction,similar to superconducting electrons.The analogy has been extended to nomenclature,with the QCD version called colour superconductivity.It has been known for some years that perturbative,single gluon exchange between quarks is attractive and will generate a pairing gap around the Fermi surface [1].More re-cently,it was suggested that colour superconductivity might also arise by non-perturbative means at moderate quark density [2].Since then,more detailed studies using models in-spired by that of Nambu and Jona-Lasinio [3]and instantons [4]have supported this idea.This talk describes how diquark formation restores broken chiral symmetry in the con-text of the QCD instanton vacuum,an approach which has accounted for many hadronic observables through the use of fundamental degrees of freedom (quarks and gluons)in a microscopic approximation.2.Quark Eﬀective ActionThe derivation of an eﬀective action for chiral quarks in N f ﬂavours has been discussed in detail in other publications.Here we concentrate on the two ﬂavour case,which is often adequate for low energy phenomenology.Growing quark chemical potential naturally makes the strange quark more relevant,as has been studied in other models [5].These authors conclude that the two-ﬂavour superconducting state is likely to be present at moderate values of the quark chemical potential for a realistic strange mass.The basic idea is to replace the partition function of QCD with an eﬀective form which divides the low and high energy contributions.The high momentum part is taken to be2SS;FFigure 1.Schwinger-Dyson-Gorkov diagrams to ﬁrst order in λ.perturbative and as such the gluons here are assumed to be small corrections to the stable,low-energy conﬁgurations of the gauge ﬁelds –the instantons.Each (anti-)instanton in turn induces a quark zero mode of (right)left chirality,and averaging over all possible instanton backgrounds results in a delocalization of the zero modes which spontaneously breaks chiral symmetry.This picture of the vacuum is supported by various lattice studies and has a long history of successful phenomenology.Following this procedure,the expected ’t Hooft interaction is obtained and one has an eﬀective quark action which is suitable for practical calculations.We have reformulated this eﬀective action for ﬁnite quark chemical potential.The result can be expressed asS [ψ,ψ†]=−d 4p ψ†(p /+iµγ4)ψ+λdUN ff(d 4p f d 4k f )3combine in the physical quantities:M 1=(5−4/N c )g 1+(2N c −5+2/N c )g 2,M 2=2(2−1/N c )g 1+2(N c −2)g 2,and ∆=(1+1/N c )f .The M 1,2are measures of chiral symmetry breaking and act as an eﬀective mass.Meanwhile the diquark loop 2∆plays the role of twice the single-quark energy gap formed around the Fermi surface.The solution of the gap equations depends on the vertex coupling constant,λ,which itself is determined by balancing the instanton background with the condensates through its saddle-point value.This minimization of the partition function leads to [4]N/V =λ Y ++Y − =4(N 2c −1)[2g 1M 1+(N c −2)g 2M 2+4f ∆]/λ.(2)This joins the gap equations to close a system of equations,numerically solvable.Once this is done,the chiral condensate proper may be computed as an integral over the resummed propagator.For any given chemical potential,multiple solutions can be obtained for the gaps.These correspond to diﬀerent phases of quark matter,and they are summarized as follows:(0)Free massless quarks:g 1=g 2=f =0;(1)Pure chiral symmetry breaking:g 1=g 2=0,f =0;(2)Pure diquark condensation:g 1=g 2=0,f =0;and (3)Mixed symmetry breaking:g 1=g 2=0,f =0.The free energy,calculated to ﬁrst order in λ,is minimized in order to resolve the stable solution.The phase corresponding to the lowest coupling λis the thermodynamically favoured [4].No solutions were found matching Phase (0),and the Phase (3)solution obtained disap-pears at relatively low chemical potential (µ≈80MeV)and is never thermodynamically competitive [4].The remaining phase competition is then between Phases (1)and (2).In the vacuum,where µ=0,one ﬁnds Phase (1)preferred –this is the standard picture.However,at a critical chemical potential µc ,deﬁned by the ratio of superconductive gap to chiral eﬀective masses ∆/M = 3/4,a ﬁrst-order phase transition occurs.With the standard instanton parameters N/V =1fm −4and ¯ρ=0.33fm,we ﬁnd µc ≃340MeV.The ﬁrst-order nature of the phase transition is clearly seen in Fig.2.Physically,the quark density is more relevant than the chemical potential.As an intermediate step and in order to demonstrate the microscopic diﬀerences between the two phases,we have calculated the occupation number density for quarks.This is nontrivial50100150200250300350010*******400500600700800(M e V )µ(MeV)M∆−<ψψ>1/3Figure 2.Condensates for N c =3as a function of µ.0.020.040.060.080.10.120.1400.20.40.60.81n ρ3µρFigure 3.The quark density n q vs.µ.424681012140120240360480600720840n (|p |)|p| (MeV)Figure 4.Occupation number n (p )vs.p for Phase (1)for µ=1/¯ρ=600MeV.024*********120240360480600720840n (|p |)|p| (MeV)Figure 5.Occupation number n (p )vs.pfor Phase (2)for µ=1/¯ρ=600MeV.and here we present only numerical results in Figs.4and 5.In Phase (1),there isclearly an eﬀective mass brought about by spontaneous symmetry breaking,indicated by the reduced Fermi radius.We stress that,despite the complicated four-momentum dependence of the interaction,the resulting occupation number density appears as a perfect Fermi step function.Cooper pairing,however,smears the Fermi surface,and this is evinced in the second plot.The residual discontinuity at | p |=µis the contribution from the free,colour-3quarks which do not participate in the diquark.Integrating over momenta and recalling the critical chemical potential,the quark density proﬁle as a function of chemical potential is plotted in Fig.3for the equilibrium states.We see a discontinuity at the phase transition,where the horizontal line signiﬁes the quark density of stable nuclear matter.The phase transition occurs at an extremely low quark density,which remains a conceptual conundrum.4.ConclusionsBeginning from the instanton picture of the QCD vacuum,we have extended the model for ﬁnite density and found chiral symmetry restoration due to the onset of colour super-conductivity.This phase transition is strongly ﬁrst order and in agreement with other quark-based approaches.REFERENCES1. D.Bailin and A.Love,Phys.Rep.107(1984)325.2.M.Alford,K.Rajagopal and F.Wilczek,Phys.Lett.B422(1998)247;R.Rapp,T.Sch¨a fer,E.V.Shuryak and M.Velkovsky,Phys.Rev.Lett.81(1998)53.3.J.Berges and K.Rajagopal,Nucl.Phys.B538(1999)215.4.G.W.Carter and D.Diakonov,Nucl.Phys.A642(1998)78c;Phys.Rev.D60(1999)016004;R.Rapp,T.Sch¨a fer,E.V.Shuryak and M.Velkovsky,hep-ph/9904353.5.M.Alford,J.Berges and K.Rajagopal,hep-ph/9903502;T.Sch¨a fer and F.Wilczek,hep-ph/9903503.。

马奇和西蒙模型的参考文献

马奇和西蒙模型的参考文献马奇和西蒙模型的参考文献包括以下几篇研究论文：1. March, J. G., & Simon, H. A. (1958). Organizations. John Wiley & Sons.这是马奇和西蒙最早的合著之一，论述了他们对组织行为的研究。

2. Simon, H. A. (1955). A behavioral model of rational choice. The Quarterly Journal of Economics, 69(1), 99-118.这篇论文提出了马奇和西蒙的决策理论框架，并介绍了理性选择模型。

3. March, J. G. (1991). Exploration and exploitation in organizational learning. Organization science, 2(1), 71-87.这篇论文重点讨论了马奇和西蒙模型中的探索与利用的概念，并探讨了组织学习的过程和机制。

4. Cohen, M. D., March, J. G., & Olsen, J. P. (1972). A garbage can model of organizational choice. Administrative science quarterly, 1-25.这篇论文介绍了马奇和西蒙的“垃圾桶”模型，用于解释组织内部的决策过程和选择行为。

5. Cyert, R. M., & March J. G. (1963). A Behavioral Theory of the Firm. Prentice-Hall.这本书是马奇和西蒙合写的，详细讨论了组织行为和决策的行为理论，并提出了“增量决策”的概念。

以上是马奇和西蒙模型的一些经典参考文献，供进一步研究和学习使用。

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

(2)
Here η = eiδ represents the CP violating phase δ in the quark sector, while the values of the constants ad ≃ 2, ae ≃ 3 2 correspond to the best ﬁt of all masses of quarks and electrons (their central values, see below). ˆa : The diagonalization of the matrices (2) and (3) by means of an unitary matrix U ˆ diag = U ˆa M ˆ aU ˆ+ M a a 1 a = u, d, e, (4)

0
1 3 + √ σ η 2 2 +
ˆu = M
σ η
1 3 √ σ η 2 1 2 −2σ 1 √ σ 2
σ2 η
1 √ σ 2

1

mt ,
ˆd = ad σ 3 M 0

0
ad σ 3 0 m 2 b σ −2 0 σ 0 σ

1
′
(5)
Here, according to experimental data [1,6], one has
:ห้องสมุดไป่ตู้
s13
√ The factors 1/ 2, 1/2, 2, 3/2, etc. in Eq.(2),(3) were adjusted so as to reproduce all these mixing angles and all the masses of quarks and leptons. As has been mentioned, for quarks the mixing angles are small, i.e. s13 ≪ s23 ≪ s12 ≪ 1 and therefore the cosines c13 , c23 , c12 of them can be put approximately equal to unity. The major ideas of Stech’s approach were: 1) to consider the running masses of all quarks at the same scale (instead of the pole ones) and 2) to obtain these masses and mixing angles in terms of few parameters. However, the matrices (2) and (3) have been written phenomenologically, just by hand. Below we develop an approach to quark and lepton masses and mixings through the following steps: a) A general method is suggested which allows one to construct the quark mass matrices consistently (without using Eq.(2)) in terms of their masses and mixing angles (6). The resulting mass matrices coincide with those given by Eq.(2). They are diagonalized in an analytical form of decomposition in powers of small parameter σ and the quark CKM matrix is reproduced analytically. ˆν. b) The same approach is applied to the reconstruction the neutrino mass matrices M ˆν . Preliminary results on masses This is performed using the Eq.(4) and the unitary matrix U of neutrinos and their mixing angles, extracted from neutrino oscillation experiments have been used. ′ ˆl ˆ+ ˆ c) The leptonic CKM matrix V CKM = Uν Ue containing the leptonic CP-phase δ is introduced. It leads to νµ ντ oscillation of the type observed at Super-Kamiokande and also to some suppression of νe νµ oscillation.
0 ae σ 3 0 ˆ e = ae σ 3 −σ 2 0 mτ M σ 0 0 σ
(3)
ˆa U ˆ+ = U ˆ +U ˆ (where U a a a = 1), reproduces simultaneously both the experimental (central) values of running masses of quarks and electrons and all quark mixing angles (their sines): q q s12 = sin ϑq 12 , s23 = sin ϑ23 , s13 = sin ϑ13 in the CKM matrix: ˆCKM V c12 c13 s12 c13 s13 e−iδ ′ +ˆ iδ′ ˆu =U Ud = c12 c23 − s12 s23 s13 eiδ s23 c13 . −s12 c13 − c12 s23 s13 e ′ ′ s12 s23 − c12 c23 s13 eiδ −c12 s23 − s12 c23 s13 eiδ c23 c13
with a very small σ 2 ≃ 1/300, σ ≃ 0.058. He has also introduced the following mass matrices which reproduced approximately the masses of all the quarks as well as theirs mixing angles:
values of νe νµ , νe ντ transition probabilities, averaged over oscillations, on the leptonic
∗ †
ryzhikh@vitep5.itep.ru termarti@heron.itep.ru
1. Introduction Serious eﬀorts have been invested recently in the natural understanding of experimental results on neutrino oscillations. They have shown that neutrinos of three generations have, perhaps, non vanishing small masses. The heaviest of them, the neutrino of the third generation, seems to have a mass of the order of ∼ (1/20)eV and, as the Super Kamiokande data on atmospheric neutrinos show, has the maximal possible mixing with the neutrino of the second generation, and may be, also not too small mixing with that of the ﬁrst generation. This was not expected a priori, since all similar mixing angles of quarks are small. It is the challenge of modern particle physics to include naturally these results into the framework of Grand Uniﬁcation Theory together with the data on quark masses and mixing angles. For the quarks these angles are small and are known already [1]. We begin this paper by reminding the well-known picture of masses, CP-phase and mixings for the quark sector of a theory. A general method will be developed which allows one to construct consistently the 3 × 3 mass-matrix and the CKM mixing matrix for quarks. The same general approach will be used later for the electron-neutrino sector of a theory. Let us consider, as a useful introduction to a consistent theory of quark and lepton masses and mixings, a simple phenomenological approach suggested by B. Stech [2]. He has noticed the following quark and charged lepton (”electrons”) mass hierarchies: mt : mc : mu ≃ 1 : σ 2 : σ 4 , 1 mb : ms : md ≃ 1 : σ : 8σ 3 , 2 3 mτ : mµ : me ≃ 1 : σ : σ 3 , (1) 2