Theory of quark mixing matrix and invariant functions of mass matrices. Cecilia jarlskog proposed this invariant already in 1973 and it was. Pdf cpviolation phase analysis via nontrivial correlation of. Implications of maximal jarlskog invariant and maximal cp violation. Cpsymmetry, often called just cp, is the product of two symmetries. The preferred scheme leads to simple mass textures and allows us to compute the ckm mixing matrix, the jarlskog invariant j, and the three inner angles of the unitarity triangle in terms of four quark mass ratios and only one free parameter. The jarlskogs invariant the most important property of qaib j that follows from the unitarity of the ckm matrix is. In the standard model cp violation is related to the jarlskog invariant j, which can be obtained. The jarlskog invariant 14, j, contains the dependence on the ckm elements. In recent years there has been a large amount of work on invariant subspaces, motivated by interest in the structure of nonselfadjoint of the results have been obtained in operators on hilbert space. The strong interaction and electromagnetic interaction seem to be invariant under the combined cp transformation operation, but this symmetry is slightly. Volume of the set of unistochastic matrices of order 3 and the mean jarlskog invariant charles dunkl1,a and karol zyczkowski2,b 1department of mathematics, university of virginia. Over the complex numbers the j jinvariant is a modular function on the upper half plane which serves to characterize most of the properties of the moduli stack of elliptic curves in this case.
Invariant approach to flavourdependent cpviolating. Edm free supersymmetric cpviolation with nonuniversal soft terms 2000 nucl. We find that all squared moduli of the quark mixing elements have a singular point when the cp violation phase phi takes the value phi90 degrees. The most commonly used unitarity triangle arises from vud v. She is known for her work on cp violation in the electroweak sector of the standard model, introducing what is known as the jarlskog invariant, and for her work on. The number of invariants of a given degree is encoded in the hilbert series. Matt has given a nice explanation for why one should care about the jinvariant up to a constant, but the precise form of the jinvariant has to do with its numbertheoretic properties and is not easy to explain in an introduction. Librivox free audiobook chris seits s podcast frecuencia clandestina bigg impact audio experience mickael haller podcast queer as fiction in the name of game a gaming podcast gaming nerds. Invariance of jarlskog determinant above the gut scale. In this article we derive the conditions to have hermitian mass matrices which give maximal jarlskog invariant j and maximal cp violating phase phi. Fortunately, the number of free parameters can be greatly reduced by. The results for the amount of cp violation in the lepton sector can be quantified with the jarlskog invariant j cp and the majorana invariants i 1 and i 2 34 see. For neutrinos, using the standard parameterization of the pmns matrix, this invariant is given by j s 23c 23s c 2 s 12c 12 sin.
Nonzero jarlskog invariant impliesthat angles are not 0 or. Bednyakov x winter school on theoretical physics, february, 2012. Covariant extremisation of flavoursymmetric jarlskog invariants and the neutrino mixing matrix. Full list of physics courses physics brown university. Brief history of flavor physics fermi theory of beta decay e. Algebraic structure of lepton and quark flavor invariants. The origin of jarlskog invariant and applications sylwester kornowski abstract. The courses are listed by the term in which they are usually offered.
Pdf the existence and mysterious nature of sterile neutrinos are revolutionizing. Index 843 breit frame, 696 breitwigner distribution, 462 brick wall frame, 696 brown muck, 760 brst invariance, 499502 bubbles feynman diagrams, 91. The different size scales in nature and cp violation lead to very simple our precosmos with lefthanded polarization which has transformed into the lefthanded internal helicity of the baryons it solves the matterantimatter asymmetry. It describes our latest reserach on the connection between weak basis. Cecilia jarlskog born in 1941 is a swedish theoretical physicist, working mainly on elementary particle physics jarlskog obtained her doctorate in 1970 in theoretical particle physics at the technical university of lund. On the other hand books on elliptic curves dont want to give you the wrong definition. Quark and lepton masses and mixing in the landscape. Implications of maximal jarlskog invariant and maximal cp. Next, we investigate some formulas for invariant phases of unitary matrices and derive some explicit relations for them. In our previous paper,4 we proposed a phenomenological formula, md ms mb. We describe a calculation of jarlskogs determinant in the case of n 4 in detail. Newest invarianttheory questions mathematics stack. Abstract we explore generic features of the leptonic cp violation in the framework of the seesaw type i mechanism with similarity of the dirac lepton and quarks mass matrices m d.
This course introduces the fundamental elements of. Contour plots showing the dependence of jarlskog invariant j. This special feature of the jarlskog invariant j and the quark mixing matrix is a clear and precise indication that cp violating phase phi is maximal in order to let nature treat democratically. Steven robertson canadian institute of particle physics mcgill university triumf summer institute vancouver b. The invariant of abiyevs balanced squares and balanced. What is called the j jinvariant is an invariant of cubic curves and hence of elliptic curves, partly characterizing them. She is known for her work on cp violation in the electroweak sector of the standard model, introducing what is known as the jarlskog invariant, and for her work on grand. Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Hilario perez ramirez rephasing invariant functions of the ckm matrix. Theory of quark mixing matrix and invariant functions of. However, one is free to rephase any unitary matrix without a. Jarlskog invariant and its mathematical origin physics stack. Fo example, in the parameterization of wolfenstein the jarlskogs in variant8 j. The discovery of an invariant, the jarlskog invariant 1, that controls the size of cp violation in both quark and neutrino sectors was a monumental step in the understanding of avor physics.
If the only origin of cp violation is the lefthanded lh. In this paper, we show that the ratio of the effective jarlskog invariant j. For this, we elaborate on the standard parametrization conditions which allow to simultaneously obtain the dirac and majorana phases. It has been shown that jarlskog determinant remains nearly invariant above the gut scale. This is a problem about vacuum orbits in an su3 symmetric model, suggested by the penultimate. Covariant extremisation of flavoursymmetric jarlskog. Some formulas for invariant phases of unitary matrices by. Invariant approach to flavourdependent cpviolating phases in the mssm. August 20, 2015, presusy 2015 abstract this is a set of latexed notes from cornell universitys physics 7661 special topics in theoretical high energy physics course by yuval grossman in fall 2010. Jarlskog didnt derive it constructively it was a clever guess, a convention were only looking for a quantity that is nonzero when it should be, and the exact value of such a quantity is clearly not unique and there is no canonical generalization of it to larger matrices. Results for different assumptions concerning the analysis of data from reactor experiments are shown.
Here is how a proof goes, if you can find an invariant that is in the initial state, and that this invariant persists regardless of any legal transformation applied to the state, then you can prove that if a certain state does not have this invariant then it can never occur, no matter what sequence of transformations are applied to the. The invariant of abiyevs balanced squares and balanced cubes of odd order 1asker ali abiyev, 2yusif alizada 1institute of radiation problems of anas 9b vakhabzadeh street, baku, azerbaijan republic, az1143. A mathematical model of the quark and lepton mixing angles 2011 update j. Pdf some formulas for invariant phases of unitary matrices by. Simple factorization of the jarlskog invariant for. Quark and lepton masses and mixing in the landscape john f. The decomposition separates the complex phase information from the mixing angle information of the matrices and leads to a new type of parameterization. The jarlskog s invariant the most important property of qaib j that follows from the unitarity of the ckm matrix is.
Rephasing invariants of quark and lepton mixing matrices. The breaking of the flavour permutational symmetry. Steven robertson canadian institute of particle physics. Donoghue, koushik dutta and andreas ross department of physics university of massachusetts amherst, ma 01003, usa february 2, 2008 abstract even if quark and lepton masses are. In this letter we will show that extremisation of flavoursymmetric jarlskog invariants, leads naturally to jarlskogcovariant constraints. We describe a calculation of jarlskogs determinant in the case of n4 in detail. These relations were proposed by georgi and jarlskog and are known as the gj relations. Pdf implications of maximal jarlskog invariant and. Planck scale effects on neutrino mixing, we have derived the mixing angles of neutrino flavour due to planck scale effects.
In this paper we modify the previously proposed formula. Please note that some courses are not offered every year. The normalization of reactor fluxes is left free and data from shortbaseline less than 100 m reactor experiments are included. This presnetation was give at dae symposium in high energy physics at benaras hindu university, varanasi in december 2008 by surender verma, one of my collaborators. Jarlskog invariant in turn, observables can only depend on the combination that is invariant under righthanded transformations the simplest cp odd quantity that does not depend on the flavor basis is the jarlskog determinant where j is a rephasing invariant combination of. That is, the jarlskog invariant jas a function of the cp violating phase. Next, we investigate some formulas for invariant phases of unitary matrices and. A mathematical model of the quark and lepton mixing angles. Cp odd weak basis invariants free download as powerpoint presentation. If omitted, invariant will throw regardless of the truthiness of the condition. January, 2012 abstract we present a new decomposition of unitary matrices particularly useful for mixing matrices. On the properties of the effective jarlskog invariant for threeflavor. Nowosci zaklad optyki atomowej, atomic optics department. Leptonic cp violation phases, quarklepton similarity and.
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