Invariants Describing Quark Mixing

arXiv:hep-ph/9207230v1 9 Jul 1992ITP-SB-92-38July 6, 1992Invariants Describing Quark Mixingand Mass MatricesAlexander Kusenko1Institute for Theoretical PhysicsState University of New YorkStony Brook, N. Y. 11794-38402AbstractWe introduce two new sets of invariant functions of quark mass matrices, which expressthe constraints on these mass matrices due to knowledge of the quark mixing matrix.

Theseinvariants provide a very simple method to test candidate forms for mass matrices.1Supported in part by NSF contract PHY-89-084952email: kusenko@sunysbnp.bitnet

In recent years the increasingly accurate data on quark mixing has stimulated one’sinterest in possible predictions concerning quark mass matrices.The parameters of theCabibbo-Kobayashi-Maskawa mixing matrix V are now known reasonably well [1]. Thisdetermination has been made possible partly by the finding that there are only three gen-erations of usual standard-model fermions (with corresponding light or massless neutrinos).Since the diagonalization of the quark matrices in the up and down sectors determines V ,one can work back from the knowledge of V to put constraints on the possible forms of(original, nondiagonal) quark mass matrices.

However, the data on quark mixing determinesthese mass matrices only up to an arbitrary unitary similarity transformation. This is aresult of the fact that if the up and down quark mass matrices, Mu and Md, are both actedon by the same unitary operator U0 according toMu,d →U0 Mu,d U†0(1)then the mixing matrix V remains unchanged.

There have been many attempts to studyspecific assumed forms for quark matrices. While this is worthwhile, it is desirable to expressthe constraints from data on V on the quark mass matrices in an invariant form.

In Refs. [2, 3] certain invariant functions of the quark mass matrices Ipq were introduced, which areexpressed in terms of the quark masses squared and the |Vij|:Ipq = Tr(Hpu Hqd) =Xij(m(u)i )2p(m(d)j )2q |Vij|2(2)where Hq = MqM†q , and m(u)iand m(u)iare the masses of quarks in the “up” and “down”charge sectors.However, in practice, these are somewhat awkward to use because of the high powers ofelements of the actual quark mass matrices which are involved.In this paper we introduce two new sets of invariants of the quark mass matrices (withrespect to the transformation (1)) which can be expressed in terms of the measurable quan-tities only (up to the ± sign ambiguity in fermion masses, which, as we will show, can bedealt with by considering all choices of these signs).

These invariants have an important1

advantage relative to (2) that they involve lower powers of the elements of the quark massmatrices and hence yield much less complicated analytic expressions in practice.In the standard model with only left-handed charged weak currents one may choose Muand Md both to be hermitian without any loss of generality, by re-phasing the right-handedcomponents of quarks.We introduce the following new invariants of the transformation (1):Kpq = Tr( Mpu Mqd )(3)Lpq(α, β) = det(αMpu + βMqd)(4)where p, q, α, β ̸= 0.The hermitian matrices Mu and Md can be diagonalized by a unitary similarity transfor-mation:UuMuU†u = DuUdMdU†d = Dd(5)where Dq = diag(m(q)1 , m(q)2 , m(q)3 ) are the diagonal matrices of the quark masses.The mixing matrix V can be written then as:V = UuU†d(6)We can now express the invariants Kpq in terms of the squares of absolute values of theelements of the mixing matrix Uij = |Vij|2 (the latter are measurable quantities [1]):Kpq = Tr(U†uDpuUpU†dDqdUd) = Tr(V †DpuV Dqd) =Pij (m(u)i )p (m(d)j )q Uij(7)In order to find an expression for Lpq in terms of Uij we will need the following2

Theorem 1.If A and B are two 3×3 matrices such that det(A) ̸= 0 and det(B) ̸= 0 then the followingrelation holds:det(A + B) = det(A) + det(B) + det(A) Tr(A−1B) + det(B) Tr(AB−1)(8)Proof:We denote the elements of matrices A and B by Aij and Bij correspondingly. Theirco-factors (which are equal to the corresponding minors, up to sign) will be written as ˆAijand ˆBij.

Then each determinant may be decomposed in a sum (Laplace expansion):det(A) =XiAij ˆAij =XjAij ˆAijdet(B) =XiBij ˆBij =XjBij ˆBijBy definition, the determinant of a 3 × 3 matrix is a sum of n! terms:det(A + B) =X(−1)r(A1k1 + B1k1)(A2k2 + B2k2)(A3k3 + B3k3)(9)where r is the sign of the permutation (1k12k23k3).The terms in the sum (9) which contain only the elements of A can be arranged as det(A).Similarly, the terms containing only B’s give det(B).

The terms containing one element ofA multiplied by two elements of B, or visa versa, can be rewritten as:Xi,j(Aij ˆBij + Bij ˆAij)(10)We can now use an identity:(A−1)ij =1det(A)ˆAjito rewrite (10) as:Xi,j(Aij ˆBij + Bij ˆAij) = det(B)XijAij(B−1)ji + det(A)Xij(A−1)jiBij =3

det(A) Tr(A−1B) + det(B) Tr(AB−1)All together we getdet(A + B) = det(A) + det(B) + det(A) Tr(A−1B) + det(B) Tr(AB−1)which is the statement of Theorem 1. This completes the proof.Theorem 1 may be easily generalized to the case of 2 × 2 matrices, in which case the lasttwo terms in (8) are equal and correspond to a redundant counting of the same terms in asum similar to (9).

Thus for the 2 × 2 matrices we get:det(A + B) = det(A) + det(B) + det(A) Tr(A−1B) ≡det(A) + det(B) + det(B) Tr(AB−1)(11)The immediate consequence of Theorem 1 and equation (7) is the following relation:Lpq(α, β) ≡det(αMpu + βMqd) =α3 (m(u)1 m(u)2 m(u)3 )p [ 1 + (β/α)Xij[(m(d)j )q/(m(u)i )p] Uij ] +(12)β3 (m(d)1 m(d)2 m(d)3 )q [ 1 + (α/β)Xij[(m(u)j )p/(m(d)i )q] Uij ]We also notice thatLpq(1, ±1) = (m(u)1 m(u)2 m(u)3 )p(1 ± K(−p) q) ± (m(d)1 m(d)2 m(d)3 )q(1 ± Kp (−q))(13)In summary, we have introduced and studied two new sets of invariants of the quarkmass matrices which are model- and weak basis-independent. The identities (12) and (13)involving these invariants which we have derived above provide important constraints on thepossible forms of quark mass matrices Mu and Md since they directly relate the elements of4

these matrices to the measurable parameters |Vij|2 and quark masses, and thereby enable oneto avoid the explicit calculation of the eigenvectors of Mu and Md. (For another approachconcerned with obtaining information on these mass matrices from V , see, e.g., Ref.

[4]. )Although these invariants involve ambiguities stemming from the fact that the signs offermion masses are not physical, these ambiguities can be dealt with by considering allpossible sign combinations.

In practice, we have found that this is not a serious complication.Applications of these invariants will be discussed as part of a separate paper [5].The author is grateful to Professor Robert E. Shrock for helpful discussions and for hiscomments.References[1] Particle Data Group, Review of Particle Properties, Phys. Rev.

D 45 (1992) III.65[2] C. Jarlskog, Phys. Rev.

D35 (1987) 1685. (See also C. Jarlskog, Phys.

Rev. Lett.

55(1985) 1039. )[3] G. C. Branco and L. Lavoura, Phys.

Lett. B 208 (1988) 123.

[4] A. Kusenko, Phys. Lett.

B 284 (1992) 390. [5] A. Kusenko and R. E. Shrock, in preparation.5

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