Natural flavour conservation in the neutral currents and the determination of the Cabibbo angle

Volume 80B, number 3

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1 January 1979

NATURAL F L A V O U R CONSERVATION IN THE NEUTRAL CURRENTS AND THE DETERMINATION OF THE CABIBBO ANGLE ~ R. GATTO Ddpartement de Physique Thdorique, Universitd de Gendve, Geneva, Switzerland G. MORCHIO Istituto di Fisica dell'Universit& Pisa, Italy and F. STROCCHI CERN, Geneva, Switzerland Received 27 October 1978

We point out a conflict between the requirement of natural flavour conservation in the neutral currents and the possibility of determining the generalized Cabibbo angles as a function of the quark masses in the SU(2) X U(1) and in the SU (2) L X SU(2) R X U(I) models. The general implications of natural flavour conservation in the neutral currents on the fermion mass matrices are also discussed.

The possibility of determining the Cabibbo angle within the current frame o f a gauge theory o f weak and electromagnetic interactions has received much attention recently [1,2]. As pointed out by Weinberg [1 ], the possibility of determining the Cabibbo angle (more generally relations between mixing angles and masses) as a result o f radiative corrections faces some difficulties and it has been suggested that such a deterruination should rather arise as a property o f the quark mass matrix at lowest order (tree approximation). The determination o f the Cabibbo angle in terms o f the quark masses could be obtained by suitably adjusting the expectation values of the Higgs fields, but the so obtained relations are in general spoiled by renormalization and radiative corrections. One is then led to look for a natural determination (i.e. without an ad hoc choice of the parameters), guaranteed by symmetry properties (for instance discrete) o f the Higgs-quark coupling and therefore automatically stable under Partly supported by the National Swiss Science Foundation.

renormalization * 1. We shall call such additional symmetries K symmetries. Clearly a complete understanding of this problem will shed light on the less established part o f weak interactions, namely the Higgs structure. A reasonable constraint on the Higgs sector is that it leads to natural flavour conservation in the neutral currents [3]. A sufficient condition for such natural conservation is that quarks o f given charge receive their mass through the coupling of precisely one neutral Higgs, as discussed by Glashow and Weinberg [3]. This, however, may look as too strong a requirement, considering the possible r61e of discrete symmetries whose judicious introduction can be crucial for the determination o f the Cabibbo angle. We will therefore consider the constraint of natural flavour conservation in the neutral currents in a weaker form, namely that the diagonalization of the quark mass matrix (which is the way by which the various flavours are defined) .1 Many examples of this type have been discussed in the literature during the last year, mainly by using SU (2) L X SU (2) R X U(1) as a gauge group [1,2]. 265

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automatically leads to the diagonalization of the Yukawa couplings o f the quarks with the neutral ttiggs [4]. The aim o f the present note is to investigate whether a determination o f the Cabibb0 angle (more generally, or mass matrix relations) in terms of the quark masses is possible, consistently with the naturality requirements discussed above (i.e. a natural relation obtained with a Higgs structure leading to natural flavour conservation in neutral currents). For concreteness we will discuss the gauge groups SU(2) × U ( 1 ) , 2 and SU(2)L X SU(2)R X U(1) (the generalizations being without difficulties). In the case of SU(2) × U(1) we will show that if flavour conservation occurs naturally, namely it is implied by a K symmetry group (and the Higgs representation), the Cabibbo angle is directly determined by such symmetry and not as a function of the mass ratios. In particular, if the mass ratios are free parameters of the theory the Cabibbo angle is not a function of them. In the case of SU(2)L X SU(2)R × U(1) natural flavour conservation leads to vanishing mixing angles.

(i) SU{2) X U(1). We group quarks into left-handed doublets ffL and right-handed singlets ff~, t~d . The most general Higgs-quark coupling is then of the form = u F~ u d a d .~OqH ~iLtPa( ij)u~jR + dJiL~a(Fij)d~/R + h . c . ,

(I)

where SU(2) indices are not spelled out explicitly and are supposed to be summed in an invariant way, a = 1 .... , N labels the various Higgs multiplets, i, j label the fermion representations and Fund are the coupling matrices. The mass matrix M -= (~%) C a ,

(2)

,2 For the group SU(2) × U(1) with an arbitrary number of left-handed quark doublets and right-handed quark singlets, the impossibility of determining the Cabibbo angle with the standard Higgs structure (two doublets) has been pointed out in a previous paper [4]. It has also been shown that this difficulty persists in the case of two quark doublets and an arbitrary Higgs structure if the Higgs fields transform as one-dimensional representations of the discrete symmetry [4]. An attempt to determine the Cabibbo angle in an SU(2) × U(1) model has been discussed by Pakvasa and Sugawara [5]. However, in their model charm is not conserved in neutral couplings and the obtained relation for the Cabibbo angle does not emerge naturally in the sense that the stability of the particular vacuum parameters is not guaranteed under renormalization.

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splits into two blocks: Mu, d corresponding to charge 2/3 (u) and charge - 1 / 3 (d), respectively, m

-/,~t.',d\r,a

u,d - '~

''u,d

-

(3)

In order to guarantee natural flavour conservation in neutral currents we require that there exist unitary matrices (U L ' UR) , (DE, DR) , which diagonalize F au,d for any a:

r.%,

(4)

=rad , D '

(5)

=

IX

D F DR

simultaneously with the mass matrix Mu, d (D stands for diagonal). We explore the consequences of the invariance of "~qH under a setZ[K (i)} of symmetries ~L _>K(i) L t~L ,

~ , d -+ (Kt~))u,dt~'d " ,

lpau,d ~ (Q) (i) 3u,d .u,d a~, w3 '

(6) (7)

(K0),R, cDa~-(i) unitary matrices). The P matrices must then satisfy the following equations: K(/)t a t/d (i)'~ L p*u,dt*'R Ju,d

/,N

,~

(c-/)~'d)Iu ~ ,d r au, d ,

(8)

u, 3 = 1 ..... N.

A solution of eqs. (8) is a set of 2 N matrices {P~}, {P~} and in general one may have more than one solution. In the latter case, since eqs. (8) are linear equations any linear combination of two solutions would also be a solution. For simplicity we consider the case in which the mass matrix is naturally non-degenerate, i.e. there are possible values of (~0a) minimizing the Higgs potential and solutions o f eqs. (8) which leads to a non-degenerate Mu, d (otherwise eqs. (8) should imply a degenerate Mu,d). First we note that each solution of eqs. (8) uniquely determines the physical quark basis. This basis is defined as the basis in which the mass matrix is diagonal and positive. The requirement o f natural flavour conservation in the neutral currents implies that this basis does not depend on the set of vacuum expectation values (s%), but only on the set {Fu) , a {Fd} . ~ In fact, if there are two biunitary transformations (VL, VR) , ( V / , Vt~), which simultaneously diagonalize all the P matrices, they will also diagonalize M = (~o~)P a, MM'f and M'~M. Now MM t is a hermitean non-degenerate

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(positive) matrix and therefore the only arbitrariness involved in the unitary transformations VL, VI~, which diagonalize it, are permutations and/or multiplications by phases. The same argument applies to VR, VI~, which diagonalize MtM. Since the mixing angles depend only on the physical basis of the up and down quarks, each solution {U~}, {Pd~ uniquely determines the generalized Cabibbo angles, independently of the values of the quark masses. One can further argue that actually the generalized Cabibbo angles can only depend on the given K symmetry (and the representation of the Higgs fields) and not on the particular pair of solutions {F~), {P~}, since natural flavour conservation in the neutral currents requires that any solution of eqs. (8) (in particular any linear combination of two given solutions) should satisfy eqs. (4), (5), Otherwise the property of flavour conservation would in general be spoiled by renormalization. All the solutions of eqs. (8) must therefore be simultaneously diagonalizable and therefore define the same physical basis. In conclusion, if flavour conservation in the neutral currents has to occur naturally it must be implied by the symmetry properties of the couplings, namely by eqs. (8), and therefore, under this condition, the gene-

ralized Cabibbo angles are uniquely determined by the K symmetry group and the representation of the Higgs fields, independently of the values of the quark masses. Concerning now the determination of the quark masses it is useful to distinguish the following cases: (i) eqs. (8) have more than one solution; then not all the quark masses are determined and some of them remain free parameters; (ii) eqs. (8) have only one solution but the Higgs field stransform as a reducible representation of the group of K symmetries; then again not all the quark masses are determined; (iii) eqs. (8) have only one solution and the Higgs fields transform as an irreducible representation of the group of K symmetries; then the quark mass ratios are determined by the K symmetry group and the Higgs representation ,3. In fact, if <~> is a set of vacuum expectation values which minimize the Higgs potential, it is not difficult to see that <(g¢)a> = ~a#<¢~>, where g is any element of the K symmetry group (the set of v.e.v, of such a form are called the "orbit" of <¢~>), determine the same set of quark masses, since <(gtp)a>P a = c/),~KLtF#KR ,

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and obviously {F a} and (KtLFaKR} determine the same set of quark masses. Clearly,within this framework the mass scale is not determined and one can only speak of mass ratios. Now, given a K symmetry group and an irreducible representation of it, the set o f "critical" orbits, i.e. those orbits which can be obtained by minimizing the most general invariant potential, are determined [6]. Thus, the K symmetry group determines, through the representation of the Higgs fields, the critical orbits and the possible values of the mass ratios (different critical orbits give rise to different mass ratios, in general). In this case, since both the generalized Cabibbo angles and the quark mass ratios are determined by the K symmetry group, one might of course write down relations which are numerically satisfied. However, these relations should not be regarded as the solution o f the problem of determining the generalized Cabibbo angles as functions of the quark masses, since the quark masses are not free parameters in this case. The above analysis suggests that the general strategy should be changed: the problem is not that of determining the generalized Cabibbo angles as functions of the quark masses, but rather that of finding a K symmetry group which ensures natural flavour conservation and determines the quark mass ratios; the generalized Cabibbo angles will also be independently determined by K. It is very easy to extend the above considerations to the case in which the K symmetry group is a continuous group, for example by considering an "horizontal" extension [7] of SU(2) × U(1). Especially in this case one is led to use the same type of representations for both Cu and Cd ,4 ,3 Clearly, if only one type of Higgs fields, e.g. ~d, transforms as an irreducible representation of K the down angles and the down mass ratios are determined, whereas the up angles cannot be functions of the up masses. It is easy to see that if ~d transforms as a one-dimensional representation of K, then the K symmetry group must be abelian to allow a nondegenerate Md; therefore if ~0u transforms as an irreducible representation of K one can apply the analysis of ref. [4], whereas if Cu transforms as a reducible representation of K the up mass ratios are free parameters. ,4 The "horizontal" group cannot distinguish between the left-handed up and down quarks and it looks rather unnatural that it could do so for the right-handed quarks. Moreover, in order to determine the quark mass ratios, the Higgs representations should be irreducible. 267

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One can see in general that under the constraint of natural flavour conservation in the neutral currents, the choice K ~ = K~R, (@ a#)u = (c-/)a~)d leads to zero mixing angles. The argument is the same as the one used to conclude that different solutions determine the same physical basis. Finally, we would like to remark that all of the above conclusions apply also to the case in which one has n generations of quarks and natural flavour conservation in the neutral currents is required only for the first m < n generations. This means that conditions (4), (5) are replaced by the requirement that in the basis in which the mass matrix M u (Md) is diagonal all the P~ (Pda) are block diagonal, consisting of a diagonal m X m block and a (n - m) X (n - m) block. Again the K symmetry group uniquely determines the mixing angles between the first m quarks, independently of the values of the quark masses and the results discussed so far can be generalized to this case. It is also clear that the considerations presented in this note apply equally well to any gauge group with the same sequential structure of SU(2) X U(1) with independent up and down mass matrices.

(ii) SU(2) L X SU(2) R X U(1). The Higgs-quark coupling has the following general form: .~OqH= ~LtPal~C~R + h . c . ,

(9)

where ~L denote the left-handed quarks transforming trivially under SU(2)R and as doublets under SU(2) L and f i r denote the right-handed quarks transforming trivially under SU(2)L and as doublets or singlets under SU(2)R. Natural flavour conservation in the neutral currents requires that all the quarks of a given charge and given helicity must have the same value of T3L,R and of T2,R . Thus, with the standard quark charges 2/3, - 1 / 3 , the Higgs fields which may occur in eq. (9) are either of the type (1/2, 1/2), if all the right-handed quarks are SU(2)R doublets, or of the type (1/2, 0) if the right-handed quarks are SU(2)R singlets. The latter case is equivalent to the SU(2) X U(1) case and therefore we have to discuss only the first case. By giving a vacuum expectation value to the Higgs fields (¢~)

=

,

ks 268

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the mass matrix splits into M u and Md,

M u=kc~p °~,

Md=kar !

O/

.

Natural fiavour conservation in the neutral currents now requires that all the P a be simultaneously diagonalizable,

VL*P°UR = P ; , UL, U R being independent of a. This implies that also M u and M d are simultaneously diagonalizable and therefore all the mixing angles vanish. In conclusion, in the SU(2)L X SU(2)R X U(1) case, nonvanishing mixing angles cannot arise at the lowest order and one can only appeal to some hypothetical mechanism induced by radiative corrections.

References [1 ] S. Weinberg, The problem of mass, in the Festschrift for I.I. Rabi, Trans. NY Acad. Sci. II (1977) 38; F. Wilczek and A. Zee, Phys. Lett. 70B (1977) 418; H. Frizsch, Phys. Lett. 70B (1977) 436; A. De Rfijula, H. Georgi and S.L. Glashow, Ann. Phys. (NY), to be published; for a formulation before gauge theories aee: R. Gatto, G. Sartori and M. Tonin, Phys. Lett. 28B (1968) 128; N. Cabibbo and L. Maiani, Phys. Lett. 28B (1968) 131; H. Pagels, Phys. Rev. D l l (1975) 1213. [2] R. Mohapatra and G. Senjanovid, Phys. Lett. 73B (1978) 317; H. Fritzsch, Phys. Lett. 73B (1978) 602; T. Hagiwara, T. Kitazoe, G.B. Mainland and K. Tanaka, Phys. Lett. 76B (1978) 602; T. Kitazoe and K. Tanaka, to be published; C. Branco, Phys. Lett. 76B (1978) 70; W. Kummer, Lectures at the Triangle Seminar on Quarks and gauge fields (Matraffired, Sept. 1978). [3] S.L. Glashow and S. Weinberg, Phys. Rev. D15 (1977) 1958. [4] R. Barbieri, R. Gatto and 1:. Strocchi, Phys. Lett. 74B (1978) 344; a generalization has been discussed by D. Wyler, to be published. [5 ] S. Pakvasa and H. Sugawara, Phys. Lett. 73B (1978) 61. [6] For an illustration of the method of determining the critical orbits see: G. Cicogna, F, Strocchi and R. Vergara-Caffarelli, Phys. Rev. Lett. 22 (1969) 494; Phys. Rev. D1 (1970) 1197; and more generally: L. Michel and L.A. Radicati, Ann. Phys. (NY) 66 (1971) 758. [7] F. Wilczek and A. Zee, Princeton preprint, to be published.