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Discrete symmetries, natural flavor conservation and weak mixing angles
Volume 82B, number 2
PHYSICS LETTERS
26 March 1979
DISCRETE SYMMETRIES, NATURAL FLAVOR CONSERVATION AND WEAK MIXING ANGLES 1 G. SARTORI Ddpartement de Physique Thdorique, Universitd de Gendve, 1211 Geneva 4, Switzerland Received 21 December 1978
Through their physically relevant representations, I characterize the class of finite symmetry groups insuring natural conservation of all quark flavors in neutral current effects, in generalized Weinberg-Salam models. As pointed out by Gatto, Morchio and Strocchi, these situations lead to a natural determination of all the generalized Cabibbo angles. A necessary and sufficient condition for avoiding a trivial generalized Cabibbo matrix is also given.
The striking observed suppression of strangenesschanging weak neutral currents has induced Glashow and Weinberg [1] to explore the conditions under which this effect may be justified in a natural way, that is as a consequence of the group structure and representation content of the model, rather than of the particular values taken by the free parameters of the theory. They find in particular that in a sequential extension of the standard Weinberg-Salam model all flavorchanging neutral current effects are naturally supressed if and only if there exists a (physical) basis for the quark fields in which all the coupling matrices of the scalar fields to the quarks of given helicity and charge are simultaneously diagonal, independently of the values of the arbitrary scalar quark coupling constants. This is certainly the case when all the quarks of fixed charge and helicity receive their contribution, in the quark mass matrix, from the vacuum expectation value of a single neutral scalar meson. As remarked by Glashow and Weinberg [1], this condition is also necessary, unless one can prove that all the scalar-quark coupling matrices of each charge sector are simultaneously diagonalizable for particular values of the coupling constants prescribed by some additional discrete s y m m e t r y o f the lagrangian. 1 On leave of absence from lstituto di Fisica dell'Universitfi di Padova, 35100 Padova, Italy.
My aim in this letter is to characterize the possible groups D of such symmetries. A sufficient reason to justify the study of generalized Weinberg-Salam models with a richer Higgs structure, constrained by additional discrete symmetries, is that in so doing one might be led to a significant determination of the Cabibbo angle in the tree approximation, thus solving an old problem [3] that has recently been the object of renewed interest [4]. Solutions have been tried in the direction of relations between the Cabibbo angle and quark mass ratios [3,4]. Weinberg [2] in particular has argued that the Cabibbo angle should not be the result of radiative corrections, it should rather be present already in the lowest order of perturbation theory and determined in a natural way in order to be stable against renormaiization. As pointed out by Gatto et al. [5], there is, however, a conflict between the condition of natural conservation of flavor and the possibility of expressing the Cabibbo angle in terms of quark mass ratios. They noted that it is the additional discrete symmetries insuring the flavor conservation which must fix the Higgs-quark coupling matrices in such a way that they become simultaneously diagonalizable, independently of the values of the remaining free parameters in the model. Therefore, the physical basis chosen by the quarks, and as a consequence the generalized Cabibbo angles, are determined by the discrete sym255
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26 March 1979
metries, independently of the values of the quark masses. The quark masses are only determined by the critical orbits of the global symmetry group *1. Looking for a realistic model in which the Cabibbo angle and eventually the quark masses can be determined in a "natural way", along the lines indicated above, a preliminary problem to be solved is evidently to single out under which conditions on the additional symmetry structure of the model one can get scalar-quark coupling matrices which insure natural flavor conservation +2, being all diagonal in the physical basis, independently of the values of the free parameters in the model. In the following I shall state and prove some necessary and/or sufficient conditions in this connection. I shall also give a necessary and sufficient condition to be satisfied in order to get a nontrivial generalized Cabibbo matrix. I shall consider a sequential extension of the standard Weinberg Salam model invariant with respect to a global symmetry group G = SU(2) × U(1) × D, where D is a finite set of discrete symmetries. Given the direct product structure of G the q u a r k Higgs interaction lagrangian can be written
tions which I shall comment on briefly. (1) Nondegeneration of the fixed charge quark masses. (2) Irreducibility of the group D representations L, R(q) and H(q) (q = 2/3, - 1 / 3 ) . The first assumption is justified on phenomeno logical grounds. It is essentially a simplifying hypothesis and it will be tacitly assumed in the sequel. The second assumption is not solely technical. It leads in fact to a reduction of the number of arbitrary parameters in the theory and should be accepted if one marries the ambitious proposal of determining the largest possible number of parameters from the D-symmetry. It has also to be noted that the irreducibility of L and R(q) (q = 2/3, 1/3) excludes the possibility that natural conservation of flavor, while seemingly violated, may on the contrary be insured by the existence of a discrete symmetry group larger than D. In view of the possibility that the incompleteness of the present model requires some adjustable parameters I shall mitigate, when possible, assumption (2) in the statement of my results. Let n L and nil(q) be the number of quark and qHiggs generations,
Lfh=
n L = dim L = dim R_(q),
q=2/3,-1/3
-
nil(q) nL
X h~=l ~ ~~UL,ilif P(q)'~(q)J'(q)+h.c. = i,j=l V'h ~R,j
(1)
where q is the quark charge index, while i, j and h are vector indices of the unitary representations of the group D, L, R(q) and/2/(q), according to which the left-handed SU(2)-do.ublet quark fields, fiT,i, the :~
rhHnd;sdfqld: k~b£q~1~q. Hfiggha~ge]~si ,~r~ ' e ; t d %
transform. To prove, or to simplify the proof of some of my results, I shall need in the sequel the following assump,1 Owing to the finiteness of D, the critical orbits are generally infinite in number. This leaves a sensible freedom when fitting the quark masses. ,2 Only the requirement of natural conservation of strangeness is for the moment phenomenologically justified. The stronger condition of natural conservation of all the quark flavors has to be seen as a reasonable working assumption in order to determine "naturally" the generalized Cabibbo angles [5]. 256
q = 2/3,-1/3,
nil(q) = dim _H(q) , and let t_(s), Fl(q)(s) and I-t(q)(s) be the unitary matrices representating the elements of D in L, R(q) and _H(q). The D-symmetry leads to the following conditions [5] on the coupling matrices ph, for each charge sector +3 : nH
D
Hh'h(s) I-+(s)Ph' FI(s) = F h , for all s @O.
(2)
h'=l The solutions of eqs. (2) can be written in terms of (fixed) Clebsch-Gordan coefficients for the group D and a number of arbitrary (Higgs-quark coupling) constants which depend on the number and the multiplicities of the irreducible representations contained in L_, R_ and _H. The requirement of natural conservation of all the quark flavors in the model is equivalent to the state4:3 Matrices in a physical basis will always be denoted by a ^
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ment that, for each arbitrary choice of the coupling parameters, there exist four unitary matrices o}q ) and • • a physical .. basis • +5 , such U(Rq) (q - 2/3, --1/3), defining that +4
26 March 1979 nH
~=~
ehp h .
(5)
h=l Then from eqs. (5) and (2) one gets
U(Lq)r (q)h U(Rq)+ = ~ ( q ) h , diagonal for all h ' s .
(3)
C+(s) ~ 9~ + ((s)
The generalized Cabibbo matrix will then be UC = U(2/3), ,(-1/3)+ " L "%
nn (4)
It is not uniguely determined and will be said to be trivial when U(L2/3) and U (-1/3) can be chosen in such a way that U C = 1. The statement of the main result of this letter requires some preliminary definitions. A unitary matrix representation _M of a group D
s -~ Mi/(s ) , for all s E D is said to be a monomial representation, if all the representative matrices have but one nontrivial element (of modulus 1) in each line and column. Given two irreducible monomial representations M (1) and M (2) of the same group and dimensions, one can define a product M = M (1) *_34(2) in the following way:
Mii(s )-=M~O.1)(s)M~i2)(s),
s ED.
Since irreducible monomial representations are always induced monomial representations [6], M is again a monomial representation. In particular, for any monomial M, P_ -- M * M * will be a permutation representation (Pi](s) = 0 or 1). The following theorem allows a first characterization o l D . T h e o r e m 1. I f L and R_ are irreducible representations, the D-symmetry will insure natural conservation of all the quark flavors if and only if ~ and are monomial representations of D #5 such that L_'*£* =/~ */~* and _HC £ . / ~ * , for each charge sector. Proof. Let _//be irreducible and call ch the VEV's of the fields ~bh . The mass matrix o f the quarks in a fixed charge sector is: :1-4Here and in the following it is tacitly assumed that the multiplicity of any irreducible component _Hi c _His equal to the multiplicity of_H in L X R_*, or to zero.
=
h,k,h',k'=l
(6)
• ..h'h*~s,Hk'k~s,ph ch,ek,ll ( ) ~ ) pk+ ,
The r.h.s, of eq. (6) is a diagonal matrix unitarily equivalent to the diagonal matrix _¢~c-~+. As a consequence, given the non degeneracy of this last matrix and eq. (2), ~ must be a monomial representation of D. In a similar way, the transformation properties and the non degeneracy of c,~ + c ~ imply that ~ must be a monomial representation. When reinserting these pieces of information into eq. (2) one finds:
I£ij(s)l = It~i](s)l ,
(7)
and nL
nH
~
i'=1 h ' = l
{£~i(S)t~i,i(s)Hh'h(s)
--6i,i6h'h )
(s)
I•/t' = 0 ,
for all s ~ D .
The number v of independent solutions { I ~ ' ) i = t ..... n L , h = l ..... n i l '
of this system of linear equations is v = nLn H - r. where r is the rank o f the coefficient matrix C:
Csih, i'h' -~ ['~'i (s)t~i'i (s) Hh'h (s) -- 6i, i 6 h'h .
(9)
The rank of C can be calculated as the rank of the matrix Q -= (1/2d) C + C ,
(10)
where d is the number of elements in D. It is not difficult to verify that Q is a projector,
s -+ Ui,h," ih(S) = £;i(s)t~i,i(s)ah'h(s)
(1 l )
being a unitary representation of D as follows from 257
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the remark preceding the statement of the theorem. The rank of Q is therefore its trace:
r= TrQ= nLnH
1~ x(LL**k)(s)x(H_)(s ) d s~ D
(12)
where XA (s) is the character of s E D in the representation A. Thus v > 0 if and only if H C £ •/~ *. When _H is reducible the same agruments can be applied to each irreducible component in its Clebsch - G o r d a n series. Theorem 1 has an interesting corollary. Corollary. Under the same assumptions as in theorem 1, D will insure natural conservation of all the quark flavors only if it is homorphic to a transitive 4-6 subgroup of the symmetric group ShE and the kernel of the homomorphism is abelian. Proof. F r o m the p r o o f of theorem 1 one realizes that it is possible to homomorphically associate to each s C D a permutation of the masses of the chargeq quarks. Thus D is homorphic to a subgroup of SnL" This subgroup has to be transitive owing to the irreducibility of L_, requiring at least n 2 linearly independent matrices in/£. The kernel of the homomorphism is made up only of the diagonal matrices o f [ , it is therefore abelian. Theorem 1 and its colollary are quite selective and turn out to be rather effective in practical applications. In fact, given the number n L of quark generations, simple methods can be found in the mathematical literature in order to single out the transitive subgroups of SnL. Given one of such subgroups, group extension theory allows to construct a class of groups among which one has to single out those admitting irreducible monomial representations o f dimension n L. I would like to recall that nilpotency is (unfortunately only) a sufficient condition for a group to admit irreducible monomial representations [6]. Nilpotent groups are among the best classified in group theory. A simple example illustrating theorem 1 is the following. The symmetric group S4 has five classes of irreducible representations: 1, 1', 2, 3, -3'. They are all monomial in a convenient basis and real. 4-6 A group of permutations of n symbols is said to be transitive, if, given any two symbols, it contains at least one permutation sending the first into the second. 258
26 March 1979
The choice L = R = 3 and H = 2 + 1 (H = 3) is good (not good) for natural conservation of one charge sector quark flavors, since 2 + 1 (3) is (is not) equivalent to 3*3. Note that both 2 + 1 and 3 are contained in 3 * 3, and can therefore couple to L = 3 and R = 3. S4 has an abelian invariant subgroup V4, and S4/V4 = $3,, which is evidently a transitive subgroup of itself. It is possible to prove a further very simple sufficient condition for natural conservation of flavor. To this purpose I need the following lemma which is a generalization of a well known result on sets of hermitean matrices. Lemma 1. Let { A i } i = 1 ..... n be a set of square matrices. Then the sets {A/A/.+ )i,/= 1..... n and {A+Ai}i,/= 1 ..... n being abelian is equivalent to the existence of a biunitary transformation which diagonalizes simultaneously all the matrices A i, i = 1,..., n. Proof. The necessity of the condition is evident. In order to prove the sufficiency note that in the hypotheses of the lemma the set
S~L =-- {AiA;}i,j= 1 ..... n
(S~R =
{A+Aj}i,j = 1 ..... n)
is an abelian set of normal matrices, therefore there exists a unitary martix UL (UR) diagonalizing all the matrices in the set simultaneously. It can be shown that U L and U R can be chosen in such a way that Ai -= ULAiU ~ , i = 1 .... ,n is diagonal. Lemma 1 is an advantageous basis-independent reformulation of the Glashow-Weinberg criterium for natural conservation of flavors. An immediate obvious application is the following: Theorem 2. In a SU(2) × U(1) × D generalized Weinberg-Salam model all the charge-q quark flavors are naturally conserved if and only if the set of matrices
.o~(Lq) ---- ~I "(q)h ["(q)k+t
Jh, k = 1..... nil(q) ,
-~)
=- {I'(q)h+p(q)k)h , k = 1 ..... nil(q) ,
are abelian. Besides, the generalized Cabibbo matrix will be non trivial iff s~ L = -~(L2/3) U M(L-1/3) is nonabelian. Proof. The first part of the theorem coincides with the lemma. For the second part it is sufficient to note that iff -~L is abelian, according to the arguments sketched in the proof of the lemma, one can choose
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U(L2/3) = U (-1/3). Therefore, the generalized Cabibbo matrix defined in eq. (4) turns out to be trivial. I am read to prove Theorem 3. If L and R(q) are irreducible, then the following conditions 1 and 2 are sufficient to insure natural conservation of all the charge-q quark flavors: (1) nil(q) ~< 2; (2) the multiplicity of each irreducible component of H(q) in its C l e b s c h - G o r d a n series is one. Proof. The theorem is non-trivial only in the following two cases: (1)_His reducible in the sum o f two singlets: H = 1 + 1'; (2) H i s irreducible and n H = 2. In the first case it is known from the properties o f the C l e b s c h - G o r d a n coefficients that in the hypotheses of the theorem F 1- and F 1' are proportional to unitary matrices and therefore simultaneously diagonalizable by means o f a biunitary transformation. In case (2), from the properties o f the C l e b s c h - G o r d a n coefficients one gets 2
2
. p h i ' h + =/~lnL = ~ r h + r h , h=l h=l
(13)
and it is a simple matter of matrix algebra to show that eqs. (13) imply the abelianity o f the sets S~(Lq) and S~(Rq) defined in theorem 3, which is then sufficient to conclude. It has to be noted that in the trivial case n H = 1, p ! is proportional to a unitary matrix, therefore the charge-q quark masses are all degenerate. Before concluding I would like to add some remarks. Remark 1. Nowhere in this letter have I used the assumption that D is finite, except in the p r o o f of the last condition in theorem I, therefore all the other results hold equally well for continuous compact groups. Such groups should however be gauged and this would modify the gauge structure o f the model. Remark 2. The problem of determining one D, insuring, through a convenient choice of its representa-
26 March 1979
tions, in a non-trivial way (n H ~> 1) the natural conservation o f all the quark flavors, while giving a nontrivial Cabibbo matrix, has certainly solutions. For instance i f D is S 3 or the quaternion group one can get 0C = 45 °. The real problem is evidently to find out phenomenologically acceptable solutions. It is clearly not possible to say a priori if such solutions exist. Natural conservation of all the quark flavors might also be too strong a condition, but for the moment it is a very appealing one.
References [1] S.L. Glashow and S. Weinberg, Phys. Rev. D15 (1977) 1958. [2] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in: Elementary particles theory: relativistic groups and analiticity (Nobel Syrup. No. 8), ed. N. Svartholm (Almquist and Wiksells, Stockholm, 1968), p. 367; S.L. Glashow, J. lliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. [3] 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. [4] S. Weinberg, The problem of mass, in: Festschrift for I.I. Rabi, Trans. NY Acad. Sci. II (1977) 38; F. Wilczek and A. Zee, Phys. Lett. 70B (1977) 418; H. Fritzsch, Phys. Lett. 70B (1977) 436; 73B (1978) 602; A. De Rujula, H. Georgi and S.L. Glashow, Ann. Phys. (NY), to be published; R. Mohapatra and G. Senjanovich, Phys. Lett. 73B (1978) 317; T. Hagiwara, T. Kitazoe, G.B. Mainland and T. 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 (Matrafured, Sept. 1978); S. Pakvasa and H. Sugawara, Phys. Lett. 73B (1978) 61; F. Wilczek and A. Zee, Princeton preprint. [5] R. Gatto, G. Morchio and F. Strocchi, Phys. Lett. 80B (1979) 265; and private communication. [6] C.W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras (Wiley, 1962) p. 314.
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26 March 1979
DISCRETE SYMMETRIES, NATURAL FLAVOR CONSERVATION AND WEAK MIXING ANGLES 1 G. SARTORI Ddpartement de Physique Thdorique, Universitd de Gendve, 1211 Geneva 4, Switzerland Received 21 December 1978
Through their physically relevant representations, I characterize the class of finite symmetry groups insuring natural conservation of all quark flavors in neutral current effects, in generalized Weinberg-Salam models. As pointed out by Gatto, Morchio and Strocchi, these situations lead to a natural determination of all the generalized Cabibbo angles. A necessary and sufficient condition for avoiding a trivial generalized Cabibbo matrix is also given.
The striking observed suppression of strangenesschanging weak neutral currents has induced Glashow and Weinberg [1] to explore the conditions under which this effect may be justified in a natural way, that is as a consequence of the group structure and representation content of the model, rather than of the particular values taken by the free parameters of the theory. They find in particular that in a sequential extension of the standard Weinberg-Salam model all flavorchanging neutral current effects are naturally supressed if and only if there exists a (physical) basis for the quark fields in which all the coupling matrices of the scalar fields to the quarks of given helicity and charge are simultaneously diagonal, independently of the values of the arbitrary scalar quark coupling constants. This is certainly the case when all the quarks of fixed charge and helicity receive their contribution, in the quark mass matrix, from the vacuum expectation value of a single neutral scalar meson. As remarked by Glashow and Weinberg [1], this condition is also necessary, unless one can prove that all the scalar-quark coupling matrices of each charge sector are simultaneously diagonalizable for particular values of the coupling constants prescribed by some additional discrete s y m m e t r y o f the lagrangian. 1 On leave of absence from lstituto di Fisica dell'Universitfi di Padova, 35100 Padova, Italy.
My aim in this letter is to characterize the possible groups D of such symmetries. A sufficient reason to justify the study of generalized Weinberg-Salam models with a richer Higgs structure, constrained by additional discrete symmetries, is that in so doing one might be led to a significant determination of the Cabibbo angle in the tree approximation, thus solving an old problem [3] that has recently been the object of renewed interest [4]. Solutions have been tried in the direction of relations between the Cabibbo angle and quark mass ratios [3,4]. Weinberg [2] in particular has argued that the Cabibbo angle should not be the result of radiative corrections, it should rather be present already in the lowest order of perturbation theory and determined in a natural way in order to be stable against renormaiization. As pointed out by Gatto et al. [5], there is, however, a conflict between the condition of natural conservation of flavor and the possibility of expressing the Cabibbo angle in terms of quark mass ratios. They noted that it is the additional discrete symmetries insuring the flavor conservation which must fix the Higgs-quark coupling matrices in such a way that they become simultaneously diagonalizable, independently of the values of the remaining free parameters in the model. Therefore, the physical basis chosen by the quarks, and as a consequence the generalized Cabibbo angles, are determined by the discrete sym255
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PHYSICS LETTERS
26 March 1979
metries, independently of the values of the quark masses. The quark masses are only determined by the critical orbits of the global symmetry group *1. Looking for a realistic model in which the Cabibbo angle and eventually the quark masses can be determined in a "natural way", along the lines indicated above, a preliminary problem to be solved is evidently to single out under which conditions on the additional symmetry structure of the model one can get scalar-quark coupling matrices which insure natural flavor conservation +2, being all diagonal in the physical basis, independently of the values of the free parameters in the model. In the following I shall state and prove some necessary and/or sufficient conditions in this connection. I shall also give a necessary and sufficient condition to be satisfied in order to get a nontrivial generalized Cabibbo matrix. I shall consider a sequential extension of the standard Weinberg Salam model invariant with respect to a global symmetry group G = SU(2) × U(1) × D, where D is a finite set of discrete symmetries. Given the direct product structure of G the q u a r k Higgs interaction lagrangian can be written
tions which I shall comment on briefly. (1) Nondegeneration of the fixed charge quark masses. (2) Irreducibility of the group D representations L, R(q) and H(q) (q = 2/3, - 1 / 3 ) . The first assumption is justified on phenomeno logical grounds. It is essentially a simplifying hypothesis and it will be tacitly assumed in the sequel. The second assumption is not solely technical. It leads in fact to a reduction of the number of arbitrary parameters in the theory and should be accepted if one marries the ambitious proposal of determining the largest possible number of parameters from the D-symmetry. It has also to be noted that the irreducibility of L and R(q) (q = 2/3, 1/3) excludes the possibility that natural conservation of flavor, while seemingly violated, may on the contrary be insured by the existence of a discrete symmetry group larger than D. In view of the possibility that the incompleteness of the present model requires some adjustable parameters I shall mitigate, when possible, assumption (2) in the statement of my results. Let n L and nil(q) be the number of quark and qHiggs generations,
Lfh=
n L = dim L = dim R_(q),
q=2/3,-1/3
-
nil(q) nL
X h~=l ~ ~~UL,ilif P(q)'~(q)J'(q)+h.c. = i,j=l V'h ~R,j
(1)
where q is the quark charge index, while i, j and h are vector indices of the unitary representations of the group D, L, R(q) and/2/(q), according to which the left-handed SU(2)-do.ublet quark fields, fiT,i, the :~
rhHnd;sdfqld: k~b£q~1~q. Hfiggha~ge]~si ,~r~ ' e ; t d %
transform. To prove, or to simplify the proof of some of my results, I shall need in the sequel the following assump,1 Owing to the finiteness of D, the critical orbits are generally infinite in number. This leaves a sensible freedom when fitting the quark masses. ,2 Only the requirement of natural conservation of strangeness is for the moment phenomenologically justified. The stronger condition of natural conservation of all the quark flavors has to be seen as a reasonable working assumption in order to determine "naturally" the generalized Cabibbo angles [5]. 256
q = 2/3,-1/3,
nil(q) = dim _H(q) , and let t_(s), Fl(q)(s) and I-t(q)(s) be the unitary matrices representating the elements of D in L, R(q) and _H(q). The D-symmetry leads to the following conditions [5] on the coupling matrices ph, for each charge sector +3 : nH
D
Hh'h(s) I-+(s)Ph' FI(s) = F h , for all s @O.
(2)
h'=l The solutions of eqs. (2) can be written in terms of (fixed) Clebsch-Gordan coefficients for the group D and a number of arbitrary (Higgs-quark coupling) constants which depend on the number and the multiplicities of the irreducible representations contained in L_, R_ and _H. The requirement of natural conservation of all the quark flavors in the model is equivalent to the state4:3 Matrices in a physical basis will always be denoted by a ^
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ment that, for each arbitrary choice of the coupling parameters, there exist four unitary matrices o}q ) and • • a physical .. basis • +5 , such U(Rq) (q - 2/3, --1/3), defining that +4
26 March 1979 nH
~=~
ehp h .
(5)
h=l Then from eqs. (5) and (2) one gets
U(Lq)r (q)h U(Rq)+ = ~ ( q ) h , diagonal for all h ' s .
(3)
C+(s) ~ 9~ + ((s)
The generalized Cabibbo matrix will then be UC = U(2/3), ,(-1/3)+ " L "%
nn (4)
It is not uniguely determined and will be said to be trivial when U(L2/3) and U (-1/3) can be chosen in such a way that U C = 1. The statement of the main result of this letter requires some preliminary definitions. A unitary matrix representation _M of a group D
s -~ Mi/(s ) , for all s E D is said to be a monomial representation, if all the representative matrices have but one nontrivial element (of modulus 1) in each line and column. Given two irreducible monomial representations M (1) and M (2) of the same group and dimensions, one can define a product M = M (1) *_34(2) in the following way:
Mii(s )-=M~O.1)(s)M~i2)(s),
s ED.
Since irreducible monomial representations are always induced monomial representations [6], M is again a monomial representation. In particular, for any monomial M, P_ -- M * M * will be a permutation representation (Pi](s) = 0 or 1). The following theorem allows a first characterization o l D . T h e o r e m 1. I f L and R_ are irreducible representations, the D-symmetry will insure natural conservation of all the quark flavors if and only if ~ and are monomial representations of D #5 such that L_'*£* =/~ */~* and _HC £ . / ~ * , for each charge sector. Proof. Let _//be irreducible and call ch the VEV's of the fields ~bh . The mass matrix o f the quarks in a fixed charge sector is: :1-4Here and in the following it is tacitly assumed that the multiplicity of any irreducible component _Hi c _His equal to the multiplicity of_H in L X R_*, or to zero.
=
h,k,h',k'=l
(6)
• ..h'h*~s,Hk'k~s,ph ch,ek,ll ( ) ~ ) pk+ ,
The r.h.s, of eq. (6) is a diagonal matrix unitarily equivalent to the diagonal matrix _¢~c-~+. As a consequence, given the non degeneracy of this last matrix and eq. (2), ~ must be a monomial representation of D. In a similar way, the transformation properties and the non degeneracy of c,~ + c ~ imply that ~ must be a monomial representation. When reinserting these pieces of information into eq. (2) one finds:
I£ij(s)l = It~i](s)l ,
(7)
and nL
nH
~
i'=1 h ' = l
{£~i(S)t~i,i(s)Hh'h(s)
--6i,i6h'h )
(s)
I•/t' = 0 ,
for all s ~ D .
The number v of independent solutions { I ~ ' ) i = t ..... n L , h = l ..... n i l '
of this system of linear equations is v = nLn H - r. where r is the rank o f the coefficient matrix C:
Csih, i'h' -~ ['~'i (s)t~i'i (s) Hh'h (s) -- 6i, i 6 h'h .
(9)
The rank of C can be calculated as the rank of the matrix Q -= (1/2d) C + C ,
(10)
where d is the number of elements in D. It is not difficult to verify that Q is a projector,
s -+ Ui,h," ih(S) = £;i(s)t~i,i(s)ah'h(s)
(1 l )
being a unitary representation of D as follows from 257
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the remark preceding the statement of the theorem. The rank of Q is therefore its trace:
r= TrQ= nLnH
1~ x(LL**k)(s)x(H_)(s ) d s~ D
(12)
where XA (s) is the character of s E D in the representation A. Thus v > 0 if and only if H C £ •/~ *. When _H is reducible the same agruments can be applied to each irreducible component in its Clebsch - G o r d a n series. Theorem 1 has an interesting corollary. Corollary. Under the same assumptions as in theorem 1, D will insure natural conservation of all the quark flavors only if it is homorphic to a transitive 4-6 subgroup of the symmetric group ShE and the kernel of the homomorphism is abelian. Proof. F r o m the p r o o f of theorem 1 one realizes that it is possible to homomorphically associate to each s C D a permutation of the masses of the chargeq quarks. Thus D is homorphic to a subgroup of SnL" This subgroup has to be transitive owing to the irreducibility of L_, requiring at least n 2 linearly independent matrices in/£. The kernel of the homomorphism is made up only of the diagonal matrices o f [ , it is therefore abelian. Theorem 1 and its colollary are quite selective and turn out to be rather effective in practical applications. In fact, given the number n L of quark generations, simple methods can be found in the mathematical literature in order to single out the transitive subgroups of SnL. Given one of such subgroups, group extension theory allows to construct a class of groups among which one has to single out those admitting irreducible monomial representations o f dimension n L. I would like to recall that nilpotency is (unfortunately only) a sufficient condition for a group to admit irreducible monomial representations [6]. Nilpotent groups are among the best classified in group theory. A simple example illustrating theorem 1 is the following. The symmetric group S4 has five classes of irreducible representations: 1, 1', 2, 3, -3'. They are all monomial in a convenient basis and real. 4-6 A group of permutations of n symbols is said to be transitive, if, given any two symbols, it contains at least one permutation sending the first into the second. 258
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The choice L = R = 3 and H = 2 + 1 (H = 3) is good (not good) for natural conservation of one charge sector quark flavors, since 2 + 1 (3) is (is not) equivalent to 3*3. Note that both 2 + 1 and 3 are contained in 3 * 3, and can therefore couple to L = 3 and R = 3. S4 has an abelian invariant subgroup V4, and S4/V4 = $3,, which is evidently a transitive subgroup of itself. It is possible to prove a further very simple sufficient condition for natural conservation of flavor. To this purpose I need the following lemma which is a generalization of a well known result on sets of hermitean matrices. Lemma 1. Let { A i } i = 1 ..... n be a set of square matrices. Then the sets {A/A/.+ )i,/= 1..... n and {A+Ai}i,/= 1 ..... n being abelian is equivalent to the existence of a biunitary transformation which diagonalizes simultaneously all the matrices A i, i = 1,..., n. Proof. The necessity of the condition is evident. In order to prove the sufficiency note that in the hypotheses of the lemma the set
S~L =-- {AiA;}i,j= 1 ..... n
(S~R =
{A+Aj}i,j = 1 ..... n)
is an abelian set of normal matrices, therefore there exists a unitary martix UL (UR) diagonalizing all the matrices in the set simultaneously. It can be shown that U L and U R can be chosen in such a way that Ai -= ULAiU ~ , i = 1 .... ,n is diagonal. Lemma 1 is an advantageous basis-independent reformulation of the Glashow-Weinberg criterium for natural conservation of flavors. An immediate obvious application is the following: Theorem 2. In a SU(2) × U(1) × D generalized Weinberg-Salam model all the charge-q quark flavors are naturally conserved if and only if the set of matrices
.o~(Lq) ---- ~I "(q)h ["(q)k+t
Jh, k = 1..... nil(q) ,
-~)
=- {I'(q)h+p(q)k)h , k = 1 ..... nil(q) ,
are abelian. Besides, the generalized Cabibbo matrix will be non trivial iff s~ L = -~(L2/3) U M(L-1/3) is nonabelian. Proof. The first part of the theorem coincides with the lemma. For the second part it is sufficient to note that iff -~L is abelian, according to the arguments sketched in the proof of the lemma, one can choose
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PHYSICS LETTERS
U(L2/3) = U (-1/3). Therefore, the generalized Cabibbo matrix defined in eq. (4) turns out to be trivial. I am read to prove Theorem 3. If L and R(q) are irreducible, then the following conditions 1 and 2 are sufficient to insure natural conservation of all the charge-q quark flavors: (1) nil(q) ~< 2; (2) the multiplicity of each irreducible component of H(q) in its C l e b s c h - G o r d a n series is one. Proof. The theorem is non-trivial only in the following two cases: (1)_His reducible in the sum o f two singlets: H = 1 + 1'; (2) H i s irreducible and n H = 2. In the first case it is known from the properties o f the C l e b s c h - G o r d a n coefficients that in the hypotheses of the theorem F 1- and F 1' are proportional to unitary matrices and therefore simultaneously diagonalizable by means o f a biunitary transformation. In case (2), from the properties o f the C l e b s c h - G o r d a n coefficients one gets 2
2
. p h i ' h + =/~lnL = ~ r h + r h , h=l h=l
(13)
and it is a simple matter of matrix algebra to show that eqs. (13) imply the abelianity o f the sets S~(Lq) and S~(Rq) defined in theorem 3, which is then sufficient to conclude. It has to be noted that in the trivial case n H = 1, p ! is proportional to a unitary matrix, therefore the charge-q quark masses are all degenerate. Before concluding I would like to add some remarks. Remark 1. Nowhere in this letter have I used the assumption that D is finite, except in the p r o o f of the last condition in theorem I, therefore all the other results hold equally well for continuous compact groups. Such groups should however be gauged and this would modify the gauge structure o f the model. Remark 2. The problem of determining one D, insuring, through a convenient choice of its representa-
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tions, in a non-trivial way (n H ~> 1) the natural conservation o f all the quark flavors, while giving a nontrivial Cabibbo matrix, has certainly solutions. For instance i f D is S 3 or the quaternion group one can get 0C = 45 °. The real problem is evidently to find out phenomenologically acceptable solutions. It is clearly not possible to say a priori if such solutions exist. Natural conservation of all the quark flavors might also be too strong a condition, but for the moment it is a very appealing one.
References [1] S.L. Glashow and S. Weinberg, Phys. Rev. D15 (1977) 1958. [2] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in: Elementary particles theory: relativistic groups and analiticity (Nobel Syrup. No. 8), ed. N. Svartholm (Almquist and Wiksells, Stockholm, 1968), p. 367; S.L. Glashow, J. lliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. [3] 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. [4] S. Weinberg, The problem of mass, in: Festschrift for I.I. Rabi, Trans. NY Acad. Sci. II (1977) 38; F. Wilczek and A. Zee, Phys. Lett. 70B (1977) 418; H. Fritzsch, Phys. Lett. 70B (1977) 436; 73B (1978) 602; A. De Rujula, H. Georgi and S.L. Glashow, Ann. Phys. (NY), to be published; R. Mohapatra and G. Senjanovich, Phys. Lett. 73B (1978) 317; T. Hagiwara, T. Kitazoe, G.B. Mainland and T. 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 (Matrafured, Sept. 1978); S. Pakvasa and H. Sugawara, Phys. Lett. 73B (1978) 61; F. Wilczek and A. Zee, Princeton preprint. [5] R. Gatto, G. Morchio and F. Strocchi, Phys. Lett. 80B (1979) 265; and private communication. [6] C.W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras (Wiley, 1962) p. 314.
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