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in the absence of yukawa interactions the global flavour symmetry of the standard model ( sm ) is @xmath6 . it was realized a long time ago @xcite that these sort of flavour symmetries forbid flavour changing neutral currents ( fcnc ) at tree - level .
the idea behind minimal flavour violation ( mfv ) @xcite is that the yukawa matrices are the sole sources , @xmath7 that break this flavour symmetry . n.b .
the further breaking of this group down to @xmath8 due to the chiral anomaly @xcite is not central to this work . for notational simplicity we shall focus in this work on the quark sector .
results can easily be transfered to the lepton sector .
it is observed that the flavour symmetry is restored when the following transformation properties , @xmath9 , are assigned to the yukawa matrices , where @xmath10 is the quark flavour symmetry group .
an _ effective theory _ constructed from the sm fields and the yukawa matrices is then said to obey the principle of _ minimal flavour violation _
@xcite , if all operators are invariant under @xmath11 .
the mfv approach is in part motivated by the , so far , pertinent absence of fcnc .
relative bounds from @xmath12 , @xmath13-oscillations on the wilson coefficients @xmath14 . the reader is referred to the talks on mfv @xcite and mfv - bounds @xcite for further reference .    even in the absence of the knowledge of the exact dynamics
one delicate question might be raised : how is the @xmath15 symmetry broken ? if the symmetry is broken spontaneously , which is what has been proposed so far e.g. @xcite , this would then give rise to @xmath16 cp - odd massless goldstone bosons , bearing in mind possible u(1 ) anomalous contributions , in the quark sector associated with the breaking of @xmath17 .
those goldstone bosons set bounds on new flavour structure many orders of magnitude above the tev - scale @xcite .    in this paper we aim to ameliorate this situation by replacing @xmath15 by a discrete symmetry .
spontaneous breaking of discrete symmetries do _ not _ lead to goldstone bosons .
the absence of the latter in this framework constitutes the _ primary _ motivation of our work .
the main remaining issue is then to investigate whether the reduced symmetry provides enough protection for a discrete tev - scale mfv - scenario . on the technical side
the analysis of the effective field theory boils down to the classification of invariants of discrete su(3 ) subgroups .
replacing the continuous flavour symmetry with a discrete flavour symmetry requires the following additional information or assumptions : @xmath18 where @xmath19 denotes a discrete groups .
the three families transform under a 3d irreducible representation ( irrep ) of @xmath19 , which has to be specified since some groups have more than one ( modulo the complex conjugate ) .
the yukawa expansion linear mfv versus @xmath20 non - linear mfv where resummation ( non - linear @xmath21-model techniques ) become imperative . ] allows higher dimensional operators to be controlled , which otherwise give rise to rather anarchic flavour transitions @xcite .
the effective lagrangian , in the absence of the knowledge of the dynamics of the underlying model , is parametrized as , @xmath22 the sum of all combinations invariant under @xmath23 .
the dimension of the operator ( invariant ) brings in a certain hierarchy in the infinite sum above with the scale of new physics is generally obscured by loop factors , mixing angles and renormalization group effects as in any bottom - up effective field theory approach . ] .
the crucial technical point is that _ finding the invariants is equivalent to finding the constant tensors of the symmetry group_. let us here introduce the following ( standard ) notation for tensors : an index transforming under a @xmath24 or @xmath25 representation shall be denoted by lower and upper indices respectively @xmath26 non - constant tensors will be denoted by @xmath27 . in principle this tensor classification is not sufficient for our general problem since there are three different group factors .
it will though prove sufficient here to contract the other indices invariants and those groups are not of interest to us anyway . ] .
we therefore contract the @xmath28 index and directly write @xmath29 the subscript @xmath30 shall be dropped when there is no reason for confusion . in the reminder we shall use the following notation : @xmath31 the following operator classification @xmath32 directly connects to the mfv operators @xcite @xmath33 before entering into the realm of invariants
let us briefly discuss the discrete su(3 ) subgroups .
the discrete su(3 ) subgroups were classified a long time ago @xcite and further analysed as alternatives to su(3)@xmath34 in the context of the eightfold way @xcite .
explicit representations and clebsch - gordan coefficients were systematically worked out in a series of papers around 1980 @xcite , partly motivated as alternatives to @xmath35 for lattice qcd ; and further elaborated very recently @xcite in the context of family symmetries .
the discrete su(3 ) subgroups are of two kinds .
the analogues of crystal groups , of which we list here the maximal subgroups : @xmath0 , @xmath3 and @xmath2 .
the factor @xmath36 can in general either be one or three depending on whether the center of su(3 ) is divided out or not .
for the maximal subgroups it is three .
the second kind are the infinite sequence of groups , sometimes called `` dihedral like '' or `` trihedral '' , @xmath37 and @xmath38 for @xmath39 .
the symbol `` @xmath40 '' denotes a semidirect product .
the largest irreps of @xmath41 and @xmath42 are 3- , respectively 6-dimensional ; independent of @xmath43 .    that the catalogue of @xcite as compared to @xcite is not complete already surfaced in the 1980 @xcite and it has recently been reemphasized @xcite that the so - called @xmath44-groups @xcite have not been discussed systematically in the literature . in appendix
@xcite we argue that the @xmath44-groups , or more precisely the six - parameter @xmath45 matrix groups , are subgroups of @xmath46 , where @xmath47 is the lowest common multiple of @xmath43 , @xmath48 and @xmath49 .
this is sufficient for our consideration .      in discussing invariants
we are going to use the fact , based on the orthogonality theorem , that the number of times the identity appears in a kronecker product @xmath50 , denoted by @xmath51 , is equal to the number of invariants that can be formed out of the irreps @xmath52@xcite .
+ * new invariants @xmath53-level  no @xmath54 : * the problem of finding all invariants of the @xmath4 operator is equivalent to finding the invariants of the following kronecker product : @xmath55 .
the symbol @xmath56 stands for the symmetric part .
the restriction to the symmetric part can be justified by first considering the tensor products of the yukawas and the quarks separately .
the logic that we shall employ is that if the kronecker product of the @xmath57 decomposes any different from su(3 ) then there are necessarily further invariants .
the following kronecker products will prove sufficient to convey our argument : @xmath58 the two equations above make it evident that a necessary condition for an identical decomposition is that the discrete group contains a 27d irrep .
the trihedral groups @xmath41 and @xmath42 are not in this category since their largest irreducible representations ( irreps ) are at most 3- , respectively 6-dimensional . going through the character tables in @xcite and
the more recent work @xcite we realize that there is _ no _ discrete subgroup of su(3 ) which has a 27d irrep ! note , on even more general grounds that @xmath59 almost saturates the relation between the order of the group and the sum of the dimension of its irreps squared @xcite and leaves @xmath60 as the only hypothetical candidate among the crystal - like groups .
+ * no new invariants @xmath61-level for four groups : * the good news is though that the four crystal - like groups @xmath0 , @xmath1 , @xmath2 and @xmath3 do have representations that decompose as and are going to be interesting under the technical assumption of `` family irreducibility '' to be discussed below . here
we shall give an overview of the number of complex conjugate 3d irreps and the number of invariants ( under certain symmetrizations ) :    .__number of invariants for tensors of type @xmath62 , whose definition can be inferred from eq .  .
the subscripts @xmath63 indicate symmetrizations of @xmath64 and @xmath65 pairs of @xmath66 indices . @xmath67 and
@xmath68 correspond to the ( symmetric ) contractions of @xmath5 and @xmath69 in .
_ _ [ cols="<,>,^,^,^,^,^,^,^ " , ]     note that we implicitly stated that there are groups where @xmath69 operators are possible with @xmath61 solely ; without any yukawa transitions .
the group @xmath70 is an example which is discussed in appendix a of reference @xcite .
it is argued , with concrete examples , that new invariants necessarily upset the flavour hierarchy of mfv @xcite . in particular
the breaking of the continuous flavour group down to a discrete group implies that the mass - flavour basis transformation become observable and deprive the approach of its predictivity . to conclude , that the necessity of new invariants at the @xmath53-level and its connection with the well - tested @xmath4 transitions , imply that no discrete flavour group is suitable appears too hasty .
the generation mechanism of @xmath4 operators has to be reflected upon .
we distinguish the two cases where the @xmath69 process is generated via two subsequent @xmath5 parts and where this is not the case .
we shall call the former `` family irreducible '' and the latter `` family reducible '' , c.f . fig .
[ fig : deltaf12 ] .
the sm and the r - parity conserving mssm are examples of the `` family reducible''-type whereas the r - parity violating mssm is of the `` family irreducible''-type .
the composite technicolor model @xcite , in the absence of the knowledge of its non - perturbative dynamics , have to be counted into the latter class as well .
`` family reducability '' essentially implies @xmath71 factorization of the invariant and proves to be a _ sufficient _ condition for a discrete tev - scale scenario for the four crystal - like groups listed in tab .
[ tab : candidates ] . to this end
we would like to discuss two further points :    * @xmath72 model - independent : the most suitable candidate , for a tev - scale dmfv scenario is @xmath72 , since the first new invariants appear only at the @xmath53-level c.f .
tab  [ tab : candidates ] . yet : _ how small does @xmath73 need to be in order for @xmath74 to satisfy the same kind of experimental bounds as for @xmath75 found in reference @xcite ? _ the discussion in section 5.1 @xcite suggests that @xmath76 could be induced at first order in @xmath77 for new invariants , as compared to order @xmath78 in mfv .
a @xmath79 transition could therefore be @xmath80 as compared to @xmath81 in mfv . according to our reflection above we have to balence : @xmath82 .
* it has to be kept in mind that even when a model is in the `` family irreducible''-class , the fact that the vertices are @xmath23-invariant prevents the generation of non factorizable @xmath53-invariants .
we argue that this is indeed the case for the r - parity violating mssm ( at least to leading order ) @xcite .
our aim , in this work , was to point out general issues of implementing mfv via a discrete group .
we would hope that this work would be of some help for further investigations towards more specific models .
it also has to be said that although mfv has very attractive features , e.g. a sufficiently stable proton in the r - parity violating mssm @xcite , so far no explicit model for mfv has appeared in the literature .
moreover the formulation could be refined by constructing a discrete subgroup of @xmath11 which does not factor into direct products of su(3 ) subgroups @xcite .
one might wonder what the consequences are , of such a non - trivial embedding , for the invariants .
* acknowledgements : * rz would like to thank many colleagues for inspiring discussions and remarks and at last i would like to thank the organizers if the kazimierz workshop for the organization and a very pleasant atmosphere .
rz acknowledges the support of an advanced stfc fellowship .
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we investigate the consequences of replacing the continuous flavour symmetry of minimal flavour violation by a discrete group . 
 goldstone bosons , resulting from the breaking of the continuous flavour symmetry , generically lead to bounds on new flavour structure by many orders of magnitude above the tev - scale . 
 the absence of goldstone bosons for discrete symmetries constitutes the _ primary _ motivation of our work . 
 the four crystal - like groups @xmath0 , @xmath1 , @xmath2 and @xmath3 provide enough protection for a discrete tev - scale mfv scenario in the case where @xmath4 processes are generated by two subsequent @xmath5 transitions .