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the origin of the flavor structure is a big puzzle in particle physics . in the standard model ,
the masses and mixing of quarks and leptons are just parameters given by hand .
obviously this is unsatisfactory , and there have been many attempts to explain the fermion mass matrices in the literature @xcite .
there are some characteristic features in the quark and lepton mass matrices . as for the mixing among different generations , the quark sector has all small mixing angles whereas the neutrino sector has two large mixings and one small one . another important property concerns the mass hierarchy of the quarks and leptons .
the third generation of the quarks and charged leptons are heavier than the other generations .
in particular , the top quark is much heavier than the others .
here we shall point out that the large top quark mass can be accounted for by a singular normalization of its kinetic term .
after rescaling the fields to get the canonical normalization , one can obtain a large yukawa coupling and thus a large mass of the top quark .
we shall demonstrate this interesting possibility in the framework of a democratic fermion mass matrix model @xcite . in the democratic @xmath0 symmetry ,
the fermions belong to 3 dimensional reducible representations , which are decomposed into 1 and 2 dimensional representations .
the trivial representation is identified with the field in the third generation .
this group theoretical structure allows the third generation to have different normalization from the other two generations@xcite . we will show that , with diagonal breaking of the @xmath0 symmetry , the singular normalization of the top quark is essential to reproduce observed quark masses and mixing angles .
we will also discuss other examples where this mechanism may be important .
the model we are considering was described in refs .
non - canonical khler potential was introduced in ref .
we would like to briefly review essential points of our model . here
we focus on the quark sector .
@xcite for the successful extensions to the lepton sector , including neutrino masses and mixing .    in the quark sector
, there are a product of permutation symmetry groups @xmath1 . under @xmath2 , the doublet quarks @xmath3 ( @xmath4 2 , 3 ) transform as 3 dimensional representation , @xmath5 , which is decomposed into two irreducible representations : @xmath6 .
here @xmath7 is a trivial representation and @xmath8 is two dimensional one .
in fact , @xmath9 does not change under the permutation .
the other two combinations will constitute a basis of @xmath8 .
similar arguments can apply for @xmath10 singlet quarks @xmath11 and @xmath12 .    in the following
we consider the minimal supersymmetric standard model ( mssm ) to illustrate our points , and use the terminology of supersymmetry , such as khler potential and superpotential .
supersymmetry is , however , not essential for our subsequent argument .    to begin with
we will develop some formalism .
let us introduce a @xmath5 representation , @xmath13 , which will be identified with three families of the mssm matter fields later .
the @xmath0 invariant khler potential is written @xmath14_{ij } x^\dag_i x_j ,     \label{eq : kahler}\end{aligned}\ ] ] where @xmath15 reflecting the fact that the above khler potential is a bilinear function of @xmath13 , there arise two invariants : the universal matrix @xmath16 and the democratic matrix @xmath17 . here
@xmath18 and @xmath19 are functions of fields in general and we omit terms which have no relation to mass matrices .
we assume that some dynamics fixes vacuum expectation values ( vevs ) , @xmath20 existence of the democratic part @xmath21 plays an essential role in our arguments .
in fact , the non - universal kinetic terms stem from @xmath22 , @xmath23_{ij}.    \label{eq : metric}\end{aligned}\ ] ] using the following matrix : @xmath24 @xmath25 is diagonalized as @xmath26 where @xmath27 hereafter we will call the field basis obtained this way the diagonal basis .
the kinetic terms are written explicitly @xmath28 in this basis , it becomes clear that the reducible @xmath5 representation consists of the trivial @xmath7 representation and the irreducible @xmath8 representation .
after rescaling the fields by using the diagonal matrix @xmath29 we obtain canonically normalized kinetic terms : @xmath30 we will call this field basis the canonical basis .
let us consider yukawa interaction which consists of two matter fields @xmath31 and a higgs field @xmath32 , @xmath33 where @xmath34 represent a yukawa coupling matrix .
the @xmath35 symmetry allows @xmath34 to have only the democratic matrix : @xmath36 where @xmath37 is a constant .
it is obvious that @xmath38 in the diagonal basis and that @xmath39 in the canonical basis , and thus no mixing angle arises .
notice that only the @xmath40 element in @xmath41 and @xmath42 survives since both of the @xmath7 representation of @xmath43 and the one of @xmath44 must be involved in order for couplings to be invariant .
therefore , there appear two massless fields and one massive field after the higgs develops a vev , which , roughly speaking , simulates our world .
@xmath0 breaking parameters must be involved in order to make our model realistic .
let us assume that the small @xmath0 breaking matrix possesses a diagonal form : @xmath45 .
\label{eq : yukawa_dem}\end{aligned}\ ] ] diagonalization of the kinetic terms using @xmath46 followed by rescaling the fields using @xmath47 drives the yukawa matrix to the following form : @xmath48 c_r      \label{eq : yukawa_c}\end{aligned}\ ] ] in the canonical basis .
one is convinced that @xmath42 is almost diagonal and that resulting mixing angles will be small .
after diagonalization of @xmath42 , @xmath49 we obtain @xmath50 where @xmath51 here we choose @xmath52 and @xmath53 to be real and ignore higher orders of @xmath52 and @xmath53 .
the unitary matrices @xmath54 and @xmath55 are given by @xmath56 where @xmath57 notice that in the case where @xmath58 , a hierarchical spectrum is derived : @xmath59 hereafter we assume this type of hierarchy since it matches with the observed quark mass hierarchy as we will see shortly .
let us apply the above discussion to the quark sector in the mssm , in which the left - handed quark doublets @xmath60 , the charge - conjugated right - handed up - type quarks @xmath61 and the charge - conjugated right - handed down - type quarks @xmath62 exist .
the up - type yukawa coupling matrix and the down - type one are diagonalized as @xmath63 respectively .
the cabibbo - kobayashi - maskawa ( ckm ) matrix is given by @xmath64 where @xmath65 one finds that the small @xmath0 breaking parameters induce small mixing angles . notice that our model naturally includes the model discussed in ref .
@xcite , where non - universal kinetic terms are absent .
we can express the ckm matrix elements in terms of the mass eigenvalues .
the cabibbo angle is determined solely by the observed values as @xmath66 this fritzsch relation have been known to be successful @xcite .
@xmath67 and @xmath68 are given by @xmath69 where @xmath70 notice that in eq .
( [ eq : ckm ] ) values of @xmath71 and @xmath72 are almost fixed by the empirical masses as @xmath73 ( see eqs .
( [ eq : masses ] ) and ( [ eq : ulr ] ) ) .
thus , parameters determined from @xmath67 and @xmath74 are @xmath75 and @xmath76 .
@xmath77 does not play an essential role .
let us perform crude estimation before making numerical analysis , elucidating the importance of unusual structure of kinetic terms . for the present , we adopt the following representative values : @xmath78 taking into account that @xmath79 the area where @xmath80 itself is close to @xmath81 is preferable . from the constraint @xmath82
one finds that the most favorable range is @xmath83 and @xmath84 we discard the case where @xmath85 since rather large @xmath80 is demanded . combining this inequality ( eq .
( [ eq : x ] ) ) with the constraint @xmath86 , we obtain @xmath87 for @xmath88 and @xmath89 .
thus , we conclude that considerably suppressed @xmath76 is necessary in order that our @xmath0 model with the diagonal breaking matrices explains the empirical masses and mixings . recalling that @xmath90 this conclusion highlights a possibility that extremely large top mass is attributed to singular structure of normalization of the fields in the case of small @xmath91 .
put another way , we draw a plot of the allowed value of the top quark mass @xmath92 as a function of @xmath76 when the other quantities are fit with their experimental values as above .
the allowed region is depicted in fig . [
fig : mt ] .
one finds that the allowed top mass is rather sensitive to the choice of @xmath76 .
it is interesting to observe that the case of universal kinetic terms examined in ref .
@xcite is completely ruled out and thus the inclusion of the singular normalization with extremely small @xmath76 is essential to make the model realistic .
we now exhibit numerical analysis based on the diagonal breaking matrices . for simplicity we set @xmath93 , which does not alter our conclusion .
the following parameter set , @xmath94 reproduces mass eigenvalues @xmath95 and magnitudes of the ckm matrix elements @xmath96 which should be compared with the experimental data evaluated at the @xmath97-boson mass scale @xcite .
we have made the comparison at the @xmath97-boson mass scale .
the same conclusion that extremely small @xmath76 is required holds even if we made this comparison at the gut scale in the mssm , though the value @xmath76 itself is different .
this is because the effect of running comes from the top yukawa coupling , which appears in the wave - function renormalization of the top quark in supersymmetry .
although we have so far supposed that the parameters which are responsible for quark masses and mixing angles are real , the observed @xmath98 asymmetry can be also explained in this framework by taking complex @xmath53 and @xmath52 .
in this paper , we have pointed out the possibility that the wave function normalization of the top quark can be singular , which makes it very heavy .
we have illustrated this in the framework of the democratic fermion ansatz . there
the fermion mass structure is controlled by the @xmath0 permutation symmetries , with diagonal breaking introduced . in this case
, we can make even a stronger statement : the singular normalization is essential to reproduce the top quark mass and the other quark masses as well as their mixing .
this investigation presented here is indeed based on a particular form of the @xmath0 breaking . however , we emphasize that the large top quark mass can be realized by invoking singular structure of the corresponding kinetic terms .
this argument is irrespective of the form of the @xmath0 breaking matrices .
this new idea of highly different normalization of fields can save theories in which yukawa couplings would otherwise be suppressed and thus would fail to reproduce the large top mass .
examples include    * brane - world scenarios in which quark and/or higgs fields propagate in the bulk . in this case
yukawa coupling constants in four dimensions are volume - suppressed by some powers of @xmath99 , where @xmath100 denotes the compactification scale and @xmath101 the fundamental one ; * @xmath102 guts .
an attractive feature of the @xmath102 guts is that the mssm higgs doublets can arise as pseudo - goldstone multiplets @xcite .
a natural extension to @xmath102 from @xmath103 suggests to introduce @xmath104 , @xmath105 and @xmath106 as quarks and leptons , and @xmath107 and @xmath105 as higgs multiplets .
however with these matter contents one can not write down @xmath102 invariant yukawa couplings for up - quark masses at the renormalizable level .
up - type yukawa couplings arise from non - renormalizable operators and are suppressed by powers of @xmath108 , where @xmath109 represents the gut scale .
thus one conventionally introduce * 20 * to obtain the large top yukawa coupling .
our mechanism of the yukawa enhancement due to the singular wave - function normalization , however , may allow us to construct a simpler model without introducing the * 20 * multiplet .    throughout this paper
, we have assumed that the singular normalization for the top quark is realized by some means , but we have not specified a possible mechanism .
a particularly interesting possibility is that the singular normalization is realized dynamically associated with , for example , the electroweak symmetry breaking .
although a naive inspection suggests that one needs a very flat potential for a field responsible for the wave - function normalization , which is thus unlikely , further study along this line is interesting and should be encouraged .
this work was supported in part by the grant - in - aid from the ministry of education , culture , sports , science and technology , japan , no.12047201 .
mk thanks the japan society for the promotion of science for financial support .
it is pointed out that heavy top quark mass can be attributed to a singular normalization of its kinetic term , in which rescaling into canonical one yields large top yukawa coupling . 
 we pursue this novel possibility in a democratic mass matrix model where only the normalization of the third generation can be different from that of the other two generations . with diagonal breaking of democratic @xmath0 symmetry , we show that the singular normalization for the top quark is essential to reproduce observed quark masses and mixing angles . 
 we also briefly argue other applications of this mechanism .