The Mystery of Flavor

Home , Family symmetries

View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by CERN Document Server

R. D. Peccei

Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547

Abstract. After outlining some of the issues surrounding the avor problem,

I present three sp eculative ideas on the origin of families. In turn, families are

conjectured to arise from an underlying preon dynamics; from random dynamics

at very short distances; or as a result of compacti cation in higher dimensional

theories. Examples and limitations of each of these sp eculative scenarios are

discussed. The twin roles that family symmetries and GUTs can have on the

sp ectrum of quarks and leptons is emphasized, along with the dominant role

that the top mass is likely to play in the dynamics of mass generation.

I THE QUESTION OF FLAVOR

Flavor is an old problem. I. I. Rabi's famous question ab out the muon:

\who ordered that?" has now b een replaced by an equally dicult question

to answer: \why do we have three families of quarks and leptons?" Although

qualitatively we understand the issues connected to avor a lot b etter now,

quantitatively we are as puzzled as when the muon was discovered.

When thinking of avor, it is useful to consider the standard mo del La-

grangian in a sequence of steps. At the roughest level, neglecting b oth gauge

and Yukawa interactions, the Standard Mo del Lagrangian L = L (g =

o SM i

0; =0) has a U (48) global symmetry corresp onding to the freedom of be-

ing able to interchange any of the 16 fermions of the 3 families of quarks and

leptons with one another. If we turn on the gauge interactions, the Lagrangian

L = L (g 6=0; =0)hasamuch more restricted symmetry [U (3))] cor-

1 SM i ij

resp onding to interchanging fermions of a given typ e (e.g. the (u; d) doublet)

from one family to the other. When also the Yukawa interactions are turned

on, L = L (g 6= 0; 6= 0), then the only remaining symmetry of the

2 SM i ij

Lagrangian is U (1) U (1) . In fact, b ecause of the chiral anomaly [1], at

B L

the quantum level the symmetry of L is just U (1) .

2 BL

The ab ove classi cation scheme serves to emphasize that there are really

three distinct avor problems. There is a matter problem ,afamily prob-

lem and a mass problem. The rst of these problems is simply that of

understanding the origin of the di erent sp ecies of quarks and leptons (i.e.

c c

why do es one have a and a u state?). The second problem is related to

L L

the triplication of the quarks and leptons. What physics forces such a tripli-

cation? Finally, the last problem is related to understanding the origin of the

observed p eculiar mass pattern of the known fermions.

The usual approach when thinking ab out avor is to try to decouple the

ab ove three problems from one another. Thus, for example, one assumes the

existence of the quarks and leptons in the Standard Mo del and asks for the

physics b ehind the replication of families. Although it is dicult to argue

cogently on this p oint, it is certainly true in the examples which we will

discuss that the matter problem seems to be unrelated to the question of

family replication. Indeed, quite often one also assumes the reverse, namely,

that the family replication question is indep endent of the typ es of quarks and

leptons one has. In fact, it is p ossible that there is other matter b esides the

known quarks and leptons and that this matter is also replicated. Certainly,

even in the minimal Standard Mo del there is other matter b esides the quarks

and leptons, connected to the symmetry breaking sector. This raises a host of

questions including that of p ossible family replication of the ordinary Higgs

doublet. One knows, empirically, that this cannot happ en if one is to avoid

avor changing neutral currents (FCNC) [2]. However, some replication is

needed if there is sup ersymmetry, but the two di erent Higgs doublets needed

in sup ersymmetry are connected with di erent quark charges and need not

replicate as families.

The ab ove remarks suggests that there are some p erils asso ciated with trying

to seek the origin for family replication indep endently from that of the quarks

and leptons themselves. Nevertheless, that is the approach usually taken

and the one I will follow here. Similarly, one also usually tries to disconnect

the problem of mass from that of matter and family. That is, one generally

assumes the existence of the three observed families of quarks and leptons,

and then tries to p ostulate (approximate) symmetries of the mass matrices

for quarks and leptons which will give interrelations among the masses and

mixing parameters for some of these states.

This approach usually involves some kind of family symmetry and is

sensible provided that:

i) There is some misalignmentbetween the mass matrix basis and the gauge

interaction basis for the quarks and leptons. Only through such a mis-

alignment will there result a nontrivial mixing matrix: V 6=1.

CKM

ii) The family symmetries of the mass matrices are broken (otherwise

V 1) either explicitly or sp ontaneously. Furthermore, if the break-

CKM

ing is sp ontaneous, it must o ccur at a suciently high scale to have

escap ed detection so far.

Although the origin of avor remains a mystery,Iwant to discuss here three

sp eculative ideas for the origin of families. These ideas are realized up to now

only in incomplete ways, in what amount essentially to toy mo dels. Thus, for

instance, the issue of family generation is in general disconnected from the

question of SU (2) U (1) breaking and, often, also from trying to explicitly

calculate the Yukawa couplings. As a result, in all of these attempts at trying

to understand avor, the question of mass is approached from a much more

phenomenological viewp oint. One guesses certain family or GUT symmetries,

and their p ossible patterns of breaking, and then one checks out these guesses

by testing their predictions exp erimentally. In all of these considerations, the

top mass, b ecause it is the dominant mass in the sp ectrum, plays a funda-

mental role.

In my lectures [3], I will b egin by describing three sp eculative ideas for

the origin of families. Sp eci cally, I will consider in turn the generation of

families dynamically; through short distance chaotic dynamics; and as a result

of geometry. After this sp eculative tour, I will discuss brie y the issue of mass

generation. In particular, I will illustrate the twin roles that family symmetries

and GUTs can have for the sp ectrum of quarks and leptons. I will conclude

by commenting on the profound role that the top mass is likely to have on the

detailed dynamics of mass generation.

II GENERATING FAMILIES DYNAMICALLY

The underlying idea b ehind this approach to the avor problem is that

familiies of quarks and leptons result b ecause they are themselves comp osites

of yet more fundamental ingredients{preons. There is a nice isotop e analogy

[4] which serves to illustrate this p oint. Think of the three isotop es of Hydro-

gen as three distinct families. Just like the families of quarks and leptons, all

three isotop es have the same interactions{their chemistry b eing determined by

the electromagnetic interactions of the proton. Deuterium and tritium, how-

ever, have di erent masses than the proton b ecause they have, resp ectively,

1 3

1 and 2 neutrons. Of course, the analogy is not p erfect since H and H are

fermions and H is a b oson! Nevertheless, it is tempting to supp ose that the

3 families of quarks and leptons, just like the Hydrogen isotop es, result from

the presence of di erent \neutral" constituents.

I will illustrate how to generate families dynamically by using as an ex-

ample some recent work of Kaplan, Lep eintre and Schmaltz [4]. By using

essentially the isotop e analogy, these authors constructed an interesting toy

mo del of avor. Their simplest toy mo del is based on an underlying sup ersym-

metric gauge theory based on the symplectic group Sp(6). The fundamental

constituents in this mo del are 6 preons Q transforming according to the fun-

damental representation of Sp(6) and one preon A transforming according

to the 2-rank antisymmetric representation. Such a theory has three families

of b ound states distinguished by their A content, plus a pair of (neutral)

exotic states. To wit, the b ound states of the mo del are the 15 avor states

[i;j ] [i;j ] [i;j ]

i j i i j

F Q Q ; F Q A Q ; F Q A A Q (1)

3 2 1

plus the two neutral exotic states

2 3

T TrA ; T TrA : (2)

2 3

The six Q preons act as the protons in the isotop e analogy. In principle,

one could imagine having the SU (3) SU (2) U (1) interactions act on the

Q states, while the A preons act as the neutrons. Furthermore, there is

clearly a family U (1) in the sp ectrum which counts the number of A elds.

Finally, one should note that, b ecause of the sup ersymmetry, each of the states

in Eqs. (1) and (2) contain b oth fermions and b osons.

Although the numb er of b ound states p er family (15) is encouraging, these

states cannot really be the ordinary quarks and leptons (minus the right-

handed neutrinos). It turns out that one cannot prop erly incorp orate the

SU (3) SU (2) U (1) gauge interactions with only 6 Q preons. To do that,

in fact, one has to at least triplicate the underlying gauge theory [4] from

Sp(6) to Sp(6) Sp(6) Sp(6) . Each of these Sp(6) groups has again

L R H

six Q and one A preon. To obtain the desired quarks and leptons the Q

preons are assumed to have the following SU (3) SU (2) U (1) assignments:

Q :(3;1) (1; 2) (1; 1)

L 0 1=6 1=3

Q :(3;1) (1; 1) 2(1; 1)

R 0 2=3 1=3

Q :(1;2) (1; 1) (1; 1) 2(1; 1) (3)

1=6 1=3 2=3 1=3

[i;j ]

Because of the preon group triplication, instead of having 15 F b ound

states per family, one now has 45 such states. Per family, these states now

include 16 states with the quantum numb ers of the observed quarks and lep-

tons, plus 29 exotic states which, however, sit in vector-like representations of

the Standard Mo del group. Sp eci cally, the quark doublet (u; d) is a b ound

c c

state of Sp(6) ; u and d are b ound states of Sp(6) ; while the lepton states

L R

L L

c c

(; e) , and e are b ound states of Sp(6) . Among the exotic states one

L H

L L

nds as b ound states of Sp(6) two states with the quantum numb ers of the

o o

Higgs doublets of a sup ersymmetric theory: H (h ;h ) and H (h ;h ).

1 2

So, in this mo del, there is a natural family rep etition of the Higgs states.

Naively, this could cause problems with FCNC. It turns out, however, that

when one calculates the dynamical sup erp otential of the theory [5] one can

show [4] that there is a ground state where only one of the three families of

Higgs states are left light. So, in fact, there are no FCNC problems.

This nice result is temp ered by other troublesome features of the mo del

which render it unrealistic{but not uninteresting. For example, to break the

[U (1)] family symmetry of the mo del, it is necessary to intro duce by hand

some heavy elds (with masses > {the dynamical scale of the preon theo-

ries) which serve to couple the preon groups together. The simplest p ossibility

1 2 3

is a orded byhaving 3 such elds: v ; v ; v with indices spanning

H R R L L H

2ofthe preon groups, interacting through a sup erp otential

1 2 3 1 1 1 1 1 1 2 2 2 2 2 2

W = av v v + b v v A + b v v A + b v v A + b v v A

H R R L

3 3 3 3 3 3

+ b v v A + b v v A (4)

L H

The a-term ab ove ties the preon theories together, while the various b-terms

serve to break the family symmetries. Although Eq. (4) is intro duced by

hand, integrating out the e ects of the heavy v elds gives e ective Yukawa

couplings of di erent strengths, much in the way originally suggested byFrog-

gatt and Nielsen [6]. This is illustrated schematically in Fig. 1 for the Yukawa

(3)

coupling of u with (c; s) via the Higgs state H of the third family{which

is the only one which is assumed to get a VEV. One nds [4]

(3)

1 2 3 6 6

a b b b (=) (5)

R R L

Although the various elements in the up-and-down quark mass matrices are

hierarchial, unfortunately there is no resulting quark mixing since M M .

u d

This follows b ecause the mo del has an unbroken global SU (2) symmetry at

the preon level corresp onding to the interchange of the (1; 1) and (1; 1)

2=3 1=3

assignments in Eq. (3). Furthermore, for the lepton sectors there is a dynam-

ically generated set of Yukawa couplings [5] which are typically unsuppressed.

As a result, naively, one exp ects m m . Both of these results make the

In the mo del [4] the lightest family has the most A elds{c.f. Eq. (1).

FIGURE 1. E ectiveYukawa coupling generated in the [Sp(6)] preon mo del.

[Sp(6)] mo del as presented ab ove unrealistic. By further complicating the

mo del, Kaplan, Lep eintre and Schmaltz [4] are able to obtain b oth a non-

trivial CKM matrix and re-establish the top as the heaviest b ound state.

However, these \improved" mo dels are not particularly attractive and rep-

resent, more than anything else, a \pro of of principle". In addition, even

after these problems are resolved, the mo dels still lack mechanisms for break-

ing SU (2) U (1) and sup ersymmetry, features which must b e understo o d to

make contact with reality.

These negative remarks should not obscure the considerable achievement

of these dynamical mo dels for understanding the origin of avor. Families in

these mo dels arise as a result of hidden degrees of freedom in some underlying

con ning dynamics. Furthermore, the presence of heavy excitations in this

same dynamics can result in hierarchial patterns of Yukawa couplings, once

all family symmetries are explicitly broken. Unfortunately, it is dicult to

see how one can obtain real evidence for these kinds of schemes, barring the

discovery of some of the exotic b ound states they predict{in the example

discussed, the T and T states or the vector-like partners of the quarks and

2 3 leptons.

III FAMILIES FROM SHORT-DISTANCE

RANDOM DYNAMICS

A radically di erent scheme for the origin of families has b een prop osed

and elab orated by Holgar Nielsen and his collab orators [7]. The basic idea

that Nielsen has put forth is that there exist b oth order and chaos at very

short distances. He imagines that at scales much smaller than the inverse

of the Planck mass there is actually a lattice structure of scale length a

1=M . However, b oth the dynamics on the lattice as well as the structure

Planck

of the lattice is random. In particular, the lattice is amorphous with sites at

random p ositions. Furthermore, characteristic of the random dynamics, the

interactions on each of the links are governed by di erent groups, with the

groups varying from link to link.

Remarkably, even starting from these very general assumptions, one can

arrive at some conclusions. Generally, one naively would imagine that no

group could survive the random dynamics. That is, that the gauge group

will end up by breaking down sp ontaneously, pro ducing sup ermassive elds of

mass M a M . In fact, as Brene and Nielsen [8] showed, there are

Planck

sp ecial groups G on the links which survive the random dynamics{i.e., the

surv :

asso ciated vector b osons are massless. What Brene and Nielsen [8] showed

is that the groups which survive must have a center which is non-trivial and

connected. By taking values in the center the links are e ectively gauge-

invariant. However, the center cannot be simply the unit matrix b ecause the

random nature of the dynamics would then end up byaveraging out the e ects

of all links. The connectedness of the center, nally, is necessary to insure that

the Bianchi identities are satis ed. Sp eci cally, it turns out that G is a

surv :

pro duct of \prime" groups with a certain discrete group D , generated

prime

from the center, removed:

G = U (1) SU (2) SU (3) SU (5) ::: SU (prime)=D (6)

surv : prime

From the ab ove, it app ears that Nielsen's random dynamics allows the Stan-

dard Mo del group to survive, with a restriction:

G = SU (3) SU (2) U (1)=D : (7)

Here the discrete group D is given by powers of the center element h =

n o

2i 2i

3 6

e I ; I ;e :

3 2

D = fh jn Z g (8)

3 6

In practice, this imp oses a restriction on the matter states which are placed

on the random lattice sites

h = ; (9)

which xes the hyp ercharge of the quarks relative to the leptons. Eq. (9)

e ectively imp oses the familiar charge quantization, giving the quarks third-

integral charges. This is a very nice result!

In this scheme the origin of family replication o ccurs through what Bennett,

Nielsen and Picek [9] call \confusion" in the random dynamic pro cesses. This

can be understo o d as follows. At some step in the random dynamics what

survives is not simply the group G but a number N of copies of G ,

SM SM

each with one family of quarks and leptons. Subsequently, this pro duct group

collapses to its diagonal subgroup [G ] . This collapse, through \confu-

diag

sion", results in N replicas of a Standard Mo del family of quarks and leptons.

Thus, schematically, family generation o ccurs in random dynamics when N

Standard Mo del surviving groups collapse:

G G ::: G ! [G ] (10)

diag

SM SM SM SM

Bennett, Nielsen and Picek [9] try to estimate N {the number of families{

which arise from random dynamics confusion by making a numb er of assump-

tions. Although some of these assumptions are questionable, they are not

unreasonable. First, Bennett et al. supp ose that the lattice scale asso ciated

with the random dynamics is of order of the Planck scale: a = M . This

allows the calculation of the coupling constants of the Standard Mo del group

[G ] from their low energy values via the renormalization group:

diag

[g ] ' g [M ] (11)

i diag i P

Second, by identifying the gauge elds in [G ] with the individual elds

diag

in each of the SM groups in Eq. (10), it follows that the individual couplings

of each of the groups in the \confused" con guration G G ::: G

SM SM SM

is given by

conf

g = N [g ] (12)

F i diag

conf

A knowledge of g then provides an estimate for N . What Bennett, Nielsen

and Picek [9] assume is that

conf

g = g ; (13)

i i

with g b eing the mean eld theory critical coupling for each of the groups in

the Standard Mo del. This assumption guarantees that in the confusion stage

there is no con nement of quark and lepton states at Planck length scales{a

reasonable b oundary condition.

The result for N which follows from the three assumptions (11)-(13) are

rather remarkable, given the spare theoretical framework! One nds [7]

3:4 U(1)

3:5 SU(2)

N = (14)

3:1 SU(3)

This result notwithstanding, however, it is not clear how one pro ceeds further

in developing a consistent theoretical framework from random dynamics. For

instance, it is totally unclear how through this scheme one induces the break-

down of the SU (2) U (1) electroweak group at scales of O (100 GeV ), or how

one even generates the Yukawa couplings which can provide the quarks and

leptons eventually with some mass.

IV A GEOMETRICAL ORIGIN FOR FAMILIES

Perhaps the most interesting way to get family replications is through the

compacti cation of extra dimensions. One starts with a theory in d > 4 di-

mensions but then assumes that the extra dimensions somehow compactify,

leaving a 4-dimensional theory. The earliest example of such a theory was

the 5-dimensional Kaluza-Klein theory of gravity [10], which when compacti-

ed to 4 dimensions gave rise, in addition to gravity, also to electromagnetic

interactions. More mo dern examples are sup erstring theories [11] which are

known to be consistent only in d > 4 dimensions, but where again the extra

dimensions can compactify leaving an e ective 4-dimensional theory.

It is quite easy to understand how one can generate families in these typ es

of theories. The general idea was rst sketched out by Wetterich [12] and

Witten [13] in the early 1980's. Consider chiral fermions in a d-dimensional

space-time. Such fermions, by de nition, ob ey a massless Dirac equation

D =0 (15)

Here =1;2;:::d and D is the Dirac op erator in the background of whatever

other elds (gravity,Yang-Mills) are present in the theory. Supp ose now(d4)

dimensions compactify. Then Eq. (15) can be written as

( D + D ) =0 (16)

Chiral fermions o ccur naturally in d = 2 mo d 4 dimensions.

with =0;1;2;3 and a =4; ::: ; d1. Clearly the (d 4) op erator D in

Eq. (16) acts as a 4-dimensional mass [ D M ] unless it vanishes when

applied on :

D =0 (17)

If Eq. (17) holds, corresp onding to a chirality constraint on the (d 4)-

dimensional compact space K , then also the 4-dimensional fermions will be

chiral. If (17) do es not hold, then the resulting 4-dimensional fermions have

a mass.

Since the quarks and leptons are chiral, if they are pro duced from d> 4chi-

ral fermions via compacti cation of the extra dimension, a constraint equation

like (17) on the compact space K must hold. Now, in general, such constraint

equations have anumber of solutions, which dep end on the intrinsic prop er-

ties of the compact space K . So, in these kinds of theories, families and family

numb er are intrinsically related to the top ological prop erties of compact spaces

asso ciated with the original d> 4 theory.

Perhaps the b est known example of family replication using these ideas is

the one considered by Candelas, Horowitz, Strominger, and Witten [14] in-

volving the Calabi-Yau compacti cation of the d = 10 heterotic sup erstring.

This string theory [15] has an asso ciated E E gauge symmetry and is

8 8

sup ersymmetric. The chiral fermions in the d = 10 theory are gauginos of

one of the E groups (the other E acts as a hidden sector), sitting in the

8 8

248 dimensional adjoint representation. Candelas et al. [14] assumes that the

10-dimensional space of the theory compacti es down to d = 4 Minkowski

space times a 6-dimensional Calabi-Yau space, whose principal prop erty for

our purp oses is that it p ossesses an SU (3) holonomy. This means that the

chiral zero mo des in K {those that ob ey the constraint equations (17){have

non-trivial SU (3) prop erties, even though this SU (3) is broken in the com-

pacti cation. By decomp osing E into its E SU (3) subgroup one identi es

8 6

the chiral zero mo des in the gauginos which are candidates for the surviving

chiral matter in 4 dimensions. Since, under this decomp osition,

248 = (78; 1) (27; 3) (27; 3)+(1;8) (18)

one sees that, after Calabi-Yau compacti cation, the 4-dimensional chiral mat-

ter involve fermions in either the 27 or 27 reprentations of E . So, in general,

one exp ects to have n 27 plus (27+27) states in the sp ectrum. The num-

b ers n and are related to top ological indices characteristic of the Calabi-Yau

Think of solving a di erential equation in a p erio dic b ox.

Ma jorana fermions exist in d = 2 mo d 8 dimensions.

space K which compacti ed. In particular, Candelas et al. [14] showed that

n {the number of families{is connected to the Euler number of K :

n = n (19)

F Euler

Note that, in this example, the families one obtains have the right stu .

The 27-dimensional representation of E when decomp osed in terms of its

SO(10) subgroup contains the 16-dimensional representation, appropriate for

a family of quarks and leptons, plus a 10 and a singlet. The 10 itself, since the

theory is sup ersymmetric, contains the two needed Higgs doublets, which in

this case also come in family rep etitions. In principle, the (27+27) states (as

well as the 10 and 1) are vectorlike, and one can imagine these states getting

masses of the order of the compacti cation scale{presumably of O (M ). So

in this example, the light states are just n replications of the chiral quarks

and leptons!

Connecting family replication to the geometry of a compact space is a b eau-

tiful idea. Furthermore, there is another advantage. Through compacti ca-

tion, Yukawa couplings are naturally pro duced, arising from the fermion-gauge

eld interactions in d>4 dimension along the gauge eld comp onents in the

(d 4) compact dimensions. Unfortunately, however, one cannot in general

compute these couplings explicitly. Nevertheless, often one can infer some

useful symmetry restrictions among the Yukawa couplings in these schemes

[16].

In my view, obtaining families from compacti cation is the most app ealing

solution to the origin of the mysterious rep etitions we see in nature. It is

not, however, easy to arrive at the correct theory. Basically, even b elieving

that sup erstrings are the right theory, we still do not understand how to

cho ose among the many p ossible compacti cations available for these theories,

since we have no idea of what is the underlying physics principle that drives

the compacti cation. At the same time, we are also ignorant of how these

schemes can give rise to terms which break sup ersymmetry and eventually

SU (2) U (1). Until such problems are solved, these ideas will just remain

ideas which are app ealing but untested.

V NAVIGATING THROUGH THE MASS MAZE

Even if one were to eventually understand the origin for families and their

matter content, the mystery of avor will not be solved until one is able also

to decipher the physics which leads to the p eculiar mass sp ectrum of quarks

and leptons. Lacking a complete theory, most physicists have taken a very

pragmatic approach to the mass generation problem. Basically, what has b een

assumed is that this problem is essentially decoupled from that of families and

matter. Therefore, it makes sense to pursue a quite phenomenological strategy

to get some insights into the problem of mass.

Following the lead set by some early work of Weinb erg [17] and Fritzsch [18],

the strategy has b een to assume that the mass matrices for the fermions have

certain \textures", imp osed on them by some underlying symmetries. These

textures, in turn, allow one to derive some interesting \predictions" which can

then b e compared with exp eriment. Typically, one obtains in this way certain

interrelations among the quark mixing angles and the quark masses{relations

which go b eyond the standard mo del.

Perhaps the most famous \prediction" of this typ e of approach is the fol-

lowing formula for the Cabibb o angle, expressed as a function of quark mass

ratios:

s s

m m

d u

sin = (20)

m m

s c

Equation (20) follows directly, in the case of two generations, if the quark

mass matrices have the Fritzsch pattern [18]

! !

O A O C

M = ; M = (21)

u D

A B C D

which display a texture zero in the 11 matrix element. Given that Eq. (20)

is rather successful phenomenologically, the natural question to ask in this

context is the underlying reason for the app earance of the texture zero in Eq.

(21).

The app earance of texture zero or other interrelations b etween the elements

in the quark and lepton mass matrices are generally assumed to arise at some

high scale M where new physics connected with mass generation comes into

play. In mo dels where the breakdown of SU (2) U (1) is dynamical likeTech-

nicolor [19], the scale M is generally assumed to b e not to o far from the TeV

scale. However, in general, one has to b e careful with FCNC induced through

the pro cess of mass generation [20] and one must app eal to dynamical prop er-

ties of the underlying theory [21] to avoid contradiction with exp eriment. The

resulting theories are quite complicated [22] and, as a result, many physicists

think it more likely that the scale M connected with mass generation is likely

to b e of order of the Planck or GUT scale. In what follows, I shall concentrate

only on this latter p ossibility and discuss two di erent, but complementary,

mechanisms which can provide mass matrices with interesting textures: family

symmetries and GUTs.

I will illustrate the rst of these p ossibilities by discussing brie y a mo del

intro duced by Ibanez ~ and Ross [23], which makes use of a U (1) family

symmetry. In the Ibanez-Ross ~ mo del, the quarks and antiquarks of each

generation have the opp osite U (1) charge, while the two Higgs b osons H

F 1

and H of the mo del carry twice the U (1) charge of the third generation:

2 F

! !

i i

c i c 2i

u (u ) ! e (u ) H ! e H u

1 1

L L

! e (22) ; ;

i i

c i c

d H ! e H d

(d ) ! e (d )

2 2

L L

As a result of this symmetry, clearly the quark mass matrices only have a

non-zero 33 element:

1 0

0 0 0

C B

u;d

0 0 0

(23) M = m

A @

t;b

0 0 1

This provides a reasonable starting p oint for mo del building.

To pro ceed, Ibanez ~ and Ross [23] need to intro duce b oth a way to break the

U (1) symmetry and some interactions whichphysically will serve to generate

the integenerational mass splittings. They accomplish the second p oint by

imagining that at some high scale M some SU (2) U (1) singlet elds

and , with U (1) charges of +1 and -1, resp ectively, acquire some e ective

interactions with the quark and Higgs elds. How these e ective interactions

come ab out need not b e sp eci ed, but the U (1) symmetry will x their form.

Ibanez ~ and Ross essentially make use of the Froggatt-Nielsen [6] mechanism

we illustrated earlier with the [Sp(6)] preon mo del. For example, there will

be a U (1) preserving e ectiveinteraction among the u and t quarks and the

two scalar elds H and of the form

1 3

e c

L u H t (24)

L 2

ut L

Ibanez ~ and Ross [23], in addition, assume that the U (1) family symmetry is

itself sp ontaneously broken at high scales by VEVs of the and elds. Once

acquires a VEV a term like (24) b ecomes an e ective Yukawa interaction,

h i

which can give rise to small corrections to the mass matrix (23) if = 1.

Of course, within this approach, one is not able to predict precisely these

corrections. Nevertheless, if one assumes that the prop ortionality constants

In the literature, there are many mo dels which use a U (1) symmetry as a family sym-

metry [24], starting from the original pap er on avor textures by Froggatt and Nielsen [6].

in the Froggatt-Nielsen terms are of O (1), the magnitude of the di erent

matrix elements will b e governed bypowers of , re ecting the original U (1)

symmetry. In the case of the Ibanez ~ and Ross mo del, for example, the mo di ed

up-quark mass matrix under these assumptions takes the form

0 1

j4 2 j j3 j j 2 j

1 2 1 2 1

B C

j3 j 2j j j j

1 2 1 2 1

(25) M m

@ A

j 2 j j j

2 1 2 1

Not only is this mass matrix hierarchial if 1, but there are interesting

interrelations among the matrix elements; for example

u 2 u

[(M ) ] [(M ) ]

1 13 1

u u

(M ) ' ; (M ) ' (26)

11 22

1 1

u u

(M ) (M )

33 33

1 1

In general, the detailed comparison of a mass matrix like that in Eq. (25),

which holds at a large scale M , with exp eriment is complicated by the evo-

lution of the Yukawa couplings with energy. This evolution, for example, can

change zeros in a mass matrix at a given scale into small, but non-vanishing,

contributions at a di erent scale. This is easy to understand since, for ex-

ample, a non-vanishing Yukawa coupling to quarks of the second generation

can induce at one lo op an e ective Yukawa coupling to quarks of the rst

generation, provided there is also a non-vanishing Yukawa coupling between

the rst two generations.

The e ect of the renormalization group evolution of the couplings is to fur-

ther obscure p ossible mass matrix patterns. For instance, the mass matrix M

of Eq. (25) is connected to the Yukawa couplings via the familiar equation

hH i. However, at one scale is di erent from at another scale, M =

2 u u

with the evolution b etween scales governed by the renormalization group. At

one lo op, this evolution is given by the equation:

y y

16 = Tr (3 +3a +a )

u d `

d `

d ln M=

2 y

G + (b +c ) (27)

u d u

u u d

In the ab ove, the co ecients a; b; c and G dep end on the assumptions one

makes on the matter contentbetween the scale M and the scale . As a result,

patterns set byphysics at a high scale M O (M )atlower energies are

Planck

smudged. In particular, the measured mass matrix at M is in uenced by

the assumptions one makes on the matter content in the region b etween and

One can take + + = 0, without loss of generality

1 2 3

M {a region for which one has no information! Nevertheless, once one xes

this matter content through some mo del assumptions, one can either validate

or exclude sp eci c mass matrix mo dels by comparing their, renormalization

group evolved, predictions with precision electroweak data.

Rather than illustrating this for the mo del of Ibanez ~ and Ross sketched

ab ove, I prefer instead to examine the predictions of a sp eci c class of SUSY

GUT mo dels, based on SO(10) studied by Anderson, Dimop oulos, Hall, Raby

and Starkman [25]. These mo dels illustrate a second way in which theoretical

input at a high scale may help x the patterns of the quark and lepton masses

and mixing we observe. These SUSY SO(10) mo dels have texture zeros and

matrix element interrelations set by SO(10), with hierarchies among these

elements pro duced by insertion of higher dimensional op erators much along

the lines of what transpired in the Ibanez ~ and Ross mo del. The b est mo del

of [25] with three texture zeros has the following Yukawa couplings:

0 1 0 1

0 C 0 0 27C 0

B C B C

B B

C 0 27C Ee

= ; = ;

@ A @ A

u d

3 9

4B 4B

0 0 A A

3 9

0 1

0 27C 0

B C

27C 3Ee B

= (28)

@ A

0 2B A

The parameters A; B ; C and E have the following hierarchy

A B E >C (29)

and the mo del Clebsch-Gordan co ecients have a Georgi-Jarlskog [26] pattern

for : [u; d; `] = [0; 1; 3], which pro duces the nice result

m m

' (30)

m 9m

d e

Although these and other results provide an acceptable t to the low-energy

data, a recent analysis by Blacek, Carena, Raby and Wagner [27] has shown

that these mo dels do not t the data as well as mo dels with just one texture

zero, like the Lucas-Raby mo del [28].

The Lucas-Raby mo del adds two non-zero matrix elements to the Yukawa

coupling of Eq. (28), with strength D O (C ). Sp eci cally one has:

4D 2D

i i

e ;( ) = e ; ( ) = e ;( ) =

u 31 u 13 d 13

3 3

i i

( ) = 9De ;( ) = 54De ;( ) = De (31)

d 31 ` 13 ` 31

Observable Central Value Result of [25] Result of [28]

M 175.0 6.0 173.9 175.7

m (M ) 4.26 0.11 4.360 4.287

b t

M M 3.4 0.2 3.146 3.440

b c

m 180 50 162.6 189.0

m =m 0.05 0.015 0.0461 0.0502

d s

Q 0.00203 0.00020 0.00173 0.00204

M 1.777 0.0089 1.777 1.776

M 105.66 0.53 105.6 105.7

M 0.5110 0.0026 0.5113 0.5110

V 0.2205 0.0026 0.2215 0.2205

V 0.0392 0.003 0.0450 0.0400

V =V 0.08 0.02 0.0463 0.0772

ub cb

B 0.8 0.1 0.9450 0.8140

TABLE 1. Comparison of the Mo dels of Ref. [25] and Ref. [28] with Exp er-

iment: Adapted from [27]

Table 1 displays a comparison of the ts provided by these two mo dels of

all the extant low-energy data. As can be seen, p erhaps not surprisingly,

the data clearly favors the Lucas-Raby mo del with all predictions within

one standard deviation from the data. In contrast, the b est of the Ander-

son et al. [25] mo dels has four observables ab out 3 away from the data:

2 2

m m

Q = ; V ; V =V and B {the parameter characterizing the

cb ub cd K

o o

strength of the K K matrix element which enters in the CP violating

parameter.

I should remark that, b esides tting the extant data, these mo dels are

predictive. Once all the parameters (A-E and the phases) in Eqs. (28) and

(31) are xed from the global t, one can extract further information from

these ansatze, b oth on the CKM matrix as well as on the sp ectrum of Higgs

and sup ersymmetric states. For example, the Lucas-Raby mo del predicts the

following values for the parameters asso ciated with the CKM unitary triangle,

so imp ortant for studies of CP violation in the B system [28].

sin 2 =0:96 ; sin 2 =0:52 ; sin =0:93 ; = 0:125 ; =0:32 (32)

The angles ; and will be so on measured and Eq. (32) will be tested

exp erimentally. At the same time, the Lucas-Raby mo del also predicts that

the lightest Higgs b osons are a pair of CP odd and CP even states nearly

degenerate with each other with mass around 74 GeV{a prediction which

should b e tested already by LEP 200.

VI LESSONS FROM THE TOP MASS.

Because m m , many of the imp ortant features of the Yukawa matrices

t i

are crucially dep endentonhow the top coupling b ehaves. Theoretically, rather

than considering the physical mass for the top measured by the CDF and DO

Collab orations [29] M = (175 6) GeV, it is more useful to consider the

running mass

m (M ) = (167 6) GeV (33)

t t

1+ (m )

s t

The running mass is directly related to the diagonal Yukawa coupling of the

top (m )

t t

m (m )= (m )hH i (34)

t t t 1 2

This coupling, keeping only the dominant 3rd generation couplings in Eq. (27)

ob eys the RG equation [30]

n o

1 d ()

2 2 2

= a ()+ a ()+a () 4c () () (35)

t b i i t

t b

d ln (4 )

The co ecients a ;c , as we discussed earlier, dep end on the matter content

i i

of the theory. For instance, for the SM one has

3 9 9 17

; a = ; a =1 ; c = ; ; 8 ; (36) a =

b i t

2 2 20 4

while for the minimal sup ersymmetric extension of the Standard Mo del

(MSSM) one has

16 13

a =6 ; a =1 ; a =0 ; c = ; 3; (37)

t b i

5 3

Knowing the value for (m ) one can compute the value for () at any

t t t

scale by using the RG equation (35). Because a > 0, for large enough

eventually () ! 1. The lo cation of this, so called, Landau p ole [31] is

theory-dep endent. As we shall see, is an uninteresting scale in the SM,

Landau

but for the MSSM M {a result which may be quite signi cant.

Landau Planck

We use the same convention as Table 1 in whichlower case masses are running masses

and capital case masses are physical (p ole) masses.

At any rate, it is worthwhile to explore these di erences in a bit more detail

[32].

For the Standard Mo del, b ecause one has only one Higgs b oson, one has

hH ihHi' ' 174 GeV and (m )=0:96 0:04. Furthermore, in

2 t t

1=2

3=4

2 G

2 2 2

this case, b ecause top is so heavy ; and these other couplings can

t b

be safely dropp ed from Eq. (35). The solution of the RG equation

n o

d () 1

a () 4c () () (38) =

t i i t

d ln (4 )

is easily found to b e

() (m )

h i

()= : (39)

1 (m )I()

2 t

Here () and I () are functions determined by the running of the gauge

coupling constants

b d ()

i i

= () (40)

d ln 2

41 19

with b ; ; 7 and one nds

10 6

# "

c=b

i i

ln()

(m )

i t

0 0

; I ()= ( )d ln : (41) ()=

()

ln(m )

From Eq (39) one sees that the Landau p ole for the SM mo del o ccurs at a

value of where

I ( ) = ' 18:9 (42)

Landau

a (m )

t t

Using Eq. (36) one nds ' 10 GeV a scale well beyond the

Landau

Planck mass, which has clearly no physical signi cance. At the Planck mass,

I (M ) 12:2, so one is still far away from the Landau p ole. Indeed

Planck

(M ) 0:7 < (m ). So the top Yukawa coupling in the SM remains

t Planck t t

p erturbative ( =4 1) up to the Planck scale.

The situation is quite di erent if there is sup ersymmetry. Because there are

now two Higgs doublets, (m ) is not xed by m (m ) but dep ends also on

t t t t

the ratio of the two Higgs VEV, tan . One has hH i = sin ,so that

1=2

3=4

2 G F

0:96 0:04

(m )= (43)

t t

sin

There are twointeresting regions for tan . The rst of these has tan O (1),

in which case one can again neglect with resp ect to in Eq. (35). In the

b t

second region tan 1, but (m ) ' (m ). That is, there is a uni cation

t t b t

of the top and b ottom Yukawa couplings. In either of these two regions Eq.

(35) reduces to the same approximate form (38). However, now one has either

0:96 0:04

a =6 ; (m )= (44)

t t t

sin

if tan O (1). If tan 1 instead, one has

a =7 ; (m )=0:96 0:04 (45)

t t t

The form of the solution for () in these two cases is again given by Eq.

(39). However, here the running of the gauge couplings which enter in ()

and I () is governed by a di erent set of co ecients b and c , appropriate

i i

to having sup ersymmetric matter ab ove the weak scale. In particular, b =

; 1 ; 3 . Given Eqs. (44) and (45), it is clear that the Landau p ole

will o ccur at the same place in b oth cases, provided that in the rst case

sin = (tan ' 2:45). In what follows, therefore, we concentrate only on

the second case, corresp onding to Yukawa uni cation of couplings.

The Landau p ole in this case o ccurs at

I ( ) = =12:15 1:00 (46)

Landau

a (m )

MSSM

Using the MSSM coupling evolution, it is easy to check that such values for I

obtain when O (M ). Indeed, the error on is large enough not

Landau Planck t

to p ermit a more accurate determination. In fact, it is much more sensible to

turn the argument around. If there is sup ersymmetry and b ecomes strong

around M , then at low energy will b e driven to an infrared xed p oint

Planck t

at (m ) ' 1 This is illustrated by Fig. 2, calculated using the two-lo op

t t

MSSM RGE equation for tan =2, where the \fo cusing" e ect at low scales

of couplings which are strong around = M is clearly demonstrated.

Planck

The results displayed in Fig. 2 are p erfectly consistent with having the

ratio m =m =1, as SO(10) uni cation suggests, at scales of O (M ). An

b Planck

At one-lo op level, this xed p oint o ccurs at avalue of where the RHS of Eq. (38)

1=2

vanishes: = (m ) = [64 (m )=21] ' 1:03.

t t 3 t t 4

3.5

175 ± 6 3 mt = tan β = 2 α 2.5 s (MZ ) = 0.117 t λ

1.5

0.5 2.5 5 7.5 10 12.5 15 17.5 20

Log µ[GeV]

FIGURE 2. Fo cusing of the Yukawa couplings as ! .

analysis similar to the one we did for Eq. (38) relates this ratio at a scale of

m to that at the uni cation scale M M :

t X Planck

" # " #

1=2 1=6

m (M ) m (m ) (M ) (M )

b X b t X t X

= : (47)

m (m ) m (M ) ^(M ) (m )

t X X t t

Here ^() is a quantity similar to (), detailing the running of the coupling

constants in the quark to lepton mass ratio:

c^=b

i i

(m )

i t

^()= (48)

()

4 16

with the co ecients c^ = for the MSSM. The ; 0 ;

3 3

1=2

ratio [ (M )=^(M )] ' 1:68 is ab ove the exp erimental ratio

X X

m (m )=m (m ) = 1:58 0:08, suggesting that (M ) (m ). That

b t t t X t t

is, the top coupling is stronger at the GUT scale than at low energy, muchas

indicated in Fig. 2.

The upshot of this discussion is that the assumption that sup ersymmetric

matter exists ab ove the weak scale gives a consistent picture, with a large top

Yukawa coupling at the Planck scale b eing driven by an infrared xed p oint

to a value (m ) 1. This b ehavior obtains in two regimes of tan . Either

t t

tan O (1) and is the dominant coupling. Or and tan is large.

t b t

The second p ossibility is natural in the SO(10) mo dels discussed earlier where

all quarks and leptons of one family are in the 16-dimensional representation.

Furthermore, at least intuitively,having a large Yukawa coupling at the Planck

scale ts in well with the ideas that families are generated either dynamically

or through geometry in sup ersymmetric theories.

VI I CONCLUDING REMARKS

In my opinion, one probably will not be able to unravels the mystery of

avor without some new exp erimental information. In particular, I b elieve

that ascertaining whether or not low energy sup ersymmetry exists will havea

profound impact on this question. The discovery of low energy sup ersymmetry

would, of course, provide a tremendous b o ost for sup erstring theories. At the

same time, it would also signal the death knell of the random dynamics ideas

of Nielsen. These ideas, if one is to b elieve in them, require that there should

be a real desert up to M , with no physics beyond the Standard Mo del

Planck

between the weak and the Planck scale.

If sup ersymmetry is found, p erhaps it is sensible to imagine that some of

the ideas discussed in the previous section are true. That is, that there is

indeed a large Yukawa coupling of top at energies of O (M ), which results

Planck

in the mass of the top b eing determined essentially by the infrared xed p oint

of the Yukawa evolution equations. Furthermore, it is easy to imagine then

that the quark and lepton mass sp ectrum is a result of a combination of a

broken family symmetry{which sets up the hierarchy among the masses{and

of a GUT{which interrelates the quark and lepton mass tap estries.

Even in this very favorite circumstance, however, it will be dicult to get

real evidence for the origin of avor. Is it due to dynamics or to some pri-

mordial compacti cation? Perhaps the tell-tale sign will emerge from the

discovery of some exotic states, b esides the quarks and leptons and their su-

p erpartners. In fact, the most characteristic signals of mo dels for avor is

the inevitable presence of exotic states. Recall the exotic T and T states

2 3

in the Sp(6) mo del, or the extra 10 1 states in the 27 pro duced through a

Calabi-Yau compacti cation. In this resp ect, I should note that certain exotic

states seem to b e quite generic. In particular, the presence of extra (3; 1)

1=3

states is very natural.

On a more p edestrian level, our undestanding of avor and mass will be

aided by a continuous exp erimental (and theoretical) re nement of the values

for the quark and lepton masses and mixing parameters. Precise values for

these parameters are crucial if one wants to sort out alternative tap estries,

signalling di erent origins for avor. Eventually, it is going to be imp ortant

to know that V =0:038 rather than 0.040!

Acknowledgements

I am grateful to Ikaros Bigi and Luigi Moroni for their hospitality at the

Varenna Scho ol. A condensed version of these lectures was presented in

Chicago, Illinois, at the Symp osium \20 Years of Beauty Physics", while a

more p opular version was given as the rst Ab dus Salam Memorial Lecture in

Islamabad, Pakistan. I thank Dan Kaplan and Ahmed Ali, resp ectively, for

their kind invitations. This work is supp orted in part by the Department of

Energy under Grant# DE-FOO3-91ER40662, Task C.

REFERENCES

1. S. L. Adler, Phys. Rev. 177, 2426 (1969); J. Bell and R. Jackiw, Nuovo Cimento

51A, 47 (1969); W. A. Bardeen, Phys. Rev. 184, 1848 (1969).

2. S. Glashow and S. Weinb erg, Phys. Rev. D15, 1958 (1977).

3. A shorter version of these lectures was presented at the symp osium Twenty

Years of Bottom Physics, Chicago, Illinois, July 1997, and will b e published in

the pro ceedings of this symp osium.

4. D. B. Kaplan, F. Lep eintre and M. Schmaltz, hep-ph 9705411.

5. P. Cho and P. Kraus, Phys. Rev. D54, 7640 (1996); C. Czaki, W. Skiba, and

M. Schmaltz, Nucl. Phys. B487, 128 (1997).

6. C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B147, 727 (1979).

7. These ideas are discussed in considerable detail in the monograph Origin of

Symmetries byC.D.Froggatt and H. B. Nielsen (World Scienti c, Singap ore,

1991).

8. H. B. Nielsen and N. Brene, Nucl. Phys. B236, 167 (1984); see also, Pro-

ceedings of the XVIII International Ahrensho op Symp osium (Akademie der

Wissenschaften der DDR, Berlin-Zeuthen, 1985).

9. D. L. Bennett, H. B. Nielsen, and I. Picek, Phys. Lett. 208B, 275 (1988).

10. Th. Kaluza, Sitzungb er Preuss. Akad. Wiss Berlin, Math. Phys. K1, 966 (1921);

O. Klein, Nature 118, 516 (1926).

11. M. Green, J. Schwarz, and E. Witten Sup erstring Theory, Vols. 1 & 2

(Cambridge University Press, Cambridge, U.K. 1987).

12. C. Wetterich, Nucl. Phys. B223, 109 (1983).

13. E. Witten, in Pro ceedings of the 1983 Shelter Island Conference I I (MIT Press,

Cambridge, Mass., 1984).

14. A. Candelas, G. Horowitz, A. Strominger, and E. Witten, Nucl. Phys. B258,

46 (1985).

15. D. Gross, J. Harvey, E. Martinec, and R. Rohm, Nucl. Phys. B255, 257 (1985);

B267, 75 (1986).

16. See. for example, B. R. Greene, K. H. Kirklin, P. J. Miron, and G. G. Ross,

Nucl. Phys. B278, 667 (1986); B292, 606 (1987).

17. S. Weinb erg, Transaction of the New York Academy of Sciences, Series I I, 38,

185 (1977).

18. H. Fritzsch, Phys. Lett. 70B, 436 (1977).

19. S. Weinb erg, Phys. Rev. D13, 974 (1976); L. Susskind, Phys. Rev. D20, 2619

(1979).

20. S. Dimop oulos and J. Ellis, Nucl. Phys. B182, 505 (1981).

21. B. Holdom, Phys. Rev. D24, 1441 (1981); Y. Yamamoto, M. Bando, and K.

Matsumoto, Phys. Rev. Lett. 56, 1335; (1986); T. Akiba and T. Yanagida,

Phys. Lett. 169B, 432 (1986); T. App elquist, D. Karabali and L. Wijeward-

hana, Phys. Rev. Lett. 57, 982 (1986).

22. See, for example, B. Holdom, Phys. Lett. B246, 169 (1990); see also B. Holdom

in Dynamical Symmetry Breaking, ed. K. Yamawaki (World Scienti c,

Singap ore, 1992).

23. L. Ibanez ~ and G. G. Ross, Phys. Lett. B332, 100 (1994).

24. See, for example, Y. Nir and N. Seib erg, Phys. Lett. B309, 337 (1993); M.

Leurer, Y. Nir, and N. Seib erg, Nucl. Phys. B420, 468 (1994).

25. G. Anderson, S. Dimop oulos, L. J. Hall, S. Raby and G. Starkman, Phys. Rev.

D49, 3660 (1974).

26. H. Georgi and C. Jarlskog, Phys. Lett. B86, 297 (1979).

27. T. Blazek, M. Carena, S. Raby and C. E. M. Wagner, hep-ph/9611217, Phys.

Rev. (to b e published).

28. V. Lucas and S. Raby,Phys. Rev. D55, 6986 (1997).

29. Particle Data Group, R. M. Barnett et al.,Phys. Rev. D54, 1 (1995).

30. M. Olechowski and S. Pokorski, Phys. Lett. B257, 388 (1991).

31. L. Landau in Niels Bohr and the Development of Physics, ed. W. Pauli

(McGraw Hill, New York,1955).

32. B. Schrempp and M. Wimmer, Progr. in Part. and Nucl. Phys. 37, 1 (1996).