Fermion mass matrices with the maximal mass hierarchy and calculability, and family structure of leptons

Volume 152, number 1,2 PHYSICS LETTERS 28 February 1985

FERMION MASS MATRICES WITH THE MAXIMAL MASS HIERARCHY

AND CALCULABILITY, AND FAMILY STRUCTURE OF LEPTONS ~r

Michael SHIN

L yman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA

Received 12 December 1984

The fermion mass matrices with the maximal mass hierarchy and calculability are derived and presented. One of them is applicable to the charged lepton sector and can produce the huge hierarchy ratio, me/m r ~- 1/3500, from the smallest input parameter of 1/40. The complementary aspect between the fermion mass hierarchy and the observed K-M mixing angles is considered. It is conjectured that the mass matrices for the quark sector are of the Fritzsch form, while the proposed form of mass matrix with the maximal hierarchy is valid for the charged lepton sector. Composite models for the quarks are advocated from the mass hierarchy point of view while the leptons are regarded as fundamental.

In troduction One of the most puzzling aspects of the particle

physics at the present t ime is the observed family

(generat ion) s t ructure o f quarks and leptons. Al though

the existence o f the family itself may not be a severe

problem *~, the observed values o f the fermion mass

eigenvalues and the K - M mixing angles are beyond

our comprehens ion and present a great theoret ical

challenge at this time.

In an a t tempt to understand the observed K - M

mixing angles [1 ] including the strength o f the weak

CP violat ion, we conjectured in our previous article

[2] that the underlying s tructure o f the particle phys-

ics may be such that the mass matrices for the quarks are o f the Fritzsch form with the two physically rele-

vant phases, o and 7, being o = r = _+rr/2. in the subsequent article [3], a model building was a t tempted to generate these phases, o = 7 = +-rr/2, from the V E V ' s

of complex scalars. Al though the Fritzsch form o f the fermion mass

matrices, which has the at tract ive feature o f the maxi-

mal calculabili ty ( to be defined in section 1), may ex-

plain the observed K M mixing angles in terms o f the

observed quark mass eigenvalues, the observed fermion

¢~ This research is supported in part by the National Science Foundation under Grant No. PITY-82-15249.

,1 The horizontal family symmetry group may be present in the next layer of high energy physics.

mass hierarchy ( m l "~ m2 ~ m3) i tself is not satisfac- torily explained by this form since all o f the three in-

dependent scales ( IAI ~ x / m l m2, IB[ ~ x / m 2 m 3 , [CI -~ m3) have to be put in by hand to generate the

observed fermion mass hierarchy, in particular, for the

lepton sector where the mixing angles be tween fami-

lies are physically unobservable and irrelevant ,2, the

Fritzsch form of the lepton mass matr ix may not be

the most economica l choice.

In this article, therefore, we shall try to find more

economical forms of mass matrices in which all o f the attractive features ( the maximal calculabili ty) o f the

Fritzsch form are still retained, but the observed fermion mass hierarchy is more natural ,3

1. Virtues o f the Fritzsch form o f mass matrices." the maximal calculability

Since we are looking for the forms of the fermion mass matrices which are bet ter than the Fritzsch form

in all respects, here we briefly consider the virtues o f

the Fritzsch mass matr ix which make it the most at-

tractive.

A 3 X 3 fermion mass matr ix M F is said to be o f

+2 Here we are assuming that all neutrinos are massless. ,3 By this we mean that the extremely small ratio, ml/m 3, is

well generated from the form of the mass matrix, where the smallest dimensionless parameters in the mass matrix are O(m2/m3).

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the Fritzsch [4] form, i fMF takes the form

( 0 A 0 1 MF= A ' 0 B , (1)

0 B ' C

where A, A ' , B, B ' , C are in general complex with [AI = JA'[ and IBI = tB' 1. The mass 2 eigenvalues of MFM{.. are independent of the phases of (A, A ', B, B ' , C), and depend on only three real parameters IA 1, [B[, [CI. The reason for this is that all of the phases inMF can be rotated away by the redefinition of the chiral fermion fields since MF has only five non-vanishing elements. (For an n X n fermion mass matrix, all phases can be similarly rotated away, if it contains no more than ( 2 n - 1) non-vanishing elements.) Moreover, all of the real parameters ([AI = I A ' [, [BI = I B' l, I C[) are well determined in terms of mass eigenvalues of MF, and the weak charged current ( K - M ) mixing matrix involves the mininmm number of input parameters: quark mass eigenvalues and two phases o and r, which are linear combinations of the phases in M~ and MFd.. [For an n × n Fritzsch matrix, the K - M matrix will involve ( n - 1) phases besides the quark mass eigenvalues.]

From this, we now intdoduce the following definition o f " t h e maximal calculability" of a fermion mass matrix.

Definition. An n X n fermion mass matrix M is said to be of the form with the maximal calculability, if it contains no more than ( 2 n - 1) non-vanishing elements and all IMi/l's have not more than n distinct values.

For this definition of an n X n fermion mass matrix with the maximal calculability, the mass 2 eigenvalues will be expressible in terms of the absolute magnitudes of the (complex) parameters in the matrix, and the K - M mixing matrix will be calculable in terms of the quark mass eigenvalues and not more than ( n - 1) phases.

Note that the following 2 X 2 matrix (as a counter example),

has eigenvalues which depend on the phases of A, B and does not belong to the class of mass matrices with the maximal calculability.

With the above definition of maximal calculability, we now consider the fermion mass matrices with the maximal calculability for three generations of quarks and leptons. (For more generations, the analysis will be similar to the one given in the following sections and will be considered elsewhere [5].) For three generations then there are only three distinct possibilities. The number of non-zero entries in the mass matrix can be 5 or 4 or 3. If the mass matrix has less than 3 non- zero entries, then the determinant is always zero, which is not what we are looking for. (We want the mass eigenvalue of the lightest family to be non-zero and small!) We shall consider these possibilities separately in the following sections.

2. The case with five non-zero entries Given a form of a fermion mass matrix, interchang-

ing rows or colunms or both does not change the spectrum of the mass eigenvalues due to the fact that these operations are unitary ,4. Therefore, without loss of generality, we can always arrange the permutations on the columns of the matrix so that the number of non- zero elements in the ith column is not greater than that of the j th column for i < j. Moreover, each column should have at least one non-zero entry since the determinant is required to be non-zero. This means that there are only two possibilities in the number of non-zero entries in each column of the fermion mass matrix. They are (1,1,3) and (1,2,2), where the j th number represents the number of non-zero entries in the j th column. We shall consider these two possibilities separately.

2.1. The (1,1,3) case. In this case, the third column is full. For the second column, we can always perform permutations on the rows so that the non-zero entry is on the third row. Then there are two possibilities for the first column. Either the non-zero entry is on the same row as the second row (which is not all filled) or it is on a different row. If the latter is the case, then we can always choose the non-zero entry to be on the second row since the first and second row are equiva-

4:4 For example, interchanging the first two rows in the mass matrix is equivalent to the multiplication of the matrix from the left by the matrix (01 !) S=- 1 0 .

0 0

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lent to each other by the operation of interacting rows. Thus, in (1,1,3) case, we are led to two unique possibilities: (!o,

0 * , (2)

o r

0 0 * • 0 * , ( 3 )

0 * *

where the *'s mean some non-zero complex numbers. The matrix (3) can be put into the following triangular form by permutations of columns

• . ( 3 ' )

0

2.2. The ( l , 2, 2) case. In this case, the third column can be always chosen to be of the form (0, *, ,)Y by interchanging rows. Then the second column has the possibilities of (0, *, , )T or (*, 0, ,)T. Note that the possibility of (* ,* ,O) y for the second column is equivalent to that of (*, 0,*)T by interchanging the second row with the third since the " tex ture" of the third column is unchanged by this operation. By consider- ing the possible forms of the first column, we are led to the following four unique possibilities with non- zero determinants

* 0 0 ) * 0 * * , 0 0 * * 0 0.0) * 0 * ,

0 * *

0 * , (4,5)

.0) 0 * . (6,7)

The matrices (5) and (7) can be put into the following triangular forms by permutat ions of rows and columns.

* * 0 , 0 * . ( 5 ' , 7 ' ) 0 * * * *

From (2 ) - (7 ' ) , we can see that there are two kinds of fermion mass matrices with five non-vanishing elements. One kind consists of those matrices which can-

not be put into the triangular forms [(2),(4),(6)]. The other kind consists of those which can be put into the triangular forms [ (3 ' ) , (5 ' ) , (7 ' ) ] . The matrix (2)has the determinant equal to zero and should be excluded from the list. The matrix (4) is not very interesting because it has the block-diagonal form and one of the families is completely decoupled from the other two. The matrix (6) is indeed interesting and our old friend the Fritzsch mass matrix (1), is of this form.

The matrices (3'), (5 ') and (7') are all triangular and deserve further analysis. In particular, the determinants of these matrices depend on the three diagonal elements only and produce an interesting mass eigen value spectrum as we shall see later (in section 4).

3. The case with four or three non-vanishing entries. In the case with four non-vanishing entries, withoul

loss of generality, the third column can be chosen to be of the form (0,*, ,)Y. Each of the first and the second column then has only one non-vanishing element. The only possibility with non-vanishing determinant is found to be of the form

0 0 0 • (8)

For the case with three non-vanishing entries, each column has only one non-vanishing element and the following is the only possibility with non-zero determinant.

* 0 0 0 * 0 (9) 0 0 *

In matrix (8), one of the families is completely decoupled from the other two and it is not very interesting. The matrix (9) produces no mixing between the families and the fermion mass hierarchy has to be put in by hand. Thus, for the case with four or three non- vanishing elements in the fermion mass matrix, we find no interesting forms of mass matrix.

4. Realistic fermion mass matrices with maximal mass hierarchy and calculability

In section 2, we have found three interesting classe.. of fermion mass matrices, namely matrices (3'), (5'), (7') and the ones which are generated by permutations on the rows or columns from these.

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Volume 152, number t,2 PHYSICS LETTERS 28 February 1985

Here we consider whether this new class of mass matrices will help understand the observed mass hier- archies of the quarks and the leptons. To begin with, we note that the physically observed mass 2 eigenvalues are eigenvalues of the hermitian matrix M M t. Since det (MM ~) is equal to [det Mt 2, the mass eigenvalue of the lightest family would be m 1 = [det M[/ (m2m3). Thus, given the values o fm 2 and m3, m 1 would be as small as possible if [det M[ is made as small as possible. (This is our definition o f " the maximal mass hierarchy".) For the triangular matrices (3'), (5'), and (7'), this will be realized if the product of the diagonal elements is minimal. One (or more) of the non-diagonal elements has to be large (-~O(m3)) to produce the heaviest generation mass. Among all the possibilities, we have found the following mass matrices as the only ones (up to permutations of rows and columns) with the maximal mass hierarchy.

(i °° M I = A 0 0 A

(:oo M3 = A 0

B A

(!0o M2= A 0

C A , ( 3 " , 5 " )

(7")

where A, B, C (with A, B "~ C) are chosen to be real and positive by the redefinition of the phases of the chiral fermion fields.

In table 1, we give the eigenvalues ofMM ~, and the diagonalizing unitarity matrix U L [U~(MM t ) U L ~-

2 . . MD]. ci and si are cos Oi and sm Oi, respectwely.

M 1 and M 3 produce the geometrical mass hierarchy m l / m 2 = m 2 / m 3. This geometrical mass hierarchy is neither realized in the up-quark sector nor in the charged lepton sector, although it may be realized in the down-quark sector if the mass of the s-quark is around 250 MeV (150 MeV ~ ms(1 GeV) ~ 300 MeV) so that 1/20 -~ md/ms ~ m s / m b for mb(1 GeV) -~ 5 GeV.

As far as the mass spectrum is concerned, M 2 is good for the charged lepton sector and the up-quark sector and may probably be also good for the down- quark sector [for large ms(l GeV) ~- 300 MeV] since it is valid as long as m l / m 2 < m 2 / m 3. For instance, numerical values of

A = ( m e m u m r ) 1/3 = 45.8 MeV

B = mu[1 - (rnemr/m2) 2/3] 1/2 = 95.2 MeV,

C = m r = 1.78 GeV, (10)

Table 1 Fermion mass matrices with the maximal mass hierarchy and calculability for three generations.

Mass matrix Mass eigenvalues Parameters U L

( i 0 O) m21~lAl4/m 2 IAl~-m2 Ml = A 0 tn2~lAi 2 IBI ~lCI

0 A m2~ICL2+IBI2+21AI 2 ICI ~-m3

ml/m 2 ~- m2/m 3

M 2 = A C

A 0 0 ) M3= 0 A 0

\ C B A

m21 ~- [A 16Imam23 m~ ~ IB[ 2 + [A [2

m23 ~ ICl2 + 21AI 2

m~ ~- IA 14/m~ m~ = IA I z m 2 ~ IC[2+ 1BI2+ 21AI 2

ml/m 2 ~-- rrt2/m 3

[AI = (m lm2m3) 1]3 I B l = m 2 [ 1 - ( m l m 3 / m 2 ) 2 / 3 ] 1[2

ICI ~- m3

IAI ~-m2 IBI < ICI ICI ~-m3

c 1 0 sl ) U L ~ s is 2 c2 c1s2

\--SIC2 --S 2 CLC2!

01 ~ tan-I (m2/m3) ~- tan-1 (m l/t 02 ~ tan -1(tBl[ICI)

UL ~ I - s 1 ¢ 2 ¢1c2 \ s1s 2 - s 2 c 1

Ol ~ tan--1 (.[1 ( m l m 3 / m 2 ) l / 3 - (m I m 3/rn~)2/311

02 ~- tan -1 [(mlrnz/m23) 1/31

CIC2 --S 1 ¢IS2

UL ~-| $IC2 ¢I SiS2 ) \ -s 2 0 c 2

0 1 ~ tan- l ( IBI /[CI) 02 ~- tan -1 (m2/m3) ~ tan - I (rnl/i

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A/C ~-1/39, B/C~-1/18.7 ( 1 0 c o n t ' d )

produce the huge hierarchy ratio,

me/m r ~- 1/3490. (11)

We find this very interesting. (This is much better than the Fritzsch form.) A detailed investigation of the K M mixing angles produced by all pairs of matrices in table 1 was performed. The result, however, shows that no realistic K - M mixing angles can be obtained from these matrices. (At least one of the three real angles in the K - M matrix turned out to be too large!) This is a very unfortunate aspect of these matrices, leavingM 2 to be valid only for the charged lepton sector.

5. Reflections, conjectures and conclusion In section 4, we have seen that the triangular form

of mass matrix M 2 may be the correct form for the charged lepton sector with the attractive feature of the maximal mass hierarchy produced by the minimal hierarchy in the input parameter. However, all of the matrices in table 1, which produce the maximal mass hierarchy, do not produce realistic K - M angles, if used for the quark sector. Thus, the Fritzsch form of the mass matrix is the most realistic one for the quarks although the feature of the maximal mass hierarchy is not realized in this particular case. The reason for this is that in order to produce the maximal hierarchy in the mass eigenvalues from the minimal hierarchy of input parameters, the mass matrix is required to produce large mixing angles whereas the observed K - M mixing angles are all small (the largest one is the Cabib- bo angle, sin 0 C ~ 0.23). We find this a puzzling aspect of the family structure. The strengths of the K - M angles and the fermion mass hierarchy seem to have this complementary aspect, at least in the quark sector. The charged lepton sector does not have this problem only because the mixing angles are irrelevant for massless neutrinos (massless neutrinos are all degenerate and have no mass hierarchy!). Why nature has chosen small K - M angles instead of large ones? We do not know. It is interesting to conjecture that nature has chosen two different mechanisms for the mass genera-

tions of the quarks and the leptons, and two different forms of mass matrices for them. The quark sector may be dictated by the Fritzsch form (with appropri- ate phases), while M 2 in table 1 dictates the lepton sector. Quarks may be composite particles while the leptons are not. "['he question of the fermion mass hierarchy may be better answered for quarks in composite models, where the small parameters in tile mass matrices can be naturally generated by some power of coupling constants [6]. The leptons may be fundamental (from the beginning of tile history of particle physics, the electron has been a fundamental particle, while the hadrons have not) and some horizontal family symmetry may answer the origin of the " tex ture" *" of the mass matrix M 2.

To conclude, we have considered the possible forms of the fermion mass matrices for three generations of the quarks and leptons from the calculability and the mass hierarchy. One realistic mass matrix for the charged leptons was found, in which the maximal mass hierarchy is produced with the minimum hierarchy in the input parameters. Some complementary aspects between the fermion mass hierarchy and the observed K - M mixing angles were considered and can be used as a guide in the subsequent theoretical challenges to the family problem. Composite models tbr quarks have been advocated.

I would like to thank Howard Georgi for useful comments and suggestions.

+s This terminology is due to S.L. Glashow.

References

[ 1 ] M. Shin, ltarvard preprint HUTP-84/A024 (1984). [2] M. Shin, Phys. Lett. 145B (1984) 285; ltarvard preprint

tlUTP-84/A070 (1984). [3] It. Georgi, A. Nelson and M. Shin, Phys. Lett. 150B

(1985) 306. [4] fl. Fritzsch, Nucl. Phys. B155 (1979) 189; Phys. Lett.

73B (1978) 317; L.F. Li, Phys. Lett. 84B (1979) 461.

[5] M. Shin, in preparation. [6] It. Georgi, Phys. Lett. 151B (1985) 57.

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Fermion mass matrices with the maximal mass hierarchy and calculability, and family structure of leptons