On the proposition that all fermions are created equal

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PHYSICS LETTERS

30 June 1983

ON THE PROPOSITION THAT ALL FERMIONS ARE CREATED EQUAL ~ Howard GEORGI, Ann NELSON i and Aneesh MANOHAR Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Received 2 May 1983

We discuss the possibility that the light fermion mass ratios and mixing angles are all quantum renormalization effects. At some large mass scale, the different flavors are indistinguishable. We show that this idea can account for many features of the observed masses. But in a simple realization we fail to obtain large enough flavor mixing angles.

What is flavor? For all the spectacular success o f the standard six-quark SU(3) × SU(2) X U ( 1 ) m o d e l of describing particle physics at currently accessible energies, we cannot answer that simple question. We can describe the masses and mixing angles of the leptons and quarks as independent parameters in SU(3) × SU(2) × U ( I ) , but we cannot, as yet, calculate them from a more fundamental theory. We do not even understand why three (or any) families of quarks and leptons exist. There appear to be at least two fundamental questions here: (A) Why do the flavors exist? (B) What determines the flavor mass matrices? In the effective SU(3) × SU(2) × U ( I ) theory which appears to describe physics up to energies of ~ 1 0 0 GeV, we cannot even address these questions. The number of families and the Yukawa couplings which determine the mass matrix are free parameters. We can hope to understand them only if the low energy effective theory is embedded in some more restrictive theory at higher energies. But it is not obvious that the two questions will be answered in the same way or at the same energy scale. What is clear is that the answer to (A) will require a radical departure from conventional quantum field theory (QFT). In any ordinary, sensible theory, the fermion content is not fixResearch supported in part by the National Science Foundation under Grant No. PHY-82-15249. I Research supported also by an NSF graduate fellowship. 0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland

ed by any requirements of self-consistency, but must simply be put in by hand. Thus if it is to determine the fermion content o f the world, the high energy theory cannot be an ordinary, sensible QFT. The rules must change. One might hope that the same structure which determines the fermion content will also determine the gauge dynamics and incorporate gravity, because these issues also seem to require a radical change in the rules for their resolution. Presumably, this change occurs at a very large mass scale such as the Planck mass Mp or the unification scale M G. Some o f the wilder current theoretical ideas, such as N = 8 supergravity and generalized K a l u z a - K l e i n theories are attempts in this direction. But none have had much success in explaining the structure o f our world. This is not surprising. A radical change is not accomplished easily. Perhaps we have not been radical enough. It is possible that the answer to question (B), the origin of the mass matrices, also lies beyond the realm of conventional quantum field theories at very high energies. But it is also possible that the key to the masses can be found at an intermediate energy scale, beyond the domain o f validity of SU(3) × SU(2) × U(l ), but below the scale at which QFT breaks down. In this case, we may hope to gain some insights by clever use of conventional ideas. If this latter possibility is to be interesting, the Yukawa couplings which are ultimately responsible for the fermion masses must be related in a very simple way at the large scale, in order that the intermediate scale physics produces all the interesting structure in 169

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the mass matrix. Two simple relations come to mind immediately. One possibility is that only one fermion or family of fermions get mass from the large scale physics, while the physics at intermediate scales allows mass to cascade down from one family to the next, each time suppressed by a power of some small parameter (such as a gauge coupling constant). This idea has a long history, going back to attempts to relate the elect r o n - m u o n mass ratio to the electromagnetic coupling [ 1 ] * ~. It still looks attractive because it could explain the apparently hierarchical nature of the flavor masses. Nevertheless, it has not led to any convincing results for the mass matrices and we believe that it is worth exploring the other possibility, that all fermions are created equal. Here the idea is that at the large scale, the families are not distinguished. Only the interactions at intermediate energies differentiate among the various fermions and produce the nontrivial mass matrix. Thus in such a picture, all mass ratios are quantum renormalization effects like m b / m r in the simplest SU(5) model [3]. In this paper, we will investigate a simple version of this idea in some detail. We will find that it can naturally account for many features of the fermion mass matrix, but that it cannot produce a large enough mixing between the families in a satisfactory way. We wilt conclude by discussing modifications of the idea which could produce a fully realistic model. Our first task is to write down a model of the high energy physics. As discussed above, in the real world we do not expect physics at high energy (~Mp) to be described by conventional QFT. However, in this paper, we will model the high energy world with purely conventional ideas for two reasons. We do not know anything better and we want to concentrate on the physics of intermediate energies that produces the nontrivial structure of the mass matrix. One of the simplest models that produces equal masses for all the fermions is a left-right symmetric model based on a gauge group SU(2)L X SU(2)R X G ,

(1)

under which the fermions are a left-handed (LH) multiplet ~bL which transforms as (2, l, R ) , * 1 See ref. [21 for more recent work. 170

(2a)

and a right-handed (RH) multiplet forms as

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fiR which

(1,2, R),

trans-

(2b)

where R is some complex irreducible representation of G. Here SU(2)L is the SU(2) of weak interactions. If the dimension of R is N, (2) describes N SU(2) doublets of LH fermions with the corresponding righthanded (RH) particles in singlets. Note that G must contain color SU(3) because some of these doublets must be quarks. For simplicity, we will take G = SU(N), with R as the defining representation. Now if the only scalar mesons in the theory which transform as SU(2)L doublets are in a real four-component representation

(2, 2, 1),

(3)

the symmetry G requires that all the Yukawa couplings between (2) and (3) are equal. If (3) got a vacuum expectation value (vev) with no breaking of G symmetry, all the fermions would be degenerate. What we have in mind is breaking (1) at the large scale, MG, down to SU(2)L X U(I)I 1 X H ,

(4)

where H is a subgroup o f G and the U(1)H is a combination of the neutral generator of SU(2)R and a U ( I ) subgroup of G. At a much smaller scale MI, perhaps 100 TeV, (4) breaks further down to SU(2) X U(1) X SU(3). It is in this intermediate region between M G and M I that the flavor masses can split apart. A nontrivial mass matrix results if the different generations transform differently under H. For example, H might contain an SU(5) factor under which R transforms reducibly. Suppose that some of the components of R transform like SU(5) 10's (or 10's), some transform like 5's, and some transform like singlets. Each of these components is renormalized differently by the SU(5) gauge interactions. Just as the SU(3) gauge interactions make the b quark heavier than the r in the simplest SU(5) model, so in a model of this kind, the 10's are heavier than the 5's which are heavier than the singlets. To leading order the renorrealization due to some factor h of the group H has the form [4]:

[g(Ml)/g(MG) ] - 3(T[a + T~ta)/16 ~r2 B ,

(5)

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where TLa (TRa) are the generators o f h on the LH (RH) and g(la) = g/(l - 2Be 2 In/a//a0) 1/2

(6)

is the gauge coupling at scale/a. Can we generate mass renormalizations in this way which are large enough to account for the large ratio (at least ~105) between the heaviest and lightest Dirac fermions? In the familiar example o f m b / m r in SU(5), the renormalization factor is only about a factor of 3. But here the renormalization is due primarily to color SU(3) which is asymptotically free. Most of the effect comes from small momenta, near the b mass, where the SU(3) coupling constant is not very small. Likewise, for a U(1 ) gauge group, or any other gauge group whose coupling grows with increasing energy, the coupling must be very small at low energies, otherwise the system becomes nonperturbative at high energies. Then one gets a sizable renormalization only from the high energy region and the total effect cannot be very large. To get the maximum effect, we want at least one factor of H whose coupling is approximately asymptotically flat. Call the factorf. Then the renorrealization due to f c o m e s from the entire region from M G to M I. In the limit B ~ 0, g is constant and renormalization factor (5) becomes 2

2

+

2

(MG/MI)3g (gLa TRa)/16rr

2

.

(7)

Clearly, ifMG[M I is enormous, the renormalization due to f c a n be sizable, even if the coupling is not large. On the other hand it is also clear that the mechanism will not work unless MG]M 1 is very large. This is exciting because it suggests that the onset of flavor physics cannot be too far away. An M i of 100 TeV might be detectable in experiments to detect GIM violating flavor changing neutral current processes, such as K L ~ / a e , / a ~ e7 or 3e, etc..2 In this simple model the nonabelian components of H (including f ) come exclusively from G. Thus TLa = TRa is the same for both members of an SU(2)L [or SU(2)R ] doublet. These renormalizations do not split charge 2/3 quarks from charge - 1 / 3 quarks or neutrinos from charged leptons. However, the U(I)H ,2 Note, however, that the requirement that the SU(2)L coupling constant not blow up below MG gives a bound on (MG/MI) , (_MG/MI)< {(MG/MW)1° exp[6rr/ ot2(Mw) ] )I[(N-12). Thus N cannot be too large.

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coupling does distinguish ups from downs. If the U ( l ) r i charge of some LH doublet is q, the U(1)H charges of the corresponding RH singlets are q -+ 1/2. Thus the U(1)H renormalization splits the up and down components of each SU(2) doublet. Up is heavier than down i f q > 0 and down is heavier than up for q < 0. Because the U(I)H coupling is not asymptotically free, these splittings within doublets are not as large as the splittings between families which come from f. This picture can account very well for the mass spectrum of the quarks and charged leptons. However, it is incomplete in three ways. (1) The neutrinos are Dirac particles with masses similar to those of the charged leptons. (2) There are extra fermions, not associated with the light families, but which so far have only ordinary SU(2) × U(1) breaking masses. (3) There is no flavor mixing. The first two problems can be solved by enlarging the Higgs structure of the model. Problem (1) can be eliminated if the RH neutrinos in the (1,2, R) get a large Majorana mass or if they get a Dirac mass that involves not the LH neutrino, but some SU(2)L X U(1)singlet fermion. The first can be realized if there is a Higgs which transforms as ( 1 , 3 , N ( N + 1)/2),

(8)

and couples to (~kR)2. The second can be realized if there are several Higgs which transform as (2b) and which couple ~bR to neutral singlet leptons. Both mechanisms produce RH neutrino masses which are (in general) of the order Of Ml, because most of the RH neutrinos transform nontrivially under H and cannot get mass until it is broken. The Majorana mass possibility is dangerous (or interesting) because it produces Majorana mass for the LH neutrinos of order m2/Ml, where m is a corresponding lepton mass. The extra fermions of problem (2) must be present for two reasons. The heavier families must transform under fairly large representations o f f , such as the 10 of SU(5) in the example we discussed above. These representations contain extra fermions. Extra fermions are also required in order that f b e asymptotically flat. On the other hand, we cannot rely on the Higgs which gives mass to the light families to give mass to these extra fermions. They would be too light, and some would already have been observed. They must get a 171

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larger mass. The most attractive possibility is that these extra fermions get SU(2) × U(1) conserving masses of order M! when H is broken. Some of these could come from the couplings of the Higgs, (8), but this cannot give mass to all the extra fermions since it does not couple to ffL. There must be a Higgs representation which transforms as (1,

1,N(N-

1)/2).

(9)

This can couple both to (~OL)2 and to (fiR) 2. To understand what is happening, it is convenient to think abom the SU(3) X U(I)G subgroup of R, where this U(1)G is the component of the weak U ( I ) which comes from G. An ordinary family of quarks and leptons comes from a piece of R which transforms as (3)1/6 + (1)._ 1/2

(10)

under SU(3) X U(I)G [the subscript is the U(1)G charge]. The weak U(1) of the LH fields is equal to the U(I)G charge. The weak U(! ) charges of the RH fields are equal to the U(1)G charge plus and minus 1[2 [from the RH SU(2)]. Thus (10)produces a normal family plus a RH neutrino. What we want in order for the couplings o f ( 9 ) to give mass to all the extra fermions is for R to decompose into three copies of (10) plus a (reducible) representation which is real with respect to SU(3) X U(1)G. Then the extra fertalons break up into pairs transforming like complex conjugates of one another under SU(3) X U(1)G. When (9) develops the most general vev consistent with SU(3) X U(I)G atMl, these pairs get put together into Dirac doublets. For example, a pair (3)7/6 + (3)--7/6 in R leads to two Dirac doublets of quarks, each with charges 5/3 and 2/3. One transforms like a doublet under SU(2)L, the other like a doublet under the (badly broken) SU(2)R. All these fermions have mass of order M I. Now, finally, we can address the question of flavor mixing angles. First consider the situation in which (a) there is no Higgs of the form (8); and (b) the Rtt neutrinos get their mass through coupling to neutral singlet leptons via a Higgs which transforms like (2b). In this case, the only source of flavor violation in the charged fermion mass matrix is the coupling of the Higgs (9) to (¢/L) 2 and 0kR) 2. The form of these couplings at the scale M G is determined by the G = SU(N) gauge symmetry and is therefore identical for (qJL) 2 172

30 June 1983

a n d (~R) 2, up to a possible overall scale factor. Of

course, these Yukawa couplings are renormalized in coming from M G to M I according to (5). The most interesting sectors of the fermion mass matrix are those which describe the charge 2/3 and - 1 / 3 quarks and the charge - 1 leptons. Here we will describe the quark mass matrices. Suppose that, in addition to the three light families, there are n Dirac pairs in R with the conventional charge [(3)1/6 + (3-)-1/6]" There are then 2n + 3 charge 2/3 (and - 1 / 3 ) quarks which can get mixed up by the Higgs, (9) [Note that charge 2/3 quarks coming from (3)7/6 + ( 3 ) - 7/6 pairs do not mix with the conventional quarks unless the Higgs, (8), is present] .Let us arrange the 2n + 3 LH and RH charge 2/3 quark fields into column vectors XjL, X/'R where[ = 1 ..... 2n + 3, such that• = I ..... n + 3 in X/L [XjR] and j = n + 4 ..... 2n + 3 in XjR /X/L[ refer to the fields which are doublets under SU(2)L [SU(2)R ] . Then the mass term has the following structure: X LMUx R + h.c.,

(11)

where

M~ = Kjsj(6jkm

* f L 4 - n - 3 + fRv~J-n-3)Kkrr

" (12)

Here m is the common SU(2) X U(1) breaking mass, not including (: dependent) renormalization. 4 is defined to be nonzero only for[ = 1 ..... n and k = 1..... n + 3. It represents the vev's of (9) which produce the order M ! Dirac masses. 1(7,sy and r/are the renormalization factors. K[ is the factor which is common to charge 2/3 and charge - 1/3 quarks.

K! = I-I [g,,(M1)/g~(Mo) l - 3Q'~ 2/16'~2B'~ ot

× [gtI(MI)/gH(MG)]-3qH2/16~r2BH.

(13)

Here c~ runs over all the factors of H with ga, Ta~ and the corresponding gauge coupling, gauge generators and 13-function, and

Bag3 are

k __

4)

The last factor, where gH, BHg~ are the U(1)H gauge coupling and/3-function and q/~is the U(1)H charge of theBh doublet (the LH doublet for/' = 1..... n + 3 and the RH doublet f o r [ = n + 4 ..... 2n + 3).

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rj = [gl_l(Ml)/gll(MG)l- 3qff /16,~2 aH for/. = 1..... n + 3 =1

(15)

( n + 4 ..... 2 n + 3 )

for/,= n + 4 ..... 2 n + 3

(1 ..... n + 3 )

is the part of the U(l)lt renormalization which is specific to charge 2/3 quarks. Tile point is that the U(I)t 1 charges of the LH and RH components of thejth quark field are qG and qG + 1/2 for charge 2/3 quarks and qG and q/G --1/2 for charge - 1 / 3 quarks. The mass matrix for the charge - 1 / 3 quarks has exactly the same form except that

sj--->l/sj,

r/~l/r/,

(16)

because of the different U(I)H charges. The mass matrices described by (11)- (16) have a very appealing property. Suppose that the renormalization factors K and r were absent. Then the o and v* terms determine which linear combinatons of ×t, and XR get large Dirac masses. For m "~ M i, it is the same linear combination for both and the left over light quarks are all degenerate with no flavor mixing up to terms of order m2/M 1. But as the renormalizations grow, so also do mass ratios between families and flavor mixing angles. If

K 1 ~K2,~Kj,

/ ' = 3 ..... n + 3 ,

(17)

there will be a natural family hierarchy, with/. = 1 associated with the lightest family and j = 2 with the intermediate mass family. Projecting out the heavy Dirac quarks does not affect the hierarchy very much for reasonable ~ (that is if all components of v/~ are of the same order of magnitude). If the r~ renormalizations are significant, there will be interesting splittings within families between the charge 2/3 and - 1 / 3 quarks. Flavor mixing arises because of (15). The charge 2/3 and - 1 / 3 quark mass matrices are different, and therefore tile unitary transformations required to diagonalize them will be different. Unfortunately, they are not different enough, for reasonable choices of the parameters. We can write MU=sbo ",

MD=s

l g r '1

(18)

Ia = ULDUtR , D/k = d/Sjk .

(19)

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eigenvalues d/,j = 1..... 3, much smaller than MI, which are, in some sense, "'averages" over up and down masses in each generation. The other eigenvalues are of order M 1. Because of (17), the dj's satisfy d 1 ,~ d 2 ,~ d 3 .

(20)

Now we can write the mass-squared matrices for the U and D quark LH fields as

MUM U*' = sU L DUtR r2 U R DU Es, ~ --2URDUtLS- 1 MDMDt =s -I ULDURr

(21)

The point is that by making the same unitary transformation (UL) on both ups and downs, we can bring these into the form

U~ MUM U* U L = U~SULDUtRr2URDU~sUL " u [ MDM D~fUL= ULS - I ULDU?Rr- 2URDUtLS -1UL . (22) These matrices are almost diagonal on the light quark fields because o f ( 1 8 ) ! Simply because of the factor of D on the outside, the off-diagonal terms are suppressed relative to the relevant diagonal terms by one power Ofdl/d 2 or d2/d3, which is a measure of the mass ratio between neighboring families, something like 1/10--1/100 in the real world .3. In fact, the situation is even worse for generic o/, because the unitary matrix U R is nearly diagonal. The moral is that we do not produce large enough mixing, at least not unless we introduce such peculiar u~'s that they spoil the hierarchy, (18). This is too high a price to pay. To us, it indicates that this simple model is inadequate. There are some minor modifications of the scheme discussed above which produce large mixings. For example, if we also have Higgs of the form (8), then quarks in doublets with unconventional charges (5/3 and 2/3 or - 1 / 3 and - 4 / 3 ) c a n mix with the normal doublets. This produces mixing of the right order of magnitude (the square-root of the mass ratio between families). But we do not like it. In this scheme, the

and

where rik = riS/k and gt is the matrix obtained by setting s~ and r~ = 1 in (1 2). Tile diagonal matrixD has three

4,3 The matrices u~sU L and its inverse do not affect this argument because s is equal to the identity on all the LH doublet directions. Note that the light quark mass matrix has no dependence on the s's at all, for M I ~, MW.

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angles are (in a sense) present in the couplings at MG, without any renormalizations at all, because they come directly from the interplay between the couplings of the two types of Higgs, (8) and (9). We feel that this violates the spirit of flavor democracy. It may be, however, that angles can be produced in an attractive way in models in which the SU(2) X U ( I ) breaking Higgs is a linear combination of Higgs in several representations of H. In this case, as discussed in ref. [4], the calculation of the masses involves not only the renormalization of the Yukawa couplings, (5), but also renormalization of the parameters in the Higgs potential which determines the vev. We are presently studying this possibility and hope to report on it in a future publication.

174

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References [1] T. Hagiwara and B.W. Lee, Phys. Rev. D7 (1973)459; S. Weinberg, Phys. Rev. Lett. 29 (1973) 388; B.W. Lee, Proc. 16th Intern. Conf. on High-energy physics (Chicago-Batavia, IL, 1972) eds. J.D. Jackson and A.Roberts, Vol. 4 (NAL, Batavia, 1L, 1973) p. 249. H. Georgi and S.L. Glashow, Phys. Rev. D6 (1972) 2977; D7 (1973) 2457; H. Georgi and A. Pals, Phys. Rev. DI0 (1974) 539. [2] S. Barr, Phys. Rev. D21 (1980) 1424, and references thereill.

[3] M. Chanowitz, J. Ellis and M.K. Gaillard, Nucl. Phys. B128 (1977) 506; A. Buras, J. Ellis, M.K. GaiUard and D.V. Nanopoulos, Nucl. Phys. B135 (1978) 66. [4] H. Georgi and M. Machacek, Nucl. Phys. B173 (1980) 32.