A theory of flavor mixing

ANNALS

OF PHYSICS

109, 258-313

(1977)

A Theory A. DE R~JULA, Lyman

Laboratory

of Flavor

HOWARD

of Physics,

Harvard

Mixing* AND S. L. GLASHOW

GEORGI,’ University,

Cambridge,

Massachusetts

02138

Received June 9, 1977

We investigate the origin of the Cabibbo angle and other flavor-mixing angles in an ambidextrous SU(2)r x SU(2)a x U(1) gauge theory of weak and electromagnetic interactions involving 2n quark flavors. We show how a discrete symmetry of the Lagrangian leads to flavor-diagonal weak currents where all flavor-mixing angles vanish. Each quark is assigned a position on the “clock of flavor” and the charged-current weak interactions are identified as nearest-flavor couplings. Soft breakdown of the discrete symmetry leads to a one-parameter family of flavor-mixing angles in the charged weak currents. All angles are expressed in terms of this parameter and quark mass ratios. The neutral currents remain flavor diagonal. Upper and lower bounds on 2n, the number of flavors, are obtained by constraining the theory to flt what is known from experiment. Departures from Cabibbo universality become intolerable if 2n < 8. It is impossible to obtain enough CP violation if 2n 2 12. We conclude that the number of flavors must be 8,10, or 12. We construct a more ambitious theory in which the Cabibbo angle and its analogs are O(z) calculable radiative corrections. For reasonable values of the quark mass ratios, we fail to obtain a Cabibbo angle of the right order of magnitude. Our theory involves an equal number of lepton flavors and quark flavors. A large number of flavors is required if neutrinos are to be sufficiently light. We explore the case of 2tz = 12, and find a novel mechanism for the neutrino induction of trimuon events.

1. INTRODUCTION Until the late 1950s it was generally believed that the weak interactions were characterized by a “universal” coupling strength common to muon decay, nuclear beta decay, and muon capture. This regularity was apparently disrupted when it was discovered that strange particles decay by weak interactions that are about a fifth as strong as other weak interactions. In the early sixties, Cabibbo and Gell-Mann and Levy [1] proposed a simple schemewhich both resurrected the idea of universality and gave it a plausible raison d’etre. The weak current was assumedto display a peculiar simplicity; the weak charge and its Hermitian adjoint were required to generate under commutation the Lie algebra SU(2). The weak current was to be composed of a canonically normalized leptonic current JL , a dS = 0 hadronic current Jh , and a dS = 1 hadronic current * Research supported in part by the National + Sloan Foundation Fellow.

Science Foundation

under Grant PHY75-20427.

258 Copyright All rights

0 1977 by Academic Press, Inc. of reproduction in any form reserved.

ISSN

0003-4916

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J’r, according to J = JI + cos B,& + sin 0J’h. Thus, the apparent failure of universality for dS = 1 processes was linked to a smaller (and observed) failure of universality for dS = 0 processes. The idea was as elegant and mysterious as it was phenomenologically successful. With the advent of unified gauge theories of weak and electromagnetic interactions the understanding of “Cabibbo universality” improved considerably. Closure of the commutation ‘algebra of the weak charges is the essential precondition for the existence of such a theory [2]. In what we shall call the “standard” SU(2) x U(1) four-quark, four-lepton gauge model [3], the existence of a Cabibbo angle is natural. The hadronic charged weak currents are not flavor diagonal, and the pattern of d-s flavor mixing is precisely of the form that Cabibbo proposed. The GIM [4] mechanism is also natural and the neutral weak currents are flavor diagonal to O(G) and O(aG). The actual value of the Cabibbo angle, however, is arbitrary, and must be put into the theory by hand. We briefly review these facts in Section 2. The standard model is not only elegant and simple but has also been incredibly successful in predicting several of the important experimental discoveries of the past few years (neutral currents, charmonium, charmed particles). More recently, however, experiment has jumped ahead of theory. Candidates for new leptons are accumulating at a fast pace [5]. Experiments on parity-violating effects in Bismuth atoms seem to rule out the standard model [6]. Anomalies in antineutrino scattering off nuclei hint at the existence of more than four quarks [7, 81. All of these are indications that Nature may have chosen a richer structure than the standard model. We review one such possible structure, the “ambidextrous” model [9], in Section 3, where we also comment on the phenomenological problems of the standard model. The ambidextrous model is based on SU(2)L x SU(2), K U(1) [IO] as the weak-electromagnetic gauge group. In such theories, there must be more than four quarks, and there must be left- and right-handed charged weak currents. The number of Cabibbo-like flavor-mixing angles increases quadratically with the number of quarks [I I]. The new angles are not known experimentally, but stringent limits are imposed on some of them by Cabibbo universality and the V-A nature of conventional particle weak decays. A situation analogous to the standard model, wherein all flavor-mixing angles are free parameters, becomes very unnatractive. Many mixing angles must be arbitrarily chosen with care and the benefit of hindsight. In Section 4 we solve this problem. We construct an ambidextrous model in which all flavor-mixing angles vanish identically. This is a consequence of a discrete symmetry of the theory, not the result of an arbitrary choice of parameters. The discrete symmetry forces the left- and right-handed charged weak currents to be as depicted in Fig. I, in a six-quark model. In the figure, R and L stand for right- and left-handed charged weak currents. All charged currents must be “nearest-flavor” couplings. The discrete symmetry enforces the condition that the quark mass matrix and the charged weak currents be simultaneously flavor diagonal; all flavor-mixing angles vanish. A theory without flavor-mixing angles has fewer parameters than a

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general theory, but is unrealistic for, after all, the Cabibbo angle may be small but it is not zero. In Section 5 we softly break the discrete symmetry of our theory in such a way that a one-parameter family of flavor-mixing angles is generated. When the discrete symmetry is softly broken, the quark mass matrix is no longer flavor diagonal in the basis where the weak currents are flavor diagonal. In this basis, the quark mass matrix (in a six-quark model) is diagrammatically depicted in Fig. 2. The figure means that there is a left-right entry llLcR in the quark mass matrix, with a coefficient cmd, E being the parameter that measures the breakdown of the discrete symmetry. Similar flavor-mixing terms between other quark pairs can be read off the figure. All mixings are functions of quark masses and the single parameter E. When the quark mass matrix is rediagonalized to properly define quark flavor, flavor-mixing angles appear in the charged weak currents. All the angles are functions of E and quark mass ratios. We devote Sections 6 to 9 to the calculation of these angles.

S

FIG. 1. The clock of flavor with six quarks. Solid (open) circles designate Q = Q (-9) Labeled arrows denote contributions to the left- and right-handed charged weak currents.

quarks.

FIG. 2. Diagrammatic representation of the quark mass matrices for Q = g and Q = -4 quarks.

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Notice in Fig. 1 that the discrete symmetry which is crucial to our approach has forced us to take as a starting point (before flavors mix), a weak current that seems phenomenologically unsuitable: It includes a right-handed (C& coupling. This would seem to threaten the whole approach from the very start, but the theory automatically evades this problem through a phenomenon that we call “flavor flip.” After we break the discrete symmetry and rediagonalize the quark mass matrix, the original (Ed), coupling turns into a current that is mainly (&)a . This is discussed in detail in Section 8. In Section 10 we discuss the curious possibility that the down and strange quarks are degenerate in mass before flavor-mixing. This leads to the prediction that tan2 8, is the ratio of down to strange quark masses. We construct Yukawa couplings that naturally ensure the “curious” possibility. In Section 11 we compare our predicted flavor-mixing angles with experiment, and run into trouble. In the six-quark theory, the strength of the induced righthanded (z1~& coupling is too large to agree with experiment. We generalize our theory, and the discrete symmetry on which it is based, to describe more than six quark flavors. We are led to the conclusion that eight or more quark flavors must exist for our theory to agree with Cabibbo universality to its observed degree of precision. In Section 12 we discuss CP violation. We find that our theory naturally leads to the observed size of CP-violating effects, provided there exist no more than 12 quark flavors. The theory reproduces the superweak phenomenology for K---f 2x decays when we restrict ourselves to the “curious” possibility discussed in Section 10. We devote Sections 13, 14, and 15 to the discussion of a more ambitious approach wherein all mixing angles, and the Cabibbo angle in particular, are calculable, rather than just related to each other and to the quark masses [12]. In this, we are not successful. Our conclusions are given in Section 16, together with a discussion of some experimental consequences of our theory. The final chapter is devoted to a study of the leptons. No fewer than 12 leptons are needed to prevent neutrinos from becoming too massive. We encounter a novel mechanism for the production of multimuon events by neutrinos and antineutrinos. The fact that neutrinos are light is shown to be related to the exact and separate conservation of electron and muon numbers.

2. THE CABIBBO ANGLE

IN THE STANDARD

MODEL

In the standard four-quark, four-lepton SU(2), x U(1) unified gauge theory, all right-handed fermion fields are assigned to SU(2), singlets. All left-handed fermion fields are assigned to SU(2), doublets: (2.1)

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where d, =dcosB,+ssinO,, so =SCOS&-ddsin8,.

These choices correspond to a charged weak current (2.2)

where L=

cos e, sin ec ( - sin ec cos 8, >*

A property of a theory is called “natural” [13] if it is independent of the choice of parameters, as opposed to a property that is imposed by hand by a judicious but arbitrary choice of parameters. The occurrence of the “Cabibbo-rotated” quark dO and the orthogonal “GIM-rotated” quark se in Eq. (2.1) are, in the standard model, natural rather than mysterious. The reason is as follows. A quark, or a quark flavor, is defined as an eigenvector of the quark mass matrix.l The quark mass matrix arises from renormalizable Yukawa couplings of the quarks to scalar Higgs fields. These acquire nonzero vacuum expectation values (vev’s) as a consequence of spontaneous symmetry breakdown of the gauge symmetry. The gauge invariance and renormalizability of the theory do not impose the requirement that the charged weak currents and the quark mass matrix be simultaneously flavor diagonal. Let us denote the quark doublets
d OR'

'OR

9

(2.3b)

'OR -

Then, the hadronic charged current is 0.4)

The general form of the quark mass terms in the Lagrangian is h.c., 1Jf the quark massterm in the Lagrangian is $&& value equation

(2.5)

+ h.c., the quark massesm satisfy the eigen-

where Y and 1are right- and left-handed mass eigenstates.

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with MU0 and MD0 arbitrary matrices. The mass matrices Muo, MD0 can always be made diagonal, real, and nonnegative, by biunitary transformations (2.6a) (2.6b) Let the basis where the quark mass matrix is flavor diagonal be denoted by quark names without subscripts. In this basis flavor is well defined and the charged weak currents are (2.7a) with (2.7b) The presence of the arbitrary unitary matrix L in the weak current means that the charged weak currents need not be flavor diagonal. With only four quarks, the phasesin L can be reabsorbed into innocuous redefinitions of quark fields, so that all L can be is a two-dimensional rotation, asin Eq. (2.1). In this argument the parameters (the entries in the quark massmatrix) were left arbitrary. Thus, the Cabibbo angle and the GIM mechanism are “natural.” If neutrinos are massless,no mixing angle occurs in the lepton sector: the entries v,, and v, in Eq. (2.1) are definitions, there being no neutrino mass matrix to distinguish the two neutrinos by specifying their masses.The occurrence of the Cabibbo angle is natural. Its value, and the values of the massesof the quarks and charged leptons, is entirely arbitrary.

3. AMBIDEXTROUS

THEORIES. NEED WE Go BEYOND THE STANDARDMODEL?

In the simplest version of the standard model, the massesof intermediate vector bosons, quarks, and leptons are all generated by couplings to a single Higgs doublet. The ratios of charged current to neutral current neutrino- and antineutrino-induced cross sections are functions of a single parameter: the weak mixing angle Bw . The observed ratios, for elastic and deep inealstic cross sections on nuclear targets agree with the expectations of the standard model, for a common value of sin28, N 0.3 [14]. This success,and the simplicity of the theory on which it is based, is unmatched by any other unified gauge theory of weak and electromagnetic interactions proposed so far. Yet, there are experimental indications that the standard model (or its generalization to more left-handed quarks and lepton doublets) may not be the correct theory. These indications are the “y-anomaly” and the apparent suppression of parity-violating effects in the atomic levels of Bismuth.

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Two related facts are generally referred to as the “y-anomaly.” One of them is the apparent increase with energy of the ratio of antineutrino-induced to neutrino-induced charged-current inclusive cross sections off nuclei. The other is the increase with antineutrino energy of ( y), where y is defined as the fractional energy loss y IZ= (E, - E,)/& in inclusive antineutrino scattering off nuclei. These effects can be easily explained if there exist “fancy” [7] right-handed couplings (u, !J)~ of up quarks to “bottom” quarks, couplings that are beyond the purely left-handed couplings of the standard model. The experiments aimed at detecting parity-violating neutral-current weak effects in the interaction between atomic electrons and nuclei are still at the pioneering stage. The observed effects, however, are compatible with zero and are well below the standard model expectations (the “atomic physics” part of the theoretical estimates, however, is still a subject of debate) [6]. “Vector” SU(2) x U(1) gauge theories have been proposed that predict the existence of right-handed couplings such as (u, b)R , and no observable parity violation in the “atomic physics” experiments [15]. Vector-like theories, however, predict that o(v -+ v) = u(c --f c) for any given process. This disagrees with experiment. The standard and vector theories are the only SU(2) x U(1) theories with conventionally charged quarks (Q = $, -i) in which the GIM suppression of AS = 2 effects and dS = 1 neutral currents is “natural.” [16] But, as we have seen, both standard and vector theories seem to have trouble with experiment. Thus, if one wants to stick to theories where the GIM mechanism is natural, one must abandon conventional charges or enlarge the gauge group. The second possibility has been discussed by Mohapatra and Sidhu, and by us, in the form of “ambidextrous” theories [9]. The ambidextrous theories are in a sense “in between” vector and standard theories. They have right-handed couplings that may explain the y-anomaly, neutrino-coupled neutral currents that are not parity conserving, and small or vanishing parity-violation effects in atomic physics. The ambidextrous gauge group is SU(2), x SU(2), x U(1). The SU(2), and W(2), gauge couplings are made identical by the requirement that the Lagrangian, before spontaneous symmetry breakdown, be invariant under the parity operation W, f-) W, . There must be at least six quarks. The quark multiplet structure is as follows. The left-handed quark fields are (2, 1)‘s: doublets under W(2), and singlets under X7(2),: (3.1)

The right-handed

quark fields are (1,2)‘s: (3.2)

In the above expressions the quark labels could in general stand for arbitrary orthonormal combinations of quark fields. Yet, Cabibbo universality and the V-A nature

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of neutron and lambda b-decays have forced us into a series of unnatural choices: all flavor mixings that have not been specified in Eqs. (3.1) and (3.2) must be kept very small. It is the purpose of this paper to investigate the origin of all flavor-mixing angles in the ambidextrous context, to attempt to relate them, and to investigate whether the choices implicit in Eqs. (3.1) and (3.2) can be made natural. Quark masses, as we shall see in much detail in the following sections, are generated via the Yukawa couplings of left- and right-handed quark doublets to a Higgs field 4 with a nonzero vacuum expectation value. The field 4 belongs to a complex (2,2) representation of SU(2)L x SU(2), and acquires a vacuum expectation value

(4, = a0(:, fj.

(3.3)

The field rj that is necessary to generate quark masses also contributes to the masses of the intermediate vector bosons. Let the gauge couplings of SU(2), x SU(2), x U(1) be defined by the covariant derivative D” = a@+ i + I

W

WL.“TL + W,“TrJ

+ (cos 2”s )l,z Yxw\ W

(3.4)

where XJ‘ is the U(1) field and Y is the weak hypercharge, defined so that TsL + TfR + Y = Q, the electric charge. As the gauge symmetry is spontaneously broken, intermediate vector boson masses arise from the kinetic terms in the Higgs Lagrangian: -Y&in = $tr D”~D,c$‘. (3.5) The transformation

properties of C$under SU(2),

x SU(2),

x U(1) are

where the T are the Pauli matrices. The contribution of 4 to the masses of intermediate vector bosons is computed in the standard fashion by inserting Eqs. (3.6), (3.3), and (3.4) into Eq. (3.5) and selecting the terms quadratic in the intermediate vector boson fields. We will be interested in the result for the charged intermediaries, a matrix in W, , W, space: m.2 =

e2a02 1 + E2 2E 4 sin2 Bw ( 2~ 1 + l 21 ’

(3.7)

Because the field 4 transforms nontrivially both under SU(2)L and SU(2), , the matrix Eq. (3.7) is not diagonal. Right- and left-handed intermediate vector bosons mix and there will be effective four-fermion interactions involving the left-handed currents of Eq. (3.1) coupling to the right-handed currents of Eq. (3.2).

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In general, the Higgs field 4 is not the only necessary one. In the ambidextrous theory of Ref. [9], for example, a field belonging to the (1, 2) representation of SU(2), x SU(2), was found to be necessary to give masses to neutral heavy leptons. Higgs fields belonging to (N1, 1) or (I, NJ representations, with Ni > 2 contribute only to the diagonal entries of the mass matrix Eq. (3.7), which in the more general case is of the form (3.8) Let R(y) be the orthogonal

matrix that diagonalizes Eq. (3.8)

R-lm2R = M,2 0

0 Mh2’1

R=

(3.9b)

Denote by W, and wh the eigenfields with masses Ml and Mh , where I and h stand for lighter and heavier. The left-handed charged weak currents couple to the combination (3.10) COSywL +sinyWh, while the right-handed

currents

couple to the orthogonal -SinywL

The left-left, are

left-right,

+

and right-right

combination

cosywh.

effective current-current

LL:

e2 cos2 y sin2 y -t r 8 sin2 Bw ( Ml” )’

LR:

e2 ~cos y sin y 8 sin2 8,

RR:

e2 8 sin2 BW

(3.11) Fermi couplings

(3.12a)

(3.12c)

The nonvanishing of the Fermi coupling Eq. (3.12b), can produce the y-anomaly. In Section 11 we will need to discuss the ratio of LR to LL couplings, LR -zzz LL

2E < 25 1 + E2 + 62

(3.13)

which follows readily from Eq. (3.8). Our evaluations of E will show that the y-anomaly must be small.

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4. How TO GENERATE FLAVOR-DIAGONAL

CHARGED

WEAK

CURRENTS

We construct theories whose symmetry properties are such that the right- and left-handed charged weak currents are flavor diagonal in the quark field basis where the quark mass matrix is flavor diagonal. The Cabibbo angle and all other flavormixing anglesvanish. At a subsequentstage (discussedin the next section) we break the symmetry in such a way that a one-parameter family of flavor-mixing angles is generated in the charged weak currents, while the neutral weak currents remain flavor diagonal. We demand that the relevant properties of the theory at every stage be “natural,” independent of the specific values of the parameters, and stable under renormalization. Because of our naturalness requirement, we cannot olympically state, “Let the currents and the quark massmatrix be simultaneously flavor diagonal.” As we saw in the example of the standard model, nondiagonal terms in the quark massmatrix are not forbidden by the gauge symmetry and renormalizability of the theory. They must in general be present as arbitrary parameters, and they result in arbitrary flavor-mixing angles. We construct theories in which nondiagonal massterms vanish as a consequence of a symmetry of the theory, not by arbitrary choice. In gauge theories with spontaneously broken symmetries, massesarise via couplings to scalar fields that acquire nonzero vacuum expectation values. Thus, we are after Higgs couplings whose symmetry properties are such as to forbid flavor-mixing terms in the quark massmatrix. Consider the six-quark version of the ambidextrous SU(2)L Y SU(2), x U(1) theory. Let z+$+,&, k = 1, 2, 3, denote left- and right-handed quark doublets: (4.la) (4.lb) The fields #kR[#kL] belong to the (2, 1) [(I, 2)] representation of SU(2), x SU(2), . At this point the quark labels are just conventions: we have not written that piece of the Lagrangian that specifies“who’s who”: the quark massmatrix. What we do below is to show that we can construct a theory in which the massmatrix naturally has the form muoiiLouRO + m,“?LocRo+ mtoiLotRo+ nzdo;iLodRo + msoSLosRo + mbO&ObRO. (4.2) We could have chosen the right-handed doublets to be identical to the left-handed doublets. Then it would be trivial to obtain Eq. (4.2). We would simply demand that our theory conserve the u + d, c + s, and t + b numbers. We do not study this simpler casebecausethe resulting theory has no Cabibbo angle and there is no obvious way to get any. Once we demand that the right-handed doublets be different from the

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left-handed doublets, the form Eq. (4.1) is essentially unique. As we shall see, it is then only a small step to the natural appearance of a set of “Cabibbo angles.” Figure 1 is a pictorial representation of the structure of the charged weak interactions implied by Eq. (4.1). The right- and left-handed charged weak currents are all “nearest-flavor” interactions. Quark masses will be generated by couplings to a Higgs field 4. Because quark mass terms are &z/R + h.c. and because I& - (2, l), & - (1,2), the field 4 must belong to a (2,2) representation of SU(2)L x SU(2), . Charge conservation requires that 4 carry no U(1) charge. Thus #Jis of the form (4.3a) with all fields complex. Define, as a device to simplify the coming notation, the field

where x+ = -x-+ and & = -$++. We demand that the theory be invariant under the discrete symmetry (4.4a)

D:

(4.4b) (4.4c)

This, as we shall see by construction and demonstration, is the symmetry that makes currents and masses simultaneously flavor diagonal. Notice that the field 4 transforms like a reducible representation of the SU(2), x SU(2)a group because we can write

4 = 4 + iz2,

(4.5)

where G.2 =

01.2

+ iT . x1,2

(4.6)

with the u’s and r’s real. Each of the Z’s is a “real” (2, 2), transforming into itself under SU(2), x SU(2)R like the fields in the linear sigma model. The 27s are real in the sense that ‘E = r,.z:*r, = z. It is only the discrete symmetry which transforms Z; into Z2 .

(4.7)

THEORY

The most general gauge-invariant,

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D-invariant

LZH = &tr D&DY#+ - p2 tr($+& - g I tr(#+6N2 -fl

269

Higgs coupling is - h tr(++&$+$)

tr(4++l12,

(4.8)

where D, are covariant derivatives. The stability of the vacuum imposes the conditions h +f > 0, h + 2f + 2g > 0 on the quartic self-couplings. For imaginary p, the usual Higgs phenomenon takes place and 4 acquires a nonzero vacuum expectation value. If g > 0 (4.9a) a, = j I* j/(2/? + 2f)l’“.

(4.9b)

It is the term g I tr($+J)12 w h’ic h constraint the vev to have the form of Eq. (4.9a). This term is

g I tr(~+4%2= 4g I 96 I2 I x0 12.

(4. IO)

For g > 0, this is minimized when either $. or x0 vanishes. The discrete symmetry forbids the appearance of any terms linear in $. or x0 (i.e., xo+o3) which would spoil the form of Eq. (4.9a). The most general gauge-invariant, D-invariant Yukawa coupling of fermions to Higgs fields is =% = - i

[muo&,hh,

+ ~bV2,N2,

+ nQ$3LN3Ll

- $ [m~“$lL6#2R+ msO$2L&3R + n~~“$3L&+4Rl + kc.

(4.11)

The quantities m,O, etc., are arbitrary constants with dimensions of mass and suggestive names. The field 4 develops a vev of the form shown in Eq. (4.9) and the gU develops the fermion mass terms muOtiOuO+ mCo~,co + m,Oi,t, + mdoJodo + msoSOSO + mb060bo.

(4.12)

Lo ! We have generated diagonal and only diagonal quark masses. There is no mystery in this; the discrete symmetry D forbids the occurrence of terms such as $2L~#2r, that would have become id nondiagonal mass terms. If $ were to acquire a vev in both of its diagonal entries, nondiagonal quark mass terms would appear despite the simplicity of the Yukawa couplings. However, our discrete symmetry also prevents this from occurring. It is essential to the success of our scheme that the right-handed doublets be completely different from the left-handed doublets. The right-handed doublet (CO,S”>R>which is so often inserted into vector theories, is here simply not permitted.

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Since charmed D mesons decay very preferentially into strange final states, there is a potential difficulty here. We shall see anon how deftly our theory skirts this problem. Now that the quark mass matrix has decided who’s who among quarks, the labels in Eq. (4.1) acquire the meanings they conventionally signify. The gauge bosons couple to the doublets of Eq. (4. I), so that the weak currents are indeed flavor diagonal in the basis wherein quark masses are diagonal. All Cabibbo-like angles are zero. Flavor diagonality of the charged weak current has been implemented in a natural way: It results from a symmetry of the theory and not from a choice of parameters. It is a corollary that the diagonal character of the currents will not be destroyed by renormalization counterterms, since these must share the symmetry properties of the original theory. The above argument shows that any Cabibbo-like angle which appears in this theory to any order must be finite and calculable, because there are no counterterms available to absorb infinities in such an angle. But, do such angles appear at all? For example, can the s-quark decay (nonleptonically) into nonstrange quarks due to higher-order effects ? One might think that it can. After all, the discrete symmetry D that leads to the flavor-diagonal character of the charged currents is spontaneously broken. There are charged currents coupled both to ?d and to Ts, so why could not the s-quark virtually emit a W- and become a c, then absorb the W- and become a du Alas, this does not occur because the left-handed W which couples to ?s is not the same as and does not mix with the right-handed W which couples to Ed. Let us be more precise. There is a discrete symmetry of this theory, not spontaneously broken, which prevents the appearance of a Cabibbo angle in any order. It is

exp ii (4k - 1)37. 1 ( --i + (

i

(4k 6 3) 6 “!

exp (i+-

TV) AL , (4.13)

T3) hRy

+-+exp(i+-)exp(-i$-~p)+exp(-i-+-~:,). D’ is a product of the discrete symmetry D with a finite gauge transformation. Since D’ leaves the vev of Eq. (4.9a) invariant, this symmetry remains after spontaneous symmetry breakdown. Both the L and R components of each flavor transform the same way under D’ [as they must for the mass Eq. (4.12) to be invariant]. The flavors transform as follows:

) (tO+exp

(i+)

to,

2Tr do + exp i ( 3 1 do, 4Tl so---f exp i-i 3 > so, b” ---f exp (i+)

bO.

(4.14)

THEORY OF FLAVOR MIXING

271

This has the general form qn -+ ein*13qn, wheren=1,2,3,4,5,6forq=u,d,c,s, t, b. The discrete symmetry suggestively associates each flavor with a different discrete phase rotation (see Fig. 1). From Eq. (4.14) it is clear that so cannot mix with do in any order, because a mixing term is not invariant under D’. An amusing but probably irrelevant comment is that strangeness is only multiplicatively conserved in this theory. Three s-quarks decaying into three d-quarks is consistent with D’ symmetry, and readily proceeds in fourth order of weak interactions.

5. BREAKING

THE DISCRETE SYMMETRY

We saw in the previous section how the discrete symmetry defined by Eq. (4.4) leads naturally to charged currents and a quark mass matrix which are simultaneously flavor diagonal. The discrete symmetry makes all flavor-mixing angles vanish. In order to generate angles, and in particular the empirically needed Cabibbo angle, the discrete symmetry must be broken. The quartic Higgs meson self-couplings in Eq. (4.8) and the Yukawa couplings in Eq. (4.11) are of dimension 4. Should we break the discrete symmetry in either of these couplings, renormalization effects would force us to break it in the other. Flavor-mixing angles would appear, and they would all be quite arbitrary. We would be back to the unpalatable situation where all quark masses and mixing angles are adjustable parameters. There is, however, another possibility. We may introduce a “soft” breakdown of the discrete symmetry in the mass terms of the Higgs mesons. These terms are of dimension 2, quadratic in scalar fields. It is possible for these superrenormalizable interactions to destroy the discrete symmetry without the appearance of quartic or Yukawa counterterms [17]. There are finite calculable D-violating Yukawa couplings induced by higher-order effects, but these are of order 01.The most important effect of soft D breakdown is to change the vev of the Higgs field 4, as we now discuss. Let us write the (2, 2) Higgs scalars of Eq. (4.3) as (5.1)

where we have used the gauge freedom of the theory to eliminate the charged Higgs fields (in favor of longitudinal components for the charged intermediate vector bosons) and to make the fields a and b relatively real. The Higgs meson Lagrangian of Eq. (4.8) is 9u = $tr(D%$+D,,& - V(4), (5.2) where the potential

V(4)>, written in terms of the fields defined in Eq. (5.1), is

~(4) = +$(a2 + b2) + (h + f>(a” + b2j2 + 2G’g - f4 a2b2.

(5.3)

272

DE RtiJULA,

GEORGI, AND GLASHOW

V(4) is minimized when CI = a,, [the value quoted in Eq. (4.9b)] and b = 0. We now add a term that solftly breaks the discrete symmetry V(4) --+ V’(4) = V(4) + /Pei~ tr(d+j?) + h.c. In terms of the fields of Eq. (5.1) the new potential

(5.4)

is

v’(4) = p2(u2+ b2) + @ + f>(a” + W2 + 2(2g - tz) a2b2- 2fi2 cos(j3 + 201)ab.

For imaginary p, the minimum that satisfy the equations

(5.5)

of this potential now occurs at values of a and b and c1

(5.6a)

uo2+ b,2 = I P 12/W + f>,

(5.6b)

u,,b, = /T2/2(2g- h), a0 = -p/2.

(5.6~)

Thus, the field 4 acquires nonzero vev’s in both of its diagonal entries:

(4) = ewi(B/2)ao (i t 1,

(5.7a)

(4)= ei(O/2)uo (i v),

(5.7b)

where we have defined E = bo/uo. Invariance under CP corresponds to the choice /3 = 0. In the next several chapters we neglect CP violation and take /3 = 0. In Section 12, we relax this condition and study the question of CP violation in detail.

6. A ONE-PARAMETER

FAMILY

OF FLAVOR-MIXING

ANGLES

We have broken the discrete symmetry that allowed us to construct a flavordiagonal quark mass matrix in such a “soft” way that the only significant change in the theory is in the vev’s of the Higgs fields

do = a0(:, ;) -+ do = a, (:, t,.

(6.1)

THEORY

OF

FLAVOR

273

MIXING

Insertion of Eq. (6.1) into the Higgs couplings of Eq. (4.11) now leads to nondiagonal quark masses: + h.c.,

(6.2a)

(6.2b)

(6.2~)

Equations (6.2b) and (6.2c), where we have denoted quark masses by quark names, have the structure depicted in Fig. 2. The left- and right-handed charged weak currents can be read off Eqs. (4.1). They are

0 d0

3

(6.3a)

J,“-o = 4(Uo, zo , fo), y,,RO so , 0 b0 R

(6.3b)

J,‘*”

=

4(Uo 9 co 2 fO)L Y&O

so b0

L

do

where (6.3~)

(6.3d)

To define the quark flavors we must diagonalize the quark mass matrices. In the CPconserving case we are now discussing, this can always be done by biorthogonal transformations: (6.4a)

(6.4b)

274

DE RljJULA,

GEORGI,

AND

GLASHOW

where quark names with no subscripts stand for quark eignemasses and RD , LD , RU , Lu are three-dimensional orthogonal matrices. In the basis where the quark mass matrix is flavor diagonal, the charged weak currents are

JuL = 4(U, (‘, i)L y,L

)

J,” = 4(?, i, ii)R y,,R

(6.5a)

,

(6.5b)

where L = LuLOLD1,

(6.5~)

R = RQTRURoR$.

(6.5d)

Notice that in Eq. (6.5b) we have reordered the Q =I $ quarks. This will be convenient when we examine the explicit form of the matrix R. The quark mass matrices, Eq. (6.2) depend only on the six “bare” quark masses, a,, sO, b, , uU , and t, and the single parameter E. The currents, Eq. (6.5), are determined by the same seven parameters, or equivalently by the physical quark masses and the parameter E. All mixing angles are determined by any one angle, say the Cabibbo angle, and the quark masses. We shall confront the question of whether the remaining angles describe a phenomenology which is compatible with experiment. First, however, we must find the matrices L and R.

7.

THE

FAILURE

OF PERTURBATION

THEORY

IN

E

Observed left-handed charged weak currents are rather close to the flavor-diagonal currents we expect for E = 0. This suggests that we might be able to treat E as a small parameter and determine the mixing matrices of Eq. (6.4) in perturbation theory in E. To first order in E the quark masses are just the “bare” quark masses because the E terms are all off diagonal. To compute the mixing angles to first order in E, it is sufficient to study separately the 2 x 2 submatrices for each quark pair. As an example, study the U-C system described by the submatrix (7.1) We

wish to diagonalize this with a biorthogonal

transformation (7.2)

THEORY

OF

FLAVOR

275

MIXING

To this order, the matrices have the form

ru= (‘,, Equation

‘J,

c = (Jez ;I).

(7.2) becomes (;

g = ( --Eruo -“:~co

with u = U, and c = co as promised,

+ Edo EzUo2; ,,,O) + WE%

(7.4)

and

luo -I- rco = 0, (7.5)

YUO+ zc, = do )

or r = dorro/(uo2 -

I = doco~(c~- u$),

co”).

(7.6)

These perturbation theory estimateswill be reliable so long as EZand ETare small. We can perform the same analysis for each pair, and put the results together to construct the matrices L and R which determine the charged weak currents. We find 1 L,

R

=

(7.7)

-?1L,R -rL.R

with Ecodo v-

=

uo2

TL = U02

_

=odo c 2 + 0

so2

-

TR

=

do2 ’

EUobo EUobo - to2 + b,2 - d:’ -of0

UL

-

co2

-

c-Jot0 to2

+ bo2- so2’

(7.8a)

and ECOSO

Q-R =

‘SR

=

so2Eyd

0

cuodo co2

-

-

co2

-

to2

7

cuodo uo2

+ bo2- do2’

uo2

+ bo2 - so2 ’

Etobo to2

2 +

etobo

(7.8b)

These results have a simple graphical interpretation. Consider, for example, the

276

DE RtiJULA,

GEORGI,

AND

GLASHOW

Q term in the matrix of Eq. (7.7). When insreted into Eq. (65a) it yields a term in the charged weak current, ?1LQYuU + Y&.

(7.9)

We can compute qL by doing ordinary covariant perturbation theory treating the E = 0 theory as our “free” Lagrangian and the off-diagonal terms in the mass matrices [Eq. (6.2)] as a perturbation. The first term in qL in Eq. (7.8a) corresponds to the diagram shown in Fig. 3. This diagram shows a Us W coupling induced by the perturbation, marked by a cross. The amplitude for the process is Cd (1 - 7%) F + cn 0 p2 _ cn2 Y”(l + 7%) 2

(7.10)

(with the CsW coupling normalized to 1). Thep term vanishes because of the (1 f y5)‘s. Because p2 = uo2 for an on-mass-shell u quark, the amplitude is

The coefficient is precisely the first term in rL , corresponding of Eq. (7.6).

to the u. - co mixing

FIG. 3. A Feynman diagram contributing to a Wits coupling to first order in e. The order-e u-c mass mixing is designated by a cross.

How good is this perturbation estimate for the problem at hand? To answer this question, we must discuss the sizes of the mixing terms in Eq. (7.8) in the physically interesting case, where to , b, , co > so , do , u,, . The 7L , qL , and 7R terms are manifestly small, of order ER or cR2, where R is a ratio of a light quark mass to a heavy quark mass. The uR term presents a potential problem; it satisfies c+ > 2~ and will be large if b, > to or to 3 b, . But, certainly the terms 71~and qR which have light quark masses in denominators are large. Unless E is very small, rlL and qR will be large. In fact, 7L is just the Cabibbo angle to this order in E, since it is the coefficient of the Us, term in the charged weak current: We know experimentally that vL N 0.2. Since the

THEORY OF FLAVOR MIXING

277

terms in 9, and vR with heavy quark massesin the denominator are small compared to those with so2- do2, qL and r], satisfy

Equation (7.12) is a disaster for the perturbation theory becausewe believe that the s mass(in the Lagrangian) is much larger than the d mass.Standard current algebra estimates [18] give s/d = 20. Clearly qR is much too large to admit a useful perturbation estimate of this kind. Despite its ultimate failure, this exercise in perturbation theory is encouraging. It is clear that many of the Cabibbo-like anglesin this theory are small, of order ER. One useful approach to a more accurate analysis is to treat mixing in the q-d, system exactly and all other mixings perturbatively. This is done in Section 9. A slightly simpler approach is suggestedby the fact that only qR is large. Perhaps the field that we have been calling doRis mostly sRwhile suRis mostly dR . In that case, we can use perturbation theory, but we must start with a mass matrix that treats d,,, and soL as the chiral components of the s field.

8. FLAVOR

FLIP

In the exact discrete-symmetry limit of our theory (e = 0) the right-handed doublets are

A right-handed coupling cTdRwas proposed [19] some time ago as a straightforward and economical way of enhancing the AI = 4 amplitudes in AS = 1 nonleptonic decays. The coupling EdR, however, soon faced an avalanche of theoretical criticism [20]. Meanwhile, charmed particles have been discovered, and an experimental upper limit on their decay branching ratios into nonstrange and strange final states now exists [21]: F(D -+ nonstrange) = 0.065 + 0.040. r(D + strange) Thus, either a right-handed EdRcoupling is unacceptable, or the su(2), intermediate vector bosons do not mix much with the SU(2)L ones. The second possibility, in our framework, would disagreewith the large observed high y-anomaly. On the other hand, as we have already emphasized in Section 4, we can only build our crucial discrete symmetry if the right-handed couplings are as in Eq. (8.1). We seemto be facing an unsurmountable problem. This is not so. We devote the rest of this section to proving that the theory automatically solves this problem. The key observation was made at the end of Section 7. The doR field is misnamed; it is actually primarily sR. The reason is that for any choice of parameters that yields

278

DE

RtiJULA,

GEORGI,

AND

GLASHOW

the correct Cabibbo angle and physical quark masses, ECUis actually larger than do and .s, in Moo [Eq. (6.2b)l. To see that this is corerct, we redo the perturbative analysis of Section 7, writing MDO== i&

cr

lj+(1

;

cIo)

(8.3)

and treating the second term as a perturbation. Now the lowest-order s mass is EC,. The d mass is nonzero only in second-order perturbation theory, which gives d N doso/ecO N dose/s.

(8.4)

We find that (8.5a) with Ecodo TL

=

7L

=

UL

=

;

uo2

-

co2

uo2

EUobo _

to2

do EC0

+

2

-

(8.5b)

;

ESOfO co2

’

@oto

to2 + bo2 -

E2C02’

and 1 R =

TR -Us

pR

-f” i -OR

(8.6a)

1

-7R

with

5-R

=

Euodo

2--

CO

Euodo uo2

ctobo *R

Again, the potentially

-

to2 -

(8.6b)

+ bo2 - E2C02’ ++

Uo2

0

large terms are yL (=e), TR , and OR . But now, since and

?)R+,

0

(8.7)

we tind 1 doso TR

=-

7jL

(‘Co)”

1d =

e s

*

(8.8)

279

THEORY OF FLAVOR MIXING

If d/s is as small as current algebra suggests, about l/20, then qR is not large. Instead, 71~and TR are roughly equal, so that this perturbation estimate is consistent. Qualitatively, what has happened is the following. In order to obtain 6’ N 0.2 and d/s N l/20, we were forced to take EC,,larger than d,, and s, (by about a factor of l/e ‘v 5). In this situation the physical s quark mass term comes primarily from the cc0 term in the mass matrix. The system looks very different from its E = 0 limit. In particular, the dORfield is mostly & while the soRfield is mostly dR . We refer to this phenomenon asfluvorflip. It completely obviates the problem posed at the beginning of this chapter. In the flavor-flipped theory, the charmed quark has right-handed couplings to soR = -sR + qRdR(ignoring terms of order ER and vR2), where 9R is a mixing angle of the same order of magnitude as the Cabibbo angle. Both the rightand left-handed currents cause charmed particle decays primarily into strange final states.

9. EXACT

OF d-s MIXING

TREATMENT

The validity of the perturbation expansion used in Section 8 depends on the small value d/s ‘v l/20 usually obtained from current algebra. The assumptions required to obtain this result have recently been called into question by Gunion et a/. [22]. They estimate d/s N l/5. If this is correct, yR in Eq. (8.6) is large, and perturbation theory is unreliable. Fortunately, there is another way to do the analysis which is more general and only slightly more complicated. We can treat d - s mixing exactly and all other mixings perturbatively. The matrix L appearing in the left-handed current, Eq. (6.5a), is most conveniently expressed as a product of two matrices:

i

pL 1 -TL

-PL 1 -GIL

oL 1 ii TL

- cos sin eL 6L cos sin BL 0, 0 1 . 0 0 1

(9.la)

We find (9.lb)

l cdo PL

=

rL

=

’

OL

=

Et%

c2 -

w

u2

E--,

do

uob ( b2

(9.lc)

C bou

_

d2

b2

_

c2

_

u2

1

1 (

NN ’

(

1 s2

-

t2

_

c2

1 ’

_--uo b

bou c2

1’

(9.ld) (9. le)

280

DE

RZjJULA,

GEORGI,

AND

GLASHOW

The matrix R appearing in the right-handed current in Eq. (6.5b) is also most conveniently expressed as a product of two matrices: --t-b

TR

OR

1 ---OR

1

-

cos

8,

sin 8,

Sin

eR

COS

0

I(

&+ 0

0 0

(9.2a)

1

where

?-R =

Cd” c2 - 28

tab

(9.2e)

hot

Notice that Eqs. (9.lb) and (9.2b) imply tan eR = (s/d) tan eL ,

(9.3)

which shows that flavor flip occurs whenever s/d is larger than l/e, . The undiagonalized quark masses U, d, etc., appearing in Eqs. (9.1) and (9.2) are themselves expressible in terms of eigenmasses and angles. Heavy quark masses are negligibly affected by the nondiagonal terms in the original quark mass matrix c N co ) tCrlto,

(9.4)

b N b,,

while the corresponding relations for the lighter quarks are do =

COS

6,

COS

&d +

Sin

8,

Sin

eR$,

(9.5a)

So =

COS

eL

COS

eRs +

Sin

eL

Sin

t&d,

(9.5b)

u, N 24+ E2 (c:yd~2)2 (

+

(t2t”“x2

)

c: 24.

(9.5c)

Insertion of Eqs. (9.4) and (9.5) into Eqs. (9.1) and (9.2) gives the desired result: All flavor mixing angles as functions of the single parameter E and the quark eigenmasses. Let us reexpress the currents in the representation where the quark mass matrix

THEORY

OF

FLAVOR

281

MIXING

is diagonal in the form of the doublets to which the intermediate couple. Define the rotated fields do =

cos

vector bosons

eLd + sin B,s,

se = -sin 8,d +

de = -sin e,d +

(9.6b)

e,s,

cos

die = cos eRd + sin

(9.6a)

(9.6~)

eRS,

cos

(9.6d)

eRs.

The left-handed fields that are (2, 1)‘s under SU(2), x SU(2)R are

(

de

-

Pose’

The right-handed

+

4

)L ’

(se + pL:e + oLb)L ’

(b - TLL

%se

1.

(9.7)

1

(9.8)

L

fields that are (1, 2)‘s are

+ URb R’

10.

A

CURIOUS

POSSIBILITY

In this section, we consider the curious possibility of a degeneracy between the down quark and the strange quark masses in the limit E -+ 0 wherein the discrete symmetry becomes exact. Thus, the empirical splitting in mass between d and s and the appearance of the Cabibbo angle have a common origin. We investigate the discrete symmetries of the theory that make the requirement do = s,, natural. We have no a priori reasons for doing any of this; only the interest of the results justifies the means. Recall Eqs. (9.5a), (9.5b), and (9.3):

cos8,d + sin t& sin eLs, so = cos8, coseLs+ sin eRsin e,d,

do = cos 6,

tan Off = (s/d) tan 8, .

(lO.la) (lO.lb) (lO.lc)

Assume s,, = d,, . The above equations can then be solved for tQR as functions of the eigenmasses s and d: tan2 8, = d/s,

(10.2a)

tan2 8, = s/d.

(10.2b)

282

DE

RtiJULA,

GEORGI,

AND

GLASHOW

This shows that flavor flip has become “exact,” with OR = r/2 - 0, . Recall from the previous chapter that BCN 8, and rewrite Eq. (9.2a) in the form

where use has been made of current algebra estimates of quark mass ratios. 1s it an accident that the observed value of sin2 13,is also 4.04 ? Or, conversely, if the current algebra estimate of d/s is correct, then we conclude that d and s must be degenerate in the E + 0 limit. Equation (10.3) is not new; it has been discussed in the past by many authors [23]. The unsatz s0 = do can also be used to fix the magnitude of the parameter E. Invert the expression Eq. (9.lb) for 8, to obtain, with no loss of generality, s2 - d2

’ = 7 Substitution

sin eL (d2 + s2 tan2 BL)1/2 ’

(10.4)

of Eq. (9.2a) in the above yields s-d c(so = do) = -N-* CO

s c

A determination of E is relevant, since the L-R etries in the mass matrix of the intermediate vector bosons are also functions of E; see Eq. (3.13). The following remarks are of a technical nature, and may be disregarded by nonspecialists. In order to construct a theory in which the relation do = so is natural, we must invent some discrete symmetry which interchanges the do and so fields. The obvious choice is &, t) J+G~, and ti2, t) zj+ , but it is easy to see that these interchanges cannot be consistent with D symmetry. To stay consistent with D invariance and interchange do and so, we must simultaneously interchange right-handed with left-handed fields; that is, we must make a parity transformation. The Yukawa couplings in a theory with such a parity invariance (and the single Higgs field 4) are

with all couplings real. These couplings are invariant peculiar parity operation, p, defined by

#l(X) + YOvMX’L $44

-

f#J’W),

$w - YOvwo, WLW

-

Wrdx’),

under D symmetry and the

#2(x) wl&>

-

Y0~2(~‘),

-

WL(X’),

(1o

7a) .

where x’ is the parity-transformed x, x’O = x0, x’ = -x. Having written this down in detail, we hasten to point out that its effect is simply to interchange all L and R

283

THEORY OF FLAVOR MIXING

fields (both fermions and gauge bosons), all 1 and 3 subscripts, and 4 and 4’. Thus, we can condense our description of P to read L t) R,

(10.7b)

1 t) 3,

Unfortunately, the simple forms Eqs. (10.6) and (10.7b) cannot be the whole story, becauseEq. (10.6) implies that t, = u,, in addition to d, = sO. We could, of course, break the degeneracy of u and t with the symmetry-breaking term (~b,), as in the c--s system. But then right-handed flavor flip would occur in the 21-f system as well as in the c--s system. This would lead to disaster in the form of a large right-handed 21-d coupling (seeSection 11). We must complicate the Yukawa couplings to allow for 21-t nondegeneracy. One possible set of Yukawa couplings is

Here x1 and x3 are two Higgs multiplets transforming exactly like $ under gauge transformations and D symmetry. The parity operation p is given by Eq. (10.7b). It also interchanges the x’s Finally these Yukawa couplings are invariant under the discrete reflection symmetry, #I,-+

-94,9

A+

-AL3

Xk

-

-x7;

.

(10.9)

These symmetries force the Yukawa couplings to have the form of Eq. (10.8). Now if the x’s develop different vev’s,

the P symmetry is spontaneously broken and u0 f t, . The theory described by Eqs. (10.7)-(10.10) lacks, perhaps, the compelling uniqueness of truth. Still, we find it amusing that in this class of theories, parity invariance of the initial Lagrangian is not something imposed gratuitously for reasons of elegance. Rather we were led to P” invariance by an experimental fact, the observed relation between the Cabibbo angle and the s and d masses.

Il. Do WE NEED MORE THAN

SIX QUARKS?

We now turn to the discussion of mixing angles other than the left- and righthanded Cabibbo angles. The successof Cabibbo universality and the observed V - A character of ,4 and neutron P-decay impose stringent conditions on some of these angles. The couplings of the currents Eb, , fd, , UdR, and i& to YreL must be very small, relative to the Fermi coupling iid,L~ce, . All of the flavor-mixing angles in our

284

DE

RljJULA,

GEORGI,

AND

GLASHOW

theory are fixed by the Cabibbo angle and quark mass ratios. The question is whether our theory automatically ensures that those couplings that are observed to be small are indeed small, for reasonable values of the quark masses. The relative magnitude of the iibL coupling can be read off the first doublet in Eqs. (9.8). Use Eqs. (9.ld), (9.4), and (9.5~) to obtain the ratio of amplitudes:

The angle BL(u, b) is of order ER - (s/c)R, where R is the ratio of a light quark mass (u) to a heavy quark mass (b or t). For reasonable values of quark masses 8,(u, b) -C 0.01. The iibL term in the current affects the normalization of the usual G& current. It leads to departures from Cabibbo universality of order &(u, b)2 < 10-4. There is certainly no phenomenological difficulty with this angle. The id, coupling is equally negligible. The situation with the angles in the right-handed couplings is more challenging. The magnitude of the iL& and UsR mixing angles can be read off Eqs. (9.8). Use Eqs. (9.2d), (9.2c), (9.4), (9.5a), (9.6~) and (9.6d) to obtain (11.2a) e&i, s) z - cos eRE 4 (b

+%

(11.2b)

Because t/b + b/t > 2, at least one of these angles may be large and potentially dangerous. Recall, however, that we are interested in the strength of the effective interactions of the form iidRFvL relative to the conventional Fermi couplings iidLi?vL . The right-left interactions are, in our theory, suppressed by an extra factor of O(E). [See Eq. (3.13).] Thus the relevant ratios of amplitudes are

We cannot estimate these ratios without knowledge of the ratio b/t. To develop a feeling for the gravity of the potential phenomenological problems, let us first discuss the most optimistic situation: b M t, in which case R1-

sin eR , 1+ 43 3 + 62

R2-

1+

4G

E2 + 82 cos 8, .

(11.4b)

THEORY

OF

FLAVOR

MIXING

285

It is RI that represents the bigger threat. Cabibbo universality of the vector currents in neutron and muon decay implies R, < 1O-2. We cannot make sin OR small without destroying the phenomenologically necessary flavor flip discussed in Section 8. Thus, one is tempted to demand that 4e2/(1 + c2 + a2) < 0.01. But, Eq. (10.5) says E = s/c. Most reasonable estimates of quark masses suggest that s/c N 0.2, which would lead to an unacceptably large value of R, , with any reasonable value of 8. The problem we have encountered is rather simply and elegantly solved. All that is necessary is the introduction of additional quarks. The observed suppression of R, is a hidden clue that there exist at least eight quark flavors. We devote the rest of this section to proving this point and to sketching the generalization to more than six quarks. Let there be n Q = $ quarks u1 ... u, and n Q = -+ quarks dl *.. d,, . Let the following left-handed quark fields be (2, 1)‘s under SU(2), x SU(2),:

and let the right-handed

fields (11.5b)

belong to the (1,2) representation. Quark masses are again generated by couplings to a Higgs field 4 transforming as a (2,2) representation of SU(2), x SU(2)R . We demand that the self-couplings of the Higgs field and its Yukawa couplings to quarks be invariant under the discrete symmetry operation D,:

(11.6)

This is a straightforward generalization of the analysis in Section 4. Again, the resulting quark mass matrix is flavor diagonal, and the charged weak currents are nearest-flavor interactions. The only difference is that there are 2n hours on the clock of flavor. A “soft” breakdown of the discrete symmetry D, can again be allowed with the result that a one-parameter family of weak mixing angles is generated. If we assign the b, , uO, dO , c,, , and s,, fields to consecutive hours on the clock of flavor, most of the analysis of Sections 7-10 can be taken over unchanged. The Cabibbo angle is generated in order e by dO - s,, mixing. Flavor flip occurs. If do = s,, then tan2 8, = d/s, etc. But the right-handed iiRdR current is suppressed. Let us compare the 6-quark model (and its flavor clock shown in Fig. 1) with a 12-quark model whose clock (with the conventional number of hours) is shown in Fig. 4. In both cases, the b, and d, quarks surround the U, quark and are next-nearest neighbors. The order mixing of b, and d,, is small because it is proportional to the u,,

286

DE RdJULA,

GEORGI, AND GLASHOW 12(b)

6 : Q=2/3 0 : Q=-l/3

l

FIG. 4. The clock of flavor with 12 quarks. The contiguity of the U, d, c, and s entries is dictated by experiment.

mass. In the 6-quark model, there is an order E mixing between the b, and s,, , which are also next-nearest neighbors. This mixing gives rise to the troublesome U,d, current discussedabove. But in the 12-quark clock, the b, and s,, are not next-nearest neighbors. We can get b, - sOmixing by going clockwise or counterclockwise from b, to s,, . The clockwise mixing is order e2 (one factor of b for each 2-hour step) and also proportional to the uOmass, so it is completely negligible. The counterclockwise mixing is not suppressedby light-quark masses,but it is order c4. In general in a 2n-quark model, the (counterclockwise) b, - s,, mixing is order l n--2 and the strength of the effective iiRdR current is order 8-l. For n 3 4, this automatically recaptures Cabibbo universality to an adequate level of accuracy.

12. CP VIOLATION So far we have assumedthat all couplings in our model are real, so that CP invariance is exact. If we relax this assumption, as is certainly indicated by the experimental fact that CP is violated, phases will creep into the Yukawa couplings and thence into the quark mass matrices, ultimately to appear in the charged weak currents. Since the weak currents in our model are quite complicated with left- and right-handed pieces and mixing angles everywhere, there are many potential sources of CP-violating effects. Fortunately, the analysis of the CP properties of our model is greatly simplified by the existence of the softly broken discrete symmetry D, . Consider the 2n-quark model with softly broken D, symmetry, but with arbitrary phasesin the Yukawa and Higgs couplings. We can use the gauge invariance of the theory to choosethe vev of 4 to have the form

= eie(; $2

(12.1)

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where a and b are real and positive. Now we can redefine the 4 field to eliminate the phase in Eq. (12.1) at the cost of changing the phasesof the Yukawa couplings and the D, breaking mass term in the Higgs couplings. All other Higgs couplings are unaffected by this redefinition because they are invariant under the replacement 4 --t e-ie~. In fact, it is clear that the redefined theory must have a real coefficient for the symmetry breaking term tr(&+) + h.c. in order to give a real vev (see Eqs. (5.5)-(5.7). Thus, this redefinition eliminates all phasesfrom the Higgs meson sector of the theory. The Yukawa couplings have the form

We can eliminate the phaseseverywhere except in Pn by redefinition of the I,!J’sas follows: Redefine &, to make CQreal; then Redefine &, to make & real; then Redefine &, to make CX~ real; then . ..

(12.3)

Redefine I/,~ to make &PI real; then Redefine $J,~to make cylzreal. The circle finally closes,and we cannot redefine #1, to make Pn real without changing the phase of 01~, so in general /3nwill be complex after this procedure is carried out. But that is the only remaining CP-violating phase in the theory. It goes without saying that by a similar procedure, we could move the phase onto any of the other Yukawa couplings, at our convenience. We are interested in the effects of the CP-violating phase on weak interaction processesinvolving quarks alone, or quarks and leptons, but no Higgs mesons. Such effects arise becausethe phase in the Yukawa couplings introduces phasesin the quark massmatrix and thus in the charged weak currents. For example, in the sixquark model, we can put the phase on the coupling (12.4) With this definition, the usual left-handed charged weak currents are real to first order in E. However, the right-handed u-d current has a complex coefficient of order (12.5) where 7 is the phase of the coupling in Eq. (12.4). Becauseof the d - s mixing, there 59.5/109/1-‘9

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is also a complex right-handed u-s coupling. In the language of Section 9, in which the d - s mixing is treated exactly but other mixings perturbatively, the relevant piece of the right-handed current is &@(l

- y,)(-sin

8,d + cos f&s).

(12.6)

To proceed, we must construct the weak Hamiltonian responsible for nonleptonic decays. In our model, it does not have a simple currentcurrent form; rather it is a sum of two current-current interactions (for the two W mass eigenstates). To simplify the discussion, we consider first the contribution from the lighter W, (G/2’/“) J;,Jlu,

(12.7)

where the relevant piece of the J1 current is J1u = Qu((l

+ y5) cos ~(~0s B,d + sin eLs) + (1 - y&t sin Y( -sin egd + coseRs)j

+ . ...

(12.8)

Equations (12.7) and (12.8) have the structure of the “complex weak current” model of CP violation [24]. In general, this model may not reproduce the superweak phenomenology for K + ~TT~T decays. But for the special choice do = so , it does. The do = s0 possibility was discussed in Section 10 and was shown to imply the famous relation e2 g d/s. If do = s,, , the right-handed d-s mixing angle is t$, = n/2 - eL , so the current J1 becomes Jf = uyu’((l + r5) cos ~(~0s B,d + sin eLs) + (1 - y5)[ sin y(-cos l&d + sin eLs)j + . .. = cos O,{(cos y - 4 sin 7) iiyud

(12.9)

+ (cos y + 5 sin Y) &y%dl

+ sin e,{(cos y + [ sin r) Uy+ + (cos y - 5 sin 7) UyUyss} + *-*. Redefining the d and s fields to make the vector parts real, we can write Jp =

BL(aiiyud + bei*iiyuysd) + sin BL(bEyus + ae-%iy”y6s) + . . .3

cos

(12.10)

289

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where a = Icosy-[ssinrl, b = 1cos y + f sin y j, abeid = (cos y - f* sin y)(cos y -t t sin y).

(12.11)

In Eq. (12.10) the phase of the dS = 0 axial current is the negative of the phase of the dS = 1 axial current. This is precisely what Mohapatra and Pati call an “isoconjugate” model [25]. It automatically gives rloO = T+- because the two terms AS = 1 decays (Ily@d:y,y5u and in JIUJl which contribute to parity-violating izyu”y5d$,u) have the same phase. The isoconjugacy persists when we include the effect of the heavier W, so that d,, = s,, implies q+- = rloO. The phase 4 in Eq. (12.10) is a direct measure of the size of observed CP violation in the theory, and we expect 4 N 10-3. Such a result is consistent with the observed experimental limit on the relative phase of V and A couplings in nuclear beta decay

WI.

In a theory involving n doublets of quarks, the analysis is virtually the same. In general, $ is of order ~“-l sin 7. In an 8-quark model (n = 4), and with the plausible estimate E N 0.2, the observed magnitude of CP violation corresponds to a relatively large value of the fundamental CP-violating phase, 7 N 0.1. The percolation of CP violation from the Lagrangian to the phenomena is seen to be rather indirect and inefficient. On the other hand, if IZwere substantially larger than 4, it would simply be impossible to account for the observed magnitude of CP violation even if CP violation in the Lagrangian were maximal, 77 = m/2. In the context of our theory, in which the 6-quark model is ruled out, we may conclude that n = 4, 5, or 6. Thus, there exist either 8, 10, or 12 quarks. Flavor flips among heavy quarks could modify this result by introducing large quark mass ratios in the coefficient of the CP-violating current. Since CP violation in this theory is not superweak, the electric dipole moment of the neutron may be measurably large. We expect a dipole moment of the order of (m/mp) 10-23e cm, where m is the light quark mass and mp the proton mass. We regard the CP properties of this class of theories as evidence that we may be on the right track toward a theory of flavor. The observable CP-violating effects are automatically small. With 10 or 12 quarks, maximal CP violation in the Lagrangian leads to effects of about the observed size. Furthermore, the experimentally valid valid relation qoO= v+- is related to the relation o2 = d/s. Ultimately, both arise because of the D, symmetry and the parity invariance of the initial Lagrangian. 13. THE CABIBBO ANGLE AS A RADIATIVE

CORRECTION

So far, we have been able to express the Cabibbo angle and the analogous flavormixing angles as functions of an arbitrary parameter E, and quark masses. Given one angle, and the quark masses, all other angles are fixed. In this and the subsequent

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two sections we discuss a more ambitious approach, wherein all angles are computed as functions of quark masses and other independent parameters of the theory. In the language of the preceding chapters, we attempt to compute the parameter E. We begin by stating the result. The flavor-mixing angles arise as O(U) radiative corrections to the original theory where all the angles vanish. The value of E, in a particular case to be discussed in detail, is (13.1) Here DLis the fine structure constant, Bw is the weak mixing angle of the SU(2)L x SU(2), x U(1) model, y is the angle in the two-dimensional orthogonal matrix that diagonalizes the squared mass matrix of the charged intermediate vector bosons, and Mh, M2 are their eigenmasses. These quantities have been defined in Section 3, Eqs. (3.4) and (3.9). If E is to arise as a radiative correction, we must go back to the situation described in Section 4. The discrete symmetry D is exact, so that the field 4 acquires a vev (13.2) and the quark mass matrix is flavor diagonal. The mass matrix of the charged intermediate the form “”

= 4

e2ao 1+p’ ew i 0

sin2

vector bosons, Eq. (3.8), is now of 0 1 + y’ 1 .

(13.3)

Alas, the discrete symmetry D of the theory has forced this matrix to be diagonal in W, , W, space. The W, and W, gauge bosons are eigenstates of the mass matrix. There are now two disconnected worlds, a world where the left-handed quark and lepton currents interact via the intermediate vector bosons of SU(2), and the corresponding right-handed world. These worlds are entirely separate and there are no ;LuR ---f pL-bR transitions to generate a y-anomaly. Even sadder, we are in the D-symmetric situation where there are no flavor nondiagonal terms in the mass matrix of the original quark fields. Flavor-mixing angles vanish to all orders. Should we softly break the discrete symmetry D as in Section 5, angles would appear, but they would depend on an arbitrary E. We seem to be at an impasse in attempting to have a calculable E. There is a successful, though somewhat baroque way [27] to mix the WL and WR intermediaries without breaking the discrete symmetry D essential to our analysis. This involves the introduction of an extra “mediator” vector boson WM , whose purpose is to bridge the gap between the separate left- and right-handed worlds. We enlarge the gauge group to SU(2), x SU(2), x SU(2), x U(1). All quarks and leptons are SU(2), singlets. To generate quark masses the usual field C$is necessary, which now is a (2, 1, 2) representation of SU(Z), x SU(2), x SU(2), . An extra

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Higgs field belonging to the (1, 1,2) representation is necessary to give masses to neutral heavy leptons. The field 4 and its Yukawa couplings to quarks are D-symmetric as usual. As for other Higgs fields that may give mass to W, and mix W, with both W, and W, we may be rather open-minded. We assume only that $ is the unique Higgs multiplet which transforms nontrivially under both SU(2), and SU(2), . The squared mass matrix of the charged intermediate vector bosons is now a 3 x 3 dimensional object in W, , W, , W, space. Its general form is ABO W=

B

i0

C D

D. E1

(13.4)

The vanishing of the LR and RL entries is again a consequence of the discrete symmetry D of the couplings of the (2, 1, 2) Higgs field. Notice that the separateness of the left- and right-handed worlds has been bridged. This can most simply be checked in perturbation theory in the nondiagonal entries Band D of Eq. (13.4). As an example, an S,d, mixing (and a Cabibbo angle) will be generated by the process shown in Fig. 5, where the name and role of the mediator field W, are mcde abundantly clear.

FIG. 5. Finite contribution

to s-d mass mixing induced by the mediator field WM .

Perturbation theory in the nondiagonal entries of Eq. (13.4) may not be a good approximation. More correctly, we should diagonalize Eq. (13.4) and study the couplings of the eigenfields. Rather than study the general formalism right now, we postpone its discussion to Section 15, and proceed to study a specific case. This has the advantage of displaying the gist of our argument in a simplified context. Let the gauge-coupling ghI of the mediator gauge fields be much bigger than the gauge coupling g = e sin-l BL of the L and R gauge fields. In this situation, the diagonalization of the mass matrix in Eq. (13.4) is greatly simplified. The central entry C is O( gMz) and there is an eigenvalue M2 N O( gM2) which is much larger than the other two and corresponds to an eigenvector that is approximately W, . The nondiagonal entries B and D are O(g,g) and the A and E entries are U(g2). The lighter eigenvectors are mixtures of W, and W, , with very small W, contamination. They are the eigenvectors of a matrix of the form (13.5)

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in W, , W, space, with A, E, F - 0(g2). It is very easy to understand what has taken place, in terms of the diagram of Fig. 2. In the large gM limit, the propagator of the WM field tends to m-“(WM) N g;. The momentum dependence associatedwith this propagator disappears. The W,--W, mass-mixing term B and the W,-W, mass mixing term D are both proportional to g, g. All g, dependencedisappearsfrom the diagram in Fig. 5. The product of B, D, and the W, propagator becomesan effective L-R mixing of order g2, the entry F in Eq. (13.5). In practice, the field W, has disappeared from this sector of the theory. We have arranged matters so that the self-couplings and the Yukawa couplings of the Higgs field 4 are D-symmetric. The right- and left-handed charged weak currents are simultaneously flavor-diagonal. But, the W, , W, mass matrix is not diagonal. In this situation, as we now proceed to prove, finite, calculable flavormixing angles arise as radiative corrections. To understand how flavor-mixing angles originate, recall that the charged weak currents of our theory are constrained to have a cyclic structure, so that the currents involving the charmed quark, forjinstance, are CsLand CdR. Let us, for the sake of definiteness, concentrate on how radiative corrections induce an s-d mixing term in the quark mass matrix. The relevant diagrams are shown in Fig. 6. In the figure, W, and Wh stand for the charged intermediate vector bosons that are eigenvectors of the massmatrix, Eq. (13.5). The angle y is defined in Eq. (3.9). The couplings of W, and wh to left- and right-handed quark currents are as in Eqs. (3.10) and (3.1 l), respectively, The L-R combination of couplings forces the contribution of the diagrams in Fig. 6 to the quark mass matrix to be proportional to the charmed

mc

-siny

dR

(a)

(b) FIG.

is finite.

6. Contributions

to s-d mass mixing in the limit of very heavy W, . The sum of (a) and (b)

THEORY OF FLAVOR MIXING

293

quark mass c. Each of the diagrams is logarithmically divergent but the divergences cancel, as they must, as a consequence of the orthogonality of the couplings in Eqs. (3.10) and (3.11). The net contribution of the sum of the two diagrams is an effective aRsL interaction, (13.6) Notice that L&r, a weak radiative effect, is O(Z), not O(01/ML2). L&r is a aRsL mixing proportional to the charmed quark mass. Similar mixings appear between other pairs of quark flavors, whenever the two flavors have a third quark in common in their left- and right-handed charged weak couplings, i.e., when they are 2 hours apart on the clock of flavor. The mixing patetrn in the quark mass matrix is identical to the mixing pattern in Eqs. (6.2b) and (6.2~). The difference is that the mixing parameter E has now been calculated: (13.7)

E--z-ln$$.

We have succeeded in constructing a theory where the flavor-mixing angles are finite, calculable O(a) radiative corrections. We devote the next section to some technical subtleties, and Section 15 to the discussion of whether the calculated angles compare favorably with experiment. 14. WHY THE MEDIATOR

FIELD IS NECESSARY

We discussed in the previous section a simplified situation where the mass of the IV, field was made much bigger than the masses of the other two intermediate vector bosons. One may be tempted to believe that we could have sent the WM masses all the way to infinity, effectively eliminating the extra SU(2), group from the theory. Why then, could we not find a scheme with calculable nonzero flavor-mixing angles, not involving the mediator field ? The answer to this question lies in the Higgs sector of the theory. Remember that the Higgs field 4

acquires a nonzero vacuum expectation value only in its upper-left entry, in the D-symmetric SlJ(2), x SU(2), x U(1) theory. But in the SU(2), x SU(2), x SU(2)R x U(1) context, as we shall presently show, the lower-right entry x0 also acquires a (finite, calculable) vev as a consequence of quantum effects. This is an extra contribution to the flavor-mixing angles that will turn out to be small. However, if we try to restrict the theory to SU(2)L x SU(2)R x U(l), we find that (x0> becomes infinite, a renormalization counterterm becomes necessary, and we are back to the

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situation where the discrete symmetry is broken and the flavor-mixing angles are not calculable. The lowest-order diagram that generates a nonzero (x0) in the SU(2), x SU(2), x SU(2)a x U(1) is shown in Fig. 7. The diagram is superficially quadratically divergent. Yet, both the quadratic and logarithmic divergences cancel, as a necessary and automatic consequence of the discrete symmetry D. We proceed to prove this point explicitly.

FIG.

7. Finite vector-boson

loop contribution

to x0 vacuum expectation value.

Let R be the orthogonal matrix that diagonalizes the (3 mass matrix of Eq. (13.4),

x

3)-dimensional

squared

(14.2a)

(14.2b) RTR = 1.

(14.2~)

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The discrete symmetry forces the upper-right yields the condition

295

MIXING

entry in Eq. (14.2a) to vanish, and

(14.3)

We now come back to the computation of the diagram in Fig. 7. The relevant couplings, bilinear in Higgs and gauge fields, can be read off the kinetic terms (D,# in the Higgs field Lagrangian. Let A be a cutoff in the Feynman integral corresponding to Fig. 7. The divergent terms in this integral are proportional to a * yf12

-

i

oliyiMi2

In -$? .

(14.4)

z

i=l

The quadratic divergence automatically vanishes, because of the orthogonality condition, Eq. (14.2~). The logarithmic divergence also cancels, as a consequence of Eq. (14.3), imposed by the discrete symmetry. The field x0 acquires a finite vacuum expectation value b proportional to the vacuum expectation value a of the field c$,,. The finite result of the computation of the diagram of Fig. 7 is

1 ’ = XQ

3cx

8~ sin2 8, cwlK2

(14Sb)

ln

where Mn is the mass of the x0 Higgs field. Notice that if we send any of the eigenvalues Mi to infinity, EIbecomes infinite. The quantity Z plays the same role as Ein the study of flavor-mixing angles. We now have two additive contributions to what we used to call E, the one in Eq. (14.5b) and the one due to flavor mixing through intermediate vector boson radiative corrections, that we computed in the previous chapter in a specific case M, > Ml,3. In the general case where the intermediate vector bosons of SU(2)r x SU(2), x SU(2), x U(1) are all treated on equal footing, the generalization of the result Eq. (13.7) of the previous chapter is a:

’ = 4~ sin2

8, t wh ln

-$$-

+

a2y2 In

-)

MZ2

(14.6)

M22

Since C/E N O(Mw2/MH2), where Mw and MH are vector boson and Higgs masses, the standard reflex is to expect z > E. This is not so. These particular Higgs fields must be very heavy, lest they mediate disastrous AS = 2 effects. The effective “Fermi” coupling strength for Sd -+ Higgs ---f dS transitions is (sin2 e,c2)/MH2(f$o)2.

(14.7)

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To avoid disaster this must be smaller than 10-5GF - 1/(4J2. Thus A4n > 100~ 100 GeV. Since Mw - eG;l” sin-l 8, - 100 GeV, we have Z/e < O(1). In practice, we can forget about the contribution of (x0) to the weak mixing angles.

15. DID WE SUCCEED IN CALCULATING

THE CABIBBO ANGLE?

In this section, we argue that the attempt described in Sections 13 and 14 to generate the flavor-mixing angles as finite radiative corrections is probably doomed to failure. The problem is that for reasonable choices of the parameters of the theory, the d - s mixing angle generated by radiative corrections is much smaller than the observed Cabibbo angle. The argument is nontrivial because the predicted angle depends upon quark mass ratios evaluated at large momenta. Before we enter the discussion of quark masses and the effect of the strong interaction corrections on the computed value of the Cabibbo angle, we state the result

with Ethe quantity computed in Section 14, Eq. (14.6). The symbols d, s, c in Eq. (15.1) stand for quark masses renormalized at a Euclidean momentum M of the order of the mass of the charged intermediate vector bosons of our theory (Mw - 100 GeV). We shall argue that c/s - 5 and that d/s is the “current algebra” ratio (d/s - l/20, according to the “classical” analysis [18]; d/s - l/5, secundumGunion et al. [22]). The quantity E of Eq. (14.6) is formally of O(a), so that 0, - O(a), to be compared with the experimental result & - l/5. It is difficult to believe that the parameters of the theory can be chosen so as to reproduce the observed angle. To be more specific, consider the ambidextrous limit of the SU(2), x SU(2), x SU(2), x U(1) theory discussed in Section 13. In this limit (15.2)

In the particular version of the ambidextrous theory that we compared with experiment in Ref. [9], the values of the relevant parameters were found to be: sin2 0, - 0.25, sin 2y - 0.9, ME - 70 GeV,

Mh -

(15.3)

190 GeV.

The corresponding value of E is -11220 N m,lm, . The predicted value of tic is too small by about two orders of magnitude, unless our estimate of c/s at M .- Mw

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is a gross underestimate. We now come back to our discussion of quark masses and strong interaction effects. The quark masses are not directly measurable quantities. Determination of the quark masses from the observed spectrum of ordinary hadrons requires some insight into the dynamics of the strong interactions. We assume that the strong interactions are described by quantum chromodynamics (QCD). The theory is asymptotically free. All the coupling constants and mass parameters can be defined at Euclidean momenta of the order of M, the renormalization point. If in a particular process, all momenta are of the order of M (and Euclidean) no large logarithms occur in the Feynman diagrams describing the process. This is the basic idea of renormalization group improved perturbation theory. To apply these ideas to the question of radiative generation of the Cabibbo angle, we must make two technical comments: (1) from strong at M)

In the flavor-mixing diagram of Fig. 6, the dominant contribution comes loop momenta of the order of M w, so we avoid large corrections from the interactions by choosing M N Mw . Then the Cabibbo angle (renormalized is determined by Eq. (15.1), where the quark masses are renormalized at M.

(2) The Cabibbo angle is almost independent of M for Mw 2 M > s, so that the Cabibbo angle renormalized at Mw is essentially the same as the Cabibbo angle observed in ordinary weak decays. We could prove these statemenst in the technical language of the operator product expansion, but as the issues involved are rather simple we will adopt a less formal stance. Comment (1) is rather obvious. The loop integration is cut off for values of the loop momenta large compared with the W masses. If a renormalization point were chosen much smaller than Mw , the zeroth-order (in the QCD gauge-coupling gJ diagram would not accurately describe d - s mixing. There would be large contributions from diagrams such as that shown in Fig. 6, but with internal gluon loops. These pitfalls are avoided by choosing M ‘v Mw , a value at which gs2/4x is expected to be very small. Comment (2) follows from two observations. For momenta less than or of the order the diagram shown in Fig. 6 is equivalent to a d - s mixing term in the ofMw> Lagrangian. This may sound surprising since if a d - s mixing term existed in the Lagrangian, there would be an infinite counterterm associated with it. But in the model of Sections 13 and 14 there is no such counter-term; indeed that is just what is meant by the statement that the Cabibbo angle is calculable. To see how the equivalence comes about, consider the situation for M N Mw . Suppose that the diagram shown in Fig. 6 appears inside some more complicated diagram involving gluon loops, as in Fig. Sa, and compare that to an analogous diagram in which the d - s mixing comes from a mass term in the Lagrangian, as in Fig. 8b. The diagram in Fig. Sb is divergent and we must cut it off at some momentum scale (1. Renormalization then replaces n with the renormalization mass M. But the diagram in Fig. 8a is finite. It is automatically cut off at momenta of the order of Mw N M. In both cases, the

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d

C

d

(a)

P S

\/

s

A

d

d

(b) FIG. 8. (a) A QCD correction to radiatively induced d-s mass mixing. correction to direct d-s mass mixing.

(b) An analogous QCD

effective cutoff is M. This equivalence is valid only if the external momentum in the diagrams is smaller than Mw . At much larger momenta, the d - s mixing from Fig. 6 goes away, but since we are not very interested in momenta large compared to Mw , that does not concern us. To complete the justification of comment (2), we must show that the Cabibbo angle is independent of M for A4 > s in the simpler theory in which a d - s mixing term appears in the Lagrangian. The point is that the strong interactions and the weak interactions are rather independent. It makes no difference in QCD how the quarks participate in the weak currents. The strong interactions simply induce M dependence in each of the diagonalized quark masses separately. QCD affects the Cabibbo angle only by acting differently on the strangeness-violating and strangeness-conserving weak currents. But the difference is negligible for momenta much larger than the s-quark mass, because both currents are almost conserved. Since we measure the Cabibbo angle at momenta of the order of the s-quark mass, the total effect of the strong interactions on 8, is small. There are no large logarithms of order In(mw/m,). We have now convinced ourselves that the Cabibbo angle is given by Eq. (15. l), where the quark masses are renormalized at M cv Mw . We must now ask, “What are the quark mass ratios at A4 N Mw ?” This question has been discussed in Refs. [28, 291, using various combinations of current algebra techniques, constituent quark mass determinations, and the renormalization group. Plausible values are d/s N l/20 and s/c N l/5. There is still considerable theoretical uncertainty, but

THEORY OF FLAVOR MIXING

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these ratios are reasoned guesses.This makes it unlikely that the Cabibbo angle in our model arises as a radiative correction, and renders Sections 13-15 irrelevant to the main thrust of our work.

16. THE

CLOCK OF FLAVOR

The central issuesof elementary-particle physics once included hadron spectroscopy, hadron structure, and hadron-hadron scattering. Now, it appearsthat these questions are not so fundamental as we had thought. The heart of the matter is to know how many quarks there are, and why there are so many. We must learn the origin of quark masses,and be able to predict the massesof quarks not yet seen. Just as important is the question of the structure of weak interactions: How are the weak currents built up out of quark and lepton fields, and what gauge group is involved? For a brief time, it seemedthat Nature needed just four kinds of quarks, and four kinds of leptons as well. Unified weak interactions based upon spontaneously broken SU(2), x U(1) and using a chirally pure (left-handed) charged current gave an adequate description of weak phenomena, and quantitatively predicted neutrino(and antineutrino-) induced weak neutral-current phenomena. The existence of a fourth charmed quark was demanded by the absenceof strangeness-changingneutral currents, and arguments at second order in weak interactions showed that it could weigh no more than a few gigaelectronvolts [30]. So it was that the discovery of .I/# and its interpretation as charmonium confirmed the entire picture of elementaryparticle physics: chromodynamics, asymptotic freedom, unified weak and electromagnetic interactions, quarks as hadron constituents, and charm. In a world with only four quarks, the structure of the weak gauge theory is tightly constrained: the group must be SU(2), x U(1) and the charged current must be purely left-handed. The only arbitrary parameters in the hadronic domain are the values of the quark massesand of the Cabibbo angle:j& parameters all told. Cabibbo universality need not be imposed on the theory; it is automatic. There is no room for CP violation to be built into the structure of the weak current; phasesmay be eliminated by redefinitions of quark fields. The neutral current is necessarily flavor diagonal; there are neither strangeness-changingnor charm-changing neutral current effects to order G, nor indeed to order olG. Recent data show convincingly that this simple picture is simply wrong. An early warning comes from the normally quiescent discipline of lepton spectroscopy. Today we know of the existenceof at least five and perhaps as many as nine leptons [5]. Following the historically successfulargument of quark-lepton symmetry, reinforced by the need to avoid triangle anomalies, we are led to conclude that there must exist more than the “observed” four quarks. With more than four quarks, we are faced with many choicesfor the structure of the weak interactions. Let us continue to demand that the GIM mechanism be natural and that the neutral current be diagonal in all quark flavors. For simplicity, let us

300

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assume that all quarks have electrical charges of -4 X42) x U(1) then admits only two possibilities.

or +$. The gauge group

A. An extended left-handed theory, involving n each of Q = g quarks, Q = -4 quarks, Q = -1 leptons, and neutrinos. The charged weak current is chirally pure: All left-handed fermions transform as weak doublets, while all right-handed fermions are singlets. B. A vector model, in which there are again n doublets of quarks. Not only are all left-handed quarks in weak doublets, but so also are all right-handed quarks. Rightand left-handed quark pairings are different, so that an adequate description of most weak phenomena is obtained with n 3 3. It now seems quite clear that neither of these two possible models can describe all available data. It is this dilemma that compelled several groups to devise the ambidextrous model [9] based on the gauge group SU(2), x W(2), x U(l), and at the same time, to restore a certain right-left symmetry to the structure of weak interactions. In such a rich theory, involving two charged w’s and two neutral W’s, currently available data is easily fit. A minimum of three quark doublets is required, with left-handed quarks put into SU(2)r doublets and right-handed quarks put into &T(2), doublets. As the number of quarks grows, the number of parameters characterizing the hadronic current grows more rapidly, and a theory in which all these parameters are arbitrary becomes aesthetically untenable. With IZ weak doublets of quarks, the left-handed weak current is characterized by an n x n unitary matrix telling which linear combinations of Q = -6 quarks couple to the Q = +j quarks. Since CP invariance is not an exact symmetry of Nature, we cannot limit ourselves merely to a real orthogonal matrix. A second II x II untiary matrix prescribes the structure of the right-handed hadronic current. Counting the 2n quark masses as additional parameters, and allowing for 2n - 1 innocuous redefinitions of quark phases, we find that the hadronic domain of the weak interaction theory involves a grand total of 2n2 + 1 arbitrary and independent parameters, a considerably less pleasing result than the five parameters characterizing the original theory. Not only are there too many parameters, but there are too many constraints among them which must be satisfied if the theory is to agree with experiment: (a) leptonic (b) (c) (d) (e)

Cabibbo universaity for the (Ud)L and (US)~ couplings couplings. Virtual absence of (U& and (US)~ couplings. Smallness of the (Cd)R coupling. Correct size of observable CP-violating effects. Superweak mimicry: we must obtain T+- = qOO.

Surely, these conditions

must be satisfied in an automatic

relative to the

and natural fashion,

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301

MIXING

and this can only be if not a11of the 29 + 1 parameters in the ambidextrous theory are arbitrary. Our philosophy has been to reduce as much as is possible the number of free parameters in the theory. With our least ambitious viewpoint, the only one we are so far able to implement rigorously, we are able to cut down the number of parameters to the 2n quark masses, and to one additional complex number Eeinwhich is ultimately responsible for all quark mixing angles and for CP violation. Our effectiveness in reducing parameters is shown in the following table: Number of parameters Number of quarks 6 8 10 2n

General ambidextrous 19 33 51 29 + 1

Our theory 8 10 12 2n + 2

Further information on the structure of the theory comes from a study of conditions (a) to (e) listed above. Reassuringly, (a) is automatically true, depending only upon what is already known about the quark mass spectrum. Condition (b) provides us with a lower bound to the number of quarks that must exist. The troublesome couplings come out to be of order E”-~, so that agreement with experiment requires that 2n 3 8. Condition (c), potentially a source of difficulty, satisfies itself in a truly remarkable way: by the mechanism of flavor flip. Even though the starting point of the theory involves the couplings (CS)~and (E& , the theory that emerges when flavor mixing is taken into account no longer involves a large (Ed) coupling. The validity of the wellknown relationship Be2 cz d/s N l/20

(16.1)

tells us that the d and s quarks are at least approximately degenerate in the E + 0 limit wherein there is no flavor mixing. The d - s mass splitting as well as the Cabibbo angle is induced by the soft breaking of the discrete symmetry D. Consistency of this point of view then requires the relationship E CT? s/c N 0.2,

where the numerical estimate of the quark mass ratios in Eqs. (16.1) and (I 6.2) are those of conventional current aIgebra and asymptotic freedom. Condition (d) allows us to obtain an upper limit on the number of quarks. CP violation in observable phenomena results from the appearance of complex phases in the iid and US weak couplings, which are of order en-l sin q. This estimate should be com-

302

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pared to the ~2 i: lop3 phase which is required to describe observed CP violation. Even if our theory violates CP as much as it can, with 7 .= r/2, the observable consequences will be too small unless IZ < 6. If our theory is essentially correct, we conclude that the total number of quarkjavors can only be 8, 10, or 12. Miraculously, condition (e) is also automatically satisfied. The CP ciolating phases in the iid and 17s couplings satisfy the Mohapatra-Pati isoconjugacy condition, a sufficient and probably necessary condition to ensure q+- = qoO. Let us return to the alleged high y-anomaly, which was instrumental in motivating this work. Although such an effect must be present at sufficiently high energy, we must conclude that the high y-anomaly simply cannot be large. From Eq. (3.13) we find that +, + p+b) < 12~~ . a(;,~ --f p+d). We have not abandoned the search for further relations among the 2n + 2 parameters characterizing the quarks in the ambidextrous theory. We hope to devise a scheme whereby both E and 7 are calculable effects (presumably, E being of order CY and 7 being 7r/2). Were this accomplished, we should be able to express all mixing parameters in terms of just the 2n quark masses. The degeneracy of d and s in the E -+ 0 limit suggests the existence of further relations among quark masses. In our ambidextrous model with “nearest-flavor” weak couplings, we have precisely quantified each quark as an hour on the clock of flavors. Perhaps it is only a small step to learn how to compute quark masses in terms of position on the clock. The reader will by now have realized that whatever strengths this work has are more in the direction of aesthetic accommodation of known phenomenology than prediction. True, we have argued that there remain quarks to be “discovered,” between four and eight more, to be exact. And, there must exist two W*‘s and two ZO’s, all with masses less than 200 GeV. However, such predictoins are not unique to our model. We have refrained from comment on the detailed structure of the neutral currents. There are simply too many parameters in the theory to make such a discussion worthwhile at the present time. The quarks other than U, d, c and s are probably all quite heavy, for otherwise they should already have been detected. Our theory does make predictions about the weak decays of hadrons containing heavy quarks. Let x denote the lightest of the unobserved quarks. The Xx state corresponding to J/$ must be heavier than 7.8 GeV, and the xi.& xa states corresponding to D must be heavier than 4 GeV. D, decays by first-order weak interactions, suppressed by a power of E which depends upon where x lies on the flavor clock. We now give some simple consequences of our model without proof. The D,+ will have quark composition td or zlb depending upon whether x = t (a quark with Q = s), or x = b (a quark with Q = -8). Curiously, our results are the same in either case. The dominant decays of the D,+ will arise either from RR couplings or LL couplings, depending on where x lies on the clock of flavor. The LL decays will produce semileptonic decays into hadron final states with Q = S = C (charm) = 0. The nonleptonic Q = 1 final states will be primarily S = C = 0 or S = C = 1 with comparable probabilities. The amplitude for LL decays is smaller than the universal value by at least a factor of RE (R is a light-to-heavy quark mass ratio). Typically,

303

THEORY OF FLAVOR MIXING

RR decays will dominate. Then the principal decay will be nodeptonic into C = S = 1 states. There may also be copious semileptonic decays into states involving heavy leptons (i.e., D,+ + p+fi” + hadrons with C = S = 0; see Section 17). For either LL or RR decays, interesting cascade decays involving charmed particles, such as D,+ + D,OK+, are expected. On the other hand, the lifetime of D, depends sensitively on the clock position of the new quark. If the D, mass is -5 GeV, its a priori lifetime is -lo-l5 sec. This must be modified by a factor of wcP, where P depends on how many hours separate the x quark from the domain of the known quarks 11,d, c, s. For a l-hour separation, P = 0; for a 2-hour separation, P =: 2n ~ 6 (where 2n is the total number of quark flavors: 8, 10 or 12); for a 3-hour separation P = 2; and for a 4-hour separation P = 2n - 8. The lifetime of D, could be as long as IO-l1 set! There are some reasons to believe that the next two new quarks to be found, i.e., the lightest missing quarks, will be contiguous with the four observed quarks whose clock positions are known. The high y-anomaly suggests that a relatively light (5-6 GeV) Q = -$ b quark adjoins u on the clock. The Q = -$ t quark lying adjacent to s simply cannot be very heavy. It is coupled right-handedly to d, and will yield an RR contribution to dS = 2 proportional to sin2 20(t2 - c2)*. The arguments that forced the c quark to be “not too heavy” apply equally well to this t quark [30]. Unfortunately, we have no real theory of quark flavor. From the masses of the four observed quarks, we are unable to deduce the remaining quark masses or even their number. Perhaps, with the discovery of one or more quarks beyond charm, we will find out whether it makes sense to put the quarks around the clock.

17. POSTLUDE: THE CLOCK OF LEPTON FLAVOR In the previous 16 sections we have been concerned primarily with a theory of quark flavor and have more or less ignored the leptons. A year ago we might have offered a plausible excuse for this omission: Leptonic weak interactions were observed to have a very simple structure and the lepton sector did not seem to offer any mystery. No longer. The discovery of a charged heavy lepton (the 7) at SPEAR was an early indication of trouble with the standard model. The preliminary results of atomic physics parity-violation experiments provided a major impetus for the development of ambidextrous theories. The observation of trimuon events in neutrino scattering may signal the existence of a new family of heavy letpons with peculiar interactions. Can we shed any light on this complicated situation by applying our theory of flavor mixing to the lepton sector? Before answering this question in detail, we discuss some general considerations. The fact that parity-violating effects in the interactions of electrons with heavy nuclei are much smaller than expected on the basis of the standard model can be explained in two simple but different ways. (a) In an SU(2) x U(1) model, the right-handed 5951'091'-20

e- field may be a member of

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a weak doublet (e”, e-)n so that the electron neutral current is pure vector (even though the quark neutral currents are not). (b) In an ambidextrous SU(2)r x SU(2)u x U(1) model, the spontaneous symmetry breakdown may conserve parity so that the two z’s couple respectively to pure vector and axial-vector currents in both the quark and lepton sectors [9]. Eventually, atomic physics experiments will distinguish between these possibilities [(a) gives observable parity-violating effects in hydrogen, while (b) gives none]. For the time being, we have only theory to guide us. In our present framework, we could implement (a) by assigning the right-handed eR- field to a doublet (e”, e-)n of the left-handed gauge group, SU(2), . That would be a bizarre thing to do. In the quark sector, all the left-handed quark fields are in doublets under SU(2), while all the right-handed quark fields are in doublets under SU(2), . Arguments on the basis of lepton-quark symmetry, cancellation of anomalies and superunification, all suggest that we should treat the leptons in the same way as the quarks. We are then forced to choose possibility (b). To implement (b), we must arrange that the spontaneous symmetry breaking preserve the parity-conserving character of the neutral current. The Higgs field 4 transforming like a (2, 2) under SU(2), x SU(2), satisfies this constraint automatically, but it leaves an unbroken U(1) x U(1) gauge symmetry. There must be other Higgs fields to avoid a doubling of the photon. The simplest possibility, the addition of a (1, 2) Higgs field, was analyzed by us in Ref. [9]. This is an attractive choice because the mass terms generated by this Higgs field couple the right-handed doublet lepton fields to left-handed singlet neutral heavy lepton fields. The left-handed neutrinos remain naturally massless. But the vev of the (1, 2) field spoils parity conservation in the neutral currents (in our phenomenological analysis, parity violation was found to be smaller than the standard-model prediction, but still too large to agree with recent, unpublished [31], atomic physics data). We could restore parity conservation by introducing a (2, 1) in addition to the (1,2) with precisely the same vev. We reject this possibility because it is unnatural. Parity is violated somewhere in the theory (after all, the world is not parity conserving) and this implies that there are independent counterterms for the (2, 1) and (1, 2) vev’s. We choose to break the extra unwanted U(1) symmetry with a single Higgs field whose vev is automatically parity conserving, for example, a second Higgs field p, which is also a (2, 2) but which transforms nontrivially under the U(1) factor of the gauge group;

p = (PIp” ). -P

We now have a natural explanation2

(17.1)

Pz-

for the failure to observe parity violation

in

2 There is still one sticky point. We must ensure that gL = ga by imposing a parity invariance on the Lagrangian which is then softly and/or spontaneously broken. An example of such a parity invariance is given at the end of Section 10.

THEORY

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FLAVOR

305

MIXING

atomic physics but we face another mystery. Why are the neutrinos light? The answer that appears below is related to the separate conservation of electron and muon numbers. Unlike the clock of quark flavors, the clock of lepton flavors is broken into two halves, one-half with muon number, the other with electron number. The neutrinos appear at the “cracks” in the clock. They are strictly massless when E = 0 as a consequence of muon-number conservation. For nonzero E, their masses are small. The resulting system is very tightly constrained if it is to give small enough neutrino masses and simultaneously explain T decays and trimuon events. We proceed to imitate in the lepton domain the steps that led us to a theory of quark-flavor mixing. The starting point is the D, symmetry of Eq. (11.6). We will take n = 6 because we will need 12as large as possible to suppress the neutrino masses. The lepton fields are defined as follows: lkL , k = l,..., 6: (::“,L

3 oL

>--., (g),

(17.2a)

2 (F)R

,..., (F),

(17.2b)

are doublets under SU(2), and IkR , k = I,..., 6: (g),

are doublets under SlJ(2)R . This is the analog of Eq. (11.5). Under D, , the lepton fields transform in the same way as the quarks: IkL + eiwzk161kL , 1kR + ei”‘2”-1”“/kR . We identify IIL and &, with the conventional

(17.3)

left-handed doublets

up to small angles. The breaking of the clock is enforced by the discrete symmetry M: liL ---f -I-

ZL

Ij, + -ljR

with the remaining

for

i=

1,2,3

for

j = 2, 3, 4,

(17.5)

lepton fields unchanged. This forbids the couplings L#Ll

and

74,dZ, Y

(17.6)

which would lead to neutrino mass terms. It also implies separate electron-number and muon-number conservation. The reflection symmetry Eq. (17.5) divides the leptons into two families, one containing TV-, v, and other fields which are odd under

306

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the symmetry, the other containing e-, v, and other fields which are even. Clearly p + ey decay is absolutely forbidden by Eq. (17.5). Likewise there is absolutely no mixing between v, and v, , and thus no neutrino oscillations. To make sense of this 1Zlepton system, we will have to make use of the flavor-flip mechanism discussed in Section 8. In the quark sector, the c,, quark mass is much larger than the masses of the neighboring d,, and s, quarks and the EC,,mixing term is the largest contribution to the d, - s0 mass matrix. Thus the d, field becomes primarily dL + sR while the s0 is sr + dR (up to angles of the order of 0,) becuase the physical s mass comes primarily from the mixing term. Tn this situation the masses satisfy s N EC. In the lepton case, we will need two very heavy neutral leptons (which we call MO and TO). The corresponding charged fields M- and 7- are flavor flipped and their masses satisfy M- ‘v EMO,

r- 5% s-0.

(17.7)

In a sense, M- and T- are antithetical to p- and e-, respectively. The conventional leptons are associated with much lighter, indeed almost massless, neutrinos. The heavy charged leptons are associated with neutral leptons which are of order ten times heavier. In Fig. 9, we exhibit the clock of leptons after flavor flip. In terms of the fields defined in Eq. (17.2), the left-handed doublets are (up to small angles) (17.8a)

FIG. 9. The broken clock of lepton flavor (after flavor flip). Left- (right-) sociated with left- (right-) handed currents, as indicated.

handed field are as-

THEORY

and the right-handed

OF

FLAVOR

307

MIXING

doublets are (17.8c) (17.8d)

The fields carrying muon number [Eqs. (17.8a) and (17.8c)l do not mix with those carrying electron number [Eqs. (178b) and (17.8d)l. The jagged line in Fig. 9 represents the break in the clock between these two families. We discuss the electron sector first. The mass matrix for the charged electronic leptons, e-, r-, and E- is

(So-,To--,E,-)R

(17.9)

i Flavor flip has occurred because the bare masses of e and 7 are small, e,-, To- -g EToO.

(17.10)

E- N E,-,

(17.11)

Then the actual masses are T- c? CToO,

e- ‘v e,-7,-/T-.

Flavor flip ensuresthat the electron is light, e- < T-. The massmatrix for the neutral electronic leptons is

(Ce, > FOOD, EO”>R

0 Fro-

0

0

eEo-

TO0

eeo-

ve,

0 O + h.c. I( iL ;o Eo”

(17.12)

The massesare TOc1 TOO,

E” CI’ Eoo,

c3e0-7.,-E,- N v, = ToOEoO -

c4e-

(17.13)

E Eoa

The neutrino is light becausee4is small, but is it light enough? For E = 0.2, it is not. Equation (17.13) predicts v, N 800(E-/E”) eV. The experimental upper limit is ve < 35 eV [32]. Consistency requires a very small value of E-/E”. But the general form of the massmatrices Eqs. (17.9) and (17.12) requires E-/E0 3 E. Thus E = 0.2 implies Y, > 160 eV which is ruled out. We must take a smaller value for c. For E = 0.1. we predict V,

= 50(E-/E”) eV > 5eV,

which is consistent with experiment for E-/E0 < 0.7.

(17.14)

308

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Back in the quark sector of the theory, E ‘v 0.1 would require cjs ru 10, which is not unreasonable. Such a small value of E would imply that the contribution of the right-handed tib current to the high l--anomaly is small, at most about 10 7; of the total V cross section. We will take the view that this, plus the logarithmic corrections due to QCD, is enough to explain the data [33]. There is another reason why E must be small. There is a right-handed ijeeR- current with coefficient (calculated perturbatively) (17.15) This current gives a small V + A correction in /3- and p-decay, which becomes ~,yW

to the electronic charged weak current

+ ys) + 41 - yd e-,

(17.16)

where (17.17) [6 is defined in Eq. (3.8).] The current in p decay

in Eq. (17.16) yields a Michel parameter

p = $(l/(l The experimental

+ r”)).

p

(17.18)

result p = 0.7518 f 0.0025 [34], implies r2 < 0(10-3).

(17.19)

For E = 0.1 (and reasonable values of 8) this is possible only if E” CY E-.

(17.20)

Improvement of the limit on the mass of v, or in the determination of p could rule out our theory. On the other hand, we are predicting measurable mass for V, and a detectable effect on p. As the name suggests, we identify the r- field with the heavy lepton discovered at SPEAR [5]. This is the only assignment consistent with I’ - A weak decay of T-. The or- field mixes with the eL- with &(T-, e-) c1 -~~-/.r-. The T- decays through an ordinary left-handed weak interaction suppressed by 1 &(T-, e-)1”. If j dL(T-, e-)1 N 0.1, the lifetime is -lo-11 sec. In neutrino scattering the 7 must be produced at the 1 % level by electron neutrinos but not at all by muon neutrinos. Equation (17.7) implies that the To is heavier than T- by a factor of I/E. Unfortunately a 20-GeV neutral lepton will be difficult to observe experimentally. The E- and E” are presumably

THEORY

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309

MIXING

heavy and as yet unobserved. It is also possible that it is the E- (or N-) which has been discovered at SPEAR and that the T- is heavier. In this case the decay E- + v, + ... (or N- -+ v,) involves a right-handed iJeER- (or ;JVR-) current. The current has full strength, but the decay is suppressed because L-R mixing is small [see Eq. (3.13)]. Again we expect a lifetime of the order of lo-l1 sec. It is essential that experiment determine whether T-decay is V - A or I’ 4 A in order to firmly pin down its assignment to the clock of lepton flavor. The muon half of the clock leads to a possible explanation of the trimuon events recently observed in v scattering [5]. If the ,K - M- mass difference comes entirely from the cM,,O mixing term (that is, the p--Msystem is analogous to d - s in the quark sector), then the mixing angle between p- and M- is the same for both leftand right-handed fields, &(p-,

AC-) = e&L-,

The relevant doublets are the left-handed

(cos [pand the right-handed

(-

M-)

N (/A-/M-y”

= 5.

(17.21)

doublets

VU

+ sin fM- bL

( cos [M- Mo - sin fp- 1L

(17.22)

doublets

PLO 17 (

(17.23)

cos fp- + sin EM- R

Muon neutrinos produce M-‘s. Through the left-handed current, Eq. (17.22), well above threshold, the ratio of M- to PL- production by muon neutrinos is tan2 5 = p-/M-. Violations of Cabibbo universality occur at the same level. As in conventional small-angle models of trimuon production [35], 5” measures both the Mproduction probability and universality breakdown. But whereas in conventional models, the small angle is a free parameter, in our model it is related to the M- mass. For a 6-GeV M-, tan2 f = p-/M- N 0.0175. How does an M- produce trimuons ? The important decay modes of the M- are the following (with approximate branching ratios indicated).

M--v,

(8 ‘lb),

+ hadrons with C = S = 0

(25 09,

M- ---f

V,

M- -

vW -I- hadrons with C = S = -1

M-+p”+ M-+ The branching

+ e- -I- fic3

hadronswithC=S= pa + p- + pi0

-1

(17.24)

(25 ::I)> (25 “Q, (8 O,>.

ratios were estimated crudely by counting leptonic and semihadronic

310

DE RfJJULA,

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decay modes. The decays involving v, occur through the ordinary left-handed charged weak current. The decays involving p” (which we assume is a light neutral lepton with a mass of -1 GeV) occur through the right-handed charged weak current. We have ignored decays such as M- + p” + p- + fU which arise from mixing between W, and W, because they are suppressed by a factor of 49 N l/25. The last decay in Eq. (17.24), M- - ,u” + p- + ,JZOalways produces an event containing at least three muons. This is because the p” must decay into a p-. The decay goes through the right-handed current [Eq. (17.23)]. Since there are no other right-handed currents involving only light particles, p” decay must proceed through W, - W, mixing. We expect the decays of the p” to be

pO--+p-+e++v, p”+p-+p++vLL

%>, (20 %>? (60 %>.

(20

t~O-+p-- + had rons with C = S = 0

(17.25)

To underline the essential features and predictions of our model of trimuon production, we compare it with the standard small-angle model [35]. The small-angle model can be implemented in an SU(2) x U(1) theory with the left-handed doublets of Eq. (16.13). In this case it is the MO field which is light (-2 GeV). Trimuons are produced when the M- decays are as follows (see Fig. IO): M--+MO+p-+Cu (17.26) L p- + I*+ + VP *

In our model, trimuons are produced in the M- decays M- -+ p” + i;” + p-

L p+e-

+ rie or hadrons

(17.27a)

-+ p- + ef + v, or hadrons or (less copious) M-+pO+hadronswithC=S= I

I 4 p- + I’U

-1 (17.27b)

i---f p- + /.L+ + vu .

In the small-angle model, the branching ratio for M- decay into trimuons is expected to be 0.04 (because the CL+and one p- are produced through the ordinary weak current with branching ratio 0.2). In our model, the branching ratio is about 0.07, bigger by almost a factor of 2. This may be important since it appears difficult to get a large enough rate for trimuon production in the small-angle model.

THEORY OF FLAVOR MIXING

311

FIG. 10. Tri-, tetra-, and pentamuon production. M- production by muon neutrinos proceeds via a calculable small.angle. M- decays into ,? and a p$, pair; the p0 decays into pt, e& or hadrons in the ratios 1:1:3.

A more spectacular difference is that our model predicts four- and five-muon events. The M- decays into p+p-p-t.- + a.* and p+p+p-p- + *** with a branching ratio of about 1%. Pentamuon events can arise from the decay M- --f p” + ,ii” + p(17.28)

with branching ratio -0.2 %. Another important difference is that in the small-angle model, one ~+JL- pair must come from the decay of the MO. The observed trimuon events are consistent with this only if MO is relatively heavy, at least -2 GeV. In our model there is no such constraint. The p” mass can be 1.5 GeV or even less.In both models, the light neutral lepton is relatively long-lived. In the small-angle model, the decay rate is suppressed by the small angle suqared. In our model, it is suppressedby the L-R mixing factor of order 4~~. For a 1.5-GeV PO,we estimate a lifetime of lo-l1 sec. Observation of a long-lived neutral particle produced in neutrino scattering and decaying into p-e+ has been reported by Baranov et al [5]. Returning to the clock, we note that the N- must also be relatively light. The mixing angle eR(v, , PO) is ~EN-/P~. If the N- mass is much larger than PO, this mixing will suppresstrimuon production. The muon neutrino masssatisfies (17.29) The massesmust satisfy N-/p0 < l/c. Also, N-/p0 > 1, since either the N- is the

312

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AND

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particle discovered at SPEAR with a mass of -2 GeV or it is heavier. Thus for E = 0.1, we predict that the muon neutrino mass, vu N E4(N-/po) p-,

(17.30)

10 keV < 11~< 100 keV.

(17.31)

satisfies This range is well below the present experimental limit, vii < 650 keV [36]. We have postponed our discussion of leptons to this final section because we are the least confident in the validity of our leptonic scenario. Our scheme is subject to experimental test: indeed, to immediate disproof. We absolutely forbid the decays p + ey and ,u --f eeZ. We say there must be y10 neutrino oscillations, and that the electron neutrino and muon neutrino neuer interchange roles. On the other hand, our neutrinos are not massless, and we cannot tolerate an upper limit to the electron neutrino an order of magnitude improved. Furthermore, our entire picture depends on the fact that CP violation is milliweak and not superweak. An order of magnitude improvement in the experimental upper limit of the neutron electric dipole moment would destroy our theory. REFERENCES 1. N. CABIBBO, Phys. Rev. Lett. 10 (1963), 531; M. CELL-MANN AND M. LEVY, Nuovo Cimento 16 (1960), 705. 2. S. L. FLASHOW, as presented by M. GELL-MANN, in “Proceedings of 1960 Annual International Conference on High Energy Physics, University of Rochester” also Nucl. Phys. 10 (1959), 107. 3. J. SCHWINGER, Ann. Physics 2 (1957), 407; S. L. GLASHOW, Nucl. Phys. 22 (1961), 579; A. SALAM AND J. C. WARD, Nuovo Cimento 11 (1959), 568; S. WEINBERG, Phys. Rev. Left. 19 (1967), 1264; A. SALAM, in “Elementary Particle Theory” (V. Svartholm, Ed.), p. 387, Almquist and Wiksell, Stockholm, 1968; C. BOUCHIAT, J. ILIOPOULOS, AND PH. MEYER, Phys. Left. B38 (1972), 519. 4. S. L. GLASHOW, J. ILIOPOULOS, AND L. MAIANI, Phys. Rev. D 2 (1970), 1285. 5. M. PERL et al., Phys. Rev. Lett. 35 (1975), 1489; A. BENVENUTTI et al., Phys. Rev. Lett. 38 (1977), 1110; D. S. BARANOV et al., Preprint IHEP77-30 (1977). 6. P. E. G. BAIRD, M. W. S. M. BRIMICOMBE, G. J. ROBERTS, P. G. H. SANDARS, D. C. SOREIDE, E. N. FORTSON, L. L. LEWIS, E. G. LINDAHL, AND D. C. SOREIDE, Letf. Nature 264 (1976), 528; M. A. BOUCHIAT AND C. C. BOUCHIAT, Phys. Lett. B 48 (1974), 111; M. W. S. M. BRIMICOMBE, C. F. LOVING, AND P. G. H. SANDARS, J. Phys. B 1 (1976), 237; E. M. HENLEY AND L. WILETS, Phys. Rev. A 14 (1976), 1411; I. GRAND, N. C. PYPER, AND P. G. H. SANDARS, to appear; I. B. KHRIPLOVICH, Sov. Phys. JETP, in press; D. C. SOREIDE et al., Phys. Rev. Lett. 36 (1976), 352; S. ME~HKOV AND S. P. ROSEN, “Configuration Mixing and Tests of Parity Violation,” Nat. Bur. Stand. U.S.A. Preprint. For a review see I. B. KHRIPLOVICH, talk at the XVIII International Conference on High Energy Physics, Tbilisi, USSR, July 1976. 7. A. DE R~~JULA, H. GEORGI, S. L. GLASHOW, AND H. QUINN, Rev. Mod. Phys. 46 (1974), 391. 8. B. AUBERT et al., Phys. Rev. Lett. 33 (1974), 984. See “Proceedings of the June 1976 Neutrino Conference, Aachen, FDR”; A. Benvenutti et al., Phys. Rev. Lett. 36 (1976), 1478. 9. R. N. MOHAPATRA AND P. P. SIDHU, to appear; A. De R~JULA, H. GEORGI, AND S. L. GLASHOW, Ann. Physics 109 (1977), 242. 10. Such SU(2) x SC/(2) x Or(l) models were constructed by J. PATI AND A. SALAM, Phys. Rev. D 10 (1974), 275, in order to explain quantization of electric charge. They were first introduced by S. WEINBERG, Phys. Rev. Left. 29 (1972), 388, in an attempt to explain approximate isospin invariance. More recent references, closer in spirit to the present work are: H. FRITZSCH AND

THEORY

11. 12. 13.

14. 15.

16. 17. 18.

19. 20. 21. 22. 23.

24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

36.

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P. MINKOWSKI, Nuci. Phys. B103, 61 (1976); R. N. MOHAPATRA AND J. C. PATI, Phys. Rev. D 11 (1975), 566, 2558; G. SENJANOVIC AND R. N. MOHAPATRA, Phys. Rev. D 12 (1975), 1502; E. MA, OITS-68 Preprint. M. KOBAYASHI AND K, MASKAWA, Progr. Theor. Phys. 49 (1973), 652. A. earlier version of this work was presented by S. L. GLASHOW, in “Proceedings of the 1976 Coral Gables Conference, Orbis Scientiae, University of Miami, 1976.” T. HAGIWARA AND B. W. LEE, Phys. Rev. D 7 (1973), 459; S. WEINBERG, Phys. Rev. Lett. 29 (1973), 388; B. W. LEE, in “Proceedings of the XVI International Conference on High Energy Physics, Chicago-Batavia, Illinois, 1972” (J. D. Jackson and A. Roberts, Eds.), Vol. 4, p. 249, NAL, Batavia, Ill., 1973; H. GEORGI AND S. L. GLASHOW, Phys. Rev. D 6 (1972), 2977,7 (1973), 2457; H. GEORGI AND A. PAIS, Phys. Rev. D 10 (1974), 539. For a review, see S. S. GERSHTEIN, Plenary talk at the XVIII International Conference on High Energy Physics, Tbilisi, USSR, July 1977. A. DE R~~JULA, H. GEORGI, AND S. L. GLASHOW, Phys. Reo. D 12 (1975), 3589; G. BRANCO, T. HAGIWARA, AND R. N. MOHAPATRA, Phys. Rev. D 13 (1976), 104; E. GOLOWICH AND B. R. HOLSTEIN, Phys. Rev. Lett. 35 (1975), 831; A. FERNANDEZ-PACHECO, A, MORALES, R. NUI%NEZLAGOS, AKD J. SANCHEZ-GUILLEN, GIFT report; H. FRITZCH, M. GELL-MANN, AND P. MINKOWSKI, Phys. Lett. B59 (1973, 256; F. WILCSEK, A. ZEE, R. L. KINGSLEY, AND S. B. TREIMAN, Phys. Rev. D 12 (1975), 2768; S. PAKVASA, W. A. SIMMONS, AND S. F. TUAN, Phys. Rev. Lett. 35 (1975), 702. S. L. GLASHOW AND S. WEINBERG, Phys. Rev. D 15 (1977), 1958. K. SYMANZIK, in “Coral Gables Conference on Fundamental Interactions at High Energies II,” (A. Perlmutter, G. J. Iverson, and R. M. Williams, Eds.), Gordon & Breach, New York, 1970. S. L. GLASHOW AND S. WEINBERG, Phys. Rev. Lett. 20 (1968), 224; M. GELL-MANN, R. J. OAKES, AND B. RENNER, Phys. Rev. 175 (1968), 2195; R. JACKIW AND H. J. SCHNITZER, Phys. Rev. D 5 (1972), 2008. A. DE R~~IULA, H. GEORGI, AND S. L. GLASHOW, Phys. Rev. Lett. 35 (1975), 69. E. GOLOWICH AND BARRY R. HOLSTEIN, Phys. Rev. Let& 35 (1975), 831. See also Ref. [15]. G. GOLDHABER, in “Proceedings of the Summer Institute on Particle Physics, SLAC, 1976.” J. F. GUNION, P. C. MCNAMEE, AND M. D. SCACHON, SLAC-Pub-1847, November 1976. R. GATTO, G. SARTORI, AND M. TONIN, Phys. Left. B 28 (1968), 128; N. CABIBBO AND L. MAIANI, Phys. L&t. B28 (1968), 131; R. JACKIW AND H. J. SCHNITZER, Phys. Rev. D 5 (1972), 2008; S. WEINBERG, contribution to a Festschrift in Honor of I. I. Rabi, 1977, unpublished. S. L. GLASHOW, Phys. Rev. Lett. 14 (1964), 35; W. ALLES, Phys. Lett. 15 (1965), 348; A. MORALES, R. NU&EZ-LAGOS, AND M. SOLER, Nuoco Cimetzto 38 (1965), 1607. R. N. MOHAPATRA AND J. C. PATI, Phys. Rev. D 8 (1973), 2317. F. P. CALAPRICE et a/., Phys. Rev. 184 (1969), 1117. H. GEORGI AND S. L. GLASHOW, Phys. Rev. D 7 (1973), 2457. H. GEORGI AND H. D. POLI~ZER, Phys. Rev. D 14 (1976), 1829. H. D. POLITZER, Nucl. Phys. B 117 (1976), 397. See, for example, M. K. GAILLARD AND B. W. LEE, Phys. Rev. D 10 (1974), 897. N. FORTSON AND P. SANDARS, Harvard University Colloquia (1977), unpublished. F. F. TRETJAKOV et al., Preprint ITEF No. 15 (1976). G. ALTARELLI, R. PETRONZIO, AND G. PARISI, Phys. Lett. B 63 (1976) 182. p. 47, McGraw-Hill, New York, 1973. See E. D. COMMINS, “Weak Interactions,” P. LANGACKER AND G. SEGRE, Pennsylvania Reports, Nos. UPR-0072T and UPR-0073T; C. H. ALBRIGHT, J. SMITH, AND J. A. M. VERMASEREN, Phys. Rev. Lett. 38 (1977), 1187; V. BARGER, T. GOTTSCHALK, D. V. NANOPOULOS, J. ABAD, AND R. J. N. PHILLIPS, Phys. Rev. Lett. 38 (1977), 1190. Mechanisms not involving small angles, in which the heavy lepton is produced along with a new quark have been investigated by B. W. LEE AND S. WEINBERG, Rept. No. Fermi-lab-Pub77/31-Thy (1977); R. M. BARNETT AND L. N. CHANG, SLAC-Pub-1932, May 1977. A. R. CLARK et al., Phys. Rev. D 9 (1974), 533.