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Generations of massless composite quarks and leptons
Volume 113B, number 4
PHYSICS LETTERS
24 June 1982
GENERATIONS OF MASSLESS COMPOSITE QUARKS AND LEPTONS Koichi YAMAWAKI and Terufumi YOKOTA 1
Department of Physics, Nagoya University,Nagoya 464, Japan Received 5 March 1982
We present a QCD-like composite model in which quarks, leptons and technifermions are three-body systems made out of three kinds of massless elementary fermions t, c and w, each carrying technicolor, color and weak gauge interactions, respectively. Discrete symmetries, remnants of the U(1) A of the original lagrangian, are responsible for the masslessness of all the quarks and leptons and give the precise meaning of the generations. The model exhibits three generations for both quarks and leptons. Small but non-zero masses of the quarks and leptons are produced by the technicolor condensate of the composite technifermions, which thereby leads to the non-trivial Cabibbo mixing. Proton decays are all forbidden at the mass scale of the QCD-like theory.
Recently composite models for quarks and leptons have been considered by many authors [1,2]. However, one of the peculiar features of the composite quarks and leptons is the smallness of their masses compared with the size [2]. This is in sharp contrast to all the known composite systems; hadrons [mass (size)-I ], atoms [mass ~> (size) -1 ] and so forth, Another peculiar aspect is the presence of the generations which are simple repetitions and are almost degenerate in mass compared with the composite size. This is actually a mystery as far as they are considered as structureless particles. However, if we naively mock up the familiar composite systems and regard the generations as some excitation levels, orbital or radial, we would obtain the generation gap of roughly ~(size) -1, which is not what we want for quarks and leptons. So the crucial problem for the composite quarks and leptons should be to find out some mechanism which naturally guarantees: (i) The masslessness and/or the locality of the composite quarks and leptons. (ii) Tile distinction among and enough number (at least three) of the almost degenerate generations. In addition, proton decay imposes a severe restric1 Present address: Japan Atomic Energy Research Institute, Tokai-mura, Ibaraki 319-11, Japan.
0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
tion on the model [3,4]. In fact, if quarks and leptons are built up out of some common entities, there would in general be transitions between quarks and leptons, thereby leading to catastrophic proton decays at the mass scale of (size) -1 which is expected to be very much smaller than, say, 1015 GeV - the GUTs scale. So we further need some mechanism or symmetry which guarantees: (iii) No proton decays. In view of the conditions (i), (ii) and (iii), there seem to be no satisfactory models available so far. Yet other conditions would be required for a model to be realistic somehow. That is, we should provide the composite quarks and leptons with: (i)' Small but non-zero masses, and: (ii)' Considerable mixings among generations (at least in the low-energy region), in such a way that the mechanism responsible for (i)-(iii) is not completely destroyed. These conditions are very peculiar to the composite model for quarks and leptons and seem to be calling for a completely new mechanism which has not been known for the familiar composite systems. There have been considerable efforts toward satisfying condition (i). Probably the most natural way is to assume some symmetry for the underlying dynamics. 't Hooft [5] investigated a possibility for the unbroken anomaly-free chiral symmetriesl It is an elegant idea to consider the Wigner (spontaneously unbroken) 293
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phase of the chiral symmetries in an underlying QCDlike theory. However, it would be a serious problem whether the Wigner phase is really possible in the same gauge theory (except for the scale parameter) as the QCD in which Nature seems to choose the NambuGoldstone (spontaneously broken) phase of the chiral symmetry. Apart from the dynamical problem, the Wigner phase might lead us to 't Hooft's anomaly conditions [5], which, if taken seriously, would lead to sets of highly unrealistic composite states; abundance of unwanted states ("exotics") and lack of the necessary quarks and leptons [6] * a. As to condition (ii), the 't Hooft conditions tell us nothing. So let us turn to another possibility that the discrete symmetries originated from the U(1)A of the underlying QCD-like theory may be responsible for the masslessness of the composites [9,10]. Massless elementary fermions superficially possess U(1)A symmetries, which, however, are broken by the instantons down to the discrete subgroups without (physical) Nambu-Goldstone particles. It is very important to notice that the discrete symmetries alone can keep massless some or all o f the composite states as well as the elementary fermions, even when the anomaly-free continuous chiral symmetries are absent [9]. So the
't Hooft conditions are totally irrelevant to the masslessness of the composites, when some suitable discrete symmetries remain unbroken. We will return to the details of this point later. Another advantage of this idea over others is that condition (ii) may be easily satisfied. It has been explicitly demonstrated in the rishon model that the discrete quantum number associated with discrete symmetry might be used to label some part of the different generations [11 ]. A much simpler possibility has been considered by Guadagnini and Konishi [10], whose model, though quite unrealistic because of the lack of the color indices, exhibits several generations for three-body composites alone.
24 June 1982
In this article we present a model as a possible framework which seems to satisfy all the conditions (i), (ii), (iii), (i)' and (ii)'. We assume an SU(3) subcolor [1,3] *2 gauge theory for the underlying dynamics of the massless elementary fermions. One might be tempted to identify [5,7, 1] this theory with the technicolor [ 12] which has been invented to break the SU(2)L × U(1)y and has a mass scale, ATC, of order ~1 TeV in order to reproduce row, m z ~ 102 GeV. However, the locality of leptons would require the mass scale to be >~103 TeV [2] and the composite fermions would in general acquire the dynamical mass of this order, unless some part of the chiral symmetry remains unbroken ,3. We then adopt another viewpoint [10,14] that the technifermions (techniquarks and technileptons) also are composites on the same footing as quarks and leptons, the underlying dynamics being the SU(3) subcolor gauge theory with a mass scale, ASC (>>ATc), which is expected to bind the elementary fermions, 3 of the SU(3) subcolor, into the subcolor-singlet three-body composites; quarks, leptons and technifermions. So we presume, besides subcolor, three kinds of "flavour" gauge interactions; technicolor, color and weak gauges +4. An orthodox and traditional or the simplest attitude toward the composite model would be to regard each independent degree of freedom as being due to the different matter elements. Given three kinds of "flavour" gauge interactions, we are naturally led to a model with three kinds of elementary fermions with spin 1/2, each corresponding to the different gauge interactions. We choose t, c and w, each assigned as 2, 3* and 2 of SU(2)Technicolor , SU(3)Color and SU(2)Weak, respectively. All the elementary fermions t, c, and w are assumed to be massless. c carries the baryon number B = 1/6, t the lepton number L = 1/2 and w carries none of them. Their electric charge, given by Q =/3L + I3R + (B - L ) / 2 ,
*1 Even if we had luckily succeeded in finding a solution which only accommodates the existing quarks and leptons, it would not mean that the 't Hooft conditions by themselves guarantee the absence of other unrealistic solutions which are full o f exotics and/or lacking of quarks and leptons. If one further requites questionable [ 61 decoupling or persistent mass conditions [6 ], one is left with nothing interesting [ 5 ]. The dynamical rishon model [ 7 ] also gets in trouble with the 't Hooft conditions [8], besides the difficulties of the proton decays [3,4].
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is - 1 / 4 , 1/12 and -+1/2 for t, c and w, respectively. Let us disregard the weak gauge interaction for a .2 Metacolor [5] or hypercolor [7] are often used in the
literature. t3 The case of the remaining unbroken chital symmetry has been discussed [ 13]. ,4 Electromagnetismis treated differently. See the later discussions.
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moment. We thus have a global symmetry [SU(2)L X SU(2)R]W X U(1)t X U(1)c X U(1)w , plus discrete symmetries [9,10] as follows: Due to the subcolor, technicolor and color instantons, the generators Xt, X c and X w of the U(1)A'S can only change by the amounts AX t = 2(2 VSC + 3PTC)
[-'I/t --n(t L) -- rt(tR)],
2~Xc = 2(3 VSC+ 3 VC)
IXc = n(c L) - n(cR)], (1)
AXw = 2" 2PSC
[Xw = n(WL) -- n(wR)],
where Use, t'TC and vC stand for the instanton number minus antiinstanton number for the subcolor, technicolor and color gauge fields, respectively and n(tL) means the number of t L. To make the discrete symmetries more transparent, we rewrite eq. (I) as x -~X w ,
Ax = 4Use = 0, _+4, -+8.... ,
y=-2Xe-3X w,
Ay=12v c=0,-+12 +24 .... ,
z =-X t - X w ,
Az = 6~,TC = 0, -+6, -+12. . . . . (2)
These discrete symmetries forbid masses of all the elementary fermions. In fact mass terms of t, c and w would have (0, 0, -+2), (0, +4, 0) and (-+2, T6, T2), respectively, for (Ax, ky, Az) and thus are forbidden. It is then natural [9J to set the masses of t, c and w equal to zero, even when there are no anomaly-free chiral symmetries for t and c. Also forbidden by the discrete symmetries are masses of almost all the three-body composites * s The model exhibits three generations ,6 of quarks (q) and leptons (£), each generation being characterized by the set of discrete quantum numbers (x, y, z) as
follows: ql L = (c R c RwL)
with
(1, - 7 , - I ) ,
q2L = (c L c LwL)
with
(1,
q3E = (c L c R WE)
with
(1, - 3 , - 1 ) ,
~1L = (t R t R WL)
with
(1, - 3 , - 3 ) ,
~2L = (t L t L WL)
with
(1,-3,
J~3E = (t L t R WE)
with
(1, - 3 , - 1 ) ,
1,-1),
Oa)
l), (3b)
*s Only (tLtLtL) L or R can have a mass term. ,6 The same quark contents and possible generation interpretation have been considered [ 15], however, without discrete quantum numbers.
24 June 1982
where (cc) in the quark is 3 c and (tt) in the lepton is 1TC. Their parity partners are given as ql L ~ qlR = (CL CLWR) with ( - 1 , 7, 1), etc. It is easily seen that mass terms of quarks, ?:/1ql, c12q2, ~t3q3, ?:11q2, ~11q3 and ~12q3 would have discrete quantum numbers (+2, T14, T2), (+2, +2, ~2), (+2, ~-6, -7-2),(-+2, -7-6,~2), (-+2, ¢-10, T-2) and (+2, -7-2,-7-2),respectively, and are not allowed by the discrete symmetries. The same arguments are a/so applied to leptons. We thus have obtained three generations of massless quarks and leptons, each generation being distinguished from each other by discrete quantum numbers. Other possible three-body composites include technifermions and "exotic" states having anomalous baryon number, lepton number, color and/or weak charges. Among others, only technifermions in 2 of the SU(2)W are relevant to the later discussions: We have four generations of techniquarks Q1L --=(tRCR WE), Q2 L =(tLCLWL), Q3 L =- (tLCRWL) and Q 4 L ~ (t R CRWL) with discrete quantum numbers (1, - 5 , - 2 ) , (1, - 1 , 0), (1, - 5 , 0) and (1, - 1 , - 2 ) , respectively, plus three generations of technileptons L1E =-(tR tR WL), L2E = (tttLWL) and L3E - (tEtRWL) with (1, - 3 , - 3 ) , (1, - 3 , 1) and (1, - 3 , - 1 ) , respectively, where (t o in L is 3TC. We may have "mirrors" Qm and L m instead, which are defined as, for instance, OlL = (t L c L WR) for Q1E = (t R c R WE). The technifermions and the exotics are also massless for the same reason as the quarks and leptons, unless the "mirrors" co-exist with the corresponding partners in the same number * 7,8. We have seen that our model satisfies the conditions (i) and (ii). How about the conditions (i)' and (ii)'? Small but non-zero masses of quarks and leptons may be generated by the condensation of the composite technifermions in the residual Fermi interactions among th e composites [ 10,14]. Only the subcolor scale ASC is available for the mass dimension of the • 7 We could imagine "mirrors" for quarks (leptons) which would have opposite L- or R- SU(2) W charges to the ordinary quarks (leptons); e.g., q]nL =- (CLCLWR) and i~ts parity partner q]nR = (CRCRWL), etc. If they existed, they would have formed "mass terms" of order ASC together with the corresponding quarks (leptons), e.g., q~nRq1L' etc. ,8 Besides discrete symmetries, our model possesses the SU(2) W (L and/or R) symmetry which also forbids a mass term of the composites including quarks, leptons and those technifermions mentioned above. The same argument is also applied to the w itself.
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Fermi coupling constants and four-fermion interactions are expected to dominate multifermion vertices with more than six composite fermions. The fourfermion interactions which are relevant to the mass of quarks and leptons have forms like f ~IqTT and f ££TT, respectively, where T denotes generically the technifermions and f the Fermi coupling constants of order O(1/A2c). The discrete symmetries only allow special combinations of quarks or leptons with the technifermions. For the reason to be mentioned later, we may pick up only terms that are invariant under the discrete symmetries, eq. (2) with vc set equal to zero. As to quarks, these terms are q2Rq2LQ4RQ4L (or --m m q2Rq2LQ4LQ4R ) with discrete quantum numbers (4, 0 , - 6 ) , ~3Rq3LL3LL3R (or gtaRq3LL3mRL3mL) with (0, 0, 0) and C:I1Rq2LL3LL3R (or ~tl Rq2L --m m • X L3RL3L) with (0, 0, 0) ,9, where the [SU(2)L X SU(2)R ] W invariant form is understood as, e.g., -i k --1 t
ei/eklq2R q2L Q 4 R Q4 L.
When the technifermion condensation * 1o takes place, we may expect (T-T) ~ O(A~c ). Thus the masses of quarks and leptons are -O(A3Tc/A2c), which are of order ~100 MeV if we assume Asc 102 TeV and ATC ~ 1 TeV. The technifermion condensation which spontaneously breaks the discrete symmetries also implies mixings among the generations which we have distinguished by discrete quantum numbers. Corresponding to the allowed four-fermion interactions mentioned above, the quark mass matrix takes the form
0 5)
m=
a*
b
0
0
,
(4)
where a, c ~ (L3L3) and b oc (Q4Q4). Assuming ~u g= o d for (TiT/) = 6ilai [i = l(down), 2(up)I, we may diagonalize the matrix, considering the fact that m s >>m d and rn c >> m u. A nontrivial Cabibbo angle then is obtained as [16,10]
0 c = tan-l((rnd/ms)l/2) -- tan-l((mu/mc)l/2).
(5)
,9 Oft-diagonal combinations of technifermions like QaRQbL (a ~ b) are irrelevant to the following discussions. 4:10 We will later discuss the electro-weak gauge symmetry which might be broken by this condensation. 296
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The peculiar form of the quark mass matrix, eq. (4), is essentially due to the discrete symmetries, eq. (2) with vc = 0, imposed on the residual four-fermion interactions. No such peculiarity happens to the lepton mass matrix, all the matrix elements remaining nonzero in general without further assumptions. If we take account of the vc # 0 effects and the effective interactions among more than six fermions, the quark mass matrix would have nothing particular. However, these effects are extremely small, since the four-fermion interactions which are allowed only for vc =P 0 should be proportional to the instanton density (or some power of it) which is vanishingly small for the energy scale >>Ac and the residual six-fermion vertex is ex3 2 pected to give a mass of order O(AdTc/ASsc) ~ ATc/Asc. So eq. (5) would not be changed by the inclusion of these effects. These small effects may be responsible for the Kobayashi-Maskawa mixings of the first or the second generations with the third one. However, it is not satisfactory that a and e in eq. (4) are due to the same kind of condensation, (L3L3), so are expected to be of the same order of magnitude, unless for some reason the Fermi coupling constants for ~tl q2 and Ct3q3 are different by one or two orders. It is also noted that the instanton effects in our model may be responsible for CP violation. We thus have seen that the conditions (i)' and (ii)' are also satisfied, though not completely. Now to condition (iii). The model possesses a U(1) symmetry corresponding to c-number conservation, which absolutely forbids proton decay. This is in sharp contrast to some models [3,4] which, for instance, take both 3 c and 3c as the elementary fermions, so that proton decay can easily take place at the relevant energy scale to the model unless further selection rules are imposed by hand. It might be interesting to consider a "GUT" also including subcolor and technicolor in some energy region much higher than ASC , in which case the proton decay may be possible also in our model but at the new "GUT" scale. So far we have disregarded the gauge interaction of the weak-electromagnetism. Inclusion of this gauge interaction affects none of the above results. We will consider two possible ways of introducing this gauge. First we take SU(2)L X U(1)y. There are no anomalies for the elementary fermion WL, since Y = B - L + 213R = 0 for w L. As to the composite level, quarks and leptons cancel the anomaly each other but the technifer-
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24 June 1982
mions should be arranged so as to cancel the anomaly. m A simple example is to take a set {Q4L, L3L, two out m m m of (Q 1L, Q2L , Q3L), one of (LIL, L2L)) , with their parity partners. The same set with the change (mirror +~ non-mirror) is also possible. The symmetry breaking SU(2)L X U(1)y -+ U(1)e m occurs via the usual technicolor dynamics among the composite technifermions. Secondly, we may take SU(2)L X SU(2)R X U(1)B__L . We may assume that the subcolor dynamics break SU(2)L X SU(2)R X U(1)B_ L into SU(2)L X U ( l ) y . In order to preserve the discrete symmetries eq. (2), symmetry breaking should be caused by the multibody condensate of the elementary fermions, e.g.
X SU(2)RIw × U(1)t × U(1)c X U(1)w. The solutions always cancel the SU(2)L × U(1)y anomaly. If one further assumes that all the "flavour" gauge interactions can be switched off without affecting the composite spectrum, one will obtain a global symmetry G F = SU(7)L X SU(7)R × U(1)t+c+w, because there is nothing to distinguish among seven elementary fertalons t, c and w in that limit. Solutions of the 't Hooft conditions in this case also cancel the SU(2)L × U(1)y anomaly. Indices [5] for the representations of G F are defined as
((tLtL)(tRte)(~RW/L)(r 1 -- ir2)/cl(W~W~))~ 0 ,
([],
where ( ) means a Lorentz scalar, i, j, k and l are SU(2)w indices and r ~ is a Pauli matrix. The gauge bosons acquire mass of order ASC , i.e., mWR, m z ' O(Asc), which are much larger than mWL, m z ,- O(ATc ). The electromagnetic interactions among quarks or leptons are subject to the same selection rules as the mass terms [9]. Magnetic-dipole anomalous moments and M1 transitions of form Cta ouvqb FUr (a - 1 , 2 , 3) are forbidden by the discrete symmetries ((TT) ¢ 0 does not spoil this result). The anomalous gyromagnetic ratio is expected to be ~m2/A2c ~ (ATc/Asc) 6 "- 10 -12 if A T c / A s c ~ 10 - 2 . Decays like p ~ e7 are also forbidden. Our model also forbids electric-monopole transitions like ~a7~12b O,FU~, (a 4=b) if there are no generation mixings for leptons. As we have emphasized before, we need not the 't Hooft conditions for the masslessness of quarks and leptons. Can we then avoid the 't Hooft conditions at all? In the case at hand, it may be possible that technicolor and color forces, no matter how small they may be, affect the spectrum of composites in an essential manner, so that the 't Hooft conditions, if at all, could only be applied to [SU(2)L X SU(2)R]W X U(1)t X U(1)c X U(1)w. However, this chiral symmetry may be spontaneously broken by the subcolor dynamics via the multifermion condensation mentioned before, in which case we have massless quarks and leptons but no 't Hooft conditions. If, on the other hand, one assumes a Wigner phase for this possibility and accepts the assumptions [5] made by 't Hooft, then one should look for solutions of the 't Hooft conditions imposed on [SU(2)L
One of the solutions is ~1+ = - 3 , ~1 = 1, ~2+ = - 4 , ~2_ = 1 and ~3 = 0. The solution contains three generations of quarks and leptons (the third generation is threefold degenerate * al), plus technifermions and many exotics. The quark mass matrix is the same as eq. (4) except for the threefold degeneracy of the third generation. Finally, there are some unpleasant features with our model which are subject to future studies. The discrete symmetries cannot suppress the flavour changing neutral processes such as K ° - K 0 mixing. Residual fourfermion interactions nKe f SR 3z aRSL')'/scl L carry zero discrete quantum numbers, thus are not forbidden by the discrete symmetries alone. The next problem is that SU(2)T C is not asymptotically free at the composite level unless only one technifermion is involved. One way out of this problem might be to assume that the effective theory of the composite technifermions has an ultraviolet fixed point (g = go) and the region g > g 0 is identified with the confining phase [17] in which case the problem of the flavour changing neutral processes may also be avoided [17,18]. Among others, presumably the most serious problem * a2 in our approach is to eliminate many unwanted states, in particular, various exotics and some "mirrors" * 13.
(~-~3, 1)~1+ ,
%,
( ~ , 1) ~1_,
([D, [i~l~])~2+,
1)
qql The third generation is contained in ~22+= -4, i.e. fourfold. However, one of them can form a mass term (-Asc) with a "mirror" contained in ~2_ = 1. 4-12This also applies to any other chiral symmetric approaches. See footnote one. • t3 There may be a possibility that exotics would be accompanied by their "mirrors" in the same number as themselves, so that they would form mass terms of order ASC. 297
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At this moment, we have no way to handle it, mainly due to the lack o f dynamical arguments * le. In order to resolve this problem, there seems to be required more dynamical knowledge o f the model, for example, wave functions of the composite fermions. One o f the authors (K.Y.) would like to thank Professor Z. Maki for warm hospitality at the Research Institute for Fundamental Physics, K y o t o University where this work was initiated. His sincere thanks also go to Dr. K. Konishi for stimulating discussions. The authors are indebted to many colleagues at Nagoya University, in particular, Professor Y. Ohnuki, Professor S. Kitakado and Dr. T. Matsuoka for helpful discussions. Professor S. Kitakado is acknowledged for reading the manuscript.
Note added in proof. It is argued in this article that the technilepton may either be "non-mirror" L L = (ttwL) or "mirror" L ~ = (ttwR). However, we now realize that it should be "mirror", since otherwise the l e p t o n - t e c h n i l e p t o n transition via technigluon emission is possible, so that the lepton may acquire a mass of order ATC due to the technilepton condensation, which is disastrous.
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References [11 For a review, see: H. Terazawa, Phys. Rev. D22 (1979) 184; in Proc. 1981 INS Syrup. on Quark and lepton physics, eds. K; Fujikawa, U. Terazawa and A. Ukawa. [21 For a review, see: S. Brodsky and S. Drell, Phys. Rev. D22 (1980) 2236. [31 R. Casalbuoni and R. Gatto, Phys. Lett. 100B (1981) 135. [4] E.J. Squires, Phys. Lett. 102B (1981) 127; H. Harari, R.N. Mohapatra and N. Seiberg, preprint ref. TH-3123-CERN. I5] G. 't Hooft, Lectures Carg~se Summer Institute (1979). [61 J. PreskiU and S. Weinberg, Phys. Rev. D24 (1981) 1059. [71 H. Harari and N. Seiberg, Phys. Lett. 98B (1981) 269. [81 S.F. King, Phys. Lett. 107B (1981) 201. [91 S. Weinberg, Phys. Lett. 102B (1981) 401. [101 E. Guadagnini and K. Konishi, Nucl. Phys. B196 (1982) 165. [111 H. Harari and N. Seiberg, Phys. Lett. 102B (1981) 263. [121 S. Weinberg, Phys. Rev. D13 (1974) 974; D19 (1980) 2236; L. Susskind, Phys. Rev. D20 (1979) 2619. [131 See for a review, e.g., E. Fahri and L. Susskind, Phys. Rep. 74C (1981) 277. [14] P. Sikivie, Phys. Lett. 103B (1981) 437. [151 K. Matumoto and K. Kakazu, Prog. Theor. Phys. 65 (1981) 390. [161 H. Fritzsch, Phys. Lett. 70B (1977) 436;73B (1978) 317. [17l B. Holdom, Phys. Rev. D24 (1981) 1441. [181 K. Yamawaki and T. Yokota, in preparation.
PHYSICS LETTERS
24 June 1982
GENERATIONS OF MASSLESS COMPOSITE QUARKS AND LEPTONS Koichi YAMAWAKI and Terufumi YOKOTA 1
Department of Physics, Nagoya University,Nagoya 464, Japan Received 5 March 1982
We present a QCD-like composite model in which quarks, leptons and technifermions are three-body systems made out of three kinds of massless elementary fermions t, c and w, each carrying technicolor, color and weak gauge interactions, respectively. Discrete symmetries, remnants of the U(1) A of the original lagrangian, are responsible for the masslessness of all the quarks and leptons and give the precise meaning of the generations. The model exhibits three generations for both quarks and leptons. Small but non-zero masses of the quarks and leptons are produced by the technicolor condensate of the composite technifermions, which thereby leads to the non-trivial Cabibbo mixing. Proton decays are all forbidden at the mass scale of the QCD-like theory.
Recently composite models for quarks and leptons have been considered by many authors [1,2]. However, one of the peculiar features of the composite quarks and leptons is the smallness of their masses compared with the size [2]. This is in sharp contrast to all the known composite systems; hadrons [mass (size)-I ], atoms [mass ~> (size) -1 ] and so forth, Another peculiar aspect is the presence of the generations which are simple repetitions and are almost degenerate in mass compared with the composite size. This is actually a mystery as far as they are considered as structureless particles. However, if we naively mock up the familiar composite systems and regard the generations as some excitation levels, orbital or radial, we would obtain the generation gap of roughly ~(size) -1, which is not what we want for quarks and leptons. So the crucial problem for the composite quarks and leptons should be to find out some mechanism which naturally guarantees: (i) The masslessness and/or the locality of the composite quarks and leptons. (ii) Tile distinction among and enough number (at least three) of the almost degenerate generations. In addition, proton decay imposes a severe restric1 Present address: Japan Atomic Energy Research Institute, Tokai-mura, Ibaraki 319-11, Japan.
0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
tion on the model [3,4]. In fact, if quarks and leptons are built up out of some common entities, there would in general be transitions between quarks and leptons, thereby leading to catastrophic proton decays at the mass scale of (size) -1 which is expected to be very much smaller than, say, 1015 GeV - the GUTs scale. So we further need some mechanism or symmetry which guarantees: (iii) No proton decays. In view of the conditions (i), (ii) and (iii), there seem to be no satisfactory models available so far. Yet other conditions would be required for a model to be realistic somehow. That is, we should provide the composite quarks and leptons with: (i)' Small but non-zero masses, and: (ii)' Considerable mixings among generations (at least in the low-energy region), in such a way that the mechanism responsible for (i)-(iii) is not completely destroyed. These conditions are very peculiar to the composite model for quarks and leptons and seem to be calling for a completely new mechanism which has not been known for the familiar composite systems. There have been considerable efforts toward satisfying condition (i). Probably the most natural way is to assume some symmetry for the underlying dynamics. 't Hooft [5] investigated a possibility for the unbroken anomaly-free chiral symmetriesl It is an elegant idea to consider the Wigner (spontaneously unbroken) 293
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phase of the chiral symmetries in an underlying QCDlike theory. However, it would be a serious problem whether the Wigner phase is really possible in the same gauge theory (except for the scale parameter) as the QCD in which Nature seems to choose the NambuGoldstone (spontaneously broken) phase of the chiral symmetry. Apart from the dynamical problem, the Wigner phase might lead us to 't Hooft's anomaly conditions [5], which, if taken seriously, would lead to sets of highly unrealistic composite states; abundance of unwanted states ("exotics") and lack of the necessary quarks and leptons [6] * a. As to condition (ii), the 't Hooft conditions tell us nothing. So let us turn to another possibility that the discrete symmetries originated from the U(1)A of the underlying QCD-like theory may be responsible for the masslessness of the composites [9,10]. Massless elementary fermions superficially possess U(1)A symmetries, which, however, are broken by the instantons down to the discrete subgroups without (physical) Nambu-Goldstone particles. It is very important to notice that the discrete symmetries alone can keep massless some or all o f the composite states as well as the elementary fermions, even when the anomaly-free continuous chiral symmetries are absent [9]. So the
't Hooft conditions are totally irrelevant to the masslessness of the composites, when some suitable discrete symmetries remain unbroken. We will return to the details of this point later. Another advantage of this idea over others is that condition (ii) may be easily satisfied. It has been explicitly demonstrated in the rishon model that the discrete quantum number associated with discrete symmetry might be used to label some part of the different generations [11 ]. A much simpler possibility has been considered by Guadagnini and Konishi [10], whose model, though quite unrealistic because of the lack of the color indices, exhibits several generations for three-body composites alone.
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In this article we present a model as a possible framework which seems to satisfy all the conditions (i), (ii), (iii), (i)' and (ii)'. We assume an SU(3) subcolor [1,3] *2 gauge theory for the underlying dynamics of the massless elementary fermions. One might be tempted to identify [5,7, 1] this theory with the technicolor [ 12] which has been invented to break the SU(2)L × U(1)y and has a mass scale, ATC, of order ~1 TeV in order to reproduce row, m z ~ 102 GeV. However, the locality of leptons would require the mass scale to be >~103 TeV [2] and the composite fermions would in general acquire the dynamical mass of this order, unless some part of the chiral symmetry remains unbroken ,3. We then adopt another viewpoint [10,14] that the technifermions (techniquarks and technileptons) also are composites on the same footing as quarks and leptons, the underlying dynamics being the SU(3) subcolor gauge theory with a mass scale, ASC (>>ATc), which is expected to bind the elementary fermions, 3 of the SU(3) subcolor, into the subcolor-singlet three-body composites; quarks, leptons and technifermions. So we presume, besides subcolor, three kinds of "flavour" gauge interactions; technicolor, color and weak gauges +4. An orthodox and traditional or the simplest attitude toward the composite model would be to regard each independent degree of freedom as being due to the different matter elements. Given three kinds of "flavour" gauge interactions, we are naturally led to a model with three kinds of elementary fermions with spin 1/2, each corresponding to the different gauge interactions. We choose t, c and w, each assigned as 2, 3* and 2 of SU(2)Technicolor , SU(3)Color and SU(2)Weak, respectively. All the elementary fermions t, c, and w are assumed to be massless. c carries the baryon number B = 1/6, t the lepton number L = 1/2 and w carries none of them. Their electric charge, given by Q =/3L + I3R + (B - L ) / 2 ,
*1 Even if we had luckily succeeded in finding a solution which only accommodates the existing quarks and leptons, it would not mean that the 't Hooft conditions by themselves guarantee the absence of other unrealistic solutions which are full o f exotics and/or lacking of quarks and leptons. If one further requites questionable [ 61 decoupling or persistent mass conditions [6 ], one is left with nothing interesting [ 5 ]. The dynamical rishon model [ 7 ] also gets in trouble with the 't Hooft conditions [8], besides the difficulties of the proton decays [3,4].
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is - 1 / 4 , 1/12 and -+1/2 for t, c and w, respectively. Let us disregard the weak gauge interaction for a .2 Metacolor [5] or hypercolor [7] are often used in the
literature. t3 The case of the remaining unbroken chital symmetry has been discussed [ 13]. ,4 Electromagnetismis treated differently. See the later discussions.
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moment. We thus have a global symmetry [SU(2)L X SU(2)R]W X U(1)t X U(1)c X U(1)w , plus discrete symmetries [9,10] as follows: Due to the subcolor, technicolor and color instantons, the generators Xt, X c and X w of the U(1)A'S can only change by the amounts AX t = 2(2 VSC + 3PTC)
[-'I/t --n(t L) -- rt(tR)],
2~Xc = 2(3 VSC+ 3 VC)
IXc = n(c L) - n(cR)], (1)
AXw = 2" 2PSC
[Xw = n(WL) -- n(wR)],
where Use, t'TC and vC stand for the instanton number minus antiinstanton number for the subcolor, technicolor and color gauge fields, respectively and n(tL) means the number of t L. To make the discrete symmetries more transparent, we rewrite eq. (I) as x -~X w ,
Ax = 4Use = 0, _+4, -+8.... ,
y=-2Xe-3X w,
Ay=12v c=0,-+12 +24 .... ,
z =-X t - X w ,
Az = 6~,TC = 0, -+6, -+12. . . . . (2)
These discrete symmetries forbid masses of all the elementary fermions. In fact mass terms of t, c and w would have (0, 0, -+2), (0, +4, 0) and (-+2, T6, T2), respectively, for (Ax, ky, Az) and thus are forbidden. It is then natural [9J to set the masses of t, c and w equal to zero, even when there are no anomaly-free chiral symmetries for t and c. Also forbidden by the discrete symmetries are masses of almost all the three-body composites * s The model exhibits three generations ,6 of quarks (q) and leptons (£), each generation being characterized by the set of discrete quantum numbers (x, y, z) as
follows: ql L = (c R c RwL)
with
(1, - 7 , - I ) ,
q2L = (c L c LwL)
with
(1,
q3E = (c L c R WE)
with
(1, - 3 , - 1 ) ,
~1L = (t R t R WL)
with
(1, - 3 , - 3 ) ,
~2L = (t L t L WL)
with
(1,-3,
J~3E = (t L t R WE)
with
(1, - 3 , - 1 ) ,
1,-1),
Oa)
l), (3b)
*s Only (tLtLtL) L or R can have a mass term. ,6 The same quark contents and possible generation interpretation have been considered [ 15], however, without discrete quantum numbers.
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where (cc) in the quark is 3 c and (tt) in the lepton is 1TC. Their parity partners are given as ql L ~ qlR = (CL CLWR) with ( - 1 , 7, 1), etc. It is easily seen that mass terms of quarks, ?:/1ql, c12q2, ~t3q3, ?:11q2, ~11q3 and ~12q3 would have discrete quantum numbers (+2, T14, T2), (+2, +2, ~2), (+2, ~-6, -7-2),(-+2, -7-6,~2), (-+2, ¢-10, T-2) and (+2, -7-2,-7-2),respectively, and are not allowed by the discrete symmetries. The same arguments are a/so applied to leptons. We thus have obtained three generations of massless quarks and leptons, each generation being distinguished from each other by discrete quantum numbers. Other possible three-body composites include technifermions and "exotic" states having anomalous baryon number, lepton number, color and/or weak charges. Among others, only technifermions in 2 of the SU(2)W are relevant to the later discussions: We have four generations of techniquarks Q1L --=(tRCR WE), Q2 L =(tLCLWL), Q3 L =- (tLCRWL) and Q 4 L ~ (t R CRWL) with discrete quantum numbers (1, - 5 , - 2 ) , (1, - 1 , 0), (1, - 5 , 0) and (1, - 1 , - 2 ) , respectively, plus three generations of technileptons L1E =-(tR tR WL), L2E = (tttLWL) and L3E - (tEtRWL) with (1, - 3 , - 3 ) , (1, - 3 , 1) and (1, - 3 , - 1 ) , respectively, where (t o in L is 3TC. We may have "mirrors" Qm and L m instead, which are defined as, for instance, OlL = (t L c L WR) for Q1E = (t R c R WE). The technifermions and the exotics are also massless for the same reason as the quarks and leptons, unless the "mirrors" co-exist with the corresponding partners in the same number * 7,8. We have seen that our model satisfies the conditions (i) and (ii). How about the conditions (i)' and (ii)'? Small but non-zero masses of quarks and leptons may be generated by the condensation of the composite technifermions in the residual Fermi interactions among th e composites [ 10,14]. Only the subcolor scale ASC is available for the mass dimension of the • 7 We could imagine "mirrors" for quarks (leptons) which would have opposite L- or R- SU(2) W charges to the ordinary quarks (leptons); e.g., q]nL =- (CLCLWR) and i~ts parity partner q]nR = (CRCRWL), etc. If they existed, they would have formed "mass terms" of order ASC together with the corresponding quarks (leptons), e.g., q~nRq1L' etc. ,8 Besides discrete symmetries, our model possesses the SU(2) W (L and/or R) symmetry which also forbids a mass term of the composites including quarks, leptons and those technifermions mentioned above. The same argument is also applied to the w itself.
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Fermi coupling constants and four-fermion interactions are expected to dominate multifermion vertices with more than six composite fermions. The fourfermion interactions which are relevant to the mass of quarks and leptons have forms like f ~IqTT and f ££TT, respectively, where T denotes generically the technifermions and f the Fermi coupling constants of order O(1/A2c). The discrete symmetries only allow special combinations of quarks or leptons with the technifermions. For the reason to be mentioned later, we may pick up only terms that are invariant under the discrete symmetries, eq. (2) with vc set equal to zero. As to quarks, these terms are q2Rq2LQ4RQ4L (or --m m q2Rq2LQ4LQ4R ) with discrete quantum numbers (4, 0 , - 6 ) , ~3Rq3LL3LL3R (or gtaRq3LL3mRL3mL) with (0, 0, 0) and C:I1Rq2LL3LL3R (or ~tl Rq2L --m m • X L3RL3L) with (0, 0, 0) ,9, where the [SU(2)L X SU(2)R ] W invariant form is understood as, e.g., -i k --1 t
ei/eklq2R q2L Q 4 R Q4 L.
When the technifermion condensation * 1o takes place, we may expect (T-T) ~ O(A~c ). Thus the masses of quarks and leptons are -O(A3Tc/A2c), which are of order ~100 MeV if we assume Asc 102 TeV and ATC ~ 1 TeV. The technifermion condensation which spontaneously breaks the discrete symmetries also implies mixings among the generations which we have distinguished by discrete quantum numbers. Corresponding to the allowed four-fermion interactions mentioned above, the quark mass matrix takes the form
0 5)
m=
a*
b
0
0
,
(4)
where a, c ~ (L3L3) and b oc (Q4Q4). Assuming ~u g= o d for (TiT/) = 6ilai [i = l(down), 2(up)I, we may diagonalize the matrix, considering the fact that m s >>m d and rn c >> m u. A nontrivial Cabibbo angle then is obtained as [16,10]
0 c = tan-l((rnd/ms)l/2) -- tan-l((mu/mc)l/2).
(5)
,9 Oft-diagonal combinations of technifermions like QaRQbL (a ~ b) are irrelevant to the following discussions. 4:10 We will later discuss the electro-weak gauge symmetry which might be broken by this condensation. 296
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The peculiar form of the quark mass matrix, eq. (4), is essentially due to the discrete symmetries, eq. (2) with vc = 0, imposed on the residual four-fermion interactions. No such peculiarity happens to the lepton mass matrix, all the matrix elements remaining nonzero in general without further assumptions. If we take account of the vc # 0 effects and the effective interactions among more than six fermions, the quark mass matrix would have nothing particular. However, these effects are extremely small, since the four-fermion interactions which are allowed only for vc =P 0 should be proportional to the instanton density (or some power of it) which is vanishingly small for the energy scale >>Ac and the residual six-fermion vertex is ex3 2 pected to give a mass of order O(AdTc/ASsc) ~ ATc/Asc. So eq. (5) would not be changed by the inclusion of these effects. These small effects may be responsible for the Kobayashi-Maskawa mixings of the first or the second generations with the third one. However, it is not satisfactory that a and e in eq. (4) are due to the same kind of condensation, (L3L3), so are expected to be of the same order of magnitude, unless for some reason the Fermi coupling constants for ~tl q2 and Ct3q3 are different by one or two orders. It is also noted that the instanton effects in our model may be responsible for CP violation. We thus have seen that the conditions (i)' and (ii)' are also satisfied, though not completely. Now to condition (iii). The model possesses a U(1) symmetry corresponding to c-number conservation, which absolutely forbids proton decay. This is in sharp contrast to some models [3,4] which, for instance, take both 3 c and 3c as the elementary fermions, so that proton decay can easily take place at the relevant energy scale to the model unless further selection rules are imposed by hand. It might be interesting to consider a "GUT" also including subcolor and technicolor in some energy region much higher than ASC , in which case the proton decay may be possible also in our model but at the new "GUT" scale. So far we have disregarded the gauge interaction of the weak-electromagnetism. Inclusion of this gauge interaction affects none of the above results. We will consider two possible ways of introducing this gauge. First we take SU(2)L X U(1)y. There are no anomalies for the elementary fermion WL, since Y = B - L + 213R = 0 for w L. As to the composite level, quarks and leptons cancel the anomaly each other but the technifer-
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mions should be arranged so as to cancel the anomaly. m A simple example is to take a set {Q4L, L3L, two out m m m of (Q 1L, Q2L , Q3L), one of (LIL, L2L)) , with their parity partners. The same set with the change (mirror +~ non-mirror) is also possible. The symmetry breaking SU(2)L X U(1)y -+ U(1)e m occurs via the usual technicolor dynamics among the composite technifermions. Secondly, we may take SU(2)L X SU(2)R X U(1)B__L . We may assume that the subcolor dynamics break SU(2)L X SU(2)R X U(1)B_ L into SU(2)L X U ( l ) y . In order to preserve the discrete symmetries eq. (2), symmetry breaking should be caused by the multibody condensate of the elementary fermions, e.g.
X SU(2)RIw × U(1)t × U(1)c X U(1)w. The solutions always cancel the SU(2)L × U(1)y anomaly. If one further assumes that all the "flavour" gauge interactions can be switched off without affecting the composite spectrum, one will obtain a global symmetry G F = SU(7)L X SU(7)R × U(1)t+c+w, because there is nothing to distinguish among seven elementary fertalons t, c and w in that limit. Solutions of the 't Hooft conditions in this case also cancel the SU(2)L × U(1)y anomaly. Indices [5] for the representations of G F are defined as
((tLtL)(tRte)(~RW/L)(r 1 -- ir2)/cl(W~W~))~ 0 ,
([],
where ( ) means a Lorentz scalar, i, j, k and l are SU(2)w indices and r ~ is a Pauli matrix. The gauge bosons acquire mass of order ASC , i.e., mWR, m z ' O(Asc), which are much larger than mWL, m z ,- O(ATc ). The electromagnetic interactions among quarks or leptons are subject to the same selection rules as the mass terms [9]. Magnetic-dipole anomalous moments and M1 transitions of form Cta ouvqb FUr (a - 1 , 2 , 3) are forbidden by the discrete symmetries ((TT) ¢ 0 does not spoil this result). The anomalous gyromagnetic ratio is expected to be ~m2/A2c ~ (ATc/Asc) 6 "- 10 -12 if A T c / A s c ~ 10 - 2 . Decays like p ~ e7 are also forbidden. Our model also forbids electric-monopole transitions like ~a7~12b O,FU~, (a 4=b) if there are no generation mixings for leptons. As we have emphasized before, we need not the 't Hooft conditions for the masslessness of quarks and leptons. Can we then avoid the 't Hooft conditions at all? In the case at hand, it may be possible that technicolor and color forces, no matter how small they may be, affect the spectrum of composites in an essential manner, so that the 't Hooft conditions, if at all, could only be applied to [SU(2)L X SU(2)R]W X U(1)t X U(1)c X U(1)w. However, this chiral symmetry may be spontaneously broken by the subcolor dynamics via the multifermion condensation mentioned before, in which case we have massless quarks and leptons but no 't Hooft conditions. If, on the other hand, one assumes a Wigner phase for this possibility and accepts the assumptions [5] made by 't Hooft, then one should look for solutions of the 't Hooft conditions imposed on [SU(2)L
One of the solutions is ~1+ = - 3 , ~1 = 1, ~2+ = - 4 , ~2_ = 1 and ~3 = 0. The solution contains three generations of quarks and leptons (the third generation is threefold degenerate * al), plus technifermions and many exotics. The quark mass matrix is the same as eq. (4) except for the threefold degeneracy of the third generation. Finally, there are some unpleasant features with our model which are subject to future studies. The discrete symmetries cannot suppress the flavour changing neutral processes such as K ° - K 0 mixing. Residual fourfermion interactions nKe f SR 3z aRSL')'/scl L carry zero discrete quantum numbers, thus are not forbidden by the discrete symmetries alone. The next problem is that SU(2)T C is not asymptotically free at the composite level unless only one technifermion is involved. One way out of this problem might be to assume that the effective theory of the composite technifermions has an ultraviolet fixed point (g = go) and the region g > g 0 is identified with the confining phase [17] in which case the problem of the flavour changing neutral processes may also be avoided [17,18]. Among others, presumably the most serious problem * a2 in our approach is to eliminate many unwanted states, in particular, various exotics and some "mirrors" * 13.
(~-~3, 1)~1+ ,
%,
( ~ , 1) ~1_,
([D, [i~l~])~2+,
1)
qql The third generation is contained in ~22+= -4, i.e. fourfold. However, one of them can form a mass term (-Asc) with a "mirror" contained in ~2_ = 1. 4-12This also applies to any other chiral symmetric approaches. See footnote one. • t3 There may be a possibility that exotics would be accompanied by their "mirrors" in the same number as themselves, so that they would form mass terms of order ASC. 297
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At this moment, we have no way to handle it, mainly due to the lack o f dynamical arguments * le. In order to resolve this problem, there seems to be required more dynamical knowledge o f the model, for example, wave functions of the composite fermions. One o f the authors (K.Y.) would like to thank Professor Z. Maki for warm hospitality at the Research Institute for Fundamental Physics, K y o t o University where this work was initiated. His sincere thanks also go to Dr. K. Konishi for stimulating discussions. The authors are indebted to many colleagues at Nagoya University, in particular, Professor Y. Ohnuki, Professor S. Kitakado and Dr. T. Matsuoka for helpful discussions. Professor S. Kitakado is acknowledged for reading the manuscript.
Note added in proof. It is argued in this article that the technilepton may either be "non-mirror" L L = (ttwL) or "mirror" L ~ = (ttwR). However, we now realize that it should be "mirror", since otherwise the l e p t o n - t e c h n i l e p t o n transition via technigluon emission is possible, so that the lepton may acquire a mass of order ATC due to the technilepton condensation, which is disastrous.
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References [11 For a review, see: H. Terazawa, Phys. Rev. D22 (1979) 184; in Proc. 1981 INS Syrup. on Quark and lepton physics, eds. K; Fujikawa, U. Terazawa and A. Ukawa. [21 For a review, see: S. Brodsky and S. Drell, Phys. Rev. D22 (1980) 2236. [31 R. Casalbuoni and R. Gatto, Phys. Lett. 100B (1981) 135. [4] E.J. Squires, Phys. Lett. 102B (1981) 127; H. Harari, R.N. Mohapatra and N. Seiberg, preprint ref. TH-3123-CERN. I5] G. 't Hooft, Lectures Carg~se Summer Institute (1979). [61 J. PreskiU and S. Weinberg, Phys. Rev. D24 (1981) 1059. [71 H. Harari and N. Seiberg, Phys. Lett. 98B (1981) 269. [81 S.F. King, Phys. Lett. 107B (1981) 201. [91 S. Weinberg, Phys. Lett. 102B (1981) 401. [101 E. Guadagnini and K. Konishi, Nucl. Phys. B196 (1982) 165. [111 H. Harari and N. Seiberg, Phys. Lett. 102B (1981) 263. [121 S. Weinberg, Phys. Rev. D13 (1974) 974; D19 (1980) 2236; L. Susskind, Phys. Rev. D20 (1979) 2619. [131 See for a review, e.g., E. Fahri and L. Susskind, Phys. Rep. 74C (1981) 277. [14] P. Sikivie, Phys. Lett. 103B (1981) 437. [151 K. Matumoto and K. Kakazu, Prog. Theor. Phys. 65 (1981) 390. [161 H. Fritzsch, Phys. Lett. 70B (1977) 436;73B (1978) 317. [17l B. Holdom, Phys. Rev. D24 (1981) 1441. [181 K. Yamawaki and T. Yokota, in preparation.