Volume 159B, number 2,3
19 September 1985
THE G I M M E C H A N I S M F O R T E C l l N I C O L O R
Shen-Chang C H A O and Kenneth L A N E Department of Phvsws. Ohio State Untt'ersto'. Columbus. Ott 43210. USA
Received 17 May 1985
We present conditions for a nonahelian' generalization of the Glashow--lliopoulos-Maiani mechanism to theories of extended technicolor. When these conditions are met. flavor-changing neutral-current interactions of quarks are absent in the tree approximation. We illustrate our mechanism in a model of quarks and technifermions.
Extended technicolor (ETC) is the dynamical construct in which the electroweak interactions are spontaneously broken by a new strong force, technicolor, whose characteristic scale is O(1 TeV) [1,2] and in which quark and lepton masses are generated by chiral-synnnetry breaking gauge interactions mediated by heavy ETC bosons [3,4]. The constraints on building realistic ETC theories were set down in ref'. . The phenomenological desiderata underlying these constraints were to avoid (1) too much chiral symmetry, which results in very light pseudoGoldstone bosons coupling to fermions with strength ~ G l / 2 m f and (2) trivial quark masses and mixing angles. These resulted in the following condition on the ETC representations of fermions *t. All technifermions, quarks and leptons must belong to at most three irreducible representations of Get c ® SU(2)e w. One, called (O~ L, 2), contains the left-handed electroweak isodoublets, a second, called (O/R, I), contains (among other things) righthanded up-quarks, and the third, ( ~ R, 1) contains right-handed down quarks. "~R and c-DR must be inequivalent ETC representations to avoid identical up- and down-quark mass matrices. It was also stressed in ref.  that these constraints on ETC representations could be expected to lead to couplings between quarks and/or leptons that generate effective flavor-changing neutral-current (FCNC) interactions. It was found there that the only obviously disastrous FCNC interaction is ETC boson exchange resulting in IASI = 2 effects. Such an interaction is severely constrained by the K L - K S mass difference: to avoid conflict with experiment, an effective ETC mass ~>1000 TeV is required. This appears unnaturally large compared to the "typical" masses of 1 0 - I00 TeV of those ETC bosons which couple quarks to technifermions. That, in a nutshell, is the FCNC problem of technicolor, and little progress has been made on it since 1980. Clearly, some elegant symmetry mechanism, akin to that of Glashow, lliopoulos and Maiani  (GIM), is required for ETC. But this techni-GIM mechanism has been elusive. In 1981, one of us proposed a scenario for avoiding FCNC based on the general phenomenon of vacuum alignment in ETC theories [ 6 - 8 ] . In ref. , it was argued, but not proved, that if one took R = c)~ L (the so-called vectorial down sector), then all mixing angles occurred in the up-sector of the theory where one had only to deal with the much less-constrained problem of D 0 - D 0 mixing. There have been other attempts at a techni-GIM mechanism (see ref.  and references therein.) No one, however, has shown that a techni-GIM mechanism survives even in tree approximation when vacuum alignment is properly carried out; indeed, there are good reasons to expect that some proposals will fail on this account. In this letter we propose a techni-GIM mechanism which is a natural nonabelian generalization of the ordinary *t For simplicity, we assume here that the ETC group Getc commutes with SU(2)ew. It was also assumed in reL  that the same technicolor interaction is responsible for both quaxk and lepton masses. It follows that Get e cannot commute with U(l)e w nor with Gc = SU(3). 0370-2693/85/S 03.30 © Elsevier Science Publishers B.V. (North-llolland Physics Publishing Division)
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one. We formulate our mechanism as a set of five conditions on an ETC theory of quarks and technifermions. These conditions appear to be consistent with the constraints set forth in ref. . We then show how these conditions imply that, even after vacuum alignment has been performed, a generic FCNC interaction for quarks automatically becomes flavor-conserving in tree approximation. Finally, we illustrate our techni-GIM mechanism in a toy model of quarks and technifermions.
Conditions for techni-GIM 22 (1) The ETC representations c~ L, 9 / R and ~ R must be "quasi-equivalent", i.e., the ETC generator matrices of left-handed weak isodoublet quarks and the technifermions to which they couple, those of the right-handed up-quarks and associated technifermions, and of the right-handed down-quarks and their technifermions must be equal up to an overall (coupling) constant. (2) The FCNC interactions of all fermions will form a subgroup, Gfc, of Get c. Gfc must not be simple and the FCNC interactions of quarks must form a proper subgroup of Gfc. The quark-mass-generating ETC bosons then belong to the coset space Getc/(Gtc @ G c ® Gfc ). (3) All FCNC bosons that couple to quarks and belong to a single irreducible representation of Gfc must have the same mass in tree approximation. (4) Not all quark-mass-generating ETC bosons (those in Getc/(Gtc ® G c ® Gfc)) can have the flavor symmetry of the Gfc bosons that couple to quarks. (5) All quarks must transform according to a complex representation of Gfc. Condition (4) is necessary to obtain flavor-asymmetric quark mass matrices, without which the FCNC problem is trivial. We shall see below that conditions (3) and (4) are incompatible unless the FCNC interactions of quarks arise from a proper subgroup of Gfc. We now show that our conditions eliminate FCNC interactions of quarks in the tree approximation. To lowest order in ETC couplings, the FCNC interaction of quarks has the generic form ~ f c = ~i,J=L, R C~lJ, where 1
~ l J = "~ GfclR's (getc/gfc)abqlrTg(W~ t
Wl)rsql s ~tjtT~(W;tJbl~J)tuqju .
Here, t La and t Ra are the FCNC generators for left- and right-handed quarks; r, s, ... = 1, ..., 2N~ are quark flavor indices; and the quark fields refer to mass eigenstates. Ordinary color indices are suppressed in (la). WL and I4/R are 2Nq X 2Nq unitary matrices resulting from vacuum alignment; they reflect the chiral rotation from the "standard vacuum", with diagonal quark condensates, to the correct (i.e., lowest energy) chiral-perturbative vacuum, with diagonal quark mass matrices and condensates. Under the assumption in footnote 2, minimizing the vacuum energy determines only I¢ = 1~R I¢i~. The remaining ambiguity in WL and I4,'R is removed by diagonalizing the ETC-generated quark mass matrix via an SU(2Nq) v transformation, the vectorial symmetry of the chiral-perturbative vacuum . Now, conditions (1) and (3) imply that we trivially can do the group sum over Gfc indices. Since the Gfc representation of quarks is complex, this sum has the form 1" 2 ~ tla.Jb
/lafc)ab rs ttu = CIJ~ru ~ ts + DIJ6rs6 tu ' (2) where CI] and DIj (I, J = L, R) are flavor-independent constants. Inserting this into eq. (1), we see that FCNC interactions can occur only in ~t'LR + ~ R L " Now comes a miracle: By virtue of the quasi-equivalent nature of the ETC matrices, the effective hamiltonian 16] used in vacuum alignment is vectorial in each fermion sector when electric charge and ordinary color remain conserved ,3. It follows that I#L = I4,'R in each fermion sector minimizes the vacuum energy [7,8] and leads to positive definite quark and technifermion mass matrices. Thus, ~ L R + ,2 In addition to the assumption of footnote 1, we assume that all relevant quarks and technifermions are in complex representations of Gtc ® Go, so that all nonabelian chiral symmetries are of the SU(Nf) ® SU(Nf) type, breaking down in the "standard vacuum" to the diagonal, vectorial SU(Nf). See ref. . *3 This is true for almost all reasonable ETC boson mass matrices. 136
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RL is also flavor-conserving! We now demonstrate this miracle in a toy ETC model. A model with techni-GIM. The simplest model we have invented with techni-GIM satisfies all our conditions except that the quark flavor group is identical with Gfe. We shall indicate how to circunwent this problem later. As our main concern now is to demonstrate that I4'L = I4,'R in each fermion sector, we shall discuss but then ignore the consequences of not satisfying condition (2) in our toy model. Our model is based on Get c = SU(6)I ® SU(6)2 with fermions in the quasi-equivalent representations ~4 Q~L = (6, 1),
~ R = (6, 1),
(J9R = (1, 6).
We imagine that, at some scale of order 1 0 - 1 0 0 TeV, Get c is broken down to the diagonal SU(6)d = SU(6)I SU(6)2. We decompose this SU(6)d into SU(3)t c ® (SU(3) ® U(1))fc. Technicolor acts on the first three indices o f the 6 and flavor on the second three. Under this decomposition, 6 = (3, I, Y/2) ~ (1, 3, -Y/2) and the adjoint representation is 35 = (8, 1, 0) ¢9 (1,8, 0) * (3, 3, Y) * (3, 3, - Y ) * (1, 1, 0), where Y = 1/X/~. We then suppose that, at a possibly not much lower scale, SU(6)d is broken down to SU(3)t e. This model has 2 flavors of technifermions in two (3, 1)'s and 6 quark flavors in two (1, 3)'s *s. We label our fermion fields as follows: Let A, B .... = 1.... 6; a,/3 .... = 1,2, 3;a, b .... = 4, 5 , 6 ; i,] .... = U, D; and denote.a fermion in the 6 by F,~, where primed fields refer to the standard vacuum with condensates given by eq. (6) below. Then i
=~L, Rio~E(3, I,Y/2),
FL, RiA =qL, RiaE(I,3,--Y/2).
The tree-approximation propagator matrices of the ETC bosons which embody our techni-GIM conditions ~3) and (4) are most conveniently given in terms of the gauge fields V~[AB] = ( g 2 B / l u + g l W l u ) ' ( ~ / 2 ) A B / ( g ~ l +g~2) 1/2 ,
Au[AB] = ( g l W l u -g2W2u)'(k/2)AB/(g 2 +g~2) 1/2 •
Wi and gi are the gauge bosons and coupling constants of SU(6)i and ~,/2 represents the 6. Then, with (VxIAB 1 Vu[CD1)= --igxuAV[AB, CDI (k), the massive ETC propagators are Here,
(8, 1,0)A: AAic~t3,.r61 = ~(6c~66.y# -- ~6c~#6,~6Xk2 - / a 2 A ) - 1 I
Av,g[ab,cd] = ~(6ad6cb -- ~)ab6cdX k2 -.u2WV A )-1
Av,g[aa,b~] =½6ct#(( k2 --Id2V,A)-l)ab ,
(1, 1,0)V,A: AV,A[AB, CD] =
2 -1 l(SAo~6B -6Aa~BaX~C~D(3-~Cb~DbXk2 --/JBv,A ) .
Thus, we have taken Gfc (W) boson masses to be flavor-symmetric and the masses of ETC (X) bosons (3, 3, Y) (3, 3, - Y ) connecting q to ~b to be represented by hermitian matrices. In accord with our assumed order of Get c breakdown, we generally expect V-masses in (5) to be somewhat smaJler than A-masses. Actually, it is not possible to achieve in a natural way the pattern of W- and X-masses assumed in eq. (5), even in tree approximation *6. Rather, the global Gfc symmetry of W-masses together with the local Gtc symmetry implies that the coset X-bosons necessarily are degenerate also. Conversely, if the X-bosons are made nondegenerate, then so are the W's. The way out of this problem is stated in condition (2): enlarge Gfc to include separate flavor groups for quarks and technifermions. We discuss how this works at the end of the paper. Now we may turn to the problem of vacuum alignment and current-algebra mass generation in our model. We proceed as usual: We define the flavor symmetry group, Gf, of technifermions and quarks in the neglect of chiral-
*4 The anomalies of these and any other gauge interactions are assumed to be cancelled by other, undisplayed fermions. *s For simplicity, we have included no ordinary color SU(3) in this model, although we may do so trivially by replicating the representations (3) three times. Another way is to expand Get c to SU(9) ® SU(9), with fermions in the antisymmetric 36-dimensional representation. In any case, we sh',dl suppose that quarks condense and get a dynamical mass of order 300 MeV. *6 We thank V. Baluni for emphasizing this point to us. 137
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symmetry breaking ETC (and electroweak) interactions. We assume that Gf is spontaneously broken in the vacuum to a subgroup. We choose a standard vacuum, I~), whose invariance group is Sf and do all calculations in I~). Then the problem of vacuum alignment is to determine the chiral rotation matrices, WL and WR, in the chiralsymmetry breaking hamiltonian, ~ ( I ¢ ) , which minimize its expectation value E(I¢) = (~lc3t'(W)l~2). In the neglect of ETC and electroweak interactions, the chiral symmetry group of our model is Gf = (SU(2) ® SU(2))~ ® (SU(6) ® SU(6))q. We assume that Gf is broken in the standard vacuum to its vectorial subgroup Sf = SU(2)~ ® SU(6)q, with the nonzero condensates *s --r
([21qLiaqRjb[[2) = -~i/~abAc.
(~2[~Lic~Rj~l[2) = _~i/~c~Atc,
Here, Ate > 0 is of order A~c and A c is order A 3. Simple scaling suggests that Ate/A c = (F/f) 3, where F = 250 G e V , f = 95 MeV *6. These condensates break electroweak symmetry down to the usual electromagnetism [1,2[. The vacuum-aligning hanfiltonian is ~ (It,') = W- 1 ~ I4I, where ~ is the sum of the second-order current X current ETC interaction and the fourth-order electroweak interaction ,7. To show that ~ ( W ) is effectively vectorial, we note first that we need keep only those terms in it which are products of a left-handed current with a righthanded one. This is because vacuum alignment determines only those axial-vector transformations in Gf/Sf. Also, we need keep only those left-right products which will have expectation values which potentially are Gf/Sf noninvariant. The effective hamiltonian is now given by C'~eff(W) = I4/- 1 ~ e f f W , where, with flu = (gl/g2) 2 and I/D g~l g~2 d4k ~ ( e f f = 2(~11 + - - g 2 ) f (2--~)4 -,
f d4xe-ik'x id=U,D ~
× T [~ Li~uqLia(x)qR/b~'~R/~(O)] + h.c. + [(8,~8 8~a - ~8~a8.r~)/(/aGA -- k2)] -, , -, ., 1 2 1 ,7/(~A_k2)-I
X rlj T[~bLia~/~kLio(x)qlRj.r7 fiR/f(0)] + Z [(/aB v -- k2) - + --t
X T[~Lie~Tt~bLio~(X)~R/fjT~Rj~(O) + qLiaT~qLia(X)qRjb7 qRjb(0)] +(6ad6cb _ 36ab~cd)[(laWvl 2 _ k 2 ) - I + r//(/a2 A - k 2 ) -1 ] T[qLiaTaqLib(X)qRjcTa qRjd(0)] . ' }
Now we can see that the minimum of E(W) occurs when the mass-eigenstate (unprimed) fields are merely a vectorial Sf-rotation of the standard-vacuum fields: ~L, Ric~ = ~L, Ria '
qL, R/a =
(i = U, D ) .
Given the chiral structure of Gf, we know that there is a conserved, vectorial electric charge after vacuum alignment [7,8]. This implies that we can find the minimum of E(W) in the subspace of Gf in which the chiral rotation matrices are diagonal in up- and down-blocks. (Strictly speaking, this is true only up to an irrelevant (SU(2) ® U(1))ew rotation.) Thus, terms in eq. (7) which involve products of an up-current with a down-current contribute only Gf-invariant constants to E(W), and may be scratched out of ~(eff(W). Then, because the ETC representations are quasi-equivalent, c'Y(eff(W) is exactly left-right symmetric. Since we expect V-masses to be lighter than A-masses, the propagator'matrices in ~ e f f ( [ ¢ ) are positive definite (after the Wick rotation). Thus, C~(eff is truly vectorial and E(I¢) is minimized by a solution of the form (8) [7,8]. E(W) is completely independent of the S tmatrices Wi, but it is obvious that they are the unitary matrices that diagonalize precisely those hermitian integrals over the Av(aa, b#) + r~tAA(aa, b#) that contribute to the current-algebra masses of quarks. Indeed, an equivalent way of saying this is that ~ e f f generates positive definite current-algebra mass matrices, diagonalized by eq. (8), and this corresponds to the minimum of E(W) . And, as we showed earlier, no FCNC interactions occur in the second-order ETC hamiltonian. *'z We will not exhibit the electroweak part, but this omission does not alter our argument; see ref. . 138
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In this letter we have presented a natural generalization of the GIM mechanism to technicolor, illustrating how it works in tree approximation for a simple model of quarks and technifermions. We mention here several questions, which we shall treat more fully in forthcoming publications. (1) How can we implement conditions (3) and (4) naturally? We require that Gfc is the product Gtf c ® Gqf c o f technifermion and quark flavor groups. The Gqfc-bosons ar to be degenerate in tree approximation, but the Gtfc-bosons are not. It then should be possible to find Higgs scalars (or dynamical condensates) which give Gtf c ® Gqfc-noninvariant masses to the coset bosons connecting quarks to technifermions, but which cannot contribute to the Gtfc-invariant mass term of the Gqfc-bosons. As a simple example, suppose that both Gtf c and Gqf c are SU(3) groups. Then, a mass term transforming like (8, 8) under Gtf c ® Gqf e breaks the Gqf c symmetry o f the coset bosons, but does not affect the Gqfc-boson masses *s (2) How large are the FCNC effects in higher orders of the ETC interaction? For box graphs and W-propagator 2 2 times a power of A/a2X//a2 ; here, A/a2 meaand vertex corrections, we expect these to be at most of order %tc//aW sures the amount o f G q f c breaking in the appropriate X-boson masses. This may well be as small as (1/1000 TeV) 2 without doing violence to our notion of a "natural" hierarchy of ETC boson masses. (3) Can we find an ETC model with techni-GIM which yields a realistic pattern of quark masses and mixing angles? If so, we can go on to determine the technifermion mass matrices and, from there, the masses of pseudoGoldstone technipions and their couplings to ordinary matter. Note that these couplings are needed to estimate any FCNC effects due to technipion exchange. (4) Do models - f the type we have described, extended to include ordinary color, have a strong-CP problem ? If not, are there unwanted axions ? This question can only be investigated by the method of vacuum alignment. (5) Finally, how do we include leptons in a model with our techni-GIM conditions? If there are only lefthanded neutrinos, the ETC representations r-~ L, cff R and c/) R necessarily are asymmetric. If quarks and leptons get their mass from technifermions which transform according to the s a m e technicolor group, these three representations necessarily are irreducible . Furthermore, ordinary SU(3) color now must be incorporated nontrivially into Get c. It will be challenging to generalize our conditions to this situation. Another possibility is to include right-handed neutrinos, but to render them harmless somehow (e.g., massless and non-interacting, or else, heavy and ustable). We do not know yet which is the more interesting path, but we are confident that an elegant solution to this problem, and the others, exists. Once leptons are included, of course, we can attack the problem of FCNC involving leptons, such as K L ~/ae,/a ~ e~' and/a ~ eee [3,6]. We have benefitted from conversations with V. Baluni, H. Georgi, L. McLerran and M.J.G. Veltman. K.L. thanks the high-energy theory groups at the University of Michigan and Harvard University for profitable visits during the final stages of this work. This research was supported in part by the US Department of Energy under Contract No. EY-76-C-02-1545. *8 We are aware that this scenario requires either a strongly interacting U(l)t c or quark-mass generation in two-loop order.  [21     [71   [ t0]
S. Weinberg, Phys. Rev. D13 (1976) 974; Phys. Rev. D19 (1979) 1277. L. Susskind, Phys. Rev. D20 (1979) 2610. E. Eichten and K. Lane, Phys. Lett. 90B (1980) 125. S. Dimopoulos and L. Susskind, Nucl. Phys. B155 (1979) 237. S.L. Glashow, J. lliopoulos and L. Maiani, Phys. Rev. D2 (1970) 1285. K. Lane, Phys. Scripta 23 (1981) 1005; E. Eichten, K. Lane and J. Preskill, Phys. Rev. Lett. 45 (1980) 225. J. PreskiU, NucL Phys. B177 (1981) 21. M. Peskin, Nuci. Phys. B175 (1980) 197. S. Dimopoulos, H. Georgi and S. Raby, Phys. Lett. 127B (1983) 101. R. Peccei and H. Quinn, Phys. Rev. Lett. 38 (1977) 1440; S. Weinberg, Phys. Rev. Lett. 40 (1978) 223; F. Wilezek, Phys. Rev. Lett. 40 (1978) 279. 139