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Flavor changing suppression in technicolor
Volume 143B, number 1, 2, 3
PHYSICS LETTERS
9 August 1984
FLAVOR C H A N G I N G S U P P R E S S I O N IN T E C H N I C O L O R Bob H O L D O M 1 Institute for Theoretical Physics, Department of Physics, Stanford University, Stanford, CA 94305, USA Received 20 December 1983 Revised manuscript received 4 May 1984
A S = 2 effects are suppressed by a factor (~tw/2'n')202 while allowing for Cabibbo mixing. A model realizing this mechanism has no severe flavor changing problems./~ ~ e't is suppressed while K --, e+/.t - is not. Charged pseudo-Goldstone masses are first order rather than second order in the weak coupling. And there are two stable pseudo-Goldstone bosons.
Technicolor theories [1] are based on the assumption that elementary scalar fields are not necessary to break gauge symmetries. Elementary scalar fields are removed and with them the arbitrary parameters in the form of Yukawa couplings. Fermion masses are related instead to dynamical mass scales arising from strong gauge interactions. In particular, sideways interactions [2] with a mass scale above the weak scale are required to induce ordinary fermion masses from the dynamical mass of technifermions. But sideways interactions also typically cause transitions between different families of ordinary fermions. This leads directly to the problem of flavor changing neutral currents [2,3], which appears not to have a simple resolution
[4,5]. A sideways induced term of the form QQF:Iq where Q is a techniquark will generate a suitable down quark mass if the coefficient of this term is roughly 10 -3 TeV -2. But ~qTqq terms are expected with similar coefficients, and among these terms are AS = 2 terms. To be consistent with K K mixing the coefficients of such terms must be less than 10 - 6 TeV -2 [3]. And if the A S = 2 terms carry phases then the number is 10 - 9 TeV -2 in order to be consistent with the C P violating part of K K i Work supported by the National Science Foundation under Grant PHY-78-26847.
mixing. Clearly a large suppression of the AS = 2 terms is required and this is by far the most serious of the flavor changing problems in technicolor theories [3]. A mechanism will be proposed which suppresses A S = 2 effects in particular. It will be described briefly and then incorporated in an explicit model. At this point all flavor changing effects can be considered and it appears that they pose no major challenge to the model. The pseudo-Goldstone bosons (PGBs) are discussed in this context; they also provide the best way of experimentally identifying the model. The model illustrates how the charged PGBs can receive mass in the first order of weak interactions instead of the experimentally troublesome second order mass [2]. In addition, two PGBs are stable. The model also predicts a stable proton. The initial idea is to place the (~, d) quarks and the (c, s) quarks in separate sideways representations. Then the theory has an exact vectorial symmetry which assigns the quantum numbers/~(+), d ( + ) , c ( - ) , s ( - ) . But this would prohibit mass mixing of the form tic or ds, as well as AS = 2 terms. Since it will be assumed that the sideways gauge group commutes with the weak gauge group, the above mass eigenstates are also weak eigenstates. The result is no Cabibbo mixing. It must therefore be the case that this vectorial
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symmetry is spontaneously broken. Now consider the possibility that another symmetry is not spontaneously broken which assigns the quantum numbers /~(0), d ( + ) , c(0), s ( - ) . This symmetry can only be approximate; it is explicitly broken by the weak interactions. But it has the desirable feature of allowing ~c mass mixing while at the same time prohibiting AS = 2 terms in the limit a w ~ 0. In this context it is useful to compare AS = 2 effects with AC = 2 effects. Both effects are induced from sideways generated AS = 0 and AC = 0 four quark terms in the presence of ds and ~c mass mixing effects, respectively. For example, a term of the form (~D,~CL)2 where c is a sideways 2 eigenstate implies a AC = 2 term, (CL't~ttL) when expressed in terms of mass eigenstates. The latter term is suppressed by (0L) 2 where 0uLR is the mixing angle between the sideways and mass eigenstates of #L.R and CL.R. The essential point is that AC = 2 and AS = 2 effects are proportional to the square of mixing angles. Due to the previous approximate symmetry almost all of the mass mixing responsible for Cabibbo mixing has to occur in the (/~, c) sector; thus 0~L is nearly the Cabibbo angle, 0c. Conversely odL'R~ 0 as a w ---, 0; it will turn out that OdL'R is induced with an extra weak loop giving 0L = (Ctw/2~r)Oc. The corresponding AS = 2 term is suppressed by (ODE)2 -- 1 0 - 6 . This is a welcome suppression since the AS = 2 terms may well carry phases and contribute to the C P violating part of K K mixing. A complementary problem in this type of picture is the issue of Goldstone bosons (GBs). A spontaneously broken vector symmetry has already been mentioned. But the model will be such that this potential GB is absorbed by a gauge boson, as are all other exact GBs. Thus the proposed mechanism for suppressing AS = 2 does not automatically imply GBs. The model is constructed without the aid of an arbitrary scalar sector. Instead, strong non-QCDlike gauge interactions are simply assumed to give the desired symmetry breakings. But despite this new arbitrariness, the previous remarks and other phenomenological constraints greatly restrict the structure of the theory. And this structure leads to predictions. This is especially true in the technicol228
9 August 1984
or sector which yields an easily testable PGB spectrum. The underlying gauge group and fermion content is the following. SUs(4 ) X SUp(n) x SUE(2 ) X SUR(2 ) x UA(1), ~IL ~
(4, 4, 2, 1, - 1 ) L ,
+ I R ~-~ (4,
4, 1, 2, + 1)a,
Ip2 L ~- ( 4 ,
4, 2, 1, + 1 ) L ,
~bZR-- (4, 4, 1, 2, --1)a.
(1)
The sideways SUs(4 ) is a family symmetry while leptons are the fourth color under the P a t i - S a l a m SUp(4). The (4, 2, 1)L, (4, 1, 2)R structure is one P a t i - S a l a m family; eight of these have been placed in a 4 + 4 under SUs(4 ). The B-functions for SUs(4 ) and SUp(4) are identical, making it possible to set the couplings equal. The same is done for SUL(2 ) and SUR(2 ). The SU(4)'s are asymptotically free and at the sideways scale, A s , they are assumed to have grown sufficiently strong to initiate a breakdown of gauge symmetries. The unbroken gauge symmetries just above A TC ----300 GeV must be SUTc(2) X SUe(3 ) X SUL(2 ) X U r ( 1 ). The 4 + 4 under SUs(4 ) becomes 2(2) + 4(1) under S U T c ( 2 ) ; this gives four ordinary families and two technifamilies. Majorana masses for the righthanded neutrinos and technineutrinos serve as order parameters for this symmetry breaking. This ensures SUp(4) x SUR(2 ) --* SUc(3 ) X Ur(1 ) where Uy(1) is the standard hypercharge. This is convenient since p a is the one member of a Pati-Salam family not observed in nature. The breaking of UA(1 ) above ATC will be discussed below. The precise pattern of gauge symmetry breakdown between ATC and A s is more relevant for fermion masses than for the present discussion. But a few remarks will be made. The various neutrino condensates are ~ P l R P l R ) ' ~ P l R P 2 R ) ' ~P2RP2R) '
(CabN,aRN2R), b
(2)
where a, b are SUTc(2 ) indices and the other subscripts refer to those in (1). (All CP violation originates in these condensates; this is relevant
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when trying to find contributions to the neutron electric dipole moment for example.) Note that these condensates transform as (4 × 4)s or 4 × under SUs(4 ) and as (4 X 4)s under SUp(4). Different relative magnitudes give different breaking patterns. For example, it is useful to assume that an SUI(3 ) subgroup of SUs(4 ) remains unbroken down to some intermediate scale Al, A x c < A I < A s. This gives two heavy families (cx A~-2) and two light families (cx As2). Finally SUTc(2 ), the unbroken subgroup of SUI(3 ), eventually grows strong enough at A wc to produce the following condensates and break SUL(2 ) × Ur(1 ) to UEM(1 ).
(U,u,), (U~u~), (%bNILN2L), ° ( D , D , ) , (E)2D2), ( E , E , ) , (F.2E2).
(3)
Note that it must be assumed that the SUTc(2 ) coupling is on the strong coupling side of a nontrivial ultraviolet fixed point [5]. There are various effects due to the physics above ATC which cannot be fully discussed here. For example, effects can be identified which cause mass splittings within each pair of families and between quarks and leptons. And the nonrenormalizable part of the theory at ATC can influence the vacuum alignment and relative magnitudes of the condensates in (3). This is important for an acceptable W, Z mass ratio, which is only ensured if technilepton condensates are smaller than techniquark condensates. The former do not have a global vector SU(2) symmetry due to the absence of N R at ATe. (On the other hand, the absence of N R will be responsible for increasing the charged PGB masses.) Returning to the main line of argument, generators of various phase rotations of the four representations are identified in table 1. QA generates the gauged UA(1). Qv is the exact global vector symmetry alluded to before. The fermion number
LETTERS
generator QF is exact up to SUL(2) XSUR(2) anomalies. Qw is an approximate global symmetry broken explicitly by SUE(2) X SUR(2); the ~3's act in the doublet space of each representation. Qv + Qw is a generator with the required properties to suppress AS = 2, as described before. It is this combination of Qv and Qw which is not spontaneously broken. For the lightest two families it assigns the quantum numbers /z(0), d ( + ) , re(0), e ( + ) , c(0), s ( - ) , v,(0),/~(-). Since Qv + Qw is only broken weakly any ds mass mixing must involve weak effects. The lowest order diagram for dLSa in fig. 1 has a W R boson (mass - As); this loop gives a aw/2~r factor. A similar diagram with W R removed contributes to #LCR mass mixing and L this must be 8~c - 8c times the CLCRmass diagram. (And incidentally, if 8uc. ) The AS = 2 suppression factor of (osL) 2 where OL = (aw/2~r)0 c now follows. From the quantum numbers of Qv + Qw for and e it is seen that the process /~--* ey is also suppressed. Typically in technicolor theories this process proceeds without weak effects with an amplitude an order of magnitude or two too large [3]. The dominant diagram (fig. 2) now requires an extra SUR(2 ) loop which suppresses the amplitude by at least aw/2~'. On the other hand, the process K---, e+/~ - is not suppressed by this mechanism. For example, the operator d~,,e~y"s is induced directly via an SUp(4) boson exchange, and the resulting lower bound on the mass of the gauge boson is = 300 TeV [6]. DD and BB mixing also does not involve a w suppression. The contributions here depend on the broken sideways generators and it is easy to check that the most important contributions come from
su s (4) sup(4)
Table 1
~ I L ~ (4, 4, 2, 1)L ~ m - = (4, 4, 1, 2 ) R
9 August 1984
Q^
Qv
Qv
Qw
-+
+ +
+ +
-- ~'3 -~'3
~/'2L ~- (4, 4, 2, 1)L
+
+
--
+ "r3
tk2R-= (4, 4, 1, 2 ) R
-
+
-
+r 3
sR
/
x
x
_
dL
(D2L D2R >
F i g . 1. s, d m a s s m i x i n g .
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~R
)(
Fig. 2./~ --*ey. diagonal generators. For example, a linear combination of the (1, 1, 1, - 3 ) generators of SUs(4 ) and SUp(4) can remain unbroken down to some scale Asp, A~ < Asp < A s. The resulting contribution to D D mixing is only suppressed by 02 C, implying that Asp >_ 30 TeV [3]. Another diagonal generator lies in SU~(3); its mass cannot be greater than A~. It couples only to the heavy families, but it does produce the largest contribution to BB mixing. The coefficient of the resulting AB = 2 operator is 02d/A 2. O b d is not a w suppressed but OEd _< m d / m b due to a nearly symmetrical mass matrix in the down sector. The conclusion is that BB mixing ~< D D mixing. It has been mentioned that there are no exact GBs in the theory. To see this, note that the number of GBs not absorbed by gauge bosons is given by the total number of broken generators (global or gauge or linear combinations) minus the number of broken gauge generators. This is the same as the number of original global symmetries minus the number of unbroken global symmetries after all spontaneous symmetry breaking. The original global symmetries are generated by Q v a n d QF" L e t Qp a n d Qs be the ( 1 / 3 , 1/3, 1/3, - 1 ) and (1, 1, - 1 , - 1 ) diagonal generators of SUp(4) and SUs(4 ) respectively. Then it can be checked that the unbroken symmetries of the condensates in (2) and (3) are QF + QP and Qv + Qs. OF + QP is proportional to quark plus techniquark number, broken only to the extent that QF is SUE(2 ) × SUR(2 ) anomalous. This implies a stable proton. Qv + Qs assigns opposite quantum numbers to the two different technifamilies and zero quantum number to ordinary fermions. The number of unbroken global symmetries and original global symmetries is the same and thus there is no exact GB. 230
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Note that the combinations QF -- QP and Qv Qs are spontaneously broken, but the potential GBs are absorbed by the Qp and Qs gauge bosons. Thus there is no GB associated with the vectorial symmetry Qv, which in the combination Qv + Qw was responsible for suppressing AS = 2. Having a suitable Qs for this to happen can be traced to the appearance of both 4 and 4's under SUs(4 ) in (1). It still remains to discuss the breaking of UA(1). The neutrino condensates are not sufficient since they leave unbroken the gauged combination QA + Qs; the associated gauge boson would give too much D D mixing unless sufficiently massive. But the neutrino condensates are the only bilinears allowed by the unbroken gauge symmetries above ATc. Therefore, a suitable breaking of QA above ATC can only o~cur via a four-fermion condensate. It will be assumed that strong SUI(3)× SUc(3 ) forces are responsible at some scale AAS, A~ < AAS < A s- The suitable condensate invariant under all global and_gauge symmetries at AAS except UA(1 ) --p p p p is ( ~ I L ~ I R ~ 2 R ~ 2 L ) where the primes indicate fields nontrivial under S U I ( 3 ) X S U c ( 3 ). By the same argument as before, AAS >~ 30 TeV. It is of interest that the fl-functions for SUI(3 ) × SUc(3 ) are extremely flat; perturbatively fl(g) = - g 3 ( 3 3 - 32)/24~r2 + . . . . And it can be assumed that the SUc(3 ) fl-function reverses sign between A TC and A~ due to strongly interacting techniquarks. This can give a weak SUc(3 ) at ATC. The last topic concerns the PGBs arising from the spontaneous breaking of numerous approximate symmetries at ATC. The PGB spectrum, which is below the weak scale, provides the best experimental probe of the structure of the theory above the weak scale. The relevant features of the present structure include the existence of two technifamilies and the absence of the NR'S at ATC. The most interesting departure from standard predictions comes for the mass of the charged PGBs. The pair corresponding to the standard charged pair now have first order weak masses rather than second order masses. This follows from the fact that the corresponding currents contain a term [,Ly~ ±LL, L = (N, E). Due to the absence of NR, the current cannot be written as a sum of a current which is a symmetry of SUE(2) and a current which is not spontaneously broken. The
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result is a mass as high as 50-70 GeV, rather than the 5-10 GeV mass which has come into conflict with experiment. To simplify the following discussion it is assumed that the condensates in (3) are equal. Then there is a total of 27 color singlet PGBs, 11 neutral and 16 charged. 8 of the latter are technileptonic. All charged PGBs have first order weak masses or masses = A~c/AAsThe PGBs which carry net Qv + Qs charge are rather novel. Remember that Qv + Qs is an unbroken global symmetry under which all ordinary fermions are neutral. Thus there are no PGB-fermion-fermion couplings for these PGBs. The neutral PGBs of this type couple to the currents U~,~ys~"+U, D3,~'/5~ ±D, and F.~,.3,5~ +E where the z ' s act in the flavor space of technifermions. And the lightest pair is stable! They couple to ET~Vs~'±E, have a mass A ~ c / A s p , and carry no SUE(2) >(Uy(1) charge. Their possible role as dark matter in the universe will be discussed elsewhere. The tightest two neutral PGBs' couple to the currents D'/~'fsr0D-3Ey~Ys~0 E and E),~75~3E. Their mass is -.~ A ~ c / A s. Neither couple to charge 2 / 3 quarks. Flavor nondiagonal couplings to the charge - 1 / 3 quarks are suppressed for the following reasons. (1) The couplings to first order in quark masses and the quark mass matrix itself are simultaneously diagonalizable in the limit that EEclq terms can be neglected. In other words these terms violate " m o n o p h a g y " [3], but they require SUp(4) exchange. Thus monophagy violation becomes more negligible for heavier quarks. (2) The ~d coupling is suppressed by the symmetry argument discussed. It must involve a factor aw/2~. (3) The b d coupling to second order in the b mass must be checked because of (1). But in the limit that all interactions in the charge - 1 / 3 sector are vectorial, these contributions are also simultaneously diagonalizable with the mass matrix. Thus they also involve weak effects. Two other PGBs couple to U3,~3,5~3U and E)3,~,s~-3D with mass = A~rc/AAs. Their flavor nondiagonal couplings are larger and reflect the breakdown of monophagy due to the fact that each quark can potentially receive mass from two
9 August 1984
flavors of techniquarks. But argument (2) still holds and their larger mass further suppresses their flavor changing effects. The conclusion is that flavor changing effects induced by PGB exchange are likely smaller than the sideways induced effects. The remaining neutral PGB is the axion. A Peccei-Quinn symmetry arises in this model in the same way as described elsewhere (e.g. ref. [7]). Basically, it is another result of the large N R mass. Described also [7] was the way by which strong SUc(3 ) above Axc could yield a heavy axion. (It turns out that four-fermion condensates above ATc are also interesting in this connection.) Unfortunately, this mechanism yields a very uncertain mass for the axion. And then there is the issue of technicolor induced anomalous scaling. It has been proposed [5] that enough anomalous scaling could cure the flavor changing problem by itself. Although anomalous scaling is now not the important factor in the present model, it would be difficult to argue that none occurs at all. This adds some uncertainty to various mass scales. Modifications of the model can also be considered. For example, if the UA(1 ) gauge field was removed the theory would have an exact GB. This could only be acceptable if a four-fermion condensate broke QA at a sufficiently high scale to render the GB invisible. The other possibility to consider is maximal gauging, where Qv is gauged as well as QA. Then the theory would have a massless gauge boson coupling to Qv + Qs- The new photon would only have renormalizable couplings to technifermions and the induced nonrenormalizable couplings to ordinary fermions do not appear disastrous. In summary it has been proposed that AS = 2 effects in technicolor theories are suppressed via an approximate symmetry, broken only by weak effects. This occurs in a model having a suitable exact vectorial symmetry spontaneously broken by right-handed neutrino condensates. This does not give a Goldstone boson and it does give Cabibbo mixing. The basic prediction is that if CP violation in K K mixing is a AS = 2 superweak effect induced by sideways physics then the AC = 2 terms should be (2~r/aw) 2 larger. This would give D D mixing at an experimentally accessible level. 231
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I a m g r a t e f u l t o S. D i m o p o u l o s , M . P e s k i n , J. P r e s k i l l , a n d L. S u s s k i n d f o r c r u c i a l d i s c u s s i o n s .
References [1] L. Susskind, Phys. Rev. D20 (1979) 2619; S. Weinberg, Phys. Rev. D13 (1976) 974; D19 (1978) 1277 [2] E. Eichten and K. Lane, Phys. Lett. 90B (1980) 1251; S. Dimopoulos and L. Susskind, Nucl. Phys. B155 (1979) 237. [3] S. Dimopoulos and J. Ellis, Nucl. Phys. B182 (1981) 505. [4] J. Ellis and P. Sikivie, Phys. Lett. 104B (1981) 141;
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K. Lane, Phys. Scr. 23 (1981) 1005; A. Masiero, E. Papantonopoulos and F. Yanagida¢ Phys. Lett. l15B (1982) 229; A. Buras, S. Dawson and A. Schellekens, Phys. Rev. D27 (1983) 1171; W. Konetschny, Lett. Nuovo Cimento 37 (1983) 221; S. Dimopoulos, H. Georgi and S. Raby, Phys. Lett. 127B (1983) 101. [51 B. Holdom, Phys. Rev. D24 (1981) 1441. [61 S. Dimopoulos, G. Kane and S. Raby, Nucl. Phys. B182 (1981) 77. [7] B. Holdom and M. Peskin, Nuci. Phys. B208 (1982) 397.
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9 August 1984
FLAVOR C H A N G I N G S U P P R E S S I O N IN T E C H N I C O L O R Bob H O L D O M 1 Institute for Theoretical Physics, Department of Physics, Stanford University, Stanford, CA 94305, USA Received 20 December 1983 Revised manuscript received 4 May 1984
A S = 2 effects are suppressed by a factor (~tw/2'n')202 while allowing for Cabibbo mixing. A model realizing this mechanism has no severe flavor changing problems./~ ~ e't is suppressed while K --, e+/.t - is not. Charged pseudo-Goldstone masses are first order rather than second order in the weak coupling. And there are two stable pseudo-Goldstone bosons.
Technicolor theories [1] are based on the assumption that elementary scalar fields are not necessary to break gauge symmetries. Elementary scalar fields are removed and with them the arbitrary parameters in the form of Yukawa couplings. Fermion masses are related instead to dynamical mass scales arising from strong gauge interactions. In particular, sideways interactions [2] with a mass scale above the weak scale are required to induce ordinary fermion masses from the dynamical mass of technifermions. But sideways interactions also typically cause transitions between different families of ordinary fermions. This leads directly to the problem of flavor changing neutral currents [2,3], which appears not to have a simple resolution
[4,5]. A sideways induced term of the form QQF:Iq where Q is a techniquark will generate a suitable down quark mass if the coefficient of this term is roughly 10 -3 TeV -2. But ~qTqq terms are expected with similar coefficients, and among these terms are AS = 2 terms. To be consistent with K K mixing the coefficients of such terms must be less than 10 - 6 TeV -2 [3]. And if the A S = 2 terms carry phases then the number is 10 - 9 TeV -2 in order to be consistent with the C P violating part of K K i Work supported by the National Science Foundation under Grant PHY-78-26847.
mixing. Clearly a large suppression of the AS = 2 terms is required and this is by far the most serious of the flavor changing problems in technicolor theories [3]. A mechanism will be proposed which suppresses A S = 2 effects in particular. It will be described briefly and then incorporated in an explicit model. At this point all flavor changing effects can be considered and it appears that they pose no major challenge to the model. The pseudo-Goldstone bosons (PGBs) are discussed in this context; they also provide the best way of experimentally identifying the model. The model illustrates how the charged PGBs can receive mass in the first order of weak interactions instead of the experimentally troublesome second order mass [2]. In addition, two PGBs are stable. The model also predicts a stable proton. The initial idea is to place the (~, d) quarks and the (c, s) quarks in separate sideways representations. Then the theory has an exact vectorial symmetry which assigns the quantum numbers/~(+), d ( + ) , c ( - ) , s ( - ) . But this would prohibit mass mixing of the form tic or ds, as well as AS = 2 terms. Since it will be assumed that the sideways gauge group commutes with the weak gauge group, the above mass eigenstates are also weak eigenstates. The result is no Cabibbo mixing. It must therefore be the case that this vectorial
0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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symmetry is spontaneously broken. Now consider the possibility that another symmetry is not spontaneously broken which assigns the quantum numbers /~(0), d ( + ) , c(0), s ( - ) . This symmetry can only be approximate; it is explicitly broken by the weak interactions. But it has the desirable feature of allowing ~c mass mixing while at the same time prohibiting AS = 2 terms in the limit a w ~ 0. In this context it is useful to compare AS = 2 effects with AC = 2 effects. Both effects are induced from sideways generated AS = 0 and AC = 0 four quark terms in the presence of ds and ~c mass mixing effects, respectively. For example, a term of the form (~D,~CL)2 where c is a sideways 2 eigenstate implies a AC = 2 term, (CL't~ttL) when expressed in terms of mass eigenstates. The latter term is suppressed by (0L) 2 where 0uLR is the mixing angle between the sideways and mass eigenstates of #L.R and CL.R. The essential point is that AC = 2 and AS = 2 effects are proportional to the square of mixing angles. Due to the previous approximate symmetry almost all of the mass mixing responsible for Cabibbo mixing has to occur in the (/~, c) sector; thus 0~L is nearly the Cabibbo angle, 0c. Conversely odL'R~ 0 as a w ---, 0; it will turn out that OdL'R is induced with an extra weak loop giving 0L = (Ctw/2~r)Oc. The corresponding AS = 2 term is suppressed by (ODE)2 -- 1 0 - 6 . This is a welcome suppression since the AS = 2 terms may well carry phases and contribute to the C P violating part of K K mixing. A complementary problem in this type of picture is the issue of Goldstone bosons (GBs). A spontaneously broken vector symmetry has already been mentioned. But the model will be such that this potential GB is absorbed by a gauge boson, as are all other exact GBs. Thus the proposed mechanism for suppressing AS = 2 does not automatically imply GBs. The model is constructed without the aid of an arbitrary scalar sector. Instead, strong non-QCDlike gauge interactions are simply assumed to give the desired symmetry breakings. But despite this new arbitrariness, the previous remarks and other phenomenological constraints greatly restrict the structure of the theory. And this structure leads to predictions. This is especially true in the technicol228
9 August 1984
or sector which yields an easily testable PGB spectrum. The underlying gauge group and fermion content is the following. SUs(4 ) X SUp(n) x SUE(2 ) X SUR(2 ) x UA(1), ~IL ~
(4, 4, 2, 1, - 1 ) L ,
+ I R ~-~ (4,
4, 1, 2, + 1)a,
Ip2 L ~- ( 4 ,
4, 2, 1, + 1 ) L ,
~bZR-- (4, 4, 1, 2, --1)a.
(1)
The sideways SUs(4 ) is a family symmetry while leptons are the fourth color under the P a t i - S a l a m SUp(4). The (4, 2, 1)L, (4, 1, 2)R structure is one P a t i - S a l a m family; eight of these have been placed in a 4 + 4 under SUs(4 ). The B-functions for SUs(4 ) and SUp(4) are identical, making it possible to set the couplings equal. The same is done for SUL(2 ) and SUR(2 ). The SU(4)'s are asymptotically free and at the sideways scale, A s , they are assumed to have grown sufficiently strong to initiate a breakdown of gauge symmetries. The unbroken gauge symmetries just above A TC ----300 GeV must be SUTc(2) X SUe(3 ) X SUL(2 ) X U r ( 1 ). The 4 + 4 under SUs(4 ) becomes 2(2) + 4(1) under S U T c ( 2 ) ; this gives four ordinary families and two technifamilies. Majorana masses for the righthanded neutrinos and technineutrinos serve as order parameters for this symmetry breaking. This ensures SUp(4) x SUR(2 ) --* SUc(3 ) X Ur(1 ) where Uy(1) is the standard hypercharge. This is convenient since p a is the one member of a Pati-Salam family not observed in nature. The breaking of UA(1 ) above ATC will be discussed below. The precise pattern of gauge symmetry breakdown between ATC and A s is more relevant for fermion masses than for the present discussion. But a few remarks will be made. The various neutrino condensates are ~ P l R P l R ) ' ~ P l R P 2 R ) ' ~P2RP2R) '
(CabN,aRN2R), b
(2)
where a, b are SUTc(2 ) indices and the other subscripts refer to those in (1). (All CP violation originates in these condensates; this is relevant
Volume 143B, number
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PHYSICS
when trying to find contributions to the neutron electric dipole moment for example.) Note that these condensates transform as (4 × 4)s or 4 × under SUs(4 ) and as (4 X 4)s under SUp(4). Different relative magnitudes give different breaking patterns. For example, it is useful to assume that an SUI(3 ) subgroup of SUs(4 ) remains unbroken down to some intermediate scale Al, A x c < A I < A s. This gives two heavy families (cx A~-2) and two light families (cx As2). Finally SUTc(2 ), the unbroken subgroup of SUI(3 ), eventually grows strong enough at A wc to produce the following condensates and break SUL(2 ) × Ur(1 ) to UEM(1 ).
(U,u,), (U~u~), (%bNILN2L), ° ( D , D , ) , (E)2D2), ( E , E , ) , (F.2E2).
(3)
Note that it must be assumed that the SUTc(2 ) coupling is on the strong coupling side of a nontrivial ultraviolet fixed point [5]. There are various effects due to the physics above ATC which cannot be fully discussed here. For example, effects can be identified which cause mass splittings within each pair of families and between quarks and leptons. And the nonrenormalizable part of the theory at ATC can influence the vacuum alignment and relative magnitudes of the condensates in (3). This is important for an acceptable W, Z mass ratio, which is only ensured if technilepton condensates are smaller than techniquark condensates. The former do not have a global vector SU(2) symmetry due to the absence of N R at ATe. (On the other hand, the absence of N R will be responsible for increasing the charged PGB masses.) Returning to the main line of argument, generators of various phase rotations of the four representations are identified in table 1. QA generates the gauged UA(1). Qv is the exact global vector symmetry alluded to before. The fermion number
LETTERS
generator QF is exact up to SUL(2) XSUR(2) anomalies. Qw is an approximate global symmetry broken explicitly by SUE(2) X SUR(2); the ~3's act in the doublet space of each representation. Qv + Qw is a generator with the required properties to suppress AS = 2, as described before. It is this combination of Qv and Qw which is not spontaneously broken. For the lightest two families it assigns the quantum numbers /z(0), d ( + ) , re(0), e ( + ) , c(0), s ( - ) , v,(0),/~(-). Since Qv + Qw is only broken weakly any ds mass mixing must involve weak effects. The lowest order diagram for dLSa in fig. 1 has a W R boson (mass - As); this loop gives a aw/2~r factor. A similar diagram with W R removed contributes to #LCR mass mixing and L this must be 8~c - 8c times the CLCRmass diagram. (And incidentally, if
su s (4) sup(4)
Table 1
~ I L ~ (4, 4, 2, 1)L ~ m - = (4, 4, 1, 2 ) R
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Q^
Qv
Qv
Qw
-+
+ +
+ +
-- ~'3 -~'3
~/'2L ~- (4, 4, 2, 1)L
+
+
--
+ "r3
tk2R-= (4, 4, 1, 2 ) R
-
+
-
+r 3
sR
/
x
x
_
dL
(D2L D2R >
F i g . 1. s, d m a s s m i x i n g .
229
Volume 143B, number 1, 2, 3
PHYSICS LETTERS SUs(~,)
~R
)(
Fig. 2./~ --*ey. diagonal generators. For example, a linear combination of the (1, 1, 1, - 3 ) generators of SUs(4 ) and SUp(4) can remain unbroken down to some scale Asp, A~ < Asp < A s. The resulting contribution to D D mixing is only suppressed by 02 C, implying that Asp >_ 30 TeV [3]. Another diagonal generator lies in SU~(3); its mass cannot be greater than A~. It couples only to the heavy families, but it does produce the largest contribution to BB mixing. The coefficient of the resulting AB = 2 operator is 02d/A 2. O b d is not a w suppressed but OEd _< m d / m b due to a nearly symmetrical mass matrix in the down sector. The conclusion is that BB mixing ~< D D mixing. It has been mentioned that there are no exact GBs in the theory. To see this, note that the number of GBs not absorbed by gauge bosons is given by the total number of broken generators (global or gauge or linear combinations) minus the number of broken gauge generators. This is the same as the number of original global symmetries minus the number of unbroken global symmetries after all spontaneous symmetry breaking. The original global symmetries are generated by Q v a n d QF" L e t Qp a n d Qs be the ( 1 / 3 , 1/3, 1/3, - 1 ) and (1, 1, - 1 , - 1 ) diagonal generators of SUp(4) and SUs(4 ) respectively. Then it can be checked that the unbroken symmetries of the condensates in (2) and (3) are QF + QP and Qv + Qs. OF + QP is proportional to quark plus techniquark number, broken only to the extent that QF is SUE(2 ) × SUR(2 ) anomalous. This implies a stable proton. Qv + Qs assigns opposite quantum numbers to the two different technifamilies and zero quantum number to ordinary fermions. The number of unbroken global symmetries and original global symmetries is the same and thus there is no exact GB. 230
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Note that the combinations QF -- QP and Qv Qs are spontaneously broken, but the potential GBs are absorbed by the Qp and Qs gauge bosons. Thus there is no GB associated with the vectorial symmetry Qv, which in the combination Qv + Qw was responsible for suppressing AS = 2. Having a suitable Qs for this to happen can be traced to the appearance of both 4 and 4's under SUs(4 ) in (1). It still remains to discuss the breaking of UA(1). The neutrino condensates are not sufficient since they leave unbroken the gauged combination QA + Qs; the associated gauge boson would give too much D D mixing unless sufficiently massive. But the neutrino condensates are the only bilinears allowed by the unbroken gauge symmetries above ATc. Therefore, a suitable breaking of QA above ATC can only o~cur via a four-fermion condensate. It will be assumed that strong SUI(3)× SUc(3 ) forces are responsible at some scale AAS, A~ < AAS < A s- The suitable condensate invariant under all global and_gauge symmetries at AAS except UA(1 ) --p p p p is ( ~ I L ~ I R ~ 2 R ~ 2 L ) where the primes indicate fields nontrivial under S U I ( 3 ) X S U c ( 3 ). By the same argument as before, AAS >~ 30 TeV. It is of interest that the fl-functions for SUI(3 ) × SUc(3 ) are extremely flat; perturbatively fl(g) = - g 3 ( 3 3 - 32)/24~r2 + . . . . And it can be assumed that the SUc(3 ) fl-function reverses sign between A TC and A~ due to strongly interacting techniquarks. This can give a weak SUc(3 ) at ATC. The last topic concerns the PGBs arising from the spontaneous breaking of numerous approximate symmetries at ATC. The PGB spectrum, which is below the weak scale, provides the best experimental probe of the structure of the theory above the weak scale. The relevant features of the present structure include the existence of two technifamilies and the absence of the NR'S at ATC. The most interesting departure from standard predictions comes for the mass of the charged PGBs. The pair corresponding to the standard charged pair now have first order weak masses rather than second order masses. This follows from the fact that the corresponding currents contain a term [,Ly~ ±LL, L = (N, E). Due to the absence of NR, the current cannot be written as a sum of a current which is a symmetry of SUE(2) and a current which is not spontaneously broken. The
Volume 143B, number 1, 2, 3
PHYSICS LETTERS
result is a mass as high as 50-70 GeV, rather than the 5-10 GeV mass which has come into conflict with experiment. To simplify the following discussion it is assumed that the condensates in (3) are equal. Then there is a total of 27 color singlet PGBs, 11 neutral and 16 charged. 8 of the latter are technileptonic. All charged PGBs have first order weak masses or masses = A~c/AAsThe PGBs which carry net Qv + Qs charge are rather novel. Remember that Qv + Qs is an unbroken global symmetry under which all ordinary fermions are neutral. Thus there are no PGB-fermion-fermion couplings for these PGBs. The neutral PGBs of this type couple to the currents U~,~ys~"+U, D3,~'/5~ ±D, and F.~,.3,5~ +E where the z ' s act in the flavor space of technifermions. And the lightest pair is stable! They couple to ET~Vs~'±E, have a mass A ~ c / A s p , and carry no SUE(2) >(Uy(1) charge. Their possible role as dark matter in the universe will be discussed elsewhere. The tightest two neutral PGBs' couple to the currents D'/~'fsr0D-3Ey~Ys~0 E and E),~75~3E. Their mass is -.~ A ~ c / A s. Neither couple to charge 2 / 3 quarks. Flavor nondiagonal couplings to the charge - 1 / 3 quarks are suppressed for the following reasons. (1) The couplings to first order in quark masses and the quark mass matrix itself are simultaneously diagonalizable in the limit that EEclq terms can be neglected. In other words these terms violate " m o n o p h a g y " [3], but they require SUp(4) exchange. Thus monophagy violation becomes more negligible for heavier quarks. (2) The ~d coupling is suppressed by the symmetry argument discussed. It must involve a factor aw/2~. (3) The b d coupling to second order in the b mass must be checked because of (1). But in the limit that all interactions in the charge - 1 / 3 sector are vectorial, these contributions are also simultaneously diagonalizable with the mass matrix. Thus they also involve weak effects. Two other PGBs couple to U3,~3,5~3U and E)3,~,s~-3D with mass = A~rc/AAs. Their flavor nondiagonal couplings are larger and reflect the breakdown of monophagy due to the fact that each quark can potentially receive mass from two
9 August 1984
flavors of techniquarks. But argument (2) still holds and their larger mass further suppresses their flavor changing effects. The conclusion is that flavor changing effects induced by PGB exchange are likely smaller than the sideways induced effects. The remaining neutral PGB is the axion. A Peccei-Quinn symmetry arises in this model in the same way as described elsewhere (e.g. ref. [7]). Basically, it is another result of the large N R mass. Described also [7] was the way by which strong SUc(3 ) above Axc could yield a heavy axion. (It turns out that four-fermion condensates above ATc are also interesting in this connection.) Unfortunately, this mechanism yields a very uncertain mass for the axion. And then there is the issue of technicolor induced anomalous scaling. It has been proposed [5] that enough anomalous scaling could cure the flavor changing problem by itself. Although anomalous scaling is now not the important factor in the present model, it would be difficult to argue that none occurs at all. This adds some uncertainty to various mass scales. Modifications of the model can also be considered. For example, if the UA(1 ) gauge field was removed the theory would have an exact GB. This could only be acceptable if a four-fermion condensate broke QA at a sufficiently high scale to render the GB invisible. The other possibility to consider is maximal gauging, where Qv is gauged as well as QA. Then the theory would have a massless gauge boson coupling to Qv + Qs- The new photon would only have renormalizable couplings to technifermions and the induced nonrenormalizable couplings to ordinary fermions do not appear disastrous. In summary it has been proposed that AS = 2 effects in technicolor theories are suppressed via an approximate symmetry, broken only by weak effects. This occurs in a model having a suitable exact vectorial symmetry spontaneously broken by right-handed neutrino condensates. This does not give a Goldstone boson and it does give Cabibbo mixing. The basic prediction is that if CP violation in K K mixing is a AS = 2 superweak effect induced by sideways physics then the AC = 2 terms should be (2~r/aw) 2 larger. This would give D D mixing at an experimentally accessible level. 231
Volume 143B, number 1, 2, 3
PHYSICS LETTERS
I a m g r a t e f u l t o S. D i m o p o u l o s , M . P e s k i n , J. P r e s k i l l , a n d L. S u s s k i n d f o r c r u c i a l d i s c u s s i o n s .
References [1] L. Susskind, Phys. Rev. D20 (1979) 2619; S. Weinberg, Phys. Rev. D13 (1976) 974; D19 (1978) 1277 [2] E. Eichten and K. Lane, Phys. Lett. 90B (1980) 1251; S. Dimopoulos and L. Susskind, Nucl. Phys. B155 (1979) 237. [3] S. Dimopoulos and J. Ellis, Nucl. Phys. B182 (1981) 505. [4] J. Ellis and P. Sikivie, Phys. Lett. 104B (1981) 141;
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K. Lane, Phys. Scr. 23 (1981) 1005; A. Masiero, E. Papantonopoulos and F. Yanagida¢ Phys. Lett. l15B (1982) 229; A. Buras, S. Dawson and A. Schellekens, Phys. Rev. D27 (1983) 1171; W. Konetschny, Lett. Nuovo Cimento 37 (1983) 221; S. Dimopoulos, H. Georgi and S. Raby, Phys. Lett. 127B (1983) 101. [51 B. Holdom, Phys. Rev. D24 (1981) 1441. [61 S. Dimopoulos, G. Kane and S. Raby, Nucl. Phys. B182 (1981) 77. [7] B. Holdom and M. Peskin, Nuci. Phys. B208 (1982) 397.