Walking technicolor models

Nuclear Physics B320 (1989) 487-540 North-Holland, Amsterdam

WALKING TECHNICOLOR

MODELS

Stephen F. KING

Physics Department, The University, Southampton S09 5NH, UK Received 8 August 1988

Recent work on technicolor theories with small fl-functions has shown that the flavour changing neutral current problem which besets any realistic extended techaicolor model might be solved by Holdom's original suggestion of raising the extended technicolor scales. In this paper we apply these field theoretic ideas to the problem of constructing a realistic model of the quark and lepton mass spectrum. We discuss two closely related models: (1) An extended technicolor model based on the gauge group SO(10)ETC × SO(10)~UT; (2) A composite/elementary extended technicolor model based on the gauge group SO(10)Mc × SO(10)ETC × SU(5)CJUT. Model (1) is relatively simple, and contains three families of quarks and leptons plus an SO(7)T c family of teehnifermions. The technicolor sector corresponds to one of the examples of walking technicolor discussed by Appelquist et al. The model is fully discussed with particular emphasis on the resulting quark and lepton mass spectrum. Charged lepton masses are adequately described, but the quark masses are degenerate in pairs with zero mixing angles. Model (2) shares the desirable low energy spectrum of Model (1) but in addition provides a mechanism for enhancing the mass of u-type quarks relative to d-type quarks, based on non-perturbative compositeness corrections. We discuss these compositeness corrections, as far as a perturbative treatment allows, and develop techniques for calculating quark masses and mixing angles. We apply these techniques to the first two families of quarks, and are encouraged to find that we can reproduce the observed features of u-d mass inversion for the first family, and Cabibbo mixing. Model (2) leads to the prediction of D0-Y)o mixing, K u ---,e+~ +, K +---, ~r+e ~+, all at rates close to current experimental limits. The model also predicts three families and a top quark mass m~ ~ 50 GeV.

1. Introduction T h e s t a n d a r d m o d e l d o e s n o t a d d r e s s the q u e s t i o n s o f f l a v o u r : W h y are there t h r e e f a m i l i e s o f q u a r k s a n d l e p t o n s ? W h y are the q u a r k a n d l e p t o n m a s s e s a n d q u a r k m i x i n g a n g l e s w h a t t h e y are? W h y is the H i g g s scalar m a s s s t a b l e u n d e r r a d i a t i v e c o r r e c t i o n s ? U n i f i c a t i o n of S U ( 3 ) × S U ( 2 ) x U ( 1 ) i n t o S U ( 5 ) [1] o r SO(10) [2] g r a n d u n i f i e d theories ( G U T s ) o n l y p u t s t h e s e q u e s t i o n s i n t o s h a r p relief. C o n v e n t i o n a l w i s d o m alleviates the H i g g s m a s s s t a b i l i t y p r o b l e m b y i n t r o d u c i n g a n N = 1 s u p e r s y m m e t r y w h i c h is b r o k e n n o t t o o far a b o v e the F e r m i m a s s scale G v 1/2. T h e p r o b l e m o f the f e r m i o n m a s s s p e c t r u m is p o s t p o n e d to the P l a n c k scale m p = 1019 G e V w h e r e g r a v i t a t i o n a l effects b e c o m e i m p o r t a n t . A n a d d i t i o n a l a d v a n 0550-3213/89/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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tage of supersymmetry is sometimes said to be that the grand unified predictions based on the desert hypothesis MGUT= (2.0-{-12:1) X 1 0 1 4 G e V ,

sin 2 0W= 0.214 + 0.004, become modified so that they achieve consistency with the experimental requirements [3]: Mou T >/7 × 1014 GeV, sin 2 0w = 0.228 +_ 0.004. However it should be emphasised that any new physics threshold which violates the desert hypothesis may have a similar effect. A different example of a new physics threshold is compositeness. The basic compositeness idea is that the naturalness problem is symptomatic of the fact that Higgs scalars are some sort of fermion-antifermion bound state. This idea has to be taken seriously since the Higgs boson is not yet discovered, and the mechanism which breaks SU(2) × U(1) remains unprobed. A simple corollary of the compositeness option is that the fermion spectrum becomes a low energy problem. This follows because Higgs scalars not only break S U ( 2 ) x U(1) but are also responsible for fermion masses in the standard model. Of course it is far from clear if any composite model is consistent with the idea of GUTs. A particular scheme for breaking S U ( 2 ) × U(1) dynamically is the technicolor mechanism [4-6]. The idea is that QCD-like dynamics with a confinement scale equal to GF 1/2 will break S U ( 2 ) × U(1) in such a way as to preserve the mass relation M w / M z = cos 0w [4]. The quarks and leptons remain massless, however. Quark and lepton masses can be achieved in principle by enlarging the technicolor (TC) gauge group G x c to some extended technicolor (ETC) gauge group GETc G x c [7,8]. If GETc ~ G x c at mass scales M > GF1/:= 300 GeV then the gauge bosons in G E T c / G T c connect technifermions to fermions, allowing the latter to

N

Fig. 1. The radiative diagram responsible for fermion masses in ETC.

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receive mass from the diagram in fig. 1 [7, 8]. The shortcomings of ETC were well known by the original perpetrators [8] and will become apparent shortly. The most notorious of these shortcomings is the flavour-changing neutral-current (FCNC) problem [8, 9]. Consider the exchange of the heavy ETC gauge bosons in G Evc/G vc- Below the ETC scales M such exchanges will generate four-fermion non-renormalisable terms in the low energy effective lagrangian of the form ( g 2 / M 2 ) ~b~b~, where g is a gauge coupling constant. Such terms involving two fermion fields and two technifermion fields ~b'+'++ will lead to fermion masses - g2A3c/M2(fig. 1) when the technifermions condense ( ~ ' ~ ' ) - A3c. Unfortunately terms involving four-fermion fields such as (gZ/M2)(gd)2lead to excessive FCNC's, once estimates of fermion masses are used to fix M [8, 9]. The seeds of a solution to this problem were sown by Holdom [10] who pointed out that M may be raised in theories with a non-trivial ultraviolet stable fixed point (NTUVSFP). More recently Holdom showed that a similar effect was possible in asymptotically free theories with a small B-function [11]. Very recently both N T U V S F P theories and asymptotically free theories with a slowly running or walking technicolor coupling have been exploited [12, 13]. In the walking technicolor approach one envisages an ordinary asymptotically free theory with a large number of technifermions, or a small number in a large representation, so that the/?-function is small but asymptotic freedom is maintained [13]. In such a theory the dynamical technifermion mass 2;(p) falls more slowly with increasing momentum p over a large range of p before the asymptotic form takes over. The physical consequence is to increase the value of ( + ' + ' ) , and hence increase fermion masses and pseudo-Goldstone boson (PGB) masses, without changing F,~ appreciably [13]. Numerical estimates lead to typical fermion masses of m f ~ 20 MeV for an ETC scale M ~- 300 TeV, as compared to the naive estimate mf ~ 0.1 MeV for the same value of M. This may be sufficient to avoid the FCNC problems*. Assuming that the idea of walking technicolor is capable of solving the FCNC problems in principle, it is natural to return to the long-standing problem of finding a realistic model of fermion masses based on ETC. That is the subject of this paper. It might be argued that the problem of finding a realistic ETC model has already been fully explored [5,6] and that no further progress is possible until new experimental input is available. We would challenge this point of view for three reasons. To begin with, one cannot claim to have a solution to the FCNC problems unless one has a specific model which describes masses and mixing angles. It is only within the context of a realistic model that the FCNC problem can be properly addressed. Secondly, the experimental clues if and when they come are bound to be

* The recent claim that the walking technicolor mechanism provides insufficient enhancement of the fermion mass [14] is based on an erroneous assumption concerning boundary conditions, as pointed out in ref. [15].

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difficult to interpret without a specific model in mind. Thirdly, the requirement that the technicolor theory must have a walking coupling narrows down the possibilities of technicolor gauge groups and numbers of technifermions to some extent. Although there has been much recent work on the field theoretic aspects of technicolor theories with small fl-functions, there has been much less work concerned with the application of these ideas to realistic models. This paper is an attempt to repair this deficiency. In this paper we discuss two closely related models: (1) An ETC model based on the gauge group SO(10)ETC × SO(10)~UT. (2) A composite/elementary ( C O M P / E L ) ETC model based on the larger gauge group SO(10)M c × SO(10)ETC × SU(5)~UT. We shall shortly outline both of these models, and explain the function of the above gauge groups. We first remark that Model (1) and (2) were proposed in refs. [16,17], respectively. One specific purpose of the present paper is to discuss the problem of quark masses and mixing angles in Model (2): a problem which was only discussed very briefly in ref. [17]. It is apparent from ref. [17] that Model (2) is a rather complicated model. Moreover Model (2) can be viewed as an extension of Model (1). For both these reasons it makes sense to first discuss Model (1), which is less complicated and serves to illustrate the problems which motivate Model (2). By so doing we are able to develop many of the basic techniques we shall need for Model (2) within the simpler framework of Model (1). There is another reason why we discuss Model (1) in such detail. Although there is no "standard model" of ETC, a good deal of theoretical attention has been focussed on the possibility that the technifermions come in a single complete family, i.e. the technifermions have the same SU(3) × SU(3) × U(1) quantum numbers as a family of quarks and leptons, but in addition carry a technicolor index r which labels the technicolor representation RTC of the technicolor gauge group GTC. This simple assumption has underpinned much of the model building, vacuum alignment analyses, and phenomenological studies mentioned in refs. [5, 6]. The result is that the one techni-family scenario is the best studied example in the literature. Until now the choice of GTC and RTC for the techni-family has been unconstrained although vacuum alignment analyses requires that G TC be either an orthogonal or unitary gauge group (not symplectic). If one requires that the technicolor theory have a walking technicolor coupling with the conditions of (1) asymptotic freedom, and (2) reasonable convergence, then there are only a finite number of possible choices of G TC, R TC and n f , the number of Dirac technifermion flavours. Appelquist, Wijewardhana and Carrier [13] have performed a computer search, and have tabulated a list of 37 such walking technicolor theories. Of these theories only o n e has n r = 8 corresponding to a complete techni-family: GTC = SO(7)T c, RTC = 7. The remarkable feature of Model (1) is that it generates this unique walking technicolor family together with precisely three families of quarks and leptons. For this reason it is worthwhile exploring Model (1) in full detail for its own sake.

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Model (1) is a relatively simple ETC model based on the ETC gauge group SO(10)ET¢ which commutes with the grand unified gauge group SO(10)~uT. Whereas in the usual G U T scenario the fermions form three copies of the spinorial 16 rep of SO(10)~uv, here the fermions are assigned to a single copy of the SO(10)ETC × SO(10)ovT rep (10, 16) [16]. The idea is that SO(10)c~v v is broken to SU(3) × SU(2) × U(1) at M~u v as usual, and that SO(10)ETC is broken to SO(9) at a scale M 1, SO(8) at a scale M 2 and finally SO(7)Tc at a scale M3, where SO(7)Tc is unbroken and becomes the technicolor gauge group which confines at A Tc = 1 TeV. The scales M 1 - M 3 are in the range 10-10,000 TeV. In the course of the ETC symmetry breakings the decomposition 10 ~ 7 + 1 + 1 + 1 yields a family of technifermions in the 7 rep of SO(7)Tc plus three singlets identified as the three quark and lepton families. We find it quite remarkable that the permutation symmetric SO(10) 2 model yields the desired walking technicolor theory plus three quark and lepton families. This model is fully discussed in sect. 2, including a new mechanism for generating off-diagonal mass matrix elements based on a one-loop vacuum polarisation diagram. The model adequately describes the charged lepton masses by the simple expedient of using them to set the scales M 1, M 2, M 3. However there is no mechanism for obtaining small neutrino masses, and the quark masses turn out to be degenerate in pairs, with zero mixing angles. Model (1) clearly has several attractive features, but ultimately fails to account for the complexities of the quark spectrum observed in experiments. Our desire to preserve the attractive low energy spectrum of the model while simultaneously accounting for the quark masses and mixing angles leads us to Model (2) in which some of the fermions and technifermions are composite on a scale A = 100 TeV. The idea is to replace the SO(10)~vT group by SU(5)~vT, and suppose that the "1 + 10" are composite while the "5" is elementary [17]. The reason for this peculiar hypothesis is that it leads to the u-type quarks being fully composite, with the d-type quarks having composite left-handed chiral components and elementary right-handed chiral components. The leptons also have a similar hybrid nature to the d-type quarks but with left-handed components being elementary and righthanded components composite. In sect. 3 we discuss Model (2) including the compositeness corrections which are important in determining the fermion masses. We argue that the fully composite u-type quarks become significantly heavier as a result of such compositeness corrections. We also postulate a neutrino condensate which will lead to small physical neutrino masses. The new mechanism for off-diagonal masses discussed in the context of the Model (1) is shown to lead to non-zero quark mixing angles in the presence of the compositeness corrections because of the momentum dependence of the vacuum polarisation bubble graph. The phenomenology of both models turns out to be quite interesting and is discussed in sect. 4. Model (1) has the least exotic phenomenology, although it does predict a slew of PGB's with masses upwards of 50 GeV. Model (2) predicts a similar spectrum of PGB's but is distinguished by its further prediction of

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the compositeness-induced rare processes: D 0 - D 0 mixing, K L ~ e - + ~ ;, K + ~ ~r+e-/~ +, all at rates close to current limits. The model also predicts a top quark mass m t = 5 0 G e V . Finally we note that although this paper is rather lengthy, it is not meant to be a review of technicolor. There are already excellent reviews in the literature [5]. Unfortunately ETC model building has declined in popularity, mainly as a result of the F C N C problem and so in order to make this paper accessible we have attempted to discuss Model (1) in a pedagogical way. The detailed structure of the remainder of the paper is as follows. In sect. 2 we discuss Model (1) [16]. In section 2.1 we introduce the model, in sect. 2.2 we discuss the unification aspects of the model, and in sect. 2.3 we explore the low energy theory which results. In sect. 2.4 we describe how fermion masses arise in the model, and in sect. 2.5 we present a new mechanism for calculating off-diagonal fermion masses based on the vacuum polarisation diagram. In sect. 3 we discuss Model (2) [17]. In sect. 3.1 we describe the C O M P / E L hypothesis and in sect. 3.2 we present Model (2) which was first proposed in ref. [17]. In sect. 3.3 we discuss the calculation of diagonal fermion masses in the presence of compositeness corrections, and in sect. 3.4 we discuss how the mechanism for off-diagonal masses in sect. 2.5 leads to non-zero quark mixing angles when compositeness corrections are included. In sect. 4 we discuss the phenomenology arising from the two models. In sect. 4.1 we outline the phenomenology from Model (1), while in sect. 4.2, we discuss the phenomenology of Model (2). Sect. 5 concludes the paper.

2. Extended technicolor (ETC) 2.1. M O D E L (1)

The model [16] is based on the gauge group G = SO(10)ETC X SO(10)OUT.

(1)

The second factor is just the usual grand unified group [2[ which we break at the unification mass MGu x down to the conventional low energy group SU(3) × SU(2) × U(1). We resist the temptation to break the group via SU(4) × SU(2) × SU(2) as is done in similar models of this kind [18-20]. The first factor in eq. (1) is an ETC gauge group which is broken sequentially by Higgs scalars down to technicolor SO(7)T o as follows, SO(10)ET C M1 SO(9)

M2 SO(8)

M3 SO(7)TC,

(2)

with the 10 rep decomposing as 10 --* 7 + 1 + 1 + 1. The scales M1, M2, M 3 will be estimated later to lie in the range 10-104 TeV (104-107 GeV) compared to M G u x ~ 1015 GeV.

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G in eq. (1) is not a simple group but it can in principle be unified by imposing an additional permutation symmetry P to ensure that there is only one gauge coupling constant. The gauge bosons of the model are in the adjoint representation " 9 0 " = (45,1) + (1,45).

(3)

The twelve light gauge bosons, the gluon, photons, W -+ and Z are in the (1,45) above, embedded in the usual way [2]. The Higgs scalars required to perform the symmetry breakings above are 3["20"] =3[(10,1) + (1,10)],

(4)

" 2 4 0 " = (120,1) + (1,120).

(5)

The Higgs scalars in eq. (5) of the form (1,120) are assumed to develop a vacuum expectation value (vev) and break SO(10)~uT to SU(3)× SU(2)× U(1) at M(~vT. The Higgs scalars in eq. (4) of the form 3[(10,1)] are assumed to develop vev's 341, M 2, M 3 leading to the symmetry breakings in eq. (2). The scalars 3[(1,10)] will be assumed to have masses O(/14,) although they do not develop vev's. They play no part in SU(2) × U(1) breaking. The left-handed Weyl fermions are assigned to the representation " 3 2 0 " = (16,10) + (10,16).

(6)

The (10,16) quite clearly decomposes under the symmetry breaking in eq. (2) into three families of quarks and leptons plus a family of technifermions in the 7 representation of SO(7)xc. All these fermions have zero intrinsic masses since the Higgs scalars 3(1,10) to which they may couple do not develop vev's. By contrast the fermions (16,10) may couple to the Higgs fields (10, 1) which do develop vev's, as follows, 3

£PvuK = ~'~ X~b (16, 10)~p (16, 10)4~ (10, 1).

(7)

x=l

The coupling is invariant since 16 × 16 × 10 D 1. The fermion masses which arise when q~x(10, 1) develop vev's ~ effectively removes the fermions in the (16, 10) rep from the low energy theory. 2.2. U N I F I C A T I O N OF THE MODEL

Although the model has been presented as a unified model, we must consider the evolution of the coupling constants up to Mc, v.r in order to substantiate this claim. In fact when this is done it becomes apparent that the model does not lead to a sensible unification at Mou T. We need only consider the evolution of the coupling

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constants between M~u T (taken to be 10 ~5 GeV) and M 1 (taken to be 10 7 GeV). We shall consider the SO(10)ETC coupling aETc and the SU(3) coupling a 3 (and also the SU(4) coupling a 4 for comparison). We use the usual definitions I T a, T b ] ----i / a b c r c '

Tr(eaR b) = r(R)~ ob

(8)

where T" is a generator of the group and R a is a representation of the generators and the one loop results da

fl - ~ - ~

=

- h a 2 +

...

,

1

b= ~

a

( ~ T(adjoint) - } T(Weyl fermion)),

1

~a(M1 - a(Maua.)

(Mo~T 1 bln

M1 ] .

(9)

where/3 is the/3-function, a is the coupling and the index T(R) takes the values below SO(10):

SU(3):

adjoint --- 45,

T(45) = 4,

r ( 1 0 ) = ~,

r ( 1 6 ) = 1,

adjoint = 8,

T(8_) = 3,

T(3) = T ( ~ ) - ~ SU(4):

adjoint = 15,

T(15) = 4,

T(4) = r(4~)= ~,

r(6_) = 1.

(lo)

Consider first the evolution of aETc between MGu T and M v The fermions in eq. (6) all contribute to the fl-function in this region (with a small contribution from the scalars 3(10,1) which we ignore). We find, using the above results the value for the coefficient of the ETC fl-function bET C =

0.42,

so that S O ( 1 0 ) E T C maintains its asymptotic freedom. If we take definiteness, then we find OtETc(MGuT)-= 0.11.

(11) a E T c ( M 1 ) -----I

for

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NOW consider SU(3). The fermions in eq. (6) constitute 72 Weyl fermions in either 3 or 3 reps, which corresponds to 36 Dirac quarks or 18 quark generations. Asymptotic freedom is clearly lost and we find b3 = - 2 . 0 7 .

(12)

If we assume a unification in which a3(M~uv) = aETc(M~us) = 0.11, then we find a3(M1) = 0.021, which is obviously much too small. SU(4) would not fare much better. The equivalent results here are b4 = - 1.48

(13)

with a4(M1) = 0.028. Thus, the loss of asymptotic freedom of SU(3) (or SU(4)) above M 1 appears to preclude a full unification of the model. Henceforth we shall not impose the permutation symmetry on the model, but instead regard the couplings a E T c ( M G u T ) , Ot3(MGuT) as free parameters. SU(3)× S U ( 2 ) × U(1) might still be unifiable within SO(10)~ur. However the severe loss of colour asymptotic freedom between M1 and MGu v is still cause for concern. For example, if we set a3(Mctor ) = 1, we find ag(M1) = 0.025 which is still too small to yield ag(AQcD) = 1. It may be necessary to assume that the QCD coupling constant enters a region of strong coupling below M~u v. Obviously this problem could be avoided by discarding the fernfions (16,10) which are no longer required now that the permutation symmetry has been relaxed. We choose to retain these fermions, however, for aesthetic reasons. 2.3. THE LOW ENERGY THEORY Below the scales M i in eq. (2) the effective low energy theory is based on the gauge group

SO(7)-rc × SU(3) × SU(2) × O(1).

(14)

There are only fermions (no scalars) in the effective theory, corresponding to the components of the (10, 16), since the fermions in (16, 10) are removed by the masses arising from eq. (7). In left-handed Weyl notation, the three quark and lepton families, plus the techni-family corresponding to the decomposition of the (10, 16) transform under the gauge group in eq. (14) as q: = (7,3,2, ~),

qi = (1,3,2, g),l

d r c ' -- -

(7,3, 1 ~),

d~ = (1,3,1, ~),

/d rc'=

(7,3,1,

u : = (1,3,1, - 2),

2),

l~' = (7, 1,2, - ½),

l, = (1,1,2, - ½ ) ,

(7,1,1,1),

e2= (1,1,1,1),

v~' = ( 7 , 1 , 1 , 0 ) ,

~c= ( 1 , 1 , 1 , o ) .

e r~'

(15)

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Technicolor

Our notation is that the index r = 1 . . . . . 7 is a technicolor index, and primed fields are technifermions, while i = 1 . . . . . 3 is a family index. The label c denotes charge conjugation. Electric charge is given by Q = T3 + Y, where T3 is the third generator of SU(2) and Y is the generator of U(1). Orthogonal technicolor groups have been studied [21] and shown to lead to condensates which break S U ( 2 ) × U(1) in the correct manner. To be explicit we write out the technifermion family in eq. (15), dropping the technicolor index r, replacing left-handed charge conjugated fields by right-handed fields, and writing out the SU(2) doublet structure explicitly, but suppressing colour indicies,

d£ ' u~' d~'

e[ ]' v~, e~.

(16,

Then the technifermion condensates are [21] U - ,L U R, )

-~ t = ( d- tt d Rt ) = ( e- tt e Rt ) = (VLVR) =A3c "

(17)

With SU(3) X SU(2) x U(1) forces neglected, the techifermions in eq. (15) appear as 16 Weyl fields in the 7 rep of SO(7)x c. The global flavour symmetry is therefore SU(16), which is spontaneously broken to SO(16) by the condensates in eq. (17), releasing 135 PGB's. In fact most of these PGB's are not massless as would be expected from the Goldstone theorem since the presence of SU(3) X SU(2) x U(1) forces explicitly breaks the original SU(16) symmetry. The resultant spectrum has been estimated [21]. The lightest technipions are given below: ,ITi = q " y 5 ~ t q t -~-

]'y5Ti/' '

p += ?/'~,5,r+-q' _ 3i'ys,r +1', p3 = ~,ys,r3q, _ 3~,ys,r3/,, p 0 = q'Tsq' - 3]'Ysl', where q' = ( " ' ]

d']'

(18)

l' = [ ~'] and r i are the Pauli matrices. P + have been estimated to \e'J

have masses in the mass range 8-14 GeV which is essentially excluded by experiment [22]. ~i are massless and are eaten by W -+ and Z to produce masses in the correct ratio:

M w / M z = cos 0w,

(19)

which is guaranteed by a techni-isospin SU(2) symmetry corresponding to the equality of the first and second pairs of condensates in eq. (17). The remaining P G B ' s p3 and p0 appear to be massless in this model [21]. Such particles should have already been seen in axion searches.

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In order to generate non-zero masses for p0, p3 and to raise the mass of P + above the experimental limit one requires four-technifermion operators such as £t~q~t][~l{_ where q' are techniquarks (carry colour) and 1' are technileptons (colourless), which break the relevant chiral symmetries, but respect the gauge symmetry in eq. (14). In the present model such operators arise from the Yukawa couplings 3

"~flYUK= ~ Xx~ (10, 16) ~ (10, 16) q]x(1, 10),

(20)

x~l

which is the permutation conjugate to the coupling in eq. (7). Exchange of ~ generates the desired operators. The resulting masses follow from Dashen's theorem

Mp - F_2M2 (

{21)

where M G is the heavy scalar mass. The technifermions in eq. (16) consist of 16 Weyl fermions in the 7 rep of SO(7)T c. This is precisely one of the examples of Appelquist et al. of a walking technicolor theory [13]. In this theory the technicolor /}-function is small since b = 0.61 and b a c = 0.43 where a c = 0.70 is the critical value of the technicolor coupling at which chiral symmetry breaking is supposed to take place [13]. Such a walking technicolor theory has several interesting features. The value of ( ~ ' ~ ' ) is exponentially enhanced while the value of F. remains virtually unchanged. In this model there is a complete technifermion family, and hence four electroweak doublets, so F , ~ - - ( 2 5 0 / ~ ) GeV = 125 GeV. The SU(2) techni-isospin symmetry ensures F~+= F~0, which leads to eq. (19). The value of ~0, defined by 27(p) for p ~< At where /~ is the chiral symmetry breaking scale, is approximately given by scaling up QCD, ignoring the differences between the gauge groups SO(7) and SU(3), 125 GeV X0 ~ 93 MeV × 300 MeV =- 400 GeV.

(22)

The enhanced condensates lead to enhanced fermion masses from the result m r ~ (tf'~b')/M 2. Typically one expects [13] m r -- 20 MeV corresponding to M ~ 300 TeV to be compared to the naive estimate of mf --- 0.1 MeV. The PGB masses in eq. (21) are similarly enhanced which raises the masses of P0, P3, P -+ to about 50 GeV [13]. In the extreme limit that the technicolor coupling is flat out to the scale M one would expect an even larger fermion mass [10-15], m f ~- ~ , 2 / M = 500 MeV,

(23)

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u s i n g 2~0 i n eq. (22) a n d M = 300 TeV. I n such a theory the P G B masses would be

O( A Tc ). A p a r t from the condensates in eq. (17) all being exponentially e n h a n c e d relative to their naive expected strength = A 3T C , the techniquark condensates receive a n a d d i t i o n a l e n h a n c e m e n t from Q C D effects [23]. I n walking technicolor theories the t e c h n i q u a r k condensates receive an e n h a n c e m e n t m u c h greater t h a n a naive c~QcD correction w o u l d imply [23]. F o r a value of bc~c = 0.43 one would typically expect !

--p t -' ' = c(eLee. =
--!

t

---- C < . L V R )

=

CXZ30

(24)

where x is some exponential e n h a n c e m e n t due to walking technicolor [13], a n d c = 2 - 5 is a n e n h a n c e m e n t due to colour [23]. 2.4. FERMION MASSES F e r m i o n masses originate from four-fermion operators generated dynamically by the exchange of heavy E T C gauge bosons (fig. 1). I n order to write down the f o u r - f e r m i o n operators we need to u n d e r s t a n d the mass spectrum of the heavy bosons. T h e symmetry breaking p a t t e r n in eq. (2) is i n d u c e d by three Higgs m u l t i p l e t s i n the 10 rep of SO(10)ET c which develop v a c u u m expectation values (vev's) - M 1, M 2, M 3 respectively. We regard the r e - i n t r o d u c t i o n of Higgs scalars as a p a r a m e t r i s a t i o n of u n k n o w n physics at these inaccessible energy scales*. A single Higgs field ~ in the 10 rep of SO(10)ET c has a l a g r a n g i a n

(D.<,). (D.<,) -

(25)

where ~ - q~, i = 1 . . . . . 10 is a c o l u m n vector. The potential V(~* ~) :

~1 2~,- ~, + ¼X(~,- ~,)2

(26)

* The use of Higgs scalars at some high mass scale M>> GF1/2 will be necessary in the models discussed in this paper. This seems to go against the technicolor philosophy that all scalars should be banished to MGUT. In our approach the "scalars" should be regarded as fiction to parametrise some unknown dynamics involving extra gauge groups and fermions. A full and immediate resolution of the fine-tuning problem appears to be virtually impossible at the present time [24]. We believe that progress in particle physics is like climbing a ladder in energy, with unknown and unprobed regions of energy corresponding to the rungs above us being parametrised by Higgs scalars. When future experiments probe the TeV energy scale we believe the Higgs "scalars" of the standard model will need to be replaced by fermion dynamics. We will then have moved up to the next rung on the energy ladder. In this scenario it is perfectly natural to expect any new theory which replaces the standard model to once again involve Higgs "scalars" on a yet higher scale M >> GF1/2 whose role will once again be to parametrise an unexplored region of energy. Thus within our bottom ~ up approach Higgs scalars are tolerated provided their masses are >> G{ 1/2

S.F. King / Technicolor

499

is minimised by qb2 = --/.£2/~ and the most general vev can be rotated into the form shown below by suitable SO(10) transformations,

0 0 0 0 0 0 0 0

ic ~d I (~,) =

f g h

(27)

Gauge boson masses arise from the kinetic term ½(Dug) • (D%) where ij DI, = Or, + tgLijA ~ = O~,+ gA~,,

(28)

•

where

(L,j)k,=--;[~,k~j,--~,,~jk]

(;,;,k,;=l

. . . . . lo)

(29)

is a 10 × 10 representation of the Lie algebra of SO(10) which satisfies the usual commutation relations

[ Lid', Lkl ] = --i(~jkLil- {~ikny.l- ~jlLik q- ~ilLjk )

(30)

with Lij = - L j i . From eqs. (28) and (29) we see that 0 At=

A12

Ag13

...

A t19

0

A 23

...

A 29

(31)

0

From eqs. (27), (28) and (3l) the kinetic term yields gauge boson masses 1 2( & ' ~ ) - ( A ~ , ) + . . . I ( D , , ) . D r , = 3g

= ½g2u2((A12)2q-(A13)2

+ ...

q - ( A 1 9 ) 2)

(32)

showing that the single Higgs field ~ gives equal mass to the first row (or column) of the A t in eq. (31).

500

S.F. King / Technicolor

Now consider the case of two Higgs fields in the 10 rep, ~'1, and 0z- The lagrangian is now more complicated .L~ ° = ½(D~,,1) • D~'q~l + ~ (D~,*2)" (D~'q~2) - Vt(*, "q~l) - V2('2" *2) - Vt2(*t" q~2).

(33)

The full potential V = V1 + V2 + V12 now depends on I~b1[, ]~21 and I~bl. ~21 and is minimised without the loss of generality by the vev's • 1= a(1,0,0 .... ,0), 02 = b (sin 7, cos 7,0 . . . . . 0),

(34)

where 7 is the angle between ~1 and 02- The kinetic terms in eq. (33) are a sum of squares of lengths of the two vectors and does not depend on the angle between them so we are free to rotate ~ and 4'2 in eq. (34) into two new vectors ~] and ~ where ~ " ~ = 0, • i = a(1,0,0,...,0), • 2' = b ( O , l , O , . . "~0).

(35)

Since (D~q~l)2 + (D~qb2)2= (D~qb])2 + (D~qb~)2 it is convenient to work in the basis in eq. (35). In this basis the kinetic terms yield the gauge boson masses

1 (D,tt~])2 = ~1_2_2[[ ,d12~2+ - " + ( A ~ g a k~a~,)

9) 2) + " "

1 , 2 = I_2LZ/[A12~2 + ( A ~ ' ) 2 + - . . ~(D~<) +(A~912)+ . . -

(36)

Thus, the first row/column of the gauge boson matrix in eq. (31) receives equal mass contributions from 4~], while the second row/column receives equal mass contributions from q~. Thus the gauge boson A~2 receives mass from both Higgs, but the other gauge bosons only receive mass from one source. For the case of three Higgs fields ~1, ~2, #3, each in the 10 rep, they can again be made mutually orthogonal leading to the masses in eq. (36) plus 1

23

A~ ) +

+

where ,;=c(0,0,1,0,...,0)

(37)

S.F King / Technicolor

501

leading to equal masses in the third row/column apart from A~3, A 23 which also receive masses from eq. (36). Eqs. (36) and (37) contain all the information we require about the heavy gauge boson spectrum in order to write down all the relevant four-fermion operators. It is worth pointing out that the heavy ETC gauge bosons which connect fermions to technifermions all have identical masses within a given row/column. Also the gauge boson masses are given as a sum of squares, so the gauge boson mass matrices are automatically diagonalised by our choice of Higgs basis, and there is no gauge boson mixing in flavour space*. Consider ETC gauge boson exchange between fermions in the (10,16) rep which contains the quarks, leptons and technifermions in eq. (15). Since ETC commutes with S U ( 3 ) × S U ( 2 ) x U(1), the bosons are colourless and carry no electroweak quantum numbers, so when considering such exchanges we shall write the fermions as ~ki, i = 1 . . . . ,10, dropping all indices except S O ( 1 0 ) E T C . The fermion kinetic term ~D~b, with D r given by eq. (28) and L u in eq, (29), contains the interaction

g~d+ = gAuiJ(~iY"~j- f jY"~Pi)

(38)

The fermion fields ~ki with i - - 1 , 2 , 3 represent quark and lepton families while i --- 4 . . . . . 10, are the technicolor components of the technifermion family. Thus it is convenient to change labels to i, j = 1,2, 3, and r, s = 4 , . . . , 10 where i, j label flavour and r, s, label technicolor. In terms of the new labels, eq. (38) becomes

q- gA rs( ~yr'~l~,Cs-- ~s'~l~dr ) .

(39)

In this form we clearly see the division of labour into the massless gauge boson exchanges A, r' which generate the technicolor dynamics, the massive gauge boson exchanges A, ir which connect fermions to technifermions, and other massive gauge boson exchanges A j j which connect fermions to fermions. Defining the currents

j~ = fiyu~bj - ~gV"~, ,

(41)

the exchange of heavy gauge bosons A~r generates four-fermion interactions in the effective theory below the mass scales M of

Y~(g2/M~2) Jj,~l 2 ,

(42)

i,r * This conclusion differs from that presented in earlier literature [19,20]. The existence of a basis in which all the Higgs fields can be made mutually orthogonal was pointed out to me by Georgi. The point of the rather lengthy discussion from eq. (25) to eq. (37) is to clear up this confusion.

502

S.I~ King / Technicolor

where m 2 = 1 _ 2_ 2 7g c from eqs. (36, 37). The current in eq. = M3 (40) when expanded in terms of the fields in eq. (15) gives

j~ = ?liy~q; + @,{~'dC/+ fi~7~'u~'+ ]i~'l; + Y~'y~'e~'+ ?,c'y"vc'- (i ~, r)

(43)

Expanding the interaction in eq. (42), and writing out the SU(2) doublets explicitly, gives

Y'. ~.? _(u.v -, i,r

.,urY ' vC + er.g~,eier -, -c' ~.i+-' c dr gt~didr Y die + ~r Y~Vi~,c.gt, Y ei +

~

D

•

)

i

(44) After charge conjugation and Fierz rearrangement these operators become g2

2~

- , , -

~ltrLUtrR~liRUiL +

drLdrRdiRdiL

l,t"

PrtL/YrtR~iRViLq- ~Le',.R~iReit + . . . ).

(45)

It is these operators in eq. (45) which lead to quark and lepton masses when technifermions condense according to eq. (24). The fermion masses can be read off directly,

Y',(2g2x2,3o/Mi2)[c(fiiRUic + diRdie) + (~a~Vic + YiReik)] .

(46)

i

The fermion masses are enhanced by a factor x due to the walking technicolor analysis [13], and the quark masses are enhanced by an additional factor c due to Q C D corrections [23]. The fermion spectrum in eq. (46) adequately describes the charged lepton masses, which are naturally lighter than the quark masses by a factor of c. The three free parameters M 1, M 2, M 3 in our model may be adjusted to fit the charged lepton masses m e = 0.511 MeV, m r = 105.66 MeV, m~ = (1784 + 3) MeV. However, the neutrino masses are equal to the charged lepton masses, family by family. In a similar way the quark masses are degenerate in pairs. The mass matrices of the quarks are equal and diagonal. In order to generate off-diagonal elements one requires some mass mixing of the heavy gauge bosons in flavour space. But, as already noted, the gauge bosons A ir /L in eq. (31) correspond to mass eigenstates from eqs. (36), (37). As a result the operators in eq. (44) are diagonal in flavour space, leading to diagonal mass matrices.

S.F. King / Technicolor

503

The four-fermion operators of the form I/~1z contain AS = 2 operators,

g M ? + M 2 (dl~/PLd2 + d~"/'Ud~ -- d2"~dl -2

_

g2 _ "~ ~ 1 2 ( ( N L , / ~ S L ) 2 +

_

d~'~#d~)2

_

NL"[tZSLNR"[~SR+ (jR"[/~SR)2+ . - . )

(47)

Thus this model has the remarkable feature that, although the quark mass matrices are diagonal, there are still AS = 2 operators which mediate sd ~ dg mixing! In subsect. 2.5 we discuss a mechanism which allows the ETC gauge bosons to mix in flavour space, and so produce off-diagonal four-fermion operators, and hence off-diagonal entries in the quark mass matrices. Within the present model, this mechanism is not particularly interesting since it will simply lead to M

= M D,

(48)

where the u-type and d-type quark mass matrices now contain off-diagonal elements. It is straightforward to see that it is possible to simultaneously diagonalise the quark mass matrices, which returns us to the original situation. This should not be too surprising since we still have not broken isospin degeneracy. The importance of the mechanism discussed in subsect. 2.5 will become apparent in sect. 3 when we discuss a composite/elementary version of the present model. However it is convenient to introduce the mechanism now.

2.5. MECHANISM FOR OFF-DIAGONAL MASSES

The problem of generating off-diagonal masses is really one of coupling the ETC gauge bosons Airr, and A~r where i ~ j . Consider the fermions (16, 10)+ (10, 16) in eq. (6). We have already considered the fermions in (10, 16) which we represented as +i, i = 1 . . . . . 10, dropping SU(3) × SU(2) × U(1) indices. In a similar way we write the fermions in (16,10) as ~a, a = 1 . . . . . 16. These fermions couple to the Higgs fields ~1, ~z, ~3, according to eq. (7), and are expected to be very heavy, being essentially removed from the low energy theory. However we show that these fermions do still play a role in the low energy theory by allowing the ETC gauge bosons A~, ir and A~r to couple via a vacuum polarisation diagram involving a loop of the +~ fermions, as shown in fig. 2. A crucial ingredient in this diagram is the Yukawa couplings of the fermion ~a to the Higgs fields '~k - ' and ~l - ' with the vev's in eqs. (35) and (37). These vev's break the SO(3) symmetry of the kinetic terms of the Higgs lagrangian, allowing the gauge bosons to mix in flavour space.

S.F. King / Technicolor

504

<~°k> I I

,

10

J

-<5>--

F

tI I

10 r

X

<~'t> Fig. 2. The vacuum polarisation diagram responsible for off-diagonal fermion masses. The external legs correspond to massless fermions and technifermions in the ETC rep 10, while the internal fermion loop involves heavy fermions in the ETC rep 16.

To see this explicitly we need to introduce a little more group theory, using the notation of Georgi and Nanopoulos [2]. We construct the 16-dimensional rep in terms of 16 x 16 matrices which are tensor products of four independent sets of Pauli matrices, a, "r, ~1 and 0, acting on different two-dimensional spaces. The generators of the spinorial-16 rep are then given by -~j,~ ( j , k = 1 . . . . . 10), where /~j, ~ = -/~k, j and the ~j, k are the following tensor products: ]~ i, j "~" Eijk~k ' ~i+6,

j+6

~i+3,0

~£i + 3,

j + 3 =

=

Eijk'gk,

~ti,O = ~ i P 3 ,

=

°iPl,

~i+6,0

~ i + 3 , j + 6 =13i'rjP3 ,

=

EijkOk '

TIP2,

~ti+6,j='ri~jPl

~i,j+3=~i~O2,

,

(49)

where i, j in eq. (49) run from 1 , . . . , 3. Given these generators we could write down fermion fields in a basis in which o acts on the smallest space, T next smallest, next, 0 largest space, then define a particular basis I1),..., L16) and hence write out the matrix elements of/~j, k in eq. (49) explicitly. Since each matrix contains 256 elements, and there are 45 such matrices this would be somewhat cumbersome! Fortunately we do not need to do this. To simplify the structure of the Yukawa couplings, it is convenient to write the Higgs fields ~1, ~2, ~3, in the 10-rep as 16 × 16 matrices. Consider one such Higgs field ~ -= q~0,~kj ( j = 1 , . . . , 9). This can be written [2], ~_ = cb° - ilxj,o~ j

(50)

where ~ is a 16 x 16 matrix, somewhat analogous to writing the Higgs field of the

S.F. King / Technicolor

505

s t a n d a r d m o d e l as a 2 × 2 matrix 2~ in the G e l l - M a n n and Levy ~Y model. The Y u k a w a couplings in eq. (7), again dropping colour and electroweak q u a n t u m numbers, become

(51)

~kx¢CR~x~YL

where x = 1, 2, 3 labels the three Higgs fields reps, and tpc = C~ ~_ where C is the usual charge conjugation matrix, and L, R label the chirality of the field. Suppose ~1 develops a vev in the j -- 1 component, ~2 has a vev for j = 2, 43 for j = 3, all other c o m p o n e n t s being zero. In this basis, the gauge b o s o n masses are as in eqs. (36, 37) and we write the fermions in the 10 rep as q~ - ~b°, q J ( j = 1 , . . . , 9) where j = 1,2, 3 labels flavour and the other c o m p o n e n t s label technicolor. Thus in this basis the group theoretical factor associated with the diagram in fig. 2 is

(52) where r = 0, 4 . . . . . 9 is a technicolor index which must be equal on both fermion legs of the d i a g r a m in order that a technicolor singlet condensate can be constructed, i, j = 1, 2, 3 are flavour labels and l, k = 1, 2, 3 label the three Higgs fields with the vev's indicated above. The trace in eq. (52) can be readily evaluated using eq. (49) for any technicolor component r = 0, 4 . . . . . 9. For r = 0 we find

TrIt~,,0t~,0t~j,0~k,0l

= 24(ai,Sj,

- ~,,~,.,)

(53)

with a similar result differing only by a change of sign for r = 4 , . . . ,9. Eq. (53) clearly illustrates how the diagram in fig. 2 allows two gauge bosons A~" and A~" of different flavour to couple. The term 6i/6jk requires the Higgs field ~t with the vev in the l c o m p o n e n t to be equal to the flavour index i, and similarly we must have ~k with k = j . T h e second term in eq. (53) allows other possibilities. H a v i n g shown explicitly that the graph in fig. 2 allows the ETC gauge bosons A ~" and A~" to couple when i ¢ j we now turn to the problem of calculating the m a g n i t u d e of the contribution. The effect of the Y u k a w a couplings in fig. 2 is to give the fermion circulating round the loop an effective mass of )tkv k on the upper leg and ~/vt on the lower leg, where v I = a, v 2 = b, v3 = c are the vev's of the Higgs fields in eqs. (35, 37). Clearly these effective masses are of the same order as the E T C bosons masses M 1, M z, M 3, depending on the unknown Yukawa couplings Xl, Xz, X3. The simplest assumption is X l - - X z = X 3 = g / v ~ - , in which case the effective fermion masses become precisely equal to the E T C boson masses. More generally, since the effective fermion masses mr, m z , m 3 are proportional to M 1, M 2, M3, respectively, we would expect a hierarchy of fermion masses m 1 >> m 2 >> m 3- As discussed in the appendix, the leading contribution to the diagram in fig. 3 is analogous to that which occurs in the calculation of the W and Z mass corrections. This diagram has been calculated by several authors in the limit of zero

S.F.King/ Technicolor

506

Ink x

P-q

mt

Fig. 3, The fermion bubble part of fig. 2.

m o m e n t u m transfer q [25]. However this well-known calculation is not quite a d e q u a t e for our needs, since we also require the low q dependence of this diagram, as we shall see in sect. 3. It is therefore pertinent to calculate this diagram afresh, including its m o m e n t u m dependence. T h e leading contribution to the E T C gauge boson v a c u u m polarisation from the d i a g r a m in fig. 3 is H~d(q ), where

H~,7(q )

4

[ (1-vs) (z~+m,) . . . .

2-4 1" d p Tr[7. = g z n j (2w)----X

2

p2-m2

¥.

(1-vs) (~-4+m,) 2

(p-q)a-m

7 (54)

w h e r e 24 is the group theoretical factor in eq. (53), and n = 10 is the n u m b e r of

flavours of fermions circulating round the loop. Using dimensional regularisation with e = 2 - n/2, we find, after introducing a F e y n m a n p a r a m e t e r y,

H~7(q ) =

2vr n/zC(e)24n

d y y ( y - 1)

(2~r)"/"(2) _.~gp.vLldy [(~2--m2)y--lYl2k!

m 2--- ( r n ~ - m~)y + m 2, and we have d r o p p e d 2ie,,~v#p~(p - q)# whose exclusion we will justify later.

where

a

numerator

,

(55)

term

If we are interested in the low q2 behaviour we can expand the denominators in eq. (55), keeping the terms up to O(q2/m2), [ m 2 + qZy(y _ 1)]

~ --- 1 -- eln m 2 -- e ( q Z / m 2 ) y ( y -- 1).

(56)

S.F. King /

507

Technicolor

To this approximation the integral over the Feynman parameter can be done to yield

II~7(q) =

2w2_F(e) g224n (2~')4F(2)

_~(q~,G_g~,.q2)_~g~,.(mk+m~)_~ ~eg~,~qz 1

[

m2

(m~+m])

+g~,~a (m~ + m~) In ,tt2 2

-2e(q~,G-g~vq 2)

+ (m2_mk

1

4

m2

---ln 6 ~

3 m2 - m2 +

+

2

+

m4 2(m 2 - m 2 )

5

1

36

3 m,Z--2_mk

In

m~ [

~1+

m!] mT~]

m2 (m~-m~)

:t m ]l

In m---~k .

The result for the case of equal effective masses k = 1 is to the same approximation,

2~-zF(e)g224n //~(q)

=

~qz)-g~m~+g~e[m21n(m~/l~2)]

(2~r)4F(2)(-~(q~G-g.

-2e(q~,G-g~,~q2)[-~lnm2]-~eg~,.q=}.

(58)

The results in eqs. (57, 58) are divergent in the physical limit e ~ 0, since F(e) contains a pole at e = 0. We are interested in the renormalised ETC gauge boson propagator which involves the renormalisation of aETc. However the physics we are interested in can be extracted by a simple and straightforward procedure. In the case of the analogous electroweak calculations of Ap it turns out that M w2 + - M23 is finite and/~-independent [25]. Similarly we can obtain a finite and #-independent result by forming the combination 1

2~r2eF(e)g224n { g"" [ 2(-~12---m--2) lnm2--m2+ (m2 +m2) 14

- 2( G,q~- gl,~q2)

( 1 12

22

5

l (mkml)

36

3 (m~ -- m~) 2

m6 3 ( m~ -- m~)3 1

])

1

2

m4 ---22 (m~-m,)

l n ~m2

] rnk (59)

S.F. King /

508

Technicolor

Multiplying by g"~ and summing on ~ and v we obtain, taking e ~ O, Z [ !2( \ ±l~ttts ~ k k -~- II~lV) -- IVlkl ] ~v

ng 2 ,1T2

2(m~m~) --3--~

m2 ] ~ + ("I +"~)

(m, --mk) lnrn~

+6q2[{ 1

1

rn 6

12

3 (m~_ml) ~

m 4 K

1

] In m2

2 (rn2_rnkl2 21 5

1

_ __

Jr_

36

m k2m !

m~ 2

3 (m~_m~) 2

(60)

The result in eq. (60), summed over G,, is the analogue of M2w+ - M23 in ref. [25], to which it reduces in the limit q2 __. 0. However, the physical interpretation of this quantity in our case is somewhat different. For us, the result represents an off-diagonal contribution AMZt to the ETC gauge boson mass squared matrix,

AMk2= Z t2~'kk[l(~'~+ H~') - H~] g,~.

(61)

p.,v

The full mass matrix in flavour space is

aM?~ M~ aM~ ,~M23 AM23

(62)

M32

Strictly speaking, this interpretation is only valid when the effective masses of the fermions ink, m t in the loop exceed the masses of the external gauge bosons Mg, M / in figs. 2, 3. Consider single ETC gauge boson exchange. In the low energy theory the propagator in 't H o o f t - F e y n m a n gauge is Taylor expanded, and the leading term gives the coefficient of the four-fermion operator,

q2_M2

3//"2 1 + M--5 +

.-..

(63)

To lowest order in g2 (the ETC coupling constant), the masses M i appearing in eq. (63) are just the diagonal masses M 1, M 2, M 3 leading to the four-fermion terms discussed earlier. However to next order in g2 we have seen that the gauge boson

S.F. King /

509

Technicolor

J t

j

Fig. 4. If the fermions in the loop are heavier than the ETC gauge bosons, then their effect may be regarded as the mass insertion shown here.

mass matrix is not diagonal (eq. (62)). This leads to off-diagonal four-fermion operators analogous to eq. (63). The coefficients of the resulting off-diagonal four-fermion operators can be estimated directly from eqs. (60)-(62). Consider a graph such as the one shown in fig. 4 where i ~ j and the cross represents a mass insertion AM, 2, given by eqs. (60)-(62). It is clear that if i :~j then the group theoretical factor in eq. (53) implies that either l = i, k = j , or l = j , k = i (from the e term), so the effective internal fermion masses are dictated by the flavour of the external fermion legs. In the effective low energy theory the off-diagonal four-fermion operator corresponding to this graph is given by expanding in low q2, analogous to eq. (63), g

q2

g

Mi2 "AMi2j(q2) " q2

~r2 M2M/ F,( m2'

M2

j

m,2

+ __

+ .-. ]

(64)

where we have written AM, 2 in eqs. (60) and (61) m terms of

2mZrn 2 D'I2 )

m2 +

(65)

+

,,,j

1

2m~

2

( m ~ - rn 2)3

3m~

m j2_ j _ _5 +

(rn.~- m2) 2 In m~

2 2 2mirn )

6

(66)

510

S.F. King / Technicolor

It makes sense to compare the coefficient of the off-diagonal operators in eq. (64) to the coefficient of the diagonal operator in eq. (63), even though we have only worked to lowest order in g2 in eq. (63). This is because aMi 2 is defined in eq. (61) as the difference between the diagonal and off-diagonal vacuum polarisation diagrams. By comparing eqs. (63) and (64) it is apparent that the q2 dependence, for small q2 of the diagonal four-fermion operators differs from that of the off-diagonal operators. In the present model this feature is not important since typical values of q2 relevant for calculating fermion masses are always much smaller than Mi2, so we may drop the q2 terms of eqs. (63) and (64). However our effort in calculating these small q2 terms has not been wasted, since they will play a role in the calculation of quark mixing angles in the composite/elementary model discussed in the next section. However for the moment we drop the q2 terms, in which case the diagonal four-fermion operators simply reduce to those discussed in eqs. (44) and (45), and the additional off-diagonal operators take the form, ng4 fl(m2i' m2) ( ldrLUrRUiRbtjL --t t -2 ~ q72 Mi2M 2 -' ' + drLdrRdjRdiL is j, r + ~;L~rPR~iR~'yL+ e;Le;ReirejL q- . . . ).

(67)

When the technifermions condense, the off-diagonal fermion masses analogous to eq. (46) are

E i 4:.1

ng4 3 F1( m2, m2) 9.1.--~'-X~-~O Mi2~My~

[£(~liRUjL q- diRdjL) q- (~iRb'jL q- eiRejL)] " (68)

The mass dependence in eq. (68), with F 1 given in eq. (65) can be approximated by

Fl(m2, =

M,2M2

-

-

(69)

M,2M2

if m 2 >> mj. 2 The quark and lepton mass matrices are then given approximately as,

M U = M D ..~ 2g2xcZ3o

1

g2m~

g~m2~

gZm2

1

g2m2

M2M22

M2

M2M~

g2m21

g2m2

1

M?M32

2 M22M3

M2

S.F. King / Technicolor

511

Fig. 5. The contribution to fermion masses from scalar exchange.

with

MN = M E = (i/c)

t v = ( a / c ) M D,

(70)

where we have made the approximation, n = 10 : ~2.

(71)

As expected, eq. (70) shows that M u = M I~ and so the quark mass matrices are simultaneously diagonalised. This leads us back to the original situation in which the quark masses are degenerate in pairs with zero mixing angles. And, to add insult to injury, the neutrino masses are still degenerate with the charged lepton masses. H o w can we improve on this? Perhaps the most obvious way is to use the scalar fields that appear to be necessary in this model to fix-up the spectrum. We have already used Yukawa couplings of the fermions in the (16, 10) rep in our mechanism for generating off-diagonal operators. The permutation symmetric Yukawa couplings in eq. (20) will lead to fermion masses (72) from the diagram in fig. 5. Decomposing eq. (20) under SU(5) we see that the charged lepton and d-quark masses from this diagram are equal, and similarly there is a mass equality between neutrinos and u-quarks from this source. Moreover the SO(10) Evc symmetry of the Yukawa couplings implies equal masses for each family. Thus the mechanism is inappropriate for explaining why m t > rob, for example. In fact we require mf from eq. (72) to be quite small, say mf < m e, so that we do not spoil the spectrum completely. This leads to a constraint on the lightest PGB masses in eq. (21), which can be re-written using eq. (72) as m~x m f

m~

me

(73)

Thus the idea of using Higgs scalars to fix up the fermion spectrum does not appear to work. It is time to abandon this model in favour of a related model in

S.F. King / Technicolor

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which some of the fermions are regarded as composites of preons with a compositeness scale A ~< M 3.

3. Composite/ elementary extended technicolor (COMP / EL ETC) 3.1. T H E BASIC IDEA

We have seen that the model in sect. 2 has certain attractive features, but ultimately fails to account for the lightness of neutrinos and the complexities of the quark spectrum. The six masses and four angles of the quark sector, as measured experimentally, contrast sharply with the three quark masses that can be accommodated in this model. Clearly the model predicts an over-simplified picture of the quark spectrum compared to the richness observed in Nature. The question naturally arises: can this model be modified in order to become more realistic? We have already seen that it is rather difficult to fix-up the spectrum by using heavy Higgs scalars. And we reject the idea of using Higgs scalars with light masses ~< GF 1/2, since then we have not improved the naturalness of the standard model. One possibility is to assume that the SO(10)c;u T is not broken directly to SU(3) × SU(2) × U(1), but breaks to SU(4) × SU(2) × SU(2) which survives down to the ETC scales M. If the SU(4) gauge group is broken to QCD SU(3) at M, then the exchange of heavy gauge bosons corresponding to the broken generators will break the quark mass matrix degeneracy M u ~ M D. To explain the gross violations of isospin that are observed in Nature one must assume that the SU(4) coupling constant is O(1). It is then natural to speculate that SU(4) is broken dynamically by some sort of "tumbling" mechanism. Unfortunately such a mechanism would necessarily give the singlet techni-neutrino ~[' a large mass O(M). One must then argue that the result to = 1 can be maintained without a custodial SU(2) symmetry to protect it [18]. In this section we shall discuss a different possibility based on the notion of composite fermions. We are well aware that in making the transition from elementary fermions to composite fermions we are not on completely solid theoretical ground. The concept of a massless or light (compared to its inverse radius A) composite fermion has been addressed in the literature [26-28]. 't Hooft [26] has formulated a set of necessary conditions under which a set of composite fermions can be kept massless by unbroken chiral symmetry. Other authors have addressed the question of when such systems make dynamical sense [27, 28]. Assuming that the notion of light composite fermions makes sense, how might it help us to improve the mass spectrum in this model? In ref. [16] it was argued that if the entire spectrum of eq. (15) were composite, then one would expect additional four-fermion operators arising from composite boson exchange. These additional operators would take the form (h2/A2)q'q"t"'t " where A is the confinement scale of the constituent preons from which the composite fields g" are constructed, and the

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S.F. King / Technicolor

is a strong coupling. The idea is that such operators would reproduce the observed pattern of fermion masses without leading to excessive FCNC's. However it is difficult to see how a value of A which is low enough to account for the top quark mass would not lead to excessive K 0 - K 0 mixing, for example. In this section we shall discuss a more promising possibility which was proposed in ref. [17]. The basic idea can be stated very simply: we shall suppose that some of the fermions in the low energy spectrum of eq. (15) are composite, and some of them are elementary. To be explicit we write out the low energy fermions in eq. (15), in the notation of eq. (16)

h2

/dR dR

¢R sR

IR~ ba:

U~

PeR

U~,R

P,R:.

l'~,

eR

/~R

~'R

eR

d~

eL/ (74)

This is just the usual three family fermion spectrum, supplemented by right-handed neutrinos, plus a family of technifermions distinguished by primes. We shall suppose that

dL , UR, eeR and e a are composite while d R and

eL are elementary,

with the pattern repeated for the second and third families and the corresponding technifermions. In other words rows 1, 2, 5 and 6 of eq. (74) are composite, while rows 3 and 4 are elementary. At first sight the C O M P / E L hypothesis above seems rather counter-intuitive. However, with this hypothesis it becomes possible to account for such gross violations of isospin symmetry as r o t > mb, m c > m s. Quark mixing angles can occur from the mechanism discussed in sect. 2.5. It becomes possible to obtain small neutrino masses by assuming that the composite right-handed neutrinos P~R, e~,~, UTR (but not v~) condense close to the compositeness scale A. The basic features of Model 1 are retained, with the ETC bosons of mass M 1, M z, M 3 coupling to a constituent preon of the composite fermion. This is illustrated in figs. 6, 7 for the diagrams which will lead to d-type and u-type quark masses. Since the u-type quarks and techniquarks are fully composite there will be compositeness corrections present in fig. 7 which are not present in fig. 6 since d R and d~ are elementary. Such corrections will lead to a violation of isospin symmetry.

S.F. King / Technicolor

514

Ot

O;

dCL

dCr'

Fig. 6. The graph responsible for d-type quark masses in the COMP/EL approach. The basic mechanism is still ETC gauge boson exchange leading to fermion masses as in fig. 1.

If the compositeness scale A lies above the ETC scales, A > M1, then the effects of compositeness would be expected to correspond to some small form-factor correction to the ETC boson exchange diagrams. However if A ~< M 3 then the compositeness corrections are significant. This will be discussed shortly, but for the moment the following analogy might prove helpful. In the standard model we have a similar situation in which heavy gauge bosons, namely W and Z, are exchanged between quarks which are confined within hadrons of inverse radius AQcr) < Mw, z. Here we are well aware that in a low-energy Fermi theory of weak interactions the four-fermion operator involving two proton and two neutron fields will have a very different coefficient from the operator involving one proton, one neutron, and two lepton fields. Such strong-interaction corrections are non-perturbative in nature and hence difficult to calculate. But we are in no doubt that they have the potential to be very large indeed. Thus, returning to our mechanism in figs. 6, 7, one might hope that if A ~< M 3 the strong-interaction corrections present in fig. 7 might be sufficient to account for the ratio m c / m s ~- 10, for example. Since we require quite a low value for the compositeness scale A ~< M 3 the compositeness-induced four-fermion operators ( h 2 / A 2 ) ' t ' f f ' g ' q • will have large coefficients. Any operators involving two composite fermion and two composite tech-

U~ Zu ~ q'i

Z,, ~ t U~ *;

~-Ur Fig. 7, The graph responsible for u-type quark masses in the COMP/EL approach. Whereas the d-type quarks have a hybrid COMP/EL nature, u-type quarks are fully composite. Thus there will be extra compositeness corrections such as the metagluon exchange shown here.

S.F. King /

Technicolor

515

nifermion fields which lead to fermion masses would spoil the fermion spectrum, since the value of the resulting fermion mass will be quite large. However the top quark mass m a y in principle require a contribution in this way. Similarly AS = 2 operators with coefficients h2/A 2 must be avoided since they are inconsistent with K o - K o physics. The existence and structure of such non-renormalisable four-fermion operators are governed by the global symmetries respected by the vacuum of the composite theory. In the example we are about to give the composite theory is assumed to respect a global SU(10) × SU(16) symmetry. This symmetry is sufficient to forbid all fermion mass-giving operators, and all AS = 2 operators, arising from the composite theory. However AC = 2 operators are not forbidden leading to the prediction of D 0 - D 0 mixing at an enhanced rate.

3.2. M O D E L (2)

Consider a model [17] based on the gauge group

= SO(IO)Mc X SO(IO) TC X SU(5)O T.

(75)

The new ingredient relative to the model in subsect. 2.1 is the metacolour gauge group SO(10)MC. Metacolour is an unbroken, asymptotically free gauge group which becomes strong and confining at a scale A ~< M 3. It is used to bind the fermionic subconstituents (preons) into composite fermiol~ , The choice of SO(10)MC as a binding force for preons with three-fermion bound states corresponding to the metacolour singlet contained in 16 X 16 x 10 D 1 was discussed in ref. [29]. It was shown [29] that there are dynamical arguments which support the formation of a massless composite fermion in such a theory, at least in the case of a simple prototype model with two Weyl fermions 10 + 16. The choice of SO(10)~ c was initially motivated by the desire to extend Model (1) to a permutation symmetric model based on SO(10) 3. However in order to implement the C O M P / E L hypothesis the G U T group must be SU(5)~UT, as will become clear shortly. The gauge group SO(10)ETc is again broken sequentially by Higgs scalars as in eq. (2). We then envisage a hierarchy of scales in this model,

GF 1/2 ~ ATC < A < M 3 < M~ < M 1 <

MGu T .

(76)

We introduce the scalar fields

3(1,ao,a),

(77)

(1,1,24).

(78)

The Higgs scalars in eq. (77) are introduced in order to break SO(10)ETC sequen-

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S.F. King / Technicolor

tiaUy as described in sect. 2. Those in eq. (78) break SU(5)ouT to SU(3) × SU(2) × U(1) at Mou T. In addition we shall require the following scalars O ~ (10,10,1).

(79)

The scalars in eq. (79) do not develop vev's and are assumed to have mass O(M;). The left-handed Weyl fermions transform under eq. (75) as + ~ (16,10,1) + X ~ (10,1,1 + 5 + 10) + ¢ ~ (1,16,5 + 7 ) .

(80)

Using the results in sect. 2.2 we see that with these fermions the metacolour theory is asymptotically free with b = 0.42. If the model were a permutation symmetric SO(10) 3 model, then the above fermions would represent the cyclic "480" rep. However P-symmetry is broken since we have SU(5)~uT. The first two fermions in eq. (80) carry metacolour and are called preons. The other fermion is a metacolour singlet and acts as a spectator in the metacolour dynamics. Since we have broken the P-symmetry we are free to introduce a further fermion ~

(16,1,1).

(81)

The reason for introducing the ~ preon in eq. (81) is to assist the formation of a composite fermion which is constructed as follows ~+X--*(1,10,1 + 7 + 1 0 ) .

(823

We shall argue shortly that the composite fermion in eq. (82) is massless. If this is true then, below the scale A, the low energy fermion spectrum is just that of the model in sect. 2, now reproduced as composites. This is clear since at the scale A both SO(10)ETC and SU(5)GUT are broken, according to eq. (76), so we must decompose the composite fermion in eq. (82) which when done leads to the spectrum in eq. (15). But, according to the discussion in sect. 3.1, this is not what we desire. In order to obtain a C O M P / E L low energy theory we need to introduce the spectator fermion f-= (1,10, 5 + 3).

(83)

We shall assume that f remains massless below the scale MGu T. The idea behind adding the fermion f is that the elementary 5 component from eq. (83) will marry the composite 5 component in eq. (82) (i.e. form a large gauge invariant mass term) leaving a C O M P / E L spectrum at low energies of the kind discussed below eq. (74). Namely: composite (1,10,10 + 1) plus elementary (1,10, 5).

S.F. King / Technicolor

517

The gauge invariant Yukawa couplings include 3

~'YUK = ~ Xx~(1,16,5)~(1,16,5)q~(1,10,1 )

(84)

x=l

which is the analogue of eq. (7) and effectively removes ~ from the low energy theory when q~:,develop their vev's O(M i). The couplings involving q~ in eq. (79) are "~YUK =

091@ (16,

10, 1)~ (16, 1, l)q~ (10, 10, 1)

+ w:fs(1,10, 5)X~(10, 1,5-) q~(10, 10, 1),

(85)

which are precisely the desired Yukawa couplings we need in order to provide the gauge invariant mass terms mentioned above. Below the mass M e of the scalar, q) exchange generates four-fermion operators

(

Mg ) ;

.

(86)

Clearly this operator generates a mass term which corrects the 5 component of the composite fermion +T/X~ = (1,10, 5) in eq. (82) to the 5 component of the elementary fermion -/5 = (1,10, 5) in eq. (83) as required. Below the compositeness scale, the mass is m = Wl~O2A3/M~.

(87)

This is the reason for introducing the scalar qJ in eq. (79). Clearly this mechanism will only work for SU(5)~UT, since the required couplings in eq. (85) would violate SO(10)~uT. The mechanism also requires the existence of the preon ~7 in eq. (81). We have succeeded in reproducing the same low energy spectrum as in eq. (74) with the C O M P / E L nature discussed below eq. (74), providing the composite fermion in eq. (82) can be produced massless. There is an argument based on complementarity which supports the idea that two Weyl preons - 10 + 16 will bind to form a massless composite fermion [29]. This argument will not be repeated here. In the present case we have ~,, X and 7/from eqs. (80, 81). Suppose we ignore ETC, colour and electroweak coupling constants, then the symmetry at A is SO(10)M c X S U ( l l ) X SU(16) × U(1)

(88)

corresponding to (~b, ~) - 11(16) and X - 16(10). One of the two U(1)'s is broken by metacolour instantons and the anomaly-free combination survives as shown. Let us suppose that the following condensate forms at A, (~)

~ o.

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S.F. King / Technicolor

Then the symmetry in eq. (89) is spontaneously broken to SC'(10)Mc × SU(10) X SU(16) ×

[u(1)l

(89)

under which the preons rotate as ~ = (16,10,1),

~ = (16,1,1),

X = (10,1,16).

(90)

Ignoring the U(1) symmetry in eq. (89), the SU(10) 3 and SU(16) 3 anomalies calculated using the preons in eq. (90) are matched by the massless composite fermion ~ X = (1,10,16)

(91)

in accordance with 't Hooft's anomaly constraint [26]. This is of course exactly the composite fermion we assumed to be massless in eq. (82). In this model there are two chiral U(1)'s which are broken. The first U(1) acts only on ~ and is broken by the condensate ( ~ ) v~ 0. The spontaneous breaking produces a Goldstone boson, whose mass is finite because of the Yukawa coupling %+~q} in eq. (85) which explicitly breaks the symmetry. The other U(1) is that which appears in eq. (89), which must be broken since it would prevent anomaly matching. There are two questions: how is it broken, and how can the Goldstone boson associated with its breaking be given a mass to avoid phenomenological problems? We have no immediate answer to these questions. However we would remark that SO(10)ETC instanton effects explicitly violate the U(1) symmetry in eq. (89), so that if strong ETC dynamics play a more subtle role in the energy range A - M 1 than we have assumed, then it could in principle solve both problems. The model outlined above and in ref. [17] provides an existence proof of the type of C O M P / E L model envisaged in sect. 3.1. However many features of the calculation of quark masses and mixing angles are independent of the details of the preon model, as we now show.

3.3. DIAGONAL FERMION MASSES AND COMPOSITENESS CORRECTIONS

Consider a general C O M P / E L ETC model based on the gauge group GMC X SO(10)ETC X SU(5)OUT ,

(92)

where G~ac is unbroken and confines at a scale A, SO(10)ETCis broken at scales M 1, M 2, M 3 to SO(7)Tc and SU(5)GuT is broken at Mov T to SU(3) × SU(2) × U(1), with the hierarchy of scales in eq. (76).

S.F. King / Technicolor

519

T w o essential ingredients are the spectator fermion f = (1,10, 5),

(93)

which gives the elementary fermions in rows 3 and 4 of eq. (74) and a set of preons which includes

q,-- ( R , 1 0 , 1 ) .

(94)

This preon, in some general representation R of G~ac, must be in the 10 rep of SO(10)E-ro and a singlet of SU(5)OUT. It is responsible for the SU(10) part of the global s y m m e t r y of the previous model. Additional preons are required as follows

= (R,,1,1),

(95)

x = (R2,1,10).

(96)

Model (2) is one example but it is not too difficult to invent other models in this class*. Therefore we discuss fermion masses within the more general framework above. The preon ~b in eq. (94) interacts with ETC gauge bosons as in eqs. (38)-(42). The m e t a c o l o u r carried by the preon is just a d u m m y index as far as ETC is concerned. Similarly the spectators in eq. (93) exchange ETC gauge bosons in the usual way. Thus we get four-fermion operators in the low energy effective theory analogous to eq. (44) but involving the preon ~p, and the spectator f (figs. 8, 9), 2

2

--t

--*

p-

--p -* #+ ¢,y~¢,d, y d, + ¢- - ' y ~ ¢ , F ' y " v i + ¢r'Yl~¢ierV e, + . . . )

i,r

(97) We have d e c o m p o s e d the spectator f into the d c singlet and the lepton doublet (~), which just corresponds to the contents of the SU(5)6OT 5, where each field is in the 10 rep of SO(10)ETC. There are not yet any fermions in the theory corresponding to the 10 or singlet of SU(5)ov. r. These will be constructed from the preons ~p and XT o be precise, in writing eq. (97) we are considering an effective field theory at a scale ~ where A < / , < M 3. At a lower scale t* < A, the preon fields ~p in eq. (97) *Another example of a model in this class is a supersymmetric model based on SU(10)Mc X SO(]0)ETCX SIO(5)GtJT,with the supermultiplets q, = (1-0,10,1), X = (10,1,10), f = (1,10, 5). At the compositeness scale the approximate symmetry is SU(10)~ac X SU(10) x SU(10), again assuming the U(1) factors are broken and the 't Hooft anomalies are matched by the composite supermuhiplet X+= (1,10,10). This extremely simple model satisfies all the general requirements of anomaly freedom and asymptotic freedom. However it does not yield a composite neutrino singlet.

520

S.F. King /

Technicolor

Fig. 8. The effective four-fermion interaction between two preon fields ~b and two spectator fields f generated by heaW ETC gauge boson exchange.

must be replaced by composite fields, +, ~ q'i -- n + , X = Q , , U~c, E ? , N~c ,

(98) \

where Q - - [ ~ ), and all fields are left-handed as usual with composite fields written in capital letters. The operators in eq. (97) in the effective theory at A
f

i,r

~c~',,"v + ~r~ N ' '~¢'-, r rM'i rr i + C E E-¢' r Y ~ E i~-' e r Y ~ei + " ' "

),

(99)

which are similar to those in eq. (44), but which now involve some composite helicity components. The coefficients Cu, CD, CN, C E represent the non-perturbatire corrections which the operators undergo in making the traumatic transition from the preonic theory to the composite theory. We shall not be able to calculate these coefficients analytically. However our naive expectation is C u :~ C D = C N ~ C E .

(100)

This is simply because C u is the coefficient of an operator involving four composite fields, whereas C D, CN, C e multiply operators involving two composite fields and two elementary fields. Thus we have a potential mechanism for explaining why u-type quarks are so heaw, at least qualitatively. Although we cannot calculate the non-perturbative corrections in reducing the scale/~ through the compositeness scale A, we can discuss the perturbative corrections in the region A < ~ < M i. These perturbative corrections are not expected to be dominant, but we may gain some qualitative insight by studying them. For

S.F. King / Technicolor

521

Fig. 9. The effective four-fermion interaction between four preon fields ~b generated by heavy ETC gauge boson exchange. The coefficient of such operators receives large logarithmic corrections from metagluon exchange between incoming and outgoing fermion legs.

simplicity we consider only a single ETC scale M,, and ignore any effects of the boundaries between the effective theories in the regions M 1 - M 3. We then have essentially two types of four-fermion operators to consider, from eq. (97)

Mi2Cl(~)~r'~p.~i@rYlt~i q- M2i2c2(~ - '

-, p,

(10 )

where the boundary condition is CI( ~ = mi) = c2( ~ = mi) = 1, and f denotes d c, v or e. The perturbative metagluon corrections incurred in reducing/, from M i down to A are illustrated in fig. 9 for the first operator in eq. (101) involving four preon fields. The procedure we are following is well known. In the full theory the one loop correction in fig. 9 would be finite. In the effective theory the graph is logarithmically divergent, and we must add a counterterm to the effective theory to compensate for this. Although a~ac = O(1), these leading logarithms can be summed to all orders using the renormalisation group equations (RGE) of the effective theory. We note that the exchange of metagluons across legs of opposite handedness does not produce a logarithmic renormalisation. Thus the diagram in fig. 8 does not receive any substantial corrections. Therefore c 2 ( . = A ) = 1.

(102)

It is well known from the study of, for example, charmed baryon decay that exchanges between legs of the same handedness increases the coupling as we go down in /~ if it produces an attractive force, and decreases the coupling if it produces a repulsive force. In the case of charm the gluon exchange is neither purely repulsive nor purely attractive because the fields of the same handedness are not in states of definite colour. The point is that the two initial quarks are in a mixed colour state 3 × 3 = 3 + 6, and so the colour state could be 3 (attractive potential) or 6 (repulsive potential). In the model of sect. 3.2 we have the analogous situation with SO(10)M o 16 × 16 = 10 s + 120A + 126 s, where the smallest symmetric combi-

522

S.F. King / Technicolor

nation corresponds to 16 x 16-~ 10 s, which is an attractive channel. Since ETC boson exchange conserves metacolour, we can write the first operator in eq. (101) (103) where a, fl = 1 , . . . , 16 label metacolour in this particular model. Since all the fields are left-handed the operator in eq. (103) is equivalent by a Fierz transformation to

Mi 2

(104)

2

In this form, the operator is symmetric in the metacolour indices of the preon fields +,. Thus we expect the channels 16 × 16 ~ l0 s or 126 s. Renormalisation group scaling of the coefficient gives

q(M,)

'

where/~ < M,, the anomalous dimension a - - 27/8, + 5 / 8 in the channels 16 × 16 10 S, 126 s, respectively, and in Model (2), 2~rb = 8/3. Thus in the smallest symmetric channel 16 × 16 --* 10 s the coefficient c1(/~) grows as ~t is decreased, and

]

cl(A ) =

aMc(M, )

) - 81/128 > 1,

(105)

which may be compared to eq. (102). The perturbative results in eqs. (102) and (105) motivate our assumed values for the non-perturbative coefficients, C U = 10,

C D -~ C N = C E = 1.

(106)

Following the steps analogous to those in eqs. (44-46), the operators in eq. (99) lead to the diagonal fermion masses:

~., 2gZx~,3o/Mi2 [cCuUipUiL + cC D dia OiL + CNN/RFiL -~- CEEiReiL], (107) i

where we continue to write composite fields as capital letters. We are assuming the low energy theory is precisely that of eq. (74), which is the case for Model (2). The technicolor dynamics know nothing of compositeness, of course, since A >> A TO so the discussion of sect. 2.3 culminating in eq. (24) remains unchanged. The mirror ferm[ons in eq. (86) may influence the technicolor B-function at large momentum, however.

S.F. King / Technicolor

523

Eq. (107) shows that we have broken the isospin degeneracy of the original model in sect. 2. The quarks are heavier than the leptons because of the colour correction factor c. The u-type quarks are heavier than the d-type quarks because of the compositeness corrections parametrised by C U in eq. (106). The neutrinos are still apparently degenerate with the charged leptons. However this may be remedied by assuming that the composite neutrino singlets NSR (but not the techni-neutrinos N ~ ) condense close to the composite scale (N,.NiRZ,,.)

= A .

(a0s)

This assumption leads to small physical neutrino masses by the see-saw mechanism. The quark and lepton spectrum is beginning to resemble reality. However, so far we have not generated any quark mixing angles. To do this we need the mechanism introduced in subsect. 2.5 for generating off-diagonal masses.

3.4. OFF-DIAGONAL FERMION MASSESAND COMPOSITENESSCORRECTIONS In addition to the diagonal operators in eq. (97), the mechanism of subsect. 2.5 will generate off-diagonal operators analogous to those in eq. (67),

g4 Fl(m , 2 Z

i~-d, r

.~5

9 2 Mi'M)

( -~Pr~l~il~r~ . - . . ~pj + -~brY.~pid , - rc .y. d)c ,~_ - - t

-t --t -t ,~ ~brv.~biv; y ~ pj + ~bry~b,erv e, +

. . .

)

(109)

where we have dropped q2 terms. As before we expect the operators in eq. (109) in the effective theory at an intermediate scale ~, where A
M D - - , CDM D "

From eq. (70) it is clear that such a re-scaling implies that the quark mass matrices are proportional to each other. However if M o = k M D, where k is the constant of proportionality, the quark mass matrices can be simultaneously diagonalised leading to zero quark mixing angles.

S.F. King / Technicolor

524

The above conclusion is true if the q2 terms in eqs. (63) and (64) are neglected. What is a typical q2 flowing through the ETC gauge boson propagators likely to be? For a four-fermion operator involving four elementary fermions, as in sect. 2, it is reasonable to drop the terms O(q2/M;2). The calculation of fermion masses in the case of elementary fermions involves a loop integral as shown in fig. 1. The loop momentum varies from 0 to the cut-off 34,.. However the dynamical technifermion mass X(p) represented as a cross in the diagram falls like 1/p initially, then 1/p 2 asymptotically. The bulk of the contribution to the fermion mass therefore arises from p << M;, even for the case of walking technicolor theories in which the 1/p region is extended. Therefore the average momentum flowing through the gauge boson propagator is q << M;, and it is reasonable to drop terms O(q2/M;2) for both the diagonal propagator in eq. (63) and the off-diagonal propagator in eq. (64) and fig. 4. Now consider the four fermion operators involving preons. They are of two kinds: those involving four preon fields and those involving two preon fields and two spectator fermions, as in eq. (101). As shown in figs. 8, 9 the four preon operator has metagluon corrections not shared by the two preon operator. One effect of the metagluon corrections present in the case of the four preon operator is that the average momentum squared q2 flowing through the ETC gauge boson propagator can no longer be considered small. Thus when considering four preon operators we cannot neglect the qZ/M~2 terms as we did in eq. (109). The differing momentum dependences of the diagonal and non-diagonal four preon operators leads to the conclusion that M v is no longer related to M D by a simple re-scaling. Consider the effective theory at a scale/x < A. According to the above discussion the average value of q2 flowing through the ETC gauge boson propagators may be quite large for operators involving four preon fields. Consequently the average value of q2 may be quite large for operators involving four composite fields. How large is the average momentum transfer q? For the composite fields, the quantum motion of the preons implies typical preon energies and momenta - A, and hence the typical momentum transfer between two preons is also - A . Therefore we might expect q2 = A 2 for the operators involving four composite fields, as compared to q2 << A 2 for operators involving two composite fields plus two elementary fields. Setting q2 = A 2, the diagonal operator in eq. (99) involving four composite fields is modified according to eq. (63) g2 [ i, r

"

A2

11+-Mi 2

--,

--c,

g

c

y,u; v u,

(11o)

keeping terms up to O(A/M;2). The remaining operators in eq. (99) are unchanged. The off-diagonal operator involving four composite fields is also modified relative to the off-diagonal operators involving two composite fields and two elementary

S.F King / Technicolor

525

fields by the terms O ( A / M i 2) in eq. (64):

E

i4=j, r

vr2 Mi2Mf

F1 + A2F2 + A F 1

--!

--Ct ~ C

C u U r "~[zUiU; "V Uj q- E

+

i~j r

- - c~' "ynDsd~ -c' "y~d s.c + C N ~ C , T N f ~ , y n v j + × (CDD --c, Yt~EiG7 c-, nej + . .. ) C£Er

71.2Mi2Mj (111)

where F a = Fl(rn i, mi), F 2 = F2(m i, ms) are given in eqs. (65) and (66). As discussed initially, apart from the differing momentum dependences, the compositeness corrections in the case of the off-diagonal operators are expected to be the same as for the diagonal operators. Thus we make the simplifying approximation that CU, CD, CN, C E in eq. (111) are still given by eq. (106). F r o m eq. (111) we readily deduce the off-diagonal fermion masses in the usual way, 2g4n

{1[

~CuGU,~

Z 77.2 XX~o M?Ms------~rl + a~F~+ A:F1 i,.--~ + i~=.j, r F1

q- ~

-

F1

--

F,

_

)

cC D djr DiL q- Mi2~2~ CNNiR.jL -b Mi2Ms~ CEEiRCj L ) . (112)

For convenience we also list the diagonal masses, which are slightly modified from eq. (107) due to eq. (110),

~2g2x~,3o

CCuBRUiL q- ~2CCDdiRDiL

1 q-

Mi

-/ " -t- mi12 CNN/RV, L q- Mi] 2 CEEiReiL]

(113)

If m i >> rnj then eq. (66) approximates to

rl=m 2, r2=

_~ln|m) l\ {

5

6.

(114)

S.F. King / Technicolor

526

Assuming both m~ >> rnj and M~ >> Mj, the momentum dependent term in eq. (112) is given by

2(, ,)]

1[

M2M) FI + A2F2 + A F1 Mi2 + -~f

~-775;'75~.2 mi 1 +

+ A2 - ~ In m/

Mi M ;

m2i

6

"

(115)

Eqs. (112) and (113) are the predictions of the quark and lepton masses m the C O M P / E L ETC approach in general and in Model (2) in particular. Before exploiting these results it is worthwhile summarising their range of validity and the approximations we have made in deriving them: (1) We have made a small q2 approximation, q 2 / M i 2 << 1. While this approximation does not affect the validity of the charge - _} quark and the charged lepton mass estimates (for which we set q2 = 0) it does affect our estimates of the charge quark masses. For these fully composite u-type quarks we have argued that q2 ~ A 2 where A is the scale of compositeness. As we shall see later the strong phenomenological requirements arising from the non-observation (so far) of D0-D 0 mixing, for example, imply that A cannot be too low. We shall be led to conclude that A - - M 3 = 100 TeV. Thus the above estimates are unreliable for the off-diagonal elements of M u which involve the top quark mixing with either up or charm quarks. (2) We have made two assumptions concerning the values of the effective fermion masses m~, one explicit and one implicit. Although m i are free parameters of the model, we have conveniently assumed that there are no hierarchical Yukawa couplings so that ml

m2

- -

=

M1

-

-

3//2

m3 ~

-

-

-

-

~

.

M3

(116)

Since the gauge boson masses are hierarchical M 1 >> M 2 >> M 3, eq. (116) implies that the effective fermion masses are also rnl >> rn2 >> rn 3.

(117)

Besides being a very reasonable assumption, eq. (117) enabled us to simplify the discussion by allowing us to discard the extra terms which arise from the Majorana nature of the fermions, as discussed in the appendix. Implicit in our discussion is the assumption that the fermions are heavy, me>> M i or e>> 1. This assumption allowed us to use the language of effective field theories. For instance eqs. (62) and (64) and fig. 4 and the interpretation which follows them is based upon this

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S.F. King / Technicolor

assumption. Nevertheless we believe that the problem is largely one of language. At any rate we shall assume that our results are a reasonable approximation even for the case of very light fermions e << 1. (3) We have parametrised the effects of walking technicolor on the fermion masses according to m = x Z 3 / M 2, where x is an exponential enhancement factor. A possible alternative estimate might be m = X 2 / M corresponding to a zero fl-function and a perfectly flat coupling constant. We should bear in mind that Model (2) possesses additional technicolored fermions in the mass range m = A wc - 343, corresponding to those in eqs. (85-87). Above their threshold the value of b is reduced to b = + 0.08 which maintains asymptotic freedom, but spoils the good convergence properties of the walking technicolor analysis. Similarly the fermions in eq. (80) with effective masses m~ will contribute to the fl-function above threshold. On the other hand the heavy ETC gauge bosons in the mass range M 1 - M 3 contribute with opposite sign to the fernfions and will tend to restore asymptotic freedom. Clearly there is much structure in the fi-function as various thresholds are crossed. Our parametrisation of the fermion masses is overly simplistic, but it will suffice for the present discussion. Having assessed the results in eqs. (112) and (113), let us now explore them. According to these results, M D is given by 1

g2e2

g2g2

1

g262

M? g282

M D ~ 2g2xcCDZ3o

(118)

g2~.2 g282

1

which is similar to eq. (70), where we have again used eq. (71). trivially related to M ° by 1 M E= C--~-z1 - M D = - M CD c

D.

M E

is apparently

(119)

c

Eq. (119) implies that, after the matrices are diagonalised, me - -

FHd

rn~ =

-

-

FF/s

rn~ ~

-

-

/T/b

1 =

-

.

C

(120)

This result is not too bad for the second and third families, but does not work for the lightest family. Since the lightest family is the most sensitive to the effects of

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S.F. King / Technicolor

matrix diagonalisation, perhaps this is an indication that our parametrisation of the effects of compositeness corrections in terms of the overall factors C E and C D is just too naive. In discussing M u we are limited to the case of the lightest two families by our small q2 approximation. However the top quark mass itself is not subject to such uncertainties and we expect the following mass relation to hold, mc

.

.

.

mt

.

.

CU

ms mb CD

10,

(121)

from eq. (106). Eq. (121) predicts a top quark mass m t = 50 GeV. The mass ratio mJrn d is sensitive to quark mixing effects and is discussed separately below. In discussing M U for the lightest two families we shall assume eq. (116) and consider three extreme values of ~: ~ >> 1, e = 1, e << 1. For e >> 1, we can neglect the second term in eq. (115). In this case,

M D ,~ 2g2X¢CD~,3o

M u = 2g2xcCuZ3o

1/M2 g2e2/Md I g2e2/M~ 1/M~ ] '

1(A2) 1+ 0

(122)

A2) M2 11+O

g2E2(1 + A~222 ) m--~(1 + A~22)

(123)

The matrices are distinguished by having rather large off-diagonal entries. In writing eq. (123) we have neglected the second term in eq. (115), proportional to A 2. Since A 2 = M 2, we can also drop terms like A2/M 2, AZ/M 2 compared to unity in eq. (123). If this is done we again return to the case M U = kM D with an approximately zero Cabibbo angle. This argument neglects any effect of the third family on the Cabibbo angle, but things do not look very promising. Similar comments apply to the case with e = 1, for which we may still neglect the second term in eq. (115) to give

l/m? MD~ 2g2XCCDZgg2/Md

g 2/M22 1/M 2

MU= 2gZxcCur'30(1/M?g2/M22 g2/M221/M22 where we have explicitly made the approximations eq. (125).

(124)

(125)

A2/M ff << 1, A2/M 2 << 1

in

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529

T h e choice e << 1 is m o r e promising. In this case, for sufficiently small e, we m a y neglect the first term in eq. (115) p r o p o r t i o n a l to m~, a n d also we m a y neglect the o f f - d i a g o n a l elements of M D,

MD

~

2g2xcCDY"~

a/M? o ) 0

MU = 2g~-xcCuZ3° g2A

1/M~

(126)

'

//M2 g2a2K/M?M¢I M2

1/M~

]'

(127)

w h e r e K = - 91n(m2/m 2) - 56. I n this case all the C a b i b b o mixing is g e n e r a t e d in the up-sector. The values of m d and rn s can be read off directly from eq. (126). Since the o f f - d i a g o n a l entries in eq. (127) are not too large we expect r n c ~ ( C v / C D ) m s = 10rn s. Similarly the bare up quark mass is given as rn Ou -~ ( C u / C D ) m a -- 10rn a. F o r example, if m a = 8 MeV, then rnuo 80 MeV. H o w e v e r the eigenvalue m u is given b y m u ---m 0u - b2/mc where b ~ m ° g 2 ( A 2 / M 2 ) K , a n d the C a b i b b o angle associated with the d i a g o n a l i s a t i o n is 0c ~ b/rn c. In o r d e r to achieve the small physical value of m u the value of b2/mc must cancel rn 0u to within 5%. F o r e x a m p l e , if 0c ~ 0.225, m e -- 1.5 GeV, m~0 _ 80 MeV, then m u ~ rn 0u _ 0 2 m c 4 MeV. I n o r d e r to o b t a i n the desired C a b i b b o angle we require b ~- O~mc ,-~ 337 MeV. This i m p l i e s g2(A2/M22)K ~ 4.2. W e k n o w that A2

M32

m s

M22

M2

mb

rn~

M~

rn a

ml2

M?

ms

0.03.

A l s o K is k n o w n since

-- 0.05

l e a d s to K ~ 12. T a k e n together these estimates i m p l y that we require g2 ~ 11 in o r d e r to achieve the physical C a b i b b o angle. Thus it w o u l d a p p e a r that we require the E T C c o u p l i n g c o n s t a n t to be rather strong g2 _ 4~r, or a E r c -----1. This m a y not b e u n r e a s o n a b l e since as discussed in p o i n t (3) a b o v e there are a d d i t i o n a l techn i f e r m i o n s in this m o d e l which c o n t r i b u t e to the fl-function a b o v e threshold. H o w e v e r it d o e s i m p l y that estimates of fermion masses b a s e d on E T C p e r t u r b a t i o n t h e o r y suffers f r o m a d d i t i o n a l uncertainties. T o s u m m a r i s e the results of this section, we have shown that for e << 1 the q u a r k m a s s m a t r i c e s are those given in eqs. (126,127). Providing that the E T C c o u p l i n g c o n s t a n t is strong, these mass matrices successfully r e p r o d u c e the observed features of u - d m a s s inversion in the first family, a n d C a b i b b o mixing.

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S.F. King / Technicolor

4. Phenomenology 4.1. MODEL (1)

The ETC model of sect. 2 does not successfully describe the fermion masses. However we describe the phenomenology of this model here as an introduction to the phenomenology of Model (2). The phenomenology of the PGB's in this model has been well studied. Let us recall the salient features. The technifermions of this model correspond to the primed fields in eq. (74). The general case of a single family of technifermions which transform as a real representation of an orthogonal technicolor group has been discussed extensively [21, 31]. The basic feature of such a model is that it predicts 135 PGB's corresponding to the global symmetry breaking SU(16) ~ SO(16). The lightest PGB's were listed in eq. (18), and were shown to have their masses raised to the 50 GeV range due to eq. (21) with an enhanced condensate from the walking technicolor mechanism [13]. Similarly we must add about 50 GeV to the mass values estimated for the heavier states [21]. Thus we would expect the PGB spectrum summarised in table 1. As discussed elsewhere [31] it will be a formidable experimental challenge to discover any of these states with pre-SSC accelerators. The additional 50 GeV mass increment does not help the situation.

TABLE 1 PGB

Colour

Charge

Approx. mass (GeV)

Diquarks u'u', d'd', u'd' u'u', d'd', u'd'

6 3

~,--~,42 31

~,4 _ 2, ~

- 400 - 200

Colour octets fi'd',d'u',Wu' T- d'd'

8

1,

1,0

- 300

Lep t oquarks 6'u',fi'e'

3,3

4~ ,

4~ 32

~'u',fi'v' Ud', d'e' ~'d', a,v, e'u',fi'~'

3,3 3,3 3,3 3,3

2, ~ l ~-,2 32 ~ l -~,~11

- 200

v'u',W~' e'd', d'~' v'd', d'~'

3,3 3,3 3,3

7,-2 32I - ~4, g a 1

- 200

Dileptons v'v', e'v', e'e'

1

Lightest states p+ p3,pO

1

O, - 1, - 2 _+1,0,0

- 100-150 -50

S.F. King / Technicolor

531

Optimistically one might hope to discover the colour octet fi'u'+ d'd' at a pre-SSC hadron collider. This "technieta" can be produced singly by gluon-gluon fusion at the Tevatron with a cross-section of about 0.0l nb. If it decays into ti pairs which can be reconstructed, the signal would be equal to the standard model background, making its discovery difficult but possible. Another hope might be to discover some of the myriad leptoquarks whose cross-section for production at H E R A energies could be as large as 1 pb [32]. Pessimistically none of these states will be seen until the SSC is running. Worse still, if the Higgs boson of the standard model is discovered at LEP on the Z pole in the decays Z ~ g'+g H ° (where f = e, #) then this would be the death knoll for technicolor. The standard model diagram involves a tree-level coupling of the form Z Z H °, whereas in technicolor the analogous couplings ZZP °, ZZP 3, are mediated by a technifermion loop and are highly suppressed. The two body decay Z--0 P + P and recently discussed three body decay Z ~ p + p - p 3 [33] are expected to be phase-space suppressed in walking technicolor. The most sensitive probe of FCNC's is the K L - K s mass difference [8,9]. The coefficients of A S = 2 operators which mediate s d ~ d~ mixing must be smaller than 10 -6 TeV -2. The standard model satisfies this condition in a remarkable way by virtue of the GIM mechanism. In the ETC model in sect. 2 there is no " G I M - t y p e " mechanism. Instead the AS = 2 operators are suppressed by the simple expedient of increasing the mass of the mediating bosons. The AS = 2 operators in this model are given in eq. (47). These are the operators arising from tree-level ETC boson exchange. The box diagram which would normally give a /IS = 2 operator with coefficient sin20~g4/16~rZM~ is not relevant here since 0~ = 0. This of course emphasises the point that FCNC estimates are of little use unless one has a model with realistic values of quark mixing angles. The claims of walking technicolor that a value of M ~ 300 TeV can lead to a fermion mass mr ~ 20 MeV, rather than the naive estimate of 0.1 MeV gives reason for optimism, however [13].

4.2. MODEL (2) The new feature of the C O M P / E L ETC model of sect. 3 compared to the ETC model of sect. 2 is that some of the fields are composite on a scale A. Also, in the specific model of subsect. 3.2 [17] there are additional mirror fermions in the mass range A r c - A . However if these mirror fermions are sufficiently heavy they will not directly affect the low energy fermion spectrum. For instance, they will not mix with the ordinary fermions unless they are sufficiently light that the mirror technifermions play a role in the technicolor dynamics at Arc- Above the mass scale of the mirror fermions and technifermions, the effective theory will be subject to an influx of extra technifermions. Whereas the walking technicolor analysis assumes that eventually the asymptotic form of the technifermion dynamical mass Z ( p ) 1/p 2 takes over, this assumption may no longer be true. The additional tech-

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S.F. King / Technicolor

nifermions result in b = +0.08 above the mirror mass scale m, as compared to b = + 0.61 below m. The convergence of the fl-function is threatened in such a situation, and it is difficult to draw any conclusions. One might expect, however, that the behaviour ~ ( p ) - 1/p is extended over a larger range in p, resulting in an even greater enhancement of ( ~ ' q / ) and hence larger fermion masses, and larger values of PGB mass than those quoted in table 1. In the limit that the technicolor coupling remains flat to large momenta, the fermion masses are given by eq. (23). Another interesting difference between Model (2) and Model (1) is that some of the AS = 2 operators in eq. (47), induced by ETC gauge boson exchange, now involve composite fields and consequently have enhanced coefficients. In particular there will be a A S = 2 operator, Cu~

(DLT"SL) 2 ,

(1281

which involves four composite fields and will therefore be enhanced by the same factor C u as enhances the u-type quark mass operator in eq. (99) for example. The AS = 2 operator from the box diagram will have a similarly enhanced coefficient Cusin 2 0¢ (g4/16~rZ)M2. The scalars q~ in eq. (79) can be exchanged between composite fermions, leading to additional non-renormalisable operators in the low energy theory. The Yukawa couplings involving these scalars are given in eq. (85). One can draw diagrams in which a single scalar is exchanged between two composite fermions, leading to operators like E)i3'u DiDy7" Dj. Such operators have an in-built GIM mechanism and are harmless. However the SO(3) symmetry of the ~b~@ couplings in eq. (85) is insufficient to enforce a full GIM mechanism (this requires SU(3) × U(1)) and so we expect G I M violation to occur at some level. The offending diagram is the preon box diagram in fig. 10 which induces the operator ~rV~b,~br7 ~b,)

(129)

The structure of the preon operators in eq. (129) is identical to that of the operators generated by ETC gauge boson exchange in eq. (97). After the replacements in eq. (98), the first operator in eq. (129) leads to AS = 2 operators of the kind in eq. (128) but with a universal coefficient Cu(~4/16vr2M2). The value of this coefficient is related to the masses of the mirror fermions m = ¢01~2A3/M 2 (eq. 87). Assuming ~2 = 1, we can eliminate co1, so that the coefficient in eq. (129) becomes w~

m 4M6

16~rZM2 - 16qr2A12.

(130)

The second preon operator in eq. (129) after the substitution in eq. (98) leads to --Ct operators such as Urt 7~U~Ur 7 ~ UiC which contribute to u-type quark masses. Since

S.F. K i n g /

533

Technicolor

>

Fig. 10. The scalar box diagram which generates GIM violating couplings involving four preon fields ~b.

the second operator gives a contribution to M u proportional to the unit matrix, its effect is felt most sensitively for the up quark mass. By comparison with the bare up quark mass 2g2xcCu2~/M ( in eq. (119) if we require that the new contribution is less than this, then we have the b o u n d

m4M 6 -

ga

-

16~r 2A12

<

-

(131)

-

M( '

F o r sufficiently small values of m, this b o u n d can be satisfied. By far the most important effect of compositeness is to introduce a new type of four fermion operator arising from the exchange of composite bosons. The effective lagrangian at a scale ff where m
rL r.rL,a

r~ - --Lib

+

-,.

-

~ g'L Y~,g'Lj.'/'{hY "~L#, + h.c.

# x

)

+ .,.,, --it

--i

--

.,.L,

"P' R l

t

p.

j

--Is

--i

r

--

p.

"P' R3

I

1r

--.'r --t i~ j ~¢~tr., A t ~ ! .,,a4~ts ~_ ] - q~ID',q'Lj~RrY ~R -- "~L r,~'L,'eRrr "eR -- L ~ R + h.c.,

(132)

S.F. King / Technicolor

534

where

q,ii.,e, =

,Il u

J

[iv N p

,

o'£

,R

=

LD L,e,

E'

Ut

L,R"

(133)

D'

The primed fields indicate technifermions. A number of features are apparent. To begin with, there are no four-fermion operators which contribute to fermion masses. Such operators are forbidden by the SU(10) × SU(16) approximate global symme-i I, J try. All the flavour changing physics results from the single term ffcV,0r_jq~e,i7 0R. Expanding this term, we find after removing heavy fields -,

--

U{'~ULj +

DLYt~

DLj)(Ue.iTP'U{~ + kSR/y"EJ ) + h.c.

(134)

In eqs. (132)-(134) we have traded the labels i, j = 1 . . . . . 10, for i, j = 1,2,3 and r, s = 4 , . . . , 1 0 corresponding to flavour and technicolor labels, respectively. In writing eq. (134) we have kept only the composite fields which appear in eq. (98) since the heavy mirror fermions are removed from the low energy spectrum below the mass scale m. Eq. (134) leads to exciting predictions of rare processes at or near current experimental bounds. In this model we require A ~ M 3, the lowest ETC scale, in order that compositeness effects are important in determining the top quark mass. Note that M 3 is determined by m~ rather than m b since the latter receives an additional enhancement from QCD effects (eq. (46) and ref. [23]). Good estimates of M 1_3 are not possible due to the existence of mirror technifermions of arbitrary mass, as discussed above. However a crude estimate of M 3 using eq. (23), yields M 3 = 100 TeV for Z 0 --- 400 GeV. Using this ball-park estimate of M 3 the restriction A ~ M 3, combined with the expectation that the coefficient g2 is a strong coupling analogous to the hadronic coupling pNN leads to large values for the coefficients in eq. (134). Taking g2 = 4~r, A = 100 TeV gives,

(135)

g 2 / A 2 = 1 0 - 3 TeV 2.

There are no A S = 2 operators in eq. (134). The heavy composite De., Se,, Be, are absent from the low energy spectrum, replaced by elementary de,, se,, be,. The remaining composite DL, So, B L respect an SU(3)× U(1) global symmetry contained in SU(10), and this symmetry enforces a GIM-mechanism. The S U ( 3 ) × U(1) symmetry of eq. (134) is insufficient to prevent A C = 2 and A S = 1, ALe. ~ = 1 operators. The AC = 2 operators are contained in --i

j--i

j _

--

--

--

--

UcT~ULUgTye, - CLy~,UcCe,7~,Ue, + ULT~,CcUe,7~,Ce, + . . .

(136)

w

are not Cabibbo suppressed. The operators in eq. (136) induce D0-D 0 mixing. The

S.F. King / Technicolor

535

contribution to the mass difference between D L and D s is

Am D = BDf2mD( g 2 / a 2 ) .

(137)

B D = 1 if the vacuum insertion approximation is perfect, fD is the meson decay constant and rnr~ is the mass of the meson. A recent A R G U S measurement places a bound [35], A m p < 3 × 10 -13 GeV. (138) Using B D = 1, fD = 100 MeV, m D -- 1.86 GeV, eqs. (137) and (138) gives a bound

g 2 / A 2 ~ 10 -s TeV -e.

(139)

This bound is in apparent conflict with our estimate of the coefficient in eq. (135) by two orders of magnitude. We would argue that this mismatch between eqs. (135) and (139) does not rule out the model, since we have made many approximations in order to arrive at the stated results. Indeed, slightly different assumptions lead to grossly different results. For example, Bars [34] places a limit from D 0 - D 0 mixing on A of A > Ig2J50 TeV. If for some dynamical reason ]g21 = 1 rather than Pg21 = 4 ~ , then there is no conflict. The major uncertainties include the unknown factors B D f 2, and our estimate of M 3 and A. Of course the reason why D 0 - D 0 is so exciting is that in the three family standard model, the box diagram gives an utterly negligible contribution to Am D, and long distance effects are expected to give a contribution to Am D an order of magnitude smaller than the bound in eq. (138). Similar bounds on A result from the AS = 1, ALe, ~ = 1 operators contained in --, . ER i DLT~DLjERi2 = Sfff~DL~trC/"eR + •...

(140)

The operator mediates the rare process K t ---, e-+/~:~ whose non-observance bounds A > Ig2]40 TeV, and K+---* ¢r+e-/x + whose non-observance bounds A > ]gz)60 TeV [34]. Failure to observe these processes at rates close to current experimental limits will exclude Model (2). Since this model has a better chance of providing a realistic description of quark masses and mixing angles it now makes more sense to make numerical estimates for the whole range of AS = 2 operators. The leading AS = 2 operators are summarised below:

(i)

g2 )2 CU ~12 (DL~//zS L

(ii)

sin20cg 4 -2 (from the ETC boson box diagram), CU ~'-~'~22 ( DLy~'SL)

(iii)

Cure 4M6 16vr2A~2 (DL~/~SL)2

(from tree-level ETC boson exchange),

(from the preon and scalar box diagram in fig. 10).

(141)

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S.F. King / Technicolor

For our estimates we shall use the values C u = 1 0 , g = l , M 1 = 3 × 1 0 5 TeV, M 2 = 1000 TeV, A = M 3 = M~ = 100 TeV, rn = 10 TeV. We then find the following values for the coefficients of the AS = 2 operators in eq. (141): (i)

g2 C u ~ 2 12 = 10 -1° TeV -2,

(ii)

C U 16vr2M22

(iii)

m4m~ C u 16~2A1------5 = 6 × 10 -1° TeV -2 .

sin 2 0cg 4 3X

10 _9

TeV -2,

(142)

The standard model box diagram generates AS = 2 operators with real coefficients - - 1 0 - 6 TeV -2 and imaginary coefficients - 10 9 TeV 2. The above operators give a negligible contribution to A m K, but may contribute to e. Finally the simplest prediction made by this model is that there are no more than three families of quarks and leptons. The missing top quark is expected to have a mass m t ~ 50 GeV from eq. (127), and therefore should be discovered soon at either the C E R N collider or the Tevatron. Such a low value of m t is marginally consistent with the observed rate of B ° - B ° mixing [36].

5. Conclusions

To summarise, we have discussed two closely related models: (1) An E T C model based on the gauge group SO(10)ETC × SO(10)Gvv. (2) A C O M P / E L ETC model based on the gauge group SO(10)M c × SO(10)zvc × SU(5)ou v. Model (1) was first proposed in ref. [16]. In this paper we have studied this model in much greater detail, including the problem of unification and the mechanism for providing PGB masses. We have given a thorough discussion of the Higgs sector and shown that the resulting ETC gauge boson mass matrices are diagonal in flavour space. In order to generate off-diagonal fermion masses we have proposed a new mechanism in which a vacuum polarisation diagram involving a loop of heavy permutation symmetric fermions allows the ETC gauge bosons to mix in flavour space (fig. 2). We have calculated this diagram including its momentum dependence for small q2/Mi2. Model (1) cannot be regarded as a candidate for a realistic ETC model. The quark mass matrices in eq. (70) can be simultaneously diagonalised leading to quark masses which are degenerate in pairs with zero mixing angles, and a similar degeneracy between charged leptons and neutrinos. Despite these failings, we find it quite remarkable that the model yields precisely three families of quarks and leptons, plus an SO(7)T c family of technifermions corresponding to one of the examples of a walking technicolor theory studied by Appelquist et al. [13].

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Model (2) was first proposed in ref. [17]. Here the model is explored in much greater depth, drawing on many of the techniques developed for Model (1). Compositeness corrections which are responsible for the enhancement of u-type quark masses are discussed as far as a perturbative treatment allows. The mechanism for generating off-diagonal masses in Model (1) is then applied to Model (2), where the momentum dependence of the vacuum polarisation graph is shown to lead to non-zero quark mixing angles. There are two aspects of Model (2) that are not discussed explicitly: the question of unification, and the problem of the PGB masses for P -+, p3, p0. Both these issues were addressed for Model (1), and the lessons learned there are carried over implicitly to Model (2). In other words, unification is a problem for these models, but masses for PGB's can be achieved from Yukawa couplings which explicitly violate the unwanted chiral symmetries. Model (2) involves the speculative notion of massless composite fermions, Our discussion of 't Hooft's anomaly matching conditions is no substitute for hard dynamical evidence that such composites do indeed form, although arguments based on complementarity suggest that SO(10)M c with two fermions 10 + 16 might be used to construct massless composite fermions [29]. We have no immediate resolution for the "U(1)" problem discussed in sect. 3.2. One resolution of this problem would be for SO(10)ETC to become strongly interacting in the range A - M 1. Perhaps SO(10) ETC is broken by some "tumbling" dynamics [37] parametrised by our choice of Higgs scalars? Model (2) succeeds in providing a fair description of the quark and lepton mass spectrum in terms of a smaller number of parameters than in the standard model. The three parameters M1, 3//2, M 3 are set by the charged lepton masses. The Yukawa couplings in eq. (84), are chosen to be approximately equal X1 = ?~2 ~ 2~3 ~ e g / V ~ . This is equivalent to supposing that the effective fermion masses running round the loop in fig. 2 are proportional to the ETC gauge boson masses M 1, M z, M 3 as in eq. (116). Of the parameters ¢0l, ~02, M~, only the combination in eq. (87) which determines the mirror fermion mass m is important. Similarly the scalar masses M,~ are unimportant in determining fermion masses. The mass scales in this model are: AQCD, Ave, m, A, M3, M2, M1, Me~uv. The coefficients c, C U, C D, C N, C E are all of dynamical origin and although we cannot calculate them they are not free parameters. The colour enhancement factor c is what makes quarks heavier than leptons. The compositeness correction factor C U is what makes u-type quarks so much heavier than everything else. The standard model offers no explanation of these features of the mass spectrum. We have not attempted to reproduce the quark mixing angles involving the third family. The mixing angles involving the third family are beyond the range of our small q2 approximation since q2 ~ A 2 _ m 2 for composite u-type quarks. For the first two families we found that the C O M P / E L ETC approach with e << 1 successfully reproduced the features of u - d mass inversion for the first family and Cabibbo

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mixing. It remains to be seen if a large q2 treatment of the vacuum polarisation diagram could reproduce the entire pattern quark mixing angles. The ultimate success of this approach, however, depends on the top quark being found at its expected mass of m t ~ 50 GeV. From a phenomenological point of view, Model (2) is quite interesting. The prediction of the compositeness induced rare processes: D0-D 0 mixing, K L --* e -+tt;, K + ~ ~r+e-lz +, all at rates close to current experimental limits is quite exciting. The requirement that A <~M 3 --100 TeV implies that unless these rare processes are observed soon then Model (2) will be excluded. Regardless of the ultimate fate of Model (2), it does represent a serious attempt to apply the recent field theoretical ideas concerning technicolor theories with small B-functions to the problem of the quark and lepton mass spectrum. We have developed a COMP/EL ETC approach which represents a radical departure from conventional ETC, and which could in principle be applied to a large class of models. I wish to thank my colleagues at Southampton University for discussions.

Appendix Fig. 3 is a typical one loop diagram which, in configuration space, involves the following Wick contractions, -

M(x)`/~(1 2"/5) ?M(X)~M(y)`/~

(1 - `/5) CM(y)

2

J

'

(A.1)

which leads to the usual momentum space result

f~-~

Tr `/~

p--:--m--~k` / , ~

~

(p_q)2_m~].

(A.2)

Note that +~t represents a Majorana spinor, and eq. (A.2) involves the momentum space propagators appropriate to Majorana fermions with majorana masses, which happen to have the same form as Dirac propagators. However for Majorana fermions there are further Wick contractions which contribute other than those in eq. (A.1), namely (a -

~M(X)`/~'(1--`/5) I

I

I

2

`/,)

q M(y).

(A.3)

i

To understand why the additional contractions in eq. (A.3) contribute, observe that

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Majorana fermions obey the relations (A.4) where C is the charge conjugation matrix. Eqs. (A.4) imply the identity fM(x)`1"(1 -2YS) ~pU(X)= (~pM(x))'r1T(1 - 2Y[)

(~bM(x))"r

(A.5)

from which it is manifest that the contractions in eq. (A.3) must also be considered. Taking the transpose of eq. (A.5), then substituting this into eq. (A.3) yields - 2/5)

_jTM(x)`1~ (1 + `15) ~M(x)j77(y)`1 (1 2 t 2

~pM(y), j

(A.6)

which is equivalent to eq. (A.3), with the minus sign coming from the anticommutation of two Fermi fields. Eq. (A.6) leads to the momentum space result r d4p Tr[1 ( 1 + ' 1 5 ) ( / ~ + m k ) -J(2~r) 4 " 2 p2 m 2

[

(1-75)

r~

2

(~-4+m;)

]

- -, - - 2 (- -p --q- )- - -2m

] "

(A.V)

Eqs. (A.2) and (A.7) must both be evaluated for Majorana fermions circulating round the loop in fig. 3. The term in eq. (A.2) is calculated in the text below eq. (54). It is analogous to the fermion loop corrections to the W and Z masses in the standard model. The result for the other term arising from eq. (A.7) is stated below, tip Hk;(q)

- 2v"/ZF(r)

(2=)"r/2 g224 × { gu~mkm/-- gu~ernkm/[ln(m~) - 1 + mkm l 1 -gu~eqZm2- m~ 2

m~ + m~- m~

m2 win m7 2 2 mkml

m 2 ]]

l

( m ~ - rn~) In '-7~ m k If "

(A.8)

Since we expect the effective fermion masses to be hierarchical m 1 >> m 2 >> m3, it is apparent that the result in eq. (A.8) which is O(mkrnl) is negligible compared to the result in eq. (57) which is O(m~ + m~). Henceforth the result in eq. (A.8) will be neglected. References

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