(Warning! The equations have not been converted yet, proceed with caution.) Chapter 7 The Techni-Higgs 7.1 - Scale Invariance The idea of scale invariant field theories was first proposed by Weyl [68] in 1918. Weyl was attempting to unify gravity and electromagnetism, by incorporating a local scale invariance into Einstein's field equations. Weyl referred to the local scale invariance as local "gauge" invariance, in the same sense as the gauge of railroad tracks. The idea of local scale invariance quickly died, but the name gauge invariance survived, although its meaning was generalized to refer to invariance of the theory under any local symmetry. Scale invariance resurfaced in the late 60's, when the famous SLAC- MIT experiments uncovered Bjorken scaling in electroproduction. An underlying theory with global scale invariance seemed to be a natural explanation of this phenomena, and many theorists were involved in studying the implications of scale invariance1. It was found [69] that, for a wide class of renormalizable theories, the scale transformation Y'(x) = edaY(eax) (7.1) (where d = 1 for Bose fields, and d = 32 for Fermi fields) is associated with a Noether current, known as the scale current2. The scale current for a general Lagrangian is given by sm(x) = xnQmn, (7.2) where Qmn is the "new improved energy-momentum tensor" of Coleman and Jackiw [70] (as opposed to the symmetric energy-momentum tensor of Belinfante [71]). Since, by Poincare invariance, the energy-momentum tensor is conserved, we have msm = gmnQmn + xnmQmn = Qmm. (7.3) That is, the scale current is classically conserved, if the trace of the energy- momentum tensor vanishes, which is precisely what happens when there are no dimensionful parameters in the Lagrangian. In simpler language, a theory is scale invariant if there is no mass scale in the theory. The possibility that scale invariance was spontaneously broken was considered in order to explain the appearance of mass scales in strong interactions (i.e., mproton). This implied the existence of an associated Goldstone Boson (GB). The GB was known as the dilaton, and the analogs of soft-pion theorems were investigated, and speculations about PCDC (partially- conserved-dilatation current) were made [69,72]. Despite all this effort and enthusiasm, the study of scale invariance came to a quick halt. In retrospect, it is easy to see that classical scale invariance would run into problems when quantum effects are taken into account. For example, we know that even in massless QCD, a scale appears in the theory3 (the running coupling is a function of momentum and LQCD), even though the only parameter in the Lagrangian (eq. 2.1) is the dimensionless coupling g. In general, the quantum failure of a classical symmetry is referred to as an anomaly, and in the case of scale invariance the non-conservation of the scale current is referred to as the trace anomaly, since the vacuum expectation value of the trace of the energy-momentum tensor usually does not vanish. For QCD, it has been shown that [73]: 0|Qmm|0 = b2g Famn Famn + i=1N(1+gi)miyi(x)yi(x) (7.4) where b is defined in terms of the running gauge coupling g, b = m gm , (7.5) and gi is the anomalous dimension of the ith quark flavour. Thus even if the quark masses are small, scale invariance is explicitly broken due to the fact that the coupling runs. The only way out is if, for some reason, b is small. In QCD, asymptotic freedom allows b to be small---at high energies g is small, perturbation theory applies, and b is of order g3. Thus approximate scale invariance is recovered for strong interactions at short distances. 7.2 - Technicolour Technicolour models attempt to explain spontaneous electroweak symmetry breaking without introducing fundamental scalar fields [74]. Instead new fundamental fermions (technifermions) are postulated to exist and to possess technicolour charges. Particles with technicolour charges interact with a non-Abelian technicolour gauge field (i.e., they exchange technigluons). In other words, technicolour models are analogs of QCD, the main difference being that, in order to explain the masses of the W and Z0, the scale where technicolour interactions become strong must be much larger than the corresponding scale in QCD. In the Standard Model [75], a Higgs doublet acquires a vacuum expectation value (VEV) which spontaneously breaks the electroweak gauge symmetry. In technicolour models, the gauge symmetry is broken by a VEV of a technifermion bilinear (eg. 0|TT|0 0, where T is the technifermion field). If the technifermions are massless, then 0|TT|0 0 implies that the technifermions have developed a dynamical mass. Thus a chiral symmetry is spontaneously broken, and therefore there must exist pseudoscalar GB's. Three electroweak gauge bosons gain a mass by eating three of these GB's. The gauge boson mass is equal to the GB decay constant multiplied by the electroweak gauge coupling. In order to explain the masses of the ordinary quarks and leptons, additional (extremely heavy) gauge bosons are introduced (such models are referred to as extended technicolour models). The heavy gauge bosons are often called sideways gauge bosons, since they couple ordinary fermions to technifermions, as opposed to horizontal gauge bosons which couple ordinary fermions to ordinary fermions. We will call the mass of the sideways gauge bosons the sideways mass, Ms. At energies far below Ms, sideways interactions are effectively represented by four-Fermi interactions, eg. gs2Ms2 TT yy (7.6) where gs is the sideways gauge coupling. These interactions couple two ordinary fermions to two technifermions. At energies far below the dynamical technifermions mass the effect of such interactions is to produce a mass term for ordinary fermions, eg. gs2Ms2 0|TT|0 yy . (7.7) Thus sideways exchange can produce a "current" mass for the ordinary fermions from the dynamical mass of the technifermions. Unfortunately, the same processes also produce flavour changing neutral currents, which are known to be small. This experimental constraint requires that Msgs be very large. The value of 0|TT|0 is related to the mass of the W+ through the value of the GB decay constant. It turns out that is extremely difficult to get reasonable current quark masses while suppressing flavour changing neutral currents, without introducing some new ideas into technicolour models. One way out of this problem is the walking technicolour scenario [76,77]. As we have seen, the value of a fermion condensate (eq. 2.7) is sensitive to the high momentum behaviour of the dynamical mass S(p) (in fact with the expected asymptotic behaviour (eq. 6.6) the condensate diverges4). However, the pseudoscalar GB decay constant (eq. 3.56) is not sensitive to the high momentum behaviour of S(p); the integrand is efficiently damped at large loop momenta. Thus if the behaviour of S(p) could be modified so that it fell off slower for some range of momentum, then we would expect that the condensate would be enhanced with little change in the GB decay constant. This has been borne out by numerical studies [77]. This slowly falling behaviour for S(p) is expected to occur when the technicolour gauge coupling runs slowly, that is when the gauge coupling walks [78]. A walking gauge coupling implies a small b (eq. 7.5), and since the technifermions are massless we expect from equation 7.4 that the trace anomaly for walking technicolour models will be small. That is, walking technicolour theories should be approximately scale invariant, and since dynamical masses appear (thus spontaneously breaking scale invariance) we would naively expect to find a light dilaton in such theories [79]. Such a dilaton should correspond to fluctuations in the scale of the condensate, as the physical Higgs does in the Standard Model. Also, the coupling of a single dilaton, like the coupling of a single Higgs, to a quark or lepton should be proportional to the mass of the quark or lepton5. Thus the dilaton is the technicolour analogue of the Higgs, and may be referred to as the techni-Higgs. Given this motivation, we will construct a non-local model that couples technifermions to dilatons in a manner analogous to that of chapter 3. 7.3 - A Local Dilaton Model It will be instructive to first consider the local generalization of the nonlinear s model to the dilaton case. Consider the action for N identical fermions and a dilaton [69] given by Ss = ??d4xy(x)/y(x)+y(x)mexp????s(x)fsy(x) + fs22 mexp????s(x)fsmexp????s(x)fs. (7.8) where s is the dilaton field and fs is the dilaton decay constant. This action is invariant under scale transformations where the fermion transforms as in equation 7.1, and the dilaton transforms nonlinearly as s'(x) = s(eax) + fs a . (7.9) However, the fermion has a mass (at tree order), so the scale invariance is spontaneously broken, and the massless s field is the associated GB. Note that even though the symmetry group is Abelian, the interaction term cannot be made locally invariant. Local invariance would require the presence of a factor ()1+xlls(x) to cancel part6 of the Jacobian of a local scale transformation of the coordinates. However, such a factor violates translational invariance which we would like to maintain. Model actions like that of equation 7.8 were considered at tree order in the early 70's [69] in order to gain some insight into theories with spontaneously broken scale invariance. We will integrate out the fermions to obtain an effective action for the dilaton field; thus we will be incorporating loop effects. We write the quadratic term of the effective action (in momentum space) as 12 F(q)s(q)s(-q). As usual, F(q) is given by the graphs in Figure 7.1. Figure 7.1; Quadratic terms in the effective action for the dilaton. The dashed lines represent the dilaton. Taking q0 we can calculate the dilaton mass squared: ms2 = F(0) = 8Nm2fs2 ??d4p(2p)41()p2+m22 0 . (7.10) Thus it would appear that our GB has a mass. This, of course, is not the case. To see what has gone wrong, we calculate the coefficient of the linear term in the effective action (in momentum space Ls(0)). The linear term is given by the tadpole graph in Figure 7.2. Figure 7.2; Linear term in the effective action. We find L = 4Nm2fs ??d4p(2p)41p2+m2 0 . (7.11) Thus 0|s|0 = 0 does not correspond to a stable vacuum. Since L is positive we expect s to acquire a negative VEV. It should be noted that we cannot absorb this VEV into a counterterm without explicitly breaking scale invariance. To find the VEV of s we can perform a background field calculation. We replace s by s + s0 in the action (eq. 7.8), s0 represents the VEV of s. We can then adjust s0 so that 0|s|0 = 0. For the effective action the effect of the background field is to replace m by m exp????s0(x)fs. Making this substitution in equation 7.11 we find that L will vanish if we let s0 - . Making the substitution for m in equation 7.10 we find that the dilaton mass vanishes as s0 - , as expected. In fact, all the interactions between the dilaton and fermions vanish in the limit s0 - , the fermion becomes massless, and the dilaton completely decouples. Thus the vacuum expectation value of the GB field has restored the symmetry that (we thought) was spontaneously broken at tree order. This unusual phenomenon may be referred to a "spontaneous symmetry restoration". It should be noted that "spontaneous symmetry restoration" is only possible when the symmetry group is non-compact. For a compact symmetry group, a VEV of the GB field simply corresponds to a another (physically indistinguishable) vacuum. In the case of a non-compact group a VEV of the GB field may correspond to a physically different vacuum since (for a non-compact group) the limit of a sequence of group elements may not be a group element. In the case at hand the scaling ea is a group element for any finite a. However, taking the limit a - , does not yield a group element; 0 is not a group element since it does not possess an inverse. Our conclusion is that such simple local models cannot describe spontaneous breaking of scale invariance. In the next section we will attempt to construct a nonlocal model for dilatons. 7.4 - A Simple Nonlocal Dilaton Model Our model will couple a dilaton to N technifermions with NTC "technicolours"; the technicolour symmetry will not be gauged. The model action is S = d4xd4yy(x)d(x-y)/y(y) + ??d4xd4yy(x)exp52????s(x)+s(y)fsS????exp12????s(x)+s(y)fs()x-yy(y). (7.12) It can be checked that this action is scale invariant under the transformations given in equations 7.1 and 7.9. For the special case S(x-y) = md(x-y) this model reduces to that of section 7.3, without the dilaton kinetic and self-interaction term however. In order to maintain translational invariance, the x and y in the argument of S must be multiplied by the same function of s. There are not as many possibilities for the scale invariance breaking model as there were in the chiral symmetry breaking model because the group of scale transformations is Abelian. The only remaining freedom we have is in choosing the coefficients of s(x) and s(y) in the exponentials. This freedom is constrained by the requirement of global scale invariance so that the sum of the coefficients must add up to 5 in the first exponential, and to 1 in the second exponential. We have made the symmetrical choice. Since the interaction term is not locally scale invariant (local scale invariance would break translational invariance) we cannot absorb the non- trivial contributions to the scale current vertex into a GB pole as we did for the axial-vector current in chapter 3 in order to motivate the form of the one GB coupling. We can however calculate the vertices coupling dilatons to two quarks. The one dilaton, two quark vertex is: Gs(p,q,p+q) = - 12fs []S(p)+S(p+q) + p2fs S'(p) + (p+q)2fs S'(p+q), (7.13) where ' denotes ddp2 . The two dilaton, two quark vertex is: Gs2(p,q1,q2,p+q1+q2) = - 14fs2 []S(p)+S(p+q1)+S(p+q2)+S(p+q1+q2) (7.14) -1fs2[]p4S''(p)+(p+q1)4S''(p+q1)+(p+q2)4S''(p+q2)+(p+q1+q2)4S''(p+q1+q2). We again denote the quadratic term of the effective action (in momentum space) as 12 F(q)s(q)s(-q) and the linear term as Ls(0). For the linear term we find L = 4NNTCfs ??d4p(2p)4S(p)2-2p2S(p)S'(p)p2+S(p)2 . (7.13) If L vanishes, then 0|s|0 = 0 corresponds with a stable vacuum. This obviously places a constraint on the form of S(p). Without deriving S(p) from a Schwinger-Dyson equation we cannot say whether or not L = 0, we will simply assume we have an appropriate S(p) to form a stable vacuum. We can now calculate the mass squared of the dilaton, after an integration by parts we find ms2 = F(0) = 8NNTCfs2 ??d4p(2p)4p2S(p)2-2p4S(p)S'(p)()p2+S(p)22-4p2S(p)S'(p)p2+S(p)2. (7.14) Thus the dilaton mass does not, in general, vanish. However, if we add and subtract a times the last term in the integrand from the integral, and integrate the first of these new terms by parts we find ms2 = 8NNTCfs2??d4p(2p)4(1-a)p2S(p)2-(6+2a)p4S(p)S'(p)-4p2S(p)3S'(p)-2aS(p)4()p2+S(p)22. (7.15) Using our assumption that L vanishes we have -4 ??d4p(2p)4p2S(p)3S'(p)(p2+S(p)2)2 =??d4p(2p)4-2p2S(p)2-2S(p)4+4p4S(p)S'(p)(p2+S(p)2)2 . (7.16) Substituting this in for the third term in the integrand in equation 7.15 we find ms2 = 8NNTCfs2??d4p(2p)4-(1+a)p2S(p)2-(2+2a)p4S(p)S'(p)-(2+2a)S(p)4()p2+S(p)22 , (7.17) which vanishes for a = -1. Thus the dilaton mass does vanish, provided that the dynamical mass is such that the vacuum is stable around s = 0. That is the dilaton tadpole graph must vanish. This should be contrasted with the case of the GB corresponding to spontaneous chiral symmetry breaking (described in chapter 3), where the GB mass vanished identically, independent of the form of S(p). By Taylor expanding7 F(q) to second order in momentum we can pick out the dilaton kinetic term. As in chapter 3, requiring that the kinetic term be properly normalized determines the value of the decay constant fs = -F'(0). We find fs2 = NNTC16p2??0dp2p22H(p)(p2+S(p)2)3+p2G(p)(p2+S(p)2)4 . (7.18) where H(p)=4S(p)5p4S(p)'''-8S(p)3p6S(p)'''-4S(p)p8S(p)'''- 8S(p)3p4S(p)'3 + 24S(p)p6S(p)'3 + 12p2S(p)4S(p)'2- 32p4S(p)'2S(p)2 + 4p6S(p)'2 + 8S(p)4p4S(p)''S(p)'- 8p8S(p)''S(p)'- 5S(p)5S(p)'+ 12S(p)3p2S(p)' - 7S(p)p4S(p)' - 12S(p)5p2S(p)'' - 16S(p)3p4S(p)'' - 4S(p)p6S(p)'' - 2S(p)4 + 2S(p)2p2 , (7.19) and G(p) =176S(p)3p4S(p)'3+ 6S(p)'4S(p)'-36S(p)3p2S(p)'+ 71p4S(p)'2S(p)2 + 6S(p)4 - 2S(p)2p2 - 5p6S(p)'2 - 56S(p)'6S(p)'3 - 17S(p)5p2S(p)'' - 7S(p)3p4S(p)''- 7S(p)'6S(p)''- 107p2S(p)4S(p)'2 - 8p8S(p)''S(p)' + 22S(p)5S(p)' - 40S(p)4p4S(p)''S(p)' - 52S(p)5p4S(p)''' - 44S(p)3p6S(p)''' - 12S(p)'8S(p)''' + 16p8S(p)'4 - 4p10S(p)'4 +9S(p)6S(p)'2-17S(p)7S(p)''-80p6S(p)2S(p)'S(p)''-24p2S(p)'3S(p)5 + 32p2S(p)'S(p)''S(p)6 - 20S(p)7p2S(p)''' - 96p6S(p)'4S(p)2 + 72p8S(p)'2S(p)S(p)'' + 48p6S(p)'2S(p)3S(p)'' + 16p4S(p)'4S(p)4 - 24p4S(p)'2S(p)5S(p)'' - 4p8S(p)'4S(p)2 + 4p6S(p)'4S(p)4 + 4p4S(p)'4S(p)6 - 8p10S(p)'S(p)''' - 8p8S(p)'S(p)'''S(p)2 + 8p6S(p)'S(p)'''S(p)4 + 8p4S(p)'S(p)'''S(p)6 - 4S(p)'10S(p)'''' - 12S(p)3p8S(p)'''' - 12S(p)5p6S(p)'''' - 4S(p)7p4S(p)'''' . (7.20) In equations 7.18 through 7.20 we have not used the fact that the dilaton tadpole should vanish. Note that if S(p) falls like 1p2 then the integral in equation 7.18 is convergent. Since the equation for fs is quite complicated, it is perhaps interesting to compare some numerical results with those obtained in chapter 6 for the decay constant8 f of the GB associated with spontaneous chiral symmetry breaking. For this comparison we will assume that there are three flavours9 and three technicolours. Using the form of the self-energy given in equation 6.4 (which is given in terms of a dynamical mass m, and a parameter A) we find fsm = (0.30, 0.30) and fm = (0.25, 0.28) for A = 1 or 13 respectively. We may consider calculating the dilaton mass in the walking technicolour scenario, but since the explicit scale breaking comes from the slowly running gauge coupling we would have to properly incorporate technigluons into the model. This is beyond the scope of this thesis. Footnotes for Chapter 7: 1 For a pleasant review see reference [69]. 2 Also known as the dilation, or dilatation current. 3 This is an example of dimensional transmutation [87]. 4 The behaviour of Z(p) in eq. 2.7 is irrelevant, since we can choose the Landau gauge where Z(p) approaches 1 for large p [88]. 5 The ordinary fermion mass is proportional to 0|TT|0, which should be proportional to the technifermion mass. The single dilaton coupling is proportional to the technifermion mass, since this mass is the order parameter for spontaneous scale invariance breaking (see eqs. 7.8 and 7.12). 6 The remaining factor of exp()-4a cancel the scale change of the Lagrangian. 7 The Taylor expansions were done using Maple. 8 In the limit of massless quarks. 9 Note that fs depends on the number of flavours, while f does not.