Comment on Technicolor/Extended Technicolor Models

Robert Shrock of Stony Brook sent me the following to post as a response to one of the comments on the latest posting. With his permission I’m putting it here as a separate posting, since I think it’s a valuable informed summary of the current state of technicolor/extended technicolor models.

I would like to respond to Eric’s recent comment on Oct. 23 in which he said that “technicolor models were..eventually rejected due to some serious shortcomings. Namely, in order to generate fermion mass hierarchies for the SM fermions, one ends up with serious problems with FCNC’s.” and that these theories “led to a plethora of technimesons, for which there is absolutely no evidence.”.

While it is true that FCNC’s are a relevant constraint on technicolor and extended technicolor (TC/ETC) theories and were viewed as very serious before the development of walking TC in the mid-1980’s, they do not obviously exclude TC/ETC models where the TC sector has walking behavior. The walking (slow running of the TC gauge coupling over an extended range), results naturally from the presence of an approximate IR fixed point in the renormalization group equation for the TC gauge coupling. This walking has the effect of enhancing SM fermion masses for a fixed set of ETC breaking scales. Indeed, since the mid-1980’s, the only viable TC models have been those with walking behavior. This walking allows one to use higher ETC breaking scales and still obtain the same SM fermion masses. It also enhances the masses of (pseudo)Nambu-Goldstone bosons. This was discussed in T. Appelquist and L. C. R. Wijewardhana, Phys. Rev. D 35, 774 (1987); Phys. Rev. D36, 568 (1987) and reviewed already a number of years ago, e.g., in R. S. Chivukula, hep-ph/9503202, hep-ph/9803219 and K. Lane, hep-ph/0202255, as well as more recent reviews such as C. Hill and E. Simmons, hep-ph/0203079 (published in Phys. Repts.) and my brief SCGT06 review, hep-ph/0703050. PNGB’s in one-family TC/ETC models may still be a phenomenological concern, but the early estimates of their masses were substantially increased by walking TC.

Let me explain in more detail how TC/ETC models may be able to satisfy FCNC constraints. ETC models generically gauge the generational index and combine it with TC, so a simple SU(N_ETC) model has N_ETC=N_gen+N_TC. With three generations of SM fermions, N_gen=3 and using the minimal value of N_TC, namely N_TC=2, this yields an SU(5) ETC theory. The SU(5) ETC symmetry can break to an exact residual vectorial SU(2) TC gauge group in three stages, characterized by three different mass scales, $\Lambda_j$, j=1,2,3. The ETC gauge bosons with masses $\Lambda_1$ mediate transitions between SM fermions of the first generation and the technifermions, and so forth for the other scales. With the values
$$\Lambda_1 \simeq 10^3\ TeV$$
$$\Lambda_2 \simeq 10^2\ TeV$$
and
$$\Lambda_3 \simeq few\ TeV$$
this model appears to be able to fit constraints on FCNC processes. Consider, for example, one of the most severe such constraints, arising from $K^0 – \bar K^0$ mixing. In early studies in the 1980’s, in the absence of an explcit ultraviolet ETC completion, one simply wrote down a generic form for the low-energy effective Lagrangian for this process, $\simeq (c/\Lambda_{ETC}^2)$ times the relevant four-quark operators, where $\Lambda_{ETC}$ was taken to be “the” ETC breaking scale. But the key observation is the following: in the initial $\bar K^0$, the d sbar pair can annhilate to produce a $V^2_1$ ETC gauge boson, where the indices are the gauged generational indices. But this cannot directly produce the s dbar pair of the K⁰, which requires a $V^1_2$ ETC gauge boson. Hence, in order for the K-Kbar transition to proceed, the the actual ETC propagator factor is not 1 over the mass squared of the $V^2_1$ gauge boson, $1/\Lambda_1^2$, but instead
$$(1/\Lambda_1^2)\ \Pi\ (1/\Lambda_1^2)$$
where Π denotes the requisite nondiagonal propagator insertion that takes a $V^2_1$ to a $V^1_2$. Using a reasonably ultraviolet-complete ETC theory, in the paper hep-ph/0308061, published as Phys. Rev. D 69, 015002 (2003), Appelquist, Piai, and I showed, via explicit calculation of the ETC gauge boson mixings, that the nondiagonal ETC gauge boson mixing term is generically of order the square of a low ETC breaking scale, essentially as a consequence of a residual approximate generational symmetry in the ETC theory. This suppresses the $K^0 -\bar K^0$ mixing strongly, by a factor like $(\Lambda_3/\Lambda_1)^2$, i.e., the coefficient of the four-quark operator is not $1/\Lambda_1^2$ but the much smaller $\Lambda_3^2/\Lambda_1^4$, which is sufficient to satisfy the constraint from the experimentally measured mixing and resultant $K_L – K_S$ mass difference. In this and a series of other papers, taking account of the mixing between ETC group eigenstates of fermions to form mass eigenstates, we also examined many other FCNC constraints on TC/ETC theories and showed that they appear to be able to be satisfied.

There are also FCNC processes that do not involve mixing of ETC gauge bosons. For example, the (conjugate of the) process $s \rightarrow d\ \mu^-\ e^+$ via exchange of a virtual $V^2_1$ ETC gauge boson gives rise to $K^+ \rightarrow \pi^+\ \mu^+ e^-$, for which the upper bound on the branching ratio (from the E865 experiment at BNL) is $BR(K^+ \rightarrow \pi^+\ \mu^+\ e^-) < 1.3 \times 10^{-11}$ (90 % CL). This is satisfied with the above value, $\Lambda_1 = 10^3\ TeV$.

Although Eric did not mention the effect that technifermions have on Z and W boson propagators, these also serve as a stringent constraint on technicolor models, especially the electroweak S parameter. However, because technicolor is strongly interacting at the scale of a few hundred GeV, it is not possible to use a perturbative estimate of S, and nonperturbative estimates based, e.g., on spectral function integrals, are difficult to make reliably for a walking TC theory since one cannot just scale up results from QCD. (There have been a number of papers over the years giving estimates on this, and I can send a list to anyone who is interested, but the issue is not resolved yet.)

In the SM, the electroweak symmetry breaking (EWSB) is produced by the vacuum expectation value of the hypothesized Higgs field. But this breaking is simply put in by hand, via an ad hoc choice of a negative coefficient for the quadratic Higgs term. No explanation is given in the SM of why this coefficient was not positive, when, a priori, it could just as well have been. Since the SM give no explanation for this negative sign of the coefficient, it does not provide a satisfactory fundamental explanation for EWSB. Indeed, it is interesting to recall that in both of the previous two main cases where a scalar field was used in phenomenological models for spontaneous symmetry breaking, namely in the Ginzburg-Landau free energy functional for superconductivity and the Gell-Man Levy sigma model for chiral symmetry breaking in hadronic physics, the microscopic physics did not involve the vacuum expectation value of a fundamental scalar field, but instead a bilinear fermion condensate – the Cooper pair in the BCS theory and the quark condensate in the case of QCD. In technicolor theories, it is precisely this type of bilinear fermion condensate – now involving technifermons,- which is responsible for electroweak symmetry breaking. Furthermore, the quark condensate in QCD already breaks electroweak symmetry. Thus, the original construction of technicolor models by Weinberg and Susskind was quite well motivated.

This is, of course, not to say that TC/ETC theories do not face many challenges. They are very ambitious, since they try to explain both EWSB and the spectrum of SM fermion masses, and no fully realistic model of this type has been constructed. Moreover, it is certainly true that far more people are currently working on various variants of SUSY models than on dynamical EWSB approaches such as TC/ETC. But at least readers should know that Eric’s comment refers to the old TC theories of the early 1980’s, which were, indeed, rejected; the results of more recent work indicate that modern walking TC theories appear to be able to satisfy FCNC constraints. PNGB’s and the S parameter are concerns, but, in the opinion of a number of us who work in this area, they do not obviously exclude these theories. In any case, we should know soon from the LHC whether dynamical EWSB via TC/ETC or some other possibility like low-energy SUSY is realized in nature.