Technicolor (physics) - Walking Technicolor

Walking Technicolor

Since quark and lepton masses are proportional to the bilinear technifermion condensate divided by the ETC mass scale squared, their tiny values can be avoided if the condensate is enhanced above the weak-α_TC estimate in Eq. (2), .

During the 1980s, several dynamical mechanisms were advanced to do this. In 1981 Holdom suggested that, if the α_TC(μ) evolves to a nontrivial fixed point in the ultraviolet, with a large positive anomalous dimension γ_m for, realistic quark and lepton masses could arise with Λ_ETC large enough to suppress ETC-induced mixing. However, no example of a nontrivial ultraviolet fixed point in a four-dimensional gauge theory has been constructed. In 1985 Holdom analyzed a technicolor theory in which a “slowly varying” α_TC(μ) was envisioned. His focus was to separate the chiral breaking and confinement scales, but he also noted that such a theory could enhance and thus allow the ETC scale to be raised. In 1986 Akiba and Yanagida also considered enhancing quark and lepton masses, by simply assuming that α_TC is constant and strong all the way up to the ETC scale. In the same year Yamawaki, Bando and Matumoto again imagined an ultraviolet fixed point in a non-asymptotically free theory to enhance the technifermion condensate.

In 1986 Appelquist, Karabali and Wijewardhana discussed the enhancement of fermion masses in an asymptotically free technicolor theory with a slowly running, or “walking”, gauge coupling. The slowness arose from the screening effect of a large number of technifermions, with the analysis carried out through two-loop perturbation theory. In 1987 Appelquist and Wijewardhana explored this walking scenario further. They took the analysis to three loops, noted that the walking can lead to a power law enhancement of the technifermion condensate, and estimated the resultant quark, lepton, and technipion masses. The condensate enhancement arises because the associated technifermion mass decreases slowly, roughly linearly, as a function of its renormalization scale. This corresponds to the condensate anomalous dimension γ_m in Eq. (3) approaching unity (see below).

In the 1990s, the idea emerged more clearly that walking is naturally described by asymptotically free gauge theories dominated in the infrared by an approximate fixed point. Unlike the speculative proposal of ultraviolet fixed points, fixed points in the infrared are known to exist in asymptotically free theories, arising at two loops in the beta function providing that the fermion count N_f is large enough. This has been known since the first two-loop computation in 1974 by Caswell. If N_f is close to the value at which asymptotic freedom is lost, the resultant infrared fixed point is weak, of parametric order, and reliably accessible in perturbation theory. This weak-coupling limit was explored by Banks and Zaks in 1982.

The fixed-point coupling α_IR becomes stronger as N_f is reduced from . Below some critical value N_fc the coupling becomes strong enough (> α_{χ SB}) to break spontaneously the massless technifermions' chiral symmetry. Since the analysis must typically go beyond two-loop perturbation theory, the definition of the running coupling α_TC(μ), it’s fixed point value α_IR, and the strength α_{χ SB} necessary for chiral symmetry breaking depend on the particular renormalization scheme adopted. For ; i.e., for N_f just below N_fc, the evolution of α_TC(μ) is governed by the infrared fixed point and it will evolve slowly (walk) for a range of momenta above the breaking scale Λ_TC. To overcome the -suppression of the masses of first and second generation quarks involved in mixing, this range must extend almost to their ETC scale, of . Cohen and Georgi argued that γ_m = 1 is the signal of spontaneous chiral symmetry breaking, i.e., that γ_m(α_{χ SB}) = 1. Therefore, in the walking-α_TC region, γ_m ≅ 1 and, from Eqs. (2) and (3), the light quark masses are enhanced approximately by M_ETC/Λ_TC.

The idea that α_TC(μ) walks for a large range of momenta when α_IR lies just above α_{χ SB} was suggested by Lane and Ramana. They made an explicit model, discussed the walking that ensued, and used it in their discussion of walking technicolor phenomenology at hadron colliders. This idea was developed in some detail by Appelquist, Terning and Wijewardhana. Combining a perturbative computation of the infrared fixed point with an approximation of α_{χ SB} based on the Schwinger-Dyson equation, they estimated the critical value N_fc and explored the resultant electroweak physics. Since the 1990s, most discussions of walking technicolor are in the framework of theories assumed to be dominated in the infrared by an approximate fixed point. Various models have been explored, some with the technifermions in the fundamental representation of the gauge group and some employing higher representations.

The possibility that the technicolor condensate can be enhanced beyond that discussed in the walking literature, has also been considered recently by Luty and Okui under the name "conformal technicolor". They envision an infrared stable fixed point, but with a very large anomalous dimension for the operator . It remains to be seen whether this can be realized, for example, in the class of theories currently being examined using lattice techniques.