The Mass of the Heavy Axion

If electroweak dynamical symmetry breaking is due to a chiral condensate of color sextet quarks, dynamics analagous to “walking technicolor” will enhance the condensate by orders of magnitude compared to the electroweak chiral scale. This enhancement compensates for the exponential suppression of electroweak scale color instanton interactions. As a result the η6 axion can naturally aquire an electroweak scale mass. Contributed to the XVI International Symposium on Lepton-Photon Interactions, Cornell University, Aug. 10-15th, 1993. ∗Work supported by the U.S. Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38 The existence of a massive color sextet quark sector could be an essential factor in the very high-energy consistency of QCD [1] and in the dynamics of the electroweak sector [2, 3]. In particular, low-energy Strong CP conservation in the normal triplet quark sector can be directly due to a sextet quark axion state[3], the η6. If this is the case the QCD interactions of the sextet sector are not CP conserving. As a result the η6 has large CP violating couplings to the electroweak sector that could be responsible for its production at LEP as an intermediate state in Z → γγ +μμ events [3]. Mixing of the color triplet and sextet sectors could also underlie CP violation in general. At first sight[4], the η6 is a conventional Peccei-Quinn axion[5]. However, because of its higher color constituents, color instanton interactions provide an additional contribution to its mass [6] not envisaged in the original Peccei-Quinn mechanism. It is very important to understand how large this contribution can be. In this paper we shall outline how the dynamics[7] of “walking technicolor”(WTC), as applicable to sextet electroweak symmetry breaking, can compensate for the normal suppression of instanton interactions. As a result it is natural to expect the η6 to have an electroweak scale mass. We shall suppose that the entry of a flavor doublet (U,D) of color sextet quarks into QCD above the electroweak scale can be described by an effective β-function. If we write β(α) = −β0α (q)/2π − β1α (q)/8π + ... (1) then for six color triplet flavors the normal two-loop calculation gives β0 = 11− 2nf/3 = 7, β1 = 102− 38nf/3 = 26 (2) whereas when the two sextet flavors are included we obtain[8] β0 = 7− 4T (R)n 6 f/3 = 7− 4( 5 2 )2/3 = 1/3, (3) and β1 = 26− 20T (R)n 6 f − 4C2(R)T (R)n 6 f = 26− 100− 66 2 3 = − 140 2 3 (4) where we have used T (R) = 5/2 and C2(R) = 10/3 for sextet quarks. The resulting β-function, β, for six flavors of light triplet quarks (we ignore subtleties associated with a heavy top quark) is shown in Fig. 1(a) and compared, in Fig. 1(b), 1 with β, the β-function obtained with the sextet quarks added. Noting the greatly expanded vertical scale in Fig. 1(b), it is clear that as soon as the sextet sector enters the theory the evolution of αs essentially comes to a halt. If, as we assume, the theory nevertheless evolves smoothly, but very slowly, into the small coupling asymptotically-free region then a natural way to connect the evolution before and after the sextet sector enters is via a β-function of the form shown in Fig. 2. Such an evolution (presumably) provides an oversimplified picture of the physics involved but will allow us to give an order of magnitude discussion of quantities which we hope is not too unrealistic. We assume, therefore, that above the electroweak scale the β-function can be taken to be a small (almost) momentum independent constant, which we denote as βc. This provides the essential prerequisite[7] for the application of WTC dynamics. A major ingredient of WTC, in addition to the existence of a small βc, is the assumption[7] that the linearised Dyson-Schwinger “gap equation” for the quark dynamical mass Σ(p), gives a semi-quantitive description of the dynamics of chiral symmetry breaking. The gap equation has a solution corresponding to spontaneous chiral symmetry breaking for αs ≥ αc, where αc is determined by an equation of the form C2(R)αc = constant (5) C2(R) is the Casimir operator already referred to above and, since C2 for sextet quarks is 5/2 the corresponding triplet Casimir, (5) is consistent with the sextet chiral scale being the electroweak scale provided that αs evolves logarithmically from the triplet chiral scale up to this scale, in the usual manner. If αs is momentum independent above the electroweak scale, as will be approximately the case if βc is small enough, then the solution of the gap equation for αs ∼ αc has the form[7] Σ(p) ∼ μ (p) (6) where μ is determined at the electroweak scale and should be essentially the sextet quark constituent quark mass i.e. μ ≥ 300 GeV. When this behavior is inserted into the perturbative formula[7] for the high-momentum component of the sextet condensate 〈