PHYSICAL REVIEW D 100, 095007 (2019) Crawling technicolor O. Cat`a* Theoretische Physik 1, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany † R. J. Crewther CSSM and ARC Centre of Excellence for Particle Physics at the Tera-scale, Department of Physics, University of Adelaide, Adelaide SA 5005, Australia ‡ Lewis C. Tunstall Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH–3012 Bern, Switzerland (Received 19 July 2019; published 8 November 2019) We analyze the Callan-Symanzik equations when scale invariance at a nontrivial infrared (IR) fixed α ’ α point IR is realized in the Nambu-Goldstone (NG) mode. As a result, Green s functions at IR do not scale in the same way as for the conventional Wigner-Weyl (WW) mode. This allows us to propose a new mechanism for dynamical electroweak symmetry breaking where the running coupling α “crawls” α α towards(butdoesnotpass) IR in the exact IR limit. The NG mechanism at IR implies the existence of a σ ϵ ≡ α − α massless dilaton , which becomes massive for IR expansions in IR andisidentifiedwiththe Higgs boson. Unlike “dilatons” that are close to a WW-mode fixed point or associated with a Coleman- Weinberg potential, our NG-mode dilaton is genuine and hence naturally light. Its ðmassÞ2 is ϵβ0 4 β0 −2 ˆ 2 β0 α proportional to ð þ ÞFσ hG ivac,where is the (positive) slope of the beta function at IR, ˆ 2 Fσ is the dilaton decay constant and hG ivac is the technigluon condensate. Our effective field theory for this works because it respects Zumino’s consistency condition for dilaton Lagrangians. We find a closed form of the Higgs potential with β0-dependent deviations from that of the Standard Model. Flavor- α ≲ α changing neutral currents are suppressed if the crawling region IR includes a sufficiently large range of energies above the TeV scale. In Appendix A, we observe that, contrary to folklore, condensates protect fields from decoupling in the IR limit. DOI: 10.1103/PhysRevD.100.095007 I. WW OR NG MECHANISM gauge theories [1–5] in that we have a true dilaton: it does AT FIXED POINTS? not decouple in the relevant conformal limit. Modern approaches to the conformal properties of field The discovery of the Higgs boson has focussed attention theories depend on a key assertion from long ago: renorm- on strongly coupled electroweak theories that can produce a alization destroys the conformal invariance of a theory at all light scalar. Crawling technicolor (TC) is a new proposal couplings α except at fixed points where the ψ function of for this. Gell-Mann and Low or the related β function of Callan and The main idea of crawling TC is that there is a conformal Symanzik (CS) vanishes. limit of dynamical electroweak theory at which the Higgs At a fixed point, exact conformal invariance corresponds boson corresponds to a zero-mass dilaton. This differs μ to the limit θμ → 0, where θμν is the energy-momentum fundamentally from recent work on “dilatonic” walking tensor (improved [6] when scalar fields are present). Like other global symmetries, this symmetry can be realized in *[email protected] two ways [7]: † [email protected] (1) The Wigner-Weyl (WW) mode, where conformal ‡ [email protected] symmetry is manifest, Green’s functions exhibit power-law behavior, and all particle masses go Published by the American Physical Society under the terms of to zero. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to (2) The Nambu-Goldstone (NG) mode, where there is a the author(s) and the published article’s title, journal citation, massless scalar boson of the NG type (a genuine and DOI. Funded by SCOAP3. dilaton) that allows other masses to be nonzero. 2470-0010=2019=100(9)=095007(42) 095007-1 Published by the American Physical Society CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019) There are no theoretical grounds for preferring one nor dimensional transmutation can be used to prove any- mode over the other: consistent model field theories that thing about the IR running of α (Sec. II and Appendix A). exhibit scale invariance in either the WWor NG mode exist. Indeed, there have been many attempts (reviewed in The choice ultimately depends on phenomenological Ref. [28]) to find IR fixed points for small Nf, but the requirements. outcome is unclear: there is no reliable theory of non- Dilaton Lagrangians were invented long ago [8–12]. perturbative gauge theory beyond the lattice, and lattice They were used recently to construct chiral-scale pertur- investigations of IR behavior for small Nf are in their bation theory [13–15] for three-flavor quantum chromo- infancy. The signal for a fixed point in the NG mode α α dynamics (QCD) with a nonperturbative infrared (IR) would be either tending to a constant value IR, or better fixed point. (given the scheme dependence of α), the presence of a light 2 Nevertheless, most theoretical discussions of IR fixed scalar particle (a pseudodilaton σ) with Mσ linearly points, such as all work on dynamical electroweak sym- dependent on the techniquark mass mψ as the TC limit metry breaking since 1997 [16], implicitly assume that the mψ → 0 is approached. Then conformal symmetry is WW mode of exact scale invariance is realized at the fixed hidden, so particle masses and scale condensates such 2 ψψ¯ point. This is natural if perturbation theory is the guide, as h ivac can be generated dynamically in the conformal α ⇁ α since the NG mode is necessarily nonperturbative. This limit IR, as in the left-hand diagram of Fig. 1. choice was also influenced by Wilson’s pioneering work on The result is a new theoretical possibility which we ultraviolet (UV) fixed points [17]. As he noted in footnote call “crawling technicolor.” The Higgs boson corresponds α 21 of Ref. [17], the NG scaling mode is a phenomeno- to the dilaton of the scaling NG mode at IR. Its small mass α α logical possibility but he had no way of applying his is due to the proximity of to IR at the Standard Model methods to that case. Accordingly, he designed his theo- (SM) energy scale, which is IR relative to the TC scale. retical framework for the WW mode and required [18] that Unlike all other Higgs-boson theories, crawling TC is an the nonlocality of rescaled interaction Hamiltonians be expansion about a limit D_ → 0 with Djvaci ≠ 0, where D short range. Subsequent observations in lattice QCD of generates dilatations. This theory is unique in being an long-range effects such as pions, which are not an obvious expansion about a genuine scaling limit in the NG mode consequence of Wilson’s method, indicate that a self- with a genuine dilaton. It has its own phenomenology consistent procedure to replace the Wilsonian framework (Secs. III–VIII), distinct from all others. when dynamics chooses the NG scaling mode may not be Unlike WW-mode fixed points, it is not possible to 1 necessary after all. understand this scenario from a perturbative point of view. There is extensive theoretical and phenomenological Already, small-Nf lattice studies have shown that the interest in the possibility that α runs to an IR fixed point dynamics of gauge bosons and fermions with zero in non-Abelian gauge theories. Investigations of this type Lagrangian (or “current”) masses can drive α while should be distinguished according to the manner in which producing hadronization, a nonperturbative effect. These conformal symmetry is realized. dynamical variables do not drop out of the analysis because The WW mode is associated with the conformal window, TC hadrons are created. They will remain the basic where the signal for a fixed point is the scaling of Green’s α variables of any nonperturbative method to show that IR functions. For Nf fermion gauge triplets, WW-mode fixed exists, irrespective of whether one has written down an points are seen in lattice studies [20–24] in the range effective low-energy theory or not. The effective theory will 9 ⪅ ≤ 16 α Nf . The lower edge of the conformal window is be the result, not the cause, of the existence of IR. thought to lie between Nf ¼ 8 and Nf ¼ 12, with the value The dynamical setting of our theory requires an analysis Nf ¼ 12 being debated currently [25–27]. At a WW-mode of the CS equations near IR fixed points in the NG mode, α something that, to the best of our knowledge, has not been fixed point ww, massive particles and all types of NG bosons are forbidden. attempted before. This topic is introduced in Sec. II. The main result is that conventional scaling equations are The NG mode corresponds to small values of Nf outside the conformal window. Much of this article is devoted to replaced by soft-dilaton theorems. That is why NG-mode explaining why this possibility is so often overlooked. In scale invariance produces scale-dependent amplitudes. particular, i) the lattice results above are not applicable because Green’s functions do not scale at a fixed point in 2A misconception that fermion condensates decouple in the IR the NG mode (Secs. II and VII), and ii) neither confinement limit has crept into the literature; reasons why that idea fails are given in Appendix A. Having the chiral condensate act as a scale condensate was proposed for strong interactions in Refs. [10,29]. 1Wilson’s framework has recently been used to analyze the NG This was later extended to QCD in chiral-scale perturbation mode at a UV fixed point in the OðNÞ model in three dimensions theory [13–15], of which crawling TC is a technicolored [19].