sign in sign up
<<
Home , Hypercolor (physics), Infrared fixed point, Technicolor (physics), Unparticle physics

D 100, 095007 (2019)

Crawling

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 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 , University of Bern, Sidlerstrasse 5, CH–3012 Bern, Switzerland

(Received 19 July 2019; published 8 November 2019)

We analyze the Callan-Symanzik equations when 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 . 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 at IR, ˆ 2 Fσ is the dilaton decay constant and hG ivac is the technigluon condensate. Our 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 . Flavor- α ≲ α changing neutral currents are suppressed if the crawling region IR includes a sufficiently large range of 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- dilaton. This differs μ to the limit θμ → 0, where θμν is the - 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 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 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 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 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 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 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]. In practice, the NG mode is more practical for IR fixed analogue. Reference [5] cited Refs. [13–15] as forerunners for points because soft-dilaton theorems are derived from low-energy their TC theory, but the IR fixed point considered in Ref. [5] is expansions. actually in the WW mode, as in walking TC.

095007-2 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019)

FIG. 1. Crawling and walking scenarios for the TC β function in SUð3Þ gauge theories with Nf Dirac flavors. Our proposal is shown in ψψ¯ ≠ 0 the left diagram: for small Nf values outside the conformal window, a fermion condensate h ivac forms at nonzero coupling and α remains nonvanishing at the IR fixed point IR (NG mode). The solid curve in the right diagram is for a walking gauge theory on the ψψ¯ ≠ 0 lower edge of the conformal window (large Nf consistent with h ivac ). The dashed curve is for a theory with a still larger Nf inside α ψψ¯ 0 α the conformal window; it has an IR fixed point ww where scale invariance is manifest (WW mode): h ivac ¼ . The gauge coupling α α ≫ α β of the solid curve walks past ww and continues into its IR region ww, where its j j is assumed to be large and (say) linear.

It reinforces a key point made above: lattice investigations invariance. That avoids problems with the conformal of the conformal window [16,20–22,24,27] assume a NG mode of λϕ4 theory found by Fubini 6 years power-law behavior for Green’s functions at fixed points, later [31]—a point largely overlooked since then. so they do not exclude NG-mode fixed points occurring at (4) Zumino’s condition is stable under NG-mode re- normalization of the nonlinear theory, where NG small Nf values [14,23]. Section III introduces crawling TC, a new dynamical bosons couple via derivative interactions. Following brief remarks about phenomenology in mechanism for electroweak theory. As indicated above, we Sec. V, the construction of the low-energy effective field assume the existence of an IR fixed point α in the NG IR theory (EFT) for crawling TC is considered in Sec. VI. The mode at which both electroweak and conformal symmetry 2 resulting EFT looks like an electroweak chiral Lagrangian are hidden . This happens if there is a fermion condensate [32–34] with a generic Higgs-like scalar field h [35–45], ψψ¯ ≠ 0 α h ivac at IR. Crawling TC differs from the standard but in our theory, the NG mode for exact scale invariance — — theory walking TC in several key ways that are sum- requires us to constrain h and verify that the equivalence marized in Fig. 1 (left diagram for crawling TC and right theorem permits our change of field variables σ → h.Asa diagram for walking TC). In crawling TC, the Higgs boson result, we obtain a closed form for the Higgs potential as a is identified as the pseudodilaton for α ≲ α . Unlike other IR function of h. It differs from the SM Higgs potential by “dilaton” proposals, our pseudodilaton does not decouple at terms depending on β0. α and so we can legitimately argue that it is naturally light IR Section VII contains a discussion of signals for NG- for α ≲ α . An explicit formula is derived for the pseudo- IR mode fixed points which may be seen in lattice inves- dilaton mass in terms of the parameters of the underlying tigations. In particular, we note that observations [46–49] of gauge theory at the fixed point; this follows from a direct 8 application of the CS analysis of Sec. II. a light scalar particle for Nf ¼ flavors may indicate the 8 This leads to the following general observations (Sec. IV): presence of an NG-mode IR fixed point in the Nf ¼ (1) A careful distinction must be made between a theory theory. Prompted by recent work [5,44,45] on “dilaton- like crawling TC where exact scale invariance is based” potentials, we consider testing our Higgs potential realized in the NG mode, and a large class of theories on the lattice in order to determine β0. based on scalons [30]. Scalons are not genuine dilatons The main text concludes in Sec. VIII with a brief review because the scale-invariant limit in which they become of the key points and an analysis showing that the effects of exactly massless is in the WW mode characteristic of flavor-changing neutral currents (FCNCs) can be naturally an unconstrained polynomial Lagrangian. suppressed in crawling TC. (2) In a Lagrangian formalism, a scaling NG mode is There are five appendices. Appendix A shows that the possible if a real scalar field χ that scales homo- assertion2 that condensates decouple in the IR limit is geneously obeys the scale-invariant constraint underivable and contradicts QCD. Appendix B reviews the χ > 0, e.g., when written ∼ expðσ=FσÞ in terms of original current-algebraic approach to soft-pion theorems an unconstrained field σ. However, a scaling NG and their extension to scale [29,50,51] and conformal mode is guaranteed to exist only if amplitudes are [52–54] symmetry. Appendix C examines how shown to depend on dimensionful constants in the and technigluon condensates may be defined without scale-invariant limit. relying on perturbative subtractions. Appendix D describes α (3) In 1970, Zumino (on page 472 of Ref. [11]) ob- the NG-mode scale-invariant world at IR: most amplitudes served that dilaton Lagrangians are consistent only depend on dimensionally transmuted masses, but coeffi- if ϕ4 interactions disappear in the limit of scale cient functions in short-distance expansions are shown to

095007-3 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019) obey the same scaling and conformal rules as leading standard prescription3 for a connected insertion of the singularities in WW-mode theories. Finally, Appendix E renormalized action into Green’s functions: reviews formulas for the anomalous dimension of the trace- Z operator. ∂ 1 α ⟷ − 4 ˆ 2 In standard terminology, a symmetry realized in the NG i d xG : ð3Þ ∂α μ J⃗ 4 mode is said to be “spontaneously broken.” As noted by ; conn Dashen long ago [55], this can be misleading: a global Here α is the renormalized TC coupling, μ is the renorm- symmetry in the NG mode is hidden, not broken. Similarly, J⃗ the term “electroweak symmetry breaking” misleads, since alization scale, and are source functions for renormalized O gauge-invariant physical quantities are necessarily invariant spectator operators f ng. This prescription is valid pro- O under the global chiral subgroup of the local gauge group. vided that each n is constructed from covariant derivatives α Of course, all of this is well known. However, in crawling but is otherwise independent in the following sense. TC, we have to deal with the NG mode not just for chiral Briefly, ignoring details of and ghosts, invariance but also for the less familiar case of scale the rule (3) is a consequence [62] of absorbing the bare invariance as well. In this paper, we take care to avoid gB into the functional measure the terms “spontaneous” and “electroweak symmetry ˆ ˆ ˆ ˆ breaking” because, in a scaling context, they are so easily DAμB → DAμB; AμB ¼ gBAμB: ð4Þ confused with explicit symmetry breaking. α O Throughout, the gauge constant g and coupling ¼ Then all operators ð nÞB constructed from covariant g2=ð4πÞ refer to TC. Our notation for the gluon and derivatives alone, including A a electroweak gauge fields will be Gμ ;Wμ;Bμ, with gs, 0 A a ˆ ˆ gw, gw and Gμν;Wμν;Bμν for the corresponding coupling GμνB ¼ gBGμνB; ð5Þ constants and field-strength tensors. To indicate TC fields, ˆ A ˆ A we will add a hat, i.e., Gμ and Gμν. For symbols like θμν and are gB independent. In the action, all dependence on gB σ 1 2 − 1 Gˆ 2 the dilaton , we will let the context distinguish between appears as a constant source =gB for 4 B. A textbook TC and QCD. Dilaton decay constants are Fσ for TC and fσ 1 2 argument [63] relates terms linear in the sources =gB and for QCD [13–15]: ⃗ J B to their renormalized counterparts: θ σ 3 − 2 Z hvacj μνj ðqÞiTC ¼ðFσ= Þðqμqν gμνq Þ; 1 4 − Gˆ 2 J⃗ O⃗ θ σ 3 − 2 d x 2 B þ BðxÞ · BðxÞ hvacj μνj ðqÞiQCD ¼ðfσ= Þðqμqν gμνq Þ: ð1Þ 4g Z B 1 2 The phases of jσi are chosen such that Fσ and fσ are 4 − Gˆ J⃗ O⃗ J 2 TC;QCD ¼ d x 4 2 þ ðxÞ · ðxÞ þ Oð Þ: ð6Þ positive. g

2 2 II. NG-MODE SOLUTIONS OF THE Then the rule follows from α ¼ g =ð4πÞ. The term OðJ Þ CS EQUATIONS represents subtractions of quadratic or higher order in J for multiple insertions of the composite operators On. The basic idea of this section is to understand the CS In our analysis, the operator in the technigluon con- equation as a Ward identity for scale transformations near densate appears as a spectator, so an α-independent choice an IR fixed point in the NG mode. The method is similar to such as the original non-Lagrangian procedure for analyzing chiral condensates; see Appendix B for a review. 1 α O Gˆ 2 Gˆ 2 Let us begin with TC where the Lagrangian is chiral ¼ 4π2 ¼ π ð7Þ SUðNfÞL × SUðNfÞR symmetric. For scale transforma- tions, the relevant operator is the divergence of the is appropriate when using Eq. (3) (the normalization is D αθ dilatation current μ ¼ x αμ. It is governed by the trace that originally chosen [64] for the gluon condensate). anomaly [56–59], which for massless fermion fields takes In Appendix C, we show that there is a multiplicatively the form renormalizable version of O, i.e., one which does not mix with the identity operator I. These twin requirements are β α ∂μD θμ ð Þ ˆ A ˆ Aμν − ˆ A ˆ Aμν μ ¼ μ ¼ 4α fGμνG hGμνG ivacg; ð2Þ 3This is an example of the renormalized action principle. The ˆ 2 ˆ 2 simplest version of it [60] is for minimal schemes such as where hG ivac is the technigluon condensate hvacjG jvaci dimensional , where gauge-invariant composite and jvaci is the nonperturbative vacuum state. We apply operators have block-diagonal renormalization matrices. See the Eq. (2) at zero momentum transfer, where there is a discussion in Ref. [61].

095007-4 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019)

α essential if ambiguities in the definitions of gluon and If scale invariance is realized in the NG mode at IR,as technigluon condensates are to be avoided. we propose, there are amplitudes for which the right-hand α θμ → 0 Consider the CS equation for (say) the vacuum expect- side of Eq. (9) does not vanish at IR as μ . That can ation value (VEV) of OðxÞ: occur if the sum over physical states jni in the dispersion θμ O 0 integral for Th μðxÞ ð Þisubtr includes the exchange of a ∂ ∂ σ μ β α γ α O 0 0 pseudodilaton : ∂μ þ ð Þ ∂α þ Oð Þ hvacj ð Þjvaci¼ : ð8Þ X X I ¼ jnihnj¼jσihσjþ jnihnj: ð11Þ Let us move the β∂=∂α term to the right-hand side of this n n≠σ formula. Then Eqs. (2) and (3) imply that the right-hand side is given by a suitably renormalized zero-momentum P μ Here includes multi-NG boson states and states insertion of −iθμ: n≠σ containing non-NG particles; the latter have invariant mass ≠ 0 → 0 ∂ Mnon-NG in the scale-symmetry limit Mσ . The μ γ α O 0 σ ∂μ þ Oð Þ hvacj ð Þjvaci exchange of produces a pole term Z Z 4 iq·x μ ¼ −i lim d xe ThvacjθμðxÞOð0Þjvaci : ð9Þ 4 μ σ →0 subtr iq·x θ O 0 pole q d xe Thvacj μðxÞ ð Þjvacisubtr

μ i The notation hisubtr indicates that small-x singularities have θ 0 σ σ O 0 ¼hvacj μð Þj ðqÞi 2 2 h ðqÞj ð Þjvaci; ð12Þ been subtracted to renormalize the answer minimally with a q − Mσ μ counterterm of order θμ; it will not affect our conclusions. 2 The result (9) remainsQ valid [modulo OðJ Þ terms in which does not depend on the subtraction procedure. O γQ α γ Eq. (6)] for a product if Oð Þ is the sum of Taking the limit q → 0 with Mσ ≠ 0, we see that the functions of individual spectator operators. Note that the zero-momentum μ limit q → 0 in Eq. (9) is taken for θμ ≠ 0 when there are no μ massless states to which θμ can couple. 2 2 2 i=ðq − MσÞj 0 ¼ −i=Mσ; ð13Þ Having taken the limit q → 0, what happens to the right- q¼ hand side of Eq. (9) if there is an IR fixed point which 4 μ 2 allows a second limit θμ → 0 to be taken? cancels the Mσ dependence of the matrix element The standard procedure is to set all amplitudes involving θμ μ 2 μ to zero. In effect, this assumes that there is no NG hvacjθμjσi¼−FσMσ; ð14Þ mechanism, i.e., that scale invariance is realized in the WW mode: where Fσ is defined in Eq. (1). In the scale-invariant limit, σ ∂ Fσ remains nonzero because is a dilaton, and so Eq. (9) μ γO α vac O 0 vac 0: 10 implies our key result: ∂μ þ ð wwÞ h j ð Þj iww ¼ ð Þ ∂ Then the theory at a WW fixed point α is manifestly scale ww μ þ γOðαÞ hvacjOð0Þjvaci and conformal invariant. Green’s functions scale according ∂μ NG to power laws, with μ dependence reduced to trivial factors → σ 0 O 0 θμ → 0 Fσh ðq ¼ Þj ð ÞjvaciNG; μ : ð15Þ −γOðα Þ μ ww . There is no , so particles (if they exist) are massless. Dimensional transmutation does not occur. ≠ σ α States jn i do not affect this result: at most, relative to In particular, fermions cannot condense at ww if scale 2 the σ-pole term, their contributions are OðMσ ln MσÞ for invariance is in the WW mode. Instead, it must be assumed 2 that fermion condensation is possible only when scale two-dilaton states and OðMσÞ for other states, including O symmetry is explicitly broken. For example, in walking the subtraction. EquationQ(15) remains valid if is replaced O O gauge theories [16], α is thought to vary rapidly after it by unordered products n nðynÞ of operators n with μ γ α α θ scaling functions nð Þ, walks past ww because, by assumption, a large μ is necessary for the region where hψψ¯ i ≠ 0 (Fig. 1, right vac ∂ X Y diagram). μ γ α vac O y vac ∂μ þ nð IRÞ nð nÞ n n NG 4Care must be taken with the order of limits, as noted in Y σ 0 O α ⇁ α Appendix B for the chiral case. The analysis, but not the final ¼ Fσ ðq ¼ Þ nðynÞvac ; IR ð16Þ answer, depends on which limit is taken first. n NG

095007-5 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019) provided that light-like momenta in 0þþ channels with σ Connecting this with the physical region involves poles are avoided. a subtlety: in contrast with UV fixed points, the dyna- Two features of Eqs. (15) and (16) are unfamiliar: mical dimension of an operator may change at an IR (1) They are soft-meson theorems which have not been fixed point. That is because operator dimension is deter- derived directly from an effective Lagrangian. That mined by short-distance behavior: in the physical region 0 α α reflects the fact that effective Lagrangians for scale < < IR, asymptotic freedom requires it to take its invariance were not constructed with the CS equa- canonical value tion in mind. (2) The CS equation cannot be formulated at α ¼ α in can IR dynamical dimension of O ¼ dO ; 0 < α < α the presence of a dilaton. Results such as Eq. (15) IR α ⇁ α ð20Þ refer to the limit IR. The rest of this section examines the peculiarities of IR fixed points in the NG scaling mode. up to renormalized Schwinger terms (Appendix C1), α The most important point is that the world at IR is whereas dO is determined by the short-distance properties 0 α α α not the same as the physical world on < < IR.In of the world at IR (Appendix D). α α particular, short-distance behavior at IR is not governed by In the limit of scale invariance at IR, there is a continuum because α is fixed: it cannot run of vacua related by scale transformations. In the first half of α 0 5 α towards the origin ¼ . The theory at IR is exactly Appendix D, we explain why physics does not depend on 0 α α scale invariant but in the NG mode, amplitudes may be which vacuum is chosen. For < < IR, scale invariance complicated functions of dynamically transmuted scales. is broken explicitly, and there is a unique vacuum to which Exceptions are coefficient functions of operator product quantities like γOðαÞ refer. can expansions at short distances, which are manifestly scale The relation between dO and dO can be easily seen by and conformal covariant; the proof (Appendix D) is similar considering the connected two-point function to that for chiral symmetry [65]. Consider what happens at α when the conserved Δþ O O 0 IR ðxÞ¼hvacj ðxÞ ð ÞjvaciNG; conn ð21Þ dilatation current Dμ carries momentum q ≠ 0 in a scaling → 0 Ward identity (Appendix B3), and then the limit q is at short distances x ∼ 0, where the effects of dimensional θμ → 0 taken, i.e., after the limit of scale invariance μ . That transmutation are nonleading. In the physical region yields a soft-dilaton formula 0 α α < < IR,wehave σ 0 O 0 O 0 Fσh ðq ¼ Þj ð ÞjvaciNG ¼ dOhvacj ð ÞjvaciNG ð17Þ 2 − can 2 2 2γ β ΔþðxÞ ∼ fconstantgðx Þ dO ðlnðμ x ÞÞ 1= 1 ; ð22Þ where where O α dO ¼ dynamical dimension of at IR: ð18Þ 2 βðαÞ ∼−β1α and γOðαÞ ∼ γ1α; α → 0 ð23Þ In an effective Lagrangian formalism for dilatons with O represented by an external effective operator define the one-loop coefficients β1 > 0 and γ1 > 0 for an asymptotically free theory. At α , according to Eq. (16), O O σ IR eff ¼h ivac expðdO =FσÞ ΔþðxÞ satisfies the relation 2 ¼hOi f1 þ dOσ=Fσ þ Oðσ Þg; ð19Þ vac ∂ þ μ þ 2γOðα Þ Δ ðxÞ Eq. (17) arises from the term linear in σ. The Käll´en- ∂μ IR Lehmann representation requires dO ≥ 1 for all local opera- ¼ FσhσjOðxÞOð0Þjvaci : ð24Þ tors O ≠ I [66], so every soft-σ amplitude hσjOjvaci which NG; conn θμ → 0 does not vanish in the limit μ corresponds toQ a scale Consider contributions to each side from the operator condensate O ≠ 0 O → O h ivac (and similarly for n n). product expansion for OðxÞOð0Þ. Clearly, the term propor- Not all scale condensates are chiral condensates, but if tional to the dimension-0 identity operator I that contrib- ψψ¯ ≠ 0, the vacuum at α breaks both chiral and h ivac IR utes to the left-hand side of Eq. (24) dominates the leading scale invariance. contribution to the right-hand side from a dimension ≥ 1 operator ≠ I: 5 In chiral-scale perturbation theory for three-flavor QCD [13,14], the asymptotic value RIR of the Drell-Yan ratio for ∂ þ − α α μ 2γ α Δþ ∼ 0 e e annihilation at s ¼ sIR is not the same as the QCD value þ Oð Þ ðxÞ : ð25Þ 2 2 4 ≲ ≲ 3 1 ∂μ IR RUV ¼ . The most recent estimate is . RIR . [15].

095007-6 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019)

So the leading singularity of Δþ has μ dependence observed, in agreement with Eq. (10). The results −2γOðα Þ 6 ∼μ IR . Since x is the only other dimensionful quantity define what is meant by the conformal window [22]. α α which can appear in the result, we have for ¼ IR (2) For smaller values of Nf outside the conformal window, where dimensional transmutation occurs, it can þ 2 −d 2 2 −γOðα Þ Δ ðxÞ ∼ fconstantgðx Þ O ðμ x Þ IR ; ð26Þ remains to be seen if there are NG-mode IR fixed points [23]. If so, scale invariance is not manifest which corresponds to because of Eqs. (15) and (16). Signals for this on the lattice will be considered in Sec. VII. can Let us recall how, despite the absence of fermion mass dO ¼ dO þ γOðα Þ: ð27Þ IR terms in the Lagrangian, dimensional transmutation can M The role of dimensional transmutation requires some arise in massless QCD and TC. Observable constants discussion. Often it is regarded as a one-loop phenomenon with dimensions of mass, such as decay constants and which, in that context, breaks scale invariance explicitly, as non-NG masses, are permitted because renormalization Coleman and Weinberg [67] discovered for scalar quantum group (RG) invariance electrodynamics (QED) before the discovery of asymptotic ∂ ∂ μ β α M 0 freedom. But in non-Abelian gauge theories, the one-loop ∂μ þ ð Þ ∂α ¼ ð30Þ approximation makes sense only in the UV limit, where Λ there is a dimensionally transmuted scale QCD=TC which is consistent with M being proportional to the sole scale in 2 Λ2 μ normalizes arguments of UV logarithms lnðq = QCD=TCÞ the theory, the renormalization scale : as q → ∞. Of course, dimensional transmutation persists Z α outside the UV region because it is necessary to incorporate M μ − β 0 κ α ¼ exp dx= ðxÞ ; < M < IR: ð31Þ nonperturbative effects like fermion condensation into κM QCD and (by analogy) TC. Chiral perturbation theory and EFTs for TC are low-energy expansions with their own Here κM is a dimensionless constant which depends on M dimensionally transmuted scales but not on α or μ. As is well known [69], the non- perturbative nature of M can be verified by considering Λ 4π α ∼ 0 μ χPT ¼ ffπ or Fπg and non-NG masses ð28Þ the limit at fixed : from Eq. (23), there is an essential singularity due to the factor expf−1=ðβ1αÞg which ensures Λ α which have nothing to do with QCD=TC. They normalize the absence of a Taylor series in . arguments of IR logarithms In the IR limit, two points of view are possible. One, to be discussed below, is to treat amplitudes A as functions of 0 2 2 α, μ and various momenta fpg and consider what happens lnðq · q or fm or Mgπ=Λ Þð29Þ χPT α α as tends to IR. The other is to note that, since observable 0 2 constants M are annihilated by the CS differential operator for q · q ∼ fm or Mgπ ∼ 0. In chiral-scale perturbation in Eq. (30), they act as constants of integration in CS theory or crawling TC, dimensional transmutation persists equations for amplitudes, i.e., the CS equations allow any at α or α through dependence on the dilaton decay sIR IR dependence on M consistent with engineering dimensions. constants 4πfσ or 4πFσ. There are no theoretical reasons, Therefore, if an amplitude is observable and hence RG beyond a disregard for old but well-established work on the invariant, its dependence on α and μ can be entirely scaling NG mode for strong interactions [51], to suppose replaced by a dependence on the transmuted masses M that dimensional transmutation a) necessarily “turns itself alone. This : we want to apply approximate scale off” as a fixed point is approached, or b) prevents an NG- λ invariance to physical amplitudes, so the limit θλ → 0 is mode fixed point from forming anywhere outside the M μ conformal window, with scale invariance hidden and not taken at fixed , not fixed . explicitly broken.7 At present, lattice calculations are the Alternatively, as shown by the analysis leading to only guide: Eq. (27), it can be useful to consider amplitudes depending on operators O such as ðα=πÞGˆ 2 which are not RG (1) If Nf is large enough, WW-mode IR fixed points with manifestly scale-invariant Green’s functions are invariant. Such amplitudes can be treated as functions of M and μ, with residual dependence on μ being retained in

6 the scale-invariant theory at the fixed point. For example, By convention, the normalization of composite field operators O can the amplitude h ivac appearing in Eqs. (15) and (17) can be excludes dimensionful factors, so dO is the engineering dimen- sion of O. written as 7 “ ” As observed in a note added in Ref. [14], footnote 20 of can γ γ O 0 ∼ MdO M μ O MdO μ O Ref. [4] missed these points. Contrary to footnote 8 of Ref. [68],it hvacj ð ÞjvaciNG cO ð = Þ ¼ cO = is not possible to deduce anything about IR fixed points from the one-loop formula for the beta function. ð32Þ

095007-7 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019)

λ for θλ → 0, where the dimensionless constant cO does not depend on μ or M. The result (32) must not be confused with hyperscaling relations in mass-deformed theories [70–72], where the conformal invariance of a gauge theory in the WW mode is explicitly broken by a mass term −mψψ¯ in the Lagrangian. Hyperscaling is a property of the scaling WW mode. All FIG. 2. Generation of the mass M of a non-NG particle P via O “ ” μ P ¯ VEVs h ivac and hadron masses Minside inside the the dominant σ pole in hPjθμjPi, where −gσPPPP defines the 8 μ conformal window scale with fractional powers of the σPP coupling. In the scale-invariant limit θμ → 0, MP remains Lagrangian parameter m: nonzero.

can γ α 1 γ α O ∼ fdO þ Oð wwÞg=f þ mð wwÞg h ivac m ; dependence of any amplitude within the interval 0 < α < 1 1 γ α ∼ =f þ mð wwÞg α Minside m : ð33Þ IR would be a disaster: it would indicate a lack of analyticity, such as a Landau pole, at a finite space-like O Not surprisingly, both h ivac and Minside in Eq. (33) momentum. μ α → α vanish in the limit m → 0 of scale invariance. That is not Similarly, the fixed- limit IR applied to Eq. (31) is the case for Eq. (32) because, unlike m, M is not a variable singular: current fermion mass in a Lagrangian. Rather, M is a fixed M ∼ μ α − α −1=β0 μ non-Lagrangian constant associated with a condensate, ð IR Þ fconstantg; fixed : ð35Þ such as a decay constant or a non-NG technihadron mass arising from nonzero constituent masses of m ¼ 0 fermions Note that this implies (Appendix A). The key property of a scaling NG-mode dO fixed point is that amplitudes depend on the nonzero scales ∂ dOM μ MdO ∼ ð36Þ M in the scale-invariant limit θμ → 0 (Appendix D), so ∂α α − α β0 ð IR Þ unlike Eq. (33), the right-hand side of Eq. (32) does not vanish in that limit. Furthermore, if we add −mψψ¯ to the and hence, from Eq. (32), Lagrangian, nonzero results such as Eq. (32) are corrected by terms linear in m, as expected in chiral perturbation ∂ dO vac O 0 vac ∼ vac O 0 vac ; theory or its chiral-scale extension [13–15]; fractional ∂α h j ð Þj iNG α − α β0 h j ð Þj iNG ð IR Þ powers of m are never seen. There is no such thing as hyperscaling in crawling TC. ð37Þ Now let us check what happens when amplitudes are α μ α α μ which shows that Eq. (34) is consistent with the soft-dilaton treated as functions of ; ; fpg as tends to IR at fixed . β∂=∂α theorem (17). Comparison with the term in the CS equation (8) M shows that the right-hand side of Eq. (15) arises from the Equation (35) implies that, for to remain finite in the θλ → 0 μ singular α dependence of the condensate as α approaches scaling limit λ , tends to 0 according to the rule α for fixed μ: IR μ ∝ α − α 1=β0 ð IR Þ : ð38Þ ∂ 1 Fσ vac O 0 vac ∼ σ O 0 vac : 34 ∂αh j ð Þj iNG α −α β0 h j ð Þj iNG ð Þ Singularities are removed when the μ, α dependence of IR amplitudes is eliminated in terms of physical quantities.10 This singularity is to be expected.9 The operator ∂=∂α A simple example of μ dependence being related to a inserts O at zero-momentum transfer, so a pole due to a soft-dilaton amplitude is when M in Eq. (31) is the mass zero-mass particle (the dilaton) coupled to O will produce a MP of a non-NG particle P. Then the scalar analogue singular result. Note that it is only at the fixed point that this [51,74] of the Goldberger-Treiman relation (Fig. 2) applies: is allowed. A singularity or lack of smoothness in the α ∂ μ ∂μ MP ¼ MP ¼ FσgσPP: ð39Þ 8In walking TC, there must be a at the sill of the conformal window [73] that causes fermions to condense and hence create NG and non-NG technihadrons outside the con- We close this section with a discussion of the analogue of formal window, but the distinction between spectra inside and the low-energy theorem (15) for QCD. There the relevant outside the window is usually left unclear. We reserve the term “condensate” for VEVs which are nonzero in a symmetry limit 10 such as m → 0. This is similar to what happensffiffiffiffiffiffi in the large-Nc limit of QCD, 9 p Perhaps this could be exploited in searches for NG-mode where the singularity fπ ∼ Nc is eliminated by writing every- fixed points on the lattice (Sec. VII). thing in terms of the pion decay constant fπ.

095007-8 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019) equations have extra terms because q have mass and the CS equation m ≠ 0. When the trace anomaly [56–59] (with the vacuum q X expectation value subtracted)11 ∂ ∂ ∂ μ þ βðαsÞ − γmðαsÞ mq þ γOðαsÞ ∂μ ∂αs ∂mq β α X q θμ ð sÞ A Aμν 1 γ α ¯ − μ ¼ 4α GμνG þð þ mð sÞÞ mqqq fVEVg × hvacjOð0Þjvaci¼0 ð41Þ s q ð40Þ are compared, we find

Z ∂ X 4 iq·x μ μ þ γOðαsÞ hvacjOð0Þjvaci¼i lim d xe T vac mqqq¯ ðxÞ − θμðxÞ Oð0Þvac : ð42Þ ∂μ q→0 q subtr → 0 If heavy quarks have been decoupled, and the limit Pmq is taken for the light quarks q ¼ u, d, s as the IR fixed point is θμ ¯ – approached, dilaton pole terms from both μ and q mqqq may survive the limit [13 15]:   ∂ 2 2 1 2 2 μ γO α vac O 0 vac → fσ σ O 0 vac 1 − 3 − γ α fπ=fσ m mπ mσ : 43 ∂μ þ ð sÞ h j ð Þj i h j ð Þj i ð mð sIRÞÞð Þ K þ 2 ð Þ

≃ 441 – III. CRAWLING TC: HIDDEN mf0 MeV [79 81] (evidence which seems to have ELECTROWEAK-SCALE SYMMETRY been mostly overlooked in the TC literature), and also for a ≃ 125 – narrow Higgs boson h at mh GeV [82,83]. Given TC is based on the idea [76 78] that electroweak 500 ? symmetry “breaking” is the dynamical effect of a gauge these facts, can h be the TC version of the f0ð Þ At first theory which resembles QCD but whose coupling becomes sight, the answer to this question is negative. An applica- strong at scales of a few TeV. The trigger for this effect tion of the scaling rules mentioned above requires the TC ψψ¯ ≠ 0 analogue f0T of f0 to have a large mass [84] is a techniquark condensate h ivac . The resulting technipions become the longitudinal components of the 0 m ≈ ðv=fπÞm ¼ OðTeVÞ; ð44Þ W and Z bosons, while the masses and couplings of the f0T f0 other technihadrons are estimated by scaling up QCD also, they seem to imply an OðTeVÞ width except for the quantities, where the v ≃ 246 GeV plays fact that the f0ð500Þ has plenty of phase space for its decay the role of the pion decay constant fπ ≃ 93 MeV. into two pions, whereas there are no technipions for f0T to An attractive feature of TC is that the decay into and (for a mass of 125 GeV) no phase space for is avoided: the mechanism for does not it to decay into WþW− or Z0Z0. But it is evident that this rely on elementary Higgs-like scalars. Instead, masses are estimate for the mass is much too large. generated dynamically through dimensional transmutation A convincing explanation for why the observed mass [67], as in QCD. mh ≃ 125 MeV is so small relative to TeV scales is hard to When TC was invented, the Particle Data Group (PDG) find. That is a key problem shared by all theories of PC 0þþ tables did not include QCD scalar J ¼ resonances dynamical Higgs mass generation, including TC and its ≈1 12 below GeV, so for many years, it was thought, by extensions. The most promising strategy is to suppose that analogy with QCD, that TC scalar particles would not be the Higgs is a pseudo-NG (pNG) boson of a hidden seen below the TeV scale. symmetry. Then the mass acquired by the pNG boson 0þþ There is now strong evidence for a light, broad due to explicit symmetry breaking is protected by the 500 resonance f0ð Þ in the QCD meson spectrum with mass underlying symmetry [85,86]. A light Higgs mass can arise if explicit symmetry breaking is due to physics at the 11 electroweak scale and hence small relative to the scale of For consistency, the γm terms in Eqs. (40) and (41) must have opposite signs (unlike Ref. [14] where conventions dynamical symmetry breaking. were changed during review). Here we choose the definition In [87–91], where the hidden γ −μ∂ ∂μ ¯ m ¼ ln mq= [75]. Then qq has dynamical dimension symmetry is internal, this mechanism is well understood: 3 − γ α at a QCD fixed point α , and similarly for ψψ¯ in mð sIRÞ sIR the Higgs boson and all would-be NG bosons are placed in crawling TC, where the notation becomes γ α and α . mð Þ IR 5 12The ϵð700Þ was excluded from the PDG tables in 1974. Its the same multiplet of an extended group such as SOð Þ – successor f0ð500Þ was first mentioned in 1996, but became a [92 94]. For a recent review of these models, see chapter III well-defined resonance only in the 2008 tables. of Ref. [95].

095007-9 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019)

Our focus is on the main alternative: broken scale and A precise formula for the pseudodilaton mass can be conformal invariance with a “dilatonic” Higgs boson. A obtained as an important application of Eq. (15). The result dilaton, or NG boson for conformal invariance, has the is an analogue of the Gell-Mann–Oakes–Renner relation property that it couples to particle mass [74]. At first, this [99] for 0− mesons. idea was applied to strong interactions, as reviewed in To see this, consider the case O ¼ Gˆ 2 with each side of 1 Ref. [51]. A few years later, it was noted [96] that, in the Eq. (15) multiplied by the factor 4 β=α. The result is SM, tree-level couplings of the Higgs field are dilaton-like, i.e., they couple to mass. The literature on dynamical Higgs β α ∂ ð Þ μ γ α ˆ 2 → σ θμ bosons spawned by this observation is unfortunately not 4α ∂μ þ Gˆ 2 ð Þ hG ivac Fσh j μjvaci; ð50Þ consistent about the meaning of “dilaton” and overlooks the need to hide conformal invariance as it becomes exact. where a simple derivation [13,14,61,100] (discussed in The clearest examples of this are walking TC theories Appendix E) implies with dilatonic modifications [1–5]. Consider the walking region shown in the right-hand diagram of Fig. 1. The WW- 0 0 γ ˆ 2 ðαÞ¼β ðαÞ − βðαÞ=α; β ðαÞ¼∂βðαÞ=∂α; ð51Þ mode fixed point lies within the conformal window where G dilatons cannot exist, but it is supposed that in the walking 2 for the anomalous scaling function of Gˆ . Equation (14) region at the edge of the window, dynamics is affected by implies that the right-hand side of Eq. (50) is given by “dilatons” due to a field dependence ∼ expðσ=FσÞ in an 2 2 −MσFσ. For an IR expansion in ϵ ¼ α − α ≳ 0 about the effective Lagrangian. It is then argued (following a sug- IR fixed point, the left-hand side reads gestion in Ref. [97]) that these “dilatons” couple to an operator which is small near the scale-symmetry limit βðαÞ ∂ 2 μ γ 2 α ˆ þ Gˆ ð IRÞ hG ivac θμ α − α 4α ∂μ μ ¼ Oð wwÞ; ð45Þ ϵβ0 4 β0 − ð þ Þ ˆ 2 ϵ2 ¼ 4α hG ivac þ Oð Þ; ð52Þ and so they have a small mass protected by scale symmetry IR α at ww. β0 β0 α The flaw in this argument becomes evident when the where the ¼ ð IRÞ is positive (Fig. 1, relation left diagram) and we have used dimensional analysis to trade the μ∂=∂μ term for the engineering dimension of 2 − θμ σ α − α Gˆ 2 . Equations (50) and (52) imply the desired mass MσFσ ¼ hvacj μj i¼Oð wwÞð46Þ h ivac relation is considered. In walking TC, the so-called “dilaton” ϵβ0 4 β0 2 ð þ Þ ˆ 2 2 decouples from the theory as the WW-mode fixed point Mσ G O ϵ ; ¼ 4α 2 h ivac þ ð Þ ð53Þ is approached, IRFσ

13 ∼ 0 α ∼ α which exhibits the pseudo-NG nature of σ explicitly. Fσ for ww ð47Þ 2 The requirement Mσ ≥ 0 fixes the sign of the condensate: ˆ 2 ≥ 0 because there can be no scales at α . Therefore, no hG ivac . ww α conclusion can be drawn about Mσ from Eq. (46). The This mass is protected by scale invariance at IR because only general theorem governing particle decoupling is that the condition (48) ensures that our dilaton is a genuine NG of Appelquist and Carazzone [98] for heavy particles. boson. That is what allows us to identify the pseudodilaton α In crawling TC (left diagram in Fig. 1), the IR fixed point in the crawling region near IR as the Higgs boson with is in the NG mode, not the WW mode. As noted above mass much smaller than the TeV scale of TC. Eq. (15), the (pseudo)dilaton does not decouple as the fixed This conclusion also applies if the techniquarks are given point is approached, a current mass mψ , as in the case of TC lattice simulations where an extrapolation to the chiral limit mψ → 0 must be → ≠ 0 α → α performed. Since the fermion mass is an additional source Fσ constant as IR ð48Þ of explicit scale symmetry breaking, the IR expansion in ϵ so from must be augmented by powers of mψ .

2 μ 13 MσFσ ¼ −hvacjθμjσi¼Oðα − α Þ; ð49Þ A similar formula in Refs. [1,3] lacks the anomalous IR 4 β0 dimensionμ þ . The main problem is that its derivation assumes θμ ∼ 0 near a WW fixed point α , where condensates 2 α − α ww we can safely conclude that Mσ is Oð IRÞ and tend to zero and σ is not a pseudodilaton because it decouples hence small. (Fσ ∼ 0).

095007-10 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019)

Repeating the same steps that led to our mass for- We are revisiting this topic because the NG and WW mula (53), but this time for the operator11 scaling modes are still being confused and Zumino’s condition is not being respected. This seems to stem from β α X O ð Þ ˆ A ˆ Aμν 1 γ α ψψ¯ two 1976 papers, both of which a) referred to the NG mode ¼ 4α GμνG þð þ mð ÞÞ mψ ; ð54Þ ψ of conformal invariance but not to the 1969–1970 literature, and b) have attracted a lot of interest since then: we find (1) Gildener and Weinberg [30] used the term “scalon” μ to describe a scalar particle which couples to θμ but ϵβ0 4 β0 μ ˜ 2 ˜ 2 ð þ Þ ˆ 2 − 3 − γ 1 γ ψψ¯ where the limit θμ → 0 is in the WW scaling mode. MσFσ ¼ 4α hG ivac ð mÞð þ mÞmψ h ivac IR It is therefore not a dilaton, contrary to remarks in an 2 2 þ Oðϵ ; ϵmψ ;mψ Þ; ð55Þ early paragraph of Ref. [30] and to assertions in subsequent literature [103–110]. ˜ ˜ ’ “ ” where Mσ and Fσ are the dilaton mass and decay constant (2) Fubini s new approach to conformal invariance γ γ α λϕ4 in the presence of mψ , m is shorthand for mð IRÞ, and we [31] is limited to and its generalizations. There- made use of the homogeneity equation fore it cannot be used to disprove the existence of the NG mode for (exact) scale invariance, contrary to ∂ ∂ – can subsequent claims [108 110]. μ þ mψ − dO hvacjOð0Þjvaci¼0: ð56Þ ∂μ ∂mψ A. Flat directions? If hGˆ 2i can be reliably estimated (Appendix C), the vac If a symmetry is realized in the NG mode, it follows that leading-order result (55) may be used to test candidate there are directions in field space, one for each NG boson, theories of crawling TC on the lattice; see also Sec. VII. for which the action is flat. Often this is used as a shortcut Returning to the mψ ¼ 0 case, we note that the explicit to search for NG modes of complex Lagrangians. scale symmetry breaking responsible for the dilaton mass So, if a Lagrangian L is scale invariant, it is tempting to arises from renormalization and is entirely nonperturbative. suppose that, when the action is varied, a flat direction That should be contrasted with necessarily corresponds to a dilaton. The classic counter- (1) the pion mass due to (chiral) symmetry breaking by 1 2 example is the Lagrangian L ¼ ð∂ϕÞ for a massless current -mass terms in the bare QCD Lagran- free 2 spin-0 field ϕ. gian, and As is well known, ϕ describes a genuine NG boson, but (2) the “scalon” mass [30] of a Coleman-Weinberg that is for invariance under field translations potential [67] generated by explicit scale breaking from one-loop renormalization of gauge theories ϕ → ϕ constant ; 59 whose tree-level amplitudes lack massive parameters. þf g ð Þ not for scale transformations. The theory is exactly soluble IV. PECULIARITIES OF DILATON with amplitudes which do not depend on a scale, so scale LAGRANGIANS invariance is realized in the WW mode and ϕ is not a Compared with chiral Lagrangians, the conformal case dilaton. This is entirely different from exact scale invari- ance in the NG mode, where amplitudes depend on a involves some subtleties which caused problems when first nonzero dilaton decay constant Fσ and hence other dimen- encountered in 1969 [101,102]: the would-be NG bosons sionful constants (Appendix D). seemed to be massive in the limit of conformal invariance. If a scale-invariant L depends on many field compo- By late 1970, these puzzles had been resolved: just one NG nents, there can be many flat directions. One of them may field (the dilaton) is needed for the entire conformal group be associated with the NG mode of scale transformations, [9] (Appendix B3), and the class of consistent dilaton but not necessarily. If amplitudes do not depend on Lagrangians is specified by Zumino’s condition [11] dimensionful constants in the scale-invariant limit, as in scalon theories (Sec. IV E), the theory is dilaton-free. λ ¼ OðϵÞð57Þ

4 ’ if there is a term λϕ in the potential. The unusual feature of B. Zumino s consistency condition Eq. (57) is the requirement that a symmetry-preserving Zumino’s condition (57) is necessary for scale invariance operator ϕ4 have a symmetry-breaking coefficient λ, i.e., to be realized in the NG mode. Its genesis was the work of Salam and Strathdee [101], λ → 0 ð58Þ who sought to extend the nonlinear theory of chiral Lagrangians to the conformal case. They introduced the in the limit ϵ → 0 of conformal invariance. now-standard parametrization

095007-11 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019)

ϕðxÞ¼Fσ expfσðxÞ=Fσgð60Þ is not allowed because the subtraction would violate scale invariance. for the scalar field ϕ in terms of a would-be dilaton field σ The subtlety exposed by Zumino is that writing ϕ in with the transformation property terms of σ does not necessarily force a theory into the NG scaling mode, and, for λ0 ≠ 0 in the symmetry limit, it is 0 Fσ ∂x not legitimate to do so. That is connected with the fact that σ → σ − ln det ;x→ x0 conformal: ð61Þ ϕ 4 ∂x Eq. (60) constrains :

(There was also a vector field Aμ for special conformal ϕ > 0: ð68Þ transformations, but that was subsequently abandoned [102] in favor of ∂μσ.) Then, imitating the procedure for The conclusion (65) is incorrect because it was derived chiral Lagrangians, they wrote down the most general without first finding a minimum about which to expand in 4σ=Fσ Lagrangian consistent with symmetry requirements, the unconstrained field σ, and e has no minimum for finite variations of σ.14 1 Given that λ must vanish for scale invariance in the L ∂ϕ 2 − λ ϕ4 L˜ ϕ; ρ ¼ 2 ð Þ 0 þ ð Þð62Þ NG mode, why is there an apparent clash with the principle learned from chiral Lagrangians that the most 1 general Lagrangian consistent with symmetry should be ∂σ 2 2σ=Fσ − κ 4σ=Fσ L˜ σ ρ ¼ 2 ð Þ e e þ ð ; Þ; ð63Þ considered? The answer is that the principle needs to be more carefully stated. When constructing an effective with λ0 > 0 in the scale-invariant limit [unlike λ in Lagrangian, the most general result consistent with sym- Eq. (57)]. Here ρðxÞ denotes chiral and non-NG metry and NG-mode requirements must be sought. 4 fields and κ ¼ λ0Fσ is a positive constant. Consider any continuous symmetry, either compact or The result of applying these apparently general princi- noncompact. Then the set of all possible Lagrangians con- ples was puzzling. When the ϕ4 term is expanded in σ, sistent with the symmetry will include a subset in the WW mode, another subset in an NGmode, and otherswhich cannot

4σ=Fσ 2 2 3 κe ¼ κ þ 4κσ=Fσ þ 8κσ =Fσ þ Oðσ Þð64Þ be expanded about a point in field space because of a poor choice offield variables orLagrangian coefficients. So,having “ ” the Oðσ2Þ term seems to give the would-be dilaton a mass written down a general Lagrangian, it is necessary to check by hand that it can be expanded in all NG fields about a ? pffiffiffi stationary point. Only then can it be treated as an effective 4 κ mσ ¼ =Fσ ð65Þ Lagrangian for the desired NG mode(s). Let us contrast noncompact scale symmetry with sym- in the scale-invariant limit [101]. Terms in metry under global compact Uð1Þ transformations X φ → eiθφ of a complex spin-0 field. Consider the class L˜ O χ ð4−dÞσ=Fσ ¼ dð Þe ð66Þ of symmetric Lagrangians d 1 O L ¼ j∂φj2 − Ajφj2 − Bjφj4 ð69Þ cannot compensate for this: the dimension-d operators d A;B 2 do not have vacuum expectation values because of their dependence on ρðxÞ. A massive σ cannot be an NG boson, parametrized by constants A, B. If free-field theory is but could its mass have arisen from a Higgs-style mecha- excluded, B lies in the range B>0. Then both modes of the nism [101], despite the fact that the conformal symmetry theory are determined by inequalities, i.e., by continuous being investigated is global, not local? ranges of the ðmassÞ2 A: Zumino observed that these puzzles were symptoms of a more basic problem: scale-invariant ϕ4 theories and the NG NG mode∶ A<0; WW mode∶ A ≥ 0: ð70Þ scaling mode are not compatible. If one tries to use the parametrization (60) to force the theory into the scaling NG Thus, when the choice of coefficients in a chiral Lagrangian mode, a low-energy expansion cannot be performed: is said to be “arbitrary,” there is an understanding that this is (1) The requirement σ → 0 as xμ → ∞ for the fluc- tuation field σðxÞ produces infinite action if there is a 14 ’ 4σ We have been asked if Zumino s condition, when extended term ∼e =Fσ in L. to include gravity, is consistent with having a cosmological (2) Modifying constant Λ ≠ 0. The discussion above concerns the limit ϵ → 0, but Λ breaks scale invariance explicitly, so there is no contra- diction. For example, Zumino’s OðϵÞ example (74) would allow 4σ=Fσ → 4σ=Fσ − 1 e e ð67Þ Λ ≠ 0.

095007-12 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019) not entirely so, especially for the model (69) where the diagrams. Since ϕ4 counterterms occur, the point λ ¼ 0 is familiar constraints A<0 and B>0 apply. For the scale- unstable under WW-mode RG flow. invariant Lagrangian (63), the free-field case is avoided by However, in the NG scaling mode, we are dealing with a requiring L˜ ≠ 0. As we have seen, one of the two modes of nonrenormalizable loop expansion in powers of NG-boson scale invariance is specified by an equality: momenta q ∼ 0 and explicit symmetry breaking ϵ,asin Appendix A of Ref. [14]. The constraint (68) occurs at NG mode∶ λ ¼ 0; WW mode∶ λ > 0: ð71Þ ϕ ¼ 0, so fluctuations ϕ ∼ 0 to form ϕ are not allowed. Instead, we expand in the unconstrained field σ This difference between Eqs. (70) and (71) is hardly and form loops with σ propagators and vertices. The surprising, given that degenerate minima for scale trans- outcome resembles that for nonlinear chiral theories formations have to lie on a half-line to infinity in field [111–113]: each new loop order produces a new set of space, unlike the periodic orbits characteristic of compact coupling constants because ϵ-independent counterterms group symmetries. have more derivatives than before. All RG mixing of A feature shared by chiral and dilaton Lagrangians is that coupling constants of a given order is OðϵÞ, as in Eq. (57). in the symmetry limit, NG bosons do not interact at zero For example, let σ be coupled to the matrix field U momentum: [113,114] for chiral NG-bosons as follows [13–15]: π π → π π; σ σ → σ σ: 72 1 1 þ þ þ þ ð Þ L ∂σ 2 2 ∂ ∂μ † 2σ=Fσ ϵ 0 ¼ 2ð Þ þ4Fπtrð μU U Þ e þOð Þ; ð75Þ In both cases, this follows from the flatness requirement for degenerate minima. For dilatons, it is obviously consistent where OðϵÞ denotes terms which break scale and chiral with Eq. (58). invariance explicitly, and for ϵ → 0, we have chosen λ ¼ 0, Now let scale symmetry be broken explicitly by adding i.e., κ ¼ 0 in Eq. (63). Then for ϵ ¼ 0, all NG-boson OðϵÞ terms to the Lagrangian (63). Zumino observed that interactions (dilatons and chiral bosons) involve a field one of these terms could be e4σ=Fσ with a coefficient derivative, and so there can be no nonderivative counter- proportional to ϵ, as in Eq. (57), and that subtractions terms like mass counterterms δm2fσ2 or π2g or four-point 4 4 such as Eq. (67) are then allowed. By itself, e4σ=Fσ still does interactions δλfσ or π g which would violate the mass- not allow a minimum at any finite value of σ, but when lessness of NG bosons and no-interaction conditions like combined with d ≠ 4 terms which break scale symmetry Eq. (72). Instead, there are higher-derivative counterterms explicitly, the resulting dilaton potential such as the scale-invariant four-point interaction

−4 4 VðσÞ¼OðϵÞð73Þ Fσ ð∂σÞ ð76Þ may have a minimum and produce a genuinely light which is Oðq2Þ in NG-boson momenta q relative to leading 2 15 dilaton: mσ ¼ OðϵÞ. Zumino gave an example order. In the presence of explicit scale breaking, as in pffiffiffi Eq. (74), σ propagators carry a small mass mσ ¼ Oð ϵÞ. 4 2σ=Fσ 2 VðσÞ¼ϵFσfe − 1g ð74Þ Then there can be a d ¼ 4 counterterm in VðσÞ, but the correction to λ is clearly OðϵÞ. Therefore Zumino’s con- related to the model of Freund and Nambu [8]; it implies dition (57) is stable under NG-mode RG flow. 2 2 mσ ¼ 8ϵFσ and λ ¼ ϵ. So far, the discussion has been restricted to the NG- There may be a concern that renormalization violates the boson sector. The result is an expansion in powers of constraint λ ¼ 0 in the limit of scale invariance. Here it is 4π 4π important to distinguish loop expansions in WW and NG fq or mNGg=f Fσ or Fπgð77Þ scaling modes: they are not equivalent. 4 μ In a renormalizable theory with a λϕ interaction, the with coefficients depending on logarithms lnðmNG= Þ; loop expansion is a series in a finite set of coupling the renormalization scale μ provides the sole UV cutoff constants (including λ) which mix under RG flow, to all for integrals. For dimensional regularization in n complex orders in the expansion. The perturbation series is obtained dimensions, include the OðϵÞ terms (otherwise all loop via small-field fluctuations such as ϕ ∼ 0, as in the WW integrals vanish), and in L0, replace scaling mode, where ϕ is unconstrained. Then ϕ propa- gators can be formed and used to construct tree and loop e2σ=Fσ → eðn−2Þσ=Fσ : ð78Þ

15For early work consistent with Eq. (57), see Ref. [9] The inclusion of non-NG particles such as fermions with [formula below Eq. (3.11)] and Ref. [10] [Eq. (4.6)]. Compare mass M ≠ 0 for ϵ → 0 presents difficulties already familiar Eqs. (45)–(50) of Ref. [14]. from baryonic chiral perturbation theory [115,116]: for

095007-13 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019) fermion fields, the expansion is in ði=∂ − MÞ,noti∂,so 4 p=ðc1p þ c2ÞþM ∼ ip= → ∞ SFðpÞ¼i 2 4 2 2 6 ;p higher-derivative fermionic terms can be of leading order. p ðc1p þ c2Þ − M c1p Consequently, extending the NG-mode renormalization procedure to massive fermions is not obvious. Special ð82Þ techniques have been invented to deal with loops containing at least one NG boson [117–119], but little can be said about which makes all closed fermion loops converge. pure non-NG particle dynamics such as effects due to closed Of course, this procedure is arbitrary, but that is the fermion loops. Instead, it must be assumed that all non-NG point: nothing can be said about dynamics in the non-NG dynamics can be contained in the low-energy constants of sector. We must follow the example of chiral perturbation loop expansions involving NG bosons, where chiral and (in theory, and start from the basic hypothesis, well supported our case) conformal symmetry provide some guidance. by experiment in the chiral case, that non-NG particle We mention closed fermion loops because it might be dynamics does not force the theory out of the NG mode. thought that they should be part of the renormalization procedure. Could they produce counterterms which give NG C. Digression: Fubini’s “new approach” ’ bosons mass and violate Zumino s condition? If so, non-NG Modern investigators of light Higgs bosons often cite dynamics would force the theory out of the NG mode. Fubini’s 1976 paper [31] as evidence that the NG scaling σ ∼ 0 Consider a toy model such as the expansion of the mode cannot be realized in the limit of conformal sym- scale-invariant Lagrangian metry. A cursory reading of Ref. [31] can easily produce   this wrong conclusion, especially if earlier work leading to 1 i ↔ L ∂σ 2 2σ=Fσ ψ¯ ∂ − σ=Fσ ψ Zumino’s condition (57) (to which Fubini does not refer) is toy ¼ ð Þ e þ = Me : ð79Þ 2 2 not known. Fubini’s approach was not just “new”: it was radically L In the tree approximation, for which toy is designed, one different from the standard theory of dilaton Lagrangians can read off relations such as the scalar analogue of the described above. Conformal invariance is imposed on λϕ4 L Goldberger-Treiman relation [Eq. (39) and Fig. 2]. If toy is theory and, more generally, on polynomial scalar-field supposed to produce a renormalizable perturbation series Lagrangians in D space-time dimensions with no depend- in the Yukawa coupling −M=Fσ, closed fermion loops ence on dimensionful constants. Scale breaking due to certainly do produce divergent self-energy, triangle and box renormalization is ignored. All fields are unconstrained: diagrams. nonlinear chiral or scale fields depending on Fπ or Fσ are The flaw in this picture is the assertion that, for momenta not present. Then Fubini considered introducing a funda- ≳ 0 M, non-NG particle dynamics can be represented by the mental scale a via a state j iF which he called the perturbative series of a local renormalizable theory for “vacuum” but which looks more like a coherent state; it baryon and meson fields or their TC counterparts. There is corresponds to a classical field BðxÞ: no hint of this from QCD or experiment. Interactions 0 ϕ 0 between non-NG hadrons are strong and produce higher Fh j ðxÞj iF ¼ BðxÞ: ð83Þ resonances which could not all be represented by separate fields. He observed (correctly) that BðxÞ cannot be constant for λ ≠ 0 0 Instead, it must be recognized that there can be non- , and so j iF does not preserve translation invariance. renormalizable higher-derivative fermionic terms in leading Instead, it preserves a linear combination order, as in the modified toy example 1 −1 Rμ ¼ ðaPμ þ a KμÞð84Þ 1 i ↔ 2 L ∂σ 2 2σ=Fσ ∂μ∂νΨ¯ ∂ ∂ ∂ Ψ mod ¼ 2 ð Þ e þ c1 2 = μ ν   ↔ of the momentum components Pμ and special conformal ¯ i σ 4σ Ψ ∂ − =Fσ Ψ =Fσ generators Kμ. To restore translation invariance, Fubini þ c2 2 = Me e ; ð80Þ proposed a “statistical” average over the continuum of “ ” 4 degenerate vacua where c1M þ c2 ¼ 1 and we have chosen a new fermion variable μ 0 jhiF ¼ expðiPμh Þj iF; ð85Þ

ΨðxÞ¼expf−2σðxÞ=FσgψðxÞð81Þ but the properties of the resulting theory and its true vacuum (if it has one) are not known. 1 which carries dimension − 2. In the tree approximation, Fubini’s conclusions do not exclude the existence of this model also produces Eq. (39), but the corresponding dilaton Lagrangians which preserve translation invariance, fermion propagator has asymptotic behavior because his choice of conformal models excludes the set of

095007-14 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019) known dilaton Lagrangians, all of which obey Zumino’s As an example of the equivalence theorem in the tree condition (57). Fubini considered Eq. (62) but not Eq. (63): approximation, consider the toy Lagrangian (79). The field he avoided the error of assuming Eq. (60) for λ ≠ 0.His σ can be expanded about a point σ0 determined by the limit 2 analysis leaves ϕ unconstrained, contrary to Eq. (68), and ϵ → 0 of a scale-violating perturbation ∼ϵðσ − σ0Þ .Ifwe so yields an x-dependent result (83). In contrast, genuine choose σ0 ¼ 0, the fermion ψ has mass M in lowest order, L dilaton Lagrangians involve constrained scale fields (60) so clearly, toy is a dilaton Lagrangian: its amplitudes with constant vacuum expectation values exhibit the NG scaling mode in the limit ϵ → 0.Isit equivalent to a polynomial Lagrangian? The answer is “ ” only hvacjFσ expðσðxÞ=FσÞjvaci¼Fσ: ð86Þ yes, but if the new field variable is constrained, e.g.,

σ=Fσ σ → ϕc ¼ Fσðe − 1Þ; ϕc ≥−Fσ: ð87Þ Fubini’s interests were semiclassical, with apparently no intention that his work be compared with the literature This change of variables is permitted by the equivalence on nonlinear dilaton Lagrangians of six years earlier theorem because the constraint on ϕc does not interfere [8–12,51]. He was not known to be against the existence with fluctuations ϕc ∼ 0 corresponding to σ ∼ 0: of the NG mode for global scale transformations, nor was X n his work seen in that light when it was published. σ 2 ϕ ¼ σ þ ¼ σ þ Oðσ Þ; jϕ j ≪ Fσ: ð88Þ c ! n−1 c n>1 n Fσ D. Changing field variables Unlike nonlinear chiral Lagrangians, dilaton The result is a polynomial Lagrangian in the constrained 16 ϕ Lagrangians can be linearized by a change of variable field c consistent with the equivalence theorem if renormalization 1 is ignored and noninteger dimensions are absent. On L0 ∂ ϕ ∂μϕ Ψ¯ ∂ − − −1ϕ Ψ toy ¼ μ c c þ ði= M MFσ cÞ ð89Þ dimensional grounds, the nonlinear Lagrangian necessarily 2 depends on a dimensionful quantity, the dilaton decay L giving the same tree-diagram S matrix as toy. As noted constant Fσ, but that dependence tends to be hidden in L L0 the linear version. This may mask the presence of an for toy at the end of Sec. IV B, toy is not a good basis for NG scaling mode; if so, it certainly obscures NG-mode NG-mode renormalization. renormalization. Alternatively, in the absence of other When renormalizing in the NG mode, it is not a priori fields such as chiral bosons, it may indicate a theory obvious that parametrizations of the chiral matrix field U actually in the WW scaling mode with all Fσ dependence and the scalar field (60) in terms of unconstrained NG fields transformed away. survive the process. Furthermore, not all Lagrangians The equivalence theorem17 was originally derived equivalent at tree level are equally amenable, because [120–122] without regard to renormalization, so it was the process can be upset by terms proportional to the explicitly valid only in the tree approximation. Sub- equations of motion. The most undesirable scenario is sequently, a renormalized version of the theorem was having to subtract convergent as well as divergent loop proven for renormalizable theories [123,124], but not diagrams by hand to enforce the masslessness of NG generally for NG-mode renormalization of nonlinear chiral bosons and the no-interaction requirement (72) generalized models [111,125]. We believe that an equivalence theorem to amplitudes with many NG-boson legs: can be formulated and proven for nonlinear NG-boson A 0 0 ϵ → 0 Lagrangians with derivative interactions in the limit of π…πσ…σjall q ¼ 0 ¼ ;mπ ¼ ¼ mσ; : ð90Þ exact symmetry, all renormalized in the NG mode as outlined in Sec. IV B, but an all-order analysis remains In each order of the loop expansion, that would require an to be done. infinite set of counterterms, i.e., the renormalization pro- cedure would be nonlocal. Note that by itself, 16This terminology is standard, but what is really meant is that the Lagrangian becomes a polynomial in the field variables. 1 Similarly, read “nonpolynomial” for “nonlinear.” 2 L0 ¼ ð∂σÞ expð2σ=FσÞð91Þ 17In statements of the theorem, a Lagrangian theory is defined 2 by the all-order loop expansion due to small-field fluctuations about a local minimum of the potential. Modulo renormalization, is not a dilaton Lagrangian. The theory appears to be Lagrangians related by an invertible point transformation map- interacting, with a loop expansion which requires renorm- ping one fluctuation region to the other, as in Eq. (88) below, are equivalent: their S matrices agree. The mapping σ ↔ ϕ of alization. However, when renormalized by subtracting Eq. (60) is forbidden because the constraint (68) disallows about any point in momentum space which is not IR fluctuations ϕ ∼ 0. singular, seemingly complicated sets of diagrams at each

095007-15 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019) loop order sum to zero on shell [126]. Evidently, L0 for an implicit assumption that “conformality” is always in 1 2 σ ∼ 0 is equivalent to 2 ð∂ϕcÞ for ϕc ∼ 0, so tree-level the WW mode. amplitudes sum to zero on shell; then cutting rules can be A key element of this belief is that the way to elevate any used to extend the result to loops. The conclusion is that all theory to dilaton status is to write f expðσ=fÞ for a scalar dependence on Fσ is absorbed by the change of varia- field close to a fixed point and avoid discussing what this ble (87). This shows that merely writing a scalar field as means for the fixed point itself. In the scale-invariant limit, f expðσ=fÞ is not enough to ensure the existence of there are four main possibilities: dilatons: it must be shown that amplitudes of the scale- (1) The WW mode is produced because f → 0. That is invariant theory depend on dimensionful constants. the origin of the “fine-tuning” problem of scalon theories [106–110], where f2 is proportional to the E. Scalons are not dilatons magnitude of explicit scale breaking. Approximate ≪ 1 ∼ In their influential work on scalons, Gildener and scale invariance requires f=v contrary to f v Weinberg [30] considered a scale-invariant limit for poly- experimentally. More generally, the expansion wanted nomial Lagrangians, but unlike Fubini, they to σ σ σ2 2 σ3 6 2 … produce amplitudes with no dependence on a dimensionful f expð =fÞ¼f þ þ =ð fÞþ =ð f Þþ constant. They did this by retaining translation invariance ð93Þ and assuming the tree approximation for unshifted fields. All dependence on dimensionful constants would be fails: it would produce singularities ∼f−p in effec- generated by an explicit breaking of scale invariance due tive Lagrangian vertices. to renormalization corrections depending on a scale μ. (2) A phase transition causes the scale-violating expan- Scalon theories are constructed as follows. First, a sion to fail. In walking TC, the walking coupling α is L α polynomial Lagrangian gauge is constructed for a scale- separated from the WW-mode fixed point ww in the invariant gauge theory involving one [67] or more WW mode by a chiral phase transition [73] at the sill [30,103–106] scalars. In the tree approximation, all of of the conformal window. Nevertheless, the small these scalars are massless, but none of them can be a dilaton value of the Higgs mass is claimed to be a first-order α α because, by construction, amplitudes do not depend on consequence of the expansion in about ww.That dimensionful constants. So scale invariance is realized in creates severe conceptual difficulties [5,127] for L the WW mode, which (as for free above) is entirely “dilatonic” walking TC theories (Sec. VI). consistent with the presence of flat directions. Then one- (3) The constant f can be transformed away via the V loop quantum corrections CW [67] are calculated and used equivalence theorem, allowing the fixed point to be L to perturb gauge: in the WW mode. That may circumvent the fine- tuning or phase-transition problems, but then there L L − V L − V one-loop ¼ gauge CW ¼ K:E: eff: ð92Þ would be no soft-dilaton theorems: any effective Lagrangian could be rendered independent of f,as The explicit breaking of scale invariance by logarithmic in the example (91) above. ϕ2 μ2 V ≠ 0 factors lnð = Þ in CW gives rise to two scale-violating (4) At the fixed point, f is the decay constant Fσ effects, viz. a compact set of chiral- (not scale-) degenerate given by Eq. (1), so soft-dilaton theorems (Appen- V minima of eff, and masses for one or more scalons. dix B3) exist and amplitudes do not scale at the Despite the third paragraph of Ref. [30], none of these fixed point (Sec. II). Then the fixed point is in the scalons can be a pseudodilaton because, in the scale- NG mode, which excludes walking TC and scalons. V → 0 invariant limit CW , amplitudes have no scales and Theoretical ambiguity about whether the fixed point is in hence there are no dilatons. Scalon theories deserve to be the NG or WW mode is popular but untenable: a choice studied in their own right, but must not be confused with must be made. Physically, the NG mode is far closer to dilaton theories. reality and hence a far better candidate for theories of This may be the origin of a pervasive belief that the NG approximate scale invariance: the particle spectrum in the mode for scaling is possible only in the presence of explicit scale-invariant limit (Appendix D) resembles that of the scale violation [95], as in oft-repeated references to real world. Compare that with the WW mode, where there “spontaneous breaking of approximate scale invariance.” are no thresholds except for branch cuts and poles at zero This sounds odd because it is not correct: only in the limit momentum, and particles may not even exist [128]. of exact scale invariance can the distinction between the NG and WW scaling modes be made. The most obvious V. COMMENTS ON PHENOMENOLOGY cause of this is the misunderstanding of Fubini’s work [31] discussed in Sec. IV C. In walking TC or scalon theory, Since our Higgs-boson theory differs fundamentally which is generally not dependent on Ref. [31], it may stem from all others (they are not expansions about a scale- either from the third paragraph of Ref. [30] or simply from invariant theory with a scale-dependent vacuum), its

095007-16 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019) phenomenology cannot be inferred from a subclass of VI. ELECTROWEAK EFT existing theories: a new analysis is necessary. We begin By analogy with QCD, where at energies below the with remarks about the width of pseudodilatons, the relative confinement scale one can use EFT methods to describe magnitudes of pNG boson decay constants in QCD and pion dynamics, an EFT for dynamical electroweak sym- crawling TC, and the electroweak S parameter [129,130]. metry is the most efficient way to describe physics at QCD and crawling TC borrow an idea from broken scale energies ranging from a few GeV to several hundred GeV. invariance for strong interactions that a chiral condensate In this range, all SM interactions are relatively weak. can also act as a scale condensate [10,29], implying a σ → ππ Perturbation theory is possible not only in the electroweak relation for the coupling 0 couplings gw and gw but also in the gluon coupling constant 2 gs because of asymptotic freedom for QCD. The upper limit fσgσππ ≃−mσ ð94Þ of several hundred GeV is chosen so that interactions presumed to be strong at the TeV scale which remains valid in chiral-scale perturbation theory [13–15]. Eq. (94) implies a width of a few hundred MeV for 18 Λv ∼ 4πv ¼ 4πFπ ð95Þ σ, which is consistent with data for the f0ð500Þ reso- nance, the obvious candidate for the QCD pseudodilaton.19 become sufficiently weak in the SM sector to justify a Here fσ and fπ are observed to have similar orders of perturbative EFT approach. At energies ∼Λ , hadronic magnitude within a factor of ∼2. Given that both arise from v bound states from the TC interactions are expected to having hqq¯ i ≠ 0 in the scale-invariant limit, this was to vac populate the spectrum and be responsible for the Higgs be expected. Note that we could not use a symmetry sector seen at lower energies. The EFT is constructed by argument to fix the ratio fσ=fπ, because the Coleman- requiring SUð3Þ × SUð2Þ × Uð1Þ gauge invariance and Mandula theorem [131] does not permit internal chiral and c L Y including the currently observed particle content, with space-time scale symmetry to be unified. the Higgs identified as a pseudodilaton instead of a weak Since this works for QCD, there is good reason to let doublet. The resulting theory is an effective chiral hψψ¯ i be a condensate for both chiral and scale trans- vac Lagrangian (augmented with gauge bosons and fermions), formations in crawling TC, with similar orders of magni- which for crawling TC is extended [10] to include the NG tude for the electroweak scale v Fπ and the TC dilaton ¼ mode of scale invariance. decay constant Fσ. This avoids the fine-tuning problem of Electroweak EFT was originally developed [32,33] with scalon theories noted above, where the strength of explicit a heavy Higgs boson in mind. Although no longer valid, scale breaking f must be artificially adjusted to match the some basic features of that work survived subsequent scale v of the chiral condensate [106–109]. developments [35–39] and remain in low-energy EFTs It is often suggested that TC theories have trouble for light Higgs bosons [34,40–43]. In all of these theories, generating a small enough value of the S parameter the effective Lagrangian has a chiral component for the (defined such that S 0 in the SM) that is compatible ¼ would-be NG bosons which give (conveniently in Landau with the experimental number S ¼ 0.05 0.10 [80]. 0 gauge) mass to the weak W and Z bosons. The standard Quoted values of S typically include the estimates S ≈ procedure is to choose a nonlinear chiral Lagrangian 0.32 obtained originally by Peskin and Takeuchi [130] and [122,136–138] based on (say) a unitary matrix field U S ¼ 0.42ð2Þ in recent two-flavor lattice calculations [113,114]; linear models are inconvenient because they [132,133]. But the prescription [130] used to obtain these depend on extraneous non-NG fields such as the sigma estimates involves subtracting the contribution of a heavy field of the linear sigma model. The advantage of the SM Higgs boson, and must be amended [134] if the TC effective Lagrangian formalism is that, with symmetries spectrum contains a light scalar. In Ref. [135], TC scenarios implemented at an operator level, radiative corrections which include a generic light scalar resonance were are easily computed, and contact can be made with the confronted with electroweak precision data. Figure 6 of SM Lagrangian in order to spot potential deviations in the Ref. [135], which plots the deviation κ from the SM (f=v W phenomenology. or Fσ=v in our notation) against the technirho mass M , V As noted in Sec. IV D, the extension to dilatons is shows that the experimental constraints on S require v ≃ Fσ necessarily nonlinear: the spin-0 field which transforms and M ≃ 1 TeV. Both requirements are naturally satisfied V with scale dimension 1 enters linearly but produces the NG in crawling TC. scaling mode only if it is suitably constrained and hence a nonlinear function of unconstrained fields. In analogy with 18 The dilaton-Higgs of crawling TC is relatively narrow Eq. (87), we use a special notation χc to distinguish our because (unlike the case of QCD), the pions are eaten, and there χ field from the WW-mode fields implicitly used in walking is no phase space for σ to decay strongly into other particles. This −4 TC or scalon theories. The key feature of our theory is that is consistent with the current [80] upper bound Γh=mh ≲ 10 . 19 χ This provides a clear counterexample to the claim [108–110] c is constrained in the exact limit of scale invariance as that no light dilaton is expected in QCD. well as when there is explicit scale symmetry breaking.

095007-17 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019)

By definition, the fields U and χc transform linearly We consider crawling TC for a QCD-like SUð3Þ gauge under the electroweak gauge group and scale transforma- theory but with only Nf ¼ 2 flavors of massless Dirac tions. It is convenient to choose constraints which are techniquarks so that, at low energies, all technipions are manifestly symmetry preserving eaten giving SM gauge bosons and fermions their masses.20 We stress that the form of the EFT to be derived below does U ¼ SUð2Þ matrix and χc > 0 ð96Þ not depend on Nf, as long as one is outside the conformal window. The choice Nf ¼ 2 simply avoids having to and for which there are standard parametrizations in terms justify the absence of light physical technipions. of unconstrained Goldstone fields φa [113,114] and σ As noted in Sec. I, the possibility that IR fixed points [10,11,101]: occur at small values of Nf has been studied extensively [28], but currently there is little direct evidence for or a a iφ τ =v σ=Fσ U ¼ e and χc ¼ Fσe : ð97Þ against their existence (see Sec. VII). If present, they are almost certainly in the NG scaling mode, as indicated in the Here τa are Pauli matrices. left diagram of Fig. 1. That is because they lie outside the The next step is to specify the theory responsible for conformal window: dimensional transmutation can occur, crawling TC and how its effects are to be incorporated with the WW-mode scaling laws (10) replaced by the soft- into our EFT. As for all TC theories, we assume it to be a dilaton theorems (15) and (16). gauge theory which exhibits asymptotic freedom in the UV We make the standard assumption that TC theory mimics massless QCD. At the TeV scale and below, the technigluon limit, i.e., well above Λv. Since the range of energies being considered is well below the strongly interacting TeV scale, coupling α is strong, techniquarks and technigluons are the result is controlled by the IR limit of whatever TC confined and bound states and resonances are expected to be theory is held responsible for those effects. In that limit, the produced. All technihadrons in the non-NG sector are heavy, α α i.e., in the TeV range. Unlike QCD, the would-be techni- TC coupling either runs to a fixed point IR, as in the left diagram of Fig. 1 (crawling TC), or it runs to ∞. pions are unphysical, but in crawling TC there is a In crawling TC, the Higgs boson is light because it pseudodilaton (the Higgs particle), which plays a role similar 500 corresponds to a small OðϵÞ term in the IR expansion of the to that of the QCD resonance f0ð Þ in chiral-scale ϵ α − α 0 perturbation theory [13,14].AtenergieswellbelowΛ , continuous variable ¼ IR > about the NG-mode v α one can build an EFT where the dynamical degrees of fixed point IR. This is a great advantage over walking TC, where the small value of α − α in the walking region is freedom are the quarks, and gauge fields of the SM ww a said to be responsible for the small Higgs mass. That and the unconstrained Goldstone fields ϕ and σ. Effects due ψψ¯ ≠ 0 assumes that the walking region of the solid curve in the to TC fields such as h ivac are still present, but hidden right diagram of Fig. 1 can be approximated by the dashed inside the low-energy coefficients of the EFT. The gauge A a A α potentials are Gμ , Wμ and Bμ with field-strength tensors Gμν, curve in that diagram near ww. The problem is that the solid a 3 2 1 and dashed curves are separated by a strong phase disconti- Wμν and Bμν for SUð Þc and SUð ÞL and Uð ÞY nuity [73] at the critical number of flavors Nf defining the electroweak bosons, respectively. The SM fermions have the sill of the conformal window (see footnote 8, and item 2 on usual charge assignments under the SM gauge group, page 16). Confinement, a light scalon and a large chiral 3 3 2 1 condensate are presumed to exist in the walking region for SUð Þ SUð Þc SUð ÞL Uð ÞY N

095007-18 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019)     u νl The masses for the gauge bosons and fermions are q ¼ and l ¼ : ð98Þ L d L e contained in the last line when U ¼ I (unitary gauge). L L 1 In terms of the Uð ÞY hypercharges Yf tabulated In crawling TC, the SM Higgs doublet is replaced with above, the gauge-covariant derivatives of quark fields q ¼ a chiral-singlet dilaton field σ and a triplet of Goldstone u or d are fields ϕa, so the EFT combines the loop expansion of a renormalizable theory with that of an effective Goldstone ∂ 0 DμqL ¼ð μ þ igsGμ þ igwWμ þ igwYqL BμÞqL; Lagrangian. This is in close analogy with, e.g., what 0 Dμq ¼ð∂μ þ ig Gμ þ ig Y BμÞq ; ð103Þ happens when pion dynamics is coupled to QED. It is R s w qR R understood that all mass is to be produced by a Higgs-style mechanism, so the relevant renormalizable Lagrangian is with analogous expressions for leptons obtained by 3 that for a massless version of the SM with terms depending omitting the SUð Þc terms. The covariant derivative on massive constants like v omitted. It is convenient to associated with the gauge property (100) is [33] postpone including the dilaton field σ; first we add to the massless SM Lagrangian the lowest-order nonlinear chiral i ∂ − 0 τ3 and Yukawa terms constructed from U such that SM gauge DμU ¼ μU þ igwWμU 2 gwBμU : ð104Þ 2 1 invariance is preserved. Under SUð ÞL × Uð ÞY, U must transform to a new matrix U˜ which also satisfies the Equation (101) can be made scale invariant by constraint (96), i.e., it is unitary and obeys the condition ˜ multiplying each operator by an appropriate power of det U ¼ 1: σ the dimension-1 field e =Fσ and adding a dilaton kinetic ˜ term [10] U → U ¼ VLUVY with det VL ¼ 1 ¼ det VY: ð99Þ 1 1 It follows that V is not proportional to I and so U does not 2 ∂ σ=Fσ 2 2σ=Fσ ∂ σ∂μσ Y 2 Fσð e Þ ¼ 2 e μ : ð105Þ have a unique value of Y. Instead, VY must belong to the Uð1Þ subgroup of SUð2Þ generated by τ3 [for consistency with the charge assignments in Eq. (98)]. That yields a More generally, approximate scale invariance implies that a familiar result chiral Lagrangian operator Q with dynamical dimension dQ is replaced by U → U˜ ¼ eiω·τUeiητ3 ð100Þ 0 ð4−dQÞσ=Fσ ð4−dQþβ Þσ=Fσ Qσ ¼ Q × fcQe þð1 − cQÞe g originally obtained [33] from the gauge property for the cQQ 1 − cQ Qβ0 : matrix field for a heavy Higgs boson. Our presentation ¼ inv þð Þ ð106Þ shows that there is no need to introduce a Higgs field to Q determine the gauge property of U. Here inv has dimension 4 (the scale-invariant part), Then invariance under the SM gauge group gives the while Qβ0 accounts for explicit scale symmetry breaking α well-known EFT Lagrangian for Higgsless dynamical by the trace anomaly near IR and so has dimension 0 electroweak symmetry in leading order (LO) [32–34] 4 þ β (Appendix E). The coefficient of Qβ0 is fixed by requiring that the original operator Q be recovered in 1 1 1 A Aμν a aμν μν the absence of dilaton interactions. The dimensions dQ L ¼ − GμνG − WμνW − BμνB no Higgs 4 4 4 take the naive values implied by canonical dimensions, ¯ ¯ ¯ ¯ 3 þ qLiDq= L þ uRiDu= R þ dRiDd= R þ lLiD= lL i.e., 1 and 2 for gauge and fermion fields and 0 for the ¯ ¯ ˆ ¯ ˆ unitary field U. þ eRiDe= R − vfqLYuUUR þ qLYdUDR The values of the low-energy constants cQ depend on 1 l¯ ˆ E 2 μ † dynamics and are not fixed by symmetry arguments alone. þ LYeU R þ H:c:gþ4 v trðDμUD U Þ; However, scale invarianceP imposes constraints on them. For Qj ð101Þ a Lagrangian of the form j σ, the trace of the improved energy-momentum tensor is where the doublet notation X θμ − 4 Qj − Qj U¯ ¯ 0 D¯ ¯ E¯ 0 ¯ μjeff ¼ ðdQj Þf σ h σivacg R ¼ðuR Þ; R ¼ð0 dR Þ; R ¼ð eR Þð102Þ σ jX 2 2 β0 1 − Qj − Qj for right-handed fermions matches the × matrix U, ¼ ð cQjÞf β0 h β0 ivacg; ð107Þ ˆ j and Yu;d;e are 3 × 3 Yukawa matrices in generation space.

095007-19 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019)

Qj ≠ 4 where only operators β0 with dynamical dimension must be adopted. As for any EFT with an underlying contribute. The requirement that this expression vanish in strongly coupled dynamics, the expansion is organized by μ the number of loops, with each order absorbing the the scale-invariant limit θμ → 0 implies [15] divergences of the previous one. This ensures that the EFT is renormalized order by order. In the following we cQj ¼ 1 þ OðϵÞ; ð108Þ will rely on v ∼ Fσ (see the end of Sec. V). L where the correction OðϵÞ is due to the explicit breaking of The pure-dilaton part of LO is easily found, being so scale invariance by the trace anomaly in the low-energy similar to a standard nonlinear chiral Lagrangian. We seek a α ≲ α simultaneous expansion in momenta p and masses of pNG region IR. In the limit ϵ → 0, a potential ∼e4σ=Fσ is forbidden by bosons (just Mσ in our case): Zumino’s consistency condition (57). If its coefficient is ∼ ≪ 4π ∼ Λ OðϵÞ, e4σ=Fσ by itself is still not acceptable because it has no p Mσ Fσ v: ð113Þ σ minimum for finite variations of the unconstrained field 2 (Sec. IV B). However, a dilaton potential V of first order in The scale-invariant kinetic term (105) is already Oðp Þ,so 2ϵ Q → 1 ∂σ 2 ϵ is possible because there can be a term of dimension extra Oðp Þ terms generated by Eq. (106) for 2 ð Þ 4 þ β0 as well, are NLO. The Oðp2Þ term (105) is of the same order as the 2 OðMσÞ dilaton potential V, so the LO contribution is 0 4σ=Fσ ð4þβ Þσ=Fσ VðσÞ¼c1Vfe − 1gþc2Vfe − 1g; ð109Þ 1 L 2σ=Fσ ∂ σ∂μσ − σ σ;LO ¼ e μ Vð Þ: ð114Þ where both c1V and c2V are OðϵÞ, with constant terms 2 subtracted as in Zumino’s example (74). The function VðσÞ has a minimum for c1V < 0 and c2V > 0, which we assume. This Lagrangian is suitable for the tree approximation: the σ σ 2 − 2 The value h ivac of at the minimum is a matter of p; Mσ dependence of each propagator i=ðp MσÞ is 2 2 convention because field translations compensated by the Oðp Þ or OðMσÞ behavior of the next vertex. σ → σ L þ constant ð110Þ The remaining terms in LO are obtained by making the Higgsless Lagrangian (101) scale invariant. The result merely affect which of the physically equivalent scale- invariant worlds is chosen as ϵ → 0 (Appendix D). 1 1 1 L − A Aμν − a aμν − μν ¯ = σ σ 0 LO ¼ GμνG WμνW BμνB þqLiDqL Our convention for is h ivac ¼ . Minimizing V at 4 4 4 σ ¼ 0, we have ¯ ¯ þu¯ RiDu= R þdRiDd= R þlLiD= lL þe¯RiDe= R 4 4 β0 0 1 1 c1V þð þ Þc2V ¼ ð111Þ 2σ=Fσ ∂ σ∂μσ − σ 2 μ † 2σ=Fσ þ2e μ Vð Þþ4v trðDμUD U Þe 1 2 2 ¯ ˆ ¯ ˆ ¯ ˆ σ=Fσ so (say) c V can be eliminated in favor of c V. In turn, cV −vfqLYuUUR þqLYdUDR þlLYeUER þH:c:ge can be eliminated in terms of the pseudodilaton mass Mσ 1 2 2 ð115Þ by equating the second-order term of V to 2 Mσσ . The result is an explicit formula for the LO Higgs potential in crawling TC: describes the low-energy behavior of strongly interacting TC plus weak interactions of the dilaton with the SM gauge 2 2 bosons and fermions. As we saw for the dilaton kinetic MσFσ V σ ð Þ¼ β0 energy, not all terms from Eq. (106) are needed, but the   reasons for this are less obvious and require a discussion. 1 1 β0 4σ 4 β0 σ Equation (115) contains covariant derivatives Dμ given × − e =Fσ þ eð þ Þ =Fσ þ : 4 4 þ β0 4ð4 þ β0Þ by Eq. (103), so gauge invariance requires products like ∂ ð112Þ gwWμ in Eq. (103) to be counted like μ, i.e., as OðpÞ.In the original version of this rule [112,113], the gauge field The form of this potential is fixed solely by the presence of was taken to be OðpÞ. That choice works if the field is an IR fixed point and the fact that the explicit breaking of external, but it is not suitable when gauge propagators scale symmetry occurs through the operator Gˆ 2. appear; then it is necessary to require [143] L The full LO Lagrangian LO is obtained by collecting from above all modifications of the Higgsless Lagrangian gauge field ¼ Oð1Þ; charge ¼ OðpÞð116Þ (101) and discarding terms considered to be next-to- leading order (NLO). It is at this point that consistent so that gauge-boson kinetic energies are Oðp2Þ, corre- rules for the expansion into LO, NLO, next-to-NLO, … sponding to Oðp−2Þ behavior for the gauge propagators.

095007-20 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019)

In bosonic diagrams, this Oðp−2Þ behavior is compensated chiral-scale perturbation theory (the forerunner of crawling by the next vertex being Oðp2Þ because of the OðpÞ rule TC). However, as noted in Appendix A of Ref. [14],a (116) for gauge coupling constants.21 There are separate general NLO analysis depends on the next order of Taylor −1 β γ α α rules for fermions because propagators are Oðp Þ [146]: expansions of and functions in about IR, which would take us far afield. Therefore, for the remainder of this 1 2 article, we restrict our attention to the LO approximation. fermion field ¼ Oðp = Þ; Yukawa coupling ¼ OðpÞ: An immediate comparison of the Higgs sector of ð117Þ crawling TC with that of the SM is obscured by the σ L complicated dependence of the formula (115) for LO. The rules (116) and (117) should be understood purely as However, in the tree approximation, there is a field tagging devices to ensure correct power counting for l-loop redefinition [43] chiral amplitudes. Numerical estimates for the gauge Z 0 σ couplings gw, gw;gs in various energy regimes should 0 σ =Fσ 0 σ=Fσ h ¼ e dσ ¼ Fσðe − 1Þ;h≥−Fσ ð120Þ not be inferred beyond the requirement that perturbation 0 theory remains applicable. The rules for the tree approximation and beyond can be efficiently described in terms of chiral dimensions which simplifies the structure of Higgs vertices and importantly, satisfies the requirements of the equivalence [147,148]. Fields or coupling constants counted as 17 … theorem ; indeed, h is just the constrained field ϕ already Oðp½ Þ above are said to have chiral dimension ½…: c discussed in Eqs. (87) and (88). The constraint on h refers to the fact that its scale transformations 1 σ ϕa 0 ψ ½Gμ;Wμ;Bμ; ; ¼ ; ½ ¼2 ; → ρ−1 ρ−1 − 1 → ρ 0 ˆ 2 h h þ Fσð Þ;xx ð121Þ ½gs;gw;gw; Yu;d;e; ∂μ¼1; ½Mσ¼2: ð118Þ − ≤ ∞ The construction of the LO Lagrangian is summarized by are restricted to the region Fσ h< . The change of σ → the rule that it be homogeneous in chiral dimension field variables h is permitted by the theorem because L 2 fluctuations σ ∼ 0 about the minimum of V are mapped to with ½ LO¼ . ∼ 0 The utility of Eq. (118) becomes evident beyond LO, h . where rules for the low-energy expansion of EFT loop With this redefinition, the EFT Lagrangian (115) diagrams are needed. Despite the complications presented becomes by the structure of loop diagrams, the rule for the NlLO 1 1 1 Lagrangian at l-loop order implied by low-energy power A Aμν a aμν μν L ¼ − GμνG − WμνW − BμνB þ q¯ iDq= counting is simple: construct all operators with chiral LO 4 4 4 L L dimension 2l 2, i.e., ¯ ¯ þ þ u¯ RiDu= R þ dRiDd= R þ lLiD= lL þ e¯RiDe= R X 1 1 ∂ 2 − 2 μ † 1 2 L L l L l 2l 2 þ ð hÞ VðhÞþ v trðDμUD U Þð þ h=FσÞ EFT ¼ N LO with ½ N LO¼ þ : ð119Þ 2 4 l≥0 ¯ ˆ ¯ ˆ ¯ ˆ − vfqLYuUUR þ qLYdUDR þ lLYeUER þ H:c:g

At each order, the set of operators includes renormalization × ð1 þ h=FσÞ: ð122Þ counterterms needed to render all loop integrals UV l convergent. Similarly, on-shell N LO amplitudes are Apart from the Higgs potential V, this LO result O p2lþ2 p ∼ 0 2 ð Þ modulo logarithms, where refers to resembles the SM, where the factors ð1 þ h=FσÞ and momenta, NG boson masses and the rule (118) for coupling 2 ð1 þ h=FσÞ in the last two terms become ð1 þ h=vÞ and constants. ð1 þ h=vÞ respectively. Similar results are often quoted for NLO amplitudes are interesting [149,150], especially for “dilatonic” or scalon-type theories, despite the WW mode → γγ electroweak processes like h . The QCD analogues of being chosen in the limit of scale invariance. The difference these are two- reactions such as f0ð500Þ → γγ in is that the experimental result Fσ ∼ v indicated by mea- surements [151] of the decays h → ττ;WW;ZZ and bb¯ is 21In Sec. VI and Appendices A and B of Ref. [14], the old expected in crawling TC but requires “fine tuning” in OðpÞ rule for an external photon field Aμ was used. To adapt that scalon theories (Secs. IV E and V). discussion to fit Eq. (116), simply regard eAμ as the photon source, where −e is the ’s charge. Then everything works, What clearly distinguishes our result from the SM and including chiral power counting for UV-convergent one-loop other theories is the unique dependence of the Higgs 0 amplitudes such as KS → γγ [144,145]. potential V on the nonperturbative constant β :

095007-21 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019)       2 2 1 4 1 4þβ0 β0 MσFσ − 1 h 1 h VðhÞ¼ 0 þ þ 0 þ þ 0 : ð123Þ β 4 Fσ 4 þ β Fσ 4ð4 þ β Þ In the expansion   X h n VðhÞ¼ an ; ð124Þ n≥2 Fσ the coefficients an are given by 2 2 0 MσFσ 1 Γ½4 þ β 3! 1 1;n¼ 4; a ¼ − ; with ≡ ð125Þ n β0 n! Γ½5 þ β0 − n ð4 − nÞ! ð4 − nÞ! 0;n>4:

4þβ0 We have retained the notation Mσ for the LO mass of h as for the operator ð1 þ h=FσÞ in Eq. (128), so we can a reminder that our Higgs boson is a genuine pseudodi- ˆ 2 conclude that hG ivac corresponds to the remaining term in laton. Expanding the gamma functions in Eq. (125) yields Eq. (128):     1 2 5 β0 3 2 2 h þ h 0 2 2 VðhÞ¼MσFσ þ ϵβ 2 MσFσ 2 2 3! Gˆ O ϵ : 130 Fσ Fσ 4α h ivac ¼ 4 β0 þ ð Þ ð Þ   IR þ 11 þ β0ðβ0 þ 6Þ h 4 þ þ Oðh5Þ : ð126Þ 4! Fσ This yields the LO formula

ϵβ0 4 β0 The corresponding SM formula is 2 ð þ Þ ˆ 2 2 Mσ G O ϵ ; ¼ 4α 2 h ivac þ ð Þ ð131Þ 1 IRFσ V h m2h2 1 h= 2v 2: SMð Þ¼2 h f þ ð Þg ð127Þ in agreement with the result (53) derived from the CS Unlike the SM, our effective theory is not renormalizable, equations. At NLO, the correction to Eq. (131) is of the 5 4 so powers ∼h and higher are present in Eq. (126). form ∼Mσ lnðMσ=Λσ;vÞ; scale invariance forbids quadratic Although a determination of the Higgs quartic self- dependence on the scales Λσ;v. coupling appears to be out of the LHC’s reach and very challenging even for future colliders [152], measurements VII. SIGNALS FOR CRAWLING TC of Higgs double production during the high-luminosity ON THE LATTICE phase of the LHC should make it possible to place bounds on the cubic coupling [153,154]. Currently, this is the most Throughout this paper we have been careful to distin- promising way to determine β0. guish IR fixed points according to whether scale invariance Subject to the NG-mode requirements for h noted above, is realized in the WW or NG mode. This distinction is it is evident that our Lagrangian (122) belongs to a class of especially important for the search for IR fixed points on the Lagrangians [35–43] proposed for dynamical electroweak lattice, since these investigations typically rely on criteria or theories with the Higgs field treated as a generic scalar. techniques that apply to the WW mode only. In particular, General formulas are given in Sec. 2 of Ref. [43]. the existence of NG-mode IR fixed points cannot be inferred Given the explicit form of the potential, we can check the from spectral studies based on hyperscaling relations [22], relation (53) between the dilaton mass and the technigluon because these scaling laws are forbidden by the soft-dilaton condensate. Let us compare the trace anomaly in the EFT results of Eqs. (15) and (16): conformal symmetry is hidden.   Another key point [noted below Eq. (29) in Sec. II]isthat 2 2 4 β0 μ MσFσ h þ θμ − 1 − 1 128 NG-mode IR fixed points are theoretically possible for any jeff ¼ 4 β0 þ ð Þ þ Fσ value of Nf outside the conformal window. When con- formal symmetry is hidden, there is no need to tune N to lie with that of the underlying theory f on the edge of the conformal window (which is standard ϵβ0 practice in “dilatonic” walking gauge theories [1–5]). This θμ − ˆ 2 − ˆ 2 ϵ2 μ ¼ 4α fG hG ivacgþOð Þ: ð129Þ suggests that the set of IR fixed points may be larger IR than previously envisioned; a sketch for SUð3Þ gauge As specified in Appendices C and E, the operator Gˆ 2 scales theories is shown in Fig. 3. homogeneously (no mixing with the identity operator I) The most direct ways to look for candidate theories of and with dynamical dimension 4 þ β0. That is also true crawling TC on the lattice are as follows:

095007-22 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019)

mψ ≠ 0, what is actually measured is the mψ ≠ 0 version ˜ Fσ of the decay constant [see Eq. (55)], from which Fσ is found by extrapolating in mψ to mψ ¼ 0. γ ψψ¯ α Then the anomalous dimension m of at IR can be deduced from the LO soft-σ theorem hσjψψ¯ jvaci 3 − γ ¼ Fσ þ Oðmψ Þ: ð132Þ m hvacjψψ¯ jvaci σ ψψ¯ ψψ¯ 0 This involves the -pole residues of h ðxÞ ð Þivac and θ ψψ¯ 0 h αβðxÞ ð Þivac, where the latter is required to check magnitudes and fix the sign of 3 − γm. An analysis [5] of mψ dependence in effective Lagrangians has led to proposals [44,45] that “dilaton- 3 FIG. 3. Search for IR fixed points in SUð Þ gauge theories with based” potentials for walking TC and conformally deformed Nf Dirac fermions in the triplet representation. The diagram theories be tested on the lattice. Can a similar approach be assumes that IR fixed points of some kind exist from N ¼ 2 to f applied to our Higgs potential (109) and hence determine β0? Nf ¼ 16, with chiral-scale perturbation theory χPTσ at Nf ¼ 3. 2 − 4 The first step of Refs. [5,44,45] is to account for mψ ≠ 0 Crawling TC with Nf physical technipions (none for Nf ¼ 2) is possible anywhere outside the conformal window. by including in the EFT a chiral mass operator which shifts α α χ χ For simplicity of presentation, we choose to have IR (or ww the VEV of the dimension-1 scalar field from h imψ ¼0 to for Nf ≳ 12) be a decreasing function of Nf for a given χ h imψ ≠0. Constraints on decay constants and spin-0 masses renormalization scheme. are found by minimizing the mψ ≠ 0 potential and evalu- ating its curvature at the minimum. The mψ dependence of (1) Search for the “freezing” of α [28] for Nf values outside the conformal window, where dimensional the results appears to be entirely determined by dependence χ → χ transmutation and chiral condensation can occur. on the shift h imψ ¼0 h imψ ≠0. Note that the UV expansions of α typically used to Applying the same procedure to Eq. (112), one finds measure nonperturbative corrections to asymptotic that physical results do not depend on the value chosen freedom for QCD amplitudes are not applicable. The for hχi > 0, or equivalently, for hσi. Invariance under energy scale must be lowered beyond the region σ → σ þ constant is expected for a true dilaton because where UVexpansions break down and into the far IR the equivalence theorem allows us to choose anyreal value of α ≲ α σ σ → 0 region IR. h i; indeed, h i does not have to vanish even for mψ . (2) Test whether amplitudes exhibit the singular behav- The equations are simplest if we use this freedom to retain the ior displayed in Eq. (34). choice hσi¼0 as mψ is turned on. Even with that choice, it is (3) Confirm the presence of a pseudodilaton with a necessary to distinguish observable masses and decay 2 ˜ ˜ small value of Mσ, especially if Eq. (53) or Eq. (131) constants such as Mσ and Fσ for values of mψ ≠ 0 from can be tested. those for mψ ¼ 0 (Mσ and Fσ). Similarly, we replace the Lattice calculations for Nf ¼ 8 triplet fermions [46–49] and mψ ¼ 0 coefficients c1V and c2V of Eq. (109) by their Nf ¼ 2 sextet fermions [155] suggest that, in the IR region, α counterparts c˜1V and c˜2V for mψ ≠ 0. Since c1V and c2V varies slowly and chiral condensation produces technipions are counted as LO, we must also count Oðmψ Þ corrections to (63 for N ¼ 8 triplets and 3 for N ¼ 2 sextets) plus a light f f them as LO, as in chiral-scale perturbation theory [13–15]: scalar boson. These examples have been taken as support for walking gauge theories, but they could actually point to 2 2 c˜ ¼ c þmψ d þOðmψ ;mψ ϵ;ϵ Þ;n¼ 1;2: ð133Þ crawling scenarios. At present, definite conclusions cannot nV nV nV be drawn because current lattice investigations can be Here dnV do not depend on the scale-breaking parameters mψ matched to chiral constraints only for large fermion masses or ϵ. In terms of σ, the effective LO mass operator for N [133,156–158]. Consequently these investigations may be f degenerate flavors is [13–15] too far from the chiral limit for soft dilaton theorems to apply. σ If a candidate pseudodilaton is seen on the lattice for a 1 2 † 3−γ σ ˜ 2 L ð mÞ =Fσ ϵ mass ¼ mψ BπFπTrðU þ U Þe þ Oðmψ ;mψ Þ; small number Nf of fermion triplets, it may be possible to 2 2 σ determine Fσ from the -pole residue of a component of ð134Þ θ θ 0 “ ” h αβðxÞ μνð Þivac. (We say may, because explicit break- ing of space-time symmetries by the lattice regulator makes where to first order in mψ , we can use the mψ ¼ 0 values of θαβ hard to study on the lattice [159].) Since lattice the decay constants Fπ and the condensate constant Bπ ψ¯ jψ i − 2 δij simulations are performed with massive techniquarks appearing in h ivac ¼ FπBπ .

095007-23 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019)

The mψ -dependent potential to be minimized is VIII. FINAL REMARKS: SUPPRESSION OF FCNCs 4σ ˜ 4 β0 σ ˜ 2 3−γ σ ˜ ˜ σ ˜ =Fσ ˜ ð þ Þ =Fσ − ð mÞ =Fσ Vð Þ¼c1Ve þ c2Ve Nfmψ BπFπe Many papers have been written about the idea that the 2 2 “ ” þ Oðmψ ;mψ ϵ; ϵ Þ: ð135Þ Higgs boson is some sort of dilaton, but unlike crawling TC, there is a lack of commitment to the NG-mode requirement that the limit of exact scale invariance produce Keeping just LO terms OðϵÞ and Oðmψ Þ, the result of scale-dependent amplitudes (Appendix D). As explained in V˜ σ 0 minimizing at ¼ is Sec. IV E, schemes like walking TC and deformed con- formal potentials are not dilaton theories: they follow the ˜ 0 ˜ 2 0 ¼ 4c1V þð4 þ β Þc2V − Nfmψ BπFπð3 − γmÞ: ð136Þ LO example of scalon theory [30] by assuming manifest scale invariance (i.e., no scaling-NG mechanism) in the scale- 2 1 ˜ 2 2 invariant limit. Only in crawling TC, where there is a We now set the Oðσ Þ term equal to 2 Mσσ and find genuine dilaton with a nonzero decay constant in the scale- ˜ 2 ˜ 2 0 2 2 2 invariant limit, can it be argued that approximate scale MσFσ ¼ 16c˜1V þð4 þ β Þ c˜2V − Nfmψ BπFπð3 − γmÞ LO invariance protects the small mass of the Higgs boson. ð137Þ To satisfy the NG-mode requirement, crawling TC assumes the existence of a nonperturbative IR fixed point α at which conformal invariance is exact, and a con- which, from Eqs. (133) and (136), corresponds to mψ IR densate hψψ¯ i for both electroweak and scale trans- dependence vac formations that is nonvanishing in the conformal limit α → α ˜ 2 ˜ 2 2 2 0 0 IR (left diagram of Fig. 1). Both of the decay MσFσ ¼ MσFσ þ mψ ½d2V β ð4 þ β Þ LO constants v and Fσ arise from this condensate in the limit 2 of scale invariance, so their ratio v=Fσ is allowed to be of þ N BπFπð3 − γ Þð1 þ γ Þ: ð138Þ f m m order unity without the fine-tuning problem of scalon-type theories. Equations (136) and (137) can be solved for c˜1 and c˜2 : V V Hidden scale invariance corresponds to new solutions for the CS equations near α , with scaling laws for Green’s 0 ˜ 2 ˜ 2 2 0 IR 4β c˜1V ¼ − MσFσ þ Nfmψ BπFπð3 − γmÞðβ þ γm þ 1Þ; LO functions replaced by the soft-dilaton theorems (15) and 0 0 ˜ 2 ˜ 2 2 (16). Since the scaling-law criteria used to find IR fixed β ð4 þ β Þc˜2V ¼ MσFσ − Nfmψ BπFπð3 − γmÞðγm þ 1Þ: LO points inside the conformal window are not valid for ð139Þ NG-mode fixed points, they may appear at small Nf values (Fig. 3). Unlike Refs. [5,44,45], we do not find any additional The distinctive feature of our theory is the dependence of – β0 constraints at this stage. Without input from higher-order the Higgs potential of Eqs. (123) (126) on , the slope of ˜ βðαÞ at α (left diagram of Fig. 1). We look forward to terms in V, we have no independent information about c˜1 IR V determinations of β0 via experiment (Sec. VI) or the lattice or c˜2 . V (Sec. VII). A determination of β0 from V˜ is difficult because it Finally, we note that standard explanations of the mass involves calculating a three-point function on the lattice and hierarchy of quarks and leptons and the suppression of then going on shell to measure the cubic Higgs coupling. FCNCs can be naturally adapted to fit crawling TC. Given Eq. (139), we find According to the theory of extended technicolor (ETC) [160–163], there is a unification scale Λ ≫ Λ at which 1 3! − ∂σ 2σ ˜ σ3 ˜ V v gσσσ ¼ ð = Þh ð Þ =Fσ þ Oð Þ terms in Vion shell LO the SM and TC gauge groups combine to form an ETC 0 ˜ 2 ˜ 2 gauge group with an intermediate boson X of mass ¼ ð5 þ β ÞMσ=Fσ − Nfmψ BπFπð3 − γmÞðγm þ 1Þ LO 0 ˜ 3 ∼ Λ 4π ≫ × ðγm þ β þ 1Þ=Fσ: ð140Þ MX V=ð Þ W;Z masses: ð141Þ

→ 0 ψ l In the limit mψ , this agrees (as it should) with the The ETC coupling gX of SM fermions SM ¼ q; to TC 3 22 ψ Oðh Þ coupling in Eq. (126). fermions TC and directly or indirectly to other SM α Otherwise, if IR can be isolated, it may be easier to fermions induces FCNCs via effective four-fermion inter- β0 α α obtain directly from the running of near IR.Ifan actions such as independent value of the technigluon condensate becomes ≠ 0 22 available (Appendix C), the mψ version (55) of the Previously denoted as ψ, as in Fig. 1 and Eq. (54). ETC ψ dilaton mass formula may be tested. fermions ETC do not play a major role in this analysis.

095007-24 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019)

L 2 ¯ γμψ ψ¯ γ (N ≠ 3) is much smaller than β0 for QCD where23 N ¼ qi↔qj ¼ cijðgX=MXÞ qiL TCR TCL μqjR þ H:c:; f f 3 after decoupling t, b, c. That would allow TC resonances ð142Þ Λ to appear up to an energy Mmax much larger than v and explain the delay in the onset of asymptotic freedom in TC L 2 ¯ γμ ¯ γ jΔSj¼2 ¼ cΔS¼2ðgX=MXÞ dL sRsL μdR þ H:c:; ð143Þ compared with QCD. The IR end of the integral in Eq. (145) is sensitive to the proximity of αðmhÞ to the 1 fixed point α ,som is the relevant lower limit. Given where cij and cΔS¼2 are Oð Þ numerical coefficients. The IR h 0 0 3 α α α M observed bound on K ↔ K¯ requires M =g ≳ 10 TeV that varies little between IR and ð maxÞ,wehave X X γ α ≃ γ α , so the exponential factor in Eq. (145) in Eq. (143), but then, rough estimates of the contributions mð Þ mð IRÞ becomes of L ↔ and its leptonic analogue Ll ↔l to the SM- qi qj i j Z fermion mass matrix tend to be orders of magnitude too M max dμ γ α γ α μ ≈ mð IRÞ small to fit the observed quark- spectrum. This exp μ mð ð ÞÞ ðMmax=mhÞ : ð148Þ ψ mh assumes vacuum insertion for the TC-dependent part of Eq. (142) renormalized at the ETC scale, γ γ α Like m, mð IRÞ is a nonperturbative number. If we take γ α 1 102 (say) mð IRÞ¼ , an enhancement of corresponds ψ ψ¯ → ψ¯ ψ 12 5 TCR TCL hvacj TCL TCRjvaciETC: ð144Þ to Mmax ¼ . TeV. In conclusion, crawling TC is a consistent theory which The conclusion then follows from the relation avoids the conceptual difficulties of walking TC while sharing its benefits. Crawling TC (left diagram of Fig. 1) does not suffer from walking TC’s phase discontinuity hvacjψ¯ ψ jvaci TCL TCR ETC Z (between the solid and dashed lines of the right diagram), Λ V dμ vac ψ¯ ψ vac exp γ α μ 145 and allows the number of physical technipions to be ¼h j TCL TCRj iTC μ mð ð ÞÞ ð Þ Λv minimized by taking Nf small. between ETC- and TC-scale amplitudes implied by the ACKNOWLEDGMENTS O ψ¯ ψ CS equation (8) for ¼ TCL TCR, and from the obser- We thank Tom Appelquist, Georg Bergner, Simone vation that Biondini, Luigi Del Debbio, C´esar Gómez, Kieran Holland, Andrei Kataev, David Mesterházy, Germano ψ¯ ψ Λ3 hvacj TCL TCRjvaciTC ¼ Oð vÞð146Þ Nardini, Elisabetta Pallante, Mannque Rho, Francesco Sannino, David Schaich, Koichi Yamawaki, and Roman is very small compared with the ETC amplitude. If Zwicky for useful correspondence and discussions. Eliezer Rabinovici and Paolo Di Vecchia are thanked for comments asymptotic freedom in TC sets in above Λv as rapidly as it does in QCD, the exponential factor in Eq. (145) is at on Fubini’s work [31] which helped us to reformulate Sec. IV. The work of L. C. T. was supported by the Swiss most logarithmic in ΛV=Λv and thus much too small to fit the SM-fermion spectrum. National Science Foundation. The work of O. C. is sup- Walking TC dispenses with part of the QCD/TC analogy ported in part by the Bundesministerium for Bildung und by assuming that the TC β function is close to zero over Forschung (BMBF FSP-105), and by the Deutsche the large range of energies between the TC and ETC Forschungsgemeinschaft (DFG FOR 1873). scales [164–170]. That corresponds to the walking region of the right diagram of Fig. 1. Since β ≈ 0 APPENDIX A: DECOUPLING: THE DIFFERENCE BETWEEN CURRENT AND implies γmðαÞ ≈ constant, γmðαÞ can be approximated by CONSTITUENT MASSES a constant value γm in the integral. The result is power enhancement While writing this paper and its predecessors [13–15], Z we were puzzled by a reluctance in the literature to consider Λ μ V d γ the NG mode for scale invariance, especially at an IR fixed exp γ α μ ≈ Λ =Λ m 147 μ mð ð ÞÞ ð V vÞ ð Þ point of a gauge theory. Possible reasons for this are Λv suggested in the text of this paper. However, our attention in Eq. (145). A minimal enhancement ≳102 is obtained for γ ≈ 1. That gives an order-of-magnitude fit to the SM- 23Precocious asymptotic freedom for QCD is observed in the m 3 ≫ fermion spectrum (apart from the and , Nf ¼ region, i.e., for momenta Q ms up to the charm threshold. It has nothing to do with the candidate Nf ¼ 2 theory which require special treatment). for crawling TC; in particular, s does not decouple in the chiral In crawling TC, power enhancement can occur in the 2 2 SUð ÞL × SUð ÞR limit taken at fixed ms. On the lattice, the 0 crawling region (left diagram of Fig. 1)ifβ for TC cases Nf ¼ 2 þ 1 and Nf ¼ 2 are clearly distinct.

095007-25 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019) has just been drawn to a reason we did not take seriously: a m=q2 is finite, so decoupling does not occur. It is also belief that condensates decouple in the IR limit because reassuring that the existence of gluonic condensates does they give their constituent fields “mass,” effectively making not cause gluons to decouple. More generally, dimension- them heavy relative to IR scales. Fermion condensates are ally transmuted scales M ≠ 0 are allowed in the extreme usually mentioned in this regard [16], presumably because IR limit of QCD and (by analogy) TC. These observations a gluonic analogue of the Lagrangian mass term −mψψ¯ hold irrespective of whether an IR fixed point exists or not, cannot be constructed without introducing extra field so if it exists, nothing prevents it from realizing scale variables (perturbative ). invariance in the NG mode. The gap in this argument is that it does not distinguish Consider QCD at low energies, with the t, b and c quarks ’ → 0 3 between a light fermion s small current mass m and its decoupled. In the limit mu;d;s of chiral SUð ÞL × Σ 0 Σ 2 ’ 3 large constituent mass ð Þ. Here ðq Þ is the fermion s SUð ÞR symmetry, low-energy theorems can be derived self-energy dynamically generated by the nonperturba- for amplitudes involving the eight chiral NG bosons π, K, η ¯ tive mechanism responsible for fermionic condensation and local operators such as chiral currents and qLqR.For ψψ¯ ≠ 0 3 h ivac and (in QCD) the SUð ÞLþR-invariant part of amplitudes with non-NG states excluded, soft-meson the- the masses of non-NG hadrons. While m violates chiral orems are derivable when all external momenta q tend to symmetry explicitly, Σ does not: it remains nonzero in the zero. The key point is that this soft-meson limit and the IR → 0 limit m of chiral SUðNfÞL × SUðNfÞR symmetry. In limit of the running of αs for Nf ¼ 3 are indistinguishable: general, m and Σ are matrices in flavor and spinor space. the results of applying chiral perturbation theory and the The current mass appears in the covariant operator iD= − m RG must match. An assumption that u, d, s decouple in in the Schwinger-Dyson equation the IR limit would imply decoupling of π, K and η and so contradict the well-known soft-meson theorems for = − ψ ψ¯ δ4 − T hðiDx mÞ ðxÞ ðyÞivac ¼ i ðx yÞ; ðA1Þ these NG bosons required by chiral perturbation theory. That would be a disaster for QCD. where the time-ordering operation T preserves chiral Instead, what happens is that π, K and η exist in the IR SUðNfÞL × SUðNfÞR and gauge covariance, and limit of QCD. There are two main possibilities: ∂ … ≡∂ … Σ T f μ g μT f g by definition. The self energy (a) There is no IR fixed point; rather, αs runs to ∞ with appears in the solution for the dressed propagator scale invariance explicitly broken. Standard chiral 3 3 Z SUð ÞL × SUð ÞR perturbation theory is applicable. 4 i (b) There is an IR fixed point α which is necessarily in d xeiq:xThψðxÞψ¯ ð0Þi ¼ : sIR vac Aðq2Þ=q − m − Σðq2Þ the scaling NG mode because of the quark condensate responsible for chiral NG bosons. There are nine NG ðA2Þ bosons, a dilaton σ as well as π, K and η, and chiral- – Equations (A1) and (A2) imply the standard self-consistent scale perturbation theory [13 15] is applicable. 1 It is important to check that these conclusions are conditions for ð1 γ5ÞΣ shown in Fig. 4. 2 consistent with the Appelquist-Carazzone theorem. It states The argument above contains an implicit assumption that that, to all orders in a perturbative gauge theory, a field Σ sets the scale for decoupling, i.e., that for momenta with a large Lagrangian mass M decouples in the limit q ≪ Σ, the fermions are very “heavy” relative to q and so M → ∞ taken at fixed renormalization scale μ, with finite decouple. We show below that QCD is not tenable if that changes in other renormalization constants such as the assertion is believed. gauge coupling α. Since dimensionally transmuted con- Fortunately for QCD, a natural extension of the pertur- stants M such as those associated with fermion condensa- bative Appelquist-Carazzone theorem [98] to include tion are nonperturbative, as is evident from the discussion dimensionally transmuted scales such as Σð0Þ indicates of Eq. (31), the proof of Appelquist and Carazzone that it is the current mass which matters. In chiral effectively assumes that all M vanish. So at first sight, perturbation theory, both m and q tend to zero such that the theorem is irrelevant, and nothing needs to be checked. However, it is reasonable to suppose that the Appelquist- Carazzone theorem could be extended to include M constants with the conditions of the theorem otherwise unchanged. First, it is necessary to identify RG invariants M associated with the field with a large current mass M; either they tend to ∞ with M, such as particle thresholds for FIG. 4. Equation (A1) written as a relation between one- heavy-quark production in QCD, or they vanish because ’ ¯ particle-irreducible (1PI) two-point and three-point functions in they involve Green s functions like hbbivac which depend M momentum space. The fermion and gauge-boson propagators on that field. Then, for RG invariants res that remain in within the loop are fully dressed. the residual theory after decoupling, there would be a finite

095007-26 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019) change due e.g., to the finite renormalization of α and β in but as long as this lack of uniformity is respected, the Eq. (31). This is just a consistency argument, not a answer ends up being the same. Scale and conformal derivation, but we suggest that a reasonable derivation invariance are considered in Appendix B3. may be possible using Landau’s diagrammatic analysis of the nonperturbative behavior of vertex functions [171]. 1. Chiral-symmetric theory Although this extension of the theorem appears not to have been stated, much less derived, it is implicit in non- Consider a chiral SUðNfÞL × SUðNfÞR-symmetric a perturbative applications such as the decoupling of t, b TC theory with conserved axial-vector currents Jμ5 ¼ a and c in the presence of light-quark condensates such ψγ¯ μγ5T ψ and axial charges ¯ ≠ 0 as huLuRivac . Z Following Appelquist and Carazzone [98], let us scale 3 2 Qa ¼ d xJa ðxÞ; Tr Ta ¼ 0;a¼ 1; …;N − 1: the theorem’s conditions such that the limit M → ∞ for 5 05 f → 0 finite momenta q is replaced by the IR limit q at fixed ðB2Þ M, with μ fixed in both cases (mass-independent renorm- alization). In perturbation theory, that works if all masses For any operator (or operator product) O which is not chiral ≠ M are scaled with q, so that, in the IR-limit version of the invariant, there is another operator theorem, they tend to zero with q. The problem is that the same argument applied to the a a δ5O ¼ i½Q5; O ≠ 0: ðB3Þ M-dependent extension of the theorem would require M res to scale with q, and not with the heavy current mass A nonzero VEVof δaO can occur only if jvaci is not chiral M 5 M. In the finite-q version of the theorem, all remain invariant. Then the amplitude finite or vanish as M → ∞, hδaOi ≠ 0 ðB4Þ M → 0 5 vac res=M ; ðA3Þ is called a chiral condensate. In the standard case, O is the so decoupling in the IR limit q ∼ 0 with M fixed can be b pseudoscalar operator ψγ¯ 5T ψ: M concluded only if it is assumed that all res vanish. Clearly, decoupling is a consequence of a current mass a b a b b a hδ5fψγ¯ 5T ψgi ¼ −hψ¯ ðT T þ T T Þψi ≠ 0: ðB5Þ M being large relative to dimensionally transmuted scales vac vac M res in the residual theory. Light-quark condensates and Let O be a local spin-0 operator OðxÞ. Then the Ward their TC analogues do not decouple in the chiral IR limit. identity for the time-ordered amplitude Z Aa 4 iq·x a O 0 APPENDIX B: NON-LAGRANGIAN METHODS μ5ðqÞ¼ d xe ThJμ5ðxÞ ð Þivac ðB6Þ FOR CHIRAL AND CONFORMAL NG BOSONS The NG mode for a symmetry is usually explained in is given by terms of symmetric Lagrangians with potential functions Z μAa 4 iq·xδ a O 0 that have flat directions. This obscures the more general q μ5ðqÞ¼i d xe ðx0Þh½J05ðxÞ; ð Þivac: ðB7Þ understanding developed in the 1960s [172,173] that currents Jμ5 and their divergences are all that is needed. At zero momentum q → 0, Eq. (B7) reduces to As preparation for the scaling application in Sec. II,we present a brief review of the analysis for chiral SUðN Þ × μ f L lim q Aa ðqÞ¼ih½Qa;Oð0Þi ¼hδaOð0Þi ≠ 0; ðB8Þ →0 μ5 5 vac 5 vac SUðNfÞR symmetry, and extend it to scale and conformal q invariance, where the currents are nonlocal. Aa 1 The aim is to derive theorems for NG mesons carrying which is possible only if μ5ðqÞ has an Oð =qÞ singularity. μ soft momenta q → 0 as the symmetry limit ∂ Jμ5 → 0 for This implies Goldstone’s theorem: such a singularity can 2 − 1 πa current divergences is taken. There are two ways of arise only if there are Nf massless technipions a proceeding (Appendices B1 and B2 below); the choice coupled to Jμ5, depends on the order in which these limits are taken. This matters because there can be factors involving the pseudo- a qμ a A ðqÞ¼− Fπhπ ðq ¼ 0ÞjOð0Þjvaci NG mass m for which the limits do not commute, e.g., μ5 q2 þ terms finite at q ¼ 0; ðB9Þ m2 m2 lim lim ¼ 0; lim lim ¼ 1 ðB1Þ →0 →0 2 − 2 →0 →0 2 − 2 q m m q m q m q where the decay constant Fπ is defined by

095007-27 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019)

a b ab hvacjJμ5ð0Þjπ ðqÞi ¼ iδ Fπqμ: ðB10Þ currents. A separate analysis is necessary for nonlocal operators such as the dilatation and conformal currents Equation (B8) fixes the residue of the q2 0 pole in ¼ μ 2 λ Eq. (B9). The result is a standard soft-π theorem Dν ¼ x θμνðxÞ; Kμν ¼ð2xμxλ − x gμλÞθνðxÞðB16Þ

πa 0 O 0 − δaO 0 which correspond to generators Fπh ðq ¼ Þj ð Þjvaci¼ h 5 ð Þivac: ðB11Þ Z Z 3 3 DðtÞ¼ d xD0ðt;xÞ;KμðtÞ¼ d x Kμ0ðt;xÞ: ðB17Þ 2. Chiral currents partially conserved The alternative 1960s procedure is to give the currents Given these definitions, the partial conservation equations small divergences X ν λ ν λ ∂ Dν θ and ∂ Kμν 2xμθ B18 μ a a a ¼ λ ¼ λ ð Þ ∂ Jμ5 ¼ D5 ¼ 2i mψ ψγ¯ 5T ψ ðB12Þ ψ λ show that scale invariance θλ → 0 ensures conformal by letting each techniquark have a small renormalized mass invariance. The result that only one NG boson is needed—the dilaton mψ and then take the symmetry limit mψ → 0. This is the of scale invariance—was not obvious at first [101,102], forerunner of our approach in Sec. II, where the scale- μ but it was quickly realized [9] that conformal-invariant breaking divergence θμ tends to zero as α approaches the α Lagrangians can be constructed by having the derivatives fixed point IR. ∂μσ of the dilaton field σðxÞ act as Goldstone fields for the For massive TC fermions, the Ward identity (B7) is replaced by four conformal generators Kμ. Also, Eq. (B18) shows that a Z pseudo-NG boson for either scale or conformal invariance must have spin 0. The absence of extra NG bosons has been μAa 4 iq·xδ a O 0 q μ5ðqÞ¼i d xe ðx0Þh½J05ðxÞ; ð Þivac attributed [174–176] to the failure of Kμ to commute with Z the translation generators Pμ in the limit of conformal 4 iq·x a O 0 þ i d xe ThD5ðxÞ ð Þivac: ðB13Þ invariance,

−2 ≠ 0 θλ → 0 The traditional derivation of the soft-π theorem then runs as ½Kμ;Pν¼ iðgμνD þ MμνÞ ; λ ðB19Þ a follows [173]. The NG bosons acquire mass, so Aμ5ðqÞ where Mμν generate Lorentz transformations. cannot have a 1=q singularity, and the q → 0 limit of The literature on the NG mode for scale and conformal Eq. (B13) is invariance is dominated by Lagrangian models of scale Z invariance. Unlike the chiral case [122,137,177], the model 0 δaO 0 4 a O 0 ¼h 5 ð Þivac þ i d xThD5ðxÞ ð Þivac; ðB14Þ independence of their predictions for multiple soft-dilaton emission has yet to be proven explicitly, and they have not where the commutators (B3) are now understood to be been used at all to obtain conformal theorems. Instead, taken at equal times. The second term is a zero-momentum soft-dilaton results for special conformal transformations a [52–54] were obtained by the non-Lagrangian method, insertion of the current divergence D5. It can be nonzero a which we consider now. It is model independent and in the limit D5 → 0 only if there exists a single-particle intermediate state which becomes massless as resembles the chiral version discussed in Appendices B1 and B2, but there are some interesting differences which 2 2 2 λ − − → are best seen for the symmetric case θλ ¼ 0. imψ =ðq MπÞjq¼0 ¼ imψ =Mπ finite: ðB15Þ We begin with the analogue of Appendix B1 for scale When the residue of this π pole is evaluated via symmetry (Dν conserved), excluding for a moment the a b 2 ab O O 0 hvacjD ð0Þjπ i¼MπFπδ , the soft-π result (B11) is special case of a single spin-0 operator . Instead of ð Þ, 5 O 0 recovered. Note that pole dominance is not assumed: in let us consider a T-ordered product of 1ð Þ (not neces- the symmetry limit, branch cuts are less singular than poles. sarily spin-0) and Fourier transforms Z 4 O˜ ipn·xn O 1 3. Soft-dilaton theorems for scale nðpnÞ¼ d xne nðxnÞ;n> ðB20Þ and conformal invariance

Goldstone’s theorem, that the number of NG bosons of other local operators OnðxnÞ, and hence connected … δ4 equals the number of independent group generators which momentum-space amplitudes h ic with the function transform the vacuum, is generally valid only for local for momentum conservation removed.

095007-28 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019)

The general amplitude involving the dilatation current can be written as Z D E Y ∂ 4 iq·x ˜ Bν ¼ d xe T vacDνðxÞO1ð0Þ OnðpnÞvac ¼ −i μ Γμνðq; fpgÞ; ðB21Þ c ∂ n>1 q where Γμν is constructed entirely from local operators, including the traceless tensor θμν: Z Z D E Y Y 4 iq·x 4 ipm·xm Γμν ¼ d xe d xme T vacθμνðxÞO1ð0Þ OnðxnÞvac : ðB22Þ c m>1 n>1

Then the scaling Ward identity for Bν takes the form Z Z   Y X Y ν ν ∂ 4 4 0 0 B Γ ; − iq·x ipm·xm δ − D O O iq ν ¼q ∂ μ μνðq fpgÞ¼ d xe d xme T vac ðx xlÞ 0ðxÞ; lðxlÞ nðxnÞvac : q 1 l≥1 ≠l c m> n x1¼ 0 n≥1 ðB23Þ

For scale invariance in the NG mode, the vacuum state is not scale invariant, so there will be operators fOng for which the right-hand side of Eq. (B23) does not vanish in the limit of zero momentum q:

Y ∂ ˜ ν FσT σ q 0 O1 0 O p vac lim q μ Γμνðq; fpgÞ h ð ¼ Þj ð Þ nð nÞj ic q→0 ∂q 1 Y n> − O 0 O˜ X ¼ Thvacj½D; 1ð Þ nðpnÞjvacic ¼ d1 þ ðdm − 4 − pm · ∂=∂pmÞ n>1 m>1 ≠ 0 Y ¼ FðfpgÞ : ðB24Þ O 0 O˜ ×Thvacj 1ð Þ nðpnÞjvacic ðB27Þ n>1

σ 24 Equation (B24) can be satisfied only if there is a singularity which is a standard soft- theorem.Q ∼ 2 Γ → 0 O O˜ qμqν=q in μν as q : The case where Tf 1 n>1 ng is just a single spin-0 operator O is special. Consider the unordered amplitude

qμqν Z Γμνðq; fpgÞ ¼ FðfpgÞ þ GμνðfpgÞ þ OðqÞ: ðB25Þ 3 2 þ 4 iq·x q ΓOμνðqÞ¼ d xe hvacjθμνðxÞOð0Þjvaci: ðB28Þ

It has the remarkable property that momentum conserva- Since this result includes amplitudes FðfpgÞ where μ þ μν þ tion q ΓOμν ¼ 0 and scale invariance g ΓOμν ¼ 0 deter- internal momentum transfers are not light-like, the q−2 mine its nonperturbative dependence on q: pole must be due to a massless spin-0 particle, the dilaton σ. The residue of the pole can be determined from Eq. (1) for þ 2 ΓOμνðqÞ¼2πkqμqνδðq Þθðq0Þ;k¼ const: ðB29Þ the decay constant Fσ. Given that On has dynamical dimension dn, 25 Time ordering introduces a constant ambiguity cgμν,

Γ 2 ϵ i½D;O1ð0Þ ¼ d1O1ð0Þ; OμνðqÞ¼ikqμqν=ðq þ i Þþcgμν; ðB30Þ ˜ ˜ i½D;OmðpmÞ ¼ ðdm −4−pm ·∂=∂pmÞOmðpmÞ;m>1 24The earliest example appeared in Sec. 5 of Ref. [50]. The ðB26Þ model in Sec. IV is an almost scale-invariant version of the linear sigma model [74,178] (clarified in the Appendix of Ref. [179]); it was not used to derive the soft-dilaton theorem. 25 Notice that c cannot be chosen such that Γμν is conserved and the pole term in Eq. (B25) implies the result has zero trace. That will not matter.

095007-29 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019)

μ but the dependence on c drops out when DνðxÞOð0Þ is i½D; OðxÞ ¼ ðdO þ x ∂μÞOðxÞ; ðB33Þ time ordered: Z ∂ 4 iq·x we see that Eq. (B32) fixes the constant k in Eq. (B29): i ΓOμνðqÞ¼ d xe ThvacjDνðxÞOð0Þjvaci ∂qμ 3 2 ϵ 2 ¼ ikqν=ðq þ i Þ: ðB31Þ Γþ π O 0 δ 2 θ OμνðqÞ¼3 dOhvacj ð Þjvaciqμqν ðq Þ ðq0Þ: ðB34Þ The Ward identity which follows P ∂ When the completeness sum I ¼ jnihnj is inserted ν Γ 3 O 0 n iq OμνðqÞ¼ ik ¼ ihvacj½D; ð ÞjvaciðB32Þ between θμν and O in Eq. (B28), only single-dilaton ∂qμ states jni¼jσi can reproduce this q dependence. So Γþ has a q-independent right-hand side, so there is no need without approximating, we can relate Oμν to the dilaton 26 to expand about q ¼ 0. From decay constant Fσ of Eq. (1):

Z d3p Γþ 2π 4δ4 − θ 0 σ σ O 0 OμνðqÞ¼ 3 ð Þ ðq pÞhvacj μνð Þj ðpÞih ðpÞj ð Þjvaci 2p0ð2πÞ 2 2πδ 2 θ θ 0 σ σ O 0 π δ 2 θ σ O 0 ¼ ðq Þ ðq0Þhvacj μνð Þj ðqÞih ðqÞj ð Þjvaci¼3 Fσqμqν ðq Þ ðq0Þh ðqÞj ð Þjvaci: ðB35Þ

Comparison of Eqs. (B34) and (B35) yields the soft- which is nonzero only if there is a singular term λ 2 λ dilaton formula (17). ∼q qνqβ=q in Γν: If scale invariance is in the NG mode, so also is conformal invariance: Kμjvaci ≠ 0 because of the identity   ∂2 ∂2 qλq q Y Y ν − 2 ν β −6 q gμλ α μ λ 2 ¼ gμβ: ðB39Þ O O˜ 2 O O˜ ∂q ∂qα ∂ ∂ q h½Kμ; ½Pν; 1 nivac ¼ igμν D; 1 n q q n>1 n>1 vac ≠ 0 ðB36Þ Therefore, Eqs. (B24) and (B25) can be extended to include 27 implied by Eq. (B19) and Poincar´e invariance of the the OðqÞ pole term: vacuum. For the conformal current Kμν, the analogue of Eq. (B21) is Γμν q; p Z ð f gÞσ pole Y  Y  4 iq·x ˜ qμqν Bμν ¼ d xe ThvacjKμνðxÞO1ð0Þ O ðp Þjvaci − 2 β O 0 O˜ n n c ¼ 2 Thvacj D þ iq Kβ; 1ð Þ nðpnÞ jvacic 1 6q n> n>1 ∂2 ∂2 2 − 2 Γλ ; þ Oðq Þ: ðB40Þ ¼ gμλ α μ λ νðq fpgÞ: ðB37Þ ∂q ∂qα ∂q ∂q

νB β Then q μν gives a conformal Ward identity similar to Although Kβ appears as a projection q Kβ in a light-like Eq. (B23) but with D0 replaced by Kμ0 in equal-time 2 0 direction (q ¼ 0, q0 > 0), all space-like directions q − q → 0 commutators. In the limit q , the result is and hence individual Kβ components can be obtained by comparing σ states with small on-shell momenta q and q0. ∂2 ∂2 ν − 2 Γλ ; The general soft-σ result for special conformal transforma- lim q gμλ α μ λ νðq fpgÞ q→0 ∂q ∂qα ∂q ∂q tions is therefore Y O 0 O˜ ¼ iThvacj½Kμ; 1ð Þ nðpnÞjvacic; ðB38Þ 27 n>1 The connection between OðqÞ terms and special conformal transformations was noted in Ref. [180] and used to derive soft- dilaton theorems [52] long ago. The subject has been revived very 26Spin-0 operators O are defined such that an extra term on the recently [53,54]; note the important distinction they made right-hand side ∝ I does not appear. See Appendix C1. between NG-mode dilatons and “gravitational dilatons.”

095007-30 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019) Y σ − σ 0 O 0 O˜ ½D; O ð0Þ ¼ −id O ð0Þ; Fσðh ðqÞj h ðq ÞjÞ T 1ð Þ nðpnÞ jvacic n n n n>1   ½Mμν;Onð0Þ ¼ −ΣnμνOnð0Þ; 1 Y β ˜ 0 − O 0 O Kμ;O 0 0 B42 ¼ 2 ðq qÞ Thvacj Kβ; 1ð Þ nðpnÞ jvacic ½ nð Þ ¼ ð Þ n>1 þ Oððq or q0Þ2Þ: ðB41Þ where Mμν generate the Lorentz group and Σnμν are the corresponding spin matrices for On. Translating with As is well known [51,179,181], the Kμ commutators are μ expðiP xμÞ yields the standard formula best classified via the little group at x ¼ 0. Each time Kμ commutes with a dimension-d operator, it reduces the − 1 dimension to d . So there are towers of local operators i½Kμ;OnðxÞ (mostly derivatives of other operators) above familiar 2 ρ∂ − 2∂ − 2 ρΣ spin-J operators of minimal dimension such as chiral ¼f xμðdn þ x ρÞ x μ ix nμρgOnðxÞ: ðB43Þ currents and θμν whose Kμ commutators necessarily vanish at x ¼ 0. For all operators On of this type, we have So for operators On → On, Eq. (B41) implies

Y σ − σ 0 0 ˜ Fσðh ðqÞj h ðq ÞjÞ T O1ð Þ OnðpnÞ jvacic 1  n>  X ∂ ∂ 1 ∂2 ∂ Y 0 μ ˜ q − q 4 − d p − p μ iΣ μρ O1 0 O p ¼ð Þ m þ m · ∂ ∂ μ 2 m ∂ 2 þ m ∂ Thvacj ð Þ nð nÞjvacic m>1 pm pm pm pmρ n>1 þ Oððq orq0Þ2Þ: ðB44Þ

The soft-dilaton theorems (B27) and (B44) will be considered. The analysis refers to the physical region, 0 α α α needed in Appendix D. which is < < IR if there is an IR fixed point IR, and 0 α ∞ α → ∞ < < if not ( IR ). When the gauge coupling α is finite, functional integrals APPENDIX C: NONPERTURBATIVE DEFINITION for Green’s functions are dominated by nonperturbative OF GLUON AND TECHNIGLUON gauge [183] and fermion fields. Evidently these fields are CONDENSATES hard to characterize analytically, i.e., beyond numerical 2 lattice methods. Instead, attention is focused on a few Definitions of the gluon condensate hG ivac and its TC 2 O analogue hGˆ i are problematic because physical operators that form nonperturbative condensates vac O (1) they are perturbatively divergent nonperturbative h ivac which can be given theoretical and phenomenological O quantities, and meaning. The problem is to define without introducing (2) they involve operators like Gˆ 2 which are hard to ambiguities proportional to the identity operator I. Only then separate from the identity operator I under renorm- does the condensate acquire a physical meaning. alization or within operator product expansions. Even before QCD was invented, it was known how to do a The issue arises in the discussion following Eq. (7), where this for divergences D5 of partially conserved currents [99]: the spectator operator O ¼ðα=πÞGˆ 2 is used to obtain soft- they belong to an irreducible representation of an equal- dilaton results such as Eq. (15) and hence Eq. (53) for the time non-Abelian chiral group which distinguishes them a technigluon condensate. To avoid ambiguity, we require that from the chiral-invariant operator I. For example, if D5 and O be multiplicatively renormalizable28 and (in the sense of I appear in an operator product expansion, their contribu- that discussion) α independent. tions to its VEV can be distinguished, provided that other In this Appendix, we explain the need for these require- operators with poorly defined condensates are known to ments, noting that, while they serve our purposes and are have less singular coefficient functions. consistent with various proposals to define hOi , the In gauge theories, these chiral condensates are formed vac O ¯ 1 γ results still lack sufficient precision for unambiguous when is a fermion bilinear: qið 5Þqj for QCD and ψ¯ 1 γ ψ calculations, e.g., on the lattice. The extent to which a ið 5Þ j for TC. As long as the renormalization definitive nonperturbative definition is possible is then procedure respects chiral SUðNfÞL × SUðNfÞR symmetry, these operators do not have counterterms proportional to I 28 and so belong to an irreducible chiral representation. The In QCD with mq ≠ 0, mixing with mqqq¯ must also be considered [100,182]. corresponding condensates are necessarily nonperturbative,

095007-31 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019) so they produce power corrections to short-distance expan- d μ T ¼ F ðμ; αμÞI; ðC3Þ sions of operator products. dμ R R However, any discussion of power corrections at short distances in QCD would be incomplete without including where FR is an ordinary function. However, the I- terms induced by the gluon condensate [64,69] formed dependent term can be absorbed into the definition of 2 when O is the operator ðαs=πÞG . Then group theory the trace operator cannot be used to distinguish O from I, so the definition of Z 2 μ dμ0 hG ivac remains ambiguous due to counterterms propor- − μ0 α TR0 ¼ TR 0 FRð ; μ0 ÞI ðC4Þ tional to I. The same problem arises for the technigluon c μ condensate hGˆ 2i . vac μ α A practical approach still in use [184–186] is to impose a where c is a constant independent of and . Note that the μ0 μ0 α regulator such as the lattice and then identify and subtract integral over takes account of the dependence of μ0 . ˆ 2 Evidently the resulting operator is RG invariant: perturbative contributions to hG ivac up to some high but finite order. The series is not expected to converge because d of renormalons, but perturbative coefficients can be μ 0 μ TR0 ¼ : ðC5Þ checked numerically to see if their behavior is consistent d with Borel summability. The theoretical argument for this is that, if all orders of perturbation theory can in principle be We are not done, because the solution of Eq. (C5) is not unique. Any ordinary function fðMÞ whose dependence summed by a well-defined technique and the result is then μ α M subtracted, the remainder will be the nonperturbative on and is carried solely by the RG invariant mass of amplitude being sought [187]. However, even if Borel Eq. (31) is itself RG invariant. Therefore all operators summability can be proven to all orders, Borel’s method is M not unique: nonperturbative dynamics may choose another TR00 ¼ TR0 þ fð ÞI ðC6Þ well-defined procedure. Generally, there is no guarantee that nonperturbative amplitudes can be deduced from are RG invariant. To preserve engineering dimensions, M M4 purely perturbative considerations [188,189],soitisnot fð Þ can be chosen to be times a constant indepen- μ α surprising that these issues remain a source of unease [3]. dent of and . This ambiguity does not affect the ˆ 2 β α 4α multiplicative renormalizability of operators obtained by Another idea is to multiply G by ð Þ=ð Þ and use RG α invariance. Since purely nonperturbative constants can multiplying by -dependent factors. have no Taylor series about α ∼ 0 [as noted for the constant Nevertheless, it is necessary to eliminate the ambiguity M (C6) if soft-dilaton results such as Eq. (15) are to be in Eq. (31)], perturbative terms cannot be invariant. But α that is not sufficient, because nothing has been done to derivable. That happens when we apply the -independence distinguish the desired operator O from I. For example, if criterion to O   and I appear in the expansion of a product of physical α 4α2 currents, hOi can mix with the VEV of the I term under O ˆ 2 vac R00 ¼ π G ¼ πβ TR00 ðC7Þ RG transformations. R00 Our proposal is to consider the scaling properties of θμ operators, not just amplitudes. We consider mainly the TC for use as a spectator operator in the μ insertion rule (9), case where chiral symmetry of the Lagrangian stops Gˆ 2 because from mixing with techniquark bilinears. ∂ ∂ Let μ set the scale for an arbitrary renormalization β α M −μ M −M ð Þ ∂α ¼ ∂μ ¼ ðC8Þ prescription R for composite operators O, including the trace operator implies ˆ 2     T ¼ðβðαÞ=4αÞG : ðC1Þ 2 4 ∂ 4α 4α d β 4 M4 − M4 − ≠ 0: ∂α πβ ¼ πβ2 α α2 α ðC9Þ We need to distinguish the results of variations μ∂=∂μ at d fixed coupling α and the total variation Then O has no ambiguity ∝ I, in principle. d ∂ ∂ In practice, we would like to be able to test our soft- μ ≡ μ þ βðαÞ : ðC2Þ dμ ∂μ ∂μ dilaton results by comparison with experimental or lattice data. For that, our minimal requirements on Gˆ 2 Because subtractions necessarily include perturbative are necessary but not sufficient. It is not even clear how to terms, we do not expect the result TR to be exactly RG implement them for prescriptions currently on offer ˆ 2 invariant: there must be mixing with I, for hG ivac.

095007-32 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019)

Phenomenology is based on the original QCD prescrip- Equation (C13) is equivalent to a condition [75,194] arising tion [64,69,190], where m -independent power corrections from the Feynman-Hellmann theorem. q α in the small-x expansion of two electromagnetic currents In crawling TC, where there is a fixed point IR, the α technigluon condensate at IR appears in results such as α α ⇁ α s 2 Eqs. (50) and (53). It is obtained as a limit IR of the Jμ x Jν 0 ∼ Cμν x I C 2 x G … ð Þ ð Þ Ið Þ þ μνG ð Þ π þ ðC10Þ amplitude

ˆ 2 β α 4α −1 are by definition contained entirely within the gluonic hvacjðG ÞR00 jvaci¼f ð Þ= g hvacjTR00 jvaci; ðC14Þ C 2 coefficient function μνG ðxÞ. Dispersive sum rules for the 0 α α ˆ 2 operator product (C10) are reasonably consistent with each where < < IR. The operator ðG ÞR00 is local, so the α π 2 ≈ 0 012 4 one other for hð s= ÞG ivac . GeV , but there is no amplitude (C14) is a -point function where intermediate σ ˆ 2 reason to suppose that a similar definition of the gluon states such as j i cannot occur. It follows that hðG ÞR00 ivac is condensate for a different operator product would be continuous in the scaling limit: equivalent. ˆ 2 ˆ 2 hvacjðG Þ 00 jvaci α ¼ lim hvacjðG Þ 00 jvaci0 α α : The idea of this definition is to suppose that the I term in R at IR μ R < < IR θμ→0 a short-distance expansion is purely perturbative. That is a C15 difficult concept: even if CμνIðx; αs; μÞ is truncated to a ð Þ polynomial in αs, the running of αs depends on dimen- sionally transmuted masses M which can produce power 1. Relation to commutators with the corrections.29 And it is unclear how this proposal can be dilatation generator D related to purely theoretical definitions, where perturbative A conventional soft-dilaton theorem such as Eq. (17) is truncation is also a problem. Examples are directly estimat- valid for operators O which scale homogeneously with ˆ 2 ing the one-point function hG ivac on the lattice (noted operator dimension dO, i.e., above), or adding a heavy fermion Ψ and taking its mass M μ to ∞: i½D; OðxÞ ¼ ðdO þ x ∂μÞOðxÞðC16Þ

β α with other operators absent. As seen above, if O has spin 0, ΨΨ¯ ð Þ ˆ 2 lim hvacjM jvaciM ¼ hvacjG jvaci: ðC11Þ mixing with the identity operator I is hard to control, M→∞ def 12παβ1 producing ambiguities such as Eq. (C6). Since Eq. (C16) is ˇ A way around this impasse may be to use experimental not invariant under the shift O → O ¼ O þ cI (c ¼ const), data to extend α beyond the UV region where asymptotic ˇ μ ˇ freedom is applicable. If α can be measured at finite values i½D; OðxÞ ¼ ðdO þ x ∂μÞOðxÞ − cdOI; ðC17Þ where the small-α expansion is no longer valid, perturba- α tive truncation would not be needed. This would have to be it could act as an alternative to independence as a criterion done within a renormalization scheme suitable for match- for resolving the ambiguity in Eq. (C7). These commuta- α α ing to lattice calculations at these intermediate non-UV tors (equal-time for < IR where D is not conserved) are θ O 0 energies. Then, if the thermodynamic limit can be dem- determined by short-distance expansions of μνðxÞ ð Þ.As onstrated for the Euclidean partition function on the lattice, noted in Sec. II [footnote 5 and Eq. (27)], short-distance 0 α α behavior for < < IR is determined by the fixed point α 0 (asymptotic freedom), whereas, when α is Z ∼ expf−V4Γðα; μÞg; Euclidean volume V4 → ∞; ¼ α first fixed at IR, it cannot run: short-distance behavior is ðC12Þ α then controlled by the nonperturbative world at IR α α α α (Appendix D). Therefore the cases ¼ IR and < IR a practical nonperturbative definition of the Euclidean must be considered separately. α condensate would be At IR, the condition (C16) would resolve the ambiguity in spin-0 operators O, e.g., the operator Gˆ 2 in the mass ∂Γ 0 ˆ 2 − 4α formula (53) with dO identified as 4 þ β [Eqs. (51) and hvacjG jvaciEucl ¼ : ðC13Þ def ∂α (E9)]. However, we have been unable to relate this to the α- independence criterion for the operator (C7) which defines ˆ 2 29Sometimes the presence of nonperturbative power correc- G in the mass formula. In Eq. (9), we considered replacing μ μ tions is attributed to a “breakdown” of the Wilson expansion. It is θμ by ∂ Dμ in order to obtain a scaling Ward identity, but true that a fully rigorous proof [191] has so far been possible only could not circumvent the facts that Eq. (9) is valid only for within perturbation theory, but Wilson and Zimmermann [192] α α → 0 α ⇁ α gave convincing arguments for operator product expansions to be < IR and the limits x and IR do not commute. valid in any nonperturbative theory consistent with axiomatic In Appendices B and D, we assume that spin-0 operators field theory. See also Ref. [193]. can be classified according to the condition (C16).

095007-33 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019)

α ≤ α O In all cases IR, the relevant operator product expansion for spin-0 operators takes the form α θμνðxÞOð0Þ ∼ CμνIðxÞI þ CμνOðxÞfdOOð0ÞþkOIgþCμνα∂OðxÞ∂ Oð0Þþfless singular; or other operatorsg; ðC18Þ where the constant kO has mass dimension 4, and where the leading singularities 1 1 CμνOðxÞ¼ ∂μ∂ν ; ðC19Þ 12π2 x2 1 1 2 −2 1 Cμνα∂OðxÞ¼− gμα∂ν þ gνα∂μ − gμν∂α − ∂μ∂ν∂ ∂α ; ðC20Þ 4π2 3 3 x2

−8 −2β0 2 have x dependence determined by conservation and trace- CμνIðxÞ¼Oðx ln ðx ÞÞ; ðC26Þ lessness in the indices μν [compare Eq. (B29)]. Here the 2 2 −4 −2β0 2 iϵ prescription x → x − iϵx0 for unordered products which is Oðx ln ðx ÞÞ compared with Eq. (C19) for ∂2 2 −1 0 should be understood, so we have ðx Þ ¼ with no CμνO ðxÞ as x ∼ 0. δ4 25 ∂−2∂ 2 −1 1 2 −1 ¼T ðxÞ term and can define αðx Þ to be 2 xαðx Þ . Since G is RG invariant, it can be written as a function C The coefficient function μνα∂O is normalized to produce of x2 and a dimensionally transmuted scale M: the correct commutators of Oð0Þ with the Poincar´e gen- erators Pμ and Mμν. It also corresponds to the term x · G ¼ðx2Þ−3fðx2M2Þ: ðC27Þ ∂OðxÞ in Eq. (C16). α θμ 0 C At IR where μ ¼ , μνI is both traceless and conserved It is therefore likely that an x ∼ 0 expansion of G contains a ∂ ∂ 1 2 4 2 and hence proportional to μ νð =x Þ. Therefore, given the nonleading term ∝ M =x whose contribution to CμνI in presence of the term kOCμνOI in Eq. (C18), we can set C Eq. (C18) cannot be distinguished from fkO μνOgO→TI. C 0 μνI ¼ . The commutator condition (C16) is reproduced Then the latter term should be absorbed into CμνI by setting if we set kT ¼ 0. We conclude that a study of x ∼ 0 behavior for α < α 0 IR does not produce a criterion to resolve the kO ¼ : ðC21Þ ambiguity (C6). All we can say is that asymptotic freedom 4 On the lattice, it would be hard to insulate a test of this requires dO¼T ¼ , as noted in Eq. (20), and hence α ≠ α condition from IR effects. −4 −2β α α ½Dðx0 þ ϵÞ;TðxÞ ∼ ð4 þ x · ∂ÞTðxÞþOðϵ ln 1 ðϵÞÞI For < IR, the problem is that asymptotic freedom and the trace anomaly require the coefficient function ðC28Þ 1 C ∂2 − ∂ ∂ G 2 in the equal-time limit ϵ → 0. μνIðxÞ¼3 ðgμν μ νÞ ðx ÞðC22Þ to be far more singular than the Oðx−4Þ coefficient APPENDIX D: NG-MODE α function CμνOðxÞ. CONFORMAL-INVARIANT WORLD AT ir O For example, let be the RG-invariant trace operator T Unlike a scale-invariant theory in the WW mode, the discussed in Eqs. (C1) and (C6). Then asymptotic freedom α world at IR is somewhat similar to the physical world (TC requires the trace amplitude 0 α α or QCD) for < < IR. It has a particle spectrum with Fðx2Þ¼hvacjθλðxÞT ð0Þjvaci¼∂2Gðx2ÞðC23Þ (1) non-NG masses close to their physical values, λ inv because in the physical world, scale invariance is to have the following short-distance behavior, approximate at low energies, and (2) an NG sector which is massless because at α , scale 1 IR 2 2 2 2 −2β1 ˆ 2 ˆ 2 and chiral invariance are exact. Fðx Þ ∼ β ðln μ x Þ hvacjG ðxÞG ð0Þjvaciα 0 16 1 ¼ α This situation is allowed because amplitudes at IR can have ∼ Kðln μ2x2Þ−2β1 =ðx2Þ4; ðC24Þ a complicated dependence on scales set by the dilaton decay constant (Fσ or fσ ≠ 0) and other dimensionally where K ≠ 0 is a constant and β1 > 0 is the one-loop transmuted masses M and condensates. As we emphasize β-function coefficient (23). That corresponds to in various sections of this paper, the effects of M 1 dependence must be carefully distinguished from those G 2 ∼ μ2 2 −2β1 2 3 ðx Þ 12 Kðln x Þ =ðx Þ ðC25Þ due to an explicit breaking of scale invariance in the Lagrangian, e.g., by fermion mass parameters or and hence Coleman-Weinberg potentials.

095007-34 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019)

In case this picture seems counterintuitive, recall the fact GeV → GeV0: ðD5Þ α that the NG scaling mode for the ground state at IR requires it to have a noncompact scaling degeneracy in addition to Since no observer is able to compare measurements in compact chiral degeneracies. Under a finite scaling trans- different worlds, all experimental data in one world formation, the ground state jvaci of a scale-invariant world would be exactly the same as in another world. 0 W is transformed to the ground state jvaci of another Therefore these worlds must be physically equivalent, 0 scale-invariant world W : as is the case for any other symmetry with a vacuum degeneracy. Each observer could rely on the same scale- jvaci → jvaci0 ¼ eiDρjvaci;x→ x0 ¼ e−ρx: ðD1Þ invariant version of the PDG tables, i.e., with particle α masses at IR differing slightly from those of our world 0 α α The same is true for all members of a complete set of states < < IR,asdescribedabove. fjnig for W, Crawling TC picks out one of these scale-degenerate ϵ α − α vacua via tiny values of the parameter ¼ IR which n → n 0 ;eiDρ n n 0; cause scale invariance to be broken explicitly by the fj ig fj i g j i¼j i ðD2Þ 0 α α nonvanishing trace anomaly on < < IR. M 0 0 Clearly, conformal group theory fails for -dependent where fjni g span the state space of W . The well-known α amplitudes at IR. However, it has always been accepted identity [10,179] that symmetries of the Hamiltonian, whether hidden or approximate, become exact for coefficient functions iDρ 2 −iDρ 2ρ 2 e P e ¼ e P ðD3Þ in short-distance expansions. These rules were originally proposed [66] for approximate scale and chiral M 0 implies that if jni has mass , the mass of jni is SUðNfÞL × SUðNfÞR symmetry (hidden in the chiral case), and later applied to hidden scale and conformal M0 ¼ eρM: ðD4Þ invariance [196]. Subsequently, asymptotic chiral invari- ance was derived from soft-pion identities [65].Herewe Clearly this applies generally: all dimensionally transmuted extend this method to derive asymptotic scale and masses M associated with a given world W are scaled up conformal invariance from the soft-dilaton theorems or down to M0 in the transformed world W0. (B27) and (B44). Identities such as Eq. (D4) are often quoted as a reason We begin with the coordinate-space version of the for supposing that a scale- or conformal-invariant world scaling identity (B27) for a general operator product: must be entirely unphysical. How can there be a particle Y spectrum if for every massive particle, there is a continuum FσThσðq ¼ 0Þj O ðx Þjvaci of particles [195] with the same quantum numbers m m c X m Y except for their mass, which ranges from infinitesimal ∂ O values to infinity? Must we conclude that we have an ¼ ðdl þ xl · lÞ Thvacj mðxmÞjvacic: ðD6Þ l unparticle theory [128] with power-law branch cuts at zero- m momentum thresholds, or that particles, if they exist, are necessarily massless, as in free-field theory? In what follows, neighborhoods of coinciding points are To answer these concerns, note that such arguments excised to avoid time-ordering ambiguities involving 4 depend on an implicit assumption that the ground state is δ ðxm − xm0 Þ and its derivatives, as in Ref. [66]. For a either unique or that, while it may exhibit a compact chiral subset m ∈ S of the operators Om in Eq. (D6), there is an degeneracy, it lacks the scaling degeneracy specified by operator-product expansion Eq. (D1). Scaling degeneracy changes the picture com- pletely, because different members of each particle con- Y X O ∼ C O 0 tinuum belong to different worlds. mðxmÞ nðfxl∈SgÞ nð ÞðD7Þ ∈ Consider observers O and O0 in their respective uni- m S n verses, W and W0. Assume that these observers choose (say) natural units when making measurements. Since these for the short-distance limit xl∈S → 0 with other coordinates units involve reference to a dimensionally transmuted mass, xl∉S held fixed at values ≠ 0. When Eq. (D7) is inserted a scale transformation necessarily scales the units used in into each side of Eq. (D6), the result is an equivalence W to those in W0, e.g., between asymptotic expansions:

095007-35 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019) X Y C σ 0 O 0 O nðfxl∈SgÞFσTh ðq ¼ Þj nð Þ mðxmÞjvacic ∉ n X X m S Y ∼ ∂ C O 0 O ðdl þ xl · lÞ nðfxl∈SgÞ Thvacj nð Þ mðxmÞjvacic l∈ ∉ n XS X m YS C ∂ O 0 O þ nðfxl∈SgÞ ðdl þ xl · lÞ Thvacj nð Þ mðxmÞjvacic: ðD8Þ n l∉S m∉S

The soft-σ amplitude on the left-hand side can be eliminated via Eq. (D6) Y X Y σ 0 O 0 O ∂ O 0 O FσTh ðq ¼ Þj nð Þ mðxmÞjvacic ¼ dn þ ðdl þ xl · lÞ Thvacj nð Þ mðxmÞjvacic; ðD9Þ m∉S l∉S m∉S with the result

X X Y − ∂ C O 0 O ∼ 0 dn þ ðdl þ xl · lÞ nðfxl∈SgÞ Thvacj nð Þ mðxmÞjvacic : ðD10Þ n l∈S m∉S

Q O Since m∉S m can be chosen at will, this asymptoticQ The same procedure works for special conformal trans- O O expansion is valid only if all coefficients of h n m∉S mi formations. We give details for operators Om of the vanish: type (B42) which commute with Kμ at x ¼ 0: X Y X ∼ C 0 ∉ −dn þ ðdl þ xl · ∂lÞ Cnðfxl∈SgÞ ¼ 0: ðD11Þ OmðxmÞ nðfxl∈SgÞOnð Þþoperators fOmg: l∈S m∈S n C ðD13Þ ThereforeP all coefficient functions n scale with dimension − l∈S dl dn: P Let the action of an infinitesimal Lorentz transformation on d − dl the tensor or spinor indices of an operator Ol inside a field C ρxl∈ ρ n l∈S C xl∈ : D12 nðf SgÞ ¼ nðf SgÞ ð Þ product be denoted as follows: ’ This is the same as Wilson srule[66] for leading Y singularities in a theory of WW-mode scale invariance Σlμν OmðxmÞ explicitly broken by generalized mass terms such as m∈S current fermion masses. The difference is that our result Y  Y  is for all coefficient functions in a theory of exact ≡ OmðxmÞ ΣlμνOlðxlÞ OnðxnÞ : ðD14Þ scale invariance in the NG mode. All dependence on ml dimensionally transmuted “constituent” masses arises fromQ scale condensates formed from vacuum amplitudes In coordinate space, the conformal soft-σ theorem (B44) O O ≠ 0 h n m∉S mivac . becomes

  Y 0 Fσ σðqÞ − σðq Þ T OmðxmÞ vac m c X Y i ρ − 0 − μ 2 ∂ − 2 ∂ − 2 Σ 0 2 ¼ 2 ðq qÞ f xlμðdl þ xl · lÞ xl lμ ixl lμρgT vac OmðxmÞvac þ Oððq or q Þ Þ: ðD15Þ l m c

When the expansion (D13) is applied to both sides of Eq. (D15), the result for the coefficient function of On is X 2 ρ f2xlμðdl þ xl · ∂lÞ − xl∂lμ − 2ixlΣlμρgCnðfxl∈SgÞ ¼ 0: ðD16Þ l∈S

095007-36 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019) Here we made use of the observation above Eq. (B41) that ∂ ∂ μ þ βðαÞ þ γ ˆ 2 ðαÞ A ˆ 2 ¼ 0; ðE5Þ the four components of q − q0 can be chosen independently. ∂μ ∂α G G Together with Eq. (D12) and Poincar´e symmetry, this shows that exact conformal invariance can be applied to asymptotic where coefficient functions despite the M dependence of ampli- β α tudes outside the short-distance region. γ α β0 α − ð Þ Gˆ 2 ð Þ¼ ð Þ α ðE6Þ APPENDIX E: SCALING DIMENSION OF THE TECHNIGLUON (FIELD-STRENGTH OPERATOR)2 is the anomalous scaling function (51) in the form given in Refs. [13,14] and confirmed in Ref. [198]. [There is an This appendix concerns the origins and derivation of incorrect factor of 2 in Eq. (13) of Ref. [197].] This result formulas for the anomalous scaling function (51) and was given originally for massless QCD in the form [100] 4 β0 dimension þ of the trace anomaly. These formulas,   derived in Refs. [13,14], were common private knowledge ∂ β¯ðgÞ γ 2 ðgÞ¼g ðE7Þ as far back as the 1970s. Only recently, we rediscovered the G ∂g g original version which corresponds to Eq. (E6) because of the fixed-μ ¯ 0 fscaling dimensiong¼4 þ β ðg∞Þ; identity g∞ ¼ g at fixed point ðE1Þ     ∂ β¯ðgÞ ∂ βðαÞ g ¼ α : ðE8Þ below Eq. (28) of Ref. [58], where β¯ and g are given by ∂g g ∂α α

2 ¯ dg g The dynamical dimension of the technigluon operator in βðgÞ¼μ and α ¼ : ðE2Þ the IR limit α → α is therefore dμ 4π IR

2 4 γ 2 α 4 β0 α For Eq. (51), see also Refs. [61] and [100,197] (drawn to dGˆ ¼ þ Gˆ ð IRÞ¼ þ ð IRÞ: ðE9Þ our attention by R. Zwicky and E. Pallante, respectively). First, let us review the derivation in Refs. [13,14] of This corresponds to the rule 4 þ β0 found for QCD [13,14]. Eq. (51), simplified for the case of TC with massless The same rule holds for a UV fixed point g∞ or α∞, fermions. Construct CS equations for RG invariant ampli- which was the context of the original version (E1); the ¯ 0 0 tudes A relation β ðg∞Þ¼β ðα∞Þ is a consequence of Eq. (E8).In 4 β0 the UV case, the plus sign in dGˆ 2 ¼ þ is crucial [58] ∂ ∂ β0 μ β α A 0 because is negative: scaling corrections have dimension þ ð Þ ¼ ðE3Þ 4 ∂μ ∂α dGˆ 2 < and so do not upset the leading short-distance behavior of operator product expansions. and apply the operator α∂=∂α: For an IR fixed point, where β0 > 0, such as in chiral- scale perturbation theory or crawling TC, the expansion is ∂ ∂ βðαÞ ∂A at low energies, so the dimension of scale-breaking terms μ þ βðαÞ þ β0ðαÞ − α ¼ 0: ðE4Þ ∂μ ∂α α ∂α can be either d>4 due to the trace anomaly or (if there are 4 fermion mass terms) dmass < . The main proviso is to α∂A ∂α A 2 Since = is proportional to the amplitude Gˆ 2 where ensure the condition ðmassÞ ≥ 0 for all particles appearing Gˆ 2 is inserted at zero momentum into A, Eq. (E4) implies in the expansion.

[1] T. Appelquist and Y. Bai, Light dilaton in walking gauge Division of Particles and Fields, (American Physical theories, Phys. Rev. D 82, 071701 (2010). Society, Minneapolis MN, 2013), arXiv:1309.1206. [2] M. Hashimoto and K. Yamawaki, Technidilaton at con- [4] S. Matsuzaki and K. Yamawaki, Dilaton Chiral Perturbation formal edge, Phys. Rev. D 83, 015008 (2011). Theory: Determining the Mass and Decay Constant of the [3] T. Appelquist et al., Proceedings of the 2013 Community Technidilaton on the Lattice, Phys. Rev. Lett. 113,082002 Summer Study on the Future of U.S. , (2014).

095007-37 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019)

[5] M. Golterman and Y. Shamir, Low-energy effective action [25] C.-J. D. Lin, K. Ogawa, and A. Ramos, The Yang-Mills for pions and a dilatonic meson, Phys. Rev. D 94, 054502 gradient flow and SUð3Þ gauge theory with 12 massless (2016). fundamental fermions in a colour-twisted box, J. High [6] C. G. Callan, Jr., S. Coleman, and R. Jackiw, A new Energy Phys. 12 (2015) 103. improved energy-momentum tensor, Ann. Phys. (N.Y.) 59, [26] Z. Fodor, K. Holland, J. Kuti, S. Mondal, D. Nogradi, and 42 (1970). C. H. Wong, Fate of the conformal fixed point with twelve [7] M. Gell-Mann, Symmetry Violation in Hadron Physics, massless fermions and SUð3Þ gauge group, Phys. Rev. D 1969 Hawaii Topical Conference on Particle Physics 94, 091501 (2016). (Western Periodicals Co., Los Angeles, CA, 1970), p. 168. [27] A. Hasenfratz and D. Schaich, Nonperturbative beta [8] P. G. O. Freund and Y. Nambu, Scalar fields coupled to the function of twelve-flavor SUð3Þ gauge theory, J. High trace of the energy-momentum tensor, Phys. Rev. 174, Energy Phys. 02 (2018) 132. 1741 (1968). [28] A. Deur, S. J. Brodsky, and G. F. de T´eramond, The QCD [9] C. J. Isham, A. Salam, and J. A. Strathdee, Broken chiral running coupling, Prog. Part. Nucl. Phys. 90, 1 (2016). and conformal symmetry in an effective-Lagrangian for- [29] R. J. Crewther, Broken scale invariance and the width of a malism, Phys. Rev. D 2, 685 (1970). single dilaton, Phys. Lett. B 33, 305 (1970). [10] J. R. Ellis, Aspects of conformal symmetry and , [30] E. Gildener and S. Weinberg, Symmetry breaking and Nucl. Phys. B22, 478 (1970). scalar bosons, Phys. Rev. D 13, 3333 (1976). [11] B. Zumino, Effective Lagrangians and broken symmetries, [31] S. Fubini, A new approach to conformal invariant field in Lectures on Elementary Particles and Quantum Field theories, Nuovo Cimento A 34, 521 (1976). Theory, 1970 Brandeis University Summer Institute in [32] T. Appelquist and C. W. Bernard, Strongly interacting Theoretical Physics, Vol. 2 (M.I.T. Press, Cambridge, MA, Higgs bosons, Phys. Rev. D 22, 200 (1980). 1970), pp. 437–500. [33] A. C. Longhitano, Heavy Higgs bosons in the Weinberg- [12] J. Ellis, Approximate scale and chiral invariance, 1971 Salam model, Phys. Rev. D 22, 1166 (1980). Coral Gables Conference Fundamental Interactions at [34] G. Buchalla and O. Cat`a, Effective theory of a dynamically High Energy, Vol. 2, Broken Scale Invariance and the broken electroweak Standard Model at NLO, J. High Light Cone, M. Gell-Mann and K. Wilson (Gordon and Energy Phys. 07 (2012) 101. Breach, New York 1971), p. 77. [35] F. Feruglio, The chiral approach to the electroweak [13] R. J. Crewther and L. C. Tunstall, Origin of ΔI ¼ 1=2 rule interactions, Int. J. Mod. Phys. A 08, 4937 (1993). for kaon decays: QCD , arXiv:1203 [36] J. Bagger, V. D. Barger, K. Cheung, J. F. Gunion, T. Han, .1321. G. A. Ladinsky, R. Rosenfeld, and C.-P. Yuan, The [14] R. J. Crewther and L. C. Tunstall, ΔI ¼ 1=2 rule for kaon strongly interacting WW system: gold-plated modes, Phys. decays derived from QCD infrared fixed point, Phys. Rev. Rev. D 49, 1246 (1994). D 91, 034016 (2015). [37] V. Koulovassilopoulos and R. S. Chivukula, The phenom- [15] R. J. Crewther and L. C. Tunstall, Status of chiral-scale enology of a nonstandard Higgs boson in WLWL scatter- perturbation theory, Proc. Sci., CD15 (2015) 132 [arXiv: ing, Phys. Rev. D 50, 3218 (1994). 1510.01322]. [38] C. P. Burgess, J. Matias, and M. Pospelov, A Higgs or not a [16] T. Appelquist, J. Terning, and L. C. R. Wijewardhana, Higgs? What to do if you discover a new scalar particle, Postmodern Technicolor, Phys. Rev. Lett. 79, 2767 Int. J. Mod. Phys. A 17, 1841 (2002). (1997). [39] S.-Z. Wang and Q. Wang, Electroweak chiral Lagrangian [17] K. G. Wilson, The and strong for neutral Higgs boson, Chin. Phys. Lett. 25, 1984 (2008). interactions, Phys. Rev. D 3, 1818 (1971). [40] B. Grinstein and M. Trott, A Higgs-Higgs bound state [18] K. G. Wilson and J. Kogut, The renormalization group and due to new physics at a TeV, Phys. Rev. D 76, 073002 the ϵ expansion, Phys. Rep. 12, 75 (1974), Sec. 12. (2007). [19] E. Marchais, P. Mati, and D. F. Litim, Fixed points and the [41] R. Contino, C. Grojean, M. Moretti, F. Piccinini, and R. spontaneous breaking of scale invariance, Phys. Rev. D 95, Rattazzi, Strong double Higgs production at the LHC, 125006 (2017). J. High Energy Phys. 05 (2010) 089. [20] T. Appelquist, G. T. Fleming, and E. T. Neil, Lattice Study [42] R. Alonso, M. B. Gavela, L. Merlo, S. Rigolin, and J. of the Conformal Window in QCD-Like Theories, Phys. Yepes, The effective chiral Lagrangian for a light dynami- Rev. Lett. 100, 171607 (2008); Erratum, Phys. Rev. Lett. cal “Higgs particle”, Phys. Lett. B 722, 330 (2013); 102, 149902(E) (2009). Erratum, Phys. Lett. B 726, 926(E) (2013). [21] T. Appelquist, G. T. Fleming, and E. T. Neil, Lattice study [43] G. Buchalla, O. Cat`a, and C. Krause, Complete electro- of conformal behavior in SU(3) Yang-Mills theories, Phys. weak chiral Lagrangian with a light Higgs at NLO, Nucl. Rev. D 79, 076010 (2009). Phys. B880, 552 (2014). [22] L. Del Debbio, The conformal window on the lattice, Proc. [44] T. Appelquist, J. Ingoldby, and M. Piai, Dilaton EFT Sci., LATTICE2010 (2010) 004 [arXiv:1102.4066]. framework for lattice data, J. High Energy Phys. 07 (2017) [23] L. Del Debbio, IR fixed points in lattice field theories, Int. 035. J. Mod. Phys. A 29, 1445006 (2014). [45] T. Appelquist, J. Ingoldby, and M. Piai, Analysis of a [24] T. DeGrand, Lattice tests of beyond Standard Model dilaton EFT for lattice data, J. High Energy Phys. 03 dynamics, Rev. Mod. Phys. 88, 015001 (2016). (2018) 039.

095007-38 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019)

[46] Y. Aoki et al. (LatKMI Collaboration), Light composite [67] S. Coleman and E. Weinberg, Radiative corrections as the scalar in eight-flavor QCD on the lattice, Phys. Rev. D 89, origin of spontaneous symmetry breaking, Phys. Rev. D 7, 111502(R) (2014). 1888 (1973). [47] Y. Aoki et al. (LatKMI Collaboration), Light flavor-singlet [68] M. Bando, K. Matumoto, and K. Yamawaki, Technidila- scalars and walking signals in Nf ¼ 8 QCD on the lattice, ton, Phys. Lett. B 178, 308 (1986). Phys. Rev. D 96, 014508 (2017). [69] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, QCD [48] T. Appelquist et al. (LSD Collaboration), Strongly inter- and resonance physics. Theoretical foundations, Nucl. acting dynamics and search for new physics at the LHC, Phys. B147, 385 (1979). Phys. Rev. D 93, 114514 (2016). [70] L. Del Debbio and R. Zwicky, Hyperscaling relations in [49] T. Appelquist et al. (LSD Collaboration), Nonperturbative mass-deformed conformal gauge theories, Phys. Rev. D investigations of SUð3Þ gauge theory with eight dynamical 82, 014502 (2010). flavors, Phys. Rev. D 99, 014509 (2019). [71] L. Del Debbio and R. Zwicky, Scaling relations for the [50] G. Mack, Partially conserved dilatation current, Nucl. entire spectrum in mass-deformed conformal gauge theo- Phys. B5, 499 (1968). ries, Phys. Lett. B 700, 217 (2011). [51] P. Carruthers, Broken scale invariance in particle physics, [72] L. Del Debbio and R. Zwicky, Conformal scaling and the Phys. Rep. 1, 1 (1971). size of m-hadrons, Phys. Rev. D 89, 014503 (2014). [52] R. J. Crewther, Spontaneous breakdown of conformal and [73] T. Appelquist et al. (LSD Collaboration), Toward TeV chiral invariance, Phys. Rev. D 3, 3152 (1971); Erratum, 4, Conformality, Phys. Rev. Lett. 104, 071601 (2010). 3814 (1971). [74] M. Gell-Mann, Symmetries of baryons and mesons, Phys. [53] P. Di Vecchia, R. Marotta, M. Mojaza, and J. Nohle, New Rev. 125, 1067 (1962), Footnote 38. soft theorems for the gravity dilaton and the Nambu- [75] L. Del Debbio and R. Zwicky, Renormalisation group, Goldstone dilaton at subsubleading order, Phys. Rev. D 93, trace anomaly and Feynman-Hellmann theorem, Phys. 085015 (2016). Lett. B 734, 107 (2014). [54] P. Di Vecchia, R. Marotta, and M. Mojaza, Double-soft [76] S. Weinberg, Implications of dynamical symmetry break- behavior of the dilaton of spontaneously broken conformal ing, Phys. Rev. D 13, 974 (1976). invariance, J. High Energy Phys. 09 (2017) 001. [77] S. Weinberg, Implications of dynamical symmetry break- [55] R. F. Dashen, Chiral SUð3Þ ⊗ SUð3Þ as a symmetry of the ing: An addendum, Phys. Rev. D 19, 1277 (1979). strong interactions, Phys. Rev. 183, 1245 (1969), end of [78] L. Susskind, Dynamics of spontaneous symmetry breaking sec. IIC. in the Weinberg-Salam theory, Phys. Rev. D 20, 2619 [56] P. Minkowski, On the anomalous divergence of the (1979). dilatation current in gauge theories, Berne Report [79] I. Caprini, G. Colangelo, and H. Leutwyler, Mass and No. PRINT-76-0813, 1976. Width of the Lowest Resonance in QCD, Phys. Rev. Lett. [57] S. L.Adler,J. C.Collins,andA.Duncan,Energy-momentum- 96, 132001 (2006). tensor trace anomaly in spin 1=2 , [80] M. Tanabashi et al. (Particle Data Group), Review of Phys. Rev. D 15, 1712 (1977). particle physics, Phys. Rev. D 98, 030001 (2018). [58] N. K. Nielsen, The energy momentum tensor in a non- [81] J. R. Pela´ez, From controversy to precision on the sigma Abelian quark gluon theory, Nucl. Phys. B120, 212 meson: A review on the status of the non-ordinary f0ð500Þ (1977). resonance, Phys. Rep. 658, 1 (2016). [59] J. C. Collins, A. Duncan, and S. D. Joglekar, Trace and [82] G. Aad et al. (ATLAS Collaboration), Observation of a dilatation anomalies in gauge theories, Phys. Rev. D 16, new particle in the search for the Standard Model Higgs 438 (1977). boson with the ATLAS detector at the LHC, Phys. Lett. B [60] P. Breitenlohner and D. Maison, Dimensional renormali- 716, 1 (2012). zation and the action principle, Commun. Math. Phys. 52, [83] S. Chatrchyan et al. (CMS Collaboration), Observation of 11 (1977). a new boson at a mass of 125 GeV with the CMS 2 [61] V. P. Spiridonov, Anomalous dimension of Gμν and β- experiment at the LHC, Phys. Lett. B 716, 30 (2012). function, Institute for Nuclear Research (Moscow) Report [84] F. Sannino, Conformal dynamics for TeV physics and No. IYal-P-0378, 1984. , Acta Phys. Pol. B 40, 3533 (2009), Subsec- [62] V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. tions E.2 and E.3.3. Zakharov, Are all hadrons alike?, Nucl. Phys. B191, 301 [85] S. Weinberg, Approximate Symmetries and Pseudo- (1981). Goldstone Bosons, Phys. Rev. Lett. 29, 1698 (1972). [63] J. C. Collins, Renormalization (Cambridge University [86] H. Georgi and A. Pais, Vacuum symmetry and the pseudo- Press, Cambridge, England, 1984). Goldstone phenomenon, Phys. Rev. D 12, 508 (1975). [64] A. I. Vainshtein, V. I. Zakharov, and M. A. Shifman, Gluon [87] D. B. Kaplan and H. Georgi, SUð2Þ × Uð1Þ breaking by condensate and leptonic decays of vector mesons, Pis’ma vacuum misalignment, Phys. Lett. B 136, 183 (1984). Eksp. Teor. Fiz. 27, 60 (1978); JETP Lett. 27, 55 (1978). [88] D. B. Kaplan, H. Georgi, and S. Dimopoulos, Composite [65] C. Bernard, A. Duncan, J. LoSecco, and S. Weinberg, Higgs scalars, Phys. Lett. B 136, 187 (1984). Exact spectral-function sum rules, Phys. Rev. D 12, 792 [89] H. Georgi, D. B. Kaplan, and P. Galison, Calculation of the (1975), Appendix. composite Higgs mass, Phys. Lett. B 143, 152 (1984). [66] K. G. Wilson, Non-Lagrangian models of current algebra, [90] H. Georgi and D. B. Kaplan, Composite Higgs and Phys. Rev. 179, 1499 (1969). custodial SUð2Þ, Phys. Lett. B 145, 216 (1984).

095007-39 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019)

[91] M. J. Dugan, H. Georgi, and D. B. Kaplan, Anatomy [113] J. Gasser and H. Leutwyler, Chiral perturbation theory: of a composite Higgs model, Nucl. Phys. B254, 299 Expansions in the mass of the strange quark, Nucl. Phys. (1985). B250, 465 (1985). [92] K. Agashe, R. Contino, and A. Pomarol, The minimal [114] H. Georgi, Weak Interactions and Modern Particle Theory composite Higgs model, Nucl. Phys. B719, 165 (2005). (Benjamin/Cummings, Menlo Park, CA, 1984). [93] R. Contino, L. Da Rold, and A. Pomarol, Light custodians [115] J. Gasser, M. E. Sainio, and A. Švarc, Nucleons with chiral in natural composite Higgs models, Phys. Rev. D 75, loops, Nucl. Phys. B307, 779 (1988). 055014 (2007). [116] S. Scherer and M. R. Schindler, A Primer for Chiral [94] G. F. Giudice, C. Grojean, A. Pomarol, and R. Rattazzi, Perturbation Theory, Lecture Notes in Physics, Vol. 830 The strongly-interacting light Higgs, J. High Energy Phys. (Springer, Berlin Heidelberg, 2012). 06 (2007) 045. [117] E. Jenkins and A. V. Manohar, Baryonic chiral perturba- [95] C. Csáki, C. Grojean, and J. Terning, Alternatives to an tion theory using a heavy fermion lagrangian, Phys. Lett. B elementary Higgs, Rev. Mod. Phys. 88, 045001 (2016). 255, 558 (1991). [96] J. R. Ellis, M. K. Gaillard, and D. V. Nanopoulos, A [118] T. Becher and H. Leutwyler, Baryon chiral perturbation phenomenological profile of the Higgs boson, Nucl. Phys. theory in manifestly Lorentz invariant form, Eur. Phys. J. C B106, 292 (1976). 9, 643 (1999). [97] D. D. Dietrich, F. Sannino, and K. Tuominen, Light [119] T. Fuchs, J. Gegelia, G. Japaridze, and S. Scherer, composite Higgs from higher representations versus Renormalization of relativistic baryon chiral perturbation electroweak precision measurements: Predictions for theory and power counting, Phys. Rev. D 68, 056005 CERN LHC, Phys. Rev. D 72, 055001 (2005). (2003). [98] T. Appelquist and J. Carazzone, Infrared singularities and [120] J. S. R. Chisholm, Change of variables in quantum field massive fields, Phys. Rev. D 11, 2856 (1975). theories, Nucl. Phys. 26, 469 (1961). [99] M. Gell-Mann, R. J. Oakes, and B. Renner, Behavior of [121] S. Kamefuchi, L. O’Raifeartaigh, and A. Salam, Change of current divergences under SUð3Þ × SUð3Þ, Phys. Rev. 175, variables and equivalence theorems in quantum field 2195 (1968). theories, Nucl. Phys. 28, 529 (1961). [100] B. Grinstein and L. Randall, The renormalization of G2, [122] S. R. Coleman, J. Wess, and B. Zumino, Structure of Phys. Lett. B 217, 335 (1989). phenomenological Lagrangians. 1, Phys. Rev. 177, 2239 [101] A. Salam and J. A. Strathdee, Nonlinear realizations. 2. (1969). Conformal symmetry, Phys. Rev. 184, 1760 (1969). [123] Y.-M. P. Lam, Equivalence theorem on Bogolyubov- [102] C. J. Isham, A. Salam, and J. A. Strathdee, Spontaneous Parasiuk-Hepp-Zimmermann renormalized Lagrangian breakdown of conformal symmetry, Phys. Lett. B 31, 300 field theories, Phys. Rev. D 7, 2943 (1973). (1970). [124] M. C. Berg`ere and Y.-M. P. Lam, Equivalence theorem and [103] K. A. Meissner and H. Nicolai, Conformal symmetry and Faddeev-Popov ghosts, Phys. Rev. D 13, 3247 (1976). the standard model, Phys. Lett. B 648, 312 (2007). [125] C. Arzt, Reduced effective Lagrangians, Phys. Lett. B 342, [104] W. F. Chang, J. N. Ng, and J. M. S. Wu, Shadow Higgs 189 (1995). 1 boson from a scale-invariant hidden Uð Þs model, Phys. [126] D. Kreimer and K. Yeats, Diffeomorphisms of quantum Rev. D 75, 115016 (2007). fields, Math. Phys. Anal. Geom. 20, 16 (2017). [105] R. Foot, A. Kobakhidze, and R. R. Volkas, Electroweak [127] M. Golterman and Y. Shamir, Effective action for pions Higgs as a pseudo- of broken scale and a dilatonic meson, Proc. Sci., LATTICE2016 (2016) invariance, Phys. Lett. B 655, 156 (2007). 205 [arXiv:1610.01752]. [106] W. D. Goldberger, B. Grinstein, and W. Skiba, Distinguish- [128] H. Georgi, , Phys. Rev. Lett. 98, 221601 ing the Higgs Boson from the Dilaton at the Large Hadron (2007). Collider, Phys. Rev. Lett. 100, 111802 (2008). [129] M. E. Peskin and T. Takeuchi, New Constraint on a strongly [107] L. Vecchi, Phenomenology of a light scalar: The dilaton, interacting Higgs Sector, Phys. Rev. Lett. 65, 964 (1990). Phys. Rev. D 82, 076009 (2010). [130] M. E. Peskin and T. Takeuchi, Estimation of oblique [108] B. Bellazzini, C. Csáki, J. Hubisz, J. Serra, and J. electroweak corrections, Phys. Rev. D 46, 381 (1992). Terning, A Higgs-like dilaton, Eur. Phys. J. C 73, 2333 [131] S. R. Coleman and J. Mandula, All possible symmetries of (2013). the S matrix, Phys. Rev. 159, 1251 (1967). [109] B. Bellazzini, C. Csáki, J. Hubisz, J. Serra, and J. Terning, [132] T. Appelquist et al. (LSD Collaboration), Parity Doubling A naturally light dilaton and a small cosmological con- and the S Parameter below the Conformal Window, Phys. stant, Eur. Phys. J. C 74, 2790 (2014). Rev. Lett. 106, 231601 (2011). [110] F. Coradeschi, P. Lodone, D. Pappadopulo, R. Rattazzi, [133] T. Appelquist et al. (LSD Collaboration), Lattice simu- and L. Vitale, A naturally light dilaton, J. High Energy lations with eight flavors of domain wall fermions in Phys. 11 (2013) 057. SUð3Þ gauge theory, Phys. Rev. D 90, 114502 (2014). [111] Y.-M. P. Lam, Nonlinear chiral symmetry and partial [134] R. Foadi and F. Sannino, S and T parameters from a light conservation of axial-vector current—Bogoliubov- nonstandard Higgs particle, Phys. Rev. D 87, 015008 Parasiuk-Hepp-Zimmermann renormalization, Phys. Rev. (2013). D 7, 2950 (1973). [135] A. Pich, I. Rosell, and J. J. Sanz-Cillero, Oblique S and T [112] J. Gasser and H. Leutwyler, Chiral perturbation theory to constraints on electroweak strongly-coupled models with a one loop, Ann. Phys. (N.Y.) 158, 142 (1984). light Higgs, J. High Energy Phys. 01 (2014) 157.

095007-40 CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019)

[136] S. Weinberg, Nonlinear realizations of chiral symmetry, [156] A. D. Gasbarro and G. T. Fleming, Examining the low Phys. Rev. 166, 1568 (1968). energy dynamics of walking gauge theory, Proc. Sci., [137] C. G. Callan, Jr., S. R. Coleman, J. Wess, and B. Zumino, LATTICE2016 (2017) 242 [arXiv:1702.00480]. Structure of phenomenological Lagrangians. 2, Phys. Rev. [157] A. Gasbarro, Can a linear sigma model describe walking 177, 2247 (1969). gauge theories at low energies?, EPJ Web Conf. 175, [138] S. Weinberg, Phenomenological Lagrangians, Physica 08024 (2018). (Amsterdam) 96A, 327 (1979). [158] T. Appelquist et al. (LSD Collaboration), Linear sigma [139] W. E. Caswell, Asymptotic Behavior of Non-Abelian EFT for nearly conformal gauge theories, Phys. Rev. D 98, Gauge Theories to Two-Loop Order, Phys. Rev. Lett. 114510 (2018). 33, 244 (1974). [159] H. Suzuki, Energy-momentum tensor on the lattice: [140] T. Banks and A. Zaks, On the phase structure of vector-like Recent developments, Proc. Sci., LATTICE2016 (2017) gauge theories with massless fermions, Nucl. Phys. B196, 002 [arXiv:1612.00210]. 189 (1982). [160] S. Dimopoulos and L. Susskind, Mass without scalars, [141] G. Veneziano, Some aspects of a unified approach to Nucl. Phys. B155, 237 (1979). gauge, dual and Gribov theories, Nucl. Phys. B117, 519 [161] E. Eichten and K. D. Lane, Dynamical breaking of weak (1976). interaction symmetries, Phys. Lett. B 90, 125 (1980). [142] R. S. Chivukula and T. P. Walker, Technicolor cosmology, [162] K. D. Lane, Two lectures on technicolor, l’Ecole de GIF at Nucl. Phys. B329, 445 (1990). LAPP, Annecy-le-Vieux, France, arXiv:hep-ph/0202255. [143] R. Urech, Virtual in chiral perturbation theory, [163] C. T. Hill and E. H. Simmons, Strong dynamics and Nucl. Phys. B433, 234 (1995). electroweak symmetry breaking, Phys. Rep. 381, 235 [144] G. D’Ambrosio and D. Espriu, Rare decay modes of the K (2003); Erratum, 390, 553 (2004). mesons in the chiral Lagrangian, Phys. Lett. B 175, 237 [164] B. Holdom, Raising the sideways scale, Phys. Rev. D 24, (1986). 1441(R) (1981). [145] J. L. Goity, The decays K0ðsÞ → γγ and K0ðlÞ → γγ in the [165] B. Holdom, Techniodor, Phys. Lett. B 150, 301 (1985). chiral approach, Z. Phys. C 34, 341 (1987). [166] K. Yamawaki, M. Bando, and K. I. Matumoto, Scale- [146] M. Knecht, H. Neufeld, H. Rupertsberger, and P. Talavera, Invariant Model and a Dilaton, Phys. Rev. Lett. Chiral perturbation theory with virtual photons and lep- 56, 1335 (1986). tons, Eur. Phys. J. C 12, 469 (2000). [167] T. Akiba and Y. Yanagida, Hierarchic chiral condensate, [147] G. Buchalla, O. Cat`a, and C. Krause, On the power Phys. Lett. B 169, 432 (1986). counting in effective field theories, Phys. Lett. B 731, [168] T. Appelquist, D. Karabali, and L. C. R. Wijewardhana, 80 (2014). Chiral Hierarchies and the Flavor Changing Neutral [148] G. Buchalla, O. Cat`a, A. Celis, and C. Krause, Comment Current Problem in Technicolor, Phys. Rev. Lett. 57, on “Analysis of general power counting rules in effective 957 (1986). field theory”, arXiv:1603.03062. [169] T. Appelquist and L. C. R. Wijewardhana, Chiral hierar- [149] G. Buchalla, O. Cat`a, A. Celis, M. Knecht, and C. Krause, chies and chiral perturbations in technicolor, Phys. Rev. D Complete one-loop renormalization of the Higgs- 35, 774(R) (1987). electroweak chiral Lagrangian, Nucl. Phys. B928,93 [170] T. Appelquist and L. C. R. Wijewardhana, Chiral hierar- (2018). chies from slowly running couplings in technicolor theo- [150] R. Alonso, K. Kanshin, and S. Saa, Renormalization group ries, Phys. Rev. D 36, 568 (1987). evolution of Higgs effective field theory, Phys. Rev. D 97, [171] L. D. Landau, On analytic properties of vertex parts in 035010 (2018). , Nucl. Phys. 13, 181 (1959). [151] G. Aad et al. (ATLAS and CMS Collaborations), Mea- [172] Y. Nambu, Axial Vector Conservation in Weak Inter- surements of the Higgs boson production and decay rates actions, Phys. Rev. Lett. 4, 380 (1960). and constraints on its couplings from a combined ATLAS [173] S. L. Adler and R. F. Dashen, Current Algebras and pffiffiffi and CMS analysis of the LHC pp collision data at s ¼ 7 Applications to Particle Physics (Benjamin, New York, and 8 TeV, J. High Energy Phys. 08 (2016) 045. 1968). [152] B. Fuks, J. H. Kim, and S. J. Lee, Scrutinizing the Higgs [174] D. V. Volkov, Phenomenological Lagrangians, Fiz. Elem. quartic coupling at a future 100 TeV proton-proton collider Chastits At. Yadra 4, 3 (1973). with taus and b-jets, Phys. Lett. B 771, 354 (2017). [175] E. A. Ivanov and V. I. Ogievetsky, The inverse Higgs [153] M. J. Dolan, C. Englert, N. Greiner, K. Nordstrom, and M. phenomenon in nonlinear realizations, Teor. Mat. Fiz. Spannowsky, hhjj production at the LHC, Eur. Phys. J. C 25, 164 (1975). 75, 387 (2015). [176] I. Low and A. V. Manohar, Spontaneously Broken Space- [154] Q.-H. Cao, Y. Liu, and B. Yan, Measuring trilinear Higgs time Symmetries and Goldstone’s Theorem, Phys. Rev. coupling in WHH and ZHH productions at the HL-LHC, Lett. 88, 101602 (2002). Phys. Rev. D 95, 073006 (2017). [177] R. Dashen and M. Weinstein, Soft pions, chiral symmetry, [155] Z. Fodor, K. Holland, J. Kuti, D. Nogradi, and C. H. and phenomenological Lagrangians, Phys. Rev. 183, 1261 Wong, Can a light Higgs impostor hide in composite (1969). gauge models?, Proc. Sci., LATTICE2013 (2014) 062 [178] M. Gell-Mann and M. L´evy, The axial-vector current in [arXiv:1401.2176]. beta decay, Nuovo Cimento 16, 705 (1960).

095007-41 CATA,` CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019)

[179] G. Mack and A. Salam, Finite component field represen- [190] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, QCD tations of the conformal group, Ann. Phys. (N.Y.) 53, 174 and resonance physics. Applications, Nucl. Phys. B147, (1969). 448 (1979). [180] D. J. Gross and J. Wess, Scale invariance, conformal [191] W. Zimmermann, Local operator products and renor- invariance, and the high-energy behavior of scattering malization in quantum field theory, in Lectures on amplitudes, Phys. Rev. D 2, 753 (1970). Elementary Particles and Quantum Field Theory, 1970 [181] Scale and Conformal Symmetry in Hadron Physics, edited Brandeis University Summer Institute in Theoretical by R. Gatto (Wiley-Interscience, New York 1973). Physics, Vol. 1 (M.I.T. Press, Cambridge, MA, 1970), [182] R. Tarrach, The renormalization of FF, Nucl. Phys. B196, pp. 397–589. 45 (1982). [192] K. G. Wilson and W. Zimmermann, Operator product [183] G. K. Savvidy, Infrared stability of the vacuum state of expansions and composite field operators in the general gauge theories and asymptotic freedom, Phys. Lett. B 71, framework of quantum field theory, Commun. Math. Phys. 133 (1977). 24, 87 (1972). [184] T. Lee, Renormalon subtraction from the average pla- [193] V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. quette and the gluon condensate, Phys.Rev.D82, Zakharov, Wilson’s operator product expansion: Can it 114021 (2010). fail?, Nucl. Phys. B249, 445 (1985). [185] G. S. Bali, C. Bauer, and A. Pineda, Model Independent [194] V. Prochazka and R. Zwicky, Gluon condensates from Determination of the Gluon Condensate in Four Dimen- the Hamiltonian formalism, J. Phys. A 47, 395402 sional SUð3Þ Gauge Theory, Phys. Rev. Lett. 113, 092001 (2014). (2014). [195] J. Wess, The conformal invariance in quantum field theory, [186] T. Lee, Extracting gluon condensate from the average Nuovo Cimento 18, 1086 (1960). plaquette, Nucl. Part. Phys. Proc. 258-259, 181 (2015). [196] R. J. Crewther, Nonperturbative Evaluation of the Anoma- [187] M. A. Shifman, Snapshots of hadrons, Prog. Theor. Phys. lies in Low-Energy Theorems, Phys. Rev. Lett. 28, 1421 Suppl. 131, 1 (1998); reprinted in ITEP Lectures on (1972). Particle Physics and Field Theory (World Scientific, [197] S. S. Gubser, A. Nellore, S. S. Pufu, and F. D. Rocha, Singapore 1999), Vol. I, Chapter 2. and Bulk Viscosity of Approximate [188] M. Beneke and V. M. Braun, Heavy quark effective Black Hole Duals to Finite Temperature Quantum theory beyond perturbation theory: Renormalons, the pole Chromodynamics, Phys. Rev. Lett. 101, 131601 (2008). mass and the residual mass term, Nucl. Phys. B426, 301 [198] T. Nunes da Siva, E. Pallante, and L. Robroek, The scalar (1994). glueball operator, the a-theorem, and the onset of con- [189] M. Beneke, Renormalons, Phys. Rep. 317, 1 (1999). formality, Phys. Lett. B 778, 316 (2018).

095007-42