Genuine Dilatons in Gauge Theories

A genuine dilaton $\sigma$ allows scales to exist even in the limit of exact conformal invariance. In gauge theories, these may occur at an infrared fixed point (IRFP) $\alpha_\text{IR}$ through dimensional transmutation. These large scales at $\alpha_\text{IR}$ can be separated from small scales produced by $\theta^\mu_\mu$, the trace of the energy-momentum tensor. For quantum chromodynamics (QCD), the conformal limit can be combined with chiral $SU(3) \times SU(3)$ symmetry to produce chiral-scale perturbation theory $\chi$PT$_\sigma$, with $f_0(500)$ as the dilaton. The technicolor (TC) analogue of this is crawling TC: at low energies, the gauge coupling $\alpha$ goes directly to (but does not walk past) $\alpha_\text{IR}$, and the massless dilaton at $\alpha_\text{IR}$ corresponds to a light Higgs boson at $\alpha \lesssim \alpha_\text{IR}$. Unlike crawling TC, in walking TC, $\theta^\mu_\mu$ produces \emph{all} scales, large and small, so it is hard to argue that its ``dilatonic'' candidate for the Higgs boson is not heavy.


Introduction
It is surprising how far the notion of a "dilaton" has strayed from the original version of 1968-70. Now it can be any of the following: I. a scalar Nambu-Goldstone (NG) boson for exact conformal invariance of a Hamiltonian H which has a scale-dependent ground state |vac ; or II. a scalar component of the gravitational field; or III. a scalar particle for a system where scale-dependent effects such as fermion condensation exist only in the presence of a scale-violating term δH in the Hamiltonian H = H 0 + δH. Both H 0 and its ground state |vac 0 are conformal invariant, i.e. in the Wigner-Weyl (WW) mode.
Evidently, I and III contradict each other and may have little to do with II. For many years, type-III theories such as walking TC have been promoted as the way to explain why the Higgs boson is so light. I observe that these theories are very unlikely to achieve this and should be replaced by a type-I theory called crawling TC [1]. ADP-20-10/T1120 Most papers on "dilatons" consider type II or type III. The problem is that some of them claim to be also talking about type I. So I begin with a quick summary of the fundamental type-I theory in its original setting, strong interactions (Section 2).
Gauge theories are then considered in Section 3. For quantum chromodynamics (QCD), there is only a type-I theory, chiral-scale perturbation theory χPT σ [2][3][4], where the dilaton at the IRFP corresponds to the light resonance f 0 (500). The TC analogue of this is crawling TC, where the Higgs boson is the type-I TC analogue of f 0 . The scale-dependent IRFP lies outside the conformal window [5].
Section 4 compares crawling TC with type-III theories for the Higgs boson. The type-III concept is due to Gildener and Weinberg [6]. They called their spin-0 + particle a "scalon" -a good name -but that morphed into the term "dilaton" in "dilatonic" walking TC [7][8][9][10][11][12][13][14][15] and deformed conformal potential theory [16][17][18][19][20][21][22][23]. I will reserve the term "genuine dilaton" for type I dilatons. The key observation is that in type-III theories, the trace θ µ µ produces large scales as well as small scales. For example, in walking TC, the sill of the conformal window produces the fermion condensate ψψ vac . That indicates a large mass for the would-be Higgs boson h III , no matter how slowly α walks. A similar conclusion was drawn in early work on walking TC [7,9,10]. It is not affected by whether one assumes a dilaton-like Lagrangian below the sill [19] or not. By contrast, in a type-I theory, all large scales arise from the scale-dependence of amplitudes in the exact conformal limit -that is what is meant by the NG mode for conformal invariance. In crawling TC, the fermion condensate ψψ vac sets the scale at the IRFP α IR . As α moves a small distance below α IR , θ µ µ appears as a small perturbation which produces small scales, such as the mass acquired by the type-I dilaton: a light Higgs boson.
The concept that there can be scale dependence in the conformal limit, which seemed so simple in 1968-70, has become a major sticking point. In Sections 5 and 6, I offer possible reasons for this.
Possible tests of these proposals are considered in Section 7. A light scalar boson has been observed in lattice data for SU(3) gauge theory with N F = 8 triplet fermions [24][25][26][27] and two sextet fermions [28,29]. In each case, this is being interpreted as a type-III dilaton for walking TC, but it is more likely to be a genuine dilaton, and hence evidence for an IRFP just outside the conformal window.

Hadronic Physics
The idea that scale and conformal invariance may be spontaneously (i.e. not explicitly) broken dates from 1962 (footnote 38 of [30]). An analogy was drawn with the partial conservation of the axial-vector currents F µ5 for chiral SU(3) L × SU(3) R symmetry, where π, K,K, η pole dominance of amplitudes of the divergences ∂ µ F µ5 yields soft-meson relations such as the Goldberger-Treiman relation for the pion-nucleon coupling constant g πNN . Similarly, a spin-0 + particle σ tied to the trace θ µ µ of the energy-momentum tensor θ µν couples universally to particle mass. For a nucleon N with mass M N , the σ-nucleon coupling constant g σNN is given by where f σ is the scalar analogue of the pion decay constant f π : The currents for scale and conformal transformations can be written in terms of θ µν (improved [31] if spin-0 fields are present) as follows: As for chiral SU(3) L × SU(3) R currents, explicit breaking of the symmetry is measured by current divergences: Therefore scale and conformal invariance correspond to the limit The question [32][33][34][35] is, does the vacuum state respect the symmetry, or break it? Are scale and conformal invariance realised in the WW mode or the "spontaneous" NG mode? A comparison with the chiral SU(3) L × SU(3) R group in hadronic physics is instructive [35]. In that case, both modes occur. The subgroup SU(3) L+R associated with vector currents F µ has a symmetry limit in the WW mode -its generators F annihilate the vacuum state: The symmetry is manifest: its representations can be seen in the particle spectrum. The rest of the group, represented by cosets SU(3) L × SU(3) R SU(3) L+R and generated by axial charges F 5 , has its symmetry realised in the NG mode: As a result, the axial part of the symmetry is hidden, |vac becomes a member of a degenerate set of physically equivalent vacua |vac α = exp{i α· F 5 }|vac α=0 , and there is a massless NG boson for each independent direction in α space -for SU(3) L × SU(3) R , eight 0 − NG bosons π, K,K, η. A unique vacuum state can be picked out by perturbing the Hamiltonian with a term which breaks the axial part of the symmetry and gives the NG bosons mass. Similarly, for the limit of scale and conformal invariance, "there are two possibilities: either all particle masses go to zero, or there is a massless scalar boson of the NG type that allows other masses to be non-zero" [35].
The first possibility refers to the WW scaling mode, where scale and conformal invariance are manifest. Let generate scale and special conformal transformations. In the symmetry limit, D and K µ become time independent, and their commutators with the translation and Lorentz generators P µ and M µν simplify, e.g.
Given that |vac is the only state annihilated by both P µ and M µν , it follows from (11) that K µ |vac = 0 implies D|vac = 0 and vice versa. Conformal invariance of the vacuum state implies that the theory lies within the conformal window: Green's functions exhibit power-law behavior characteristic of representations of the conformal group SO(4, 2). Dimensional couplings vanish, e.g. scalar particles decouple from θ µν : Particles are massless or do not exist [36], and the rest of the mass spectrum is empty or continuous. Consequently, the WW-mode scaling limit is nothing like the real world. Key physical properties such as a massive spectrum can arise only as dominant contributions from terms in the Hamiltonian which break scale invariance explicitly. The other possibility is the NG scaling mode, where there is a non-compact degeneracy of Poincaré invariant vacua |vac ρ = exp{iρD}|vac ρ=0 .
As for the chiral case (9), these vacua are physically equivalent; one of them is picked out if a small symmetry breaking term is added to the Hamiltonian. Equation (11) remains valid, so D|vac = 0 implies K µ |vac = 0. Conformal symmetry is hidden by the dependence of amplitudes on dimensional constants such as masses. This is allowed if there is a massless 0 + NG boson σ for scale and conformal invariance: the dilaton 1 .
The key property of a dilaton is that the decay constant f σ in (3) remains non-zero in the scale-symmetric limit: Equation (2) becomes exact, and f σ has dimensions of mass, so particles such as nucleons N can remain massive in a theory with NG-mode scale invariance. The pion decay constant f π can also remain non-zero: the NG mode for conformal invariance is compatible with the NG mode for chiral invariance [37][38][39][40][41][42].
Evidently, compared with the WW mode, the NG scaling mode offers the great advantage that there is a chance that it approximates the real world. A small scale-violating perturbation of the Hamiltonian may be sufficient to give the dilaton and other NG bosons their observed small masses and make small corrections to large masses in the non-NG particle sector. The consistency of assuming an NG mode for scale invariance was confirmed via effective Lagrangians [32,34,37,38,40], and by the end of 1970, a complete understanding had been achieved [40,41]. However, dilaton phenomenology at that time did not go far: the only candidate for σ was a vague 0 + resonance (700) which disappeared from the Particle Data Tables in 1974. The main results were for the σ → ππ coupling [38,39], and for the σ → γγ coupling due to the electromagnetic trace anomaly [43][44][45].

Gauge Theories
This line of investigation was resumed almost 40 years later. It was prompted by the discovery in 2006 of a broad low-mass 0 + resonance f 0 (500) at [46] 441 − 272 i MeV, with small experimental and theoretical uncertainties -unlike the (700). A perfect candidate for the hadronic dilaton of 1968-72 had appeared: The mass of f 0 is close to K(495) and η(549), so it makes sense to extend standard chiral SU(3) × SU(3) perturbation theory χPT 3 to chiral-scale perturbation theory [2][3][4] χPT σ , with NG bosons π, K,K, η, σ in the combined limit of chiral and conformal symmetry. To achieve a combined scale-chiral limit in QCD, heavy quark masses are first decoupled: 1 This term was coined in 1969, and first appeared in print in [37].
The result is QCD for N f = 3 flavors q = u, d, s. Since QCD is a gauge theory, there is a trace anomaly [47][48][49][50] proportional to the N f = 3 Callan-Symanzik function β(α s ), where α s is the gauge coupling for strong interactions and G a µν is the field-strength tensor. As NG-boson momenta and light quark masses tend to zero, conformal symmetry can occur if α runs simultaneously to an infrared fixed point α IR s : The infrared limit causing α s to run to α IR s is the same as the low-energy limit for soft π, K,K, η theorems, so quark condensation survives at α IR s , and conformal invariance is realised in NG mode. -1 Figure 1. Alternatives for the N f = 3 QCD β-function. Conventional N f = 3 soft-meson theory χPT 3 involves a large breaking of scale invariance at α s ∼ ∞ to ensure that heavy hadrons such as nucleons acquire sufficient mass, but then it is hard to explain why f 0 is relatively light: m f 0 ∼ m K m N . That problem is solved in χPT σ : the massless dilaton σ at α IR s allows nucleons to be heavy (Equation (2)), and it becomes the pseudodilaton f o as it acquires (mass) 2 to first order in = α IR s − α s . Claims that there can be no light "dilatons" in gauge theories [9,10] are made for a type-III definition of "dilaton", to be discussed further in Section 4 below. They do not affect the identification above of the f 0 (500) as a genuine dilaton σ for QCD.

PT
In TC literature, where chiral NG bosons are called "technipions" and not (say) "technikaons", it is often stated that there is no light dilaton in hadronic physics because there is no scalar particle nearly degenerate with pions. That overlooks the role of the light s quark: 1. pions are unusually light because m 2 π is proportional to m u,d m s , and 2. decoupling the not-heavy s quark would be a bad approximation.
However, it is a good approximation to regard u, d, s as light, corresponding to the light 0 − mesons π, K, η associated with chiral SU(3) × SU(3) symmetry. The scale at which α s is evaluated is set by the soft are similar in magnitude to those of m s . That is why f 0 (500) is as light as K and η ( Figure 2).
The assumption that α IR s exists produces a low-energy mesonic theory which explains the ∆I = 1/2 rule for nonleptonic kaon decays. Its consistency is supported by the fact that its effective Lagrangian is of a type studied carefully in 1970 [37,38,40]. Equation (15) remains valid [4], while R in Equation (16) is replaced by the high-energy ratio R IR for the scale-invariant theory at α IR [2,3].
The most recent application of the idea of an NG mode at an IRFP α IR is crawling TC [1]. It adopts the standard TC viewpoint [51][52][53] that the Higgs mechanism is the dynamical effect of a gauge theory which resembles QCD, with a TC coupling α which is nonperturbative at scales of a few TeV. Where it differs from other TC theories is that, in analogy with (21), the TC gauge coupling runs to an infrared fixed point α IR with conformal invariance in NG mode. So at α IR , there is a massless dilaton -a feature unique to crawling TC. As α moves from α IR to a slightly smaller value, the dilaton acquires a relatively small mass M σ and at mass 125 GeV, it becomes the Higgs boson.
The characteristic feature of crawling TC is its dependence on the slope of the TC β function at the fixed point: In particular, the Higgs potential in leading order is a nonpolynomial function where h = h(x) is a fluctuating Higgs field, F σ is the TC analogue of f σ , and M σ is identified as the Higgs boson mass m h .

Comparison of Crawling and Walking TC
While writing [1], we became aware that the 1968-70 concepts of "dilaton" and "spontaneous breaking of conformal invariance", on which our work relies, have lost their original meaning (Section 1). Most of the thousands of papers on the subject written since 1972 do not recognise the type-I concept that scale-dependent amplitudes can occur in the limit of conformal invariance. For most authors, the label "dilaton" is just a fancy name for a scalar field appearing in a conformal theory. As a result, there was a lot to explain -too much to summarise here. Instead I will try to distinguish and clarify the competing concepts which have arisen.
Most definitions of "dilaton" on the internet are of type II: they refer to a scalar component of the gravitational field. There is now a vast literature on this. The term was first used in that context in 1971 [54]. At the time, it drew the remark [55] (quoted in [56]) that "Brans-Dickeon" would be a better name.
There is a third meaning for "dilaton" (type III), also with an extensive literature, which unfortunately contradicts the 1968-70 definition reviewed above. This re-working of the subject started in 1976: 1. Fubini [57] noted problems with the conformal NG mode for λφ 4 theory which subsequent authors incorrectly interpreted as an inconsistency of type-I theories in general; see Section 5 below. 2. Gildener and Weinberg (GW) [6] introduced the concept of a spin-0 + "scalon" associated with a flat direction of the potential of a massless gauge theory in the tree approximation. Scale invariance is broken explicitly by one-loop corrections of the Coleman-Weinberg (CW) [58] type. The analysis is entirely consistent, except for a remark that the result is an example of a "spontaneous breaking" of scale invariance 2 . That is not so: the tree approximation is scale-free by construction, so the invariance is realised in the WW mode. In that limit, the "scalon" is massless but is not a genuine dilaton because it lacks a decay constant connecting θ µν to the vacuum. All breaking of conformal invariance is explicit: the one-loop corrections violate scale invariance of the Hamiltonian.
A general definition for the type-III dilaton 3 ϕ for walking TC and CW-deformed potentials is as follows. It is a 0 + particle in a theory which approximates a system with exact conformal invariance in the WW mode, and obeys Equation (12). All scales, large and small, are "triggered" when the Hamiltonian is perturbed by a term which breaks conformal invariance explicitly. These large scales include a fermion condensate ψψ vac and hence chiral NG bosons.
Walking TC assumes that all infrared fixed points lie within the conformal window, where deep infrared dynamics is scale-free and Green's functions exhibit the power-law scaling expected for the WW mode. The gauge coupling α for a theory just outside the conformal window is supposed to walk slowly when it passes the IRFP α WW of a theory just inside the window. The result is then a small β-function which (it is hoped) can be held responsible for small-scale effects such as the mass of the Higgs boson. Physics outside the conformal window is vastly different from physics inside, so there has to be a discontinuity or phase transition in N f at a sill [59][60][61] produced by a term δH in θ 00 which breaks conformal symmetry explicitly. Despite being proportional to β, δH has to produce effects ∼ several TeV, such as ψψ vac .
So, even though θ µ µ is formally small, its effects are ∼ a few TeV, as foreshadowed in Section 1 and in general remarks below Equation (12). Therefore it is hard to argue that the sill produces a small-mass Higgs boson. That can be done only if an explicit model for the sill can be formulated with unusual properties. The model would have to specify a large-scale mechanism for the gauge theory to produce the chiral condensate ψψ vac without affecting the mass of the 0 + boson. Early attempts in that direction [7,9,10] came to the conclusion that this is not possible: type-III dilatons are heavy 4 . Of course, the self-consistency of these gauge-theory models for fermion condensation is far from obvious, but that does not mean that 2 Coleman and E. Weinberg [58] stick to the textbook definition of the term "spontaneous", i.e. for breaking which is not explicit, and apply it only to the breaking of chiral invariance. In footnote 8, they note that scale invariance is broken explicitly by the one-loop trace anomaly. 3 In [1], we suggested retaining the GW term "scalon" but with no effect, so far. 4 The analysis does not appear to depend on their having an ultraviolet (UV) fixed point instead of an IRFP. the difficulty can be circumvented by assuming (as in [19]) that a type-I dilaton Lagrangian from 1970 [37,38,40] may be valid below the sill but not above it. For that to be convincing, the self-consistent model for the sill would have to produce a dilaton-like Lagrangian.
This should be contrasted with crawling TC, which relies on a single assumption that there is an IRFP α IR in the NG mode for conformal invariance, Support for this comes from evidence noted in Section 3 for the analogue theory [2][3][4] for QCD. Assumptions about the detailed dynamics of the sill are not neededindeed, the sill plays no role in the extrapolation from α IR to α. That extrapolation accounts for small-scale corrections to the scale set at α IR , including the mass of the Higgs boson See [1] for an explicit O( ) formula for m h in terms of the gluon condensate at α IR . A diagrammatic comparison of the two theories is shown in Figure 3.
crawling sill @ @ R q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq qq q qq q qq qq q q qq q q qq qq q qq q qq qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q f < N f 16) is scale-free and has an IRFP α WW with conformal invariance in WW mode. The sill generates all scales in the physical theory, both large and small, no matter how closely the red line in the walking region approaches α WW . Therefore a type-III "dilaton" is unlikely to be light.

Scale Dependence in the Conformal Limit
Evidently the proposition that amplitudes at α IR can be scale-dependent requires further explanation. In the limit of exact conformal invariance, a) is scale dependence of the ground state generally possible, and b) can it occur at an IRFP of a massless gauge theory (Section 6 below)?
The argument against a) typically refers to Fubini [57] and runs as follows [22]: "if a theory is exactly conformal, it either does not break scale invariance, or the breaking scale is arbitrary (a flat direction)." In effect, it is being argued that the NG mode for exact conformal invariance with a type-I dilaton is absolutely impossible. That cannot be so: 1. Fubini's analysis is restricted to λφ 4 theories and therefore does not constitute a general proof that strict conformal invariance must be manifest, i.e. in WW mode. To obtain the NG mode for conformal invariance, simply omit the φ 4 term and add other invariants to 1 2 (∂φ) 2 such as couplings to chiral-NG bosons or (say) the 4-point self interaction and [1] constrain φ, e.g. to a half line φ > const. (28) 2. As noted in [1], Fubini's conclusion was anticipated in 1970 by Zumino (page 472 of [40]), who observed that a dilaton Lagrangian is consistent only if the quartic term vanishes in the conformal limit: Here is a measure of the explicit breaking of conformal symmetry. Equation (29) reflects the fact that, like other genuine NG bosons, type-I dilatons for → 0 are massless and cannot self-interact at zero momentum: they correspond to a flat direction of the dilaton potential. 3. All dilaton Lagrangians from 1968-70 which obey Zumino's rule (29) are counterexamples [32][33][34]37,38,40]: they exist in the limit of exact conformal symmetry and produce amplitudes which depend on a non-arbitrary scale, the dilaton decay constant f σ of Equations (3) and (14). All except [32] allow chiral condensates to exist in the conformal limit → 0. 4. The "flat direction" is not associated with a continuum of scales. Instead, it corresponds to the continuum of degenerate vacuum states (13).
The quote continues: "Thus an explicit breaking has to be present to trigger and stabilize the spontaneous breaking of scale invariance." 5. Again, the effective Lagrangians above are counterexamples. A tiny 0( ) scale-violating perturbation δH tiny can pick out one of the degenerate vacua (stabilization) and produce tiny corrections to the scale-dependent amplitudes and masses of the type-I theory at = 0. 6. Implicit in this quote is the type-III assumption that there are no scales in the = 0 theory, so it is necessary to have a large discontinuity appear "spontaneously" at a small or infinitesimal value of = 0 to produce large scales. Since the = 0 theory is in the WW scaling mode, it does not have degenerate vacua, so there is nothing to stabilize. 7. The large discontinuity is a problem for type-III phenomenology, because θ µ µ ∼ 0 is such a bad approximation.
A formal argument that "the breaking scale is arbitrary" in a conformal invariant theory was first given by Wess [62]. It is most simply derived from the identity [34] e iDρ P 2 e −iDρ = e 2ρ P 2 , θ µ µ → 0 . (30) which implies that mass-M eigenstates |M obey the relation That implies a spectrum of zero-mass particles, or a continuum 0 M < ∞, or both -provided that the ground state is unique (WW mode of conformal invariance). However, for vacua (13) degenerate under scale transformations, this conclusion is not valid because states related by e iDρ belong to different worlds, W and W . A discrete scale M can exist in W and correspond to a discrete scale M in W . Since dimensional units are also scaled up or down in the same way, e.g.
experimental data in W and W are identical. Therefore these worlds are physically equivalent, as for any other symmetry in the NG mode. See Appendix D of [1] for details.
Sometimes type-I dilaton Lagrangians are written in a form such that scales do not appear explicitly in the conformal limit. That happens when all fields are chosen to transform homogeneously under scale transformations. The result is a polynomial Lagrangian with dimensionless coupling constants which is easily confused with the conformal WW-mode considered by Gildener and Weinberg [6]. The difference for the NG mode is that the scale emerges from a constraint (28) implicit in non-linear field transformations to unconstrained variables. The simplest example is the constraint φ > 0 which is not changed by scale transformations and seems to have no scale dependence. However, to implement it, a scale must be introduced. The classic example is the mapping [63] to the unconstrained Goldstone field σ.
Flat directions for conformal-invariant Lagrangians are also possible for type-III theories, as noted by Gildener and Weinberg [6], but they do not correspond to the vacuum degeneracy (13) because a type-III vacuum state is conformal invariant. Instead, the flat direction corresponds to field-translation invariance which forbids definitions like (33) which introduce a scale. In particular, Equation (33) cannot be used above the sill of the conformal window.

Scale Dependence at an IRFP
There is an extensive literature on IRFP's, but almost all of it assumes "conformality", i.e. a lack of scale dependence at an IRFP: 1. The initial work [64] was perturbative with a scale-free IRFP. That implied manifest chiral symmetry, so an IRFP of that type would presumably be close to a discontinuous transition to a phase where fermions can condense [65]. That became the model for walking TC. 2. It is relatively easy to search for scale-free IRFPs on the lattice: Green's functions exhibit power-law behavior in the conformal window. 3. There is a belief that dimensional transmutation, which produces nonperturbative scales like Λ QCD or Λ TC , implies θ µ µ = 0. If true, that would exclude scale dependence at IRFPs.
It is certainly true that, at one-loop order 2 , dimensional transmutation is produced by the trace anomaly, but a valid argument that this applies nonperturbatively does not exist. For example, for nonperturbative values of α, which may well include α IR , dimensional transmutation may be responsible for f σ = 0. Scales like f σ , Λ QCD for massless N f = 3 QCD and their TC counterparts F σ , Λ TC are all renormalization-group invariants of the form where κ M is a dimensionless constant that depends on M but not on α or µ. The formula (35) makes perfect sense 5 in the presence of α IR . Evidently items 1-3 are assumptions characteristic of type-III theories. For the type-I theories χPT σ and crawling TC, the problem is to find a satisfactory replacement for item 2. If scales are present at α IR , Green's functions do not exhibit power-law behavior -rather, they behave much like amplitudes observed in the real world.

Nonperturbative Tests of Type-I Theories
The obvious tactic is to define α non-perturbatively outside the conformal window and see if it stops increasing as the infrared limit is approached. There are two difficulties: 1. For non-Abelian theories, there is no true analogue of the Gell-Mann-Low function ψ(x) for quantum electrodynamics, where the effective charge is nonperturbative, runs smoothly and monotonically, and tests whether the dynamics chooses to have an IRFP or not. The method of effective charges [66][67][68] is nonperturbative, but there are as many definitions as there are physical processes, and it is not obvious which of them (if any) has the desired properties all the way to the far infrared. 2. On the lattice, it is hard to reach infrared scales well below the particle spectrum.
A less ambitious procedure is to look for a light scalar particle in the particle spectrum. The problem then is to decide whether this is evidence for a type-III or a type-I theory.
In the context of walking TC, the most interesting cases are those just under the sill of the conformal window. Evidence for a light scalar particle almost degenerate with technipions has been found in lattice data for SU (3) with N f = 8 Dirac fermions in the fundamental representation [24][25][26][27] and two Dirac fermions in the sextet representation [28]. In each case, the particle is identified as a "dilaton". In this type-III interpretation, the small mass is considered to be due to α being close to a WW-mode fixed point α WW just inside the conformal window.
However, as explained above, that involves unlikely assumptions about the dual character of δH, the term in H which breaks scale invariance explicitly. Type-III theories require δH to generate large-scale effects such as Λ TC and the fermion condensate at the sill, but to desist in cases where that is inconvenient, e.g. the scalar-boson mass.
The most likely explanation [1] of the light scalar particle is that there exists an IRFP α NG just outside the conformal window. Since there is scale dependence at α NG , a genuine type-I dilaton and hence all large-scale effects exist at that point. At α NG , both conformal and chiral invariance are in the NG mode, so massless technipions exist there as well as the type-I dilaton. Large-scale effects cannot be due to δH, because α can run smoothly to α NG in the conformal limit δH → 0. The sill does not get in the way, so there is no need to assume anything about its dynamics. The small scale of the scalar-particle mass M σ corresponds to α being in the infrared region close to α NG (Figure 4).   r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r sill Figure 4. Competing explanations for the appearance of a scalar particle in lattice data for SU(3) gauge theory with N f = 8 triplet fermions. Two IRFPs are shown: a) the closest scale-free IRFP α WW just inside the conformal window, N c f < N f 16, and b) a scale-dependent IRFP α NG for N f = 8 N c f . Walking TC assumes that α NG is not present. Instead, the small scalar mass is supposed to arise at an intermediate energy where the curve is closest to the axis, and then the theory chooses the blue line labelled "walking" in order to approach the infrared region. In crawling TC, the N f = 8 theory enters the infrared region as it approaches the axis and chooses the red line labelled "crawling". The short length of the red line accounts for the small mass acquired by the type-I dilaton.
In walking TC, IRFPs outside the conformal window are thought to be forbidden. Instead, an explanation for the light scalar particle is sought by appending dilaton Lagrangians to the type-III framework [29,[69][70][71][72]. In fact, these effective Lagrangians are type-I theories developed in 1970 [37,38,40]: they generate asymptotic expansions in δH eff ∼ 0 about a conformal limit with scale-dependent amplitudes depending on the decay constant f σ or its TC analogue F σ . The question is: if α NG is not available, about what point is the expansion to be performed?
Since the emphasis in walking TC has been to minimize |β|, presumably the understanding has been that the expansion should be carried out about α WW . The trouble with that is the lack of scale dependence at α WW . An alternative has just been suggested [73], that the dilaton Lagrangian expansion should correspond to expanding in "the distance to the conformal window", i.e. about the sill, where there is certainly a large scale. In a type-III theory, the sill acquires its scale dependence from a large scale violation δH in the Hamiltonian. The problem is then that the expansion of L dil is about a point where the corresponding effective Hamiltonian is conformally invariant.
The conclusion is that type-I and type-III theories should not be mixed -they are based on contradictory assumptions. A type-I effective Lagrangian introduced to discuss the light scalar boson should be given a type-I point about which it can be expanded: α NG . Why this should be such a fearsome prospect is puzzling.
Finally, I should comment on the perception that lattice data for N f = 4 implies that the f 0 (500) is heavy, contrary to my rermarks below Figure 2. A comparison of data for N f = 4 and N f = 8 (Figure 1 of [27]) shows that, as the fermion mass m ψ becomes small, the light scalar particle is almost degenerate with (techni-)pions for N f = 8 but not for N f = 4. This indicates that the gluonic contribution to the scalar mass is negligible for N f = 8 but not for N f = 4. A type-III interpretation of this is that the gluonic contribution is a large-scale effect due to δH -a point of view similar to that of [7,9,10].
In type-I theories, there is no problem. A gluonic contribution to the scalar mass is a small-scale effect due to the coupling being close to but not at the NG-mode IRFP. For N f = 8, apparently = α NG − α is so small that the scalar mass is dominated by the masses m ψ used in the lattice analysis. For N f = 4 (a lattice-friendly approximation to the physical case of N f = 3 light flavors), the effects of appear similar in magnitude to those of m ψ , within fairly large errors.
So I maintain that the Higgs boson is the direct TC analogue of f 0 (500) for QCD: both of them are derived from type-I dilatons at scale-dependent IRFPs.The main difference is in the ratio r of small-scale to large-scale effects, r QCD ≈ 500 MeV 4π(92 MeV) = 0.4 and r TC ≈ 125 GeV 4π(246 GeV) = 0.04 .
This is permissible in type-I theories because large-and small-scale effects have separate origins.
Acknowledgments: I thank Lewis Tunstall and Oscar Catà for their continuing advice after a long and fruitful collaboration.
Funding: This research received no external funding.

Conflicts of Interest:
The author declares no conflict of interest.

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