Origin of Mass - Horizons Expanding from the Nambu’s Theory

# Origin of Mass - Horizons Expanding from the Nambu’s Theory

Koichi Yamawaki
###### Abstract

The visible (Matter) has no difference from the invisible (Vacuum), the invisible has no difference from the visible. The visible is nothing but the invisible, the invisible is nothing but the visible.

- Heart Sutra (translation by KY)

Origin of mass may be strong dynamics of matter in the vacuum. Since the initial proposal of Nambu for the origin of the nucleon mass, the dynamical symmetry breaking in the strongly coupled underlying theories has been expanding the horizons in the context of the modern version of the origin of mass beyond the Standard Model (SM).

The Nambu-Jona-Lasinio (NJL) model is a typical strong coupling theory with the non-zero critical coupling and a large anomalous dimension , in sharp contrast to its precedent model, the Bardeen-Cooper-Schrieffer theory for the superconductor. The non-zero critical coupling is also hidden in the asymptotically free gauge theories including QCD and walking technicolor: it reveals itself in the chiral symmetry restoration where the coupling cannot grow above the “hidden” critical coupling in the infrared region (infrared conformality).

As is well known, the NJL model can be cast into the SM Higgs Lagrangian. We show that the SM Higgs Lagrangian is simply rewritten into a form of the (approximately) scale-invariant nonlinear sigma model, with both the chiral symmetry and scale symmetry realized nonlinearly, with the SM Higgs being nothing but the (pseudo-) dilaton. The SM Higgs Lagrangian is further gauge equivalent to the scale-invariant Hidden Local Symmetry (HLS) Lagrangian, s-HLS, having spin 1 bosons hidden in the SM.

As the simplest possible underlying theory for the SM Higgs Lagrangian we first discuss the top quark condensate (“top-mode SM”) based on the (scale-invariant) NJL model with only top (plus possibly bottom) coupling larger than the critical coupling, where the top-mode dilaton is the 125 GeV Higgs and the HLS gauge boson (“top-mode rho meson”) (and the top-mode axion) may be detected at LHC.

We then discuss the walking technicolor having near infrared conformality and large anomalous dimension . Its effective theory is the s-HLS model precisely the same as the SM Higgs Lagrangian (with larger chiral symmetry), where the 125 GeV Higgs is successfully identified with the techidilaton. The 2 TeV diboson and 750 GeV diphoton excesses at LHC are identified with the HLS technirho and the technipion, respectively.

## 1 Introduction

Professor Nambu made great achievements in so much inexhaustible depth and wideness, and thus it may be something like the picture of “The Blind Men and the Elephant” to talk about only a single aspect of his physics. But the subject I am going to talk about is not just one of them, but probably his most influential one. In fact the 2018 Nobel Prize announcement is “for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics” [1]. His spontaneous symmetry breaking (SSB) [2] was the theory for the origin of mass of the nucleon, then an elementary particle, generated dynamically from nothing (the vacuum) through the nucleon-antinucleon pair condensate. Although the nucleon is no longer an elementary particle, the essence of his mechanism to generate mass of composite particles, hadrons including the nucleon, as well as the near masslessness of the composite pion, is now realized through the quark-antiquark condensate in the underlying theory QCD. This mass constitutes 99 % of mass of the nucleon, namely of the ordinary matter made out of the atoms, and thus the Nambu’s theory already accounted for the origin of the dominant part of the mass of the visible world.

The problem of the origin of mass in the modern particle physics is only for the rest 1% of the mass of the matter, the elementary particles of the Standard Model (SM), which is attributed to the Higgs boson whose origin is still mysterious. I will discuss that this 1% may also be explained by the dynamical symmetry breaking in some underlying theory, similarly to the Nambu’s theory.

Origin of Mass of all the SM particles is the Higgs VEV or the Higgs mass read from the SM Higgs Lagrangian:

 \@fontswitchLHiggs = |∂μh|2−μ20|h|2−λ|h|4 (1) = 12[(∂μ^σ)2+(∂μ^πa)2]−12μ20[^σ2+^πa2]−λ4[^σ2+^πa2]2 (2) = 12tr(∂μM∂μM†)−[μ202tr(MM†)+λ4(tr(MM†))2], (3)

where we have rewritten the conventional form in Eq.(1) into the Gell-Mann-Levy (GL) linear sigma model [3] in Eq.(1) through

 h=(ϕ+ϕ0)=1√2(i^π1+^π2^σ−i^π3), (4)

and further into Eq.(1) with the matrix

 M=(iτ2h∗,h)=1√2(^σ⋅12×2+2i^π)(^π≡^πaτa2), (5)

which transforms under as:

 M→gLMg†R,(gR,L∈SU(2)R,L). (6)

Then the origin of mass is attributed to the mysterious input mass parameter of the tachyons, and (not physical particles), with the mass such that

 μ20<0 (7)

as a free parameter. But why the tachyon? How is the tachyon mass determined? SM cannot answer to these questions, even though the Higgs boson has been discovered with the mass near 125 GeV.

Historically, the GL linear sigma model in the form of Eq.(1) as the prototype of the Higgs Lagrangian Eq.(1) was proposed for phenomenologically describing the pion (as well as the nucleon) without concept of the SSB, while the Nambu-Jona-Lasinio (NJL) model [2] explained the same property at the deeper level in terms of the dynamical symmetry breaking due to the vacuum property. Here we should recall that the SSB was born as a dynamical symmetry breaking (DSB), where the tachyons are in fact generated as composites of the dynamical consequence of the strong dynamics, but not ad hoc inputs as in the GL theory. Actually, the GL theory is now regarded as an effective theory (macroscopic theory) for the NJL model as a microscopic theory, as is the Ginzburg-Landau (GL) theory for the Bardeen-Cooper-Schrieffer (BCS) theory [4] for the superconductor. So history may repeat itself.

I first discuss that although his idea was motivated by the BCS theory, it was not just a copy of it but essentially new in the most important aspect, namely it created a new dynamics, although based on the same kind of four-fermion interaction: the NJL dynamics is the strong coupling theory having non-zero critical coupling to separate the SSB phase with (above the critical coupling) from the non-SSB phase with (below the critical coupling). It is in sharp contrast to the BCS theory which is a weak coupling theory having the zero critical coupling, always in the SSB phase even for infinitesimal (attractive) coupling, due to the Fermi surface. The Fermi surface reduces the effective dimensions by 2 so as to make the theory in effectively dimensions like the Thirring model and/or Gross-Neveu model.

The non-zero critical coupling is also hidden in the asymptotically free gauge theories including the QCD: it reveals itself in the chiral symmetry restoration when the system is in the extreme condition such as the high temperature, high density, and large number of light fermions, where the coupling cannot grow above the hidden critical coupling in the infrared region as strong enough to form the fermion-antifermion condensate. The existence of the non-zero critical coupling in the gauge theory was first recognized by Maskawa-Nakajima [5] in the ladder Schwinger-Dyson (SD) equation, a gauge theory analogue of the NJL gap equation. The solution of the ladder SD equation in the weak coupling region (existing even for the infinitesimal coupling) [6] disappears at zero fermion bare mass at finite cutoff, which is actually the explicit chiral symmetry breaking solution vanishing when the cutoff is removed. 222 At TV interview after announcement of the Nobel prize together with Professor Nambu in 2008, Toshihide Maskawa confessed that the paper he most studied was the Nambu’s paper on SSB, “I exhausted it”. He in fact discovered the non-zero critical coupling for SSB in the gauge theory [5] not just in the NJL four-fermion model. At that time I was a graduate student at Kyoto University to hear it first hand and have been influenced by this work, strong coupling gauge theory (SCGT) with non-zero criticality, ever since. See Nagoya SCGT workshops, http://www.kmi.nagoya-u.ac.jp/workshop/SCGT15/ Although the asymptotically-free gauge theory like QCD has no explicit critical coupling to divide the SSB phase from non-SSB phase (having only a single phase of SSB), the running coupling always becomes strong in the infrared region where the coupling exceeds a hidden critical coupling to trigger the SSB having the condensate of order of the scale of this mass region [7].

The main purpose of this article is to describe the expanding horizon of such a strong coupling dynamics characterized by the non-zero critical coupling initiated by Professor Nambu in view of the modern version of the origin of mass, namely the composite Higgs models having large anomalous dimension. First, the (weakly gauged) strong coupling four-fermion models like the top quark condensate model [8, 9, 10], where only the top quark has the strong coupling above the criticality (anomalous dimension [11]) so as to be responsible for the electroweak symmetry breaking [8]. Second, the gauge theories such as the walking technicolor based on the near conformal gauge theory just above the criticality having anomalous dimension and a composite dilaton (technidilaton) as the composite Higgs [12, 13]. The technidilaton in the walking technicolor has been shown to be consistent with the 125 GeV Higgs at the present LHC experimental data [14, 15, 16].

Before discussing possible underlying theory for SM, I show that the SM Higgs Lagrangian itself already has some hints for the theory beyond the SM. It was shown [17] that the SM Higgs Lagrangian itself possesses nonlinearly realized “hidden” symmetries (scale symmetry and Hidden Local Symmetry (HLS) [18, 19, 20], both spontaneously broken), in addition to the well-known symmetry, nonlinearly realized global chiral symmetry (also spontaneously broken) to be gauged by the electroweak symmetry. It is in fact straightforward to show [17] that the SM Higgs Lagrangian is cast into the scale-invariant nonlinear chiral Lagrangian [14], and then further shown to be gauge equivalent to the scale-invariant HLS (s-HLS) Lagrangian [21]. The SM Higgs is nothing but a (pseudo-) dilaton! [17] This is the very nature of the SM Higgs Lagrangian, quite independent of details of the possible underlying theory as the UV completion. Also the HLS can naturally accommodate the vector bosons, analogues of the rho mesons in the QCD, into the SM (“SM rho meson”) [22]: It would be the simplest extension of the SM to account for the 2 TeV diboson events at LHC [23].

Then I elaborate [24] on the well-known fact that the NJL model can be regarded as the microscopic theory (underlying theory or ultraviolet (UV) completion) for the SM Higgs Lagrangian, or the GL linear sigma model, as the macroscopic theory (effective theory) at composite level. With the coupling larger than the non-zero critical coupling, the NJL model equivalent to the SM Higgs has also the nonlinearly realized hidden (approximate) scale symmetry for the SM Higgs as a composite pseudo-dilaton (“NJL dilaton”), together with the HLS for the dormant composite spin 1 boson (“NJL rho meson”) as a possible candidate for the LHC diboson events [23]. Although both are trivial theories having no interaction in the infinite cutoff limit (Gaussian fixed point), I will discuss possible way out, one [25] being the gauged NJL model in combination with the walking gauge theory, another [24] the recently suggested different way of the continuum limit where the composite Higgs becomes massless (up to the trace anomaly) as the pseudo-dilaton in the same sense as the SM Higgs.

The simplest possibility for such a composite model would be the top quark condensate model (“top-mode SM”) [8, 9, 10], where crucial is the non-zero criticality [8]: only top (may also bottom) has the coupling larger than the non-zero critical coupling to acquire the dynamical mass due to SSB. Near the scale-invariant limit, the top-mode dilaton may be the 125 GeV Higgs, and the HLS gauge boson (“top-mode rho meson”) may be identified with the recent 2 TeV diboson excess (and the top-mode axion, bound state, may be identified with the 750 GeV diphoton excess at LHC [26] which was reported after this symposium).

We then discuss the walking technicolor proposed based on the SSB solution of the ladder SD equation to have a large anomalous dimension and technidilaton as a pseudo-Nambu-Goldstone (NG) boson of the approximate scale symmetry [12, 13]. Such a scale-symmetric walking gauge theory may be realized when flavor number of massless fermions is large in the asymptotically free gauge theory (“large QCD”) [27, 28], with slightly smaller than that having an infrared fixed point (conformal window) where the coupling in the infrared region is almost constant and below the critical coupling so that the SSB does not take place. The effective theory of the walking technicolor is the s-HLS Lagrangian with a larger chiral symmetry , with typically (one-family model), precisely the same type of the s-HLS as in the case for the SM Higgs Lagrangian with . The technidilaton as a composite Higgs has been shown [14, 15, 16] to be consistent with the present LHC 125 GeV Higgs, and the HLS vector mesons (walking technirhos) have also been shown [29] to be consistent with the LHC diboson events [23]. (We also showed [30] that one of the technipions can be identified consistently with the 750 GeV diphoton events at LHC [26] reported after the symposium).

Several theoretical issues are discussed such as the recent lattice studies of the walking theories, as well as the ladder, and the renormalizability of the gauged-NJL model and the conformal phase transition, etc.

## 2 NJL the Strong Dynamics vs. BCS the Weak Dynamics

It is widely believed that the NJL model is a copy of the BCS. Here I emphasize that they are essentially different dynamics, NJL as the strong coupling with critical coupling no-zero, while the BCS as a weak coupling with the critical coupling zero. The difference comes from the Fermi surface in BCS which reduces the effective phase space from 3+1 to 1+1, while the NJL case is in the free space of full 3+1 dimensions. The attractive forces are more efficient in smaller phase space.

Let us start with the NJL model [2] for 2-flavored Dirac fermions :

 \@fontswitchLNJL=¯ψiγμ∂μψ+G2[(¯ψψ)2+(¯ψiγ5τaψ)2]. (8)

When the fermion-antifermion condensate in the vacuum takes place, , it reads

 \@fontswitchLNJL−¯ψiγμ∂μψ= G⟨¯ψψ⟩¯ψψ+⋯=−mF¯ψψ+⋯. (9)

At the leading order this yields the self-consistent NJL gap equation for the dynamical mass of :

 mF=−G⟨¯ψψ⟩=GTr[SF(p)]=G⋅4NC∫d4pi(2π)4mFm2F−p2 (10)

which has an SSB solution :

 1G−1Gcr=Λ24π2(1g−1gcr)=−14π2NCm2Fln(Λ2m2F)<0, (11)

only for the strong coupling

 G>Gcr=4π2NCΛ2≠0(g≡GΛ24π2>gcr=1Nc≠0). (12)

We shall later discuss that this in fact corresponds to the tachyon mass in Eq.(1): , where . The (composite) tachyon has been induced dynamically by the term due to the loop effects in the large limit. Of course the tachyon is not a physical particle, which simply implies instability of the trivial vacuum with (no SSB). For the weak coupling there exists only the non SSB solution where no tachyon exists.

Note that “strong coupling” as defined by non-zero critical coupling does not necessarily mean numerically strong, particularly in the large limit . However, the attractive forces in the condensate channel are not from a single fermion but actually from sum of all the fermions coherently, which ends up with really strong . [As we discuss later, this also applies to the strong coupling gauge theory where , while the gauge coupling criticality itself is negligibly small (but non-zero) in the large limit].

In contrast to the non-zero critical coupling of the NJL model, the BCS theory for the superconductor has the zero critical coupling (“weak coupling theory”) due to the electron Fermi surface (: electron mass in the free space), which affects the fermion-fermion condensate (dynamical Majorana mass as the gap) instead of fermion-antifermion condensate . The essence can be read from the effective dimension of the momentum in the integral of the gap equation of the BCS :

 ∫d3p(2π)3=∫(4πp2)dp(2π)3⇒4πp2F∫|E(p)−EF|<ω/2dp(2π)3=N2∫EF+ωD/2EF−ωD/2dE(p), (13)

where and constant: The dimensional electron momentum is confined to a one-dimensional direction normal to the Fermi surface in the narrow energy shell bounded by the Debye energy (cutoff). After integral , with the fermion propagator ( instead of ) ), the BCS gap equation corresponding to Eq.(2) (Majorana mass without factor 4) reads

 |Δ|=GN2∫EF+ωD/20dE(p)|Δ|√|Δ|2+E(p)2∼|Δ|[NG2ln|Δ|ωD]. (14)

Then the SSB solution with exists even for infinitesimal coupling :

 |Δ|∼ωDexp(−2NG),(1≫NG>NGcr=0), (15)

in contrast to the SSB solution in NJL model in Eq.(2) with in Eq.(2).

The result is intuitively obvious: The fermion pair in the one dimensional space is “bound” even for infinitesimal coupling, since there is no way to escape from each other, while that in the higher dimensional space can freely move from each other and hence needs strong attractive forces to bind it together. This is the effective dimensional reduction. The situation that the lower dimensional theory lowers the critical coupling can be viewed explicitly by the dimensional four-fermion theory, the Gross-Neveu model, with changed continuously [31]. The gap equation is simply changed as in Eq. (2). Similarly to Eq.(2), the SSB solution exists:[31]

 1g−1gcr=−NCξD2−D2⋅(mFΛ)D−2<0, (16)

only for the strong coupling;333 In dimensions, the form remains the same, in accord with the above intuitive picture for the required binding force strength depending on the phase volume, while the gap equation takes a similar but different form: [49] .

 g>gcr=1NC(D2−1)→0(D→2), (17)

where and for ( for ) 444 If we take the limit, on the other hand, the gap equation is reduced to Eq.(2) except for the logarithmic factor. This log factor is a crucial difference between the and the four-fermion theories. As we discuss later, the former is renormalizable in expansion having the nontrivial fixed point at in the beta function, [31], while the latter is not, a trivial theory, with the beta function having the Gaussian fixed point at . ( is an infrared fixed point defining the infrared free theory, with being the repulsive forces.) . The critical coupling indeed decreases as does to vanish at . This yields for the well-known result:

 mF=Λexp(−12NCg),(NCg>NCgcr=0), (18)

which is of the same form as Eq.(2).

Thus the BCS dynamics in some sense is similar to the four-fermion theories such as the Thirring model and the Gross-Neveu model. There is a caveat [33], however: the genuine dimensional theory is not actually in the SSB phase in accord with the Merwin-Wagner-Coleman theorem, although it has a massless bound state and a massive fermion with mass of the form of Eq.(2) in the large limit, similarly to the SSB phase. However, the absence of the NG boson and lack of SSB does not apply to the BCS theory in contrast to the Thirring model and Gross-Neveu model, since the BCS theory is not a genuine dimensional model but rather a brane model: only fermions (not anti-fermions) are confined to the -brane, the Fermi surface, a consequence of the Fermi statistics, while the fermion-fermion pair composite NG boson as a boson lives freely from the Fermi surface in the full dimensional bulk, and hence SSB and NG boson do exist, in accord with the superfluidity and superconductor.

To summarize the Nambu’s approach to the origin of mass, the theory having intrinsic mass scale may or may not produce the particle mass , depending on the coupling strength: the strong coupling dynamics for creates the composite tachyon with negative in such a way that the particle mass is generated from the intrinsic mass scale . By fine tuning the strong coupling as , we can arrange a big hierarchy (near chiral symmetry restoration). On the other hand, for the weak coupling there exists no particle mass , although the theory has an intrinsic mass scale . This is an essential difference from the BCS theory which has a zero critical coupling, producing always a non-zero gap even for the infinitesimal coupling.

As discussed later, this non-BCS phase structure of the NJL dynamics was in fact the original motivation of the top quark condensate model of Ref. [8], where the top quark having a coupling larger than the critical coupling is discriminated from others having those smaller than the critical coupling, so that only the top has mass of order of weak scale in such a way as to produce only three NG bosons responsible for the electroweak symmetry breaking. This is in contrast to the “bootstrap symmetry breaking” [9] which is based on the BCS dynamics without the notion of the non-zero criticality.

## 3 Strong Coupling Gauge Theories for the Origin of Mass

The dynamical mass of the fermion picks up the intrinsic scale (cutoff) which regularizes the theory and brings the explicit breaking of the scale symmetry corresponding to the trace anomaly in the renormalized quantum theory. In the asymptotically free gauge theory can be identified with the renormalization-group invariant intrinsic scale such as induced by the perturbative trace anomaly, as we discuss later.

There also exists a non-zero critical coupling for SSB in the gauge theory with massless fermion, as first noted [5] in the ladder SD equation, with non-running coupling in the Landau gauge, a straightforward extension of the NJL gap equation Eq. (2), this time for the fermion mass function instead of the constant mass (For details see e.g., Ref.[16]):

 S−1F(p) = S−1(p)+∫d4k(2π)4 C2g2Dμν(p−k)γμ SF(k) γν, (19)

where and are the full and bare fermion inverse propagators, respectively, and the bare gauge boson propagator in the Landau gauge, and is the quadratic Casimir of the fermion of the gauge theory, with for the fundamental representation in . After the angular integration, the ladder SD equation in Landau gauge for reads:

 Σ(x)=m0+3C24πα∫Λ2dy[θ(x−y)x+θ(y−x)y]yΣ(y)y+Σ2(y),(Z−1(x)≡1). (20)

This form is reduced back to the form of the NJL gap equation with , Eq.(2), if the kernel is local: , such as in the case of the massive gauge boson, (See also Eq.(70)).

Eq.(3) is converted into a differential equation plus IR and UV boundary conditions [34]:

 (xΣ(x))′′+α3C24πΣ(x)x+Σ2(x) = 0, (21) limx→0x2Σ′(x) = 0, (22) (xΣ(x))′∣∣x=Λ2 = m0. (23)

The asymptotic solution of Eq.(3) at takes the form , with a conventional normalization , which is plugged back into the equation to yield , i.e., .

For , either solution, dominant () or non-dominant (), has a power behavior, which does not satisfy the UV boundary condition Eq.(3) for the chiral limit , where . The solution exists only at the presence of the explicit breaking , namely the explicit breaking solution with the renormalized mass , which yields the anomalous dimension in the unbroken phase [35]:

 m0=mR(ΛmR)−1+ω,γm=Λ∂lnZ−1m∂Λ=1−√1−ααcr<1(α<αcr=π3C2). (24)

The result is written in terms of the one-loop anomalous dimension as which coincides with for :

On the other hand, the SSB solution does exist for

 α>π3C2=αcr, (25)

where with , and the solution is of the oscillating form , , which satisfies the UV boundary condition as

 0=m0∼m2FΛ~ωsin(~ωln(4ΛmF)), (26)

for (numerically ). In the large limit the critical coupling itself is numerically small, , although the effective coupling in the condensate channel is as was the case in the NJL coupling. This is the reason why the ladder approximation yields reasonable result.

The ground state solution is , which yields the dynamical mass of the Berezinsky-Koterlitz-Thouless (BKT) form of essential-singularity (“Miransky scaling”) [37]:

 mF ≃ 4Λexp⎛⎜ ⎜⎝−π√ααcr−1⎞⎟ ⎟⎠,(α>αcr=π3C2≠0), (27) = 0,(α<αcr). (28)

This is compared with the NJL gap equation Eq. (2) and D-dimensional NJL Eq.(2), and also with the BCS Eq.(2) and 2-dimensional model Eq.(2). Again the large hierarchy

 mF≪Λ(α/αcr−1≪1) (29)

can be realized near criticality (near chiral symmetry restoration).

The essential-singularity scaling yields a peculiar phase transition, dubbed “conformal phase transition” [28], which is different from the typical 2nd order phase transition as the Ginzburg-Landau phase transition. While the order parameter such as is continuously changed as to from to , the spectrum changes discontinuously, since there is no light spectrum in (conformal, unparticle) in contrast to the SSB phase where mass spectrum all goes to zero as . It reflects the fact that the essential singularity is not analytic at . The light spectrum is possible for only when which violates the conformality. Thus all the mass spectrum for the conformal phase scales like the explicit breaking renormalized mass , which is given by Eq.(3) as [36], in conformity with the hyperscaling relation frequently used in the lattice analyses for the conformal signals.

Eq.(3) implies that the coupling is a function of with the nonperturbative beta function:

 β(NP)(α) = Λ∂α(Λ)∂Λ=−2π2αcrln3(4ΛmF)=−2αcrπ(ααcr−1)32, (30) α(μ) = αcr⎡⎢⎣1+π2ln2(4μmF)⎤⎥⎦, (31)

with being now regarded as a nontrivial ultraviolet fixed point (approached much faster than the asymptotic freedom ). The asymptotic form of the SSB solution is which is compared with the Operator Product Expansion to yield a large anomalous dimension: [12, 13]

 γm=1(α>αcr). (32)

This ladder result is the characteristic feature of the walking technicolor.

Due to this mass generation which breaks the scale symmetry spontaneously, the ladder scale symmetry is also broken explicitly producing the new nonperturbative trace anomaly besides the perturbative trace anomaly induced by the cutoff regularization (See Ref. [16] and references cited therein):

 ⟨∂μDμ⟩=⟨θμμ⟩(NP) ≡ ⟨θμμ⟩(full)−⟨θμμ⟩(perturbative)=β(NP)(α)4α⟨G2μν⟩(NP), (33) ≃ −NFNC4ξ2π4m4F,(ξ≃1.1),

where is the nonpertubative gluon condensate and is a number of flavors of massless fermions (besides color ). Note that although and are depending on the renormalization point , the trace anomaly is not as it should be (the energy-momentum tensor is a conserved current and is not renormalized), with both dependence being cancelled each other precisely. [16]

This ladder dynamics was the basis for the walking technicolor [12, 13] where the coupling is almost non-running for even after the SSB takes place to produce the nonperturbative running.

The non-zero critical coupling also exists in the asymptotically free gauge theory including the QCD in a more sophisticated way, in spite of no explicit non-zero critical coupling separating the SSB phase and the non-SSB phase, namely the QCD is in one phase always in the SSB similarly to the BCS. The theory is classically scale-invariant but actually has an intrinsic mass scale due to the trace anomaly by the quantum effects (regulator). The intrinsic scale is usually given by the one-loop beta function: with given in Eq.(35). Although this looks like the BCS mass generation in Eq.(2), should not be confused with the mass generation . The existence of the intrinsic scale does not necessarily imply the SSB as in the NJL model where the intrinsic scale does not necessarily imply the mass .

The QCD coupling runs depending on the renormalization scale in units of to grow in the infrared region. The fermion mass is dynamically generated due to the fermion-antifermion condensate which takes place in the infrared region where the coupling becomes strong as to exceed the “hidden” critical coupling of order 1: . In the usual QCD it so happens that . In a wider parameter space, however, we can see the cases (chiral restoration) and (near chiral restoration), where the non-zero critical coupling is actually essential.

The “hidden” non-zero critical coupling become “visible” when the system is put in the medium with finite temperature and density with baryon chemical potential , where the running coupling in the infrared region is no longer growing indefinitely and levels off at the relevant energy scale of order of or . Then for such that the SSB would not take place, namely the chiral symmetry restoration occurs as has been studied actively. In contrast to the disappearance of the fermion-antifermion condensate, the BCS dynamics for fermion-fermion condensate instead can be operative in the finite density even with the weakest coupling due to the Fermi surface, which is called color superconductor.

Here I discuss another case to visualize the non-zero critical coupling in the QCD-like vector-like gauge theory with massless technifermions, still in the asymptotically free theory with the running coupling vanishing in the ultraviolet region. This is the basis for the walking technicolor to be discussed later and I denote the intrinsic scale as hereafter.

When one increases , the vacuum polarization due to the virtual fermion-antifermion pairs (loop effects) increases the screening of the charges in the long distance (infrared energy region), which is operative opposite to the asymptotically free anti-screening effects of the gluon loops: in the ultraviolet region the coupling is small and running is essentially one-loop dominated, while in the infrared region where the coupling grows, the higher loop effects particularly by the fermion loop screening effects are getting dominant, which then balances the anti-screening effects to tend to make the coupling level off. Then the dynamical mass such that will be getting smaller, as we increase :

 mFΛTC↘forNFNC↗, (34)

in contrast to the ordinary QCD with for . It then eventually could realize at certain large an infrared fixed point ), which implies that no SSB takes place and no bound states exist (“unparticle”), the phase called “conformal window”.555 Here we are talking about the phase transition in the parameter by changing the theory. It does not imply the existence of two phases in one theory with fixed . See the discussions below and Fig. 1 The approximate scale symmetry is operative with almost nonrunning coupling in the infrared region , although it is violated explicitly by due to the trace anomaly in the ultraviolet region where the coupling is running as in the usual asymptotically free theory.

The existence of the conformal window in fact can been seen explicitly at two-loop beta function, which is scheme-independent while higher loops are not: [38].

 β(2−loop)(α) = −b0α2−b1α3, b0 = 16π(11NC−2NF),b1=124π2(34N2C−10NCNF−3N2C−1NCNF), α∗ = α∗(NF,NC)=−b0b1, (35)

where we have by balancing the one-loop ( as far as asymptotically free, i.e., ) with the two-loop contributions at the infrared limit , which is realized only when s.t. is satisfied. Note that as and exists for ( for ).

In the context of large limit, such a situation corresponds to the “anti-Veneziano limit” (in distinction to the original Veneziano limit with ):[16]

 NC→∞andNC⋅α=fixed,withr≡NF/NC=fixed≫1. (36)

The anti-Veneziano limit in fact realizes a situation very close to the ladder approximation, with the behaving as a continuous parameter. Then the theory has two phases in the parameter space : SSB phase for such that and the non-SSB phase otherwise.

In the case , there in fact exists no SSB and no bound states (“unparticle”). The coupling is almost constant for all the infrared region (infrared conformality), while it is running in the ultraviolet region essentially as the one-loop running, in accord with the scale symmetry violation due to the perturbative trace anomaly. 666For the region , there might exist SSB, in which case there might exist two phases separated by the ultraviolet fixed point at in the sense similar to the conjecture on the asymptotically non-free gauge theory such as the strong coupling QED for , although such a fixed point may be a Gaussian fixed point (trivial theory) [39].

On the other hand, for the SSB takes place with mass generated similarly to the ladder SD result in Eq. (3) with replaced by [27],

 mF∼ΛTCexp⎛⎜ ⎜⎝−π√α∗αcr−1⎞⎟ ⎟⎠(≪ΛTCforNFNCs.t.α∗(NF,NC)αcr(NC)−1≪1), (37)

where the phase transition in the parameter space has a characteristic essential singularity scaling (Miransky-BKT scaling), which takes the same type of the “conformal phase transition” as the ladder one [28].

Once is generated, the scale symmetry is explicitly broken so as to yield the nonperturbative trace anomaly, Eq.(33), responsible for the nonperturbative running of the coupling, Eq.(3), and the would-be infrared fixed point at two-loop is actually washed out. The resultant coupling would have a form with (quasi) ultraviolet fixed point similarly to Eq.(3) for (), while it still has a remnant of infrared fixed point (quasi fixed point) for (). Thus the theory is in one phase, which is not separated by . The beta function has no exact zero at and the coupling runs through continuously. See Fig.1.

To summarize the origin of mass in the strong coupling theories, the mass originates from the intrinsic scale (Lagrangian parameter or the trace anomaly through SSB which takes place only in the strong coupling phase with the coupling larger than the non-zero critical coupling. There is no mass generation in the weak coupling phase, even though the theory has intrinsic scale .

## 4 Hidden Symmetries in the SM Higgs Lagrangian [17]

Here we recapitulate Ref. [17] to show that the SM Higgs Lagrangian Eq.(1) in the form of the linear sigma model, Eqs.(1) and (1), is rewritten into precisely the form equivalent to the scale-invariant version of the chiral nonlinear sigma model based on the manifold , with and , as far as it is in the broken phase, with both the chiral and scale symmetries spontaneously broken due to the same Higgs VEV , and thus are both nonlinearly realized.

The SM Higgs Lagrangian is further shown to be gauge equivalent to the scale-invariant version [21] of the Hidden Local Symmetry (HLS) Lagrangian [18, 19, 20], which contains possible new vector bosons, analogues of the mesons, as the gauge bosons of the (spontaneously broken) HLS hidden behind the SM Higgs Lagrangian.

Let us discuss the Higgs Lagrangian in the form of Eqs.(1) and (1): The potential minimum exists at the chiral-invariant circle:

 ⟨σ2(x)⟩=−μ20λ≡v2,σ2(x)≡^σ2(x)+^πa2(x). (38)

In Eq.(1) any complex matrix can be decomposed into the Hermitian (always diagnonalizable) matrix and unitary matrix as ( “polar decomposition” ):

 M(x)=H(x)⋅U(x),H(x)=1√2(σ(x)00σ(x)),U(x)=exp(2iπ(x)Fπ), (39)

with and . The chiral transformation of is inherited by ,777The nonlinear realization was first introduced by K. Nishijima (then at Osaka City University) [40] (in case) to make the nucleon massive in a chiral invariant way using the NG boson as , where the physical (massive) nucleon transforms as , (), while the original nucleon field does as . Here the nonlinear base is defined by with the transformation, , , in accord with Eq.(40). See, e.g., Ref.[19]. while is a chiral singlet such that:

 U→gLUg†R,H→H, (40)

where and implies , namely the spontaneous breaking of the chiral symmetry is taken granted in the polar decomposition. Note that the radial mode is a chiral-singlet in contrast to which is a chiral non-singlet transformed into the chiral partner by the chiral rotation. The physical particles are and which are defined by the nonlinear realization, in contrast to the tachyons and .

We further parametrize as

 σ(x)=