# Nambu and Compositeness111Invited talk, Miami 2003,Topical Conference on
Elementary Particles, Astrophysics and Cosmology, organizer: Thom Curtright,
University of Miami.

Christopher T. Hill<EMAIL_ADDRESS>Fermi National Accelerator Laboratory, P.
O. Box 500, Batavia, IL 60510, USA  Department of Physics, University of
Wisconsin-Madison, Madison, WI, 53706<EMAIL_ADDRESS>
###### Abstract

The Nambu–Jona-Lasinio model (NJL) NJL  is the simplest field theory of a
composite scalar boson, consisting of a pair of chiral fermions. A bound state
emerges from an assumed point-like 4-fermion interaction and is described by
local effective field, $\Phi(x)$. We review this in the context of the
renormalization group and show some its phenomenological successes, such as
the QCD chiral dynamics and the prediction of stable heavy-light resonances
that reveal the single light quark chiral dynamics. We also review some
NJL–inspired composite Higgs models. We then describe a novel UV completion of
the NJL model, to an extended interaction, where the solution is described by
a bilocal effective, Yukawa field, $\Phi(x,y)$. I conclude with a personal
recollection of Yoichiro Nambu.

††preprint: FERMILAB-CONF-24-0010-T

##  I Introduction

This is an expanded version of an invited talk given in an honorary session
dedicated to Yoichiro Nambu at the Miami 2023 Topical Conference on Elementary
Particles, Astrophysics and Cosmology, Dec. 13–19, Fort Lauderdale, Florida.
The session featured two talks, Pierre Ramond, “Nambu and Strings,” and me,
“Nambu and Compositeness.” Based upon discussions with participants I was
inspired to collect things together and present them in this form. In
particular, I want to emphasize that the Nambu–Jona-Lasinio model, NJL NJL ,
often considered a toy model for chiral dynamics, is a powerful tool and has
led to bona fide discoveries in physics.

For example: thirty years ago Bill Bardeen and I applied the NJL model to
heavy–light mesons BHHQ . This predicted a universal mass gap between a ground
state, e.g., the $0^{-}$, charm-light $D_{u,d}$ meson, $\overline{c}(u,d)$,
and it’s “chiral partner,” the $0^{+}$, $\widetilde{D}_{u,d}$. Hence, these
two particles would be degenerate if we could turn off chiral symmetry
breaking. We estimated the mass gap in the NJL model to be $\Delta M\sim 338$
MeV. This is enough to make $\widetilde{D}_{u,d}$ hopelessly broad due to the
main decay mode $\widetilde{D}_{u,d}\rightarrow D+\pi$, so this seemed at
first to be an uninteresting study. However, we realized that the
$\overline{c}s$ state, the $\widetilde{D}_{s}$, would decay as
$\widetilde{D}_{s}\rightarrow D_{u,d}+K$. If we do the arithmetic with the
estimated $\Delta M\sim 338$ MeV we see the decay mode is kinematically
blocked, owing to the large mass of the K meson. Hence the chiral partner of
the $\widetilde{D}_{s}$, would be a very narrow resonance and potentially
observable!

Ten years after our original study the $\widetilde{D}_{s}(2317)$ was
discovered in the BABAR experiment BABAR  and the mass gap is measured to be
$\Delta M\sim 349$ MeV, which agrees with our NJL estimate to a few percent.
We then joined up with Estia Eichten, BEH , and showed that this is a generic
phenomenon and applies to the $(0^{+},1^{+})$ states, $\overline{b}s$ and
$\overline{c}s$, and the $\small(\frac{1}{2}^{-},\frac{3}{2}^{-})$ baryons,
$[cc]_{J=1}s$, $[bb]_{J=1}s$, $[cb]_{J=1}s$, and $[cb]_{J=0}s$.

So, if you count them all, we have theoretically discovered twelve new long-
lived resonances, all precisely predicted and observable in experiments.
Remarkably, these provide a unique view of chiral dynamics of single light
quarks, and all of this followed by using the Nambu–Jona-Lasinio model as a
guide.

In the present note I review the chiral dynamics of QCD through the NJL model,
including the heavy-light phenomena.

I then go further to discuss the possibility of composite Higgs bosons. To me,
ascertaining whether the known Higgs boson is point-like or not is perhaps the
most important thing the LHC (or future colliders, such as an $e^{+}e^{-}$ LC
or a muon collider) can pursue.

I will also discuss a new effort to extend the NJL model beyond the point-like
limit and to formulate the non-point-like internal dynamics of the bound
states it approximates. The NJL model serves as an anchor to the extended
dynamics, and I will show a surprise emerges here as well. I think this may be
an important venue for future research.

I conclude with a personal recollection, a story of lunch with Nambu in
Hiroshima.

##  II  Nambu–Jona-Lasinio Model

The Nambu–Jona-Lasinio model (NJL) NJL  is the simplest field theory of a
composite scalar boson, consisting of chiral fermions. An effective point-like
bound state emerges from an assumed point-like 4-fermion interaction. We begin
with a lightning review of the modern solution of the NJL model using concepts
of the renormalization group.

We assume chiral fermions, each with $N_{c}$ “colors” labeled by $(a,b,...)$.
A non-confining, chiral invariant $U(1)_{L}\times U(1)_{R}$ action, then takes
the form:

$\displaystyle
S_{NJL}=\\!\int\\!d^{4}x\;\biggl{(}i\overline{\psi}^{a}_{L}(x)\partial\\!\\!\\!/\\!\,\,\psi_{aL}(x)+i\overline{\psi}^{a}_{R}(x)\partial\\!\\!\\!/\\!\,\,\psi_{aR}(x)$
$\displaystyle\qquad\qquad+\;\frac{g^{2}}{M_{0}^{2}}\overline{\psi}^{a}_{L}(x)\psi_{aR}(x)\;\overline{\psi}^{b}_{R}(x)\psi_{bL}(x)\biggr{)}.$
(1)

This can be readily generalized to a $G_{L}\times G_{R}$ chiral symmetry as in
Section III below.

We can rewrite eq.(II) in an equivalent form by introducing the local
auxiliary field $\Phi(x)$:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{NJL}=\\!\int\\!d^{4}x\;\biggl{(}i\overline{\psi}^{a}_{L}(x)\partial\\!\\!\\!/\\!\,\,\psi_{aL}(x)+i\overline{\psi}^{a}_{R}(x)\partial\\!\\!\\!/\\!\,\,\psi_{aR}(x)$
$\displaystyle-
M_{0}^{2}\Phi^{\dagger}(x)\Phi(x)+g\overline{\psi}^{a}_{L}(x)\psi_{aR}(x)\Phi(x)+h.c.\biggr{)}.$
(2)

Using the $\Phi$ “equation of motion” in eq.(II) reproduces the 4-fermion
interaction of eq.(II).

Figure 1:  Diagrams contributing to the point-like NJL model effective
Lagrangian, eqs.(II,II). External lines are $\Phi$ and internal lines are
fermions $\psi$.

Following Wilson, Wilson , we view eq.(II) as the action defined at the high
energy (short-distance) scale $\mu\sim M$. We then integrate out the fermions
to obtain the effective action for the composite field $\Phi$ at a lower scale
$\mu<\\!\\!<M$. The calculation in the large-$N_{c}$ limit is discussed in
detail in BHL ; CTH . The leading $N_{c}$ fermion loop yields the result for
the $\Phi$ terms in the action at a new scale $\mu$:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{\mu}=\\!\int\\!d^{4}x\;\biggl{(}i\overline{\psi}^{a}_{L}\partial\\!\\!\\!/\\!\,\,\psi_{aL}+i\overline{\psi}^{a}_{R}\partial\\!\\!\\!/\\!\,\,\psi_{aR}+Z\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-m^{2}\Phi^{\dagger}\Phi-\frac{\lambda}{2}(\Phi^{\dagger}\Phi)^{2}+g\overline{\psi}^{a}_{L}\psi_{aR}\Phi(x)+h.c.\biggr{)}.$
(3)

where the diagrams of Fig.(1) yield,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!m^{2}=M_{0}^{2}\\!-\\!\frac{N_{c}g^{2}}{8\pi^{2}}(M_{0}^{2}-\mu^{2}),$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!Z=\frac{N_{c}g^{2}}{8\pi^{2}}\ln(M_{0}/\mu),\;\;\;\lambda=\frac{N_{c}g^{4}}{4\pi^{2}}\ln(M_{0}/\mu).$
(4)

Here $M_{0}^{2}$ is the UV loop momentum cut-off, and we include the induced
kinetic and quartic interaction terms. The one-loop result can be improved by
using the full renormalization group (RG) BHL ; CTH . Hence the NJL model is
driven by fermion loops, which are $\propto\hbar$ intrinsically quantum
effects.

Note the behavior of the composite scalar boson mass, $m^{2}$, of eq.(II) in
the UV. The $-{N_{c}g^{2}M_{0}^{2}}/{8\pi^{2}}$ term arises from the negative
quadratic divergence in the loop involving the pair
$\left({\psi}_{R},\psi_{L}\right)$ of Fig.(1), with UV cut-off $M_{0}^{2}$.
Therefore, the NJL model has a critical value of its coupling defined by the
cancellation of the large $M_{0}^{2}$ terms,

$\displaystyle
g_{c}^{2}=\frac{8\pi^{2}}{N_{c}}+{\cal{O}}\biggl{(}\\!\frac{\mu^{2}}{M_{0}^{2}}\\!\biggr{)}.$
(5)

Note that $\mu$ is the running RG mass, and comes from the lower limit of the
loop integrals and can in principle be small, $\mu^{2}<\\!\\!<m^{2}$.

We can renormalize, $\Phi\rightarrow\sqrt{Z}^{-1}\Phi$, hence:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{\mu}=\\!\int\\!d^{4}x\;\biggl{(}i\overline{\psi}^{a}_{L}\partial\\!\\!\\!/\\!\,\,\psi_{aL}+i\overline{\psi}^{a}_{R}\partial\\!\\!\\!/\\!\,\,\psi_{aR}+\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-m_{r}^{2}\Phi^{\dagger}\Phi-\frac{\lambda_{r}}{2}(\Phi^{\dagger}\Phi)^{2}+g_{r}\overline{\psi}^{a}_{L}\psi_{aR}\Phi(x)+h.c.\biggr{)}.$
(6)

where,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!m_{r}^{2}=\frac{1}{Z}\biggl{(}M_{0}^{2}\\!-\\!\frac{N_{c}g^{2}}{8\pi^{2}}(M_{0}^{2}-\mu^{2})\biggr{)}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!g^{2}_{r}\\!=\\!\frac{g^{2}}{Z}\\!=\\!\frac{8\pi^{2}}{N_{c}\ln(M_{0}/\mu)},\;\;\lambda_{r}\\!=\\!\frac{\lambda}{Z^{2}}\\!=\\!\frac{16\pi^{2}}{N_{c}\ln(M_{0}/\mu)}.$
(7)

For super-critical coupling, $g^{2}>g^{2}_{c}$, we see that $m_{r}^{2}<0$ and
there will be a vacuum instability. The effective action, with the induced
quartic $\sim\lambda_{r}(\Phi^{\dagger}\Phi)^{2}$ term, is then the usual
sombrero potential. The chiral symmetry is spontaneously broken and the field
$\Phi$ acquires a VEV,

$\displaystyle\langle\Phi\rangle=v=\frac{m_{r}}{\sqrt{\lambda_{r}}},$ (8)

and the chiral fermions acquire mass,

$\displaystyle m_{f}=g_{r}v$ (9)

The physical radial mode (“Higgs” boson), defined as $\sqrt{2}Re(\Phi)$, has a
mass $m_{h}$ given by,

$\displaystyle m_{h}^{2}=2\lambda v^{2}.$ (10)

while the Nambu-Goldstone mode, $Im(\Phi)$, is massless. Hence we see that the
NJL model yields a prediction for the radial mode,

$\displaystyle m_{h}=\frac{\sqrt{2\lambda_{r}}}{g_{r}}m_{f}=2m_{f}.$ (11)

The predicted mass, $m_{h}=2m_{f}$, may seem like a free field relation, but
in fact the binding involves the fermion loops that extend from $\mu$ to
$M_{0}$, so the composite Higgs boson is tightly bound and effectively point-
like up to the scale $M_{0}$. The Nambu-Goldstone bosons also emerge as point-
like bound states.

Fine-tuning of $g^{2}\rightarrow g_{c}^{2}$ is possible if we want a theory
with a hierarchy, $|m_{r}^{2}|<\\!\\!<M_{0}^{2}$. In that case we can appeal
to the behavior of the renormalized couplings as $\mu\rightarrow M_{0}$. We
see then that both $g_{r}$ and $\lambda_{r}$ diverge in the ratio
$\lambda_{r}/g^{2}_{r}\rightarrow 2$. This can be used as a boundary condition
on the full RG evolution of $g_{r}$ and $\lambda_{r}$ including gauge fields
and scalar interactions themselves.

When applied to a Higgs boson composed of top and anti-top this yields
$m_{r}\sim m_{top}\sim 220$ GeV (the quasi-infrared fixed point PR FP ), and
$m_{Higgs}\sim 260$ GeV BHL . While the top mass prediction isn’t too far off
(about $20$% too high), the Higgs mass is significantly larger than the
observed $125$ GeV. It’s possible to get the top mass concordant with the
observed value of $175$ GeV by extending the model to include more Higgs
doublets, but the Higgs mass is difficult to accommodate in the NJL model
approximation. More promising may be a seesaw with a second sequential
composite Higgs doublet composed of the b-quark (see Section IV.)

##  III Application to QCD Chiral Dynamics

###  III.1 Low mass states

The NJL model can be used to understand the chiral dynamics of QCD BHHQ
Bijnens Bardeen:1993ae . The effective Lagrangian in the light quark sector is
approximated by:

${\cal{L}}=\overline{\psi}(i\partial\\!\\!\\!/\\!\,\,-m_{q})\psi-\frac{g^{2}}{\Lambda^{2}}\overline{\psi}\gamma_{\mu}\frac{\lambda^{A}}{2}\psi\;\overline{\psi}\gamma^{\mu}\frac{\lambda^{A}}{2}\psi$
(12)

where $\psi$ is an $N_{c}=3$ color triplet and $\lambda^{A}$ are $SU(3)$ color
matrices.

The interaction in eq.(12) can be viewed as a single gluon exchange potential,
where we assume a fake “gluon mass” $\Lambda/\sqrt{2}$, where $\Lambda\sim 1$
GeV. We integrate out the massive gluon and truncate on dim $=6$ operators,
yielding the effective Lagrangian at $\mu^{2}\sim\Lambda^{2}$. For
concreteness we will take a flavor isodoublet, $\psi=(u,d)$, and in the limit
that the quark mass matrix $m_{q}\rightarrow 0$, we have an exact chiral
$SU(2)\times SU(2)\times U(1)$ invariant Lagrangian.

Upon Fierz–rearrangement of the interaction Lagrangian, keeping only leading
terms in $1/N_{c}$, eq.(12) takes the form:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal{L}}_{L}=\overline{\psi}(i\partial\\!\\!\\!/\\!\,\,-m_{q})\psi$
$\displaystyle+\frac{g^{2}}{\Lambda^{2}}\left(\overline{\psi}_{L}\psi_{R}\overline{\psi}_{R}\psi_{L}+\overline{\psi}_{L}\tau^{A}\psi_{R}\overline{\psi}_{R}\tau^{A}\psi_{L}\right)$
(13)

where $\psi_{L}=(1-\gamma_{5})\psi/2$, $\psi_{R}=(1+\gamma_{5})\psi/2$ and
$\tau^{A}$ are isospin matrices acting. Here we have truncated eq.(III.1) on
the pure NJL terms which represent the most attractive channels.222Here we
have dropped the the (vector)2 and (axial-vector)2 terms which are also
generated, but play no significant role in the chiral dynamics. These are
associated with the formation of virtual $\rho$ and $A_{1}$ vector mesons in
the model.

We can solve the light–quark dynamics in large–$N_{c}$ by introducing the
local auxiliary field $\Sigma(x)$, as we did with $\Phi(x)$ in eq.(II), thus
yielding an equivalent effective Lagrangian of the form:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal{L}}=\overline{\psi}(i\partial\\!\\!\\!/\\!\,\,-m_{q})\psi-g\overline{\psi}_{L}\Sigma\psi_{R}-g\overline{\psi}_{R}\Sigma^{\dagger}\psi_{L}$
$\displaystyle-\frac{1}{2}\Lambda^{2}\mathop{\rm Tr}(\Sigma^{\dagger}\Sigma)$
(14)

where:

$\displaystyle\Sigma=\frac{1}{2}\sigma I_{2}+i\pi^{a}\frac{\tau^{a}}{2}$ (15)

$\Sigma$ is a $2\times 2$ complex field, i.e., both $\sigma$ and $\pi^{a}$ are
complex.333Otherwise, with $\sigma$ and $\pi$ real there would be unwanted
contributions from
$\langle{T\Sigma\;\Sigma}\rangle=\langle{T\Sigma^{\dagger}\;\Sigma^{\dagger}}\rangle\neq
0$ in integrating out $\Sigma$. Hence, there is parity doubling at this stage,
and note that $Im(\sigma)$ is the $\eta$ meson in the $SU(2)\times SU(2)\times
U(1)$ model (it would become the $\eta^{\prime}$ in an $SU(3)\times
SU(3)\times U(1)$ generalization).

We now integrate out the fermion fields on scales $\mu^{2}<\Lambda^{2}$,
keeping only the leading large–$N_{c}$ fermion loop contributions of Fig.(1).
We use the massless fermion propagator, treating $\Sigma$ as a classical
background field. Thus we arrive at an effective field theory at the scale
$\mu$:

$\displaystyle{\cal{L}}$ $\displaystyle=$
$\displaystyle\overline{\psi}(i\partial\\!\\!\\!/\\!\,\,-m_{q})\psi-g\overline{\psi}_{L}\Sigma\psi_{R}-g\overline{\psi}_{R}\Sigma^{\dagger}\psi_{L}$
(16) $\displaystyle+Z_{2}\mathop{\rm
Tr}(\partial_{\mu}\Sigma^{\dagger}\partial^{\mu}\Sigma)-V(\Sigma)$

where:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!Z_{2}=\frac{g^{2}N}{16\pi^{2}}\ln(\Lambda^{2}/\mu^{2}),\;\;\;\;\lambda=\frac{g^{4}N}{16\pi^{2}}\ln(\Lambda^{2}/\mu^{2}),$
(17)

and,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!V(\Sigma)=\biggl{(}\frac{1}{2}\Lambda^{2}-\frac{g^{2}N}{8\pi^{2}}(\Lambda^{2}-\mu^{2})\biggr{)}\mathop{\rm
Tr}(\Sigma^{\dagger}\Sigma)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\frac{gN}{8\pi^{2}}(\Lambda^{2}-\mu^{2})\mathop{\rm
Tr}(m_{q}\Sigma+h.c.)+\lambda\mathop{\rm
Tr}(\Sigma^{\dagger}\Sigma\Sigma^{\dagger}\Sigma).$ (18)

We see that $Z_{2}\rightarrow 0$ as $\mu\rightarrow\Lambda$, reflecting the
compositeness of the $\Sigma$ field. Let us now renormalize the $\Sigma$
field:

$\Sigma\rightarrow(Z_{2})^{-1/2}\Sigma,$ (19)

and we have the effective Lagrangian at the scale $\mu$:

$\displaystyle{\cal{L}}=\overline{\psi}(i\partial\\!\\!\\!/\\!\,\,-m_{q})\psi-\widetilde{g}\overline{\psi}_{L}\Sigma\psi_{R}-\widetilde{g}\overline{\psi}_{R}\Sigma^{\dagger}\psi_{L}$
$\displaystyle\qquad+\mathop{\rm
Tr}(\partial_{\mu}\Sigma^{\dagger}\partial^{\mu}\Sigma)-\widetilde{V}(\Sigma),$
(20)

where:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\widetilde{g}\equiv
1/\sqrt{Z_{2}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!m^{2}_{\sigma}=\left(\frac{1}{Z_{2}}\right)\left[\frac{1}{2}\Lambda^{2}-\frac{g^{2}N_{c}}{8\pi^{2}}(\Lambda^{2}-\mu^{2})\right]$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\omega=\frac{\widetilde{g}gN_{c}}{8\pi^{2}}(\Lambda^{2}-\mu^{2}),\;\;\;\widetilde{\lambda}=\widetilde{g}^{2}=\frac{16\pi^{2}}{N_{c}\ln(\Lambda^{2}/\mu^{2})},$
(21)

and,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\widetilde{V}(\Sigma)$
$\displaystyle=$ $\displaystyle m_{\sigma}^{2}\mathop{\rm
Tr}(\Sigma^{\dagger}\Sigma)-\omega\mathop{\rm Tr}(m_{q}\Sigma+h.c.)$ (22)
$\displaystyle+\widetilde{\lambda}\mathop{\rm
Tr}(\Sigma^{\dagger}\Sigma\Sigma^{\dagger}\Sigma).$

The effective Lagrangian is seen to be a linear $\sigma$–model at scales
$\mu<\Lambda$. The theory will develop a chiral instability, hence a
constituent quark mass, provided that $m_{\sigma}^{2}$ becomes tachyonic
(negative) at some scale $\mu_{0}$.

By the choice of the bare coupling constant, $g^{2}$, we can put the model in
a symmetric phase, $m_{\sigma}^{2}>0$ where ${g^{2}N_{c}}/{4\pi^{2}}<1$, or in
a chiral symmetry breaking phase: $m_{\sigma}^{2}<0$ where
${g^{2}N_{c}}/{4\pi^{2}}>1$, where the critical bare coupling corresponds to
$m_{\sigma}^{2}=0$ as $\mu_{0}\rightarrow 0$.

In the broken phase (ignoring $m_{q}$) the $\sigma$ field develops a vacuum
expectation value,
$\langle{\sigma}\rangle=f_{\pi}=\sqrt{2}|m_{\sigma}|/\sqrt{\lambda}$, given
by:

$\langle\sigma\rangle=\frac{Z_{2}\Lambda^{2}}{\lambda}\left(\frac{g^{2}N_{c}}{4\pi^{2}}-1\right)=\left(\frac{\Lambda^{2}}{g^{2}}\right)\left(\frac{g^{2}N_{c}}{4\pi^{2}}-1\right)$
(23)

In the broken phase we can then write $\sigma=f_{\pi}+\hat{\sigma}$, and the
physical mass2 of the $\hat{\sigma}$ is readily seen to be
$m_{\hat{\sigma}}^{2}=2|m^{2}_{\sigma}|$, while the fermion mass becomes
$m_{0}=\frac{1}{2}f_{\pi}\widetilde{g}$. Thus, using
$\widetilde{g}^{2}=\lambda$, we obtain the usual Nambu–Jona–Lasinio result:
$m_{\hat{\sigma}}=2m_{0}$.

Note that the induced interaction term $\sim\mathop{\rm
Tr}(\Sigma^{\dagger}\Sigma\Sigma^{\dagger}\Sigma)$, has lifted the degeneracy
of the $Re(\pi)$ and $Im(\pi)$ fields. The $Im(\pi)$ states have become heavy
resonances and the $Re(\pi)$ states will emerge as Nambu-Goldstone bosons
dynamically at low energies.444We must go beyond the NJL model and add a
nonperturbative $e^{i\theta}\det(\Sigma)+h.c.$ term to lift the low mass
$Im(\sigma)$ mode in an $SU(3)\times SU(3)$ theory, which is the heavy
$\eta^{\prime}$. The solution to the theory can then be written as a chiral
quark model in which we have both constituent quarks described by $\psi$ and
the mesons described by $\Sigma$. In the broken phase it is useful to pass to
a nonlinear $\sigma$–model and write:

$\Sigma\rightarrow\frac{1}{2}f_{\pi}\exp(i\pi^{a}\tau^{a}/f_{\pi}),$ (24)

and:

$\displaystyle{\cal{L}}=\overline{\psi}(i\partial\\!\\!\\!/\\!\,\,-m_{q})\psi-
m_{0}\overline{\psi}_{L}\exp(i\pi^{a}\tau^{a}/f_{\pi})\psi_{R}$
$\displaystyle-
m_{0}\overline{\psi}_{R}\exp(-i\pi^{a}\tau^{a}/f_{\pi})\psi_{L}$
$\displaystyle+\mathop{\rm
Tr}(\partial_{\mu}\Sigma^{\dagger}\partial^{\mu}\Sigma)+\omega\mathop{\rm
Tr}(m_{q}\Sigma+h.c.),$ (25)

where $m_{0}=\frac{1}{2}\widetilde{g}f_{\pi}$ is the constituent quark mass.
Note, in our present normalization conventions that $f_{\pi}=93$ MeV. In the
broken phase we can replace $\sigma=f_{\pi}$ and
$\Sigma\rightarrow\frac{1}{2}f_{\pi}\exp(i\pi^{a}\tau^{a}/f_{\pi})$. We can
often replace ${g}\widetilde{\sigma}/2={g}\sigma/2+m_{q}\sqrt{Z_{2}}$ since it
easy to restore the explicit chiral symmetry breaking quark mass terms.

By a chiral redefinition of the fields, $\psi_{R}\rightarrow\xi\psi_{R}$ and
$\psi_{L}\rightarrow\xi^{\dagger}\psi_{L}$, where $\Sigma\equiv\xi^{2}$, we
arrive at the Georgi–Manohar Lagrangian (Manohar , e.g., their eq.(2.9)).

###  III.2  Heavy-Light Mesons

Heavy-light (HL) systems involving a single valence light–quark, and a heavy-
quark (or a pair of heavy quarks). These systems can be viewed as a
constituent–quark “tethered” to the inertially fixed heavy quark. In QCD the
light–quark chiral symmetry is spontaneously broken as described via chiral
Lagrangians with nearly massless pions.

The full dynamics can be treated in an NJL model that contains both heavy
quarks with heavy quark symmetry HQ Georgi1 Wise , together with light quark
chiral symmetry as in the previous section, and the four fermion interaction
inspired by QCD BHHQ . We solve the theory by factorizing the heavy–light
interaction into auxiliary background fields with Yukawa couplings to heavy
and light quark vertices. Upon integrating out the quarks on scales $\Lambda$
to $\mu$, the auxiliary fields acquire induced kinetic terms and thus become
dynamical heavy–light mesons, in parallel to the previous examples.

An essential feature of the dynamical chiral symmetry breaking is a
mass–splitting in heavy-light fields, between normal parity $(0^{-},1^{-})$
ground states and abnormal parity, $(0^{+},1^{+})$, partner states. Heavy
quark symmetry implies that spin-0 and spin-1 states of a given parity are
approximately degenerate, and thus form multiplets. The key idea presently is
that a $|s^{+}\rangle+|s^{-}\rangle$ is a “right-handed” chiral state,
transforming as $(0,3)$ under $SU(3)_{R}$, while $|s^{+}\rangle-|s^{-}\rangle$
is “left-handed,” $(3,0)$ under $SU(3)_{L}$, where $s=(0,1)$. The mass
splitting is then caused by the light quark $(\overline{3},3)$ condensate
$\langle\sigma\rangle\sim f_{\pi}$, discussed in section III.A.

This leads to broad resonances for the heavier $(0^{+},1^{+})$ states in
$c\overline{u}$ and $c\overline{d}$ mesons as they transition to the lighter
$(0^{-},1^{-})$ groundstates, emitting a pion. But in the case of heavy-
strange systems, the heavier $c\overline{s}$ $(0^{+},1^{+})$ states would have
a principle decay mode to the lighter $(0^{-},1^{-})$ non-strange
groundstates, $c\overline{s}\rightarrow c(\overline{u},\overline{d})+K$. This
is kinematically blocked by the large Kaon mass, hence the $c\overline{s}$,
$(0^{+},1^{+})$ states becomes a set of narrow, observable resonances. This
prediction was subsequently confirmed by the BABAR experiment BABAR , and the
predicted mass splitting from the NJL model agrees to within a few percent
with the experimental observation. Taken together this predicts four narrow
$(0^{+},1^{+})$ heavy-strange resonances in the $cs$, $bs$. It also applies to
the eight heavy-heavy-strange $(\frac{1}{2}^{-},\frac{3}{2}^{-})$ baryons555 A
heavy $QQ$ pair, which has color $3\times 3=\overline{3}$, behaves like a
single heavy $\overline{Q}$., and the
$\small(\frac{1}{2}^{-},\frac{3}{2}^{-})$ “heavy-heavy-strange” baryons,
$[cc]_{J=1}s$, $[bb]_{J=1}s$, $[cb]_{J=1}s$, and $[cb]_{J=0}s$ BEH .

A heavy–light fermion sector can be written as an NJL model extending the
interaction of eq.(12) to include a heavy quark $Q$:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal{L}}_{HL}=\overline{Q}(i\partial\\!\\!\\!/\\!\,\,-M)Q-\frac{2g^{2}}{\Lambda^{2}}\overline{Q}\gamma_{\mu}\frac{\lambda^{A}}{2}Q\;\overline{\psi}\gamma^{\mu}\frac{\lambda^{A}}{2}\psi.$
(26)

Here we may generally take $Q=(b,c..)$, $M$ to be the heavy quark mass, and
$g$ the effective coupling at the scale $\Lambda$. Upon Fierz–rearrangement of
the interaction, (again keeping only leading terms in $1/N_{c}$ and writing in
terms of color singlet densities and isospin indices, $i$), eq.(26) takes the
form:

$\displaystyle{\cal{L}}_{HL}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\overline{Q}(i\partial\\!\\!\\!/\\!\,\,-M)Q+\frac{g^{2}}{\Lambda^{2}}\left(\overline{Q}\psi_{i}\overline{\psi}^{i}Q-\overline{Q}\gamma^{5}\psi_{i}\overline{\psi}^{i}\gamma^{5}Q\;\right.$
(27)
$\displaystyle\left.-\frac{1}{2}\overline{Q}\gamma_{\mu}\psi_{i}\overline{\psi}^{i}\gamma^{\mu}Q-\frac{1}{2}\overline{Q}\gamma_{\mu}\gamma_{5}\psi_{i}\overline{\psi}^{i}\gamma_{\mu}\gamma_{5}Q\right)$

Following Georgi, we introduce a projection operator on the heavy quark field
with a well defined four–velocity $v_{\mu}$, Georgi1

$\displaystyle P^{\pm}=\frac{1\pm v\\!\\!\\!/\\!\,\,}{2}$ $\displaystyle
Q_{v}=P^{+}\exp(-iMv\cdot x)Q(x)$ (28)

The HQ kinetic term then takes the form:

$\overline{Q}_{v}iv^{\mu}{\partial}_{\mu}Q_{v}$ (29)

and eq.(27) takes the form:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\;\;{\cal{L}}_{HL}=\overline{Q}_{v}iv^{\mu}{\partial}_{\mu}Q_{v}+\frac{g^{2}}{2\Lambda^{2}}\biggl{(}\overline{Q}_{v}\psi_{i}\overline{\psi}^{i}Q_{v}-\overline{Q}_{v}\gamma^{5}\psi_{i}\overline{\psi}^{i}\gamma^{5}Q_{v}\;$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\;\;-\overline{Q}_{v}\gamma_{\mu}P^{-}\\!\psi_{i}\overline{\psi}^{i}P^{-}\\!\gamma^{\mu}Q_{v}+\overline{Q}_{v}\gamma_{\mu}P^{-}\\!\gamma_{5}\psi_{i}\overline{\psi}^{i}\gamma_{5}P^{-}\\!\gamma_{\mu}Q_{v}\biggr{)}$
(30)

The interaction can be factorized by introducing heavy auxiliary fields,
$(B,B^{\prime})$ which represent heavy-light bound states of four velocity
$v_{\mu}$. To do so we must introduce four independent fields, $B\;(B^{5})$
are $0^{+}$ ($0^{-}$) scalars, while $B_{\mu}\;(B_{\mu}^{5})$ are $1^{-}$
($1^{+}$) vectors. We assemble the auxiliary fields into complex $\bf 4$
multiplets under $O(4)=SU(2)_{h}\times SU(2)_{l}$, where $SU(2)_{h}$
($SU(2)_{l}$) is the little group of rotations which preserves $v_{\mu}$. One
heavy spin 4 multiplet consists of the $0^{+}$ scalar together with the
abnormal parity ($1^{+}$) vector as $(B,B^{5\mu})$ (isospin indices are
understood):

$\displaystyle{\cal{B}}^{\prime}=(i\gamma^{5}B+\gamma_{\mu}B^{5\mu})P^{+}$
(31)

The other $\bf 4$ multiplet consists of the usual $0^{-}$ scalar and a $1^{-}$
vector $(B^{5},B^{\mu})$:

$\displaystyle{\cal{B}}=(i\gamma^{5}B^{5}+\gamma_{\mu}B^{\mu})P^{+}$ (32)

Under heavy spin $O(4)=SU(2)_{h}\times SU(2)_{l}$ rotations the $(B,B^{5\mu})$
mix analogously to $(B^{5},B^{\mu})$. Note that $v_{\mu}B^{\mu}=0$ always.

The factorized heavy–light interaction Lagrangian then takes the compact form:

$\displaystyle{\cal{L}}_{HL}$ $\displaystyle=$
$\displaystyle\overline{Q}_{v}iv^{\mu}{\partial}_{\mu}Q_{v}+{g}\overline{Q}_{v}\;(-i\overline{\cal{B}^{\prime}}^{i}\gamma^{5}+\overline{\cal{B}}^{i})\;\psi_{i}+h.c.$
(33) $\displaystyle+\Lambda^{2}\left[\mathop{\rm
Tr}({\cal\overline{B}B})+\mathop{\rm
Tr}({\cal\overline{B}^{\prime}{B}^{\prime}})\right]$

Eq.(33) is exactly equivalent to the full four–fermion theory eq.(27) upon
integrating out the ${\cal{B}}$ fields.

For weak coupling, $g$, the linear chiral invariance is realized and the
theory must contain parity doubled meson states. The chiral representations of
the composite fields in the effective theory are the following combinations:

$\displaystyle{\cal{B}}_{1}=\frac{1}{\sqrt{2}}\left({\cal{B}}-i{\cal{B}}^{\prime}\right)\qquad{\cal{B}}_{2}=\frac{1}{\sqrt{2}}\left({\cal{B}}+i{\cal{B}}^{\prime}\right)$
(34)

The full effective Lagrangian for the heavy mesons is derived at loop level by
integrating out the heavy and light quarks, $Q$ and $\psi$, in Eq.(33). The
loop integrals involve the heavy quark propagator $\sim i/v\cdot\ell$ and run
over momentum scales $\mu<\ell<\Lambda$, and we keep the leading large-$N_{c}$
induced terms. The details of the explicit calculations are given in BHHQ .

With $\Sigma=\frac{1}{2}(\sigma+i\pi\cdot\tau)$, which emerges upon including
the the light quark sector (as discussed above in the section III.A),
performing a conventional wave–function renormalization and several field
redefinitions, we arrive at the full effective action valid to
O$(\mu/\Lambda)^{2}$ BHHQ :

$\displaystyle{\cal{L}}_{LH}$ $\displaystyle=$
$\displaystyle-i\frac{1}{2}\mathop{\rm
Tr}(\overline{{\cal{B}}}_{1}v\cdot\partial{\cal{B}}_{1})-i\frac{1}{2}\mathop{\rm
Tr}(\overline{{\cal{B}}}_{2}v\cdot\partial{\cal{B}}_{2})$ (35)
$\displaystyle-\frac{{g_{r}}}{2}\left[\mathop{\rm
Tr}(\overline{{\cal{B}}_{1}}\Sigma^{\dagger}{\cal{B}}_{2})+\mathop{\rm
Tr}(\overline{{\cal{B}}_{2}}\Sigma{\cal{B}}_{1})\right]$
$\displaystyle+\frac{ik_{r}}{2f_{\pi}}\left[\mathop{\rm
Tr}(\overline{{\cal{B}}_{1}}\gamma^{5}(\partial\\!\\!\\!/\\!\,\,\Sigma^{\dagger}){\cal{B}}_{2})-\mathop{\rm
Tr}(\overline{{\cal{B}}_{2}}\gamma^{5}(\partial\\!\\!\\!/\\!\,\,\Sigma){\cal{B}}_{1})\right]$
$\displaystyle+\left(\Delta+\frac{h_{r}}{\Lambda}\Sigma^{\dagger}\Sigma\right)\left[\mathop{\rm
Tr}(\overline{{\cal{B}}_{1}}{\cal{B}}_{1})+\mathop{\rm
Tr}(\overline{{\cal{B}}_{2}}{\cal{B}}_{2})\right]\ $

The parameters of this Lagrangian are determined by loop integrals as:

$\displaystyle g_{r}$ $\displaystyle=$
$\displaystyle\frac{g}{\sqrt{Z_{2}}};\qquad
h_{r}=\frac{2g^{2}\sqrt{Z_{2}}\Lambda}{Z_{1}};\qquad
k_{r}=\frac{2gf_{\pi}\sqrt{Z_{2}}}{Z_{1}}$ $\displaystyle\Delta$
$\displaystyle=$
$\displaystyle\frac{1}{Z_{1}}\left(\Lambda^{2}-Z_{1}(\Lambda+\mu)/2\pi\right),$
(36)

where:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!Z_{1}$ $\displaystyle=$
$\displaystyle\frac{g^{2}N_{c}}{8\pi}(\Lambda-\mu);\qquad
Z_{2}=\frac{g^{2}N_{c}}{16\pi^{2}}\left[\ln(\Lambda^{2}/\mu^{2})\right].$ (37)

Note $Z_{1}$ is linear in $\Lambda$ and $\mu$. The parameters $g_{r}$, $h_{r}$
and $k_{r}$ are dimensionless, and are determined by fitting the observables
of the model. We view $\mu$ to be of order the light quark constituent mass,
and henceforth will be neglected in the expression for $Z_{1}$. Note that
terms like
$\overline{\cal{B}^{\prime}}\gamma^{5}(\Sigma^{\dagger}\partial\\!\\!\\!/\\!\,\,\Sigma){\cal{B}}$
are potentially induced, but they are subleading as $\sim
O(1/\ln(\Lambda/\mu)$, relative to the terms we keep.

Eq.(35)) is manifestly invariant under $SU(2)_{L}\times SU(2)_{R}$ where the
fields transform as:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal{B}}_{1}=\biggl{(}0,\frac{1}{2}\biggr{)}\qquad{\cal{B}}_{2}=\biggl{(}\frac{1}{2},0\biggr{)}\qquad\Sigma=\biggl{(}\frac{1}{2},\frac{1}{2}\biggr{)}$
(38)

In the spontaneously broken chiral symmetry phase we can pass to the the
nonlinear realization,

$\Sigma=\frac{1}{2}f_{\pi}\exp(i\pi\cdot\tau/f_{\pi})\qquad\Sigma=\frac{1}{2}f_{\pi}\xi^{2}$
(39)

We can then pass to the “current form” by performing the chiral field
redefinitions:

$\displaystyle{\cal{B}}_{1}\rightarrow\xi^{\dagger}{\cal{B}}_{1}\qquad{\cal{B}}_{2}\rightarrow\xi{\cal{B}}_{2}$
(40)

where the chiral currents are BHHQ :

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal{J}}_{\mu,L}=i\xi\partial_{\mu}\xi^{\dagger},\qquad{\cal{J}}_{\mu,R}=i\xi^{\dagger}\partial_{\mu}\xi\qquad$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal{A}}_{\mu}=\frac{1}{2}({\cal{J}}_{\mu,R}-{\cal{J}}_{\mu,L}),\;\;\;\;{\cal{V}}_{\mu}=\frac{1}{2}({\cal{J}}_{\mu,R}+{\cal{J}}_{\mu,L}).$
(41)

As usual the ${\cal{J}}_{A}$ are matrices acting on the isospin indices of
meson fields. The mass matrix is diagonal in the current form by:

$\widetilde{\cal{B}}=\frac{1}{\sqrt{2}}({\cal{B}}_{1}+{\cal{B}}_{2}),\qquad\widetilde{\cal{B}}^{\prime}=\frac{i}{\sqrt{2}}({\cal{B}}_{1}-{\cal{B}}_{2}).$
(42)

The Lagrangian then becomes:

$\displaystyle{\cal{L}}_{LH}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\mathop{\rm
Tr}(\overline{\widetilde{\cal{B}}}v\cdot(i{\partial}+{{\cal{V}}}){\widetilde{\cal{B}}})-\frac{1}{2}\mathop{\rm
Tr}(\overline{\widetilde{\cal{B}}}^{\prime}v\cdot(i{\partial}+{{\cal{V}}}){\widetilde{\cal{B}}}^{\prime})$
(43) $\displaystyle+\left(\Delta_{r}-\frac{g_{r}f_{\pi}}{4}\right)\mathop{\rm
Tr}\overline{\widetilde{\cal{B}}}\widetilde{\cal{B}}+\left(\Delta_{r}+\frac{g_{r}f_{\pi}}{4}\right)\mathop{\rm
Tr}\overline{\widetilde{\cal{B}}}^{\prime}\widetilde{\cal{B}}^{\prime}$
$\displaystyle-\frac{1}{2}\mathop{\rm
Tr}(\overline{\widetilde{\cal{B}}}(v\cdot{{\cal{A}}}){\widetilde{\cal{B}}}^{\prime})-\frac{1}{2}\mathop{\rm
Tr}(\overline{\widetilde{\cal{B}}}^{\prime}(v\cdot{{\cal{A}}}){\widetilde{\cal{B}}})$
$\displaystyle-i\frac{k_{r}}{2}\mathop{\rm
Tr}\overline{\widetilde{\cal{B}}}\gamma^{5}{\cal{A}}\\!\\!\\!/\\!\,\,\widetilde{\cal{B}}+i\frac{k_{r}}{2}\mathop{\rm
Tr}\overline{\widetilde{\cal{B}}}^{\prime}\gamma^{5}{\cal{A}}\\!\\!\\!/\\!\,\,\widetilde{\cal{B}}^{\prime}$

where:

$\Delta_{r}=\left(\Delta+\frac{h_{r}}{4\Lambda}f_{\pi}^{2}\right)$ (44)

and the heavy meson mass eigenvalues are:

$\displaystyle
2\Delta_{r}-gf_{\pi}/2,\;\;\makebox{and}\;\;2\Delta_{r}+gf_{\pi}/2$ (45)

respectively (our normalization conventions imply the physical mass shift is
$\delta M$ if the Lagrangian contains $\frac{1}{2}\delta M(\mathop{\rm
tr}{\overline{\cal{B}}{\cal{B}}})$; see BHHQ ). The mass eigenfields are
nontrivial functionals of the pions through the absorbed $\xi$,
$\xi^{\dagger}$ factors.

Note the appearance of the off–diagonal pionic transition terms of the form
$\overline{\widetilde{\cal{B}}}(v\cdot{{\cal{A}}}){\widetilde{\cal{B}}}^{\prime}$.
At this stage it can be seen that these terms are associated with a
“Goldberger–Trieman relation,” by taking
${\cal{A}}_{\mu}=\partial_{\mu}\pi\cdot\tau/f_{\pi}$, integrating by parts,
and using the equations of motion. One finds that the
${\cal{B}}^{\prime}{\cal{B}}\pi$ amplitude has a coupling strength $g_{r}$,
and this is seen to be given by $\Delta M/f_{\pi}$.666 Note that we can
decouple the heavier field $\widetilde{\cal{B}}^{\prime}$ to leading order in
the mass gap $g_{r}f_{\pi}$ by “integrating it out” which amounts to setting
it to zero in leading order, equivalent to the point of departure taken by
Wise, Wise , justified so long as $q$ is small compared to the mass gap.

The central observation of this analysis, however, is that the underlying
chiral representations of the full HL meson theory is a parity doubled scheme.
The mass gap between the parity partners arises from $VEV{\sigma}$ and leads
to a splitting between the even and odd parity multiplets:

$\displaystyle\Delta M=g_{r}f_{\pi}$ (46)

Here $g_{r}$ is the $BB^{\prime}\pi$ transition coupling constant and is the
analogue of the $g_{NN\pi}$ in the nucleon system.

Note that to match the formalism of III.A we have written things in terms of
$SU(2)_{L}\times SU(2)_{R}$, but this readily applies to $SU(3)_{L}\times
SU(3)_{R}$ with the octet of Nambu-Goldstone bosons. The axial vector current
describes the main transitions between parity partners,
$(0^{+},1^{+})\rightarrow(0^{-},1^{-})+``\pi"$ where $``\pi"$ is the $SU(3)$
matching member of the octet, e.g.
$\overline{c}s\rightarrow\overline{c}u+K^{-}$, etc. The parity degeneracy of
the $(0^{+},1^{+})$ and $(0^{-},1^{-})$ states is lifted by
$\langle\sigma\rangle$.

The BABAR collaboration reported the observation of a narrow resonance in
$D^{+}_{s}\pi^{0}$ with a mass of $2317$ MeV BABAR . There was also a hint of
a second state (the $1^{+}$) in $D_{s}\pi^{0}\gamma$ with a mass $2460$ MeV.
The mass difference between the $D^{*}_{s}(2317)$ and the well established
lightest charm-strange meson, $D_{s}$, is $\Delta M=349$ MeV. With this mass
gap the decay $D_{s}^{*}(2317)\rightarrow D_{u,d}+K$ is kinematically
forbidden.

Both of these states are members of the heavy quark $(0^{+},1^{+})$ spin
multiplet. The multiplet is then established to a good approximation as the
chiral partner of the $(0^{-},1^{-})$ ground state predicted by QCD in the NJL
approximation discussed above. The observed magnitude of $\Delta M$ is in
remarkably good agreement with the NJL model estimate of ref.BHHQ

The suppressed decays of all of these states must proceeds through $SU(3)$
breaking effects, e.g., $D_{s}(2317)\rightarrow
D_{u,d}+(\eta\rightarrow\pi^{0})$, emitting a virtual $\eta$ that then mixes
with the $\pi^{0}$ through isospin violating effects. We have tabulated the
spectrum and all decay widths for all analogue processes inBEH . Moreover, we
show in BEH  that electromagnetic transitions are indeed, and somewhat
remarkably, suppressed, since they involve cancellations between the heavy and
light magnetic moments. The overall picture of the chiral structure of the
heavy-light systems works quite well. The $D_{s}(2317)$ is the first clear
observation of a general phenomenon that applies to the “heavy-strange”
$(0^{+},1^{+})$ mesons, $\overline{b}s$ and $\overline{c}s$, and the
$\small(\frac{1}{2}^{-},\frac{3}{2}^{-})$ “heavy-heavy-strange” baryons,
$[cc]_{J=1}s$, $[bb]_{J=1}s$, $[cb]_{J=1}s$, and $[cb]_{J=0}s$ BEH .

##  IV Nambu–Jona-Lasinio Inspired Composite Higgs Bosons

###  IV.1 Minimal Dynamical Symmetry Breaking of Electroweak Interactions

Prior to the top quark discovery, following suggestions of Nambu, Nambutc ,
and pioneering work of Miransky, Tanabashi and Yamawaki, Yama , a predictive
minimal theory of a composite Higgs boson was constructed by Bardeen, Hill and
Lindner BHL . This is based upon a Nambu-Jona-Lasinio model where the Higgs is
composed of $\overline{t}t$ (for a review see CTH NSD ).

We assume chiral fermions, each with $N_{c}=3$ labeled by $(a,...)$, where we
identify $\psi_{L}^{a}=(t,b)_{L}$ and $\psi_{R}=t_{R}$ A non-confining,
$SU(3)\times SU(2)_{L}\times U(1)_{R}$ invariant action, then takes the form:

$\displaystyle
S_{NJL}=\\!\int\\!d^{4}x\;\biggl{(}i\overline{\psi}^{ai}_{L}(x)\partial\\!\\!\\!/\\!\,\,\psi_{aiL}(x)+i\overline{\psi}^{a}_{R}(x)\partial\\!\\!\\!/\\!\,\,\psi_{aR}(x)$
$\displaystyle\qquad\qquad+\;\frac{g^{2}}{M_{0}^{2}}\overline{\psi}^{ai}_{L}(x)\psi_{aR}(x)\;\overline{\psi}^{b}_{R}(x)\psi_{biL}(x)\biggr{)}.$
(47)

where $i$ is an $SU(2)$ index, summed from 1 to 2. The interaction is
factorized by a isodoublet Higgs auxiliary field $H$, and loops are computed
theory in section II, yielding an unrenormalized action, with an induced
kinetic term for $H$. We renormalize, $H\rightarrow\sqrt{Z}^{-1}H$, hence:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{\mu}=\\!\int\\!d^{4}x\;\biggl{(}i\overline{\psi}^{a}_{L}\partial\\!\\!\\!/\\!\,\,\psi_{aL}+i\overline{\psi}^{a}_{R}\partial\\!\\!\\!/\\!\,\,\psi_{aR}+\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-m_{r}^{2}\Phi^{\dagger}\Phi-\frac{\lambda_{r}}{2}(\Phi^{\dagger}\Phi)^{2}+g_{r}\overline{\psi}^{a}_{L}\psi_{aR}\Phi(x)+h.c.\biggr{)}.$
(48)

where the renormalized parameters are similar to those in section II,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!m_{r}^{2}=\frac{1}{Z}\biggl{(}M_{0}^{2}\\!-\\!\frac{N_{c}g^{2}}{8\pi^{2}}(M_{0}^{2}-\mu^{2})\biggr{)}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!g^{2}_{r}\\!=\\!\frac{g^{2}}{Z}\\!=\\!\frac{8\pi^{2}}{N_{c}\ln(M_{0}/\mu)},\;\;\lambda_{r}\\!=\\!\frac{\lambda}{Z^{2}}\\!=\\!\frac{16\pi^{2}}{N_{c}\ln(M_{0}/\mu)}.$
(49)

We see that the $g_{r}^{2}$ and $\lambda_{r}$ each approach a Landau pole as
the RG scale $\mu$ approaches the composite scale $M_{0}$. The NJL theory thus
makes two predictions:

$\displaystyle(1)\qquad
g^{2}_{r}\rightarrow\infty\qquad\makebox{as}\qquad\mu\rightarrow M_{0}$
$\displaystyle(2)\qquad\frac{\lambda_{r}}{g_{r}^{2}}\rightarrow
2\qquad\makebox{as}\qquad\mu\rightarrow M_{0}$ (50)

To obtain a theory that is close to the observed top quark mass we must tune a
significant hierarchy in $M_{0}/m_{t}$, and the logs become large (this allows
us to neglect small additive constant corrections to the logs ). If the logs
are large, however, we must go beyond the fermion loop approximation of NJL
and include the full renormalization group evolution, assuming the standard
model. The remnant of the NJL dynamics is then the compositeness conditions,
which are boundary conditions on the full RG. It turns out that the solution
for the renormalized Higgs-Yukawa coupling of the minimal top condensation
theory is given by the RG “infrared quasi-fixed point.”

Indeed, the idea that the renormalization group would yield a large top quark
mass predates the composite $\overline{t}t$ Higgs boson scheme and was first
discussed by Pendleton and Ross PR  and Hill FP . The skeletal one-loop RG
equation for the top quark Higgs-Yukawa coupling, $g_{t}$, is:

$Dg_{t}=g_{t}\left(\left(N_{c}+\frac{3}{2}\right)g_{t}^{2}-\left(N_{c}^{2}-1\right)g_{3}^{2}\right)$
(51)

where $D=16\pi^{2}\partial/\partial\ln(\mu)$, $\mu$ is the running mass scale,
$g_{3}$ is the QCD coupling, and $N_{c}=3$ is the number of colors. For
illustrative purposes we suppress electroweak corrections. Pendleton and Ross
emphasized a “tracking solution” in which the ratio of $g_{t}/g_{3}$ is
constant. The tracking solution predicts a top mass of order $100$ GeV.

In FP  the quasi-IR fixed point was emphasized, in which the initial value of
$g_{t}$ in the extremely far UV is arbitrary, in fact very large, e.g. a
Landau pole. It was then noted that $g_{t}$ runs down to a relatively constant
fixed value determined by $g_{3}$, hence “quasi” fixed point. This leads to a
predicted top mass $\sim 220$ GeV, which depends softly upon the initial scale
$\Lambda$ as $\sim\ln(\ln(M_{0}/m_{t})$. The quartic coupling of the Higgs,
$\lambda(\mu)$, is also governed by a similar infrared quasi-fixed point (see
the detailed RG discussion in CTH ). The large value of $g_{t}$ at $M_{0}$
suggests a composite Higgs boson theory.

In Fig.(2), starting the running of $g_{t}(\mu)$ at very large mass scales,
$\mu=M_{X}$, with large initial values, i.e., $g_{t}(M_{X})>>1$ (effectively a
Landau pole at $M_{X}$), it is seen that $g_{t}(\mu)$ flows into an “infrared
quasi-fixed point.” This is “quasi” in the sense that, if the QCD coupling,
$g_{3}$, was a constant then $g_{t}$ would flow to an exact conformal fixed
point. Scale invariance is therefore broken by the QCD $\beta$-function, and
dictates a slow evolution of $g_{3}$, and $g_{t}$ locks into the quasi-fixed
point trajectory. The low energy prediction of the top quark Higgs-Yukawa (HY)
coupling from the infrared quasi-fixed point is very insensitive to its
precise, large initial values and mass scales.777This is often nowadays called
a “focus point,” but referred to as a “fixed point” is the traditional
nomenclature inherited from condensed matter physics. Since a focusing occurs
in the infrared, the top quark mass is determined to lie within a narrow range
of values.

The effective top mass $m_{top}=g_{t}(\mu)v$ (where $v=175$ GeV is the
electroweak scale) is plotted vs. renormalization scale $\mu$ (in the figure
the physical top mass corresponds $\log(\mu)\sim 2$). Assuming the initial
mass scale of the Landau pole (LP) is of order $10^{15}-10^{19}$ GeV, the
quasi-fixed point translates into values of the top quark mass that are heavy,
as seen in Table I. One obtains $m_{top}\sim 220$ GeV starting at the Planck
mass.

Figure 2: The top quark quasi–fixed point in the top mass $m_{top}=g_{t}v$
where $v=175$ GeV is the Higgs VEV, plotted vs the running scale $\mu$, The
focusing in the infrared and its relative insensitivity to initial values is
indicated.

Initially this prediction for an ultra-heavy top mass was greeted with some
derision. The favored top quark mass values in the early 1980’s ranged from
$\sim 15$ GeV to $\sim 26$ GeV and it was evidently considered absurd that an
elementary fermion in the standard model (SM) would have a mass comparable to
the weak scale. However, as the experimental and indirect lower limits for
$m_{top}$ climbed above $\sim 100$ GeV, the possibility of the ultra-heavy top
quark became a high probability.

Table 1: Top quark mass for initial Landau pole $M_{0}$. $M_{0}$ GeV | $10^{19}$ | $10^{15}$ | $10^{11}$ | $10^{7}$ | $10^{5}$
---|---|---|---|---|---
$m_{top}$ GeV | 220 | 230 | 250 | 290 | 360

Ultimately, the top quark was discovered in 1995 at Fermilab by CDF and D-Zero
CDF Dzero , with a mass of $m_{t}=174$ GeV. The top mass, though significantly
larger than the original expectations, is shy of the naive SM infrared quasi-
fixed point with $M_{0}=10^{19}$ GeV by about 20%. I view as success of the
infrared quasi-fixed point, given that the prediction assumed only SM physics
extending all the way up to Planck scale, however the whole story is yet to be
unraveled.

What would convincingly bring the top mass quasi-fixed point prediction into
concordance with experiment? It turns out that it isn’t easy to do this with
minor modifications of the SM. For example, one might consider embedding
$SU(3)\rightarrow SU(3)\times SU(3)...$, at intermediate scales in the RG
running, such as a perturbative version of “topcolor,” or “flavor universal
colorons,” or “latticized extra dimensions,” etc. Topcolor bdob1 Pokorski .
However, this generally causes the effective $g_{3}$ to become larger and the
quasi-fixed point of $g_{t}(m_{t})$ moves up and the discrepancy with
experiment gets larger.

Moreover, in the $\overline{t}t$ composite scheme the quartic Higgs coupling
is too high $\lambda\sim 1$ vs. the SM value, $\lambda\sim 0.25$, and leads to
an unacceptably large Higgs boson mass BHL . This may also be informing us of
required modifications of the renormalization group beyond the simple SM
inputs at higher energy scales to bring the result into concordance with
experiment.

###  IV.2 Scalar Democracy

We have considered a maximal scalar field extension of the SM HMTT  (and a
less drastic minimal version, HMTT2 ). This was largely motivated by
curiosity: How obstructive are the rare weak decay flavor constraints on a
rich spectrum of Higgs bosons? How many scalars might exist, given the fermion
composition of the SM, and what patterns might be suggested, etc.? Here we
embody the principle that every chiral fermion pair in the SM has a
corresponding scalar bound state of the same quantum numbers. This is
suggested by thinking along the lines of the NJL model.

We call this model “Scalar Democracy.” We count the vertex operators for every
chiral fermion pair in the SM and find that this leads to a vast spectrum of
color octet isodoublets and triplet leptoquarks, etc. We assume these exotic
new scalar bosons have ultra-large masses and do not affect low energy RG
running. The model contains eighteen Higgs doublets in the quark sector and
likewise in the lepton sector that will have masses extending up to $\sim
10^{5}$ TeV. These two subsectors resemble an $SU(6)_{L}\times SU(6)_{R}$
linear $\Sigma$-model Lagrangian Manohar  (as in Fig.(2)), where the
interaction is subcritical and ultimately only the SM Higgs condenses.

The theory has one universal HY coupling $g$ defined at the Planck scale,
which we view as the compositeness scale of all scalars. This universal $g$ is
renormalized as we flow into the infra-red and symmetry breaking effects lead
to slightly different values in the different subsectors of the theory. For
quarks this coupling near the weak scale is identified with the top quark,
$g_{t}=g$, while for leptons we obtain $g_{\ell}\sim 0.7g$. Hence, our theory
is calibrated by the known value of $g_{t}\approx 1$ HMTT . This buys us some
predictivity.

Figure 3:  $SU(6)_{L}\times SU(6)_{R}$ chiral Lagrangian structure of the
scalar democracy in the quark sector. $T=(t,b)$, $C=(c,s)$, $U=(u,d)$ and
$\Sigma_{xy}=(H_{x},H_{y}^{C})$. The quark Higgs subsector becomes a
subcritical $SU(6)_{L}\times SU(6)_{R}$ linear $\Sigma$-model Lagrangian,
where $\Sigma$ is a $6\times 6$ complex matrix that can be viewed as nine
$2\times 2$ complex $\Sigma$ fields where each $\Sigma_{xy}=(H_{x},H^{C}_{y})$
is a pair of Higgs doublets. There is one universal coupling $g$ which is
subject to RG effects.

Here the flavor physics and mass hierarchy problems are flipped out of the
Higgs-Yukawa coupling texture and into the mass matrix of the many Higgs
fields. The Higgs mass matrix is input as $d=2$ gauge invariant operators. We
have no theory of these, but we must choose the inputs to fit the quark and
lepton sector masses and CKM physics, as well as maintain consistency with
rare weak decays, etc. It is not obvious a priori that there exists a
consistent solution with the flavor constraints.

The RG equation for the universal HY coupling in the quark subsector now takes
the one-loop form HMTT :

$Dg_{t}=g_{t}\left((N_{c}+N_{f})g_{t}^{2}-(N_{c}^{2}-1)g_{3}^{2}\right)$ (52)

Note the enhanced coefficient of $g_{t}^{2}$ in eq.(52) relative to eq.(1)
where $N_{f}$ is the number of flavors, i.e., $N_{f}=6$ in the quark sector of
the model.

Figure 4: The top quark fixed point is shifted down in the $SU(6)\times SU(6)$ model and becomes concordant with the observed top mass, with residual corrections coming from sensitivity to the extended Higgs spectrum (see Table II).  Table 2: Top quark mass for initial Landau pole $M_{0}$. $M_{0}$ GeV | $10^{19}$ | $10^{15}$ | $10^{11}$ | $10^{7}$ | $10^{5}$
---|---|---|---|---|---
$m_{top}$ GeV | $156+\Delta$ | $162+\Delta$ | $177+\Delta$ | $205+\Delta$ | $254+\Delta$

The resulting RG evolution is shown in Fig.(4) and the top quark mass
predictions are given in Table II. There is residual sensitivity to the
decoupling scale of the heavy Higgs bosons, summarized by the parameter
$\Delta$, which reduces the effective $N_{f}$ below the Higgs decoupling scale
as discussed in HMTT HMTT2 . There we estimate
$\Delta\approx(2.8\;GeV)\ln(\langle M_{H}\rangle/10^{2})$ hence $\Delta\approx
19$ GeV for an average heavy doublet mass $M_{H}\approx 100$ TeV. Hence, the
top mass can be accommodated for $M_{X}\sim 10^{19}$ GeV, and is indirectly
probing the new physics of the additional heavy Higgs bosons.

Higgs field |  Fermion mass |  Case (1) [TeV] | Case (2) [TeV]
---|---|---|---
$H^{\prime}_{0}=v+\frac{h}{\sqrt{2}}$ |  $m_{t}=gv=175$ GeV |  $m_{H}=0.125$ | $m_{H}=0.125$
$H^{\prime}_{b}=v\frac{\mu^{2}}{M_{b}^{2}}+H_{b}$ |  $m_{b}=gv\frac{\mu^{2}}{M_{b}^{2}}=4.5$ GeV |  $M_{b}=3.5$ | $M_{b}=0.620$
$H^{\prime}_{\tau}=v\frac{\mu^{2}}{M_{\tau}^{2}}+H_{\tau}$ |  $m_{\tau}=g_{\ell}v\frac{\mu^{2}}{M_{\tau}^{2}}=1.8$ GeV |  $M_{\tau}=6.8$ | $M_{\tau}=0.825$
$H^{\prime}_{c}=v\frac{\mu^{2}}{M_{b}^{2}}+H_{c}$ |  $m_{c}=gv\frac{\mu^{2}}{M_{c}^{2}}=1.3$ GeV |  $M_{c}=13.5$ | $M_{c}=1.2$
$H^{\prime}_{\mu}=v\frac{\mu^{2}}{M_{\mu}^{2}}+H_{\mu}$ |  $m_{\mu}=g_{\ell}v\frac{\mu^{2}}{M_{\mu}^{2}}=106$ MeV |  $M_{\mu}=1.2\times 10^{2}$ | $M_{\mu}=3.4$
$H^{\prime}_{s}=v\frac{\mu^{2}}{M_{s}^{2}}+H_{s}$ |  $m_{s}=gv\frac{\mu^{2}}{M_{s}^{2}}=95$ MeV |  $M_{s}=1.8\times 10^{2}$ | $M_{s}=4.3$
$H^{\prime}_{d}=v\frac{\mu^{2}}{M_{d}^{2}}+H_{d}$ |  $m_{d}=gv\frac{\mu^{2}}{M_{d}^{2}}=4.8$ MeV |  $M_{d}=3.6\times 10^{3}$ | $M_{d}=19$
$H^{\prime}_{u}=v\frac{\mu^{2}}{M_{u}^{2}}+H_{u}$ |  $m_{u}=gv\frac{\mu^{2}}{M_{u}^{2}}=2.3$ MeV |  $M_{u}=7.6\times 10^{3}$ | $M_{u}=27$
$H^{\prime}_{e}=v\frac{\mu^{2}}{M_{e}^{2}}+H_{e}$ |  $m_{e}=g_{\ell}v\frac{\mu^{2}}{M_{e}^{2}}=0.5$ MeV |  $M_{e}=2.45\times 10^{4}$ | $M_{e}=49$
Table 3: The estimates for diagonal Higgs bosons masses in the limit of no CKM
mixing, and assuming (1) the level-repulsion feedback on the Higgs mass term
is limited to $(100\,\text{GeV})^{2}$ for each of the quarks and leptons,
hence $M_{q}=(100$ GeV$)(m_{t}/m_{q})$ and $M_{\ell}=(100$
GeV$)(g_{\ell}m_{t}/m_{\ell})$. (2) $\mu=100$ GeV for all mixings, hence
$M_{q}=\mu(m_{t}/m_{q})^{1/2}$ and $M_{\ell}=\mu(m_{t}g_{\ell}/gm_{q})^{1/2}$.
Here $g=1$, $g_{\ell}=0.7$ and $v=175$ GeV. For a more detailed discussion see
HMTT .

The CKM physics is generated by off-diagonal Higgs field and their mixings. We
ran numerous benchmark tests of hypothetical Higgs mass matrices to study
consistency with flavor physics (rare weak decay constraints) and the
generation of quark and lepton masses and some of these are discussed and
displayed in is HMTT ; HMTT2 . The results are nontrivial, yet we find there
are consistent solutions where the masses and mixings of a spectrum of new
Higgs doublets can explain the entire fermionic mass and CKM-mixing structure
of the SM with the single universal HY coupling. Some results for mass limits
are quoted in Table III.

The generic idea of a maximally large spectrum of Higgs bosons, “scalar
democracy,” is proposed in HMTT ; HMTT2 . Related prior literature includes
models of Ben Tov, Porto, and Zee, et.al., Zee Bentov  which considers only
diagonal Higgs fields (typically $6$ in each subsector) and expects new Higgs
states in the TeV mass range. Related enhanced spectroscopy of composite Higgs
bosons arises in various other models, e.g., Marco Osipov Rode Ishida . A
similar large number of Higgs doublets has been invoked in an interesting
model of Weinberg’s asymptotic safety San . This may preclude a conventional
GUT picture if the unification of gauge interactions occurs in tandem with the
non-perturbative asymptotically safe fixed point. Scalars go hand-in-hand with
Weyl invariance, and recent models suggest a unified picture of inflation
occurring with the dynamical generation of the Planck scale Weyl , which may
provide a larger context for this scenario. Also, the idea of universal large
Higgs-Yukawa coupling, e.g., $H_{b}$ with $g=1$, was previously introduced in
the context of a Coleman-Weinberg Higgs potential CWH  (see also Rao thesis ).

###  IV.3 Counting Scalars

If we assume one bound state in nature per $s$-wave fermion pair at some high
scale $M_{X}$, then we can count the number of composite scalars. All of the
SM matter fields can be represented by $48$ two-component left-handed spinors,
$\psi_{A}^{i}$. This includes all the left-handed and anti-right-handed
fermions. We can collect these into a large global $SU(48)\times U(1)$
multiplet, the new dynamics that is blind to the SM gauge interactions. We
emphasize that this is a dynamical symmetry, and familiar GUT theories that
contain only the SM fermions will be gauged subgroups of this $SU(48)$. Here
the indices $\left(i,j\right)$ run over all the $48$ flavor, doublet, and
color degrees of freedom of the SM fermions.

The most general non-derivative ($s$-wave) scalar-field bilinear we can
construct of these fields is through the “vertex operator,”
$\epsilon^{AB}\psi_{A}^{i}\psi_{B}^{j}$:

$\displaystyle\epsilon^{AB}\psi_{A}^{i}\psi_{B}^{j}\Theta_{ij}+h.c.,$ (53)

where $\Theta_{ij}$ transforms as the symmetric $\mathbf{1176}$ representation
of $SU(48)$ (this is analogue to the sextet representation of $SU(3)$). The
field $\Theta_{ij}$ contains many complex scalar fields with assorted quantum
numbers, including baryon and lepton number, color, and weak charges.

To make contact with the SM fields, we consider the usual $24$ left-handed
quarks and leptons, $\Psi_{Li}$, and the $24$ right-handed counterparts,
$\Psi_{R\widehat{i}}$. The index $i$ now runs over the chiral $SU(24)_{L}$ and
$\widehat{i}$ over the chiral $SU(24)_{R}$ subgroups of $SU(48)$. We thus
have:

$\displaystyle\Phi_{i\widehat{j}}\overline{\Psi}_{L}^{i}\Psi_{R}^{\widehat{j}}+\Omega_{ij}\overline{\Psi}_{L}^{i}\Psi_{R}^{jC}+\widehat{\Omega}_{\widehat{ij}}\overline{\Psi}_{R}^{\widehat{i}}\Psi_{L}^{\widehat{j}C}+\text{h.c}.,$
(54)

where $\Phi_{i\widehat{j}}$ is the $(\mathbf{24}_{L},\,\mathbf{24}_{R})$
complex scalar field with $24^{2}=576$ complex degrees of freedom. $\Omega$
and $\widehat{\Omega}$ are the symmetric $\mathbf{300}$ representation of
$SU(24)_{L}$ and $SU(24)_{R}$ respectively These match the degrees of freedom
of $\Theta_{ij}$. Here $\Omega_{ij}$ and $\widehat{\Omega}_{ij}$ are the
analogues of Majorana masses and carry fermion number, while $\Phi$ contains
fermion number neutral fields, such as Higgs fields, in addition to ($B-L$)
leptoquark multiplets.

The $\Phi,\Omega$ and $\widehat{\Omega}$ fields can be viewed as the
“composite fields” arising from a NJL model effective description of the new
forces. Consider just the $SU(24)_{L}\times SU(24)_{R}\times U(1)\times
U(1)_{A}$ invariant NJL model:

$\displaystyle-\frac{g^{2}}{M^{2}}(\overline{\Psi}_{L}^{i}\Psi_{R}^{j})(\overline{\Psi}_{R,i}\Psi_{L,j}),$
(55)

where the negative sign denotes an attractive interaction in the potential. It
should be noted that we can equally well write current-current (and tensor-
tensor) interactions, mediated by heavy spin-1 bosons (or Pauli-Fierz spin-2
gravitons); these will generally contain scalar channels and will Fierz
rearrange to effectively reduce to eq.(55) with the attractive signs. There
also exists the possibility of the following NJL models:

$\displaystyle-\frac{g^{2}}{M^{2}}(\overline{\Psi}_{L}^{i}\Psi_{R}^{Cj})(\overline{\Psi}_{R,i}^{C}\Psi_{L,j})\;\;\makebox{or}\;\;\left(R\leftrightarrow
L\right),$ (56)

which lead to the composite bosons $\Omega$ and $\widehat{\Omega}$. The first
step to solving an NJL theory would be to factorize the interaction of eq.(55)
by introducing auxiliary scalar fields. This leads to the equation we started
with, eq.(54), where $\Phi,\Omega$ and $\widehat{\Omega}$ are auxiliary
fields.

A universal flavor and color blind interaction will bind fermion pairs into
scalars that are bound states of ordinary quarks and leptons and will generate
a plethora of Higgs doublets. E.g., for any fermion pair we expect there is a
mini-black-hole with mass of order $M_{Planck}$ BHole . These bound states
will have a universal Yukawa coupling $g$ at the scale $M^{2}$. Moreover, with
$g$ taking on a near-but-subcritical value, these bound states will generally
have large positive masses but can be tuned to be lighter than $M$. Symmetry-
breaking effects are required to split the spectroscopy, including the SMH
down to its observed negative mass term. All other doublets remain heavy, but
will mix.

If $g$ is supercritical then some or all multiplets will acquire negative
renormalized masses, $M^{2}<0$ and the theory develops an overall vacuum
instability. For example, the field $\Phi^{ij}$ with a supercritical coupling
will generally condense into a diagonal VEV,
$\left\langle\Phi_{ij}\right\rangle=V\delta_{ij}$. This would become a
spontaneously broken $\Sigma$-model of $SU(24)_{L}\times SU(24)_{R}\times
U(1)\times U(1)_{A}$. In this supercritical case, all the fermions would
acquire large, diagonal constituent masses of order $gV$, grossly inconsistent
with observation.

However, the structure we have just outlined can be subcritical. It will then
contain many composite Higgs doublets with a spectrum of positive $M^{2}$’s
and all fermions would be massless. This can come from a scale invariant
theory with extended quartic interactions and additional scalars. This in fact
could connect up to Weyl invariant theories in which even the Planck scale is
generated spontaneously Weyl , and will be intimately intertwined with
inflation. We can consider such a spectrum of masses and mixings between the
bound state scalars that allows for a light sector from the SMH to multi-TeV
scales exists and extends up to the highest scales. We would assume that
$\Omega_{ij}$, $\widehat{\Omega}_{ij}$ and all color-carrying weak doublets
have very large positive $M^{2}$ and therefore we will ignore them. They will
be inactive in the RG evolution (though they may be welcome when unification
is included). There is fine-tuning required to engineer the light $M^{2}$ as
$d=2$ operators, however many of these terms are technically natural,
protected by the $SU(48)$ symmetry structure (see HMTT  for further details).

###  IV.4 The Top-Bottom Subsector

What might convince us that an extended Higgs boson spectrum exists in nature?
The discovery of the first sequential Higgs boson, the $H_{b}$ with a coupling
$g\approx g_{t}\sim O(1)$ to $\overline{b}b$ would lend some credibility to
the scenario. $H_{b}$ has an upper bound on its mass of about $5.5$ TeV based
on a “no unnatural cancellations” hypothesis. It may be discovered up to 3 TeV
with the LHC, and conclusively so with the energy doubled LHC. Indeed, the LHC
already has the capability of placing useful limits on the $H_{b}$.

The top-bottom subsystem is a subsector of the full scalar democracy which can
be defined in a self-contained way. It can also be easily extended to include
the $(\tau,\nu_{\tau})$ leptons, and we will only quote results HMTT2 . We
presently assume the top-bottom subsystem is approximately invariant under a
simple extension of the Standard Model symmetry group structure

$G=SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}\times U(1)_{A}.$ (57)

By “approximately” we mean that, if we turn off the $U(1)_{Y}$ gauging,
$g_{1}\rightarrow 0$, the symmetry $G$ is exact in the $d=4$ operators
(kinetic terms, Higgs-Yukawa couplings, and potential terms). The SM gauging
is the usual $SU(2)_{L}\times U(1)_{Y}$, and is a subgroup of $G$. The
$U(1)_{Y}$ generator is now $I_{3R}+(B-L)/2$. This electroweak gauging weakly
breaks the symmetry $SU(2)_{R}\times U(1)_{B-L}\rightarrow U(1)_{Y}\times
U(1)_{B-L}$, however, the $SU(2)_{R}$ remains as an approximate global
symmetry of the $d=4$ operators. In addition, a global $U(1)_{A}$ arises as
well. $G$ provides custodial symmetry so that new symmetry breaking effects,
in $d=2$ operators, are technically natural. Beyond the usual naturalness
issue of the Higgs boson mass, our present model is no less natural than the
Standard Model.

To implement $G$ in the $(t,b)$ sector we require that the SM Higgs doublet,
$H_{0}$, couples to $t_{R}$ with coupling $y_{t}\equiv g_{t}$ in the usual
way, and a second Higgs doublet, $H_{b}$, couples to $b_{R}$ with coupling
$g_{b}$. The symmetry $G$ then dictates that there is only a single Higgs-
Yukawa coupling $g=g_{t}=g_{b}$ in the quark sector. This coupling is thus
determined by the known top quark Higgs-Yukawa coupling, $y_{t}=g\simeq 1$.

The assumption of the symmetry, $G$, of Eq. (57) for the top-bottom system
leads to a Higgs-Yukawa (HY) structure that is reminiscent of the chiral
Lagrangian of the proton and neutron (or the chiral constituent model of up
and down quarks Manohar ; in a composite Higgs scenario this model was
considered by Luty Luty ):

$\displaystyle
V_{HY}=g\overline{\Psi}_{L}\Sigma\Psi_{R}+h.c.\quad\text{where}\quad\Psi=\left(\begin{array}[]{c}t\\\
b\end{array}\right).$ (60)

In Eq. (60), $\Sigma$ is a $2\times 2$ complex matrix and $V_{HY}$ is
invariant under $\Sigma\rightarrow U_{L}\Sigma U^{\dagger}_{R}$ with
${\Psi}_{L}\rightarrow U_{L}{\Psi}_{L}$ and ${\Psi}_{R}\rightarrow
U_{R}{\Psi}_{R}$. Note that the $U(1)_{Y}$ generator $Y$ of the SM now becomes
$Y=I_{3R}+(B-L)/2$ and furthermore $\Sigma$ is neutral under $B-L$ which
implies its weak hypercharge $[Y,\Sigma]=\Sigma\,I_{3R}$.

An additional $U(1)_{A}$ axial symmetry arises as an overall phase
transformation of $\Sigma\rightarrow e^{i\theta}\Sigma$, accompanied by
${\Psi}_{L}\rightarrow e^{i\theta/2}{\Psi}_{L}$ and ${\Psi}_{R}\rightarrow
e^{-i\theta/2}{\Psi}_{R}$. $\Sigma$ can be written in terms of two column
doublets,

$\Sigma=(H_{0},H^{c}_{b}),$ (61)

where $H^{c}=i\sigma_{2}H^{*}$. Under $SU(2)_{L}$ we have $H\rightarrow
U_{L}H$, and $H^{c}\rightarrow U_{L}H^{c}$ where we note that the weak
hypercharge eigenvalues of $H$ and $H^{c}$ have opposite signs. The HY
couplings of Eq. (60) become

$\displaystyle
V_{HY}=g_{t}\left(\overline{t},\,\overline{b}\right)_{L}H_{0}t_{R}+g_{b}\left(\overline{t},\,\overline{b}\right)_{L}H^{c}_{b}b_{R}+h.c.,$
(62)

where the $SU(2)_{R}$ symmetry has forced $g=g_{t}=g_{b}$.

Note that this becomes identical to the SM if we make the identification

$H_{b}\rightarrow\epsilon H_{0},\qquad\epsilon=\frac{m_{b}}{m_{t}}=0.0234.$
(63)

The HY coupling of the $b$-quark to $H_{0}$ is then $y_{b}=\epsilon g_{b}$,
but with $\epsilon\ll 1$ the $SU(2)_{R}$ symmetry is then lost. The SM Higgs
boson (SMH), $H_{0}$, in the absence of $H_{b}$, has the usual SM potential:

$\displaystyle
V_{\text{Higgs}}=M_{0}^{2}H_{0}^{\dagger}H_{0}+\frac{\lambda}{2}(H_{0}^{\dagger}H_{0})^{2},$
(64)

where $M_{0}^{2}\simeq-(88.4)$ GeV2 (note the sign), and $\lambda\simeq 0.25$.
Minimizing this, we find that the Higgs field $H_{0}$ acquires its usual VEV,
$v=|M_{0}|/\sqrt{\lambda}=174$ GeV, and the observed physical Higgs boson,
$h$, acquires mass $m_{h}=\sqrt{2}|M_{0}|\simeq 125$ GeV. In our present
scheme Eqs. (63) and (64) arise at low energies dynamically.

Given Eqs. (60)and (61), we postulate a new potential,

$\displaystyle V$ $\displaystyle=$ $\displaystyle
M_{1}^{2}Tr\left(\Sigma^{\dagger}\Sigma\right)-M_{2}^{2}Tr\left(\Sigma^{\dagger}\Sigma\sigma_{3}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\mu^{2}\left(e^{i\theta}\det\Sigma+h.c.\right)+\frac{\lambda_{1}}{2}Tr\left(\Sigma^{\dagger}\Sigma\right)^{2}+\lambda_{2}|\det\Sigma|^{2}.$

We have written the potential in the $\Sigma$ notation in order to display the
symmetries more clearly. In the above, $\sigma_{3}$ acts on the $SU(2)_{R}$
side of $\Sigma$, hence in the limit $M_{2}^{2}=0$ the potential $V$ is
invariant under $SU(2)_{R}$, while the $\mu^{2}$ term breaks the additional
$U(1)_{A}$. The associated CP-phase can be removed by field redefinition in
the present model. The global symmetries of $G$ restrict the $d=4$ terms of
this model. They also make the $d=2$ symmetry breaking terms technically
natural. In fact, our theory is no less natural than the SM and this is true
for the entire scalar democracy!

Using Eq. (61), $V$ can be written in terms of $H_{0}$ and $H_{b}$:

$\displaystyle V$ $\displaystyle=$ $\displaystyle
M_{H}^{2}H_{0}^{\dagger}H_{0}+M_{b}^{2}H_{b}^{\dagger}H_{b}+\mu^{2}\left(e^{i\theta}H_{0}^{\dagger}H_{b}+h.c.\right)$
(66)
$\displaystyle+\frac{\lambda}{2}\left(H_{0}^{\dagger}H_{0}+H_{b}^{\dagger}H_{b}\right)^{2}+\lambda^{\prime}\left(H_{0}^{\dagger}H_{b}H_{b}^{\dagger}H_{0}\right),$

where $\lambda_{2}=\lambda^{\prime}+\lambda$, $M_{H}^{2}=M_{1}^{2}-M_{2}^{2}$,
and $M_{b}^{2}=M_{1}^{2}+M_{2}^{2}$. In the limit $M_{b}^{2}\rightarrow\infty$
the field $H_{b}$ decouples, and Eq. (66) reduces to the SM if
$M_{H}^{2}\rightarrow M_{0}^{2}$, and $\lambda\rightarrow 0.25$ thereby
recovering Eq. (64).

We assume the quartic couplings are all of order the SM value $\lambda\sim
0.25$. Those associated with the new heavy Higgs bosons, such as
$\lambda^{\prime}\sim\lambda$, will therefore contribute negligibly small
effects since they involve large, positive, $M^{2}$ heavy Higgs fields. We
also set $\theta=0$.

Then, varying the potential with respect to $H_{b}$, the low momentum
components of $H_{b}$ become locked to $H_{0}$:

$H_{b}=-\frac{\mu^{2}}{M_{b}^{2}}H_{0}+O(\lambda,\lambda^{\prime}).$ (67)

Substituting back into $V$ we recover the SMH potential

$\displaystyle V$ $\displaystyle=$ $\displaystyle
M_{0}^{2}H_{0}^{\dagger}H_{0}+\frac{\lambda}{2}\left(H_{0}^{\dagger}H_{0}\right)^{2}+O\left(\frac{\mu^{2}}{M_{b}^{2}}\right),$
(68)

with

$M_{0}^{2}=M_{H}^{2}-\frac{\mu^{4}}{M_{b}^{2}}.$ (69)

Note that, even with $M_{H}^{2}$ positive, $M_{0}^{2}=-(88.4)^{2}$ GeV2 can be
driven to its negative value by the mixing with $H_{b}$ (level repulsion). We
minimize the SMH potential and define

$\displaystyle H_{0}=\left(\begin{array}[]{c}v+\frac{1}{\sqrt{2}}h\\\
0\end{array}\right),\qquad v={174}\;{GeV},$ (72)

in the unitary gauge. The minimum of Eq. (72) yields the usual SM result

$v^{2}=-M_{0}^{2}/\lambda,\qquad m_{h}=\sqrt{2}|M_{0}|=125\;\makebox{GeV},$
(73)

where $m_{h}$ is the propagating Higgs boson mass. We can then write

$H_{b}\rightarrow
H_{b}-\frac{\mu^{2}}{M_{b}^{2}}\left(\begin{array}[]{c}v+\frac{1}{\sqrt{2}}h\\\
0\end{array}\right),$ (74)

for the full $H_{b}$ field. This is a linearized (small angle) approximation
to the mixing, and is reasonably insensitive to the small
$\lambda,\lambda^{\prime}\ll 1$.

Note the effect of “level repulsion” of the Higgs mass, $M_{0}^{2}$, downward
due to the mixing with heavier $H_{b}$. The level repulsion in the presence of
$\mu^{2}$ and $M_{b}^{2}$ occurs due to an approximate “seesaw” Higgs mass
matrix

$\begin{pmatrix}M_{H}^{2}&\mu^{2}\\\ \mu^{2}&M_{b}^{2}\end{pmatrix}.$ (75)

The input value of the mass term $M_{H}^{2}$ is unknown and in principle
arbitrary, and can have either sign. We can presumably bound
$M_{H}^{2}\gtrsim{1}\;{GeV^{2}}$ from below, since QCD effects will mix
glueballs and the QCD-$\sigma$-meson with $H_{0}$ in this limit. As
$M_{H}^{2}$ is otherwise arbitrary, we might then expect that the most
probable value is $M_{H}^{2}\sim{1}\;{GeV^{2}}$, and
$|M_{H}^{2}|\ll(\mu^{2},M_{b}^{2})$.

Let us consider the case $M_{b}^{2}>>M_{H}^{2}$. Then Eq. (75) has eigenvalues
$M_{0}^{2}=-\mu^{4}/M_{b}^{2}$, and $M_{b}^{2}$. Thus, in the limit of small,
nonzero $|M_{H}^{2}|$, we see that a negative $M_{0}^{2}$ arises naturally,
and to a good approximation the physical Higgs mass is generated entirely by
this negative mixing term.

The mass mixing causes the neutral component of $H_{b}$ to acquire a small VEV
(“tadpole”) of $-v(\mu^{2}/M_{b}^{2})$. This implies that the SM HY-coupling
of the $b$-quark is induced with the small value
$y_{b}=g_{b}(\mu^{2}/M_{b}^{2})$ (note $g_{b}$ is positive with a phase
redefinition of $b_{R}$). The $b$-quark then receives its mass from $H_{b}$,

$m_{b}=g_{b}(m_{b})v\frac{\mu^{2}}{M_{b}^{2}}=m_{t}\frac{g_{b}(m_{b})\mu^{2}}{g_{t}(m_{t})M_{b}^{2}},$
(76)

where we have indicated the renormalization group (RG) scales at which these
couplings should be evaluated.

In a larger scalar democracy framework both $g_{t}(m)$ and $g_{b}(m)$ have a
common renormalization group equation modulo $U(1)_{Y}$ effects. This implies
$g(M_{P})=g_{t}(M_{P})=g_{b}(M_{P})\mathrel{\vbox{\hbox{$>$}\nointerlineskip\hbox{$\sim$}}}1$.
Then, we will predict $g(M_{b})\simeq g_{t}(M_{b})\simeq g_{b}(M_{b})$, e.g.,
where the values at the mass scale $M_{b}$ are determined by the RG fixed
point.

Furthermore, we find that $g_{t}(m)$, and more so $g_{b}(m)$, increase
somewhat as we evolve downward from $M_{b}$ to $m_{t}$ or $m_{b}$. The top
quark mass is then $m_{t}=g_{t}(m_{t})v$ where $v$ is the SM Higgs VEV. From
these effects we obtain the ratio

$R_{b}=\frac{g_{b}(m_{b})}{g_{t}(m_{t})}\simeq 1.5.$ (77)

The $b$-quark then receives its mass from the tadpole VEV of $H_{b}$.

$m_{b}=g_{b}(m_{b})v\frac{\mu^{2}}{M_{b}^{2}}=m_{t}R_{b}\frac{\mu^{2}}{M_{b}^{2}}.$
(78)

In the case that the Higgs mass, $M_{0}^{2}$, is due entirely to the level
repulsion by $H_{b}$, i.e.. $M_{H}^{2}=0$, (see also Ishida  and references
therein) using Eqs. (69), (77) and (78), we obtain a predicted mass of the
$H_{b}$,

$M_{b}=\frac{m_{t}}{m_{b}}R_{b}|M_{0}|\simeq{5.5}\;{TeV},$ (79)

with $m_{b}={4.18}\;{GeV}$, $m_{t}={173}\;{GeV}$, and $|M_{0}|={88.4}\;{GeV}$.
We have ignored the effects of the quartic couplings $\lambda$, which we
expect are small.

This is a key prediction of the model. In fact, we can argue that with
$M_{H}^{2}$ nonzero, but with small fine-tuning the result
$M_{b}\mathrel{\vbox{\hbox{$<$}\nointerlineskip\hbox{$\sim$}}}{5.5}\;{TeV}$ is
obtained. This mass scale is accessible to the LHC with luminosity and energy
upgrades, and we feel represents an important target for discovery of the
first sequential Higgs Boson.

This simple $(t,b)$ system described above can be extended to the third
generation leptons $(\nu_{\tau},\tau)$. Remarkably the predictions for the
mass spectrum are sensitive to the mechanism of neutrino mass generation. The
next sequential massive Higgs iso-doublet, in addition to $H_{b}$, is likely
to include an $H_{\tau}$, and possibly also $H_{\nu}$, which is dependent upon
whether neutrino masses are Majorana or Dirac in nature. If neutrino masses
are Dirac then $H_{\nu}$ is very heavy, $\sim 10^{16}$ GeV, and then have a
Dirac seesaw and we can ignore $H_{\nu}$.

The details of this and phenomenology of the sequential Higgs bosons are
discussed in detail in are given in HMTT ; HMTT2 . Our estimated discovery
luminosities for $h^{0}_{b}$ are seen to be attainable at the LHC or its
upgrades, and we thus encourage and plan more detailed studies. Moreover, the
lower mass range $\lesssim 1$ TeV is currently within range of the LHC and
collaborations should attempt to place limits.

##  V Construction of Bilocal Composites in a Local Scalar Field Theory

The NJL model is a point-like interaction generating a point-like bound state
in field theory. However, realistic theories, such as QCD or electrodynamics,
feature extended interactions and non-point-like bound states. What happens if
we extend the NJL model to a non-point-like UV completion theory? How can we
formulate the formation of non-point-like bound states, yet remain close to
NJL to garner intuition? This section is based upon ref.HComp .

Many years ago Yukawa proposed a multilocal field theory for the description
of relativistic bound states Yukawa . For a composite scalar field, consisting
of a pair of constituents, he introduced a complex bilocal field, $\Phi(x,y)$.
This is factorized, $\Phi(x,y)\rightarrow\chi(X)\phi(r)$ where
$X^{\mu}=(x^{\mu}\\!+\\!y^{\mu})/2$ where $r^{\mu}=(x^{\mu}\\!-\\!y^{\mu})/2$,
and $\chi(X)$ describes the center-of-mass motion like any conventional point-
like field. Then, $\phi(r)$ describes the internal structure of the bound
state. The formalism preserves Lorentz covariance, though we typically “gauge
fix” to the center-of mass frame and Lorentz covariance is not then manifest.
Here we must confront ths issue of “relative time.”

Each of the constituent particles in a relativistic bound state carries its
own local clock, e.g., $x^{0}$ and $y^{0}$. These are in principle
independent, so the question arises, “how can a description of a multi-
particle bound state be given in a quantum theory with a single time variable,
$X^{0}$?” Yukawa introduced an imaginary “relative time”
$r^{0}=(x^{0}-y^{0})/2$, but this was not effective and abandon it.

We show that a bilocal field theory formalism can be constructed in an action
from a local constituent field theory. The removal of relative time is
associated with the canonical normalization of the constituent fields $\chi$
and $\phi$. In the center-of-mass frame the internal wave-function reduces to
a static field, $\phi(\vec{r})$, where $\vec{r}=(\vec{x}-\vec{y})/2$. This
leads to a fairly simple solution to the problem of relative time, matching
the conclusions one gets from the Dirac Hamiltonian constraint theory Dirac
Reltime . For us, the resulting $\phi(\vec{r})$ appears as a straightforward
result.

After first considering a bosonic construction, we apply this to a theory of
chiral fermions with an extended interaction mediated by a perturbative
massive gluon, i.e., the “coloron model” Topcolor Bijnens Simmons NSD . This
provides a UV completion for the Nambu–Jona-Lasinio model, recovered in the
point-like limit, $\vec{r}\rightarrow 0$ and leads to an effective (mass)2
Yukawa potential with coupling $g$. We form bound states with mass, $m^{2}$,
determined as the eigenvalue of a static Schrödinger-Klein-Gordon (SKG)
equation for the internal wave-function $\phi(\vec{r})$.

A key result of this analysis is that the coloron model has a nontrivial
classical critical behavior, $g>g_{c}$, leading to a bound state with a
negative $m^{2}$. The classical interaction is analogous to the Fröhlich
Hamiltonian interaction in a superconductor and has a BCS-like enhancement of
the coupling by a factor of $N_{c}$ (number of colors) Cooper . Remarkably we
find the classical $g_{c}$ is numerically close to the NJL critical coupling
constant which arises in fermion loops. Moreover, the scalar bound state will
have an effective Yukawa coupling to its constituent fermions, distinct from
$g$, that is emergent in the theory.

The description of a relativistic bound state in the rest frame is similar to
the eigenvalue problem of the nonrelativistic Schrödinger equation and some
intuition carries over. However, the eigenvalue of the static Schrödinger-
Klein-Gordon (SKG) equation, is $m^{2}$, rather than energy. Hence, a bound
state with positive $m^{2}$ is a resonance that can decay to its constituents
and has a Lorentz line-shape in $m^{2}$ (we give an example in HComp ), and
thus has a large distance radiative component in its solution that represents
incoming and outgoing open scattering states.

If the eigenvalue for $m^{2}$ is negative, or tachyonic, then we have chiral
vacuum instability, which requires a quartic interaction of the composite
field, $\sim\lambda\Phi^{4}$. In the broken symmetry phase the composite field
$\Phi(x,y)$ acquires a vacuum expectation value (VEV), $\langle\Phi\rangle=v$.
In the perturbative ($\lambda$) solution, in the broken phase $\phi(\vec{r})$
remains localized and the Nambu-Goldstone modes and Brout-Englert-Higgs (BEH)
boson retain the common localized solution for their internal wave-functions.

###  V.1 Scalar Theory

Consider local scalar fields $\varphi(x)$ (complex) and $A(x)$ (real) and
action:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S=\int_{x}\biggl{(}|\partial\varphi|^{2}\\!+\\!\frac{1}{2}(\partial
A)^{2}\\!-\\!\frac{1}{2}M^{2}A^{2}\\!-\\!gM|\varphi|^{2}A\\!-\\!\frac{\lambda}{2}|\varphi|^{4}\biggr{)},$

where we abbreviate,
$|\partial\varphi|^{2}=\partial_{\mu}\varphi^{\dagger}\partial^{\mu}\varphi$
and $(\partial A)^{2}=\partial_{\mu}A\partial^{\mu}A$. Here $g$ is
dimensionless and we refer all mass scales to the single scale $M$. We will
discuss the quartic term separately below, and presently set it aside,
$\lambda=0$.

If we integrate out $A$ we obtain an effective, attractive, bilocal potential
interaction term at leading order in $g^{2}$,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S=\\!\\!\int_{x}\\!|\partial\varphi|^{2}+\frac{{g^{2}M^{2}}}{2}\\!\\!\\!\int_{xy}\\!\\!\\!\\!\varphi^{\dagger}\\!(y)\varphi(y)D_{F}(y\\!-\\!x)\varphi^{\dagger}\\!(x)\varphi(x),$
(81)

where the two-point function is given by $(i)\times$ the Feynman propagator,

$\displaystyle
D_{F}(x-y)=-\\!\\!\int\frac{e^{iq_{\mu}(x^{\mu}-y^{\mu})}}{(q^{2}-M^{2})}\frac{d^{4}q}{(2\pi)^{4}}.$
(82)

In the action eq.(81), the kinetic term is still local while the interaction
is bilocal, and the theory is still classical in that this only involved a
tree diagram that is ${\cal{O}}(\hbar^{0})$.

We now define a bilocal field of mass dimension $d=1$,

$\displaystyle\Phi(x,y)=M^{-1}\varphi(x)\varphi(y).$ (83)

The free particle states described by the bilocal field trivially satisfy an
equation of motion,

$\displaystyle\partial_{x}^{2}\Phi(x,y)=0,$ (84)

and this is generated by a bilocal action,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S=M^{4}\\!\\!\int_{xy}\\!\\!Z|\partial_{x}\Phi(x,y)|^{2},$
(85)

where we will specify the normalization, $Z$ and scale $M$ subsequently. With
the bilocal field the interaction of eq.(81) becomes,

$\displaystyle\frac{{g^{2}M^{4}}}{2}\\!\\!\\!\int_{xy}\\!\\!\\!\\!\Phi^{\dagger}(x,y)D_{F}(x\\!-\\!y)\Phi(x,y).$
(86)

We can therefore postulate a bilocalized action as a free particle part plus
the interaction,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S=\\!\int_{xy}\\!\\!\biggl{(}ZM^{4}|\partial_{x}\Phi(x,y)|^{2}\\!$
$\displaystyle\qquad+\\!\frac{1}{2}{g^{2}M^{4}}\Phi^{\dagger}(x,y)D_{F}(x-y)\Phi(x,y)\biggr{)}.$
(87)

In the limit $g=0$ the field $\Phi(x,y)$ and the action faithfully represents
two particle kinematics.

We see that a $U(1)$ conserved Noether current is generated by
$\Phi(x,y)\rightarrow e^{i\theta(x)}\Phi(x,y)$,

$\displaystyle J_{\Phi\mu}(x)=iZ\int
d^{4}y\biggl{(}\Phi^{\dagger}(x,y)\frac{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}}{\partial
x^{\mu}}\Phi(x,y)\biggr{)}$ (88)

where
$A\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}B=A{\partial}B-({\partial}A)B$.
We can demand this match the conserved $U(1)$ current in the constituent
theory,

$\displaystyle
J_{\varphi\mu}(x)=i\varphi^{\dagger}(x)\frac{\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}}{\partial
x^{\mu}}\varphi(x)$ (89)

Substituting eq.(83) into $J_{\Phi\mu}(x)$ we see that the matching requires

$\displaystyle
J_{\Phi\mu}(x)=J_{\varphi\mu}(x)\;ZM^{2}\\!\\!\int\\!d^{4}y|\varphi(y)|^{2}$
(90)

hence

$\displaystyle 1=ZM^{2}\int\\!d^{4}y\;|\varphi(y)|^{2}$ (91)

This is a required constraint for the bound state sector of the theory. Note
that the square of the constraint is the 4-normalization of $\Phi$

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!1=Z^{2}M^{4}\int\\!d^{4}y\;d^{4}y\;|\varphi(y)|^{2}|\varphi(x)|^{2}$
$\displaystyle=Z^{2}M^{6}\int\\!d^{4}y\;d^{4}y\;|\Phi(x,y)|^{2}$ (92)

This implies that the presence of a correlation in the two particle sector,
$\Phi(x,y)$ acts as a constraint on the single particle action in that sector.
We can now see how the underlying $\varphi$ action of eq.(81) leads to the
$\Phi$ action by inserting the constraint of eq.(91) onto the kinetic term of
eq.(81) and rearranging to obtain,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S=\\!\\!\int_{xy}\\!\\!\biggl{(}ZM^{2}|\varphi(y)\partial_{x}\varphi(x)|^{2}$
$\displaystyle\qquad\\!\\!\\!+{\frac{g^{2}M^{2}}{2}}\varphi^{\dagger}(y)\varphi(x)D_{F}(x-y)\varphi^{\dagger}(x)\varphi(y)\biggr{)}$
(93)

and $S$ remains dimensionless. With the bilocal field of eq.(83) the
bilocalized action becomes eq.(V.1).

We go to barycentric coordinates $(X,r)$,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!X=\frac{1}{2}(x+y),\qquad
r=\frac{1}{2}(x-y).$ (94)

where $r^{\mu}=(r^{0},\vec{r})$, where $\vec{r}$ is the radius and $r^{0}$ is
the relative time.

Hence we write,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Phi(x,y)=\Phi(X\\!+\\!r,X\\!-\\!r)\equiv\Phi(X,r).$
(95)

Let $S=S_{K}+S_{P}$ and can then rewrite the kinetic term, $S_{K}$, using the
derivative $\partial_{x}=\frac{1}{2}(\partial_{X}\\!+\\!\partial_{r})$,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{K}=\frac{JM^{4}}{4}\\!\\!\int_{Xr}\\!\\!Z|(\partial_{X}\\!+\\!\partial_{r})\Phi(X,r)|^{2}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\frac{JM^{4}}{4}\\!\\!\int_{Xr}\\!\biggl{(}Z|\partial_{X}\Phi|^{2}\\!+\\!Z|\partial_{r}\Phi|^{2}$
$\displaystyle\qquad\qquad\\!+\\!Z(\partial_{X}\Phi^{\dagger}\partial_{r}\Phi+h.c.)\biggr{)}.$
(96)

Note the Jacobian $J=16$,

$\displaystyle
J^{-1}=\left|\frac{\partial(X,r)}{\partial(x,y)}\right|=\biggl{(}\frac{1}{2}\biggr{)}^{4}.$
(97)

Likewise, the potential term is,

$\displaystyle
S_{P}=\frac{JM^{4}}{2}\\!\int_{Xr}\\!{g^{2}}D_{F}(2r)|\Phi(X,r)|^{2}.$ (98)

We will treat the latter term in eq.(V.1),
$Z(\partial_{X}\Phi^{\dagger}\partial_{r}\Phi+h.c.),$ as a constraint, with
its contribution to the equation of motion,

$\displaystyle\frac{\partial}{\partial X^{\mu}}\frac{\partial}{\partial
r_{\mu}}\Phi=0.$ (99)

We can redefine this term in the action as a Lagrange multiplier while
preserving Lorentz invariance,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\rightarrow\int_{Xr}\eta\biggl{(}\frac{\partial\Phi^{\dagger}}{\partial
X^{\mu}}\frac{\partial\Phi}{\partial
r_{\mu}}+h.c.\biggr{)}^{2}\;\;\makebox{hence,}\;\;\delta S/\delta\eta=0,$
(100)

which also enforces the constraint on a path integral in analogy to gauge
fixing. In the following we assume the constraint is present in the total
action though not written explicitly.

We therefore have the bilocal action,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S=\frac{JM^{4}}{4}\\!\\!\int_{Xr}\\!\biggl{(}Z|\partial_{X}\Phi|^{2}+Z|\partial_{r}\Phi|^{2}\\!$
$\displaystyle\qquad+2g^{2}D_{F}(2r)|\Phi(X,r)|^{2}\biggr{)}.$ (101)

Following Yukawa, we assume we can factorize $\Phi$,

$\displaystyle\sqrt{J/4}\;\Phi(X,r)=\chi(X)\phi(r).$ (102)

${\phi(r)}$ is the internal wave-function which we define to be dimensionless,
$d=0$, while $\chi$ is an ordinary local field with mass dimension $d=1$.
$\chi(X)$ determines the center-of-mass motion of the composite state. The
action for the factorized field takes the form,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S={M^{4}}\\!\\!\int_{Xr}\\!\biggl{(}\\!Z|\partial_{X}\chi|^{2}|\phi^{2}|$
$\displaystyle+|\chi|^{2}(Z|\partial_{r}\phi|^{2}\\!+2g^{2}D_{F}(2r)|\phi(r)|^{2}\\!)\biggr{)}.$
(103)

Matching the $U(1)$ current generated by $\chi\rightarrow e^{i\theta(X)}\chi$
to that of the underlying constituent theory we see that the normalization of
the world-scalar 4-integral is,888 The current is now generated by
$e^{i\theta(x)}\Phi=e^{i\theta((X+r)/2)}\Phi$. The matching is exact for the
zero component (charge) and implies eq.(104). The currents will be discussed
in detail elsewhere.

$\displaystyle 1=ZM^{4}\\!\int\\!d^{4}r\;|\phi(r)|^{2}.$ (104)

This then leads to a canonical normalization of $\chi(X)$.

We can then represent $S$ in terms of two “nested” actions. For the field
$\chi$,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S=\\!\int_{X}\\!\biggl{(}|\partial_{X}\chi|^{2}-m^{2}|\chi|^{2}\biggr{)}\;\;\;\makebox{where,}\;\;\;m^{2}=-S_{\phi},$
(105)

and $S_{\phi}$ is an action for the internal wave-function,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{\phi}=M^{4}\\!\\!\int_{r^{0},\vec{r}}\\!\biggl{(}\\!Z|\partial_{r}\phi|^{2}\\!+{2g^{2}}D_{F}(2r)|\phi|^{2}\\!\biggr{)}.$
(106)

Eq.(105) then implies,

$\displaystyle\partial^{2}_{X}\chi=-m^{2}\chi\;\;\;\makebox{hence},\;\;\chi\sim\exp(iP_{\mu}X^{\mu}).$
(107)

$\chi(X)$ has free plane wave solutions with $P^{2}=m^{2}$.

In the center of mass frame of the bound state we can choose $\chi$ to have
4-momentum $P_{\mu}=(m,0,0,0)$ where we then have,

$\displaystyle\Phi(X,r)=\chi(X)\phi(r)\propto\exp(imX^{0})\phi(r).$ (108)

$\phi(r)$ must then satisfy the Lagrange multiplier, constraint

$\displaystyle P^{\mu}\frac{\partial}{\partial r^{\mu}}\phi(r)=0,$ (109)

and therefore becomes a static function of $r^{\mu}=(0,\vec{r})$.

While we have specified $Z$ in eq.(104), we still have the option of
normalizing the internal wave-function $\phi(\vec{r})$. This can be
conveniently normalized in the center of mass frame as,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!M^{3}\\!\int\\!\\!d^{3}r\;|\phi(\vec{r})|^{2}=1.$
(110)

Note that in eq.(110) we have implicitly defined the static internal wave-
function $\phi(\vec{r})$ to be dimensionless, $d=0$.

We see that the relative time now emerges in the 4-integral over
$|\phi(r)|^{2}$ of eq.(104) together with eq.(110),

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!1=ZM^{4}\\!\\!\int\\!\\!d^{4}r|\phi(r)|^{2}=ZM^{4}\\!\\!\int\\!\\!dr^{0}\\!\\!\\!\int\\!\\!d^{3}r|\phi(\vec{r})|^{2}$
$\displaystyle=ZMT,$ (111)

where $\int dr^{0}=\int dr^{\mu}P_{\mu}/M\equiv T$. Then from eq.(V.1) we
have,

$\displaystyle TZ=M^{-1}.$ (112)

With static $\phi(r)\rightarrow\phi(\vec{r})$ the internal action of eq.(106)
becomes,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{\phi}=M^{4}\\!\\!\int_{r^{0},\vec{r}}\\!\biggl{(}\\!-Z|\partial_{\vec{r}}\phi(\vec{r})|^{2}\\!+{2g^{2}}D_{F}(2r)|\phi(\vec{r})|^{2}\\!\biggr{)},$
(113)

where
$|\partial_{\vec{r}}\phi(\vec{r})|^{2}=\partial_{\vec{r}}\phi^{\dagger}\cdot\partial_{\vec{r}}\phi$.
Note $\partial_{\vec{r}}\phi$ is spacelike, and the arguments of the
constrained $\phi(\vec{r})$ are now 3-vectors, however $D_{F}(2r^{\mu})$ still
depends upon the 4-vector $r^{\mu}$.

There remains the integral over relative time $r^{0}$ in the action. For the
potential, we have by residues,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!V(r)=-2\\!\\!\int\\!\\!dr^{0}D_{F}(2r)=-\\!\int\\!\\!\frac{e^{2i\vec{q}\cdot\vec{r}}}{{\vec{q}}^{2}+M^{2}}\frac{d^{3}q}{(2\pi)^{3}}$
$\displaystyle=-\\!\frac{e^{-2M|\vec{r}|}}{8\pi|\vec{r}|},$ (114)

and the potential term becomes the static Yukawa potential,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{P}=-M^{3}\\!\\!\int_{\vec{r}}\\!{g^{2}}MV(\vec{r})|\phi(\vec{r})|^{2},\;\;\;$
(115)

The $\phi(\vec{r})$ kinetic term in eq.(113) becomes,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{K}=-M^{4}\\!\\!\int_{r^{0},\vec{r}}\\!\\!Z|\partial_{\vec{r}}\phi(\vec{r})|^{2}\\!=-M^{4}ZT\\!\\!\int_{\vec{r}}\\!|\partial_{\vec{r}}\phi(\vec{r})|^{2}\\!$
$\displaystyle=-M^{3}\\!\\!\int_{\vec{r}}\\!|\partial_{\vec{r}}\phi(\vec{r})|^{2}\\!,$
(116)

where we use eq.(112). The action $S_{\phi}$ thus becomes,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!m^{2}=-S_{\phi}=M^{3}\int_{\vec{r}}\biggl{(}|\partial_{\vec{r}}\phi|^{2}+{g}^{2}MV(|\vec{r}|)|\phi|^{2}\biggr{)}.$
(117)

Note $S_{\phi}$ has dimension $d=2$, as it must for $m^{2}$. We thus see, as
previously mentioned, that the combination $ZT$ occurs in the theory, and the
relative time has disappeared into normalization constraints, eqs.(104, 110).

The extremalization of $S_{\phi}$ leads to the Schrödinger-Klein-Gordon (SKG)
equation,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\nabla_{\vec{r}}^{2}\phi(r)-{g^{2}M}\frac{e^{-2M|\vec{r}|}}{8\pi|\vec{r}|}\phi(r)=m^{2}{\phi}(r),$
(118)

where, for spherical symmetry in a ground state,

$\displaystyle\nabla_{r}^{2}=\partial_{r}^{2}+\frac{2}{r}\partial_{r}.$ (119)

We see that the induced mass2 of the bound state, $m^{2}$, is the eigenvalue
of the SKG equation. We can compare this to a non-relativistic Schrödinger
equation (NRSE),

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\frac{1}{2M}\nabla_{r}^{2}{\phi}(r)-{g^{2}}\frac{e^{-2Mr}}{16\pi
r}{\phi}(r)=E{\phi}(r).$ (120)

In the next section we will obtain a similar results for a bound state of
chiral fermions and use the known results for the Yukawa potential in the NRSE
to obtain the critical coupling. The negative eigenvalue of $E$ in the NRSE,
which signals binding, presently implies a vacuum instability.

Integrating by parts we then have from eq.(117),

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!m^{2}=M^{3}\int_{\vec{r}}\biggl{(}\phi^{\dagger}(-\nabla^{2}_{\vec{r}}\phi+{g^{2}M}V(r)\phi(\vec{r}))\biggr{)}.$
(121)

Note consistency, using eq.(120), and the normalization of the dimensionless
field $\phi$ of eq.(110).

More generally, by promoting $\chi$ to a $(1+3)$ time dependent field while
maintaining a static $\phi$ we have the full joint action:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\;S=\\!M^{3}\\!\\!\int_{X\vec{r}}\\!\biggl{(}\\!|\phi|^{2}\\!\left|\frac{\partial\chi}{\partial
X}\right|^{2}\\!\\!\\!-\\!|\chi|^{2}\biggl{(}|\partial_{\vec{r}}\phi|^{2}\\!+\\!g^{2}MV(\vec{r})|\phi|^{2}\biggr{)}\biggr{)}.$

In summary, we have constructed, by “bilocalization” of a local field theory,
a bilocal field description $\Phi(x,y)$ for the dynamics of binding a pair of
particles. The dynamics implies that, in barycentric coordinates,
$\Phi(x,y)\sim\Phi(X,r)\sim\chi(X)\phi(\vec{r})$, where the internal wave-
function, $\phi(\vec{r})$, is a static function of $\vec{r}$ and satisfies an
SKG equation with eigenvalue $m^{2}$, which determines the squared-mass of a
bound state. This illustrates the removal of relative time in an action
formalism, which is usually framed in the context of Dirac Hamiltonian
constraints Dirac .

###  V.2 The Coloron Model

The point-like NJL model can be viewed as the limit of a physical theory with
a bilocal interaction. An example that motivates the origin of the NJL is the
“coloron model,” deployed in Sections III and IV to describe chiral
constituent quark models Bijnens BHHQ Topcolor NSD Simmons .

We assume chiral fermions, each with $N_{c}$ “colors” labeled by $(a,b,...)$
with the local Dirac action,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{F}=\int_{x}\biggl{(}i\overline{\psi}^{a}_{L}(x)\partial\\!\\!\\!/\\!\,\,\psi_{aL}(x)+i\overline{\psi}^{a}_{R}(x)\partial\\!\\!\\!/\\!\,\,\psi_{aR}(x)\biggr{)},$
(123)

The single coloron exchange interaction then takes a bilocal current-current
form:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\;S_{C}=-g^{2}\\!\\!\\!\int_{xy}\\!\\!\overline{\psi}_{L}(x)\gamma_{\mu}T^{A}\psi_{L}(x)$
$\displaystyle\qquad\times\;D^{\mu\nu}(x-y)\;\overline{\psi}_{R}(y)\gamma_{\nu}T^{A}\psi_{R}(y),$
(124)

where $T^{A}$ are generators of $SU(N_{c})$. The coloron propagator in a
Feynman gauge yields:

$\displaystyle
D_{\mu\nu}(x-y)=\int\frac{-ig_{\mu\nu}}{q^{2}-M^{2}}e^{iq(x-y)}\frac{d^{4}q}{(2\pi)^{4}}.$
(125)

A Fierz rearrangement of the interaction to leading order in $1/N_{c}$ leads
to an attractive potential Topcolor :

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{C}=g^{2}\\!\\!\int_{xy}\\!\\!\;\overline{\psi}^{a}_{L}(x)\psi_{aR}(y)\;D_{F}(x-y)\;\overline{\psi}^{b}_{R}(y)\psi_{bL}(x),$
(126)

where $D_{F}$ is defined in eq.(82). Note that if we suppress the $q^{2}$ term
in the denominator of eq.(125),

$\displaystyle{D}_{F}(x-y)\rightarrow\frac{1}{M^{2}}\delta^{4}(x-y),$ (127)

and we immediately recover the point-like NJL model interaction.

Consider spin-$0$ fermion pairs of a given color $[\overline{a}b]$
$\overline{\psi}^{a}_{R}(x)\psi_{bL}(y)\\!$. We will have free fermionic
scattering states, $\sim\;:\\!\overline{\psi}^{a}_{R}(x)\psi_{bL}(y)\\!:$,
coexisting in the action with bound states $\Phi(x,y)$,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\overline{\psi}^{a}_{R}(x)\psi_{bL}(y)\\!\rightarrow\;:\\!\overline{\psi}^{a}_{R}(x)\psi_{bL}(y)\\!:+\;M^{2}\Phi^{a}_{\;b}(x,y).\;\;$
(128)

This is analogous to the shift done to introduce the auxiliary field in the
NJL model in eq.(II). The main differences with the NJL model are that
presently $\Phi^{a}_{\;b}(x,y)$ is bilocal and it has a bare two body kinetic
term. The normal ordering $:...:$ signifies that we have subtracted the bound
state from the product.

We see that $\Phi^{a}_{b}(X,r)$ is an $N_{c}\times N_{c}$ complex matrix that
transforms as a product of $SU(N_{c})$ representations,
$\overline{N}_{c}\times{N}_{c}$, and therefore decomposes into a singlet plus
an adjoint representation of $SU(N_{c})$. We write $\Phi^{a}_{b}$ it as a
matrix $\widetilde{\Phi}$ by introducing the $N_{c}^{2}-1$ adjoint matrices,
${T}^{A}$, where $\mathop{\rm Tr}({T}^{A}{T}^{B})=\frac{1}{2}\delta^{AB}$. The
unit matrix is ${T}^{0}\equiv\;$diag$(1,1,1,..)/\sqrt{2N_{c}}$, and
$\mathop{\rm Tr}({T}^{0}{})^{2}=1/2$, hence we have,

$\displaystyle\widetilde{\Phi}=\sqrt{2}({T}^{0}\Phi^{0}+\sum_{A}{T}^{A}\Phi^{A}).$
(129)

The $\sqrt{2}$ is present because $\Phi^{0}$ and $\Phi^{A}$ form complex
representations since they also represent the $U(1)_{L}\times U(1)_{R}$ chiral
symmetry.

For the bilocal fields, we have a bosonic kinetic term which, for the singlet
representations, takes the form,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{K}=\frac{{J}ZM^{4}}{2}$
$\displaystyle\times\int_{Xr}\\!\biggl{(}|\partial_{X}\Phi^{0}|^{2}+|\partial_{r}\Phi^{0}|^{2}\\!+\\!\eta|\partial_{X}\Phi^{0}{}^{\dagger}\partial_{r}\Phi^{0}|^{2}\biggr{)}.$
(130)

We assume the constraint in the barycentric frame, and integrate out relative
time with $ZMT=1$,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{K}=(J/2)\\!\\!\int_{X}\int^{\prime}_{\vec{r}}\\!\biggl{(}|\partial_{X}\Phi^{0}(X,\vec{r})|^{2}-|\partial_{\vec{r}}\Phi^{0}(X,\vec{r})|^{2}\biggr{)}$
(131)

(where $\int^{\prime}_{\vec{r}}=M^{3}\int d^{3}r$).

The full interaction of eq.(126) thus becomes,

$\displaystyle S_{C}\rightarrow
g^{2}\\!\\!\int_{xy}\\!\\!:\\!\overline{\psi}^{a}_{L}(x)\psi_{aR}(y)\\!:D_{F}(x-y):\\!\overline{\psi}^{b}_{R}(y)\psi_{bL}(x)\\!:$
$\displaystyle+g^{2}JM^{2}\sqrt{N_{c}}\\!\\!\int_{X,r}\\!\\!\\!\\!\\!:\\!\overline{\psi}^{a}_{L}(X\\!-\\!r)\psi_{aR}(X\\!+\\!r)\\!:\\!\\!D_{F}(2r)\;\Phi^{0}{+h.c.}$
$\displaystyle+\;g^{2}JM^{4}N_{c}\\!\\!\int_{X,r}\\!\\!\\!\\!\;\Phi^{0}{}^{\dagger}(X,r)\;D_{F}(2r)\;\Phi^{0}(X,r)$
(132)

where,

$\displaystyle{D}_{F}(2r)=-\\!\int\frac{1}{(q^{2}-M^{2})}e^{2iq_{\mu}r^{\mu}}\frac{d^{4}q}{(2\pi)^{4}}.$
(133)

The leading term $S_{C}$ of eq.(V.2) is just a free 4-fermion scattering state
interaction and has the structure of the NJL interaction in the limit of
eq.(127). This identifies $g^{2}$ as the NJL coupling constant. This is best
treated separately by the local interaction of eq.(126). We therefore omit
this term in the discussion of the bound states.

The second term $\sim\mathop{\rm Tr}(\psi^{\dagger}\psi)\Phi^{0}+h.c.$ in
eq.(V.2) determines the Yukawa interaction between the bound state $\Phi^{0}$
and the free fermion scattering states. We will treat this below.

Note that the third term in eq.(V.2) is the binding interaction and it
involves only the singlet, $\mathop{\rm
Tr}\widetilde{\Phi}=\sqrt{N_{c}}\Phi^{0}$. The adjoint representation
$\Phi^{A}$ are decoupled from the interaction and remain as unbound, two body
massless scattering states. Moreover, the singlet field $\Phi^{0}$ therefore
has an enhanced interaction by a factor of $N_{c}$. This is analogous to a BCS
superconductor, where the $N_{c}$ color pairs are analogues of $N$ Cooper
pairs and the weak 4-fermion Fröhlich Hamiltonian interaction is enhanced by a
factor of $N_{Cooper}$ Cooper . The color factor enhancement also occurs in
the NJL model, but at loop level. Here we see that the color enhancement is
occurring in the semi-classical (no loop) coloron theory by this coherent
mechanism. To our knowledge this semi-classical enhancement of the underlying
coupling strength in a coloron model, or its approximation to QCD chiral
dynamics, has not been previously noted.

Factorizing $\Phi^{0}$,

$\displaystyle\sqrt{J/2}\;\Phi^{0}(X,r)=\chi(X)\phi(r),$ (134)

the kinetic term action becomes identical to the bosonic case,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{K}=\\!\\!\int_{X}\biggl{(}|\partial_{X}\chi(X)|^{2}-|\chi(X)|^{2}\int^{\prime}_{\vec{r}}\\!|\partial_{\vec{r}}\phi(\vec{r})|^{2}\biggr{)},$
(135)

with,

$\displaystyle\int^{\prime}_{\vec{r}}|\phi(\vec{r})|^{2}=1.$ (136)

The interaction term can then be written from eq.(V.1) as,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{C}\rightarrow
g^{2}JM^{4}N_{c}\\!\\!\int_{X,r}\\!\\!\\!\\!\;\Phi^{0}{}^{\dagger}(X,r)\;D_{F}(2r)\;\Phi^{0}(X,r)$
$\displaystyle=g^{2}N_{c}\\!\int_{X}|\chi(X)|^{2}\int^{\prime}_{\vec{r}}|\phi(\vec{r})|^{2}M\frac{e^{-2M|\vec{r}|}}{8\pi|\vec{r}|}.$
(137)

Hence, the removal of relative time is identical procedure as in the bosonic
model, and we obtain to the same action, $S=S_{K}+S_{C}$ in the compact
notation of eqs.(LABEL:compact1) with the interaction now enhanced by $N_{c}$.

The extremalization of $\phi$ then leads to the SKG equation,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-\nabla_{\vec{r}}^{2}\phi(\vec{r})-{g^{2}N_{c}M}\frac{e^{-2M|\vec{r}|}}{8\pi|\vec{r}|}\phi(\vec{r})=m^{2}{\phi}(\vec{r}).$
(138)

###  V.3 Classical Criticality of the Coloron Model

The coloron model furnishes a direct UV completion of the NJL model. However,
in the coloron model we do not need to invoke large-$N_{c}$ quantum loops to
have a critical theory. Rather, it leads to an SKG potential of the Yukawa
form which has a classical critical coupling, $g_{c}$. For $g<g_{c}$ the
theory is subcritical and produces resonant bound states that decay into
chiral fermions. For $g>g_{c}$ the theory produces a tachyonic bound state
which implies a chiral instability and $\Phi$ must develop a VEV. This
requires stabilization by, e.g., quartic interactions and a sombrero
potential. All of this is treated bosonically in our present formalism.

The criticality of the Yukawa potential in the nonrelativistic Schrödinger
equation is discussed in the literature in the context of “screening.” The
nonrelativistic Schrödinger equation $r=|\vec{r}|$ is:

$\displaystyle-\nabla^{2}\psi-2m\alpha\frac{e^{-\mu r}}{r}\psi=2mE$ (139)

and criticality (eigenvalue $E=0$) occurs for $\mu=\mu_{c}$ where a numerical
analysis yields, Edwards ,

$\displaystyle\mu_{c}=1.19\;\alpha m.$ (140)

For us the spherical SKG equation is now eq.(138). Comparing, gives us a
critical value of the coupling constant, when $\mu_{c}\rightarrow 2M$,
$m\rightarrow M/2$ and $\alpha\rightarrow g^{2}N_{c}/8\pi$, then:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!2M=(1.19)\biggl{(}\\!\frac{M}{2}\\!\biggr{)}\biggl{(}\\!\frac{g^{2}N_{c}}{8\pi}\\!\biggr{)},\;\;\makebox{hence:}\;\;{g^{2}}/{4\pi}=6.72/N_{c}.$

We can compare the NJL critical value of eq.(II),

$\displaystyle g_{cNJL}^{2}/4\pi=2\pi/N_{c}=6.28/N_{c}.$ (142)

Hence, the NJL quantum criticality is a comparable effect, with a remarkably
similar numerical value for the critical coupling.

Note that we can rewrite eq.(138) with dimensionless coordinates,
$\vec{u}=M\vec{r}$,

$\displaystyle
M^{2}\biggl{(}-\nabla_{u}^{2}\phi(u)-{g^{2}N_{c}}\frac{e^{-2u}}{8\pi
u}\biggr{)}\phi(u)=0,$ (143)

and then $M^{2}$ only appears as an overall scale factor. Hence we see that
critical coupling is determined by eq.(143), and the scale $M$ cancels out at
criticality. The mass scale $M$ is dictated in the exponential $e^{-2Mr}$ of
the Yukawa potential. However, in general, we can start with the dimensionless
form of the SKG equation, as in eq.(143), and infer the scale $M$ by matching
to any desired potential.

However, it is important to realize that the NJL model involves the Yukawa
coupling, $g_{NJL}$ while the present criticality involves the coloron
coupling constant. The Yukawa coupling is emergent in the coloron model, and
we need to compute it.

The second term in eq.(V.2) is the induced the Yukawa interaction $S_{Y}$, and
can be written as:

$\displaystyle S_{Y}=g^{2}(\sqrt{{2JN_{c}}})M^{2}\times$
$\displaystyle\int_{Xr}\\!\overline{\psi}^{a}_{L}(X-r)\psi_{aR}(X+r)D_{F}(2r)\chi(X)\phi(\vec{r}){+h.c.}$
(144)

This is the effective Yukawa interaction between the bound state $\Phi^{0}$
and the free fermion scattering states.

We can’t simply integrate out the relative time here. However, we can first
connect this to the point-like limit by suppressing the $q^{2}$ term in the
denominator of $D(2r)$ with $z\rightarrow 0$ in:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{D}_{F}(2r)\rightarrow\\!\\!\int\\!\\!\frac{1}{M^{2}}e^{2iq_{\mu}r^{\mu}}\frac{d^{4}q}{(2\pi)^{4}}\rightarrow\frac{1}{JM^{2}}\delta^{4}(r),$
(145)

where $\delta^{4}(2r)=J^{-1}\delta^{4}(r)$, hence with $J^{-1}=1/16$,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{Y}=g^{2}(\sqrt{{N_{c}}/{8}})\int_{X}\\!\overline{\psi}^{a}_{L}(X)\psi_{aR}(X)\chi(X)\phi(0)+h.c.$

This gives the Yukawa coupling,

$\displaystyle g_{Y}=g^{2}(\sqrt{{N_{c}}/{8}})\phi(0).$ (147)

The wave-function at the origin, $\phi(0)$ in the NJL limit is somewhat
undefined. However, if we consider a spherical cavity of radius $R$ where
$MR=\pi/2$, with a confined, dimensionless, $\phi(r)$, then $\phi(0)$ is
obtained:

$\displaystyle\phi(0)=\frac{1}{\pi}.$ (148)

Plugging this into the expression for $g_{Y}$ in eq.(147) gives,

$\displaystyle g_{Y}=g^{2}\sqrt{{N_{c}}/{8\pi^{2}}}=\frac{g^{2}}{g_{cNJL}},$
(149)

where $g^{2}_{cNJL}=(8\pi^{2}/N_{c})$ is the critical coupling of the NJL
model, as seen in eq.(II).

Hence, if the coloron coupling constant, $g^{2}$, is critical, as in
eqs.(V.3,142). then we have seen in that $g^{2}\approx g^{2}_{cNJL}$, and the
induced Yukawa coupling from eq.(149) is then $g_{Y}\approx g_{cNJL}$. The
coloron model is then consistent with the NJL model in the point-like limit
where the NJL model coupling is the Yukawa coupling, as seen in eq.(II).
However, the induced Yukawa coupling in the bound state, $g_{Y}$, may be
(significantly?) different than the coloron coupling $g$ in realistic extended
$\vec{r}$ models.

For subcritical coupling there are resonance solutions with positive $m^{2}$
that have large distance tails of external incoming and outgoing radiation,
representing a steady state of resonant production and decay.

With super-critical coupling, $g>g_{c}$, the bilocal field $\Phi(X,r)$ has a
negative squared mass eigenvalue (tachyonic), with a well-defined localized
wave-function. In the region external to the potential (forbidden zone) the
field is exponentially damped. At exact criticality with $g=g_{c}-\epsilon$
there is a $1/r$ (quasi-radiative) tail that switches to exponential damping
for $g=g_{c}+\epsilon$. The supercritical solutions are localized and
normalizable over the entire space $\vec{r}$, but with $m^{2}<0$ they lead to
exponential runaway in time of the field $\chi(X^{0})$, and must be
stabilized, typically with a $|\Phi|^{4}$ interaction.

We then treat the supercritical case as resulting in spontaneous symmetry
breaking. In the point-like limit, $\Phi(X)\sim\Phi(X,0)$, the theory has the
“sombrero potential”,

$\displaystyle V(\Phi)=-|M^{2}\Phi^{2}|+\frac{\lambda}{2}|\Phi|^{4}.$ (150)

The point-like field develops a VEV, $\langle\Phi\rangle=|M|/\sqrt{\lambda}$.
In this way the bound state theory will drive the usual chiral symmetry
breaking from the underlying dynamics of a potential induced by new physics.

The simplest sombrero potential can be modeled as,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\;S=\\!\\!\int_{X\vec{r}}^{\prime}\\!\biggl{(}\\!|\phi|^{2}\\!\left|\frac{\partial\chi}{\partial
X}\right|^{2}\\!\\!\\!-\\!|\chi|^{2}(|\nabla_{r}\phi|^{2}\\!+\\!g^{2}N_{c}MV(r)|\phi(r)|^{2})$
$\displaystyle\qquad\qquad-|\chi|^{4}\frac{\hat{\lambda}}{2}|\phi(\vec{r})|^{4}\biggr{)}.$
(151)

In the case of a perturbatively small $\lambda$ we expect the eigensolution of
$\phi$ to be essentially unaffected,

$\displaystyle\int^{\prime}_{r}\\!\\!\biggl{(}\\!|\nabla_{\vec{r}}\phi|^{2}\\!+\\!{g^{2}N_{c}\\!M}V(\vec{r})|\phi(\vec{r})|^{2}\\!\biggr{)}\approx
m^{2}.$ (152)

The effective quartic coupling is then further renormalized by the internal
wave-function,

$\displaystyle\frac{\hat{\lambda}}{2}|\chi|^{4}\int^{\prime}_{r}|\phi(\vec{r})|^{4}=|\chi|^{4}\frac{\widetilde{\lambda}}{2}.$
(153)

In this case we see that $\chi$ develops a VEV in the usual way:

$\displaystyle\langle|\chi|^{2}\rangle=|m^{2}|/\widetilde{\lambda}=v^{2}.$
(154)

This is consequence of $\phi(\vec{r})$ remaining localized in its potential

The external scattering state fermions, $\psi^{a}(X)$, will then acquire mass
through the emergent Yukawa interaction described in the previous section,
$\sim g_{Y}\langle|\chi|\rangle.$ For general $\widetilde{\lambda}$ and
possibly large (as in a nonlinear sigma model), the situation is potentially
more complicated. While perturbative solutions maintain locality in
$\phi(\vec{r})$, it is unclear what solutions exist for non-perturbative
$\lambda$.

##  VI A Personal recollection of Yoichiro Nambu

In the 1990’s Nambu had taken a considerable interest in the idea of top quark
condensation, and we became friends. During this era, and on several
occasions, he would surprisingly appear at my office door at Fermilab. I
sensed he enjoyed an occasional sojourn away from the city, and perhaps the
open space of cornfields and the lab had some appeal. On these visits we would
have discussions on far ranging topics, including historical Japanese baseball
games, bridge building, and his basement experiments (Bruce Winstein told me
years earlier that Nambu had a sophisticated optics lab in his basement and a
penchant for experiment).

Nambu’s thinking about physics was subtle and connections that interested him
were often seemingly remote to others. Some examples were smoke rings and
strings, Bode’s law of planetary orbits and naturalness, and that he thought
there was a hidden supersymmetry in the NJL model (which inspired me to study
this question and led to an “approximate SUSY” in super  which may be the
leading terms in a power series in $\Lambda^{-2}$, but I consider the question
still open). Nambu considered that the heavy top quark may be special and that
it may play a significant role in electroweak physics, which is certainly a
view I shared and still have to this day.

In 1992 I attended a meeting in Hiroshima Hiroshima  on electroweak symmetry
breaking and gave a talk on top quark condensation. My talk was a long one,
and was last of the morning session, followed by a bus that would take the
speakers to a fine local restaurant for lunch. At the conclusion of my talk,
however, I was surrounded by a number of students and post-docs with many
questions about technical details. Mainly, there was extant confusion as to
whether a particular bubble-sum calculation was better than another. I was
mainly trying to dispel this, arguing that the renormalization group is the
unique and far superior way to view the dynamics of NJL. The discussion ran on
for almost a half an hour and I missed the bus.

I finally exited the large conference center onto a plaza and studied the
surrounding streets in search of a convenient restaurant for lunch (and a
beer). I couldn’t spot one, and I expected I’d have to walk some distance from
the plaza, when suddenly someone tapped me on the shoulder…it was Nambu!

Nambu said, “I am taking you to lunch to a special place for the Japanese
delicacy, called ‘okonomyaki,’ which is particularly good here.” So we hiked
several blocks to a subway, which took us to a transfer where we boarded a
street tram. I was struck by the tram, which was old, dark green, and wooden
and resembled photos of those I had seen of Hiroshima after the war. The tram
progressed to the coast and to the harbor, which was modern and active with
large container ships and a buzz of loading–unloading operations underway.
Here we came upon a diner, crowded with workers, and we were seated at a small
table. Nambu did the ordering, and soon our plates of okonomiyaki arrived with
Asahi beers.

The okonomiyaki was for me a bit problematic, though possibly it could become
an acquired taste. It seemed to be a kind of pizza with the parts of the fish
partially grilled on top, some more or less so, and all of the parts of fish
(including innards) seemed part of the recipe. I mustered a slight smile and
nod to Nambu who could clearly tell my relishing of the delicacy was somewhat
muted. So I struggled for some conversational gratuity, and I asked him about
his early days in physics.

He talked about the end of the war and his first post-doc at the Institute for
Advanced Study. Oppenheimer was in charge and Albert Einstein was in
residence. Nambu had nearly completed his post-doctoral appointment and had
befriended Einstein’s secretary. She was interested in Japanese culture, and
practiced a little of the language with him (I am not sure if this was Helen
Dukas or someone else).

As the end of his appointment approached, Nambu asked if it would be possible
to meet Professor Einstein? His secretary said she could arrange it, but it
was very important that Oppenheimer not find out. Oppenheimer had put out an
edict that “no-one shall bother Prof. Einstein,” and Oppenheimer could be
quite imperious in his management style. So, an appointment was scheduled late
in the day, 5 PM or so, when Einstein would still be working quietly in his
office.

On the appointed day Nambu arrived at Einstein’s office door and gently
knocked. From within came the distinctive voice, “come in.” Nambu entered and
bowed in Japanese style to the professor who was seated at his desk, pipe in
hand. After introduction, Einstein said, “So what are you working on?”999I
later heard from Leon Lederman of his two encounters with Einstein, that this
was the standard greeting. You could then expect about a minute to explain
what you were doing and Einstein would then reply. In Leon’s case, he
described the K-mesons on one occasion, and multiple neutrinos on another.
Each time Einstein’s reply was ”achh, so many particles. If I wanted to study
so many species I would have been a botanist!”

Nambu had been working on collective phenomena in condensed matter physics,
superconductivity, Ising models, etc., topics which clearly influenced his
later thinking about symmetry breaking in particle physics. I didn’t quite
catch specifically what problem he described to Einstein but after a minute
Einstein reacted.

The Moon was rising in the early evening and visible through Einstein’s office
window. Einstein was tamping tobacco into his pipe with a piece of chalk. He
then pointed the chalk toward the Moon and said, “Tell me…do you think that
does not simultaneously have a position and a momentum?” There was silence.
Einstein then launched into an animated diatribe against the quantum theory.
He became rather exercised in this, and Nambu was terrified. After a few
minutes Nambu stood up, bowed, humbly backed away, thanking the great
professor, and made a hasty retreat.

Nambu was mortified by this experience, but to add to his misery Oppenheimer
found out about the encounter and was furious and summoned Nambu to his office
for a stern admonishment. Nambu became depressed in his last few weeks at the
IAS, and packed his bags and prepared to return, jobless, to Japan. Literally,
however, the last time he went to collect his mail at the IAS, there was a
letter… from The University of Chicago and it was an offer for him to join the
faculty.

We finished our okonomyaki and made the reverse trip back to the conference
center. I wondered then, and have ruminated ever since, about the strange
irony of this event:

In Hiroshima I had lunch with the great physicist, Yoichiro Nambu, who
explained the broken chiral symmetry of strong interaction mass and the
special properties of nearly massless pi-mesons as a general phenomenon, who
had a remarkable encounter with Albert Einstein, who created modern physics,
gave us $E=mc^{2}$, and even created the idea of quanta (though rejected the
formulation of the quantum theory), only to infuriate J. Robert Oppenheimer,
who managed the construction of the bomb that destroyed … Hiroshima.

Years later I was fortunate to attend the Nobel luncheon for Nambu at The
University of Chicago in 2008. Nambu was unable to travel to Stockholm due to
his wife’s illness, and the Swedish ambassador arranged to present the medal
to him in Chicago. The luncheon delightfully turned out to be a quaint
gathering of about a dozen people, including Nambu’s son, John.

At the Nobel luncheon the conversation lulled. I could tell Nambu was happy to
achieve this award, but at a deeper level there was a visible sadness at the
prospect of his wife’s passing. So I asked him to tell the Einstein story to
the luncheon table. He did so, and I felt that momentarily he was lifted by
the diversion.

The Nobel luncheon was my last encounter with Nambu, who retired and returned
to Japan, and passed away in 2015. Ironically, Nambu had also won the
Oppenheimer Memorial Prize in 1977.

## Acknowledgments

I thank the Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359
with the U.S. Department of Energy, Office of Science, Office of High Energy
Physics, and the University of Wisconsin, Madison.

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