disposition disposition

CERN-TH-2023-201

QCD with an Infrared Fixed Point and a Dilaton

Roman Zwickya,b
a Higgs Centre for Theoretical Physics, School of Physics and Astronomy,
The University of Edinburgh
b Theoretical Physics Department, CERN,
Esplanade des Particules 1, Geneva CH-1211, Switzerland

E-Mail<EMAIL_ADDRESS>

###### Abstract

Following previous work we further explore the possibility that the chirally
broken phase of gauge theories admits an infrared fixed point interpretation.
The slope of the $\beta$ function, $\beta^{\prime}_{*}$, is found to vanish at
the infrared fixed point which has several attractive features such as
logarithmic running. We provide a more in-depth analysis of our previous
result that the mass anomalous assumes $\gamma_{*}=1$ at the fixed point. The
results are found to be consistent with ${\cal N}=1$ supersymmetric gauge
theories. In a second part the specific properties of a dilaton, the (pseudo)
Goldstone, due to spontaneous symmetry breaking are investigated. Dilaton soft
theorems indicate that a soft dilaton mass can only originate from an operator
of scaling dimension two. In the gauge theory this role is taken on by the
$\bar{q}q$-operator. The QCD dilaton candidate, the $\sigma=f_{0}(500)$ meson
is investigated. Singlet-octet mixing is found to be important. We briefly
discuss the dilaton as a candidate for the Higgs boson, which relies on the
ratio of dilaton to pion decay constant being close to unity. In QCD this is
approximately satisfied but it is remains unclear if this is accidental or
whether there is unknown reason behind it.

###### Contents

  1. 1 Introduction
  2. 2 The Gauge Theory and its Low Energy Effective Theory
    1. 2.1 The gauge theory and its renormalisation group quantities
    2. 2.2 Synopsis of the conformal window
    3. 2.3 Low energy effective theory – dilaton-$\chi$PT at LO
  3. 3 Scalar Correlators with Goldstones in the Deep IR ($m_{q}=0$)
    1. 3.1 Renormalisation group analysis of correlators
    2. 3.2 Spectral analysis and matching to (dilaton-)$\chi$PT
      1. 3.2.1 Multi-Goldstone case: the operator ${S^{a}}$
      2. 3.2.2 Single-Goldstone case: the operators ${P^{a}}$ and ${S}$
      3. 3.2.3 Multi-Goldstone case II: the trace of the energy momentum tensor $T^{\rho}_{\;\;\rho}$
      4. 3.2.4 Renormalisation group derivation of $\beta^{\prime}_{*}=0$
    3. 3.3 Consequences of $\beta^{\prime}_{*}=0$
      1. 3.3.1 Logarithmic running
      2. 3.3.2 The vanishing of $\beta^{\prime}_{*}$ implies the vanishing of $\gamma^{\prime}_{*}$
  4. 4 Comparison with ${\cal N}=1$ Supersymmetric Gauge Theories
  5. 5 Dilaton Soft Theorems
    1. 5.1 Consequences of the single and the double soft dilaton theorem
  6. 6 Massive or Massless Dilaton?
    1. 6.1 Soft-theorem perspective
    2. 6.2 Large-$N_{c}$ consideration
    3. 6.3 Is dilaton-$\chi$PT consistent with a massless dilaton?
  7. 7 The Dilaton Candidate in QCD: $\sigma\equiv f_{0}(500)$
    1. 7.1 The width of the $\sigma$-meson
    2. 7.2 The mass of the $\sigma$-meson
  8. 8 Outlook
    1. 8.1 Quark mass dependence of the $\sigma$-meson(s) on the lattice and beyond
    2. 8.2 The Higgs as a dilaton?
  9. 9 Conclusions and Summary
  10. A Conventions
  11. B More on the soft theorems
    1. B.1 The double soft theorem from dilaton-$\chi$PT
    2. B.2 The mass operator in the deep infrared
    3. B.3 The single soft theorem, $\langle 0|\bar{q}q|D\rangle$ and the the dilaton decay constant
  12. C On Singlet-Octet Mixing

##  1 Introduction

That the strong interaction could be described by an infrared (IR) fixed point
is an idea [1, 2, 3, 4, 5, 6] that predates QCD itself. Applied in our
previous work Ref. [7], it was found in three different ways, that pion
physics can be reproduced when the quark mass anomalous dimension at the IR
fixed point assumes $\gamma_{*}=1$. The methods included the hyperscaling
relation for the pion mass, compatibility of the Feynman-Hellmann theorem with
the trace of the energy momentum tensor (TEMT) and matching a long distance
correlator. The main idea that underlies the scenario, is that the gauge
theory flows into an IR fixed point and that it is the quark condensate
$\langle\bar{q}q\rangle\neq 0$ that breaks _both_ the chiral and the conformal
symmetry spontaneously.111Throughout this paper we do not distinguish
conformal and scale symmetry. In $d=4$ it is believed that these are
equivalent for non-trivial unitary theories (cf. the review [8] or [7] for
more comments and references). This leads, besides the pion, to an additional
(pseudo) Goldstone, the dilaton. The fact that the conformal symmetry is only
emergent and that the dilaton has vacuum quantum numbers complicates matters.
Let us turn to the assumptions. The minimal implementation of IR conformality,
adopted in [7], is that the TEMT on single particle IR-states $\varphi$, which
include the vacuum, the pion and possibly the dilaton

$\langle\varphi^{\prime}(p^{\prime})|T^{\rho}_{\;\;\rho}|\varphi(p)\rangle_{q=0}=0\;,$
(1.1)

vanishes for zero momentum transfer $q=p-p^{\prime}$. In a true CFT (1.1)
holds on any physical state and momentum transfer and can be see as its
definition.222In the presence of a massless dilaton (1.1) holds for the LO
EFT: $T^{\rho}_{\;\;\rho}|_{{\text{LO}}}=0$ [9]; and extends to the case of
single nucleon states $\langle
N(p^{\prime})|T^{\rho}_{\;\;\rho}|N(p)\rangle_{q=0}=0$ despite $m_{N}\neq 0$
[10]. We make the reasonable assumption that if (1.1) holds true that there
exists a scheme with vanishing $\beta$ function in the IR:
$\beta_{*}\equiv\beta|_{\mu=0}=0$. This technical assumption is necessary
since it is the product with the field strength tensor
$\frac{\beta(\mu)}{2g(\mu)}G^{2}(\mu)$ which is physical and not the $\beta$
function by itself. Additionally, we assume that correlators obey conformal
field theory (CFT) scaling in the deep IR

$\langle{\cal O}(x){\cal O}(0)\rangle\propto\frac{1}{(x^{2})^{\Delta_{{\cal
O}}}}+\text{GB-corrections}$ (1.2)

corrected by terms due to Goldstone boson (GB). Above $\Delta_{{\cal
O}}=d_{{\cal O}}+\gamma_{{\cal O}}$ is the scaling dimension, defined as the
sum of the engineering and the anomalous dimension. We will see that (1.2)
essentially emerges by combining RG and EFT reasoning. Following the
terminology in [10] we will refer to this scenario as conformal dilaton (CD)
QCD.

Let us turn to the dilaton mass. It is generally believed that the dilaton
mass in the chiral limit $m_{q}\to 0$ (no explicit symmetry
breaking)333Throughout this work we refer to the dilaton as $D$ when generic
matters are discussed and to $\sigma(\leftrightarrow f_{0}(500))$ when
referring to the dilaton candidate in QCD and to $h$ in the context of the
Higgs boson.

$\overline{m}_{D}\equiv m_{D}|_{m_{q}=0}\;,$ (1.3)

is non-zero for an IR-emergent conformal symmetry. To the best of our
knowledge this issue has never been fully settled. We take a pragmatic
attitude by studying both cases in our investigations. For
$\overline{m}_{D}=0$ we use dilaton-$\chi$PT (d$\chi$PT ) and for
$\overline{m}_{D}\neq 0$ we use standard chiral perturbation theory ($\chi$PT
) since the dilaton can be integrated out. Phenomenologically a dilaton
approach might still be interesting if
$\overline{m}_{D}\ll\Lambda_{\text{hadron}}={\cal O}(m_{N})$ which is what
most approaches have in mind in their respective frameworks. This includes
lattice Monte Carlo investigation of gauge theories [11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24], EFT descriptions thereof [25, 26, 27, 28];
the dilaton as a Higgs boson within [29, 30] and beyond gauge theories [31,
32, 33, 34, 35]. Or the dilaton as a driving force of inflation [36], in dense
nuclear interactions [37, 38, 39], in cosmology [40, 41, 42] and the
$\sigma$-meson in QCD [43, 44]. The name dilaton is also used in composite
models for a light $0^{++}$-partner of pseudo Goldstones [45]. In the context
of gauge theories, it is generally believed that $\overline{m}_{D}$ becomes
larger with respect to the other hadronic scales as one moves away from the
conformal window by lowering the number of flavours.

Let us briefly summarise the main findings of this work. By RG methods it is
inferred that the slope of the $\beta$ function vanishes at the fixed point:
$\beta^{\prime}_{*}=0$. A result with many attractive features, as it is
compatible with ${\cal N}=1$ supersymmetric gauge theories, makes a light
dilaton more likely and implies logarithmic running near the fixed point as
one would expect from the EFT itself. A more thorough justification for
$\gamma_{*}=1$ in the context of matching correlation function is given
combining RG and EFT methods. Applying a double-soft dilaton theorem, it is
found that an operator giving a mass to the dilaton is of necessarily of
scaling dimension $\Delta_{{\cal O}}=2$. The second part of the paper is more
qualitative and contains a discussion on whether a dilaton can be massless,
applies d$\chi$PT to the $\sigma$-meson in QCD, and concerns the dilaton as a
Higgs boson. The main outstanding questions are: i) the size of the chiral
mass $\overline{m}_{D}$ (as the distance from the conformal window) and ii)
the ratio of decay constants $r_{N_{f}}={F_{\pi}}/{F_{D}}$ as a function of
the number of flavours $N_{f}$. If $r_{2}\approx 1$ was true, then this would
provide the rationale for coupling the dilaton like a Higgs and quite possibly
make it compatible with large hadron collider (LHC) constraints.

The paper is organised as follows. In Sec. 2 the gauge theory and d$\chi$PT
are defined including a conformal window review. QCD correlators in the deep
IR are matched to the EFTs in Sec. 3, establishing: i) $\gamma_{*}=1$ with
more detail and care than in our previous work and ii) the vanishing of the
slope of the $\beta$ function: $\beta^{\prime}_{*}=0$. Consistency with ${\cal
N}=1$ supersymmetric gauge theories is discussed in Sec. 4 . Double-soft
theorems are studied in Sec. 5. Arguments in favour and disfavour of a
massless dilaton are addressed in Sec. 6. The dilaton as the $\sigma$-meson in
QCD and as the Higgs boson are discussed in Secs. 7 and Sec. 8 respectively.
The paper ends with summary and conclusions in Sec. 9. Conventions, more on
soft theorems and details on mixing are deferred to Apps. A, B and C
respectively.

##  2 The Gauge Theory and its Low Energy Effective Theory

The core ingredients to this work are the gauge theory, its RG quantities and
the low energy EFT which corresponds to $\chi$PT and d$\chi$PT when a dilaton
is added. In this section we introduce them in minimal form.

###  2.1 The gauge theory and its renormalisation group quantities

The gauge theory Lagrangian is given by

${\cal
L}_{QCD}=-\frac{1}{4}G^{2}+\sum_{q=1}^{N_{f}}\bar{q}(i\not{D}-m_{q})q\;,$
(2.1)

where $G^{2}=G_{\mu\nu}^{a}G^{a\,\mu\nu}$ is the field strength tensor with
$a$ denoting the adjoint index of the gauge group,
$D_{\mu}=(\partial+igA)_{\mu}$ is the gauge-covariant derivative and the
quarks are in some unspecified representation of the gauge group. For the more
formal part of the paper the $N_{f}$ flavours are assumed to be degenerate.
There are two parameters in the Lagrangian, the gauge coupling $g$ and the
quark mass $m_{q}$ whose RG-behaviour we must follow. The gauge coupling is
relevant and irrelevant in the UV and IR respectively whereas for the quark
mass it is the other way around. Aspects of the global flavour symmetries are
discussed in the next section.

The anomalous dimensions of the parameters and their conjugate operators are
defined from the renormalised quantities as follows

$\displaystyle{\beta}$ $\displaystyle\;\equiv\;$
$\displaystyle\phantom{-}\frac{d}{d\ln\mu}g\;,\qquad$
$\displaystyle\gamma_{G^{2}}$ $\displaystyle\;\equiv\;$
$\displaystyle-\frac{d}{d\ln\mu}\ln G^{2}\;,$ $\displaystyle\gamma_{m}$
$\displaystyle\;\equiv\;$ $\displaystyle-\frac{d}{d\ln\mu}\ln m_{q}\;,\qquad$
$\displaystyle\gamma_{\bar{q}q}$ $\displaystyle\;\equiv\;$
$\displaystyle-\frac{d}{d\ln\mu}\ln\bar{q}q\;.$ (2.2)

A central quantity to this work is the TEMT [46, 47, 48, 49, 50, 51, 52]

$T^{\rho}_{\;\;\rho}|_{\text{phys}}=\frac{\beta}{2g}G^{2}+\sum_{q}m_{q}(1\\!+\\!\gamma_{m})\bar{q}q\;,$
(2.3)

since its vanishing on physical states signals conformality. The subscript
“phys” indicates that we have omitted terms which are proportional to the
equation of motion (and BRST exact terms which arise upon gauge fixing) which
nota bene vanish on physical states [50, 51, 52] . It is very useful to note
that (2.3) consists of two separately RG-invariant terms
$T^{\rho}_{\;\;\rho}={\cal O}_{1}+{\cal O}_{2}$

${\cal
O}_{1}=\frac{\beta}{2g}G^{2}+\sum_{q}\gamma_{m}m_{q}\bar{q}q\;,\quad{\cal
O}_{2}=\sum_{q}m_{q}\bar{q}q\;.$ (2.4)

Imposing $\frac{d}{d\ln\mu}{\cal O}_{1,2}=0$ results in

$\gamma_{G^{2}}=\beta^{\prime}-\frac{\beta}{g}\;,\quad\gamma_{\bar{q}q}=-\gamma_{m}\;,$
(2.5)

if one considers a quark mass independent scheme for the $\beta$ function.
Quantities at the IR fixed point are denoted by a star

$(\gamma_{G^{2}})_{*}=\beta^{\prime}_{*}\;,\quad\gamma_{*}\equiv(\gamma_{m})_{*}\;,$
(2.6)

where $\beta_{*}=0$ has been assumed as stated earlier. In the context of
conformal field theories (CFT) e.g [53, 54, 55, 56, 57, 58], the important
quantities are the operator scaling dimensions, which consist of the sum of
the engineering and the anomalous dimension $\Delta_{\cal O}\equiv d_{\cal
O}+\gamma_{\cal O}$,

$\displaystyle\Delta_{G^{2}}$ $\displaystyle\;=\;$ $\displaystyle
d+\beta^{\prime}_{*}$ $\displaystyle\;\to\;$ $\displaystyle\quad
4+\beta^{\prime}_{*}\;,$ $\displaystyle\Delta_{\bar{q}q}$ $\displaystyle\;=\;$
$\displaystyle(d-1)-\gamma_{*}$ $\displaystyle\;\to\;$ $\displaystyle\quad
3-\gamma_{*}\;,$ (2.7)

and the $d$-dimensional expressions have been given for later use.

###  2.2 Synopsis of the conformal window

For the reader not acquainted with the basic notation of the conformal window
we briefly summarise its minimal content necessary to understand what is to
follow. The well-understood ${\cal N}=1$ supersymmetric case is described in
Sec. 4. The conformal window is the study of different phases of gauge
theories, cf. Fig. 1, as a function of the the gauge group e.g. $SU(N_{c})$
and the quark representation of $N_{f}$ massless fermions. The conformal
window has first been intensely studied within technicolor model building
(reviewed in [59, 60, 61]). The current understanding is based on the ${\cal
N}=1$ supersymmetric case and non-supersymmetric lattice studies.

Figure 1: Sketch of the conformal window for $G=SU(N_{c})$ with $N_{c}=3$. For
$N_{f}>16$ asymptotic freedom is lost and those theories are not considered.
Below there is the perturbative Caswell-Banks-Zaks IR fixed point. As $N_{f}$
is lowered the IR fixed point coupling becomes stronger until chiral symmetry
breaking sets in (for some unknown critical $\bar{N}_{f}$). In this paper we
explore the possibility that the theory admits an IR fixed point
interpretation in parts or all of the broken phase as well. The evolution of
the IR fixed point quantities $\gamma_{*}$ and $\beta^{\prime}_{*}$, to be
deduced in Sec. 3, are shown as a function of $N_{f}$. We stress that the
evolution between $\bar{N}_{f}$ and $16$ is only schematic.

The main point is that in the $(N_{f},N_{c})$-plane, in the domain of
asymptotic freedom, there are two phases: one where the theory flows into an
IR fixed point and the QCD-phase. In the latter there is confinement and
chiral symmetry is spontaneously broken by the quark condensate
$\langle\bar{q}q\rangle\neq 0$ (the global flavour symmetry breaks from
$SU(N_{f})_{L}\otimes SU(N_{f})_{R}\to SU(N_{f})_{I}$ to the diagonal subgroup
accompanied by $N_{f}^{2}-1$ massless pions). If we fix $N_{c}=3$ then we know
that around $N_{f}\approx 16$ the theory admits a perturbative IR fixed point
(so-called Caswell-Banks-Zaks FP [62, 63]) and that somewhere in between
$N_{f}=16$ and $N_{f}=3$ (QCD) there is the transition into the QCD phase.
Where this happens exactly is a matter of intense debate but known exactly in
${\cal N}=1$ supersymmetric gauge theories.

###  2.3 Low energy effective theory – dilaton-$\chi$PT at LO

In this section the dilaton EFT is discussed, its starting point is the well-
understood $\chi$PT [64, 65, 66, 67] which is the EFT of $N_{f}^{2}-1$ pions
describing a wealth of data at low energy. Based on this successful framework
the dilaton, the Goldstone due to the scale symmetry breaking, is added. The
Goldstone bosons are parameterised in exponential form

$U=e^{i2\pi^{a}T^{a}/F_{\pi}}\;,\quad\chi\equiv F_{D}e^{-D/F_{D}}\;,$ (2.8)

where the quantities $F_{\pi,D}$ are fundamental in that they are the order
parameters of the symmetry breaking

$\langle\pi^{b}(q)|J_{5\mu}^{a}(x)|0\rangle=iF_{\pi}q_{\mu}\delta^{ab}e^{iqx}\;,\quad\langle
D(q)|J^{D}_{\mu}(x)|0\rangle=i{F_{D}}q_{\mu}e^{iqx}\;,$ (2.9)

defined as the matrix elements of the global symmetry currents

$J_{5\mu}^{a}(x)=\bar{q}(x)T^{a}\gamma_{\mu}\gamma_{5}q(x)\;,\quad
J^{D}_{\mu}(x)=x^{\nu}T_{\mu\nu}(x)\;,$ (2.10)

between the vacuum and the corresponding Goldstone. It is noted that the
equation for the dilaton in (2.9) is equivalent to

$\langle
D(q)|T_{\mu\nu}|0\rangle=\frac{F_{D}}{{d-1}}(m_{D}^{2}\eta_{\mu\nu}-q_{\mu}q_{\nu})\;,$
(2.11)

when taking into account that $x^{\nu}=-i\partial_{q_{\nu}}$. From the zeroth
component of the currents one can see heuristically that the charge of the
symmetry does not annihilate the vacuum and thus signals SSB. The symmetries
of the Goldstones are said to be realised non-linearly

$U\to LUR^{\dagger}\;,\qquad\chi\to\chi e^{\alpha(x)}\;,$ (2.12)

where $(L,R)\in SU(N_{f})_{L}\otimes SU(N_{f})_{R}$ and
$\alpha(x)\in\mathbb{R}$ parameterise the standard chiral and the Weyl
transformation respectively. The latter can be interpreted as the
implementation of scale transformations on the metric $g_{\mu\nu}$, rather
than the coordinates, and the fields $\varphi$

$g_{\mu\nu}\to e^{-2\alpha(x)}g_{\mu\nu}\;,\quad\varphi\to
e^{s_{\varphi}\alpha(x)}\varphi\;,$ (2.13)

where $s_{\varphi}$ is called the Weyl-weight which is not to be confused with
the engineering dimension $d_{\varphi}$ of the field. The Weyl-weight of the
metric is $s_{g_{\mu\nu}}=-2$ since it naturally contracts to coordinate
vectors $x^{2}=g_{\mu\nu}x^{\mu}x^{\nu}$ which would transform as $x^{\mu}\to
e^{\alpha(x)}x^{\mu}$. Local Weyl invariance,

$\sqrt{-g}{\cal L}\to(e^{-\alpha d}\sqrt{-g})(e^{\alpha d}{\cal
L})=\sqrt{-g}{\cal L}\;,$ (2.14)

is the guiding principle for the low energy EFT. The $\alpha$ space-dependence
has and will be suppressed occasionally hereafter. In the case of explicit and
anomalous symmetry breaking the spurion technique will have to be adapted.

The leading order (LO) Lagrangian consists of

${\cal L}_{\text{LO}}={\cal L}^{(\pi)}_{\text{kin}}+{\cal
L}^{(D)}_{\text{kin}}+{\cal L}^{R}_{d}+{\cal L}_{m_{q}}-V(\chi)+{\cal
L}_{\text{anom}}\;,$ (2.15)

two kinetic-, an improvement-, a quark mass-, an anomaly- and a dilaton
potential-term. The kinetic terms read

${\cal
L}^{(\pi)}_{\text{kin}}=\frac{F_{\pi}^{2}}{4}\hat{\chi}^{d-2}\mathrm{Tr}[\partial^{\mu}U\partial_{\mu}U^{\dagger}]\;,\quad{\cal
L}^{(D)}_{\text{kin}}=\frac{1}{2}\chi^{d-4}(\partial\chi)^{2}\;,$ (2.16)

where hatted quantities are, hereafter, understood to be dimensionless divided
by the appropriate power of $F_{D}$, e.g. $\hat{\chi}\equiv{\chi/F_{D}}$. The
pion kinetic term is standard and the prefactor $\chi^{2}$ indicates that the
pion has zero Weyl-weight which can be deduced directly from the conformal
algebra [3]. In Ref. [9] the following coupling of the dilaton to the Ricci
tensor was introduced

${\cal
L}^{R}_{d}=\frac{1}{2(d-2)(d-1)}\,R\,\chi^{d-2}\to\frac{1}{12}\,R\,\chi^{2}\;.\quad$
(2.17)

It renders the dilaton kinetic term locally Weyl-invariant which has many
advantages.444The term (2.17) is the adaption of the improved EMT [68] to the
Goldstone case for which the presence of the dilaton is needed to improve the
pion. It leads to
$T^{\rho}_{\;\;\rho}|^{\text{d}\chi\text{PT}}_{{\text{LO}}}=0$ for $m_{q}=0$.
It improves flow theorems [9]. The quark mass term reads (${\cal
M}\equiv\text{diag}(m_{q_{1}},\dots,m_{q_{N_{f}}})$)

${\cal L}_{m_{q}}=\frac{B_{0}F_{\pi}^{2}}{2}\mathrm{Tr}[{\cal
M}U+U^{\dagger}{\cal M}^{\dagger}]\hat{\chi}^{\Delta_{\bar{q}q}}\;,\quad
B_{0}\equiv-\frac{\langle\bar{q}q\rangle}{F_{\pi}^{2}}\;,$ (2.18)

where $B_{0}$ assures that the Gell-Mann Oakes Renner (GMOR)-relation
$m_{\pi}^{2}F_{\pi}^{2}=-2m_{q}\langle\bar{q}q\rangle$ [99, 67] is reproduced.
The chiral symmetry is formally restored by assigning the spurious
transformation rule ${\cal M}\to L^{\dagger}{\cal M}R$ which in turn dictates
the form in (2.18). The quantities $B_{0}$ and $F_{\pi}$ are the two
(independent) low energy constants of LO $\chi$PT. The quark mass corresponds
to explicit Weyl symmetry breaking but the latter can be restored by assigning
the spurious transformation $m_{q}\to e^{(1+\gamma_{m})\alpha}m_{q}$ and the
extra factor $\hat{\chi}^{\Delta_{\bar{q}q}}$ is then required to render
$\sqrt{-g}{\cal L}_{m_{q}}$ Weyl invariant. Alternatively we may regard
$m_{q}$ as the source for $\bar{q}q$ with the factor
$\hat{\chi}^{\Delta_{\bar{q}q}}$ capturing its scaling. The remaining two
terms are more delicate and need more elaboration.

Let us first turn to the anomalous term for which the $m_{q}\to 0$ limit is to
considered such that $T^{\rho}_{\;\;\rho}=\frac{\beta}{2g}G^{2}$. It is not
well understood whether a term needs to be added for the trace anomaly in the
EFT in analogy with the WZW-term for the chiral anomaly in $\chi$PT (e.g.
[65]). Clearly, the leading effect would be captured by $\beta^{\prime}_{*}$
and is sometimes parameterised in the EFT as $\beta^{\prime}_{*}$ times
operators in the LO TEMT cf. [44, 30] (and also [33]). Using RG methods we
will establish $\beta^{\prime}_{*}=0$ in Sec. 3 which implies that $\delta
g=g-g_{*}$ runs logarithmically instead of power-like close the the fixed
point. We would therefore think that the EFT can capture these effects through
its own loops and matching of NLO low energy constants with QCD. Hence, we
will drop ${\cal L}_{\text{anom}}$ from the LO Lagrangian.

We turn to the discussion of the potential. It has been recognised by Zumino a
long time ago [69] that, at least in the absence of anomalous breaking, the
only permissible term in the potential is $V_{Z}(\chi)=\lambda\chi^{d}$. Such
a term on its own is troublesome, cf. Ref. [30] for detailed discussion, as it
generates a linear as well as a mass term. Hence it was concluded that
$\lambda$ must be a function of a symmetry breaking parameter (such as $m_{q}$
in the discussion below). A simple and frequently used form of a potential is
given by adding another power $\chi^{\Delta}$ which can be written as
($V^{\prime}(1)=0$ and $V^{\prime\prime}(1)=m_{D}^{2}$)555In the literature
this potential is often discussed without specific reference to the
$\bar{q}q$-operator. For $\Delta\to 4$ this potential assumes the famous
logarithmic form used as an ansatz for an EFT of pure Yang-Mills [70].

$\displaystyle V_{\Delta}(\hat{\chi})$ $\displaystyle\;=\;$
$\displaystyle\frac{m_{D}^{2}F_{D}^{2}}{\Delta-d}\left(\frac{1}{\Delta}\hat{\chi}^{\Delta}-\frac{1}{d}\hat{\chi}^{d}\right)$
$\displaystyle\;=\;$
$\displaystyle\text{const}+m_{D}^{2}F_{D}^{2}\big{(}\frac{1}{2}\hat{D}^{2}-\frac{d+\Delta}{3!}\hat{D}^{3}+\frac{(d+\Delta)^{2}}{4!}\hat{D}^{4}+{\cal
O}(D^{5})\big{)}\;,$ (2.19)

with a well-defined minimum. The extra term is usually associated with the
presence of an operator of scaling dimension $\Delta$ which breaks conformal
invariance. In Sec. 5 we will see that in our framework the soft theorem seems
to indicate that $\Delta=d-2$. Indeed the quark mass term in ${\cal
L}_{m_{q}}|_{\pi=0}$ (2.18) corresponds to the
$\Delta={\Delta_{\bar{q}q}}=d-2$ term for which the Zumino-term (2.20) assumes
the form

$V_{Z}(\hat{\chi})=\frac{{\Delta_{\bar{q}q}}}{d}\sum_{q=1}^{N_{f}}m_{q}\langle
qq\rangle\hat{\chi}^{d}\;,$ (2.20)

where $\lambda\propto m_{q}$ is a special case of the symmetry breaking
parameter discussed above. It assures positivity of the dilaton

$F_{D}^{2}m_{D}^{2}=(d-{\Delta_{\bar{q}q}}){\Delta_{\bar{q}q}}\frac{N_{f}}{2}F_{\pi}^{2}m_{\pi}^{2}\Big{|}_{d=4,{\Delta_{\bar{q}q}}=2}=2N_{f}F_{\pi}^{2}m_{\pi}^{2}\;,$
(2.21)

as it contributes the ${\Delta_{\bar{q}q}}d$-term. In analogy to the pion case
we will refer to (2.21) as the dilaton GMOR-relation. It will be derived from
a double-soft theorem in Sec. 5. To this end let us summarise the LO d$\chi$PT
Lagrangian considered for $d=4$ (${\Delta_{\bar{q}q}}=2$)

$\displaystyle{\cal L}^{\text{d}\chi\text{PT}}_{\text{LO}}$
$\displaystyle\;=\;$
$\displaystyle\frac{F_{\pi}^{2}}{4}\hat{\chi}^{2}\mathrm{Tr}[\partial^{\mu}U\partial_{\mu}U^{\dagger}]+\frac{1}{2}(\partial\chi)^{2}+\frac{1}{12}\chi^{2}R\quad+$
$\displaystyle\frac{B_{0}F_{\pi}^{2}}{2}\big{(}\mathrm{Tr}[{\cal
M}U^{\dagger}+U{\cal M}^{\dagger}]\,\hat{\chi}^{2}-\mathrm{Tr}[{\cal M}+{\cal
M}^{\dagger}]\,\hat{\chi}^{4}\big{)}\;,$ (2.22)

where the anomalous part has been dropped as discussed above. We refrain from
parameterising further potential terms at LO, as they are not really required
for our work and neither is it clear what form they would take. If we want to
derive the insertion of the $\bar{q}q$-operator then it is important to
realise that its source $s(x)$ is only to be substituted in the term
containing $U$ where it acts as a true spurion, ${\cal M}U^{\dagger}\to({\cal
M}+s)U^{\dagger}$, and not in the Zumino-term. We further note that similar
Lagrangians have frequently appeared in the literature [71, 29, 44, 25, 27].
The novelty in our case is the justification for the absence of the anomaly
term, the ${\Delta_{\bar{q}q}}=3-\gamma_{*}=2$ requirement and with respect to
many references the inclusion of the $R\chi^{2}$-term.

As we sometimes consider the $\chi$PT Lagrangian on its own, we quote its LO
version

${\cal
L}^{\chi\text{PT}}_{\text{LO}}=\frac{F_{\pi}^{2}}{4}\mathrm{Tr}[\partial^{\mu}U\partial_{\mu}U^{\dagger}]+\frac{B_{0}F_{\pi}^{2}}{2}\mathrm{Tr}[{\cal
M}U^{\dagger}+U{\cal M}^{\dagger}]\;,$ (2.23)

which follows from (2.3) by setting $\chi\to 1$ and dropping constant terms.

##  3 Scalar Correlators with Goldstones in the Deep IR ($m_{q}=0$)

In CFTs correlation functions are determined by a minimal amount of
information. For example, two-point functions are characterised by a single
parameter

$\langle{\cal O}(x){\cal
O}^{\dagger}(0)\rangle_{\text{CFT}}\propto\frac{1}{(x^{2})^{\Delta_{\cal
O}}}\;,$ (3.1)

the scaling dimension $\Delta_{{\cal O}}$ introduced earlier and
$\langle\dots\rangle$ stands for the vacuum expectation value (VEV) hereafter.
For theories which flow into an IR fixed point, the so-called conformal window
cf.Fig. 1, (3.1) represents the leading behaviour in the deep IR,
$x^{2}\to\infty$. The aim of this section is to investigate how this picture
is affected in the presence of Goldstone bosons (due to scale and chiral
symmetry breaking). Generally, one may expect the massless Goldstones to
affect the long distance behaviour. Our basic tool is that the CDQCD and the
EFT description requires that the following

$\langle{\cal O}(x){\cal O}^{\dagger}(0)\rangle_{\text{CDQCD}}=\langle{\cal
O}(x){\cal O}^{\dagger}(0)\rangle_{\text{d}\chi\text{PT}}\;,\quad\text{for
}x^{2}\to\infty\;,$ (3.2)

must hold, as it does for any EFT interpretation since these theories describe
the same IR-physics. We will first analyse the two-point functions from the RG
viewpoint combining it with some knowledge from the EFT described in Sec. 2.3.
This will provide a more profound understanding of the analysis in our
previous work [7] and the reason why the scalar adjoint correlation function
was singled out.

Let us introduce some notation. Operators ${\cal O}$ split into ones

${S}=\bar{q}q\;,\quad{P^{a}}=\bar{q}i\gamma_{5}\,T^{a}q\;,$ (3.3)

which couple to single Goldstones and those

${S^{a}}=\bar{q}T^{a}q\;,\quad T^{\rho}_{\;\;\rho}=\frac{\beta}{2g}G^{2}\;,$
(3.4)

that do not. The quantum numbers of the operators are $J^{PC}({S})=0^{++}$,
$J^{P}({P^{a}})=0^{-}$, $J^{P}({S^{a}})=0^{+}$ and
$J^{PC}(T^{\rho}_{\;\;\rho})=0^{++}$ and their scaling dimensions,
$\Delta_{\cal O}\equiv d_{\cal O}+\gamma_{\cal O}$, are given by

$\Delta_{{P^{a}}}=\Delta_{{S}}=\Delta_{{S^{a}}}=3-\gamma_{*}\;,\quad\Delta_{T^{\rho}_{\;\;\rho}}=4+\beta^{\prime}_{*}\;.$
(3.5)

The reason of why (3.5) hold true for the currents can be found in [7], and
for $T^{\rho}_{\;\;\rho}$ it is given in App. A. It is noted that the scaling
dimension of $G^{2}$ and $T^{\rho}_{\;\;\rho}$ are the same,
$\Delta_{T^{\rho}_{\;\;\rho}}=\Delta_{G^{2}}$, since multiplying by the
$\beta$ function which is a scalar does not alter the operators’ behaviour.
Two-point functions of ${\cal O}$ will sometimes be abbreviated as

$\Gamma_{{\cal O}}(x^{2})\equiv\langle{\cal O}(x){\cal O}(0)\rangle\;,$ (3.6)

and refer to Euclidean space unless otherwise stated.

###  3.1 Renormalisation group analysis of correlators

It is intuitively clear that Goldstones will alter and complicate the IR-
behaviour. From the formal RG viewpoint the main change is the presence of an
additional scale [10]. There are in fact two, the pion and the dilaton decay
constant but as they are of the same order we may group them into one single
quantity $F=F_{D,\pi}$. Under these circumstances the RG equation for the
correlators of the type (3.6) assume the following form, e.g. [72],

$(2\partial_{\ln x^{2}}-y_{F^{2}}\partial_{\ln F^{2}}+\Delta_{{\cal
O}})\Gamma_{{\cal O}}(x^{2},F^{2})=0\;,$ (3.7)

where we have neglected the $\beta\,\partial_{\ln g}$-term since its effect is
subleading as will become clearer later. Note that
$y_{F^{2}}=d_{F^{2}}+\gamma_{F^{2}}=2$ since $F$ has vanishing anomalous
dimension. The solution of this equation reads

$\Gamma_{{\cal O}}(x^{2},F^{2})\propto\frac{1}{(x^{2})^{\Delta_{\cal
O}}}h_{{\cal O}}(x^{2}F^{2})\;,$ (3.8)

where in general $h_{{\cal O}}$ is arbitrary such that predictiveness is
essentially lost. However, we can improve this situation by matching to
(d)$\chi$PT , as in (3.2), taking advantage of its Lagrangian (2.3). We make
the following observations:

  * a)

The EFT cannot produce any fractional powers of $1/x^{2}$. Whereas the EFT
expansion is in powers of $1/(x^{2}F^{2})$ it could be that $\ln
x^{2}$-corrections, related to the neglect of the $\beta$ function, resum to
fractional powers ($n$ integer and $\eta$ not)

$\frac{1}{(x^{2})^{(n+\eta)}}=\frac{1}{x^{2n}}e^{-\eta\ln
x^{2}}=\frac{1}{x^{2n}}(1-\eta\ln x^{2}+\frac{\eta^{2}}{2}\ln x^{2}+\dots)\;.$
(3.9)

The answer is however negative since $\eta$ itself must be proportional to
inverse powers of $F$, but then there is no other scale in the chiral limit to
make $\eta$ dimensionless. Hence we conclude that $\ln x^{2}$-terms can only
appear in subleading terms which is fairly natural from an EFT perspective
(cf. Sec. 3.2.3 for a more explicit discussion of this aspect). Thus we may
write

$h_{{\cal O}}(z)=a_{k_{{{\cal O}}}}z^{k_{{{\cal
O}}}}+\dots+a_{0}+a_{-1}\frac{1}{z}+{\cal O}(z^{-2})+\ln\text{-terms}$ (3.10)

where $k_{{\cal O}}$ is some positive integer and subleading terms contain
$\ln x^{2}$-corrections which we have not indicated explicitly.

  * b)

Hence we have learned that the overall powers are integers. There is an
ambiguity left and that is the interpretation of the $\Delta_{{\cal O}}$
coefficient (or the actual number $k_{\cal O}$). If the operator ${\cal O}$
shares the quantum numbers with the Goldstone boson $\pi$ and
$\langle\pi|{\cal O}|0\rangle\neq 0$, then $\Gamma_{{\cal O}}(x^{2})$ scales
as

$\Gamma_{{\cal O}}(x^{2})\propto\frac{1}{x^{2}}+{\cal O}(x^{-4})\;,$ (3.11)

as a consequence of the spectral representation. Are we to interpret
$\Delta_{{\cal O}}$ with this contribution, that is $\Delta_{{\cal O}}=1$? The
answer is negative since then ${\cal O}$ would correspond to a free field e.g.
[73]. This is however not the case since all operators considered are capable
of producing multiparticle states as we shall see shortly.

  * c)

There is a second exception. It could be that in the leading term
$\Gamma_{{\cal O}}(x^{2})=c/(x^{2})^{\Delta_{\cal O}}$, $c=0$ due to some
symmetry. In this case $\Delta_{{\cal O}}$ is to be identified from the
leading correction

$\Gamma_{{\cal O}}(x^{2})\propto\frac{1}{(x^{2})^{\Delta_{\cal
O}+n_{{\text{LO}}}}}\;.$ (3.12)

This is the case for the $T^{\rho}_{\;\;\rho}$-correlator in the case of
d$\chi$PT as we shall see.

Finally, we conclude that it is the leading multiparticle state contribution
which is to be identified with $\Delta_{{\cal O}}$ provided the exceptional
case under item c) does not occur.

###  3.2 Spectral analysis and matching to (dilaton-)$\chi$PT

Here we aim to underlay the previous discussion from the viepoint of the
spectral representation which provides a more intuitive insight into the
matter. Below we give a brief summary and refer to e.g. [74, 75] for further
reading. In Minkowski space the spectral density $\rho(s)$

$\bar{\Gamma}_{{\cal O}}(p^{2})\equiv i\int d^{4}x\,e^{ipx}\langle T{\cal
O}(x){\cal O}^{\dagger}(0)\rangle=\int_{0}^{\infty}ds\,\frac{\rho_{{\cal
O}}(s)}{s-p^{2}-i0}\;,$ (3.13)

of the two-point function is proportional to the imaginary part

$\rho_{{\cal O}}(s)=\frac{1}{\pi}\textrm{Im}\bar{\Gamma}_{{\cal O}}(s)\;.$
(3.14)

The same spectral function enters the Euclidean correlation function

$\int d^{4}x\,e^{ipx}\langle{\cal O}(x){\cal
O}^{\dagger}(0)\rangle=\int_{0}^{\infty}ds\,\frac{\rho_{{\cal
O}}(s)}{s+p^{2}}\;.$ (3.15)

It is instructive to first consider an example of a QCD and a CFT spectral
function before turning to the Goldstone case. At last a technical remark. The
representation (3.13) will in general need subtraction terms which are not
important since polynomial $(p^{2})^{n}$-terms corresponds to local
$\delta^{(n)}(x)$-terms which are irrelevant for the deep IR.

#### Spectral function in QCD with a heavy $b$-quark

In QCD a typical spectral function consists of a few (stable) bound states,
characterised by $\delta$-functions, and a continuum, beginning at some branch
point $s_{0}$,

$\rho^{\text{QCD}}_{{\cal O}}(s)=\sum_{n}|f_{{\cal
O}}^{(n)}|^{2}\delta(s-m_{n}^{2})+\theta(s-s_{0})\sigma_{{\cal O}}(s)\;,$
(3.16)

including resonances as well as other multiparticle states. The
$\delta$-function prefactor is $f_{{\cal O}}^{(n)}\equiv\langle 0|{\cal
O}|n\rangle$, sometimes referred to as the decay constants since such a
quantity governs the pion decay. We consider pure QCD, QED and weak
interactions turned off, with a heavy $b$-quark flavour ${\cal
O}(x)\to\bar{b}i\gamma_{5}q(x)$ for which: $m_{1}^{2}\to m_{B}^{2}$ with no
further stable states and the continuum threshold is given by
$s_{0}=(m_{B}+2m_{\pi})^{2}$. The $x$-dependence due to $\sigma_{{\cal O}}$
cannot be evaluated without knowing the function but the $\delta$-term part is
simply given by the Fourier transform of the propagator

$\rho_{{\cal O}}(s)\propto\delta(s-m^{2})\quad\Leftrightarrow\quad\langle{\cal
O}(x){\cal
O}^{\dagger}(0)\rangle\propto\int\frac{d^{4}x\,e^{ipx}}{p^{2}+m^{2}}\big{|}_{m^{2}=0}\propto\frac{1}{x^{2}}\;,$
(3.17)

which exhibits the $1/x^{2}$-scaling stated earlier.

#### Spectral function in a CFT without spontaneous symmetry breaking

It is straightforward to deduce that the following identification holds

$\rho^{\text{CFT}}_{{\cal O}}(s)\propto s^{\Delta_{{\cal
O}}-\frac{d}{2}}\quad\Leftrightarrow\quad\langle{\cal O}(x){\cal
O}^{\dagger}(0)\rangle\propto\frac{1}{(x^{2})^{\Delta_{{\cal O}}}}\;,$ (3.18)

either by direct computation [76] or on grounds of dimensional analysis. This
function is to be interpreted as belonging to the multiparticle threshold
$\sigma_{{\cal O}}$. It is noted that the limit $\Delta_{\cal O}\to 1$ for
$d=4$ is pathological (IR-divergent spectral integral) since the operator
really describes a free field rather than a multiparticle spectrum.

#### Spectral function in spontaneous scale symmetry breaking

The case of spontaneous symmetry breaking has elements of both the QCD- and
the CFT-case. The stable particle becomes the massless Goldstones and the
continuum threshold moves to zero assuming a simple power law. The
$\delta$-functions are very singular and have no counterpart in the CFT-case
and it is intuitively clear that they are not to be identified with
$\Delta_{{\cal O}}$. In Sec. 3.1 this has been argued more formally, that
$\Delta_{{\cal O}}=1$ does strictly imply a free field and is therefore not
associated with $s$ raised to some power. In conclusion the spectral function
in the case of SSB generically reads

$\rho^{\text{SSB}}_{{\cal O}}\propto s^{{\Delta}_{{\cal
O}}-\frac{d}{2}}\left(F^{2}\delta(s)+c\,\theta(s)\right)+\delta\rho(s)\;,$
(3.19)

where $c$ is a constant of proportionality and the $\delta$-term is only
present if the Goldstone couples to ${\cal O}$. If $c=0$ by a symmetry then
one has to substitute $c\to c^{\prime}(s/F^{2})^{n_{\text{LO}}}$ in accordance
with the discussion under item c) in Sec. 3.1. The quantity $\delta\rho(s)$
correspond to suppressed e.g nucleons thresholds and the remaining NLO-terms.

####  3.2.1 Multi-Goldstone case: the operator ${S^{a}}$

The case of the operator ${S^{a}}$ (3.4) is the simplest as its quantum
numbers do not allow for the propagation of a single Goldstone. It was chosen
for this reason in our earlier work [7] which we recapitulate here in the
language of the spectral function. Dropping the $\delta$-function piece in
(3.19) one gets the standard CFT-scaling

$\rho_{{S^{a}}}(s)\propto
s^{\Delta_{{S^{a}}}-\frac{d}{2}}\quad\Leftrightarrow\quad\langle{{S^{a}}}(x){{S^{a}}}(0)\rangle_{\text{CDQCD}}\propto\frac{1}{(x^{2})^{\Delta_{{S^{a}}}}}\;.$
(3.20)

In d$\chi$PT these operators are matched at leading order to

$S^{a}|_{{\text{LO}}}=-\frac{F_{\pi}^{2}B_{0}}{2}\mathrm{Tr}[T^{a}U^{\dagger}+UT^{a}]\hat{\chi}^{2}=\frac{B_{0}}{2}d^{abc}\pi^{b}\pi^{c}+{\cal
O}(1/F)\;,$ (3.21)

to two pions using the method of sources [7]. Its evaluation involves the
propagation of two free pions
$\langle\pi^{a}(x)\pi^{b}(0)\rangle=\frac{\delta^{ab}}{4\pi^{2}x^{2}}$ only,
as illustrated in Fig. 2, and one gets

$\langle{{S^{a}}}(x){{S^{a}}}(0)\rangle_{\chi\text{PT}}\;\propto\;\frac{1}{x^{4}}\;,$
(3.22)

where SL stands for subleading. Matching (3.2) one finds

$\Delta_{{S^{a}}}=3-\gamma_{*}=2\quad\Leftrightarrow\quad\gamma_{*}=1\;,$
(3.23)

that the mass anomalous dimension at the IR fixed point is unity [7].

####  3.2.2 Single-Goldstone case: the operators ${P^{a}}$ and ${S}$

The cases of the operators ${P^{a}}$ and ${S}$ (3.3) differ in that they
couple to a single Goldstone, the pion and the dilaton respectively, and thus
both terms in (3.19) are present. Omitting the truly subleading terms we have

$\rho_{{P^{a}}}(s)\propto
s^{{\Delta}_{{P^{a}}}-\frac{d}{2}}(F_{\pi}^{2}\delta(s)+c)\;,$ (3.24)

implying

$\langle{{P^{a}}}(x){{P^{a}}}(0)\rangle_{\text{CDQCD}}\propto\frac{F_{\pi}^{2}}{(x^{2})^{\Delta_{{P^{a}}}-1}}+\frac{c^{\prime}}{(x^{2})^{\Delta_{{P^{a}}}}}\;,$
(3.25)

where $c$ and $c^{\prime}$ are constants. The case of ${S}$ is analogous with
$F_{\pi}\to F_{D}$, cf. Fig. 2. In d$\chi$PT the operators are given by

$\displaystyle S$ $\displaystyle\;=\;$
$\displaystyle-\frac{F_{\pi}^{2}B_{0}}{2}\mathrm{Tr}[U^{\dagger}+U]\hat{\chi}^{2}$
$\displaystyle\;=\;$
$\displaystyle\phantom{-}2B_{0}N_{f}F_{\pi}^{2}\hat{D}+\frac{B_{0}}{4}(\pi^{a}\pi^{a}-N_{f}{F}_{\pi}^{2}\hat{D}^{2})+{\cal
O}(1/F)\;,$ $\displaystyle P^{a}$ $\displaystyle\;=\;$
$\displaystyle-\frac{F_{\pi}^{2}B_{0}}{2}i\mathrm{Tr}[T^{a}U^{\dagger}-UT^{a}]\hat{\chi}^{2}$
$\displaystyle\;=\;$ $\displaystyle-B_{0}F_{\pi}\pi^{a}(1-2\hat{D})+{\cal
O}(1/F)\;.$ (3.26)

From these expressions the following correlation functions result

$\displaystyle\langle{{S}}(x){{S}}(0)\rangle_{\text{d}\chi\text{PT}}$
$\displaystyle\;=\;$
$\displaystyle\frac{c_{S}^{(2)}}{x^{2}}+\frac{c_{S}^{(4)}}{x^{4}}\;,$
$\displaystyle\langle{{P^{a}}}(x){{P^{a}}}(0)\rangle_{\text{d}\chi\text{PT}}$
$\displaystyle\;=\;$
$\displaystyle\frac{c_{P^{a}}^{(2)}}{x^{2}}+\frac{c_{P^{a}}^{(4)}}{x^{4}}\;.$
(3.27)

with

$c_{S}^{(2)}=\frac{B_{0}^{2}F_{\pi}^{2}\hat{F}_{\pi}^{2}N_{f}^{2}}{\pi^{2}}\;,\quad
c_{S}^{(4)}=\frac{B_{0}^{2}((N_{f}^{2}-1)+N_{f}^{2}\hat{F}_{\pi}^{4})}{32\pi^{4}}\;,\quad
c_{P^{a}}^{(2)}=\frac{B_{0}^{2}F_{\pi}^{2}}{4\pi^{2}}\;,\quad
c_{P^{a}}^{(4)}=\frac{B_{0}^{2}\hat{F}_{\pi}^{2}}{4\pi^{4}}\;,$

and here and thereafter the flavour index is understood to be held fixed.
Matching the two expressions, as in (3.2), one deduces

$\Delta_{{S}}=\Delta_{{P^{a}}}=3-\gamma_{*}=2\quad\Leftrightarrow\quad\gamma_{*}=1\;,$
(3.28)

the same result as in the previous section. This should not be taken for
granted but as a sign of the consistency of the approach.

Figure 2: Two-point functions of operators defined in (3.3,3.4) evaluated in
the deep IR. (top) scalar adjoint correlator with no coupling to Goldstones
scaling as $1/x^{4}$. (centre) scalar and adjoint pseudoscalar correlators
with a single and a two dilaton intermediate state. (bottom)
$T^{\rho}_{\;\;\rho}$-correlator in $\chi$PT (the one for d$\chi$PT is zero at
LO).

####  3.2.3 Multi-Goldstone case II: the trace of the energy momentum tensor
$T^{\rho}_{\;\;\rho}$

The case of the TEMT is interesting as it brings several new elements. It is
tempting to think that the dilaton would couple to
$T^{\rho}_{\;\;\rho}=\frac{\beta}{2g}G^{2}$ but it does in a way (2.11)

$\langle D|T^{\rho}_{\;\;\rho}|0\rangle=F_{D}m_{D}^{2}\;,$ (3.29)

which vanishes when the dilaton is massless. Note that if it had a mass then
it would simply decouple from the IR physics. The next candidates in question
are two pions and two dilatons. For pedagogical reasons it is instructive to
first consider

$\rho_{G^{2}}(s)\propto
s^{\Delta_{G^{2}}-\frac{d}{2}}\quad\Leftrightarrow\quad\langle{G^{2}}(x){G^{2}}(0)\rangle_{\text{CDQCD}}\propto\frac{1}{(x^{2})^{4+\beta^{\prime}_{*}}}\;,$
(3.30)

the gluon correlator for which $\Delta_{G^{2}}=4+\beta^{\prime}_{*}$ (2.1) is
determined by the slope of the $\beta$ function at the fixed point. Adapting
it to $T^{\rho}_{\;\;\rho}$ one multiplies two powers of the $\beta$ function

$\langle{T^{\rho}_{\;\;\rho}}(x){T^{\rho}_{\;\;\rho}}(0)\rangle_{\text{CDQCD}}\propto\frac{\beta^{2}}{(x^{2})^{4+\beta^{\prime}_{*}}}\;$
(3.31)

which exhibits the same scaling as the $G^{2}$-correlator since $\beta$ is
just a scalar. In the EFT it makes a difference whether one uses $\chi$PT
(2.23) or d$\chi$PT in (2.3). From the LO EFTs one obtains [9], neglecting
terms of higher powers in $\pi$,

$T^{\rho}_{\;\;\rho}|^{\text{LO}}_{\chi\text{PT}}=-\frac{1}{2}\partial^{2}\pi^{a}\pi^{a}\;,\quad
T^{\rho}_{\;\;\rho}|^{\text{LO}}_{d\chi\text{PT}}=0\;.$ (3.32)

Thus at LO $\chi$PT is only scale invariant but not conformal contrary to
d$\chi$PT . One gets

$\langle{T^{\rho}_{\;\;\rho}}(x){T^{\rho}_{\;\;\rho}}(0)\rangle^{\text{LO}}_{\chi\text{PT}}\;\propto\;\frac{1}{x^{8}}\;,\quad\langle{T^{\rho}_{\;\;\rho}}(x){T^{\rho}_{\;\;\rho}}(0)\rangle^{\text{LO}}_{\text{d}\chi\text{PT}}\;\propto\;0\;.$
(3.33)

In $\chi$PT the matching with (3.31) implies, in straightforward manner, that
the slope of the $\beta$ function vanishes at the IR fixed point:
$\beta^{\prime}_{*}=0$. The case of d$\chi$PT is exceptional and corresponds
to the case discussed under item c) in Sec. 3.1 in that the LO vanishes by
conformal symmetry. In fact one has that $\langle G^{2}(x)G^{2}(0)\rangle$ has
no $1/x^{8}$-term such that not even $\beta^{\prime}_{*}=0$ is a necessary
condition in reproducing the result in (3.33). We proceed in a different
manner using RG arguments.

####  3.2.4 Renormalisation group derivation of $\beta^{\prime}_{*}=0$

We first consider the GMOR-relation and the TEMT to deduce
$\langle\pi|G^{2}|\pi\rangle={\cal O}(m_{q})$ and by equating to the RG
version of the matrix element we deduce $\beta^{\prime}_{*}=0$. We may
decompose the TEMT as $T^{\rho}_{\;\;\rho}={\cal O}_{1}^{\prime}+2{\cal
O}_{2}$ with ${\cal O}_{1}^{\prime}\equiv{\cal O}_{1}-{\cal O}_{2}$ (2.4).
Note that ${\cal O}_{1}^{\prime}$ inherits the RG invariance of ${\cal
O}_{1,2}$. Since $2{\cal O}_{2}$ saturates the GMOR-relation
$\langle\pi|T^{\rho}_{\;\;\rho}|\pi\rangle=\langle\pi|2{\cal
O}_{2}|\pi\rangle+{\cal O}(m_{q}^{2})$, one gets that the ${\cal
O}_{1}^{\prime}$ matrix element must vanish to linear order in $m_{q}$.
Expanding in $\delta g$, assuming $\gamma_{*}=1$ and $\beta_{*}=0$, we get

$0=\langle\pi|{\cal O}_{1}^{\prime}|\pi\rangle|_{m_{q}}=\delta
g\Big{(}\frac{\beta^{\prime}_{*}}{2g_{*}}\langle\pi|G^{2}|\pi\rangle+\sum_{q}\gamma^{\prime}_{*}m_{q}\langle{\pi}|\bar{q}q|{\pi}\rangle\Big{)}|_{m_{q}}+{\cal
O}((\delta g)^{2})\;.$ (3.34)

Without making any assumptions on $\beta^{\prime}_{*}$ and
$\gamma^{\prime}_{*}$ we deduce that $\langle\pi|G^{2}|\pi\rangle={\cal
O}(m_{q})$ since we know that $\langle{\pi}|\bar{q}q|{\pi}\rangle={\cal
O}(m_{q}^{0})$ from the GMOR-relation. From general RG arguments, similar to
hyperscaling, e.g. [72, 77, 78, 79] and also [10], one infers the $m_{q}$ and
$F$ depenence

$\langle\pi|G^{2}|\pi\rangle=m_{q}^{\frac{\Delta_{G^{2}}}{1+\gamma^{\prime}_{*}}}h({F}^{-y_{m}}{m}_{q},\mu)\;,$
(3.35)

where $\mu$ is the RG scale, $y_{m}=1+\gamma_{*}$ and $h$ an a priori unknown
function.666Unlike in the conformal window the pion states do not alter the
scaling. By the soft theorem we know that $|\pi\rangle\to\frac{1}{F_{\pi}}$
and $F_{\pi}$ does not depend on $m_{q}$. As before, predictivity seems lost
unless we can identify the functional dependence of $h$. We can do so by using
that $\langle\pi|G^{2}|\pi\rangle$ and $\langle\pi|\bar{q}q|\pi\rangle$ are of
the same order in $F_{\pi}$. That power is $F_{\pi}^{2}$ since by the
footnote,
$\langle\pi|\bar{q}q|\pi\rangle\propto\langle\bar{q}q\rangle=-B_{0}F_{\pi}^{2}$
which we know to be true by the GMOR-relation. Therefore we get

$\langle\pi|G^{2}|\pi\rangle\propto
m_{q}^{\frac{4+\beta^{\prime}_{*}}{1+\gamma_{*}}}(F_{\pi}^{2}m_{q}^{\frac{-2}{1+\gamma_{*}}})\;.$
(3.36)

Equating with $\langle\pi|G^{2}|\pi\rangle={\cal O}(m_{q})$ deduced above and
using $\gamma_{*}=1$, consistency emerges if and only if the slope vanishes

$\beta^{\prime}_{*}=0\;,$ (3.37)

which is the same result as for $\chi$PT above. This leads us to the
implications thereof.

###  3.3 Consequences of $\beta^{\prime}_{*}=0$

In our view $\beta^{\prime}_{*}=0$ is an important result.

  * a)

It is supported by smooth matching to ${\cal N}=1$ supersymmetry (cf. Sec. 4).

  * b)

It is the promised justification for dropping the anomaly term in the LO
Lagrangian (2.15). For example, in Ref. [30] the LO anomaly terms become
degenerate with the kinetic terms in the limit $\beta^{\prime}_{*}\to 0$, and
may therefore be dropped. As $\chi$PT is an EFT the EMT needs renormalisation
[80]. In that case three new counterterms, parameterised as a function of the
Riemann tensor, have been found at NLO. We expect the same program to apply in
d$\chi$PT such that the breaking of scale invariance emerges naturally at
higher orders.777In d$\chi$PT EFT the divergent parts of the counterterms have
been computed very recently [28] but not the ones for the EMT. We would though
expect that the terms from the Weyl or conformal anomaly [81], which are the
analogue of the Wess-Zumino-Witten term in QCD, would need to be matched at
NLO.

  * c)

The slow logarithmic running (discussed below) may be interpreted in terms the
mass gap amongst the hadrons. Once the $\rho$, $a_{0}$-meson, depending on the
channel, decouple the theory asymptotes slowly into the free Goldstone EFT
(accompanied by $1/(x^{2}F^{2})$ power correction at NLO in accordance with
the earlier discussion).

  * d)

It seems to make the existence of a QCD Seiberg dual in the conformal window
[82, 83, 84] more likely since the turnaround of the $\beta^{\prime}_{*}$ in
the conformal window would go well with a weakly coupled dual IR fixed point.

####  3.3.1 Logarithmic running

We may expand the $\beta$ function around the fixed point coupling $\delta
g\equiv g-g_{*}$

$\beta=\beta^{\prime}_{*}\delta g+\frac{1}{2}\beta^{\prime\prime}_{*}{(\delta
g)^{2}}+{\cal O}((\delta g)^{3})\;.$ (3.38)

By the very assumption of an IR fixed point we have $\beta^{\prime}_{*}\geq 0$
and the value in (3.37) saturates this constraint. Using
$\beta^{\prime}_{*}=0$ gives an RG equation

$\frac{d}{d\ln\mu}\delta g=\frac{1}{2}\beta_{*}^{{}^{\prime\prime}}\delta
g^{2}\;,$ (3.39)

which solves to a logarithmic

$\delta
g(\mu)=\frac{4}{|\beta_{*}^{{}^{\prime\prime}}|\ln\frac{\mu^{2}}{\lambda_{\text{QCD}}^{2}}}\;,$
(3.40)

instead of to a power-like form $\delta g\propto\mu^{\beta^{\prime}_{*}}$.
Above, the boundary condition $\delta g(0)=0$ was imposed and the expression
is maximally valid for $\mu<\lambda_{\text{QCD}}$ where it runs into a Landau
pole from below. The scale $\lambda_{\text{QCD}}$ is the analogue of
$\Lambda_{\textrm{QCD}}$ in the IR. Note that whereas $\beta^{\prime}_{*}$ is
scheme independent under regular coupling redefinitions this is not the case
for $\beta^{\prime\prime}_{*}$. For the sake of concreteness we may assume it
to be non-vanishing, in which case it needs to be negative to assure
negativity of the $\beta$ function (3.38). In retrospect the logarihtmic
running is assuring as it is difficult to see how (d)$\chi$PT could reproduce
power-like scaling as per item c) above.

####  3.3.2 The vanishing of $\beta^{\prime}_{*}$ implies the vanishing of
$\gamma^{\prime}_{*}$

In ${\cal N}=1$ supersymemtric gauge theories $\beta^{\prime}_{*}$ is
proportional to $\gamma^{\prime}_{*}=0$ due to the NSVZ $\beta$ function [85].
Hence, the vanishing of $\beta^{\prime}_{*}$ implies the vanishing of
$\gamma^{\prime}_{*}$ and it is natural to ask whether this extends to the
non-supersymmetric case. Using Eq. (3.34) one infers that zero
$\beta^{\prime}_{*}$ implies zero $\gamma^{\prime}_{*}$ and vice versa:

$\beta^{\prime}_{*}=0\quad\Leftrightarrow\quad\gamma^{\prime}_{*}=0\;.$ (3.41)

We note that this result might be more universal since they both govern the
correction to hyperscaling e.g. [79].

##  4 Comparison with ${\cal N}=1$ Supersymmetric Gauge Theories

It is instructive to reflect on the results obtained in the context of ${\cal
N}=1$ supersymmetric gauge theories and its Seiberg dualities, for which many
excellent reviews exist [86, 87, 88, 89]. Our brief summary below, extends on
the non-supersymmetric conformal window in Sec. 2.2. For $SU(N_{c})$ gauge
theories with $N_{f}$ fermions, which is referred to as the electric theory,
the conformal window extends over the following region

$\text{conformal window:}\qquad\frac{3}{2}N_{c}\leq N_{f}\leq 3N_{c}\;.$ (4.1)

In this window there exists a Seiberg dual, referred to as the magnetic
theory, which is also asymptotically free with gauge group $SU(N_{f}-N_{c})$
and a Yukawa-superpotential interaction ${\cal W}=\lambda
M_{\bar{i}}^{j}q_{i}\tilde{q}^{\,\bar{j}}$ which couples $N_{f}$ chiral
$q_{i}$ and $N_{f}$ antichiral $\tilde{q}^{\bar{j}}$ matter fields to a colour
neutral chiral meson superfield $M_{\bar{i}}^{j}$ (where $i$ and $\bar{j}$ are
$SU(N_{f})_{L}$ and $SU(N_{f})_{R}$ flavour indices respectively). The duality
[90], in part based on non-trivial matching of anomalies, is the statement
that both of these theories flow to the same CFT in the IR. The relation of
the dilatation and non-anomalous $R$-current in the same superconformal
multiplet allows to determine the scaling dimension of the bilinear (s)quark
fields in terms of the known $R$-charges888In the supersymmetry literature
$\gamma_{Q}$, the matter-field anomalous dimension, is used rather than
$\gamma_{m}$. They are related by $\gamma_{m}=-\gamma_{Q}$ through non-
renormalisation theorems to all order in perturbation theory.

$\Delta_{\tilde{Q}Q}=2-\gamma_{*}=3-\frac{3N_{c}}{N_{f}}\;,\quad\Delta_{\bar{q}q}=3-\gamma_{*}=4-\frac{3N_{c}}{N_{f}}\;,$
(4.2)

valid in the conformal window. Alternatively, this formula can be deduced by
requiring the NSVZ $\beta$ function [91, 92] to vanish in the domain (4.1).

Below the conformal window squark bilinears can take on VEVs and break the
superconformal $R$-symmetry and the relations (4.2) cease to be valid. In the
range

$\text{IR-free magnetic:}\qquad N_{c}+1<N_{f}<\frac{3}{2}N_{c}\;,$ (4.3)

the magnetic theory plays the role of the weakly coupled IR EFT of the
strongly coupled electric theory. The magnetic theory is not asymptotically
free anymore and the electric theory can be seen as its UV completion. We may
think of this magnetic description as a meson EFT of the hybrid type, as it
still contains coloured degrees of freedom, and as such resembles the
phenomenologically successful chiral quark model [93]. There are three further
phases below: s-confinement for $N_{f}=N_{c}+1$ (confinement without chiral
symmetry breaking), $N_{f}=N_{c}$ anomalies are matched by mesons and baryons
only, $N_{c}>N_{f}>1$ has a runaway potential (no stable vacuum). We will not
focus on these phases as they are quite possibly peculiar to supersymmetry
itself. Further note, that for $N_{f}=N_{c}+1$ the dual gauge group in
particular ceases to make sense but interestingly the anomalies can still be
matched. We discuss $\gamma_{*}=1$, $\beta^{\prime}_{*}=\gamma^{\prime}_{*}=0$
under items a-c) below from the supersymmetric viewpoint and contemplate
whether it makes sense to assume that this value remains the same in the IR-
free magnetic phase (4.3).

  * a)

The squark bilinear $\tilde{Q}Q$ is believed to match onto the free meson
field $M_{\bar{i}}^{j}$ for which $\langle M(x)M(0)\rangle\propto 1/x^{2}$ for
$x^{2}\to\infty$ must hold in the IR-free magnetic range (4.1). This implies
$\Delta_{M}=\Delta_{\tilde{Q}Q}=2-\gamma_{*}=1$ and thus $\gamma_{*}=1$ must
hold throughout the IR-free magnetic phase.999As a side remark, it is curious
that in ${\cal N}=1$ the end of the conformal window coincides with the
unitarity bound whereas for QCD gauge theories the unitarity bound
$\gamma_{*}\leq 2$ is not reached. The interpretation that suggests itself [7]
is that chiral symmetry breaking sets in once the scaling dimension allow the
operator $S^{a}$ ($\Delta_{S^{a}}=2$) to produce two free pions as predicted
by (d)$\chi$PT .

  * b)

In the conformal window $\beta^{\prime}_{*}$ is non-zero. This can established
close to the electric Caswell-Banks-Zaks perturbative fixed point. The very
idea of the Caswell-Banks-Zaks procedure is based on $\beta^{\prime}_{*}$
being small. When the coupling increases $\beta^{\prime}_{*}$ does equally so
e.g. [94]. Since $\beta^{\prime}_{*}$ is also the scaling exponent of the
TEMT-correlator (3.31) in the conformal window [85] this implies that the
slopes of the electric and magnetic theory must equal each other

$\beta_{*}^{\prime}|_{\text{el}}=\beta_{*}^{\prime}|_{\text{mag}}\quad\Leftrightarrow\quad\langle{T^{\rho}_{\;\;\rho}}(x){T^{\rho}_{\;\;\rho}}(0)\rangle_{\text{el}}=\langle{T^{\rho}_{\;\;\rho}}(x){T^{\rho}_{\;\;\rho}}(0)\rangle_{\text{mag}}\;,$
(4.4)

since the latter holds in the deep IR.101010This result was derived earlier by
matching the Konishi currents via an involved Kutasov construction in [95].
The value of $\beta_{*}^{\prime}|_{\text{mag}}$ corresponds to the minimal
eigenvalue of the gradient matrix of the $\beta$ function matrix in the gauge-
Yukawa coupling space. Since the magnetic theory is weakly coupled at the
lower edge of the conformal window, $N_{f}$ towards $\frac{3}{2}N_{c}$, where
it assumes its own perturbative fixed point this means that

$\beta^{\prime}_{*}|_{N_{f}=3}=\beta^{\prime}_{*}|_{N_{f}=\frac{3}{2}}=0\;,\quad\beta^{\prime}_{*}|_{\frac{3}{2}N_{c}<N_{f}<N_{c}}>0\;,$
(4.5)

$\beta^{\prime}_{*}$ is zero at both edges of the conformal window and
positive in between. Note that, positivity follows from the very assumption of
an IR fixed point. This is curious as either very weak and very strong
coupling seems to makes $\beta^{\prime}_{*}$ vanish. However, since
$\beta^{\prime}_{*}=0$ also implies the logarithmic running implicit in the
EFT cf. Sec. 3.2.3, this can be seen as a necessary result and matches the
expectation of the IR-free magnetic phase (4.3). This is also the reason it
should continue to hold in the IR-free magnetic phase which is though not
argument inherent to supersymmetry.

  * c)

The NSVZ $\beta$ function [91, 92] is expressed in terms of known quantities
and the anomalous dimension $\gamma_{m}$. Hence, differentiating, and using
$\beta_{*}=0$, give a relation [85]

$\beta^{\prime}_{*}=\frac{\alpha^{2}_{*}}{2\pi}\,\,\frac{N_{f}}{1-\frac{\alpha_{*}}{2\pi}N_{c}}\gamma^{\prime}_{*}\;,$
(4.6)

which must hold at least in the conformal window. ($\alpha_{*}$ denotes the IR
fixed point gauge coupling $\alpha=g^{2}/4\pi$ of the electric theory).
Therefore, $\beta^{\prime}_{*}=0$ at $N_{f}=\frac{3}{2}N_{c}$ implies the same
for $\gamma^{\prime}_{*}=0$ at this point. In IR-free magnetic phase,
$\gamma^{\prime}_{*}=0$ might continue to hold by the same reasoning as given
in Sec. 3.3.2. This argument is not specific to supersymmetry.

In summary, we have argued that for ${\cal N}=1$ supersymmetry $\gamma_{*}=1$
and $\beta^{\prime}_{*}=\gamma^{\prime}_{*}=0$ hold at the boundary of the
conformal window. Using Seiberg duality we provided an argument of why
$\gamma_{*}=1$ carries over into the IR-magnetic phase. For
$\beta^{\prime}_{*}=\gamma^{\prime}_{*}=0$ we referred to non-supersymmetric
arguments with regards to the IR-magnetic phase. It is worthwhile to emphasise
that there have been interesting attempts to understand the magnetic dual in
terms of hidden local symmetry and low energy Goldstone physics [96, 97]. It
may well be that the massless $0^{++}$ flavour-singlet found in these cases is
the dilaton.

##  5 Dilaton Soft Theorems

In this section we apply the double soft theorem to the matrix element

$2m_{D}^{2}=\langle D|T^{\rho}_{\;\;\rho}|D\rangle\;,$ (5.1)

which will provide some surprising results and the dilaton GMOR-relation
(2.21). Eq. (5.1) is usually considered a first principle formula but in the
presence of a massless dilaton it does not hold for a standard hadronic state
$\varphi$. The dilaton pole cancels the $2m_{\varphi}^{2}$-contribution from
the kinetic term such that
$\langle\varphi|T^{\rho}_{\;\;\rho}|\varphi\rangle=0$ [10]. However, since
here $m_{D}\neq 0$ is kept, this situation does not arise (which we have
explicitly checked at LO). All the information needed from the potential is
the mass term $V\supset\frac{1}{2}m_{D}^{2}D^{2}$.

The idea of soft theorems is that one has a pseudo Goldstone with a light
mass, $m_{\text{pGB}}\ll\Lambda_{\textrm{QCD}}$, whose momenta can be
approximated to be soft $q\to 0$ while keeping $m_{\text{pGB}}\neq 0$ [65]
(which is automatic in an EFT treatment).111111That the soft theorem is
encoded in the Lagrangian is of no coincidence as effective Lagrangians are
considered a more transparent way to organise them [98]. The procedure assumes
that the original matrix element does not change significantly in the soft
limit. The gain is that the evaluation then proceeds by a simpler matrix
element where the Goldstone emerges in a computable commutator. For the
dilaton and pion they read

$\displaystyle\langle D(q)\beta|{\cal O}(0)|\alpha\rangle$
$\displaystyle\;=\;$ $\displaystyle-\frac{1}{F_{D}}\langle\beta|i[Q_{D},{\cal
O}(0)]|\alpha\rangle$ $\displaystyle\;+\;$ $\displaystyle\lim_{q\to 0}iq\cdot
R\;,$ $\displaystyle\langle\pi^{a}(q)\beta|{\cal O}(0)|\alpha\rangle$
$\displaystyle\;=\;$
$\displaystyle-\frac{i}{F_{\pi}}\langle\beta|[Q_{5}^{a},{\cal
O}(0)]|\alpha\rangle$ $\displaystyle\;+\;$ $\displaystyle\lim_{q\to 0}iq\cdot
R^{a}\;,$ (5.2)

where the one for pions can be found in the textbook [65]. The $R$’s are the
so-called remainders

$\displaystyle R_{\mu}$ $\displaystyle\;=\;$
$\displaystyle-\frac{i}{F_{D}}\int d^{d}xe^{iq\cdot
x}\langle\beta|TJ^{D}_{\mu}(x){\cal O}(0)|\alpha\rangle\;,$ $\displaystyle
R^{a}_{\mu}$ $\displaystyle\;=\;$ $\displaystyle-\frac{i}{F_{\pi}}\int
d^{d}xe^{iq\cdot x}\langle\beta|TJ_{5\mu}^{a}(x){\cal O}(0)|\alpha\rangle\;,$
(5.3)

which vanish unless there are intermediate states degenerate with either
$\alpha$ or $\beta$.121212We have checked that they vanish in the cases at
hand for which it is important to keep $m_{D,\pi}\neq 0$. An example of where
the $R$ matters is sketched in the appendix of Ref. [7]. Let us focus on an
operator present in the TEMT ${\cal O}\subset T^{\rho}_{\;\;\rho}$ which is
responsible for generating the mass. We will refer to this operator as the
mass operator. Applying the soft theorem (5.1) one gets

$2m_{D}^{2}=\langle D|{\cal O}(x)|D\rangle=-\frac{1}{F_{D}}\langle
0|i[Q_{D},{\cal O}(x)]|D\rangle=-(\Delta_{{\cal O}}+x\cdot\partial)\langle
0|{\cal O}(x)|D\rangle\;.$ (5.4)

Above we used the CFT definition of applying $Q_{D}$ to a primary operator
which is an assumption similar to (1.1). It is, in our opinion, crucial to
keep the derivative term since the matrix element, $\langle 0|{\cal
O}(x)|D(q)\rangle=F_{\cal O}e^{-iqx}$, carries $x$-dependence. We may regard
this matrix element as a test function in a distribution space and are
therefore led to integrate over space by

$\mathbb{1}_{V}=\frac{1}{V}\int_{V}d^{d}x\;.$ (5.5)

This allows for integration by parts such that131313The proper way to do this
would be to form a wave packet within a finite region in $x$-space. In
addition this makes it clear that the boundary terms that arise upon
integration by parts do not contribute.

$\mathbb{1}_{V}[x\cdot\partial\langle 0|{\cal
O}(x)|D\rangle]=-d\,\frac{1}{V}\int_{V}d^{d}x\langle 0|{\cal
O}(x)|D\rangle\;,$ (5.6)

and assembling we get the single soft dilaton theorem

$m_{D}^{2}F_{D}=\frac{1}{2}(d-\Delta_{\cal O})\langle 0|{\cal O}|D\rangle\;,$
(5.7)

where the integral has been removed since the second dilaton is to be made
soft as well. Applying the procedure once more one gets the double soft
dilaton theorem

$m_{D}^{2}F_{D}^{2}=\frac{1}{2}(\Delta_{\cal O}-d)\Delta_{\cal O}\langle{\cal
O}\rangle\;,$ (5.8)

where this time the derivative term can be dropped since $\langle{\cal
O}(x)\rangle$ has no $x$-dependence by translation invariance.

###  5.1 Consequences of the single and the double soft dilaton theorem

There are a number of things that one can learn from these two relations.

  * a)

One has $\langle{\cal O}\rangle\leq 0$ necessarily, such that $\langle
T^{\rho}_{\;\;\rho}\rangle$ is lowered (${\cal O}\subset
T^{\rho}_{\;\;\rho}$). Hence, the mass squared (5.8) is indeed positive thanks
to the $d$-term that originates from the derivative term. This gives us
confidence that this term is present (cf. item c for a further comment).

  * b)

One can get a very similar relation to (5.7) by contracting (2.11)

$m_{D}^{2}F_{D}=\langle 0|{\cal O}|D\rangle\;,$ (5.9)

where we have assumed $T^{\rho}_{\;\;\rho}\to{\cal O}$ since ${\cal O}$ is the
operator that generates the mass. Inspecting (5.9) and (5.7) one infers that

$\Delta_{\cal O}=d-2\;.$ (5.10)

Hence, the soft dilaton theorem indicates that the dilaton can only get a mass
from an operator of dimension $\Delta_{\cal O}=2$ in $d=4$. This correspond to
the scaling dimension of the free field mass term $\frac{1}{2}m^{2}\phi^{2}$
and this role is taken on by $\bar{q}q$ in CDQCD as noticed earlier. In App.
B.2 this result is derived in alternative manner from a scaling analysis. It
is tempting to read the result in (5.10) backwards and interpret it as yet
another demonstration that $\gamma_{*}=1$ (${\Delta_{\bar{q}q}}=3-\gamma_{*}$)
has to necessarily hold.

  * c)

Let us consider the special case

${\cal O}_{\bar{q}q}=(1+\gamma_{*})\sum_{q}m_{q}\bar{q}q\;,\quad$ (5.11)

which obeys (5.10) and figures in the TEMT: $T^{\rho}_{\;\;\rho}={\cal
O}_{\bar{q}q}+{\cal O}(\delta g)$. Using (5.8) we obtain

$F_{D}^{2}m_{D}^{2}={\Delta_{\bar{q}q}}({\Delta_{\bar{q}q}}-d)\frac{N_{f}}{2}(1+\gamma_{*})m_{q}\langle\bar{q}q\rangle=-4N_{f}m_{q}\langle\bar{q}q\rangle\;,$
(5.12)

where $\gamma_{*}=1$ (2.6) and $d=4$ have been used in the last equality.
Since the effective Lagrangian is rather standard, this relation has been
obtained previously e.g. [3, 71, 30]. What is new is the result $\gamma_{*}=1$
(${\Delta_{\bar{q}q}}=2$), that there are no $\beta^{\prime}_{*}$-terms and
the derivation in itself from the double-soft theorem. In App. B (5.8) is
shown to follow from the effective Lagrangian (2.3) provided that
${\Delta_{\bar{q}q}}=d-2$, which is the result expected on grounds of
consistency.

The dilaton GMOR-relation quoted in (2.21) can be obtained by going through
the analogous process for the pion which has been done in [7] but we shall
repeat it here for completeness. Starting with the analogue of (5.1) one gets

$\displaystyle 2m_{\pi}^{2}$ $\displaystyle\;=\;$
$\displaystyle\langle\pi^{a}|T^{\rho}_{\;\;\rho}|\pi^{a}\rangle=\langle\pi^{a}|{\cal
O}_{\bar{q}q}|\pi^{a}\rangle=\frac{-(1+\gamma_{*})m_{q}}{F_{\pi}}\langle
0|i[Q_{5}^{a},\bar{q}\mathbb{1}_{N_{f}}q]|\pi^{a}\rangle$ $\displaystyle\;=\;$
$\displaystyle\frac{2m_{q}(1+\gamma_{*})}{F_{\pi}}\langle
0|P^{a}|\pi^{a}\rangle=\frac{-2m_{q}(1+\gamma_{*})}{F_{\pi}^{2}}\langle\bar{q}q\rangle\;,$
(5.13)

upon using $\langle\pi^{a}|P^{b}|0\rangle=-\frac{1}{F_{\pi}}\langle
i[Q_{5}^{a},P^{b}]\rangle=-\frac{1}{F_{\pi}}\langle\bar{q}q\rangle\delta^{ab}$
with $P^{b}$ in (3.3). Rewriting

$m_{\pi}^{2}F_{\pi}^{2}=-(1+\gamma_{*})m_{q}\langle\bar{q}q\rangle=-2m_{q}\langle\bar{q}q\rangle\;,$
(5.14)

the GMOR-relation emerges [99, 67] upon using $\gamma_{*}=1$. Combining (5.12)
and (5.14)

$F_{D}^{2}m_{D}^{2}={\Delta_{\bar{q}q}}(d-{\Delta_{\bar{q}q}})\frac{N_{f}}{2}F_{\pi}^{2}m_{\pi}^{2}\;,$
(5.15)

we get the dilaton GMOR-relation (2.21). There is a further insight hidden
here. Namely the $d$-term, which originated from the derivative term in (5.4),
in fact corresponds to the Zumino-type term in (2.20) (to see that one needs
to expand to second order in $D$). This underlines the necessity of the
integration by parts procedure applied once more.

  * d)

At last we turn to a delicate point. Under item b) we learned that the soft
theorem demands $\Delta_{\cal O}=2$. It is more often than not inferred from
the soft theorem that the field strength tensor squared
$T^{\rho}_{\;\;\rho}=\frac{\beta}{2g}G^{2}$ generates a dilaton mass in the
chiral limit. However, this is not done from $\langle D|G^{2}|D\rangle$ but
from $\langle D|G^{2}|0\rangle$ which gives the relation

$m_{D}^{2}F_{D}^{2}=-\frac{2\beta}{g}\langle G^{2}\rangle\;,$ (5.16)

upon using $\Delta_{G^{2}}=4$, sometimes referred to as the partially
conserved dilatation current (PCDC) relation. Since $\Delta_{G^{2}}=4$ the
condition (5.10) then raises a question mark over the applicability of this
procedure.

In summary we have learned that the dilaton can only obtain a soft mass from
an operator of scaling dimension two which is $\bar{q}q$ in our case at hand.
In addition, the soft theorems reproduce the dilaton GMOR-relation (5.15).
Single soft theorems are equivalent to results found in more formal
considerations e.g. [100]. It could be interesting to extend the techniques of
both approaches to each other.

##  6 Massive or Massless Dilaton?

It is commonly believed that there are CFTs for which conformal symmetry is
spontaneously broken, leading to a massless dilaton and massive states e.g.
[81]. The situation of when there is an RG flow into an IR fixed point cannot
be regarded as settled. The problem is that the symmetry is only emergent and
we are not aware of a systematic treatment of this case. Investigations in
holograpic approaches argue for a light but not a parametrically light dilaton
[35, 101, 102, 103]. However, since the answer to the question might be model-
dependent we focus on our framework.141414There are examples of massless
dilatons in lower dimensions e.g. $d=2$ [104] at finite temperature and $d=3$
[105] (cf. also [106]) but they do not involve an RG flow. An example with a
flow is given by a Yukawa theory in $d=3$ where spontaneous scale symmetry,
emerges for certain initial conditions [107], accompanied with a massless
scalar and massive fermions. Explicit studies with fundamental scalars
indicate that they cannot take on the role of a dilaton [108, 109, 110]
although they share some of these features. In [30] it was stressed that
scalars, called scalons in [108], are not to be regarded as dilatons.
Parametrically, the leading effect is expected to come from the slope of the
$\beta$ function, which is however zero in our framework (3.37) and thus the
parametric expectation becomes

$m^{2}_{D}\propto{\cal O}(\beta^{\prime\prime}_{*})\;.$ (6.1)

This finding can be taken as an indication that the dilaton mass could at
least be small. We are going to reflect on the question from three different
points of view: the soft theorems, the large-$N_{c}$ limit and the EFT-
perspective. Whereas the discussion is not conclusive we hope that the reader
finds it instructive.

###  6.1 Soft-theorem perspective

The form of the TEMT (2.3) is correct to all orders in perturbation theory and
believed to hold beyond. If we set $m_{q}=0$ then there is only
$T^{\rho}_{\;\;\rho}=\frac{\beta}{2g}G^{2}$ and thus we have
$2m_{D}^{2}=\langle D|\frac{\beta}{2g}G^{2}|D\rangle$ in line with (5.1).
However, the double-soft dilaton theorem indicates that the dilaton mass ought
to originate from an operator ${\cal O}\subset T^{\rho}_{\;\;\rho}$ of scaling
dimension $\Delta_{{\cal O}}=2$ a result underlined by an alternative
derivation from scaling in App. B.2. Hence, this role cannot be taken by
$G^{2}$ since
$\Delta_{T^{\rho}_{\;\;\rho}}=\Delta_{G^{2}}=4+\beta^{\prime}_{*}=4$.151515 As
stated earlier this puts into question the use of the PCDC-type relation
$m_{D}^{2}F_{D}^{2}=-2\beta/g\langle G^{2}\rangle$ (5.16). To the best of our
knowledge the relationship of $\langle G^{2}\rangle$ in (5.16) to the gluon
condensate introduced in phenomenology [111, 112] has never been clarified.
The latter has been determined empirically [114, 115] and its existence is
underlined by an elegant renormalon analysis within perturbation theory [113].
A similar quantity has been studied for pure SU(3) Yang Mills on the lattice
[116, 117, 118] and is found to be non-zero by eleven standard deviation in
the lattice scheme. The analysis is consistent with the renormalon picture. As
there is no other operator than $G^{2}$ present in the EMT (for $m_{q}=0$)
this would then seem to imply that the dilaton is massless.161616It is in
principle conceivable that another operator becomes relevant and mixes into
$G^{2}$ by renormalisation just as $m_{q}qq$ does. If then the lowest
eigenvalue were $\Delta=2$ then this could give rise to a mass. We consider
this possibility rather unlikely as lattice studies of four fermi operators
for example do not indicate large anomalous dimensions [119]. We would not
want to go as far as stating that this prooves that the dilaton is massless,
as for example there are assumptions involved such as the use of the soft
dilaton formula (5).

###  6.2 Large-$N_{c}$ consideration

The large-$N_{c}$ limit [120, 121, 65] is a useful tool for QCD as it leads to
simplifications [122, 65]. Examples are the two-point functions of flavoured
and non-flavoured operators. The following relations, with notation as in Eqs.
(3.3) and (3.4),

$\displaystyle\langle{S}(x){S}(0)\rangle_{c}$ $\displaystyle\;=\;$
$\displaystyle\frac{2}{N_{f}}\langle{S^{a}}(x){S^{a}}(0)\rangle(1+{\cal
O}(1/N_{c}))\propto{\cal O}(N_{c})\;,$
$\displaystyle\langle{P}(x){P}(0)\rangle$ $\displaystyle\;=\;$
$\displaystyle\frac{2}{N_{f}}\langle{P^{a}}(x){P^{a}}(0)\rangle(1+{\cal
O}(1/N_{c}))\propto{\cal O}(N_{c})\;,$ (6.2)

must hold since they are only distinguished by quark-disconnected diagrams
(i.e. connected by gluons only) which are suppressed in the ’t Hooft
large-$N_{c}$ counting. The factor $2/N_{f}$ takes into account the
normalisation $\mathrm{Tr}[T^{a}T^{b}]=\delta^{ab}/2$. The subscript $c$
stands for the connected part in order to remove the disconnected part
$\langle{S}(x){S}(0)\rangle\supset\langle\bar{q}q\rangle^{2}\propto{\cal
O}(N_{c}^{2})$ which is exceptional due to the vacuum quantum numbers of $S$.
The leading graphs in (6.2) are the connected planar ones of ${\cal
O}(N_{c})$. Non-planar corrections are $g^{4}\propto 1/N_{c}^{2}$ suppressed
($g^{2}N_{c}$ is held fixed). The disconnected non-flavoured case is ${\cal
O}(N_{c}^{0})$ and falls in between the $N_{c}$-counting of the flavoured
case. These contributions arise when in the operators ${S}$ and ${P}$ the
quark propagators in the external gluon field close amongst themselves.
Famously, one can deduce in this way that the $\eta^{\prime}$ mass must go to
zero in the large-$N_{c}$ limit. Clearly in deep-IR the ${P^{a}}$-correlator
is dominated by the pions and one has

$\langle{P^{a}}(x){P^{a}}(0)\rangle\propto\frac{|\langle
0|P^{a}|\pi^{a}\rangle|^{2}}{x^{2}}\propto{\cal O}(N_{c})\;,$ (6.3)

with only leafing leading terms in $x^{2}\to\infty$ shown. The scaling above
follows from soft pion theorem $|\langle
0|P^{a}|\pi^{a}\rangle|^{2}\propto\langle\bar{q}q\rangle/F_{\pi}\propto{\cal
O}(N_{c}^{1/2})$ where $\langle\bar{q}q\rangle\propto{\cal O}(N_{c})$ and
$F_{\pi}\propto{\cal O}(N_{c})$ are well-known [80, 67]. Since it is part of
the leading contribution, as established above, this behaviour must be
mirrored by the $P$-correlator. One concludes that $m_{\eta^{\prime}}\to
m_{\pi}=0$ for $N_{c}\to\infty$ and also that $F_{\pi}\to F_{\eta^{\prime}}$
in that limit.

Does the same thing happen in the scalar channels? If the answer is yes, then
a massless dilaton would imply that the corresponding flavoured states would
approach zero in the large-$N_{c}$ limit. This is however in contradiction
with a lattice $SU(N_{c})$-study as a function of $N_{c}$ where the
$a_{0}$-meson, the $0^{++}$-cousin of the pion, shows no sign of becoming
massless in that limit (cf. [123] Sec. 3.5 and also Fig. 14 in [122]). The
only caveat is that these studies are performed in the quenched approximation
but one cannot expect these qualitative features to be overturned by
unquenching. We therefore conclude that either the dilaton cannot be massless
or that the dilaton must be subleading in the connected $S$-correlator. We
will argue that the latter is the case.

In clarifying at what order the dilaton appears in the large-$N_{c}$ counting
we consider the analogue of (6.3), for which we use the soft theorem (B.11),

$\langle{S}(x){S}(0)\rangle\propto\frac{|\langle
0|S|D\rangle|^{2}}{x^{2}}\propto\frac{\langle\bar{q}q\rangle^{2}}{F_{D}^{2}}\frac{1}{x^{2}}\;.$
(6.4)

Hence, if we know the scaling of the dilaton decay constants the question can
be answered. As the latter is defined from the coupling to the EMT (2.11), we
may consider the correlator

$\langle
T_{\mu\nu}(x)T_{\rho\lambda}(0)\rangle=t^{(0)}_{\mu\nu\rho\lambda}\,\Gamma^{(0)}_{TT}(x^{2})+t^{(2)}_{\mu\nu\rho\lambda}\,\Gamma^{(2)}_{TT}(x^{2})\;,$
(6.5)

where $t^{(0,2)}_{\mu\nu\rho\lambda}$ are the tensor structures for spin $0$
and $2$ respectively which depend on $x$ and the metric. Since the dilaton is
of spin $0$ we have

$\Gamma^{(0)}_{TT}(x^{2})\propto\frac{F_{D}^{2}}{x^{2}}\;.$ (6.6)

Thus one can determine the large-$N_{c}$ behaviour of $F_{D}$ through the
corresponding behaviour of the correlator. To infer this we must have a look
at the EMT of a non-abelian gauge theory which assumes the form

$T_{\mu\nu}=\left(\frac{1}{4}g_{\mu\nu}G^{2}-G_{\mu\lambda}G^{\lambda}_{\nu}\right)+\frac{i}{4}\bar{q}\left(\gamma_{\\{\mu}\overset{\rightarrow}{D}_{\nu\\}}-\gamma_{\\{\mu}\overset{\leftarrow}{D}_{\nu\\}}\right)q+\dots\;,$
(6.7)

where
$\overset{\rightarrow}{D}_{\nu}=(\overset{\rightarrow}{\partial}+igA)_{\nu}$,
$\overset{\leftarrow}{D}_{\nu}=(\overset{\leftarrow}{\partial}-igA)_{\nu}$ and
the dots stand for terms which vanish on physical states. The main point is
that the gluonic part is in the adjoint and the quark part in the fundamental
representation of the $SU(N_{c})$ gauge group hence we do expect

$\Gamma_{TT}=AN_{c}^{2}+BN_{c}+{\cal O}(N_{c}^{0})\;,$ (6.8)

and matching with (6.6) we infer that in the canonical case

$F_{D}^{2}=aN_{c}^{2}+bN_{c}+{\cal O}(N_{c}^{0})\;,$ (6.9)

is expected. This means that in the large-$N_{c}$ limit one has
$F_{D}\propto{\cal O}(N_{c})$.171717This scaling is identified with glueballs
as opposed to $F\propto\sqrt{N_{c}}$ which are referred to as
$\bar{q}q$-states. With (6.4) it follows

$\langle{S}(x){S}(0)\rangle\propto\frac{{\cal O}(N_{c}^{0})}{x^{2}}\;,$ (6.10)

that the dilaton contribution is subleading in the large-$N_{c}$ limit. We
therefore conclude that a finite value in $m_{a_{0}}$ at large $N_{c}$ does
not exclude a massless dilaton.

###  6.3 Is dilaton-$\chi$PT consistent with a massless dilaton?

Another way to assess whether a dilaton could be massless is to seek for
contradictions with the EFT. Are quantum fluctuations going to induce a mass
term? In the chiral limit of $\chi$PT  it is simply impossible to write down
a potential for the pion respecting the symmetries within the coset
construction $U$ (2.8) (shift symmetry of the Goldstone). Hence the zero pion
mass is built into $\chi$PT naturally since chiral symmetry is present at all
scales. Returning to the dilaton we may observe that since there is no scale
in the LO Lagrangian, in the chiral limit, no dilaton mass can be generated
either. This conclusion is however too quick since scales enter through hadron
masses. The dilaton couples to the nucleon mass term, e.g. [10],

$\delta{\cal L}_{m_{N}}=-\hat{\chi}m_{N}\bar{N}N\;.$ (6.11)

Generically such a term will induce a dilaton mass term (e.g. a single nucleon
loop). There are several loopholes in this argument. First of all nucleon
chiral perturbation theory [124, 125, 67] is designed to compute nucleon
properties due to a light pion cloud and not the other way around. Whereas the
nucleon is the only other stable hadron made out of light quarks there are of
course many other resonances such as the $\rho,\omega,\dots$ with light quark
content. That opens the door to potential cancellations. This does in fact
happen since bosons and fermions contribute with opposite sign as exploited in
supersymmetry and the Veltman condition on the Higgs mass [126]. Moreover, in
the case where a CFT is spontaneously broken, e.g. [81], the same problems
would be apparent, but only apparent, as the dilaton is believed to be a true
massless Goldstone in that case. Hence, one has to conclude that the EFT is
not capable of making a definite statement about the dilaton mass due to
(potential) hidden cancellations.

##  7 The Dilaton Candidate in QCD: $\sigma\equiv f_{0}(500)$

Let us turn to the dilaton candidate in QCD, the $\sigma$-meson or
$f_{0}(500)$ by official PDG-terminology [127]. We replace $D\to\sigma$
honouring the name used by many particle physicists. The $\sigma$-meson has
captured the interest and imagination of particle physicists for long as
testified by its history and properties in a dedicated physics reports [128]:
it is very broad, it does not fit well into a nonet structure, it defies Regge
trajectories as well as qualitative aspects of the large-$N_{c}$ limit
[128].181818By nonet one means the union of the $SU(3)_{F}$-octet and -singlet
which mix when $SU(3)_{F}$ is broken by non-degenerate quark masses. The goal
of this section is to apply d$\chi$PT at LO and try to see whether one can
understand its mass and width semi-quantitatively.

The meaning of the mass and the width are given by its pole
$\sqrt{{s_{\sigma}}}=m_{\sigma}-\frac{i}{2}\Gamma_{\sigma}$, on the second
Riemann sheet, in $\pi\pi$-scattering. Its current PDG-value [127] is

$\sqrt{s_{\sigma}}=(400-550)-i(200-350)\,\mbox{MeV}\;.$ (7.1)

This range is noticeably larger than
$\sqrt{{s_{\sigma}}}=(441^{+16}_{-8}-i272^{+9}_{-12.5})\,\mbox{MeV}$ the
specific determination in 2006 [129] from Roy equations, which is considered
to have settled the issue of its existence. Earlier determinations were not
far off but suffered larger uncertainties e.g. [130]. It is often noted with
regard to the $\sigma$ being a pseudo Goldstone that its mass is notably
heavier than that of the pion. However, this is not the right comparison since
the $\sigma$, just like the $\eta$, is an $SU(3)_{F}$-singlet for degenerate
quark masses and thus retains sensitivity to the strange quark mass. This can
be seen in the dilaton GMOR-relation (2.21).

As we shall see that the decay rate singlet-octet mixing is important, It is
therefore instructive to discuss the nonet structure and compare it with the
$\rho$-meson family (cf. Tab. 1). The qualitative differences are apparent: i)
the $u\bar{d}$-mesons ($I=1$) are lighter than the $u\bar{s}$-mesons ($I=1/2$)
for the vectors but heavier for the scalars. ii) the ratio of the $I=1$ octet
to $I=0$ singlet is roughly one for the vectors but a factor of two for
scalars. These aspects challenge the quark model picture and can be seen as
one the motivations for introducing the phenomenologically successful
tetraquark model [131]. The widths also follow interesting patterns. The
decays $a_{0}(968),f_{0}(980)\to\pi\pi$ are suppressed by $G$-parity and being
mostly an $\bar{s}s$-state respectively. This is analogous to the
$\omega(780)$ and the $\phi(1200)$-meson in the vector channel. The $\kappa$
is the $K^{*}(895)$-analogoue and indeed rather broad $\Gamma_{\kappa}\approx
600(80)\,\mbox{MeV}$, an aspect which we will understand better when
considering the mixing in the next section.

We now turn to the decay constant $F_{\sigma}$. Its definition from the matrix
element (2.11) is challenging as the $\sigma$-meson is broad. The residue at
the $\sigma$-pole of this quantity is well-defined and generally complex. It
has been extracted in QCD with $\langle\pi|\bar{q}q|\pi\rangle$ and
$\langle\pi|T^{\rho}_{\;\;\rho}|\pi\rangle$ form factor input up to ${\cal
O}(q^{2})$ [132, 133]. To precisely understand how to make use of this
information would require a more careful investigation as the slope of the
form factors change once a dilaton is included. We may get an indirect
estimate from relation $g_{\sigma NN}=\frac{m_{N}}{F_{\sigma}}$ e.g [10] (akin
to the Goldberger-Treiman relation $g_{\pi NN}=\frac{m_{N}}{F_{\pi}}$). In
nuclear physics there approaches using $g_{\sigma NN}$ in their LO Lagrangian
such as the one boson exchange model describing nucleon-nucleon scattering
[134]. The $\sigma$-meson, amongst $\pi$, $\rho$ and $\omega$, describe the
scattering phase shifts reasonably well. From the range in Tab. III [134] one
infers $g_{\sigma NN}=10(2)$ as a reasonable estimate. Remarkably, using
$m_{N}=0.93\,\mbox{GeV}$,

$F_{\sigma}=\frac{m_{N}}{g_{\sigma NN}}\approx 93(19)\,\mbox{MeV}\;,$ (7.2)

a value close to $F_{\pi}=93\,\mbox{MeV}$ emerges. However, one ought to be
cautious as this is a very difficult subject where systematics are difficult
to estimate. Presumably, (7.2) is to be interpreted as a LO value and the
larger value in [132] indicates that corrections are important.

$\begin{array}[]{ l | l || lll}J^{PC}\backslash I&0&0&1/2&1\\\ \hline\cr 0^{++}&\sigma\equiv f_{0}(500)&f_{0}(980)&\kappa\equiv K^{*}_{0}(700)&a_{0}(980)\\\ \Gamma_{0^{++}}&550(150)\,\mbox{MeV}&55(15)\,\mbox{MeV}&600(80)\,\mbox{MeV}&90(50)\,\mbox{MeV}\\\ \hline\cr 1^{--}&\omega(780)&\phi(1020)&K^{*}(895)&\rho(770)\\\ \Gamma_{1^{--}}&6.86(13)\,\mbox{MeV}&0.016\,\mbox{MeV}&52(12)\,\mbox{MeV}&145(3)\,\mbox{MeV}\end{array}$ Table 1: Nonet of $SU(3)_{F}$ which illustrates the special character of the $J^{PC}=0^{++}$ mesons versus the more familiar and understood $J^{PC}=1^{--}$ vector mesons, given in PDG-notation [127]. The double bar separates the singlet from the octet states in the $SU(3)_{F}$ limit.

###  7.1 The width of the $\sigma$-meson

The width of the $\sigma$-meson has been one of the early qualitative
successes of the dilaton approaches. It is well approximated by
$\Gamma_{\sigma}\approx\Gamma_{\sigma\to\pi\pi}$. The photon channel
$\Gamma_{\sigma\to\gamma\gamma}=1.7(4)\,\mbox{keV}$ [135] is highly suppressed
as one would expect. Its interest lays more in the possibility to learn about
the $\sigma$-meson substructure [128]. The amplitude into two pions is
described the effective coupling, ${\cal
L}_{\text{eff}}=\frac{1}{2}g_{\sigma\pi\pi}\sigma\pi^{a}\pi^{a}$, which we can
read off from the Lagrangian (2.3)191919In an EFT framework (for off-shell
$\sigma$ e.g. [9]) $m_{\sigma}^{2}\to q^{2}$ where $q^{2}$ is the momentum
entering the $\sigma$-field.

$g_{\sigma\pi\pi}=\frac{1}{F_{\sigma}}(m_{\sigma}^{2}+(1-\gamma_{*})m_{\pi}^{2}+{\cal
O}(\beta^{\prime\prime}_{*},m_{q}^{2}))\to\frac{m_{\sigma}^{2}}{F_{\sigma}}(1+{\cal
O}(\beta^{\prime\prime}_{*},m_{q}^{2}))\;,$ (7.3)

and resembles earlier expressions [69, 3, 44]. Differences are that
$\gamma_{*}=1$ is an open parameter and that $\beta^{\prime}_{*}$-corrections
with unknown coefficients are parameterised e.g. [44]. The rate

$\Gamma_{\sigma\to\pi\pi}|_{SU(2)}=\frac{3|g_{\sigma\pi\pi}|^{2}}{32\pi
m_{\sigma}}\sqrt{1-4\hat{m}_{\pi}^{2}}\;,$ (7.4)

follows from the $1\to 2$ decay
$d\Gamma=\frac{1}{32\pi^{2}}\frac{|\vec{p}\,|}{m_{\sigma}}d\Omega$ [127] where
$\frac{|\vec{p}\,|}{m_{\sigma}}=\frac{1}{2}\sqrt{1-4\hat{m}_{\pi}^{2}}$ is the
velocity in the frame of the $\sigma$-meson, $\int d\Omega\to 2\pi$ as the
pions are identical particles and the factor of $3$ results from the three
pion channels. The case with and without mixing are denoted by $SU(3)_{F}$ and
$SU(2)_{F}$ respectively.

##### No mixing

Using $m_{\sigma}=440\,\mbox{MeV}$ and $F_{\sigma}=93\,\mbox{MeV}$ one gets

$\Gamma_{\sigma\to\pi\pi}|_{SU(2)}=227\,\mbox{MeV}$ (7.5)

where uncertainties are not given since the mixing is neglected. The rate is a
factor of $\approx 2.5$ lower compared to (7.1) corresponding to correction
factor $\approx 1.5$ in the rate. Given the crudeness of the approach this is
not bad but the question we will try to answer is whether refining it will
bring it closer or further away.

##### Singlet-octet mixing

The effect of singlet-octet mixing is driven by the breaking of $SU(3)_{F}$
and that means by $m_{s}\gg m_{u,d}$. It is known that this effect is not
negligible from the $\eta$-$\eta^{\prime}$ system. In the same way the
$\sigma$-$f_{0}(980)$ system is parameterised by a single angle $\theta$

$\displaystyle|\sigma\rangle$ $\displaystyle\;=\;$
$\displaystyle\phantom{-}\,\cos\theta|S_{1}\rangle+\sin\theta|S_{8}\rangle\;,$
$\displaystyle|f_{0}(980)\rangle$ $\displaystyle\;=\;$
$\displaystyle-\sin\theta|S_{1}\rangle+\cos\theta|S_{8}\rangle\;,$ (7.6)

in terms of the $SU(3)_{F}$ eigenstates. The isospin breaking mixing with the
$a_{0}$, the analogue of $\pi^{0}$ in the $\eta$-$\eta^{\prime}$ system, is
neglected. We follow the approach by Oller [136] based on a Wigner-Eckart
decomposition with two reduced matrix element $g_{1}$ and $g_{8}$ for which
one has

${\cal A}_{\sigma\to(\pi\pi)_{0}}=-\frac{\sqrt{3}}{4}\cos\theta
g_{1}-\sqrt{\frac{3}{10}}\sin\theta g_{8}\;,$ (7.7)

and similarly for $f_{0}(980)$ including all open channels. The zero subscript
stands for $I=0$. The three unknowns $g_{1,8}$ and $\theta$ when fitted to
experiment are (Eq. 2.30 [136])202020There is a second determination
$\theta\approx 21^{\circ}$ in an $U(3)\times U(3)$ $\sigma$-model [137].
Whereas no error is given in this determination, presumably due to model-
dependence, the agreement with (7.8) is encouraging.,212121We have used the
singlet-octet to $\bar{s}s$-$\text{n}\bar{s}s$ basis conversion
$\theta=\phi+35.264^{\circ}$ for $\phi\approx-14^{\circ}$. I am grateful to
Oller in assisting in the conversion which incidentally is not the same as in
the standard $\eta^{\prime}$-$\eta$ mixing.

$g_{1}=3.9(8)\,\mbox{GeV}\;,\quad
g_{8}=8.2(8)\,\mbox{GeV}\;,\quad\theta=19(5)^{\circ}\;.$ (7.8)

Crucially, we note the remarkable enhancement of the octet component
$g_{8}/g_{1}\approx 2$ which could alter the result considerably. Its effect
may be estimated by the the ratio of

$r_{18}=\left|\frac{{\cal
A}_{\sigma\to(\pi\pi)_{0}}^{\theta\phantom{=0}}}{{\cal
A}_{\sigma\to(\pi\pi)_{0}}^{\theta=0}}\right|=1.81\,^{-0.18}_{+0.20}\;,$ (7.9)

mixing versus no mixing. Using this value we get

$\Gamma_{\sigma\to\pi\pi}|_{SU(3)}=r^{2}_{18}\Gamma_{\sigma\pi\pi}|_{SU(2)}=744\,^{-108}_{+146}\pm
40\%\,\mbox{MeV}\;,$ (7.10)

where uncertainties were obtained by adding the ones due to $g_{1}$ and
$\theta$ in quadrature, and we have added the one for $F_{\sigma}$ which make
up the $40\%$ as well. Whereas the estimate is crude that is errors are large,
it is fair to state that its central value is considerably improved due to the
mixing and compares favourable with $\Gamma_{\sigma\to\pi\pi}\approx
544^{+18}_{-25}\,\mbox{MeV}$ from the Roy equation [129]. The enhanced octet
versus singlet matrix element provides an explanation of the large $\kappa$
rate (Tab. 1). It is enhanced by $(g_{8}/g_{1})^{2}$ and its natural value
when compared with the $\sigma$-meson is therefore $140\,\mbox{MeV}$ rather
than $600\,\mbox{MeV}$. It would be interesting to understand the octet
enhancement qualitatively.

###  7.2 The mass of the $\sigma$-meson

In QCD the GMOR-type mass relation, which are the $\chi$PT  LO expressions
[65, 67],

$m_{\pi}^{2}=(m_{u}+m_{d})B_{0}\;,\quad
m_{K^{+}}^{2}=(m_{u}+m_{s})B_{0}\;,\quad m_{K^{0}}^{2}=(m_{d}+m_{s})B_{0}\;,$
(7.11)

work rather well for the pions and the kaons (recall
$B_{0}=-\langle\bar{q}q\rangle/F_{\pi}^{2}$). This raises hopes that that the
dilaton GMOR-relation (5.12) will give a good for value of the $\sigma$-mass.
We shall see and understand that this is not necessarily the case. Let us
first adapt (5.12) to non-degenerate quark flavours. Using (7.11) may express
in terms of physical quantities

$F_{\sigma}^{2}m_{\sigma}^{2}=(1+\gamma_{*})(3-\gamma_{*})F_{\pi}^{2}\sum_{q=u,d,s}m_{q}B_{0}|_{\gamma_{*}=1}=2F_{\pi}^{2}(m_{K^{0}}^{2}+m_{K^{+}}^{2}+m_{\pi}^{2})\;.$
(7.12)

With $m_{\pi}=140\,\mbox{MeV}$, $m_{K}=495\,\mbox{MeV}$,
$F_{\pi}=93\,\mbox{MeV}$ and $F_{\sigma}=93\,\mbox{MeV}$ as input on gets,

$m_{\sigma}|_{\text{LO}}\approx
1\,\mbox{GeV}\left(\frac{93\,\mbox{MeV}}{F_{\sigma}}\right)\;,$ (7.13)

a value which is about a factor of two larger than the in the real world.
There is some irony here as often the $\sigma$-meson is regarded as being too
heavy to be considered a pseudo Goldstone. Formula (7.12) has been obtained in
[138] but the difference is that $\gamma_{*}$ and
$\beta^{\prime}_{*}$-corrections were undetermined and thus the large value
has not come to their attention.

One might wonder whether d$\chi$PT would be convergent with such a large LO
value since it is well-known that $SU(3)$-$\chi$PT is not as efficient as
$SU(2)$-$\chi$PT because of the proximity of $m_{K}$ to $m_{\rho}\approx
770\,\mbox{MeV}$. The separation of the Goldstone sector is not strong in
actual numbers. To get a an idea we might want to use the NLO formula for
$SU(2)$-$\chi$PT [67, 139]

$m_{\pi}^{2}|_{{\text{NLO}}}=m_{\pi}^{2}(1-\frac{1}{32\pi^{2}}\frac{m_{\pi}^{2}}{F_{\pi}^{2}}\bar{\ell}_{3})\;,$
(7.14)

with $\bar{\ell}_{3}\equiv\ln{\Lambda_{3}^{2}}/{m_{\pi}^{2}}=3.53(26)$ [140]
is sizeable due to a chiral logarithm. One infers NLO-corrections factors of
$(0.025,0.21,0.25,1.4)$, using LO masses
$(m_{\pi},m_{K},m_{\sigma},1\,\mbox{GeV})$.222222This procedure only gives
realistic numbers for the pions since the Kaon requires $SU(3)$-$\chi$PT [141]
and for the $\sigma$ d$\chi$PT is required which has not been fully developed
at NLO yet. We believe however that the numbers give reasonable estimate of
the size of the NLO corrections. The large correction factor $1.4$ for
$m_{\sigma}|_{{\text{LO}}}$ is telling us that convergence cannot be expected.
The situation is unsatisfactory but deserves some more contemplation.

The large value obtained is driven by $m_{K}$ which is comparatively large due
the $m_{s}\gg m_{u,d}$. The GMOR-type formulae are expansions $m_{q}$ (or
Goldstone masses) and not in $1/m_{q}$ . That is, there is no built-in
decoupling limit but rather one decouples by hand in excluding a quark from
the sum in (7.12). Can we assess this in another way? Yes, through the mixing
if we are willing to commit to a quark mass picture. In App. C we have argued
the mixing angle supports the $\bar{q}q$-state interpretation. Eq. (C)
indicates suppression of the $\bar{s}s$ versus the
$\bar{u}u$,$\bar{d}d$-contribution. The strange quark decoupling angle
$\theta_{\text{dec}}\approx 35.7^{\circ}$ is close but not too close to
$\theta=19^{\circ}$.232323There is indirect evidence for strangeness in the
$\sigma$-meson. The $\langle\bar{s}s(x)\bar{u}u(0)\rangle$-correlator is non-
zero suggesting a light state which could correspond to the $\sigma$-meson
[142]. Indeed if we were to completely decouple the strange quark we would
expect to replace $2m_{K}^{2}+m_{\pi}^{2}\to 2m_{\pi}^{2}$ in (7.12) which
does yield $m_{\sigma}\approx 2m_{\pi}=280\,\mbox{MeV}$ an underestimate.

In summary the situation remains inconclusive but we pointed out why we cannot
expect formula (7.12) to give us a good number. First from the convergence in
the EFT and second the formula overestimates the role of the strange quark
since it has no decoupling built in.

##  8 Outlook

There are a few directions in which the work begun in this paper can be
extended, which is to investigate the $m_{q}$ dependence of the $\sigma$-mass,
extending the LO Lagrangian (2.3) to include $\overline{m}_{D}$-mass and
investigate whether the Higgs boson could be a dilaton. Below we give a brief
outlook on some of these matters.

###  8.1 Quark mass dependence of the $\sigma$-meson(s) on the lattice and
beyond

Investigating the $m_{q}$-dependence of the $\sigma$-meson in QCD and other
gauge theories with chiral symmetry breaking is interesting and still largely
open. There are several avenues and it is to be hoped that efforts continue
and for new ones to start.

Before the advent of the LHC investigations in lattice gauge theory started in
order to obtain reliable information for walking technicolor [59, 60, 143] and
composite Higgs models [144]. In recent years light scalars with $\sigma$
quantum numbers were reported for decreasing quark masses [11, 12, 13, 14, 15,
16, 17, 18, 19, 21, 22, 23, 24] and in particular [20]. There are also studies
of the QCD $\sigma$-meson on the lattice [145, 146, 147, 148] which is
challenging because of its large width. As far as we know all simulations are
performed at physical strange quark mass but larger $m_{u,d}$ masses.
Concretely, in [145, 146] a pion mass $m_{\pi}=391\,\mbox{MeV}$ leads to
$m_{\sigma}=745(5)\,\mbox{MeV}$ and $m_{f_{0}(980)}=1166(45)\,\mbox{MeV}$.

Amongst the analytic methods the Roy equations [149, 129], which give the most
precise determination of the $\sigma$-pole, would seem the most promising
method. They do rely though, as many dispersive methods, on a mixture of
experimental and theoretical hadronic input. Hence the success would depend on
how well one could control the input as a function of the quark masses. The
lattice could play an important and indirect role in providing input for the
Roy equations. For example, recently the data from [145, 146] has been used in
Roy equations [150] and the $m_{\sigma}=745(5)\,\mbox{MeV}$ consistent with
[145, 146] has been found. Another possible avenue is the analytic $S$-matrix
bootstrap [151, 152] which would equally depend on input. Methods based on
elastic unitarity, $\pi\pi$-scattering $s<4m_{\pi}^{2}$, have been applied
(reviewed in [128]). Concrete applications include the inverse amplitude
method with NLO $\chi$PT input for fixed $m_{s}$ and varying pion mass [153],
unitarised chiral perturbation theory [154] with varying $m_{s}$-dependence
and the $N/D$-method with LO $\chi$PT -input [136] where the the nonet has
been deformed to a $SU(3)_{F}$-symmetric point,

The numbers from lattice and analytic methods indicate that $\sigma$ could
decrease by ${\cal O}(100\,\mbox{MeV})$ for in $m_{u,d}\to 0$. Hence, the role
of the strange quark mass might be important for which information is sparse.
The work of Oller [136] gives most concrete information where the nonet, in
mass and widths, are continuously deformed to become $SU(3)_{F}$-symmetric at
$m_{\pi,K}=350\,\mbox{MeV}$ (corresponding to a degenerate quark mass $\approx
23\,\mbox{MeV}$, an increase in $m_{u,d}$ and a decrease in $m_{s}$). At this
point $m_{\sigma}\approx 300\,\mbox{MeV}$ as can be inferred from the plot in
Fig. 2 in [136]. This is a significant reduction in view of the expected
increase due to $m_{u,d}$.242424Note that for small deformation it is the
width rather than the mass that decreases. This also explains why the
$\sigma$-mass shows little variation for small $m_{s}$ in the in Ref. [154].
This is somewhat at variation with [154] where (small) $m_{s}$-changes in mass
and width were found to be very small. Hence the situation is not entirely
conclusive but rather motivate to further investigate $m_{\sigma}$ for varying
light quark mass.

###  8.2 The Higgs as a dilaton?

It has been appreciated for a long time that in the absence of the Higgs VEV
the SM is conformal up to the logarithmic running of the couplings. The
dilaton therefore fits the role of the Higgs naturally as it couples to mass
and is associated with a VEV. There are several realisations of this scenario
but they all have in common that the SM Higgs sector is replaced by a
(strongly coupled) sector which undergoes spontaneous scale and electroweak
symmetry breaking at a scale $F_{D}$ (not necessarily equal to the Higgs VEV
$v\approx 250\,\mbox{GeV}$). The most relevant change is that in the LO-SM
Lagrangian the dilaton replaces the Higgs as follows

$1+\frac{h}{v}\to e^{-\frac{D}{F_{D}}}\to 1+\frac{h}{F_{D}}\;,$ (8.1)

where in the second arrow the freedom of field redefinition has been made use
of.252525This is legitimate as long as we are interested in on-shell matrix
elements and for small fluctuations ($h<-F_{D}$ would violate the positivity
of the exponential). This particle behaves like a SM Higgs with all couplings
rescaled by the ratio of the two scales: $r_{v}=\frac{v}{F_{D}}$. Post LHC we
know that this ratio has to be close to one within approximately ten percent
which would equally tame tensions with electroweak precision observables (e.g.
[155] where $r_{v}=1$ implies $\kappa_{W}=1$). This raises the question: is
$r_{v}\approx 1$ natural? Other important low energy parameters are the
dilaton mass and more generally its potential. The answer to all of these
questions is model-dependent and we thus restrict attention to our framework.

We consider a gauge theory with a gauge group $G^{\prime}$ which undergoes
chiral symmetry breaking close to the electroweak scale. This means that the
$W$\- and $Z$-boson masses are generated in the same way as in technicolor.
The Higgs VEV and the pion decay constant of the new sector are related by
$v^{2}=n_{d}F_{\pi}^{2}$, where $n_{d}$ is the number of techniquark
electroweak doublets. Since the Higgs width has been measured to be narrow
$\Gamma_{h}=3.2^{+2.8}_{-2.2}\,\mbox{MeV}$ [127] there can only be one
doublet, which takes on the role of the longitudinal degrees of freedom of the
gauge bosons. Otherwise the Higgs/dilaton would disintegrate fast into the
additional $\pi^{\prime}\pi^{\prime}$-pairs giving rise to a width
$\Gamma_{h}={\cal O}(100\,\mbox{GeV})$. Hence, $v=F_{\pi}$ and therefore the
difference of the dilaton versus SM-Higgs coupling are parameterised by the
ratio

$r_{N_{f}}=\frac{F_{\pi}}{F_{D}}\;,\quad$ (8.2)

of pion to dilaton decay constants. Besides the indicated $N_{f}$-dependence
there is also an implicit $G^{\prime}$-dependence which is less important but
has to be kept in mind. Whereas it is likely that $r_{N_{f}}={\cal O}(1)$
there is no known reason why it should be numerically close to one. The
quantity we are interested in is $r_{2}$ and we resort to actual QCD for
guidance. With $F_{D}$ as in (7.2) and $F_{\pi}=93\,\mbox{MeV}$ one finds a
number $r_{2-3}|_{\text{QCD}}=1.0(2)$ which is compatible with one within
uncertainties. Cautionary remarks apply. First it is difficult to estimate the
systematics of (7.2). Second whereas we are interested in $N_{f}=2$ massless
quarks, in QCD we have two light quarks and a light but sizeable strange
quark.262626The $N_{f}$ dependence comes from the $\sigma$-meson being a
singlet $SU(N_{f})$ as manifested in (2.21). Hence on expects
$r_{N_{f}}\propto 1/\sqrt{N_{f}}$. In a lattice fit [156] $r_{8}\approx 0.33$
found for which the scaling gives $r_{2}=2r_{8}=0.66$ a value slightly lower
than $1$. There could be many reasons, one of which is that the (2.21) is only
a LO relation and the analysis in Sec. 6.2 indicates that the counting is not
that straightforward. The most pragmatic interpretation is that this motivates
trying to find a reason for $r_{2}$ being close to one.

So far we have not addressed how the coupling (8.1) arises.272727In the most
commonly used dilaton scenarios this follows from the conformality of the
total SM and extended sector. This is not the route we have in mind as this
scenario has sizeable contributions in $gg\to h$ for example, in tension with
the findings at the LHC [157]. The starting point is the higgsless SM
developed as an EFT for a heavy Higgs [158, 159, 160]. The would-be Goldstone
bosons $U=\exp(i2T^{a}\pi^{a}/F_{\pi})$ transform as $U\to V_{L}UV_{Y}$ under
the SM gauge group $SU(2)_{L}\times U(1)_{Y}$ with $V_{Y}=e^{iyT_{3}}$ such
that the condition $\det U=1$ is preserved. The effective Lagrangian contains
terms of the form

${\cal
L}\supset\frac{1}{4}v^{2}\mathrm{Tr}[D^{\mu}UD_{\mu}U^{\dagger}]-v\bar{q}_{L}Y_{d}U{\cal
D}_{R}+\dots\;,$ (8.3)

where $\bar{\cal D}_{R}\equiv(0,\bar{d}_{R})$ and the covariant derivatives
assure $SU(2)_{L}\times U(1)_{Y}$ gauge invariance. The dots correspond to
further and similar fermionic and gauge kinetic terms. Eq. (8.3) resembles the
corresponding d$\chi$PT term in (2.16) but lacks the dilaton nota bene. It
seems to us that one cannot resort to the compensator argument in coupling the
dilaton since it is only the scale symmetry of $G^{\prime}$ which is broken.
However, the same rationale that would imply $r_{2}\approx 1$ ought to group
the pions and the dilaton under one and the same symmetry. Assuming this to be
true then gives

${\cal
L}\supset\frac{1}{4}v^{2}e^{-2D/F_{D}}\mathrm{Tr}[D^{\mu}UD_{\mu}U^{\dagger}]-ve^{-D/F_{D}}\bar{q}_{L}Y_{d}U{\cal
D}_{R}+\dots\;,$ (8.4)

an effective Lagrangian equivalent to the SM at LO modulo the Higgs potential.
Hence, finding an (approximate) symmetry assuring $r_{2}\approx 1$ would
therefore help in both ways, explaining the required closeness to the SM and
giving a rationale for coupling the dilaton in the same way as the SM-Higgs.
The reasoning above is the same as in [30], except that we give provide a
reason for how the dilaton is coupled to the SM.

Let us turn to the dilaton potential which consists of the pure
$G^{\prime}$-part and the one induced by the mixed Lagrangian (8.4). For the
former the situation is similar to CDQCD, we do not know anything for certain
other than how to incorporate mass terms of the $q^{\prime}$-quarks. However,
since $m_{q^{\prime}}\neq 0$ breaks $SU(2)_{L}$, these terms are absent in the
most straightforward setting. If the dilaton acquires a mass in the chiral
limit $\overline{m}_{D}\neq 0$ then $V_{\Delta}$ (2.3) is a simple potential
describing it. By the findings of the double soft theorems (in Sec. 5)
$\Delta=2$ but it yet remains to be understood what operator takes on this
role or whether the assumptions do not apply. The value $\Delta=2$ precisely
corresponds to the SM Higgs potential when linearised as in (8.1). Let us turn
to the potential due to the coupling to the SM in (8.3). As is the case in
composite Higgs models the coupling to the $W,Z$-boson and the top quark would
induce sizeable corrections to the Higgs mass which are quadratically
sensitive to the cut-off $\Lambda^{\prime}=4\pi F_{D}$ of the
$G^{\prime}$-confinement scale. It is difficult to say anything concrete other
than parametrically these contribution are of order ${\cal O}(v^{2})$ subject
to further sizeable NLO corrections. In view of the generally large and
negative contribution of the top mass a sizeable chiral dilaton mass could
potentially be required. A more thorough assessment might necessitate to find
a UV completion at some scale $M^{2}$, explaining the origin of the coupling
(8.4) through terms of the type ${\cal L}_{\text{eff}}\supset\frac{{\cal
O}(1)}{M^{2}}\bar{q}^{\prime}q^{\prime}\bar{t}t$.

There are further phenomenological aspects that need attention. In the
standard dilaton scenarios the $\beta$ function contributions to the SM
radiative processes $gg\to h$ and $h\to\gamma\gamma$ give rise to severe
constraints e.g. [157]. In our scenario the $q^{\prime}$-fermions are not
charged under $SU(3)_{c}$, implying that $gg\to h$ is truly loop-suppressed
(with respect to the SM) and therefore we do not expect tensions with the LHC.
The process $h\to\gamma\gamma$ is more subtle as the $q^{\prime}$-fermions are
generally electrically charged (since $q^{\prime}_{L}$ is charged under
$SU(2)_{L}$). Its calculation is a formidable task as non-perturbative. Early
assessments within QCD go back to the discovery of the trace anomaly [46, 47,
48]. This LO evaluation gives the correct order of magnitude e.g. [43]. In QCD
the rate is extracted indirectly from scattering data [135, 161]. Since there
is no precise method of direct computation and the rate might well be
$m_{\sigma}/m_{N}$-dependent, it seems that one cannot easily borrow an
estimate from QCD. The QCD-value
$\Gamma_{\sigma\to\gamma\gamma}=1.7(4)\,\mbox{keV}$ indicates however that it
could be sizeable and therefore deserves attention. There is another challenge
and that is the non-discovery of hadronic $G^{\prime}$-resonances. Whereas the
dilaton itself might be light, the basic scale is set by the Higgs VEV
$v=F_{\pi}\approx 250\,\mbox{GeV}$. A $2\,\mbox{TeV}$-benchmark gives a ratio
$2\,\mbox{TeV}/F_{\pi}|_{G^{\prime}}=8$ which is comparable to the one in QCD:
${m_{\rho}/F_{\pi}}|_{\text{QCD}}\approx 8.3$. Without a phenomenological
analysis and a more precise assessment of the ratio it seems difficult to make
a statement other than that these resonances may be within the future LHC-
reach.

In summary the by far most important task is investigate whether there is a
reason for $r_{2}\approx 1$ in (8.2). If this were true then our rough
assessment indicates that the model might pass current constraints. We hope to
return to these topics in future work.

##  9 Conclusions and Summary

In this paper we further explored the possibility that gauge theories in the
chirally broken phase admit an IR fixed point interpretation (cf. Fig. 1).
Scaling dimensions were deduced by matching the gauge theory fixed-point
behaviour with the effective (dilaton)-$\chi$PT (Sec. 3). The mass anomalous
dimension at the fixed point is found to be $\gamma_{*}=1$, providing a more
in-depth analysis of our earlier findings [7]. The vanishing of the $\beta$\-
and $\gamma_{m}$-slopes at the fixed point:
$\beta^{\prime}_{*}=\gamma^{\prime}_{*}=0$, cf. (3.37) and (3.41), are new
results. In particular, $\beta^{\prime}_{*}=0$ has many attractive features:
i) it suits a mass-gap interpretation, ii) it is consistent with ${\cal N}=1$
supersymmetry (Sec. 4)282828Furthermore, we argue that $\gamma_{*}=1$ in the
chirally broken phase provides a correct description of ${\cal N}=1$ Seiberg-
duality aspects. and, iii) it implies logarithmic running of the fixed-point
coupling $\delta g\equiv g-g^{*}$ (3.40). The latter makes it plausible that
the trace anomaly is reproduced by the generalisation of the energy-momentum
tensor for chiral theories (to include a dilaton).

In the second part of the paper we focus on the dilaton $D$, the (pseudo)
Goldstone boson due to spontaneous scale symmetry breaking. The scaling
dimension of an operator ${\cal O}\subset T^{\rho}_{\;\;\rho}$, generating the
dilaton mass, is found to be $\Delta_{{\cal O}}=2$ (using a double-soft
dilaton theorem cf. Sec. 5). An important role is played by the partial
derivative $x\cdot\partial$-term in the dilatation commutator (5.4): i) it
renders the mass positive and ii) it is the counterpart of the Zumino-term
(2.20) in d$\chi$PT . In the gauge theory, ${\cal O}=m_{q}\bar{q}q$ takes on
the role of this operator and can be viewed as an alternative reasoning for
$\Delta_{\bar{q}q}=3-\gamma_{*}=2$. Our findings suggest that one should be
careful when interpreting dilaton soft theorems in conjunction with the field
strength tensor $G^{2}$ (as is done in PCDC-type relations (5.16)). In Sec. 6
we contemplate whether the dilaton could be massless or not. In the large
large-$N_{c}$ limit degeneracy the dilaton is not mass degenerate with its
nonet partners because of the $F_{D}\propto N_{c}$ scaling. This is relevant
as otherwise $\overline{m}_{D}=0$ would be in contradiction with lattice
simulation. Comparison with spontaneously broken conformal theories, where the
dilaton is believed to be massless, suggest the EFT inherits subtle
cancellations.

In the third part applications of the dilaton are explored. In Sec. 7 we
consider whether the $\sigma$-meson in QCD could be a dilaton. Singlet-octet
mixing is found to be important as it considerably improves the prediction of
the width (7.10). The leading order $\sigma$-mass, which follows from the
dilaton GMOR-relation (7.12), is rather large and indicates convergence issues
in the EFT (due to the large strange quark mass). The dilaton as the low
energy part of a new gauge sector, can take on the role of the Higgs boson
provided that the ratio of pion to dilaton decay constants $r_{N_{f}}\equiv
F_{\pi}/F_{D}$ is unity for two flavours (cf. Sec. 8). Whereas in QCD
indications are that $r_{2}\approx 1$ there is no underlying principle known
why this ought to happen. An assessment beyond leading order would give rise
to a potential and corrections to radiative processes such as $gg\to h$ and
$h\to\gamma\gamma$ which might pass current LHC-constraints.

There are many directions to explore but among the unresolved questions the
most outstanding ones are: i) what is the dilaton (or $\sigma$-meson) mass in
the chiral limit? Is it zero or at least considerably smaller than the nucleon
mass and how does it depend on the distance to the lower boundary of the
conformal window? ii) is there a principle that would imply $r_{2}\approx 1$?
These questions are far from straightforward to answer by pure reasoning.
Hopefully lattice Monte Carlo simulations and methods using analyticity and
unitarity can be of help.

##### Acknowledgments:

During large periods of this work RZ was supported by a CERN associateship and
an STFC Consolidated Grant, ST/P0000630/1. I am grateful to Claude Amsler,
Gunar Bali, Latham Boyle, Jorge Camalich, Gilberto Colangelo, Matthew
McCullough, Poul Damgard, Luigi Del Debbio, Shoji Hashimoto, Martin
Hoferichter, James Ingoldby, Andreas Jüttner, Max Hansen, Georgios Karananas,
Heiri Leutwyler, Daniel Litim, Tom Melia, José-Antonio Oller, Bashir
Moussalam, Agostino Patella, José-Ramon Pelaez, Werner Porod, Géraldine
Servant, Misha Shifman, Christopher Smith, Yuji Tachikawa, Neil Turok f and
the participant of “asymptotic safety meets particle physics” at DESY for
useful discussions. At last apologies if I missed your work.

##  Appendix A Conventions

The Minkowski metric $\eta_{\mu\nu}$ reads $\textrm{diag}(1,-1,-1,-1)$,
$g=\det(g_{\mu\nu})$ is the determinant. Weyl transformations are defined by

$g_{\mu\nu}\to e^{-2\alpha}g_{\mu\nu}\;,\quad\hat{D}\to\hat{D}-\alpha\;,$
(A.1)

where the normalised dilaton field is $\hat{D}\equiv D/F_{D}$. Kinetic terms
are shortened as
$(\partial\varphi)^{2}\equiv\partial_{\mu}\varphi\partial^{\mu}\varphi$. The
$\beta$ function is defined in (2.1) and is given by

$\frac{\beta}{g}=-(\beta_{0}\frac{g^{2}}{(4\pi)^{2}}+{\cal
O}(g^{4}))\;,\quad\beta_{0}=(\frac{11}{3}C_{A}-\frac{4}{3}N_{F}T_{F})\;,$
(A.2)

where for QCD with $N_{f}=3$ and $G=SU(3)$ we have $\beta_{0}=9$ since
$T_{F}=\frac{1}{2}$ and $C_{A}=N_{c}$. Further to that we derive
$\gamma_{G^{2}}$ in (2.5). It departs from the observation that
$T^{\rho}_{\;\;\rho}=\frac{\beta}{2g}G^{2}$ (for $m_{q}=0$) is an RG invariant

$0=\frac{d}{d\ln\mu}T^{\rho}_{\;\;\rho}=\frac{d}{d\ln\mu}\left(\frac{\beta}{2g}G^{2}\right)=\left(\frac{\beta}{2g}\right)^{\prime}\beta
G^{2}+\frac{\beta^{2}}{2g}(G^{2})^{\prime}\;,$ (A.3)

where the prime denotes differentiation with respect to $g$ and from (2.1) one
then gets

$\gamma_{G^{2}}=2g\left(\frac{\beta}{2g}\right)^{\prime}=\beta^{\prime}-\frac{\beta}{g}\;,$
(A.4)

in accordance with (2.5).

##  Appendix B More on the soft theorems

In this appendix we take a look at the results in Sec. 5 from a slightly
different angle. First in deriving formula (5.8) directly from d$\chi$PT
and second by considering the soft theorem for a single dilaton. The common
theme is the matching of the QCD operator

$\displaystyle{\cal O}_{\bar{q}q}$ $\displaystyle\;=\;$
$\displaystyle(1+\gamma_{*})\sum_{q}m_{q}\bar{q}q=\partial_{\ln m_{q}}{\cal
L}^{\text{d}\chi\text{PT}}_{\text{LO}}$ $\displaystyle\;=\;$
$\displaystyle(1+\gamma_{*})\sum_{q}m_{q}\langle\bar{q}q\rangle\hat{\chi}^{{\Delta_{\bar{q}q}}}=(-N_{f}m_{\pi}^{2}F_{\pi}^{2})\hat{\chi}^{{\Delta_{\bar{q}q}}}\;,$
(B.1)

to the d$\chi$PT . In the last equality was used (5.14) and we further
notice consistency for $\langle{\cal O}_{\bar{q}q}\rangle$.

###  B.1 The double soft theorem from dilaton-$\chi$PT

We consider it important to make contact with the result (5.8) directly from
the Lagrangian. Whereas in a general CFT setup (5.8) holds for any
$\Delta_{{\cal O}}$ this is not the case here as it was already concluded that
$\Delta_{{\cal O}}=d-2$. Hence, the aim is to find this result again. Using
d$\chi$PT (2.3) we find

$\displaystyle{\langle D|{\cal O}_{\bar{q}q}|D\rangle}$ $\displaystyle\;=\;$
$\displaystyle\left(\langle
D|\frac{1}{2}\hat{D}^{2}({\Delta_{\bar{q}q}}(d+{\Delta_{\bar{q}q}})-{\Delta_{\bar{q}q}}^{2})-\frac{1}{2}\frac{(\partial\hat{D})^{2}}{m_{D}^{2}}{\Delta_{\bar{q}q}}(d-2)|D\rangle\right){\langle{\cal
O}_{\bar{q}q}\rangle}$ (B.2) $\displaystyle\;=\;$
$\displaystyle\frac{1}{F_{D}^{2}}({\Delta_{\bar{q}q}}(d+{\Delta_{\bar{q}q}})-{\Delta_{\bar{q}q}}^{2}-{\Delta_{\bar{q}q}}(d-2)){\langle{\cal
O}_{\bar{q}q}\rangle}|_{{\Delta_{\bar{q}q}}=d-2}=\frac{1}{F_{D}^{2}}2(d-2){\langle{\cal
O}_{\bar{q}q}\rangle}\;.$

Indeed, this result is consistent with Eqs. (5.1) and (5.8) provided that
${\Delta_{\bar{q}q}}=d-2$ holds. Hence, we have achieved our goal. It is
instructive to comment on the origin of the terms as there is an unusual
element due to the presence of the tadpole in

$\hat{\chi}^{{\Delta_{\bar{q}q}}}=1-{\Delta_{\bar{q}q}}\hat{D}+\frac{1}{2}{\Delta_{\bar{q}q}}^{2}\hat{D}^{2}+{\cal
O}(\hat{D}^{3})\;.$ (B.3)

Let us focus on the first line in (B.2). The ${\Delta_{\bar{q}q}}^{2}$-term is
straightforward as it corresponds to the $D^{2}$-term in (B.3). The remaining
two originate from tadpole diagrams due to the linear $\hat{D}$-term in (B.3).
For those the dilaton propagator assumes the form
$\Delta_{F}(q^{2}=0)\to-i/m_{D}^{2}$ and the mass term gets cancelled against
a corresponding term in the Lagrangian. Specifically, the
$\hat{D^{2}}\Delta(d+{\Delta_{\bar{q}q}})$\- and
$(\partial\hat{D})^{2}{\Delta_{\bar{q}q}}(d-2)$-terms are due to the two
interaction terms

${\cal
L}^{\text{d}\chi\text{PT}}_{\text{LO}}\supset(d+{\Delta_{\bar{q}q}})\,m_{D}^{2}F_{D}^{2}\frac{\hat{D}^{3}}{3!}-(d-2)\,F_{D}^{2}\hat{D}\frac{(\partial\hat{D})^{2}}{2}\;.$
(B.4)

###  B.2 The mass operator in the deep infrared

The goal of this section is to sketch a derivation of the result (5.10) on the
scaling dimension of the mass operator $\lambda{\cal O}\subset
T^{\rho}_{\;\;\rho}$ using methods similar to Sec. 3. The parameter $\lambda$
is analogous to $m_{q}$ for ${\cal O}=\bar{q}q$. Using RG arguments similar to
Sec. 3.2.4, the TEMT correlator assumes the form

$\langle{T^{\rho}_{\;\;\rho}}(x){T^{\rho}_{\;\;\rho}}(0)\rangle_{\text{CDQCD}}=\frac{\mu_{0}^{8}}{(\hat{x}^{2})^{d}}\,g(\hat{x}^{y_{\lambda}}\hat{m}_{q},\hat{F}^{-y_{m}}\hat{m}_{q},\mu)\;,$
(B.5)

where $\Delta_{T^{\rho}_{\;\;\rho}}=d$ with $\beta^{\prime}_{*}=0$ has been
assumed, $y_{\lambda}=d-\Delta_{\cal O}$, $g$ is a dimensionless function and
$\mu_{0}$ some arbitrary reference scale dividing all hatted quantities to
render them dimensionless. Predictivity is restored by analysing the
$g$-behaviour. Since the TEMT is proportional to $m_{\pi,D}^{2}={\cal
O}(\lambda)$ it follows that
$\langle{T^{\rho}_{\;\;\rho}}(x){T^{\rho}_{\;\;\rho}}(0)\rangle={\cal
O}(\lambda^{2})$ and thus the scaling exponent effectively changes from $d\to
d-y_{\lambda}=\Delta_{\cal O}$ as we would expect. If there is no $F$
dependence involved such as is the case for the two-pion and -dilaton
intermediate states then this is all there is. The single dilaton intermediate
state is ${\cal O}(F_{D}^{2})$ which can be formed as the ratio of the first
and the second entry in $g$ to yield $x^{2}F^{2}$ and thus $d\to
d-y_{\lambda}-1=\Delta_{{\cal O}}-1$ in that case. In summary

$\eta=\left\\{\begin{array}[]{ll}\Delta_{\cal O}-1&\text{single GB}\\\
\Delta_{\cal O}&\text{two GBs}\end{array}\right.\;,$ (B.6)

where $\eta$ is the $1/(x^{2})^{\eta}$ scaling exponent. Hence, one gets

$\langle{T^{\rho}_{\;\;\rho}}(x){T^{\rho}_{\;\;\rho}}(0)\rangle_{\text{CDQCD}}\propto\frac{c_{D}e^{-m_{D}x}}{(x^{2})^{\Delta_{\cal
O}-1}}+\frac{1}{(x^{2})^{\Delta_{\cal
O}}}(c_{2\pi}e^{-2m_{D}x}+c_{2D}e^{-2m_{D}x})\;,$ (B.7)

where $c_{D}={\cal O}(F_{D}^{2})$ and $c_{2\pi,D}$ are $F_{\pi,D}$-independent
as of above. The exponential behaviour follows from dominance of the lowest
state in the specific channel and is well known from the study of Euclidean
correlation functions in lattice applications. Let us turn to the EFT where
the TEMT at LO assumes the form

$T^{\rho}_{\;\;\rho}|^{\text{LO}}_{d\chi\text{PT}}=m_{D}^{2}D^{2}+m_{\pi}^{2}\pi^{2}+F_{D}m_{D}^{2}D+\dots\;,$
(B.8)

which of course is zero for $\lambda\to 0$. Dots stand for terms which give
suppressed contributions in the deep IR. Assembling all the information one
gets

$\langle{T^{\rho}_{\;\;\rho}}(x){T^{\rho}_{\;\;\rho}}(0)\rangle^{\text{LO}}_{\text{d}\chi\text{PT}}\propto\frac{F_{D}^{2}}{(x^{2})^{d-3}}e^{-m_{D}x}+\frac{1}{(x^{2})^{d-2}}(c^{\prime}_{2\pi}e^{-2m_{D}x}+c^{\prime}_{2D}e^{-2m_{D}x})\;,$
(B.9)

where the exponential behaviour follows from the asymptotic limit of the
scalar propagator $\Delta_{E}(x,m)\propto e^{-mx}/x^{2}$. Equating with (B.7)
one gets the result in accordance with (5.10)

$\displaystyle\Delta_{\cal O}=d-2\;.$ (B.10)

.

###  B.3 The single soft theorem, $\langle 0|\bar{q}q|D\rangle$ and the the
dilaton decay constant

Let us turn to the single dilaton soft theorem. Applying (5) yields

$\langle 0|{\cal O}_{\bar{q}q}|D\rangle=-\frac{1}{F_{D}}\langle[Q_{D},{\cal
O}_{\bar{q}q}]\rangle=-\frac{{\Delta_{\bar{q}q}}}{F_{D}}\langle{\cal
O}_{\bar{q}q}\rangle$ (B.11)

as the remainder vanishes and we notice that the same result would follow from
expanding (B) to linear order in the dilaton field. We may now make contact
with [156, 162] where the following matrix element was defined

$\langle 0|\sum_{q}m_{q}\bar{q}q|D\rangle=F_{S}m_{D}^{2}\;.$ (B.12)

By matching with (B.11) one obtains

$F_{S}=N_{f}\,\frac{{\Delta_{\bar{q}q}}}{(1+\gamma_{*})}\left(\frac{m_{\pi}^{2}F_{\pi}^{2}}{m_{D}^{2}F_{D}^{2}}\right)F_{D}\;.$
(B.13)

It is the analogue of Eq. 9 in [156] derived from their Lagrangian with the
identification $y={\Delta_{\bar{q}q}}$ and $1+\gamma_{*}\to 2$ above. A
distinctive feature is that in [156] a generic potential (2.3) is assumed on
top of $m_{q}\neq 0$ with ${\Delta_{\bar{q}q}}=y$ and this leads to
$f_{\pi,D}$ and $F_{\pi,D}$ where the lower case quantities are the one in the
chiral limit. Using that $F_{\pi}/F_{D}=f_{\pi}/f_{D}$ [156] we can escape
these difficulties in principle and if we use the values in [162] we of get an
agreement of the right- and left-hand within $7\%$ which is well within the
errors. This is of no surprise as according to our understanding $F_{S}$ went
into the fit in [156]. This is just a confirmation of their numbers. However,
if we make the assumption that $\overline{m}_{D}=0$ and use the dilaton GMOR-
relation (2.21) with the formula above then we get a remarkable relation
(using $\gamma_{*}=1$)

$F_{S}|_{\overline{m}_{D}=0}=\frac{1}{2}F_{D}\;,$ (B.14)

between the dilaton decay constant and the dilaton coupling to $\bar{q}q$. We
we take $F_{\pi}^{2}/F_{D}^{2}=0.1089(41)$ [156] and $F_{\pi}=0.021677(40)$
[162] and get $F_{S}\approx 0.033$ from (B.14) which is $30\%$ off from
$F_{S}=0.0254(17)$ [162] which is possibly too good in view of the crude
treatment here: i) $m_{\pi}/F_{\pi}\approx 4$ which is a similar ratio to the
kaon mass in QCD for which one would expect $30\%$ NLO-corrections in the EFT
and ii) the fact that those references assume the potential (2.3) for which
they fit $\Delta\approx 3.5$.

##  Appendix C On Singlet-Octet Mixing

The aim of this appendix is to give some more interpretation with regards to
the quark content of the $\sigma$-meson. First of all it should be noted that
unlike for flavoured mesons there is a higher level of model ambiguity
inherent in this question. There are many ways to think about it as summarised
in the excellent review [128].

An instructive starting point is the following ratio of rates [127]

$r_{f_{0}\to KK/\pi\pi}=\frac{\Gamma(f_{0}(980)\to
K^{+}K^{-})}{\Gamma(f_{0}(980)\to\pi^{+}\pi^{-})}=0.69(32)\;,$ (C.1)

which on grounds of phase space ought to be $r_{f_{0}\to KK/\pi\pi}\ll 1$. The
surprisingly large rate to kaons could be explained by the $f_{0}$ having a
large mount of $\bar{s}s$-quarks versus $\bar{q}q$-quarks ($q=u,d$). It is the
commonly accepted view that $f_{0}(980)$ has a high strange quark content.

It is helpful to consider the octet and singlet in a $\bar{q}q$\- and a
tetraquark-basis

$\displaystyle|S^{I=0}_{1}\rangle_{\bar{q}q}$ $\displaystyle\;=\;$
$\displaystyle\frac{1}{\sqrt{6}}(\bar{u}u+\bar{d}d+\bar{s}s)\;,\quad$
$\displaystyle|S^{I=0}_{1}\rangle_{\bar{q}q\bar{q}q}$ $\displaystyle\;=\;$
$\displaystyle\frac{1}{\sqrt{6}}(\bar{s}s\bar{u}u+\bar{s}s\bar{d}d+\bar{u}u\bar{d}d)\;,$
$\displaystyle|S^{I=0}_{8}\rangle_{\bar{q}q}$ $\displaystyle\;=\;$
$\displaystyle\frac{1}{\sqrt{6}}(\bar{u}u+\bar{d}d-2\bar{s}s)\;,\quad$
$\displaystyle|S^{I=0}_{8}\rangle_{\bar{q}q\bar{q}q}$ $\displaystyle\;=\;$
$\displaystyle\frac{1}{\sqrt{6}}(\bar{s}s\bar{u}u+\bar{s}s\bar{d}d-2\bar{u}u\bar{d}d)\;.$
(C.2)

The difference in role that the strange quark takes in the $\bar{q}q$\- and
the tetraquark-cases is immediately recognisable. With the central value from
the $SU(3)_{F}$-analysis by Oller one finds the following quark decompositions

$\displaystyle|\sigma\rangle_{\bar{q}q}$ $\displaystyle\;\approx\;$
$\displaystyle 0.67(\bar{u}u+\bar{d}d)$ $\displaystyle\;+\;$ $\displaystyle
0.28\bar{s}s\;,\quad$ $\displaystyle|\sigma\rangle_{\bar{q}q\bar{q}q}$
$\displaystyle\;\approx\;$ $\displaystyle
0.67(\bar{s}s\bar{u}u+\bar{s}s\bar{d}d)$ $\displaystyle\;+\;$ $\displaystyle
0.28\bar{u}u\bar{d}d\;,$ $\displaystyle|f_{0}\rangle_{\bar{q}q}$
$\displaystyle\;\approx\;$ $\displaystyle 0.20(\bar{u}u+\bar{d}d)$
$\displaystyle\;-\;$ $\displaystyle 0.95\bar{s}s\;,\quad$
$\displaystyle|f_{0}\rangle_{\bar{q}q\bar{q}q}$ $\displaystyle\;\approx\;$
$\displaystyle 0.20(\bar{s}s\bar{u}u+\bar{s}s\bar{d}d)$ $\displaystyle\;-\;$
$\displaystyle 0.95\bar{u}u\bar{d}d\;.$ (C.3)

We see that the strange quark content in the $f_{0}(980)$ is enhanced in the
$\bar{q}q$-states and suppressed for the tetraquarks. Hence the
$\bar{q}q$-state interpretation harmonises with the commonly accepted picture
that the $f_{0}(980)$ has a large strange quark content.292929In fact the
angle where this is picture is extremised is $\theta\approx 35.7^{\circ}$,
also known as ideal mixing. The angle, in the original proposal of Jaffe, is
$\theta\approx-54.7^{\circ}$, where
$|\sigma\rangle_{\bar{q}q\bar{q}q}=\bar{u}u\bar{d}d$. Other analyses finding
support for the $\bar{q}q$-interpretation, which is not the common view, are
for example [163, 164, 132].

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