Power counting of the pion-dilaton effective field theory
Maarten Goltermana and Yigal Shamirb

aDepartment of Physics and Astronomy, San Francisco State University,

San Francisco, CA 94132, USA

bRaymond and Beverly Sackler School of Physics and Astronomy,

Tel Aviv University, 69978, Tel Aviv, Israel

> Confining QCD-like theories close to the conformal window have a “walking”
> coupling. This is believed to lead to a light singlet scalar meson in the
> low-energy spectrum, a dilaton, which is the pseudo Nambu–Goldstone boson
> for the approximate scale symmetry. Extending chiral perturbation theory to
> include the dilaton requires a new small parameter to control the dilaton
> mass and its interactions. In our previous work we derived a systematic
> power counting for the dilaton couplings by matching the effective low-
> energy theory to the underlying theory using mild assumptions. In this paper
> we examine two alternative power countings which were proposed in the
> literature based on a phenomenological picture for the conformal transition.
> We find that one of these power countings fails, in fact, to generate a
> systematic expansion; the other coincides with the power counting we
> derived. We also point out that the so-called $\Delta$-potential coincides
> with the tree-level potential of the former, invalid, power counting.

##  I  Introduction

QCD-like asymptotically free gauge theories in four dimensions have two
sources of explicit breaking of scale invariance. The fermion mass provides a
soft breaking, while the running of the gauge coupling provides a hard
breaking at the quantum level.

Asymptotic freedom restricts the number of Dirac fermions in the fundamental
representation, $N_{f}$, to be smaller than $\frac{11}{2}N_{c}$, where $N_{c}$
is the number of colors. It is generally believed that this range is divided
into two phases: a confining and chirally broken phase111The $N_{f}=1$ theory
confines, but its only chiral symmetry ${\rm U}(1)_{A}$ is anomalous.  for
$2\leq N_{f}<N_{f}^{*}$, and a phase with an infrared fixed point (IRFP), the
conformal window, for $N_{f}^{*}\leq N_{f}\leq\frac{11}{2}N_{c}$. The critical
value $N_{f}^{*}=N_{f}^{*}(N_{c})$ is the sill of the conformal window. The
actual value of $N_{f}^{*}$ has been the subject of numerous lattice studies.
For $N_{c}=3$, there is strong evidence that $N_{f}^{*}\leq 10$ [2].

When $N_{f}$ is close to, but below, the sill $N_{f}^{*}$, the beta function
is small; this is the region of a walking coupling. When the theory eventually
confines, the smallness of the hard breaking of scale invariance at the
chiral-symmetry breaking scale is believed to lead to the presence of a
“dilaton” in the light hadron spectrum. This is a flavor-singlet scalar meson,
which is interpreted as a pseudo Nambu–Goldstone boson (NGB) arising from the
breaking of the approximate scale symmetry.222 Notably, a light singlet scalar
was found in the 8-flavor theory [3, 4, 5, 6, 7]. It is unclear, however, if
this theory is inside or below the conformal window.

In addition, the light meson spectrum contains the familiar pions, which are
the NGBs of spontaneously broken chiral symmetry. Chiral perturbation theory
(ChPT) is the well-founded effective field theory (EFT) providing a systematic
description of the pion sector. The small parameter controlling the low-energy
expansion is $m/\Lambda$, where $m$ is the fermion mass, and $\Lambda$ is the
confinement scale of the (massless) theory.

In order to extend ordinary ChPT to includes the light dilatonic meson, we
need an additional small parameter to control the dilaton mass and its
interactions. What drives the smallness of the beta function in the walking
region is the proximity of the conformal window. Hence, one expects the new
small parameter to be proportional to the distance to the sill in theory
space, $N_{f}^{*}-N_{f}$. One can furthermore invoke the Veneziano limit [8]
which turns this quantity into a continuous parameter $n_{f}^{*}-n_{f}$, where
$n_{f}=N_{f}/N_{c}$ and
$n_{f}^{*}=\lim_{N_{c}\to\infty}N_{f}^{*}(N_{c})/N_{c}$.

In a series of papers we developed [9, 10, 11, 12] and applied [13, 14, 15]
dilaton chiral perturbation theory (dChPT), an extension of ordinary ChPT
which contains the dilatonic meson. Any EFT should produce correlation
functions that match those in the underlying theory (order by order in the
expansion in the small parameters of the EFT). What this means is that
concrete information about how scale invariance works in the underlying theory
is an essential ingredient of the EFT construction. By matching the
correlation functions of dChPT to the underlying theory we showed that the new
small parameter controlling the dilaton sector is indeed $n_{f}^{*}-n_{f}$.
The power counting of dChPT is333Strictly speaking, $1/N_{c}$ is an additional
small parameter which controls the approach to the Veneziano limit. See also
footnote 15 below.

$p^{2}\sim m\sim n_{f}^{*}-n_{f}\ .$ (1)

In comparison with ordinary ChPT, the theoretical foundations of dChPT are
less rigorous. In particular, a few assumptions of a technical nature are
needed in order to set it up [9, 12]. We have tested dChPT by applying it to
numerical data [13, 14, 15]. However, it turns out that such tests are, at
present, limited by the available data. It is thus important to continue
scrutinizing the foundations of dChPT also at the theoretical level.

The goal of this paper is to explore alternative power countings that have
been proposed in the literature as a starting point for the construction of a
low-energy EFT of pions and a dilatonic meson and compare them with the power
counting on which dChPT is based. As we will discuss in more detail in the
final section, a different power counting will in general yield a different
EFT lagrangian at each order, and will thus lead to different physical
predictions. It is thus important to determine the correct power counting for
a pion-dilaton EFT.

One of the reasons why a pion-dilaton EFT is more complicated than ordinary
ChPT is the following. Scale transformations act on both fields and
coordinates. As a result, a quartic potential for the (effective) dilaton
field is consistent with scale invariance. Because the quartic potential does
not violate any symmetry of the pion-dilaton EFT, the power counting will
necessarily allow this potential to occur in the EFT with an $O(1)$ coupling.
That the quartic potential is not suppressed by any small parameter is a fact
that threatens the existence of any systematic expansion for the pion-dilaton
EFT, and will turn out to play a key role. Another important open question is
what the physical mechanism is that drives the conformal transition. Ignorance
about the correct answer creates an additional difficulty when trying to
identify the correct power counting for the pion-dilaton EFT, as we will see.

This paper is organized as follows. In Sec. II we discuss dChPT. We briefly
review the derivation of its power counting (1), with emphasis on how the
general principles for the construction of an EFT are applied. We also explain
how dChPT copes with the $O(1)$ quartic potential.

In Sec. III, which contains the main results of this paper, we begin by
introducing a phenomenological picture for the onset of the conformal
transition [16, 17]. As it turns out, this physical picture suggests two
possible power countings (rather than a unique one) for the pion-dilaton EFT,
which we will refer to below as the $G\to 0$ and the $\Delta\to 4$ power
countings. We show that the $G\to 0$ power counting actually fails to provide
a systematic expansion. The $\Delta\to 4$ power counting turns out to coincide
with the power counting of dChPT. Finally, we discuss the so-called
$\Delta$-potential [6, 18, 19, 20, 21], which has been claimed [20, 21, 22] to
be the leading-order dilaton potential arising from the $G\to 0$ power
counting. As a result of the failure of the $G\to 0$ power counting, it
follows immediately that the $\Delta$-potential fails to generate a systematic
expansion as well. Once again, the scale invariant quartic potential turns out
to play a key role. In Sec. IV we summarize and discuss our results.444For
other related low-energy approaches see Refs. [23, 24, 25].

As we will see, the questions we need to address in this paper can be answered
by considering the pure dilaton sector of the EFT in the chiral limit; the
pions do not play any role. Hence, throughout this paper we will set the
fermion mass to zero unless otherwise stated.

##  II  dChPT and its power counting

Spurion fields play a pivotal role in the construction of an EFT. The first
step is to introduce the spurions into the underlying theory, in such a way
that the relevant symmetries are formally restored. In the familiar case of
ChPT one begins by promoting the fermion mass $m$ to a matrix-valued spurion
field $\chi(x)$. The fermion mass term in the underlying theory then reads
$\overline{\psi}(P_{R}\,\chi+P_{L}\,\chi^{\dagger})\psi$ where
$P_{R,L}={1\over 2}(1\pm\gamma_{5})$ are the chiral projectors. With suitable
transformation rules for the chiral spurion, formal invariance under ${\rm
SU}(N_{f})_{L}\times{\rm SU}(N_{f})_{R}$ is achieved. Explicit breaking of the
chiral flavor symmetry by the fermion mass is recovered by setting the spurion
field to its “expectation value”
$\left\langle\chi\right\rangle=\left\langle\chi^{\dagger}\right\rangle=m$.555We
consider only the mass degenerate case to avoid irrelevant technicalities.
The full chiral symmetry is truly restored if we then take the chiral limit
$m\to 0$.

The EFT is constructed using effective fields for the low-energy degrees of
freedom, but in addition it must depend on the same set of spurions that we
have introduced into the underlying theory. When we transform simultaneously
the effective and the spurion fields, the EFT is formally invariant under the
same symmetries as the underlying theory. Explicit breaking of the relevant
symmetries is recovered once again by fixing the spurions to the same
expectation values as in the underlying theory. All correlation functions that
can be matched between the underlying theory and the EFT are obtained by
taking derivatives with respect to the spurion fields. Since the dependence of
the underlying theory on the spurions is always analytic by construction, so
must be the dependence of the EFT on the spurion fields as well.

Spurions can play a role in the power counting in two different ways. In
ordinary ChPT, the small expansion parameter is $m/\Lambda$, and the fermion
mass $m$ is just the expectation value of the chiral spurion. The underlying
theory and the EFT both become chirally symmetric by letting $m\to 0$. But
spurions can be involved in the power counting in a more subtle way as well.
As we discussed in Ref. [9], the relevant example is the extension of ordinary
ChPT to include the $\eta^{\prime}$ meson.

The $\eta^{\prime}$ meson is the would-be NGB for the spontaneous breaking of
the axial ${\rm U}(1)_{A}$ symmetry. Unlike the fermion mass, in any given
theory one cannot “turn off” the axial anomaly. However, the axial anomaly
becomes parametrically small in the large-$N_{c}$ limit.666Here the limit
$N_{c}\to\infty$ is taken at fixed $N_{f}$. This is different from the
Veneziano limit where the ratio $N_{f}/N_{c}$ is held fixed.  This leads to an
extension of ChPT [26, 27] that includes an effective field for the
$\eta^{\prime}$ meson, which we will denote simply as $\eta(x)$. At the level
of the underlying theory, the key step is to promote the vacuum angle $\theta$
to a spurion field, and to augment the gauge-field lagrangian by the term
$i\theta(x)q(x)$, where $q(x)$ is the topological charge density

$q=\frac{c\lambda N_{f}}{N_{c}}\,{\rm tr}\,F\tilde{F}\ .$ (2)

Here $\lambda=g^{2}N_{c}$ is the ’t Hooft coupling, and $c$ is a numerical
constant. The fact that $q(x)\propto 1/N_{c}$ (when expressed in terms of the
’t Hooft coupling) means that it is parametrically small in large-$N_{c}$
counting.

Under a ${\rm U}(1)_{A}$ transformation, both the effective field $\eta(x)$
and the spurion field $\theta(x)$ transform by a common shift. The shift is
chosen such that the variation of the topological term in the gauge-field
lagrangian cancels the anomalous ${\rm U}(1)_{A}$ transformation of the
fermion determinant. Unlike the case of the fermion mass, where chiral
symmetry is restored for $m\to 0$, here there is no fixed value for the vacuum
angle $\theta$ (including zero) that would lead to the restoration of ${\rm
U}(1)_{A}$ invariance in the underlying theory; the axial anomaly is always
present. Instead, differentiating the partition function of the underlying
theory with respect to $i\theta(x)$ yields an insertion of the topological
charge density $q(x)$. Since $q(x)$ is parametrically small for large $N_{c}$,
this ultimately allows for a systematic expansion for the EFT.777 The power
counting of the EFT is usually taken to be $p^{2}\sim m\sim 1/N_{c}$.

The main difficulty that must be overcome at the level of the EFT is the
following. Because the $\eta$ and $\theta$ fields are shifted by the same
amount under ${\rm U}(1)_{A}$ transformations, the difference $\eta-\theta$ is
invariant. Hence, on symmetry grounds alone, any term in the EFT lagrangian
can be multiplied by a function of $\eta-\theta$. If that function was
completely arbitrary, the EFT would lose predictability. What allows for a
systematic expansion in the EFT is that, as in the underlying theory, each
differentiation with respect to $\theta$ must produce a factor of the small
parameter $1/N_{c}$. It follows that every function $V(\eta-\theta)$ that
occurs in the EFT must have the Taylor expansion

$\displaystyle V$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}c_{n}(\eta-\theta)^{n}\ ,$ (3) $\displaystyle
c_{n}$ $\displaystyle=$ $\displaystyle O(1/N_{c}^{n})\ .$

At each order in the low-energy expansion of the EFT, the hierarchy of the
$c_{n}$ coefficients implies that we must truncate the Taylor series to a
finite number of terms. In particular, at leading order, the functions of
$\eta-\theta$ that multiply the pion kinetic term and the pion mass term in
the tree-level lagrangian must be truncated to their leading constant term
$c_{0}$, and can thus be disregarded altogether. Once all the relevant
functions of $\eta-\theta$ have been truncated appropriately, one can set
$\theta=0$ and proceed with the calculation of correlation functions to a
given order in the low-energy expansion.

The systematic expansion of dChPT works in a similar way.888For full details,
see Ref. [9].  Much like the $\eta$ and $\theta$ fields associated with the
${\rm U}(1)_{A}$ symmetry, we introduce a spurion field $\sigma(x)$ and an
effective dilatonic meson field $\tau(x)$, both dimensionless, that transform
by a common shift under dilatations. To be precise,

$\displaystyle\sigma(x)$ $\displaystyle\to$ $\displaystyle\sigma(\lambda
x)+\log\lambda\ ,$ (4a) $\displaystyle\tau(x)$ $\displaystyle\to$
$\displaystyle\tau(\lambda x)+\log\lambda\ .$ (4b)

Using dimensional regularization the bare action of the underlying theory
takes the form

$S=\mu_{0}^{d-4}\int d^{d}x\,e^{(d-4)\sigma(x)}{\cal L}_{0}(x)\ .$ (5)

The bare lagrangian ${\cal L}_{0}(x)$ is defined in terms of bare fields and
couplings that all transform according to their canonical mass dimension.999
As mentioned in the introduction we set $m=0$ to avoid irrelevant
technicalities.  The mass dimension of ${\cal L}_{0}(x)$ is 4, and it thus
transforms under dilatations as ${\cal L}_{0}(x)\to\lambda^{4}{\cal
L}_{0}(\lambda x)$. The fixed reference scale $\mu_{0}$ does not transform
under dilatations by construction,101010See, however, the generalized
discussion in Ref. [11].  and can be thought of as an ultraviolet cutoff.
Thanks to the spurion transformation rule (4a), the action (5) is formally
invariant under dilatations.

Much like the case of the axial anomaly, setting $\sigma(x)$ to any fixed
value will not restore scale invariance of the underlying theory; it is always
broken at the quantum level by the running of the coupling. However,
differentiating the underlying partition function with respect to $\sigma(x)$
again produces an insertion of a parametrically small operator. Expressing the
action in terms of renormalized fields and couplings, it takes the
form111111Equation (6) is true up to local operators that are multiplied by
higher powers of $\sigma(x)$, that do not affect the construction of the EFT
[9].

$S=\int d^{d}x\,\Big{(}{\cal L}_{\rm ren}+\sigma T\big{)}\ ,$ (6)

where ${\cal L}_{\rm ren}$ is the usual renormalized lagrangian, and

$T=\frac{\beta(g)}{2g}\,F^{2}$ (7)

is the trace anomaly [28]. Here $\beta(g)=\mu(\partial g/\partial\mu)$ is the
beta function, and $F^{2}=F_{\mu\nu}^{a}F_{\mu\nu}^{a}$. It follows that the
differentiation with respect to $\sigma(x)$ yields an insertion of the trace
anomaly, which is parametrically small for walking theories close to the sill
of the conformal window,

$T\sim n_{f}^{*}-n_{f}\ .$ (8)

In analogy with the case of ${\rm U}(1)_{A}$ transformations, any function $V$
of the field difference $\tau(x)-\sigma(x)$ transforms homogeneously under
dilatations, $V(x)\to V(\lambda x)$. Thus, $V(\tau-\sigma)$ can multiply any
existing term in the EFT which is already allowed by the symmetries. As
before, this could lead to the loss of predictability, but again, the
requirement that each differentiation with respect to $\sigma(x)$ generates
one power of the small parameter $n_{f}^{*}-n_{f}$ implies that any such
function occurring in the lagrangian of dChPT must satisfy (compare Eq. (3))

$\displaystyle V$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}c_{n}(\tau-\sigma)^{n}\ ,$ (9) $\displaystyle
c_{n}$ $\displaystyle=$ $\displaystyle O((n_{f}^{*}-n_{f})^{n})\ .$

Thus, the function $V(\tau-\sigma)$ that multiplies a particular term in the
EFT must again be truncated at a finite order of its Taylor series, according
to the order of the low-energy expansion we are interested in. In particular,
at leading order in the low-energy expansion, which is always $O(p^{2})$, the
functions that multiply the pion and dilaton kinetic terms in the tree-level
dChPT lagrangian (as well as the function that multiplies the pion mass term
when $m\neq 0$) must be truncated to their leading constant term, and can be
disregarded once again.

The only exception is the pure dilaton potential, to which we now turn. First,
the quartic dilaton potential discussed in the introduction takes the form
$e^{4\tau}$. It transforms as $e^{4\tau(x)}\to\lambda^{4}e^{4\tau(\lambda
x)}$, and thus the corresponding term in the action is indeed scale invariant.
Next, according to the previous discussion, this term should be multiplied by
a function of $\tau-\sigma$, which in turn should be truncated appropriately.
Finally we set $\sigma=0$. The pure-dilaton part of the tree-level lagrangian
of dChPT is

$\frac{f_{\tau}^{2}}{2}\,e^{2\tau}(\partial_{\mu}\tau)^{2}+V_{d}\ .$ (10)

We will shortly see how the parameter $f_{\tau}$ is related to the physical
dilaton decay constant. Guided by the power counting (1), and remembering that
the tree-level lagrangian of the EFT is always $O(p^{2})$, it follows from the
discussion above that the resulting tree-level dilaton potential can be
expressed as

$\displaystyle V_{d}$ $\displaystyle=$ $\displaystyle f_{\tau}^{2}B_{\tau}U\
,$ (11a) $\displaystyle U$ $\displaystyle=$ $\displaystyle
e^{4\tau}(c_{0}+c_{1}\tau)\ ,$ (11b)

where $c_{0}=O(1)$ and $c_{1}=O(n_{f}^{*}-n_{f})$. The parameter $B_{\tau}$
has mass dimension 2. Note that the pure $e^{4\tau}$ potential indeed occurs
with an $O(1)$ coupling, in agreement with our general reasoning.

Solving the classical equation, the minimum of the potential is

$v=\left\langle\tau\right\rangle=-\frac{1}{4}-\frac{c_{0}}{c_{1}}\ ,$ (12)

and the curvature at the minimum is

$U^{\prime\prime}=4c_{1}e^{4v}\ ,$ (13)

where $v$ is given by Eq. (12). Now, virtually all terms occurring in the
dChPT lagrangian involve some power of $e^{\tau}$. As a result, low-energy
constants get “dressed” by a corresponding power of $e^{v}$. In particular, it
follows from Eq. (10) that the dilaton decay constant121212Since $m=0$ this is
the dilaton decay constant in the chiral limit. is $F_{\tau}=f_{\tau}e^{v}$.
The dressed, physical $B_{\tau}$ parameter is $\hat{B}_{\tau}=B_{\tau}e^{2v}$.
The dilaton mass is given by

$M_{\tau}^{2}=\frac{V_{d}^{\prime\prime}}{F_{\tau}^{2}}=4c_{1}\hat{B}_{\tau}\
.$ (14)

Equation (14) implies that the dilaton mass (squared) is parametrically small,
because $c_{1}$ is, as expected from a pseudo NGB. This has the following
immediate implication. The dimensionless ratio which controls the low-energy
expansion in the dilaton sector is

$\frac{M_{\tau}^{2}}{(4\pi F_{\tau})^{2}}=\frac{4c_{1}\hat{B}_{\tau}}{(4\pi
F_{\tau})^{2}}=\frac{4c_{1}B_{\tau}}{(4\pi f_{\tau})^{2}}\ .$ (15)

This ratio is proportional to $c_{1}$, making it parametrically small as well,
as it must be for a systematic low-energy expansion.

Of the four parameters that occur in the tree-level lagrangian of the dilaton
sector, the ratio (15) depends on $f_{\tau}$, $B_{\tau}$ and $c_{1}$. But it
is independent of $c_{0}$, the troublesome $O(1)$ coefficient of the quartic
$e^{4\tau}$ potential. This remarkable property is intimately related to a
unique feature of dChPT: the freedom to shift the $\tau$ field by a constant.

As we have discussed extensively in Refs. [9, 12], the effect of shifting the
$\tau$ field can be absorbed into redefinitions of the parameters of the
theory. The behavior of the parameters of the pure dilaton sector is as
follows. $f_{\tau}$, $B_{\tau}$ and $c_{0}$ are affected by the $\tau$-shift,
while $c_{1}$ is invariant. The dressed parameters $F_{\tau}$ and
$\hat{B}_{\tau}$ are invariant as well. Thus, the dilaton mass (14) and the
ratio (15) are invariant under the $\tau$-shift, as all physical quantities
must be. On the other hand, $c_{0}$ is affected by the $\tau$-shift, and this
explains why the dimensionless ratio (15) cannot contain $c_{0}$. In the next
section we will see that the situation is different for another power counting
that has been proposed for the pion-dilaton EFT.

The above discussion requires one refinement. Because $c_{0}=O(1)$ while
$c_{1}=O(n_{f}^{*}-n_{f})$, the expectation value $v$ of the dilaton field in
Eq. (12) is parametrically $O(1/(n_{f}^{*}-n_{f}))$. Considering the Taylor
series (9) for the case of the dilaton potential, if we substitute $\tau=v$
(and $\sigma=0$) then all terms in the series become $O(1)$ within the power
counting (1). As a result, certain technical assumptions about the global
features of the dilaton potential are required in order to set up dChPT. The
most important one is that, taken as a function of the product
$(n_{f}^{*}-n_{f})\tau$, the dilaton potential vanishes for some value of its
argument [29]. Once this condition is satisfied, there exists a $\tau$-shift
that eliminates the $O(1)$ part of $c_{0}$. After that, the new value of
$c_{0}$ is $O(n_{f}^{*}-n_{f})$, at which point the previous discussion
applies.131313An additional $\tau$-shift can be done to set the
$O(n_{f}^{*}-n_{f})$ part of $c_{0}$ to some desired value, see Ref. [12].
This is proven in detail in Appendix A of Ref. [12].

In summary, thanks to its behavior under $\tau$-shifts, dChPT copes with the
$O(1)$ coefficient of the quartic potential and successfully generates a
systematic low-energy expansion, at the price of a few mild assumptions about
the global features of the dilaton potential.

We conclude this section with a technical comment. Non-perturbatively, the
underlying non-abelian theory is always defined on the lattice, where the
breaking of spacetime symmetries, including in particular dilatations, takes a
complicated form. However, the derivation of an EFT always involves the
intermediate step of first constructing the Symanzik effective theory (SET),
which lives in the continuum. Thus, the bare continuum theory (5) can be
identified with the leading order of the (dimensionally regularized) Symanzik
effective theory, assuming that the underlying lattice theory is close enough
to the continuum limit.

##  III  Phenomenology of the conformal transition
and related power countings

In this section we turn to alternative power countings for the pion-dilaton
EFT which have been proposed in the literature. The starting point is a
certain physical picture for the mechanism that drives the transition into the
conformal window. According to this physical picture, (massless) theories near
the sill of the conformal window are governed in the infrared by a
phenomenological lagrangian of the form [16, 17]

${\cal L}={\cal L}_{\rm conf}+G{O_{\rm sill}}\ .$ (16)

By assumption, ${\cal L}_{\rm conf}$ describes a conformal theory. Hence, in
itself it would be suitable to describe the low-energy physics of a theory
inside the conformal window, whose infrared behavior is governed by an IRFP.
The conformal transition is then attributed to the presence of ${O_{\rm
sill}}$ in the lagrangian. Under a scale transformation, we assume that

${O_{\rm sill}}(x)\to\lambda^{\Delta}\,{O_{\rm sill}}(\lambda x)\ .$ (17)

The coupling $G$ and the scaling dimension $\Delta$ of ${O_{\rm sill}}$ are
both functions of $N_{c}$ and $N_{f}$.

This starting point raises numerous questions. A crucial question is whether
the phenomenological lagrangian (16) can be derived from the standard
renormalizable lagrangian that defines the underlying gauge theory. We will
defer all such questions to the discussion section, whereas in this section we
will discuss the implications of this physical picture for the power counting
of the pion-dilaton EFT.

As was realized in Refs. [16, 17], there are two ways by which conformality of
the phenomenological lagrangian in Eq. (16) can be restored, and this suggests
two alternative power counting for the pion-dilaton EFT:

$G\to 0$ power counting. Here one envisages that $G>0$ below the conformal
sill while $G=0$ at (and above) the sill, where the theory has an IRFP. By
assumption, $G$ tends to zero as the sill is approached from below in flavor
space. The corresponding power counting is

$p^{2}\sim m\sim G\ .$ (18)

We will discuss this power counting in Sec. III.1. We note that, like the
fermion mass $m$, also the coupling $G$ is dimensionful. Hence, the
dimensionless expansion parameters are $m/\Lambda$ and $G/\Lambda^{4-\Delta}$.
The infrared scale $\Lambda$ may be identified with the vacuum expectation
value (VEV) of the effective dilaton field $\chi$ that will be introduced in
Sec. III.1 below.

$\Delta\to 4$ power counting. Alternatively, one assumes that $G$ is $O(1)$ on
both sides of the conformal sill. Instead, it is assumed that the scaling
dimension $\Delta$ tends to 4 at the sill. Usually it is further assumed that
$\Delta<4$ below the sill, so that ${O_{\rm sill}}$ is a relevant operator,
while $\Delta\geq 4$ above the sill, and thus ${O_{\rm sill}}$ is marginal or
irrelevant. At the sill $\Delta=4$, and ${O_{\rm sill}}$ is marginal (at the
quantum level), thereby restoring scale invariance. The power counting is then

$p^{2}\sim m\sim|4-\Delta|\ .$ (19)

We discuss this option in Sec. III.2. Finally, in Sec. III.3 we offer a brief
summary, while in Sec. III.4 we discuss the so-called $\Delta$ potential [6,
18, 19, 20, 21, 22].

###  III.1  $G\to 0$ power counting

Using $G$ as a small parameter works in a way similar to the familiar chiral
expansion. Taking the phenomenological lagrangian (16) to represent the
underlying theory, one begins by promoting the coupling $G$ to a spurion field
${\cal S}(x)$. The lagrangian becomes

${\cal L}={\cal L}_{\rm conf}+{\cal S}{O_{\rm sill}}\ ,$ (20)

and with the spurion transformation rule

${\cal S}(x)\to\lambda^{4-\Delta}\,{\cal S}(\lambda x)\ ,$ (21)

the lagrangian is formally invariant under dilatations. The original
lagrangian, in which scale invariance is explicitly broken by ${O_{\rm
sill}}$, is recovered by fixing the spurion field to $\left\langle{\cal
S}\right\rangle=G$.

As in Sec. II, at the level of the EFT we will focus on the pure dilaton
sector. Its lagrangian is

${\cal L}_{d}={1\over 2}\partial_{\mu}\chi\partial_{\mu}\chi+V(\chi)\ .$ (22)

The scale transformation of the dilaton field $\chi(x)$ is

$\chi(x)\to\lambda\,\chi(\lambda x)\ .$ (23)

The dilaton potential has the expansion

$V=\chi^{4}\sum_{n=0}^{\infty}b_{n}\Big{(}{\cal S}\chi^{\Delta-4}\Big{)}^{n}\
.$ (24)

The low-energy constants $b_{n}$ are invariant under dilatations, and they are
$O(1)$ in the power counting, just like the low-energy constants in standard
ChPT. With the transformation rules of the dilaton and spurion fields
introduced above, the EFT action defined by the lagrangian (22) is formally
invariant under dilatations.

After fixing the spurion to $\left\langle{\cal S}\right\rangle=G$ the
potential becomes

$V=\chi^{4}\sum_{n=0}^{\infty}b_{n}\Big{(}G\chi^{\Delta-4}\Big{)}^{n}\ ,$ (25)

making the expansion in $G$ manifest. At the same time, since the spurion has
been set to a fixed value, the potential breaks scale invariance explicitly.
As usual, the exception is the leading term in the sum: this is the by-now
familiar quartic potential $\chi^{4}$, whose coupling $b_{0}$ is
parametrically $O(1)$.

According to the power counting (18), at tree level we keep terms up to
$O(G)$, hence

$V=b_{0}\chi^{4}-\tilde{b}_{1}\chi^{\Delta}\ ,$ (26)

where we have introduced $\tilde{b}_{1}=-b_{1}G$ (the minus sign is for
convenience). We will assume that both $b_{0}$ and $\tilde{b}_{1}$ are
positive. We also require $0<\Delta<4$. The upper bound $\Delta<4$ is needed
to ensure boundedness from below of the potential. The lower bound $0<\Delta$
is to avoid a singularity of the potential for $\chi\to 0$.

We comment that, in analogy with the cases discussed in Sec. II, every term in
the lagrangian of the EFT which is (a) allowed by the symmetries, and (b)
independent of the ${\cal S}$ spurion, should be multiplied by a function
$U=U({\cal S}\chi^{\Delta-4})$. Actually, the dilaton potential (24) has this
form, where the original potential allowed by the symmetries is the pure
$\chi^{4}$. Similarly, the dilaton kinetic term in Eq. (22) should be
multiplied by another function of the argument ${\cal S}\chi^{\Delta-4}$,
which, after setting $\left\langle{\cal S}\right\rangle=G$, becomes a function
of the argument $G\chi^{\Delta-4}$. At tree level, however, that function
would again have to be truncated to its constant leading term. Hence, we
omitted it from Eq. (22).

We expand the dilaton field as

$\chi=v+\phi,$ (27)

where the VEV is $\left\langle\chi\right\rangle=v$, and $\phi$ is the quantum
part. The saddle-point equation can be written as

$0=vV^{\prime}(v)=4b_{0}v^{4}-\Delta\,\tilde{b}_{1}v^{\Delta}\ .$ (28)

The absolute minimum is the $v>0$ solution,

$v^{4-\Delta}=\frac{\Delta}{4}\frac{\tilde{b}_{1}}{b_{0}}\ .$ (29)

Upon substituting the field expansion (27) into the potential (26) and using
the classical solution (29), the potential becomes

$V=\frac{\Delta-4}{\Delta}\,b_{0}v^{4}+\frac{M_{\tau}^{2}}{2}\,\phi^{2}+M_{\tau}^{2}v^{2}\sum_{n\geq
3}g_{n}\frac{\phi^{n}}{v^{n}}\ ,$ (30)

where the dilaton mass squared is141414Equations (29) and (31) were previously
derived in Ref. [17].

$M_{\tau}^{2}=4b_{0}v^{2}(4-\Delta)\ .$ (31)

The first few coefficients in the sum in Eq. (30) are

$\displaystyle g_{3}$ $\displaystyle=$ $\displaystyle(\Delta+1)/3!\ ,$ (32)
$\displaystyle g_{4}$ $\displaystyle=$
$\displaystyle(\Delta^{2}-2\Delta+3)/4!\ ,$ $\displaystyle g_{5}$
$\displaystyle=$ $\displaystyle(\Delta-1)(\Delta-2)(\Delta-3)/5!\ ,$
$\displaystyle g_{6}$ $\displaystyle=$
$\displaystyle(\Delta-1)(\Delta-2)(\Delta-3)(\Delta-5)/6!\ ,$

and so on. Note that the terms with $n\geq 5$ arise from the Taylor series of
$\chi^{\Delta}=(v+\phi)^{\Delta}$ only, hence all these terms contain an
explicit factor of $\Delta-4$. After we absorb this factor into $M_{\tau}^{2}$
it follows that $g_{n}$ tends to a non-zero value in the limit $\Delta\to 4$.

The reader who is familiar with ordinary ChPT will recognize that the
expansion in Eq. (30) is qualitatively similar to the expansion of the pion
mass term in the standard chiral lagrangian, if the identifications
$\phi\leftrightarrow\pi$ and $v\leftrightarrow F_{\pi}$ are made, where $\pi$
is the pion field and $F_{\pi}$ its decay constant. In ordinary ChPT, the
systematic expansion is effectively an expansion in the ratio

$\frac{M_{\pi}^{2}}{(4\pi F_{\pi})^{2}}\ .$ (33)

Since $M_{\pi}^{2}$ is proportional to the quark mass $m$ at tree level, the
same is true for this ratio. It follows that by taking $m$ small enough, the
ratio (33) can be made arbitrarily small. Since this is the ratio that
controls the low-energy expansion, this expansion is indeed systematic.

Similarly, under the power counting (18), the ratio that should control the
low-energy expansion in the dilaton sector is (compare Eq. (31))

$\frac{M_{\tau}^{2}}{(4\pi v)^{2}}=\frac{b_{0}(4-\Delta)}{4\pi^{2}}\ .$ (34)

We illustrate this fact via several examples of dilaton self-energy diagrams
at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO):

  * •

NLO tadpole. This diagram comes from a single quartic vertex, which provides
an explicit factor of $M_{\tau}^{2}/v^{2}$. An additional factor of
$(M_{\tau}/4\pi)^{2}$ comes from the loop integral.

  * •

NLO one-loop diagram with two cubic vertices. Together, the two vertices
provide a factor of $M_{\tau}^{4}/v^{2}$, while a factor $1/(4\pi)^{2}$ comes
from the loop, as the diagram is dimensionless.

  * •

NNLO double tadpole. This diagram comes from a single $\phi^{6}$ vertex. The
explicit factor is $M_{\tau}^{2}/v^{4}$, and two additional factors of
$(M_{\tau}/4\pi)^{2}$ come from the two tadpole loops.

The crucial question is thus the following. Given the power counting (18), is
the ratio (34) parametrically small, as is the ratio (33) in ordinary ChPT, or
the ratio (15) in dChPT? The answer is No! The reason why the ratio (34) fails
to be parametrically small is that it involves the parameter $b_{0}$. As
explained above, $b_{0}$ is the coupling of the $\chi^{4}$ term in the dilaton
potential (25), and it is $O(1)$ because a pure $\chi^{4}$ potential is scale
invariant. If we reexamine the expression for the dilaton mass (31), we see
that the only reason why it is parametrically small is the presence of the
factor $v^{2}$, since $v$ itself is parametrically small. However, the factor
of $v^{2}$ drops out in the ratio (34). Thus, would-be subleading EFT
contributions, such as the examples listed above, are all parametrically of
the same size as the leading order terms.

This negative conclusion implies that, in fact, there is no systematic
expansion for the dilaton sector of the low-energy effective theory if the
power counting (18) is assumed, thereby ruling out this power counting for the
pion-dilaton EFT.

There is, however, one exception to this conclusion. The ratio (33) will
become parametrically small, if $\Delta$ is close to 4. However, for this to
be the case, we need to assume that $\Delta-4$ is parametrically small, in
other words, we must replace the power counting (18) by the power counting
(19), to which we turn next.

###  III.2  $\Delta\to 4$ power counting

The power counting (19) is based on the assumption that ${O_{\rm sill}}$ is an
irrelevant operator inside the conformal window, which become relevant below
the conformal sill, with $\Delta=4$ precisely at the sill. We will thus assume
that, below the sill,151515Both here and in Eq. (8), which provides the basis
for the power counting (1) of dChPT, one can replace $n_{f}^{*}-n_{f}$ by
$|n_{f}^{*}-n_{f}|^{\eta}$ for some $\eta>0$ without changing the basic
physical features. The actual value of $\eta$ is not known, and to avoid
cumbersome notation we set $\eta=1$.

$4-\Delta\sim n_{f}^{*}-n_{f}\ .$ (35)

The ${\cal S}$ spurion we have introduced for the $G\to 0$ power counting is
not appropriate for the $\Delta\to 4$ power counting, and we trade it with the
$\sigma$ spurion of Sec. II according to

${\cal S}(x)=Ge^{(4-\Delta)\sigma(x)}\ .$ (36)

Using the $\sigma$ transformation rule (4a) it is easy to check that the
transformation rule of the ${\cal S}$ spurion is reproduced, and thus that the
action defined by the lagrangian (20) remains formally invariant, as it
should. As for the dilaton effective field of the EFT, it is convenient to
similarly switch to the $\tau$ field, via

$\chi(x)=f_{\tau}e^{\tau(x)}\ .$ (37)

Once again, using Eq. (4b) it immediately follows that the correct
transformation rule of the $\chi$ field is reproduced.

The reason why the $\sigma$ spurion is now the appropriate one is that,
according to Eq. (36), this spurion comes multiplied by the small parameter
$4-\Delta$. It follows that the $\Delta\to 4$ power counting works in the same
way as the power counting of dChPT. As explained in Sec. II, in dChPT the
differentiation with respect to the $\sigma$ spurion of the renormalized
lagrangian (6) of the underlying theory generates an insertion of the trace
anomaly, which in turn is parametrically small according to Eq. (8).
Similarly, with the substitution (36), here the differentiation with respect
to the $\sigma$ spurion of the phenomenological lagrangian (20) generates one
power of the small parameter controlling the dilaton sector, which is now
$4-\Delta$. We recall that the large-$N_{c}$ extension of ChPT that
accommodates the anomalous U(1)A symmetry works similarly: the $\theta$
spurion plays the role of the $\sigma$ spurion, and the differentiation with
respect to $\theta$ generates one power of the small expansion parameter
$1/N_{c}$ (see Sec. II).

In fact, as we will now show, the $\Delta\to 4$ power counting is completely
equivalent to the power counting of dChPT, and the EFT it generates is just
dChPT itself. Starting from the dilaton potential (24), which was valid for
the $G\to 0$ power counting, we expand out the spurion field ${\cal S}$ as a
power series in $\sigma$, and the effective field $\chi$ as a power series in
$\tau$. The outcome is that the dilaton potential gets rearranged as

$V=f_{\tau}^{2}B_{\tau}e^{4\tau}\sum_{n=0}^{\infty}c_{n}(\tau-\sigma)^{n}\ ,$
(38)

where the expansion coefficients satisfy

$c_{n}\propto(4-\Delta)^{n}=O(|n_{f}^{*}-n_{f}|^{n})\ .$ (39)

Once again it is easy to check that the (formal) scaling dimension of the
potential remains 4. As always, the explicit breaking of scale invariance is
recovered by setting $\sigma=0$.

The similarity of Eq. (38) to Eq. (9) is evident. We have thus established
that the $\Delta\to 4$ power counting reproduces the dilaton potential of
dChPT. The same reasoning can be applied to every term in the effective
lagrangian, and the conclusion is that the resulting EFT of the $\Delta\to 4$
power counting is indeed dChPT itself.

###  III.3  Summary

In this section we considered the two power countings that can be motivated by
the phenomenological lagrangian (16). For the $G\to 0$ power counting we found
that both the VEV of the $\chi$ field and the dilaton mass $M_{\tau}$ are
proportional to some (fractional) power of the small parameter
$\tilde{b}_{1}$. However, the parametric smallness of $M_{\tau}^{2}$
originates solely from its dependence on $v^{2}$. Thus, in the ratio (34) that
should control the low-energy expansion in the dilaton sector, the dependence
on the small parameter $\tilde{b}_{1}$ drops out. The ratio comes out to be
$O(1)$ as a direct consequence of the fact that the coupling $b_{0}$ of the
scale invariant dilaton potential is $O(1)$, and this invalidates the $G\to 0$
power counting.

As for the $\Delta\to 4$ power counting, we showed that this is the same power
counting as in dChPT. However, the power counting of dChPT is better motivated
because it is derived directly from the underlying theory, rather than from a
phenomenological lagrangian. The VEV of the $\tau$ field in dChPT behaves in a
qualitatively different way. The $O(1)$ coupling of the scale invariant
dilaton potential, $c_{0}$, leads to a VEV that is inversely proportional to
the small parameter $n_{f}^{*}-n_{f}$ (see Eq. (12)). For this reason, one
needs to make (mild) assumptions about the global form of the potential, as we
have discussed in detail in Ref. [12]. Once these assumptions are satisfied
one can use the $\tau$ shift to eliminate the $O(1)$ part of $c_{0}$, a
feature which implies that $c_{0}$ is not physical in dChPT. The resulting
low-energy expansion is systematic.

Two EFTs that are so qualitatively different cannot both describe the same
underlying theory. We have been able to rule out the EFT of the $G\to 0$ power
counting, because it does not generate a systematic low-energy expansion. We
conclude that the correct EFT with the correct power counting is dChPT.

###  III.4  $\Delta$-potential

The so-called $\Delta$-potential was used in a numbers of publications as the
leading-order dilaton potential in fits to numerical data [6, 18, 19, 20, 21,
22]. It was argued in Appendix A of Ref. [20] that the $\Delta$-potential
arises from the $G\to 0$ power counting.161616See also Ref. [22].  The
arguments in Ref. [20] are rather opaque to us, as we have pointed out in Ref.
[14]. Nonetheless, as it turns out it is easy to check the claim that the
$\Delta$-potential can be used at leading order in a systematic expansion. In
our notation, the $\Delta$-potential reads

$V_{\Delta}=\frac{M_{\tau}^{2}}{4(4-\Delta)v^{2}}\left(\chi^{4}-\frac{4}{\Delta}\frac{\chi^{\Delta}}{v^{\Delta-4}}\right)\
.$ (40)

Upon substituting Eq. (31) for the dilaton mass, and Eq. (29) for the VEV of
the $\chi$ field, the leading-order dilaton potential of the $G\to 0$ power
counting, Eq. (26), is readily recovered.

As we showed above, the $G\to 0$ power counting does not lead to a systematic
low-energy expansion, and thus this is true for the $\Delta$-potential as
well. The only way out is to take $4-\Delta$ small, but this amounts to
reverting to the $4-\Delta$ power counting, at which point the pion-dilaton
EFT becomes dChPT.

We comment that one might postulate that the VEV of the $\chi$ field is
parametrically $O(1)$ while the dilaton mass is parametrically small. As we
have shown above (note in particular Eq. (34)), if $4-\Delta$ is not
parametrically small this is equivalent to the ad hoc assumption that the
coupling $b_{0}$ of the scale-invariant quartic potential in the EFT is
parametrically small; whereas physically, there is nothing in the underlying
theory to prevent $b_{0}$ from being $O(1)$. This assumption would thus turn
the $\Delta$-potential into a model. While this model might be useful, it does
not qualify as an effective field theory.

##  IV  Discussion

An effective low-energy field theory must reproduce the correlation functions
of the underlying theory, and it must do so systematically, order by order in
the expansion in its small parameters. These stringent requirements explain
why the construction of an EFT must follow very strict rules, as we briefly
reviewed in Sec. II.

The underlying theories we have been dealing with in this paper are QCD-like.
Such theories are well understood theoretically. What we mean by this is that
they are defined by a renormalizable lagrangian, and, thanks to their
asymptotic freedom, they can be defined non-perturbatively on the lattice by
approaching the continuum limit at the gaussian fixed point. The safe approach
to the construction of a pion-dilaton EFT is thus to match the EFT directly to
the underlying theory, as we have done. The approximate scale symmetry stems
from the smallness of the beta function in “walking” theories which are close
to the conformal window. The distance in theory space to the sill of the
conformal window thus naturally serves to provide the small parameter which
controls the mass and the interactions of the dilatonic meson.

The construction of a pion-dilaton EFT is significantly more complicated than
ordinary ChPT. As it turns out, a key difficulty is that a purely quartic
potential for the (effective) dilaton field is allowed by scale invariance.
Hence, the power counting of the pion-dilaton EFT must allow the quartic
potential to be present with an $O(1)$ coupling in the low-energy lagrangian.
The question is thus whether the EFT can cope with the quartic potential and
still generate a systematic low-energy expansion. As we have shown in Refs.
[9, 12], in dChPT the answer is positive. At the technical level, the freedom
to shift the dilatonic meson field $\tau$ plays an important role in ensuring
that the $O(1)$ coupling of the quartic potential does not jeopardize the
systematic expansion. As an example, we reviewed how the ratio controlling the
low-energy expansion in the dilaton sector, Eq. (15), is parametrically small,
as it should be.

Our main goal in this paper was to explore power countings for the pion-
dilaton EFT which are motivated by the phenomenological picture of the
conformal transition presented in Sec. III. We note that, from the start,
there are serious objections to this program. First, we have already shown
that the pion-dilaton EFT can be constructed by matching it directly to the
underlying renormalizable theory. Why should we, then, trade the actual
underlying theory with a speculative phenomenological lagrangian which may or
may not reflect the physics of the underlying theory?

The second objection is that this phenomenological approach is in fact
ambiguous, in that it suggests two alternative power countings for the pion-
dilaton EFT. In one sentence, one assumes that the phenomenological lagrangian
consists of a conformal part, plus a single term $G{O_{\rm sill}}$. The
conformal transition is triggered either by the coupling $G$ of ${O_{\rm
sill}}$ tending to zero, which leads to the first power counting; or,
alternatively, by the scaling dimension $\Delta$ of ${O_{\rm sill}}$ tending
to 4, which leads to the alternative power counting, with a potentially
different EFT. However, since the underlying theory is well defined, there
cannot exist two different EFTs describing the low-energy physics. Any
assumption that leads to two different possible EFT constructions thus leads
us to an apparent paradox: which EFT is the correct one? Only one EFT can
match the correlation functions of the underlying theory.

To gain more insight, in Sec. III we went ahead and investigated the resulting
EFTs that follow from these two power countings, setting aside the questions
raised above. We found that the $G\to 0$ power counting fails to generate a
systematic expansion. This failure is directly related to the presence of the
quartic dilaton potential with the unsuppressed $O(1)$ coupling. Unlike in
dChPT there is no mechanism to “tame” this $O(1)$ coupling. As a result, the
ratio (34) which defines the would-be expansion parameter of the dilaton
sector turns out to be $O(1)$, instead of being parametrically small as
required. This implies an inconsistency of the EFT defined by the $G\to 0$
power counting. Furthermore, since, as we have shown the $\Delta$-potential is
completely equivalent to the leading-order potential of the $G\to 0$ power
counting, this conclusion applies to the $\Delta$-potential as well.

As for the $\Delta\to 4$ power counting, we found that it is in fact the same
as the power counting of dChPT, and the resulting EFT is just dChPT itself.

Thus, setting aside more fundamental possible objections to the
phenomenological lagrangian (16), we see that it does not lead to any new
option. This is as one might have expected. Truly different power countings
would lead to different EFTs, and thus to different physical predictions,
leading to the paradox described above. Our results resolve this paradox,
because what we find is that there is only a single consistent power counting
for the pion-dilaton EFT, even if we start from the phenomenological
lagrangian (16), and the resulting EFT is dChPT.

Acknowledgments

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of High Energy Physics, Office of Basic Energy
Sciences Energy Frontier Research Centers program under Award Number DE-
SC0013682 (MG). YS is supported by the Israel Science Foundation under grant
no. 1429/21.

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