Radiative symmetry breaking from the on-shell perspective

Bartosz Sójka***Electronic address<EMAIL_ADDRESS>and Bogumiła
Świeżewska†††Electronic address<EMAIL_ADDRESS>

Faculty of Physics, University of Warsaw,
Pasteura 5, 02-093 Warsaw, Poland

Models with classical scale symmetry, which feature radiative symmetry
breaking, generically lead to a supercooled first-order phase transition in
the early Universe resulting in a strong gravitational-wave signal,
potentially observable by LISA. This is thanks to the absence of mass terms in
the potential and the resulting logarithmic structure of the zero-temperature
effective potential. It is known that imposing a symmetry at classical level
does not prohibit breaking it by quantum corrections. In the case of scale
symmetry, a mass term can in principle appear through renormalisation. This is
not the case in the commonly used $\overline{\mathrm{MS}}$ or Coleman–Weinberg
schemes. In this work we renormalise a scale invariant model in the on-shell
scheme to check whether parameterising it with the physical masses will
introduce mass terms to the potential. We find that indeed mass terms appear
for an arbitrary choice of the physical masses. However, we formulate an on-
shell condition for radiative symmetry breaking, sufficient and necessary for
the cancellation of mass terms in the renormalised potential, yielding a
logarithmic potential needed for supercooled phase transitions.

###### Contents

  1. 1 Introduction
  2. 2 Massless scalar quantum electrodynamics
  3. 3 Different renormalisation schemes
  4. 4 On-shell renormalisation
  5. 5 Relating on-shell and $\overline{\textrm{MS}}$ results
  6. 6 Loop-generated vs tree-level mass term
  7. 7 Summary and discussion

##  1 Introduction

Cosmological phase transitions (PT) are among the early-Universe phenomena
that will likely be probed via gravitational waves (GW), in particular by LISA
[1, 2, 3]. The electroweak symmetry breaking proceeds via a crossover within
the Standard Model (SM) [4, 5, 6], therefore no observable GW signature is
expected in this scenario. However, first-order PTs can be realised in
extensions of the SM, which are needed to solve the long-standing puzzles of
dark matter, baryon asymmetry of the Universe, etc. Numerous models of beyond-
SM (BSM) physics are subject to detailed scrutiny to determine the nature of
the predicted PT and the associated GW signal. Such a signal would be a clear
sign of BSM physics and a source of information about fundamental
interactions, complementary to collider searches. Even a negative outcome,
with no PT-like signal, would give us important information, constraining or
excluding certain BSM models.

A supercooled PT is a scenario well-suited for being tested with LISA. It
proceeds at temperatures much below the critical temperature, leading to a
substantial energy release and a strong GW signal. This kind of transition is
accommodated by models with nearly conformal dynamics or classical scale
symmetry [7, 8, 9, 10, 11]. Results for various models with classical scale
symmetry indicate that GW from a supercooled PT will be well within
experimental reach [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25]. A particular example is a classically scale invariant model with a
new SU(2) gauge group for which the whole allowed parameter space yields a GW
signal with a significant signal-to-noise ratio [23, 25]. Thus, in case of
non-observation of a PT-like signal, the whole model can be falsified.
Moreover, a supercooled PT has been one of the suggested explanations [26, 27,
28, 29, 30, 31] of the stochastic GW background recently observed by the
pulsar timing array collaborations [32, 33, 34]. However, it is not easy to
construct a concrete model that would fit the data and would not contradict
other observations [35, 36].

The characteristic feature that makes models with classical scale invariance
particularly suited for strong first-order PTs is a logarithmic potential. The
lack of a term quadratic in the field (that would be associated with a mass
term in the potential), makes the potential very flat around the origin as it
is dominated by the term quartic in the field. In such a setting, even small
thermal corrections introduce a barrier that delays the tunnelling from the
symmetric to the broken phase until very low temperatures. This causes
supercooling and results in a large energy release and a strong GW signal. It
is known, however, that if a classical symmetry is present in a theory,
nothing prevents it from being broken by loop corrections. Indeed, within
classically conformal models, masses of particles are generated via radiative
symmetry breaking (RSB). This can be realised without an explicit mass term in
the potential. But there is no reason, in principle, why such a term would not
arise at one-loop. If it does, it could entirely change the physical picture
of a strong, supercooled PT.

In the widely used modified minimal subtraction ($\overline{\textrm{MS}}$)
renormalisation scheme and in the scheme introduced in the seminal paper
studying RSB by Coleman and Weinberg [37], a mass term does not appear at one-
loop level. But, as we explain in detail in the main body of the text, these
schemes are tailored to preserve the scaling symmetry at the level of the
Lagrangian. However, in the physical world scales exist and in particular,
masses exist. Then a question arises – does parameterising the model with
physical masses require the inclusion of a mass counterterm that would
explicitly break scale invariance? The aim of the present work is to
renormalise a classically scale-invariant model in the on-shell (OS) scheme
and study the consequences of such parameterisation for the PT and the
resulting GW. Moreover, this analysis will allow us to evaluate how well the
commonly used running masses (computed within the $\overline{\textrm{MS}}$
scheme) approximate the physical masses. This may seem to be a trivial
question that should have been long answered given the number of works on
models with classical scale symmetry. However, to our knowledge, it has not
been so far addressed in the literature. Motivated by the important
phenomenological consequences of supercooled PTs we aim to fill this gap in
the present work.

In this article, we will analyse in detail the renormalisation of massless
scalar quantum electrodynamics (QED) as a minimal toy model allowing for RSB
[37]. The model is introduced in section 2. Further, we will review different
renormalisation schemes in section 3. Our approach to the on-shell
renormalisation of scalar QED is presented in section 4. Next, we relate the
obtained results to the more familiar $\overline{\textrm{MS}}$-results in
section 5 and discuss the relation between loop-generated and tree-level mass
terms in section 6. Finally, we summarise our findings and discuss their
implications for GWs from supercooled PTs in section 7.

##  2 Massless scalar quantum electrodynamics

Starting from the seminal paper of Coleman and Weinberg [37], massless scalar
quantum electrodynamics (QED), also known as the massless Abelian Higgs model,
has been an archetypal model for the mechanism of RSB (also known as
dimensional transmutation). It consists of a complex scalar charged under a
U(1) gauge group and a gauge field – a photon – originating thereof. The
Lagrangian is given by:

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\left(D_{\mu}\phi\right)^{*}D^{\mu}\phi-V^{(0)},$
(2.1)

where $F_{\mu\nu}$ is the usual energy-momentum tensor for the gauge field,
the covariant derivative is given by $D_{\mu}=\partial_{\mu}+ieA_{\mu}$, $e$
is the U(1) charge and $\phi$ denotes the complex scalar field. The tree-level
potential is given by

$V^{(0)}(\phi)=\lambda(\phi^{*}\phi)^{2}.$ (2.2)

We can use the global U(1) symmetry to rotate the complex field such that the
vacuum expectation value (VEV) is located along the real direction and write
the classical contribution to the effective potential as

$V^{(0)}(\varphi)=\frac{1}{4}\lambda\varphi^{4},$ (2.3)

where $\varphi$ denotes the classical (background) field. The Abelian Higgs
model possesses classical scale (conformal) symmetry as no dimensionful
parameters are present at tree level. Classically, the model features only a
symmetric minimum at $\varphi=0$.

To study the vacuum structure beyond tree level, one needs to consider the
effective potential. While the scalar contributions to the effective potential
are negligible for perturbative couplings [37], including loop corrections
from the gauge boson modifies the structure of the potential. The one-loop
correction to the effective potential reads (in the Landau gauge and using
dimensional regularisation in $D=4-\varepsilon$ dimensions)

$V^{(1)}(\varphi)=\kappa
e^{4}\varphi^{4}\left(\log\frac{e^{2}\varphi^{2}}{\mu^{2}}-\frac{5}{6}+\eta\right),$
(2.4)

where we introduced a short-hand notation for the loop factor
$\kappa=\frac{3}{64\pi^{2}}$, the term $e\varphi$ corresponds to the field-
dependent mass of the photon and the infinite part is contained in
$\eta=-\frac{2}{\varepsilon}+\gamma_{E}-\log(4\pi)$ with $\gamma_{E}=0.5772$
being the Euler constant. One should note, that the one-loop term computed
using dimensional regularisation does not contain infinities proportional to
$\varphi^{2}$. This would not be the case if cutoff regularisation was used
[37]. To make the potential finite, we need to split the bare couplings into
the sum of renormalised ones and the counterterms that will cancel infinities
as

$\lambda_{\mathrm{bare}}=\lambda+\delta\lambda,\quad
m^{2}_{\mathrm{bare}}=m^{2}+\delta m^{2}.$ (2.5)

Both the renormalised couplings and the counterterms depend on the
renormalisation scheme. Even if we assume that the renormalised mass term
vanishes, the dimensionful mass counterterm $\frac{1}{2}\delta
m^{2}\varphi^{2}$ has to be included since the scale symmetry is only imposed
on the model at classical level. Since there is no infinity proportional to
$\varphi^{2}$, $\delta m^{2}$ can serve to perform finite renormalisation of
the mass. In principle, also the field strength has to be renormalised by
performing the shift $\varphi^{2}\to Z\varphi^{2}$. However, the effective
potential is the zero-momentum part of the effective action so we can assume
that $Z=1$ at the renormalisation scale at work. Also note that we do not need
to renormalise $e$, as it only appears in the effective potential at one-loop
level.

The full renormalised one-loop potential reads

$V(\varphi)=V^{(0)}(\varphi)+V^{(1)}(\varphi)+V_{\mathrm{CT}}(\varphi),\;V_{\mathrm{CT}}(\varphi)=\frac{1}{2}\delta
m^{2}\varphi^{2}+\frac{1}{4}\delta\lambda\varphi^{4}.$ (2.6)

The values of the counterterms depend on the renormalisation scheme. In the
next section, we review the popular choices for the renormalisation schemes.

##  3 Different renormalisation schemes

The infinities present in loop computations can be removed by imposing
renormalisation conditions. Different sets of conditions can be chosen,
leading to diverse renormalisation schemes. The most popular one, due to its
convenience, is the modified minimal subtraction scheme
($\overline{\textrm{MS}}$). In the context of RSB, the original scheme
introduced by Coleman and Weinberg [37] is also used sometimes. The parameters
defined within these schemes cannot, however, be considered physical. The
scheme that relates the parameters of the potential to the on-shell masses of
physical particles is the so-called on-shell (OS) scheme. In this section, we
review these renormalisation schemes.

### Modified minimal subtraction scheme

Within the $\overline{\textrm{MS}}$ scheme the counterterms are required to
simply cancel infinities (plus some constant terms) proportional to respective
powers of $\varphi$. Since, when dimensional regularisation is used, no
divergent terms proportional to $\varphi^{2}$ are generated at one-loop level
(see eq. (2.4)), the $\delta m^{2}$ counterterm is equal to zero. We have

$\begin{dcases}\delta m^{2}_{\overline{\textrm{MS}}}=0,\\\
\delta\lambda_{\overline{\textrm{MS}}}=-4\kappa e^{4}\eta,\end{dcases}$ (3.1)

and the one-loop renormalised potential takes the following form

$V(\varphi)=\frac{1}{4}\lambda_{\overline{\textrm{MS}}}\varphi^{4}+\kappa
e^{4}\varphi^{4}\left(\log\frac{e^{2}\varphi^{2}}{\mu^{2}}-\frac{5}{6}\right).$
(3.2)

We introduced a subscript on the coupling which indicates the scheme in which
it is defined.

We can now study the symmetry breaking. The minimisation condition, with
$\langle\varphi\rangle=v_{\overline{\textrm{MS}}}$ yields two solutions

$\lambda_{\overline{\textrm{MS}}}=-4\kappa
e^{4}\left(\log\frac{e^{2}v_{\overline{\textrm{MS}}}^{2}}{\mu^{2}}-\frac{1}{3}\right)$
(3.3)

or $v_{\overline{\textrm{MS}}}=0$. The special property of models with
classical scale invariance is that from the minimisation of the potential we
do not obtain the value of the VEV, but rather a relation between the
couplings. Setting $\mu=ev_{\overline{\textrm{MS}}}$ it further simplifies to

$\lambda_{\overline{\textrm{MS}}}=\frac{4}{3}\kappa
e^{4}=\frac{e^{4}}{16\pi^{2}}.$ (3.4)

This relation is at the heart of understanding RSB – the one-loop correction
can modify the potential quantitatively if the tree-level (parameterised by
$\lambda$) and the one-loop (parameterised by $e^{4}$) contributions belong to
the same order in perturbation theory. In this case, one should not expand in
loops, but rather in powers of couplings, using the scaling of eq. (3.4). This
is important also to preserve gauge independence of the effective potential at
the minimum in perturbative computations, see refs. [38, 39].

Using the relation of eq. (3.3) we can simplify the
$\overline{\textrm{MS}}$-renormalised effective potential of eq. (3.2) to the
following form

$V(\varphi)=\kappa
e^{4}\varphi^{4}\left(\log\frac{\varphi^{2}}{v_{\overline{\textrm{MS}}}^{2}}-\frac{1}{2}\right).$
(3.5)

Note that formally we have not set the scale to $ev_{\overline{\textrm{MS}}}$.
The scale dependence in $\lambda_{\overline{\textrm{MS}}}$ in eq. (3.3)
cancelled the scale dependence in the effective potential. The resulting
potential is parameterised using just two quantities: $e$ and
$v_{\overline{\textrm{MS}}}$ – the presence of a dimensionful parameter
signals dimensional transmutation.

Using the effective potential we can approximate the mass of the scalar by the
second derivative of the potential evaluated at the minimum (the running mass,
denoted with a bar as $\overline{M}_{S}^{2}$),

$\overline{M}_{S}^{2}=V^{\prime\prime}(v_{\overline{\textrm{MS}}})=8\kappa
e^{4}v_{\overline{\textrm{MS}}}^{2}.$ (3.6)

Now, knowing the mass of the vector,
$\overline{M}_{V}=ev_{\overline{\textrm{MS}}}$, we can compute the ratio of
masses

$\frac{\overline{M}_{S}^{2}}{\overline{M}_{V}^{2}}=8\kappa
e^{2}=\frac{3e^{2}}{8\pi^{2}}\equiv r_{{\rm{CW}}}.$ (3.7)

This ratio of masses is said to be a prediction of the massless scalar QED
[37]. We will check whether it holds for the physical masses (not just the
running ones).

### Coleman-Weinberg (CW) scheme

The CW scheme [37] consists of the following conditions

$\begin{dcases}\left.\frac{\differential^{2}V}{\differential\varphi^{2}}\right|_{\varphi=0}=0,\\\
\left.\frac{\differential^{4}V}{\differential\varphi^{4}}\right|_{\varphi=M}=6\lambda_{\rm{CW}}.\end{dcases}$
(3.8)

One should note that the first condition explicitly enforces that the
renormalised mass term vanishes. The other condition defines the coupling at a
certain energy scale $M$. One cannot define it at $\varphi=0$ in analogy to
the first one, since the logarithmic term is ill-defined there. The
counterterms read111In ref. [37] the cutoff regularisation was used, thus a
quadratic infinity arose which had to be cancelled by a non-zero mass
counterterm. Here we use dimensional regularisation, thus $\delta
m^{2}_{\rm{CW}}=0$.

$\begin{dcases}\delta m^{2}_{\rm{CW}}=0,\\\ \delta\lambda_{\rm{CW}}=-4\kappa
e^{4}\left(\log\frac{e^{2}M^{2}}{\mu^{2}}+\frac{10}{3}+\eta\right).\end{dcases}$
(3.9)

The potential, after expressing $\lambda_{\rm{CW}}$ in terms of the coupling
$e$ with the use of the minimisation condition has the same form as in the
$\overline{\textrm{MS}}$ case, eq. (3.2), with $v_{\overline{\textrm{MS}}}$
replaced by $v_{{\rm{CW}}}$. Moreover, the predicted ratio of masses is the
same as in $\overline{\textrm{MS}}$, hence it is denoted by $r_{{\rm{CW}}}$ in
eq. (3.7).

### On-shell scheme

The on-shell scheme is considered to be the physical renormalisation scheme as
it requires the renormalised mass to correspond to the pole of the propagator,
with residue equal to one. A condition requiring that the VEV of the scalar
field corresponds to the physical one is often added. The general
renormalisation conditions can be summarised as:

$\begin{dcases}\left.\frac{\differential
V}{\differential\varphi}\right|_{\varphi=v}=0,\\\\[4.0pt]
G^{(2)}(p^{2}=M^{2})=0,\\\\[4.0pt] \left.\frac{\differential
G^{(2)}}{\differential p^{2}}\right|_{p^{2}=M^{2}}=1,\end{dcases}$ (3.10)

where $G^{(2)}(p^{2})$ is the two-point function (the inverse of the
propagator) and $M$ corresponds to the physical mass. Most commonly, in the
case of one-loop renormalisation, $M$ and $v$ are identified with the tree-
level parameters. In this case, the conditions of eq. (3.10) can be further
simplified. Using the expression for the loop-corrected two-point function

$G^{(2)}(p^{2})=p^{2}-M^{2}-\Sigma(p^{2}),$ (3.11)

with $\Sigma(p^{2})$ being the one-loop self-energy and assuming that the
tree-level mass corresponds to the physical mass $M$ and that the physical VEV
is the tree-level one, we obtain

$\begin{dcases}\left.\frac{\differential
V^{(1)}}{\differential\varphi}\right|_{\varphi=v}=0,\\\\[4.0pt]
\Sigma(p^{2}=M^{2})=0,\\\\[4.0pt]
\left.\frac{\differential\Sigma}{\differential
p^{2}}\right|_{p^{2}=M^{2}}=0.\end{dcases}$ (3.12)

If one is interested in the effective potential, one can set the wave-function
renormalisation constant $Z$ to one, at some given renormalisation scale $\mu$
and neglect the last condition. Furthermore, one can use the fact that the
second derivative of the one-loop correction to the effective potential
corresponds to the zero-momentum part of the self-energy as

$\Sigma(p^{2})=\Sigma(0)+\Sigma(p^{2})-\Sigma(0)=\left.\frac{\differential^{2}V^{(1)}}{\differential\varphi^{2}}\right|_{\varphi=v}+\Delta\Sigma(p^{2}).$
(3.13)

If $\Delta\Sigma(M^{2})$ can be considered negligible, the set of
renormalisation conditions can be rewritten as

$\begin{dcases}\left.\frac{\differential
V^{(1)}}{\differential\varphi}\right|_{\varphi=v}=0,\\\\[4.0pt]
\left.\frac{\differential^{2}V^{(1)}}{\differential\varphi^{2}}\right|_{\varphi=v}=0.\end{dcases}$
(3.14)

This is the form of the on-shell renormalisation conditions commonly used in
computations making use of the effective potential. We cannot, though, apply
these conditions in the case of classically scale-invariant models since there
is no tree-level VEV or mass to refer to. We devote the next section to
studying the OS renormalisation of the massless scalar QED.

##  4 On-shell renormalisation

In the absence of tree-level symmetry breaking, we will introduce
renormalisation conditions which refer to the full one-loop effective
potential, not just the one-loop correction as discussed in section 3. In the
former approach, one could use the tree-level relations given by the
minimisation condition and the second derivative of the potential to determine
the VEV and the masses using Lagrangian parameters. Here, the one-loop mass
and VEV will be used to fix the counterterms so we will not be able to express
the physical masses in terms of the Lagrangian parameters. Nonetheless, we
will be able to obtain the full one-loop effective potential parameterised by
the physical masses. As the input parameters, we choose the mass of the vector
$M_{V}$, the mass of the scalar $M_{S}$ and the electric charge $e$. One
should note that we parameterise the model with 3 free parameters, not 2 as
was in the case of $\overline{\textrm{MS}}$ and CW schemes.

The OS renormalisation conditions read

$\begin{dcases}\left.\frac{\differential
V}{\differential\varphi}\right|_{\varphi=M_{V}/e}=0,\\\\[4.0pt]
\left.\frac{\differential^{2}V}{\differential\varphi^{2}}\right|_{\varphi=M_{V}/e}=M_{S}^{2}.\end{dcases}$
(4.1)

As already explained in the previous section, they state that $M_{V}/e$
corresponds to the VEV of the scalar field and $M_{S}$ corresponds to the pole
of the scalar propagator in the zero-momentum limit. In the OS scheme, we
obtain the following relations

$\begin{dcases}\delta m^{2}_{\mathrm{OS}}=-\frac{1}{2}M_{S}^{2}+4\kappa
e^{2}M_{V}^{2},\\\
\lambda_{\mathrm{OS}}+\delta\lambda_{\mathrm{OS}}=\frac{e^{2}}{2}\frac{M_{S}^{2}}{M_{V}^{2}}-4\kappa
e^{4}\left(\log\frac{M_{V}^{2}}{\mu^{2}}+\frac{2}{3}+\eta\right).\end{dcases}$
(4.2)

Introducing the ratio of the physical masses

$r=\frac{M_{S}^{2}}{M_{V}^{2}}$ (4.3)

and using the CW mass ratio of eq. (3.7) we obtain the following expression
for the renormalised one-loop effective potential

$V(\varphi)=\kappa
e^{4}\varphi^{4}\left(\log\frac{e^{2}\varphi^{2}}{M_{V}^{2}}-\frac{3}{2}+\frac{r}{r_{{\rm{CW}}}}\right)+2\kappa
e^{2}M_{V}^{2}\varphi^{2}\left(1-\frac{r}{r_{{\rm{CW}}}}\right).$ (4.4)

We cannot further simplify this potential using the stationarity condition, as
was done in the $\overline{\textrm{MS}}$ and CW cases, since we have already
used this condition for fixing the counterterms.

The first observation is that the ratio of the physical masses $r$ turns out
to be the key input parameter determining the properties of the potential, in
contrast to the prediction for the mass ratio in the
$\overline{\textrm{MS}}$/CW schemes. Second, in the OS procedure, a
nonvanishing (finite) mass counterterm is generated, which explicitly breaks
the classical scale invariance. Since the scale symmetry in the analysed model
is imposed only at classical level, there is no fundamental reason why it
should be preserved once quantum corrections are taken into account. In fact,
the CW and $\overline{\textrm{MS}}$ schemes are tailored to preserve the
explicit scale invariance at loop level. The former explicitly demands that
the renormalised mass term is zero via one of the renormalisation conditions.
The other is based on the property of dimensional regularisation, which
prevents the appearance of quadratic divergences. This is also the reason why
in the $\overline{\textrm{MS}}$ and CW schemes one less parameter was needed
to fully specify the model – the mass term was “invisible” since it was set to
zero.

We can now study the structure of the renormalised potential depending on the
value of $r/r_{{\rm{CW}}}$. This is illustrated in the left panel figure 1,
where we show the potential normalised by the mass scale set by $M_{V}$

$\tilde{V}(\tilde{\varphi})=\frac{V(\varphi)}{\kappa
M_{V}^{4}}=\tilde{\varphi}^{4}\left(\log\tilde{\varphi}^{2}-\frac{3}{2}+\frac{r}{r_{{\rm{CW}}}}\right)+2\tilde{\varphi}^{2}\left(1-\frac{r}{r_{{\rm{CW}}}}\right),$
(4.5)

where $\tilde{\varphi}$ is a dimensionless field
$\tilde{\varphi}=e\varphi/M_{V}$. In this parameterisation, the shape of the
potential only depends on the ratio $r/r_{{\rm{CW}}}$, while changing the
values of $M_{V}$ and $e$ does not change the picture.

Figure 1: Left panel: the rescaled potential of eq. (4.5) for
$r/r_{{\rm{CW}}}=0.75$ (long-dashed, green), $r/r_{{\rm{CW}}}=1$ (solid,
blue), $r/r_{{\rm{CW}}}=1.25$ (short-dashed, red). Right panel: the ratio of
the running mass to the physical one for the vector (solid line) and the
scalar (dashed line).

If $r=r_{{\rm{CW}}}$ (solid blue line), the mass term vanishes at one-loop
level. The potential of eq. (4.4) exactly coincides with the
$\overline{\textrm{MS}}$ and CW result and has the familiar logarithmic shape,
very flat around the origin, dominated by the term quartic in the field. When
$r/r_{{\rm{CW}}}<1$ (long-dashed green curve), the mass term in the potential
is positive and it induces a minimum at the field-space origin and a barrier
separating the symmetric and the broken (radiatively generated) minima. In
this regime, symmetry breaking is even harder to achieve than in the standard
RSB scenario, as the loop corrections have to cancel the tree-level quadratic
and quartic terms. For $r/r_{{\rm{CW}}}<0.5$ the symmetry-breaking minimum
becomes only a local minimum of the potential. When $r/r_{{\rm{CW}}}>1$ (red
dotted curve), the mass term is negative and contributes to symmetry breaking
as in the classical version of the Brout–Englert–Higgs mechanism. If
$r/r_{{\rm{CW}}}$ is close to one, it is only a small correction to the RSB
mechanism but if it dominates over the logarithmic term, the potential has a
parabolic shape around the origin. For $r/r_{{\rm{CW}}}>1.5$, the sign of the
$\tilde{\varphi}^{4}$ term becomes positive, contrary to what is observed in
the common RSB analyses.

##  5 Relating on-shell and $\overline{\textrm{MS}}$ results

To further explore the relation between the $\overline{\textrm{MS}}$ and the
OS schemes we will now relate the potential parameters defined in both
schemes. To do that, we use the fact that the bare couplings are scheme-
independent. We thus have

$\lambda_{\mathrm{bare}}=\lambda_{\overline{\textrm{MS}}}+\delta\lambda_{\overline{\textrm{MS}}}=\lambda_{\mathrm{OS}}+\delta\lambda_{\mathrm{OS}}.$
(5.1)

From this relation, we obtain $\lambda_{\overline{\textrm{MS}}}$ in terms of
the physical masses

$\lambda_{\overline{\textrm{MS}}}=\lambda_{\mathrm{OS}}+\delta\lambda_{\mathrm{OS}}-\delta\lambda_{\overline{\textrm{MS}}}=\frac{e^{2}}{4}\frac{M_{S}^{2}}{M_{V}^{2}}-4\kappa
e^{4}\left(\log\frac{M_{V}^{2}}{\mu^{2}}+\frac{2}{3}\right),$ (5.2)

where the expressions for the counterterms from eqs. (3.1) and (4.2) have been
used. The same reasoning can be employed to rewrite the
$\overline{\textrm{MS}}$ mass parameter in terms of the physical quantities.
Using the fact that $m^{2}_{\mathrm{OS}}=\delta
m^{2}_{\overline{\textrm{MS}}}=0$ we get

$m^{2}_{\overline{\textrm{MS}}}=\delta m^{2}_{\mathrm{OS}}.$ (5.3)

If we now substitute $m^{2}_{\overline{\textrm{MS}}}$ and
$\lambda_{\overline{\textrm{MS}}}$ from eqs. (5.3) and (5.2) to the
$\overline{\textrm{MS}}$-renormalised potential of eq. (3.2), we recover the
expression for the OS-renormalised potential of eq. (4.4), as expected. The
relation of eq. (5.3), however, tells us that if we want to capture the same
physics that is described by the model in the OS, in the
$\overline{\textrm{MS}}$ scheme, a mass term appears. That is not visible from
inside the $\overline{\textrm{MS}}$ scheme because in dimensional
regularisation no quadratic divergences appear. Also, no direct link to the
physical masses is made during the procedure of $\overline{\textrm{MS}}$
renormalisation so there is no need to introduce a mass counterterm.

It is likely more insightful to explore the relation between the OS and
$\overline{\textrm{MS}}$ masses. This can be achieved by equating the
expression for $\lambda_{\overline{\textrm{MS}}}$ from eq. (5.2) with the one
obtained from the minimisation condition in the $\overline{\textrm{MS}}$
scheme, eq. (3.3), using the fact that $e$ needs not to be renormalised and
hence it is the same in both schemes. This leads to the following relation
between the OS and $\overline{\textrm{MS}}$ vector masses

$\frac{\overline{M}_{V}^{2}}{M_{V}^{2}}=\exp\left(1-\frac{r}{r_{{\rm{CW}}}}\right).$
(5.4)

Using eq. (3.7) we obtain a similar result for the scalar mass

$\frac{\overline{M}_{S}^{2}}{M_{S}^{2}}=\frac{r_{{\rm{CW}}}}{r}\exp\left(1-\frac{r}{r_{{\rm{CW}}}}\right).$
(5.5)

The behaviour of these ratios is illustrated in the right panel of figure 1
(solid line for the vector and dashed line for the scalar). If the physical
mass ratio coincides with the CW value, the running masses exactly coincide
with the physical ones (in the zero-momentum approximation) and the
$\overline{\textrm{MS}}$ and CW descriptions work perfectly. On the other
hand, if the mass ratio is different but we try to use a model that has no
mass term, our predictions will quickly deviate from the physical values. In
particular, the OS scalar mass to the running mass ratio diverges as $r\to 0$.
This is due to the disappearance of the global symmetry-breaking minimum in
this limit (see figure 1). For $r/r_{{\rm{CW}}}\gg 1$, the OS to running mass
ratios tend to zero. In this case, our description should contain a mass term
already at classical level.

##  6 Loop-generated vs tree-level mass term

One may ask the following question – if a mass term is generated in the
renormalisation procedure, could we, equally well, include it in the potential
from the very beginning? To understand that, we can conduct the same analysis
starting from the potential of eq. (2.3) supplemented with a mass term

$V^{(0)}(\varphi)=\frac{1}{4}\lambda\varphi^{4}+\frac{1}{2}m^{2}\varphi^{2}.$
(6.1)

We can apply the same renormalisation conditions as before, see eq. (4.1).
Then it is only possible to determine the combinations
$\lambda+\delta\lambda$, $m^{2}+\delta m^{2}$. The result is the same as in
the massless case given in eq. (4.2), with $m^{2}$ added on the left-hand side
of the first equation. Hence, it does not matter whether a nonzero mass term
had been included at tree level, if we impose the on-shell renormalisation
conditions on the full one-loop potential.

Having a (negative) tree-level mass term that allows for symmetry breaking
already at tree level, we can relate the physical masses, $M_{S}$ and $M_{V}$,
to Lagrangian parameters already at tree level, using the first and the second
derivative of the classical potential. Applying the relation
$\langle\varphi\rangle=M_{V}/e$ we have

$\begin{dcases}m^{2}=-\frac{1}{2}M_{S}^{2},\\\
\lambda=\frac{e^{2}}{2}\frac{M_{S}^{2}}{M_{V}^{2}}.\end{dcases}$ (6.2)

Now we can renormalise the potential using the more standard conditions listed
in eq. (3.14), which state that the one-loop corrections should not alter the
tree-level values of the scalar mass and VEV. This yields

$\begin{dcases}\delta m^{2}=4\kappa e^{2}{M_{V}^{2}},\\\
\delta\lambda=-4\kappa
e^{4}\left(\log\frac{M_{V}^{2}}{\mu^{2}}+\eta+\frac{2}{3}\right).\end{dcases}$
(6.3)

The relations of eqs. (6.3) and (6.2) correspond to a split of eq. (4.2)
(supplemented with the $m^{2}$ term as described above) into tree-level and
one-loop parts. This approach allows us to relate the tree-level Lagrangian
parameters with the physical parameters but the final potential, expressed in
terms of the physical parameters, will look the same.

The discussion of this subsection shows that the split between the tree-level
and loop-generated mass term is to some extent artificial. Nonetheless, to
describe a model of scalar electrodynamics with arbitrary masses for the
scalar and the vector, we need a potential with a term quadratic in the field.
If the ratio of the masses is very special and equal to $r_{{\rm{CW}}}$ as
“predicted” in the $\overline{\textrm{MS}}$ or CW scheme (see eq. (3.7)), one
can describe it with a potential that is classically scale invariant. In this
case, at loop level no mass term is generated and the scale symmetry is only
broken by logarithmic effects. On the other hand, in the very same case of the
CW mass ratio, one can start at tree level with a description that is not
scale invariant and contains a mass term as in eq. (6.2). Then, the loop
corrections will cancel this mass term and in the one-loop potential the
symmetry will only be broken by logarithmic effects – an effective “radiative
(explicit) symmetry restoration”. This shows that a logarithmic one-loop
potential can be attained if tree-level and one-loop terms are of the same
order (this condition is expressed by $r=r_{{\rm{CW}}}$), irrespective of
whether there is a mass term in the tree-level potential.

Of course, if the mass ratio is far away from $r_{{\rm{CW}}}$, the last
approach to renormalisation of the effective potential presented above is the
most suitable. In such a case, the tree-level term dominates, and the one-loop
contribution gives only a small correction to the leading-order results. Then
there is a natural split between the leading tree-level order and the
subleading one-loop order. The perturbative expansion should follow the loop
expansion.

This leads us to the conclusion that in the parameter space of scalar QED,
there is a subspace, parameterised by $r=r_{{\rm{CW}}}$, where tree-level and
one-loop terms conspire to produce a logarithmic potential at one-loop level.
In this subclass, one should use the scaling of couplings as a guideline to
conduct perturbative expansions, instead of the loop counting. Is it fair to
say that this subclass corresponds to classical scale invariance? Not exactly,
since, as we have seen the same result is obtained even with a tree-level mass
term, if the ratio of physical masses is equal to $r_{{\rm{CW}}}$. It is,
nonetheless, true that this case corresponds to radiative symmetry breaking,
as the one-loop and tree-level contributions conspire to produce a potential
with a non-trivial minimum, generated by the loop-induced logarithm and not a
term quadratic in the field. Therefore, the condition $r=r_{{\rm{CW}}}$ can be
considered as an on-shell condition for the realisation of RSB.

One remark is in order. In the whole reasoning above, we have neglected the
scalar contribution to the effective potential. This is well justified in the
case of RSB when $\lambda$ scales as $e^{4}$. Then the scalar contribution
scales as $\lambda^{2}\sim e^{8}$ and is a higher-order effect. This should
hold as long as the mass ratio is close to the CW one. If we drift far away
from it, the scalar effects may become more important. They are, however,
usually expected to be subleading compared to the vector ones.

##  7 Summary and discussion

In this article, we have studied an archetypal model of radiative symmetry
breaking – massless scalar QED – from the on-shell perspective. We
parameterised the model with the physical masses of the vector and scalar
particles, and the electric charge. The renormalisation conditions were chosen
such that the scalar mass corresponds to the pole of the propagator (in the
zero-momentum approximation) and the vector mass is defined via the tree-level
relation at the one-loop minimum. We have compared the obtained results with
the ones from the $\overline{\textrm{MS}}$ and CW schemes. Below we summarise
the main findings:

  * –

If we choose arbitrary masses for the vector and scalar, then at one-loop
level a mass (counter)term is generated. This mass term is controlled by the
ratio of the physical masses. If the ratio coincides with the value predicted
in $\overline{\textrm{MS}}$/CW schemes, $r_{{\rm{CW}}}$, the mass term
vanishes.

  * –

For a given ratio of masses we can equally well use a potential containing a
mass term. If the ratio of masses is equal to $r_{{\rm{CW}}}$, in the
renormalised potential the mass term is cancelled by a counterterm and the
resulting potential has the logarithmic form, characteristic of scale-
invariant models.

  * –

In general, the scalar-to-vector mass ratio controls the properties of the
renormalised potential. For $r<r_{{\rm{CW}}}$, a positive mass term is
present, which impedes symmetry breaking. For $r<0.5r_{{\rm{CW}}}$, the
symmetry-breaking minimum becomes a local one. For $r>r_{{\rm{CW}}}$, there is
a negative mass term that facilitates symmetry breaking. For $r$ deviating
slightly from $r_{{\rm{CW}}}$ the quadratic term is only a small correction to
the logarithmic potential, but for larger $r$, the potential becomes
polynomial as in the usual Brout–Englert–Higgs mechanism.

  * –

If $r$ is close to $r_{{\rm{CW}}}$ the running masses approximate the physical
masses well. The approximation becomes exponentially worse with $r$ deviating
from $r_{{\rm{CW}}}$.

  * –

Models with logarithmic potentials are not necessarily classically scale
symmetric. They are rather a subclass of more general massive models, in which
tree-level and one-loop terms conspire to cancel mass terms. In this sense,
the term radiative symmetry breaking is more general than classical scale
symmetry. In the case of scalar QED the sub-model with RSB is defined by
$r=r_{{\rm{CW}}}$. Therefore, the condition $r=r_{{\rm{CW}}}$ should not be
considered a prediction of massless scalar QED [37] but rather an assumption
necessary for it to be realised, an on-shell version of the typical condition
necessary for radiative symmetry breaking, $\lambda\sim e^{4}$.

These observations can contribute to the discussion on whether classical scale
symmetry is a solution to the hierarchy problem, dating back to the paper by
Bardeen [40]. The common argument is simple – with classical scale symmetry
and dimensional regularisation, no quadratic divergence is generated so the
hierarchy problem of the SM is cured. However, as we show in the present
paper, to attain a potential with vanishing renormalised mass term, one needs
to tune the physical masses to give the CW ratio, $r_{{\rm{CW}}}$. Therefore,
the tuning of the Higgs (scalar) mass is traded for tuning the scalar-to-
vector mass ratio.

The main motivation for performing this exercise in renormalisation was to
verify whether the extremely promising predictions for GW signals from
supercooled phase transitions in classically scale-invariant models are robust
to the renormalisation scheme change. The appearance of a strong GW signal
relies on the absence of the mass term in the renormalised potential, thus,
generating such a term in the OS scheme could change the predictions
dramatically. A vast majority (if not all) of works on classically scale-
invariant models utilise the $\overline{\textrm{MS}}$ (or CW) scheme, we are
not aware of a reference that would use the OS scheme. This paper aims to fill
this gap.

Our findings are reassuring. If one assumes $r=r_{{\rm{CW}}}$, no mass term is
generated in the renormalised potential and the results of
$\overline{\textrm{MS}}$ and CW schemes coincide with the OS ones. One should
be aware, however, that $r=r_{{\rm{CW}}}$ is rather an assumption in the
procedure than a prediction of the model. This means that to find a
logarithmic potential suitable for supercooling one needs to tune the physical
masses such that their ratio is close to $r_{{\rm{CW}}}$.222Of course, in more
complicated models the criterion of the scalar-to-vector mass ratio has to be
substituted with an adequate analogue. Having that, one can proceed with
$\overline{\textrm{MS}}$ or OS schemes, and the results should be the same. In
particular, the predictions of strong GW signals originating from supercooled
phase transitions obtained within the $\overline{\textrm{MS}}$ scheme hold and
such scenarios are prime candidates for being observed with LISA. Furthermore,
as shown in ref. [41], models with small deviations from the exact scale
symmetry, in the form of a non-vanishing mass term or cubic coupling, allow
for supercooled PTs. Therefore, if the ratio of physical masses is not exactly
equal to $r_{{\rm{CW}}}$, we can still expect a supercooled PT. Moreover, we
have shown that if the ratio of masses is close, but not equal to
$r_{{\rm{CW}}}$, the scale-invariant description will yield good approximate
results.

## Acknowledgements

We thank Gabriel Lourenço and Rui Santos for numerous fruitful discussions and
for reading the manuscript. We are grateful to Mikołaj Misiak for an
interesting discussion. This research was funded by the National Science
Centre, Poland, through the SONATA project number 2018/31/D/ST2/03302 and the
OPUS project number 2023/49/B/ST2/02782.

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