Perturbative running of the topological angles

Alessandro Valenti$\,{}^{a,b}$, Luca Vecchi$\,{}^{b}$

a Dipartamento di Fisica e Astronomia “G. Galilei”, Università di Padova,
Italy

b Istituto Nazionale di Fisica Nucleare, Sezione di Padova, I-35131 Padova,
Italy

###### Abstract

We argue that in general renormalizable field theories the topological angles
may develop an additive beta function starting no earlier than 2-loop order.
The leading expression is uniquely determined by a single model-independent
coefficient. The associated divergent diagrams are identified and a few
independent methods for extracting the beta function in dimensional
regularization are discussed. We show that the peculiar nature of the
topological angles implies non-trivial constraints on the anomalous dimension
of the CP-violating operators and discuss how a non-vanishing beta function
affects the Weyl consistency conditions. Some phenomenological considerations
are presented.

00footnotetext<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. 1 Motivations
  2. 2 $\theta$ in perturbation theory
    1. 2.1 $\beta_{\theta}$ from (extra)-ordinary diagrams
    2. 2.2 Extracting $\beta_{\theta}$ from operator mixing
  3. 3 $\beta_{\theta}$ in renormalizable QFTs
    1. 3.1 CP and flavor symmetry
    2. 3.2 The 3-loop diagrams
  4. 4 Implications
    1. 4.1 Consistency conditions
    2. 4.2 (In)Stability of UV solutions of the strong CP problem
  5. 5 Outlook
  6. A CP-odd flavor-invariants

##  1 Motivations

Relativistic four-dimensional quantum field theories feature several
renormalizable operators: Yukawa, gauge, scalar self-interactions and the
topological terms. The latter stand out of this list because they are total
derivatives. At the classical level they have no impact since they do not
alter the equations of motion; at the quantum mechanical level they do not
introduce Feynman vertices. Nevertheless, the topological angles $\theta$ can
affect physical observables if non-perturbative effects are taken into account
[1, 2, 3, 4] or if topological defects are present [5]. In a semi-classical
analysis one studies the quantum fluctuations around a certain background
field configuration, and $\theta$ shows up in the path integral via a factor
$e^{i\theta\nu}$ whenever the background gauge fields are associated to a non-
trivial topological charge $\nu$.

Because $\theta$ can appear in observables, it is only natural to wonder about
its renormalization group evolution. Since $\theta$ does not parametrize
vertices for the quantum fluctuations, it is clear that the perturbative
correlators and beta functions cannot depend on it. At most, perturbatively
$\theta$ represents a _counterterm_ necessary to reabsorb (finite and
divergent) corrections to CP-violating diagrams with external background gauge
fields. In other words, in perturbation theory $\theta$ can only get
additively renormalized because of contributions induced by the other
couplings.

That $\theta$ renormalizes is out of the question. We can distinguish between
_finite_ and _infinite_ renormalization effects. Finite (threshold)
corrections to $\theta$ are very common. In generic theories they occur at
tree-level, when crossing a CP-violating fermion mass threshold, and even more
often at loop level. In the case of the QCD $\theta$ angle, for example, ref.
[9] found that the leading order finite correction is a 3-loop effect when
matching to the effective field theory below the $W^{\pm}$ mass. Infinite
corrections are more rare. A necessary condition for these contributions to
actually occur in a mass independent renormalization scheme is that the
quantum field theory under consideration possesses polynomial CP-odd flavor-
invariant combinations of the couplings (other than $\theta$) that can
contribute to

$\displaystyle\beta_{\theta}=\mu\frac{d\theta}{d\mu}.$ (1.1)

In the Standard Model the first such combination appears at a very high power
in the Yukawa couplings [6, 7] indicating that, if present, the UV divergence
that renormalizes $\theta$ should occur at a prohibitive perturbative order.
In view of this one may ask if such divergence exists at all, and whether this
high perturbative order is a general feature of renormalizable quantum field
theories. To better assess these questions it would be useful to find “toy”
field theories in which $\theta$ develops a beta function at a sufficiently
low order as to allow an explicit computation, which may provide an indirect
confirmation of our expectations in the Standard Model. It is not difficult to
find non-renormalizable field theories that induce divergent corrections to
$\theta$. For example, adding $cg^{2}|H|^{2}G\widetilde{G}/\Lambda^{2}$ to the
Standard Model Lagrangian gives the minimal subtraction result
$\beta_{\theta}=-4cm_{H}^{2}/\Lambda^{2}$, see [8]. However, the latter effect
is power-law suppressed and is therefore practically irrelevant in the
presence of a large gap between the IR scales and the UV cutoff. In addition,
we will see there are a few non-trivial challenges that a calculation of
divergent contributions to $\theta$ faces within renormalizable theories,
including the fact that the regularization scheme adopted must be able to
consistently deal with the so-called $\gamma_{5}$ problem. In the non-
renormalizable theory we mentioned above those challenges are not encountered
because the operator $G\widetilde{G}$ was already present on the outset. The
questions we are interested in therefore better be addressed within
_renormalizable_ field theories. Yet, surprisingly enough, to the best of our
knowledge nobody has ever found divergent corrections to $\theta$ in that
context.

While no concrete renormalizable example where $\theta$ gets infinitely
renormalized is on the market, there exists well-known instances in which one
can rigorously prove that $\theta$ _is not_ infinitely renormalized at any
order. An example is pure Yang-Mills, where this property follows trivially
because that theory does not satisfy the necessary condition stated above:
there are no flavor-invariant CP-odd phases other than $\theta$, and hence
there is nothing that can contribute perturbatively to $\beta_{\theta}$. The
same holds for QCD with massive fermions, since once the phases in the quark
mass matrix are removed via anomalous chiral rotations of the fermions, CP-
violation is entirely encoded in $\theta$. Another popular instance is
provided by supersymmetric gauge theories, where the exact one-loop running of
the holomorphic gauge coupling reveals that the theta angle does not run. The
reason here is again the same: there is no available CP-odd (holomorphic)
combination of the other marginal couplings.

The main goals of this paper are to present concrete examples of four-
dimensional renormalizable field theories that can induce infinite corrections
to $\theta$, as well as to identify the leading order structure of the beta
function $\beta_{\theta}$ in any mass-independent renormalization scheme (see
Section 3); to discuss the subtleties encountered in a perturbative treatment
of $\theta$ as well as to show how to concretely approach the calculation of
$\beta_{\theta}$ in dimensional regularization (see Sections 2); and finally
to analyze the relevance of $\beta_{\theta}$ (Section 4). 111It is worth
dissipating a possible source of confusion right away. It is well-known that
physical observables must depend on a flavor-invariant combination
$\bar{\theta}$ of the topological angle and the other couplings of the theory
(e.g. the quark masses in QCD or the Yukawa couplings in the Standard Model).
In a generic field basis the beta function of $\bar{\theta}$ receives
contributions from corrections to both $\theta$, which represent the main
subject of this paper, and the other couplings as well (these latter
corrections are those estimated in [6], strictly speaking). Throughout the
paper we will be mostly concerned with the basis-dependent parameter $\theta$,
the coefficient of the topological term. The physics of the QCD parameter
$\bar{\theta}$ will be discussed in Section 4.2.

Studying the RG evolution of $\theta$ in a general renormalizable theory is
not only of academic interest. There are technical as well as potentially
phenomenological reasons for doing it. At a genuinely phenomenological level,
studying the beta function of $\theta$ in general theories might potentially
shed light on the absence of CP-violation in QCD, or more precisely on the
necessary properties that its UV completion must satisfy to account for
experimental observations, which in fact was the original motivation of our
precursors [10, 6]. Yet another reason is that we do not know what theory will
eventually be found to complete the SM at shorter distances. It is perhaps not
completely unconceivable that in such a UV-completion topological angles occur
in some particular observable, say because of the presence of magnetic
monopoles or because small-instanton effects cannot be ignored. In these
situations a renormalization group evolution of $\theta$ may become
phenomenologically relevant. At a more technical level, $\beta_{\theta}$ is an
obvious target for explicit calculations. The beta functions of the “ordinary”
couplings of general renormalizable field theories have been explicitly
calculated up to 2-loop order (see [11, 12, 13] and more recent updates). Yet,
strictly speaking, this program cannot be viewed as fully complete until
$\beta_{\theta}$ is also computed. On a completely different note, the
peculiar nature of $\theta$ encodes important information about the
renormalization properties of the theory. For example we will see that the
independence of perturbative correlators on $\theta$ translates into a
constraint on the anomalous dimension of the CP-odd operators of the theory.
These and other observations may turn out to be useful in concrete
calculations.

##  2 $\theta$ in perturbation theory

In a semi-classical approach to quantum field theory, the bare fields
$\phi_{0}$ are split into a classical finite-action background $\phi_{0c}$
overlapping with the vacuum, plus a quantum fluctuation $\delta\phi_{0}$ that
vanishes sufficiently fast at the boundary. The path integral is defined to
include a sum over all inequivalent background configurations (e.g. collective
coordinates) along with the functional integration over the fluctuations
around each background. Singling out the topological term from the total
gauge-fixed action, 222Throughout the paper we assume the gauge-fixing
preserves the background gauge invariance. which we schematically write as
$S_{\rm tot}=S+({g_{0}^{2}\theta_{0}})/({32\pi^{2}})\int
G_{0}\widetilde{G}_{0}$, this recipe produces the following generating
functional

$\displaystyle{\cal Z}[J_{0}]$ $\displaystyle=$
$\displaystyle\sum_{\phi_{0c}}~{}e^{i\frac{g_{0}^{2}\theta_{0}}{32\pi^{2}}\int
G_{0c}\widetilde{G}_{0c}+i\int J_{0}\phi_{0c}}\int{\cal
D}\delta\phi_{0}~{}e^{iS[\phi_{0c}+\delta\phi_{0}]+i\int J_{0}\delta\phi_{0}}$
$\displaystyle=$
$\displaystyle\sum_{\phi_{0c}}~{}e^{i\frac{g_{0}^{2}\theta_{0}}{32\pi^{2}}\int
G_{0c}\widetilde{G}_{0c}}~{}\widehat{\cal Z}[J_{0},\phi_{0c}].$

The topological term is special because it is a total derivative. By
construction the quantum fluctuations vanish at the boundary, so that any
fluctuation-dependent contribution to such a term vanishes and $\int
G_{0}\widetilde{G}_{0}=\int G_{0c}\widetilde{G}_{0c}$ reduces to an integral
of the sole external background fields, which we can take outside the
functional integral in (2): the angles $\theta$ do not appear in any
interaction of the quantum fluctuations but can act as counterterms in
computations with external background fields.

So far our discussion has been rather general. Yet, an actual evaluation of
(2) is necessarily regularization-scheme dependent. In the following we will
specialize on dimensional regularization (Dim-Reg), in which space-time is
continued to $d$ dimensions with coordinates
$x^{\mu}=\left\\{x^{\bar{\mu}},x^{\hat{\mu}}\right\\}$, where
$\bar{\mu},\bar{\nu},\cdots=0,1,2,3$ and $\hat{\mu},\hat{\nu},\cdots$ denote
the $(d-4)$-dimensional indices.

In Dim-Reg the very definition of topological term forces us to define the
Levi-Civita tensor and deal with the famous $\gamma_{5}$ problem. So far the
only known consistent prescription is the ’t Hooft-Veltman-Breitenlohner-
Maison scheme [14, 15], where the Levi-Civita tensor is a formal object
$\epsilon^{\bar{\mu}\bar{\nu}\bar{\alpha}\bar{\beta}}$ that carries only
4-dimensional indices. In other words, the $(d-4)$-dimensional indices
$\hat{\mu},\hat{\nu},\cdots$ of an arbitrary vector do not contribute when
contracted with this tensor. An important implication is that

$\displaystyle
G_{0}\widetilde{G}_{0}\equiv\frac{1}{2}G_{0}^{\bar{\mu}\bar{\nu}}G_{0}^{\bar{\alpha}\bar{\beta}}\epsilon_{\bar{\mu}\bar{\nu}\bar{\alpha}\bar{\beta}}=\partial_{\bar{\mu}}K_{0}^{\bar{\mu}}$
(2.2)

is $4$-dimensional divergence of a ($x^{\mu}$-dependent) vector. Hence the
regularized quantity $\int d^{d}x~{}G_{0c}\widetilde{G}_{0c}$ contains a non-
trivial residual $(d-4)$-dimensional integral and is not a topological term in
$d$ dimensions.

The $d$-dimensional continuation of (2) formally represents a set of
regularized Green’s functions. Such a path integral violates two of the
familiar properties of the topological angle, namely its periodicity in $2\pi$
and its role as compensator (spurion) of the abelian axial symmetry. The
technical reason for the first loss boils down to the fact that, as a
consequence of (2.2), the bare angle $\theta_{0}$ is not the coefficient of a
topological operator in the regularized theory. The second loss occurs because
anomalies are $d$-dependent; as a result, in $d$-dimensions a shift of the
coefficient of $G\widetilde{G}$ does not fully compensate an axial rotation.
An intuitive way of arriving to the same conclusions is provided by
dimensional analysis: the engineering dimension of the bare coupling in Dim-
Reg is $[\theta_{0}]=d-4$, and it is therefore impossible for $\theta_{0}$ to
be periodic in $2\pi$ or even to shift via the dimensionless parameter of the
axial transformation while retaining its $\mu$-independence in $d$-dimensions.

To recover the topological nature of the theta angle, as well as its role as a
compensator for abelian axial transformations, one has to derive the
renormalized 4-dimensional version of the path integral. In general this
procedure requires a renormalization of theta as well. Renormalization renders
$\theta$ a genuinely (4-dimensional) topological term and the background-
dependence in the 4-dimensional limit of the path integral (2) reduces to a
dependence on the topological index $\nu$. The renormalized coupling $\theta$
is periodic in $2\pi$ and transforms via a shift under abelian axial
rotations. This ensures in particular that physical amplitudes are invariant
under unitary field re-definitions.

###  2.1 $\beta_{\theta}$ from (extra)-ordinary diagrams

For completeness we recall the standard prescription to extract the beta
function within the Minimal Subtraction scheme. The relation between the bare
couplings $\theta_{0}$, $\xi_{0i}$ (the latter symbol denotes all couplings
except $\theta_{0}$) and the renormalized couplings $\theta$ and $\xi_{i}$
read (in $d=4-\epsilon$ dimensions)

$\displaystyle\theta_{0}=\mu^{-\epsilon}[\theta+Z_{\theta}],$ (2.3)

and $\xi_{0i}=\mu^{\rho_{i}\epsilon}[\xi_{i}+Z_{\xi_{i}}]$. By definition
$Z_{\theta}=\sum_{n=1}^{\infty}\epsilon^{-n}Z_{\theta,n}(\xi_{i})$ contains no
finite term, and similarly for $Z_{\xi_{i}}$. The $4$-dimensional beta
functions $\beta_{\xi}\equiv\lim_{d\to 4}\mu{d\xi}/{d\mu}$ read
$\beta_{\theta}=\rho_{i}\xi_{i}{\partial
Z_{\theta,1}}/{\partial\xi_{i}}+Z_{\theta,1}$ and similarly
$\beta_{\xi_{i}}=\rho_{j}\xi_{j}{\partial
Z_{i,1}}/{\partial\xi_{j}}-\rho_{i}Z_{i,1}$. In the above $\partial
Z_{\theta}/\partial\theta=\partial Z_{\xi_{i}}/\partial\theta=0$ because
$\theta$ does not appear in Feynman diagrams. As customary for ordinary
couplings, also the beta function of $\theta$ is controlled by the simple pole
$Z_{\theta,1}/\epsilon$. Diagrammatically, the latter counterterm is defined
to subtract the divergent contributions to the connected, CP-odd _vertices_
with external background gauge fields in $\widehat{\cal
Z}[J_{0},\phi_{c}]$($\widehat{\cal Z}[J_{0},\phi_{0c}]$?), see (2):

$\displaystyle{\cal
Z}[J_{0},\phi_{c}]\supset-\frac{Z_{\theta,1}}{\epsilon}i\frac{g_{0}^{2}}{32\pi^{2}}\mu^{-\epsilon}\int
d^{d}x\,G_{c0}\widetilde{G}_{c0}.$ (2.4)

The divergent corrections to the external legs are removed via a
renormalization of the background field, $A^{\mu}_{0c}=(g/g_{0})A^{\mu}_{c}$,
so that $g_{0}^{2}G_{0c}\widetilde{G}_{0c}=g^{2}G_{c}\widetilde{G}_{c}$. The
divergence remaining in (2.4), if any, must be subtracted by $Z_{\theta,1}$.

A word regarding the actual computation of $Z_{\theta,1}$ is now needed. After
all, we have argued that the topological term does not represent an ordinary
vertex, so how can we now claim that combining the ordinary vertices of the
theory one can find divergent corrections to it? The key point is that (2.4),
in order to be non-vanishing, must be a functional of classical backgrounds
with _non-trivial_ asymptotic behavior. This implies the calculation of
$Z_{\theta,1}$ must be dealt with care. In particular, integration by parts
cannot be performed lightly, as opposed to what is customary done when dealing
with external sources for asymptotic states. 333If integration by parts is
performed neglecting boundary terms, the structure (2.4), say the part with
two external legs for definiteness, would be written in the familiar form
$\displaystyle\widehat{\cal Z}[J_{0},\phi_{0c}]\supset\int
d^{d}xA_{0c}^{\mu}(x)\int d^{d}yA_{0c}^{\nu}(y)~{}G_{\mu\nu}(x-y),$ (2.5) with
$G_{\mu\nu}(x-y)$ an ordinary Feynman diagram. This cannot contribute to the
beta function of $\theta$, since it is impossible for $G_{\mu\nu}(x-y)$ to
contain two derivatives contracted with a Levi-Civita tensor, as required to
reproduce the topological term in (2.4). Any CP-violating structure of the
type (2.5) identically vanishes and cannot describe (2.4). One way to proceed
is to extract the divergent piece in (2.4) directly in position space while
allowing an arbitrary background $A^{\mu}_{c}$ [16]. This is the approach
followed in [9] and [17] to compute the induced $\theta$ term in the Standard
Model below the weak scale.

An alternative way to calculate the divergent diagrams contributing to
$Z_{\theta,1}$ may be via the trick proposed in [18]. The idea is to promote
the CP-odd couplings to non-propagating fields, i.e. “axions”, in the
intermediate steps and then send them to constant values at the very end of
the computation. In this way $\int d^{d}x~{}\theta\,G\widetilde{G}$ would
describe an ordinary vertex with $\theta(x)$ and a number of gluons, which is
non-vanishing as long as the external $\theta(x)$ carries a momentum. It is
now plausible that ordinary Feynman diagrams with external gluons and non-
dynamical axions contain divergent contributions that need to be subtracted by
$\int d^{d}x~{}\theta\,G\widetilde{G}$, very much like ordinary diagrams were
shown in [18] to be capable of generating finite threshold corrections to
$\theta$. Amusingly, though, now that the topological term behaves as an
ordinary vertex one may naively expect $\theta$ to be able to show up in
matrix elements as well as in the beta functions. It turns out however that
this cannot happen. One way to see it is that the renormalized S-matrix
amplitudes must be periodic functions of $\theta$. Yet, the new vertex
described by the topological term is measured by the strength $\theta
g^{2}/32\pi^{2}$. Hence the only way it can contribute to renormalized
amplitudes and beta functions is via powers of $\theta g^{2}/32\pi^{2}$.
However there is no way that a perturbative function of $\theta
g^{2}/32\pi^{2}$ and the other couplings be invariant under
$\theta\to\theta+2\pi$ unless the dependence on $\theta g^{2}/32\pi^{2}$ is
actually absent altogether. The situation is completely different when non-
perturbative effects are taken into account, of course, since in that case
inverse powers of the gauge coupling cannot be excluded a priori and a
dependence on $\theta g^{2}/32\pi^{2}$ can be turned into a periodic function
of the sole $\theta$.

We see that the divergent contributions to $\theta$ can be seen via non-
standard perturbative calculations, either by directly evaluating (2.4) in
position space [16] or via the method proposed in [18]. In Section 2.2 we will
present a third alternative approach to calculate directly $\beta_{\theta}$,
based on more conventional perturbative methods.

Yet, whatever method is adopted one has to pay particular attention to how
chirality is implemented. The divergent contribution we are interested in is
proportional to the Levi-Civita tensor and must arise from a fermion trace
involving the $\gamma_{5}$ matrix. A consistent treatment of the latter has to
be implemented, and at present in Dim-Reg the unique option seems to be given
by the ’t Hooft-Veltman-Breitenlohner-Maison scheme. Unfortunately, using this
scheme might complicate the calculation a bit, due to the non-standard anti-
commutation properties that $\gamma_{5}$ satisfies and the necessity of
introducing ad-hoc symmetry-restoring counterterms in order to preserve non-
anomalous (global and gauge) chiral symmetries. Fortunately the results of
this paper are completely general and hold independently of the approach
followed.

###  2.2 Extracting $\beta_{\theta}$ from operator mixing

The insertion of composite operators in correlator functions can be
systematically described by introducing in the bare Lagrangian appropriate
spacetime-dependent sources for them (and adding the appropriate
counterterms). After having integrated out the dynamical fields one is left
with a generating functional for the time-ordered, connected correlation
functions of the renormalized operators. The local renormalization group [19,
20, 21, 22] is a very powerful incarnation of this general prescription. In
that approach the Lagrangian is expanded in a complete basis of operators
${\cal O}$ and all couplings of the theory become functions of spacetime,
including the metric that sources the energy momentum tensor. A local version
of the renormalization group equation for the operators ${\cal O}$ can be
derived. Once all couplings are sent to their constant ($x^{\mu}$ independent
but $\mu$-dependent) background values, this reads [23]

$\displaystyle\mu\frac{d}{d\mu}\left(\begin{matrix}\partial_{\mu}J_{\dot{A}}^{\mu}\\\
{\cal
O}_{\bar{I}}\end{matrix}\right)=-\left(\begin{matrix}\gamma_{\dot{A}\dot{B}}&0\\\
\gamma_{\bar{I}\dot{B}}&\frac{\partial}{\partial\xi_{\bar{I}}}\beta_{{\bar{J}}}\end{matrix}\right)\left(\begin{matrix}\partial_{\mu}J_{\dot{B}}^{\mu}\\\
{\cal O}_{\bar{J}}\end{matrix}\right).$ (2.6)

Here $J_{\dot{A}}^{\mu}$ are the (conserved or anomalous [24]) currents of the
global symmetries of the theory, ${\cal O}_{\bar{I}}$ denote the marginal
interactions, $\xi_{\bar{I}}$ the associated couplings, and
$\beta_{{\bar{I}}}=\mu d\xi_{\bar{I}}/d\mu$. We considered a theory without
mass operators for simplicity, but the introduction of mass terms is
straightforward and does not affect our discussion. 444A complete basis of
${\cal O}_{\bar{I}}$’s includes two sets of marginal operators: ${\cal
O}_{\bar{I}}=\left\\{E_{\bar{I}^{\prime\prime}},O_{\bar{I}^{\prime}}\right\\}$.
The interactions $O_{\bar{I}^{\prime}}$ are those that define the bare action.
These are multiplied by sources $\xi_{\bar{I}^{\prime}}$ whose background
values represent the ordinary couplings of the theory. The
$E_{\bar{I}^{\prime\prime}}$’s denote instead redundant marginal interactions.
A complete basis of $E_{\bar{I}^{\prime\prime}}$ can be taken to be the set of
operators that vanish via the equations of motion. They are not associated to
any actual coupling of the theory. However they must be included in the
functional integral multiplied by spacetime-dependent sources
$\xi_{\bar{I}^{\prime\prime}}$ in order to obtain a closed set of composite
operators under renormalization. The background value of such sources
vanishes. Because the $\xi_{\bar{I}^{\prime\prime}}$’s are not actual
couplings, their (background-value) beta functions are proportional to the
$\xi_{\bar{I}^{\prime\prime}}$’s themselves times functions of the true
couplings $\xi_{\bar{I}^{\prime}}$, i.e. the operators
$E_{\bar{I}^{\prime\prime}}$ are multiplicatively renormalized. With this
observation we see that (2.6) describes a familiar pattern of RG-mixing: the
anomalous dimension matrix for
$\left\\{E_{\bar{I}^{\prime\prime}},O_{\bar{I}^{\prime}}\right\\}$ has a lower
triangular form in which the redundant operators renormalize among each other
whereas the $O_{\bar{I}^{\prime}}$ renormalize via a mixture of themselves,
$\partial_{\mu}J_{\dot{A}}^{\mu}$, and $E_{\bar{I}^{\prime\prime}}$.

Let us begin by discussing the consequence of (2.6) on the CP-odd sector of
QCD. The marginal CP-odd operators are the divergence of the singlet axial
current, $\partial_{\mu}J_{5}^{\mu}$, and the topological term
$G\widetilde{G}$ (modulo operators that vanish via the equations of motion).
By CP-invariance the anomalous dimension matrix contains a block diagonal 2 by
2 subgroup involving these two operators only (the operators that vanish on-
shell do not affect the following discussion). From (2.6) one confirms that
this takes the form discussed in [25]. The vanishing
$\gamma_{\dot{A}\bar{J}}=0$ entry is understood as a result of the fact that
in Dim-Reg the current $J_{5}^{\mu}$ renormalizes multiplicatively, being the
unique gauge-invariant axial current of the theory, and the same is true for
its derivative. Eq. (2.6) reveals that the $G\widetilde{G}$-$G\widetilde{G}$
element of the anomalous dimension, often denoted by $\gamma_{G\widetilde{G}}$
in the literature, is given by
$\gamma_{G\widetilde{G}}={\partial}\beta_{{G\widetilde{G}}}/{\partial\xi_{G\widetilde{G}}}$,
with $\xi_{G\widetilde{G}}$ being the renormalized coupling

$\displaystyle\xi_{G\widetilde{G}}\equiv\frac{\theta g^{2}}{32\pi^{2}}.$ (2.7)

Crucially, $\theta$ cannot appear in any perturbative calculation, so all our
expressions are understood as being evaluated at
$\xi_{G\widetilde{G}}=0=\theta$. As expected, in the external source formalism
the latter can only occur with derivatives [26]. Because $\mu d({\theta
g^{2}})/d\mu=\beta_{\theta}g^{2}+\theta\beta_{g^{2}}$ we find

$\displaystyle\gamma_{G\widetilde{G}}=\left.\frac{\partial}{\partial\xi_{G\widetilde{G}}}\beta_{{G\widetilde{G}}}\right|_{\xi_{G\widetilde{G}}=0}=\left.\frac{\partial}{\partial\xi_{G\widetilde{G}}}\left[\beta_{\theta}\frac{g^{2}}{32\pi^{2}}+\xi_{G\widetilde{G}}\frac{\beta_{g^{2}}}{g^{2}}\right]\right|_{\xi_{G\widetilde{G}}=0}=\frac{\beta_{g^{2}}}{g^{2}}.$
(2.8)

In the last equality it is crucial that $\beta_{\theta}$ nor $\beta_{g^{2}}$
depend on $\theta$, or equivalently $\xi_{G\widetilde{G}}$. The resulting
relation between $\gamma_{G\widetilde{G}}$ and the beta function of the gauge
coupling is consistently observed in explicit calculations [27]. 555We use a
different convention for the anomalous dimension than the author of [27]. Note
that $\beta_{\theta}$ disappears from this relation and (2.8) would still hold
even if it was non-trivial. Yet, we know that $\beta_{\theta}=0$ in QCD
because of the argument given in the introduction: by CP-invariance of the
theory the only quantity that could appear in $\beta_{\theta}$ is $\theta$
itself, but this can never happen in perturbation theory.

Suppose now we extend QCD introducing in the Lagrangian new CP-violating
dimension-4 operators $O_{\bar{I}}$ with $\bar{I}\neq G\widetilde{G}$, e.g.
Yukawa couplings. A priori these may mix with the topological term as well as
the derivative of the axial current. Yet, according to (2.6) the anomalous
dimension matrix has a lower triangular form. This is because by dimensional
analysis the renormalized axial and Chern-Simons currents cannot contain a
component of the bare dimension-4 interactions $O_{\bar{I}0}$, they can only
mix among each other following the pattern described above. On the other hand,
nothing forbids the renormalized $O_{\bar{I}}$ to contain a linear combination
of $\partial_{\mu}J_{5\,0}^{\mu},G_{0}\widetilde{G}_{0}$. The lower-triangular
structure can also be understood as a consequence of the independence of the
anomalous dimensions and beta functions on $\theta$, i.e.
$\partial\beta_{\bar{I}\neq G\widetilde{G}}/\partial\xi_{G\widetilde{G}}=0$
[24].

Interestingly, eq. (2.6) demonstrates that (2.8) remains valid even in the
presence of the new interactions $O_{\bar{I}}$ (${\bar{I}}\neq
G\widetilde{G}$). More importantly for us, the new CP-violating couplings make
it possible for $\theta$ to get additively renormalized, and (2.6) turns out
to contain important information on $\beta_{\theta}$. The component of $-\mu
dO_{\bar{I}}/d\mu$ proportional to $G\widetilde{G}$ is (for $\bar{I}\neq
G\widetilde{G}$)

$\displaystyle\gamma_{{\bar{I}}G}=\left.\frac{\partial}{\partial\xi_{\bar{I}}}\beta_{{G\widetilde{G}}}\right|_{\xi_{G\widetilde{G}}=0}=\left.\frac{\partial}{\partial\xi_{\bar{I}}}\left[\beta_{\theta}\frac{g^{2}}{32\pi^{2}}+\xi_{G\widetilde{G}}\frac{\beta_{g^{2}}}{g^{2}}\right]\right|_{\xi_{G\widetilde{G}}=0}=\frac{g^{2}}{32\pi^{2}}\frac{\partial}{\partial\xi_{\bar{I}}}\beta_{\theta}.$
(2.9)

Eq. (2.9) may be interpreted as an indirect procedure for deriving
$\beta_{\theta}$. The latter may indeed be extracted by integrating the off-
diagonal element $\gamma_{\bar{I}G}$ of the anomalous dimension matrix of the
CP-violating operators, roughly measured by the amount of
$G_{0}\widetilde{G}_{0}$ contained in $O_{\bar{I}}$, with respect to the
couplings $\xi_{\bar{I}}$ that contribute additively to $\beta_{\theta}$.
Interestingly, this procedure does not require the background field method nor
promoting the CP-odd phases to spurion fields. It can be carried out using
ordinary perturbation theory because the anomalous dimension matrix is to be
calculated assuming a non-vanishing momentum is flowing into the relevant
operators, such that even $G\widetilde{G}$ describes an ordinary vertex.

A generalization of eq. (2.9) to arbitrary renormalizable theories is
straightforward.

##  3 $\beta_{\theta}$ in renormalizable QFTs

In the previous section we have shown how to extract the beta function of the
topological angle. In this section we are interested in identifying which
renormalizable theories have enough CP-violating phases to allow, at least in
principle, the presence of divergent contributions to $\theta$. More
precisely, we will identify the explicit form of the leading non-trivial order
beta function $\beta_{\theta}$ as well as the Feynman diagrams responsible for
generating it.

The most general ($d=4$) renormalizable gauge theory can be compactly written
in terms of Weyl fermions $\psi_{i}$ and real scalars $\phi_{a}$ as

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle-\frac{1}{4g^{2}_{AB}}F^{A}_{\mu\nu}F^{B\mu\nu}+\frac{1}{2}(D_{\mu}\phi)_{a}(D_{\mu}\phi)_{a}+\psi^{\dagger}_{i}i\bar{\sigma}^{\mu}({D}_{\mu}\psi)_{i}$
$\displaystyle-$
$\displaystyle\left(\frac{1}{2}Y_{a\,ij}\psi_{i}\psi_{j}\phi_{a}+{\rm
hc}\right)-\frac{\lambda_{abcd}}{4!}\phi_{a}\phi_{b}\phi_{c}\phi_{d}+\frac{\theta^{AB}}{64\pi^{2}}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}^{A}F_{\rho\sigma}^{B}$
$\displaystyle+$ $\displaystyle({\rm relevant~{}couplings})+({\rm
gauge~{}fixing})+({\rm ghosts}),$

where
$(D_{\mu}\psi)_{i}=\partial_{\mu}\psi_{i}-i{A}^{A}_{\mu}T^{A}_{ij}\psi_{j}$
and
$(D_{\mu}\phi)_{a}=\partial_{\mu}\phi_{a}-i{A}^{A}_{\mu}S^{A}_{ab}\phi_{b}$.
The gauge generators $T^{A}$ are hermitian whereas $S^{A}$ are purely
imaginary hermitian, and hence anti-symmetric. The fermions and scalars are in
general in a reducible representation of the gauge group, and the indices
$i,j,\cdots$ (ranging from $1$ to some integer $N_{\psi}$) and $a,b,\cdots$
(ranging from $1$ to $N_{\phi}$) include both gauge and flavor components. The
coupling $\lambda$ is fully symmetric, and $Y$ is symmetric in the fermionic
indices. The gauge symmetry is an arbitrary product of abelian factors and
simple groups ${\cal G}_{\rm gauge}=\Pi_{\cal G}{\cal G}$. The indices
$A,B,\cdots$ run over the adjoint representation of ${\cal G}_{\rm gauge}$ and
the (real) gauge coupling $g^{2}_{AB}$ is to be interpreted as the direct sum
of identities in the non-abelian part of that space plus a symmetric part for
the abelian factors. Note that our normalization of the gauge fields is non-
canonical, so that the gauge boson propagators are proportional to
$g_{AB}^{2}$.

Gauge invariance restricts the form of some of the couplings in (3). Most
importantly for us, the topological term must be proportional to the identity
$\delta^{AB}_{\cal G}$ in any non-abelian components ${\cal G}$ and symmetric,
but potentially with off-diagonal entries, in the abelian factors:

$\displaystyle\theta^{AB}=\sum_{{\cal G}={\rm non-Ab.}}\theta_{\cal
G}\delta^{AB}_{\cal G}+\sum_{{\cal G}_{1},{\cal G}_{2}={\rm
Ab.}}\theta_{{{\cal G}_{1},{\cal G}_{2}}}^{AB}.$ (3.2)

Another implication of gauge invariance is

$\displaystyle S^{A}_{ab}Y_{b}=Y_{a}T^{A}+(T^{A})^{*}Y_{a}.$ (3.3)

We will make use of this relation extensively throughout the text. There is
also a constraint on $\lambda$, though such a relation will not be relevant
here. Note that none of these relations depend on the space-time dimension and
hence can be assumed to hold in Dim-Reg as well.

###  3.1 CP and flavor symmetry

We want to identify which combinations of the couplings can appear in
$\beta_{\theta}$. In Dim-Reg the renormalization group evolution of $\theta$
is necessarily controlled by the marginal couplings of the theory. Classically
relevant interactions are therefore not important in the present discussion
and have not been explicitated in (3). To proceed we first have to familiarize
with the approximate symmetries of our theory.

In the absence of interactions Eq. (3) enjoys a large global symmetry that
includes a flavor symmetry that rotates the matter fields, one that rotates
vectors, and CP:

$\displaystyle{\cal G}_{\rm flav}\times{\cal G}_{\rm glob}\times CP.$ (3.4)

The flavor symmetry ${\cal G}_{\rm flav}\equiv U(N_{\psi})\times O(N_{\phi})$
rotates all fermions among each other and acts similarly on scalars, the group
${\cal G}_{\rm glob}$ rotates the vectors leaving $g_{AB}^{2}$ in (3)
invariant and finally CP acts as usual up to unitary rotations. The combined
action on the fields reads:

$\displaystyle\psi_{i}(x)\to U_{ij}\epsilon\psi^{\dagger}_{j}({\cal
P}x),~{}~{}~{}~{}\phi_{a}(x)\to O_{ab}\phi_{b}({\cal
P}x),~{}~{}~{}~{}A^{A}_{\mu}(x)\to R^{AB}{\cal P}_{\mu}^{\nu}A^{B}_{\nu}({\cal
P}x)$ (3.5)

where $U,O,R$ are matrices respectively of $U(N_{\psi})$, $O(N_{\phi})$,
${\cal G}_{\rm glob}$ and we defined ${\cal P}_{\mu}^{\nu}x_{\nu}=x^{\mu}$.
The symmetry (3.4) is explicitly broken by the gauge generators, the Yukawa
and scalar couplings as well as anomalies. Yet, it can be formally restored by
promoting $T^{A}$, $S^{A}$, $Y_{a}$, $\lambda,\theta$ to spurions with
transformations designed to exactly compensate (3.5) so that the theory is
manifestly invariant under the full group (3.4). 666In Dim-Reg the addition of
counterterms is generically necessary to render the theory formally invariant
under the axial part of $U(N_{\psi})$. We assume this is done order by order
in perturbation theory. Explicitly, the transformations of the spurions
$T^{A}$, $S^{A}$, $Y_{a}$, $\lambda$ under ${\cal G}_{\rm flav}\times{\cal
G}_{\rm glob}\times CP$ are given by

$\displaystyle T^{A}_{ij}$ $\displaystyle\to$
$\displaystyle-R^{AB}U_{im}U^{*}_{jn}[T^{B}]^{*}_{mn}$ (3.6) $\displaystyle
S^{A}_{ij}$ $\displaystyle\to$
$\displaystyle-R^{AB}O_{im}O_{jn}[S^{B}]^{*}_{mn}$ $\displaystyle Y_{a\,ij}$
$\displaystyle\to$ $\displaystyle U^{*}_{im}U^{*}_{jn}O_{ab}[Y_{b\,mn}]^{*}$
$\displaystyle\lambda_{abcd}$ $\displaystyle\to$ $\displaystyle
O_{am}O_{bn}O_{co}O_{dp}\lambda_{mnop}.$

The renormalized coupling $\theta^{AB}$ should also be interpreted as a
spurion. It is a complete singlet of $O(N_{\phi})$, a 2-index symmetric tensor
of ${\cal G}_{\rm glob}$, and is also $U(N_{\psi})$-invariant except for its
anomalous subgroup. Denoting with $T_{r}^{A}$ the generators of each
irreducible representation $r$ of the gauge group modulo flavor degeneracies,
and by $U_{r}$ the flavor rotation among the fermions in $r$, the spurious
${\cal G}_{\rm flav}\times{\cal G}_{\rm glob}\times CP$ transformation of
$\theta$ explicitly reads

$\displaystyle\theta^{AB}\to-R^{AM}R^{BN}\left\\{\theta^{MN}-2\sum_{r}{\rm
Arg}~{}{\rm Det}[U_{r}]~{}{\rm Tr}[T_{r}^{M}T_{r}^{N}]\right\\}.$ (3.7)

Here the overall minus is necessary to compensate the P-odd nature of
$\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}^{A}F_{\rho\sigma}^{B}$.

The beta function $\mu d\theta^{AB}/d\mu$ is not affected by the anomalous
shift of the topological term and therefore is a complete singlet of the
flavor symmetry ${\cal G}_{\rm flav}$ and CP-odd in the sense that

$\displaystyle\mu\frac{d}{d\mu}\theta^{AB}\to-R^{AM}R^{BN}\mu\frac{d}{d\mu}\theta^{MN}.$
(3.8)

The beta function is a linear combination of functions $I^{AB}$ of the
spurions (3.6) multiplied by real numerical coefficients, since its
calculation involves no branch cuts. The quantities $I^{AB}$ are obtained by
contracting the indices of the spurions with the invariant tensors
$\delta_{ij},\delta_{ab},g^{2}_{AB}$ that define the kinetic terms. In the
language of Feynman diagram, this is just a consequence of the fact that any
diagram contributing to the beta function is obtained by contracting vertices
with propagators. In order to appear in $\beta_{\theta}$ the tensors should
transform precisely as in (3.8). Yet, from (3.6) follows that under the full
symmetry (3.4) any 2-index tensor function of (3.6) transforms as

$\displaystyle I^{AB}(T,S,Y,\lambda)\to
R^{AM}R^{BN}I^{MN}(T^{*},S^{*},Y^{*},\lambda^{*})$ (3.9)

The minus signs of (3.6) cancel out because ${\cal G}_{\rm glob}$-covariance
forces $I^{AB}$ to be built out of an even number of (scalar plus fermion)
gauge generators. Now, to reproduce the transformation of $\mu
d\theta^{AB}/d\mu$ the tensors must satisfy
$I^{AB}(T^{*},S^{*},Y^{*},\lambda^{*})=-I^{AB}(T,S,Y,\lambda)$, and of course
be real. We thus conclude that it is the imaginary part of the invariant that
has the correct CP-odd property, namely 777The relations
$I^{AB}(T^{*},S^{*},Y^{*},\lambda^{*})=[I^{AB}(T,S,Y,\lambda)]^{*}=-I^{AB}(T,S,Y,\lambda)$,
the first equality following from the fact that all coefficients are real and
the second from the requirement that (3.9) reproduces (3.8), are equivalent to
${\rm Re}[I^{AB}(T^{*},S^{*},Y^{*},\lambda^{*})]={\rm
Re}[I^{AB}(T,S,Y,\lambda)]=0$ and ${\rm
Im}[I^{AB}(T^{*},S^{*},Y^{*},\lambda^{*})]=-{\rm Im}[I^{AB}(T,S,Y,\lambda)]$.

$\displaystyle\mu\frac{d}{d\mu}\theta^{AB}=\sum_{\alpha}c_{\alpha}~{}{\rm
Im}\left[I_{\alpha}^{AB}\right],$ (3.10)

where $c_{\alpha}$ are real numbers and $\alpha$ is some label. In other
words, the CP-odd invariants that define $\beta_{\theta}$ are the _imaginary_
parts of the ${\cal G}_{\rm flav}$-singlet, 2-index tensors of ${\cal G}_{\rm
glob}$. A theory that does not possess any such quantity cannot renormalize
$\theta$. This is what happens in Yang-Mills theories as well as QCD. In the
next subsection we will show the explicit form of the leading order
contribution to $\beta_{\theta}$ in arbitrary renormalizable theories of the
form (3).

###  3.2 The 3-loop diagrams

Following the method described above we identified all structures that can
potentially contribute to $\beta_{\theta}$ at the first few perturbative
orders. Our analysis is summarized in Appendix A. We find that there are no
1-loop-sized 2-index tensors that are CP-odd, and therefore that the 1-loop
beta function must vanish. The first non-trivial contribution to
$\beta_{\theta}$ potentially arises at 2-loops and is controlled by a unique
structure

$\displaystyle\mu\frac{d}{d\mu}\theta^{AB}=c\frac{\hbar^{2}}{(16\pi^{2})^{2}}{\rm
Im}\left[I^{AB}_{(2)}\right]+{\cal O}(\hbar^{3}),$ (3.11)

where $c$ is an ordinary number expected to be of order unity and

$\displaystyle\begin{aligned} I_{(2)}^{AB}&={\rm
Tr}\left\\{(Y_{a}^{*}[T^{A}]^{*}Y_{b}Y_{a}^{*}Y_{b}-Y^{*}_{a}Y_{b}Y^{*}_{a}[T^{A}]^{*}Y_{b})T^{B}\right\\}\\\
&={\rm
Tr}\left\\{(Y^{*}_{a}Y_{c}Y_{b}^{*}Y_{c}-Y^{*}_{c}Y_{a}Y_{c}^{*}Y_{b})T^{B}\right\\}S^{A}_{ab}\\\
&=\frac{1}{2}{\rm
Tr}\left\\{Y^{*}_{a}Y_{c}Y_{b}^{*}Y_{c}-Y^{*}_{c}Y_{a}Y_{c}^{*}Y_{b}\right\\}\,(S^{A}S^{B})_{ab}\end{aligned}$
(3.12)

is the only invariant with non-vanishing CP-odd component (i.e. an imaginary
part) at ${\cal O}(\hbar^{2})$. Note that symmetry under the exchange
$A\leftrightarrow B$ in the first and third lines of (3.12) is a consequence
of cyclicity of the trace as well as symmetry of the Yukawa under the exchange
of the fermionic indices and hermiticity of the gauge generators, whereas in
the second line of (3.12) also (3.3) is needed.

The invariant $I^{(2)}_{AB}$ has been written in three different forms
employing the identity (3.3). These expressions provide complementary
information about the properties that a theory must possess in order to
renormalize $\theta^{AB}$ at this order. For instance, from the second and
third lines in (3.12) it is evident that there would be no 2-loop beta
function if the scalars were gauge-singlets, i.e. if $S^{A}=0$. To show this
using the expression in the first line is less immediate: one has to use (3.3)
with $S^{A}=0$ to prove that $Y_{a}^{*}Y_{b}T^{A}=T^{A}Y_{a}^{*}Y_{b}$, from
which consistently follows that the first line of (3.12) vanishes. Actually, a
more careful inspection reveals that _the scalar fields have to belong to at
least two different representations of $\mathcal{G}$_. To prove this we
distinguish between non-abelian and abelian gauge groups. In the case the
indices $A,B$ are associated to a non-abelian gauge group, from (3.2) we know
that the relevant part of the beta function is the one proportional to
$\delta^{\cal G}_{AB}$. If all the scalars belonged to the same
representation, then contracting the latter with the expression in the third
line of (3.12) would give a Casimir times the identity in the scalar index
space. This implies that the invariant would vanish as ${\rm
Tr}\left\\{Y^{*}_{a}Y_{c}Y_{a}^{*}Y_{c}-Y^{*}_{c}Y_{a}Y_{c}^{*}Y_{a}\right\\}=0$.
In the case $A,B$ refer to abelian gauge groups the generators can all be
taken diagonal, i.e. $S^{A}_{ab}=q^{A}_{a}\delta_{ab}$ and so
$(S^{A}S^{B})_{ab}=q^{A}_{a}q^{B}_{a}\delta_{ab}$. As before, we see that when
the charges $q_{a}^{A},q^{B}_{a}$ do not depend on the index $a$ then the
combination of scalar generators is again proportional to the identity
$\delta_{ab}$ and the third line of (3.12) is identically zero.

The three equivalent forms of (3.12) also help us identify the diagrams that
contribute to the 2-loop beta function, e.g. if we adopted the background
field method. These are illustrated in Fig. 1, with the $\otimes$ indicating
the insertion of the external background gauge field. The topology in Fig.
1(a) is responsible for generating the invariant in the first line of (3.12),
the one in Fig. 1(b) is associated to the second line of (3.12) and finally
the topology of Fig. 1(c) to the third form of the CP-odd invariant. The
overall correction to $\beta_{\theta}$ must include a sum over all three
topologies. We emphasize that these are effectively 3-loop diagrams, and yet
they contribute to the 2-loop order beta function because $\theta$ appears in
the action multiplied, in the canonically normalized field basis, by a loop
factor $g^{2}\hbar/32\pi^{2}$.

(a)

(b)

(c)

Figure 1: Topologies of the diagrams generating the CP-odd invariant (3.12).
Crossed circles represent insertions of the external gauge field. Topology (a)
refers to the form in (3.12) written in terms of two fermionic generators,
topology (b) to the one in terms of one fermionic and one scalar generator,
and topology (c) to the one in terms of two scalar generators. It is intended
that each diagram within a given topology should be properly symmetrized.

The coefficient $c$ in (3.11) is model-independent and may be derived
calculating the above diagrams in any model with a non-vanishing
$I_{(2)}^{AB}$, as for example in the toy scenario discussed below. We will
not be able to compute it here because far beyond our current technical
abilities. Yet, we think there is circumstantial evidence that an explicit
computation would find $c\neq 0$. Indeed, ref. [28] presents a calculation of
the beta function of the gauge coupling at 3-loop order using the background
field method. The class of diagrams considered there includes those of Fig. 1.
888In practice the calculation is very different, though, because here we are
interested in the CP-odd contributions proportional to the Levi-Civita tensor
(see the discussion in Section 2), whereas in [28] the authors were allowed to
take $\epsilon_{\bar{\mu}\bar{\nu}\bar{\alpha}\bar{\beta}}=0$. Consistently,
those authors find that the beta function of the gauge coupling is controlled
by the set of CP-even invariants of Appendix A, including in particular the
CP-even component of our $I_{(2)}^{AB}$ (namely its real part). This gives us
confidence that the evaluation of the diagrams in Fig. 1 will not cancel
against each other. The result $c=0$ would thus be rather surprising to us,
and would indicate the presence of an accidental cancellation.

##  4 Implications

In the introduction we mentioned a few implications of $\beta_{\theta}$, both
technical and phenomenological. Here we elaborate on two of them. First we
show the impact of a non-vanishing $\beta_{\theta}$ on the Weyl consistency
conditions. Next we discuss the relevance of $\beta_{\theta}$ on UV solutions
of the strong CP problem.

###  4.1 Consistency conditions

Jack and Osborn pointed out that the beta functions in a general P-conserving
renormalizable theory must satisfy a constraint (see e.g. [22])

$\displaystyle\frac{\partial\widetilde{A}}{\partial\xi_{\bar{I}}}=T_{\bar{I}\bar{J}}\beta_{\bar{J}},$
(4.1)

where $\widetilde{A}$ and $T_{\bar{I}\bar{J}}$ are functions of the couplings
(except $\theta$) that appear in the Weyl anomaly when all couplings are
promoted to space-time dependent functions. Such conditions relate terms of
the beta functions of different couplings and different orders in perturbation
theory, and can serve as non-trivial consistency checks of multi-loop
calculations. For example, eq. (4.1) has been employed to resolve an ambiguity
in the 4-loop beta function of the strong gauge coupling in the Standard Model
[29]. The very same tool can potentially be used to extract information about
the renormalization group evolution in general renormalizable theories like
(3) (see e.g. [30]). In that case however one cannot a priori ignore the
topological angles. Irrespective of whether they vanish in the bare action,
they may be needed as counterterms and therefore generically possess a beta
function. Let us see how $\theta$ qualitatively impacts (4.1).

To start, according to [24] the formal structure of (4.1) remains unchanged
when considering P-violating theories. The absence of an explicit dependence
on $\theta$ then implies that its $\bar{I}=\theta$ component becomes [24]

$\displaystyle 0=T_{\theta\theta}\beta_{\theta}+T_{\theta
Y}\beta_{Y}+T_{\theta g}\beta_{g}+T_{\theta\lambda}\beta_{\lambda},$ (4.2)

where $\theta,Y,g,\lambda$ schematically denote the couplings in (3). The same
considerations developed in Section 3 allow us to identify the tensorial form
of $T_{\bar{I}\bar{J}}$. At leading order we have

$\displaystyle
T_{\theta^{AB}\theta^{CD}}=c_{1}\frac{g_{AC}^{2}g_{BD}^{2}}{(16\pi^{2})^{3}}$
(4.3)

and

$\displaystyle T_{\theta^{AB}Y}dY$ $\displaystyle=$ $\displaystyle
c_{2}\frac{g_{AC}^{2}g_{BD}^{2}}{(16\pi^{2})^{3}}{\rm
Tr}\left[(Y^{\dagger}(T^{C})^{*}dY-dY^{\dagger}(T^{A})^{*}Y)T^{D}\right]$
$\displaystyle+$ $\displaystyle
c_{3}\frac{g_{AC}^{2}g_{BD}^{2}}{(16\pi^{2})^{3}}{\rm Tr}\left[(Y^{\dagger}dY-
dY^{\dagger}Y)\left\\{T^{C},T^{D}\right\\}\right]$

with $c_{1,2,3}$ numerical coefficients. Note that $T_{\theta Y}dY$ must be
CP-odd. In extracting its form a key role is played by the fact that CP-odd
quantities can here depend on the derivative of the couplings, as opposed to
Section 3 when we discussed the form of $\beta_{\theta}$. Analogously, one can
see that the CP-odd structures $T_{\theta g}dg$ and
$T_{\theta\lambda}d\lambda$ inevitably arise at a higher perturbative order.
Within a 2-loop accuracy the contributions proportional to
$\beta_{g},\beta_{\lambda}$ can therefore be neglected and (4.2) gives

$\displaystyle\mu\frac{d\theta^{AB}}{d\mu}$ $\displaystyle=$
$\displaystyle\frac{c_{2}/c_{1}}{16\pi^{2}}{\rm
Tr}\left[(Y_{a}^{\dagger}(T^{A})^{*}\beta_{Y_{a}}-\beta_{Y_{a}}^{\dagger}(T^{A})^{*}Y_{a})T^{B}\right]$
$\displaystyle+$ $\displaystyle\frac{c_{3}/c_{1}}{16\pi^{2}}{\rm
Tr}\left[(Y_{a}^{\dagger}\beta_{Y_{a}}-\beta_{Y_{a}}^{\dagger}Y_{a})\left\\{T^{A},T^{B}\right\\}\right]$
$\displaystyle+$ $\displaystyle{\cal O}(\hbar^{3}).$

Plugging in the 1-loop beta function of the Yukawa coupling $Y_{a}$ we find
that the fermionic trace proportional to $c_{3}$ does not contribute at
2-loops whereas the one proportional to $c_{2}$ consistently reproduces (3.11)
provided $2c_{2}/c_{1}=c$. We thus find a fourth independent method for
computing the coefficient $c$ of $\beta_{\theta}$. Establishing its numerical
value would require knowing the leading order $T_{\theta\theta}$ and
$T_{\theta Y}$, which means performing a 1-loop and again a 3-loop calculation
respectively.

The remaining components of (4.1) are also quite consequential: they imply
that, in employing the consistency relations of Jack and Osborn _the 2-loop
$\beta_{\theta}$ cannot be ignored compared to the 3-loop $\beta_{g}$, the
4-loop $\beta_{Y}$, and the 3-loop $\beta_{\lambda}$_. Again, quantitative
statements would require an accurate inspection of ${\widetilde{A}}$ up to
6-loops, as well as some of the $T_{\bar{I}\bar{J}}$’s.

###  4.2 (In)Stability of UV solutions of the strong CP problem

In the Standard Model the first contribution to $\beta_{\theta}$ is expected
to arise at 7-loops [6, 7]. This implies that the renormalization group
evolution of $\theta$ is numerically negligible, leaving open the possibility
that the absence of CP violation in low energy QCD be the result of some
unknown mechanism at very short distances $\sim 1/\Lambda$, even close to the
Planck scale, that sets $\bar{\theta}(\Lambda)\approx 0$. Among the proposed
UV solutions of the strong CP problem we can find heavy axion scenarios or
models with approximate CP [31, 32] and P [33, 34].

The numerical stability of $\theta=0$ is however not a generic property in
field theory. We have seen that arbitrary renormalizable theories potentially
develop a non-trivial renormalization of the theta angle already at 2-loops.
This demonstrates that UV solutions of the strong CP problem cannot work if
the effective field theory below $\Lambda$ is completely generic and has
unsuppressed Yukawa interactions. This has important phenomenological
implications. If no QCD axion is ever to be found, the possibility that some
yet unknown UV solution is at play would become increasingly more likely.
Tools like those developed here should then be used to identify the class of
extensions of the Standard Model that may consistently describe the low energy
physics below $\Lambda$.

The radiative instability of UV solutions to the strong CP problem is
typically worse than naively expected, actually. First of all, arbitrary
extensions of the Standard Model would necessarily feature new mass scales,
and it is well-known that threshold corrections can affect the topological
angles at tree and 1-loop order. What we want to stress here is another subtle
effect: despite the fact that the beta function of $\theta$ is generically at
2-loop order, _physical_ phases typically renormalize at 1-loop. By _physical_
phases we mean those special combinations of the renormalized couplings,
namely those combinations that are invariant under unitary rotations of the
fields, that enter observables. We have seen that $\theta$ is not invariant
under anomalous rotations of the fermions, see (3.7), and therefore the
_physical_ rescaling invariant quantity of interest must take the form

$\displaystyle\bar{\theta}=\theta+f(\left\\{\xi_{i}\right\\}),$ (4.6)

where $f$ is a model-dependent function. Here $\bar{\theta}$ is what we may
call the physical, rescaling invariant “topological” angle. When no $f$ with
the required properties exists then no invariant can be built out of $\theta$
and that parameter is unphysical. The physical angle runs according to

$\displaystyle\beta_{\bar{\theta}}=\beta_{\theta}+\frac{\partial
f}{\partial\xi_{i}}\beta_{\xi_{i}}.$ (4.7)

The symmetry properties of $\beta_{\bar{\theta}}$ are precisely the same as
those of $\beta_{\theta}$. In particular, both must be proportional to the
imaginary parts of the invariants $I^{AB}$. Yet, while $\beta_{\theta}$ and
$\beta_{\xi_{i}}$ are polynomial in the couplings, it may happen that
$\partial f/\partial\xi_{i}$ brings inverse powers of the $\xi_{i}$’s that are
not compensated by $\beta_{\xi_{i}}$. In such a situation
$\beta_{\bar{\theta}}$ develops negative powers of the couplings in front of
the invariants $I^{AB}$, which in practice indicates that
$\beta_{\bar{\theta}}$ arises at an order in perturbation theory that is lower
compared to $\beta_{\theta}$. We will see explicitly below how this can occur
in a concrete model. Importantly, the Standard Model turns out to be special
even in this respect because this enhancement cannot appear. Indeed, adopting
the standard definition $\bar{\theta}=\theta-{\rm Im}\,{\rm Tr}\,{\log}[{\cal
Y}_{u}{\cal Y}_{d}]$, where ${\cal Y}_{u,d}$ denote the Standard Model
couplings, we find $\beta_{\bar{\theta}}=\beta_{\theta}-{\rm Im}{\rm
Tr}\left[{\cal Y}_{u}^{-1}\beta_{{\cal Y}_{u}}+{\cal Y}_{d}^{-1}\beta_{{\cal
Y}_{d}}\right]$. Since by the accidental flavor symmetries $\beta_{\cal
Y}=\beta_{\cal Y}^{\prime}{\cal Y}$, for some polynomial $\beta_{\cal
Y}^{\prime}$, then $\beta_{\bar{\theta}}$ is made up of the very same
polynomial structures of couplings as $\beta_{\theta}$. This fact for example
ensures that the estimates of [6, 7] are robust.

The faster renormalization group evolution of the physical theta angles and
the presence of important threshold corrections put significant stress on
whatever low energy theory emerges from possible UV solutions of the strong CP
problem. Rather than carrying out a general analysis of the challenges these
scenarios have to face, we find it convenient to discuss a simple
representative model.

#### An example

Consider the following extension of the Standard Model. We introduce two Weyl
fermions with charges $\psi\sim{\bf 3}$ and $\psi^{c}\sim\overline{\bf 3}$
under the QCD group $SU(3)_{c}$, a singlet scalar $\phi_{1}$ and a scalar
octet $\phi_{2}\sim{\bf 8}$. These fields are all neutral under the
electroweak group. The Lagrangian reads

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle{\cal L}_{\rm
SM}+\frac{1}{2}(\partial\phi_{1})^{2}+\frac{1}{2}(D\phi_{2})^{2}+\psi^{\dagger}i{\not{D}}\psi+{\psi^{c}}^{\dagger}i{\not{D}}\psi^{c}$
$\displaystyle-$
$\displaystyle[(m+y_{1}\phi_{1})\psi\psi^{c}+y_{2}\psi\psi^{c}\phi_{2}+{\rm
hc}]-V(\phi_{1},\phi_{2}^{2},|H|^{2}).$

In addition to the CP-odd parameters of the Standard Model, this theory
features two more phases. On top of that, the rescaling-invariant QCD theta
angle contains a new contribution because $\theta_{\rm QCD}$ is now affected
by chiral rotations of $\psi,\psi^{c}$ as well. A natural definition of the
phases (the definition of the Jarlskog invariant is the standard one) is given
by

$\displaystyle\varphi_{y}$ $\displaystyle=$ $\displaystyle{\rm
Arg}\left[(y_{1}y_{2}^{*})^{2}\right]$ (4.9) $\displaystyle\varphi_{m}$
$\displaystyle=$ $\displaystyle{\rm
Arg}\left[y_{1}^{2}y_{2}^{2}[m^{*}]^{4}\right]$ $\displaystyle\bar{\theta}$
$\displaystyle=$ $\displaystyle\theta_{\rm QCD}-{\rm Arg}\,{\rm det}[{\cal
Y}_{u}{\cal Y}_{d}]-{\rm Arg}[m].$

All these CP-odd parameters run already at 1-loop:

$\displaystyle\mu\frac{d\varphi_{y}}{d\mu}$ $\displaystyle=$
$\displaystyle-\frac{4}{16\pi^{2}}\sin\varphi_{y}\left[|y_{1}|^{2}+\frac{4}{3}|y_{2}|^{2}\right]$
$\displaystyle\mu\frac{d\varphi_{m}}{d\mu}$ $\displaystyle=$
$\displaystyle+\frac{4}{16\pi^{2}}\left[|y_{1}|^{2}\left(\sin\varphi_{y}-2\sin\frac{\varphi_{y}+\varphi_{m}}{2}\right)+\frac{4}{3}|y_{2}|^{2}\left(-\sin\varphi_{y}+2\sin\frac{\varphi_{y}-\varphi_{m}}{2}\right)\right]$
$\displaystyle\mu\frac{d\bar{\theta}}{d\mu}$ $\displaystyle=$
$\displaystyle-\frac{2}{16\pi^{2}}\left[|y_{1}|^{2}\sin\frac{\varphi_{y}+\varphi_{m}}{2}+\frac{4}{3}|y_{2}|^{2}\sin\frac{-\varphi_{y}+\varphi_{m}}{2}\right].$

Note that, as opposed to the Standard Model, this is also true for
$\bar{\theta}$ because the phase of the exotic fermion mass has a non-trivial
renormalization group evolution. As a result $\beta_{\bar{\theta}}$ arises at
1-loop while our result (3.11) says that

$\displaystyle\beta_{\theta}$ $\displaystyle=$
$\displaystyle\frac{c}{(16\pi^{2})^{2}}(+3){\rm
Im}\left[(y_{1}y_{2}^{*})^{2}\right]+{\cal O}(\hbar^{3})$ $\displaystyle=$
$\displaystyle\frac{3c}{(16\pi^{2})^{2}}|y_{1}|^{2}|y_{2}|^{2}\sin\varphi_{y}+{\cal
O}(\hbar^{3}).$

Therefore, in the present model the latter beta function is always subdominant
and can be neglected. Indeed it would be enough to have either a small $y_{1}$
or a small $y_{2}$ in order to suppress $\beta_{\theta}$, but this would not
stop $\bar{\theta}$ from running. Similarly, $\sin\varphi_{y}=0$ would
suppress $\beta_{\theta}$ but not $\beta_{\bar{\theta}}$ if
$\sin\varphi_{m}\neq 0$. Only under the simultaneous conditions
$\sin\varphi_{y}=\sin\varphi_{m}=0$ the parameter $\bar{\theta}$ becomes
approximately stable under the renormalization group, but this is obvious
since in that situation our theory introduces no new CP-odd phases compared to
the Standard Model.

It is interesting to verify our earlier claim that the 2-loop $\beta_{\theta}$
requires at least two scalars with different gauge representations. In the
present model

$\displaystyle I^{AB}_{(2)}=3\,{\rm
Im}\left[(y_{1}y_{2}^{*})^{2}\right]\delta^{AB}.$ (4.11)

If we replace $\phi_{2}$ with another scalar singlet (or $\phi_{1}$ with
another scalar in the adjoint) that invariant would not be generated by any
diagram. The reason is that in such a scenario the kinetic term of the scalars
would possess a $O(2)$ symmetry that rotates $(\phi_{1},\phi_{2})$. As usual
this flavor symmetry is explicitly broken by the interactions, but may be
formally resurrected by promoting all the couplings to spurions, as described
in Section 3. The pair $(y_{1},y_{2})$ would be formally a doublet of $O(2)$
and $\beta_{\theta}$ should be a singlet. However (4.11) does not meet this
requirement. Manifestly $O(2)$ invariants are $y_{a}y_{a}$, which is not
invariant under axial rotations of the fermions, and $y_{a}y_{a}^{*}$, which
is CP-even. There is no combination of the former that is simultaneously CP-
odd and invariant under axial rotations, besides of course the one involving
$\theta$, which we know has no perturbative effect. Note that the quantity
$y_{a}\epsilon_{ab}y_{b}=i{\rm Im}[y_{1}y_{2}^{*}]$ is $SO(2)$-symmetric and
CP-odd, but in order to have an invariant under $Z_{2}\subset O(2)$ one would
need an even power of it, thus resulting in a CP-even combination. We conclude
that $\phi_{1,2}$ cannot be in the same representation if (4.11) is to be
generated.

An important point to stress is the following. The definitions (4.9) (and more
generally (4.6)) are ambiguous because any combination of invariants is also
invariant. One may then object that the claim that the rescaling-invariant
topological angle runs at 1-loop is not sufficiently general: maybe there
exists a rescaling invariant combination that is also approximately
renormalization group invariant, and one may decide to call it $\bar{\theta}$.
Yet, the ambiguity in defining the physical theta angle arises only from a UV
perspective. Its IR definition is basically fixed by experiments. In the
Standard Model one usually takes $\theta_{\rm QCD}-{\rm Arg}\,{\rm det}[{\cal
Y}_{u}{\cal Y}_{d}]$ because at leading order in a perturbative expansion in
Yukawas (masses) this is precisely the quantity observed in low energy
processes. The situation is similar here. Indeed, because the new fermions
$\psi,\psi^{c}$ are colored, phenomenologically we expect $|m|\gtrsim 1$ TeV;
at scales below the mass threshold $|m|$ we can thus integrate them out such
that our theory reduces to the Standard Model with topological angle given by

$\bar{\theta}_{\rm SM}=\left\\{\theta_{\rm QCD}-{\rm Arg}\,{\rm det}[{\cal
Y}_{u}{\cal Y}_{d}]-{\rm Arg}[m]\right\\}_{\mu=|m|}+{\cal O}(\hbar),$

where the small corrections are due to 1-loop threshold effects. We thus see
that the UV ambiguity is resolved: within the accuracy we are working the
rescaling-invariant topological angle defined in (4.9) is precisely the one
constrained by low energy experiments. The distinction between
$\bar{\theta}_{\rm SM}$ and $\bar{\theta}$ is 1-loop order, and so the
difference in their beta functions starts at 2-loops. Hence the statement that
$\bar{\theta}$ runs already at 1-loop in fact also applies to the physically
relevant CP-odd parameter measured in experiments. An UV solution of the
strong CP problem that enforces the condition $\bar{\theta}(\Lambda)=0$ would
not be able to explain the absence of CP violation in low energy QCD. To
ensure $|\bar{\theta}_{\rm SM}|\lesssim 10^{-10}$ one has to invoke a non-
trivial conspiracy among different UV parameters.

We conclude by commenting on what would happen if we set $m=0$. Such an
alternative scenario may still be phenomenologically viable provided the
exotic fermion gets a mass $y_{1}v_{1}$ from the vacuum expectation value of
the singlet $\langle\phi_{1}\rangle=v_{1}$. This model introduces a unique new
phase $\varphi_{y}$ whereas the QCD topological angle may be defined as
$\bar{\theta}=\theta-{\rm Arg}\,{\rm det}[{\cal Y}_{u}{\cal Y}_{d}]-{\rm
Arg}\left[y_{1}^{2}\right]/2$ as well as in an infinite number of inequivalent
ways. Again, the ambiguity in its definition is resolved by observing that the
QCD angle at the matching scale reads

$\displaystyle\bar{\theta}_{\rm SM}=\left\\{\theta_{\rm QCD}-{\rm Arg}\,{\rm
det}[{\cal Y}_{u}{\cal Y}_{d}]-{\rm
Arg}[y_{1}v_{1}]\right\\}_{\mu=|y_{1}v_{1}|}+{\cal O}(\hbar).$

Again, in order to have a small $|\bar{\theta}_{\rm SM}|$ there should be a
cancellation between the phase of $y_{1}$, which runs at 1-loop, and
$\theta_{\rm QCD}-{\rm Arg}\,{\rm det}[{\cal Y}_{u}{\cal Y}_{d}]$, which does
not run before 7-loops. This cancellation is not stable unless
$\sin\varphi_{y}=0$, i.e. unless we recover the same amount of CP-violation as
in the Standard Model.

The lesson to be learned is clear. Generic extensions of the Standard Model
feature several flavor-invariant CP-odd phases with non-trivial
renormalization group evolution. In such a situation there are no simple UV
conditions that ensure $|\bar{\theta}_{\rm SM}|$ stays small.

##  5 Outlook

It has been known for quite some time that topological angles have measurable
consequences. Re-summing the logs associated to their renormalization group
evolution might therefore be quantitatively relevant in some case. In this
paper we discussed the perturbative running of the $\theta$ angles in general
renormalizable theories as well as the techniques necessary to compute it in
dimensional regularization.

We found that $\theta$ can acquire an additive beta function no earlier than
2-loop order. This effect is proportional to a unique combination of the
ordinary couplings we called $I^{AB}_{(2)}$. The effect is present only in
sufficiently complex theories with (CP-violating) Yukawa interactions
involving scalars with more than one irreducible representation of the gauge
group. Within the background field method the corresponding divergent diagrams
are shown in Fig. 1.

There are a number of directions in which our work can be extended. An obvious
one would be to perform the explicit calculation of the numerical coefficient
$c$ of the 2-loop beta function (3.11). This can be achieved with one, or
more, of the four techniques discussed in this paper, namely the background
field method [16], the CP-odd spurion trick of [18], or the two new techniques
proposed here: via inspection of off-diagonal elements of the anomalous
dimension matrix of the CP-odd interactions (see Section 2.2) or via the Weyl
consistency conditions (see Section 4.1). The calculation may be carried out
in any model with a non-vanishing $I^{AB}_{(2)}$, like for example the toy
model discussed in Section 4.2. In all cases it requires the computation of
3-loop diagrams.

Another direction that needs to be explored is the consistency conditions
implied by the local renormalization group approach of Jack and Osborn. In
Section 4.1 we have shown that the 2-loop $\beta_{\theta}$ cannot be ignored
compared to the 3-loop $\beta_{g}$, the 4-loop $\beta_{Y}$, and the 3-loop
$\beta_{\lambda}$. Yet, in order to draw more quantitative conclusions a
systematic generalization of the local renormalization group technique to
P-violating theories with anomalous currents, as initiated in [24], as well as
a (re)analysis of the beta functions within a consistent scheme for
$\gamma_{5}$, appear necessary.

The phenomenological implications of $\beta_{\theta}$ also deserve to be
further scrutinized. We have seen that the radiative stability of the QCD
angle is a peculiar accident of the Standard Model. Quantitatively, a small
strong CP phase represents a renormalization-invariant feature of the Standard
Model, but certainly not of arbitrary extensions. This observation becomes
phenomenologically relevant if the strong CP problem is solved by some
mechanism at some high UV cutoff (e.g. models based on approximate P or CP
symmetries). In those cases the UV theory would typically predict
$\bar{\theta}(\Lambda)=0$ and the experimental observation
$|\bar{\theta}(1~{}{\rm GeV})|<10^{-10}$ would translate into a constraint on
the renormalization evolution below $\Lambda$. Our considerations can help
identify which extensions of the Standard Model at renormalization scales
$1~{}{\rm GeV}<\mu<\Lambda$ are compatible with such a picture. For example,
we found that a 2-loop $\beta_{\theta}$ can only be generated if there are at
least two scalars with different representations of the gauge group. Thus, a
two-Higgs doublet model would not induce that effect, despite the many
additional sources of CP violation. The claim that such an extension features
the same radiative stability as the Standard Model is however too naive. We
have demonstrated that the _rescaling-invariant_ angles $\bar{\theta}$ often
run at lower order in perturbation theory than $\theta$ itself and that
sizable (finite) effects on $\bar{\theta}$ usually arise at the mass
thresholds that extensions of the Standard Model inevitable possess. UV
completions where all these effects are under control are rather rare. Only a
very limited number of options would be available if Nature decided to address
the strong CP problem in the far UV. The identification of these options
represents an interesting open problem.

## Acknowledgments

We have enjoyed conversations with M. Fael, M. Pospelov, A. E. Thomsen, G.
Villadoro. This research was partly supported by the “iniziativa specifica”
Physics at the Energy, Intensity, and Astroparticle Frontiers (APINE) of
Istituto Nazionale di Fisica Nucleare (INFN), and the European Union’s Horizon
2020 research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 860881-HIDDeN.

##  Appendix A CP-odd flavor-invariants

In this appendix we build the flavor-invariants introduced in Section 3 and
identify the CP-odd ones. We approach the problem perturbatively. By counting
the powers of $\hbar$ in the flavor-singlet structures one sees that
corrections to the 1PI vertex with two external gauge bosons proportional to
$g^{2c_{g}}Y^{2c_{Y}}\lambda^{c_{\lambda}}$ correspond to diagrams with
$n=c_{g}+c_{Y}+c_{\lambda}$ loops. Yet, the associated contribution to
$Z_{\theta,1}$ is a 1-loop factor smaller because the definition of the
$\theta$ vertex includes a factor $g^{2}\hbar/16\pi^{2}$. This implies that in
order to find a correction of $n$-loop size to the beta function, hereafter
denoted by $\beta_{\theta}^{(n)}$, one should calculate an $n+1$-loop diagram.

On pure dimensional grounds, therefore, corrections to $\beta_{\theta}$ are
expected to be controlled by

$\displaystyle\beta_{\theta}^{(1)}:~{}~{}~{}~{}~{}g^{4},g^{2}\lambda,g^{2}Y^{2}$
(A.1)
$\displaystyle\beta_{\theta}^{(2)}:~{}~{}~{}~{}~{}g^{6},g^{4}\lambda,g^{2}\lambda^{2},g^{4}Y^{2},g^{2}Y^{2}\lambda,g^{2}Y^{4}$
$\displaystyle\cdots$

In this paper we will content ourselves with 1- and 2-loop size effects,
namely $\beta_{\theta}^{(1,2)}$, though the formalism we adopt can be extended
up to arbitrary order. Up to this order it is rather straightforward to argue
that only the terms involving the Yukawa couplings have some real chance of
being CP-odd, as we now show.

A general renormalizable gauge theory without scalar quartics and Yukawas
always conserves CP (if the topological angles can be neglected, which is the
case perturbatively). Hence there cannot be any CP-odd invariant $I^{AB}$
built out of $T^{A},S^{A}$ only. Purely bosonic flavor-invariants cannot work
either. They depend on $S^{A},\lambda_{abcd}$ and are automatically CP-even.
Indeed, $\lambda$ is real whereas $S^{A}$ are purely imaginary. To build a CP-
odd combination we would need an odd number of $S^{A}$, which cannot be
covariant under rotations of the adjoint index because the only invariant
tensor at our disposal for contractions is $g_{AB}^{2}$. This conclusion was
expected on account of that the Feynman diagrams we are interested in must be
proportional to the Levi-Civita tensor, and therefore fermion traces are
strictly necessary to generate them.

We can explicitly demonstrate that even the combinations $g^{2}\lambda$,
$g^{2}\lambda^{2}$ and $g^{4}\lambda$ can be discarded. Recall that a fermion
loop is necessary, so such invariant must include traces of the fermion
generator. The only $g^{2}\lambda$ invariant we can have is ${\rm
Tr}[T^{A}T^{B}]\lambda_{aabb}$ and is manifestly real, i.e. CP-even, since the
fermionic trace gives the direct sum of identities in the adjoint index
multiplied by (real) fermion Casimirs. Similar considerations apply to the
$g^{2}\lambda^{2}$ invariants, which are of the form ${\rm
Tr}[T^{A}T^{B}]\lambda^{2}$ with all possible contractions of the scalar
indices. Whatever contraction is taken the invariant is CP-even. Finally, at
order $g^{4}\lambda$ we can have structures of the form ${\rm
Tr}[TTTT]\lambda$, ${\rm Tr}[TT]{\rm Tr}[TT]\lambda$, and ${\rm
Tr}[TT]SS\lambda$. The former two are manifestly real because the potential
CP-violating contribution would have to come exclusively from the fermionic
generators, and we have recalled above that this is not possible. The last one
may either contain $[S^{M}S^{N}]_{ab}\lambda_{abcc}$ or
$[S^{M}S^{N}]_{aa}\lambda_{bbcc}$, which are both real. We did not include
structures with a single scalar generator because those vanish: the (anti-
symmetric) scalar indices in $S^{A}$ should necessarily be contracted with
those (symmetric) of $\lambda$. Similarly, three scalar generators would
require the fermionic trace ${\rm Tr}T=0$, which vanishes in the absence of
mixed gravity-gauge anomalies.

We conclude that, at least up to the perturbative order considered here, the
Yukawa couplings are strictly necessary to build the CP-odd flavor-invariants
$I^{AB}$ of the theory (3). For convenience it is useful to introduce the
basic covariant combinations in which they can appear:

$\displaystyle\Upsilon^{A_{1}\cdots A_{n}}_{ab;ij}\equiv
Y_{a\,ik}^{*}(T^{A_{1}})^{*}\cdots(T^{A_{n}})^{*}Y_{b\,kj}.$ (A.2)

These objects transform under fermion rotations precisely as $T^{A}$, under
scalar rotations as the product of two Yukawas, whereas under gauge boson
rotations in an obvious way. We will use this compact notation to write the
possible invariants appearing in $\beta_{\theta}^{(1,2)}$.

#### Absence of $\beta^{(1)}$: 2-loop diagrams

At lowest order we have a very limited number of flavor-invariant structures
that can contribute. They are so few that we can write them explicitly:

$\displaystyle g^{2}Y^{2}:$ $\displaystyle{\rm
Tr}[T^{(A}T^{B)}\Upsilon_{aa}],~{}~{}{\rm
Tr}[T^{(A}\Upsilon^{B)}_{aa}],~{}~{}{\rm Tr}[T^{(A}T^{B)}]{\rm
Tr}[\Upsilon_{aa}],~{}~{}S^{(A}_{mn}S^{B)}_{mn}{\rm Tr}[\Upsilon_{aa}],$ (A.3)

Here and in the following ${\rm Tr}[\cdots]$ denotes a trace over the
fermionic indices and $(~{})$ imply symmetrization. We did not include
invariants in which the scalar indices of $S^{A}_{ab}$ are contracted with
those of the Yukawas because thanks to (3.3) these can be written in terms of
the first and second invariants of (A.3). Furthermore, we did not include
structures of the form ${\rm Tr}[\Upsilon^{(AB)}_{aa}]$ because ${\rm
Tr}[\Upsilon^{A_{1}\cdots A_{n}}_{ab}]={\rm Tr}[\Upsilon_{ab}T^{A_{n}}\cdots
T^{A_{1}}]$ due to the trace transposition property. This relation will also
be exploited later in enumerating the invariants of higher order.

We find that none of the invariants in (A.3) is CP-odd. To see this note that
the matrices $\Upsilon_{aa}$, $\Upsilon^{A}_{aa}$ are hermitian and therefore
their trace is real. This immediately tells us that the last two structures
are real. Similarly, the structures ${\rm Tr}[T\Upsilon^{A}]$ are necessarily
CP-even because they are the trace of the product of two hermitian matrices,
which is real. The first structure in (A.3) is CP-even for the same reason,
because $T^{(A}T^{B)}$ is also hermitian.

#### Terms in $\beta^{(2)}$: 3-loop diagrams

At the next order, the relevant structures are:

$\displaystyle g^{2}Y^{2}\lambda:$ $\displaystyle{\rm
Tr}[TT\Upsilon_{0}]\lambda,~{}~{}{\rm Tr}[T\Upsilon_{1}]\lambda,~{}~{}SS{\rm
Tr}[\Upsilon_{0}]\lambda,$ $\displaystyle{\rm Tr}[TT]{\rm
Tr}[\Upsilon_{0}]\lambda$ $\displaystyle g^{4}Y^{2}:$ $\displaystyle{\rm
Tr}[TT\Upsilon_{2}],~{}~{}{\rm Tr}[TTT\Upsilon_{1}],~{}~{}{\rm
Tr}[TTTT\Upsilon_{0}]$ $\displaystyle{\rm Tr}[TT]{\rm
Tr}[\Upsilon_{2}],~{}~{}SS{\rm Tr}[\Upsilon_{2}],~{}~{}{\rm Tr}[TT]{\rm
Tr}[T\Upsilon_{1}],~{}~{}{\rm Tr}[TTT]{\rm Tr}[\Upsilon_{1}],~{}~{}SSS{\rm
Tr}[\Upsilon_{1}],$ $\displaystyle{\rm Tr}[TTTT]{\rm
Tr}[\Upsilon_{0}],~{}~{}{\rm Tr}[TT]{\rm Tr}[TT]{\rm
Tr}[\Upsilon_{0}],~{}~{}SSSS{\rm Tr}[\Upsilon_{0}]$ $\displaystyle
g^{2}Y^{4}:$ $\displaystyle{\rm Tr}[\Upsilon_{2}\Upsilon_{0}],~{}~{}{\rm
Tr}[\Upsilon_{1}\Upsilon_{1}],~{}~{}{\rm Tr}[\Upsilon_{1}\Upsilon_{0}T],$
$\displaystyle{\rm Tr}[\Upsilon_{2}]{\rm Tr}[\Upsilon_{0}],~{}~{}{\rm
Tr}[\Upsilon_{1}]{\rm Tr}[\Upsilon_{1}],~{}~{}{\rm Tr}[TT]{\rm
Tr}[\Upsilon_{0}\Upsilon_{0}],~{}~{}SS{\rm Tr}[\Upsilon_{0}\Upsilon_{0}]$
$\displaystyle{\rm Tr}[\Upsilon_{0}]{\rm Tr}[\Upsilon_{0}]{\rm
Tr}[TT],~{}~{}SS{\rm Tr}[\Upsilon_{0}]{\rm Tr}[\Upsilon_{0}]$

where to make our notation more compact the $n$ in the expression
$\Upsilon_{n}$ indicates the number of adjoint indices in $\Upsilon$. The
indices are left implicit because many contractions are possible, and again
invariants where one scalar index must be contracted with a Yukawa have been
omitted because of (3.3).

####  $g^{2}Y^{2}\lambda$ terms

Most of the $g^{2}Y^{2}\lambda$ terms are manifestly real once we take into
account that the symmetry of $\lambda$ forces the scalar indices in
$[\Upsilon_{n}]_{ab}$ to be symmetrized or contracted among themselves, i.e.
$\Upsilon_{(ab)}$ or $\Upsilon_{aa}$. In either case the resulting $\Upsilon$
tensor is hermitian in the fermion indices. Hence all traces are inevitably
real and the invariants CP-even.

####  $g^{4}Y^{2}$ terms

Consider next the $g^{4}Y^{2}$ terms. Here the scalar indices are contracted
among themselves; it is the possible contractions of the gauge indices that
increases the number of independent structures. However all the invariants
turn out to be CP-even because of properties of the gauge generators and gauge
invariance. Let us show this explicitly by considering the invariants of the
form ${\rm Tr}[TT\Upsilon_{2}]$. These are

$\displaystyle g^{2}_{CD}{\rm Tr}[T^{(A}T^{B)}\Upsilon^{CD}_{aa}]$ (A.5)
$\displaystyle g^{2}_{CD}{\rm Tr}[T^{C}T^{D}\Upsilon^{(AB)}_{aa}]\ $

and

$\displaystyle g^{2}_{CD}{\rm Tr}[T^{C}T^{(A}\Upsilon^{B)D}_{aa}]$ (A.6)
$\displaystyle g^{2}_{CD}{\rm Tr}[T^{C}T^{(A|}\Upsilon^{D|B)}_{aa}]$
$\displaystyle g^{2}_{CD}{\rm Tr}[T^{(A|}T^{C}\Upsilon^{|B)D}_{aa}]$
$\displaystyle g^{2}_{CD}{\rm Tr}[T^{(A|}T^{C}\Upsilon^{D|B)}_{aa}]$

where the contraction among adjoint indices is consistently performed with the
metric $g^{2}_{AB}$, (3). In the first two structures, hermiticity of
$T^{(A}T^{B)}$ and $\Upsilon_{aa}^{(MN)}$ ensures they are all CP-even. The
other structures are also CP-even. To see this let us look for a CP-odd
version of the first invariant in (A.6), namely

$\displaystyle g^{2}_{CD}{\rm
Tr}[T^{(A|}T^{C}\Upsilon^{|B)D}_{aa}]-g^{2}_{CD}{\rm
Tr}[T^{C}T^{(A|}\Upsilon^{D|B)}_{aa}]$ (A.7)

The indices $C,D$ run over all possible adjoint indices. On the other hand,
$A,B$ are restricted to a certain non-abelian group or to any two of the
abelian factors. Consider first the case $A,B$ refer to a certain non-abelian
group. Then $T^{A,B}$ commute with all the $T^{C,D}$ that are associated to
the other groups. In addition, $\theta^{AB}\propto\delta^{AB}_{\cal G}$ and
hence we should not simply symmetrize but actually sum over $A,B$ as well. As
a result the above invariant identically vanishes. Consider next the case in
which $A,B$ refer to the abelian groups. Then $T^{A,B}$ commute with $T^{C,D}$
and the expression again identically vanishes. Similar considerations show
that all invariants in (A.6) are CP-even.

What about invariants of the form ${\rm Tr}[TTT\Upsilon_{1}]$? Here we have

$\displaystyle g_{CD}^{2}{\rm Tr}[T^{(A|}T^{C}T^{D}\Upsilon_{aa}^{|B)}]$ (A.8)
$\displaystyle g_{CD}^{2}{\rm Tr}[T^{C}T^{D}T^{(A}\Upsilon^{B)}_{aa}]$

and

$\displaystyle g_{CD}^{2}{\rm Tr}[T^{(A|}T^{C}T^{|B)}\Upsilon_{aa}^{D}]$ (A.9)
$\displaystyle g_{CD}^{2}{\rm Tr}[T^{C}T^{(A|}T^{D}\Upsilon_{aa}^{|B)}]$
$\displaystyle g_{CD}^{2}{\rm Tr}[T^{(A}T^{B)}T^{C}\Upsilon_{aa}^{D}]$
$\displaystyle g_{CD}^{2}{\rm Tr}[T^{C}T^{(A}T^{B)}\Upsilon_{aa}^{D}]$

The expressions with the Casimir $g_{CD}^{2}T^{C}T^{D}=\oplus_{\cal G}C_{\cal
G}$ are manifestly real. The first and second in (A.9) are real because they
are the trace of the product of two hermitian matrices. A potential CP-odd
combination with the third structure is

$\displaystyle g_{CD}^{2}{\rm
Tr}[\left\\{T^{A},T^{B}\right\\}[T^{C},\Upsilon_{aa}^{D}]].$ (A.10)

An identical one is obtained from the last structure in (A.9). Again, these
identically vanish when imposing gauge-invariance. Namely, if $A,B$ refer to
indices of a non-abelian group, then we should include a sum $\delta_{\cal
G}^{AB}$ and find that $\left\\{T^{A},T^{B}\right\\}$ is proportional to the
Casimir $C_{\cal G}$ of that group. Hence it commutes with all $T^{C}$ and we
get $g_{CD}^{2}{\rm Tr}[C_{\cal G}[T^{C},\Upsilon_{aa}^{D}]]=g_{CD}^{2}{\rm
Tr}[[C_{\cal G},T^{C}]\Upsilon_{aa}^{D}]=0$. The same result trivially applies
also when $A,B$ refer to the abelian groups.

The arguments just exposed can be applied to all the other invariants
appearing in (A), including the ones with the scalar generators. Therefore our
conclusion is that all the $g^{4}Y^{2}$ structures are CP-even.

####  $g^{2}Y^{4}$ terms

The structures $g^{2}Y^{4}$ are more involved because the scalar indices can
be contracted in several different ways. For example ${\rm
Tr}[\Upsilon_{2}\Upsilon_{0}]$ can be ${\rm
Tr}[\Upsilon^{(AB)}_{ab}\Upsilon_{ab}]$, ${\rm
Tr}[\Upsilon^{(AB)}_{ab}\Upsilon_{ba}]$, ${\rm
Tr}[\Upsilon^{(AB)}_{aa}\Upsilon_{bb}]$. Here it is crucial to observe that
$[\Upsilon_{ab}]^{\dagger}=\Upsilon_{ba}$ if $\Upsilon$ has less than two
gauge indices, or even more provided they are symmetrized. This way one finds
that ${\rm Tr}[\Upsilon_{2}\Upsilon_{0}]$ are all CP-even. Similar
considerations apply to ${\rm Tr}[\Upsilon_{1}\Upsilon_{1}]$ as well as all
the multi-trace expressions in (A). Including the generator $T$, however,
changes the game and leads finally to some interesting candidates. We are thus
left with the structure ${\rm Tr}[\Upsilon_{1}\Upsilon_{0}T]$. The CP-odd
invariants of this form are

$\displaystyle{\rm Tr}\left[\Upsilon^{A}_{ab}[T^{B},\Upsilon_{ab}]\right]$
(A.11) $\displaystyle{\rm
Tr}\left[\Upsilon^{(A}_{ab}[T^{B)},\Upsilon_{ba}]\right]$ $\displaystyle{\rm
Tr}\left[\Upsilon^{(A}_{aa}[T^{B)},\Upsilon_{bb}]\right]$

where the first invariant is already symmetric in the adjoint indices, as
follows from the trace transposition property. The same property also shows
that the second and third invariants are actually equivalent. Furthermore,
using (3.3) it is straightforward to prove that $[\Upsilon_{aa},T^{A}]=0$, so
the second and third expressions actually vanish. This ensures that the only
non-vanishing CP-odd invariant in (A.11) is the first one, which is therefore
the only possible contribution to the $\beta-$function of $\theta^{AB}$ at
3-loops. This is precisely $I_{(2)}^{AB}$ of eq. (3.12).

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