#  Spurious gauge-invariance and $\gamma_{5}$
in Dimensional Regularization

Pablo Olgoso Ruiz<EMAIL_ADDRESS>_Istituto Nazionale di Fisica
Nucleare, Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italy_ Luca
Vecchi<EMAIL_ADDRESS>_Istituto Nazionale di Fisica Nucleare, Sezione
di Padova, Via F. Marzolo 8, 35131 Padova, Italy_

###### Abstract

Dimensional regularization is arguably the most popular and efficient scheme
for multi-loop calculations. Yet, when applied to chiral (gauge) theories like
the Standard Model and its extensions, one is forced to deal with the infamous
“$\gamma_{5}$ problem”. The only formulation that has been demonstrated to be
consistent at all orders in perturbation theory, known as Breiteinlhoner-
Maison-’t Hooft-Veltman scheme, is rather cumbersome because of the lack of
manifest chiral gauge-invariance. In this paper we point out that this
drawback can be alleviated by the introduction of auxiliary fields that
restore a spurious version of gauge-invariance. If combined with the
background field method, all 1PI amplitudes and the associated counterterms
are formally covariant and thus severely constrained by the symmetries. As an
illustration we evaluate the symmetry-restoring counterterms at 1-loop in the
most general renormalizable gauge theory with Dirac fermions and scalar
fields, the Standard Model representing a particular example.

###### Contents

  1. 1 Introduction
  2. 2 Regularized theory
    1. 2.1 The standard theory
    2. 2.2 Spurious chiral-invariance
  3. 3 Structure of the radiative corrections
    1. 3.1 Symmetry-restoring counterterms: general considerations
    2. 3.2 Symmetry-restoring counterterms: 1-loop calculation
      1. 3.2.1 Basis of counterterms
  4. 4 Conclusions
  5. A The Standard Model case
  6. B d-dimensional Lorentz and gauge invariance

##  1 Introduction

Dimensional regularization (Dim-Reg) [1, 2] is a very popular scheme in
perturbative calculations. This is so for very good reasons: Dim-Reg is
compatible with vector-like gauge symmetries and Lorentz invariance, it
simultaneously regulates both UV and IR divergences, and it is
straightforwardly applicable to higher orders. For all these reasons Dim-Reg
has by now become the standard regularization scheme for multi-loop
calculations.

Already at its conception, however, ’t Hooft and Veltman noted a complication
in dealing with chiral theories like the Standard Model [1]. At the heart of
Dim-Reg is the fact that convergence of the loop integrals is ensured by
extending the kinetic terms of all fields to $d$-dimensions. Yet, in
$d$-dimensions there is no notion of chirality and hence chiral symmetries are
inevitably broken.

In the approach of ’t Hooft and Veltman, and subsequently formalized by
Breitenlohner and Maison [3], the Dirac gamma matrices are assumed to satisfy
the $d$-dimensional Clifford algebra and split into a four-dimensional set
($\gamma^{\bar{\mu}}$) and a $d-4$-dimensional set ($\gamma^{\hat{\mu}}$).
Instead, the $\gamma_{5}$ matrix and the Levi-Civita tensor remain genuinely
four-dimensional objects and satisfy (we use $\epsilon_{0123}=1$)

$\gamma_{5}=\frac{i}{4!}\epsilon_{\bar{\mu}\bar{\nu}\bar{\alpha}\bar{\beta}}\gamma^{\bar{\mu}}\gamma^{\bar{\nu}}\gamma^{\bar{\alpha}}\gamma^{\bar{\beta}}.$
(1)

From these definitions follows that $\gamma_{5}$ anti-commutes only with the
four-dimensional gamma matrices, but commutes with the others

$\left\\{\gamma^{\bar{\mu}},\gamma_{5}\right\\}=0,~{}~{}~{}~{}~{}~{}~{}\bigl{[}\gamma^{\hat{\mu}},\gamma_{5}\bigr{]}=0.$
(2)

As a result of (2), the $d$-dimensional kinetic term of fermions mixes left
and right chiralities, violating all chiral symmetries, global and local (see
[4] for a review and more references). Eq. (1), and its consequence (2),
define what we will refer to as the _BMHV scheme_.

The breaking of the chiral symmetries implied by the BMHV scheme is of course
only fictitious and has no physical consequences, if properly handled.
Consider for instance a theory with a local chiral symmetry. A regularization-
independent condition guaranteeing that our theory has no gauge anomalies can
be formulated as a requirement on the fermionic generators. Specifically,
denoting by $T_{L,R}^{A}$ the gauge generators of the left and right
chiralities, a well-defined (anomaly-free) gauge theory must satisfy

$D^{ABC}=\mathrm{tr}\left(T^{A}_{L}\left\\{T^{B}_{L},T^{C}_{L}\right\\}\right)-\mathrm{tr}\left(T^{A}_{R}\left\\{T^{B}_{R},T^{C}_{R}\right\\}\right)=0.$
(3)

When (3) holds, if the regularization scheme violates the gauge symmetry one
can always add appropriate _gauge non-invariant_ counterterms that eliminate
unphysical degrees of freedom order by order in perturbation theory. Those
counterterms must be included in addition to the more familiar ones that
remove the $1/(d-4)$ poles, which in such non-invariant schemes also contain
gauge non-invariant pieces in general. In an anomaly-free gauge theory
satisfying (3), therefore, the BMHV scheme is a perfectly viable one; it just
requires more work. In the last years the explicit form of the BRST-restoring
counterterms for various chiral gauge theories regularized via the BMHV scheme
has been calculated at the first non-trivial orders in [5, 6, 7, 8, 9, 10].
Similarly, the counterterms necessary to restore gauge-invariance when using
the background field method have been calculated for various theories at the
one-loop order in [11, 12].

Even though the fictitious violation of chiral (global or local) invariance
induced by the BMHV scheme is not a conceptual obstacle, it is certainly a
practical nuisance. Bare correlators do not satisfy the generalized Ward
identities prior to the introduction of the symmetry-restoring counterterms,
and so a priori there is no manifest guiding principle that controls the
structure of the radiative corrections. This lack of symmetry complicates
calculations especially in chiral gauge theories like the electroweak sector
of the Standard Model. For this reason, alternative approaches to the
“$\gamma_{5}$ problem” in Dim-Reg have been proposed in the last decades with
the goal of limiting as much as possible the regularization-induced violation
of chiral symmetries. Particularly popular are the so-called “naive Dim-Reg”
[13] and “KKS” [14, 15] schemes (see also [16]). Yet, none of those
alternative approaches rely on rigorous mathematical definitions of
$\gamma_{5}$, and as such are inherently ambiguous. 111A concrete indication
of possible shortcomings of the KKS scheme is found in the evaluation of the
ABJ anomaly at 2-loops [17] or in the beta functions of the Standard Model
couplings at 4-loops [18, 19]. Nevertheless, the vast majority of groups in
our community still favors the technical simplicity of the alternatives over
the cumbersome rigour of the BMHV scheme.

In this paper we will adopt the BMHV scheme for $\gamma_{5}$, the only
approach demonstrated to be fully consistent at all orders in perturbation
theory. Our goal is to find some “order” in the apparent chaos associated to
that formalism. Specifically, we will see that the structure of the
counterterms and the correlators themselves are constrained by the way Dim-Reg
explicitly breaks the chiral symmetry: a _spurious gauge-invariance_ will
guide our calculation and replace the broken gauge (or BRST) invariance.

We begin in Section 2.1 with an introduction of the standard regularization
procedure. This preliminary discussion is a summary of known results and
mostly serves the purpose of introducing our notation. A version of the
regularized theory that preserves the spurious chiral invariance is introduced
in Section 2.2. In Section 3 we discuss a few of the advantages deriving from
the spurious invariance. As a concrete application we identify the complete
set of symmetry-restoring counterterms for an arbitrary renormalizable gauge
theory with Dirac fermions and scalars at 1-loop, generalizing the earlier
work of [11]. The case of the Standard Model can be obtained using the
expressions in Appendix A. In Appendix B we point out that, if desired, one
can build a regularized theory that also respects a spurious version of
$d$-dimensional Lorentz invariance. We conclude in Section 4.

##  2 Regularized theory

In this section we introduce the regularized action for a chiral gauge theory
with charged fermionic fields $\Psi_{i}$, where $i,j,\dots$ collectively
denote both gauge and flavor indices, and scalars $\phi_{a}$, with $a,b,\dots$
again running over both gauge and flavor indices. The gauge symmetry is
assumed to be the direct product of simple and abelian unitary groups, but is
otherwise completely arbitrary. The representations of the fermions and
scalars are in general reducible. We describe fermions via a Dirac field
$\Psi$ because more suitable for Dim-Reg; theories of Weyl fermions may be
obtained via the introduction of unphysical dummy components of $\Psi$, that
are effectively decoupled in four dimensions. The scalar fields $\phi_{a}$ are
taken to be real without loss of generality.

###  2.1 The standard theory

As reviewed in Section 1, in Dim-Reg the kinetic terms of all fields are
extended to $d$ dimensions. In the case of bosons this can be trivially
achieved while preserving all symmetries, whereas some subtlety arises when
considering chiral fermions, as we will review shortly.

The regularized action is the sum of various components:

$S^{\text{Reg}}_{\text{std}}[\xi]\equiv\int\text{d}^{d}x\left[{\cal
L}^{\text{Reg}}_{\text{Bos}}+{\cal L}^{\text{Reg}}_{\text{Yuk}}+{\cal
L}_{\text{Kin}}^{\text{Reg}}\right],$ (4)

with $\xi$ collectively denoting all fields. The first one involves the
bosonic degrees of freedom and includes the kinetic terms of the gauge fields
and the scalars as well as the scalar potential: 222We use mostly-minus
signature.

${\cal
L}^{\text{Reg}}_{\text{Bos}}=-\frac{1}{4}F^{A}_{\mu\nu}F^{A\,\mu\nu}+(D_{\mu}\phi)_{a}^{\dagger}(D^{\mu}\phi)_{a}-V(\phi).$
(5)

These terms are obtained by simply promoting the four-dimensional theory to a
$d$-dimensional one, so $\mu$ runs over both the four-dimensional coordinates
$\bar{\mu}=0,1,2,3$ and the evanescent ones $\hat{\mu}$. We define
$D_{\mu}\phi_{a}=\partial_{\mu}\phi_{a}-iA^{A}_{\mu}[T_{\phi}^{A}]_{ab}\phi_{b}$,
with $T_{\phi}^{A}$ the (hermitian and purely imaginary) generators of the
local transformations of $\phi$.

The second and third terms in (4) involve the fermionic fields $\Psi$. Before
writing them explicitly let us make some preliminary considerations. In four
dimensions the left and right fermion chiralities form independent Lorentz
components and can thus be rotated independently. In that case the fermionic
kinetic term possesses a large $SU(N_{f})\times SU(N_{f})$ global symmetry (in
general broken by the weak gauging and the Yukawa interactions) and the left
and right chiralites may carry different representations under the gauge
group, generated by hermitian matrices $T_{L}^{A}$ and $T_{R}^{A}$
respectively. Ideally, we would like to retain the same features also in
$d$-dimensions. Thus we define the projectors

$P_{L}=\frac{1}{2}(1-\gamma_{5}),~{}~{}~{}~{}~{}~{}P_{R}=\frac{1}{2}(1+\gamma_{5})$
(6)

analogously to what done in four dimensions, but here in terms of the object
$\gamma_{5}$ given in (1). Conventionally we still call $P_{L}\Psi$ the “left”
and $P_{R}\Psi$ the “right” chiralities, respectively. However, these are no
more independent fields in $d$-dimensions. Nevertheless, in the basis of the
orthogonal subspaces defined by $P_{L}$, $P_{R}$, we can postulate that under
a $d$-dimensional gauge rotation fermions transform as

$\Psi_{i}\to
U_{ij}\Psi_{j},~{}~{}~{}~{}~{}~{}\overline{\Psi}_{i}\to{\overline{\Psi}}_{j}\,[\overline{U}^{\dagger}]_{ji},$
(7)

where

$U=e^{i\alpha^{A}T^{A}},~{}~{}~{}~{}~{}~{}\overline{U}=\gamma^{0}U\gamma^{0}=e^{i\alpha^{A}\overline{T}^{A}}$
(8)

with $\alpha^{A}$ the gauge parameter and

$T^{A}=P_{L}T^{A}_{L}+P_{R}T^{A}_{R},~{}~{}~{}~{}~{}~{}\overline{T}^{A}=\gamma^{0}T^{A}\gamma^{0}=P_{R}T^{A}_{L}+P_{L}T^{A}_{R}.$
(9)

Having clarified what we mean by chiral gauge transformations in $d$
dimensions, we can now have a look at the fermionic interactions in (4). We
start with the second term, which includes the couplings between fermions and
scalars. For definiteness we assume that the global fermion number is
conserved and focus on Dirac-type interactions. Our results can be
straightforwardly generalized to Majorana-type Yukawas, but we feel no urgency
to do so at present given that no technical complication would emerge in that
case and, moreover, the Standard Model belongs to the class of scenarios we
consider. We define

${\cal
L}^{\text{Reg}}_{\text{Yuk}}=-Y^{a}_{ij}\overline{\Psi_{i}}P_{R}\Psi_{j}\phi_{a}+{\rm
h.c.},$ (10)

where gauge-invariance in $d$-dimensions is ensured if

$[T^{A}_{L}]_{ik}Y^{a}_{kj}-Y^{a}_{ik}[T^{A}_{R}]_{kj}=Y^{b}_{ij}[T^{A}_{\phi}]_{ba}.$
(11)

The structure of the regularized Yukawa operator is formally the same as in
four dimensions. The story, however, complicates when considering the
fermionic kinetic term.

A natural generalization of the Dirac Lagrangian to $d$ dimensions leads to

$\displaystyle{\cal
L}_{\text{Kin}}^{\text{Reg}}=\overline{\Psi}i\gamma^{\bar{\mu}}D_{\bar{\mu}}\Psi+\frac{1}{2}\left\\{\overline{\Psi}\gamma^{\hat{\mu}}i\left(\partial_{\hat{\mu}}-icA^{A}_{\hat{\mu}}T^{A}\right)\Psi+{\rm
h.c.}\right\\},$ (12)

where $D_{\mu}\Psi=(\partial_{\mu}-iA^{A}_{\mu}T^{A})\Psi$ and $c$ is an
arbitrary number (more on this below). Crucially, (12) inevitably breaks the
chiral symmetries. Indeed, under (7) the fermion current transforms as

$\overline{\Psi}\gamma^{\mu}\Psi\to\overline{\Psi}\,\overline{U}^{\dagger}\gamma^{\mu}U\Psi=\begin{cases}\overline{\Psi}\gamma^{\bar{\mu}}\Psi&{\text{for}}~{}\mu=\bar{\mu}\\\
\overline{\Psi}\,\overline{U}^{\dagger}U\gamma^{\hat{\mu}}\Psi&{\text{for}}~{}\mu=\hat{\mu},\end{cases}$
(13)

where in the last step we used (2), according to which the gauge matrices
satisfy $\gamma^{\hat{\mu}}U=U\gamma^{\hat{\mu}}$ and
$\gamma^{\bar{\mu}}U=\overline{U}\gamma^{\bar{\mu}}$. Therefore, the
evanescent part of the kinetic term in (12) cannot be invariant if
$\overline{U}\neq U$, namely if the transformation is chiral. The problem
obviously applies to both global and local chiral symmetries.

Because chiral invariance is anyway lost, we can appreciate the meaning of the
parameter $c$ introduced in (12). For $c=1$, the evanescent derivative in the
brackets of (12) is covariant under the entire $d$-dimensional gauge
rotations, i.e. under local transformations (7) with arbitrary
$\alpha^{A}(x)$. Yet, since the chiral symmetry is broken there is no reason
to prefer $c=1$ over any other choice. Hence the value of $c$ can be taken to
be arbitrary, it merely defines a scheme-dependence. We will work with $c=0$
in our explicit calculations.

To conclude, the action given in (4), with the three components shown in (5),
(10) and (12), is the dimensionally regularized version of the theory we are
interested in. For this reason we will refer to it as the _standard_
regularized theory. Regrettably, it is not invariant under chiral
transformations because of the evanescent part in (12). Hence one is forced to
introduce symmetry-restoring counterterms order by order to ensure the theory
satisfies the generalized Ward identities for chiral symmetries, as already
anticipated below Eq. (3). There is instead no need of symmetry-restoring
counterterms for vector-like internal symmetries, as these are preserved by
our regularization.

It is also important to stress that the presence of the chirality projectors
in (10) and (12) breaks $d$-dimensional Lorentz invariance. Only the four-
dimensional subgroup $O(1,3)$ and rotations $O(d-4)$ in the extra dimensions
are preserved. This is a consequence of the fact that $\gamma_{5}$ is not
covariant since the $\bar{\mu}\hat{\nu}$ components of the Lorentz generators
do not commute with it. But this is not a serious problem in practice because
we are ultimately interested in the four-dimensional physics, and our
regularized Lagrangian is invariant under the four-dimensional Lorentz
symmetry. There is thus no need to add Lorentz-restoring counterterms. The
only drawback is that the bare correlator functions derived from (4) are not
$d$-dimensional Lorentz covariant but rather $O(1,3)\times O(d-4)$ tensors.

###  2.2 Spurious chiral-invariance

The violation of the chiral symmetries appearing in (13) is somehow
reminiscent of the explicit breaking of the chiral $SU(N_{f})\times SU(N_{f})$
symmetry of QCD induced by a quark mass proportional to the identity. In that
context a way to describe low-energy QCD while still enjoying the full power
of $SU(N_{f})\times SU(N_{f})$ is well known: one introduces an appropriate
Nambu-Goldstone matrix and “dresses” the quark mass term with them. This way
one can interpret the quark mass as a field transforming covariantly under
chiral transformations and the full chiral symmetry is formally restored. This
trick proves extremely useful when analyzing the dependence of physical
observables on quark masses.

Borrowing from that experience we introduce in our theory an auxiliary, non-
dynamical scalar field matrix $\Omega$ transforming as

$\Omega\to e^{i\alpha^{A}T_{L}^{A}}\Omega e^{-i\alpha^{A}T_{R}^{A}}$ (14)

under the gauge symmetry (or analogously for global chiral rotations). With
$\Omega$ at our disposal, we can formally recover invariance under the full
chiral symmetry and retain control over the symmetry-breaking consequences of
our regularization. It is sufficient to introduce the combination

$(\Omega^{\dagger}P_{L}+\Omega
P_{R})\to\overline{U}(\Omega^{\dagger}P_{L}+\Omega P_{R})U^{\dagger}$ (15)

in the evanescent component of the fermion kinetic term and replace the
regularized action (4) with

$\displaystyle S^{\text{Reg}}[\xi,\Omega]$
$\displaystyle\equiv\int\text{d}^{d}x\left[{\cal
L}_{\text{Bos}}^{\text{Reg}}+{\cal L}_{\text{Yuk}}^{\text{Reg}}\right]$ (16)
$\displaystyle+\frac{1}{2}\int\text{d}^{d}x\left\\{\overline{\Psi}\gamma^{\bar{\mu}}iD_{\bar{\mu}}\Psi+\overline{\Psi}\gamma^{\hat{\mu}}\left(\Omega^{\dagger}P_{L}+\Omega
P_{R}\right)i\left(\partial_{\hat{\mu}}-icA^{A}_{\hat{\mu}}T^{A}\right)\Psi+{\rm
h.c.}\right\\}.$

The non-symmetric standard theory of Section 2.1 is a particular case of (16)
in which the auxiliary field is set to a background equal to the identity
matrix

$\Omega\to{\mathbb{1}},$ (17)

namely $S^{\text{Reg}}[\xi,{\mathbb{1}}]=S_{\text{std}}^{\text{Reg}}[\xi]$.
Yet, when $\Omega$ is turned on, (16) is formally invariant under _spurious_
$d$-dimensional gauge transformations (namely gauge transformations with
$\alpha^{A}=\alpha^{A}(x^{\mu})$ functions of $x^{\mu}$) if $c=1$, or only
invariant under the _spurious_ four-dimensional restriction (namely gauge
transformations with $\alpha^{A}=\alpha^{A}(x^{\bar{\mu}})$ functions of
$x^{\bar{\mu}}$ alone) if $c\neq 1$. The local rotation of the dynamical
fields is exactly compensated by the rotation (14) of the auxiliary field
$\Omega$. For this reason we will refer to the combined action of ordinary
gauge rotations plus (14) as a _spurious gauge transformation_. Similarly, if
$\Omega$ is also assumed to transform appropriately under global chiral
rotations we are guaranteed that a spurious version of the global chiral
transformations preserved by the Yukawa interactions remain conserved as well.
(A generalization with spurious $d$-dimensional Lorentz and gauge invariance
is also presented in Appendix B.)

From the defining property (14) and by analogy with the theory of pions in
QCD, we know that we can take $\Omega$ to be a special unitary matrix, a
condition obviously consistent with the background value reported in (17). We
will make heavy use of the convention
$\Omega^{\dagger}\Omega=\Omega\Omega^{\dagger}=1$ in Section 3.

Besides being invariant under spurious chiral transformations, (16) is also
endowed with the same global symmetries of the standard regularized theory: it
is invariant under $d$-dimensional translations, four-dimensional Lorentz
transformations as well as rotations in the extra dimensions; moreover, it is
symmetric under spurious versions of P and C, under which

$\begin{split}A^{A}_{\mu}(x)&\xrightarrow[]{\text{P}}{\cal
P}_{\mu}^{\nu}A^{A}_{\nu}({\cal P}x)\\\
\Psi_{i}(x)&\xrightarrow[]{\text{P}}\gamma^{0}\Psi_{i}({\cal P}x)\\\
\phi_{a}(x)&\xrightarrow[]{\text{P}}\phi_{a}({\cal P}x)\\\
\Omega_{ij}(x)&\xrightarrow[]{\text{P}}\Omega^{*}_{ji}({\cal P}x)\\\
T^{A}_{ij}&\xrightarrow[]{\text{P}}\overline{T}^{A}_{ij}\\\
\overline{T}^{A}_{ij}&\xrightarrow[]{{P}}{T}^{A}_{ij}\\\
[T_{\phi}^{A}]_{ab}&\xrightarrow[]{\text{P}}[T_{\phi}^{A}]_{ab}\\\
Y_{ij}^{a}&\xrightarrow[]{\text{P}}{Y_{ji}^{a}}^{*}\end{split}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\begin{split}A^{A}_{\mu}(x)&\xrightarrow[]{\text{C}}-A^{A}_{\mu}(x)\\\
\Psi_{i}(x)&\xrightarrow[]{\text{C}}C\overline{\Psi}_{i}^{t}(x)\\\
\phi_{a}(x)&\xrightarrow[]{\text{C}}\phi_{a}(x)\\\
\Omega_{ij}(x)&\xrightarrow[]{\text{C}}\Omega_{ij}^{*}(x)\\\
T^{A}_{ij}&\xrightarrow[]{\text{C}}{T_{ij}^{A}}^{*}\\\
\overline{T}_{ij}^{A}&\xrightarrow[]{\text{C}}{\overline{T}_{ij}^{A}}^{*}\\\
[T_{\phi}^{A}]_{ab}&\xrightarrow[]{\text{C}}[T_{\phi}^{A}]_{ab}^{*}\\\
Y^{a}_{ij}&\xrightarrow[]{\text{C}}{Y^{a}_{ij}}^{*}\end{split}$ (18)

where ${\cal P}_{\mu}^{0}x^{\mu}=x^{0}$ and ${\cal P}_{\mu}^{\nu\neq
0}x^{\mu}=-x^{\nu\neq 0}$. These relations implicitly involve a unitary
rotation of the fields, which for brevity is not displayed. Note that a
definition of the operator $C$ with the same properties of its four-
dimensional version can be generalized to $d\neq 4$ dimensions [7].

##  3 Structure of the radiative corrections

In this section we will see why working with (16) is more convenient than with
the standard theory in (4). The basic reason is that calculations carried out
with (16) are symmetric under spurious chiral rotations. This significantly
constrains the structure of the radiative corrections and provides a precious
guide throughout any computation.

In this paper we will adopt the background gauge formalism and a background-
gauge invariant gauge-fixing [20, 21, 22]

$S_{\text{gf}}[a,A_{c}]=-\frac{1}{2}(D_{c}^{\mu}a_{\mu})^{A}(D_{c}^{\nu}a_{\nu})^{A}$
(19)

where $D_{c}^{\mu}a_{\mu}=\partial^{\mu}a_{\mu}-i[A_{c}^{\mu},a_{\mu}]$
denotes the covariant derivative of the quantum fluctuation
$a_{\mu}=A_{\mu}-A_{c\,\mu}=a_{\mu}^{A}T^{A}$ with respect to the background
field $A_{c\,\mu}=A_{c\,\mu}^{A}T^{A}$. Note that (19) corresponds to a
Feynman-’t Hooft gauge for all (abelian and simple) group factors.

The central observation of our paper is the following. The generating
functional of 1-particle irreducible diagrams constructed with (16) and (19)
is invariant under the spurious chiral symmetry. It can hence be split into
two terms: the first involves amplitudes that are truly invariant and the
second contains amplitudes that are invariant _only_ because of $\Omega$. The
latter class is associated to the “symmetry-violating” contributions
characterizing the standard theory of Eq. (4). The explicit dependence on
$\Omega$ thus allows us to straightforwardly isolate the offending terms, both
the finite and singular parts. The spurious chiral-invariance is not only a
very useful bookkeeping device, though, but in fact it also leads to practical
advantages. One important advantage is in the extraction of the symmetry-
restoring counterterms, which will be discussed in detail below. Another is
that the “symmetry-violating” terms may be determined by inspecting diagrams
with external $\Omega$ lines, a procedure which may simplify their evaluation
in some cases, as one for example is dispensed from having to deal with the
different polarizations of the vector fields. 333We thank Sergey Sibiryakov
for emphasizing this aspect.

###  3.1 Symmetry-restoring counterterms: general considerations

The standard procedure used to identify the symmetry-restoring counterterms in
a theory satisfying (3) consists in computing a naive one-particle irreducible
(1PI) effective action with the standard bare action (4), and taking its
symmetry variation; this gives the “anomaly”, which order by order in
parturbation theory is a local functional. Alternatively one can directly
compute the anomaly order by order via the insertion of appropriate local
operators. Subsequently one needs to formally integrate the anomaly in order
to find an appropriate set of symmetry-restoring counterterms whose variation
precisely cancels the anomaly order by order in the $\hbar$ expansion.

In this subsection we show how the determination of the symmetry-restoring
counterterms is considerably simplified by the introduction of $\Omega$. The
basic object we are interested in is the 1PI effective action for the
regularized theory $S^{\text{Reg}}$ given in (16), defined with the
(background) gauge-invariant gauge-fixing (19) and the associated ghost term
$S_{\text{ghost}}$. Treating $\Omega$ as a non-dynamical external source, that
quantity is defined as:

$e^{i\Gamma_{\text{inv}}[\xi_{c},\Omega]}=\begin{subarray}{c}{\text{LIM}}\\\
{d\to 4}\end{subarray}\int_{\rm 1PI}{\cal
D}\xi~{}e^{iS^{\text{Reg}}_{\text{tot}}[\xi,\xi_{c},\Omega]+S_{\text{ct}}^{\text{Div}}[\xi+\xi_{c},\Omega]+S_{\text{ct}}^{\text{Fin}}[\xi+\xi_{c},\Omega]},$
(20)

where $\xi$ collectively denotes all quantum degrees of freedom (including
ghosts), $\xi_{c}$ their backgrounds, and

$S^{\text{Reg}}_{\text{tot}}[\xi,\xi_{c},\Omega]=S^{\text{Reg}}[\xi+\xi_{c},\Omega]+S_{\text{gf}}[a,A_{c}]+S_{\text{ghost}}[\xi,\xi_{c}].$
(21)

By $\begin{subarray}{c}{\text{LIM}}\\\ {d\to 4}\end{subarray}$ we indicate the
limit in which all evanescent terms are dropped and $d\to 4$, whereas the
counterterms are defined as follows. The “divergent” counterterms
$S_{\text{ct}}^{\text{Div}}$ are introduced to absorb all (evanescent and non-
evanescent) $1/(d-4)$ poles according to the $\overline{\text{MS}}$
renormalization scheme. Their determination is standard. The only novelty
added by our formalism is that $S_{\text{ct}}^{\text{Div}}$ can be taken to be
invariant under the spurious chiral symmetry. Indeed,
$S^{\text{Reg}}_{\text{tot}}$ in (21) is symmetric under a backgound gauge
transformation (combined with a linear field redefinition of the quantum
field); as a result all divergences induced by it are chiral-invariant order
by order, and this in turn guarantees that $S_{\text{ct}}^{\text{Div}}$ can be
taken to be invariant as well. Analogously, finite counterterms
$S_{\text{ct}}^{\text{Fin}}$ invariant under the spurious chiral symmetry can
be added at will. The resulting expression
$\Gamma_{\text{inv}}[\xi_{c},\Omega]$ is therefore invariant by construction:
a chiral transformation of the background fields $\xi_{c}$ gets exactly
compensated by a transformation of the spurion $\Omega$.

We define $S_{\text{ct}}^{\text{Fin}}$ such that chiral-invariance of (20) is
also ensured in the limit $\Omega={\mathbb{1}}$, where (20) reduces to the
invariant 1PI effective action of the standard theory, i.e.
$\Gamma_{\text{inv}}[\xi_{c},{\mathbb{1}}]$. Hence
$S_{\text{ct}}^{\text{Fin}}[\xi_{c},{\mathbb{1}}]$ are the symmetry-restoring
counterterms we alluded to below (3). To identify them we follow an iterative
procedure. Assume we have removed all divergences up to the $n$-th order in
perturbation theory, but that chiral-invariance of the standard theory is only
satisfied up to the order $n-1$. In other words, assume we have computed

$e^{i\Gamma[\xi_{c},\Omega]|_{(n)}}=\begin{subarray}{c}{\text{LIM}}\\\ {d\to
4}\end{subarray}\int_{\rm 1PI}{\cal
D}\xi~{}e^{iS^{\text{Reg}}_{\text{tot}}[\xi,\xi_{c},\Omega]+iS_{\text{ct}}^{\text{Div}}[\xi+\xi_{c},\Omega]|_{(n)}+iS_{\text{ct}}^{\text{Fin}}[\xi+\xi_{c},\Omega]|_{(n-1)}},$
(22)

where $|_{(n)}$ indicates that the corresponding counterterm contains all loop
effects up to the $n$-th order. Under our assumptions
$\Gamma[\xi_{c},\Omega]|_{(n)}$ is finite. Furthermore,
$S_{\text{ct}}^{\text{Fin}}[\xi,\Omega]|_{(n-1)}$ ensures that
$\Gamma[\xi_{c},{\mathbb{1}}]|_{(n-1)}$ be chiral-invariant, though in general
$\Gamma[\xi_{c},{\mathbb{1}}]|_{(n)}$ is not. Our goal is to see how chiral-
invariance of the standard theory is restored at order $n$, namely we want to
find $S_{\text{ct}}^{\text{Fin}}[\xi,\Omega]|_{(n)}$.

In general we can split our functional (22) in a $\Omega$-dependent part

$\Gamma_{\Omega}[\xi_{c},\Omega]|_{(n)}$ (23)

and a $\Omega$-independent part

$\Gamma_{\not{\Omega}}[\xi_{c}]|_{(n)}\equiv\Gamma[\xi_{c},\Omega]|_{(n)}-\Gamma_{\Omega}[\xi_{c},\Omega]|_{(n)}.$
(24)

Because a chiral transformation cannot mix $\Omega$-dependent and
$\Omega$-independent terms, both $\Gamma_{\Omega}|_{(n)}$ and
$\Gamma_{\not{\Omega}}|_{(n)}$ must be separately invariant under the spurious
chiral symmetry. An infinitesimal symmetry variation of the background fields
leaves the latter completely invariant, i.e.

$\delta_{\xi_{c}}(\Gamma_{\not{\Omega}}[\xi_{c}]|_{(n)})=0,$ (25)

whereas generically acts non-trivially on $\Gamma_{\Omega}|_{(n)}$. In other
words, $\Gamma_{\not{\Omega}}|_{(n)}$ is genuinely chiral-invariant while
$\Gamma_{\Omega}|_{(n)}$ is only invariant because of $\Omega$. The quantity

$\delta_{\xi_{c}}(\Gamma_{\Omega}[\xi_{c},{\mathbb{1}}]|_{(n)})\equiv{\cal
A}[\xi_{c}]|_{(n)}$ (26)

corresponds to the chiral anomaly of the $n$-th order 1PI action of the
standard theory. From this immediately follows that
$\Gamma_{\Omega}[\xi_{c},{\mathbb{1}}]|_{(n)}$ must be local, since the
anomaly is local and the variation $\delta_{\xi_{c}}$ cannot turn a non-local
functional into a local one.

Now, the part of the symmetry-restoring counterterm of order $n$

$\Delta
S_{\text{ct}}^{\text{Fin}}[\xi_{c},{\mathbb{1}}]|_{(n)}=S_{\text{ct}}^{\text{Fin}}[\xi_{c},{\mathbb{1}}]|_{(n)}-S_{\text{ct}}^{\text{Fin}}[\xi_{c},{\mathbb{1}}]|_{(n-1)}$
(27)

is implicitly defined by

$\delta_{\xi_{c}}(\Delta
S_{\text{ct}}^{\text{Fin}}[\xi_{c},{\mathbb{1}}]|_{(n)})=-{\cal
A}[\xi_{c}]|_{(n)}.$ (28)

Comparing (26) and (28) we see that the $n$-th order contribution to the
symmetry variation of
$S_{\text{ct}}^{\text{Fin}}[\xi_{c},{\mathbb{1}}]|_{(n)}$ must be the opposite
of the variation of $\Gamma_{\Omega}[\xi_{c},{\mathbb{1}}]|_{(n)}$, see (26).
This implies that the two functionals coincide up to a term that is invariant
and $\Omega$-independent — and hence by construction residing in
$\Gamma_{\not{\Omega}}|_{(n)}$. Conventionally taking the $\Omega$-independent
chiral-invariant term equal to zero, without loss of generality we can thus
identify

$\Delta
S_{\text{ct}}^{\text{Fin}}[\xi_{c},{\mathbb{1}}]|_{(n)}=-\Gamma_{\Omega}[\xi_{c},{\mathbb{1}}]|_{(n)}.$
(29)

Our conclusion is therefore that the $n$-th order correction $\Delta
S_{\text{ct}}^{\text{Fin}}[\xi_{c},{\mathbb{1}}]|_{(n)}$ necessary to restore
chiral invariance in the standard theory can be obtained iteratively by
computing (the opposite of) the finite, $\Omega$-dependent part of the $n$-th
loop order 1PI functional (22) and sending $\Omega\to{\mathbb{1}}$ at the very
end. There is no need of taking an infinitesimal symmetry variation and
subsequently integrating it.

Armed with the finite counterterm one can construct the chiral-invariant 1PI
effective action of the standard theory at $n$-th order:
$\Gamma_{\text{inv}}[\xi_{c},{\mathbb{1}}]|_{(n)}=\Gamma[\xi_{c},{\mathbb{1}}]|_{(n)}+\Delta
S_{\text{ct}}^{\text{Fin}}[\xi_{c},{\mathbb{1}}]|_{(n)}$. Iterating the above
procedure one obtains (20) at the desired order in the perturbative expansion.

At 1-loop the analysis is particularly simple. In fact, because the classical
anomaly is evanescent, its effect can appear only as a $1/(d-4)$ singular
evanescent amplitude or as a finite amplitude resulting from the compensation
of a $1/(d-4)$ pole and evanescent terms. As a result
$\begin{subarray}{c}{\text{LIM}}\\\ {d\to
4}\end{subarray}\;S_{\text{ct}}^{\text{Div}}$ is genuinely symmetric (even
without $\Omega$) and the 1-loop anomaly is manifestly local, as expected on
general grounds.

###  3.2 Symmetry-restoring counterterms: 1-loop calculation

As an application of our spurion trick we now derive
$S_{\text{ct}}^{\text{Fin}}$ at the first non-trivial order using the
identification (29).

Strictly speaking, $\Gamma_{\Omega}|_{(1)}$ should be evaluated with a space-
time dependent $\Omega$. Yet, in our analysis we found it convenient to assume
$\Omega=\Omega_{0}$ is an arbitrary constant unitary matrix. This way the
fermionic propagator can be easily found for arbitrary $\Omega_{0}$,

$\frac{i}{p^{2}}\left[\gamma^{\bar{\mu}}p_{\bar{\mu}}+\left(\Omega_{0}P_{L}+\Omega_{0}^{\dagger}P_{R}\right)\gamma^{\hat{\mu}}p_{\hat{\mu}}\right],$
(30)

and $\Gamma[\xi_{c},\Omega_{0}]|_{(1)}$ straightforwardly computed. However,
with such a choice the covariant derivatives of $\Omega$ that appear in
$\Gamma_{\Omega}$ do not display the $\partial_{\mu}\Omega$ term that would
allow us to immediately identify the corresponding gauge-invariant operators.
In practice, only a subset of the terms in $\Gamma_{\Omega}|_{(1)}$ depend
explicitly on $\Omega_{0}$: the operators that are not invariant under the
_chiral_ global symmetry when $\Omega_{0}\to{\mathbb{1}}$. Nevertheless,
identifying them turns out to be sufficient to fully reconstruct all symmetry-
restoring counterterms because of the underlying spurious gauge-invariance
enjoyed by $\Gamma|_{(1)}$. In fact, we can extract the symmetry-restoring
counterterms as follows:

  * Step 1:

First we compute the $\Omega_{0}$-dependent terms in
$\Gamma_{\Omega}[\xi_{c},\Omega_{0}]|_{(1)}$. For this, it is enough to
compute the hard region of $\Gamma[\xi_{c},\Omega_{0}]|_{(1)}$, because all
the interactions of $\Omega$ are evanescent and therefore can only contribute
if multiplied by the UV poles of the divergent integrals. In other words,
$\Gamma_{\Omega}[\xi_{c},\Omega_{0}]|_{(1)}\equiv\Gamma^{\mathrm{(hard)}}_{\Omega}[\xi_{c},\Omega_{0}]|_{(1)}$.

  * Step 2:

We then write a basis of counterterms
$S_{\text{ct}}^{\text{Fin}}[\xi_{c},\Omega]|_{(1)}$ invariant under the
spurious chiral symmetry and the spurious P and C of (18), with arbitrary
coefficients, and evaluate the basis for $\Omega=\Omega_{0}$.

  * Step 3:

According to (29), the $\Omega_{0}$-dependent terms in
$\Gamma_{\Omega}[\xi_{c},\Omega_{0}]|_{(1)}$ must exactly match (minus) those
in $S_{\text{ct}}^{\text{Fin}}[\xi_{c},\Omega_{0}]|_{(1)}$. Consistently, this
is indeed the case: all the coefficients of the counterterm basis are uniquely
determined.

The 1-loop expression of $\Gamma_{\Omega}[\xi_{c},\Omega_{0}]|_{(1)}$ is
derived for definiteness assuming $c=0$ in (16) and the Feynman-’t Hooft gauge
of (19). The calculation was performed with Matchmakereft [23], in which the
BMHV prescription has been implemented in a version that will be made public
in the future. $\Gamma_{\Omega}^{({\text{hard}})}[\xi_{c},\Omega_{0}]|_{(1)}$
is a local object, and thus can be expressed in a basis of local operators,
via a procedure completely parallel to the matching of UV poles during the
renormalization of an effective field theory. The only difference is that in
the present case the coefficients are finite. Since Matchmakereft is prepared
to compute the hard region of a 1PI action to calculate the renormalization
group equations, it is by construction well suited also to compute
$\Gamma_{\Omega}^{({\text{hard}})}[\xi_{c},\Omega_{0}]|_{(1)}$ automatically.

The list of relevant counterterms
$S_{\text{ct}}^{\text{Fin}}[\xi_{c},\Omega]|_{(1)}$ is discussed below and
summarized in Tables 1, 2, 3, and Eq. (35). Matching the two sets of
$\Omega_{0}$-dependent operators, as explained in Step 3 above, we obtain the
1-loop coefficients reported in the right hand side of the tables. As a
consistency check of our calculation we verified that exactly the same
counterterms and the same coefficients are obtained via the standard method
reviewed in the first paragraph of Section 3.1. Our results also reduce to
those of [11] in the limit in which the scalar fields are decoupled. The
explicit case of the Standard Model can be derived as a particular limit using
the expressions for the generators and the Yukawa coupling collected in
Appendix A.

In conclusion, we see two main concrete advantages of our approach. First, the
symmetry-restoring counterterms are automatically found by computing a subset
of 1-particle irreducible diagrams using (22), there is no need to first
evaluate the quantum anomaly and subsequently integrate it. Second, since we
compute the effective action instead of its gauge variation, the number of
diagrams is smaller. This is because in the calculation of the anomaly using
the standard approach one would need to compute an insertion of the gauge
variation of the classical action for each fermionic propagator, so that the
number of diagrams to compute scales as the number of fermionic propagators in
the amplitude.

####  3.2.1 Basis of counterterms

The counterterms $S_{\text{ct}}^{\text{Fin}}[\xi_{c},\Omega]|_{(1)}$ are
invariants built out of the basic (four-dimensional) building blocks444To
avoid cluttering we do not write the subscript c in the fields. It is
understood from the general analysis of Section 3.1 that these counterterms
are expressed in terms of the backgound fields.

$\Omega,A^{A}_{\bar{\mu}},\Psi_{i},\overline{\Psi_{i}},\phi_{a},D_{\bar{\mu}},\gamma^{\bar{\mu}}$
(31)

as well as the vertex spurions

$T^{A},\overline{T}^{A},T_{\phi}^{A},Y^{a}$ (32)

appropriately contracted among each other with $\delta_{ij}$, $\delta_{AB}$,
$\delta_{ab}$ from the fermion, gauge, scalar propagators. Note that in our
convention the gauge couplings are included in the generators, so the number
of (32) in a given operator effectively counts the number of coupling
insertions.

The full list of the four-dimensional $\Omega$-dependent structures can be
divided in four classes: “Operators with only vectors and no Levi-Civita”
(Table 1), “Operators with scalars and no fermions” (Table 2), “Operators with
fermions” (Table 3), and finally “Operators with the Levi-Civita tensor” (see
Eq. (35)). All of them are invariant under the spurious chiral symmetry and
also symmetric under the spurious P and C of (18). It is important to
emphasize that not all counterterms must be included in a given theory. To
better appreciate this we distinguish among three classes of theories:

  * •

Theories with chiral gauge symmetries. In this case all the counterterms in
the tables are in general necessary (though some might identically vanish in a
specific theory). This is the case of the Standard Model, which can be
recovered with the help of Appendix A.

  * •

Theories with vector-like gauge symmetries and global chiral symmetries. Now
all counterterms containing a covariant derivative of $\Omega$ vanish and only
the two on the left of Table 2 and the one on the right of Table 3 can
survive. The presence of a $\psi^{2}\phi$ term in particular implies that in
the Standard Model we have a Yukawa-like counterterm proportional to the QCD
and Yukawa couplings.

  * •

Theories with no chiral symmetries. Obviously, when no chiral symmetry (global
nor local) is present one does not need to add any symmetry-restoring
counterterm. $\Omega$ is not necessary in the first place.

In identifying the structure of the finite counterterms one essentially
follows the same rules used to construct the chiral Lagrangian for the pion
matrix, whose role in the present context is played by $\Omega$. There are
only a couple of important differences. First, the would-be kinetic term
$\langle\partial_{\mu}\Omega^{\dagger}\partial^{\mu}\Omega\rangle$, where
$\langle\,\dots\rangle$ indicates the matrix trace, does not appear at any
order because of dimensional considerations. Hence, as opposed to what is done
in the chiral Lagrangian, one cannot remove some of the higher-derivative
interactions making field redefinitions of $\Omega$.

Second, here the underlying theory is assumed to be perturbative, and this
fact constrains the type of operators that can be generated at each order in
the perturbative expansion. The set of counterterms that are necessary is
therefore a very small subgroup of all operators compatible with the spurious
symmetries. To start, at 1-loop order and with $c=0$ the counterterms contain
at most a trace over the flavor indices; double-trace operators will be
generated starting at two loops. Furthermore, an inspection of the topology of
the 1PI diagrams immediately suggests which structures can be relevant in each
class. Let us discuss this more explicitly.

“Operators with only vectors and no Levi-Civita” are generated by a single
fermionic loop with no virtual bosons. Hence, no explicit dependence on the
Yukawa coupling nor the gauge generators can appear. Taking this into account
the complete set of invariant $\Omega$-dependent single-trace dimension-4
operators without (32) is shown in Table 1. The 1-loop coefficients of the
counterterms, in units of $1/(16\pi^{2})$, is shown for each of the operators
in the second column of Table 1. We note that a linear combination of those
operators is also gauge-invariant (up to a total derivative of some vector
$T^{\mu}$) in the limit $\Omega\to{\mathbb{1}}$, since

$\displaystyle\langle
L_{\bar{\mu}\bar{\nu}}L^{\bar{\mu}\bar{\nu}}+R_{\bar{\mu}\bar{\nu}}R^{\bar{\mu}\bar{\nu}}\rangle+\partial_{\mu}T^{\mu}=$
(33) $\displaystyle\langle
R_{\mu\nu}\Omega^{\dagger}L^{\mu\nu}\Omega-{i}\left(R_{\mu\nu}D^{\mu}\Omega^{\dagger}D^{\nu}\Omega+L_{\mu\nu}D^{\mu}\Omega
D^{\nu}\Omega^{\dagger}\right)+D_{\mu}D_{\nu}\Omega
D^{\mu}D^{\nu}\Omega^{\dagger}-D_{\mu}D^{\mu}\Omega
D_{\nu}D^{\nu}\Omega^{\dagger}\rangle_{\Omega={\mathbb{1}}},$

where we defined $L_{\bar{\mu}}=A^{A}_{\bar{\mu}}T^{A}_{L}$ and
$L_{\bar{\mu}\bar{\nu}}=i[D_{\bar{\mu}},D_{\bar{\nu}}]$, and analogously for
$R_{\bar{\mu}}$. The coefficient of the combination (33) therefore depends on
the renormalization scheme.

Similarly, “Operators with only scalars” are generated by a fermionic loop
with external $\phi$ or $\Omega$. Hence they can only depend on the external
$\phi$ via the gauge-covariant combination

$\Phi_{ij}=Y^{a}_{ij}\phi_{a}.$ (34)

All hermitian, single-trace and dimension-4 operators involving $\Phi$ and
compatible with our symmetries have non-vanishing coefficients at 1-loop, see
Table 2.

Analogous considerations demonstrate that only a very limited number of
“Operators with fermions” is generated. The topology of the 1-loop diagrams
evidently selects what string of couplings and spurions is chained within the
external $\overline{\Psi}$ and $\Psi$ lines. The counterterms induced in our
1-loop analysis are shown in Table 3 along with their coefficients.

$\boldsymbol{D^{4}}$
---
$\langle L_{\bar{\mu}\bar{\nu}}\Omega R^{\bar{\mu}\bar{\nu}}\Omega^{\dagger}\rangle$ | $0$
$i\langle L_{\bar{\mu}\bar{\nu}}D^{\bar{\mu}}\Omega D^{\bar{\nu}}\Omega^{\dagger}+R_{\bar{\mu}\bar{\nu}}D^{\bar{\mu}}\Omega^{\dagger}D^{\bar{\nu}}\Omega\rangle$ | $-\frac{1}{2}$
$\langle D_{\bar{\mu}}\Omega D^{\bar{\mu}}\Omega^{\dagger}D_{\bar{\nu}}\Omega D^{\bar{\nu}}\Omega^{\dagger}+D_{\bar{\mu}}\Omega^{\dagger}D^{\bar{\mu}}\Omega D_{\bar{\nu}}\Omega^{\dagger}D^{\bar{\nu}}\Omega\rangle$ | $-\frac{1}{6}$
$\langle D_{\bar{\mu}}\Omega D_{\bar{\nu}}\Omega^{\dagger}D^{\bar{\mu}}\Omega D^{\bar{\nu}}\Omega^{\dagger}\rangle$ | $+\frac{1}{12}$
$\langle D_{\bar{\mu}}D^{\bar{\mu}}\Omega D_{\bar{\nu}}D^{\bar{\nu}}\Omega^{\dagger}\rangle$ | $0$
$\langle D_{\bar{\mu}}D_{\bar{\nu}}\Omega D^{\bar{\mu}}D^{\bar{\nu}}\Omega^{\dagger}\rangle$ | $+\frac{1}{6}$
Table 1: Counterterms and 1-loop coefficients (in units of ${1}/({16\pi^{2}})$) in the class “Operators with only vectors and no Levi-Civita”. The symbol $\langle\,\dots\rangle$ stands for the trace in the fermionic indices. We take $c=0$ in (16) and work in the Feynman-’t Hooft gauge (19). $\boldsymbol{\phi^{4}}$ | $\boldsymbol{\phi^{2}D}$
---|---
$\langle(\Phi\Omega^{\dagger})^{4}\rangle+{\text{h.c.}}$ | $-\frac{1}{12}$ | $\langle\Phi D_{\bar{\mu}}\Omega^{\dagger}\Phi D^{\bar{\mu}}\Omega^{\dagger}\rangle+{\text{h.c.}}$ | $+\frac{1}{3}$
$\langle(\Phi\Omega^{\dagger})^{2}\Phi\Phi^{\dagger}\rangle+{\text{h.c.}}$ | $-\frac{2}{3}$ | $\langle(\Phi\Omega^{\dagger})^{2}D_{\bar{\mu}}\Omega D^{\bar{\mu}}\Omega^{\dagger}\rangle+{\text{h.c.}}$ | $-\frac{1}{3}$
|  | $\langle\Phi\Phi^{\dagger}D_{\bar{\mu}}\Omega D^{\bar{\mu}}\Omega^{\dagger}+\Phi^{\dagger}\Phi D_{\bar{\mu}}\Omega^{\dagger}D^{\bar{\mu}}\Omega\rangle$ | $-\frac{1}{3}$
|  | $\langle\Phi\Omega^{\dagger}D_{\bar{\mu}}\Omega\Phi^{\dagger}\Omega D^{\bar{\mu}}\Omega^{\dagger}\rangle$ | $+\frac{1}{3}$
|  | $\langle D_{\bar{\mu}}\Phi D^{\bar{\mu}}\Omega^{\dagger}\Phi\Omega^{\dagger}+D_{\bar{\mu}}\Phi\Omega^{\dagger}\Phi D^{\bar{\mu}}\Omega^{\dagger}\rangle+{\text{h.c.}}$ | $+\frac{2}{3}$
|  | $\langle D_{\bar{\mu}}\Phi\Omega^{\dagger}D^{\bar{\mu}}\Phi\Omega^{\dagger}\rangle+{\text{h.c.}}$ | $+\frac{1}{6}$
|  | $\langle\Phi\overleftrightarrow{D}_{\bar{\mu}}\Phi^{\dagger}\Omega D^{\bar{\mu}}\Omega^{\dagger}+\Phi^{\dagger}\overleftrightarrow{D}_{\bar{\mu}}\Phi\Omega^{\dagger}D^{\bar{\mu}}\Omega\rangle$ | $+\frac{1}{3}$
Table 2: Same as in Table 1 for the class “Operators with scalars and no fermions”. $\boldsymbol{\psi^{2}D}$ | $\boldsymbol{\psi^{2}\phi}$
---|---
$\overline{\Psi}\gamma^{\bar{\mu}}T_{L}^{A}\Omega iD_{\bar{\mu}}\Omega^{\dagger}T_{L}^{A}P_{L}\Psi+{\text{P.c.}}$ | $1$ | $[\overline{\Psi}T_{R}^{A}\Omega^{\dagger}\Phi\Omega^{\dagger}T_{L}^{A}P_{L}\Psi+{\text{P.c.}}]+{\text{h.c.}}$ | $-2$
$\overline{\Psi}\gamma^{\bar{\mu}}Y^{a}\Omega^{\dagger}iD_{\bar{\mu}}\Omega[Y^{a}]^{\dagger}P_{L}\Psi+{\text{P.c.}}$ | $\frac{1}{2}$ |  |
Table 3: Same as in Table 1 for the class “Operators with fermions”. By P.c.
we mean “parity conjugate”.

The set “Operators with the Levi-Civita tensor” is special. As well known, not
all $\Omega$-dependent terms in the chiral Lagrangian can be written as
invariant operators of a four-dimensional functional. There is only one
exception: the Wess-Zumino-Witten term [24, 25]. In a theory satisfying (3),
the gauged version of this term can be written as555The form of the gauged
Wess-Zumino-Witten term originally presented in [25] was not completely
correct. We use the corrected version subsequently found in [26, 27, 28] (see
also [29]).

$\left.S_{\text{ct}}^{\rm
Fin}[\xi,\Omega]\right|_{\text{WZW}}=\frac{n}{48\pi^{2}}\left\\{\int\text{d}^{4}x~{}\epsilon^{\bar{\mu}\bar{\nu}\bar{\alpha}\bar{\beta}}\,Z_{\bar{\mu}\bar{\nu}\bar{\alpha}\bar{\beta}}+\dots\right\\},$
(35)

where

$\displaystyle Z_{\bar{\mu}\bar{\nu}\bar{\alpha}\bar{\beta}}=$
$\displaystyle\,\langle-\Omega^{\dagger}\partial_{\bar{\mu}}L_{\bar{\nu}}L_{\bar{\alpha}}\Omega
R_{\bar{\beta}}+\Omega\partial_{\bar{\mu}}R_{\bar{\nu}}R_{\bar{\alpha}}\Omega^{\dagger}L_{\bar{\beta}}$
(36) $\displaystyle-$
$\displaystyle\partial_{\bar{\mu}}R_{\bar{\nu}}\Omega^{\dagger}L_{\bar{\alpha}}\Omega
R_{\bar{\beta}}+\partial_{\bar{\mu}}L_{\bar{\nu}}\Omega
R_{\bar{\alpha}}\Omega^{\dagger}L_{\bar{\beta}}$ $\displaystyle+$
$\displaystyle
i\Omega^{\dagger}L_{\bar{\mu}}L_{\bar{\nu}}L_{\bar{\alpha}}\Omega
R_{\bar{\beta}}-i\Omega
R_{\bar{\mu}}R_{\bar{\nu}}R_{\bar{\alpha}}\Omega^{\dagger}L_{\bar{\beta}}$
$\displaystyle+$ $\displaystyle\frac{i}{2}\Omega^{\dagger}L_{\bar{\mu}}\Omega
R_{\bar{\nu}}\Omega^{\dagger}L_{\bar{\alpha}}\Omega R_{\bar{\beta}}+{\cal
O}(\partial\Omega)\rangle,$

with $\epsilon^{0123}=-1$. The dots in (35) indicate terms that cannot be
written as a four-dimensional functional, and that are proportional to the
derivatives of the Goldstones. They vanish when setting $\Omega=1$ and are
therefore not relevant to us.

Thanks to the spurious chiral-invariance we see that the
$\epsilon^{\bar{\mu}\bar{\nu}\bar{\alpha}\bar{\beta}}$-dependent finite
counterterm is fully determined by a unique coefficient $n$. Our 1-loop
computation gives

$n=1,$ (37)

a result that agrees with the earlier calculation of [11]. Importantly, $n$
must be an integer in order for (35) to describe a consistent gauge-invariant
four-dimensional theory [25]. This implies that (37) cannot receive
perturbative corrections. The result $n=1$ is therefore exact at all orders in
perturbation theory.

To conclude we stress that while the focus of this subsection is on
$S_{\text{ct}}^{\text{Fin}}$, our spurious symmetries severely constrain the
counterterms appearing in $S_{\text{ct}}^{\text{Div}}$ as well. Because the
divergent counterterms in general include evanescent contributions, in
identifying the list of operators in $S_{\text{ct}}^{\text{Div}}$ the set of
building blocks (31) should be extended to include also the evanescent
counterparts.

##  4 Conclusions

In concrete quantum field theory calculations it is always preferable to adopt
a regularization scheme that respects as many symmetries as possible. This way
quantum corrections are severely constrained and our efforts can be guided by
symmetry considerations and be subject to powerful crosschecks. Unfortunately,
a consistent treatment of the $\gamma_{5}$ matrix in Dimensional
Regularization results in an explicit breaking of all chiral symmetries, like
those defining the Standard Model gauge symmetry.

In this paper we emphasized that the introduction of appropriate auxiliary
Goldstone fields allows to formally restore the chiral symmetry in
dimensionally-regularized theories with arbitrary gauge group. If combined
with the background gauge technique, chiral-invariance is therefore retained
at all steps. This has the practical advantage that the structure of the
radiative corrections becomes more transparent. The additional, seemingly
unstructured counterterms appearing in Dim-Reg simply emerge because a chiral
dimensionally regularized theory secretly has more fields than its four-
dimensional cousin, i.e. the non-dynamical “Goldstone spurions”. Once one
explicitates those fields, order is re-established.

A straightforward consequence of our formalism is that the symmetry-restoring
counterterms are automatically delivered by the computation of 1PI diagrams,
without the need of performing operator insertions and a subsequent
“integration” of the chiral anomaly, as instead done with the more standard
methods. As a concrete application, we carried out a complete 1-loop
calculation of the finite four-dimensional counterterms necessary to restore
all (non-anomalous) chiral symmetries in renormalizable gauge theories with
scalars and Dirac fermions, the Standard Model being a particular example. A
sharp implication of our formalism is that the finite counterterms
proportional to the Levi-Civita tensor are 1-loop exact.

It would be interesting to apply our formalism to more general theories, like
for example the Standard Model effective field theory, as well as to go beyond
the 1-loop approximation. In this latter case, restoring also full
$d$-dimensional Lorentz invariance might turn out to be especially convenient.

## Acknowledgments

We would like to thank C. Cornella for discussions in the early stages of the
project and S. Sibiryakov for renewing our interest on the topic. The work of
LV was partly supported by the Italian MIUR under contract 202289JEW4
(Flavors: dark and intense), 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. The work was also partially supported by grant FPU19/06813, an
Arqus mobility grant, by MICIU/AEI/10.13039/501100011033 and FEDER/UE (grant
PID2022-139466NB-C21), the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie grant agreement n.
101086085 – ASYMMETRY by the INFN Iniziativa Specifica APINE, and by the
Italian MUR Departments of Excellence grant 2023-2027 ”Quantum Frontiers”.

##  Appendix A The Standard Model case

The Standard Model is a particular case of the theories considered in the
present paper. In this appendix we present the explicit expressions for all
the couplings and tensors necessary to compute our counterterms in such a
theory.

The gauge group is $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ and
the vectors are embedded in a $12$-dimensional field. We can describe the
particle content using an eight-dimensional fermionic multiplet $\Psi_{i}$
(with $i=1,\dots,8$) and a four-component scalar multiplet $\phi_{a}$ (with
$a=1,\dots,4$) given by:

$\Psi=\begin{pmatrix}u\\\ d\\\ \nu\\\ e\\\ \end{pmatrix},\qquad
H=\frac{1}{\sqrt{2}}\begin{pmatrix}\phi_{3}+i\phi_{4}\\\
\phi_{1}+i\phi_{2}\end{pmatrix}.$ (38)

The left-handed components of the up-type ($u$) and the down-type ($d$) quarks
form the familiar quark doublet $q_{L}=(u_{L}\ d_{L})$, and similarly for the
left-handed leptons $\ell_{L}=(\nu_{L}\ e_{L})$. The right-handed neutrinos
$\nu_{R}$ are included to keep the notation symmetric; they eventually
decouple because neutral under
$\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ and because their
Yukawa interactions are assumed to be absent. The fermion family index is
suppressed for simplicity throughout this appendix. Summation over the three
Standard Model generations can be straightforwardly taken into account.

The Higgs doublet $H$ is taken to have hypercharge $+1/2$. The (opposite of
the) Yukawa Lagrangian thus reads

$-\mathcal{L}^{\mathrm{(SM)}}_{\mathrm{Yuk}}=\overline{q_{L}}y_{u}u_{R}\widetilde{H}+\overline{q_{L}}y_{d}d_{R}H+\overline{\ell_{L}}y_{e}e_{R}H+\mathrm{h.c.},$
(39)

with all gauge and flavor indices left as understood and
$\widetilde{H}=i\sigma^{2}H^{*}$.

The SU(3) gauge generators of the left(right)-handed fermion representations
are

$\left.T_{L}\right|_{\text{SU(3)}}=\left.T_{R}\right|_{\text{SU(3)}}=\begin{pmatrix}\frac{1}{2}\lambda&&&\\\
&\frac{1}{2}\lambda&&\\\ &&0&\\\ &&&0\end{pmatrix},$

with $\lambda$ denoting the Gell-Mann matrices (empty space indicates
vanishing components). Similarly, the generators for the fermions of the
electroweak ${\text{SU(2)}}\times{\text{U(1)}}$ gauge group are

$\displaystyle\left.T_{L}^{\left\\{1,2,3\right\\}}\right|_{\text{SU(2)}}=\left\\{\frac{1}{2}\begin{pmatrix}&{\mathbb{1}}&&\\\
{\mathbb{1}}&&&\\\ &&&1\\\
&&1&\end{pmatrix},\frac{1}{2}\begin{pmatrix}&-i{\mathbb{1}}&&\\\
+i{\mathbb{1}}&&&\\\ &&&-i\\\
&&i&\end{pmatrix},\frac{1}{2}\begin{pmatrix}{\mathbb{1}}&&&\\\
&-{\mathbb{1}}&&\\\ &&1&\\\ &&&-1\end{pmatrix}\right\\},$
$\displaystyle\left.T_{R}^{\left\\{1,2,3\right\\}}\right|_{\text{SU(2)}}={\mathbb{0}}$

and

$\left.T_{L}\right|_{\text{U(1)}}=\begin{pmatrix}\frac{1}{6}{\mathbb{1}}&&&\\\
&\frac{1}{6}{\mathbb{1}}&&\\\ &&-\frac{1}{2}&\\\
&&&-\frac{1}{2}\end{pmatrix},\qquad\left.T_{R}\right|_{\text{U(1)}}=\begin{pmatrix}\frac{2}{3}{\mathbb{1}}&&&\\\
&-\frac{1}{3}{\mathbb{1}}&&\\\ &&0&\\\ &&&-1\end{pmatrix},$ (40)

with ${\mathbb{1}}$ the identity matrix in three dimensions and ${\mathbb{0}}$
the vanishing matrix in eight dimensions. The
${\text{SU(2)}}\times{\text{U(1)}}$ generators $T_{\phi}$ for the scalars
$\phi_{a}$ can be written as:

$\displaystyle\left.T_{\phi}^{\left\\{1,2,3\right\\}}\right|_{\text{SU(2)}}=\left\\{\frac{1}{2}\begin{pmatrix}&&&i\\\
&&-i&\\\ &i&&\\\ -i&&&\end{pmatrix},\frac{1}{2}\begin{pmatrix}&&i&\\\ &&&i\\\
-i&&&\\\ &-i&&\end{pmatrix},\frac{1}{2}\begin{pmatrix}&-i&&\\\ i&&&\\\ &&&i\\\
&&-i&\end{pmatrix}\right\\},$
$\displaystyle\left.T_{\phi}\right|_{\text{U(1)}}=\frac{1}{2}\begin{pmatrix}&i&&\\\
-i&&&\\\ &&&i\\\ &&-i&\end{pmatrix}.$ (41)

Finally, the Yukawa couplings $Y^{a}$ acquire the following form:

$\displaystyle Y^{1}=\begin{pmatrix}+y_{u}{\mathbb{1}}&&&\\\
&+y_{d}{\mathbb{1}}&&\\\ &&0&\\\ &&&+y_{e}\end{pmatrix},~{}~{}~{}$
$\displaystyle~{}~{}~{}Y^{2}=\begin{pmatrix}-iy_{u}{\mathbb{1}}&&&\\\
&+iy_{d}{\mathbb{1}}&&\\\ &&0&\\\ &&&+iy_{e}\end{pmatrix},$ $\displaystyle
Y^{3}=\begin{pmatrix}&+y_{d}{\mathbb{1}}&&\\\ -y_{u}{\mathbb{1}}&&&\\\
&&&+y_{e}\\\ &&0&\end{pmatrix},~{}~{}~{}$
$\displaystyle~{}~{}~{}Y^{4}=\begin{pmatrix}&+iy_{d}{\mathbb{1}}&&\\\
+iy_{u}{\mathbb{1}}&&&\\\ &&&+iy_{e}\\\ &&0&\end{pmatrix}.$ (42)

##  Appendix B d-dimensional Lorentz and gauge invariance

The formalism we have invoked in Section 2.2 is familiar from the study of
non-linearly realized symmetries. The original global symmetry
$G=SU(N)_{L}\times SU(N)_{R}$ of the fermionic kinetic term is broken by
$\gamma^{\hat{\mu}}$ down to its diagonal subgroup of vector-like
transformations $H=SU(N)_{V}$. Formally, we can view $\gamma^{\hat{\mu}}$ as a
$G$-tensor invariant under $H$: $\gamma^{\hat{\mu}}\to
h\gamma^{\hat{\mu}}h^{\dagger}=\gamma^{\hat{\mu}}$. To restore the complete
$G$ we must introduce Nambu-Goldstone bosons $\Pi$ and a coset representative
$\sqrt{\Omega}(\Pi)$ transforming as

$\sqrt{\Omega}\to
L\sqrt{\Omega}h^{\dagger}(\Pi)=h(\Pi)\sqrt{\Omega}R^{\dagger},$

with $L\in SU(N)_{L}$ and $R\in SU(N)_{R}$. Then we can define

$\sigma\equiv P_{L}\sqrt{\Omega}^{\dagger}+P_{R}\sqrt{\Omega},$ (43)

such that $\sigma\to h(\Pi)\sigma U^{\dagger}=\overline{U}\sigma
h^{\dagger}(\Pi)$, and use it to construct the $G$-covariant object

$\sigma\gamma^{\hat{\mu}}\sigma=\gamma^{\hat{\mu}}(P_{L}\Omega^{\dagger}+P_{R}\Omega)\to\overline{U}\sigma\gamma^{\hat{\mu}}\sigma
U^{\dagger}.$ (44)

On the other hand $\sigma\gamma^{\bar{\mu}}\sigma=\gamma^{\bar{\mu}}$ remains
unchanged. This argument explains the $\Omega$-dependence of (16).

We can introduce additional auxiliary fields to restore full $d$-dimensional
Lorentz invariance. In such case the symmetry breaking pattern is
$SO(1,d-1)\to SO(1,3)\times SO(d-4)$. What breaks the original $d$-dimensional
Lorentz invariance is $\gamma_{5}$, which is a singlet only under the four-
dimensional subgroup, namely $\gamma_{5}\to
S(\Lambda_{4})\gamma_{5}S^{-1}(\Lambda_{4})=\gamma_{5}$, where
$S(\Lambda_{4})$ denotes the spinorial representation and $\Lambda_{4}\in
SO(1,3)$. On the other hand
$\gamma^{\mu}=\Lambda^{\mu}_{~{}\nu}S(\Lambda)\gamma^{\nu}S^{-1}(\Lambda)$, as
usual, with $S(\Lambda)$ the spinorial representation associated to
$\Lambda\in SO(1,d-1)$. Introducing a new set of Nambu-Goldstone bosons
$\Pi^{\prime}$ and a new coset representative

$\Omega^{\prime}\to\Lambda\Omega^{\prime}\Lambda_{4}^{-1}(\Pi^{\prime})$ (45)

we can build a $G$-covariant version of $\gamma_{5}$:

$\Gamma_{5}\equiv S(\Omega^{\prime})\gamma_{5}S^{-1}(\Omega^{\prime})\to
S(\Lambda)\Gamma_{5}S^{-1}(\Lambda).$ (46)

The Clifford algebra implies $S^{\dagger}\gamma^{0}=\gamma^{0}S^{-1}$. Hence,
by construction we have
$\gamma^{0}\Gamma_{5}^{\dagger}\gamma^{0}=-\Gamma_{5}$. Because
$\Gamma_{5}^{2}=1$, we can define a Lorentz-covariant version of the chirality
projectors

$\displaystyle{\cal
P}_{L}\equiv\frac{1}{2}(1-\Gamma_{5})~{}~{}~{}~{}~{}~{}~{}{\cal
P}_{R}\equiv\frac{1}{2}(1+\Gamma_{5})$ (47)

and of (43):

$\Sigma\equiv{\cal P}_{L}\sqrt{\Omega}^{\dagger}+{\cal P}_{R}\sqrt{\Omega}.$
(48)

These satisfy $\gamma^{0}{\cal P}^{\dagger}_{L}\gamma^{0}={\cal P}_{R}$, and
similarly $\gamma^{0}{\cal P}^{\dagger}_{R}\gamma^{0}={\cal P}_{L}$, as well
as $\gamma^{0}\Sigma^{\dagger}\gamma^{0}=\Sigma$. Importantly, because the
symmetry we are restoring is only global, the new Goldstones can be taken to
be space-time independent. This allows us to treat ${\cal P}_{L,R}$ as a
constant field.

With our new machinery ${\cal P}_{L}\Psi$ and ${\cal P}_{R}\Psi$ are
$d$-dimensional Lorentz spinors. We postulate they transform under the chiral
gauge symmetry as

${\cal P}_{L}\Psi\to e^{i\alpha^{A}T_{L}^{A}}{\cal
P}_{L}\Psi,~{}~{}~{}~{}~{}~{}~{}~{}{\cal P}_{R}\Psi\to
e^{i\alpha^{A}T_{R}^{A}}{\cal P}_{R}\Psi.$

The covariant derivatives read666Here it is crucial that ${\cal P}_{L,R}$ are
constant fields.

${\cal D}_{\mu}\Psi\equiv\left(\partial_{\mu}-iA^{A}_{\mu}{\cal
T}^{A}\right)\Psi,~{}~{}~{}~{}~{}{\cal
D}_{\mu}\overline{\Psi}\equiv\partial_{\mu}\overline{\Psi}+iA^{A}_{\mu}\overline{\Psi}\,\overline{\cal
T}^{A},$

with ${\cal T}^{A}={\cal P}_{L}T^{A}_{L}+{\cal P}_{R}T^{A}_{R}$ and
$\overline{\cal T}^{A}={\cal P}_{R}T^{A}_{L}+{\cal P}_{L}T^{A}_{R}$ Lorentz-
covariant space-time independent gauge generators. An incarnation of (16) that
satisfies (a spurious) global $d$-dimensional Lorentz invariance as well as
gauge-invariance may finally be written as

$\displaystyle\int\text{d}^{d}x\left\\{{\cal
L}_{\text{Bos}}^{\text{Reg}}+\left[\frac{1}{2}\overline{\Psi}_{j}\Sigma\gamma^{\mu}\Sigma
i{\cal D}_{\mu}\Psi_{j}+Y^{a}_{ij}\overline{\Psi_{i}}{\cal
P}_{R}\Psi_{j}\phi_{a}+{\rm h.c.}\right]\right\\}.$ (49)

This expression reduces to (16) when $\Omega^{\prime}$ becomes the identity
matrix, i.e. when ${\cal P}_{L,R}\to P_{L,R}$.

Intuitively, a regularization that is invariant under the full $d$-dimensional
Lorentz group should be more constraining than one, like (16), that realizes
explicitly only its four-dimensional subgroup. A manifestation of this
tendency is that dimension-3 counterterms like $\overline{\Psi}\Omega
P_{R}\Psi$, in principle allowed by the spurious symmetries of Section 2.2, in
the formulation (49) cannot occur at any order in perturbation theory. Indeed,
such an hypothetical operator should be of the form $\overline{\Psi}O\Psi$,
with $O$ an object constructed out of $\Sigma$, $\gamma^{\mu}$, $\Gamma_{5}$,
and the couplings (32), and no derivatives; but it is easy to see that there
exists no structure of that form that is simultaneously gauge and Lorentz
covariant.

The higher degree of constraining power is however partially compensated by
the fact that (49) has more vertices involving non-dynamical Goldstones. In
the calculation presented in Section 3.2 the formalism used in Sections 2 and
3 seemed to us more convenient. Yet, this may not be the case in more involved
calculations.

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# The Mass Hyperboloid as a Poisson-Lie Group

S. G. Rajeev<EMAIL_ADDRESS>Department of Physics and Astronomy,
Department of Mathematics,University of Rochester, Rochester, NY 14627, USA
Patrizia Vitale<EMAIL_ADDRESS>Dipartimento di Fisica “Ettore
Pancini”, Università di Napoli Federico II, Napoli, Italy  INFN, Sezione di
Napoli, Italy

###### Abstract

The light cone formalism of a massive scalar field has been shown by Dirac to
have many advantages. But it is not manifestly Lorentz invariant. We will show
that this is a feature not a bug: Lorentz invariance is indeed a symmetry, but
in a different sense defined by Drinfel’d. The key idea is that the mass shell
(mass hyperboloid) is a Poisson-Lie group: there is a non-abelian group
multiplication and non-zero Poisson brackets between components of four-
momentum. Rotations form the dual group of the hyperboloid in the sense of
Drinfel’d. Infinitesimal Lorentz transformations form a Lie bi-algebra.

##  1 Introduction

Quantum groups, originally introduced in relation with quantum integrability
more than thirty years ago (see for example [1]), have acquired a central role
in modern literature as appropriate symmetries of non-commutative space-time,
non-commutative dynamics and non-commutative models of matter and gauge
fields. All this activity is relevant in view of a consistent theory of
quantum gravity which would imply noncommutative models of spacetime as
effective theories, at scales which are large in comparison with Planck
length. Poisson-Lie groups are semi-classical approximations of quantum
groups, where non-commutativity of the algebra of functions on the groups
manifold is replaced by non-zero Poisson brackets. As such, they are mainly
studied in the same perspective as above, namely to grasp hints on the
deviation from the standard pseudo-Riemannian framework, which is foreseen by
most of the QG models on the market.

In this contribution, we wish to show that these structures do not necessarily
need to depart from standard physics in order to be detected, but may play a
role in conventional, undeformed dynamical systems. We will discuss a specific
example, the Poisson-Lie group $SU(2)$ and its Drinfel’d double [2, 3]
$SL(2,\mathbb{C})$, namely the (double covering) Lorentz group, and show that
they are intimately related with the light-front formalism of particle
physics, being indeed the most natural framework where to describe the mass
hyperboloid of massive scalar fields. The example will give us the opportunity
to make explicit the geometric structures of Poisson-Lie groups, such as
Poisson-Lie brackets, dressing actions, Poisson-Lie duality and to place them
into a familiar context.

##  2 The Hyperboloid

The four-momentum of a particle with unit mass belongs to one-sheet of a
hyperboloid. There are equivalent descriptions of this manifold:

$p_{0}^{2}-p_{1}^{2}-p_{2}^{2}-p_{3}^{2}=1,\quad p_{0}>0\iff
p_{+}p_{-}-(p_{2}^{2}+p_{3}^{2})=1,\quad p_{+}>0$

where $p_{+}=p_{0}+p_{1},p_{-}=p_{0}-p_{1}$. We can solve for

$p_{-}=\frac{1+p_{2}^{2}+p_{3}^{2}}{p_{+}},\quad
p_{0}=\frac{1+p_{+}^{2}+p_{2}^{2}+p_{3}^{2}}{2p_{+}},\quad
p_{1}=\frac{-\left(1+p_{2}^{2}+p_{3}^{2}\right)+p_{+}^{2}}{2p_{+}}$

in terms of $p_{+},p_{2},p_{3}$. Let us refer to the latter as
${\mathcal{P}}_{+}$. Thus the hyperboloid ${\mathcal{P}}_{+}$ can be
identified with the half-space
$\mathbb{R}^{+}\times\mathbb{R}^{2}\ni(p_{+},p_{2},p_{3})$. Notice that, if we
were to use $p_{1},p_{2},p_{3}$ as independent variables, solving for $p_{0}$
would involve a square root. This is a disadvantage: we would have to somehow
project out the negative energy solutions. Dirac was the first to point the
convenience of the light–cone (more accurately light–front) framework for
field theory.

Let $\mathcal{H}=\left\\{A\mid A^{\dagger}=A,\ A>0,\ \det A=1\right\\}$ be the
set of $2\times 2$ positive Hermitian matrices with unit determinant. There is
a one-to-one correspondence between ${\mathcal{P}}_{+}$ and $\mathcal{H}$. In
light-front coordinates the correspondence goes as follows

${\mathcal{P}}_{+}\ni(p_{+},p_{2},p_{3})\rightarrow{\mathcal{P}}(p)=\left(\begin{array}[]{cc}p_{+}&p_{2}+ip_{3}\\\
p_{2}-ip_{3}&p_{-}\end{array}\right)\in\mathcal{H}$ (2.1)

with $p_{+}p_{-}-(p_{2}^{2}+p_{3}^{2})=1$. Moreover, a positive Hermitian
matrix with unit determinant has a unique decomposition in terms of upper
triangular matrices with real diagonal and unit determinant (Cholesky
factorisation), ${\mathcal{P}}(p)=\ell^{\dagger}\ell$ with

$\ell=\frac{1}{\sqrt{p_{+}}}\left(\begin{array}[]{cc}p_{+}&p_{2}+ip_{3}\\\
0&1\end{array}\right).$ (2.2)

The latter form a group, which we shall refer to as $SB(2,C)$, it being the
Borel subgroup of $SL(2,\mathbb{C})$, as we shall explain in a moment.
Therefore, the Cholesky “square root” of a positive Hermitian matrix with unit
determinant is a Lie group.

Let us remark that:

  * •

The Lorentz group acts on ${\mathcal{P}}(p)\in\mathcal{H}$ by
${\mathcal{\mathcal{P}}}(p)\mapsto\Lambda{\mathcal{P}}(p)\Lambda^{\dagger},\Lambda\in
SL(2,C)$;

  * •

The subgroup $SU(2)$ acts with $1$ as fixed point, i.e., rotations;

  * •

The trace of ${\mathcal{P}}(p)$ is invariant under $SU(2)$: $\>{\rm
Tr}\,\mathcal{P}(p)=2p_{0}$, twice the energy.

Besides the standard realization of the Lie algebra of the Lorentz group,
which we shall identify with $\mathfrak{sl}(2,C)$ from now on, in terms of
rotations and boosts, it is possible to choose a different basis, which
emphasizes its bi-algebra structure. Let us indicate with $e_{a},f^{a}$ dually
related generators with respect to the inner product $\mathrm{Im}\>{\rm
Tr}\,$. There are two maximally isotropic, non-commuting sub-algebras,
$\mathfrak{su}(2)$ (spanned by $e_{a}$) and the Borel algebra,
$\mathfrak{sb}(2,C)$ (spanned by $f^{a}$). The latter is the Lie algebra of
the group $SB(2,C)$. As a vector space
$\mathfrak{sl}(2,\mathbb{C})=\mathfrak{su}(2)\oplus\mathfrak{sb}(2,\mathbb{C})$.
All together they form a Manin triple. As a basis we may choose

$\begin{array}[]{ccc}e_{1}=\left(\begin{array}[]{cc}\frac{i}{2}&0\\\
0&-\frac{i}{2}\end{array}\right),&e_{2}=\left(\begin{array}[]{cc}0&\frac{i}{2}\\\
\frac{i}{2}&0\end{array}\right),&e_{3}=\left(\begin{array}[]{cc}0&-\frac{1}{2}\\\
\frac{1}{2}&0\end{array}\right)\\\ \\\ f^{1}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),&f^{2}=\left(\begin{array}[]{cc}0&2\\\
0&0\end{array}\right),&f^{3}=\left(\begin{array}[]{cc}0&2i\\\
0&0\end{array}\right)\end{array}$ (2.3)

which verify $\mathrm{Im\ Tr}\,e_{a}f^{b}=\delta_{b}^{a}$ with Lie brackets

$[e_{i},e_{j}]={\epsilon_{ij}}^{k}e_{k},\;\;\left[f^{i},f^{j}\right]={f^{ij}}_{k}f^{k},\;\;\left[e_{i},f^{j}\right]={\epsilon_{ki}}^{j}f^{k}+{f_{i}}^{jk}e_{k}$
(2.4)

and ${f_{i}}^{jk}=2\epsilon_{1i\ell}\epsilon^{\ell jk}$ the structure
constants of the Lie algebra $\mathfrak{sb}(2,\mathbb{C})$. Notice the mutual
adjoint action of one subalgebra onto the other in the last Lie bracket.

The corresponding dual subgroups of $SL(2,\mathbb{C})$, which realize the
latter as a Drinfel’d double, are $SU(2)=\left\\{u\mid u^{\dagger}u=1,\ \det
u=1\right\\}$ and
$SB(2,\mathbb{C})=\left\\{\lambda=\left(\begin{array}[]{cc}a&b+ic\\\
0&a^{-1}\end{array}\right),a>0,b+ic\in\mathbb{C}\right\\}$. Therefore an
element of the Lorentz group $\gamma\in SL(2,\mathbb{C})$, may be locally
parametrized as a product $\gamma=u\cdot\lambda$ or alternatively
$\gamma=\lambda^{\prime}\cdot u^{\prime}$. Notice that this is nothing but the
familiar Iwasawa decomposition of $SL(2,\mathbb{C})$, $g=k\cdot a\cdot n$,
with $k\in SU(2),\,a\cdot n=b\in SB(2,\mathbb{C})$ (see [5] for a review).

##  3 The Hyperboloid Group

As we have seen, the hyperboloid ${\mathcal{P}}_{+}$ is itself a group, it
being $\ell\in SB(2,\mathbb{C})$. Therefore $SB(2,\mathbb{C})$ plays multiple
roles: on one hand it is a factor of the Lorentz group in the Iwasawa
decomposition, and, as such, it acts with the appropriate representation; on
the other hand, it models the manifold of momenta for particles of unit mass,
therefore, it is acted upon by the Lorentz group and its factors; moreover, it
being the dual group of $SU(2)$, its group elements replace standard angular
momentum which is usually associated with the Abelian dual Lie algebra of
$\mathfrak{su}(2)$ through momentum map. Let us see these structures in more
detail.

The left-invariant Maurer-Cartan form can be easily calculated:

$\alpha=\lambda^{-1}d\lambda=\frac{da}{a}f^{1}+\frac{1}{2}\left(\frac{db}{a}+\frac{b}{a^{2}}da\right)f^{2}+\frac{1}{2}\left(\frac{dc}{a}+\frac{c}{a^{2}}da\right)f^{3}$
(3.1)

This yields, under the identification of $\lambda\rightarrow\ell$, namely
$a\rightarrow\sqrt{p_{+}},\,b\rightarrow\frac{p_{2}}{\sqrt{p_{+}}},\,c\rightarrow\frac{p_{3}}{\sqrt{p_{+}}}$,
a basis of left-invariant one-forms

$\alpha_{1}=\frac{1}{2}\frac{dp_{+}}{p_{+}},\;\;\alpha_{2}=\frac{1}{2}\frac{dp_{2}}{p_{+}},\;\;\alpha_{3}=\frac{1}{2}\frac{dp_{3}}{p_{+}}$
(3.2)

with left-invariant vector fields

${Y}^{1}=2p_{+}\frac{\partial}{\partial
p_{+}},\;\;\;{Y^{2}}=2p_{+}\frac{\partial}{\partial
p_{2}},\;\;\;{Y^{3}}=2p_{+}\frac{\partial}{\partial p_{3}}$ (3.3)

satisfying $[Y^{i},Y^{j}]={f^{ij}}_{k}Y^{k}$. The latter generate the right
action of $SB(2,\mathbb{C})$ on the momentum hyperboloid. The left action is
obtained by repeating the above for the right-invariant Maurer-Cartan form. We
get the right-invariant forms

$\theta_{1}=\frac{1}{2}\frac{dp_{+}}{p_{+}},\;\;\theta_{2}=\frac{1}{2}\left(dp_{2}-\frac{p_{2}}{p_{+}}dp_{+}\right),\;\;\theta_{3}=\frac{1}{2}\left(dp_{3}-\frac{p_{3}}{p_{+}}dp_{+}\right)$
(3.4)

with right-invariant vector fields

${Z}^{1}=2\left(p_{+}\frac{\partial}{\partial
p_{+}}+p_{2}\frac{\partial}{\partial p_{2}}+p_{3}\frac{\partial}{\partial
p_{3}}\right),\;\;\;{Z^{2}}=2\frac{\partial}{\partial
p_{2}},\;\;\;{Z^{3}}=2\frac{\partial}{\partial p_{3}}$ (3.5)

satisfying $[{Z}^{i},{Z}^{j}]=-f^{ij}_{k}Z^{k}$.

###  3.1 Poisson-Lie brackets

In terms of a generalisation of the Kirillov-Souriau-Konstant bracket (KSK),
which is naturally defined on the dual of any Lie algebra, dually related Lie
groups possess a natural Poisson bracket (that reduces to the KSK bracket when
evaluated at the identity), which is compatible with the group action, namely
the group multiplication results to be a Poisson morphism. It has been shown
(see [4] for a review) that Poisson-Lie brackets for a given Lie group $G$ may
always been written as

$\\{\gamma_{1},\gamma_{2}\\}=[r,\gamma_{1}\gamma_{2}].$ (3.6)

with $\gamma\in G$ and $r\in\mathfrak{g}\otimes\mathfrak{g}$ the so called
classical $r$-matrix, solution of the classical (possibly modified) Yang-
Baxter equation (CYB) [4]. It is standard notation to denote
$\gamma_{1}=\gamma\otimes 1,\gamma_{2}=1\otimes\gamma$ with
$\gamma_{1}\gamma_{2}=\gamma\otimes\gamma$. The Lorentz group itself is a
Poisson-Lie group with

$r=e_{i}\otimes
f^{i},\;\;\;\;\;[r_{12},r_{13}+r_{23}]+[r_{13},r_{23}]=0\;\;\rm{(CYB)}.$ (3.7)

When specialised to $\lambda\in SB(2,\mathbb{C})$ Eq. (3.6) yields

$\displaystyle\\{\lambda\stackrel{{\scriptstyle\otimes}}{{,}}\lambda\\}$
$\displaystyle=$ $\displaystyle e_{i}\lambda\otimes f^{i}\lambda-\lambda
e_{i}\otimes\lambda f^{i}$ (3.8)
$\displaystyle\\{\lambda\stackrel{{\scriptstyle\otimes}}{{,}}\lambda^{\dagger}\\}$
$\displaystyle=$ $\displaystyle
e_{i}\lambda\otimes(f^{i}\lambda)^{\dagger}-\lambda e_{i}\otimes(\lambda
f^{i})^{\dagger}$ (3.9)

namely

$\\{a,b\\}=ac,\;\;\;\\{a,c\\}=-ab,\;\;\;\\{b,c\\}=a^{2}-\frac{1}{a^{2}}$
(3.10)

The latter may be rewritten in terms of the hyperboloid coordinates to give

$\\{p_{+},p_{2}\\}=2p_{+}p_{3},\;\;\;\\{p_{3},p_{+}\\}=2p_{+}p_{2},\;\;\;\\{p_{2},p_{3}\\}=p_{+}^{2}-p_{2}^{2}-p_{3}^{2}-1$
(3.11)

The Poisson algebra (3.11) admits a Casimir function which corresponds to the
energy $p_{0}$,
$C=\frac{1}{2}\left(p_{+}+\frac{1}{p_{+}}+\frac{p_{2}^{2}+p_{3}^{2}}{p_{+}}\right).$
Notice however that the group $SB(2,C)$ has no Casimir intended as a
polynomial in the center of its universal enveloping algebra (see for example
[6], where $\mathfrak{sb}(2,\mathbb{C})$ is classified as a type B algebra).

###  3.2 Dressing action of $SU(2)$ on the hyperboloid

The Drinfel’d double $SL(2,C)$ is a deformation of the semi-direct product
$SU(2)\ltimes\mathbb{R}^{3}$, with $\mathbb{R}^{3}$ the dual algebra of
$SU(2)$. In standard Poisson geometry the momentum map associated with the
Hamiltonian action of $SU(2)$ on $\mathbb{R}^{3}$ gives rise to Hamiltonian
functions which close the algebra of $SU(2)$ with respect to the KSK Poisson
bracket. In classical mechanics these functions are the three components of
the angular momentum, $\ell_{i}$, according to $\Lambda(d\ell_{i})=-V_{i}$,
where $\Lambda$ is the KSK Poisson tensor and $V_{i}$ the vector fields
realizing the infinitesimal action of $SU(2)$ on $\mathbb{R}^{3}$.

The generalization of the above construction to the Drinfel’d double
$SL(2,\mathbb{C})$ yields the infinitesimal dressing action of $SU(2)$ on
$SB(2,\mathbb{C})$. The Maurer-Cartan forms $\alpha^{i}$ derived in Eq. (3.2)
play now the role of the exact one-forms $d\ell_{i}$ whereas $\Lambda$ is
replaced by the Poisson-Lie tensor computed in Sec. 3.1,

$P=2p_{+}p_{3}\frac{\partial}{\partial p_{+}}\wedge\frac{\partial}{\partial
p_{2}}-2p_{+}p_{2}\frac{\partial}{\partial
p_{+}}\wedge\frac{\partial}{\partial
p_{3}}-(1-p_{+}^{2}+p_{2}^{2}+p_{3}^{2})\frac{\partial}{\partial
p_{2}}\wedge\frac{\partial}{\partial p_{3}}.$ (3.12)

This yields $\;L_{i}=-P(\alpha_{i}),\;$ that is, in the light-front
coordinates of the hyperboloid

$\begin{array}[]{lll}L_{1}&=&-p_{2}\frac{\partial}{\partial
p_{3}}+p_{3}\frac{\partial}{\partial p_{2}}\\\
L_{2}&=&-p_{3}\frac{\partial}{\partial
p_{+}}-\frac{1}{2p_{+}}(1-p_{+}^{2}+p_{2}^{2}+p_{3}^{2})\frac{\partial}{\partial
p_{3}}\\\ L_{3}&=&p_{2}\frac{\partial}{\partial
p_{+}}+\frac{1}{2p_{+}}(1-p_{+}^{2}+p_{2}^{2}+p_{3}^{2})\frac{\partial}{\partial
p_{2}}\end{array}$ (3.13)

They satisfy the commutation relations of the Lie algebra $\mathfrak{su}(2)$,
$[L_{i},L_{j}]={\varepsilon_{ij}}^{k}L_{k}$. That is, they generate the
dressing action of $SU(2)$ on the hyperboloid. The latter is the dressing
action generated through left-invariant one-forms. Analogously, we have
another one, generated by right-invariant forms (3.4). We have
$\;\widehat{L}_{\ell}=-P(\theta_{\ell}),\;$ that is, in the light-front
coordinates of the hyperboloid

$\begin{array}[]{lll}\widehat{L}_{1}&=&p_{3}\frac{\partial}{\partial
p_{2}}-p_{2}\frac{\partial}{\partial p_{3}}\\\
\widehat{L}_{2}&=&-p_{3}(p_{+}\frac{\partial}{\partial
p_{+}}+p_{2}\frac{\partial}{\partial
p_{2}})-\frac{1}{2}(1-p_{+}^{2}-p_{2}^{2}+p_{3}^{2})\frac{\partial}{\partial
p_{3}}\\\ \widehat{L}_{3}&=&p_{2}(p_{+}\frac{\partial}{\partial
p_{+}}+p_{3}\frac{\partial}{\partial
p_{3}})+\frac{1}{2}(1-p_{+}^{2}+p_{2}^{2}-p_{3}^{2})\frac{\partial}{\partial
p_{2}}\end{array}$ (3.14)

They close the same $SU(2)$ algebra,
$[\widehat{L}_{i},\widehat{L}_{j}]={\varepsilon_{ij}}^{k}\widehat{L}_{k}$.

##  4  The group manifold of $SU(2)$

As well as the group manifold of $SB(2,\mathbb{C})$, which we have regarded as
the mass hyperboloid, the group manifold $SU(2)$, namely the three-sphere
$S^{3}$, is an interesting configuration space for classical dynamics(see for
example the rigid rotor and all its applications, or the non-linear sigma
model with target space the sphere $S^{3}$). Therefore it is worth to
duplicate the previous construction, where now the role of the two groups is
exchanged.

A convenient parametrization of $u\in SU(2)$ is
$u=y^{0}\mathbf{1}+2y^{i}e_{i}$ with $y^{\mu}\in\mathbb{R}^{4},\sum
y^{\mu}y^{\mu}=1$ and $e_{i}$ the generators of $SU(2)$. given by the first
line of Eq. (2.3). Explicitly

$u=\left(\begin{array}[]{cc}y^{0}+iy^{1}&iy^{2}-y^{3}\\\
iy^{2}+y^{3},&y^{0}-iy^{1}\end{array}\right)\;\;\;\;{\rm with}\;\det u=1$
(4.1)

From the left-invariant Maurer-Cartan one-form $u^{-1}du$ the three basis
left-invariant one-forms are obtained

$\eta^{i}=2\left(y^{0}dy^{i}-y^{i}dy^{0}-{\varepsilon^{i}}_{jk}y^{j}dy^{k}\right)$
(4.2)

with dual vector fields

$W_{i}=\frac{1}{2}\left(y^{0}\partial_{i}-y^{i}\partial_{0}-{\varepsilon_{ij}}^{k}y^{j}\partial_{k}\right)$
(4.3)

Analogously, from $du\,u^{-1}$ the right-invariant forms are found to be

$\omega^{i}=2\left(y^{0}dy^{i}-y^{i}dy^{0}+{\varepsilon^{i}}_{jk}y^{j}dy^{k}\right).$
(4.4)

with dual vector fields

$V_{i}=\frac{1}{2}\left(y^{0}\partial_{i}-y^{i}\partial_{0}+{\varepsilon_{ij}}^{k}y^{j}\partial_{k}\right)$
(4.5)

###  4.1 Poisson-Lie bracket on $SU(2)$

The Poisson-Lie bracket (3.6) is defined on the whole group
$SL(2,\mathbb{C})$. For

$SL(2,\mathbb{C}))\ni\gamma=\left(\begin{array}[]{cc}\alpha&\beta\\\
\delta&\zeta\end{array}\right)$ (4.6)

with $\alpha\zeta-\beta\delta=1$, $\alpha,\beta,\delta,\zeta$ complex
variables we obtain

$\displaystyle\\{\alpha,\beta\\}=-i\alpha\beta,\;\;$
$\displaystyle\\{\alpha,\delta\\}=-i\alpha\delta,\;\;$
$\displaystyle\\{\alpha,\zeta\\}=-2i\beta\delta$
$\displaystyle\\{\beta,\delta\\}=0,\;\;\;\;\;\;\;\;\;$
$\displaystyle\\{\beta,\zeta\\}=-i\beta\zeta,\;\;$
$\displaystyle\\{\delta,\zeta\\}=-i\delta\zeta\,.$ (4.7)

Specialising it to $u\in SU(2)$, we find

$\displaystyle\\{y_{0},y_{1}\\}=-(y_{2}^{2}+y_{3}^{2}),\;\;$
$\displaystyle\\{y_{0},y_{2}\\}=y_{1}y_{2},\;\;\;\;$
$\displaystyle\\{y_{0},y_{3}\\}=y_{1}y_{3}$
$\displaystyle\\{y_{1},y_{2}\\}=-y_{0}y_{2},\;\;\;\;\;\;\;\;\;$
$\displaystyle\\{y_{1},y_{3}\\}=-y_{0}y_{3},\;\;$
$\displaystyle\\{y_{2},y_{3}\\}=0\,$ (4.8)

The Poisson tensor which endows $SU(2)$ with a Poisson-Lie structure is
therefore

$Q=(y_{1}\partial_{0}-y_{0}\partial_{1})\wedge(y_{2}\partial_{2}+y_{3}\partial_{3})-(y_{2}^{2}+y_{3}^{2})\partial_{0}\wedge\partial_{1}.$
(4.9)

###  4.2 Dressing actions of $SB(2,\mathbb{C})$ on $SU(2)$

By repeating the calculation performed in Sec. 3.2 for the group
$SB(2,\mathbb{C})$ it is possible to get the dressing actions (left and right)
of the latter on $SU(2)$. In comparison with the standard momentum map
picture, the left-invariant Maurer-Cartan one-forms $\eta^{i}$ (resp. right-
invariant) derived above play now the role of the exact one-forms $d\ell_{i}$
whereas the Konstant-Souriau-Kirillov bracket $\Lambda$ is replaced by the
Poisson-Lie tensor (4.9). The generators of the right dressing action
(respectively left dressing action) of $SB(2,\mathbb{C})$ on $SU(2)$ are
therefore retrieved according to $\;B^{\ell}=-Q(\eta^{\ell}),\;$ which gives

$B^{1}=-2(y_{2}\partial_{2}+y_{3}\partial_{3}),\;\;B^{2}=2(y_{2}\partial_{1}-y_{3}\partial_{0}),\;\;B^{3}=2(y_{2}\partial_{0}+y_{3}\partial_{1})$
(4.10)

where use has been made of the constraints on the parameters of $SU(2)$,
$y_{\mu}y^{\mu}=1$ implying $y^{\mu}\partial_{\mu}=0$. They satisfy the
commutation relations of the Lie algebra $\mathfrak{sb}(2)$,

$[B^{1},B^{2}]=-2B^{2},\;\;[B^{1},B^{3}]=-2B^{3},\;\;[B^{2},B^{3}]=0$ (4.11)

Analogously, from the right-invariant one-forms

$\omega^{i}=2\left(y^{0}dy^{i}-y^{i}dy^{0}+{\varepsilon^{i}}_{jk}y^{j}dy^{k}\right).$
(4.12)

We get the vector fields $\;\widehat{B}^{\ell}=-Q(\omega^{\ell}),\;$ so that

$\widehat{B}^{1}=B^{1},\;\;\widehat{B}^{2}=2(y_{2}\partial_{1}+y_{3}\partial_{0}),\;\;\widehat{B}^{3}=2(y_{3}\partial_{1}-y_{2}\partial_{0})$
(4.13)

which close the same Lie algebra as the previous one, therefore, they generate
the left dressing action of $SB(2,\mathbb{C})$ on $SU(2)$.

##  5 Dressing actions of $SL(2,\mathbb{C})$ on the hyperboloid

In order to obtain the action of the Lorentz group on the mass hyperboloid, it
is natural to exploit the isomorphism of its Lie algebra with
$\mathfrak{sl}(2,\mathbb{C})$ and look for its dressing action on the dual.
The group $SL(2,\mathbb{C})$ is itself a Poisson-Lie group with dual group
another copy of $SL(2,\mathbb{C})$, which shall be indicated by
$SL(2,\mathbb{C})^{*}$ to keep track of the doubling (and to stress that it is
the exponentiation of the dual Lie algebra). Therefore, the Poisson-Lie
bracket (3.6) may be used to derive the left and right dressing actions of
$SL(2,\mathbb{C})$ on its dual. The hyperboloid (identified with the group
manifold of $SB(2,\mathbb{C})$) may be regarded as the quotient
$SL(2,\mathbb{C})^{*}/SU(2)$. Depending on the decomposition chosen, namely
$SL(2,\mathbb{C})\sim SB(2,\mathbb{C})\cdot SU(2)$ or $SL(2,\mathbb{C})\sim
SU(2)\cdot SB(2,\mathbb{C})$, the hyperboloid is a right coset or a left
coset.111Dually, $SU(2)$ is a left or right quotient with respect to
$SB(2,\mathbb{C})$. This remark and the next are relevant when, as in previous
section, it is the group manifold of $SU(2)$ which is associated with the
carrier space of dynamics, and, therefore, one is interested in the action of
the Lorentz group on it.  Therefore, once a decomposition chosen, the
infinitesimal generators of the dressing action of $SL(2,\mathbb{C})$ on its
dual have to be computed consistently. In this way, they will pass to the
quotient and give the appropriate (left or right) dressing action of the whole
Lorentz group on the hyperboloid.222 Analogously, the appropriate dressing
action of the Lorentz group on $SU(2)$ is obtained.

First of all, we choose to represent $\gamma\in SL(2,\mathbb{C})$ as the
product $\gamma=\lambda u$, with $\lambda\in SB(2,\mathbb{C})$ and $u\in
SU(2)$, parametrized as above. Then, in order to write the Poisson-Lie bi-
vector field in the chosen parametrization, we compute the LHS of (3.6)

$\begin{array}[]{ll}&\\{\gamma_{1},\gamma_{2}\\}=\lambda_{1}\\{u_{1},\lambda_{2}\\}u_{2}+\lambda_{1}\lambda_{2}\\{u_{1},u_{2}\\}+\\{\lambda_{1},\lambda_{2}\\}u_{1}u_{2}+\lambda_{2}\\{\lambda_{1},u_{2}\\}u_{1}\\\
&=\lambda_{1}\\{u_{1},\lambda_{2}\\}u_{2}+\lambda_{1}\lambda_{2}(ru_{1}u_{2}-u_{1}u_{2}r)+(r\lambda_{1}\lambda_{2}-\lambda_{1}\lambda_{2}r)u_{1}u_{2}+\lambda_{2}\\{\lambda_{1},u_{2}\\}u_{1}\end{array}$

where use has been made of the Poisson brackets for the $SU(2)$ and
$SB(2,\mathbb{C})$ variables, $\\{u_{1},u_{2}\\}=[r,u_{1}u_{2}]$ and
$\\{\lambda_{1},\lambda_{2}\\}=[r,\lambda_{1}\lambda_{2}]$. The RHS of (3.6)
yields instead

$r\gamma_{1}\gamma_{2}-\gamma_{1}\gamma_{2}r=r\lambda_{1}u_{1}\lambda_{2}u_{2}-\lambda_{1}u_{1}\lambda_{2}u_{2}r\,.$
(5.1)

Comparing the two we get
$\;\lambda_{1}\\{u_{1},\lambda_{2}\\}u_{2}+\lambda_{2}\\{\lambda_{1},u_{2}\\}u_{1}=0,\;$
from which, by consistency,

$\\{u_{1},\lambda_{2}\\}=\lambda_{2}r_{12}u_{1}=e_{i}u\otimes\lambda
f^{i},\;\;\;\\{\lambda_{1},u_{2}\\}=-\lambda_{1}r_{12}u_{2}=-\lambda
e_{i}\otimes f^{i}u.$

Explicitly they yield

$\begin{array}[]{llll}\\{y_{0},a\\}=-\frac{a}{2}y_{1}&\\{y_{1},a\\}=\frac{a}{2}y_{0}&\\{y_{2},a\\}=-\frac{a}{2}y_{3}&\\{y_{3},a\\}=\frac{a}{2}y_{2}\\\
\\{y_{0},b\\}=\frac{b}{2}y_{1}-ay_{2}&\\{y_{1},b\\}=-\frac{b}{2}y_{0}+ay_{3}&\\{y_{2},b\\}=\frac{b}{2}y_{3}+ay_{0}&\\{y_{3},b\\}=-\frac{b}{2}y_{2}-ay_{1}\\\
\\{y_{0},c\\}=\frac{c}{2}y_{1}-ay_{3}&\\{y_{1},c\\}=-\frac{c}{2}y_{0}-ay_{2}&\\{y_{2},c\\}=\frac{c}{2}y_{3}+ay_{1}&\\{y_{3},c\\}=-\frac{c}{2}y_{2}+ay_{0}\end{array}$
(5.2)

Therefore the Poisson-Lie tensor for $SL(2,\mathbb{C})$ may be written as
$\;\mathbb{P}=P+Q+M,\;$ with $P$ and $Q$ respectively given by Eq. (3.12) and
Eq. (4.9), while $M$ is to be read from (5.2), that is

$\begin{array}[]{lllll}M&=&\left[\frac{a}{2}y_{1}\partial_{a}+(ay_{2}-\frac{b}{2}y_{1})\partial_{b}+(ay_{3}-\frac{c}{2}y_{1})\partial_{c}\right]&\wedge&\partial_{y_{0}}\\\
&-&\left[\frac{a}{2}y_{0}\partial_{a}+(ay_{3}-\frac{b}{2}y_{0})\partial_{b}-(ay_{2}+\frac{c}{2}y_{0})\partial_{c}\right]&\wedge&\partial_{y_{1}}\\\
&+&\left[\frac{a}{2}y_{3}\partial_{a}-(ay_{0}+\frac{b}{2}y_{3})\partial_{b}-(ay_{1}+\frac{c}{2}y_{3})\partial_{c}\right]&\wedge&\partial_{y_{2}}\\\
&-&\left[\frac{a}{2}y_{2}\partial_{a}-(ay_{1}+\frac{b}{2}y_{2})\partial_{b}+(ay_{0}-\frac{c}{2}y_{2})\partial_{c}\right]&\wedge&\partial_{y_{3}}\end{array}$
(5.3)

Notice that, for simplicity, Eqs. (5.2) and (5.3) are not explicitly written
in the light-front coordinates of the hyperboloid. We shall see, however, that
the parametrization will homogenize after projection.

We now observe that a basis of left- (resp. right-) invariant one-forms for
$SL(2,\mathbb{C})$ is represented by $(\eta^{i},\alpha_{i})$ (resp.
$(\omega^{i},\theta_{i})$). Therefore we may compute the dressing action of
the Lorentz group on its dual by following the same procedure (the generalised
momentum map) as in previous sections. Then, by considering the quotient with
respect to each sub-group we shall obtain the dressing action of the Lorentz
group on each of its factors.

Notice that, having chosen the parametrization $\gamma=\lambda u$, the group
$SB(2,\mathbb{C})$ is a right coset with respect to the $SU(2)$ action, while
$SU(2)$ is a left coset with respect to the $SB(2,\mathbb{C})$ action.
Therefore, we have to consider the pair $(\omega^{i},\alpha_{i})$ namely,
right-invariant forms of $SU(2)$ and left-invariant forms of
$SB(2,\mathbb{C})$ in order to retrieve the dressing action correctly, upon
projection. Let us start with $\mathbb{P}(\omega^{i})$. We have to evaluate

$\mathbb{P}(\omega^{i})=P(\omega^{i})+Q(\omega^{i})+M(\omega^{i})$ (5.4)

$P(\omega^{i})=0$ trivially while $Q(\omega^{i})=-\hat{B}^{i}$ according to
Eq. (4.13), Eqs. (LABEL:hatBri1)-(LABEL:hatBri3), namely the generators of the
left dressing action of $SB(2,\mathbb{C})$ on $SU(2)$. Moreover,
$\;M(\omega^{i})=-Y^{i},\;$ with $Y^{i}$ the left-invariant vector fields of
$SB(2,\mathbb{C})$ already computed in (3.3). Namely,
$\mathbb{P}(\omega^{i})=-\hat{B}^{i}-Y^{i},\;$ yielding in turn

$\pi_{*}^{R}\mathbb{P}(\omega^{i})=-Y_{i},\;\;\;\pi_{*}^{L}\mathbb{P}(\omega^{i})=-\hat{B}_{i}$
(5.5)

where we have indicated with $\pi^{R},\pi^{L}$respectively the projection of
$SL(2,\mathbb{C})$ to the quotients $SL(2,\mathbb{C})/SU(2)$,
$SL(2,\mathbb{C})/SB(2,\mathbb{C})$. Analogously,

$\mathbb{P}(\alpha_{i})=P(\alpha_{i})+Q(\alpha_{i})+M(\alpha_{i})$ (5.6)

with now $Q(\alpha_{i})=0$ trivially, because $Q$ is the Poisson-Lie tensor on
$SU(2)$ while $\alpha_{i}\in\Omega^{1}(SB(2,\mathbb{C}))$. As for
$M(\alpha_{i})$, we find
$\;M(\alpha_{i})=y^{i}\partial_{0}-y^{0}\partial_{i}-\varepsilon^{i}_{jk}y^{j}\partial_{k},\;$
that is, $M(\alpha_{i})=-V_{i}$, the right-invariant vector fields of $SU(2)$
computed in Eq. (4.5). $P(\alpha_{i})$ in turn yields the generators of the
right dressing action of $SU(2)$ on $SB(2,\mathbb{C})$ that we have already
computed with Eq. (3.13), $\;P(\alpha_{i})=L_{i}.\;$ Namely,
$\;\mathbb{P}(\alpha_{i})=L_{i}-V_{i},\;$ yielding in turn

$\pi_{*}^{R}\mathbb{P}(\alpha_{i})=L_{i},\;\;\;\pi_{*}^{L}\mathbb{P}(\alpha_{i})=-V_{i}.$
(5.7)

Summarising, by contracting the Poisson-Lie tensor $\mathbb{P}$ of
$SL(2,\mathbb{C})$ with an appropriate basis of one-forms of its dual,
$(\alpha_{i},\omega^{i})\in\Omega^{1}(SL(2,\mathbb{C})^{*})$, we have obtained
a set of vector fields, with right and left projections given respectively by
$(Y^{i},L_{i})\in\mathfrak{X}(SB(2,\mathbb{C}))$ and
$(\hat{B}^{i},V_{i})\in\mathfrak{X}(SU(2))$.

It may be verified that $Y^{i},L_{i}$ verify the Lie bi-algebra relations

$[{L}_{i},Y^{j}]=Y^{k}{\epsilon_{ki}}^{j}+{2\epsilon_{1i\ell}{\epsilon}^{\ell
jk}}{L}_{k}$ (5.8)

besides
$\;\;[{L}_{i},{L}_{j}]={\epsilon_{ij}}^{k}L_{k},\;\;\;[Y^{i},Y^{j}]={f^{ij}}_{k}Y^{k},\;\;$
where ${f^{ij}}_{k}=2{\epsilon^{1i}}_{\ell}{\epsilon^{\ell j}}_{k}$, therefore
representing the infinitesimal action of the Lorentz group on the hyperboloid.

Dually, $\hat{B}_{i},V_{i}$ obey the same bi-algebra relations, but are vector
fields on the group manifold of $SU(2)$; therefore, they represent the
infinitesimal action of the Lorentz group on the three sphere.

##  6 Conclusions

$SL(2,\mathbb{C})$, the universal covering of the Lorentz group, is a
fantastic arena where to probe our understanding of Poisson-Lie geometry,
Manin triples and dually related groups. We have proposed in this short
contribution, with no pretention of rigor, a hopefully pedagogical approach.

Besides describing in detail the geometric structures involved, we have
singled out two physically interesting applications of the formalism: one is
already well known and widely studied, the other is new, to our knowledge. The
former deals with the identification of the group manifold of $SU(2)$ with the
target space of some dynamical system (such as the rigid rotor, or the sigma
model with and without Wess-Zumino term, see for example [7, 8, 9, 10]). The
latter explores the lesser known group manifold of $SB(2,\mathbb{C})$, which
may be naturally associated with the mass hyperboloid of relativistic
particles, in the light-front formalism. This second example might have
interesting applications which are still under investigation.

## Acknowledgments

This contribution is dedicated to A. P. Balachandran, on the occasion of his
eighty-fifth birthday.

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