# Enlarging the symmetry of pure $R^{2}$ gravity, BRST invariance and its
spontaneaous breaking

Ariel Edery<EMAIL_ADDRESS>
###### Abstract

Pure $R^{2}$ gravity was considered originally to possess only global scale
symmetry. It was later shown to have the larger restricted Weyl symmetry where
it is invariant under the Weyl transformation
$g_{\mu\nu}\to\Omega^{2}(x)\,g_{\mu\nu}$ when the conformal factor $\Omega(x)$
obeys the harmonic condition $\Box\Omega(x)=0$. Restricted Weyl symmetry has
an analog in gauge theory. Under a gauge transformation $A_{\mu}\to
A_{\mu}+\frac{1}{e}\partial_{\mu}f(x)$, the gauge-fixing term
$(\partial_{\mu}A^{\mu})^{2}$ has a residual gauge symmetry when $\Box f=0$.
In this paper, we consider scenarios where the symmetry of pure $R^{2}$
gravity can be enlarged even further. In one scenario, we add a massless
scalar field to the pure $R^{2}$ gravity action and show that the action
becomes on-shell Weyl invariant when the equations of motion are obeyed. We
then enlarge the symmetry to a BRST symmetry where no on-shell or restricted
Weyl condition is required. The BRST transformations here are not associated
with gauge transformations (such as diffeomorphisms) but with Weyl (local
scale) transformations where the conformal factor consists of a product of
Grassmann variables. BRST invariance in this context is a generalization of
Weyl invariance that is valid in the presence of the Weyl-breaking $R^{2}$
term. In contrast to the BRST invariance of gauge theories like QCD, it is not
preserved after quantization since renormalization introduces a scale (leading
to the well-known Weyl (conformal) anomaly). We show that the spontaneous
breaking of the BRST symmetry yields an Einstein action; this still has a
symmetry which is also anomalous. This is in accord with previous work that
shows that there is conformal anomaly matching between the unbroken and broken
phases when conformal symmetry is spontaneously broken.

##  1 Introduction

Pure $R^{2}$ gravity ($R^{2}$ alone with no additional $R$ term) is unique
among quadratic gravity theories as it is unitary and moreover has been shown
to be conformally equivalent to Einstein gravity with non-zero cosmological
constant and massless scalar field [1, 2, 3, 4, 5] (though in a Palatini
formalism one can avoid having a massless scalar [6]). It has been known for a
long time that it is invariant under the global scale transformation
$g_{\mu\nu}\to\lambda^{2}\,g_{\mu\nu}$ where $\lambda$ is a constant. It was
later discovered to possess a larger symmetry than global scale symmetry
called restricted Weyl symmetry [7] where it is invariant under the
transformation $g_{\mu\nu}\to\Omega^{2}(x)\,g_{\mu\nu}$ when the conformal
factor $\Omega(x)$ obeys the harmonic condition
$\Box\Omega=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\Omega=0$. The conformal factor
$\Omega(x)$ is therefore not limited to being a constant. The aforementioned
equivalence between pure $R^{2}$ gravity and Einstein gravity with
cosmological constant was then interpreted in a new light: it occurs when the
restricted Weyl symmetry is spontaneously broken [3, 5]. In the broken sector,
the Ricci scalar of the background (vacuum) spacetime has $R\neq 0$ which
excludes a flat background. This is why the equivalence requires a non-zero
cosmological constant on the the Einstein side. The unbroken sector which has
an $R=0$ vacuum (background) has no relation to Einstein gravity. In fact, it
has been shown that a linearization of pure $R^{2}$ gravity about Minkowski
spacetime does not yield gravitons but only a propagating scalar [4]; simply
put, pure $R^{2}$ gravity does not gravitate about a flat background [4].
However, it was later shown that if one includes a non-minimally coupled
scalar field in the restricted Weyl-invariant action and the field acquires a
non-zero VEV, then the theory can gravitate about flat spacetime [5, 8].
Various aspects of restricted Weyl symmetry, it spontaneous breaking as well
as its role in critical gravity were then explored further in [5, 7, 9, 6, 10,
11]

Restricted Weyl symmetry has an analog in gauge theory. The gauge-fixing term
$(\partial_{\mu}A^{\mu})^{2}$ is invariant under the gauge transformation
$A_{\mu}\to A_{\mu}+\frac{1}{e}\partial_{\mu}f(x)$ only when the arbitrary
smooth function $f(x)$ obeys the condition $\Box f=0$ where $\Box$ here
represents the flat space d’Alembertian. Therefore, the gauge-fixing term has
a residual gauge symmetry when $\Box f=0$ is satisfied [12]. This is the
analog to the restricted Weyl symmetry of pure $R^{2}$ gravity when the
conformal factor $\Omega(x)$ satisfies $\Box\Omega=0$. As we will see, this
analogy is fruitful as it provides a bridge to the BRST symmetry of pure
$R^{2}$ gravity. Recent work on the BRST invariance of other gravitational
theories can be found in [13, 14, 16].

In this paper, we consider scenarios where the symmetry of pure $R^{2}$
gravity is enlarged further. We show that when a massless scalar field is
added to pure $R^{2}$ gravity, the action becomes Weyl invariant when the
equations of motion are satisfied. No separate external condition is required
to be imposed on the conformal factor $\Omega(x)$ as this occurs naturally via
the equations of motion. One passes from restricted Weyl invariance to on-
shell Weyl invariance. One can then enlarge the symmetry further to include
BRST symmetry. In analogy with the BRST invariance in gauge theories in the
presence of a gauge-fixing term, we establish BRST invariance in the presence
of the Weyl-breaking pure $R^{2}$ gravity term. The BRST transformations here
are not associated with gauge transformations (such as diffeomorphisms) but
are a generalization of Weyl (local scale) transformations where the conformal
factor is composed of Grassmann variables. Therefore, in contrast to the BRST
invariance in gauge theory, it is anomalous since renormalization introduces a
scale (leading to the well-known Weyl (conformal) anomaly). We show that the
spontaneous breaking of the BRST symmetry yields an Einstein action with its
own symmetry that is also anomalous. This is in agreement with previous work
where it was shown that when conformal symmetry is spontaneously broken there
is conformal anomaly matching in the unbroken and and broken phases [18, 19].

The paper is organized as follows. In section 2, we obtain the on-shell Weyl
invariance of pure $R^{2}$ gravity when a massless scalar field is included in
the action. In section 3, we obtain the BRST invariance of pure $R^{2}$
gravity. In section 4, we show that the spontaneous breaking of the BRST
symmetry yields an Einstein action and that there is a quantum anomaly in both
the unbroken and broken sectors. We conclude with section 5 where we summarize
our results, provide further physical insights and discuss directions for
future work. We relegate to Appendix A some technical details on the symmetry
of the Einstein action.

##  2 Pure $R^{2}$ gravity plus a massless scalar: from restricted to on-shell
Weyl invariance

The action of pure $R^{2}$ gravity is given by

$S=\int\sqrt{-g}\,d^{4}x\,\alpha\,R^{2}$ (1)

where $R$ is the Ricci scalar and $\alpha$ a dimensionless constant. This
action is restricted Weyl invariant i.e. it is invariant under the Weyl
transformation $g_{\mu\nu}\to\Omega^{2}(x)\,g_{\mu\nu}$ if the conformal
factor $\Omega(x)$ obeys the condition $\Box\Omega(x)=0$. This invariance
stems from the fact that $R\to R/\Omega^{2}$ when $\Box\Omega(x)=0$. As
already mentioned, this implies that pure $R^{2}$ gravity has a greater
symmetry than global scale symmetry (where $\Omega(x)$ would have to be a
constant).

We now show that pure $R^{2}$ gravity can be Weyl-invariant on-shell when a
minimally coupled real massless scalar field is added to the action. Here, the
condition $\Box\Omega(x)=0$ is not imposed as an external condition but
satisfied automatically by the equations of motion. The action of pure $R^{2}$
gravity with a minimally coupled real massless scalar field $\phi$ is given by

$S_{a}=\int\sqrt{-g}\,d^{4}x\,\big{(}\alpha\,R^{2}-\dfrac{1}{2}\,g^{\mu\nu}\,\partial_{\mu}\phi\,\partial_{\nu}\phi\big{)}$
(2)

where $\phi(x)$ is a real scalar field. Under the Weyl transformation
$g_{\mu\nu}\to e^{-2\,\epsilon\,\phi}\,g_{\mu\nu}$, where $\epsilon$ is a real
constant, the Ricci scalar transforms as

$R\to R\,e^{2\epsilon\phi}-6\,e^{3\epsilon\,\phi}\,\Box(e^{-\epsilon\,\phi})$
(3)

and $\sqrt{-g}\to e^{-4\,\epsilon\,\phi}\,\sqrt{-g}$ so that action (2)
transforms to

$\displaystyle
S_{b}=\int\sqrt{-g}\,d^{4}x\,\Big{(}\alpha\,\big{(}R^{2}-12\,R\,e^{\epsilon\,\phi}\,\Box(e^{-\epsilon\,\phi})+36\,e^{2\epsilon\,\phi}\,(\Box(e^{-\epsilon\,\phi}))^{2}\big{)}-\dfrac{1}{2}\,e^{-2\,\epsilon\,\phi}\,g^{\mu\nu}\,\partial_{\mu}\phi\,\partial_{\nu}\phi\Big{)}\,.$
(4)

The equations of motion yield $\Box(e^{-\epsilon\,\phi})=0$. Therefore, when
the equations of motion are satisfied, the above action reduces to

$\displaystyle
S_{c}=\int\sqrt{-g}\,d^{4}x\,\Big{(}\alpha\,R^{2}-\dfrac{1}{2}\,g^{\mu\nu}\,\partial_{\mu}{\psi}\,\partial_{\nu}\psi\Big{)}$
(5)

where $\psi$ is a real massless scalar field (related to the old scalar $\phi$
via $\psi=e^{-\epsilon\,\phi}/\epsilon$). Note that the equation of motion for
$\psi$ is $\Box\psi=0$ which is equivalent to $\Box(e^{-\epsilon\,\phi})=0$
and consistent with what we previously obtained. We therefore recover pure
$R^{2}$ gravity with a minimally coupled real massless scalar field $\psi$.
What happened here is that the restricted Weyl condition $\Box\,\Omega=0$ with
$\Omega=e^{-\epsilon\,\phi}$ did not have to be imposed as a separate
condition because it was satisfied automatically by the equations of motion.
In short, pure $R^{2}$ gravity became Weyl invariant on-shell in the presence
of a massless scalar field. It passed from restricted Weyl invariance to on-
shell Weyl invariance.

##  3 BRST invariance of pure $R^{2}$ gravity

Before discussing BRST invariance in the case of pure $R^{2}$ gravity, let us
first recall how BRST invariance works in gauge theories in Minkowski
spacetime. For illustrative purposes, we will consider the case of scalar QED.
The Abelian version of the Faddeev-Popov Lagrangian is then given by [12]

$\displaystyle\mathcal{L}=-\dfrac{1}{4}\,F_{\mu\nu}^{2}-(D^{\mu}\phi_{a}^{*})(D_{\mu}\phi_{a})-m^{2}\,\phi_{a}^{*}\,\phi_{a}-\dfrac{1}{2\,\xi}(\partial_{\mu}\,A^{\mu})^{2}+\bar{c}\,\Box
c$ (6)

where $c(x)$ and $\bar{c}(x)$ are independent Grassmann-valued fields,
$\phi_{a}$ are a set of complex scalar fields and $D^{\mu}$ is the usual
covariant derivative. The gauge fixing term,
$\frac{1}{2\,\xi}(\partial_{\mu}\,A^{\mu})^{2}$ breaks the gauge symmetry
since it is not invariant under the transformation $A_{\mu}\to
A_{\mu}+\frac{1}{e}\,\partial_{\mu}f(x)$ where $f(x)$ is an arbitrary
function. However, it has a residual symmetry: it is invariant if $f(x)$ obeys
the condition $\Box f=0$. As previously mentioned, this residual symmetry is
the analog of restricted Weyl symmetry in pure $R^{2}$ gravity.

The equation of motion for $c(x)$ is $\Box c=0$. Consider the gauge
transformation with $f(x)=\theta\,c(x)$ for arbitrary Grassmann number
$\theta$. Then, if the equation of motion for $c$ is satisfied, the scalar QED
Lagrangian (6) is invariant under the following transformations

$\displaystyle A_{\mu}\to A_{\mu}+\frac{1}{e}\,\theta\,\partial_{\mu}c(x)$
$\displaystyle\phi_{a}(x)\to
e^{i\theta\,c(x)}\,\phi_{a}(x)=\phi_{a}(x)+i\theta\,c(x)\phi_{a}(x)\,.$ (7)

In other words, the equation $\Box f=\theta\,\Box c=0$ is automatically
satisfied on-shell and does not have to be imposed as a separate condition.
This is similar to what we saw in the previous section for pure $R^{2}$
gravity which was invariant under $g_{\mu\nu}\to\Omega^{2}\,g_{\mu\nu}$ with
$\Omega=e^{-\epsilon\phi}$ when the equations of motion were satisfied.

If the equation of motion for $c$ is not used, the only term in the Lagrangian
(6) which is not invariant under the transformation (7) is
$(\partial_{\mu}A^{\mu})^{2}$ which transforms as

$\displaystyle(\partial_{\mu}A^{\mu})^{2}\to(\partial_{\mu}A^{\mu})^{2}+\dfrac{2}{e}(\partial_{\mu}A^{\mu})(\theta\Box
c)$ (8)

where we used the fact that $\theta^{2}=0$ since $\theta$ is Grassmann. Now,
if under (7) we also have $\bar{c}$ transforming as

$\displaystyle\bar{c}(x)\to\bar{c}(x)-\dfrac{\theta}{e\,\xi}\,(\partial_{\mu}A^{\mu})$
(9)

then the scalar QED Lagrangian (6) is invariant without having to use the
equation of motion for $c$. This is BRST invariance. The crucial point is that
under the BRST transformations given by (7) and (9), the Lagrangian is
invariant despite the presence of the gauge-fixing term
$(\partial_{\mu}A^{\mu})^{2}$.

We now turn to pure $R^{2}$ gravity. Consider the action

$\displaystyle S=\int d^{4}x\sqrt{-g}\,(\alpha\,R^{2}+\bar{c}\,\Box c)$ (10)

where again $c(x)$ and $\bar{c}(x)$ are independent Grassmann-valued fields.
This action is not Weyl-invariant i.e. it is not invariant under the
transformation $g_{\mu\nu}\to\Omega^{2}(x)g_{\mu\nu}$ where $\Omega(x)$ is an
arbitrary smooth function. Consider now the Weyl transformation

$g_{\mu\nu}\to e^{2\,\theta\,c(x)}g_{\mu\nu}=(1+2\,\theta\,c)\,g_{\mu\nu}$
(11)

where $\theta$ is again an arbitrary Grassmann number. Under this
transformation we have

$\displaystyle\sqrt{-g}\,\alpha\,R^{2}\to\sqrt{-g}\,(\alpha\,R^{2}-12\,\alpha\,R\,\theta\,\Box
c\,)$ (12)

where the following transformations were used:
$\sqrt{-g}\to(1+4\,\theta\,c)\,\sqrt{-g}$ and
$R\to(1-2\,\theta\,c)\,R-6\,\theta\,\Box c$. Again, we used that
$\theta^{2}=0$. Under the transformation (11), $\Box c$ transforms as

$\Box c\to(1-2\,\theta\,c)\,\Box c$ (13)

where $g^{\mu\nu}\partial_{\mu}c\,\partial_{\nu}c=0$ was used (this stems from
the fact that $g^{\mu\nu}$ is symmetric and $c$ is Grassmann). The equation of
motion for $c$ is $\Box c=0$ and we see from (12) that
$\sqrt{-g}\,\alpha\,R^{2}$ is Weyl invariant on-shell. However, we can
dispense with the on-shell condition if we also allow $\bar{c}$ to transform
as

$\displaystyle\bar{c}\to(1-2\,\theta\,c)\,\bar{c}+12\,\alpha\,R\,\theta\,.$
(14)

We then obtain

$\displaystyle\sqrt{-g}\,\bar{c}\,\Box c\to\sqrt{-g}\,(\bar{c}\,\Box
c+12\,\alpha\,R\,\theta\,\Box c)\,.$ (15)

The last term on the right hand side of (15) above cancels precisely the last
term on the right hand side of (12). Therefore, the action (10) is invariant
under the combined transformations of (11) and (14) (which we refer to to as
BRST transformations).  This is the BRST invariance of pure $R^{2}$ gravity.
Note that BRST invariance does not require any on-shell or restricted Weyl
condition. It is a generalization of Weyl (conformal) invariance that is valid
in the presence of the Weyl-breaking $R^{2}$ term.

Let us now take a closer look at what is common and what is different between
the BRST invariance of pure $R^{2}$ gravity and the BRST invariance in the
gauge theories of particle physics (for concreteness and simplicity, we will
consider scalar QED again but the main points apply also to QCD). The BRST
invariance in scalar QED can be viewed as a generalization of gauge invariance
in the presence of the gauge-fixing (and hence gauge-breaking) term
$(\partial_{\mu}\,A^{\mu})^{2}$. The are two points in common between the
scalar QED and $R^{2}$ cases. First, the Ricci scalar $R$ under a Weyl
transformation and the term $\partial_{\mu}\,A^{\mu}$ under a gauge
transformation both pick up an extra $\Box\Phi(x)$ term (where $\Phi(x)$
represents either a conformal factor $\Omega(x)$ in a Weyl transformation or a
function $f(x)$ in a gauge transformation). Recall that in a BRST
transformation, $\Phi(x)$ is a product of a Grassmann number $\theta$ with a
Grassmann field (the product yields a commuting (bosonic) quantity). The
second point in common is that $R$ and $\partial_{\mu}\,A^{\mu}$ are both
squared. The squaring yields a $(\Box\Phi(x))^{2}$ term which is zero since
$\theta^{2}=0$. The squaring still leaves one extra $\Box\Phi(x)$ term and
this is cancelled out in both cases via the transformation property of a
Grassmann field. These two common points render the analogy between the two
cases quite strong. However, there is one important difference. In scalar QED
(and in QCD) , the BRST transformations are associated with gauge
transformations. The BRST invariance of pure $R^{2}$ gravity that we are
considering here is not associated with gauge transformations (such as
diffeomorphisms) but with Weyl (local scale) transformations. We will see that
this difference plays an important role when the theory is quantized.

##  4 Spontaneous breaking of BRST symmetry

We now show that the BRST-invariant action

$\displaystyle S=\int d^{4}x\sqrt{-g}\,(\alpha R^{2}+\bar{c}\,\Box c)$ (16)

is conformally equivalent to an action that involves the Einstein-Hilbert
term; this will involve the spontaneous breaking of BRST symmetry. The
starting point is to introduce a auxiliary field $\sigma(x)$ to rewrite the
above action into the equivalent form

$\displaystyle S_{1}$ $\displaystyle=\int
d^{4}x\sqrt{-g}\,(-\alpha(b\,\sigma+R)^{2}+\alpha R^{2}+\bar{c}\,\Box c)$
$\displaystyle\int
d^{4}x\sqrt{-g}\,(-\alpha\,b^{2}\,\sigma^{2}-2\,\alpha\,b\,R\,\sigma+\bar{c}\,\Box
c)$ (17)

where $b$ is a real non-zero constant with dimensions of mass squared and
$\sigma(x)$ is dimensionless. Action (17) is equivalent to the original action
(16) since adding the squared term in the first line of (17) does not alter
anything (classically, the equations of motion are unaffected and quantum
mechanically, the path integral over $\sigma$ is a Gaussian which yields a
constant). The equivalent action (17) is also BRST invariant; it is invariant
under the following transformations:

$\displaystyle
g_{\mu\nu}\to(1+2\,\theta\,c)\,g_{\mu\nu}\quad;\quad\bar{c}\to(1-2\,\theta\,c)\,\bar{c}-12\,\theta\,\alpha\,b\,\sigma\quad;\quad\sigma\to(1-2\theta\,c)\,\sigma$
(18)

where $\theta$ is again a Grassmann number. Note that the BRST invariance
requires the auxiliary field $\sigma$ to transform besides the fields
$g_{\mu\nu}$ and $\bar{c}$. We now perform the following conformal (Weyl)
transformation:

$\displaystyle g_{\mu\nu}\to\sigma^{-1}\,g_{\mu\nu}$
$\displaystyle\bar{c}\to\sigma\,\bar{c}$ (19)

which leads to $\sqrt{-g}\to\sigma^{-2}\,\sqrt{-g}$ and
$R\to\sigma\,R-6\,\sigma^{3/2}\Box(\sigma^{-1/2})$. Under the above conformal
transformation, action (17) becomes

$\displaystyle S_{2}$ $\displaystyle=\int
d^{4}x\sqrt{-g}\,(-\alpha\,b^{2}-2\,\alpha\,b\,R+\dfrac{3\alpha\,b}{\sigma^{2}}\,\partial_{\mu}\sigma\,\partial^{\mu}\sigma+\bar{c}\,\Box
c-\dfrac{1}{\sigma}\,\bar{c}\,\,\partial^{\mu}c\,\partial_{\mu}\sigma)\,.$
(20)

The above action is no longer invariant under the BRST transformations given
by (18). The BRST symmetry has been spontaneously broken. The factor
$\sigma^{-1}$ appearing in the conformal transformation (19) is valid only for
non-zero $\sigma$ so that the VEV (vacuum expectation value) of the field
$\sigma$ must be non-zero. The VEV is therefore not invariant under the BRST
transformation $\sigma\to(1-2\theta\,c)\,\sigma$ leading to the spontaneous
breaking of the BRST symmetry.

We can identify $-2\,\alpha\,b\,R$ as an Einstein-Hilbert term if we equate
$-2\,\alpha\,b$ with $\frac{1}{16\pi\,G}$ where $G$ is Newton’s constant. The
constant term $-\alpha\,b^{2}$ can then be associated with a cosmological
constant $\Lambda=-b/4$. Note that though $-2\,\alpha\,b$ is positive, the
constant $b$ can be either positive or negative (but not zero). This implies
that the cosmological constant can be either positive corresponding to a de
Sitter (dS) background or negative corresponding to an anti-de Sitter (AdS)
background but it cannot be identically zero. We can then express (20) as the
following Einstein action,

$\displaystyle S_{E}$ $\displaystyle=\int
d^{4}x\sqrt{-g}\,\Big{(}\frac{1}{16\pi\,G}(R-2\,\Lambda)+\dfrac{3\alpha\,b}{\sigma^{2}}\,\partial_{\mu}\sigma\,\partial^{\mu}\sigma+\bar{c}\,\Box
c-\dfrac{1}{\sigma}\,\bar{c}\,\,\partial^{\mu}c\,\partial_{\mu}\sigma)\Big{)}\,.$
(21)

We have left the constant $3\,\alpha\,b$ in the action for simplicity but it
is not an independent constant; it is equal to $\tfrac{-3}{32\,\pi G}$. We
therefore obtain an Einstein-Hilbert action with non-zero cosmological
constant, a kinetic term for the scalar $\sigma$ (which we will express in
canonical form later) and an interaction term. Recall that $\sigma$ is non-
zero so that divisions by $\sigma$ pose no issue. It is well-known that in
spontaneously broken theories, the vacuum breaks the symmetry but it is not
actually broken in the Lagrangian but manifested or realized in a different
way [12]. It can be directly verified (see Appendix A) that the Einstein
action (21) is invariant under the following transformations:

$\displaystyle\sigma\to(1-2\theta\,c)\,\sigma\,\,,\,\,g_{\mu\nu}\to
g_{\mu\nu}\text{ and }\bar{c}\to\bar{c}-12\,\theta\,\alpha\,b\,.$ (22)

The BRST symmetry of action (17) manifests itself in the Einstein action (21)
via its symmetry under the above transformations (22). We now show how
transformation (22) stems from the BRST transformations (18). In the Einstein
action and transformation (22) label the metric and the barred Grassmann field
with a subscript E i.e. $g_{{\mu\nu}_{E}}$ and $\bar{c}_{E}$. In action (17)
and transformation (18) we leave $g_{\mu\nu}$ and $\bar{c}$ as is. Then the
conformal transformation (19) yields $g_{{\mu\nu}_{E}}=\sigma\,g_{\mu\nu}$ and
$\bar{c}_{E}=\sigma^{-1}\,\bar{c}$. Under the BRST transformations (18) we
obtain
$g_{{\mu\nu}_{E}}=\sigma\,g_{\mu\nu}\to(1-2\,\theta\,c)\,\sigma\,(1+2\,\theta\,c)\,g_{\mu\nu}=\sigma\,g_{\mu\nu}=g_{{\mu\nu}_{E}}$
and
$\bar{c}_{E}=\sigma^{-1}\,\bar{c}\to(1+2\,\theta\,c)\,\sigma^{-1}\big{(}(1-2\,\theta\,c)\,\bar{c}-12\,\theta\,\alpha\,b\,\sigma\big{)}=\sigma^{-1}\bar{c}-12\,\theta\,\alpha\,b=\bar{c}_{E}-12\,\theta\,\alpha\,b$.
We have therefore obtained the transformations $g_{{\mu\nu}_{E}}\to
g_{{\mu\nu}_{E}}$ and $\bar{c}_{E}\to\bar{c}_{E}-12\,\theta\,\alpha\,b$ which
correspond to those in (22). Note that we used
$\sigma\to(1-2\theta\,c)\,\sigma$ in (18) to derive this, so the
transformation of $\sigma$ is also part of (22).

We can define a real massless scalar field
$\psi(x)=\sqrt{-3\alpha\,b}\,\ln{\sigma(x)}$ so that the kinetic term for
$\sigma$ is expressed in canonical form. The Einstein action (21) expressed in
terms of the field $\psi$ is

$\displaystyle S$ $\displaystyle=\int
d^{4}x\sqrt{-g}\,\Big{(}\frac{1}{16\pi\,G}(R-2\,\Lambda)-\partial_{\mu}\psi\,\partial^{\mu}\psi+\bar{c}\,\Box
c-\dfrac{1}{\sqrt{-3\alpha\,b}}\,\bar{c}\,\partial^{\mu}c\,\partial_{\mu}\psi\Big{)}\,.$
(23)

The massless scalar field $\psi$ corresponds to the Nambu-Goldstone boson of
the broken sector. Under transformation (22), the field $\psi$ transforms as a
shift $\psi\to\psi-\sqrt{-3\alpha\,b}\,2\,\theta\,c$ (whereas
$\bar{c}\to\bar{c}-12\,\theta\,\alpha\,b$ and $g_{\mu\nu}\to g_{\mu\nu}$). The
above action (23) is invariant under those transformations (see Appendix A).
This is in accord with what we expect from spontaneously broken theories: the
original symmetry in the Lagrangian manifests itself in the broken sector as a
shift symmetry of the Goldstone bosons [12].

###  4.1 Quantum anomaly

We saw that the action (17) is BRST invariant under the following
transformations:
$g_{\mu\nu}\to(1+2\,\theta\,c)\,g_{\mu\nu}\,\,,\,\,\bar{c}\to(1-2\,\theta\,c)\,\bar{c}-12\,\theta\,\alpha\,b\,\sigma\,\,,\,\,\sigma\to(1-2\theta\,c)\,\sigma$.
Each transformation involves a Weyl transformation where the conformal factor
is expressed in terms of of a product of two Grassmann variables The BRST
symmetry is therefore a generalization of Weyl (conformal) symmetry. After
quantization, renormalization introduces a scale which breaks the BRST
symmetry since it automatically breaks Weyl symmetry (leading to the well-
known Weyl (conformal) anomaly). So the BRST symmetry of pure $R^{2}$ gravity
is anomalous. This is in contrast to the BRST invariance of gauge theories
like QCD which have no anomaly.

After the BRST symmetry is spontaneously broken and we obtain the Einstein
action (21), we saw that the BRST symmetry manifests itself now in the
Einstein action as a symmetry under the transformations (22). This symmetry is
also anomalous since the transformation of the field $\sigma$ is a Weyl
transformation and renormalization breaks this symmetry (leading again to the
Weyl (conformal anomaly). Another way to see this is to note that the only
fields that transform in (22) are $\bar{c}$ and $\sigma$. The transformation
for $\bar{c}$ is simply a constant shift so that its path integral measure
$\mathcal{D}\bar{c}$ is invariant. However, $\sigma$ undergoes a Weyl
transformation and this introduces a non-trivial Jacobian J (i.e. $J\neq 1$)
to the measure $\mathcal{D}\sigma$. Since the measure is not invariant, this
implies there is an anomaly [17]. So the symmetry in the unbroken phase and
its associated symmetry in the broken phase are both anomalous. Our finding
here is in accord with previous work that shows that when the Weyl or
conformal symmetry is spontaneously broken there is conformal anomaly matching
between the unbroken and broken phases [18, 19].

##  5 Conclusion

In the last six years or so, we have kept discovering new aspects of pure
$R^{2}$ gravity. A non-exhaustive list includes its unitarity among quadratic
gravity theories [4], its conformal equivalence to Einstein gravity with non-
zero cosmological constant and massless scalar field [1, 2, 3, 4, 5], its
restricted Weyl symmetry [7, 10, 11], its spontaneous symmetry breaking to
Einstein gravity [3, 5] and the lack of a propagating graviton when the theory
is linearized about a Minkowski background [4] (where there is no Einstein
equivalence since the cosmological constant is zero).

In this paper, we have gained further insights into this theory. We saw that
pure $R^{2}$ gravity has an analog with the gauge-fixing term
$(\partial_{\mu}A^{\mu})^{2}$ in gauge theory. $R^{2}$ is not invariant under
the Weyl transformation $g_{\mu\nu}\to\Omega^{2}(x)\,g_{\mu\nu}$ just like
$(\partial_{\mu}\,A^{\mu})^{2}$ is not invariant under the gauge
transformation $A_{\mu}\to A_{\mu}+\frac{1}{e}\,\partial_{\mu}f(x)$. However,
each have a residual symmetry (when $\Box\Omega=0$ is satisfied in the gravity
case and $\Box f=0$ is satisfied in the gauge case). This analogy opened the
door towards enlarging the symmetry of pure $R^{2}$ gravity to include BRST
symmetry. We first showed that when a massless scalars field was included in
the pure $R^{2}$ action, the condition $\Box\Omega=0$ could be met
automatically when the equations of motion were satisfied i.e. we went from
restricted Weyl to on-shell Weyl invariance. Finally, we obtained the BRST
invariance of pure $R^{2}$ gravity where no restricted Weyl or on-shell
condition is required. The BRST transformations involve Weyl transformations
where the conformal factor is composed of products of Grassmann variables (the
conformal factor itself is commutative). The important point is that the BRST
invariance exists despite the Weyl-breaking $R^{2}$ term.

There is one important difference between the BRST symmetry in gauge theories
like QCD and the BRST symmetry that we have considered here for pure $R^{2}$
gravity. Gauge invariance in particle physics is preserved after quantization.
The BRST invariance of QCD is a generalization of gauge invariance so that it
is also preserved after quantization; there is no anomaly. In contrast to
gauge symmetry, global scale or Weyl (local scale) symmetry is broken after
quantization since renormalization introduces a scale. The BRST symmetry of
pure $R^{2}$ gravity is a generalization of Weyl (conformal) symmetry so that
it is also broken after quantization leading to the well-known Weyl
(conformal) anomaly. After the spontaneous breaking of the BRST symnmetry, we
obtained an Einstein action. We showed that this action has its own symmetry
and that it is also anomalous. This is in accord with previous work that shows
that when the Weyl (conformal) symmetry is spontaneously broken there is
conformal anomaly matching between the unbroken and broken sectors [18, 19].

The focus of this paper was pure $R^{2}$ gravity because of its many special
and attractive features that we previously mentioned. All other quadratic
gravity theories (like Weyl-squared, Riemann-squared, etc.), apart from
boundary terms, can be expressed as a linear combination of $R^{2}$ and
$R^{\mu\nu}R_{\mu\nu}$. The latter term, the square of the Ricci tensor,
appears in quantum corrections to General Relativity (GR) and even though it
does not constitute a valid UV completion of GR due to its non-unitarity
(yields a massive spin two ghost [4, 20]), it still makes a well-known
calculable short-range correction to the Newtonian potential [21, 12]. Like
$R^{2}$, the term $R_{\mu\nu}R^{\mu\nu}$ is not Weyl-invariant so it would be
of interest to see if it can be BRST invariant. It is not in the form of a
scalar squared like $(\partial_{\mu}A^{\mu})^{2}$ or $R^{2}$, so one may be
inclined to think that the BRST formalism would not apply here. However, like
pure $R^{2}$, it was shown in [7] that $R_{\mu\nu}R^{\mu\nu}$ is restricted
Weyl invariant (up to a boundary term). This suggests that the procedure used
to establish the BRST invariance of pure $R^{2}$ gravity might in the end also
work for this quadratic theory. It would therefore be worthwhile and
interesting to investigate this further.

## Acknowledgments

A.E. acknowledges support from a discovery grant of the National Science and
Engineering Research Council of Canada (NSERC).

##  Appendix A Symmetry of Einsten Action

In this appendix we show that the Einstein action (21) given by

$\displaystyle S_{E}$ $\displaystyle=\int
d^{4}x\sqrt{-g}\,\Big{(}\frac{1}{16\pi\,G}(R-2\,\Lambda)+\dfrac{3\alpha\,b}{\sigma^{2}}\,\partial_{\mu}\sigma\,\partial^{\mu}\sigma+\bar{c}\,\Box
c-\dfrac{1}{\sigma}\,\bar{c}\,\,\partial^{\mu}c\,\partial_{\mu}\sigma)\Big{)}$
(A.1)

is invariant under the transformations (22) given by

$\displaystyle\sigma\to(1-2\theta\,c)\,\sigma\,\,,\,\,g_{\mu\nu}\to
g_{\mu\nu}\text{ and }\bar{c}\to\bar{c}-12\,\theta\,\alpha\,b\,.$ (A.2)

Under the above transformation, the metric $g_{\mu\nu}$ does not change so
that $\sqrt{-g}$ as well as the term
$\sqrt{-g}\,\frac{1}{16\pi\,G}(R-2\,\Lambda)$ does not change. The other terms
in the above Einstein action transform as

$\displaystyle\dfrac{3\alpha\,b}{\sigma^{2}}\,\partial_{\mu}\sigma\,\partial^{\mu}\sigma\to\dfrac{3\alpha\,b}{\sigma^{2}}\,\partial_{\mu}\sigma\,\partial^{\mu}\sigma-\frac{12\,\theta\,\alpha\,b}{\sigma}\,\partial^{\mu}c\,\partial_{\mu}\sigma$
$\displaystyle-\dfrac{1}{\sigma}\,\bar{c}\,\,\partial^{\mu}c\,\partial_{\mu}\sigma)\to-\dfrac{1}{\sigma}\,\bar{c}\,\,\partial^{\mu}c\,\partial_{\mu}\sigma+\frac{12\,\theta\,\alpha\,b}{\sigma}\,\partial^{\mu}c\,\partial_{\mu}\sigma$
$\displaystyle\bar{c}\,\Box c\to\bar{c}\,\Box c-12\,\theta\,\alpha\,b\,\,\Box
c$ (A.3)

where we used that $\theta^{2}=0$ (since $\theta$ is a Grassmann number) and
that $g^{\mu\nu}\,\partial_{\mu}c\,\partial_{\nu}c=0$ since $g^{\mu\nu}$ is
symmetric and $c(x)$ and its derivatives are Grassmann fields. We see that the
extra term
$-\frac{12\,\theta\,\alpha\,b}{\sigma}\,\partial^{\mu}c\,\partial_{\mu}\sigma$
in the first line of (A.3) is canceled exactly by the extra term in the second
line. The extra term in the third line, $-12\,\theta\,\alpha\,b\,\,\Box c$,
where $-12\,\theta\,\alpha\,b$ is a constant, does not cancel out with any
other extra term in (A.3). However, $\sqrt{-g}\,\Box c$ is a total derivative
that yields an inconsequential boundary term in the action. We have therefore
shown that action (A.1) is invariant under transformations (A.2).

We saw in section 4 that the Einstein action (A.1) can be expressed in terms
of a real massless scalar field $\psi(x)=\sqrt{-3\alpha\,b}\,\ln{\sigma(x)}$
as action (23):

$\displaystyle S$ $\displaystyle=\int
d^{4}x\sqrt{-g}\,\Big{(}\frac{1}{16\pi\,G}(R-2\,\Lambda)-\partial_{\mu}\psi\,\partial^{\mu}\psi+\bar{c}\,\Box
c-\dfrac{1}{\sqrt{-3\alpha\,b}}\,\bar{c}\,\partial^{\mu}c\,\partial_{\mu}\psi\Big{)}$
(A.4)

where $\psi$ was identified as the Nambu-Goldstone boson of the broken sector.
We stated in section 4 that the action (A.4) was invariant under the following
transformations:

$\displaystyle\psi\to\psi-\sqrt{-3\alpha\,b}\,2\,\theta\,c\text{ ,
}\bar{c}\to\bar{c}-12\,\theta\,\alpha\,b\text{ and }g_{\mu\nu}\to
g_{\mu\nu}\,.$ (A.5)

We now verify this statement. Under (A.5) the last three terms in action (A.4)
transform as:

$\displaystyle-\partial_{\mu}\psi\,\partial^{\mu}\psi\to-\partial_{\mu}\psi\,\partial^{\mu}\psi+4\,\theta\sqrt{-3\,\alpha\,b}\,\,\partial_{\mu}\psi\,\partial^{\mu}c$
$\displaystyle-\dfrac{1}{\sqrt{-3\alpha\,b}}\,\bar{c}\,\partial^{\mu}c\,\partial_{\mu}\psi\to-\dfrac{1}{\sqrt{-3\alpha\,b}}\,\bar{c}\,\partial^{\mu}c\,\partial_{\mu}\psi-4\,\theta\,\sqrt{-3\,\alpha\,b}\,\,\partial_{\mu}\psi\,\partial^{\mu}c\,$
$\displaystyle\bar{c}\,\Box c\to\bar{c}\,\Box c-12\,\theta\,\alpha\,b\,\Box
c\,.$ (A.6)

We see that the extra term
$+4\,\theta\,\sqrt{-3\,\alpha\,b}\,\,\partial_{\mu}\psi\,\partial^{\mu}c$ in
the first line above is cancelled by the extra term on the second line which
is equal to its negative. The only extra term that is not cancelled is the
term $-12\,\theta\,\alpha\,b\,\Box c$ appearing in the last line. However,
$\sqrt{-g}\,\,\Box c$ is a total derivative which yields a boundary term with
no physical consequence. We have therefore verified that the Einstein action
(A.4) is indeed invariant under the transformations (A.5).

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# Towards high-energy few-cycle optical vortices with minimized topological
charge dispersion

Federico J. Furch1,∗, Gunnar Arisholm2
1Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Max-
Born-Strasse 2a, 12489 Berlin, Germany
2Norwegian Defence Research Establishment (FFI), PO Box 25, 2027 Kjeller,
Norway
∗Corresponding author<EMAIL_ADDRESS>

A simple approach to generate high-energy few-cycle optical vortices with
minimized topological charge dispersion is introduced. By means of numerical
simulations it is shown that, by leveraging the intrinsic properties of
optical parametric chirped pulse amplification (OPCPA), clean transfer of
topological charge from a high energy narrowband pump pulse to a broadband
idler is feasible under certain particular conditions, enabling the generation
of high-energy few-cycle vortex pulses with extremely low topological charge
dispersion.

In 1989 Coullet et al., proposed the existence of optical vortices in laser
cavity modes [1]. A few years later Allen and co-workers demonstrated that
Laguerre-Gaussian (LG) spatial amplitude distributions with azimuthal index
different from zero, which constitute one type of optical vortex, carry
orbital angular momentum (OAM) [2]. Since then, optical vortices have emerged
as powerful tools across a wide range of applications including but not
limited to, optical trapping and manipulation, optical communications, high-
dimensional quantum cryptography, laser micromachining and microscopy [3].
Moreover ultrashort optical vortices with high energy have been proposed for
diverse applications such as hollow plasma drilling [4] or acceleration of
attosecond electron slices [5]. In particular, it has been demonstrated that
high harmonic generation driven by ultrashort IR fields carrying a topological
charge can produce extreme ultraviolet light with OAM [6, 7, 8]. These novel
laser fields can lead, among other interesting effects, to spatially-dependent
modified selection rules during single photon ionization [9, 10] and enable
OAM-induced x-ray dichroism [11].

An optical vortex is characterized by a point of zero intensity and an
azimuthal phase variation that integrates to $2\pi m$ on a closed path around
that point, for a nonzero integer $m$. While vortices can appear randomly in
speckle patterns, the term vortex beam describes a beam with a well-defined
complex amplitude pattern and a single vortex in the center. This form of
phase variation means that the wavefront is helical. $m$ is called the
topological charge, and in the context of LG modes it is indicated by the
azimuthal index of the mode. Although a variety of methods for generating
vortex beams have been explored, most notably, a combination of cylindrical
lenses [2], spiral phase plates [12], liquid-crystal-based spatial light
modulators [13], and holographic plates [14], these approaches rely on
imparting a specific spatial phase at a particular target wavelength. Since
the spatial spiral phase varies for different wavelengths, implementation of
these methods with broadband sources leads to what is known as topological
charge dispersion, i.e., different content of topological charges for
different wavelengths of the spectrum. Topological charge dispersion leads
among other things to a non-preserving spatial distribution during propagation
and complex spatio-temporal structures [15].

A number of methods have been proposed and implemented in order to address the
topological charge dispersion challenge. Bock et al. employed a diffractive-
refractive element to obtain few-cycle vortices at a specific plane from
Ti:Sapphire oscillator pulses [16, 17]. Tokizane et al. combined achromatic
wave plates and a segmented half wave plate for broadband vortex beam
generation [18]. The same group further extended this approach, amplifying a
broadband vortex beam produced in such way in a 2-stage optical parametric
amplifier [19]. Amidst these methods, Atencia et al. developed a compound
holographic optical element to generate achromatic vortices [20], while
Swartzlander designed achromatic spiral phase plates from two different
glasses, providing a perfect topological charge at two wavelengths, akin to
achromatic lenses [21].

With the notable exception of Naik et al., who introduced an elegant approach
based on a Sagnac interferometer to generate achromatic vortices [22], these
techniques often prove complex or challenging to implement, and/or
incompatible with the generation of high-energy ultra broadband vortex pulses.
An alternative to generating a wide-band optical vortex directly (i.e.,
transforming a field without OAM), is to use an optical parametric chirped
pulse amplifier (OPCPA)[23] to transfer the OAM of a narrow-band pump beam to
a wide-band idler beam. The phase-sensitive nature of the parametric
amplification process implies that the idler produced during the amplification
of a weak signal wave carries the phase difference between the pump and signal
waves. This also holds for helical wavefronts, so the difference in
topological charge between the pump and the signal will be transferred to the
idler. The concept of transferring the topological charge from the pump to the
idler was originally demonstrated in an optical parametric amplifier (OPA)
with continuous wave sources [24] and later implemented in pulsed narrowband
optical parametric oscillators where the selective transfer of the topological
charge of the pump to the signal or idler was demonstrated [25]. Camper et al.
demonstrated the generation of a high energy femtosecond optical vortex in the
idler of a noncollinear OPA pumped by a vortex beam [26]. This OPA featured
relatively long pulses and consequently their analysis did not address
topological charge dispersion. In this work it is shown that this concept can
be extended to ultra broadband sources supporting few-cycle pulses. The
topological charge content of the wide-band idler generated in an OPCPA pumped
by a narrowband vortex is analyzed and it is shown that, under certain
conditions, the topological charge dispersion can be minimized over the entire
bandwidth.

Figure 1: Proposed experimental scheme. The beam profiles illustrate Gaussian
or Laguerre-Gaussian (LG) modes. The prism (P) illustrates an optical element
introduced to achieve pulse-front matching. The LG mode of the output idler is
also indicated. M represents a set of mirrors to transport/re-size the
amplified signal into the second amplification stage.

The system under study is the two-stage noncollinear OPCPA schematically shown
in Fig. 1. The first stage had Gaussian beams for both pump and seed, and the
second stage was pumped by an LG mode. The 3D numerical simulations of the
parametric processes were performed utilizing the code Sisyfos (SImulation
SYstem For Optical Science) [27, 28, 29]. The code allows simulating second-
order frequency mixing processes including the effects of pump depletion,
back-conversion, walk-off, dispersion, diffraction, parasitic effects, and
thermal effects. The coupled-wave equations in three dimensions are solved in
Fourier space. This code has been utilized before to simulate OPCPAs under
realistic conditions [30, 31, 32].

All the simulations presented in this work are based on noncollinear type-I
phase-matching in BBO crystals, specifically with a noncollinear angle
$\alpha=2.5^{\circ}$. The first stage was under the walk-off compensation
(WOC) geometry ($\theta_{signal}=0.4702$ rad, $\theta_{pump}=0.4265$ rad, with
$\theta$ the angle between the crystal axis and the k-vector of the signal or
the pump). The WOC geometry minimizes space-time couplings in the amplified
signal but suffers from parasitic second harmonic of signal and idler [33]
(taken into account in the simulations). Meanwhile, the second stage was under
tangential phase-matching or non-walk-off compensation geometry
($\theta_{signal}=0.3829$ rad, $\theta_{pump}=0.4265$ rad), which is better
suited for large beams for which the lateral walk-off between the beams is
small compared to the beam sizes. The pulse parameters employed herein are
similar to a near-infrared OPCPA system developed at the Max Born Institute
[34, 35, 36]. A measured pulse spectrum from a Ti:Sapphire oscillator was
utilized to build the input seed pulses, stretched by applying positive group
delay dispersion (800 fs2), and energy of 1 nJ. The pump pulses were assumed
to have a Gaussian temporal shape with Fourier-Transform limited duration of 1
ps, centered at 515 nm.

It was assumed that the pump beam is transformed into an LG mode by means of a
spiral phase plate, which are nowadays commercially available for a target
wavelength of 515 nm. Spiral phase plates provide high damage threshold and
are easy to implement experimentally. For the simulations shown in this work
it was assumed that the spatial distribution of the pump field is described by
an LG mode with a topological charge of $m=1$ and no topological charge
dispersion. It is important to remark that it was also assumed that in the
second stage pulse front matching was implemented [37] (illustrated by the
prism (P) in Fig. 1). The results can degrade substantially without pulse
front matching (see Supplementary material).

Figure 2: Frequency resolved idler beam profiles at selected wavelengths of
the spectrum. The color scale indicates the normalized intensity

Figure 3: Frequency resolved idler spatial phase profiles at selected
wavelengths of the spectrum. The color scale indicates phase in radians

In the first OPCPA stage the pump pulse energy was 0.4 mJ and the peak
intensity incident on the crystal was 97 GW/cm2. The pump and signal beams in
the first OPCPA stage had matching sizes (700 µm diameter $1/e^{2}$). The
signal pulses were amplified in a 1.5 mm thick crystal to 14.9 µJ. These
amplified signal pulses were utilized to seed the second amplification stage,
pumped by an LG mode with a topological charge of $m=1$, pulse energy of 5 mJ
and peak intensity of 50 GW/cm2 incident on a 1.2 mm thick BBO crystal. In
this case the signal beam diameter was 4.8 mm, approximately 20$\%$ larger
than the pump. At the output of this second OPCPA stage the idler carries the
same topological charge as the pump. This is illustrated in Fig. 2 and Fig. 3
showing frequency resolved beam profiles and spatial phases at selected
wavelengths. The beam profiles show a clear annular shape with the amplitude
vanishing in the center. The noncollinear geometry and spatial walk-off cause
an uneven spatial intensity distribution (left/right asymmetry), especially
towards the edges of the spectrum. The phase profiles show in all cases
illustrated in Fig. 3 the characteristic azimuthal phase variation. The
topological charge dispersion of the idler was quantified applying the method
of Wang et al., who studied the effects of parametric amplification over the
topological charge content of a broadband optical vortex used as seed of the
amplifier [38]. In order to calculate the topological charge content they
expanded the spatial part of the laser electric field in helical harmonics.
Here it is assumed that this is done at a particular frequency component:

$E(r,\phi)=\frac{1}{\sqrt{2\pi}}\sum_{m=-\infty}^{\infty}a_{m}(r)e^{im\phi}$
(1)

where the functions $a_{m}(r)$ can be calculated as

$a_{m}(r)=\frac{1}{\sqrt{2\pi}}\int_{0}^{2\pi}E(r,\phi)e^{-im\phi}d\phi$ (2)

To obtain a coefficient corresponding to a particular topological charge $m$
the absolute value square of the function $a_{m}(r)$ is integrated:

$b_{m}=\int_{0}^{\infty}|a_{m}(r)|^{2}rdr$ (3)

Finally, the proportion of a particular coefficient relative to all others in
the expansion can be simply calculated as

$P_{m}=\frac{b_{m}}{\sum_{n=-\infty}^{\infty}b_{n}(r)}$ (4)

Figure 4: Topological charge dispersion in the idler. (a) Relative topological
charge content ($P_{m}$ coefficients) at selected wavelengths across the idler
spectrum. (b) Relative topological charge content assuming a spiral phase
plate (SPP) has been utilized to convert the broadband Gaussian field into a
Laguerre-Gaussian (c) Idler spectrum (red), relative proportion of topological
charge $m=1$ across the spectrum for the optical vortex generated during
parametric amplification (dashed blue) or utilizing an SPP (dashed green
line).

Equation 4 offers insight into the topological charges forming the spatial
profile at a particular frequency component. Figure 4 (a) shows the
distribution of topological charges calculated at selected wavelengths over
the idler spectrum. It is observed that the topological charge of the pump has
been transferred to the idler with remarkable purity. In addition, Fig. 4 (c)
shows the idler spectrum (red) and illustrates the proportion of topological
charge $m=1$ across the entire idler spectrum (dashed blue). For comparison
Fig. 4 (b) shows the distribution of topological charge in the case in which a
fused silica spiral phase plate designed for 1600 nm is utilized to produce a
vortex beam with topological charge $m=1$ for an initial Gaussian spatial
distribution. In addition, the dashed green line in Fig. 4 (c) illustrates the
proportion of topological charge $m=1$ across the spectrum in this case.
Converting a broadband OPCPA seed into a vortex beam using a spiral phase
plate (before amplification) is an approach that has been reported in the
literature [39, 38], and in particular the effect of saturation over the
topological charge dispersion has been observed.

Under the amplification conditions previously described the signal field was
further amplified to an energy of 1.02 mJ and the resulting idler had a pulse
energy of 0.59 mJ. This corresponds to a total extraction efficiency
(considering both, signal and idler) of 32.2 $\%$ and represents a reasonable
compromise between efficient energy extraction and minimization of space-time
couplings that degrade the peak intensity of the amplified pulses [32]. The
spectrum of Fig. 4 (c) supports a Fourier-transform limited pulse duration of
8.5 fs, corresponding to approximately 1.67 optical cycles at the carrier
frequency 195.9 THz (1531 nm) (see Supplementary material). The angular
dispersion of the idler can be compensated with established methods [37, 40]
producing high energy, sub-2 cycle optical pulses in the short-wave infrared
with a clean topological charge content. Alternatively, a controlled angular
dispersion could be introduced in the signal beam seeding the second stage to
obtain an angular-dispersion-free idler [41].

The scenario proposed here represents just one example of many different
variations that can be implemented around the same idea. Other approaches
include seeding a last OPCPA stage with the idler (with the appropriate
angular dispersion) from a previous stage and pumped by an LG mode to obtain
an angular-dispersion-free signal with controlled topological charge,
combining optical vortices in both pump and seed fields (signal or idler) to
control the topological charge of the third field generated during
amplification, or designing a multi-stage, high energy OPCPA system in which
the last amplification stage is pumped by an optical vortex to scale up the
pulse energy as already proposed in [26] for a femtosecond OPA. Additionally,
similar scenarios based on sum frequency generation [42] can be implemented.
Furthermore, nonlinear propagation in thin glass plates could be utilized to
push the pulse duration towards the single-cycle limit [43, 44]. Generating
few- and near-single-cycle signal and idler pulses with controlled topological
charge can be exploited in ultrafast two-color pump-probe spectroscopic
techniques and XUV attosecond pulse generation with controlled OAM [7, 8].

In conclusion, this study demonstrates the remarkable capability to transfer
the topological charge of a narrowband, high-energy laser pulse with
exceptional fidelity to broadband idler pulses during optical parametric
chirped pulse amplification. The proposed approach opens the door for the
generation of high-energy few-cycle optical vortices with fine control over
the topological charge. Follow up work will focus on the experimental
demonstration of the proposed idea.

Funding. European Union Horizon 2020 program Laserlab Europe (871124).

The authors thank Achut Giree for useful discussions.

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## Supplementary material

In the main manuscript it has been shown that the topological charge of an
energetic narrowband optical vortex pumping an optical parametric chirped
pulse amplification (OPCPA) stage can be transferred to the generated idler
with high fidelity over the whole resulting ultra broad spectrum which
supports few-cycle pulse duration. But amplification of extremely broadband
pulses requires a noncollinear phase-matching geometry, which inherently
introduces space-time couplings [45, 33, 46, 32]. In this additional material
the influence of pulse front matching on the fidelity of the topological
charge transferred is briefly explored, complementing the results presented in
the main manuscript. In addition, the compressibility of the idler pulses is
corroborated by compensating the group delay dispersion.

##  Appendix A Influence of pulse front matching

One of the well studied and well understood effects of the noncollinear phase-
matching geometry in OPCPAs is the emergence of spatial chirp in the amplified
signal. This effect originates in the effective tilted gain experienced by the
seed (signal) field due to the angle between the pulse fronts of the signal
and pump (see for example Fig. 1 in [46]). As a result, the idler not only
experiences angular dispersion, essential for satisfying phase-matching over a
broad spectral range, but also suffers from additional spatial chirp. One way
of visualizing the spatial chirp is by observing the frequency resolved beam
profile at particular frequencies over the spectrum. The center of gravity of
the beam moves with frequency in the walk-off plane of the OPCPA stage. Of
course, in the case of the idler, this also happens due to angular dispersion.

In order to explore this effect more closely in the case of OPCPAs pumped with
an optical vortex a set of additional simulations were performed. In all of
them, a noncollinear stage was simulated in the non-walk-off compensation
(NWOC) geometry and a seed almost identical to the seed in the first stage
described in the main manuscript (i.e., same spectrum, chirp, energy). The
pump field was characterized by a Laguerre-Gaussian (LG) mode with a
topological charge $m=1$, with the same spectrum and pulse duration of the
pump fields utilized in the simulations of the main manuscript, and pulse
energy of 5 mJ. The pump peak intensity incident on the 2 mm thick BBO crystal
was 50 GW/cm2 and the seed beam size was slightly larger ($\approx$ 20$\%$)
than the pump beam. In the choice of input parameters it was intended to
minimize other effects inducing space-time couplings such as saturation
(combined extraction efficiency of signal plus idler below 1$\%$) and the
lateral translation of the beams during propagation.

Figure S.1: Frequency resolved beam profiles at selected frequencies of the
idler spectrum. The pulse fronts are not tilted, that is, the pulse front of
each beam is perpendicular to the corresponding k-vector.

Figure S.1 shows the frequency resolved beam profiles for the resulting idler.
In the case of a vortex beam pumping the amplifier, the spatial chirp leads to
an interesting effect, namely the beam profile is a ring only around the
center of the spectrum, while towards the edges of the spectrum it turns into
a half-ring shape, with opposite halves of the ring-like mode appearing in
opposite ends of the spectrum.

In stark contrast, the beam profiles at different frequencies is relatively
homogeneous when pulse front matching [37] is implemented. Fig. S.2 shows the
results of the simulation for the case of pulse-front matching between signal
and pump.

Figure S.2: Frequency resolved beam profiles at selected frequencies of the
idler spectrum. The pulse front of the pump is tilted to be parallel with the
pulse front of the signal (pulse-front matching).

Figure S.3: Frequency resolved phase profiles at selected frequencies of the
idler spectrum. Pulse-front matching between signal and pump has been
implemented.

Figure S.4: Frequency resolved phase profiles at selected frequencies of the
idler spectrum. The pulse fronts are not tilted. The pulse front of each beam
is perpendicular to the corresponding k-vector.

The frequency resolved phase profiles are also affected. In this case the
characteristic homogeneous azimuthal phase variation that can be observed when
the pulse fronts are matched (Fig. S.3) acquires a top/bottom asymmetry when
the pulse-fronts of signal and pump subtend an angle (Fig. S.4). The azimuthal
variation of the phase is preserved, but the change is not constant (or
linear) over the azimuthal coordinate. As it can be observed in Fig. S.4, the
phase variation is faster in the upper half-plane ($y>0$). Interestingly this
top/bottom asymmetry reverses when the sign of the topological charge changes
or when the phase-matching geometry changes (walk-off compensation or non-
walk-off compensation). A detailed investigation of these effects is outside
the scope of the present work and will be addressed in future work.

Figure S.5: Relative topological charge content at selected wavelengths of the
idler spectrum for an OPCPA stage in which pulse-front matching between signal
and pump has been implemented.

Figure S.6: Relative topological charge content at selected wavelengths of the
idler spectrum for an OPCPA stage in which the pulse-fronts of signal and pump
subtend an angle equal to the noncollinear angle.

The consequence of the effects shown in Fig. S.4 and Fig. S.1 is a loss of
fidelity in the transfer of topological charge during the amplification
process. By simple inspection of equations 1-4 in the main manuscript, it is
clear that the combination of the top/bottom asymmetry in the phase profiles
and the left/right asymmetry in the amplitude (beam profiles) result in a
broader distribution of topological charge. In other words, additional modes
are necessary to describe the resulting spatial distribution. Figures S.5 and
S.6 show the relative topological charge content for the case of pulse-front
matching and when no pulse-front matching is implemented respectively.

##  Appendix B Pulse compression to few-cycles

In order to confirm compressibility to few-cycles, the idler pulses
corresponding to the simulation shown in the main manuscript have been
numerically compressed by imposing a global quadratic spectral phase across
the whole spatial distribution. First, the spectral phase was extracted at a
particular point of the near field distribution and approximated by a second
order polynomial. Subsequently this second order phase was subtracted over the
whole spatio-spectral distribution and the spatio-temporal distribution
calculated by applying the inverse Fourier transform. Finally, the pulse shape
was integrated over the entire spatial distribution
($\int|E(x,y,t)|^{2}dxdy$). The procedure was repeated at four different
positions on the near field distribution (marked with crosses in Fig. S.7
(a)). The resulting pulse shapes corresponding to the four positions marked in
S.7 (a) are shown in S.7 (b), together with the Fourier-transform limited
(FTL) pulse.

Figure S.7: (a) Near field beam profile of the idler. The crosses correspond
to the positions from which the spectral phase was extracted. (b) Spatially
integrated pulse shapes corresponding to second-order phase compensation with
the phases extracted at the positions marked in (a).

The remarkable agreement between the pulse shapes obtained by extracting the
phase from four different positions across the spatial distribution shows that
the spatio-temporal distortions are low, as expected from the moderate
extraction efficiencies and the pulse-front matching scheme. In all cases the
pulse duration obtained by compensating the second-order of the spectral phase
amounts to 10.2 fs, corresponding to 2 optical cycles, while the Fourier-
limited pulse duration of 8.5 fs corresponds to 1.67 cycles. Compensation of
the spectral phase up to third order leads to a pulse duration only $0.6\%$
longer than the Fourier-transform limit (not shown in Fig. S.7).