# All-optical beam steering using the polariton lighthouse effect

Samuel M.H. Luk  [  Hadrien Vergnet  [  Ombline Lafont  [  Przemyslaw
Lewandowski  [  Nai H. Kwong  [  Elisabeth Galopin  [  Aristide Lemaitre  [
Philippe Roussignol  [  Jérôme Tignon  [  Stefan Schumacher  [  Rolf Binder  [
Emmanuel Baudin  [<EMAIL_ADDRESS>
###### Abstract

We demonstrate theoretically and experimentally that a specifically designed
microcavity driven in the optical parametric oscillation regime exhibits
lighthouse-like emission, i.e., an emission focused around a single direction.
Remarkably, the emission direction of this micro-lighthouse is continuously
controlled by the linear polarization of the incident laser, and angular beam
steering over $360\text{\,}\mathrm{\SIUnitSymbolDegree}$ is demonstrated.
Theoretically, this unprecedented effect arises from the interplay between the
nonlinear optical response of microcavity exciton-polaritons, the difference
in the subcavities forming the microcavity, and the rotational invariance of
the device.

###### keywords:

Nonlinear Optics, All-Optical Signal Processing, Semiconductor Microcavity,
Polaritons

UoA]Department of Physics, University of Arizona, Tucson, AZ 85721, USA
LPENS]Laboratoire de Physique de l’Ecole Normale Supérieure, ENS, Université
PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité,
24 rue Lhomond 75005 Paris, France LPENS]Laboratoire de Physique de l’Ecole
Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université
Paris-Diderot, Sorbonne Paris Cité, 24 rue Lhomond 75005 Paris, France
Paderborn]Physics Department and Center for Optoelectronics and Photonics
Paderborn (CeOPP), Universität Paderborn, Warburger Strasse 100, 33098
Paderborn, Germany Paderborn]Wyant College of Optical Sciences, University of
Arizona, Tucson, AZ 85721, USA C2N]Centre de Nanosciences et de
Nanotechnologies (C2N), CNRS, Université Paris Sud, Université Paris-Saclay,
91120 Palaiseau, France C2N]Centre de Nanosciences et de Nanotechnologies
(C2N), CNRS, Université Paris Sud, Université Paris-Saclay, 91120 Palaiseau,
France LPENS]Laboratoire de Physique de l’Ecole Normale Supérieure, ENS,
Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne
Paris Cité, 24 rue Lhomond 75005 Paris, France LPENS]Laboratoire de Physique
de l’Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université,
Université Paris-Diderot, Sorbonne Paris Cité, 24 rue Lhomond 75005 Paris,
France Paderborn]Physics Department and Center for Optoelectronics and
Photonics Paderborn (CeOPP), Universität Paderborn, Warburger Strasse 100,
33098 Paderborn, Germany [Paderborn]Wyant College of Optical Sciences,
University of Arizona, Tucson, AZ 85721, USA UoA]Department of Physics,
University of Arizona, Tucson, AZ 85721, USA [Paderborn]Wyant College of
Optical Sciences, University of Arizona, Tucson, AZ 85721, USA
LPENS]Laboratoire de Physique de l’Ecole Normale Supérieure, ENS, Université
PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité,
24 rue Lhomond 75005 Paris, France

##  1 Introduction

Lighthouses have been used for millennia to inform ships on their relative
position on the sea. The lighthouse design possesses two advantages (Fig. 1a):
Its highly directive radiation pattern allows limiting the required power to
reach remote locations, and the dynamic control of the emission allows
converting spatial information into time information, and vice versa. Such
design is used in our everyday life in bar code readers, phased array radars
or beamformer sonars.

Non-mechanical lighthouse designs are attractive because they allow reducing
mechanical fatigue limitations, and increasing the lifetime, simplicity and
speed of devices. There are two strategies to reach such goal: The first one
relies on using the interference of a controllable array of antennas or light
modulators. This method is inspired from the phased array radar technology 1
and usually requires complex photonic circuits in the visible wavelengths 2,
3, 4, 5, controlled electronically by phase shifters.

Figure 1: Geometry of the lighthouse emission (a) and of the planar double
microcavity (b) composed of three distributed Bragg reflectors (DBR)
surrounding two sets of quantum wells (QWs). The microcavity is excited at
normal incidence from the top and the emission direction is governed by the
incident linear polarization of the resonant excitation. (c) Dispersion of the
two lower polariton branches of the planar double microcavity, and
representation of the triply degenerate optical parametric oscillation process
at work in the lighthouse effect between LP1 and the LP2 elastic circle.

Another approach relies on using the nonlinear light-matter response to steer
light, opening the road to all-optical operation. Steering can be achieved by
using the light field intensity - usually of an auxiliary pump field - to
modify the field propagation within the device. Beam steering has been
achieved in this way to steer solitons 6, 7, intense fields in biased
photorefractive crystals 8, femtosecond pulses on metallic mirrors 9, or to
control the emission of spatial multimodal lasers by using the nonlinear gain
response 10, 11.

The nonlinear approach has the advantage of much simpler and robust designs,
it is also usually much faster. But since direction is not controlled
independently of intensity, it is hard to imagine practical applications,
e.g., in information technology, ranging or microscopy. In this letter we show
that, rather surprisingly, the linear polarization is sufficient to achieve
continuous control of the beam emission. We demonstrate the lighthouse-like
emission of a planar multiple microcavity device (Fig. 1). This simple device
is composed of two coupled microcavities containing multiple quantum wells 12.
The strong coupling between cavity photons and quantum well excitons is
reached and the collective excitations of the device are microcavity exciton-
polaritons 13, 14, 15, which are part-light, part-matter quasiparticles and
therefore exhibit strong nonlinear interactions. 16, 17, 18, 19, 20, 21 The
dispersion of the two lower polariton branches (LP) is shown in figure 1c:
When we resonantly pump the LP2 at normal incidence, a triply degenerate
$\chi^{3}$ parametric scattering process occurs towards LP1, and light is
emitted at a finite emission angle $\theta$.

In principle, due to the rotational invariance of the polaritonic dispersion,
parametric scattering can occur in any direction, but we predict and observe
that in the nonlinear system the emission direction is controlled by the
linear pump polarization in the optical parametric oscillation (OPO) regime
allowing beam steering over an interval of the azimuth angle $\phi$ of
$360\text{\,}\mathrm{\SIUnitSymbolDegree}$.

##  2 Theory

To describe the scattering of LP2 polaritons at zero in-plane wave vector onto
the LP1 elastic circle, we utilize the nonlinear coupled cavity and exciton
field theory discussed in Ref. 22.

The core ingredients to this theory are interband polarization functions,
obtained from a fermionic quantum-field theory and specialized to excitonic
polarizations. The Coulomb interaction between charge carriers gives rise to
spin-dependent exciton-exciton interactions and thus to two-exciton
correlations, as for example obtained in the dynamics-controlled truncation
formalism 23, 24. The nonlinear optical response of the two subcavities
follows from a numerical solution of a coupled mode theory including the
excitonic interband polarizations coupled to single-mode equations for the
light fields at the positions of the quantum wells, Eqs. (1)-(4) in Ref. 22,
with parameter values, in the notation of Ref. 22,
$\gamma_{x}=\gamma_{c}=$0.2\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$,
$m_{TM}=$0.23\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\,\mathrm{p}\mathrm{s}^{2}\,\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-2}$$,
$m_{TE}=1.001~{}m_{TM}$, $\Omega_{x}=6.35~{}\text{meV}$,
$\Omega_{c}=$5.05\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$,
$T^{++}=$5.69\times
10^{-3}\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$$,
$T^{+-}=-T^{++}/3$, $A_{\mathrm{PSF}}=$2.594\times
10^{-4}\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}$$; detuning of the
cavity mode from the bare exciton frequency
$\Delta_{P}=$-4.3086\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$.

The nonlinearities entering the theory are phase-space filling and spin-
dependent exciton-exciton T-matrices, which are different in the co-circular
$T^{++}$ and cross-circular $T^{+-}$ scattering channels. The transverse-
electric (TE) and transverse-magnetic (TM) bare cavity modes are assumed to
have parabolic dispersion with different curvatures (or cavity effective
masses). This results in polaritonic TE-TM splitting 25. In the experimental
conditions, the two subcavities are slightly different, which leads to a
different coupling with the external pump. A quantitative estimate of this
difference follows from a linear transfer matrix simulation of the entire
structure and as a result the external pump field is chosen to be larger (by a
factor of $z=-1.52$) in the first subcavity than in the second subcavity. The
factor $z$ is defined by the relation
$p^{x}_{\text{pump},1}=zp^{x}_{\text{pump},2}$, where $p^{x}_{\text{pump},i}$
are pump parameters for the two subcavities $i=1,2$. The relation between
$p^{x}_{\text{pump},j}$ and $R^{\pm}_{\text{pump},i}$ in Eqs. (1) of Ref. 22
is as follows. We first solve the steady-state versions of Eqs. (1) and (2) of
Ref. 22, Fourier-transform from real (configuration) to wave vector space and
take $k=0$ to describe the spatially homogeneous pump process. These equations
read
$E^{\pm}_{\text{pump},i}=-(\Delta_{p}+i\gamma_{x}-T^{++}|p^{\pm}_{\text{pump},i}|^{2}-T^{+-}|p^{\mp}_{\text{pump},i}|^{2})p^{\pm}_{\text{pump},i}/[\Omega_{x}(1-2A_{PSF}|p^{\pm}_{\text{pump},i}|^{2})]$
and
$R^{\pm}_{\text{pump},i}=-(\Delta_{p}+i\gamma_{c})E^{\pm}_{\text{pump},i}+\Omega_{c}E^{\pm}_{\text{pump},j}+\Omega_{x}p^{\pm}_{\text{pump},i}$.
For given $p^{\pm}_{\text{pump},i}$ they determine $R^{\pm}_{\text{pump},i}$.
For our linearly polarized pump, we have
$p^{\pm}_{\text{pump},i}=\frac{1}{\sqrt{2}}e^{\mp i\phi}p^{x}_{\text{pump},i}$
where $\phi$ is the pump polarization angle with respect to the x-axis. More
details of the pump simulation are given in Sec. 2.3 of Ref. 26. We use
$p^{x}_{\text{pump},1}=$12\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{-1}$$
and $p^{x}_{\text{pump},2}=p^{x}_{\text{pump},1}/z$. For a linearly polarized
pump with negative detuning from the bare exciton resonance, because of the
spin-dependent exciton-exciton interactions, the polarization channel cross-
linear to the pump’s polarization is favored for instability (OPO) of pump-
induced LP1 polaritons. Together with the TE-TM cavity mode splitting this
leads to a spatial anisotropy for polariton scattering. Taking also into
account the effect of the cavity difference in the double-cavity system, here
we find stable 2-spot patterns at pump powers not too far above the OPO
threshold. The spatial orientation of these 2-spot patterns is found to be
parallel to the vectorial polarization direction of the pump. A representative
numerical solution that has reached a steady state for each pump polarization
is shown in Fig. 2.

Figure 2: Theoretical simulation of the polaritonic lighthouse effect. (a)
Two-spot far-field pattern detected in the cross-linear polarization channel
for resonant excitation with a linearly polarized cw laser (polarization
direction in this example is 108∘ relative to the x-axis). (b) Relative
orientation (azimuth) of the two emission intensity maxima with respect to the
linear vectorial polarization of the pump. When rotating the pump polarization
orientation, the 2-spot emission rotates with it.

The pattern shape depends on the choice in the cavity asymmetry and pump
power: The 2-spot pattern of figure 2 is obtained for a particular choice in
the cavity asymmetry ($z\neq 1$) and the pump power. If we choose a symmetric
cavity or larger pump powers, we can also obtain a 6-spot pattern with a
$60\text{\,}\mathrm{\SIUnitSymbolDegree}$ angle between the output fields.
Which angles can occur in principle is determined by the nonlinearity, in our
case a 3rd\- order (or $\chi^{(3)}$) nonlinearity. The physical processes
giving rise to the $180\text{\,}\mathrm{\SIUnitSymbolDegree}$ (2-spot) and
$60\text{\,}\mathrm{\SIUnitSymbolDegree}$ angles (6-spot) are illustrated in
Ref. 27. For a conventional single-cavity design, we have also found the
possibility of $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ angles 28 (4-spot) in
the case of linearly polarized pump beams. These
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$ angles patterns are made possible by
the interplay between the exciton-exciton interaction and TE-TM splitting.
Interestingly, Whittaker et al. report on the experimental observation of a
diverse family of polariton patterns, including ones with an odd number of
lobes, by using circularly polarized excitation29. In their experiment as
well, a pattern rotation was observed by changing from left to right
circularly polarized pumping. We finally note that our model is for an
optically isotropic crystal without strain or defects, so that the crystal
orientation does not come into play in the selection of patterns and in their
orientation.

##  3 Experimental setup and observations

Figure 3: Experimental observation of the polaritonic lighthouse effect. (a)
Scheme of the experimental setup: the linear pump polarization is controlled
with a half-wave plate, and a field stop is used in the Fourier space (F.S.)
to filter the direct pump reflection on the sample. (b),(c) Examples of far
field observed for an incoming linear polarization pump beam oriented at
$32\text{\,}\mathrm{\SIUnitSymbolDegree}$ (resp.
$136\text{\,}\mathrm{\SIUnitSymbolDegree}$) with respect to the [100] axis.
Two spots are observed collinear to the polarization direction. (d), (e)
Corresponding radiation pattern at $32\text{\,}\mathrm{\SIUnitSymbolDegree}$
(gray) (resp. $136\text{\,}\mathrm{\SIUnitSymbolDegree}$). Beam width is about
$14\text{\,}\mathrm{\SIUnitSymbolDegree}$ and extinction reaches up to
$10\text{\,}\mathrm{d}\mathrm{B}_{\mathrm{i}}$, the direction of emission is
controlled with a $5\text{\,}\mathrm{\SIUnitSymbolDegree}$ accuracy. The beige
disk represents the direction-independent background due to elastic
scattering. (f) Map of the emission amplitude as a function of the pump linear
polarization angle and of the emission angle relative to the pump
polarization.

The sample, which has been used in several studies 12, 30, 31, 32, consists of
two coupled $\lambda/2$ Ga0.05Al0.95As cavities embedded between three
Ga0.05Al0.95As/Ga0.8Al0.2As distributed Bragg reflectors with 25 (back), 17.5
(middle), and 17.5 (front) pairs respectively (fig. 1b). The nominal Q factor
is $10^{5}$, and the middle Bragg mirror induces a
$10\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ coupling between bare cavity
modes. In each cavity, three sets of four $7\text{\,}\mathrm{n}\mathrm{m}$
GaAs QWs are inserted at the antinodes of the field, resulting in a
$13\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ Rabi splitting. Experiments are
performed at $T=$6\text{\,}\mathrm{K}$$, the sample is excited at normal
incidence, in resonance with the lower polariton branch 2 (see fig. 1c), with
a linearly polarized single-mode Ti:Sapphire laser. The pump polarization is
controlled by a half-wave plate, and the far field is recorded in reflection
geometry in the polarization orthogonal to the pump. The pump reflection is
spatially filtered.

With a $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$-wide resonant excitation
beam, we observe the optical parametric oscillation (OPO) regime above an
incoming laser power of $60\text{\,}\mathrm{m}\mathrm{W}$ and the
structuration of the far field in a variety of light patterns 12. The two-spot
patterns are the simplest of them and are observed just above OPO threshold,
when the LP2 branch is slightly negatively detuned with respect to the bare
exciton energy ($-2\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$).

Figure 3 represents the observed far-field emission of the microcavity (3a)
and the corresponding radiation pattern (3b) that is observed at a radial
angle $\theta$ of $20\text{\,}\mathrm{\SIUnitSymbolDegree}$. This lighthouse-
like emission is always directed along the linear pump polarization,
independently of the crystal orientation, as evidenced by the direction
emission map in Fig. 3c.

##  4 Discussion

Let us now discuss the experimental observations in light of the theoretical
prediction of a lighthouse-like emission. Since the polariton lighthouse
effect implies both a directional and controllable emission, we are going to
analyze these two aspects.

The radiation patterns are approximately symmetric and highly directional: In
the radial direction, the lighthouse emission is well focused with a beam
width $\Delta\theta$ of only $0.4\text{\,}\mathrm{\SIUnitSymbolDegree}$. This
feature results from the strong constraint on radial emission set by the
phase-matching conditions on resonant parametric scattering between the two
lower polaritonic branches. In the azimuthal direction, the beam width is
$\Delta\phi=14^{\circ}$ independent of the excitation linear polarization
configuration.

In contrast, the theoretical prediction on figure 2 suggests much tighter
azimuthal focusing is possible. This discrepancy probably originates from the
effect of elastic scattering on line defects within the DBRs, which are
signaled by a characteristic speckle signature in the elastically scattered
signal (not shown). Indeed, whereas polariton-polariton parametric scattering
in a uniform planar microcavity preserves total momentum, additional random
elastic scattering lifts this restriction.

Most of the emission is cast in the two main lobes. The main lobes apparent
imbalance results from a partial field stop within the large NA collecting
objective. Stray light accounts for
$[-7;-10]\text{\,}\mathrm{d}\mathrm{B}_{\mathrm{i}}$ and has two origins:
First, the resonant elastic scattering on DBR defects, which forms a direction
independent background (yellow disk on fig. 3b) 31. Second, the competing
parametric scattering pathways which are responsible for the formation of
structured side lobes. Figure 3b) gives a representative example of radiation
pattern (incoming polarization at $32\text{\,}\mathrm{\SIUnitSymbolDegree}$
with the [100] crystalline axis), as well as the most selective configuration
($136\text{\,}\mathrm{\SIUnitSymbolDegree}$). From figure 3c), we see that
side lobes are most pronounced when the incoming polarization is oriented
along the [100] direction. This anisotropic response is most probably the
result of residual built-in strain effects within the heterostructure.33, 34
We note that a clear lighthouse effect is evidenced over 90% of the incoming
polarization angles and that the two-spot emission always dominate side lobes
emission.

The angular precision of the lighthouse emission by the linear polarization is
achieved with a remarkably high accuracy of
$5\text{\,}\mathrm{\SIUnitSymbolDegree}$. Previous work using InGaAs/GaAs
microcavities presented strong anisotropic line defects and mosaicity which
pinned the OPO emission on the semiconductor crystalline axes35. Therefore, we
conclude that the small lattice mismatch of AlGaAs/GaAs microcavities is
critical to achieve controllability of the emission.

The maximum emitted power in the main pattern reaches
$200\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{W}$, which is only $0.4\%$ of
the incoming power ($60\text{\,}\mathrm{m}\mathrm{W}$, necessary to reach OPO
threshold). This currently low external power efficiency (emitted
power/incident power) is due to the fact that a strong Kerr effect occurs at
resonance. As a result, the microcavity LP2 resonance is repelled from the
excitation laser energy and most of the incoming power gets reflected on the
first DBR and does not enter the microcavity. 36 By taking into account the
low injection efficiency, we estimate that the internal power efficiency
(emitted power/transmitted power) is in the $25\%$ range, whereas the
theoretical limit is $50\%$ due to power balance between parametrically
scattered beams in the OPO regime.

##  5 Conclusion

We have shown that an optical microcavity can present lighthouse-like emission
continuously controlled by the linear vectorial polarization of the pump laser
itself. The use of vectorial polarization as a control parameter implies that
this function can only be implemented in a nonlinear optical device.

We used a coupled cavity theory combined with a many-particle theory
describing exciton-exciton interactions to demonstrate the polariton
lighthouse effect if a minimal set of physical ingredients are included,
namely the nonlinear optical response of microcavity exciton-polaritons, the
difference in the subcavities forming the microcavity, and the rotational
invariance of the device. We observed the lighthouse effect with a device made
of quantum wells embedded in a double microcavity and evidenced that a two-
spot pattern oriented along the linear incoming pump polarization is observed
for most azimuth angles with good control on the emission direction.

With our present cavity, we observe some deviations from the ideal lighthouse
emission, such as parasitic stray light, originating from elastic scattering
on line defects and the effect of built-in strain, and inherent to real world
devices. Progresses in heterostructures quality and power efficiency could
turn this polariton lighthouse effect into an original and useful all-optical
beam-steering method which can be of great interest for microscopy or LIDARs.

We acknowledge that this project has received funding from the Agence
Nationale de la Recherche (ANR) (ANR-16-CE24-0023 TeraMicroCav), the US
National Science Foundation (NSF) (DMR 1839570), and the Deutsche
Forschungsgemeinschaft (SCHU 1980/5-2 and Heisenberg grant 270619725). RB
acknowledges CPU time at HPC (University of Arizona); SS acknowledges
computing time at Paderborn Center for Parallel Computing ($\mathrm{PC^{2}}$).

The microcavity geometry is detailed in section 3. During fabrication by
molecular beam epitaxy, a wedge in the cavity thicknesses is introduced by
stopping the cavity rotation during growth of the cavity layers. As a
consequence, the sample has a natural gradient in the bare cavity energy,
while the confined exciton energy is mostly unaffected. Note that upper and
lower cavities have a parallel gradient in energy so that the coupled cavity
modes inherit the same energy gradient. By recording the photoluminescence of
lower polariton branches 1 and 2 on the sample surface, we can reconstruct the
polariton anticrossing curve and characterize the polariton key properties of
the sample. Figure 4a shows a typical dispersion curve obtained by
photoluminescence, where only the two lower polariton branches (labeled LPB1
and LPB2) are visible. Lower polariton energies at normal incidence are
represented function of the position on the sample along the gradient
direction on figure 4b, the position corresponding to panel a is indicated as
green points.

Figure 4: Characterization of the double microcavity exciton-polaritons (a)
Typical energy dispersion for the double microcavity sample. Only the two
lower polaritons branches are visible (LPB1 and LPB2). The pump is resonant
with LPB2 at normal incidence. The white lines correspond to a fourth order
polynomial fit of the branches to obtain as precisely as possible the energy
minima of LPB1 and LPB2. (b) Anticrossing curve obtained from
photoluminescence spectra at various positions onto the sample. The two green
points correspond to the position of panel (a). The dashed black line
corresponds to the bare exciton energy as a function of the position on the
sample.The blue and red dashed lines correspond to the optical cavity modes’
energies as a function of the position on the sample.

The two lower polariton branches of figure 4b are fitted by

$E_{\mathrm{pol}}^{(i)}=\frac{1}{2}(E_{\mathrm{cav}}^{(i)}+E_{\mathrm{exc}}+\sqrt{(E_{\mathrm{cav}}^{(i)}-E_{\mathrm{exc}})^{2}+4\Omega_{R}^{2}})$

where $E_{\mathrm{cav}}^{(i)}$ is the coupled cavity mode forming the
polariton of interest, and both $E_{\mathrm{cav}}$ and $E_{\mathrm{exc}}$
evolves linearly with position on the sample. From these data we infer the
cavity energy gradient of $3.6\pm
0.1\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\,\mathrm{m}\mathrm{m}{-1}$, the
exciton energy $1.606\pm 0.003\text{\,}\mathrm{e}\mathrm{V}$, and the Rabi
coupling $\Omega_{R}=$6.4\pm 0.8\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$,
which is half the Rabi splitting between the lower and upper polaritons. From
the coupled cavity splitting, we also deduce the coupling between the upper
and lower microcavities $\Omega_{c}=$5.1\pm
0.7\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$.

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[a,]Mathieu Giroux

# Looping the loops: a tale of elliptic dual Feynman integrals

0000-0002-7005-9652 Andrzej Pokraka  Franziska Porkert  Yoann Sohnle

###### Abstract

In this talk, we review a loop-by-loop approach used to generate differential
equations for multi-scale (dual) Feynman integrals. We illustrate the method
on a well-established example: the unequal mass elliptic sunrise.

##  1 Introduction

It is a long-standing fact, reminiscent of the golden age of string theory,
that (hyper-)elliptic curves and Calabi-Yau manifolds play a central role in
various corners of mathematics and physics. Over the past few decades, a
heroic effort has been made by various authors to demonstrate this in the
context of particle physics (see [1, 2, 3, 4, 5, 6, 7, 8] and references
therein.) In this context, Feynman diagrams are used to compute scattering
amplitudes, which yields observables such as cross-sections measurable in
collider experiments or waveforms detected in gravitational wave
observatories.

For diagrams with loops, the corresponding _Feynman integrals_ over the off-
shell loop momenta determine the probabilities of various particle
interactions. Beyond the 1-loop level, Feynman integrals depending on multiple
kinematic scales (like energies, angles, and masses) often involve complex
underlying geometric structures. Fig. 1 shows a few phenomenologically
relevant 2-loop examples that involve one or more elliptic curves.

$m_{1}$$m_{2}$$m_{3}$$m$$m$$m$$m$$Z$$m$$m$$m$$m$

Figure 1: Examples of graphs in which one (or more) elliptic curves is (are)
lurking. These include self-energy kite integrals, electro-weak form-factors
and Bhabha scatterings.

As our understanding of these integrals deepened, it became evident that many
could be totally understood and even evaluated through inherent properties of
these rich geometries.

An effective approach employed to accomplish this is the method of
_differential equations_. However, a significant hurdle in using this approach
is the path to a so-called _canonical basis_. Such a basis, denoted by
$\vec{I}$, satisfies a differential equation that we can solve order-by-order
in the dimensional regularization parameter $\varepsilon\in\mathbb{C}$ [9]. It
typically takes the following form:

$\text{d}\vec{I}=\varepsilon~{}\boldsymbol{\Omega}\cdot\vec{I}\,,$ (1)

where $\boldsymbol{\Omega}$ is a matrix of kinematic one forms independent of
$\varepsilon$. For many integrals (e.g., those without internal masses), a
canonical basis is systematically obtained normalizing by _leading
singularities_ (in the polylogarightmic case, one can think of leading
singularities simply as maximal residues or as powers of Gram determinants in
the external kinematics.)

As soon as we turn on some internal masses, the story often changes. Here, we
argue that this is because the precise definition of leading singularities in
some of these examples is somewhat blurry (i.e., not uniquely defined), and so
are systematic methods used to achieve a canonical form. In that sense, one of
the most pressing problems in this program is how to construct such a basis
$\vec{I}$ from first principles, i.e., without relying too heavily on _ad-hoc_
ansatz-based techniques. Interesting progress on this issue was made recently
in [10].

While we do not claim to solve this problem fully, we believe that we can make
significant progress using a concise set of analytical tools: unitarity and
geometry. These tools are intrinsically integrated into the _dual forms_
framework of [11, 12]. For this reason, dual forms seems to be the appropriate
objects to leverage the intuitive, yet subtle, notion that a multi-loop
problem is simply a bunch of coupled 1-loop problems [13]. Furthermore, when
working with elliptic Feynman integrals, it is to our advantage to foreground
a key symmetry: the _modular symmetry_. When combined into one toolbox, the
above provides streamlined instructions to break down the construction of a
canonical basis into a minimal sequence of algebraic steps. Below, we will
demonstrate this approach using a proof-of-concept example: the unequal mass
elliptic sunrise.

##  2 From Feynman integrals to the dual paradigm

To set the stage, let us revisit the definition of Feynman integrals within
dimensional regularization. At the most basic level, these are the integrals
we obtain by applying Feynman rules on loop diagrams. However, as is the case
with any other function, there isn’t a unique way to represent it as an
integral. Over the years, various representations have been introduced, each
tailored for a specific purpose (for an extensive pedagogical account, see
[14].) Most representations take the form of _twisted periods_ :

$I=\int_{X}u~{}\phi\,,$ (2)

where the _twist_ $u$ typically represents a multi-valued function through its
dependence on the (non-integer) spacetime dimension. Meanwhile, $X$ denotes an
integration domain that incorporates the Feynman $i0$ prescription. Both $u$
and $X$ are common to all Feynman integrals in a given family, whereas the
algebraic differential $n$-form $\phi$ is not shared among all integrals, and
its form depends on the propagator structure of each graph. Thus, the purpose
behind expressing Feynman integrals as shown in (2) is to highlight the
distinction between quantities such as $u$ and $X$ that are universal to a
family of integrals, and those tied to a particular diagram such as $\phi$.

An important property of Feynman integrals is that they can consistently be
represented as a _finite_ combination of simpler integrals [15]. This finite
collection of integrals is commonly referred to as the set of _master
integrals_. It follows that any family of Feynman integrals forms a vector
space. This vector space also possesses the additional property of being
closed under differentiation with respect to the kinematic variables [16].

Now, one can invoke a standard fact from linear algebra: for any given finite-
dimensional vector space, there exists a _dual vector space_ , from which the
original space can be studied. Our next objective is to understand how the
dual space for Feynman integrals is defined and its practical use in
computations.

#### Dual forms

In the most direct manner, the integrand $\widecheck{\phi}$ dual to the
Feynman integrand $\phi$ (as given in (2)) is defined such that the
_intersection paring_ :

$\langle\widecheck{\phi}|\phi\rangle\propto\int_{\Gamma}~{}(\widecheck{u}\times
u)~{}\widecheck{\phi}{{\wedge}}\phi\,,$ (3)

makes sense.111See [12] for the exact constant of proportionality. Here,
$\Gamma\equiv\mathbb{C}^{n}\setminus\\{u=0\\}\cup\\{\text{propagators}=0\\}$
and $\widecheck{u}\equiv u|_{\varepsilon\to-\varepsilon}$ denotes the _dual
twist_.

Let us unpack this further. For the intersection number (3) to be meaningful,
it must be unique. This uniqueness requires that the product of the dual twist
and the twist is an algebraic function (not necessarily one!) Furthermore, we
need this unique number to be finite. This condition is met when the zeros of
the dual form are paired with the poles of the Feynman integrands, which are
the propagators. In other words, $\widecheck{\phi}$ is supported only away
from $\phi$’s _unregulated_ (_non-twisted_) poles.

In practice, we implement this condition by attaching strings of
$\delta$-functions to dual forms for each propagator on the Feynman side: the
form $\widecheck{\phi}$ dual to the Feynman form $\phi$ is proportional to a
wedge product of $\text{d}\theta(x)=\delta(x)\text{d}x$’s (in the sense of
distributions), each of which accounts for a propagator that can be put on-
shell. The overall function multiplying this wedge product varies depending on
the situation, as we will see later. For a complete treatment of dual forms,
see [11, 12, 13].

##  3 Looping the loops and differential equations

Now that the basics of dual forms are established, let us describe in detail
the method used below to generate differential equations for dual Feynman
integrals. This method draws from the naive idea that a multi-loop problem can
be broken down into a collection of simpler, yet coupled, 1-loop problems. We
will see below how this can be used to construct differential equations for
(dual) Feynman integrals, one loop at a time. This approach is interesting
from the standpoint of multi-scales problems because it comes with the
immediate reward of having to deal with less variables at a time.

#### Setup

To turn a multi-loop problem into a series of more manageable 1-loop problems,
it is essential to get a grip on the fibre bundle structure of the multi-loop
integrals we want to study. Given that a 2-loop integrand effectively
illustrates this concept and that the pattern is evident for $(L>2)$-loop
integrands, we focus on 2-loop integrals.

Denoting by $\ell_{1}$ and $\ell_{2}$ the loop momenta, one may rewrite any
2-loop integrand as the wedge product of some (fibered) 1-loop basis:

$\displaystyle\underbracket{\widecheck{\varphi}^{\text{\tiny(2-loop)}}_{a}(\ell_{1},\ell_{2},\\{p_{ij}^{2}\\},\\{m_{j}\\})}_{\text{2-loop
integrand (total
space)}}=\underbracket{\widecheck{\varphi}^{\text{\tiny(1-loop)}}_{b}(\ell_{2},\\{q_{ij}^{2}\\},\\{m_{j}\\})}_{\text{First
loop (fibre)}}{{\wedge}}\underbracket{\widecheck{\varphi}^{\text{\tiny(left-
over)}}_{ba}(\ell_{1},\\{p_{ij}^{2}\\},\\{m_{j}\\})}_{\text{Second loop
(base)}}\,,$ (4)

with $p_{ij}=p_{i}{+}p_{j}$, where the $p$’s denote the external momenta in
the 2-loop integrand (total space). Similarly, $q_{ij}=q_{i}{+}q_{j}$, where
the $q$’s label the external momenta in the first loop integrand (fibre),
which can include some of the $p$’s as well as $\ell_{1}$.

The splitting given in (4) imposes constraints on the left-over pieces (base)
and guides our selection of “good” $\widecheck{\varphi}^{\text{\tiny(left-
over)}}_{ba}$. In addition, once the differential equation
$\boldsymbol{\Omega}^{\text{\tiny(1-loop)}}$ for the 1-loop basis is known, we
can commute
$\widecheck{\nabla}\equiv\text{d}+\text{d}\log\widecheck{u}^{\text{\tiny(1-loop)}}{{\wedge}}$
across the 1-loop basis to get a new covariant derivative acting on the left-
over part:

$\displaystyle\widecheck{\nabla}\left(\widecheck{\varphi}^{\text{\tiny(1-loop)}}_{b}{{\wedge}}\widecheck{\varphi}^{\text{\tiny(left-
over)}}_{ba}\right)\simeq\widecheck{\varphi}^{\text{\tiny(1-loop)}}_{b}{{\wedge}}\widecheck{\nabla}_{bc}^{\text{\tiny(new)}}\left(\widecheck{\varphi}^{\text{\tiny(left-
over)}}_{ca}\right)\,.$ (5)

Above,$\simeq$ stands for equal mod total derivatives. Moreover, the knowing
$\boldsymbol{\Omega}^{\text{\tiny(1-loop)}}$ gives us the covariant derivative
$\widecheck{\nabla}_{ab}^{\text{\tiny(new)}}$ for the left-over part. Indeed,
after the dust settles, one finds from (5)

$\displaystyle\begin{split}\widecheck{\nabla}_{ab}^{\text{\tiny(new)}}&=\delta_{ab}\big{(}\textnormal{d}+\text{d}\log\widecheck{u}^{\text{\tiny(left-
over)}}\big{)}+\Omega_{ab}^{\text{\tiny(1-loop)}}{{\wedge}}\\\
&\equiv\delta_{ab}\textnormal{d}+\widecheck{\omega}_{ab}{{\wedge}}\end{split}\quad,\text{where}\quad\widecheck{\nabla}\widecheck{\varphi}^{\text{\tiny(1-loop)}}_{a}\simeq\widecheck{\varphi}^{\text{\tiny(1-loop)}}_{b}{{\wedge}}\Omega_{ba}^{\text{\tiny(1-loop)}}\,.$
(6)

We expect that it should be easier to obtain a good (but not necessarily
$\varepsilon$-form) 2-loop differential equations from the action of
$\widecheck{\nabla}_{bc}^{\text{\tiny(new)}}$ on
$\widecheck{\varphi}^{\text{\tiny(left-over)}}_{ca}$ than from the action of
$\widecheck{\nabla}$ on $\widecheck{\varphi}^{\text{\tiny(2-loop)}}$:

$\text{\emph{Computing}}~{}~{}\widecheck{\nabla}\widecheck{\varphi}^{\text{\tiny(2-loop)}}_{a}\simeq\widecheck{\varphi}^{\text{\tiny(2-loop)}}_{b}{{\wedge}}\Omega_{ba}^{\text{\tiny(2-loop)}}~{}~{}\text{\emph{is
harder
than}}~{}~{}\widecheck{\nabla}_{bc}^{\text{\tiny(new)}}\widecheck{\varphi}^{\text{\tiny(left-
over)}}_{ca}\simeq\widecheck{\varphi}^{\text{\tiny(left-
over)}}_{bc}{{\wedge}}\Omega_{ca}^{\text{\tiny(2-loop)}}\,.$ (7)

The rationale for this expectation is that this approach yields two strong
constraints on the fibered dual bases, such that there are not too many
options to start the problem with. We refer to these constraints below as
_loop-by-loop constraints_. The first imposes that the fibre basis is
normalized such that $\widecheck{\boldsymbol{\omega}}$ in (6) factorizes
linearly in $\varepsilon$ – i.e., that
$\widecheck{\boldsymbol{\omega}}\propto\varepsilon$. The second requires the
wedge product in (4) to be single-valued (algebraic). Note that when this
constraint is applied, the fibre forms have _already_ been fixed by the first
constraint. Therefore, only the base form basis is affected by this
constraint.

While the expectation given in (7) is quite general, there is an important
technical caveat we would like address here. Indeed, although the (dual) IBP
reduction resulting in $\boldsymbol{\Omega}^{\text{\tiny(2-loop)}}$ works well
using a _global_ set of coordinates in some of the examples we examined (e.g.,
Sec. 4), we found instances (e.g., the 5-mass kite integral family) with
obstructions to this global approach [17]. One way out of this is to use
multiple local coordinate systems, which allow for a reduction in each sector,
instead of relying solely on a global one. The results can then be pulled back
to a global set of coordinates post reduction. More optimistically, in view of
the recent progress discussed in [18], one might soon be able to bypass the
cumbersome use of local coordinates and IBPs by employing directly cutting-
edge intersection numbers algorithms.

##  4 The 3-mass sunrise

Let us now exemplify the discussion above and consider the scalar 2-loop
3-mass elliptic sunrise. From the Feynman rules, we find (up to an overall
normalization):

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}}}}}}$}}}}=\int\frac{\delta^{\text{D}}(\ell_{3}{-}\ell_{1}{+}\ell_{2}{-}p)}{(\ell_{1}^{2}{+}m_{1}^{2}{-}i0)(\ell_{2}^{2}{+}m_{2}^{2}{-}i0)(\ell_{3}^{2}{+}m_{3}^{2}{-}i0)}\prod_{a=1}^{3}\frac{\text{d}^{\text{D}}\ell_{a}}{i\pi^{\text{D}/2}}\qquad(\text{D}=4{-}2\varepsilon)\,.$
(8)

#### The _elliptic_ sunrise

From now on, it will be convenient to work directly in momentum space and, in
particular, to consider the cylindrical-like parameterization for the internal
loop momenta:

$\ell_{i}=\ell_{i\|}+\ell_{i\perp}\,,\quad\ell_{i\|}\cdot\ell_{i\perp}=0\,,\quad\ell_{1\|}={\color[rgb]{0.68,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0,0}\pgfsys@color@cmyk@stroke{0}{0.87}{0.68}{0.32}\pgfsys@color@cmyk@fill{0}{0.87}{0.68}{0.32}x}~{}p\,,\quad\ell_{2\|}=y~{}(\ell_{1}+p)\,,\qquad\text{where}~{}i=1,2\,.$
(9)

In these variables, the propagators (boundaries) are given by

$\displaystyle\textsf{D}_{1}=\ell_{1\perp}^{2}+{\color[rgb]{0.68,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0,0}\pgfsys@color@cmyk@stroke{0}{0.87}{0.68}{0.32}\pgfsys@color@cmyk@fill{0}{0.87}{0.68}{0.32}x}^{2}p^{2}+m_{1}^{2}\,,\qquad\textsf{D}_{2}=\ell_{2\perp}^{2}+y^{2}~{}(\ell_{1\perp}^{2}+(1+{\color[rgb]{0.68,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0,0}\pgfsys@color@cmyk@stroke{0}{0.87}{0.68}{0.32}\pgfsys@color@cmyk@fill{0}{0.87}{0.68}{0.32}x})^{2}p^{2})+m_{2}^{2}\,,$
(10)
$\displaystyle\textsf{D}_{3}=\ell_{2\perp}^{2}+\left(y+1\right)^{2}\ell_{1\perp}^{2}+\left(1+y\right)^{2}\left(1+{\color[rgb]{0.68,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0,0}\pgfsys@color@cmyk@stroke{0}{0.87}{0.68}{0.32}\pgfsys@color@cmyk@fill{0}{0.87}{0.68}{0.32}x}\right)^{2}p^{2}+m_{3}^{2}\,.$

We are now in a good position to understand why the sunrise integral is
referred to as elliptic. If we compute the maximal cut
($\textsf{D}_{i}^{-1}{\to}2\pi i\delta(\textsf{D}_{i})$ $\forall~{}i$) of (8)
in the critical dimension ($\varepsilon=0$), we obtain a 1-fold integral over
the square root
$Y({\color[rgb]{0.68,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0,0}\pgfsys@color@cmyk@stroke{0}{0.87}{0.68}{0.32}\pgfsys@color@cmyk@fill{0}{0.87}{0.68}{0.32}x})$
of the irreducible quartic polynomial:

$E(\mathbb{C}):Y^{2}-({\color[rgb]{0.68,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0,0}\pgfsys@color@cmyk@stroke{0}{0.87}{0.68}{0.32}\pgfsys@color@cmyk@fill{0}{0.87}{0.68}{0.32}x}-r_{1})({\color[rgb]{0.68,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0,0}\pgfsys@color@cmyk@stroke{0}{0.87}{0.68}{0.32}\pgfsys@color@cmyk@fill{0}{0.87}{0.68}{0.32}x}-r_{2})({\color[rgb]{0.68,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0,0}\pgfsys@color@cmyk@stroke{0}{0.87}{0.68}{0.32}\pgfsys@color@cmyk@fill{0}{0.87}{0.68}{0.32}x}-r_{3})({\color[rgb]{0.68,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0,0}\pgfsys@color@cmyk@stroke{0}{0.87}{0.68}{0.32}\pgfsys@color@cmyk@fill{0}{0.87}{0.68}{0.32}x}-r_{4})=0\,.$
(11)

where the roots $r_{i}$ are explicitly recorded in [13, Eq. (4.32)]. In
mathematics, the object $E(\mathbb{C})$ given in (11) is known as an _elliptic
curve_ and has a rich and well-documented geometric structure. In particular,
for a detailed discussion on its relationship with tori, refer to [13, §4.3].

###  4.1 Looping the sunrise’s loops: dual bases and differential equation

The next step involves constructing an explicit 2-loop basis based on the
loop-by-loop splitting presented in (4). This procedure is schematically
summarized in Fig. 2.

,,,,,,$2$$3$$3$$1$$2$$1$$2$$3$$1$$\lor$$\lor$$\lor$$\lor$$(1)$$(2)$$(3)$$(4)$$\\{\widecheck{\varphi}_{j}^{\text{\tiny~{}(2-loop)}}\\}_{j=1}^{7}=$$2$$3$$1$$\lor$$2$$3$$1$$\lor$$2$$3$$1$$\lor$
,,$3$$2$$3$$\lor$$\lor$$\big{\\{}{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\widecheck{\varphi}_{i}^{\text{\tiny~{}(1-loop)}}}\big{\\}}_{i=1}^{3}=$$1$$1$$1$$1$$3$$\lor$$\lor$$\lor$$\lor$$(1)$$\big{\\{}{\color[rgb]{0.68,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.68,0,0}\pgfsys@color@cmyk@stroke{0}{0.87}{0.68}{0.32}\pgfsys@color@cmyk@fill{0}{0.87}{0.68}{0.32}\widecheck{\varphi}_{ij}^{\text{\tiny~{}(left-
over)}}}\big{\\}}_{j=1}^{7}=$$2$$\lor$$\begin{pmatrix}1\\\ 0\\\
0\end{pmatrix}$ ,$\begin{pmatrix}0\\\ 1\\\ 0\end{pmatrix}$
,$\begin{pmatrix}0\\\ 0\\\ 1\end{pmatrix}$ ,$\begin{pmatrix}0\\\ 0\\\
1\end{pmatrix}$ ,$1$$3$$\lor$$(2)$$\begin{pmatrix}0\\\ 0\\\ 1\end{pmatrix}$
,$1$$3$$\lor$$(3)$$\begin{pmatrix}0\\\ 0\\\ 1\end{pmatrix}$
,$1$$3$$\lor$$(4)$$\begin{pmatrix}0\\\ 0\\\ 1\end{pmatrix}$

Figure 2: The splitting in (4) is schematically shown for the 2-loop sunrise.
Top: The 2-loop (total space) dual basis, where both $\ell_{1}$ and $\ell_{2}$
are active integration variables. Middle: The 1-loop (fibre) basis, where only
$\ell_{2}$ is active. Bottom: The left-over (base) basis, where $\ell_{2}$ is
integrated out already and $\ell_{1}$ is active.

We’ve divided the discussion into two steps, and further details can be found
in [13].

#### Step I: fixing the 1-loop (fibre) basis

Examining the middle panel of Fig. (2), we notice that the fibre basis is
essentially a dual bubble basis (up to normalization). As established in [11],
a canonical basis for the dual bubble, in the sense of [9], is

$\widecheck{\varphi}_{1}^{\text{\tiny~{}(1-loop)}}=N\
\frac{2\varepsilon~{}\text{d}\theta_{2}{\wedge}{\text{d}}\ell_{2\parallel}}{\sqrt{(p{+}\ell_{1})^{2}}\left.\ell_{2\perp}^{2}\right|_{2}}\,,~{}\widecheck{\varphi}_{2}^{\text{\tiny~{}(1-loop)}}=N\frac{2\varepsilon~{}\text{d}\theta_{3}{\wedge}{\text{d}}\ell_{2\parallel}}{\sqrt{(p{+}\ell_{1})^{2}}\left.\ell_{2\perp}^{2}\right|_{3}}\,,~{}\widecheck{\varphi}_{3}^{\text{\tiny~{}(1-loop)}}=N\frac{\text{d}\theta_{2}{\wedge}\text{d}\theta_{3}}{\sqrt{(p{+}\ell_{1})^{2}\left.\ell_{2\perp}^{2}\right|_{23}}}\,,$
(12)

where $N$ is a constant. Here, we have abbreviated $\theta(\text{D}_{i})$ as
$\theta_{i}$, with $\text{D}_{i}$ specified in (10).

Subsequently, the $\ref{Word:fLBL}^{\text{st}}$ loop-by-loop constraint
instructs us to promote $N$ to a function of the kinematics. This constraint
uniquely determines $N$ as the Gram determinant on the left-over loop – i.e.,
$N=(\ell_{1\perp}^{2})^{-1/2}$. Consequently, when localized on the boundary
where all propagators are on-shell, the bubble form denominator becomes
proportional to $Y$ as given in (11). This renders the fibre basis non-
algebraic. Consequently, through the $\ref{Word:sLBL}^{\text{nd}}$ loop-by-
loop constraint, further restrictions are imposed on the remaining (base)
basis.

#### Step II: fixing the left-over (base) basis

In addition to the $\ref{Word:sLBL}^{\text{nd}}$ loop-by-loop constraint, one
might want the base basis to exhibit certain properties that feel simply
natural. For instance, we may ask the base basis to be as close as possible of
being uniformly transcendental (readers with exposure to polylogarithmic
Feynman integrals will likely find this assumption legitimate.) Furthermore,
we can narrow our attention to bases satisfying a differential equation with a
specific modular transformation rule: it must be independent of the modular
parameters $a$ and $b$ under a modular transformation (defined in [13, Eq.
(4.52)].) Such a condition is crucial if one aims to rewrite/pullback the
differential equation, initially written in terms of Mandelstam invariants and
masses, into a modular form spanned by modular and Kronecker forms (a
comprehensive review aimed at physicists can be found in [14, §13].)

The simplest basis that satisfies all these assumptions is presented in of
[13, Eqs. (5.2-5.3)]. Notably, the elliptic sector is both surprisingly
compact and natural:

$\big{\\{}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widecheck{\varphi}_{ij}^{\text{\tiny~{}(left-
over)}}}\big{\\}}_{j=4}^{7}=\left\\{\frac{\psi_{1}^{2}}{\pi~{}\varepsilon~{}W_{X}}\widecheck{\nabla}_{X}^{\text{\tiny(new)}},\frac{(x{-}r_{1})\psi_{1}}{\pi},\frac{Y(c)\psi_{1}}{\pi(x-c)},1\right\\}\frac{\pi\textnormal{d}\theta_{1}{\wedge}\textnormal{d}x}{m_{1}^{4\varepsilon}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\psi_{1}}Y}\tiny\begin{pmatrix}0\\\
0\\\ 1\end{pmatrix}\,.$ (13)

This basis is considered natural because the first and last forms span the
cohomology of the bare (unpunctured) elliptic curve. The other two meromorphic
forms have poles at twisted poles ($x=\infty$ and $x=c$, respectively) in
$\text{D}=4$, which accurately accounts for the two punctures present on the
given elliptic curve, as illustrated in [13, Fig. 3]. In the basis above,
$X=\frac{p^{2}}{m_{3}^{2}}$, $\psi_{1}$ denotes one period of the elliptic
curve and the term $W_{X}$ stands for a Wronskian. The various factors of
$\pi$ and $\varepsilon$ ensure the basis is as close as possible of being of
uniform transcendental weight.

Interestingly, this basis possesses another property that we did not
explicitly enforce: its differential equation is linear and strictly lower-
triangular. In other words, it takes the form:

$\boldsymbol{\Omega}^{\text{\tiny(2-loop)}}=\boldsymbol{\Omega}^{\text{\tiny(2-loop)}}_{(0)}+\varepsilon~{}\boldsymbol{\Omega}^{\text{\tiny(2-loop)}}_{(1)}\,,\quad\text{with}~{}\boldsymbol{\Omega}^{\text{\tiny(2-loop)}}_{(1)}~{}\text{lower-
triangular}\,.$ (14)

Notice that $\boldsymbol{\Omega}^{\text{\tiny(2-loop)}}$ is _not_ in
$\varepsilon$-form. To achieve this form, a gauge transformation
$\boldsymbol{U}$ must be performed on (13) such that the new differential
equation reads

$\varepsilon~{}\tilde{\boldsymbol{\Omega}}^{\text{\tiny(2-loop)}}_{(1)}=(\boldsymbol{U}\cdot\boldsymbol{\Omega}^{\text{\tiny(2-loop)}}+\text{d}\boldsymbol{U})\cdot\boldsymbol{U}^{-1}\,.$
(15)

As highlighted in [13] and further verified with more general examples in
[17], the matrix $\boldsymbol{U}$ is entirely determined by modular symmetry.
This implies that the differential equation

$\boldsymbol{U}\cdot\boldsymbol{\Omega}^{\text{\tiny(2-loop)}}_{(0)}+\text{d}\boldsymbol{U}=0\,,$
(16)

can be solved analytically for $\boldsymbol{U}$ without performing any
integration (!), simply by asking that (16) is modular covariant. The matrix
$\boldsymbol{U}$ specific to the sunrise example is recorded in [13, App. E].
The corresponding differential equation is provided in [13, Eq. (5.81)] and is
demonstrated to have a simple relation to that of Feynman integrals considered
in [19].

##  5 Conclusion

Driven by the idea that multi-loop Feynman integrals are an iteration over
simpler 1-loop problems, we developed a loop-by-loop method for the
computation of multi-loop dual Feynman integrands’ differential equations.
Through the intersection number, dual Feynman integrands are in one-to-one
correspondence with conventional Feynman integrands. These dual integrands,
supported on generalized unitarity cuts, are inherently simpler.

In this talk, the primary benefit of working with dual integrands was the
ability to localize to generalized unitarity cuts, which later on informed on
an optimal choice of basis. The geometry tied to a Feynman integrand is often
concealed within its cuts, and dual integrands bring this to the fore.
Moreover, because dual integrands aren’t constrained to look like traditional
Feynman integrands, we considered a loop-by-loop basis that is entirely
motivated from the underlying elliptic geometry.

As a simple yet non-trivial example, we constructed an $\varepsilon$-form dual
basis for the three mass 2-loop elliptic sunrise family and the associated
differential equation. In doing so, we saw that breaking up the problem into
simpler 1-loop problems yields several advantages. First, we can reuse the
known $\varepsilon$-form basis and differential equation at 1-loop to
construct the 2-loop basis and differential equation. Second, at each step
only a small subset of variables are active on the fibre simplifying algebraic
manipulations as well as integration by-parts.

While the starting base basis (13) was geometrically and physically well
motivated, it did not have an $\varepsilon$-form differential equation. A
gauge transformation was needed to bring the system into $\varepsilon$-form,
and we saw that it was entirely fixed by modular covariance. We anticipate
that this idea will prove useful for future cutting-edge computations of
differential equations for topologies with elliptic-like geometries, whether
one uses the dual form framework or not.

#### Acknowldgments

This work was co-funded by the National Science and Engineering Council of
Canada (NSERC) (MG), the Simons Investigator Award $\\#$376208 (AP), and the
European Union (ERC Consolidator Grant LoCoMotive 101043686 (FP)).

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