Group invariants for Feynman diagrams

Idrish Huet1, Michel Rausch de Traubenberg2 and Christian Schubert3$\star$

1 Facultad de Ciencias en Física y Matemáticas, Universidad Autónoma de
Chiapas

Ciudad Universitaria, Tuxtla Gutiérrez 29050, Mexico

2 Université de Strasbourg, CNRS, IPHC UMR7178, F-67037 Strasbourg Cedex,
France

3 Centro Internacional de Ciencias A.C. Campus UNAM-UAEM Avenida Universidad
1001 Cuernavaca Morelos Mexico C.P. 62100 *<EMAIL_ADDRESS>
34th International Colloquium on Group Theoretical Methods in Physics
Strasbourg, 18-22 July 2022 10.21468/SciPostPhysProc.?

## Abstract

It is well-known that the symmetry group of a Feynman diagram can give
important information on possible strategies for its evaluation, and the
mathematical objects that will be involved. Motivated by ongoing work on
multi-loop multi-photon amplitudes in quantum electrodynamics, here I will
discuss the usefulness of introducing a polynomial basis of invariants of the
symmetry group of a diagram in Feynman-Schwinger parameter space.

##  1 Introduction: Schwinger parameter representation of Feynman diagrams

The most universal approach to the calculation of Feynman diagrams uses
Feynman-Schwinger parameters $x_{i}$, introduced through the exponentiation of
the (Euclidean) scalar propagator,

$\frac{1}{p^{2}+m^{2}}=\int_{0}^{\infty}dx\,{\rm e}^{-x(p^{2}+m^{2})}$

For scalar diagrams, one finds the following universal structure for an
arbitrary graph $G$ with $n$ internal lines and $l$ loops in $D$ dimensions:

$I_{G}=\Gamma(n-lD/2)\int_{x_{i}\geq
0}d^{n}x\,\delta\Bigl{(}1-\sum_{i=1}^{n}x_{i}\Bigr{)}\frac{{\cal
U}^{n-(l+1)\frac{D}{2}}}{{\cal F}^{n-lD/2}}$

${\cal U}$ and ${\cal F}$ are polynomials in the $x_{i}$ called the first and
second Symanzyk (graph) polynomials. There exist graphical methods for their
construction.

For more general theories (involving not only scalar particles) the same graph
will involve, in addition to these two polynomials, also a numerator
polynomial ${\cal N}(x_{1},\ldots,x_{n})$.

##  2 Symmetries of Feynman diagrams

Many Feynman diagrams possess a non-trivial symmetry group, generated by
interchanges of the internal lines that leave the topology of the graph
unchanged. Then all its graph polynomials must be invariant under the natural
action of the group on the set of polynomials ${\rm
I\\!R}[x_{1},\ldots,x_{n}]$, $g.P(X)\equiv P(g.X)$.

A nice example is the $l-loop$ banana graph shown in Fig. 1.

Figure 1: $l-loop$ banana graph.

This graph has full permutation symmetry in all the internal lines, so the
symmetry group is $S_{l+1}$, and its graph polynomials must be symmetric
functions of $x_{1},\ldots,x_{l+1}$. As is well-known, this implies that they
can be rewritten as polynomials in the elementary symmetric polynomials
$S_{1},\ldots,S_{n}$.

Perhaps less known is that this generalizes to the case of a general symmetry
group as follows [1]:

Theorem 1: Let $G$ be a finite group and let $\Gamma$ be an $n-$dimensional
(real) representation.

  1. 1.

There exist $n=\dim(\Gamma)$ algebraically invariant polynomials
$P_{1},\cdots,P_{n}$, called the primitive invariants, such that the Jacobian
$\frac{\partial(P_{1},\ldots,P_{n})}{\partial(x_{1},\ldots,x_{n})}\neq 0$.

  2. 2.

Denote $d_{k}=\deg(P_{k})$ and ${\cal R}=\mathbb{R}[P_{1},\cdots,P_{n}]$ the
subalgebra of polynomial invariants generated by the primitive invariants.

  3. 3.

There exists $m=d_{1}\cdots d_{n}/|G|$ secondary invariants polynomials
$S_{1},\cdots,S_{m}$.

  4. 4.

The subalgebra of invariants $\mathbb{R}[x_{1},\cdots,x_{n}]^{\Gamma}$ is a
free ${\cal R}-$module with basis $(S_{1},\cdots,S_{m})$. In particular this
means that any invariant $I\in\mathbb{R}[x_{1},\cdots,x_{n}]^{\Gamma}$ can be
uniquely written as $I=\sum\limits_{i=1}^{m}f_{i}(P_{1},\cdots,P_{n})S_{i}$,
where $f_{i}(P_{1},\cdots,P_{n}),i=1,\cdots,m$ belong to ${\cal R}$, i.e., are
polynomials in $(P_{1},\cdots,P_{n})$.

There exist computer algebra systems for the computation of
$P_{1},\ldots,P_{n}$ such as SINGULAR [2].

##  3 The Euler-Heisenberg Lagrangian at one loop

In 1936 Heisenberg and Euler obtained their following famous representation of
the one-loop QED effective Lagrangian in a constant field (“ Euler-Heisenberg
Lagrangian”)

$\displaystyle{\cal L}^{(1)}(a,b)$ $\displaystyle=$ $\displaystyle-{1\over
8\pi^{2}}\int_{0}^{\infty}{dT\over T^{3}}\,\,{\rm
e}^{-m^{2}T}\biggl{[}{(eaT)(ebT)\over{\rm tanh}(eaT)\tan(ebT)}-{e^{2}\over
3}(a^{2}-b^{2})T^{2}-1\biggr{]}$ (1)

Here $m$ is the electron mass, and $a,b$ are the two invariants of the Maxwell
field, related to $\bf E$, $\bf B$ by $a^{2}-b^{2}=B^{2}-E^{2},\quad ab={\bf
E}\cdot{\bf B}$.

This effective Lagrangian hold the information on the QED $N$ \- photon
amplitudes in the low-energy limit where all photon energies are small
compared to the electron mass, $\omega_{i}\ll m$. It corresponds to the
Feynman diagrams shown in Fig. 2.

Figure 2: Feynman diagrams corresponding to the Euler-Heisenberg Lagrangian.
The photon legs are in the low-energy limit.

For the extraction of the amplitudes from the effective Lagrangian, one
expands it in powers of the Maxwell invariants,

$\displaystyle{\cal L}(a,b)=\sum_{k,l}c_{kl}\,a^{2k}b^{2l}$ (2)

then fixes a helicity assignment and uses spinors helicity techniques [3].

##  4 Imaginary part of the effective action

Except for the purely magnetic case where $b=0$, the proper-time integral in
(1) has poles on the integration contour at $ebT=k\pi$ which create an
imaginary part. For the purely electric case one gets [4]

$\displaystyle{\rm Im}{\cal L}^{(1)}(E)$ $\displaystyle=$
$\displaystyle\frac{m^{4}}{8\pi^{3}}\beta^{2}\,\sum_{k=1}^{\infty}\frac{1}{k^{2}}\,\exp\left[-\frac{\pi
k}{\beta}\right]$

($\beta=eE/m^{2}$). We note:

  * •

The $k$th term relates to coherent creation of $k$ pairs in one Compton
volume.

  * •

In the weak-field limit $\beta\ll 1$ the terms with $k\geq 2$ can be
neglected.

  * •

${\rm Im}{\cal L}(E)$ depends on $E$ non-perturbatively (non-analytically),
which is consistent with Sauter’s [5] interpretation of pair creation as
vacuum tunnelling (Fig. 3).

Figure 3: Pair creation by an external field as vacuum tunnelling.

##  5 Beyond one loop

The two-loop correction to the Euler-Heisenberg Lagrangian due to one internal
photon exchange (Fig. 4) has been analyzed [6, 7, 8], and turned out to
contain important information on the Sauter tunnelling picture [9], on-shell
versus off-shell renormalization[6, 10], and the asymptotic properties of the
QED photon S-matrix [11].

Figure 4: Feynman diagrams contributing to the 2-loop EHL.

It leads to rather intractable two-parameter integrals. However, in the
electric case its imaginary part ${\rm Im}{\cal L}^{(2)}(E)$ permits a
decomposition analogous to Schwinger’s (LABEL:schwinger) [9]. For single-pair
production, this is now interpreted as a a tunnelling process where, in the
process of turning real, the electron-positron pair is already interacting at
the one-photon exchange level.

Even for the imaginary part no completely explicit formulas are available.
However, it simplifies dramatically in the weak-field limit, where it just
becomes an $\alpha\pi$ correction to the one-loop contribution:

$\displaystyle{\rm Im}{\cal L}^{(1)}(E)+{\rm Im}{\cal
L}^{(2)}(E)\,\,\,\,{\stackrel{{\scriptstyle\beta\to
0}}{{\sim}}}\,\,\,\,\frac{m^{4}\beta^{2}}{8\pi^{3}}\bigl{(}1+\alpha\pi\bigr{)}\,{\rm
e}^{-{\pi\over\beta}}$

This suggests that higher loop orders might lead to an exponentiation, and
indeed Lebedev and Ritus [9] provided strong support for this hypothesis by
showing that, assuming that

$\displaystyle{\rm Im}{\cal L}^{(1)}(E)+{\rm Im}{\cal L}^{(2)}(E)+{\rm
Im}{\cal L}^{(3)}(E)+\ldots\,\,\,\,{\stackrel{{\scriptstyle\beta\to
0}}{{\sim}}}\,\,\,\,\frac{m^{4}\beta^{2}}{8\pi^{3}}\,{\rm
exp}\Bigl{[}-{\pi\over\beta}+\alpha\pi\Bigr{]}={\rm Im}{\cal
L}^{(1)}(E)\,\,{\rm e}^{\alpha\pi}$

then the result can be interpreted in the tunnelling picture as the
corrections to the Schwinger pair creation rate due to the pair being created
with a negative Coulomb interaction energy

$\displaystyle m(E)\approx m+\delta m(E),\quad\delta
m(E)=-\frac{\alpha}{2}\frac{eE}{m}$

Moreover, the resulting field-dependent mass-shift $\delta m(E)$ is identical
with the Ritus mass shift, originally derived by Ritus in [13] from the
crossed process of one-loop electron propagation in the field (Fig. 5).

$\hskip 20.0pt\Longleftrightarrow\hskip 20.0pt$

Figure 5: Photon-corrected pair-creation vs. electron propagation in the
field.

Unbeknownst to the authors of [9], for scalar QED the corresponding conjecture
had already been established two years earlier by Affleck, Alvarez and Manton
[12] using Feynman’s worldline path integral formalism and a semi-classical
worldline instanton approximation.

Figure 6: Feynman diagrams contributing to the exponentiation hypothesis.

Diagrammatically, we note the following features of the exponentiation formula
(see Fig. 6):

  * •

It Involves diagrams with any numbers of loops and legs.

  * •

Although not shown, also all the counter-diagrams from mass renormalization
must contribute.

  * •

It does not include diagrams with more than one fermion loop (those get
suppressed in the weak-field limit [12]).

  * •

Horizontal summation produces the Schwinger exponential ${\rm
e}^{-\frac{\pi}{\beta}}$.

  * •

Vertical summation produces the Ritus-Lebedev/Affleck-Alvarez-Manton
exponential ${\rm e}^{\alpha\pi}$.

##  6 QED in 1+1 dimensions

The exponentiation conjecture has so far been verified only at two loops. A
three-loop check is in order, but calculating the three-loop EHL in $D=4$ is
presently hardly feasible. Motivated by work by Krasnansky [14] on Euler-
Heisenberg in various dimensions, in 2010 two of the authors with D.G.C.
McKeon started investigating the analogous problem in 2D QED. In [15] we used
the worldline instanton method to generalize the exponentiation conjecture to
the 2D case, resulting in

$\displaystyle{\rm Im}{\cal L}^{(all-loop)}_{2D}\sim\,{\rm
e}^{-\frac{m^{2}\pi}{eE}+\tilde{\alpha}\pi^{2}\kappa^{2}}$ (3)

where $\kappa=m^{2}/(2ef)$, $f^{2}={1\over 4}F_{\mu\nu}F^{\mu\nu}$, and
$\tilde{\alpha}=\frac{2e^{2}}{\pi m^{2}}$ is the two-dimensional analogue of
the fine-structure constant. Defining the weak-field expansion coefficients in
$2D$ by

$\displaystyle{\cal L}^{(l)(2D)}(\kappa)$ $\displaystyle=$
$\displaystyle\frac{m^{2}}{2\pi}\sum_{n=1}^{\infty}(-1)^{l-1}c_{2D}^{(l)}(n)(i\kappa)^{-2n}$
(4)

we then used Borel analysis to derive from (3) a formula for the limits of
ratios of $l$ \- loop to one - loop coefficients:

$\displaystyle{{\rm lim}_{n\to\infty}}{c^{(l)}_{2D}(n)\over
c^{(1)}_{2D}(n+l-1)}$ $\displaystyle=$
$\displaystyle{(\tilde{\alpha}\pi^{2})^{l-1}\over(l-1)!}$ (5)

Moreover, we calculated the 2D Euler-Heisenberg Lagrangian at one and two
loops,

$\displaystyle{\cal L}^{(1)}(f)$ $\displaystyle=$ $\displaystyle-{m^{2}\over
4\pi}{1\over\kappa}\Bigl{[}{\rm ln}\Gamma(\kappa)-\kappa(\ln\kappa-1)+{1\over
2}\ln\bigl{(}{\kappa\over 2\pi}\bigr{)}\Bigr{]}$ (6) $\displaystyle{\cal
L}^{(2)}(f)$ $\displaystyle=$ $\displaystyle{m^{2}\over
4\pi}\frac{\tilde{\alpha}}{4}\Bigl{[}\tilde{\psi}(\kappa)+\kappa\tilde{\psi}^{\prime}(\kappa)+\ln(\lambda_{0}m^{2})+\gamma+2\Bigr{]}$
(7)

where $\tilde{\psi}(x)\equiv\psi(x)-\ln x+{1\over 2x}$,
$\psi(x)=\Gamma^{\prime}(x)/\Gamma(x)$, and the constant $\lambda_{0}$ comes
from an IR cutoff. One finds from (6) and (7) that

$\displaystyle c_{2D}^{(1)}(n)$ $\displaystyle=$
$\displaystyle(-1)^{n+1}\frac{B_{2n}}{4n(2n-1)}$ (8) $\displaystyle
c_{2D}^{(2)}(n)$ $\displaystyle=$
$\displaystyle(-1)^{n+1}\frac{\tilde{\alpha}}{8}\frac{2n-1}{2n}B_{2n}$ (9)

Using properties of the Bernoulli numbers $B_{n}$ it is then easy to verify
that

$\displaystyle\lim_{n\to\infty}{c_{2D}^{(2)}(n)\over c_{2D}^{(1)}(n+1)}$
$\displaystyle=$ $\displaystyle\tilde{\alpha}\pi^{2}$

in accordance with (5).

##  7 Three-loop EHL in 2D: diagrams

At three loops, we face the task of computing the two diagrams shown in Fig. 7
(there are also diagrams involving more than one fermion-loop, including
several that involve Gies-Karbstein tadpoles [16], but those can be shown to
be subdominant in the asymptotic limit).

Figure 7: Three-loop diagrams contributing to the exponentiation conjecture.

Due to the super-renormalizability of $2D$ QED these diagrams are already UV
finite. They suffer from spurious IR - divergences, but those can be removed
by going to the traceless gauge $\xi=-2$ [17]. The calculation of diagram A is
relatively straightforward, thus we focus on the much more substantial task of
computing diagram B and its weak-field expansion coefficients.

Figure 8: Parametrization of diagram B.

Introducing Schwinger parameters for this diagram as shown in Fig. 8 leads to
the integral representation [17]

$\displaystyle{\cal L}^{3B}(f)$ $\displaystyle=$
$\displaystyle\frac{\tilde{\alpha}^{2}m^{2}}{128\pi}\int_{0}^{\infty}dwdw^{\prime}d\hat{w}d\bar{w}~{}I_{B}~{}\,{\rm
e}^{-a}$ $\displaystyle I_{B}$ $\displaystyle=$
$\displaystyle\frac{\rho^{3}}{\cosh^{2}\rho w\cosh^{2}\rho
w^{\prime}\cosh^{2}\rho\hat{w}\cosh^{2}\rho\bar{w}}\frac{B}{A^{3}C}$
$\displaystyle-\rho\frac{\cosh(\rho\tilde{w})}{\cosh\rho w\cosh\rho
w^{\prime}\cosh\rho\hat{w}\cosh\rho\bar{w}}\Bigl{[}\frac{1}{A}-\frac{C}{G^{2}}\ln\Bigl{(}1+\frac{G^{2}}{AC}\Bigr{)}\Bigr{]}$

where

$\displaystyle B$ $\displaystyle=$ $\displaystyle(\rm tanh^{2}z+\rm
tanh^{2}\hat{z})(\rm tanhz^{\prime}+\rm tanh\bar{z})+(\rm
tanh^{2}z^{\prime}+\rm tanh^{2}\bar{z})(\rm tanhz+\rm tanh\hat{z})$
$\displaystyle C$ $\displaystyle=$ $\displaystyle\rm tanhz\,\rm
tanhz^{\prime}\,\rm tanh\,\hat{z}+\rm tanhz\,\rm tanhz^{\prime}\,\rm
tanh\bar{z}+\rm tanhz\,\rm tanh\hat{z}\,\rm tanh\bar{z}+\rm
tanhz^{\prime}\,\rm tanh\hat{z}\,\rm tanh\bar{z}$ $\displaystyle G$
$\displaystyle=$ $\displaystyle\rm tanhz\,\rm tanh\hat{z}-\rm
tanhz^{\prime}\,\rm tanh\bar{z}$

($z=\rho w$ etc.). Although for a three-loop diagram this is a fairly compact
representation, an exact calculation is out of the question, and a
straightforward expansion in powers of the external field to get the weak-
field expansion coefficients turns out to create huge numerator polynomials.
To deal with those, we will now take advantage of the high symmetry of the
diagram.

##  8 Integration-by-parts algorithm

Introduce the operator $\tilde{d}\equiv\frac{\partial}{\partial
w}-\frac{\partial}{\partial
w^{\prime}}+\frac{\partial}{\partial\hat{w}}-\frac{\partial}{\partial\bar{w}}$
which acts simply on the trigonometric building blocks of the integrand.
Integrating by parts with this operator, it is possible to write the integrand
of the $n$-th coefficient $\beta_{n}$ as a total derivative
$\beta_{n}=\tilde{d}\theta_{n}$. Then, using once more the symmetry of the
graph,

$\displaystyle\int_{0}^{\infty}dwdw^{\prime}d\hat{w}d\bar{w}\,\,{\rm
e}^{-a}\beta_{n}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dwd\bar{w}d\hat{w}dw^{\prime}\tilde{d}\,\,{\rm
e}^{-(w+w^{\prime}+\hat{w}+\bar{w})}\theta_{n}$ $\displaystyle=$
$\displaystyle 4\int_{0}^{\infty}dwdw^{\prime}d\hat{w}\,\,{\rm
e}^{-(w+w^{\prime}+\hat{w})}\,\theta_{n}|_{\bar{w}=0}$

The remaining threefold integrals are already of a fairly standard type.

##  9 Using the polynomial invariants of $D_{4}$

Diagram $B$ has the symmetries

$\displaystyle\qquad w$ $\displaystyle\leftrightarrow$ $\displaystyle\hat{w}$
$\displaystyle\quad w^{\prime}$ $\displaystyle\leftrightarrow$
$\displaystyle\bar{w}$ $\displaystyle\quad(w,\hat{w})$
$\displaystyle\leftrightarrow$ $\displaystyle(w^{\prime},\bar{w})$

Those generate the dihedral group $D_{4}$. After a slight generalization to
the inclusion of semi-invariants (invariants up to a sign) [18], Theorem 1 can
be used to deduce that the numerator polynomials can be rewritten as
polynomials in the variable $\tilde{w}=w-w^{\prime}+\hat{w}-\bar{w}$ with
coefficients that are polynomials in the four $D_{4}$ \- invariants $a,v,j,h$,

$\displaystyle a$ $\displaystyle=$ $\displaystyle
w+w^{\prime}+\hat{w}+\bar{w}$ $\displaystyle v$ $\displaystyle=$
$\displaystyle 2(w\hat{w}+w^{\prime}\bar{w})+(w+\hat{w})(w^{\prime}+\bar{w})$
$\displaystyle j$ $\displaystyle=$ $\displaystyle
a\tilde{w}-4(w\hat{w}-w^{\prime}\bar{w})$ $\displaystyle h$ $\displaystyle=$
$\displaystyle
a(ww^{\prime}\hat{w}+ww^{\prime}\bar{w}+w\hat{w}\bar{w}+w^{\prime}\hat{w}\bar{w})+(w\hat{w}-w^{\prime}\bar{w})^{2}$

These invariants are moreover chosen such that they are annihilated by
$\tilde{d}$. Thus they are well-adapted to the integration-by-parts algorithm.
This rewriting leads to a very significant reduction in the size of the
expressions generated by the expansion in the field.

##  10 Results

In this way we obtained the first two coefficients of the weak-field expansion
analytically,

$\displaystyle\Gamma^{B}_{0}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}+\frac{7}{4}\zeta(3)$ $\displaystyle\Gamma^{B}_{1}$
$\displaystyle=$ $\displaystyle-\frac{251}{120}+\frac{35}{16}\zeta(3)$

and five more coefficients numerically. For a definite conclusion concerning
the exponentiation conjecture this is still insufficient, and the computation
of further coefficients is in progress.

##  11 Outlook

  * •

Writing Feynman graph polynomials in terms of invariant polynomials is a
universal option that, to the best of our knowledge, has not previously been
used, but we expect that it will be found very useful for multiloop
calculations involving a large number of external legs.

  * •

In particular, this is the case for the weak-field expansion of the QED
effective Lagrangian starting from three loops (in any dimension).

## References

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  * [18] I. Huet, M. Rausch de Traubenberg and C. Schubert, Proc. of Group 32, Prague, Czech Republic, July 9 - 13, 2018, publ. in J. Phys. Conf. Ser. 1194 (2019) no. 1, 012046 (10 pages), arXiv:1906.01041[hep-th].


11institutetext: Physikalisches Institut, Universität Bern, CH-3012 Bern,
Switzerland
11email<EMAIL_ADDRESS><EMAIL_ADDRESS>22institutetext:
Center for Space and Habitability, Universität Bern, CH-3012 Bern, Switzerland
33institutetext: Département d’Astronomie, Université de Genève, CH-1290
Versoix, Switzerland 44institutetext: Institute of Plant Biology, Biological
Research Centre Szeged, Temesvári krt. 62, 6726 Szeged, Hungary
55institutetext: Lund Observatory, Lund University, P.O. Box 43, SE-22100
Lund, Sweden 66institutetext: Leiden Observatory, Leiden University, P.O. Box
9513, 2300 RA Leiden, The Netherlands 77institutetext: Faculty of Aerospace
Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The
Netherlands 88institutetext: Université Grenoble Alpes, CNRS, IPAG, 38000
Grenoble, France

# Biosignatures of the Earth I. Airborne spectropolarimetric detection of
photosynthetic life

C.H. Lucas Patty 11 Jonas G. Kühn 2233 Petar H. Lambrev 44 Stefano Spadaccia
11 H. Jens Hoeijmakers 55 Christoph Keller 66 Willeke Mulder 66 Vidhya
Pallichadath 77 Olivier Poch 88 Frans Snik 66 Daphne M. Stam 77 Antoine
Pommerol 11 Brice O. Demory 22

###### Abstract

Context. Homochirality is a generic and unique property of life on Earth and
is considered a universal and agnostic biosignature. Homochirality induces
fractional circular polarization in the incident light that it reflects.
Because this circularly polarized light can be sensed remotely, it can be one
of the most compelling candidate biosignatures in life detection missions.
While there are also other sources of circular polarization, these result in
spectrally flat signals with lower magnitude. Additionally, circular
polarization can be a valuable tool in Earth remote sensing because the
circular polarization signal directly relates to vegetation physiology.

Aims. While high-quality circular polarization measurements can be obtained in
the laboratory and under semi-static conditions in the field, there has been a
significant gap to more realistic remote sensing conditions.

Methods. In this study, we present sensitive circular spectropolarimetric
measurements of various landscape elements taken from a fast-moving
helicopter.

Results. We demonstrate that during flight, within mere seconds of
measurements, we can differentiate (S/N¿5) between grass fields, forests, and
abiotic urban areas. Importantly, we show that with only nonzero circular
polarization as a discriminant, photosynthetic organisms can even be measured
in lakes.

Conclusions. Circular spectropolarimetry can be a powerful technique to detect
life beyond Earth, and we emphasize the potential of utilizing circular
spectropolarimetry as a remote sensing tool to characterize and monitor in
detail the vegetation physiology and terrain features of Earth itself.

##  1 Introduction

Terrestrial biochemistry is based on chiral molecules which, in their most
simple form, occur in a left-handed (L-) and right-handed (D-) version and are
non-superimposable mirror images of each other. Unlike abiotic chemistry,
where these molecules occur in roughly equal concentrations of both versions,
living nature almost exclusively (Grishin et al. 2020) utilizes these
molecules in only one of the mirror image configurations. This is called
homochirality.

Homochirality also manifests itself within biological macromolecules and
biomolecular architectures. The $\alpha$-helix, which is the most prevalent
secondary structure of proteins, is almost exclusively right-hand-coiled
(Novotny & Kleywegt 2005). The biochemical homochirality is essential for
processes ranging from enzymatic functioning to self-replication. The latter
also implies the prerequisite of homochirality for life as we know it (Popa
2004; Bonner 1995; Jafarpour et al. 2015). Homochirality is likely a universal
feature of life (Wald 1957; MacDermott 2012; Patty et al. 2018c; Sparks et al.
2009b); and owing to its high exclusivity, homochirality may serve as an
agnostic trait of life itself.

The molecular dissymmetry of chiral molecules and architectures endows them
with a specific response to electromagnetic radiation (Fasman 2013; Patty et
al. 2018c). This response is most evident upon interaction with polarized
light, but also modifies the polarization properties of incident unpolarized
light. Circular dichroism is the differential absorbance of left- and right-
handed circularly polarized incident light, which only yields a net signal
when the molecules are homochiral. Circular polarizance is related to circular
dichroism; circular polarizance is the fractionally induced circular
polarization resulting from the interaction with unpolarized incident light,
such as is essentially emitted from the Sun (Kemp et al. 1987). It has been
demonstrated that circular polarizance carries the same information as
circular dichroism (Patty et al. 2017, 2018b), but unlike the latter, cicular
polarizance can be sensed remotely (i.e., without controlling the source)
(Pospergelis 1969; Wolstencroft et al. 2004; Sparks et al. 2009b).

Remote sensing of the Earth’s surface with spectropolarimetry could offer
information not accessible with the reflectance alone. The circular
polarization response of vegetation, albeit small and typically less than 1%
around the chlorophyll absorbtion band, is an extremely sensitive indicator of
the membrane macro-organization of the photosynthetic apparatus and its
functioning (Garab & van Amerongen 2009; Garab 1996; Lambrev & Akhtar 2019).
The polarization spectra could, in principle, be used to detect various
physiological stress responses and could provide sensitive monitoring of
vegetation physiology (Lambrev & Akhtar 2019). It has for instance been
demonstrated that circular polarization can be more indicative of stress
factors such as drought than the scalar reflectance (Patty et al. 2017). Other
phototrophic organisms can also yield clear circular polarization signals and
some of these can produce relatively large signals: up to 2% for certain
multicellular algae (Patty et al. 2018a). Additionally, nonzero polarization
signals can be observed for various (an)oxygenic phototrophic microorganisms
and microbial mats (Sparks et al. 2009a, b, 2021). Polarization remote sensing
could prove to be a very powerful complementary tool in assessing, for example
the effects of climate change, desertification, deforestation, and the
monitoring of vegetation physiological state in vulnerable regions.

As a result of its dependence on homochirality and the three-dimensional
molecular asymmetry of life, the circular polarization induced by life is a
serious contender to be a generic agnostic (universal) surface biosignature.
While most research efforts on remotely detectable biosignatures have focused
on the spectroscopic characterization of planetary atmospheres, detection of
gases related to the possible presence of a planetary biosphere is not free of
false-positive scenarios (Meadows et al. 2018; Schwieterman et al. 2016;
Schwieterman 2018). Surface biosignatures, such as circular polarization, are
a direct observable characteristic of life and as such have a potential to
reduce the model dependence (Schwieterman 2018). While abiotic materials can
also create circular polarization, such as through multiple scattering by
clouds or aerosol particles, these interactions result in a weaker and
spectrally flat circular polarization signal (Rossi & Stam 2018).
Additionally, incident unpolarized light cannot become circularly polarized by
anisotropic Rayleigh scattering alone, but it can become circularly polarized
if scattered at least once by a Rayleigh scatterer and consequently by a large
aerosol or particle (Rossi & Stam 2018). Furthermore, as the direction of
circular polarization is directly determined by the chirality of the molecular
assembly, the signal is not averaged out in a planetary disk average.
Averaging out does occur for the circular polarization signal resulting from
scattering by clouds or aerosol particles (Rossi & Stam 2018).

Additionally, circular spectropolarimetric measurements on various minerals,
artificial grass fields, and on the surface of Mars, and the absence of
nonzero signals measured in these studies attest to the general lack of false
positive scenarios and abiotic mimicry (Sparks et al. 2009b, a; Pospergelis
1969; Sparks et al. 2012; Patty et al. 2019). Other suggested remotely
detectable (surface) biosignatures, such as the vegetation red edge or other
pigment signatures (Seager et al. 2005; Kiang et al. 2007; Schwieterman et al.
2015), have a higher risk of possible false positives by mineral reflectance.

Research on circularly polarized biosignatures or possible remote sensing
applications have been developing quickly in recent years. Previous works have
demonstrated the potential of high-sensitive circular polarization
measurements obtained in the laboratory (Patty et al. 2017, 2018a, 2018b;
Sparks et al. 2009b, a, 2021; Martin et al. 2010, 2016; Martin 2020) and in
the field (Patty et al. 2019). All of these measurements, however, were taken
under stable conditions in the laboratory or are measurements of semi-static
(allowing integration over longer durations) scenes in the field.

In the present study we show, for the first time, sensitive circular
spectropolarimetry from a fast-moving aerial platform using the
spectropolarimeter FlyPol, based on the TreePol design (Patty et al. 2017,
2018c, 2019). We report the observation of circularly polarized signals for
various landscape features, such as forests and grass fields, and we
demonstrate that these signals can be readily distinguished from abiotic
scenes such as roads and urban area merely using nonzero circular polarization
spectral features as a discriminator.

##  2 Materials and methods

Circular spectropolarimetric measurements were carried out using FlyPol.
FlyPol is an adaptation of the TreePol instrument (see (Patty et al. 2017,
2018c, 2019) for more information), which was specifically designed to measure
the fractionally induced circular polarization ($V/I$) as a function of
wavelength (400–900 nm) at high sensitivity ($<10^{-4}$) and accuracy
($<10^{-3}$). For FlyPol we upgraded the spectrographs (Avantes, The
Netherlands) and amended the design with active temperature control of the
optics (Thorlabs, USA) and electronics, which greatly improved the temporal
stability of the instrument. An additional feature is the possibility of
mechanically actuating optical elements, such as inserting and removing
optical elements, which will facilitate the sequential acquisition of full-
Stokes measurements in the future. The angular field of view of FlyPol is
approximately 1.2∘. The instrument was oriented using a calibrated, parallax-
free telescope pointer mounted on top of the instrument casing (53x33x13cm).

All the measurements were taken from an Enstrom 280C helicopter and were
carried out in the districts of Val-de-Travers and Le Locle of the canton of
Neuchâtel, Switzerland. These districts lie in the Swiss Jura, which is part
of the Alpine foreland and is characterized by a hilly terrain and a series of
sub-parallel valleys and ridges. Measurements were taken with an angle to the
vertical of approximately 45∘, at a ground speed of 20-39 kts (10-20 m/s) and
up to a altitude of 5500 ft AMSL (1.7 km). A GPS receiver was used to track
the flight trajectories.  All measurements were performed under clear sky
conditions on September 17, 2019 at 1600-1620 hr. During the flight, the solar
zenith and azimuth angle were approximately $57^{\circ}$ and $233^{\circ}$,
respectively. The measurements, however, were averaged over different phase
angles, which could not be recorded accurately. Instrumental calibration was
performed on a spectralon target on the ground prior and post flight. The
reflectance data displayed are qualitative and merely shown as support for the
polarization spectra. The integration time for all measurements was 12 ms to
prevent detector saturation. For several scenes, the average photon count
(¡10000 counts) was too low and these measurements were discarded.

Laboratory measurements of vegetation (Graminae sp., Acer platanoides,
Platanus sp.) and concrete were taken in reflection using an integrating
sphere. For the polarimetric spectra of vegetation, three different leaves or
grass patches per species were measured. The measurements on concrete were
taken from three different regions on the same slab. The results were averaged
over a 30 s measurement per sample.

Figure 1:  Enstrom 280C taking off at Aérodrome de Môtiers (A) and a closeup
of the cockpit with FlyPol inside (B).

##  3 Results

###  3.1 Landscape element differentiation

Figure 2: A: One of the measurement trajectories (in red) during a flight near
Aérodrome de Môtiers, Môtiers, flown in E direction. The inset indicates the
vegetation classification using NDVI where the white is classified as non-
grass (same color bar and scale as B). Satelite image by Swiss Federal
Authorities, CNES; Spot Image, Swisstopo, NPOC. B: Qualitative reflectance
data for the trajectory per wavelength in time [s]. C: V/I for the trajectory
per wavelength in time [s]. D: Time-averaged reflectance of the grass and
road, respectively. E: Time-averaged V/I of the grass and road, respectively.
Shaded areas denote the standard deviation between measurements.

In order to differentiate between the different landforms and landscape
elements we used the differences in reflectance as measured by FlyPol and we
calculated the normalized difference vegetation index (NDVI) (Rouse Jr 1973)
as follows:

$NDVI=\frac{I_{FR}-I_{R}}{I_{FR}+I_{R}},$ (1)

where $I_{R}$ is the mean of the reflectance between 650 nm - 680 nm and
$I_{FR}$ is the mean between 750 nm - 780 nm. A comparison with the flight
trajectories showed that using the NDVI was accurate in differentiating
between the elements per scene.

In Figure 2 we show the results of a trajectory recorded near Aérodrome de
Môtiers. The approximate trajectory is shown in Figure 2 A, where the inset
shows the mask based on the NDVI data used to filter out the two intersecting
roads. Figure 2 B and C show the reflectance and circular polarization,
respectively, binned to one second measuring steps. Interestingly, while the
major part of the trajectory in Figure 2 (B) exhibits a red edge, a circular
polarization signal is not continuously detected (C). However, by averaging
over the whole scene, a clear signal can be obtained. The results averaged
over the whole scene per different landscape element, grass, and roads are
shown in Figure 2 D and E for the reflectance and $V/I$.

Using the NDVI, we were able to separate four different landscape elements:
grass, trees, urban, and water. The resulting average circular polarization
spectra for these categories are shown in Figure 3. It is clearly visible that
the average over the urban area has a very low signal, where
$V/I_{max}=2.6\cdot 10^{-4}$ at $\sim$700 nm. For water containing algae and
grass, which both show a positive band with a maximum at $\sim$680 nm, the
magnitudes are $V/I_{max}=4.1\cdot 10^{-4}$ and $V/I_{max}=1.1\cdot 10^{-3}$,
respectively. The average taken over the trees shows a positive maximum of
$V/I_{max}=6.1\cdot 10^{-4}$ at $\sim$ 700 nm and a negative band with a value
of $V/I_{min}=-1.4\cdot 10^{-3}$ at $\sim$660 nm.

Figure 3: Circular polarization spectra for various landscape features: grass,
urban area, trees, and water containing algae. The red shaded area represents
the red edge.

###  3.2 Lacustrine aerial spectropolarimetry

Figure 4: A: One of the measurement trajectories (in red) over the lake Lac
des Taillères in La Brévine, flown in W direction. The inset indicates the
filtered classification using NDVI (same color bar and scale as B). Satelite
image by: Swiss Federal Authorities, CNES, Spot Image, Swisstopo, NPOC. B:
Qualitative reflectance data for the trajectory per wavelength in time [s]. C:
V/I for the trajectory per wavelength in time [s]. D: Time-averaged
reflectance for the whole lake and filtered NDVI. E: Time-averaged circular
polarization for the whole lake and filtered NDVI. The shaded areas denote the
standard deviation between measurements.

A trajectory was flown over the lake Lac des Taillères, which is shown in
Figure 4. A red edge is only visible along part of the trajectory (Figure 4 B
) and, for instance, is absent after 49 seconds. The lack of a visible red
edge does not necessarily indicate the absence of photosynthetic organisms but
could also be the effect of the lower total photon count, which might be
related to the lake depth. The average fractional circular polarization for
the whole scene as well as for the filtered scene (for low NDVI and/or unusual
high and low counts) are given in Figure 4 E.

###  3.3 Validation

In Figure 5 we show the circular polarization over shorter measurement
durations and the signal-to-noise ratio (S/N) for the different landscape
elements: A, E, and I indicate grass; B, F, and J represent trees; C, G, and K
indicate urban; and D, H, and L indicate water. The upper panels show the
circular polarization spectra for different time intervals: per 1, 2, 3, 4, 5,
and 10 s. The center panels show the average of three subsequent one second
measurements and the standard deviation between those scenes is denoted by the
shaded area. The top panels show the detection S/N for the center panels using
the offset substracted measurements on spectralon as sigma. In general, clear
features are already present for measurements of one second duration and the
repeatability is high as is demonstrated by the center panels of Figure 5 with
a S/N larger than 5 for vegetation.

Figure 5: Circular polarimetric spectra of grass (A, E), forest (B and F)
urban (C and G), and water (D and H). The upper panels show the fractional
circular polarization per wavelength for different measuring time intervals.
The middle panel shows the average of three subsequent one second
measurements; the standard deviation between them is denoted by the shaded
area and the corresponding S/N is shown in I, J, K, and L.

The grass does not exhibit significant differences for measurements longer
than two seconds (Figure 5 A) with a positive band maximum $V/I_{max}=1.0\cdot
10^{-3}$ at 680 nm. For tree canopies (Figure 5 B), the variations are larger;
where the measurement of shorter duration start with a positive band maximum
$V/I_{max}=1.3\cdot 10^{-3}$ this reduces to $V/I_{max}=3.2\cdot 10^{-4}$ for
10 seconds. The measurements over the urban area (Figure 5 C) show no clear
features for the longer measurement durations. The observations of the water
surface (Figure 5 D) generally display a signal amplitude that is smaller on
average than that of vegetation. A nonzero signal can, however, be observed
around the chlorophyll absorbance band.

Figure 6 shows polarimetric measurements carried out in the laboratory on
three different vegetation samples and concrete. In reflection, leaves often
display two main spectral types of circular polarizance. One is characterized
by the spectrum in Figure 6 B, which is similar to A with both a positive and
a negative band. Additionally, in many instances a signal with only a negative
band is observed (Figure 6 C). The measurements on concrete show a clear zero
signal, where $V/I_{max}=2.5\cdot 10^{-5}$ and $V/I_{min}=2\cdot 10^{-5}$.

##  4 Discussion and conclusions

Figure 6: Circular polarimetric spectra measured (in reflection) in the
laboratory of grass (Gramineae sp.) (A), tree leaves,  Acer platanoides (B),
Platanus sp. (C), and concrete (D). The inset provides a close-up of the same
measurement. The shaded area denotes the standard deviation, n=3.

We show for the first time, to our knowledge, the possibility of acquiring
high-quality circular polarization spectra from a fast-moving aerial platform
with a total measurement duration down to one second. We were able to
distinguish vegetation from abiotic surfaces and demonstrate for the first
time the possibility of detecting life in lakes through spectropolarimetry,
underlining the potential use of life’s circular polarization as an agnostic
biosignature for the detection of life beyond Earth.

With the average of three subsequent one-second duration measurements (Figure
5), we were able to distinguish between the forest, grass, and urban areas
with a S/N of almost 14 for forests and 5 for grass. While our measurements
over water bodies display a spectropolarimetric signal that is indicative of
the presence of phototrophic organisms, the signal is low in amplitude. The
noise in the urban area measurements is relatively large compared to the
measurements on concrete in the laboratory (shown in Figure 6 D). This is
likely a consequence of the large number of different geometries (e.g., by
roofs and walls), the presence of objects with high linear polarization (e.g.,
windows and cars) and the possible presence of vegetation (i.e., in gardens
and along roads). Nonetheless, the spectrum is essentially free from any
significant spectral features compared to areas covered by phototrophic life.

The spectral characteristics of the measurements in this work are
qualitatively similar in shape to those obtained in transmission in the
laboratory, that is, showing negative and positive bands that are
characteristic of the chloroplast thylakoid membranes of plants. These two
bands are termed “psi-type” (polymer- and salt-induced) circular polarizance
or dichroism and they are associated with the long-range chiral order of
chlorophyll molecules in ordered macroarrays of the chlorophyll-protein
complexes in the membrane (Garab & van Amerongen 2009). The two bands have
different physical origins, where the negative band is associated
preferentially with the stacking of the thylakoid membranes inside the
chloroplasts, whereas the positive band is mainly associated with the lateral
organization of the chiral macro-domains (Lambrev & Akhtar 2019). We note that
although it appears that the negative band is absent in for instance the
grass, this is likely due to a slight offset in the zero point. The positive
band is often less well characterized in measurements taken at a distance (cf.
(Patty et al. 2019)) and is also characteristic for certain trees such as the
Platanus shown in Figure 6 C. We observed similar variations for some of the
forest trajectories, shown in Figure 5 B, where the initially larger positive
band decreases in magnitude with more captured frames. In other trajectories
this effect was less pronounced, which was also observed for different scenes
in Patty et al. (2019). Similar spectral phenomena occur around the veins of
leaves, where it was suggested to have been caused by the orientation of the
chloroplasts (Patty et al. 2018b). It is at this point unclear what the
underlying mechanisms are for the results on whole leaf canopies and how this
might vary between different species.

We also would like to emphasize that it might be possible that, to some
extent, the phase angles involved play a role in this phenomenon;  we did not
accurately record this during the flight campaign. Additionally, as of yet, no
systematic experimental study exists on the effect of incidence and reflection
angles on vegetation polarizance. Such a study would additionally aid in the
development of an (exo)planetary model with realistic circular polarization
surface components (Groot et al. 2020).

One of the key limiting factors in terms of measuring time and sensitivity is
the accurate scrambling of linear polarization to prevent linear-to-circular
polarization crosstalk ($Q,U\rightarrow V$ components). Especially from a
fast-moving platform, large fluctuations on a short timescale can be expected.
The data (see also Figure 5) indicate that a one-second measurement time
generally ensures that the scrambling of the linear polarization allows for
sub-sequential subtraction. It would be interesting to observe the variability
in linear polarizance originating from these different landscape elements. In
terms of remote sensing it has been suggested that linear polarization offers
no additional information compared to the scalar reflectance (Peltoniemi et
al. 2015), although it has been demonstrated that linear polarization can
inform plant physiology as well (Vanderbilt et al. 2017, 2019). We intend to
include linear polarization observations in future flight campaigns. Capturing
linear and circular polarization simultaneously, however, would require an
additional second polarimeter or a full-Stokes polarimeter that is capable of
acquiring the polarization on a single data frame (see, e.g., (Snik et al.
2019; Sparks et al. 2019; Keller et al. 2020)).

While subsequent measurements are often very repeatable (as shown in Figure
5), it is shown in Figure 2 C that in some cases no clear signal is measured
while generally a clear red edge is visible for the whole scene with the
exception of the roads (Figure 2 B). It is possible that the lack of clear
$V/I$ signals in this case might be due to local variations in the grass
physiology. The measurements were taken at the end of summer and it is visible
in 1 A that there are various dry patches in the grass. It has been
demonstrated before that drought can greatly decrease the circular polarizance
of the grass, while it does not decrease the red edge in a similar manner
(Patty et al. 2017). Nonetheless, for this particular campaign we did not take
ground $V/I$ observations that could elucidate.

Sensitive spectropolarimetry could prove to be a very valuable tool for the
remote sensing of vegetation and other photosynthetic organisms on Earth. The
psi-type circular polarizance of vegetation in particular is a sensitive
indicator of the photosynthetic membrane macro-organization, which can show
changes and variations as a result of various stress factors (Lambrev & Akhtar
2019). It has additionally been demonstrated that various species display
different spectral characteristics. As such, circular polarimetry could be
used, for instance, to measure deforestation and diseases in forests and it
might, in addition, be possible to distinguish between tree species and to
monitor evasive species. The first results on water bodies in this study look
promising. It could be valuable to implement circular polarization in the
monitoring of toxic algal blooms, of coral reefs, and the effects of
acidification thereon. Additionally, it could aid as a sensitive indicator of
pollution, such as by spills, by measuring the polarization response of
phytoplankton.

While promising in terms of biosignature exclusivity, the magnitude of the
circular polarization signal is low and typically much less than 1% in the
field. As such the detection of similar circular polarization signals
originating from exoplanets will be challenging. Additionally, in a planetary
disk average it can be expected that the signal is further diluted by surfaces
with a reflectance that is higher than those creating the circular
polarization. This is also the case for clouds, which are highly reflective
and can hide the circular polarization signal and other surface biosignatures
such as the red edge. In general, however, the atmospheric circular
polarization signal is very low and allows for the detection of circular
polarization signals originating from surface biological sources (Rossi & Stam
2018). It is unlikely that the next generation of telescopes could measure
circular polarization signals with a similar magnitude originating from Earth-
like planets around solar-type stars. In theory, such signals could however be
detected with only 100 hours of observations on a terrestrial super Earth in
close orbit (0.05 AU) to the star based on the instrumental performance of
ground-based telescopes and the precision of POLLUX (Sparks et al. 2021). In
the end, a confident detection of extraterrestrial life will naturally have to
consist of a consistent range of spectroscopic biosignatures (Seager 2014), of
which the detection of circular polarization signals will be very strong
evidence of homochirality and hence life.

We have successfully demonstrated the possibility of using sensitive circular
spectropolarimetry from a fast-moving aerial platform. An important next step
will be to measure the circular polarization of the Earth surface from an
orbiter such as from the International Space Station. Even with a one second
measurement time; this would result in roughly a 6-7 km resolution and will
provide valuable information for future polarization remote sensing and will
also provide crucial input on the potential use of life’s circular
polarization as an agnostic biosignature for the detection of life beyond
Earth.

## Acknowledgments

This work has been carried out within the framework of the National Centre of
Competence in Research (NCCR), PlanetS, supported by the Swiss National
Science Foundation (SNSF). We thank Jean-Claude Pointet for flying the
helicopter. The research of FS leading to these results has received funding
from the European Research Council under ERC Starting Grant agreement 678194
FALCONER.

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