# Universal scaling and ferroelectric hysteresis regimes in the giant squid
axon propagating action potential: a Phase Space Approach

Nikola K. Jurisic<EMAIL_ADDRESS>Jurisic Consultants, 12013 Navy
St, Los Angeles, CA 90066, USA  Fred Cooper<EMAIL_ADDRESS>The Santa Fe
Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA  Theoretical Division
and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos,
NM 87545

###### Abstract

The experimental data for the giant squid axon propagated action potential is
examined in phase space. Plots of capacitive and ionic currents vs. potential
exhibit linear portions providing temperature dependent time rates and maximum
conductance constants for sodium and potassium channels. First order phase
transitions of ionic channels are identified. Incorporation of time rates into
Avrami equations for fractions of open channels yields for each channel a
temperature independent dimensionless constant that is remarkably close in
value to the fine structure constant. It also yields temperature independent
scaling exponents $\theta_{x}$. Evidence is presented that the action
potential traverses a ferroelectric hysteresis loop starting with a linear
outgoing current followed by incoming sodium current that is accompanied with
a channels lattice morphing in the negative resistance region. This results in
a second order phase transition polarization flip at the peak of the action
potential, followed by closing of sodium and opening of potassium channels,
and finally closing the loop by reversing the polarization flip as the resting
potential is reached. The existence of this hysteresis loop for the giant
squid action potential suggests the possibility of neurons with two stable
states, the basis for memory storage and retrieval.

††preprint: LA-UR 17-30245,

##  I Introduction

The initiation and propagation of the action potential in the giant squid axon
has been described by the Hodgkin-Huxley 1 ; 2 empirical equations in terms of
sodium, potassium and leakage currents traversing the membrane. However, a
description of the squid axon behavior based on first-principles physics has
yet to be achieved. In the present work, using the Rosenthal-Bezanilla RB
experimental data, we present a phase space based phenomenological theory for
the steady propagation of the action potential as described by the charge
conserving cable equation:

$C_{m}\frac{dV}{dt}+J_{I}(V)=\frac{R}{2v^{2}R_{i}}\frac{d^{2}V}{dt^{2}},$ (1)

where the experiment provides the action potential $V(t)$ as function of time
(See Fig. 1). $C_{m}$ is the membrane capacitance per $cm^{2}$, $R$ is the
axon radius, $R_{i}$ is the temperature dependent resistivity of the axoplasm
and $v$ is the temperature dependent velocity of propagation of the action
potential. The first and the third term of Eq. (1) are the capacitive and the
so-called membrane current respectively. $J_{I}$ is the total ionic current
traversing the membrane of the axon.

Figure 1: The Experimental data $V(t)$ vs. time is presented as colored
segments that will be elucidated in phase space.

Since $V(t)$ is provided by the experiment, all three terms of Eq. (1) are
known. The different regions of the graph in Fig. 1 correspond to specifically
identifiable regions in the phase space phase of the currents vs. action
potential V. “As a whole, the phase diagram represents all that the system can
be, and its shape can easily elucidate qualities of the system that might not
be obvious otherwise” phase1 . In particular we shall identify and describe
the diverse colored segments of $V(t)$ vs. $t$ from Fig. 1 in the phase space
diagrams Fig. 2a-2d of currents vs. potential and in Fig. 6b and Fig. 6d of
the hysteresis phase space trajectory. The colored segments correspond to

  1. 1.

Linear outgoing current, later identified as potassium current.

  2. 2.

Sodium M-channel $1^{st}$ order phase transition exhibiting scaling across a
range of temperatures.

  3. 3.

Sodium lattice morphing with a $2^{nd}$ order phase polarization flip from
M-channel to H-channel.

  4. 4.

Sodium H-channel $1^{st}$ order phase transition exhibiting scaling across a
range of temperatures.

  5. 5.

Potassium N-channel $1^{st}$ order phase transition exhibiting scaling across
a range of temperatures.

We shall fit ionic and capacitive currents and also show that the fractions of
completed $1^{st}$ order phase transitions scale across a temperature range
yielding dimensionless temperature independent constants very close to the
value of the fine-structure constant.

Figure 2: Capacitive and ionic currents, and linear segments. Arrows
$\rightarrow$ indicate the direction of time. Recovery polarization (morphing)
current is displayed in Fig. 3. Note that $C_{m}=1\mu F/cm^{2}$. The linear
slope of the $J_{H}$ ionic current and the linear slope of the corresponding
capacitive current intercept the potential axis at different points. P4 fit is
4th order polynomial fit. a, c, The slope of the capacitive current linear
segment at the foot of the action potential is the time rate constant
$\mu_{0}.$ The slope of the capacitive current corresponding to $J_{M}$ linear
segment is -$\mu_{M}$ where $\mu_{M}$ is the time rate constant of the sodium
M-channel. The slope of the current $J_{0}$ is the conductance $g_{0}$. The
slope of the linear segment of the ionic current $J_{M}$ is the maximum
conductance $g_{M}$ for the sodium M-channel. b, d, The slopes of sodium s and
potassium s capacitive current linear segments correspond to rate constants
$\mu_{H}$ and $\mu_{N}$. The slope of $J_{H}$ linear segment is $-g_{H}$ where
$g_{H}$ is the maximum conductance of the sodium H-channel. The slope of the
ionic current $J_{N}$ is the maximum conductance $g_{N}$.  Figure 3:
Polarization ionic, ionic, and capacitive currents detail around action
potential peak. Arrows $\rightarrow$ indicate time direction. The rising edge
ionic, capacitive and membrane currents intersect the zero-current axis at the
point $V=V_{M}$. The linear segment of the recovery ionic current, $J_{H}$,
intersects the axis at $V=V_{H}$ that is about 6 mV smaller than $V_{M}$. The
rising edge polarization ionic current is first incoming and then outgoing.
Recovery polarization ionic current is incoming.

Eq. (2) lists the ionic currents. Ionic currents $J_{M}$, $J_{H}$ and $J_{N}$
have the familiar structure as the product of maximum conductance $g_{X}$,
fraction of open channels $Xo(V)/X$, and the driving force $(V-V_{X}$).

$J_{0}(V)=g_{0}V$ (2a) $J_{M}(V)=g_{M}(Mo/M)(V-V_{M})$ (2b)
$J_{PM}(V)=J_{I}(V)-J_{M}(V)~{}~{}\text{in the negative resistance region of
the rising edge}$ (2c) $J_{PH}(V)=J_{I}(V)-J_{H}(V)~{}~{}\text{in the recovery
region near the peak of the action potential }$ (2d)
$J_{H}(V)=-g_{H}(Ho/H)(V-V_{H})$ (2e) $J_{N}(V)=g_{N}(No/N)(V-V_{N})$ (2f)

Experimental data shows that $g_{X}$ grows exponentially with temperature (see
Fig. 4c), $Xo/X$ is temperature independent (see Fig. 5), and $(V-V_{X}$) is
slightly temperature dependent.

Figures 2a and 2b display the capacitive current $C_{m}dV/dt$ and the ionic
current $J_{I}$ as functions of $V$ at $4.5^{o}C$ for the rising edge and the
recovery regions of the action potential respectively. Similarly, Figures 2c
and 2d display the currents at $19.8^{o}C$. Also displayed are fits for
$J_{0},J_{M},J_{H}$ and $J_{N}$, respectively linear potassium current at the
foot of the action potential, incoming sodium M-channel current, polarization
current caused by the morphing of sodium channels lattice, at first incoming
and then outgoing sodium H-channel current and outgoing potassium N-channel
current. Note that the rising edge currents $J_{0}$ and $J_{M}$ are cancelling
each other at the foot of the action potential. $J_{0}$ is outgoing and
$J_{M}$ is incoming. We shall argue later that $J_{0}$, negligible at low
temperature, is potassium current. In the recovery region, the currents
$J_{N}$ and $J_{H}$ overlap adding to total outgoing current.

Note also that in the negative resistance region of the rising edge of the
action potential the total ionic current is parsed into the phase transition
current $J_{M}$ and the small current $J_{PM}$. Similarly, immediately after
the peak of the action potential, the total ionic current is parsed into
$J_{H}$ and $J_{PH}$ (see Fig. 3). Later, we shall argue that the currents
$J_{PM}$ and $J_{PH}$ are caused by the morphing of the sodium channels
lattice. Of the two, only $J_{PM}(V)$ is displayed in Fig. 2. Linear segments
of $J_{0}$ and $J_{N}$ and $J_{H}$, apparent at once in Fig. 2, correspond to
the fraction of open channels equal to unity. Fig. 2 also displays the fits
for the linear portions of the capacitive and ionic currents. The slopes of
the capacitive linear segments correspond to time rate constants $\mu_{0}$,
$\mu_{M}$, $\mu_{H}$, $\mu_{N}$, and the slopes of the ionic current linear
segments correspond to maximum conductance $g_{0}$, $g_{M}$, $g_{H}$ and
$g_{N}$. Temperature dependence of rate constants are plotted in Fig. 4a and
Fig. 4b. Temperature dependence of maximum conductance are plotted in Fig. 4c.

The linear segments of ionic currents $J_{0}$ and $J_{M}$ and $J_{N}$
intersect the zero current axis at the same point as the corresponding linear
segments of capacitive and membrane currents. The cable Eq. 1.1 yields for
them the relationships between maximum conductance and corresponding time
rate. See also Scott :

$g_{0}=\mu_{0}\left[\frac{\mu_{0}}{k}-1\right]C_{m}$ (3a)
$g_{M}=\mu_{M}\left[\frac{\mu_{M}}{k}+1\right]C_{m}$ (3b)
$g_{N}=\mu_{N}\left[\frac{\mu_{N}}{k}+1\right]C_{m}$ (3c) where the so called
propagation constant is $k=\frac{2C_{m}R_{i}v^{2}}{R}$ (3d)

Figure 4:  Rate constants [1/mSec] and conductance [mS/cm2] vs. Temperature
[${}^{o}C$]. Fig. 4a shows the rising edge rate constants for sweeps 181,170,
220, 327, 425, 525, 630, 695, 754, 764 and 790. In the first linear region,
the rate constant $\mu_{0}$ is essentially equal to $k$ at temperatures of up
to $12-14^{o}C$. As temperature increases, $\mu_{0}$ and $k$ increase, with
$\mu_{0}$ increasing faster than $k$. Ionic current is outgoing in the region
where $\mu_{0}>k$. The value of $k$ was determined by using Rosenthal s
measures of both the axon s radius $R=238mm$ and the distance between the two
electrodes and the time of the action potential passage to give the velocity.
Sweeps 764 and 790 are outliers at temperatures $28.5^{o}C$ and $28.7^{o}C$
respectively. Rate constants are fitted with the exponentials for points up to
$19.8^{o}C$: $\mu_{M}=10.5e^{0.0634{\rm Temp}}$, $k=3.3e^{0.0635{\rm Temp}}$,
$\mu_{0}=3.1e^{0.077{\rm Temp}}$. Fig. 4b displays the recovery region rate
constants fits: $\mu_{N}=2.04e^{0.11{\rm Temp}}$ and $\mu_{H}=2.77e^{0.092{\rm
Temp}}$. Fig. 4c displays exponential fits for maximum conductance:
$g_{M}=45e^{0.062{\rm Temp}}$, $g_{H}=3.1e^{0.119{\rm Temp}}$, and
$g_{N}=3.2e^{0.141{\rm Temp}}$.

We have fitted the experimental data with two different version of $Xo/X$, the
first one directly in the phase space:

$\frac{Mo(V)}{M}=\left[\frac{(V-V_{oM})}{(V_{p}-V_{oM})}\right]^{\delta_{M}}$
(4a)
$\frac{Ho(V)}{H}=\left[\frac{(V-V_{cH})}{(V_{p}-V_{cH})}\right]^{\delta_{H}}$
(4b)
$\frac{No(V)}{N}=\left[\frac{(V_{oN}-V)}{(V_{oN}-V_{N})}\right]^{\delta_{N}}$
(4c)

where $V_{p}$ is the value of the potential at the peak of the action
potential, $V_{oM}$ and $V_{oN}$ are the respective potentials at which sodium
M channels and potassium N channels start to open, $V_{cH}$ is the potential
at which sodium H channels end up closing. The fits of experimental data show
that expressions Eqs. (4) scale with the exponents $\delta_{X}$ being
independent of temperature. The average values of exponents are: $\delta_{M}$
= 3.79 for ten sweeps from $1^{o}C$ to $28.5^{o}C$, $\delta_{H}$ = 3.10 and
$\delta_{N}$ = 3.12 for 6 sweeps from $4.5^{o}C$ and $19.8^{o}C$. (see Fig. 4
for M-channel phase fits at $4.5^{o}C$ and $19.8^{o}C$).

We have also fitted the fraction of open channels with the Avrami equation
avrami1 , avrami2 , avrami3 . In particular, the Avrami equation for fraction
of open M-channels is

$\frac{Mo(t)}{M}=1-e^{-\alpha(t-t_{oM})^{\theta_{M}}}$ (5)

where $t_{oM}$ is the time when M-channels start opening and, $\alpha$ and
$\theta_{M}$ are Avrami parameters. The Avrami equation is best known for
describing isothermal phase changing in solids and crystallization processes.
The particular value of the exponent $\theta=4$ is said to have contributions
from three dimensions of growth and one representing a constant nucleation
rate. The Avrami parameter $\alpha$ is typically very temperature dependent.
In particular, for the M-channel it varies exponentially from about 40 at
$1^{o}C$ to 50000 at $28.5^{o}C$. The modified Avrami (mAvrami) Eqs. (6) are
more meaningful for the propagated action potential:

$\frac{Mo(t)}{M}=1-e^{-\alpha_{M}[\mu_{M}(t-t_{oM})]^{\theta_{M}}}$ (6a)
$\frac{Ho(t)}{H}=1-e^{-\alpha_{H}[\mu_{H}(t_{cH}-t)]^{\theta_{H}}}$ (6b)
$\frac{No(t)}{N}=1-e^{-\alpha_{N}[\mu_{N}(t-t_{oN})]^{\theta_{N}}}$ (6c)
$\frac{PMo(t)}{PM}=1-e^{-\alpha_{PM}[\mu_{M}(t-t_{oPM})]^{\theta_{PM}}}$ (6d)

The incorporation of the temperature dependent time rate constants $\mu_{X}$
into Avrami equations yields temperature independent dimensionless constants
$\alpha_{X}$ for each ionic channel. The average values of mAvrami parameters
over a range of temperatures, $\alpha_{M}=0.0063,\alpha_{H}=0.0067$ and
$\alpha_{N}=0.0084$, are very close to the value of the fine structure
constant $\alpha^{*}\approx 0.0073$. The average values of mAvrami exponents
are $\theta_{M}=3.72,\theta_{H}=3.20$ and $\theta_{N}=3.09$.

Figure 5: Rising edge fraction of open M-channels and fraction of completed
M-phase lattice morphing. Sweep 170 at $4.5^{o}C$, and Sweep 695 at
$19.8^{o}C$. Parameters $\mu_{M},g_{M},Vo_{M},\delta_{M},\alpha_{M},to_{M}$,
and $\theta_{M}$ for different temperatures come from fitting the experimental
data in RB . Parameters $\alpha_{M},\theta_{M}$and $\delta_{M}$, are fairly
constant across different temperatures informing about sodium channel and
structure of the channels lattice and their interaction with the electric
field. $Po/P$ is interpreted as the completed fraction of the M-lattice
morphing process. For simplicity, we have dropped the subscript M from the
M-phase lattice morphing variables. a, d, Plots of $Mo(V)/M$ and $Po(V)/P$ vs.
Potential [mV]. The fraction of open channels $Mo(V)/M$ is fitted with the Eq.
1.4a. b, e, Plots of $Mo(t)/M$ and $Po(t)/P$ vs. Time[mSec]. The fraction of
open channels $Mo(V)/M$ is fitted with mAvrami Eq. 1.6a. c, Plots of
$\ln[-\ln(1-Mo(t)/M)]$ and $\ln[-\ln(1-Po(t)/P]$ vs. $\ln[\mu_{M}(t-to_{M})]$.
$\ln[-\ln[e^{-\alpha_{M}\left(\mu_{M}(t-to_{M})\right)^{\theta_{M}}}]]=3.74\ln[\mu_{M}(t-to_{M})]-5.1.$
However, other fits are possible, such as seeding the value of $\alpha_{M}$
with the value of the fine structure constant $\alpha^{*}$ = 0.007297352:
$\ln[-\ln[e^{-\alpha_{M}(\mu_{M}(t-to_{M}))^{\theta_{M}}}]]=3.6597\ln[\mu_{M}(t-to_{M})]-4.9204$
Note that: $-4.9204=\ln(\alpha_{M})=\ln(0.007296016)$.
$\ln[-\ln(1-Po/P)]=11.9\ln[\mu_{P}(t-to_{P})]-20.3$. The mAvrami exponent
value, $\theta_{P}=11.9$, points to a more complex process. The process
described by $Po/P$ comes from the morphing of the M-phase lattice and
$to_{P}$ is the time when morphing begins. f, Plots of $\ln[-\ln(1-Mo(t)/M)]$
and $\ln[-\ln(1-Po(t)/P)]$ vs. $\ln[\mu_{M}(t-to_{M})]$.

Time frame (laboratory) mAvrami fits require four parameters:
$\alpha_{X},\mu_{X},\theta_{X}$ and $t_{yX}$ of which $\mu_{X}$, is determined
from the derivative of the action potential trace. Our fits show that
$\theta_{M}\approx\delta_{M}$, $\theta_{H}\approx\delta_{H}$, and that
$\theta_{N}\approx\delta_{N}$ within few % points. The phase space fits for
fractions of open channels given by Eq. (4) coincide with the lab fits given
by Eq. (6) except in the region of very few channels open. It is obvious that
potential segments $|V-V_{yX}|$ in phase space correspond to time segments
$|(t-t_{yX}|$ in the lab frame. Should then one expect that
$\theta_{X}=\delta_{X}$ as the fitting results suggest?

Figure 5 displays the fits for the fraction of open sodium channels $Mo/M$ and
sodium s lattice polarization current caused by sodium channels lattice
morphing at two different temperatures. Fig. 5a and Fig. 5d display almost
identical $Mo/M$ and $Po/P$ curves. Increasing the temperature from $4.5^{o}C$
to $19.8^{o}C$ downshifts the curves along the potential axis by 8 mV.
($V_{p}-Vo_{M})=79.4mV$ at $4.5^{o}C$ and $(V_{p}-Vo_{M})$ = 81.0 mV at
$19.8^{o}C$. Fig. 5b and Fig. 5e display almost identical $Mo/M$ curves given
by mAvrami functions. The effect of the temperature is that all channels open
about three times faster at $19.8^{o}C$ than at $4.5^{o}C$. Comparing Fig. 5c
with Fig. 5f shows that the factor $\mu_{M}(t-to_{M})$ is practically
temperature independent and that both straight lines intercept the vertical
axis at - 5.1 which corresponds to $\alpha_{M}=0.0060$. The fraction of phase
transition completed for M-channels scales across the whole range of
temperatures. And so do fractions for the H-channels and N-channels. Fig. 5
also displays the the fraction of sodium lattice morphing completed as
function of the potential and time. Fig. 5c and Fig. 5f display
$\ln[-\ln(1-Po(t)/P)]$ vs. $\ln[\mu_{M}(t-to_{M})]$. The larger value of the
Avrami exponent points to a participation of a larger number of channels
contributing to the morphing process. The time rate for the morphing process
was assumed to be $\mu_{M}$.

Figure 3 displays a detail around the peak of the action potential. The linear
segment of the recovery capacitive segment is displayed showing that it
intersects the zero current axis at the same point as the rising edge linear
segments of $J_{M},$ $J_{m},$ and capacitive currents. However, the linear
segment of the current $J_{H}$ intersects the zero current axis at a
significantly lower potential. We shall later attribute this difference to the
polarization flip at the peak of the action potential.

Figure 3 shows that both, the rising edge and the recovery polarization
currents have quasi-linear segments. Therefore, they can be represented
approximately as follows:

$J_{PM}(V)\approx g_{PM}[Po(V)/P]_{M}(V-V_{PM})$ (7a) $J_{PH}(V)\approx
g_{PH}[Po(V)/P]_{H}(V-V_{PH})~{}~{}$ (7b)

During the rising edge of the action potential the polarization current is
first incoming and then outgoing. During recovery the polarization current is
only incoming and it only present in the region of less than 2 mV from the
peak of the action potential. The morphing of sodium’s lattice and the
concurrent polarization current change sodiums equilibrium potential from
$V_{M}$ to sodium’s effective equilibrium potential Meep(V) starting in the
negative resistance region to sodium’s effective equilibrium potential Heep(V)
and then to eqilibrium potential $V_{H}$.

The rising edge portion of the polarization current $J_{PM}(V)$ displays two
distinct segments. The outgoing current’s quasi-linear part intercepts the
current axis with the slope $g_{PM}\approx 58mS/cm^{2}$ and the value of the
intercept is $V_{PM}=106.6mV$. Note that $g_{PM}$ and $g_{M}$ are almost
identical. This segment of the polarization current traverses the $M$-channel
and the overall incoming ionic current is decreased. This implies the decrease
of sodium’s $M$-channel effective equilibrium potential (Meep). Since the
$M$-channel maximum conductance is reached at the peak of the action
potential, the increase of the overall ionic current is solely accounted for
by an increase of Meep. Thus, the $M$-channel total ionic current $J_{IM}(V)$
and Meep(V) can be written as follows:

$J_{IM}(V)=g_{M}[Mo(V)/M][V-Meep(V)]=J_{M}(V)+J_{PM}(V)=J_{I}(V)|_{Experiment}$
(8a) $Meep(V)=V-\frac{J_{I}(V)|_{Experiment}}{g_{M}[Mo(V)/M]}$ (8b) It follows
that in the limit of approaching the peak of the action potential $V_{p}$
$J_{IM}(V_{p})=g_{M}[V_{p}-Meep(V_{p})]=\frac{R}{2v^{2}R_{i}}\frac{d^{2}V}{dt^{2}}|_{V=V_{p}}<0\implies~{}V_{p}<Meep(V_{p})$
(8c)

The recovery polarization current is all incoming, its segment covers about
1.5 mV below the peak of the action potential and all the H channels are open
in this short regon, $Ho/H=1.$ The relation equivalent to Eq. (8a) is:

$J_{IH}(V)=-g_{H}[V-Heep(V)]=-g_{H}[V-V_{H})]+J_{PH}(V)=J_{I}(V)|_{Experiment}$
(9a) $Heep(V)=V+\frac{J_{I}(V)|_{Experiment}}{g_{H}}$ (9b) It follows that in
the limit of approaching the peak of the action potential $V_{p}$
$J_{IH}(V_{p})=-g_{H}[V_{p}-Heep(V_{p})]=\frac{R}{2v^{2}R_{i}}\frac{d^{2}V}{dt^{2}}|_{V=V_{p}}<0\implies~{}V_{p}>Heep(V_{p})$
(9c) and in the limits of reaching $V_{p}$ from recovery and rising edge
regions
$-g_{H}(V_{p})[V_{p}-Heep(V_{p})]=g_{M}[V_{p}-Meep(V_{p})]\approx-64\mu
A/cm^{2}~{}\text{ at }~{}4.5^{o}C~{}~{}$ (9d)
$-g_{H}(V_{p})[V_{p}-Heep(V_{p})]=g_{M}[V_{p}-Meep(V_{p})]\approx-442\mu
A/cm^{2}\text{at}~{}19.8^{o}C~{}~{}$ (9e)

Figure 6:  Schematic ferroelectric hysteresis. a, $(V_{M}-V_{H})$ [mV] vs.
$(V_{0}-V_{N})$ [mV]. b, Schematic ferroelectric hysteresis loop at
$4.5^{o}C,$ Polarization $=y(V_{M}-V_{H})$ [y][mV] vs. Electric field [x][mV].
$y$ is the temperature dependent proportionality constant. Time direction is
given by the arrow $\rightarrow$ . On the left hand side we have potassium s
effective equilibrium potential going from -14.2 mV to 0 mV. c,
$(V_{0}-V_{N})$ and $(V_{M}-V_{H})$[mV] vs. Temp.[${}^{o}C$]. See Appendix A
for a possible explanation for the gap between the two slopes. d, Schematic
ferroelectric hysteresis loop at $19.8^{o}C$, Polarization$=y(V_{M}-V_{H})$
[y][mV] vs. Electric field [x][mV]. $y$ is the temperature dependent
proportionality constant. Time direction is given by the arrow $\rightarrow$ .

Eq. (9d) implies $Meep(V_{p})\sim V_{p}+1mV$ and $Heep(V_{p})\sim V_{p}-10mV$,
and Eq. (9e) implies $Meep(V_{p})\sim V_{p}+2.6mV$, and $Heep(V_{p})\sim
V_{p}-16mV$.

Hippel, A. R. von. Hippel and Leuchtag Leuchtag1 have suggested that
ferroelectricity plays a role in biological excitability. A hypothesis that a
single sodium channel exhibits ferroelectric behavior has been advanced
Leuchtag2 based on measurements Palti of the axon capacitance in the
temperature region with membrane excitability. Figure 6b and Figure 6d display
the phase space trajectory of the hysteresis loop traversed by the action
potential in terms of sodium and potassium effective equilibrium potentials at
$4.5^{o}C$ and $19.8^{o}C$. Potassium channels are open at the resting
potential $V_{0}$, albeit with a maximum conductance $g_{0}$ one order of
magnitude smaller than $g_{N}$. Potassium channels close and sodium channels
start to open as potential increases to about 30-40 mV. It is hard to estimate
the extent of the overlap between the two currents because both are very small
in this region and potassium is outgoing while sodium is incoming.

Here we introduced the quantity $K$ which is the density of potassium channels
in the resting membrane and $Ko/K$ as the fraction of open potassium channels
at the foot of the action potential, although at $4.5^{o}C$ the conductance
$g_{0}$ is negligible. The polarization flip at the peak of the action
potential changes $g_{M}$ to $g_{H}$ while sodium channels are open as
observed in Fig. 3, and silently changes $g_{0}$ to $g_{N}$ while potassium
channels are closed. The reverse polarization flip changes $g_{N}$ to $g_{0}$
while potassium channels are open and silently changes $g_{H}$ to $g_{M}$
while sodium channels are closed. It remains to be seen if potassium s K-phase
can be fitted with similar functions as M, H and N phases. While the
polarization flip is ultimately the cause of the eep discontinuity, the
explicit mechanism connecting the two is not obvious. Note that $V_{H}$
decreases by 13 mV as temperature increases from 4.5 ${}^{o}C$ to
19.8${}^{o}C$ while $V_{M}$ decreases by less than 6 mV. Fig. 6a shows the
linear correlation between $(V_{M}-V_{H})$ and $(V_{0}-V_{N})$ over a range of
temperature supporting the argument that the reverse polarization flip,
completing the hysteresis loop occurs, albeit slowly, in the region between
$V_{N}$ and $V_{0}$ preventing excitation until the reverse polarization flip
is completed. Fig. 6c shows that the linear fits for $(V_{N}-V_{0})$ and
$(V_{M}-V_{H})$ vs. potential have the same slope and differ for about 1.5 mV.
This difference comes from one of two possible causes. The first is that the
values of $(V_{M}-V_{H})$ have been systematically underestimated
or$(V_{N}-V_{0})$ values overestimated and the second is that there is an
additional, temperature independent, process occurring along the hysteresis
loop that has not been accounted for.

Sodium’s M-channel maximum conductance $g_{M}$ and H-channel maximum
conductance $g_{H}$ are different and so are the respective time rates
$\mu_{M}$ and $\mu_{H}$. Sodium’s M-phase lattice and H-phase lattice have
different symmetry. There is a discontinuity of sodium’s effective equilibrium
potential at the peak of the action potential. These facts are a signature of
a second order phase transition Landau2 .

##  II Discussion and Conclusion

Nature has designed the giant squid axon to propagate a steady action
potential along its length. The axon exists to provide communication between
the external stimulus and squid’s physiological response. At the same time
Nature provides various ways to initiate the all or nothing pulse. At the
synapse, the release of acetylcholine by the action potential at the end of
the nerve initiates the signal at the connecting nerve. In the lab, injecting
current into the axon will generate an all or nothing action potential
depending on the stimulus’ magnitude. Hodgkin and Huxley used the detailed
voltage clamp experimental data for sodium and potassium currents in
conjunction with the cable equation to describe the initiation and propagation
of the action potential caused by a sufficient injection of charge into the
axon. This approach, while providing an empirical description for the
generation of the action potential did not provide deeper meaningful physical
insight into it. It also features a significant cancellation of opposing
sodium and potassium currents during action potential. The extensive overlap
of incoming sodium and outgoing potassium is a feature of the Hodgkin and
Huxley model resulting in decrease of velocity of propagation and waste of
energy [19]. Increasing the delay of potassium conductance onset and delaying
sodium channel inactivation reduces the discrepancy between Hodgkin and
Huxley’s predicted and observed velocity of propagation [19]. In addition, the
recovery portion of the H-H action potential is not quite right when compared
with the experiment. Furthermore, H-H model doesn’t say anything about the
temperature dependence and thus precludes applying thermodynamics to the study
of the action potential and by extension it has nothing to say about possible
phase changes involved. Voltage clamp experiments can’t possibly detect second
order phase change, even if present in a normally functioning axon.

The excitation process involves time and electrical potential evolution of the
non-linear all or nothing event. However, once initiated and propagating, the
action potential has no memory of how it was generated. Steady state
propagation is a much simpler physical phenomenon to analyze than the all or
nothing particular way of excitation leading to propagation.

In the present work we have analyzed the simplest possible giant squid axon
experimental data that measures the steady propagation of the action potential
at two separate points at a given temperature. The time elapsed between the
signals at the two points provides the values of the velocity of propagation.
All other results stem from the shape of the action potential in space along
the axon and from the time evolution of the action potential when observed at
a fixed point along the axon in conjunction with the charge conserving cable
equation.

Steady state propagating action potential allows the display of all three
currents from the cable equation \- capacitive, ionic and membrane - as
functions of the potential. This display clearly exhibits three linear
portions of the total ionic current, potassium’s $J_{0}$ at the foot of the
action potential, sodium’s $J_{H}$ at the beginning of the recovery and
potassium’s $J_{N}$ ending in the undershoot of the action potential. The
rising edge of the propagating action potential carries its own ongoing
stimulus of advancing capacitive charge accompanied by a very small outgoing
linear current followed by opening of sodium M-channels and by a correlated
non linear incoming current. The opening of M channels, i.e. the progress of
the M-phase, is driven by the increase of the potential across the membrane
caused by the incoming sodium current. The outgoing linear current at the foot
of the action potential, negligible at low temperature, increases with
temperature up to about 30 mV and then decreases crossing the current axis at
about 40 mV. The total ionic current in the recovery region, with two linear
segments, was parsed into sum of sodium’s H-channel and potassium’s N-channel
currents. The two linear portions of the total current correspond respectively
to all sodium and potassium channels open. The small polarization current
given by the difference between experimental ionic current and the linear fit
over few millivolts following the peak of the action potential is interpreted
as a result of sodium’s H-cannel lattice conformational change or lattice
morphing. The analysis of the recovery region, hereto the less interesting
region, offers the key to understand that the incoming ionic current in the
rising edge region is the sum of sodium’s M-channel current and polarization
current in the negative resistance segment.

We have presented two different fittings for fractions of open channels. The
phase space fitting by Equations 1.3 with three parameters, two of which are
physically observable potentials and one dimensionless exponent exhibit
scaling across a wide range of temperatures. The first order phase transitions
of M-channel, H-channel and N-channel lattices scale across a wide range of
temperature.

The morphing of the sodium channel lattice and the accompanying polarization
current, proceeding in the negative resistance region, are manifested by the
change in the effective ionic equilibrium potential. At the peak of the action
potential the sodium channel maximum conductance changes abruptly from $g_{M}$
to $g_{H}$ with $g_{H}$ about one order of magnitude smaller than $g_{M}$ and
the effective equilibrium potential change is completed by an abrupt decrease
from $V_{M}$ to $V_{H}$ while keeping the ionic current continuous. The
maximum conductance is usually assumed to be the product of channel density
and the conductivity of a single channel, i.e. $g_{X}$ = $Xg^{*}_{X}$. At this
point it is not clear if the abrupt change of $g_{M}$ to $g_{H}$ is caused by
the change in M or change in $Mg^{*}_{M}$ or change in both. The abrupt
decrease in sodium’s equilibrium potential is caused by the polarization flip
put in evidence by the polarization current discontinuity. The sodium’s
M-phase and H-phase lattices have different symmetry and different structure.
The transition from M-phase to H-phase is a second-order or rather continuous
phase transition and it is interpreted as a ferroelectric polarization switch.
The fact that the morphing of sodium’s channel lattice with symmetry change is
associated with the polarization flip is consistent with the view that in
ferroelectric phenomena “The mechanism of switching is understood to take
place on scales longer than the unit-cell scale [20].”

To close the ferroelectric hysteresis loop the membrane structure must undergo
an equivalent switch reversal before it can be excited again from the resting
potential. Fig. 6a shows a temperature dependent correlation between the
increase of $(V_{0}-V_{N})$ and the decrease of $(V_{M}-V_{H}).$ Polarization
flip reversal must be accomplished as potential changes from potassium
reversal potential $V_{N}$ to resting potential $V=0$ while sodium channels
are closed and while the potassium-sodium transport mechanism is operating.

The fitting of fractions of open channels by the Avrami equation, originally
derived to describe the phase changing isothermal crystallization of metals,
and by the mAvrami Equations 1.6, introduced here by incorporating explicitly
temperature dependent time rates, yield temperature independent dimensionless
constants associated with ionic channels. These constant are very close in
value to the fine structure constant. The fraction of open channels is
described by temperature dependent time rate, elapsed time, a dimensionless
Avrami exponent almost identical to the dimensionless exponent obtained by the
phase space fit and by a dimensionless temperature independent constant very
close in value to fine structure constant. As a matter of fact, we have
obtained fits by seeding the values of the $\alpha_{X}$ constants with the
fine structure constant $\alpha^{*}$. The time evolution of fractions of open
channels is interpreted as time evolution of isothermal sodium’s M and H
phases and potassium’s N-phase driven by changing potential across the
membrane.

The fine structure constant $\alpha^{\star}$ is associated with quantum
electrodynamics rather than materials science. However, unexpectedly, it has
been shown that the opacity of suspended graphene22 is defined solely by it,
with authors saying: ”It is remarkable that the fine structure constant can so
directly be assessed practically by the naked eye.” In the present work,
unexpectedly, we were able to fit the fractions of open channels by seeding
the value of the fine structure constant $\alpha^{\star}$ for the
dimensionless and temperature independent constants $\alpha_{M},\alpha_{H}$
and $\alpha_{N}$. The final clarification rests with improved experimental
data, and possibly with independent determination of parameters $\theta_{X}$
and $\delta_{X}$, and their relationship. It would also help to eliminate the
discrepancies, a result of the present fitting, between $V_{toM},V_{toN}$ and
$V_{tcH}$ and $V_{oM},V_{oN}$ and $V_{cH}$ respectively.

Which one is Nature’s choice, one temperature independent dimensionless
constant revealed by the two dimensional structure of the membrane or a
different, possibly temperature independent constant for each channel? To be
certain which scenario is the right one we need more information, and improved
treatment of the data. In particular, fitting of sodium and potassium currents
in recovery region should be done enforcing current conservation taking into
account both currents simultaneously. Eventually one should also take into
account the observed granularity of the sodium H-channel current $J_{H}$.

However, the fact that that the $M-channel$, $H-channel$, and $N-channel$ have
very different maximum conductance and time rates across all temperature, and
the fact that $M-channel$ Avrami exponent is very different from $H-channel$
and $N-channel$ exponents while all three mAvrami constants $\alpha_{X}$ are
very close in value to the fine structure constant $\alpha^{*}$ strongly
supports the argument that all mAvrami constants $\alpha_{X}$ are in fact the
fine structure constant $\alpha^{*}$.

It is expected that presented results will provide a framework for further
analysis of biological excitability, ionic channels lattice structure,
thermodynamic phase changing behavior, and the physical basis for memory
formation and retrieval.

##  Appendix A  Fitting Procedure

Rosenthal’s data [3] provides the action potential $V$ at two points along the
axon at different temperatures. The action potential data was analyzed at the
point furthest away from the stimulus. Rosenthal measured the resistivity at
$18.5^{o}C$ and determined that his measurement was consistent with the
generally used expression for resistivity, also used in this work,

$R_{i}=51.05\times 1.35^{[-\frac{(T-6.3^{o}C)}{10}]}[Ohm-cm].$ (10)

Rosenthal’s experimental data of the action potential at different
temperatures are discrete values taken at fixed time intervals. The amount of
noise in the data depends on its rate of change and on the temperature. To
improve fitting wee have used Gauss smoothing while trying to minimize the
loss of accuracy. At lower temperatures there is more noise and there are many
more data points than at high temperature. Larger Gauss sigma smoothing values
are required at low temperature than what is required at higher temperatures.
The recovery region, being noisier, required more smoothing than the rising
edge at a given temperature.

We systematically fitted the data for fractions of open channels two different
ways using Excel’s Solver software. First we fitted the phase space Eq. 1.4.
Then we fitted the mAvrami Eq. 1.6 by seeding the exponent $\theta_{X}$ with
the value of the corresponding $\delta_{X}$. The average results for 10
M-channel sweeps are: $\delta_{M}=3.79$ , $\theta_{M}=3.72$ ,
$\alpha_{M}=0.0066$. The average results for 6 H-channel and N-channel sweeps
are: $\delta_{H}=3.10$ , $\theta_{H}=3.20$ , $\alpha_{H}=0.0067$;
$\delta_{N}=3.12$ , $\theta_{N}=3.09$ , $\alpha_{N}=0.0084$

The previous fits of mAvrami Eq. 1.4 yielded values of dimensionless constants
$\alpha_{X}$ hovering around the value of the fine-structure constant,
$\alpha^{*}=0.007329...$, so we used this value to seed the fit and kept it
fixed until the last Solver optimization when it was allowed to vary. The
average results for 9 M-channel sweeps are: $\theta_{M}=3.74$ ,
$\alpha_{M}=0.007361$. The average results for 6 H-channel and N-channel
sweeps are: $\theta_{H}=3.11$ , $\alpha_{H}=0.007294$; $\theta_{N}=3.13$ ,
$\alpha_{N}=0.007262$

The start of the polarization current was chosen to coincide with the start of
the negative resistance region or equivalently, the current $J_{M}$ was fitted
with data points up to the start of the negative resistance region. This
choice resulted in the effective value for sodium’s reversal potential $V_{M}$
.

All fittings are sensitive to the number of points included, the sensitivity
increasing with temperature since fewer and fewer points were recorded. The
fits for the H-channel vary with most latitude, as they are one step removed
from N channel data fits. In addition, $J_{H}$, the outgoing sodium current
displays a granularity of at least three distinct regions for sweeps at lower
temperatures. In particular there is a prominent bump just above 55 mV (See
Fig. 1b and Fig. 1d). Although each fit individually may present some
uncertainty and latitude, the cumulative picture gathered over several
temperatures presents a coherent and self-consistent scenario.

Polarization current, being much smaller than M-channel sodium current,
displays fits with larger uncertainty than fits for the M-channel. Given the
scarcity of points, the values for $g_{P}$ are difficult to read from figures
at higher temperatures other than $4.5^{o}C$. Given the fact that $g_{P}$ is
almost equal to $g_{M}$ at $4.5^{o}C$ we have assume that the equality
$g_{P}=g_{M}$ holds at all temperatures. In other words, lattice morphing
doesn’t affect the maximum conductance until the peak of the action potential
where $g_{M}$ switches to $g_{H}$.

###### Acknowledgements.

NJ wants to thank Prof. J. J. C. Rosenthal making the present work possible by
giving him in 1999 the electronic copy of Wa05097a.dat file containing the
experimental data collected in Reference [3]. He is greatly indebted to him
and to Prof. Francisco Bezanilla for sharing their data. NJ wants to thank Dr.
Bogdan Mihaila for recommending further study of the role and the meaning of
temperature independent dimensionless constants $\alpha_{X}$. NJ wants to
thank Dr. Andrew Perlman for asking a question about Hodgkin-Huxley equations
close to 50 years ago. NJ and FC want to thank the Santa Fe Institute for its
hospitality during the completion of this paper.

## References

  * (1) Hodgkin, A. L., Huxley, A. F. and Katz, B. 1952. Measurement of current-voltage relations in the membrane of the giant axon of Loligo. J. Physiol. (London)
  * (2) Hodgkin, A. L., and A. F. Huxley. 1952. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (Lond.). 117: 500-544.
  * (3) Rosenthal, J. J. C. and Francisco Bezanilla. Seasonal variation in conduction velocity of action potentials in squid giant axons. 2000. Biol. Bull. 199: 135-143.
  * (4) Landau L. D. and Lifshitz E. M., Statistical Physics, Third Edition, Pergamon Press, 1980. pg. 2.
  * (5) See https://en.wikipedia.org/wiki/Phase space.
  * (6) Scott, A. C., 1975. The electrophysics of a nerve fiber. Rev. Mod. Phys. 47: 487-533.
  * (7) Avrami, M. 1939. Kinetics of Phase Change. I. General Theory. Journal of Chemical Physics. 7 (12): 1103-1112.
  * (8) Avrami, M. 1940. Kinetics of Phase Change. II. Transformation-Time Relations for Random Distribution of Nuclei. Journal of Chemical Physics. 8 (2): 212-224.
  * (9) Avrami, M. 1941. Kinetics of Phase Change. III. Granulation, Phase Change, and Microstructure. Journal of Chemical Physics. 9 (2): 177-184.
  * (10) Hippel, A. R. von. 1970. Do we really understand ferroelectricity? Proceedings, Second Int. Meeting on Ferroelectricity 1969. J. Phys. Soc. Japan 28 (supl), 1.
  * (11) Leuchtag, H. R. 1987. Indications of the Existence of Ferroelectric Units in Excitable-membrane Channels. JJ. theor. Biol. 127: 321-340.
  * (12) Leuchtag, H. R. 1995. Fit of the dielectric anomaly of squid axon membrane near heat-block temperature to the ferroelectric Curie-Weiss law. Biophysical Chemistry. 53: 197-205.
  * (13) Palti, Y. and W. J. Adelman Jr.. (1969). Measurements of Axonal Membrane Conductances and Capacity by Means of a Varying Potential Control Voltage Clamp. J. Membr. Biol. 1: 431-458.
  * (14) Landau L. D. and Lifshitz E. M., Statistical Physics, Third Edition, Pergamon Press, 1980. pg. 445.
  * (15) Sangrey, T. D., Friesen, W. O. and Levy, W. B. 2004. Analysis of the Optimal Channel Density of the Squid Axon Using a Reparameterized Hodgkin-Huxley Model. J. Neurophysiol. 91: 2541-2550.
  * (16) K. Rabe, C. H. Ahn, J.-M. Triscone (Eds.). Physics of Ferroelectrics: A Modern Perspective, Topics Appl. Physics. 105: 1 30 (2007).
  * (17) See https://en.wikipedia.org/wiki/Fine-structure constant.
  * (18) R. R. Nair, et al., 2008. Fine Structure Constant Defines Visual Transparency of Graphene. Science. Vol. 320, Issue 5881, pp. 1308.


aainstitutetext: Department of Physics, Brown University,
182 Hope Street, Providence, RI 02912, U.S.A.bbinstitutetext: Brown
Theoretical Physics Center, Brown University,
340 Brook Street, Providence, RI 02912, U.S.A.

# Landau Singularities of the 7-Point Ziggurat I

Luke Lippstreu a,b Marcus Spradlin a and Anastasia Volovich
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The (first-type Landau) singularities of any $n$-point massless planar Feynman
integral, at any loop order, are encoded in a certain “ziggurat graph”. We
compute the leading Landau singularities of the 7-point ziggurat graph and
find perfect agreement with a subset of the “heptagon symbol alphabet” that
has appeared in the context of planar $\mathcal{N}=4$ super-Yang-Mills theory.
The remaining heptagon symbol letters are found in subleading Landau
singularities, which we address in a companion paper.

##  1 Introduction

A key aspect of the S-matrix program is the expectation that scattering
amplitudes should be largely determinable from a thorough understanding of
their analytic structure. In recent years planar $\mathcal{N}=4$ super-Yang-
Mills (SYM) theory has served as an exemplar of this approach Arkani-
Hamed:2022rwr . In particular, the assumption that the singularities of all
six- and seven-point amplitudes are encoded in symbol letters that are cluster
variables Goncharov:2010jf ; Golden:2013xva , together with physical input
from near-collinear and multi-Regge kinematics, has allowed them to be
determined to high loop order (see Caron-Huot:2020bkp  for a review).

While it is known that higher-point amplitudes in SYM theory (and certainly
those in other, less “simple” field theories) can have significantly more
complicated analytic structure, a general criterion for determining the
locations of singularities of Feynman integrals was formulated over 60 years
ago by Landau Landau:1959fi . Recently it has proven fruitful to explore
general implications of the Landau equations and other classic work on
discontinuities of amplitudes in the context of the modern amplitudes program;
see for example Dennen:2015bet ; Dennen:2016mdk ; Prlina:2017azl ;
Prlina:2017tvx ; Bourjaily:2020wvq ; Mizera:2021fap ; Mizera:2021icv ;
Hannesdottir:2021kpd ; Hannesdottir:2022xki .

In Prlina:2018ukf  it was shown that the Landau singularities of an arbitrary
$n$-point massless planar Feynman integral (at any loop order) are a subset of
those of a certain $n$-point ziggurat graph (see Fig. 1). Knowing the
locations of an amplitude’s singularities is closely related to, but not quite
the same as, knowing its symbol alphabet. Knowledge of the former only
provides information about where symbol letters _vanish_ (or where the letters
themselves have algebraic branch points, as a function of the kinematic data),
not necessarily what the symbol letters _are_ (although in many cases it gives
enough information to make a natural guess). Indeed, the results of
Prlina:2018ukf  do not directly involve the notion of symbol at all; they
capture the singularities of arbitrary integrals not necessarily limited to
those that can be expressed in terms of any particular special class of
functions (such as multiple polylogarithms).

Also in Prlina:2018ukf  the singularities of the 6-point ziggurat were
determined and found to comprise precisely those encoded in the “hexagon
symbol alphabet” which is expected to suffice for expressing all 6-point
amplitudes in SYM theory. Consequently, no massless planar 6-point Feynman
integral, in any field theory, can have (first-type) Landau singularities
anywhere other than on the vanishing locus of the hexagon symbol alphabet. In
this paper we initiate a study of the singularities of the 7-point ziggurat
and its connection to the “heptagon symbol alphabet” that, according to all
evidence available to date Drummond:2014ffa ; Dixon:2016nkn ; Caron-
Huot:2020bkp , seems to suffice for expressing all 7-point amplitudes in SYM
theory. However we emphasize that just like Prlina:2018ukf , our present
analysis is completely general and is in no way tied to SYM theory.

In Sec. 2 we review the Landau equations, and in particular their formulation
in momentum twistor space where on-shell conditions can be solved via simple
geometric considerations in many nontrivial examples. As illustrative examples
we discuss in detail how to determine the singularities of the 4- and 5-point
ziggurat graphs before reviewing the 6-point case from Prlina:2018ukf . In
Sec. 3 we find the leading singularities of (a graph equivalent to) the
7-point ziggurat, which are all consistent with the heptagon symbol alphabet.
The graph has numerous subgraphs that need to be analyzed separately, which we
defer to a companion paper.

(a)

(b)

(c)

(d)

Figure 1: The four-, five-, six- and seven-point ziggurat graphs. A massless
external leg attaches to each white vertex. The pattern continues by adding
another column to the right each time $n$ increases by 2.

$7$$4$ (a)

$\mathcal{L}_{1}$$\mathcal{L}_{2}$$\mathcal{L}_{3}$$\mathcal{L}_{4}$$1$$2$$3$$4$$5$$6$$7$$y_{2}$$y_{3}$$y_{4}$$y_{5}$$y_{6}$$y_{7}$$y_{1}$
(b)

Figure 2: Six- and seven-point graphs that are “Landau equivalent” to the
corresponding graphs shown in Fig. 1(c) and (d). The labeling on (b) will be
used in Sec. 3. The arrows indicate the four propagators carrying loop
momentum $\ell_{a}$ (adjacent to the corresponding $\mathcal{L}_{a}$).

##  2 Warm-Up

In Landau:1959fi  Landau showed that a Feynman integral can have singularities
only if certain polynomial equations are satisfied. In this section we review
(largely following Prlina:2017azl ) the formulation of the Landau equations in
momentum twistor space. A significant advantage of working in momentum twistor
space is that some of the Landau equations (the on-shell conditions) can be
solved analytically, in many non-trivial cases (such as the showcase of this
paper: Fig. 2(b)), via simple geometric considerations (see for example
Arkani-Hamed:2010pyv  for some simpler examples). We also review the
equivalence of solutions to the Landau equations under certain graphical moves
and the role of ziggurat graphs for classifying solutions of the Landau
equations in massless planar theories.

###  2.1 Landau equations in momentum twistor space

In momentum twistor space each external leg of a planar $n$-point graph is
associated to a point $Z_{i}$ in $\mathbb{P}^{3}$ (with an implied cyclic
ordering of $1,\ldots,n$) and the external face bounded by legs $Z_{i}$,
$Z_{i+1}$ is associated to the line containing those points, which we denote
by $(Z_{i},Z_{i+1})$ or simply $(i\,i{+}1)$. Momentum twistors are related to
the standard Mandelstam variables by

$\displaystyle(p_{i}+p_{i+1}+\cdots+p_{j})^{2}=\frac{\braket{i{-}1\,i\,j\,j{+}1}}{\braket{i{-}1\,i\,\mathcal{I}}\braket{j{-}1\,j\,\mathcal{I}}}$
(1)

where $\braket{ABCD}$ denotes the $4\times 4$ determinant of the homogeneous
coordinates of four points in $\mathbb{P}^{3}$ and
$\mathcal{I}=(\mathcal{I}_{1},\mathcal{I}_{2})$ represents the “line at
infinity”, the choice of which necessarily breaks dual conformal invariance.
To put it another way, any quantities involving $\mathcal{I}$ must cancel out
in SYM theory since there is no invariant notion of “infinity” in momentum
space.

Each internal face of a graph is associated to a line $\mathcal{L}_{\ell}$
representing the loop integration degrees of freedom. Often we parameterize
$\mathcal{L}_{\ell}$ as $(A_{\ell},B_{\ell})$ for a choice of two distinct
points on the line. We label a propagator bounded by two faces $(A,B)$ and
$(C,D)$ by $\braket{ABCD}$. While the numerical value of this quantity is
ambiguous, depending on the choices of representative points $(A,B)$ and
$(C,D)$ on the two lines, it vanishes if and only if the two lines intersect,
which is all that we will be interested in: this corresponds to the propagator
going on-shell.

Consider an $L$-loop planar graph with $p$ propagators labeled
$f_{1},\ldots,f_{p}$. The Landau equations come in two types. First we have
the on-shell conditions

$\displaystyle f_{J}=0\qquad J=1,2,\ldots,p\,,$ (2)

(a solution of which we naturally call a cut), and next we have the Kirchhoff
conditions

$\displaystyle\sum_{J=1}^{p}\alpha_{J}\frac{\partial f_{J}}{\partial
c_{A}}=0\qquad A=1,\ldots,4L\,,$ (3)

where the $\alpha_{J}$’s are Feynman parameters and the $c_{A}$ stand for $4L$
independent variables in terms of which we choose to parameterize the $L$ loop
momenta. For the sake of definiteness we can choose for example

$\displaystyle\mathcal{L}_{1}=\begin{pmatrix}1&0&c_{1}&c_{2}\\\
0&1&c_{3}&c_{4}\end{pmatrix},\qquad\mathcal{L}_{2}=\begin{pmatrix}1&0&c_{5}&c_{6}\\\
0&1&c_{7}&c_{8}\end{pmatrix},\qquad\text{etc.}$ (4)

We always exclude trivial solutions to (3) having all $\alpha_{J}=0$. The
singularities encoded in the Landau equations (2) and (3) are sometimes called
_first-type_ singularities, in contrast to _second-type_ singularities which
arise from pinch singularities at infinite loop momentum ELOP ; Fairlie:1962-1
; Fairlie:1962-2 . The latter commonly arise from triangle subdiagrams; see
for example the discussion in Sec. V of Dennen:2015bet . Henceforth whenever
we talk about Landau singularities, we only mean first-type singularities.

To find the Landau singularities of any given graph it is necessary to analyze
the Landau equations for the full graph itself as well as for any subgraph
that can be obtained by contracting any subset of its propagators. Because
this is tantamount to setting various $\alpha$’s to zero, the on-shell
conditions (2) are traditionally written as $\alpha_{J}f_{J}=0$ to emphasize
that one can consider the two cases $\alpha_{J}=0$ or $f_{J}=0$ separately.
However for massless diagrams, it is common (as we will see below) to have
solutions with both $\alpha_{J}=0$ and $f_{J}=0$ for one or more $J$’s. For
bookkeeping purposes we find it more convenient to demand that all propagators
of a given graph must be put on-shell, and then remember to analyze all
possible subgraphs separately. Note that (other than excluding trivial
solutions) we are never interested in the values of the $\alpha$’s (or $c$’s),
only in the binary question: what constraints must the external $Z_{i}$
satisfy in order for nontrivial solutions of the Landau equations to exist?
This is the locus of (potential) singularities for any Feynman integral
involving the propagators indicated in the graph under consideration.
(Specific choices of numerator factors in a Feynman integral may conspire to
cancel some of the potential singularities; the Landau equations are
manifestly blind to numerators and only care about the propagator structure,
which is encoded in the graph topology.)

###  2.2 Ziggurat graphs

In Dennen:2016mdk ; Prlina:2018ukf  it was shown that when all propagators are
_massless_ , the locus of solutions to the Landau equations associated to a
graph is invariant under the graphical moves familiar from electrical circuit
theory: series reduction, parallel reduction, and most importantly the wye-
delta transform.

The problem of classifying all _planar_ graphs under these graphical moves has
been solved in terms of what were called “ziggurat” graphs in Prlina:2018ukf .
Specifically, any planar $n$-point graph is equivalent to the $n$-point
ziggurat graph or a minor thereof. A minor of a graph is any graph that can be
obtained by any combination of edge contractions or edge deletions. Therefore,
the possible locations of any Landau singularity of any $n$-point massless
planar Feynman integral are the same as those of the $n$-point ziggurat graph
and its minors. Singularities associated to a given initial graph are called
its _leading_ singularities, while those associated to a minor are called
subleading singularities.

###  2.3 Four-Point

In order to demonstrate the procedure of solving Landau equations in momentum
twistor space we begin with the massless box graph shown in Fig. 1(a). The on-
shell conditions are

$\displaystyle\braket{\mathcal{L}12}=\braket{\mathcal{L}23}=\braket{\mathcal{L}34}=\braket{\mathcal{L}41}=0\,,$
(5)

where we use the shorthand
$\braket{\mathcal{L}ij}=\braket{\mathcal{L}Z_{i}Z_{j}}$. These admit two
distinct solutions

$\displaystyle\mathcal{L}=(Z_{1},Z_{3})\qquad\text{or}\qquad\mathcal{L}=(Z_{2},Z_{4})\,.$
(6)

The Kirchhoff conditions take the form of a $4\times 4$ matrix multiplying
$(\alpha_{1}\,\alpha_{2}\,\alpha_{3}\,\alpha_{4})$ to give zero. Nontrivial
solutions exist only when the determinant of this matrix, which evaluates to
$\braket{1234}^{2}$ on either of the two on-shell solutions, vanishes. This
agrees with the usual momentum space analysis which reveals that (in terms of
$s=(p_{1}+p_{2})^{2}$, $t=(p_{2}+p_{3})^{2}$) the massless box integral has
leading Landau singularities only when

$\displaystyle
s\,t=\frac{\braket{1234}^{2}}{\braket{12\mathcal{I}}\braket{23\mathcal{I}}\braket{34\mathcal{I}}\braket{41\mathcal{I}}}=0\,.$
(7)

Interestingly we don’t see any sign of IR divergences in the leading Landau
singularity. However if we contract (for example) the fourth edge, the three
remaining on-shell conditions then admit two one-parameter families of
solutions

$\displaystyle\mathcal{L}=(Z_{2},\alpha
Z_{3}+(1-\alpha)Z_{4})\qquad\text{or}\qquad\mathcal{L}=(Z_{3},\alpha
Z_{1}+(1-\alpha)Z_{2})\,.$ (8)

The Kirchhoff conditions are no longer equivalent to a vanishing determinant
since they are non-linear in the remaining variables
($\alpha_{1},\alpha_{2},\alpha_{3},\alpha$). Nevertheless it is easy to check
that the Landau equations admit the nontrivial solution
$\mathcal{L}=(Z_{2},Z_{3})$, $\alpha_{1}=\alpha_{3}=0$ for arbitrary external
kinematics $Z_{i}$. We interpret solutions that exist for all kinematics as
signalling the presence of IR singularities, arising from the soft/collinear
region of loop momentum space. Going forward we are interested in classifying
solutions of the Landau equations that exist only on codimension-one surfaces
in the space of external kinematics, since these determine the locus of
(potential) branch points (“branch surfaces”, really) of an integral. (See
Bourjaily:2022vti  for an interesting recent discussion of higher codimension
singularities.)

Altogether, after checking the Landau equations for all (triangle, bubble, or
tadpole) subdiagrams one can obtain from the box by any combination of edge
contractions, and discarding all solutions corresponding to IR singularities,
one finds no additional singularities beyond the one at $\braket{1234}=0$
present already in the box’s leading singularity.

###  2.4 Five-Point

We label the external edges of the five-point ziggurat graph shown in Fig 1(b)
with $Z_{1}$ on the lower right corner, increasing in clockwise order, and we
label the lower (upper) loop with $\mathcal{L}_{1}$ ($\mathcal{L}_{2}$)
respectively. It is well-known that the seven on-shell conditions (i.e., the
double box heptacut) admit six distinct one-parameter families of solutions
(see for example Caron-Huot:2012awx  for a nice discussion):

$\displaystyle\mathcal{L}_{1}$ $\displaystyle=(Z_{1},\alpha
Z_{2}+(1{-}\alpha)Z_{3})$ $\displaystyle\mathcal{L}_{1}=(Z_{1},\alpha
Z_{2}+(1{-}\alpha)Z_{3})$ $\displaystyle\mathcal{L}_{2}$
$\displaystyle=(Z_{1},Z_{4})\qquad$
$\displaystyle\mathcal{L}_{2}=(Z_{5},(\alpha\braket{1245}+(1{-}\alpha)\braket{1345})Z_{3}-\alpha\braket{1235}Z_{4})$
$\displaystyle\mathcal{L}_{1}$
$\displaystyle=(Z_{1},\braket{1345}Z_{2}-\braket{1245}Z_{3})\qquad$
$\displaystyle\mathcal{L}_{1}=(Z_{2},\alpha Z_{1}+(1{-}\alpha)Z_{5})$
$\displaystyle\mathcal{L}_{2}$ $\displaystyle=(Z_{4},\alpha
Z_{1}+(1{-}\alpha)Z_{5})$
$\displaystyle\mathcal{L}_{2}=(Z_{5},\braket{1245}Z_{3}-\braket{1235}Z_{4})$
$\displaystyle\mathcal{L}_{1}$ $\displaystyle=(Z_{2},Z_{5})$
$\displaystyle\mathcal{L}_{1}=(Z_{2},\alpha Z_{1}+(1{-}\alpha)Z_{5})$
$\displaystyle\mathcal{L}_{2}$ $\displaystyle=(Z_{5},\alpha
Z_{3}+(1{-}\alpha)Z_{4})$ $\displaystyle\mathcal{L}_{2}=(Z_{4},\alpha
Z_{1}+(1{-}\alpha)Z_{5})$

Plugging (for example) the first on-shell solution into the Kirchhoff
conditions gives eight equations in eight variables (the seven Feynman
parameters $\alpha_{i}$ and the on-shell parameter $\alpha$). There are seven
non-trivial solutions which are at most codimension one in the external
kinematics. One of these solutions has $\alpha=1$ (so that
$\mathcal{L}_{1}=(Z_{1},Z_{2})$) and all Feynman parameters $\alpha_{i}$
vanishing except for the one associated to the propagator
$\braket{\mathcal{L}_{1}12}$. This solution exists for all external kinematics
and we interpret as an IR singularity, as discussed in the previous section.
In addition to such uninteresting solutions, there are other solutions that
only exist when $\braket{1345}\braket{1245}\braket{1235}=0$.

After repeating this analysis for the other on-shell solutions, and
considering also all cyclic relabelings of the ziggurat graph, one finds that
leading Landau singularities of five-point integrals can exist when
$\braket{i\,i{+}1\,i{+}2\,i{+}3}=0$ for some $i$. In momentum space this
corresponds to $(p_{i}+p_{i+1})^{2}=s_{i,i+1}=0$. As in the four-point case, a
thorough analysis of the Landau equations for every minor of the ziggurat
graph Fig. 1(b) reveals no additional singularities beyond those of the form
$\braket{i\,i{+}1\,i{+}2\,i{+}3}$ already present in the ziggurat’s leading
singularities. We conclude that no massless planar five-point Feynman integral
can have Landau singularities anywhere other than on the locus where
$\prod_{i}s_{i,i+1}=0$.

Let us pause here to emphasize that there is no tension between this result
and the fact that massless planar five-point Feynman integrals with more
complicated singularities are certainly known. For example, the two-loop
master integrals relevant to five-point functions in massless QCD have a
30-letter symbol alphabet Gehrmann:2015bfy  that indicates the presence of
branch point singularities at $s_{i,i+1}+s_{i+1,i+2}=0$ (and other more
complicated functions of the Mandelstam variables). This is not a
contradiction because our claim, as mentioned above, only applies to the so-
called first-type Landau singularities, which can always be expressed in
momentum twistor space without reference to a choice of infinity twistor
$\mathcal{I}$. The fact that the locus $s_{i,i+1}+s_{i+1,i+2}=0$ cannot be
expressed without introducing $\mathcal{I}$ is a giveaway that these more
complicated letters indicate the presence of second-type singularities.

###  2.5 Six-Point

The six-point case was analyzed in Prlina:2018ukf . It is qualitatively more
similar to the four-point case than to the five-point case. The problem can be
simplified by using a sequence of graphical moves to transform the six-point
ziggurat graph shown in Fig. 1(c) into the equivalent graph in Fig. 2(a). The
latter has the advantage of having precisely four times as many propagators as
loops. This means that like in the four-point case, the solutions to the on-
shell conditions are discrete (here there are 16 instead of 2) and the
Kirchhoff conditions can be expressed as a determinant (here it is $12\times
12$ instead of $4\times 4$). Also like in the four-point case, the analysis of
the leading singularity is not clouded by the need to isolate and discard
solutions corresponding to IR singularities.

When evaluated on any one of the on-shell solutions, the Kirchhoff determinant
factors into a product of various four-brackets $\braket{i\,j\,k\,l}$. By
scanning over all 16 solutions, and considering all independent cyclic images
of the graph, one encounters all $\binom{6}{4}=15$ distinct four-brackets,
leading to the conclusion that all (first-type) Landau singularities of any
massless planar six-point Feynman integral lie on the locus where a four-
bracket vanishes.

This bolsters the expectation—consistent with all results available to date,
which now extends to seven loop order Caron-Huot:2019vjl —that the 15 four-
brackets constitute the symbol alphabet for all six-point amplitudes in SYM
theory, to any loop order. However, we pause again to emphasize that this
analysis cannot be considered to provide a proof of this fact, since the
Landau equations can only tell us where symbol letters _vanish_ , not their
exact functional form away from the vanishing locus.

##  3 The Seven-Point Ziggurat

In this section we outline the calculation of the (leading) Landau
singularities of the seven-point ziggurat graph shown in Fig. 1(d). We will
see that this analysis is computationally similar to the five-point
calculation reviewed in Sec. 2.4. Our first step is to reduce the six-loop
ziggurat graph to a simpler four-loop graph by the sequence of graphical moves
shown in Fig. 3.

Figure 3: A sequence of graphical moves (see Prlina:2018ukf ) that transforms
the 7-point ziggurat graph Fig. 1(d) into the wheel graph Fig. 2(b). YD
indicates wye-delta transformation(s) on the node(s) shaded in grey; DY
indicates delta-wye transformations(s) on the triangle(s) shaded in grey, and
FP indicates a trivial contraction of external edges.

###  3.1 Solving the on-shell conditions

We now turn to the first set of Landau equations–the on-shell conditions.
Since the graph Fig. 2(b) has 14 propagators, and there are 16 degrees of
freedom in the four loop momenta, we expect solutions to come in 2-parameter
families.

To organize the calculation we first consider the two propagators

$\displaystyle\braket{\mathcal{L}_{1}71}=\braket{\mathcal{L}_{1}12}=0$ (9)

which are of “one-mass bubble” type, referring to the standard terminology
(see for example Tab. 1 of Prlina:2017azl ), and have two solutions: one for
which the line $\mathcal{L}_{1}$ passes through the point $Z_{1}$, and one for
which $\mathcal{L}_{1}$ lies in the plane $\bar{1}$. (Here we use the standard
notation $\bar{i}=(i{-}1\,i\,i{+}1)$, with $(ijk)$ denoting the plane
containing $Z_{i}$, $Z_{j}$ and $Z_{k}$.) Since the Landau equations are
parity invariant, the parity conjugate of any solution is again a solution, so
it suffices to consider only the case where $\mathcal{L}_{1}$ passes through
$Z_{1}$ and obtain the remaining solutions by parity conjugation.

Altogether we find a total of 40 two-dimensional solutions to the on-shell
equations. Half of these solutions are listed in Tables 1 and 2; the rest are
their parity conjugates.

|
---
#
| ${}\begin{pmatrix}\mathcal{L}_{1}&&\mathcal{L}_{4}\\\
\mathcal{L}_{2}&&\mathcal{L}_{3}\end{pmatrix}\qquad{}$
---
| Letter
---
Classes
1 | $(Z_{1},l_{24}(\alpha))$ | $(Z_{6},\beta Z_{1}+(1-\beta)l_{24}(\alpha))$ | $b_{0},b_{2},b_{4}$
$(Z_{2},Z_{4})$ | $(Z_{4},Z_{6})$
2 | $(Z_{1},l_{24}(\alpha))$ | $(Z_{6},(Z_{1},l_{24}(\alpha))\cap\bar{5})$ | $b_{0},b_{1},b_{2},$
$(Z_{2},Z_{4})$ | $(Z_{4},l_{56}(\beta))$ | $b_{4}$
3 | $(Z_{1},P)$ | $(Z_{6},(1,P)\cap(356))$ | $b_{0},b_{1},b_{3}$
$(Z_{3},Z_{1})$ | $(Z_{5},Z_{3})$
4 | $(Z_{1},l_{23}(\alpha))$ | $(Z_{6},(l_{23}(\alpha),Z_{1})\cap(356))$ | $b_{0},b_{1},b_{3}$
$(Z_{3},l_{12}(\beta))$ | $(Z_{5},Z_{3})$
5 | $(Z_{1},Z_{2}(1{-}\beta)+\beta l_{34}(\alpha))$ | $\Big{(}Z_{6},\big{(}Z_{1},Z_{2}(1{-}\beta)+\beta l_{34}(\alpha)\big{)}\cap\big{(}l_{34}(\alpha),Z_{5},Z_{6}\big{)}\Big{)}$ | $b_{0},b_{1},b_{2},$
$(Z_{2},l_{34}(\alpha))$ | $(Z_{5},l_{34}(\alpha))$ | $b_{3},b_{5}$
6 | $(Z_{1},l_{23}(\alpha))$ | $(Z_{6},\beta Z_{1}+(1-\beta)l_{23}(\alpha))$ | $b_{0},b_{3}$
$(Z_{3},(12)\cap\bar{4})$ | $(Z_{5},(34)\cap(Z_{5},Z_{6},\beta Z_{1}+(1-\beta)l_{23}(\alpha)))$
7 | $(Z_{1},\beta Z_{2}+(1-\beta)l_{34}(\alpha))$ | $(Z_{6},(Z_{1},\beta Z_{2}+(1-\beta)l_{34}(\alpha))\cap\bar{5})$ | $b_{0},b_{1},b_{2}$
$(Z_{2},l_{34}(\alpha))$ | $(Z_{4},(56)\cap\bar{3})$
8 | $(Z_{1},P)$ | $(Z_{6},(Z_{1},P)\cap\bar{5})$ | $b_{0},b_{1},b_{3},$
$(Z_{3},Z_{1})$ | $(Z_{4},(56)\cap(134))$ | $b_{4}$
9 | $(Z_{1},l_{23}(\alpha))$ | $(Z_{6},\beta Z_{1}+(1-\beta)l_{23}(\alpha))$ | $b_{0},b_{2},b_{3},$
$(Z_{3},(12)\cap(346))$ | $(Z_{4},Z_{6})$ | $b_{4}$
10 | $(Z_{1},l_{23}(\alpha))$ | $(Z_{6},(Z_{1},l_{23}(\alpha))\cap\bar{5})$ | $b_{0},b_{1},b_{2},$
$(Z_{3},l_{12}(\beta))$ | $(Z_{4},(56)\cap(l_{12}(\beta),Z_{3},Z_{4}))$ | $b_{3},b_{4}$
Table 1: The first ten solutions to the on-shell conditions associated to Fig. 2(b). These solutions all have the line $\mathcal{L}_{1}$ passing through $Z_{1}$ and the line $\mathcal{L}_{4}$ passing through $Z_{6}$. Here $\bar{i}$ indicates the plane $(i{-}1\,i\,i{+}1)$, $l_{ij}(\alpha)=\alpha Z_{i}+(1-\alpha)Z_{j}$ is a point on the line $(ij)$, and $P$ denotes an arbitrary twistor. Each solution has two degrees of freedom, manifested in most cases by the arbitrary parameters $\alpha$ and $\beta$. The arbitrary point $P$ in solutions #3 and #8 has three degrees of freedom, but shifting $P$ in the direction of $Z_{1}$ leaves the solution unchanged so there are effectively only two degrees of freedom. The third column indicates which symbol letters (see main text) appear as Landau singularities for each cut. |
---
#
| ${}\begin{pmatrix}\mathcal{L}_{1}&&\mathcal{L}_{4}\\\
\mathcal{L}_{2}&&\mathcal{L}_{3}\end{pmatrix}\qquad{}$
---
| Letter
---
Classes
11 | $(Z_{1},l_{24}(\alpha))$ | $\bar{6}\cap(Z_{1},l_{24}(\alpha),l_{56}(\beta))$ | $b_{0},b_{1},b_{2}$
$(Z_{2},Z_{4})$ | $(Z_{4},l_{56}(\beta))$
12 | $(Z_{1},(23)\cap(Z_{1},Z_{5},P))$ | $\bar{6}\cap(Z_{1},Z_{5},P)$ | $b_{0},b_{1},b_{3}$
$(Z_{3},l_{12}(\alpha))$ | $(Z_{5},Z_{3})$
13 | $(Z_{1},P)$ | $\bar{6}\cap(Z_{1},Z_{5},P)$ | $b_{0},b_{1},b_{3}$
$(Z_{3},Z_{1})$ | $(Z_{5},Z_{3})$
14 | $(Z_{1},l_{2,(34)\cap\bar{6}}(\alpha))$ | $\bar{6}\cap(Z_{1},l_{2,(34)\cap\bar{6}}(\alpha),P)$ | $b_{0},b_{3},b_{5}$
$(Z_{2},(34)\cap\bar{6})$ | $(Z_{5},(34)\cap\bar{6})$
15 | $(Z_{1},(2,l_{34}(\alpha))\cap(Z_{1},Z_{5},P))$ | $\bar{6}\cap(Z_{1},Z_{5},P)$ | $b_{0},b_{1},b_{3}$
$(Z_{2},l_{34}(\alpha))$ | $(Z_{5},l_{34}(\alpha))$
16 | $(Z_{1},l_{23}(\alpha))$ | $\bar{6}\cap(Z_{1},l_{23(\alpha)},P)$ | $b_{0},b_{3}$
$(Z_{3},(12)\cap\bar{4})$ | $(Z_{5},(34)\cap\bar{6})$
17 | $(Z_{1},l_{23}(\alpha))$ | $\bar{6}\cap(Z_{1},l_{23}(\alpha),5)$ | $b_{0},b_{1},b_{3}$
$(Z_{3},(12)\cap\bar{4})$ | $(Z_{5},l_{34}(\alpha))$
18 | $(Z_{1},\beta Z_{2}+(1-\beta)l_{34}(\alpha))$ | $\bar{6}\cap\Big{(}Z_{1},\beta Z_{2}+(1-\beta)l_{34}(\alpha),(56)\cap\bar{3}\Big{)}$ | $b_{0},b_{1},b_{2}$
$(Z_{2},l_{34}(\alpha))$ | $(Z_{4},(56)\cap\bar{3})$
19 | $(Z_{1},l_{23}(\alpha))$ | $\bar{6}\cap\Big{(}Z_{1},l_{23}(\alpha),(56)\cap(l_{12}(\beta),Z_{3},Z_{4})\Big{)}$ | $b_{0},b_{1},b_{2},$
$(Z_{3},l_{12}(\beta))$ | $\Big{(}Z_{4},(56)\cap(l_{12}(\beta),Z_{3},Z_{4})\Big{)}$ | $b_{3},b_{4}$
20 | $(Z_{1},P)$ | $\bar{6}\cap\Big{(}Z_{1},P,(56)\cap(134)\Big{)}$ | $b_{0},b_{1},b_{3},$
$(Z_{3},Z_{1})$ | $\Big{(}Z_{4},(56)\cap(134)\Big{)}$ | $b_{4}$
Table 2: The second ten solutions to the on-shell conditions associated to
Fig. 2(b). These solutions all have the line $\mathcal{L}_{1}$ passing through
$Z_{1}$ and the line $\mathcal{L}_{4}$ lying in the plane $\bar{6}$. Each
solution has two degrees of freedom.

###  3.2 Solving the Kirchhoff conditions

When evaluated on any one of the on-shell solutions, the Kirchhoff conditions
provide a system of 16 equations in 16 variables: the 14 Feynman parameters
$\alpha_{i}$ (which appear linearly) and the 2 parameters $\alpha,\beta$
associated to the cut (which in general appear nonlinearly). These equations
are much more difficult to solve analytically than the on-shell conditions,
though we report some very helpful intermediate results below. In general we
find it necessary to adopt a “numerical experimentation” approach.
Specifically, we populate the $4\times 7$ matrix $Z$ of momentum twistors
describing the external kinematics with 28 random integers, except for a
single parameter “$z$” in some position. We then evaluate the Kirchhoff
conditions on this one-parameter family of kinematic configurations, and find
all solutions that exist only for certain values of $z$. Like the five-point
calculation, this analysis is complicated by the fact that there are branches
of solutions that exist for all values of $z$, which must be excluded. By
iterating over all possible positions of the parameter $z$, and by repeating
the calculation for many choices of random integer values for the other
entries of $Z$, we can be sure that we have identified all codimension-one
loci in kinematic space where the Landau equations admit solutions.

The last step is to make the connection between the Landau singularities found
in this way and the vanishing of symbol letters; and specifically to test the
expectation that the heptagon symbol alphabet captures the singularities of
all seven-point amplitudes in SYM theory. The 49 symbol letters of the
heptagon alphabet (see Caron-Huot:2011zgw ; Drummond:2014ffa ) fall into seven
classes under the $Z_{i}\to Z_{i+1}$ cyclic group. Let us denote the letters
by

$\displaystyle b_{01}=\braket{1234},\qquad b_{11}=\braket{1256},\qquad
b_{21}=\braket{1456},\qquad b_{31}=\braket{1236},$ (10) $\displaystyle
b_{41}=\braket{1346},\qquad b_{51}=\braket{1(23)(45)(67)},\qquad
b_{61}=\braket{1(34)(56)(72)},$ (11)

with $b_{ij}$ obtained from $b_{i1}$ by cyclically relabeling $Z_{m}\to
Z_{m+j-1}$. Here
$\braket{a(bc)(de)(fh)}=\braket{bade}\braket{cafg}-(b\leftrightarrow c)$. The
letters of type $b_{0j}$, $b_{1j}$ and $b_{6j}$ are individually invariant
under parity while the others are related under parity by

$b_{2,j}\leftrightarrow b_{3,j-1}\,,\qquad b_{4,j}\leftrightarrow
b_{5,j-1}\,.$ (12)

In the third column of Tables 1 and 2 we indicate the families of symbol
letters encountered for each of the on-shell solutions, using $b_{i}$ as
shorthand for the cyclic family $\\{b_{i1},\ldots,b_{i7}\\}$. Specifically,
the Kirchhoff equations associated to a given cut admit (codimension-one)
solutions only if the parameter $z$ takes a value that sets one or more symbol
letters in an indicated family to zero. The union of singularities found in
all cyclic relabelings of the original ziggurat graph comprises complete
cyclic families. From the tables (and the parity conjugate cuts, which lead to
analogous results with $b_{2}\leftrightarrow b_{3}$ and $b_{4}\leftrightarrow
b_{5}$) we see that all heptagon symbol letters except for family $b_{6}$ are
found as (leading) singularities of the seven-point graph Fig. 2(b).
Singularities of type $b_{6}$ certainly appear in relaxations, as discussed in
Sec. 3.4.

###  3.3 Some analytic details

Let us provide some insight into solving the Kirchhoff equations analytically
using momentum twistors, and in doing so demonstrate how to translate the
twistor space results into kinematic configurations occurring in momentum
space. The Kirchhoff equations associated to Fig. 2(b), written in momentum
space read,

$\displaystyle\alpha_{1}(l_{1}-p_{1})+\alpha_{2}l_{1}+\alpha_{11}(l_{1}+p_{2}-l_{2})+\alpha_{14}(l_{1}-p_{1}-l_{4}-p_{6}-p_{7})=0\,,$
(13)
$\displaystyle\alpha_{9}l_{4}+\alpha_{10}(l_{4}+p_{6})-\alpha_{13}(l_{3}+p_{5}-l_{4})-\alpha_{14}(l_{1}-p_{1}-l_{4}-p_{6}-p_{7})=0\,,$
$\displaystyle\alpha_{4}l_{2}+\alpha_{5}(l_{2}+p_{3})+\alpha_{12}(l_{2}+p_{3}-l_{3}+p_{4})-\alpha_{11}(l_{1}+p_{2}-l_{2})+\alpha_{3}(l_{2}-p_{2})=0\,,$
$\displaystyle\alpha_{7}l_{3}+\alpha_{8}(l_{3}+p_{5})+\alpha_{13}(l_{3}+p_{5}-l_{4})-\alpha_{12}(l_{2}+p_{3}-l_{3}+p_{4})+\alpha_{6}(l_{3}-p_{4})=0\,.$

Each momentum $p_{a\dot{a}}$ in the graph is associated to four twistors
$Z_{A},Z_{B},Z_{C},Z_{D}$ (for external legs only three of the four twistors
are distinct) with the explicit mapping given by

$p_{a\dot{a}}(A,B,C,D)=\mathcal{I}^{\alpha\beta}\mathcal{I}_{\gamma\delta}\frac{\epsilon_{\beta}(\cdot,A,B,C)D^{\delta}-\epsilon_{\beta}(\cdot,A,B,D)C^{\delta}}{\braket{\mathcal{I}AB}\braket{\mathcal{I}CD}}\,,$
(14)

where $\mathcal{I}^{\alpha\beta}\mathcal{I}_{\gamma\delta}$ denotes the
infinity twistor and it’s dual, and the four twistors associated to a loop
momentum occurring between the loop region $\mathcal{L}_{i}$ and the zone
$y_{j}$ are $(\mathcal{L}_{i}Z_{j}Z_{j-1})$. For example
$(l_{1})_{a\dot{a}}=p_{a\dot{a}}(\mathcal{L}_{1},Z_{2},Z_{1})$. The reader can
verify that using (14) in (13) and then going to the gauge (4) precisely
reproduces the twistor space formulation of the Kirchhoff equations described
in (3).

Note that the first Kirchhoff equation in (13) requires four four-dimensional
vectors to be linearly dependent, which only occurs if their determinant
vanishes. We can translate this vanishing determinant condition to
$(a,\dot{a})$ indices via the relation

$4i\epsilon_{\mu\nu\rho\sigma}\sigma^{\mu}_{\dot{a}_{1}a_{1}}\sigma^{\nu}_{\dot{a}_{2}a_{2}}\sigma^{\rho}_{\dot{a}_{3}a_{3}}\sigma^{\sigma}_{\dot{a}_{4}a_{4}}=\epsilon_{a_{1}a_{2}}\epsilon_{\dot{a}_{2}\dot{a}_{3}}\epsilon_{a_{3}a_{4}}\epsilon_{\dot{a}_{4}\dot{a}_{1}}-\epsilon_{\dot{a}_{1}\dot{a}_{2}}\epsilon_{a_{2}a_{3}}\epsilon_{\dot{a}_{3}\dot{a}_{4}}\epsilon_{a_{4}a_{1}}\,,$
(15)

where $\sigma^{\mu}_{\dot{a}a}$ are the Pauli matrices. Using this we deduce
that the first Kirchhoff equation admits solutions if either all of its
Feynman parameters are zero, or if

$\displaystyle\text{Det}\big{[}l_{1}-p_{1},l_{1},l_{1}+p_{2}-l_{2},l_{1}-p_{1}-l_{4}-p_{6}-p_{7}\big{]}\Big{|}_{\mathcal{L}_{1}=(Z_{1},B)}=0$
(16)
$\displaystyle\implies\braket{72\mathcal{L}_{1}}\braket{\mathcal{I}(\mathcal{L}_{2}1)\cap(\mathcal{L}_{4}1)}=0$
(17)

where the implication is understood only to hold on the support of our twenty
cut solutions in Tables 1 and 2 which all take the form
$\mathcal{L}_{1}=(Z_{1},B)$. Similar constraints apply for the second
Kirchhoff equation in (13). For example on the support of the first ten cut
solutions (Table 1) which have $\mathcal{L}_{4}=(Z_{6},H)$ we deduce that
either

$\displaystyle\text{Det}\big{[}l_{4},l_{4}+p_{6},l_{3}+p_{5}-l_{4},l_{1}-p_{1}-l_{4}-p_{6}-p_{7}\big{]}\Big{|}_{\mathcal{L}_{4}=(6,H)}=0$
(18)
$\displaystyle\implies\braket{57\mathcal{L}_{4}}\braket{\mathcal{I}(\mathcal{L}_{3}6)\cap(\mathcal{L}_{1}6)}=0$
(19)

or that all the Feynman parameters in the second Kirchhoff equation vanish.

Let us exemplify the kinematic configurations which solve the Landau equations
by using the aforementioned results to find a solution to the Kirchhoff
equations for the second cut in Table 1, which has
$\mathcal{L}_{1}=(Z_{1},l_{24}(\alpha))$. If we seek a solution where not all
$\alpha$’s in the first Kirchhoff equation are zero, then we must necessarily
satisfy (17). A simple branch of solutions can be obtained by choosing
$l_{24}=Z_{2}$ so that the first bracket in (17) vanishes. Plugging this
solution into the first Kirchhoff equation we find that this sets
$(l_{1})_{a\dot{a}}$ soft, in particular

$(l_{1})_{a\dot{a}}=p_{a\dot{a}}(Z_{1},Z_{2},Z_{1},Z_{2})=0\,,$ (20)

where we used that (14) vanishes when only two distinct twistors occur in its
arguments. Let us continue on this branch of solutions and seek a solution
where not all Feynman parameters in the second Kirchhoff equation are zero, in
which case (19) must also be satisfied. A particular branch of solutions can
be obtained by setting the first bracket in (19) to zero,

$\braket{576(1,2)\cap\bar{5}}=\braket{4567}\braket{1256}\overset{!}{=}0\,.$
(21)

By choosing either of the four brackets in (21) to vanish we obtain two
branches of solutions. Let us examine the $\braket{1256}=0$ branch. Using
$\braket{1256}=0$ in the second Kirchhoff equation we find that
$(l_{4})_{a\dot{a}}$ is set to zero,

$\displaystyle(l_{4})_{a\dot{a}}$
$\displaystyle=p_{a\dot{a}}(\mathcal{L}_{4},5,6)$ (22)
$\displaystyle=-\braket{1256}\mathcal{I}^{\alpha\beta}\mathcal{I}_{\gamma\delta}\frac{\epsilon_{\beta}(\cdot,4,5,6)Z_{6}^{\delta}}{\braket{\mathcal{I}56}\braket{\mathcal{I}6(12)\cap\bar{5}}}\,,$
(23)

where we used that
$\mathcal{L}_{4}=(6,(12)\cap\bar{5})=(6,Z_{4}\braket{5612}+Z_{5}\braket{6412})$.
We therefore observe that

$\braket{1256}=0\,\,\implies\,\,(l_{4})_{a\dot{a}}=0\,.$ (24)

We still have one degree of freedom left in our second cut solution which has
$\mathcal{L}_{3}=(4,l_{56}(\beta))$. One can readily show that for codimension
one solutions to the Landau equations our choice $\mathcal{L}_{1}=(1,2)$
requires $\alpha_{11}=0$. One way to see this is that all of the momenta in
the first Kirchhoff equation are proportional to
$\mathcal{I}_{\gamma\delta}Z_{1}^{\delta}$ except for
$(l_{1}+p_{2}-l_{2})\propto I_{\gamma\delta}Z_{2}^{\delta}$; hence either this
latter momentum must be zero, which we do not have the remaining degrees of
freedom available to achieve, or $\alpha_{11}=0$. Setting $\alpha_{11}=0$
requires that either all of the remaining $\alpha$’s in the third Kirchhoff
equation are zero, or that the remaining vectors satisfy the determinant
condition

$\displaystyle\text{Det}\big{[}l_{2}-p_{2},l_{2},l_{2}+p_{3},l_{1}+p_{2}-l_{2}\big{]}\Big{|}_{\mathcal{L}_{2}=(2,4)}=0$
(25) $\displaystyle\implies\braket{1234}\braket{234l_{56}}=0\,.$ (26)

Let us consider the case where not all of the remaining of $\alpha$’s in the
third Kirchhoff equation are zero, in which case (26) applies, and we can
solve this requirement by choosing

$l_{56}=(56)\cap\bar{3}\,.$ (27)

Plugging this solution into the third Kirchhoff equation we find that this
choice sets two of the momenta collinear,

$\mathcal{L}_{2}=(2,4)\And\mathcal{L}_{3}=(4,(56)\cap{3})\implies
l_{2}+p_{3}\parallel l_{2}+p_{3}-l_{3}+p_{4}\,.$ (28)

Plugging in our now fully localized cut solution into the fourth Kirchhoff
equation we find that two of the momenta are automatically collinear,

$\mathcal{L}_{2}=(2,4)\And\mathcal{L}_{3}=(4,(56)\cap{3})\implies
l_{3}-p_{4}\parallel l_{2}+p_{3}-l_{3}+p_{4}\,,$ (29)

which is as expected from momentum conservation at the vertex where the
momenta (28) and (29) meet. In summary we have found the solution

$\begin{gathered}\mathcal{L}_{1}=(1,2),\quad\mathcal{L}_{2}=(2,4),\quad\mathcal{L}_{3}=(4,(56)\cap\bar{3}),\quad\mathcal{L}_{4}=(6,(12)\cap\bar{5})\\\
\text{with the constraint: }\braket{1256}=0\end{gathered}$ (30)

which solves the Kirchhoff equations,

$\begin{gathered}l_{1}=l_{4}=0\\\
\alpha_{5}(l_{2}+p_{3})+\alpha_{12}(l_{2}+p_{3}-l_{3}+p_{4})=0,\quad\alpha_{6}(l_{3}-p_{4})-\alpha_{12}(l_{2}+p_{3}-l_{3}+p_{4})=0\end{gathered}$
(31)

by setting the momenta on the first line soft, and all three momenta on the
second line collinear. Lastly, we must verify that our solution (30) is not a
subcase of a solution which should be associated to an IR-divergence. As
explained in the previous sections, IR divergences are associated to solutions
to the Landau equations which do not impose any constraints on the external
kinematics. We note that it is impossible to solve $l_{1}=l_{4}=0$ on the
second cut solution in Table 1 without imposing $\braket{1256}=0$, thus our
solution (31) cannot be associated to an IR divergence. This concludes our
prototypical example of a leading solution to the Landau equations.

###  3.4 Relaxations

In order to complete the enumeration of all possible (first-type Landau)
singularities of seven-point amplitudes in massless planar theories, it
remains to consider all possible relaxations of graph Fig. (2)(b). By
relaxation, we mean any subgraph that can be obtained by contracting any
subset of propagators (which, at the level of the Landau equations, amounts to
setting the corresponding Feynman parameters to zero). We don’t need to
explicitly consider “edge deletions”, mentioned at the end of Sec. 2.2, since
solutions associated to such subgraphs come along for the ride as solutions of
the parent graph with identically zero momentum flowing through the deleted
edge.

It is interesting to note that the analysis of four-, five- and six-particle
ziggurats reviewed in Sec. 2 does not reveal any additional singularities in
relaxations beyond those already encountered at leading order. This certainly
cannot be the case for graph Fig. (2)(b) since it does not have any leading
singularities corresponding to symbol letters of type $b_{6}$, yet it contains
(after relaxing 10 propagators) subgraphs of one-loop three-mass box type,
which do have such singularities (see for example Dennen:2015bet ). Therefore
we know that the seven-point ziggurat has singularities (leading and/or
subleading) corresponding to _all_ elements of the heptagon symbol alphabet,
but our analysis is not yet enough to conclude that no _other_ singularities
are possible. Since Fig. (2)(b) has quite a few nontrivial graphs as
relaxations, we postpone a full analysis of this question to a companion
paper.

###### Acknowledgements.

We are grateful to I. Prlina and S. Stanojevic for collaboration in the early
stages of this work, to S. Mizera for correspondence, and to A. Ghandikota for
independently checking the graphical reduction shown in Fig. 3 as part of his
summer 2022 SPRINT—UTRA project. This work was supported in part by the US
Department of Energy under contract DE-SC0010010 (Task F) and by Simons
Investigator Award #376208.

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