# Cloud Continents: Terraforming Venus Efficiently by Means of a Floating
Artificial Surface

Alex R. Howe<EMAIL_ADDRESS>
###### Abstract

The similarity of Venus and Earth in bulk properties make Venus an appealing
target for future colonization. Several proposals have been put forward for
colonizing and even terraforming Venus despite the extreme conditions on the
planet’s surface. Such a terraforming project would face large challenges
centered around removing Venus’s massive carbon dioxide atmosphere and
replacing it with a habitable environment. I review past proposals and propose
a new method for terraforming Venus by building an artificial surface in the
much more hospitable upper atmosphere where the temperature and pressure are
both Earth-like. Such a surface could be built with locally produced materials
and would float above the bulk of the atmosphere using nitrogen as a lifting
gas. This would allow the engineering of a breathable atmosphere above the
surface and would remove the need to import or export extreme amounts of mass,
except for comparatively modest quantities of water. The engineering,
logistical, and energy requirements of this method are surveyed. I find that
such a terraforming project could be completed in a minimum of 200 years in a
best-case scenario, comparable to other proposals, with significantly lower
resource costs.

NASA Goddard Space Flight Center

##  1 Introduction

The terraforming of Mars to produce a habitable environment on the surface is
a frequent topic of discussion in long-term human spaceflight (e.g. Zubrin &
McKay (1997); Fogg (1998); Jakosky & Edwards (2018); Mansfield (2018)). The
reason for this is clear: the surface conditions on Mars are the most similar
to those on Earth of any body in our solar system and thus would require the
least modification to adapt its surface for human habitation. Venus, in
contrast, appears to be a much less likely candidate for terraforming due to
its extremely thick atmosphere and a very high surface temperature produced by
a runaway greenhouse effect. Nonetheless, the potential terraforming of Venus
has received some consideration, beginning with Sagan Sagan (1961). Venus also
provides some advantages over Mars for colonization with its near-Earth-like
surface gravity, an atmosphere thick enough to provide robust protection from
cosmic rays Zeitlin et al. (2004) and UV radiation Mogul et al. (2021)
compared with Mars, and shorter travel time from Earth.

The challenges for terraforming Venus’s surface are great, requiring a
significantly higher degree of speculative planetary engineering than Mars.
The main requirements for producing a habitable environment on the surface
are:

  * •

Removing or sequestering the $\sim$90 bars of carbon dioxide from the
atmosphere.

  * •

Producing a breathable nitrogen-oxygen atmosphere.

  * •

Cooling Venus from its current surface temperature of 735 K.

  * •

Replacing the planet’s lost water.

  * •

Speeding up the planet’s rotation from its current value of a solar day
lasting 117 Earth days. (However, this is probably not essential for
habitability if the albedo can be artificially controlled, cf. Eager et al.
(2020)).

An alternative to terraforming the surface of Venus would be to colonize the
upper atmosphere (Landis, 2011). Taking advantage of the twin facts that the
upper atmosphere of Venus at an altitude of 50-55 km has a similar temperature
and pressure to Earth’s surface, and that nitrogen and oxygen are lifting
gases in Venus’s CO2 atmosphere, it would be possible to construct “cloud
cities” using aerostats filled with breathable air that could be inhabited by
humans. This method sidesteps most of the problems of colonizing the surface;
it would be achievable with current technology, and it has reached the level
of a mission concept (Arney & Jones, 2015).

In this paper, I propose a method of terraforming Venus that similarly
sidesteps most of the problems posed by the hostile conditions on the surface
by expanding the cloud city concept to “cloud continents,” building an entire
artificial surface floating in the upper atmosphere. While still speculative,
this could be done far more efficiently than other proposed methods for
terraforming Venus’s surface while allowing a far more robust and expansive
habitable environment than could be achieved by conventional aerostats. Nearly
all of the requisite materials for the project would be produced locally on
Venus, while water could be delivered from Mars via space elevator. If the
requisite technology were sufficiently developed, the process could plausibly
be energy-limited, and if all available solar energy resources are exploited
at currently available efficiencies, it could potentially be completed in as
little as 200 years, comparable to the fastest timescales for other proposed
methods (Birch, 1991).

Certainly, this estimate of 200 years is an extreme best-case scenario and
would require substantial development of technology to be possible at all. All
of the materials needed for such an effort have been demonstrated in the
laboratory, but cannot yet be manufactured on an industrial, let alone
planetary scale. The geoengineering techniques involved would be immensely
more efficient on Earth (or even Mars), so it would likely be a long time
before we had reason to turn them to Venus rather than applying them at home.
However, the goal of this paper is to provide an outline for how the
terraforming of Venus could be accomplished using foreseeable technology on
something close to a human timescale, and to demonstrate that it is a more
tractable problem than is usually assumed.

This paper is organized as follows. In Section 2, I give an overview of past
proposals for terraforming Venus and their limitations. Section 3 outlines the
design and construction process for the artificial surface. Section 4 explores
options for building an Earth-like environment on the new surface. Section 5
examines the costs of construction, the logistics of importing water, and a
potential timetable. I summarize my conclusions in Section 6.

##  2 Previous Proposals

The first major proposal for terraforming Venus was made by Sagan Sagan
(1961), even before the extreme surface conditions were known. At that time, a
“carboniferous swamp” was considered to be one of several possible models for
the surface environment, and microwave emissions had only raised the
possibility of a strong greenhouse environment in excess of 600 K. However, it
was known that Venus’s atmosphere was primarily made of CO2, which would need
to be removed to make the planet habitable for humans. Sagan suggested that
this could be achieved by seeding the cloud layer with photosynthetic algae to
break down the CO2 into oxygen and organic material. However, as we understand
Venus today, it is too dry and its atmosphere too thick for changing the
atmospheric chemistry alone to work; even if the atmospheric composition were
successfully changed, the problem would remain of removing the tens of bars of
oxygen that would be produced by photosynthetically breaking down all of the
CO2.

Adelman Adelman (1982) suggested colliding asteroids with Venus to strip it of
its excess atmosphere. This would be a difficult process, as the mass of the
impactors would need to be greater than the mass of the atmosphere itself
(Schlichting & Mukhopadhyay, 2018), which is on the order of $5\times 10^{20}$
kg. Like many such proposals, this carries the implicit challenge of moving a
dwarf planet’s worth of mass across interplanetary distances, itself a very
expensive process.

Birch Birch (1991) lays out a detailed program for terraforming Venus with the
claim that it could theoretically be accomplished in as little as 200 years.
Most of this time would be needed to cool the planet by placing a large,
space-based sunshade in front of Venus to block all sunlight from reaching it
and waiting for the atmosphere to condense and freeze. The frozen atmosphere
could then be artificially sequestered by a thermally-insulating cover.
However, long-term, the CO2 would need to be exported off the planet.
Meanwhile, a water ocean would be provided by breaking up Enceladus and
delivering it in pieces to the planet. This also has the problem of needing to
move large amounts of mass, and even if Enceladus were replaced with a
different body, we may be reluctant to scuttle an entire large moon for the
project. Birch does raise the possibility of constructing an artificial
surface on Venus, but considers creating a “natural” geosphere to be more
desirable.

Bullock & Grinspoon Bullock & Grinspoon (1996), though not addressing
terraforming directly, suggest that it may be possible to sequester the CO2 in
Venus’s atmosphere by chemically reacting it with calcium minerals on the
surface. This would remove the need to export the atmosphere, but the natural
timescale for this process is estimated at 107–108 years, so speeding it up to
a practical timescale would be challenging.

More recently, Landis Landis (2011) surveyed the various proposals for
terraforming Venus and found colonizing the upper atmosphere to be much more
feasible. This would most naturally be done with aerostats, but he also
suggests building foundations up from the surface and using genetically-
modified organisms to produce a breathable atmosphere in the cloud layer.
However, building up 50 km from the surface presents significant engineering
challenges, and it would be necessary to engineer the whole atmosphere at once
due to vertical mixing. An artificial surface dividing the upper and lower
layers of the atmosphere would be needed to limit the terraforming
requirements to the habitable-pressure zone.

##  3 Surface Design

An artificial surface on Venus must perform two tasks: (1) reliably separate
the bulk of the CO2 atmosphere below it from the eventual oxygen-nitrogen
atmosphere above and (2) support its own weight and any colonies built on it
by buoyant force both during and after construction. (For a fully enclosed
surface, the supporting force is the uniform upward pressure of the atmosphere
rather than buoyancy, but as there are several advantages to maintaining
“neutral buoyancy” of the finished surface as discussed below, both cases are
described using buoyant forces for simplicity.) The first requirement means
that the planet’s surface must be fully enclosed before the atmospheric
engineering can take place. This requires a large initial investment, but,
unlike most other proposals, it is the only part of the terraforming process
that is dependent on a single, large step, whereas the rest of the operations
can be done piecemeal.

The second requirement means that the artificial surface should be neutrally
buoyant. While a fully enclosing surface could be heavier and still be
supported by gas pressure once completed, neutral buoyancy will be essential
during construction, and it would offer a more stable platform for the uneven
weight distribution of a partially built-up surface. A neutrally buoyant
surface would also decrease the rate of leakage from the lower atmosphere in
case of damage, thus reducing maintenance costs.

These considerations suggest an artificial surface built in two parts: first,
a lightweight surface enclosing the entire planet that need only support its
own weight and be resistant to tearing. Second, larger floating “islands” and
“continents” could be constructed embedded in the surface with enough lifting
power to support full-scale colonies with a robust infrastructure including
housing, manufacturing capacity, croplands, and an imported biosphere.

Both parts of the surface could be constructed with local materials, using
solar power to break CO2 into carbon and oxygen. The carbon would be used to
manufacture carbon nanotubes or similar structures for construction material,
while the oxygen could be used as a lifting gas or stored for future use. The
3.5 bars of nitrogen already present in the atmosphere would be the primary
lifting gas. Any metals or other minerals needed for construction could be
mined robotically from the ground level. Additionally, both parts of the
surface could be constructed in parallel with each other, and settling the
planet could occur at the same time through enclosed, “conventional” aerostat
colonies.

###  3.1 Enclosing the Surface

The first step of the terraforming process will be to fully enclose Venus with
a lightweight surface at an altitude of $\sim$50 km to separate the two parts
of the atmosphere. The main structural limitation at this stage will be wind
forces. In Venus’s atmosphere, zonal wind speeds at the 1-bar level are 40-60
m/s (Gubenko & Kirillovich, 2018; Sánchez-Lavega et al., 2017), indicating
that the surface must withstand horizontal wind shear of tens of meters per
second. Both meridional and vertical winds are significantly weaker at
$\lesssim$5 m/s, although vertical winds due to convection will be a source of
unwanted mixing of the upper and lower atmosphere through any holes in the
surface.

A surface made as a single piece will not be able to withstand these wind
forces over thousands of kilometers. Instead, we may envision large,
interlocking hexagonal tiles, a few tens of meters thick, with the joints
between them providing the slack to resist the wind forces. These tiles would
only need to support their own weight, so they could be manufactured with
similar materials and techniques to conventional airships (perhaps using
carbon fiber if sufficient nanotube production is not yet available). The top
surface of the tiles could be mirrored to maintain the planet’s Bond albedo of
0.76, which will help with later climate control. Tears in the surface could
be repaired by replacing damaged tiles. For tiles 100 meters in width
diagonally, comparable to a large airship, $7.2\times 10^{10}$ of them would
be sufficient to enclose the planet at an altitude of 50 km.

Maintenance of this tile surface would be essential to maintaining the
chemical composition of the upper atmosphere, as any holes would allow CO2 to
leak from below. A neutrally buoyant surface would produce the least leakage,
but it could still be significant in severe cases. Assuming Earth-like
conditions for the upper atmosphere and a 5 m/s vertical flow, a tear with an
area of 1 km2 would leak CO2 at a rate of $8.04\times 10^{11}$ kg/day. This
rate of leakage would increase the CO2 concentration in the upper atmosphere
by 0.101 ppm/day until the tear is closed. Adequate maintenance systems would
need to detect and deflect any asteroids capable of causing tears this large
before they reach the planet, while being able to replace the tiles fast
enough to keep up with lesser damage. If limited to aircraft speeds, it may
take several days to dispatch repair equipment to remote areas, so it may be
preferable to build a surplus of replacement tiles and station them at depots
around the planet.

###  3.2 Floating Landmasses

Building floating landmasses large enough to support cities, infrastructure,
and ecosystems similar to those on Earth requires more careful consideration.
These landmasses will be limited by the inventory of chemicals available on
Venus and particularly in the atmosphere for construction. The inventory of
atmospheric species of greater than 1 ppmv abundance (based on surface
abundances) is shown in Table 1. For this purpose, I will assume a surface
pressure of 92.0 bars. Clearly, CO2 and N2 are the species of interest, and
the problem of removing the sulfur dioxide (and indeed, sulfuric acid) from
the upper atmosphere will be secondary to changing it to a nitrogen-oxygen
composition.

Species | Mole Fraction | Areal Density
---|---|---
|  | (kg m-2)
CO2 | 96.5% | 1027000
N2 | 3.5% | 23710
SO2 | 200 ppm | 310
Ar | 70 ppm | 67.8
COS | 55 ppm | 79.8
H2O | 30 ppm | 13.1
CO | 12-40 ppm | 8-27
He | 12 ppm | 1.16
Ne | 7 ppm | 3.39
Total | 100.0% | 1051000
Table 1: Chemical inventory of Venus’s atmosphere, assuming a surface pressure
of 92.0 bar. (Based on surface abundances.) (Basilevsky & Head, 2003)

The floating landmasses would be constructed with a honeycomb structure, not
too dissimilar from the tiles, but much taller, kilometers high rather than
tens of meters. As with the lighter tiled surface, the main construction
material will be carbon nanomaterials made with carbon extracted from the
atmosphere. Because carbon nanotubes are much stronger in tension than
compression, it will likely be better to use a different structure such as
aggregated diamond nanorods (Dubrovinskaia et al., 2006) or some form of
diamond-nanotube composite for the load-bearing walls. Diamond is reported to
have a compressive strength of 250 GPa and a shear strength of 75 GPa at nano-
scales (Wheeler et al., 2016). This is comparable with the reported tensile
strength of carbon nanotubes of 100 GPa (Peng et al., 2008) and well above the
tensile strength reported for carbon nanotube bundles of 17 GPa (Filleter et
al., 2011). As we will see, the stress placed on the load-bearing structures
of the artificial surface will be $\sim$10.8 GPa in the most extreme case and
could be made significantly lower, so it is plausible that adequate carbon
nanostructures could be found.

In principle, a large fraction of the Venusian atmosphere could be converted
to building materials, the limitation being how much surplus oxygen could be
safely stored in the honeycomb as lifting gas before becoming a fire hazard.
However, for definiteness, I will focus on the most efficient design in terms
of lift-to-weight ratio, in which exactly as much CO2 is spent as is needed to
produce the same partial pressure of oxygen as Earth in the upper atmosphere,
while still using all of the surplus nitrogen for lift. This design will
require 0.32 bar of CO2, and it will be able to support an additional 7440
kg/m2 of mass loading.

The general design of the floating landmasses is illustrated in Figure 1. Due
to the difference in scale height between N2 and CO2, the honeycomb should be
built in layers, with each layer pressurized to the same level as the ambient
atmosphere, in order to prevent a large pressure differential from developing
at either the top or bottom of the column. The thickness of the honeycomb is
dependent on the amount of lifting gas used, but in all cases will be several
kilometers, and the load on top of the landmass must equal the lifting
capacity in order to maintain neutral buoyancy.

Figure 1: Illustration of the floating landmass design. The load-bearing
surface is supported by a honeycomb filled with lifting gas (either nitrogen
or a nitrogen-oxygen mixture) with each layer of cells pressurized to the
ambient pressure to minimize stress during construction. In all designs, the
honeycomb will be several kilometers tall and will support several tons of
mass loading per square meter.

In order to produce a pressure of 1 bar above the artificial surface, we
require an areal air density of $\frac{1bar}{g_{V}}=11620$ kg/m2, where
$g_{V}$ is the gravity on Venus (noting that at altitude, the gravity will be
slightly lower than at ground level: 872 cm/s2 rather than 887 cm/s2). To have
a molar fraction of 79% N2 and 21% O2 requires this atmosphere to include 8910
kg/m2 of nitrogen. This is 37.6% of the atmosphere’s nitrogen inventory. The
remaining nitrogen, 14800 km/m2, will be pumped into the honeycomb to serve as
lifting gas. With the layers of the honeycomb pressurized to the ambient
pressure, the lift is given simply by the ratio in molecular weights between
N2 and CO2. The nitrogen will thus displace 23260 kg/m2 of CO2 extending down
to a pressure level of 3.002 bar. Assuming an average air temperature the same
as Earth’s, 288 K (see Section 4), the scale height of CO2 will be 6240 m, and
the thickness of the honeycomb will be 6860 m. (Given the temperature
difference over such a change in altitude, the actual thickness may need to be
several hundred meters greater.)

The buoyancy of this volume of nitrogen is 8460 kg/m2, which must include the
weight of the honeycomb itself plus anything built on top of it. The mass of
the honeycomb is equal to the amount of carbon extracted from CO2 to produce
the oxygen in the atmosphere, giving it a mass of 1020 kg/m2, leaving 7440
kg/m2 of available lifting capacity for the floating landmass. This value also
determines the strength required for the load-bearing structure, as the walls
of the honeycomb must extend for its whole height. The internal structure of
the honeycomb is not certain, and cross supports will be needed between the
cells, if not elsewhere, but for definiteness, I will assume that two thirds
of the carbon goes into the vertical walls, as would be the case for a stack
of uniform cubes. In the worst-case scenario, the carbon nanostructure will
have the density of diamond as opposed to lighter allotropes of carbon,
$\rho=3.515$ g/cm3. In this configuration, a horizontal cross section of the
honeycomb would have a fill fraction of $f=\frac{2\sigma_{C}}{3\rho
H}=2.82\times 10^{-5}$, where $\sigma_{C}$ is the area density of carbon, and
$H$ is the height of the honeycomb. The base of the honeycomb will be under a
pressure of 3.002 bar, so the necessary compressive strength to support it
will be $P/f=10.8$ GPa.

The mechanical stresses on the honeycomb could be reduced by processing more
CO2 into carbon and adding the surplus oxygen to the lifting gas supply. The
disproportionately greater carbon supply would allow the walls of the
honeycomb to be thicker, reducing compressive stress. Alternatively, the
height of the entire column could be made shorter, supporting less weight and
using less nitrogen for lift. This would be an ideal solution if the full
lifting capacity of the honeycomb is not needed.

Table 2 provides a breakdown of the mass loading and other mechanical
properties for three possible designs of floating landmasses: a “standard”
design using the full nitrogen budget of the atmosphere, a “heavy” design that
adds enough surplus oxygen to fill the entire honeycomb with breathable air,
and a “light” design that provides the minimum amount of lift estimated to
support a viable biosphere on the surface. Both the heavy and light designs
require significantly less structural strength.

| Standard | Heavy | Light
---|---|---|---
Mass Loading |  |  |
(kg m-2) |  |  |
N2 | 23710 | 23710 | 16310
Atmospheric | 8910 | 8910 | 8910
Lifting | 14800 | 14800 | 6000
O2 | 2710 | 7210 | 2710
Atmospheric | 2710 | 2710 | 2710
Lifting | 0 | 4500 | 0
Structure | 1020 | 2710 | 1020
Total Mass | 27440 | 33630 | 18640
Displacement | 23260 | 29450 | 9430
Net Lift | 7440 | 7440 | 2410
Base Pressure (bar) | 3.002 | 3.534 | 1.810
Column Height (m) | 6860 | 7880 | 3700
Compressive Stress (GPa) | 10.8 | 5.5 | 3.5
Table 2: Mass loading for three floating landmass designs. The “Standard”
design uses all of the nitrogen in the atmosphere, but only as much CO2 as is
needed to produce a breathable nitrogen-oxygen upper atmosphere. The “Heavy”
design uses enough CO2 to use breathable air as a lifting gas throughout the
honeycomb. The “Light” design is based on as estimate of the minimum lift
needed to support a viable biosphere (including soil, water, and biomass).

###  3.3 Heat Balance of the Atmosphere

One complication to this design is that enclosing the surface with these
floating landmasses will change the temperature profile of the lower region of
the atmosphere. The upper surface of the landmasses and the adjacent topmost
layer of the gas-filled honeycomb would be at approximately the same
temperature as the present atmosphere of Venus at the same altitude. However,
the loss of solar heating to the deeper layers would cause them to cool and
contract, diminishing the pressure support for the surface. The cooling rate
would depend on the rate of thermal transport in Venus’s atmosphere. This rate
would likely be slowed by the honeycomb itself cutting off convection in the
top several kilometers and may be slowed more if the honeycomb is designed to
be insulating, but it would still likely not maintain a steady state over long
periods.

To restore the thermal equilibrium of the lower region of the atmosphere, a
simple solution would be to place “windows” of transparent material in the
artifical surface. This could be done by building the aerostat tiles of parts
of the initial enclosing surface our of transparent plastics, either imported
directly, or made by reacting carbon with hydrogen extracted from the sulfuric
acid clouds and the small supply of water vapor. These would then allow solar
radiation to reach the lower atmosphere and maintain it in thermal
equilibrium. The floating landmasses could not be extended across the windows,
but if the honeycomb is sufficiently insulating, the windows would likely need
to cover only a small percentage of the surface. Further research is needed to
more accurately predict the thermal transport in such an enclosed atmosphere.

###  3.4 Parallel Colonization

Unlike the initial enclosing of the planet with the tile surface, building the
floating landmasses could be done gradually because the other steps in the
terraforming process (delivery of water, soil, and building material and
changing the upper atmospheric chemistry) can be done in parallel with it and
do not rely on the honeycomb structure being completed. However, open-air
colonies and ecosystems do depend on the upper atmosphere being changed to an
Earth-like composition. This is expected to be a protracted process requiring
extracting 37.6% of the nitrogen from the lower atmosphere and splitting all
of the needed carbon dioxide to supply oxygen, as well as pumping out all of
the CO2 initially occurring in the upper atmosphere. Thus, these open-air
colonies could be constructed only after the terraforming process is largely
complete.

However, the human colonization of Venus could proceed unimpeded in parallel
with these efforts. Once small (a few km2) floating islands are constructed,
large habitat domes could be built on top of them. Rather than freestanding
domes, these habitats could be enclosed by the same aerostat tiles covering
the rest of the planet, allowing much larger enclosed volumes than can be
achieved with self-supporting structures. While the aerostat tiles would be
floating above a lighter nitrogen-oxygen atmosphere, lift could be maintained
by partially filling the tiles with helium. The partial pressure of 1.1 mbar
of helium at ground level could provide an ample supply. Light could be let in
by building the tiles out of the same transparent plastics as the surface
windows. Eventually, once the exterior air becomes breathable, the domes would
be dismantled and recycled.

##  4 The Terraformed Environment

###  4.1 Temperature

With a breathable atmosphere in place, the next question will be how to
regulate the climate of the terraformed Venus. Being much closer to the Sun
than Earth, a precisely Earth-like atmosphere and surface (with much lower
albedo than the present Venus) will become too hot for human habitation. Many
proposals suggest a sunshade to reduce the sunlight reaching the surface to
Earth-like levels, but this is unnecessary for temperature regulation. With
its high albedo, Venus’s current equilibrium temperature is already colder
than Earth’s. Thus, the planet’s surface albedo should be decreased just
enough to raise its equilibrium temperature to that of Earth. Then, an Earth-
like atmosphere will induce the correct amount of greenhouse warming to
produce an Earth-like climate. This will involve lowering Venus’s Bond albedo
from its current value of 0.76 to 0.62, which could be accomplished by keeping
roughly half of the surface mirrored to reflect incoming sunlight. (Some of
this mirrored surface would need to be inside colony areas to combat the urban
heat island effect.)

Controlling Venus’s temperature in this way would result in Earth-like climate
patterns without the need for a space-based sunshade, and the greater sun
exposure would be beneficial for solar power production and agricultural
yields.

###  4.2 Rotation

Venus’s slow rotation is often listed as one of the problems of colonizing the
planet, with a solar day lasting 117 Earth days. While this is not necessarily
an obstacle to a functioning biosphere (Eager et al., 2020), a floating
artificial surface removes this issue by taking advantage of the superroation
of Venus’s atmosphere and moving with the prevailing winds. While not as fast
as Earth’s rotation, if a typical wind speed of $\sim$50 m/s holds at the
equator, the artificial surface will rotate roughly once every 9 Earth days.
Further study is needed to more accurately predict the interaction of a rigid,
floating shell with the superrotating atmosphere.

###  4.3 Ecosystems and Infrastructure

Once floating islands are in place, the next question will be what sorts of
natural and urban environments are to be built there. Notably, the full 7440
kg/m2 of lifting capacity must be used to maintain neutral buoyancy. If this
much lift is not needed, a thinner surface should be constructed, but it
allows the option of having more built-up regions.

Just as on Earth, much of Venus’s surface will likely be dedicated to
farmland, so we will want to cover most of the surface with soil. One meter of
typical arable soil will have a mass of $\sim$2000 kg/m2 (Håkansson & Lipiec,
2000). Much of this material can be made from regolith mined from the ground.
However the water content ($\sim$500 kg/m2) and a seed stock of organic
material will need to be imported. Options for sources of water are considered
in Section 5.2.

The shallowness of this soil layer means that it will be vital to include
drainage channels and barriers against soil erosion in the surface design.
Hills and valleys could be built into the surface simply by building contours
into the honeycomb supporting it, at a negligible cost to lifting capacity.
Properly maintained, drainage channels and basins could serve the role of
rivers and lakes in the terraformed ecosystem, although they would need to be
very shallow, no more than $\sim$8 meters deep.

For urban environments, building single-family homes would pose no problem,
nor would smaller high-rise buildings. Typical building regulations for a
weight loading of $\sim$500-1000 kg/m2 of floor space in office buildings
suggest that they could be built up at least a few floors high, and likely
higher because their footprints do not cover all of the land area. If properly
supported (e.g. by arches and buttresses in the top layer of the honeycomb,
some individual buildings could plausibly be built as tall as the tallest
buildings on Earth. Instead, the largest limitation on urban environments is
likely to be weight-intensive infrastructure such as subway tunnels and water
reservoirs. Thus, construction typical of medium-sized cities will be viable
on a terraformed Venus.

##  5 Logistics and Time Scales

###  5.1 Energy Requirements

Any terraforming operation will require very large amounts of energy. These
energy requirements may be the limiting factor on terraforming speed. The
challenges of industrial production on a planetary scale (e.g. for manufacture
of tile aerostats) can be assumed to be solved by automation; self-replicating
machines, if used, are more likely to be limited by energy availability than
by raw industrial capacity. By estimating the energy requirements of each step
in the process, we can compute a minimum timescale for terraforming
operations.

Terraforming Venus using the strategy outlined in this paper will require four
major, potentially energy-limited operations:

  * •

Electrolyzing CO2 into carbon and oxygen.

  * •

Separating nitrogen from the atmosphere.

  * •

Mining regolith from ground level and delivering it to the artificial surface.

  * •

Importing enough water to produce arable soils.

CO2 can be electrolyzed at high efficiencies (Chen et al., 2012). Breaking CO2
into carbon (graphite) and oxygen requires an input of 393.5 kJ/mol, or 8.94
MJ/kg of CO2. In order to process 0.32 bar of CO2 in this way to obtain oxygen
and building materials will require $3.33\times 10^{10}$ J/m2 of energy flux,
or as much sunlight as falls on the planet in 5.8 years. Thus, solar power
production will need to proceed on a planetary scale to complete the process
on a feasible timescale. With the tile surface already in place, this will be
relatively straightforward to implement, but the efficiency of solar power
(the likely primary source of energy on Venus) will limit the production rate.
With a plausible end-to-end efficiency of 20%, the oxygen production will take
nearly 30 years in total. This of course would require most or all of the
initial enclosing surface to be covered with sufficiently lightweight solar
panels.

Extracting the nitrogen from the atmosphere will be a much greater task
because virtually the whole atmosphere must be processed. The heat of
sublimation alone for that much CO2 is more than an order of magnitude greater
than that needed to produce the oxygen: 26 kJ/mol, amounting to $5.99\times
10^{11}$ J/m2. As a best case scenario, a condenser running at the sublimation
point of CO2 at the 1-bar level of 195 K and operating at the Carnot limit
would have a coefficient of performance of
$\frac{195\,\rm{K}}{288\,\rm{K}-195\,\rm{K}}=2.06$. Based on the measured
temperature-pressure profile (Jenkins et al., 1994), it may be possible to
increase this to $\sim$3 by operating the condenser higher in the atmosphere
at $\sim$75 km, but even in this case, the energy requirements to extract all
of the nitrogen from the atmosphere are on the order of $2\times 10^{11}$
J/m2. With a 20% efficient source of solar power, this would require all of
the solar energy falling on Venus for $\sim$170 years.

It is assumed that the processing of the atmosphere will occur at altitude,
where it is needed, to remove the energy costs of lifting that much mass from
lower down. (Manually lifting the mass of the atmosphere 75 km from ground
level to the condenser would require more energy than the condenser itself.)
However, the delivery of soil to the artificial surface will require a lifting
operation of regolith mined from the ground. The requirement of 1500 kg/m2 of
regolith lifted 50 km corresponds to an energy cost of $6.65\times 10^{8}$
J/m2, much less than the other steps in the terraforming operation.

###  5.2 Sources of Water

The water content of Venus’s atmosphere is 20 ppmv, or 8.6 kg/m2. To produce
arable soils over the entire planet would require 500 kg/m2 or $2.30\times
10^{17}$ kg of water, equal to a cube 61.3 km on a side. (This is an order of
magnitude greater than the water vapor content of Earth’s atmosphere, so it
will be sufficient for a complete hydrosphere). This requirement necessitates
consideration of how this water is to be delivered.

If the water is delivered before building the tile surface, it would be
possible to deliver all of it by colliding a few large comets with the planet.
However, this goes against the design philosophy of having as few large,
indivisible steps as possible that could hold up the terraforming process,
along with facing the problems of safety and security for an operation to
deliberately navigate large cometary bodies through the Inner Solar System.
During or after construction, water must be delivered either by soft landing
or (more likely) by delivering the ice in small pieces that would burn up in
the atmosphere a safe distance above the new surface.

With the large amount of mass to be moved, both the fuel and energy costs will
be extremely high, and all of the possible sources of water require trade-offs
in this area. The inner planets offer greater access to solar energy, which
will likely be needed to export water in large quantities. However, exporting
water from either Earth or Mars would mean a high $\Delta v$ cost to lift it
from the planet’s surface. Icy moons in the Outer Solar System suffer similar
problems due to the gravity of their host planets. Mining ice from a large
centaur like 10199 Chariklo would eliminate the need to lift it out of a
planetary gravity well, but it would mean a larger $\Delta v$ cost to move it
to the Inner Solar System. Table 3 lists the $\Delta v$ costs to export water
from various possible sources to Venus. As an average, I will use a (one-way)
Hohmann transfer between circular orbits to estimate the cost. (The exact cost
will depend on the orbital configuration at the time of launch and may be
significantly larger or smaller at different times.) The $\Delta v_{\rm tot}$
column lists the total $\Delta v$ needed to make the delivery with a single
impulse, ignoring any more complicated gravity assists.

Object |  $v_{\rm esc}$ (m/s) |  $v_{\rm esc,pl}$ (m/s) |  $\Delta v_{\rm Hoh}$ (m/s) |  $\Delta v_{\rm tot}$ (m/s)
---|---|---|---|---
Earth | 11180 |  | 2500 | 11460
Moon | 2380 | 1440 | 2500 | 3730
Mars | 5020 |  | 4770 | 6920
Ceres | 510 |  | 6380 | 6400
Callisto | 2440 | 11600 | 6610 | 13570
Iapetus | 400 | 4610 | 6020 | 7590
Chariklo | $<200$ |  | 5270 | 5270
Pluto | 1210 |  | 3840 | 4026
Eris | 1380 |  | 3090 | 3844
Table 3: The $\Delta v$ budget needed to transport water from various solar
system bodies to Venus. Because the exact requirements vary with orbital
configuration, a one-way Hohman transfer (with uncontrolled reentry) between
circular orbits is used to compute an estimated average value.

Even going far into the Kuiper belt where large amounts of solar energy are
not available, the $\Delta v$ costs are significant. Earth’s Moon has the
lowest $\Delta v$ cost, but the water reserves on the Moon are estimated at
only $6\times 10^{11}$ kg (Spudis et al., 2010), far short of the required
amount. Other potential sources of water nearer than the Kuiper belt have a
$\Delta v$ cost of at least 5.3 km/s. This would almost certainly require a
fuel-to-payload ratio greater than 1, which further compounds the problem of
moving large amounts of mass. With such large fuel costs, it may be preferable
to build a railgun powerful enough to launch a foil-wrapped projectile to
escape velocity, without expending significant fuel. To deliver 500 kg/m2 of
water to Venus by this method would require on the order of $10^{10}$ J/m2,
albeit not expended on Venus itself. As this would require a large fraction of
a planet’s energy budget, much like the in situ terraforming processes, space-
based solar arrays around Earth or Mars may be the only feasible way to
produce this much energy.

However, for fast-rotating bodies, a more energy-efficient way to move mass
over interplanetary distances would be a space elevator. Unlike either a
chemical rocket or a railgun, a space elevator does not suffer significant
losses from air resistance and does not need such high power densities. Also,
unlike a rotating electromagnetic tether, a space elevator can be powered from
the ground and mostly does not need to account for momentum exchange (but see
below), allowing it to lift much larger payloads. Most importantly, the energy
required to launch a payload via space elevator is only what is needed to lift
a payload vertically to the synchronous orbit. Beyond that, centrifugal forces
will accelerate the payload outward to the end of the tether, drawing from the
planet’s rotational energy, giving it large amounts of both rotational and
radial velocity.

We can compute the engineering requirements for the space elevator by
integrating the apparent gravitational force. In the rotating frame, the
apparent gravitational force is given by:

$g_{\rm app}(r)=-\frac{GM}{r}+\omega^{2}r,$ (1)

where $\omega$ is the planet’s rotation speed. The synchronous orbit occurs
where this apparent force is zero. The energy needed to lift the payload from
the surface to the synchronous orbit is:

$E=\int_{r_{s}}^{r_{sync}}\left(-\frac{GM}{r^{2}}+\omega^{2}r\right)\,dr=GM\left(\frac{1}{r_{sync}}-\frac{1}{r_{s}}\right)+\frac{\omega^{2}}{2}(r_{sync}^{2}-r_{s}^{2}).$
(2)

If the far end of the tether is at radius $r_{f}$, and the payload is able to
slide without friction (perhaps by magnetic levitation), then the departure
speed will be:

$v_{d}=\left[2GM\left(\frac{1}{r_{f}}-\frac{1}{r_{sync}}\right)+\omega^{2}(r_{f}^{2}-r_{sync}^{2})+(\omega
r_{f})^{2}\right]^{1/2}.$ (3)

Solving this expression for $r_{f}$ and finding the elevator length needed to
reach the Hohmann transfer orbit tells us the mechanical requirements for the
space elevator. These results are shown for Earth, Mars, and Ceres in Table 4.
$\Delta v_{\rm eq}$, the velocity corresponding to an equivalent amount of
kinetic energy, is also provided for comparison.

Object | Elift (MJ/kg) | $\Delta v_{\rm eq}$ |  $r_{f}$ (km) |  $v_{\rm tip}$ (m/s)
---|---|---|---|---
Earth | 62.36 | 9840 | 57100 | 4150
Mars | 12.58 | 4360 | 53800 | 3810
Ceres | 0.13 | 330 | 23500 | 4520
Table 4: Energy and design requirements for a space elevator that can lift a
payload from Earth, Mars, or Ceres directly to a Hohman transfer orbit to
Venus. The $\Delta v$ corresponding to an equal amount of kinetic energy is
included for comparison. Ceres is much more efficient, but lacks enough
angular momentum to transfer adequate amounts of water to Venus.

All three options require a similar tip velocity (the speed of the tip of the
elevator relative to the planet’s rest frame) and thus have similar
engineering requirements. These velocities are not attainable with current
materials, but are plausibly possible with carbon nanotubes. The tensile
stress on a rotating, untapered tether is given by $\sigma=v_{\rm
tip}^{2}\rho/2$. Again using the density of diamond as a worst-case estimate,
all of these space elevators have tensile stresses of 25-35 GPa, or likely a
bit less because the part of their length that would pass through the planet
is not needed. This is within the range reported for carbon nanotubes, and
using tapered elevators could reduce this stress significantly.

The energy requirement for this system is much lower for Ceres than for Mars
or Earth, which may be useful for rapid delivery of water in the early stages
of colonization on Venus. The surface of Ceres is estimated to be 5%-10% water
ice (Beck et al., 2021), so the supply is sufficient. However Ceres is not
able to deliver enough water to terraform the whole surface of Venus by this
method because it does not have enough angular momentum. The moment of inertia
of Ceres is $I_{\rm Ceres}\approx\frac{2}{5}Mr^{2}=8.73\times 10^{31}$ kg m2,
while the moment of inertia of $2.30\times 10^{17}$ kg of ice lifted to the
end of the tether is $I_{\rm ice}=M_{\rm H_{2}O}r_{f}^{2}=1.27\times 10^{32}$
kg m2. When released, the ice would carry away angular momentum, slowing
Ceres’s rotation to zero before the process can be completed.

It appears that the majority of the water used to terraform Venus will have to
be delivered from either Earth or Mars. Of the two, we may be reluctant to
build the planetary infrastructure needed to export water in bulk on Earth
(not least because of the environmental impact), and Mars has lower energy
costs and less strenuous requirements for a space elevator, so Mars will be
the optimal source. (Being much larger than Ceres, the angular momentum
exchange would lengthen Mars’s day by $\sim$20 seconds.) With the energy
requirement to export water from Mars of 12.58 MJ/kg, delivering the requisite
water to Venus at 20% energy efficiency would require all of the solar energy
resources available on Mars for $\sim$22 years. This is only a tenth of the
investment needed on Venus itself, so it is not a limiting factor in the
process. Moreover, water need only be delivered at the rate that the colonized
land is built out, which may extend long after the atmospheric engineering is
completed. Nonetheless, it is apparent that a large-scale terraforming effort
on Mars would likely need to be underway to undertake the task on Venus.

##  6 Conclusion

The prospect of terraforming Venus initially seems like an impractical one
given the extreme conditions on its surface, the most hostile in the solar
system. What proposals have been made face serious logistical challenges given
the difficulty of shedding 90 bars of CO2 in a reasonable amount of time and
the importation of enough water for conventional oceans. However, the physical
similarity of the planet to Earth, its closeness to Earth, and the surprising
near-habitable conditions in its upper atmosphere make the planet an enticing
target for other forms colonization, particularly “cloud cities” that avoid
the problems of trying to colonize the surface.

In this paper I lay out a potential program for terraforming Venus that uses
the cloud city concept not as an alternative to terraforming, but a stepping
stone to it. Expanding the proposed cloud cities to “cloud continents”
comprising a complete artificial surface could lay the foundation for a
terraformed biosphere similar to Earth without the problems of building on the
natural ground. A honeycomb of aerostats filled with nitrogen extracted from
the atmosphere a few kilometers thick would generate enough lift to support
most types of natural and human-made environments, and a breathable atmosphere
could be installed above it, with Venus’s original atmosphere sequestered
below.

This strategy avoids the need to lift Venus’s massive atmosphere out of its
gravity well or somehow sequester it chemically. The artificial surface could
be built with materials that have been demonstrated in the laboratory, and it
could support enclosed city-sized colonies before it is completed if so
desired. Powered by Venus’s abundant solar energy resources, construction
could take as little as 200 years in a best-case scenario, similar to other
proposals for fast terraforming, with all necessary materials except for water
acquired locally. Sufficient water ice to produce arable soils and a
functioning hydrosphere could be delivered from Mars via space elevator within
the construction time frame. Thus, over the very long term, terraforming Venus
could become comparable to terraforming Mars in terms of costs and benefits
for human space colonization.

## Acknowledgements

This research was supported by an appointment to the NASA Postdoctoral Program
at NASA Goddard Space Flight Center, administered by Universities Space
Research Association under contract with NASA. I thank Avi Mandell, Michael
McElwain, Fred Adams, and Michael Way for helpful conversations.

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# Confronting perturbative QCD with the hardest exclusive reactions: kaon
electromagnetic form factors

Long-Bin Chen<EMAIL_ADDRESS>School of Physics and Materials Science,
Guangzhou University, Guangzhou 510006, China  Wen Chen
<EMAIL_ADDRESS>Key Laboratory of Atomic and Subatomic Structure and
Quantum Control (MOE), Guangdong Basic Research Center of Excellence for
Structure and Fundamental Interactions of Matter, Institute of Quantum Matter,
South China Normal University, Guangzhou 510006, China  Guangdong-Hong Kong
Joint Laboratory of Quantum Matter, Guangdong Provincial Key Laboratory of
Nuclear Science, Southern Nuclear Science Computing Center, South China Normal
University, Guangzhou 510006, China  Feng Feng<EMAIL_ADDRESS>China
University of Mining and Technology, Beijing 100083, China  Institute of High
Energy Physics, Chinese Academy of Sciences, Beijing 100049, China  Yu Jia
<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China  School of Physical Sciences, University of
Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

Among countless channels of hard exclusive reactions, the kaon electromagnetic
form factors (EMFFs) are of special interest, which have been measured up to
$Q^{2}\sim 50\;{\rm GeV}^{2}$ in the timelike domain. The kaon EMFFs thereby
serve an ideal platform to critically examine the validity and effectiveness
of perturbative QCD (pQCD) in accounting for hard exclusive processes. In this
work we confront the pQCD predictions that incorporate the next-to-next-to-
leading-order (NNLO) perturbative corrections, with the available kaon EMFFs
data set from experimental measurements and from lattice predictions. The
inclusion of the NNLO corrections turns out to have a substantial and positive
impact. If the profiles of the kaon light-cone distribution amplitudes (LCDAs)
are taken from the recent lattice QCD prediction by LPC Collaboration, the
satisfactory agreement between theory and data can be reached for both charged
and neutral kaons, in both spacelike and timelike large-$Q^{2}$ domains.

##  I Introduction

The electromagnetic form factors (EMFFs) of hadrons, which probe the internal
charge distributions inside a hadron, play a fundamental role in advancing our
knowledge toward Quantum Chromodynamics (QCD) and hadron properties. To date,
the most intensively studied examples reside in the light hadron sector,
exemplified by the EMFFs of the nucleons and the light pseudoscalar mesons
such as pion and kaons Gross:2022hyw .

The kaons, comprising the charged $K^{\pm}$ and neutral
$K^{0}/\overline{K}^{0}$, are the next-to-lightest mesons within the
$SU_{f}(3)$ octet pseudoscalar family. As unique carriers of the strange
quark, the kaons have historically played a vital role in the making of the
Standard Model, in particular in discovering the parity and $CP$ violation in
the weak interaction sector. On the other hand, the revelation of the
approximate $SU(3)$ flavor symmetry in strong interaction sector stands as a
cornerstone in the eventual formulation of QCD. Owing to the hierarchy
$m_{u,d}\ll m_{s}\lesssim\Lambda_{\text{QCD}}$, the conservation of $U$ and
$V$ spins in static properties, such as masses, magnetic moments, and decay
constants, are satisfied with an accuracy of approximately $20\%$, in stark
contrast with the nearly perfect (percent level) precision of isospin
symmetry.

Experimental investigations of the charged kaon EMFFs have been continuously
pursued since 70s Bernardini:1973pe ; DM2:1988obi ; CLEO:2005tiu ;
CMD-2:2008fsu ; Seth:2012nn ; BaBar:2013jqz ; BaBar:2015lgl ;
Serednyakov:2015ooa ; BESIII:2018ldc , while the $K^{0}/\bar{K}^{0}$ EMFFs
have also been the subject of numerous studies since the beginning of this
century Akhmetshin:2002vj ; Seth:2013eaa ; BaBar:2014uwz ; CMD-3:2016nhy ;
BESIII:2021yam ; BESIII:2023zsk . On the theoretical front, the kaon EMFFs
have also been extensively studied using various approaches Wu:2008yr ;
Raha:2008ve ; Raha:2010kz ; Gao:2017mmp ; Krutov:2016luz ; Stamen:2022uqh ;
Ahmed:2023zkk ; Xu:2023izo ; Bruch:2004py ; Miramontes:2022uyi . Governed by
the soft strong interactions, the kaon EMFFs at low $Q^{2}$ have been explored
within chiral perturbation theory Gasser:1983yg , which enables the extraction
of the kaon charge radius. Conversely, at large momentum transfer, the kaon
EMFFs are anticipated to be accurately captured by perturbative QCD within the
context of collinear factorization Lepage:1979zb ; Lepage:1979za ;
Lepage:1980fj ; Efremov:1978rn ; Efremov:1979qk ; Duncan:1979ny ;
Duncan:1979hi  (for a comprehensive review on application of collinear
factorization to the hard exclusive reactions, see Chernyak:1983ej ).

It is interesting to realize that the largest $Q^{2}$ in kaon EMFFs
measurement has surpassed 50 ${\rm GeV}^{2}$ BaBar:2015lgl , which is
significantly greater than 10 ${\rm GeV}^{2}$ in the measured pion EMFFs
Bebek:1977pe ; BaBar:2012bdw , also considerably larger than 30 ${\rm
GeV}^{2}$ in the measured proton EMFFs Sill:1992qw . We note that,
historically the applicability of pQCD at moderate momentum transfer has been
subject to controversy Isgur:1989cy ; Bolz:1996sw ; Radyushkin:1998rt ;
Braun:1999uj , with the crux centered on whether the soft nonfactorizable
mechanisms may dominate the pQCD contributions or not. With $Q^{2}$ reaching
about 50 ${\rm GeV}^{2}$, the kaon EMFFs may be viewed as the hardest
exclusive reaction ever measured experimentally, so that the pQCD should be
reliably applied to such situation. Consequently the kaon EMFFs might serve an
ideal arena to critically testify the validity and effectiveness of pQCD for
hard exclusive processes.

One may wonder whether the kaon EMFFs are identical to the pion EMFFs within
the projected 20% accuracy. Intriguingly, the effects of $SU(3)$ flavor
symmetry breaking in these observables turn to be more pronounced than those
in static properties. Actually there has existed a long-standing puzzle
concerning the kaon EMFFs: although the energy dependence of higher-energy
data aligns with the asymptotic form predicted by pQCD, their magnitudes
exceed the predicted asymptotic value by a factor of approximately four
BaBar:2013jqz ; BaBar:2015lgl .

Recently the next-to-next-to-leading-order (NNLO) perturbative corrections to
pion EMFFs at large momentum transfer have become available in the context of
collinear factorization Chen:2023byr . It was found that the effect of the
NNLO correction is positive and substantial. In comparison with the data, some
strong constraint has been placed on the profile of the leading-twist light-
cone distribution amplitude (LCDA) of pion, especially on the second
Gegenbauber moment of the pion LCDA.

Due to the mass hierarchy between $u$, $d$ and $s$ quark, one anticipates that
the profiles of the kaon LCDAs significantly deviate from those of the pions,
which are symmetric under the exchange $x\leftrightarrow 1-x$. Thereby the
kaon EMFFs provide a prime opportunity to advance our understanding of the
intricate interplay between quark masses and $SU_{f}(3)$ breaking and the
underlying QCD dynamics. It is the very goal of this work to extend the
preceding analysis of pion EMFFs Chen:2023byr  to the kaon EMFFs, by
incorporating the NNLO QCD corrections.

The rest of the paper is distributed as follows. In Sec. II, we sketch the
pQCD formula for kaon EMFFs in collinear factorization. With the virtue of
turning a convolution task into an algebraic one, we present the master
formula for pQCD predictions to kaon EMFFs through NNLO accuracy. In Sec. III,
we perform a comprehensive comparison between the state-of-the-art pQCD
predictions and the available large-$Q^{2}$ data and recent lattice
predictions for the EMFFs of the charged and neutral kaons, in both spacelike
and timelike domains. Finally we summarize in Sec. IV. In Appendix A, we
present the analytic expressions of the lowest-lying short-distance
coefficient matrix elements through NNLO accuracy, which appear in the master
formula given in Sec. II.

##  II Two-loop pQCD corrections to kaon EMFFs

To be specific, let us start with the EMFF of the $K^{+}$ meson. It is defined
via

$\langle K^{+}(P^{\prime})|J_{\rm
em}^{\mu}|K^{+}(P)\rangle=F_{K}(Q^{2})(P^{\mu}+P^{\prime\mu}),$ (1)

where $J^{\mu}_{\rm em}=\sum_{f}e_{f}\bar{\psi}_{f}\gamma^{\mu}\psi_{f}$
signifies the electromagnetic current, and $Q^{2}\equiv-(P^{\prime}-P)^{2}$.

At large momentum transfer, the collinear factorization demands that, at the
lowest order in $1/Q$, the $F_{K}(Q^{2})$ can be expressed in the following
convolution integral:

$F_{K}(Q^{2})=\int\\!\\!\\!\int
dx\,dy\,\Phi_{K}^{*}(x,\mu_{F})\,T\left(x,y,\frac{\mu_{R}^{2}}{Q^{2}},\frac{\mu_{F}^{2}}{Q^{2}}\right)\,\Phi_{K}(y,\mu_{F})+\cdots,$
(2)

where $\mu_{R}$ and $\mu_{F}$ represent the renormalization and factorization
scales, respectively. $T(x,y)$ signifies the perturbatively calculable hard-
scattering kernel, and $\Phi_{K}(x,\mu_{F})$ represents the nonperturbative
yet universal leading-twist kaon LCDA, $i.e.$, the probability amplitude of
finding the valence $u$ and $\bar{s}$ quarks inside $K^{+}$ carrying the
momentum fractions $x$ and $\bar{x}\equiv 1-x$, respectively.

The leading-twist kaon LCDA in (2) assumes the following operator definition:

$\displaystyle\Phi_{K}(x,\mu_{F})$ $\displaystyle=\int\frac{dz^{-}}{2\pi
i}e^{iz^{-}xP^{+}}\left\langle 0\left|\bar{s}(0)\gamma^{+}\gamma_{5}{\cal
W}(0,z^{-})u(z^{-})\right|K^{+}(P)\right\rangle,$ (3)

with ${\cal W}$ signifies the light-like gauge link to ensure gauge
invariance. $\Phi_{K}(x,\mu_{F})$ obeys the celebrated Efremov-Radyushkin-
Brodsky-Lepage (ERBL) evolution equation Lepage:1980fj ; Efremov:1979qk :

$\frac{d\Phi_{K}(x,\mu_{F})}{d\ln\mu^{2}_{F}}=\int_{0}^{1}\\!\\!dyV(x,y)\,\Phi_{K}(y,\mu_{F}).$
(4)

The hard-scattering kernel can be expanded in the power series of
$\alpha_{s}$:

$T(x,y,\mu_{R}^{2}/Q^{2},\mu_{F}^{2}/Q^{2})=\frac{16C_{F}\pi\alpha_{s}}{Q^{2}}\left\\{T^{(0)}+{\alpha_{s}\over\pi}T^{(1)}+\left({\alpha_{s}\over\pi}\right)^{2}T^{(2)}+\cdots\right\\}.$
(5)

The hard-scattering kernel can be determined via standard perturbative
matching strategy. Luckily, there requires no any extra work except one
directly transplants the $T^{(i)}$ ($i=0,1,2$) for the $\pi^{+}$ EMFF
Chen:2023byr  by making the substitution $e_{d}\to e_{s}$. For technical
details, we refer the interested readers to our preceding paper Chen:2023byr .

The leading-twist kaon LCDA is customarily expanded in the basis of the
Gegenbauer polynomials:

$\displaystyle\Phi_{K}(x,\mu_{F})=\frac{f_{K}}{2\sqrt{2N_{c}}}{\sum_{n=0}}a_{n}(\mu_{F})\psi_{n}(x),$
(6a) $\displaystyle\psi_{n}(x)=6x\bar{x}\,C_{n}^{3/2}(2x-1),$ (6b)

where $N_{c}=3$ is the number of colors, $f_{K}=0.160$ GeV signifies the kaon
decay constant ParticleDataGroup:2022pth . All the nonperturbative dynamics is
encoded in the Gegenbauer moments. At a given factorization scale, the running
Gegenbauer moment $a_{n}(\mu_{F})$ is known to three-loop accuracy
Strohmaier:thesis .

Since the V-spin symmetry has been badly broken, the sum in (6a) entails all
integers. This feature should be contrasted with that for the pion LCDA, where
the sum is only extended over even integers due to nearly perfect isospin
symmetry.

To facilitate phenomenological analysis, it is advantageous to turn the
convolution problem into an algebraic one Chen:2023byr . Substituting (6) into
(2), conducting two-fold integration, one can reexpress the kaon EMFF as

$Q^{2}F_{K}(Q^{2})=\frac{2C_{F}\pi^{2}f_{K}^{2}}{3}\sum_{k=0}\left(\alpha_{s}\over\pi\right)^{k+1}\,{\sum_{m,n}}[e_{u}-(-1)^{m+n}e_{s}]\,{\cal
T}^{(k)}_{mn}(\mu/Q)\,a_{m}(\mu)a_{n}(\mu).$ (7)

The short-distance matrix element ${\cal T}^{(k)}_{mn}$ is defined by

${\cal T}^{(k)}_{mn}(\mu/Q)={1\over e_{u}-(-1)^{m+n}e_{s}}\int\\!\\!\\!\int
dx\,dy\,\psi_{m}(x)\,T^{(k)}\left(x,y,\mu^{2}/Q^{2}\right)\,\psi_{n}(y),$ (8)

where we have fixed the number of light quarks to three, and for simplicity
set $\mu_{R}=\mu_{F}=\mu$ in (7).

$(m,n)$ | $c_{1}$ | $c_{2}$ | $d_{1}$ | $d_{2}$ | $d_{3}$
---|---|---|---|---|---
(0,0) | 20.25 | 59.25 | 45.5625 | 302.936 | 514.600
(0,1) | 28.25 | 88.7893 | 76.1181 | 540.224 | 1018.02
(0,2) | 32.75 | 112.473 | 96.4306 | 735.637 | 1498.75
(0,3) | 35.95 | 131.413 | 112.244 | 901.650 | 1933.00
(0,4) | 38.45 | 147.638 | 125.390 | 1049.88 | 2346.49
(1,1) | 36.25 | 130.750 | 113.785 | 908.006 | 1925.92
(1,2) | 40.75 | 160.199 | 138.097 | 1184.52 | 2697.02
(1,3) | 43.95 | 183.855 | 156.755 | 1417.99 | 3398.57
(1,4) | 46.45 | 203.376 | 172.123 | 1619.63 | 4034.36
(2,2) | 45.25 | 192.871 | 164.660 | 1513.95 | 3690.60
(2,3) | 48.45 | 218.864 | 184.918 | 1788.70 | 4575.65
(2,4) | 50.95 | 240.181 | 201.536 | 2024.03 | 5371.58
(3,3) | 51.65 | 246.688 | 206.313 | 2096.55 | 5620.90
(3,4) | 54.15 | 269.327 | 223.820 | 2358.13 | 6548.85
(4,4) | 56.65 | 292.970 | 242.021 | 2640.85 | 7589.17
Table 1: The numerical values of ${\cal T}^{(1,2)}_{mn}(\mu/Q)$ for $0\leq
m,n\leq 4$. We have parameterized ${\cal
T}^{(1)}_{mn}(\mu/Q)=c_{1}L_{\mu}+c_{2}$ and ${\cal
T}^{(2)}_{mn}(\mu/Q)=d_{1}L_{\mu}^{2}+d_{2}L_{\mu}+d_{3}$, with
$L_{\mu}\equiv\ln Q^{2}/\mu^{2}$.

The two-fold convolution integrals in (8) can be worked out in closed form at
various perturbative orders. At tree level, one simply has ${\cal
T}^{(0)}_{mn}=9$ for all $m,n$. The analytic expressions of some lowest-lying
short-distance matrix elements ${\cal T}^{(1,2)}_{mn}$ have been presented In
Appendix A. For reader’s convenience, we enumerate in Table 1 the numerical
values of ${\cal T}^{(1,2)}_{mn}$ for $0\leq m,n\leq 4$, which appears to
suffice for most phenomenological applications at current stage. It is
straightforward to adapt the master formula for the charged kaon EMFF from
spacelike domain to the timelike one, provided that one makes the replacement
$L_{\mu}\to L_{\mu}+i\pi$ in Table 1, with $Q^{2}$ now indicating the squared
invariant mass of the $K^{+}K^{-}$ pair.

In the exact $U$-spin limit, the neutral kaon EMFF vanishes due to
$e_{d}=e_{s}$. The magnitudes of the neutral kaon EMFFs thereby signal the
$SU(3)$ flavor symmetry breaking effect. For the EMFF of neutral $K$ mesons,
one simply makes the replacement $e_{u}\to e_{d}$ in (7). In this case, the
short-distance matrix element becomes nonvanishing only when $m+n$ is an odd
number. Consequently, the leading term of ${\cal T}^{(k)}_{mn}$ starts with
$(m,n)$ equal to $(0,1)$ rather than $(0,0)$.

Although it is the $K^{0}/\bar{K}^{0}$ that are produced via strong force, it
is the approximate $CP$ eigenstates $K_{S}/K_{L}$
($K_{S}={1\over\sqrt{2}}(|K^{0}\rangle+|\overline{K}^{0}\rangle)$,
$K_{L}={1\over\sqrt{2}}(|K^{0}\rangle-|\overline{K}^{0}\rangle)$) which are
reconstructed via weak decay. By charge conjugation symmetry, the following
different kinds of kaon EMFFs are related to each other:

$F_{K^{+}}(Q^{2})=-F_{K^{-}}(Q^{2}),\quad
F_{K^{0}}(Q^{2})=-F_{\bar{K}^{0}}(Q^{2}),\quad|F_{K_{L}}(Q^{2})|=|F_{K_{S}}(Q^{2})|=|F_{K^{0}}(Q^{2})|.$
(9)

These relations hold both in spacelike and timelike domains.

##  III Confronting the state-of-the-art pQCD predictions with data

As the key nonperturative input, our knowledge of the leading-twist kaon LCDA
is still far from complete. Early phenomenological analysis simply uses the
asymptotic form and CZ parametrization Chernyak:1983ej . In recent years the
kaon LCDA have been investigated from different theoretical approaches, such
as light-cone sum rule Khodjamirian:2004ga ; Ball:2006wn , instanton vacuum
model Nam:2006mb ; Nam:2006sx ; Nam:2006au , holographic QCD Chang:2016ouf ,
light-front quantization Lan:2019rba , and recently from the lattice
simulations Braun:2006dg ; RQCD:2019osh ; LatticeParton:2022zqc . The
$SU_{f}(3)$ symmetry breaking in kaon LCDA has also been studied within the
QCD sum rule approach Ball:2003sc ; Chetyrkin:2007vm . The kaon’s Gegenbauer
moments predicted from various approaches are scattered in a wide range.

$a_{i}$(2 GeV) | $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$
---|---|---|---|---
RQCD RQCD:2019osh  | $-0.0525^{+31}_{-33}$ | $0.106^{+15}_{-16}$ | - | -
LPC LatticeParton:2022zqc  | $-0.108\pm 0.014\pm 0.051$ | $0.170\pm 0.014\pm 0.044$ | $-0.043\pm 0.006\pm 0.022$ | $0.073\pm 0.008\pm 0.021$
Table 2: The values of $a_{i}$ ($i=1,\cdots,4$) at 2 GeV evaluated by RQCD and
LPC.

In the phenomenological study, we take the recent lattice results by RQCD
RQCD:2019osh  and LPC LatticeParton:2022zqc  Collaborations as the inputs for
the Gegenbauer moments of the kaon LCDAs in $\overline{\rm MS}$ scheme, whose
values are tabulated in Table 2 555It appears that the convention in defining
the kaon LCDA differs between RQCD RQCD:2019osh  and LPC LatticeParton:2022zqc
. We have inserted an additional minus sign to $a_{1}$ in RQCD prediction to
be in conformity with the LPC prediction.. These two groups of lattice studies
are based on quite different algorithms. RQCD has presented precise
predictions of the first and second Gegenbauer moment of the kaon LCDA. LPC
has presented the profile of the kaon LCDA, from which various Gegenbauer
moments can be extracted, and relatively larger $SU_{f}(3)$-symmetry breaking
effects have been observed.

We use three-loop evolution equation Braun:2017cih ; Strohmaier:thesis  to
evolve each $a_{n}$ evaluated initially at 2 GeV by lattice simulation to any
desired scale $\mu$. We retain only those $a_{n}$ with $n$ up to $4$ in
phenomenological exploration. The package FAPT Bakulev:2012sm  is invoked to
evaluate the running QCD coupling constant to three-loop accuracy.

Figure 1: pQCD predictions vs. experimental data for $Q^{2}F_{K^{\pm}}(Q^{2})$
in the space-like (left) and timelike (right) domains. We take the central
values of $a_{1,2}$ provided by RQCD as input in making predictions. The red,
green and blue bands correspond to the LO, NLO and NNLO predictions with the
uncertainty estimated by sliding $\mu$ from $Q/2$ to $Q$. Due to lack of
experimental measurement in the spacelike domain, we take the lattice data
points from the very recent lattice study Ding:2024lfj  as substitution. In
the timelike domain, we take the data points from BaBar BaBar:2015lgl  (black)
and BESIII BESIII:2018ldc (red) experiments.

Figure 2: Same as Fig. 1, except the predictions are made by taking the
central values of $a_{i}$ from LPC as inputs.

In Fig. 1 and Fig. 2 we confront the state-of-the-art pQCD predictions with
the available data for charged kaon EMFFs, with the predictions made by taking
the central values of $a_{i}$ given by RQCD and LPC, respectively, as
nonperturbative input parameters. Since the experimental measurements of kaon
EMFFs at spacelike domain are rather sparse, we invoke the very recent
accurate lattice determination of the spacelike kaon EMFFs up to quite large
$Q^{2}$ Ding:2024lfj  as an experimental substitute. The charged kaon EMFFs in
the timelike domain have recently been measured quite precisely by BaBar
BaBar:2015lgl  and BESIII BESIII:2018ldc . It is impressive that the covered
$Q^{2}$ range in the former experiment has surpassed 50 ${\rm GeV}^{2}$, which
implies that $e^{+}e^{-}\to K^{+}K^{-}$ can be viewed as the hardest hard
exclusive reaction, an ideal platform to test the validity of the pQCD and
collinear factorization.

Figure 3: pQCD predictions vs. BESIII measurement of
$Q^{2}F_{K_{S}K_{L}}(Q^{2})$ in the timelike domain. The predictions are made
by taking the central values of the Gegenbauer moments of the kaon LCDA from
RQCD (left) an LPC (right) as inputs, respectively. The red, green and blue
bands correspond to the LO, NLO and NNLO predictions, with the uncertainty
estimated by sliding $\mu$ from $Q/2$ to $Q$.

In Fig. 3, we also compare the pQCD predictions of the neutral kaon EMFF in
timelike domain at various perturbative accuracy with the very recent BESIII
data BESIII:2021yam ; BESIII:2023zsk . Clearly the measured
$F_{K_{S}K_{L}}(Q^{2})$ is nonvanishing in a wide range of $Q^{2}$, thereby
indicating a notable $SU_{f}(3)$ flavor symmetry breaking effect.

From Fig. 1 to Fig. 3, we observe a unified pattern that the NNLO perturbative
corrections are always positive and substantial for both charged and neutron
kaons. Under all the circumstances, including the two-loop QCD corrections
further increase the NLO predictions by a factor of two, thereby bearing a
significant phenomenological impact. We also observe that, the NNLO
predictions based on the RQCD input appear to be significantly lower than the
lattice predictions in spacelike domain Ding:2024lfj  and the measured
timelike form factors in $e^{+}e^{-}$ collision experiments at large $Q^{2}$
range, for both charged and neutral kaons. On the contrary, the NNLO
predictions based on the LPC input can satisfactorily account for the lattice
prediction and measured values for kaon EMFFs at large $Q^{2}$. The reason may
be largely attributed to the fact that the value of $a_{1}$ given by LPC is
about twice greater than that by RQCD 666It is worth reminding that, in the
pion case, it is the NNLO prediction based on the RQCD input that gives a
better account for the measured EMFFs than that based on the LPC input
Chen:2023byr ..

To strengthen our understanding, we further confront the NNLO predictions with
the measurement of the timelike neutral kaon EMFF at
$Q^{2}=17.4\;\text{GeV}^{2}$ by CLEO-c Seth:2013eaa :

$\displaystyle F_{K_{S}K_{L}}(17.4\;\text{GeV}^{2})|_{\tt CLEO\\!-\\!c}$
$\displaystyle=$ $\displaystyle 5.3\times 10^{-3},\qquad\left[(3.1-7.9)\times
10^{-3}\;{\rm at}\;90\%\;{\rm C.L.}\right]$ (10a) $\displaystyle
F_{K_{S}K_{L}}(17.4\;\text{GeV}^{2})|^{\texttt{RQCD}}_{\text{NNLO}}$
$\displaystyle=$ $\displaystyle(1.4-2.1)\times 10^{-3},$ (10b) $\displaystyle
F_{K_{S}K_{L}}(17.4\;\text{GeV}^{2})|^{\texttt{LPC}}_{\text{NNLO}}$
$\displaystyle=$ $\displaystyle(5.2-8.5)\times 10^{-3},$ (10c)

where the theoretical uncertainties are estimated by varying the
renormalization and factorization scales in the range $Q/2<\mu<Q$. Again the
NNLO predictions based on the LPC input agrees with the CLEO-c measurement
to a decent degree.

CLEO-c also presents the ratio of the timelike EMFFs of the neutral kaon to
that of the charged kaon at $Q^{2}=17.4\;\text{GeV}^{2}$ Seth:2013eaa , which
gauges the degree of the $SU(3)_{f}$ symmetry breaking. This measurement can
be contrasted with the NNLO predictions based on the RQCD and LPC inputs, with
uncertainties estimated by varying $\mu$ in the range $Q/2<\mu<Q$:

$\displaystyle{F_{K_{S}K_{L}}(17.4\;\text{GeV}^{2})\over
F_{K^{+}K^{-}}(17.4\;\text{GeV}^{2})}\Big{|}_{\tt CLEO\\!-\\!c}$
$\displaystyle=$ $\displaystyle 0.12,\qquad\left[0.07-0.19\;{\rm
at}\;90\%\;{\rm C.L.}\right]$ (11a)
$\displaystyle\frac{F_{K_{S}K_{L}}(17.4\text{GeV}^{2})}{F_{K^{+}K^{-}}(17.4\text{GeV}^{2})}\Big{|}^{\tt
RQCD}_{\text{NNLO}}$ $\displaystyle=$ $\displaystyle 0.076-0.087,$ (11b)
$\displaystyle\frac{F_{K_{S}K_{L}}(17.4\text{GeV}^{2})}{F_{K^{+}K^{-}}(17.4\text{GeV}^{2})}\Big{|}^{\tt
LPC}_{\text{NNLO}}$ $\displaystyle=$ $\displaystyle 0.21-0.24.$ (11c)

The same ratio of the timelike EMFFs of the neutral to the charged kaons has
also been measured at BESIII in the interval
$12\;\text{GeV}^{2}<Q^{2}<25\;\text{GeV}^{2}$ BESIII:2023zsk :

${F_{K_{S}K_{L}}(12\;\text{GeV}^{2}<Q^{2}<25\;\text{GeV}^{2})\over
F_{K^{+}K^{-}}(12\;\text{GeV}^{2}<Q^{2}<25\;\text{GeV}^{2})}\Big{|}_{\tt
BESIII}=0.21\pm 0.01.$ (12)

Figure 4: Ratio of the EMFFs of neutral to charged kaons in the timelike
domain as the function of $Q^{2}$. For definiteness, we have chosen the
various Gegenbauer moments of kaon LCDA with the central values of LPC
predictions. The red, green and blue bands correspond to the LO, NLO and NNLO
prediction, with the errors estimated by sliding $\mu$ from $Q/2$ to $Q$. The
black rectangles represent the ratio measured by BESIII measurement, (12).
However in the theoretical calculation, we extend the upper limit of $Q^{2}$
from $25\;\text{GeV}^{2}$ to $60\;\text{GeV}^{2}$.

In Fig. 4 we also juxtapose the ratio predicted by pQCD using the LPC input
and measured by BESIII, (12). It is evident from (11) and Fig. 4 that, the
pQCD predictions at NNLO accuracy, when armed with the Gegenbauber moments
provided by LPC, gives a satisfactory account of the CLEO-c and BESIII data.

##  IV Summary

The timelike kaon EMFFs, viz., $e^{+}e^{-}\to\gamma^{*}\to K\overline{K}$, are
arguably the hardest exclusive reactions ever measured experimentally, with
the momentum transfer exceeding $50\;{\rm GeV}^{2}$. Therefore, the kaon EMFFs
may be regarded as an ideal testing ground for the validity of pQCD when
applied to hard exclusive processes. Moreover, the magnitude of the neutral
kaon’s EMFF also affords a sensitive probe to the $SU(3)$ flavor symmetry
breaking effect.

In this work we conduct a comprehensive comparison between the state-of-the-
art pQCD predictions in collinear factorization and the available kaon EMFFs
data sets with large momentum transfer. For the lacking of kaon EMFF
measurement in the spacelike domain, we compare our pQCD predictions with the
recent lattice determinations of the kaon EMFF at large $Q^{2}$. In the
timelike domain, we confront our predictions with the BaBar, BESIII and
CLEO-c data. The key finding is that, the inclusion of the NNLO perturbative
corrections has a substantial and positive impact on the theoretical
predictions. We also observe that, if the values of Gegenbauber moments of the
kaon LCDAs are taken from the lattice results given by LPC, the satisfactory
agreement between theory and data (including lattice predictions to spacelike
kaon EMFFs) can be reached for both charged and neutral kaons, in both
spacelike and timelike domains with large momentum transfer. Obviously it is
highly desirable if the kaon LCDAs can be ascertained to a higher precision.

###### Acknowledgements.

We thank Heng-Tong Ding and Qi Shi for providing us with the lattice
predictions for the spacelike kaon EMFF in Ding:2024lfj . We are also indebted
to Jun Hua for clarifying the sign discrepancy in $a_{1}$ between RQCD and LPC
predictions. The work of L.-B. C. is supported by the NNSFC under Grant No.
12175048. The work of W. C. is supported by Guangdong Major Project of Basic
and Applied Basic Research (No. 2020B0301030008). The work of F. F. is
supported by the NNSFC under Grant No. 12275353. The work of Y. J. is
supported in part by the NNSFC under Grant No. 11925506.

##  Appendix A The analytical expressions of the lowest-lying ${\cal
T}^{(k)}_{mn}$

The two-fold convolution integrals in (8) can be worked out in the closed
form. At one-loop level, the lowest-lying short-distance coefficient matrix
elements read

$\displaystyle{\cal T}^{(1)}_{01}$ $\displaystyle=$
$\displaystyle\frac{113}{4}L_{\mu}-\frac{18}{5}\zeta_{3}+\frac{5587}{60},$
(13a) $\displaystyle{\cal T}^{(1)}_{02}$ $\displaystyle=$
$\displaystyle\frac{131}{4}L_{\mu}-9\zeta_{3}+\frac{2959}{24},$ (13b)
$\displaystyle{\cal T}^{(1)}_{03}$ $\displaystyle=$
$\displaystyle\frac{719}{20}L_{\mu}-9\zeta_{3}+\frac{85339}{600},$ (13c)
$\displaystyle{\cal T}^{(1)}_{04}$ $\displaystyle=$
$\displaystyle\frac{769}{20}L_{\mu}-9\zeta_{3}+\frac{47537}{300},$ (13d)
$\displaystyle{\cal T}^{(1)}_{11}$ $\displaystyle=$
$\displaystyle\frac{145}{4}L_{\mu}+\frac{523}{4},$ (13e) $\displaystyle{\cal
T}^{(1)}_{12}$ $\displaystyle=$
$\displaystyle\frac{163}{4}L_{\mu}-\frac{351\zeta_{3}}{35}+\frac{48231}{280},$
(13f) $\displaystyle{\cal T}^{(1)}_{13}$ $\displaystyle=$
$\displaystyle\frac{879}{20}L_{\mu}-27\zeta_{3}+\frac{259573}{1200},$ (13g)
$\displaystyle{\cal T}^{(1)}_{14}$ $\displaystyle=$
$\displaystyle\frac{929}{20}L_{\mu}-27\zeta_{3}+\frac{141499}{600},$ (13h)
$\displaystyle{\cal T}^{(1)}_{22}$ $\displaystyle=$
$\displaystyle\frac{181}{4}L_{\mu}+\frac{54\zeta_{3}}{7}+\frac{20563}{112},$
(13i) $\displaystyle{\cal T}^{(1)}_{23}$ $\displaystyle=$
$\displaystyle\frac{969}{20}L_{\mu}-\frac{108\zeta_{3}}{7}+\frac{997123}{4200},$
(13j) $\displaystyle{\cal T}^{(1)}_{24}$ $\displaystyle=$
$\displaystyle\frac{1019}{20}L_{\mu}-54\zeta_{3}+\frac{732221}{2400},$ (13k)
$\displaystyle{\cal T}^{(1)}_{33}$ $\displaystyle=$
$\displaystyle\frac{1033}{20}L_{\mu}+30\zeta_{3}+\frac{2274761}{10800},$ (13l)
$\displaystyle{\cal T}^{(1)}_{34}$ $\displaystyle=$
$\displaystyle\frac{1083}{20}L_{\mu}-\frac{180\zeta_{3}}{11}+\frac{22888579}{79200},$
(13m) $\displaystyle{\cal T}^{(1)}_{44}$ $\displaystyle=$
$\displaystyle\frac{1133}{20}L_{\mu}+\frac{810\zeta_{3}}{11}+\frac{674701}{3300}.$
(13n)

At two-loop level, the lowest-lying short-distance coefficient matrix elements
read

$\displaystyle{\cal T}^{(2)}_{00}$ $\displaystyle=$
$\displaystyle\frac{729L_{\mu}^{2}}{16}-(8\zeta_{3}+\frac{35\pi^{2}}{6}-\frac{2961}{8})L_{\mu}+205\zeta_{5}-\frac{3\pi^{4}}{20}-\frac{651\zeta_{3}}{2}-\frac{275\pi^{2}}{24}+821,$
(14a) $\displaystyle{\cal T}^{(2)}_{01}$ $\displaystyle=$
$\displaystyle\frac{10961L_{\mu}^{2}}{144}+\left(-\frac{137\zeta_{3}}{5}+\frac{704879}{1080}-\frac{145\pi^{2}}{18}\right)L_{\mu}-\frac{1577\zeta_{5}}{2}-\frac{4\pi^{2}\zeta_{3}}{5}-\frac{251\pi^{4}}{1350}$
(14b) $\displaystyle+$
$\displaystyle\frac{16009\zeta_{3}}{100}-\frac{20857\pi^{2}}{1080}+\frac{30154739}{16200},$
$\displaystyle{\cal T}^{(2)}_{02}$ $\displaystyle=$
$\displaystyle\frac{6943L_{\mu}^{2}}{72}+\left(-61\zeta_{3}+\frac{387371}{432}-\frac{80\pi^{2}}{9}\right)L_{\mu}+\frac{155\zeta_{5}}{2}-2\pi^{2}\zeta_{3}-\frac{67\pi^{4}}{270}-\frac{107491\zeta_{3}}{180}$
(14c) $\displaystyle-$
$\displaystyle\frac{5921\pi^{2}}{216}+\frac{254500067}{103680},$
$\displaystyle{\cal T}^{(2)}_{03}$ $\displaystyle=$
$\displaystyle\frac{202039L_{\mu}^{2}}{1800}+\left(-\frac{321\zeta_{3}}{5}+\frac{58020191}{54000}-\frac{436\pi^{2}}{45}\right)L_{\mu}+\frac{165\zeta_{5}}{2}-2\pi^{2}\zeta_{3}-\frac{343\pi^{4}}{1350}$
(14d) $\displaystyle-$
$\displaystyle\frac{12785497\zeta_{3}}{18900}-\frac{28237\pi^{2}}{900}+\frac{3042907421}{1008000},$
$\displaystyle{\cal T}^{(2)}_{04}$ $\displaystyle=$
$\displaystyle\frac{451403L_{\mu}^{2}}{3600}+\left(-\frac{667\zeta_{3}}{10}+\frac{33226643}{27000}-\frac{917\pi^{2}}{90}\right)L_{\mu}+\frac{355\zeta_{5}}{4}-2\pi^{2}\zeta_{3}-\frac{1397\pi^{4}}{5400}$
$\displaystyle-$
$\displaystyle\frac{76336859\zeta_{3}}{100800}-\frac{740213\pi^{2}}{21600}+\frac{257785535129}{72576000},$
$\displaystyle{\cal T}^{(2)}_{11}$ $\displaystyle=$
$\displaystyle\frac{16385L_{\mu}^{2}}{144}+\left(-8\zeta_{3}+\frac{220117}{216}-\frac{185\pi^{2}}{18}\right)L_{\mu}+3546\zeta_{5}-\frac{3\pi^{4}}{20}-\frac{56177\zeta_{3}}{20}-\frac{2711\pi^{2}}{108}+\frac{4892983}{2592},$
(14e) $\displaystyle{\cal T}^{(2)}_{12}$ $\displaystyle=$
$\displaystyle\frac{9943L_{\mu}^{2}}{72}+\left(-\frac{2659\zeta_{3}}{35}+\frac{20948881}{15120}-\frac{100\pi^{2}}{9}\right)L_{\mu}-\frac{11037\zeta_{5}}{2}-\frac{78\pi^{2}\zeta_{3}}{35}-\frac{869\pi^{4}}{3150}$
(14f) $\displaystyle+$
$\displaystyle\frac{81199421\zeta_{3}}{29400}-\frac{294449\pi^{2}}{7560}+\frac{140650728527}{25401600},$
$\displaystyle{\cal T}^{(2)}_{13}$ $\displaystyle=$
$\displaystyle\frac{31351L_{\mu}^{2}}{200}+\left(-\frac{1003\zeta_{3}}{5}+\frac{15990101}{9000}-\frac{536\pi^{2}}{45}\right)L_{\mu}+\frac{495\zeta_{5}}{2}-6\pi^{2}\zeta_{3}-\frac{38\pi^{4}}{75}$
(14g) $\displaystyle-$
$\displaystyle\frac{39593063\zeta_{3}}{25200}-\frac{82603\pi^{2}}{1350}+\frac{208836835727}{36288000},$
$\displaystyle{\cal T}^{(2)}_{14}$ $\displaystyle=$
$\displaystyle\frac{619643L_{\mu}^{2}}{3600}+\left(-\frac{2081\zeta_{3}}{10}+\frac{107582681}{54000}-\frac{1117\pi^{2}}{90}\right)L_{\mu}+\frac{1035\zeta_{5}}{4}-6\pi^{2}\zeta_{3}-\frac{937\pi^{4}}{1800}$
(14h) $\displaystyle-$
$\displaystyle\frac{59475599\zeta_{3}}{33600}-\frac{470201\pi^{2}}{7200}+\frac{1450125210497}{217728000},$
$\displaystyle{\cal T}^{(2)}_{22}$ $\displaystyle=$
$\displaystyle\frac{23711L_{\mu}^{2}}{144}+\left(\frac{337\zeta_{3}}{7}+\frac{9519367}{6048}-\frac{215\pi^{2}}{18}\right)L_{\mu}+\frac{281325\zeta_{5}}{14}+\frac{12\pi^{2}\zeta_{3}}{7}-\frac{29\pi^{4}}{630}$
(14i) $\displaystyle-$
$\displaystyle\frac{20454703\zeta_{3}}{1470}-\frac{37249\pi^{2}}{1512}-\frac{977775959}{5080320},$
$\displaystyle{\cal T}^{(2)}_{23}$ $\displaystyle=$
$\displaystyle\frac{73967L_{\mu}^{2}}{400}+\left(-\frac{4402\zeta_{3}}{35}+\frac{28919281}{14000}-\frac{1147\pi^{2}}{90}\right)L_{\mu}-\frac{44685\zeta_{5}}{2}-\frac{24\pi^{2}\zeta_{3}}{7}-\frac{773\pi^{4}}{2100}$
(14j) $\displaystyle+$
$\displaystyle\frac{1162960399\zeta_{3}}{88200}-\frac{1365911\pi^{2}}{25200}+\frac{3573525754441}{285768000},$
$\displaystyle{\cal T}^{(2)}_{24}$ $\displaystyle=$
$\displaystyle\frac{90691L_{\mu}^{2}}{450}+\left(-\frac{2176\zeta_{3}}{5}+\frac{48201937}{18000}-\frac{596\pi^{2}}{45}\right)L_{\mu}+\frac{915\zeta_{5}}{2}-12\pi^{2}\zeta_{3}-\frac{847\pi^{4}}{900}$
(14k) $\displaystyle-$
$\displaystyle\frac{25432877\zeta_{3}}{8400}-\frac{2216813\pi^{2}}{21600}+\frac{1065085696391}{108864000},$
$\displaystyle{\cal T}^{(2)}_{33}$ $\displaystyle=$
$\displaystyle\frac{742727L_{\mu}^{2}}{3600}+\left(\frac{695\zeta_{3}}{3}+\frac{1897099387}{972000}-\frac{1219\pi^{2}}{90}\right)L_{\mu}+\frac{144475\zeta_{5}}{2}+\frac{20\pi^{2}\zeta_{3}}{3}+\frac{119\pi^{4}}{405}$
(14l) $\displaystyle-$
$\displaystyle\frac{91161661\zeta_{3}}{1890}-\frac{74287\pi^{2}}{16200}-\frac{773409858259}{68040000},$
$\displaystyle{\cal T}^{(2)}_{34}$ $\displaystyle=$
$\displaystyle\frac{11191L_{\mu}^{2}}{50}+\left(-\frac{1576\zeta_{3}}{11}+\frac{1585362629}{594000}-\frac{632\pi^{2}}{45}\right)L_{\mu}-\frac{1465225\zeta_{5}}{22}-\frac{40\pi^{2}\zeta_{3}}{11}-\frac{793\pi^{4}}{1980}$
(14m) $\displaystyle+$
$\displaystyle\frac{1689813617\zeta_{3}}{40656}-\frac{44720293\pi^{2}}{712800}+\frac{2603110257632537}{98794080000},$
$\displaystyle{\cal T}^{(2)}_{44}$ $\displaystyle=$
$\displaystyle\frac{871277L_{\mu}^{2}}{3600}+\left(\frac{6833\zeta_{3}}{11}+\frac{605198897}{297000}-\frac{1309\pi^{2}}{90}\right)L_{\mu}+\frac{4380975\zeta_{5}}{22}+\frac{180\pi^{2}\zeta_{3}}{11}+\frac{67\pi^{4}}{66}$
(14n) $\displaystyle-$
$\displaystyle\frac{8897578969\zeta_{3}}{67760}+\frac{511331\pi^{2}}{9900}-\frac{11028087335760793}{263450880000}.$

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