###### Abstract

This paper studies liquid staking tokens (LSTs) on automated market makers
(AMMs), both theoretically and empirically. LSTs are tokenized representations
of staked assets on proof-of-stake blockchains.

First, we theoretically model LST-liquidity on AMMs. This includes
categorizing suitable AMM types for LST liquidity, as well as deriving
formulas for the necessary returns from trading fees to adequately compensate
liquidity providers under the particular price trajectories of LSTs. Two
relevant metrics are considered. Firstly, losses compared to holding the
liquidity outside the AMM (loss-versus-holding, or “impermanent loss”).
Secondly, the relative profitability compared to fully staking the capital
(loss-versus-staking) which is a metric is specifically introduced for the
case of LST-liquidity.

Subsequently, we empirically measure these metrics for Ethereum LSTs across
the most relevant AMM pools. We find that, while trading fees often compensate
for impermanent loss, fully staking is more profitable for many pools, putting
the sustainability of current LST allocation to AMMs into question.

Liquid Staking Tokens in
Automated Market Makers

Krzysztof Gogol Robin Fritsch Malte Schlosser Johnnatan Messias Benjamin
Kraner Claudio Tessone

##  1 Introduction

Liquid Staking Tokens (LSTs) have become the preferred method of staking, now
constituting over a third of total staked ETH [6]. Moreover, liquid staking
has grown to become the largest category in decentralized finance (DeFi) by
allocated capital. DeFi itself has experienced remarkable growth in recent
years with over 5 million private wallet addresses interacting with DeFi
protocols [4], and investment and central banks are experimenting with its
application in traditional finance [20, 19]. While the emergence of liquid
staking has been one of the DeFi ecosystem’s most significant recent
developments, it has so far seen limited academic attention.

In traditional staking on proof-of-stake (PoS) blockchains, stakers lock
(“stake”) native blockchain tokens with a validator, typically subject to a
lock-up period. Validators maintain the blockchain network and receives
staking rewards in exchange. Stakers have the option to run their own
validators or become customers of a staking pool operator. In the case of the
Ethereum blockchain, the minimum collateral of 32 Ether (ETH) is required to
set up and operate a validator.

In contrast, liquid staking tokens (LSTs) allow for more simple and flexible
staking. LSTs are tokenized representations of the staked underlying asset,
and can be minted and redeemed for the corresponding amount of the underlying
asset directly from the LST protocol. Additionally, since LSTs are fungible
tokens, they can be bought and sold on exchanges, in particular on
decentralized automated market makers (AMMs).

LST protocols manage a network of validators and take a percentage of the
staking rewards as compensation for their services [26]. Holders of LSTs can
earn staking rewards without directly locking any tokens in their own
validator or with an external validator operator. Moreover, LSTs enable
combining the benefits of staking with other functionalities within DeFi.

The emergence of LSTs has already transformed decentralized lending, with LSTs
becoming the dominant collateral [9], and has also reshaped investment
strategies within DeFi. A unique advantage and feature of LSTs which is also
vital to the intended functioning of LSTs is the possibility to trade them on
AMMs, which enables instant entering or exiting of staking positions.
Precisely this aspect, i.e. LST liquidity on AMMs, is the subject of this
research.

###  1.1 Methodology and Contribution

To the best of our knowledge, this is the first work to systematically study –
both theoretically and empirically – the functioning of LST-liquidity on AMMs.
A central aspect in this regard is the profitability of providing LSTs as
liquidity to AMMs.

On AMMs, LSTs are commonly traded against their underlying asset. Since the
price of LSTs is by design either pegged to a unit of the underlying asset or
experiences a constant increase compared to the underlying asset as a result
of staking rewards, such LST trading pairs exhibit a specific behavior: Apart
from short-term price fluctuations, their price changes are constant and
predictable.

This paper studies AMM liquidity in this particular setting. We begin by
categorizing suitable AMM types for LST liquidity. Subsequently, we
concentrate on LP profitability. We analyze empirically whether, historically,
trading fees sufficiently compensated liquidity provider (LPs) for the costs
of providing liquidity. Moreover, we theoretically derive formulas for the
necessary returns from trading fees, as a function of the staking rate.

For this purpose, we consider the following two metrics measuring the losses
to LPs on AMMs.

First, we consider a well-established metric for LP profitability on AMMs:
loss-vs-holding (LVH) – also known as“impermanent loss” or “divergence loss”.
It measures how much a liquidity position loses from changing prices compared
to keeping the initial position outside the AMM.

A second relevant measure, that we introduce specifically for LSTs, is
comparing proving LST liquidity on an AMM to staking. We coin the term _loss-
versus-staking_ (LVS) for this metric. Instead of providing an LST and its
underlying assets as liquidity on an AMM, LP can stake its full capital. In
the latter case, the LP earns staking rewards on their full capital compared
to trading fees from the AMM and staking rewards on part of their capital in
the first case.

This work begins theoretically by deriving formulas for the necessary returns
from trading fees required to adequately compensate LPs for each of the losses
under the particular price trajectory of LSTs. This results in expressions for
the required returns as a function of the staking rate $r$.

Subsequently, we empirically measure these metrics for the largest Ethereum
LSTs: stETH and wstETH by Lido [2], Coinbase’s cbETH [3] and Rocketpool’s rETH
[11]. Together, they represent over 90% of TVL in liquid
staking111https://defillama.com/lsd. For these tokens, we consider the largest
AMM pools in terms of TVL, which are on the decentralized exchanges Curve (v1
[22] and v2 [23]) and Uniswap (v3 [17]) on the Ethereum blockchain [21].

We find that simply holding LSTs is more profitable than allocating LSTs to
AMM liquidity pools: trading fees on AMMs did not compensate LPs for not
staking their capital fully (by allocating it to LSTs) during the studied time
period. Yet, impermanent loss was sufficiently compensated for most of the
analyzed liquidity pools. Nonetheless, our result put the sustainability of
the current allocation of LST liquidity on AMMs into question.

##  2 Related Work

Liquid staking has received limited attention in the literature so far. The
price difference between LSTs and their underlying cryptocurrency has been
investigated by Scharnowski et al. [33]. A taxonomy for LSTs, including the
differentiation between rebase, reward and dual token models, can be found in
the SoK [26] LSTs’ tracking of staking rewards has been empirically examined
in [25]. Furthermore, leveraging LSTs on lending protocols and the principal-
agent problem in liquid staking have been discussed in [36] and [35],
respectively.

Liquidity provider returns have been discussed in the literature since the
first theoretical analysis of the current form of AMMs [18]. A comprehensive
theoretical treatment of slippage and impermanent loss in different types of
AMMs, including Uniswap v3 and Curve v1 which are of interest for this paper,
can be found in the SoK by Xu et al. [37]. Furthermore, a general framework
for impermanent loss has been laid out in [34].

On the empirical side, the general profitability of liquidity providers on
AMMs has been studied in the literature for different AMMs, token pairs, and
time periods: for Uniswap v2 in [28] and for Uniswap v3 in [24, 27, 29]. Most
of these works focus on impermanent loss (loss-versus-holding) as a measure of
LP profitability. As an alternative metric, loss-versus-rebalancing (LVR) that
benchmarks AMM liquidity against a rebalancing portfolio that hedges the
position’s market risk was introduced in [31], [30].

##  3 Background

###  3.1 Liquid Staking Tokens

LSTs are tokenized representations of staked native PoS blockchain tokens.
They can be redeemed for the underlying token at any time (possibly with a
delay) at the LST protocol, or traded on AMMs. Consequently, their market
price on AMMs oscillates around the value of the staked assets they represent.
Like other pegged tokens such as stablecoins (pegged to fiat currency) and
wrapped tokens (pegged to tokens from other blockchains), LSTs offer a means
to represent staked tokens in a liquid and tradable form. LSTs can be
categorized into two main types based on their staking distribution models:
rebase or reward [25]. The third model - dual token LSTs - is disappearing
[12]. LSTs also vary in validator selection processes and governance [26].
Additionally, liquid staking protocols typically charge a fee, usually around
10-25% of the staking rewards, to cover operational costs and provide
incentives for the platform’s maintenance and development.

####  3.1.1 Rebase-LST

In the rebase model, LST holders receive staking rewards in the form of
additional LSTs on a daily basis. Meanwhile, the rebase-LST tokens always
remain redeemable for 1 ETH (or 1 unit of its underlying asset), meaning its
price stays close to 1 ETH. A prominent example is stETH from the Lido
Protocol [2], which was the first LST and currently holds the largest market
capitalization among liquid staking tokens. The historical values of rebase-
LST, denominated in ETH, are presented in Figure 5.

The rebase model is not compatible with certain DeFi protocols such as lending
pools like Aave [1] and Compound [32]. These platforms do not support stETH
since lenders to stETH pools would only earn staking rewards from stETH tokens
in the pool that have not been borrowed. Moreover, rebase-LSTs are not
compatible with some AMMs, Uniswap being the most notable example [13, 8].

####  3.1.2 Reward-LST

For reward-LSTs, the staking rewards are accumulated in the value of the
staking tokens. That is, one reward-LST token is always redeemable for its
initial underlying amount (e.g. 1 ETH) as well as all staking rewards earned
up to this time. Hence, the value of reward-LSTs measured in ETH will increase
over time. Examples of reward-LSTs include wstETH from Lido[2], rETH from
RocketPool [11] and cbETH from CoinBase [3]. The historical prices of reward-
LSTs are presented in Figure 6 and compared to the returns from staking ETH.

###  3.2 Risks of LSTs

LSTs carry a risk of slashing, associated with the staking process. However,
this risk is mitigated to some extent since the total staked tokens are
distributed across multiple validators, making it less pronounced compared to
traditional staking. Notably, certain liquid staking protocols like RocketPool
mandate validators to deposit collateral, which is then used to cover losses
from slashing[10].

The market prices of LSTs exhibit fluctuations around the peg price (derived
from the reserves of staked tokens)[25]. As with any pegged token, there
exists a de-peg risk, e.g., in the aftermath of significant market events. The
effects of the Terra/Luna crash and the FTX insolvency in 2022 on LST prices
can be seen in Figure 5 for rebase-LSTs, and Figure 6 for reward-LSTs.

In general, redemptions of LSTs for the underlying assets are limited by the
unstaking process of the PoS blockchain, which can include waiting periods and
withdrawal limits per time period. Hence, large demand for unstaking can lead
to delays in LST redemptions through the LST protocol, and therefore cause LST
prices to depeg.

###  3.3 Constant Function Market Makers (CFMMs)

AMMs are protocols deployed on a blockchain to allow traders to exchange
assets in a fully noncustodial manner. Their main component are liquidity
pools for a number (most commonly two) of traded tokens, into which liquidity
providers (LPs) deposit reserves. Traders can then swap these assets against
the pool at an exchange rate determined by an AMM-specific trade function,
depending on the reserves currently in the pool. For every trade, a trading
fee in form of a fixed percentage of the trade is paid. These fees (or, for
some AMMs, part of them) are distributed pro rata among liquidity providers in
the pool.

Constant Function Market Makers (CFMM), the most common form of AMMs, have a
trading function $\varphi:\mathbb{R}^{N}_{+}\rightarrow\mathbb{R}$, often
called a conservation function or reserve curve, associated to them, which
maps their token reserves to a real number, often referred to as invariant,
which is kept constant for each trade. Here $N$ denotes the number of tokens
in the liquidity pool, which for all relevant LST pools (and almost all
relevant AMM pools in general) is $N=2$.

A wide range of different CFMMs has been implemented and suggested [37]. In
the following, we describe the major types of CFMMs, which also include those
being used for LST trading, at the time of writing. For all, the pools
reserves are denotes by $x_{1},\ldots,x_{N}$.

##### Constant Product (CPMM)

The first-used and most simple invariant, used in Uniswap v1 and v2 [16]:

$\prod_{i=1}^{N}x_{i}=L^{N}.$

##### Constant Product with Concentrated Liquidity (CLMM)

Introduced by Uniswap v3 [17]), it allows LPs to specify a price range
$[p_{a},p_{b}]$ to which they provide liquidity. Then the trade invariant in
this interval is

$\left(x_{1}+\frac{L}{\sqrt{p_{b}}}\right)\left(x_{2}+L\cdot\sqrt{p_{a}}\right)=L^{2}.$

##### Stableswap Invariant

Introduced with Curve v1 [22], this CFMM is specifically designed for assets
that trade at a constant price to each other. The invariant is

$K\cdot
D^{N-1}\cdot\sum\limits_{i=1}^{N}x_{i}+\prod\limits_{i=1}^{N}x_{i}=K\cdot
D^{N}+\left(\frac{D}{N}\right)^{N}$ (1)

with

$K=\frac{A\cdot\prod_{i=1}x_{i}}{D^{N}}\cdot N^{N},$

where $A$ is a concentration parameter.

##### Cryptoswap Invariant

Curve v2 [23], an adaptation of the Stableswap invariant for assets with non-
constant prices. It uses the same equation as (1), but with a different $K$,
defined as

$K=A\cdot\underset{\text{ =:
}K_{0}}{\underbrace{\frac{A\cdot\prod_{i=1}x_{i}}{D^{N}}\cdot
N^{N}}}\cdot\frac{\gamma^{2}}{(\gamma+1-K_{0})^{2}}.$

##  4 The Theory of LSTs on AMMs

While LSTs can potentially be paired and traded against any other token on
AMMs, the majority of LST liquidity lies in liquidity pools containing the LST
and the underlying token. This supports the main purpose of LSTs on AMMs:
Entering or exiting a staking positions by buying or selling the LST for the
underlying. Pairing LSTs against their underlying is also advantageous, since
the volatility between the two should be low (if the LST functions well).

###  4.1 Suitable types of AMMs for LSTs

Different types of AMMs come with different properties and limitations,
rendering them more or less well suited for specific liquid staking token
pairs. Refer to Table 1 for which AMMs are most suitable for rebase and reward
LSTs to achieve high capital efficiency for LPs and low price impact for
traders. While LSTs could potentially be traded on CPMMs, their low capital
inefficiency for assets that move little in price, such as LSTs, makes them an
unattractive choice.

Rebase-LSTs maintain a price pegged to 1 ETH, aligning perfectly with the
Stableswap invariant, which was specifically designed for pairs with stable
prices. Although CLMMs are theoretically also well-suited for trading pairs
with stable prices, they are currently not widely used for trading rebase-LSTs
due to the technical incompatibility of Uniswap v3 – the market-leading CLMM -
with rebase-LSTs [7].

Reward-LSTs, on the other hand, increase their value every day, according to
the staking rate. Consequently, they are ideally traded on AMMs employing the
Cryptoswap invariant or a CLMM. It is important to note that with CLMMs,
periodic rebalancing of liquidity positions is necessary.

Token 1 |  Token 2 |  AMMs
---|---|---
rebase-LST |  ETH |  Stableswap
rebase-LST |  rebase-LST |  Stableswap
reward-LST |  ETH |  Cryptoswap, CLMM(*)
reward-LST |  reward-LST |  Cryptoswap, CLMM
Table 1: Suitable (in terms of liquidity efficiency) AMMs for trading pairs
with different LST types. (*) regular rebalancing required

###  4.2 Losses and Required Returns for LPs

Within any AMM pool, it is imperative that trading fees adequately reward
liquidity providers for their capital contribution. On the one hand, LPs earn
trading fees, on the other hand, the value of their liquidity position
fluctuates with token prices. In particular, when prices change, LPs suffer a
loss when comparing the value of their liquidity position to holding their
liquidity elsewhere than in the AMM pool.

At the very least, returns from trading fees should offer sufficient
compensation to offset potential losses of a liquidity position compared to a
number of basic benchmark strategies. In this context, a generally relevant
metric is the loss versus holding the initial positions outside the AMM (loss-
vs-holding, LVH, sometimes referred to as “impermanent loss”) Specifically for
AMM pools involving liquid staking tokens and their underlying assets, it is
furthermore highly relevant to measure the loss compared to a fully staked
portfolio (loss-vs-staking, LVS).

Each of these losses can be quantified specifically for the particular price
trajectory of LST trading pairs. This allows us to calculate the required
returns to compensate LPs as a function of the staking rate $r$. Moreover, in
broad terms, a liquidity provider’s return from trading fees can be expressed
as follows:

$r_{LP}=\frac{\text{trading volume}\cdot\text{trading fee}}{\text{liquidity in
pool}}$

This implies that, based on the required returns, the necessary trading
volumes (relative to the pool size) to render the pools profitable for LPs can
be inferred.

It is important to note that for CLMMs, LPs earn different returns depending
on the concentration of their positions and whether their liquidity is
consistently within range. Consequently, one must factor in the concentration
factor of the position and consider only the amount of liquidity in range when
determining the denominator in the equation above.

In the following, we theoretically derive closed-form expressions for the
different LP losses – and thereby the required returns to compensate them –
for the particular case of LST liquidity as a function of the staking rate
$r$.

We consider a liquidity pool containing an LST and the underlying asset (such
as ETH), and use the underlying asset as unit of measurement. Let $P(t)$
denote the price of the LST (denominated in the underlying asset) at time
$t\geq 0$. As a baseline, we consider an AMM liquidity position without taking
earnings from trading fees into account. Let the liquidity position consist of
$x(t)$ LST units and $y(t)$ units of the underlying token at time $t$. Then
its value is given by

$\displaystyle V_{LP}(t)=x(t)P(t)+y(t).$

Furthermore, let $V_{HOLD}(t)$ denote the value of holding the initial
position:

$\displaystyle V_{HOLD}(t)=x(0)P(t)+y(0)$

Next, we consider the value $V_{LST}(t)$ of a fully staked position (i.e.
holding the initial position fully in the LST):

$\displaystyle
V_{LST}(t)=V_{LP}(0)\frac{P(t)}{P(0)}=\left(x(0)+\frac{y(0)}{P(0)}\right)P(t)$

Comparing the AMM position value to these benchmarks (holding and staking)
reveals the size of the losses for LPs:

$\displaystyle LVH(t)$ $\displaystyle=V_{HOLD}(t)-V_{LP}(t)$ $\displaystyle
LVS(t)$ $\displaystyle=V_{LST}(t)-V_{LP}(t)$

Moreover, the required returns from trading fees $rr_{LVH}$ and $rr_{LVS}$
needed to compensate for LVH and LVS, are given as follows:

$\displaystyle rr_{LVH}(t)$ $\displaystyle=\frac{V_{HOLD}(t)}{V_{LP}(t)}-1$
$\displaystyle rr_{LVS}(t)$ $\displaystyle=\frac{V_{LST}(t)}{V_{LP}(t)}-1$

###  4.3 LST prices

##### Reward LSTs

Reward-LSTs increase their value every day, according to the staking rate $r$.
Therefore, we model their price process as a geometric Brownian motion with
drift $r$ and some volatility $\sigma\geq 0$:

$\displaystyle P(t)=P(0)\exp\left(\left(r-\frac{\sigma^{2}}{2}\right)t+\sigma
B(t)\right),$

where $\\{B(t)\\}_{t\geq 0}$ is a standard Brownian motion. A detailed
treatment of the price process can be found in Appendix 0.A. In particular,
note that under ideal circumstances where the price of the LST solely reflects
the added staking rewards, i.e. the variance $\sigma=0$, the LST’s price is
given by

$\displaystyle P(t)=P(0)e^{rt}.$

In later sections (starting Section 4.4), we derive expressions for LP losses
under this price process for certain AMMs.

##### Rebase LSTs

For trading pairs between rebase LSTs and the underlying asset, the matter is
more straightforward, as the price of rebase LSTs is pegged to the price of
the underlying asset. We only briefly discuss this case at a high level.

First, note that in most AMMs half of the initial position is allocated to the
LST and will be earning rewards when holding. Consequently, if the value of
the initial position is $V_{LP}(0)$, the value of holding becomes
$V_{HOLD}(t)=V_{LP}(0)(1/2+e^{rt}/2)$.

On the other hand, due to the stable price, the pool maintains an equal
allocation between LST and the underlying. Therefore, half of the liquidity in
the pool is earning rewards, resulting in $V_{LP}(t)=V_{LP}(0)(1/2+e^{rt}/2)$.
As a result, no loss-versus-holding occurs, and $rr_{LVH}(t)=0$.

Finally, the value of the fully staked portfolio is
$V_{LST}(t)=V_{LP}(0)e^{rt}$, leading to

$rr_{LVS}(t)=\dfrac{V_{LP}(0)e^{rt}}{V_{LP}(0)(1/2+e^{rt}/2)}-1=\dfrac{2e^{rt}}{1+e^{rt}}.$

###  4.4 Required returns for CPMM

For the reward-LST price process, as described in the previous section, the
required returns for trading pairs involving reward-LSTs can be calculated for
various AMMs. For a constant product AMM, we show in Appendix 0.B, that

$\displaystyle
rr_{LVH}(t)=\frac{1}{2}\left(e^{(\frac{r}{2}-\frac{\sigma^{2}}{4})t+\frac{\sigma}{2}B(t)}+e^{(-\frac{r}{2}+\frac{\sigma^{2}}{4})t-\frac{\sigma}{2}B(t)}\right)-1.$

which in expectation is

$\displaystyle\mathbb{E}[rr_{LVH}]=\frac{1}{2}\left(e^{(\frac{r}{2}-\frac{\sigma^{2}}{8})t}+e^{(-\frac{r}{2}+\frac{3\sigma^{2}}{8})t}\right)-1.$

For the ideal LST price trajectory with $\sigma=0$, this implies
$\mathbb{E}[rr_{LVH}]=(e^{r/2}+e^{-r/2})/2-1$.

For loss-versus-staking, we derive the following expression in Appendix 0.B:

$\displaystyle
rr_{LVS}(t)=e^{(\frac{r}{2}-\frac{\sigma^{2}}{4})t+\frac{\sigma}{2}B(t)}-1.$

In particular, in expectation, LVS is

$\displaystyle\mathbb{E}\left[rr_{LVS}(t)\right]=e^{(r/2-\sigma^{2}/8)t}-1.$

Finally, for the ideal case, where the LST perfectly tracks the staking rate,
i.e. $\sigma=0$, we have $\mathbb{E}\left[rr_{LVS}(t)\right]=e^{rt/2}-1$.

###  4.5 Required Returns for Concentrated Liquidity CPMMs

For CLMMs, we consider symmetric positions around the expected price movement
of the LST during the time interval $[0,t]$. Under the assumption that the LST
follows the ideal price trajectory, i.e. $P(t)=P(0)e^{rt}$, we derive the
closed-form expressions for LVH and LVS in Appendix 0.C. For LVH, we find that

$\displaystyle rr_{LVH}(t)$
$\displaystyle=\frac{1}{2}\left(e^{rt/2}+e^{-rt/2}\right)-1.$

As Figure 1 shows, this value is small for most practical staking rates. Note
also, that this metric is independent of the width of the chosen price range.

Next, we calculate LVS as

$\displaystyle rr_{LVS}(t)=\frac{V_{LST}(t)}{V_{LP}(t)}-1=e^{rt/2}-1.$

As could be expected, $rr_{LVS}\approx r/2$ (see Figure 1) stemming from the
fact that LPs lose out on the staking rewards on about half of their
liquidity. In other words, returns from fees need to be at least half the
staking rate, to make it more attractive to LP in a pool over fully staking
one’s capital.

Again, a notable discovery is that $rr_{LVS}$ remains unaffected by the chosen
width of the position range, provided that the position is symmetric. This is
highly relevant, since it means that the range of the liquidity position can
be chosen by solely focussing on maximizing the amount of fees earned.
Achieving this requires optimizing the liquidity concentration versus the
duration the position remains within range.

Figure 1: Required returns for different staking rates.

##  5 Empirical Analysis

In this section, we empirically measure the previously described losses to LPs
in the largest AMM pools containing liquid staking tokens, and analyze if
trading fees have historically compensated LPs adequately. For each pool, we
historically track the value of an LP position that earns fees (i.e. $V_{LP}$
\+ fees). Moreover, given historical prices, we calculate the values of the
HOLD portfolio ($V_{HOLD}$) and a fully staked portfolio ($V_{LST}$). By
comparing $V_{LP}$ with $V_{HOLD}$ and $V_{LST}$, we assess whether LPs earned
sufficient compensation to balance LVH and LVS, as described in the previous
section.

###  5.1 Methodology

##### Data Origin

The nine analyzed pools include pools on Uniswap v3 and Curve v1 and v2 on the
Ethereum blockchain.

For the Curve pools, historical daily token prices and AMM-pool compositions
are sourced from an Ethereum archive node. More specifically, for the last
block of each day, we retrieved the token pool reserves via the _balances_
method of the Curve pool smart contract and the amount of LP tokens in
circulation via the _totalSupply_ method of the LP token smart contract. Based
on this data, the value of a single LP token for each day and thereby the
value of a liquidity position in the pool can be calculated. These numbers
already include earning from trading fees, since these are paid into the pool.
On top of trading fees, LPs can earn extra rewards paid in CRV (Curve’s
protocol token) in certain Curve pools. We query the value of these rewards
from a subgraph using The Graph
protocol.222https://api.thegraph.com/subgraphs/name/messari/curve-finance-
ethereum The returns from the CRV rewards are then added to the value of the
liquidity position.

For Uniswap v3 pools, we obtain all swap transaction, i.e. the amount traded
and the amount of active liquidity at the time of the swap, from an Ethereum
node. This allows us to historically calculate the fees a liquidity position
in the pool earns.

As a reference for the historical staking rate, we use Beacon Chain’s _Ether
Staking Offered Rate_ (ETH.STORE), as it represents the average financial
return validators on the Ethereum network have achieved over the last 24
hours. It includes MEV rewards, which are also earned by holders of LSTs [15].

##### Pool Description

We analyze nine dominant ETH-LST liquidity pools – a full list of which can be
found in Table 2 in the appendix. Figure 7 in the appendix shows that stETH-
ETH on Curve v1 has by far the highest TVL among the analyzed pools with about
$250m at the end of 2023, followed by another stETH pool on Curve v1 with
about $70m. The highest daily traded volume on the other hand, is observed for
the wstETH-ETH pool on Uniswap v3. The higher trading volume for wstETH in
comparison to stETH confirms the higher utilization of wstETH (reward-LST) in
DeFi, whereas stETH (reward-LST) tokens seem to be primarily bought to hold
and accumulate rewards from staking and consequently less actively traded.

##### Computation

The daily wealth of LPing ($V_{LP}$ \+ fees) is calculated using the data
described above. Moreover, the value of HOLD ($V_{HOLD}$) and fully staking
($V_{LST}$) can be computed from the price of the LST. For each, we assume an
initial capital of 1 ETH, and measure the value of the portfolio in ETH. It is
assumed that the staking rewards are re-staked on a daily basis.

For rebase tokens, such as the stETH pools on Curve v1, we adjust the HOLD and
LST portfolios daily to account for the distributions of staking rewards in
the form of newly minted LST tokens.

For Uniswap v3 it is assumed that the LP (and HOLD) position is re-balanced on
a monthly basis, i.e. the price range is reset at the beginning of each month.
On each reset, the price range of the liquidity position is set to be
$[-0.25\%,+0.75\%]$ around the pool price at the beginning of the month. This
range is based on an expected monthly return from staking of 0.5% which has
approximately been observed on average during our observation period.
Additionally, the range adds $\pm 0.25\%$ around the expected monthly price
change to accommodate for some price volatility of the LST. All studied LSTs
pools on Uniswap v3 are pools between reward-LSTs and ETH.

###  5.2 Analysis

Figure 2: Historical portfolio values for the stETH-ETH pool on Curve v1:
LPing in the pool (blue line), holding the initial position (orange line), and
fully holding the LST (blue line). The red line indicates the average staking
rate.

We begin the analysis with the stETH-ETH pool on Curve v1 - the largest pool
in terms of TVL. The development of the considered portfolio values – LP
(including fees), HOLD, and LST (full LST position) – as well as the reference
staking rate are depicted in Figure 2. We additionally plot the wealth of an
LP including CRV rewards (LP+CRV). In Curve, LPs earn additional rewards for
liquidity provision on top of trading fees. These rewards are distributed in
Curve governance tokens - CRV according to its emissions [14]. While we
observe that the additional CRV rewards did not significantly impact the
wealth of LPs in this pool, they do make a difference in other pools (see
Figure 8 in the appendix).

We find that during most of the observation period, the value of the liquidity
position stays slightly above the value of HOLD. This indicates that fees
sufficiently compensate LPs for LVH. On the other hand, the LP’s wealth
evolves significantly lower than the value of fully staking, meaning that LVS
was higher than the returns from fees, and the LPs would have been better off
fully investing their portfolio in the LST instead of providing liquidity to
the AMM.

##### Moving Average

Figure 3: Historical 7 day moving average of LVH and LVS for the stETh-ETH
pool on Curve v1. Color green marks periods with no loss-versus-staking (LVS)
and color yellow periods with no loss-versus-holding (LVH)

Whereas Figure 2 shows the wealth of LP, HOLD, and LST portfolios, assuming
they enter the pool at the beginning of the observation period, Figure 3
presents the 7-day moving average of the difference in daily returns of LPs on
the one hand, and HOLD and LST on the other hand. The differences indicate
whether providing liquidity to the AMMs yielded higher returns than HOLD or
LST in the past week. The periods during which LPs’ wealth (including fees)
grew more than both HOLD and LST are marked in green (fees compensated for
LVS). The periods when LP accumulated more wealth than HOLD but less than LST
are marked in yellow (fees compensated for LVH, but not for LVS). Periods
marked in red denote that LP accumulated less wealth than both HOLD and LST
(fees did not compensate for LVH).

We find that during most of the analyzed periods, liquidity provision resulted
in both a loss-versus-holding and a loss-versus-staking over a 7-day period
during about half the observation period. Figure10 in the appendix, displaying
the same comparison, but for a 30-day moving average, shows a similar picture.

##### Impact of CRV Rewards

Figure 4: Historical portfolio value of Staker (red line), HOLD (orange line),
LPs (blue line) and LST holder (green line) for the cbETH-ETH pool on Curve
v2, denominated in ETH

Figure4 exhibits a significantly larger impact of CRV rewards on the
profitability of the LP strategy in the cbETH-ETH pool on Curve v2: without
CRV rewards, the wealth of LP would not exceed the wealth of HOLD and LST,
resulting in loss-versus-holding and loss-versus-staking. With CRV rewards, on
the other hand, LPing out earns both the HOLD and LST portfolio.

##### LST pools on Curve

Figures 8, 9 and 10 show the development of the wealth of LP, HOLD, and LST,
as well as the 7-day and 30-day moving averages of their differences, for the
five LST pools on Curve.

Three of the pools allow swapping stETH for ETH but differ in the parameters
used and, consequently, in the behavior of the bonding curve. While the
leverage parameter A for the largest pool in terms of TVL is 30, the
“concentrated” (A=1000) and “ng” (A=1500) pools that have later inception
dates, offer more concentrated liquidity. Figure 10 shows that the more
concentrated pools almost always compensate LPs for loss-versus-holding, but
trading fees are not sufficient to compensate for loss-versus-staking.

The analysis also includes two pools with reward-LSTs – rETH and cbETH –
deployed to Curve v2. As reward-LSTs, rETH and cbETH are not pegged to 1 ETH
and their pools use the Cryptoswap Invariant - an AMM that concentrates
liquidity around their current prices. The returns in the rETH and cbETH pools
were significantly impacted by the distribution of CRV token rewards. As
depicted in Figure 8 (blue dotted lines), the LP wealth including CRV tokens
actually significantly exceeded the staking returns for rETH and cbETH.
Without rewards, the picture is similar

##### LST pools on Uniswap

Figures 11, 12 and 13 show the wealth of LP, HOLD, and LST, as well as 7-day
and 30-day moving average of wealth difference, for the four LST pools on
Uniswap v3. Due to incompatibility with rebase tokens, all LSTs listed at
Uniswap are reward-based (wstETH, rETH, and cbETH). The moving average for the
weekly positions in Figure 12 shows that the pools with the higher fee rate of
5 bp were able to generate sufficient amount of fees to compensate for LVS
during certain time periods, while the 1 bp wstETH-ETH pool consistently
failed to achieve this.

Over the whole observation period, total fee returns were generally high
enough to compensate LPs for LVS in the 5bp pools, especially in the more
recent past (see right plots in Figure 11). However, the actual profitability
of liquidity positions depends on their holding period, as it is affected by
price fluctuations of cbETH and rETH. In particular, depositing into the pool
during the depegs following the FTX insolvency in November 2022 (rETH being
overpriced while cbETH was underpriced), can lead to different results.

###  5.3 Discussion

The primary objective of the empirical part was to assess the profitability of
allocating LSTs to AMMS. We found that the majority of LST liquidity is
currently not sufficiently compensated compared to the possibility of fully
staking their capital. This puts the sustainability of the current allocation
of LST liquidity on AMMs into question. If return from trading fees do not
increase, e.g. as a result of increased trading volume, the current amount of
liquidity will likely not remain in these pools in the long term.

##### CRV rewards

We find that most Curve pools currently are only profitable for LPs,
especially compared to fully staking, when taking the CRV token rewards to LPs
into account. The amounts of CRV token rewards emitted to pools are decided by
the Curve governance, and can change every week. A drop in their distribution
to the LST pools, could lead to lower returns for LPs and subsequently a
decrease of liquidity in these pools.

##### CLMM Rebalancing

LP returns on CLMMs such as Uniswap v3 can be increased by choosing a smaller
price range to provide liquidity to and rebalance more frequently. However,
this strategy requires active monitoring of the LP position and incurs gas
fees, thereby reducing LP’s profits. Additionally, we observed the emergence
of new pools featuring the same LST and ETH pairs but offering lower trading
fees (1 basis point instead of 5 basis points). Despite attracting higher
trading volumes, these new pools so far do not generate a sufficient
compensation for LPs.

##### Reward-LST token pairs

One possible approach to reduce LP losses is to utilize pools involving two
reward-LSTs, such as wstETH and rETH. Such pools would reduce LVH, since the
LST prices should move in parallel with staking rewards, and LVS, since the
full liquidity position earns trading rewards. Moreover, one reward-LST in
such pools could potentially profit from the deep liquidity of the other
reward-LST to the underlying. However, at the time of writing, these pools
have so far not attracted much liquidity or trading volume.

##### Limitation of Rebase-LST.

It is notable that stETH, the 7th largest cryptocurrency by marketcap at the
time of writing [5], is not traded on Uniswap. As a rebase-LST, stETH’s price
is pegged to 1 ETH, meaning an stETH-ETH AMM pool would potentially not suffer
impermanent loss. However, the current implementation of Uniswap protocol does
not support pools with rebase tokens, and listing of stETH is not possible.

##  6 Conclusions

Liquid staking tokens as liquidity on AMMs has grown to a vital part of the
DeFi landscape, but has so far seen liquidated academic attention. This work
systemizes liquidity provision of LSTs on AMMs, and both theoretically and
empirically studies its returns.

In addition to LVH and LVR, the two most commonly-used metrics to measure LP
losses on AMM, we introduce loss-versus-staking (LVS) which is specifically
tailored to the case of LST pools on AMMs.

Our empirical results show that most AMM pools with LSTs have historically not
compensated for loos-versus-staking. Nevertheless, most pools did compensate
for LVH (“impermanent loss”).

## Acknowledgements

The first author is a research fellow at Matter Labs. The fourth author is a
researcher at Matter Labs. This research article is a work of scholarship and
reflects the authors’ own views and opinions. It does not necessarily reflect
the views or opinions of any other person or organization, including the
authors’ employer. Readers should not rely on this article for making
strategic or commercial decisions, and the authors are not responsible for any
losses that may result from such use.

## References

  * [1] Aave - Protocol Whitepaper V1.0. Tech. rep. (2020)
  * [2] Lido: Ethereum Liquid Staking Whitepaper (2020), https://lido.fi/static/Lido:Ethereum-Liquid-Staking.pdf
  * [3] CoinBase cbETH Whitepaper (2022), https://www.coinbase.com/cbeth/whitepaper
  * [4] Statista DeFi Users (2022), https://www.statista.com/statistics/1297745/defi-user-number/
  * [5] CoinGecko (2023), https://www.coingecko.com/
  * [6] Dune Analytics - Ethereum Staking (12 2023), https://dune.com/hildobby/eth2-staking
  * [7] Lido Documentation (2023), https://help.lido.fi/en/
  * [8] Lido Technical Documentation: Lido Token Integration Guides (2023), https://docs.lido.fi/guides/lido-tokens-integration-guide
  * [9] Messari Report - State of L1s Q2 2023 (11 2023), https://messari.io/report/state-of-l1s-q2-2023
  * [10] Rocket Pool Documentation (2023), https://docs.rocketpool.net/
  * [11] RocketPool (2023), https://docs.rocketpool.net/guides/staking/overview.html
  * [12] StakeWise (2023), https://docs.stakewise.io/
  * [13] Uniswap Technical Documentation: Token Integration Issues (2023), https://docs.uniswap.org/concepts/protocol/integration-issues
  * [14] Curve Resources: Calculating Yield (2024), https://resources.curve.fi/lp/calculating-yield/
  * [15] ETH.STORE Documentation (2024), https://staking.ethermine.org/
  * [16] Adams, H., Zinsmeister, N., Robinson, D.: Uniswap v2 Core. Tech. rep. (2020)
  * [17] Adams, H., Zinsmeister, N., Salem moody, M., River Keefer, u., Robinson, D.: Uniswap v3 Core. Tech. rep. (2021)
  * [18] Angeris, G., Kao, H.T., Chiang, R., Noyes, C., Chitra, T.: An Analysis of Uniswap markets. Cryptoeconomic Systems 0(1) (4 2021), https://cryptoeconomicsystems.pubpub.org/pub/angeris-uniswap-analysis
  * [19] Bank for International Settlements.: Lessons learnt on CBDCs. Report submitted to the G20 Finance Ministers and Central Bank Governors (2023)
  * [20] Bank for International Settlements.: The crypto ecosystem key elements and risks (2023)
  * [21] Buterin, V.: Ethereum: A Next-Generation Smart Contract and Decentralized Application Platform. Tech. rep. (2014)
  * [22] Egorov, M.: StableSwap-efficient mechanism for Stablecoin liquidity. Tech. rep. (2019)
  * [23] Egorov, M.: Automatic market-making with dynamic peg. Tech. rep. (2021)
  * [24] Fritsch, R.: Concentrated Liquidity in Automated Market Makers. In: DeFi 2021 - Proceedings of the 2021 ACM CCS Workshop on Decentralized Finance and Security, co-located with CCS 2021. pp. 15–20. Association for Computing Machinery, Inc (11 2021). https://doi.org/10.1145/3464967.3488590
  * [25] Gogol, K., Kraner, B., Schlosser, M., Yan, T., Tessone, C., Stiller, B.: Empirical and Theoretical Analysis of Liquid Staking Protocols (2023)
  * [26] Gogol, K., Velner, Y., Kraner, B., Tessone, C.: SoK: Liquid Staking Tokens (LSTs) (2024)
  * [27] Heimbach, L., Schertenleib, E., Wattenhofer, R.: Risks and Returns of Uniswap V3 Liquidity Providers (5 2022), http://arxiv.org/abs/2205.08904
  * [28] Heimbach, L., Wang, Y., Wattenhofer, R.: Behavior of Liquidity Providers in Decentralized Exchanges (5 2021), http://arxiv.org/abs/2105.13822
  * [29] Loesch, S., Hindman, N., Welch, N., Richardson, M.B.: Impermanent Loss in Uniswap v3 (2021)
  * [30] Milionis, J., Moallemi, C.C., Roughgarden, T.: Automated market making and arbitrage profits in the presence of fees (2023)
  * [31] Milionis, J., Moallemi, C.C., Roughgarden, T., Zhang, A.L.: Automated Market Making and Loss-Versus-Rebalancing (8 2022), http://arxiv.org/abs/2208.06046
  * [32] Robert Leshner, Geoffrey Hayes: Compound: The Money Market Protocol. Tech. rep. (2019), https://docs.google.com/document/d/1KoXEEYg4YAaPacS4dudPuFZwgAX0Swv9Yi7-iE4C5JU/edit#
  * [33] Scharnowski, S., Jahanshahloo, H.: Liquid Staking: Basis Determinants and Price Discovery. Tech. rep. (2022), https://ssrn.com/abstract=4180341
  * [34] Tiruviluamala, N., Port, A., Lewis, E.: A General Framework for Impermanent Loss in Automated Market Makers (3 2022), http://arxiv.org/abs/2203.11352
  * [35] Tzinas, A., Zindros, D.: The principal–agent problem in liquid staking. In: Financial Cryptography and Data Security. FC 2023 International Workshops: Voting, CoDecFin, DeFi, WTSC, Bol, Brač, Croatia, May 5, 2023, Revised Selected Papers. p. 456–469. Springer-Verlag, Berlin, Heidelberg (2023), https://doi.org/10.1007/978-3-031-48806-1˙29
  * [36] Xiong, X., Wang, Z., Chen, X., Knottenbelt, W., Huth, M.: Leverage staking with liquid staking derivatives (lsds): Opportunities and risks (2023)
  * [37] Xu, J., Paruch, K., Cousaert, S., Feng, Y.: SoK: Decentralized Exchanges (DEX) with Automated Market Maker (AMM) Protocols (3 2021), http://arxiv.org/abs/2103.12732

##  Appendix 0.A Continuous-time Loss-versus-staking

Assume there exists an infinitely deep centralized exchange, where the token
$T$ and the LST can be traded without any fees. The price of the token is
given by a geometric Brownian motion that is a $\mathbb{Q}$-martingale, i.e.

$\frac{dP_{T}(t)}{P_{T}(t)}=\mu_{T}(t)dt+\sigma_{T}(t)dW_{T}\qquad\forall
t\geq 0,$

with $\mu_{T}(t)$ being the long-term drift of the token and $\sigma_{T}$ the
volatility of the Token. For the LST we have

$\frac{dP_{LST}(t)}{P_{LST}(t)}=\mu_{LST}(t)dt+\sigma_{LST}(t)dW_{LST}\qquad\forall
t\geq 0.$

Assuming the LST works as intended on the long run, its drift $\mu^{LST}(t)$
would be composed of $T$’s drift $\mu_{T}(t)$ plus the staking rate $r$, i.e.
$\mu_{LST}(t)=\mu_{T}(t)+r$. In addition, the two tokens are clearly not
independent and have some sort of correlation $\rho$, resulting in two
correlated Brownian motions $W_{T}$ and $W_{LST}$ with quadratic variation
$[W_{T},W_{LST}](t)=\rho(t)$.

Then we can rewrite the two Brownian motions as

$\displaystyle W_{T}$ $\displaystyle=B_{T}$ $\displaystyle W_{LST}$
$\displaystyle=\rho B_{T}+\sqrt{1-\rho^{2}}B_{LST}$

and get the analytic solution (It$\hat{o}$’s formula on $ln(P_{T/LST}(t))$):

$\displaystyle P_{T}(t)$
$\displaystyle=P_{T}(0)\exp((\mu_{T}-\frac{1}{2}\sigma_{1}^{2})t+\sigma_{1}B_{T}(t))$
$\displaystyle P_{LST}(t)$
$\displaystyle=P_{LST}(0)\exp((\mu_{LST}-\frac{1}{2}\sigma_{1}^{2})t+\sigma_{2}(\rho
B_{T}(t)+\sqrt{1-\rho^{2}}B_{LST}(t)))$

In our paper we focus only on pairs of LST and ETH as the underlying token T.
So we don’t compare it to a third currency and therefore can omit the GBM from
the token, resulting in the much simpler stochastic process for LST:

$\frac{dP_{LST}(t)}{P_{LST}(t)}=\underbrace{\mu_{LST/T}}_{=r}dt+\sigma_{LST/T}(t)dB(t)\qquad\forall
t\geq 0,$

where $\mu_{LST/T}$ is the drift of the LST against the token, which is given
by the staking rate $r$, $\sigma_{LST/T}$ is the historical standard deviation
of the price of LST in T and $B(t)$ a standard Brownian motion.

We then can define a self-financing trading strategy for our staking portfolio
with LST that starts holding only LST (1,0) at the beginning and keeps holding
them by

$V_{LST}(t)=V(0)+\int\limits_{0}^{t}1dP_{LST}\qquad\forall t\geq 0.$

Since the portfolio only consists of LST this can be simplified to the initial
value times the change in price:

$V_{LST}(t)=V(0)\exp((r-\frac{1}{2}\sigma_{LST/T}^{2})t+\sigma_{LST/T}B(t))$

leading to an expected value of

$\mathbb{E}[V_{LST}(t)]=V(0)\exp((r-\frac{1}{2}\sigma_{LST/T}^{2})t)+\frac{1}{2}\sigma_{LST/T}^{2}t=V_{0}\exp(rt),$

where we used that $\mathbb{E}[\exp(uZ)]=\exp(\frac{u^{2}}{2})$ for
$u\in\mathbb{R}$ and $Z\sim\mathcal{N}(0,1)$.

For the traditional staking we assume a fixed compounding staking rate $r$
that leads to $V_{stake}(t)=V(0)\cdot\exp(rt)$.

The LVS is the difference between the value of the liquidity position at time
$t$ and the staking position until time $t$.

For the traditional staking this is therefore given by

$LVS_{stake}(t)=V(0)\exp(rt)-V_{LP}(t)$

and for an LST by

$LVS(t)=V_{LST}(t)-V_{LP}(t)=V(0)\exp((r-\frac{1}{2}\sigma_{LST/T}^{2})t+\sigma_{LST/T}B(t))-V(t).$

The value of $V_{LP}(t)$ depends on the chosen AMM and corresponding
parameters. We discuss two examples in the following sections.

##  Appendix 0.B LP Losses in CPMMs

For the standard CPMM with 2 assets, as described in [16], the holdings of a
liquidity position are

$\displaystyle x(t)$ $\displaystyle=\frac{L}{\sqrt{P(t)}}$ $\displaystyle
y(t)$ $\displaystyle=L\sqrt{P(t)},$

where $L$ is the size of the position. Hence, we have the following portfolio
values:

$\displaystyle V_{LP}(t)$ $\displaystyle=x(t)P(t)+y(t)=2L\sqrt{P(t)}$
$\displaystyle V_{HOLD}(t)$
$\displaystyle=x(0)P(t)+y(0)=\frac{L}{\sqrt{P(0)}}P(t)+L\sqrt{P(0)}$
$\displaystyle V_{LST}(t)$
$\displaystyle=\left(x(0)+\frac{y(0)}{P(0)}\right)P(t)=\frac{2L}{\sqrt{P(0)}}P(t)$

##### Loss-versus-staking

The required return to compensate loss-versus-staking is

$\displaystyle
rr_{LVS}(t)=\frac{V_{LVS}(t)}{V_{LP}(t)}-1=\frac{\sqrt{P(t)}}{\sqrt{P(0)}}-1.$

To derive $\sqrt{P(t)}$, we apply Ito’s formula with
$f(t,P_{LST}(t))=\sqrt{P_{LST}(t)}$:

$\displaystyle d\sqrt{P(t)}$
$\displaystyle=\frac{1}{2\sqrt{P(t)}}dP(t)+\frac{1}{2}\cdot\frac{-1}{4}P(t)^{\frac{-3}{2}}(dP(t))^{2}$
$\displaystyle=\frac{1}{2\sqrt{P(t)}}(rP(t)dt+\sigma
P(t)dB(t))-\frac{1}{8}P(t)^{\frac{-3}{2}}(\sigma^{2}\cdot P(t)^{2})dt$
$\displaystyle=(\frac{r}{2}\sqrt{P(t)}-\frac{\sigma^{2}}{8}\sqrt{P(t)})dt+\frac{\sigma}{2}\sqrt{P(t)}dB(t))$

This implies

$\displaystyle\sqrt{P(t)}$
$\displaystyle=\sqrt{P(0)}\cdot\exp((\frac{r}{2}-\frac{\sigma^{2}}{8}-\frac{1}{2}(\frac{\sigma}{2})^{2})t+\frac{\sigma}{2}B(t))$
$\displaystyle=\sqrt{P(0)}\cdot\exp((\frac{r}{2}-\frac{\sigma^{2}}{4})t+\frac{\sigma}{2}B(t)).$

Inserting this into loss-versus-staking, yields

$\displaystyle
rr_{LVS}(t)=e^{(\frac{r}{2}-\frac{\sigma^{2}}{4})t+\frac{\sigma}{2}B(t)}-1.$

In particular, in expectation, LVS is

$\displaystyle\mathbb{E}\left[rr_{LVS}(t)\right]=e^{(\frac{r}{2}-\frac{\sigma^{2}}{4})t}\mathbb{E}[\exp(\frac{\sigma}{2}B(t))]-1=e^{(\frac{r}{2}-\frac{\sigma^{2}}{8})t}-1.$

Finally, note that in the ideal case, where the LST perfectly tracks the
staking rate, i.e. $\sigma=0$, we have
$\mathbb{E}\left[rr_{LVS}(t)\right]=e^{rt/2}-1$.

##### Loss-versus-holding

$\displaystyle
rr_{LVH}=\frac{1}{2}\left(\frac{\sqrt{P(t)}}{\sqrt{P(0)}}+\frac{\sqrt{P(0)}}{\sqrt{P(t)}}\right)-1$

To calculate the change in LVH we need the differential
$\frac{1}{\sqrt{P(t)}}.$ We use Ito’s formula with
$f(t,P(t))=\frac{1}{\sqrt{P(t)}}$ and get:

$\displaystyle d\frac{1}{\sqrt{P(t)}}$
$\displaystyle=(-\frac{1}{2}P(t)^{-\frac{3}{2}}dP(t)+\frac{1}{2}\cdot\frac{3}{4}P(t)^{-\frac{5}{2}})(dP(t))^{2}$
$\displaystyle=(-\frac{1}{2}P(t)^{-\frac{3}{2}}(rP(t)dt+\sigma
P(t)dB(t))+\frac{3}{8}P(t)^{-\frac{5}{2}}(\sigma^{2}P(t)^{2})dt$
$\displaystyle=(\frac{3\sigma^{2}}{8\sqrt{P(t)}}-\frac{r}{2\sqrt{P(t)}})dt-\frac{\sigma}{2\sqrt{P(t)}}dB(t)$

which implies:

$\displaystyle\frac{1}{\sqrt{P(t)}}$
$\displaystyle=\frac{1}{\sqrt{P(0)}}\cdot\exp((-\frac{r}{2}+\frac{3\sigma^{2}}{8}-\frac{\sigma^{2}}{8})t-\frac{\sigma}{2}B(t)))$
$\displaystyle=\frac{1}{\sqrt{P(0)}}\cdot\exp((-\frac{r}{2}+\frac{\sigma^{2}}{4})t-\frac{\sigma}{2}B(t)))$

This leads to the following expression for the required return:

$\displaystyle
rr_{LVH}=\frac{1}{2}\left(e^{(\frac{r}{2}-\frac{\sigma^{2}}{4})t+\frac{\sigma}{2}B(t)}+e^{(-\frac{r}{2}+\frac{\sigma^{2}}{4})t-\frac{\sigma}{2}B(t)}\right)-1$

which in expectation is

$\displaystyle\mathbb{E}[rr_{LVH}]=\frac{1}{2}\left(e^{(\frac{r}{2}-\frac{\sigma^{2}}{8})t}+e^{(-\frac{r}{2}+\frac{3\sigma^{2}}{8})t}\right)-1.$

And for $\sigma=0$, we have $\mathbb{E}[rr_{LVH}]=(e^{r/2}+e^{-r/2})/2-1$.

##  Appendix 0.C LP Losses in Concentrated Liquidity CPMMs

In the following, we examine LP’s losses and required returns for CPMM with
concentrated liquidity as introduced in [17].

Consider a concentrated liquidity position that is opened at time $t=0$ with
the prospect of maintaining it until time $t=T$. The range of the positions is
set symmetric around the expected price movement from $P(0)$ to $P(0)e^{rT}$
during that period, i.e. as $[P(0)e^{-d},P(0)e^{rT+d}]$ for some $d\geq 0$.
Hence, if the price is range at time $t\geq 0$, the position has the following
holdings, where $L$ denotes the size of the liquidity position.

$\displaystyle x(t)$
$\displaystyle=L\left(\frac{1}{\sqrt{P(t)}}-\frac{1}{\sqrt{P(0)e^{rT+d}}}\right)$
$\displaystyle y(t)$
$\displaystyle=L\left(\sqrt{P(t)}-\sqrt{P(0)e^{-d}}\right)$

In particular, if the LST functions as intended and its price has increased
according to the staking rate at the end of the period, such that
$P(T)=P(0)e^{rT}$, we have

$\displaystyle
V_{LP}(T)=x(T)P(T)+y(T)=L\sqrt{P(0)}\left(2e^{rT/2}-e^{rT/2-d/2}-e^{-d/2}\right).$

##### Loss-versus-staking

On the other hand, holding only the LST results in a final wealth of

$\displaystyle
V_{LST}(T)=\left(x(0)+\frac{y(0)}{P(0)}\right)P(T)=L\sqrt{P(0)}\left(2-e^{-rT/2-d/2}-e^{-d/2}\right)e^{r}T.$

This implies

$\displaystyle rr_{LVS}(T)=\frac{V_{LST}(T)}{V_{LP}(T)}-1=e^{rT/2}-1.$

In particular, note that $rr_{LVS}$ is independent of the position width $d$,
and depends only on $r$ (under the assumption that the price of LST increases
according to the staking rate).

##### Loss-versus-holding

Note that the initial position was chosen to take the expected price change of
the LST into account. It was set symmetric around the expected price, and not
symmetric around the initial price, as would have been the default without an
expected price movement. With the default range symmetric around the initial
price, both the initial position and the expected holding of the liquidity
position over a future time period will be split 50-50 in value between the
two tokens in the pool. In particular, considering LVH means comparing the
liquidity position to a portfolio with the same expected holdings.

To achieve the same for our setting with an expected price change and an
accordingly adjusted initial position, we also adjust the HOLD portfolio: We
consider the position $(\overline{x}(0),\overline{y}(0))$ whose initial
holdings equal the expected holdings of the liquidity position
$(\overline{x}(P(0)),\overline{y}(P(0))$. Let $\overline{V_{HOLD}}$ denote the
value of this position. Specifically, for a symmetric position around the
expected movement of the LST, the average (expected) holdings of the liquidity
position over the time period is 50-50. Hence, for adjusted LVH, we compare to
holding a position that starts with 50% value in each ETH and the LST. Simple
calculation show that the initial composition of such a 50-50 position (that
has the same value as the liquidity position at $t=0$) is

$\displaystyle\overline{x}(0)=\frac{L}{\sqrt{P(0)}}\frac{1}{2}\left(2-e^{-rT/2-d/2}-e^{-d/2}\right),$
$\displaystyle\overline{y}(0)=L\sqrt{P(0)}\frac{1}{2}\left(2-e^{-rT/2-d/2}-e^{-d/2}\right).$

From this, the wealth of holding

$\displaystyle\overline{V_{HOLD}}(T)=\overline{x}(0)P(T)+\overline{y}(0)=L\sqrt{P(0)}\frac{1}{2}\left(2-e^{-rT/2-d/2}-e^{-d/2}\right)\left(1+e^{rT}\right)$

and, in turn, the loss-versus-holding can be calculated:

$\displaystyle rr_{LVH}(T)$
$\displaystyle=\frac{\overline{V_{HOLD}}(T)}{V_{LP}(T)}-1=\frac{1}{2}\left(e^{rT/2}+e^{-rT/2}\right)-1$

Again, note that $rr_{LVH}$ simplifies to be independent of $d$, and depends
solely on $r$.

##  Appendix 0.D Extra Figures and Tables

Figure 5: Historical market value of stETH - rebase-LSTs pegged to 1 ETH, denominated in ETH Figure 6: Historical daily returns of reward LSTs, compared to daily returns from staking ETH (dashed line) Table 2: Overview of liquidity pools analyzed in this work. Total value locked (TVL) and 24h trading volume denominated in USD on 31st December 2023. The parameters A and Gamma are constants in Stableswap and Cryptoswap Invariant AMMs Pool |  AMM | A |  Gamma |  TVL | 24h Volume
---|---|---|---|---|---
wstETH_WETH_100 |  Uniswap v3 | - |  - |  39.66m | 58.65m
wstETH_WETH_500 |  Uniswap v3 | - |  - |  0.77m | 0.02m
cbETH_WETH_500 |  Uniswap v3 | - |  - |  4.01m | 0.7m
rETH_WETH_500 |  Uniswap v3 | - |  - |  4.53m | 0.3m
stETH_WETH |  Curve v1 | 30 |  - |  251.64m | 17.72m
stETH_WETH_ng |  Curve v1 | 1500 |  - |  74.61m | 1.25m
stETH_WETH_con |  Curve v1 | 1000 |  - |  4.35m | 0.12m
rETH_WETH |  Curve v2 | 20 000 000 |  0.01 |  8.18m | 0.6m
cbETH_WETH |  Curve v2 | 20 000 000 |  0.01 |  4.29m | 0.30m

Figure 7: Hisotrical daily TVL and daily volume (30 day moving average) for
analyzed liquidity pools with LSTs, denominated in ETH

Figure 8: Historical portfolio value of LPs (blue line), LPs with CRV token
rewards (dotted blue line), HOLD (orange line), LST holder (green line), and
the average staker (red line) for selected Curve LST pools, denominated in
ETH. Pools with a longer history are additionally plotted after the Shanghai
upgrade.

Figure 9: Historical 7 day moving average for the selected Curve liquidity
pools with LST. Color green marks periods with no loss-versus-staking (LVS)
and color yellow periods with no loss-versus-holding (LVH, impermanent loss).
Pools with a longer history are additionally plotted after the Shanghai
upgrade.

Figure 10: Historical 30 day moving average for the selected Curve liquidity
pools with LST. Color green marks periods with no loss-versus-staking (LVS)
and color yellow periods with no loss-versus-holding (LVH, impermanent loss).
Pools with a longer history are additionally plotted after the Shanghai
upgrade.

Figure 11: Historical portfolio value of LPs (blue line), HOLD (orange
line),LST holder (green line), and the average staker (red line) for selected
Uniswap v3 LST pools, denominated in ETH. Pools with a longer history are
additionally plotted after the Shanghai upgrade.

Figure 12: Historical 7 day moving average for the selected Uniswap v3
liquidity pools with LST. Color green marks periods with no loss-versus-
staking (LVS) and color yellow periods with no loss-versus-holding (LVH,
impermanent loss). Pools with a longer history are additionally plotted after
the Shanghai upgrade.

Figure 13: Historical 30 day moving average for the selected Uniswap v3
liquidity pools with LST. Color green marks periods with no loss-versus-
staking (LVS) and color yellow periods with no loss-versus-holding (LVH,
impermanent loss). Pools with a longer history are additionally plotted after
the Shanghai upgrade.


1em captionlabel caption

# Electroweak Corrections to the Charm-Top-Quark Contribution to
$\epsilon_{K}$

Joachim Brod<EMAIL_ADDRESS>Department of Physics, University of
Cincinnati, Cincinnati, OH 45221, USA Sandra Kvedaraitė<EMAIL_ADDRESS>Department of Physics, University of Cincinnati, Cincinnati, OH 45221, USA
Zachary Polonsky<EMAIL_ADDRESS>Department of Physics, University of
Cincinnati, Cincinnati, OH 45221, USA Ahmed Youssef<EMAIL_ADDRESS>Department of Physics, University of Cincinnati, Cincinnati, OH 45221, USA

###### Abstract

We calculate the leading-logarithmic and next-to-leading-logarithmic
electroweak corrections to the charm-top-quark contribution to the effective
$|\Delta S|=2$ Lagrangian, relevant for the parameter $\epsilon_{K}$. We find
that these corrections lead to a $-0.5\%$ shift in the corresponding Wilson
coefficient. Moreover, our calculation removes an implicit ambiguity in the
standard-model prediction of $\epsilon_{K}$, by fixing the renormalization
scheme of the electroweak input parameters.

##  1 Introduction

Indirect CP violation in the neutral kaon system, parameterized by
$\epsilon_{K}$, is one of the most sensitive precision probes of new physics.
The parameter $\epsilon_{K}$ can be expressed to excellent approximation as
[1]

$\epsilon_{K}\equiv
e^{i\phi_{\epsilon}}\sin\phi_{\epsilon}\frac{1}{2}\arg\bigg{(}\frac{-M_{12}}{\Gamma_{12}}\bigg{)}\,.$
(1)

Here, $\phi_{\epsilon}=\arctan(2\Delta M_{K}/\Delta\Gamma_{K})$, with $\Delta
M_{K}$ and $\Delta\Gamma_{K}$ the mass and lifetime difference of the weak
eigenstates $K_{L}$ and $K_{S}$. $M_{12}$ and $\Gamma_{12}$ are the Hermitian
and anti-Hermitian parts of the Hamiltonian that determines the time evolution
of the neutral kaon system. The short-distance contributions to $\epsilon_{K}$
are then contained in the matrix element $M_{12}=-\langle
K^{0}|\mathcal{L}^{\Delta\text{S}=2}_{f=3}|\bar{K}^{0}\rangle/(2\Delta
M_{K})$, up to higher powers in the operator-product expansion.

Experimentally, $|\epsilon_{K}|=(2.228\pm 0.0011)\times 10^{-3}$ [2], with an
uncertainty at the permil level. From the theory side, recent progress
indicates that we will be able to predict $\epsilon_{K}$ in the Standard Model
(SM) with an uncertainty at the percent level in the not-so-far future.
Currently, the combined perturbative uncertainty is of the order of $3\%$,
while the non-perturbative uncertainty is of the order of $3.5\%$ [3].
Interestingly, both these errors can in principle be reduced by perturbative
calculations, the first by computing the three-loop QCD corrections to the
top-quark contribution to $\epsilon_{K}$, and the second by computing the two-
loop conversion to the $\overline{\text{MS}}$ scheme of the hadronic matrix
element. The non-local long-distance contributions to $\epsilon_{K}$,
estimated in Refs. [4] and [5], can be improved in the future with lattice
calculations (see Ref. [6] for recent results).

With theory uncertainties approaching the percent level, also parametrically
smaller corrections have been taken into consideration recently. The power
corrections to the effective Lagrangian [7] have been revisited in an extended
analysis [8], leading to a one-percent increase of the SM prediction of
$\epsilon_{K}$. On the perturbative side, the electroweak corrections to the
top-quark contribution to $\epsilon_{K}$ have been calculated by some of us
[9].

In this work, we complete the analysis of the leading perturbative electroweak
and QED corrections to $\epsilon_{K}$ by considering the mixed charm-top
contributions. The paper is organized as follows. The analytic results,
including the details of our calculation, are presented in Sec. 2. The
numerical evaluation, as well as a discussion of the results, can be found in
Sec. 3, where we also give an updated SM prediction for $\epsilon_{K}$. App. A
contains the definition of evanescent operators used in our calculation.

##  2 Electroweak corrections in the charm-top sector

In this section we provide the details of the renormalization-group (RG)
analysis. We will show the analytic results only for the electroweak and QED
corrections; the QCD corrections to $\tilde{C}_{S2}^{ut}$ have already been
presented in Refs. [10, 11] and can be transcribed to our convention for the
effective Lagrangian as explained in Ref. [3]. In particular, the NNLL QCD
results are not needed as an ingredient of our calculation. Of course, they
are included in our final numerics.

###  2.1 The effective Lagrangians

As shown in Ref. [3], it is advantageous to choose the effective Lagrangian
describing the $|\Delta S|=2$ transition in the three-flavor theory as

$\begin{split}\mathcal{L}^{|\Delta
S|=2}_{f=3}=-\frac{G_{F}^{2}M_{W}^{2}}{4\pi^{2}}\big{[}\lambda_{u}^{2}\tilde{C}_{S2}^{uu}(\mu)+\lambda_{t}^{2}\tilde{C}_{S2}^{tt}(\mu)+\lambda_{u}\lambda_{t}\tilde{C}_{S2}^{ut}(\mu)\big{]}\tilde{Q}_{S2}+\textrm{h.c.}+\dots\,,\end{split}$
(2)

because then the higher-order QCD corrections are small. Here, $G_{F}$ denotes
Fermi’s constant, and $M_{W}$ the $W$-boson mass. The parameters
$\lambda_{i}\equiv V_{is}^{*}V_{id}^{\phantom{*}}$ comprise the Cabibbo-
Kobayashi-Maskawa (CKM) matrix elements. The $|\Delta S|=2$ transition is
induced by the local operator

$\tilde{Q}_{S2}=(\bar{s}_{L}\gamma_{\mu}d_{L})(\bar{s}_{L}\gamma_{\mu}d_{L})\,,$
(3)

where $s_{L}$, $d_{L}$ denote the left-handed strange- and down-quark fields,
respectively. The ellipsis in Eq. (2) denotes non-local contributions as well
as the contribution of higher dimension operators [7, 8]. In the PDG phase
convention for the CKM matrix, $\lambda_{u}$ is real and the coefficient
$\tilde{C}_{S2}^{uu}$ does not affect $\epsilon_{K}$ (it does contribute to
the kaon mass difference $\Delta M_{K}$). The coefficient
$\tilde{C}_{S2}^{tt}$ depends on the top-quark mass and is independent of the
charm-quark mass to excellent approximation. It is known including next-to-
leading-logarithmic (NLL) QCD corrections [12], while the electroweak
corrections have been presented in Ref. [9]. The coefficient
$\tilde{C}_{S2}^{ut}$, on the other hand, depends on both the charm and top
masses and has been predicted including next-to-next-to-leading-logarithmic
(NNLL) QCD corrections [10, 11, 3]. Here, we calculate the electroweak
corrections to $\tilde{C}_{S2}^{ut}$.

The Lagrangian (2) is valid below the charm-quark scale. Its Wilson
coefficients are obtained by matching from the effective four- and five flavor
Lagrangians

$\begin{split}\mathcal{L}_{f=4,5}^{\text{eff}}&=-\frac{4G_{F}}{\sqrt{2}}\Bigg{[}\sum_{q,q^{\prime}=u,c}\\!\\!\\!V_{qs}^{\ast}V_{q^{\prime}d}(C_{+}Q_{+}^{qq^{\prime}}+C_{-}Q_{-}^{qq^{\prime}})-\lambda_{t}\sum_{i=3,6}C_{i}Q_{i}\Bigg{]}\\\
&\quad-\frac{G_{F}^{2}M_{W}^{2}}{4\pi^{2}}\lambda_{t}^{2}C_{S2}^{tt}Q_{S2}-8G_{F}^{2}\big{(}\lambda_{u}\lambda_{t}+\lambda_{t}^{2}\big{)}\tilde{C}_{7}\tilde{Q}_{7}+\text{h.c.}\,,\end{split}$
(4)

after the appropriate RG evolution, as described below. The first line in Eq.
(4) contains the $|\Delta S|=1$ current-current operators, defined as

$Q_{\pm}^{qq^{\prime}}=\frac{1}{2}\big{(}(\bar{s}_{L}^{\alpha}\gamma_{\mu}q_{L}^{\alpha})(\bar{q}_{L}^{\prime\beta}\gamma^{\mu}d_{L}^{\beta})\pm(\bar{s}_{L}^{\alpha}\gamma_{\mu}q_{L}^{\beta})(\bar{q}_{L}^{\prime\beta}\gamma^{\mu}d_{L}^{\alpha})\big{)}\,.$
(5)

Here, $\alpha$, $\beta$ are $SU(3)$ color indices. The QCD-penguin operators
$Q_{i}$, $i=3,\ldots,6$, are defined, e.g., in Ref. [11]. They are neglected
in this work as they constitute a percent-level correction to our numerically
small results (see Sec. 3 below; cf. also Ref. [13]). The $|\Delta S|=1$
operators mix, via bilocal insertions, into the local $|\Delta S|=2$ operator.
For the contributions proportional to $\lambda_{u}\lambda_{t}$, the Glashow-
Iliopoulos-Maiani mechanism ensures that the mixing starts at order
$m_{c}^{2}$; it is therefore convenient to define a rescaled version of the
$Q_{S2}$ operator as

$\tilde{Q}_{7}=\frac{m_{c}^{2}}{g_{s}^{2}\mu^{2\epsilon}}(\bar{s}_{L}\gamma_{\mu}d_{L})(\bar{s}_{L}\gamma_{\mu}d_{L})\,.$
(6)

This operator is formally of dimension eight. The appearance of the strong
coupling constant in the denominator takes account of the large logarithm in
the LO result.

###  2.2 RG Analysis of the charm-top contribution

To begin, we briefly discuss the structure of the RG-improved perturbation
series. Recall that the leading QCD RG evolution of $\tilde{C}_{S2}^{ut}$
reproduces the large logarithm $\log(m_{c}^{2}/M_{W}^{2})$ that appears in the
(fixed-order) Inami-Lim function [14], and sums this logarithm to all orders.
As these leading-order boxes involve no gluon exchange, it is conventional to
rescale the $|\Delta S|=2$ effective operator (3) with an inverse power of
$g_{s}^{2}$ (see Eq. (6)). In this way, the leading-logarithmic (LL) series
has the standard form, with terms proportional to $(\alpha_{s}\log)^{n}$,
where $n=1,\ldots$. The terms in the NLL and NNLL series are then proportional
to $\alpha_{s}(\alpha_{s}\log)^{n}$ and $\alpha_{s}^{2}(\alpha_{s}\log)^{n}$,
respectively.

Here, we will sum the two series whose terms are proportional to
$\alpha\alpha_{s}^{n}\log^{n+1}$ (“LL QED”) and $\alpha(\alpha_{s}\log)^{n}$
(“NLL QED”). The former series receives contributions only from the one-loop
QED running of the current-current operators (see below), while the latter
requires the calculation of the one-loop electroweak initial conditions in the
current-current sector, the mixed QED-QCD RG evolution of the current-current
operators, and the QED corrections to the anomalous dimension tensor, encoding
the mixing of current-current operators into the $|\Delta S|=2$ sector. In
contrast to the case of $C_{S2}^{ut}(\mu)$, the two-loop electroweak initial
condition of the Wilson coefficient in the $|\Delta S|=2$ sector is not
needed, due to the presence of the large logarithm in the leading-order (LO)
result. In addition to summing these series, our calculation fixes the
renormalization scheme of the electroweak input parameters. That is achieved
by normalizing the initial conditions of the current-current Wilson
coefficients to the muon decay constant [15]. In this way, a large part of the
radiative corrections is absorbed into the measured value of the muon decay
rate [16], and $G_{F}$ is the only requisite electroweak input parameter for
our calculation.

The actual RG analysis involves the determination of the initial conditions of
the Wilson coefficients at the electroweak scale, and the subsequent RG
evolution down to the hadronic scale where the local hadronic matrix element
is evaluated. The steps of this analysis have been discussed extensively in
the literature and no new conceptual questions arise in our analysis, so we
can afford to be brief in our exposition.

Expanding all Wilson coefficients in the five- and four-flavor effective
theories as

$C_{i}(\mu)=C_{i}^{(0)}(\mu)+\frac{\alpha}{\alpha_{s}(\mu)}C_{i}^{(e)}(\mu)+\frac{\alpha}{4\pi}C_{i}^{(es)}(\mu)\,,$
(7)

we find by performing an explicit matching calculation at the electroweak
scale

$\begin{split}C_{\pm}^{(0)}(\mu_{W})&=1\,,\quad
C_{\pm}^{(e)}(\mu_{W})=0\,,\quad
C_{\pm}^{(es)}(\mu_{W})=-\frac{22}{9}-\frac{4}{3}\log\frac{\mu_{W}^{2}}{M_{Z}^{2}}\,,\end{split}$
(8)

consistent with the results in the literature [17, 18]. The initial conditions
for the Wilson coefficients in the $|\Delta S|=2$ sector vanish at this order,
i.e., we have $\tilde{C}_{7}^{(e)}(\mu_{W})=\tilde{C}_{7}^{(es)}(\mu_{W})=0$.

In order to evolve the Wilson coefficients down to the hadronic scale, we need
to solve the set of RG equations

$\mu\frac{d}{d\mu}C_{i}(\mu)=C_{j}(\mu)\gamma_{ji}\,,\quad i,j=+,-\,,$ (9)

and

$\mu\frac{d}{d\mu}\tilde{C}_{7}(\mu)=\tilde{C}_{7}(\mu)\tilde{\gamma}_{77}+\sum_{k,l=+,-}C_{k}(\mu)C_{l}(\mu)\hat{\gamma}_{kl,7}\,.$
(10)

Here, $\tilde{\gamma}_{77}=\tilde{\gamma}_{S2}+2\gamma_{m}+2\beta$ is given in
terms of the anomalous dimension $\tilde{\gamma}_{S2}$ of the local operator
$\tilde{Q}_{S2}$. The quark anomalous dimension and the beta function appear
because of the explicit factors of $m_{c}$ and $g_{s}$ in the definition of
$\tilde{Q}_{7}$. Further, $\gamma_{ij}$ denotes the anomalous dimension matrix
in the current-current sector, and $\hat{\gamma}_{kn,7}$ is the anomalous
dimension tensor, describing the mixing of the dimension-six operators into
$\tilde{Q}_{7}$. Defining $dg_{s}/d\log\mu=\beta$, with

$\beta(g_{s},e)=-\beta_{0}\frac{g_{s}^{3}}{16\pi^{2}}-\beta_{1}\frac{g_{s}^{5}}{(16\pi^{2})^{2}}-\beta_{es}\frac{e^{2}g_{s}^{3}}{(16\pi^{2})^{2}}+\ldots\,,$
(11)

and $dm/d\log\mu=-m\gamma_{m}$, with

$\gamma_{m}(g_{s},e)=\gamma_{m}^{(0)}\frac{g_{s}^{2}}{16\pi^{2}}+\gamma_{m}^{(1)}\frac{g_{s}^{4}}{(16\pi^{2})^{2}}+\gamma_{m}^{(e)}\frac{e^{2}}{16\pi^{2}}+\gamma_{m}^{(es)}\frac{e^{2}g_{s}^{2}}{(16\pi^{2})^{2}}+\ldots\,,$
(12)

we have [19, 9]

$\tilde{\gamma}_{S2}^{(0)}=4\,,\quad\tilde{\gamma}_{S2}^{(e)}=\frac{4}{3}\,,\quad\tilde{\gamma}_{S2}^{(es)}=-\frac{148}{9}\,,$
(13)

and

$\gamma_{m}^{(0)}=8\,,\qquad\gamma_{m}^{(e)}=\frac{8}{3}\,,\qquad\gamma_{m}^{(es)}=\frac{32}{9}\,,$
(14)
$\beta_{0}=11-\frac{2}{3}f\,,\qquad\beta_{e}=0\,,\qquad\beta_{es}=-\frac{8}{9}(f_{u}+\frac{f_{d}}{4})\,,$
(15)

where $f_{u}$ and $f_{d}$ denote the number of up- and down-type quark
flavours, and $f=f_{u}+f_{d}$. Moreover [20]

$\gamma^{(0)}=\begin{pmatrix}4&0\\\
0&-8\end{pmatrix}\,,\quad\gamma^{(e)}=\begin{pmatrix}-\frac{8}{3}&0\\\
0&-\frac{8}{3}\end{pmatrix}\,,\quad\gamma^{(es)}=\begin{pmatrix}\frac{107}{9}&-18\\\\[4.30554pt]
-9&\frac{38}{9}\end{pmatrix}\,,$ (16)

and

$\hat{\gamma}_{kl,7}^{(0)}=\begin{pmatrix}3&-1\\\
-1&1\end{pmatrix}\,,\quad\hat{\gamma}_{kl,7}^{(e)}=\begin{pmatrix}0&0\\\
0&0\end{pmatrix}\,,\quad\hat{\gamma}_{kl,7}^{(es)}=\begin{pmatrix}43&-\frac{43}{3}\\\\[4.30554pt]
-\frac{43}{3}&\frac{43}{3}\end{pmatrix}\,,$ (17)

with an expansion defined in analogy to Eq. (12). The result for
$\hat{\gamma}_{kl,7}^{(es)}$ is new. It has been calculated in terms of the
renormalization constants [21] for bilocal insertions of current-current
operators (see Fig. 1 for sample Feynman diagrams). All diagrams have been
calculated using self-written FORM [22] routines, implementing the two-loop
recursion presented in Refs. [23, 24]. The amplitudes were generated using
qgraf [25]. We used the algorithm in Ref. [26] to isolate the UV divergences.

Figure 1: Sample Feynman diagrams with bilocal insertions of the current-
current operators $Q_{\pm}^{qq^{\prime}}$. Solid lines denote appropriate
quark flavors, and wavy lines denote photons.

Solving the inhomogeneous system of differential equations (9) and (10) is
tedious. It is, however, straightforward to verify that Eqs. (9) and (10) are
equivalent to the homogeneous system of equations

$\mu\frac{d}{d\mu}D(\mu)=D(\mu)\gamma\,,$ (18)

with111All penguin contributions to the QCD RG evolution are included in our
final numerics, and have been evaluated using a straightforward generalization
of these definitions. We checked explicitly that we reproduce the QCD results
in the literature, up to NNLL.

$D(\mu)=\begin{pmatrix}C_{+}(\mu)^{2}\\\ C_{+}(\mu)C_{-}(\mu)\\\
C_{-}(\mu)^{2}\\\ \tilde{C}_{7}\end{pmatrix}$ (19)

and

$\gamma=\begin{pmatrix}2\gamma_{++}&\gamma_{+-}&0&\hat{\gamma}_{++,7}\\\
2\gamma_{-+}&\gamma_{++}+\gamma_{--}&2\gamma_{+-}&\hat{\gamma}_{+-,7}+\hat{\gamma}_{-+,7}\\\
0&\gamma_{-+}&2\gamma_{--}&\hat{\gamma}_{--,7}\\\
0&0&0&\tilde{\gamma}_{77}\end{pmatrix}\,.$ (20)

This can be solved using standard methods (see, for instance, Reference [27]).
The RG evolution can be conveniently written in terms of an evolution matrix,
such that $D(\mu)=D(\mu_{0})U(\mu_{0},\mu,\alpha)$. We expand

$\begin{split}U(\mu_{0},\mu,\alpha)=U^{(0)}(\mu_{0},\mu)+\frac{\alpha}{\alpha_{s}(\mu)}U^{(e)}(\mu_{0},\mu)+\frac{\alpha}{4\pi}U^{(se)}(\mu_{0},\mu)+\ldots\,.\end{split}$
(21)

Here, $\mu_{0}$ and $\mu$ denote the generic “high” and “low” scale of the RG
evolution (i.e., $\mu_{0}=\mu_{t}$ and $\mu=\mu_{b}$ in the five-flavor, and
$\mu_{0}=\mu_{b}$ and $\mu=\mu_{c}$ in the four-flavor theory). We find the
following contributions to the Wilson coefficient at the low scale:

$\displaystyle D^{(0)}(\mu)$
$\displaystyle=D^{(0)}(\mu_{0})U^{(0)}(\mu_{0},\mu)\,,$ (22) $\displaystyle
D^{(e)}(\mu)$
$\displaystyle=D^{(0)}(\mu_{0})U^{(e)}(\mu_{0},\mu)+\eta^{-1}D^{(e)}(\mu_{0})U^{(0)}(\mu_{0},\mu)\,,$
(23) $\displaystyle\begin{split}D^{(se)}(\mu)&=\eta
D^{(1)}(\mu_{0})U^{(e)}(\mu_{0},\mu)+\eta^{-1}D^{(e)}(\mu_{0})U^{(1)}(\mu_{0},\mu)\\\
&\quad+D^{(se)}(\mu_{0})U^{(0)}(\mu_{0},\mu)+D^{(0)}(\mu_{0})U^{(se)}(\mu_{0},\mu)\,,\end{split}$
(24)

where we have introduced the ratio $\eta=\alpha_{s}(\mu_{0})/\alpha_{s}(\mu)$.
The explicit expression for the evolution matrix, in terms of the anomalous
dimensions of the Wilson coefficients, can be found in Ref. [28].

At the bottom threshold, $\mu_{b}\sim m_{b}$, the bottom quark is removed as a
dynamical degree of freedom. Numerically, the impact of this threshold
correction is small. In fact, since we neglect the contribution of penguin
operators for the QED and electroweak corrections, the only effect is the
decoupling of $\alpha_{s}$ from $f=5$ to $f=4$:

$\alpha_{s}^{(5)}=\alpha_{s}^{(4)}\bigg{(}1+\frac{2}{3}\frac{\alpha_{s}^{(4)}}{4\pi}\log(\mu_{b}^{2}/m_{b}(\mu_{b})^{2})\bigg{)}\,,$
(25)

leading to an additional logarithmic contribution to all Wilson coefficients.
Requiring the equality of all Green’s functions at the matching scale and
writing $\delta C(\mu_{b})=C_{f=5}(\mu_{b})-C_{f=4}(\mu_{b})$ we find, for the
dimension-six Wilson coefficients,

$\delta C_{i}^{(0)}=0\,,\quad\delta C_{i}^{(e)}=0\,,\quad\delta
C_{i}^{(es)}=\frac{2}{3}C_{i}^{(e)}\log\bigg(\frac{\mu_{b}^{2}}{m_{b}(\mu_{b})^{2}}\bigg{missing})\,,$
(26)

and for the dimension-eight Wilson coefficient (taking into account the factor
$m_{c}^{2}/g_{s}^{2}$ in the definition of the operator)

$\delta\tilde{C}_{7}^{(0)}=0\,,\quad\delta\tilde{C}_{7}^{(e)}=0\,,\quad\delta\tilde{C}_{7}^{(es)}=\frac{4}{3}\tilde{C}_{7}^{(e)}\log\bigg(\frac{\mu_{b}^{2}}{m_{b}(\mu_{b})^{2}}\bigg{missing})\,.$
(27)

At the scale $\mu_{c}\sim m_{c}$ the charm quark is removed from the theory as
a dynamical degree of freedom, and the effective Lagrangian is now given by
Eq. (2). Requiring the equality of the Green’s functions in both the four-
flavor and three-flavor theories at the charm-quark scale leads to the
matching condition

$\sum_{k,l=+,-}C_{k}(\mu_{c})C_{l}(\mu_{c})\langle
Q_{k}Q_{l}\rangle(\mu_{c})+{\tilde{C}}_{7}(\mu_{c})\langle{\tilde{Q}}_{7}\rangle(\mu_{c})=\frac{M_{W}^{2}}{32\pi^{2}}{\tilde{C}}_{S2}^{ut}(\mu_{c})\langle{\tilde{Q}}_{S2}\rangle(\mu_{c})\,,$
(28)

where the angle brackets denote the partonic $|\Delta S|=2$ matrix elements.
We parameterize these matrix elements in the following way:

$\langle{\tilde{Q}}_{7}\rangle=r_{S2}\langle{\tilde{Q}}_{7}\rangle^{(0)}\,,\quad\langle{\tilde{Q}}_{S2}\rangle=r_{S2}\langle{\tilde{Q}}_{S2}\rangle^{(0)}\,,\quad\text{and}\,\,\langle
Q_{i}Q_{j}\rangle(\mu_{c})=\frac{m_{c}^{2}(\mu_{c})}{32\pi^{2}M_{W}^{2}}r_{ij,S2}^{ut}\langle{\tilde{Q}}_{S2}\rangle^{(0)}\,.$
(29)

Taking into account the explicit factor of $m_{c}^{2}/g_{s}^{2}$ in the
definition of $\tilde{Q}_{7}$, we expand the Wilson coefficient
$\tilde{C}_{S2}^{ut}$ as

$\tilde{C}_{S2}^{ut}=\frac{4\pi}{\alpha_{s}^{(3)}}\tilde{C}_{S2}^{ut(0)}+\frac{4\pi\alpha}{(\alpha_{s}^{(3)})^{2}}\tilde{C}_{S2}^{ut(e)}+\frac{\alpha}{\alpha_{s}^{(3)}}\tilde{C}_{S2}^{ut(es)}+\ldots\,,$
(30)

and find the following contributions to the matching:

$\displaystyle\tilde{C}_{S2}^{ut(0)}(\mu_{c})$
$\displaystyle=2\frac{m_{c}^{2}(\mu_{c})}{M_{W}^{2}}\tilde{C}_{7}^{(0)}(\mu_{c})\,,$
(31) $\displaystyle\tilde{C}_{S2}^{ut(e)}(\mu_{c})$
$\displaystyle=2\frac{m_{c}^{2}(\mu_{c})}{M_{W}^{2}}\tilde{C}_{7}^{(e)}(\mu_{c})\,,$
(32) $\displaystyle\tilde{C}_{S2}^{ut(es)}(\mu_{c})$
$\displaystyle=2\frac{m_{c}^{2}(\mu_{c})}{M_{W}^{2}}\bigg{[}\tilde{C}_{7}^{(es)}(\mu_{c})-\frac{4}{3}\tilde{C}_{7}^{(e)}(\mu_{c})\log\frac{\mu_{c}^{2}}{m_{c}(\mu_{c})^{2}}\bigg{]}$
$\displaystyle\quad+\frac{m_{c}^{2}(\mu_{c})}{M_{W}^{2}}\Big{[}C_{i}^{(0)}(\mu_{c})C_{j}^{(e)}(\mu_{c})+C_{i}^{(e)}(\mu_{c})C_{j}^{(0)}(\mu_{c})\Big{]}r_{ij,S2}^{ut,(0)}\,.$
(33)

Here, $m_{c}(\mu_{c})$ denotes the running charm-quark mass, including the
leading QED running. We see that for the electroweak corrections, the LO
matching result is sufficient. According to Ref. [3], it can be taken as
$r_{ij,S2}^{ut}=2r_{ij,S2}^{cc}-r_{ij,S2}^{ct}$ in terms of the results in
Ref. [10, 11, 3]. We find

$r_{ij,S2}^{ut,(0)}=\begin{pmatrix}\frac{9}{2}-3\log\frac{\mu_{c}^{2}}{m_{c}^{2}(\mu_{c})}&-\frac{3}{2}+\log\frac{\mu_{c}^{2}}{m_{c}^{2}(\mu_{c})}\\\\[4.30554pt]
-\frac{3}{2}+\log\frac{\mu_{c}^{2}}{m_{c}^{2}(\mu_{c})}&\frac{3}{2}-\log\frac{\mu_{c}^{2}}{m_{c}^{2}(\mu_{c})}\end{pmatrix}\,.$
(34)

Finally, the RG evolution in the effective three-flavor theory involves only
the single physical operator $\tilde{Q}_{S2}$, with anomalous dimension given
in Eq. (13). As discussed in detail in Ref. [9], the mixed two-loop anomalous
dimension $\tilde{\gamma}_{S2}^{(es)}$ is renormalization-scheme independent,
which prevents us from extending the definition of the scheme-independent
correction factors $\eta_{ut}$ to include electroweak corrections. Instead, we
have to work with Wilson coefficients directly. In particular, our result for
$\tilde{C}_{S2}^{ut}$ is not independent of the renormalization scheme.

Of course, this residual scheme dependence will cancel once we multiply
$\tilde{C}_{S2}^{ut}$ by the hadronic matrix element of the local operator
$\tilde{Q}_{S2}$, evaluated including the leading QED corrections. While this
matrix element is a non-perturbative quantity, and the QED corrections are not
(yet) available, it is easy to calculate the scheme-dependent part [9]. As a
cross check of our calculation, we kept the definition of all contributing
evanescent operators arbitrary (see App. A) and verified that all scheme
dependence completely cancels in the product of the Wilson coefficient and
(the scheme-dependent part of) the matrix element. In particular, we verified
that the only left-over dependence of $\tilde{C}_{S2}^{ut}$ is on the
parameter $a_{11}$; all other parameters cancel.222In this context we note
that one of the statements made below Eq. (C.12) in Ref. [9] is not correct:
even for scheme-independent $\tilde{\gamma}_{S2}^{(es)}$, the corresponding
evolution matrix $U^{(se)}$ does depend on the renormalization scheme via its
dependence on the two-loop QCD anomalous dimension,
$\tilde{\gamma}_{S2}^{(1)}$. However, our analytic check shows that this
dependence drops out completely in the product of Wilson coefficient and the
(known) QCD part of the hadronic matrix element. The same is true in the top-
quark sector. We will discuss the numerical size of the scheme-dependent term
in the following section.

##  3 Discussion and Conclusion

To obtain a numerical estimate of the size of the electroweak corrections, as
well as an estimate of the remaining perturbative uncertainties, we evaluate
the Wilson coefficient $\tilde{C}_{S2}^{ut}(2\,\text{GeV})$, including now all
known QCD corrections, and varying the electroweak and charm-threshold
matching scales in the intervals $40\,\text{GeV}\leq\mu_{t}\leq
320\,\text{GeV}$ and $1\,\text{GeV}\leq\mu_{c}\leq 2\,\text{GeV}$. (The
dependence on the bottom-quark matching scale is negligible in comparison.)
The resulting residual scale variation is displayed in Fig. 2.

Figure 2: Residual dependence of the Wilson cofficient
$\tilde{C}_{S2}^{ut}(2\,\mathrm{GeV})$ on the electroweak (left panel) and
charm-threshold (right panel) matching scales. The short-dashed, dotted, and
dash-dotted lines show the LL, NLL, and NNLL QCD results, respectively. The
long-dashed and solid lines show the results including also the LL and NLL
electroweak corrections.

To obtain a final value, we fix $\mu_{t}=m_{t}$ and take the average of the
highest and lowest value of $\tilde{C}_{S2}^{ut}$ in the interval for the
variation of $\mu_{c}$, and half the difference between the highest and lowest
values as the uncertainty. Retaining only the QCD corrections up to NNLL, we
find $\tilde{C}_{S2}^{ut,\text{QCD}}=-13.84\pm 0.17$. Including also the LL
and NLL electroweak corrections gives $\tilde{C}_{S2}^{ut}=-13.92\pm 0.16$.
This amounts to a $-0.5\%$ shift, while the uncertainty is essentially
unchanged.

What is the numerical impact of the unmatched scheme-dependent term on this
result? First, recall that the only dependence is on the parameter $a_{11}$
(see App. A); all other scheme dependence fully cancels against the
corresponding terms in the hadronic matrix element.333In practice, the
hadronic matrix element is only converted to the $\overline{\text{MS}}$ scheme
at NLO, such that part of the NNLO QCD scheme dependence is not included.
However, in the conventional formalism of the $\eta$ correction factors, the
perturbative part of the result is scheme independent including NNLO, as far
as QCD is concerned. Part of the dependence on $a_{11}$ of our result arises
from the dependence on the two-loop QCD anomalous dimension; any such
dependence is also canceled by the corresponding scheme dependence of the
hadronic matrix element. The only residual dependence on $a_{11}$ is that
related to the leading QED corrections to the matrix element; numerically, it
is tiny:

$\tilde{C}_{S2}^{ut}(2\,\text{GeV})=-14.075+0.001a_{11}\,.$ (35)

We obtained this number by setting all threshold matching scales to their
respective quark masses.

While a consistent estimate of the full electroweak and QED corrections can be
obtained only once a lattice calculation (or another systematic estimate) of
the QED correction to the hadronic matrix element becomes available, we point
out that this correction is not enhanced by a large logarithm and thus of
order $\alpha/(4\pi)\sim 10^{-4}$, the same as the residual scheme dependence.
It is expected to be numerically negligible compared to the $-0.5\%$ shift
found above. It follows that for our numerics we can safely adopt the standard
definition of evanescent operators with $a_{11}=4$.

To summarize, given the uncancelled (but small) residual scheme dependence of
our result, we propose a temporary prescription in analogy to the case of the
top-quark contribution [9]: we rescale the NNLL QCD value of
$\eta_{ut}=0.402(5)$ [3] by a factor of $1.005$, to take account of the
electroweak corrections.444Recall that $\eta_{ut}$ is defined via
$\tilde{C}_{S2}^{ut}=2\eta_{ut}\mathscr{S}_{ut}(x_{c},x_{t})$, with the
modified Inami-Lim function [3] being negative. Including also the power
correction presented in Ref. [8], this leads to an updated SM prediction of

$|\epsilon_{K}|=\big{(}2.170\pm 0.065_{\text{pert.}}\pm
0.076_{\text{nonpert.}}\pm 0.153_{\text{param.}}\big{)}\times 10^{-3}\,.$ (36)

Here, the quoted errors correspond to the residual perturbative, non-
perturbative, and parametric uncertainties, respectively; see Ref. [3] for
details. We obtained this number by employing the phenomenological expression
in Ref. [19], including the long-distance corrections presented in Refs. [4,
5].

All parametric inputs are taken from PDG [2]. In particular, as input for the
top-quark mass we use the $\overline{\text{MS}}$ mass
$m_{t}(m_{t})=162.92(67)\,$GeV, obtained by converting the pole mass
$M_{t}=172.5(7)\,$GeV [2] to $\overline{\text{MS}}$ at three-loop accuracy
using RUNDEC [29].

In summary, we calculated the leading and next-to-leading electroweak
corrections to the charm-top contribution $\tilde{C}_{S2}^{ut}$ to the
effective $|\Delta S|=2$ effective Lagrangian, using RG-improved perturbation
theory. We find a small negative shift of the Wilson coefficient, and a
corresponding small positive shift of $\epsilon_{K}$. A systematic estimate of
the QED corrections to the hadronic matrix element would complete our
analysis.

As consistency checks, we performed the calculation in generalized $R_{\xi}$
gauge for gluons and photon and verified the gauge-parameter independence of
our results. We analytically checked that our results are independent of all
matching scales, and that the dependence on the renormalization scheme is
canceling by the corresponding scheme dependence of the hadronic matrix
element.

The aim of this work is to provide a further step in the prediction of
$\epsilon_{K}$ with residual theoretical uncertainty at the percent level.
Further important directions of improvement are the calculation of the three-
loop QCD corrections in the top-quark sector of the effective Lagrangian, and
the NLO scheme conversion from RI/SMOM to $\overline{\text{MS}}$ for the
hadronic matrix element of the local $|\Delta S|=2$ operator.

## Acknowledgements

The authors acknowledge support in part by DoE grant de-sc0011784. J.B. thanks
Martin Gorbahn and Emmanuel Stamou for discussions.

##  Appendix A Definition of evanescent operators

In the context of dimensional regularization, evanescent operators arise in
intermediate stages of the calculation because certain relations (such as
Dirac algebra and Fierz transformations) are valid only in four space-time
dimensions. In the dimension-six sector, we define them as

$\displaystyle E_{1}^{qq^{\prime}(1)}$
$\displaystyle=(\bar{s}_{L}\gamma_{\mu_{1}\mu_{2}\mu_{3}}T^{a}q_{L})\otimes(\bar{q}_{L}^{\prime}\gamma^{\mu_{1}\mu_{2}\mu_{3}}T^{a}d_{L})-(16-A_{11}\epsilon-
A_{12}\epsilon^{2})Q_{1}^{qq^{\prime}}\,,$ (37) $\displaystyle
E_{2}^{qq^{\prime}(1)}$
$\displaystyle=(\bar{s}_{L}\gamma_{\mu_{1}\mu_{2}\mu_{3}}q_{L})\otimes(\bar{q}_{L}^{\prime}\gamma^{\mu_{1}\mu_{2}\mu_{3}}d_{L})-(16-B_{11}\epsilon-
B_{12}\epsilon^{2})Q_{2}^{qq^{\prime}}\,,$ (38) $\displaystyle
E_{1}^{qq^{\prime}(2)}$
$\displaystyle=(\bar{s}_{L}\gamma_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}}T^{a}q_{L})\otimes(\bar{q}_{L}^{\prime}\gamma^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}}T^{a}d_{L})-\bigg{(}256-A_{21}\epsilon-
A_{22}\epsilon^{2}\bigg{)}Q_{1}^{qq^{\prime}}\,,$ (39) $\displaystyle
E_{2}^{qq^{\prime}(2)}$
$\displaystyle=(\bar{s}_{L}\gamma_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}}q_{L})\otimes(\bar{q}_{L}^{\prime}\gamma^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}}d_{L})-\bigg{(}256-B_{21}\epsilon-
B_{22}\epsilon^{2}\bigg{)}Q_{2}^{qq^{\prime}}\,.$ (40)

while the evanescent operators in the dimension-eight sector have been chosen
as

$\displaystyle\tilde{E}_{F}$
$\displaystyle=\frac{m_{c}^{2}}{g^{2}\mu^{2\epsilon}}(\bar{s}_{L}^{\alpha}\gamma_{\mu}d_{L}^{\beta})\otimes(\bar{s}_{L}^{\beta}\gamma^{\mu}d_{L}^{\alpha})-\tilde{Q}_{7}\,,$
(41) $\displaystyle\tilde{E}_{7}^{(1)}$
$\displaystyle=\frac{m_{c}^{2}}{g^{2}\mu^{2\epsilon}}(\bar{s}_{L}^{\alpha}\gamma_{\mu_{1}\mu_{2}\mu_{3}}d_{L}^{\alpha})\otimes(\bar{s}_{L}^{\beta}\gamma^{\mu_{1}\mu_{2}\mu_{3}}d_{L}^{\beta})-(16-a_{11}\epsilon-
a_{12}\epsilon^{2})\tilde{Q}_{7}\,,$ (42) $\displaystyle\tilde{E}_{8}^{(1)}$
$\displaystyle=\frac{m_{c}^{2}}{g^{2}\mu^{2\epsilon}}(\bar{s}_{L}^{\alpha}\gamma_{\mu_{1}\mu_{2}\mu_{3}}d_{L}^{\beta})\otimes(\bar{s}_{L}^{\beta}\gamma^{\mu_{1}\mu_{2}\mu_{3}}d_{L}^{\alpha})-(16-b_{11}\epsilon-
b_{12}\epsilon^{2})(\tilde{Q}_{7}+\tilde{E}_{F})\,,$ (43)
$\displaystyle\tilde{E}_{7}^{(2)}$
$\displaystyle=\frac{m_{c}^{2}}{g^{2}\mu^{2\epsilon}}(\bar{s}_{L}^{\alpha}\gamma_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}}d_{L}^{\alpha})\otimes(\bar{s}_{L}^{\beta}\gamma^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}}d_{L}^{\beta})-(256-a_{21}\epsilon-
a_{22}\epsilon^{2})\tilde{Q}_{7}\,,$ (44) $\displaystyle\tilde{E}_{8}^{(2)}$
$\displaystyle=\frac{m_{c}^{2}}{g^{2}\mu^{2\epsilon}}(\bar{s}_{L}^{\alpha}\gamma_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}}d_{L}^{\beta})\otimes(\bar{s}_{L}^{\beta}\gamma^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}\mu_{5}}d_{L}^{\alpha})-(256-b_{21}\epsilon-
b_{22}\epsilon^{2})(\tilde{Q}_{7}+\tilde{E}_{F})\,.$ (45)

Note that, to facilitate an additional check on our calculation, we have kept
the coefficients in front of the $\epsilon$ terms arbitrary. In the
conventional definition of these operators, $A_{1i}=B_{1i}=a_{1i}=b_{1i}=4$
where $i=1,2$, $A_{21}=B_{21}=a_{21}=b_{21}=224$, $a_{22}=5712/25$,
$B_{22}=10032/25$, and $a_{22}=b_{22}=108\,816/325$. The evanescent operators
related to $\tilde{Q}_{S2}$ are defined with the same coefficients. We have
checked explicitly that the terms quadratic in $\epsilon$ do not contribute to
the two-loop anomalous dimensions. All results quoted in the main body of the
paper correspond to the conventional definition of evanescent operators.

## References

  * [1] _$B$ physics at the Tevatron: Run II and beyond_, 12, 2001.
  * [2] Particle Data Group collaboration, R. L. Workman, _Review of Particle Physics_ , _PTEP_ 2022 (2022) 083C01.
  * [3] J. Brod, M. Gorbahn and E. Stamou, _Standard-Model Prediction of $\epsilon_{K}$ with Manifest Quark-Mixing Unitarity_, _Phys. Rev. Lett._ 125 (2020) 171803, [1911.06822].
  * [4] _$B$ physics at the Tevatron: Run II and beyond_, 12, 2001.
  * [5] A. J. Buras, D. Guadagnoli and G. Isidori, _On $\epsilon_{K}$ Beyond Lowest Order in the Operator Product Expansion_, _Phys. Lett. B_ 688 (2010) 309–313, [1002.3612].
  * [6] RBC, UKQCD collaboration, R. Abbott et al., _Direct CP violation and the $\Delta I=1/2$ rule in $K\to\pi\pi$ decay from the standard model_, _Phys. Rev. D_ 102 (2020) 054509, [2004.09440].
  * [7] O. Cata and S. Peris, _Long distance dimension eight operators in B(K)_ , _JHEP_ 03 (2003) 060, [hep-ph/0303162].
  * [8] M. Ciuchini, E. Franco, V. Lubicz, G. Martinelli, L. Silvestrini and C. Tarantino, _Power corrections to the CP-violation parameter $\varepsilon$K_, _JHEP_ 02 (2022) 181, [2111.05153].
  * [9] J. Brod, S. Kvedaraitė and Z. Polonsky, _Two-loop electroweak corrections to the Top-Quark Contribution to $\epsilon$K_, _JHEP_ 12 (2021) 198, [2108.00017].
  * [10] S. Herrlich and U. Nierste, _Enhancement of the K(L) - K(S) mass difference by short distance QCD corrections beyond leading logarithms_ , _Nucl. Phys. B_ 419 (1994) 292–322, [hep-ph/9310311].
  * [11] J. Brod and M. Gorbahn, _Epsilon_K at Next-to-Next-to-Leading Order: The Charm-Top-Quark Contribution_ , _Phys. Rev. D_ 82 (2010) 094026, [1007.0684].
  * [12] A. J. Buras, M. Jamin and P. H. Weisz, _Leading and Next-to-leading QCD Corrections to $\epsilon$ Parameter and $B^{0}-\bar{B}^{0}$ Mixing in the Presence of a Heavy Top Quark_, _Nucl. Phys. B_ 347 (1990) 491–536.
  * [13] S. Herrlich and U. Nierste, _The Complete $|\Delta S|=2$ Hamiltonian in the next-to-leading order_, _Nucl. Phys. B_ 476 (1996) 27–88, [hep-ph/9604330].
  * [14] T. Inami and C. S. Lim, _Effects of Superheavy Quarks and Leptons in Low-Energy Weak Processes $K_{L}\to\mu\bar{\mu}$, $K^{+}\to\pi^{+}\nu\bar{\nu}$, and $K^{0}\leftrightarrow\bar{K}^{0}$_, _Prog. Theor. Phys._ 65 (1981) 297.
  * [15] A. Sirlin, _Large m(W), m(Z) Behavior of the O(alpha) Corrections to Semileptonic Processes Mediated by W_ , _Nucl. Phys. B_ 196 (1982) 83–92.
  * [16] J. Brod and M. Gorbahn, _Electroweak Corrections to the Charm Quark Contribution to $K^{+}\to\pi^{+}\nu\bar{\nu}$_, _Phys. Rev. D_ 78 (2008) 034006, [0805.4119].
  * [17] P. Gambino and U. Haisch, _Electroweak effects in radiative B decays_ , _JHEP_ 09 (2000) 001, [hep-ph/0007259].
  * [18] P. Gambino and U. Haisch, _Complete electroweak matching for radiative B decays_ , _JHEP_ 10 (2001) 020, [hep-ph/0109058].
  * [19] G. Buchalla, A. J. Buras and M. E. Lautenbacher, _Weak decays beyond leading logarithms_ , _Rev. Mod. Phys._ 68 (1996) 1125–1144, [hep-ph/9512380].
  * [20] A. J. Buras, M. Jamin and M. E. Lautenbacher, _Two loop anomalous dimension matrix for Delta S = 1 weak nonleptonic decays. 2. O(alpha-alpha-s)_ , _Nucl. Phys. B_ 400 (1993) 75–102, [hep-ph/9211321].
  * [21] J. Brod, _Higher-order corrections to precision observables in kaon decays_ , Ph.D. thesis, Karlsruhe U., TTP, 2009.
  * [22] J. A. M. Vermaseren, _New features of FORM_ , math-ph/0010025.
  * [23] A. I. Davydychev and J. B. Tausk, _Two loop selfenergy diagrams with different masses and the momentum expansion_ , _Nucl. Phys._ B397 (1993) 123–142.
  * [24] C. Bobeth, M. Misiak and J. Urban, _Photonic penguins at two loops and $m_{t}$ dependence of $BR[B\to X_{s}l^{+}l^{-}]$_, _Nucl. Phys. B_ 574 (2000) 291–330, [hep-ph/9910220].
  * [25] P. Nogueira, _Automatic Feynman graph generation_ , _J. Comput. Phys._ 105 (1993) 279–289.
  * [26] K. G. Chetyrkin, M. Misiak and M. Munz, _Beta functions and anomalous dimensions up to three loops_ , _Nucl. Phys. B_ 518 (1998) 473–494, [hep-ph/9711266].
  * [27] M. Gorbahn and U. Haisch, _Effective Hamiltonian for non-leptonic $|\Delta F|=1$ decays at NNLO in QCD_, _Nucl. Phys. B_ 713 (2005) 291–332, [hep-ph/0411071].
  * [28] A. J. Buras, M. Jamin and M. E. Lautenbacher, _The Anatomy of $\epsilon^{\prime}/\epsilon$ beyond leading logarithms with improved hadronic matrix elements_, _Nucl. Phys. B_ 408 (1993) 209–285, [hep-ph/9303284].
  * [29] K. G. Chetyrkin, J. H. Kuhn and M. Steinhauser, _RunDec: A Mathematica package for running and decoupling of the strong coupling and quark masses_ , _Comput. Phys. Commun._ 133 (2000) 43–65, [hep-ph/0004189].