# Finite-key Analysis for Quantum Conference Key Agreement with Asymmetric
Channels

Zhao Li  Xiao-Yu Cao  Chen-Long Li  Chen-Xun Weng  Jie Gu  Hua-Lei Yin
<EMAIL_ADDRESS>Zeng-Bing Chen<EMAIL_ADDRESS>National Laboratory of
Solid State Microstructures, School of Physics and Collaborative Innovation
Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

###### Abstract

As an essential ingredient of quantum networks, quantum conference key
agreement (QCKA) provides unconditional secret keys among multiple parties,
which enables only legitimate users to decrypt the encrypted message.
Recently, some QCKA protocols employing twin-field was proposed to promote
transmission distance. These protocols, however, suffer from relatively low
conference key rate and short transmission distance over asymmetric channels,
which demands a prompt solution in practice. Here, we consider a tripartite
QCKA protocol utilizing the idea of sending-or-not-sending twin-field scheme
and propose a high-efficiency QCKA over asymmetric channels by removing the
symmetry parameters condition. Besides, we provide a composable finite-key
analysis with rigorous security proof against general attacks by exploiting
the entropic uncertainty relation for multiparty system. Our protocol greatly
improves the feasibility to establish conference keys over asymmetric
channels.

##  I INTRODUCTION

Remote distribution of secret key is an essential task of quantum
cryptographic network. Quantum key distribution (QKD) [1, 2, 3, 4, 5] which
allows two remote users to share unconditionally secure key has achieved
relatively mature development [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20]. To extend QKD to multiparty scenarios, one intuitive way is to refer
to classical conference key agreement [21, 22]. Therefore, quantum conference
key agreement (QCKA) [23, 24, 25, 26] has been proposed and become a
significant pillar of quantum cryptography. QCKA can offer information-
theoretic secure conference key which promises group encryption and decryption
for all legitimate users. Specifically, the conference key can be naturally
established with distribution of Greenberger-Horne-Zeilinger (GHZ) entangled
states [27, 28, 23] and local projection measurement, which has been proved to
precede simple bipartite QKD links [29]. Basic laws of quantum mechanics
combined with one-time pad encryption [30] guarantee the security of
multiparty quantum communication [24, 31]. Currently, QCKA has been
generalized to other variants both in theory [32, 33, 34, 35, 36, 37, 38, 39,
40] and experiment [41].

Although some progresses have been made, the conference key rates of most QCKA
protocols are lower than the capacity of quantum channels [42, 43]. To achieve
higher conference key rate and longer transmission distance, some protocols
inspired by the idea of twin-field QKD were proposed [44, 45, 46]. However,
all of these protocols neglect the case of asymmetric channels, which makes
them impractical. Our paper focuses on the tripartite QCKA protocol [46]
employing the sending-or-not-sending scheme [13]. The precondition for
security requires two senders having the same source parameters (e.g. the
intensity and the probability of sending varied coherent pulses). Asymmetric
channels with symmetric source parameters will introduce additional intrinsic
errors, which inevitably reduce the conference key rate and the transmission
distance. In practical, such as intercity communication, the transmission
losses of two channels are usually asymmetric.

Inspired by the variants of twin-field QKD over asymmetric channels [19, 47,
48, 49, 50, 51], we design a high-efficiency QCKA protocol for asymmetric
channels. With the similar devices as twin-field QKD [52, 53], our protocol
has a longer transmission distance, a higher key rate and is more practical
than other QCKA protocols [40]. We provide a security proof for this protocol
by utilizing the entanglement distillation [31] of virtual GHZ-class state. In
addition, with the extension of twin-field QKD to QCKA, we perform a finite-
key analysis [54, 55, 35, 56] with multiparty composable security to make our
work more practical.

##  II Protocol description

Figure 1: The schematic of the asymmetric QCKA protocol. There are two
senders, Alice and Bob, including some primary devices as follows. Continuous-
wave laser: generation of the global phase stabilized coherent light;
intensity modulator (IM): pulse preparation; phase modulator (PM): phase
randomization; attenuator (Att): weak-light modulation. They send the pulses
to Charlie over their own channels. Charlie, as a receiver, possesses the
following devices. Beam splitters BS1 and BS2: passive-basis choice; beam
splitter BS3: interference between two pulses; detector D1 and D2: the Z basis
measurement; detector D3 and D4: the X basis measurement.

As shown in figure 1, our asymmetric QCKA protocol includes three legitimate
users: Alice, Bob and Charlie with the same devices as the original scheme
[46]. We utilize the decoy-state method with three intensities for both
senders. We present the detailed protocol as follows.

Preparation.— For each time window, Alice and Bob independently and randomly
choose one basis from $\\{\mathrm{Z},\ \mathrm{X}\\}$. If Alice (Bob) selects
Z basis, she (he) encodes the logic bit value $1\ (0)$ with phase-randomized
weak coherent pulses $\ket{e^{i\omega_{a}}\sqrt{\mu_{a}}}\
(\ket{e^{i\omega_{a}}\sqrt{\mu_{b}}})$, whose probability is $t_{a}\ (t_{b})$.
She (He) encodes the logic bit value $0\ (1)$ with sending nothing, whose
probability is $1-t_{a}\ (1-t_{b})$. If Alice (Bob) selects X basis, she (he)
generates phase-randomized weak coherent pulses
$\ket{e^{i\theta_{a}}\sqrt{k_{a}}}\ (\ket{e^{i\theta_{b}}\sqrt{k_{b}}})$, with
random global phase $\theta_{a},\ \theta_{b}\in\left[0,\ 2\pi\right)$ and
intensity $k_{a}\in\\{\mu_{a},\ \nu_{a},\ 0\\},\ k_{b}\in\\{\mu_{b},\
\nu_{b},\ 0\\}$. They send the pulses to Charlie through two insecure quantum
channels that may have different losses. Besides, a constraint for source
parameters is required [47],

$\frac{\nu_{a}}{\nu_{b}}=\frac{t_{a}\left(1-t_{b}\right)\mu_{a}e^{-\mu_{a}}}{t_{b}\left(1-t_{a}\right)\mu_{b}e^{-\mu_{b}}},$
(1)

which will be given by security analysis.

Measurement.— For each round, Charlie passively selects a basis from Z and X
bases by two beam splitters and performs measurement. For Z basis, Charlie
gets logic bit 1 when detector D1 clicks only, and logic bit 0 when detector
D2 clicks only. The bit value is 0 when D3 clicks only in X basis while the
bit value is 1 when D4 clicks only.

Reconciliation.— Alice, Bob, and Charlie announce the basis information for
each successful measurement event, which means that only one detector clicks
in each round of measurement. Alice and Bob publicly announce the intensity
information unless Alice, Bob, and Charlie all select Z basis. In the case
where Alice and Bob choose intensities $\nu_{a}$, $\nu_{b}$ respectively and
only D3 or D4 clicks, Alice and Bob disclose the global phase. Then they
perform post-selected phase-matching, which screens out the effective events
satisfying $-\delta+r\pi\leq\theta_{a}-\theta_{b}-\varphi_{ab}\leq\delta+r\pi\
(\mathrm{mod}~{}2\pi)$, where $\delta$ is a selected value for a small phase
range and $r\in\\{0,\ 1\\}$. Here $\varphi_{ab}$ is the phase difference
between the reference frames of Alice and Bob, which can be obtained by phase
post-compensation methods. All classical information is transmitted via the
authenticated classical channels.

Parameter estimation.— Alice, Bob, and Charlie form raw keys from the data
when they all selected Z basis. Utilizing the decoy-state method [6, 7], the
other cases are exploited to estimate parameters and the information leakage.

Postprocessing.— They implement classical error correction, error verification
and privacy amplification to distill the final conference key.

##  III Security analysis

In our protocol, Alice randomly chooses a basis for each round. For Z basis,
she prepares $\ket{0}$ and $\ket{e^{i\omega_{a}}\mu_{a}}$ by using the vacuum
and phase-randomized coherent state with probabilities $1-t_{a}$ and $t_{a}$,
where $\omega_{a}\in\left[0,\ 2\pi\right)$ is the randomized phase. For X
basis, she randomly generates phase-randomized coherent state
$\ket{e^{i\theta_{a}}\sqrt{\nu_{a}}}$. Likewise, Bob does the same operations
as Alice.

Alice and Bob only reserve joint single-photon states in Z basis, i.e.
$\ket{10}_{ab}$ and $\ket{01}_{ab}$, to extract the conference key. The
density matrix is

$\displaystyle\rho_{z}^{1}=$
$\displaystyle\left[t_{a}\left(1-t_{b}\right)\mu_{a}e^{-\mu_{a}}\ket{10}_{ab}\bra{10}\right.$
(2)
$\displaystyle\left.+t_{b}\left(1-t_{a}\right)\mu_{b}e^{-\mu_{b}}\ket{01}_{ab}\bra{01}\right]/C,$

where
$C=t_{a}\left(1-t_{b}\right)\mu_{a}e^{-\mu_{a}}+t_{b}\left(1-t_{a}\right)\mu_{b}e^{-\mu_{b}}$
is the normalization factor. They implement post-selected phase-matching and
only consider the case of joint one-photon states in X basis. The density
matrix is

$\rho_{x}^{1}=\frac{1}{\nu_{a}+\nu_{b}}(\nu_{a}\ket{10}_{ab}\bra{10}+\nu_{b}\ket{01}_{ab}\bra{01}),$
(3)

which needs to meet $\rho_{x}^{1}=\rho_{z}^{1}=\rho^{1}$, so we get equation
(1). Note that we introduce the phase-matching here to obtain low phase error
rate. The decoy state method can guarantee the feasibility of retaining joint
single-photon states. Our security analysis only considers perfect sources
which can prepare pulses with precise intensity for meeting the constraint of
equation (1). However, due to the source imperfection in practice, the
intensity fluctuations [57] will affect Alice’s and Bob’s state preparation,
which is still an issue for consideration. The security analysis for more
practical cases should be further considered.

Here, we consider a entanglement-swapping virtual protocol [16], which has a
same density matrix as the one mentioned above. We provide an security proof
for our protocol by exploiting the entanglement distillation argument of GHZ-
class entangled state [31].

### Virtual protocol.

($i$) Alice prepares entangled state

$\displaystyle\ket{\Phi}_{Aa}$
$\displaystyle=\sqrt{1-t_{a}}\ket{0}_{A}\ket{0}_{a}+\sqrt{t_{a}\mu_{a}e^{-\mu_{a}}}\ket{1}_{A}\ket{1}_{a},$
(4)

where $A$ is the virtual qubit system and $a$ is the optical mode. $\ket{0}$
and $\ket{1}$ are the eigenstates of Z basis while $\ket{+}$ and $\ket{-}$ are
the eigenstates of X basis for both virtual qubit system and optical mode,
where $\ket{\pm}=\frac{1}{\sqrt{2}}(\ket{0}\pm\ket{1})$. Bob has the same
entangled state. Alice and Bob send the optical modes $a$ and $b$ to Charlie
respectively.

Here, we introduce an additional virtual step: Alice and Bob jointly perform
quantum non-demolition measurement on the state of optical modes $a$ and $b$
to implement photon-number-resolving before they send the state to Charlie.
Alice and Bob only reserve the cases of joint single-photon states of modes
$a$ and $b$. The joint quantum state between Alice and Bob after this step is

$\displaystyle\ket{\psi}_{ABab}$
$\displaystyle=\frac{\sqrt{\nu_{a}}\ket{1010}_{ABab}+\sqrt{\nu_{b}}\ket{0101}_{ABab}}{\sqrt{\nu_{a}+\nu_{b}}},$
(5)

where $\nu_{a}$ and $\nu_{b}$ meet the constraint of equation (1). Note that
Charlie can use optical modes $\\{\ket{01}_{ab},\ \ket{10}_{ab}\\}$ to
represent the virtual qubit system $\\{\ket{0}_{C},\ \ket{1}_{C}\\}$.
Therefore, we can rewrite the joint quantum state

$\displaystyle\ket{\phi}_{ABC}=\frac{1}{\sqrt{\nu_{a}+\nu_{b}}}$
$\displaystyle\left(\sqrt{\nu_{a}}\ket{1}_{A}\ket{0}_{B}\ket{1}_{C}\right.$
(6)
$\displaystyle\left.+\sqrt{\nu_{b}}\ket{0}_{A}\ket{1}_{B}\ket{0}_{C}\right),$

which is the GHZ-class entangled states.

($ii$) For each round, Alice (Bob) independently and randomly chooses a basis
from $\\{\mathrm{Z},\ \mathrm{X}\\}$ corresponding to the states
$\\{\ket{0}_{A(B)},\ \ket{1}_{A(B)}\\}$ and $\\{\ket{+}_{A(B)},\
\ket{-}_{A(B)}\\}$. They then perform local projective measurements on their
kept virtual qubits. Charlie measures the qubit system $C$ in the Z or X basis
to obtain his logic bit. ($iii$) Alice, Bob, and Charlie announce the basis
information for each round of measurement. ($iv$) Alice, Bob, and Charlie
extract raw keys from the data of Z basis while the data of X basis for
estimation of information leakage. ($v$) They distill final key by exploiting
the classical postprocessing.

For the above virtual protocol, the joint single-photon state of modes $a$ and
$b$ has the same density matrix as $\rho^{1}$ when Alice and Bob both measures
the kept qubit systems in the $Z$ or $X$ basis. Therefore, for the single-
photon state, one cannot distinguish the practical protocol as shown in figure
1 and the virtual protocol. We let the probability of Alice and Bob choosing Z
basis $p_{z_{a}},\ p_{z_{b}}\approx 1$ in the asymptotic limit. The secret key
rate of our protocol is

$\displaystyle R=$
$\displaystyle\left[t_{a}(1-t_{b})e^{-\mu_{a}}+t_{b}(1-t_{a})e^{-\mu_{b}}\right]Y_{0}^{z}$
(7)
$\displaystyle+\left[t_{a}(1-t_{b})\mu_{a}e^{-\mu_{a}}+t_{b}(1-t_{a})\mu_{b}e^{-\mu_{b}}\right]$
$\displaystyle\times
Y_{1}^{z}\left[1-h\left(e_{1}^{\mathrm{ph}}\right)\right]-\xi_{\mathrm{EC}},$

where $Y_{0}^{z}$ and $Y_{1}^{z}$ are the yields when Alice and Bob both send
the vacuum state and joint single-photon states given that all users choose Z
basis respectively. $e_{1}^{\mathrm{ph}}$ is the phase error rate with all
users choosing Z basis, which can be estimated by the bit error rate of X
basis. $\xi_{\mathrm{EC}}=Q^{z}fh(E^{z})$ is the information leakage in
classical error correction, where $Q^{z}$ is the gain of they all selecting Z
basis, $f$ is the efficiency of error correction, and $E^{z}$ is the bit error
rate of Z basis. $h(x)=-x\log{(x)}-(1-x)\log{(1-x)}$ is the binary Shannon
entropy function.

The yields $Y_{0}^{d}=Q_{00}^{d}$ and $Y_{1}^{d}$ can be estimated by

$Y_{1}^{d}\geq\frac{\nu_{a}}{\nu_{a}+\nu_{b}}Y_{10}^{d}+\frac{\nu_{b}}{\nu_{a}+\nu_{b}}Y_{01}^{d},$
(8)

where $Y_{10}^{d}$ is the yield that Alice sends single-photon state while Bob
sends the vacuum state, and $Y_{01}^{d}$ is the yield that Alice sends the
vacuum state while Bob sends the single-photon state. $d$ denotes all users
choose D basis, with $\mathrm{D}\in\\{\mathrm{Z},\ \mathrm{X}\\}$. The yields
$Y_{10}^{d}$ and $Y_{01}^{d}$ can be bounded by exploiting the decoy-state
method [6, 7],

$Y_{10}^{d}\geq\frac{\mu_{a}}{\mu_{a}\nu_{a}-\nu_{a}^{2}}\left[e^{\nu_{a}}Q_{\nu_{a}0}^{d}-\frac{\nu_{a}^{2}}{\mu_{a}^{2}}e^{\mu_{a}}Q_{\mu_{a}0}^{d}-\frac{\mu_{a}^{2}-\nu_{a}^{2}}{\mu_{a}^{2}}Q_{00}^{d}\right],$
(9)
$Y_{01}^{d}\geq\frac{\mu_{b}}{\mu_{b}\nu_{b}-\nu_{b}^{2}}\left[e^{\nu_{b}}Q_{0\nu_{b}}^{d}-\frac{\nu_{b}^{2}}{\mu_{b}^{2}}e^{\mu_{b}}Q_{0\mu_{b}}^{d}-\frac{\mu_{b}^{2}-\nu_{b}^{2}}{\mu_{b}^{2}}Q_{00}^{d}\right],$
(10)

where $Q_{k_{a}k_{b}}^{d}$ is the gain of Charlie’s D basis given that Alice
and Bob send pulses with intensities $k_{a}$ and $k_{b}$ respectively. The
phase error rate can be bounded by

$e_{1}^{\mathrm{ph}}\leq\frac{1}{(\nu_{a}+\nu_{b})Y_{1}^{x}}\left(e^{\nu_{a}+\nu_{b}}E_{\nu_{a}\nu_{b}}^{\mathrm{pm}}Q_{\nu_{a}\nu_{b}}^{\mathrm{pm}}-\frac{1}{2}Y_{0}^{x}\right),$
(11)

where $E_{\nu_{a}\nu_{b}}^{\mathrm{pm}}$ and
$Q_{\nu_{a}\nu_{b}}^{\mathrm{pm}}$ are the bit error rate and gain of Charlie
selecting X basis when Alice chooses intensity $\nu_{a}$ while Bob chooses
intensity $\nu_{b}$ and they successfully perform the postselected phase-
matching.

##  IV Finite-key analysis

Here, we give the key rate formula of our asymmetric QCKA protocol considering
the effect of the finite key size [54, 55, 56]. The protocol is
$\varepsilon_{\rm sec}$-secret if the secret key of length $l$ satisfies [58,
31, 35]

$l=\underline{s}_{0}^{z}+\underline{s}_{1}^{z}[1-h(\overline{e}_{1}^{\mathrm{ph}})]-\lambda_{\rm
EC}-\log_{2}\frac{4}{\varepsilon_{\rm
cor}}-6\log_{2}\frac{26}{\varepsilon_{\rm sec}},$ (12)

where $s_{0}^{z}$ and $s_{1}^{z}$ are the numbers of successful events that
Alice and Bob both send the vacuum state and joint single-photon states when
all users choose Z basis respectively. Note that we define $x$ as the observed
value, $x^{*}$ as the expected value, $\underline{x}$ and $\overline{x}$ as
the lower and upper bound of $x$. Here, the protocol is $\varepsilon_{\rm
cor}$-correct [35], where $\varepsilon_{\rm cor}$ is the probability that the
error correction fails. $\lambda_{\rm EC}=n^{z}fh\left(E^{z}\right)$ is the
error correction leakage, with the number of event $n^{z}$ that all users
choose Z basis.

We can use the announced event to estimate our unknown parameters in equation
(12). When Alice and Bob sent pulses with intensities $k_{a}$ and $k_{b}$, we
let $N_{k_{a}k_{b}}$ be the number of the announced event, e.g.
$\begin{aligned}
N_{00}=&\left[\left(1-p_{z_{a}}\right)\left(1-p_{z_{b}}\right)p_{0_{a}}p_{0_{b}}+\left(1-p_{z_{a}}\right)p_{z_{b}}p_{0_{a}}(1-t_{b})\right.\\\
&\left.+p_{z_{a}}\left(1-p_{z_{b}}\right)(1-t_{a})p_{0_{b}}\right]N\end{aligned}$,
where $p_{0_{a}}$ and $p_{0_{b}}$ are the probabilities of choosing the vacuum
state given that Alice and Bob select X basis. $n_{k_{a}k_{b}}^{d}$ is the
observed number of effective event measured by basis
$\mathrm{D}\in\\{\mathrm{X},\ \mathrm{Z}\\}$. By exploiting the decoy-state
method for finite sample sizes, we can derive the lower bound on the expected
numbers of vacuum events

$\displaystyle s_{0}^{z*}\geq\underline{s}_{0}^{z*}=$ $\displaystyle
p_{z_{a}}p_{z_{b}}\left[t_{a}(1-t_{b})e^{-\mu_{a}}+t_{b}(1-t_{a})e^{-\mu_{b}}\right]$
(13) $\displaystyle\times N\frac{\underline{n}_{00}^{z*}}{N_{00}},$

and joint single-photon events

$s_{1}^{z*}\geq\underline{s}_{1}^{z*}=\underline{s}_{10}^{z*}+\underline{s}_{01}^{z*},$
(14)

where $N$ is the total number of the all events. Here, the lower bound on
expected numbers of two kinds of joint single-photon events,
$\underline{s}_{10}^{z*}$ and $\underline{s}_{01}^{z*}$, can be given by

$\displaystyle s_{10}^{z*}\geq\underline{s}_{10}^{z*}=$
$\displaystyle\frac{t_{a}(1-t_{b})\mu_{a}^{2}e^{-\mu_{a}}p_{z_{a}}p_{z_{b}}N}{\mu_{a}\nu_{a}-\nu_{a}^{2}}$
(15)
$\displaystyle\times\left[e^{\nu_{a}}\frac{\underline{n}_{\nu_{a}0}^{z*}}{N_{\nu_{a}0}}-\frac{\nu_{a}^{2}}{\mu_{a}^{2}}e^{\mu_{a}}\frac{\overline{n}_{\mu_{a}0}^{z*}}{N_{\mu_{a}0}}-\frac{\mu_{a}^{2}-\nu_{a}^{2}}{\mu_{a}^{2}}\frac{\overline{n}_{00}^{z*}}{N_{00}}\right],$
$\displaystyle s_{01}^{z*}\geq\underline{s}_{01}^{z*}=$
$\displaystyle\frac{t_{b}(1-t_{a})\mu_{b}^{2}e^{-\mu_{b}}p_{z_{a}}p_{z_{b}}N}{\mu_{b}\nu_{b}-\nu_{b}^{2}}$
(16)
$\displaystyle\times\left[e^{\nu_{b}}\frac{\underline{n}_{0\nu_{b}}^{z*}}{N_{0\nu_{b}}}-\frac{\nu_{b}^{2}}{\mu_{b}^{2}}e^{\mu_{b}}\frac{\overline{n}_{0\mu_{b}}^{z*}}{N_{0\mu_{b}}}-\frac{\mu_{b}^{2}-\nu_{b}^{2}}{\mu_{b}^{2}}\frac{\overline{n}_{00}^{z*}}{N_{00}}\right].$

The successful probability of post-selected phase-matching is
$p_{\mathrm{pm}}=\frac{2\delta}{\pi}$. We can also work out the lower bound on
the expected number of joint single-photon events after successful post-
selected phase-matching $\underline{s}_{1}^{\mathrm{pm}*}$,

$s_{1}^{\mathrm{pm}*}\geq\underline{s}_{1}^{\mathrm{pm}*}=\underline{s}_{10}^{\mathrm{pm}*}+\underline{s}_{01}^{\mathrm{pm}*}.$
(17)

Similarly, the lower bound on expected numbers of two kinds of joint single-
photon events, $\underline{s}_{10}^{\mathrm{pm}*}$ and
$\underline{s}_{01}^{\mathrm{pm}*}$, can be given by

$\displaystyle s_{10}^{\mathrm{pm}*}\geq\underline{s}_{10}^{\mathrm{pm}*}=$
$\displaystyle\frac{(1-p_{z_{a}})(1-p_{z_{b}})p_{\nu_{a}}p_{\nu_{b}}p_{\mathrm{pm}}\mu_{a}\nu_{a}e^{-(\nu_{a}+\nu_{b})}N}{\mu_{a}\nu_{a}-\nu_{a}^{2}}$
(18)
$\displaystyle\times\left[e^{\nu_{a}}\frac{\underline{n}_{\nu_{a}0}^{x*}}{N_{\nu_{a}0}}-\frac{\nu_{a}^{2}}{\mu_{a}^{2}}e^{\mu_{a}}\frac{\overline{n}_{\mu_{a}0}^{x*}}{N_{\mu_{a}0}}-\frac{\mu_{a}^{2}-\nu_{a}^{2}}{\mu_{a}^{2}}\frac{\overline{n}_{00}^{x*}}{N_{00}}\right],$
$\displaystyle s_{01}^{\mathrm{pm}*}\geq\underline{s}_{01}^{\mathrm{pm}*}=$
$\displaystyle\frac{(1-p_{z_{a}})(1-p_{z_{b}})p_{\nu_{a}}p_{\nu_{b}}p_{\mathrm{pm}}\mu_{b}\nu_{b}e^{-(\nu_{a}+\nu_{b})}N}{\mu_{b}\nu_{b}-\nu_{b}^{2}}$
(19)
$\displaystyle\times\left[e^{\nu_{b}}\frac{\underline{n}_{0\nu_{b}}^{x*}}{N_{0\nu_{b}}}-\frac{\nu_{b}^{2}}{\mu_{b}^{2}}e^{\mu_{b}}\frac{\overline{n}_{0\mu_{b}}^{x*}}{N_{0\mu_{b}}}-\frac{\mu_{b}^{2}-\nu_{b}^{2}}{\mu_{b}^{2}}\frac{\overline{n}_{00}^{x*}}{N_{00}}\right],$

Where we denote the probability of Alice and Bob choosing intensities
$\nu_{a}$ and $\nu_{b}$ given that they all select X basis as $p_{\nu_{a}}$
and $p_{\nu_{b}}$. The upper and lower bound of all the above expected values
can be obtained by using the variant of Chernoff bound with the announced
observed values [59] for each parameter with failure probability
$\varepsilon_{\rm sec}/26$. Once acquiring the lower bound of expected values
$\underline{s}_{0}^{z*}$, $\underline{s}_{1}^{z*}$ and
$\underline{s}_{1}^{\mathrm{pm}*}$, one can exploit the Chernoff bound [59] to
calculate the lower bound of corresponding observed values
$\underline{s}_{0}^{z}$, $\underline{s}_{1}^{z}$ and
$\underline{s}_{1}^{\mathrm{pm}}$ for each parameter with failure probability
$\varepsilon_{\rm sec}/26$.

By utilizing the decoy-state method, we can also estimate the upper bound of
the amount of bit error $\overline{t}_{1}^{\mathrm{pm}}$ associated with the
joint single-photon events after successful post-selected phase-matching [60],

$t_{1}^{\mathrm{pm}}\leq\overline{t}_{1}^{\mathrm{pm}}=m_{\nu_{a}\nu_{b}}^{\mathrm{pm}}-\underline{t}_{0}^{\mathrm{pm}},$
(20)

with

$\underline{t}_{0}^{\mathrm{pm}*}=(1-p_{z_{a}})(1-p_{z_{b}})p_{\nu_{a}}p_{\nu_{b}}p_{\mathrm{pm}}e^{-(\nu_{a}+\nu_{b})}N\frac{\underline{n}_{00}^{x*}}{2N_{00}},$
(21)

where $m_{\nu_{a}\nu_{b}}^{\mathrm{pm}}$ is the observed number of bit error
after the post-selected phase-matching. Note that expected value of bit errors
on vacuum events satisfies $m_{00}^{x*}=n_{00}^{x*}/2$. The lower bound on the
observed number of bit error in the vacuum state
$\underline{t}_{0}^{\mathrm{pm}}$ can also be obtained by exploiting the
Chernoff bound. The hypothetically observed phase error rate associated with
the joint single-photon events in Z basis can be obtained by using the random
sampling without replacement [59],

$\overline{e}_{1}^{\mathrm{ph}}=\frac{\overline{t}_{1}^{\mathrm{pm}}}{\underline{s}_{1}^{\mathrm{pm}}}+\gamma\left(\underline{s}_{1}^{z},\underline{s}_{1}^{\mathrm{pm}},\frac{\overline{t}_{1}^{\mathrm{pm}}}{\underline{s}_{1}^{\mathrm{pm}}},\varepsilon_{\rm
sec}/26\right),$ (22)

where $\gamma$ is the parameter of the random sampling without replacement
with failure probability $\varepsilon_{\rm sec}/26$. The variant of Chernoff
bound is used twelve times, the Chernoff bound is used four times and the
random sampling without replacement is used one time. Setting each error term
to a common value, we get the factor is 26, including 9 error terms due to the
smooth min-entropy estimation [55].

##  V Simulation details

Figure 2: Comparison of the secure conference key rate between our protocol
and the original protocol in infinite-key asymptotic limit. $L=L_{a}+L_{b}$ is
the total transmission distance. Here, the difference in length between two
channels is fixed at 50 km, i.e. $L_{b}-L_{a}=50\mathrm{km}$. The typical
experimental parameters are listed in table 1.

Figure 3: Comparison of the secure conference key rate between our protocol
and the original protocol considered finite-key analysis. The abscissa axis
$L=L_{a}+L_{b}$ means transmission distance between three parties. (a) Here,
the difference in length between two channels is fixed at 50 km. The typical
experimental parameters are listed in table 1. Numerically optimized secret
key rates with logarithmic scale are obtained for a predetermined signals
$N=10^{14}$ with $\varepsilon_{\rm sec}=\varepsilon_{\rm cor}=10^{-10}$. (b)
Here, the difference in length between two channels is fixed at 100 km with
the same experimental parameters as (a).

Here, we present the simulation results of different QCKA protocols over
asymmetric channels in figure 2 and 3. The experimental parameters used in the
simulation are listed in table 1. In the asymptotic case, the superiority of
our asymmetric protocol is demonstrated in figure 2. In figure 3, the finite-
key effect has been considered.

In the original protocol, the intensities of the pulses interfering at BS3
will differ a lot under different transmission distances, which will result in
a relatively high bit error rate in X basis. Therefore, the phase error rate
of joint single-photon in Z basis, $\overline{e}_{1}^{\mathrm{ph}}$, will
increase a lot.

In figure 3, we show the conference key rates of two protocols with the length
difference between the two channels $L_{b}-L_{a}$ fixed at a constant length.
Here, we assume that the efficiency and dark count rate of Charlie’s detectors
are the same and define the channel transmittance $\eta$ as $\eta_{d}\times
10^{-\alpha L/10}$. We numerically optimize the conference key rate over the
free parameters $p_{z_{a}}$, $p_{z_{b}}$, $\nu_{a}$, $\nu_{b}$, $\mu_{a}$,
$\mu_{b}$, $t_{a}$, $p_{0_{a}}$, $p_{0_{b}}$, $p_{v_{a}}$, $p_{v_{b}}$,
$q_{z}$ and $\delta$, where $q_{z}$ is the beam splitting ratio of BS1 and BS2
for one photon to Z basis. And $t_{b}$ will be restricted by equation (1).
Numerically optimized secret key rates are obtained for a predetermined number
of signals $N=10^{14}$ with $\varepsilon_{\rm sec}=\varepsilon_{\rm
cor}=10^{-10}$. It is easy to find that in the asymmetric channels, our
asymmetric protocol outperforms the original one. Moreover, comparing figure 3
with figure 3, we can notice that the advantage of our asymmetric protocol
enhances performance significantly as the difference in distance between two
channels increases.

Table 1: Simulation parameters. $\eta_{d}$ and $p_{d}$ are the detector efficiency and dark count rate. $e_{d}^{x}$ is the misalignment rate of X basis. $\alpha$ is the attenuation coefficient of the ultralow-loss fiber. $f$ is the error correction efficiency.  $\eta_{d}$ | $p_{d}$ | $e_{d}^{x}$ | $\alpha$ | $f$
---|---|---|---|---
$56\%$ | $10^{-8}$ | $3.5\%$ |  $0.167$dB/km | $1.1$

##  VI Conclusion

In summary, we propose a QCKA protocol with different source parameters to
increase the conference key rate in asymmetric channels. We provide a
mathematical constraint in equation (1) for the intensity and probability of
sending on both sides of Alice and Bob to ensure security in the asymmetric
case. Furthermore, we perform finite key analysis for practical scenarios.
Note that our protocol is not a measurement-device-independent scheme because
Charlie will randomly select the measurement basis. And generalizing our
protocol to more than three parties is also a worthwhile work. We hope that
this issue will be addressed felicitously in the future.

Simulation results of two protocols show that our asymmetric protocol has a
significantly higher key rate than the original protocol under the condition
of asymmetric channels. When the difference in distance between two channels
is 100 km, the key rate of the asymmetric protocol is at least one order of
magnitude higher than that of the original protocol. The transmission distance
is approximately 200 km longer. Meanwhile, our protocol can be directly
implemented with current available twin-field QKD devices. This work paves the
way for large scale deployment of quantum cryptographic network.

## Acknowledgments

We gratefully acknowledge support from the National Natural Science Foundation
of China (under Grant No. 61801420); the Key-Area Research and Development
Program of Guangdong Province (under Grant No. 2020B0303040001); the
Fundamental Research Funds for the Central Universities (under Grant No.
020414380182).

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# First concurrent extraction of the leading-order scalar and spin proton
polarizabilities

E. Mornacchi<EMAIL_ADDRESS>Institut für Kernphysik, Johannes
Gutenberg-Universität Mainz, D-55099 Mainz,Germany  S. Rodini  Institut für
Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany  B.
Pasquini  Dipartimento di Fisica, Università degli Studi di Pavia, I-27100
Pavia, Italy  INFN Sezione di Pavia, I-27100 Pavia, Italy  P. Pedroni  INFN
Sezione di Pavia, I-27100 Pavia, Italy

###### Abstract

We performed the first simultaneous extraction of the six leading-order proton
polarizabilities. We reached this milestone thanks to both new high-quality
experimental data and an innovative bootstrap-base fitting method. These new
results provide a self-consistent and fundamental benchmark for all future
theoretical and experimental polarizability estimates.

##  I  Introduction

Understanding the hadron structure in the non-perturbative regime of quantum
chromodynamics (QCD) is one of the major challenge of modern physics. We can
classify hadrons in terms of their global properties, such as mass and spin,
but we can not fully explain how these properties emerge from the underlying
dynamics of the hadron’s interior. A clean probe to investigate the internal
structure of hadrons is the Compton scattering process that gives access to
observables with a clear interpretation in terms of structure-dependent
properties of the hadrons. In particular, real Compton scattering (RCS) at low
energies is parametrized by polarizabilities that describe the response of the
charge and magnetization distributions inside the nucleon to an applied quasi-
static electromagnetic field. These structure constants are fundamental
properties of the nucleon and their determination has driven a relevant
experimental effort in the last few years Martel  _et al._ (2015); Sokhoyan
_et al._ (2017); Paudyal  _et al._ (2020); Mornacchi  _et al._ (2022); Li  _et
al._ (2022).

The effective multipole interactions for the coupling of the electric
($\vec{E}$) and magnetic ($\vec{H}$) fields of the photon with the internal
structure of the nucleon is described at leading order in terms of the
electric ($\alpha_{E1}$) and magnetic ($\beta_{M1}$) scalar polarizabilities
Babusci  _et al._ (1998); Holstein  _et al._ (2000):

$H^{(2)}_{eff}=-4\pi\Bigl{[}\frac{1}{2}\alpha_{E1}\vec{E}^{2}+\frac{1}{2}\beta_{M1}\vec{H}^{2}\Bigr{]},$
(1)

while the four spin polarizabilities ($\gamma_{EiMj}$) show up in the
subleading terms:

$\displaystyle H^{(3)}_{eff}$
$\displaystyle=-4\pi\Bigl{[}\frac{1}{2}\gamma_{E1E1}\vec{\sigma}\cdot(\vec{E}\times\dot{\vec{E}})+\frac{1}{2}\gamma_{M1M1}\vec{\sigma}\cdot(\vec{H}\times\dot{\vec{H}})$
$\displaystyle\quad-\gamma_{M1E2}E_{ij}\sigma_{i}H_{j}+\gamma_{E1M2}H_{ij}\sigma_{i}E_{j}\Bigr{]},$
(2)

where $\vec{\sigma}$ are the proton’s Pauli spin matrices,
$\dot{\vec{E}}=\partial_{t}\vec{E}$ and
$E_{ij}=\frac{1}{2}(\nabla_{i}E_{j}+\nabla_{j}E_{i}$) are partial derivatives
with respect to time and space, respectively.

In addition to being fundamental properties of the nucleon, polarizabilities
play a profound role in precision atomic physics — in the evaluation of the
nuclear corrections to atomic energy levels Drell and Sullivan (1967);
Bernabeu and Ericson (1983); Faustov and Martynenko (2000); Martynenko (2006);
Carlson and Vanderhaeghen (2011) — and in astrophysics — influencing neutron
star properties Bernabeu  _et al._ (1974).

Despite their evident importance in a broad range of physics topics, up to now
a self-consistent experimental extraction of all the different polarizability
values has not been possible, due to the scarce quality of the available
database (see, for instance, Ref. Pasquini  _et al._ (2019)). In all existing
fits of the RCS data, some of the polarizabilities have been fixed either
using theoretical calculations Griesshammer  _et al._ (2012); McGovern  _et
al._ (2013); Lensky and McGovern (2014); Griesshammer  _et al._ (2016) or
empirical evaluations from other reactions Krupina  _et al._ (2018), or, at
most, have been constrained to vary within certain intervals Pasquini  _et
al._ (2019). The situation has recently improved with the first measurements
of the double-polarization observables $\Sigma_{2x}$ Martel  _et al._ (2015)
and $\Sigma_{2z}$ Paudyal  _et al._ (2020) and new data for the unpolarized
differential cross sections and the single-polarized $\Sigma_{3}$ asymmetry
Sokhoyan  _et al._ (2017); Mornacchi  _et al._ (2022); Li  _et al._ (2022)
(see Ref. Babusci  _et al._ (1998) for the definition of the different
observables). In particular, the work from Mornacchi et al. Mornacchi  _et
al._ (2022) provides the highest statistics proton RCS single dataset ever
obtained below the pion photoproduction threshold, with 60 unpolarized
differential cross section points and 36 beam asymmetry points measured over a
large angular range and with small statistical and systematic errors.
Therefore, it represents a significant improvement for a more accurate
extractions of all the different polarizability values.

In this paper we present the first consistent and simultaneous fit of the six
leading-order static proton polarizabilities. It has been obtained thanks to
both the new recent experimental data and an innovative bootstrap-base fitting
method Pedroni and Sconfietti (2020). This algorithm has already been deployed
successfully for the extraction of the proton scalar dynamical and static
polarizabilities from low-energy RCS data Pasquini  _et al._ (2018, 2019). The
theoretical framework used for this extraction is based on fixed-$t$
subtracted dispersion relations (DRs) Drechsel  _et al._ (1999); Holstein  _et
al._ (2000); Pasquini  _et al._ (2007). The theoretical uncertainties
associated to the model dependence of our results are also evaluated by using,
as input, pion photoproduction amplitudes obtained from three different
partial wave analyses (PWAs) of the available experimental data: BnGa-2019
Anisovich  _et al._ (2016), MAID-2021 Drechsel  _et al._ (2007); Kashevarov
and Tiator (2021), and SAID-MA19 Briscoe  _et al._ (2019). This is the very
first time that such a comprehensive and self-consistent study on the
simultaneous extraction of all the six leading-order proton polarizabilities
from RCS data is performed.

##  II  Database selection and fit procedure

The proton RCS database used for this work consists of two main sets: the
unpolarized differential cross section data, and the (single and double)
polarization asymmetries. The former can be further divided into low- and
high-energy data, namely data for which the incoming photon energy
$\omega_{\gamma}$ is below or above the pion photoproduction threshold ($\sim
150$ MeV), respectively.

For the low-energy set, in addition to the new data from Refs. Mornacchi  _et
al._ (2022); Li  _et al._ (2022), we used the same selection extensively
discussed in a previous work by Pasquini et al. Pasquini  _et al._ (2019),
which includes datasets from Refs. Oxley (1958); Hyman  _et al._ (1959);
Goldansky  _et al._ (1960); Bernardini  _et al._ (1960); Pugh  _et al._
(1957); Baranov  _et al._ (1974, 1975); Federspiel  _et al._ (1991); Zieger
_et al._ (1992); Hallin  _et al._ (1993); MacGibbon  _et al._ (1995); de León
_et al._ (2001). For the high-energy set, thanks to the DR model used for the
theoretical framework, we were able to consider data measured up to
$\omega_{\gamma}=300$ MeV (corresponding to the $2\pi$ photoproduction
threshold), thus extending the range used in the fits of Refs. McGovern  _et
al._ (2013); Pasquini  _et al._ (2019). Furthermore, we decided to narrow the
selection to the new-generation experiments, namely the measurements performed
using tagged photon facilities (see, for instance, the review of Ref.
Schumacher (2005)). Their main advantage, in addition to a more reliable
photon flux determination, is that the incoming photon energy is known with a
resolution of a few MeV. This is an essential ingredient to reject the
overwhelming background coming from the single $\pi^{0}$ photoproduction
channel that has a cross section by two orders of magnitude higher than
Compton scattering process, and can mimic the Compton signature when one of
the two photons coming from the $\pi^{0}$ decay escapes the particle
detection. The available datasets for the unpolarized cross section come from
two different facilities: MAMI Peise and Schumacher (1996); Molinari  _et al._
(1996); Wissmann  _et al._ (1999); Wolf  _et al._ (2001); Galler  _et al._
(2001); Camen  _et al._ (2002) (with also few data above threshold from Ref.
de León  _et al._ (2001)) and LEGS Tonnison  _et al._ (1998); Blanpied  _et
al._ (2001). However, the latter is in significant disagreement with the MAMI
dataset, and these differences led to an extraction of $\gamma_{\pi}$ from the
LEGS data in strong disagreement from all the other fits based on a different
dataset Schumacher (2005); Griesshammer  _et al._ (2012). Very similar
discrepancies were also noticed in the unpolarised differential $\pi^{0}$
photoproduction cross section Beck  _et al._ (1997, 2000). For all these
reasons, we conformed to the selection of Refs. Schumacher (2005);
Griesshammer  _et al._ (2012) and excluded the LEGS dataset from the database
of this work.

The polarization observables have enhanced sensitivity to the spin
polarizabilities Pasquini  _et al._ (2007); Griesshammer  _et al._ (2018),
hence are crucial for the extraction of these structure constants. The adopted
datasets include three different polarization observables: $\Sigma_{2x}$
Martel  _et al._ (2015), $\Sigma_{2z}$ Paudyal  _et al._ (2020), and
$\Sigma_{3}$ both below Sokhoyan  _et al._ (2017); Mornacchi  _et al._ (2022);
Li  _et al._ (2022) and above Blanpied  _et al._ (2001) pion photoproduction
threshold. Although the data for the photon asymmetry $\Sigma_{3}$ above
threshold are from the same dataset of the LEGS unpolarized cross section,
they are consistent with the available MAMI results for both the $p\pi^{0}$
and the $n\pi^{+}$ photoproduction channels Beck  _et al._ (2000). Such an
agreement is not surprising since most of the possible systematic biases
cancel out in an asymmetry observable.

The fit to extract the leading-order scalar and spin polarizabilities was
performed using a bootstrap-based method Pedroni and Sconfietti (2020) that
consists of randomly generating $N$ Monte Carlo replicas of the fitted
experimental database, where each data point $e_{i,j}^{(0)}$ is replaced by:

$e_{i,j}^{(0)}\rightarrow
e_{i,j}^{(b)}=(1+\delta_{j,b})(e_{i,j}^{(0)}+r_{i,j,b}\sigma^{(0)}_{i,j}).$
(3)

The indices $i$, $j$, and $b$ run over the number of data points in each
dataset, the number of datasets, and the bootstrap replica, respectively;
$r_{i,j,b}$ is a random number extracted from the normal distribution
$\mathcal{N}(0,1)$, and $\delta_{j,b}$ is a random variable that accounts for
the effect of the common systematic errors, independently for each dataset.
From each of these simulated databases, a set of fitted parameters is
extracted. The mean and standard deviation of the obtained distributions give
then the best value and the error for each of the fitted parameters. This
technique offers several advantages compared to other fitting procedures,
especially when different datasets are used together, as in the present work:
i) a straightforward inclusion of common systematic uncertainties without any
a priori assumption on their distributions and without introducing any
additional fit parameter; ii) the probability distribution of the fit
parameters (often non-gaussian) is obtained directly by the procedure itself;
iii) the uncertainties on possible nuisance model parameters are easily and
directly taken into account in the sampling procedure; iv) the correct fit
$p$-value is always provided when systematic uncertainties are present and in
all the other cases when the goodness-of-fit distribution is not given by the
$\chi^{2}$-distribution.

In the first step of our analysis, we checked the consistency of the selected
database by looking at the distribution of the normalized residual for each of
the largest datasets (e.g., with more than 40 data points): the unpolarized
cross section from the A2 Mornacchi  _et al._ (2022), the TAPS de León  _et
al._ (2001), and the LARA Wolf  _et al._ (2001) Collaborations; and the beam
asymmetry from the LEGS Blanpied  _et al._ (2001) Collaboration. For this
purpose, we performed a preliminary test including the entire database where
all six polarizabilities were simultaneously fitted using the MAID-2021
Drechsel  _et al._ (2007); Kashevarov and Tiator (2021) multipole solution.
Taking the output polarizability best-values, we calculated the residual
distribution for each of those datasets, and produced a probability plot
Chambers  _et al._ (1983) for assessing the residual normal distribution. The
residuals were found to follow fairly well the expected normal distribution
for all the selected datasets, but the unpolarized cross section from the LARA
Collaboration, as shown in Fig. 1. We repeated the same test using the SAID-
MA19 Briscoe  _et al._ (2019) and BnGa-2019 Anisovich  _et al._ (2016) PWAs
and obtained very similar results. Given also the small fit $p$-value ($\simeq
1\cdot 10^{-2}$) obtained in all cases, we excluded this set from the database
of the present work. The probability plots for the other datasets can be found
in the Supplemental Material sup .

Taking all these considerations together, 25 datasets for a total of 388
points were included in the fit. The angular and energy coverage of all the
used datasets are listed in the Supplemental Material sup .

Figure 1: Probability plot using the normalized residuals for the unpolarized
cross section from the LARA Collaboration Wolf  _et al._ (2001), obtained from
a global fit using the MAID-2021 multipole solution Drechsel  _et al._ (2007);
Kashevarov and Tiator (2021). The points are expected to lie on the bisector
(in black) if their residuals are normally distributed. The red line shows a
linear regression fit to the points to help the comparison with the expected
distribution.

##  III  Results and Discussion

(a) $\alpha_{E1}$ (b) $\beta_{M1}$ (c) $\gamma_{E1E1}$ (d) $\gamma_{M1M1}$ (e)
$\gamma_{E1M2}$ (f) $\gamma_{M1E2}$

Figure 2: Values of the leading-order static proton polarizabilities obtained
from the fit procedure. In each panel, our new results are reported in blue.
The blue error bars represent the fit error only, the increase in the total
error due to the inclusion of the systematic contribution from the model
dependency is shown in red. Some of the existing extractions and theoretical
predictions are also shown below and above the dotted line, respectively. For
the two scalar polarizabilities, DR Pasquini  _et al._ (2019) and HBChPT
McGovern  _et al._ (2013) refer to two extractions within DR Drechsel  _et
al._ (1999); Holstein  _et al._ (2000); Pasquini  _et al._ (2007) and HBChPT
McGovern  _et al._ (2013) frameworks. In both cases, most of the spin
polarizabilities were fixed to a given value. BChPT refers to a full EFT
calculations to $\mathcal{O}(p^{4})$ Lensky and Pascalutsa (2010). For the
four spin polarizabilities, A2 refers to the most recent experimental
extraction Paudyal  _et al._ (2020), where both scalar polarizabilities were
fixed to the PDG values Zyla  _et al._ (2020). DR Holstein  _et al._ (2000),
HBChPT McGovern  _et al._ (2013), and BChPT Lensky and McGovern (2014) refer
to theoretical predictions using different approaches.

A total of $N=10^{4}$ bootstrapped samples of the database was generated and
the minimization was performed at the end of each iteration with all six
proton polarizabilities treated as free parameters. For convenience, we used
as actual fit parameters some linear combinations of the scalar and spin
polarizabilities: $\alpha_{E1}\pm\beta_{M1}$, $\gamma_{E1E1}$,
$\gamma_{M1M1}$,
$\gamma_{0}=-\gamma_{E1E1}-\gamma_{M1M1}-\gamma_{E1M2}-\gamma_{M1E2}$, and
$\gamma_{\pi}=-\gamma_{E1E1}+\gamma_{M1M1}-\gamma_{E1M2}+\gamma_{M1E2}$. The
last term, $\gamma_{\pi}$, is the sum of the dispersive contribution
$\gamma_{\pi}^{{\rm disp}}$, to be fitted to the data, and the pion-pole
contribution, fixed to $\gamma_{\pi}^{{\rm\pi^{0}-pole}}=-46.7$ Schumacher
(2005). The choice of the fit parameters allows for a direct comparison, as a
consistency check, of the fit results for $\alpha_{E1}+\beta_{M1}$ and
$\gamma_{0}$ to the available experimental predictions of the Baldin de León
_et al._ (2001); Gryniuk  _et al._ (2015); F.  _et al._ (2016); Strakovsky
_et al._ (2022) and Gellmann-Goldberger-Thirring (GGT) Pasquini  _et al._
(2010); Gryniuk  _et al._ (2015); F.  _et al._ (2016); Strakovsky  _et al._
(2022) sum rule values, respectively, obtained using pion photoproduction
data. When present, point-to-point systematic errors were added in quadrature
to the statistical errors, while common systematic scale factors were treated
as in the previous bootstrap extractions by Pasquini et al. Pasquini  _et al._
(2018, 2019), namely they are assumed to follow a uniform distribution, unless
otherwise specified in the original publication. Moreover, when multiple
systematic sources are given, the final error is the product of all the
generated random uniform variables.

We performed the minimizations by using the nonlinear least-squares fitting
routines of the GSL library Galassi  _et al._ (2009). As a consistency check
of this procedure, both the gradient and the simplex methods were used as
best-fit algorithms and identical results were obtained. The entire procedure
was performed for the three PWAs. The obtained distributions for each
parameters are reported in the Supplemental Material sup . The parameter
distributions obtained using the three different PWA inputs have the same
shape and differ only for a shift in their central values. For this reason, we
evaluated the central polarizability values as the mathematical average of the
three different sets of fit values. Additionally, the largest of the
differences between each set of fit values and the average was used to
estimate an additional model error (conservatively considered as a standard
deviation) due to dependence on the PWA used as input in the DRs. The
resulting best-fit values are:

$\displaystyle\alpha_{E1}+\beta_{M1}$ $\displaystyle{}=15.1\pm 0.7\
\hbox{(fit)}\ \pm 0.1\ \hbox{(model)},$ $\displaystyle\alpha_{E1}-\beta_{M1}$
$\displaystyle{}=10.3\pm 1.2\ \hbox{(fit)}\ \pm 0.2\ \hbox{(model)},$
$\displaystyle\gamma_{E1E1}$ $\displaystyle{}=-3.0\pm 0.6\ \hbox{(fit)}\ \pm
0.4\ \hbox{(model)},$ (4) $\displaystyle\gamma_{M1M1}$ $\displaystyle{}=3.7\pm
0.5\ \hbox{(fit)}\ \pm 0.1\ \hbox{(model)},$ $\displaystyle\gamma_{0}$
$\displaystyle{}=-1.6_{-1.4}^{+1.3}\ \hbox{(fit)}\ \pm 0.9\ \hbox{(model)},$
$\displaystyle\gamma_{\pi}^{\rm disp}$
$\displaystyle{}=9.9_{-2.0}^{+1.9}\hbox{(fit)}\ \pm 0.5\ \hbox{(model)}.$

From here on, the values of scalar and spin polarizabilities are reported in
units of $\rm 10^{-4}fm^{3}$ and $\rm 10^{-4}fm^{4}$, respectively. The quoted
fit errors are the 68% confidence level (CL) and include the contribution of
both the statistical and systematic uncertainties of the experimental data.
The additional model-dependent systematic uncertainties, evaluated as
explained above, are given in rms units, and have to be added in quadrature to
the previous ones to get the overall values of the estimated systematic
uncertainties.

All different fits gave, within rounding errors, a minimum value of the fit
function equal to $\rm\hat{\chi}^{2}_{red}=1.13$. As mentioned before, the
expected goodness-of-fit distribution is not given by the $\chi^{2}$ function,
because of the correlations between points of a same dataset introduced by the
systematic uncertainties. This density was then evaluated in the framework of
the bootstrap technique Pedroni and Sconfietti (2020) and a $p$-value = 24%
was estimated from its cumulative distribution. A plot of this function and
the fit correlation matrix are reported in Fig. 3 and Table 3 of the
Supplemental Material sup , respectively.

The obtained value of both $\alpha_{E1}+\beta_{M1}$ and $\gamma_{0}$ are in
agreement, within the quoted errors, with the available estimates of the
Baldin and GGT sum rule values listed in Refs. F.  _et al._ (2016); Strakovsky
_et al._ (2022). This gives a confirmation of the validity of our overall
procedure.

From the fit results reported in Eq. 4, we obtained the $68\%$ CL intervals of
the six proton static polarizabilities as:

$\displaystyle\alpha_{E1}$ $\displaystyle{}=12.7\pm 0.8\ \hbox{(fit)}\ \pm
0.1\ \hbox{(model)},$ $\displaystyle\beta_{M1}$ $\displaystyle{}=2.4\pm 0.6\
\hbox{(fit)}\ \pm 0.1\ \hbox{(model)},$ $\displaystyle\gamma_{E1E1}$
$\displaystyle{}=-3.0\pm 0.6\ \hbox{(fit)}\ \pm 0.4\ \hbox{(model)},$ (5)
$\displaystyle\gamma_{M1M1}$ $\displaystyle{}=3.7\pm 0.5\ \hbox{(fit)}\ \pm
0.1\ \hbox{(model)},$ $\displaystyle\gamma_{E1M2}$ $\displaystyle{}=-1.2\pm
1.0\ \hbox{(fit)}\ \pm 0.3\ \hbox{(model)},$ $\displaystyle\gamma_{M1E2}$
$\displaystyle{}=2.0\pm 0.7\hbox{(fit)}\ \pm 0.4\ \hbox{(model)}\ ,$

where the meaning of the fit and model errors is the same as in Eq. 4.

The proton polarizability values reported in Eq. 5 are shown as blue points in
Fig. 2 (bottom row in each panel). The blue horizontal bars represent the
errors given by the fit procedure. The increase of the overall systematic
uncertainties due to the inclusion of the model errors is shown in red. Our
new results are compared with some of the available experimental extractions
and theoretical predictions. The black dotted line separates experimental
extractions (on the bottom) from the theoretical predictions (on the top). The
vertical gray bands in Fig. 2a and Fig. 2b gives the average $68\%$ CL
interval on $\alpha_{E1}$ and $\beta_{M1}$ as evaluated by the Particle Data
Group (PDG) Zyla  _et al._ (2020). In the same figures, the new results are
compared with two global extractions of $\alpha_{E1}$ and $\beta_{M1}$ using
DRs Pasquini  _et al._ (2019) and HBChPT McGovern  _et al._ (2013),
respectively, where at least three spin polarizabilities were kept fixed. The
points above the dotted line are from a full EFT calculation up to
$\mathcal{O}(p^{4})$ Lensky and Pascalutsa (2010). In Fig. 2c, Fig. 2d, Fig.
2f, and Fig. 2e our new results are compared with the last experimental
extraction Paudyal  _et al._ (2020) — where the two scalar polarizabilities
were fixed —, and three theoretical calculations within DRs Holstein  _et al._
(2000), HBChPT McGovern  _et al._ (2013), and ChPT Lensky and McGovern (2014).
The high quality, as well as the importance, of these new results is
highlighted in all the plots. In fact, they provide a self-consistent
extraction of the six leading-order static proton polarizabilities without any
fitting approximation, and with errors that are competitive with those of all
the existing evaluations, that were all performed by constraining some of the
polarizabilities to reduce the uncertainty on the ones of interest. In
particular, the new results from the A2 Collaboration on the unpolarized cross
section Mornacchi  _et al._ (2022) were fundamental in reducing the
correlations between $\alpha_{E1}-\beta_{M1}$ and $\gamma_{\pi}$, and
$\alpha_{E1}-\beta_{M1}$ and $\gamma_{M1M1}$.

These new results then are a significant benchmark for all future theoretical
and experimental polarizability estimates. However, the fit parameters have
still relevant uncertainties and, as it can be seen from the values reported
in the Supplemental Material sup , there is still a slightly high correlation
between $\gamma_{M1M1}$ and $\gamma_{\pi}$
($\rho_{(\gamma_{M1M1}-\gamma_{\pi})}=\simeq 0.80$. These are clear
indications of the need for new dedicated measurements, that will be discussed
in detail in a dedicated forthcoming publication.

In summary, we presented the first simultaneous and self-consistent extraction
of the six leading-order static proton polarizabilities. The fit was performed
using a bootstrap-based technique combined with a fixed-$t$ subtracted
Dispersion Relation model for the theoretical calculation, using three
different PWA solutions as input. The obtained values have an error that is
competitive with the existing extractions, that were all obtained with the
inclusion of constraints on some of the fit parameters. These results provide
new important information to our understanding of the internal electromagnetic
proton structure, and should be used as input for further experimental and
theoretical extractions.

###### Acknowledgements.

We are grateful to Igor Strakovsky and Viktor Kashevarov, and who provided us
with the SAID-MA19 and MAID-2021 multipole values, respectively. The work of
E. Mornacchi is supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation), through the Collaborative Research Center [The Low-
Energy Frontier of the Standard Model, Projektnummer 204404729 - SFB 1044]. S.
Rodini acknowledges the support from the DFG grant under the research Unit FOR
2926, “Next Generation pQCD for Hadron Structure: Preparing for the EIC”,
project number 430824754.

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$\displaystyle+4\pi\varepsilon^{2}+4\pi\varepsilon\sqrt{\varepsilon^{2}-h^{2}}+\operatorname{O}\left(\frac{h^{4}}{\varepsilon^{2}},h^{2}\right).$
(6.79)

and from our volume constraint given in (6.39) we obtain

$\displaystyle
2=2c^{3}+\frac{3h^{2}c^{2}}{\varepsilon}+\operatorname{O}\left(\varepsilon^{3},h^{2}\right).$
(6.80)

Thus, from (6.80) we have

$c\geq
1-\frac{h^{2}c^{2}}{2\varepsilon}+\operatorname{O}\left(\varepsilon^{3},h^{2}\right).$
(6.81)

Lastly, from (6.81), (6.49), and (6.2) we obtain

$\displaystyle E_{\varepsilon}\left(\Omega,X\right)>4\pi+\frac{\pi
h^{4}}{\varepsilon^{2}}\log{\left(\frac{\varepsilon^{4}c^{2}}{2h^{6}}\left(1-\frac{h^{2}}{\varepsilon^{2}}\right)\right)}$
$\displaystyle+4\pi\varepsilon^{2}+4\pi\varepsilon\sqrt{\varepsilon^{2}-h^{2}}+\operatorname{O}\left(\varepsilon^{3},h^{2},\frac{h^{4}}{\varepsilon^{2}}\right)$
(6.82) $\displaystyle\geq 4\pi+4\pi\varepsilon^{2},$ (6.83)

for $\varepsilon<\varepsilon_{0}$ with $\varepsilon_{0}$ small. Thus, from
Lemma 6.1 we conclude that $(\Omega,X)$ cannot be a minimizer. ∎

######  Theorem 6.15.

There exist universal constants $C,\;C_{1},\;\varepsilon_{0}>0$ such that if
$\varepsilon<\varepsilon_{0}$ and $\gamma<\frac{8\pi}{\varepsilon}-C$ then
there exists a unique, up to translations and rotations, minimizer
$(\Omega^{\ast},\\{(-\frac{L^{\ast}}{2},0,0),(\frac{L^{\ast}}{2},0,0)\\})$ to
(6.1). Furthermore, it is a case 1 configuration, and if
$\frac{C_{1}}{\log{\varepsilon^{-1}}}<\gamma<\frac{8\pi}{\varepsilon}-C$, then

$\displaystyle E_{\varepsilon}$
$\displaystyle\left(\Omega^{\ast},\left\\{\left(-\frac{L^{\ast}}{2},0,0\right),\left(\frac{L^{\ast}}{2},0,0\right)\right\\}\right)$
$\displaystyle=4\pi+\frac{\gamma\varepsilon^{3}}{2}(1+\varepsilon+\varepsilon^{2})+\frac{\gamma^{2}\varepsilon^{6}}{64\pi}\log{(\gamma\varepsilon^{4})}+\operatorname{O}\left(\varepsilon^{6}\gamma^{\frac{7}{4}}(1+\gamma)^{\frac{1}{4}}(\log{\varepsilon^{-1}})^{\frac{3}{4}}\right),$
(6.84) $\displaystyle L^{\ast}$
$\displaystyle=2-2\varepsilon-\frac{\gamma\varepsilon^{3}}{8\pi}\log{(\gamma\varepsilon^{4})}+\operatorname{O}\left(\gamma^{\frac{3}{4}}(1+\gamma)^{\frac{1}{4}}\varepsilon^{3}(\log{\varepsilon^{-1}})^{\frac{3}{4}}\right),$
(6.85)

and $(\Omega^{\ast},\\{(-\frac{L^{\ast}}{2},0,0),(\frac{L^{\ast}}{2},0,0)\\})$
has contact height $h^{\ast}$ satisfying

$h^{\ast}=\sqrt[]{\frac{\gamma\varepsilon^{4}}{8\pi}}+\operatorname{O}\left(\frac{\varepsilon^{2}\gamma(\gamma+1)}{\log{{\varepsilon}^{-1}}}^{\frac{1}{4}}\right).$
(6.86)

###### Proof.

First note that the existence and rotational symmetry of a minimizer follows
directly from Theorem 3.2, Lemma 6.3 and Lemma 6.4. Furthermore, Lemma 6.7
implies that this minimizer is either a case 1, case 2, or case 3
configuration. Thus, Propositions 6.13 and 6.14 rule out all but a case 1
minimizer
$(\Omega^{\ast},\\{(-\frac{L^{\ast}}{2},0,0),(\frac{L^{\ast}}{2},0,0)\\})$.

Let $h^{\ast}$ be the contact height of the minimizer above. Then from (6.40)
and Lemma 6.1 we conclude that

$h^{\ast}\leq\frac{\varepsilon}{2}.$ (6.87)

Thus, from (6.41) we conclude that

$E_{\varepsilon}\left(\Omega^{\ast},\left\\{\left(-\frac{L^{\ast}}{2},0,0\right),\left(\frac{L^{\ast}}{2},0,0\right)\right\\}\right)\\\
=4\pi+\frac{\gamma\varepsilon^{3}}{2}(1+\varepsilon+\varepsilon^{2})-\left(\frac{2\pi{h^{\ast}}^{4}}{\varepsilon^{2}}-\frac{\gamma{h^{\ast}}^{2}\varepsilon^{2}}{2}\right)\log{(h^{\ast})}\\\
+\operatorname{O}\left(\gamma\varepsilon^{6},(\gamma+1){h^{\ast}}^{2}\varepsilon^{2},\frac{{h^{\ast}}^{4}}{\varepsilon^{2}}\right).$
(6.88)

Now let
$\left(\Omega,\left\\{\left(-\frac{L}{2},0,0\right),\left(\frac{L}{2},0,0\right)\right\\}\right)$
be a case 1 candidate minimizer with a contact height $h$ given by

$h=\sqrt[]{\frac{\gamma\varepsilon^{4}}{8\pi}}.$ (6.89)

From the minimality of
$\left(\Omega^{\ast},\left\\{\left(-\frac{L^{\ast}}{2},0,0\right),\left(\frac{L^{\ast}}{2},0,0\right)\right\\}\right)$
and from (6.41) and (6.88) we obtain

$\displaystyle
4\pi+\frac{\gamma\varepsilon^{3}}{2}(1+\varepsilon+\varepsilon^{2})-\left(\frac{2\pi{h^{\ast}}^{4}}{\varepsilon^{2}}-\frac{\gamma{h^{\ast}}^{2}\varepsilon^{2}}{2}\right)\log{(h^{\ast})}+\operatorname{O}\left(\gamma\varepsilon^{6},(\gamma+1){h^{\ast}}^{2}\varepsilon^{2},\frac{{h^{\ast}}^{4}}{\varepsilon^{2}}\right)$
(6.90) $\displaystyle\leq
4\pi+\frac{\gamma\varepsilon^{3}}{2}(1+\varepsilon+\varepsilon^{2})+\frac{\gamma^{2}\varepsilon^{6}}{32\pi}\log{\left(\sqrt[]{\frac{\gamma\varepsilon^{4}}{8\pi}}\right)}+\operatorname{O}\left(\gamma(\gamma+1)\varepsilon^{6}\right).$

Thus

$-\left(\frac{2\pi{h^{\ast}}^{4}}{\varepsilon^{2}}-\frac{\gamma{h^{\ast}}^{2}\varepsilon^{2}}{2}\right)\log{(h^{\ast})}+\operatorname{O}\left(\gamma(\gamma+1)\varepsilon^{6},(\gamma+1){h^{\ast}}^{2}\varepsilon^{2},\frac{{h^{\ast}}^{4}}{\varepsilon^{2}}\right)\leq\frac{\gamma^{2}\varepsilon^{6}}{32\pi}\log{\left(\sqrt[]{\frac{\gamma\varepsilon^{4}}{8\pi}}\right)}.$
(6.91)

Now note that (6.91) implies that

$h^{\ast}\geq\frac{\sqrt[]{\gamma\varepsilon^{4}}}{8},$ (6.92)

since otherwise (6.91) implies

$\frac{\gamma^{2}\varepsilon^{6}}{128}\log{\left(\frac{\sqrt[]{\gamma\epsilon^{4}}}{8}\right)}+\operatorname{O}\left(\gamma(\gamma+1)\varepsilon^{6}\right)\leq\frac{\gamma^{2}\varepsilon^{6}}{32\pi}\log{\left(\sqrt[]{\frac{\gamma\varepsilon^{4}}{8\pi}}\right)}.$
(6.93)

However, this implies

$\frac{\gamma^{2}\varepsilon^{6}}{256}\log{\left(\gamma\varepsilon^{4}\right)}+\operatorname{O}\left(\gamma(\gamma+1)\varepsilon^{6}\right)\leq\frac{\gamma^{2}\varepsilon^{6}}{64\pi}\log{\left(\gamma\varepsilon^{4}\right)}.$
(6.94)

Now pick $\varepsilon_{0}>0$ so that
$|\log{(\gamma\varepsilon^{4})}|>\operatorname{O}\left(\frac{\gamma+1}{\gamma}\right)$,
then (6.94) provides a contradiction. Thus, (6.92) holds.

Finally, let

$h^{\ast}=\sqrt[]{\frac{\gamma\varepsilon^{4}}{8\pi}}+h_{e},$ (6.95)

then from (6.91) we obtain

$-\left(\gamma\varepsilon^{2}h_{e}^{2}+\sqrt{8\pi\gamma}h_{e}^{3}+\frac{2\pi}{\varepsilon^{2}}h_{e}^{4}\right)\log{h^{\ast}}\\\
\leq-\frac{\gamma^{2}\varepsilon^{6}}{32\pi}\log{\left(1+\sqrt{\frac{8\pi}{\gamma\varepsilon^{4}}}h_{e}\right)}+\operatorname{O}\left(\gamma(\gamma+1)\varepsilon^{6},(\gamma+1){h^{\ast}}^{2}\varepsilon^{2},\frac{{h^{\ast}}^{4}}{\varepsilon^{2}}\right)$
(6.96)

If $h_{e}>0$, then (6.95) implies
$h^{\ast}>\sqrt[]{\frac{\gamma\varepsilon^{4}}{8\pi}}$, and (6.96) gives

$-\frac{2\pi}{\varepsilon^{2}}h_{e}^{4}\log{h^{\ast}}\leq\operatorname{O}\left(\frac{{h^{\ast}}^{4}(\gamma+1)}{\gamma\varepsilon^{2}}\right).$
(6.97)

Thus,

$h_{e}\leq\operatorname{O}\left(h^{\ast}\left(\frac{\gamma+1}{\gamma\log{\left({h^{\ast}}^{-1}\right)}}\right)^{\frac{1}{4}}\right).$
(6.98)

Now (6.98) and (6.95) imply that

$h^{\ast}\leq\sqrt[]{\frac{\gamma\varepsilon^{4}}{8\pi}}\left(1+\operatorname{O}\left(\left(\frac{\gamma+1}{\gamma\log{({h^{\ast}}^{-1})}}\right)^{\frac{1}{4}}\right)\right),$
(6.99)

whenever $h^{\ast}\leq\varepsilon<\varepsilon_{0}$ is sufficiently small.

If $h_{e}<0$, then from (6.95) we have
$0<h^{\ast}<\sqrt[]{\frac{\gamma\varepsilon^{4}}{8\pi}}$ and (6.96) implies

$-\frac{2\pi}{\varepsilon^{2}}h_{e}^{4}\log{h^{\ast}}\leq-\frac{\gamma^{2}\varepsilon^{6}}{32\pi}\log{\left(1+\sqrt{\frac{8\pi}{\gamma\varepsilon^{4}}}h_{e}\right)}+\operatorname{O}\left(\gamma(\gamma+1)\varepsilon^{6}\right).$
(6.100)

Furthermore, (6.92) implies that
$h_{e}\geq\frac{\sqrt{\gamma\varepsilon^{4}}}{8}-\frac{\sqrt{\gamma\varepsilon^{4}}}{\sqrt{8\pi}}$.
Thus, from (6.100) we obtain

$h_{e}\geq\operatorname{O}\left(\varepsilon^{2}\left(\frac{\gamma(\gamma+1)}{\log{({h^{\ast}}^{-1})}}\right)^{\frac{1}{4}}\right).$
(6.101)

Hence, (6.99) and (6.101) imply that

$h^{\ast}=\sqrt[]{\frac{\gamma\varepsilon^{4}}{8\pi}}+\operatorname{O}\left(\varepsilon^{2}\left(\frac{\gamma(\gamma+1)}{\log{{\varepsilon}^{-1}}}\right)^{\frac{1}{4}}\right).$
(6.102)

Thus, from (6.88) and (6.102) we obtain

$E_{\varepsilon}\left(\Omega^{\ast},\left\\{\left(-\frac{L^{\ast}}{2},0,0\right),\left(\frac{L^{\ast}}{2},0,0\right)\right\\}\right)\\\
=4\pi+\frac{\gamma\varepsilon^{3}}{2}(1+\varepsilon+\varepsilon^{2})+\frac{\gamma^{2}\varepsilon^{6}}{64\pi}\log{(\gamma\varepsilon^{4})}+\operatorname{O}\left(\varepsilon^{6}\gamma^{\frac{7}{4}}(1+\gamma)^{\frac{1}{4}}(\log{\varepsilon^{-1}})^{\frac{3}{4}}\right).$
(6.103)

Lastly, from (6.50), (6.51), and (6.30) we obtain

$L^{\ast}=2-2\varepsilon-\frac{\gamma\varepsilon^{3}}{8\pi}\log{(\gamma\varepsilon^{4})}+\operatorname{O}\left(\gamma^{\frac{3}{4}}(1+\gamma)^{\frac{1}{4}}\varepsilon^{3}(\log{\varepsilon^{-1}})^{\frac{3}{4}}\right).$
(6.104)

To conclude the proof, observe that
$(\Omega^{\ast},\\{(-\frac{L^{\ast}}{2},0,0),(\frac{L^{\ast}}{2},0,0)\\})$ is
unique, since expanding the second derivative of (6.30) gives

$\hat{E}_{\varepsilon}^{\prime\prime}(h)=\left(\frac{24\pi
h^{2}}{\varepsilon^{2}}-\gamma\varepsilon^{2}\right)\log
h^{-1}+\operatorname{O}\left(\frac{h^{2}}{\varepsilon^{2}},\gamma\varepsilon^{2}\right),$
(6.105)

which implies that $\hat{E}_{\varepsilon}(h)$ is strictly convex in the
neighborhood of the minimum. ∎

###### Proof of Theorem 3.6.

The proof is obtained by combining the statements of Propositions 6.13 and
6.14 with that of Theorem 6.15. ∎

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