# Asymptotically Optimal Closed-Form Phase Configuration of $1$-bit RISs via
Sign Alignment

Kyriakos Stylianopoulos1, Panagiotis Gavriilidis1, and George C.
Alexandropoulos1,2
1 Department of Informatics and Telecommunications, National and Kapodistrian
University of Athens, Greece
2 Department of Electrical and Computer Engineering, University of Illinois
Chicago, IL, USA
{kstylianop, pangavr<EMAIL_ADDRESS>This work has been supported by the
SNS JU project TERRAMETA under the EU’s Horizon Europe research and innovation
programme under Grant Agreement No 101097101, including top-up funding by UKRI
under the UK government’s Horizon Europe funding guarantee.

###### Abstract

While Reconfigurable Intelligent Surfaces (RISs) constitute one of the most
prominent enablers for the upcoming sixth Generation (6G) of wireless
networks, the design of efficient RIS phase profiles remains a notorious
challenge when large numbers of phase-quantized unit cells are involved,
typically of a single bit, as implemented by a vast majority of existing
metasurface prototypes. In this paper, we focus on the RIS phase configuration
problem for the exemplary case of the Signal-to-Noise Ratio (SNR) maximization
for an RIS-enabled single-input single-output system where the metasurface
tunable elements admit a phase difference of $\pi$ radians. We present a novel
closed-form configuration which serves as a lower bound guaranteeing at least
half the SNR of the ideal continuous (upper bound) SNR gain, and whose mean
performance is shown to be asymptotically optimal. The proposed sign alignment
configuration can be further used as initialization to standard discrete
optimization algorithms. A discussion on the reduced complexity hardware
benefits via the presented configuration is also included. Our numerical
results demonstrate the efficacy of the proposed RIS sign alignment scheme
over iterative approaches as well as the commonplace continuous phase
quantization treatment.

###### Index Terms:

Reconfigurable intelligent surfaces, discrete optimization, phase
configuration, SNR maximization.

##  I Introduction

The technology of Reconfigurable Intelligent Surfaces (RISs) is posed as one
of the key enablers for next sixth Generation (6G) of wireless networks due to
its dynamic connectivity enhancements with minimal deployment and operation
costs [1, 2]. In fact, theoretical results have shown that an RIS of $N$ unit
cells can offer an $N^{2}$-fold increase in the end-to-end channel magnitude
by optimally configuring the phase shifts of its comprising elements [3]. Such
benefits are further amplified by the metasurfaces’ near-passive energy
consumption and minute manufacturing cost per unit element. In an effort to
reduce hardware complexity, however, a vast majority of developed prototypes
has been designed with elements admitting discrete phase shift values, often
with a discretization of a single bit. Not only are the optimal gains of
finite discrete RIS configurations subpar to the theoretical $N^{2}$ gain
(e.g., [4, 5, 6]), but also this design principle casts the optimization
problem of the surface’s phase shifts into the area of discrete optimization
[7, 8]. In fact, most problem formulations associated with RIS phase
configuration are NP-hard and the approximate optimization procedures employed
often lead to degraded performance, while requiring impractical computational
costs [3, 9], considering the fact that the number of elements of RISs is
expected to be in the order of thousands, notwithstanding their lack of
theoretical guarantees.

Motivated by the above, this work specifically tackles the problem of
maximization of the Signal-to-Noise Ratio (SNR) for an RIS-enabled Single
Input Single Output (SISO) system, when each element of the RIS admits $1$-bit
quantized phase shifts with a $\pi$ phase difference between them. By re-
expressing the SNR maximization problem in this particular case, we derive a
closed-form configuration vector for the RIS elements that is guaranteed to
always achieve more than half of the achievable gain, while its expected gain
is proved to be at least $N^{2}/4$ when Rayleigh channel fading or Line of
Sight (LoS) conditions are encountered. We furthermore proceed to use this
configuration as initial point to a standard discrete optimization approach
offering both performance improvements and faster convergence. Our numerical
evaluation demonstrates that the proposed closed-form approach outperforms the
standard baseline of discretizing the optimal continuous RIS configuration, as
commonly used.

Notation: Matrices (vectors) are expressed in bold uppercase (bold lowercase)
typeface. ${\rm sign}(\cdot)$ returns the sign of its argument in
$\\{-1,+1\\}$. Element-wise multiplication and expectation are denoted by
$\odot$ and $\mathbb{E}[\cdot]$, respectively. The real (complex) normal
distribution is expressed by $\mathcal{N}$ ($\mathcal{CN}$) and the central
Chi-squared distribution with $k$ degrees of freedom is denoted as
$\mathcal{\chi}^{2}_{k}$. $\mathfrak{Re}(\cdot)$ and $\mathfrak{Im}(\cdot)$
return the real and imaginary parts of a complex quantity, $\angle$ denotes
the phase of a complex number in Euler form, while $\jmath$ denotes the
imaginary unit. The transpose, conjugate, and Hermitian transpose of
$\mathbf{A}$, are denoted as $\mathbf{A}^{\rm T}$, $\mathbf{A}^{\ast}$, and
$\mathbf{A}^{\rm H}$, respectively. Finally, lowercase subscripts denote
elements of vectors or matrices.

##  II System Model and Design Objective

Consider a SISO system in the presence of an RIS with $N$ unit cells, where
the channel responses for the Transmitter (TX) $\to$ RIS and the RIS $\to$
Receiver (RX) links are denoted by $\boldsymbol{h}\in\mathbb{C}^{N\times 1}$
and $\boldsymbol{g}^{\rm H}\in\mathbb{C}^{1\times N}$, respectively. For
simplification of presentation, and to present theoretical results in line
with the relevant literature, assume there is no direct TX $\to$ RX link. Let
the diagonal matrix $\boldsymbol{\Phi}\in\mathbb{C}^{N\times N}$ include the
RIS phase profile, so that each $n$-th unit cell ($n=1,2,\ldots,N$) with
configuration $\boldsymbol{\theta}_{n}$ is modeled as
$\boldsymbol{\Phi}_{n,n}=\exp(\jmath\boldsymbol{\theta}_{n})$. By further
denoting as $\rho$ the value of the transmit SNR and as ${\rm PL}$ the
multiplicative path loss of the two links, then the SISO SNR optimization
objective can be formulated as follows [4]:

$\displaystyle\mathcal{OP}_{1}:$
$\displaystyle~{}\underset{\boldsymbol{\theta}}{\max}\left\\{\rho{\rm
PL}|\boldsymbol{g}^{\rm
H}\boldsymbol{\Phi}\boldsymbol{h}|^{2}\right\\}=\underset{\boldsymbol{\theta}}{\max}\left\\{{\rho{\rm
PL}|(\boldsymbol{h}\odot\boldsymbol{g}^{\ast})^{\rm
T}\boldsymbol{\phi}|^{2}}\right\\}$ $\displaystyle\hskip 4.55254pt{\rm
s.t.}~{}~{}\boldsymbol{\theta}_{n}\in\mathcal{F},~{}n=1,2,\ldots,N,$

where $\boldsymbol{\phi}$ is the $N$-element vectorized form of the main
diagonal of $\boldsymbol{\Phi}$, and $\boldsymbol{h}$ and $\boldsymbol{g}$ are
normalized such that $\mathbb{E}[\boldsymbol{h}^{\rm
H}\boldsymbol{h}]=\mathbb{E}[\boldsymbol{g}^{\rm H}\boldsymbol{g}]=N$. In
addition, $\mathcal{F}$ represents the set of available quantized phase shift
values for each RIS element.

In the remainder of the paper, we will be assuming the availability of Channel
State Information (CSI) and, for ease of notation, that
$\mathcal{F}=\\{0,\pi\\}$, so that the phase shift for each RIS element aligns
with the axis of the real numbers of the complex plane. To this end,
$\phi^{0}\triangleq\exp(\jmath 0)=1$ and
$\phi^{\pi}\triangleq\exp(\jmath\pi)=-1$, therefore,
$\boldsymbol{\phi}\in\\{-1,+1\\}^{N}$. It is noted that the derivations hold
for any two phase shift values with a difference of $\pi$ radians, through a
rotation of axes. Based on the above and by denoting the concatenated channel
vector as
$\boldsymbol{c}\triangleq(\boldsymbol{h}\odot\boldsymbol{g}^{\ast})\in\mathbb{C}^{N\times
1}$, $\mathcal{OP}_{1}$ can be re-formulated as follows:

$\displaystyle\mathcal{OP}_{2}:$
$\displaystyle~{}\underset{\boldsymbol{\phi}}{\max}\,\gamma(\boldsymbol{\phi})\triangleq|\boldsymbol{c}^{\rm
T}\boldsymbol{\phi}|^{2}$ $\displaystyle~{}\hskip
31.2982pt=\underbrace{\left(\sum_{i=1}^{N}\boldsymbol{\phi}_{i}\mathfrak{Re}(\boldsymbol{c}_{i})\right)^{2}}_{\triangleq
A(\boldsymbol{\phi})}+\underbrace{\left(\sum_{i=1}^{N}\boldsymbol{\phi}_{i}\mathfrak{Im}(\boldsymbol{c}_{i})\right)^{2}}_{\triangleq
B(\boldsymbol{\phi})}$ $\displaystyle\hskip 4.55254pt{\rm
s.t.}~{}~{}\boldsymbol{\phi}\in\\{-1,+1\\}^{N},$

where $\rho$ and ${\rm PL}$ have been dropped without affecting the
equivalence between the $\mathcal{OP}_{1}$ and $\mathcal{OP}_{2}$ problems.

##  III Closed-Form RIS Phase Configuration

We commence with the following remark elaborating on the optimization of the
two terms appearing in $\mathcal{OP}_{2}$’s objective.

######  Remark 1.

The maximum values of functions $A(\boldsymbol{\phi})$ and
$B(\boldsymbol{\phi})$ in $\mathcal{OP}_{2}$ for the feasible values of
$\boldsymbol{\phi}$ appearing in its constraint are given respectively by
$A_{\max}\triangleq(\sum_{i=1}^{N}|\mathfrak{Re}(\boldsymbol{c}_{i})|)^{2}$
and
$B_{\max}\triangleq(\sum_{i=1}^{N}|\mathfrak{Im}(\boldsymbol{c}_{i})|)^{2}$,
which would be respectively achieved via the RIS phase configurations
$\boldsymbol{\phi}^{\text{Re}}\triangleq{\rm
sign}(\mathfrak{Re}(\boldsymbol{c}))$ and
$\boldsymbol{\phi}^{\text{Im}}\triangleq{\rm
sign}(\mathfrak{Im}(\boldsymbol{c}))$.

In Algorithm 1, we present a simple approach that selects between the two
configurations $\boldsymbol{\phi}^{\text{Re}}$ and
$\boldsymbol{\phi}^{\text{Im}}$ the one that maximizes the dual-summand
objective of $\mathcal{OP}_{2}$, i.e., the received SNR. The computational
time of this approach is $\Theta(N)$, which is optimal. In the sequel, we
provide a theoretical analysis of this Sign Alignment (SA) scheme for RIS
phase configuration along with guarantees on its performance.

###  III-A Theoretical Analysis

######  Theorem 1 (Lower bound for the achievable instantaneous SNR).

When the RIS phase configuration is set to $\boldsymbol{\phi}^{\text{SA}}$ via
Algorithm 1, the achievable instantaneous SNR
$\gamma(\boldsymbol{\phi}^{\text{SA}})$ is lower bounded by the quantity:

$f_{\rm
LB}(\boldsymbol{\phi}^{\text{SA}})\\!\triangleq\\!\max\\!\big{\\{}\\!\big{(}\sum\nolimits_{i=1}^{N}|\mathfrak{Re}(\boldsymbol{c}_{i})|\big{)}^{\\!\\!2},\big{(}\sum\nolimits_{i=1}^{N}|\mathfrak{Im}(\boldsymbol{c}_{i})|\big{)}^{\\!\\!2}\\!\big{\\}}\\!.$
(1)

###### Proof.

It follows directly from the selection criterion in Step $3$ of Algorithm 1
that either $A(\boldsymbol{\phi})$ or $B(\boldsymbol{\phi})$ are maximized,
and the maximum values of $A(\boldsymbol{\phi})$ and $B(\boldsymbol{\phi})$
are obtained using the results in Remark 1. ∎

Algorithm 1 RIS Phase Configuration via Sign Alignment

1:The CSI vector $\boldsymbol{c}$.

2:Set
$\boldsymbol{\phi}^{\text{Re}}\leftarrow\text{sign}(\mathfrak{Re}(\boldsymbol{c}))$.

3:Set
$\boldsymbol{\phi}^{\text{Im}}\leftarrow\text{sign}(\mathfrak{Im}(\boldsymbol{c}))$.

4:Select $\boldsymbol{\phi}^{\text{SA}}={\rm
arg}\max_{\boldsymbol{\phi}\in\\{\boldsymbol{\phi}^{\text{Re}},\boldsymbol{\phi}^{\text{Im}}\\}}\gamma(\boldsymbol{\phi})$.

######  Corollary 1 (Lower bound for the achievable SNR w.r.t. optimal
configuration).

Let $\boldsymbol{\phi}^{\star}$ be the RIS phase configuration obtained by
optimally solving $\mathcal{OP}_{2}$, then
$\gamma(\boldsymbol{\phi}^{\text{SA}})\geq
0.5\gamma(\boldsymbol{\phi}^{\star})$.

###### Proof.

It follows from $\mathcal{OP}_{2}$’s dual-summand objective, Theorem 1, and
the fact that $A(\boldsymbol{\phi}),B(\boldsymbol{\phi})>0$. ∎

######  Theorem 2 (Lower bound for the expected achievable SNR under Rayleigh
fading).

When both the links TX $\to$ RIS and RIS $\to$ RX experience Rayleigh fading
conditions, i.e.,
$\boldsymbol{h}_{i},\boldsymbol{g}_{i}\overset{i.i.d.}{\sim}\mathcal{CN}(0,1)$
$\forall$$n=1,2,\ldots,N$, the expectation of the achievable SNR,
$\mathbb{E}[\gamma(\boldsymbol{\phi}^{\text{SA}})]$, is lower bounded by the
quantity:

$\bar{f}_{\rm LB}(\boldsymbol{\phi}^{\text{SA}})\triangleq 0.25N^{2}.$ (2)

###### Proof.

Without loss of generality, let us assume that
$(\sum_{i=1}^{N}|\mathfrak{Re}(\boldsymbol{c}_{i})|)^{2}>(\sum_{i=1}^{N}|\mathfrak{Im}(\boldsymbol{c}_{i})|)^{2}$.
Then, following Step $3$ of Algorithm 1 results in
$\boldsymbol{\phi}^{\text{SA}}=\boldsymbol{\phi}^{\text{Re}}$, yielding:

$\displaystyle\mathbb{E}\left[\gamma(\boldsymbol{\phi}^{\text{SA}})\right]=\mathbb{E}\left[A(\boldsymbol{\phi}^{\text{Re}})+B(\boldsymbol{\phi}^{\text{Re}})\right]\geq\mathbb{E}\left[A(\boldsymbol{\phi}^{\text{Re}})\right]$
(3)
$\displaystyle=\mathbb{E}\left[A_{\max}\right]=\mathbb{E}\left[\left(\sum_{i=1}^{N}|\mathfrak{Re}(\boldsymbol{c}_{i})|\right)^{2}\right]$
(4)
$\displaystyle\geq\mathbb{E}^{2}\left[\sum_{i=1}^{N}|\mathfrak{Re}(\boldsymbol{c}_{i})|\right]=\sum_{i=1}^{N}\mathbb{E}^{2}\left[|\mathfrak{Re}(\boldsymbol{c}_{i})|\right].$
(5)

We next express $\boldsymbol{h}_{i}=x_{i}+\jmath y_{i}$ and
$\boldsymbol{g}_{i}^{\ast}=z_{i}+\jmath w_{i}$ with
$x_{i},\,y_{i},\,z_{i},w_{i}\overset{i.i.d.}{\sim}\mathcal{N}(0,0.5)$, which
leads to the formulation:

$\displaystyle\mathbb{E}\left[|\mathfrak{Re}(\boldsymbol{c}_{i})|\right]=\mathbb{E}\left[|\mathfrak{Re}(\boldsymbol{h}_{i}\boldsymbol{g}^{*}_{i})|\right]=\mathbb{E}\left[|x_{i}z_{i}-y_{i}w_{i}|\right].$
(6)

It is well known that the product of the two normally distributed Random
Variables (RVs) $x_{i}$ and $z_{i}$ follows the distribution of the difference
of two chi-squared RVs, since
$x_{i}z_{i}=0.25\left(\left(x_{i}+z_{i}\right)^{2}-\left(x_{i}-z_{i}\right)^{2}\right)$.
Furthermore, since both terms inside the parentheses have the same variance,
it holds that they can be thought of as i.i.d. $\mathcal{\chi}^{2}_{1}$ RVs,
i.e., their Pearson correlation coefficient in [10, Sec. 4.B] is zero. By
introducing the terms $Q_{i}\triangleq(x_{i}+z_{i})^{2}+(y_{i}-w_{i})^{2}$ and
$R_{i}\triangleq(x_{i}-z_{i})^{2}+(y_{i}+w_{i})^{2}$, (6) can be re-expressed
as:

$\mathbb{E}\left[|x_{i}z_{i}-y_{i}w_{i}|\right]=0.25\mathbb{E}\left[|Q_{i}-R_{i}|\right],$
(7)

where $Q_{i}$ and $R_{i}$ are i.i.d. $\mathcal{\chi}^{2}_{2}$ RVs. Then,
capitalizing on the results in [10, Sec. 4.C], the probability density
function of the RV $S_{i}\triangleq Q_{i}-R_{i}$ is obtained as follows:

$f_{S_{i}}(x)\triangleq 0.25\exp\left(-|x|/2\right).$ (8)

Finally, returning to (6) and exploiting (7) and (8), we can derive the lower
bound for expected achievable SNR as:

$\mathbb{E}\left[|x_{i}z_{i}-y_{i}w_{i}|\right]=0.25\int_{-\infty}^{\infty}|x|f_{S}(x)\,dx=0.5.$
(9)

The proof is concluded by substituting (9) into (6). ∎

######  Theorem 3 (Lower bound for the achievable SNR under pure LoS
channels).

Assume pure LoS channel conditions for all wireless links and let
$\boldsymbol{h}_{i}=\exp{(\jmath\boldsymbol{\vartheta}^{\boldsymbol{h}}_{i})}$
and
$\boldsymbol{g}_{i}^{*}=\exp{(-\jmath\boldsymbol{\vartheta}^{\boldsymbol{g}}_{i}})$.
The achievable deterministic SNR value $\gamma(\boldsymbol{\phi}^{\text{SA}})$
is again lower bounded by $\bar{f}_{\rm LB}(\boldsymbol{\phi}^{\text{SA}})$,
similar to Theorem 2.

###### Proof.

Let
$\boldsymbol{\vartheta}_{i}\triangleq\angle\boldsymbol{c}_{i}=\boldsymbol{\vartheta}^{\boldsymbol{h}}_{i}-\boldsymbol{\vartheta}^{\boldsymbol{g}}_{i}$
and assume, without loss of generality, that
$\sum_{i=1}^{N}|\mathfrak{Re}(\boldsymbol{c}_{i})|>\sum_{i=1}^{N}|\mathfrak{Im}(\boldsymbol{c}_{i})|$,
which is equivalently expressed as
$\sum_{i=1}^{N}|\cos(\boldsymbol{\vartheta}_{i})|>\sum_{i=1}^{N}|\sin(\boldsymbol{\vartheta}_{i})|$.
The following derivations hold:

$\displaystyle\cos^{2}(\boldsymbol{\vartheta}_{i})+\sin^{2}(\boldsymbol{\vartheta}_{i})\\!=\\!1\Rightarrow\\!\sum_{i=1}^{N}\\!\cos^{2}(\boldsymbol{\vartheta}_{i})+\sum_{i=1}^{N}\\!\sin^{2}(\boldsymbol{\vartheta}_{i})\\!=\\!N$
$\displaystyle\Rightarrow\\!\sum_{i=1}^{N}\\!|\cos(\boldsymbol{\vartheta}_{i})|+\sum_{i=1}^{N}\\!|\sin(\boldsymbol{\vartheta}_{i})|\geq
N\Rightarrow\\!\sum_{i=1}^{N}\\!|\cos(\boldsymbol{\vartheta}_{i})|\geq\frac{N}{2}$
$\displaystyle\Rightarrow\\!\left(\sum_{i=1}^{N}\\!|\cos(\boldsymbol{\vartheta}_{i})|\right)^{\\!\\!2}\geq
0.25N^{2}\overset{\eqref{eq:lb1}}{\Rightarrow}\gamma(\boldsymbol{\phi}^{\text{SA}})\geq
0.25N^{2}.$ (10)

∎

######  Remark 2.

The lower bound $\bar{f}_{\rm LB}(\boldsymbol{\phi}^{\text{SA}})$ in Theorems
2 and 3 has an optimal asymptotic growth of $\Theta(N^{2})$. This is due to
the fact that the continuous case of $\mathcal{OP}_{1}$, i.e., when
$\boldsymbol{\theta}_{n}\in[0,2\pi]$ $\forall$$n$, can be optimally solved via
Phase Alignment (PA) [11], yielding the upper bound SNR value $N^{2}$ for
normalized $\rho{\rm PL}$.

##  IV Iterative Discrete Optimization Schemes

As previously mentioned, for the perfect CSI availability case, the continuous
version of $\mathcal{OP}_{1}$ can be optimally solved via PA [11], i.e., by
setting $\boldsymbol{\phi}^{\rm PA}_{n}$ to
$-(\angle\boldsymbol{h}_{n}-\angle\boldsymbol{g}_{n})$ $\forall$$n$. The most
common approach in the literature for treating the discretized case is to
optimally solve the continuous problem via PA, and then quantize each element
so that $\boldsymbol{\phi}^{QPA}_{n}\triangleq{\rm
arg}\min_{\boldsymbol{\phi}_{n}\in\mathcal{F}}\|\boldsymbol{\phi}_{n}-\boldsymbol{\phi}^{\rm
PA}_{n}\|$ [11, 7, 4, 12]; this will be referred to as Quantized Phase
Alignment (QPA) in the sequel. It has been shown [4] that, under Rayleigh
conditions, QPA achieves an asymptotic expected gain of about $0.4N^{2}$.

Other categories of discrete optimization algorithms include Branch and Bound
(BnB) approaches that treat the quantization problem as mode selection [13,
4]. Despite their effectiveness, BnB methods have exponential computational
requirements which makes them less attractive for real-time RIS configuration
with instantaneous CSI. Black-box optimization algorithms, such as Genetic
Optimization (GO), have also been used in discrete RIS tuning problems, such
as in [3, 9, 14], with encouraging performance. However, their population-
based search entails many times more SNR evaluations than $N$ during a fixed
CSI frame, which is infeasible for instantaneous RIS tuning. Those methods are
in principal better tailored for statistical CSI setups [15] or codebook
search approaches [16].

From the perspective of deep learning, reinforcement learning algorithms for
discrete [17] and $1$-bit [5] RIS phase configuration have been lately
proposed, as well as a two-step supervised learning methodology that couples
the problems of SNR prediction and optimization [6]. Despite the fact that
such approaches have demonstrated promising results in real-time RIS control,
their performance is dependent on collected data that are scenario specific,
and often imply considerable measurement overheads. For this reason, these are
not considered in the evaluation framework of this paper.

###  IV-A Hill Climbing (HC)

One of the typical baseline approaches to solve the discrete optimization
problem $\mathcal{OP}_{2}$ is to: i) start with a random candidate solution;
ii) flip each RIS element’s phase shift from $\phi^{0}$ to $\phi^{\pi}$ or
vice-versa; iii) evaluate the objective function (i.e., SNR) after the flip is
performed; and v) discard the change if no SNR improvement was obtained. This
procedure is one of the standard generic discrete optimization approaches,
commonly termed as “coordinate hill climbing” [18], which is described in
Algorithm 2 specifically for the problem of the $1$-bit RIS phase
configuration selection. In this algorithm, we propose to use
$\boldsymbol{\phi}^{\text{SA}}$ from Algorithm 1 as a starting RIS
configuration. This has the two-fold benefit of starting the search from an
already improved point, compared to random initialization, as well as stronger
performance guarantees. Since the Hill Climbing (HC) algorithm may only
improve on any given configuration, the performance bounds of Section III-A
hold for Algorithm 2, as well. We will be referring to the randomly
initialized approach as HC, and as HC&SA when initialized by
$\boldsymbol{\phi}^{\text{SA}}$. Notice that HC&SA will outperform SA by
design, albeit at an iterative computational cost.

Algorithm 2 Iterative RIS Phase Configuration

1:The CSI vector $\boldsymbol{c}$.

2:Obtain $\boldsymbol{\phi}^{(0)}\leftarrow\boldsymbol{\phi}^{\text{SA}}$ from
Algorithm 1.

3:Set $\gamma^{\rm max}\leftarrow\gamma(\boldsymbol{\phi}^{(0)})$.

4:for $k=1,2,\ldots$ until convergence do

5: for $n=1,2,\ldots,N$ do

6: Set $\boldsymbol{\phi}^{(k)}_{n}\leftarrow-\boldsymbol{\phi}^{(k-1)}_{n}$.

7: if $\gamma(\boldsymbol{\phi}^{(k)})>\gamma^{\rm max}$ then

8: Set $\gamma^{\rm max}\leftarrow\gamma(\boldsymbol{\phi}^{(k)})$.

9: else

10: Set $\boldsymbol{\phi}^{(k)}_{n}\leftarrow\boldsymbol{\phi}^{(k-1)}_{n}$.

11: end if

12: end for

13:end for

14:return $\boldsymbol{\phi}^{(k)}$

##  V Numerical Evaluation

In this section, we evaluate the performance of the proposed SA RIS
configuration methodology and compare it against: i) QPA, as the main
candidate closed-form solution available in the literature; ii) the HC
iterative algorithm of Section IV-A that is comparably less computationally
demanding than the relevant works discussed in Section IV; and iii) HC&SA, to
assess the improvements brought by the initialization via SA. We have also
evaluated the SNR obtained via the upper bound approach of the continuous case
through PA, which has been used for normalizing the achievable SNRs of all
previous methods. For the experimentation setup, we used the generic Ricean
system model in [17], while the RIS was modeled as a $N^{\rm vert}\times
N^{\rm hor}$ Uniform Rectangular Array (URA) so that $N=N^{\rm vert}N^{\rm
hor}$ with $|N^{\rm vert}-N^{\rm hor}|$ selected to be minimized. In the
Cartesian $(x,y,z)$ frame, the TX was positioned at $(0,0,6)$ m, the RX at
$(0,20,1.5)$ m, and the RIS at $(2,2,2)$ m, with the latter’s orientation
being perpendicular to the $yz$ plane. The performance curves included in this
section are reported as averages over $1000$ random channel realizations,
apart from Fig. 2b, where realizations were averaged over $900$ deterministic
channels of different angles, and from Fig. 2c, where $200$ realizations were
averaged to alleviate from the consuming computations of exhaustive search.

Figure 1: Average achievable SNR versus increasing Ricean $\kappa$-factors
from Rayleigh fading to LoS conditions for an RIS with $N=100$ unit cells.

In Fig. 1, the performance comparison of the considered methods is reported
under different Ricean $\kappa$-factors, namely, from $-30$dB (rich
scattering) to $25$dB (almost pure LoS). As shown, SA achieves about $8$%
improvement on the normalized SNR over QPA under Rayleigh fading and around
$17$% improvements under LoS conditions. HC offers a mere $2$% increase over
SA under Rayleigh, while this difference is diminished for LoS, where SA
performs equally well. HC&SA offers a small improvement over HC, however, it
comes at a faster convergence cost, as will be discussed in the sequel.

The results of Fig. 1 could be specific to the considered setup parameters,
hence, we have performed averages over Ricean fading and all possible
elevation/azimuth angles in Fig. 2, where the performances are evaluated as
$N$ increases. Similar gains among all considered methods are observed in Fig.
2a, as in Fig. 1, however, the mean SNR values among all methods are more
closely comparable under LoS (Fig. 2b), when all possible angles are averaged
out, indicating that the problem of selecting effective RIS configuration
schemes is tightly coupled with its positioning under LoS conditions. It is
important to highlight that in Fig. 2a, QPA converges to its asymptotic
normalized expected gain of $0.4$ for larger $N$, as stated in Section IV,
however, SA immensely exceeds the lower bounds given in Section III-A. The
convergence iterations for Figs. 2a and 2b for the two proposed iterative
approaches are given in Table I, from which it can be inferred that, in RISs
with $N\geq 100$, the SA initialization offers $15$%-$25$% convergence speed
improvements, on top of marginally improved performance.

The performance evaluation displayed so far has been compared against the SNR
achievable by an RIS with continuous phase shifts, which calls for
investigation on the performance of the considered methods with respect to the
optimal $1$-bit-quantized solution of $\mathcal{OP}_{1}$. For $N$ up to $26$,
where evaluating all $2^{N}$ combinations is computationally tractable, this
comparison is given in Fig. 2c for Rayleigh fading conditions. Interestingly,
in the small RIS regime, SA obtains around $95$% of the optimal performance,
despite being a closed-form approximate solution, which hints that the results
in larger RIS sizes may not be too far from the optimal value. Evidently, the
proposed iterative approaches obtain near-optimal performance in these
scenarios, however, there is a maintained $13$%-$15$% performance difference
between SA and QPA, indicating that the proposed closed-form RIS configuration
is offering firm benefits with negligent computational overhead.

(a) Rayleigh fading

(b) Pure LoS conditions

(c) Rayleigh fading (few-element RIS)

Figure 2: Average achievable SNR versus the number of RIS unit cells under different fading conditions. The results are normalized, similar to Fig. 1, w.r.t.: i) the upper bound SNR obtainable by non-quantized phase shifts in (a) and (b); and ii) the SNR obtained by exhaustively solving $\mathcal{OP}_{1}$ in (c). TABLE I: Average Number of HC Iterations for Convergence. $N$ | $16$ | $36$ | $64$ | $100$ | $144$ | $196$
---|---|---|---|---|---|---
| Rayleigh
HC | $47.52$ | $120.78$ | $254.72$ | $421.5$ | $648.0$ | $1030.96$
HC&SA | $47.68$ | $113.22$ | $209.6$ | $355.5$ | $551.52$ | $814.38$
| LoS
HC | $48.79$ | $126.0$ | $232.49$ | $372.53$ | $635.11$ | $851.15$
HC&SA | $40.44$ | $103.44$ | $191.80$ | $267.59$ | $478.22$ | $715.04$

##  VI Implementation of RIS Sign Alignment

Notwithstanding its performance improvements over conventional RIS
configurations, SA can offer benefits over the related QPA closed-form
approach also in terms of hardware requirements. To compute
$\boldsymbol{\phi}^{\rm Re}$ and $\boldsymbol{\phi}^{\rm Im}$ from Algorithm
1, one needs only the signs of each channel gain $\boldsymbol{c}_{i}$, which
is equivalent to measuring the quadrant of $\angle\boldsymbol{c}_{i}$. For
each single-hop channel, those signs can be acquired through a simple
Reception (RX) Radio-Frequency chain (RFC) comprising a pair of $1$-bit
Analog-to-Digital Converters (ADCs), tasked to measure the signs of each
$\boldsymbol{c}_{i}$’s I/Q components [19]. Note that the power consumption of
ADCs grows exponentially with their bit resolution, hence, an $1$-bit ADC
offers an important improvement over traditional high-resolution ADCs.
Secondly, since no amplitude information is required to compute the SA
configurations, the detection process can be further simplified and will be
less susceptible to errors, equivalently to how Quadrature-Phase-Shift-Keying
(QPSK) constellations are easier to detect compared to high-order Quadrature
Amplitude Modulation (QAM) [20].

On a conceptual level, such simplified and low-power RX RFCs can be envisioned
to be endowed onto the metasurface, toward autonomous RIS solutions [21]. In
fact, the two output bits of the two ADCs, could be directly connected to the
RIS element switches, so that the surface’s state may be set to
$\boldsymbol{\phi}^{\rm Re}$ or $\boldsymbol{\phi}^{\rm Im}$ without the need
for involved digital controllers. Since each element’s configuration is
independent of the other channel coefficients, the operation can take place at
the unit-cell level. Furthermore, the design is straightforwardly related to
the output of the ADCs, thus no detection algorithm is needed at the RIS
controller [1]. Notice that the SNR computations implied by Step $3$ of
Algorithm 1 could be performed at the receiver, where high-resolution RX RFCs
can be readily available, and the result of the comparison could be fed back
to the RIS if a control channel is present. However, describing a detailed RX
RFC and its integration to the RIS, that can facilitate SA on the RIS side
with reduced hardware complexity, lies beyond the scope of this paper.

It is finally noted that, from a communications system perspective, the
considered SISO system may be extended to the general case of Multiple Input
Multiple Output (MIMO) systems by leveraging a capacity analysis on the
eigenvalues and eigenvectors of the covariance matrices of the wireless
channels, as it has been recently studied in [15]. We delegate such analysis
to the journal version of this work.

##  VII Conclusion

In this paper, we considered an RIS-enabled SISO communication system and
presented a novel closed-form configuration for the RIS responses for the case
of $1$-bit-phase-quantized unit cells where the phase difference is of $\pi$
radians. Lower bounds of the proposed SA configuration under Rayleigh fading
and LoS conditions, which are asymptotically optimal, were derived. Our
numerical evaluation showcased that SA outperforms the state-of-the-art
configuration scheme, which is obtained by quantizing the optimal continuous
phase shifts. It was also demonstrated that when SA is used as initial point
in the iterative HC algorithm, it can provide further performance increase
with faster convergence. We finally discussed how the proposed closed-form RIS
configuration can be realized at the RIS side in a standalone operation
fashion via low-power RX units each comprising a pair of $1$-bit ADCs.

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contains two occurrences of $\mathcal{K}_{3,\textrm{df}}^{(u,u)}$, both of
which are dressed on the external legs by $\mathcal{L}$ and $\mathcal{R}$ in
the same way as the leading order. We have at this order

$\displaystyle\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})\Big{\rvert}_{\mathrm{NLO}}$
$\displaystyle=\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{L}^{(u,u)\,\prime}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{T}_{\textrm{df},1}^{(u,u)\,\prime}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{R}^{(u,u)\,\prime}(\mathbf{k}^{\prime},\mathbf{k})\,,$
$\displaystyle=-\int_{p^{\prime}}\int_{q^{\prime\prime}}\int_{q^{\prime}}\int_{q}\int_{k^{\prime}}\,\mathcal{L}^{(u,u)\,\prime}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p}^{\prime},\mathbf{q}^{\prime\prime})\,\Gamma\,^{\prime}(\mathbf{q}^{\prime\prime},\mathbf{q}^{\prime})$
$\displaystyle\qquad\times\,\mathcal{L}^{(u,u)\,\prime}(\mathbf{q}^{\prime},\mathbf{q})\,\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{q},\mathbf{k}^{\prime})\,\mathcal{R}^{(u,u)\,\prime}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(64)

which we can apply the same identities to both $K$ matrices. We find that we
can rewrite the next-to-leading order term in the symmetric form as

$\displaystyle\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})\Big{\rvert}_{\mathrm{NLO}}$
$\displaystyle=-\int_{p^{\prime}}\int_{q^{\prime}}\int_{q}\int_{k}\,\mathcal{L}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{K}_{3,\textrm{df}}(\mathbf{p}^{\prime},\mathbf{q}^{\prime})\,\widetilde{\rho}(\sigma_{q^{\prime}})$
$\displaystyle\qquad\times\,\mathcal{L}(\mathbf{q}^{\prime},\mathbf{q})\mathcal{K}_{3,\textrm{df}}(\mathbf{q},\mathbf{k}^{\prime})\,\mathcal{R}(\mathbf{k}^{\prime},\mathbf{k})\,,$
$\displaystyle\equiv\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{L}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{T}_{\textrm{df},1}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{R}(\mathbf{k}^{\prime},\mathbf{k})\,,$

where $\mathcal{L}$ and $\mathcal{R}$ are as above and we have defined
$\mathcal{T}_{\textrm{df}}$ as

$\displaystyle\mathcal{T}_{\textrm{df}}(\mathbf{p},\mathbf{k})=-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}(\mathbf{p},\mathbf{p}^{\prime})\,\widetilde{\rho}(\sigma_{p^{\prime}})\,\mathcal{L}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\mathcal{K}_{3,\textrm{df}}(\mathbf{k}^{\prime},\mathbf{k}^{\prime})\,.$

After applying these operations on all occurrences of
$\mathcal{K}_{3,\textrm{df}}$ in $\mathcal{M}_{3,\textrm{df}}^{(u,u)}$, the
symmetric $K$ matrix form of the $\mathcal{M}_{3,\textrm{df}}^{(u,u)}$
equation reads

$\displaystyle\mathcal{M}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})\equiv\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{L}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{T}_{\textrm{df}}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{R}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(65)

where $\mathcal{T}_{\textrm{df}}$ is a symmetric amputated amplitude
satisfying the linear integral equation driven by the symmetric $K$ matrix

$\displaystyle\mathcal{T}_{\textrm{df}}(\mathbf{p},\mathbf{k})=\mathcal{K}_{3,\textrm{df}}(\mathbf{p},\mathbf{k})-\int_{p^{\prime}}\int_{k^{\prime}}\,\mathcal{K}_{3,\textrm{df}}(\mathbf{p},\mathbf{p}^{\prime})\,\widetilde{\rho}(\sigma_{p^{\prime}})\,\mathcal{L}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{T}_{\textrm{df}}(\mathbf{k}^{\prime},\mathbf{k})\,.$
(66)

Equation (66) is identical to the one derived in Ref. Hansen and Sharpe (2015)
provided the cutoff functions are chosen to be the same. We can combine Eq.
(66), (65), and (33) into a single integral equation similar to its asymmetric
form (II). Alternatively, we can arrive at the same relation by applying the
symmetry operations to Eq. (II) directly. We find that the connected
$\mathbf{3}\to\mathbf{3}$ amplitude satisfies the integral equation

$\displaystyle\mathcal{M}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\frac{1}{9}\,\mathcal{K}_{3,\textrm{df}}(\mathbf{p},\mathbf{k})-\mathcal{K}_{2}(\sigma_{p})\,\mathcal{G}(\mathbf{p},\mathbf{k})\,\mathcal{M}_{2}(\sigma_{k})-\frac{1}{3}\mathcal{K}_{3,\textrm{df}}(\mathbf{p},\mathbf{k})\,\widetilde{\rho}(\sigma_{k})\,\mathcal{M}_{2}(\sigma_{k})\,$
$\displaystyle\qquad-\mathcal{K}_{2}(\sigma_{p})\,\int_{k^{\prime}}\,\Gamma(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{M}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})-\frac{1}{3}\int_{k^{\prime}}\mathcal{K}_{\textrm{df},3}(\mathbf{p},\mathbf{k}^{\prime})\,\widetilde{\rho}(\sigma_{k^{\prime}})\,\mathcal{M}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(67)

where
$\Gamma(\mathbf{p},\mathbf{k})=\delta(\mathbf{p},\mathbf{k})\,H(\sigma_{k})\,\mathcal{I}(\sigma_{k})+\mathcal{G}(\mathbf{p},\mathbf{k})\equiv\delta(\mathbf{p},\mathbf{k})\,\widetilde{\rho}(\sigma_{k})+\mathcal{G}(\mathbf{p},\mathbf{k})$
as before. We have reduced the original set of integral equations from Ref.
Hansen and Sharpe (2015) to a simple single integral equation, where we
calculated $\mathcal{M}_{3}^{(u,u)}$ directly given $\mathcal{K}_{2}$ and
$\mathcal{K}_{3,\textrm{df}}$. The numerical factors arise from over-counting
the states as we write the relation in terms of the unsymmetrized
$\mathcal{M}_{3}^{(u,u)}$.

As a final remark, the finite-volume equivalents can be found by following the
same arguments in the beginning of Sec. V, which we simply state the result

$\displaystyle\det\Big{[}\,1+\mathcal{K}_{3,\textrm{df}}(P)\cdot
F_{3,L}(P)\,\Big{]}_{E=E_{\mathfrak{n}}}=0\,,$ (68)

where $F_{3,L}$ is a symmetrized version of $F_{3,L}^{(u,u)}$ introduced
earlier,

$\displaystyle
F_{3,L}(P)\equiv\widetilde{F}_{2,L}\cdot\left(\,\frac{1}{3}-\frac{1}{{\widetilde{\mathcal{K}}_{2,L}}^{\,-1}+\widetilde{F}_{2,L}+\widetilde{G}_{L}(P)}\cdot\widetilde{F}_{2,L}\right)\,,$

which agrees with the original quantization condition presented in Ref. Hansen
and Sharpe (2014).

##  VI Conclusions

We have presented a general way to derive relativistic
$\mathbf{3}\to\mathbf{3}$ scattering integral equations and finite volume
quantization conditions using $S$ matrix unitarity. The method uses the fact
that singularities from on-shell physics is responsible for the dominant
finite-volume effects and $S$ matrix unitarity constrains all physical
singularities in the kinematic region of interest. The method is non-
perturbative, and considers only on-shell amplitudes, avoiding the need to
connect these quantities to quantum fields. This approach differs from
previous FVU studies Mai  _et al._ (2017); Mai and Doring (2017) by
considering the $T$ matrix elements directly, which give a simpler path toward
both the scattering equations and the finite-volume quantization conditions.

While the methods were shown to be equivalent Jackura  _et al._ (2019b);
Blanton and Sharpe (2020b), this work provides a direct connection between the
RFT and FVU approaches. We found that the TOPT version of the RFT approach is
most directly related to the FVU method, which was noted in Ref. Blanton and
Sharpe (2020a) in their derivation of the quantization conditions. The
connection manifest itself by examining the on-shell representation of the $T$
matrix elements, not the fully connected amplitude, and embedding the system
in a confined volume. Making this connection more explicit allowed us to find
a new expression for the $\mathbf{3}\to\mathbf{3}$ $T$ matrix integral
equation shown in Eq. (2), as well as gave new physical insight the
$\mathcal{T}_{\textrm{df}}$ object which can be interpreted as the
kinematically singular free $T$ matrix element which was shown in Sec. V. We
were able to recover the quantization conditions of Ref. Blanton and Sharpe
(2020a) directly, shown in Eq. (7). By using various identities and
manipulations, we were able to transform the simple on-shell form for the $T$
matrix element and the resulting quantization conditions to all previously
derived results for three identical particles, including new versions for the
asymmetric three-body $K$ matrix given in Eq. (II) for the amplitude and Eq.
(8) for the associated quantization condition.

Our approach here is rather efficient in generating the scattering equations
and quantization conditions, requiring only that unitarity condition for the
$T$ matrix element and the kinematic functions which characterize the on-shell
propagation of intermediate state particles. The results can easily be
generated to particles of different species, coupled two and three channel
systems, and particles with arbitrary spin once the state space is fully
defined. Moreover, we find this direction promising for extending the
framework to higher number of particles, e.g. four particle systems. Combined
with the recent efficient version of the all-orders approach using TOPT
Blanton and Sharpe (2020a), this framework provides a unified approach to
linking lattice QCD observables to few-body phenomena.

## Acknowledgements

The author would like to thank R. Briceño, S. Dawid, and F. Romero-López for
helpful discussions and comments on the manuscript, and would also like to
thank M. Hansen, S. Sharpe, and A. Szczepaniak for useful discussions
regarding this work. A.W.J. acknowledges supported from U.S. Department of
Energy contract DE-AC05-06OR23177, under which Jefferson Science Associates,
LLC, manages and operates Jefferson Lab. A.W.J. also acknowledges support of
the USDOE Early Career award, contract DE-SC0019229.

##  Appendix A Three pariticle unitarity relation

In this appendix, we provide some details regarding the derivation of Eq.
(III.1), which has been presented in Ref. Jackura  _et al._ (2019a), with some
modifications to agree with the conventions set in this manuscript. Since we
are interested in amplitudes within the pair-spectator basis, we find
immediately for these states that the unitarity relation has the form

$\displaystyle
2\operatorname{Im}\bra{P,\mathbf{p},\ell^{\prime}m_{\ell^{\prime}}}T\ket{P,\mathbf{k},\ell
m_{\ell}}=\bra{P,\mathbf{p},\ell^{\prime}m_{\ell^{\prime}}}T^{{\dagger}}T\ket{P,\mathbf{k},\ell
m_{\ell}}\,,$ (69)

where Hermitian analyticity is assumed to write the left hand side as the
imaginary part of the amplitude Olive (1962); Eden  _et al._ (1966). We insert
the identity between $T^{{\dagger}}T$ in the form of the completeness relation
for identical three particle states,

$\displaystyle\mathbbm{1}=\frac{1}{3!}\int_{k_{1}}\int_{k_{2}}\int_{k_{3}}\,\ket{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3}}\\!\bra{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3}}\,,$
(70)

where the division by $3!$ is the symmetry factor since the particles are
identical. No other multiparticle states contribute in the completeness
relation since energies in the elastic three particle scattering region are
considered only.

In terms of the pair-spectator basis (11), the identity splits into two
distinct topologies: one where the spectator is remains the same in the
intermediate state, and one where it exchanges

$\displaystyle\mathbbm{1}$
$\displaystyle=\frac{1}{2!}\sum_{\ell,m_{\ell}}\int_{k}\int_{0}^{\infty}\\!\textrm{d}p_{k}^{\star}\,\frac{p_{k}^{\star\,2}}{(2\pi)^{2}\,\omega_{p_{k}}^{\star}}\int_{P}\,\ket{P,\mathbf{k},\ell
m_{\ell}}\\!\bra{P,\mathbf{k},\ell m_{\ell}}$
$\displaystyle\qquad+\sum_{\ell^{\prime},m_{\ell^{\prime}}}\sum_{\ell,m_{\ell}}\int_{k}\int_{p}\int_{P}\,4\pi
Y_{\ell^{\prime}m_{\ell^{\prime}}}(\mathbf{\hat{\mathbf{k}}}_{p}^{\star})Y_{\ell
m_{\ell}}^{*}(\mathbf{\hat{\mathbf{p}}}_{k}^{\star})\,\ket{P,\mathbf{p},\ell^{\prime}m_{\ell^{\prime}}}\\!\bra{P,\mathbf{k},\ell
m_{\ell}}\,.$ (71)

The first term arises when the spectators in both states are equal, giving an
overall factor of 3 since the particles are identical and leaving a factor
$1/2!$ which is the symmetry factor for the two particles composing the pair.
Additionally, the measure is transformed into the pair CM frame, i.e. the
frame where $\mathbf{P}^{\star}-\mathbf{k}^{\star}=\mathbf{0}$, which
integrates away the angular contribution via the spherical harmonic
orthonormality relation leaving the magnitude of the relative momentum of the
pair $p^{\star}$. The shorthand for integral labeled by $P$ follows the
definition

$\displaystyle\int_{P}\equiv\int\\!\frac{\textrm{d}^{3}\mathbf{P}}{(2\pi)^{3}\,2\omega_{Pkp}}\,,$
(72)

with
$\omega_{Pkp}=\sqrt{m^{2}+\lvert\mathbf{P}-\mathbf{k}-\mathbf{p}\rvert^{2}}$.
Note for the first term of Eq. (A), we have boosted to the
$\mathbf{P}^{\star}-\mathbf{k}^{\star}=\mathbf{0}$ frame, so the energy factor
in Eq. (72) is
$\omega_{Pkp}=\sqrt{m^{2}+\lvert\mathbf{p}_{k}^{\star}\rvert^{2}}$. In the
second term, the spectators $\mathbf{p}$ and $\mathbf{k}$ are not the same,
which leaves 6 identical terms that cancels the symmetry factor.

Inserting Eq. (A) into (69), along with the definition of the asymmetric
amplitude Eq. (12), we find the unitarity relation

$\displaystyle
2\operatorname{Im}\mathcal{T}_{\ell^{\prime}m_{\ell^{\prime}},\ell
m_{\ell}}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\frac{1}{2!}\sum_{\ell^{\prime\prime},m_{\ell^{\prime\prime}}}\int_{k^{\prime}}\int_{0}^{\infty}\\!\textrm{d}p_{k^{\prime}}^{\prime\star}\,\frac{p_{k^{\prime}}^{\prime\star\,2}}{(2\pi)^{2}\,\omega_{p_{k^{\prime}}^{\prime}}^{\star}}\int_{P^{\prime}}\,(2\pi)^{4}\delta^{(4)}(P^{\prime}-P)$
$\displaystyle\qquad\times\mathcal{T}_{\ell^{\prime\prime}m_{\ell^{\prime\prime}},\ell^{\prime}m_{\ell^{\prime}}}^{(u,u)\,*}(\mathbf{p},\mathbf{k}^{\prime})\mathcal{T}_{\ell^{\prime\prime}m_{\ell^{\prime\prime}},\ell
m_{\ell}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\sum_{\ell^{\prime\prime\prime},m_{\ell^{\prime\prime\prime}}}\sum_{\ell^{\prime\prime},m_{\ell^{\prime\prime}}}\int_{k^{\prime}}\int_{p^{\prime}}\int_{P^{\prime}}\,4\pi
Y_{\ell^{\prime\prime\prime}m_{\ell^{\prime\prime\prime}}}(\mathbf{\hat{\mathbf{k}}}_{p^{\prime}}^{\prime\star})Y_{\ell^{\prime\prime}m_{\ell^{\prime\prime}}}^{*}(\mathbf{\hat{\mathbf{p}}}_{k^{\prime}}^{\prime\star})\,(2\pi)^{4}\delta^{(4)}(P^{\prime}-P)$
$\displaystyle\qquad\times\mathcal{T}_{\ell^{\prime\prime\prime}m_{\ell^{\prime\prime\prime}},\ell^{\prime}m_{\ell^{\prime}}}^{(u,u)\,*}(\mathbf{p},\mathbf{p}^{\prime})\mathcal{T}_{\ell^{\prime\prime}m_{\ell^{\prime\prime}},\ell
m_{\ell}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$ (73)

The integration over $\mathbf{P}^{\prime}$ in both terms is trivially
evaluated using $(2\pi)^{3}\,\delta^{(3)}(\mathbf{P}^{\prime}-\mathbf{P})$,
leaving a single delta function over the energy. Since the spectator energy is
conserved in the first term, the remaining delta function has an argument
$E^{\prime}-E=(E^{\prime}-\omega_{k^{\prime}})-(E-\omega_{k^{\prime}})$, and
therefore the integral over $p_{k^{\prime}}^{\prime\star}$ evaluates to
$2\rho(\sigma_{k^{\prime}})$ given in Eq. (14) since the amplitudes are
independent of $p_{k^{\prime}}^{\prime\star}$. The energy delta function in
the second term of Eq. (A) is most readily evaluated by using the identity

$\displaystyle\frac{1}{2\omega_{Pk^{\prime}p^{\prime}}}\delta(E-\omega_{p^{\prime}}-\omega_{k^{\prime}}-\omega_{Pk^{\prime}p^{\prime}})=\delta((P-k^{\prime}-p^{\prime})^{2}-m^{2})\Theta(E-\omega_{k^{\prime}}-\omega_{p^{\prime}})\,,$

which reduces to $2\Delta(\mathbf{p},\mathbf{k})$ since the Heaviside function
enforces that the intermediate state is a physical process, which is
equivalent to the definition given in Eq. (III.1). Given these evaluations,
Eq. (A) reduces to (III.1), as claimed in Sec. III.

##  Appendix B Proof of unitarity for scattering equations

We show how the various ansatz for the $\mathbf{3}\to\mathbf{3}$ scattering
equations satisfy the unitarity relations. The approach presented here is more
efficient than those presented in Refs. Jackura  _et al._ (2019a); Briceño
_et al._ (2019c). It is instructive to illustrate the method by demonstrating
that the simpler $\mathbf{2}\to\mathbf{2}$ on-shell representation satisfies
Eq. (16). First, we take the Hermitian conjugate of Eq. (19), and use
$\mathcal{K}_{2}^{{\dagger}}=\mathcal{K}_{2}$ in the physical region to find

$\displaystyle\mathcal{K}_{2}=\mathcal{M}_{2}^{{\dagger}}+\mathcal{M}_{2}^{{\dagger}}\,\mathcal{I}^{{\dagger}}\,\mathcal{K}_{2}\,.$
(74)

The, we substitute Eq. (74) into the second term of Eq. (19) to find

$\displaystyle\mathcal{M}_{2}=\mathcal{K}_{2}-\mathcal{M}_{2}^{{\dagger}}\,\mathcal{I}\,\mathcal{M}_{2}-\mathcal{M}^{{\dagger}}\,\mathcal{I}^{{\dagger}}\,\mathcal{K}_{2}\,\mathcal{I}\,\mathcal{M}_{2}\,.$
(75)

Finally, we take the Hermitian conjugate of Eq. (75), and subtract this from
(75) to find

$\displaystyle\mathcal{M}_{2}-\mathcal{M}_{2}^{{\dagger}}=-\mathcal{M}_{2}^{{\dagger}}\,\left[\,\mathcal{I}-\mathcal{I}^{{\dagger}}\,\right]\,\mathcal{M}_{2}\,,$
(76)

which with
$\mathcal{I}-\mathcal{I}^{{\dagger}}=2i\,\operatorname{Im}\mathcal{I}=-2i\,\rho$
and
$\mathcal{M}_{2}-\mathcal{M}_{2}^{{\dagger}}=2i\,\operatorname{Im}\mathcal{M}_{2}$
since $\mathcal{M}_{2}^{\top}=\mathcal{M}_{2}$ by time-reversal invariance,
recovers Eq. (16) and completes the demonstration.

We follow the same strategy for the $\mathbf{3}\to\mathbf{3}$ amplitude,
applying it to the integral equation (2) for the full $T$ matrix element,
given here for convenience

$\displaystyle\mathcal{T}^{(u,u)}(\mathbf{p},\mathbf{k})=\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{T}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$

Note that while there is a cluster decomposition in $\mathcal{T}^{(u,u)}$ and
$\mathcal{K}_{3}^{(u,u)}$, Eqs. (3) and (5) respectively, it does not spoil
the proof. Taking now the Hermitian conjugate and using the fact that the $K$
matrices are real symmetric matrices, we find

$\displaystyle\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})=\mathcal{T}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{k})+\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{T}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma^{{\dagger}}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{K}_{3}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$
(77)

This equation is substituted into the second term of Eq. (2), giving

$\displaystyle\mathcal{T}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{T}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{T}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$
$\displaystyle-\int_{p^{\prime}}\int_{q^{\prime}}\int_{q}\int_{k^{\prime}}\mathcal{T}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma^{{\dagger}}(\mathbf{p}^{\prime},\mathbf{q}^{\prime})\,\mathcal{K}_{3}^{(u,u)}(\mathbf{q}^{\prime},\mathbf{q})\,\Gamma(\mathbf{q},\mathbf{k}^{\prime})\,\mathcal{T}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,.$
(78)

We can now take the difference between $\mathcal{T}^{(u,u)}$ and its Hermitian
conjugate to arrive at

$\displaystyle\mathcal{T}^{(u,u)}(\mathbf{p},\mathbf{k})-\mathcal{T}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{k})=-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{T}^{(u,u)\,{\dagger}}(\mathbf{p},\mathbf{p}^{\prime})\,\left[\,\Gamma(\mathbf{p}^{\prime},\mathbf{k}^{\prime})-\Gamma^{{\dagger}}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\right]\,\mathcal{T}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$

which is the unitarity relation (III.1) since
$\mathcal{T}^{(u,u)}-\mathcal{T}^{(u,u)\,{\dagger}}=2i\,\operatorname{Im}\mathcal{T}$
and $\Gamma-\Gamma^{{\dagger}}=2i\,\operatorname{Im}\Gamma=-2i\Phi$. We
conclude that the linear integral equation of the form (2) is a valid on-shel
representation. This method works similarly to the other forms of the integral
equations, such as for $\mathcal{T}_{\textrm{df}}^{(u,u)}$ in Eq. (38).

##  Appendix C Relating $K$ matrices

In this appendix we outline the procedure for relating two different $K$
matrices defined by their respective kinematic functions. We work with the
integral equation (2) for $\mathcal{T}^{(u,u)}$, with one scheme defined by

$\displaystyle\mathcal{T}^{(u,u)}(\mathbf{p},\mathbf{k})=\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{T}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(79)

and the other defined with

$\displaystyle\mathcal{T}^{(u,u)}(\mathbf{p},\mathbf{k})=\mathcal{K}_{3}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{T}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma\,^{\prime}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{K}_{3}^{(u,u)\,\prime}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(80)

where $\Gamma^{\prime}=\Gamma+\delta\Gamma^{\prime}$ with
$\delta\Gamma\,^{\prime}$ being the difference in the schemes, and we have
reordered the second equation which can be seen by expanding the Fredholm
series and resumming on the final state. Note that both the disconnected and
connected $K$ matrices may be affected by the scheme, i.e.
$\mathcal{K}_{3}^{(u,u)\,\prime}=\delta(\mathbf{p},\mathbf{k})\,\mathcal{K}_{2}\,^{\prime}+\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}$.
We now form two new equations, one by substituting Eq. (80) into the second
term of (79),

$\displaystyle\mathcal{T}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{K}_{3}^{(u,u)\,\prime}(\mathbf{k}^{\prime},\mathbf{k})\,$
$\displaystyle+\int_{p^{\prime}}\int_{q^{\prime}}\int_{q}\int_{k^{\prime}}\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{q}^{\prime})\mathcal{T}^{(u,u)}(\mathbf{q}^{\prime},\mathbf{q})\,\Gamma\,^{\prime}(\mathbf{q},\mathbf{k}^{\prime})\,\mathcal{K}_{3}^{(u,u)\,\prime}(\mathbf{k}^{\prime},\mathbf{k})$
(81)

and the second by substituting (79) into the second term of (80),

$\displaystyle\mathcal{T}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\mathcal{K}_{3}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma\,^{\prime}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{K}_{3}^{(u,u)\,\prime}(\mathbf{k}^{\prime},\mathbf{k})\,$
$\displaystyle+\int_{p^{\prime}}\int_{q^{\prime}}\int_{q}\int_{k^{\prime}}\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\Gamma(\mathbf{p}^{\prime},\mathbf{q}^{\prime})\mathcal{T}^{(u,u)}(\mathbf{q}^{\prime},\mathbf{q})\,\Gamma\,^{\prime}(\mathbf{q},\mathbf{k}^{\prime})\,\mathcal{K}_{3}^{(u,u)\,\prime}(\mathbf{k}^{\prime},\mathbf{k})\,.$
(82)

Notice that the last term in Eqs. (C) and (C) are identical. We can now
subtract Eq. (C) from (C), giving the relation

$\displaystyle\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{k})=\mathcal{K}_{3}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})-\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\delta\Gamma\,^{\prime}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{K}_{3}^{(u,u)\,\prime}(\mathbf{k}^{\prime},\mathbf{k})\,,$
(83)

where we have used $\Gamma\,^{\prime}=\Gamma+\delta\Gamma\,^{\prime}$. Given
two different schemes, the master equation (83) relates the resulting $K$
matrices.

As an example, let’s consider
$\delta\Gamma\,^{\prime}=-\mathcal{G}_{\textrm{pv}}$ as in Sec. V.2. In this
case, $\mathcal{K}_{2}$ will be unchanged, and the only difference is
$\mathcal{K}_{3,\textrm{df}}^{(u,u)}$. We can manipulate the master equation
(83) into a form for just $\mathcal{K}_{3,\textrm{df}}^{(u,u)}$. Using
$\mathcal{K}_{3}^{(u,u)\,\prime}=\delta(\mathbf{p},\mathbf{k})\,\mathcal{K}_{2}+\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}$,
we find

$\displaystyle\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p},\mathbf{k})+\mathcal{K}_{2}(\sigma_{p})\,\mathcal{G}_{\textrm{pv}}(\mathbf{p},\mathbf{k})\,\mathcal{K}_{2}(\sigma_{k})$
$\displaystyle+\int_{k^{\prime}}\mathcal{K}_{2}(\sigma_{p})\,\mathcal{G}_{\textrm{pv}}(\mathbf{p},\mathbf{k}^{\prime})\,\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{k}^{\prime},\mathbf{k})+\int_{p^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{G}_{\textrm{pv}}(\mathbf{p}^{\prime},\mathbf{k})\,\mathcal{K}_{2}(\sigma_{k})$
$\displaystyle+\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{G}_{\textrm{pv}}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{k}^{\prime},\mathbf{k})\,.$

This relation can be reorganized into a more compact form as

$\displaystyle\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{p},\mathbf{k})$
$\displaystyle=\int_{p^{\prime}}\int_{k^{\prime}}\mathcal{C}_{L}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p}^{\prime},\mathbf{k}^{\prime})\,\mathcal{C}_{R}(\mathbf{k}^{\prime},\mathbf{k})$
$\displaystyle+\int_{p^{\prime}}\int_{q^{\prime}}\int_{k^{\prime}}\mathcal{C}_{L}(\mathbf{p},\mathbf{p}^{\prime})\,\mathcal{K}_{3,\textrm{df}}^{(u,u)\,\prime}(\mathbf{p}^{\prime},\mathbf{q}^{\prime})\mathcal{G}_{\textrm{pv}}(\mathbf{q}^{\prime},\mathbf{k}^{\prime})\,\mathcal{K}_{3,\textrm{df}}^{(u,u)}(\mathbf{k}^{\prime},\mathbf{k})\,,$

where

$\displaystyle\mathcal{C}_{L}(\mathbf{p},\mathbf{k})$
$\displaystyle=\delta(\mathbf{p},\mathbf{k})+\mathcal{K}_{2}(\sigma_{p})\,\mathcal{G}_{\textrm{pv}}(\mathbf{p},\mathbf{k})\,,$
$\displaystyle\mathcal{C}_{R}(\mathbf{p},\mathbf{k})$
$\displaystyle=\delta(\mathbf{p},\mathbf{k})+\mathcal{G}_{\textrm{pv}}(\mathbf{p},\mathbf{k})\,\mathcal{K}_{2}(\sigma_{k})\,.$

which is the result presented in Eq. (V.2) in Sec. V.2.

##  Appendix D Symmetrization identities

This appendix serves to summarize important properties of the asymmetric
amplitudes, which have been detailed in Refs. Hansen and Sharpe (2014, 2015);
Blanton and Sharpe (2020a). For ease of demonstration, let us consider a
generic function $X(\mathbf{k},\mathbf{a})$ which suppresses the momenta of
the final state, and $\mathbf{k}$, $\mathbf{a}$, and
$\mathbf{b}=\mathbf{P}-\mathbf{k}-\mathbf{a}$ are the momenta of the initial
three particle state. From the symmetrization definition Eq. (12), the
asymmetric amplitude is defined as

$\displaystyle X(\mathbf{k},\mathbf{a})$
$\displaystyle\equiv\sum_{\mathbf{k}\in\\{\mathbf{k},\mathbf{a},\mathbf{b}\\}}X^{(u)}(\mathbf{k},\mathbf{\hat{\mathbf{a}}}_{k}^{\star})\,,$
$\displaystyle=X^{(u)}(\mathbf{k},\mathbf{\hat{\mathbf{a}}}_{k}^{\star})+X^{(u)}(\mathbf{a},\mathbf{\hat{\mathbf{k}}}_{a}^{\star})+X^{(u)}(\mathbf{b},\mathbf{\hat{\mathbf{k}}}_{b}^{\star})\,,$
(84)

The $(u)$ superscript indicates that the amplitude is asymmetric since the
spectator momentum is distinguished. In addition to this definition, Ref.
Hansen and Sharpe (2014) introduced an additional notion where the spectator
is labeled with alternative indices $s$ and $\widetilde{s}$, i.e.

$\displaystyle X^{(u)}(\mathbf{a},\mathbf{\hat{\mathbf{k}}}_{a}^{\star})$
$\displaystyle=X^{(s)}(\mathbf{k},\mathbf{\hat{\mathbf{a}}}_{k}^{\star})\,,$
(85a) $\displaystyle
X^{(u)}(\mathbf{b},\mathbf{\hat{\mathbf{k}}}_{b}^{\star})$
$\displaystyle=X^{(\widetilde{s})}(\mathbf{k},\mathbf{\hat{\mathbf{a}}}_{k}^{\star})\,.$
(85b)

Given these relations, Eq. (D) can also take the form

$\displaystyle
X(\mathbf{k},\mathbf{a})=X^{(u)}(\mathbf{k},\mathbf{\hat{a}}_{k}^{\star})+X^{(s)}(\mathbf{k},\mathbf{\hat{a}}_{k}^{\star})+X^{(\widetilde{s})}(\mathbf{k},\mathbf{\hat{a}}_{k}^{\star})\,.$
(86)

Each of these individual amplitudes have an expansion in terms of the pair
angular momentum, which can be expressed in terms of any of the $u$, $s$, or
$\widetilde{s}$ amplitudes, e.g.

$\displaystyle X^{(u)}(\mathbf{k},\mathbf{\hat{\mathbf{a}}}_{k}^{\star})$
$\displaystyle=\sqrt{4\pi}\,\sum_{\ell,m_{\ell}}X_{\ell
m_{\ell}}^{(u)}(\mathbf{k})\,Y_{\ell
m_{\ell}}^{*}(\mathbf{\hat{\mathbf{a}}}_{k}^{\star})\,,$
$\displaystyle=\sqrt{4\pi}\,\sum_{\ell,m_{\ell}}X_{\ell
m_{\ell}}^{(s)}(\mathbf{a})\,Y_{\ell
m_{\ell}}^{*}(\mathbf{\hat{\mathbf{k}}}_{a}^{\star})\,.$ (87)

In the second line, we used the relation Eq. (85a) to express the expansion in
a different pair angular momentum. Since interchange of the particles in the
pair affect the orientation by flipping the direction, e.g.
$\mathbf{\hat{a}}_{k}^{\star}\to-\mathbf{\hat{a}}_{k}^{\star}$ under
interchange, the resulting partial wave amplitudes exhibit the symmetry
condition

$\displaystyle X_{\ell
m_{\ell}}^{(\widetilde{s})}(\mathbf{k})=(-1)^{\ell}X_{\ell
m_{\ell}}^{(s)}(\mathbf{k})\,.$ (88)

For $X_{\ell m_{\ell}}^{(u)}$ representing contributions from short-distance
physics, e.g. the $\mathcal{K}_{3,\textrm{df}}^{(u,u)}$ matrix, it is
convenient to define a _symmetric_ function in the $k\ell m_{\ell}$ basis

$\displaystyle X_{\ell m_{\ell}}(\mathbf{k})\equiv X_{\ell
m_{\ell}}^{(u)}(\mathbf{k})+X_{\ell m_{\ell}}^{(s)}(\mathbf{k})+X_{\ell
m_{\ell}}^{(\widetilde{s})}(\mathbf{k})\,.$ (89)

The last two terms of Eq. (89) can be simplified by using Eq. (88) as

$\displaystyle X_{\ell m_{\ell}}^{(s)}(\mathbf{k})+X_{\ell
m_{\ell}}^{(\widetilde{s})}(\mathbf{k})=\begin{cases}\,\,2X_{\ell
m_{\ell}}^{(s)}(\mathbf{k})&\ell\mathrm{\,\,even}\\\
\,\,0&\mathrm{otherwise}\,.\end{cases}$ (90)

Introducing the symmetric version of $X_{\ell m_{\ell}}$ allows us too
manipulate quantities which sum over all spectators in the following manner,

$\displaystyle\sqrt{4\pi}\sum_{\mathbf{k}\in\mathcal{P}_{k}}\sum_{\ell,m_{\ell}}X_{\ell
m_{\ell}}^{(u)}(\mathbf{k})Y_{\ell
m_{\ell}}^{*}(\mathbf{\hat{\mathbf{a}}}_{k}^{\star})$
$\displaystyle=X^{(u)}(\mathbf{k},\mathbf{\hat{a}}_{k}^{\star})+X^{(s)}(\mathbf{k},\mathbf{\hat{a}}_{k}^{\star})+X^{(\widetilde{s})}(\mathbf{k},\mathbf{\hat{a}}_{k}^{\star})\,,$
$\displaystyle=\sqrt{4\pi}\sum_{\ell,m_{\ell}}X_{\ell
m_{\ell}}(\mathbf{k})Y_{\ell
m_{\ell}}^{*}(\mathbf{\hat{\mathbf{a}}}_{k}^{\star})\,,$
$\displaystyle=\sqrt{4\pi}\sum_{\mathbf{k}\in\mathcal{P}_{k}}\sum_{\ell,m_{\ell}}\frac{1}{3}X_{\ell
m_{\ell}}(\mathbf{k})Y_{\ell
m_{\ell}}^{*}(\mathbf{\hat{\mathbf{a}}}_{k}^{\star})\,.$ (91)

In the first line, we identified the sum over all partial wave expanded
asymmetric functions with Eq. (86), while in the second line we partial wave
expand each kernel in terms of spherical harmonics of
$\mathbf{\hat{\mathbf{a}}}_{k}^{\star}$, and equate the sum of all projected
functions with Eq. (89). The last line follows from the second, using the
redundancy of the symmetric $X_{\ell m_{\ell}}$, where the factor of $1/3$
accounts for the over counting from summing over all possible spectators.
Therefore, whenever the asymmetric function is contracted with the appropriate
spherical harmonics and all spectators are summed over, then we may replace
the asymmetric kernel with the symmetric function via the interchange

$\displaystyle X_{\ell
m_{\ell}}^{(u)}(\mathbf{k})\longrightarrow\frac{1}{3}X_{\ell
m_{\ell}}(\mathbf{k})\,.$ (92)

A final identity concerning the asymmetric functions concerns with trading a
$(u)$ function convoluted with an exchange operator with and $(s)$ function
conjuncted with a direct channel phase space factor. The identity is given by

$\displaystyle\int_{p}\pi\delta$
$\displaystyle\left((P_{k}-p)^{2}-m^{2}\right)H(\sigma_{p})H(\sigma_{k})\sum_{\ell,m_{\ell}}X_{\ell
m_{\ell}}^{(u)}(\mathbf{p})\,4\pi Y_{\ell
m_{\ell}}^{*}(\mathbf{\hat{\mathbf{k}}}_{p}^{\star})Y_{\ell^{\prime}m_{\ell^{\prime}}}(\mathbf{\hat{\mathbf{p}}}_{k}^{\star})$
$\displaystyle=\sum_{\ell,m_{\ell}}X_{\ell
m_{\ell}}^{(s)}(\mathbf{k})H(\sigma_{k})\int_{p}\pi\delta\left((P_{k}-p)^{2}-m^{2}\right)H(\sigma_{p})\,4\pi
Y_{\ell
m_{\ell}}^{*}(\mathbf{\hat{\mathbf{p}}}_{k}^{\star})Y_{\ell^{\prime}m_{\ell^{\prime}}}(\mathbf{\hat{\mathbf{p}}}_{k}^{\star})\,,$
$\displaystyle=2iX^{(s)}(\mathbf{k})\,\widetilde{\rho}(\sigma_{k})\,.$ (93)

where $\widetilde{\rho}(\sigma_{k})=H(\sigma_{k})\,\mathcal{I}(\sigma_{k})$
with $\mathcal{I}$ being defined in Eq. (20). In going from the first to
second equality, the symmetry relation Eq. (85a) is used to convert the $X$
function from a spectator in $p$ to $k$. The last line is found by evaluating
the integrals in the CM frame of the system of the pair associated with
spectator $k$, and setting $H(\sigma_{p})=1$ since the integration of the
delta function picks out the physical point.

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