aainstitutetext: Department of Physics, Chongqing University,
Chongqing 401331, People’s Republic of Chinabbinstitutetext: Chongqing Key
Laboratory for Strongly Coupled Physics, Chongqing University,
Chongqing 401331, People’s Republic of China

# Photoproduction of $P$-wave doubly charmed baryon at future $e^{+}e^{-}$
collider

Xi-Jie Zhan,111Corresponding author. a,b Xing-Gang Wu a,b Xu-Chang Zheng
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The photoproduction of $P$-wave doubly charmed baryon ($\Xi_{cc}$) is
investigated in the context of future high-energy and high-luminosity
$e^{+}e^{-}$ colliders. The direct photoproduction via the sub-process
$\gamma+\gamma\rightarrow\Xi_{cc}+\bar{c}+\bar{c}$ and the resolved channel
$\gamma+g\rightarrow\Xi_{cc}+\bar{c}+\bar{c}$ are considered. Within the
framework of non-relativistic QCD, the calculation encompasses four $P$-wave
$(cc)$-diquark configurations: $(cc)_{\bar{\textbf{3}}}[{}^{1}P_{1}]$,
$(cc)_{\textbf{6}}[{}^{3}P_{0}]$, $(cc)_{\textbf{6}}[{}^{3}P_{1}]$ and
$(cc)_{\textbf{6}}[{}^{3}P_{2}]$. The two $S$-wave states,
$(cc)_{\bar{\textbf{3}}}[{}^{3}S_{1}]$ and $(cc)_{\textbf{6}}[{}^{1}S_{0}]$,
are also included for comparison. The cross sections, as well as the
differential distributions involving transverse momentum, rapidity, and
angular variables, have been computed. Numerical results reveal that the
resolved photoproduction process plays a significant role and can provide
dominant contributions. The photoproduction rate of the $P$-wave $\Xi_{cc}$ is
approximately one order of magnitude lower than that of the $S$-wave.

##  1 Introduction

A doubly heavy baryon is a system composed of two heavy quarks and one light
quark. It possesses a simple structure akin to that of a heavy quarkonium,
which enables rigorous theoretical analysis. As a result, investigating its
production properties is believed to aid in understanding and validating the
theory of Quantum Chromodynamics (QCD). In 2002 and 2005, the SELEX
Collaboration reported a possible discovery of the $\Xi_{cc}^{+}$
baryonMattson:2002vu ; Ocherashvili:2004hi . More recently, in 2017, the LHCb
Collaboration confirmed the existence of another type of doubly heavy baryon,
$\Xi_{cc}^{++}$, through the decay channel
$\Xi_{cc}^{++}\rightarrow\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+}$
($\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+}$). The measured properties of mass
and lifetime for this $\Xi_{cc}^{++}$ baryon were found to be in excellent
agreement with theoretical calculations. Subsequently, the LHCb Collaboration
further confirmed its existence in the decay
$\Xi_{cc}^{++}\rightarrow\pi^{+}\Xi_{c}^{+}$ Aaij:2018gfl ; Aaij:2018wzf . The
genuine discovery of doubly heavy baryons is poised to trigger a new wave of
theoretical research.

Due to the inherent nonrelativistic characteristics coupled with the
confinements attributed to strong interactions, the production of doubly heavy
baryons encompasses intricate nonperturbative effects that elude computation
through conventional perturbative QCD methods. The study conducted by Ma et
al. Ma:2003zk  undertook the task of describing this production process
through the utilization of the nonrelativistic QCD (NRQCD) factorization
framework Bodwin:1994jh . This ingenious framework dissects the procedure into
two sequential stages: firstly, the perturbative formation of a heavy-quark
pair within a distinct quantum state, often referred to as a diquark, followed
by its subsequent nonperturbative transformation into a baryon. By employing
an expansion rooted in the heavy quark’s diminished velocity ($v_{Q}$) within
the rest frame of the baryon, the study was able to pinpoint two foremost
states of $(cc)$-diquarks at the leading order:
${}_{\bar{\textbf{3}}}[{}^{3}S_{1}]$ and ${}_{\textbf{6}}[{}^{1}S_{0}]$. These
diquark states respectively correspond to the ${}^{3}S_{1}$ and ${}^{1}S_{0}$
S-wave configurations, while existing in the $\overline{\textbf{3}}$ and 6
color states. Accompanying these states are the associated long-distance
matrix elements (LDMEs), denoted as $h_{\bar{\textbf{3}}}$ and
$h_{\textbf{6}}$, which encapsulate the nonperturbative likelihood of their
transition into the baryonic state.

Numerous comprehensive theoretical investigations have delved into the realm
of producing doubly heavy baryons Baranov:1995rc ; Berezhnoy:1996an ;
Jiang:2012jt ; Jiang:2013ej ; Chen:2014frw ; Yang:2014ita ; Yang:2014tca ;
Zheng:2015ixa ; Bi:2017nzv ; Sun:2020mvl ; Chen:2014hqa ; Chen:2019ykv ;
Chen:2018koh ; Martynenko:2014ola ; Koshkarev:2016acq ; Koshkarev:2016rci ;
Groote:2017szb ; Berezhnoy:2018bde ; Brodsky:2017ntu ; Berezhnoy:2018krl ;
Wu:2019gta ; Qin:2020zlg ; Niu:2018ycb ; Niu:2019xuq ; Zhang:2022jst ;
Luo:2022jxq ; Luo:2022lcj ; Ma:2022cgt . These investigations have encompassed
a range of production mechanisms, including direct processes occurring in
$pp$, $ep$, $\gamma\gamma$, and $e^{+}e^{-}$ collisions, as well as indirect
channels involving the decays of Higgs bosons, $W$ and $Z$ bosons, and top
quarks. For the purpose of simulating hadroproduction in $pp$ collisions, a
dedicated generator known as GENXICC Chang:2007pp ; Chang:2009va ; Wang:2012vj
has been meticulously developed.

Next-generation $e^{+}e^{-}$ colliders have been put forth in recent times,
among them being the FCC-ee FCC:2018evy , the CEPC CEPCStudyGroup:2018rmc ;
CEPCStudyGroup:2018ghi , and the ILC ILC:2007bjz ; Erler:2000jg . These
cutting-edge colliders are designed to operate at high collision energies and
are projected to achieve unparalleled luminosities. As a result of their
capabilities, these advanced $e^{+}e^{-}$ colliders hold immense potential to
serve as exceptional platforms for a wide array of research topics. The
$e^{+}e^{-}$ collider provides two main pathways for the direct production of
the doubly heavy baryon $\Xi_{cc}$: production through $e^{+}e^{-}$
annihilation and via the photoproduction mechanism. In this work, $\Xi_{cc}$
denotes the baryon $\Xi_{ccq}$, where $q$ corresponds to an up ($u$), down
($d$), or strange ($s$) quark. Regarding photoproduction, the $\Xi_{cc}$
baryon can be generated through direct photon-photon fusion, such as
$\gamma+\gamma\rightarrow\Xi_{cc}+\bar{c}+\bar{c}$. Beyond direct
photoproduction, another category of processes, known as resolved
photoproduction Klasen:2001cu , exists. In these cases, the photon undergoes a
process of resolution, leading to its parton’s involvement in the subsequent
hard processes. The resolved photoproduction channels share a comparable order
of perturbative expansion with the direct approach, underscoring the need for
their incorporation in calculations. Earlier investigations Klasen:2001cu ;
Li:2009zzu ; Zhan:2020ugq ; Zhan:2021dlu ; Zhan:2022nck ; Zhan:2022etq  have
indicated that the single resolved channel($\gamma+g$) tend to exert a
dominant influence on the photoproduction of heavy quarkonium and doubly heavy
baryon at $e^{+}e^{-}$ colliders. The contributions stemming from double
resolved photoproduction channels are generally negligible and can be safely
ignored.

In this work, we offer an analysis of the photoproduction of $P$-wave doubly
charmed baryon at future $e^{+}e^{-}$ collider. Based on the NRQCD
factorization framework, we will consider two types of photoproduction
processes: $\gamma+\gamma\rightarrow\Xi_{cc}+\bar{c}+\bar{c}$ and
$\gamma+g\rightarrow\Xi_{cc}+\bar{c}+\bar{c}$. For $(cc)$-diquark, the quantum
number is $(cc)_{\bar{\textbf{3}}}[{}^{3}S_{1}]$,
$(cc)_{\textbf{6}}[{}^{1}S_{0}]$, $(cc)_{\bar{\textbf{3}}}[{}^{1}P_{1}]$ or
$(cc)_{\textbf{6}}[{}^{3}P_{J}]$ with $J=0,1,2$. Section 2 provides the
formulation of the calculation, while Section 3 presents the numerical results
and subsequent discussions. Section 4 gives a brief summary.

##  2 Formulation and calculation

At future $e^{+}e^{-}$ colliders, the initial photons possess the capacity to
attain both high energy and luminosity via laser back-scattering(LBS). This
phenomenon is expounded upon in the spectrum detailed by Ginzburg et al.
Ginzburg:1981vm ,

$f_{\gamma/e}(x)=\frac{1}{N}\left[1-x+\frac{1}{1-x}-4r(1-r)\right],$ (1)

where $x=E_{\gamma}/E_{e}$, $r=x/(x_{m}(1-x))$, and the normalization factor
is given by:

$N=(1-\frac{4}{x_{m}}-\frac{8}{x_{m}^{2}})\log(1+x_{m})+\frac{1}{2}+\frac{8}{x_{m}}-\frac{1}{2(1+x_{m})^{2}}.$
(2)

Here, $x_{m}=4E_{e}E_{l}\cos^{2}\frac{\theta}{2}$, with $E_{e}$ and $E_{l}$
representing the energies of the incident electron and laser beams,
respectively, and $\theta$ denoting the angle between them. The range of
energy for the laser back-scattering (LBS) photon is constrained by:

$0\leq x\leq\frac{x_{m}}{1+x_{m}},$ (3)

with the optimal value of $x_{m}$ being $4.83$ Telnov:1989sd .

Within the framework of non-relativistic QCD (NRQCD), the photoproduction
cross-section of $\Xi_{cc}$ at the $e^{+}e^{-}$ collider can be represented as
follows:

$\displaystyle\mathrm{d}\sigma(e^{+}e^{-}\rightarrow
e^{+}e^{-}\Xi_{c}+\bar{c}+\bar{c})$
$\displaystyle=\int\mathrm{d}x_{1}f_{\gamma/e}(x_{1})\int\mathrm{d}x_{2}f_{\gamma/e}(x_{2})$
$\displaystyle\times\sum_{i,j}\int\mathrm{d}x_{i}f_{i/\gamma}(x_{i})\int\mathrm{d}x_{j}f_{j/\gamma}(x_{j})$
$\displaystyle\times\sum_{n}\mathrm{~{}d}\hat{\sigma}(ij\rightarrow(cc)[n]+\bar{c}+\bar{c})\left\langle\mathcal{O}^{\Xi_{cc}}[n]\right\rangle.$
(4)

Here, $f_{\gamma/e}(x)$ represents the energy spectrum of the photon.
$f_{i/\gamma}$ ($i=\gamma,g,u,d,s$) corresponds to the Glück-Reya-Schienbein
(GRS) distribution function of parton $i$ within the photon Gluck:1999ub .
$f_{\gamma/\gamma}(x)=\delta(1-x)$ is utilized for the direct photoproduction
process. $\mathrm{~{}d}\hat{\sigma}(ij\rightarrow(cc)[n]+\bar{c}+\bar{c})$
denotes the differential partonic cross-section, which is evaluated
perturbatively. For the baryons $\Xi_{cc}$,
$(cc)[n]=(cc)_{\bar{\textbf{3}}}[{}^{3}S_{1}]$,
$(cc)_{\textbf{6}}[{}^{1}S_{0}]$, $(cc)_{\bar{\textbf{3}}}[{}^{1}P_{1}]$ or
$(cc)_{\textbf{6}}[{}^{3}P_{J}]$. $\langle{\cal
O}^{\Xi_{cc}}[n]\rangle=h_{\bar{\textbf{3}}(\textbf{6})}$ denotes the long-
distance matrix element (LDME). Typically, people adopt a potential model
approach, drawing parallels with the heavy quarkonium scenario, and introduce
and correlate a wave function with $h_{\bar{\textbf{3}}}$Falk:1993gb ;
Kiselev:1994pu ; Bagan:1994dy ; Baranov:1995rc ; Berezhnoy:1998aa ,

$h_{\bar{\textbf{3}}}\simeq|\Psi_{cc}(0)|^{2}\,\mathrm{or}\,|\Psi^{\prime}_{cc}(0)|^{2}.$
(5)

Regarding $h_{\textbf{6}}$, there exists no specific relation, and for the
sake of simplicity, it is assumed to be equal to $h_{\bar{\textbf{3}}}$. This
assumption is rooted in NRQCD’s power counting relative to $v_{c}$, where both
$h_{\textbf{6}}$ and $h_{\bar{\textbf{3}}}$ are assigned equivalent orders
Ma:2003zk . In accordance with NRQCD, the bound state $\Xi_{cc}$ can be
expanded into a series of Fock states:

$\displaystyle\left|\Xi_{cc}\right\rangle=c_{1}(v)|(cc)q\rangle+c_{2}(v)|(cc)qg\rangle+c_{3}(v)|(cc)qgg\rangle+\cdots.$
(6)

Because a light quark can easily emit gluons, the constituents in Eq. (6)
carry equivalent significance, specifically, $c_{1}\sim c_{2}\sim c_{3}$.
Consider a $cc$ pair in the ${}_{\bar{\textbf{3}}}[{}^{3}S_{1}]$ state; one of
the heavy quarks can emit a gluon without altering the spin of the heavy
quark. Subsequently, this emitted gluon can undergo a splitting process,
leading to the formation of a light quark-antiquark pair $q\bar{q}$. This
allows the heavy $cc$ pair to interact with the light $q$ and form the
composite particle $\Xi_{cc}$. Similarly, for a $cc$ pair in the
${}_{\textbf{6}}[{}^{1}S_{0}]$ state, one of the heavy quarks can emit a gluon
while preserving the spin of the heavy quark. This emitted gluon then
separates into a light $q\bar{q}$ pair, and the light quarks also have the
capability to emit gluons. Consequently, this heavy $cc$ pair can capture a
light quark and a gluon to assemble into $\Xi_{QQ}$. This is the reason for
$h_{\textbf{6}}$ and $h_{\bar{\textbf{3}}}$ holding the same order in $v_{c}$.
For the sake of simplicity, we assume $h_{\textbf{6}}=h_{\bar{\textbf{3}}}$ in
the subsequent calculations. It is worth noting that the long-distance matrix
elements (LDMEs) function as overarching parameters beyond the perturbative
components, indicating that the results can be refined with the acquisition of
new LDMEs.

Figure 1: Some Feynman diagrams of the partonic processes for $\Xi_{cc}$
photoproduction. The diagrams are drawn by JaxoDraw Binosi:2003yf .

For the partonic processes in leading order of $\alpha_{s}$, there are 40
Feynman diagrams for $\gamma+\gamma\rightarrow\Xi_{cc}+\bar{c}+\bar{c}$ and 48
diagrams for $\gamma+g\rightarrow\Xi_{cc}+\bar{c}+\bar{c}$, as exemplified in
Fig. 1. In practical calculations, we apply charge conjugation
$C=-i\gamma^{2}\gamma^{0}$ to reverse the one of the $c\sim\bar{c}$ fermion
chains, such as
$L_{1}=\bar{u}_{s_{1}}(k_{12})\Gamma_{i+1}S_{F}(q_{i},m_{i})\cdots
S_{F}(q_{1},m_{1})\Gamma_{1}v_{s_{2}}(k_{2})$. Here, $\Gamma_{i}$ represents
the interaction vertex, $S_{F}(q_{i},m_{i})$ denotes the fermion propagator,
where $q_{i}$ and $m_{i}$ are the respective momentum and mass parameters. The
subscripts $s_{1}$ and $s_{2}$ are used for spin indices, while the index $i$
enumerates the fermion propagators $(i=0,1,\ldots)$ along this fermion line.
The conversion obeys:

$\displaystyle v_{s_{2}}^{T}(k_{2})C$ $\displaystyle=-\bar{u}_{s_{2}}(k_{2}),$
(7) $\displaystyle C^{-}\bar{u}_{s_{1}}(k_{12})^{T}$
$\displaystyle=v_{s_{1}}(k_{12}),$ $\displaystyle
C^{-}S_{F}^{T}(-q_{i},m_{i})C$ $\displaystyle=S_{F}(q_{i},m_{i}),$
$\displaystyle C^{-}\Gamma_{i}^{T}C$ $\displaystyle=-\Gamma_{i}.$

$L_{1}$ is reversed to:

$\displaystyle L_{1}$
$\displaystyle=L_{1}^{T}=v_{s_{2}}^{T}(k_{2})\Gamma_{1}^{T}F_{F}^{T}(q_{1},m_{1})\cdots
S_{F}^{T}(q_{i},m_{i})\Gamma_{i+1}^{T}\bar{u}_{s_{1}}^{T}(k_{12})$ (8)
$\displaystyle=v_{s_{2}}^{T}(k_{2})CC^{-}\Gamma_{1}^{T}CC^{-}S_{F}^{T}(q_{1},m_{1})CC^{-}\cdots
CC^{-}S_{F}^{T}(q_{i},m_{i})CC^{-}\Gamma_{i+1}^{T}CC^{-}\bar{u}_{s_{1}}^{T}(k_{12})$
$\displaystyle=(-1)^{(n+1)}\bar{u}_{s_{2}}(k_{2})\Gamma_{1}S_{F}(-q_{1},m_{1})\cdots
S_{F}(-q_{i},m_{i})\Gamma_{i+1}v_{s_{1}}(k_{12}),$

here $n$ is the number of vector vertices in the fermion chain. Let us take
the first diagram in Fig. 1 as an example, its amplitudes read:

$\displaystyle M_{1}\sim$
$\displaystyle\frac{1}{(k_{12}+k_{2})^{2}}\bar{u}_{s12}(k_{12})\gamma^{\mu}v_{s2}(k_{2})\bar{u}_{s11}(k_{11})\notin(p_{1})$
(9) $\displaystyle\frac{\not p_{2}-\not k_{12}-\not k_{2}-\not
k_{3}+m_{c}}{(\not p_{2}-\not k_{12}-\not k_{2}-\not
k_{3})^{2}-m^{2}_{c}}\gamma_{\mu}\frac{\not p_{2}-\not k_{3}+m_{c}}{(\not
p_{2}-\not k_{3})^{2}-m^{2}_{c}}\notin(p_{2})v_{s3}(k_{3}),$

Where $k_{11}$ and $k_{12}$ are momenta of the two c quarks. In the diquark
state, their relative momentum $q$ is small and we can set
$k_{11}=\frac{m_{c}}{M_{cc}}k_{1}+q$ and $k_{12}=\frac{m_{c}}{M_{cc}}k_{1}-q$
with $k_{1}$ being the momentum of the diquark. Here the diquark mass
$M_{cc}=2m_{c}$ is adopted in order to ensure the gauge invariance of the
amplitude. After reverse the first fermion line, the amplitude becomes:

$\displaystyle M_{1}\sim$
$\displaystyle-\frac{1}{(k_{12}+k_{2})^{2}}\bar{u}_{s2}(k_{2})\gamma^{\mu}v_{s12}(k_{12})\bar{u}_{s11}(k_{11})\notin(p_{1})$
(10) $\displaystyle\frac{\not p_{2}-\not k_{12}-\not k_{2}-\not
k_{3}+m_{c}}{(\not p_{2}-\not k_{12}-\not k_{2}-\not
k_{3})^{2}-m^{2}_{c}}\gamma_{\mu}\frac{\not p_{2}-\not k_{3}+m_{c}}{(\not
p_{2}-\not k_{3})^{2}-m^{2}_{c}}\notin(p_{2})v_{s3}(k_{3}).$

Now we can replace $v_{s{12}}(k_{12})\bar{u}_{s{11}}(k_{11})$ by the spin
projector for $(c\bar{c})[n]$ to get the amplitude of
$\gamma+\gamma\rightarrow(cc)[n]+\bar{c}+\bar{c}$. The spin projector takes
the form of

$\Pi_{k_{1}}(q)=\frac{-\sqrt{M_{cc}}}{4m^{2}_{c}}(\not
k_{12}-m_{c})\gamma^{5}(\not k_{11}+m_{c}),$ (11)
$\Pi_{k_{1}}^{\beta}(q)=\frac{-\sqrt{M_{cc}}}{4m^{2}_{c}}(\not
k_{12}-m_{c})\gamma^{\beta}(\not k_{11}+m_{c}),$ (12)

for $n={}^{1}S_{0}$ and ${}^{3}S_{1}$ respectively. The amplitudes of the
$P$-wave production can be obtained via the derivation of the $S$-wave
expression, i.e.,

$\displaystyle M_{1}[{}^{1}P_{1}]\sim$
$\displaystyle\varepsilon_{\alpha}^{l}(k_{1})\frac{d}{dq_{\alpha}}[-\frac{1}{(k_{12}+k_{2})^{2}}\bar{u}_{s2}(k_{2})\gamma^{\mu}\frac{-\sqrt{M_{cc}}}{4m^{2}_{c}}(\not
k_{12}-m_{c})\gamma^{5}(\not k_{11}+m_{c})\notin(p_{1})$ (13)
$\displaystyle\frac{\not p_{2}-\not k_{12}-\not k_{2}-\not k_{3}+m_{c}}{(\not
p_{2}-\not k_{12}-\not k_{2}-\not k_{3})^{2}-m^{2}_{c}}\gamma_{\mu}\frac{\not
p_{2}-\not k_{3}+m_{c}}{(\not p_{2}-\not
k_{3})^{2}-m^{2}_{c}}\notin(p_{2})v_{s3}(k_{3})]|_{q=0},$ $\displaystyle
M_{1}[{}^{3}P_{J}]\sim$
$\displaystyle\varepsilon_{\alpha\beta}^{J}(k_{1})\frac{d}{dq_{\alpha}}[-\frac{1}{(k_{12}+k_{2})^{2}}\bar{u}_{s2}(k_{2})\gamma^{\mu}\frac{-\sqrt{M_{cc}}}{4m^{2}_{c}}(\not
k_{12}-m_{c})\gamma^{\beta}(\not k_{11}+m_{c})\notin(p_{1})$ (14)
$\displaystyle\frac{\not p_{2}-\not k_{12}-\not k_{2}-\not k_{3}+m_{c}}{(\not
p_{2}-\not k_{12}-\not k_{2}-\not k_{3})^{2}-m^{2}_{c}}\gamma_{\mu}\frac{\not
p_{2}-\not k_{3}+m_{c}}{(\not p_{2}-\not
k_{3})^{2}-m^{2}_{c}}\notin(p_{2})v_{s3}(k_{3})]|_{q=0},$

Here, $\varepsilon_{\beta}^{s}(k_{1})$ or $\varepsilon_{\alpha}^{l}(k_{1})$
represents the polarization vector associated with the spin or orbital angular
momentum of the diquark in the spin triplet $S$-state or spin singlet
$P$-state. $\varepsilon_{\alpha\beta}^{J}(k_{1})$ corresponds to the
polarization tensor for the spin triplet $P$-wave states, where $J$ can be 0,
1, or 2. To determine the suitable total angular momentum, we appropriately
perform the polarization sum. The summation over polarization vectors is
carried out as follows:

$\sum_{r_{z}}\varepsilon_{\alpha}^{r}\varepsilon_{\alpha^{\prime}}^{r*}=\Pi_{\alpha\alpha^{\prime}}$
(15)

The summation over polarization tensors is conducted as:

$\varepsilon_{\alpha\beta}^{0}\varepsilon_{\alpha^{\prime}\beta^{\prime}}^{0*}=\frac{1}{3}\Pi_{\alpha\beta}\Pi_{\alpha^{\prime}\beta^{\prime}}$
(16)
$\sum_{J_{z}}\varepsilon_{\alpha\beta}^{1}\varepsilon_{\alpha^{\prime}\beta^{\prime}}^{1*}=\frac{1}{2}(\Pi_{\alpha\alpha^{\prime}}\Pi_{\beta\beta^{\prime}}-\Pi_{\alpha\beta^{\prime}}\Pi_{\alpha^{\prime}\beta})$
(17)
$\sum_{J_{z}}\varepsilon_{\alpha\beta}^{2}\varepsilon_{\alpha^{\prime}\beta^{\prime}}^{2*}=\frac{1}{2}(\Pi_{\alpha\alpha^{\prime}}\Pi_{\beta\beta^{\prime}}+\Pi_{\alpha\beta^{\prime}}\Pi_{\alpha^{\prime}\beta})-\frac{1}{3}\Pi_{\alpha\beta}\Pi_{\alpha^{\prime}\beta^{\prime}},$
(18)

with the definition

$\Pi_{\alpha\beta}=-g_{\alpha\beta}+\frac{k_{1\alpha}k_{1\beta}}{M_{cc}^{2}}.$
(19)

For the processes $\gamma+\gamma\rightarrow(cc)[n]+\bar{c}+\bar{c}$, the color
factor $\mathcal{C}_{ij,k}$ is universal for all the Feynman diagrams and
$\mathcal{C}_{ij,k}=\mathcal{N}\times\sum_{a,m,n}(T^{a})_{mi}(T^{a})_{nj}\times
G_{mnk}$. Here $\mathcal{N}=1/\sqrt{2}$ is the normalization constant,
$i,j,m,n$ are color indices of four heavy quarks and $k$ denotes the color
index of the diquark. $G_{mnk}$ corresponds to either the antisymmetric
function $\varepsilon_{mnk}$ for the color anti-triplet state, or the
symmetric function $f_{mnk}$ for the color sextuplet state, and they obey

$\varepsilon_{mnk}\varepsilon_{m^{\prime}n^{\prime}k}=\delta_{mm^{\prime}}\delta_{nn^{\prime}}-\delta_{mn^{\prime}}\delta_{nm^{\prime}},$
(20)
$f_{mnk}f_{m^{\prime}n^{\prime}k}=\delta_{mm^{\prime}}\delta_{nn^{\prime}}+\delta_{mn^{\prime}}\delta_{nm^{\prime}}.$
(21)

The color factors of diagrams for $\gamma+g\rightarrow(cc)[n]+\bar{c}+\bar{c}$
are not the same; they need to be calculated individually.

##  3 Numerical results and discussions

In the calculation, we adopt the wave functions at the origin from
Kiselev:2002iy  as $|\Psi(0)|^{2}=0.0218\,\mathrm{GeV^{3}}$ and
$|\Psi^{\prime}(0)|^{2}=2.48\times 10^{-3}\,\mathrm{GeV^{5}}$. The quark mass
is set as $m_{c}=M_{\Xi_{cc}}/2=1.8\,\mathrm{GeV}$. The fine structure
constant is assigned the value $\alpha=1/137$. Regarding the strong coupling
constant, we utilize the one-loop running formulation. The renormalization
scale is typically taken as the transverse mass of $\Xi_{cc}$, specifically
$\mu=\sqrt{M^{2}_{\Xi_{cc}}+p^{2}_{t}}$, where $p_{t}$ denotes the transverse
momentum of the particle.

Table 1 lists cross sections of different photoproduction processes, where
three collision energies, $\sqrt{S}=250,500,1000\mathrm{GeV}$, are adopted.
From the table, it is evident that the contribution from production channel
$\gamma+\gamma$ decreases as the collision energy increases, while the
contribution from production channel $\gamma+g$ increases with higher energy
levels. As a cumulative result, the total production cross-section increases
with the growth of collision energy. At $\sqrt{S}=250\,\mathrm{GeV}$, the
contribution from the $\gamma+\gamma$ channel is comparable to that of the
$\gamma+g$ channel. However, as the collision energy increases, the scenario
changes. The $\gamma+g$ channel begins to dominate the photoproduction
process. Specifically, at $\sqrt{S}=500\,\mathrm{GeV}$, the $\gamma+\gamma$
channel provides 20% of the contribution, while the $\gamma+g$ channel
contributes 80%. Table 1 clearly emphasizes the significance of the single
resolved photoproduction channel.

$\sqrt{S}(\mathrm{GeV})$ | $\gamma+\gamma\rightarrow\Xi_{cc}+\bar{c}+\bar{c}$ | $\gamma+g\rightarrow\Xi_{cc}+\bar{c}+\bar{c}$ | total
---|---|---|---
250 | $228.61$ | $238.90$ | $467.50$
500 | $101.34$ | $411.50$ | $512.84$
1000 | $40.94$ | $659.81$ | $700.75$
Table 1: The integrated cross sections (in unit of fb) from different channels for $\Xi_{cc}$ photoproduction under various collision energies at future $e^{+}e^{-}$ collider. The contributions from the $S$-wave and $P$-wave have been combined. $\sqrt{S}(\mathrm{GeV})$ | $(cc)_{\bar{\textbf{3}}}[{}^{3}S_{1}]$ | $(cc)_{\textbf{6}}[{}^{1}S_{0}]$ | $(cc)_{\bar{\textbf{3}}}[{}^{1}P_{1}]$ | $(cc)_{\textbf{6}}[{}^{3}P_{0}]$ | $(cc)_{\textbf{6}}[{}^{3}P_{1}]$ | $(cc)_{\textbf{6}}[{}^{3}P_{2}]$
---|---|---|---|---|---|---
250 | $407.43$ | $36.32$ | $8.81$ | $1.75$ | $2.44$ | $10.75$
500 | $442.71$ | $41.24$ | $9.42$ | $2.12$ | $3.03$ | $14.32$
1000 | $603.0$ | $57.05$ | $12.66$ | $2.97$ | $4.33$ | $20.72$
Table 2: The integrated cross sections (in unit of fb) of different
intermediate diquark states for $\Xi_{cc}$ photoproduction under various
collision energies at future $e^{+}e^{-}$ collider.

Similarly, for the three collision energies, cross sections of various
intermediate diquark states are provided in Table 2. These results reveal that
the contribution of $S$-wave diquarks is significantly larger than that of the
$P$-wave diquarks. At $\sqrt{S}=250\,\mathrm{GeV}$, the ratio between the
contributions from the $S$-wave and $P$-wave is 18.7:1. At
$\sqrt{S}=500\mathrm{\,GeV}$, the ratio becomes 16.8:1, meaning that the
$P$-wave contributions constitute 5.6% of the total production. Among the
$P$-wave states, the contribution of the $(cc)_{\textbf{6}}[{}^{3}P_{2}]$
diquark is the largest. Their ratio is as follows:
$(cc)_{\bar{\textbf{3}}}[{}^{1}P_{1}]~{}$:$~{}(cc)_{\textbf{6}}[{}^{3}P_{0}]~{}$:$~{}(cc)_{\textbf{6}}[{}^{3}P_{1}]~{}$:$~{}(cc)_{\textbf{6}}[{}^{3}P_{2}]$=1:0.23:0.32:1.52
at $\sqrt{S}=500\,\mathrm{GeV}$. Assuming an integrated luminosity of
$\mathcal{O}(10^{4})\mathrm{~{}fb^{-1}}$ at future $e^{+}e^{-}$ colliders and
summing up the contributions from all $P$-wave excited baryons, approximately
$2.9\times 10^{5}$ $P$-wave $\Xi_{cc}$ baryons would be generated, given a
collision energy of $\sqrt{S}=500\mathrm{~{}GeV}$. The $P$-wave doubly charmed
baryons are likely to decay to the ground state with almost 100% probability,
making them additional sources of ground-state baryons.

Figure 2:  Kinematic distributions for the photoproduction of $\Xi_{cc}$ at
future $e^{+}e^{-}$ collider($\sqrt{S}=$$500$ $\mathrm{GeV}$). Contributions
from different channels are displayed individually.

Figure 3:  Kinematic distributions for the photoproduction of $\Xi_{cc}$ at
future $e^{+}e^{-}$ collider($\sqrt{S}=$$500$ $\mathrm{GeV}$). Contributions
from different diquark states are displayed individually. The $y$ and
$\cos\theta$ curves use same legends as those of $p_{t}$.

To reveal more characteristics of $\Xi_{cc}$ photoproduction at the
$e^{+}e^{-}$ collider, we have computed the differential distributions at
$\sqrt{S}=500\,\mathrm{GeV}$, as illustrated in Fig.2 and Fig.3. Fig. 2
depicts the transverse momentum($p_{t}$), the rapidity($y$) and $cos\theta$
distributions, featuring distinct representations of contributions originating
from different channels. Here, $\theta$ represents the angle between
$\Xi_{cc}$ and the $e^{+}e^{-}$ beams. The $\gamma+g$ channels exert their
dominance in the lower $p_{t}$ region, gradually passing the torch to the
$\gamma+\gamma$ channels as the $p_{t}$ values increase. In practical
experiments, there may not be a sufficient number of events in the high
$p_{t}$ region to attain precise measurements. Consequently, it becomes
necessary to consider the single resolved channel $\gamma+g$ when performing
photoproduction calculations. In contrast to the curves for the $p_{t}$
distribution, in the rapidity and angular distributions, the curves for the
two production channels do not intersect. Throughout the entire rapidity
distribution range, the contributions of $\gamma+g$ consistently surpass those
of $\gamma+\gamma$. The same pattern is also evident in the angular
distribution curves. Fig.3 displays the contribution curves for different
intermediate diquark states. In each $p_{t}$ distribution, a noticeable peak
emerges at approximately several GeV, followed by a logarithmic decline in the
high $p_{t}$ region. In all three distinct kinematic distributions, it is
consistently evident that the ${}_{\bar{\textbf{3}}}[{}^{3}S_{1}]$
configurations maintain prominence across the entire range, while
contributions from other states are small.

$m_{c}(\mathrm{GeV})$ | $(cc)_{\bar{\textbf{3}}}[{}^{3}S_{1}]$ | $(cc)_{\textbf{6}}[{}^{1}S_{0}]$ | $(cc)_{\bar{\textbf{3}}}[{}^{1}P_{1}]$ | $(cc)_{\textbf{6}}[{}^{3}P_{0}]$ | $(cc)_{\textbf{6}}[{}^{3}P_{1}]$ | $(cc)_{\textbf{6}}[{}^{3}P_{2}]$
---|---|---|---|---|---|---
1.7 | $609.62$ | $56.85$ | $14.50$ | $3.28$ | $4.67$ | $22.27$
1.8 | $442.71$ | $41.24$ | $9.42$ | $2.12$ | $3.03$ | $14.32$
1.9 | $328.38$ | $30.53$ | $6.24$ | $1.40$ | $2.0$ | $9.45$
Table 3: The total cross sections (in unit of fb) under different $m_{c}$ at $\sqrt{S}=500\mathrm{~{}GeV}$. ${\cal C}$ | $(cc)_{\bar{\textbf{3}}}[{}^{3}S_{1}]$ | $(cc)_{\textbf{6}}[{}^{1}S_{0}]$ | $(cc)_{\bar{\textbf{3}}}[{}^{1}P_{1}]$ | $(cc)_{\textbf{6}}[{}^{3}P_{0}]$ | $(cc)_{\textbf{6}}[{}^{3}P_{1}]$ | $(cc)_{\textbf{6}}[{}^{3}P_{2}]$
---|---|---|---|---|---|---
0.5 | $531.74$ | $50.46$ | $11.44$ | $2.47$ | $3.57$ | $16.40$
1.0 | $442.71$ | $41.24$ | $9.42$ | $2.12$ | $3.03$ | $14.32$
2.0 | $376.96$ | $35.12$ | $8.0$ | $1.84$ | $2.59$ | $12.61$
Table 4: The total cross sections (in unit of fb) under various $\mu(={\cal
C}\sqrt{M^{2}_{\Xi_{cc}}+p^{2}_{t}}$ with ${\cal C}=0.5,1,2$) at
$\sqrt{S}=500\mathrm{~{}GeV}$.

Finally, we delve into a brief discussion of the theoretical uncertainties
inherent in our calculations, stemming from three main sources: the heavy
quark mass $m_{c}$, the renormalization scale $\mu$, and the LDMEs. It’s worth
noting that uncertainties originating from $h_{\bar{\textbf{3}}}$ and
$h_{\textbf{6}}$ have been omitted due to the lack of reported errors in the
literature. As mentioned earlier, these coefficients represent global factors,
and their influence on production outcomes can be further refined with more
precise values. Table 3 demonstrates the impact of changing the value of
$m_{c}$ within the range of $1.8\pm 0.1\mathrm{~{}GeV}$ while keeping
$\mu=\sqrt{M^{2}_{\Xi_{cc}}+p^{2}_{t}}$ constant. As observed in the table,
even small variations in the heavy quark mass can result in substantial
fluctuations in cross-section values. For example, in Table 3, the cross
section for $(cc)_{\bar{\textbf{3}}}[{}^{3}S_{1}]$ varies by approximately 46%
with only a 12% change in $m_{c}$. This notable sensitivity becomes evident
when analyzing the relevant Feynman diagrams, as exemplified in Fig. 1. For
photoproduction of $\Xi_{cc}$ considered here, it is noted that the final
particles involved in the short-distance processes consist exclusively of $c$
and $\bar{c}$, while the internal lines are composed solely of charm and gluon
propagators. Therefore, the significant impact of heavy quark masses on the
cross section appears to be a reasonable outcome.

Table 4 evaluates the sensitivity to the renormalization scale ($\mu={\cal
C}\sqrt{M^{2}_{\Xi_{cc}}+p^{2}_{t}}$, where ${\cal C}=0.5,1,2$), while keeping
the value of $m_{c}$ fixed at $1.8\mathrm{~{}GeV}$. Clearly, there is a
significant dependence on the renormalization scale, which could suggest the
importance of next-to-leading order corrections in $\alpha_{s}$. As we
confront real-world measurements in the future, high-order calculations become
imperative. Taking into account the aforementioned uncertainties, our leading-
order calculation results may vary by approximately one order of magnitude.
Despite this range of variability, the photoproduction rates of doubly charmed
baryons remain significant.

##  4 Summary

In this work, we have investigated the $\Xi_{cc}$ photoproduction within the
framework of non-relativistic QCD specifically focusing on future $e^{+}e^{-}$
colliders. Two dominant photoproduction processes are considered, i.e.,
$\gamma+\gamma\rightarrow\Xi_{cc}+\bar{c}+\bar{c}$ and
$\gamma+g\rightarrow\Xi_{cc}+\bar{c}+\bar{c}$. Four $P$-wave diquark states
are included in the calculation and they are
$(cc)_{\bar{\textbf{3}}}[{}^{1}P_{1}]$, $(cc)_{\textbf{6}}[{}^{3}P_{0}]$,
$(cc)_{\textbf{6}}[{}^{3}P_{1}]$ and $(cc)_{\textbf{6}}[{}^{3}P_{2}]$. Upon
assuming $h_{\textbf{6}}=h_{\bar{\textbf{3}}}$, the results demonstrate the
photoproduction of $P$-wave $\Xi_{cc}$ is about one order lower than that of
the $S$-wave. Specifically, at a center-of-mass energy of
$\sqrt{S}=500\,\mathrm{GeV}$, the cross section for $P$-wave $\Xi_{cc}$
production is approximately 6% of that for $S$-wave production. The numerical
results further emphasize the crucial role played by the single resolved
photoproduction channel $\gamma+g$ in the overall photoproduction process. It
gains increasing significance as the collision energy rises. Assuming an
integrated luminosity for future $e^{+}e^{-}$ collisions on the order of
$\mathcal{O}(10^{4})\mathrm{~{}fb^{-1}}$, approximately $4.8\times 10^{6}$
$S$-wave $\Xi_{cc}$ and $2.9\times 10^{5}$ $P$-wave $\Xi_{cc}$ baryons would
be expected to be produced at a collision energy of
$\sqrt{S}=500\mathrm{~{}GeV}$. The excited doubly charmed baryons are likely
to decay into the ground state with nearly 100% probability. Therefore, when
faced with precise real-world measurements, their contributions should be
thoroughly examined and taken into account.

###### Acknowledgements.

This work was supported in part by the Natural Science Foundation of China
under Grants No. 12305083, No. 12147116, No. 12175025, No. 12005028 and No.
12147102, and by the Fundamental Research Funds for the Central Universities
under Grant No. 2020CQJQY-Z003.

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authors in recent years (Sollich, 2001; Bordelon et al., 2020; Jacot et al.,
2020a; Simon et al., 2021; Loureiro et al., 2021; Dobriban & Wager, 2018; Wu &
Xu, 2020; Hastie et al., 2020; Richards et al., 2021) — a result which we
actually bootstrapped to derive our RF eigenframework (Appendix E). We also
recall this same eigenframework at the start of Appendix E. This risk estimate
as follows:

Let $\kappa\geq 0$ be the unique nonnegative solution to

$\sum_{i}\frac{\lambda_{i}}{\lambda_{i}+\kappa}+\frac{\delta}{\kappa}=n.$ (65)

Then test risk is given approximately by

$\texttt{MSE}_{\textnormal{te}}\approx\mathcal{E}_{\textnormal{te}}\vcentcolon=\mathcal{E}_{0}\mathcal{B},$
(66)

where the overfitting coefficient $\mathcal{E}_{0}$ is given by

$\mathcal{E}_{0}\vcentcolon=\frac{n}{n-\sum_{i}\left(\frac{\lambda_{i}}{\lambda_{i}+\kappa}\right)^{2}}$
(67)

and the bias is given by

$\mathcal{B}=\sum_{i}\left(\frac{\kappa}{\lambda_{i}+\kappa}\right)^{2}v_{i}^{2}+\sigma^{2}.$
(68)

As discussed in Appendix E, the “$\approx$” in 66 can be given several
meanings. Firstly, it becomes an equivalence in an asymptotic limit in which
$n$ and the number of eigenmodes in a given eigenvalue range (or the number of
duplicate copies of any given eignemode) both grow large proportionally
(Hastie et al., 2020; Bach, 2023). This is often phrased as sampling a
proportional number of new eigenmodes from a fixed measure. Secondly, with
fixed task eigenstructure, the error incurred can be bounded by a decaying
function of $n$ (Cheng & Montanari, 2022). Thirdly, numerical experiments find
small error even at quite modest $n$ (Canatar et al., 2021; Simon et al.,
2021). As with the RF eigenframework, in this paper we will simply treat it as
an equivalence, formally proving facts about the risk estimate
$\mathcal{E}_{\textnormal{te}}$.

Recall that all sums run from $i=1$ to $\infty$. Train risk is given by

$\texttt{MSE}_{\textnormal{tr}}\approx\mathcal{E}_{\textnormal{tr}}\vcentcolon=\frac{\delta^{2}}{n^{2}\kappa^{2}}\mathcal{E}_{\textnormal{te}},$
(69)

and so the fitting error ratio is given roughly by

$\frac{\texttt{MSE}_{\textnormal{tr}}}{\texttt{MSE}_{\textnormal{te}}}\approx\mathcal{R}_{\textnormal{tr}/\textnormal{te}}\vcentcolon=\frac{\mathcal{E}_{\textnormal{tr}}}{\mathcal{E}_{\textnormal{te}}}=\frac{\delta^{2}}{n^{2}\kappa^{2}}.$
(70)

Recall that the noise level is defined relative to the zero-noise-zero-ridge
risk as

$\sigma^{2}=\sigma_{\textnormal{rel}}^{2}\cdot\mathcal{E}_{\textnormal{te}}|_{\sigma^{2}=\delta=0}.$
(71)

We will assume throughout that $\alpha>1$ and $\beta\in(1,2\alpha+1)$. We will
also assume $n\geq 1$. We will generally use an asterisk to demarcate
quantities which occur at the optimal ridge w.r.t. test risk. For example,
$\delta_{*}=\arg\min_{\delta}\mathcal{E}_{\textnormal{te}}$, and
$\kappa_{*}=\kappa|_{\delta=\delta_{*}}$, and
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}^{*}=\mathcal{R}_{\textnormal{tr}/\textnormal{te}}|_{\delta=\delta_{*}}$.

It is worth noting that we have two varying parameters in our system: the
sample size $n$ and the ridge $\delta$. Other quantities, like $\kappa$ or
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}$, may be seen as functions of
$n$ and $\delta$. When we state scaling results using big-$O$-style notation,
they will describe scaling with respect to $n$ — that is, $x=O(f(n,\delta))$
means that there exist constants $n_{0},C>0$ such that, for all $n\geq n_{0}$,
it holds that $x\leq Cf(n,\delta)$. We will allow $\delta$ to vary
arbitrarily, including as a function of $n$.

###  I.2 Continuum approximations to sums

Our main trick will be approximating the sums appearing in the eigenframework
by integrals. The primary technical difficulty will be in bounding the error
of these approximations (though we will ultimately find that these errors do
not present a problem). Upon a first reading of this appendix, a reader may
wish to simply ignore all error terms to grasp the overall flow of the
argument.

Various quantities in this appendix will have complicated prefactors depending
on the exponents $\alpha$ and $\beta$. These prefactors often cancel and
simplify in the final accounting.121212The authors are grateful for the
existence of computer algebra systems. It is useful at first to pay less
attention to these prefactors and more attention to how quantities scale —
with respect to, for example, $n$ (which will be large) and $\kappa$ (which
will be small).

Here we state continuum approximations to the three eigensums of the
eigenframework.

######  Lemma 1 (Continuum approximations to eigensums).

The sums appearing in the KRR eigenframework can be approximated as follows:

$\sum_{i=1}^{\infty}\frac{\lambda_{i}}{\lambda_{i}+\kappa}=\frac{\pi}{\alpha\sin\lparen\pi/\alpha\rparen}\kappa^{-1/\alpha}+O\lparen
1\rparen,$ (72)

$\sum_{i=1}^{\infty}{\left\lparen\frac{\lambda_{i}}{\lambda_{i}+\kappa}\right\rparen}^{2}=\frac{\pi\lparen\alpha-1\rparen}{\alpha^{2}\sin\lparen\pi/\alpha\rparen}\kappa^{-1/\alpha}+O\lparen
1\rparen,$ (73)

$\sum_{i=1}^{\infty}{\left\lparen\frac{\kappa}{\lambda_{i}+\kappa}\right\rparen}^{2}v_{i}^{2}=\frac{\pi\lparen\alpha-\beta+1\rparen}{\alpha^{2}\sin{\left\lparen\pi\frac{\lparen\beta-1\rparen}{\alpha}\right\rparen}}\kappa^{\frac{\beta-1}{\alpha}}+O{\left\lparen\kappa^{2}+\kappa^{\frac{\beta}{a}}\right\rparen}.$
(74)

Proof. First, we argue that we may disregard the first $i_{0}=O(1)$ terms in
these sums and replace them with their ideal powerlaw values
$\lambda_{i}=i^{-\alpha}$ and $v_{i}^{2}=i^{-\beta}$. For the first two sums,
note that replacing the first $i_{0}$ terms (or neglecting the first $i_{0}$
terms entirely) only incurs an $O(1)$ error, which is the size of the error
terms in Equations 72 and 73 anyways. For the third sum, note that replacing
the first $i_{0}$ terms (or neglecting the first $i_{0}$ terms entirely) only
incurs an $O(\kappa^{2})$ error, which is at most the size of the error term
in Equation 74 anyways. We may thus proceed assuming perfect powerlaw
structure, with $i_{0}=1$.

To prove the first clause, we note that the summand is monotonically
decreasing, and thus use integrals to bound the sum as

$\int_{0}^{\infty}\frac{i^{-\alpha}}{i^{-\alpha}+\kappa}\mathrm{d}i>\sum_{i=1}^{\infty}\frac{\lambda_{i}}{\lambda_{i}+\kappa}>\int_{1}^{\infty}\frac{i^{-\alpha}}{i^{-\alpha}+\kappa}\mathrm{d}i>\int_{0}^{\infty}\frac{i^{-\alpha}}{i^{-\alpha}+\kappa}\mathrm{d}i-1.$
(75)

The LHS integral evaluates to
$\frac{\pi}{\alpha\sin\lparen\pi/\alpha\rparen}\kappa^{-1/\alpha}$, which is
sufficient to give Equation 72.

Equation 73 is obtained in exactly the same way.

Equation 74 is obtained in the same way, except that the summand is no longer
monotonically-decreasing if $\beta\in(2\alpha,2\alpha+\beta)$, instead
monotonically increasing to a maximum of size $\Theta(\kappa^{\beta/\alpha})$
at index
$i_{\text{max}}=\beta^{-1/\alpha}(2\alpha-\beta)^{1/\alpha}\kappa^{-1/\alpha}+O(1)$
before monotonically decreasing to zero. Splitting the sum into increasing and
decreasing parts and again using integrals to bound the sum gives Equation 74.
∎

Remark. We will more or less never again need to worry about the constant
cutoff index $i_{0}$ and can forget about it now. Our conclusions will live in
the regime of large $n$, and in this regime, it will be sufficient to have
powerlaw tails, and we can simply neglect a constant number of eigenmodes at
low index. One should usually expect the sub-leading-order error terms we will
carry around to be smaller the smaller $i_{0}$ is and the less the task
deviates from perfect powerlaw eigenstructure, however (though one likely
ought to consider some measure of total deviation rather than simply the
cutoff $i_{0}$).

Remark. In several places in this appendix, including Equation 74, we will
encounter the fraction $(\alpha-\beta+1)/\sin(\pi(\beta-1)/\alpha)$. This is
nominally undefined when $\beta=\alpha+1$. However, this discontinuity
disappears with an application of L’Hopital’s rule, simplifying to
$\pi/\alpha$. (Indeed, when evaluating the integral to arrive at the RHS of
Equation 74, if we specify that $\beta=\alpha+1$, then e.g. Mathematica
informs us that the integral evaluates to the result of applying L’Hopital’s
rule to the general form.) Rather than treat the case $\beta=\alpha+1$
specially, we will simply gloss over it with the understanding that
L’Hopital’s rule ought to be used to resolve this undefined fraction.

###  I.3 The zero-noise-zero-ridge test risk

Because the noise level is defined in a normalized fashion by
$\sigma^{2}=\sigma_{\textnormal{rel}}^{2}\cdot\mathcal{E}_{\textnormal{te}}|_{\sigma^{2}=\delta=0}$,
we need to find the zero-noise-zero-ridge risk
$\mathcal{E}_{\textnormal{te}}|_{\sigma^{2}=\delta=0}$ before we can study the
noise in the general case. The zero-noise-zero-ridge risk is also an
interesting object in its own right.

######  Lemma 2 (Zero-ridge implicit regularization).

At $\delta=0$, the implicit regularization $\kappa$ is given by

$\kappa|_{\delta=0}={\left\lparen\frac{\pi}{\alpha\sin\lparen\pi/\alpha\rparen}\right\rparen}^{\alpha}n^{-\alpha}+O{\left\lparen
n^{-\lparen\alpha+1\rparen}\right\rparen}.$ (76)

Proof. This follows straightforwardly from inserting Equation 72 into Equation
65. ∎

######  Lemma 3 (Zero-noise-zero-ridge test risk).

At $\sigma^{2}=0$ and $\delta=0$, the test risk
$\mathcal{E}_{\textnormal{te}}$ is given by

$\mathcal{E}_{\textnormal{te}}|_{\sigma^{2}=\delta=0}=\frac{\pi^{\beta}\lparen\alpha-\beta+1\rparen}{\alpha^{\beta}\sin{\left\lparen\frac{\pi\lparen\beta-1\rparen}{\alpha}\right\rparen\left(\sin\lparen\pi/\alpha\rparen\right)^{\beta-1}}}n^{-\lparen\beta-1\rparen}+O{\left\lparen
n^{-2\alpha}+n^{-\beta}\right\rparen}.$ (77)

Proof. This follows from inserting Equation 76 into the continuum
approximations of Lemma 1, then inserting the results into the the
eigenframework to get $\mathcal{E}_{\textnormal{te}}$, and finally
simplifying. ∎

Remark. In the process of doing this, one finds (after a surprising
cancellation) that the overfitting coefficient is simply
$\mathcal{E}_{0}|_{\delta=0}=\alpha+O(n^{-1})$, as previously reported by
Mallinar et al. (2022).

###  I.4 Test risk in terms of $\kappa$

We now turn back to the general case in which noise and regularization are
nonzero. The following lemma gives test risk in terms of the implicit
regularization $\kappa$.

######  Lemma 4.

$\mathcal{E}_{\textnormal{te}}=\frac{n}{n-\frac{\pi\lparen\alpha-1\rparen}{\alpha^{2}\sin\lparen\pi/\alpha\rparen}\kappa^{-1/\alpha}}\\\
\times\Bigg{(}\frac{\pi\lparen\alpha-\beta+1\rparen}{\alpha^{2}\sin{\left\lparen\pi\frac{\lparen\beta-1\rparen}{\alpha}\right\rparen}}\kappa^{\frac{\beta-1}{\alpha}}+\sigma_{\textnormal{rel}}^{2}\cdot\frac{\pi^{\beta}\lparen\alpha-\beta+1\rparen}{\alpha^{\beta}\sin{\left\lparen\frac{\pi\lparen\beta-1\rparen}{\alpha}\right\rparen\left(\sin\lparen\pi/\alpha\rparen\right)^{\beta-1}}}\cdot
n^{-\lparen\beta-1\rparen}\Bigg{)}\\\ +O\lparen
n^{-\beta}+\kappa^{2}+\kappa^{\beta/\alpha}\rparen.$ (78)

Proof. This lemma follows from inserting Lemmas 1 and 3 into the definition of
$\mathcal{E}_{\textnormal{te}}$. ∎

######  Lemma 5.

$\kappa=\Omega\left(n^{-\alpha}\right),$
$\mathcal{E}_{\textnormal{te}}=\Omega\left(\min\left(\kappa^{\frac{\beta-1}{\alpha}},1\right)\right),$
and thus $\mathcal{E}_{\textnormal{te}}=\Omega\left(n^{-(\beta-1)}\right).$

Proof. The fact that $\kappa=\Omega\left(n^{-\alpha}\right)$ follows from the
fact that $\kappa|_{\delta=0}=\Theta(n^{-\alpha})$ and $\kappa$ is a
monotonically increasing function of $\delta$.

To see that
$\mathcal{E}_{\textnormal{te}}=\Omega\left(\min\left(\kappa^{\frac{\beta-1}{\alpha}},1\right)\right),$
we return to the definition of $\mathcal{E}_{\textnormal{te}}$. It is easily
seen that $\mathcal{E}_{0}=\Theta(1)$, because
$\mathcal{E}_{0}|_{\delta=0}=\Theta(1)$, $\mathcal{E}_{0}\geq 1$, and
$\mathcal{E}_{0}$ monotonically decreases with $\delta$, so we need only
examine the bias $\mathcal{B}$ to understand the size of
$\mathcal{E}_{\textnormal{te}}$ in a scaling sense. Because
$\sigma_{\textnormal{rel}}^{2}=O(1)$, we have that
$\sigma^{2}=\sigma_{\textnormal{rel}}^{2}\cdot\mathcal{E}_{\textnormal{te}}|_{\sigma^{2}=\delta=0}=O(n^{-(\beta-1)})$.
Examining the eigensum $\sum_{i}(1-\mathcal{L}_{i})^{2}v_{i}^{2}$, it is
easily seen that at all modes $i>i^{\prime}$ for some
$i^{\prime}=\Theta(\max(1,\kappa^{-1/\alpha}))$, we will have that
$\mathcal{L}_{i}=\frac{i^{-\alpha}}{i^{-\alpha}+\kappa}\leq\frac{1}{2}$, say,
which tells us that
$\mathcal{E}_{\textnormal{te}}=\Omega(\int_{i^{\prime}}^{\infty}i^{-\beta}di)=\Omega\left((i^{\prime})^{-(\beta-1)}\right)$131313It
might seem that we need to be mindful of the cutoff index $i_{0}$ up to which
we do not necessarily have powerlaw eigenstructure, but we do not actually:
because $i_{0}=\Theta(1)$, we will still have
$i^{\prime}=\Theta(\max(1,\kappa^{-1/\alpha}))$.. The second clause of the
lemma follows from this.

The third clause of the lemma follows from the insertion of the first
statement into the second. ∎

######  Lemma 6.

At optimal regularization, we have that
$\mathcal{E}_{\textnormal{te}}^{*}=\Theta(n^{-(\beta-1})$,
$\kappa_{*}=\Theta(n^{-\alpha})$, and thus $\delta_{*}=O(n^{-(\alpha-1)})$

Proof. The first clause of this lemma follows from the lower bound on
$\mathcal{E}_{\textnormal{te}}$ given by Lemma 5 and the fact that we can
saturate this lower bound in a scaling sense because we do so already at zero
ridge, $\mathcal{E}_{\textnormal{te}}|_{\delta=0}=\Theta(n^{-(\beta-1)})$
(which itself follows from inserting Lemma 2 into Lemma 4). Given this, the
second clause of Lemma 5 assures us that $\kappa_{*}=\Theta(n^{-\alpha})$ as
desired.141414Seeing this is made easier by writing out the scaling notation
explicitly — e.g., $\mathcal{E}_{\textnormal{te}}=\Omega(n^{-(\beta-1)})$
means that there exists a constant $A_{1}>0$ such that
$\mathcal{E}_{\textnormal{te}}>_{1}\cdot n^{-(\beta-1)}$ for sufficiently
large $n$, and so on — and then solving for constants which are sufficient to
give the desired new statements. The bound on $\delta_{*}$ follows from
Equation 72 and Equation 65. ∎

Remark. Lemma 5 is useful because it tells us that, at optimal ridge, the
error terms in Lemma 4 will indeed decay faster with $n$ than
$\mathcal{E}_{\textnormal{te}}$ itself.

###  I.5 Relating $\kappa$ and $\mathcal{R}_{\textnormal{tr}/\textnormal{te}}$

######  Lemma 7.

The fitting ratio is given in terms of $\kappa$ by

$\sqrt{\mathcal{R}_{\textnormal{tr}/\textnormal{te}}}=1-n^{-1}\frac{\pi}{\alpha\sin(\pi/\alpha)}\kappa^{-1/\alpha}+O(n^{-1}).$
(79)

Solving for $\kappa$, we have

$\kappa=\left(\frac{\pi}{\alpha\sin(\pi/\alpha)}\right)^{-\alpha}n^{\alpha}\left(1-\sqrt{\mathcal{R}_{\textnormal{tr}/\textnormal{te}}}+O(n^{-1})\right)^{-\alpha}.$
(80)

Proof. This lemma follows directly from Equations 65, 70 and 72. ∎

Remark. A minor annoyance in the usage of Equation 80 is that, when
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}$ is too close to $1$, the
$O(n^{-1})$ term can affect $\kappa$ to leading order. As a patch to avoid
this, we will initially restrict the domain of study to
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}\in[0,1-c]$ for some
$n$-independent constant $c>0$. We are then assured that

$\kappa=\left(\frac{\pi}{\alpha\sin(\pi/\alpha)}\right)^{-\alpha}n^{\alpha}\left(1-\sqrt{\mathcal{R}_{\textnormal{tr}/\textnormal{te}}})\right)^{-\alpha}+O(n^{-(\alpha+1)}).$
(81)

The fact that $\kappa$ increases monotonically with
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}$ (Lemma 8) will let us extend
results of interest to the small region
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}\in(1-c,1)$. As far as the study
of the optimal regularization point go, we need not consider this small edge
region in the following sense:

######  Lemma 8.

The fitting error ratio
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}\in[0,1)$ and implicit
regularization $\kappa\in[\kappa|_{\delta=0},\infty)$ are monotonically-
increasing functions of each other.

Proof. This lemma is apparent from the fact that
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}=\left(1-n^{-1}\sum_{i}\frac{\lambda_{i}}{\lambda_{i}+\kappa}\right)^{2}$.
∎

######  Lemma 9.

The optimal value of the fitting error ratio is bounded away from $1$ in the
sense that there exists a constant $c^{\prime}>0$ such that
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}^{*}\leq
1-c^{\prime}+O(n^{-1})$.

Proof. We know from Lemma 6 that $\kappa_{*}=\Theta(n^{-\alpha})$. We then
obtain the desired result from Equation 79. ∎

###  I.6 Putting it all together: test risk in terms of
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}$

The following lemma gives the test risk $\mathcal{E}_{\textnormal{te}}$ in
terms of the fitting error ratio
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}$ and is the basis for our
ultimate conclusions about the optimal fitting error ratio.

######  Lemma 10.

Let $c>0$ be an $n$-independent constant. Then for
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}\in[0,1-c]$, we have that
$\mathcal{E}_{\textnormal{te}}=\pi^{\beta}\alpha^{-\beta}(\alpha-\beta+1)\left(\sin\left(\frac{\pi}{\alpha}\right)\right)^{-(\beta-1)}\csc\left(\frac{\pi(\beta-1)}{\alpha}\right)\\\
\times\frac{1}{1+(\alpha-1)\sqrt{\mathcal{R}_{\textnormal{tr}/\textnormal{te}}}}\left(\alpha\sigma_{\textnormal{rel}}^{2}+\left(1-\sqrt{\mathcal{R}_{\textnormal{tr}/\textnormal{te}}}\right)^{-(\beta-1)}\right)n^{-(\beta-1)}\\\
+O(n^{-\beta}+n^{-2\alpha}),$ (82) where the suppressed constants in the
$O(n^{-\beta}+n^{-2\alpha})$ term may depend on $c$.

Proof. This lemma follows from insertion of Equation 81 into Lemma 4. ∎

######  Lemma 11.

Let us abbreviate $r=\sqrt{\mathcal{R}_{\textnormal{tr}/\textnormal{te}}}$,
and let $c$ be the $n$-independent constant from Lemma 10. Then for
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}\in[0,1-c]$, we have that the
second derivative of the log of the train error obeys

$\displaystyle\frac{d^{2}}{dr^{2}}\log\mathcal{E}_{\textnormal{te}}$
$\displaystyle=\frac{\lparen\alpha-1\rparen^{2}}{{\left\lparen 1+\lparen
1-\alpha\rparen r\right\rparen}^{2}}+\frac{\beta-1+\alpha\beta(\beta-1)\lparen
1-r\rparen^{\beta-1}\sigma_{\textnormal{rel}}^{2}}{{\left\lparen
1-r+\alpha\lparen
1-r\rparen^{\beta}\sigma_{\textnormal{rel}}^{2}\right\rparen}^{2}}+O(n^{-1}+n^{-2\alpha+\beta-1})$
(83)
$\displaystyle\geq\frac{(\alpha-1)^{2}}{\alpha^{2}}+O(n^{-1}+n^{-2\alpha+\beta-1}),$
(84)

and thus at sufficiently large $n$, $\mathcal{E}_{\textnormal{te}}$ is
strongly logarithmically convex w.r.t. $r$.

Proof. This lemma follows from taking two derivatives of
$\mathcal{E}_{\textnormal{te}}$ as given by Lemma 10. ∎

######  Lemma 12.

For all $\mathcal{R}_{\textnormal{tr}/\textnormal{te}}\in[0,1)$, we have that

$\frac{\mathcal{E}_{\textnormal{te}}}{\mathcal{E}_{\textnormal{te}}^{*}}\geq
1+\frac{(\alpha-1)^{2}}{2\alpha^{2}}\left(\sqrt{\mathcal{R}_{\textnormal{tr}/\textnormal{te}}}-\sqrt{\mathcal{R}_{\textnormal{tr}/\textnormal{te}}^{*}}\right)^{2}+O(n^{-1}+n^{-2\alpha+\beta-1}).$
(85)

Proof. It is easy to see that this holds for all
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}\in[0,1-c]$ for any constant
$c>0$. Indeed, it follows directly from Lemma 11. Note that

$\log\mathcal{E}_{\textnormal{te}}-\log\mathcal{E}_{\textnormal{te}}^{*}\geq\frac{(\alpha-1)^{2}}{2\alpha^{2}}\left(\sqrt{\mathcal{R}_{\textnormal{tr}/\textnormal{te}}}-\sqrt{\mathcal{R}_{\textnormal{tr}/\textnormal{te}}^{*}}\right)^{2}+O(n^{-1}+n^{-2\alpha+\beta-1}),$
(86)

so

$\frac{\mathcal{E}_{\textnormal{te}}}{\mathcal{E}_{\textnormal{te}}^{*}}\geq\exp\left\\{\frac{(\alpha-1)^{2}}{2\alpha^{2}}\left(\sqrt{\mathcal{R}_{\textnormal{tr}/\textnormal{te}}}-\sqrt{\mathcal{R}_{\textnormal{tr}/\textnormal{te}}^{*}}\right)^{2}\right\\}+O(n^{-1}+n^{-2\alpha+\beta-1}),$
(87)

which yields Equation 85 using the fact that $e^{x}>1+x$ for $x\in\mathbb{R}$.
We now need to assure ourselves that this bound still holds for
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}\in(1-c,1)$. Intuitively, the
test tisk will generally explode as
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}\rightarrow 1$, growing from
$\Theta(n^{-(\beta-1)})$ to $\Theta(1)$, so we should expect this lower bound
to hold near $\mathcal{R}_{\textnormal{tr}/\textnormal{te}}=1$ with little
trouble. Since the error terms we have been carrying around grow large in this
region, we will need to rely on Lemmas 5 and 8 for support.

First, observe that there exist constants $B_{1}$, $n_{1}$ such that, if

$\mathcal{E}_{\textnormal{te}}\geq B_{1}n^{-(\beta-1)}\qquad\text{for all
}n\geq n_{1}$ (88)

on the region $\mathcal{R}_{\textnormal{tr}/\textnormal{te}}\in(1-c,1)$, then
Equation 85 will hold on this region. (For example, one might set
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}=1$ on the RHS of Equation 85,
then use the fact that
$\mathcal{E}_{\textnormal{te}}^{*}=\Theta(n^{-(\beta-1)})$ (Lemma 6), then
increase the constant a small amount to absorb the
$O(n^{-1}+n^{-2\alpha+\beta-1})$ error term. ) Then note that, by Lemma 5,
there exist constants $B_{2}$, $n_{2}$ such that

$\mathcal{E}_{\textnormal{te}}\geq
B_{2}\kappa^{-(\beta-1)/\alpha}\qquad\text{for all }n\geq n_{2}.$ (89)

Thus, provided that $n>\max(n_{1},n_{2})$, we are assured of Equation 88 so
long as $c$ is small enough that
$\kappa\geq(B_{2}/B_{1})^{\alpha/(\beta-1)}n^{-\alpha}$. We can identify such
a sufficiently small $c$ by looking at Equation 79, which tells us that so
long as $n\geq n_{3}$ for some $n_{3}$, we can be assured that $\kappa$ is
sufficiently large when $\mathcal{R}_{\textnormal{tr}/\textnormal{te}}=1-c$
for $c=(B_{1}/B_{2})^{1-\beta}\pi/(\alpha\sin(\pi/\alpha))$. The fact that
$\kappa$ monotonically increases with
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}$ (Lemma 8) assures us that
$\kappa$ will remain sufficiently large for all
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}\in(1-c,1)$, and thus Equation
88 will continue to hold. This patches over our edge case and completes the
proof. ∎

######  Lemma 13.

The value of the fitting error ratio that minimizes the test error is

$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}^{*}=r_{*}^{2}+O(n^{-1}+n^{-2\alpha+\beta-1}),$
(90)

where $r_{*}$ is either the unique solution to

$\alpha-\beta-(\alpha-1)\beta
r+\alpha(\alpha-1)(1-r_{*})^{\beta}\sigma_{\textnormal{rel}}^{2}=0$ (91)

over $r_{*}\in[0,1)$ or else zero if no such solution exists.

Proof. First, let the constant $c$ in Lemma 10 be less than $c^{\prime}/2$,
where $c^{\prime}$ is the constant prescribed by Lemma 9, so that we are
assured that $\mathcal{R}_{\textnormal{tr}/\textnormal{te}}^{*}\in[0,1-2c]$.
Because the error terms are small relative to the quantity itself in the
region $\mathcal{R}_{\textnormal{tr}/\textnormal{te}}\in[0,1-c]$, we may
simply take a derivative of $\mathcal{E}_{\textnormal{te}}$ in terms of
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}$ as given by Lemma 10, setting
$\frac{d\mathcal{E}_{\textnormal{te}}}{dr}=0$. This yields Equations 90 and
91. We are assured that this equation can have only one solution on the domain
of interest because the function we differentiated to obtain it is strongly
log-convex for $r\in[0,1)$, as shown in proving Lemma 11. If this equation has
no solution, this implies that there must be no local minimum on the domain,
so the minimum over the domain must lie at (or more precisely, because we have
error terms, close to) an endpoint — that is, at either
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}=0+O(n^{-\gamma})$ or
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}=1-c+O(n^{-\gamma})$, where
$\gamma=\min(1,2\alpha-\beta+1)$. However, we chose $c$ to be small enough
that $\mathcal{R}_{\textnormal{tr}/\textnormal{te}}<1-2c+O(n^{-\gamma})$, so
we can eliminate the right endpoint, and we have that
$\mathcal{R}_{\textnormal{tr}/\textnormal{te}}^{*}=O(n^{-\gamma})$ as desired.
∎

Remark. It is worth emphasizing that, because we were free from the beginning
to choose $c$ to be quite small, the rather technical patch business in the
preceding proof is not really all that important and can be glossed over on a
first reading. It is, however, nice to have for completeness, as covering the
whole region $\mathcal{R}_{\textnormal{tr}/\textnormal{te}}\in[0,1)$ permits
the final theorem statement to be simpler.

###  I.7 Stating the final theorem

Putting together Lemmas 12 and 13 gives us Theorem 2.