# Beam-Beam Interaction in a Dielectric Laser Accelerator Electron-Positron
Collider

R. Joel England<EMAIL_ADDRESS>SLAC National Accelerator
Laboratory, 2575 Sand Hill Rd, Menlo Park, CA 94025-7015 USA  Levi Schächter
<EMAIL_ADDRESS>Technion - Israel Institute of Technology, Haifa 32000,
Israel

###### Abstract

We examine through numerical calculation the collision of counter-propagating
trains of optically spaced electron/positron microbunches in a 1 TeV collider
scenario for a dielectric laser accelerator (DLA). A time-dependent envelope
equation is derived for arbitrary number of bunches in the classical limit,
with inclusion of the radiation reaction force (RRF). Example parameters are
examined based on a constrained luminosity relation that takes into account
the bunch charge for optimal efficiency, material damage limits, and power
constraints. We find that for initially identical counter-propagating Gaussian
bunch trains the periodic temporal structure leads to a peak in luminosity
with number of bunches. For longer bunch trains, the enhancement then
decreases inversely with number of bunches. The corresponding fractional
energy loss of the beam is found to be of order 1.75%, which is reduced to
0.35% when the nonlinear radial dependence of the transverse force is
included, with an average beamstrahlung parameter of 0.075, an important
result considering that beamstrahlung losses are a critical concern for future
TeV colliders.

††preprint: APS/123-QED

##  I Introduction

A future electron-positron linear collider is one of the primary proposed
tools for experimental investigation of high energy physics beyond the limits
of the Large Hadron Collider (LHC). At present, the two main proposals for a
next generation electron/positron collider are the International Linear
Collider (ILC) and the Compact Linear Collider at CERN (CLIC). Both approaches
are based on well established radio frequency (RF) technology, with the first
(ILC) relying on superconducting microwave technology and the second (CLIC) on
a normal-conducting two-beam acceleration scheme [1, 2]. However, heat-induced
quenching of superconducting cavities and surface breakdown for metallic RF
structures limit the achievable gradients in these approaches to of order 30
and 100 MeV/m respectively. Consequently, many kilometers of real estate (and
tens of billions of U.S. dollars) are required to reach TeV scale energies.
The enormous cost and footprint requirements for future colliders has prompted
interest in the development of new acceleration techniques with the potential
for higher gradients and lower cost.

Of these alternative schemes, the most heavily explored to date is the plasma
wakefield accelerator, which uses either a high-peak-power laser pulse (laser
wakefield accelerator or LWFA) or a high-density particle beam (plasma
wakefield accelerator or PWFA) to generate a longitudinal plasma wave that can
accelerate one or more trailing bunches [3, 4]. Plasma-based accelerators have
shown enormous progress, with demonstrated acceleration gradients in excess of
40GeV/m [5, 6, 7]. Consequently, the length of such a machine could
potentially be on the order of a hundred meters. However, efficiency of the
drive laser and/or plasma instabilities at the required repetition rates (of
order 10 kHz) pose significant technical challenges [8, 9, 10].

Table 1: Parameters leading to ILC [1] and DLA [11, 12, 13] geometrical (unenhanced) luminosity rate of $\bar{\mathcal{L}}_{\text{ref}}$ = $2\times 10^{34}$ cm${}^{-2}s^{-1}$ at 1 TeV center of mass energy.  Parameter | ILC | DLA
---|---|---
Bunch length (mm) | 0.3 | 0.0004
Beam Size X (nm) | 474 | 0.76
Beam Size Y (nm) | 5.9 | 0.76
Bunch population $N$ | $2\times 10^{10}$ | $2\times 10^{6}$
Bunches (M) per crossing | 1 | 100
Bunches per train | 1312 | 100
Bunch train repetition rate (Hz) | 5 | $168\times 10^{6}$

We consider here a different approach, based on dielectric laser acceleration
(DLA). The technique utilizes a slow wave hollow-core dielectric micro-
structure for acceleration, driven by an infrared laser pulse. In recent
experiments, measured acceleration gradients up to 850 MeV/m have been
demonstrated with sustained axial fields of 1.8 GV/m [14, 15, 16]. Compatible
laser-driven focusing methods for transverse particle confinement over long
distances have also been developed [17, 18, 19]. With the increase in
efficiency of wall-plug to laser light over the past two decades, the
projected machine efficiency for a DLA collider is comparable with that of ILC
or CLIC [20]. A conceptual roadmap for a future multi-TeV DLA collider is
considered among various other advanced schemes in Ref. [21], with outlines of
the various technical constraints pertaining to each approach. While the
stringent demands of a HEP collider pose significant challenges for any
accelerator technology, we note that the scope of the present work is not to
address all of the technical issues of a DLA collider, but rather to study one
particular concern, namely the multi-bunch luminosity and energy loss at the
interaction point (IP).

One of the main concerns associated with a DLA based collider is whether it
can deliver the required luminosity, since the beam parameters are very
different than for either conventional RF or plasma based schemes [11, 12,
13]. For the DLA case, beam-loading and efficiency considerations limit the
macrobunch population to order of $10^{6}$ particles whereas space-charge
limitations require splitting the macro-bunch into a train of microbunches
with separation determined by the laser wavelength (here $\lambda$ = 2
$\mu$m). These microbunches naturally have sub-fs bunch duration, as has
recently been demonstrated in net acceleration experiments [22, 23].
Consequently, the laser repetition rate must be of order 10 to 100 MHz in
order to deliver the required average current. This requirement is readily met
by current solid state fiber lasers, which have compact footprint (e.g. rack-
mounted), $\mu$J pulse energies, and multi-MHz repetition rates, in contrast
to the high peak power lasers used for plasma acceleration [10]. In Table 1,
we present a typical set of parameters for the ILC [1] and DLA cases such that
in both cases the bare luminosity is the same. The superscripted DLA1 and DLA2
denote the two test cases (Case 1 and Case 2) discussed in Section III.

In the DLA scenario, the entire train of $M$ microbunches cross together at
the IP, producing a total of $M^{2}$ crossings per laser pulse, whereas in RF
or plasma-based schemes, the bunches in each train are sufficiently far apart
that only one pair of bunches interact at a given time. This difference in
bunch format has significant implications for the beam-loaded efficiency,
luminosity, and energy loss dynamics. For a given repetition rate and number
of particles the primary way to enhance the luminosity is to reduce the
transverse beam size at the IP. This happens naturally as electrons and
positrons collide, producing a pinching effect due to the lensing of each
beam’s charge distribution on the other.

It is our goal in this study to investigate this process for colliding
electron-positron bunch trains with DLA like parameters. Both electrons and
positrons are assumed to be equally affected (strong-strong interaction) as
each microbunch experiences the quasi-periodic force experienced by the
counter-propagating species. As each microbunch is deflected by the counter-
propagating train, radiation (beamstrahlung) is emitted. We note that our
analytical formulation is agnostic as to which of the two beams contains
electron vs. positrons. Since the colliding beams are identical, the resulting
beam envelope analysis applies to both beams, given that the relative charge
between them has the product $-e^{2}N^{2}$. Our analysis considers the impact
of the bunch train structure on the emitted radiation, but we ignore $e^{+}$
$e^{-}$ pair production.

In the following section, we derive the relation for the luminosity as a
function of number of microbunches $M$ corresponding to optimal beam-loading
efficiency. We then subject this relation in Section III to constraints upon
the material damage limits, power consumption, and bunch charge to obtain two
sets of example collider parameters for further analysis. We then derive the
multi-bunch equations of motion and energy loss relations in Sections V, VI,
and VII. In Section VIII we present numerical solutions for the considered
beam parameters up to $M=150$, followed by analysis and concluding remarks in
Section IX.

##  II Microbunch Efficiency

For a multi-bunch beam with a constant unloaded gradient $G_{0}$, the
corresponding loaded gradient $G_{L}$ at a given bunch position will vary from
bunch to bunch due to the wakes from prior bunches, resulting in energy spread
between the bunches. In order to mitigate this effect and produce a constant
loaded gradient, the laser pulse may be tapered so as to counteract the bunch
wakes and level out the loaded gradient. As discussed in [13], the axial
longitudinal field in this scenario has the form

$E_{z}(z,t)=\mathcal{E}(z,t)\cos(\omega t)$ (1)

where the laser field amplitude function $\mathcal{E}(z,t)$ is a flat pulse of
amplitude $\mathcal{E}_{0}$ over a duration $\tau_{D}$ followed by a step of
duration $\tau_{B}$ and magnitude $(1+\delta)\mathcal{E}_{0}$. The wake
correction factor $\delta\equiv 2kqL/(\mathcal{E}_{0}\lambda)$ is derived in
Ref. [13]. Here $q$ is the micro-bunch charge, $\lambda$ is the laser
wavelength and microbunch spacing, and $L$ is the length of a single
acceleration stage. We additionally introduce the fundamental mode loss factor
$k$ and the Cherenkov wake loss factor $h$, given by [24, 25]

$k=\frac{1}{4}\frac{cZ_{C}}{\lambda^{2}}\frac{\beta_{g}}{1-\beta_{g}}\ \ ;\ \
h=\frac{cZ_{H}}{\lambda^{2}}$ (2)

where $Z_{C}$ is the characteristic impedance of the fundamental mode, $Z_{H}$
is the Cherenkov wake impedance, and $\beta_{g}=v_{g}/c$ is the group velocity
normalized to light speed. The interaction time is limited by the number of
bunches $M$ and laser pulse duration $\tau_{P}=\tau_{B}+\tau_{D}$ where

$\tau_{B}=(M-1)\lambda/c\ \ ;\ \
\tau_{D}=\frac{1-\beta_{g}}{\beta_{g}}\frac{L}{c}\ .$ (3)

The corresponding unloaded and loaded gradients seen by the $i$’th bunch are
then given by

$G_{0}(i)=\mathcal{E}_{0}+(i-1)2kq\ ;\ G_{L}=\mathcal{E}_{0}-hq$ (4)

where the extra factor of $(i-1)2kq$ is the wake correction to flatten the
loaded gradient. The unloaded gradient averaged over all $M$ bunches is then
$\langle G_{0}\rangle=\mathcal{E}_{0}+kq(M-1)$. A detailed treatment is
discussed in [13]. We will not repeat the derivation there but merely note
that the corresponding $M$-dependent beam-loaded energy transfer efficiency is
given by

$\eta(q,M)=\frac{4Mkq(\mathcal{E}_{0}-hq)}{\mathcal{E}_{0}^{2}[1+\frac{\beta_{g}}{1-\beta_{g}}(M-1)\frac{\lambda}{L}(1+\frac{2kq}{\mathcal{E}_{0}}\frac{L}{\lambda})^{2}]}$
(5)

We can obtain from this the optimal charge $q_{\text{opt}}$ corresponding to
maximum efficiency $\eta_{\text{opt}}$ by solving $\partial\eta/\partial q=0$
for $q$, which gives

$q_{\text{opt}}=\frac{\mathcal{E}_{0}}{h(1+\chi)}\ ,$ (6)
$\eta_{\text{opt}}=\frac{4M}{1+\chi}\frac{\eta_{\text{sc}}(\frac{\chi}{\chi+1})}{1+\kappa\xi(1+\frac{2\eta_{\text{sc}}}{\xi(1+\chi)})^{2}}$
(7)

where $\eta_{\text{sc}}\equiv k/h$ is the single-bunch efficiency,
$\xi\equiv\lambda/L$ and for the sake of compactness we define the
dimensionless parameters

$\kappa\equiv\frac{\beta_{g}}{1-\beta_{g}}(M-1)\ ;\
\chi\equiv\sqrt{\frac{\xi+\kappa(2\eta_{\text{sc}}+\xi)^{2}}{\xi(1+\kappa\xi)}}$
(8)

We see that in the limit $M=1$, $\kappa\rightarrow 0$ and $\chi\rightarrow 1$,
and we recover the single bunch optimal efficiency and charge
$q_{\text{opt}}=\mathcal{E}_{0}/2h$ and $\eta_{\text{opt}}=\eta_{\text{sc}}$.

For a given number of bunches $M$, the value of $q_{\text{opt}}$ determines a
corresponding total macrobunch population
$N_{\text{opt}}(M)=Mq_{\text{opt}}(M)/e$. Let us assume that we want the
average geometrical luminosity rate to match some reference value
$\bar{\mathcal{L}}_{\text{ref}}$. Then for a round beam we require

$\bar{\mathcal{L}}_{0}(M)=\frac{N_{\text{opt}}(M)^{2}f_{\text{rep}}}{2\pi\sigma_{0}^{2}}=\bar{\mathcal{L}}_{\text{ref}}\
\ ;\ \sigma_{0}^{2}=\epsilon_{n}\beta^{*}/\gamma\beta$ (9)

where $\sigma_{0}$ is the incident radial beam size, $\beta^{*}$ is the beta
function at the IP, $f_{\text{rep}}$ is the repetition rate of the laser, and
$\epsilon_{n}$ is the radial normalized transverse emittance. Here
$N_{\text{opt}}^{2}(M)$ highlights the fact that $N_{\text{opt}}$ is a
function of bunch number $M$.

##  III Parameter Optimization

In evaluating Eq. (9), we may treat $\mathcal{E}_{0}$, $f_{\text{rep}}$,
$\epsilon_{n}$, and $\lambda$ as free parameters to be varied within a
reasonable range. Our choice of parameters must also be constrained by at
least three additional concerns: (1) the peak field in the structure should
not exceed the damage limit of the material, (2) the required wallplug power
should be within reasonable bounds, and (3) the single-bunch charge $N/M$
should not exceed $10^{5}$ electrons/positrons to avoid space-charge blowup of
the beam, as discussed in [21]. We emphasize that we do not here attempt to
determine a global optimum solution over the space of all free parameters.
Rather, we evaluate Eq. (9) with the above constraints in mind and present a
self-consistent parameter set satisfying them which we use in subsequent
sections to investigate microbunch luminosity enhancement. For the accelerator
structure itself, we use the design of [26] which is a silicon dioxide
photonic crystal, having a cylindrical vacuum bore, leading to a cylindrical
cavity mode, which is favorable to the case of a linear collider [21]. The
relevant structure parameters are given in Table 2.

Table 2: Structure parameters corresponding to a cylindrical Bragg geometry [27].  Parameter | Symbol | Value
---|---|---
Normalized Group Velocity | $\beta_{g}$ | 0.5
Laser Wavelength ($\mu$m) | $\lambda$ | 2
Single-Stage Length (mm) | $L$ | 0.2
Channel Radius ($\mu$m) | $a$ | 1.0
Characteristic Impedance ($\Omega$) | $Z_{C}$ | 115
Cherenkov Impedance ($\Omega$) | $Z_{H}$ | 240
Fundamental Loss Factor ($\times 10^{20}$ V/m) | $k$ | 21.5
Cherenkov Loss Factor ($\times 10^{20}$ V/m) | $h$ | $180$
Damage Factor | $f_{D}$ | 2

To evaluate the damage limit, we use the empirical formula for laser induced
damage fluence $F(\tau)$ of silicon dioxide, as reported originally in [28]
and interpreted by [12]. For the range of pulse lengths of interest (0.4 ps
$<\tau<10$ ps), $F(\tau)\approx(2.51\
\text{J/cm}^{2}/\text{s}^{1/4})\tau^{1/4}$. The corresponding peak field for a
Gaussian mode profile is

$E_{\text{th}}=\sqrt{\frac{2Z_{0}\epsilon_{r}F(\tau)}{\tau}}$ (10)

where $Z_{0}$ is the impedance of free space and $\epsilon_{r}=2.13$ is the
dielectric constant of silicon dioxide. We take $E_{\text{th}}$ to correspond
to the peak field in the structure $E_{\text{peak}}$ which is related to the
maximum axial field amplitude $\mathcal{E}_{\text{max}}$ by the so-called
”damage factor” $f_{D}=E_{\text{peak}}/\mathcal{E}_{\text{max}}$. We equate
the maximum field tolerated by the structure to the value $E_{\text{th}}$
above evaluated at the laser pulse duration $\tau_{P}$ and thereby obtain the
threshold value $\mathcal{E}_{\text{max}}=E_{\text{th}}(\tau_{P})/f_{D}$,
which the axial field must not exceed in order to avoid damage to the
structure. For the cylindrical Bragg geometry of Table 2, $f_{D}\approx 2$.

The wall-plug power is given by

$P_{\text{wall}}=\frac{P_{\text{beam}}}{\eta_{L}\ \eta_{c}\
\eta_{\text{opt}}}\ ;\ P_{\text{beam}}=N_{\text{opt}}\gamma
mc^{2}f_{\text{rep}}$ (11)

where $\eta_{L}$ is the laser wall-plug efficiency, $\eta_{c}$ is the optical
coupling efficiency to the structure, $\eta_{\text{opt}}$ is the optimized
coupling efficiency of the field to the beam as given by Eq. (5), and
$P_{\text{beam}}$ is the beam power. Thulium fiber lasers have demonstrated
$>50\%$ wall plug efficiency and efficiencies as high as 82% are theoretically
possible [29]. For our calculations, we take $\eta_{L}=75\%$. Since the
optical coupling of the laser to the structure must be repeated many times
(once for each stage of acceleration), optical losses must be carefully
controlled, with coupling efficiency close to 100%. We here assume 10% optical
coupling losses or $\eta_{c}=90\%$.

In Table 3 we list a self-consistent set of accelerator parameters satisfying
the constraints imposed above on damage limits, power consumption, and
microbunch charge. We note that these are similar to the parameters from [20]
in terms of assumed bunch charge, gradient, and efficiency. The top portion of
the table represents starting parameters, and the bottom portion are the
calculated results arising from solution of Eq. (9).

Table 3: Parameters and bunch number $M_{\text{opt}}$ for matching to the reference luminosity $\bar{\mathcal{L}}_{\text{ref}}$ in Eq. (9).  Parameter | Symbol | Value
---|---|---
Beam Energy (GeV) | $\gamma mc^{2}$ | 500
Field Intensity (GV/m) | $\mathcal{E}_{0}$ | 1.1
Train Repetition Rate (MHz) | $f_{\text{rep}}$ | 168
IP Normalized Emittance (nm) | $\epsilon_{n}$ | 4.65
IP Spot Size (nm) | $\sigma_{0}$ | 0.76
Laser Pulse Duration (ps) | $\tau_{P}$ | 1.3
Laser Wallplug Efficiency | $\eta_{L}$ | 0.75
Optical Copuling Efficiency | $\eta_{c}$ | 0.90
Wake Correction (GV/m) | $\mathcal{E}_{0}(1+\delta)$ | 2.51
Field Threshold (GV/m) | $\mathcal{E}_{\text{max}}$ | 2.85
Unloaded Gradient (GeV/m) | $\langle G_{0}\rangle$ | 1.80
Loaded Gradient (GeV/m) | $G_{L}$ | 1.04
Optimal Number of Bunches | $M_{\text{opt}}$ | 100
Microbunch Population ($\times 10^{4}$) | $q_{\text{opt}}/e$ | 2.05
Bunch Train Population ($\times 10^{6}$) | $N_{\text{opt}}$ | 2.05
IP Beta Function ($\mu$m) | $\beta^{*}$ | 121
Normalized Plasma Frequency | $\Omega^{2}$ | 0.042
Normalized Emittance | $\Theta^{2}$ | 0.00027
Coupling Efficiency | $\eta_{\text{opt}}$ | 39%
Wallplug Power (MW) | $P_{\text{wall}}$ | 103

In Fig. 1(a) we plot the constrained luminosity given by Eq. (9) as a function
of $M$ for the test case in Table 3. The horizontal orange line corresponds to
the ILC reference $\bar{\mathcal{L}}_{\text{ref}}$ = $2\times 10^{34}$ cm-2
s-1, and the vertical dashed line marks the intersection point with the
luminosity curves. These points indicate the corresponding number of bunches
$M_{\text{opt}}$ that satisfy both equalities in Eq. (9). The same points
marked in Fig. 1(b) and (c) indicate the corresponding values for optimal
number of particles $N_{\text{opt}}$ and the resulting averaged unloaded and
loaded gradients $\langle G_{0}\rangle$ and $G_{L}$. Although higher
luminosities are indicated in Fig. 1(a) for $M>M_{\text{opt}}$ for either
case, these higher luminosity values do not necessarily satisfy the imposed
constraints on damage limits, wallplug power, and microbunch charge. The
parameters in Table 3 are summarized by the two dimensionless quantities
$\Omega$ and $\Theta$ which are the normalized plasma frequency and emittance
parameters to be explained in Section V. These two quantities will represent
the primary input variables for the beam dynamics equations.

Figure 1: Constrained parameters from Eq. (9) as function of number $M$ of
bunches, showing (a) macrobunch population (black) and optimal efficiency
(gray); (b) corresponding field intensity parameters; and (c) corresponding
laser repetition rate (black) and wallplug power (gray). All points on the
curves are matched to $\bar{\mathcal{L}}_{\text{ref}}$ = $2\times
10^{34}\text{cm}^{-2}\text{s}^{-1}$. Vertical blue dashed lines mark the value
$M=100$ corresponding to the parameters of Table 3.

##  IV Luminosity Enhancement

The single-collision geometrical luminosity $\mathcal{L}$ for a pair of beams
with number density distributions $\rho_{1}(\textbf{r},t)$,
$\rho_{2}(\textbf{r},t)$ and velocities $\pmb{\beta}_{1}$, $\pmb{\beta}_{2}$
(normalized by speed of light $c$) may be written in relativistically
invariant form [30]

$\mathcal{L}=\int S(\textbf{r},t)d^{3}rdt$ (12)

where $S=\mathcal{K}\rho_{1}\rho_{2}$ and $\mathcal{K}$ is the Moller
luminosity factor.

$\mathcal{K}=c\sqrt{|\pmb{\beta}_{1}-\pmb{\beta}_{2}|^{2}-|\pmb{\beta}_{1}\times\pmb{\beta}_{2}|^{2}}$
(13)

For counter-propagating ultra-relativistic Gaussian beams, $\mathcal{K}=2c$
and Eq. (12) yields the familiar form

$\mathcal{L}=\frac{N_{1}N_{2}}{4\pi\frac{1}{2}[(\sigma_{1x}^{2}+\sigma_{2x}^{2})(\sigma_{1y}^{2}+\sigma_{2y}^{2})]^{1/2}}$
(14)

where $N_{1}$ and $N_{2}$ are number of particles in beams 1 and 2 and
$\sigma_{1x}$, $\sigma_{2x}$, $\sigma_{1y}$, $\sigma_{2y}$ are the Gaussian
widths at the IP. The corresponding average luminosity per unit time is then
$\bar{\mathcal{L}}=\mathcal{L}f_{\text{rep}}$, where $f_{\text{rep}}$ is the
bunch repetition rate. If the two colliding beams each consist of a train of
$M$ microbunches with $n=N/M$ particles per microbunch, it has previously been
assumed for DLA like scenarios [20], where the full train of microbunches is
generated by a single sub-picosecond laser pulse, that $N\rightarrow Mn$ in
Eq. (15). Since the microbunches are closely spaced at the laser wavelength
$\lambda$ within each macrobunch, the constituent bunch trains effectively
collide in unison. This differs from bunch trains in conventional RF
accelerator scenarios, where instead one makes the replacements $N\rightarrow
n$ and $f_{\text{rep}}\rightarrow Mf_{\text{rep}}$ and so the average
luminosity scales as $Mn^{2}$ rather than $(Mn)^{2}$.

The luminosity and corresponding enhancement for the collision of microbunch
$i$ of one beam with microbunch $j$ of the other beam may be written,
following from Eq. (14),

$\mathcal{L}_{ij}\equiv\frac{n^{2}}{4\pi}\frac{4}{(\sigma_{i}^{2}+\sigma_{j}^{2})}\
;\
\mathcal{H}_{ij}\equiv\frac{\mathcal{L}_{ij}}{\mathcal{L}_{0}}=\frac{2\sigma_{0}^{2}}{\sigma_{i}^{2}+\sigma_{j}^{2}}$
(15)

where we assume initially round beams with the same number of particles
($N_{1}=N_{2}=N=Mn$) but different radial Gaussian beam sizes
($\sigma_{ix}=\sigma_{iy}=\sigma_{i}/\sqrt{2}$,
$\sigma_{jx}=\sigma_{jy}=\sigma_{j}/\sqrt{2}$). During each crossing, space
charge attraction will tend to reduce the transverse sizes of the two beams.
If both beams are assumed initially identical and round (but Gaussian and with
opposite charge) in the absence of the other, with nominal IP spot size
$\sigma_{0}$, then the corresponding geometrical luminosity is
$\mathcal{L}_{0}=N^{2}/2\pi\sigma_{0}^{2}$. The enhanced luminosity,
accounting for the possibility of different sizes of the crossing beams, is of
the form of Eq. (15), and so the multi-bunch luminosity and enhancement factor
may be written

$\mathcal{L}=\sum_{i,j=1}^{M}\mathcal{L}_{ij}\ \ ;\
\mathcal{H}_{D}=\frac{\mathcal{L}}{\mathcal{L}_{0}}=\frac{1}{M^{2}}\sum_{i,j=1}^{M}\frac{2\sigma_{0}^{2}}{(\sigma_{i}^{2}+\sigma_{j}^{2})}$
(16)

where $\sigma_{0}$ is the nominal radial RMS IP spot size of either beam in
the absence of the other and $\sigma_{i}$ and $\sigma_{j}$ are assumed to be
evaluated within the sum at the corresponding time at which bunches $i$ and
$j$ cross. Figure 2 illustrates the schematics of the collision process for
the simple example of three microbunches ($M=3$) in each train. We note by
virtue of Eq. (12) that in principle the crossing of each microbunch should be
integrated in time, since the spot sizes are functions of $t$. However, due to
the small bunch charges and short (sub-optical) time scales of each
microbunch, the resulting correction is small. In the following section, we
derive an analytical formulation for the transverse dynamics for colliding
microbunch trains to numerically evaluate Eqs. (16).

Figure 2: Schematic of the $e^{+}e^{-}$ collision process with $M=3$
microbunches per train. At $t=0$ the first microbunches of each train collide.
For an arbitrary $M$, the full duration of the interactions is $\Delta
t=(M-1)\lambda/c$ with a total of $M^{2}$ microbunch crossings.

##  V Equations of Motion - Constant Energy

The theory of beam-beam interaction at the IP of a collider has been
extensively developed by prior authors [31, 32, 33, 34]. We extend these
results to consider trains of colliding electron and positron bunches. We
initially consider the case of constant energy, to be followed in the
subsequent section by inclusion of the beamstrahlung energy losses due to the
radiation reaction force (RRF). We begin by considering a single particle
interacting with the transverse fields produced by the counter-propagating
train of $M$ microbunches in the other beam, which we may represent by the
number density function

$\rho(\textbf{r},t)=\frac{1}{M}\sum_{j=1}^{M}F_{j}(\textbf{r}_{\perp})G_{j}(z,t)$
(17)

where
$\textbf{r}_{\perp}=r\hat{\textbf{r}}=x\hat{\textbf{x}}+y\hat{\textbf{y}}$ is
the radial coordinate and $F_{j}$, $G_{j}$ denote the transverse and
longitudinal distribution functions for a round beam

$\displaystyle F_{j}(\textbf{r}_{\perp})\equiv
2[\mathcal{G}(r,\sigma_{j})]^{2},$ (18) $\displaystyle
G_{j}(z,t)\equiv\mathcal{G}(z+\beta ct-(j-1)\lambda,\sigma_{z}).$ (19)

Here
$\mathcal{G}(\xi,\sigma)\equiv[1/\sqrt{2\pi}\sigma]\exp[-\xi^{2}/(2\sigma^{2})]$
is a generic Gaussian, and $\sigma_{j}$, $\sigma_{i}$ are assumed to be time-
dependent functions, while the length of the pulses $\sigma_{z}$ is assumed
constant in time and the same for all bunches. A particle of charge $e$ sees a
time-dependent transverse focusing force due to $\rho(\textbf{r},t)$ of the
counter-propagating beam, which may be written (see Derivation in Appendix A)

$\textbf{F}_{\perp}(\textbf{r}_{\perp},z,t)=-\gamma_{0}mK^{2}(z,t)\
\textbf{r}_{\perp}\ ,$ (20)

where $m$ is the electron/positron mass, $\gamma_{0}$ is the initial particle
Lorentz energy factor, $K^{2}$ is the linear focusing term,

$K^{2}(z,t)\equiv\frac{c}{\lambda}\frac{\Omega^{2}}{M}\sum_{j=1}^{M}\frac{\sigma_{0}^{2}}{\sigma_{j}^{2}}\
\mathcal{G}\left(\frac{t}{2}+\frac{z-(j-1)\lambda}{2c},\sigma_{t}\right)$ (21)

and we define the normalized plasma frequency

$\Omega^{2}\equiv\frac{2Nr_{e}\lambda}{\gamma_{0}\sigma_{0}^{2}}\ .$ (22)

Here $r_{e}$ is the classical electron radius and
$\sigma_{t}\equiv\sigma_{z}/2c$ is the interaction time of the bunches. The
factor of $2$ here is due to the fact that in the lab frame, the beams have a
relative velocity of $2c$. We have also made use of the fact that the generic
Gaussian function satisfies the transformation property
$\mathcal{G}(\xi,\sigma)=(1/a)\mathcal{G}(a\xi,a\sigma)$. Invoking the
relativistic form of Newton’s second law, $\partial_{t}(\gamma
m\dot{\textbf{r}}_{\perp})=\textbf{F}_{\perp}$, we thus obtain for the case of
constant energy ($\gamma=\gamma_{0}$) the equations of motion

$\ddot{\textbf{r}}_{\perp}+K_{i}^{2}\textbf{r}_{\perp}=0\ ,$ (23)

where overhead dots denote $t$ derivatives and

$K_{i}^{2}\equiv\frac{c}{\lambda}\frac{\Omega^{2}}{M}\sum_{j=1}^{M}\frac{\sigma_{0}^{2}}{\sigma_{j}^{2}}\
\mathcal{G}\left(t-\frac{\lambda}{c}\left[\frac{i+j}{2}-1\right],\sigma_{t}\right)\
.$ (24)

We note that $K_{i}$ is merely $K(z,t)$ evaluated at $z=ct-(i-1)\lambda$,
thereby taking the test particle to lie in the $i$’th microbunch of the
incident beam. To simplify future calculations, let us define the normalized
time coordinate $u\equiv ct/\lambda$ and cast Eqs. (23,24) into the forms

$\textbf{r}_{\perp}^{\prime\prime}+\hat{K}_{i}^{2}\textbf{r}_{\perp}=0$ (25)
$\hat{K}_{i}^{2}\equiv\frac{\Omega^{2}}{M}\sum_{j=1}^{M}\frac{\sigma_{0}^{2}}{\sigma_{j}^{2}}\
\mathcal{G}(u-u_{ij},\sigma_{u})\\\ $ (26)

where $u_{ij}\equiv(i+j)/2-1$ is the crossing time of bunches $i$ and $j$, and
$\sigma_{u}\equiv\sigma_{t}c/\lambda$ is the microbunch duration in our
normalized units. By analogy with the usual equation [35] for the evolution of
the transverse beam size under the action of a linear focusing force, we can
write the corresponding envelope equation

$\bar{\sigma}_{i}^{\prime\prime}+\hat{K}_{i}^{2}\bar{\sigma}_{i}-\frac{\Theta^{2}}{\bar{\sigma}_{i}^{3}}=0\
\ ;\ \ i=1,2,...,M$ (27)

with the associations

$\Theta\equiv\frac{\lambda\epsilon_{0}}{\sigma_{0}^{2}}\ \ ;\ \
\bar{\sigma}_{i}\equiv\frac{\sigma_{i}}{\sigma_{0}}$ (28)

where $\epsilon_{0}$ is the initial transverse RMS emittance. Eqs. (27)
represent $M$ equations in the $M$ unknown functions $\bar{\sigma}_{i}(u)$.
Due to the symmetry of the problem, the solutions for $\bar{\sigma}_{i}$ apply
to both the electron and positron beam. This is referred to as a strong-strong
interaction since the dynamics of both species vary in the course of the
interaction. We can associate the plasma frequency appearing in this equation
with $\Omega^{2}=D/(M-1)$ where $D=Nr_{e}\ell/(\gamma\sigma_{0}^{2})$ is the
disruption parameter of the macrobunch of length $\ell=(M-1)\lambda$. The
dimensionless quantity $\Theta$ is a normalized angular momentum term which
gives rise to the transverse beam emittance. Its definition in Eq. (28)
ensures that if the independent variable in Eq. (27) is cast into units of
path length $s=u\lambda=ct$ that the $\Theta^{2}$ term assumes the usual form
of the emittance pressure $\epsilon_{0}^{2}/\sigma^{3}$. Obviously this
formulation is only valid close to the IP, and it suffices to set the initial
conditions $\bar{\sigma}_{i}(0)=1$ and $\bar{\sigma}_{i}^{\prime}(0)=0$.
Evidently, in the framework of this formulation, subject to the assumption
that initially all the microbunches are identical, there are three degrees of
freedom: the normalized emittance $\Theta$, plasma frequency $\Omega$ and
number of microbunches $M$.

##  VI Equations of Motion with RRF

We now extend the theory to include the radiation reaction force (RRF) and
thus the particle energy loss. Our analysis is limited to the classical regime
under the assumption that the beamstrahlung parameter $\Upsilon\ll 1$ for
cases of interest. The correctness of this assumption will be shown later. The
relativistic theory of radiation reaction of classical point electrons derived
by Dirac has the following Lorentz invariant 4-vector form:

$\frac{dp_{\mu}}{d\tau}=\mathcal{F}_{\mu}^{\text{rad}}+\mathcal{F}_{\mu}^{\text{ext}}$
(29)

where $\tau$ is the proper time, $p_{\mu}$ is the 4-vector momentum, and
$\mathcal{F}_{\mu}^{\text{rad}}$ and $\mathcal{F}_{\mu}^{\text{ext}}$ are the
Lorentz invariant forms of the radiation reaction force and the externally
applied force respectively. By using the requirement that any force must
satisfy $\mathcal{F}_{\mu}p^{\mu}=0$, the following form for the radiation
force in the $(1,-1,-1,-1)$ metric may be written:

$\mathcal{F}_{\mu}^{\text{rad}}=\frac{2}{3}\frac{r_{e}}{c}\left[\frac{d^{2}p_{\mu}}{d\tau^{2}}+\frac{p_{\mu}}{(mc)^{2}}\left(\frac{dp_{\nu}}{d\tau}\frac{dp^{\nu}}{d\tau}\right)\right]$
(30)

where $r_{e}$ is the classical electron radius. Defining the normalized
momenta $q^{\nu}\equiv p^{\nu}/(mc)=(\gamma,\textbf{q})$, where
$\textbf{q}\equiv\gamma\pmb{\beta}$, Eq. (29) may be expressed to good
approximation by (see derivation in Appendix C)

$\displaystyle\gamma^{\prime}$
$\displaystyle\simeq\pmb{\beta}\cdot\bar{\textbf{F}}_{\perp}-\alpha_{e}|\textbf{q}^{\prime}|^{2}|\bar{\textbf{F}}_{\perp}|^{2}$
(31) $\displaystyle\textbf{q}^{\prime}$
$\displaystyle\simeq\bar{\textbf{F}}_{\perp}-\alpha_{e}\gamma\textbf{q}|\bar{\textbf{F}}_{\perp}|^{2}$
(32)

where primes denote derivatives with respect to the normalized time
$u=ct/\lambda$, $\bar{\textbf{F}}\equiv\lambda\textbf{F}/(mc^{2})$ is the
normalized external force and $\alpha_{e}\equiv c\tau_{e}/\lambda$, with
$\tau_{e}\equiv 2r_{e}/(3c)$. Noting by the arguments of the prior section
that our form for the external force is
$\bar{\textbf{F}}_{\perp}=-(\gamma_{0}/\lambda)\hat{K}_{i}^{2}\textbf{r}_{\perp}$,
these equations of motion lead to the following modified envelope equations
for the $i$’th microbunch:

$\displaystyle\bar{\sigma_{i}}^{\prime\prime}+\frac{1}{\bar{\gamma}_{i}}\hat{K}_{i}^{2}\bar{\sigma}_{i}-\frac{\Theta^{2}}{\bar{\sigma}_{i}^{3}}+\frac{\bar{\gamma}_{i}^{\prime}}{\bar{\gamma}_{i}}\bar{\sigma}_{i}^{\prime}+$
$\displaystyle\bar{\alpha}_{e}\hat{K}_{i}^{4}\bar{\sigma}_{i}^{\prime}\bar{\sigma}_{i}^{2}\bar{\gamma}_{i}=0$
(33)
$\displaystyle\bar{\gamma}_{i}^{\prime}+\bar{\alpha}_{e}\hat{K}_{i}^{4}\bar{\sigma}_{i}^{2}\bar{\gamma}_{i}^{2}$
$\displaystyle=0$

where we define the additional dimensionless parameters

$\bar{\alpha}_{e}\equiv\alpha_{e}\gamma_{0}^{2}\left(\frac{\sigma_{0}}{\lambda}\right)^{2}\
\ ;\ \ \bar{\gamma}_{i}\equiv\gamma_{i}/\gamma_{0}$ (34)

Equations (33), derived in detail in Appendix D, represent $2M$ equations in
$2M$ unknowns ($\bar{\sigma}_{i}$ and $\bar{\gamma}_{i}$) subject to the
initial conditions $\bar{\sigma}_{i}(0)=1$, $\bar{\sigma_{i}}^{\prime}(0)=0$,
and $\bar{\gamma}_{i}(0)=1$. The primary assumptions underlying these
equations are: (1) that the interaction is classical ($\Upsilon\ll 1$), (2)
that the external beam-beam force is linear in $\textbf{r}_{\perp}$, and (3)
the approximation of Landau and Lifshitz, whereby momentum derivatives
appearing in $\mathcal{F}_{\mu}^{\text{rad}}$ may to first order be
approximated by their usual relativistic forms in the absence of RRF. The
quantities multiplied by $\bar{\alpha}_{e}$ represent the RRF contributions.
We see further that if these terms are neglected in the limit
$\bar{\gamma}_{i}^{\prime}\rightarrow 0$, Eq. (33) reduce to the form of Eq.
(27) for constant energy $\gamma_{i}=\gamma_{0}$.

An additional observation is that the normalized emittance under the action of
the RRF is not a constant of motion, as it is in the case of simple
acceleration under linear focusing, but rather varies in proportion to
$\gamma$. By virtue of this observation, a third set of equations is implicit
in Eq. (33) of the form $\epsilon_{n,i}(u)=\gamma_{i}(u)\epsilon_{0}$, where
$\epsilon_{n,i}$ is the normalized emittance of the $i$’th microbunch and
$\epsilon_{0}$ is the initial RMS (un-normalized) emittance. This result is
also derived and explained in Appendix D. To our knowledge this effect has not
been previously noted.

##  VII Energy Loss

The beamstrahlung parameter is a measure of the degree to which an
electromagnetic interaction may be described classically, given by the Lorentz
invariant

$\Upsilon=\frac{e\hbar}{m^{3}c^{4}}[p_{\mu}F^{\mu\xi}p^{\nu}F_{\xi\nu}]^{1/2}$
(35)

where $p_{\mu}$ is the 4-momentum and $F_{\mu\nu}$ is the mean energy-momentum
stress tensor of the beam field. Letting $E_{c}=(m^{2}c^{3})/(e\hbar)$ denote
the Schwinger critical field, we find for a purely transverse force
$\textbf{F}_{\perp}$ that

$\Upsilon=\gamma\frac{|\textbf{F}_{\perp}|}{eE_{c}}$ (36)

Noting that the transverse force in the lab frame is given by
$\textbf{F}_{\perp}=-\gamma_{0}mK_{i}^{2}\textbf{r}_{\perp}$, averaging over
the transverse beam distribution $F_{i}(\textbf{r}_{\perp})$, and defining the
normalized transverse focusing for interaction $(i,j)$ by

$\hat{K}_{ij}^{2}\equiv\frac{\Omega^{2}}{M}\frac{1}{\bar{\sigma}_{j}^{2}}\mathcal{G}(u-u_{ij},\sigma_{u})$
(37)

where $u_{ij}\equiv(i+j)/2-1$, we obtain the following form for the spatially
averaged beamstrahlung parameter for interaction $(i,j)$:

$\Upsilon_{ij}(u)=\bar{\gamma}_{i}\bar{\sigma}_{i}\frac{\lambdabar_{c}}{\lambda}\frac{\sqrt{\pi}}{2}\left(\frac{\sigma_{0}}{\lambda}\right)\gamma_{0}^{2}\hat{K}_{ij}^{2}$
(38)

where $\lambdabar_{c}=\hbar/(mc)$ is the reduced Compton wavelength. Let us
define the time averaged beamstrahlung parameter by integrating over $u$:

$\langle\Upsilon_{ij}\rangle\equiv\frac{5}{6}\frac{1}{\sqrt{2\pi}2\sigma_{u}}\int_{-\infty}^{\infty}\Upsilon_{ij}(u)du$
(39)

Here we include the extra factor of $5/6$ which appears in some formulations,
noting that this factor is merely for convenience and has no physical
significance. For short bunches we may approximate the function
$\mathcal{G}(u,\sigma_{u})$ by a Dirac delta function and simply evaluate the
integrand at the time $u_{ij}$ when bunches $i$ and $j$ are overlapped. We
thus obtain

$\langle\Upsilon_{ij}\rangle\simeq\frac{5}{6}\frac{\gamma_{0}^{2}}{4\sqrt{2}\sigma_{u}}\frac{\lambdabar_{c}}{\lambda}\left(\frac{\sigma_{0}}{\lambda}\right)\frac{\Omega^{2}}{M}\frac{\bar{\gamma}_{i}\bar{\sigma}_{i}}{\bar{\sigma}_{j}^{2}}\bigg{|}_{u=u_{ij}}$
(40)

For comparison with the usual form of this parameter, we may recast the above
relation into the equivalent form

$\langle\Upsilon_{ij}\rangle\simeq\Upsilon_{\text{sc}}\times\frac{\bar{\gamma_{i}}\bar{\sigma}_{i}}{\bar{\sigma}_{j}^{2}}\bigg{|}_{u=u_{ij}}\
;\ \
\Upsilon_{\text{sc}}\equiv\frac{5}{6}\frac{\gamma_{0}r_{e}^{2}(N/M)}{\alpha\sigma_{z}\sqrt{2}\sigma_{0}}$
(41)

We recognize the expression for $\Upsilon_{\text{sc}}$ as the familiar form
for the single-crossing beamstrahlung parameter for two bunches of equal size
containing $N/M$ particles. Since $\sigma_{0}$ is here the radial RMS beam
size, for comparison with the usual formula we would make the replacement
$\sigma_{0}\rightarrow(\sigma_{x}+\sigma_{y})/\sqrt{2}$. We further see that
in the limit of a single crossing ($M\rightarrow 1$) that the correction term
is unity at $u=0$ and hence
$\langle\Upsilon_{ij}\rangle\rightarrow\Upsilon_{\text{sc}}$ as expected. The
average beamstrahlung parameter may be obtained by summing over $i$ and $j$:

$\langle\Upsilon\rangle=\frac{1}{M^{2}}\sum_{i,j=1}^{M}\langle\Upsilon_{ij}\rangle\
.$ (42)

The corresponding fractional energy loss for the $(i,j)$ interaction may be
obtained by integrating $\bar{\gamma}^{\prime}$ in Eq. (33), yielding

$\delta\bar{\gamma}_{ij}=\frac{\bar{\alpha}_{e}}{2\sqrt{\pi}\sigma_{u}}\frac{\Omega^{4}}{M^{2}}\frac{\bar{\gamma_{i}}^{2}\bar{\sigma}_{i}^{2}}{\bar{\sigma}_{j}^{4}}\bigg{|}_{u=u_{ij}}\
.$ (43)

Here we use the fact that for a Gaussian function $\mathcal{G}$, $\int
f(\xi)[\mathcal{G}(\xi-\xi_{0},\sigma)]^{2}d\xi\approx
f(\xi_{0})/(2\sqrt{\pi}\sigma)$. Combining Eqs. (43) and (40), we obtain:

$\delta\bar{\gamma}_{ij}=\frac{192}{25\sqrt{\pi}}\frac{\alpha}{\Gamma_{0}}\langle\Upsilon_{ij}\rangle^{2}$
(44)

where $\Gamma_{0}\equiv\lambdabar_{c}\gamma_{0}/\sigma_{z}$ and $\alpha$ is
the fine structure constant. Eq. (44) is correct to the same level of
approximation as our envelope equations (33) and thus should provide a close
match to its predictions of energy loss of the bunches. The fractional energy
loss of the entire bunch train is then given by

$\langle\delta\bar{\gamma}\rangle=\langle\frac{\delta\gamma}{\gamma_{0}}\rangle\simeq\frac{1}{M}\sum_{i,j=1}^{M}\delta\bar{\gamma}_{ij}$
(45)

We additionally note that because our form of the transverse force was
truncated to lowest order in the transverse coordinate, the energy loss given
by Eq. (45) is actually an overestimate. The correction term
$\mathcal{Z}_{ij}=(25\pi/288)\bar{Z}_{ij}$ is derived in Appendix E.
Multipyling it with Eq. (44), we obtain the reduced energy loss

$\langle\delta\bar{\gamma}\rangle_{\text{red}}=\frac{2\sqrt{\pi}}{3}\frac{\alpha}{\Gamma_{0}}\frac{1}{M}\sum_{i,j=1}^{M}\langle\Upsilon_{ij}\rangle^{2}\bar{Z}_{ij}$
(46)

where

$\bar{Z}_{ij}\equiv\frac{\mu^{4}}{\ln(2/\sqrt{3})}\ln[\frac{1+\mu^{2}}{\mu\sqrt{2+\mu^{2}}}]\
;\ \mu\equiv\frac{\bar{\sigma}_{j}}{\bar{\sigma}_{i}}\bigg{|}_{u=u_{ij}}$ (47)

The results of Eqs. (33) and (45) therefore represent conservative estimates
of the energy loss. We see that in the limit of a single bunch crossing
($M=1$) where the bunches are of equal size ($\mu=1$) then
$\langle\Upsilon_{ij}\rangle\rightarrow\Upsilon_{\text{sc}}$ and
$\bar{Z}_{ij}\rightarrow 1$, and thus Eq. (46) reduces to

$\langle\delta\bar{\gamma}\rangle_{\text{red}}\rightarrow\frac{2\sqrt{\pi}}{3}\frac{\alpha}{\Gamma_{0}}\Upsilon_{\text{sc}}^{2}$
(48)

which is the usual formula for single bunch energy loss.

##  VIII Numerical Results

We evaluate numerically the solutions to Eq. (33) for some typical parameters
consistent with a DLA based collider presented in Table 3. To solve the
normalized envelope equations (33), we require the two dimensionless input
parameters $\Omega^{2}$=0.042 and $\Theta^{2}$ = 0.00027. For the normalized
plasma frequency in both cases with $M=100$, the disruption parameter is of
order $D=M\Omega^{2}\approx 4.2$. In the numerical evaluations that follow we
keep the disruption as specified, keeping the simulation and its accuracy at a
reasonable level but without affecting the physical process.

Figure 3: Frames (a) and (b) show the normalized spot size
$\bar{\sigma}_{i}=\sigma_{i}/\sigma_{0}$ and energy
$\bar{\gamma}_{i}=\gamma_{i}/\gamma_{0}$ respectively from solving Eq. (33)
for three representative microbunches $i$ = 1, 50, 100 in a train of $M=100$
bunches. The calculations are performed in the time range $0\leq
t\leq(M-1)\lambda/c$. The parameters correspond to those of Table 3 with
$\Omega^{2}$ = 0.042 and $\Theta^{2}=2.7\times 10^{-4}$. Part (c) shows the
partial luminosity enhancement factor $\mathcal{H}_{ij}$ along each
trajectory.

Let us start by examining the solutions for three microbunches in a bunch
train with $M$ = 100 bunches. We consider the first microbunch $i=1$, the
central one $i=50$ and the last one $i=100$. In Figure 3(a) we present the
normalized envelope of the three microbunches. According to this plot, the
first microbunch pinches to half of its initial transverse size after a time
$t=50\lambda/c$ which is half of the total interaction time $\Delta
t=(M-1)\lambda/c$. Beyond this point the repelling angular momentum term
$\Theta$ becomes dominant. As a result of the pinching effect, later bunches
see a higher density in the counter-propagating beam and thus are focused more
strongly, as shown by the envelopes of the middle ($i=50$) and final ($i=100$)
bunches.

The initial transverse emittance, here represented by the normalized variable
$\Theta$, sets the minimum envelope of each microbunch. If we reduce it, the
pinching is more pronounced. A lower $\Theta$ facilitates stronger pinching
but a fraction of the microbunches expand back to beam sizes much larger than
the initial condition of the envelope. In part (b) of Fig. 3, we plot the
solutions for $\bar{\gamma}_{i}$ for each of the bunches, which show
increasing energy loss with later bunches in the train, due to the increased
deflection of the particle trajectories and hence increased RRF-induced energy
loss. In Fig. 3(c) we plot the partial luminosity enhancement
$\mathcal{H}_{ij}$ evaluated for each of the three values of $i$ at the
crossing times $u=u_{ij}=(i+j)/2-1$. We see that another byproduct of the
pinching effect is that the luminosity enhancement increases for later
interactions.

Figure 4: Computed solutions for (a) luminosity enhancement, (b) beamstrahlung
parameter, and (c) fractional energy loss of the entire bunch train for
various number of bunches $M$ for the case $\Omega^{2}=0.042$,
$\Theta^{2}=0.00027$. Note that the values of Table 3 correspond to $M$ = 100
on these plots (vertical dashed blue markers). In part (c), numerical
solutions of fractional energy loss (black dots) are shown for every 5th $M$
value to aid visualization. Including nonlinearity of the transverse force
yields approximately a factor of 4 reduction in energy loss (blue curve).

In Fig. 4, we plot the luminosity enhancement, beamstrahlung parameter, and
energy loss of the entire bunch train as we vary the number of bunches from
$M=1$ to $M=150$ while keeping fixed the other parameters of Table 3. Hence,
the points on these curves correspond to the same geometrical (un-enhanced)
luminosity $\mathcal{L}_{0}$, but with the macrobunch population $N$
distributed between different numbers of bunches. In Fig. 4(a), the luminosity
enhancement is seen to peak at around $M=70$. We find that the peak of the
enhancement scales approximately as
$\mathcal{H}_{D}\propto\Theta^{-1}\propto(\epsilon/\sigma_{0}^{2})^{-1}$.
Beyond its peak value, $\mathcal{H}_{D}$ decreases monotonically with the
number of microbunches. In fact, according to our calculations, the product
$M\mathcal{H}_{D}^{(M\gg 1)}$ is independent of $M$, thus we conclude that
roughly, $\mathcal{H}_{D}^{(D>5)}\propto D^{-1}$. In the range $1\leq M\leq
30$, before the peak of $\mathcal{H}_{D}$, the enhancement factor has a weak
linear dependence on the disruption parameter ($1\leq D\leq 1.5$). This linear
dependence is virtually independent of $\Theta$ and thus of the emittance. In
Fig. 4(b), the corresponding average beamstrahlung parameter
$\langle\Upsilon\rangle$ from Eq. (42) is plotted, and compared with the
canonical single-crossing $\Upsilon_{\text{sc}}$ defined in Eq. (40).

Table 4: Summary of luminosity enhancement and corresponding energy loss for the two test cases of Table 3.  Parameter | Symbol | Value
---|---|---
Plasma Frequency | $\Omega^{2}$ | 0.042
Angular Momentum | $\Theta^{2}$ | $0.00027$
Optimal Number of Bunches | $M_{\text{opt}}$ | 100
Luminosity Enhancement Factor | $\mathcal{H}_{D}$ | 2.0
Geo. Luminosity ($10^{34}\text{cm}^{-2}\text{s}^{-1}$) | $\bar{\mathcal{L}}_{0}$ | 2
Enh. Luminosity ($10^{34}\text{cm}^{-2}\text{s}^{-1}$) | $\mathcal{H}_{D}\bar{\mathcal{L}}_{0}$ | 4.0
Avg Beamstrahlung Param | $\langle\Upsilon\rangle$ | 0.075
Average Energy Loss | $\langle\delta\bar{\gamma}\rangle$ | 1.75 %
Reduced Energy Loss | $\langle\delta\bar{\gamma}\rangle_{\text{red}}$ | 0.35 %

In Table 4 we report the corresponding energy loss values for the parameters
of Table 3. The results support one of the fundamental assumptions of our
analysis, which is that the beamstrahlung parameter $\langle\Upsilon\rangle\ll
1$, and therefore that a classical treatment is approprite. The total
fractional energy loss $\langle\delta\bar{\gamma}\rangle$ is $1.75\%$, or
$0.35\%$ with the inclusion of the reduction term $Z_{ij}$ that accounts for
the nonlinear dependence of the external force on transverse coordinate, as in
Eq. (46). This is a very important result since a factor of 2 in luminosity
enhancement is by far more important than a fraction of one percent reduction
of the particle energy.

##  IX Conclusions

In this study, we have numerically calculated the luminosity enhancement and
radiative energy loss for a 1 TeV center of mass $\text{e}^{+}\text{e}^{-}$
collider scenario with a set of self-consistent DLA beam parameters. We find
that the optically microbunched DLA beam leads to a significant luminosity
enhancement $\mathcal{H}_{D}$ due to the contraction (pinch) of the two
counter-propagating trains of microbunches. We find that for a given plasma
frequency and emittance, there is a particular value $M\approx 70$ where the
luminosity enhancement $\mathcal{H}_{D}$ is maximal, and that this value is
independent of the emittance, at least in the regime of interest.

However, the optimal $\mathcal{H}_{D}$ value is inversely proportional to the
emittance. For small number of microbunches in the train ($M<30$)
$\mathcal{H}_{D}$ varies linearly with $M$, while for larger values ($M>70$),
$\mathcal{H}_{D}$ scales inversely with $M$. For the considered parameter
case, we determined an optimal number of bunches $M_{\text{opt}}=100$ that
matches the desired geometrical luminosity while satisfying constraints on
beam loading efficiency, power consumption, and bunch charge. We find that
there is a tradeoff between having $M_{\text{opt}}$ matched to the peak of the
luminosity curve and the overall amplitude of this peak. Although the scenario
of Table 3 requires $M_{\text{opt}}=100$ microbunches and so does not sit at
the peak of the $\mathcal{H}_{D}$ curve, it also leads to lower required
repetition rate, lower maximal field in the structure, lower single-bunch
charge, higher efficiency, and reduced wallplug power consumption, making it
generally favorable.

We see that the overall beamstrahlung energy losses are 1.75% for the
unreduced case and 0.35% for the reduced case (the latter taking into account
the full transverse shape of the beam potential $\Psi$). We note that our
analysis predicts similar values of the luminosity enhancement and
beamstrahlung energy loss to prior estimates (e.g. Ref. [20]), which are on
the order of $\mathcal{H}_{D}=2.7$ and $\langle\delta\bar{\gamma}\rangle=1\%$
when scaled to 1 TeV center of mass energy. This is in spite of the fact that
here we treat the microbunches as distinct, whereas prior calculations assumed
a “washed out” microstructure, treating the entire bunch train as a single
longer bunch. The small energy loss here is an important potential benefit of
the DLA approach for a linear collider, particularly in the multi-TeV regime,
where the energy losses for more conventional bunch formats can be in the tens
of percents.

###### Acknowledgements.

This work was supported by the Israel Science Foundation (ISF), the Moore
Foundation (GBMF4744), and the U.S. Department of Energy (DE-AC02-76SF00515).

##  Appendix A Transverse Potential

Let O denote the lab frame and $\tilde{\text{O}}$ the frame of a counter-
propagating charge distribution moving at relative velocity $v=-\beta c$ along
the $z$-axis such that the origins of O and $\tilde{\text{O}}$ overlap at time
$t=0$. This is illustrated in Fig. 5.

Figure 5: Coordinate system for single particle of velocity $v$ colliding with
the charge density function $\rho$ with velocity -$v$.

Let r be the position of a test charge $q$ in the lab frame. Then its position
in $\tilde{\text{O}}$ is obtained by transforming only the longitudinal
component $\tilde{z}=\gamma(z+vt)$. Hence, the gradient of the electrostatic
potential in this frame is

$\tilde{\bm{\nabla}}\tilde{\Phi}=\bm{\nabla}\tilde{\Phi}(\textbf{r}_{\perp},\tilde{z})\big{|}_{\tilde{z}=\gamma(z+\beta
ct)}$ (49)

In [33] the potential for a tri-Gaussian beam distribution is found to have
the following form

$\tilde{\Phi}(\tilde{\textbf{r}})=\frac{Ne}{4\pi\varepsilon_{0}}\mathcal{G}(\tilde{z},\tilde{\sigma}_{z})\Psi(\tilde{\textbf{r}}_{\perp})$
(50)

where $\mathcal{G}(\tilde{z},\tilde{\sigma}_{z})$ is the Gaussian longitudinal
charge distribution normalized to unit integral over $\tilde{z}$ and

$\Psi(\tilde{\textbf{r}}_{\perp})=\int_{0}^{\infty}\frac{\exp(-\frac{\tilde{x}^{2}}{2\sigma_{x}^{2}+\xi}-\frac{\tilde{y}^{2}}{2\sigma_{y}^{2}+\xi})}{\sqrt{(2\sigma_{x}^{2}+\xi)(2\sigma_{y}^{2}+\xi)}}d\xi$
(51)

In [36] the following approximate form for the gradient of this potential is
derived for a round beam
($\sigma\equiv\sqrt{2}\sigma_{x}=\sqrt{2}\sigma_{y}$):

$\bm{\nabla}_{\perp}\Psi=\frac{2\textbf{r}_{\perp}}{r^{2}}\left[1-\exp(\frac{r^{2}}{\sigma^{2}})\right]\approx\frac{2\textbf{r}_{\perp}}{\sigma^{2}}$
(52)

In the second step, we expand the term in square brackets to lowest order in
$r/\sigma^{2}$ and we remove the tildes, since the transverse coordinates are
not affected by the Lorentz transformation and so are the same in both frames.

##  Appendix B The External Force

Referencing the coordinates of Appendix A, consider now a static charge
density $\tilde{\rho}$ that is stationary in frame $\tilde{\text{O}}$. In this
frame, the charge density $\tilde{\rho}$ gives rise to the electrostatic
potential $\tilde{\Psi}$ with corresponding electric and magnetic fields

$\tilde{\textbf{E}}=-\tilde{\bm{\nabla}}\tilde{\Phi}\ \ ;\ \
\tilde{\textbf{B}}=0.$ (53)

Transforming from the $\tilde{\text{O}}$ to O frame, we have that
$\textbf{E}_{\perp}=\gamma\tilde{\textbf{E}}_{\perp}$ and
$\textbf{B}_{\perp}=\gamma\beta(\tilde{E}_{y}\hat{\textbf{x}}-\tilde{E}_{x}\hat{\textbf{y}})$.
The resulting transverse Lorentz force on the particle of charge $-e$ in the
lab frame with normalized velocity $\pmb{\beta}=\beta\hat{\textbf{z}}$ is thus

$\textbf{F}_{\perp}=-e(\textbf{E}+\pmb{\beta}\times\textbf{B})_{\perp}=-e\gamma(1+\beta^{2})\tilde{\bm{\nabla}}_{\perp}\tilde{\Phi}$
(54)

Regarding the potential as a 2D function with separable longitudinal part, it
was shown in Appendix A that

$\tilde{\bm{\nabla}}_{\perp}\tilde{\Phi}=\frac{Ne}{4\pi\varepsilon_{0}}\mathcal{G}(\tilde{z},\tilde{\sigma}_{z})\bm{\nabla}_{\perp}\Psi(\textbf{r}_{\perp})$
(55)

Note that since we have already performed the Lorentz transformation of the
fields, under the variable substitution $\tilde{z}=\gamma(z+\beta ct)$ we can
write

$\mathcal{G}(\tilde{z},\tilde{\sigma}_{z})=\frac{1}{\gamma}\
\mathcal{G}(z+\beta ct,\sigma_{z})$ (56)

where $\mathcal{G}(z,\sigma_{z})$ is now the longitudinal charge density as
seen in the lab frame O. Further, since
$\tilde{\textbf{r}}_{\perp}=\textbf{r}_{\perp}$, the force seen by a particle
going through the potential $\Phi$ in the laboratory frame is thus

$\textbf{F}_{\perp}=-\frac{Ne^{2}(1+\beta^{2})}{4\pi\varepsilon_{0}}\mathcal{G}(z+\beta
ct,\sigma_{z})\bm{\nabla}_{\perp}\Psi$ (57)

Taking $\beta\simeq 1$ for ultrarelativistic beams, we convert from spatial
($z$) to temporal ($t$) coordinate using

$\mathcal{G}(z+ct,\sigma_{z})=\frac{1}{2c}\
\mathcal{G}\bigg{(}\frac{t}{2}+\frac{z}{2c},\sigma_{t}\bigg{)}$ (58)

where $\sigma_{t}=\sigma_{z}/(2c)$. Extending this result to a sum over
multiple such potentials spaced in $z$ at intervals of $j\lambda$, and making
the substitution of Eq. (52), we have

$\textbf{F}_{\perp}=-\frac{2Ne^{2}}{4\pi\varepsilon_{0}c}\sum_{j=1}^{M}\mathcal{G}\bigg{(}\frac{t}{2}+\frac{z-(j-1)\lambda}{2c},\sigma_{t}\bigg{)}\frac{\textbf{r}_{\perp}}{\sigma_{j}^{2}}$
(59)

With the parameter definition of $\Omega^{2}$ in Eq. (22) we see that this is
equivalent to Eq. (20).

##  Appendix C RRF Equations of Motion

Starting from the relativistic Lorentz-Abraham-Dirac (LAD) equation of motion
given by Eq. (29-30), we recognize that the right-most term in the expression
for $\mathcal{F}_{\mu}^{\text{rad}}$ is the Lorentz-invariant form of the
radiated Larmor power

$\mathcal{P}=-\tau_{e}\left[\frac{dp_{\mu}}{d\tau}\frac{dp^{\nu}}{d\tau}\right]$
(60)

where $p_{\mu}=(U/c,-\textbf{p})$, $U=\gamma mc^{2}$ is the particle energy,
$\tau_{e}\equiv 2r_{e}/(3c)$, and p is the usual 3-vector momentum. The
external 4-vector force is given in component form by

$\mathcal{F}_{\mu}^{\text{ext}}=\gamma(\pmb{\beta}\cdot\textbf{F},-\textbf{F})$
(61)

where $\textbf{F}=-e(\textbf{E}+\pmb{\beta}\times\textbf{B})$ is the usual
3-vector Lorentz force in the lab frame and $\pmb{\beta}=\textbf{v}/c$ is the
normalized velocity. In component form, the full LAD equations (29) then take
the following form, with no approximations:

$\displaystyle\dot{U}$
$\displaystyle=[\tau_{e}(\gamma\ddot{U}+\dot{\gamma}\dot{U})-\mathcal{P}]+\textbf{v}\cdot\textbf{F}$
(62) $\displaystyle\dot{\textbf{p}}$
$\displaystyle=[\tau_{e}(\gamma\ddot{\textbf{p}}+\dot{\gamma}\dot{\textbf{p}})-\frac{\pmb{\beta}}{c}\mathcal{P}]+\textbf{F}$

where we have employed $d\tau=dt/\gamma$ and overdots denote derivatives with
respect to $t$. The Larmor power is given by

$\mathcal{P}=\gamma^{2}\tau_{e}[\dot{\textbf{p}}^{2}-\dot{U}^{2}/c^{2}]$ (63)

Let us introduce the following normalized variables:

$q^{\nu}\equiv\frac{p^{\nu}}{mc}=(\gamma,\textbf{q})\ \;\ \
\textbf{q}\equiv\gamma\pmb{\beta}\ \ ;\ \
\bar{\mathcal{P}}\equiv\frac{\lambda}{c}\frac{\mathcal{P}}{mc^{2}}$ (64)
$\bar{\textbf{F}}\equiv\frac{\lambda}{c}\frac{\textbf{F}}{mc}\ \ ;\ \
\alpha_{e}\equiv\frac{c\tau_{e}}{\lambda}$ (65)

Switching to derivatives with respect to the normalized time $u=ct/\lambda$,
which we will denote by primes instead of dots, we thus have in place of Eq.
(62),

$\displaystyle\gamma^{\prime}$
$\displaystyle=[\alpha_{e}(\gamma\gamma^{\prime\prime}+\gamma^{\prime
2})-\bar{\mathcal{P}}]+\pmb{\beta}\cdot\bar{\textbf{F}}$ (66)
$\displaystyle\textbf{q}^{\prime}$
$\displaystyle=[\alpha_{e}(\gamma\textbf{q}^{\prime\prime}+\gamma^{\prime}\textbf{q}^{\prime})-\pmb{\beta}\bar{\mathcal{P}}]+\bar{\textbf{F}}$

with

$\bar{\mathcal{P}}=\gamma^{2}\alpha_{e}[\textbf{q}^{\prime 2}-\gamma^{\prime
2}]$ (67)

Thus far no approximations have been made and these equations are quite
general. We note that the left-most expressions within Eq. (66) are small
corrections. Even in the extreme case where the particle loses all of its
energy, neglecting these terms produces only about a 10% error. We thus omit
them and combine Eqs. (66) and (67), employing the approximation of Landau and
Lifshitz, whereby we assume to lowest order that on the right side we may
substitute the relativistic forms for the momentum and energy derivatives with
the expressions in the absence of RRF ($\textbf{q}^{\prime}=\bar{\textbf{F}}$,
$\gamma^{\prime}=\pmb{\beta}\cdot\bar{\textbf{F}}$), yielding

$\displaystyle\gamma^{\prime}$
$\displaystyle\simeq\pmb{\beta}\cdot\bar{\textbf{F}}-\alpha_{e}\gamma^{2}[\bar{\textbf{F}}^{2}-(\pmb{\beta}\cdot\bar{\textbf{F}})^{2}]$
(68) $\displaystyle\textbf{q}^{\prime}$
$\displaystyle\simeq\bar{\textbf{F}}-\alpha_{e}\pmb{\beta}\gamma^{2}[\bar{\textbf{F}}^{2}-(\pmb{\beta}\cdot\bar{\textbf{F}})^{2}]$

Taking the force to be predominantly transverse, in accordance with our
earlier prescription ($\textbf{F}=\textbf{F}_{\perp}$) and noting that in the
limit of small transverse velocity $\beta_{\perp}\ll\beta_{z}\simeq 1$, to
good approximation,

$\bar{\textbf{F}}_{\perp}^{2}-(\pmb{\beta}\cdot\bar{\textbf{F}}_{\perp})^{2}\simeq\bar{\textbf{F}}_{\perp}^{2}\beta^{2}$
(69)

Upon insertion of (69) into Eqs. (68), we immediately obtain Eqs. (32).

##  Appendix D Envelope Equation with RRF

The RRF equations of motion Eq. (32) cast into unnormalized variables read as
follows:

$\displaystyle\dot{\gamma}=$ $\displaystyle\
\frac{\textbf{v}\cdot\textbf{F}_{\perp}}{mc^{2}}-\tau_{e}\frac{p^{2}}{(mc)^{2}}\left(\frac{\textbf{F}_{\perp}}{mc}\right)^{2}$
(70) $\displaystyle\dot{\textbf{p}}=$ $\displaystyle\
\textbf{F}_{\perp}-\tau_{e}\gamma\textbf{p}\left(\frac{\textbf{F}_{\perp}}{mc}\right)^{2}$

where $\textbf{p}=\gamma m\textbf{v}$ is the usual 3-vector momentum. Taking
the external force to have the previously derived form
$\textbf{F}_{\perp}=-\gamma_{0}mK^{2}\textbf{r}_{\perp}$ where
$\textbf{r}_{\perp}=x\hat{\textbf{x}}+y\hat{\textbf{y}}$, the above equations
take the component forms

$\displaystyle\ddot{x}$
$\displaystyle=-\frac{\gamma_{0}}{\gamma}K^{2}x-\frac{\dot{\gamma}}{\gamma}\dot{x}-\dot{x}\frac{\tau_{e}\gamma\gamma_{0}^{2}K^{4}(x^{2}+y^{2})}{c^{2}}$
(71) $\displaystyle\ddot{y}$
$\displaystyle=-\frac{\gamma_{0}}{\gamma}K^{2}y-\frac{\dot{\gamma}}{\gamma}\dot{y}-\dot{y}\frac{\tau_{e}\gamma\gamma_{0}^{2}K^{4}(x^{2}+y^{2})}{c^{2}}$
$\displaystyle\dot{\gamma}$
$\displaystyle=-\gamma_{0}\frac{K^{2}}{c^{2}}(\dot{x}x+\dot{y}y)-\tau_{e}\gamma^{2}\beta^{2}\left[\frac{\gamma_{0}^{2}K^{4}(x^{2}+y^{2})}{c^{2}}\right]$

We now switch to cylindrical coordinates, taking $x=\sigma\cos\phi$,
$y=\sigma\sin\phi$. We here take the particle to correspond to the RMS
particle at radial position $\textbf{r}_{\perp}=\sigma\hat{\textbf{r}}$. We
ignore the factor of $\beta^{2}$ in the equation for $\gamma$ as it has a
negligible effect. Multiplying the $x$ equation by $\cos\phi$ and the $y$
equation by $\sin\phi$ and adding them, we obtain

$\displaystyle\ddot{\sigma}+\left(\frac{\gamma_{0}}{\gamma}K^{2}-\dot{\phi}^{2}\right)\sigma+\frac{\dot{\gamma}}{\gamma}\dot{\sigma}+\gamma\left[\frac{\tau_{e}\gamma_{0}^{2}K^{4}}{c^{2}}\right]\dot{\sigma}\sigma^{2}=0$
(72)
$\displaystyle\dot{\gamma}+\gamma_{0}\frac{K^{2}}{c^{2}}\dot{\sigma}\sigma+\frac{\tau_{e}\gamma_{0}^{2}K^{4}}{c^{2}}\sigma^{2}\gamma^{2}=0$

We additionally note that the angular momentum
$\textbf{L}=\textbf{r}\times\textbf{p}$ has longitudinal component
$L_{z}=xp_{y}-yp_{x}=\gamma m\sigma^{2}\dot{\phi}$. Taking the time
derivative, we may obtain a third equation of motion:

$\dot{L}_{z}=\gamma
mx\left[\frac{\dot{\gamma}}{\gamma}\dot{y}+\ddot{y}\right]-\gamma
y\left[\frac{\dot{\gamma}}{\gamma}\dot{x}+\ddot{x}\right]$ (73)

Inserting into this expression from the equations of motion for $x$ and $y$ in
Eqs. (71), we find that this equation is equivalent to

$\dot{L}_{z}=-\left[\frac{\tau_{e}\gamma\gamma_{0}^{2}K^{4}\sigma^{2}}{c^{2}}\right]L_{z}$
(74)

Thus, in the absence of RRF, this becomes $\dot{L}_{z}=0$ and hence the
longitudinal angular momentum is a constant of the motion. We associate this
constant with the normalized transverse emittance, which for linear focusing
and acceleration in the absence of RRF is a conserved quantity:
$L_{z}/(mc)=\gamma\epsilon=\epsilon_{n}$. In accordance with Eq. (74) this
invariance can no longer be assumed, and in fact $\epsilon_{n}$ can be seen to
decrease during the interaction. Converting Eqs. (72,74) to normalized
derivatives with respect to $u=(\lambda/c)t$ and invoking the normalized
variables,

$\displaystyle\bar{\sigma}\equiv\frac{\sigma}{\sigma_{0}}\ \ ;\ \
\bar{\gamma}\equiv$ $\displaystyle\frac{\gamma}{\gamma_{0}}\ \ ;\ \
\hat{K}=\frac{\lambda}{c}K$ (75) $\displaystyle\Lambda\equiv\frac{\lambda
L_{z}}{\gamma_{0}mc\sigma_{0}^{2}}=\frac{\lambda\epsilon_{n}}{\gamma_{0}\sigma_{0}^{2}}$
$\displaystyle;\ \
\bar{\alpha}_{e}\equiv\frac{c\tau_{e}}{\lambda}\gamma_{0}^{3}\left(\frac{\sigma_{0}}{\lambda}\right)^{2}$

we may recast Eqs. (72,74) into the form

$\displaystyle\bar{\sigma}^{\prime\prime}+\frac{1}{\bar{\gamma}}\hat{K}^{2}\bar{\sigma}-\frac{\Lambda^{2}}{\bar{\gamma}^{2}\bar{\sigma}^{3}}+\frac{\bar{\gamma}^{\prime}}{\bar{\gamma}}\bar{\sigma}^{\prime}+\bar{\alpha}_{e}\hat{K}^{4}\bar{\sigma}^{\prime}\bar{\sigma}^{2}\bar{\gamma}=0$
(76)
$\displaystyle\bar{\gamma}^{\prime}+\left(\frac{\sigma_{0}}{\lambda}\right)^{2}\bar{\sigma}^{\prime}\bar{\sigma}\hat{K}^{2}+\bar{\alpha}_{e}\hat{K}^{4}\bar{\sigma}^{2}\bar{\gamma}^{2}=0$
$\displaystyle\Lambda^{\prime}+\bar{\alpha}_{e}[\bar{\gamma}\hat{K}^{4}\bar{\sigma}^{2}]\Lambda=0$

subject to the initial conditions

$\bar{\sigma}(0)=1\ ;\ \bar{\sigma}^{\prime}(0)=0\ ;\ \bar{\gamma}(0)=1\ ;\
\Lambda(0)=\Theta$ (77)

Here $\Theta$ is the normalized emittance term defined in Eq. (28). We observe
that the second term containing $(\sigma_{0}/\lambda)^{2}$ in the equation for
$\bar{\gamma}$ in Eq. (76) is small and has negligible effect on the solution.
(Neglecting it produces errors on the order of 1 part in $10^{6}$). Hence,
from the second and third relations of (76), we find that

$\displaystyle\bar{\gamma}^{\prime}+[\bar{\alpha}_{e}\hat{K}^{4}\bar{\sigma}^{2}]\bar{\gamma}^{2}=0$
(78)
$\displaystyle\Lambda^{\prime}+[\bar{\alpha}_{e}\hat{K}^{4}\bar{\sigma}^{2}]\Lambda\bar{\gamma}=0$

Subject to the initial conditions (77), these yield the solution
$\Lambda(u)=\Theta\bar{\gamma}(u)$. From the definitions of the parameters, we
see that this is equivalent to $\epsilon_{n}(u)=\epsilon_{0}\gamma(u)$, where
$\epsilon_{0}$ is the initial RMS emittance. Hence we obtain the remarkable
result that the normalized emittance is not conserved under the action of the
RRF but instead varies directly with $\gamma$. This observation negates the
need for the third equation in (76). Upon making the substitution
$\Lambda=\Theta\bar{\gamma}$ into the first of Eq. (76), neglecting the second
term in the equation for $\bar{\gamma}$ per the discussion above, and
restoring the subscripted $i$’s to denote that the equations are to be solved
separately for each microbunch in the train, we immediately obtain Eqs. (33).

##  Appendix E Energy Correction Factor

We note that the energy equation of motion (33) with the
$\pmb{\beta}\cdot\bar{\textbf{F}}$ terms neglected amounts to equating the
energy derivative with the Larmor radiation:

$\gamma^{\prime}=-\bar{\mathcal{P}}\ \ ;\ \ \text{where}\ \
\bar{\mathcal{P}}=\gamma^{2}\alpha_{e}\bar{\textbf{F}}_{\perp}^{2}\ .$ (79)

If we consider the full form of the potential $\Psi$ from Eq. (52), the
transverse force for the $(i,j)$ bunch crossing is

$\bar{\textbf{F}}_{\perp}=-\frac{1}{2}(\gamma_{0}/\lambda)\hat{K}_{ij}^{2}\sigma_{j}^{2}\nabla_{\perp}\Psi_{j}$
(80)

where
$\nabla_{\perp}\Psi_{j}=\nabla_{\perp}\Psi|_{\sigma\rightarrow\sigma_{j}}$
from Eq. (52). Upon insertion into the energy equation we obtain the energy
loss of interaction $(i,j)$

$\bar{\gamma}_{ij}^{\prime}=-\bar{\alpha}_{e}\bar{\gamma}_{i}^{2}\bar{\sigma}_{i}^{2}\hat{K}_{ij}^{4}\times\frac{1}{4}\frac{\sigma_{j}^{2}}{\sigma_{i}^{2}}\left|\sigma_{j}\bm{\nabla}_{\perp}\Psi_{j}\right|^{2}$
(81)

We recognize this is the same as our form of the energy equation of motion in
Eq. (33) but with an extra factor on the right. Integrating over the
normalized time $u$ and the transverse distribution
$F_{i}(\textbf{r}_{\perp})$ of the $i$’th incident bunch, the average energy
loss for the $(i,j)$ interaction is

$\delta\bar{\gamma}_{ij}=-\int\bar{\gamma}_{ij}^{\prime}F_{i}(\textbf{r}_{\perp})d^{2}\textbf{r}_{\perp}du$
(82)

With the form of Eq. (81), we see that this will yield the same result as Eq.
(43) but with an extra dimensionless factor

$\mathcal{Z}_{ij}\equiv\frac{1}{2}\int[\mathcal{G}(r,\sigma_{j})]^{2}\frac{\sigma_{j}^{2}}{\sigma_{i}^{2}}\left|\sigma_{j}\bm{\nabla}_{\perp}\Psi_{j}\right|^{2}d^{2}\textbf{r}_{\perp}$
(83)

Making the variable substitutions $\zeta=\sigma_{j}r$,
$\mu=\sigma_{j}/\sigma_{i}$ and taking
$d^{2}\textbf{r}_{\perp}=rdrd\phi\rightarrow 2\pi\sigma_{j}^{2}\zeta d\zeta$
this integral becomes

$\mathcal{Z}_{ij}=2\mu^{4}\int_{0}^{\infty}e^{-\mu^{2}\zeta^{2}}\frac{1}{\zeta}\left[1-e^{-\zeta^{2}}\right]^{2}d\zeta$
(84)

which evaluates explicitly to

$\mathcal{Z}_{ij}=2\mu^{4}\ln\left[\frac{1+\mu^{2}}{\mu\sqrt{2+\mu^{2}}}\right]\
\ ;\ \ \mu\equiv\frac{\sigma_{j}}{\sigma_{i}}$ (85)

To reconcile this with our prior result, Eq. (44), in the absence of the extra
factor of $\mathcal{Z}_{ij}$, we note that the linear force approximation
amounts to taking the term in square brackets in Eq. (84)
$[...]\approx\zeta^{2}$ to lowest order and the integral in this approximation
evaluates to $\mathcal{Z}_{ij}=1$. We further note that in the case of equal
bunch sizes ($\mu=1$), $\mathcal{Z}_{ij}=2\ln(2/\sqrt{3})\simeq 25\pi/288$.
For convenience of other calculations, we may thus write the correction factor
in the form

$\mathcal{Z}_{ij}\simeq\frac{25\pi}{288}\bar{Z}_{ij}\ ;\ \text{where}\
\bar{Z}_{ij}\equiv\frac{\mathcal{Z}_{ij}}{2\ln(2/\sqrt{3})}$ (86)

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