# LHCb: Status and Prospects on the $b$ Anomalies

LHCb Collaboration / Federal University of Rio de Janeiro.
E-mail also at U. Constantine, Algeria.

###### Abstract:

Since the start of the Large Hadron Collider program, direct searches for
Beyond Standard Model (BSM) particles have constrained their mass scale to
limits which are now above the energy reach of the current collider. As a
result, studies of indirect probes of BSM physics have gained a considerable
momentum, both experimentally and theoretically. The flavour anomalies in $b$
hadron decays are now recognized as an important laboratory for the indirect
detection of BSM physics. This short review presents several key analyses in
this area, and some prospects with future data.

##  1 Introduction

Heavy Flavour decays are usually described by low energy effective
Hamiltonians forming an Effective Field Theory (EFT) (see, e.g. reference [1]
for a review). The Hamiltonians are written as:

$H=\sum_{i}V_{CKM}^{i}C_{i}(\mu)O_{i}(\mu),$ (1)

where $C_{i}(\mu)$ are the Wilson coefficients integrating out the physics
above the scale $\mu$ (short range), $O_{i}(\mu)$ are current operators which
matrix elements represent the low energy (non-perturbative/long range)
hadronic physics, and $\mu$ is the renormalization scale (typically $\sim
1\mathrm{\,Ge\kern-1.00006ptV}$) distinguishing the two regimes. $V_{CKM}^{i}$
represents the flavour coupling associated to an operator $O_{i}$, i.e.,
Cabibbo-Kobayashi-Maskawa (CKM) matrix elements for SM operators. The Wilson
coefficients thus represent the quantities which are impacted by the
intervention of BSM physics.
For the semileptonic tree decays, the coupling of the mediating $W$ boson does
not discriminate between lepton flavours in the SM. On the contrary, a BSM
mediator might exhibit different couplings between light and heavy leptons.
This is referred to as Lepton Flavour Universality Violation (LFUV). Such an
effect could also occur for the semileptonic loop decays $b\rightarrow
s\ell\ell$ where the dominant operators are $O_{7},~{}O_{9},~{}O_{10}$. Loop
decays could also be the ground of new dynamics involving Lepton Flavour
Violation (LFV) where leptons of different flavours are produced together. A
review of various LHCb analyses is proposed. Several of them exhibit
deviations from SM that can be explained consistently with theoretical models.

##  2 Semileptonic tree decays

The dominant decays ($b\rightarrow c\ell^{-}\bar{\nu}$) of this kind are
generically written as $H_{b}\rightarrow H_{c}\ell^{-}\bar{\nu}$ where $H_{b}$
is a $b$ hadron and $H_{c}$ is a charm hadron. The search for a possible LFUV
is performed through the measurement of the ratio:

$R(H_{c})=\frac{\mathcal{B}(H_{b}\rightarrow
H_{c}\tau^{-}\bar{\nu})}{\mathcal{B}(H_{b}\rightarrow
H_{c}\mu^{-}\bar{\nu})},$ (2)

where $\mathcal{B}$ denotes the branching fraction. A BSM mediating heavy
boson might couple preferentially to the tau lepton, as in Fig.1, and thus
produce a $R(H_{c})$ ratio different from the expected SM-based calculations.

Figure 1: $b\rightarrow c\tau^{-}\bar{\nu}$ semileptonic transition with a
mediating $W$ or charged Higgs boson as derived from 2HDM models discussed
e.g. in Ref.[2].

The modes $\bar{B}^{0}\rightarrow D^{+(*)}\ell^{-}\bar{\nu}$ have drawn
attention both at the $b$ factories and the LHCb experiment. The LHCb
collaboration has been focusing so far on $R(D^{*+})$, where $D^{*+}$ is
reconstructed via $D^{*+}\rightarrow
D^{0}(\rightarrow{{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$. The $\tau$ lepton is
reconstructed in the muonic mode,
$\tau^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\nu_{\tau}$ [3], or the hadronic
mode $\tau\rightarrow\pi\pi\pi(\pi^{0})\nu_{\tau}$ [4]. The discriminating
variables include the missing mass,
$m_{miss}^{2}=(P_{B}-P_{D^{*}}-P_{\mu})^{2}$, the momentum transfer
$q^{2}=(P_{B}-P_{D^{*}})^{2}$, the muon energy $E_{\mu}^{*}$, the $\tau$
lifetime (for the hadronic mode) and a Boost Decision Trees classifier [5] to
reject double-charm decays of the type $B\rightarrow DDX$ (for the hadronic
mode). The muonic tau analysis [3] obtained a measurement of
$R(D^{*+})=0.336\pm 0.027(stat)\pm 0.030(syst)$ while the hadronic tau study
[4] gives $R(D^{*+})=0.291\pm 0.021(stat)\pm 0.026(syst)\pm 0.013(BF)$, where
the last uncertainty is due to the external branching fraction of the
normalizing channel $\bar{B}^{0}\rightarrow
D^{*+}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$. The latest HFLAV averaging [6] in
the $R(D)-R(D^{*})$ plane, including the recent Belle collaboration
$R(D^{(*)})$ measurements [7], is shown in Fig.2. Compared to an averaged
series of SM-based predictions [8], a discrepancy of 3.1$\sigma$ is observed.

Figure 2: Average of $R(D)$ and $R(D^{*})$ [6].

A similar measurement with the decays
$B_{c}^{-}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\ell^{-}\bar{\nu}$, $R({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})$, has been performed recently by the LHCb collaboration for the
$\tau$ muonic mode [9], leading to $R({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}})=0.71\pm 0.17(stat)\pm 0.18(syst)$ which lies 2$\sigma$ above the
range of the known theoretical estimates [10].

##  3 $b\rightarrow s\ell\ell$ transitions

At quark level, these transitions proceed through the diagrams shown in Fig.3.
The operators contributing to these decays are not evenly distributed in the
$q^{2}=m_{\ell\ell}^{2}$ range: at low $q^{2}$, $O_{7}$ dominates (for
transitions to non-scalar hadrons), in the central $q^{2}$ region below the
charmonium resonances, $O_{7}$ and $O_{9}$ interfere, and at high $q^{2}$
$O_{9}$ and $O_{10}$ interfere. At the hadron level, the modes investigated by
LHCb are ${{{B}^{+}}}\rightarrow K^{*+}\ell^{+}\ell^{-}$, $B^{0}\rightarrow
K^{0}\ell^{+}\ell^{-}$, $B^{0}\rightarrow K^{*0}\ell^{+}\ell^{-}$,
${{B}^{0}_{s}}\rightarrow\phi\ell^{+}\ell^{-}$, and ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{\mathchar 28931\relax}\ell^{+}\ell^{-}$.

A series of studies [11] have dealt with the dynamics of the muonic modes,
$\ell=\mu$, to infer the differential decay rate $\frac{d\Gamma}{dq^{2}}$, as
illustrated in Fig.4. The data is systematically below the SM-based
theoretical predictions, with local discrepancies exceeding 3$\sigma$.
Attempting to explore this intriguing behaviour, angular analyses were
performed for $B^{0}\rightarrow K^{*0}{\mu^{+}}{\mu^{-}}$ [11](c),
${{B}^{0}_{s}}\rightarrow\phi{\mu^{+}}{\mu^{-}}$ [11](b) and ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{\mathchar 28931\relax}{\mu^{+}}{\mu^{-}}$
[12]. Quantities such as $P_{5}^{\prime}=\frac{S_{5}}{\sqrt{F_{L}(1-F_{L})}}$
have been built to reduce the hadronic uncertainties [13] from the
coefficients $S_{5}$ and $F_{L}$ (fraction of the $K^{*}$ longitudinal
polarization) of the angular distribution. For $B^{0}\rightarrow
K^{*0}{\mu^{+}}{\mu^{-}}$, the discrepancy reported in previous studies for
$P_{5}^{\prime}$ seems to be persistent as shown in Fig.5(left). The fit for
the deviations from SM to the real parts of the $C_{9}$ and $C_{10}$ Wilson
coefficients gives the results depicted in Fig.5(right). Considering only
$C_{9}$, the deviation from SM is determined to be 3.3$\sigma$.

Figure 3: $b\rightarrow s\ell\ell$ (left) penguin and (right) box transitions.

Figure 4: $\frac{d\Gamma}{dq^{2}}$ distribution for (left) $B^{+}\rightarrow
K^{+}{\mu^{+}}{\mu^{-}}$ and (right)
${{B}^{0}_{s}}\rightarrow\phi{\mu^{+}}{\mu^{-}}$. The
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ $q^{2}$
regions are vetoed. The points represent the data measurements and the
rectangles or band represent the SM-based predictions.

Figure 5: (left) Evolution of $P_{5}^{\prime}$ (see text) as a function of
$q^{2}$ for $B^{0}\rightarrow K^{*0}{\mu^{+}}{\mu^{-}}$ and (right) resulting
1,2,3 $\sigma$ contours, using all the angular variables, of the deviations
from SM of the real parts of the $C_{9}$ and $C_{10}$ Wilson coefficients.

Another way to probe the presence of New Physics is to measure the ratio
$R_{X}=\frac{\mathcal{B}(H_{b}\rightarrow
X\mu^{+}\mu^{-})}{\mathcal{B}(H_{b}\rightarrow Xe^{+}e^{-})}$, where $X$
denotes a hadronic system comprising a strange quark. The LHCb collaboration
studied $R_{K}$ (${{{B}^{+}}}\rightarrow{{K}^{+}}\ell^{+}\ell^{-}$) [14],
$R_{K^{*}}$ ($B^{0}\rightarrow K^{*0}\ell^{+}\ell^{-}$) [15] and $R_{pK}$ with
the decay ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow
p{{K}^{-}}\ell^{+}\ell^{-}$ [16]. For $R_{K}$, the explored $q^{2}$ range is
$[1.1,6]~{}{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, i.e. below the
charmonium radiative tails, and above backgrounds of the type
${{{B}^{+}}}\rightarrow{{K}^{+}}\phi(\rightarrow\ell^{+}\ell^{-})$. The
$R_{K^{*}}$ analysis uses two bins in $q^{2}$,
$[0.045,1.1]~{}{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ (above the photon
pole) and $[1.1,6]~{}{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. Finally,
$R_{pK}$ is measured with the requirements
$q^{2}\in[0.1,6.0]~{}{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
$m(pK)<2.6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.

The obtained measurements are
$R_{K}=0.846^{+0.060}_{-0.054}(stat)^{+0.016}_{-0.014}(syst)$
($1.1<q^{2}<6{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$);
$R_{K^{*}}=0.66^{+0.11}_{-0.07}(stat)\pm 0.03(syst)$ for
$0.045<q^{2}<1.1{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
$0.69^{+0.11}_{-0.07}(stat)\pm 0.05(syst)$ for
$1.1<q^{2}<6{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. All these values
are systematically below the SM-based predictions by 2.2$\sigma$ to
2.5$\sigma$ [17]. With the ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow
p{{K}^{-}}\ell^{+}\ell^{-}$ decay, a first observation of ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow p{{K}^{-}}e^{+}e^{-}$ is obtained with a
similar Run 1 and part of Run 2 data set, as illustrated in Fig.6, leading to
the measurement of the ratio $R_{pK}=0.86^{+0.14}_{-0.11}(stat)\pm
0.05(syst)$. For all these numbers, the uncertainty will be reduced soon with
the addition of the remainder of Run 2 data.

Figure 6:  Invariant mass distributions of (left) ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow p{{K}^{-}}{\mu^{+}}{\mu^{-}}$ and (right)
${{\mathchar 28931\relax}^{0}_{b}}\rightarrow p{{K}^{-}}e^{+}e^{-}$
candidates. The fit shapes of the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow p{{K}^{-}}\ell^{+}\ell^{-}$ signals and the
main backgrounds are overlaid.

Figure 7 shows the impact of the $R_{K^{*}}$ measurements by BaBar and LHCb,
as well as the combination of the LFUV and angular parameters from all the
experiments, on the New Physics contributions to the Wilson coefficients
$C_{9}$ and $C_{10}$, as derived in Ref.[18]. Combining all anomalous data,
$C_{9}$ departs by more than 6$\sigma$ from its SM-based prediction.

Figure 7:  The 1,2,3 $\sigma$ contours for the New Physics contributions to
the $C_{9}$ and $C_{10}$ Wilson coefficients using (left) only the LFUV data
from Belle and LHCb and (right) combining all data from angular analyses and
LFUV. The fits are provided in Ref.[18].

##  4 $B\rightarrow\ell\ell$

For these purely leptonic modes, the combination of the most recent results of
the ATLAS [19], CMS [20] and LHCb [21] experiments, giving
$\mathcal{B}({{B}^{0}_{s}}\rightarrow{\mu^{+}}{\mu^{-}})=(2.69^{+0.37}_{-0.35})\times
10^{-9}$ and $\mathcal{B}(B^{0}\rightarrow{\mu^{+}}{\mu^{-}})<1.9\times
10^{-10}$ at 95% confidence level (CL), shows that
${{B}^{0}_{s}}\rightarrow{\mu^{+}}{\mu^{-}}$ is 2$\sigma$ below the SM-based
predictions. First attempts by LHCb of measuring the ditauon [22] and
dielectron [23] modes lead to the results
$\mathcal{B}({{B}^{0}_{s}}\rightarrow\tau^{+}\tau^{-})<6.3\times 10^{-3}$,
$\mathcal{B}(B^{0}\rightarrow\tau^{+}\tau^{-})<2.1\times 10^{-3}$,
$\mathcal{B}({{B}^{0}_{s}}\rightarrow e^{+}e^{-})<11.2\times 10^{-9}$ and
$\mathcal{B}(B^{0}\rightarrow e^{+}e^{-})<3.0\times 10^{-9}$ at 95% CL.

##  5 LFV searches

The hints of LFUV in $b\rightarrow s\ell\ell$ decays have motivated recent LFV
searches, seeking to observe decays of the type $b\rightarrow
s\ell\ell^{\prime}$ or $B\rightarrow\ell\ell^{\prime}$. A first study of
${{{B}^{+}}}\rightarrow{{K}^{+}}\mu^{\pm}e^{\mp}$ [24] lead to the
establishment of the 95% CL limits:
$\mathcal{B}({{{B}^{+}}}\rightarrow{{K}^{+}}\mu^{-}e^{+})<9.5\times 10^{-9}$
and $\mathcal{B}({{{B}^{+}}}\rightarrow{{K}^{+}}\mu^{+}e^{-})<8.8\times
10^{-9}$. Another analysis on ${{{B}^{+}}}\rightarrow{{K}^{+}}\mu^{-}\tau^{+}$
[25], characterized by the original use of the decay
$B_{s2}^{*0}\rightarrow{{{B}^{+}}}{{K}^{+}}$ to constraint the $\tau$ four-
momentum, obtained the less stringent limit
$\mathcal{B}({{{B}^{+}}}\rightarrow{{K}^{+}}\mu^{-}\tau^{+})<4.5\times
10^{-5}$. For what concerns the LFV leptonic modes, the decays
$B^{0}_{(s)}\rightarrow e^{\pm}\mu^{\mp}$ [26] and
$B^{0}_{(s)}\rightarrow\tau^{\pm}\mu^{\mp}$ [27] have been studied , setting
the 95% CL limits to $\mathcal{B}(B^{0}_{s}\rightarrow
e^{\pm}\mu^{\mp})<6.3\times 10^{-9}$, $\mathcal{B}(B^{0}\rightarrow
e^{\pm}\mu^{\mp})<1.3\times 10^{-9}$,
$\mathcal{B}(B^{0}_{s}\rightarrow\tau^{\pm}\mu^{\mp})<4.2\times 10^{-5}$ and
$\mathcal{B}(B^{0}\rightarrow\tau^{\pm}\mu^{\mp})<1.4\times 10^{-5}$.

##  6 Prospects and summary

Most analyses presented have been published on a partial LHCb data set and are
currently being updated. The second column of Table 1 shows the expected
precisions for the $R_{X}$ measurements for the full Run1+Run2 statistics.

Table 1: Expected precisions for $R_{X}=\mathcal{B}(H_{b}\rightarrow X\mu^{+}\mu^{-})/\mathcal{B}(H_{b}\rightarrow Xe^{+}e^{-})$. The numbers are taken from Ref.[28] $R_{X}$ | Run 1&2 (9$\mbox{\,fb}^{-1}$) | Run 3 (23$\mbox{\,fb}^{-1}$) | Run 4 (50$\mbox{\,fb}^{-1}$) | Run 5 (300$\mbox{\,fb}^{-1}$)
---|---|---|---|---
$R_{K}$ | 0.043 | 0.025 | 0.017 | 0.007
$R_{K^{*0}}$ | 0.052 | 0.031 | 0.020 | 0.008
$R_{\phi}$ | 0.130 | 0.076 | 0.050 | 0.020
$R_{pK}$ | 0.105 | 0.061 | 0.041 | 0.016
$R_{\pi}$ | 0.302 | 0.176 | 0.117 | 0.047

On the longer term, the third, fourth and fifth columns of Table 1 show the
evolution of the expected sensitivities for the future runs of data taking.
For the tree semileptonic decays, the LFU ratios $R(D^{0})$, $R(D^{+})$,
$R(D_{s}^{(*)})$, $R(\Lambda_{c}^{(*)})$,
$R({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}})$ and $R(p)$ (from
${{\mathchar 28931\relax}^{0}_{b}}\rightarrow p\tau\nu$) are foreseen during
the first phase of Run 3. Figure 8 shows the evolution of the $R(H_{c})$
ratios throughout the periods of data taking.

Figure 8: Evolution of the sensitivity on the semileptonic ratios $R(H_{c})$
as reported in Ref.[29].

The available results on the anomalies in the $b$-hadron decays show a
combination of deviations, which has triggered an intense activity on the
phenomenological side in studies aiming at constraining the Wilson
coefficients and probing possible contributions of New Physics [18, 31, 30].
The 6 to 7$\sigma$ deviation from SM derived for $C_{9}$ is subject to
interpretations, which try to account for what is observed in both
$b\rightarrow c$ tree transitions and $b\rightarrow s$ loop decays.
Explanations based on vector Leptoquarks [32] and the “4321” model [33] have
become popular. Any explored paradigm will have to satisfy the constraints of
the ${B}^{0}_{s}$ meson mixing and ${B}_{c}^{+}$ meson lifetime.

## References

  * [1] G. Buchalla et al., Rev. Mod. Phys. 68 (1996) 1125-1144
  * [2] J.F. Gunion, E.H. Haber, G.L. Kane, S. Dawson, The Higgs Hunter’s Guide, Front.Phys.80 (2000) 1-404
  * [3] LHCb collaboration, R. Aaij et al., Phys. Rev. Lett. 115 (2015) 111803
  * [4] LHCb collaboration, R. Aaij et al., Phys. Rev. Lett. 120 (2018) 171802
  * [5] L. Breiman, J.H. Friedman, R.A. Olshen and C.J. Stone, Classification and regression trees, Wadsworth international group (1984) Belmont, California, USA
  * [6] Heavy Flavor Averaging Group, Y. Amhis et al., arXiv:1909.12524, latest results and plots available at https://hflav.web.cern.ch/
  * [7] Belle Collaboration, A. Abdesselam et al., arXiv:1904.08794
  * [8] D. Bigi, P. Gambino, Phys. Rev. D94 (2016) 094008; F.Bernlochner et al., Phys. Rev. D95 (2017) 115008 ; D.Bigi et al., JHEP 1711 (2017) 061; S.Jaiswal et al., JHEP 1712 (2017) 060
  * [9] LHCb collaboration, R. Aaij et al., Phys. Rev. Lett. 120 (2018) 121801
  * [10] A.Yu.Anisimov et al., Phys. Lett. B452 (1999) 129; M.A.Ivanov et al., Phys. Rev. D73 (2006) 054024; E.Hernandez et al., Phys. Rev. D74 (2006) 074008
  * [11] LHCb collaboration, R. Aaij et al., a: JHEP 06 (2014) 133; b: JHEP 09 (2015) 179; c: PRL 125 (2020) 011802; d: JHEP 06 (2015) 115
  * [12] LHCb collaboration, R. Aaij et al., JHEP 09 (2018) 146
  * [13] Descotes-Genon et al., JHEP 05 (2013) 137
  * [14] LHCb collaboration, R. Aaij et al., Phys. Rev. Lett. 122 (2019) 191801
  * [15] LHCb collaboration, R. Aaij et al., JHEP 08 (2017) 055
  * [16] LHCb collaboration, R. Aaij et al., JHEP 05 (2020) 040
  * [17] Non-exhaustive: C. Bobeth et al. JHEP 07 (2007) 040; M.Bordone et al., Eur. Phys. J. C76 (2016) 440; W. Altmannshofer et al., Phys. Rev. D96 (2017) 055008
  * [18] M. Alguero et al., Eur. Phys. J. C79 (2019) 8, 714; Eur. Phys. J.C80 (2020) 6, 511 (addendum).
  * [19] ATLAS collaboration, ATLAS-CONF-2020-049
  * [20] CMS collaboration, CMS-PAS-BPH-20-003
  * [21] LHCb collaboration, LHCb-CONF-2020-002
  * [22] LHCb collaboration, R. Aaij et al., Phys. Rev. Lett. 118 (2017) 251802
  * [23] LHCb collaboration, R. Aaij et al., Phys. Rev. Lett. 124 (2020) 211802
  * [24] LHCb collaboration, R. Aaij et al., Phys. Rev. Lett. 123 (2019) 241802
  * [25] LHCb collaboration, R. Aaij et al., JHEP 06 (2020) 129
  * [26] LHCb collaboration, R. Aaij et al., JHEP 03 (2018) 078
  * [27] LHCb collaboration, R. Aaij et al., Phys. Rev. Lett. 123 (2019) 211801
  * [28] LHCb collaboration, R. Aaij et al., arXiv:1808.08865, LHCB-PUB-2018-009, CERN-LHCC-2018-027
  * [29] S. Bifani et al., J. Phys. G: Nucl. Part. Phys. 46 (2019) 023001
  * [30] J. Aebischer et al., arXiv:1903.10434
  * [31] D. M. Straub, flavio package, arXiv:1810.08132
  * [32] D. Buttazzo et al., JHEP 11 (2017) 044
  * [33] L. Di Luzio et al., Phys. Rev. D96, (2017) 115011; JHEP 11 (2018) 081


# Mean-field limit for particle systems with topological interactions

Dario Benedetto  Dario Benedetto Dipartimento di Matematica, Università di
Roma ‘La Sapienza’ P.le Aldo Moro 2, 00185 Roma, Italy
<EMAIL_ADDRESS>,  Emanuele Caglioti  Emanuele Caglioti Dipartimento
di Matematica, Università di Roma ‘La Sapienza’ P.le Aldo Moro 2, 00185 Roma,
Italy<EMAIL_ADDRESS>and  Stefano Rossi  Stefano Rossi Dipartimento
di Matematica, Università di Roma ‘La Sapienza’ P.le Aldo Moro 2, 00185 Roma,
Italy<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

The mean-field limit for systems of self-propelled agents with “topological
interaction” cannot be obtained by means of the usual Dobrushin approach. We
get a result on this direction by adapting to the multidimensional case the
techniques developed by Trocheris in 1986 to treat the Vlasov-Poisson equation
in one dimension.

###### Key words and phrases:

mean-field limit, topological interaction, Cucker-Smale model

###### 2020 Mathematics Subject Classification:

35Q92, 35Q83, 82B40,

##  1\. Introduction

Many interesting physical systems can be described at the microscopic level as
particle dynamics and at the mesoscopic level with kinetic equations. In the
wide field of two-body interactions, the link between these two regimes is
mathematically well understood in the case of the mean-field limit, i.e. when
the density of the particles diverges with their number $N$, the mean free
path vanishes as $1/N$ and the interaction intensity scales with $1/N$. In
this limit, each particle feels the interaction with the others as a mean.

A rigorous mathematical proof of this result can be done in the case of two-
body interactions with sufficiently regular potentials. This classical
achievement has been obtained independently by several authors in the ’$70$s
(see [5, 14, 27]) and its explanation is particularly clear in the Dobrushin’s
argument [14] where the result follows by noticing that the empirical measure
associated with the particle system is a weak solution of the mean-field
equation; the proof follows by showing the weak continuity, w.r.t the initial
datum, of the weak solutions.

Although the theory for regular pairwise interactions is sufficiently well
understood, going beyond it considering singular potentials, is instead a
harder task. This is the case of the three-dimensional Vlasov-Poisson
equation, which is the most important equation of plasma physics and of
galactic dynamics, based on the choice of the Coulomb or Newton potential,
respectively. In this equation, the potential $1/r$ is singular at the origin
and does not belong to any $L^{p}$ space. Although the mean-field limit for
the Vlasov-Poisson equation remains an open problem, there has been important
progress in recent years, see the works [21, 22] where the mean-field limit is
proven for potentials with singularities “weaker than $1/r$” and also [25,
26]. However, in the case of the one-dimensional Vlasov-Poisson equation, the
problem has been solved in [30, 29] and with a simpler proof in [20], being
the force discontinuous, but not diverging.

The mean-field limit is a case of propagation of chaos, i.e. the $j$-particle
distribution function factorize in the limit. This property is the key for
obtaining a kinetic description of the particle dynamics (see for instance
[28] and [10, 16, 23] for some reviews on this point of view).

In recent years, the conceptual and mathematical apparatus of kinetic
equations has been used in the study of self-propelled particle systems of
biological nature, in particular for the motion of swarms and other animals.
Starting with the pioneering paper in [31], several models have been proposed
to explain the evolution of these systems. In the simplest [11, 12, 31], a
bird is modeled as a self-propelling particle that interacts with its
neighbors. The interaction is such that neighboring birds tend to align their
velocities. For many of these models, the mean-field limit has often been used
to obtain a kinetic description of the dynamics (see, for instance, [18, 7, 6,
8, 17, 3]).

A few years ago, supported by observational data ([2, 9, 1]), “topological”
models for interaction were introduced: an agent reacts to the presence of
another not according to the distance, but according to the proximity ranking
(see eq.s (1.1), (1.2), (1.3) below for a rigorous formulation). These models
come out of the case of two-body interaction, and present various problems in
their kinetic treatment. In particular, the solutions of the kinetic equation
are not weakly continuous w.r.t. the initial datum and there are also some
difficulties in defining the particle motion.

In this paper we prove a result on the mean-field limit for topological
models. We focus our attention on the topological Cucker-Smale model, but,
with the same ideas, it is possible to consider more general cases. A first
result in this direction has been proved in [19], for a smoothed version of
the model in which the weak continuity in the initial datum is recovered. We
also mention that a kinetic Boltzmann equation for a stochastic particle model
with rank-based interaction has been obtained in [13]; the introduced
stochasticity allows to consider the system as a two-body interaction system,
which can be faced with the study of the BBGKY hierarchy.

We formulate the problem and summarize our results. A Cucker-Smale type model
for the motion of $N$ agents, in the mean-field scaling, is the system

$\left\\{\begin{aligned} &\dot{X}_{i}(t)=V_{i}(t)\\\
&\dot{V}_{i}(t)=\frac{1}{N}\sum_{j=1}^{N}p_{ij}(V_{j}(t)-V_{i}(t)),\end{aligned}\right.$
(1.1)

where the “communication weights” $\\{p_{ij}\\}_{i,j=1}^{N}$ are positive
functions that take into account the interactions between agents. In classical
models, $p_{ij}$ depends only on the distance $|X_{i}-X_{j}|$ between the
agents. In topological models the weights depend on the positions of the
agents by their rank

$p_{ij}\coloneq K\bigl{(}M(X_{i},|X_{i}-X_{j}|)\bigr{)},$ (1.2)

where $K\colon[0,1]\to\mathbb{R}^{+}$ and, for $r>0$, the function

$M(X_{i},r)\coloneq\frac{1}{N}\sum_{k=1}^{N}\mathcal{X}\\{|X_{k}-X_{i}|\leq
r\\}$ (1.3)

counts the number of agents at distance less than or equal to $r$ from
$X_{i}$, normalized with $N$. Note that in this case $p_{ij}$ is a stepwise
function of the positions of all the agents. In the sequel we assume that $K$
is a positive decreasing function, Lipschitz continuous, and such that
$\int_{0}^{1}K(z)\,\mathop{}\\!\mathrm{d}z=\gamma$.

In the mean-field limit $N\to+\infty$, the one-agent distribution function
$f_{t}=f(t,x,v)$ is expected to verify the equation

$\partial_{t}f_{t}+v\cdot\nabla_{x}f_{t}+\nabla_{v}\cdot\left(W[Sf_{t},f_{t}](x,v)f_{t}\right)=0,$
(1.4)

where $Sf_{t}(x)\coloneq\int f_{t}(x,v)\mathop{}\\!\mathrm{d}v$ is the spatial
distribution and where, given a probability density $f$ in
$\mathbb{R}^{d}\times\mathbb{R}^{d}$ and a probability density $\rho$ in
$\mathbb{R}^{d}$,

$W[\rho,f](x,v)\coloneq\int
K\left(M[\rho](x,|x-y|)\right)\,(w-v)f(y,w)\,\mathop{}\\!\mathrm{d}y\,\mathop{}\\!\mathrm{d}w,$
(1.5)

with

$M[\rho](x,r)\coloneq\int_{|x^{\prime}-x|\leq
r}\rho(x^{\prime})\,\mathop{}\\!\mathrm{d}x^{\prime}.$ (1.6)

A weak formulation of this equation is given requiring that the solution
$f_{t}$ fulfills

$\int\alpha(x,v)\mathop{}\\!\mathrm{d}f_{t}(x,v)=\int\alpha\left(X_{t}(x,v),V_{t}(x,v)\right)\mathop{}\\!\mathrm{d}f_{0}(x,v)$

for any $\alpha\in C_{b}(\mathbb{R}^{d}\times\mathbb{R}^{d})$, where $f_{0}$
is the initial probability measure and $(X_{t}(x,v),V_{t}(x,v))$ is the flow
defined by

$\left\\{\begin{aligned} &\dot{X}_{t}(x,v)=V_{t}(x,v)\\\
&\dot{V}_{t}(t,x,v)=W[Sf_{t},f_{t}](X_{t}(x,v,),V_{t}(x,v))\\\ &X_{0}(x,v)=x,\
\ V_{0}(x,v)=v.\\\ \end{aligned}\right.$ (1.7)

In other words, $f_{t}$ is the push-forward of $f_{0}$ along the flow
generated by the velocity field, determined by $f_{t}$ itself.

It is easy to verify that the empirical measure

$\mu^{N}_{t}\coloneq\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{i}^{N}(t)}\,\delta_{V_{i}^{N}(t)}$

associated with the solution of (1.1), (1.2) and (1.3) is a weak solution of
(1.4). Namely, $M[S\mu^{N}_{t}](X,r)$ is exactly $M(X,r)$ defined in (1.3)
(from now on we use the more complete notation $M[S\mu^{N}_{t}](X,r)$). Thus,
we can rewrite the agent evolution in (1.1) as

$\left\\{\begin{aligned} &\dot{X}_{i}^{N}(t)=V_{i}^{N}(t)\\\
&\dot{V}_{i}^{N}(t)=W[S\mu^{N}_{t},\mu^{N}_{t}](X^{N}_{i}(t),V^{N}_{i}(t)).\end{aligned}\right.$
(1.8)

In the Dobrushin approach to the mean-field limit, the result is achieved from
this fact and from the weak continuity, w.r.t the initial datum, of the weak
solutions of (1.4). We cannot use this approach in presence of topological
interaction, since in general the solutions of (1.7) are not weakly continuous
w.r.t the initial datum (see Section 3). We can overcome this difficulty if
the solution of (1.4) has a bounded density. To obtain our result, we adapt
the ideas used in [30] for the derivation of the one-dimensional Vlasov
equation in presence of discontinuity of the force. In particular we prove
that

[Thm. 3.1] the $N$-particle dynamics is well defined, except for a set of
measure zero;

[Thm. 4.1] if $f_{0}$ is bounded, there exists a unique weak solution $f_{t}$
of the topological Cucker-Smale equation, which is bounded;

[Thm. 5.1] if $\mu^{N}_{t}$ solves (1.8) and
$\mu^{N}_{0}\rightharpoonup\mu_{0}$, then $\mu^{N}_{t}\rightharpoonup f_{t}$.

We divide the work as follows: in Section 2 we discuss some properties of the
“discrepancy distance”, the main tool for dealing with topological
interactions. In Section 3 we discuss existence, uniqueness and regularity of
the agent dynamics (1.8), proving Thm. 3.1. In Section 4 we discuss existence,
uniqueness and regularity of the weak solutions of the mean-field equation
(1.4) with bounded initial datum, proving Thm. 4.1. In Section 5 we prove Thm.
(5.1).

##  2\. Distances and weak convergence

We recall that the 1-Wasserstein distance ${\mathscr{W}}$ of two probability
measures $\rho_{1}$ and $\rho_{2}$ on $\mathbb{R}^{d}$ can be defined by
duality with Lipschitz functions:

$\displaystyle{\mathscr{W}}(\rho_{1},\rho_{2})$ $\displaystyle=\sup_{\phi\in
C_{b}(\mathbb{R}^{d}),\text{Lip}(\phi)\leq
1}\int\phi(\mathop{}\\!\mathrm{d}\rho_{1}-\mathop{}\\!\mathrm{d}\rho_{2})$
$\displaystyle=\sup_{\phi\in
C^{1}_{b}(\mathbb{R}^{d}),\|\nabla\phi\|_{\infty}\leq
1}\int\phi(\mathop{}\\!\mathrm{d}\rho_{1}-\mathop{}\\!\mathrm{d}\rho_{2}),$

where $\text{Lip}(\phi)$ is the Lipschitz constant of $\phi$.

The counter of the number of particles in (1.6) is not continuous w.r.t.
${\mathscr{W}}$, so we work with the weaker topology induced by another
distance, the discrepancy, defined as

${\mathscr{D}}(\rho_{1},\rho_{2})\coloneq\sup_{x,r>0}\left|\int_{B_{r}(x)}\mathop{}\\!\mathrm{d}\rho_{1}-\int_{B_{r}(x)}\mathop{}\\!\mathrm{d}\rho_{2}\right|.$

Here and after, we denote by $B_{r}(x)$ the closed ball of center $x$ and
radius $r$ in $\mathbb{R}^{d}$. In the sequel, we also indicate by $B_{R}$ the
closed ball $B_{R}(0)$. The discrepancy distance is mostly used in one
dimension to quantify the uniformity of sequence of points (see [24, 15]), but
its multidimensional version is cited in [27], in the contest of kinetic
limits.

By definition, it holds the following proposition.

######  Proposition 2.1 (Lipschitzianity of $M$ w.r.t. ${\mathscr{D}}$).

Let $\rho_{1}$ and $\rho_{2}$ be two probability measures on $\mathbb{R}^{d}$.
Then, for any $x\in\mathbb{R}^{d}$ and $r>0$,

$|M[\rho_{1}](x,r)-M[\rho_{2}](x,r)|\leq{\mathscr{D}}(\rho_{1},\rho_{2}).$

We can also define ${\mathscr{D}}$ in terms of regular functions. Let $X$ be
the subset of $C_{b}^{1}([0,+\infty);\mathbb{R})$, and define

$\|\phi\|_{X}\coloneq\int_{0}^{+\infty}|\phi^{\prime}(r)|\mathop{}\\!\mathrm{d}r.$

Then

${\mathscr{D}}(\rho_{1},\rho_{2})=\sup_{\phi\in X:\,\|\phi\|_{X}\leq
1}\sup_{x}\int\phi\bigl{(}|x-y|\bigr{)}\bigl{(}\mathop{}\\!\mathrm{d}\rho_{1}(y)-\mathop{}\\!\mathrm{d}\rho_{2}(y)\bigr{)}.$

This assertion is an easy consequence of the following lemma.

######  Lemma 2.1.

Let $g_{1}$ and $g_{2}$ be two probability measures on $[0,+\infty)$. Then

$\sup_{r\geq
0}\left|\int_{[0,r]}\mathop{}\\!\mathrm{d}g_{1}-\int_{[0,r]}\mathop{}\\!\mathrm{d}g_{2}\right|=\sup_{\phi\in
X:\,\|\phi\|_{X}\leq
1}\int_{0}^{+\infty}\phi\left(\mathop{}\\!\mathrm{d}g_{1}-\mathop{}\\!\mathrm{d}g_{2}\right).$
(2.1)

###### Proof.

Fix $r>0$, there exists $\phi_{r,\varepsilon}\in X$ with
$\|\phi_{r,\varepsilon}\|_{X}=1$ and such that $\phi_{r,\varepsilon}(s)=1$ if
$0\leq s\leq r$ and $\phi_{r,\varepsilon}(s)=0$ if $s\geq r+\varepsilon$. For
any measure $g$,

$\lim_{\varepsilon\to
0}\int_{0}^{+\infty}\bigl{(}\phi_{r,\varepsilon}(s)-\mathcal{X}\\{s\in[0,r]\\}\bigr{)}\mathop{}\\!\mathrm{d}g(s)=0,$

then

$\int_{[0,r]}(\mathop{}\\!\mathrm{d}g_{1}-\mathop{}\\!\mathrm{d}g_{2})=\lim_{\varepsilon\to
0}\int_{0}^{+\infty}\phi_{r,\varepsilon}(\mathop{}\\!\mathrm{d}g_{1}-\mathop{}\\!\mathrm{d}g_{2})\leq\sup_{\phi\in
X:\,\|\phi\|_{X}\leq
1}\int_{0}^{+\infty}\phi(\mathop{}\\!\mathrm{d}g_{1}-\mathop{}\\!\mathrm{d}g_{2}).$

To prove the opposite inequality, we denote by $G_{1}$ and $G_{2}$ the
distribution functions of $g_{1}$ and $g_{2}$:

$G_{i}(r)\coloneq\int_{[0,r]}\mathop{}\\!\mathrm{d}g_{i}.$

Then, integrating by parts,

$\int_{0}^{+\infty}\phi(\mathop{}\\!\mathrm{d}g_{1}-\mathop{}\\!\mathrm{d}g_{2})=-\int_{0}^{+\infty}\phi^{\prime}(r)\bigl{(}G_{1}(r)-G_{2}(r)\bigr{)}\mathop{}\\!\mathrm{d}r\leq\|\phi\|_{X}\|G_{1}-G_{2}\|_{\infty}.$

We conclude the proof by noticing that $\|G_{1}-G_{2}\|_{\infty}$ is exactly
the left-hand-side of (2.1). ∎

For our purposes, we need the equivalence of ${\mathscr{D}}$ and
${\mathscr{W}}$ in the case in which one of the two measures has bounded
density. We note that in the general case the equivalence is false, as can be
easily checked by considering two Dirac measures $\delta_{x_{1}}$ and
$\delta_{x_{2}}$: ${\mathscr{W}}$ vanishes when $|x_{1}-x_{2}|\to 0$, while
${\mathscr{D}}$ is one whenever $x_{1}\neq x_{2}$. Nevertheless, using the
covering principles as in [4], for measures on a compact set, it can be proved
the continuity of the Wasserstein distance ${\mathscr{W}}$ w.r.t. the
discrepancy distance ${\mathscr{D}}$. For the sake of completeness, we give a
proof in the appendix, although this property is not really necessary for our
results.

In the sequel, in the definition of ${\mathscr{D}}$ we choose functions in
$\phi\in C([0,+\infty),\mathbb{R})$, with first derivative continuous up to a
finite number of jumps. With abuse of notation, we keep calling this set of
functions $X$. Let us expose some technical properties.

Given $\phi\in X$, we define some useful regularizations, $\phi^{\pm}$,
$\phi_{\varepsilon}$ and $\psi_{\varepsilon}$, with $\varepsilon>0$, as
follows. Denoting by $\tilde{\phi}$ the function

$\tilde{\phi}(r)\coloneq\int_{0}^{r}|\phi^{\prime}(s)|\mathop{}\\!\mathrm{d}s,$

we define

$\phi^{\pm}(r)\coloneq\left\\{\begin{aligned}
&\frac{1}{2}(\tilde{\phi}(r)\pm\phi(r)),\ &\text{ if }r\geq 0,\\\
&\pm\frac{1}{2}\phi(0),&\text{ if }r<0,\end{aligned}\right.$

and

$\phi_{\varepsilon}(r)\coloneq\phi^{+}(r+\varepsilon)-\phi^{-}(r-\varepsilon).$
(2.2)

Finally, fixed a regular mollifier $\eta$ supported in $(0,1)$, we define

$\psi_{\varepsilon}(r)\coloneq\int_{0}^{\varepsilon}\eta_{\varepsilon}(s)\phi^{+}(r+s)\mathop{}\\!\mathrm{d}s-\int_{0}^{\varepsilon}\eta_{\varepsilon}(s)\phi^{-}(r-s)\mathop{}\\!\mathrm{d}s.$
(2.3)

where $\eta_{\varepsilon}(s)\coloneq\varepsilon^{-1}\eta(s/\varepsilon)$.

We summarize the properties of these regularizations in the following lemma,
where we indicate with $c$ any constat which does not depends on $\phi$ and
$\varepsilon$.

######  Lemma 2.2.

a

  1. i)

$\phi^{\pm}$ are not decreasing. Moreover

$\int_{0}^{+\infty}(\phi^{\pm})^{\prime}(r)\,\mathop{}\\!\mathrm{d}r\leq\|\phi\|_{X}$
(2.4)

and $\phi(r)=\phi^{+}(r)-\phi^{-}(r)$ for $r\geq 0$.

  2. ii)

$\phi_{\varepsilon}\in X$, $\phi(r)\leq\phi_{\varepsilon}(r)$ and

$\int_{0}^{+\infty}\bigl{(}\phi_{\varepsilon}(r)-\phi(r)\bigr{)}\mathop{}\\!\mathrm{d}r\leq
2\varepsilon\|\phi\|_{X}.$ (2.5)

  3. iii)

$\psi_{\varepsilon}(r)\geq\phi(r)$. Moreover $\psi_{\varepsilon}$ is a $C^{1}$
function in $X$,

$\|(\psi_{\varepsilon})^{\prime}\|_{\infty}\leq\frac{2}{\varepsilon}\|\eta\|_{\infty}\|\phi\|_{X}$
(2.6)

and

$\int_{0}^{+\infty}|\psi_{\varepsilon}(r)-\phi(r)|\mathop{}\\!\mathrm{d}r\leq
c\varepsilon\|\phi\|_{X}.$ (2.7)

###### Proof.

The proof is elementary, we only describe how to get the bounds in ii) and
iii). Since $\phi=\phi^{+}-\phi^{-}$, we rewrite the l.h.s. of (2.5) as

$\displaystyle\int_{0}^{+\infty}\bigl{(}\phi^{+}(r+\varepsilon)-\phi^{+}(r)\bigr{)}+\bigl{(}\phi^{-}(r)-\phi^{-}(r-\varepsilon)\bigr{)}\mathop{}\\!\mathrm{d}r$
$\displaystyle=\int_{0}^{+\infty}\left(\int_{0}^{\varepsilon}\left((\phi^{+})^{\prime}(r+\xi)+(\phi^{-})^{\prime}(r-\xi)\right)\mathop{}\\!\mathrm{d}\xi\right)\mathop{}\\!\mathrm{d}r\leq
2\varepsilon\|\phi\|_{X}.$

The estimate in (2.6) is immediate while, regarding (2.7), we rewrite
$\psi_{\varepsilon}(r)-\phi(r)$ as

$\displaystyle\int_{0}^{1}\eta(s)\bigl{(}\phi^{+}(r+\varepsilon
s)-\phi^{+}(r)+\phi^{-}(r)-\phi^{-}(r-\varepsilon
s)\bigr{)}\mathop{}\\!\mathrm{d}s$
$\displaystyle=\varepsilon\int_{0}^{1}s\eta(s)\left(\int_{0}^{1}(\phi^{+})^{\prime}(r+\varepsilon
s\xi)\mathop{}\\!\mathrm{d}\xi+\int_{0}^{1}(\phi^{-})^{\prime}(r-\varepsilon
s\xi)\mathop{}\\!\mathrm{d}\xi\right)\mathop{}\\!\mathrm{d}s.$

We conclude by integrating in $r$, switching the order of integration and
using (2.4). ∎

Now we can prove the following proposition.

######  Proposition 2.2.

Let $\rho$ and $\nu$ be two probability measures on $\mathbb{R}^{d}$ with
support in a ball $B_{R}$ and such that $\rho\in L^{\infty}(\mathbb{R}^{d})$.
Then

${\mathscr{D}}(\nu,\rho)\leq
C(\|\rho\|_{\infty},R)\sqrt{{\mathscr{W}}(\nu,\rho)},$

where $C$ is a constant that depends on the dimension $d$, as well as on
$\|\rho\|_{\infty}$ and on $R$.

###### Proof.

Let $\phi$ be in $X$ and consider $\psi_{\varepsilon}$ as in (2.3). Fixed
$x\in\mathbb{R}^{d}$, let $\Phi$ and $\Psi_{\varepsilon}$ be the functions

$\Phi(y)\coloneq\phi(|x-y|)\ \ \text{and}\ \
\Psi_{\varepsilon}(y)\coloneq\psi_{\varepsilon}(|x-y|).$

Then, from iii) of Lemma 2.2,

$\int\Phi\mathop{}\\!\mathrm{d}\nu-\int\Phi\mathop{}\\!\mathrm{d}\rho\leq\int\Psi_{\varepsilon}\mathop{}\\!\mathrm{d}\nu-\int\Phi\mathop{}\\!\mathrm{d}\rho=\int\Psi_{\varepsilon}\mathop{}\\!\mathrm{d}(\nu-\rho)+\int(\Psi_{\varepsilon}-\Phi)\mathop{}\\!\mathrm{d}\rho.$

From (2.6) of Lemma 2.2, the first term is bounded by
$\frac{c}{\varepsilon}\|\phi\|_{X}{\mathscr{W}}(\nu,\rho)$. Regarding the
second term, denoting by $\sigma_{r}$ the uniform measure on
$\mathop{}\\!\partial B_{r}(x)$, we have

$\displaystyle\int(\Psi_{\varepsilon}-\Phi)\mathop{}\\!\mathrm{d}\rho$
$\displaystyle\leq\|\rho\|_{\infty}\int_{0}^{+\infty}\mathop{}\\!\mathrm{d}r\left(\psi_{\varepsilon}(r)-\phi(r)\right)\int_{\mathop{}\\!\partial
B_{r}(x)}\mathcal{X}\\{z\in B_{R}\\}\sigma(\mathop{}\\!\mathrm{d}z)$ (2.8)
$\displaystyle\leq c\varepsilon R^{d-1}\|\phi\|_{X}\|\rho\|_{\infty},$

where in the last inequality we have used (2.7). Optimizing on $\varepsilon$
and passing to the supremum in $\phi$, we get the proof. ∎

Note that if $\mu^{N}$ is an empirical measure and $\nu$ a probability measure
that does not give mass to the atoms of $\mu^{N}$,
${\mathscr{D}}(\mu^{N},\rho)\geq 1/N$. With this constraint, the discrepancy
between two empirical measures is “small” if the measures are close in the
sense specified in the following proposition.

######  Proposition 2.3.

Let

$\mu^{N}=\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}\ \ \text{and}\ \
\nu^{N}=\frac{1}{N}\sum_{i=1}^{N}\delta_{y_{i}}$

be two empirical measures on $\mathbb{R}^{d}$ and take $\delta>0$ such that
$|x_{i}-y_{i}|\leq\delta$ for all $i=1,\dots,N$. Then, for any probability
measure $\rho\in L^{\infty}(\mathbb{R}^{d})$ supported on a ball $B_{R}$,

${\mathscr{D}}(\mu^{N},\nu^{N})\leq
cR^{d-1}\delta\|\rho\|_{\infty}+c{\mathscr{D}}(\mu^{N},\rho).$

###### Proof.

Given $\phi\in X$ with $\|\phi\|_{X}\leq 1$, we construct $\phi_{\delta}$ as
in (2.2) and, fixed $x\in\mathbb{R}^{d}$, we consider
$\Phi(y)\coloneq\phi(|x-y|)$, $\Phi_{\delta}(y)\coloneq\phi_{\delta}(|x-y|)$.

Since $|x-x_{i}|-\delta\leq|x-y_{i}|\leq|x-x_{i}|+\delta$, we have that

$\Phi(y_{i})=\phi^{+}(|x-y_{i}|)-\phi^{-}(|x-y_{i}|)\leq\Phi_{\delta}(x_{i}).$

Then

$\int\Phi\mathop{}\\!\mathrm{d}(\nu^{N}-\mu^{N})\leq\int(\Phi_{\delta}-\Phi)\mathop{}\\!\mathrm{d}\mu^{N}=\int(\Phi_{\delta}-\Phi)\mathop{}\\!\mathrm{d}(\mu^{N}-\rho)+\int(\Phi_{\delta}-\Phi)\mathop{}\\!\mathrm{d}\rho.$

Since $(\phi_{\delta}-\phi)\in X$, the first term is bounded by
$c{\mathscr{D}}(\mu^{N},\rho)$. Using (2.5) and reasoning as in (2.8) we
estimate the second term with $c\delta R^{d-1}\|\rho\|_{\infty}.$ ∎

##  3\. Agent dynamics

One of the difficulties in handling (1.8) is that the dynamic is not
continuous w.r.t the initial datum. For instance, consider three agents
$\\{X_{i}\\}_{i=1}^{3}$ on a line, such that

$\begin{array}[]{lll}X_{1}(0)=-1,&X_{2}(0)=\varepsilon,&X_{3}(0)=1,\\\
\,V_{1}(0)=-1,&\,V_{2}(0)=0,&\,V_{3}(0)=1,\end{array}$ (3.1)

with $\varepsilon\in(-1,1)\setminus\\{0\\}$. Then
$p_{i,j}=M(X_{i},|X_{i}-X_{j}|)$ takes the values $1/3,2/3,1$. Suppose for
simplicity that $K(2/3)=3$ and $K(1)=0$, then the equations for $V_{1}$ and
$V_{3}$ read as

$\left\\{\begin{aligned} &\dot{V}_{1}(t)=V_{2}(t)-V_{1}(t)\\\
&\dot{V}_{3}(t)=V_{2}(t)-V_{3}(t),\end{aligned}\right.$

while

$\dot{V}_{2}(t)=\begin{cases}V_{3}(t)-V_{2}(t)&\text{ if
}\varepsilon\in(0,1)\\\ V_{1}(t)-V_{2}(t)&\text{ if
}\varepsilon\in(-1,0).\end{cases}$

It follows that

$\left\\{\begin{aligned} &V_{1}(t)=-(1+\mathrm{e}^{-2t})/2\\\
&V_{2}(t)=-(1-\mathrm{e}^{-2t})/2\\\
&V_{3}(t)=(-1+4\mathrm{e}^{-t}-\mathrm{e}^{-2t})/2\\\ \end{aligned}\right.$

if $\varepsilon\in(-1,0)$, while

$\left\\{\begin{aligned}
&V_{1}(t)=-(-1+4\mathrm{e}^{-t}-\mathrm{e}^{-2t})/2\\\
&V_{2}(t)=(1-\mathrm{e}^{-2t})/2\\\ &V_{3}(t)=(1+\mathrm{e}^{-2t})/2\\\
\end{aligned}\right.$

if $\varepsilon\in(0,1)$, so that $\\{X_{i}(t),V_{i}(t)\\}_{i=1}^{3}$ is
discontinuous in $\varepsilon=0$. Note that the discontinuity of the
trajectories in the phase space is easily translated in the weak discontinuity
of the empirical measure at time $t$, w.r.t the initial measure.

This discontinuity reflects the fact that, for data as in (3.1) with
$\varepsilon=0$, there is not a unique way to define the dynamics.
Nevertheless, we can prove that the system (1.8) is well-posed for almost all
initial data. To do so, let us define some subsets of the phase space

$\left\\{(X,V)\coloneq(x_{1},\dots,x_{N},v_{1},\dots,v_{N})\in\mathbb{R}^{Nd}\times\mathbb{R}^{Nd}\right\\},$

where $d\geq 1$ is the dimension of the configuration space of the agents.

######  Definition 3.1.

a

  * ${\mathcal{R}}$

is the set of “the regular points”, i.e. the set of points $(X,V)$ such that
for each triad of different indices it holds that
$|x_{i}-x_{k}|\neq|x_{j}-x_{k}|$.

  * ${\mathcal{S}}$

is the “iso-rank” manifold, i.e. the set of points $(X,V)$ such that there
exists a triad of different indices $i,j,k$ for which
$|x_{i}-x_{k}|=|x_{j}-x_{k}|$, i.e. the agents $i$ and $j$ have the same rank
with respect to the agent $k$.

  * ${\mathcal{S}}_{r}$

is the set of the “regular points” of the iso-rank manifold, i.e.. the subset
of points $(X,V)\in{\mathcal{S}}$ such that if $|x_{i}-x_{k}|=|x_{j}-x_{k}|$
then $x_{i}$, $x_{j}$, $x_{k}$ are different and
$(v_{i}-v_{k})\cdot\hat{n}_{ik}\neq(v_{j}-v_{k})\cdot\hat{n}_{jk}$, where
$\hat{n}_{ab}\coloneq(x_{a}-x_{b})/|x_{a}-x_{b}|$.

We can define the dynamics locally in time, not only for initial data in
${\mathcal{R}}$, but also in ${\mathcal{S}}_{r}$. Namely, if initially the
agents $i$ and $j$ have the same rank with respect to the agent $k$, we can
redefine the force exerted on the agent $k$ accordingly to the velocities: if
$(v_{i}-v_{k})\cdot\hat{n}_{ik}>(v_{j}-v_{k})\cdot\hat{n}_{jk}$ we evaluate
the rank as if $|x_{i}-x_{k}|>|x_{j}-x_{k}|$ for $t>0$ and as if
$|x_{i}-x_{k}|<|x_{j}-x_{k}|$ for $t<0$. In other words, the different speeds
of change of the distances among the agents allow the dynamics to leave
${\mathcal{S}}$ instantaneously.

We discuss the existence of the dynamics, so redefined.

######  Lemma 3.1.

If $(X,V)\in{\mathcal{R}}\cup{\mathcal{S}}_{r}$, there exists $\tau>0$ such
that the system (1.8) has a unique solution for $t\in(-\tau,\tau)$, with
initial datum $(X,V)$. Moreover the solution is locally Lipschitz in $t$ and
in $(X,V)$.

We omit the proof.

In ${\mathcal{R}}$ the solution is regular, so we can compute the determinant
of the Jacobian of the flow $J(t)\equiv J(X,V,t)$. It verifies the equation

${\dfrac{\mathrm{d}\phantom{t}}{\mathrm{d}{t}}}J(t)=-\left(\frac{d}{N}\sum_{i,j:i\neq
j}p_{ij}\right)J(t)=-dN\gamma_{N}J(t),$ (3.2)

where

$\gamma_{N}\coloneq\frac{1}{N}\sum_{n=2}^{N}K\left(n/N\right).$

Thus, volumes of the phase space are shrunk in time at a constant rate,
therefore their measure cannot vanish in finite time. This implies the
following fact, of which we omit the proof.

######  Lemma 3.2.

The subset of initial data $(X,V)\in{\mathcal{R}}$ such that the trajectory,
at a first time in the future or in the past, intersects
${\mathcal{S}}\setminus{\mathcal{S}}_{r}$, has Lebesgue measure zero. Namely,
${\mathcal{S}}\setminus{\mathcal{S}}_{r}$ has dimension $2Nd-2$.

This lemma guarantees that, except for a subset of Lebesgue measure zero, we
can prolong the dynamics with initial data in ${\mathcal{R}}$ also after a
crossing in ${\mathcal{S}}$. To define the dynamics for all times, we need to
control the number of crossings.

######  Lemma 3.3.

The subset of initial data $(X,V)\in{\mathcal{R}}$ such that the trajectory
intersect ${\mathcal{S}}_{r}$ infinitely many times in finite time, has
Lebesgue measure zero.

###### Proof.

Fix $T>0$ and suppose to take $(X,V)\in{\mathcal{R}}$ such that the solution
$\left(X^{N}(t),V^{N}(t)\right)=(X_{1}(t),\dots,X_{N}(t),V_{1}(t),\dots,V_{N}(t))$
with initial data $(X,V)$ intersects ${\mathcal{S}}_{r}$ a finite number of
times in $[0,T-\varepsilon)$ and infinitely many times in $[0,T)$. The number
of particles is finite, so we can assume that there exists a triad of indices
such that $|X_{i}-X_{k}|=|X_{j}-X_{k}|$ infinitely many times. Since the
velocities $V_{i}$ are bounded by a constant, as follows by simple
considerations (see also Thm. 3.1), from the equation we have that
$|X_{i}-X_{k}|$ and $|X_{j}-X_{k}|$ are $C^{1}$ functions, with time
derivatives uniformly Lipschitz, if $|X_{i}-X_{k}|$ and $|X_{j}-X_{k}|$ remain
far from $0$. Then, as $t\to T$, either $|X_{i}-X_{k}|\to 0$ or
$(V_{i}-V_{k})\cdot\hat{n}_{ik}$ and $(V_{j}-V_{k})\cdot\hat{n}_{jk}$ converge
to the same limit. In both the cases, the trajectory reaches ${\mathcal{S}}$
at a point that is not in ${\mathcal{S}}_{r}$. We conclude the proof observing
that the intial point with these properties lives in a subset of dimension
$2Nd-1$. ∎

From these lemmas and other few considerations, we obtain the following
theorem.

######  Theorem 3.1.

Except for a set of measure zero, given
$(X,V)\in\mathbb{R}^{Nd}\times\mathbb{R}^{Nd}$, there exists a unique global
solution

$\left(X^{N}(t,X,V),V^{N}(t,X,V)\right)\in
C^{1}(\mathbb{R}^{+},\mathbb{R}^{2dN})\times
C(\mathbb{R}^{+},\mathbb{R}^{2dN})$

with initial datum $(X,V)$.

Moreover, given $R_{x}>0$ and $R_{v}>0$, we have that

$|X_{i}(t)|\leq R_{x}+tR_{v},\ |V_{i}(t)|\leq R_{v}$

for any $i$, if $|x_{i}|\leq R_{x}$ and $|v_{i}|\leq R_{v}$. Therefore
$V_{i}(t,X,V)$ has Lipschitz constant bounded by $2R_{v}K(0)$.

###### Proof.

The proof follows easily from Lemma (3.1), Lemma (3.2) and Lemma (3.3).

The a-priori bound on the support follows from (3.2) and by noticing that

${\dfrac{\mathrm{d}\phantom{t}}{\mathrm{d}{t}}}|V_{i}(t)|^{2}=-2\sum_{j\neq
i}p_{ij}\left(|V_{i}(t)|^{2}-V_{i}(t)\cdot V_{j}(t)\right)$

is null or negative if $|V_{i}|^{2}$ is maximum in $i$.

∎

##  4\. The mean-field equation in $L^{\infty}$

In this section we show how to get an existence and uniqueness result for
bounded weak solutions of equation (1.4). We start by stating some elementary
facts.

######  Lemma 4.1.

Let $\rho\in L^{\infty}(\mathbb{R}^{d})$ be a probability density.

  1. i)

Given $r_{1},r_{2}>0$,

$\left|M[\rho](x,r_{1})-M[\rho](x,r_{2})\right|\leq
c\|\rho\|_{\infty}\left|r_{1}^{d}-r_{2}^{d}\right|.$

  2. ii)

Given $x_{1},x_{2}\in\mathbb{R}^{d}$ and $r>0$,

$\left|M[\rho](x_{1},r)-M[\rho](x_{2},r)\right|\leq
c\|\rho\|_{\infty}r^{d-1}|x_{1}-x_{2}|.$

###### Proof.

The proof of the first assertion is immediate. For the second, we use the
following splitting

$\displaystyle\mathcal{X}\\{|x_{1}-y|<r\\}-\mathcal{X}\\{|x_{2}-y|<r\\}$
$\displaystyle=\mathcal{X}\\{|x_{1}-y|<r\\}\mathcal{X}\\{|x_{2}-y|\geq r\\}$
$\displaystyle-\mathcal{X}\\{|x_{2}-y|<r\\}\mathcal{X}\\{|x_{1}-y|\geq r\\}$

and we note that, if $|x_{1}-x_{2}|\geq r$,

$\int_{|x_{1}-y|<r}\mathcal{X}\\{|x_{2}-y|\geq r\\}\mathop{}\\!\mathrm{d}y\leq
cr^{d}\leq cr^{d-1}|x_{1}-x_{2}|,$

while, if $|x_{1}-x_{2}|<r$,

$\displaystyle\int_{|x_{1}-y|<r}$ $\displaystyle\mathcal{X}\\{|x_{2}-y|\geq
r\\}\mathop{}\\!\mathrm{d}y\leq\int\mathcal{X}\left\\{r-|x_{1}-x_{2}|<|x_{1}-y|<r\right\\}\mathop{}\\!\mathrm{d}y$
$\displaystyle=cr^{d}\left(1-\left(1-|x_{1}-x_{2}|/r\right)^{d}\right)\leq
cdr^{d-1}|x_{1}-x_{2}|.$

∎

In the following, we denote by $\mathcal{B}_{r}$ the closed ball of center $0$
and radius $r$ in $L^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d})$ and by
$C_{w}\left([0,+\infty);L^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d})\right)$
the set of families of bounded probability densities $\\{f_{t}\\}_{t\geq 0}$
which are weakly continuous in time in the sense of measures.

######  Lemma 4.2.

Let $\\{f_{t}\\}_{t\geq 0}$ be a family of probability densities such that
$\\{f_{t}\\}\in C_{w}\left([0,+\infty);\mathcal{B}_{r(t)}\right)$, with $r(t)$
a continuous nondecreasing function. Suppose that

$\text{supp}(f_{t})\subset B_{R_{x}(t)}\times B_{R_{v}(t)},$ (4.1)

where $R_{v}(t)$ and $R_{x}(t)$ are two continuous non decreasing functions.
Then, for any initial datum $(x,v)\in\mathbb{R}^{d}\times\mathbb{R}^{d}$,
there exists a unique global solution of (1.7).

###### Proof.

From the classical Cauchy-Lipschitz theory, we only have to verify that
$W[Sf_{t},f_{t}](x,v)$ is bounded on compact sets, locally Lipschitz and
continuous in $t$.

Recalling 1.5, the boundness on compact sets follows from

$|W[Sf_{t},f_{t}](x,v)|\leq\|K\|_{\infty}\left(R_{v}(t)+|v|\right).$

Since from i) and ii) of Lemma 4.1

$\begin{aligned} |M[Sf_{t}](x_{1},|x_{1}-y|)&-M[Sf_{t}](x_{2},|x_{2}-y|)|\\\
&\leq
c\|Sf_{t}\|_{\infty}(|x_{1}|+|x_{2}|+|y|)^{d-1}|x_{1}-x_{2}|\end{aligned},$

we have that, if $(x_{1},v_{1})$ and $(x_{2},v_{2})$ belong to a compact
subset of $\mathbb{R}^{d}\times\mathbb{R}^{d}$,

$|W[Sf_{t},f_{t}](x_{1},v_{1})-W[Sf_{t},f_{t}](x_{2},v_{2})|\leq
C(|x_{1}-x_{2}|+|v_{1}-v_{2}|),$

where $C$ depends on $R_{x},R_{v}$ and on the diameter of the compact set.

In order to prove that $W[Sf_{t},f_{t}](x,v)$ is continuous in $t$, we first
observe that ${\mathscr{W}}(Sf_{t},Sf_{s})\leq{\mathscr{W}}(f_{t},f_{s})$ and
that, from the Lipschitzianity of $K$ and Propositions 2.1 and 2.2,
$K\left(M([Sf_{t}](x,|x-y|)\right)$ is continuous in $t$. Since
$K(M([Sf_{t}](x,|x-y|))$ is Lipschitz in $y$, also

$\int
K\left(M([Sf_{t}](x,|x-y|)\right)(v-w)\left(f_{t}(y,w)-f_{s}(y,w)\right)\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}w$

vanishes when ${\mathscr{W}}(f_{t},f_{s})\to 0$. ∎

Now we can prove the main theorem of this section.

######  Theorem 4.1.

Let $f_{0}(x,v)\in L^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d})$ be a
probability density such that $\text{supp}(f_{0})\subset B_{R_{x}}\times
B_{R_{v}}$. Given $T>0$, there exists a unique weak solution $f\in
C_{w}\left([0,T];L^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d})\right)$ of the
topological Cucker-Smale equation. Moreover

$\text{supp}(f_{t})\subset B_{R_{x}+tR_{v}}\times B_{R_{v}}.$ (4.2)

###### Proof.

We first note that, if the solution exists, (4.2) follows from an argument
similar to the one used in the discrete case (see Theorem 3.1).

We now prove the existence. As in Lemma 4.2, consider a family of probability
densities $\\{g_{t}\\}_{t\geq 0}\in C_{w}\left([0,T];\mathcal{B}_{M}\right),$
with $M\coloneq\|f_{0}\|_{\infty}\mathrm{e}^{d\gamma T}$ and such that (4.1)
holds with $R_{x}(t)=R_{x}+tR_{v}$ and $R_{v}(t)=R_{v}$. The push-forward of
$f_{0}$ along the flow generated by $g_{t}$, denoted by $\tilde{g}_{t}$, is
weakly continuous in $t$, uniformly in $g_{t}$, with $t\in[0,T]$. Moreover,
the determinant of the Jacobian of the flow $J(t)=J(t,x,v)$ verifies

${\dfrac{\mathrm{d}\phantom{t}}{\mathrm{d}{t}}}J(t)=-J(t)d\gamma.$

So the push-forward $\tilde{g}_{t}$ is bounded by
$\|f_{0}\|_{\infty}\mathrm{e}^{d\gamma t}$.

With a standard construction we can prove that, for $T$ sufficiently small,
the map $\\{g_{t}\\}\mapsto\\{\tilde{g}_{t}\\}$ is a contraction in
$C_{w}\left([0,T];\mathcal{B}_{M}\right)$, with the distance defined by the
supremum on time of the Wasserstein distance; in this way we prove local
existence and uniqueness. Using the a-priori estimate on the supremum and on
the support, we get the global result. ∎

##  5\. The mean-field limit

In this section we prove the main result regarding the mean-field limit for
the topological Cucker-Smale equation. In the sequel, $f_{t}$ is the fixed
global solution of eq. (1.7) as in Theorem 4.1, with initial datum $f_{0}$,
and $\mu^{N}_{t}$ is the global solution of eq. (1.8) in the sense of Theorem
3.1, with initial datum

$\mu_{0}^{N}=\frac{1}{N}\sum_{i=0}^{N}\delta_{x_{i}}\delta_{v_{i}}.$

We assume that $f_{0}$ and $\mu^{N}_{0}$ are supported in $B_{R_{x}}\times
B_{R_{v}}$. Fixed $T$, we indicate by $C(T)$ any constant that depends only on
$T$, $R_{x}$, $R_{v}$ and $\|f_{0}\|_{\infty}$.

In order to get the result, we compare the $N$-agent dynamics with the
“intermediate” dynamics given by

$\begin{cases}\dot{X}^{f}_{i}(t)=V^{f}_{i}(t)\\\
\dot{V}^{f}_{i}(t)=W[Sf_{t},\nu^{N}_{t}](X^{f}_{i},V_{i}^{f}),\end{cases}$
(5.1)

where

$\nu^{N}_{t}\coloneq\frac{1}{N}\sum_{k=1}^{N}\delta_{X^{f}_{k}(t)}\,\delta_{V^{f}_{k}(t)}$

is the empirical measure. The initial datum is $\nu^{N}_{0}=\mu^{N}_{0}$, i.e.

$\\{(X^{f}_{i}(0),V^{f}_{i}(0))\\}_{i=1}^{N}=\\{(x_{i},v_{i})\\}_{i=1}^{N}.$

######  Proposition 5.1.

Given $T>0$, it holds that

  1. i)

For $t\in[0,T]$,

${\mathscr{W}}(f_{t},\nu_{t}^{N})\leq C(T){\mathscr{W}}(f_{0},\mu^{N}_{0}).$
(5.2)

  2. ii)

For $t\in[0,T]$, the distance

$\delta(t)\coloneq\max_{i=1,\dots,N}\left(|X^{f}_{i}(t)-X^{N}_{i}(t)|+|V^{f}_{i}(t)-V^{N}_{i}(t)|\right)$

verifies

$\delta(t)\leq C(T)\sqrt{{\mathscr{W}}(f_{0},\mu^{N}_{0})}.$ (5.3)

###### Proof.

Since $f_{t}$ is bounded, $K\left(M[Sf_{t}](x,|x-y|)\right)$ is locally
Lipschitz in $x$ and $y$ (see i) and ii) of Lemma 4.1) and then
$W[Sf_{t},\nu](x,v)$ is weakly continuous in $\nu$ in the sense that

$\sup_{x,v}|W[S_{f},\nu_{1}](x,v)-W[S_{f},\nu_{2}](x,v)|\leq
C(T){\mathscr{W}}(\nu_{1},\nu_{2}).$

It is straightforward to prove that the solution $\nu_{t}$ of the system

$\left\\{\begin{aligned} &\dot{X}_{t}=V_{t}\\\
&\dot{V}_{t}=W[Sf_{t},\nu_{t}](X_{t},V_{t})\\\ &\nu_{t}=\text{ push-forward of
}\nu_{0}\text{ along the flow }(X_{t},V_{t})\end{aligned}\right.$

is continuous in ${\mathscr{W}}$ w.r.t. the initial datum $\nu_{0}$. Taking
$\nu_{0}=f_{0}$ and $\nu_{0}=\mu_{0}^{N}$ we get the proof of i).

In order to estimate $\delta(t)$, we need to evaluate, for $0\leq s\leq t$ and
for $i=1,\dots,N$, the difference $|\dot{V}^{f}_{i}(s)-\dot{V}^{N}_{i}(s)|$
given by

$|W[Sf_{s},\nu_{s}^{N}](X^{f}_{i},V^{f}_{i})-W[S\mu^{N}_{s},\mu_{s}^{N}](X^{N}_{i},V^{N}_{i})|.$

We estimate this quantity with the sum of three terms:

$\displaystyle(a)$
$\displaystyle|W[Sf_{s},\nu_{s}^{N}](X^{f}_{i},V^{f}_{i})-W[Sf_{s},\nu_{s}^{N}](X^{N}_{i},V^{N}_{i})|,$
$\displaystyle(b)$
$\displaystyle|W[Sf_{s},\nu_{s}^{N}](X^{N}_{i},V^{N}_{i})-W[Sf_{s},\mu_{s}^{N}](X^{N}_{i},V^{N}_{i})|,$
$\displaystyle(c)$
$\displaystyle|W[Sf_{s},\mu_{s}^{N}](X^{N}_{i},V^{N}_{i})-W[S\mu^{N}_{s},\mu_{s}^{N}](X^{N}_{i},V^{N}_{i})|.$

Since $K\left(M[Sf_{s}](x,|x-y|)\right)$ is Lipschitz in $x$, from the
definition of $W$ it is easy to prove that (a) is bounded by

$\left(c\,\text{Lip}(K)\|Sf_{s}\|_{\infty}R_{x}^{d-1}(s)R_{v}+c\|K\|_{\infty}\right)\delta(s)$

and that (b) is estimated by

$c\,\text{Lip}(K)\|Sf_{s}\|_{\infty}R_{x}^{d-1}(s)R_{v}\delta(s).$

Note that $\|Sf_{s}\|_{\infty}\leq cR_{v}^{d}\|f_{s}\|_{\infty}$. From
Proposition 2.1 we have that (c) is bounded by

$c\text{Lip}(K)R_{v}{\mathscr{D}}(Sf_{s},S\mu^{N}_{s}).$

Since

${\mathscr{D}}(Sf_{s},S\mu^{N}_{s})\leq{\mathscr{D}}(Sf_{s},S\nu^{N}_{s})+{\mathscr{D}}(S\nu^{N}_{s},S\mu^{N}_{s}),$

by Proposition 2.3 with $\rho=Sf_{s}$, $\mu^{N}=S\nu^{N}_{s}$ and
$\nu^{N}=S\mu^{N}_{s}$, we get

${\mathscr{D}}(S\nu^{N}_{s},S\mu^{N}_{s})\leq
c\delta(s)+c{\mathscr{D}}(Sf_{s},S\nu_{s}^{N}).$

Writing $\delta(t)$ in terms of the time integral of $\delta(s)$ and the
difference of the interaction terms and using the Gronwall lemma, we readily
get the estimate

$\delta(t)\leq
C(T)\int_{0}^{t}{\mathscr{D}}(Sf_{s},S\nu^{N}_{s})\mathop{}\\!\mathrm{d}s,$

valid for $0\leq t\leq T$. We conclude the proof by using Proposition 2.2, eq.
(5.2) and the fact that
${\mathscr{W}}(Sf_{s},S\nu^{N}_{s})\leq{\mathscr{W}}(f_{s},\nu^{N}_{s})$. ∎

######  Theorem 5.1.

Fixed $T>0$, let $f_{t}$ be a solution of eq. (1.7) as in Theorem 4.1 with
initial datum $f_{0}$ and let $\mu^{N}_{t}$ be a solution of eq. (1.8) in the
sense of Theorem 3.1 with initial datum $\mu_{0}^{N}$. Then, for $0\leq t\leq
T$,

${\mathscr{W}}(f_{t},\mu^{N}_{t})\leq
C(T)\text{max}\left\\{{\mathscr{W}}(f_{0},\mu^{N}_{0}),\sqrt{{\mathscr{W}}(f_{0},\mu^{N}_{0})}\right\\}.$

###### Proof.

By the triangular inequality,

${\mathscr{W}}(f_{t},\mu^{N}_{t})\leq{\mathscr{W}}(f_{t},\nu^{N}_{t})+{\mathscr{W}}(\nu^{N}_{t},\mu^{N}_{t}).$

From (5.2), using that ${\mathscr{W}}(\nu^{N}_{t},\mu^{N}_{t})\leq\delta(t)$
and (5.3), we get the thesis. ∎

## Appendix: Continuity of ${\mathscr{W}}$ w.r.t. ${\mathscr{D}}$

In this appendix we prove the continuity of the Wasserstein distance
${\mathscr{W}}$ w.r.t. the discrepancy distance ${\mathscr{D}}$ for compactly
supported measures.

Consider two probability measures $\mu$ and $\nu$, both with support in the
ball $B_{R}$ of $\mathbb{R}^{d}$. Fix $\varepsilon>0$ and consider a Lipschitz
test function $\phi$; it is sufficient to consider $\phi$ with support of
diameter less than $cR$, so that $\|\phi\|_{\infty}\leq cR$. Given such
$\phi$, take $\delta_{1}>0$ such that
$\text{Lip}(\phi)\delta_{1}<\varepsilon$.

By the Besicovitch covering principle (see [4]), there exist $N_{\varepsilon}$
disjoint closed balls $\\{B_{i}\\}_{i=1}^{N_{\varepsilon}}$ of radius at most
$\delta_{1}$ such that

$\mu\left(\bigcup_{i=1}^{N_{\varepsilon}}B_{i}\right)\geq 1-\varepsilon.$

We estimate

$\int\phi\mathop{}\\!\mathrm{d}(\mu-\nu)=\int_{\mathbb{R}^{d}\setminus\bigcup
B_{i}}\phi\mathop{}\\!\mathrm{d}(\mu-\nu)+\int_{\bigcup
B_{i}}\phi\mathop{}\\!\mathrm{d}(\mu-\nu)\equiv A+B.$

We have that

$\displaystyle A$
$\displaystyle\leq\|\phi\|_{\infty}\left(\mu\left(\mathbb{R}^{d}\setminus\cup_{i=1}^{N_{\varepsilon}}B_{i}\right)+\nu\left(\mathbb{R}^{d}\setminus\cup_{i=1}^{N_{\varepsilon}}B_{i}\right)\right)$
$\displaystyle\leq\|\phi\|_{\infty}\left(2\varepsilon+N_{\varepsilon}{\mathscr{D}}(\mu,\nu)\right),$

while

$\displaystyle B$
$\displaystyle\leq\sum_{i=1}^{N_{\varepsilon}}\int_{B_{i}}(\sup\phi-\inf\phi)\mathop{}\\!\mathrm{d}\nu+\sum_{i=1}^{N_{\varepsilon}}\int_{B_{i}}\sup\phi\mathop{}\\!\mathrm{d}(\mu-\nu)$
$\displaystyle\leq
2\text{Lip}(\phi)\delta_{1}+N_{\varepsilon}\|\phi\|_{\infty}{\mathscr{D}}(\mu,\nu).$

Hence we obtain

$\int\phi\mathop{}\\!\mathrm{d}(\mu-\nu)\leq
cR\varepsilon+cRN_{\varepsilon}{\mathscr{D}}(\mu,\nu).$

Taking ${\mathscr{D}}(\mu,\nu)<\delta_{2}$ such that
$N_{\varepsilon}\delta_{2}<\varepsilon$, we get the thesis.

## References

  * [1] A. Attanasi, A. Cavagna, L. Del Castello, I. Giardina, T.S. Grigera, A. Jelić, S. Melillo, L. Parisi, O. Pohl, E.Shen, M. Viale Information transfer and behavioural inertia in starling flocks Nat. Phys. 10 (2014) 691–696.
  * [2] M. Ballerini, N. Cabibbo, R. Andelier, A. Cavagna, A. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study Proc. Nat. Acad. Sci. USA 105 (2008) 1232–1237.
  * [3] D. Benedetto, P. Buttà, E. Caglioti Some aspects of the inertial spin model for flocks and related kinetic equations Math. Mod. Meth. Appl. Sci. 30 (10) (2020) 1987–2022.
  * [4] A. S. Besicovitch A general form of the covering principle and relative differentiation of additive functions I Proc. Cambridge Philos. Soc, 41 (1945), 103-110; II, Proc. Cambridge Philos. Soc, 42 (1946), 1-10, with corrections in Proc. Cambridge Philos. Soc, 43 (1947), 590.
  * [5] W. Braun, W.K. Hepp The Vlasov dynamics and its fluctuations in the $1/N$ limit of interacting classical particles Commun. Math. Phys 56 (1977) 101–113.
  * [6] J.A. Cañizo, J.A. Carrillo, J. Rosado A well-posedness theory in measures for some kinetic models of collective motion Math. Mod. Meth. Appl. Sci. 21 3 (2011) 515–539.
  * [7] J.A. Carrillo, M. Fornasier, G. Toscani, F. Vecil Particle, kinetic, and hydrodynamic models of swarming in: Naldi G., Pareschi L., Toscani G. (eds) Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston (2010)
  * [8] J.A. Carrillo, Y.-P.Choi, M. Hauray The derivation of swarming models: mean-field limit and Wasserstein distances in: Muntean A., Toschi F. (eds) Collective Dynamics from Bacteria to Crowds. CISM International Centre for Mechanical Sciences, vol. 553, Springer, Vienna (2014)
  * [9] A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. Santagati, F. Stefanini, R. Tavarone From empirical data to inter-individual interactions: Unveiling the rules of collective animal behavior Math. Mod. Meth. App. Sci. 20 (2010) 1491–1510.
  * [10] L.-P. Chaintron, A. Diez Propagation of chaos: a review of models, methods and applications arXiv:2106.14812v1 (2021)
  * [11] F. Cucker, S. Smale On the mathematics of emergence Japan. J. Math. 2 (2007) 197–227.
  * [12] F. Cucker, S. Smale Emergence behavior in flocks IEEE Trans. Automat. Control 52 (2007) 852–862.
  * [13] P. Degond, M. Pulvirenti Propagation of chaos for topological interaction Ann. Appl. Prob. 29 (2019) 2594–2612.
  * [14] R. Dobrushin Vlasov equations Funct. Anal. Appl. (1979) 13 115–123
  * [15] A.L. Gibbs, F.E. Su On choosing and bounding probability metrics International Statistical Review / Revue Internationale De Statistique, vol. 70, no. 3 (2002) 419–435.
  * [16] F. Golse On the Dynamics of Large Particle Systems in the Mean Field Limit in: Muntean A., Rademacher J., Zagaris A. (eds) Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity. Lecture Notes in Applied Mathematics and Mechanics, vol 3. Springer, (2016).
  * [17] F. Golse, S.-Y. Ha A mean-field limit of the Lohe matrix model and emergent dynamics Arch. Rational Mech. Anal. 234 (2019) 1445–1491.
  * [18] S.-Y. Ha, J.-G. Liu A simple proof of the Cucker-Smale flocking dynamics and mean-field limit Comm. Math. Sci. 7 (2009) 297–325.
  * [19] J. Haskovec Flocking dynamics and mean-field limit in the Cucker-Smale-type model with topological interactions Phys. D 261 (2013) 42–51.
  * [20] M. Hauray Mean field limit for the one dimensional Vlasov-Poisson equation Séminaire Laurent Schwartz — EDP et applications, Exposé no. 21 (2012-2013).
  * [21] M. Hauray, P.-E. Jabin N-particle approximation of the Vlasov equations with singular potential Arch. Ration. Mech. Anal. 183 (2007) 489–524.
  * [22] M. Hauray, P.-E. Jabin Particle approximation of Vlasov equations with singular forces Ann. Sci. Ecol. Norm. Sup. 48 (2015) 891–940.
  * [23] P.-E. Jabin A review of the mean field limits for Vlasov equations Kinet. Relat. Models 7 (2014) 661–711.
  * [24] L. Kuipers, H. Niederreiter Uniform Distribution of Sequences Wiley, New York (1974).
  * [25] D. Lazarovici The Vlasov–Poisson dynamics as the mean-field limit of rigid charges Commun. Math. Phys. 347 (2016) 271–289.
  * [26] D. Lazarovici, P. Pickl A mean-field limit for the Vlasov–Poisson system Arch. Ration. Mech. Anal. 225 (2017) 1201–1231.
  * [27] H. Neunzert An introduction to the nonlinear Boltzmann Vlasov equation in: Kinetic Theories and the Boltzmann Equation (C. Cercignani ed.), Lecture Notes in Math., vol. 1048, Springer, Berlin, Heidelberg, (1984) 60–110.
  * [28] A.-S. Sznitman Topics in propagation of chaos in: Ecole d’Eté de Probabilités de Saint-Flour XIX (P.-L. Hennequin ed.), Lecture Notes in Math., vol. 1464, Springer-Verlag, Berlin, (1991).
  * [29] M. Trocheris Continuite entre une solution de l’equation de Vlasov a une dimension et le mouvement d’un systeme de points (EUR-CEA-FC–1222) France (1984).
  * [30] M. Trocheris On the derivation of the one dimensional Vlasov equation Transport Theory and Statistical Physics 15, 5 (1986) 597–628.
  * [31] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, O. Shochet Novel type of phase transition in a system of self-driven particles Phys. Rev. Lett. 75 (1995) 1226.


# Technologies for Future Vertex and Tracking Detectors at CLIC

S. Spannagel<EMAIL_ADDRESS>on behalf of the CLICdp Collaboration
CERN, Geneva, Switzerland

###### Abstract

CLIC is a proposed linear
$\mathup{{{e}}^{\scriptstyle{+}}}\mathup{{{e}}^{\scriptstyle{-}}}$ collider
with center-of-mass energies of up to $3\text{\,}\mathrm{TeV}$. Its main
objectives are precise top quark and Higgs boson measurements, as well as
searches for Beyond Standard Model physics. To meet the physics goals, the
vertex and tracking detectors require not only a spatial resolution of a few
micrometers and a very low material budget, but also timing capabilities with
a precision of a few nanoseconds to allow suppression of beam-induced
backgrounds.

Different technologies using hybrid silicon detectors are explored for the
vertex detectors, such as dedicated readout ASICs, small-pitch active edge
sensors as well as capacitively coupled High-Voltage CMOS sensors. Monolithic
sensors are considered as an option for the tracking detector, and a prototype
using a CMOS process with a high-resistivity epitaxial layer is being
designed. Different designs using a silicon-on-insulator process are under
investigation for both vertex and tracking detector.

All prototypes are evaluated in laboratory and beam tests, and newly developed
simulation tools combining Geant4 and TCAD are used to assess and optimize
their performance. This contribution gives an overview of the R&D program for
the CLIC vertex and tracking detectors, highlighting new results from the
prototypes.

###### keywords:

CLIC , Silicon Detectors , HR-CMOS , HV-CMOS , Allpix Squared

††journal: Nucl. Instr. Meth. A

##  1 Introduction

The Compact Linear Collider (CLIC) [1, 2] is a proposed linear
$\mathup{{{e}}^{\scriptstyle{+}}}\mathup{{{e}}^{\scriptstyle{-}}}$ collider to
be built at CERN, Geneva, Switzerland. It uses a novel two-beam acceleration
scheme achieving field gradients of more than
$100\text{\,}\mathrm{MV}\text{\,}{\mathrm{m}}^{-1}$, which enables the design
of a compact accelerator. The construction is foreseen in three stages with
center-of-mass energies of $380\text{\,}\mathrm{GeV}$,
$1.5\text{\,}\mathrm{TeV}$ and $3\text{\,}\mathrm{TeV}$, each with a different
physics focus ranging from precision measurements of the Higgs boson and Top
quark to exotica and beyond the Standard Model physics.

The design of the vertex and tracking detectors is strongly influenced by the
beam structure at CLIC. Trains of 312 bunches with a bunch spacing of
$0.5\text{\,}\mathrm{ns}$ are repeated with a frequency of
$50\text{\,}\mathrm{Hz}$. To obtain luminosities of up to
$5.9\text{\times}{10}^{34}\text{\,}{\mathrm{cm}}^{-2}\text{\,}{\mathrm{s}}^{-1}$,
the bunches are focused to a size of a few nanometers at the interaction
point. The resulting high bunch density leads to interactions between
colliding bunches which create background particles. At the highest collision
energy, about 100 particles per bunch crossing are expected within the
acceptance of the tracking systems, leading to pixel occupancies of up to 3%
per bunch train in the innermost layers. Timing information is required to
reduce the effect of these pile-up hits on the reconstruction.

The current model for a detector at CLIC [3] features an all-silicon vertex
and tracking system with very stringent requirements on resolution and
material budget. To limit multiple scattering, the material budget is limited
to about 0.2% X0 per vertex detector layer and 1% X0 per layer of the tracking
detector. A low power consumption of below
$50\text{\,}\mathrm{mW}\text{\,}{\mathrm{cm}}^{-2}$ in the vertex detector is
required in order to enable passive air-flow cooling.

Single-point resolutions of about $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in
the vertex and $7\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in the tracking
detector are required for optimal flavor tagging and track reconstruction
performance. To allow efficient reduction of background, time stamping
capabilities with a precision of about $5\text{\,}\mathrm{ns}$ are required.

The CLIC vertex and tracking detectors thus need to achieve an order of
magnitude reduction both in material budget and cell size, while maintaining a
similar hit-time resolution, compared to the current LHC detectors.

##  2 Silicon Technologies

A variety of silicon technologies are currently explored in order to find the
best match with the requirements for a detector at CLIC. This section provides
a brief overview of the considered technologies and showcases the investigated
prototypes.

###  2.1 Hybrid Pixel Detectors

Hybrid detectors represent the traditional design of silicon pixel detectors
in high-energy physics and consist of two independent parts: the high-
resistivity sensor and the CMOS readout chip. The connection between the
individual channels is established by solder bumps. With this design,
extensive functionality can be placed in the pixels of the readout chip, while
the sensor can be fully depleted to maximize charge collection.

Pixel cell sizes of $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ –
$250\text{\,}\mathrm{\SIUnitSymbolMicro m}$ have been achieved, but the bump
bonding process is both the main cost driver and the limiting factor for the
pixel pitch and the device thickness.

The _CLICpix2_ prototype [4] is a readout ASIC designed to meet the
requirements for the CLIC vertex detector. It features a $128\times 128$ pixel
matrix with a total active area of $3.2\text{\times}3.2\text{\,}\mathrm{mm}$,
implemented in a $65\text{\,}\mathrm{nm}$ CMOS process with a pixel pitch of
$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ x
$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The pixel cells feature
simultaneous measurement of the charge and time-of-arrival. The data
acquisition is shutter-based, and the pixel matrix can be powered down between
shutters.

\begin{overpic}[width=216.81pt]{clicpix2_source.pdf}
\put(20.0,23.0){{\boldmath CLICdp}} \put(20.0,20.0){work in progress}
\end{overpic}

Figure 1: Sr-90 measurement of a CLICpix2 prototype bonded to a planar sensor
with $130\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thickness. A notch can be
observed in the top-right corner, where 55 bumps were lost when lifting the
chip from the tape. The isolated unresponsive pixels have been identified as
dead channels of the readout ASIC.

Promising results have been achieved recently for single-chip bump bonding
using a support wafer process with SnAg bumps, which is a challenge at the
pixel pitch of $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$, due to the
requirements on the diameter and uniformity of the solder balls and on the
alignment precision. Figure 1 shows a Sr-90 source measurement of a _CLICpix2_
prototype bonded to a planar silicon sensor, indicating a high bump yield of
more than $99.6\text{\,}\mathrm{\char 37\relax}$. The prototype has been
successfully tested in laboratory and beam tests, its detailed
characterization is still ongoing.

###  2.2 Monolithic High-Voltage CMOS Sensors

In monolithic High-Voltage (HV) CMOS sensors, electronics and sensor are
placed on the same wafer [5]. This facilitates fully integrated designs
comprising amplification as well as discrimination and readout, which feature
lower mass and do not require bump bonding. The readout electronics is
surrounded by a deep collection diode which also acts as shield from the
electric field in the sensor bulk. By this, higher bias voltages of about
$60\text{\,}\mathrm{V}$ can be applied to increase the depleted volume. The
large collection diode however entails a large input capacitance.

\begin{overpic}[width=216.81pt]{atlaspix_clustersize.png}
\put(20.0,13.0){{\color[rgb]{0.9,0.9,0.9}\definecolor[named]{pgfstrokecolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@stroke{0.9}\pgfsys@color<EMAIL_ADDRESS>CLICdp}}}
\put(20.0,10.0){{\color[rgb]{0.9,0.9,0.9}\definecolor[named]{pgfstrokecolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@stroke{0.9}\pgfsys@color<EMAIL_ADDRESS>in progress}} \end{overpic}

Figure 2: Mean cluster size as a function of the incidence position of the
particle for the _ATLASpix_Simple_ prototype. Here, tracks from the full pixel
matrix are plotted with respect to their relative position within a $2\times
2$ pixel area.

For the CLIC tracking detector, the _ATLASpix_Simple_ prototype [6] is being
investigated. This fully integrated chip designed for the ATLAS ITk upgrade
has been produced in an AMS $180\text{\,}\mathrm{nm}$ HV-CMOS process with
substrate resistivities of
$20-$1000\text{\,}\mathrm{\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$$. The chip
has 25 x 400 pixels with a pitch of $130\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ x $40\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and features in-pixel charge
amplifier and discriminators, while the charge and arrival time measurement is
performed in the periphery. Promising results have been achieved in first beam
tests. Figure 2 shows the mean cluster size as a function of the particle
position within a $2\times 2$ pixel area. The expected charge sharing is
clearly visible at the boundaries between pixel cells, where the mean cluster
size increases while staying close to one inside the individual pixels. A
position resolution of $13\text{\,}\mathrm{\SIUnitSymbolMicro m}$ has been
reached, the efficiency of $99.5\text{\,}\mathrm{\char 37\relax}$ has been
measured at a charge threshold of $1500\text{\,}\mathrm{e}$.

The prototype is currently being integrated into the CaRIBou DAQ system [7]
which allows to characterize the timing performance and provides the capacity
to record data at higher rates.

###  2.3 Monolithic High-Resistivity CMOS Sensors

Monolithic High-Resistivity (HR) CMOS sensors represent an alternative to HV-
CMOS sensors. Here, the electronics is placed outside the charge-collection
diode and is separately shielded. Depletion is achieved by using a high-
resistivity substrate and a lower bias voltage, and no dedicated high-voltage
design rules by the foundry have to be taken into account. The small
collection diode reduces the input capacitance and thus reduces the noise.
Recent process modifications allow full lateral depletion [8].

The _Investigator_ chip designed by the ALICE collaboration has been used to
evaluate this technology in view of the CLIC tracking detector. The chip is an
analog prototype, where the digitization of the signals is performed off-chip
in the data acquisition system. Two variants of a TowerJazz
$180\text{\,}\mathrm{nm}$ HR-CMOS process have been investigated and
resolutions of $4\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in space and
$5\text{\,}\mathrm{ns}$ in time have been measured for a pitch of
$28\text{\times}28\text{\,}\mathrm{\SIUnitSymbolMicro m}$ [9].

A prototype of a fully integrated sensor dedicated to the CLIC tracking
detector, _CLICTD_ , is currently being designed in this technology exploiting
the very low noise and minimum detection threshold. Given the excellent
achievable resolution, this technology is also of interest for the CLIC vertex
detector.

###  2.4 Monolithic Silicon-on-Insulator Sensors

Another option to isolate the sensitive CMOS electronics from the high-field
region in the sensor is the silicon-on-insulator (SOI) approach, where the
electronics is separated by an insulation oxide layer from the sensor on a
single high-resistivity wafer.

\begin{overpic}[width=216.81pt]{soi_resolution.png} \put(16.0,40.0){{\boldmath
CLICdp}} \put(16.0,37.0){work in progress} \end{overpic}

Figure 3: Position resolution of the SOI test chip as a function of the
applied back bias voltage. The three data sets correspond to the different
pixel circuit designs, two with a charge pre-amplifier (CPA) and one source
follower (SF) design.

To evaluate this technology, an SOI test chip has been manufactured in a LAPIS
$200\text{\,}\mathrm{nm}$ SOI process [10]. It combines different parameters
on a single chip, such as different pixel pitches as well as different
amplification and readout schemes. First test beam results for a sensor with a
thickness of $500\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and a pitch of
$30\text{\times}30\text{\,}\mathrm{\SIUnitSymbolMicro m}$ show an excellent
position resolution of about $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ as
indicated in Figure 3.

Analysis of prototype test beam data is ongoing, and a dedicated prototype
chip for the CLIC vertex detector, _CLIPS_ , has been designed and is
currently in production.

###  2.5 Capacitively Coupled Detectors

In order to overcome the limitations of bump bonding in traditional hybrid
detectors, capacitively coupled hybrids are being investigated, where an
active HV-CMOS sensor with analog signal amplification is glued to a CMOS
readout chip and the amplified signal coupled capacitively through the glue
layer.

This approach permits using the full feature set of standard CMOS processes
for the readout chip while keeping the required circuitry in the HV-CMOS chip
to a minimum. Investigations have shown that precise alignment and uniform
glue dispensation are key to a good assembly performance.

\begin{overpic}[width=216.81pt]{c3pd_glued2.png}
\put(75.0,30.0){\hbox{\pagecolor{black!10}CLICpix2}}
\put(75.0,15.0){\hbox{\pagecolor{black!10}C3PD}}
\put(38.0,15.0){{\color[rgb]{0.9,0.9,0.9}\definecolor[named]{pgfstrokecolor}{rgb}{0.9,0.9,0.9}\pgfsys@color@gray@stroke{0.9}\pgfsys@color@gray@fill{0.9}$25\text{\,}\mathrm{\SIUnitSymbolMicro
m}$}} \end{overpic}

Figure 4: Cross-section of a glue assembly with _CLICpix2_ and _C3PD_ with a
pixel pitch of $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

_CLICpix2_ prototypes have been glued to _C3PD_ sensors [11], a second-
generation active sensor designed in an AMS $180\text{\,}\mathrm{nm}$ HV-CMOS
process with a matching pixel pitch of
$25\text{\times}25\text{\,}\mathrm{\SIUnitSymbolMicro m}$. A cross-section of
such an assembly is shown in Figure 4. First test beam measurements show a
temporal and spatial resolution of approximately $7\text{\,}\mathrm{ns}$ and
$8\text{\,}\mathrm{\SIUnitSymbolMicro m}$, respectively. Extensive simulations
of the sensor using TCAD [12] and of the capacitive coupling via finite-
element analysis [13] have been conducted for comparison with the measured
performance.

Further assemblies with high-resistivity substrate sensors are under
investigation, and the gluing process is being optimized further.

###  2.6 Enhanced Lateral Drift Detectors

A novel sensor concept explored for the CLIC vertex detector are Enhanced
Lateral Drift (ELAD) sensors [14], which are an attempt to break the paradigm
of ever-decreasing pixel pitch. In thin sensors, the position resolution is
limited to $p/\sqrt{12}$, where $p$ is the pixel pitch, since almost no charge
sharing takes place.

By introducing additional deep implants in the sensor bulk, the electric field
is altered such that the lateral spread of charge carriers is increased during
their drift. The implants are designed in a way that the resulting charge
sharing is close to the theoretical optimum of a linear charge distribution.
Challenges to overcome for this type of sensor are the complex production
process and the exact placement of the implants to avoid low-field regions and
the resulting recombination of charge carriers.

This sensor design performs very well in finite-element device simulation, and
a first production batch with test structures as well as sensors with
$55\text{\,}\mathrm{\SIUnitSymbolMicro m}$ pitch is expected still in 2018.

##  3 Simulation Tools

Advanced simulation tools help in understanding the performance of new
prototypes and to optimize the design of new detectors. The Monte-Carlo
approach is widely used to account for the stochastic nature of the processes
under investigation, and a new software framework has been developed to
combine this with information obtained from detailed device modeling.

The Allpix Squared silicon detector simulation framework [15] combines the
detailed description of particle interactions provided by Geant4 [16, 17, 18]
with electric fields simulated using TCAD, and implements fast algorithms for
charge carrier propagation in silicon as well as front-end electronics
simulations. It provides access to the main detector characteristics such as
position resolution, charge collection efficiency, or tracking efficiency.

Figure 5: Comparison of the cluster size distribution from test beam
measurements with a $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick planar
silicon sensor and a simulation of the device using Allpix Squared. Figure
taken from [15].

The simulation framework has been validated using test beam data, and very
good agreement between data and simulation has been found. An example is shown
in Figure 5, comparing the cluster size distribution of a
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick planar silicon sensor as
simulated and measured in test beam data.

The software sees continuous development and a new feature version has
recently been released [19]. Among other improvements, it adds support of
magnetic fields and Lorentz drift, new particle source types and the
possibility to record particle tracks as Monte Carlo truth information.

Currently, a detailed transient simulation using weighting fields is under
development with the goal to better model the timing behavior of silicon
detectors.

##  4 Summary and Outlook

The beam conditions and physics goals at the proposed CLIC linear
$\mathup{{{e}}^{\scriptstyle{+}}}\mathup{{{e}}^{\scriptstyle{-}}}$ collider
pose challenges to the vertex and tracking detectors. An excellent spatial and
temporal resolution is required while maintaining a minimum material budget
and power consumption.

A comprehensive R&D program is underway to qualify silicon technologies for
the vertex and tracking detectors at CLIC. Many different concepts are being
investigated and most initial requirements have been shown to be achievable.
However, the desired position resolution of
$3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in the vertex detector remains a
challenge, and prototypes fulfilling the full set of requirements are yet to
be built.

A dedicated prototype using the HR-CMOS technology is currently being
designed. Productions of the SOI technology prototype as well as the first
ELAD silicon sensor are ongoing.

New and validated simulation tools combining precise device modeling and the
Monte-Carlo approach ease the R&D and prototyping processes.

##  5 Acknowledgements

This project has received funding from the European Union’s Horizon 2020
Research and Innovation programme under Grant Agreement no. 654168.

## References

## References

  * [1] L. Linssen, A. Miyamoto, M. Stanitzki, H. Weerts, Physics and Detectors at CLIC: CLIC Conceptual Design Report, Tech. Rep. CERN-2012-003, Geneva (2012).
URL https://cds.cern.ch/record/1425915

  * [2] M. J. Boland, et al., Updated baseline for a staged Compact Linear Collider, Tech. Rep. CERN-2016-004, CERN, Geneva (Aug 2016).
URL https://cds.cern.ch/record/2210892

  * [3] N. Alipour Tehrani, et al., CLICdet: The post-CDR CLIC detector model, Tech. rep., CERN (Mar 2017).
URL https://cds.cern.ch/record/2254048

  * [4] S. Spannagel, Silicon Technologies for the CLIC Vertex Detector, JINST 12 (06) (2017) C06006.  arXiv:1706.00224, doi:10.1088/1748-0221/12/06/C06006.
  * [5] I. Perić, A novel monolithic pixelated particle detector implemented in high-voltage cmos technology, Nucl. Instr. Meth. A 582 (3) (2007) 876 – 885.  doi:10.1016/j.nima.2007.07.115.
  * [6] I. Perić, et al., High-voltage pixel sensors for atlas upgrade, Nucl. Instr. Meth. A 765 (2014) 172 – 176.  doi:10.1016/j.nima.2014.06.035.
  * [7] A. Fiergolski, A multi-chip data acquisition system based on a heterogeneous system-on-chip platform, Tech. Rep. CLICdp-Conf-2017-012, CERN, Geneva (Jun 2017).
URL https://cds.cern.ch/record/2272077

  * [8] W. Snoeys, et al., A process modification for CMOS monolithic active pixel sensors for enhanced depletion, timing performance and radiation tolerance, Nucl. Instr. Meth. A 871 (2017) 90 – 96.  doi:https://doi.org/10.1016/j.nima.2017.07.046.
  * [9] R. M. Munker, et al., Study of the ALICE Investigator chip in view of the requirements at CLIC, Tech. rep., CERN (Sep 2017).
URL https://cds.cern.ch/record/2284145

  * [10] R. Bugiel, et al., Test-beam results of a SOI pixel-detector prototype, Nucl. Instr. Meth. A 901 (2018) 173 – 179.  doi:10.1016/j.nima.2018.06.017.
  * [11] I. Kremastiotis, et al., Design and standalone characterisation of a capacitively coupled HV-CMOS sensor chip for the CLIC vertex detector, JINST 12 (09) (2017) P09012.  arXiv:1706.04470, doi:10.1088/1748-0221/12/09/P09012.
  * [12] M. D. Buckland, Analysis and simulation of HV-CMOS assemblies for the CLIC vertex detector, Tech. Rep. CLICdp-Conf-2017-014, CERN, Geneva (Jun 2017).
URL https://cds.cern.ch/record/2272217

  * [13] M. Vicente Barreto Pinto, Finite-element simulations of coupling capacitances in capacitively coupled pixel detectors, Tech. rep., CERN (Jun 2017).
URL https://cds.cern.ch/record/2267848

  * [14] H. Jansen, Lateraler-Drift-Halbleitersensor, DE Patent App. DE201,510,116,270 (Mar. 30 2017).
  * [15] S. Spannagel, et al., Allpix2: A modular simulation framework for silicon detectors, Nucl. Instr. Meth. Phys. A 901 (2018) 164 – 172.  arXiv:1806.05813, doi:10.1016/j.nima.2018.06.020.
  * [16] S. Agostinelli, et al., Geant4–a simulation toolkit, Nucl. Instr. Meth. A 506 (3) (2003) 250 – 303.  doi:10.1016/S0168-9002(03)01368-8.
  * [17] J. Allison, et al., Geant4 developments and applications, IEEE Transactions on Nuclear Science 53 (1) (2006) 270–278.  doi:10.1109/TNS.2006.869826.
  * [18] J. Allison, et al., Recent developments in Geant4, Nucl. Instr. Meth. A 835 (Supplement C) (2016) 186 – 225.  doi:10.1016/j.nima.2016.06.125.
  * [19] Allpix Squared 1.2 released, accessed 6 2018.
URL https://cern.ch/allpix-squared/post/2018-06-13-allpix-
squared-1.2-released/