# Cross section measurement of $e^{+}e^{-}\to K_{S}^{0}K_{L}^{0}$ at
$\sqrt{s}=2.00-3.08$ GeV

M. Ablikim1, M. N. Achasov10,c, P. Adlarson64, S. Ahmed15, M. Albrecht4, A.
Amoroso63A,63C, Q. An60,48, Anita21, X. H. Bai54, Y. Bai47, O. Bakina29, R.
Baldini Ferroli23A, I. Balossino24A, Y. Ban38,k, K. Begzsuren26, J. V.
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Biernat64, J. Bloms57, A. Bortone63A,63C, I. Boyko29, R. A. Briere5, H. Cai65,
X. Cai1,48, A. Calcaterra23A, G. F. Cao1,52, N. Cao1,52, S. A. Cetin51B, J. F.
Chang1,48, W. L. Chang1,52, G. Chelkov29,b, D. Y. Chen6, G. Chen1, H. S.
Chen1,52, M. L. Chen1,48, S. J. Chen36, X. R. Chen25, Y. B. Chen1,48, Z. J
Chen20,l, W. S. Cheng63C, G. Cibinetto24A, F. Cossio63C, X. F. Cui37, H. L.
Dai1,48, J. P. Dai42,g, X. C. Dai1,52, A. Dbeyssi15, R. B. de Boer4, D.
Dedovich29, Z. Y. Deng1, A. Denig28, I. Denysenko29, M. Destefanis63A,63C, F.
De Mori63A,63C, Y. Ding34, C. Dong37, J. Dong1,48, L. Y. Dong1,52, M. Y.
Dong1,48,52, S. X. Du68, J. Fang1,48, S. S. Fang1,52, Y. Fang1, R.
Farinelli24A, L. Fava63B,63C, F. Feldbauer4, G. Felici23A, C. Q. Feng60,48, M.
Fritsch4, C. D. Fu1, Y. Fu1, X. L. Gao60,48, Y. Gao38,k, Y. Gao61, Y. G. Gao6,
I. Garzia24A,24B, E. M. Gersabeck55, A. Gilman56, K. Goetzen11, L. Gong37, W.
X. Gong1,48, W. Gradl28, M. Greco63A,63C, L. M. Gu36, M. H. Gu1,48, S. Gu2, Y.
T. Gu13, C. Y Guan1,52, A. Q. Guo22, L. B. Guo35, R. P. Guo40, Y. P. Guo9,h,
Y. P. Guo28, A. Guskov29, S. Han65, T. T. Han41, T. Z. Han9,h, X. Q. Hao16, F.
A. Harris53, K. L. He1,52, F. H. Heinsius4, T. Held4, Y. K. Heng1,48,52, M.
Himmelreich11,f, T. Holtmann4, Y. R. Hou52, Z. L. Hou1, H. M. Hu1,52, J. F.
Hu42,g, T. Hu1,48,52, Y. Hu1, G. S. Huang60,48, L. Q. Huang61, X. T. Huang41,
Y. P. Huang1, Z. Huang38,k, N. Huesken57, T. Hussain62, W. Ikegami
Andersson64, W. Imoehl22, M. Irshad60,48, S. Jaeger4, S. Janchiv26,j, Q. Ji1,
Q. P. Ji16, X. B. Ji1,52, X. L. Ji1,48, H. B. Jiang41, X. S. Jiang1,48,52, X.
Y. Jiang37, J. B. Jiao41, Z. Jiao18, S. Jin36, Y. Jin54, T. Johansson64, N.
Kalantar-Nayestanaki31, X. S. Kang34, R. Kappert31, M. Kavatsyuk31, B. C.
Ke43,1, I. K. Keshk4, A. Khoukaz57, P. Kiese28, R. Kiuchi1, R. Kliemt11, L.
Koch30, O. B. Kolcu51B,e, B. Kopf4, M. Kuemmel4, M. Kuessner4, A. Kupsc64, M.
G. Kurth1,52, W. Kühn30, J. J. Lane55, J. S. Lange30, P. Larin15, L.
Lavezzi63C, H. Leithoff28, M. Lellmann28, T. Lenz28, C. Li39, C. H. Li33,
Cheng Li60,48, D. M. Li68, F. Li1,48, G. Li1, H. B. Li1,52, H. J. Li9,h, J. L.
Li41, J. Q. Li4, Ke Li1, L. K. Li1, Lei Li3, P. L. Li60,48, P. R. Li32, S. Y.
Li50, W. D. Li1,52, W. G. Li1, X. H. Li60,48, X. L. Li41, Z. B. Li49, Z. Y.
Li49, H. Liang1,52, H. Liang60,48, Y. F. Liang45, Y. T. Liang25, L. Z.
Liao1,52, J. Libby21, C. X. Lin49, B. Liu42,g, B. J. Liu1, C. X. Liu1, D.
Liu60,48, D. Y. Liu42,g, F. H. Liu44, Fang Liu1, Feng Liu6, H. B. Liu13, H. M.
Liu1,52, Huanhuan Liu1, Huihui Liu17, J. B. Liu60,48, J. Y. Liu1,52, K. Liu1,
K. Y. Liu34, Ke Liu6, L. Liu60,48, Q. Liu52, S. B. Liu60,48, Shuai Liu46, T.
Liu1,52, X. Liu32, Y. B. Liu37, Z. A. Liu1,48,52, Z. Q. Liu41, Y. F. Long38,k,
X. C. Lou1,48,52, F. X. Lu16, H. J. Lu18, J. D. Lu1,52, J. G. Lu1,48, X. L.
Lu1, Y. Lu1, Y. P. Lu1,48, C. L. Luo35, M. X. Luo67, P. W. Luo49, T. Luo9,h,
X. L. Luo1,48, S. Lusso63C, X. R. Lyu52, F. C. Ma34, H. L. Ma1, L. L. Ma41, M.
M. Ma1,52, Q. M. Ma1, R. Q. Ma1,52, R. T. Ma52, X. N. Ma37, X. X. Ma1,52, X.
Y. Ma1,48, Y. M. Ma41, F. E. Maas15, M. Maggiora63A,63C, S. Maldaner28, S.
Malde58, Q. A. Malik62, A. Mangoni23B, Y. J. Mao38,k, Z. P. Mao1, S.
Marcello63A,63C, Z. X. Meng54, J. G. Messchendorp31, G. Mezzadri24A, T. J.
Min36, R. E. Mitchell22, X. H. Mo1,48,52, Y. J. Mo6, N. Yu. Muchnoi10,c, H.
Muramatsu56, S. Nakhoul11,f, Y. Nefedov29, F. Nerling11,f, I. B. Nikolaev10,c,
Z. Ning1,48, S. Nisar8,i, S. L. Olsen52, Q. Ouyang1,48,52, S. Pacetti23B,23C,
X. Pan46, Y. Pan55, A. Pathak1, P. Patteri23A, M. Pelizaeus4, H. P. Peng60,48,
K. Peters11,f, J. Pettersson64, J. L. Ping35, R. G. Ping1,52, A. Pitka4, R.
Poling56, V. Prasad60,48, H. Qi60,48, H. R. Qi50, M. Qi36, T. Y. Qi2, S.
Qian1,48, W.-B. Qian52, Z. Qian49, C. F. Qiao52, L. Q. Qin12, X. S. Qin4, Z.
H. Qin1,48, J. F. Qiu1, S. Q. Qu37, K. H. Rashid62, K. Ravindran21, C. F.
Redmer28, A. Rivetti63C, V. Rodin31, M. Rolo63C, G. Rong1,52, Ch. Rosner15, M.
Rump57, A. Sarantsev29,d, Y. Schelhaas28, C. Schnier4, K. Schoenning64, D. C.
Shan46, W. Shan19, X. Y. Shan60,48, M. Shao60,48, C. P. Shen2,9, P. X. Shen37,
X. Y. Shen1,52, H. C. Shi60,48, R. S. Shi1,52, X. Shi1,48, X. D Shi60,48, J.
J. Song41, Q. Q. Song60,48, W. M. Song27, Y. X. Song38,k, S. Sosio63A,63C, S.
Spataro63A,63C, F. F. Sui41, G. X. Sun1, J. F. Sun16, L. Sun65, S. S. Sun1,52,
T. Sun1,52, W. Y. Sun35, X Sun20,l, Y. J. Sun60,48, Y. K. Sun60,48, Y. Z.
Sun1, Z. T. Sun1, Y. H. Tan65, Y. X. Tan60,48, C. J. Tang45, G. Y. Tang1, J.
Tang49, V. Thoren64, B. Tsednee26, I. Uman51D, B. Wang1, B. L. Wang52, C. W.
Wang36, D. Y. Wang38,k, H. P. Wang1,52, K. Wang1,48, L. L. Wang1, M. Wang41,
M. Z. Wang38,k, Meng Wang1,52, W. H. Wang65, W. P. Wang60,48, X. Wang38,k, X.
F. Wang32, X. L. Wang9,h, Y. Wang60,48, Y. Wang49, Y. D. Wang15, Y. F.
Wang1,48,52, Y. Q. Wang1, Z. Wang1,48, Z. Y. Wang1, Ziyi Wang52, Zongyuan
Wang1,52, D. H. Wei12, P. Weidenkaff28, F. Weidner57, S. P. Wen1, D. J.
White55, U. Wiedner4, G. Wilkinson58, M. Wolke64, L. Wollenberg4, J. F.
Wu1,52, L. H. Wu1, L. J. Wu1,52, X. Wu9,h, Z. Wu1,48, L. Xia60,48, H. Xiao9,h,
S. Y. Xiao1, Y. J. Xiao1,52, Z. J. Xiao35, X. H. Xie38,k, Y. G. Xie1,48, Y. H.
Xie6, T. Y. Xing1,52, X. A. Xiong1,52, G. F. Xu1, J. J. Xu36, Q. J. Xu14, W.
Xu1,52, X. P. Xu46, L. Yan63A,63C, L. Yan9,h, W. B. Yan60,48, W. C. Yan68, Xu
Yan46, H. J. Yang42,g, H. X. Yang1, L. Yang65, R. X. Yang60,48, S. L.
Yang1,52, Y. H. Yang36, Y. X. Yang12, Yifan Yang1,52, Zhi Yang25, M. Ye1,48,
M. H. Ye7, J. H. Yin1, Z. Y. You49, B. X. Yu1,48,52, C. X. Yu37, G. Yu1,52, J.
S. Yu20,l, T. Yu61, C. Z. Yuan1,52, W. Yuan63A,63C, X. Q. Yuan38,k, Y. Yuan1,
Z. Y. Yuan49, C. X. Yue33, A. Yuncu51B,a, A. A. Zafar62, Y. Zeng20,l, B. X.
Zhang1, Guangyi Zhang16, H. H. Zhang49, H. Y. Zhang1,48, J. L. Zhang66, J. Q.
Zhang4, J. W. Zhang1,48,52, J. Y. Zhang1, J. Z. Zhang1,52, Jianyu Zhang1,52,
Jiawei Zhang1,52, L. Zhang1, Lei Zhang36, S. Zhang49, S. F. Zhang36, T. J.
Zhang42,g, X. Y. Zhang41, Y. Zhang58, Y. H. Zhang1,48, Y. T. Zhang60,48, Yan
Zhang60,48, Yao Zhang1, Yi Zhang9,h, Z. H. Zhang6, Z. Y. Zhang65, G. Zhao1, J.
Zhao33, J. Y. Zhao1,52, J. Z. Zhao1,48, Lei Zhao60,48, Ling Zhao1, M. G.
Zhao37, Q. Zhao1, S. J. Zhao68, Y. B. Zhao1,48, Y. X. Zhao25, Z. G. Zhao60,48,
A. Zhemchugov29,b, B. Zheng61, J. P. Zheng1,48, Y. Zheng38,k, Y. H. Zheng52,
B. Zhong35, C. Zhong61, L. P. Zhou1,52, Q. Zhou1,52, X. Zhou65, X. K. Zhou52,
X. R. Zhou60,48, A. N. Zhu1,52, J. Zhu37, K. Zhu1, K. J. Zhu1,48,52, S. H.
Zhu59, W. J. Zhu37, X. L. Zhu50, Y. C. Zhu60,48, Z. A. Zhu1,52, B. S. Zou1, J.
H. Zou1
(BESIII Collaboration)
1 Institute of High Energy Physics, Beijing 100049, People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s
Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
8 COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road,
54000 Lahore, Pakistan
9 Fudan University, Shanghai 200443, People’s Republic of China
10 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
11 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
12 Guangxi Normal University, Guilin 541004, People’s Republic of China
13 Guangxi University, Nanning 530004, People’s Republic of China
14 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
15 Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz,
Germany
16 Henan Normal University, Xinxiang 453007, People’s Republic of China
17 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
18 Huangshan College, Huangshan 245000, People’s Republic of China
19 Hunan Normal University, Changsha 410081, People’s Republic of China
20 Hunan University, Changsha 410082, People’s Republic of China
21 Indian Institute of Technology Madras, Chennai 600036, India
22 Indiana University, Bloomington, Indiana 47405, USA
23 (A)INFN Laboratori Nazionali di Frascati, I-00044, Frascati, Italy; (B)INFN
Sezione di Perugia, I-06100, Perugia, Italy; (C)University of Perugia,
I-06100, Perugia, Italy
24 (A)INFN Sezione di Ferrara, I-44122, Ferrara, Italy; (B)University of
Ferrara, I-44122, Ferrara, Italy
25 Institute of Modern Physics, Lanzhou 730000, People’s Republic of China
26 Institute of Physics and Technology, Peace Ave. 54B, Ulaanbaatar 13330,
Mongolia
27 Jilin University, Changchun 130012, People’s Republic of China
28 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
29 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
30 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
31 KVI-CART, University of Groningen, NL-9747 AA Groningen, The Netherlands
32 Lanzhou University, Lanzhou 730000, People’s Republic of China
33 Liaoning Normal University, Dalian 116029, People’s Republic of China
34 Liaoning University, Shenyang 110036, People’s Republic of China
35 Nanjing Normal University, Nanjing 210023, People’s Republic of China
36 Nanjing University, Nanjing 210093, People’s Republic of China
37 Nankai University, Tianjin 300071, People’s Republic of China
38 Peking University, Beijing 100871, People’s Republic of China
39 Qufu Normal University, Qufu 273165, People’s Republic of China
40 Shandong Normal University, Jinan 250014, People’s Republic of China
41 Shandong University, Jinan 250100, People’s Republic of China
42 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
43 Shanxi Normal University, Linfen 041004, People’s Republic of China
44 Shanxi University, Taiyuan 030006, People’s Republic of China
45 Sichuan University, Chengdu 610064, People’s Republic of China
46 Soochow University, Suzhou 215006, People’s Republic of China
47 Southeast University, Nanjing 211100, People’s Republic of China
48 State Key Laboratory of Particle Detection and Electronics, Beijing 100049,
Hefei 230026, People’s Republic of China
49 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
50 Tsinghua University, Beijing 100084, People’s Republic of China
51 (A)Ankara University, 06100 Tandogan, Ankara, Turkey; (B)Istanbul Bilgi
University, 34060 Eyup, Istanbul, Turkey; (C)Uludag University, 16059 Bursa,
Turkey; (D)Near East University, Nicosia, North Cyprus, Mersin 10, Turkey
52 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
53 University of Hawaii, Honolulu, Hawaii 96822, USA
54 University of Jinan, Jinan 250022, People’s Republic of China
55 University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
56 University of Minnesota, Minneapolis, Minnesota 55455, USA
57 University of Muenster, Wilhelm-Klemm-Str. 9, 48149 Muenster, Germany
58 University of Oxford, Keble Rd, Oxford, UK OX13RH
59 University of Science and Technology Liaoning, Anshan 114051, People’s
Republic of China
60 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
61 University of South China, Hengyang 421001, People’s Republic of China
62 University of the Punjab, Lahore-54590, Pakistan
63 (A)University of Turin, I-10125, Turin, Italy; (B)University of Eastern
Piedmont, I-15121, Alessandria, Italy; (C)INFN, I-10125, Turin, Italy
64 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
65 Wuhan University, Wuhan 430072, People’s Republic of China
66 Xinyang Normal University, Xinyang 464000, People’s Republic of China
67 Zhejiang University, Hangzhou 310027, People’s Republic of China
68 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at Bogazici University, 34342 Istanbul, Turkey
b Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia
c Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
d Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
e Also at Istanbul Arel University, 34295 Istanbul, Turkey
f Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
g Also at Key Laboratory for Particle Physics, Astrophysics and Cosmology,
Ministry of Education; Shanghai Key Laboratory for Particle Physics and
Cosmology; Institute of Nuclear and Particle Physics, Shanghai 200240,
People’s Republic of China
h Also at Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443, People’s
Republic of China
i Also at Harvard University, Department of Physics, Cambridge, MA, 02138, USA
j Currently at: Institute of Physics and Technology, Peace Ave.54B,
Ulaanbaatar 13330, Mongolia
k Also at State Key Laboratory of Nuclear Physics and Technology, Peking
University, Beijing 100871, People’s Republic of China
l School of Physics and Electronics, Hunan University, Changsha 410082, China

###### Abstract

The cross sections of the process $e^{+}e^{-}\to K_{S}^{0}K_{L}^{0}$ are
measured at fifteen center-of-mass energies $\sqrt{s}$ from $2.00$ to $3.08$
GeV with the BESIII detector at the Beijing Electron Positron Collider
(BEPCII). The results are found to be consistent with those obtained by BaBar.
A resonant structure around $2.2$ GeV is observed, with a mass and width of
$2273.7\pm 5.7\pm 19.3~{}{\rm MeV}/c^{2}$ and $86\pm 44\pm 51~{}{\rm MeV}$,
respectively, where the first uncertainties are statistical and the second
ones are systematic. The product of its radiative width
($\Gamma_{e^{+}e^{-}}$) with its branching fraction to $K_{S}^{0}K_{L}^{0}$
($Br_{K_{S}^{0}K_{L}^{0}}$) is $0.9\pm 0.6\pm 0.7$ eV.

###### pacs:

13.60.Le, 13.66.Jn

##  I Introduction

Among the light unflavored mesons, the strangeonium-like state $\phi(2170)$ is
particularly interesting. It was first reported in
$e^{+}e^{-}\to\gamma_{\mathrm{I}SR}\phi f_{0}(980)$ by the BaBar collaboration
Aubert:2006bu , and then confirmed in $J/\psi\to\eta\phi f_{0}(980)$ by the
BESII collaboration Ablikim:2007ab  and in the $e^{+}e^{-}\to\phi f_{0}(980)$
and $\phi\pi^{+}\pi^{-}$ processes by the Belle collaboration Shen:2009zze .
Subsequently, the $\phi(2170)$ has been studied extensively by BaBar
Lees:2011zi ; Aubert:2007ym ; Aubert:2007ur ; Aubert:2006bu , Belle
Shen:2009zze , BESII Ablikim:2007ab , and BESIII Ablikim:2014pfc ;
Ablikim:2019tpp ; Ablikim:2020pgw ; Ablikim:2020coo ; Ablikim:2017auj ;
Ablikim:2020das ; Ablikim:2018iyx ; Ablikim:2020wyk .

Initially, the strangeonium-like state $\phi(2170)$ was only observed in
hidden-strange decays, which makes its nature mysterious. Different
interpretations have been proposed. In Refs. Wang:2006ri ; Agaev:2019coa ;
Ke:2018evd ; Dong:2020okt ; Liu:2020lpw ; Deng:2010zzd ; Drenska:2008gr , the
$\phi(2170)$ is considered to be a tetraquark, while in Refs. Ding:2006ya ;
Ding:2007pc , it is considered as an $s\bar{s}g$ hybrid state. Lattice QCD
Dudek:2011bn  and QCD sum rule Ho:2019org  investigations disfavor the
$s\bar{s}g$ hybrid interpretation. Considering the near threshold location of
the $\phi(2170)$, various hadronic molecular possibilities have been proposed,
such as $\Lambda\bar{\Lambda}$ baryonium Klempt:2007cp ; Zhao:2013ffn ;
Yang:2019mzq , a $\phi K\bar{K}$ MartinezTorres:2008gy  or a $\phi f_{0}(980)$
AlvarezRuso:2009xn  resonance. Besides these exotic interpretations, the
$\phi(2170)$ has been considered to be conventional strangeonium,
corresponding to $3^{3}S_{1}$ Barnes:2002mu ; Pang:2019ttv  or $2^{3}D_{1}$
Ding:2007pc ; Ding:2006ya ; Wang:2012wa ; Pang:2019ttv ; Li:2020xzs  states.
The predicted decay rates of $\phi(2170)\to K\bar{K}$ differ among these
theoretical interpretations. For example, the branching fraction is predicted
to be $5\%-10\%$ under the assumption of a $2^{3}D_{1}$ state Wang:2012wa ;
Ding:2007pc  but close to zero in the case of an $s\bar{s}g$ or
$3^{3}Ss\bar{s}$ Ding:2006ya  state. Therefore, an experimental measurement of
the branching fraction of $\phi(2170)\to K\bar{K}$ provides crucial
information to distinguish between the different interpretations.

Recently, the cross sections for $e^{+}e^{-}\to K^{+}K^{-}$ were measured by
the BESIII and BaBar collaborations Ablikim:2018iyx ; BABAR:2019oes . A
structure near 2.2 GeV was reported with a mass (width) differing from the
world averaged parameters of the $\phi(2170)$ by $3\sigma$ ($2\sigma$). The
structure is not supported by Babar based on the measurements of the process
$e^{+}e^{-}\to{K_{S}^{0}K_{L}^{0}}$ BABAR:2019oes , though the uncertainties
are very large which are more than $100\%$ in most of the energy intervals. On
the other hand, a theoretically guided fit to the BESIII cross sections for
$e^{+}e^{-}\to K^{+}K^{-}$ provided consistent results with respect to the
$\phi(2170)$ parameters Chen:2020xho . The structure observed in the cross
section measurements can also be explained as an $\omega$-like state
Wang:2019jch . In general, considering the interferences between resonance and
non-resonance contributions, additional information from other processes, such
as $e^{+}e^{-}\to K_{S}^{0}K_{L}^{0}$, is needed. Although, this process has
been investigated in the past by the DM1 Mane:1980ep , OLYa Ivanov:1982cr ,
CDM2 Akhmetshin:1999ym ; Akhmetshin:2002vj ; Akhmetshin:2003zn , SND
Achasov:2000am ; Achasov:2006bv  and BaBar BABAR:2019oes ; Lees:2014xsh
collaborations, these measurements mainly focused on the energy region below
$2.0~{}$GeV.

In this work, we present Born cross section measurements of the process
$e^{+}e^{-}\to K_{S}^{0}K_{L}^{0}$. The results obtained in the overlapping
center-of-mass region from $2.00-2.54$ GeV are compared to previous
measurements by BaBar BABAR:2019oes . Moreover, we present, for the first
time, Born cross section measurements taken in the interval from $2.54$ to
$3.08$ GeV. A fit is applied to the cross section measurements of the
$e^{+}e^{-}\to K_{S}^{0}K_{L}^{0}$ process, and the resonant structure result
is compared with that found by BESIII Ablikim:2018iyx  and BaBar BABAR:2019oes
in $e^{+}e^{-}\to K^{+}K^{-}$.

##  II Detector and Monte Carlo Simulation

The BESIII detector is a magnetic spectrometer Ablikim:2009aa  located at
BEPCII Yu:2016cof . The cylindrical core of the BESIII detector consists of a
helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-
flight system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC), which
are all enclosed in a superconducting solenoidal magnet providing a 1.0 T
magnetic field. The solenoid is supported by an octagonal flux-return yoke
with resistive plate counter muon identifier modules interleaved with steel.
The acceptance of charged particles and photons is 93% over $4\pi$ solid
angle. The charged-particle momentum resolution at $1~{}{\rm GeV}/c$ is
$0.5\%$, and the $dE/dx$ resolution is $6\%$ for the electrons from Bhabha
scattering. The EMC measures photon energies with a resolution of $2.5\%$
($5\%$) at $1$ GeV in the barrel (end cap) region. The time resolution of the
TOF barrel part is 68 ps, while that of the end cap part is 110 ps.

The data samples used in this work are collected by the BESIII detector at
fifteen center-of-mass (c.m.) energies between $2.00$ and $3.08$ GeV with an
integrated luminosity of $583~{}{\rm pb}^{-1}$ Ablikim:2017wlt ;
Ablikim:2017duf .

Monte Carlo (MC) samples simulated with a model of the complete detector are
used to determine detection efficiency, optimize event selection criteria, and
estimate backgrounds. Detector geometry, material description, propagation and
interactions with the detector of the final-state particles are handled by
GEANT4-based Agostinelli:2002hh  simulation software, BESIII Object Oriented
Simulation Tool BOOSTBESIII .

Signal and background samples are generated at each c.m. energy ($\sqrt{s}$).
Signal MC samples of $e^{+}e^{-}\to{K_{S}^{0}K_{L}^{0}}$ and
$K_{S}^{0}\to\pi^{+}\pi^{-}$ are generated with ConExc Ping:2013jka . Non-
hadronic backgrounds including continuum processes of $e^{+}e^{-}\to
e^{+}e^{-}$, $e^{+}e^{-}\to\gamma\gamma$ and $e^{+}e^{-}\to\mu^{+}\mu^{-}$ are
generated with Babayaga Balossini:2006wc . Inclusive hadronic samples
($e^{+}e^{-}\to q\bar{q}$) are generated with Luarlw Andersson:1999ui . Two-
photon samples are generated with BesTwoGam Nova:1990prh .

##  III Event selection and background analysis

The momentum of the $K_{S}^{0}$ meson is reconstructed from its
$K_{S}^{0}\to\pi^{+}\pi^{-}$ decay. Events containing the reconstructed
$K_{S}^{0}$ candidates are retained for further analysis. The $K_{L}^{0}$
meson is not detected directly; because of the two-body decay, its presence is
inferred by a requirement on the $K_{S}^{0}$ candidate momentum. To select
signal candidates, the following criteria are applied:

  * •

Exactly two oppositely-charged tracks are required without any requirement on
neutral tracks. The distance of closest approach of the track with respect to
the interaction point is required to be less than $20$ cm along the beam
direction ($z-$axis of the BESIII coordinate system), while no requirement is
made with respect to the transverse direction. Tracks are required to be
within the acceptance of the MDC, $i.e.$ $|\cos\theta|<0.93$, where $\theta$
is the polar angle between the track and the $z-$axis. A vertex fit is applied
to constrain the two tracks to a common vertex, and subsequently a secondary
vertex fit is performed to determine the flight distance $L$ and corresponding
uncertainty $\delta L$, where $L$ corresponds to the separation between the
secondary vertex and the interaction point. We require $L/\delta L$ to be
larger than $2$, as illustrated by the green vertical line in Figure 1. The
invariant mass of the two tracks ($m_{\pi^{+}\pi^{-}}$), where the tracks are
treated as $\pi^{+}$ and $\pi^{-}$ candidates, is required to satisfy
$|m_{\pi^{+}\pi^{-}}-m_{K_{S}^{0}}|<35$ MeV$/c^{2}$, where $m_{K_{S}^{0}}$ is
the mass of $K_{S}^{0}$ taken from the Particle Data Group (PDG)
Tanabashi:2018oca . The signal yields are determined from fits to the
invariant-mass distributions, as discussed in Section IV.

Figure 1: $L/{\delta L}$ distribution for data taken at $\sqrt{s}=2.125$ GeV.
Dots refer to data and the shaded area corresponds to simulated signal events
normalized to the integrated luminosity of the data. The (green) vertical line
indicates the requirement that is applied to select signal candidate events.

  * •

To reject backgrounds from the $e^{+}e^{-}\to e^{+}e^{-}$ and
$e^{+}e^{-}\to\gamma\gamma$ processes, we require the ratio $E/cp$ between the
deposited energy in the EMC ($E$) and the momentum measured by the MDC ($p$)
to be less than $0.8$.

  * •

$|p_{\pi^{+}\pi^{-}}-p_{K_{S}^{0}}|<\sigma_{p}$ must be satisfied to suppress
backgrounds from three (or more) body decays, where $p_{\pi^{+}\pi^{-}}$ is
the momentum reconstructed from the $\pi^{+}\pi^{-}$ system,
$p_{K_{S}^{0}}=\sqrt{\frac{s}{4}-({m_{K_{S}^{0}}})^{2}}$ is the expected
$K_{S}^{0}$ momentum, and $\sigma_{p}=15$ MeV$/c$ is the momentum resolution
of the reconstructed $K_{S}^{0}$ determined using the signal MC. The
$p_{\pi^{+}\pi^{-}}$ distribution is shown in Figure 2.

Figure 2: $p_{\pi^{+}\pi^{-}}$ momentum distribution taken at $\sqrt{s}=2.125$
GeV. Dots refer to data and the shaded area depicts simulated signal events
normalized to the integrated luminosity of the data. The wide peak on the left
side in the simulated signal distribution stems from events that undergo
initial-state radiation. The vertical lines indicate the window of the signal
region.

MC studies indicate that the non-hadronic background and two-photon process
contribute less than $5\%$ in the region
$|m_{\pi^{+}\pi^{-}}-m_{K_{S}^{0}}|<3\sigma_{K_{S}^{0}}$ at low c.m. energies
($<2.396~{}{\rm GeV}$), where $\sigma_{K_{S}^{0}}=4$ MeV$/c^{2}$ is the mass
resolution of the pion pair determined by fitting the signal MC shape, and it
dominates at $3.080$ GeV with a contribution of less than $20\%$. No peaking
backgrounds were found from non-hadronic processes after applying the
previously described criteria at all c.m. energies. Detailed event type
analysis with a generic tool, TopoAna Zhou:2020ksj , shows that the dominant
hadronic background channels are $e^{+}e^{-}\to K_{S}^{0}K_{L}^{0}\pi^{0}$,
$e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$,
$e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}$ and
$e^{+}e^{-}\to(\gamma)\pi^{+}\pi^{-}$. A study using exclusive hadronic MC
samples showed that only at $3.080$ GeV a peaking background can be expected
from the process $e^{+}e^{-}\to K_{S}^{0}K_{L}^{0}\pi^{0}$. This will be
further discussed in Section V.

##  IV Cross section

Born cross sections ($\sigma_{B}$) are obtained at each energy point by:

$\displaystyle\sigma_{B}=\frac{N_{sig}}{\epsilon(1+\delta)\mathcal{L}}\,,$ (1)

where $N_{sig}$ is the signal yield, $\epsilon$ is the detection efficiency,
$1+\delta$ is the correction factor including vacuum polarization (VP) and
initial-state radiation (ISR) effects, and $\mathcal{L}$ is the integrated
luminosity measured using large-angle Bhabha scattering events with the method
elucidated in Ref. Ablikim:2017wlt . The branching ratio of
$K_{S}^{0}\to\pi^{+}\pi^{-}$ has been incorporated into $\epsilon$.

The signal yields are determined with an unbinned maximum-likelilood fit to
the invariant-mass distribution of $\pi^{+}\pi^{-}$ pairs of the selected
events obtained for each c.m. energy point, where the signal shape is
described by a Gaussian function and the background is represented with a
zero-order Chebychev polynomial. The fit range is taken with a window of more
than $8\sigma_{K_{S}^{0}}$ around the signal $K_{S}^{0}$. The mass and the
width of the Gaussian function is fixed to $m_{K_{S}^{0}}$ and
$\sigma_{K_{S}^{0}}$ for all the c.m. energy points, except for the two
energies with the highest statistics ($2.000$ and $2.125$ GeV). The signal and
background yields are set free for all c.m. energies. As an example, Figure 3
illustrates the $m_{\pi^{+}\pi^{-}}$ distribution together with the
corresponding fit result for data taken at $\sqrt{s}=2.125$ GeV.

Figure 3: $m_{\pi^{+}\pi^{-}}$ distribution of data taken at $\sqrt{s}=2.125$
GeV. The solid curve denotes the best fit through the data of the complete
model, whereby the dash-dotted and dashed lines are the corresponding signal
and background components, respectively.

Both $\epsilon$ and $1+\delta$ depend on the line shape of the cross sections
and are determined via an iterative procedure. In the first iteration, the
cross sections from $2.00~{}{\rm GeV}$ to $3.08~{}{\rm GeV}$ are obtained and
taken as initial inputs. The cross sections below $2.00~{}{\rm GeV}$ are
provided by previous experiments Mane:1980ep ; Ivanov:1982cr ;
Akhmetshin:1999ym ; Achasov:2000am ; Akhmetshin:2002vj ; Akhmetshin:2003zn ;
Achasov:2006bv ; Lees:2014xsh  and fitted together with our measurements above
$2.00~{}{\rm GeV}$. The parameters $\epsilon$ and $1+\delta$ are calculated
according to the fit curve at each c.m. energy and are taken as input for the
next iteration. The procedure is repeated until the measured Born cross
sections converge.

Results are summarized in Table 1 with both statistical and systematic
uncertainties given in the last column. The systematic uncertainties are
discussed in Section V.

Table 1: Born cross sections of the $e^{+}e^{-}\to K_{S}^{0}K_{L}^{0}$ process. The columns $N_{sig}$ and $N_{bkg}$ show the numbers of signal and background events determined by fitting the $m_{\pi^{+}\pi^{-}}$ distribution. The detection efficiency $\epsilon$, ISR and VP correction factor $1+\delta$, and the integrated luminosity $\mathcal{L}$ are summarized in the $4^{\rm th}$, $5^{\rm th}$, and $6^{\rm th}$ column, respectively. The values presented in the column labeled with $\sigma_{B}$ correspond to the measured Born cross section, where the first uncertainty is statistical and the second one is systematic. $\sqrt{s}$ (GeV) | $N_{sig}$ | $N_{bkg}$ | $\epsilon(\times 10^{-4})$ | ($1+\delta$) | $\mathcal{L}({\rm pb}^{-1})$ | $\sigma_{B}(\times 10^{-3})({\rm nb})$
---|---|---|---|---|---|---
2.000 | 185$\pm$18 | 341$\pm$22 | 541.2 | 6.09 | 10.1 | 53.9$\pm$5.2$\pm$4.1
2.050 | 51$\pm$9 | 115$\pm$12 | 448.8 | 7.48 | 3.34 | 44.0$\pm$7.8$\pm$3.7
2.100 | 101$\pm$13 | 252$\pm$18 | 289.6 | 11.77 | 12.2 | 23.5$\pm$3.0$\pm$3.6
2.125 | 658$\pm$34 | 1731$\pm$47 | 230.2 | 14.77 | 108. | 17.2$\pm$0.9$\pm$1.4
2.150 | 14$\pm$6 | 101$\pm$11 | 198.1 | 16.85 | 2.84 | 14.2$\pm$6.1$\pm$1.3
2.175 | 67$\pm$10 | 125$\pm$13 | 213.0 | 15.59 | 10.6 | 18.3$\pm$2.7$\pm$2.6
2.200 | 81$\pm$11 | 146$\pm$14 | 266.9 | 12.13 | 13.7 | 17.6$\pm$2.4$\pm$1.2
2.232 | 98$\pm$12 | 133$\pm$13 | 360.9 | 9.03 | 11.9 | 24.4$\pm$3.0$\pm$2.1
2.309 | 116$\pm$13 | 171$\pm$15 | 259.4 | 13.04 | 21.1 | 15.6$\pm$1.8$\pm$1.0
2.386 | 27$\pm$7 | 78$\pm$10 | 82.0 | 40.84 | 22.5 | 3.4$\pm$0.9$\pm$0.7
2.396 | 91$\pm$13 | 309$\pm$20 | 77.4 | 43.12 | 66.9 | 3.9$\pm$0.6$\pm$0.5
2.644 | 52$\pm$9 | 90$\pm$11 | 51.8 | 59.69 | 33.7 | 4.8$\pm$0.8$\pm$0.7
2.646 | 57$\pm$9 | 70$\pm$10 | 51.8 | 59.30 | 34.0 | 5.2$\pm$0.8$\pm$0.3
2.900 | 43$\pm$9 | 91$\pm$11 | 42.9 | 68.07 | 105. | 1.4$\pm$0.3$\pm$0.2
3.080 | 42$\pm$8 | 85$\pm$11 | 34.7 | 77.79 | 126. | 1.3$\pm$0.2$\pm$0.2

##  V Systematic uncertainties and line shape

###  V.1  Systematic uncertainties of the Born cross sections

Several sources of the systematic uncertainties are estimated at each c.m.
energy point, including uncertainties in the determination of the $K_{S}^{0}$
selection efficiency, in applying the $E/cp$ requirement, in the ISR and VP
correction factors, in the integrated luminosity, and in the fit procedure
that was used to determine the signal yield. The uncertainty in the
$K_{S}^{0}\rightarrow\pi^{+}\pi^{-}$ branching ratio is only $0.07\%$
Tanabashi:2018oca , which is considered to be negligible in this study.

The systematic uncertainty of the $K_{S}^{0}$ selection efficiency is obtained
using the control samples $J/\psi\to
K^{*}(892)^{\mp}K^{\pm},K^{*}(892)^{\mp}\to K_{S}^{0}\pi^{\mp}$ and
$J/\psi\to\phi K_{S}^{0}K^{\mp}\pi^{\pm}$, and the uncertainties are between
$2.2\%$ and $4.8\%$ depending on the reconstructed $K_{S}^{0}$ momentum
Ablikim:2012sf . The uncertainty from the $E/cp$ requirement is estimated by
changing the momentum $p$ of each track to its value before applying the
secondary-vertex fit. The uncertainties of the signal model, background model
and fit range determine the uncertainties of the signal yields. The
uncertainty from the signal model is estimated by changing the signal model to
the shape predicted by the MC data. The uncertainty due to the background
model is determined by replacing the background function with a first-order
Chebychev polynomial. The uncertainty associated to the fit range is estimated
by enlarging or reducing the fit range with an amount corresponding to the
mass resolution.

The systematic uncertainty of $\epsilon\times(1+\delta)$ is obtained by
fluctuating randomly all the fit parameters within the iteration procedure by
one $\sigma$ and taking into account the correlations among the parameters.
The distribution of the randomly produced $\epsilon\times(1+\delta)$ is fitted
by a Gaussian function, and the width of the fitted parameter is defined as
the systematic uncertainty of $\epsilon\times(1+\delta)$. The uncertainty due
to the luminosity is estimated using large-angle Bhabha scattering events,
which is about $0.9\%$ Ablikim:2017duf ; Ablikim:2017wlt .

A MC study shows a peaking background from the process
$K_{S}^{0}K_{L}^{0}\pi^{0}$ at a center-of-mass energy of $3.080$ GeV.
However, the contribution normalized according to the integrated data
luminosity is expected to be only $2.6$ events. To compensate for a possible
incomplete simulation, such as an incorrect angular distribution, the
systematic uncertainty from the possible $K_{S}^{0}K_{L}^{0}\pi^{0}$
background is increased to $3.1\%$ assuming the background level might be
higher by $50\%$.

All the systematic uncertainties are listed in Table 2. The total systematic
uncertainty is obtained by summing the individual contributions in quadrature.

Table 2: The relative systematic uncertainties (in $\%$) from the $K_{S}^{0}$ selection ($\epsilon(K_{S}^{0})$), $E/cp$, the ISR and VP correction factor ($1+\delta$), the luminosity ($\mathcal{L}$) and the fit on the invariant mass of $\pi^{+}\pi^{-}$ pair ($\rm{Fit}$). The column $\rm{peak}$ denotes the source from the peaking background and it has been estimated only at the c.m. energy of $3.080$ GeV as elucidated in the text. The total systematic uncertainty ($\rm{syst.}$) is calculated by summing the individual contributions in quadrature. The relative statistical uncertainty ($\rm{stat.}$) is shown in the last column. $\sqrt{s}$ (GeV | $\epsilon(K_{S}^{0})$ | $E/cp$ | $\epsilon(1+\delta)$ | $\mathcal{L}$ | $\rm{Fit}$ | $\rm{peak}$ | $\rm{syst.}$ | $\rm{stat.}$
---|---|---|---|---|---|---|---|---
2.000 | 2.99 | 0.53 | 0.63 | 0.89 | 6.87 | – | 7.6 | 9.7
2.050 | 3.02 | 0.01 | 0.42 | 0.90 | 7.74 | – | 8.4 | 17.7
2.100 | 2.92 | 0.01 | 0.52 | 0.89 | 15.14 | – | 15.5 | 12.9
2.125 | 2.82 | 0.15 | 0.67 | 0.69 | 7.54 | – | 8.1 | 5.2
2.150 | 2.82 | 0.03 | 0.82 | 0.89 | 8.93 | – | 9.4 | 42.9
2.1750 | 3.47 | 0.03 | 0.65 | 0.90 | 13.47 | – | 13.9 | 14.9
2.200 | 3.47 | 1.24 | 0.52 | 0.89 | 5.42 | – | 6.6 | 13.6
2.232 | 4.12 | 0.02 | 0.72 | 0.90 | 7.63 | – | 8.7 | 12.2
2.309 | 3.17 | 0.01 | 0.94 | 0.89 | 5.24 | – | 6.2 | 11.2
2.386 | 2.23 | 0.04 | 1.02 | 0.90 | 20.65 | – | 20.8 | 25.9
2.396 | 3.51 | 0.03 | 0.95 | 0.89 | 13.25 | – | 13.7 | 14.3
2.644 | 3.38 | 1.93 | 0.03 | 0.89 | 14.60 | – | 15.1 | 17.3
2.646 | 3.38 | 1.75 | 0.03 | 0.89 | 3.81 | – | 5.5 | 15.8
2.900 | 2.63 | 2.33 | 0.04 | 0.89 | 12.98 | – | 13.5 | 20.9
3.080 | 4.8 | 2.38 | 0.04 | 0.84 | 14.75 | 3.1 | 15.7 | 19.1

###  V.2  Line shape

The line shape of the Born cross section of $e^{+}e^{-}\to
K_{S}^{0}K_{L}^{0}$, obtained from the results given in Table 1, is displayed
in Figure 4. A resonance structure $R$ around $2.2$ GeV is observed. The cross
section data are fitted by

$\displaystyle\sigma_{B}=$
$\displaystyle\frac{M^{2}\beta(s)^{3}}{s\beta(M^{2})^{3}}|\sqrt{\sigma}BW(s)+P(s)e^{i\phi}|^{2}\,,$
(2)

where $\beta(s)=\sqrt{1-4m_{K_{S}^{0}}^{2}/s}$; $s$ is the square of the c.m.
energy; $BW(s)=M\Gamma/(M^{2}-s-i\sqrt{s}\Gamma)$ is a Breit-Wigner function
describing the resonance; $M$, $\Gamma$ and $\sigma$ are the mass, width and
peak cross section of the resonance, respectively;
$P(s)=c_{p_{0}}+c_{p_{1}}{\sqrt{s}}+c_{p_{2}}s$ is a second-order polynomial
function that is used to describe the nonresonant contribution, $c_{p_{i}}$
corresponds to the coefficient of the $i^{th}$-degree polynomial function, and
$\phi$ is the relative phase between nonresonant and resonant amplitudes.

The least-squares ($\chi^{2}$) method is used to perform the fit with both
statistical and systematic uncertainties taken into account. The $\chi^{2}$ is
obtained via a matrix (see Eq. (1) in Ref. Mo:2003cna  and Eq. (2) in Ref.
Mo:2006bea ) in which correlation effects of the various terms are included.
Uncertainties from the $K_{S}^{0}$-selection efficiency, $1+\delta$,
luminosity and $\epsilon$ are considered to be correlated, while the remaining
ones are treated as uncorrelated. The line shape and the individual
contributions obtained from the fit are shown in Figure 4.

Figure 4: Line shape of the process $e^{+}e^{-}\to K_{S}^{0}K_{L}^{0}$ and fit
curves. Points are data, solid curve shows the fit result, the dotted curve
denotes the signal component and the dash-dotted line is the polynomial
contribution.

The mass and width of the structure determined by the fit are $M=2273.7\pm
5.7$ MeV$/c^{2}$ and $\Gamma=86\pm 44$ MeV, respectively, where the
uncertainties are statistical. The goodness of the fit is
$\chi^{2}/NDF=4.6/8$, and the statistical significance of the structure is
$7.5\sigma$.

Various sources of systematic uncertainties of the observed structure are
considered including those associated with the choice of the model used to
describe the nonresonant component, the description of its width and the
chosen fit range. To estimate the systematic uncertainties, we changed the
description of the nonresonant component to a coherent sum of a second-order
polynomial and continuum functions

$\displaystyle P(s)=$ $\displaystyle
P^{\prime}(s)e^{i\phi}+c_{c}(\sqrt{s})^{\alpha}e^{i\phi_{c}}\,,$ (3)

where $P^{\prime}(s)$ and $\phi$ are the same as those defined in Eq. (2) but
only used in the fit when $\sqrt{s}<c_{p_{2}}$, $c_{c}$ is the coefficient of
the continuum function and $\phi_{c}$ is the relative phase between continuum
and resonant amplitudes. The differences in the values of the peak cross
section, mass, and width with respect to the nominal ones are
$\Delta\sigma=0.0150$ $\rm nb$, $\Delta m=17.7$ ${\rm MeV}$$/c^{2}$, and
$\Delta\Gamma=8.4$ MeV, respectively. By replacing the description of the
width with an energy dependent one
($\Gamma(s,m)=\Gamma\times\frac{s}{m_{R}^{2}}(\frac{\beta(s,m_{K}^{0})}{\beta(m^{2},m_{K}^{0})})^{3}$)
in Eq. (2), the peak cross section, mass, and width change by an amount of
$\Delta\sigma=0.0001$ $\rm nb$, $\Delta m=2.2$ MeV$/c^{2}$, and
$\Delta\Gamma=0.3$ MeV, respectively. Uncertainties from the fit range are
estimated by excluding the point at the c.m. energy of $2.00$ GeV or the one
at $3.08$ GeV. $\Delta\sigma_{1}$ and $\Delta\sigma_{2}$ ($\Delta m_{1}$ and
$\Delta m_{2}$, $\Delta\Gamma_{1}$ and $\Delta\Gamma_{2}$) denote the
differences of the peak cross sections (masses and widths) obtained by fitting
all energy points with a fit excluding those two energy points. Systematic
uncertainties associated with the fit range on the mass and width are
subsequently estimated by
$\sqrt{(\Delta\sigma_{1})^{2}+(\Delta\sigma_{2})^{2}}=0.0030$ nb,
$\sqrt{(\Delta m_{1})^{2}+(\Delta m_{2})^{2}}=7.5$ MeV$/c^{2}$, and
$\sqrt{(\Delta\Gamma_{1})^{2}+(\Delta\Gamma_{2})^{2}}=50.2$ MeV. Total
systematic uncertainties are obtained by taking the quadratic sum of all the
differences, which amount to $0.0153$ nb, $19.3$ MeV$/c^{2}$, and $50.9$ MeV
on the peak cross section, mass, and width, respectively. Only the statistic
uncertainty on $\phi$ is considered.

$\Gamma_{e^{+}e^{-}}Br_{K_{S}^{0}K_{L}^{0}}$ of the resonance $R$ is
calculated from the peak cross section by making use of $\sigma_{R}={12\pi
C\Gamma_{e^{+}e^{-}}Br_{K_{S}^{0}K_{L}^{0}}}/{(\Gamma M^{2})}$ Lees:2014xsh ,
where $\sigma_{R}$ represents the peak cross section obtained through Eq. 2,
$Br_{K_{S}^{0}K_{L}^{0}}$ is the branching fraction of $R\to
K_{S}^{0}K_{L}^{0}$, $\Gamma_{e^{+}e^{-}}$ is partial width of $R\to
e^{+}e^{-}$, $M$ and $\Gamma$ are the mass and width of the resonance, and
$C=0.3894~{}\times~{}10^{12}~{}{\rm nb}~{}{\rm MeV^{2}}/c^{4}$
Tanabashi:2018oca . $\Gamma_{e^{+}e^{-}}Br_{K_{S}^{0}K_{L}^{0}}$ for the
process is obtained from the fit results and listed in Eq. 4.

The $\chi^{2}$ obtained by the earlier-described matrix may cause a bias in
the fit Mo:2003cna ; Mo:2003jwa ; Mo:2006bea ; Mo:2007aea . To estimate the
bias effect, an unbiased $\chi^{2}$ definition (Eq. (7) in Ref. Mo:2007aea )
is used to fit the line shape. The differences between the two cases are
negligible in this analysis.

The parameters of the resonance around $2.2$ GeV are

$\displaystyle M=$ $\displaystyle 2273.7\pm 5.7\pm 19.3~{}{\rm{MeV}}/c^{2}\,,$
(4) $\displaystyle\Gamma=$ $\displaystyle 86\pm 44\pm 51~{}{\rm{MeV}}\,,$
$\displaystyle\sigma=$ $\displaystyle 0.0289\pm 0.0125\pm 0.0153~{}{\rm
nb}\,,$ $\displaystyle\Gamma_{e^{+}e^{-}}Br_{K_{S}^{0}K_{L}^{0}}=$
$\displaystyle 0.9\pm 0.6\pm 0.7~{}{\rm eV}\,,$ $\displaystyle\phi=$
$\displaystyle 81.1\pm 17.4~{}{\rm deg}\,,$ $\displaystyle{\rm or}-98.9\pm
23.0~{}{\rm deg}\,,$

where the quoted uncertainties are statistical and systematic, respectively.
The mass and width are consistent within $2\sigma$ with measurements of the
mass and width of a similar structure observed in $e^{+}e^{-}\to K^{+}K^{-}$
at BESIII Ablikim:2018iyx , which gave $M=2239.2\pm 7.1\pm 11.3$ MeV$/c^{2}$
and $\Gamma=139.8\pm 12.3\pm 20.6$ MeV.

##  VI  Summary

We report a measurement of the Born cross sections in $e^{+}e^{-}\to
K_{S}^{0}K_{L}^{0}$ from $\sqrt{s}=2.00$ to $3.08$ GeV obtained at fifteen
energy points with BESIII. The data are consistent within $2\sigma$ with
previous measurements by the BaBar collaboration BABAR:2019oes  in the overlap
region from $2.00$ to $2.54$ GeV, but with a significantly improved precision
as demonstrated in Figure 4. Moreover, the Born cross sections from $2.54$ to
$3.08$ GeV are reported for the first time. A structure is observed around
$2.2$ GeV, which is similar to the one observed earlier in $e^{+}e^{-}\to
K^{+}K^{-}$ Ablikim:2018iyx . The results of both processes taken with BESIII
and BaBar are shown in Figure 5 for comparison.

A fit is applied to the data, where the mass and width of the resonance are
determined to be $M=2273.7\pm 5.7\pm 19.3$ MeV$/c^{2}$ and $\Gamma=86\pm 44\pm
51$ MeV, respectively. In addition,
$\Gamma_{e^{+}e^{-}}Br_{K_{S}^{0}K_{L}^{0}}$ is found to be $0.9\pm 0.6\pm
0.7~{}{\rm eV}$. The first uncertainties in the parameters are statistical and
the second ones are systematic. The mass and width are consistent within
$2\sigma$ and $1\sigma$, respectively, with the resonance parameters obtained
by fitting the cross sections for the process $e^{+}e^{-}\to K^{+}K^{-}$
($M=2239.2\pm 7.1\pm 11.3$ MeV$/c^{2}$ and $\Gamma=139.8\pm 12.3\pm 20.6$ MeV)
Ablikim:2018iyx .

Comparing to one of the $1^{--}$ candidate of the structure $\phi(2170)$ by
looking up the PDG Tanabashi:2018oca , the mass parameter obtained in this
paper differs from the world average for more than $4\sigma$. The width
parameter is consistent within $1\sigma$ compared to the world averaged width
of $\phi(2170)$. The uncertainty on the width in this paper is large. For
another $1^{--}$ candidate of the structure $\rho(2150)$ Tanabashi:2018oca ,
the mass parameter in this paper is more than $5\sigma$ different from the
world average and the wold averaged width is not given in the PDG
Tanabashi:2018oca . But the mass and width are consistent with the individual
measurement of the process $e^{+}e^{-}\to\gamma\pi^{+}\pi^{-}$ by Babar
Lees:2012cj . The conclusions support the discussions in the $e^{+}e^{-}\to
K^{+}K^{-}$ study by BESIII Ablikim:2018iyx . Due to limit of the statistics,
especially the cross section measurements above $2.4~{}\rm{GeV}$, it is
difficult to discuss deeper on the structure found in this paper. More precise
and fine interval measurements are needed.

Figure 5: Comparison of cross-section measurements of the processes
$e^{+}e^{-}\to K_{S}^{0}K_{L}^{0}$ (top panel) and $e^{+}e^{-}\to K^{+}K^{-}$
(bottom panel) by BESIII (filled dots) Ablikim:2018iyx  and BaBar (open
circles) BABAR:2019oes .

## Acknowledgements

We appreciate Prof. Dianyong Chen from Southeast University (SEU) for the
important discussions and suggestions. The BESIII collaboration thanks the
staff of BEPCII and the IHEP computing center and the supercomputing center of
USTC for their strong support. This work is supported in part by National Key
R$\&$D Program of China under Contracts Nos. 2020YFA0406400, 2020YFA0406300,
2015CB856700; National Natural Science Foundation of China (NSFC) under
Contracts Nos. 11625523, 11635010, 11735014, 11822506, 11835012, 11935015,
11935016, 11935018, 11961141012; the Chinese Academy of Sciences (CAS) Large-
Scale Scientific Facility Program; Joint Large-Scale Scientific Facility Funds
of the NSFC and CAS under Contracts Nos. U1732263, U1832207 and U2032115; CAS
Key Research Program of Frontier Sciences under Contracts Nos. QYZDJ-SSW-
SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; INPAC and Shanghai Key
Laboratory for Particle Physics and Cosmology; ERC under Contract No. 758462;
German Research Foundation DFG under Contracts Nos. Collaborative Research
Center CRC 1044, FOR 2359; Istituto Nazionale di Fisica Nucleare, Italy;
Ministry of Development of Turkey under Contract No. DPT2006K-120470; National
Science and Technology fund; STFC (United Kingdom); The Knut and Alice
Wallenberg Foundation (Sweden) under Contract No. 2016.0157; The Royal
Society, UK under Contracts Nos. DH140054, DH160214; The Swedish Research
Council; U. S. Department of Energy under Contracts Nos. DE-FG02-05ER41374,
DE-SC-0012069. This paper is also supported by the NSFC under Contract No.
11605074, 11335008.

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††thanks: M. J. Rendell and S. D. Liles contributed equally to this work

# Spin polarisation and spin dependent scattering of holes in transverse
magnetic focussing

M. J. Rendell  School of Physics and Australian Research Council Centre of
Excellence in Future Low-Energy Electronics Technologies, University of New
South Wales, Sydney, NSW 2052, Australia  S. D. Liles  A. Srinivasan  School
of Physics, University of New South Wales, Sydney, NSW 2052, Australia  O.
Klochan  University of New South Wales Canberra, Canberra, ACT 2600, Australia
School of Physics and Australian Research Council Centre of Excellence in
Future Low-Energy Electronics Technologies, University of New South Wales,
Sydney, NSW 2052, Australia  I. Farrer  Department of Electronic and
Electrical Engineering, University of Sheffield, Sheffield, S1 3JD, UK
Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, UK  D. A.
Ritchie  Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, UK
A. R. Hamilton<EMAIL_ADDRESS>School of Physics and Australian
Research Council Centre of Excellence in Future Low-Energy Electronics
Technologies, University of New South Wales, Sydney, NSW 2052, Australia

###### Abstract

In 2D systems with a spin-orbit interaction, magnetic focussing can be used to
create a spatial separation of particles with different spin. Here we measure
hole magnetic focussing for two different magnitudes of the Rashba spin-orbit
interaction. We find that when the Rashba spin-orbit magnitude is large there
is significant attenuation of one of the focussing peaks, which is
conventionally associated with a change in the spin polarisation. We instead
show that in hole systems with a $k^{3}$ spin-orbit interaction, this peak
suppression is due to a change in the scattering of one spin state, not a
change in spin polarisation. We also show that the change in scattering length
extracted from magnetic focussing is consistent with results obtained from
measurements of Shubnikov-de Haas oscillations. This result suggests that
scattering must be considered when relating focussing peak amplitude to spin
polarisation in hole systems.

††preprint: APS/123-QED

##  I Introduction

In a magnetic focussing experiment, a collimated beam of charge is focussed
into a circular orbit by a transverse magnetic field, analogous to a mass
spectrometer. Magnetic focussing was originally proposed as a method of
studying the Fermi surface of metals [1, 2], and has also been used to measure
band structures in graphene [3], and electron-electron scattering lengths in
GaAs/AlGaAs [4].

In systems with a spin-orbit interaction (SOI), the magnetic focussing
trajectories become spin dependent as the spin states are now coupled to
momentum. If the SOI is sufficiently large, magnetic focussing can spatially
separate the spin states and create a spin-dependent mass spectrometer [5, 6,
7, 8, 9, 10, 11, 12]. The high mobility and large SOI of 2D hole systems in
GaAs has made them an ideal candidate for spin-dependent magnetic focussing
experiments. Experimental work has used magnetic focussing to measure spatial
separation of spin [5], spin filtering by quantum point contacts (QPCs) [7]
and interactions between 1D subbands in a QPC [13]. Magnetic focussing of
holes has also been proposed as a way to measure g-factor anisotropies [14],
and complex spin dynamics [9, 15], which are not visible in other measurements
of 2D systems such as Shubnikov-de Haas oscillations.

Here, we concentrate on the use of magnetic focussing peak amplitude as a
measure of the spin polarisation [16, 17, 5, 7, 18]. It has been proposed that
the relative amplitudes of the spin-split magnetic focussing peaks is
determined by the spin polarisation of the injected charge. This technique has
been used in hole systems to observe spontaneous polarisation in QPC
transmission [7] and spin-dependent transmission of QPCs [5, 13]. Despite
magnetic focussing being used for these techniques, there has been limited
study of the effect of changing the magnitude of the Rashba SOI on hole
magnetic focussing.

A recent study investigated magnetic focussing using a device where the Rashba
SOI magnitude could be tuned in situ using a top gate voltage
(V${}_{\text{TG}}$) [19]. This technique revealed an increase in the spatial
separation of the spin-split focussing trajectories as the Rashba SOI was
increased. However, there is a limit to the amount the Rashba SOI can be
changed using this method. In addition, any change to $V_{TG}$ will also
change the 2D hole density and confining potential in addition to the Rashba
SOI magnitude. As such, further study requires a different method of changing
the Rashba SOI.

In this work we study magnetic focussing in two lithographically identical
samples which differ only in the magnitude of the Rashba SOI. We change the
Rashba SOI by changing the heterostructure used to confine the 2D system,
allowing us to create a large change in the magnitude of the Rashba SOI for a
similar V${}_{\text{TG}}$ and 2D density. By comparing the two samples, we
observe a change in the amplitude of the magnetic focussing peaks, which is
typically associated with a change in the spin polarisation. However, we
instead find that the change in peak amplitude is consistent with an increase
in scattering of one spin state rather than a change in spin polarisation. We
measure the scattering length of each spin state from the focussing peak
amplitude, and find good agreement with scattering lenghts found from
Shubnikov-de Haas measurements. We conclude that the change in focussing peak
amplitude is due to the $k^{3}$ Rashba term causing a different effective mass
and hence scattering length of each spin state, rather than a change in spin
polarisation. This result suggests that care must be taken when relating the
amplitude of spin-split focussing peaks to the spin polarisation in 2D hole
systems.

##  II Magnetic focussing with a cubic Rashba spin-orbit interaction

Figure 1: a) Magnetic focussing in the presence of a spin-orbit interaction.
The red and blue lines correspond to the spin split focussing trajectories,
which result in a splitting of the first focussing peak. The dashed line
corresponds to the classical focussing trajectory. c) The first 2D subband for
a hole system. Here the Rashba SOI term depends on $k^{3}$ which causes a
change in the slope (and hence $m^{*}$) of the spin resolved subbands.

Fig 1a) shows a schematic of a hole magnetic focussing device. A constant
current is applied through an injector, where an out-of-plane perpendicular
magnetic field causes the holes to follow cyclotron orbits. The voltage across
the detector is measured, and a peak is observed when the focussing diameter
is equal to the spacing between injector and detector (black dashed line in
Fig 1a). Peaks in the focussing signal occur when the magnetic field is an
integer multiple of [20]

$B=\frac{2\hbar k_{F}}{ed}$

Where $k_{\text{F}}$ is the Fermi momentum and $d$ is the distance between
injector and collector QPC (focussing diameter). In the presence of a spin-
orbit interaction (SOI) the hole trajectories become spin dependent, resulting
in a spatial separation of spin (blue and red lines in Fig 1a). The spatial
spin separation causes the first magnetic focussing peak to split into two,
with each peak corresponding to a different spin chirality. The relative
amplitude of these spin peaks has been used as a measure of the spin
polarisation in 2D hole systems.[5]

The form of the Rashba spin-orbit term for 2D hole systems is fundamentally
different to equivalent electron systems. This difference can have a dramatic
impact on spin resolved focussing peaks. In GaAs, the subband dispersion for
2D holes with a Rashba SOI is given by [21]

$\mathcal{E}_{\text{h}}=\frac{\hbar^{2}k^{2}}{2m^{*}}\pm\frac{\beta
E_{\text{z}}}{\Delta_{\text{HH-LH}}}k^{3}$ (1)

where $E_{\text{z}}$ is the electric field in the out-of-plane direction and
$\Delta_{\text{HH-LH}}$ is the splitting between the heavy hole (HH) and light
hole (LH) subbands. Fig 1b) shows the resulting HH subband dispersion for a 2D
hole system with Rashba SOI. The SOI causes the momentum of the holes to
become spin dependent, with two values of $k$ ($k_{+}$ and $k_{-}$) at the
Fermi energy (horizontal dashed line). In a magnetic focussing measurement,
this results in separate cyclotron orbits for each spin and creates a spatial
spin separation, splitting the first focussing peak. Previous work has
demonstrated the ability to detect a change in peak splitting as the magnitude
of the Rashba SOI is changed [19].

The $k^{3}$ structure of the Rashba SOI term for holes also causes the
curvature of the 2D subbands to become spin dependent. This results in a
difference in effective mass for each spin chirality in addition to the
difference in $k$ [22]. The spin dependent effective mass has been used to
demonstrate electrical control of the Zeeman splitting [23], and proposed as a
way to detect and generate topological properties in a 2D hole system [24,
25]. The change in effective mass is also possible to detect via focussing
peaks. If the Rashba SOI term is sufficiently large, the difference in
effective mass can be observed as a difference in scattering. Since focussing
peak amplitude is exponentially sensitive to scattering [26, 27], the change
in effective mass will therefore impact the focussing peak amplitude. This
analysis does not include contributions from k-linear Rashba SOI terms for 2D
holes [28, 29, 30]. These terms do not cause a spin-dependent change in the
curvature of the 2D subbands and should not affect the difference in effective
mass between the spin chiralities.

##  III Interface dependence

To create a large change in the Rashba SOI without changing the carrier
density, we prepare two samples with different interface symmetry. One sample
uses a 15nm (100) GaAs/Al0.33GaAs0.67 quantum well (QW) heterostructure with a
square well like 2D confining potential (wafer W713). The other is a (100)
GaAs/Al0.33GaAs0.67 single heterojunction (SHJ) which creates a triangular
confining potential (wafer W640). Changing from a QW to a SHJ reduces the 2D
confinement, decreasing $\Delta_{\text{HH-LH}}$ without causing a large change
in $k$ and $E_{\text{z}}$. From Eq 1, the SHJ device (smaller
$\Delta_{\text{HH-LH}}$) will have a larger Rashba SOI term and therefore
larger focussing peak splitting. The SHJ device will also have a larger
difference in effective mass between the spin subbands.

Figure 2: Comparison of sample heterostructures. a) Heterostructure of the QW
sample and resultant band edge. b) Heterostructure and band edge of the SHJ
sample. c) A comparison of the HH1 subband dispersion for both samples.
Calculated with a 6x6 k.p solver (Nextnano).

Fig 2 compares the wafer structure and resultant confining potentials for the
QW (a) and SHJ (b) samples. The left side of each panel is the wafer structure
around the 2D interface, while the right side shows the resulting band edge
found using a Schrodinger-Poisson solver (Nextnano
111http://www.nextnano.de/). The $E(k)$ dispersion relations of both samples
are also calculated using Nextnano. This calculation uses a $6\times 6$
$k\cdot p$ solver and includes contributions from Rashba SOI terms but does
not include Dresselhaus SOI terms. Fig 2 c) shows the spin split HH1 subbands
for both samples, with a clear difference in $k$ between the HH+ and HH-
subbands at $E=0$ (horizontal dashed line). It is this difference in $k$ that
results in a splitting of the first focussing peak in both samples.

There is a significantly larger splitting visible for the SHJ sample at
$E_{\text{F}}$ (horizontal dashed line), which leads to an increase in the
focussing peak spacing. There is also a large difference in the curvature of
the HH+ and HH- subbands. The difference in curvature of the $E(k)$ dispersion
results in a spin dependent effective mass, which can also be detected in a
focussing measurement.

Figure 3: Spin-split magnetic focussing with different interface symmetry. Red
is focussing in a 15nm quantum well with weaker Rashba spin-orbit
($n_{\text{2D}}$ = 1.57$\times 10^{11}$ cm-2). Black is focussing in a single
heterojunction with stronger Rashba spin orbit at the same $V_{\text{TG}}$
($n_{\text{2D}}=1.89\times 10^{11}\text{cm}^{-2}$). Dashed vertical lines
indicate the spacing of the 800nm QW focussing peaks. X-axis in the bottom
panel has been scaled by the ratio of $n_{\text{2D}}$ to account for the
difference in density.

Fig 3 compares focussing in the QW and SHJ samples over the same focussing
diameter (800 nm). Starting with the QW sample (Fig 3a), we observe a clear
spin split focussing peak in positive $B$, with higher order peaks also
observed. No splitting of higher order peaks is observed due to spin-flip
reflections from the boundary [32, 12]. Vertical dashed lines indicate the
position of the spin-split first focussing peaks. In the opposite $B$ polarity
the holes are focussed away from the collector QPC and only Shubnikov-de Haas
oscillations are visible.

The same measurement is repeated on a lithographically identical sample,
fabricated on the SHJ wafer as shown in Fig 3b. This measurement was performed
at the same $V_{\text{TG}}$ as the QW sample, resulting in a slightly higher
hole density ($n_{\text{2D}}$ = $1.89\times 10^{11}\text{cm}^{-2}$ vs
$1.57\times 10^{11}\text{cm}^{-2}$). The increased Rashba SOI in the SHJ
sample results in focussing peaks which are further apart than the QW sample.
The focussing peaks in Fig 3b) are also significantly smaller in amplitude
than those in the QW sample, with the higher field peak attenuated and broader
compared to the lower field peak. Typically such a difference in amplitude of
spin-resolved focussing peaks is interpreted as a change in the spin
polarisation [5, 7]. However, here the spin polarisation should be
approximately equal as both QPCs are biased to the G = 2e2/h plateau to
transmit both spin states.

To determine if the change in peak amplitude in the SHJ sample is instead
related to an increase in scattering, the decay of focussing peak amplitude is
measured over a range of focussing diameters. The device geometry of both
focussing samples allows focussing to be measured for a range of focussing
diameters (d = 800, 2300 and 3100nm). By measuring the change in peak
amplitude as a function of focussing diameter, the scattering length of each
of the spin peaks can be found [26, 27].

Figure 4: Focussing over different diameters in the QW sample. The focussing
diameter is varied from 800nm (a) to 3100nm (c). The x-axis range of each
panel has been chosen based on the focussing diameter. The vertical dashed
lines indicate the position of the spin resolved focussing peaks in the 800nm
trace.

First, the focussing diameter dependence of the peak amplitude is measured on
the QW sample. Fig 4 shows focussing measured on the QW sample for all three
focussing diameters. To allow for easy comparison between the focussing
lengths, the $B$ axis range of each plot has been chosen based on the ratio of
the focussing diameters. This should result in vertical alignment of the same
focussing peaks across each diameter. Vertical dashed lines indicate the
position of the spin split focussing peaks in Fig 4a) (focussing diameter =
800nm). As the focussing diameter is increased from 800nm (Fig 4a) to 2300nm
(Fig 4b) there is good agreement in $B$ location between all of the peaks.
Multiple higher order peaks can be observed, with the amplitude of the higher
order peaks similar to the spin-split focussing peaks, indicating specular
reflections from the boundary between the injector and detector QPCs. The spin
split focussing peaks can also be clearly resolved, and align with the peaks
in Fig 4a) as indicated by the vertical dashed lines. The spin split focussing
peaks also have a similar amplitude, indicating an equal population of both
spin states (i.e. no spin polarisation). As the focussing diameter is further
increased to 3100nm, (Fig 4c) the effects of scattering begin to dominate the
focussing signal. The amplitude of the first focussing peak is significantly
reduced and both spin peaks can no longer clearly be resolved.

Figure 5: Focussing comparison for the SHJ sample. Focussing over a diameter
of 800nm (a), 2300nm (b) and 3100nm (c). The x-axis range has been chosen to
allow direct comparison of focussing peaks. Vertical dashed lines indicate the
position of the spin resolved focussing peaks for the 800nm focussing
diameter.

In Fig 5 focussing is measured in the SHJ sample for the same three focussing
diameters (800, 2300 and 3100nm). Once again, the x-axis range of each panel
has been chosen based on the focussing diameter, and the vertical dashed lines
indicate the position of the spin-split focussing peaks in Fig 5a). The HH-
peak is narrow and large in amplitude, while the HH+ peak is significantly
lower in amplitude and broader. The spacing of the peaks has also increased
compared to the QW sample (Fig 4a), as expected for a larger Rashba SOI
magnitude. As the focussing diameter is increased from 800nm (Fig 5a) to
2300nm (Fig 5b) multiple changes can be observed. First, the amplitude of all
focussing peaks decreases with the increase in focussing path length. In
particular, the HH+ spin peak becomes very broad and low in amplitude. In
comparison, the HH- spin split peak is higher in amplitude and narrower, and
both spin peaks have superimposed interference structure [20, 33, 34, 12]. The
HH- spin peak displays structure similar to a spin split focussing peak,
however this splitting is too small to be caused by spin. The structure on the
HH- peak is most likely due to an interference effect, as similar (but smaller
amplitude) oscillations are visible on the low B side of the peak, a
characteristic signature of interference due to diffraction in focussing [34,
12]. Finally, as the focussing diameter is increased to 3100nm (Fig 5c) the
amplitude of the HH+ and HH- peaks is further reduced, with the HH+ peak
barely resolved due to scattering.

The significantly lower amplitude of the HH+ spin split peak in Fig 5 is
consistent with the larger effective mass of this spin band (see Fig 2c).
Assuming the scattering time is the same, the larger effective mass should
result in a shorter scattering length for the HH+ spin state. As the total
path length travelled by both spin states is the same, being fixed by the
focussing geometry, a shorter scattering length will result in more scattering
for the HH+ state and hence a lower amplitude of the corresponding HH+
focussing peak. Since focussing measurements are exponentially sensitive to
scattering effects [35, 26, 27], a change in scattering rate can be the
dominant cause of the amplitude change, rather than a change in the spin
polarisation.

##  IV Scattering length

To understand the suppression of the HH+ peak in the SHJ focussing sample, we
extract the scattering length of both spin peaks and compare this to the
scattering length extracted from Shubnikov-de Haas oscillations. The amplitude
of focussing peaks decays exponentially as the focussing path length is
increased:

$R_{\text{Focus}}\propto A~{}e^{-\pi d/2l}$ (2)

where $l$ is the small angle scattering length [35, 26, 27] and $d$ is the
focussing diameter. By fitting a double Gaussian to the spin peaks for each
focussing diameter, the amplitude of the peaks can be found as a function of
path length.

Figure 6: Focussing decay length for the SHJ sample. a) - c) Gaussian fits to
the spin-split focussing peaks on the SHJ sample for all focussing diameters.
d) fit to the amplitude of each spin peak with error bars showing the fit
uncertainty. From this we can obtain the scattering length for each spin.

Fig 6 a) - c) shows the results of a double Gaussian fit to the SHJ focussing
peaks for all focussing lengths. The amplitude of the spin split peaks as a
function of focussing path length is plotted on semilog axes in Fig 6d). A
straight line fit to the data in Fig 6d) allows the scattering length for each
of the spin peaks to be found ($l_{+}$ and $l_{-}$). It is difficult to
directly compare the scattering lengths found from focussing to a mean free
path for two reasons. First, focussing measurements require the holes to
travel through a narrow detector QPC, where even small scattering events can
be sufficient to prevent a hole from reaching the detector. Second, the value
of the scattering length found from focussing is sensitive to the choice of
background resistance used for the peak fitting. However, the ratio of the
scattering lengths is independent of the background resistance. From the
linear fits in Fig 6 d), Eq2 can be used to find the scattering length for
each spin state ($l_{+}$ and $l_{-}$). From this we find that $l_{+}/l_{-}$ =
0.77 $\pm$ 0.01.

Finally, the ratio of the scattering lengths is compared to a ratio found from
Shubnikov-de Haas oscillations. The scattering length is given by $l\propto
v_{F}\tau$, where $\tau$ is the scattering time. Assuming that $\tau$ is
constant for both spin states, the ratio of $v_{\text{F}}$ (i.e.
$v_{+}/v_{-}$) can be used to predict the ratio of $l_{+}/l_{-}$. The ratio of
$v_{+}/v_{-}$ can be found from the ratio of $n_{+}/n_{-}$ since

$\frac{v_{+}}{v_{-}}=\frac{n_{+}}{m_{+}}\frac{m_{-}}{n_{+}}=\frac{k_{+}m_{-}}{k_{-}m_{+}}$

Assuming the subbands are approximately parabolic this expression can be
simplified to

$\frac{l_{+}}{l_{-}}=\frac{v_{+}}{v_{-}}=\sqrt{\frac{n_{-}}{n_{+}}}$

The values of $n_{+}$ and $n_{-}$ were found from measurements of the
frequency of Shubnikov-de Haas oscillations on the same sample ($f=nh/e$).
From this $n_{+}=1.21\times 10^{11}\text{cm}^{-1}$ and $n_{-}=0.68\times
10^{11}\text{cm}^{-1}$ which gives a predicted ratio of $l_{+}/l_{-}=0.75$,
almost identical to the measured value of $l_{+}/l_{-}=0.76$.

The good agreement between the predicted and measured ratios of $l$ indicates
that the increased scattering of one spin state is the likely cause of the
difference in peak amplitude observed in the SHJ focussing sample.

##  V Conclusions

In this work we have investigated magnetic focussing with a large change in
the magnitude of the Rashba spin-orbit interaction. We observed an attenuation
of one of the spin peaks that is typically associated with a change in spin
polarisation. Here we have instead shown that the difference in peak amplitude
is consistent with a difference in effective mass and hence scattering rate
between the spin chiralities. This result indicates that care must be taken
when associating the amplitude of hole focussing peaks to spin polarisation in
hole systems with $k^{3}$ spin-orbit interaction.

##  VI Acknowledgements

The authors would like to thank S. Bladwell, O.P. Sushkov, E.Y. Sherman and U.
Zülicke for many valuable discussions. Devices were fabricated at the UNSW
node of the Australian National Fabrication Facility (ANFF). This research was
funded by the Australian Government through the Australian Research Council
Discovery Project Scheme; Australian Research Council Centre of Excellence
FLEET (project number CE170100039); and by the the UK Engineering and Physical
Sciences Research Council (Grant No. EP/R029075/1).

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