# Observation of $e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c}$ and a neutral
charmoniumlike structure $Z_{c}(4020)^{0}$

M. Ablikim1, M. N. Achasov8,a, X. C. Ai1, O. Albayrak4, M. Albrecht3, D. J.
Ambrose42, A. Amoroso46A,46C, F. F. An1, Q. An43, J. Z. Bai1, R. Baldini
Ferroli19A, Y. Ban30, D. W. Bennett18, J. V. Bennett4, M. Bertani19A, D.
Bettoni20A, J. M. Bian41, F. Bianchi46A,46C, E. Boger22,g, O. Bondarenko24, I.
Boyko22, R. A. Briere4, H. Cai48, X. Cai1, O. Cakir38A, A. Calcaterra19A, G.
F. Cao1, S. A. Cetin38B, J. F. Chang1, G. Chelkov22,b, G. Chen1, H. S. Chen1,
H. Y. Chen2, J. C. Chen1, M. L. Chen1, S. J. Chen28, X. Chen1, X. R. Chen25,
Y. B. Chen1, H. P. Cheng16, X. K. Chu30, Y. P. Chu1, G. Cibinetto20A, D.
Cronin-Hennessy41, H. L. Dai1, J. P. Dai1, D. Dedovich22, Z. Y. Deng1, A.
Denig21, I. Denysenko22, M. Destefanis46A,46C, F. De Mori46A,46C, Y. Ding26,
C. Dong29, J. Dong1, L. Y. Dong1, M. Y. Dong1, S. X. Du50, P. F. Duan1, J. Z.
Fan37, J. Fang1, S. S. Fang1, X. Fang43, Y. Fang1, L. Fava46B,46C, F.
Feldbauer21, G. Felici19A, C. Q. Feng43, E. Fioravanti20A, C. D. Fu1, Q. Gao1,
Y. Gao37, I. Garzia20A, K. Goetzen9, W. X. Gong1, W. Gradl21, M. Greco46A,46C,
M. H. Gu1, Y. T. Gu11, Y. H. Guan1, A. Q. Guo1, L. B. Guo27, T. Guo27, Y.
Guo1, Y. P. Guo21, Z. Haddadi24, A. Hafner21, S. Han48, Y. L. Han1, F. A.
Harris40, K. L. He1, Z. Y. He29, T. Held3, Y. K. Heng1, Z. L. Hou1, C. Hu27,
H. M. Hu1, J. F. Hu46A, T. Hu1, Y. Hu1, G. M. Huang5, G. S. Huang43, H. P.
Huang48, J. S. Huang14, X. T. Huang32, Y. Huang28, T. Hussain45, Q. Ji1, Q. P.
Ji29, X. B. Ji1, X. L. Ji1, L. L. Jiang1, L. W. Jiang48, X. S. Jiang1, J. B.
Jiao32, Z. Jiao16, D. P. Jin1, S. Jin1, T. Johansson47, A. Julin41, N.
Kalantar-Nayestanaki24, X. L. Kang1, X. S. Kang29, M. Kavatsyuk24, B. C. Ke4,
R. Kliemt13, B. Kloss21, O. B. Kolcu38B,c, B. Kopf3, M. Kornicer40, W.
Kuehn23, A. Kupsc47, W. Lai1, J. S. Lange23, M. Lara18, P. Larin13, M. Leyhe3,
Cheng Li43, D. M. Li50, F. Li1, G. Li1, H. B. Li1, J. C. Li1, Jin Li31, K.
Li12, K. Li32, Q. J. Li1, T. Li32, W. D. Li1, W. G. Li1, X. L. Li32, X. M.
Li11, X. N. Li1, X. Q. Li29, Z. B. Li36, H. Liang43, Y. F. Liang34, Y. T.
Liang23, G. R. Liao10, D. X. Lin13, B. J. Liu1, C. L. Liu4, C. X. Liu1, F. H.
Liu33, Fang Liu1, Feng Liu5, H. B. Liu11, H. H. Liu1, H. H. Liu15, H. M. Liu1,
J. Liu1, J. P. Liu48, J. Y. Liu1, K. Liu37, K. Y. Liu26, L. D. Liu30, Q.
Liu39, S. B. Liu43, X. Liu25, X. X. Liu39, Y. B. Liu29, Z. A. Liu1, Zhiqiang
Liu1, Zhiqing Liu21, H. Loehner24, X. C. Lou1,d, H. J. Lu16, J. G. Lu1, R. Q.
Lu17, Y. Lu1, Y. P. Lu1, C. L. Luo27, M. X. Luo49, T. Luo40, X. L. Luo1, M.
Lv1, X. R. Lyu39, F. C. Ma26, H. L. Ma1, Q. M. Ma1, S. Ma1, T. Ma1, X. N.
Ma29, X. Y. Ma1, F. E. Maas13, M. Maggiora46A,46C, Q. A. Malik45, Y. J. Mao30,
Z. P. Mao1, S. Marcello46A,46C, J. G. Messchendorp24, J. Min1, T. J. Min1, R.
E. Mitchell18, X. H. Mo1, Y. J. Mo5, H. Moeini24, C. Morales Morales13, K.
Moriya18, N. Yu. Muchnoi8,a, H. Muramatsu41, Y. Nefedov22, F. Nerling13, I. B.
Nikolaev8,a, Z. Ning1, S. Nisar7, S. L. Niu1, X. Y. Niu1, S. L. Olsen31, Q.
Ouyang1, S. Pacetti19B, P. Patteri19A, M. Pelizaeus3, H. P. Peng43, K.
Peters9, J. L. Ping27, R. G. Ping1, R. Poling41, Y. N. Pu17, M. Qi28, S.
Qian1, C. F. Qiao39, L. Q. Qin32, N. Qin48, X. S. Qin1, Y. Qin30, Z. H. Qin1,
J. F. Qiu1, K. H. Rashid45, C. F. Redmer21, H. L. Ren17, M. Ripka21, G. Rong1,
X. D. Ruan11, V. Santoro20A, A. Sarantsev22,e, M. Savrié20B, K. Schoenning47,
S. Schumann21, W. Shan30, M. Shao43, C. P. Shen2, P. X. Shen29, X. Y. Shen1,
H. Y. Sheng1, M. R. Shepherd18, W. M. Song1, X. Y. Song1, S. Sosio46A,46C, S.
Spataro46A,46C, B. Spruck23, G. X. Sun1, J. F. Sun14, S. S. Sun1, Y. J. Sun43,
Y. Z. Sun1, Z. J. Sun1, Z. T. Sun18, C. J. Tang34, X. Tang1, I. Tapan38C, E.
H. Thorndike42, M. Tiemens24, D. Toth41, M. Ullrich23, I. Uman38B, G. S.
Varner40, B. Wang29, B. L. Wang39, D. Wang30, D. Y. Wang30, K. Wang1, L. L.
Wang1, L. S. Wang1, M. Wang32, P. Wang1, P. L. Wang1, Q. J. Wang1, S. G.
Wang30, W. Wang1, X. F. Wang37, Y. F. Wang1, Y. Q. Wang21, Z. Wang1, Z. G.
Wang1, Z. H. Wang43, Z. Y. Wang1, D. H. Wei10, J. B. Wei30, P. Weidenkaff21,
S. P. Wen1, U. Wiedner3, M. Wolke47, L. H. Wu1, Z. Wu1, L. G. Xia37, Y. Xia17,
D. Xiao1, Z. J. Xiao27, Y. G. Xie1, Q. L. Xiu1, G. F. Xu1, L. Xu1, Q. J. Xu12,
Q. N. Xu39, X. P. Xu35, Z. Xue1, L. Yan43, W. B. Yan43, W. C. Yan43, Y. H.
Yan17, H. X. Yang1, L. Yang48, Y. Yang5, Y. X. Yang10, H. Ye1, M. Ye1, M. H.
Ye6, J. H. Yin1, B. X. Yu1, C. X. Yu29, H. W. Yu30, J. S. Yu25, C. Z. Yuan1,
W. L. Yuan28, Y. Yuan1, A. Yuncu38B,f, A. A. Zafar45, A. Zallo19A, Y. Zeng17,
B. X. Zhang1, B. Y. Zhang1, C. Zhang28, C. C. Zhang1, D. H. Zhang1, H. H.
Zhang36, H. T. Zhang1, H. Y. Zhang1, J. J. Zhang1, J. L. Zhang1, J. Q. Zhang1,
J. W. Zhang1, J. Y. Zhang1, J. Z. Zhang1, K. Zhang1, L. Zhang1, S. H. Zhang1,
X. J. Zhang1, X. Y. Zhang32, Y. Zhang1, Y. H. Zhang1, Z. H. Zhang5, Z. P.
Zhang43, Z. Y. Zhang48, G. Zhao1, J. W. Zhao1, J. Y. Zhao1, J. Z. Zhao1, Lei
Zhao43, Ling Zhao1, M. G. Zhao29, Q. Zhao1, Q. W. Zhao1, S. J. Zhao50, T. C.
Zhao1, Y. B. Zhao1, Z. G. Zhao43, A. Zhemchugov22,g, B. Zheng44, J. P. Zheng1,
Y. H. Zheng39, B. Zhong27, L. Zhou1, Li Zhou29, X. Zhou48, X. K. Zhou43, X. R.
Zhou43, X. Y. Zhou1, K. Zhu1, K. J. Zhu1, S. Zhu1, X. L. Zhu37, Y. C. Zhu43,
Y. S. Zhu1, Z. A. Zhu1, J. Zhuang1, 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 Bochum Ruhr-University, D-44780 Bochum, Germany
4 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
5 Central China Normal University, Wuhan 430079, People’s Republic of China
6 China Center of Advanced Science and Technology, Beijing 100190, People’s
Republic of China
7 COMSATS Institute of Information Technology, Lahore, Defence Road, Off
Raiwind Road, 54000 Lahore, Pakistan
8 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090,
Russia
9 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
10 Guangxi Normal University, Guilin 541004, People’s Republic of China
11 GuangXi University, Nanning 530004, People’s Republic of China
12 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
13 Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz,
Germany
14 Henan Normal University, Xinxiang 453007, People’s Republic of China
15 Henan University of Science and Technology, Luoyang 471003, People’s
Republic of China
16 Huangshan College, Huangshan 245000, People’s Republic of China
17 Hunan University, Changsha 410082, People’s Republic of China
18 Indiana University, Bloomington, Indiana 47405, USA
19 (A)INFN Laboratori Nazionali di Frascati, I-00044, Frascati, Italy; (B)INFN
and University of Perugia, I-06100, Perugia, Italy
20 (A)INFN Sezione di Ferrara, I-44122, Ferrara, Italy; (B)University of
Ferrara, I-44122, Ferrara, Italy
21 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45,
D-55099 Mainz, Germany
22 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
23 Justus Liebig University Giessen, II. Physikalisches Institut, Heinrich-
Buff-Ring 16, D-35392 Giessen, Germany
24 KVI-CART, University of Groningen, NL-9747 AA Groningen, The Netherlands
25 Lanzhou University, Lanzhou 730000, People’s Republic of China
26 Liaoning University, Shenyang 110036, People’s Republic of China
27 Nanjing Normal University, Nanjing 210023, People’s Republic of China
28 Nanjing University, Nanjing 210093, People’s Republic of China
29 Nankai University, Tianjin 300071, People’s Republic of China
30 Peking University, Beijing 100871, People’s Republic of China
31 Seoul National University, Seoul, 151-747 Korea
32 Shandong University, Jinan 250100, People’s Republic of China
33 Shanxi University, Taiyuan 030006, People’s Republic of China
34 Sichuan University, Chengdu 610064, People’s Republic of China
35 Soochow University, Suzhou 215006, People’s Republic of China
36 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
37 Tsinghua University, Beijing 100084, People’s Republic of China
38 (A)Ankara University, Dogol Caddesi, 06100 Tandogan, Ankara, Turkey;
(B)Dogus University, 34722 Istanbul, Turkey; (C)Uludag University, 16059
Bursa, Turkey
39 University of Chinese Academy of Sciences, Beijing 100049, People’s
Republic of China
40 University of Hawaii, Honolulu, Hawaii 96822, USA
41 University of Minnesota, Minneapolis, Minnesota 55455, USA
42 University of Rochester, Rochester, New York 14627, USA
43 University of Science and Technology of China, Hefei 230026, People’s
Republic of China
44 University of South China, Hengyang 421001, People’s Republic of China
45 University of the Punjab, Lahore-54590, Pakistan
46 (A)University of Turin, I-10125, Turin, Italy; (B)University of Eastern
Piedmont, I-15121, Alessandria, Italy; (C)INFN, I-10125, Turin, Italy
47 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
48 Wuhan University, Wuhan 430072, People’s Republic of China
49 Zhejiang University, Hangzhou 310027, People’s Republic of China
50 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
b Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia and at the Functional Electronics Laboratory, Tomsk State University,
Tomsk, 634050, Russia
c Currently at Istanbul Arel University, Kucukcekmece, Istanbul, Turkey
d Also at University of Texas at Dallas, Richardson, Texas 75083, USA
e Also at the PNPI, Gatchina 188300, Russia
f Also at Bogazici University, 34342 Istanbul, Turkey
g Also at the Moscow Institute of Physics and Technology, Moscow 141700,
Russia

###### Abstract

Using data collected with the BESIII detector operating at the Beijing
Electron Positron Collider at center-of-mass energies of $\sqrt{s}=4.23$,
4.26, and 4.36 GeV, we observe $e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c}$ for the
first time. The Born cross sections are measured and found to be about half of
those of $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ within less than 2$\sigma$. In the
$\pi^{0}h_{c}$ mass spectrum, a structure at 4.02 GeV/$c^{2}$ is found. It is
most likely to be the neutral isospin partner of the $Z_{c}(4020)^{\pm}$
observed in the process of $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ is found. A fit
to the $\pi^{0}h_{c}$ invariant mass spectrum, with the width of the
$Z_{c}(4020)^{0}$ fixed to that of its charged isospin partner and possible
interferences with non-$Z_{c}(4020)^{0}$ amplitudes neglected, gives a mass of
($4023.9\pm 2.2\pm 3.8$) MeV/$c^{2}$ for the $Z_{c}(4020)^{0}$, where the
first error is statistical and the second systematic.

###### pacs:

14.40.Rt, 13.66.Bc, 14.40.Pq

In the study of $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$, a distinct charged
structure, $Z_{c}(3900)^{\pm}$, was observed in the $\pi^{\pm}J/\psi$ spectrum
by the BESIII ref1  and Belle ref2  experiments, and confirmed shortly
thereafter with CLEO-c data CLEO . A similar charged structure but with a
slightly higher mass, $Z_{c}(4020)^{\pm}$, was soon reported in
$e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ guoyp  by BESIII. As there are at least
four quarks within these two charmoniumlike structures, they are interpreted
as either tetraquark states, $D\bar{D}^{*}$ (or $D^{*}\bar{D}^{*}$) molecules,
hadrocharmonia, or other configurations ref4 . More recently, charged
structures in the same mass region were observed via their decays into charmed
meson pairs, including the charged $Z_{c}(4025)^{\pm}$ in
$e^{+}e^{-}\to\pi^{\pm}(D^{*}\bar{D}^{*})^{\mp}$ ref5  and the charged
$Z_{c}(3885)^{\pm}$ in $e^{+}e^{-}\to\pi^{\pm}(D\bar{D}^{*})^{\mp}$ ref5add .
These structures together with the recently confirmed $Z(4430)^{-}$ belle4430
; belle4430conf ; lhcb4430  and similar structures observed in the bottomonium
system ref6  indicate that a new class of hadrons has been observed. An
important question is whether all these charged structures are part of isospin
$I=1$ triplets, in which case neutral partners with $I_{z}=0$ should also be
found. Evidence for a neutral $Z_{c}(3900)$ was observed in
$e^{+}e^{-}\to\pi^{0}\pi^{0}J/\psi$ process with CLEO-c data at center-of-mass
energy (CME) $\sqrt{s}$=4.17 GeV CLEO . A neutral structure, the
$Z_{c}(4020)^{0}$, is expected to couple to the $\pi^{0}h_{c}$ final state and
be produced for in $e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c}$ processes.

In this Letter, we present the first observation of
$e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c}$ at $\sqrt{s}=4.23$ GeV, 4.26 GeV, and 4.36
GeV, and the observation of a neutral charmoniumlike structure
$Z_{c}(4020)^{0}$ in the $\pi^{0}h_{c}$ spectrum. We closely follow the
analysis of $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ guoyp  with the selection of
$\pi^{+}\pi^{-}$ replaced with the selection of a pair of $\pi^{0}$s. The data
samples were collected with the BESIII detector bepc . The CMEs and
corresponding integrated luminosities are listed in Table 1.

We use a GEANT4 based geant4  Monte Carlo (MC) simulation to optimize the
event selection criteria, determine the detection efficiency, and estimate
backgrounds. In the studies presented here, the $h_{c}$ is reconstructed via
its electric-dipole (E1) transition $h_{c}\to\gamma\eta_{c}$ with $\eta_{c}\to
X_{i}$, where $X_{i}$ denotes 16 hadronic final states: $p\bar{p}$,
$\pi^{+}\pi^{-}K^{+}K^{-}$, $\pi^{+}\pi^{-}p\bar{p}$, $2(K^{+}K^{-})$,
$2(\pi^{+}\pi^{-})$, $3(\pi^{+}\pi^{-})$, $2(\pi^{+}\pi^{-})K^{+}K^{-}$,
$K_{S}K^{\pm}\pi^{\mp}$, $K_{S}K^{\pm}\pi^{\mp}\pi^{+}\pi^{-}$,
$K^{+}K^{-}\pi^{0}$, $K^{+}K^{-}\eta$, $p\bar{p}\pi^{0}$,
$\pi^{+}\pi^{-}\eta$, $\pi^{+}\pi^{-}\pi^{0}\pi^{0}$, $2(\pi^{+}\pi^{-})\eta$,
and $2(\pi^{+}\pi^{-}\pi^{0})$. The initial state radiation (ISR) is simulated
with KKMC KKMC , where the Born cross section of
$e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c}$ is assumed to follow the
$e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ line-shape guoyp .

Table 1: Energies ($\sqrt{s}$), luminosities ($\cal L$), numbers of events ($n_{h_{c}}^{\rm obs}$), average efficiencies ($\sum\limits^{16}_{i=1}\epsilon_{i}{\cal B}(\eta_{c}\rightarrow X_{i})$), initial state radiative correction factor ($1+\delta^{r}$) guoyp , vacuum polarization factor ($1+\delta^{v}$), Born cross sections $\sigma^{\rm B}(e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c})$ and ratios ${\cal R}_{\pi\pi h_{c}}=\frac{\sigma(e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c})}{\sigma(e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c})}$, where the third errors are from the uncertainty in ${\cal B}(h_{c}\to\gamma\eta_{c})$ ref11 . $\sqrt{s}~{}(\rm GeV)$ | ${\cal L}~{}({\rm pb}^{-1})$ | $n_{h_{c}}^{\rm obs}$ | $\sum\limits^{16}_{i=1}\epsilon_{i}{\cal B}(\eta_{c}\rightarrow X_{i})$ | $1+\delta^{r}$ | $1+\delta^{v}$ | $\sigma^{\rm B}(e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c})~{}(\rm pb)$ | ${\cal R}_{\pi\pi h_{c}}$
---|---|---|---|---|---|---|---
4.230 | 1090.0 | $82.5\pm 15.6$ | $6.82\times 10^{-3}$ | 0.756 | 1.056 | $25.6\pm 4.8\pm 2.6\pm 4.0$ | $0.54\pm 0.11\pm 0.06$
4.260 | 826.8 | $62.8\pm 13.3$ | $6.54\times 10^{-3}$ | 0.831 | 1.054 | $24.4\pm 5.2\pm 3.2\pm 3.8$ | $0.63\pm 0.14\pm 0.10$
4.360 | 544.5 | $64.3\pm 11.5$ | $6.68\times 10^{-3}$ | 0.856 | 1.051 | $36.2\pm 6.5\pm 4.1\pm 5.7$ | $0.73\pm 0.14\pm 0.10$

The selection of charged tracks, photons, and $K_{S}^{0}\to\pi^{+}\pi^{-}$
candidates are described in Refs. ref13 ; guoyp . A candidate
$\pi^{0}~{}(\eta)$ is reconstructed from a pair of photons with an invariant
mass in the range $|M_{\gamma\gamma}-m_{\pi^{0}}|<$15 MeV/$c^{2}$
($|M_{\gamma\gamma}-m_{\eta}|<$15 MeV/$c^{2}$), where
$m_{\pi^{0}}~{}(m_{\eta})$ is the nominal $\pi^{0}~{}(\eta)$ mass ref14 . The
event candidates of $e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c}$,
$h_{c}\to\gamma\eta_{c}$ are required to have at least one
$\gamma\pi^{0}\pi^{0}$ combination with the mass recoiling against
$\pi^{0}\pi^{0}$, $M_{\pi^{0}\pi^{0}}^{\rm recoil}$, in the $h_{c}$ mass
region ($M_{\pi^{0}\pi^{0}}^{\rm recoil}\in[3.3,3.7]$ GeV/$c^{2}$) and with
the mass recoiling against $\gamma\pi^{0}\pi^{0}$,
$M_{\gamma\pi^{0}\pi^{0}}^{\rm recoil}$, in the $\eta_{c}$ mass region (
$M_{\gamma\pi^{0}\pi^{0}}^{\rm recoil}\in[2.8,3.2]$ GeV/$c^{2}$).

To determine the species of final state particles and to select the best
photon candidates when additional photons (and $\pi^{0}$ or $\eta$ candidates)
are found in an event, the combination with the minimum value of
$\chi^{2}=\chi^{2}_{4C}+\Sigma^{N}_{i=1}\chi^{2}_{\rm PID}(i)+\chi^{2}_{1C}$
is selected for further analysis. Here $\chi^{2}_{4C}$ is the $\chi^{2}$ of
the initial-final four-momentum conservation ($4C$) kinematic fit,
$\chi^{2}_{\rm PID}(i)$ is the $\chi^{2}$ of particle identification (PID) of
each charged track using the energy loss in the main drift chamber and the
time measured with the time-of-flight system, $N$ is the number of the charged
tracks, and $\chi^{2}_{1C}$ is the sum of the $1C$ $\chi^{2}$s of the
$\pi^{0}$s and $\eta$ in each final state with the invariant mass of the
daughter photon pair constrained to that of the parent. There is also a
$\chi^{2}_{4C}$ requirement, which is optimized by maximizing the figure of
merit $S/\sqrt{S+B}$, where $S$ and $B$ are the numbers of Monte Carlo (MC)
simulated signal and background events, respectively. The requirement
$\chi^{2}_{4C}<30$ has an efficiency of 82% for $\eta_{c}$ decays with only
charged or $K^{0}_{S}$ particles in the final states, while the requirement
$\chi^{2}_{4C}<25$ has an efficiency of 81% for the other decays corr . A
similar optimization is performed to determine the $\eta_{c}$ candidate mass
window around its nominal value, which is found to be $\pm 35$ MeV/$c^{2}$.
This mass window contains 77% of $\eta_{c}$ decays with only charged or
$K^{0}_{S}$ particles in final states and 74% for the other decays.

The inset of Fig. 1 shows the scatter plot of $M_{\gamma\pi^{0}\pi^{0}}^{\rm
recoil}$, which corresponds to the invariant mass of the reconstructed
$\eta_{c}$ candidate, versus $M_{\pi^{0}\pi^{0}}^{\rm recoil}$, which
corresponds to the invariant mass of the reconstructed $h_{c}$ candidate,
summed over the events at $\sqrt{s}=4.23$, 4.26, and 4.36 GeV, where a clear
cluster of events corresponding to the $h_{c}\to\gamma\eta_{c}$ signal is
observed. Figure 1 shows the projection of the invariant mass distribution of
$\gamma\eta_{c}$ candidates for events in the $\eta_{c}$ signal region
($M_{\gamma\pi^{0}\pi^{0}}^{\rm recoil}\in[2.945,3.015]$ GeV/$c^{2}$ ), where
a clear peak at the $h_{c}$ mass is observed. The events in the sideband
regions, 2.865 GeV/c2 $<M_{\gamma\pi^{0}\pi^{0}}^{\rm recoil}<$ 2.900 GeV/c2
and 3.050 GeV/c2$<M_{\gamma\pi^{0}\pi^{0}}^{\rm recoil}<$ 3.085 GeV/c2 are
used to study the background. To extract the number of $\pi^{0}\pi^{0}h_{c}$
signal events, the $M^{\rm recoil}_{\pi^{0}\pi^{0}}$ mass spectrum is fitted
with a MC simulated signal shape convolved with a Gaussian function to
represent the data-MC mass resolution difference, together with a linear
background term. A simultaneous fit to the $M^{\rm recoil}_{\pi^{0}\pi^{0}}$
mass spectrum summed over the 16 $\eta_{c}$ decay modes at the three CME
points yields the numbers of $\pi^{0}\pi^{0}h_{c}$ signal events
($n_{h_{c}}^{\rm obs}$) listed in Table 1. Figure 1 also shows the fit results
summed over the three CME points.

Figure 1: The $M^{\rm recoil}_{\pi^{0}\pi^{0}}$ distribution for the events
with an $\eta_{c}$ candidate. The plot shows the sum over three CME points.
Dots with error bars are data; the solid curve is the best fit; the dashed
black line is the background; the green shaded histogram shows the normalized
$\eta_{c}$ sideband events. The inset shows the scatter plot of
$M_{\gamma\pi^{0}\pi^{0}}^{\rm recoil}$ versus $M_{\pi^{0}\pi^{0}}^{\rm
recoil}$. The two red dashed lines represent the signal region of $\eta_{c}$.

The Born cross section $\sigma^{\rm B}(e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c})$ is
calculated with the formula

$\displaystyle\sigma^{\rm B}$
$\displaystyle(e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c})=$
$\displaystyle\textstyle\frac{n^{\rm obs}_{h_{c}}}{{\cal
L}(1+\delta^{r})(1+\delta^{v})\sum\limits^{16}_{i=1}\epsilon_{i}{\cal
B}(\eta_{c}\to X_{i}){\cal B}(h_{c}\to\gamma\eta_{c})},$ (1)

where $n^{\rm obs}_{h_{c}}$ is the number of observed $h_{c}$ signal events;
${\cal L}$ is the integrated luminosity; $(1+\delta^{r})$ is the initial
radiative correction factor, which is taken to be the same as that for the
analysis of $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ guoyp ; $(1+\delta^{v})$ is the
vacuum polarization factor vacu ; $\epsilon_{i}$ is the detection efficiency
for the $i^{th}$ $\eta_{c}$ decay mode in the decay
$e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c}$ without consideration of any possible
intermediate structures and with ISR and vacuum polarization effects
considered in the MC simulation; ${\cal B}(\eta_{c}\rightarrow X_{i})$ is the
corresponding $\eta_{c}$ branching fraction; ${\cal
B}(h_{c}\to\gamma\eta_{c})$ is the branching fraction of
$h_{c}\rightarrow\gamma\eta_{c}$.

The measured Born cross sections are listed in Table 1. The ratios of the Born
cross sections for the neutral and charged $e^{+}e^{-}\to\pi\pi h_{c}$ modes
are also listed in Table 1; the cross sections for the charged channel are
taken from Ref. guoyp , where vacuum polarization effects were not taken into
account. A corresponding correction factor $(1+\delta^{v})$ is applied to the
previous Born cross section. The common systematic uncertainties in the two
measurements cancel in the ratio calculation. The combined ratio ${\cal
R}_{\pi\pi h_{c}}$ is obtained with a weighted least squares method xx  and
determined to be $(0.63\pm 0.09)$, which is within 2$\sigma$ of the
expectation of isospin symmetry, 0.5.

Systematic uncertainties in the cross section measurement mainly come from the
luminosity measurement ($\delta_{\cal L}$), branching fraction of
$h_{c}\to\gamma\eta_{c}$, branching fractions of $\eta_{c}\to X_{i}$,
detection efficiencies ($\delta_{\epsilon_{i}\cdot{\cal B}(\eta_{c}\rightarrow
X_{i})}$), radiative correction factors ($\delta_{\rm ISR}$), vacuum
polarization factors ($\delta_{\rm Vac}$) vacu , and fits to the mass
spectrum. The integrated luminosity at each CME points is measured using
large-angle Bhabha events and has an estimated uncertainty of 1.0%. The
$h_{c}\to\gamma\eta_{c}$ and $\eta_{c}\to X_{i}$ branching fractions are taken
from Refs. ref11 ; ref13 , and the uncertainties in the radiative correction
are the same as those used in the analysis of
$e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ guoyp . The uncertainties in the vacuum
polarization factor are 0.5% vacu . The detection efficiency uncertainty
estimates are done with the same way as described in Refs. ref13 ; ref16 . The
uncertainty due to the $\eta_{c}$ mass ($\delta_{\eta_{c}-\rm mass}$) is
estimated by changing its mass by $\pm$1$\sigma$ of its world average value
ref14 ; the uncertainties due to the background shapes ($\delta_{\rm bkg}$)
are estimated by changing the background function from a first-order to a
second-order polynomial; the uncertainty from the mass resolution
($\delta_{\rm res}$) is estimated by varying the mass resolution difference
between data and MC simulation by one standard deviation; the uncertainty from
fit range ($\delta_{\rm fit}$) is estimated by extending the fit range; the
uncertainty from the $\pi^{0}\pi^{0}h_{c}$ substructure ($\delta_{\rm sub}$)
is estimated by considering the efficiency with and without the inclusion of a
$Z_{c}(4020)^{0}$. The contribution from each source of systematic error are
listed in Table 2.

Assuming all of the above uncertainties are independent, the total systematic
uncertainties in the $e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c}$ cross section
measurements are determined to be between 10% and 13%. The uncertainty in
${\cal B}(h_{c}\to\gamma\eta_{c})$, not listed in Table 2 but common to all
CME points, is 15.7% ref14  and is quoted separately in the cross section
measurement.

Table 2: The systematic uncertainties (%) in $\sigma^{\rm B}(e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c})$. $\sqrt{s}$ (GeV) | $\delta_{\cal{L}}$ | $\delta_{\rm fit}$ | $\delta_{\rm res}$ | $\delta_{\rm bkg}$ | $\delta_{\eta_{c}-\rm mass}$ | $\delta_{\rm sub}$ | $\delta_{\rm ISR}$ | $\delta_{\rm Vac}$ | $\delta_{\epsilon_{i}{\cal B}(\eta_{c}\rightarrow X_{i})}$
---|---|---|---|---|---|---|---|---|---
4.230 | 1.0 | 1.3 | 0.9 | 5.9 | 1.6 | 2.1 | 2.2 | 0.5 | 7.2
4.260 | 1.0 | 0.9 | 0.4 | 9.5 | 4.8 | 1.6 | 2.3 | 0.5 | 7.3
4.360 | 1.0 | 1.0 | 0.1 | 7.1 | 4.6 | 0.6 | 0.4 | 0.5 | 7.2

Intermediate states are studied by examining the $M^{\rm recoil}_{\pi^{0}}$
distribution (which corresponds to the reconstructed $\pi^{0}h_{c}$ invariant
mass) for the selected $\pi^{0}\pi^{0}h_{c}$ candidate events. The $h_{c}$
signal events are selected by requiring $M_{\pi^{0}\pi^{0}}^{\rm recoil}$ in
the range of [3.51, 3.55], and events in the sideband regions [3.45, 3.49] and
[3.57, 3.61] are used to study the background. From the two combinations of
the $\pi^{0}$ recoil mass in each event, we retain the one with the larger
$\pi^{0}$ recoil mass value, and denote this as $M^{\rm
recoil}_{\pi^{0}}|_{\rm max}$. Figure 2 shows the $M^{\rm
recoil}_{\pi^{0}}|_{\rm max}$ distribution for the signal events where there
is an obvious peak near 4.02 GeV/$c^{2}$, which corresponds to the expected
position of a $Z_{c}(4020)^{0}$ signal.

Figure 2: Sum of the simultaneous fit to the $M^{\rm recoil}_{\pi^{0}}|_{\rm
max}$ distribution at $\sqrt{s}=4.23$, 4.26 and 4.36 GeV as described in the
text. Dots with errors bars are data; the green shaded histogram shows the
normalized $h_{c}$ sideband events; the black dashed curve is the background
from the fit; the red histogram shows the result from a phase space MC
simulation. The solid blue line shows the total fit.

An unbinned maximum likelihood fit is applied to the $M^{\rm
recoil}_{\pi^{0}}|_{\rm max}$ distribution summed over all 16 $\eta_{c}$ decay
modes. The data at $\sqrt{s}=4.23$, 4.26, and 4.36 GeV are fitted
simultaneously with the same signal function with common mass and width. The
signal shape is parametrized with a constant-width relativistic Breit-Wigner
function convolved with a Gaussian-distributed mass resolution, where the mass
resolution is determined from a fit to a MC sample with the width set to zero.
Because of the limited statistics of the $Z_{c}(4020)^{0}$ signal, its width
is fixed to that of its charged partner, ($7.9\pm 2.6$) MeV guoyp . Assuming
the spin and parity of the $Z_{c}(4020)^{0}$ are $1^{+}$, a phase space factor
$pq^{3}$ is included in the partial width, where $p$ is the $Z_{c}(4020)^{0}$
momentum in the $e^{+}e^{-}$ rest frame and $q$ is the $h_{c}$ momentum in the
$Z_{c}(4020)^{0}$ rest frame.

There are two types of backgrounds in the $M^{\rm recoil}_{\pi^{0}}|_{\rm
max}$ distribution. One is the non-$h_{c}$ background in the $h_{c}$ signal
region, which can be represented by the $h_{c}$ sideband events, and the other
is the non-$Z_{c}(4020)^{0}$ $\pi^{0}\pi^{0}h_{c}$ events that may come from
three-body $\pi^{0}\pi^{0}h_{c}$ decays or from production of intermediate
scalar states, such as the $f_{0}(980)$, that decay into $\pi^{0}\pi^{0}$.
Since the widths of the low-mass scalar particles are large, these
non-$Z_{c}(4020)^{0}$ $\pi^{0}\pi^{0}h_{c}$ events can be reasonably well
described with a phase space distribution. For the non-$h_{c}$ background, a
comparison of the $h_{c}$ sideband events with the simulated phase space
events indicates that it can also be described with a three-body phase space
distribution. Thus, in the fit all of the background sources are described
with a single MC-simulated phase space shape with a total normalization that
is left as a free parameter. In the fit, the signal shape mentioned above is
multiplied by the efficiency, which depends on $M^{\rm recoil}_{\pi^{0}}|_{\rm
max}$. Interference between the signal and background is neglected.

The solid curve in Fig. 2 shows the fit results, which yields a
$Z_{c}(4020)^{0}$ mass of $4023.9\pm 2.2$ MeV/$c^{2}$. The mass difference
between neutral and charged $Z_{c}(4020)$ is $1.0\pm 2.3$ (stat.) MeV/$c^{2}$,
which agrees with zero within error. By projecting the events into a histogram
with 50 bins, the goodness of the fit is calculated from the combined
$\chi^{2}$ values, the number of bins and the number of free parameters at
three CME points, and found to be $\chi^{2}/n.d.f.=28.6/33$. Here the event
number in each bin used in the $\chi^{2}$ evaluation is required to be larger
than 7. The statistical significance of the $Z_{c}(4020)^{0}$ signal is
determined from a comparison of the fit likelihoods with and without the
signal. Additional fit are also performed with different signal shapes, and
background shapes. In all cases, the minimum significance is found to be above
$5\sigma$. The numbers of $Z_{c}(4020)^{0}$ signal events are listed in Table
3.

The Born cross section $\sigma^{\rm
B}(e^{+}e^{-}\to\pi^{0}Z_{c}(4020)^{0}\to\pi^{0}\pi^{0}h_{c})$ is calculated
with eq. 1, with the measured numbers of observed signal and MC-determined
detection efficiencies for the $\pi^{0}Z_{c}(4020)^{0}$ channel.

Table 3: Energies ($\sqrt{s}$), numbers of events ($n_{Z_{c}(4020)^{0}}^{\rm obs}$), initial state radiative correction factor ($1+\delta^{r}$) guoyp , vacuum polarization factor ($1+\delta^{v}$), average efficiencies ($\sum\limits^{16}_{i=1}\epsilon_{i}{\cal B}(\eta_{c}\rightarrow X_{i})$), Born cross sections $\sigma(e^{+}e^{-}\to\pi^{0}Z_{c}(4020)^{0}\to\pi^{0}\pi^{0}h_{c})$, and ratios ${\cal R}_{\pi Z_{c}(4020)}=\frac{\sigma(e^{+}e^{-}\to\pi^{0}Z_{c}(4020)^{0}\to\pi^{0}\pi^{0}h_{c})}{\sigma(e^{+}e^{-}\to\pi^{\pm}Z_{c}(4020)^{\mp}\to\pi^{\pm}\pi^{\mp}h_{c})}$, where the third errors are from the uncertainty in ${\cal B}(h_{c}\to\gamma\eta_{c})$ ref13 . $\sqrt{s}~{}({\rm GeV})$ | $n_{Z_{c}(4020)^{0}}^{\rm obs}$ | $(1+\delta^{r})$ | $1+\delta^{v}$ | $\sum\limits^{16}_{i=1}\epsilon_{i}{\cal B}(\eta_{c}\rightarrow X_{i})$ | $\sigma^{\rm B}(e^{+}e^{-}\to\pi^{0}Z_{c}(4020)^{0}\to\pi^{0}\pi^{0}h_{c})~{}({\rm pb})$ | $R_{\pi Z_{c}(4020)}$
---|---|---|---|---|---|---
4.230 | $21.7\pm 7.4$ | 0.756 | 1.056 | $7.08\times 10^{-3}$ | $6.5\pm 2.2\pm 0.7\pm 1.0$ | $0.77\pm 0.31\pm 0.25$
4.260 | $22.5\pm 7.7$ | 0.831 | 1.054 | $6.72\times 10^{-3}$ | $8.5\pm 2.9\pm 1.1\pm 1.3$ | $1.21\pm 0.50\pm 0.38$
4.360 | $17.2\pm 7.2$ | 0.856 | 1.051 | $6.56\times 10^{-3}$ | $9.9\pm 4.1\pm 1.3\pm 1.5$ | $1.00\pm 0.48\pm 0.32$
Table 4: Systematic uncertainties (%) in the $\sigma(e^{+}e^{-}\to\pi^{0}Z_{c}(4020)^{0}\to\pi^{0}\pi^{0}h_{c}$) measurement, in addition to the common part of those in $\sigma(e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c}$). $\sqrt{s}$ (GeV) | $\delta_{\rm signal}$ | $\delta_{\rm bkg}$ | $\delta_{\rm res}$ | $\delta_{h_{c}-\rm signal}$ | $\delta_{\epsilon_{\rm curve}}$ | $\delta_{\rm MC-model}$
---|---|---|---|---|---|---
4.230 | 0.3 | 5.8 | 0.5 | 5.1 | 0.3 | 0.6
4.260 | 1.1 | 3.5 | 0.2 | 8.6 | 0.3 | 0.6
4.360 | 0.8 | 4.8 | 0.2 | 3.5 | 0.0 | 0.6

The systematic uncertainties on the $Z_{c}(4020)^{0}$ mass come from
uncertainties in the mass calibration and energy scale, parametrizations of
the signal and background shapes, mass dependence of the efficiency, width
assumption, MC modeling with a different $J^{P}$ value, and mass resolution.
The uncertainty from the mass calibration is estimated by using the
difference, $(2.3\pm 1.5)$ MeV/c2, between the measured and known $h_{c}$
mass. The uncertainty from the photon energy scale is estimated with
$\psi^{\prime}\to\gamma\chi_{c1,2},\chi_{c1,2}\to\gamma
J/\psi,J/\psi\to\mu^{+}\mu^{-}$ for photons with low energy, and with
radiative Bhabha processes for photons with high energy ref11 . After
adjusting the MC energy scale accordingly, the resulting changes in the mass
of $Z_{c}(4020)^{0}$ are negligible. The $J^{P}$ value of $Z_{c}(4020)^{0}$ is
uncertain; two possible alternatives, $J^{P}=1^{-}$ and $2^{+}$, are used to
estimate the corresponding systematic errors. A difference of 0.4 MeV/$c^{2}$
in the $Z_{c}(4020)^{0}$ mass is found under different $J^{P}$ assumptions.
The uncertainty due to the background shape is determined by changing the
phase space shape to a parametrized background function,
$f(M)=[(M-M_{a})^{1/2}+c_{1}(M-M_{a})^{3/2}]\times[(M_{b}-M)^{1/2}+c_{2}(M_{b}-M)^{3/2}]$.
Here $M$ is mass of the background, $M_{a}$ and $M_{b}$ are the two extreme
points determined by the minimal and maximal mass. $f(M)=0$ for $(M-M_{a})<0$
or $(M_{b}-M)<0$. The coefficients $c_{1}$ and $c_{2}$ are determined by the
fit ref5add . A difference of 0.1 MeV/$c^{2}$ is found and taken as the
systematic uncertainty. The uncertainty due to the mass dependence of the
efficiency is determined by assuming a uniform efficiency in the whole $M^{\rm
recoil}_{\pi^{0}}|_{\rm max}$ recoil mass region, and the difference is found
to be negligible. The uncertainty due to the mass resolution is estimated by
varying the data-MC difference in resolution by one standard deviation of the
measured uncertainty in the mass resolution of the $h_{c}$ signal; the
difference in the $Z_{c}(4020)^{0}$ mass is negligible. Similarly, the
uncertainty due to the fixed $Z_{c}(4020)^{0}$ width is estimated by varying
the width determined for its charged partner by one standard deviation. The
difference is 0.1 MeV/$c^{2}$ and is taken as the systematic error. Assuming
all the sources of the systematic uncertainty are independent, the total
systematic error is estimated to be 3.8 MeV/$c^{2}$.

The systematic uncertainties in the measured Born cross section,
$\sigma(e^{+}e^{-}\to\pi^{0}Z_{c}(4020)^{0}\to\pi^{0}\pi^{0}h_{c})$, are
estimated in the same way as for $e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c}$. In
addition to those common parts in the $e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c}$
measurement, the uncertainties due to signal parametrization ($\delta_{\rm
signal}$), background shape ($\delta_{\rm bkg}$), $h_{c}$ signal window
selection ($\delta_{h_{c}-\rm{signal}}$), mass resolution ($\delta_{\rm
res}$), efficiency ($\delta_{\epsilon_{\rm curve}}$), and MC model
($\delta_{\rm MC-model}$) are considered; their values are summarized in Table
4.

The ratios of Born cross section for $e^{+}e^{-}\to\pi Z_{c}(4020)\to\pi\pi
h_{c}$ between neutral and charged modes at three center-of-mass energies are
listed in Table 3. Similar to the calculation of the
$\sigma(e^{+}e^{-}\rightarrow\pi^{0}\pi^{0}h_{c})$ ratio, the same correction
factor $(1+\delta^{v})$ is also applied to the previously measured
$e^{+}e^{-}\rightarrow\pi^{\pm}Z_{c}(4020)^{\mp}$ Born cross section. The
common systematic uncertainty between neutral and charged mode cancel. The
combined ratio ${\cal R}_{\pi Z_{c}(4020)}$ is determined to be $(0.99\pm
0.31)$ with the same method as for the combined ${\cal R}_{\pi\pi h_{c}}$,
which is well within 1$\sigma$ of the expectation of isospin symmetry, 1.0.

In summary, we observe $e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c}$ at $\sqrt{s}=4.23$,
4.26, and 4.36 GeV for the first time. The measured Born cross sections are
about half of those for $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$, and agree with
expectations based on isospin symmetry within systematic uncertainties. A
narrow structure with a mass of $(4023.9\pm 2.2\pm 3.8)$ MeV/$c^{2}$ is
observed in the $M^{\rm recoil}_{\pi^{0}}|_{\rm max}$ mass spectrum. This
structure is most likely the neutral isospin partner of the charged
$Z_{c}(4020)$ observed in the $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ process guoyp
. This observations indicate that there is no anomalously large isospin
violations in $\pi\pi h_{c}$ and $\pi Z_{c}(4020)$ system.

The BESIII collaboration thanks the staff of BEPCII and the IHEP computing
center for their strong support. This work is supported in part by National
Key Basic Research Program of China under Contract No. 2015CB856700; Joint
Funds of the National Natural Science Foundation of China under Contracts Nos.
11079008, 11179007, U1232201, U1332201; National Natural Science Foundation of
China (NSFC) under Contracts Nos. 10935007, 11121092, 11125525, 11235011,
11079023, 11322544, 11335008; the Chinese Academy of Sciences (CAS) Large-
Scale Scientific Facility Program; CAS under Contracts Nos. KJCX2-YW-N29,
KJCX2-YW-N45; 100 Talents Program of CAS; German Research Foundation DFG under
Contract No. Collaborative Research Center CRC-1044; Istituto Nazionale di
Fisica Nucleare, Italy; Ministry of Development of Turkey under Contract No.
DPT2006K-120470; Russian Foundation for Basic Research under Contract No.
14-07-91152; U. S. Department of Energy under Contracts Nos. DE-
FG02-04ER41291, DE-FG02-05ER41374, DE-FG02-94ER40823, DESC0010118; U.S.
National Science Foundation; University of Groningen (RuG) and the
Helmholtzzentrum fuer Schwerionenforschung GmbH (GSI), Darmstadt; WCU Program
of National Research Foundation of Korea under Contract No.
R32-2008-000-10155-0.

## References

  * (1) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 110, 252001 (2013).
  * (2) Z. Q. Liu et al. (Belle Collaboration), Phys. Rev. Lett. 110, 252002 (2013).
  * (3) T. Xiao, S. Dobbs, A. Tomaradze and K. K. Seth, Phys. Lett. B 727, 366 (2013).
  * (4) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 111, 242001 (2013).
  * (5) J. Z. Bai et al. (BES Collaboration), Nucl. Instrum. Meth. A 344, 319 (1994); Nucl. Instrum. Meth. A 458, 627 (2001).
  * (6) S. Agostinelli et al. (GEANT4 Collaboration), Nucl. Instrum. Meth. A 506, 250 (2003).
  * (7) S. Jadach, B.F.L. Ward and Z. Was, Comput. Phys. Commun. 130, 260 (2000); Phys. Rev. D 63, 113009 (2001).
  * (8) Q. Wang, C. Hanhart, and Q. Zhao, Phys. Rev. Lett. 111, 132003 (2013); F.-K. Guo, C. Hidalgo-Duque, J. Nieves, and M. PavonValderrama, Phys. Rev. D 88, 054007 (2013); G. Li, Eur. Phys. J. C 73, 2621 (2013); C.-Y. Cui, Y.-L. Liu, W.-B. Chen, and M.-Q. Huang, Eur. Phys. J. C 73, 12 (2013); J.-R. Zhang, Phys. Rev. D 87, 116004 (2013); J.M. Dias, F. S. Navarra, M. Nielsen, and C. M. Zanetti, Phys. Rev. D 88, 016004 (2013); M.B. Voloshin, Phys. Rev. D 87, 091501 (2013); E. Braaten, Phys. Rev. Lett. 111, 162003 (2013); E. Wilbring, H.W. Hammer, and U. G. Meißner, Phys. Lett. B 726, 326 (2013); D.-Y. Chen, X. Liu, and T. Matsuki, Phys. Rev. D 88, 036008 (2013); K. Terasaki, arXiv:1304.7080; Y.-R. Liu, Phys. Rev. D 88, 074008 (2013); Q. Wang, C. Hanhart, and Q. Zhao, Phys. Lett. B 725, 106 (2013); Y. Dong, A. Faessler, T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D 88, 014030 (2013); X.-H. Liu and G. Li, Phys. Rev. D 88, 014013 (2013); S. Prelovsek and L. Leskovec, Phys. Lett. B 727, 172 (2013). C.-F. Qiao and L. Tang, Eur. Phys. J. C 74, 2810 (2014). F. Aceti, M. Bayar and E. Oset, Eur. Phys. J. A 50, 103 (2014).
  * (9) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 112, 132001 (2014).
  * (10) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 112, 022001 (2014).
  * (11) S. K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. 100, 142001 (2008).
  * (12) K. Chilikin et al. (Belle Collaboration), Phys. Rev. D 88, 074026 (2013).
  * (13) R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 112, 222002 (2014).
  * (14) A. Bondar et al. (Belle Collaboration), Phys. Rev. Lett. 108, 122001 (2012).
  * (15) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 86, 092009 (2012).
  * (16) J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, 010001 (2012).
  * (17) A track parameter correction procedure ref16  is applied to the MC simulated charged tracks to improve the agreement between the data and MC simulation of the $\chi^{2}_{4C}$ distribution of the kinematic fit. The correction factors are obtained from a high purity control sample $J/\psi\to\phi f_{0}(980)$ with $\phi\rightarrow K^{+}K^{-}$ and $f_{0}(980)\rightarrow\pi^{+}\pi^{-}$. Reasonable agreement between the data and MC simulation is observed for the channels analyzed in this work. The systematic error of the $\chi^{2}_{4C}$ requirement is taken as half of the correction in the efficiency, which is much larger than the effect due to the uncertainties in the correction factors.
  * (18) S. Actis et al., Eur. Phys. J. C 66, 585 (2010).
  * (19) J. Mandel, The Statistical Analysis of Experimental Data, Dover Publications (1964).
  * (20) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. Lett. 104, 132002 (2010).
  * (21) M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 87, 012002 (2013).


# Belief Revision from Probability

Jeremy Goodman School of Philosophy
University of Southern California, USA<EMAIL_ADDRESS>Faculty of
Philosophy
University of Oxford, UK Bernhard Salow Faculty of Philosophy
University of Oxford, UK<EMAIL_ADDRESS>
###### Abstract

In previous work ([6, 7]), we develop a question-relative, probabilistic
account of belief. On this account, what someone believes relative to a given
question is (i) closed under entailment, (ii) sufficiently probable given
their evidence, and (iii) sensitive to the relative probabilities of the
answers to the question. Here we explore the implications of this account for
the dynamics of belief. We show that the principles it validates are much
weaker than those of orthodox theories of belief revision like AGM [2], but
still stronger than those valid according to the popular Lockean theory of
belief [5], which equates belief with high subjective probability. We then
consider a restricted class of models, suitable for many but not all
applications, and identify some further natural principles valid on this
class. We conclude by arguing that the present framework compares favorably to
the rival probabilistic accounts of belief developed by Leitgeb [14, 15] and
Lin and Kelly [18].

##  1 Probability Structures

We will work with the following simplification of the models in [6]:

######  Definition 1.1.

A _probability structure_ is a tuple $\langle S,\mathcal{E},Q,Pr,t\rangle$
such that:

  1. 1.

$S$ is a non-empty set (of _states_),

  2. 2.

$\mathcal{E}\subseteq\mathcal{P}(S)\backslash\\{\emptyset\\}$ (the _possible
bodies of evidence_),

  3. 3.

$Q$ (the _question_) is a partition of $S$,

  4. 4.

$Pr$ (the _prior_) is a probability distribution over $S$, and

  5. 5.

$t\in[0,1]$ (the _threshold_)

Propositions are modeled as subsets of $S$, where $p$ is true in $s$ if and
only if $s\in p$. We say that $E^{\prime}\in\mathcal{E}$ is the result of
discovering $p$ in $E\in\mathcal{E}$ just in case $E^{\prime}=E\cap p$; this
will allow us to talk about how beliefs evolve in response to changes in one’s
evidence.

Which propositions an agent believes is a function of their evidence and is
also given by a set of states, so that an agent with evidence $E$ believes $p$
if and only if $B(E)\subseteq p$. This ensures that their beliefs are closed
under entailment, and thus already marks a departure from popular ‘Lockean’
accounts of belief [5], according to which one believes a proposition if and
only if its probability exceeds a particular threshold. But it is compatible
with the more plausible direction of Lockeanism, namely:

  * threshold: You believe $p$ only if $p$ is sufficiently probable given your evidence.

If $B(E)\subseteq p$, then $Pr(p|E)\geq t$.

We can think of the members of the question $Q$ as its answers; we write
$[s]_{Q}$ for the member of $Q$ containing $s$. The proposal in [6] then boils
down to claiming that $s\in B(E)$ if and only if $s\in E$ and the answers to
$Q$ that are more probable than $[s]_{Q}$ have total probability less than the
threshold $t$. Writing $Pr_{E}$ for $Pr(\cdot|E)$, this can be formalized as
follows:

######  Definition 1.2.

$B(E)=\\{s\in
E:Pr_{E}(\\{s^{\prime}:Pr_{E}([s^{\prime}]_{Q})>Pr_{E}([s]_{Q})\\})<t\\}$

This means that one believes as much as possible subject to two constraints:
(i) threshold, and (ii) that the totality of one’s beliefs corresponds to the
conjunction of one’s evidence with a disjunction of answers to $Q$ that
includes any answer at least as probable (given one’s evidence) as any other
it includes. One notable attraction of this proposal is that what one believes
corresponds to the discrete analogue of the highest posterior-density region
typically used to define ‘credible intervals’ from probability density
functions in Bayesian statistics. A logically significant feature of the
proposal, to which we will return later, is that it involves not only local
probability comparisons between different answers to $Q$, but also a global
probability comparison between a collection of such answers and the threshold
$t$.

##  2 Principles and Results

A core idea behind the orthodox AGM [2] theory of belief revision is that
belief revisions are trivial whenever what you learn is compatible with your
initial beliefs: you should simply add the discovery to your beliefs, draw out
the logical consequences of these beliefs, and leave everything else
unchanged. Here we will focus on five principles that encode various aspects
of this idea. Exploring when and how these principles can fail will be a
useful way of exploring the extent to which our account of belief requires
departing from orthodoxy when it comes to belief dynamics. These principles
are:111The $\Diamond$ indicates that the discovery is compatible with your
initial beliefs, while the $\Box$ indicate that it is something you initially
believe. $\Diamond-$ is often referred to as ‘preservation’; [20] call $\Box-$
‘weak preservation’ and $\Box R$ ‘very weak preservation’. If we interpret the
non-monotonic consequence relation $p\,\mid\\!\sim\,q$ as saying that
$B(p)\subseteq q$, then $\Diamond-$ corresponds to ‘rational monotony’,
$\Box+$ to ‘cut’, and $\Box-$ to ‘cautious monotony’ in the standard
terminology from [13].

  * $\Diamond-$

If you don’t believe not-$p$ and then discover $p$, you shouldn’t give up any
beliefs.

If $B(E)\cap p\neq\emptyset$, then $B(E\cap p)\subseteq B(E)$.

  * $\Diamond R$

If you don’t believe not-$p$ and then discover $p$, you shouldn’t reverse any
of your beliefs (i.e. go from believing something to believing its negation).

If $B(E)\cap p\neq\emptyset$, then $B(E)\cap B(E\cap p)\neq\emptyset$.

  * $\Box+$

If you believe $p$ and then discover $p$, you shouldn’t form any new beliefs.

If $B(E)\subseteq p$, then $B(E)\subseteq B(E\cap p)$.

  * $\Box-$

If you believe $p$ and then discover $p$, you shouldn’t give up any beliefs.

If $B(E)\subseteq p$, then $B(E\cap p)\subseteq B(E)$

  * $\Box R$

If you believe $p$ and then discover $p$, you shouldn’t reverse any of your
beliefs.

If $B(E)\subseteq p$, then $B(E)\cap B(E\cap p)\neq\emptyset$.

These principles are not logically independent: the $\Diamond$ principles
entail the corresponding $\Box$ principles, and the $+$ and $-$ principles
each entail the corresponding $R$ principles. All of them are valid according
to AGM. By contrast, only $\Box R$ is valid according to Lockean theories that
equate believing a proposition with assigning it a sufficiently high
probability (for some probability threshold less than 1), and it is valid only
if this probability threshold is above $\frac{\sqrt{5}-1}{2}\approx.62$ (as
discussed in [20]).

The present account falls in between these extremes:

######  Proposition 1.

$\Box-$ and $\Box R$ are valid on the class of probability structures.

######  Proposition 2.

$\Diamond-$, $\Diamond R$, and $\Box+$ can all fail in probability structures.

We will illustrate Proposition 2 with two examples. Consider a much discussed
thought experiment:

> Flipping for Heads
> A coin flipper will flip a fair coin until it lands heads.

A natural model of this case is as follows:

> $S=\\{s_{1},s_{2},\ldots\\}$ | $\mathcal{E}=\\{\\{s_{i},s_{i+1},s_{i+2},\dots\\}:s_{i}\in S\\}$
> ---|---
> $Q=\\{\\{s_{i}\\}:s_{i}\in S\\}$ | $Pr(\\{s_{i}\\})=\frac{1}{2^{i}}$
> $t=.99$ |

Here $s_{i}$ is the state in which the coin lands heads on the $i$th flip, and
$\\{s_{i},s_{i+1},s_{i+2},\dots\\}$ is your evidence if you have just observed
the coin land tails on the first $i-1$ flips. The question $Q$ is maximally
fine-grained, and the probabilities match the known objective chances.

In this model,
$B(\\{s_{i},s_{i+1},s_{i+2},\dots\\})=\\{s_{i},s_{i+1},\ldots,s_{i+6}\\}$: you
always believe that the coin will land heads within the next seven flips.
$\Diamond-$ is violated whenever you observe the coin land tails. For example,
let $p=\\{s_{2},s_{3},\dots\\}$. Then $B(S)\cap p\neq\emptyset$, but $B(S\cap
p)=\\{s_{2},\dots,s_{8}\\}\not\subseteq\\{s_{1},\dots,s_{7}\\}=B(S)$. We think
this is exactly the right prediction.

To turn this into a counterexample to $\Box+$, we add new body of evidence
$E^{\prime}=\\{s_{1},s_{2},\ldots,s_{7}\\}$ to $\mathcal{E}$. Intuitively, we
can think of this as the evidence you receive if you walk away from the
experiment before the first flip, and are later told that the coin landed
heads within the first seven flips. It is easy to verify that
$B(E^{\prime})=\\{s_{1},\dots,s_{6}\\}$. So $B(S)\not\subseteq B(S\cap
E^{\prime})$, even though $B(S)\subseteq E^{\prime}$. That this can happen
should be unsurprising in a framework like ours in which agents have
‘inductive’ beliefs that go beyond what is strictly entailed by their
evidence: discovering something that you previously believed only inductively
will strengthen your evidence, putting you in a position to draw further
inductive conclusions.

Counterexamples to $\Diamond R$ are subtler, for reasons we will explain in
the next section. But here is one:

> Drawing a Card
> You are holding a deck of cards, which is either a fair deck consisting of
> 52 different cards or a trick deck consisting of 52 Aces of Spades. Your
> background evidence makes it 90% likely that the deck is fair. You draw a
> card at random; it is an Ace of Spades.

Here is a possible model of the example:

> $S=\\{F_{1},F_{2},\ldots,F_{52},T\\}$ | $\mathcal{E}=\\{S,\\{F_{1}\\},\ldots,\\{F_{51}\\},\\{F_{52},T\\}\\}$
> ---|---
> $Q=\\{\\{F_{1},F_{2},\ldots,F_{52}\\},\\{T\\}\\}$ | $Pr(\\{F_{i}\\})=\frac{.9}{52}\approx.017,Pr(\\{T\\})=.1$
> $t=.85$ |

The states $F_{i}$ are all states in which the deck is fair; they are
distinguished only by which card you will draw, with $F_{52}$ being the one
where you draw the Ace of Spades. State $T$ is the state in which the deck is
the trick deck (and you thus draw an Ace of Spades). Your evidence settles all
and only what card you drew; so when you draw an Ace of Spades, it leaves open
both that you did so by chance and that you did so because it is a trick deck.
The question is simply whether the deck is fair. It is easy to see that,
according to this model, you should initially believe only that the deck is
fair. Your initial beliefs are thus compatible with it being fair and you
drawing the Ace of Spades by chance. Yet when you discover that you drew an
Ace of Spades, you should reverse your opinion and conclude that you’re
holding the trick deck, since
$Pr(\\{T\\}|\\{F_{52},T\\})\approx\frac{.1}{.1+.017}\approx.855>t$.

Note that, in this model, your discovery is not a disjunction of answers to
the question $Q$. If we changed the question to a more fine-grained one, so
that your discovery was a disjunction of its answers, then the case would no
longer yield a counterexample to $\Diamond R$. For example, relative to the
question is the deck fair and will I draw an Ace of Spades – i.e. relative to
$Q^{\prime}=\\{\\{F_{1},F_{2},\ldots,F_{51}\\},\\{F_{52}\\},\\{T\\}\\}$ – you
will initially believe that you won’t draw an Ace of Spades, in which case
your subsequent discovery isn’t compatible with your initial beliefs. And
relative to the question is the deck fair and what will I draw – i.e. relative
to the maximally fine-grained $Q^{\prime\prime}=\\{\\{s\\}:s\in S\\}$ – you
will initially have no non-trivial beliefs, and in particular you won’t start
out believing that the deck is fair. In the next section, we will see that
this is part of a more general pattern about counterexamples to $\Diamond R$.

##  3 Orthogonality

In the previous section, we saw that some of the surprising belief dynamics in
probability structures depended on discoveries that cross-cut the question.
Notice that structures in which this cannot happen, because every member of
$\mathcal{E}$ is the union of some subset of $Q$, satisfy the following
constraint:

  * orthogonality: $\frac{Pr([s]_{Q})}{Pr([s^{\prime}]_{Q})}=\frac{Pr([s]_{Q}|E)}{Pr([s^{\prime}]_{Q}|E)}$ for all $s,s^{\prime}\in E\in\mathcal{E}$ s.t. $Pr([s^{\prime}]_{Q}|E)>0$

This says that the only way that getting new evidence can change the relative
probability of two answers to $Q$ is by completely ruling out one of those
answers. While we can ensure orthogonality by making the question fine-grained
enough to capture all possible discoveries, this isn’t always necessary. For
example, we could fine-grain the states and bodies of evidence in our model of
Flipping for Heads to capture the fact that you discover where on the table
the coin lands. The bodies of evidence in such a fine-grained model will
cross-cut the question _how many time will the coin be flipped_ ; but,
plausibly, orthogonality will still hold for this question, since the added
information about where the coin lands is probabilistically independent from
how many times it will be flipped.

orthogonality is interesting because it leads to a stronger logic of belief
revision. Firstly,

######  Proposition 3.

$\Diamond R$ is valid on the class of probability structures satisfying
orthogonality.

Secondly, consider the following principle. It says (roughly) that if you’re
sure that, whatever you’re about to discover, you won’t believe a given
proposition afterwards, then you already don’t believe it:

  * $\Pi-$

If $\Pi$ is a partition any member of which you could discover, then there is
a $p\in\Pi$ such that you shouldn’t give up any beliefs upon discovering $p$.

    * If $\Pi\subseteq\mathcal{E}$ is a partition of $E$, then $B(E\cap p)\subseteq B(E)$ for some $p\in\Pi$.

We then have the following result:

######  Proposition 4.

$\Pi-$ can fail in probability structures. But it is valid on the class of
probability structures satisfying orthogonality.

It is also worth noting that $\Diamond-$ and $\Box+$ can still fail in
structures satisfying orthogonality. In particular, orthogonality holds in the
structures we used in the last section to argue that Flipping for Heads yields
counterexamples to $\Diamond-$ and $\Box+$.

In our view, a good deal of ordinary talk about what people believe is well-
modelled by structures satisfying orthogonality. This is because we think that
the question $Q$, to which attributions of belief are implicitly relativized,
typically coincides with the question under discussion in the conversational
context in which those attributions are made. Moreover, when a discovery is
salient, it is natural to consider a question that is sufficiently fine-
grained to capture all the aspects of this discovery that are relevant to its
answers. Counterexamples to orthogonality (and thus to $\Diamond R$ and
$\Pi-$) therefore tend to be ‘elusive’ in Lewis’s [17] sense: attending to
these cases often changes the context in such a way that they can no longer be
described as counterexamples.

That being said, we do not think that orthogonality is plausible as a general
constraint. This is because, very often, the only way to ensure orthogonality
is to adopt a very fine-grained question; and, often, such fine-grained
questions make overly skeptical predictions about what we can believe.
Consider, for example, the following case:

> One Hundred Flips
> You will flip a fair coin 100 times and watch how it lands each time.

There are natural contexts in which you can be correctly described as
initially believing that the coin will not land heads more than 90 times. Our
theory predicts this for various natural questions, even for very high
thresholds $t$ – for example, the polar question will the coin land heads more
than 90 times or the slightly more fine-grained question how often will the
coin lands heads. But neither of these questions satisfies orthogonality. For
example, discovering that coin lands tails on the first flip will favor ‘no’
over ‘yes’ for the first question, and ‘51’ over ‘49’ for the second question,
without ruling out any of these answers. In fact, the only natural question
that satisfies orthogonality is the maximally fine-grained question what will
the exact sequence of heads and tails be. But all answers to this question are
equally likely, and so this question prevents you from having any non-trivial
beliefs about what will happen.

We conclude that orthogonality should be rejected as a general constraint,
even if it will often hold when we are considering a particular case with a
limited range of discoveries. $\Diamond R$ and $\Pi-$ are thus not fully
general principles of belief revision; but counterexamples are likely to be
difficult to pin down.

orthogonality is also a fruitful principle in that it helps to facilitate
comparisons between our framework and other probabilistic theories of belief.
Let us now turn to these.

##  4 Comparisons

In this section, we consider two influential probabilistic accounts from the
literature, and compare them with our own account. The first can be seen as a
version of our account with an additional constraint imposed on probability
structures, and validates $\Diamond-$ but not $\Box+$; the second can be seen
as defining belief from probability structures in a related but different way,
and validates $\Box+$ but not $\Diamond-$.

###  4.1 A Stability Theory

The first theory we want to consider is inspired by Leitgeb’s _stability
theory of belief_ [14, 15]. The guiding idea behind this theory is a
probabilistic analogue of $\Diamond-$ that Leitgeb calls the _Humean thesis_.
However, despite the ‘stablity’ moniker, the constraints imposed by Leitgeb’s
theory are _synchronic_ ones relating probabilities, partitions, and
thresholds at a single time. So both to facilitate comparison with our
framework, and to be (in our view) more faithful to its motivating idea, we
will consider a strengthening of Leitgeb’s theory according to which the
requirements it imposes on one’s beliefs prior to a discovery continue to hold
after one has made that discovery. We can then interpret the view as proposing
the following constraint on probability structures:222Our stability
strengthens Leitgeb’s theory in two ways: first, by identifying the threshold
that characterizes the minimal probability of anything one believes with the
threshold in terms of which stability is defined, and, second, by not allowing
this threshold to be different for different possible bodies of evidence. It
also departs from his formulation in quantifying over $\mathcal{E}$ rather
than $\\{\bigcup Y:Y\subseteq Q\\}$; however, we read him as identifying
$\mathcal{E}$ with $\\{\bigcup Y:\emptyset\neq Y\subseteq Q\\}$, so this is
not a substantive departure.

  * stability: For all $E\in\mathcal{E}$ and $X\subseteq Q$, if $Pr(\bigcup X)\geq t$ and $E\cap\bigcup X\neq\emptyset$, then $Pr(\bigcup X|E)\geq t$.

We then have the following results:

######  Proposition 5.

$\Diamond-$ is valid in probability structures satisfying stability and
orthogonality. But $\Box+$ can still fail; and $\Diamond-$ can fail in
structures satisfying stability but not orthogonality.

This illustrates how a kind of qualitative stability of belief can be secured
by a kind of probabilistic stability (given orthogonality), without entailing
the full strength of AGM.333Leitgeb [15, chapter 4] describes his theory as
compatible with AGM (and thus with $\Box+$) since, upon getting new evidence,
one may adopt a different, higher threshold than before. But doing so is in no
way required by the demands of stability.

We reject stability because we reject $\Diamond-$ (even in cases where
orthogonality holds), and along with it the informal idea that rational belief
should be stable in anything like the way that Leitgeb claims it should be.
stability also places implausible constraints on what agents can believe at a
given time. For example, [12] show, in effect, that in Flipping for Heads
stability entails that the only way to have any non-trivial beliefs about how
many times the coin will be flipped is to believe that it will be flipped only
once. (This argument depends only on the symmetries of the example, and
doesn’t depend on whether the coin is fair, biased towards heads, or biased
towards tails.) See also [19] and [4].

###  4.2 The Tracking Theory

Lin and Kelly [18] defend a theory which (for reasons we can’t explain here)
they call the ‘tracking theory’ of belief. This theory can be seen as an
alternative way of defining belief in probability structures, with the
parameter $t$ playing a rather different role. Put informally, a state $s$ is
compatible with your LK-beliefs if there is no answer to $Q$ that is more than
$\frac{1}{t}$ times more likely than $[s]_{Q}$. Formally:

######  Definition 4.1.

$B_{LK}(E)=\\{s\in E:(\forall q\in Q)(Pr_{E}([s]_{Q})\geq t\times
Pr_{E}(q))\\}$

In many cases – such as Flipping for Heads – the subject will have similar
beliefs according to our theory and according to Lin and Kelly’s (provided $t$
is chosen judiciously: low values of $t$ for Lin and Kelly correspond to high
values of $t$ for us). However, there are important structural differences
between the theories. In particular, LK-beliefs are sensitive only to local
comparisons of probability between particular answers, while beliefs as we
understand them depend also the probabilities of sets of answers. A
consequence of this locality is that, as Lin and Kelly note, their theory
validates a reasonably strong theory of belief revision (assuming
orthogonality, which they essentially build in):

######  Proposition 6.

$\Box+$, $\Box-$, $\Box R$, $\Diamond R$, and $\Pi-$ are all valid for LK-
belief on the class of probability structures satisfying orthogonality.

The major shortcoming of the tracking theory, in our view, is that it fails to
entail threshold. Consider a case like Drawing a Card, in which one state
initially has very low probability ($.1$) but every other state has even lower
probability ($.017$). Then relative to a fine-grained question such as is the
deck fair and which card will you draw, you will LK-believe that the deck is a
trick deck even for reasonably low values of $t$ (such as $.2$). But this
belief is only $.1$ likely on your evidence! And we can, of course, make the
case more extreme by increasing the number of distinct cards in the fair deck;
so the believed proposition can be arbitrarily improbable for any fixed value
of $t$.

One might defend the tracking theory against such cases by insisting that we
choose a more coarse-grained question; while the theory still fails to entail
threshold, this response at least prevents it from recommending the extreme
violations just discussed. However, moving to coarser-grained questions is
often in conflict with orthogonality. Moreover, the reasons we gave previously
for rejecting orthogonality as a general constraint applies to the tracking
theory as well: just like our theory, the tracking theory will make
implausibly skeptical predictions in One Hundred Flips unless combined with an
orthogonality-violating question such as how many heads will there be.

Without orthogonality, the dynamics of LK-belief are substantially less
constrained:

######  Proposition 7.

$\Box+$ and $\Box-$ are valid for LK-belief on the class probability
structures. $\Diamond R$ and $\Pi-$ can both fail in such structures.

$\Box+$ is then the only principle valid for LK-beliefs but not for beliefs as
we understand them.

Moreover, without orthogonality, the tracking theory invalidates a new
principle that holds for belief as we understand it (assuming we restrict to
probability structures with $t>.5$). Consider the following variant of Drawing
a Card (taken from [9], who also makes parallel observations as an objection
to Levi’s [16] account of belief):

> Drawing a Card v.2
> You are holding a deck which could be either a ‘fair’ deck of 52 different
> cards, or one of 52 different ‘trick’ decks that just contain the same card
> 52 times. Given your background evidence, the probability that you are
> holding the fair deck is $\frac{1}{5}$, with the remaining $\frac{4}{5}$
> distributed evenly across the 52 trick decks. You are about to draw and turn
> over one card from your deck.

Let us assume that $Q$ is _which of the 53 possible decks am I holding_ and
$t>.25$. According to the tracking theory, you initially believe that you hold
the fair deck, but after drawing a card you believe that you are holding the
relevant trick deck. So we have a failure of the following principle, which
says (roughly) that if you’re sure that, whatever you’re about to discover,
you’ll believe that a given proposition is false, then don’t currently believe
that the proposition is true:

  * $\Pi R$

If $\Pi$ is a partition any member of which you could discover, there is a
$p\in\Pi$ such that you shouldn’t reverse any of your beliefs upon discovering
$p$.

    * If $\Pi\subseteq\mathcal{E}$ is a partition of $E$, then $B(E)\cap B(E\cap p)\neq\emptyset$ for some $p\in\Pi$.

By contrast, if belief requires probability over a threshold greater than $.5$
(as it does on our account), this principle cannot fail.444Failures of $\Pi R$
are to be expected for certain notions of belief that are weaker than the one
we are operating with here. For example, your ‘best guess’ about what deck you
are holding plausibly does change no matter what card you draw; and arguably
what we ‘believe’ (in ordinary English) often aligns with our best guesses.
See [11] and [3] for discussion.

Overall, then, we see few advantages for the tracking theory over our own.
Given orthogonality, which Lin and Kelly essentially build into their
formalism, the tracking theory offers a stronger theory of belief revision.
However, the theory violates threshold, often in dramatic ways. Moreover, to
make reasonable predictions in cases like One Hundred Flips, both theories
need to appeal to coarse-grained questions that conflict with orthogonality.
Having done so, both theories invalidate many principles of belief revision,
although the details differ slightly (with our theory invalidating $\Box+$ and
Lin and Kelly’s invalidating $\Pi R$).

##  5 Further work

We conclude with three directions for further work. One concerns nonmonotonic
consequence, where $p\,\mid\\!\sim\,q$ is interpreted as $B(p)\subseteq q$. We
think that distinguishing one’s evidence from one’s beliefs that go beyond
one’s evidence offers a productive way of thinking about nonmonotonic
consequence, and that the logic resulting from our framework contrasts in
interesting ways with the one resulting from Lockean theories of belief
(explored in [10]).

The second direction concerns constraints on $\mathcal{E}$. Consider, for
example, the Monty Hall problem, in which it is crucial that when one gets new
evidence about one’s environment, one also gets evidence that one has gotten
such evidence. We argue in [8] that such cases motivate a _nestedness_
requirement on $\mathcal{E}$: if two possible bodies of evidence are mutually
consistent, then one entails the other. This requirement induce new subtleties
in the resulting nonmonotonic logic.

A third question for future work concerns what happens when probability
structures are generalized by making the relevant question a function of one’s
evidence. [6, Appendix C] motivate this generalization, in order to vindicate
certain judgments about a family of examples discussed in [7]. We hope to
explore these models in future work; one notable feature is that they
invalidate $\Box-$ but still validate $\Pi R$.

## Acknowledgements

We thank Kevin Dorst, Josh Pearson, and three anonymous referees for TARK for
very helpful comments on earlier versions of this material.

##  Appendix A Proofs

######  Proposition 1.

$\Box-$ and $\Box R$ are valid on the class of probability structures.

###### Proof.

Since $\Box-$ entails $\Box R$, it’s sufficient to prove the former. We
suppose that $B(E)\subseteq p$, and show that $B(E\cap p)\subseteq B(E)$.

Note that if $s\in B(E)$, $[s]_{Q}\subseteq B(E)\subseteq p$. So for any $q\in
Q$, if $Pr([s]_{Q}|E)\geq Pr(q|E)$, then also $Pr([s]_{Q}|E\cap p)\geq
Pr(q|E\cap p)$. Contraposing, this means that if $Pr(q|E\cap p)\geq
Pr([s]_{Q}|E\cap p)$ and $s\in B(E)$, then $Pr(q|E)\geq Pr([s]_{Q}|E)$, and so
$q\subseteq B(E)$.

Moreover, since $B(E)\subseteq p$, $Pr(B(E)|E\cap p)\geq Pr(B(E)|E)\geq t$.

Now, note that $B(E\cap p)$ is the minimal $X\subseteq E\cap p$ such that (i)
if $s\in X$ and $Pr(q|E\cap p)\geq Pr([s]_{Q}|E\cap p)$ for $q\in Q$, then
$q\subseteq X$, and (ii) $Pr(X|E\cap p)\geq t$. By the above, $B(E)$ satisfies
both (i) and (ii); so it contains the minimal such $X$ as a subset. So
$B(E\cap p)\subseteq B(E)$, as required.

∎

######  Proposition 2.

$\Diamond-$, $\Diamond R$, and $\Box+$ can all fail in probability structures.

###### Proof.

Counter-models are given in the main text. ∎

######  Proposition 3.

$\Diamond R$ is valid in probability structures satisfying orthogonality.

###### Proof.

Suppose that $B(E)\cap p\neq\emptyset$. Let $s\in B(E)\cap p$ be such that
$Pr_{E}([s]_{Q})\geq Pr_{E}([s^{\prime}]_{Q})$ for any $s^{\prime}\in B(E)\cap
p$. We will show that, given orthogonality, there can be no $q\in Q$ such that
$Pr_{E\cap p}(q)>Pr_{E\cap p}([s]_{Q})$. It follows that $s\in B(E\cap p)$,
thus establishing $B(E)\cap B(E\cap p)\neq\emptyset$.

By orthogonality, if $q\in Q$ and $Pr_{E\cap p}(q)>Pr_{E\cap p}([s]_{Q})$,
then either $Pr_{E}(q)>Pr_{E}([s]_{Q})$ or else $Pr_{E\cap p}([s]_{Q})=0$. But
$s\in E\cap p$, so $Pr_{E\cap p}([s]_{Q})\neq 0$. So suppose
$Pr_{E}(q)>Pr_{E}([s]_{Q})$. By the way $s$ was chosen, it follows that
$q\cap(B(E)\cap p)=\emptyset$. But $q\cap p\neq\emptyset$, since $Pr_{E\cap
p}(q)>0$. So $q\cap B(E)=\emptyset$. But since $s\in B(E)$, this contradicts
the assumption that $Pr_{E}(q)>Pr_{E}([s]_{Q})$.

∎

######  Proposition 4.

$\Pi-$ can fail in probability structures. But it is valid in probability
structures satisfying orthogonality.

###### Proof.

To see that $\Pi-$ can fail, consider

> $S=\\{s_{1},s_{2},s_{3},s_{4},s_{5},s_{6}\\}$ | $\mathcal{E}=\\{S,\\{s_{1},s_{3},s_{5}\\},\\{s_{2},s_{4},s_{6}\\}\\}$
> ---|---
> $Q=\\{\\{s_{1},s_{2}\\},\\{s_{3},s_{4}\\},\\{s_{5}\\},\\{s_{6}\\}\\}$ |  $Pr$ is uniform
> $t=.65$ |

Let $p=\\{s_{1},s_{3},s_{5}\\}$ and $\Pi=\\{p,S\setminus p\\}$. Then $B(S\cap
p)=\\{s_{1},s_{3},s_{5}\\}\not\subseteq\\{s_{1},s_{2},s_{3},s_{4}\\}=B(S)$ and
$B(S\cap S\setminus p)=\\{s_{2},s_{4},s_{6}\\}\not\subseteq B(S)$.

Now suppose $\langle S,\mathcal{E},Q,Pr,t\rangle$ satisfies orthogonality. To
show that $\Pi-$ holds, we suppose that $B(E\cap p_{i})\not\subseteq B(E)$ for
each $p_{i}\in\Pi$, and deduce a contradiction.

For each $i$, let $s_{i}\in B(E\cap p_{i})\setminus B(E)$ be such that
$Pr_{E\cap p_{i}}([s_{i}]_{Q})\geq Pr_{E\cap p_{i}}([s]_{Q})$ for every $s\in
B(E\cap p_{i})\setminus B(E)$. Since $s_{i}\in B(E\cap p_{i})$, $Pr_{E\cap
p_{i}}(\\{s:Pr_{E\cap p_{i}}([s]_{Q})\geq Pr_{E\cap p_{i}}([s_{i}]_{Q})\\}<t$.
By orthogonality, $Pr_{E}([s]_{Q})\geq Pr_{E}([s_{i}]_{Q})$ entails that
either $Pr_{E\cap p_{i}}([s]_{Q})\geq Pr_{E\cap p_{i}}([s_{i}]_{Q})$ or
$Pr_{E\cap p_{i}}([s]_{Q})=0$. So $Pr_{E\cap p_{i}}(\\{s:Pr_{E}([s]_{Q})\geq
Pr_{E}([s_{i}]_{Q})\\})=Pr_{E\cap p_{i}}(\\{s:Pr_{E\cap p_{i}}([s]_{Q})\geq
Pr_{E\cap p_{i}}([s_{i}]_{Q})\\})<t$.

Now let $k$ be such that, for every $i$, $Pr_{E}([s_{k}]_{Q})\geq
Pr_{E}([s_{i}]_{Q})$. Then $\\{s:Pr_{E}([s]_{Q})\geq
Pr_{E}([s_{k}]_{Q})\\}\subseteq\\{s:Pr_{E}([s]_{Q})\geq
Pr_{E}([s_{i}]_{Q})\\}$, and so $Pr_{E\cap p_{i}}(\\{s:Pr_{E}([s]_{Q})\geq
Pr_{E}([s_{k}]_{Q})\\})\leq Pr_{E\cap p_{i}}(\\{s:Pr_{E}([s]_{Q})\geq
Pr_{E}([s_{i}]_{Q})\\})<t$ for every $i$. But then by the law of total
probability, $Pr_{E}(\\{s:Pr_{E}([s]_{Q})\geq Pr_{E}([s_{k}]_{Q})\\})<t$,
contradicting the assumption that $s_{k}\notin B(E)$.

∎

######  Proposition 5.

$\Diamond-$ is valid in probability structures satisfying stability and
orthogonality; but $\Box+$ can fail in such structures. Moreover, $\Diamond-$
can fail in probability structures satisfying stability in which orthogonality
fails.

###### Proof.

To see that $\Diamond-$ holds, note that $B(E\cap p)$ is the minimal
$X\subseteq E\cap p$ such that (i) if $s\in X$ and $Pr(q|E\cap p)\geq
Pr([s]_{Q}|E\cap p)$ for $q\in Q$, then $q\subseteq X$, and (ii) $Pr(X|E\cap
p)\geq t$. Then if $B(E)\cap p\neq\emptyset$, $Pr(B(E)\cap p|E\cap
p)=Pr(B(E)|E\cap p)\geq t$ by stability, so $B(E)\cap p$ meets condition (ii).
Moreover, it meets condition (i) by orthogonality. So $B(E)\cap p$ contains
the minimal $X$ meeting (i) and (ii) as a subset. So $B(E\cap p)\subseteq
B(E)\cap p\subseteq B(E)$, as required.

To see how $\Box+$ can fail, let $S=\\{a,b,c\\}$,
$\mathcal{E}=\\{S,\\{a,b\\}\\}$, $Q=\\{\\{s\\}:s\in S\\}$, $Pr(\\{a\\})=.9$,
$Pr(\\{b\\})=.09$, $Pr(\\{c\\})=.01$, and $t=.9001$. This structure satisfies
stability. $\Box+$ fails, since $B(S)=\\{a,b\\}\not\subseteq
B(\\{a,b\\})=\\{a\\}$.

To see how $\Diamond-$ can fail in the absence of orthogonality, consider a
probability structure in which $Q=\\{A,B,C\\}$, $\mathcal{E}=\\{S,E\\}$,
$Pr(A)=\frac{1}{2},Pr(B)=\frac{1}{4}+\epsilon,Pr(C)=\frac{1}{4}-\epsilon$,
$Pr_{E}(A)=Pr_{E}(B)=Pr_{E}(C)=\frac{1}{3}$, and $t=\frac{1}{2}+\epsilon$.
stability hold, but $\Diamond-$ fails: $B(S)\cap E\neq\emptyset$, but
$B(E)=E\not\subseteq B(S)=A\cup B$. ∎

######  Proposition 6.

$\Box+$, $\Box-$, $\Box R$, $\Diamond R$, and $\Pi-$ are valid for LK-belief
on the class of probability structures satisfying orthogonality

###### Proof.

For $\Diamond R$, see the proof of Proposition 3; For $\Box+$ and $\Box-$ (and
hence $\Box R$), see the proof of Proposition 7; for $\Pi-$, see [8]. ∎

######  Proposition 7.

$\Box+$ and $\Box-$ are valid for LK-Belief on the class of probability
structures. $\Diamond R$ and $\Pi-$ can both fail in such structures.

###### Proof.

The failures of $\Diamond R$ and $\Pi-$ follow from the failure of $\Pi R$
described in the main text.

Suppose that $B_{LK}(E)\subseteq p$. We will show that $B_{LK}(E\cap
p)=B_{LK}(E)$, thus establishing $\Box+$ and $\Box-$.

Since $B_{LK}(E)\subseteq p$, we have that if $s\in B_{LK}(E)$, then
$[s]_{Q}\subseteq p$. So if $s\in B_{LK}(E)$ then $\frac{Pr([s]_{Q}|E\cap
p)}{Pr(q|E\cap p)}\geq\frac{Pr([s]_{Q}|E)}{Pr(q|E)}$ for any $q\in Q$ such
that $Pr(q|E\cap p)>0$. So if $s\in B_{LK}(E)$, then for any $q\in Q$ with
$Pr(q|E\cap p)>0$, $\frac{Pr([s]_{Q}|E\cap p)}{Pr(q|E\cap
p)}\geq\frac{Pr([s]_{Q}|E)}{Pr(q|E)}\geq t$. And if $Pr(q|E\cap p)=0$, then
trivially $Pr([s]_{Q}|E)\geq t\times Pr(q|E\cap p)$. So $s\in B_{LK}(E\cap
p)$.

Moreover, if $s\notin B_{LK}(E)$, then $t>0$ and there is a $q\in Q$ such that
(i) $Pr(q|E)\times t>Pr([s]_{Q}|E)$ and (ii) $q\cap B_{LK}(E)\neq\emptyset$.
Assuming $Pr([s]_{Q}|E\cap p)>0$ then, by the above, $\frac{Pr(q|E\cap
p)}{Pr([s]_{Q}|E\cap p)}\geq\frac{Pr(q|E)}{Pr([s]_{Q}|E)}>\frac{1}{t}$. In
that case $s\notin B_{LK}(E\cap p)$. And if $Pr([s]_{Q}|E\cap p)>0$ then also
$t\times Pr(q|E\cap p)>\times Pr([s]_{Q}|E\cap p)$. So in that case also
$s\notin B_{LK}(E\cap p)$.

So $B_{LK}(E\cap p)=B_{LK}(E)$, as required.

∎

## References

  * [1]
  * [2] Carlos Alchourrón, Peter Gärdenfors & David Makinson (1985): _On the Logic of Theory Change: Partial Meet Contraction and Revision Functions_.  Journal of Symbolic Logic 50, pp. 510–530, 10.2307/2274239.
  * [3] Kevin Dorst & Matthew Mandelkern (2023): _Good Guesses_.  Philosophy and Phenomenological Research 105(3), pp. 581–618, 10.1111/phpr.12831.
  * [4] Igor Douven & Hans Rott (2018): _From Probabilities to Categorical Beliefs: Going Beyond Toy Models_.  Journal of Logic and Computation 28(6), pp. 1099–1124, 10.1093/logcom/exy017.
  * [5] Richard Foley (1993): _Working without a net: A study of egocentric epistemology_.  Oxford University Press.
  * [6] Jeremy Goodman & Bernhard Salow (2021): _Knowledge from Probability_.  In Joseph Y. Halpern & Andrés Perea, editors: Proceedings Eighteenth Conference on Theoretical Aspects of Rationality and Knowledge, TARK 2021, Beijing, China, June 25-27, 2021, EPTCS 335, pp. 171–186, 10.4204/EPTCS.335.15.
  * [7] Jeremy Goodman & Bernhard Salow (2023): _Epistemology Normalized_.  Philosophical Review 132(1), pp. 89–145, 10.1215/00318108-10123787.
  * [8] Jeremy Goodman & Bernhard Salow (ms): _Belief revision normalized_.  Unpublished manuscript.
  * [9] Ian Hacking (1967): _Isaac Levi, “Gambling with Truth: An Essay on Induction and the Aims of Science” (Book Review)_.  Synthese 17, pp. 444–448, 10.2307/20114579.
  * [10] James Hawthorne (1996): _On the Logic of Nonmonotonic Conditionals and Conditional Probabilities_.  Journal of Philosophical Logic 25(2), pp. 185–218, 10.1007/bf00247003.
  * [11] Ben Holguín (2022): _Thinking, Guessing, and Believing_.  Philosophers’ Imprint 22(1), pp. 1–34, 10.3998/phimp.2123.
  * [12] Kevin Kelly & Hanti Lin (2021): _Beliefs, Probabilities, and Their Coherent Correspondence_.  In: Lotteries, Knowledge, and Rational Belief: Essays on the Lottery Paradox, Cambridge University Press, pp. 185–222, 10.1017/9781108379755.011.
  * [13] Sarit Kraus, Daniel Lehmann & Menachem Magidor (1990): _Nonmonotonic reasoning, preferential models, and cumulative logics_.  Artificial Intelligence 44, pp. 167–207, 10.1016/0004-3702(90)90101-5.
  * [14] Hannes Leitgeb (2014): _The Stability Theory of Belief_.  Philosophical Review 123, pp. 131–171, 10.1215/00318108-2400575.
  * [15] Hannes Leitgeb (2017): _The Stability of Belief: How Rational Belief Coheres with Probability_.  Oxford University Press, 10.1093/acprof:oso/9780198732631.001.0001.
  * [16] Isaac Levi (1967): _Gambling with Truth: An Essay on Induction and the Aims of Science_.  MIT Press.
  * [17] David Lewis (1996): _Elusive Knowledge_.  Australasian Journal of Philosophy 74, pp. 549–567, 10.1080/00048409612347521.
  * [18] Hanti Lin & Kevin Kelly (2012): _Propositional Reasoning That Tracks Probabilistic Reasoning_.  Journal of Philosophical Logic 41(6), pp. 957–981, 10.1007/s10992-012-9237-3.
  * [19] Hans Rott (2017): _Stability and scepticism in the modelling of doxastic states: Probabilities and plain beliefs_.  Minds and Machines 27, pp. 167–197, 10.1007/s11023-016-9415-0.
  * [20] Ted Shear & Branden Fitelson (2019): _Two Approaches to Belief Revision_.  Erkenntnis 84(3), pp. 487–518, 10.1007/s10670-017-9968-1.