Measurement of branching fractions for D meson decaying into ϕ meson and a pseudoscalar meson

(1)

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Measurement

of

branching

fractions

for

D meson

decaying

into

φ

meson

and

a

pseudoscalar

meson

BESIII

Collaboration

M. Ablikim

a

,

M.N. Achasov

j

,

4

,

P. Adlarson

bn

,

S. Ahmed

o

,

M. Albrecht

d

,

M. Alekseev

bk

,

bm

,

A. Amoroso

bk

,

bm

,

F.F. An

a

,

Q. An

bh

,

as

,

Y. Bai

ar

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O. Bakina

ac

,

R. Baldini Ferroli

w

,

I. Balossino

y

,

Y. Ban

ak

,

12

,

K. Begzsuren

aa

,

J.V. Bennett

e

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N. Berger

ab

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M. Bertani

w

,

D. Bettoni

y

,

F. Bianchi

bk

,

bm

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_{J. Biernat}

bn

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_{J. Bloms}

be

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_{I. Boyko}

ac

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bo

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X. Cai

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as

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A. Calcaterra

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G.F. Cao

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az

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N. Cao

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az

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S.A. Cetin

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J. Chai

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J.F. Chang

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W.L. Chang

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G. Chelkov

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2

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D.Y. Chen

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H.S. Chen

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J.C. Chen

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M.L. Chen

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S.J. Chen

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Y.B. Chen

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W. Cheng

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G. Cibinetto

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M.Y. Dong

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J.Z. Fan

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J. Fang

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S.S. Fang

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Y. Fang

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R. Farinelli

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L. Gong

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W.X. Gong

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W. Gradl

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F.A. Harris

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K.L. He

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F.H. Heinsius

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Y.K. Heng

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M. Himmelreich

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Y.R. Hou

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Z.L. Hou

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H.M. Hu

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J.F. Hu

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W. Ikegami Andersson

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W. Imoehl

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J. Libby

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C.X. Lin

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D.X. Lin

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Huanhuan Liu

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https://doi.org/10.1016/j.physletb.2019.135017

(2)

J.G. Messchendorp

ae

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G. Mezzadri

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J. Min

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T.J. Min

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R.E. Mitchell

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Y.J. Mo

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C. Morales Morales

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a

a_Institute_of_High_Energy_Physics,_Beijing_100049,_People’s_Republic_of_China b_Beihang_University,_Beijing_100191,_People’s_Republic_of_China

c_Beijing_Institute_of_{Petrochemical}_Technology,_Beijing_102617,_People’s_Republic_of_China d_Bochum_{Ruhr-University,}_D-44780_Bochum,_Germany

e_Carnegie_Mellon_University,_Pittsburgh,_{PA 15213,}_USA

f_Central_China_Normal_University,_Wuhan_430079,_People’s_Republic_of_China

g_China_Center_of_Advanced_Science_and_Technology,_Beijing_100190,_People’s_Republic_of_China

h_COMSATS_University_Islamabad,_Lahore_Campus,_Defence_Road,_Off_Raiwind_Road,₅₄₀₀₀_Lahore,_Pakistan i_Fudan_University,_Shanghai_200443,_People’s_Republic_of_China

j

G.I.BudkerInstituteofNuclearPhysicsSBRAS(BINP),Novosibirsk630090,Russia k_GSI_{Helmholtzcentre}_for_Heavy_Ion_Research_GmbH,_D-64291_Darmstadt,_Germany l_Guangxi_Normal_University,_Guilin_541004,_People’s_Republic_of_China

m_Guangxi_University,_Nanning_530004,_People’s_Republic_of_China n_Hangzhou_Normal_University,_Hangzhou_310036,_People’s_Republic_of_China o_Helmholtz_Institute_Mainz,_{Johann-Joachim-Becher-Weg}_45,_D-55099_Mainz,_Germany p_Henan_Normal_University,_Xinxiang_453007,_People’s_Republic_of_China

q_Henan_University_of_Science_and_Technology,_Luoyang_471003,_People’s_Republic_of_China r_Huangshan_College,_Huangshan_245000,_People’s_Republic_of_China

s_Hunan_Normal_University,_Changsha_410081,_People’s_Republic_of_China t_Hunan_University,_Changsha_410082,_People’s_Republic_of_China u_Indian_Institute_of_Technology_Madras,_Chennai_600036,_India v_Indiana_University,_Bloomington,_{IN 47405,}_USA

w_INFN_Laboratori_Nazionali_di_Frascati,_I-00044,_Frascati,_Italy x_INFN_and_University_of_Perugia,_I-06100,_Perugia,_Italy

(3)

y_INFN_Sezione_di_Ferrara,_I-44122,_Ferrara,_Italy z_University_of_Ferrara,_I-44122,_Ferrara,_Italy

aa_Institute_of_Physics_and_Technology,_Peace_Ave._54B,_Ulaanbaatar_13330,_Mongolia

ab_Johannes_Gutenberg_University_of_Mainz,_{Johann-Joachim-Becher-Weg}_45,_D-55099_Mainz,_Germany ac_Joint_Institute_for_Nuclear_Research,₁₄₁₉₈₀_Dubna,_Moscow_region,_Russia

ad_{Justus-Liebig-Universitaet}_Giessen,_II._{Physikalisches}_Institut,_{Heinrich-Buff-Ring}_16,_D-35392_Giessen,_Germany ae_KVI-CART,_University_of_Groningen,_NL-9747_AA_Groningen,_the_Netherlands

af_Lanzhou_University,_Lanzhou_730000,_People’s_Republic_of_China ag_Liaoning_University,_Shenyang_110036,_People’s_Republic_of_China ah_Nanjing_Normal_University,_Nanjing_210023,_People’s_Republic_of_China ai_Nanjing_University,_Nanjing_210093,_People’s_Republic_of_China aj_Nankai_University,_Tianjin_300071,_People’s_Republic_of_China ak_Peking_University,_Beijing_100871,_People’s_Republic_of_China al_Shandong_Normal_University,_Jinan_250014,_People’s_Republic_of_China am_Shandong_University,_Jinan_250100,_People’s_Republic_of_China

an_Shanghai_Jiao_Tong_University,_Shanghai_200240,_People’s_Republic_of_China ao_Shanxi_University,_Taiyuan_030006,_People’s_Republic_of_China

ap_Sichuan_University,_Chengdu_610064,_People’s_Republic_of_China aq_Soochow_University,_Suzhou_215006,_People’s_Republic_of_China ar

SoutheastUniversity,Nanjing211100,People’sRepublicofChina

as_State_Key_Laboratory_of_Particle_Detection_and_Electronics,_Beijing_100049,_Hefei_230026,_People’s_Republic_of_China at_Sun_Yat-Sen_University,_Guangzhou_510275,_People’s_Republic_of_China

au_Tsinghua_University,_Beijing_100084,_People’s_Republic_of_China av_Ankara_University,₀₆₁₀₀_Tandogan,_Ankara,_Turkey aw_Istanbul_Bilgi_University,₃₄₀₆₀_Eyup,_Istanbul,_Turkey ax_Uludag_University,₁₆₀₅₉_Bursa,_Turkey

ay_Near_East_University,_Nicosia,_North_Cyprus,_Mersin_10,_Turkey

az_University_of_Chinese_Academy_of_Sciences,_Beijing_100049,_People’s_Republic_of_China ba_University_of_Hawaii,_Honolulu,_{HI 96822,}_USA

bb_University_of_Jinan,_Jinan_250022,_People’s_Republic_of_China

bc_University_of_Manchester,_Oxford_Road,_Manchester,_M13_9PL,_United_Kingdom bd_University_of_Minnesota,_Minneapolis,_{MN 55455,}_USA

be_University_of_Muenster,_{Wilhelm-Klemm-Str.}_9,₄₈₁₄₉_Muenster,_Germany bf_University_of_Oxford,_Keble_Rd,_Oxford,_OX13RH,_United_Kingdom

bg_University_of_Science_and_Technology_Liaoning,_Anshan_114051,_People’s_Republic_of_China bh_University_of_Science_and_Technology_of_China,_Hefei_230026,_People’s_Republic_of_China bi_University_of_South_China,_Hengyang_421001,_People’s_Republic_of_China

bj_University_of_the_Punjab,_{Lahore-54590,}_Pakistan bk_University_of_Turin,_I-10125,_Turin,_Italy

bl_University_of_Eastern_Piedmont,_I-15121,_Alessandria,_Italy bm_INFN,_I-10125,_Turin,_Italy

bn_Uppsala_University,_Box_516,_SE-75120_Uppsala,_Sweden bo_Wuhan_University,_Wuhan_430072,_People’s_Republic_of_China bp_Xinyang_Normal_University,_Xinyang_464000,_People’s_Republic_of_China bq_Zhejiang_University,_Hangzhou_310027,_People’s_Republic_of_China br_Zhengzhou_University,_Zhengzhou_450001,_People’s_Republic_of_China

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Articlehistory:

Received29July2019

Receivedinrevisedform30September 2019

Accepted7October2019 Availableonline10October2019 Editor:L.Rolandi Keywords: BESIII D meson Hadronicdecays Branchingfractions

Thefourdecaymodes D0_{→ φ}

_π

0_,_D0_{→ φ}

_η

_,_D+_{→ φ}

_π

+_,_and _D+_{→ φ}_K+ _are_studied_by_using_a_data

sampletakenatthecentre-of-massenergy√s=3.773 GeV withtheBESIIIdetector,correspondingtoan integratedluminosityof2.93fb−1_._The_branching_fractions_of_the_ﬁrst_three_decay_modes_are_measured

tobeB(D0_{→ φ}

_π

0₎_{= (}₁_.₁₆₈_±₀_.₀₂₈_±₀_.₀₂₈₎_×₁₀−3_,_B(_D0_{→ φ}

_η

₎_{= (}₁_.₈₁_±₀_.₄₆_±₀_.₀₆₎_×₁₀−4_,_and

B(D+→ φ

π

+)= (5.70±0.05±0.13)×10−3,respectively, wheretheﬁrstuncertainties arestatistical and the secondare systematic. Inaddition,the upper limitofthe branchingfraction for D+→ φK+ isgiventobe2.1_×10−5_at_the_{90% conﬁdence}_level._The_ratio_of_B(_D0_{→ φ}

_π

0₎_to_B(_D+_{→ φ}

_π

+₎ _is

calculated to be(20.49±0.50±0.45)%, whichisconsistentwith the theoreticalprediction basedon isospinsymmetrybetweenthesetwodecaymodes.

*

Correspondingauthors.

E-mailaddresses:houyingrui16@mails.ucas.ac.cn(Y.R. Hou),linchx5@mail2.sysu.edu.cn(C.X. Lin).

1 _Also_at_Bogazici_University,₃₄₃₄₂_Istanbul,_Turkey.

2 _Also_at_the_Moscow_Institute_of_Physics_and_Technology,_Moscow_141700,_Russia.

3 _Also_at_the_Functional_Electronics_Laboratory,_Tomsk_State_University,_Tomsk,_634050,_Russia. 4 _Also_at_the_Novosibirsk_State_University,_Novosibirsk,_630090,_Russia.

5 _Also_at_the_NRC_“Kurchatov_{Institute”,}_PNPI,_188300,_Gatchina,_Russia. 6 _Also_at_Istanbul_Arel_University,₃₄₂₉₅_Istanbul,_Turkey.

7 _Also_at_Goethe_University_Frankfurt,₆₀₃₂₃_Frankfurt_am_Main,_Germany.

8 _Also_at_Key_Laboratory_for_Particle_Physics,_Astrophysics_and_Cosmology,_Ministry_of_Education;_Shanghai_Key_Laboratory_for_Particle_Physics_and_Cosmology;_Institute_of

(4)

Fig. 1. Feynman diagrams of four D→ φP decay modes.

Table 1

CurrentresultoftheratioofB(D0_{→ φ}_π0₎_to_B(D+_{→ φ}_π+₎_.

The ratio Experiment result (%) Prediction (%)

B(D0_→φπ0₎

B(D+→φπ+) 24.6±1.8 [5,6] 19.7±0.2 [11]

1. Introduction

Comprehensive andprecise measurements of hadronic D me-son decays provide important inputs for the experimental stud-ies of both charm andbeauty decays [1]. One category of decay modesD

→ φ

P ( P representsa pseudoscalarparticle) hassimple FeynmandiagramsasdepictedinFig.1.Thisfacilitatestheoretical predictionsandtheircomparisons [2,3] to experimental measure-ments.However, the experimental measurements of D

→ φ

P are still limited [4] due to the relative low branching fractions (BF) whichare suppressed by phasespace dueto the

φ

meson mass. The singly Cabibbo-suppressed (SCS) decays of D+

→ φ

π

+ [5], D0

_{→ φ}

_π

0_[₆_],_and_D0

_{→ φ}

_η

_[₇_{] have}_been_studied_by_CLEO,_BaBar

andBelle, respectively. The BF ofthe doubly Cabibbo-suppressed (DCS)decay D+

→ φ

K+isderivedoutaccordingtotwo measure-ments of the total BF for D+

→

K−K+K+ andthe intermediate fractionof

φ

K+atLHCb [8,9].

According to isospin symmetry between u and d quarks, the BFsforD0

_{→ φ}

_π

0_and_D+

_{→ φ}

_π

+_are_{connected [}₂_,₃_{] as}_follows:

B

(

D0

→ φ

π

0

₎

B

(

D+

→ φ

π

+

)

=

1 2

D+

_D0

=

1 2

τ

_D0

τ

D+

.

(1)

However,thecurrentexperimental resultfortheBFratiodeviates from prediction value of Eq. (1) by 2

.

7σ as shown in Table 1. Therefore, improved measurement is necessary to further test it andhelptounderstandthestronginteractioninD mesonhadronic decays.

Inthisanalysis, westudyfourtwo-bodydecaymodesofD

→

φ

P ,whichareD+

→ φ

K+,D+

→ φ

π

+,D0

_{→ φ}

_π

0_,_and_D0

_{→ φ}

_η,

basedonadatasetof2.93fb−1 [10] takenat

√

s

=

3

.

773 GeV with the BESIII detector. Due to energy conservation, the D and D

¯

mesons from e+e−

→ ψ(

3770

)

→

DD are

¯

always producedin a

9 _Also_at _Government_College_Women_University,_Sialkot_{- 51310,}_Punjab,

Pak-istan.

10 _Also_at_Key_Laboratory_of_Nuclear_Physics_and_Ion-beam_Application_(MOE)_and

InstituteofModernPhysics,FudanUniversity,Shanghai200443,People’sRepublic ofChina.

11 _Also_at_Harvard_University,_Department_of_Physics,_Cambridge,_MA,_02138,_USA. 12 _Also_at_State_Key_Laboratory_of_Nuclear_Physics_and_Technology,_Peking

Univer-sity,Beijing100871,People’sRepublicofChina.

pairwithoutanyotheraccompanyinghadrons.Throughoutthis pa-per,charge-conjugatemodesareimplied.

2. BESIIIdetectorandMonteCarlosimulation

The BESIII detectorisa magneticspectrometer [12] locatedat theBeijingElectronPositronCollider(BEPCII) [13].Thecylindrical core of the BESIII detector consistsof a helium-based multilayer drift chamber (MDC), a plastic scintillator time-of-flight system (TOF),andaCsI(Tl) electromagneticcalorimeter(EMC),whichare all enclosed in a superconducting solenoidal magnet providing a 1.0 T magnetic field. The solenoid is supported by an octagonal flux-returnyokewithresistiveplatecountermuonidentifier mod-ulesinterleavedwithsteel.Theacceptanceofchargedparticlesand photonsis93%ofthe4πsolidangle.Thecharged-particle momen-tum resolution at 1 GeV

/

c is 0

.

5%, and the speciﬁc energy loss (dE

/

dx)resolutionis6% forelectronsfromBhabhascattering.The EMC measures photon energies witha resolutionof 2

.

5% (5%)at 1 GeV inthe barrel(end cap) region. The time resolutionof the TOFbarrelsectionis68 ps,whilethatoftheendcapis110 ps.

Simulatedsamplesproducedwiththe geant4-based [14] Monte Carlo(MC)package,whichincludesthegeometricdescription [15,

16] of the BESIII detector and the detector response, are used to determine the detection eﬃciency and to estimate the back-grounds.Thesimulationincludesthebeamenergyspreadand ini-tialstate radiation(ISR) inthee+e− annihilationsmodelledwith thegenerator kkmc [17].TheinclusiveMC samplesconsist ofthe production of DD pairs,

¯

the non-DD decays

¯

of the

ψ(

3770

)

, the ISRproductionofthe J

/ψ

and

ψ(

3686

)

states,andthecontinuum processes incorporated in kkmc [17]. The equivalent luminosity of the inclusive MC samples is about 10 times that of the data. The known decay modes are modelled with evtgen [18] using branching fractions taken from the Particle Data Group [4], and theremaining unknowndecaysfromthecharmoniumstateswith lundcharm [19].The ﬁnalstate radiations (FSR)fromcharged ﬁ-nalstateparticlesareincorporatedwiththe photos (version2.02) package [20,21].Thesignalprocessesaregeneratedseparately tak-ing the spin-matrix elements into account in evtgen. For each signalchannel,200000eventsaresimulated.

3. Eventselection

Candidatesof the decaymodes D

→ φ

P are reconstructed by combining the ﬁnal states of K±,

π

±,

π

0_, _and

_η

_particles _with

BESIIIoﬄinesoftwaresystem [1,22],where

φ

mesonsaredetected via decaysto K+K−.Candidatesfor

π

0 _and

_η

_are_identiﬁed_from

(5)

Table 2

Foreachsignalmode,therequirementonE,signalyieldsNi

sig,MC-determineddetectioneﬃciencyεi,branchingfractionBiinthiswork,andthecorrespondingworld

resultsBext.

Decay mode E (GeV) Ni

sig ε i_(%) _Bi (×10−4 ) Bext(×10−4) D+→ φπ+ [−0.020,0.019] 17527±152 37.7±0.1 57.0±0.5±1.3 53.7±2.3 [4] D+→ φK+ [−0.019,0.018] 12+28 −12 23.7±0.1 0.062+0.144 −0.062±0.002 ₀_.₀₈₅_±₀_._{011 [}₄_,₈_,₉_] <0.21 at 90% CL D0_{→ φ}_π0 _[−₀_.₀₇₇_,₀_.₀₃₅_] ₃₃₃₃_±₇₆ ₂₇_.₇_±₀_.₁ ₁₁_.₆₈_±₀_.₂₈_±₀_.₂₈ ₁₃_.₂_±₀_._{8 [}₄_] D0_{→ φ}_η _[−₀_.₀₄₀_,₀_.₀₃₈_] ₁₀₂_±₂₆ ₁₃_.₇_±₀_.₁ ₁_.₈₁_±₀_.₄₆_±₀_.₀₆ ₁_.₄_±₀_._{5 [}₄_]

Fig. 2. Two-dimensional distributions of MBCand MKKin data for the four signal modes. Selectedchargedtracks mustsatisfy

|

cos

θ

|

<

0

.

93,where

θ

is

thepolaranglewithrespect tothebeamaxis.Thedistanceof clos-est approach of the track to the interaction point is required to be less than 10 cm in the beam direction and less than 1 cm in the plane perpendicular to the beam. Separation of charged kaons fromcharged pions isimplemented by combining the en-ergy loss (dE

/

dx) in the MDC andthe time-of-ﬂight information fromtheTOF.We calculatetheprobabilities P

(

K

)

andP

(

π

)

with the hypothesis of K or

π

, and require that K candidates have P

(

K

)

>

P

(

π

)

,while

π

candidateshaveP

(

π

)

>

P

(

K

)

.

Photoncandidatesareselectedfromneutralshowersdeposited intheEMCcrystals,withenergieslargerthan25MeVinthebarrel (

|

cos

θ

|

<

0

.

8) and50MeVintheendcap(0

.

86

<

|

cos

θ

|

<

0

.

92). To reduce fake photons due to beam background or electronic noise, the shower clusters are required to be within [0, 700] ns fromtheeventstarttime.Furthermore,thephotoncandidatesare requiredtobeatleast10◦awayfromanychargedtrackstoremove fakephotonscausedbytheinteractionsofhadronsintheEMC.

The

π

0

₍

_η

₎

_candidates_are _formed_with_pairs _of_photon

candi-dates,whoseinvariantmass, Mγ γ , isrequiredtobewithin[0.115, 0.150]([0.500,0.560]) GeV/c2.Toimprovemomentumresolution, a 1C kinematic ﬁtconstraining the reconstructed

π

0

₍

_η

₎

_mass _to

thenominalmass [4] isperformedandtheﬁttedfour-momentum ofthe

π

0

₍

_η

₎

_is_used_in_further_analysis.

4. Dataanalysis

In the rest frame of the initial e+e− system, the total colli-sionenergyissharedequallybytheDD pair.

¯

Hence,inthisframe twovariables,theenergydifference

E andthebeamconstrained massMBC relatedtoenergyandmomentumconservation,

respec-tively,aredeﬁnedas

E

≡

ED

−

√

s

/

2

,

MBC

≡

s

/

4c4

_{− |}

_p D

|

2

/

c2

,

where

pDisthemomentumoftheD candidate.

Signals for the four

φ

P decay modes are expected to peak aroundzeroin

E distributions andthe D nominalmassin MBC

distributions.Tosuppresscombinatorialbackground,the

E ofthe D candidatesarerequiredtobewithintheregionslistedinTable2

forthedifferentsignalmodes,whichcorrespondtoabout3σ cov-erage.The asymmetric boundariesofthe

E region forthe

φ

π

0

and

φ

η

modesareduetoenergyleakageintheEMCwhen recon-structingthephotonenergy.IfthereismorethanoneD candidate left for one signal decay mode in an event, the candidate with the smallest

|

E

|

is chosen for further analysis. More than 60% ofevents havemultiple candidatesfor D0 decaymodesand20% forD+ decaymodes.AccordingtothestudiesonMCsamples,the probability to select thecorrect candidateby choosingthe mini-mum

|

E

|

is more than 90%. In addition, the credibility of this methodisveriﬁedandproventoberobust bystudyingthe high-statisticsinclusiveMCsamples.

AsshowninFig.2andFig.3,clearpeaksareseeninthe MBC

andMKKdistributionsforthefoursignalmodes,whichcorrespond

tothe D

→

K+K−P signalsand

φ

→

K+K− signals,respectively. AccordingtothestudiesbasedontheinclusiveMCsamples,three typesofbackgroundeventswill passthroughaboveselection cri-teria. The ﬁrst one is a true D mesondecaying to K+K−P ﬁnal stateswithouta

φ

mesoninvolved(D

→

K+K−P ),thesecondone isatrue

φ

mesonnotfromthecorrespondingsignalmode (Cont.

φ

P X )andthethirdoneisthecombinatorialbackgroundfrom nei-theroftheprevioustwosources(Comb.bkg).

Two-dimensional unbinned extended maximumlikelihood ﬁts to the obtained distributions of MBC and MKK are performed to

extract yields of signals, asshown inFig. 3. The MKK variable is

employed heretodiscriminatethe

φ

mesonsignal fromthe non-resonantK+K−ﬁnalstate.Theprobabilitydensityfunctionsofthe D mesonand

φ

mesonsignalsare modeledbythe MC-simulated signalshapesconvolutedwithGaussianfunctionsthatdescribethe resolutiondifferencesbetweenMCsimulationsanddata.The com-binatorialbackgroundsinMBC(MKK)aredescribedwith(inverted)

ARGUS [23] functions based on the studies on the inclusive MC sample. Since the correlation between MKK and MBC can be

ne-glected,thesetwovariablesareconsidereduncorrelatedinthefit. Theparametersofthe(inverted)ARGUSandGaussianfunctionsin two-dimensionalfitsarefixedaccordingtoone-dimensionalfitsto thecorrespondingMBC andMKKdistributions.Theobtainedsignal

(6)

Fig. 3. Two-dimensional unbinnedmaximumlikelihoodﬁtstothedistributionsofMBCandMKKindataforthefoursignalmodes.Thepointswitherrorbarsaredata,the

(red)thickcurvesarethetotalﬁts,the(blue)longdashedcurvesdescribethesignals,the(violet)dottedcurvesrepresentbackgroundsoftrueφmesonsnotfromD→ φP

decaymodes,the(black)dashedcurvesdescribebackgroundsfromD→K+K−P withoutaφmeson,andtheshadedareashowthecombinatorialbackgrounds.

5. Branchingfraction

The branchingfractions forthe D

→ φ

P decayscan be calcu-latedby

B

i

₌

N i sig 2

·

N_D_D_¯

· ε

i

_·

_B

i sub

,

(2)

wherei denotesasignalmodeofD

→ φ

P ,N_sigi isthesignalyield extractedindata,N_DD¯ isthenumberof DD event

¯

indata,which

is

(

8296

±

31

±

64

)

×

103 forD+D−and

(

10597

±

28

±

89

)

×

103 for D0D

¯

0 [24] in thedata set we analyzed,

ε

i _is _the

reconstruc-tioneﬃciencydeterminedfromMCsimulationofthesignalmode, and

B

_subi are the branching fractions of the intermediate decay processes

φ

→

K+K− and

π

0

_/

_η

_→

_{γ γ}

_, _quoted _from _{PDG [}₄_].

The branching fractionfor each decay mode is calculated in Ta-ble2.

The statisticalsigniﬁcanceofthe D0

_{→ φ}

_η

_signal_is_evaluated,

takenas

−

2 ln

(

L

stat₀

/

L

statmax

)

where

L

statmax and

L

stat0 arethe

max-imum likelihood values with and without signal, respectively, to be 4

.

2σ. Since the signiﬁcance ofthe observed D+

→ φ

K+ sig-nal is0

.

8σ,the upperlimit of

B(

D+

→ φ

K+

)

is estimatedby a likelihood scan method, whichtakes into account the systematic uncertaintiesasfollows

L

i

(

B

i

)

=

1

−1

L

stat

_[(

₁

_{+ )}

_B

i

_]

_exp

−

2 2

σ

2 i,syst

d

.

(3)

Here,

is therelativedeviationoftheestimatedbranching frac-tion from the nominal value and

σ

i,syst is the total systematic

uncertaintygiveninTable3.

ThelikelihoodcurvecalculatedaccordingtoEq. (3) isshownin Fig. 4. The upper limit on

B(

D+

→ φ

K+

)

atthe 90% conﬁdence level(CL) isestimatedto be 2

.

1

×

10−5 _by _integrating_the

likeli-hoodcurveinthephysicalregion,

B

i

_>

_0.

Fig. 4. Likelihoodcurve as the functionofassumed B(D+→ φK+). The arrow pointstothepositionofupperlimitatthe90% CL.

6. Systematicuncertainties

The following sources of systematicuncertainties, asgiven in Table 3,areconsidered. Thetotal systematicuncertainty is deter-minedbyaddingallcontributionsinquadrature.

Theuncertaintiesoftrackingandparticleidentiﬁcation(PID)for charged kaonand pionmesons, aswell as

π

0

₍

_η

₎

_{reconstruction,}

havebeenstudied inprevious worksbyusingcontrol samplesof D hadronic events [25].Theuncertainties are weightedaccording to the kinematicsof thecandidates. Furthermore,inorder to es-timate the systematic uncertainty caused by the selected

π

0

₍

_η

₎

signal regions, the requirements on Mγ γ are varied and the re-sultant changes on the BFs are 0

.

7% (1

.

1%). This uncertainty is combined withthat of

π

0

₍

_η

₎

_{reconstruction,} _the_quadrature_sum

of which is given as 1

.

2%

(

1

.

8%

)

. Requirements on

E are stud-ied by smearing the corresponding

E distribution in inclusive MC samples withGaussian functionsandre-calculatingdetection eﬃciencies.Thechangesoftheeﬃcienciesareassignedasthe cor-respondinguncertainties.

Systematic uncertainty related to the two-dimensional fit in-cludesparametersofGaussianandARGUSfunctions,fitrangeand backgroundmodels.ForthefixedparametersintheGaussianand ARGUS functions, their valuesare varied by

±

1σ from the

(7)

one-Table 3

Summaryofsystematicuncertaintiesinpercentage.

Source D+→ φπ+ D+→ φK+ D0_{→ φ}_π0 _D0_{→ φ}_η B(D0_→φπ0₎ B(D+→φπ+) Tracking 1.0 1.1 0.8 1.0 0.3 PID 1.2 1.0 0.6 0.6 0.4 π0_{reconstruction} ₋ ₋ ₁_.₂ ₋ ₁_.₂ ηreconstruction − − − 1.8 − E requirement 0.2 0.2 0.2 0.2 0.3 2D ﬁt 0.4 2.5 0.4 2.0 0.6 ND Duncertainty 0.9 0.9 0.9 0.9 1.3 B(φ→K+K−) 1.0 1.0 1.0 1.0 − B(π0_,_η_→_{γ γ}₎ ₋ ₋ ₀_.₁ ₀_.₅ ₀_.₁ QC effect − − 1.0 1.0 1.0 Total 2.2 3.3 2.4 3.5 2.2

dimensionalfitresultsandthelargestresultantchangeisassigned asthesystematicuncertainty.Theuncertaintyduetothefitrange isestimatedbyrepeatingthefitswithaseriesofvariedrangesand thecorrespondingchangesarefoundtobenegligible.Forthe back-groundmodels,potentialbackgroundofD

→

f0

(

980

)

P isincluded

inthe ﬁt and the change on the number ofsignal events is as-signedasuncertainty.Thisuncertaintyislargerfor

B(

D+

→ φ

K+

)

and

B(

D0

→ φ

η

)

duetothesmallersignalyields.

The uncertainties of the quoted N_D_D_¯ from Ref. [24],

B(φ →

K+K−

)

and

B(

π

0

_/

_η

_→

_{γ γ}

₎

_from_{PDG [}₄_{] are}_taken_into_account

for the relevant signal modes. Since D0 _and _D0 _are _coherently

producedinthe processe+e−

→ ψ(

3770

)

→

D0_D0_,_quantum

co-herence (QC) [26] should be considered according to the equa-tion

Nobs_{C P}

=

yC P

·

NobsC P

.

Theuncertainty dependsonthe D0

−

D0 mixingparameter yC P,

andistakentobe1

.

0% [27] conservatively.

Forthe systematicuncertaintiesof _B(B(_DD+0_→φ→φπ_π0+)),theeffects

re-latedto K± trackingandPIDaremostlycancelled,owingtotheir samekinematicphasespace.The remainingsystematic uncertain-tiesin Table 3are considered independently andsummed up in quadrature.

7.Summary

The decaysof D+

→ φ

π

+, D0

→ φ

π

0_, _D0

_{→ φ}

_η,

_and _D+

_→

φ

K+ are studied by analyzing 2

.

93 fb−1 data taken at

√

s

=

3

.

773 GeV with the BESIII detector.The obtainedBFsare consis-tentwithpreviousresults,aslistedinTable2,whiletheprecisions of the BFs for the ﬁrst three modes are improved. In addition, theupperlimiton

B(

D+

→ φ

K+

)

of2

.

1

×

10−5 _at_{90% CL}_is

re-ported.

Ourresultsof

B(

D

→ φ

π

)

and

B(

D0

→ φ

η

)

areconsistentwith the previous measurements. Furthermore, the ratio of

B(

D0

_→

φ

π

0

₎

_to

_B(

_D+

_{→ φ}

_π

+

₎

_is_calculated_to_be

₍

₂₀

_.

₄₉

_±

₀

_.

₅₀

_±

₀

_.

₄₅

₎

_%,

whichissmallerthanthepreviousresult

(

24

.

6

±

1

.

8

)

% [5,6]. Mean-while,thedeviationfromthe predictedvalue of

(

19

.

7

±

0

.

2

)

% in Eq. (1) isreducedfrom2

.

7σ to1

.

2σ, whichshowsbetter agree-ment than the previous measurement. Hence, our results sup-port the isospin symmetry between these two D meson decay modes.

Acknowledgements

The BESIII collaboration thanks the staff of BEPCII and the IHEPcomputingcenterfortheir strongsupport.Thisworkis

sup-ported in partby NationalKey Basic Research Programof China underContractNo.2015CB856700;NationalNaturalScience Foun-dationofChina (NSFC)underContractsNos.11625523,11635010, 11735014,11822506,11835012;theChineseAcademyofSciences (CAS)Large-ScaleScientiﬁcFacilityProgram;JointLarge-Scale Sci-entiﬁc Facility Funds of the NSFC and CAS under Contracts Nos. U1532257, U1532258, U1732263, U1832207,U1932101; CAS Key ResearchProgramofFrontierSciencesunderContractsNos. QYZDJ-SSW-SLH003, QYZDJ-SSW-SLH040; 100 Talents Program of CAS; INPAC andShanghai Key Laboratory forParticle Physicsand Cos-mology; ERCunderContractNo.758462;German Research Foun-dationDFGunderContractsNos.CollaborativeResearchCenterCRC 1044,FOR2359;IstitutoNazionalediFisicaNucleare,Italy; Konin-klijke Nederlandse Akademie van Wetenschappen (KNAW) under ContractNo.530-4CDP03;MinistryofDevelopmentofTurkey un-derContractNo.DPT2006K-120470;NationalScienceand Technol-ogyfund;STFC(UnitedKingdom);TheKnutandAliceWallenberg Foundation(Sweden)underContractNo.2016.0157;TheRoyal So-ciety,UKunderContractsNos.DH140054,DH160214;TheSwedish ResearchCouncil;U.S.DepartmentofEnergyunderContractsNos. DE-FG02-05ER41374, DE-SC-0010118, DE-SC-0012069; University ofGroningen(RuG)andtheHelmholtzzentrumfuer Schwerionen-forschungGmbH(GSI),Darmstadt.

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