Table Of ContentUnderstanding the property of η(1405/1475) in the J/ψ radiative decay
Xiao-Gang Wu1, Jia-Jun Wu1,2, Qiang Zhao1∗, and Bing-Song Zou1,3†
1) Theoretical Physics Center for Science Facilities, Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049, China
2) Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
3) State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China
(Dated: January 8, 2013)
Inthisworkwemakeasystematicanalysisofthecorrelated processesJ/ψ γη(1440)/f (1420)
1
with η(1440)/f (1420) KK¯π, ηππ and 3π, where the role played by th→e so-called “triangle
1
→
singularity mechanism” (TSM) is clarified. Our results agree well with the experimental data and
suggest a small fraction of f (1420) contributions in these processes. This study confirms our
1
conclusion in [Phys. Rev. Lett. 108, 081803 (2012)] that the dynamic feature of the TSM can be
recognized bythestrongnarrowpeakobservedin theππ invariantmassspectrumofη(1440) 3π
→
with anomalously large isospin violations. Nevertheless, we explicitly demonstrate that the TSM
can produce obvious peak position shifts for the same η(1440) or f (1420) state in different decay
1
channels. This is a strong evidence that the η(1405) and η(1475) are actually the same state, i.e.
3 η(1440). We also make an analysis of the radiative decays of η(1440) γV (V = φ, ρ0 or ω)
→
1 which showsthat such aone-stateprescription seemsnot tohaveaconflict with theso-far existing
0 experimental data. Our analysis may shed a light on the long-standing puzzling question on the
2 natureof η(1405) and η(1475).
n
PACSnumbers: 13.75.Lb,14.40.Rt,13.20.Gd
a
J
5
I. INTRODUCTION
]
h
Thecharmoniumhadronicandradiativedecaysintolighthadronshaveprovidedanimportantwaytoprobethelight
p
hadron structures. In particular, with high statistics of J/ψ and ψ′ events produced in e+e− annihilation, the light
-
p hadronspectracanbestudiedcloselyanddynamicinformationconcerningthelighthadronpropertiescanbeextracted
e
from their production and decays. During the past few years, there have been several new resonance structures with
h
JPC = 0−+ observed by BESII and BESIII in J/ψ and ψ′ decays. They could be candidates of radial excitation
[
states of the pseudoscalar mesons η and η′, or exotic states such as glueball, multiquark state or hadronic molecule.
2 For instance, the BES-II Collaboration first reported a resonance structure in J/ψ γX(1835) γη′π+π− [1],
v → →
whichwaslaterconfirmedbytheBESIIImeasurement[2]withhighstatistics. Nevertheless,twoadditionalresonance
8
structures were identified as X(2120) and X(2370) in the η′ππ invariant mass spectrum [2, 3].
4
In fact, our understanding of the isoscalar spectrum is still far from well-established. Historically, the study of
1
2 the nature of η(1405) and η(1475) has been a hot topic and closely related to the effort of searching for the ground
. state pseudoscalarglueball in experiment. Since the firstradialexcitation states of η and η′ are generally assignedto
1
η(1295)andη(1475)takingintoaccounttheirproductionanddecayproperties[4],itleavesouttheabundantη(1405)
1
2 as a possible candidate for the pseudoscalarglueball. However,we wouldlike to emphasize that suchanarrangement
1 still needs further studies, and it is still controversial whether η(1405) and η(1475) are two separated states or just
: one state of 0−+ in different decay modes [5].
v
With the availability of high-statistic J/ψ and ψ′ events from the BESIII Collaboration, it allows us to tackle
i
X the question on the nature of η(1405) and η(1475). One important experimental progress is that the BESIII
r Collaboration [6] report the observation of anomalously large isospin violations of the η(1405/1475) 3π in
a J/ψ γη(1405/1475) γπ0f (980) γ + 3π which, however, can hardly be understood by treating→them as
0
→ → →
either glueball or qq¯state. Interestingly, this decay process also explicitly involves the issue of a (980)-f (980) mix-
0 0
ings. The BESIIIdatashowthatthef (980)signalisonlyabout10MeVinwidthandthe lineshapeisdifferentfrom
0
the Breit-Wigner width of about 40 100 MeV [4]. Moreover, the isospin violation turns out to be significant with
BR(η(1405) f (980)π0 3π)/BR∼(η(1405) a0(980)π0 ηππ) = (17.9 4.2)%, which cannot be explained by
→ 0 → → 0 → ±
the a f mixing intensity measured in other channels [7].
0 0
−
AnimmediatetheoreticalinterpretationisgivenbyRef.[8],whereweproposethatatrianglesingularitymechanism
∗ [email protected]
† [email protected]
2
(TSM) via the intermediate K∗K¯ +c.c. rescatterings would lead to significant enhancement of the isospin violating
decay, i.e. η(1405/1475) K∗K¯ +c.c. f (980)π. In this transition, the dominant contributions would come
0
→ →
from such a specific kinematic region that all the intermediate mesons in the triangle loop are literally on-shell. The
identificationofsuchamechanismseemstobenontrivialsinceitcannaturallyexplainthenarrowwidthofthef (980)
0
observed in the two pion invariant mass spectrum. Consequently, it raises an essential issue concerning the nature
of η(1405/1475) since the TSM can also contribute to the decays of η(1405/1475) KK¯π and ηππ, and distort
→
the lineshapes and shift the peak positions of the η(1405/1475) in those decay channels. As a result, a coherent
study of η(1405/1475) KK¯π, ηππ, and 3π is necessary and could be a key towards a better understanding of the
→
η(1405/1475)puzzle.
In this work, we shall provide a detailed analysis of η(1405/1475) KK¯π, ηππ, and 3π. We shall show that only
one 0−+ isoscalar state, namely η(1440), is needed in this mass regi→on. With this “one state” assumption, we shall
demonstrate thatthe TSM canleadto different massspectra for η(1440) KK¯∗+c.c., a (980)π0, andf (980)π. In
0 0
J/ψ γη(1440), with η(1440) KK¯∗+c.c., a (980)π0, and f (980)π,→another possible contribution to the same
0 0
→ →
final states is via f (1420). Since the mass of f (1420) is similar to that of η(1440), we should investigate the role
1 1
played by f (1420) in these processes. In particular, due to the similar masses between f (1420) and η(1440), the
1 1
decayoff (1420)wouldalsoexperiencetheTSM.Therefore,ahelicityanalysisoftheinvariantmassspectrumforthe
1
overlappingf (1420)andη(1440)isnecessary. IncomparisonwiththeresultsreportedinRef.[8],wehavedetailedall
1
the analysisbyincluding the f (1420)contributions. We confirmthe BESIIIresultsbydetailedhelicityanalysisfrom
1
which we canextract the invariantmass spectra for η(1440)in different channels. These features as a consequence of
the TSM could be a natural solution for the long-standing puzzle about the nature of η(1405/1475)in experimental
analyses.
Wealsomentionthattheη(1405/1475) 3πdecaywasalsostudiedinRef.[9]recentlyinachiralunitaryapproach.
→
Byexhaustingseveralmodelsandtakingconstraintsfromthemeson-mesonscatterings,theauthorsconfirmthatonly
a (980)-f (980)mixingcannotexplaintheBESresult[6]andtheinclusionofthetriangulardiagramsisnecessary[8].
0 0
The rest part of this paper is organized as follows: the formalism is presented in Sec. II. Section III is devoted to
the numerical results and discussions. Our conclusion is given in Sec. IV.
II. FORMALISM
A. Effective Lagrangians and transition amplitudes
TheeffectiveLagrangiansforthe η(1440)productionhavebeenpresentedinRef.[8]. Here,weincludethe f (1420)
1
contribution and list the effective Lagrangiansas the following:
= g ε pµ pν ψρ ψσ ψ , (1)
LV1V2P V1V2P µνρσ V1 V2 V1 V2 P
= g (ψ ∂ ψ ψ ∂ ψ )ψµ , (2)
LVP1P2 VP1P2 P1 µ P2 − P2 µ P1 V
= g ψ ψ ψ , (3)
LSP1P2 SP1P2 S P1 P2
= g ψµψ ψ , (4)
LAVP AVP A Vµ P
= g ε ∂µψνψρψσ +g ε ∂µψλ∂λ∂νψρψσ , (5)
Lψγf1 1 µνρσ ψ γ f1 2 µνρσ ψ γ f1
where S, P, V and A stand for four types of fields: scalar, pseudoscalar, vector and axialvector, respectively. For
η(1440) the same diagrams as in Ref. [8] are calculated, while for f (1420) the similar diagrams are listed in Fig. 1.
1
Figure1(1a)-(1b)areforf (1420) KK¯πthroughK∗K¯ anda (980)πchannels. Figure1(2a)-(2b)areforf (1420)
1 0 1
π+π−π0 throughthe TSManda →f mixing. Figure1(3)isforf (1420) ηπ0π0,wherewe assumethata0(980)π→0
0− 0 1 → 0
gives the main contribution.
Our kinematics conventions are shown in Fig. 2. Some common functions are defined as follows:
1
G = , (6)
f s m2 +i√sΓ (s)
− f f
1
G = , (7)
a s m2 +i√sΓ (s)
− a a
Γ (s) = ga2KK¯(ρ(√s,mK0,mK¯0)+ρ(√s,mK+,mK−)) + ga2πηρ(√s,mπ0,mη) , (8)
a
16π√s 16π√s
gf2KK¯(ρ(√s,mK0,mK¯0)+ρ(√s,mK+,mK−))
Γ (s) =
f
16π√s
3
K+(K−)
π0
K∗(K¯∗)
f1(1420) f1(1420)
K−(K+) K(K¯)
K+(K−)
K¯∗(K∗) K¯(K) a0(980)
π0
(1a) (1b) K−(K+)
π0 π0
K∗(K¯∗)
f1(1420) f1(1420)
K(K¯) K(K¯)
π+ π+
K¯(K) f0(980) a0 f0
K¯(K)
π− π−
(2a) (2b)
π0 γ
K∗(K¯∗)
f1(1420) J/ψ
K(K¯)
η
K¯(K) a0(980)
f1(1420)
π0
(3) (4)
FIG. 1: Feynman diagrams for the f (1420) decays and its production in J/ψ radiative decay. Similar diagrams for η(1440)
1
havebeen given in Ref. [8].
p1,m1 p1,m1
k,ma
p,mi
p,mi p2,m2 k−p1,mc p2,m2
p23,m23 p−k,mb p23,m23
p3,m3 p3,m3
(a) (b)
FIG. 2: Kinematics definedin ourformalism.
gf2ππ(ρ(√s,mπ0,mπ0)+2ρ(√s,mπ+,mπ−))
+ , (9)
16π√s
1
ρ(√s,m ,m ) = (s (m +m )2)(s (m m )2) . (10)
A B A B A B
s − − −
p
4
Then, the typical loop integrals can be expressed as
Iˆ = i d4k (2p k)µ(2p k)ν(−gµν + kmµk2aν) 1 1 , (11)
η1 (2π)4 − 1− k2 m2 (p k)2 m2(k p )2 m2
Z − a − − b − 1 − c
Iˆ = i d4k ǫµ(2p k)ν(−gµν + kmµk2aν) 1 1
f1 (2π)4 p 1− k2 m2 (p k)2 m2(k p )2 m2
Z − a − − b − 1 − c
= ǫµ(c p +c p ) , (12)
p p µ p1 1µ
Iˆ = Iˆ (1 2)=ǫµ(d p +d p ) . (13)
f1b f1 ↔ p p µ p2 2µ
Taking into account that the relative signs between the charged and neutral loops are positive in isospin-conserving
processes but negative in isospin-violating processes, it is convenient to define
c+p ≡ ccpgkc∗Kπ +cnpgkn∗Kπ , (14)
c−p ≡ ccpgkc∗Kπ −cnpgkn∗Kπ , (15)
c+p1 ≡ ccp1gkc∗Kπ+cnp1gkn∗Kπ , (16)
c−p1 ≡ ccp1gkc∗Kπ−cnp1gkn∗Kπ , (17)
where the superscripts ‘c’ and ‘n’ denote the charged and neutral loops, respectively. We also define Iˆ±, d± and d±
η1 p p2
for Iˆ in a similar way.
f1b
The invariant amplitudes in Fig. 1 can then be expressed as
( g +p p )
M1a = gf1K∗KgK∗Kπ ǫµps− µνm2 +23iµm23Γν (p3−p2)ν +(2↔1)
(cid:20) 23− V V V (cid:21)
= gf1K∗KgK∗Kπ×ǫµp(c1p1µ+c2p2µ+c3p3µ) , (18)
1 pµpν
spin|M1a|2 = 3gf21K∗KgK2∗Kπ −gµν + m2f1 !(c1p1µ+c2p2µ+c3p3µ)(c∗1p1ν +c∗2p2ν +c∗3p3ν) , (19)
X
M1b = 2gf1K∗Kga2KK¯Ga(s23) gKc ∗KπIˆfc1+gKn∗KπIˆfn1
= 2gf1K∗Kga2KK¯Ga(s23)(cid:16)×ǫµp pµc+p +p1µc+p1 , (cid:17) (20)
4 (cid:0) pµpν (cid:1)
spin|M1b|2 = 3gf21K∗Kga4KK¯|Ga(s23)|2 −gµν + m2f1 !(pµc+p +p1µc+p1)(pνc+p∗+p1νc+p1∗) , (21)
X
M2a = 2√2gf1K∗KgfKK¯gfππGf(s23) gKc ∗KπIˆfc1−gKn∗KπIˆfn1
= 2√2gf1K∗KgfKK¯gfππGf(s23)(cid:16)×ǫµp pµc−p +p1µc−p1 , (cid:17) (22)
8 (cid:0) pµpν (cid:1)
spin|M2a|2 = 3gf21K∗Kgf2KK¯gf2ππ|Gf(s23)|2 −gµν + m2f1 !(pµc−p +p1µc−p1)(pνc−p∗+p1νc−p1∗) , (23)
X
M2b = M1b×√2gfKK¯gfππGf(s23)(loop2c−loop2n) , (24)
2 = 2 2g2 g2 G (s )2 loop2c loop2n 2 , (25)
|M2b| |M1b| × fKK¯ fππ| f 23 | | − |
spin spin
X X
M3 = 2gf1K∗KgaKK¯gaπη Ga(s23) gKc ∗KπIˆfc1+gKn∗KπIˆfn1 +(2↔1)
= 2gf1K∗KgaKK¯gaπηh×ǫµp Ga((cid:16)s23) pµc+p +p1µc+p1 +(cid:17)(2↔1) , i (26)
2 (cid:2) pµp(cid:0)ν (cid:1) (cid:3)
spin|M3|2 = 3gf21K∗Kga2KK¯ga2πη −gµν + m2f1 ![|Ga(s23)|2(pµc+p +p1µc+p1)(pνc+p∗+p1νc+p1∗)
X
+G (s )2(p d++p d+)(p d+∗+p d+∗)
| a 13 | µ p 2µ p2 ν p 2ν p2
+G (s )G (s )∗(p c++p c+)(p d+∗+p d+∗)
a 23 a 13 µ p 1µ p1 ν p 2ν p2
+G (s )∗G (s )(p c+∗+p c+∗)(p d++p d+)] , (27)
a 23 a 13 µ p 1µ p1 ν p 2ν p2
5
= g ǫ pµǫνǫρǫσ +g ǫ pµpν ǫρǫσ ǫ p B (Q) . (28)
M4 1 µνρσ ψ ψ γ f1 2 µνρσ ψ f1 γ f1 ψ· f1 2
In Eq. (27), we have put the identical factor 1/2 in the squared amplitude. The parametrization of Eq. (28) is taken
from Ref. [10] and B (Q) is the Blatt-Weisskopf barrier factor
2
1
B (Q)= , (29)
2 sQ4+3Q2Q20+9Q40
where Q is the decay momentum, and Q is a hadron scale parameter Q = 0.197321/R GeV with R the radius of
0 0
the centrifugal barrier in fermi. In this paper we adopt R=0.35 fm which is about the radius of J/ψ.
1. Helicity amplitudes
In experiment, the quantum number of an intermediate state X is generally determined by measuring the angular
distribution of the X decays. To proceed, we first make a model-independent analysis of the helicity structure of the
transition matrix element which would allow us to separate different partial waves. Then, by comparing with the
angular distributions measured by experiment, we can extract the dynamic coupling strengths for different partial
waves.
For a decay process a b+c with spin, helicity and parity (s ,λ ,η ) , the decay amplitude in the rest frame
i i i i=a,b,c
→
of a can be expressed as [11],
sa (θ,φ;λ ) Dsa∗ (φ,θ,0)Fsa , (30)
Mλbλc a ∝ λa,λb−λc λbλc
where Dsa∗ (φ,θ,0) is the rotation function, and Fsa is the helicity-coupling amplitude which is independent
λa,λb−λc λbλc
of angular variables. It satisfies two constraints taking into account angular momentum conservation and Parity
conversation:
λ λ s , (31)
b c a
| − | ≤
Fsa = η η η ( )sa−sb−scFsa . (32)
λbλc a b c − −λb−λc
From these relations, we can find out the independent helicity-coupling amplitudes.
Asfollows,wewillre-analyzetheangulardistributionsoftherecoiledphotoninJ/ψ γX andtherecoiledf (980)
0
→
in X f (980)π decays which are measured by the BESIII experiment [6]. Now as a preparation, we derive the
0
→
helicity amplitudes of each vertex from the Lagrangiansin Eqs. (1)-(5).
In J/ψ γf , the helicity amplitude with transversely polarized f can be expressed as
1 1
→
λ ,λ = 1Sˆλ = g ǫ pµǫνǫ∗ρǫ∗σ+g B (Q)ǫ pµpνǫ∗ρǫ∗σǫ p
h γ f1 ± | | ψi 1 µνρσ ψ ψ γ f1 2 2 µνρσ ψ γ γ f1 ψ· γ
= ig λ m D1∗ (φ,θ,0)+ig B (Q)λ m Q2D1∗ (φ,θ,0)
− 1 γ ψ λψ0 2 2 γ ψ λψ0
= D1∗ (φ,θ,0)F1 , (33)
λψ0 λγλf1
which is nonvanishing with λ =λ , and
γ f1
F1 = ig λ m +ig λ m Q2B (Q) . (34)
λγλf1 − 1 γ ψ 2 γ ψ 2
So the independent amplitude is
F1a =im [ g +g Q2B (Q)] . (35)
11 ψ − 1 2 2
When f is longitudinally polarized, the g term will have no contribution. The helicity amplitude is
1 2
λ ,λ =0Sˆλ = g ǫ pµǫνǫ∗ρǫ∗σ
h γ f1 | | ψi 1 µνρσ ψ ψ γ f1
E
= ig λ m f1D1∗ (φ,θ,0)
− 1 γ ψm λψλγ
f1
= D1∗ (φ,θ,0)F1 , (36)
λψλγ λγ0
with
E
F1 = ig λ m f1 . (37)
λγ0 − 1 γ ψm
f1
6
So the independent amplitude is
E
F1a = ig m f1 . (38)
10 − 1 ψm
f1
For f f (980)π0, the helicity amplitude can be written as
1 0
→
hf0π|Sˆ|f1i = 2gf1K∗Kgf0KK gKc ∗KπIˆfc1−gKn∗KπIˆfn1
= 2gf1K∗Kgf0KKǫhµp(pµc−p +p1µc−p1) (cid:17)
= 2gf1K∗Kgf0KKc−p1ǫp·p1
= D1∗ (φ,θ,0)F1b , (39)
λf10 00
where we have used the relation ǫ p=0. Also, in the helicity frame of f (1420), we have
p 1
·
ǫ p = D1∗ (φ,θ,0)Q . (40)
p· 1 − λf0
So the independent amplitude is
F010b =−2gf1K∗Kgf0KKc−p1Q . (41)
Similarly, the helicity amplitude for J/ψ γη(1440)can be obtained with both J/ψ and γ transverselypolarized:
→
λ η(1440)Sˆλ = g ǫ pµqνǫρǫσ
h γ | | ψi ψγη1 µνρσ ψ γ ψ γ
= iλ g m ~q D1∗ (φ,θ,0)
− γ ψγη1 ψ| | λψλγ
= D1∗ (φ,θ,0)F1c , (42)
λψλγ λγ0
with
F1c = iλ g m ~q . (43)
λγ0 − γ ψγη1 ψ| |
For η(1440) f (980)π0, the helicity amplitude is
0
→
f π Sˆη(1440)
0
h | | i
= 2gη1K∗Kgf0KK(gKc ∗KπIˆfc1−gKn∗KπIˆfn1)
= F0d . (44)
00
B. Angular distribution
By combining the two-body decay amplitudes in the helicity frame, we can derive the total helicity amplitudes for
the chainprocessJ/ψ γX γf (980)π0 as shownin Fig.3 and extractthe angulardistributions to comparewith
0
→ →
the experimental data [6].
γ
J/Ψ θγ,φγ
X
f0(980)
θf0,φf0
π
FIG. 3: The kinematics for thechain process J/ψ γX γf (980)π0.
0
→ →
7
For X being η(1440), the total helicity amplitude can be expressed as
1
A (λ ,λ ) D1∗ (φ ,θ ,0)F1c F0d . (45)
η1 ψ γ ∝ λψλγ γ γ λγ0s m2 +im Γ 00
− η1 η1 η1
The angular distribution is
dσ
η1 A (λ ,λ )2
dΩ ∝ | η1 ψ γ |
λψ,Xλγ=±1
2
D1∗ (φ ,θ ,0)
∝ λψλγ γ γ
λψ,Xλγ=±1(cid:12) (cid:12)
(cid:12) (cid:12)
= 1+cos2θ(cid:12) , (cid:12) (46)
γ
where dΩ dΩ dΩ with dΩ dcosθ dφ and dΩ dcosθ dφ . One step further, we obtain
≡ γ f0 γ ≡ γ γ f0 ≡ f0 f0
dσ
η1 1+cos2θ , (47)
γ
dcosθ ∝
γ
dσ
η1 const , (48)
dcosθ ∝
f0
for the angular distributions of θ and θ , respectively. These expressions are the same as those adopted in
γ f0(980)
Ref. [6].
For X being f (1420), the total helicity amplitude of the chain process can be expressed as
1
1
A (λ ,λ ,λ ) D1∗ (φ ,θ ,0)F1a D1∗ (φ ,θ ,0)F1b . (49)
f1 ψ γ f1 ∝ λψ,λγ−λf1 γ γ λγλf1s−m2f1 +imf1Γf1 λf10 f0 f0 00
The angular distribution via f (1420) is
1
2
dσ
f1 A (λ ,λ ,λ )
dΩ ∝ (cid:12) f1 ψ γ f1 (cid:12)
λψ,Xλγ=±1(cid:12)(cid:12)λf1X=0,±1 (cid:12)(cid:12)
(cid:12) (cid:12) 2
(cid:12)αD1∗ (φ ,θ ,0)D1∗ (φ(cid:12) ,θ ,0)+D1∗ (φ ,θ ,0)D1∗(φ ,θ ,0)
∝ (cid:12) λψ,0 γ γ λγ0 f(cid:12)0 f0 λψ,λγ γ γ 00 f0 f0
λψ,Xλγ=±1(cid:12) (cid:12)
(cid:12) α (cid:12)
= α2sin2θ (cid:12)sin2θ + 1 cosφ cosφ sin2θ sin2θ +cos2θ (cos2θ +1) , (cid:12) (50)
1 f0 γ 2 α f0 f0 γ f0 γ
where α≡α1eiφα is the ratio of the λf1 =±1 amplitude to that of λf1 =0. By integrating over corresponding polar
anglesin the abovedouble distribution, one hasaccessto the angulardistributions ofθ andθ , respectively,for the
γ f0
intermediate f (1420):
1
dσ
f1 1+2α2+(1 2α2)cos2θ , (51)
dcosθ ∝ 1 − 1 γ
γ
dσ
f1 2+(α2 2)sin2θ . (52)
dcosθ ∝ 1− f0
f0
With both η(1440) and f (1420) contributing to the chain process, the total helicity amplitude can be obtained in
1
a similar way, namely,
1
A (λ ,λ ) = D1∗ (φ ,θ ,0)F1c F0d
η1+f1 ψ γ λψλγ γ γ λγ0s m2 +im Γ 00
− η1 η1 η1
1
+D1∗ (φ ,θ ,0)D1∗ (φ ,θ ,0)F1a F1b
λψ0 γ γ λγ0 f0 f0 λγλγs m2 +im Γ 00
− f1 f1 f1
1
+D1∗ (φ ,θ ,0)D1∗(φ ,θ ,0)F1a F1b
λψλγ γ γ 00 f0 f0 λγ0s m2 +im Γ 00
− f1 f1 f1
λ rD1∗ (φ ,θ ,0)+αD1∗ (φ ,θ ,0)D1∗ (φ ,θ ,0)+D1∗ (φ ,θ ,0)D1∗(φ ,θ ,0)(5,3)
∝ γ λψλγ γ γ λψ0 γ γ λγ0 f0 f0 λψλγ γ γ 00 f0 f0
h i
8
where we have applied the selection rule λf1 = λγ for the transversely polarized f1(1420); r ≡ r1eiφr is the ratio of
the η(1440) amplitude to that of f (1420) with λ =0. Then the angular distribution becomes
1 f1
dσ
η1+f1 A (λ ,λ )2
dΩ ∝ | η1+f1 ψ γ |
λψ,Xλγ=±1
= r2(cos2θ +3)+2r α sinθ sin2θ cosφ cos(φ φ )+2r cosθ cosφ (cos2θ +3)
1 γ 1 1 f0 γ f0 α− r 1 f0 r r
+2α2sin2θ sin2θ +α cosφ sin2θ sin2θ cosφ +cos2θ (cos2θ +3) . (54)
1 f0 γ 1 α f0 γ f0 f0 γ
It is easy to show that Eqs. (46) and (50) can be reproduced by setting the corresponding resonance couplings to
vanish. Similarly, the angular distributions of θ and θ can be obtained
γ f0
dσ
η1+f1 1+2α2+3r2+(1 2α2+3r2)cos2θ , (55)
dcosθ ∝ 1 1 − 1 1 γ
γ
dσ
η1+f1 α2+2r2+4r cosφ cosθ +(2 α2)cos2θ . (56)
dcosθ ∝ 1 1 1 r f0 − 1 f0
f0
Checking Eqs. (47), (51) and (55), one can see that the cosθ distribution is alwayssymmetric as a feature of a two-
γ
bodydecay. Incontrast,theangulardistributionofcosθ turnsouttobenontrivial. AsshownbyEqs.(48),(52)and
f0
(56), the contributions from different states with different quantum numbers are encoded in the angular distribution
of cosθ . By fitting the experimental data, the coupling parameters can thus be determined which alternatively
f0
would provide information about the contributing resonances. As shown by Fig. 3 of Ref. [6], the cosθ distribution
f0
is apparently asymmetric which indicates some contributions from the f (1420) production besides η(1440).
1
With the explicit total helicity amplitude in Eq. (53), we can express the differential width as
1 1
dΓ= A 2 p~ p~ d√s dΩ dΩ . (57)
(2π)516m2 | η1+f1| | γ|| f0| X γ f0
ψ
We can define the following quantities by integrating the invariant mass √sX of the f0(980)π0:
F1cF0d 2
A = d√s p~ p~ 10 00 , (58)
1 X | γ|| f0| s m2 +im Γ
Z (cid:12) X − η1 η1 η1(cid:12)
(cid:12) (cid:12)2
(cid:12) F1aF1b (cid:12)
A = d√s p~ p~ (cid:12) 11 00 (cid:12) , (59)
2 Z X | γ|| f0|(cid:12)(cid:12)sX −m2f1 +imf1Γf1(cid:12)(cid:12)
(cid:12) (cid:12)2
(cid:12) F1aF1b (cid:12)
A = d√s p~ p~ (cid:12) 10 00 (cid:12) , (60)
3 Z X | γ|| f0|(cid:12)(cid:12)sX −m2f1 +imf1Γf1(cid:12)(cid:12)
(cid:12) (cid:12)
which can thus be related to the quantities measured in ex(cid:12)periment, i.e. (cid:12)
(cid:12) (cid:12)
A A
α2 = 2, r2 = 1 . (61)
1 A 1 A
3 3
From these relations, we can extract the information about the couplings from the angular distribution analysis.
To compare with the experimental measurement of the unpolarized partial decay width in terms of the recoiled
energy s by the photon in J/ψ γX γABC, the following standard expression is adopted,
→ →
dΓJ/ψ→γX→γABC 2sΓJ/ψ→γX(s) ΓX→ABC(s)
= × , (62)
d√s π (s m2 )2+Γ2 m2
− X X X
where s is the four-momentum square of X = η(1440)/f (1420) in the reaction. A constant width Γ(f (1420)) =
1 1
0.0549GeVisadoptedforf (1420)[4],whileforη(1440),bothconstantwidthandenergy-dependentformareadopted,
1
Γη(1440)(s)=Γη(1440)→K∗K→KK¯π(s)=Γ1a(s) , (63)
where Γ corresponds to Fig. 1(1a).
1a
9
III. RESULTS AND DISCUSSIONS
In this part, we present our analyses and numerical results. First we demonstrate explicitly that the TSM is
dominant in η(1440) 3π, and the main contribution is indeed from such a kinematic region that all the internal
particles are close to→their mass shells. Then, we give the fitting results about η(1440) from KK¯π spectrum and
show the predictions of π+π−π0 and ηπ0π0 which are consistent with Ref. [8]. By including f (1420), we extract
1
the couplings of f (1420) through the analysis of the angular distribution of π+π−π0 channel [6]. Finally, we show
1
that the combined results for both η(1440) and f (1420) in comparison with the BES data would allow us to draw
1
a conclusion on the anomalously large isospin violations observed in η(1405) 3π and the nature of η(1405) and
→
η(1475).
A. Loop integral
Here we discuss in detail the calculation of Iˆ and Iˆ in Eqs. (11)-(12). What we actually need in Iˆ is the
η1 f1 f1
coefficients c and c . We use two methods to calculate the loops:
p p1
1. We directly calculate Iˆ and Iˆ by LoopTools without any form factors. The UV divergences are regularized
η1 f1
dimensionally by ∆=2/(4 D) γ +log4π, where ∆ can be adjusted and the default value is ∆=0.
E
− −
2. The exponential form factor method as applied in Ref. [12]. Namely, an exponential form factor as follows is
included in Iˆ and Iˆ to cut off the UV divergence:
η1 f1
k2 m2 (p k)2 m2 (k p )2 m2
exp − a + − − b + − 1 − c , (64)
Λ2 Λ2 Λ2
(cid:20) (cid:21)
where Λ is the cutoff energy and characterize the effective range of the interaction. In principle, other forms of
form factors can also be examined and we find the results are similar to each other.
WepresentthecalculationsbasedontheabovetwotreatmentsinFig.4. Inthekinematicregionthatalltheinternal
particles are close to their mass shells, namely the TS kinematics, these two treatments give nearly identical results
sincetheformfactorcorrectionsarenearlyunity. Inparticular,theabsorptivepartisdominatedbythecontributions
fromthe TSkinematics. The realpartturnsoutto be moresensitiveto the formfactorcorrectionswhenthe internal
particles deviate from their mass shells. Similar results are found for the f (1420) since its mass is nearly the same
1
as η(1440) and they share the same TSM. As a result, one can imagine that there should be little difference between
these two treatments in the isospin-violating decay of η(1440)/f (1420) 3π since the main contribution is from
1
→
the absorptive part in the TS kinematics and the dispersive part would largely cancel out between the charged and
neutral loop amplitudes. It is worth noting that the cancellation between the charged and neutral loop amplitudes
eventually makes the calculation almost independent of the model uncertainties as explicitly pointed out in Ref. [8].
This should be a direct way to confirm the dominance of the TSM in η 3π as a dynamic mechanism.
→
B. Angular distribution analysis
Inthenumericalcalculations,thecommoncouplingconstantspresentinthetriangleloopsforη(1440)andf (1420)
1
are adopted the same as in Ref. [8], i.e. gaKK¯ = 3.33 GeV, gaηπ = 2.45 GeV, gKn∗Kπ = 3.208, and gKc ∗Kπ = 3.268.
BES [13] and KLOE [14] give different values for the f0(980) coupling, namely, gfKK¯ = 4.18 GeV and gfππ =
1.66 GeV from BES [13]; and gfKK¯ =5.92 GeV and gfππ =2.09 GeV from KLOE [14]. Similar to the treatment in
Ref. [8], contributions from Figs.1(1b) and(2b) are neglectedsince they are only about 1/10ofFigs. 1(1a)and(2a),
respectively.
Weadoptthemassandwidthoff (1420)fromthePDG[4],i.e. m =1.4264GeVandΓ =54.9MeV,
1 f1(1420) f1(1420)
but leave the mass and width of η(1440) to be fitted by the experimental data based on the “one-state” assumption.
Thisis reasonablesince thef spectrumdoesnotsufferfromtheambiguityofpossible abundantstatesinthisenergy
1
region and as we shall see later that the f (980) angular distribution measured by BESIII [6] only requires a small
0
contribution from the f (1420).
1
Taking into account the present datum status, our analysis strategy is as follows: we first fit the BESIII data [6]
for the angular distributions of the recoiled photon and f (980) in the decay of J/ψ γX and X f (980)π0.
0 0
→ →
This allows us to extract the relative strengths between the η(1440) and f (1420) in the isospin violating decays.
1
Then, by applying for the constraint from the J/ψ γη(1405/1475) γKK¯π from DM2, MARK III, and BES [4],
→ →
10
0.03
0.01
0.00 0.02
-0.01
-0.02 1 0.01
m I -0.03 e I
I -0.04 R 0.00
-0.05
-0.01
-0.06
-0.07 -0.02
0.90 0.95 1.00 1.05 1.10 0.90 0.95 1.00 1.05 1.10
M(KK)(GeV) M(KK)(GeV)
0.06
0.12
0.10 0.04
0.08
cp cp 0.02
m 0.06 e
R
I 0.04 0.00
0.02
-0.02
0.00
0.90 0.95 1.00 1.05 1.10 0.90 0.95 1.00 1.05 1.10
M(KK)(GeV) M(KK)(GeV)
0.1
0.0 0.1
-0.1
cp1 -0.2 cp1 0.0
m e
I -0.3 R -0.1
-0.4
-0.5 -0.2
-0.6
0.90 0.95 1.00 1.05 1.10 0.90 0.95 1.00 1.05 1.10
M(KK)(GeV) M(KK)(GeV)
FIG.4: Theimaginary andrealpartsoftheloop integralsintermsoftheinvariantmassofKK¯ at√s=1.42GeV. Thereare
two sets of lines identified by those two spikes which correspond to the charged (lower mass) and neutral (higher mass) KK¯
thresholds. Foreachsetoflines,thethickandthinsolidonesrepresenttheresultswithΛ=1.0andΛ=0.5GeV,respectively,
while thedashed lines denotetheresults of LoopTools calculation without form factor.
it allows us to determine the absolute differential widths for both η(1440) and f (1420). We shall compare this
1
with the exclusive fit by η(1440) as shown in Ref. [8]. In the end, we shall output the invariant mass spectra for
η(1440)/f (1420) KK¯π,ηππand3π,fromwhichwewouldexpecttoobservedifferentlineshapesandpeakpositions
1
→
from the same state in different decay channels.
The θ and θ angular distributions of exclusive η(1440) and f (1420) have been analyzed in Ref. [6] as parame-
γ f0 1
terized in Eqs (47)-(48) and (51)-(52). Now we consider the combined contributions from both η(1440) and f (1420)
1
and fit the BESIII data [6] using Eqs. (55)-(56) which can be expressed as
dN
= b (1+ccos2θ ) , (65)
γ γ
dcosθ
γ
dN
= b (1+c cosθ +c cos2θ ) , (66)
dcosθ f0 1 f0 2 f0
f0
