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BESII ??????

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Title: BESII ??????


1
BESII ??????
? ? (for BES Collaboration) ???????????? jins_at_mail
.ihep.ac.cn ????????????? ??, 2006?10?28?
2
Preliminary Remarks
  • ??????????,????,BES??????????? 30 ??,??Phys. Rev.
    Lett. ?? 6 ? ?
  • ??????,??????????? (???????????????)?
  • ?????????(??????????????), ????????????????

3
??????
  • ????????????????????
  • psi ???12 ????
  • psi(3770)?DDbar?????????
  • J/psi ?????????

4
New Observations at BESII
  • A possible bound state
    mass threshold enhancement in
    and new
    observation of X(1835).
  • mass threshold enhancement in
  • mass threshold enhancement in
  • ?? mass threshold enhancement in J/? ? ???
  • New observation of a broad 1- - resonance in J/?
    ? KK- ?0

5
A possible ppbar bound state
6
Observation of an anomalous enhancement near the
threshold of mass spectrum at BES II
J/y?gpp
BES II
acceptance weighted BW
3 5 -10 -25
M1859 MeV/c2 G lt 30 MeV/c2 (90 CL)
c2/dof56/56
0
0.1
0.2
0.3
M(pp)-2mp (GeV)
3-body phase space
acceptance
7
Features of the enhancement near the threshold of
mass spectrum at BES II
J/y?gpp
BES II
  • Peak position
  • 0 MeV above threshold
  • Width
  • 60M eV
  • Strong (Height)
  • (SB)/B 2

The above features may help us easily to
judge whether it is observed in other processes.
0
0.1
0.2
0.3
M(pp)-2mp (GeV)
8
This narrow threshold enhancement is NOT
observed in at CLEO
  • This result cannot be explained by pure FSI
    effect, since FSI is a universal effect.
  • FSI interpretation of the narrow and strong
    ppbar threshold enhancement is disfavored.
  • This indicates that X(1860) has a production
    property similar to ? meson.
  • c.f.

No enhancement near threshold
9
This narrow threshold enhancement is NOT
observed in at BESII
  • This again disfavors FSI and indicates that
    X(1860) has a production property similar to ?
    meson.
  • c.f.
  • This also indicates X(1860) may have strong
    coupling to gluons as ? meson.

No narrow strong enhancement near threshold
10
Pure FSI disfavored
  • I0 S-wave FSI CANNOT fit the BES data.

FSI curve from A.Sirbirtsev et al. (
Phys.Rev.D71054010, 2005 ) in the fit (I0)
FSI PS eff bck
11
X(1860) has large BR to ppbar
  • We (BES) measured
  • From Crystal Ball result, we etimate
  • So we would have
  • (This would be the largest BR to ppbar among
    all known mesons)

Considering that decaying into ppbar is only
from the tail of X(1860) and the phase space is
very small, such a BR indicates X(1860) has
large coupling to ppbar !
12
Summary of the properties of the strong ppbar
mass threshold enhancement X(1860)
  • So far, it is only observed in J/? radiative
    decays
  • It has production properties similar to ? meson.
  • It could have strong coupling to gluons as ?
    meson.
  • It could have the largest decay BR to ppbar among
    all PDG particles
  • It has strong coupling to ppbar.

13
  • New Observation of X(1835) in

PRL 95, 262001 (2005)
14
Observation of X(1835) in
Statistical Significance 7.7 ?
The ??-?? mass spectrum for ?? decaying into
?????-? and ??? ??
15
Mass spectrum fitting
The ??-?? mass spectrum for ?? decaying into
?????-? and ??? ??
7.7?

BESII Preliminary

16
Re-fit to J/???p pbar including FSI
Include FSI curve from A.Sirbirtsev et al. (
Phys.Rev.D71054010, 2005 ) in the fit (I0)
M 1830.6 ? 6.7 MeV ? lt 153 MeV _at_90C.L.
In good agreement with X(1835)
17
A Possible ppbar Bound State
  • X(1835) could be the same structure as ppbar mass
    threshold enhancement.
  • It could be a ppbar bound state since it
    dominantly decays to ppbar when its mass is above
    ppbar mass threshold.
  • Its spin-parity should be 0- this would be an
    important test.
  • Other interpretations are not excluded.
    Currently all theoretical interpretations have
    some difficulties. PDG2006 takes it as a possible
    ppbar bound state.

Next talk for more detail discussions
18
Observation of mass
threshold enhancement in

19
Observation of an anomalous enhancement near the
threshold of mass spectrum at BES II
BES II
3-body phase space
For a S-wave BW fit M 2075 ?12 ? 5 MeV
G 90 ? 35 ? 9
MeV
20
K? mass threshold enhancement
21
Observation of a strong enhancement near the
threshold of mass spectrum at BES II
NX
BES II
PS, eff. corrected
(Arbitrary normalization)
22
  • A strong enhancement is observed near the mass
    threshold of MK? at BES II.
  • Preliminary PWA with various combinations of
    possible N and ? in the fits The structure
    Nxhas
  • Mass 15001650MeV
  • Width 70110MeV
  • JP favors 1/2-
  • The most important is

It has large BR(J/? ? pNX) BR(NX? K?) ?2 X 10-4
, suggesting NX has strong coupling to K?.
23
A ?K resonance predicted by chiral SU(3) quark
model
  • Based on a coupled-channel study of ?K and SK
    states in the chiral SU(3) quark model, the
    phase shift shows the existence of a ?K
    resonance between ?K and SK mass threshold.
  • ( F. Huang, Z.Y. Zhang et al.
  • Phys. Rev. C71 064001, 2005 )

Ecm ( M?MK ) (MeV)
24
  • The K? mass threshold enhancement NX(1610) could
    be a K? bound/resonant state
  • (5-quark system with hidden ssbar components).

25
Observation of ?? mass threshold enhancement
26
  • We studied DOZI process

J/ ? ? ? ? ? ?
? ??-?0 K K-
27
Clear ? and ? signals
?
?
?
recoiling against ?
28
Daliz Plot
29
A clear mass threshold enhancement is observed
Acceptance
30
  • The radiative decay of J/????? has been observed
    in the 58M J/? data.
  • A significant structure of ?? has been found near
    the mass threshold.
  • PWA shows the structure favors 0, with a mass
    , width 105?20?28 MeV, and the
    corresponding branch ration is (2.61?0.27?0.65)x10
    -4.
  • It could be a multiquark/hybrid/glueball state.
  • Its relation with f0(1710),f0(1790)?

Next talk for more detail discussions
31
Is the STRONG threshold enhancement
universal/naïve in J/? decays ? NO !
  • Actually in many other cases we do NOT see STRONG
    threshold enhancements !
  • For example In J/? decays at BES II

32
New observation of a broad 1- - resonance in J/?
? KK- ?0
33
J/? ? KK- ?0
very clean ?0 signal
34
J/? ? KK- ?0
Background
PID and kinematic fit can significantly
reduce the dominant background from J/? ? ? ?-
?0.
35
Partial Wave Analysis of J/? ? KK- ?0 events
  • Parity conservations in J/? ? KK- ?0 requires
    that
  • spin-parity of KK- should be 1--,3--,
  • PWA fit with
    and
  • phase space (PS) gives ( preliminary )
  • ( can be ruled out by much worse
    likelihood )
  • X pole position
  • big destructive interference among
    and PS

36
Broad X cannot be fit with known mesons or their
interference
  • It is unlikely to be ?(1450), because
  • The parameters of the X is incompatible with
    ?(1450).
  • ?(1450) has very small fraction to KK. From PDG
  • It cannot be fit with the interference of ?(770)
    , ?(1900) and ?(2150)
  • The log-likelihood value worsens by 85 (??2170).

37
How to understand broad X(1580)?
  • Search of a similar structure in J/? ? KSK ? will
    help to determine its isospin.
  • X(1580) could have different nature from
    conventional mesons
  • There are already many 1- - mesons nearby.
  • Width is much broader than other mesons.
  • Broad width is expected for a multiquark state.
  • (???????????????????????)

Next talk for more detail discussions
38
The observation of new N peaks in
N(1520) N(1535)
N(1650) N(1675) N(1680)
N(1440)?
?
Missing mass spectrum (GeV/c2)
39
Published in Phys. Rev. Lett.
N(2065)
BW fit yields
PWA is performed.
  • well-established Ns below 2.0 GeV are fixed
    to PDG values.
  • for N(2065), L1 is much worse than L0 in the
    fit.
  • ? 1/2 or 3/2 (improve log likelihood
    by 400)
  • 1/2 3/2 (improve log likelihood
    further by 60)

40
psi ???12 ????
41
The 12 rule
M. Appelquist and H. D. Politzer, PRL34, 43 (1975)
This is the famous (or notorious) 12
rule.
42
12 rule and ?? puzzle
  • Violation found by Mark-II , confirmed
  • by BESI at higher sensitivity.
  • Extensively studied by BESII/CLEOc
  • VP mode ? ?, KK-c.c., K0K0c.c., ??0,
  • PP mode KSKL, KK-, ??-
  • BB mode pp, ??,
  • VT mode KK2, ?f2, ?a2, ?f2
  • 3-body pp?0, pp?, ??-?0,
  • Multi-body KSKShh, ??-?0 KK- , 3(??-),

43
(No Transcript)
44
?? ? ?- ?0
BESII PLB619, 247 (2005)
Very different from J/?? 3?!
BESII
229 ?0s
?
J/?
???? is observed, it is not completely missing,
BR is at 10-5 level!
45
J/?, ?? VP
BESII PLB614, 37 (2005) PRD73, 052007 (2006)
modes BESII B(?)(10-5) CLEOc B(?)(10-5) PDG04/BESII/ B(J/?)(10-4) B(?)/B(J/?) ()
?? 5.10.71.1 2.40.8-0.70.2 23426 0.1303
?(2150)? 19.42.511.5-3.4 N/A N/A N/A
??-?0 18.11.81.9 18.81.6-1.52.8 2009 0.9211
K0K0c.c. 13.32.71.7 9.22.7-2.20.9 424 2.60.6
KK-c.c. 2.91.70.4 1.31.0-0.70.3 504 0.340.20
?? 1.870.68-0.620.28 2.51.2-1.00.2 5.380.66 3.71.2
?? 1.780.67-0.620.17 3.01.1-0.90.2 1.930.23 10.93.4
?? 1.871.64-1.110.33 N/A 1.050.18 1816
?? lt0.40 N/A lt0.064 N/A
?? 3.31.10.5 2.01.5-1.10.4 8.980.92 3.01.2
?? 3.11.40.7 N/A 5.460.64 5.73.0
?? lt3.1 lt1.1 23.52.7 lt0.53
?? 3.22.4-2.00.7 N/A 2.260.43 1411
46
Multi-body ? decays
BESII PRD71, 72006 (2005) PRD74, 12004 (2006)
BESII, PRD73, 052004 (2006)
Some modes are suppressed, some are enhanced,
while some others obey the 12 rule!
47
Summary of 12 rule
  • ? ?VP suppressed
  • ? ?PP enhanced
  • ? ?VT suppressed
  • ? ?BB obey/enh
  • Multi-body obey/sup

Seems no obvious rule to categorize the
suppressed, the enhanced, and the normal decay
modes of J/? and ?. The models developed for
interpreting specific mode may hard to find
solution for other (all) modes.
Similarly ? decays have a rule of 0.02, more
data and more sophisticated analysis are needed
to extract the branching fractions from the
observed cross sections. Here because the time
limitation, I will omit the results in this talk.
  • Model to explain J/?, ? and ? decays naturally
    and simultaneously?
  • S-D mixing in ? and ? J. L. Rosner, PRD64,
    094002 (2001)
  • DD-bar reannihilation in ? (J. L. Rosner,
    hep-ph/0405196)
  • Four-quark component in ? M. Voloshin, PRD71,
    114003 (2005)
  • Survival cc-bar in ? (P. Artoisenet et al.,
    PLB628, 211 (2005))
  • Other model(s)?

48
psi(3770)?DDbar?????????
49
?(3770) non-DD decays
  • ?(3770) decays most copiously into DD.
  • ?(3770) is a mixture of the 13D1 and 23S1,
  • other ?(2S)-like decays for ?(3770) are
  • expected. (mixing angle 12?2o).
  • Many theoretical calculations estimate the
  • partial width for ?(3770) ? ??- J/?.
  • (Lipkin, Yan, Lane, Kuang, Rosner)
  • Recently, Kuang obtained a partial width for
  • ?(3770) ? ??- J/? in the range of
  • 25 -113 keV. (Y.P. Kuang, PRD 65 (2002) 094024)

50
BES first reported ?(3770) non-DD
decay?(3770) ? ??- J/?
Open histogram is for ee-, histogram in yellow
is for ??-
The histogram is ? error bars are ??
data
MC
27.7 pb-1
20 times large than the data
mainly
hep-ex/0307028 PLB 605 (2005) 63
51
Published in PLB
With BES previously measured cross sections for
DD production.
Assuming that there are interference between the
two amplitudes
52
Obtained from fitting to the inclusive hadron and
the DD-bar production cross sections
simultaneously.
  • Branching fractions

hep-ex/0605107
Inclusive hadrons
where the first error is statistical and second
systematic, which arises from the un-canceled
systematic uncertainties in hadron cross sections
(4.4 ), neutral DD-bar cross sections (4.5 )
and charged DD-bar cross sections (7.4 ).
Mar. 2003 data set
53
Search for decays of ?(3770)
Observed Cross Sections
Preliminary !
No obvious cross section discrepancy at the two
energy points is observed. However, to extract
the non-DD-bar branching fractions of ?(3770)
decays, we need to consider the interference
between the two amplitudes of the continuum and
the resonances, and to consider the difference of
ISR vacuum polarization corrections at two
energy points.
54
Searching for ?/? ?Invisible Decays in J/????/?
  • Reconstructing ??KK? ,
  • looking at missing momentum of ? decays
  • Fast ? (1.2 GeV) will help us define the decay
  • direction of invisible decay of ?/ ?(958) .

Missing direction
Tag direction
No any hit information are required outside of
the core of ? decays.
Signal MC
58 M J/?
? missing momentum distribution
hep-ex/0607006 (accepted by PRL)
Further study will be performed at BESIII
55
?????
  • ? BES ??????????,BESII ??????????? unexpected
    discoveries?
  • ????,? BESIII ??????? unexpected discoveries?
  • ??,? BESIII ????????????????????,????????(????????
    ? BESIII ????????????)?

56
? ?!Thank You!
57
Summary (I)
  • BES II has observed several strong mass threshold
    enhancements in J/? decays.
  • Why strong mass threshold structures are
    important?
  • Multiquark states may be only observable near
    mass thresholds with limited decay phase space.
  • ? Otherwise, it might be too wide to be observed
    as a resonance since it can easily fall apart
    into two or more mesons.

I can see f0(980)
I can see broad ? under other peaks
any broad resonance under other peaks?
broad resonance or phase space?
58
Summary (II)
  • A very narrow and strong mass threshold
    enhancement is uniquely observed in
    decays at BES II
  • It is NOT observed in Y(1S) decays, nor in J/psi
    hadronic decays. FSI is strongly disfavored.
  • Its large BR to suggests it be a
    bound state.
  • X(1835) is observed in
    It could be same structure as the ppbar
    mass threshold enhancement, i.e., it could be a
    ppbar bound state.

59
Summary (III)
  • mass threshold enhancement was observed in
  • Evidence of NX(1610) was observed near K? mass
    threshold, suggesting a K? bound or resonant
    state.
  • An ?? mass threshold enhancement was observed in
    J/? ? ???.
  • A very broad 1- - resonance X(1580) is observed
    in J/? ? KK- ?0 .
  • J/? decay is an ideal place to study exotic
    structures.

60
Multi-quark State, Glueball and Hybrid
  • Hadrons consist of 2 or 3 quarks
  • Naive Quark Model
  • New forms of hadrons
  • Multi-quark states Number of quarks gt 4
  • Hybrids qqg,qqqg
  • Glueballs gg, ggg

Meson( q q ) Baryon(q q q)
How quarks/gluons form a hadron is far from being
well understood.
61
Multi-quark states, glueballs and hybrids have
been searched for experimentally for a very long
time, but none is established.However, during
the past two years, a lot of surprising
experimental evidences showed the existence of
hadrons that cannot (easily) be explained in the
conventional quark model. Most of them are
multi-quark candidates. Searching for
multi-quark states becomes one of the hottest
topics in the hadron spectroscopy.
62
What do we expect from J/psi?gamma ppbar results?
  • The baryonium interpretation of the
    ppbar mass threshold enhancement predicts a new
    particle around 1.85 GeV which should be observed
    in other decay mode with full BW resonant
    structure.

63
Possible Interpretations
  • FSI? Theoretical calculations are needed.
  • Conventional K or a multiquark resonance?
  • Search for its Kp ?Kpp decay modes would help to
    understand its nature.
  • We are now studying
  • J/? ? KKp ?KKpp

64
NO strong dynamical threshold enhancement in
cross sections (at LEAR)
  • With threshold kinematic contributions removed,
    there are very smooth threshold enhancements
    in elastic matrix element and very
    small enhancement in annihilation matrix
    element
  • ? much weaker than what BES observed !

M2
M2
BES
BES
Both arbitrary normalization
Both arbitrary normalization
65
Any inconsistency? NO!
  • For example with Mres 1859 MeV, G 30 MeV,
    J0, BR(ppbar) 10, an estimation based on
  • At Ecm 2mp 6 MeV ( i.e., pLab 150 MeV
    ), in elastic process, the resonant cross section
    is 0.6 mb much smaller than the continuum
    cross section 94 ? 20 mb .
  • ? Difficult to observe it in cross
    sections experimentally.

66
This narrow threshold enhancement is NOT observed
in B decays
  • The structure in B decays is obviously different
    from the BES observation

Belle
The structure in B decays is much wider and is
not really at threshold. It can be explained by
fragmentation mechanism.
BES II
Threshold enhancement in J/? decays is obviously
much more narrow and just at threshold, and it
cannot be explained by fragmentation mechanism.
67
Pure FSI disfavored (I)
  1. Theoretical calculation (Zou and Chiang, PRD69
    034004 (2003)) shows The enhancement caused by
    one-pion-exchange (OPE) FSI is too small to
    explain the BES structure.
  2. The enhancement caused by Coulomb interaction is
    even smaller than one-pion-exchange FSI.

M2
M2
BES
BES
Both arbitrary normalization
Both arbitrary normalization
one-pion-exchange FSI
Coulomb interaction
68
FSI Factors
  • Most reliable full FSI factors are from
    A.Sirbirtsev et al. ( Phys.Rev.D71054010, 2005
    ),which fit ppbar elastic cross section near
    threshold quite well.

ppbar elastic cross section near threshold
I1 S-wave
P-wave
I0 S-wave
69
In ppbar collision, the background is much lager
(I)
  • J/? decays do not suffer large t-channel
    background as ppbar collision.

gtgt
70
In ppbar collision, the background is much lager
(II)
In ppbar elastic scattering, I1 S-wave
dominant, while in J/? radiative decays I0
S-wave dominant.
ppbar elastic cross section near threshold
I1 S-wave
P-wave
I0 S-wave
A.Sibirtsev, J. Haidenbauer, S. Krewald, Ulf-G.
Meißner, A.W. Thomas, Phys.Rev.D71054010, 2005
71
So, the mechanism in ppbar collision is quite
different from J/? decays and the background is
much smaller in J/? decaysIt would be very
difficult to observe an I0 S-wave ppbar
bound state in ppbar collisions if it exists.
  • J/? decays (in ee- collider) have much cleaner
    environment JP, I filter

72
So, pure FSI is strongly disfavored.However, we
do not exclude the contribution from FSI.
73
From B.S. Zou, Exotics 05
pp near threshold enhancement is very likely
due to some broad sub-threshold 0- resonance(s)
plus FSI.
From A. Sirbirtsev
FSI factors should be included in BW fit.
74
Re-fit to J/???p pbar including FSI
Include FSI curve from A.Sirbirtsev et al. (
Phys.Rev.D71054010, 2005 ) in the fit (I0)
M 1830.6 ? 6.7 MeV ? 0 ? 93 MeV
FSI BW PS eff bck
75
Crystal Ball results on inclusive photon
spectrum of J/psi decays
76
Discussion on I1 S-wave FSI
77
Pure FSI disfavored (III) I 1
  • Pure I1 S-wave FSI is disfavored by more
    than 3 ?.

FSI BW
Pure FSI
M 1773 ? 21 MeV ? 0 ? 191 MeV
78
I0 dominant in J/ ? radiative decays
  • Most I 0 states have been observed in J/ ?
    radiative decays with big production rate (
    especially for 0- mesons ) such as ?, ?,
    ?(1440), ?(1760), f2(1270), f2(1525), f0(1500),
    f0(1710).
  • The only observed I1 meson in J/ ? radiative
    decays is ?0 with low production rate 410 5,
    e.g., no evidence for ?(1800) in J/? ? ? 3 ?
    process.
  • It is unlikely to be from ?(1800) .
  • I1 S-wave FSI seems unlikely.

79
ppbar bound state in NNbar potential
  • Paris NNbar potential
    ( Paris 93, B. Loiseau et al.,
    hep-ph/0411218, 0501112 )
  • For 11S0 , there is a bound state
  • E - 4.8 - i 26.3 MeV
  • quite close to the BES observation.
  • However, Julich NNbar model
    ( A. Sibirtsev et al.,
    hep-ph/0411386 )
  • For 11S0 E - 104 - i 413 MeV
  • seems quite far away from BES observation.
  • They both predict an 11S0 ppbar bound state,
    although they are quantitatively different.

80
BES II Preliminary
No ?(1800)
81
NO strong dynamical threshold enhancement in
cross sections (at
LEAR)
  • With threshold kinematic contributions removed,
    there are very smooth threshold enhancements in
    elastic matrix element and very small
    enhancement in annihilation matrix element
  • ? much weaker than what BES observed !

M2
M2
BES
BES
Both arbitrary normalization
Both arbitrary normalization
82
The large BR to ppbar suggest it could be an
unconventional meson
  • For a conventional qqbar meson, the BRs decaying
    into baryons are usually at least one order lower
    than decaying into mesons.
  • There are many examples in PDG.
  • E.g.
  • So the large BR to ppbar (with limited phase
    space from the tail of X(1860)) seems very hard
    to be explained by a conventional qqbar meson.

83
pp bound state (baryonium)?
There is lots lots of literature about this
possibility
  • E. Fermi, C.N. Yang, Phys. Rev. 76, 1739 (1949)
  • I.S. Sharpiro, Phys. Rept. 35, 129 (1978)
  • C.B. Dover, M. Goldhaber, PRD 15, 1997 (1977)
  • Datta, P.J. ODonnell, PLB 567, 273 (2003)
  • M.L. Yan et al., hep-ph/0405087
  • B. Loiseau et al., hep-ph/0411218

deuteron
baryonium
attractive nuclear force
attractive force?

n

-
loosely bound 3-q 3-q color singlets with Md
2mp- e
loosely bound 3-q 3-q color singlets with Mb
2mp-d ?
Observations of this structure in other decay
modes are desirable.
84
Analysis of
X(1835) 5.1 ?
85
Analysis of
X(1835) 6.0 ?
86
Comparison of two decay modes
  • Mass and width from
  • m1827.4?8.1MeV/c2 , ?54.2?34.5MeV/c2
  • Mass and width from
  • m1836.3?7.9MeV/c2 , ?70.3?23.1MeV/c2
  • The mass, width and branching fractions obtained
    from two different decay modes are consistent
    with each other.

87
Similar enhancement also observed in
4? away from phase space.
88
This enhancement is NOT observed in
process at SAPHIR
89
Discussion on K? mass threshold enhancement
NX(1610)
  • NX(1610) has strong coupling to K?
  • From
    (SD-wave decay) and
    is a P-wave decay, we can estimate
  • From BESII,
  • The phase space of NX to K? is very small, so
    such a big BR shows NX has very strong coupling
    to K?, indicating it has a big hidden ssbar
    component. (5-quark system)

90
  • Non-observation of NX in
    suggests an evidence of new baryon
  • It is unlikely to be N(1535).
  • If NX were N(1535), it should be observed
    in process, since

  • From PDG, for the N in the mass range 15351750
    MeV, N(1535) has the largest
    , and from previous estimation, NX would also
    have almost the largest BR to K?.
  • Also, the EM transition rate of NXto proton is
    very low.

91
Side-bands do not have mass threshold enhancement
Side-bands
92
(No Transcript)
93
Partial Wave Analysis of J/? ? KK- ?0 events
  • Four decay modes are included
  • Amplitudes are defined by
  • Covariant tensor formalism
  • B.S. Zhou and D.V. Bugg, Eur. Phys. J. A16,
    537(2003)
  • BW with energy-dependent width
  • J.H. Kuhn, A. Satamaria, Z. Phys. C48, 445
    (1990).

94
Angular distributions for events with from
PWA fit
  • Figures on the right
  • (a),(c),(e) are polar angles
  • in lab. reference frame
  • (b),(d),(f) are polar angles
  • in CM frames of
  • respectively
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