theoretical description of the charmonium structure in qcd
DESCRIPTION
Theoretical description of the charmonium structure in QCD. Gabi Hoffmeister 06.12.2007. Summary. 1. Introduction 2. Charmonium spectroscopy and theoretical potential models 3. Transitions and decays of cc 4. New states above the DD-threshold 5. Conclusion. 1. Introduction. - PowerPoint PPT PresentationTRANSCRIPT
Theoretical description of the charmonium structure in QCD
Gabi Hoffmeister06.12.2007
2
Summary
1. Introduction
2. Charmonium spectroscopy and theoretical potential models
3. Transitions and decays of cc
4. New states above the DD-threshold
5. Conclusion
3
1. Introduction
Color-suppressed b c decay Predominantly from B-meson decays
e+e- annihilation/Initial State Radiation (ISR) e+e- collision below nominal cm energy JPC = 1
Double charmonium production Typically one J/ or , plus second cc state
Two-photon production Access to C = +1 states
pp annihilation All quantum numbers available
Charmonium production
J = 0,2 J = 1
JScc
resonances…
Untagged : Charmonium states with JPC = 0+, 2+
4
1. Introduction
1974: first charmonium state J/ with mJ/ = 3096 MeV discovered (SLAC: e+e- → → e+e-, , hadrons and BNL: p + Be → J → e+e- + X)
1974: discovery of Ψ´ (excited 3S1 state) with mΨ´ = 3.686 GeV and Γ ≤ 2.7 MeV at SLAC Studying of radiative decays of Ψ´: BR (Ψ´ → J/Ψ ) = 0.32 BR (Ψ´ → J/Ψ → neutrals) = 0.25 No other narrow resonances found from reactions e+e- → hadrons 1976: c,1,2,3 (triplet states 3P0,1,2) discovered from radiative decays of ´→ c,J 1980: discovery of 1S0 singlet c with mass m = 2.98 GeV in decay ´→ c 1982: c´ (excited state of c) seen at Crystall Ball (SPEAR) with mc = 3594 MeV
1977: Discovery of upsilon meson (bottonium bb with JPC = 1--) at Fermi Lab with m ≈ 9.46 GeV via p-Cu interaction again with very narrow width ~52 keV
Many excited states of the like in case of J/Ψ (similar energy levels) Bound state tt non observed: top-quark decays before building a bound state (t → W+ + b)
History of discovered charmonium states
5
2. Charmonium spectroscopy and theoretical potential models
J/Ψ
ηc
ηc‘
hc
c
´
• Singlet S-states (spin 0): c, c´ Singlet P-states (spin 0): hc
• Triplet S-states (spin 1): J/´,´´,… Triplet p-states (spin 1): 1,2,3
Charmonia:Charmonia:
6
2. Charmonium spectroscopy and theoretical potential models
Experimental data can be used to compare results to the expected values of different theoretical potential models
Charmonium states
www.e18.physik.tu-muenchen.de/teaching/struktur-dynamik-hadronen/ charmonium_1.pdf –
7
2. Charmonium spectroscopy and theoretical potential models
c,b are heavy quarks can be treated in nonrelativisticnonrelativistic approximations (Schrödinger equation + static potential) because relativistic corrections are small
At small distances: one-gluon exchange dominates (asymptotic freedom): V ~ 1/r At large distances confining potential:
Coulomb + linear potential: krr
V s 3
4
Vector part Vv
Scalar part Vs
=> Fits to the data show that Vv is small
Contributions to the cc-potential:
„Cornell-Potential“
k (´) ≈ 0.18 GeV² is the string tension (energy density of qq pair in string model of hadrons) with typical slope ´= 1 GeV² of a hadronic Regge trajectory
8
2. Charmonium spectroscopy and theoretical potential models
Fine structur splitting (spin orbit interaction):
Vs: scalar part from confining term
Vv: vector part from one-gluon (vector boson) exchange
Spin spin (splitting of singlet and triplet states):
→ no contribution from Vs
Tensor term:
By computating the various expectation values one obtains mass splitting relations:
2
42JSL
1 (3P2)
-1 (3P1)
-2 (3P0)
4
32 2
21
SSS
¼ (3S1)
-¾ (1S0)
)32)(12(2
46122
222
LL
LSSLSLSten
-1/5 (3P2)
0 (3P1)
-2 (1P0)
9
2. Charmonium spectroscopy and theoretical potential models
The resulting mass relations for the triplet are:
tenso mmmPm5
1)( 2
3 tenso mmmPm )( 1
3tenso mmmPm 22)( 1
0
→ testing if long-range potential transforms as a 4-scalar Vs or a 4-vector Vv considering a modification of the Cornell model (V = br - a/r):
,)1( brVs r
abrVv 10 with
Inserting this potential model and setting 3
1
ra
rb
58
51916
5
2
)()(
)()(
03
13
13
23
PmPm
PmPmR
0.66 (b(1P))
0.70 (b(2P))
experimental data on -P-wave
for vector confinement ( ≈ 1) formula in accord with experimental data only for≈ 0, whereas scalar confinement ( ≈ 0) larger range 0.4 ≤ ≤ 1.0 in accord with exp. values
Conclusion: confinement produced by a long-range 4-scalar interaction
10
3. Transitions and decays of cc Annihilation:
Generally suppressed for bound state Decay to leptons is a clean experimental signal
Strong interaction: Dominant above ~3.72 GeV (D mesons) Suppressed below this mass threshold
Radiative transition: EM radiative transition emitting photon Emission of gluons producing light quarks
Features:Features: Suppression of strong decays leads to (relatively) long lifetimes, narrow widths Radiative decays are competitive; often most accessible transitions Selection rules:
Conservation of J Conservation of P,C in strong and electromagnetic decays
11
3. Transitions and decays of cc
All quarkonia are unstable and decay through: 1) annihilation processes and
2) radiative transitions
1) Annihilation processes (electromagnetic and hadronic decays):
2
2
22
01)( )0(
4)0()( nn
em
mvS
for a bound state with wavefunction n(x) in electromagnetic decay
Including QCD radiative corrections and substituting the electric charge by ec = (2/3)e for the charm-quark charge and a color factor of 3:
)(6.101
3
)0()(8)(
)(4.31
81
)0(192)(
2
22
01
2
22
01
cs
c
ncsggcc
cs
c
ncc
m
m
mSn
m
mSn
c
cgg or
c
c
ggg or gg
c
c f
f
for 3S1 state
el.mag. decay
hadronic decay
12
3. Transitions and decays of cc
Decays from the 3S1-system with 3 final particles or a lepton pair including QCD radiative corrections :
)(9.01
81
)0()9(128)(
)(6.121
2187
)0()9(1024)(
)(9.41
81
)0()()9(40)(
)(
3
161
9
)0(64)(
2
222
13
2
232
13
3
2
232
13
3
2
22
13
cs
c
sggcc
cs
c
ncc
cs
c
ncsgcc
cs
cc
nllcc
m
mSn
m
mSn
m
m
mSn
m
MSn electromagnetic decay
hadronic decay
el.mag. decay
Problems:- factor lΨn(0)l² comes from non relativistic approximation, can be modified by relativistic corrections- second order terms O(s²) could play an important role
13
3. Transitions and decays of cc hadronic transitions: J/Ψ, Ψ´→ PV, PP, VV (P: pseudoscalar and V: vector mesons)
el.magn. J/ and ´ decays into meson pairs
mixing mechanism for charmonium decays into meson pairs
G-parity and isospin violating transitions with BR ~ 10-4 - 10-3, supressed by factor ~10-2 - 10-1 compared to G-parity and isospin allowed J/Ψ decays
Charmonium state possesses Fock components of light quarks, can therefore decay through these by a soft mechanism; node in 2S radial function leads to suppression of mechanism in Ψ´decays
with quark flavor basis:
mixings:
14
3. Transitions and decays of cc
G parity violating transition
Isospin violating transition
branching ratios of decays of J/Ψ and Ψ´ into meson pairs from experimental data (Beijing Electron Spectrometer Collaboration)
flavor symmetry breaking mixing
)003.0124.0()/(
)´(
)/(
)´(
llJBR
llBR
fJBR
fBR
„12%-rule“
15
3. Transitions and decays of cc 2) radiative transitions (M1 and E1 dipole transitions):
M1transitions (no parity changespin flip:L = 0,S = 1):J/Ψ→ c Ψ→ c Ψ´ → c´ →
J/Ψ
tiii
rki
i
i
ifi eikipe
Vm
QfeH i
)(ˆ2
2
1
2**int
Dipole approximation:
Schrödinger wave function for charmonium: (r) = spin·Ylm Rnl(r)
...1 i
rki rkie i
ti
ii
i
iMfi eik
m
Qf
V
ieH
)(
22*1
with Ei Ef
2
02
32
212
3
4i
rjfJ
m
ef
c
cmag
where j0 is the spheric Bessel function (jo(x) = sin(x)/x )
ii i
i
m
Q 2
fi EmM
i
fMfimag diHf
m
EViHfEmdd
V 21int2
221
int2
2 )2()(
)2(
where
2·(phase space)
Relativistic corrections and anomalous magnetic moment for quarks are neglected!
for E1, M1
16
3. Transitions and decays of cc
E1 transitions (parity changes, no spin flip: L , S ):
titi
i
ii
i
Efi eir
m
Q
m
Qf
V
ieei
m
piQf
VieH
*
2
2
1
1*
2
ˆ
2
11
Ψ´ → c,J → J/Ψ
fifc
el SirfJe 2
32
1227
4
Jf : spin of the final state and Sfi =
1 for spin singlet transition
3 for spin triplet transitions
where
0
2 )}()({ rrRrRdrrirf if
estimation of decay width by building ratios: )/()( ,3,
3,
,
Jh Jch
cc
Jc
c
Determination of s(mc):
22
32
*1
31
3
81
)()9(1031
3
i
Ss
Q
m
llS
hadronsgS
)(
)(
)9(10
81)(
2
223
llV
hadronsVQm i
Vs
from experimental decay width one gets: s(m) ≈ 0.44 s(mJ/Ψ) ≈ 0.21, s(m) ≈ 0.18
17
3. Transitions and decays of ccHigher multipole contributions in charmoniumHigher multipole contributions in charmonium
Magnetic quadrupole (M2) amplitudes provide indirect measure of charmed quark´s anomalous magnetic moment and are sensitive to D-wave mixtures in S-wave states (Ψ´´ – Ψ´)
Affect angular distributions in decays Ψ´→ c,J and c,J → J/Ψ (experimentally accessible through interference with dominant E1 amplitudes)
Radiative widths given by helicity amplitudes A, A´ with labelling the projection of the spin of c,J parallel (A) or antiparallel (A´) to the photon
setting ≡ ·E/(4mc) where for Ψ´→ c,J and for c,J → J/Ψ
cquark anomalous magnetic moment
(deviation from Dirac magnetic moment c = ⅔ ec/(2mc))
Searching for interferences with dominant E1 amplitudes (c,J → J/): expected normalized M2/E1 ratios a2:
18
3. Transitions and decays of ccHadronic transitions [QQ Hadronic transitions [QQ → (QQ)´+ light hadrons]→ (QQ)´+ light hadrons] examples:
theoretical description uses multipole expansion for gluon emission, very similar to usual multipole expansion for photonic transitions:
(color electric and color magnetic emission from a heavy quark)
Single interaction of HI changes color singlet QQ initial state i into some color octett QQ state, second interaction HI is required to return to a color singlet QQ final state (f) -> at least two gluons have to be emitted
Ordering of amplitudes in powers of velocity with leading contribution from color electric gluon emissions:
above DD-threshold: → J/Ψ and Y(3940) → J/Ψ
ta: generator of color SU(3),(a = 1,…,8)
sum over all allowed QQ octett intermediate states nO
lowest mass light hadron state:
S-wave 2-system
D-wave 2-system
HEEEE
ftxnntxi kb
ja
n ni
bkOO
aj
O O
0
19
3. Transitions and decays of ccProperties of Ψ(2S) → c,J E1 radiative transition with tot [Ψ(2S)] = 33713 keV
Properties of transitions c,J → J/Ψradiative transition
Partial widths and BR for spin-singlet states,
O = r (GeV-1) for E1 and O = j0(kr/2) for M1 transitions
c´ → hc
hc→ c
Ψ´→ c´/ c
Ψ´→ J/Ψ
Phys. Rev. D
20
3. Transitions and decays of cc
Ψ(2S)Decay to c(1S):
• forbidden M1 transition (would vanish in the limit of E = 0 because of orthogonality of 1S and 2S wave functions) at photon energy of 638 MeV → ≠ 0 averaged BR = (3.00.5)·10-3 => [Ψ(2S) → c(1S)] = (1.00 0.16) keV
Decay to c(2S):
• allowed M1 transition characterized by ≈ 1 for small photon energies• Assumption: matrix elements for (2S) → c´(2S) and J/(1S) → c(1S) are equal
=> (2S-2S)-rate = times (1S-1S)-rate leading to a BR = (2.6 0.7)·10-4 and therefore [Ψ(2S) → c´(2S)] = (87 25) eV
Hadronic transitions to J/Ψ:• Via electric dipole emission of gluon pair followed by its hadronization into
dominating decay mode in pions
21
3. Transitions and decays of cc
Ψ(2S) → 0 hc → 0 c
CLEO data with
hc
background function plus signalMeV
22
4. New states above the DD-threshold
Discovery of a new signal X(3872) in B+X K+, XJ/Ψ at Belle in 2003 with narrow width < 2.3 MeV and mass mX = 3871.20.6 MeV
Confirmed by CDF, D0 and BaBar
• X J/Ψ radiative decay confirmed by BaBar determines C = +1• Belle/CDF dipion angular analysis in XJ/Ψ favours JPC = 1++
• not seen in X JΨ=> neutral state
23
4. New states above the DD-thresholdInterpretation of X(3872)• similar to charmonium: narrow state decaying to J/Ψ• above DD threshold should be wide and XDD dominant• Quantum numbers established: 1++
• It does not fit into the charmonium model!• m(X) ≈ m(D) + m(D*0) => X could be a bound state of 2 D mesons, a D0D*0 molecule
assumption supported by predictions of mass, decay modes, JPC, branching fractions and small binding energy (deuteron like)
• Other exotic predictions: - “tetraquark” 4 quark bound state - “glueball” gluon bound state, charmonium-gluon hybrid ccg
Further new states discovered: X(3940): - discovered by Belle in double charmonium
production e+e-J/Ψ X(3940) - Decays to DD* but not DD and J/Ψ - Likely excited charmonium state (c’’’ or c1’) - JPC = 0-+,1++ ? XDD
PRL 98, 082001 (2007)
24
4. New states above the DD-threshold Z(3930) - observed in the two-photon decays
Z(3930) DD
- Predicted mass and width match charmonium assignment of c2’
- JPC = 2++
Y(3940)- - discovered by Belle the decay BKY, Y (J/Ψ)
- Possible c1’ charmonium state but
requires further investigation
- not found in DD or DD* final states
- JPC = 1++, …
2MeV/c)13113943()( Ym
MeV)262287()( Y2MeV/c)13113943()( Ym
YJ
DD
If X=Y, difficult to explain absence of Y If X=Y, difficult to explain absence of Y open charm => Hybrid? open charm => Hybrid?
25
4. New states above the DD-threshold Y(4260) new peak in ISR events discovered at
Babar, found in decay Y(4260)J/Ψ
e+e- requires quantum numbers JPC = 1--
However, all of the 1-- charmonium states have already been discovered!
Very difficult to accommodate as cc, unless previous assignments are wrong
for Y(4260)J/Ψ, Belle reproduces BaBar’s signal:
Broad second peak at slightly lower mass:
226 MeV/c)84259()( Ym
MeV)2388()( 64 Y
21726 MeV/c)124247()( Ym
MeV)19108()( 810 Y
27228 MeV/c)404008((?) mMeV)44226((?) 87
79
26
4. New states above the DD-threshold
ΨGJPC
ΨIGJPC
Candidates for hybrids
ΨIGJPC
27
5. Conclusion Charmonium states and decay widths can be calculated quite well in NRQCD but
in order to obtain a higher precision relativistic corrections have to be included Determination of s(mc) from various rations of decay widths Charmonium model with has great success below the DD-
threshold Above DD threshold, several states remain undiscovered or need further study A recent flood of experimental results from the B-factories is challenging our
understanding of the strong force:
- What is the nature of the new “Y” states? Meson molecules? Tetraquarks? Hybrids? Glueballs? Something else? Rich new spectroscopy?
What excited unknown states do exist? => waiting for data of (upgraded) B-factories like Babar, Belle, CLEO, BES
searching for resonances with non-quarkonium JPC (1-+, …)
krr
V s 3
4
28
Thanks for your attention!
29
References „A modern introduction to Particle Physics“, chapter 8, Fayyazuddin Riazuddin, World
Scientific „Dynamics of the Standard Model“, chapter 13, J.F. Donoghue E. Golowich B.R. Holstein,
Cambridge Monographs on particle physics, Nuclear physics and cosmology Lecture notes Università di Pisa, Prof. V. Cavasinni, Particelle Elementari I, „modello a
quark“, 2006/07 www.e18.physik.tu-muenchen.de/teaching/struktur-dynamik-hadronen/ charmonium_1.pdf – www.e18.physik.tu-muenchen.de/teaching/struktur-dynamik-hadronen/ charmonium_2.pdf – theory.gsi.de/~leupold/lecture-1-13_7_07.pdf „Implications of light-quark admixtures on charmonium decays into meson pairs“, Phys. Rev.
D, Vol. 62, 074006, T. Feldmann, P. Kroll http://uk.arxiv.org/PS_cache/arxiv/pdf/0711/0711.1927v2.pdf http://arxiv.org/abs/hep-ph/0701208 www-rnc.lbl.gov/ISMD/talks/Aug9/1130_Fulsom.ppt „Production of singlet P-wave cc and bb states“, Phys. Rev. D 66, 014012 (2002), S. Godfrey,
J.L. Rosner „Two-pion transitions in quarkonium revisited“. Phys. Rew. D 74, 05022 (2006), M.B. Voloshin
30
Determinating s(mc)
Extraction from partial decay widths ratios from J/Ψ:
Extraction from c:
Extraction from c,J:
Extraction from J/Ψ:
10.005.0
2 19.0)( cs m
03.005.0
2 30.0)( cs m
≈ 1.8 => large correction => caution with the value
016.0019.0
2 296.0)( cs m forc,2 and 02.0
02.02 32.0)(
cs m for c,0
08.008.0
2 175.0)( cs m
31
New charmonium states
Y(4350)S
Further resonances observed in Further resonances observed in ee++ee-- Y YISRISR (certainly J (certainly JPCPC=1=1----))
Most of these 1-- states should preferentially decay into D(*)D(*) states. ΨΨΨ[regular charmonia] clearly visible, nothing else
J
’
Only seen in Ψ(2S)
can be fit by a tetraquark model (decaying into J/f0 …) or a hybrid (with
Ψs to place!
32
Multipole expansion in QCD
Chromo-electric dipole transition:
Chromo-magnetic dipole transition:
For→
where
so with effective Hamiltonian
mixing of 3D1 – 3S1 in 3S1 statesS-wave D-wave
for 3S1-states
for 1S0-states
where supressed by (v/c)²
coordinate partsof S-wave functions