theoretical description of the charmonium structure in qcd

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


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


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 –

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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


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)

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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

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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


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

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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:

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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“

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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

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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

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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:

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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

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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

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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

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Ψ(2S) → 0 hc → 0 c

CLEO data with

hc

background function plus signalMeV

22


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

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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)

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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?

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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

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ΨGJPC

ΨIGJPC

Candidates for hybrids

ΨIGJPC

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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

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Thanks for your attention!

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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

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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

theoretical description of the charmonium structure in qcd

Documents

e e y e e

charmonium spectroscopy

charmonium state jy

j e e x1974

theoretical potential

hc singlet pstates

triplet pstates

charmonium structure