THE PREDICTION OF LEPTONIC DECAY THE PREDICTION OF LEPTONIC DECAY WIDTHS IN THE FRAME WORK OF WIDTHS IN THE FRAME WORK OF CONSTITUENT QUARK MODELSCONSTITUENT QUARK MODELS

BYBYDR. K.B. VIJAYA KUMARDR. K.B. VIJAYA KUMAR

READER,DEPARTMENT OF PHYSICSREADER,DEPARTMENT OF PHYSICSMANGALORE UNIVERSITYMANGALORE UNIVERSITY

MANGALAGANGOTRI 574 199MANGALAGANGOTRI 574 199

AIM OF THE PROJECTAIM OF THE PROJECT

To Calculate the masses of light mesons and their excited states and to compute the leptonic and radiative decay widths

To compare the masses of S,P and D wave mesons in the frame work of NRQM and RQM

With Instanton Induced Interaction (III) In both NRQM and RQM Hamiltonian III has been included

and effect on S,P and D states have been studied Using the same set of parameters compute the charge

radii and leptonic decay widths

Existing Models:Existing Models:

Non-Relativistic Quark ModelsNon-Relativistic Quark ModelsRelativistic Quark ModelsRelativistic Quark Models

Need of Instantons

Instantons are the classical solutions of QCD in Euclidean space

Instantons were originally introduced in relation to UA(1) problem

It has been argued that III is a candidate for short-range non-perturbative gluon effect and reduce the empirical strength of strong coupling constant

a) S. Takeuchi et.al., Phys.ReV.Lett.66 (1991) 1271.b) G. `t Hooft, Phys.ReV.D14 (1976) 3432.

I Non-Relativistic Quark ModelsI Non-Relativistic Quark Models The Hamitonian

Where

OGEP CONF IIIij ij ijH K V V V

22

1 2i

i cmi i

PK M K

M

1 2( ) . 1 . ( )

4 3cent sOGEP ij i j i j ij

ij i j

V r rr M M

( ) .CONFij ij c ij i jV r a r

Bhavyashri, K.B.Vijaya Kumar , B.Hanumaiah, S. Sarangi, Shan-Gui Zhou, J. Phys.G: Nucl.Part.Phys. 31 (2005) 981.

Continued….Continued…...

,0 ,08 ( )ij S Lg r

1I

,0 ,0( ) 8 ( )CentIII ij ij S LV r g r

1

2I

,0 ,0

2 '8 ( )

2 ' 0ij S L

g gr

g

0I

2 2

3

1( ) ( )

( )ijr exp r

where

The non-central part ofThe non-central part of OGEPV

3

1 1 ˆ( ) .4 4

Ten sOGEP ij i j ij

i j ij

V r SM M r

3

3 1( ) . .

4 8SO sOGEP ij i j ij ij i j

i j ij

V r r PM M r

The non-central part of IIIThe non-central part of III Spin-orbit interaction of III

( ) ( ) ( )SOIII ij LS ij L ijV r V r L S V r L

where2 2 2 22 4

2

2 2 3 31 3 2

exp( ) exp( )1 1 1( )

( ) ( )ij k ij k

LS ij k kk ki j i jk k

r rV r

M M M M

2 264

2 2 35 4

exp( )1 1( )

( )ij k

L ij kki j k

rV r

M M

Tensor interaction of III 2 28

4

37 4

ˆ exp( )( )

( )ij ij kT

III ij kki j k

S rV r

M M

C. Semay, B. Silvestre-Brac, Nucl.Phys. A 647 (1999) 72

CalculationsCalculations

Values of the parameters used in the model

The product of the quark anti-quark wave function for each meson is expressed in terms of oscillator wave functions corresponding to the CM and relative coordinates. The Hamiltonian was diagonalized

1

2

3

4

5

6

0.213

0.091

-2.13

2.651

38.64

40.43

u,d

s

c

s

-4 -2

b 0.77 fm

M 371 MeV

M 512 MeV

a 23.0 MeV fm-1

0.3

g 0.0847 × 10 MeV

-4 -2

1

2

3

4

g' 0.0535 × 10 MeV

0.35 fm

0.194 fm

0.294 fm

0.112 fm

1.501

fm

The Pseudo scalar meson masses (in The Pseudo scalar meson masses (in MeV) for successive values of nMeV) for successive values of nmax max =11=11 in in NRQMNRQM

The vector meson masses (in MeV) for The vector meson masses (in MeV) for successive values of nsuccessive values of nmax max =11=11 in NRQMin NRQM

Meson Expt.Mass (MeV)

Calculated.Mass (MeV)

b1(1235) 1229.5 3.2 1229.14

h’1(1380) 138619 1388.06

K1(1400) 1402 7 1403.7

Pseudo-Vector P wave meson masses in NRQMPseudo-Vector P wave meson masses in NRQM

(

2 1S

JN L

Meson Experimental

Mass (MeV)

Calculated Mass (MeV)

13P0 f0(600)

f0(980)

a0(980)

K0*(1430)

a0(1450)

f0(1500)

400 - 1200980 10

984.7 1.21412 601474 191507 5

700.41979.81988.53

1412.531473.591512.28

13P1 a1(1260)

K1(1270)

f1(1285)

f1(1420)

1230 401273 701281.8 .61426.3 .9

1233.07 1274.10 1281.95 1423.37

13P2

f2(1270)

a2(1320)

K2*(1430)

f2'(1525)

1275.4 1.2 1318. 61425.61.5 1525 5

1276.16 1347.65 1432.41 1529.33

Masses of P wave scalar, axial vector and tensor meson nonetsMasses of P wave scalar, axial vector and tensor meson nonets in in NRQMNRQM

N2S+1

LJ

Meson Experimental Mass

Calculated Mass

11D2 π2 (1670) 1670±20 1696.6

K2 (1770) 1773±8 1727.4

Masses of D wave singlet mesonsMasses of D wave singlet mesons in NRQMin NRQM

(1650)

3

3

N2S+1LJ Meson Experimental Mass Calculated mass

13D1 1649 ± 24 1649.8

K*(1680) 1717 ± 27 1720.2

13D2 K2(1820) 1816 ±13 1817.2

13D3 (1670) 1667 ± 4 1667.1

K*(1780) 1776 ± 7 1778.9

(1850) 1854 ± 7 1855.3

Masses of D wave triplet mesonsMasses of D wave triplet mesons in NRQMin NRQM

II. Relativistic Harmonic ModelsII. Relativistic Harmonic Models (RHM)(RHM)• This model was highly successful in explaining This model was highly successful in explaining

various aspects of hadron spectroscopy, baryon various aspects of hadron spectroscopy, baryon magnetic moments, properties of glue balls etc.magnetic moments, properties of glue balls etc.

• In RHM quarks are Dirac particles subjected to In RHM quarks are Dirac particles subjected to Lorentz scalar plus vector potentials. The Dirac Lorentz scalar plus vector potentials. The Dirac equation with a general potential is written as, equation with a general potential is written as,

Where Where S(r)S(r) is the scalar potential and is the scalar potential and V(r)V(r) is the is the time component of the vector potential. time component of the vector potential.

• The pure vector potential would produce only The pure vector potential would produce only bound states, whereas the scalar potential bound states, whereas the scalar potential provides an attractive force for both and provides an attractive force for both and q - q q - q states. states.

• Thus, for the confinement of quarks, a scalar plus Thus, for the confinement of quarks, a scalar plus vector potential is the more appropriate choice .vector potential is the more appropriate choice .

( ) ( )m S r V r E α P

q q

q q

• In the RHM [3], quarks in a hadron are confined through In the RHM [3], quarks in a hadron are confined through the action of a Lorentz scalar plus a vector harmonic the action of a Lorentz scalar plus a vector harmonic oscillator potential,oscillator potential,

• where is the Dirac matrix, , and where is the Dirac matrix, , and MM is the is the quark mass and is the confinement strength. They quark mass and is the confinement strength. They have a different value for each quark flavor. In the RHM, have a different value for each quark flavor. In the RHM, the confined single quark wave function the confined single quark wave function is given by: is given by:

with the normalization, with the normalization,

where E is the eigen value of the Single particle Dirac where E is the eigen value of the Single particle Dirac equation with the interaction potentialequation with the interaction potential

2 20

1( ) 1

2confV r A r M

0 0

1 00 1

2A

N

E M

σ P

2( )

3

E MN

E M

( )ConfV r

• By making similarity transformation, the upper By making similarity transformation, the upper component component satisfies the harmonic oscillator satisfies the harmonic oscillator potential, potential,

• The eigen values for the above expression are given The eigen values for the above expression are given by, by,

where (n≥1) is the energy dependent oscillator size where (n≥1) is the energy dependent oscillator size parameter. parameter.

• The total mass of the hadron is obtained by adding the The total mass of the hadron is obtained by adding the individualindividual

contribution of the quarks.contribution of the quarks.

n

22 2 ( )A r E M

E M

P

2 2 2(2 1)n nE M n

1 2

n nA E M

Meson Spectra in RHM with COGEP Meson Spectra in RHM with COGEP andand III III

• The earlier phenomenological quark models The earlier phenomenological quark models discussed so far have incorporated the effect of discussed so far have incorporated the effect of confinement of quarks on mesonic states, but the confinement of quarks on mesonic states, but the effect of confinement of gluons on mesonic states effect of confinement of gluons on mesonic states has not been taken into account. has not been taken into account.

• The confinement schemes for quarks and gluons The confinement schemes for quarks and gluons have to be more closely connected to each other in have to be more closely connected to each other in QCD and the confinement of gluons has to be taken QCD and the confinement of gluons has to be taken into account. into account.

• The gluons which are the quanta of the color field The gluons which are the quanta of the color field carry color charges which interact among carry color charges which interact among themselves. themselves.

• The Confined Gluon Propagators are used to obtain The Confined Gluon Propagators are used to obtain the the confined one gluon exchange potentialconfined one gluon exchange potential ( (COGEPCOGEP). ).

The Hamiltonian is given byThe Hamiltonian is given by

Confined One Gluon Exchange Confined One Gluon Exchange Potential (COGEP)Potential (COGEP)

The COGEP is obtained from the scattering The COGEP is obtained from the scattering amplitude,amplitude,

2

( )4 2 2

bajabs i

fi i i j j

gM D q

( ) ( ) ( )centconf ij COGEP ij III ijH V V V r r r

• The central part of COGEP is given byThe central part of COGEP is given by

wherewhere

• The tensor part has the form,The tensor part has the form,

COGEP contd….COGEP contd….

• The spin orbit part of COGEP is given byThe spin orbit part of COGEP is given by

Results of S waveResults of S wavePseudo scalar meson masses (MeV)Pseudo scalar meson masses (MeV)

nn KK ηη ηη`̀

00 433.22433.22 685.26685.26 663.70663.70 1388.521388.52

11 280.56280.56 583.44583.44 600.21600.21 1009.921009.92

22 156.78156.78 503.48503.48 555.45555.45 985.87985.87

33 142.25142.25 494.37494.37 550.71550.71 967.59967.59

44 139.65139.65 492.77492.77 549.95549.95 964.41964.41

ExptExpt..

138.04138.04 495.01495.01 547.3547.3 957.78957.78

Vector meson masses (MeV)Vector meson masses (MeV)

nn ρρ ωω K*K* ΦΦ

00 771.98771.98 779.84779.84 891.92891.92 1023.061023.06

11 771.77771.77 778.92778.92 891.02891.02 1021.051021.05

22 771.74771.74 778.39778.39 890.95890.95 1020.411020.41

33 771.73771.73 778.36778.36 890.72890.72 1020.371020.37

44 771.72771.72 778.19778.19 890.72890.72 1020.361020.36

ExptExpt..

769.30769.30 783.00783.00 893.14893.14 1019.421019.42

ConclusionsConclusions

• Both OGEP,COGEP are attractive for PSM, for Both OGEP,COGEP are attractive for PSM, for VM OGEP/COGEP is repulsive and hence VM OGEP/COGEP is repulsive and hence perturbative techniques are adequateperturbative techniques are adequate

• Colour electric terms of OGEP/COGEP Colour electric terms of OGEP/COGEP contributes significantly to the massescontributes significantly to the masses

• Using III brings down the value of Using III brings down the value of ααs <1s <1

• It is required to diagonalise in a smaller basis in RHM It is required to diagonalise in a smaller basis in RHM and contribution of the terms arising out of and contribution of the terms arising out of confinement of gluons is significant. confinement of gluons is significant.

• The experimental masses of The experimental masses of ηη and and ηη` are obtained by ` are obtained by diagonalzing the interaction MEsdiagonalzing the interaction MEs

• The mixing of PS isosinlet (The mixing of PS isosinlet (ηη0 0 ) and PS iso-octet () and PS iso-octet (ηη88 ) ) turns out to be quite largeturns out to be quite large

• The III lowers the mass of the (The III lowers the mass of the (ηη0 0 ) state, pushes up ) state, pushes up the mass of (the mass of (ηη88 ) )

Results of P waveResults of P wave

2 1SJN L MesonMeson Expt. Expt.

Mass (MeV)Mass (MeV) Calculated Calculated

Mass Mass (MeV)(MeV)

1133PP00 ff00(600)(600)

ff00(980)(980)

aa00(980)(980)

KK00**(1430)(1430)

aa00(1450)(1450)

ff00(1500)(1500)

400 - 600400 - 600980 980 10 10

984.7 984.7 1.2 1.214121412 60 6014741474 19 191507 1507 5 5

700.41700.41979.81979.81988.53988.53

1412.531412.531473.591473.591512.281512.28

1133PP11 aa11(1260)(1260)

KK11(1270)(1270)

ff11(1285)(1285)

ff11(1420)(1420)

1230 1230 40 401273 1273 70 701281.81281.8 .6 .61426.3 1426.3 .9.9

1233.071233.071274.101274.101281.951281.951423.371423.37

1133PP22 ff22(1270)(1270)

aa22(1320)(1320)

KK22**(1430)(1430)

ff22'(1525)'(1525)

1275.41275.4 1.2 1.21318 1318 . 6. 6

1425.61425.61.51.51525 1525 5 5

1276.161276.161347.651347.651432.411432.411529.331529.33

2 1SJN L

MesonMeson Expt. Expt. Mass (MeV)Mass (MeV)

Calculated Calculated Mass (MeV) Mass (MeV)

1111PP11 bb1(1235)1(1235)

hh11`(1380)`(1380)

KK1b1b(1400)(1400)

1229.5±3.21229.5±3.2

1386±191386±19

1402±71402±7

1229.141229.14

1388.061388.06

1403.71403.7

ConclusionsConclusions

• The inclusion of III diminishes the importance of The inclusion of III diminishes the importance of OGEP/COGEP and restricts the value of alpha to be less OGEP/COGEP and restricts the value of alpha to be less than 1.than 1.

• The mass degeneracy of pseudo vector (K1B) and axial The mass degeneracy of pseudo vector (K1B) and axial vector(K1A) mix to give physical mass K1(1270) and vector(K1A) mix to give physical mass K1(1270) and K1(1400) observed experimentally –can be accounted by K1(1400) observed experimentally –can be accounted by anti-symmetric spin orbit term anti-symmetric spin orbit term

• VLVLΔΔ of III of III

• The near mass degeneracy of the experimentally observed The near mass degeneracy of the experimentally observed iso-doublet states of the scalar and tensor meson nonents iso-doublet states of the scalar and tensor meson nonents K0* and K2* could be accounted by the off-diagonal tensor K0* and K2* could be accounted by the off-diagonal tensor ME of OGEP/COGEP and III. ME of OGEP/COGEP and III.

Results of D waveResults of D waveNN2S+12S+1LLJJ MesonMeson Exp. Exp.

massmassCal. massCal. mass

1111DD22∏∏22(1670)(1670)

KK22(1770)(1770)

1670±201670±20

1773±81773±8

1673.81673.8

1133DD11

ωω(1650)(1650)

K*(1680)K*(1680)

1649±241649±24

1717±271717±271672.21672.2

1707.41707.4

1133DD22KK22(1820)(1820) 1816±131816±13 1812.11812.1

1133DD33ωω33(1670)(1670)

K*(1780)K*(1780)

ΦΦ33(1850)(1850)

1667±41667±4

1776±71776±7

1854±71854±7

1719.21719.2

1776.71776.7

1899.71899.7

ConclusionsConclusions

Computation of the D wave masses with only Computation of the D wave masses with only OGEP/COGEP is inadequateOGEP/COGEP is inadequate

In the K sector spin anti-symmetric part L.S part In the K sector spin anti-symmetric part L.S part of III contributes significantly to the mass of III contributes significantly to the mass difference between 1difference between 111DD22(K(K221770) and 1770) and 1133DD22(K(K221820)1820)

Substantial contribution from the tensor term Substantial contribution from the tensor term ratio of tensor forces for ratio of tensor forces for 33DD11::33DD22::33DD33=-7:7:2=-7:7:2

Leptonic Decay widths of Ground state light vector mesons

Theoretical overview

b 0.9 fm

M u,d 352 MeV

Ms 545 MeV

αs. 0.6

ac 20.0 MeV fm-1

Meson Experimental Mass[] Calculated Mass[]

775.49±0.34 MeV 771.53 MeV

ω 782.65±0.12 MeV 771.53 MeV

1019.46±0.02 MeV 1019.47 MeV

||ΨΨ(0)|(0)|22(MeV)(MeV)33

Meson Meson Exp.|Exp.|ΨΨ(0)|(0)|2 2 (MeV)(MeV)33 Cal. |Cal. |ΨΨ(0)|(0)|22(MeV)(MeV)33

ρρ 3.16 x 103.16 x 1066 2.97 x 102.97 x 1066

ωω 2.47 x 102.47 x 1066 2.97 x 102.97 x 1066

ФФ 4.44 x 104.44 x 1066 4.39 x 104.39 x 1066

Leptonic decay width of vector mesonsLeptonic decay width of vector mesons

MesoMeson n

Exp. Exp. ГГ (KeV) (KeV) Cal. Cal. ГГ (KeV) (KeV)

ρρ 7.047.04±0.06±0.06 6.76.7

ωω 0.600.60±0.02±0.02 0.740.74

ФФ 1.271.27±0.03±0.03 1.261.26

ConclusionsConclusions

• The RHM with COGEP is superior to The RHM with COGEP is superior to NRQMNRQM

• Both the models reproduce the Both the models reproduce the experimental masses but in RHM the experimental masses but in RHM the result is achieved by diagonalizing a result is achieved by diagonalizing a smaller matrix.smaller matrix.

• There is a good agreement with the There is a good agreement with the leptonic decay widthsleptonic decay widths