THE PREDICTION OF LEPTONIC DECAY WIDTHS IN THE FRAME WORK OF CONSTITUENT QUARK MODELS BY DR. K.B. VIJAYA KUMAR READER,DEPARTMENT OF PHYSICS MANGALORE UNIVERSITY MANGALAGANGOTRI

AIM 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: Non-Relativistic Quark Models Relativistic Quark Models

Need of Instantons  Instantons are the classical solutions of QCD in Euclidean space  Instantons were originally introduced in relation to U A (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) b)G. `t Hooft, Phys.ReV.D14 (1976) 3432.

I Non-Relativistic Quark Models The Hamitonian Where Bhavyashri, K.B.Vijaya Kumar, B.Hanumaiah, S. Sarangi, Shan- Gui Zhou, J. Phys.G: Nucl.Part.Phys. 31 (2005) 981.

Continued….. where

The non-central part of

The non-central part of III The non-central part of III Spin-orbit interaction of III where Tensor interaction of III C. Semay, B. Silvestre-Brac, Nucl.Phys. A 647 (1999) 72

Calculations 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

The Pseudo scalar meson masses (in MeV) for successive values of n max =11 in NRQM

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

MesonExpt. Mass (MeV) Calculated. Mass (MeV) b 1 (1235)  h’ 1 (1380) 1386  K 1 (1400) 1402  Pseudo-Vector P wave meson masses in NRQM (

MesonExperimental Mass (MeV) Calculated Mass (MeV) 1 3 P 0 f 0 (600) f 0 (980) a 0 (980) K 0 *(1430) a 0 (1450) f 0 (1500)      P 1 a 1 (1260) K 1 (1270) f 1 (1285) f 1 (1420) 1230     P 2 f 2 (1270) a 2 (1320) K 2 *(1430) f 2 '(1525)     Masses of P wave scalar, axial vector and tensor meson nonetsin NRQM Masses of P wave scalar, axial vector and tensor meson nonets in NRQM

N 2S+1 L J MesonExperimental Mass Calculated Mass 11D211D2 π2 (1670)1670± K 2 (1770)1773± Masses of D wave singlet mesonsin NRQM Masses of D wave singlet mesons in NRQM

N 2S+1 L J MesonExperimental MassCalculated mass 13D113D ± K*(1680)1717 ± D213D2 K 2 (1820)1816 ± D313D3 (1670) 1667 ± K*(1780)1776 ± (1850)1854 ± Masses of D wave triplet mesonsin NRQM Masses of D wave triplet mesons in NRQM

II. Relativistic Harmonic Models (RHM) This model was highly successful in explaining various aspects of hadron spectroscopy, baryon magnetic moments, properties of glue balls etc. This model was highly successful in explaining various aspects of hadron spectroscopy, baryon magnetic moments, properties of glue balls etc. In RHM quarks are Dirac particles subjected to Lorentz scalar plus vector potentials. The Dirac equation with a general potential is written as, In RHM quarks are Dirac particles subjected to Lorentz scalar plus vector potentials. The Dirac equation with a general potential is written as, Where S(r) is the scalar potential and V(r) is the time component of the vector potential. Where S(r) is the scalar potential and V(r) is the time component of the vector potential. The pure vector potential would produce only bound states, whereas the scalar potential provides an attractive force for both and q - q states. The pure vector potential would produce only bound states, whereas the scalar potential provides an attractive force for both and q - q states. Thus, for the confinement of quarks, a scalar plus vector potential is the more appropriate choice. Thus, for the confinement of quarks, a scalar plus vector potential is the more appropriate choice.

In the RHM [3], quarks in a hadron are confined through the action of a Lorentz scalar plus a vector harmonic oscillator potential, In the RHM [3], quarks in a hadron are confined through the action of a Lorentz scalar plus a vector harmonic oscillator potential, where is the Dirac matrix,, and M is the quark mass and is the confinement strength. They have a different value for each quark flavor. In the RHM, the confined single quark wave function  is given by: where is the Dirac matrix,, and M is the quark mass and is the confinement strength. They have a different value for each quark flavor. In the RHM, the confined single quark wave function  is given by: with the normalization, with the normalization, where E is the eigen value of the Single particle Dirac equation with the interaction potential where E is the eigen value of the Single particle Dirac equation with the interaction potential

By making similarity transformation, the upper component  satisfies the harmonic oscillator potential, By making similarity transformation, the upper component  satisfies the harmonic oscillator potential, The eigen values for the above expression are given by, The eigen values for the above expression are given by, where (n≥1) is the energy dependent oscillator size parameter. The total mass of the hadron is obtained by adding the individual The total mass of the hadron is obtained by adding the individual contribution of the quarks. contribution of the quarks.

Meson Spectra in RHM with COGEP and III The earlier phenomenological quark models discussed so far have incorporated the effect of confinement of quarks on mesonic states, but the effect of confinement of gluons on mesonic states has not been taken into account. The earlier phenomenological quark models discussed so far have incorporated the effect of confinement of quarks on mesonic states, but the effect of confinement of gluons on mesonic states has not been taken into account. The confinement schemes for quarks and gluons have to be more closely connected to each other in QCD and the confinement of gluons has to be taken into account. The confinement schemes for quarks and gluons have to be more closely connected to each other in QCD and the confinement of gluons has to be taken into account. The gluons which are the quanta of the color field carry color charges which interact among themselves. The gluons which are the quanta of the color field carry color charges which interact among themselves. The Confined Gluon Propagators are used to obtain the confined one gluon exchange potential (COGEP). The Confined Gluon Propagators are used to obtain the confined one gluon exchange potential (COGEP).

The Hamiltonian is given by Confined One Gluon Exchange Potential (COGEP) The COGEP is obtained from the scattering amplitude, The COGEP is obtained from the scattering amplitude,

The central part of COGEP is given by The central part of COGEP is given bywhere The tensor part has the form, The tensor part has the form,

COGEP contd…. The spin orbit part of COGEP is given by The spin orbit part of COGEP is given by

Results of S wave Pseudo scalar meson masses (MeV) n Kη η`η`η`η` Expt

Vector meson masses (MeV) nρ ω K* K* Φ Expt

Conclusions Both OGEP,COGEP are attractive for PSM, for VM OGEP/COGEP is repulsive and hence perturbative techniques are adequate Both OGEP,COGEP are attractive for PSM, for VM OGEP/COGEP is repulsive and hence perturbative techniques are adequate Colour electric terms of OGEP/COGEP contributes significantly to the masses Colour electric terms of OGEP/COGEP contributes significantly to the masses Using III brings down the value of αs <1 Using III brings down the value of αs <1

It is required to diagonalise in a smaller basis in RHM and contribution of the terms arising out of confinement of gluons is significant. It is required to diagonalise in a smaller basis in RHM and contribution of the terms arising out of confinement of gluons is significant. The experimental masses of η and η` are obtained by diagonalzing the interaction MEs The experimental masses of η and η` are obtained by diagonalzing the interaction MEs The mixing of PS isosinlet (η 0 ) and PS iso-octet (η 8 ) turns out to be quite large The mixing of PS isosinlet (η 0 ) and PS iso-octet (η 8 ) turns out to be quite large The III lowers the mass of the (η 0 ) state, pushes up the mass of (η 8 ) The III lowers the mass of the (η 0 ) state, pushes up the mass of (η 8 )

Results of P wave MesonExpt. Mass (MeV) Mass (MeV) Calculated Mass (MeV) Calculated Mass (MeV) 13P013P013P013P0 f 0 (600) f 0 (980) a 0 (980) K 0 * (1430) a 0 (1450) f 0 (1500)      P113P113P113P1 a 1 (1260) K 1 (1270) f 1 (1285) f 1 (1420) 1230     P213P213P213P2 f 2 (1270) a 2 (1320) K 2 * (1430) f 2 '(1525)    

MesonExpt. Mass (MeV) Mass (MeV) Calculated Mass (MeV) Calculated Mass (MeV) 11P111P111P111P1 b 1(1235) h 1 `(1380) K 1b (1400) ± ±191402±

Conclusions The inclusion of III diminishes the importance of OGEP/COGEP and restricts the value of alpha to be less than 1. The inclusion of III diminishes the importance of OGEP/COGEP and restricts the value of alpha to be less than 1. The mass degeneracy of pseudo vector (K1B) and axial vector(K1A) mix to give physical mass K1(1270) and K1(1400) observed experimentally –can be accounted by anti-symmetric spin orbit term The mass degeneracy of pseudo vector (K1B) and axial vector(K1A) mix to give physical mass K1(1270) and K1(1400) observed experimentally –can be accounted by anti-symmetric spin orbit term VLΔ of III VLΔ of III The near mass degeneracy of the experimentally observed iso- doublet states of the scalar and tensor meson nonents K0* and K2* could be accounted by the off-diagonal tensor ME of OGEP/COGEP and III. The near mass degeneracy of the experimentally observed iso- doublet states of the scalar and tensor meson nonents K0* and K2* could be accounted by the off-diagonal tensor ME of OGEP/COGEP and III.

Results of D wave N 2S+1 L J Meson Exp. mass Cal. mass 11D211D211D211D2 ∏ 2 (1670) K 2 (1770) 1670±201773± D113D113D113D1 ω(1650) K*(1680)1649±241717± D213D213D213D2 K 2 (1820) 1816± D313D313D313D3 ω 3 (1670) K*(1780) Φ 3 (1850) 1667±41776±71854±

Conclusions  Computation of the D wave masses with only OGEP/COGEP is inadequate  In the K sector spin anti-symmetric part L.S part of III contributes significantly to the mass difference between 1 1 D 2 (K ) and 1 3 D 2 (K )  Substantial contribution from the tensor term ratio of tensor forces for 3 D 1 : 3 D 2 : 3 D 3 =-7:7:2

Leptonic Decay widths of Ground state light vector mesons Theoretical overview

b0.9 fm M u,d 352 MeV MsMs 545 MeV α s. 0.6 acac 20.0 MeV fm -1 MesonExperimental Mass[]Calculated Mass[] ±0.34 MeV MeV ω782.65±0.12 MeV MeV  ±0.02 MeV MeV

|Ψ(0)| 2 (MeV) 3 Meson Meson Exp.|Ψ(0)| 2 (MeV) 3 Cal. |Ψ(0)| 2 (MeV) 3 ρ 3.16 x x 10 6 ω 2.47 x x 10 6 Ф 4.44 x x 10 6

Leptonic decay width of vector mesons Meson Exp. Г (KeV) Exp. Г (KeV) Cal. Г (KeV) ρ 7.04± ω 0.60± Ф 1.27±

Conclusions The RHM with COGEP is superior to NRQM The RHM with COGEP is superior to NRQM Both the models reproduce the experimental masses but in RHM the result is achieved by diagonalizing a smaller matrix. Both the models reproduce the experimental masses but in RHM the result is achieved by diagonalizing a smaller matrix. There is a good agreement with the leptonic decay widths There is a good agreement with the leptonic decay widths