Mass Spectrum and Decay Constants of Conventional Mesons within an Infrared Confinement Model

Transcription

1 Mass Spectru and Decay Constants of Conventional Mesons within an Infrared Confineent Model Gurjav Ganbold (BLTP, JINR; IPT MAS (Mongolia)) in collaboration with: T. Gutsche (Tuebingen) M. A. Ivanov (Dubna) V. E. Lyubovitskij (Tuebingen) 1

2 Outline Model Copositeness condition Infrared confineent Meson ground states, definition of ass Nuerical results: Electroagnetic decay widths of esons Leptonic decay constants of esons Mass spectru of esons Feri coupling - Soothing Suary and outlook

3 Motivation adron physics today: Experients: big quantity of high-precision data Challenge to theory Quark bound states: low oenta pqcd looses its applicability Lattice QCD: results proising but still not good established Standard Model (QCD) operates only with fundaental particles It is not yet clear how to explain the appearance of the nuerous nuber of the observed hadrons and elucidate the generation of their asses. The origin of the hadron asses = one of the puzzles. Models needed by theory & experient 3

4 Covariant Confined Quark Model (Short Review) Lagrangian-based forulation full Lorentz invariance Direct inclusion of higher-nuber quark states (baryons, tetraquarks,...) One free paraeter per hadron (for large nuber of hadrons) Wide application and convincing results: - calculation of the leptonic decay constants, - estiation of basic for factors needed for sei-leptonic, - non-leptonic and rare decays of B esons and Λ b baryons and etc. 4

5 Basic Assuptions: T. Branz et al., Phys. Rev. D81, (010). adrons (x) interact by quark exchanges with hadron-quark coupling g. Lint g ( x) J ( x) The atrix eleent <physical state bare state> is deterined by renoralization: 1/ bare phys, bare phys Z Z The copositeness condition eliinates the bare fields fro consideration. ' Z 1 gren ( M ) 0 Infrared confineent is introduced to guarantee the absence of all possible thresholds corresponding to quark production. It allows to use the sae values for the constituent quark asses. 5

6 Copositeness Condition Yukawa-type odel L q ( iˆ ) q ( M ) g ( q q) Y g 0 Generating functional + explicit integrations over quark fields ZY 0 q q exp i dx L ( x) Y 1/ gr Z g0 x y ( x y) i T ( qq) ( qq) i tr S( x y) S( y x) d ( p ) ( M ) ( p M ) '( M ) ren ( p ), '( p ) ( p ) dp Renoralization M M g ( M ) ; Z ; g Z g ; 1/ 1/ 0 0 r 0 r 0 Z [1 g '( M )] 1 g '( M ) 1 0 r Z i ig exp ( ) r ren M ren Y r r r r r n n i gr exp dx1... dxn r ( x1 )... r ( xn ) tr S( x1 x)... S( xn x1) n3 n 6

7 Feri-type odel G L q ( iˆ ) q ( q q) F G Gaussian representation: Z q q exp i dx L ( x) F 1 exp i G ( q q) exp i ( ) i( q q) F n 1 i exp i ( ) dx1... dxn ( x1 )... ( xn) tr S( x1 x)... S( xn x1) G n n () ren Bi-linear ters in boson fields: LF ( M ) ( M ) '( M ) G G Condition and renoralization 1 G ( M ) 0 ren '( M ) 1/ Z i i 1 exp ( ) ren M ren F ren ren ren ren ren '( M ) n/ n i 1 exp dx 1... dxn ren ( x1 )... ren( xn ) tr S( x1 x)... S( xn x1) n3 n '( M ) 7

8 Z ren Y Z int G int LF J ( x) LY g ( x) J ( x) Copositeness Condition ren F Z 1 gr '( M ) 0 gr '( M ) 1 G 1/ ( M ) 0 Z 1/ 0 r 0 The vanishing of the wave function renoralization constant (Z=0) in the Yukawa theory ay be interpreted as the condition that the bare (unrenoralized) field vanishes for a coposite boson. B. Jouvet, Nuovo Ci. 3, 1133 (1956) A. Sala, Nuovo Ci. 5, 4 (196) S. Weinberg, Phys.Rev. 130, 776 (1963) 1 G( M ) 0 Equation for Meson Mass 8

9 Model: Meson-Quark Interaction Interaction Lagrangian: Lint g ( x) J ( x) Quark currents (for esons): Vertex function (translational invariant): P ; V F ( x; x, x ) ( x x x ) ( x x ) j j 1 J ( x) dx dx F ( x; x, x ) q( x ) q( x ) ; 1 Vertex in Gaussian for (its Fourier transforation) : p ( ) exp p 1/Λ ~ hadron size Quark propagator (in the Schwinger representation): ˆ S p p ds s p 1 p ˆ 1( ) 1 1 exp 1 1 p 0 ˆ 9

10 Matrix eleents are cobinations of propagators and vertices: Leptonic decay constants Electroagnetic decay widths Mass (polarization) function Loop integrals over k (and external oenta p, too) are taken in Euclidean space: k ik ; p ip ; k k 0; p p E E Loop integrals are absolutely convergent (Gaussian exponentials). Loop oenta k (in the nuerator) ay be expressed by exponentials: 1 k exp( kp) exp( kp) p 10

11 Meson Ground-State Spectru Mass function (operator) for Pseudoscalar and Vector esons: 4 dk 5 5 PP ( p ) N c ( ) tr 4 P k S 1( k1 p1 ) S( k p ) ( ) i 4 dk VV ( p ) N c ( ) tr 4 V k S 1( k1 p1 ) S( k p ) ( ) i 5 P ; V Meson ass equation: 1 G( M ) dt t n b b ( p ) ds exp t z0 z 1 p 1 4 a 0 a 0 a a t a t / ; z s (1 s) s (1 s) p ; z s p ; n ; n 1; b t s. 0 1 P V t Branching point: : p ( 1 ) : z0 p s1 (1 s) s(1 s) 0 if then s dt t Integral... diverges and a threshold singularity appear s! a 0 11

12 Reoving Singularity by Infrared Cut-off Cut off the upper bound of t-integral (infrared cut-off in ters of k-integral) 1/ dt t for 0 : no threshold singularity:... converges! a 0 1/ T. Branz et al., Phys. Rev. D81, (010). 1 3 dt t n b b a 0 a 0 a a ( p ) ds exp t z z p A eson in the interaction Lagrangian is characterized by paraeters the coupling constant g the size paraeter Λ two constituent quark asses 1 & the infrared confineent paraeter λ universal for all hadrons. g g M ereby, the Yukawa couplings for all esons are reoved by Z 1 '( ) 0 Model paraeters: constituent quark asses, hadron size paraeters, a universal infrared cut-off (totally 4+N+1 paraeters for N hadrons 1+5/N per hadron) 1

13 Nuerical results for Decay constants and Widths Fixing odel paraeters by fitting the electroagnetic decay widths and leptonic decay constants. Fixed paraeters: ud b s c GeV, GeV, GeV, GeV, GeV M. A. Ivanov et al, Phys. Rev. D 85, (01). G.Ganbold, T.Gutsche, M.Ivanov, V.Lubovitsky J.Phys. G 4, (015). 13

14 Electroagnetic decay widths: 14

15 Leptonic decay constants: + Agreeent between our fit values and the PDG data is quite satisfactory. + The constituent quark asses and the values of Λ fall into the expected range. + The eson size ~1/ Λ shrinks as the ass grows. 15

16 Nuerical results for Feri coupling G Estiation of the Feri coupling: G G.Ganbold, T.Gutsche, M.Ivanov, V.Lubovitsky J.Phys. G 4, (015). 1/ ( M ) exp J PC M G M ( ) ~[GeV ] ( ) ~ [diensionless ] PC 0 PDG λ^ G J 1 (MeV) PDF (MeV) λ^ G π ρ K ω D K* D s Φ Η c D* B D*s B s J/ψ B c B* η b B*s

17 Plot of λ G by fitting eson physical asses 17

18 λ G after soothing 18

19 Nuerical results for eson asses G.Ganbold, T.Gutsche, M.Ivanov, V.Lubovitsky J.Phys. G 4, (015). Estiation of Meson Mass: 1 Gsooth ( M ) 0 J PDG (MeV) PC PC 0 J 1 λ^ G (sooth) M (MeV) PDG (MeV) λ^ G (sooth) M (MeV) π ρ K ω D K* D s Φ η c D* B D*s B s J/ψ B c B* η b B*s

20 Feri coupling G: coparison with α s It is interesting to copare G with the effective QCD coupling α s obtained in the relativistic odels with specific fors of analytically confined propagators. 1 L A A g f A A q ig t A q 4 ab a a abc b c a C C b f f f f 1/ ˆ S( pˆ) ip dt exp t ( p ) ; /4 0 x /4 sx e D( x) ds e. 4 x G.Ganbold, Phys. Rev. D 79, (009). Phys. Part. Nucl. 43, 79, (01) Phys. Part. Nucl. 45, 10, (014) In that odels the four-quark nonlocal interaction is induced by one-gluon exchange between bi-quark currents. Since the confined gluon propagator has the diension of ~ 1/GeV, the resulting coupling α s is diensionless. 8C d k V k p k V k p M 3 J 4 1 s ( ) (, ) ( ) 0; ( ) 3 J J J J V ( ) ( ) ( ) ikx J k dx D x U J x e 0

21 G.Ganbold, T.Gutsche, M.Ivanov, V.Lubovitsky J.Phys. G 4, (015) ud s c b GeV, GeV, GeV, GeV, GeV λ^*g (red curve, rescaled by 1.74) in coparison with α s (blue curve) G.Ganbold, Phys. Rev. D 81, (010). 345 MeV ud s c b 193 MeV 93 MeV 1848 MeV 4693 MeV Despite the different odel origins and input paraeter values, the behaviors of two curves are very siilar each other in the region above GeV. Their values at origin are ostly deterined by the confineent echaniss realized in different ways. This could explain their different behaviors in the region below GeV. 1

22 Suary and Outlook A brief sketch of an approach to the bound state proble in QFT based on the copositeness condition is represented. We have explicitly deonstrated that the four-ferion theory with the Feri coupling G is equivalent to the Yukawa-type theory if, - the wave function renoralization constant in the Yukawa theory is equal to zero, - G is inversely proportional to the eson ass function calculated at physical ass. The ass spectru and decay constants of conventional esons has been estiated. We calculated G as a function of physical ass. A soothness criterion by varying the eson asses in such a way to obtain the sooth behavior of the Feri coupling G. The eson ass spectru obtained in this anner is found to be in good agreeent with the recent experiental data (fro π(140) up to (9460). We have copared the behavior of G with the strong QCD coupling α s calculated in a QCD-inspired odels. The approach ay be extended to other sections of hadron physics (Ex: charoniu radial excitations, X-Y-Z esons, baryons, glueball-like states,.).