Covariant Vector meson-Vector meson Interactions and Dynamically - PowerPoint PPT Presentation

Covariant Vector meson-Vector meson Interactions and Dynamically Generated Resonances Dilege Gülmez HISKP , Universität Bonn 15 th International Workshop on Meson Physics Uniwersytet Jagiello´ nski, Kraków June 7-12, 2018

Overview 1 Formalism 2 ρρ Interactions The on-shell Bethe-Salpeter Equation (BSE) The N / D method 3 The extension to SU ( 3 ) The BSE method in coupled channels Single channels Coupled Channels 4 Results and Summary

The Hidden Gauge Formalism Lagrangian & On-Shell BSE � Vector-Meson Interactions: L 4 V = g 2 2 � V µ V ν V µ V ν − V µ V µ V ν V ν � L 3 V = ig � ( ∂ µ V ν − ∂ ν V µ ) V µ V ν � (3 V) (4 V) � The on-shell BSE: T ( s ) = [ 1 − V ( s ) · G ( s )] − 1 · V ( s ) V = + + + d 4 q G ii ( s ) = i � 1 ( 2 π ) 4 ( q 2 − M 2 1 + i ǫ )(( P − q ) 2 − M 2 2 + i ǫ )

Partial-wave Interaction Kernels DG, Meißner, Oller, Eur. Phys. J. C 77 (2017) no.7, arXiv:1611.00168 [hep-ph] Molina, Nicmorus, Oset, Phys. Rev. D 78 (2008) 114018, arXiv:0809.2233 [hep-ph] 150 500 100 0 50 - 500 0 V 00 V 20 - 1000 - 50 - 100 - 1500 - 150 - 2000 - 200 - 2500 1100 1200 1300 1400 1500 1600 1700 1100 1200 1300 1400 1500 1600 1700 s ( MeV ) s ( MeV ) � 3 m 2 ρ = 1343 MeV. ⊲ The LHC starts at ⊲ No/A pole is found (at 1255 MeV). ⊲ A pole is found at 1467 (1491) MeV. ⊲ It is associated with the f 2 ( 1270 ) . It is associated with the f 0 ( 1370 ) . ⊲ ⊲ f 2 ( 1270 ) fits very well within the ideal P − wave qq nonet (analyses of the high-statistics Belle data a , the Regge The width: 200 − 500 MeV in PDG. ⊲ theory b ). ⊲ Far away from the ρρ threshold (1551 MeV). a Dai, Pennington, Phys. Rev. D 90 (2014) 036004, arXiv:1404.7524 [hep-ph] b Ananthanarayan, Colangelo, Gasser, Leutwyler, Phys. Rept. 353 (2001) 207, arXiv:0005297 [hep-ph]

Partial-wave Interaction Kernels DG, Meißner, Oller, Eur. Phys. J. C 77 (2017) no.7, arXiv:1611.00168 [hep-ph] Molina, Nicmorus, Oset, Phys. Rev. D 78 (2008) 114018, arXiv:0809.2233 [hep-ph] 150 500 100 0 50 - 500 0 V 00 V 20 - 1000 - 50 - 100 - 1500 - 150 - 2000 - 200 - 2500 1100 1200 1300 1400 1500 1600 1700 1100 1200 1300 1400 1500 1600 1700 s ( MeV ) s ( MeV ) Importance of LHCs: The NN scattering a : One pion exchange starts at ⊲ No/A pole is found (at 1255 MeV). ⊲ p 2 = − m 2 π → s < 4 ( m 2 N − m 2 π / 4 ) and two pion ⊲ It is associated with the f 2 ( 1270 ) . exchange starts at p 2 = − 4 m 2 π → s < 4 ( m 2 N − m 2 π ) . ⊲ f 2 ( 1270 ) fits very well within the ideal P − wave qq nonet (analyses of the high-statistics Belle data, the Regge ⊲ Apply this to the ρρ scattering ( m π and m N are to be theory). replaced by m ρ ). ⊲ Far away from the ρρ threshold (1551 MeV). ⊲ Ignoring the ρρ exchange means ignoring LHCs of OPE like term of the interaction. The dynamics of low-energy NN scattering has the ⊲ highest contribution from OPE. a Guo, Oller, Ríos, Phys. Rev. C 89 (2014) 014002, arXiv:1305.5790 [hep-th]

Partial-wave Interaction Kernels 500 p 0 - 500 V 20 - 1000 - 1500 - 2000 - 2500 1100 1200 1300 1400 1500 1600 1700 Red Circle: Bound State ( a > 0), Green Square: Virtual state s ( MeV ) ( a < 0) ⊲ The effective range is determined by: 8 π √ s No/A pole is found (at 1255 MeV). ⊲ T ( s ) = − 1 / a + r 0 p 2 − i p . It is associated with the f 2 ( 1270 ) . ⊲ r 0 is proportional to d ( √ sT − 1 ) / dE � ⊲ f 2 ( 1270 ) fits very well within the ideal P − wave qq nonet ⊲ . � � E = √ s t h (analyses of the high-statistics Belle data, the Regge theory). ⊲ As the slope of the potential increases, r 0 decreases. ⊲ Far away from the ρρ threshold (1551 MeV). ⊲ Therefore, the effective range for the scalar sector is much larger than the tensor sector. a + ir 0  i a 2 + . . .  p = ⊲ a − ir 0 r 0 − i i a 2 + . . .  ⊲ In the non-relativistic approach, both sectors have the same energy-dependence, thus, positive effective range (follows blue solid lines: r 0 a > 0). ⊲ In the covariant form, the tensor sector has a negative effective range (follows purple dashed lines: r 0 a < 0).

The N / D Method (JI)=(00) ReD(s) 1.5 ReD, q max =1 GeV ReD U , q max =1 GeV 0 1 ⊲ T = N ( s ) / D ( s ) . 0.5 Re(D(E 2 )) ⊲ N has only LHCs and D has only RHCs. ⊲ D = 0 corresponds to resonances or bound-states. 0 ⊲ N ( s ) = V ( s ) (first iterated solution). -0.5 D ( s ) diverges as s 2 . Therefore, three subtractions in ⊲ the dispersion relation for D(s): -1 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 E (GeV) D ( s ) = γ 0 + γ 1 ( s − s t h ) + 1 2 γ 2 ( s − s t h ) 2 (JI)=(00) ImD(s) 1 + ( s − s t h ) s 2 � ∞ ρ ( s ′ ) V ( JI ) ( s ′ ) ImD, q max =1 GeV ds ′ ImD U , q max =1 GeV ( s ′ − s t h )( s ′ − s )( s ′ ) 2 . π s t h 0 0.5 0 Matching condition in the threshold region up to O ( s 3 ) : ⊲ Im(D(E 2 )) -0.5 γ 0 + γ 1 ( s − s th ) + 1 2 γ 2 ( s − s s t h ) 2 -1 = 1 − V ( s ) G c ( s ) − ( s − s t h ) s 2 � ∞ ρ ( s ′ ) V ( s ′ ) ds ′ -1.5 ( s ′ − s t h )( s ′ − s )( s ′ ) 2 . π s t h -2 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 E (GeV)

The N / D Method (JI)=(00) D(s) 1.5 q max =0.7 GeV q max =1 GeV q max =1.3 GeV 1 0 0.5 ⊲ T = N ( s ) / D ( s ) . ⊲ N has only LHCs and D has only RHCs. D(E 2 ) 0 ⊲ D = 0 corresponds to resonances or bound-states. -0.5 ⊲ N ( s ) = V ( s ) (first iterated solution). D ( s ) diverges as s 2 . Therefore, three subtractions in ⊲ -1 the dispersion relation for D(s): -1.5 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 E (GeV) D ( s ) = γ 0 + γ 1 ( s − s t h ) + 1 2 γ 2 ( s − s t h ) 2 (JI)=(20) D(s) 0 + ( s − s t h ) s 2 � ∞ ρ ( s ′ ) V ( JI ) ( s ′ ) q max =0.7 GeV ds ′ q max =1 GeV ( s ′ − s t h )( s ′ − s )( s ′ ) 2 . π q max =1.3 GeV s t h -2 -4 Matching condition in the threshold region up to O ( s 3 ) : ⊲ D(E 2 ) -6 γ 0 + γ 1 ( s − s th ) + 1 2 γ 2 ( s − s s t h ) 2 -8 = 1 − V ( s ) G c ( s ) − ( s − s t h ) s 2 � ∞ ρ ( s ′ ) V ( s ′ ) ds ′ -10 ( s ′ − s t h )( s ′ − s )( s ′ ) 2 . π s t h -12 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 E (GeV)

The BSE approach in coupled channels DG, Du, Guo, Meißner, Wang, Preliminary Results � In coupled channel calculations, g is evaluated with the average mass of vector mesons ( g = 4 . 596). ( S,I )=( 0,0 ) ( S,I )=( 1,1 / 2 ) 100 J = 0 100 J = 0 ρ K * threshold 0 _ _ * → K * K K * ϕ threshold V K * ϕ→ K * ϕ 50 - 100 V K * K � The LHC overlaps with the - 200 ρρ threshold 0 RHC. * threshold K * K - 300 - 50 - 400 1.0 1.2 1.4 1.6 1.8 2.0 1.0 1.2 1.4 1.6 1.8 2.0 s ( GeV ) s ( GeV ) 5 1.0 4 Det ρρ→ρρ Det coupled � The comparison of the 0.5 Det ( I − V · G ) in the single 3 ( S,I )= 0, J = 0 ( S,I )= 0, J = 0 ρρ threshold and coupled channel isoscalar 0.0 2 scalar channel: - 0.5 1 ρρ threshold No pole is found in the � - 1.0 0 coupled channel! Bound State - 1 The on-shell BSE method is - 1.5 � 1.2 1.3 1.4 1.5 1.6 1.7 1.8 not valid for coupled channels. 1.2 1.3 1.4 1.5 1.6 1.7 1.8 s ( GeV ) s ( GeV )

Single channels in SU ( 3 ) (BSE) Single channels in SU ( 3 ) : I = 2 ( ρρ ) , I = 3 / 2 ( K ∗ K ∗ ) , I = 1 ( K ∗ K ∗ ) , I = 0 ( K ∗ K ∗ ) , and ( I , J ) = ( 0 , 1 ) ( K ∗ K ∗ ) . � Poles: 1 pole in ( I , J ) = ( 0 , 1 ) ( K ∗ K ∗ ) , and 2 poles in ( 0 , 1 ) ( K ∗ K ∗ ) . � ( S,I,J )=( 0,0,1 ) ( S,I,J )=( 0,0,1 ) 1 1 q max = 0.775 GeV q max = 1.0 GeV 0 0 _ _ * → K * K _ _ * → K * K The pole is an artefact of the � Det K * K Det K * K - 1 LHC. - 1 It is the same scenario for � - 2 - 2 ( 0 , 1 ) ( K ∗ K ∗ ) . * threshold * threshold K * K K * K - 3 - 3 1.4 1.5 1.6 1.7 1.8 1.9 1.4 1.5 1.6 1.7 1.8 1.9 s ( GeV ) s ( GeV ) 0.4 0.2 0.0 0.2 - 0.2 _ _ * → K * K _ * → K * K _ 0.0 � The second pole is sensitive ( S,I,J )=( 0,0,1 ) Det K * K Det K * K - 0.4 to the cutoff. ( S,I,J )=( 0,0,1 ) - 0.2 q max = 1. GeV - 0.6 � For g = M ρ / f , the pole is q max = 0.775 GeV - 0.4 Virtual State Bound State virtual for q max < 1 . 03 GeV. - 0.8 - 0.6 - 1.0 1.72 1.74 1.76 1.78 1.80 1.82 1.72 1.74 1.76 1.78 1.80 1.82 s ( GeV ) s ( GeV )

Single channels in SU ( 3 ) ( N / D ) 1.5 1.5 ( S,I,J )=( 2,0,1 ) ( S,I,J )=( 0,0,1 ) 1.0 1.0 q max = 0.775 GeV � The second pole behaviour, in q max = 0.775 GeV _ * Re D K * K * → K * K * K ∗ K ∗ , does not change. _ * → K * K 0.5 0.5 Re D K * K � One of the LHC artefacts 0.0 0.0 disappears where the other RS - I N / D - 0.5 - 0.5 moves deeper on the real * threshold RS - II BSE K * K K * K * threshold - 1.0 - 1.0 axis. 1.4 1.5 1.6 1.7 1.8 1.9 1.4 1.5 1.6 1.7 1.8 1.9 s ( GeV ) s ( GeV ) 4 30 Det IN / D Det IN / D 25 � We have found three ( S,I,J )=( 1,3 / 2,0 ) Det IU 3 Det IU ( S,I,J )=( 1,3 / 2,2 ) resonance poles on the 2 nd 20 Det II N / D Det II N / D Re Det Re Det R.S. in (2,2), (3/2,2), (1,2). 2 15 Det II U Det II U � In the tensor sector, matching 10 1 does not work well for the 2 nd 5 R.S. . 0 0 1.2 1.4 1.6 1.8 2.0 1.2 1.4 1.6 1.8 2.0 s ( GeV ) s ( GeV )