Photon-fusion reactions in a dispersive effective field theory from the chiral Lagrangian with vector-meson fields Igor Danilkin (collaborators: Matthias F. M. Lutz, Stefan Leupold, Carla Terschlüsen, Laura I. R. Gil) I. V. Danilkin, M. F. M. Lutz, S. Leupold and C. Terschlusen, arxiv:1211.153 [hep-ph]. I. V. Danilkin, L. I. R. Gil and M. F. M. Lutz, Phys. Lett. B 73, 54 (211) Igor Danilkin (GSI) December 5, 212 1 / 31
Table of contents Motivation Description of the method Relevant degrees of freedom Chiral Lagrangian with vector-meson fields Partial-wave dispersion relation Conformal mapping technique Numerical results Summary and K scattering γγ, KK, ηη and η reactions Igor Danilkin (GSI) December 5, 212 2 / 31
Motivation P P γ γ PP =, KK, ηη, η J PC restricted to be (even) ++ γγ reactions probe scalar resonances (e.g. σ, f (98),...) New experimental data have been reported by the Belle Collaboration not only for γγ +, but also for γγ η, ηη Poorly known γγ K + K, K K reactions both theoretically and experimentally The total cross sections for γγ PP are very sensitive to hadronic final state interaction decent description of PP PP extend to γγ PP Our aim is to obtain a unified description of PP PP and γγ PP up to 1.1-1.2 GeV Igor Danilkin (GSI) December 5, 212 3 / 31
Running coupling The QCD coupling constant α s at high energies asymptotic freedom perturbative QCD at low energies confinement nonperturbative QCD Possible way out Effective field theories PDG, Phys. Rev. D 86, 11 (212) Igor Danilkin (GSI) December 5, 212 4 / 31
Chiral dynamics Chiral perturbation theory (χpt) Spontaneous chiral symmetry breaking Goldstone bosons (, K, K, η) Effective degrees of freedom: hadrons Systematic expansion at low energies Unitarity at perturbative level only Extension to resonance region Inclusion of vector-meson degrees of freedom in χpt (next slide) Nonperturbative effects are required exact coupled-channel unitarity analyticity EM gauge invariance This has been done in the novel unitarization scheme A. Gasparyan and M. F. M. Lutz, Nucl. Phys. A 848, 126 (21) Igor Danilkin (GSI) December 5, 212 5 / 31
Relevant degrees of freedom Importance of light vector mesons Resonance saturation mechanism: LEC of (Q 4 ) counter terms are saturated by vector-meson exchange Hadrogenesis conjecture: G.Ecker, J.Gasser, A.Pich and E.de Rafael, Nucl. Phys. B 321 (1989) 311 + 1 1 + + 1 f 1(1282), a 1(123),... M.F.M. Lutz, E. Kolomeitsev, Nucl. Phys. A 73, 392 (24) + + + f (98), a (98),... I. V. Danilkin, L. I. R. Gil and M. F. M. Lutz, Phys. Lett. B 73, 54 (211) 1 + 1 2 + 1 + 1 f 2(127), a 2(132),... (open challenge) Igor Danilkin (GSI) December 5, 212 6 / 31
Inclusion of vector mesons: power-counting scheme spectrum at large N c. Λ hard J P = ±, 1±,... J P = 1 J P = mass gap C. Terschluesen, S. Leupold and M.F.M. Lutz, arxiv:124.4125 [hep-ph] hadrogenesis conjecture Power-counting scheme expansion parameter: Λ soft /Λ hard pure χpt: Λ soft m P, Q Λ hard 4f or m V Our case: (DOF not included in the Lagrangian) Vector mesons are part of the Lagrangian Dynamical generation of resonances ( +, 1 +,...) We expect Λ hard (2 3) GeV in our power-counting scheme light vector mesons are treated as soft: m V O(Q) complete picture: need to explore vector-meson loop effects... (open challenge) Igor Danilkin (GSI) December 5, 212 7 / 31
Chiral Lagrangian with vector-meson fields The LO chiral Lagrangian for the Goldstone-boson P (, K, K, η) and vector-meson V µν (ρ µν, ω µν, K µν, K µν, φ µν) fields: L = 1 { } 48f tr [P, µ P] 2 [P, µp] + P 4 χ i f V h { } P tr µp V µν νp 8f 2 e2 2 Aµ A µ tr { P Q [ P, Q ] } e + i 2 Aµ tr { [ µp Q, P ] } e fv µa ν tr { V µν Q } i f V h P tr { µp V µν } νp + e f V 8 f 2 8 f 2 µaν tr{ V µν[ P, [ P, Q ] ] } + e f V h P A 8f 2 ν tr {[ µp, V µν] [ ] } 1 Q, P 16 f tr{ µ [[ V ] 2 µα P, νp, V να] } b D 64 f tr{ V µν [ [ ] } g 1 V 2 µν P, P, χ ]+ + 32 f tr {[ V 2 µν, ] [ αp + α P, V µν] +} g2 32 f tr {[ V 2 µν, ] [ αp α P, V µν] g 3 } 32 f tr {[ µp, ν P ] [ Vντ, V µτ ] 2 + +} g5 32 f tr {[ V µτ, ] [ µp Vντ, ν P ] h A 2 } 16 f ɛ µναβ tr {[ V µν, τ V τα] + β P } b A 16 f ɛ µναβ tr {[ V µν, V αβ] +[ χ, P ] } h O + 16 f ɛ µναβ tr {[ α V µν, V τβ] τ P} + Q - charge matrix, χ diag(m 2, m2, 2m2 K m2 ) Igor Danilkin (GSI) December 5, 212 8 / 31
Parameters Most of the parameters were determined by the two-body decays at LO (ρ 2, ρ e + e, ρ γ,...) M. F. M. Lutz and S. Leupold, Nucl. Phys. A 813, 96 (28) S. Leupold and M. F. M. Lutz, Eur. Phys. J. A 39, 25 (29) f V.14 GeV, h A 2.1, b A =.27, h P f V =.23 GeV, b D =.92, f.9 GeV for PP PP: no unknown parameters (predictive power) for γγ PP: five open parameters we fix them from γγ +,, η and η γγ empirical data predictions for γγ K K, K + K and η η Igor Danilkin (GSI) December 5, 212 9 / 31
Scattering amplitudes We calculate the following tree-level diagrams PP PP PP γγ valid at low energies only and will serve as an input for the nonperturbative coupled-channel calculations (final-state interaction) Igor Danilkin (GSI) December 5, 212 1 / 31
The coupled-channel states (I G, S) ( 1 2, 1) ( 3 2, 1) (+, ) (γ γ) ( 1 1 q σ K 3 p) ( ( q T K p) q 3 p) (η q K p) 1 2 ( K q K p + K p K q) (η q η p) (1 +, ) (1, ) (2 +, ) (γ γ) ( 1 i q p) 2 (γ γ) 1 2 ( K q σ K p K ( p σ K q η p) ( ) 1 q) 1 2 ( K q σ K p + K 2 p σ K (i q j p + q j p i ) 1 3 δ ij q p q) Consider scattering and rescattering of the type γγ PP and PP PP, respectively, but disregard PP VV, VP (important for s > 1.1 1.2 GeV) Igor Danilkin (GSI) December 5, 212 11 / 31
Partial-wave dispersion relation Unitarity and analyticity T J ab(s) = U J ab(s) + c,d µ 2 thr d s s µ 2 M s µ 2 M T J ac( s) ρ J cd( s) T J db ( s) s s iɛ separate left- and right-hand cuts the generalized potential U J ab (s) contains all left-hand cuts U J ab(s) computed in χpt in the threshold region analytically extrapolated (conformal mapping) EM gauge invariance matching with χpt The phase space function Im T J ab(s) = c,d Tac(s) J ρ J cd(s) Tdb J (s), ρ J ab(s) = 1 ( ) 2 J+1 pcm δ ab 8 s Igor Danilkin (GSI) December 5, 212 12 / 31
Approximation for the generalized potential U J ab (s) In χpt one can compute amplitudes only in the close-to-threshold region (asymptotically growing potential) The potential U(s) is needed only for energies above threshold Reliable extrapolation is possible: conformal mapping techniques Typical example: U(ω) = ln (ω) (left-hand cut at ω < ) Expansion around ω = 1? ( 1) k+1 [ω 1] k k! k=1 C k [ξ(ω)] k k= w 1 ξ(w) = 1 w 1+ w ξ 1 Igor Danilkin (GSI) December 5, 212 13 / 31
Conformal mapping for U(ω) = ln (ω) ln (ω) : N ( 1) k+1 [ω 1] k vs. k! k=1 N C k [ξ(ω)] k k= 3 2 1 Taylor [w 1 ] Taylor [ξ 1 ] Taylor expansion in ω 1 converges for < ω < 2 1 2 Taylor [w 2 ] Taylor expansion in ξ converges for < ω < 3 2 4 6 w the coefficients C k = dk U(ω(ξ)) k! dξ k ξ= are determined at ω = 1 (ξ = ) Igor Danilkin (GSI) December 5, 212 14 / 31
Conformal mapping for PP PP We need U(s) for energies above threshold N U(s) = C k [ξ(s)] k for s < Λ 2 s k= Reliable approximation is within red area s ρ 4m 2 m 2 ρ 4m 2 Λ 2 S The coefficients C k are determined at s = 4m 2, where χpt is reliable Igor Danilkin (GSI) December 5, 212 15 / 31
Conformal mapping for γγ PP We need U(s) for energies above threshold U(s) = ds T (s ) s s 4 9m2 Reliable approximation is within red area + N C k [ξ(s)] k for s < Λ 2 s k= γ s 9 m2 4 γ (m2 ρ m 2 ) 2 m 2 ρ 4 m 2 Λ 2 S γ γ The coefficients C k are determined at s = m 2, where χpt is reliable Igor Danilkin (GSI) December 5, 212 16 / 31
Results for K scattering (δ IJ ) 9 6 Mercer et. al. Estabrooks et. al. Bingham et. al. Baker et. al. 1 δ1/2 3 δ3/2 2 K K.6.9 1.2 Estabrooks et. al. Linglin at. al. K K 3.6.9 1.2 δ1/2 1 18 12 6 Mercer et. al. Estabrooks et. al. K K.6.9 1.2 U(s) = N C k [ξ(s)] k k= in p-wave CDD poles ρ, K No free parameters! Igor Danilkin (GSI) December 5, 212 17 / 31
Results for scattering (δ IJ ) δ 3 2 1 NA48/2 K Grayer Sol.A Grayer Sol.B Grayer Sol.C Grayer Sol.D Grayer Sol.E. Protopopescu et. al. Kaminski et. al..2.7 1.2 δ2 1 2 Hoogland et. al. Losty et. al. Cohen et. al. Durusoy et. al. (OPE) Durusoy et. al. (OPE DP) 3.2.7 1.2 δ11 18 12 6 Protopopescu et. al. Estabrooks et. al..2.7 1.2 U(s) = N C k [ξ(s)] k k= in p-wave CDD poles ρ, K f (98) dynamically generated No free parameters! Igor Danilkin (GSI) December 5, 212 18 / 31
Minor cutoff dependence: 1.4 GeV (dashed line) < Λ s < 1.8 GeV (solid line) 18 Protopopescu et. al. Estabrooks et. al. 18 Mercer et. al. Estabrooks et. al. δ11 12 δ1/2 1 12 6 6.2.7 1.2 9 6 Mercer et. al. Estabrooks et. al. Bingham et. al. Baker et. al. K K.6.9 1.2 1 δ1/2 3 δ3/2 2 δ K K.6.9 1.2 3 2 1 NA48/2 K Grayer Sol.A Grayer Sol.B Grayer Sol.C Grayer Sol.D Grayer Sol.E. Protopopescu et. al. Kaminski et. al..2.7 1.2 δ2 Estabrooks et. al. K K Linglin at. al. 3.6.9 1.2 1 2 Hoogland et. al. Losty et. al. Cohen et. al. Durusoy et. al. (OPE) Durusoy et. al. (OPE DP) 3.2.7 1.2 Igor Danilkin (GSI) December 5, 212 19 / 31
Results for η scattering T11 2 /1 3 4.5 3 1.5 η η.8 1 1.2 there is no elastic scattering data η channel can be populated by inelastic γγ η data J = and (I G, S) = (1, ) a (98) dynamically generated f (98) and a (98) resonances are strongly coupled to K K channel Igor Danilkin (GSI) December 5, 212 2 / 31
Photon-fusion reactions The differential cross section with two helicity amplitudes dσ d cos θ = p cm 32 s s φ ++ = φ + = even J even J 2 We evaluate J =, 2 partial-wave amplitudes ( φ ++ 2 + φ + 2) (2J + 1) t (J) (2J + 1) t (J) ++ d (J) + d (J) 2 (cos θ) (cos θ) Five unknown parameters g 1, g 2, g 3, g 5 and h O have to be determined Strategy: fix them from γγ, +, η and η γγ Cross sections γγ K K, K + K and η η: pure predictions Igor Danilkin (GSI) December 5, 212 21 / 31
Results for γγ, +, η 3 2 Mark II Belle CELLO Belle syst. error γγ + - 6 4 Crystal Ball Belle Belle syst. error γγ 1 2 6 4 2.3.7 1.1 Crystal Ball Belle Belle syst. error γγ η.7.95 1.2.3.7 1.1 g 1 =.9.2 g 3 +.38 h 2 O +.128 h O g 2 = 1.5.27 g 3 +.25 g 5 Bands correspond to g 3, g 5, h O [ 5, 5] Dashed curves: g i = h O = Igor Danilkin (GSI) December 5, 212 22 / 31
Result for η γγ decay: only g 1, g 3 and h O contribute dγ/dmγγ 2 [ev/gev 2 ] 6 4 2 AGS MAMI COSY η γγ.9.18 2 M γγ [GeV 2 ] η γγ is linked to γγ η by crossing symmetry dγ/dmγγ 2 = T η γγ 2 dmγ 2 2 1 1 (2) 3 32 mη 3 pol We find: h O = 3.27, g 3 = 4.88 Therefore: g 1 = 2.7 and g 2 =.18 +.25 g 5 The fit gives the full width for the η decay of Γ η γγ =.31 ev Γ exp η γγ.35 ±.9 ev In the following we show the results for g 5 [ 5, 5] Igor Danilkin (GSI) December 5, 212 23 / 31
Comparison with tree-level result 3 2 Mark II Belle CELLO Belle syst. error γγ + - 6 4 Crystal Ball Belle Belle syst. error γγ 1 2 6 4 2.3.7 1.1 Crystal Ball Belle Belle syst. error γγ η.7.95 1.2.3.7 1.1 Solid curves: unitarized result Dashed curves: tree-level result The bands correspond to g 5 [ 5, 5] γγ does not have one-pion exchange (dominated by loop/rescattering effects) Igor Danilkin (GSI) December 5, 212 24 / 31
Comparison with M. R. Pennington, T. Mori, S. Uehara and Y. Watanabe, Eur. Phys. J. C 56 (28) 1 45 s-wave (sol. A) s-wave (sol. B) I= 21 s-wave (sol. A) I=2 s-wave (sol. B) 3 14 15 7.2.8 1.4 45 I= 3 15 d-wave (sol. A) d-wave (sol. B) d2-wave (sol. A) d2-wave (sol. B).2.8 1.4 9 6 3 d2-wave (sol. A) I=2 d2-wave (sol. B).2.8 1.4.2.8 1.4 Results are in reasonable agreement with p.w. amplitude analysis in view of parameter free description of f (98) and absence f 2(127) Igor Danilkin (GSI) December 5, 212 25 / 31
Comparison with pure χpt 45 3 Mark II χ PT: LO χ PT: N 2 LO γγ + - 21 14 Crystal Ball χ PT: NLO χ PT: N 2 LO γγ 15 7.3.4.5.3.4.5 Results are consistent with χpt χpt:nlo χpt:n 2 LO J. Bijnens and F. Cornet, Nucl. Phys. B 296, 557 (1988) J. F. Donoghue, B. R. Holstein and Y. C. Lin, Phys. Rev. D 37, 2423 (1988) J. Gasser, M. A. Ivanov and M. E. Sainio, Nucl. Phys. B 728, 31 (25) Nucl. Phys. B 745, 84 (26) Igor Danilkin (GSI) December 5, 212 26 / 31
Predictions for γγ K K, K + K, ηη 45 ARGUS γγ K + K - 3 TASSO CELLO γγ K K 3 2 15 1 4.5 3 1.5 1 1.1 1.2 Belle γγ ηη 1.1 1.15 1.2 1 1.1 1.2 Solid curves: unitarized result Dashed curves: tree-level result The bands correspond to g 5 [ 5, 5] Reduction of a Born amplitude for γγ K + K First description of γγ ηη data Igor Danilkin (GSI) December 5, 212 27 / 31
Minor cutoff dependence: 1.4 GeV (dashed line) < Λ s < 1.8 GeV (solid line) 3 2 Mark II Belle CELLO Belle syst. error γγ + - 6 4 Crystal Ball Belle Belle syst. error γγ 1 2.3.7 1.1 6 Crystal Ball Belle γγ η Belle syst. error 4 2.3.7 1.1 45 γγ K + K - ARGUS 3 15.7.95 1.2 3 TASSO CELLO γγ K K 2 1 4.5 3 1.5 1 1.1 1.2 Belle γγ ηη 1 1.1 1.2 1.1 1.15 1.2 Igor Danilkin (GSI) December 5, 212 28 / 31
Summary Within the novel approach which embodies analyticity coupled-channel unitary EM gauge invariance we have performed a theoretical study of the Goldstone-boson scattering and photon-fusion reactions Goldstone-boson scattering: and K phase shifts up to 1.2 GeV. in s-wave + + (f (98),...); in p-wave CDD poles (ρ, K,...) Photon-fusion reactions: γγ, +, η, K K, K + K, ηη up to 1.1-1.2 GeV We conclude that the dynamics depends sensitively on the details of the vector meson exchange. Igor Danilkin (GSI) December 5, 212 29 / 31
Thank you for your attention Igor Danilkin (GSI) December 5, 212 3 / 31
N/D method We reconstruct the approximate scattering amplitude by means of the N/D technique G.F.Chew, S.Mandelstam, Phys. Rev. 119 (196) 467-477 T ab (s) = c D 1 ac (s) N cb (s), where the contributions of left- and right-hand singularities are separated respectively into N ab (s) and D ab (s) functions, N ab (s) = U ab (s) + c,d µ 2 thr d s s µ 2 M s µ 2 M N ac( s) ρ cd ( s) [U db ( s) U db (s)] s s D ab (s) = δ ab c µ 2 thr d s s µ 2 M N ac( s)ρ cb ( s) s µ 2 M s s iɛ Igor Danilkin (GSI) December 5, 212 31 / 31