Two-photon effects in charge asymmetry in annihilation channel with χ 2 intermediate state

Transcription

1 Two-hoton effects in charge asymmetry in annihilation channel with χ 2 intermediate state E.A. Kuraev 1,Yu.M.Bystritskiy 1, E. Tomasi-Gustafsson 2,3 1 Bogoliubov Laboratory of Theoretical Physics, JINR Dubna, Russia 2 CEA,IRFU,SPhN, Saclay, France 3 CNRS/IN2P3, Institut de Physique Nucléaire, Orsay, France Trento Worksho Scattering and annihilation electromagnetic rocesses 20 February 2013 Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 1 / 17 Nucl

2 Resonance enhanced two-hoton contributions In a recent aer H.-Q. Zhou and B.-S. Zou, Nucl. Phys. A883, 49 (2012) it was roosed that a large C-odd effect could arise in the electron-ositron annihilation into a hadron air through two hoton exchange which coule to resonances such as Φ=η, η C, χ 0, χ 2, ρ, a 1. In one loo level, taking into account the two-hoton intermediate state, there is the ossibility to access the 1 S 0, 3 P 0, 3 P 2 quarkonium intermediate states, which are bound states of charm quark and antiquark. (k 2 ) ( 1 ) (k 2 ) ( 1 ) ieγ μ (k 1 ) γ ψ(ψ ) +ie M 2 ψ,ψ f ψ,ψ g αβ Γ ν ( 2 ) k 2 + q q μ 1 ν 1 η c,χ c0,c2 Γ k 1 q (k 1 ) ( 2 ) It was shown in Zhou 2012 and D. Ebert, R. Faustov, and V. Galkin, Mod. Phys. Lett. A18, 601 (2003) that the contribution to the asymmetry arising from the intermediate states 1 S 0, 3 P 0 is roortional to the ratio of the electron mass to the roton mass and it is negligible. The intermediate state with quantum numbers 3 P 2 (3556) = χ 2 can be roduced by two virtual hotons, which arise from the annihilation subrocess of electron and ositron. Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 2 / 17 Nucl

3 Resonance χ 2c enhanced two-hoton contributions In Zhou 2012 the contribution to asymmetry with 3 P 2 (3556) = χ 2 intermediate quarkonium state was calculated. (k 2 ) ( 1 ) k 2 + q η c,χ μ c0,c2 1 q Γ δ dσ(1) /dω ν 1 dσ (0) /dω, k 1 q (k 1 ) ( 2 ) The was calculated and it was indicated that the absolute corrections δ are larger than 10% at small θ (θ π/10) for almost all φ M δ φ E =0 φ E =2π/3 0.2 φ E =π/3 EPJA φ M =0-0.1 φ M =π/3-0.1 φ M =2π/3-0.1 φ M =π cosθ cosθ Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 3 / 17 Nucl cosθ cosθ

4 Born aroximation At the facilities VEPP (Novosibirsk), BEPC (Beijing), and PANDA/FAIR (Darmstadt), one can measure the following rocesses in the energy range of the mass of the quarkonia bound states of charmed quark and anti-quark q q q q q q + In Born aroximation the amlitude of these rocesses has the form: M γ = 4πα [ v(q + )γ μ u(q )] [ū( + )Γ μ(q)v( )], (1) s where q = + + = q + + q, s = q 2 and electromagnetic vertex of the roton Γ μ(q) is arametrized in terms of two form factors: Γ μ(q) =γ μf 1 (q 2 )+ 1 ( / qγ μ γ μ/ q ) F 2 (q 2 ), (2) 4M where F 1 and F 2 are the Dirac and Pauli form factors of the roton in the time-like region of momentum transfer (q 2 > 4M 2). Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 4 / 17 Nucl

5 Asymmetry from χ 2 : aroximation of artial widths q The interference of the Born amlitude M γ with a single hoton in the intermediate state with the amlitude with quarkonium in the intermediate state M χ2 originates backward-forward asymmetry q + q + A = dσ(c) dσ( c) dσ(c)+dσ( c) 2 sins Re [ M γm χ2 ], (3) q where dσ(c) =dσ/dc is the angular deendent cross section. Theamlitudeoftherocesses with the χ 2 intermediate state can be written as: M χ2 = M χ 2 Mχ 2 s Mχ 2, (4) 2 + im χ2 Γ χ2 χ 2 + q + where we used the artial width aroximation which is accetable within the χ 2 resonance intermediate state vicinity.the matrix elements M χ2 and M χ2 describe the conversion of χ 2 tensor article into the electron ositron and roton antiroton airs corresondingly. Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 5 / 17 Nucl

6 Asymmetry from χ 2 : aroximation of artial widths + f This tye of matrix elements can be arametrized in the following way (J. H. Kuhn, J. Kalan, and E. G. O. Safiani, Nucl. Phys. B157, 125 (1979)): χ 2 q f M χ2 f f = g f [ v( + )γ μ u( )] ( + ) ν χ μν, (5) where g f is the corresonding couling constant and χ μν is the olarization tensor of J =2state, which has the following roerties: χ μν = χ νμ, χ μνg μν =0, χ μνq μ =0, Π αβ = g αβ + qαq β q 2. χ μνχ λσ = 1 ( ) 1 Πμλ Π νσ +Π μσπ νλ 2 3 ΠμνΠ λσ, (6) The decay width which corresonds to the amlitude M χ2 f f has the form: Γ χ2 = g2 Mχ π β3 (5 2β 2 ), Γ χ2 eē = g2 e Mχ π. (7) where β is the velocity of final roton. Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 6 / 17 Nucl

7 Asymmetry from χ 2 : aroximation of artial widths Keeing in mind ossible comlexity of the roduct g eg = g eg e iφ, we can write the interference of Born amlitude M γ with the amlitude M χ2 which gives the following result: M 2 int =2 sins Re [ M γ Mχ 2 ] = 32παM 3 χ 2 Γ χ g eg where y is the convenient variable which vanishes at the resonance region: y cos φ +sinφ y 2 βc(1 β 2 c 2 ), (8) +1 y = s M χ 2 2. (9) M χ2 Γ χ2 Thus we can estimate the asymmetry A χ as A χ = A 0 y cos φ +sinφ y 2 +1 βc(1 β 2 c 2 ) 2 β 2 (1 c 2 ), A 0 = geg M χ 3 2, (10) παγ χ2 whereweusedoint-likerotonaroximation(f 1 =1, F 2 =0). Unfortunately the artial decay width of the decay of χ 2 state into the electron ositron air as well as the hase φ is not known. Below we will calculate these hases in aroximation of two hoton and two gluon intermediate state mechanisms, for the conversion of χ 2 into electron-ositron air and into roton-anti-roton air. Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 7 / 17 Nucl

8 Asymmetry from χ 2 : aroximation of two hoton and two gluon intermediate states χ 2 Our aroach for evaluating the conversion amlitudes consists in the calculation of their s-channel discontinuities and in the restoration of the whole amlitude by means of disersion relations. In this way we can exress the results in terms of the exerimentally measurable artial width of conversion of χ 2 tensor meson state to two hotons and to hadrons. The imaginary art as: Δ M 2 int =2 Re [ M ] γδm χ2. (11) sins The amlitudes M χc2 and Mχ c2 are estimated in the aroximation of two hoton and two gluon intermediate state mechanisms: ΔM χc2 =4(4πα)2 g γχ αβ T αβλ, ΔM χc2 =4(4πα s)g gc col χ μνr μνλ, (12) where the dimensional constants g γ,g g describe the conversion of 3 P 2 state to two hotons and two gluons. Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 8 / 17 Nucl

9 Asymmetry from χ 2 : aroximation of two hoton and two gluon intermediate states χ 2 The amlitudes M χc2 e+e and M χc2 have the following forms: ΔM χc2 =4(4πα)2 g γχ αβ T αβλ, ΔM χc2 =4(4πα s)g gc col χ μνr μνλ, where T αβλ = 1 [/q 4 S γ λ/q + Oαβ e Oαβ e = ], dφ 2 1 2(q k 1 ) K αβρσγ ρ(/q /k 1 )γ σ; R μνλ = 1 [(/ 4 S M )Γ λ (/ M )Oμν Oμν = dφ 2 1 2( + k 1 ) Kμνρσγρ( / + /k 1 + M )γ σ, (13) and the tensor K αβρσ has the form: K αβρσ =(k 1 k 2 )g ασg βρ + k 1α k 2β g ρσ k 1α k 2σ g ρβ k 1ρ k 2β g ασ. (14) Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 9 / 17 Nucl ],

10 Asymmetry from χ 2 : aroximation of two hoton and two gluon intermediate states The corresonding couling constants g γ and g g can be χ 2 found from the corresonding exerimental decay widths: Γ χ2 γγ = g2 γm 3 χ 320π ; Γχ 2 gg = g2 gm 3 χ 160π. (15) Keeing in mind the duality rincile, we assume that the χ 2 hadrons decay goes through two gluons at the leading order of α s. Thus we can identify Γ χ2 2gluons with the hadronic width of 3 P 2 state This gives Γ χ2 gg =0.8Γ total =1.6 MeV, Γ χ2 γγ =0.2Γ total =0.4 MeV, (16) g γ = GeV 1, g g =0.13 GeV 1. (17) Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 10 / 17 Nucl

11 Asymmetry from χ 2 : aroximation of two hoton and two gluon intermediate states χ 2 The hase factor e iφ is defined as e iφ = e i(φe+φ) where e iφe and e iφ arise resectively from the amlitudes M χ2 and M χ2. For the amlitude of conversion of the tensor state χ 2 to the leton air through two-hoton intermediate state we obtain ( M χ2 5α 72π gγ ln M ) χ [ ] m 2 iπ e 4 S /q γ λ/q + γ α (q + q ) β. (18) We can conclude that the hase associated with conversion to two hotons φ e is small tgφ e = π ln M2 χ 2 m 2 e =0.17. (19) So we will believe further the conversion amlitude of 3 P 2 state to the leton air to be real (i.e., φ e 0). Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 11 / 17 Nucl

12 Asymmetry from χ 2 : aroximation of two hoton and two gluon intermediate states χ 2 Performing similar estimation of the amlitude M χ2 we obtain the following exression for the interference M 2 int = 10π 27 α2 α s(mχ 2 2 g M χ 2 2 γg g)ln m 2 e s (s Mχ 2 2 ) 2 + Mχ 2 2 Γ 2 χ [( s Mχ 2 ) ] Hodd 2 + M χ2 Γ χg odd, G odd = c [ 34β 4 β 3 6β 3 +54β (1 β2 ) 3 β + c 2 [ 54β 4 +10β 3 6β β H odd (β,c) = 1 2π ] L + ( 4β 5 6β 3 +7β 5 ) ] L, L =ln 1+β 1 β, (20) where we used the aroximation of oint-like roton (i.e. F 1 =1and F 2 =0) which is realistic near the threshold. The real art, corresonding to G odd, can be found using the disersion relation: 1 G odd(β, c)ln 1 β2 β xdx x 2 β 2 ( G odd(β, c)+g odd (x, c)). (21) Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 12 / 17 Nucl

13 Asymmetry from χ 2 : aroximation of two hoton and two gluon intermediate states Setting F 1 =1, F 2 =0, we obtain for the asymmetry A χ = A 0 D, D = yh odd + G odd y β 2 (1 c 2 ), y = s M 2 χ 2 M χ2 Γ χ2, (22) A 0 = 5αs 864π (M 2 g M χ 2 2 γg g)ln m 2 = e For the value β =0.85 which corresonds to the s = M χ we find A χ D s,cos Θ D s,cos Θ y cos Θ θ =10 o, 30 o, 50 o s = Mχ,M χ Γ χ,m χ 2Γ χ Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 13 / 17 Nucl

14 Asymmetry from χ 2 : aroximation of two hoton and two gluon intermediate states For the global hase we have φ(β, c) φ =arctan G odd(β, c) H odd (β, c). (23) Φ y,cos Θ, Degrees Φ y,cos Θ, Degrees cos Θ y θ =10 o, 30 o, 50 o s = Mχ,M χ 10Γ χ,m χ 100Γ χ Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 14 / 17 Nucl

15 Asymmetry from Z-boson intermediate state For comarison we calculate the contribution to the asymmetry arising from the interference of two Born amlitudes, with hoton and Z boson intermediate states (we set here F 1 =1, F 2 =0): A Z = A Z 0 Ψ Z(c), (24) where A Z 0 = s MZ 2 sin2 (2θ W ) 2.1 βc 10 3, Ψ Z (c) = 2 β 2 (1 c 2 ), (25) where θ W is the Weinberg angle, sin 2 θ W One can see that this contribution, A Z, never exceeds Z c c Z Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 15 / 17 Nucl

16 Pure QED asymmetry Besides the two considered sources of the asymmetry, there is also a ure QED source, A QED.The total contribution to the odd art of the cross section was reviously considered in E. A. Kuraev and G. V. Meledin, Nucl. Phys. B122, 485 (1977) and A. I. Ahmadov, V. V. Bytev, E. A. Kuraev, and E. Tomasi-Gustafsson, Phys. Rev. D82, (2010). The angular deendence of the asymmetry corresonding to this calculation, is lotted below for s = M χ. The dashed curve corresonds to the interference of the Born amlitude with the box ones, and it is very small. The dominant contribution arises from the soft and hard real hoton emission from initial and final states (solid line). Δ odd 0.06 A QED c c Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 16 / 17 Nucl

17 Conclusion The charge asymmetry A χ for the reactions χ 2 is evaluated It was shown that the contributions to the asymmetry is small, i.e. A χ We conclude that the main contribution to the asymmetry is rovided by ure QED contribution, i.e A QED > A Z A χ Yu.M. Bystritskiy ( Bogoliubov Laboratory of Theoretical Physics, Two-hoton JINR Dubna, effects in Russia charge CEA,IRFU,SPhN, asymmetry Saclay, France CNRS/IN2P3, 20 February Institut 2013 de Physique 17 / 17 Nucl