Vector meson dominance, axial anomaly and mixing

Transcription

1 Vector meson dominance, axial anomaly and mixing Yaroslav Klopot 1, Armen Oganesian 1,2 and Oleg Teryaev 1 1 Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia 2 Institute of Theoretical and Experimental Physics, Moscow, Russia Brazil-JINR Forum Frontiers in Nuclear, Elementary Particle, and Condensed Matter Physics JINR, Dubna, June 15-19, 2015 Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 1

2 Outline 1 Introduction 2 Anomaly Sum Rule 3 Isovector channel: π 0 4 Possible corrections to ASR 5 Octet channel:η,η 6 Mixing 7 Time-like region and VMD 8 Summary Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 2

3 Transition form factors + e + e + e k q p π 0 + e Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 3

4 π 0 TFF: theoretical and experimental status Pion transition form factor: available data 0.5 CELLO Q 2 FΠΓ GeV CLEO BABAR BELLE Q 2 GeV 2 The current experimental status of the pion transition form factor (TFF) F πγ is rather controversial. The measurements of the BABAR collaboration [Aubert et al. 09] show a steady rise of Q 2 F πγ, surpassing the pqcd predicted asymptote Q 2 F πγ 2f π,f π = MeV at Q 2 10 GeV 2 and questioning the collinear factorization. Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 4

5 Axial anomaly: real and virtual photons Axial anomaly determines the π 0 γγ decay width: a unique example of a low-energy process, precisely predicted from QCD. The dispersive approach to axial anomaly leads to the anomaly sum rule (ASR) providing a handy tool to study the meson transition form factors M γγ (even beyond the factorization hypothesis). Holds for any Q 2 and any m 2. 4m 2 A 3 (s,q 2 ;m 2 )ds = 1 2π N cc (a) (1) It has neither α s corrections (Adler-Bardeen theorem) nor non-perturbative corrections (t Hooft s consistency principle). Exact nonperturbative relation powerful tool. Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 5

6 Axial anomaly In QCD, for a given flavor q, the divergence of the axial current J (q) µ5 = qγ µγ 5 q acquires both electromagnetic and gluonic anomalous terms: An octet of axial currents µ J (q) µ5 = m q qγ 5 q + e2 8π 2e2 qn c F F + α s 4π N cg G, (2) J (a) µ5 = q qγ 5 γ µ λ a 2 q Singlet axial current J (0) µ5 = 1 3 (ūγ µ γ 5 u + dγ µ γ 5 d + sγ µ γ 5 s): µ J (0) µ5 = 1 (m u uγ 5 u +m d dγ 5 d +m s sγ 5 s)+ α em 3 2π C(0) N c F F 3αs + 4π N cg G, (3) Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 6

7 The diagonal components of the octet of axial currents J (3) µ5 = 1 2 (ūγ µ γ 5 u dγ µ γ 5 d), J (8) µ5 = 1 6 (ūγ µ γ 5 u + dγ µ γ 5 d 2 sγ µ γ 5 s) acquire an electromagnetic anomalous term only: µ J (3) µ5 = 1 2 (m u uγ 5 u m d dγ 5 d)+ α em 2π C(3) N c F F, (4) µ J (8) µ5 = 1 6 (m u uγ 5 u +m d dγ 5 d 2m s sγ 5 s)+ α em 2π C(8) N c F F. (5) The electromagnetic charge factors C (a) are C (3) = 1 2 (e 2 u e 2 d) = 1 3 2, C (8) = 1 6 (e 2 u +e 2 d 2e 2 s) = 1 3 6, C (0) = 1 3 (e 2 u +e 2 d +e 2 s) = (6) Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 7

8 Anomaly sum rule The matrix element for the transition of the axial current J α5 with momentum p = k +q into two real or virtual photons with momenta k and q is: T αµν (k,q) = d 4 xd 4 ye (ikx+iqy) 0 T{J α5 (0)J µ (x)j ν (y)} 0 ; (7) Kinematics: k 2 = 0,Q 2 = q 2 Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 8

9 The VVA triangle graph amplitude can be presented as a tensor decomposition T αµν (k,q) = F 1 ε αµνρ k ρ +F 2 ε αµνρ q ρ + F 3 k ν ε αµρσ k ρ q σ +F 4 q ν ε αµρσ k ρ q σ (8) + F 5 k µ ε ανρσ k ρ q σ +F 6 q µ ε ανρσ k ρ q σ, F j = F j (p 2,k 2,q 2 ;m 2 ), p = k +q. Dispersive approach to axial anomaly leads to [Hořejší,Teryaev 95]: 4m 2 A 3 (s,q 2 ;m 2 )ds = 1 2π N cc (a), (9) A Im(F 3 F 6 ), N c = 3; C (3) = 1 2 (e 2 u e 2 d ) = 1 3 2, C (8) = 1 6 (e 2 u +e 2 d 2e2 s) = 1 3 6, C (0) = 1 3 (e 2 u +e2 d +e2 s ) = (10) Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 9

10 ASR and meson contributions Saturating the l.h.s. of the 3-point correlation function (7) with the resonances and singling out their contributions to ASR (1) we get the (infinite) sum of resonances with appropriate quantum numbers: π f a MF Mγ = where the coupling (decay) constants f a M : 4m 2 A 3 (s,q 2 ;m 2 )ds = 1 2π N cc (a), (11) 0 J (a) α5 (0) M(p) = ip αf a M, (12) and form factors F Mγ of the transitions γγ M are: d 4 xe ikx M(p) T{J µ (x)j ν (0)} 0 = ǫ µνρσ k ρ q σ F Mγ (13) Sum of finite number of resonances decreasing F asymp Mγ (Q 2 ) f M Q 2- infinite number of states are needed to saturate ASR (collective effect). [Y.K.,A.Oganesian,O.Teryaev 10] Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 1

11 Isovector channel: π 0 J (3) µ5 = 1 2 (ūγ µ γ 5 u dγ µ γ 5 d),c (3) = 1 2 (e 2 u e2 d ) = π 0 + higher contributions ( continuum ): πf π F πγ (Q 2 )+ s 0 A 3 (s,q 2 ) = 1 2π N cc (3). (14) The spectral density A 3 (s,q 2 ) can be calculated from VVA triangle diagram: The pion TFF: A 3 (s,q 2 ) = 1 2 Q 2 2π (Q 2 +s) 2. (15) F πγ (Q 2 ) = 1 2 s 0 2π 2 f π s 0 +Q2. (16) Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 1

12 F πγ (Q 2 ) = 1 2 2π 2 f π s 0 s 0 +Q 2 (17) The limit Q 2 + pqcd prediction Q 2 F πγ = 2f π gives s 0 = 4π 2 f 2 π = 0.67GeV 2 fits perfectly the value extracted from SVZ (two-point) QCD sum rules s 0 = 0.7GeV 2 [Shifman,Vainshtein,Zakharov 79]. reproduces BL interpolation formula[brodsky,lepage 81]: F BL πγ (Q 2 ) = π 2 f π 1+Q 2 /(4π 2 fπ) 2. (18) Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 1

13 Corrections interplay Q 2 GeV 2 The full integral is exact 1 2π = 0 A 3 (s,q 2 )ds = I π +I cont The continuum contribution I cont = s 0 A 3 (s,q 2 )ds may have perturbative as well as power corrections. δi π = δi cont : small relative correction to continuum due to exactness of ASR must be compensated by large relative correction to the pion contribution! Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 1

14 Possible corrections to A 3 Perturbative two-loop corrections to spectral density A 3 are zero [Jegerlehner&Tarasov 06] Nonperturbative corrections to A 3 are possible: vacuum condensates, instantons, short strings. General requirements for the correction δi = s 0 δa 3 (s,q 2 )ds: δi = 0 at s 0 (the continuum contribution vanishes), at s 0 0 (the full integral has no corrections), at Q 2 (the perturbative theory works at large Q 2 ), at Q 2 0 (anomaly perfectly describes pion decay width). δi = 1 2 λs 0 Q 2 Q2 2π (s 0 +Q 2 ) 2(ln +σ), (19) s 0 δf πγ = 1 πf π δi π = 1 2 λs 0 Q 2 Q2 2π 2 f π (s 0 +Q 2 ) 2(ln +σ). (20) s 0 Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 1

15 Correction vs. experimental data Q 2 FΠΓ GeV CELL CLEO BABAR BELLE Q 2 GeV 2 CELLO+CLEO+BABAR: λ = 0.14, σ = 2.43, χ 2 /n.d.f. = 1.08 Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 1

16 Octet channel (η,η ) J (8) α5 = 1 6 (ūγ α γ 5 u + dγ α γ 5 d 2 sγ α γ 5 s), ASR in the octet channel: 4m 2 A 3 (s,q 2 ;m 2 )ds = 1 2π N cc (8), (21) C (8) 1 6 (e 2 u +e 2 d 2e 2 s) = f 8 η F ηγ(q 2 )+f 8 η F η γ(q 2 ) = Significant mixing. 1 2 s 0 6π 2 s 0 +Q2. (22) η decays into two real photons, so it should be taken into account explicitly along with η meson. Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 1

17 Large Q 2 Q 2 F as ηγ = 2(C(8) f 8 η +C(0) f 0 η ) 1 Q 2 F as η γ = 2(C(8) f 8 η +C(0) f 0 η ) 1 φ as (x) = 6x(1 x). Then the ASR at Q 2 : 0 0 φ as (x) dx, (23) x φ as (x) dx, (24) x 4π 2 ((f 8 η) 2 +(f 8 η )2 +2 2[f 8 ηf 0 η +f 8 η f 0 η ]) = s 0 (25) Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 1

18 Q 2 = 0 The ASR takes the form: where f 8 ηf ηγ (0)+f 8 η F η γ(0) = F Mγ (0) = 4Γ M γγ πα 2 mm (26) 6π2, Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 1

19 Additional constraint - R J/Ψ. The radiative decays J/Ψ η(η )γ are dominated by non-perturbative gluonic matrix elements, and the ratio of the decay rates R J/Ψ = (Γ(J/Ψ) η γ)/(γ(j/ψ) ηγ) can be expressed as follows [Novikov 79]: 0 G R J/Ψ = G η 0 G G η where p η(η ) = M J/Ψ (1 m 2 η(η ) /M2 J/Ψ )/2. 2( pη p η ) 3, (27) µ J 8 µ5 = 1 6 (m u uγ 5 u +m d dγ 5 d 2m s sγ 5 s), (28) µ J 0 µ5 = 1 3 (m u uγ 5 u + m d dγ 5 d + m s sγ 5 s) α s 4π G G. (29) ( f 8 η + ) 2f 0 2(mη ) 4 ( ) 3 η R J/Ψ = fη 8 + pη. (30) 2fη 0 m η p η From experiment this ratio is: R J/Ψ = 4.67±0.15 [PDG 2012]. Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 1

20 Mixing Octet-singlet basis (of currents): J (8) µ5 = 1 6 (ūγ µ γ 5 u+ dγ µ γ 5 d 2 sγ µ γ 5 s),j (0) µ5 = 1 3 (ūγ µ γ 5 u+ dγ µ γ 5 d+ sγ µ γ 5 s). Matrix of decay constants (32) (31) 0 J (a) α5 (0) M(p) = ip αf a M, (32) F = ( f 8 η fη 8 fη 0 fη 0 Octet-singlet (SU(3)) mixing scheme: fη 8f η 0 +f8 η f η 0 = 0. F = ) ( f8 cosθ f 8 sinθ f 0 sinθ f 0 cosθ (33) ). (34) Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 2

21 For quark-flavour basis one explores the definitions of axial currents with decoupled light and strange quark composition: J q µ5 = 1 2 (ūγ α γ 5 u + dγ α γ 5 d), J s µ5 = sγ αγ 5 s, (35) ( ) J 8 µ5 J 0 µ5 where tanα = 2. ( J q = V(α) µ5 J s µ5 ), V(α) = Quark-flavour mixing scheme: [Feldmann,Kroll,Stech 97] F qs = fη q f η s +fq η f η s = 0. ( fq cosφ f q sinφ f s sinφ f s cosφ ( ) cosα sinα, (36) sinα cosα ). (37) Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 2

22 4π 2 ((f 8 η) 2 +(f 8 η )2 +2 2[f 8 ηf 0 η +f 8 η f 0 η ]) = s 0 (38) f 8 η F ηγ(0)+f 8 η F η γ(0) = 1 2 6π 2 s 0 s 0 +Q 2 (39) ( f 8 η + ) 2f 0 2(mη ) 4 ( ) 3 η R J/Ψ = fη 8 + pη. (40) 2fη 0 m η p η Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 2

23 fη 0 fπ f Η 8 f Π Black curve: χ 2 /n.d.f. = 1, green curve: fηf 8 η 0 +fη 8 f η 0 = 0, orange curve: fη q fη s q η f η s = 0. Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 2

24 Octet-siglet scheme: mixing parameters ( f 8 η fη 8 fη 0 fη 0 ) = ( ) f π. (41) Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 2

25 ASR in octet channel (space-like region, Q 2 > 0 (q 2 < 0)) f 8 Η FΗΓ f 8 Η FΗ Γ Q 2, GeV BABAR CLEO Q 2, GeV 2 Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 2

26 η,η TFFs Add the hypothesis: the TFFs of the state q 1 2 ( ūu + dd ) is related to the pion form factor as F qγ (Q 2 ) = (5/3)F πγ (Q 2 ) (numerical factor comes from the quark charges (eu 2 +e2 d )/(e2 u e2 d ) = 5/3). QF mixing scheme: q = cosφ η +sinφ η, s = sinφ η +sinφ η. (42) Then one can relate the form factors: F ηγ (Q 2 ) = F η γ(q 2 ) = 5 3 F πγ = F ηγ cosφ+f η γ sinφ. (43) 5 s (3) 0 ( 2f s cosφ f q sinφ) 12π s (8) 0 sinφ f s f π s (3) 0 +Q 2 4π 2 (44) f s s (8) 0 +Q2, 5 s (3) 0 ( 2f s sinφ+f q cosφ) 12π 2 1 s (8) 0 cosφ f s f π s (3) 0 +Q 2 4π2f (45) s s (8) 0 +Q2, where s (3) 0 = 4π 2 f 2 π, s(8) 0 = (4/3)π 2 (5f 2 q 2f 2 s ). Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 2

27 η,η TFF in the space-like region (Q 2 > 0 (q 2 < 0)) FMΓQ 2, GeV BABAR,Η BABAR,Η CLEO,Η CLEO,Η Q 2, GeV 2 Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 2

28 Time-like region: q 2 > 0(Q 2 < 0) and VMD The ASR for time-like q 2 is given by the double dispersive integral: 0 ds 0 The real and imaginary parts of the ASR read: dy ρ(a) (s,y) y q 2 +iǫ = N cc (a), a = 3,8. (46) p.v. ReF πγ (q 2 ) = N cc (3) 2π 2 f π 0 ds 0 0 [ s3 p.v. ds 0 dy ρ(a) (s,y) y q 2 = N c C (a), (47) dsρ (a) (s,q 2 ) = 0, a = 3,8. (48) 0 ] dy ρ(a) (s,y) 1 y q 2 = 2 s 0 2π 2 f π s 0 q 2. The TFF in the time-like region at q 2 = s 3 has a pole, which is numerically close to the ρ meson mass squared, m 2 ρ 0.59 GeV 2 VMD model. Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 2

29 F ηγ (q 2 ) = F η γ(q 2 ) = 5 s 3 ( 2fs cosφ f q sinφ) 12π 2 f s f π s 3 q s 3 ( 2fs sinφ+f q cosφ) 12π 2 f s f π s 3 q 2 1 s 8 sinφ 4π 2 f s s 8 q2, (49) 1 s 8 cosφ 4π 2 f s s 8 q2, (50) s 3 = 4π 2 f 2 π, s 8 = (4/3)π 2 (5f 2 q 2f 2 s ). Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 2

30 η TFF in the time-like region vs. data 3.0 FΗΓ q 2 FΗΓ q 2, GeV 2 Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 3

31 Summary Meson TFFs are unique quantities which link (seemingly different) important ideas: axial anomaly, mixing and VMD model. The ASR in the isovector channel gives the anomaly-based ground for the BL interpolation formula for the pion TFF in the space-like region, and for the VMD model in the time-like region. In order to describe the BABAR data on pion TFF, ASR requires a new nonperturbative correction to the spectral density. At the same time, the BELLE data can be described well without such a correction. This correction is absent in the local OPE and possibly originates from instantons or short strings. Mixing parameters of η and η meson can be extracted from TFFs using the ASR. ASR in the time-like region of TFFs substantiate the VMD model. Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 3

32 Thank you for your attention! Y. Klopot, A. Oganesian, O. Teryaev Vector meson dominance, axial anomaly and mixing 3