Axial anomaly, vector meson dominance and mixing

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1 Axial anomaly, vector meson dominance 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 International School of Subnuclear Physics Erice, June 24 - July 3, 215

2 Outline 1 Introduction 2 Isovector channel: π 3 Possible corrections to ASR 4 Time-like region and VMD 5 η,η TFFs 6 Summary Y. Klopot, A. Oganesian, O. Teryaev Axial anomaly, vector meson dominance and mixing 2

3 Transition form factors + e + e + e k q p π + e Form factors F Mγ of the transitions γγ M (M=π,η,η ): d 4 xe ikx M(p) T{J µ (x)j ν ()} = ǫ µνρσ k ρ q σ F Mγ (1) Kinematics: k 2 =, q 2 Q 2 > (space-like region), q 2 > (time-like region) Y. Klopot, A. Oganesian, O. Teryaev Axial anomaly, vector meson dominance and mixing 3

4 π TFF: experimental status Pion transition form factor: available data.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. 9] show a steady rise of Q 2 F πγ, surpassing the pqcd predicted asymptote Q 2 F πγ 2f π,f π = 13.7 MeV at Q 2 1 GeV 2 and questioning the collinear factorization. Later BELLE collaboration data [212] are more consistent with pqcd. Y. Klopot, A. Oganesian, O. Teryaev Axial anomaly, vector meson dominance and mixing 4

5 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 () µ5 = 1 3 (ūγ µ γ 5 u + dγ µ γ 5 d + sγ µ γ 5 s): µ J () µ5 = 1 (m u uγ 5 u +m d dγ 5 d +m s sγ 5 s)+ α em 3 2π C() N c F F 3αs + 4π N cg G, (3) Y. Klopot, A. Oganesian, O. Teryaev Axial anomaly, vector meson dominance and mixing 5

6 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 () = 1 3 (e 2 u +e 2 d +e 2 s) = (6) Y. Klopot, A. Oganesian, O. Teryaev Axial anomaly, vector meson dominance and mixing 6

7 Axial anomaly: real and virtual photons Axial anomaly determines the π 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). The VVA triangle graph amplitude T αµν(k,q) = d 4 xd 4 ye (ikx+iqy) T{J α5 ()J µ(x)j ν(y)} can be presented as a tensor decomposition F j = F j (p 2,k 2,q 2 ;m 2 ), p = k + q. T αµν(k,q) = F 1 ε αµνρk ρ + F 2 ε αµνρq ρ + F 3 k νε αµρσk ρ q σ + F 4 q νε αµρσk ρ q σ (7) + F 5 k µε ανρσk ρ q σ + F 6 q µε ανρσk ρ q σ, 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), (8) A Im(F 3 F 6 ), N c = 3; Y. Klopot, A. Oganesian, O. Teryaev Axial anomaly, vector meson dominance and mixing 7

8 ASR and meson contributions Saturating the l.h.s. of the 3-point correlation function with the resonances and singling out their contributions to ASR 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), (9) J (a) α5 () M(p) = ip αf a M, (1) and form factors F Mγ of the transitions γγ M are: d 4 xe ikx M(p) T{J µ (x)j ν ()} = ǫ µνρσ k ρ q σ F Mγ (11) Sum of finite number of resonances decreasing F asymp Mγ (Q 2 ) f M Q 2- infinite number of states are needed to saturate the ASR (collective effect). [Y.K.,A.Oganesian,O.Teryaev, PLB 695 (211) 13] Y. Klopot, A. Oganesian, O. Teryaev Axial anomaly, vector meson dominance and mixing 8

9 Isovector channel: π 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 + higher contributions ( continuum ): πf π F πγ (Q 2 )+ s A 3 (s,q 2 ) = 1 2π N cc (3). (12) 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. (13) F πγ (Q 2 ) = 1 2 s 2π 2 f π s +Q2. (14) Y. Klopot, A. Oganesian, O. Teryaev Axial anomaly, vector meson dominance and mixing 9

10 F πγ (Q 2 ) = 1 2 2π 2 f π s s +Q 2 (15) The limit Q 2 + pqcd prediction Q 2 F πγ = 2f π gives s = 4π 2 f 2 π =.67GeV 2 fits perfectly the value extracted from SVZ (two-point) QCD sum rules s =.7GeV 2 [Shifman,Vainshtein,Zakharov 79]. proves BL interpolation formula[brodsky,lepage 81]: F BL πγ (Q 2 ) = π 2 f π 1+Q 2 /(4π 2 fπ) 2. (16)

11 Corrections interplay Q 2 GeV 2 The full integral is exact 1 2π = A 3 (s,q 2 )ds = I π +I cont The continuum contribution I cont = s 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!

12 Possible corrections to A 3 Perturbative two-loop corrections to spectral density A 3 are zero [Jegerlehner&Tarasov 6] Nonperturbative corrections to A 3 are possible: vacuum condensates, instantons, short strings. General requirements for the correction δi = s δa 3 (s,q 2 )ds: δi = at s (the continuum contribution vanishes), at s (the full integral has no corrections), at Q 2 (the perturbative theory works at large Q 2 ), at Q 2 (anomaly perfectly describes pion decay width). δi = 1 2 λs Q 2 Q2 2π (s +Q 2 ) 2(ln +σ), (17) s δf πγ = 1 πf π δi π = 1 2 λs Q 2 Q2 2π 2 f π (s +Q 2 ) 2(ln +σ). (18) s

13 Correction vs. experimental data Q 2 FΠΓ GeV CELL CLEO BABAR BELLE Q 2 GeV 2 CELLO+CLEO+BABAR: λ =.14, σ = 2.43, χ 2 /n.d.f. = 1.8

14 Time-like region: q 2 > (Q 2 < ) and VMD The ASR for time-like q 2 is given by the double dispersive integral: ds The real and imaginary parts of the ASR read: dy ρ(a) (s,y) y q 2 +iǫ = N cc (a), a = 3,8. (19) p.v. ReF πγ (q 2 ) = N cc (3) 2π 2 f π ds [ s3 p.v. ds dy ρ(a) (s,y) y q 2 = N c C (a), (2) dsρ (a) (s,q 2 ) =, a = 3,8. (21) ] dy ρ(a) (s,y) 1 y q 2 = 2 s 2π 2 f π s q 2. The TFF in the time-like region at q 2 = s =.67 GeV 2 has a pole, which is numerically close to the ρ meson mass squared, m 2 ρ.59 GeV 2 VMD model.

15 Octet channel of ASR ASR in the octet channel: J (8) α5 = 1 6 (ūγ α γ 5 u + dγ α γ 5 d 2 sγ α γ 5 s), f 8 ηf ηγ (Q 2 )+f 8 η F η γ(q 2 ) = Significant mixing. 1 2 s (8) (22) 6π 2 s (8) +Q2. η decays into two real photons, so it should be taken into account explicitly along with η meson. ASR at Q 2 -continuum threshold s (8) : 4π 2 ((f 8 η )2 +(f 8 η )2 +2 2[f 8 η f η +f8 η f η ]) = s(8) (23)

16 η,η TFF in the space-like region (Q 2 > (q 2 < )) FMΓQ 2, GeV BABAR,Η BABAR,Η CLEO,Η CLEO,Η Q 2, GeV 2 (3) F ηγ(q 2 5 s ) = ( 2f s cosφ f q sinφ) 12π 2 f sf π s (3) + 1 (8) s sinφ + Q 2 4π 2 f s s (8), (24) + Q 2 (3) F η γ (Q 2 5 s ) = ( 2f s sinφ + f q cosφ) 12π 2 f sf π s (3) 1 (8) s cosφ + Q 2 4π 2 f s s (8), (25) + Q 2 where s (3) = 4π 2 fπ 2, s(8) = (4/3)π 2 (5fq 2 2f s 2 ). [YK, Oganesian, Teryaev Phys.Rev. D87 (213) 3, 3613]

17 η TFF in the time-like region vs. data 3. FΗΓ q 2 FΗΓ q 2, GeV 2 F ηγ (q 2 ) = 5 s 3 ( 2fs cosφ f q sinφ) 12π 2 f s f π s 3 q s 8 sinφ 4π 2 f s s 8 q2, (26) s 3 = 4π 2 f 2 π, s 8 = (4/3)π 2 (5f 2 q 2f 2 s ). [YK, Oganesian, Teryaev JETP Lett. 99 (214) 679]

18 Summary Meson TFFs are unique quantities which link (seemingly different) physics concepts: axial anomaly, mixing and VMD model. The ASR in the isovector channel gives ground for the VMD model in the time-like region. A possible new non-ope correction (e.g. due to short strings) to spectral density is supported by BABAR and not excluded by BELLE data. More accurate data is required for definite conclusions. Due to mixing in the η η system, ASR results in shifted intervals of dualities of η and η and gives ground for VMD model for the processes involving η and η mesons in time-like region.

19 Thank you for your attention!