Anomaly, mixing and transition form factors of pseudoscalar mesons

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Nuclear Physics B (Proc. Suppl.) 219–220 (2011) 141–144 www.elsevier.com/locate/npbps

Anomaly, mixing and transition form factors of pseudoscalar mesons Yaroslav Klopota,1,∗, Armen Oganesiana,b , Oleg Teryaeva a Bogoliubov

Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna 141980, Russia b Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, Moscow 117218, Russia

Abstract We derive the exact non-perturbative QCD sum rule for the transition form factors of η and η using the dispersive representation of axial anomaly. This sum rule allows to express the transition form factors entirely in terms of meson decay constants. Using this sum rule several mixing schemes were analyzed and compared to recent experimental data. A good agreement with experimental data on η, η transition form factors in the range from real to highly virtual photons was obtained. Keywords: axial anomaly, mixing, transition form factor

1. Introduction Axial anomaly [1, 2] is one of the fundamental notions of non-perturbative QCD and hadronic physics. It is known to play a crucial role in two-photon decays of pseudoscalar mesons. At the same time the dispersive form of it [3] is applicable to the case of virtual photons also, which allows to study the processes of mesonphoton transitions, e.g. γγ∗ → π0 (η, η ). In particular, the dispersive approach to axial anomaly allows to derive a so-called anomaly sum rule (ASR) [4, 5, 6], which is an exact relation (has no corrections) due to AdlerBardeen theorem [7] and ’t Hooft’s consistency principle. Using the local quark-hadron duality hypothesis, we derive the relation between transition form factors of η, η mesons and their decay constants. Not long ago, the ASR was applied to the analysis of pion transition form factor [8] and allowed to give an

additional explanation of the Brodsky-Lepage interpolation formula for pion transition form factor [9, 10]. This form factor has attracted much attention due to unexpected data of BABAR collaboration [11]. These data motivated numerous theoretical works, which, in particular, questioned the QCD factorization [12, 13], provided the detailed analysis of perturbative and nonperturbative QCD corrections [14, 15, 16, 17] and suggested various model approaches [18, 19, 20, 21, 22]. Recently, the BABAR Collaboration extended the analysis and presented the data for η and η meson transition form factors [23]. These data motivated several recent papers [24, 25, 26, 27, 28]. In this work (see also [29]) we study η and η transition form factors by means of generalized ASR to include the effects of meson mixing. This ASR-based expression shows that there is a good agreement with experimental data for the considered mixing schemes in the whole range from real to highly virtual photons.

∗ The

speaker Email addresses: [email protected] (Yaroslav Klopot), [email protected] (Armen Oganesian), [email protected] (Oleg Teryaev) 1 On leave from Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine

0920-5632/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2011.10.084

2. Anomaly sum rule in octet channel In this section we briefly remind the dispersive representation for axial anomaly [3] (see also [30] for a re-

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view) and derive anomaly sum rule for the octet channel of axial current. The VVA triangle graph correlator  d4 xd4 ye(ikx+iqy) 0|T {Jα5 (0)Jμ (x)Jν (y)}|0 T αμν (k, q) = (8) Jα5

(1) − +

¯ α γ5 d = √16 (¯uγα γ5 u + dγ currents Jμ = (eu u¯ γμ u

contains axial current 2 s¯γα γ5 s) and two vector ¯ μ d + e s s¯γμ s); k, q are momenta of photons. ed dγ It is convenient to write the tensor decomposition [31, 32, 33] of correlator (1) in a form: F1 εαμνρ kρ + F2 εαμνρ qρ

T αμν (k, q) = +

F3 kν εαμρσ kρ qσ + F4 qν εαμρσ kρ qσ F5 kμ εανρσ kρ qσ + F6 qμ εανρσ kρ qσ ,(2)

+

where the coefficients F j = F j (k , q , p ; m ), p ≡ k + q, j = 1, . . . , 6 are the corresponding Lorentz invariant amplitudes constrained by current conservation and Bose symmetry. In this paper we are interested in the case of one real and one virtual photon (Q2 = −q2 > 0). Then for the invariant amplitude F3 −F6 the anomaly sum rule (ASR) can be derived [5]:  ∞ 1 A3a (t; q2 , m2 )dt = (3) √ , 4m2 2π 6 2

2

2

2

where A3a = 12 Im(F3 − F6 ). The ASR (3) is an exact relation, i.e. it does not have perturbative corrections to the integral [7] as well as it does not have any non-perturbative corrections too (as it is expected from ’t Hooft’s principle). Another important property of this relation is that it holds for an arbitrary quark mass m and for any q2 . For analysis of the hadron properties we additionally implement the local quark-hadron duality hypothesis. Taking into account a large η − η mixing, one can express the spectral function A3a (s, Q2 ) in a form of “η + η + continuum”: 







A3a s, Q2 = π fη8 δ(s − m2η )Fηγ Q2 +   QCD π fη8 δ(s − m2η )Fη γ Q2 + A3a θ(s − s0 ).

(4)

Here s0 is a continuum threshold in the octet channel. The form factors F Mγ of transitions γγ∗ → M are defined by the matrix elements: 

d4 xeikx M(p)|T {Jμ (x)Jν (0)}|0 = μνρσ kρ qσ F Mγ , (5)

and the coupling (decay) constants f Ma are defined from matrix elements: (a) (0)|M(p) = ipα f Ma . 0|Jα5

(6)

Note that the particles with non-zero two-photon decays cannot be included to continuum as it vanishes at Q2 = 0, so they should be taken into account explicitly. For heavy mesons the corresponding coupling constants should be suppressed [36, 37] at least as (mη /mres )2 which follows from the conservation of axial current (8) in the chiral limit. That is why we restrict ourselves Jμ5 only to η and η mesons, and other contributions are absorbed by ”continuum”. Substituting (4) into (3) and using one-loop apQCD proximation2 for spectral density [35, 33] A3a = Q2 1√ 2π 6 (s+Q2 )2

we get the anomaly sum rule in the form:

fη8 Fηγ (Q2 ) + fη8 Fη γ (Q2 ) =

1 √

2π2

6

Q2

s0 . + s0

(7)

Let us stress that this relation is correct for all Q2 due to the absence of the corrections to the Im(F3 − F6 ) [34] which allows to utilize the above expression for different Q2 . For real photons (Q2 = 0) the ASR (7) is expressed in terms of two-photon decay widths Γ M→2γ and masses m M of η, η mesons:

fη8 Fηγ (0) + fη8 Fη γ (0)

1 = √ , F Mγ (0) = 2π2 6



4Γ M→2γ

. πα2 m3M (8) For highly virtual photons (Q2 → ∞), where the QCD factorization [38, 39] is applicable, the equation (7) allows us to fix the continuum threshold s0 . The form factors at large Q2 [40, 41] are:

Q2 F as Mγ =

√ 2 √ ( f M8 + 2 2 f M0 ) 3 6



1 0

φas (x) dx x

(9)

We take into account that in the limit Q2 → ∞ the light cone distribution amplitudes of both η, η mesons are described by their asymptotical form [38, 39]: φas (x) = 6x(1 − x). Then the ASR for the octet channel at large Q2 leads to: 2 As it was shown in [34] there is no two-loop α corrections to this s expression.

√ s0 = 4π2 (( fη8 )2 + ( fη8 )2 + 2 2[ fη8 fη0 + fη8 fη0 ]).

(10)

Substituting (10) into (7) we can express ASR in terms of form factors F Mγ (Q2 ) and meson decay constants f Ma only: (11) fη8 Fηγ (Q2 ) + fη8 Fη γ (Q2 ) =  √ 2 2 (4π + Q2 (( fη8 )2 + ( fη8 )2 + 2 2[ fη8 fη0 + fη8 fη0 ])−1 )−1 3 It is instructive to compare this formula with interpolation formulas for transition form factors of η, η mesons proposed in [42]. The use of the proposed interpolation formulas leads to relation which is different from (11) but coincides with it in the limits Q2 = 0 and Q2 → ∞. Numerically, the difference is small at all Q2 (the maximal difference is 10% at ∼ 1 GeV) which proves that the proposed interpolation formulas in [42] are good approximations. 3. Mixing and experimental data The ASR relation (11) contains decay constants f Ma forming a matrix:   8 fη fη8 . (12) F≡ fη0 fη0 Different mixing schemes imply different forms of F. The simplest, one-angle mixing scheme (see e.g. [43] and references therein) leads to parametrization of decay constants in terms of three parameters: f0 , f8 , θ :   f8 cos θ f8 sin θ . (13) F= − f0 sin θ f0 cos θ For this scheme the ASR acquires a simple form: Fηγ (Q2 ) cos θ + Fη γ (Q2 ) sin θ =

2 3

4π2 f8 + Q2 / f8

, (14)

where the constant f8 defined by the anomaly sum rule at Q2 = 0 (8): ⎛ ⎞−1  ⎜⎜⎜ Γη→2γ ⎟⎟⎟ Γη →2γ α ⎜ ⎟⎟⎟ . ⎜ cos θ + sin θ f8 = √ ⎜ ⎝ ⎠ m3η m3η 4 6π3/2 (15) So, (14) and (15) determine the mixing angle in terms of physical quantities (decay widths and transition form factors).

Q2  fΗ8 FΗ  fΗ’ 8 FΗ’ f8 , GeV

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0.20 0.15 0.10 0.05 0.00 0

10

20 2

Q , GeV

30

40

2

Figure 1: ASR for one-angle mixing scheme (14): θ = −14o . See explanations in the text.

The corresponding relation (14) (both sides of which were multiplied by Q2 ) for θ = −14o is plotted in Fig. 1. The dots with error bars correspond to the l.h.s. of Eq. (14), where the form factors of η, η mesons are taken from experimental data of CLEO [44] and BABAR [23] collaborations. The r.h.s. of Eq. (14) corresponds to the curve with the shaded stripe defined by the experimental uncertainties of meson decay widths. The slope of the straight line in the origins indicates the value of the l.h.s. of Eq. (14) at Q2 = 0. We see that there is a reasonable agreement with the experimental data. The slight increase at large Q2 is related, in particular, by the tendency of η contribution to decrease at large Q2 . Due to negative mixing angle this leads to the behavior in the octet channel, qualitatively resembling the pion case, which produced the BABAR puzzle. Let us now discuss the mixing schemes suggested and developed in [45, 46, 47]. These schemes parametrize the decay constants f Ma in terms of two mixing angles θ8 , θ0 :  F=

f8 cos θ8 − f0 sin θ0

f8 sin θ8 f0 cos θ0

 .

(16)

This kind of matrices may appear when one considers the quark basis (see, e.g. [46]). For the parameters, suggested in [46] ( f0 = 1.17 fπ , f8 = 1.26 fπ , θ0 = −9.2o , θ8 = −21.2o ) the plot describing the ASR (11) is shown in Fig.2. Since the ASR is an exact model-independent relation at all Q2 it can be used as additional constraint on mixing parameters.

Y. Klopot et al. / Nuclear Physics B (Proc. Suppl.) 219–220 (2011) 141–144

Q2  fΗ8 FΗ  fΗ’ 8 FΗ’ f8 , GeV

144

0.20 0.15 0.10 0.05 0.00 0

10

20 2

Q , GeV

30

40

2

Figure 2: ASR for scheme [46]: f0 = 1.17 fπ , f8 = 1.26 fπ , θ0 = −9.2o , θ8 = −21.2o

4. Conclusions Combining the exact dispersive form of anomaly relation, quark-hadron duality hypothesis and asymptotic matching with QCD factorization we express the combination of η, η meson transition form factors in terms of meson decay constants only (11). The obtained anomaly sum rule is valid in the whole kinematical region starting from Q2 = 0. This ASR can be used as a test for different sets of mixing parameters. Our analysis shows that for a large number of mixing schemes ASR is in a good agreement with experimental data. At the same time, let us note that if one estimates the value of continuum threshold s0 from Eq. (10) it appears to be quite small. This may reflect the contradiction of local quark-hadron duality from one side, and anomaly from the other side. The possible resolution of such contradiction can be due to 1/Q2 suppressed non-perturbative correction to continuum similar to suggested earlier [8] for explanation of BABAR pion puzzle. We thank B. L. Ioffe, P. Kroll, S. V. Mikhailov, T. N. Pham, A. V. Radyushkin for useful discussions and comments. Y.K. and O.T. would like to thank the organizers of the Conference for a hospitality and support. This work was supported in part by RFBR (Grants 09-02-00732, 09-02-01149, 11-02-01538,1102-01454)and by fund from CRDF Project RUP2-2961MO-09. References [1] [2] [3] [4]

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