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The γP transition form factors in QCD
Progress in Particle and Nuclear Physics 67 (2012) 228–233
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Review
The γ P transition form factors in QCD P. Kroll Fachbereich Physik, Universität Wuppertal, 42097 Wuppertal, Germany
article
abstract
info
Keywords: Photon-meson transitions QCD higher-order corrections Transverse momentum factorization
In an analysis by Kroll (2011) [11] the form factors for the transitions from a photon to a pseudoscalar meson P = π 0 , η, η′ , ηc within QCD are reported. The sharp rise of the π γ form factor with the photon virtuality demands strong power corrections to the results from collinear factorization in contrast to the other transition form factors. The power corrections are modeled as quark transverse momenta accompanied by gluon radiation. © 2011 Elsevier B.V. All rights reserved.
The recent measurements of the photon-to-pseudoscalar-meson transition form factors by the BaBar collaboration [1–3] have renewed the interest in these form factors. The BaBar data ruined the belief that these form factors are well understood within collinear QCD. The π γ form factor in particular, measured up to about 35 GeV2 now, reveals a sharp rise with the photon virtuality, Q 2 , which is in dramatic conflict with dimensional scaling and turned previous calculations based on collinear factorization obsolete—a substantial increase of the form factor is difficult to accommodate in fixed order perturbative QCD. For large Q 2 the form factor reads
√ Q Fπγ (Q ) = 2
2
2fπ 3
1
0
αs (µR ) 2 x ln x 2 2 dxΦπ (x, µF ) − (9 + 2 ln x) ln(µF /Q ) . 1+ ln x − x 3π 1−x 1
(1)
The LO order result has been derived in [4], and the NLO result (in the MS scheme) in [5]. The lowest-order Feynman graphs are shown in Fig. 1. In (1) fπ is the familiar decay constant of the pion while µF and µR are the factorization and renormalization scales, respectively. The distribution amplitude, Φπ , or more generally that of a pseudoscalar meson, P (=π 0 , η, η′ ), possesses a Gegenbauer expansion
ΦP (x, µF ) = ΦAS 1 +
n=2,4,...
apn
α (µ ) γn s F 3/2 (µ0 ) Cn (2x − 1) , αs (µ0 ) p
(2)
where the evolution of the expansion parameters, an , from an initial scale, µ0 , to the factorization scale, µF , is controlled by the anomalous dimensions γn [4]. With µF → ∞ the distribution amplitude evolves in the asymptotic form ΦAS = 6x(1 − x). Obviously, the right-hand side of (1) only provides log Q 2 -effects arising from the running of αs and the evolution of the distribution amplitude. The sharp rise of the Fπγ data measured by the BaBar collaboration [1] cannot be reproduced, the shape is wrong, see Fig. 2. Thus, the π γ transition form factor sets an example of an exclusive observable for which collinear factorization is insufficient for Q 2 as large as 35 GeV2 . Surprisingly, and this is in addition to the sharp rise of Fπ γ a second puzzle, the other transition from factors measured by the BaBar collaboration, namely ηγ , η′ γ and ηc γ , do not exhibit an anomalous Q 2 dependence, they behave as expected, see e.g. [6,7]. Many theoretical papers have already been devoted to the interpretation of the BaBar data on the π γ form factor, e.g. [8–10] to name a few. Here in this talk, it will be reported on an analysis [11] of the transition from factors on the basis of k⊥ -factorization.
E-mail address: [email protected]. 0146-6410/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ppnp.2011.12.023
P. Kroll / Progress in Particle and Nuclear Physics 67 (2012) 228–233
229
Fig. 1. Lowest order Feynman graphs for the P γ transition form factor.
Fig. 2. Left: the scaled πγ transition form factor versus Q 2 evaluated from fit (6) (solid line). The dashed line represents the result presented in [13] which is evaluated from the asymptotic distribution amplitude. The dotted line is obtained from collinear factorization to NLO accuracy. The thin solid line indicates the asymptotic behavior. Right: the relative strength of the contributions from the n-th Gegenbauer term to the P γ form factor, scaled by the n = 0 term, at the scale 1 GeV. Intrinsic k⊥ is omitted. Source: Data taken from [1,14].
The basic idea of this approach is to retain the quark transverse degrees of freedom in the hard scattering. This however implies that quarks and antiquarks are pulled apart in the impact-parameter space, canonically conjugated to the transversemomentum space. The separation of color sources is accompanied by the radiation of gluons. The corrections to the hard scattering process due to gluon radiation have been calculated in [12] in axial gauge using resummation techniques and having recourse to the renormalization group. These radiative corrections comprising resummed leading and next-toleading logarithms which are not taken into account by the usual QCD evolution, are presented in the form of a Sudakov factor, e−S . This approach, termed the modified perturbative approach (MPA), generates power corrections to the collinear result (1) which, as will be discussed below, can explain the anomalous behavior of the BaBar data [11,10]. According to [13] the P γ transition form factor reads FP γ (Q ) = 2
dx
d2 b 4π
Ψˆ P (x, −b, µF )TˆHP (x, b, Q , µR )e−S (x,b,Q ,µR ,µF )
(3)
within the MPA. Since the Sudakov exponent S is given in the impact-parameter space [12] it is convenient to work in that space. In the convolution formula (3) TˆH is the Fourier transform of the momentum-space hard scattering amplitude evaluated to lowest order perturbative QCD from the Feynman graphs displayed in Fig. 1
√ THP
ˆ =
2 6CP
π
√
K0 ( xQb).
(4)
√
Here CP is a charge factor. For the pion, for instance, it reads Cπ = (e2u − e2d )/ 2 where ea denotes the charge of a flavor-a quark in units of the positron charge. The function K0 denotes the modified Bessel function of order zero. The last item in (3) to be explained, is Ψˆ P , the valence Fock state light-cone wave function of the meson P in the impactparameter space. As in [13] the wave function is parametrized as
f
P Ψˆ P (x, b, µF ) = 2π √ ΦP (x, µF ) exp −
6
xx¯ b2 4σP2
.
(5)
In the MPA the impact parameter, b, which represents the transverse separation of quark and antiquark, acts as an infrared cut-off parameter. Thus, 1/b in the Sudakov exponent marks the interface between the non-perturbative soft momenta which are implicitly accounted for in the hadron wave function, and the contributions from soft gluons, incorporated in a perturbative way in the Sudakov factor. Obviously, the infrared cut-off serves at the same time as the gliding factorization scale µF = 1/b to be used in the evolution of the wave function. In accord with this interpretation the entire Sudakov factor is continued to zero whenever b > 1/ΛQCD . The Sudakov function leads to an asymptotic damping of any contribution except those from configurations with small quark–antiquark separations, i.e. for√ln Q 2 → ∞ the limiting behavior of the transition from factors in collinear factorization emerges, for instance, Q 2 Fπ γ → 2fπ .
230
P. Kroll / Progress in Particle and Nuclear Physics 67 (2012) 228–233
The √ maximum of the longitudinal scale appearing in the scattering amplitude and the transverse scale µR = max( xQ , 1/b) is chosen as the renormalization scale [13]. Although to lowest order there is no αs in the hard scattering amplitude for the P γ transition form factor, it nevertheless depends on µR .√ Indeed, as discussed above, the Sudakov factor comprises the gluonic radiative corrections for scales between 1/b and ξ Q / 2. Hence, the latter scale specifies the onset of the hard scattering regime. It is to be noted that logarithmic singularities arising from the running of αs and the evolution of the wave function are canceled by the Sudakov factor. The Sudakov factor in combination with the hard scattering amplitude provides a series of power suppressed terms which come from the region of soft quark momenta (x, 1 − x → 0) and grow with the Gegenbauer index n. This property leads to a strong suppression of the higher-order Gegenbauer terms at low Q 2 implying that only the lowest few Gegenbauer terms influence the transition form factor. With increasing Q 2 the higher Gegenbauer terms become gradually more important, see Fig. 2. At very large Q 2 the evolution of the distribution amplitude again suppresses all higher-order Gegenbauer terms and the asymptotic limit of the transition form factor emerges. This is to be contrasted with the collinear approximation (1) where, to leading-order accuracy, the sum of all Gegenbauer terms (at a given factorization scale) contributes to the form factor. The intrinsic transverse momentum dependence embedded in the wave function also generates power suppressed terms which are accumulated at all x and do not depend on the Gegenbauer index. We note in passing that the suppression of higher-order Gegenbauer terms by soft non-perturbative correction has also been observed within the framework of light-cone sum rules [9]. This feature of the MPA explains why the CLEO data on the π γ transition form factor [14] are well described by the asymptotic distribution amplitude as shown in [13], see √Fig. 2. With the BaBar data [1] at disposal which extend to much larger values of Q 2 and do exceed the asymptotic limit 2fπ , higher Gegenbauer terms can no longer be ignored; they are required for a successful description of the transition form factor. What can be learned about the higher-order Gegenbauer terms from the BaBar data will be discussed next. Having specified the details of the MPA, one can analyze the transition form factor by inserting (5) and (2) into (3) and fitting the Gegenbauer coefficients to the experiment. From a detailed examination of the data it becomes apparent that besides the transverse size parameter only one Gegenbauer coefficient can safely be determined. If more coefficients are freed the fits become unstable; the coefficients acquire unphysically large absolute values between 1 and 10 and with alternating signs leading to strong compensations among the various terms. A reasonable fit to the CLEO [14] and BaBar [1] data is obtained by taking for the Gegenbauer coefficient aπ2 the face value of a lattice QCD result [15]: aπ2 (µ0 ) = 0.20 at the scale µ0 = 2 GeV (with ΛQCD = 0.181 GeV and the 1-loop expression for αs ) and fitting aπ4 and the transverse size parameter, σπ , to the data for Q 2 > 2.3 GeV2 . The resulting parameters are
σπ = 0.40 ± 0.06 GeV−1 ,
a4 (µ0 ) = 0.01 ± 0.06,
(6) π
and χ = 34.2 which appear reasonable given that 28 data points are included in the fit. A fit with just σπ and a2 leads to parameters in agreement with the lattice result and the parameters quoted in (6); the results for the form factor are practically indistinguishable from the first fit. The results for the π γ form factor of the fit (6) are shown in Fig. 2. At the largest values of Q 2 the fit seems to be a bit small as compared to the BaBar data. Partially responsible for this fact are the large fluctuations the BaBar data exhibit; the fit compromises between all the data. For comparison the fit presented in [13] which is evaluated from the asymptotic distribution amplitude and σπ = 0.861 GeV, is also shown √ in Fig. 2. This result is obviously too low at large Q 2 , it does not exceed the asymptotic limit of the scaled form factor, 2fπ , in contrast to the BaBar data and the fit (6). Also shown in Fig. 2 is a result of the collinear factorization approach to NLO accuracy taking µF = µR = Q and the Gegenbauer coefficients specified above. Apparently the shape of that result is in conflict with experiment. The difference between this result and the MPA result for the same distribution amplitude reveals the strength of the power corrections taken into account by the MPA. Let us turn now to the analysis of the ηγ and η′ γ form factors. They can be expressed as a sum of the flavor-octet and flavor-singlet contributions [6] 2
FP γ = FP8γ + FP1γ .
(7)
As is the case for the π γ form factor the functions FPi γ (i = 1, 8) are proportional to the constants fPi assigned to the decays of meson P through the SU(3)F octet or singlet axial–vector weak currents which are defined by the matrix elements
⟨0|Jµi 5 |P (p)⟩ = ifPi pµ .
(8)
Adopting the general parametrization fη8 = f8 cos θ8 ,
fη1 = −f1 sin θ1 ,
fη8′ = f8 sin θ8 ,
fη1′ = f1 cos θ1 ,
(9)
one can show [16] that on exploiting the divergences of the axial–vector currents – which embody the axial–vector anomaly – the mixing angles, θ8 and θ1 , differ considerably from each other and from the state mixing angle, θ . In [16] the mixing
P. Kroll / Progress in Particle and Nuclear Physics 67 (2012) 228–233
231
Fig. 3. The scaled ηγ and η′ γ transition form factors versus Q 2 . For notations refer to Fig. 2. Source: Data taken from [14,17,3].
parameters have been determined: f8 = 1.26fπ ,
θ8 = −21.2°,
f1 = 1.17fπ ,
θ1 = −9.2°.
(10)
The renormalization-scale dependence of the singlet-decay constant is omitted since the anomalous dimension controlling it is of order αs2 and, hence, leads to tiny effects. Assuming particle-independent [6,16] wave functions, Ψ 8 and Ψ 1 , for the valence Fock states of the respective octet and singlet η mesons and parametrizing them as in (5) with decay constants f8 and f1 instead of fπ , one can cast the transition form factors into the form Fηγ = cos θ8 F 8 − sin θ1 F 1 Fη′ γ = sin θ8 F 8 + cos θ1 F 1 .
(11)
The two form factors F and F are to be evaluated from (3) and (4) in full analogy to the π γ transition form factor. As for the π γ form factor only the transverse size parameter and one Gegenbauer coefficient for each wave function can be determined. The best fit to the CLEO [14] and BaBar [3] is obtained with the parameters: 8
1
σ8 = 0.84 ± 0.14 GeV−1 ,
a82 (µ0 ) = −0.06 ± 0.06,
σ1 = 0.74 ± 0.05 GeV−1 ,
a12 (µ0 ) = −0.07 ± 0.04.
(12)
The value of χ is 29.1 for 32 data points. In Fig. 3 the results of this fit are compared to the experiment. The quality of the fit is very good. In contrast to the π γ case the data on both Fηγ and Fη′ γ lie below the asymptotic results Q 2 FP γ → 2
√
√
2/3 fP8 + 2 2fP1 . The new fits are very close to the results presented in [6]. Instead of the octet-singlet basis one may also
√
use the quark-flavor basis in which the two basis states have the valence Fock components (uu¯ + dd¯ )/ 2 and ss¯, respectively. The physical form factors are related to the flavor ones by Fηγ = cos φ F n − sin φ F s Fη′ γ = sin φ F n + cos φ F s .
(13)
There is only one mixing angle (φ = 39.3°) in this basis [16]. Inverting (13) leads to the results on F and F shown in Fig. 4. Again very good agreement with the data is to be observed. Since the octet and singlet wave functions in (12) are very close to the asymptotic one, the power corrections generated by the MPA are small. Therefore, and in sharp contrast to the π γ case, an analysis of the ηγ and η′ γ form factors within the collinear factorization approach is also possible. In this case information on the glue–glue Fock component of the η1 may be extracted [18]. The treatment of the ηc γ from factor, also measured by the BaBar collaboration [2], differs from that of the three other cases. There is a second large scale in addition to the virtuality of one of the photons, namely the mass of the ηc (Mηc ) or that of the charm quark (mc ). It cannot be neglected in a perturbative calculation in contrast to the case of the light mesons where quark and hadron masses do not play a role. Despite this the ηc γ form factor is to be calculated from (3) but the hard scattering amplitude to lowest order perturbative QCD reads [7] n
s
√
4 6e2c
. (14) xQ 2 + (1 + 4x(1 − x))m2c + k2⊥ The symmetry of the problem under the replacement of x by 1 − x is already taken into account in (14). Due to the second large scale involved the ηc γ form factor can be calculated even at Q 2 = 0. TH =
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P. Kroll / Progress in Particle and Nuclear Physics 67 (2012) 228–233
Fig. 4. Left: the flavor form factors. For notations refer to Fig. 2. Right: the ηc γ form factor scaled by its value at Q 2 = 0. The solid (dotted, dashed) line represents the results of a calculation with the values of the parameters quoted in (16) (with mc = 1.26 GeV, Mηc /2). The dashed line is the prediction given in [7]. Source: Data taken from [14,3,2].
The light-cone wave function of the ηc is parametrized as in (5). Following [7,19] the distribution amplitude is chosen as
Φηc
(x − 1/2)2 = N (σηc )xx¯ exp −σηc Mηc xx¯ 2
2
(15)
1
where N (σηc ) is determined from the usual requirement 0 dxΦηc (x) = 1. The distribution amplitude exhibits a pronounced maximum at x = 1/2 and is exponentially damped in the end-point regions. It describes an essentially non-relativistic c c¯ bound state; quark and antiquark approximately share the meson’s momentum equally. In (3) the Sudakov factor can be set to 1 in the case at hand since it is mainly active in the end-point regions (see the above discussion) which are already strongly damped by the ηc wave function. Even the k⊥ dependence of the hard scattering amplitude plays a minor role. The evolution behavior of the ηc distribution amplitude is unknown in the range where Q 2 is of order of Mη2c and is therefore ignored here. Consequently, also the running of the charm quark mass is omitted. The normalization of the ηc γ transition form factor is fixed by its value at Q 2 = 0 which is related to two-photon decay width. However, this decay width is experimentally not well known [20]. It is therefore advisable to normalize the form factor by its value at Q 2 = 0 all the more so since the recent BaBar data [2] are also presented this way. Doing so the perturbative QCD corrections at Q 2 = 0 to the ηc γ transition form factor are automatically included. Also the αs corrections for Q 2 . Mη2c [21] cancel to a high degree in the ratio Fηc γ (Q 2 )/Fηc γ (0). Even at Q 2 = 10 GeV2 their effect is less than 5%, see the discussion in [7]. The uncertainties in the present knowledge of the ηc decay constant do also not enter the predictions for this ratio. The recent Babar data on the ηc γ form factor [2] are shown in Fig. 4. The predictions given in [7] which have been evaluated from mc = Mηc /2, are about one standard deviation too large but with regard to the uncertainties of the theoretical calculation, as for instance the exact value of the mass of the charm quark, one can claim agreement between theory and experiment. A little readjustment of the value of the charm quark mass improves the fit. Thus, with the parameters mc = 1.35 GeV,
σηc = 0.44 GeV−1 ,
(16)
a perfect agreement with experiment is achieved as is to be seen in Fig. 4. For comparison there are also shown results evaluated from mc = 1.21 GeV. In summary: in this talk an analysis of the form factors for the transitions from a photon to a pseudoscalar meson [11] is reported. The analysis is performed within the MPA which bases on k⊥ factorization. In combination with the hard scattering kernel the Sudakov suppressions which are an important ingredient of the MPA and which represents radiative corrections in next-to-leading-log approximation summed to all orders of perturbation theory, lead to a series of power suppressed terms which are accumulated in the soft regions. Since these corrections grow with the Gegenbauer index the transition form factors are only affected by the few lowest Gegenbauer terms of the distribution amplitude, the higher ones do practically not contribute. How many Gegenbauer terms are relevant depends on the range of Q 2 considered: For Q 2 < 10 GeV2 it suffices to use just the asymptotic distribution amplitude while the unprecedented large range 4 GeV2 < Q 2 < 35 GeV2 covered by the BaBar data [1,3], require the next or the next two Gegenbauer terms or, turning the argument around, can be determined from an analysis of the data on the transition form factors. Indeed this is what has been done in [11]. From this analysis it turned out that for the case of the pion a fairly strong contribution from a2 is required by the data while for the η and η′ only small deviations from the asymptotic distribution amplitude are needed. For these cases the results from a previous calculation within the MPA [6] are already in fair agreement with the BaBar data. Comparing the π γ form factor with the ηγ or more precisely the η8 γ one, one observes a strong breaking of flavor symmetry in the ground-state octet of
P. Kroll / Progress in Particle and Nuclear Physics 67 (2012) 228–233
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the pseudoscalar mesons. In other processes involving pseudoscalar mesons such large flavor symmetry violations have not been observed. With regard to the theoretical importance of the transition form factors, in particular the role of collinear factorization a remeasurement, e.g. by the BELLE collaboration, would be highly welcome. References [1] [2] [3] [4] [5] [6] [7] [8]
[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
[The BABAR Collaboration], B. Aubert, et al., Phys. Rev. D 80 (2009) 052002. [The BABAR Collaboration], J.P. Lees, et al., Phys. Rev. D 81 (2010) 052010. [BABAR Collaboration], V.P. Druzhinin, et al., arXiv:1101.1142 [hep-ex]. G.P. Lepage, S.J. Brodsky, Phys. Lett. B 87 (1979) 359. F. del Aguila, et al., Nuclear Phys. B 193 (1981) 517; E. Braaten, Phys. Rev. D 28 (1983) 524. T. Feldmann, P. Kroll, Eur. Phys. J. C 5 (1998) 327. T. Feldmann, P. Kroll, Phys. Lett. B 413 (1997) 410. A.V. Radyushkin, Phys. Rev. D 80 (2009) 094009; M.V. Polyakov, JETP Lett. 90 (2009) 228; A.P. Bakulev, A.E. Dorokhov, et al., Acta Phys. Polon. Supp. 3 (2010) 943. arXiv:1003.4693 [hep-ph]. S.S. Agaev, V.M. Braun, N. Offen, F.A. Porkert, Phys. Rev. D 83 (2011) 054020. H.-S. Li, S. Mishima, Phys. Rev. D 80 (2009) 074024. P. Kroll, Eur. Phys. J. C 71 (2011) 1623. J. Botts, G. Sterman, Nuclear Phys. B 325 (1989) 62. P. Kroll, M. Raulfs, Phys. Lett. B 387 (1996) 848. [CLEO Collaboration], J. Gronberg, et al., Phys. Rev. D 57 (1998) 33. [QCDSF/UKQCD collaboration], V.M. Braun, et al., Phys. Rev. D 74 (2006) 074501. T. Feldmann, P. Kroll, B. Stech, Phys. Rev. D 58 (1998) 114006. [L3 Collaboration], M. Acciarri, et al., Phys. Lett. B 418 (1998) 399. P. Kroll, K. Passek-Kumericki, Phys. Rev. D 67 (2003) 054017. M. Wirbel, B. Stech, M. Bauer, Z. Phys. Chem. 29 (1985) 637. [Particle Data Group], K. Nakamura, et al., J. Phys. G: Nucl. Part. Phys. 37 (2010) 075021. M.A. Shifman, M.I. Vysotsky, Nuclear Phys. B 186 (1981) 475.
Contents lists available at SciVerse ScienceDirect
Progress in Particle and Nuclear Physics journal homepage: www.elsevier.com/locate/ppnp
Review
The γ P transition form factors in QCD P. Kroll Fachbereich Physik, Universität Wuppertal, 42097 Wuppertal, Germany
article
abstract
info
Keywords: Photon-meson transitions QCD higher-order corrections Transverse momentum factorization
In an analysis by Kroll (2011) [11] the form factors for the transitions from a photon to a pseudoscalar meson P = π 0 , η, η′ , ηc within QCD are reported. The sharp rise of the π γ form factor with the photon virtuality demands strong power corrections to the results from collinear factorization in contrast to the other transition form factors. The power corrections are modeled as quark transverse momenta accompanied by gluon radiation. © 2011 Elsevier B.V. All rights reserved.
The recent measurements of the photon-to-pseudoscalar-meson transition form factors by the BaBar collaboration [1–3] have renewed the interest in these form factors. The BaBar data ruined the belief that these form factors are well understood within collinear QCD. The π γ form factor in particular, measured up to about 35 GeV2 now, reveals a sharp rise with the photon virtuality, Q 2 , which is in dramatic conflict with dimensional scaling and turned previous calculations based on collinear factorization obsolete—a substantial increase of the form factor is difficult to accommodate in fixed order perturbative QCD. For large Q 2 the form factor reads
√ Q Fπγ (Q ) = 2
2
2fπ 3
1
0
αs (µR ) 2 x ln x 2 2 dxΦπ (x, µF ) − (9 + 2 ln x) ln(µF /Q ) . 1+ ln x − x 3π 1−x 1
(1)
The LO order result has been derived in [4], and the NLO result (in the MS scheme) in [5]. The lowest-order Feynman graphs are shown in Fig. 1. In (1) fπ is the familiar decay constant of the pion while µF and µR are the factorization and renormalization scales, respectively. The distribution amplitude, Φπ , or more generally that of a pseudoscalar meson, P (=π 0 , η, η′ ), possesses a Gegenbauer expansion
ΦP (x, µF ) = ΦAS 1 +
n=2,4,...
apn
α (µ ) γn s F 3/2 (µ0 ) Cn (2x − 1) , αs (µ0 ) p
(2)
where the evolution of the expansion parameters, an , from an initial scale, µ0 , to the factorization scale, µF , is controlled by the anomalous dimensions γn [4]. With µF → ∞ the distribution amplitude evolves in the asymptotic form ΦAS = 6x(1 − x). Obviously, the right-hand side of (1) only provides log Q 2 -effects arising from the running of αs and the evolution of the distribution amplitude. The sharp rise of the Fπγ data measured by the BaBar collaboration [1] cannot be reproduced, the shape is wrong, see Fig. 2. Thus, the π γ transition form factor sets an example of an exclusive observable for which collinear factorization is insufficient for Q 2 as large as 35 GeV2 . Surprisingly, and this is in addition to the sharp rise of Fπ γ a second puzzle, the other transition from factors measured by the BaBar collaboration, namely ηγ , η′ γ and ηc γ , do not exhibit an anomalous Q 2 dependence, they behave as expected, see e.g. [6,7]. Many theoretical papers have already been devoted to the interpretation of the BaBar data on the π γ form factor, e.g. [8–10] to name a few. Here in this talk, it will be reported on an analysis [11] of the transition from factors on the basis of k⊥ -factorization.
E-mail address: [email protected]. 0146-6410/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ppnp.2011.12.023
P. Kroll / Progress in Particle and Nuclear Physics 67 (2012) 228–233
229
Fig. 1. Lowest order Feynman graphs for the P γ transition form factor.
Fig. 2. Left: the scaled πγ transition form factor versus Q 2 evaluated from fit (6) (solid line). The dashed line represents the result presented in [13] which is evaluated from the asymptotic distribution amplitude. The dotted line is obtained from collinear factorization to NLO accuracy. The thin solid line indicates the asymptotic behavior. Right: the relative strength of the contributions from the n-th Gegenbauer term to the P γ form factor, scaled by the n = 0 term, at the scale 1 GeV. Intrinsic k⊥ is omitted. Source: Data taken from [1,14].
The basic idea of this approach is to retain the quark transverse degrees of freedom in the hard scattering. This however implies that quarks and antiquarks are pulled apart in the impact-parameter space, canonically conjugated to the transversemomentum space. The separation of color sources is accompanied by the radiation of gluons. The corrections to the hard scattering process due to gluon radiation have been calculated in [12] in axial gauge using resummation techniques and having recourse to the renormalization group. These radiative corrections comprising resummed leading and next-toleading logarithms which are not taken into account by the usual QCD evolution, are presented in the form of a Sudakov factor, e−S . This approach, termed the modified perturbative approach (MPA), generates power corrections to the collinear result (1) which, as will be discussed below, can explain the anomalous behavior of the BaBar data [11,10]. According to [13] the P γ transition form factor reads FP γ (Q ) = 2
dx
d2 b 4π
Ψˆ P (x, −b, µF )TˆHP (x, b, Q , µR )e−S (x,b,Q ,µR ,µF )
(3)
within the MPA. Since the Sudakov exponent S is given in the impact-parameter space [12] it is convenient to work in that space. In the convolution formula (3) TˆH is the Fourier transform of the momentum-space hard scattering amplitude evaluated to lowest order perturbative QCD from the Feynman graphs displayed in Fig. 1
√ THP
ˆ =
2 6CP
π
√
K0 ( xQb).
(4)
√
Here CP is a charge factor. For the pion, for instance, it reads Cπ = (e2u − e2d )/ 2 where ea denotes the charge of a flavor-a quark in units of the positron charge. The function K0 denotes the modified Bessel function of order zero. The last item in (3) to be explained, is Ψˆ P , the valence Fock state light-cone wave function of the meson P in the impactparameter space. As in [13] the wave function is parametrized as
f
P Ψˆ P (x, b, µF ) = 2π √ ΦP (x, µF ) exp −
6
xx¯ b2 4σP2
.
(5)
In the MPA the impact parameter, b, which represents the transverse separation of quark and antiquark, acts as an infrared cut-off parameter. Thus, 1/b in the Sudakov exponent marks the interface between the non-perturbative soft momenta which are implicitly accounted for in the hadron wave function, and the contributions from soft gluons, incorporated in a perturbative way in the Sudakov factor. Obviously, the infrared cut-off serves at the same time as the gliding factorization scale µF = 1/b to be used in the evolution of the wave function. In accord with this interpretation the entire Sudakov factor is continued to zero whenever b > 1/ΛQCD . The Sudakov function leads to an asymptotic damping of any contribution except those from configurations with small quark–antiquark separations, i.e. for√ln Q 2 → ∞ the limiting behavior of the transition from factors in collinear factorization emerges, for instance, Q 2 Fπ γ → 2fπ .
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P. Kroll / Progress in Particle and Nuclear Physics 67 (2012) 228–233
The √ maximum of the longitudinal scale appearing in the scattering amplitude and the transverse scale µR = max( xQ , 1/b) is chosen as the renormalization scale [13]. Although to lowest order there is no αs in the hard scattering amplitude for the P γ transition form factor, it nevertheless depends on µR .√ Indeed, as discussed above, the Sudakov factor comprises the gluonic radiative corrections for scales between 1/b and ξ Q / 2. Hence, the latter scale specifies the onset of the hard scattering regime. It is to be noted that logarithmic singularities arising from the running of αs and the evolution of the wave function are canceled by the Sudakov factor. The Sudakov factor in combination with the hard scattering amplitude provides a series of power suppressed terms which come from the region of soft quark momenta (x, 1 − x → 0) and grow with the Gegenbauer index n. This property leads to a strong suppression of the higher-order Gegenbauer terms at low Q 2 implying that only the lowest few Gegenbauer terms influence the transition form factor. With increasing Q 2 the higher Gegenbauer terms become gradually more important, see Fig. 2. At very large Q 2 the evolution of the distribution amplitude again suppresses all higher-order Gegenbauer terms and the asymptotic limit of the transition form factor emerges. This is to be contrasted with the collinear approximation (1) where, to leading-order accuracy, the sum of all Gegenbauer terms (at a given factorization scale) contributes to the form factor. The intrinsic transverse momentum dependence embedded in the wave function also generates power suppressed terms which are accumulated at all x and do not depend on the Gegenbauer index. We note in passing that the suppression of higher-order Gegenbauer terms by soft non-perturbative correction has also been observed within the framework of light-cone sum rules [9]. This feature of the MPA explains why the CLEO data on the π γ transition form factor [14] are well described by the asymptotic distribution amplitude as shown in [13], see √Fig. 2. With the BaBar data [1] at disposal which extend to much larger values of Q 2 and do exceed the asymptotic limit 2fπ , higher Gegenbauer terms can no longer be ignored; they are required for a successful description of the transition form factor. What can be learned about the higher-order Gegenbauer terms from the BaBar data will be discussed next. Having specified the details of the MPA, one can analyze the transition form factor by inserting (5) and (2) into (3) and fitting the Gegenbauer coefficients to the experiment. From a detailed examination of the data it becomes apparent that besides the transverse size parameter only one Gegenbauer coefficient can safely be determined. If more coefficients are freed the fits become unstable; the coefficients acquire unphysically large absolute values between 1 and 10 and with alternating signs leading to strong compensations among the various terms. A reasonable fit to the CLEO [14] and BaBar [1] data is obtained by taking for the Gegenbauer coefficient aπ2 the face value of a lattice QCD result [15]: aπ2 (µ0 ) = 0.20 at the scale µ0 = 2 GeV (with ΛQCD = 0.181 GeV and the 1-loop expression for αs ) and fitting aπ4 and the transverse size parameter, σπ , to the data for Q 2 > 2.3 GeV2 . The resulting parameters are
σπ = 0.40 ± 0.06 GeV−1 ,
a4 (µ0 ) = 0.01 ± 0.06,
(6) π
and χ = 34.2 which appear reasonable given that 28 data points are included in the fit. A fit with just σπ and a2 leads to parameters in agreement with the lattice result and the parameters quoted in (6); the results for the form factor are practically indistinguishable from the first fit. The results for the π γ form factor of the fit (6) are shown in Fig. 2. At the largest values of Q 2 the fit seems to be a bit small as compared to the BaBar data. Partially responsible for this fact are the large fluctuations the BaBar data exhibit; the fit compromises between all the data. For comparison the fit presented in [13] which is evaluated from the asymptotic distribution amplitude and σπ = 0.861 GeV, is also shown √ in Fig. 2. This result is obviously too low at large Q 2 , it does not exceed the asymptotic limit of the scaled form factor, 2fπ , in contrast to the BaBar data and the fit (6). Also shown in Fig. 2 is a result of the collinear factorization approach to NLO accuracy taking µF = µR = Q and the Gegenbauer coefficients specified above. Apparently the shape of that result is in conflict with experiment. The difference between this result and the MPA result for the same distribution amplitude reveals the strength of the power corrections taken into account by the MPA. Let us turn now to the analysis of the ηγ and η′ γ form factors. They can be expressed as a sum of the flavor-octet and flavor-singlet contributions [6] 2
FP γ = FP8γ + FP1γ .
(7)
As is the case for the π γ form factor the functions FPi γ (i = 1, 8) are proportional to the constants fPi assigned to the decays of meson P through the SU(3)F octet or singlet axial–vector weak currents which are defined by the matrix elements
⟨0|Jµi 5 |P (p)⟩ = ifPi pµ .
(8)
Adopting the general parametrization fη8 = f8 cos θ8 ,
fη1 = −f1 sin θ1 ,
fη8′ = f8 sin θ8 ,
fη1′ = f1 cos θ1 ,
(9)
one can show [16] that on exploiting the divergences of the axial–vector currents – which embody the axial–vector anomaly – the mixing angles, θ8 and θ1 , differ considerably from each other and from the state mixing angle, θ . In [16] the mixing
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Fig. 3. The scaled ηγ and η′ γ transition form factors versus Q 2 . For notations refer to Fig. 2. Source: Data taken from [14,17,3].
parameters have been determined: f8 = 1.26fπ ,
θ8 = −21.2°,
f1 = 1.17fπ ,
θ1 = −9.2°.
(10)
The renormalization-scale dependence of the singlet-decay constant is omitted since the anomalous dimension controlling it is of order αs2 and, hence, leads to tiny effects. Assuming particle-independent [6,16] wave functions, Ψ 8 and Ψ 1 , for the valence Fock states of the respective octet and singlet η mesons and parametrizing them as in (5) with decay constants f8 and f1 instead of fπ , one can cast the transition form factors into the form Fηγ = cos θ8 F 8 − sin θ1 F 1 Fη′ γ = sin θ8 F 8 + cos θ1 F 1 .
(11)
The two form factors F and F are to be evaluated from (3) and (4) in full analogy to the π γ transition form factor. As for the π γ form factor only the transverse size parameter and one Gegenbauer coefficient for each wave function can be determined. The best fit to the CLEO [14] and BaBar [3] is obtained with the parameters: 8
1
σ8 = 0.84 ± 0.14 GeV−1 ,
a82 (µ0 ) = −0.06 ± 0.06,
σ1 = 0.74 ± 0.05 GeV−1 ,
a12 (µ0 ) = −0.07 ± 0.04.
(12)
The value of χ is 29.1 for 32 data points. In Fig. 3 the results of this fit are compared to the experiment. The quality of the fit is very good. In contrast to the π γ case the data on both Fηγ and Fη′ γ lie below the asymptotic results Q 2 FP γ → 2
√
√
2/3 fP8 + 2 2fP1 . The new fits are very close to the results presented in [6]. Instead of the octet-singlet basis one may also
√
use the quark-flavor basis in which the two basis states have the valence Fock components (uu¯ + dd¯ )/ 2 and ss¯, respectively. The physical form factors are related to the flavor ones by Fηγ = cos φ F n − sin φ F s Fη′ γ = sin φ F n + cos φ F s .
(13)
There is only one mixing angle (φ = 39.3°) in this basis [16]. Inverting (13) leads to the results on F and F shown in Fig. 4. Again very good agreement with the data is to be observed. Since the octet and singlet wave functions in (12) are very close to the asymptotic one, the power corrections generated by the MPA are small. Therefore, and in sharp contrast to the π γ case, an analysis of the ηγ and η′ γ form factors within the collinear factorization approach is also possible. In this case information on the glue–glue Fock component of the η1 may be extracted [18]. The treatment of the ηc γ from factor, also measured by the BaBar collaboration [2], differs from that of the three other cases. There is a second large scale in addition to the virtuality of one of the photons, namely the mass of the ηc (Mηc ) or that of the charm quark (mc ). It cannot be neglected in a perturbative calculation in contrast to the case of the light mesons where quark and hadron masses do not play a role. Despite this the ηc γ form factor is to be calculated from (3) but the hard scattering amplitude to lowest order perturbative QCD reads [7] n
s
√
4 6e2c
. (14) xQ 2 + (1 + 4x(1 − x))m2c + k2⊥ The symmetry of the problem under the replacement of x by 1 − x is already taken into account in (14). Due to the second large scale involved the ηc γ form factor can be calculated even at Q 2 = 0. TH =
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Fig. 4. Left: the flavor form factors. For notations refer to Fig. 2. Right: the ηc γ form factor scaled by its value at Q 2 = 0. The solid (dotted, dashed) line represents the results of a calculation with the values of the parameters quoted in (16) (with mc = 1.26 GeV, Mηc /2). The dashed line is the prediction given in [7]. Source: Data taken from [14,3,2].
The light-cone wave function of the ηc is parametrized as in (5). Following [7,19] the distribution amplitude is chosen as
Φηc
(x − 1/2)2 = N (σηc )xx¯ exp −σηc Mηc xx¯ 2
2
(15)
1
where N (σηc ) is determined from the usual requirement 0 dxΦηc (x) = 1. The distribution amplitude exhibits a pronounced maximum at x = 1/2 and is exponentially damped in the end-point regions. It describes an essentially non-relativistic c c¯ bound state; quark and antiquark approximately share the meson’s momentum equally. In (3) the Sudakov factor can be set to 1 in the case at hand since it is mainly active in the end-point regions (see the above discussion) which are already strongly damped by the ηc wave function. Even the k⊥ dependence of the hard scattering amplitude plays a minor role. The evolution behavior of the ηc distribution amplitude is unknown in the range where Q 2 is of order of Mη2c and is therefore ignored here. Consequently, also the running of the charm quark mass is omitted. The normalization of the ηc γ transition form factor is fixed by its value at Q 2 = 0 which is related to two-photon decay width. However, this decay width is experimentally not well known [20]. It is therefore advisable to normalize the form factor by its value at Q 2 = 0 all the more so since the recent BaBar data [2] are also presented this way. Doing so the perturbative QCD corrections at Q 2 = 0 to the ηc γ transition form factor are automatically included. Also the αs corrections for Q 2 . Mη2c [21] cancel to a high degree in the ratio Fηc γ (Q 2 )/Fηc γ (0). Even at Q 2 = 10 GeV2 their effect is less than 5%, see the discussion in [7]. The uncertainties in the present knowledge of the ηc decay constant do also not enter the predictions for this ratio. The recent Babar data on the ηc γ form factor [2] are shown in Fig. 4. The predictions given in [7] which have been evaluated from mc = Mηc /2, are about one standard deviation too large but with regard to the uncertainties of the theoretical calculation, as for instance the exact value of the mass of the charm quark, one can claim agreement between theory and experiment. A little readjustment of the value of the charm quark mass improves the fit. Thus, with the parameters mc = 1.35 GeV,
σηc = 0.44 GeV−1 ,
(16)
a perfect agreement with experiment is achieved as is to be seen in Fig. 4. For comparison there are also shown results evaluated from mc = 1.21 GeV. In summary: in this talk an analysis of the form factors for the transitions from a photon to a pseudoscalar meson [11] is reported. The analysis is performed within the MPA which bases on k⊥ factorization. In combination with the hard scattering kernel the Sudakov suppressions which are an important ingredient of the MPA and which represents radiative corrections in next-to-leading-log approximation summed to all orders of perturbation theory, lead to a series of power suppressed terms which are accumulated in the soft regions. Since these corrections grow with the Gegenbauer index the transition form factors are only affected by the few lowest Gegenbauer terms of the distribution amplitude, the higher ones do practically not contribute. How many Gegenbauer terms are relevant depends on the range of Q 2 considered: For Q 2 < 10 GeV2 it suffices to use just the asymptotic distribution amplitude while the unprecedented large range 4 GeV2 < Q 2 < 35 GeV2 covered by the BaBar data [1,3], require the next or the next two Gegenbauer terms or, turning the argument around, can be determined from an analysis of the data on the transition form factors. Indeed this is what has been done in [11]. From this analysis it turned out that for the case of the pion a fairly strong contribution from a2 is required by the data while for the η and η′ only small deviations from the asymptotic distribution amplitude are needed. For these cases the results from a previous calculation within the MPA [6] are already in fair agreement with the BaBar data. Comparing the π γ form factor with the ηγ or more precisely the η8 γ one, one observes a strong breaking of flavor symmetry in the ground-state octet of
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the pseudoscalar mesons. In other processes involving pseudoscalar mesons such large flavor symmetry violations have not been observed. With regard to the theoretical importance of the transition form factors, in particular the role of collinear factorization a remeasurement, e.g. by the BELLE collaboration, would be highly welcome. References [1] [2] [3] [4] [5] [6] [7] [8]
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