- Email: [email protected]
Hadronic Light-by-Light Scattering Contributions to the Muon Anomaly
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Nuclear Physics B (Proc. Suppl.) 253–255 (2014) 135–138 www.elsevier.com/locate/npbps
Hadronic Light-by-Light Scattering Contributions to the Muon Anomaly Eduardo de Rafael Aix-Marseille Universit´e, CNRS, CPT, UMR 7332, 13288 Marseille, France Universit´e de Toulon, CNRS, CPT, UMR 7322, 83957 La Garde, France
Abstract I give an overview of what we know at present about the Hadronic Ligh-by-Light Scattering Contribution to the anomalous magnetic moment of the muon in the Standard Model. Keywords: Muon Anomaly, QCD, Constituent Chiral Quark Model in Japan which plan to reduce the present experimental error to 0.14 ppm 2 .
1. Introduction I was asked by the organizers of the TAU-12 Conference to review the present status of the Hadronic Ligh-by-Light Scattering (HLbyL) Contributions to the anomalous magnetic moment of the muon (aμ ). This is motivated by the fact that the present experimental determination of aμ , which is dominated by the latest BNL experiment of the E821 collaboration [1] aμ (E821 − BNL) = 116 592 089(54)stat (33)syst × 10−11 , (1)
Table 1: Standard Model Contributions1 Contribution
QED (leptons) HVP(lo)[e+ e− ] HVP(ho) HLbyL EW Total SM
Result in 10−11 units
116 584 718.85 ± 0.04 6 923 ± 42 −98.4 ± 0.7 105 ± 26 153 ± 1 116 591 801 ± 49
when confronted with the Standard Model prediction 1 aμ (SM) = 116 591 801 ± 49 ,
(2)
shows a persistent 3.6 σ discrepancy which deserves attention. The results of the Standard Model contributions to aμ , due to the work of many people, are summarized in Table 1. These include the well known QED contribution from virtual photons and leptons, which is now known to tenth order (see ref.[3] and references therein), as well as from the Hadronic Vacuum Polarization of lowest order (HVP(lo)) and higher order (HVP(ho), from the HLbyL and from the Electroweak (EW) interactions. One observes that the largest errors in Table 1 come from the HVP and the HLbyL contributions. The size of these errors is of special concern in view of the future experiments at FNAL in the USA and at JPARC Email address: [email protected] (Eduardo de Rafael) a recent review article where earlier references can be found see [2]. 1 For
http://dx.doi.org/10.1016/j.nuclphysbps.2014.09.033 0920-5632/© 2014 Elsevier B.V. All rights reserved.
The lowest order HVP contribution to the muon anomaly has a well known integral representation in terms of the e+ e− one photon annihilation cross-section into hadrons. The integral in question is in fact dominated by the contributions from the low-energy region. Its determination has been improving through the years thanks to the advent of more and more refined measurements, the latest coming from the BaBar and Belle facilities. The error here is likely to be reduced in the near future. 2. The HLbyL Scattering Contribution Contrary to the HVP contribution, the HLbyL contribution shown in Fig. 1 cannot be written as an integral over experimentally accessible observables. The calculation from theory requires the knowledge of Quantum Chromodynamics (QCD) contributions at all energy scales, 2 See the contributions by Lee Roberts and Tsutomu Mibe at this Conference.
136
E. de Rafael / Nuclear Physics B (Proc. Suppl.) 253–255 (2014) 135–138
something which we don’t know how to do at present except via numerical lattice QCD evaluations which, unfortunately, are difficult to implement in this case 3 .
π
π
π
μ
μ
μ
Figure 2:
X
Dominant HLbyL Scattering Contribution.
Hadrons
+ Permutations
μ
Numerically, for the physical values of the constants in Eq. 3 one finds
Figure 1:
α 3
Hadronic Light-by-Light Scattering Contribution.
Nc 2
π Things, however, are not as hopeless as all that. We know some important constraints from QCD which the hadronic models one has to resort to, in order to evaluate this contribution, have to obey. 2.1. Chiral Limit and Large-Nc limit of QCD It is well known that the hadronic realization of QCD in the sector of the light u , d , s quarks is governed by the phenomena of spontaneous chiral symmetry breaking and confinement. This implies the existence of a Mass Gap between the pseudoscalar (Goldstone-like) particles and the other hadronic particles (the ρ being the lowest one in Nature). In the limit where this Mass Gap is considered to be large, and to leading order in the 1/Nc –expansion (Nc is the number of colours), the contribution of the HLbyL to aμ is dominated by the one pion exchange diagrams shown in Fig. 2 and the leading result in this limit is known analytically [5]:
(HLbyL) 0 aμ (π )
α 3
m2μ
Mρ log = Nc π mπ 48π2 fπ2 Mρ + O(1) . + O log mπ 2
m2μ 48π2 fπ2
log2
Mρ = 95 × 10−11 , mπ
(4)
which is within the ballpark of the hadronic model determinations. The ρ-mass, however, is too close to the π-mass to take seriously this number which may have large corrections. In fact, it is in principle possible to m fix the coefficient of the O log mπρ term in Eq. 3. It can be shown [7, 8] that the unknown contribution to this coefficient is related to the π0 → e+ e− decay rate, albeit with the radiative corrections included. The O(1) term is, however, not fixed by symmetry restrictions which limits the use of Eq. 3 at the level of the required accuracy. 2.2. Short-Distance Constraint from the OPE in QCD The constraint in question comes from a clever observation by Melnikov and Vainshtein [9]. The three momenta k1 , k2 , k3 in the light–by–light subdiagram of the HLbyL scattering contribution to the muon anomaly (see Fig 3 below) form a triangle. When k12 ≈ k22 k32 , q
k1
q
0
0
2
γ γγ5
H
(3)
k3
k2
k3
Figure 3:
This is phenomenologically relevant because it can be used as a check of Hadronic Model Calculations: letting the hadronic masses of a model become very large with respect to the pion mass, one must find an answer compatible with the analytic result in Eq. 3 when Mρ is also taken to be large. The fact that the HLbyL contribution in this limit is positive was crucial in fixing the (HLbyL) overall sign of aμ [6].
The OPE Constraint of Melnikov and Vainshtein. and k12 ≈ k22 m2ρ in this triangle one can apply the Operator Product Expansion (OPE) in the two vector currents which carry hard momenta with the result:
4
d x1 3 There
are, however, promising lattice projects under study, see e.g. ref. [4].
d4 x2 e−ik1 ·x1 −ik2 ·x2 Jν (x1 )Jρ (x2 ) =
1 2νρδγ kˆ δ , d4 ze−ik3 ·z J5γ (z) + O kˆ 2 kˆ 3
(5)
E. de Rafael / Nuclear Physics B (Proc. Suppl.) 253–255 (2014) 135–138
γ where j5 = q Q2q qγ ¯ γ γ5 q is the hadronic axial current and kˆ = (k1 − k2 )/2 ≈ k1 ≈ −k2 . As illustrated in Fig. 3 this OPE reduces the HLbyL amplitude, in the special kinematics under consideration, to the AVV triangle amplitude which is an object for which we have a much better theoretical insight. This observation has interesting phenomenological implications: • At large k1,2 the Pseudoscalar (and Axial-Vector) exchanges dominate. • The AVV limit also implies that the Fπ0 γ∗ γ∗ (k2 , k2 ) form factor at the vertices of Fig 2 must fall as 1/k2 . Unfortunately, the two asymptotic QCD constraints discussed above are not sufficient for a full model inde(HLbyL) pendent evaluation of aμ . This explains the relatively large error of ±26 × 10−11 for this contribution in Table 2 above 4 . (HLbyL) Most of the last decade calculations of aμ found in the literature are compatible with the QCD chiral constraints and the large-Nc limit discussed above. They all incorporate the π0 -exchange contribution modulated by π0 γ∗ γ∗ form factors correctly normalized to the Adler, Bell-Jackiw point-like coupling. They differ, however, on whether or not they satisfy the particular OPE constraint discussed above, and in the shape of the vertex form factors which follow from the different models. In spite of the different choices of these form factors there is, within errors, a reasonable agreement among the final results. An exception is the calculation reported in ref. [11] using a model based on a Dyson-Schwinger approach which, however, as we shall see contradicts generic properties which emerge from the Constituent Chiral Quark Model which we next discuss. 3. The Constituent Chiral Quark Model I have emphasized in recent meetings and workshops the need of a simple reference model to evaluate the various hadronic contributions to aμ within the same framework, and use it as a yardstick to compare with the more elaborated model evaluations in the literature. The reference model which we have proposed [12] is based on the Constituent Chiral Quark Model (CχQM) of Manohar and Georgi [13] in the presence of S U(3)L × S U(3)R external sources. It is an effective field theory which incorporates the interactions of the NambuGoldstone modes (the low-lying pseudoscalars) of the 4 See
e.g. the discussion in ref. [10].
137
spontaneously broken chiral symmetry, to lowest order in the chiral expansion and in the presence of chirally rotated quark fields which have become massive. As emphasized by Weinberg [14], the effective Lagrangian in question is renormalizable in the Large-Nc limit and, as shown in ref. [15], the number of the required counterterms is minimized for the choice gA = 1 of the axial coupling of the constituent quarks to the pseudoscalars. The model has its limitations but there is an exceptional class of low-energy observables for which the predictions of the CχQM can be rather reliable. This is the case when the leading short-distance behaviour of the underlying Green’s function of a given observable is governed by perturbative QCD. The contributions to aμ from HVP, from the HLbyL Scattering and from the Hadronic Zγγ vertex (provided that gA = 1) fall in this class. With gA = 1 the only free parameter in the model is the constituent quark mass MQ . The comparison between the CχQM prediction for the HVP and the phenomenological determination of a(HVP) shows that μ fixing MQ in the MQ = (240 ± 10) MeV , reproduces the phenomenological determination of the lowest order HVP within an error of less than 10%. This error, however, does not include the systematic error of the CχQM itself. As shown in ref. [12], with MQ fixed in this range, the higher order HVP contributions, as well as the electroweak Hadronic Zγγ contribution are reproduced rather well. When examining the HLbyL scattering contribution to the muon anomaly in the CχQM, one finds that there are two competing contributions: one is the π0 – exchange diagrams in Fig. 2 where the solid discs there are now constituent quark loops, the other one the irreducible constituent quark loop in Fig. 1 where the solid triangle becomes now a constituent quark loop. An interesting feature which emerges from the calculation in ref. [12] is the balance between the Goldstone Contribution and the Quark Loop Contribution. Indeed, as the constituent quark mass MQ gets larger and larger, the Goldstone Contribution dominates: asymptotically, M for large MQ values it reproduces the log2 mπQ behaviour of Eq. 3, while for MQ smaller and smaller it is the Constituent Quark Loop Contribution which dominates: asymptotically, for small MQ values (though still with MQ > mμ ) it behaves like
HLbyL aμ (CQL)
=
α 3
⎛ ⎞ ⎟⎟⎟ ⎜⎜⎜ 4 ⎜ Nc ⎜⎜⎝ Qq ⎟⎟⎟⎠ ×
π q=u,d,s ⎧ ⎛ 4 ⎞⎫ 2 ⎪ ⎜⎜⎜ mμ m2μ ⎟⎟⎟⎪ ⎪ ⎪ 19 mμ ⎨ 3 2 ⎜ ⎟⎟⎠⎬ ζ(3) − + O ⎜⎝ 4 log . ⎪ ⎪ ⎪ 2 2 ⎩ 2 ⎭ 16 MQ MQ MQ ⎪
(6)
138
E. de Rafael / Nuclear Physics B (Proc. Suppl.) 253–255 (2014) 135–138
References These features are illustrated by the plot of the total (HLbyL) (CχQM) versus MQ shown in Fig. 4. In fact aμ (HLbyL) the plot also shows that the value of aμ (CχQM) is quite stable for a rather large choice of reasonable MQ values. 190
180
170 HLbyL
aΜ
CΧQM 1011 160
150
140 200
250
300
350
400
MQ in MeV
Figure 4:
The HLbyL Contribution in the CχQM.
The CχQM result contradicts what is reported in ref. [11] where the equivalent contribution to the constituent quark loop is found to be: (136 ± 59) × 10−11 i.e. much larger than the contribution found by the same authors for the π0 -exchange: (81 ± 12) × 10−11 which, within errors, is compatible with previous phenomenological determinations. This casts serious doubts about the compatibility of the model used in ref. [11] (or perhaps of their calculation) with basic QCD features encoded in the CχQM. In fact, during the completion of this mini review, there has appeared a new version of this model in the archives [16] with a smaller result for the quark loop contribution: (96 ± 2) × 10−11 , where the error here does not include the systematic error of their model and, furthermore, the calculation is claimed to be incomplete as yet. We conclude that, in the absence of more refined calculations, which with some effort may hopefully become available, the number quoted in Table 1 for the HLbyL scattering contribution to aμ discussed in [10] represents the best valid estimate at present. In fact, there is a recent independent analysis in ref. [17] which also confirms that estimate.
Acknowledgements I wish to thank David Greynat for a pleasent collaboration on the CχQM calculations reported here and Marc Knecht and Laurent Lellouch for helpful discussions.
[1] G.W. Bennet et al, Phys. Rev. D 73:072003 (2006). [2] J.P. Miller, E. de Rafael, B. Lee Roberts and D. St¨ockinger, Annu. Rev. Nucl. Part. Sci. 62:237-264 (2012). [3] T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, Phys. Rev. Lett. 109:111807 (2012). [4] T. Blum, Plenary talk at The 30th International Symposium on Lattice Field Theory, Cairns, Australia (2012), and talk at the TAU-12 Conference. [5] M. Knecht, A. Nyffeler, M. Perrottet and E. de Rafael, Phys. Rev. Lett. 88:071802 (2002). [6] M. Knecht and A. Nyffeler, Phys. Rev. D 65:073034 (2002). [7] A. Nyffeler, unpublished [8] M. Ramsey-Musolf and M. Wise,Phys. Rev. Lett. 89:041601 (2002). [9] K. Melnikov and A. Vainshtein, Phys. Rev. D 70:113006 (2004). [10] J. Prades, E. de Rafael and A Vainshtein, in Lepton Dipole Moments, ed. B. Lee Roberts and W.J. Marciano, p.303, World Scientific (2010). [11] T. Goecke, Ch.S. Fisher and R. Williams, Phys. Rev. D 83:094006 (2011). [12] D. Greynat and E. de Rafael, JHEP 1207 020 (2012). [13] A. Manohar and G. Georgi, Nucl. Phys. B234 189 (1984). [14] S. Weinberg, Phys.Rev.Lett. 105 (2010) 261601. [15] E. de Rafael, Phys. Lett. B703 (2011) 60. [16] T. Goecke, Ch.S. Fisher and R. Williams, arXiv:1210.1759v1 [hep-ph]. [17] J Bijnens and M.Z. Abyaneh, arXiv:1208.3548v1 [hep-ph].
Nuclear Physics B (Proc. Suppl.) 253–255 (2014) 135–138 www.elsevier.com/locate/npbps
Hadronic Light-by-Light Scattering Contributions to the Muon Anomaly Eduardo de Rafael Aix-Marseille Universit´e, CNRS, CPT, UMR 7332, 13288 Marseille, France Universit´e de Toulon, CNRS, CPT, UMR 7322, 83957 La Garde, France
Abstract I give an overview of what we know at present about the Hadronic Ligh-by-Light Scattering Contribution to the anomalous magnetic moment of the muon in the Standard Model. Keywords: Muon Anomaly, QCD, Constituent Chiral Quark Model in Japan which plan to reduce the present experimental error to 0.14 ppm 2 .
1. Introduction I was asked by the organizers of the TAU-12 Conference to review the present status of the Hadronic Ligh-by-Light Scattering (HLbyL) Contributions to the anomalous magnetic moment of the muon (aμ ). This is motivated by the fact that the present experimental determination of aμ , which is dominated by the latest BNL experiment of the E821 collaboration [1] aμ (E821 − BNL) = 116 592 089(54)stat (33)syst × 10−11 , (1)
Table 1: Standard Model Contributions1 Contribution
QED (leptons) HVP(lo)[e+ e− ] HVP(ho) HLbyL EW Total SM
Result in 10−11 units
116 584 718.85 ± 0.04 6 923 ± 42 −98.4 ± 0.7 105 ± 26 153 ± 1 116 591 801 ± 49
when confronted with the Standard Model prediction 1 aμ (SM) = 116 591 801 ± 49 ,
(2)
shows a persistent 3.6 σ discrepancy which deserves attention. The results of the Standard Model contributions to aμ , due to the work of many people, are summarized in Table 1. These include the well known QED contribution from virtual photons and leptons, which is now known to tenth order (see ref.[3] and references therein), as well as from the Hadronic Vacuum Polarization of lowest order (HVP(lo)) and higher order (HVP(ho), from the HLbyL and from the Electroweak (EW) interactions. One observes that the largest errors in Table 1 come from the HVP and the HLbyL contributions. The size of these errors is of special concern in view of the future experiments at FNAL in the USA and at JPARC Email address: [email protected] (Eduardo de Rafael) a recent review article where earlier references can be found see [2]. 1 For
http://dx.doi.org/10.1016/j.nuclphysbps.2014.09.033 0920-5632/© 2014 Elsevier B.V. All rights reserved.
The lowest order HVP contribution to the muon anomaly has a well known integral representation in terms of the e+ e− one photon annihilation cross-section into hadrons. The integral in question is in fact dominated by the contributions from the low-energy region. Its determination has been improving through the years thanks to the advent of more and more refined measurements, the latest coming from the BaBar and Belle facilities. The error here is likely to be reduced in the near future. 2. The HLbyL Scattering Contribution Contrary to the HVP contribution, the HLbyL contribution shown in Fig. 1 cannot be written as an integral over experimentally accessible observables. The calculation from theory requires the knowledge of Quantum Chromodynamics (QCD) contributions at all energy scales, 2 See the contributions by Lee Roberts and Tsutomu Mibe at this Conference.
136
E. de Rafael / Nuclear Physics B (Proc. Suppl.) 253–255 (2014) 135–138
something which we don’t know how to do at present except via numerical lattice QCD evaluations which, unfortunately, are difficult to implement in this case 3 .
π
π
π
μ
μ
μ
Figure 2:
X
Dominant HLbyL Scattering Contribution.
Hadrons
+ Permutations
μ
Numerically, for the physical values of the constants in Eq. 3 one finds
Figure 1:
α 3
Hadronic Light-by-Light Scattering Contribution.
Nc 2
π Things, however, are not as hopeless as all that. We know some important constraints from QCD which the hadronic models one has to resort to, in order to evaluate this contribution, have to obey. 2.1. Chiral Limit and Large-Nc limit of QCD It is well known that the hadronic realization of QCD in the sector of the light u , d , s quarks is governed by the phenomena of spontaneous chiral symmetry breaking and confinement. This implies the existence of a Mass Gap between the pseudoscalar (Goldstone-like) particles and the other hadronic particles (the ρ being the lowest one in Nature). In the limit where this Mass Gap is considered to be large, and to leading order in the 1/Nc –expansion (Nc is the number of colours), the contribution of the HLbyL to aμ is dominated by the one pion exchange diagrams shown in Fig. 2 and the leading result in this limit is known analytically [5]:
(HLbyL) 0 aμ (π )
α 3
m2μ
Mρ log = Nc π mπ 48π2 fπ2 Mρ + O(1) . + O log mπ 2
m2μ 48π2 fπ2
log2
Mρ = 95 × 10−11 , mπ
(4)
which is within the ballpark of the hadronic model determinations. The ρ-mass, however, is too close to the π-mass to take seriously this number which may have large corrections. In fact, it is in principle possible to m fix the coefficient of the O log mπρ term in Eq. 3. It can be shown [7, 8] that the unknown contribution to this coefficient is related to the π0 → e+ e− decay rate, albeit with the radiative corrections included. The O(1) term is, however, not fixed by symmetry restrictions which limits the use of Eq. 3 at the level of the required accuracy. 2.2. Short-Distance Constraint from the OPE in QCD The constraint in question comes from a clever observation by Melnikov and Vainshtein [9]. The three momenta k1 , k2 , k3 in the light–by–light subdiagram of the HLbyL scattering contribution to the muon anomaly (see Fig 3 below) form a triangle. When k12 ≈ k22 k32 , q
k1
q
0
0
2
γ γγ5
H
(3)
k3
k2
k3
Figure 3:
This is phenomenologically relevant because it can be used as a check of Hadronic Model Calculations: letting the hadronic masses of a model become very large with respect to the pion mass, one must find an answer compatible with the analytic result in Eq. 3 when Mρ is also taken to be large. The fact that the HLbyL contribution in this limit is positive was crucial in fixing the (HLbyL) overall sign of aμ [6].
The OPE Constraint of Melnikov and Vainshtein. and k12 ≈ k22 m2ρ in this triangle one can apply the Operator Product Expansion (OPE) in the two vector currents which carry hard momenta with the result:
4
d x1 3 There
are, however, promising lattice projects under study, see e.g. ref. [4].
d4 x2 e−ik1 ·x1 −ik2 ·x2 Jν (x1 )Jρ (x2 ) =
1 2νρδγ kˆ δ , d4 ze−ik3 ·z J5γ (z) + O kˆ 2 kˆ 3
(5)
E. de Rafael / Nuclear Physics B (Proc. Suppl.) 253–255 (2014) 135–138
γ where j5 = q Q2q qγ ¯ γ γ5 q is the hadronic axial current and kˆ = (k1 − k2 )/2 ≈ k1 ≈ −k2 . As illustrated in Fig. 3 this OPE reduces the HLbyL amplitude, in the special kinematics under consideration, to the AVV triangle amplitude which is an object for which we have a much better theoretical insight. This observation has interesting phenomenological implications: • At large k1,2 the Pseudoscalar (and Axial-Vector) exchanges dominate. • The AVV limit also implies that the Fπ0 γ∗ γ∗ (k2 , k2 ) form factor at the vertices of Fig 2 must fall as 1/k2 . Unfortunately, the two asymptotic QCD constraints discussed above are not sufficient for a full model inde(HLbyL) pendent evaluation of aμ . This explains the relatively large error of ±26 × 10−11 for this contribution in Table 2 above 4 . (HLbyL) Most of the last decade calculations of aμ found in the literature are compatible with the QCD chiral constraints and the large-Nc limit discussed above. They all incorporate the π0 -exchange contribution modulated by π0 γ∗ γ∗ form factors correctly normalized to the Adler, Bell-Jackiw point-like coupling. They differ, however, on whether or not they satisfy the particular OPE constraint discussed above, and in the shape of the vertex form factors which follow from the different models. In spite of the different choices of these form factors there is, within errors, a reasonable agreement among the final results. An exception is the calculation reported in ref. [11] using a model based on a Dyson-Schwinger approach which, however, as we shall see contradicts generic properties which emerge from the Constituent Chiral Quark Model which we next discuss. 3. The Constituent Chiral Quark Model I have emphasized in recent meetings and workshops the need of a simple reference model to evaluate the various hadronic contributions to aμ within the same framework, and use it as a yardstick to compare with the more elaborated model evaluations in the literature. The reference model which we have proposed [12] is based on the Constituent Chiral Quark Model (CχQM) of Manohar and Georgi [13] in the presence of S U(3)L × S U(3)R external sources. It is an effective field theory which incorporates the interactions of the NambuGoldstone modes (the low-lying pseudoscalars) of the 4 See
e.g. the discussion in ref. [10].
137
spontaneously broken chiral symmetry, to lowest order in the chiral expansion and in the presence of chirally rotated quark fields which have become massive. As emphasized by Weinberg [14], the effective Lagrangian in question is renormalizable in the Large-Nc limit and, as shown in ref. [15], the number of the required counterterms is minimized for the choice gA = 1 of the axial coupling of the constituent quarks to the pseudoscalars. The model has its limitations but there is an exceptional class of low-energy observables for which the predictions of the CχQM can be rather reliable. This is the case when the leading short-distance behaviour of the underlying Green’s function of a given observable is governed by perturbative QCD. The contributions to aμ from HVP, from the HLbyL Scattering and from the Hadronic Zγγ vertex (provided that gA = 1) fall in this class. With gA = 1 the only free parameter in the model is the constituent quark mass MQ . The comparison between the CχQM prediction for the HVP and the phenomenological determination of a(HVP) shows that μ fixing MQ in the MQ = (240 ± 10) MeV , reproduces the phenomenological determination of the lowest order HVP within an error of less than 10%. This error, however, does not include the systematic error of the CχQM itself. As shown in ref. [12], with MQ fixed in this range, the higher order HVP contributions, as well as the electroweak Hadronic Zγγ contribution are reproduced rather well. When examining the HLbyL scattering contribution to the muon anomaly in the CχQM, one finds that there are two competing contributions: one is the π0 – exchange diagrams in Fig. 2 where the solid discs there are now constituent quark loops, the other one the irreducible constituent quark loop in Fig. 1 where the solid triangle becomes now a constituent quark loop. An interesting feature which emerges from the calculation in ref. [12] is the balance between the Goldstone Contribution and the Quark Loop Contribution. Indeed, as the constituent quark mass MQ gets larger and larger, the Goldstone Contribution dominates: asymptotically, M for large MQ values it reproduces the log2 mπQ behaviour of Eq. 3, while for MQ smaller and smaller it is the Constituent Quark Loop Contribution which dominates: asymptotically, for small MQ values (though still with MQ > mμ ) it behaves like
HLbyL aμ (CQL)
=
α 3
⎛ ⎞ ⎟⎟⎟ ⎜⎜⎜ 4 ⎜ Nc ⎜⎜⎝ Qq ⎟⎟⎟⎠ ×
π q=u,d,s ⎧ ⎛ 4 ⎞⎫ 2 ⎪ ⎜⎜⎜ mμ m2μ ⎟⎟⎟⎪ ⎪ ⎪ 19 mμ ⎨ 3 2 ⎜ ⎟⎟⎠⎬ ζ(3) − + O ⎜⎝ 4 log . ⎪ ⎪ ⎪ 2 2 ⎩ 2 ⎭ 16 MQ MQ MQ ⎪
(6)
138
E. de Rafael / Nuclear Physics B (Proc. Suppl.) 253–255 (2014) 135–138
References These features are illustrated by the plot of the total (HLbyL) (CχQM) versus MQ shown in Fig. 4. In fact aμ (HLbyL) the plot also shows that the value of aμ (CχQM) is quite stable for a rather large choice of reasonable MQ values. 190
180
170 HLbyL
aΜ
CΧQM 1011 160
150
140 200
250
300
350
400
MQ in MeV
Figure 4:
The HLbyL Contribution in the CχQM.
The CχQM result contradicts what is reported in ref. [11] where the equivalent contribution to the constituent quark loop is found to be: (136 ± 59) × 10−11 i.e. much larger than the contribution found by the same authors for the π0 -exchange: (81 ± 12) × 10−11 which, within errors, is compatible with previous phenomenological determinations. This casts serious doubts about the compatibility of the model used in ref. [11] (or perhaps of their calculation) with basic QCD features encoded in the CχQM. In fact, during the completion of this mini review, there has appeared a new version of this model in the archives [16] with a smaller result for the quark loop contribution: (96 ± 2) × 10−11 , where the error here does not include the systematic error of their model and, furthermore, the calculation is claimed to be incomplete as yet. We conclude that, in the absence of more refined calculations, which with some effort may hopefully become available, the number quoted in Table 1 for the HLbyL scattering contribution to aμ discussed in [10] represents the best valid estimate at present. In fact, there is a recent independent analysis in ref. [17] which also confirms that estimate.
Acknowledgements I wish to thank David Greynat for a pleasent collaboration on the CχQM calculations reported here and Marc Knecht and Laurent Lellouch for helpful discussions.
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