- Email: [email protected]
g−2 of the muon and Δα re-evaluated
Available online at www.sciencedirect.com
Nuclear Physics B (Proc. Suppl.) 218 (2011) 225–230 www.elsevier.com/locate/npbps
g − 2 of the muon and Δα re-evaluated T. Teubnera , K. Hagiwarab, R. Liaoa , A. D. Martinc , D. Nomurad a
Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, U.K.
b
KEK Theory Center and Sokendai, Tsukuba 305-0801, Japan
c
Department of Physics and Institute for Particle Physics Phenomenology, University of Durham, Durham DH1 3LE, U.K. d
Department of Physics, Tohoku University, Sendai 980-8578, Japan
We briefly review the Standard Model prediction of g − 2 of the muon and confront it with the formidably accurate measurement from BNL. The focus is on recent improvements of the hadronic vacuum polarisation contributions. We also calculate the running QED coupling at space- and time-like momentum transfer. Its value at the scale of the Z boson mass, α(MZ2 ), is an important ingredient in electroweak precision fits and its prediction is improved considerably.
1. g − 2 OF THE MUON The anomalous magnetic moment of the muon, aμ = (g − 2)μ /2, is one of the great success stories of Quantum Field Theory and the Standard Model (SM), albeit various twists and turns. The discrepancy between its measured value and the SM prediction is probably the most compelling hint of physics beyond the SM at the TeV scale, and is therefore under intense scrutiny. All sectors of the SM contribute in a significant way and have been calculated with increasing precision through the efforts of many groups. The QED contributions are known completely up to four-loop accuracy, with leading five-loop effects also included. Their value is [1] aQED = μ 116584718.08(15) · 10−11 . Although most important numerically, their error is formidably small. The weak contributions are known to two-loop accuracy [2], aEW = (154 ± 2) · 10−11 , and are μ also very well under control. While they are much smaller than the QED contributions, the high precision of the experimental measurement of g − 2 is definitely sensitive to the electroweak effects. The hadronic contributions are divided into the leading order vacuum polarisation (VP) LO VP contributions, aHad, , and higher order efμ 0920-5632/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2011.06.036
HO VP ) fects ∼ α3 , induced by both VP (aHad, μ and by so-called light-by-light scattering effects l−by−l (aHad, ). None of these can be calculated usμ ing perturbative QCD (pQCD). For the light-bylight contributions one relies on model-based calculations. There is now a fair agreement between the results obtained by different groups, see e.g. the discussion in [3,4]. Below we will use the value from the recent compilation by Prades, de Rafael l−by−l and Vainshtein [5], aHad, = (10.5 ± 2.6) · μ −10 10 , for which the authors have combined predictions of their individual works. This number is consistent with the result from Jegerlehner and l−by−l Nyffeler [3,4], aHad, = (11.6 ± 4.0) · 10−10 . μ (Note that work on predictions based on lattice QCD is under way, see e.g. [6], and recently also an alternative approach based on DysonSchwinger methods was presented in [7].) Clearly, the error of the light-by-light scattering contributions is very important and may become the limiting factor in the future. However, currently still the hadronic VP contributions are the dominant source of uncertainty of the SM prediction.
1.1. Vacuum polarisation contributions In contrast to the light-by-light scattering effects, VP contributions can be predicted with dis-
226
T. Teubner et al. / Nuclear Physics B (Proc. Suppl.) 218 (2011) 225–230
persion integrals and using experimental data for the hadronic cross sections, σHad . The leading order contribution is given by ∞ 1 Had, LO VP 0 aμ = ds σHad (s)K(s) , (1) 4π 3 m2π
BaBar 09 New Fit KLOE 10 KLOE 09
1200
+ -
800
+ -
σ(e e → π π ) [nb]
1000
600 400 200 0
0.6
0.65
0.7
0.75 0.8 √s [GeV]
0.85
0.9
0.95
Figure 1. Light (yellow) band: compilation of data for the 2π channel, including the new BaBar data [10,11] (shown as darker (green) band); also shown are the data from KLOE [12,13], as indicated on the plot. See [14] for references and a detailed discussion of the data from the experiments CMD-2 and SND in Novosibirsk, which are also very important in the fit.
200 New Fit BaBar 09 KLOE 10 CMD-2 Low 150 SND Re OLYA-VEPP2 TOF-VEPP 2M NA7 CMD-VEPP 2M 100 ChPT
+ -
+ -
σ(e e → π π ) [nb]
with a well known kernel function K which weights lowest energies most strongly (for details 0 see e.g. the review [3]), and where σHad is the hadronic cross section without effects due to the running α. Recently there have been major improvements due to better experimental cross section data. These data traditionally come from experiments at colliders with tunable centre-ofmass energy (‘direct scan’), like CMD-2 and SND at VEPP-2M in Novosibirsk, BES II at BEPC in Beijing, or CLEO at CESR in Cornell. Now also the method of ‘radiative return’ (via initial state photon radiation) provides a competitive alternative for meson factories with high luminosity. Here the experiments are KLOE at DAΦNE in Frascati, BaBar at PEP-II in Stanford and BELLE at KEKB in Tsukuba. Their cross section measurements provide important new information and are also completely independent w.r.t. the analysis tools. At low energies, the hadronic cross section is measured for exclusive hadronic final states (called exclusive ‘channels’ in the following). Before summing all channels to obtain the total hadronic cross section, data from all relevant experiments have to be combined in each channel. In our analysis this is done by first re-binning all data (of one channel) and then performing a fit of the finely binned data, assuming a piecewise constant cross section and using a non-linear χ2 minimisation. (For details of the combination procedure, the treatment of data w.r.t. radiative corrections and full data references see [8,9].) Our data compilation for the most important e+ e− → 2π channel, which contributes more than 70% to the total aHad μ , is shown in Fig. 1. Figure 2 displays the lowest energy region close to the 2π threshold, a region improved considerably by new data. As is already visible in Fig. 1, there is a slight disagreement between different data sets, especially at energies at and above the ρ peak. This disagreement is made more explicit in the
1400
50
0
0.3
0.32
0.34
0.36
0.38 √s [GeV]
0.4
0.42
0.44
Figure 2. Lowest energy part of the 2π channel: Our compilation is shown as the light (yellow) band; data as indicated. The dashed line is the prediction from chiral perturbation theory (ChPT) and used for the energy range from the threshold to the first data point.
T. Teubner et al. / Nuclear Physics B (Proc. Suppl.) 218 (2011) 225–230
2
New Fit BaBar 09 KLOE 09 KLOE 10
0.08 0.06
Fit (w/o BaBar) Fit (all sets) BaBar (08) Orsay (90) DM2 (91)
1.5
(σRadRet Sets - σFit)/σFit
227
+ - 0
σ(e e → K K π ) [nb]
0.04 0.02
1
+ -
0 -0.02
0.5
-0.04 -0.06
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0
0.95
√s [GeV]
m2π
C
1 We have checked that this global error inflation is not leading to an overly optimistic error compared to local error inflation.
1.6
1.7 √s [GeV]
1.8
1.9
2
45 Fit w/o BaBar Fit all sets BaBar (05) DM2 (90) DM1 (82) M3N (79)
40 35
+ - + + -
difference plot in Fig. 3. It is currently not understood and prevents an even higher acccuracy in the 2π channel. Note that incompatibilities of data sets are reflected in an increased χ2min of the fit. Whenever χ2min /d.o.f. > 1 we inflate the error of the respective contribution to aμ by χ2min /d.o.f..1 At higher energies, many exclusive channels have to be summed to give the total hadronic cross section, σHad (s), with sufficient accuracy. The use of exclusive channels only becomes un√ feasible above s = 2 GeV, when one uses inclusive data and/or pQCD, see below. In the energy region between 1.4 and 2 GeV there exist both inclusive and exclusive data. In the past, we have chosen to use a compilation of the inclusive data. This was partly motivated by the results of a QCD sum-rule analysis, relating the convolution integral of R(s) = σHad /σ(e+ e− → μ+ μ− ) with a suitable kernel function f (s) to a corresponding contour-integral of the perturbatively calculated Adler-D function, s0 ds R(s) f (s) = ds D(s) g(s) . (2)
1.5
Figure 4. Hadronic cross section compilation for the K + K − π 0 channel without and with the recent data from BaBar, as indicated. Also displayed are the main data sets.
σ(e e → π π π π ) [nb]
Figure 3. Normalised difference of our data compilation in the 2π channel (including all data) with the sets from BaBar and KLOE as indicated in the plot.
1.4
30 25 20 15 10 5 0
1.4
1.5
1.6
1.7 √s [GeV]
1.8
1.9
2
Figure 5. As Fig. 4 but for the 2π + 2π − channel. Here f (s) = (1 − s/s0 )m (s/s0 )n , C is a circular contour of radius s0 and g(s) is a known function once f (s) is specified, see [8] for formulae and details. We then found that the inclusive data were more compatible with perturbative QCD and the world-average of αs . However, recently many multi-hadronic channels have been measured by BaBar, using radiative return. Figures 4 and 5 show two examples. These changes have improved both the accuracy and the reliability of the exclusive channels in this energy region. Figure 6 displays the results of our new sum-rule
T. Teubner et al. / Nuclear Physics B (Proc. Suppl.) 218 (2011) 225–230
228
4.5 –
(m,n,√s0)
inclusive inclusive inclusive inclusive inclusive inclusive inclusive inclusive inclusive inclusive inclusive inclusive
(0,0,3.7) (1,0,3.7) (1,1,3.7) (2,0,3.7) (0,0,3) (1,0,3) (1,1,3) (2,0,3) (0,0,2.5) (1,0,2.5) (1,1,2.5) (2,0,2.5)
exclusive exclusive exclusive exclusive exclusive exclusive exclusive exclusive exclusive exclusive exclusive exclusive
0.1
0.12
(13.0) (20.4) (13.1) (22.7) (19.9) (26.9) (26.4) (27.2) (29.0) (30.6) (43.8) (24.2)
0.14
Figure 6. Different sum-rules translated to values of αs , for use of inclusive and exclusive data in the energy range from 1.43 to 2 GeV, and using pQCD from 2 GeV up to the charm threshold. analysis for both options of using either inclusive data or the sum over exclusive channels. Clearly the latter is preferred by the sum-rules. We therefore use, for the calculation of the hadronic VP √ contributions between s ∼ 1.4 and 2 GeV, the sum over the exclusive channels and not old inclusive data as done previously. This moves the prediciton of g − 2 slightly upwards. Note that for the sum-rule results shown in Fig. 6 we have used pQCD for energies above 2 GeV and below the charm threshold. Figure 7 shows the (inclusive) data in this energy region, compared to the prediction from pQCD. It is compelling that the most recent precise data from BES II [15] are in perfect agreement with pQCD, which has a very small error as indicated by the dark (red) band in the plot. In light of this we believe that the older data in this region may be discarded and use, for our g − 2 predictions presented below, R from four-loop QCD but with errors inflated to match those of the BES II data. This leads to a small shift downwards, compared to our previous predictions, and partly cancels the shift upwards from using the exclusive data in the region below 2 GeV.
4
3.5
R(s)
(0,0,3.7) (1,0,3.7) (1,1,3.7) (2,0,3.7) (0,0,3) (1,0,3) (1,1,3) (2,0,3) (0,0,2.5) (1,0,2.5) (1,1,2.5) (2,0,2.5)
αS(M Z2)
Inclusive Inclusive mean pQCD pQCD mean BES II (09) BES (01) BES (99)
data (fraction (%))
3
2.5
2
1.5
2
2.5
3
3.5
4
√s [GeV]
Figure 7. Data compilation for R = σ(e+ e− → hadrons)/σ(e+ e− → μ+ μ− ) and prediction from pQCD in the region from 2 GeV up to the charm threshold. √ For s = 3.7 GeV to 11.09 GeV we use a compilation of the inclusive data in the dispersion integral (1). Above 11.09 GeV we use state-ofthe-art pQCD (up to four-loop accuracy for the massless terms). The narrow resonances J/ψ, ψ and the Υ(1S − 6S) family are added as narrow resonances. (See [8,9,16] for more details of our analysis and for more references.) Adding up the leading-order hadronic VP contributions from all energies, we obtain LO VP aHad, = (695.1 ± 4.5) · 10−10 . This value μ is slightly larger than our previous prediction, mainly as a consequence of the new BaBar data in the 2π channel. The corresponding value for the hadronic higher-order VP contribution is practically unHO VP changed and reads aHad, = (−9.8 ± 0.1) · μ −10 10 . 1.2. SM prediction and BNL measurement Combining all SM contributions we obtain aSM μ
=
aQED + aEW + aHad μ μ μ
=
(11659183.0 ± 4.8) · 10−10 .
In Fig. 8 this prediction is shown together with similar ones as available at the time of the conference and confronted with the measurement from BNL [17], which has slightly changed due to a new value of the muon to proton magnetic ra-
T. Teubner et al. / Nuclear Physics B (Proc. Suppl.) 218 (2011) 225–230
perimental data for the normalised hadronic cross section R, ∞ R(s) ds αq 2 ΔαHad (q 2 ) = − P . (3) 2 3π m2π s(s − q )
HMNT (06) JN (09) Davier et al, τ (09) w/o BaBar (09)
Davier et al, e+e– w/
229
BaBar (09)
HLMNT (09) HLMNT prelim. (10) experiment BNL BNL (new from shift in λ) 170
180
190
200
210
aμSM × 1010 – 11659000
Figure 8. Various recent predictions for aμ compared to the measurement from BNL.
tio from the CODATA group. The change of our new prediction (labelled ‘HLMNT prelim. (10)’ in Fig. 8) w.r.t. our earlier result ‘HLMNT (09)’ is mainly due to the inclusion of the new 2π data from BaBar. This change is in line with the findings from Davier et al.. Despite all changes as discussed above, the recent SM predictions agree well and signal a discrepancy with the measured value of more than three standard deviations. (When using our new preliminary result we obtain a 3.2 σ discrepancy, whereas Davier et al. have reported 3.6 σ at this conference [18,19].) While the current level of significance does not allow definite conclusions, it is intriguing that the discrepancy has survived all scrutiny. It could well be explained by the existence of supersymmetric extensions of the SM at the TeV scale and already strongly constrains the allowed parameter space for physics beyond the SM. 2. ELECTROMAGNETIC COUPLING α Vacuum polarisation leads to the running of the electromagnetic coupling, α(q 2 ) = α/(1 − ΔαLep (q 2 ) − ΔαHad (q 2 )). While the leptonic contributions, ΔαLep (q 2 ), can be calculated using perturbation theory, the hadronic contributions require a dispersion integral and the use of ex-
(Here α is the fine-structure constant and P stands for the principal value of the integral.) We have used known analytical expressions for the leptonic contributions and (3) for the hadronic contributions, with the same data compilation for R(s) as in our g − 2 prediction, to obtain the running QED coupling. On request we provide a simple-to-use set of Fortran routines for α(q 2 ) for space- and time-like q 2 and also for the imaginary part of the VP. These routines are used to ‘undress’ the experimentally measured hadronic cross sections w.r.t. running coupling effects (an iteration is needed to achieve consistent results for g − 2 and Δα) and are also used by other groups for applications in Monte Carlo programs (see [20] for a detailed discussion). The value of the running coupling at MZ2 is of particular interest for the electroweak precision tests of the SM. For the hadronic five-flavour corrections at this scale we obtain (5)
ΔαHad (MZ2 ) = 0.02759 ± 0.00015 ,
(4)
which is similar in size but considerably more accurate than the default typically used in the electroweak precision fits. Together with the perturbative top-quark and leptonic contributions this translates into α(MZ2 )−1 = 128.953 ± 0.020. If our result (4) would be used in the electroweak precision fits, the parabola for the indirect determination of the Higgs mass (the famous ‘blueband’ plot) would have a minimum at roughly the same value of MH , but be narrower and hence add to the tension in the SM fit. This correlation of the precision observables aμ and Δα makes it seem unlikely that the discrepancy in g − 2 could be solved by a change of the hadronic cross section data alone [21]. 3. CONCLUSIONS AND OUTLOOK We have presented a brief status report and new evaluations of g − 2 and α(MZ2 ). For g − 2, the deviation between the SM prediction and the
T. Teubner et al. / Nuclear Physics B (Proc. Suppl.) 218 (2011) 225–230
230
2
∞
mπ
mπ
1.4
aμhad,LO
REFERENCES
(error)2
value rad.
0.6
0.9
0.6
∞2 1.4
∞ 2 Δα(5) had (M Z)
mπ
0.9
mπ 0.6 0.6 0.9 1.4 2 4
11
0.9 1.4
rad.
2
∞ 11 4
Figure 9. Pie diagrams for the contributions to the value and squared error of g − 2 and (5) ΔαHad (MZ2 ) from different energy regions. BNL measurement has been further consolidated and persists at more than 3 σ. If due to SUSY, the sparticle masses could well be in reach of the LHC. For the future, further improvements in aSM μ and Δα are foreseen through upcoming measurements of the hadronic cross sections used in the theoretical prediction. Figure 9 shows how different regions contribute to the value and the error budget of both quantities. The prospects for improvements are good: There are promising plans for KLOE-2 at an upgraded DAΦNE, with chances to also help improving the determination of the light-by-light scattering contributions through measuring meson form-factors in γγ production. VEPP-2000 in Novosibirsk has already started (see [14]) and BES-III will cover higher energies. New g − 2 experiments at Fermilab and at J-PARC are planned to start in the middle of this decade. They will hopefully become reality and provide even more accurate measurements to test the SM and to constrain physics beyond it. Acknowledgements It is a pleasure to thank the organisers of Tau 2010 for a very interesting, stimulating and perfectly organised conference.
1. T. Kinoshita and M. Nio, Phys. Rev. D73 (2006) 053007. T. Aoyama et al., Phys. Rev. D77 (2008) 053012. 2. A. Czarnecki, W. J. Marciano and A. Vainshtein, Phys. Rev. D67 (2003) 073006, Erratum-ibid. D73 (2006) 119901. M. Knecht, S. Peris, M. Perrottet and E. de Rafael, JHEP 0211 (2002) 003. 3. F. Jegerlehner and A. Nyffeler, Phys. Rept. 477 (2009) 1. 4. A. Nyffeler, Phys. Rev. D79 (2009) 073012. 5. J. Prades, E. de Rafael and A. Vainshtein, arXiv:0901.0306 [hep-ph]. 6. T. Blum and S. Chowdhury, Nucl. Phys. Proc. Suppl. 189 (2009) 251; S. Chowdhury et al., PoS LATTICE2008 (2008) 251. See also contribution from P. Rakow at the conference Lattice 2008. 7. T. Goecke, C. S. Fischer and R. Williams, arXiv:1012.3886 [hep-ph]. 8. K. Hagiwara et al., Phys. Rev. D69 (2004) 093003; Phys. Lett. B557 (2003) 69. 9. K. Hagiwara et al., Phys. Lett. B649 (2007) 173. 10. BaBar Collaboration, B. Aubert et al., Phys. Rev. Lett. 103 (2009) 231801. 11. B. Malaescu, in these proceedings. 12. KLOE Collaboration, F. Ambrosino et al., Phys. Lett. B670 (2009) 285; arXiv:1006.5313 [hep-ex]. 13. G. Venanzoni, in these proceedings. 14. B. Shwartz, in these proceedings. 15. BES Collaboration, M. Ablikim et al., Phys. Lett. B677 (2009) 239. 16. K. Hagiwara et al., Chinese Phys. C34 (2010) 728. 17. G. W. Bennett et al., Phys. Rev. D73 (2006) 072003. 18. M. Davier, in these proceedings. 19. A. Hoecker, in these proceedings. 20. S. Actis et al., Eur. Phys. J. C66 (2010) 585. 21. M. Passera, W. J. Marciano and A. Sirlin, Phys. Rev. D78 (2008) 013009; arXiv:1001.4528 [hep-ph]. 22. L. Roberts, in these proceedings. 23. T. Mibe, in these proceedings.
Nuclear Physics B (Proc. Suppl.) 218 (2011) 225–230 www.elsevier.com/locate/npbps
g − 2 of the muon and Δα re-evaluated T. Teubnera , K. Hagiwarab, R. Liaoa , A. D. Martinc , D. Nomurad a
Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, U.K.
b
KEK Theory Center and Sokendai, Tsukuba 305-0801, Japan
c
Department of Physics and Institute for Particle Physics Phenomenology, University of Durham, Durham DH1 3LE, U.K. d
Department of Physics, Tohoku University, Sendai 980-8578, Japan
We briefly review the Standard Model prediction of g − 2 of the muon and confront it with the formidably accurate measurement from BNL. The focus is on recent improvements of the hadronic vacuum polarisation contributions. We also calculate the running QED coupling at space- and time-like momentum transfer. Its value at the scale of the Z boson mass, α(MZ2 ), is an important ingredient in electroweak precision fits and its prediction is improved considerably.
1. g − 2 OF THE MUON The anomalous magnetic moment of the muon, aμ = (g − 2)μ /2, is one of the great success stories of Quantum Field Theory and the Standard Model (SM), albeit various twists and turns. The discrepancy between its measured value and the SM prediction is probably the most compelling hint of physics beyond the SM at the TeV scale, and is therefore under intense scrutiny. All sectors of the SM contribute in a significant way and have been calculated with increasing precision through the efforts of many groups. The QED contributions are known completely up to four-loop accuracy, with leading five-loop effects also included. Their value is [1] aQED = μ 116584718.08(15) · 10−11 . Although most important numerically, their error is formidably small. The weak contributions are known to two-loop accuracy [2], aEW = (154 ± 2) · 10−11 , and are μ also very well under control. While they are much smaller than the QED contributions, the high precision of the experimental measurement of g − 2 is definitely sensitive to the electroweak effects. The hadronic contributions are divided into the leading order vacuum polarisation (VP) LO VP contributions, aHad, , and higher order efμ 0920-5632/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2011.06.036
HO VP ) fects ∼ α3 , induced by both VP (aHad, μ and by so-called light-by-light scattering effects l−by−l (aHad, ). None of these can be calculated usμ ing perturbative QCD (pQCD). For the light-bylight contributions one relies on model-based calculations. There is now a fair agreement between the results obtained by different groups, see e.g. the discussion in [3,4]. Below we will use the value from the recent compilation by Prades, de Rafael l−by−l and Vainshtein [5], aHad, = (10.5 ± 2.6) · μ −10 10 , for which the authors have combined predictions of their individual works. This number is consistent with the result from Jegerlehner and l−by−l Nyffeler [3,4], aHad, = (11.6 ± 4.0) · 10−10 . μ (Note that work on predictions based on lattice QCD is under way, see e.g. [6], and recently also an alternative approach based on DysonSchwinger methods was presented in [7].) Clearly, the error of the light-by-light scattering contributions is very important and may become the limiting factor in the future. However, currently still the hadronic VP contributions are the dominant source of uncertainty of the SM prediction.
1.1. Vacuum polarisation contributions In contrast to the light-by-light scattering effects, VP contributions can be predicted with dis-
226
T. Teubner et al. / Nuclear Physics B (Proc. Suppl.) 218 (2011) 225–230
persion integrals and using experimental data for the hadronic cross sections, σHad . The leading order contribution is given by ∞ 1 Had, LO VP 0 aμ = ds σHad (s)K(s) , (1) 4π 3 m2π
BaBar 09 New Fit KLOE 10 KLOE 09
1200
+ -
800
+ -
σ(e e → π π ) [nb]
1000
600 400 200 0
0.6
0.65
0.7
0.75 0.8 √s [GeV]
0.85
0.9
0.95
Figure 1. Light (yellow) band: compilation of data for the 2π channel, including the new BaBar data [10,11] (shown as darker (green) band); also shown are the data from KLOE [12,13], as indicated on the plot. See [14] for references and a detailed discussion of the data from the experiments CMD-2 and SND in Novosibirsk, which are also very important in the fit.
200 New Fit BaBar 09 KLOE 10 CMD-2 Low 150 SND Re OLYA-VEPP2 TOF-VEPP 2M NA7 CMD-VEPP 2M 100 ChPT
+ -
+ -
σ(e e → π π ) [nb]
with a well known kernel function K which weights lowest energies most strongly (for details 0 see e.g. the review [3]), and where σHad is the hadronic cross section without effects due to the running α. Recently there have been major improvements due to better experimental cross section data. These data traditionally come from experiments at colliders with tunable centre-ofmass energy (‘direct scan’), like CMD-2 and SND at VEPP-2M in Novosibirsk, BES II at BEPC in Beijing, or CLEO at CESR in Cornell. Now also the method of ‘radiative return’ (via initial state photon radiation) provides a competitive alternative for meson factories with high luminosity. Here the experiments are KLOE at DAΦNE in Frascati, BaBar at PEP-II in Stanford and BELLE at KEKB in Tsukuba. Their cross section measurements provide important new information and are also completely independent w.r.t. the analysis tools. At low energies, the hadronic cross section is measured for exclusive hadronic final states (called exclusive ‘channels’ in the following). Before summing all channels to obtain the total hadronic cross section, data from all relevant experiments have to be combined in each channel. In our analysis this is done by first re-binning all data (of one channel) and then performing a fit of the finely binned data, assuming a piecewise constant cross section and using a non-linear χ2 minimisation. (For details of the combination procedure, the treatment of data w.r.t. radiative corrections and full data references see [8,9].) Our data compilation for the most important e+ e− → 2π channel, which contributes more than 70% to the total aHad μ , is shown in Fig. 1. Figure 2 displays the lowest energy region close to the 2π threshold, a region improved considerably by new data. As is already visible in Fig. 1, there is a slight disagreement between different data sets, especially at energies at and above the ρ peak. This disagreement is made more explicit in the
1400
50
0
0.3
0.32
0.34
0.36
0.38 √s [GeV]
0.4
0.42
0.44
Figure 2. Lowest energy part of the 2π channel: Our compilation is shown as the light (yellow) band; data as indicated. The dashed line is the prediction from chiral perturbation theory (ChPT) and used for the energy range from the threshold to the first data point.
T. Teubner et al. / Nuclear Physics B (Proc. Suppl.) 218 (2011) 225–230
2
New Fit BaBar 09 KLOE 09 KLOE 10
0.08 0.06
Fit (w/o BaBar) Fit (all sets) BaBar (08) Orsay (90) DM2 (91)
1.5
(σRadRet Sets - σFit)/σFit
227
+ - 0
σ(e e → K K π ) [nb]
0.04 0.02
1
+ -
0 -0.02
0.5
-0.04 -0.06
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0
0.95
√s [GeV]
m2π
C
1 We have checked that this global error inflation is not leading to an overly optimistic error compared to local error inflation.
1.6
1.7 √s [GeV]
1.8
1.9
2
45 Fit w/o BaBar Fit all sets BaBar (05) DM2 (90) DM1 (82) M3N (79)
40 35
+ - + + -
difference plot in Fig. 3. It is currently not understood and prevents an even higher acccuracy in the 2π channel. Note that incompatibilities of data sets are reflected in an increased χ2min of the fit. Whenever χ2min /d.o.f. > 1 we inflate the error of the respective contribution to aμ by χ2min /d.o.f..1 At higher energies, many exclusive channels have to be summed to give the total hadronic cross section, σHad (s), with sufficient accuracy. The use of exclusive channels only becomes un√ feasible above s = 2 GeV, when one uses inclusive data and/or pQCD, see below. In the energy region between 1.4 and 2 GeV there exist both inclusive and exclusive data. In the past, we have chosen to use a compilation of the inclusive data. This was partly motivated by the results of a QCD sum-rule analysis, relating the convolution integral of R(s) = σHad /σ(e+ e− → μ+ μ− ) with a suitable kernel function f (s) to a corresponding contour-integral of the perturbatively calculated Adler-D function, s0 ds R(s) f (s) = ds D(s) g(s) . (2)
1.5
Figure 4. Hadronic cross section compilation for the K + K − π 0 channel without and with the recent data from BaBar, as indicated. Also displayed are the main data sets.
σ(e e → π π π π ) [nb]
Figure 3. Normalised difference of our data compilation in the 2π channel (including all data) with the sets from BaBar and KLOE as indicated in the plot.
1.4
30 25 20 15 10 5 0
1.4
1.5
1.6
1.7 √s [GeV]
1.8
1.9
2
Figure 5. As Fig. 4 but for the 2π + 2π − channel. Here f (s) = (1 − s/s0 )m (s/s0 )n , C is a circular contour of radius s0 and g(s) is a known function once f (s) is specified, see [8] for formulae and details. We then found that the inclusive data were more compatible with perturbative QCD and the world-average of αs . However, recently many multi-hadronic channels have been measured by BaBar, using radiative return. Figures 4 and 5 show two examples. These changes have improved both the accuracy and the reliability of the exclusive channels in this energy region. Figure 6 displays the results of our new sum-rule
T. Teubner et al. / Nuclear Physics B (Proc. Suppl.) 218 (2011) 225–230
228
4.5 –
(m,n,√s0)
inclusive inclusive inclusive inclusive inclusive inclusive inclusive inclusive inclusive inclusive inclusive inclusive
(0,0,3.7) (1,0,3.7) (1,1,3.7) (2,0,3.7) (0,0,3) (1,0,3) (1,1,3) (2,0,3) (0,0,2.5) (1,0,2.5) (1,1,2.5) (2,0,2.5)
exclusive exclusive exclusive exclusive exclusive exclusive exclusive exclusive exclusive exclusive exclusive exclusive
0.1
0.12
(13.0) (20.4) (13.1) (22.7) (19.9) (26.9) (26.4) (27.2) (29.0) (30.6) (43.8) (24.2)
0.14
Figure 6. Different sum-rules translated to values of αs , for use of inclusive and exclusive data in the energy range from 1.43 to 2 GeV, and using pQCD from 2 GeV up to the charm threshold. analysis for both options of using either inclusive data or the sum over exclusive channels. Clearly the latter is preferred by the sum-rules. We therefore use, for the calculation of the hadronic VP √ contributions between s ∼ 1.4 and 2 GeV, the sum over the exclusive channels and not old inclusive data as done previously. This moves the prediciton of g − 2 slightly upwards. Note that for the sum-rule results shown in Fig. 6 we have used pQCD for energies above 2 GeV and below the charm threshold. Figure 7 shows the (inclusive) data in this energy region, compared to the prediction from pQCD. It is compelling that the most recent precise data from BES II [15] are in perfect agreement with pQCD, which has a very small error as indicated by the dark (red) band in the plot. In light of this we believe that the older data in this region may be discarded and use, for our g − 2 predictions presented below, R from four-loop QCD but with errors inflated to match those of the BES II data. This leads to a small shift downwards, compared to our previous predictions, and partly cancels the shift upwards from using the exclusive data in the region below 2 GeV.
4
3.5
R(s)
(0,0,3.7) (1,0,3.7) (1,1,3.7) (2,0,3.7) (0,0,3) (1,0,3) (1,1,3) (2,0,3) (0,0,2.5) (1,0,2.5) (1,1,2.5) (2,0,2.5)
αS(M Z2)
Inclusive Inclusive mean pQCD pQCD mean BES II (09) BES (01) BES (99)
data (fraction (%))
3
2.5
2
1.5
2
2.5
3
3.5
4
√s [GeV]
Figure 7. Data compilation for R = σ(e+ e− → hadrons)/σ(e+ e− → μ+ μ− ) and prediction from pQCD in the region from 2 GeV up to the charm threshold. √ For s = 3.7 GeV to 11.09 GeV we use a compilation of the inclusive data in the dispersion integral (1). Above 11.09 GeV we use state-ofthe-art pQCD (up to four-loop accuracy for the massless terms). The narrow resonances J/ψ, ψ and the Υ(1S − 6S) family are added as narrow resonances. (See [8,9,16] for more details of our analysis and for more references.) Adding up the leading-order hadronic VP contributions from all energies, we obtain LO VP aHad, = (695.1 ± 4.5) · 10−10 . This value μ is slightly larger than our previous prediction, mainly as a consequence of the new BaBar data in the 2π channel. The corresponding value for the hadronic higher-order VP contribution is practically unHO VP changed and reads aHad, = (−9.8 ± 0.1) · μ −10 10 . 1.2. SM prediction and BNL measurement Combining all SM contributions we obtain aSM μ
=
aQED + aEW + aHad μ μ μ
=
(11659183.0 ± 4.8) · 10−10 .
In Fig. 8 this prediction is shown together with similar ones as available at the time of the conference and confronted with the measurement from BNL [17], which has slightly changed due to a new value of the muon to proton magnetic ra-
T. Teubner et al. / Nuclear Physics B (Proc. Suppl.) 218 (2011) 225–230
perimental data for the normalised hadronic cross section R, ∞ R(s) ds αq 2 ΔαHad (q 2 ) = − P . (3) 2 3π m2π s(s − q )
HMNT (06) JN (09) Davier et al, τ (09) w/o BaBar (09)
Davier et al, e+e– w/
229
BaBar (09)
HLMNT (09) HLMNT prelim. (10) experiment BNL BNL (new from shift in λ) 170
180
190
200
210
aμSM × 1010 – 11659000
Figure 8. Various recent predictions for aμ compared to the measurement from BNL.
tio from the CODATA group. The change of our new prediction (labelled ‘HLMNT prelim. (10)’ in Fig. 8) w.r.t. our earlier result ‘HLMNT (09)’ is mainly due to the inclusion of the new 2π data from BaBar. This change is in line with the findings from Davier et al.. Despite all changes as discussed above, the recent SM predictions agree well and signal a discrepancy with the measured value of more than three standard deviations. (When using our new preliminary result we obtain a 3.2 σ discrepancy, whereas Davier et al. have reported 3.6 σ at this conference [18,19].) While the current level of significance does not allow definite conclusions, it is intriguing that the discrepancy has survived all scrutiny. It could well be explained by the existence of supersymmetric extensions of the SM at the TeV scale and already strongly constrains the allowed parameter space for physics beyond the SM. 2. ELECTROMAGNETIC COUPLING α Vacuum polarisation leads to the running of the electromagnetic coupling, α(q 2 ) = α/(1 − ΔαLep (q 2 ) − ΔαHad (q 2 )). While the leptonic contributions, ΔαLep (q 2 ), can be calculated using perturbation theory, the hadronic contributions require a dispersion integral and the use of ex-
(Here α is the fine-structure constant and P stands for the principal value of the integral.) We have used known analytical expressions for the leptonic contributions and (3) for the hadronic contributions, with the same data compilation for R(s) as in our g − 2 prediction, to obtain the running QED coupling. On request we provide a simple-to-use set of Fortran routines for α(q 2 ) for space- and time-like q 2 and also for the imaginary part of the VP. These routines are used to ‘undress’ the experimentally measured hadronic cross sections w.r.t. running coupling effects (an iteration is needed to achieve consistent results for g − 2 and Δα) and are also used by other groups for applications in Monte Carlo programs (see [20] for a detailed discussion). The value of the running coupling at MZ2 is of particular interest for the electroweak precision tests of the SM. For the hadronic five-flavour corrections at this scale we obtain (5)
ΔαHad (MZ2 ) = 0.02759 ± 0.00015 ,
(4)
which is similar in size but considerably more accurate than the default typically used in the electroweak precision fits. Together with the perturbative top-quark and leptonic contributions this translates into α(MZ2 )−1 = 128.953 ± 0.020. If our result (4) would be used in the electroweak precision fits, the parabola for the indirect determination of the Higgs mass (the famous ‘blueband’ plot) would have a minimum at roughly the same value of MH , but be narrower and hence add to the tension in the SM fit. This correlation of the precision observables aμ and Δα makes it seem unlikely that the discrepancy in g − 2 could be solved by a change of the hadronic cross section data alone [21]. 3. CONCLUSIONS AND OUTLOOK We have presented a brief status report and new evaluations of g − 2 and α(MZ2 ). For g − 2, the deviation between the SM prediction and the
T. Teubner et al. / Nuclear Physics B (Proc. Suppl.) 218 (2011) 225–230
230
2
∞
mπ
mπ
1.4
aμhad,LO
REFERENCES
(error)2
value rad.
0.6
0.9
0.6
∞2 1.4
∞ 2 Δα(5) had (M Z)
mπ
0.9
mπ 0.6 0.6 0.9 1.4 2 4
11
0.9 1.4
rad.
2
∞ 11 4
Figure 9. Pie diagrams for the contributions to the value and squared error of g − 2 and (5) ΔαHad (MZ2 ) from different energy regions. BNL measurement has been further consolidated and persists at more than 3 σ. If due to SUSY, the sparticle masses could well be in reach of the LHC. For the future, further improvements in aSM μ and Δα are foreseen through upcoming measurements of the hadronic cross sections used in the theoretical prediction. Figure 9 shows how different regions contribute to the value and the error budget of both quantities. The prospects for improvements are good: There are promising plans for KLOE-2 at an upgraded DAΦNE, with chances to also help improving the determination of the light-by-light scattering contributions through measuring meson form-factors in γγ production. VEPP-2000 in Novosibirsk has already started (see [14]) and BES-III will cover higher energies. New g − 2 experiments at Fermilab and at J-PARC are planned to start in the middle of this decade. They will hopefully become reality and provide even more accurate measurements to test the SM and to constrain physics beyond it. Acknowledgements It is a pleasure to thank the organisers of Tau 2010 for a very interesting, stimulating and perfectly organised conference.
1. T. Kinoshita and M. Nio, Phys. Rev. D73 (2006) 053007. T. Aoyama et al., Phys. Rev. D77 (2008) 053012. 2. A. Czarnecki, W. J. Marciano and A. Vainshtein, Phys. Rev. D67 (2003) 073006, Erratum-ibid. D73 (2006) 119901. M. Knecht, S. Peris, M. Perrottet and E. de Rafael, JHEP 0211 (2002) 003. 3. F. Jegerlehner and A. Nyffeler, Phys. Rept. 477 (2009) 1. 4. A. Nyffeler, Phys. Rev. D79 (2009) 073012. 5. J. Prades, E. de Rafael and A. Vainshtein, arXiv:0901.0306 [hep-ph]. 6. T. Blum and S. Chowdhury, Nucl. Phys. Proc. Suppl. 189 (2009) 251; S. Chowdhury et al., PoS LATTICE2008 (2008) 251. See also contribution from P. Rakow at the conference Lattice 2008. 7. T. Goecke, C. S. Fischer and R. Williams, arXiv:1012.3886 [hep-ph]. 8. K. Hagiwara et al., Phys. Rev. D69 (2004) 093003; Phys. Lett. B557 (2003) 69. 9. K. Hagiwara et al., Phys. Lett. B649 (2007) 173. 10. BaBar Collaboration, B. Aubert et al., Phys. Rev. Lett. 103 (2009) 231801. 11. B. Malaescu, in these proceedings. 12. KLOE Collaboration, F. Ambrosino et al., Phys. Lett. B670 (2009) 285; arXiv:1006.5313 [hep-ex]. 13. G. Venanzoni, in these proceedings. 14. B. Shwartz, in these proceedings. 15. BES Collaboration, M. Ablikim et al., Phys. Lett. B677 (2009) 239. 16. K. Hagiwara et al., Chinese Phys. C34 (2010) 728. 17. G. W. Bennett et al., Phys. Rev. D73 (2006) 072003. 18. M. Davier, in these proceedings. 19. A. Hoecker, in these proceedings. 20. S. Actis et al., Eur. Phys. J. C66 (2010) 585. 21. M. Passera, W. J. Marciano and A. Sirlin, Phys. Rev. D78 (2008) 013009; arXiv:1001.4528 [hep-ph]. 22. L. Roberts, in these proceedings. 23. T. Mibe, in these proceedings.