Application of chiral resonance Lagrangian theories to the muon g-2
aa r X i v : . [ h e p - ph ] D ec DESY 13-239, HU-EP-13/76 December 2013
Application of chiral resonance Lagrangian theories to themuon g − Fred Jegerlehner
Humboldt-Universit¨at zu Berlin, Institut f¨ur Physik, Newtonstrasse 15,D-12489 Berlin, GermanyDeutsches Elektronen-Synchrotron (DESY), Platanenallee 6,D-15738 Zeuthen, GermanyWe think that phenomenological resonance Lagrangian models, con-strained by global fits from low energy hadron reaction data, can helpto improve muon g − a LO had µ = (688 . ± . · − and we find a the µ = (11659177 . ± . · − .PACS numbers: 14.60.Ef, 13.40.Em
1. Effective field theory: the Resonance Lagrangian Approach
The Resonance Lagrangian Approach (RLA) provides an extension oflow energy effective QCD as represented by Chiral Perturbation Theory(ChPT) to energies up to about 1 GeV. Principles to be included are thechiral structure of QCD, the vector-meson dominance model and electro-magnetic gauge invariance. Specifically, we will consider the HLS version,which is considered to be equivalent to alternative variants after implement-ing appropriate high energy asymptotic conditions. ChPT is the systematicand unambiguous approach to low energy effective QCD given by sponta-neously broken chiral symmetry SU (3) ⊗ SU (3), with the pseudoscalars asNambu-Goldstone bosons, together with a systematic expansion in low mo-menta and chiral symmetry breaking (SB) effects by the light quark masses, m q , q = u, d, s . The limitation of ChPT is the fact that it ceases to convergefor energies above about 400 MeV, in particular it lacks to describe physicsinvolving the vector resonances ρ, ω and φ .The Vector-meson Dominance Model (VDM) is the effective theory im-plementing the direct coupling of the neutral spin 1 vector resonances ρ, ω, φ (1) Ustron13Pro printed on July 16, 2018 etc. to the photon. Such direct couplings are a consequence of the fact thatthe neutral spin 1 resonances like the ρ are composed of charged quarks.The effect is well modeled by the VDM Lagrangian L γρ = e g ρ ρ µν F µν or = − eM ρ g ρ ρ µ A µ , which has to be implement in low energy effective QCDin a way which is consistent with the chiral structure of QCD.The construction of the HLS model may be outlined as follows: like inChPT the basic fields are the unitary matrix fields ξ L,R = exp [ ± i P/f π ],where P = P + P is the SU (3) matrix of pseudoscalar fields, with P and P the basic singlet and octet fields, respectively. The pseudoscalar fieldmatrix P is represented by P = 1 √ √ π + 1 √ η π + K + π − − √ π + 1 √ η K K − K − r η , (1) P = 1 √ η , η , η ) ; ( π , η , η ) ⇔ ( π , η, η ′ ) . (2)The HLS ansatz is an extension of the ChPT non-linear sigma model to a non-linear chiral Lagrangian [Tr ∂ µ ξ + ∂ µ ξ ] based on the symmetry pattern G global /H local , where G = SU (3) L ⊗ SU (3) R is the chiral group of QCD and H = SU (3) V the vector subgroup. The hidden local SU (3) V requires thespin 1 vector meson fields, represented by the SU (3) matrix field V µ , to begauge fields. The needed covariant derivative reads D µ = ∂ µ − i g V µ , andallows to include the couplings to the electroweak gauge fields A µ , Z µ and W ± µ in a natural way. The vector field matrix is usually written as V = 1 √ ( ρ I + ω I ) / √ ρ + K ∗ + ρ − ( − ρ I + ω I ) / √ K ∗ K ∗− K ∗ φ I . (3)The unbroken HLS Lagrangian is then given by L HLS = L A + L V ; L A/V = − f π L ± R ] , (4)where L = [ D µ ξ L ] ξ + L and R = [ D µ ξ R ] ξ + R . The covariant derivatives read D µ ξ L = ∂ µ ξ L − igV µ ξ L + iξ L L µ D µ ξ R = ∂ µ ξ R − igV µ ξ R + iξ R R µ , (5) stron13Pro printed on July 16, 2018 with known couplings to the Standard Model (SM) gauge bosons L µ = eQA µ + g cos θ W ( T z − sin θ W ) Z µ + g √ W + µ T + + W − µ T − ) R µ = eQA µ − g cos θ W sin θ W Z µ . (6)Like in the electroweak SM, masses of the spin 1 bosons may be generated bythe Higgs-Kibble mechanism if one starts in place of the non-linear σ -modelwith the Gell-Mann–Levy linear σ -model by a shift of the σ -field.In fact the global chiral symmetry G global is well known not to be re-alized as an exact symmetry in nature, which implies that the ideal HLSsymmetry evidently is not a symmetry of nature either. It evidently has tobe broken appropriately in order to provide a realistic low energy effectivetheory mimicking low energy effective QCD. Corresponding to the strengthof the breaking, usually, this has is in two steps, breaking of SU (3) in a firststep and breaking the isospin SU (2) subgroup in a second step. Unlike inChPT (perturbed non-linear σ –model) where one is performing a system-atic low energy expansion, expanding in low momenta and the quark masses,here we introduce symmetry breaking as phenomenological parameters tobe fixed from appropriate data, since a systematic low energy expansiona l´a ChPT ceases to converge at energies above about 400 MeV, while weattempt to model phenomenology up to including the φ resonance.The broken HLS Lagrangian (BHLS) is then given by (see [1]) L BHLS = L ′ A + L ′ V + L ′ tHooft ; L ′ A/V = − f π (cid:8) [ L ± R ] X A/V (cid:9) , (7)with 6 phenomenological chiral SB parameters. The phenomenological SBpattern suggests X I = diag( q I , y I , z I ) , | q I − | , | y I − | ≪ | z I − | , I = V, A .
There is also the parity odd anomalous sector, which is needed to accountfor reactions like γ ∗ → π γ and γ ∗ → π + π − π among others.We note that this BHLS model would be a reliable low energy effectivetheory if the QCD scale Λ QCD would be large relative to the scale of about1 GeV up to which we want to apply the model, which in reality is not thecase. Nevertheless, as a phenomenological model applied to low multiplicityhadronic processes (specified below) it seems to work pretty well, as we havedemonstrated by a global fit of the available data in Ref. [1]. The majorachievement is a simultaneous consistent fit of the e + e − → π + π − data fromCMD-2 [2], SND [3], KLOE [4] and BaBar [5], and the τ → π − π ν τ decayspectral functions by ALEPH [6], OPAL [7], CLEO [8] and Belle [9]. The e + e − → π − π + channel gives the dominant hadronic contribution to themuon g −
2. Isospin symmetry π − π ⇔ π − π + allows one to include existinghigh quality τ -data as advocated long time ago in [10]. Ustron13Pro printed on July 16, 2018
We note that as long as higher order corrections are restricted to themandatory pion- and Kaon-loop effects in the vector boson self-energies,renormalizability is not an issue. These contributions behave as in a strictlyrenormalizable theory and correspond to a reparametrization only. ρ − γ mixing solving the τ vs. e + e − puzzle A minimal subset of any resonance Lagrangian is given by the VDM +scalar QED part which describes the leading interaction between the ρ thepions and the photon. In order to account for the decay of the ρ , one hasto include self-energy effects, which also affect ρ − γ mixing via pion-loopsshown in Fig. 1. Most previous calculations, considered the mixing term to − i Π µν ( π ) γρ ( q ) = + . Fig. 1. Irreducible self-energy contribution at one-loop be a constant, and were missing a substantial quantum interference effect.The properly normalized pion form factor, in our approach, has the from F π ( s ) = (cid:2) e D γγ + e ( g ρππ − g ρee ) D γρ − g ρee g ρππ D ρρ (cid:3) / (cid:2) e D γγ (cid:3) , (8)with propagators including the pion loop effects, with typical couplings Fig. 2. Ratio of the full | F π ( s ) | in units of the same quantity omitting the mixingterm (full line). Also shown is the same mechanism scaled up by the branchingfraction Γ V / Γ( V → ππ ) for V = ω and φ . In the ππ channel the effects forresonances V = ρ are tiny if not very close to resonance. g ρππ bare = 5 . g ρππ ren = 6 . g ρee = 0 . x = g ρππ /g ρ = stron13Pro printed on July 16, 2018 . , fixed from the (partial) widths g ρππ = q π Γ ρ / ( β ρ M ρ ) ; g ρee = q π Γ ρee /M ρ . The effect of taking into account or not the γ − ρ mixing is illustratedin Fig. 2. The γ − ρ interference is crucial when relating charged current τ -data to e + e − -data. Including known isospin breaking (IB) corrections v ( s ) = R IB ( s ) v − ( s ) a large discrepancy [ ∼ τ vs. e + e − puzzle since [12]. In [13] it has been shown thatthe γ − ρ mixing active in the e + e − → π + π − channel is responsible for thediscrepancy, i.e. τ -data have to be corrected as v ( s ) = r ργ ( s ) R IB ( s ) v − ( s ),before they can be used as representing an equivalent I=1 e + e − → π + π − data sample (see also [14, 15]). Note that what goes into a µ directly arethe e + e − -data. Best “proof” of the required ρ − γ correction profile is the Fig. 3. How photons couple to pions? This is obviously probed in reactions like γγ → π + π − , π π . Data infer that below about 1 GeV photons couple to pionsas point-like objects (i.e. to the charged ones overwhelmingly). At higher energiesthe photons see the quarks exclusively and form the prominent tensor resonance f (1270). The π π cross section shown has been multiplied by the isospin sym-metry factor 2, by which it is reduced in reality. ALEPH vs. BaBar fit shown in Fig. 1 of [16]. Applying the correction to the τ spectra (see Fig. 8 in [13]) implies a universal shift down by δa had µ [ ργ ] ≃ ( − . ± . · − of the contribution to the muon g −
2. This shift bringsinto agreement the τ inclusive estimates with the e + e − based ones. Isour model, treating pions as point-like objects, viable? A good “answer”to this question may be obtained by looking at the ππ production in γγ fusion. Fig. 3 shows: at the strong tensor meson resonance f (1270) in the ππ channel, photons directly probe the quarks! However, in the region of Ustron13Pro printed on July 16, 2018 our interest photons see pions (below about 1 GeV). We apply the sQEDmodel up to 0.975 GeV (relevant for a µ ), which should be rather reliable.Switching off the electromagnetic interaction of pions, is definitely not arealistic approximation in trying to describe what is observed in the e + e − → π + π − channel.
3. Global fit of BHLS parameters and prediction of F π ( s ) The simple model just considered illustrates one of the main quantuminterference effects in the isospin sector, the γ − ρ mixing. A more completeeffective theory must include the ρ − ω mixing, as well as the strangenesssector, with the Kaons as additional pseudo Nambu-Goldstone bosons, in-cluding the η and the η ′ , and the mixing with the φ . This is implementedin the BHLS model introduced before. Self-energy corrections for ρ, ω, φ and γ now include Kaon-loops as well. In addition, parity odd sector con-tributions like π → γγ and γ → π + π − π must be included. At presentthere are 45 different data sets (6 annihilation channels and 10 partial widthdecays) available below E = 1 .
05 GeV (just above the φ ), and we use themto constrain the BHLS Lagrangian couplings. The method is able to re-duce uncertainties in g − π + π − , π γ, ηγ, η ′ γ, π π + π − ,K + K − , K ¯ K which account 83.4% of the HVP contribution to the muon g −
2. Contributions from the missing channels 4 π, π, π, ηππ, ωπ andfrom higher energies we evaluate using data directly and pQCD in theperturbative region and in the tail. The resulting BHLS prediction for a LO , had µ allows us to get a BHLS driven SM prediction for a µ (see Ta-ble 1). Our favored evaluation based on selected data yields a LO had µ =(681 . ± . · − and a prediction a the µ = (11659170 . ± . · − and ∆ a µ = a exp µ − a the µ = (38 . ± . the ± . exp ) · − . The associ-ated fit probability is 94% and the significance for ∆ a µ is 4 . σ . Includingall data, applying appropriate rewighting in case of inconsistencies , we find The required rewighting concerns the e + e − → π + π − π data in the vicinity of the φ ,as well as the KLOE08 and the BaBar e + e − → π + π − data sets. stron13Pro printed on July 16, 2018
150 200 250 incl. ISRDHMZ10 ( e + e − ) . ± . [ . σ ]DHMZ10 ( e + e − + τ ) . ± . [ . σ ]JS11 ( e + e − + τ ) . ± . [ . σ ]HLMNT11 ( e + e − ) . ± . [ . σ ]DHMZ10/JS11 ( e + e − + τ ) . ± . [ . σ ]BDDJ13 ∗ ( e + e − + τ ) . ± . [ . σ ]excl. ISRDHea09 ( e + e − ) . ± . [ . σ ]BDDJ12 ∗ ( e + e − + τ ) . ± . [ . σ ]experimentBNL-E821 (world average) . ± . a µ × -11659000 ∗ HLS fits
Fig. 4. Comparison with other Results. Note: results depend on which value istaken for HLbL. JS11 and BDDJ13 includes 116(39) · − [JN [17]], DHea09,DHMZ10, HLMNT11 and BDDJ12 use 105(26) · − [PdRV [18]]. a LO had µ = (688 . ± . · − such that a the µ = (11659177 . ± . · − and ∆ a µ = a exp µ − a the µ = (31 . ± . the ± . exp ) · − . The associatedfit probability is 76% and the significance for ∆ a µ is 3 . σ . The comparisonof our global fit result with other results from DHMZ10 [16, 19], JS11 [13],DHea09 [11], HLMNT11 [20] is shown in Fig. 4. We get somewhat lowercentral values than results obtained by direct integration of the data, but allresults agree well within 1 σ . Our fits, which include the τ data, exhibit thebest fit probability for KLOE10 results, while there is some tension showingup in case of the BaBar ππ data. Our analysis has been criticized lately inRef. [21] but what is shown in that reply is that BaBar [5] and KLOE dataare not quite compatible within the given experimental errors. A differentissue is the comparison between BaBar and τ spectral data. Contrary toclaims in [21] the sizable γ − ρ mixing effect has not been taken into ac-count and one should see a substantial shift which, however, is found to beabsent in the comparison between Belle τ data and the BaBar e + e − data(see Fig. 1 in [16]).A comparison between theory and experiment [22] is given in Tab. 1 (seealso [23]). Theory results shown are updates from Ref. [17] using results onimproved 4-loop and the new 5-loop QED corrections [24], improved lepton Ustron13Pro printed on July 16, 2018
Table 1. Standard model theory and experiment comparison [in units 10 − ]. Contribution Value ErrorQED incl. 4-loops + 5-loops 116 584 718.85 0.04Leading hadronic vacuum polarization 6 886.0 42.4Subleading hadronic vacuum polarization -98.32 0.82Hadronic light–by–light 116.0 39.0Weak incl. 2-loops 154.0 1.0Theory 116 591 776.5 57.6Experiment 116 592 089.0 63.0Exp. - The. 3.7 standard deviations 312.5 85.4mass ratios [25] and using the new Higgs mass value from ATLAS and CMSin the evaluation of the weak corrections [26].
4. Lessons and Outlook
Effective field theory is the only way to understand relationships betweendifferent channels, like e + e − –annihilation cross-sections and τ –decay spec-tra. Global fit strategies allow to single out variants of effective resonanceLagrangian models. Models for individual channels can parametrize data,but do not allow to understand them and their relation to other channels.We get perfect fits for | F π ( s ) | up to just above the φ without higher ρ ’s ρ ′ , ρ ′′ , which seem to be mandatory in Gounaris-Sakurai type fits. τ datain our approach play a special role, because they are much simpler than the e + e − data, which exhibit intricate γ − ρ − ω − φ mixing effects.RLA type analyses provide analytic shapes for amplitudes, and such“physical shape information” is favorable over ad hoc data interpolations,the simplest being the trapezoidal rule, which is known to be problematicwhen data are sparse or strongly energy dependent.Limitations of the RLA are the large couplings which make system-atic higher order improved analyses problematic. As illustrated by Fig. 3,considering pions and Kaons to be point-like may be not too bad an ap-proximation, in the range we are applying the model. Also, we consider ouranalysis as a starting point to be confronted with other RLA versions andimplementations and with what happens if one tries to include higher ordereffects. Acknowledgments
Many thanks to the organizers for the invitation and support to the 2013 stron13Pro printed on July 16, 2018 “Matter to the Deepest” International Conference at Ustro´n, Poland, andfor giving me the opportunity to present this talk.REFERENCES [1] M. Benayoun, P. David, L. DelBuono, F. Jegerlehner, Eur. Phys. J. C (2012) 1848; Eur. Phys. J. C (2013) 2453.[2] R. R. Akhmetshin et al. [CMD-2 Collaboration], Phys. Lett. B (2004)285; Phys. Lett. B (2007) 28.[3] M. N. Achasov et al, J. Exp. Theor. Phys. (2006) 380 [Zh. Eksp. Teor.Fiz. (2006) 437].[4] F. Ambrosino et al. [KLOE Collaboration], Phys. Lett. B (2009) 285;Phys. Lett. B (2011) 102; D. Babusci et al. [KLOE Collaboration], Phys.Lett. B (2013) 336.[5] B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett. (2009) 231801;J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D (2012) 032013.[6] S. Schael et al. [ALEPH Collaboration], Phys. Rept. (2005) 191.[7] K. Ackerstaff et al. [OPAL Collaboration], Eur. Phys. J. C (1999) 571.[8] S. Anderson et al. [CLEO Collaboration], Phys. Rev. D (2000) 112002.[9] M. Fujikawa et al. [Belle Collaboration], Phys. Rev. D (2008) 072006.[10] R. Alemany, M. Davier, A. H¨ocker, Eur. Phys. J. C (1998) 123.[11] M. Davier et al, Eur. Phys. J. C (2010) 127.[12] M. Davier, S. Eidelman, A. H¨ocker, Z. Zhang, Eur. Phys. J. C (2003) 497.[13] F. Jegerlehner, R. Szafron, Eur. Phys. J. C (2011) 1632.[14] M. Benayoun et al, Eur. Phys. J. C (2008) 199.[15] M. Benayoun, P. David, L. DelBuono, O. Leitner, Eur. Phys. J. C (2010)211; Eur. Phys. J. C (2010) 355.[16] M. Davier et al, Eur. Phys. J. C (2010) 1.[17] F. Jegerlehner, A. Nyffeler, Phys. Rept. (2009) 1; F. Jegerlehner, SpringerTracts Mod. Phys. (2008) 1; Acta Phys. Polon. B (2009) 3097.[18] J. Prades, E. de Rafael, A. Vainshtein, [arXiv:0901.0306 [hep-ph]].[19] M. Davier, A. H¨ocker, B. Malaescu, Z. Zhang, Eur. Phys. J. C (2011) 1515[Erratum-ibid. C (2012) 1874].[20] K. Hagiwara, R. Liao, A. D. Martin, D. Nomura, T. Teubner, J. Phys. G (2011) 085003.[21] M. Davier, B. Malaescu, arXiv:1306.6374 [hep-ex].[22] G. W. Bennett et al. [Muon G-2 Collaboration], Phys. Rev. D (2006)072003.[23] J. P. Miller, E. d. Rafael, B. L. Roberts, D. St¨ockinger, Ann. Rev. Nucl. Part.Sci. (2012) 237.0 Ustron13Pro printed on July 16, 2018 [24] T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio, Phys. Rev. Lett. (2012)111808.[25] P. J. Mohr, B. N. Taylor, D. B. Newell, Rev. Mod. Phys. (2012) 1527.[26] C. Gnendiger, D. St¨ockinger, H. St¨ockinger-Kim, Phys. Rev. D88