The leading disconnected contribution to the anomalous magnetic moment of the muon
Anthony Francis, Vera Gülpers, Benjamin Jäger, Harvey Meyer, Georg von Hippel, Hartmut Wittig
TThe leading disconnected contribution to theanomalous magnetic moment of the muon
Anthony Francis , , Vera Gülpers ∗ , , Benjamin Jäger , Harvey Meyer , ,Georg von Hippel , Hartmut Wittig , PRISMA Cluster of Excellence, Institut für Kernphysik, Johannes Gutenberg Universität Mainz,55099 Mainz, Germany Helmholtz Institute Mainz, Johannes Gutenberg Universität Mainz, 55099 Mainz, Germany Department of Physics, College of Science, Swansea University, SA2 8PP Swansea, UKE-mail: [email protected]
The hadronic vacuum polarization can be determined from the vector correlator in a mixed time-momentum representation. We explicitly calculate the disconnected contribution to the vectorcorrelator, both in the N f = O ( a ) -improved Wilson fermions. All-to-all propagators are computed usingstochastic sources and a generalized hopping parameter expansion. Combining the result with thedominant connected contribution, we are able to estimate an upper bound for the systematic errorthat arises from neglecting the disconnected contribution in the determination of ( g − ) µ . The 32nd International Symposium on Lattice Field Theory,23-28 June, 2014Columbia University New York, NY ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] N ov he leading disconnected contribution to the anomalous magnetic moment of the muon Vera Gülpers
Figure 1:
The connected and the disconnected contribution to the hadronic vacuum polarization.
1. Introduction
The anomalous magnetic moment of the muon a µ is one of the most precisely measured quan-tities in particle physics. A deviation of ≈ σ between the experimental and the theoretical valuehas persisted for many years. From the theory side, the largest fraction of the error comes from thehadronic vacuum contribution (hvp), which is the leading order QCD contribution to a µ . Currently,the best estimate of the hvp relies on a semi-phenomenological approach using the cross sectionof e + e − → hadrons. In the past few years, a lot of effort has been undertaken to calculate the hvpfrom first principles using lattice techniques [1, 2, 3, 4]. However, the quark-disconnected contri-bution to the hvp is generally neglected. This may be a significant source of systematic error, sincein partially quenched chiral perturbation theory, it was estimated that the disconnected contributioncould be as large as −
10% of the connected one [5].We explicitly compute the disconnected contribution to the hvp with O ( a ) -improved Wilsonfermions using the mixed-representation method [6, 7], where the hadronic vacuum polarization iscalculated using the vector correlator G γγ ( x ) = − (cid:90) d x (cid:10) j γ k ( x ) j γ k ( ) (cid:11) with j γ k = u γ k u − d γ k d + . . . (1.1)as follows: ˆ Π ( Q ) = π ∞ (cid:90) d x G γγ ( x ) (cid:20) x − Q sin (cid:18) Qx (cid:19)(cid:21) . (1.2)The vector correlator G γγ ( x ) receives a connected and a disconnected contribution as shown infigure 1. We calculate the required disconnected quark loops using stochastic sources and a hoppingparameter expansion as described in [8].
2. Results for the vector correlator
In the following we will concentrate on the vector correlator for light and strange quarkscombined. The corresponding electromagnetic current j (cid:96) s µ = j (cid:96) µ + j s µ = ( u γ µ u − d γ µ d ) (cid:124) (cid:123)(cid:122) (cid:125) I = , j ρµ + ( u γ µ u + d γ µ d − s γ µ s ) (cid:124) (cid:123)(cid:122) (cid:125) I = (2.1)can be split into an isovector part corresponding to the ρ -current and an isoscalar part. Performingthe Wick contractions one finds for the light and strange vector current G (cid:96) s ( t ) = G (cid:96) con ( t ) + G s con ( t ) + G (cid:96) s disc ( t ) with G (cid:96) con ( t ) = G ρρ ( t ) (2.2)2 he leading disconnected contribution to the anomalous magnetic moment of the muon Vera Gülpers
For convenience, we consider the disconnected correlator G (cid:96) s disc ( t ) for light and strange quarkscombined, since one can write the disconnected Wick contractions as G (cid:96) s disc ( x ) = − (cid:90) d x (cid:68) j (cid:96) sk ( x ) j (cid:96) sk ( ) (cid:69) disc = − (cid:90) d x (cid:68) ( j (cid:96) k ( x ) − j sk ( x )) ( j (cid:96) k ( ) − j sk ( )) (cid:69) disc , (2.3)i.e. we only need differences of light and strange quark loops. Thus, we expect that stochasticnoise can be canceled when light and strange quark loops are calculated using the same stochasticsources. Figure 2 shows our results for the disconnected correlator for light quarks only in red andfor combined light and strange quarks in green for the E5 ensemble (cf. table 1). As expected,we find that the stochastic error for the combined light and strange disconnected correlator is sig-nificantly smaller than the error on the light quark correlator alone. Although we can reduce the G d i s c ( t ) t/a lightlight and strange − e − − e − − e − − e − e − e − e − e −
04 0 4 8 12 16 20 24 28 32 G l s d i s c ( t ) t/a light and strange − e − − . e − − e − − e − e − e − . e − e −
05 0 4 8 12 16 20 24 28 32
Figure 2:
The disconnected vector correlator for light quarks (red) and combined light and strange quarks(green). Note, that the scales on both plots are different. statistical error significantly when light and strange loops are calculated with the same stochasticsources, we find that the disconnected correlator G (cid:96) s disc ( x ) is still consistent with zero within ourcurrent accuracy.We can add the disconnected correlator to the connected one to obtain the total vector correla-tor. Figure 3 shows the connected (red) and the total vector correlator (yellow) for the E5 ensemble.Results for light quarks as well as light and strange quarks combined are shown on the left- and theright-hand side, respectively. The horizontal line in both plots shows the level of the statistical erroron the disconnected contribution, i.e. it indicates the point from which on our total vector correla-tor is dominated by the noise of the disconnected contribution. This point sets in for significantlylarger euclidean times in the case of the combined light and strange quark correlator.Although we do not find a non-vanishing signal for the disconnected correlator, we can stilluse our results to give a limit for the maximum possible contribution to the hadronic vacuumpolarization from quark-disconnected diagrams. Here, we will solely consider the case of combinedlight and strange quarks, for which the statistical error is significantly smaller.3 he leading disconnected contribution to the anomalous magnetic moment of the muon Vera Gülpers G l ( t ) t/a wo discw discerror on disconnected1 e − e − e − e − e − . . .
01 0 5 10 15 20 25 30 G l s ( t ) t/a wo discw discerror on disconnected1 e − e − e − e − e − . . .
01 0 5 10 15 20 25 30
Figure 3:
The connected (red) and the total (yellow) vector correlator for light quarks (left) and light andstrange quarks (right). The horizontal line in both plots shows level of the statistical error on the disconnectedcontribution.
3. The vector correlator for large euclidean times
In order to estimate the maximum possible contribution from quark-disconnected diagrams werequire information about the behavior of the vector correlator for large euclidean times in additionto our data. For large euclidean times, the vector correlator is dominated by the isovector part [6],due to its lower threshold: G γγ ( t ) = G ρρ ( t ) (cid:0) + O ( e − m π t ) (cid:1) . (3.1)If we rewrite equation (2.2) as19 G (cid:96) s disc ( t ) G ρρ ( t ) = G γγ ( t ) − G ρρ ( t ) G ρρ ( t ) (cid:124) (cid:123)(cid:122) (cid:125) → t → ∞ − (cid:18) + G s con ( t ) G (cid:96) con ( t ) (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) → t → ∞ −→ − , (3.2)we find an asymptotic value of − / G (cid:96) s disc ( t ) to the ρ -correlator for large euclidean times. This ratio (3.2) is plotted against t in figure 4.The green line on the left-hand side shows the asymptotic value − /
9. As one can see, we canclearly distinguish the ratio from its asymptotic value up to t ≈ a .To give a conservative upper limit for the disconnected contribution, we assume that the ratio(3.2) falls monotonically from zero to − / G (cid:96) s disc ( t ) has to be consistent with both our data and with its theoretical asymptotic value. Thus, the dis-connected contribution would be maximized if the the ratio were basically zero up to t ≈ a andthen suddenly dropped to − /
9, as indicated by the blue line. If we take this as an estimate of thedisconnected vector correlator, we can give a conservative upper bound for the magnitude of thedisconnected contribution to a µ .
4. Hadronic vacuum polarization and a µ From the vector correlator, one can calculate the hadronic vacuum polarization (cf. equa-tion (1.2)). We calculate ˆ Π (cid:96) s ( Q ) once only for the connected vector correlator (for the details ofthe analysis, see [9]) and once with the disconnected estimate as described above, i.e.4 he leading disconnected contribution to the anomalous magnetic moment of the muon Vera Gülpers / G l s d i s c / G ρρ t/a data-1/9 − . − . . . / G l s d i s c / G ρρ t/a data − . − . . . Figure 4:
The ratio of the disconnected correlator and the ρ -correlator. The green line on the left-hand sideshows the asymptotic value. The blue line on the right-hand side shows our conservative estimate for thedisconnected correlator. • for t ≤ a ≈ G (cid:96) s ( t ) = G (cid:96) con ( t ) + G s con ( t ) (4.1) • for t > a we use the asymptotic value G (cid:96) s disc ( t ) / G ρρ ( t ) = − / G (cid:96) s ( t ) = G (cid:96) con ( t ) + G s con ( t ) − G ρρ ( t ) . (4.2) ˆ Π l s ( Q ) Q /GeV with disconnected estimateconnected00 . . .
53 0 2 4 6 8 10 ∆ ˆ Π l s ( Q ) Q /GeV difference00 . . . . . .
03 0 2 4 6 8 10
Figure 5:
The plot on the left-hand side shows the vacuum polarization from the connected correlator(yellow) and for the correlator with an estimate for the disconnected contribution. The plot on the right-handside shows their difference.
The left-hand side of figure 5 shows the vacuum contribution for both cases. As expected, thevacuum polarization with the disconnected estimate is smaller than the vacuum polarization fromthe connected contribution only, since for large euclidean times G (cid:96) s disc ( t ) has the opposite sign thanthe connected correlator. Since the difference between the two curves is small, the right hand sideof figure 5 shows their difference, which is larger than the statistical error on ˆ Π (cid:96) s ( Q ) .5 he leading disconnected contribution to the anomalous magnetic moment of the muon Vera Gülpers
From the vacuum polarization, one can now calculate the hadronic contribution to the anoma-lous magnetic moment of the muon [10, 11], a hvp µ = (cid:16) απ (cid:17) ∞ (cid:90) d Q Q K ( Q ) ˆ Π ( Q ) , (4.3)with an electromagnetic kernel function K ( Q ) . We calculate a hvp µ once for the vacuum polarizationfor the connected part only, and once for the vacuum polarization which includes the disconnectedestimate. For the E5 ensemble, we find that with the disconnected estimate the result for a hvp µ is ≈ .
5% smaller. One has to keep in mind that this is a conservative upper limit, and that thedisconnected contribution to a hvp µ could also be much smaller. We use the 3 .
5% as an upper boundfor a systematic error that arises when the disconnected contribution is neglected. β a [ fm ] lattice m π [ MeV ] m π L Label N cnfg t cut . .
063 64 ×
451 4 . . .
063 96 ×
324 5 . . .
063 96 ×
277 4 . Table 1:
The CLS ensembles used for the calculation of the disconnected contribution to the hadronicvacuum polarization.
So far, we have done this calculation for three different gauge ensembles, which are listed intable 1. For the ensembles F6 and F7 we have less statistics than for E5, and we can not resolvethe ratio of disconnected correlator and ρ -correlator as well as for E5. Thus, we choose a slightlysmaller value t cut up to which we neglect the disconnected correlator and from which on we usethe asymptotic value. For both ensembles we find a upper limit for the disconnected contributionof ≈ Figure 6: a hvp µ plotted against the pion mass. Blue points show the results from the connected correlator.The red error bars show the maximum systematic error from neglecting the disconnected contribution. Figure 6 shows the results for a hvp µ plotted against m π . The blue points show the results forthe connected correlator and are the same as in [9]. The red error bars denote the maximum sys-tematic error from neglecting the disconnected contribution. One can see that this systematic error6 he leading disconnected contribution to the anomalous magnetic moment of the muon Vera Gülpers is larger than the statistical error on a hvp µ . Thus, improving the accuracy of the QCD predictionof the hadronic contribution to a µ requires improvements to the computation of the disconnectedcontribution.
5. Conclusions
We have explicitly calculated the disconnected vector correlator for light and strange quarks.Since the disconnected correlator depends only on the difference of light and strange propagators,the statistical error can be significantly reduced when using the same stochastic sources for lightand strange loops. However, we still find that the disconnected vector correlator is consistent withzero within our current accuracy. Using the asymptotic behavior of the vector correlator for largeeuclidean times, we are able to give an upper limit for the disconnected contribution to a hvp µ . Thelattice data shows that up to some time t cut the vector correlator is well described by the connectedcontribution only, and thus the disconnected one can be neglected. From this time on, we use theasymptotic value for G (cid:96) s disc ( t ) . This allows us to give an upper estimate for the systematic errorfrom neglecting the disconnected contribution, which we find to be of the order of 4 − Acknowledgements
Our calculations were performed on the dedicated QCD platforms “Wilson” at the In-stitute for Nuclear Physics, University of Mainz, and “Clover” at the Helmholtz-Institut Mainz. We thank DaliborDjukanovic and Christian Seiwerth for technical support. We are grateful for computer time allocated to project HMZ21on the BlueGene computer “JUQUEEN” at NIC, Jülich. This research has been supported in part by the DFG in theSFB 1044. We are grateful to our colleagues in the CLS initiative for sharing ensembles.
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