PoS(LATTICE 2015)263. The leading hadronic contribution to γ-z mixing. Vera Gülpers 1, Harvey Meyer 1,2, Georg von Hippel 1, Hartmut Wittig 1,2

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1 , Harvey Meyer,2, Georg von Hippel, Hartmut Wittig,2 PRISMA Cluster of Excellence, Institut für Kernphysik, Johannes Gutenberg Universität Mainz, 5599 Mainz, Germany 2 Helmholtz Institute Mainz, Johannes Gutenberg Universität Mainz, 5599 Mainz, Germany guelpers@kph.uni-mainz.de We study the leading hadronic contribution to γ-z mixing, which determines the leading order hadronic contribution to the running of the electroweak mixing angle θ W. The required vacuum polarization function Π γz is calculated from the appropriate correlation functions in a mixed time-momentum representation. We explicitly calculate the connected and the onnected contributions to such correlators using N f = 2 dynamical flavors of nonperturbatively O(a)-improved Wilson fermions. PoS(LATTICE 25)263 The 33rd International Symposium on Lattice Field Theory 4-8 July 25 Kobe International Conference Center, Kobe, Japan* Speaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4. International License (CC BY-NC-ND 4.).

2 . Introduction The electroweak mixing angle sin 2 θ W is a fundamental parameter in the Standard Model of particle physics, which is related to the unification of the electromagnetic and the weak interactions, and is determined by the mixing of the Z boson and the photon. Quantum corrections cause it to run with the momentum scale Q 2. The hadronic contribution to the running of the Weinberg angle is given in terms of a hadronic vacuum polarization Π γz with Π γz µν(q) d 4 x e iq x jµ Z (x) jν() γ = (Q µ Q ν δ µν Q 2 )Π γz (Q 2 ), (.) with the electromagnetic photon current j γ µ and the Z current j Z µ. The corresponding currents are given by [] jµ γ = 2 3 uγ µu 3 dγ µd 3 sγ µs cγ µc (.2) jµ Z = jµ 3 sin 2 θ W jµ γ (.3) = 4 uγ µu 4 dγ µd 4 sγ µs + 4 cγ µc. (.4) j 3 µ Since QCD is invariant under charge conjugation, only the part of the Z current has to be considered. 2. The correlator for γ-z mixing To calculate the vacuum polarization Π γz, we will use the appropriate zero-momentum correlator in mixed time-momentum representation, G γz (x ) = 3 d 3 x jk Z (x) j γ k () with k =,2,3. (2.) k The subtracted γz vacuum polarization is obtained from the correlator G γz (x ) by integrating over the remaining time direction, [ Π γz R (Q2 ) Π γz (Q 2 ) Π γz () = dx G γz (x ) x 2 4 ( )] Q 2 sin2 2 Qx. (2.2) This method has originally been developed for the calculation of the hadronic contribution to the anomalous magnetic moment of the muon [2, 3, 4]. The hadronic contribution to the running of the Weinberg angle is directly proportional to the subtracted vacuum polarization [5] had sin 2 θ W (Q 2 ) = e2 sin 2 θ W Π γz R (Q2 ). (2.3) Writing down the Wick contractions for the correlator (2.) one finds connected and onnected contributions as depicted in figure. The connected correlator for four quark flavors is given by ( G γz con(x ) = 4 5 ) ( 9 sin2 θ W G l (x ) + 2 ) ( 9 sin2 θ W G s (x ) ) 9 sin2 θ W G c (x ). (2.4) The corresponding single flavor correlators G f (x ) can be expressed as G f (x ) 3 [ Tr D f (x,) γ k D f (,x) γ k k #»x ] G (2.5) PoS(LATTICE 25)263 in terms of quark propagators. The onnected Wick contractions will be ussed in section 4. 2

3 Figure : The connected and the onnected contribution to γ-z mixing. We also tested a different strategy to only calculate a correlation function of the form G 3γ (x ) = ( 3 d 3 x jk 3 (x) 9 ) k 2 jγ k (x) j γ k () (2.6) on the lattice and take the remaining required vacuum polarization Π γγ from dispersion relation and phenomenology. The advantage of this approach is that the correlation function (2.6) does not include a light connected piece. However, in this study we resort to calculate the complete correlator (2.) using lattice QCD. 3. Results for the connected contribution For the analysis of the connected contribution we follow the steps of [6], where the correlator in time momentum representation has been used to determine the leading hadronic contribution to the anomalous magnetic moment of the muon. G γz con(t).. e 5 e 6 e 7 e 8 E5 connected e sin 2 (θw ) E5 connected F6 connected F7 connected Q 2 /GeV 2 Figure 2: The plot on the left-hand side shows the result for the correlator G γz con(x ) plotted against t for the E5 ensemble (cf. table for details of the ensembles). The green curve indicates a fit to our data with a fit range of t [5, 2]. The connected hadronic contribution to the running of the Weinberg angle is plotted against the momentum Q 2 for all three ensembles on the right. The connected correlator G γz con(x ) for the E5 ensemble (cf. table ) is plotted on the left-hand side of figure 2 against the source-sink separation t. The connected correlator has been calculated with a conserved current at the sink and a local current at the source. The required multiplicative renormalization Z v is taken from [7]. To obtain the subtracted vacuum polarization Π γz R (Q2 ), one has to integrate the correlator over Euclidean time. For this purpose we interpolate the correlator with cubic splines for t 5 as indicated by the black curve in figure 2. For large Euclidean times t > 5 we fit the correlator for every quark flavor separately to a function exp( m f t) (shown by the green line in figure 2) and use the result of this fit to integrate up to t. From the subtracted vacuum polarization we obtain the hadronic contribution to the running of the Weinberg angle had sin 2 θ W (Q 2 ) according to equation (2.3). Here we use the result for the fine structure constant α from [8] and the tree-level Weinberg angle in the MS scheme sin 2 θ W () =.2387(9) 3 PoS(LATTICE 25)263

4 β a[fm] lattice m π [MeV] m π L Label N cnfg E F F7 25 Table : Overview of the CLS ensembles that have been used for the calculation of the hadronic contribution to the running of the Weinberg angle. from [9]. Our results for had sin 2 θ W (Q 2 ) are shown in the right-hand side of figure 2 for the three gauge ensembles listed in table. We observe a clear trend in had sin 2 θ W (Q 2 ) as the pion mass is lowered. 4. The onnected contribution As already ussed, the correlator G γz (x ) receives a onnected contribution. For three quark flavors the onnected Wick contractions are given by G γz (x ) 3 k The onnected correlator G (l+as),(l s) G (l+as),(l s) (x y ) = 3 k d 3 x j Z k (x) j γ k () = sin2 θ W 9 G(l+As),(l s) (x ). (4.) is given as ( L 3 #»x Tr [ γ k D l (x,x) + Aγ k D s (x,x) ]) ( #»y Tr [ ]) (4.2) γ k D l (y,y) γ k D s (y,y) in terms of quark propagators with A = 3 4sin 2 θ W. In the second row of equation (4.2) one can see that at the source y only the difference of light and strange propagators is needed. Since we calculated the required onnected quark loops with stochastic sources, we can calculate both - light and strange quark loops - with the same sources, such that the stochastic noise is largely canceled at the source y. This noise cancellation has already been used in our calculation of the onnected contribution to a µ [], where one obtains a cancellation at the source and the sink E5 onnected.. E5 connected E5 total error on onnected PoS(LATTICE 25)263 G (l+as),(l s) (t) 5e 5 5e 5. G γz (t) e 5 e 6 e 7.5 e e Figure 3: The onnected correlator is plotted on the left-hand side. The right-hand side shows the connected correlator (red) and the total correlator (yellow). The black line indicates the level of the error on the onnected contribution. 4

5 The left-hand side of figure 3 shows the onnected correlator G (l+as),(l s). The onnected quark loops have been calculated with three stochastic sources per timeslice and a generalized hopping parameter expansion []. As ussed above, light and strange propagators are estimated with the same stochastic sources. We find that the onnected correlator is consistent with zero within the statistical errors. However, we have to add the onnected contribution to the connected one to obtain the total correlator G γz. This is shown by the yellow points in the right-hand side of figure 3. The black line indicates the level of the error on the onnected contribution. For t 6 the total correlator is dominated by the noise from the onnected contribution. 5. The correlator for large Euclidean times Since we find the onnected correlator to be consistent with zero within statistical errors, we have to resort to giving an upper limit for the maximum possible onnected contribution to had sin 2 θ W (Q 2 ). For this purpose we investigate the behavior of the correlator for large Euclidean times using the same arguments as in [3, ] for for the hadronic vacuum polarization contribution a HVP µ. Since the lowest possible iso state consists of two pions and the lowest isoscalar state of three pions, the iso channel opens at energies of 2m π whereas the isoscalar channel opens only at 3m π. Thus one expects for large Euclidean times that the correlator is dominated by the iso part, which is given by ( ) G γz (t) 2 sin2 θ W G ρρ (t) for t (5.) with the ρ-correlator G ρρ 2 Gl. On the other hand, we can rewrite the expression for the total correlator G γz (t) (neglecting charm contributions) by dividing by the ρ-correlator and rearranging the terms, G (l+as),(l s) 9 sin2 (t) θ W G ρρ (t) = GγZ (t) ( 2 ) sin2 θ W G ρρ (t) G ρρ + ( (t) 9 sin2 θ W 6 2 ) G s (t) 9 sin2 θ W G l (t). (5.2) For large Euclidean times, the first term on the right-hand side of (5.2) vanishes due to equation (5.). Additionally, the term G s /G l is exponentially suppressed and vanishes for t. Hence, we find G (l+as),(l s) (t) 2 Gl (t) = G ρρ (t) for t (5.3) for the large time behavior of the onnected correlator. Figure 4 shows the ratio of the onnected correlator and the ρ-correlator. This ratio is expected to be for t according to equation (5.3). The green line on the left-hand side indicates the asymptotic value. Clearly, the ratio can be distinguished from the asymptotic value up to t 8a. We are interested in the scenario where the onnected diagram contributes as much as possible while still agreeing with our data. If we assume that the ratio rises monotonically from to its asymptotic value, the maximum possible contribution is shown by the blue curve on the right-hand side of figure 4. PoS(LATTICE 25)263 5

6 3 2 data 3 2 data G/G ρρ G/G ρρ Figure 4: The ratio of the onnected and the ρ-correlator. The green line one the left-hand side shows the asymptotic value. The blue curve on the right is our estimate for the maximum possible onnected contribution. 6. The γz vacuum polarization with onnected estimate To obtain an upper limit for the contribution of the onnected diagram to the running of the Weinberg angle, we calculate the γz vacuum polarization once without the onnected contribution and once with an estimate for the onnected contribution that follows the blue curve in figure 4. Both results are shown on the left-hand side of figure 5. Π γz (Q 2 ) with onnected estimate connected Q 2 /GeV 2 Π γz (Q 2 )/Π γz (Q 2 ) relative difference Q 2 /GeV 2 Figure 5: The γz vacuum polarization once without onnected contribution (yellow) and once with our estimate for the onnected contribution (red). The green points on the right show the relative difference of these two curves. We find that Π γz R (Q2 ) including the onnected estimate is larger than the result from connected contributions only. Thus we can state, that the onnected contribution to Π γz R (Q2 ) is positive. However, one has to keep in mind that our estimate only gives a conservative upper limit and the onnected contribution might be much smaller. The right hand side of figure 5 shows the relative difference between the two curves at the left. One can see that at e.g. Q 2 = 4 GeV 2 the upper limit for the onnected contribution is 4% of the connected one. This 4% can be used as a systematic error that arises when the onnected contribution is neglected. Figure 6 shows the results for had sin 2 θ W (Q 2 ) at 4 GeV 2 for three different pion masses (cf. table ). Red points show the results from the connected contribution only. The asymmetric green error bar indicates the maximum possible onnected contribution. On can see that the systematic error from the onnected diagram is much larger than the statistical error on the connected result. Thus, our results for had sin 2 θ W (Q 2 ) can only be improved with a better estimate of the PoS(LATTICE 25)263 6

7 onnected contribution. The blue point in figure 6 is the result of a recent lattice calculation by Burger et al.[2], where onnected contributions have been neglected. 7 systematic error from onnected connected Burger et al sin 2 (θw )(4 GeV 2 ) m 2 π [GeV 2 ] Figure 6: Our results for had sin 2 θ W (Q 2 ) plotted against the pion mass. The blue point shows the result from Burger et al. [2]. Acknowledgements We acknowledge useful ussions with Benjamin Jäger, Anthony Francis and Gregorio Herdoíza. Our calculations were performed on the dedicated QCD platforms Wilson at the Institute for Nuclear Physics, University of Mainz, and Clover at the Helmholtz-Institut Mainz. We thank Dalibor Djukanovic and Christian Seiwerth for technical support. We are grateful for computer time allocated to project HMZ2 on the BlueGene computer JUQUEEN at NIC, Jülich. This research has been supported in part by the DFG in the SFB 44. We are grateful to our colleagues in the CLS initiative for sharing ensembles. References [] F. Jegerlehner, Z. Phys.C - Particles and Fields 32, (986) [2] D. Bernecker and H. Meyer, Eur.Phys.J. A47 48 (2), [arxiv:7.4388] [3] A. Francis, B. Jäger, H. Meyer and H. Wittig, Phys. Rev. D 88, 5452 (23), [arxiv: ] [4] X. Feng et al., Phys. Rev. D 88, 3455 (23), [arxiv: ] [5] F. Jegerlehner, Nuovo Cim. C34S, 3-4 (2), [arxiv:7.4683] [6] A. Francis, V. Gülpers, G. Herdoíza, H. Horch, B. Jäger, H. Meyer and H. Wittig, PoS LAT24, (24) 27, [arxiv:4.749] [7] M. Della Morte et al., JHEP 57, 7 (25), [arxiv:hep-lat/5526] [8] T. Kinoshita et al., Phys.Rev.Lett. 9, 87 (22), [arxiv: ] [9] K.S. Kumar et al., Ann.Rev.Nucl.Part.Sci. 63, (23), [arxiv: ] [] A. Francis, V. Gülpers, B. Jäger, H. Meyer, G. von Hippel and H. Wittig, PoS LAT24, (24) 28, [arxiv:4.7592] [] V. Gülpers, G. von Hippel and H. Wittig, Phys. Rev. D 89, 9453 (24), [arxiv:39.24] [2] F. Burger, K. Jansen, M. Petschlies, G. Pientka, [arxiv: ] PoS(LATTICE 25)263 7