Status of HVP calculation by RBC/UKQCD Christoph Lehner (BNL) July 27, 2018 – Lattice 2018
Collaborators in the RBC/UKQCD g − 2 effort Tom Blum (Connecticut) Luchang Jin (Connecticut) Peter Boyle (Edinburgh) Chulwoo Jung (BNL) Mattia Bruno (BNL) Andreas J¨ uttner (Southampton) Norman Christ (Columbia) Christoph Lehner (BNL) Vera G¨ ulpers (Southampton) Kim Maltman (York) Masashi Hayakawa (Nagoya) Aaron Meyer (BNL) James Harrison (Southampton) Antonin Portelli (Edinburgh) Taku Izubuchi (BNL/RBRC) Tobi Tsang (Edinburgh)
Outline 1. Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment T. Blum, 1 P.A. Boyle, 2 V. G¨ ulpers, 3 T. Izubuchi, 4, 5 L. Jin, 1, 5 C. Jung, 4 A. J¨ uttner, 3 C. Lehner, 4, ∗ A. Portelli, 2 and J.T. Tsang 2 (RBC and UKQCD Collaborations) 1 Physics Department, University of Connecticut, Storrs, CT 06269-3046, USA 2 School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3FD, UK 3 School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK 4 Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA 5 RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA (Dated: January 22, 2018) Phys. Rev. Lett. 121, 022003 (2018) arXiv:1801.07224v1 [hep-lat] 22 Jan 2018 2. Improved methods for reduced statistical and systematic errors 1 / 24
Time-Moment Representation Starting from the vector current J µ ( x ) = i � f Q f Ψ f ( x ) γ µ Ψ f ( x ) we may write ∞ � a HVP LO = w t C ( t ) µ t =0 with C ( t ) = 1 � � � J j ( � x , t ) J j (0) � 3 j =0 , 1 , 2 � x and w t capturing the photon and muon part of the HVP diagrams (Bernecker-Meyer 2011). The correlator C ( t ) is computed in lattice QCD+QED at physical pion mass with non-degenerate up and down quark masses including up, down, strange, and charm quark contributions. The missing bottom quark contributions are computed in pQCD. 2 / 24
Diagrams – Isospin limit FIG. 1. Quark-connected (left) and quark-disconnected (right) diagram for the calculation of a HVP LO . We do not µ draw gluons but consider each diagram to represent all orders in QCD. 3 / 24
Diagrams – QED corrections [See also talks by M. Bruno and V. G¨ ulpers] (a) V (b) S (c) T (d) D1 (e) D2 (f) F (g) D3 FIG. 2. QED-correction diagrams with external pseudo-scalar or vector operators. For diagram F we enforce exchange of gluons between the quark loops as otherwise a cut through a single photon line would be possible. This single-photon contribution is counted as part of the HVP NLO and not included for the HVP LO. Diagrams T, D1, D2, D3 are not included for the central value of the current calculation. They are suppressed by SU(3), 1 / N c , or both and we estimate their contribution in our uncertainty. Diagrams V, S, and F are included. 4 / 24
Diagrams – Strong isospin breaking [See also talks by M. Bruno and V. G¨ ulpers] x x x (a) M (b) R (c) O FIG. 3. Strong isospin-breaking correction diagrams. The crosses denote the insertion of a scalar operator. For the HVP R is negligible since ∆ m u ≈ − ∆ m d and O is SU(3) and 1 / N c suppressed. Therefore we do not include R and O for the current calculation and only estimate their contribution in our uncertainty. The leading diagram M is included. 5 / 24
Regions of precision (R-ratio data here is from Fred Jegerlehner 2017) 500 Light+Strange (64I) 400 R-ratio × 10 -10 300 200 100 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 t / fm FIG. 4. Comparison of w t C ( t ) obtained using R-ratio data [1] and lattice data on our 64I ensemble. The precision of lattice data deteriorates exponentially as we go to large t , however, is precise at intermediate distances. The R-ratio is very precise at long distances. Note: in this plot a direct comparison of R-ratio and lattice data is not appropriate. Continuum limit, infinite-volume corrections, charm contributions, and IB corrections are missing from lattice data shown here. 6 / 24
Window method We therefore also consider a window method. Following Meyer-Bernecker 2011 and smearing over t to define the continuum limit we write a µ = a SD + a W µ + a LD µ µ with � a SD = C ( t ) w t [1 − Θ( t , t 0 , ∆)] , µ t a W � µ = C ( t ) w t [Θ( t , t 0 , ∆) − Θ( t , t 1 , ∆)] , t a LD � = C ( t ) w t Θ( t , t 1 , ∆) , µ t Θ( t , t ′ , ∆) = [1 + tanh [( t − t ′ ) / ∆]] / 2 . In this version of our calculation, we use d ( √ s ) R ( s ) se −√ st with R ( s ) = � ∞ 4 πα 2 σ ( s , e + e − → had ) 1 3 s C ( t ) = 12 π 2 0 to compute a SD and a LD µ . µ 7 / 24
How does this translate to the time-like region? t / fm 450 Σ t C(t) w t C(t) w t 1E+05 Σ t C(t) w t θ (t,1.5fm,0.15fm) C(t) w t θ (t,1.5fm,0.15fm) Σ t C(t) w t [1- θ (t,0.4fm,0.15fm)] 400 C(t) w t [1- θ (t,0.4fm,0.15fm)] 1E+04 350 1E+03 300 1E+02 x 10 -10 250 1E+01 200 1E+00 150 1E-01 1E-02 100 1E-03 50 0 0.1 1 10 100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 sqrt(s) / GeV t / fm Most of ππ peak is captured by window from t 0 = 0 . 4 fm to t 1 = 1 . 5 fm, so replacing this region with lattice data reduces the dependence on BaBar versus KLOE data sets. 8 / 24
Results ( Fred’s alphaQED17 results used for window result) a ud , conn , isospin 202 . 9(1 . 4) S (0 . 2) C (0 . 1) V (0 . 2) A (0 . 2) Z 649 . 7(14 . 2) S (2 . 8) C (3 . 7) V (1 . 5) A (0 . 4) Z (0 . 1) E48 (0 . 1) E64 µ a s , conn , isospin 27 . 0(0 . 2) S (0 . 0) C (0 . 1) A (0 . 0) Z 53 . 2(0 . 4) S (0 . 0) C (0 . 3) A (0 . 0) Z µ a c , conn , isospin 3 . 0(0 . 0) S (0 . 1) C (0 . 0) Z (0 . 0) M 14 . 3(0 . 0) S (0 . 7) C (0 . 1) Z (0 . 0) M µ a uds , disc , isospin − 1 . 0(0 . 1) S (0 . 0) C (0 . 0) V (0 . 0) A (0 . 0) Z − 11 . 2(3 . 3) S (0 . 4) V (2 . 3) L µ a QED , conn 0 . 2(0 . 2) S (0 . 0) C (0 . 0) V (0 . 0) A (0 . 0) Z (0 . 0) E 5 . 9(5 . 7) S (0 . 3) C (1 . 2) V (0 . 0) A (0 . 0) Z (1 . 1) E µ a QED , disc − 0 . 2(0 . 1) S (0 . 0) C (0 . 0) V (0 . 0) A (0 . 0) Z (0 . 0) E − 6 . 9(2 . 1) S (0 . 4) C (1 . 4) V (0 . 0) A (0 . 0) Z (1 . 3) E µ a SIB 0 . 1(0 . 2) S (0 . 0) C (0 . 2) V (0 . 0) A (0 . 0) Z (0 . 0) E48 10 . 6(4 . 3) S (0 . 6) C (6 . 6) V (0 . 1) A (0 . 0) Z (1 . 3) E48 µ a udsc , isospin 231 . 9(1 . 4) S (0 . 2) C (0 . 1) V (0 . 3) A (0 . 2) Z (0 . 0) M 705 . 9(14 . 6) S (2 . 9) C (3 . 7) V (1 . 8) A (0 . 4) Z (2 . 3) L (0 . 1) E48 µ (0 . 1) E64 (0 . 0) M a QED , SIB 0 . 1(0 . 3) S (0 . 0) C (0 . 2) V (0 . 0) A (0 . 0) Z (0 . 0) E (0 . 0) E48 9 . 5(7 . 4) S (0 . 7) C (6 . 9) V (0 . 1) A (0 . 0) Z (1 . 7) E (1 . 3) E48 µ a R − ratio 460 . 4(0 . 7) RST (2 . 1) RSY µ a µ 692 . 5(1 . 4) S (0 . 2) C (0 . 2) V (0 . 3) A (0 . 2) Z (0 . 0) E (0 . 0) E48 715 . 4(16 . 3) S (3 . 0) C (7 . 8) V (1 . 9) A (0 . 4) Z (1 . 7) E (2 . 3) L (0 . 0) b (0 . 1) c (0 . 0) S (0 . 0) Q (0 . 0) M (0 . 7) RST (2 . 1) RSY (1 . 5) E48 (0 . 1) E64 (0 . 3) b (0 . 2) c (1 . 1) S (0 . 3) Q (0 . 0) M TABLE I. Individual and summed contributions to a µ multiplied by 10 10 . The left column lists results for the window method with t 0 = 0 . 4 fm and t 1 = 1 fm. The right column shows results for the pure first-principles lattice calculation. The respective uncertainties are defined in the main text. For the pure lattice number the dominant errors are (S) statistics, (V) finite-volume errors, and (C) the continuum limit extrapolation uncertainty. Updates for S,V,C in second part of talk. For the window method there are additional R-ratio systematic (RSY) and R-ratio statistical (RST) errors. 9 / 24
Window method with fixed t 0 = 0 . 4 fm 740 × 10 -10 a µ 730 720 710 700 690 680 700 600 500 a µ , Lattice 400 a µ , R-ratio 300 200 100 7 6 5 Lattice Error 4 R-ratio Error 3 Total Error 2 1 0 0.5 1 1.5 2 2.5 t 1 / fm For t = 1 fm approximately 50% of uncertainty comes from lattice and 50% of uncertainty comes from the R-ratio. Is there a small slope? More in a few slides! Can use this to check experimental data sets; see my KEK talk for more details 10 / 24
ETMC 2013 HPQCD 2016 Mainz 2017 BMW 2017 RBC/UKQCD 2018 RBC/UKQCD 2018 HLMNT 2011 DHMZ 2012 DHMZ 2017 Jegerlehner 2017 KNT 2018 No new physics 610 630 650 670 690 710 730 750 a µ × 10 10 BMW and RBC/UKQCD pure lattice are compatible both with no new physics and R-ratio, need more precision! 11 / 24
Consolidate continuum limit Adding a finer lattice
Add a − 1 = 2 . 77 GeV lattice spacing ◮ Third lattice spacing for strange data ( a − 1 = 2 . 77 GeV with m π = 234 MeV with sea light-quark mass corrected from global fit): 60 LL Sin LL 58 LC Sin LC Published 56 x 10 -10 54 52 50 -0.002 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 a 2 / fm 2 ◮ For light quark need new ensemble at physical pion mass. Proposed for early science time at Summit Machine at Oak Ridge later this year ( a − 1 = 2 . 77 GeV with m π = 139 MeV). 12 / 24
Statistical noise Improved bounding method
Bounding Method BMW/RBC/UKQCD 2016 Our correlator in finite volume |� 0 | V | n �| 2 e − E n t . � C ( t ) = n We can bound this correlator at each t from above and below by the correlators � C ( t ) t < T , C ( t ; T , ˜ ˜ E ) = C ( T ) e − ( t − T ) ˜ E t ≥ T for proper choice of ˜ E . We can chose ˜ E = E 0 (assuming E 0 < E 1 < . . . ) to create a strict upper bound and any ˜ E larger than the local effective mass to define a strict lower bound. 13 / 24
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