Phenomenology with Lattice NRQCD b Quarks

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1 HPQCD Collaboration 16 July 2015

2 Our approaches to b quarks In Glasgow, we take two complementary approaches to b quarks: Nonrelativistic QCD and heavy HISQ. Here I will focus exclusively on NRQCD (for b quarks). So why NRQCD? b quark can be simulated at its physical mass. NRQCD proceeds with a relatively straightforward evolution equation computationally inexpensive.

3 Gluon Field Configurations [ ] Gluon field configurations provided by MILC Collaboration with flavours of HISQ quarks in the sea. Those marked (*) are ensembles with physical light quark masses. Set β am l am s am c L/a T/a n cfg

4 NRQCD We use the following NRQCD Hamiltonian: ( e ah = 1 aδh ) ( 1 ah ) n 0 U t 2 2n where we include terms up to O(v 4 ) ah 0 = (2) 2am b ( 1 ah 0 2n ( (2) ) 2 aδh = c 1 8(am b ) 3 + c i ( 2 8(am b ) 2 g ( ) c 3 8 (am b ) 2 σ Ẽ Ẽ ) ( 1 aδh ) 2 ) Ẽ Ẽ ( g a c 4 σ B 2 (4) a ) (2) 2 + c 5 c 6 2am b 24am b 16n (am b ) 2 with most c i coefficients O(α s ) improved. [ ],[ ]

5 Υ Decay Constant & Leptonic Width [ ] Leptonic Width: Γ(Υ (n) e + e ) = 4π 3 α2 eme 2 b f 2 Υ (n) M Υ (n) Decay constant: 0 J V,i Υ (n) j = f Υ (n)m Υ (n)δ ij The Υ leptonic width is experimentally well measured, so a determination of this quantity on the lattice is a good test of QCD, and also of our approach to b quark physics.

6 Time moments [ ] We need to renormalise our NRQCD currents, ( ) J V = Z V J (0) V,NRQCD + k 1J (1) V,NRQCD by determining both k 1 and overall normalisation Z V. We do this by comparing NRQCD to continuum QCD perturbation theory for time moments of the correlators: G V n = gv n (α s, µ/m b ) [am b (µ)] n 2 G V n = Z 2 V G V,NRQCD n

7 Υ Leptonic Width [ ] 3 fυ M Υ (GeV 3/2 ) m l /m s = 0.2 m l physical Experiment a 2 (fm 2 ) Decay Constant f Υ = 0.649(31) GeV Leptonic Width Γ(Υ e + e ) = 1.19(11) kev

8 B Meson Decay Constants [ ] We have updated our picture of B meson decay constants: B, B s and B c as well as B, B s and B c MB s R s = f Bs f Bs MBs R s a 2 [fm 2 ]

9 f B q /f Bq [ ] u, d, s, c sea B /B B s /B s HPQCD NRQCD this paper u, d sea twisted mass Lucha et al sum rules f B q /f Bq Gelhausen et al

10 Decay Constants: summary DECAY CONSTANT [GeV] Experiment : weak decays :emdecays Lattice QCD : predictions :postdictions π K B B D Bs B s φ D s Ds ψ η c ψ B c B c Υ η b Υ

11 Determination of m b [ ] m b is an important parameter in the Standard Model, for example in accurately determining Higgs branching fraction to bb. Recall: G V n = gv n (α s, µ/m b ) [am b (µ)] n 2 so we can make a determination of m b using NRQCD time moments.

12 Determination of m b [ ] mb(µ = 4.18, nf = 4) GeV m l /m s = 0.2 m l physical a 2 (fm 2 ) m b (µ = m b, n f = 5) = 4.196(23) GeV

13 m b : lattice summary HPQCD NRQCD JJ [ ] HPQCD HISQ JJ n f = 3 [ ] HPQCD HISQ ratio n f = 4 [ ] HPQCD NRQCD E 0 [ ] ETMC ratio [ ] m b (m b,n f = 5)(GeV)

14 Semileptonic decays We use NRQCD b quarks and HISQ light quarks in our calculation of B πlν. t J π B T We pick multiple values of T and fit B and π 2-point correlators simultaneously with 3-point correlators.

15 B π at zero recoil For B and π at rest, q 2 max 26.5 GeV 2, π V 0 B = f 0 ( q 2 max ) (mb + m π ) Soft pion theorem relates this to decay constants in m π 0 limit: f 0 (q 2 max) = f B /f π This relation was previously studied in lattice QCD, but it did not seem to hold. We now have the opportunity to study with light quarks at physical mass.

16 Soft Pion Theorem ) 1 + mπ mb a 0.15 fm a 0.12 fm a 0.09 fm ( f0(q 2 max)/(fb/fπ) m π (GeV)

17 A brief look at the future... We are optimistic about being able to explore the full q 2 range in B c η c lν decays on superfine lattices (a 0.06 fm). p ηc = 2.4 GeV in the B c rest frame at q 2 = 0

18 Summary Bottomonium: Using time moments from the lattice and continuum perturbation theory we have: calculated the Υ and Υ leptonic widths using NRQCD b quarks. made an accurate determination of m b (m b ). B Physics Decay constants have been calculated for B q in addition to B q We have calculated B π at q 2 max; we find results consistent with soft pion theorem. This program is now being extended and includes a full range of q 2 for B c η c on superfine ensembles.

19 Thank you!

20 Backup Slides

21 Υ Leptonic Width [ ] fυ M Υ /f Υ M Υ Decay Constant f Υ = 0.481(39) GeV 1 A = 0 J V Υ 0 J V Υ = f Υ f Υ MΥ M Υ Experiment a 2 (fm 2 ) Leptonic Width Γ(Υ e + e ) = 0.69(9) kev

22 Z V [ ] ZV fine, J NRQCD, k 1 =-0.155, Z =1.019(8) moment n

23 Time moments [ ] Time moments of our NRQCD correlators defined as, G V,NRQCD n = 2 t (t/a) n C V,NRQCD (t)e (E0 M kin)t. for n = 4, 6, 8,...

24 Fit forms Bottomonium B π 3 h(a, m sea) =h phys 1 + b l δm sea/(10m s) + c j (aλ) 2j j=1 2 + (aλ) 2j [ ] c jb δx m + c jbb (δxm) 2 j=1 3 Γ(a, m π) =Γ phys 1 + c j (aλ) 2j + b j m j π j=1 ( ) Λ 2 [ + (aλ) 2 c b δx m + (aλ) 2 (δx m) 2] m b ( m + da 2 m 2 2 ) ] π l π log m 2 π 1.6

25 Determination of m b [ ] [ ] n 2 [ ] 1/2 Rn V = rn V mb Rn r (α MS, µ/m b ) n 2 M kin m b (µ) R n 2 r n 2m b = M Υ,η b 2m b (µ) m b (µ) = M Υ,η b 2 [ ] 1/2 Rn 2 r n 2m b R n r n 2 M kin

26 Determination of m b [ ] mb(µ = 4.18, nf = 4) (GeV) fine very coarse moment n