Relativistic heavy quarks on the lattice

Transcription

1 Relativistic heavy quarks on the lattice Christine Davies University of Glasgow HPQCD collaboration ECT* workshop April 2012

2 Charm and bottom physics Lattice QCD calculations important because: simple hadronic weak decay matrix elements are key to Unitarity Triangle constraints spectrum has many gold-plated states for accurate tests/ predictions + mq determination Precision is critical! Need more tests of errors + more results using different methods.! e! V qq IP! Bs + Ds! + + K! K B s D s e ν ALEPH

3 Issues with handling heavy quarks on the lattice: L q = ψ(d/ + m)ψ ψ(γ + ma)ψ is a finite difference on the lattice - leads to discretisation errors. What sets the scale for these? For light hadrons the scale is Λ QCD For heavy hadrons the scale can be m Q E = E a=0 (1 + A(m Q a) 2 + B(m Q a) ) hadron energy assuming O(mQa) improved m c a 0.4,m b a 2 for a 0.1fm can use improved light quark action for c on fine lattices. Less clear for b - non rel. actions have (Λa) n errors best approach to c and b not necessarily same

4 Charm quarks in lattice QCD - heavy or light? Advantages of relativistic light quark method: E sim = m PCAC relation (if enough chiral symmetry) gives and other currents can also be renormalised nonperturbatively. Z = 1 same action as for u, d, s, so cancellation in ratios Best action to date: Highly improved staggered quarks (HISQ) Errors: α s (am) 2, (am) 4 HPQCD + small taste-changing

5 HISQ based on asqtad improved staggered quarks - smeared link reduces high-momentum gluon exchange taste errors + Naik 3-link term improves derivative. asqtad fat link: u u u HISQ repeats fattening with (SU(3)/U(3)) reunitarisation (so NO tadpole-improvement). Coefficient of Naik term becomes 1+ to remove (am) 4 errors in speed of light. = (am) HPQCD hep-lat/ u 0 4

6 m Ds (GeV) Spectrum Tests masses easy + accurate using relativistic methods. Test D s, tuning from. D s, η c m c ±3MeV η c inc. em effects etc x l MILC 2+1 asqtad configs: 5 lattice spacing values: a= fm full error budget: HPQCD m Ds (GeV) a 2 (fm 2 )

7 N f =2+1 tests of D s, η c mass differences D s - c /2 spin-av. no a error PACS-CS: RHQ, a=0.09fm mu,d physical 2% FNAL/MILC: Fermilab, a=0.15,0.12,0.09fm mu,d extrapoln 0.06% HPQCD: Experiment HISQ, 5 a values to 0.04fm mu,d extrapoln (GeV) Error in binding energy diff. is one way of assessing later errors..

8 Quark mass ratios from lattice QCD mc/ms m q1 (µ) m q2 (µ) = mq1,latt m q2,latt a= a 2 (in fm 2 ) Determine mc/ms using HISQ for both - allows connection from heavy to light for first time HPQCD, m c = 11.85(16) use for accurate m s m s if known... m c

9 Decay constants (2pt amp.) < 0 cγ 0 γ 5 s D s >= f Ds M Ds or, in formalism with PCAC reln: ν V cs µ (m s + m c ) < 0 cγ 5 s D s >= f Ds M 2 D s prediction f Ds (MeV) year expt!" expt " 260(11) Fermilab 249(16) 241(3) 244(8) 248.0(2.5) Fermilab HISQ TM HISQ full lattice QCD 2-flavor lattice QCD 248(6) TM D s B(D s lν l )= G 2 F V cs 2 τ Ds fd 2 8π s m Ds m 2 l 257(5) Expt av. LQCD av. 1 m2 l m 2 D s 2 can extract from expt. if Vcs assumed exciting history!

10 )2.5) Decay constants - updates 2011 CD, LAT11, f Ds " av. of HPQCD Fermilab/MILC =248.6(2.4) MeV!" 257.3(5.3) average HFAG, Jan.11 CLEO, BaBar - BES will improve σ 248.0(2.5) 260.1(9.5) 257(5)(?) no a error u, d, s sea HPQCD HISQ FNAL/MILC PACS-CS RHQ HISQ, 5 a to 0.04fm mu,d extrapoln Fermilab, a=0.12,0.09fm mu,d extrapoln RHQ, a=0.09fm mu,d physical 248(6) u, d sea MeV ETMC TM, 4 a to 0.05fm mu,d extrapoln

11 HPQCD/HISQ f Ds / GeV Fermilab/ MILC (coarse+fine lattices) a 2 / fm 2 Error m Ds f Ds statistical/valence tuning 0.094% 0.57% r 1 =a 0.025% 0.15% r % 0.57% a 2 extrapoln 0.044% 0.40% m q;sea extrapoln 0.048% 0.34% finite volume 0% 0.10% m s 0.056% 0.13% em effects in D s 0.036% 0.10% em and annihln in m c 0.076% 0.00% em effects in c - - missing c in sea 0.01% 0% Total 0.16% 1.0%

12 f Ds /f D comparison 1.26(5) HFAG, Oct (18) u, d, s sea HPQCD HISQ / a values, updated error 1.188(25) FNAL/MILC LAT11 Neil (poster) 1.14(3) PACS-CS Namekawa (Wed) u/d at chiral point 1.17(5) u, d sea ETMC cf f K /f π =1.193(5) improved results need more chiral (not finer) lattices e.g MILC configs?

13 Decay constant of the η c Not directly accessible to experiment but good test of lattice QCD calculations. f c / GeV 0.45 HPQCD/HISQ : 0.4 f ηc = 394.7(2.4)MeV a 2 in fm 2 Surprisingly close, but less than f J/ψ = 407(5)MeV

14 Bottom quarks in lattice QCD - heavy or light? Many options: increased noise in heavy-light over charm - an issue for all Use HQET methods e.g. with step-scaling (Alpha) Use nonrelativistic method at b: Fermilab - same clover action as for c RHQ - coefficients ma-dependent NRQCD - disc. nonrel. expansion of Lq (now improved through α s vb 4 and results on MILC HISQ lattices) R. Dowdall, HPQCD, later today Use relativistic method and extrapolate to b HISQ, TM success for c makes this worth exploring HISQ on MILC 2+1: a=0.15 down to 0.05fm Data below mc to ~mb CD, C. McNeile, E. Follana, K. Hornbostel, P. Lepage

15 Determination of heavy quark masses HPQCD + Chetyrkin et al, ; HPQCD, Current-current correlator method: match time-moments of heavy-heavy meson correlators to energy-derivative moments at q 2 =0 of heavy quark vac. pol. calculated J J in continuum QCD pert. th. (thru ) In continuum use J=V, but for lattice staggered quarks take J=PS (goldstone) since local and normalised from PCAC relation. G(t) =a 6 x α 3 s (am 0h ) 2 0 j 5 (x, t )j 5 (0, 0) 0 G n = t=t/2 (t/a) n G(t) η h correlator t= T/2+1

16 R n,latt = G 4 /G (0) 4 n =4 = am η h 2am 0h (G n /G (0) n ) 1/(n 4) n =6, 8, extrapolate to a=0 (and physical sea quark masses) and then compare to continuum: R n,cont = g 4 /g 0 4 n =4 = m η h 2m h (µ) (g n/g (0) n ) 1/(n 4) n =6, 8, z(µ) extraction allows determination of m at c and b and in between g n /g 0 n =1+ i c i (µ/m(µ))α MS (µ) i fit allows determination of α s

17 b and the equation: 1.7 m b (µ, n f ) R 6 /r m c (µ, n 1.5 f ) = mexp η b w(m exp η b, 0) m exp η c w(m exp η c, 0). µ =3m (41) 1.4 h (µ) em simpler 1.4 to fit m 0h directly, rather than ; but using w significantly reduces the m ηh (and therefore our extrapolation errors), and 1.0 our results 1.7 quite insensitive to uncertainties 0.9 R 8 /r 8 s for1.6 the lattice spacing. 0.8 eterize 1.5 function w with an expansion µ =3m mode one 1.4we used to fit the moments: h (µ) m ηc m ηb 1.3 N w m ηh (GeV) n 2Λ )=Z m (a) 1+ w n / (42) 1.7 m n=1 ηh FIG. 4: Ratio R 10 /r 10 completely m 0h /m ηh divided by m 0c /m ηc (which we approximate by w(m ηc 1.6 N am N w 1.5 amηh nonpert. ratio:,a)/2 from our fit) as a function of m ηh. 2i j 2Λ µ =3m h (µ) m The solid line shows the a=0 extrapolation obtained from our 1+ c ij, m fit. This is compared with b simulation =4.51(4) results for our 4 smallest lattice spacings, i=1 j=0 ηh m together with the best fits (dashed lines) 1.3 corresponding to those lattice c spacings. The point marked by r the moments, an x is for the largest mass we tried (last line in Table II); m ηc m ηb mηh /(2m h(µ)) mηh /(2m h(µ)) mηh /(2m h(µ)) Fit results at 5 values of a, allowing for disc. errors and higher orders in pert. th. Parameterise z as a function of m. Fix c and b from m ηh and m ηc ηb. i + j max(n m ηh (GeV) am,n w ). (43). cmonday, 1: and Function 2 wapril are 2012 z(µ/m determined =3,m by ) fitting m function /(2m ) as a func- m0hmηc/(m0cmηh ) strong test of c.c. method. is too large. this was not included in the fit because am ηh 10

18 Leverage small light quark errors from heavy quark masses using ratios. m n f =4 c Quark masses (MeV/c 2 ) PDG 2010 HPQCD b c m s (2GeV) = 92.2(1.3)MeV s m d (2GeV) = 4.77(15)MeV d u m u (2GeV) = 2.01(10)MeV HPQCD, m n f =5 b mηh /(2m h(µ)) c (3GeV) = 0.986(6)GeV (10GeV) = 3.618(25)GeV m ηh µ = 3m h m h m h /2 Can define masses in between c and b also in terms of meson mass. Note how flat curve is for µ = m h b

19 Heavy-strange masses Difference between heavy-strange and heavyonium mapped out as a function of mhs (as proxy for quark mass). Sea mass dependence very weak. m Bs m ηh /2 =0.658(11)GeV error a bit worse than NRQCD 1GeV =Am Hs Di (a) m Hs D i (a) inc. even powers of am h,am s,aλ QCD allow for ms mistuning HPQCD, i f Bs m Bs m b =2 Monte Carlo statistics 1.30% 1.49% m Hs! m Bs extrapolation r 1 uncertainty a 2! 0 extrapolation m s! m s ;phys extrapolation r 1 =a uncertainties Total 1.82% 1.73%

20 Heavy-strange decay constants extrapolating to b Ditto decay constant - note no renormln needed. f Bs = 225(4)MeV b c much better error than nonrel. methods 2/β0 f Hs = A(m Hs ) b αv (m Hs ) α V (m Ds ) Ci (a) 1GeV i m Hs float b - gives b=-0.51(13) Note f <f Bs Ds - in fact f Ds a max.

21 f Bs,f B comparison CD, LAT11, fb lattice average : 190(4) MeV f B f Bs f B,expt 247(40) fb expt PDG av BR(B-> ) + PDG av Vub 191(9) 189(4) 197(9) 195(12) 172(12) 227(10) 225(4) 242(10) u, d, s sea u, d sea 232(10) HPQCD NRQCD HPQCD HISQ FNAL/MILC ETMC ALPHA most accurate fbs available - most accurate fb from combining with NRQCD ratio f Bx / MeV

22 Extrapolating heavyonium to b HPQCD, in prep. disc. errors much larger now.. f Υ,expt =0.689(5)GeV f J/ψ,expt =0.407(5)GeV f ηb = 0.661(6)GeV f ηh = A Mηh 2GeV b 2GeV D i (a) i M ηh i f ηc =0.395(2)GeV HPQCD Find again fps 3-4 % below corresponding fv, checking in NRQCD... Good check of Z factors?

23 Extrapolating heavy-charm to Bc HPQCD, in prep. - m Bc m η b + m ηc 2 expt = 72(4)MeV = 84(9)MeV after adjusting for em/annihln f Hc =0.432(4)GeV dependence on m is similar to fhs since is a heavy-light system as b mass becomes large.

24 Summary plot for decay constants UNFLAVORED expt postdictions predictions 0.7 b DECAY CONSTANT (GEV) c J/ K FLAVORED D s D B B s B c 0 More work on vectors underway...

25 Semileptonic form factors 3pt amp. t J V qq K T <K V µ D>= f + (q 2 ) D q µ = p µ D pµ K p µ D + pµ K M D 2 M K 2 q 2 q µ +f 0 (q 2 ) M 2 D M 2 K q 2 q µ <K S D >= M 2 D M 2 K m 0c m 0s f 0 (q 2 ) f 0 (0) = f + (0) abs. norm. for same c/s action HPQCD:

26 Experiment sees f+ and maps in q 2 bins dγ D Klν l dq 2 poles and cut t CLEO: allows simple series SL region parameterisation of shape - obtain Vcsf+(0) Lattice QCD : want shape f + (q 2 ), value f + (0) f 0 (0) Techniques improving: High statistics, random sources; multi-exponential fits to multiple T values; phase at boundary to tune q 2 (to 0);use of z expansion to fit and extrapolate shape to chiral/contnm limit see e.g Bourrely et al, = G2 F p3 K 24π 3 V cs 2 f + (q 2 ) 2 z Transforming q 2 to z z = D K, D π t+ q 2 t + t 0 t+ q 2 + t + t 0

27 Updates on f+(0) % f + (0) D-> f + (0) D->K 1% CLEO using unitarity: V cs = (16) V cd =0.2252(7) 4% 2.5% u, d, s sea HPQCD HISQ ; FNAL/MILC hep-ph/ a values; scalar 1 a value; vector + FNAL renorm. u, d sea ETMC a values; vector + double ratios to cancel Z f + (0) HPQCD results will improve further...

28 f 0 (q 2 ) or f + (q 2 ) Updates on f+(q 2 ) coarse D to K f 0 fine D to K f 0 coarse D s to! s f 0 fine D s to! s f 0 coarse D to K f + fine D to K f + coarse D s to! s f + fine D s to! s f + f + HPQCD PRELIMINARY J. Koponen, LAT11 Abs. norm. vector and scalar ops f 0 disc. effects very small q 2 (GeV 2 ) Very little dependence on spectator quark - implications for B/Bs form factors. (see Fermilab/MILC )

29 Updates on f+(q 2 ) 2011 f (q 2 ) D K CLEO 2009, unitarity V cs D -> K coarse D -> K fine a=0 fit f q 2 (GeV) 2 HPQCD PRELIMINARY f 0 v. high stats corrs x 4T 1-2% errors allow accurate comparison of shape to expt J. Koponen, HPQCD in prep. Need to look at more form factors... vector and axial form factors for : G. Donald, HPQCD in prep. D s φlν

30 Comparison of expt and lattice bin by bin using unitarity Vcs J. Koponen, HPQCD, in prep. 1.2 Exp/Lat HPQCD PRELIMINARY 1.15 expt best lattice best Will allow 1.5% error in Vcs Bin1 Bin2 Bin3 Bin4 Bin5 Bin6 Bin7 Bin8 Bin9 q 2 [GeV 2 /c 4 ] q 2 max move on to semileptonic analyses for heavier hisq quarks..

31 Going forward a=0.045fm a=0.03 fm lattices a=0 HPQCD making a=0.03 fm 72 3 x192 lattices f h /M h 0.1 Now m c a c b m b a M h (GeV) Preliminary results for heavyonium consistent with previous heavy HISQ analysis.

32 Conclusions Charm physics in good shape. 1-2% precision possible with improved relativistic actions such as HISQ. Need more results with such formalisms.. e.g. TM Bottom physics still a problem. NRQCD currently has sizeable renormln error Fermilab/RHQ has heavy quark disc. error HQET methods in practice hard? Should get 1% accurate s/l ratios on lattices at chiral pt.. Extrapolation to b from c with HISQ/TM. Promising for masses + decay constants. Now move to semileptonic ffs. Needs extremely fine lattices to do well and e.g. 4q operators harder... ALL need more tests of errors using known quantities. Look forward to results on Nf=2+1+1 lattices... Dowdall 3pm