Lattice QCD determination of quark masses and

Transcription

1 Lattice QCD determination of quark masses and s Christine Davies University of Glasgow HPQCD collaboration APS GHP2017 Washington Jan 2017

2 Quark masses and strong coupling are fundamental parameters of the SM but cannot be directly determined from experiment because we do not have direct access to quarks Well-defined m q and s are scheme and scale-dependent. Convention is to use. MS Compare results from multiple approaches for strong test of QCD. Lattice QCD methods are particularly accurate. Masses/ s are input to theoretical expressions for SM cross-sections e.g. H! cc

3 Lattice QCD: fields defined on 4-d discrete space-(euclidean) time. Lagrangian parameters: s,m q a 1) Generate sets of gluon fields for Monte Carlo integrn of Path Integral (inc effect of u, d, s, (c) sea quarks) 2) Calculate valence quark propagators and combine for hadron correlators. Fit for hadron masses and amplitudes Determine a to convert results in physical units. Fix from hadron mass m q a *numerically extremely challenging* cost increases as and with statistics, volume. a! 0,m u/d! phys

4 Can tune bare lattice QCD mass parameters very accurately using experimentally very well-determined ground-state meson masses. MESON MASS (GeV/c 2 ) ' b b ' ' c2 c h c1 c J/ c0 c 2011 '' ' h b (2P) h b (1P) b2 b1 (2P) b0 b2 b0 b1 (1P) ' B c B c B s B (1D) B c *' B c * B s * B * D s D s * D K expt fix params postdcns predcns * B c0 few MeV accuracy requires em effects to be considered

5 Mass parameters in Lattice QCD Lagrangian can be tuned very accurately against experimental hadron masses Issue is: Conversion of lattice quark masses to MS scheme m MS (µ) =Z m (µa)m latt Options to calculate Z: 1) lattice QCD pert. th. - hard to do beyond NLO 2) Nonperturbative calculation of a quantity that can be matched to MS using continuum QCD pert. theory * Error dominated by that of Z Note: Z cancels in mass ratios, which are completely nonperturbative in lattice QCD in a given quark formalism. Provides critical test of procedure above. various of these

6 Lattice QCD: determining quark masses and providing nonperturbative tests of the determination J p1 t J p1-p2 S m c,m b current-current correlators p2 m s,m u /m d Green s function in RI-SMOM scheme m c m s determined directly from lattice QCD

7 Current-current correlator method for mc (and mb) Time-moments of lattice QCD correlators extrapolated to the continuum limit can be related to s -1 -moments of R e+ e and to continuum QCD perturbation theory known through (NNNLO) J t G n = t 3 s J vector coupling to photon (t/a) n G(t) R(s) R(s) n =4, 6, 8, n =2k pqcd BES (2001) J/ψ ψ, MD-1 CLEO BES (2006) ψ, e.g. Kuhn et al, hep-ph/ s (GeV) R e + e (s) = (e+ e! hadrons) 4 2 /(3s) M k Z ds s k+1 R c(s) c quark contribn

8 Continuum QCD perturbation theory for the moments is a function of quark mass and known through 3 s In lattice QCD can calculate moments not available to expt. e.g. for pseudoscalar density correlator for c quarks: R n,latt = G 4 /G (0) 4 n =4 ratio to results with no gluon field improves disc. errors = am c 2am c (G n /G (0) n ) 1/(n 4) n =6, 8, R n,cont = C P k C P,0 k m c 2m c (µ) C P k C P,0 k =1+ X c i i s(µ) simultaneous fit to multiple moments - gives s,m c HPQCD + Chetyrkin et al, , C. Mcneile et al, HPQCD,

9 Current-current correlator method -HISQ HPQCD, Repeat calcln for m q m c inc. ultrafine lattices 11 m h /(2mh(µ)) m h /(2mh(µ)) m h /(2mh(µ)) (4) (4) (5) (5) (6) (7) (7) (8) (7) (8) (7) (7) (9) (9) log W 11 log W 12 log W BR log W CC log W 13 log W 14 log W 22 log W 23 R 6 /r 6 µ =3m h (µ) R 8 /r 8 µ =3m h (µ) log W 13 /W 22 log W 11 W 22 /W12 2 log W CC W BR /W R 10 /r log W CC /W BR µ =3m h (µ) log W 14 /W 23 log W 11 W 23 /W 12 W 13 m c m b (5) logm W 12 h (GeV) /u (8) log W BR /u 6 0 Agrees well with contnm. 1: Function (7) z(µ/m h log =3,m W CC h )/um 6 0 h /(2m h ) as a funcof m h results (6) using The solid line, Rlog e + plus We 13 gray /u 8 0 error envelope, shows m h /(2mh(µ)) c Can determine m h µ = 3m h m h m h /2 FIG. 6: z(µ/m h,m h ) versus m h for three di erent values of µ/m h. The curve for µ =3m h comes from the best fit heavy quarks - extrapolate to the moments. The other curves are obtained by evolving perturbatively from µ=3m h. (slightly) to b. m n f =5 b h ) m h /m h b for (m b )=4.164(23)GeV key error is now extrapoln in a

10 Example error budget for HISQ current-current method TABLE IV. Error budget [31] for the c mass, QCD coupling, and the ratios of quark masses m c /m s and m b /m c from the n f =4 simulations described in this paper. Each uncertainty is given as a percentage of the final value. The different uncertainties are added in quadrature to give the total uncertainty. Only sources of uncertainty larger than 0.05% have been listed. HPQCD, m c (3) MS (M Z ) m c /m s m b /m c Perturbation theory Statistical errors a 2! m sea uds! m sea c! m h 6= m c (Eq. (15)) Uncertainty in w 0, w 0 /a prior Uncertainty in m s m h /m c! m b /m c m c : electromag., annih m b : electromag., annih Total: 0.64% 0.63% 0.55% 1.20%

11 mc summary Good consistency between lattice actions using JJ method lattice nf=4 PDG evaluation flavours of sea quarks inc. on lattice, adjust result to 4 using pert. th. m c (m c,n f = 4) = (95) GeV

12 mb summary Several different methods here. Good consistency between different methods and b-quark formalisms lattice av. PDG evaluation flavours of sea quarks inc. on lattice. Use pert. th. to inc. to 5 Lattice average: 4.178(14) GeV

13 mb/mc from lattice QCD update to mq1,latt HPQCD, m q2,latt in QCD a=0 = m q1,ms (µ) m q2,ms (µ) completely nonperturbative determination of ratio gives: Agrees with that from currentcurrent correlator method - test of pert. th. m b m c =4.541(26) see also: HPQCD, ;! ETM, ;! HotQCD,

14 mc/ms m c /m s Mass ratio can be obtained directly from lattice QCD if same quark formalism is used for both quarks. Not possible with any other method HISQ summary from hotqcd, N f = N f = HPQCD 15 MILC 14 ETMC 14 this paper χqcd (am 0c ) 2 m c m s = (65) HPQCD, HPQCD 10 N f = 2 ETMC 10 m c /m s Durr

15 Combining mc and mc/ms leads to 1% accuracy in ms - also compare to RI-SMOM scheme determination u,d,s,c sea u,d,s sea m s (3GeV,n f = 3) HPQCD this paper ETMC RBC/UKQCD Durr et al HPQCD HPQCD (pert) m s (3GeV,n f = 3) = 84.1(5)MeV mc/ms+mc RI-MOM RI-MOM RI-SMOM mc/ms+mc lattice pert Also hotqcd: (1.5) MeV

16 Alternative method to determine (light) quark masses: RI-SMOM scheme RBC/UKQCD, Impose a MOM renormalisation scheme directly p1-p2 S on the lattice, i.e. fix an amputated vertex function to its p1 p2 tree-level value (in Landau gauge). Z O Match to MS perturbatively - O (p 1,p 2 )=O tree s Z 2 = NNLO q Z m = Z 1 Important improvement: S Non-exceptional kinematics: Sturm et al, ;! Gorbahn, Jäger, p 2 1 = p 2 2 =(p 1 p 2 ) 2 = µ 2 much smaller systematic errors from non-pert effects AND pert. matching

17 Calculate lattice Zm - multiply by tuned lattice bare mass and pert. matching to MS HPQCD, Lytle et al, preliminary, Agrees well with result expected from JJ method for mc and mc/ms

18 Lattice QCD determination of ms/mud requires consideration of em effects via charged/neutral /K m ud = m u + m d 2 m s m ud = M 2 K + + M 2 K 0 M 2 + M 2 + = 25.9 LO chi-pt [151] [162] [157] [163] [164] [161] summary from PDG average dominated by lattice = 27.3(7)

19 Lattice QCD determination of s Lattice QCD Lagrangian has parameter g 2 = lattice scheme coupling at scale /a - determination of a fixes the coupling. However, again it is conversion to MS which is issue. Q = a 0 + a 1 s (µ)+a 2 s (µ) Calculate in lattice QCD. Could be e.g. continuum limit of 4th moment of JJ correlator. Minimal exptl uncty µ Choice of depends on Q e.g. ~ 3m h in JJ. Fixing it requires determination of a using e.g. a hadron mass. Perturbative expansion - using continuum QCD pert. th. in MS if Q is cont. quantity. Needs to be highorder.

20 Baikov Davier Pich Boito SM review HPQCD (Wilson loops) HPQCD (c-c correlators) Maltmann (Wilson loops) PACS-CS (SF scheme) ETM (ghost-gluon vertex) BBGPSV (static potent.) ABM BBG JR NNPDF MMHT ALEPH (jets&shapes) OPAL(j&s) JADE(j&s) Dissertori (3j) JADE (3j) DW (T) Abbate (T) Gehrm. (T) Hoang (C) GFitter CMS (tt cross section) τ-decays lattice structure functions e+ e jets & shapes electroweak precision fits hadron collider s (MS,n f =5,M Z ), PDG small Wilson loops JJ small Wilson loops Schro. functional ghost-gluon vertex static quark potential Effect of inc. pert. theory order in JJ method mc(3gev) MS (M Z) =0.1181(11) Lattice results most precise; several different methods N

21 Conclusions Lattice QCD results available from multiple quark formalisms and methods now. - good consistency m c (m c ) m b (m b ) s (M Z ) is determined to 1% and to 0.5% from continuum and lattice methods. to 1% from lattice - multiple methods 1% accurate mc/ms ratio allows 1% in ms also, along with RI-SMOM methods Tests of perturbation theory from completely nonperturbative mass ratios and JJ/RI-SMOM comparison Future improvements from higher order pert th. (?possible) and finer lattices to push up µ values.

22 Backup slides

23 t Current-current correlator method for lattice mc HPQCD + Chetyrkin et al, , C. Mcneile et al, HPQCD, Substitute time-moment of lattice charmonium correlator for experiment. In principle can use any current J now. For HISQ quarks pseudoscalar c correlator is most accurate. J is absolutely normalised. step 1: calculate c correlators by combining lattice charm quark propagators step 2: large time - fit to exponential, J gives c mass J step 3: tune lattice quark mass so c mass correct. step 4: calculate time moments to compare to QCD pert. theory. Emphasises short-time contribns. G(t)e Mt J J J PC any current now t/a

24 Further check of JJ method: compare vector moments (after normalising current) to those extracted from R e+ e Agreement is a 1% test of (lattice ) QCD. Also gives charm quark contribution to anomalous magnetic moment of the muon a c µ = 14.4(4) HPQCD, , (n th moment) 1/(n 2) (GeV 1 ) expt lattice and expt errors similar size n = 10 n = 8 n = 6 n = (am c ) 2 see also ETMC,

25 Alternative determinations of mb HPQCD, Current-current correlator method using vector bottomonium correlators calculated with improved NRQCD b quarks 11 nf = HISQ sea quarks fine very coarse mb (mb, nf = 5) = 4.196(25)GeV moment n 20 mb (µ = 4.18, nf = 4) GeV mb (µ = 4.18, nf = 4) (GeV) 4.6 ml physical FIG. 4: The b quark mass in the M S scheme determined 4.15 from our calculation of time-moments of the vector currentcurrent correlator as a function of the moment number, n, in 4.1 eq. (27). Blue open circles are for the fine set 5 lattices and 0 the red open circles for the very coarse set 1. The errors on the points are dominated by the uncertainty in the value of Larger moment numbers more nonrelativistic - use 18 ml /ms = a (fm )

26 Update and improved method HPQCD, Use improved nf = gluon field configs, more accurate lattice spacing determination etc etc. Determine mc at higher scales by using multiple mh mc(3mh) R n = 1 G 1/(n 4) n /G (0) 1 m n! c m c (µ) tuned mc Gn at mh n = m h /m c C P k C P.0 k µ =3m h m 0h = m h(µ) m 0c m c (µ) m c (m c )= (95)GeV