What Lattice QCD can do for experiment Christine Davies University of Glasgow HPQCD collaboration Birmingham Nov 2014
QCD is a key part of the Standard Model but quark confinement is a complication/interesting feature. Cross-sections calculated at CDF high energy using QCD pert. th. with ~3% errors. Also parton distribution function and hadronisation uncertainties. But (some) properties of hadrons much more accurately known and calculable in lattice QCD - can test SM and determine parameters very accurately (1%).
Weak decays probe meson structure and quark couplings V ud V us V ub V us π ! l ν K ! l ν B ! π l ν K ! π l ν V cd V cs V cb ν D ! l ν D s ! l ν B ! Dl ν D ! π l ν D ! Kl ν K V td V ts V tb h B d | B d i h B s | B s i Br ( M → µ ν ) ∝ V 2 ab f 2 M CKM matrix Expt = CKM x theory(QCD) Need precision lattice QCD to get accurate CKM elements to test Standard Model (e.g. is CKM unitary?). If V ab known, compare lattice to expt to test QCD
Lattice QCD = fully nonperturbative, based on Path Integral formalism Z Z basic L QCD d 4 x ) D U D ψ D ψ exp( − integral • Generate sets of gluon fields for Monte Carlo integrn of Path Integral (inc effect of u, d, s (+ c) sea quarks) • Calculate averaged “hadron correlators” from valence q props. • Fit as a function of time to obtain masses and simple matrix elements a • Determine and fix to get m q results in physical units. • extrapolate to a = 0 , m u,d = phys for real world **now at m phys ** a
Hadron correlation functions (‘2point functions’) give masses and decay constants. T large X h 0 | H † ( T ) H (0) | 0 i = A n e − m n T → A 0 e − m 0 T n masses of all QCD H H hadrons with quantum | h 0 | H | n i | 2 = f 2 n m n numbers of H A n = 2 m n 2 decay constant parameterises amplitude to annihilate - a property of the meson calculable in QCD. Relate to experimental decay rate. 1% accurate experimental info. for f and m for many mesons! Need accurate determination from lattice QCD to match
Darwin@Cambridge, part of STFC’s HPC facility for theoretical particle physics www.dirac.ac.uk and astronomy - DiRAC II Allows us to calculate quark propagators rapidly and store them for flexible re-use. State-of-the-art commodity cluster: 9600 Intel Sandybridge cores, infiniband interconnect, fast switch and 2 Pbytes storage
Example parameters for calculations now being done with ‘staggered’ quarks. “2nd generation” lattices inc. c MILC HISQ, 2+1+1 quarks in sea 0.14 mass HISQ = Highly of u,d improved 0.12 quarks staggered quarks - very accurate 0.1 discretisation 2 / GeV 2 E.Follana et al, 0.08 HPQCD, hep-lat/ 0610092. m � 0.06 m u , d ≈ m s / 10 0.04 real 0.02 m u , d ≈ m s / 27 world 0 Volume: m π 0 = 0 0.005 0.01 0.015 0.02 0.025 0.03 135 MeV m π L > 3 a 2 / fm 2
Example (state-of-the-art) calculation Extract meson mass and pion at physical m u/d amplitude=decay constant from correlator for multiple correlator(T) 0.1 lattice spacings and m u/d. Very high statistics 0.01 Extrapolation in a 0 20 40 60 80 100 T Convert decay constant to GeV units using to w 0 fix relative lattice spacing. Very small discretisation errors. = m l /m s R. Dowdall et al, HPQCD, 1303.1670.
The gold-plated meson spectrum 12 expt 2011 � b2 fix params h b (2P) 2012 � b1 (2P) postdcns � '' � b0 predcns ' � ' � b2 10 � b � b1 (1P) 2008 � b � (1D) HPQCD � h b (1P) � b0 1112.2590 MESON MASS (GeV/c 2 ) 1207.5149; 2014 8 0909.4462 *' ' B c B c * B c0 * B c B c 6 * B s B s 2005 B * B � ' � c2 4 ' � c1 � c h c � c0 HPQCD � c J/ � 1008.4018 * D s D s 2 error 3 MeV D - em effects important! K � � 0 older predcns: I. Allison et al, hep-lat/0411027, A. Gray et al, hep-lat/0507013
Lattice QCD is best method to determine quark masses m q,latt determined very accurately by fixing a meson mass to be correct. e.g. for m c fix M η c *masses Issue is conversion to the scheme MS important for Higgs cross- • Direct method sections* m MS ( µ ) = Z ( µa ) m latt Calculate Z perturbatively or partly nonperturbatively. • Indirect methods: (after tuning ) match a quantity m latt from lattice QCD to contnm pert. th. in terms of mass MS e.g. q 2 -derivative moments of current-current correlators (vac. pol.function) for heavy α 3 quarks known through . J J s Calc. on lattice as time-moments of ‘local’ meson correlation function Chetyrkin et al, 0907.2110 HPQCD + Chetyrkin et al, 0805.2999, C. Mcneile et al, HPQCD,1004.4285
Most accurate to use pseudoscalar correlator time-moments: ( am c ) 2 < 0 | j 5 ( ⌦ G ( t ) = a 6 � x, t ) j 5 (0 , 0) | 0 > J J ⇤ x ratio to results with no � ( t/a ) n G ( t ) G n = gluon field improves disc. t t errors R n,latt = G 4 /G (0) n = 4 4 = am η c ( G n /G (0) n ) 1 / ( n − 4) n = 6 , 8 , 10 . . . 2 am c extrapolate first 4 moments to a=0 and fit to contnm pert. th. gives m ( µ ) AND α s ( µ ) From 2+1 configs: m b ( m b ) = 4 . 164(23)GeV m c ( m c ) = 1 . 273(6)GeV α s ( M z ) = 0 . 1183(7) *new* 2+1+1 results agree:1408.4169
Improvement in result 1 . 10 m c ( 3GeV ) 1 . 05 clear as more orders 1 . 00 added in contnm pert. 0 . 95 theory. 0 . 90 0 . 130 HPQCD,1408.4169 0 . 125 α MS ( M Z ) 0 . 120 0 . 115 0 . 110 HPQCD NRQCD JJ 0 1 2 3 1408.5768 N HPQCD HISQ JJ n f = 3 Different lattice methods HPQCD HISQ JJ n f = 4 1408.4169 for m b agree. HPQCD NRQCD E 0 Weighted average (grey 1302.3739 band): 4.178(14) GeV ETMC ratio 1311.2837 1404:0319: impact on accuracy of H → bb 4 . 0 4 . 1 4 . 2 4 . 3 4 . 4 4 . 5 m b ( m b , n f = 5 )( GeV )
Quark mass ratios HPQCD: 0910.3102; 1004.4285,1408.4169 Obtained directly from lattice QCD if same quark formalism is used for both quarks. m q 1 ,MS ( µ ) � m q 1 ,latt ⇥ = Ratio is at same and for same n f . m q 2 ,MS ( µ ) µ m q 2 ,latt a =0 Not possible any other way ... m b /m c m c /m s *new* 2+1+1 with 1 . 4 physical u.d: m 0 h m η c / ( m 0 c m η h ) 1 . 3 1 . 2 1 . 1 1 . 0 0 . 9 0 . 8 m η c m η b 4 6 8 m η h (GeV) m b /m c = 4 . 51(4) m b /m s = 52 . 90(44) m c /m s = 11 . 652(65) 6 = 3 m τ /m µ allows 1% accuracy in m s (94.0(6) MeV)
Meson decay constants Parameterises hadronic information needed Γ ∝ f 2 for annihilation rate to W or photon: Experiment : weak decays 0.7 : em decays Lattice QCD : predictions DECAY CONSTANT (GEV) 0.6 HPQCD: postdictions 1207.0994 HPQCD: 0.5 HPQCD: 1302.2644 1311.6669 HPQCD: 0.4 HPQCD: 1312.5264 1303.1670 HPQCD, 0.3 1408.5768 � *NEW* 0.2 HPQCD, 1208.2855 0.1 * � � � � D s D s B s B c K D B J/ � � � � 0 Aim for same ‘overview’ as for masses. Note different scale.
Constraining new physics with lattice QCD V us /V ud * results at physical u/d quark masses* f K /f π HISQ 2+1+1 configs 0 . 164 ` 0 . 162 f K (GeV) 0 . 160 0 . 158 Annihilation of to W K/ π 0 . 156 allows CKM element determination given decay 0 . 150 constants from lattice QCD 0 . 145 f π (GeV) Γ ( K + → ⇤� ) expt for 0 . 140 Γ ( ⇥ + → ⇤� ) 0 . 135 | V us | f K + 0 . 130 | V ud | f π + = 0 . 27598(35) Br( K + ) (25) EM 0 . 05 0 . 10 0 . 15 0 . 20 f K + m 2 π / ( 2 m 2 K − m 2 π ) R.Dowdall et from lattice gives CKM al, HPQCD: = m u,d /m s f π + 1303.1670
(note: by 0.3%) f K + < f K Comparison of results good agreement from different formalisms * results at physical u/d quark masses* f K + /f � + n f =2+1+1 HPQCD, 1303.1670 f K + HISQ f π + = 1 . 1916(21) MILC, 1301.5855 HISQ ETMC, Lattice2013 twisted mass | V us | | V ud | = 0 . 23160(29) expt (21) EM (41) latt f K /f � n f =2+1 BMW, 1001.4692 clover | V us | = 0 . 22564 HPQCD, 0706.1726 HISQ LvW, 1112.4861 (28) Br (20) EM (40) latt (5) V ud domain-wall MILC, 1012.0868 asqtad 1 − | V ud | 2 − | V us | 2 − | V ub | 2 RBC/UKQCD 1011.0892 domain-wall = − 0 . 00009(51) 1.15 1.17 1.19 1.21 1.23 1.25 β V ud from nuclear decay now needs improvement for unitarity test!
B meson decay constants: results from NRQCD b and physical u/d quarks HPQCD: R Dowdall et al, 1302.2644. 1 . 26 Set 1 Set 5 MILC HISQ 2+1+1 Set 2 Set 6 1 . 24 Set 3 Set 7 configs with u/d down to √ M B ) Set 4 Set 8 1 . 22 � M B s ) / ( f B physical values + 1 . 20 improved NRQCD 1 . 18 ( f B s meson mass difference 1 . 16 correct to 2% 1 . 14 Physical point 0 . 00 0 . 05 0 . 10 0 . 15 0 . 20 0 . 25 = m l /m s M 2 π / M 2 100 η s Set 3 Set 8 95 Set 6 PDG B s to B decay constant 90 M B s − M B (MeV) 85 ratio accurate to 0.6% - 80 since Z factors cancel. 75 Separate decay 70 65 constants to 2% 60 0 . 000 0 . 005 0 . 010 0 . 015 0 . 020 0 . 025 a 2 (fm)
185(3) 225(3) averages B, B s decay constant world averages 2014 f B s f B f B,expt PDG av. branching fraction + unitarity Vub f B + HPQCD NRQCD 1302.2644 f B 0 u, d, s, c sea HPQCD NRQCD 1202.4914 HPQCD HISQ 1110.4510 FNAL/MILC 1112.3051 u, d, s sea u, d sea ETMC Lattice2013 ALPHA 1210.7932 NOTE: f Bs < f Ds (248 MeV) 150 175 200 225 250 275 300 but by much less than LO f B x / MeV HQET expects
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