Christine Davies University of Glasgow HPQCD Collaboration

Progress in Lattice QCD Christine Davies University of Glasgow HPQCD collaboration

Physics in Collision Warwick, Sept 2015 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

Vud Vus Vub π lν K lν B πlν Vus 0 ! ! ! 1 K πlν ! B V V V C B cd cs cb C ν B D lν Ds lν B DlνC B ! ! ! C BD πlνD Klν C B ! ! C K B Vtd Vts Vtb C B C B B B B B C 2 2 B h d| dihs| si CBr(M µ⌫) V f @ A ! / ab 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 Vab known, compare lattice to expt to test QCD Applications of Lattice QCD/Lattice field theory Annual proceedings:! Particle physics http://pos.sissa.it/ QCD parameters Hadron spectrum Hadron structure Nuclear physics

Glueballs and exotica CKM elements QCD at high temperatures Theories beyond the and densities Standard Model Nuclear masses and properties

Quantum gravity

Astrophysics LAT2015 talks: ! http://indico2.riken.jp/indico/ conferenceDisplay.py?confId=1805! Lattice QCD = fully nonperturbative, based on Path Integral formalism basic U exp( d4x) integral D D D LQCD Z Z • 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 • Determine a and fix m q to get results in physical units. • extrapolate to a =0 ,m u,d = phys for real world *now* able to a calculate directly Hadron correlation functions (‘2point functions’) give masses and decay constants. T mnT large m T 0 H†(T )H(0) 0 = Ane A e 0 h | | i 0 n ! QCD H masses of all H X hadrons with 2 2 quantum 0 H n fnmn numbers of H An = |h | | i| = 2mn 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 Example (state-of-the-art) calculation

pion at physical mu/d Extract meson mass and amplitude=decay constant 0.1 from correlator for multiple lattice spacings and mu/d. correlator(T) Very high statistics

0.01 Extrapolation in a 0 20 40 60 80 100 T Convert decay constant to GeV units using w 0 to fix relative lattice spacing. Very small discretisation errors.

R. Dowdall et al, HPQCD, 1303.1670. = ml/ms The gold-plated meson mass spectrum 12 2011 expt h (2P) b2 fix params 2012 b b1(2P) postdcns '' b0 ' ' predcns 10 b b2 b1(1P) 2008 b hb(1P) b0 (1D) ) 2 8 2014

' B*' Bc c * * Bc0 Bc Bc 6 * Bs Bs 2005 B B*

' 4 ' c2 MESON MASS (GeV/c c h c1 c c0 c J/ * D Ds 2 Ds

K 0 few MeV uncertainties in many cases More detailed study of unstable and excited states important to pin down oddities now being seen (e.g. in Excitedcharmonium charmonia spectrum) Hadspec 1204.5425 The resonance in scattering [PR D87, 034505]

Y(4260) M = 854.1 ± 1.1 MeV = 12.4 ± 0.6 MeV g = 5.80 ± 0.11 [M 400 MeV] X(3872) c.f. experimentally

M = 775.49 ± 0.34 MeV = 149.1 ± 0.8 MeV g ≈ 5.9 M Experiment ρ unstable c J/ χc0 hc χc1 χc2 8 states need ‘Exotic’ multivolume analysis JPC 1411.2004 M 400 MeV [HadSpec, JHEP 07 (2012) 126] Key future aim: establish whether tetra/pentaquark states,6 hybrids, glueballs exist - needs very high stats and large basis of operators. The spectrum of decay constants for gold-plated mesons

0.7 Experiment : weak decays :emdecays 0.6 Lattice QCD : predictions :postdictions 0.5

0.4

0.3

0.2

DECAY CONSTANT [GeV] 0.1 π K B∗ B D Bs∗ Bs φ Ds Ds∗ ψ′ ηc ψ Bc∗ Bc Υ′ ηb Υ 0 key input for SM calcn of + B µ µ HPQCD, arXiv:1302.2644, 1408.5768, 1503.05762 (s) ! 11

Quark masses and strong coupling constant 11 1408.4169 HPQCD– jjthis paper Lattice QCD results have ETMC 1310.3763 transformed accuracy u,d,s,c sea u,d,s sea possible. Basavov et al 1205.6155 15 HPQCD– jj1004.4285 HPQCD–Wnm 1004.4285 nihilation that we use to determine the ⌥ leptonic width. 1408.5768 The method is a variation of that used for the J/ lep- FIG. 7. Recent lattice QCD determinations of the QCD coupling JLQCD 1002.0371 HPQCD NRQCD JJ tonic width in [8]. In that case we were working with (nf =5) evaluated at scale MZ . The gray band is the weighted a relativistic discretisation of the QCD quark action on FIG. 8. Lattice QCD determinations ofPACS-CS the ratio of the 0906.3906c and s quarks’ average of the results: 0.1185(4). We include our jj result for nf = the lattice. Since here we have a nonrelativistic discreti- 3 in the average, but not our new n =4result since systematic masses. The ratios come from this paper and references [32, 33, 36, f 0.116 0.118 0.120 HPQCD HISQ JJ n f = 3 sation there are some di↵erences in the approach that we 42, 43]. The gray band is the weighted average of the three nf =4 errors are correlated between the two results. The results shown here (M ,n = 5) lay out in this section5. come from this paper and [37–41]. results: 11MS.700(46)Z . f HPQCD HISQ ratio n = 4 1408.4169 f Time-moments of current-current correlators, being FIG. 7. Recent lattice QCD determinations of the QCD coupling 1408.4169 ultraviolet-finite quantities, can be calculated in lattice In this paper, we have redone our earlier nf =3analysis [2] HPQCD this paper (nf =5) evaluated at scale MZ . The gray band is the weighted QCD and extrapolated to the continuum limit to give a using simulations with nf =4sea quarks: u, d, s and c. Our FIG. 8. Lattice QCD determinations of theHPQCD ratio of NRQCD the c andEs0 quarks’ average of the results: 0.1185(4). WeETMC include 1403.4504our jj result for nf = continuum result that can be compared to experiment [8]. new results, masses. The ratios come from this paper and1302.3739 references [32, 33, 36, 3 inu the,d,s average,,c sea but not our new nf =4result since systematic The current used in the correlator must be matched to 42, 43]. The gray band is the weighted average of the three n =4 errors are correlated betweenu,d, thes sea two results. The results shown here f the continuum current, however. When the Highly Im- mc(3 GeV,nf = 4) = 0.9851(63) GeV (52) results: 11.700(46). ETMC ratio come from this paper and [37–41]. RBC/UKQCD 1411.7017 proved Staggered Quark discretisation [22] is used, for ↵MS(MZ ,nf = 5) = 0.11822(74), (53) example, the local pseudoscalar density is absolutely nor- agree well with our earlier results of 0.986(6) GeV and Durr et al 1011.2403 malised [14, 15] but the vector current normalisation has 4.0 4.1 4.2 4.3 4.4 4.5 0.1183(7) c In this paper, we have redone our earlier nf =3analysis [2] to be fixed. For heavy quarks this can be done using , suggesting that contributions from quarks in HPQCD 0910.3102 m (m ,n = 5)(GeV) the sea are reliably estimated using perturbation theory (as using simulations with nf =4sea quarks: u, d, s and c. Our b b f the continuum QCD perturbation theory for the vec- expected). Our c mass is about 1.8 lower than the re- new results, HPQCD (pert) 0511160 tor current-current correlator moments. The multiplica- Future: Accurate tests of Higgs bb tive renormalisation factor Z is simply determined by cent result from the ETMC collaboration, also using nf = V 75 80mc(385 GeV,n90f =95 4) = 0.9851(63) GeV (52) 4 simulations but with a different method [36]: they get requireFIG. 7: Lattice halving QCD results uncertainty for mb in the MS onscheme m!b with matching the lattice result at a given lattice spacing for ↵ (M ,n = 5) = 0.11822(74), (53) 5 flavours and evaluated at its own scale. Results are from a specific moment to the perturbative result. We can mc(mc)=1.348(42) GeV, compared with our nf =4re- ms(3GeVMS ,nZf =f3) calculations that include either 3 or 4 flavours of sea quarks sult of 1.2715(95) GeV. and ↵ s ….. choose which moment to use, since di↵erences in ZV that agree well with our earlier results of 0.986(6) GeV and and so can be perturbatively corrected to 5 flavours. All 5 Our new result for the coupling (Eq. (53)) agrees with re- arise from a di↵erent choice are discretisation e↵ects that 0FIG..1183(7) 9., Lattice suggesting QCD determinations that contributions of the fromMS s-quarkc quarks mass in results use di↵erent methods, indicated on the right. The sults from other collaborations, who use different methods must disappear in the continuum limit, along with other thems(3 sea GeV are,n reliablyf =3) estimatedin MeV. These using masses perturbation come this theory paper and (as top result is from this paper, the second from [15], the third from us (and each other). Recent results (n =3or 4) are discretisation errors that result from working at a non- f references [32, 36, 44–46]c The gray band1. is8 the weighted average of from [64], the fourth from [18] and the fifth from [65], adjusted summarized in Fig. 7. expected). Our mass is about lower than the re- zero lattice spacing. The low moments, 4–10, are known these results: 84.1(5) MeV. perturbatively to nf = 5. The grey band gives the weighted 3 cent result from the ETMC collaboration, also using nf = through (↵ ) so are clearly to be preferred over higher We updated our earlier nf =3analysis [32] of the ra- average of the lattice results: 4.178(14) GeV. s 4 simulations but with a different method [36]: they get ones. ItO is convenient to use ratios of vector to pseu- tio mc/ms of quark masses using our nf =4data. This is a relatively simple analysis of data from Table II. Our new mThisc(m bringsc)=1 the.348(42) error belowGeV, 1% compared for the firstwith time. our n Valuesf =4 forre- doscalar current-current correlator moments since then sult of 1.2715(95) GeV. factors of the quark mass cancel [8]. value is: ms(µ, nf = 4) are smaller by about 0.2 MeV. Our new result suggested in [20] as being needed for a future accurate Our new result for the coupling (Eq. (53)) agrees with re- agrees with our previous analysis and also with other recent FIG.determination 9. Lattice ofQCD Higgs determinations couplings ofto thebb.MS Thes-quark value also mass mc(µ, nf ) sults from other collaborations, who use different methods When a nonrelativistic discretisation of the QCD quark = 11.652(65). (54) nf =3or 4 analyses: magreess(3 GeV well,nf with=3) determinationsin MeV. These masses from continuum come this paper meth- and action is used, neither the pseudoscalar nor the vector ms(µ, nf ) from us (and each other). Recent results (n =3or 4) are f references [32, 36, 44–46] The+ gray band is the weighted average of ods, for example using Re e results in the b region [49]. currents is absolutely normalised and the lattice current summarized in Fig. 7. It agrees well with our previous result 11.85(16), but is much 92.4(1.5) MeV HPQCD [32], theseThe results: method84.1(5) weMeV. have given here is applicable to other is only determined to a given order in a relativistic expan- We updated our earlier nf =3analysis [32] of the ra- more accurate. We compare our new result with others in ms(2 GeV) = 99.6(4.3) MeV ETMC [36], lattice formalisms for heavy quarks, for example that of sion. Hence the match to continuum QCD perturbation tio m /m of quark8 masses using our n =4data. This Fig. 8. c s >95.5(1.9) MeV Durrfet al [44], the Fermilab Lattice Collaboration [67]. Further deter- theory has both discretisation errors and relativistic er- is a relatively simple< analysis of data from Table II. Our new We obtain a new estimate for the s mass by combining our Thisminations brings of them errorb from below other 1% formalisms for the first would time. be Values useful infor rors, which are mixed by the higher dimension operators valuems is:(3 GeV) = 81.64(1.17) MeV RBC/UKQCD [45]. m (µ, n = 4) new result for mc/ms with our new estimate of the c mass :> thes long-termf are goal smaller of reducing by about uncertainties 0.2 MeV. Our in Standard new result used to implement corrections, and so we cannot simply (56) agreesModel with parameters our previous needed analysis for precision and also characterisation with other recent (Eq. (52), converted from nf =4): mc(µ, nf ) take a value of Z from the match for a specific moment. = 11.652(65). (54) noff =3 theor Higgs 4 analyses: boson. We compare thesem nonperturbatives(µ, nf ) results in Fig. 9, together In determining the normalisation of the current we 93.6(8) MeV µ =2GeV with an earlier perturbative determination from [46]. Acknowledgements We are grateful to the MILC ms(µ, nf = 3) = (55) can, however, make use of the fact that time-moments 84.7(7) MeV µ =3GeV. It agrees well with our previous result 11.85(16), but is much collaboration for the92.4(1 use.5) of MeV their configurationsHPQCD [32], and to ( Finally, we have also updated our previous (nf =3) non- with low moment number emphasise very short times more accurate. We compare our new result with others in R.m Horgans(2 GeV) for = useful99.6(4 discussions..3) MeV ComputingETMC [36], was done 8 in the current-current correlator and are therefore sen- Fig. 8. on the Darwin supercomputer> at the University of Cam- <95.5(1.9) MeV Durr et al [44], sitive to much higher internal spatial momenta within We obtain a new estimate for the s mass by combining our bridge as part of STFC’s DiRAC facility. We are grateful tom thes(3 Darwin GeV) = support 81.64(1. sta17)↵ MeVfor assistance.RBC/UKQCD Funding [45] for. the quark-antiquark pair (the overall momentum of the new result for mc/ms with our new estimate of the c mass :> (56) pair is zero) than higher moments are [15]. Thus, as the (Eq. (52), converted from n =4): this work came from STFC, the Royal Society, the Wolf- f son Foundation, NSF and DoE. moment number changes, the sensitivity to relativistic We compare these nonperturbative results in Fig. 9, together corrections changes. This is easily seen in an analysis of 93.6(8) MeV µ =2GeV ms(µ, nf = 3) = (55) with an earlier perturbative determination from [46]. the free case. At leading relativistic order, for vector or 84.7(7) MeV µ =3GeV. Finally, we have also updated our previous (nf =3) non- pseudoscalar moments, multiplying the free quark and ( Appendix A: Determination of ZV

The perturbative analysis of heavy-heavy current- current correlators is well developed in continuum QCD 5 Note also that, in a nonrelativistic formalism, the annihilation perturbation theory [39–43] and here we make use of that and scattering currents do not have the same renormalisation to normalise the lattice NRQCD vector current for bb an- factor Quark mass ratios HPQCD: 1408.4169 Obtained directly from lattice QCD if same quark formalism is used for both quarks. m m (µ) q1,latt = q1,MS Ratio is at same and for same n . m m (µ) µ f q2,latt ⇥a=0 q2,MS Not possible any other way ... 10 the ⇥ and ⇥ and the equation: mb/mc c mb /m *new* 2+1+1 with c sexp exp

) 1.4 mb(µ, nf ) mb w(mb , 0) = . (41) h

expphysicalexp u.d: mc(µ, nf ) mc w(mc , 0) 1.3 m c

0 1.2

It might seem simpler to ﬁt m directly, rather than m 0h ( 1.1 / the ratio w; but using w signiﬁcantly reduces the mh c 1.0

dependence (and therefore our extrapolation errors), and m h also makes our results quite insensitive to uncertainties 0 0.9 in our values for the lattice spacing. m 0.8 We parameterize function w with an expansion mod- m m eled after the one we used to ﬁt the moments: c 4 6 8 b

mh (GeV) N w 2 n w(m ,a)=Z (a) 1+ w / (42) h m n mb/mc =4.51(4) m FIG. 4: Ratio m /m divided by m /m (which we ap- ⇧ n=1 ⇤ h ⌅ ⌃ 0h h 0c c ↵ proximate by w(m ,a)/2 from our fit) as a function of m . N N c h am w am 2i 2 j The solid linem showsb the/ma=0s extrapolation= 52. obtained90(44) from our 1+ c h , ⌥mc/ms ij= 112 .652(65)m fit. This is compared with simulation results for our 4 small- i=1 j=0 h ↵ ↵ ⇥ ⇤ ⌅ est lattice spacings, together with=3 the bestm fits⌧ /m (dashedµ lines) allows 1% accuracy in m⌦ s (94.0(6)corresponding MeV) to those lattice spacings.6 The point marked by where, as for the moments, an “x” is for the largest mass we tried (last line in Table II);

this was not included in the ﬁt because amh is too large. i + j max(N ,N ). (43) am w

Coe⇤cients cij and wn are determined by fitting function VI. FROM WILSON LOOPS w(mh ,a) to the values of 2am0h/(amh ) from Table II. MS The fit also determines the parameters Zm(a), one for each lattice spacing, which account for the running of the bare quark masses between di⇥erent lattice spacings. In a recent paper [26], we presented a very accurate The finite-a dependence is smaller here than for the determination of the QCD coupling from simulation results for Wilson loops. Here we want to compare those moments, because the ⇥h is nonrelativistic [8], and the results to the value we obtain from heavy-quark corre- variation with m stronger (twice that of z(µ/mh = h lators. First, however, we must update our earlier anal- 3,mh )). So here we use priors ysis to take account of the new value for r1 [10] given c =0 0.05 (44) in Eq. (10) and improved values for r1/a [13] given in Ta- ij ± ble I. (The Wilson-loop paper uses some additional con- w =0 4 n ± figuration sets: from Table II in that paper, sets 1, 6, 9, Zm(a)=1 0.5 and 11 whose new r /as are 1.813(8), 2.644(3), 5.281(8) ± 1 and 5.283(8), respectively.) We have rerun our earlier with Nw =8. We again take Nam =30, although identical analysis, updating r1, r1/a, and the c and b masses. The results are obtained with Nam = 15. results are shown in Figure 5. Combining results as in the Our fit results are illustrated by Figure 4 which plots earlier paper we obtain a final value from the Wilson-loop the ratio m0h/mh divided by m0c/mc for a range of quantities of ⇥h masses. Our data for di⇥erent lattice spacings is compared with our fit, and with the a = 0 limit of our fit 2 (solid line). The fit is excellent, with ⇤ /22 = 0.42 for MS(MZ ,nf = 5) = 0.1184(6), (46) the 22 pieces of data we fit (we again exclude cases with amh > 1.95). Using the ⇥c and ⇥b masses from Sec- tion IV B, and Eq. (41) with the best-fit values for the with ⇤2/22 = 0.3 for the 22 quantities in the figure. parameters, we obtain finally This agrees very well with the result in the earlier paper, MS(MZ )=0.1183(8), but has a slightly smaller m0b 4.49(4) as a 0 (45) error, as expected given the smaller error in r1. This m0c ⇥ ⇥ new value also agrees well with our very di⇥erent de- m (µ, n ) = b f , termination from heavy-quark correlators (Eq. (38)). A mc(µ, nf ) breakdown of the error into its di⇥erent sources can be found in Table IV of [26] (reduce the r1 and r1/a errors which agrees well with our result from the moments in that table by half to account for the improved values (Eq. (36)). used here). B ⇡l⌫ semileptonicB semileptonic decay decays and V from Lattice QCD ! | ub| l Tree-level or loop

⌫ processes - need hadronic W form factors from lattice buVub QCD to obtain SM rate

B ⇡ Theoretical Motivation Tree-level: CKM d elements, Vub and Figure : B ⇡l⌫ exclusive decay process.+ l ! l Vcb

Z W l ⌫ d 2 2 2 2 Vub f+(q ) t Exp. b u dq / | b | | | d(s) Loop: test for new ⇡ 2 2 ⇡(K) 2 2 B µ B 2 µ µ MB M⇡ µ 2physicsM m µ ⇡ V B = f (q ) p +W p q + f (q ) q h | | i + B ⇡ q2 0 q2  d d q2 =(p p )2 = M2 + M2 2M M E Tree-level diagram B ⇡ Loop-levelB diagram⇡ B ⇡ ⇡ Ran Zhou (Fermilab) 09/03/2015 15 / 34

B-meson semileptonic decays through loop-level diagram (B K(⇡)l +l , B K(⇡)⌫⌫¯) ! ! Standard Model contribution is suppressed in the loop-level diagram. (Suitable processes to detect physics BSM) Studied by many experiment groups (BABAR, Belle, CDF, LHCb, B-factory etc.)

Ran Zhou (Fermilab) 09/03/2015 3 / 34 B ⇡l⌫ semileptonic decay and Vub ! | | Fermilab/MILC B ⇡l⌫ semileptonicarXiv: 1503.07839 decay and Vub !semileptonic decay | |

Combined fit to lattice and experimental results improves determination of Vub B ⇡l⌫ semileptonic decay and V ! | ub| 1503.07839 Figure : Combinednew fit result of lattice-QCD using form factors and experimental data.This work + BaBar + Belle, B → πlν (arXiv:1503.07839) The combined fit of experimental data and lattice-Fermilab/MILC 2008 + HFAG 2014, B → πlν 2 QCD databaryon significantly decay reduced in the form factor’s error at low q . RBC/UKQCD 2015 + BaBar + Belle, B → πlν lattice QCD Imsong et al. 2014 + BaBar12 + Belle13, B → πlν Ran Zhou (Fermilab) 09/03/2015 16 /HPQCD 34 2006 + HFAG 2014, B → πlν Λ → ν Detmold et al. 2015 + LHCb 2015, b pl Conclude: tension → ν BLNP 2004 + HFAG 2014, B Xul between exclusive UTFit 2014, CKM unitarity 1503.01421 and inclusive 3.2 3.6 4.0 4.4 3 |V | × 10 Figure : Combined fit of lattice-QCD form factors and experimental data. methods remains ub (arXiv:1503.07839) The combined fit of experimental data and lattice- Figure : Comparison of the Vub results from di↵erent determinations. (arXiv:1503.07839) | | QCD data significantly reduced the form factor’s error at low q2.

Ran Zhou (Fermilab) 09/03/2015 18 / 34

Ran Zhou (Fermilab) 09/03/2015 16 / 34 + Fermilab/MILC B ⇡(K)l l semileptonic decay decayarXiv: 1507.01618 Standard! Model predictions potential sensitivity to new l+

physics through loop ]

2 b Z effects l

9 + t b B ⇡` ` d(s)

)[10 ! ⇡(K)

e, µ B Lattice QCD gives SM rate:

= W `

( + + + 9 2 Br(B ⇡ µ µ) = 20.4(2.1) 10 dB dq ! d ⇥ -9 Ratios of B ⇡ll/LHCbB Kll : 18.3(2.5)x10observables Loop-level diagramarXiv: 1509.00414

) ! !

⌧ 80 =

` 70 ( 2 3 dB dq 60 ⇢, !, J/ 0 + 10

The B ⇡l l decay was⇥ 50 ﬁrst observed in 2012 by LHCb ) ) ` ` +

!2 2 + ` q ` 40 + + ⇡ (arXiv:1210.2645). K !

! 30 + + B B ( ( + B Left:RatioThe FNAL/MILC of branching ⇡ to B K partially⇡ functionlattice data of +B 20 exp the (arXiv:1503.07839)B ⇡µ µ process is ! 10 Standard Model Right: (arXiv:1312.2523): old FNAL/MILC B ⇡ !latticeLHCb data 2015 integrated rates 0 ! (arXiv:0811.3640)+ + HPQCD’s+ + B K lattice0510152025 data(arXiv:1306.2384) 8 (B ⇡Ran µZhou:µ KITP2015!)=[2.3 q2(GeV)0.6(2 stat.) 0.1(syst.)] 10 (3) +exp+LCSR+modelB ! ± ±+ ⇥ Figure : Ratio of partially-integrated branching ratios BR(B ⇡` `)/BR(B + ! ! Ran Zhou (Fermilab) + K+` `+) in the Standard Model+ using09/03/2015 the lattice+ form 22+ / 34 factors, compared The ratio of B ⇡withµ experimentalµ to measurementsB fromK LHCb(arXiv:1509.00414).µ µ is The ! errors in the Standard-Model results! are dominated by the form-factor + + + uncertainties. (B ⇡ µ µ)

B ! Ran Zhou (Fermilab) =0.053 0.014(stat09/03/2015.) 0 26. / 34001(syst.) (4) (B+ K +µ+µ ) ± ± B ! More detailed results are in arXiv:1509.00414.

Ran Zhou (Fermilab) 09/03/2015 20 / 34 Ktopipidecays

Process in which matter-antimatter asymmetry (CP violation) wasI discovered=1 (1984/2rule Nobel Prize, RBC/UKQCD Cronin and Fitch BNL) Quantitative theoretical understanding has been clouded for decades by the strong nuclear force; solveK with DiRAC. ⇡⇡ Decades-long puzzle of why K ⇡⇡ ( I = 0) 450 x more likely ! ! than K ⇡⇡(I = 2) ! 35 resolved by lattice QCD.

j π j 3 π j i ing low-mode deﬂation with 900 Lanczos eigenvectors [36] ANALYSIS AND RESULTS K ii π K i j π with the bagel fermion matrix package [37]. A complete

set of measurements' required 20 hoursÆ onS oneS half-rack of The K ππ matrix elements of the operators Qi can an IBM Blue Gene/Q computer [38]. This was in balance → FIG. 10: Dominantbeunexpected determined contractions from contributing the cancellation time dependence to Re( ofA of2 the): differentC three-1 (left) and C2 (right). with the 24 hours needed to generate four1502.00263 time units of point functions defined in Eq. (2): gauge field evolution on this same machine. −Eππ(tππ−tQ) −MK (tQ−tK ) 1505.07863 Jππcolour(tππ)Qi(t Qcharge)JK (tK ) = contractionse e 2.5 ⟨ ⟩ 9 0 Jππ(0) ππC ππ Qi(0) K K JK (0) 0 + . (3) C ×⟨ | | 1⟩⟨ | | ⟩⟨ | | ⟩ ··· 1 -C2 8 -C2 C1+C2 C1+C2 2 The ellipses represent contributions from the vacuum fi- SM test of direct 7 CP violation in ) )

6 nal state or excited kaon or ππ states.6 For the “split- 6 pion” operator Jππ(tππ), tππ is the time closest to tQ. 1.5 The operator normalization factors 5 0 Jππ(0) ππ and K ⇡⇡ ⟨ | | ⟩ K JK (0) 0 in Eq. (3), as well as the 4 energies MK and 1 ⟨ | !| ⟩ Eππ can be determined from the two-point functions: First lattice QCD 3 result for "0/" 2 Contraction (x10 † −EX (ta−tb)Contraction (x10 2 0.5 0 J (ta)JX (tb) 0 = e 0 JX (0) X (4) ⟨ | X | ⟩ ⟨ | | ⟩ !1 ! wherewithX =physicalππ or K.ForX kinematics= ππ! the contribution! 0 0 0 5 10 of the 15 vacuum 20 intermediate 25 state to 0 the left-hand 5 10 side 15 20 25 30 35 FIG. 1. Examples of the four types of diagram contributing to t t the ∆I =1/2, K → ππ decay. Lines labeled ℓ or s represent must be subtracted. Figure 2 shows the effective energy light or strange quarks. Unlabeled lines are light quarks. of the kaon and two-pion states in lattice units as ob- 3 Technically challengingFIG. 11: Cancellation - future: of dominant significant contributions toaccuracy Re(A2)onthe48 gainensembles with a K - ππ separationtained from of these 27 and two-point the 64 functions.3 ensembles The kaon with mass separation 36. is obtained from an uncorrelated fit using 6 t 32. For the more challenging I =0,ππ energy, we≤ perform≤ While exploiting our finite lattice volume to create a a correlated, single-state fit over the interval 6 t 25, final ππ state with E = M , we must ensure that this ππ K obtaining χ2/dof = 1.56(68). Consistent results≤ are≤ also limited volume does not distort the final result. Such ′ Using this ratio, wefound can from calculate a correlated, the electroweak two-state fit using penguin 3 t contribut25. ion to ϵ /ϵ, given by finite-volume corrections take two forms. The first are We find M =490.6(2.4) and E =498(11).Forthe≤ ≤ errors which fall exponentially with the lattice size and K ππ I =2state,′ EI=2 =573.0(2.9), all in units of MeV. For result from “squeezing” the physical states. Such errors ϵ ππ ω ImA2 later reference we also give results= for6 the.6(10)I =0and2,10−4, (67) are expected at the percent level if Lmπ 4. In our case, ◦ ≥ s-waveϵ phaseEWP shifts.≡ √2 Weϵ Re findAδ20 =23−.8(4.9)(1.×2) from Lmπ =3.2anderrors 7% may result [18]. ! " | | ≈ Eππ and the Lüscher quantization condition [41, 42], a value somewhat smaller than phenomenological expecta- The second are effects falling aswhere a power we of haveL, simi- used the values ω ReA2 =0.04454(12) and ϵ =2.228(11) 10−3 from [2]. lar to the discretization of the energy that we are ex- tions [43, 44]. HereReA the0 first error is statistical and the ≡ 2 | | × ploiting. Here we apply the Lellouch-Lüscher correc-′ second an estimate of the O(a )error.ForI =2we This value for (ϵ /ϵ)EWP is consistent with our◦ previously quoted value 6.25(44)(119) 3 will use δ = 11.6(2.5)(1.2) , a corrected version of the tion [20] which removes the leading 1/L error. This 2 − − × −4 continuum result obtained in Ref. [19]. approach requires our final ππ state10 to be[2]. an Finally, “s-wave” for ImA0 we find combination of the eight approximate single-pion mo- An important aspect of type 3 and 4 diagrams is the menta ( 1, 1, 1)π/L. Ensuring this s-wave symmetry quadratic divergence resulting from the quark loop. This 5 requires± care± when± constructing pion interpolating oper- contribution is the same√ as that′ from the operator dγ s ImA2 2 ϵ ϵ 2 5 −11 ImA =ReAwith a divergent coefficient| |(ms =ml)/a5.40(64). Since dγ s10is GeV . (68) ators to minimize cubic-symmetry violations introduced 0 0 ReA − ω∼ ϵ − − × at the quark level by G-parity boundary conditions. the divergence# of an2 axial current,$ its matrix element between initial and final states with equal four-momentum Essential to this calculation is the ability to define the will vanish and it will not contribute to a physical pro- seven independent, four-quark, latticeThe operators results which in Eqs.cess (67) such and as K (68)ππ. were However, obtained for matrix using elements our be- result for ImA2/ReA2 in → correspond to those in the continuumEq. (66). Eq. (1). If instead This wetween take zero-momentum ImA from our states calculation, with unequal Eq. energies, (64), and this combine it with the ex- is accomplished by using domain wall fermions (DWF). term may be2 20 larger than the other physical terms. × The chiral symmetry of DWF ensures that the pattern Even for an approximately−8 energy conserving amplitude, −5 perimental result ReA2 =1.4787(31) 10 GeV we obtain, ImA2/ReA2 = 4.73(58) 10 , of operator mixing is the same as that in the continuum it will contribute both× noise as well as an enhanced sys- − × ′ −4 −11 and we can follow well-established(ϵ procedures/ϵ)EWP = [12,6. 26]69(82)tematic10 errorand from Im enlarged,A0 = energy5.42(63) non-conserving,10 GeV. ex- to relate our operators to the continuum operators− in cited× state contamination.− We determine× the approx- Eq. (1). Specifically we apply the Rome-Southampton imate size of such an unphysical piece from the ratio 5 method [39] at µ =1.53 GeV, to introduce RI/SMOM ri = 0 Qi(tQ) K / 0 dγ s(tQ) K and then subtract, ⟨ | | ⟩ ⟨ | | ⟩ 5 normalization [26] and then use continuum QCD pertur- time slice by time slice, the operator ridγ s(tQ) [45]. This bation theory [40] to relate this normalization with the dramatically reduces the noise for the operators Q5, Q6, MS normalization used for the Wilson coefficients [3]. Q7 and Q8. Hadronic vacuum polarisation contribution to anomalous magnetic moment of muon (g 2)µ/2 B.Chakraborty et al, HPQCD: 1403.1778 differs between expt and the SM by 10 25(9) 10 *new physics*? ⇥ q Uncertainty dominated by that from µ HVP contribution calculated from q expt for + Re e Can we improve ahead of E989 run? On lattice, calculate : ↵ 1 a(f) = dq2f(q2)(4⇡↵Q2)⇧ˆ (q2) µ,HVP ⇡ f f Z0 vacuum very steep function, polarisation so small q2 dominates function 5

55.0 TABLE IV: Contributions to aµ from s and c quark vacuum B.Chakrabortypolarization. Only connected et al, parts HPQCD of the: vacuum polariza- 54.5 1403.1778tion are included. Results, multiplied by 1010, are shown for each of the Pad´eapproximants. 10 54.0 physicalQuark [1, 0] u/d1010 [1in, 1] sea1010 [2, 1] 1010 [2, 2] 1010 10 ⇥ ⇥ ⇥ ⇥ s 57.63(67) 53.28(58) 53.46(59) 53.41(59) ⇥

s µ 53.5 c 14.58(39) 14.41(39) 14.42(39) 14.42(39) a New - accurate 53.0 determinations quarks to g 2 is: of s HVP 52.5 s 10 0.005 0.010 0.015 0.020 0.025 from latticea = QCD. 53.41(59) 10 . (12) µ ⇥ a2 (fm2) The fit to [2, 2] Padéresults from all 10 of our configu- s 10 ration sets is excellent, with a 2 per degree of freedom FIG. 4: Lattice QCD results for the connected contribution to aµ = 53.41(59) 10 of 0.22 (p-value of 0.99). In Fig. 4HVP, we no compare new physics our fit the muon anomaly a from vacuum polarization of s quarks. aµ µ ⇥ with the data from configurations with ms/m` equal 5 Results are for three lattice spacings, and two light-quark and with the physical mass ratio. 1% accuratelat lat phys masses: m` = ms/5 (lower, blue points), and m` = m` The error budget for our resultHPQCD is given in Table III. (upper, red points). The dashed lines are the corresponding The dominant error, by far, comesPRELIMINARY from the uncertainty values from the fit function, with the best-fit parameter val- ETMC in the physical value of the Wilson1308.4327 flow parameter w0, ues: ca2 =0.29(13), csea = 0.020(6) and cval = 0.61(4). Total HVP from lattice 10 which we use to set the lattice spacings. We estimate the The gray band shows our final result, 53.41(59) 10 ,with lat phys ⇥ uncertainty from QED correctionsBenayoun to the vacuumet al polar- m` = m` , after extrapolation to a =0. 1210.7184 still not accurate enough, ization to be of order 0.1% from perturbationJegerlehner+Szafron theory [20], suppressed by the small charge of1101.2872 the s quark. Our re- Hagiwara et al TABLEbut III: significant Error budgets for connected progress contributions to the sults show negligible dependence (<1105.31490.1%) on the spatial muon anomaly aµ from vacuum polarization of s and c quarks. size of the lattice, which we varied by a factor of two. Also … the convergence of successive orders of Padéapproximant as ac µ µ 640 650indicates 660 670 convergence 680 690 700 to 710 better 720 than730 0.1%; results from Uncertainty in lattice spacing (w0, r1): 1.0% 0.6% fits to di↵erentHVP approximants10 are tabulated in Table IV. aµ x 10 Uncertainty in ZV :0.4%2.5% Note that the a2 errors are quite small in our analysis. MonteCarlostatistics: 0.1% 0.1% This is because we use the highly corrected HISQ dis- a2 0extrapolation: 0.1% 0.4% !QED corrections: 0.1% 0.3% cretization of the quark action. Our final (a = 0) result Quarkmasstuning: 0.0% 0.4% is only 0.6% below our results from the 0.09 fm lattices Finite lattice volume: < 0.1% 0.0% (sets 9 and 10). The variation from our coarsest lattice to Padéapproximants: < 0.1% 0.0% a = 0 is only 1.8%. We compared this with results from Total: 1.1% 2.7% the clover discretization for quarks, which had finite-a errors in excess of 20% on the coarsest lattices. Finally we also include results for c quarks in Tables III and IV. These are calculated from the moments (and er- mistuning of the sea and valence light-quark bare masses: ror budget) published in [20]. Our final result for the con- msea mphys nected contribution to the muon anomaly from c-quark x q q (9) vacuum polarization is: sea ⌘ phys q=u,d,s ms X c 10 val phys aµ = 14.42(39) 10 . (13) ms ms ⇥ xs phys . (10) ⌘ ms The dominant source of error here is in the determination of the ZV renormalization factors. This error could be For our lattices with physical u/d sea masses xsea is very substantially reduced by using the method we used for small. a2 errors from staggered ‘taste-changing’ e↵ects the s-quark contribution [26]. will remain and they are handled by ca2 . The four fit 2 parameters are aµ, ca2 , csea and cval ; we use the following (broad) Gaussian priors for each: III. DISCUSSION/CONCLUSIONS

s 10 a =0 100 10 µ ± ⇥ The ultimate aim of lattice QCD calculations of ca2 = 0(1) csea = 0(1) cval = 0(1). (11) aµ,HVP is to improve on results from using, for exam- + ple, (e e hadrons) that are able to achieve an un- Our ﬁnal result for the connected contribution for certainty of! below 1%. We are not at that stage yet. Conclusion • Lattice QCD results for gold-plated hadron masses and decay constants now providing stringent tests of QCD/SM. • Gives QCD parameters and some CKM elements to 1%. • Provides BSM constraints, tests of sum rules/HQET etc. Future • Lattice uncertainties being improved substantially e.g. with m u,d at physical value and finer lattices/higher statistics • Analysis underway by several groups of effects from QED and m = m e.g. BMW 1406.4088 mn-mp u 6 d • The range of methods and calculations is increasing “disconnected calculations”, glueballs, unstable particles .. Spares 3

B and Bs are fit separately; priors used in the fit are described in [11]. The amplitudes and energies from the 1.26 Set 1 Set 5 3/2 (0) Set 2 Set 6 fits are given in Tables IV and V. a ⇥q is the matrix 1.24

) Set 3 Set 7

(0) 3/2 (1) B element of the leading current J0 and a ⇥q that of M 1.22 Set 4 Set 8 (1) (2) B f

J0 and J0 , whose matrix elements are equal at zero ( /

) 1.20 s

meson momentum. Notice that the statistical errors in B ⇥ do not increase on the physical point lattices, because M

s 1.18 B f

they have such large volumes. ( We take two approaches to the analysis. The ﬁrst is 1.16 to perform a simultaneous chiral ﬁt to all our results for 1.14 ⇥, ⇥s, ⇥s/⇥ and MBs MB using SU(2) chiral pertur- Physical point bation theory. The second is to study only the physical 0.00 0.05 0.10 0.15 0.20 0.25 M2/M2 u/d mass results as a function of lattice spacing. ⇥ s For the chiral analysis we use the same formula and

priors for MBs MB as in [11]. Pion masses used in FIG. 1: Fit to the decay constant ratio Bs /B .Thefit the fits are listed in Table V and the chiral logarithms, result is shown in grey and errors include statistics, and chi- 2 ral/continuum fitting. l(M), include the finite volume corrections computed in [18] which have negligible e⇤ect on the fit. For the decay constants the chiral formulas, including analytic 2 terms up to M and the leading logarithmic behaviour, 0.55 are (see e.g. [19]): fBs MBs 2 /

3 0.50

2 2 ) ⇥s = ⇥s0(1.0+bsM/⇥) (5) GeV 2 2 ( q fBMB M 1+3g 3 2 B 0.45 ⇥ = ⇥0 1.0+bl 2 + 2 l(M) (6) M ⇥ 2⇥ 2 q B Set 1 Set 5 ⇥⇥ f 0.40 Set 2 Set 6 The coe⇧cients of the analytic terms bs,bl are given Set 3 Set 7 priors 0.0(1.0) and ⇥0, ⇥s0 have 0.5(5). To allow for Set 4 Set 8 0.35 Physical point discretisation errors each fit formula is multiplied by 0.00 0.05 0.10 0.15 0.20 0.25 2 4 2 2 (1.0+d1(a) + d2(a) ), with =0.4 GeV. We ex- M⇥ /Ms pect discretisation e⇤ects to be very similar for ⇥ and ⇥s and so we take the di to be the same, but di⇤ering from FIG. 2: Fit to the decay constants Bs and B . Errors on the the di used in the MBs MB fit. Since all actions used data points include statistics/scale only. The fit error, in grey, 2 here are accurate through a at tree-level, the prior on includes chiral/continuum fitting and perturbative errors. Lookd1 atis error taken budgets to be 0.0(3) to see whereas how dthings2 is 0.0(1.0). will improve The di are in future ... allowed to have mild mb dependence as in [11]. The ratio 1302.2644:⇥ /⇥ is calculation allowed additional of B, Bs lightmasses quark and massdecay dependent constants 3/2 3/2 s 0.520(11) GeV , ⇥B =0.428(9) GeV , ⇥B /⇥B = discretisation errors that could arise, for example, from s errors divided into extrapolation and other systematics: 1.215(7). For MBs MB we obtain 86(1) MeV, in agree- staggered taste-splittings. ment with the result of [11]. Figs 3 and 4 show the results of fitting M M Bs B Error % B /B MB MB B B and decay constants from the physical point ensembles s s s EM: 0.0 1.2 0.0 0.0 only, and allowing only the mass dependent discretisation a dependence: 0.01 0.9 0.7 0.7 3/2 terms above. The results are ⇥Bs =0.515(8) GeV , chiral: 0.01 0.2 0.05 0.05 3/2 ⇥B =0.424(7) GeV , ⇥B /⇥B =1.216(7) and MB g: 0.01 0.1 0.0 0.0 s s MB = 87(1) MeV. Results and errors agree well between stat/scale: 0.30 1.2 1.1 1.1 the two methods and we take the central values from the operator: 0.0 0.0 1.4 1.4 relativistic: 0.5 0.5 1.0 1.0 chiral fit as this allows us to interpolate to the correct total: 0.6 2.0 2.0 2.1 pion mass. Our error budget is given in Table VI. The errors that are estimated directly from the chiral/continuum fit are TABLE VI: Full error budget from the chiral fit as a per- those from statistics, the lattice spacing and g and other forcentage different of the quantities final answer. different systematics are importantchiral fit parameters. The two remaining sources of error in the decay constant are missing higher order corrections The results of the decay constant chiral fits are plot- in the operator matching and relativistic corrections to ted in Figs. 1 and 2. Extrapolating to the physical the current. We estimate the operator matching error by 2 point appropriate to ml =(mu + md)/2 in the absence allowing in our fits for an amb-dependent s correction to 0 of electromagnetism, i.e. M = M ,wefind⇥Bs = the renormalisation in Eq. 4 with prior on the coe⇧cient