Lattice QCD at Fermilab Celebrating the career of Paul Mackenzie FNAL 7-8 November 2019

The History of Lattice QCD

Akira Ukawa Professor Emeritus and Fellow Center for Computational Sciences, University of Tsukuba Director Center for World Premier International Research Center Initiative JSPS v The beginning v Exploring the new frontier v Building the instrument v Getting nimble with quarks Physics v Putting them all together

Algorithms Machines

- 1 - The beginning

- 2 - Wilson 1974 n Preprint CLNS-262 (February 1974) n Gauge theory on a space-time lattice n “Wilson loop” as order parameter n 2 Confinement for large values of coupling g0 ≫ 1

See Wilson’s Lattice 2004 talk for a personal historical account of the discovery - 3 - Cornell 1975

Grad students on 5th floor Paul Mackenzie Steve Shenker Serge Rudaz Junko Shigemitsu Belal Baaqui Michael Peskin

I was on 3rd floor.

Laboratory of Nuclear Studies Cornell University

Faculty Ken Wilson John Kogut Tun-Mow Yan Tom Kinoshita Don Yennie Kurt Gottfried - 4 - Early reception

200 #citations/year of Wilson’s 1974 paper 150

100

50 Total citation to date : 3787 (WoS data base) 0 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 n The paper attracted much interest, but in fact, there was little progress in the first 5 years….. n strong-coupling expansion of hadron masses (Kogut et al) did not lead to anywhere n “theoretical ideas” like “duality” , “monopole condensation” (‘tHooft, Polyakov …) attracted more interest - 5 - Creutz 1979

n Preprint-79-0919(BNL)(September 1979) n Computer evaluation of Wilson loop (VAX 11/780!)

2314 n Non-zero string tensionMICHAEL for smallerCREUTZ values of the coupling consistent with asymptotic freedom IO behavior of Eq. (3.22) occurs rather sharply over a range of about 10% in P about P =2. This appear- Earlierance of forthe Z(2)confinement :Creutz-Jacobsmechanism-Rebbi, PRLoccurs 42, at1390 (1979) σ a2 Also for S(2):Wilson, Cargese Summer Institute (1979) = 0.16. (5.1) 4p l. PHYSICAL REVIEW 0 VOLUME 21, NUMBER 8 15 AP RI L 1980 o The rapid evolution out of the perturbative regime may be responsible for the remarkable phenome- Monte Carlo study of quantized SU(2) gauge theory a K nological successes of the bag model. " High- temperature-seriesMichael Creutzresults, "as well as semi- Department ofclassicalPhysics, Brookhaventreatments,National' haveLaboratory,alsoUpton,suggestedNew Yorkan11973 abrupt onset(Receivedof confinement.24 October 1979) O. I Using Monte Carlo techniques,Our analysiswe evaluate allowspath integralsa determinationfor pure SU(2) ofgaugethe fields.re- Wilson's . regularization procedure onnormalizationa lattice of up to 10'scalesites controlsof the ultravioletcouplingdivergences.in termsOurofrenormalizationthe prescription, based on confinement, is to hold fixed the string tension, the coefficient of the 4 string tension. Using the observed asymptotic asymptotic linear potential between sources in the fundamental representation of the gauge group. Upon reducing the 2 cutoff, we observe a logarithmicnormalizationdecrease of the bare coupling constant in a manner consistent with the g0 perturbative renormalization-group prediction. This supports the coexistence of confinement and asymptotic freedom for quantized non-Abelian fields. 0.0 I gauge (5.2) 0 |.O 2.0 - 6 - 4 2 eo we can solve for e,' to give FIG. 6. The cutoff squared times the string tensionI. INTRODUCTION regime. Such a transition is essential for the lat- function of The solid lines are the strong- and eo 37T as a P. ~-0 tice formulation of conventional(5.3) electrodynamics weak-coupling limits. Gauge theories currently dominate 4wour under-11 ln(1/aA)'where photons and electrons exist as free parti- standing of elementary particle'physics.where theIndeed,renormalizationcles. scale is low P =2.1 only loops of sidewe1nowand conceive2 are signifi-that all interactions represent Renormalization-group analysis implies that for cantly different from zero soramificationswe must includeof underlyingthe local symmetries. The 6m' short-distance1 phenomena the effective coupling of A=@ Kexp — = v K. (5.4) loop of side 0 in the fit. BelowelegantP =1.inclusion6 only ofthethe strong nuclear force into 11 non-Abelian200 gauge theories becomes' small and loop of side 1 is significant thisand picturewe assumedemandsthe the phenomenon of confine- perturbative results become valid. If this "as- Thus we see the appearance of a rather large di- area term C dominates. Fromment;Eq.indeed,(3.21)physicalwe iden-hadrons should be gauge- ymptotic freedom" is to arise in the confining bound of the fundamentalmensionless andnumber. The uncertaintyWilson's formulation,in this co- then four space- tify singlet states quark phase of gluon degrees of freedom. At ourefficientpresent islevelroughlyof a timefactordimensionsof two becausemust beofinadequatethe to support the knowledge, confinement(4.4)appears largeto playcoefficienta role in thetransitionexponential.of Ref.The7. Asrenor-evidence for this, Migdal solely for the unbroken non-Abelianmalizationgauge theorymass shouldhasbepresentedstrongly andependentapproximateon nonperturbative re- In Fig. 6 we summarize these results by plot- of the of strong interactions. both the gauge group andcursionadditionrelationof quarks.between different values ting a'K versus p. Here we alsoTheoreticalplot theevidencestrong- for quark confinement by cutoff parameter. ' He finds a close analogy We have shown the onset of asymptotic freedom coupling result of Eq. (3.24)gaugeand fieldsthe weak-couplingis remarkably sparse. Renormaliza- between d-dimensional gauge theories and (d/2) for the bare coupling constant in a renormaliza- conclusion of Eq. (3.22) withtion-groupan arbitrarilyargumentschosenimply that perturbation -dimensional nearest-neighbor spin systems of tion scheme 'based on confinement. This is normalization. From P =1.theory6 to 1.may8 webeplotinapplicableboth at large distances, statistical mechanics. On this basis he concludes strongly suggestive' that SU(2) non-Abelian gauge the least-square fit and the thusresultdismissingof assumingthe lack of perturbative evidence that four dimensions represents a critical case for confinement of quarks. ' Studiestheoryof largesimultaneouslyor- whereexhibitsgaugeconfinementtheories basedand on non-Abelian groups pure area-law behavior. For P =2.2 and 2.25 we ders in the weak-coupling expansion'asymptoticas well freedom.as onlyFurthermore,possess the byconfiningreproduc-phase, whereas the plot fits including and not including the loop of semiclassical treatments" all suggesting theimportantasymptotic-freedomAbelian prediction,group U(l) ofweelectrodynamics possesses side zero. Above P =2.5 the area law is too sub- nonperturbative effects in non-Abelianstrengthengauge ties betweena peculiarthe latticetransitionformulationsimilar andto that occurring in dominant relative to the perimeter law for accur- "XY" ' theories. the more conventional theperturbativetwo-dimensionalapproaches model. ate determination. As each temperature is treated Monte Carlo techniques have Any true nonperturbative analysisto gaugerequirestheory.a Recently, proven independently of the others, meansthe fluctuationsof controlling appar-the ultraviolet divergences of to be a powerful nonperturbative tool for analysis ent in this figure represent fieldthe statisticaltheory in a mannererror independent of Feynman of quantized gauge fields. "'" We have seen clear of this analysis. diagrams. Wilson's formulation of gauge theory onACKNOWLEDGMENTSconfinement-spin-wave phase transitions for U(1) a lattice provides such a cutoff scheme. ' This par- lattice gauge theory in four space-time dimensions I have benefited from many interesting dis- V. DISCUSSIONticular regulator also preserves an exact: local and for the SU(2) theory in five dimensions. " In symmetry. With the cutoff in place,cussionsWilsonwithde- B. Freedmancontrast,andthisR. transitionSwendsen. appearsThis to be absent for Note that the changeover rivedfrom athestrong-couplingstrong-coup- expansionresearchin terms wasof performedthe four-dimensionalunder Contract No.SU(2)EY-model. These results ling behavior of Eq. (3.24) toquarksthe weak-couplingconnected by strings. In 76-02-0016this picture con-with the U.supportS. Departmentthe MigdalofargumentsEnergy. on the existence of finement arises naturally; however, to take the phase transitions; however, the observation of continuum limit one must leave the strong-coupling first-order transitions in Z„Z„and Z, lattice domain and the expansion could fail to converge. gauge theories rather than the predicted second- Balian, Drouffe, and Itzykson have presented ar- order critical points shows that Migdal's approxi- D. Gross and F. Wilczek, Phys.gumentsRev. Lett.that in30,a 1346sufficient numberM.ofBoth,space-timePhys. Bev. Lett.mate36,recursion768 (1976);relations36, 1161(E)may misidentify the na- {1973); Phys. Rev. D 8, 3633dimensions,(1973); H. D.thePolitzer,lattice theory will (1976);exhibit Y.a phaseP. Yao, ibid.ture36,of653the(1976);transition.T. Appelquist," Phys. Bev. Lett. 30, 1343 (1973).transition between the strong-couplingM. Dine,regionandofI. Muzinich, InPhys.this paperLett. we69B,extend231 (1977).our analysis of the four- 2T. Appelquist, J. Carazzone, confinementH. Kluberg-Stern,and theandweak-coupling L.perturbativeN. Lipatov, Zh. Eksp.dimensionalTeor. Fiz. SU(2)Pis'matheory.Bed. 25,Working on lattices of

21 2808 VOLUME 47, NUMBER 25HamberPHYSICAL REVIEW -LETTERSParisi21 DECEMBER, Weingarten1981 1981 data for m between 0.3 and 0.05 (see Fig. 2), periment, and this would have given us roughly where all the masses are between the low-energy the above value for the lattice spacing and v'T cutoff at 590 MeV and the high-energy cutoff at =400+ 50 MeV. The bare quark masses turn out 3520 MeV, one finds at P = 6 to be m ~' = 6.0m/a, m v' = 0.5/a'+ 6m/a, (3a~) ""(m„+m„)= 8 MeV, n Computer evaluation of hadron(lo) masses (MeV) (VAX11/780) mz'=0. 8/a +6m/a, m„=1.2/a +6m/a, (8) (3n, ) ""(m,+m, ) = 1OO Mev, m„=0.8/a+7m, mg= 1.1/a+5m. which then gives 3, 5, and 100 MeV for the u, d, and s invariant quark masses, in agreement with Here m is, in Wilson's notation, (k, —k)/2k, ' It was a feat totallyprevious phenomenological unthinkableestimates. ' The re- in traditional field theory with k = 1/M, and k, = 0.156 at p = 6. These re- sults at z =0 and = 1 seem sults mere obtained with fifty different configura- r therefore to be com- within tions at several different values for the bare patible, statistical errors. However, the = 1 quark mass, and the error in the masses is of y approach seems to be more promising for order 10%. the study of the spectrum of hadrons since the Using a '= 1120 MeV we get the following esti- separation of operators with different spins and can be done on mates (in MeV) (we use as input m, = 140 MeV): parities a single site. Using the large number of configurations generated for the = = 800+ 100, gyes ~ 950 + 100, fermions, we have also estimated the lowest glue- = = ball mass by extracting the plaquette-plaquette m ~ 1000+ 100, ~ ~ 1300+ 100, (9) correlations from finite-size effects" at ii = 6, m„= 1200+ 100, f, = 95+ 10. with the result Ana, logous estimates can be obtained for strange m~ = (3.6+ 0.5)A = 1500+ 200 MeV. mesons and baryons. We could have alternatively We plan to improve further the accuracy of the chosen to fit both the n and the p masses to ez- 11 February 1982 spectrum calculation by increasing theVolumestatistics. 109B, number 1,2 PHYSICS LETTERS Calculations of the spectrum on larger lattices are in progress in order to check the smallness relaxation parameter. For both 84 and 124 lattices O. I 40 O. I 45 O. I 50 O. I 55 of finite-size effects on the lattice. A compari-1.2 m 2 and m~r over the K range we considered were I I I son between the results on a 5'x 8 and a 6'& 10 found to be consistent with linear functions of K to lattice seems to show that these effects are small 1. an accuracy of 2% or better. On each lattice a single when the pion mass is not too close to the high- set of gauge configurations was used for all values of m or =low-energy800 ±cutoffs.100AMmoreeVdetailed account 5.0— —2.0 ρ K to minimize point-to-point statistical fluctuations. of the spectrum calculation will be published0.8 else- m 670 100MeV ρThe= results we have± quoted were obtained from our 2 where. We are also considering the possibilityme M linear fits. The errors we have given were determined mpof =introducing950 ±the1effects00Mof closedeV fermion0.6 loops. """ One w'ay of doing it mould be to use by repeating our procedures on subensembles of the 2.0— —I.O chirally invariant fermions at x =0 with one com- set of gauge configurations. The total computer time N 0.4 ponent per site for the fermionic loops and the required to obtain the points in fig. 1 is equivalent to noninvariant formulation at y = 1 for the computa- approximately 50 hours on a CDC 7600. 0.2 tion of the propagators. The solid line in fig. 1 is the prediction for mp ob- A — 0.0 The authors thank M. Creutz, J. Kogut, and tained by applying our tuning process to mp and mTr K. Wilson for fruitful discussions, and theO. organ- I I I I I I I I 0 2 4 6 8 I0 12 14 16 18 given by Wilson's leading-order strong-coupling ex- izers of the workshop at the University of Califor- N pansion in ref. [12]. The value ofmp shown in fig. 1 nia, Santa Barbara, for their hospitality where for a 124 lattice differs ;significantly from the leading- I part of this work was done. One of usFig.(G. 1.P. The) is rho mass as a function of N with the pion mass and - 7 - 0.3 0.2 Q. l 0.0 grateful for the hospitality at Brookha, stringven Nation- tension tuned to their physical values. The physical lat- order strong-coupling calculation. Fig. 1 shows some mQ al Laboratory where this work was completed.tice size Na is fixed at (89 MeV)-1 • tendency for mp to approach a finite limit as the lat- FIG. 2. Meson masses squared and baryon masses The authors would also like to thank A. Omero tice spacing (N × 90 MeV) -1 goes to zero. The value as a function of k and the bare quark mass = m (&, and G. Martinelli for pointing out an early error —k)/2&, obtained with use of the Wilson fermion action Regge slope of (938 MeV-1) 2. The values of (K,/J) of mp begins to depart appreciably from the strong- in our thanks (x = 1) atP = 6. I', V, S, and A stand for pseudoscalar program. Finally, go alsocorrespondingto the to these results are (0.2181, 1.0515) coupling line near N = 8 corresponding to/~ of 2.25, = isabelle group and the Rome University Istituto (J 0 +), vector (1 ), scalar (0++), and axial vector forN = 4, (0.1687, 2.25) forN= 8, and (0.1545, which is the same value of/3 at which the string tension (1++) masses. N and 4 stand for nucleon (~+) and de1ta Nazionale di Fisica Nucleare group for use of (2+) masses. their VAX/11780. 2.431) for N = 12. We have not quoted errors on the [2] switches from strong-coupling to weak-coupling pion mass and string tension since statistical fluctua- behavior in pure gauge theories. The values of the crit- 1794 tions in our tuning process should be interpreted as ical hopping constant K c, on the other hand, are con- giving rise to errors in (K,/~) and corresponding errors sistent with the predictions of ref. [14]. For the 84 in the rho mass prediction. The critical values of K at and 124 lattices, K c is within 10% of the large-~ calcu- which the pion mass becomes zero are 0.220 --- 0.002 lation of ref. [14]. for 44, 0.169 -+ 0.004 for 84, and 0.155 +- 0.003 for Given that our results for mp are not too far from 124 . the predictions of the strong coupling expansion, it These results were obtained from eight gauge con- seems reasonable to assume the sensitivity of our mp fi.gurations for the 44 lattice and four configurations to changes in the value of the lattice periodicity, Na, for ithe 84 and 124 lattices. To minimize correlations will be roughly the same as the sensitivity of the among the U we carried out 128 Monte Carlo sweeps strong coupling prediction to Na. If so, our choice of between each configuration on the 44 lattice and 512 Na = (89 MeV) -1 is more than sufficient for stability. between each on the 84 and 124 lattices. Thecalcula- The strong coupling expansion in finite volume yields tions for 44 used a modified Gauss-Seidel to be de- correlation functions which, if fitted to (14) and (I 5) scribed elsewhere, run at the physical hopping con- give mp and m~r independent of Na for all N large stant of 0.218 I. For 84 and 124 the Gauss-Seidel of enough to measure a mass (N ~ 2). eq. (10) [5] was used. As we have already mentioned, In conclusion, we believe we have found a tractable performing 100 to 600 Gauss-Seidel sweeps of the method for extracting a variety of physical predictions lattice with a relaxation parameter between 0.7 and from realistic lattice gauge theories including fermions. 1, we obtained mp and m,r stable to better than 0.5%. A convincing test of QCD for low energy phenomena Masses were measured at K of 0.157, 0.161 and 0.1625 is perhaps not too far in the future. for 84 and 0.149, 0.1505, and 0.152 for 124 . ForK >0.163 on 84 and K > 0.153 on 124,the Gauss- I would like to thank the high energy experimental Seidel iteration failed to converge for any value of the physics group at Indiana University for time on their

61 Exploring the new frontier

We can understand all those puzzles of strong interactions! n Confinement n Spectroscopy of hadrons n Strong interactions at high-temperature n Weak interactions of hadrons n Heavy quark physics

- 8 - 2 Confinement for ∞ ≥ g 0 ≥ 0

2 n RG running of the coupling g (q) n Monte Carlo studies following up on K. Wilson (1979)

Volumen 211,Refined number 1,2 into the stepPHYSICS-scaling LETTERS method B in the 90’s 25 August 1988

.... l .... I q U' that maximizes ReTr(2?*U'). A simple way to 0.28 Aft rescaled Lo b x/3 check whether a finite P will lead to an improvement2 is to compare the behavior of different size loopsg on( q) 0.6 ! the first few levels. If the small loops need a much 3 4 5 4π g b g b! g 0.24 β ( ) = − 0 − 1 +! larger/(2 than large loops to match, then decreasing P awill be useful. Physically, by decreasing P one is trying to bring the behavior of short- and long-dis- tance physics0.20 closer together for RGT's that unduly <3 0.4 order the short-distance (non-universal) correla- tions at initial blocking steps. For the x~ RGT, the 0.16 short-distance physics gets relatively disordered, so P=oo turns out to be optimal. The Edinburgh group finds that for b=2 RGT P~ 30 is optimal and gives 0.12 matching one level earlier when Kt e [ 6.0, 6.6 ]. How- x,x b=x/3 MCRG /J b=2 MCRG ! ever, they also find that at weaker coupling their re- o21 suits show a strong P and loop-size dependence even 0.08 5 6 7 after three blocking steps.1 This is especially true at 10 K - 6/g 2 K~ = 7.2. Thus a simple one-parameterq[GeV] optimization Fig. I. ComparisonR. Gupta of etAft resultsal PLB211, of the b = x~ 132 and (1988)b = 2 MCRG seems to beM. insufficient. Luescher et al, NPB413, 481 (1994) - 9 - methods. The latter data have been rescaled to b= x/3. The lone Recently Land and Salmhofer [ 5 ] have compared point from a ((9xf3) 4 lattice is shown with a different symbol. different implementations of the b=2 RGT for The solid line is the two-loop perturbative prediction ofeq. [4]. SU (2). They examine matching as a function of the strengths with which paths of different length and to- continuous //-function over the range of couplings pology are summed to produce 27. They find that for [ 5.9-6.35 ], our conservative estimate for the begin- the construction used by the Edinburgh group one ning of 2a deviation from asymptotic scaling is needs more than three blocking steps to get a con- K=6.0. The results at K~ =7.25 and 7.5 are consis- verged Aft for K> 2.8. Note that, for SU (3), K~ 6.6 tent with asymptotic scaling. We performed 5000 corresponds to the same correlation length. measurements (50000 sweeps) at these two points The previous two arguments suggest that the sys- in order to estimate the statistical errors reliably. The tematic errors for the b=2 RGT are increasing at result at K~ = 7.25 corresponds to the range of cou- weak coupling. This could be an explanation for the plings [ 6.75-7.25 ], and thus our estimate for the on- deviation from asymptotic scaling observed at set of asymptotic scaling is K• 6.75. K~ = 6.9 and 7.2. The b= 2 calculation shows ~ 15% deviations from The important question is to what extend does the the perturbative result even at K~ = 7.2, i.e., over the x/3 RGT suffer from such systematic errors? Cer- range [ 6.7-7.2 ]. Even though the two MCRG results tainly it has advantages over the b= 2 RGT: the path are consistent within errors, the difference between lies symmetrically about the block link; a larger frac- them appears to be systematic. Thus it is worthwile tion of the degrees of freedom is incorporated into considering the differences between the two methods the block link; and the scale factor is smaller. In par- in some detail. ticular there is no need for an optimization along the The simplest way to preserve gauge invariance in lines of ref. [ 5 ]. Thus we think it is plausible that the the blocking procedure is to construct the block link x/3 RGT is more reliable. However, this is an asser- from the sum, X, of paths between the block sites. The tion that can be justified only by further tests. block link U' can then be selected with probability One possible problem with the x//3 data is that the exp[ -P Re Tr(XtU ' ) ], with P a free parameter to initial lattices are smaller than those needed for b = 2. be tuned to obtain matching after as few blocking This possibility has been studied for SU (2) [ 6 ], by steps as possible. For P= oo, this RGT reduces to the comparing results for Aft in the range K~ • [2.5, 3.5 ] 5-function construction implemented by finding the using a starting 184 lattice with those from a 94 lat-

135 Spectroscopy of hadrons n Ground stateR. Gu hadron p t a / Ha d r o n spect rmasses umfromthel at t i ce as a function of quark mass

1 .7 “Edinburgh/APE plot” 1 .6

Infinite quark mass Nucleon/rho Nucleon/rho

From R. Gupta, Latttice 89 review

Physical point (M,r /MP)2 (pion/rho)^2 nFi g.Experimentallyla: The APE mass pl ot for Wil sonunkown f er mion dat a . Dastates, t a at ,0 = 5.7 ise.g.,fromthe APE col l abor at i on on 12 3 x 24 l at t i ces (+) and 24' x 32 l at t i ces ( f ancy -f) l4%. The dat a at )3 = 5.85 is fromI wasaki et .al . on 16n3 xH48 dibaryonl at t i ces (o) and 243 x 60 l at t i ces ( f ancy o) f51. The dat a at 3 = 6.0 is al so fromthe APE col l abor at i on on 18 3 x 32 l at t i ces ( x) and 24 3 x 32 l at t i ces ( f ancy x) ( 41. The dat a at ,0 = 6.2 is fromthe LANL coln l aborMackenzie at i on on 183 x 42-Thackerl at t i ces ( oct agon) (no[91 1985)/Iwasaki. et al (yes 1988) (still undecided today)

n Pure glue spectrum, …… - 10 -

H H

Stagger cd fermions -

i I 1 I I - -0 .5 1 2 (Mn/MP)

Fi g . 1b : The APE mass pl ot for St agger ed fermiondat a. Da t a at Q= 5.7 is fromthe APE col l abor at i on on 24 3 x 32 l at t i ces ( f ancy f) 141. The r est of the dat a is fromthe St agger ed Co l l a b o r a t i o n ; 0 = 6.0 on 163 x 40 l at t i ces ( x) and 243 x 40 l at t i ces ( f ancy x) , and 8 = 6.2 on 18 3 x 42 l at t i ces ( oct agon) [6]. Volume 98]], number 3 PHYSICS LETTERS 8 January 1981

Uu(x,r)-exp lao.a Au(x, a r) @ SU(2), tice [8]. By virtue of the periodic boundary condition, the string is a closed loop, and L(x) is therefore gauge defined on the links {/a} at each site (a is the lattice High temperatures invariant. spacing). Then (1) is replaced by The expectation value of L(x) in the ensemble de- fined by (4) yields the free energy Fq of an isolated Z=fI1]~nksdU]expI-~p (1- ~Tr(UUUU)p)l, quark (relative to the volume) via O) e-~Fq = (L(x)) ; (6) where we have written the lattice action as a sum over nplaquettesDeconfinement {p} of the trace of the product of U's a review in of thepure elementary derivation gluon of the path in- system? around each plaquette, dU represents the invariant in- tegral (1) makes this apparent. Further, the two-point tegration measure on SU(2) at each link. Our lattice is function of L(x) is related to the free energy V(R) of a q~ pair according to comprisedn of NHagedorn’s t × N 3 sites: N x, the number of sites limiting temperature (1965) T < T ≈ m along any spatial axis, is finite only as a matter of con- e-~ v(R) = (L(x)L(x + R)) . (7) c π venience, while the finiteness ofNt, the number of We will refer to V(R) as the q~ potential• sites along the time axis, is supposed to introduce ef- n There is a global symmetry operation on the system fects of nonzeroTheoretical temperature when N t ~N x. prediction by Polyakov(1978)/Susskind(1979) (4) which reverses the sign of L. To display it, it is A convenient order parameter with which to study convenient first to do a partial gauge-fixing. The gauge the phase structure of (4) is provided by an adaptation invariance of (5) shows that it is impossible in general of the Wilsonn loopFirst integral. Consider Monte Carlo in 1980 and rapid development in ‘80s to fix A 0 = 0 (i.e., U 0 = 1) everywhere; the best we Nt can do is Uo(x, ~-) = 1 for r 4= 1. Then L(x) = Tr Uo(x, L(x) = Tr [-I V0(x, r), (5) 1). Now it is apparent that the action is invariant r=l under the global transformation U 0 --> -U0, which VOLUME NUMBER 19 which is the trace of the product of U's along a 3time- transforms L -+ -L. 55, PHYSICAL REVIEW LETTERS 4 NOVEMBER 1985 oriented string running theS temporalU 2length 7of the lat-× 3 Thus (L) is reminiscent of the magnetization in a ( ) 100 I I I I I I I I I We thank Rechenzentrun der Universitat Karlsruhe, I I I T%0—LOOP SCALING Argonne National Laboratory, and the Florida State Supercomputer Research Institute for the use of their

0.4 ...... "..~ :~ deconfined machines. %'e thank J. Cavallini, K. Fengler, R. Hag- strom, P. Hanauer, M. Rushton, D. Sandee, and 0.2 ¢ . , , Continuum scalingA. Schreiner for their advice and assistance. We T would like to thank the Deutsche Forschungsgemein- :': .. c scaft for their generous support. We gratefuly ac- 0.0 --. :.. .~. . _

n Puzzles from the 60’s such as n n CP violation and Nuclear Physics B244n (1984) 381-391 ~) North-Holland PublishingPossibilities Company of lattice elucidation pointed out in 1984

VOLUME 55, NUMBER 25 PHYSICAL REVIEW LETTERS 16 DECEMBER 1985

Lattice Calculation of Weak Matrix Elements

WEAK INTERACTIONS ON THE LATI'ICE C. Bernard, T. Draper, G. Hockney, A. M. Rushton, and A. Soni University of California, Los Angeles, California 90024, and University of California, Irvine, California 9271 7, and N. CABIBBO VOLUME 53, NUMBER 14 PHYSICAL REVIEWArgonneLETTERSTati onal Laboratory, Argonne,1 OCTOBERIlIinois 604391984 (Received 17 October 1985) Dipartimento di Fisica, H Universit,~ di Roma "'Tor Vergata", INFN, Sezione di Roma, Roma, Italy CalculationWeofpresentWeaktheTransitionsfirst results fromin Latticea small-lattice (6 && 10) calculation of nonleptonic weak matrix = QCD G. MARTINELLI elements. The AI ~ rule is studied as a test case. For a lattice meson of approximately the kaon mass we find a significantly enhanced AI = 2 amplitude and a AI = 2 amplitude compatible with INFN, Laboratori Nazionali di Frascati, Frascati, Italy Richard C. Brower and Guillermo Maturana zero within our statistics. The dominance of the 4I = amplitude appears to be due to a class of Institute for Particle Physics, University of California, Santa Cruz, California 950642 R. PETRONZIO ~ graphs called the eye graphs. Qualitatively similar results are found whether or not the charm quark is integrated out ab initio. %e also report preliminary results on other weak matrix elements. CERN, Geneva, Switzerland PACS numbers:M. Helen13.25.+Gavela~'~m, 11.15.Ha, 12.35.Eq Received 5 December 1983 Department of Physics, Brandeis University, Waltham, Massachusetts 02254 (Revised 16 April 1984) A long-standing problem in low-energy hadronic operator-product expansion and renormalization group physics has been the calculationand of nonleptonic weak (OPE/RG), lattice weak-coupling perturbation theory, ' Prominent are = —, We show that lattice QCD can be used to evaluate the matrix elementsmatrix of four-fermionelements. RajanexamplesGupta the AI and chiral perturbation theory (CPTh). The OPE/RG CI'- operators which are relevant for weak decays. A first comparison between the resultsrule,Department obtainedwhoseof Physics,originon hasNortheasternremainedUniversity,obscure,Boston,andMassachusettsthe 02115is required because the characteristic scale for weak in- the lattice and other determinations are also presented. nonconservation parameters{Received 23eFebruaryand e' 1984)in neutral-kaon teractions is the mass Mlv —80 Gev of the II'boson, decays. In fact, accurate calculations of the relevant while the lattice ultraviolet- 12 - cutoff (dictated by low- We proposehadronicthe usematrixof Monteelements,Carlo simulationscoupled ofwithQCD existingto evaluate ex-hadronic energymatrix ele-hadronic physics and computer time) is ments of perimentallocal operatorsmeasurementsencountered in ofelectroweake and e',andcouldgrand-unified-theoryprovide m.transitions./a —3 GeV in our calculation (a is the lattice spac- 1. Introduction = PreliminarynontrivialMonte Carlotestsestimatesof the arestandardmade ofmodel.the AS Lattice2 matrix Monteelements responsibleing). Thusfor the 8'field has to be integrated out from the KL-Ks mass difference and the AS = 1 operators believed to explain the AI = —, enhance- With the advent of more powerful computers, which allow the studyCarlo of(MC) gaugetechniques offer a unique opportunity for the weak Hamiltonian. When performed in the con- ment. the performance of such calculations directly from the tinuum this procedure leads to two four-quark opera- theories on large lattices (103 ×20 or larger), lattice QCD is becoming a powerful fundamental theory. However, the efforts of the past tors (0+ ) in which the charm quark appears as an ex- tool in giving quantitative predictions for hadronic physics atPACS lownumbers: energy.few 11.years 15.Ha, suggest12.35.Eq, 13.that25.+ m,practical14.40.Aq difficulties often plicit field. One can also go further by integrating out In this paper we show that lattice QCD can be used to computeseriously the valuelimit of the accuracies attainable with these the charm quark (the penguin approach). 4 6 One then matrix elements of four-fermion operators which are relevant for weakmethods. interactions.It thus seems reasonable to begin by study- has six four-quark operators Ql —Q6'. Ql —Q4 are left- In any attempt at an accurate estimate of the all orders the lattice calculation. ing effects for which even qualitative resultsby can be left (LL ) operators; Qs, Q6 are left-right (LR ) opera- The methods developed here can be applied to the weak-interactionmatrix elementsamplitudes, of four-fermionthe difficulty ' lies in ' Some of these matrix elements are known experi- physically significant. The AI = —, rule, i.e., the em- tors. Since the charm mass is not very large the relia- the hadronic matrix elements, not in the short- mentally with high precision and their evaluation operators between bosonic states using the same quark propagators usedpirical for hadronicstatement that 4I = —, amplitudes are enhanced bility of this approximation is uncertain, but we study distance behavior of the gauge theory.' The initial, provides as good a test of QCD as the calculation of spectroscopy. If the propagators are available the computation can beover carriedthe AI out= —, amplitudes by a factor of —20, is one it anyway for purposes of comparison and because it intermediate, and final hadronic effects are the the hadronic spectrum. Conversely, once confi- may be useful for coarse-grained lattices. Note that using a small amount of computer time. such effect. Because theintui-enhancement is so large, an realm of nonperturbative QCD for which only dence is gained in these techniques,for matrixit becomeselementspos-relevant to CP nonconservation in As a first application of our methods we reporttive anand evaluationclever phenomenological inaccuracyof the matrixof orderestimates50%are(whichsibleis tonotsettleunusualmajor inquestions in standard weak- current MC calculations) need not mask its origin. the standard model the top quark, at least, must be elements, between pseudoscalar states, of operatorsknown of the(quark form:models, bag models, etc.). interaction phenomenology such as the Ko-Eo mix- The same machinery can, of course, also be 'used to integrated out by a penguin-type approach (since In principle, lattice gauge theories allow rigorous ing, the AI = —, rule, etc., as well as to make new compute other nonleptonic weak-interaction matrix m, &) a ). OLc = [6, y~'½(l - Ys)~2] [63 %,~1evaluation - Ys)~,],of nonperturbative QCD effects, when predictions for future measurements such as the ra- ' In order to use the standard results for the OPE/RG simulated by Monte elements.Carlo techniques.Here we reportIn this the firsttio e'/eresultsin CP-nonconservingfrom our decays, the proton from the continuum literature, a lattice weak-coupling OLR = [6, y~'½(l -- ")'5)~b2][63 Tt,½Letter(l + 3~5)~b4].we propose to latticeextend calculation.the use(1)of Monte Carlo lifetime in grand-unified theories (GUTs), etc. To calculation, relating matrix elements -of continuum simulations of lattice QCDTo relateto evaluatematrixtheelementshadronic amenablemake thisto aprogramMC calcu-more precise, we give a brief operators to their lattice counterparts, is required. The effective weak hamiltonian including short-distancematrix elements gluonthat effects,lationarise intocanweakthe beexperimentallyprocesses. FormeasureddescriptionE ofarmtheampli-phenomenological picture for two Such a relation has the following generic appearance expressed as a combination of operators of this formthe lattice[1-5]. cutoff m/a tudes,betweenthreethe keyW masstheoreticaland the ingredientscases. are used: the ' ' (sulTl otl J, A f ): In the next section we outline the method of computation.QCD scale AQcnIn sect.(Mg 3 »we m/adescribe» A&co), the lat- AI = —, rule. —The AI = —, decay amplitudesj for coll' tice provides a natural separation between[ 1 + (g2/hard-167r')and a ""+(g'/167r') a '"' = —, the numerical results based on 14 link configurations in a 103 x20 lattice, 0for which Z„(r, p,E ) ]0,~7t are known to Z2,dominate, (r, p, over) 0, the AI soft-momentum physics. The renormalization decay amplitudes by a factor of about 20. The ef- + (g'/ 16m') Z3 (r, pa )s I —ps) t'd [u y„t'u + d y t'd + s y„t's ]. ~ On leave from Dipartimento di Fisica, Universith di Roma group"La Sapienza",is used toandsum INFN,the hardSezionegluons di into the coef- fectivey„(Hamiltonian for the AS = 1 weak interac- Roma, Roma, Italy. ficients of an effectiveHeretheory,the whileZ 's arethe finitesoft-gluonrenormalizationtions withconstants,four quarkr is flavors and without strong- contribution to the matrix elements is summed to I 381 the Wilson parameter, and p, ~ is theinteractioncontinuumcorrectionsrenor- is, to first0 (gorder,) (such as s rr""I' „d) turn out to have malization point. The 0 (g ) terms can be very impor- anomalously large matrix elements. (4GF/J2)sin&, cos&,tant[dlevenyoul foruLy„siweak—dLy~cicouplingc„y~sibecausej, naive 0 (g ) LL We thus wish to calculate matrix elements of the operators can mix at this order with I.A operators type (rrvr ~si tuul'2d ~K), where I, represents an arbi- — whose matrix where G&- I/M&. In order to includeelementsthe QCDareeffectsoftendownmuchto alarger.scale p,Note,( M~, thetraryrenormalizationDirac matrix.groupThis is a four-point function and however, that the calculations suf- can be used to sum the leading logarithms between p,toand0 (gM~.) shouldFurther,be for p, ( m„practicalthe effectiveconsiderationsHamil- make it difficult to evaluate on tonian after integrationficientout ofunlessthe effectsnewof theoperatorscharm quarkwhichis4 appear only at the lattice. An approximation scheme for light meson =' — HAP( $ C (Ma/p„m, /p, )0 (x), (2)

1984 The American Physical Society Heavy quarks and CKM

n Kobayashi-Maskawa 1973 n Experimental discoveries: 1974 1977

170 n Lattice studies beganNuclear Physics B in(Proc. Suppl.)the 4 (1988) late 170-177 80’s North-Holland, Amsterdam

HEAVY QUARKS ON THE LATTICE Nuclear Physics B (Proc. Suppl.) 4 (1988) 199-203 199 North-Holland, Amsterdam E. EICHTEN Static approximation from Proceedings of Lattice 87 at Seillac Fermi National Accelerator Laboratory*, P.O. Box 500, Batavia, IL 60510

Abstract EFFECTIVE LAGRANGIANS FOR SIMULATING OF HEAVY QUARK SYSTEMS A method of treating heavy quarks is applied to lattice Q.C.D. for heavy quark masses (ran) and lattice spacing (a) satisfying the condition mHa > 1. Explicit applications to the measurement of heavy-light meson masses, decay constants, and mixing parameters are presented. Numerical results for B mesons are obtained onG.P. a 82 ×Lepage 16 × 24 lattice withNRQCD ~ = 5.7.

Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 1485S, U.S.A. ; 1. INTRODUCTION (R) can be defined andsimply D.A.M.T.P., as: University of Cambridge, Silver Street, Cambridge CBS 9EW, U.K. Heavy quarks play an important role in studying the ~(R) = - lim 1 In (W (R,T)) . (1.1) dynamics of Q.C.D. For quarks with masses (mjT) much B.A.T--.~ TThacker larger than the typical Q.C.D. scale of strong interactions The static energy has been precisely measured on the lat- their motion can be treated non-relativistically (AQ.C.D.), tice in the quenchedNewman approximation Laboratory to Q.C.D. s,4 of and Nuclear re- Studies, Cornell University, Ithaca, NY 14853, U.S.A. and the dynamics considerably simplified. cently measured in the full theory as welP. The agreement - 13 - The first systems involving heavy quarks to be ob- between theory and experiment is excellent. served were systems consisting of mesons with one heavy The leading spin dependent relativistic corrections to quark and one heavy antiquark QH, the charmonium 1. INTRODUCTION QH the dynamics of (QH(~H) systems can also be studied. (c~) and, more recently, the bottomonium (bb) system. In As shown by F. FelnbergMesons and containing the present heavyauthor 5, quarksthe pose high precision analyses of OED bound these systems, the Q.C.D. interaction between the quarks static energy arisesspecial as the problems leading term for (independent lattice simulations. of The states i, providing a very attractive alter- is well represented by a nonrelativistic potential. Phe- liras) in a systematicheavy expansion quark inis powersgenerally of lira nonrelativistic s for and native to traditional Bethe-SaIpeter tech- nomenological potential models provide an accurate de- the heavy quark propagators,as such its while dynamics retaining termsinvolves to or- three impor- niques. scription of the data for the (c~) and (bb) systems and der 1/m~ in the tantexpansion momentum allows the scales--itsdetermination mass,of M, its thus give empirical evidence for the use of nonrelativis- the leading spin three-momentum,dependent potentials for My,(QHQ, S)and sys- its kinetic en- 2. NONRELATIVIb~rlC OCD (NRGCD) tic dynamics. The form of the potential is determined tems. The evaluationergy, of Mythese 2. potentials When von << Ithe these lattice scales are The dynamical scale M is removed by from the data for distances between .1 and 1 Fermi.[1] At requires the calculationwidely of separatedcorrelations andof gauge one electricmust use a very replacing the usual Dirac theory of the shorter distances, perturbative Q.C.D. can be used to di- and magnetic plaquetteslarge lattice along the if Wilsonone is loop.to model These accurately heavy quark (OCD) by a nonrelativistic rectly compute the potential and, furthermore, the poten- calculations havestructure been done byat a numberall of ofthese groups scales.s. In the T Schrodinger-Pauli theory (NRCK;D), cut off tial (or static energy) ~(R) can be calculated from Q.C.D. The static energymeson, and spinfor dependent example, potentials v 2 is roughlymea- 1/10, at momenta of order M--i.e., one uses the on the lattice even in the non-perturbative region. sured on the latticesuggesting contain thethat information lattices aboutof order the 50 or i00 quark lagrangian density The relation between Q.C.D. and ~(R) in Euclidean masses and propertieson a of side QH(~s might states bewhich necessary. depends on This is de- space is made explicit through the Wilson Loop W(R, T) 2. the dynamics of Q.C.D.spite theThese fact known that potential the Tfunctions has a much Z(a) ~t(iDt D2 W(R, T) is defined to be the average over all field config- e/t = then determine thesmaller spectrum radius of excited than states ordinary as well ashadrons. urations of the trace of gauge field links around a rect- + _K_ the ground state withHere given we (jpc) describe since the a Schroedingerrenormalization-group angular path of spatial extent R and temporal extent T. Cl 2M ~ta'B~b equation may be strategyused. for dealing with this situation. We • Fermilab is operated by Universities Research Association, Inc. under contracttransform with the U.S. the Department usual relativistic of Energy quark the- + ___K__8M2 ¢t(c2iD.EX,+c3V.E)¢ ory into an equivalent nonrelativistic theory D 4 0920-5632/88/$03.50 © Elsevier Scmnce Publishers B.V. by integrating out all quantum fluctuations + +... (1) (North-Holland Physics Publishing Division) at momentum scales of order M or larger. The resulting theory deals only with dy- with a lattice spacing a of order 1/M or namics at nonrelativistic scales, and there- larger. As the quark energy no longer in- fore is far simpler to simulate. This same cludes its mass, space-time oscillations of approach has been used to great effect in wavelenght I/M are no longer important

0920-5632/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) How initial results looked in ‘89-’90

n Heavy-light decay constant n decay amplitude

E~ Eic~ten / B phvsics on the lattice 48.5 S.R. Shar pe/ Lat t i c e cal cul at i ons of el ect r oweak decay aü.r: : :~~es 159 Static limit mQ → ∞ I=2 channel st a :itial di sagr eement wi t h exper i ment . Thi s di sagr ee- .8 me n t wo u l d be due to twoef f ect s. Fi r st , the use of 3.0 th(: quenched appr oxi mat i on, and, second, an i ncor r ect ext r act i on of theval ue of AT' fromthe dat a. Such .6 an er r or o L is possi bl e because i sospi n is br oken by el ect r o- ;> ? magnet i sm, and by thenon- vani shi ng val ue of mu-md. O 2? Thi s can feed par t s of Ao into A2 . The cal cul at i on of 2.0≈ this feed-downis ver y di f f i cul t , as di scussed by Ber nar d and Soni 2 and emphasi zed by theELC gr oup. Fac- tors of twoef f ect s ma y be possi bl e, al t hough expl i ci t cal cul at i ons of some cont r i but i ons, such as n - 7r o mix- x 16 40 =6 .0 (BS) ing, find themto be at the10 - 20%l evel . Thi s is 2 a ver y unsat i sf act or y si t uat i on, one that cl ear l y needs t 0 243X40 =6 .0 (BS) mo r e wo r k . 0 0.6 16 6 =6 .2 (ELC) A year ago I wo u l d have accept ed this r esol ut i on . 0 1 2 3 4 S 0 0.2 0.4 0.3 0.8 However theTor ont o gr oup' s paper on FSI pr ovi des an al t er nat e expl anat i on. The cal cul at i ons of AKr do not i ncl ude theFSI depr essi on of A2 . The required f act or of 1.75is pr eci sel y of thema g n i t u d e expect ed . As for v' AZ1' t From Eichten’s review at Lattice 1990 Fi gurFrom e 8: Resul Sharpe’s t s for A2/ A! review 2 deduced at fromLattice 1989 the r esul t s for A?° " t , they ar e suspect for thefollowing - 14 - r easons. Fi r st , the ext r act i on of At e' t is bei ng done BK( 2GeV) = 0.7 conver t s into A2 = 1. 75Azx~t . wrongl y, because theFSI ener gy shi f t Ais not bei ng Anot her wa y of sayi ng this is that l owest or der i ncl uded . On e can t est for this for by l ooki ng car ef ul l y A2xpt by usi ng di f f er ent pi on dn~l~ seum~ ;- a¢ tlbe n~ al[ ~be D mesm n~s. aL3~t ~aid ~ T_/L ~ el.alL$~ qlel a~. Ch PT conver t s thephysi cal val ue intoa pr edi ct i on at theexponent i al f al l - of f s, and sour ce times. Second, if FSI ar e large, then therewi l l de~ --a I. B_ ~ eL d.~t ~e~ C. ll_ Nlem that BK( 2GeV) = 0.4. Thi s is larger than the val ue ~m~ dm~ is a ~ ~ di~mm~ l~ be a largecor r ect i on to l owest or der Ch PT, whi ch wi l l usual l y quot ed 0 by a f act or of fK1f* = 1.5. Thi s eL dL 3r/q~me~mlle~ Iileml~e~e~lp~lmem. show up as a sl ope in the cur ves in Fi g . 8. I do not di f f er ence is a hi gher or der ef f ect , but I thinkthechoi ce know whi ch si gn thesl ope shoul d have. Thi r d, if the dmml~ mm~ m~es. Jilrm d ~ 3 r..~. Ille~l~ I have ma d e is we l l mot i vat ed. In ei t her case, thereis shoul d a largedi scr epancy wi t h exper i ment , much larger than FSI mass shi f t has not been i ncl uded, ther esul t s theer r or s, whi ch ar e of a fewper cent . depend on thespat i al vol ume . some i ndi cat i on Asi mil ar di scr epancy is appar ent in thecal cul at i ons The BS r esul t s in Fi g . 8 do show UC1A Ip~p~ s~d cird~ Tke ~ md~ dram of thesecond and thirdf eat ur es, but thest at i st i cs ar e whi ch use Ar e" . The r esul t s fromt hese cal cul at i ons ar e . The ELC r esul t s at • ~ fmile ~e dki:ls (FSE) ar fm~e ~ J MH =Km ue-~ i ~ ~ d-- i shown in Fi g . 8 as a ratiotothephysi cal val ue . I have not yet good enough to be sur e however , showno evi dence for any FSI ef f ect s . scale viohlim elec~ The W l~127 kam onl y shown r esul t s wi t h r el at i vel y smal l er r or s and wi t h ß =5 .7, encour age fur- kid IJl~ tilde ;s no SillmT~nt vdome de- #>6.0. The BS r esul t s ar e fromthe Gr a n d Ch a l l e n g e The above specul at i ons ar e me a n t to t . r esul t s cannot cal cul at i on . The ELC r esul t s ar e ol d, 25 and come from ther st udy of FSI , and of Ar e' Present penden~ chnllbl tke spatid ~ thin 0-7 twoscenar i os I have j ust a smal l er l at t i ce . The ELC gr oup al so has a newr esul t yet di scr i minat e bet ween the this ~o L~ f,,,. Ako, the W Smp2~ od~ iomd fromtheir hi gh st at i st i cs cal cul at i on at /3 = 5.7. 1 do out l i ned . We shoul d, however , be abl e to set t l e cour se, the 15~ sc~ vioi~m hom,e = S.7 to ~0. G~ty not have thenumbers toal l ow me toadd it tothegr aph, quest i on in the next coupl e of year s. Of . combi nat i on of the I~s ~1 IdbeFSE b tee cmdm~m~ Ex l~p i~m~tsm~ kern. ~ ~ ~ d but their r esul t can be expr essed as A2 = 2. 0( 2) Azpt final concl usi on ma y we l l i nvol ve a opt i ons. ~lwne~ M The r esul t s do not yet have smal l enough er r or s to try t~ B meSOL ~ dE ~ betm~ d~ ~o the AI = 1/ 2 rule todi f f er ent i at e bet ween di f f er ent val ues of #. 4.3.3. Resul t s for Ao and Super f i ci al l y, bot h me t h o d s of cal cul at i on appear to The AI = 1/ 2 rule is really twopr obl ems : howto agr ee that A2 - 2A' " t . Thi s is theconcl usi on dr awn get A2 smal ! enough, and howtomake Ao largeenough . t~o~ m~. B~ ~ UCtA ~.~ n by the ELC gr oup . In this vi ew, thequenched r esul t We have seen how we must come up wi t h an ext r a for A2 is fairly we l l known, and appear s to be in sub- f act or of - 1.75 r educt i on to sol ve thef i r st pr obl em. da~ at I~er #. At i~se~ tl~e ~ re- mains unce~ An iml~0~! ~oo f~r Wl- son fern',ions has been devdoped~ wh~ d Pr~umlnmy su~s d d~b method w~e ~ help m understand~ the first order ~ cor- rc~ns. d~d ti~~ da ho~ht m~m ~ morn La,s--e pe~,~,at.~e conection Z~ for ~F~ in the •~,,q.a~ ~an dut c~ a ~ht h~---~ and that the static e~ective theory. value of fB .r~-=~,__q~d from thls ~ is .~,~ conslstent wikh the brp value rep0c~ us~ ~ • La~e 1/m~ corrections to the m~ --~ oo limit- stoic c.4Tect~ actlon. The stud s~ of ti~ ~ m~-

From S. SSharpe’s.R. Shar pe/ Lat t i c e cal culreview at i ons of el ect r oweak atdecay Latticeampli t udes 89 147

Wh a t ? Why? Wh o ? 6 Level Nucl eon ma t r i x el ements fir/MN- Wf a check MA NY 2 Axi al vect or ma t r i x el ements : gA .. . check S6mmer 7 2 EMformf act or s: GM(g2), .. . check Wil cox, Dr a p e r / L i u 8 2 Str uct ur e f unct i ons check Rossi 9 1 Ne u t r o n El ect r i c Di p o l e Mo me n t measur e OQCD Goksch 10 1 Heavy- l i ght mesons

.fD, f B, BD, BB DD and BB mixi ng Ei cht en, Ma r t i n e l l i 11 1- 2 D-+Kev, (B-+r ev) , .. . measur e Vc Vub EI Khadr a, 12 Sachr aj da 13 1- 2 D- + Ki r check Sachr aj da, Si mone 1 K decay and mixi ng ampli t udes BK ext r act 6frome Ber nar d, Ki l cup, `+ Ma r t i n e l l i 3 K-+7r r (DI =1/ 2 rule) check Ber nar d, Ki l cup, Ma r t i n e l l i 2 e I over - det er mine 6 Ki l cup, Ber nar d 2

Tabl e 1: Wo r k done on we a k ma t r i x el ements in the year pr ecedi ng Sept ember 1989

n Level 1: exploratory, setting up methods, resolve problems of principle n Level 2: larger scale calculation, reasonably small errors (10-20%) of thest at us of var i ous cal cul at i ons is gi ven in thel ast chi r al symmet r y, but rather is a number char act er i z- n Level 3: reliable quenched calculation with small errors col umn in the table. " Level 1" me a n s an expl or at or y ingQCD . Al most ever yone wh o has-ever15 -done a hadr on cal cul at i on, ma i n l y set t i ng up the met hods, and t r yi ng spect r umcal cul at i on has, as a bypr oduct , ext r act ed the to r esol ve pr obl ems of pr i nci pl e. In some cases, such decay const ant s . If we wa n t theer r or s on Rtobe, say, as thecal cul at i ons of kaon decay ampl i t udes, this ma y one quar t er of the exper i ment al val ue, so t hat we can we l l be the most di f f i cul t st ep . The numer i cal r esul t s def i ni t el y di st i ngui sh Rfromzer o, we need a 5%me a - fromthis l evel ar e to be consi der ed qual i t at i ve onl y . sur ement of fKand f r . Level 2 me a n s a larger scal e numer i cal cal cul at i on, wi t h Wh a t w e really wa n t to do, however , is di st i ngui sh r easonabl y smal l (10-20%) er r or s . The r esul t s ar e now Rin the full theory fromRin thequenched theory. quant i t at i ve, but theer r or s ar e too largetodi st i ngui sh Wh a t the quenched theory l acks is loops of f er mions . quenched fromfull QCD . Level 3 i ndi cat es a reliable Thi s me a n s that di agr ams such as Fi g . l a ar e missi ng. quenched cal cul at i on wi t h smal l er r or s . Progr ess dur i ng Par t of t hese di agr ams cor r esponds toloopsof 7r , Kand the l ast year has r esul t ed in ma n y of the cal cul at i ons 71 mesons, as i l l ust r at ed in Fi g . l b. It turnsout that al l movi ng up a l evel . Ther e ar e now lots of 2' s, and a thedi agr ams whi ch gi ve r i se tosuch loopsrequire inter- si ngl e 3. nal fermion l oops, and thus ar e absent in thequenched To cl ose thei nt r oduct i on I wa n t todi scuss thedi f - appr oxi mat i on . The lowmo me n t u m par t of t hese loops f er ence bet ween quenched and full QCD in a par t i cul ar can be cal cul at ed usi ng chi r al per t ur bat i on theory. The 15 exampl e: fKlf, My ai mher e is threefold: (A) I wa n t r esul t i ng chi r al logarithm(for mu =and = 0) is to reiterate that havi ng good quenched r esul t s is onl y )1011 m2 ,iri 2 the f i r st st ep- much important physi cs is l ost ; (B) I -1 = -0.75 1n (4~r f )2 Az wa n t tost r ess howaccur at el y we must do thequenched fK cal cul at i ons tosee di f f er ences fromfull QCD ; and ( C) (withf* =93 Me V ) . Thi s is theleadingtermin an ex- I wa n t toi nt r oduce chi r al loops and chi r al l ogar i t hms, pansi on in m2 . The scal e of the log is undet er mined, si nce I wi l l return tothemlater. but if we t ake A = mp, a scal e above whi ch chi r al loggi ves a The decay const ant s f, and fK ar e t he si mpl est per t ur bat i on theory cer t ai nl y f ai l s, then the of al l we a k ma t r i x el ements . I am i nt er est ed her e in cont r i but i on of -1-0.12toR. I t ake this as an i ndi cat i on the quant i t y R = fK/f,r - 1, whi ch is 0.22 exper i - that thequenched appr oxi mat i on ma y be l acki ng an ef - me n t a l l y . Thi s is a quant i t y whi ch is not pr edi ct ed by f ect whi ch cont r i but es roughly hal f of theexper i ment al Building the instrument

Clearly people needed (much) more computing power…..

QCD and parallel computing

Processor array Space-time continuum Space-time lattice quark fields on lattice sites CPU1 CPU2 qn

U nµ gluon fields CPU3 CPU4 on lattice links

n QCD is a local field theory; only nearest neighbor interactions n Computation on each CPU and communication between nearest neighbor CPUs, scalable to “infinite” lattice size

n An ideal case of massively parallel computation! 17 Microprocessors in the 70’s n 1971 Intel 4004 4 bit $60/chip n 1972 Intel 8008 8 bit $120 n 1974 Intel 8080 8 bit n 1974 Motorola 6800 8 bit $360 n 1978 Intel 8086 16 bit $320 n 1979 Motorola 68000 32 bit

In the 70’s, the key element for a parallel system became affordable for a reasonable pictures from Wikipedia price even for academics!

- 18 - Parallel QCD machines in the 80’s

CPU FPU peak

Columbia 1984 Christ-Terrano PDP11 TRW 16MPY Columbia-16 1985 Christ et al Intel 80286 TRW 0.25GF APE1 1987 Cabibbo-Parisi 3081/E Weitek 1GF Columbia-64 1987 Christ et al Intel 80286 Weitek 1GF Columbia-256 1988 Christ et al M68020 Weitek 3332 16GF ACPMAPS 1991 Mackenzie et al micro VAX Weitek 8032 5GF QCDPAX 1989 Iwasaki-Hoshino M68020 LSI-logic 64133 14GF GF11 1992 Weingarten PC/AT Weitek 1032/33 11GF n Parallel array of (commodity CPU + custom made FPU) n More or less hand-made by academics n Overtook vector supercomputers in speed in the late 80’s

- 19 - Order of the deconfining phase transition

n Columbia Group (N. Christ et al 1988): physical quantities has a jump, so first-order F. R. Brown et al, PRL61, 2058 (1988) n APE Group (G. Parisi et al 1988): correlation length increases with system size, so second order P. Bacilieri et al, PRL61, 1545 (1988)

Columbia-64 computer APE-1 computer

4 (ε + P) / T ξ ∝ L 243 × 4

first order! second order!

temperature System size L - 20 - M. Fukugita eta!. / Deconfiningphase transition 207

TABLE 6 Physical mass gap extracted from the zero-momentum projected (G 0(z)) and the point-to-point (G(n)) correlation functions

size /3 n8~,/n~ phase from Go(z) from G(n)

24~x 4 5.6800 45/9 C 0.214(25) 0.222(23)1) 5.6850 45/9 C 0.186(31) 0.184(29) 5.6875 80/10 C 0.184(17) 0.182(18) 5.6900 130/10 C 0.168(12) 0.161(11) 20/5 H 0.269(24) 0.163(18) 5.6925 24/8 C 0.206(64) 0.201(51) Which one is95/9.5 right?H 0.185(13) 0.161(11) 5.6950 36/9 H 0.211(34) 0.159( 9)1) 5.7000 40/10 H 0.286(43) 0.343(73) 28~x 4 5.6920 90/10 C 0.192(29) 0.186(22) n First-order making full use of finite-size35/7 scalingH theory:0.183(50) 0.141(74) 36~x 4 5.6925 95/9.5 C 0.103(16) 0.103(16) n Susceptibility peak grows linearly with spatial80/8 volumeH 0.181(32) 0.109(39)2) n Correlation length, though large, stays finite 1) rmax = 15 2) rmax = 22

The number of sweeps Fukugitaused (n5~)andetthe al,bin PRL63,size (nb,fl) for 1768the jackknife (1989)error estimate (both in units of iO~sweeps) as well as the phase (C = confined, H = deconfined) are also listed. The fitting range is z ~ I’t~/4for G0(z) and r ~ N4/4 for G(n). 194 SusceptibilityM. Fukugita eta!. / Deconfining peakphase transitionscaling Correlation length

- 0.~

IV (0) mPhYS N5”36 d~3lsing ~ 0.3- :: ~ - max V V χΩ ( s ) ∝ s

1 ≈ finite :~ ~ ξ ~12~2428 36 48 c

5.68 ‘ 5.69 ‘ 570 102 V~N~ /3 - 21 - Fig. 13. Physical mass gap obtained from the zero-momentum projected correlation function as a (b) ~$a,1/2 function of /3. The filled and open symbols correspond to the confined and deconfined phases, respectively.

N 5~8 12 16 24 28 36 48 4 1o5 102 VrN~1o

Fig. 5. (a) Peak height ofthe susceptibility xn as a function of the volume (filled circles). The solid line represents the fit eq. (12) for N~~ 16. The open symbols and dotted line are our data for the 3-dimensional Ising model and their fit for N~~ 16. (b) Peak width of Xn defined as full width half maximum as a function of the volume. The meaning of the symbols and lines is the same as in (a).

fining transition. To illustrate the difference from the second-order case, we quote

p=0.67(i), o=0.52(i) (13)

for our data [9] for the 3-dimensional Ising model (16 ~ N 1 ~ 48), which are in agreement with the known values given above. Fig. 4 also gives information on the volume dependence of the critical coupling /3~(V).If we define it as the position of the peak of Xn’ we obtain the values 24 Quark Summary Table

QUARKS I(~') =0(& ) The u-, d-, and s-quark masses are estimates of so-called "current- 1 — — Mass m = 4. to 4.5 GeV Charge = & e Bottom = 1 quark masses, " in a mass-independent subtraction scheme such as Fundamental constantsMS at a scale p, = 1 GeV. The c-ofand 6-quark QCDmasses are estimated from charmonium, bottomonium, D, and B masses. They are the fu') = o('+) 111.53 "running" masses in the MS scheme. These can be different from the heavy quark masses obtained in potential models.RPP 1996 Charge = e Top = +1 n QCD coupling QUANTUM CHROMODYNAMICS (Cont'd)n Quark masses p

to the corresponding parton cross sections, and the agreement is impressive. As an example, the figure on "Jet Production in pp and Mass m = 180 6 12 GeV (direct observation of top events) as(Q= 5 GeV) PDG 1990 f(~ ) = '('+) 0.3 p~ Interactions" in "Plots of Cross Sections and Related Quantities" Mass m = 179 + GeV (Standard Model electroweak fit) 0.1 0.2 shows the inclusive jet cross section at zero pseudorapidity as a 8+zt li~ i l [ function of the jet transverseMass momentumm = 2forto p~8 collisions.MeV t'j The QCDCharge = e f~ = prediction combines the parton distributions with the leading-order & +~ — 25 o Deep-lnelastic 2 -* 2 parton scattering amplitudes.m„/md Data0. are to also0. available70 on the angular distribution of jets; these are also in agreement with QCD b' (4'" Generation} Quark, Searches for expectations. 37 : i QCD corrections to Drell-Yan type cross sections (i.e., the oi T decays production in hadron collisions by quark-antiquark annihilation of t(i ) = p(p+) Mass m ) 85 GeV, CL = 95% (pp, charged current decays) lepton pairs of invariant mass Q from virtual photons, or of real W Mass m & 46.0 GeV, CL = ali or Z bosons) are known. 38 These O(as) QCD corrections are sizable 95% (e+ e, decays) o Fz 7 and approximately constantMass over them =lepton-pair5 to 15 massMeV range~'~ probedCharge = — e lz = — by experiments. Thus & & m, /md —17 to 25 O R(e+e-) o" o"(0) [1 + °Ls(Q2)C + ...] (D.1) Free Quark Searches DY ~ DY 27r "

It is interesting to note that the corresponding correction to W and o E-E Corr. Z production, as measured at p~ colliders, has essentially the samef(~ ) =0(2+) All searches since 1977 have had negative results. theoretical form and is of order 30%. Total W and Z production cross sections soon will be measured accurately enough to be sensitive to ~ j — — , , [ .... I .... I ' ' ' such 30% effects and can inMass principlem =offer100 a testto 300of theMeV theory. The Charge = & e Strangeness = 1 NOTES 200 400 600 800 key ingredient which is missing(m, at— present(m„+ is themd)/2)/(md complete O(as2)— QCDmu) = 34 to 51 A (4) (in M~ scheme in MeV) correction which is potentially important in view of the large O(c~s) term. QCD effects are also observable in the production of W and Z bosons with large transverse momentum. 39 There is good qualitative [a] The ratios m„/md and m, /md are extracted from pion and kaon masses agreement, although the statistics are rather poor at present.N 40 0 MS f = f(~') = o(,'+) using chiral symmetry. The estimates of u and d masses are not without Fig.α 2. Summary5G ofe theV values= of0 A(M~).1s 7from4 ±various0. 0processes.12 E. QCD IN HEAVY QUARKONIUM DECAY s ( ) mud 2GeV = 3.6 6 MeV The deep inelastic value is an average of those shown in Fig. 1. Under the assumption that the hadronic( and leptonic) decay widths ( ) controversy and remain under active investigation. Within the literature The T result is from 28 an average of measurements. 29'30'31 The of heavy QQ resonances can be factorized into a nonperturbative Mass m = 1.0 to 1.6 GeVN f =0 Charge = &2 e Charm = +1 there are even suggestions that the u quark could be essentially massless. two-photon value is the allowed range from the results of Fig. 1 part dependent on the confining potentialm 2 Gande aV calculable pertur-9 5 16 MeV and takesEl into-Kahdra account the systematicet al, errorPRL69, from the different 729 (1992)bative part, the ratios of partial decays (widths allow )measurements= ( ) The s-quark mass is estimated from SU(3) splittings in hadron masses. models for the nonperturbative component of the structure of a~s at the heavy quark mass scale. The most precise data come function. The value from R is the average 32 of the compilation from the decay widths of the 1-- J/~ and T resonances. Potential of Ref. 24.Energy The result scalefor the energy-energy from charmonium correlations 33 is a spectrummodel dependences cancel from the ratiosGough of decay widths. et Importantal, PRL79, 1622 (1997) range of inallowed quenched values and includes lattice the systematicQCD errors due examples of such ratios are to different fragmentation models. The dashed lines give our - 22 - estimate of the possible uncertainty due to higher order QCD r(1-- -~ g99) r(1--Both ---, 7gg) simulations done(E.1) on ACPMAPS corrections; see text. For convenience, the top scale gives the r(1- ~ u+, -) ' r(1-- -~ ggg) value of a8(5 GeV) corresponding to the values of ArM--g)s . The The perturbative corrections to these ratios are rather large. 41 vertical dotted lines indicate our allowed range and central value They change the predictions by a factor of 1.64 and 0.77 respectively for A. in the case of T decay. The corrections in the J/~ case are much larger. Relativistic corrections are unknown and could be substantial D. QCD IN HIGH ENERGY HADRON COLLISIONS for the J/~ case. We will therefore assign a 20% uncertainty to the There are many ways in which perturbative QCD can be tested value of c~s obtained from T decays. in high energy hadron colliders. The most precise of these is A recent analysis28 of bottonmnium decay-width ratios from CUSB, the production of single large-transverse-momentum photons. The CLEO, and ARGUS 29'3°'31 finds leading-order QCD subprocesses are q~ ~ ~fg and q9 --* "yq. Explicit as(rob) = 0.179 ± 0.009 (E.2) expressions for the corresponding scattering amplitudes can be found, for example, in Ref. 34. If the parton distributions are taken from if the theoretical uncertainties are ignored. These uncertainties are other processes and a value of A~-g assumed, then an absolute indicated in Fig. 2. prediction is obtained. Conversely, the data can he used to extract F. PERTURBATIVE QCD IN e+e - COLLISIONS information on quark and gluon distributions and the value of AR-g. The total cross section for e+e - ---* hadrons is obtained by This is also one of the few hard scattering processes for which the multiplying the muon-pair cross section by the factor R = 3~qe~. The next-to-leading-order corrections are known, 35 and so a precision test higher order QCD corrections to this quantity have been calculated, is possible in principle. In practice, however, the residual uncertainties and the results can be expressed in terms of the factor: on the (most accurate) experimental data and in the theoretical prediction are on the order of 20-30%, and this is sufficiently large R=R (°) [1+ a~ +C2 (v) 2 + C3 + .-- ] , to limit the accuracy of an as measurement. Nevertheless a value for T7 AR'g in the range 100 300 MeV gives very satisfactory agreement with --(2 11) 365 __ 11~.(3 , (F.I) a wide range of data. 36 c~ s= C(3)-]5 ~S+5~- The production of hadrons with large transverse momentum in hadron-hadron collisions provides a direct probe of the scattering of R (0) can be obtained from the formula for da/d~ for e+e - -~ f7 by quarks and gluons: qq -* qq, q9 --* qg, g9 -* g9, etc. The present integrating over fL The formula is given in "Cross-Section__ Formulae generation of p~ colliders provide center-of-mass energies which for Specific Processes," Section B. Numerically C Ms =__1.41. Recently are sufficiently high that these processes can be unambiguously C3 has been computed; 42 numerically (for n/ = 5) C MS = 64.7. This identified in two-jet production at large transverse momentum. result is strictly only correct in the zero-quark-mass limit. The O(as) Corrected inclusive jet cross sections can he directly compared corrections are also known for massive quarks. 43 “Lattice QCD on Parallel Computers”

10-15 March 1997, CCP, Tsukuba, Japan

- 23 - QCD machines in the 90’s and later

CPU vendor peak

CP-PACS 1996 Iwasaki et al PA-RISC Hitachi 0.6TF QCDSP 1998 Christ et al TI DSP -- 0.6TF APEmille 2000 APE Collab. custom -- 0.8TF QCDOC 2005 Christ et al PPC-based IBM(BG/L) 10TF PACS-CS 2006 Ukawa et al Xeon Hitachi 14TF QCDCQ 2011 Christ et al PPC-based IBM(BG/Q) 0.5PF QPACE 2012 Wettig et al PowerXCell -- 0.2PF n Big success continued, with increasing involvement of big vendors (Hitachi, IBM) to secure necessary technology n Gradual loss of control of physicists as the HPC vendors embraced the parallel paradigm

- 24 - Lattice QCD machines and supercomputer development JPN project machines

BlueGene/Q

BlueGene/L,P Vector machines

Vector parallel machines QCDOC(USA)

Parallel machines

QCDSP(USA PACS-CS(JPN

APE100Italy

GF11(USA)

CP-PACSJPN QCD machines Columbia QCDPAX JPN Tsukuba

- 25 - Advances in the 90’s n Quenched hadron spectrum (1997) n Determination with a few % error revealed a definite discrepancy with experiment n Signaled the end of “quenched” era n Big shift to full QCD late 90’s to early 00’s n MILC (staggered) n CP-PACS (Wilson-clover) n RBRC (domain-wall) n …..

- 26 - An interlude

Building the community (I) n Lattice conference (LATTICE XX) : the meeting place n Started in 1984 and being held every year n A few dozen of review talks + hundreds of parallel talks (everyone talks!) style quickly established n Rotates among USA-Europe-Asia n LatticeNews mailing list: the communication tool n latticejobs started in 1994 n latticeannounce started in 2001/renamed as LatticeNews in 2004 n Created and maintained by S. Gottlieb n Current registrations: latticejobs 636 LatticeNews 839 n arXiv (hep-lat) n Started in 1992 - 28 - Lattice XX participants and talks

1984 Argonne USA 600 1985 Tallahassee USA 1986 BNL USA 500 1987 Seillac France 1988 FNAL USA 1989 Capri Italy 400 1990 Tallahassee USA 1991 Tsukuba Japan 300 1992 Amsterdam Netherland 1993 Dallas USA 200 1994 Bielefeld Germany 1995 Melbourne Australia 1996 St Louis USA 100 1997 Edinburgh Scotland 1998 Boulder USA 0 1999 Pisa Italy 1980 1990 2000 2010 2020 2000 Bangalore India #participants #review talks #parallel talks 2001 Berlin Germany - 29 - 200 Building the180 community#submissions to LatticeNews (I)/year 160 n Lattice conference (LATTICE140 XX) : the meeting place 120 n Started in 1984 and being held100 every year 80 n A few dozen of review talks + hundreds60 of parallel talks (everyone talks!) style quickly established 40 20 n Rotates among USA-Europe-Asia 0 2000 2004 2008 2012 2016 2020 n LatticeNews mailing list: the communication tool n latticejobs started in 1994 n latticeannounce started in 2001/renamed as LatticeNews in 2004 n Created and maintained by S. Gottlieb n Current registrations: latticejobs 636 LatticeNews 839 n arXiv (hep-lat) n Started in 1992 - 30 - Lattice XX and hep-lat

2002 Boston USA 800 2003 Tsukuba Japan 2004 FNAL USA 700 2005 Dublin Ireland 2006 Tucson USA 600 2007 Regensburg Germany 500 2008 Williamsburg USA 2009 Beijin China 400 2010 Villasimius Italy 2011 Squaw Valley USA 300 2012 Cairns Australia 2013 Mainz Germany 200 2014 New York USA 2015 Kobe Japan 100 2016 Southampton England 0 2017 Granada Spain 1980 1990 2000 2010 2020 2018 East Lansing USA #arXiv lat #arxiv cross list 2019 Wuhan China 2020 Bonn Germany - 31 - Building the community (II) n International Lattice Data Grid (ILDG):sharing the resources n Started in 2002 n Federation of regional configuration repositories n NERSC(USA)/LGT(Europe)/JLDG(Japan) n FLAG (Flavor Lattice Averaging Group):outreaching the results n Critical review and summary of lattice QCD results relevant for experiments n Started in 2007 in Europe, now a global effort across Europe, America, Asia n Publication every few years: 2011, 2014, 2017, 2019; well cited: 363, 570, 528, 94 (inspire Nov. 2019) - 32 - Lattice QCD collaborations

1990 2000 2010 2020

JLQCD Japan CP-PACS PACS-CS PACS HALQCD MILC

USA Fermilab Lattice HPQCD

RBC (RIKENBNL-Columbia)

USQCD UKQCD

Europe QCDSF

BMW

ETM - 33 - 2011 Xu Feng, Marcus Petschlies, Karl Jansen, Dru B. Renner 2012 T. Blum, P.A. Boyle, N.H. Christ, N. Garron, E. Goode, T. Izubuchi, C. Jung, C. Kelly, C. Lehner, M. Lightman, Q. Liu, A.T. Lytle, R.D. Mawhinney, C.T. Sachrajda, A. Soni, C. Sturm 2013 André Walker-Loud 2014 Gergely Endrödi 2015 Stefan Meinel 2016 Antonin Portelli 2017 Raul Briceno 2018 Zohreh Davoudi 2019 Luchang Jin

- 34 - A time of crisis n Eastern Japan Great Earth Quake 2011.3.11 n The earthquake 14:46 JST on 11 March n The tsunamis 15:00-16:00 on 11 March n The Nuclear Power Plant 11 – 15 March

n No serious damage to supercomputers for lattice, but they were likely to be down for a half year or more due to power shortage, stopping research in serious ways - 35 - A case of international support

n Generous offer of international support n 3 April mail from Paul Mackenzie for USQCD n 4 April mail from Nick Samios for RBRC n May 2011 to March 2012 6 projects in Japan benefited from the US resources n March 2012

n end of the projects and report to USQCD and<> <>

Date: Sun, 3 Apr 2011 17:36:23 -0500 Date: Thu, 7 Apr 2011 10:36:01 -0400 From: paul mackenzie From: "Samios, Nicholas P" To: Akira Ukawa To: Subject: lattice supercomputing in Japan in the crisis Cc: "Taku Izubuchi" , "Norman Christ" Dear Akira, Subject: computing Members of the USQCD Executive Committee have been deeply saddened by the events in Japan, and we would like to find a way to be of help to you and your Dear collea Akira,gues. It is our understanding that limited electricity has been restored, but facilities which consume lar ge amounts of power, such as supercomputers, are still off, and will remain so for some time. If this is the case,If useful we RBRCwould islike setting to ask aside if the 4 useracks of ofup- 36 the to- 10%QCDOC computer of our cluster resources would be helpful for Japanesecomplex lattice theorists for use inby 2011 any interestedwhile the electricityJapanese crisisphysicists. lasts. This is one sixth This would be equivalent to the continuous use of approximatelyof 10 teraflops. 2,800 This compute allotment cores. can also be increased as needed.Taku is Some thought would have to be given to logistic presentlyarrangements. in Japan In particular,and can discuss we would any havedetails. to consultWe welcome all with the administrators of the clusters at Fermilab andinterested JLab. However, individuals. I am confident Please le thatt me we know can if make this is this adequate and work. acceptable.

Best wishes, Nick Paul

P.S. If you are not the right pe rson to discuss this matter with, please let me know whom I should contact.

Paul B. Mackenzie Theoretical Physics Department Fermilab (630) 840-3347

- 1 - - 1 - Getting nimble with quarks

Quarks posed many headaches!

iθγ 5 n Light quarks q(x) → e q(x) n Chiral symmetry, while playing a fundamental role in the strong interactions, is incompatible with lattice under rather general conditions (Nielsen-Ninomiya 1981) n Heavy quarks n Large quarks masses of charm and bottom leads to serious distortions if simulated on lattices with coarse lattice spacings

æ 1 ö - åqn Dnm (U )qm - å fn ç ÷ fm n,m D(U ) dq dq e n,m = det D(U ) = df df e è ønm n Computation òÕ n n òÕ n n n n n Sea quark effects requires computation of the determinant or inverse of lattice Dirac operator, whose cost blows up for light quarks 4 −1 1 ⎛ L⎞ computational cost(D ) ∝ × ⎜ ⎟ mqa ⎝ a ⎠ - 38 - Chiral symmetry n Condition for exact (but modified) chiral symmetry

γ 5 D + Dγ 5 = a Dγ 5 D Ginsparg-Wilson ‘82 n It took a full decade before the explicit formalisms were found: n Domain-wall formalism Kaplan 1992 n Overlap formalism Narayanan-Neuberger 1995 n Perfect action Hasenfratz-Niedermeyer 1998 n Simulations started in late '90s and early ‘00s n Christ et al, RBC collaboration domain-wall n KEK group, JLQCD collaboration overlap n Fine-tuning of conventional fermions Wilson formalism (Wilson 1975) / Staggered formalism (Susskind 1977) n Add terms reducing O(an) effects (Wilson ‘79, Symanzik ‘83) n Smearing of gauge links (APE ‘85, HYP ‘99, STOUT ‘02) n Complicated but highly improved actions e.g., HISQ ’07 used today - 39 - Heavy quarks n Static approximation Eichten, Lattice 87 NPBP 4, 170 (‘88) n Systematic expansion in n NRQCD Lepage-Thacker, Lattice 87 NPBP 4, 199 (‘88) n Non-relativistic reformulation of QCD by integrating out all momenta higher than the heavy quark mass M n Systematic expansion in powers of heavy quark velocity

El-Kahdra et al, PRD55, 3933 (1997) also, Aoki et al, PTP 109, 383 (2003) n Relativistic heavy quark formalism Christ et al, PRD 76, 074505 (2007) n Systematic reformulation of action so as to reduce effects of m h a > 1 i S = q D U q + c κ q ε σ B q + ic κ q σ E q +! ∑ n Wilson ( )n,m m 2 B s ∑ n ijk ij n,k n E s ∑ n 0i n,i n n,m n n - 40 - Cost of dynamical quark simulations n Panel discussion at Lattice 2001 Berlin n Panelists: C. Bernard, N. Christ, S. Gottlieb, K. Jansen, R.Kenway(chair), T. Lippert, M. Luescher, P. Mackenzie, F. Niedermeyer, S. Sharpe, F. 196 A. Ukawa/Nuclear Physics B (Proc. Suppl.) 106 (2002) 195-196 Tripicione, A. Ukawa, H. Wittig

200 2.0 cost NI=2 Computational cost blows up i _ . , . . . , . . . 1.5 150 ~ ~ 1000 configs toward the physical point:~ ~ ~ 10Gconfigs

5 ~ 100 L cost ∝ 3 8 mqa

0.5 50 I ~=V CP-PACS/JLQCD 2001

o ~ ' 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 m,]~p m / m ra#mp π ρ - 41 - Figure 1. Cost of Nr=2 QCD on an L=3 fm Figure 3. Cost of Nf =2+1 QCD at a -1 --2 GeV lattice for 1000 independent configurations. for 100 independent configurations.

20 .... , • . • , • • . 3. Cost estimate for NE = 2-t- 1 simulations t Nr=2 15 ~ 1000 conflgs Recent development of algorithms for odd num- with L = 2fm ber of flavors [4] has opened the way toward including strange quark dynamically. The ad- ditional computational cost is not large since P strange quark is relatively heavy. We assume 5 ieV 1/2 times (2) applies for the single-flavor part of the algorithm. If one makes two runs with o ~ ' m~,/m¢ = 0.6 and 0.7 to sandwitch the strange 0.0 0.2 0.4 0.6 0.8 1.0 quark point, we find Fig. 3 for the cost of Ny = 2 + 1 simulations for 100 independent configura- Figure 2. Cost of N! = 2 QCD on an L = 2 fm tions at a -1 = 2 GeV as a function of u-d quark lattice for 1000 independent configurations. mass expressed in terms of m~/mp. These sim- ulations are clearly feasible, and worth pursuing, with the present computing resources. ever, parallel those of the CP-PACS run. In par- We thank our colleagues of the two collabora- ticular the value of C is quite similar. tions, in particular, S. Hashimoto, K. Ishikawa and K. Kanaya, for discussions. This work is 2. Cost estimate for NF = 2 simulations supported in part by Grants-in-Aid of Ministry of Education (Nos. 12304011, 13640259) and in In Figs. 1 and 2 we plot the computational cost part by the JSPS Research for the Future Pro- in Tflops.years estimated from (2) for generating gram (No. JSPS-RFTF 97P01102). 1000 independent configurations for N/= 2 QCD on an L = 3 fm or 2 fm lattice size. REFERENCES With computing resources of O(10) Tflops, if we choose to work on an L = 2 fm lattice, de- 1. CP-PACS Collaboration: A. Ali Khan et aL, tailed studies either spanning a wide fange of lat- hep-lat/0105015. tice spacings (e.g., a -1 ~ 1-3 GeV) for moder- 2. JLQCD Collaboration (presented by S. ately light quarks (m,r/mp ~ 0.6), or very light Hashimoto), these proceedings. quarks (e.g., m~/mp ,~ 0.2) on coarse lattices 3. See, e.g., T. Takaishi, Comput. Phys. Com- (a -1 ~ 1 GeV) would become feasible. Expand- mun. 133 (2000) 6. ing the lattice size to L = 3 fm, however, appears 4. M. Peardon, in these pr0ceedings; JLQCD to require another order of magnitude increase of Collaboration (presented by K. Ishikawa), in computing resources. these proceedings. Removing the critical slowing down n Cost of dynamical full QCD calculation 1 # L &4 cost ∝ Ninv × ×% ( ×τ aut δτ $ a ' n With HMC Duane et al, PLB195, 216 (1987) 4 1 L / a ⎛ L⎞ 1 L5 cost ∝ × × ⎜ ⎟ × = 3 8 Sexton-Weingarten, NPB380, mqa mqa ⎝ a ⎠ mqa mqa 665 (1992) n With UV/IR separation Hasenbusch, PLB519, 177 (2001) 1 L # L &4 L5 cost 1 ∝ × ×% ( × = 6 Luescher, CPC165, 199 (2005) mqa a $ a ' mqa n With deflation/multi-grid Babich et al, PRL 105, 201602 (2010) L # L &4 L5 cost ∝1× ×% ( ×1= Frommer et al, SIAM J 36, A1581 (2014) a $ a ' a5 Note: topology has to be still treated, e.g., by open boundary condition - 42 - Putting them all together

Begininng of maturity

“High Precision Lattice QCD Confronts Experiment” HOQCD/UKQCD/MILC/Fermilab Lattice Collaborations PRL 92, 022001 (2004)

Quenched QCD Nf=3 full QCD PHYSICAL REVIEW LETTERS week ending V(seaOLUME quarks92, ignored) NUMBER 2 (u, d, s sea quarks included) 16 JANUARY 2004 formalism, are reliable and accurate tools for analyzing heavy-quark dynamics. A serious problem in the previous work was the incon- sistency between light-hadron, B=D, and = quantities. Heavy-quark masses and inverse lattice spacings, for Before Afterexample, were routinely retuned by 10%–20% when going from an  analysis to a B analysis in the same quenched simulation [14]. Such discrepancies lead to the results shown in the left panel of Fig. 1. The results in the right panel for , K, , Ds, J= , Bs, and  physics mark the first time that agreement has been achieved among such diverse physical quantities using the same QCD parameters throughout. The dominant uncertainty in our light-quark quantities comes from- our44 - extrapolations in the sea and valence light-quark masses. We used partially quenched chiral FIG. 1. LQCD results divided by experimental results for perturbation theory to extrapolate pion and kaon masses, nine different quantities, without and with quark vacuum and the weak decay constants f and f . The s-quark polarization (left and right panels, respectively). The top three  K results are from our a 1=11 and 1=8fm simulations; all mass required only a small shift; we estimated correc- others are from a 1=8fmˆ simulations. tions due to this shift by interpolation (for valence s ˆ quarks) or from the sea u=d mass dependence (for sea s quarks).We kept u=d masses smaller than ms=2 in our fits, so that low-order chiral perturbation theory was suffi- different physical quantities. The right panel shows re- cient. Our chiral expansions included the full first-order sults from QCD simulations that include realistic vacuum contribution [15], and also approximate second-order polarization. These nine results agree with experiment to terms, which are essential given our quark masses. We within systematic and statistical errors of 3% or less— corrected for errors caused by the finite volume of our with no free parameters. lattice (1% errors or less), and by the finite lattice spacing The quantities used in this plot were chosen to test (2%–3% errors). The former corrections were determined several different aspects of LQCD. Our results for f and from chiral perturbation theory; the latter by comparing fK are sensitive to light-quark masses; they test our results from the coarse and fine lattices. Residual discre- ability to extrapolate these masses to their correct values tization errors, due to nonanalytic taste violations [7] using chiral perturbation theory. Accurate simulations that remain after linear extrapolation in a2, were esti- for a wide range of small quark masses were essential mated as 2% for f and 1% for fK. Perturbative match- here. The remaining quantities are much less sensitive to ing was unnecessary for the decay constants since they the valence u=d mass, and therefore are more stringent were extracted from partially conserved currents. Our tests of LQCD since discrepancies cannot be due to tun- final results agree with experiment to within systematic ing errors in the u=d mass. The  mass tests our ability to and statistical uncertainties of 2:8%. For the n 0 case, f ˆ analyze (strange) baryons, while the Bs mass tests our we analyzed only a 1=8fm, but corrected for discre- formalism for heavy quarks. The b rest mass cancels in tization errors by assumingˆ these are the same as in our 2M M , making this a particularly clean and sensi- n 3 analysis. Bs ÿ  f ˆ tive test. The same is true of all the  splittings, and Figure 2, which shows our fits for f and fK as func- our simulations confirm that these are also independent tions of the valence u=d mass, demonstrates that the u=d ( 1%–2%) of the sea-quark masses for our smallest masses currently accessible with improved staggered masses, and of the lattice spacing (by comparing with quarks are small enough for reliable and accurate chiral r0 and r1 computed from the static-quark potential) extrapolations, at least for pions and kaons. The valence [12]. The  P masses are averages over the known spin and sea s-quark masses were 14% too high in these states; the  1†D is the 13D state recently discovered by particular simulations; and the sea u=d masses were too † 2 CLEO [13]. large—ms=2:3 and ms=4:5for the top and bottom results Note that our heavy-quark results come directly from in each pair (fit simultaneously by a single fit function). the QCD path integral, with only bare masses and a The dashed lines show the fit functions with corrected coupling as inputs—five numbers. Furthermore, unlike valence s and sea u=d=s quark masses; these lines ex- in quark models or heavy-quark effective theory (HQET), trapolate to our final fit results. The bursts mark the  physics in LQCD is inextricably linked to B physics, experimental values. Our extrapolations are not large— through the b-quark action. Our results confirm that ef- only 4%–9%. Indeed the masses are sufficiently small that fective field theories, such as NRQCD and the Fermilab simple linear extrapolations give the same results as our

022001-3 022001-3 “Lattice QCD Simulations via International Research Network”

21-24 September 2004, Shuzenji

- 45 - Lattice calculations after 2000s

n Lattice size L ≥ 3− 5fm mπ L ≥ 4 − 5 to control finite size effects L → ∞ −1 n Lattice spacings a ≤ 0.1− 0.04fm a ≥ 2 − 5GeV to control continuum extrapolation a → 0 n Sea quark effects fully incorporated for u, d, s, c N f = 2 +1+1 with their physical masses n Heavy quarks treated by effective theory or directly simulated for small enough a n Use of non-perturbative renormalization factors n Multiple set of parameter points, each with u O(1000) independent configurations d a u n Sophisticated fitting and error estimation to extract physical results

L - 46 - Light hadron spectrum

2000

O 1500 X* S* X S D L 1000 K* N

M[MeV] r

experiment 500 K width input p QCD 0

BMW Collaboration, Durr et al, Science 322, 1224 (2008)

- 47 - Isospin breaking in hadron masses n important for understanding Nature e.g., neutron-proton mass difference is crucial for Big Bang nucleo- synthesis / stability of nuclei n History n Duncan-Eichten-Thacker, PRL76, 3894(1996) n RBC/UKQCD, Blum et al, PRD82, 094508 (2010) n BMW, Borsanyi et al, Science 347, 1452 (2015)

10 m − m = +1.51 28 MeV ΔΣ experiment n p ( ) 8 QCD+QED ΔΞ prediction

6 =+2.52 30 −1.00 16 MeV ΔD ( )QCD ( )QED 4 M [MeV]

Δ ΔΞcc cf . 1.2933321(5) MeV RPP 2018 2 ΔN + ΔCG 0

BMW 2014 HCH

Figure 2: Mass splittings in channels that are stable under the strong and electromagnetic interactions. Both of these interactions are fully unquenched in our 1+1+1+1 flavor calculation. The horizontal lines are the experimental values and the grey shaded regions represent the experimental error (2). Our results are shown by red dots with their uncertainties. The error bars are the squared sums of the statistical and systematic errors. The results for the MN , M⌃, and MD mass splittings are post-dictions, in the sense- that48 their- values are known experimentally with higher precision than from our calculation. On the other hand, our calculations

yield M⌅, M⌅cc splittings, and the Coleman-Glashow difference CG, which have either not been measured in experiment or are measured with less precision than obtained here. This feature is represented by a blue shaded region around the label.

9 QCD coupling constant

experiment

Lattice QCD filled green results: systematics under control, so included in the lattice estimate

⎧ 0.1174 16 RPP2019 nonlattice 5 ⎪ ( ) ( ) ( ) M α MS ( Z ) = ⎨ ⎪ 0.1182(8) FLAG2019 ⎩ - 49 - Quark masses

N mMS 2GeV mMS 2GeV mMS 2GeV mMS m mMS m f u ( ) d ( ) s ( ) c ( c ) b ( b ) 2 +1 2.27(9)MeV 4.67(9)MeV 92.03(88)MeV 1.275(5)GeV 4.164(23)GeV 2 +1+1 2.50(17)MeV 4.88(20)MeV 93.44(68)MeV 1.280(13)GeV 4.198(12)GeV - 50 - CKM matrix elements

Vub ,Vcb Vus ,Vud

Tension with nuclear beta decay? Tension between exclusive and inclusive determinations? - 51 - High temperatures n Phase diagram n Equation of state

still debated 20

s/T3 15

physical 10 4 point e/T

HotQCD 5 Wuppertal-Butapest

HotQCD

Wuppertal-Butapest 0 100 150 200 250 300 350 400 T(MeV) Physical point is a crossover. HotQCD Bazavov et al, PRD90, 094503 (2014) Aoki et al, Nature 443, 675 (2006) WB Borsanyi et al, PLB730, 99 (2014) - 52 - Muon g-2

exp SM aexp aSM 27.9 6.3 3.6 10−10 µ − µ = ( ) ( ) × n Huge ongoing effort by lattice community to calculate hadronic contributions Fermilab/HPQCD/MILC, RBC/UKQCD, BMW, ETMC, Mainz, PACS-CS, … n Sub % accuracy in the near future?

- 53 - Lattice QCD today n 45 years since Wilson’s seminal paper (February 1974) n Amazing progress in physics, algorithms, machines n Has matured as particle theory n Direct calculation at the physical quark masses on large lattices (L4-5fm) and small lattice spacings (a0.05fm or less) with physical sea quarks n Many important hadron properties now calculated (masses, decay constants, form factors etc), verifying the validity of QCD at % level or better, and providing valuable constraints on the CKM matrix and the Standard Model.

54 What now?

What will be the next chapter of the History of Lattice QCD, and who will be the Maker(s) of it?

- 56 - Appendix

- 57 - WPI Program (World Premier International Research Center Initiative)

Shaping the future science in Japan beyond borders and barriers

Hokkaido Univ. Four basic features: Chemical Reaction Design 1. Critical mass of outstanding PIs Toho ku Univ.

Kanazawa Univ. Kyoto Univ. Materials & 2. International environment Mathematics

NIMS Cell Biology & Nano Life Science Materials 3. Long term financial support Nanotechnology Human Biology Univ. of Tsukuba 4. Robust follow up Osaka Univ. UTokyo

Sleep Immunology Origin of Universe

Kyushu Univ. Four missions: Nagoya Univ. Tokyo Tech UTokyo

Energy Plant/Animal Biology Origin of Ori in of 1. Top-notch Science & Chemistry Earth and Life Inte igence 2. Fusion Research 3. Internationalization 13 centers as of 2019 (Planned up to 20 centers) 4. System reform