Christine Davies University of Glasgow

Lattice field theory - a European perspective

European strategy meeting Krakow September 2012

Tuesday, 11 September 2012 Applications of Lattice QCD/Lattice field theory Annual proceedings of Particle physics lattice conference: 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 condensed matter physics computational physics Astrophysics computer science ...

Tuesday, 11 September 2012 Lattice QCD = fully nonperturbative QCD calculation RECIPE • Generate sets of gluon fields for Monte Carlo integrn of Path Integral (inc effect of u, d, s, (c) sea quarks) • Calculate valence quark propagators to give “hadron correlators” • Fit for masses and 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 • cost increases as a 0 ,m l phys a and with statistics, volume.→ → Tuesday, 11 September 2012 REPORTS mud, corresponding to Mp ≅ 135 MeV, are difficult. vector meson (r, K*) octets that do not require (25, 26)and forour setup(27)], simulating di- They need computationally intensive calculations, the calculation of disconnected propagators. rectly at physical mud in large enough volumes, with Mp reaching down to 200 MeV or less. Typical effective masses are shown in Fig. 1. which would be an obvious choice, is still ex- 5) Controlled extrapolations to the contin- 3) Shifts in hadron masses due to the finite tremely challenging numerically. Thus, the stan- uum limit, requiring that the calculations be size of the lattice are systematic effects. There dard strategy consists of performing calculations performed at no less than three values of the are two different effects, and we took both of at a number of larger mud and extrapolating the lattice spacing, in order to guarantee that the them into account. The first type of volume de- results to the physical point. To that end, we use scaling region is reached. pendence is related to virtual pion exchange be- chiral perturbation theory and/or a Taylor expan- Our analysis includes all five ingredients tween the different copies of our periodic system, sion around any of our mass points (19). listed above, thus providing a calculation of the and it decreases exponentially with Mp L.Using 5) Our three-flavor scaling study (27) showed light hadron spectrum with fully controlled sys- MpL > 4 results in masses which coincide, for that hadron masses deviate from their continuum tematics as follows. all practical purposes, with the infinite volume values by less than approximately 1% for lattice 1) Owing to the key statement from renor- results [see results, for example, for pions (22) spacings up to a ≈ 0.125 fm. Because the sta- malization group theory that higher-dimension, and fore baryons (23, 24)]. Nevertheless, for one tistical errors of the hadron masses calculated in local operators in the action are irrelevant in the of our simulation points, we used several vol- the present paper are similar in size, we do not continuum limit, there is, in principle, an un- umes and determined the volume dependence, expect significant scaling violations here. This is limited freedom in choosing a lattice action. which was included as a (negligible) correction at confirmed by Fig. 2. Nevertheless, we quantified There is no consensus regarding which action all points (19). The second type of volume de- and removed possible discretization errors by a would offer the most cost-effective approach to pendence exists only for resonances. The cou- combined analysis using results obtained at three the continuum limit and to physical mud.Weuse pling between the resonance state and its decay lattice spacings (19). an action that improves both the gauge and products leads to a nontrivial-level structure in We performed two separate analyses, setting fermionic sectors and heavily suppresses non- finite volume. Basedon(20, 21), we calculated thescalewith MX and MW. The results of these physical, ultraviolet modes (19). We perform a 9.the Quantumcorrections necessary chromodynamicsto reconstruct the reso- two sets29are summarized in Table 1. The X set is series of 2 + 1 flavor calculations; that is, we nance masses from the finite volume ground- showninFig.3.Withbothscale-settingproce- 2 include degenerate u and d sea quarks and an state energy and included2 them in the analysis dures, we find that the masses agree with the overall χ toadditional the centrals sea quark. valueWe fix isms determined.to its approxi- ( If19). this initial χ is larger than thehadr numberon spectrum observed in nature (28). of degrees ofmate freedom,physical value. i.e.To largerinterpolate thanto the thephys- number4) Though of individimportantualalgorithmic inputs minusdevelop- one,Thus, thenour study strongly suggests that QCD all individualical value, errorsfour areof our enlargedsimulations bywere arepeated commonments factorhave taken suchplace thatrecentlyχ2/d.o.f.[for example equalsis unity.the theory of the strong interaction, at low with a slightly diff2erent ms.Wevary mud in a If the initialrange valuethat extends of χdownisto smallerMp ≈ 190 thanMeV. the numberTable 1. Spectrum of degreesresults in ofgiga freedom,–electron volts. anThe overall,statistical (SEM) and systematic uncertainties a-priori unknown2) QCD correlationdoes not predict coehadronfficientmasses isin introducedon the last digits andare determinedgiven in the first byand requiringsecond set thatof parentheses, respectively. Experimental the total χphysical2/d.o.f.units: ofOnly thedimensionless combinationcombinations equalsmasses unity.are Inisospin-averaged both cases,(19). theFor each resultingof the isospin finalmultiplets considered, this average is (such as mass ratios) can be calculated. To set the within at most 3.5 MeV of the masses of all of its members. As expected, the octet masses are more overall uncertaintyoverall physical ofscale, theany centraldimensionful valueobserv- of αsaccurateis largerthan the thandecuplet themasses, initialand estimatethe larger the ofstrange a content, the more precise is the Gaussian error.able can be used. However, practical issues in- result. As a consequence, the D mass determination is the least precise. fluence this choice. First of all, it should be a This procedurequantity that iscan onlybe meaningfulcalculated precisely if theand individualX measureExperimentalments(28 are) known notMX to(X set) MX (W set) be correlatedwhos toe experimental large degrees,value is w i.e.ell known. if theySec- arer not - for instance0.775 - based on the0.775 same(29) (13) 0.778 (30) (33) input data,ond, andit should if thehave inputa weak valuesdependence areon largelymud, K* compatible with0.894 each other and with0.906 the(14) (4) 0.907 (15) (8) so that its chiral behavior does not interfere with N 0.939 0.936 (25) (22) 0.953 (29) (19) resulting centralthat of oth value,er observables. withinBe theircause we assignedare con- uncertaintL ies. The1.116 list of selected individual1.114 (15) (5) 1.103 (23) (10) measurementssidering discussedspectral quantities above,here, however,these two con- violatesS both these1.191 requirements: there1.169 are(18) (15) 1.157 (25) (15) several measurementsditions should guide basedour c onhoice (partlyof the particle or fully)X identical data1.318 sets, and there are1.318 results 1.317 (16) (13) whose mass will set the scale. Furthermore, the D 1.232 1.248 (97) (61) 1.234 (82) (81) which apparentlyparticle should do notnot d agreeecay under withthe othersstrong in- and/orS* with the resulting1.385 central value,1.427 within(46) (35) 1.404 (38) (27) their assignedteraction. individualOn the one hand, uncertainty.the larger the strange ExamplesX* for the firstcaseareresultsfromthe1.533 1.565 (26) (15) 1.561 (15) (15) hadronic widthcontent ofof the theparticle,τ lepton,the more p fromrecise the DISmass processesW and from1.672 jets and event shapes1.676 in(20) (15) 1.672 + − determination and the weaker the dependence on e e finalm states.. These facts Ans exampleupport the use ofof t thehe W secondbaryon, case is the apparent disagreement between ud + − results fromthe theparticleτ widthwith the h andighest thosestrange fromcontent. DISOn Fig. [264]3. The orlig fromht hadr Thruston distributions in e e annihilationthe o [278].ther hand, the determination of baryon dec- spectrum of QCD. Hori- uplet masses is usually lessLatticeprecise than those oQCDf zontal lines andhadronbands are physics the octet. This observation would suggest that the experimental values the X baryon is appropriate. Because both the with their decay widths. W and X baryon are reasonable choices, we Our results are shown by -decays carry out two analyses, one$with MW (the W set) solid circles. Vertical error and one with MX (the X set).LatticeWe find that for all bars represent our com- three gauge couplings, 6/g2DIS= 3.3, 3.57, and 3.7, bined statistical (SEM) and both quantities give consistent results, namely systematic error estimates. + - annihilation a ≈ 0.125, 0.085, and 0.065e fm,e respectively. To p, K,and X have no error fix the bare quark masses, Zwe poleuse tfitshe mass ratio bars, because they are pairs M /M ,M /M or M /M ,M /M .We used to set the light quark p W K W p X K X mass, the strange quark determine the masses of the baryon octet (N, S, 0.11mas 0.12s and the 0.13overall L, X) and decuplet (D, S*, X*, W) and those scale, respectively. members of the light latticepseudoscalar QCD(p, K most)and ! s (" # ) EMBARaccurateGOED UN methodTIL 2PM U.S. EASTERN TIME ON THE THURSDAY BEFORE THIS DATE: Figure1226 9.3: 21 NOVEMBER22008 VOL 322 SCIENCE www.sciencemag.org Summary of1500 values of αs(MZ)obtainedforvarioussub-classes of measurements (see Fig. 9.2 (a) to (d)). The new world average value of 2 αs(M )=0.1184 0.0007 is indicated by the dashed line and the shaded band. Z ± 1000 DsDs MeV

Due to these obstacles, we have chosen to determine pre-averages for each class of c

Η DD measurements, and then to combineM those to the ﬁnal world average value of αs(MZ ),

excited using the methods of errorM treatment500 as just described. The ﬁvepre-averagesarecharmonium summarized in Fig. 9.3; we recall that these are exclusively obtained from extractions m which are based on (at least) full NNLO QCD predictions, and are publishedspectrum in s peer-reviewed journals at the time0 of completing this Review.Fromthese,wedeterminewith hybrids 93.4 1.1MeV 0 1 2 1 0 1 1 2 3 0 2 the new world average value of ± Figure 16. Charmonium spectrum up to around 4.5 GeV showing only J PC channels in which we Tuesday, 11 September2 2012 identify candidatesα fors( hybridMZ )=0 mesons..1184 Red (dark0.0007 blue), boxes are states suggested to be(9 members.23) of the lightest (ﬁrst excited) hybrid supermultiplet± as described in the text and green boxes are 3 other states, all calculatedJune on the 29, 24 2012volume. 14:54 As in Fig. 14, black lines are experimental values and the dashed lines indicate the lowest non-interacting DD¯ and DsD¯ s levels.

three states observed in the negative parity sector suggested to be non-exotic hybrids, + + + (0, 2)− , 1−−. Higher in mass there is a pair of states, (0, 2) −, and a second 2 − state + slightly above this. Not shown on the figures, an excited 1− appears at around 4.6 GeV, + there is an exotic 3− state at around 4.8 GeV and the lightest 0−− exotic does not appear until above 5 GeV. The observation that there are four hybrid candidates nearly degenerate with J PC = + (0, 1, 2)− , 1−−, coloured red in Fig. 16, is interesting. This is the pattern of states pre- dicted to form the lightest hybrid supermultiplet in the bag model [37, 38] and the P-wave quasiparticle gluon approach [39], or more generally where a quark-antiquark pair in S- + wave is coupled to a 1 − chromomagnetic gluonic excitation as shown Table 5. This is not the pattern expected in the flux-tube model [40] or with an S-wave quasigluon. In addition, + the observation of two 2 − states, with one only slightly heavier than the other, appears to rule out the flux-tube model which does not predict two such states so close in mass. The pattern of J PC of the lightest hybrids is the same as that observed in light meson sector [11, 31]. They appear at a mass scale of 1.2 1.3 GeV above the lightest conventional − charmonia. This suggests that the energy difference between the first gluonic excitation and the ground state in charmonium is comparable to that in the light meson [31] and baryon [15] sectors. To explore this hypothesis of a lightest hybrid multiplet further, we follow Ref. [31] and consider in more detail operator-state overlaps. The operators (ρ D[2] )J=0,1,2 with NR × J=1 J PC =(0, 1, 2) + and (π D[2] )J=1 with J PC =1 are discussed in that reference. − NR × J=1 −− These operators have the structure of colour-octet quark-antiquark pair in S-wave with S =1(ρNR) or S =0(πNR), coupled to a non-trivial chromomagnetic gluonic field with PgCg + Jg =1 − where Jg, Pg and Cg refer to the quantum numbers of gluonic excitation.

– 25 – 6

0 + Finally, NB µ µ− is the number of observed signal The expected and observed CLs values are shown in (s)→ 0 + 0 + + + 0 Fig. 2 for the B µ µ− and B µ µ− channels, events. The observed numbers of B J/ψK , Bs s → → 0 + → → each as a function of the assumed branching fraction. J/ψφ and B K π− candidates are 340 100 4500, → ± 0 + 19 040 160 and 10 120 920, respectively. The three The expected and measured limits for Bs µ µ− and 0 + → ± ± B µ µ− at 90 % and 95 % CL are shown in Table II. normalization factors are in agreement within the uncer- → tainties and their weighted average, taking correlations The expected limits are computed allowing the presence 0 + 21st Century Lattice QCD norm 14 10 of B µ µ− events according to the SM branching into account, gives α 0 + =(3.19 0.28) 10− (s) Bs µ µ− → norm → 11 ± × fractions, including cross-feed between the two modes. and α 0 + =(8.38 0.39) 10− . B µ µ− Lattice QCD hadron physics© 2012 Andreas Kronfeld/Fermi Natl Accelerator Lab. → ± × For each bin in the two-dimensional space formed by The comparison of the distributions of observed x (2 GeV) 0.8 0.3 " #u $ d the invariant mass and the BDT we count the number events and expected background events results in a p- 0 + Lattice QCD predictions of candidates observed in the data, and compute the ex- value (1 CLb) of 18 % (60 %) for the Bs µ µ− 0 +− → Lattice QCD postdictions (B µ µ−)decay,wheretheCL values are those cor- pected number of signal and background events. → b 0.7 [ experiment The systematic uncertainties in the background and responding to CLs+b =0.5. db signal predictions in each bin are computed by fluctu- A simultaneous unbinned likelihood fit to the mass pro- 0.2 ating the mass and BDT shapes and the normalization jections in the eight BDT bins has been performed to 0 + 0.6 factors along the Gaussian distributions defined by their determine the B µ µ− branching fraction. The sig- s → associated uncertainties. The inclusion of the systematic nal fractional yields in BDT bins are constrained to the 0 + 0 + 0 + uncertainties increases the B µ µ− and B µ µ− BDT fractions calibrated with the B h h− sam- → s → (s) → 0.5 ’ proton structure 0 + +1.8 9 upper limits by less than 5%. ple. The fit gives (B µ µ−)=(0.8 ) 10− , [ 0 ∼+ 0 + s 1.3 0.1 The results for B µ µ− and B µ µ− decays, B → − × fits to experiments s where the central value is extracted from the maximum ’’ integrated over all mass→ bins in the corresponding→ signal 0.4 J/s [ B LHP arXiv:1001.3620 x of the logarithm of the profile likelihood and the uncer- dc c RBC arXiv:1003.3387 region,( areu summarizedd) in Table I. The distribution of tainty reflects the interval corresponding to a change of ETM arXiv:1101.5540 the invariant mass for2 BDT2>0.5 is shown in Fig. 1 for 0.5. Taking the result of the fit as a posterior, with a 0 + − m0 (GeV+ ) B µ µ− and B ! µ µ− candidates. ’ s → → positive branching fraction as a flat prior, the probabil- 0.3 s UTd : the complete0.0 fit 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 ity for a measured value to fall between zero and the SM Ds B expectation is 82 %, according to the simulation. The s 2 one-sided 90 %, 95 % CL limits, and the compatibility DECAY CONSTANT (GEV) 0.2 D Figure 4: Nonsinglet average momentum fraction x − vs. m from LHP (114), B ! "u d π with the SM predictions obtained from the likelihood, are K RBC and UKQCD (115), and ETM (116). The last has 2+1+1 flavors ofsea / in agreement with the CLs results. The results of a fully quark, the others 2+1. Fits to experiment from MSTW (117) and ABM (118); unbinned likelihood fit method are in agreement within 0.1 others fall between these two. uncorrelated systematic uncertainties. The largest sys- UNFLAVORED FLAVORED tematic uncertainty is due to the parametrization of the Feynman Diagrams as Space Invaders combinatorial background BDT. 0 0 + Note that earlier work with only 2 flavors of sea quark yielded confusing results. In summary, a search for the rare decays Bs µ µ− 0 + → The low moments of quark densities from 2+1- and 2+1+1-flavor simulations are and B µ µ− has been performed on a data sam- Precision electroweak MEs → 1 approaching the point where the lattice-QCD results could beincorporatedinto ple corresponding to an integrated luminosity of 1.0 fb− . These results supersede those of our previous publica- c D to K c=coarse, f=fine the traditional fits of experimental data. For collider phenomenology, the real tion [6] and are statistically independent of those ob- 1.3 c005 D to K f D to K " challenge for lattice QCD" is to compute similar moments of thegluondensity, tained from data collected in 2010 [12]. The data are D c D to ! The fit tests the combination of SM (intended as the underlyingSM theory rates only), for consistent with both the background-only hypothesis and Vcs s s which are less well constrained by low-energy experiments.FIG. 1. Distribution of selected candidates (black points) 1.2 c005 Ds to !s experimental results and theoretical inputs (lattice-QCD, perturbative QCD) 0 + 0 + the combined background plus SM signal expectation at f in the (left) Bs µ µ− and (right) B µ µ− mass f D to ! + hadronic EW→ → ) s s the 1 σ level. For these modes we set the most stringent

2 Glaring problems are: window for BDT>0.5, and expectations for, from the top, fit D to K f 0 + 0 + 9 (q 1.1 0 8QCDThermodynamics B µ µ− SM signal (gray), combinatorial background upper limits to date: (Bs µ µ−) < 4.5 10− and + (s) fit D to K f V → 0 + B →9 × + inclusive vs exclusive ub processes need0 + (light gray), B h h− background (black), and cross- (B µ µ−) < 1.03 10− at 95 % CL. fit D to ! f (s) B → × ) or f 1 s s 0 → 2 sin(2 β ) vs BR(B τν) feed of the two modes (dark gray). The hatched area depicts We express our gratitude to our colleagues in the fit D to f The previous sections consider isolated hadrons at zero temperature. Soon after (q s !s 0 → 0 latticethe uncertaintyQCD on .... the sum of the expected contributions. f CERN accelerator departments for the excellent perfor- 0.9 Enrico Lunghi the Big Bang, however,13 the universe was much hotter than it is now, and in neutron stars, for example, the baryon density is much higherthaninnormal mance of the LHC. We thank the technical and admin- f0 The compatibility of the observed distribution of istrative staﬀ at CERN and at the LHCb institutes, K 0.8 nuclear matter. These phenomena motivatemuon theevents study g-2 with of that the expected thermodynamics for a given of branching frac- and acknowledge support from the National Agencies: QCD. Even within lattice gauge theory, thermodynamicstion hypothesis is is a computed vast subject using the (119, CLs method [15]. CAPES, CNPq, FAPERJ and FINEP (Brazil); CERN; 0.7 ! ! 120), so this review touches only on some of theThe more method fascinat providesing CL aspects.s+b, a measure of the com- NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, 0 0.5 1 1.5 2 patibility of the observed distribution with the signal HGF and MPG (Germany); SFI (Ireland); INFN (Italy); 2 2 Thermodynamics starts with thermal averages in the canonical ensemble q (GeV ) plus background hypothesis, CLb, a measure of the FOM and NWO (The Netherlands); SCSR (Poland); WIMP-nucleon scatteringcompatibility with ... the background-only hypothesis, and ANCS (Romania); MinES of Russia and Rosatom (Rus- −H/Tˆ Tuesday, 11 September 2012 Hadronic vacuum polarization Hadronic light-by-light Tr e CLs =CLs+b/CLb. sia); MICINN, XuntaGal and GENCAT (Spain); SNSF = !• ", (12) − ˆ 3 !•" Tr e H/T Thursday, March 10, 2011 where T is the temperature, and the traces Tr are over the Hilbert space of the QCD Hamiltonian Hˆ .Infact,theaverageontheleft-handsideofEquation12 is precisely that of Equation 2; the time extent N4 speciﬁes the temperature −1 T =(N4a) .TheeigenstatesofHˆ —a.k.a. hadrons—do not change with T ,but as T increases the vacuum no longer dominates the way it does in Equations 3–5, and multi-hadron states begin to play a role in the thermal average. The simplest observables are quantities like the energy, pressure, and entropy density, and order parameters sensitive to symmetry breaking. The thermal state can either restore a spontaneously broken symmetry of the vacuum or be a state QCD PHASE DIAGRAM Lattice QCDAPOSSIBILITY at high temperature, density 140 160 180 200

0.08 T The QCD equation of state 151(3)(3) c 0.06 (MeV) 4 0.04 6 8 0.02 10 (2+1)-ﬂavor QCDQ (soUARKONIA far mainly staggered) AT FINITE TEMPERATURE

bare data of Budapest/Wuppertal and HotQCDS WAVES: Υ calculations agree; Transition is a CROSSOVER • need to verify continuumconstruct extrapolations spectralat1 physical175(2)(4) function: quark temperature masses dependence

• need to improve on high-T600 limit to makeT/T =0.42 T/T =1.05 T/T =1.20 Equation of c c c contact to (screened) perturbationT/T theory=1.05 T/T =1.20 T/T =1.40 0.5 c c c state 400 4 conﬁrm (?) EoS calculations by using 6 200 8 different discretization2 schemes:

M 10 Wilson (BW, WHOT, Mainz/Frankfurt),)/ 0 Lattice 2012, June 2012 –p.2 ! 600 (

" T/T =1.40 T/T =1.68 T/T =1.86 twisted mass (DESY/Rome), 4 c c c T/T =1.68 T/T =1.86 T/T =2.09 domain wall fermions (hotQCD)400 c c c melting of 3 S (vector) 176(3)(4) 1 Upsilon 200 3 Upsilon

0 states 9 10 11 12 13 14 9 10 11 12 13 14 9 10 11 12 13 14 15 P. Petreczky (hotQCD),T (MeV) Lattice 2012 (yesterday) 2 Town Meeting HIC, 2012 – p. 8 ! [GeV] 4 Tuesday, 11 September 2012 6 8 groundstate survives1 - excited states10 suppressed

140 160 180 200 Glasgow, September 2011 –p.12 T[MeV]

2 4 Figure 3: Temperature dependence of the renormalized chiralsusceptibility(m ∆χψψ¯ /T ), the strange quark 2 number susceptibility (χs/T )andtherenormalizedPolyakov-loop(PR)inthetransitionregion.Thedifferent symbols show the results for Nt =4, 6, 8and10latticespacings(filledandemptyboxesforNt =4and6, filled and open circles for Nt =8and10).Theverticalbandsindicatethecorrespondingcritical temperatures and its uncertainties coming from the T=0! analyses. This error is given by the number in the first parenthesis, whereas the error of the overall scale determination is indicated by the number in the second parenthesis. The orange bands show our continuum limit estimates for the threerenormalizedquantitiesasafunctionofthe temperature with their uncertainties.

12 Spectral Dimension

χ2/dof=35/32, CL=37% DS ( )=4.04 0.26, DS (0) = 1.457 0.064 (includes “ﬁtting” systematic ∞ ± ± error) [Laiho + Coumbe, arXiv:1104.5505]

Beyond QCD .. 5 4

Introduction Recent results Concluding remarks 3 S D Results in 3D 2 Much like in 4D: 1 SU(N)isaconfiningtheoryinthelarge-N limit (Teper, hep-lat/9804008) • Confining flux tubes behave as Nambu-Goto0 strings (Athenodorou et al., • 0 50 100 150 200 250 300 350 1103.5854; Caselle et al., 1102.0723; Mykkänen,in progress)! Glueball masses have a smooth dependenceLattice on N (Johnson quantum and gravity Teper, • Squaw Valley, July 11, 2011 – p.18/25 hep-ph/0012287; Meyer, hep-lat/0508002)

8 search for viable ‘walking 7 M ♦ ♦ technicolour’ theory √σ 6 ♦

5 " r r r supersymmetry 4 3 on lattice ♦ 0−− 2 r 0++ large number 1 0 of colours 0 0.05 0.1 0.15 0.2 0.25 limit 1/N 2 Tuesday, 11 September 2012 Figure 2. Other Q =8supersymmetriclattices The equation of state depends only trivially on N (Caselle et al., 1105.0359 and • 1111.0580)

(+) The lattice field strength is then given by the gauged forward difference Fµν = Dµ Uν and is automatically antisymmetric in its indices. Furthermore ittransformslikealattice2-formand yields a gauge invariant loop on the lattice when contracted with χµν .Similarlythecovariant backward difference appearing in DµUµ transforms as a 0-form or site field and hence can be contracted with the site field η to yield a gauge invariant expression. This use of forward and backward difference operators guarantees that the solutions of the theory map one-to-one with the solutions of the continuum theory and hence fermion doubling problems are evaded [19]. Indeed, by introducing a lattice with half the lattice spacing one can map this Kähler-Dirac fermion action into the action for staggered fermions [22]. Notice that, unlike the case of QCD, there is no rooting problem in this supersymmetric construction since the additional fermion degeneracy is already required by thecontinuumtheory. Many other examples of supersymmetric lattices exist. Figure 2. shows two such lattices arising in the case of eight supercharges – a two dimensional triangular lattice and a generalized hypercubic lattice (including body and face links) in three dimensions. Notice that in all cases almost all fields live on links with the exception of a small number of fermion site fields – the number of those corrresponding to the number of exact supersymmetries preserved in the lattice theory. Furthermore, in all cases the number of fermions exactly fills out multiples of a basic Kähler-Dirac field in the corresponding number of dimensions.

5. Twisted N =4super Yang-Mills In four dimensions the constraint that the target theory possess 16 supercharges singles out a single theory for which this construction can be undertaken – N =4SYM. The continuum twist of N =4thatisthestartingpointofthetwistedlatticeconstruction was first written down by Marcus in 1995 [23] although it now plays a important role in the Geometric-Langlands program and is hence sometimes called the GL-twist [24]. This four dimensional twisted theory is most compactly expressed as the dimensional reduction of a five dimensional theory in which the ten (one gauge field and six scalars) bosonic fields are realized as the components of a complexified five dimensional gauge fieldwhilethe16twistedfermions naturally span one of the two Kähler-Dirac fields needed in five dimensions. Remarkably, the action of this theory contains a Q-exact piece of precisely the same form as the two dimensional theory given in eqn. 6 provided one extends the field labels to run now from one to five. In addition the Marcus twist requires a new Q-closed term which was not possible in the two dimensional theory. 1 S = − Tr #mnpqrχqrDpχmn (14) closed 8 !

The supersymmetric invariance of this term then relies on theBianchiidentity#mnpqrDpFqr =0. The four dimensional lattice that emerges from examining themodulispaceoftheresulting ∗ discrete theory is called A4 and is constructed from the set of ﬁve basis vectors va pointing Future (with increased computing power).. • lattices with physically light up and down quarks in the sea now becoming available - no chiral extrapolation! • very fine lattices (a<0.03 fm) allow b quarks to be treated relativistically rather than with effective theories • large volumes (6 fm across) allow study of hadron resonances/multi-hadron states/small nuclei • very high statistics give access to calculations with more intrinsic noise - flavour singlets, glueball spectrum etc • finite temperature QCD calculations can be extended to different quark formalisms. • the huge space of BSM theories can be explored • not all progress requires improved computational resources but it helps! • results for: LHC, BES, KEK, JLAB, DAFNE, RHIC, FAIR ...

Tuesday, 11 September 2012 map of europe - Google Maps http://maps.google.co.uk/maps?oe=utf-8&client=ﬁrefox-a&q=m...

Address

European landscape - people Many European countries active in lattice field theory*: Europe provides ~50% of worldwide lattice community• of a few thousand.

70% of top-cited papers from hep- lat have some European authors^

• judged from attendance at the annual lattice QCD conference

^ from SPIRES, sampling years 2005-2010 *input to this talk from almost all of them

Tuesday, 11 September 2012

1 of 1 20/08/2012 14:05 European landscape - collaborations

Vary from international collaborations of ~20 people (e.g. Alpha, BMW, CLS, ETM, Hadron Spectrum, HotQCD, HPQCD, QCDSF, RBC-UKQCD, StrongBSM) to smaller groups. Sociologically tricky for theorists, but necessary for access to computing resources. Significant fragmentation of European community around different discretisations of QCD Lagrangian. Good for diversity or inefficient? Some success at bringing people together with EU networks - currently StrongNET and participation in III Hadronphysics 3. Flavianet spun out FLAG (lattice averaging group).

Tuesday, 11 September 2012 European landscape - computing Europe has 7 of world’s top 20 computers* with Pflops speed (32 of top 100 vs USA: 37 of top 100) Lattice QCD has some access to these capability machines, but analysis phase needs large-scale capacity computing. Hardware is provided nationally and piecemeal but PRACE provides important Europe-wide access BUT fundamental JUGENE physics not a top priority ... Lattice QCD not chosen as an application Curie area for the EXASCALE * www.top500.org Fermi DEEP project

Tuesday, 11 September 2012 International Lattice DataGrid Global initiative allowing gluon field configurations to be made publically available for analysis. BUT no resources available to run this ...

US DoE SciDAC initiative Provides steady source of funds for computing hardware and people to develop publically available efficient parallel software for lattice calculations. Smaller European groups use this - would be good to contribute/have our own initiative BUT no resources for this ...

Lattice QCD pushes boundary of supercomputer “grand challenges” (and has led to hardware developments) so is a good technical training environment.

Tuesday, 11 September 2012 Conclusions Europe must maintain a long-term world-leading programme in lattice gauge theory both as vital input for the experimental programme and for the leading-edge technology skills base. This requires sustained investment in computing infrastructure, support and people. To be competitive, we need 10-20M€ per year for dedicated hardware + 4M€ per year for software/algorithm development across Europe. More coordination between researchers would help us argue for this investment and maximise the output from it in research, training and communication of results. We need mechanisms within Europe to encourage this to happen.

Tuesday, 11 September 2012