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1 Opportunities, Challenges, and Fantasies in Lattice QCD Frank Wilczeka∗ aCenter for Theoretical Physics Massachusetts Institute of Technology Cambridge, MA 02139-4307 3 0 Some important problems in quantitative QCD will certainly yield to hard work and adequate investment of 0 resources, others appeardifficult butmay beaccessible, andstill otherswill requireessentially newideas. HereI 2 identify several examples in each class. n a J Therehasbeenanotablerenaissanceofinterest • incremental developments in algorithms 2 andprogressinQCDoverthelastfewyears. This and hardware that have now matured to 2 has come about for many reasons, including the extentthatlatticegaugetheoryhasbe- v come a powerful tool capable of providing • a vigorous experimental program in heavy 1 reliable quantitative information genuinely 4 ion physics explicitly devoted to exhibit- useful for guiding and interpreting experi- 0 ingthefundamentaldynamicsofquarkand mental work, and 2 gluon degrees of freedom and to recreat- 1 ing conditions last seen in the Universe at • increasing anticipation of new frontier 02 ∼10−2 s following the Big Bang, hadronic accelerators, the upgraded Teva- / tron, and especially the Large Hadron Col- t • a vigorous experimental program in heavy a lider (LHC), where QCD will dominate quark physics, including measurements of l - fundamentalCPviolationandweakmixing both input and output. It will provide a p challenging “background” against which to e parameters,thatrequiresaccuratetheoreti- analyze and interpret any essentially new h calcalculationofstrongmatrixelementsto : reach its full potential, phenomena. v Xi • creativedevelopmentsinfinite temperature Lattice gauge theory has been an essential in- gredient in this ferment. This subject now ap- r theory, effective field theory, and numerical a algorithms that dovetail beautifully with pears,Ibelieve,morepromisingandmoreimpor- tantthaneverbefore. Itisblessedwithmanyat- these experimental programs, tractiveresearchprogramsonvarioustimescales. • realization that high density is a regime We can classify these roughly into: of QCD in which weak coupling but nonperturbative methods can be used to give • opportunities – significant problems that tractable yet rigorous models of confine- willalmostcertainlyyieldtohardworkand ment and chiral symmetry breaking, with adequate investment of resources in a rea- possible application to neutron star interi- sonable, predefined period of time, ors (or quark stars!? or strangelets!?) • challenges – more ambitious problems, • development of lattice regularizations that which may or may not yield to incremen- incorporate chiral symmetry, tal development of known techniques, and ∗The research is supported in part by funds provided by • fantasies – grand problems that we can ar- the U.S. Department of Energy (D.O.E.) under coopera- ticulate, but presently lack usable tools to tive research agreements Nos. DE-FC02-94-ER40818 and DE-FG02-91-ER40676. MIT-CTP-3337. address. 2 I’vehadfunusingthisframeworktothinkstrate- Average gically about the future, in general. I’ll use it Hadronic Jets here to touch on three major branches of lattice + - e e rates gauge theory – few-body, many-body, and con- + - ceptual/algorithmicaspects–byprovidingacou- e e event shapes ple of examples in each category. Fragmentation Z width 1. Fundamental Particle Physics ep event shapes 1.1. Opportunities Polarized DIS 1.1.1. The Best Determination of α s Deep Inelastic Scattering (DIS) Amajortriumph of lattice QCDis alreadyen- t decays shrined in the Particle Data Book, as shown in Figure 1. Nonperturbative calculations of heavy Lattice quark spectroscopy provide a precision determi- decay nation of α competitive with the most accurate s determinations from perturbative QCD. There is 0.1 0.12 0.14 probablymoreroomforimprovementonthenon- a (M ) perturbative side, because on that side the rela- s Z tionofthecalculationstoobservablesareinprin- ciple more tightly controlled (no structure func- Figure 1. Leading determinations of the strong tions, jet parameterizations,... ), the experimen- coupling α (M ), taken from the Particle Data s Z tal measurements are very precise,and one is ac- Book. cessing α (or αp,p > 0) effects directly, rather s s thandiggingthemoutascorrections. Itisimpor- trum, wewillbe abletoleveragethis precisionto tant to pursue this direction further, for several gain insight into how unification symmetry and reasons. supersymmetry are broken. Most obviously, a better determination of α , s Finally, I think it’s very legitimate to regard by sharpening input to the perturbative frame- precise determination of α as an end in itself. s work,supports more precise predictions for high- Togetherwiththeelectronmass,theordinaryfine energy experiments. Also, if we are to have con- structureconstantα,andtheupanddownquark fidence in using lattice QCD as a mathematical masses, α is one ofa handful ofparametersthat s tool for determining weak matrix elements, we determines the structure of ordinary matter. It need to make sure it gives consistent, accurate is a unique glory of physics that we demand pre- results for a variety of quantities in spectroscopy cise, quantitative agreementbetweenourcalcula- andelectromagnetictransitions,wheretheunder- tions and reality, whenever comparison is possi- lyingphysicsissecurelyknown. Thesequantities, ble. Enough said. of course, are just those that go into the lattice QCD determination of α . 1.1.2. Exploring Canonical Structures s Precision determination of α will also add to The fact that QCD is a tight, conceptually de- s the power of the unification of couplings calcula- fined theory implies that the structures it pro- tion, which at present provides our best quan- duces are canonical. titative handle on fundamental physics beyond Let me explainwhat I mean by this, by wayof the standard model. The central value is on the an analogy with black hole physics. A feature of low side for supersymmetric grand unified theo- black holes that makes them especially fascinat- ries, but perhaps within the margin of plausible ing andattractiveis that their theoryis soclean. threshold corrections. If and when we start to Thekeytothiscleanlinessisthat,asWheelerput pin down the low-energy supersymmetric spec- it, “blackholeshavenohair”. Specifically,classi- 3 calgeneralrelativityproducesunique predictions 1.2. Challenges for the properties of a black hole, given only its 1.2.1. Precision Values of the mass and angular momentum. This is quite dif- Quark Masses ferentfromstarsorordinarymatter,whosestruc- ture depends on many parameters including el- m and m In contrast to the beautiful u d emental composition, temperature, and process- and inspiring situation regarding α , our present s inghistory. The canonical,nearlyparameter-free determinations of m and m are an embarrass- u d structure of black holes means that it is worth- ment. They are uncertain at the factor-of-two whiletostudysolutionsoftheequationsthatgov- level, at least. Despite their crucial importance ern them minutely, because these solutions are for the structure of the world2, they are by far sharply delimited. the worst measured fundamental parameters of In QCD the structure of all the hadrons is physics. It is a major challenge for lattice gauge canonical, and the structure of protons and neu- theory to improve this situation. trons is especially so since they are stable parti- Again, a comparison may be in order. Quite cles, i.e., discrete eigenstates of the Hamiltonian. properly, tremendous effort has been put into Nucleons are much balder than black holes, for measuring neutrino masses. Yet the precise val- they have no hair, not only classically but even ues of neutrino masses are no more fundamental inquantumtheory. Furthermore,thereisnofree- than light quark masses, and they are much less dom to specify their mass and angular momen- important for the structure of the world. In any tum arbitrarily. Also protons and neutrons can case,itisunlikelywewillreachaprofoundunder- bewellmodeledinQCDLiteTM (the idealization standingofonewithouttheother,sinceinunified of QCD using only massless u and d quarks). In theories quarks and leptons come as a package. that approximation, they are entirely conceptu- Besides this general sort of motivation, there ally determined structures, allowing no continu- are several more specific reasons to be especially ous parameters whatsoever! interestedinthe valuesofm andm . We would u d And they can be accessed experimentally. In- like to be absolutely sure that m 6=0, since this u deed, the answer to many “practical” questions empowers the P- and T-violating θ parameter, relies on understanding their structure. For ex- which provides the prime motivation for Peccei- ample, structure functions are vital to the in- Quinnsymmetryandaxions,withtheirprofound terpretation of accelerator experiments, since we physicalandcosmologicalimplications. Theratio use nucleons as projectiles. Notably, the primary m /m partially determines the axion coupling, u d mechanism for Higgs particle production at the and will play a vital role in pinning down the LHC will be gluon fusion, gg →h througha top- underlying microscopic parameters if and when quark loop. To predict the rates, and to check axions are observed. The proton-neutron mass for possible deviations from standard model ex- difference, whose value makes all the difference pectations for the h coupling, we obviously need for nuclear stability and for stellar and cosmic to know the gluon distribution in the proton ac- nucleosynthesis – and thereby to the structure curately. and composition of matter – of course depends Another sort of canonical structure in QCD is quitedirectlyuponm −m . Finally,thereisthe u d thefluxtubebetweenheavyquarksources. There prospect of making precise quantitative predic- are very interesting questions concerning its spa- tionsforappropriatemeasurableisospin-violating tialstructureandmodesofexcitation,andspecif- processes. Especiallypromisinginthisregardare ically how accurately it can be modeled as a bag transitionsbetweenheavyquark-heavyantiquark or an elementary string. states bysoftπ0 emission. Conversely,these processes allow a different, and potentially cleaner, path to the determination of m −m /m +m u d u d 2See,inthisconnection, thefollowingdiscussionofquan- titativeanthropics inSection1.3.2. 4 than the traditional ways involving electromag- netic mass differences or light hadron processes. m Thestrangequarkmassplaysasingu- s lar role in QCD. For all the other quark masses, it makes good sense to expand either in m/Λ or Λ/m,whereΛistheprimaryQCDscale,vaguely in the neighborhoodof300MeV. For the strange quark neither expansion is clearly appropriate. A fundamental question that I find quite interesting, and which is quite important for the in- terpretation of numerical work on light hadron spectroscopy, is how this spectroscopy depends on m . Specifically, how does the ratio-sequence s fπ : mρ : mN : m∆ vary as ms is taken from infinity down to zero? The (un?)reasonable success of the valence-quark/bag model, and of the quenched approximation, seem to suggest very mild dependence, but the instanton liquid model seems to suggest more dramatic effects. There are several major issues in QCD, especially regardingits phase structure, that hang on Figure 2. Constraints on the standard model pa- the value of m . I’ll mention them in due course. s rametersρandη(onesigmaconfidencelevel. For the standard model to be correct, they must be 1.2.2. Weak Matrix Elements that restricted to the region of overlap of the solidly Add Value to Experiments coloredbands. Theupperfiguredisplaysthecon- ′ The precise determination of ǫ/ǫ in K meson straintsas they existtoday. The lowerfiguredis- decays, and the cornucopia of results on B me- playstheconstraintsasthe wouldbeifthe preci- son decays and CP violation emerging from the sion of the lattice gauge theory calculation were BABAR and BELLE collaborations, are beauti- reducedto3%. FiguresprovidedbyR.Patterson, fulandoutstandingachievementsinexperimental Cornell University. physics. But to extract this work’s full potential we’llneedto produce theoreticalcalculations of hadronic matrix elements with comparableac- cal precision improved by a few teraflop-years of curacy. Only after controlling this “QCD back- computation. You can see the value added. ground” can we make inferences regarding the precise values of weak mixing angles and phases, 1.3. Fantasies and the possible influence of physics beyond the standard model. 1.3.1. Pinning Down (g−2) µ Herethereisapicturethatisworthathousand Anotherrecentheroic,beautiful,andoutstand- words. Figures2a and2bboth depictallowedre- ing achievement in experimental physics is the gions for a varietyofexperimentalmeasurements precision measurement of the anomalous mag- whoseresultsdependonthe weakmixingparam- netic moment of the muon, (g − 2) , reported µ eters ρ,η, in the Wolfenstein parametrization of by the Brookhaven g −2 collaboration. In this the CKM matrix. There is a consistent determi- case too, the significance of this result would be nation, and no evidence for physics beyond the greatly enhanced if we could do a better job on standard model, in the region of overlap. Fig- theQCDingredients,herelow-energyvacuumpo- ure 2a is the contemporary situation; Figure 2b larization and light-by-light scattering. Our in- assumes the same data, but with the theoreti- ability to do these calculations brings home the 5 general point that the existing technology of lat- Ifwecouldmakeastrongcasethatsmallvaria- ticegaugetheoryisquitefragile,inthesensethat tionsinm andm , tobespecific, woulddestroy u d itcanbe usedto calculateafew things verywell, the possibility of intelligent observers, then that but most others essentially not at all. So my would be prima facie evidence for the relevance fantasy here is of techniques that degrade more of the anthropic principle, since these quantities gracefully. appeartobeveryremotelyconditionedbycentral Actually, in the specific case of (g −2) , nu- principles of unified field theories, string theory, µ merical work might supply an important input or anything else. even while falling well short of the “fantasy” of Fortunately, this subjecthas anempiricalside. first-principles calculation. At present the ma- We can searchfor variation in the physical “con- jor uncertainty in the standard model prediction stants” as a function of location or of time. For arises from a discrepancy between the value of theorists, there is the challenge to quantify what the hadronic vacuum polarization as extracted welearnaboutpossiblechangesinthefundamen- extracted from e+e− annihilation or from τ de- tal constants from cosmic nucleosynthesis and cay. Evena fairly crude numericaldetermination from measurements of hyperfine splittings at dif- of the vacuum polarization at a single Euclidean ferent times, for example. One of the most pow- momentum might tell us which to believe. erful constraints on variation of constants comes from the Oklo natural reactor,which would have 1.3.2. Quantitative Anthropics left a different residue had an accidental near- At the frontiers of where physics has come degeneracy in the levels of Sm149 + n and Sm150 under control, we can start to contemplate ex- not been present two billion years ago! Clearly pansion into metaphysics. Specifically, knowing some of the requisite calculations enter further the continuous parameters (in mathematicalese, into the realm of fantasy than others. But it moduli) of our world-theory, we can ask – and is a valid and I think profound general obser- hope to answer! – a precise scientific version of vation that the freedom to do numerical experi- thegreatquestion: Couldthingshavebeendiffer- mentswithunrealisticvaluesofmodulicouldhelp ent? It seems to me that this question has taken bring this particular branch of metaphysics into on a new urgency due to recent developments in the realm of hard science. physics, as I have discussed at length elsewhere. Therearetwoimportantrefinementsofthe ques- 2. Many-Particle Physics tion: 2.1. Opportunities • AnthropicPrinciple: Arethevaluesofsome 2.1.1. The T 6= 0 Equation of State or all of the moduli determined by the re- A major achievement of lattice gauge theories quirement that there should be intelligent in recent years has been to map out the energy observers capable of detecting them? This and pressure, as a function of temperature, for could be true if the Universe were inho- various idealizations of QCD. It is quite striking mogeneous on ultra-large scales, with vari- how the number of effective degrees of freedom able values of the moduli. Then the prin- ascends to something near the free quark-gluon ciple would emerge as a sort of converse to value, at a remarkably low temperature by the naturalselection,selecting out the environ- standards of typical hadron masses. This phe- ments to which intelligence might success- nomenonisoneofthemaininspirationsforexper- fully adapt. imentalprogramstostudyquark-gluonplasmaat • Misanthropic Principle: Are the values of RHIC and eventually ALICE. some or all of the moduli determined by A fully realistic simulation, with good imple- historical accident? This could arise in the mentationofchiralsymmetryandaccuratevalues same circumstances. It is the analogue of of the quark masses, is called for. The value of genetic drift. thestrangequarkmassisespeciallyimportant. It 6 is likelythatforsufficientlyheavystrangequarks creases the chemical potential, there is a sharp thereisafirst-orderphasetransition,whereasbe- phase transition. We are justified in calling this lowsomethresholdvaluethereisonlyacrossover. ahadron-to-quarktransition,sincetherespective Averyspecific,concrete,andachievablegoalisto phases on either side evolve without further dis- locatethecriticalstrangequarkmass,andtover- continuities into matter accurately described as ify (or not) existing indications that the physical nearly free hadrons or nearly free quarks. strange quark mass is subcritical. It is also im- As the temperature rises the distinction be- portantto quantify the differentialstrangequark tween hadronic and quark matter becomes less contribution to pressure and density. This would significant;thediscontinuitiesassociatedwiththe be an interesting measure of how nearly free first-order transition grow smaller and eventu- the quarks in the plasma are, and could provide ally vanish altogether. The precise point in the a baseline for interpreting enhanced strangeness temperature-chemical potential plane where the multiplicities observed in heavy ion collisions. distinction vanishes is called the tricritical point. It is associated with a second-order phase tran- 2.1.2. Exploring Small µ/T sition, and critical fluctuations. There is some Finite chemical potential has long been terra chance that the effects of such fluctuations could incognita for numerical simulation of QCD, be- lead to observable consequences in heavyion col- cause configuration by configuration the usual lisions. path integral for the partition function contains The challenge for lattice gauge theory is very a complex phase, andthere are big cancellations. clear and concrete: locate this point! It is a no- These cancellations are less severe for small table landmark in the QCD phase diagram, and µ/T and not too large volumes, and are absent knowingits locationwouldbeveryhelpfulforor- forimaginaryµ. Explorationoftheseregimeshas ganizing the experimental explorationof that di- begun, but much more remains to be done, espe- agram. cially in working towards realistic quark masses There is probably a close relationship between and reliable error estimates. There is a potential this physicaltricriticalpoint anda more theoret- connection to heavy ion experiments, where the ical tricriticalpoint implicit in our earlier discus- effective chemical potential varies with rapidity. sions. That one arises in considering behavior in the temperature-m (strange quark mass) plane. 2.2. Challenges s For very small m one expects a first-order tran- s 2.2.1. Locate the True Critical Point sition, which weakens as ms increases, and van- There are good reasons to think that at zero ishes altogether at some critical mcrit.. A very s temperature,asonevariesthechemicalpotential, interesting possibility is that the theoretical tri- thereisafirst-orderquantumphasetransitionbe- critical point evolves into the physical one, mov- tween nuclear and quark matter. There may in ing off into nonzero values of µ above mcrit.. If s factbeseveraltransitionsofvariouskinds,includ- so, it ought to be possible to locate the physical ing meson condensation and alternative pairings tricritical point for values of ms just above mcsrit. in color superconductivity. I expect that most of by exploiting small µ/T techniques. theseareessentiallylow-temperaturephenomena, leaving only a single first-order transition above 2.2.2. Measure Critical Behavior ∼ 50 MeV. In any case, for purposes of discus- The renormalization group analysis of QCD sion let me assume this. Within this context Lite – that is, SU(3) color gauge theory with one discovers a somewhat unconventional but I two massless quarks – suggests the existence of thinksharpandinsightfulperspectiveonthedef- a second-order phase transition at finite temper- inition of “quark-gluon plasma”. At sufficiently ature, associated with the restoration of chiral high (but not too high) chemical potential, as symmetry. Universality arguments suggest that one increases the temperature, or at sufficiently it should exhibit the critical exponents of a 4- low (but not too low) temperature, as one in- component magnet, governed by the O(4) linear 7 σ model. There are many precise predictions for other novel technique might be necessary. the singular behavior of effective meson masses, After this is done, it will still remain to ad- specific heat, magnitude of the condensate, etc., dress the relatedbut perhaps still more challeng- based on this picture. One can go on to predict ing challenge of measuring the tricritical behav- analytically the equation of state near the criti- ior. They are expected to be governed by mean cal point, also allowing for small but nonzero – field theory, up to calculable logarithmic correc- conceivably, even realistic – quark masses. tions. In addition there are crossover exponents It would be a landmark achievement to check andacriticalequationofstateawaitingmeasure- someofthesepredictions,sayspecificallytomea- ment. sure a critical exponent that can be clearly dis- 2.3. Fantasies tinguished from mean field theory. This would providewelcomeconfirmationofsomeofourdeep- 2.3.1. Reinvent Nuclear Physics est prejudices about the nature of chiral symme- The historical origin of strong-interaction try and its restoration; conversely, any deviation physics, and its main appearance in the natural from the predictions would force us to rethink world, is of course the physics of atomic nuclei. some fundamentals. Although with modern QCD we have achieved Afewyearsagosomedoubtswereraisedabout an extraordinarily beautiful and “complete” the- the applicability of universality in this context. ory of the strong interaction, this fundamental They have been convincingly addressed, but live progress has not greatly advanced our under- on in modified form as valid questions about the standing of nuclei. There is an obvious, good scopeofuniversalitythatdeservetobeaddressed reason for this. The energy scales of interest in quantitatively. While now no one doubts that nuclearphysicsareoforderafew MeV,while the thereisuniversalbehaviornearthecriticalpoint, basic QCD scale is a hundred times this. So es- it remains to quantify how wide the critical re- sentially nuclear phenomena are governed by the gion is, and how important are the critical fluc- residuaofcomplicatedcancellationsamongmuch tuations against the background of conventional larger basic QCD forces. It is probably unrealis- thermodynamicbehavior. Theseareabstractbut tic, and unfruitful, to contemplate a brute force precise forms of the very basic question, What is assault from first principles, since small errors in the hadronic fluid like near the phase transition? largecancellingquantitieswilldominatethecom- Isitdominatedbycolorgaugefields(gluedegrees puted answers. offreedom),sothatthecollectivebehaviorofthe But it would be dereliction of duty, and a lost critical π and σ fields are a minor side-show? Or opportunity,to abandonthe fieldentirely. Ama- isitbestpicturedasastronglyinteractingmeson jorgoaloflatticegaugetheoryshouldbeto com- gas? In the former case, the critical singularities putetheparametersofeffectivefieldtheories,for- will be blips on a smooth background;in the lat- mulated in terms of pion and nucleon degrees of ter case, they will be fractionally large or even freedom, from first principles. (Then the effec- dominant. The location of the critical point (Tc tive theories could be turned over to many-body well below glueball masses, or even the scale set theorists.) Specifically, we should verify that the by the QCD string tension) and that it is well theory generates such key features as the hard below the Tc′ for deconfinement in the pure glue core and the spin-orbit force, which are central theory,suggesttomethatthelatterresultismore to qualitative aspects of nuclearphysics. And we likely, but I’m not at all certain. shouldgetaconvincingqualitativeexplanationof Todojusticetotheseproblems,onemustboth these effects, backed up by exploration of QCD respect the chiral symmetry and work in large variants (see below). spatial volumes, so as to allow the appropriate It would also be amusing, and instructive, to modesto existandhavetheir proper scope. This explorethelimitsof“nuclear”physicsasweknow willbe verycostly to do directly, souse of a real- it. Why are there nuclei at all? In other words, space (infrared!) renormalization group or some Whydoesthenuclearforcesaturate? Orinmod- 8 ern terms, Why do protons and neutrons in nu- ical integration. As a result many fundamental clei retain their individual identity, rather than questions become accessible, including agglomerating into a single shared bag? Does this occur for two-color QCD, or for supersym- • quantitative spectroscopy of the pseu- metric QCD? – maybe not, since in both these doscalar mesons. Since the masses of the cases there are bosonic baryons. Does it occur octet are most sensitive to the values of in a world without light quarks? In another di- smalllightquarkmasses,themesonmasses rection: How far is our world from a (literally) can be used in principle to determine the strangeworld,inwhichthegroundstateforbary- light quark masses, which then become in- onic matter has nonzero strangeness? How much puts to calculations of flavor SU(3) break- lighter would the strange quark have to be? ing and isospin violation, including “elec- tromagnetic” mass differences; 2.3.2. Reinvent Extreme Astrophysics There is a chance to break new ground, andto • questionsofphasestructure. Ashasalready advance our understanding of some of the most appeared in our earlier discussions, chiral violent and crucial events in cosmic evolution, symmetryrestorationisamajorfeaturedis- by getting a better handle on the behavior of tinguishingdifferentphases,andconditions hadronic matter in extreme conditions. Contem- the properties of collective modes at the porary astrophysical modeling of supernova ex- phase transitions, it is important in these plosions and of the structure and evolution of contexts to implement it accurately; compact stars is largely based on poorly con- • study of axial baryon number violation. trolled extrapolations of models abstracted from Axial baryon number (more precisely, ax- nuclear and resonance physics well beyond their ial N +N ) is broken intrinsically in the region of validity, into regimes where the parti- u d quantum theory by an anomaly, and to a cles are strongly interacting or even overlapping. small extent by quark masses, and spon- In otherwords,itis sophisticatedguesswork. We taneously by q¯q condensation. Simulations havegoodasymptotictheoriesbasedonQCDfor with large artifactual quark masses mangle the highesttemperaturesanddensities,butthere this quasi-symmetry, just as they mangle ismuchuncertaintyaboutwhereandhowasymp- chiralSU(2)×SU(2). Foralltheassociated topia is approached. We may look forward to a ′ questions of η phenomenology, squeezing wealthofrelevantnew data fromx-ray,neutrino, out of instantons and approximate U(1) and gravitational wave detectors, in addition to A restoration at high temperature, and ax- dramatic enhancements of the traditional optical ion cosmology, accurate implementation of and radio windows. Can we do justice to this in- classical U(1) is essential; formation? Specifically, can we devise concrete A signatures for quark matter and color supercon- • tests of the instanton liquid model. This ductivity in“neutronstar”interiors,orfor phase model predicts significant dependence of transitions during or in the immediate aftermath certain hadronic quantities on light quark of supernova explosions or “neutron star” merg- masses. ers? 3.1.2. Explore QCD Variants 3. Conceptual and Algorithmic Issues Besides enabling the noble bread-and-butter workofcomputingpropertiesofreal-worldQCD, 3.1. Opportunities lattice gauge theory offers us the opportunity to 3.1.1. See the Effects of Chiral Fermions explorevariants,forexamplewithdifferentquark Thanks to some very ingenious recentwork we mass spectra, different representations, or differ- now know how to implement chiral symmetry in entgaugegroups. Ihavealreadymentionedmany a precisewayinlattice QCD, suitablefor numer- possibilitiesforthecreativeuseofsuchflexibility, 9 in quantitative anthropics, in testing our under- inal Wilsonian program of the renormalization standingofthephasestructure,andintestingin- group,tointegrateuptoacoarselattice,andthen stantonliquid ideas. This list is far from exhaus- solve the coarse lattice theory by other means tive. As a general remark, we can test proposed (presumably, some form of strong coupling ex- qualitative explanations of phenomena in QCD pansion) might be revisited in light of the vast by seeing whether they predict the correct direc- increase in knowledge and computer power over tion of change in the phenomena as the underly- the last 25 years. ing parameters change. For example, a question 3.2.2. Empower Reduced I find fascinating is simply: Why does the naive Continuum Theories quark model work so well? In thinking about Extremely useful andimportantphenomenolo- this question, it would be very useful to identify gies have been constructed using reductions of QCD-variants where the naive model begins to QCD. They are based in one way or another on fail! (PerhapssimplyQCDwithmanylightquark integrating out hard degrees of freedom, using flavors?) asymptotic freedom. The primeval example is Let me just mention one further potential ap- the analysisof deep inelastic scattering using the plication, to clear up a long-standing debate operator product expansion, which reduces what whose origins are in the prehistory of QCD but would otherwise be an entirely hopeless prob- that remains contentious even now. This is the lem to the evaluation of a discrete series of low- question of glueballs. When a particle is ob- energyoperatormatrixelements,oralternatively served experimentally, it does not come labeled the structurefunctions ofwhichthey arethe mo- “quark-antiquark”or“glueball”,noris there any ments. Otherapplications,similarinspirit,bring strictobjectivecriteriontodistinguishthem. The in Isgur-Wise functions and fragmentation func- whole distinction is tied up with the naive quark tions. Many of these are poorly determined, yet model, which has no firm basis in quantum field they are important for the interpretationof fron- theory, and ignores the inevitability of mixing. tier experiments. It would be a major service, as Nevertheless there is a simple objective ques- well as an intellectual achievement, to compute tioncorrelatedwiththis classification: If wevary more of them from first principles. quark masses, how does the mass of the particle respond? 3.3. Fantasies 3.2. Challenges 3.3.1. Expand the Frontiers of 3.2.1. Reduce Lattice Degrees of Freedom Effective Computation Full-scale QCD simulations tend to be un- Throughout most of this talk I have been dis- wieldy, inflexible, and opaque. One would like to cussinghownumericalworkcanaddressquantita- haveproceduresthatfallsomewherebetweenfull- tively demanding and/or sophisticated questions scale simulation of the microscopic theory and about a particular strongly interacting quantum rough semiphenomenological modeling. field theory, QCD, or slight variants of it. Of There are several suggestions regarding what course, at present this is the only precisely de- are the important degrees of freedom that could fined,stronglycoupledquantumfieldtheorywith be used as the basis for a reduced description of proven 0relevance to the description of Nature. QCD that keeps contact with the microscopics, But a lot of brainpower has been expended on includingcentralvortices,abelianprojections,in- special limits of gauge theories, more or less dis- stantons, and strings. The challenge is to pro- tant cousins of realistic QCD, for which simplifi- mote such proposals into reliable quantitative cations occur and some nonperturbative results tools, or at least to identify some limit in which can be obtained analytically. Notable among they become good, systematic approximations. these are large N limits and theories with vari- Alternatively, more directly numerically based ous degrees of supersymmetry. These and many approachesmightriseto the challenge. The orig- other theories have been discussed in the context 10 ofattemptsatunification,aspossiblecatalystsof are in play, even in the simplest of physical cir- electroweak,unified,orsupersymmetrybreaking. cumstances. It wouldbe verydesirable to haveevenrelatively Does this suggest that there are much more crude resultsaboutgeneralclassesofgaugetheo- powerfulforms of computationthat we mightas- ries. Ironically, the limits that provide analytical pire to tap into? Does it connect to the emerg- simplifications seem to be especially difficult to ing theory of quantum computers? These mus- simulatenumericallyusingcurrentlyknowntech- ings suggest some concrete challenges: Could a niques. So an attractive fantasy is to imagine quantum computer calculate QCD processes effi- numerical techniques capable of coping with ciently? Could it defeat the sign problem, that plagues all existing algorithms with dynamical • large N gauge theories, fermions? Could it do real-time dynamics, which • chiral gauge theories, and is beyond the reach of existing, essentially Eu- clidean,methods? Or,ifallthatfails,doesitsug- • supersymmetric gauge theories. gest some limitation to the universality of computation? There is, after all, no guarantee that Recently there has been remarkable progress in modelsabstractedfromour(realorimagined)im- formulatingchiralfermiontheories(gaugingthese plementations of how to compute things capture symmetries is not straightforward) and super- theprocessbywhichNaturedecideshowtooper- symmetric lattice gauge theories. The construc- ate. Maybe She just “does it”, without comput- tions are very technical and delicate, and not as ing anything. general as one might hope for. Furthermore, po- A different sort of limitation to effective com- tential Monte Carlo simulations suffer from the putation has become a major theme of recent in- fermion sign problem. So the fantasy here is to vestigations in classical dynamics. It is the ubiq- develop any usable algorithm. For large N the uityofextremesensitivitytoinitialconditions,in standard is higher. In principle, we know how to problems ranging from the double pendulum to dothecalculations. Butonewouldliketohaveal- the long-term behavior of the Solar System. Are gorithms that somehow simplify, rather than be- there hadronicpropertiesthatdepend sensitively coming increasingly unwieldy, as N increases! on the fundamental parameters we use to com- Perhaps needless to say, even bigger game putetheminQCD(specifically,quarkmasses),or would come into sight if one could develop us- onthesmall“nonrenormalizable”correctionsdue able nonperturbative methods for dealing with to other interactions, or discretization artifacts? fermions in general, finite density, or real-time This is plausible, since in an effective descrip- dynamics. tion we will have many channels with the same 3.3.2. Define the Limits of quantum numbers, or in the language of dynam- Effective Computation ical systems overlapping resonances, contribut- I find it disturbing that it takes vast computer ing to high orders in perturbation theory. So resources,andcarefullimitingprocedures,tosim- even if the parameters of the effective theory de- ulatethemassandpropertiesofaprotonwithde- pend smoothly on the fundamental inputs, phys- centaccuracy. Nature,ofcourse,getssuchresults ical quantities such as exclusive scattering am- fastandeffortlessly. Buthow,ifnotthroughsome plitudes at intermediate energies may not. This kind of computation, or a process we can mimic wouldprovidearationalexplanationfortheterri- by computation? We are accustomed to the idea bledifficultieswe’vehadindevisingreasonableal- that simulation of complex systems, or systems gorithms for computing such quantities. But our with many elementary parts, may be slow. But understandingofquestionslikethisisnotevenin here we are dealing with the simplest of objects its infancy. in an ideally simple physical theory. Of course the underlying phenomenon is that in quantum field theory many interacting degrees of freedom