Instantons in QCD

T. Scha¨fer Institute for Nuclear Theory, Department of Physics, University of Washington, Seattle, Washington 98195

E. V. Shuryak Department of Physics, State University of New York at Stony Brook, Stony Brook, New York 11794

The authors review the theory and phenomenology of instantons in quantum chromodynamics (QCD). After a general overview, they provide a pedagogical introduction to semiclassical methods in quantum mechanics and field theory. The main part of the review summarizes our understanding of the instanton liquid in QCD and the role of instantons in generating the spectrum of light hadrons. The authors also discuss properties of instantons at finite temperature and how instantons can provide a mechanism for the chiral phase transition. They give an overview of the role of instantons in some other models, in particular low-dimensional sigma models, electroweak theory, and supersymmetric QCD. [S0034-6861(98)00902-7]

CONTENTS IV. Towards a Theory of the Instanton Ensemble 355 A. The instanton interaction 355 1. The gauge interaction 355 I. Introduction 324 2. The fermionic interaction 358 A. Motivation 324 B. Instanton ensembles 358 B. Physics outlook 325 C. The mean-field approximation 359 1. Hadronic structure 325 2. Scales of nonperturbative QCD 325 D. The quark condensate in the mean-field 3. Structure of the QCD vacuum 326 approximation 361 4. QCD at finite temperature 328 E. Dirac eigenvalues and chiral symmetry breaking 362 C. The history of instantons 328 F. Effective interaction between quarks and the 1. Discovery and early applications 328 mean-field approximation 363 2. Phenomenology leads to a qualitative picture 329 1. The gap equation for Nfϭ1 364 3. Technical development during the eighties 329 2. The effective interaction for two or more 4. Recent progress 330 flavors 364 D. Topics that are not discussed in detail 330 G. Bosonization and the spectrum of pseudoscalar II. Semiclassical Theory of Tunneling 331 mesons 366 A. Tunneling in quantum mechanics 331 H. Spin-dependent interactions induced by 1. Quantum mechanics in Euclidean space 331 instantons 366 2. Tunneling in the double-well potential 333 V. The Interacting Instanton Liquid 367 3. Tunneling amplitude at one-loop order 333 A. Numerical simulations 367 4. The tunneling amplitude at two-loop order 335 B. The free energy of the instanton ensemble 368 5. Instanton/anti-instanton interaction and the C. The instanton ensemble 369 ground-state energy 336 D. Dirac spectra 370 B. Fermions coupled to the double-well potential 338 E. Screening of the topological charge 371 C. Tunneling in Yang-Mills theory 340 VI. Hadronic Correlation Functions 372 1. Topology and classical vacua 340 A. Definitions and generalities 372 2. Tunneling and the Belavin-Polyakov- B. The quark propagator in the instanton liquid 374 Schwartz-Tyupkin instanton 341 1. The propagator in the field of a single 3. The theta vacua 342 instanton 374 4. The tunneling amplitude 343 2. The propagator in the instanton ensemble 375 D. Instantons and light quarks 344 3. The propagator in the mean-field

1. Tunneling and the U(1)A anomaly 344 approximation 376 2. Tunneling amplitude in the presence of light C. Mesonic correlators 377 fermions 346 1. General results and the operator product III. Phenomenology of Instantons 347 expansion 377 A. How often does tunneling occur in the QCD 2. The single-instanton approximation 378 vacuum? 347 3. The random-phase approximation 379 B. The typical instanton size and the instanton 4. The interacting instanton liquid 380 liquid model 348 5. Other mesonic correlation functions 382 C. Instantons on the lattice 349 D. Baryonic correlation functions 382 1. The topological charge and susceptibility 349 1. Nucleon correlation functions 383 2. The instanton liquid on the lattice 351 2. Delta correlation functions 384 3. The fate of large-size instantons and the beta E. Correlation functions on the lattice 384 function 352 F. Gluonic correlation functions 385 D. Instantons and confinement 354 G. Hadronic structure and n-point correlators 387

Reviews of Modern Physics, Vol. 70, No. 2, April 1998 0034-6861/98/70(2)/323(103)/$35.60 © 1998 The American Physical Society 323 324 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

VII. Instantons at Finite Temperature 388 on the other is still huge. While some fascinating discov- A. Introduction 388 eries (instantons among them) have been made, and im- 1. Finite-temperature field theory and the portant insights have emerged from lattice simulations caloron solution 388 2. Instantons at high temperature 389 and hadronic phenomenology, a lot of work remains to 3. Instantons at low temperature 391 be done in order to unite these approaches and truly 4. Instanton interaction at finite temperature 392 understand the phenomena involved. B. Chiral symmetry restoration 393 Among the problems the field is faced with is a diffi- 1. Introduction to QCD phase transitions 393 culty in communication between researchers working on 2. The instanton liquid at finite temperature and different aspects of nonperturbative field theory, and a IA molecules 396 shortage of good introductory material for people inter- 3. Mean-field description at finite temperature 397 4. Phase transitions in the interacting instanton ested in entering the field. In this review we try to pro- model 399 vide an overview of the role of instantons in field theory, C. Hadronic correlation functions at finite with a particular emphasis on QCD. Such a review is temperature 401 certainly long overdue. Many readers certainly remem- 1. Introduction 401 ber learning about instantons from some of the excellent 2. Temporal correlation functions 401 papers (Callan, Dashen, and Gross, 1978a) or introduc- 3. U(1)A breaking 402 tory reviews (Coleman, 1977; Vainshtein, Zakharov, and 4. Screening masses 404 Shifman, 1982) that appeared in the late seventies or D. Instantons at finite temperature: Lattice studies 405 early eighties, but since then there have been very few VIII. Instantons in Related Theories 406 publications addressed to a more general audience.1 The A. Two-dimensional theories 406 1. The O(2) sigma model 406 only exceptions are the book by Rajaraman (1982), 2. The O(3) sigma model 407 which provides a general discussion of topological ob- B. Instantons in electroweak theory 408 jects but without particular emphasis on applications, C. Supersymmetric QCD 410 and a few chapters in the book by Shuryak (1988c), 1. The instanton measure and the perturbative which deal with the role of instantons in hadronic phys- beta function 410 ics. All of these publications are at least a decade old. 2. Nϭ1 supersymmetric QCD 411 Writing this review, we had several specific goals in 3. Nϭ2 supersymmetric gauge theory 412 mind, which are represented by different layers of pre- IX. Summary and Discussion 414 sentation. In the remainder of Sec. I, we provide a very A. General remarks 414 general introduction into instanton physics, aimed at a B. Vacuum and hadronic structure 414 C. Finite temperature and chiral restoration 415 very broad readership. We shall try to give qualitative D. The big picture 416 answers to questions like: What are instantons? What Acknowledgments 419 are their effects? Why are instantons important? What is Appendix: Basic Instanton Formulas 419 the main progress achieved during the last decade? This 1. Instanton gauge potential 419 discussion is continued in Sec. III, in which we review 2. Fermion zero modes and overlap integrals 420 the current information concerning the phenomenology 3. Properties of ␩ symbols 420 of instantons in QCD. 4. Group integration 421 Section II is also pedagogical, but the style is com- References 421 pletely different. The main focus is a systematic devel- opment of the semiclassical approximation. As an illus- I. INTRODUCTION tration, we provide a detailed discussion of a classic example, the quantum mechanics of the double-well po- A. Motivation tential. However, in addition to the well-known deriva- tion of the leading-order WKB result, we also deal with Quantum chromodynamics (QCD), the field theory several modern developments, like two-loop corrections, describing the strong interaction, is now more than 20 instantons and perturbation theory at large orders, and years old, and naturally it has reached a certain level of supersymmetric quantum mechanics. In addition to that, maturity. Perturbative QCD has been developed in we give an introduction to the semiclassical theory of great detail, with most hard processes calculated beyond instantons in gauge theory. leading order, compared to data, and compiled in re- Sections IV–VII make up the core of the review. views and textbooks. However, the world around us can- They provide an in-depth discussion of the role of in- not be understood on the basis of perturbative QCD, stantons in QCD. Specifically, we try to emphasize the and the development of nonperturbative QCD has connection between the structure of the vacuum, had- proven to be a much more difficult task. ronic correlation functions, and hadronic structure, at This is hardly surprising. While perturbative QCD both zero and finite temperatures. The style is mostly could build on the methods developed in the context of quantum electrodynamics, strategies for dealing with the nonperturbative aspects of field theories first had to be 1A reprint volume that contains most of the early work and a developed. The gap between hadronic phenomenology number of important technical papers was recently published on the one side and exactly solvable model field theories by Shifman (1994).

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 325 that of a review of the modern literature, but we have While all these models provide a reasonable descrip- tried to make this part as self-contained as possible. tion of static properties of the nucleon, the pictures of The last two sections, VIII and IX, deal with many hadronic structure they suggest are drastically different. fascinating applications of instantons in other theories Although it is sometimes argued that different models and with possible lessons for QCD. The presentation is provide equivalent, dual descriptions of the same phys- rather cursory, and our main motivation is to acquaint ics, it is clear that not all of these models can be right. In the reader with the significant problems and to provide a the MIT bag model, for example, everything is deter- guide to the available literature. mined by the bag pressure, while there is no such thing in the Skyrmion, and the scale is set by chiral symmetry B. Physics outlook breaking. Quark models attribute the nucleon-delta mass splitting to perturbative one-gluon exchange, while it is due to the collective rotation of the pion field in 1. Hadronic structure soliton models. In this section, we should like to provide a brief out- In order to make progress, two shifts of emphasis line of the theory and phenomenology of hadronic struc- have to be made. First, it is not enough just to reproduce ture and the QCD vacuum. We shall emphasize the role the mass and other static properties of the nucleon. A the vacuum plays in determining the structure of the successful model should reproduce the correlation func- excitations and explain how instantons come into play in tions (equivalent to the full spectrum, including excited making this connection. states) in all relevant channels—not just baryons, but There are two approaches to hadronic structure that also scalar and vector mesons, etc. Second, the structure predate QCD, the quark model and current algebra. of hadrons should be understood starting from the struc- The quark model provides a simple (and very success- ture the QCD vacuum. Hadrons are collective excita- ful) scheme based on the idea that hadrons can be un- tions, like phonons in solids, so one cannot ignore the derstood as bound states of nonrelativistic constituent properties of the ground state when studying its excita- quarks. Current algebra is based on the (approximate) tions. SU(2)LϫSU(2)R chiral symmetry of the strong interac- tion. The fact that this symmetry is not apparent in the 2. Scales of nonperturbative QCD hadronic spectrum led to the important concept that chi- ral symmetry is spontaneously broken in the ground In order to develop a meaningful strategy, it is impor- state. Also, since the ‘‘current’’ quark masses appearing tant to establish whether there is a hierarchy of scales as symmetry-breaking terms in the effective chiral La- that allows the problem to be split into several indepen- grangian are small, it became clear that the constituent dent parts. In QCD there is some evidence that such a quarks of the nonrelativistic quark model have to be hierarchy is provided by the scales for chiral symmetry effective, composite objects. breaking and confinement, ⌳␹SBӷ⌳conf . With the advent of QCD, it was realized that current The first hint comes from perturbative QCD. Al- algebra is a rigorous consequence of the (approximate) though the perturbative coupling constant blows up at chiral invariance of the QCD Lagrangian. It was also momenta given roughly by the scale parameter Ϫ1 clear that quark confinement and chiral symmetry ⌳QCDϳ0.2 GeVӍ (1 fm) (the exact value depends on breaking are consistent with QCD, but since these phe- the renormalization scheme), perturbative calculations nomena occur in the nonperturbative domain, there was are generally limited to reactions involving a scale of at no obvious way to incorporate these features into had- least 1 GeV Ӎ(0.2 fm) Ϫ1. ronic models. The most popular model in the early days A similar estimate of the scale of nonperturbative ef- of QCD was the MIT bag model (DeGrand et al., 1975), fects can be obtained from low-energy effective theories. which emphasized the confinement property of QCD. The first result of this kind was based on the Nambu and Confinement was realized in terms of a special boundary Jona-Lasino (NJL) model (Nambu and Jona-Lasinio, condition on quark spinors and a phenomenological bag 1961). pressure. However, the model explicitly violated the chi- This model was inspired by the analogy between chi- ral symmetry of QCD. This defect was later cured by ral symmetry breaking and superconductivity. It postu- dressing the bag with a pionic cloud in the cloudy (Tho- lates a four-fermion interaction which, if it exceeds a mas Theberge, and Miller, 1981) or chiral bag (Brown certain strength, leads to quark condensation, the ap- and Rho, 1978) models. The role of the pion cloud was pearance of pions as Goldstone bosons, etc. The scale found to be quite large, and it was suggested that one above which this interaction disappears and QCD be- can do away with the quark core altogether. This idea comes perturbative enters the model as an explicit UV led to (topological) soliton models of the nucleon. The cutoff, ⌳␹SBϳ1 GeV. soliton model was pioneered by Skyrme in a paper that It was further argued that the scales for chiral symme- appeared long before QCD (Skyrme, 1961). However, try breaking and confinement are very different the Skyrmion was largely ignored at the time and only (Shuryak, 1981): ⌳␹SBӷ⌳confϳ⌳QCD . In particular, it gained in popularity after Witten argued that in the was argued that constituent quarks (and pions) have large-Nc limit the nucleon becomes a soliton, built from sizes smaller than those of typical hadrons, explaining nonlinear meson fields (Witten, 1983). the success of the nonrelativistic quark model. This idea

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 326 T. Scha¨fer and E. V. Shuryak: Instantons in QCD was developed in a more systematic fashion by (Georgi cannot measure qq and q¯q scattering amplitudes as we and Manohar, 1984), who argue that ⌳␹SBӍ4␲f␲ can for the nuclear force. Instead, the main tools at our Ӎ1 GeV provides a natural expansion parameter in chi- disposal are correlation functions of hadronic currents. ral effective Lagrangians. An effective theory using The phenomenology of these functions was recently re- pions and ‘‘constituent’’ quarks is the natural description viewed by Shuryak (1993). Hadronic point-to-point cor- in the intermediate regime ⌳confϽQϽ⌳␹SB , where relation functions were first systematically studied in the models of hadronic structure operate. context of ‘‘QCD sum rules.’’ The essential point, origi- While our understanding of the confinement mecha- nally emphasized by Shifman, Vainshtein, and Zakharov nism in QCD is still very poor, considerable progress has (1979), is that the operator product expansion relates been made in understanding the physics of chiral sym- the short-distance behavior of current correlation func- metry breaking. The importance of instantons in this tion to the vacuum expectation values of a small set of context is one of the main points of this review. Instan- lowest-dimension quark and gluon operators. Using the tons are localized (␳Ӎ1/3 fm) regions of spacetime with available phenomenological information on hadronic 2 very strong gluonic fields, G␮␯ϳ1/(g␳ ). It is the large correlation functions, Shifman, Vainshtein, and Za- value of this quantity which sets the scale for ⌳␹SB .In kharov deduced the quark and gluon condensates2 the regime ⌳ ϽQϽ⌳ , the instanton-induced ef- conf ␹SB 3 2 2 4 fective interaction between quarks is of the form ͗q¯q͘ϭϪ͑230 MeV͒ , ͗g G ͘ϭ͑850 MeV͒ , (2) as well as other, more complicated, expectation values. c⌫ d⌫ Ϫm͒␺ϩ ͑␺¯ ⌫␺͒2ϩ ͑␺¯ ⌫␺͒3ϩ , (1) The significance of the quark condensate is the factץϭ␺¯͑i ” ⌳2 ⌳5 ¯ L that it is an order parameter for the spontaneous break- where ⌫ is some spin-isospin-color matrix, ⌳ϳ␳Ϫ1 is de- down of chiral symmetry in the QCD vacuum. The termined by the chiral symmetry-breaking scale and gluon condensate is important because the QCD trace higher-order terms involve more fermion fields or de- anomaly rivatives. The mass scale for glueballs is even higher, and b these states decouple in the regime under consideration. ¯ 2 2 T␮␮ϭ͚ mf͗qfqf͘Ϫ 2 ͗g G ͘ (3) The instanton-induced interaction is nonlocal, and when f 32␲ calculating higher-order corrections with the vertices in relates this quantity to the energy density ⑀0Ӎ Eq. (1), loop integrals are effectively cut off at ⌳ Ϫ500 MeV/fm3 of the QCD vacuum. Here, T is the ϳ␳Ϫ1. ␮␯ energy-momentum tensor, and bϭ11Nc/3Ϫ2Nf/3 is the In addition to determining the scale ⌳, instantons also first coefficient of the beta function. provide an organizing principle for performing calcula- Any model of the QCD vacuum should explain the tions with the effective Lagrangian (1). If the instanton 4 origin and value of these condensates, the mechanism liquid is dilute, ␳ (N/V)Ӷ1, where (N/V) is the density for confinement and chiral symmetry breaking, and its of instantons, then vertices with more than 2Nf legs are relation to the underlying parameters of the theory (the suppressed. In addition to that, the diluteness parameter scale parameter and the matter content of the theory). determines which diagrams involving the leading-order Most of the early attempts to understand the QCD vertices have to be resummed. As a result, one can de- ground state were based on the idea that the vacuum is rive another effective theory valid at even smaller scales dominated by classical gauge-field configurations, for ex- which describes the interaction of extended, massive, ample, constant fields (Savvidy, 1977) or regions of con- constituent quarks with pointlike pions (Diakonov, stant fields patched together, as in the ‘‘spaghetti 1996). This theory is of the type considered by Georgi vacuum’’ introduced by the Copenhagen group (Amb- and Manohar (1984). jorn and Olesen, 1977). All of these attempts were un- Alternatively, one can go beyond leading order in 4 successful, however, because constant fields were found ␳ (N/V) and study hadronic correlation functions to all to be unstable against quantum perturbations. orders in the instanton-induced interaction. The results Instantons are classical solutions to the Euclidean are in remarkably good agreement with experiment for equations of motion. They are characterized by a topo- most light hadrons. Only when dealing with heavy logical quantum number and correspond to tunneling quarks or high-lying resonances do confinement effects events between degenerate classical vacua in Minkowski seem to play an important role. We shall discuss these space. As in quantum mechanics, tunneling lowers the questions in detail in Sec. VI. ground-state energy. Therefore instantons provide a simple understanding of the negative nonperturbative 3. Structure of the QCD vacuum vacuum energy density. In the presence of light fermi- ons, instantons are associated with fermionic zero The ground state of QCD can be viewed as a very modes. Zero modes not only are crucial to our under- dense state of matter, composed of gauge fields and quarks that interact in a complicated way. Its properties are not easily accessible in experiments because we do 2In the operator product expansion one has to introduce a not directly observe quark and gluon fields, only the scale ␮ that separates soft and hard contributions. The conden- color-neutral hadrons. Furthermore, we cannot directly sates take into account soft fluctuations, and the values given determine the interaction between quarks because we here correspond to a scale ␮Ӎ1 GeV.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 327

FIG. 1. Schematic picture of the instanton liquid: (a) at zero temperature; (b) above the chiral phase transition. Instantons FIG. 2. Instanton contribution to hadronic correlation func- and anti-instantons are shown as open and shaded circles. The tions: (a) the pion; (b) the nucleon; (c) the rho-meson cor- lines correspond to fermion exchanges. Figures (c) and (d) relator. The solid lines correspond to zero-mode contributions show the schematic form of the Dirac spectrum in the configu- to the quark propagator. rations (a) and (b).

change. It also implies that quarks can only be ex- standing of the axial anomaly, they are also intimately changed between instantons of the opposite charge. connected with spontaneous chiral symmetry breaking. Based on this picture, we can also understand the for- When instantons interact through fermion exchanges, mation of hadronic bound states. Bound states corre- zero modes can become delocalized, forming a collective spond to poles in hadronic correlation functions. As an quark condensate. example, let us consider the pion, which has the quan- A crude picture of quark motion in the vacuum can tum numbers of the current j␲ϭu¯␥5d. The correlation then be formulated as follows [see Fig. 1(a)]. Instantons function ⌸(x)ϭ͗j␲(x)j␲(0)͘ is the amplitude for an up act as a potential well, in which light quarks can form quark and a down anti-quark with opposite chiralities bound states (the zero modes). If instantons form an created by a source at point 0 to meet again at the point interacting liquid, quarks can travel over large distances x. In a disordered instanton liquid, this amplitude is by hopping from one instanton to another, similar to large because the two quarks can propagate by the pro- electrons in a conductor. Just as the conductivity is de- cess shown in Fig. 2(a). As a result, there is a light pion termined by the density of states near the Fermi surface, state. For the ␳ meson, on the other hand, we need the the quark condensate is given by the density of eigen- amplitude for the propagation of two quarks with the states of the Dirac operator near zero virtuality. A sche- same chirality. This means that the quarks have to be matic plot of the distribution of eigenvalues of the Dirac absorbed by different instantons (or propagate in non- operator is shown3 in Fig. 1(c). For comparison, the zero-mode states); see Fig. 2(c). The amplitude is spectrum of the Dirac operator for noninteracting smaller, and the meson state is much less tightly bound. quarks is depicted by the dashed line. If the distribution Using this picture, we can also understand the forma- of instantons in the QCD vacuum is sufficiently random, tion of a bound nucleon. Part of the proton wave func- there is a nonzero density of eigenvalues near zero, and tion is a scalar ud diquark coupled to another u quark. chiral symmetry is broken. This means that the nucleon can propagate as shown in The quantum numbers of the zero modes produce Fig. 2(b). The vertex in the scalar diquark channel is very specific correlations between quarks. First, since identical to the one in the pion channel with one of the there is exactly one zero mode per flavor, quarks with quark lines reversed.4 The ⌬ resonance has the quantum different flavors (say u and d) can travel together, but numbers of a vector diquark coupled to a third quark. quarks with the same flavor cannot. Furthermore, since Just as in the case of the ␳ meson, there is no first-order zero modes have a definite chirality (left-handed for in- instanton-induced interaction, and we expect the ⌬ to be stantons, right-handed for anti-instantons), quarks flip less bound than the nucleon. their chirality as they pass through an instanton. This is The paradigm discussed here bears a striking similar- very important phenomenologically because it distin- ity to one of the oldest approaches to hadronic structure, guishes instanton effects from perturbative interactions, the Nambu–Jona-Lasinio model (Nambu and Jona- in which the chirality of a massless quark does not Lasinio, 1961). In analogy with the Bardeen-Cooper-

3We shall discuss the eigenvalue distribution of the Dirac op- 4For more than three flavors, the color structure of the two erator in some detail in the main part of the review; see Figs. vertices is different, so there is no massless diquark in SU(3) 16 and 34 below. color.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 328 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

Schrieffer theory of superconductivity, it postulates a ter, this requires an understanding of the ground state short-range attractive force between fermions (nucleons and how the rearrangement of the vacuum takes place in the original model and light quarks in modern ver- that causes chiral symmetry to be restored. A possible sions). If this interaction is sufficiently strong, it can re- mechanism for chiral symmetry restoration in the instan- arrange the vacuum, and the ground state becomes su- ton liquid is indicated in Figs. 1(b) and 1(d). At high perconducting, with a nonzero quark condensate. In the temperature, instantons and anti-instantons have a ten- process, nearly massless current quarks become effec- dency to bind in pairs that are aligned along the (Euclid- tively massive constituent quarks. The short-range inter- ean) time direction. The corresponding quark eigen- action can then bind these constituent quarks into had- states are strongly localized, and chiral symmetry is rons (without confinement). unbroken. There is some theoretical evidence for this This brief outline indicates that instantons provide at picture, which will be discussed in detail in Sec. VII. In least a qualitative understanding of many features of the particular, there is evidence from lattice simulations that QCD ground state and its hadronic excitations. How can instantons do not disappear at the phase transition, but this picture be checked and made more quantitive? only at even higher temperatures. This implies that in- Clearly, two things need to be done. First, a consistent stantons affect properties of the quark-gluon plasma at instanton ensemble has to be constructed in order to temperatures not too far above the phase transition. make quantitative predictions for hadronic observables. Second, we should like to test the underlying assumption C. The history of instantons that the topological susceptibility, the gluon condensate, the chiral symmetry breaking, etc. are dominated by in- In books and reviews, physical theories are usually stantons. This can be done most directly on the lattice. presented as a systematic development, omitting the of- We shall discuss both of these issues in the main part of ten confusing history of the subject. The history of in- this review, Secs. III–VI. stantons did not follow a straight path either. Early en- thusiasm concerning the possibility of understanding 4. QCD at finite temperature nonperturbative phenomena in QCD, in particular con- finement, caused false hopes, which led to years of frus- Properties of the QCD vacuum, like the vacuum en- tration. Only many years later did work on phenomeno- ergy density, and the quark and gluon condensates, are logical aspects of instantons lead to breakthroughs. In not directly accessible to experiment. Measuring non- the following we shall try to give a brief tour of the two perturbative properties of the vacuum requires the abil- decades that have passed since the discovery of instan- ity to compare the system with the ordinary, perturba- tons. tive state.5 This state of matter has not existed in nature since the Big Bang, so experimental attempts at studying 1. Discovery and early applications the perturbative phase of QCD have focused on recre- ating miniature Big Bangs in relativistic heavy-ion colli- The instanton solution of the Yang-Mills equations sions. was discovered by Polyakov and co-workers (Belavin, The basic idea is that at sufficiently high temperature Polyakov, Schwartz, and Tyupkin, 1975), motivated by or density QCD will undergo a phase transition to a new the search for classical solutions with nontrivial topology state, referred to as the quark-gluon plasma, in which in analogy with the ’t Hooft-Polyakov monopole (Polya- chiral symmetry is restored and quarks and gluon are kov, 1975). Shortly thereafter, a number of authors clari- deconfined. The temperature scale for this transition is fied the physical meaning of the instanton as a tunneling set by the vacuum energy density and pressure of the event between degenerate classical vacua (Callan, vacuum. For the perturbative vacuum to have a pressure Dashen, and Gross, 1976; Jackiw and Rebbi, 1976a; comparable to the vacuum pressure 500 MeV/fm3,a Polyakov, 1977). These works also introduced the con- temperature on the order of 150–200 MeV is required. cept of ␪ vacua in connection with QCD. According to our current understanding, such tempera- Some of the early enthusiasm was fueled by Polyak- tures are reached in the ongoing or planned experiments ov’s discovery that instantons cause confinement in at the Alternating-Gradient Synchrotron (AGS; about certain three-dimensional models (Polyakov, 1977). 2ϩ2 GeV per nucleon in the center-of-mass system), However, it was soon realized that this is not the case CERN SPS (about 10ϩ10 GeV) or RHIC (100 in four-dimensional gauge theories. An important devel- 6 ϩ100 GeV). opment originated with the classic paper by ’t Hooft In order to interpret these experiments, we need to (1976a), in which he calculated the semiclassical tunnel- understand the properties of hadrons and hadronic mat- ing rate. In this context he discovered the presence of ter near and above the phase transition. As for cold mat- zero modes in the spectrum of the Dirac operator. This result implied that tunneling is intimately connected with light fermions, in particular, that every instanton 5A nice analogy is given by the atmospheric pressure. In or- absorbs one left-handed fermion of every species, and der to measure this quantity directly, one has to evacuate a container filled with air. Similarly, one can measure the non- perturbative vacuum energy density by filling some volume 6In this paper, ’t Hooft also coined the term instanton; Polya- with another phase, the quark-gluon plasma. kov had referred to the classical solution as a pseudoparticle.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 329 emits a right-handed one (and vice versa for anti- on this analysis, it was realized that ‘‘all hadrons are not instantons). This result also explained how anomalies, alike’’ (Novikov et al., 1981). The operator product ex- for example the violation of axial charge in QCD and pansion (OPE) does not give reliable predictions for baryon number in electroweak theory, are related to in- scalar and pseudoscalar channels (␲,␴,␩,␩Ј as well as stantons. scalar and pseudoscalar glueballs). These are precisely In his work, ’t Hooft estimated the tunneling rate in the channels that receive direct instanton contributions electroweak theory, where the large Higgs expectation (Geshkenbein and Ioffe, 1980; Novikov et al., 1981; value guarantees the validity of the semiclassical ap- Shuryak, 1983). proximation, and found it to be negligible. Early at- In order to understand the available phenomenology, tempts to study instanton effects in QCD, where the rate a qualitative picture, later termed the instanton liquid is much larger but also harder to estimate, were summa- model, was proposed by Shuryak (1982a). In this work, rized by Callan et al. (1978a). These authors realized the two basic parameters of the instanton ensemble that the instanton ensemble can be described as a four- were suggested: the mean density of instantons is n dimensional ‘‘gas’’ of pseudoparticles that interact via Ӎ1fmϪ4, while their average size is ␳Ӎ1/3 fm. This forces that are dominantly of the dipole type. While they means that the spacetime volume occupied by instantons were not fully successful in constructing a consistent in- fϳn␳4 is small; the instanton ensemble is dilute. This stanton ensemble, they nevertheless studied a number of observation provides a small expansion parameter that important instanton effects: the instanton-induced po- we can use in order to perform systematic calculations. tential between heavy quarks, the possibility that instan- Using the instanton liquid parameters nӍ1fmϪ4, ␳ tons cause the spontaneous breakdown of chiral symme- Ӎ1/3 fm, one can reproduce the phenomenological val- try, and instanton corrections to the running coupling ues of the quark and gluon condensates. In addition to constant. that, one can calculate direct instanton corrections to One particular instanton-induced effect, the anoma- hadronic correlation functions at short distances. The re- lous breaking of U(1)A symmetry and the ␩Ј mass, de- sults were found to be in good agreement with experi- serves special attention.7 Witten and Veneziano wrote ment in both attractive (␲,K) and repulsive (␩Ј) pseu- down an approximate relation that connects the ␩Ј mass doscalar meson channels (Shuryak, 1983). with the topological susceptibility (Veneziano, 1979 Wit- ten, 1979a). This was a very important step, because it was the first quantitative result concerning the effect of 3. Technical development during the eighties the anomaly on the ␩Ј mass. However, it also caused Despite its phenomenological success, there was no some confusion because the result had been derived us- theoretical justification for the instanton liquid model. ing the large-Nc approximation, which is not easily ap- The first steps towards providing some theoretical basis plied to instantons. In fact, both Witten and Veneziano for the instanton model were taken by Ilgenfritz and expressed strong doubts concerning the relation be- ¨ 8 Muller-Preussker (1981) and Diakonov and Petrov tween the topological susceptibility and instantons, sug- (1984). These authors used variational techniques and gesting that instantons are not important dynamically the mean-field approximation to deal with the statistical (Witten, 1979b). mechanics of the instanton liquid. The ensemble was sta- bilized using a phenomenological core (Ilgenfritz and 2. Phenomenology leads to a qualitative picture Mu¨ ller-Preussker, 1981) or a repulsive interaction de- rived from a specific ansatz for the gauge-field interac- By the end of the seventies the general outlook was tion (Diakonov and Petrov, 1984). The resulting en- very pessimistic. There was no experimental evidence sembles were found to be consistent with the for instanton effects and no theoretical control over phenomenological estimates. semiclassical methods in QCD. If a problem cannot be The instanton ensemble in the presence of light solved by direct theoretical analysis, it is often useful to quarks was studied by Diakonov and Petrov (1986). This turn to a more phenomenological approach. By the early work introduced the picture of the quark condensate as eighties, such an approach to the structure of the QCD a collective state built from delocalized zero modes. The vacuum became available with the QCD sum-rule quark condensate was calculated using the mean-field method (Shifman, Vainshtein, and Zakharov, 1979). approximation and found to be in agreement with ex- QCD sum rules relate vacuum parameters, in particular, periment. Hadronic states were studied in the random- the quark and gluon condensates, to the behavior of phase approximation. At least in the case of pseudo- hadronic correlation functions at short distances. Based scalar mesons, the results were also in good agreement with experiment.

7 In parallel, numerical methods for studying the instan- There is one historical episode that we should mention ton liquid were developed (Shuryak, 1988a). Numerical briefly. Crewther and Christos (Christos, 1984) questioned the sign of the axial charge violation caused by instantons. A re- simulations allow one to go beyond the mean-field and buttal of these arguments can be found in ’t Hooft (1986). random-phase approximations, and include the ’t Hooft 8Today, there is substantial evidence from lattice simulations interaction to all orders. This means that one can also that the topological susceptibility is dominated by instantons; study hadronic channels that, like vector mesons, do not see Sec. III.C.2. have first-order instanton-induced interactions, or chan-

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 330 T. Scha¨fer and E. V. Shuryak: Instantons in QCD nels, like the nucleon, that are difficult to treat in the 1995; Scha¨fer and Shuryak, 1996a, 1996b). The corre- random-phase approximation. sponding unquenched (with fermion vacuum bubbles in- Nevertheless, many important aspects of the model cluded) correlation functions significantly improved the remain to be understood. This applies in particular to description of the ␩Ј and ␦ mesons, which are the two the theoretical foundation of the instanton liquid model. channels in which the random model fails completely. When the instanton anti-instanton interaction was stud- Finally, significant progress was made in understand- ied in more detail, it became clear that there is no clas- ing the instanton liquid at finite temperature and the sical repulsion in the gauge-field interaction. A well- mechanism for the chiral phase transition. It was real- separated instanton anti-instanton pair is connected to ized earlier that, at high temperature, instantons should the perturbative vacuum by a smooth path (Balitsky and be suppressed by Debye screening (Shuryak, 1978b; Yung, 1986; Verbaarschot, 1991). This means that the Pisarski and Yaffe, 1980). Therefore it was generally as- instanton ensemble cannot be stabilized by purely clas- sumed that chiral symmetry restoration is a consequence sical interactions because, in general, it is not possible to of the disappearance of instantons at high temperature. separate nonperturbative (instanton-induced) and per- More recently it was argued that, up to the critical turbative effects. Only in special cases, like quantum temperature, the density of instantons should not be mechanics (Sec. II.A.5) and supersymmetric field theory suppressed (Shuryak and Velkovsky, 1994). This predic- (Sec. VIII.C), has this separation been accomplished. tion was confirmed by direct lattice measurements of the topological susceptibility (Chu and Schramm, 1995), which indeed found little change in the topological sus- 4. Recent progress ceptibility for TϽTc and the expected suppression for In the past few years a great deal has been learned TϾTc . If instantons are not suppressed around Tc ,a about instantons in QCD. The instanton liquid model different mechanism for the chiral phase transition is with the parameters mentioned above, now referred to needed. It was suggested that chiral symmetry is re- as the random instanton liquid model, has been used for stored because the instanton liquid is rearranged, going large-scale, quantitative calculations of hadronic correla- from a random phase below Tc to a correlated phase tion functions in essentially all meson and baryon chan- of instanton anti-instanton molecules above Tc (Ilgen- nels (Shuryak and Verbaarschot, 1993a; Scha¨fer, fritz and Shuryak, 1994; Scha¨fer, Shuryak, and Verbaar- Shuryak, and Verbaarschot, 1994). Hadronic masses and schot, 1995). This idea was confirmed in direct simula- coupling constants for most of the low-lying mesons and tions of the instanton ensemble, and a number of baryon states have been shown to be in quantitative consequences of the scenario were explored. agreement with experiment. The next surprise came from a comparison of the cor- D. Topics that are not discussed in detail relators calculated in the random model and the first results from lattice calculations (Chu et al., 1993a). The There is a vast literature on instantons (the SLAC results agree quantitatively, not only in channels that database lists over 3000 references, which probably does were already known from phenomenology, but also in not include the more mathematically oriented works), others (such as the nucleon and delta) where no previ- and limitations of space and time, as well as of our ex- ous information (except for the particle masses, of pertise, have forced us to exclude many interesting sub- course) existed. jects from this review. Our emphasis in writing this re- These calculations were followed up by direct studies view has been on the theory and phenomenology of of the instanton liquid on the lattice. Using a procedure instantons in QCD. We discuss instantons in other mod- called cooling, one can extract the classical content of els only to the extent that they are pedagogically valu- strongly fluctuating lattice configurations. Using cooled able or provide important lessons for QCD. Let us men- configurations, the MIT group determined the main pa- tion a few important omissions and give a brief guide to rameters of the instanton liquid (Chu et al., 1994). the relevant literature: Within 10% error, the density and average size coin- cided with the values suggested a decade earlier. In the (1) Direct manifestations of small-size instantons in meantime, these numbers had been confirmed by other high-energy baryon-number-violating reactions. The calculations (see Sec. III.C). In addition to that, it was hope is that in these processes one may observe shown that the agreement between lattice correlation rather spectacular instanton effects in a regime functions and the instanton model was not a coinci- where reliable semiclassical calculations are pos- dence: the correlators were essentially unaffected by sible. In the electroweak theory, instantons lead to cooling. This implied that neither perturbative (removed baryon-number violation, but the amplitude for this by cooling) nor confinement (strongly reduced) forces reaction is strongly suppressed at low energies. It are crucial for hadronic properties. was hoped that this suppression could be overcome Technical advances in numerical simulation of the in- at energies on the order of the sphaleron barrier E stanton liquid led to the construction of a self-consistent, Ӎ10 TeV, but the emerging consensus is that this interacting instanton ensemble, which satisfied all the dramatic phenomenon will not be observable. Some general constraints imposed by the trace anomaly and of the literature is mentioned in Sec. VIII.B; see also chiral low-energy theorems (Shuryak and Verbaarschot, the recent review of Aoyama et al. (1997).

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 331

(2) The transition from tunneling to thermal activation mechanics deal with variables that are subject to random and the calculation of the sphaleron rate at high fluctuations (quantum or thermal), so that only temperature. This question is of interest in connec- ensemble-averaged quantities make sense. Formally, the tion with baryogenesis in the early universe and connection is related to the similarity between the sta- axial charge fluctuations in the quark-gluon plasma. tistical partition function tr͓exp(Ϫ␤H)͔ and the generat- A recent discussion can be found in the review of ing functional (6) (see below), describing the dynamic Smilga (1996). evolution of a quantum system. (3) The decay of unstable vacua in quantum mechanics Consider the simplest possible quantum-mechanical or field theory (Coleman, 1977). A more recent re- system, the motion of a particle of mass m in a time- view can be found in Aoyama et al. (1997). independent potential V(x). The standard approach is (4) Direct instanton contributions to deep-inelastic scat- based on an expansion in terms of stationary states tering and other hard processes in QCD. See Bal- ␺ (x), given as solutions of the Schro¨ dinger equation itskii and Braun (1993; 1995) and the review of n H␺ ϭE ␺ . Instead, we shall concentrate on another Ringwald and Schrempp (1994). n n n object, the Green’s function9 (5) Instanton-inspired models of hadrons or phenom- enological Lagrangians supplemented by the G͑x,y,t͒ϭ͗y͉exp͑ϪiHt͉͒x͘, (4) ’t Hooft interaction. These models include the which is the amplitude for a particle to go from point x Nambu–Jona-Lasinia models (Hatsuda and Kuni- at time tϭ0 to point y at time t. The Green’s function hiro, 1994), soliton models (Diakonov, Petrov, and can be expanded in terms of stationary states Pobylitsa, 1988; Christov et al., 1996), potential models (Blask et al., 1990), bag models (Dorokhov, ϱ Zubov, and Kochelev, 1992), etc. G͑x,y,t͒ϭ ␺n*͑x͒␺n͑y͒exp͑ϪiEnt͒. (5) n͚ϭ1 (6) Mathematical aspects of instantons (Eguchi, Gilkey, and Hanson, 1980), the construction of the most This representation has many nice features that are de- general n-instanton solution (Atiyah, Hitchin, Drin- scribed in standard text books on quantum mechanics. feld, and Manin, 1977), constrained instantons (Af- There is, however, another representation that is more fleck, 1981), instantons and four-manifolds (Fried useful in introducing semiclassical methods and in deal- and Uhlenbeck, 1984), and the connection between ing with systems containing many degrees of freedom, instantons and solitons (Atiyah and Manton, 1989). the Feynman path integral (Feynman and Hibbs, 1965), For a review of known solutions of the classical Yang-Mills field equations in both Euclidean and G͑x,y,t͒ϭ Dx͑t͒exp͑iS͓x͑t͔͒͒. (6) Minkowski space, we refer the reader to Actor ͵ (1979). Here, the Green’s function is given as a sum over all (7) Formal aspects of the supersymmetric instanton cal- possible paths x(t) leading from x at tϭ0toyat time t. culus, spinor techniques, etc. This material is cov- The weight for the paths is provided by the action S ered by Novikov, Shifman, Voloshin, and Zakharov ϭ͐dt͓mx˙2/2ϪV(x)͔. One way to provide a more pre- (1983), Novikov, Shifman, Vainshtein, and Za- cise definition of the path integral is to discretize the kharov (1985a), Novikov (1987), and Amati et al. path. Dividing the time axis into N intervals, aϭt/N, the (1988). path integral becomes an N-dimensional integral over (8) The strong CP problem, bounds on the theta pa- xnϭx(tnϭan) nϭ1,...,N. The discretized action is rameter, the axion mechanism (Peccei and Quinn, given by 1977), etc. Some remarks on these questions can be m found in Sec. II.C.3. 2 Sϭ͚ ͑xnϪxnϪ1͒ ϪaV͑xn͒ . (7) n ͫ2a ͬ II. SEMICLASSICAL THEORY OF TUNNELING This form is not unique; other discretizations with the same continuum limit can also be used. The path inte- A. Tunneling in quantum mechanics gral is now reduced to a multiple integral over xn , where we have to take the limit n ϱ. In general, only 1. Quantum mechanics in Euclidean space Gaussian integrals can be done exactly.→ In the case of This section serves as a brief introduction to path- the harmonic oscillator V(x)ϭm␻2x2/2 one finds integral methods and can easily be skipped by readers (Feynman and Hibbs, 1965) familiar with the subject. We shall demonstrate the use m␻ 1/2 im␻ of Feynman diagrams in a simple quantum-mechanical Gosc͑x,y,t͒ϭ exp problem, which does not suffer from any of the diver- ͩ2␲i sin ␻tͪ ͫͩ 2 sin ␻tͪ gencies that occur in field theory. Indeed, we hope that 2 2 this simple example will find its way into introductory ϫ„͑x ϩy …cos͑␻t͒Ϫ2xy͒ . (8) field theory courses. ͬ Another point we should like to emphasize in this sec- tion is the similarity between quantum and statistical 9We use natural units បϭh/2␲ϭ1 and cϭ1. Mass, energy, mechanics. Qualitatively, both quantum and statistical and momentum all have dimensions of inverse length.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 332 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

In principal, the discretized action (7) should be amen- able to numerical simulations. In practice, the strongly fluctuating phase in (6) renders this approach com- pletely useless. There is, however, a simple way to get around the problem. If one performs an analytic con- tinuation of G(x,y,t) to imaginary (Euclidean) time ␶ ϭit, the weight function becomes exp„ϪSE͓x(␶)͔… with 2 the Euclidean action SEϭ͐dt͓m(dx/d␶) /2ϩV(x)͔.In this case, we have a positive-definite weight, and nu- merical simulations, even for multidimensional prob- lems, are feasible. Note that the relative sign of the kinetic- and potential-energy terms in the Euclidean ac- FIG. 3. Feynman diagrams for the anharmonic oscillator: (a) tion has changed, making it look like a Hamiltonian. In energy; (b) Green’s function. Euclidean space, the discretized action (7) looks like the energy functional of a one-dimensional spin chain with nearest-neighbor interactions. This observation provides In addition to that, there are three- and four-point ver- the formal link between an n-dimensional statistical sys- tices with coupling constants ␣ and ␤. To calculate an tem and Euclidean quantum (field) theory in (nϪ1) di- n-point Green’s function, we have to sum over all dia- mensions. grams with n external legs and integrate over the time Euclidean Green’s functions can be interpreted in variables corresponding to internal vertices. terms of ‘‘thermal’’ distributions. If we use periodic The vacuum energy is given by the sum of all closed boundary conditions (xϭy) and integrate over x,we diagrams. At one-loop order, there is only one diagram, obtain the statistical sum the free-particle loop diagram. At two-loop order, there 2 ϱ are two O(␣ ) and one O(␤) diagram [see Fig. 3(a)]. The calculation of the diagrams is remarkably simple. dxG͑x,x,␶͒ϭ exp͑ϪEn␶͒, (9) ͵ n͚ϭ1 Since the propagator is exponentially suppressed for large times, everything is finite. Summing all the dia- where the time interval ␶ plays the role of an inverse grams, we get temperature ␶ϭTϪ1. In particular, G(x,x,1/T) has the physical meaning of a probability distribution for x at ␻ ␻␶ ͗0͉exp͑ϪH␶͉͒0͘ϭ exp Ϫ temperature T. For the harmonic oscillator mentioned ͱ␲ ͩ 2 ͪ above, the Euclidean Green’s function is 3␤ 11␣2 1/2 m␻ m␻ ϫ 1Ϫ 2 Ϫ 4 ␶ϩ¯ . (13) G ͑x,y,␶͒ϭ exp Ϫ ͫ ͩ 4␻ 8␻ ͪ ͬ osc ͩ2␲ sinh ␻␶ͪ ͫ ͩ 2 sinh ␻␶ͪ For small ␣2, ␤, and ␶ not too large, we can exponenti- ate the result and read off the correction to the ground- 2 2 ϫ„͑x ϩy ͒cosh͑␻␶͒Ϫ2xy…ͬ. (10) state energy 2 For xϭy, the spatial distribution is Gaussian at any T, ␻ 3␤ 11␣ E0ϭ ϩ 2 Ϫ 4 ϩ¯ . (14) with a width ͗x2͘ϭ1/2m␻ coth(␻/2T). If ␶ is very large, 2 4␻ 8␻ the effective temperature is small and the ground state Of course, we could have obtained the result using ordi- dominates. From the exponential decay, we can read off nary Rayleigh-Schro¨ dinger perturbation theory, but the the ground-state energy E0ϭ␻/2, and from the spatial method discussed here proves to be much more power- distribution, the width of the ground-state wave function ful when we come to nonperturbative effects and field ͗x2͘ϭ(2m␻)Ϫ1. At high T we get the classical result theory. ͗x2͘ϭT/(m␻2). One more simple exercise is worth mentioning: the Non-Gaussian path integrals cannot be done exactly. evaluation of first perturbative corrections to the As long as the nonlinearities are small, we can use per- Green’s function. The diagrams shown in Fig. 3(b) give turbation theory. Consider an anharmonic oscillator 9␣2 ␣2 with (Euclidean) action ⌬G ͑0,␶͒ϭ ϩ eϪ2␻␶ 0 4␻6 2␻6 x˙ 2 ␻2x2 3 4 2 SEϭ d␶ ϩ ϩ␣x ϩ␤x . (11) 15␣ 3␤ ͵ ͫ 2 2 ͬ ϩ ␶eϪ␻␶Ϫ ␶eϪ␻␶. (15) 4␻5 2␻3 Expanding the path integral in powers of ␣ and ␤, one can derive the Feynman rules for the anharmonic oscil- Comparing the result with the decomposition in terms of lator. The free propagator is given by stationary states ϱ 1 Ϫ͑EnϪE0͒␶ 2 G0͑␶1 ,␶2͒ϭ͗x͑␶1͒x͑␶2͒͘ϭ exp͑Ϫ␻͉␶1Ϫ␶2͉͒. (12) G͑0,␶͒ϭ ͚ e ͉͗0͉x͉n͉͘ , (16) 2␻ nϭ0

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 333 we can identify the first (time-independent) term with with minima at Ϯ␩, the two ‘‘classical vacua’’ of the the square of the ground-state expectation value ͗0͉x͉0͘ system. Quantizing around the two minima, we would (which is nonzero due to the tadpole diagram). The sec- find two degenerate states localized at xϭϮ␩.Of ond term comes from the excitation of two quanta, and course, we know that this is not the correct result. Tun- the last two (with extra factors of ␶) are the lowest-order neling will mix the two states; the true ground state is ‘‘mass renormalization,’’ or corrections to the zero-order (approximately) the symmetric combination, while the gap between the ground and first excited states, E1 first excited state is the antisymmetric combination of ϪE0ϭ␻. the two states. It is easy to solve the equations of motion in imagi- nary time and obtain the classical tunneling solution10 2. Tunneling in the double-well potential ␻ x ͑␶͒ϭ␩ tanh ͑␶Ϫ␶ ͒ , (19) Tunneling phenomena in quantum mechanics were cl ͫ 2 0 ͬ discovered by George Gamow in the late 1920’s in the Ϫ ϭϪ ϭ context of alpha decay. He introduced the exponential which goes from x( ϱ) ␩ to x(ϱ) ␩. Here, ␶0 is a free parameter (the instanton center) and ␻2ϭ8␭␩2. suppression factor that explained why a decay governed 3 by the Coulomb interaction (with a typical nuclear time The action of the solution is S0ϭ␻ /(12␭). We shall scale of 10Ϫ22 sec) could lead to lifetimes of millions of refer to the path (19) as the instanton, since (unlike a soliton) the solution is localized in time.11 An anti- years. Tunneling is a quantum-mechanical phenomenon, A I a particle penetrating a classically forbidden region. instanton solution is given by xcl (␶)ϭϪxcl(␶). It is con- Nevertheless, we shall describe the tunneling process us- venient to rescale the time variable such that ␻ϭ1 and ing classical equations of motion. Again, the essential to shift x such that one of the minima is at xϭ0. In this idea is to continue the transition amplitude to imaginary case, there is only one dimensionless parameter, ␭, and time. since S0ϭ1/(12␭), the validity of the semiclassical ex- Let us give a qualitative argument for why tunneling pansion is controlled by ␭Ӷ1. can be treated as a classical process in imaginary time. The semiclassical approximation to the path integral is The energy of a particle moving in the potential V(x)is obtained by systematically expanding around the classi- given by Eϭp2/(2m)ϩV(x), and in classical mechanics cal solution only regions of phase space where the kinetic energy is ϪH␶ ϪS positive are accessible. In order to reach the classically ͗Ϫ␩͉e ͉␩͘ϭe 0 ͵ Dx͑␶͒ forbidden region EϽV, the kinetic energy would have to be negative, corresponding to imaginary momentum 1 ␦2S ϫexp Ϫ ␦x ␦xϩ . (20) p. In the semiclassical WKB approximation to quantum 2 ␦x2 ͯ ¯ ͩ x ͪ mechanics, the wave function is given by ␺(x) cl x ϳexp͓i⌽(x)͔ with ⌽(x)ϭϮ͐ dxЈp(xЈ)ϩO(ប) where Note that the linear term is absent because xcl is a solu- p(x)ϭ(2m)1/2͓EϪV(x)͔1/2 is the local classical mo- tion of the equations of motion. Also note that we im- mentum. In the classically allowed region, the wave plicitly assume ␶ to be large, but smaller than the typical function is oscillatory, while in the classically forbidden lifetime for tunneling. If ␶ is larger than the lifetime, we region (corresponding to imaginary momenta) it is ex- have to take into account multi-instanton configurations ponentially suppressed. (see below). Clearly, the tunneling amplitude is propor- Another way to introduce imaginary momenta, which tional to exp(ϪS0). The pre-exponent requires the cal- is more easily generalized to multidimensional problems culation of fluctuations around the classical instanton so- and field theory, is by considering motion in imaginary lution. We shall study this problem in the following time. Continuing ␶ϭit, the classical equation of motion section. is given by 3. Tunneling amplitude at one-loop order d2x dV m ϭϩ , (17) d␶2 dx In order to take into account fluctuations around the classical path, we have to calculate the path integral where the sign of the potential-energy term has changed. This means that classically forbidden regions are now 1 ͓D␦x͔exp Ϫ d␶ ␦x͑␶͒O␦x͑␶͒ (21) classically allowed. The distinguished role of the classi- ͵ ͩ 2 ͵ ͪ cal tunneling path becomes clear if one considers the Feynman path integral. Although any path is allowed in quantum mechanics, the path integral is dominated by 10This solution is most easily found using energy conservation 2 paths that maximize the weight factor exp(ϪS͓xcl(␶)͔) mx˙ /2ϪV(x)ϭconst rather than the (second-order) equation or minimize the Euclidean action. The classical path is of motion mx¨ϭVЈ. This is analogous to the situation in field the path with the smallest possible action. theory, where it is more convenient to use self-duality than the Let us consider a widely used toy model, the double- equations of motion. 11 4 well potential, In (1ϩ1)-dimensional ␾ theory, there is a soliton solution with the same functional form, usually referred to as the kink Vϭ␭͑x2Ϫ␩2͒2, (18) solution.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 334 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

where O is the differential operator, x(␶)ϭ͚ncnxn(␶)] with an integral over the collective coordinate ␶ . Using 1 d2 d2V 0 OϭϪ ϩ . (22) 2 d␶2 dx2 ͯ dxcl xϭxcl dxϭ d␶0ϭϪͱS0x0͑␶͒d␶0 (27) d␶0 This calculation is somewhat technical, but it provides a and dxϭx dc , we have dc ϭͱS d␶ . The functional very good illustration of the steps that are required to 0 0 0 0 0 integral over the quantum fluctuation is now given by solve the more difficult field-theory problem. We follow here the original work (Polyakov, 1977) and the review 2␲ 1/2 (Vainshtein et al., 1982). A simpler method for calculat- ͓D␦x͑␶͔͒exp͑ϪS͒ϭ ͟ ͱS0 d␶0 , ͵ ͫ nϾ0 ͩ ⑀n ͪͬ ͵ ing the determinant is described in the appendix of (28) Coleman’s lecture notes (Coleman, 1977). The integral (21) is Gaussian, so it can be done ex- where the first factor, the determinant with the zero mode excluded, is often referred to as det O. The result actly. Expanding the differential operator O in some ba- Ј shows that the tunneling amplitude grows linearly with sis ͕x (␶)͖, we have i time. This is as it should be; there is a finite transition 1 n/2 Ϫ1/2 probability per unit time. ͟ dxn exp Ϫ ͚ xiOijxj ϭ͑2␲͒ ͑det O͒ . The next step is the calculation of the non-zero-mode ͵ ͩ n ͪ ͩ 2 ij ͪ (23) determinant. For this purpose we make the spectrum discrete by considering a finite time interval The determinant can be calculated by diagonalizing O, (Ϫ␶m/2,␶m/2) and imposing boundary conditions at Oxn(␶)ϭ⑀nxn(␶). This eigenvalue equation is just a Ϯ␶ /2: x (Ϯ␶ /2)ϭ0. The product of all eigenvalues is ¨ 12 m n m one-dimensional Schrodinger equation, divergent, but the divergence is related to large eigen- d2 3 values, independent of the detailed shape of the poten- 2 Ϫ 2 ϩ␻ 1Ϫ 2 xn͑␶͒ϭ⑀nxn͑␶͒. tial. The determinant can be renormalized by taking the ͩ d␶ ͫ 2 cosh ͑␻␶/2͒ͬͪ (24) ratio over the determinant of the free harmonic oscilla- tor. The result is There are two bound states plus a continuum of scatter- d2 Ϫ1/2 ing states. The lowest eigenvalue is ⑀0ϭ0, and the other 3 2 det Ϫ 2 ϩVЉ͑xcl͒ bound state is at ⑀ ϭ 4 ␻ . The eigenfunction of the ͫ d␶ ͬ 1 zero-energy state is d2 det Ϫ ϩ␻2 ͩ d␶2 ͪ 3␻ 1 ͫ ͬ x0͑␶͒ϭ 2 , (25) ͱ 8 cosh ͑␻␶/2͒ d2 Ϫ1/2 detЈ Ϫ ϩVЉ͑x ͒ where we have normalized the wave function, ͐d␶x2 S ͫ d␶2 cl ͬ n 0 ϭ ␻ d␶ ϭ1. There should be a simple explanation for the pres- ͱ2␲ ͵ 0 d2 ence of a zero mode. Indeed, the appearance of a zero ␻Ϫ2 det Ϫ ϩ␻2 ͩ d␶2 ͪ mode is related to translational invariance, the fact that ͫ ͬ the action does not depend on the location ␶0 of the (29) instanton. The zero-mode wave function is just the de- where we have eliminated the zero mode from the de- rivative of the instanton solution over ␶0 , terminant and replaced it by the integration over ␶0 .We d also have to extract the lowest mode from the harmonic- x ͑␶͒ϭϪSϪ1/2 x ͑␶Ϫ␶ ͒, (26) 2 0 0 d␶ cl 0 oscillator determinant, which is given by ␻ . The next 0 eigenvalue is 3␻2/4, while the corresponding oscillator where the normalization follows from the fact that the 2 2 mode is ␻ (up to corrections of order 1/␶m , that are not classical solution has zero energy. If one of the eigenval- important as ␶m ϱ). The rest of the spectrum is con- ues is zero, this means that the determinant vanishes and → tinuous as ␶m ϱ. The contribution from these states the tunneling amplitude is infinite. However, the pres- can be calculated→ as follows. ence of a zero mode also implies that there is one direc- The potential VЉ(xcl) is localized, so for ␶ Ϯϱ the tion in functional space in which fluctuations are large, eigenfunctions are just plane waves. This means→ we can so the integral is not Gaussian. This means that the in- take one of the two linearly independent solutions to be tegral in that direction should not be performed in a xp(␶)ϳexp(ip␶)as␶ ϱ. The effect of the potential is Gaussian approximation, but has to be done exactly. to give a phase shift → This can be achieved by replacing the integral over xp͑␶͒ϭexp͑ip␶ϩi␦p͒ ␶ Ϫϱ, (30) the expansion parameter c0 associated with the zero- → mode direction [we have parametrized the path by where, for this particular potential, there is no reflected wave. The phase shift is given by (Landau and Lifshitz, 1959) 12This particular Schro¨ dinger equation is discussed in many 1ϩip/␻ 1ϩ2ip/␻ textbooks on quantum mechanics, e.g., Landau and Lifshitz exp͑i␦ ͒ϭ . (31) (1959). p 1Ϫip/␻ 1Ϫ2ip/␻

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 335

The second independent solution is obtained by ␶ ␻ Ϫ␶. The spectrum is determined by the quantization→ ͗Ϫ␩͉eϪH␶m͉␩͘ϭͱ eϪ␻␶m/2 ͚ ͵ ␲ n odd Ϫ␶m/2Ͻ␶1Ͻ...Ͻ␶m/2 condition x(Ϯ␶m/2)ϭ0, which gives n 6S n pn␶mϪ␦p ϭ␲n, (32) 0 n ϫ ͟ ␻d␶i exp͑ϪS0͒ ͫ iϭ1 ͬͩͱ ␲ ͪ while the harmonic-oscillator modes are determined by p ␶ ϭ␲n. If we denote the solutions of Eq. (32) by ˜p , ␻ ␻␶ d͒n n m n Ϫ␻␶ /2 ͑ m the ratio of the determinants is given by ϭͱ e m ͚ ␲ n odd n!

2 2 2 2 ␻ ϩ˜pn ␻ ϩ˜pn ␻ ϭ eϪ␻␶m/2 sinh͑␻␶ d͒, (36) ͟ 2 2 ϭexp ͚ log 2 2 ͱ m n ͫ ␻ ϩp ͬ n ͫ ␻ ϩp ͬ ␲ n ͩ n ͪ 1/2 ϱ where dϭ(6S0 /␲) exp(ϪS0). Summing over all in- 1 2pdp␦p 1 ϭexp 2 2 ϭ , (33) stantons simply leads to the exponentiation of the tun- ͩ ␲ ͵0 p ϩ␻ ͪ 9 neling rate. Now we can directly read off the level split- where we have expanded the integrand in the small dif- ting ference ˜p Ϫp ϭ␦ /␶ and changed from summation n n pn m 6S0 ⌬Eϭ ␻ exp͑ϪS ͒. (37) over n to an integral over p. In order to perform the ͱ ␲ 0 integral, it is convenient to integrate by part and use the If the tunneling rate increases, 1/⌬EӍ1/␻, interactions result for (d␦p)/(dp). Collecting everything, we finally get between instantons become important. We shall study this problem in Sec. II.A.5.

␻ ␻␶m ͗Ϫ␩͉eϪH␶m͉␩͘ϭ exp Ϫ 4. The tunneling amplitude at two-loop order ͫͱ␲ ͩ 2 ͪͬ The WKB method can be used to calculate higher 6S0 orders systematically in 1/S . Beyond leading order, ϫ exp͑ϪS ͒ ͑␻␶ ͒, (34) 0 ͫͱ ␲ 0 ͬ m however, the WKB method becomes quite tedious, even when applied to quantum mechanics. In this subsection where the first factor comes from the harmonic- we show how the 1/S correction to the level splitting in oscillator amplitude and the second is the ratio of the 0 the double well can be determined using a two-loop in- two determinants. stanton calculation. We follow here Wohler and The result shows that the transition amplitude is pro- Shuryak (1994), who corrected a few mistakes in the portional to the time interval ␶m . In terms of stationary earlier literature (Aleinikov and Shuryak, 1987; Olejnik, states this is due to the fact that the contributions from 1989). Numerical simulations were performed by the two lowest states almost cancel each other. The Shuryak (1988b), while the correct result was first ob- ground-state wave function is the symmetric combina- tained using different methods by Zinn-Justin (1981). x ϭ x ϩ x tion ⌿0( ) ͓␾Ϫ␩( ) ␾␩( )͔/&, while the first ex- To next order in 1/S0 , the tunneling amplitude can be cited state E1ϭE0ϩ⌬E is antisymmetric, ⌿1(x) decomposed as ϭ͓␾ (x)Ϫ␾ (x)͔/&. Here, ␾ are the harmonic- Ϫ␩ ␩ Ϯ␩ 2A oscillator wave functions around the two classical ϪH␶ 2 ͗Ϫ␩͉e ͉␩͘Ӎ͉␺0͑␩͉͒ 1ϩ ϩ¯ minima. For times ␶Ӷ1/⌬E, the tunneling amplitude is ͩ S0 ͪ given by ␻␶ B ϫexp Ϫ 1ϩ ϩ ͩ 2 ͫ S ¯ͬͪ ϪH␶m ϪE0␶m 0 ͗Ϫ␩͉e ͉␩͘ϭ⌿0*͑Ϫ␩͒⌿0͑␩͒e C ϩ⌿*͑Ϫ␩͒⌿ ͑␩͒eϪE1␶mϩ ϫ⌬E 1ϩ ϩ ␶, (38) 1 1 ¯ 0ͩ S ¯ͪ 0 1 ϭ ␾* ͑Ϫ␩͒␾ ͑␩͒͑⌬E␶ ͒eϪ␻␶m/2 where we are interested in the coefficient C, the next 2 Ϫ␩ ␩ m order correction to the level splitting. The other two cor- rections, A and B, are unrelated to tunneling, and we ϩ . (35) ¯ can get rid of them by dividing the amplitude by Note that the validity of the semiclassical approximation ͗␩͉exp(ϪH␶)͉␩͘; see Eq. (15). requires ␶ӷ1/␻. In order to calculate the next-order correction to the We can read off the level splitting from Eqs. (34) and instanton result, we have to expand the action beyond (35). The result can be obtained in an even more elegant order (␦x)2. The result can be interpreted in terms of a way by going to large times ␶Ͼ1/⌬E. In this case, multi- new set of Feynman rules in the presence of an instan- instanton paths are important. If we ignore the interac- ton (see Fig. 4). The triple and quartic coupling con- tion between instantons, multi-instanton contributions stants are ␣ϭ4␭xcl(t) and ␤ϭ␭ (compared to ␣0 can easily be summed up ϭ4␭␩ϭͱ2␭ and ␤0ϭ␭ for the anharmonic oscillator).

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 336 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

␦x. This leads to a tadpole graph proportional to x¨ cl , which has no counterpart in the anharmonic-oscillator case. We get ϱ ϱ tanh͑t/2͒ c1ϭϪ9␤0 dtdtЈ 2 ͵Ϫϱ ͵Ϫϱ cosh ͑t/2͒ 49 ϫtanh͑tЈ/2͒G͑t,tЈ͒G͑tЈ,tЈ͒ϭϪ SϪ1 . (44) 60 0 FIG. 4. Feynman diagrams for the two-loop correction to the tunneling amplitude in the quantum-mechanical double-well The sum of the four diagrams is Cϭ(a1ϩb11ϩb12 potential. The first three correspond to the diagrams in Fig. ϩc1)S0ϭϪ71/72. The two-loop result for the level split- 3(a) but with different propagators and vertices, while the ting is fourth diagram contains a new vertex, generated by the collec- tive coordinate Jacobian. 6S0 71 1 ⌬Eϭ ␻ exp ϪS Ϫ ϩ , (45) ͱ ␲ ͩ 0 72 S ¯ͪ 0 The propagator is the Green’s function of the differen- tial operator (22). There is one complication due to the in agreement with the WKB result obtained by Zinn- fact that the operator O has a zero mode. The Green’s Justin (1981). The fact that the next-order correction is function is uniquely defined by requiring it to be or- of order one and negative is significant. It implies that thogonal to the translational zero mode. The result is the one-loop result becomes inaccurate for moderately (Olejnik, 1989) large barriers (Sϳ1) and that it overestimates the tun- neling probability. We have presented this calculation in 1 detail in order to show that the instanton method can be ͑2␻͒G͑x,y͒ϭg ͑x,y͒ 2Ϫxyϩ ͉xϪy͉͑11Ϫ3xy͒ 0 ͫ 4 systematically extended to higher orders in 1/S. In field theory, however, this calculation is sufficiently difficult 3 that it has not yet been performed. ϩ͑xϪy͒2 ϩ ͑1Ϫx2͒͑1Ϫy2͒ ͬ 8 11 ϫ log͓g ͑x,y͔͒Ϫ , (39) ͫ 0 3 ͬ 5. Instanton/anti-instanton interaction and the ground-state energy 1Ϫ͉xϪy͉Ϫxy g ͑x,y͒ϭ , (40) 0 1ϩ͉xϪy͉Ϫxy Up to now we have focused on the tunneling ampli- tude for transitions between the two degenerate vacua where xϭtanh(␻t/2), yϭtanh(␻tЈ/2), and g0(x,y) is the of the double-well potential. This amplitude is directly Green’s function of the harmonic oscillator (12). There related to the gap ⌬E between the ground state and the are four diagrams at two-loop order (see Fig. 4). The first excited state. In this subsection we wish to discuss first three diagrams are of the same form as the how the semiclassical theory can be used to calculate the anharmonic-oscillator diagrams. Subtracting these con- mean Ectrϭ(E0ϩE1)/2 of the two levels. In this context, tributions, we get it is customary to define the double-well potential by V 2 2 ϱ 97 ϭ(x /2)(1Ϫgx) . The coupling constant g is related to a ϭϪ3␤ dt G2͑t,t͒ϪG2͑t,t͒ ϭϪ SϪ1, the coupling ␭ used above by g2ϭ2␭. Unlike the split- 1 0 ͵ „ 0 … 1680 0 Ϫϱ ting, the mean energy is related to topologically trivial (41) paths connecting the same vacua. The simplest nonper- ϱ ϱ turbative path of this type is an instanton/anti-instanton 2 b11ϭ3␣0 dtdtЈ„tanh͑t/2͒ pair. ͵Ϫϱ ͵Ϫϱ In Sec. II.A.3 we calculated the tunneling amplitude 53 using the assumption that instantons do not interact with ϫtanh͑tЈ/2͒G3͑t,tЈ͒ϪG3͑t,tЈ͒ ϭϪ SϪ1, 0 … 1260 0 each other. We found that tunneling makes the coordi- nates uncorrelated and leads to a level splitting. If we (42) take the interaction among instantons into account, the 9 ϱ ϱ contribution from instanton/anti-instanton pairs is given 2 2 b12ϭ ␣0 dtdtЈ„␣ tanh͑t/2͒ by 2 ͵Ϫϱ ͵Ϫϱ d␶ ϫtanh t /2 G t,t G t,t G t ,t ͗␩͉eϪH␶m͉␩͘ϭ␶ exp͓S ͑␶͔͒, (46) ͑ Ј ͒ ͑ ͒ ͑ Ј͒ ͑ Ј Ј͒ m ͵ ␲g2 IA

ϪG0͑t,t͒G0͑t,tЈ͒G0͑tЈ,tЈ͒ … where SIA(␶) is the action of an instanton/anti-instanton 2 Ϫ1 39 pair with separation ␶, and the prefactor (␲g ) comes ϭϪ SϪ1 . (43) from the square of the single-instanton density. The ac- 560 0 tion of an instanton/anti-instanton (IA) pair can be cal- The last diagram comes from expanding the Jacobian in culated given an ansatz for the path that goes from one

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 337

FIG. 5. Streamline configurations in the double-well potential FIG. 6. Distribution of the action density sϭx˙ 2/2ϩV(x) for for ␩ϭ1.4 and ␭ϭ1(␻Ӎ4, S0Ӎ5), adapted from Shuryak, the streamline configurations shown in Fig. 5. 1988b. The horizontal axis shows the time coordinate and the vertical axis the amplitude x(␶). The different paths corre- spond to different values of the streamline parameter ␭ as the 1 1 S͑T͒ϭ Ϫ2eϪTϪ12TeϪ2TϩO͑eϪ2T͒ configuration evolves from a well-separated pair to an almost g2 ͫ3 ͬ perturbative path. The initial path has an action Sϭ1.99S0 , the other paths correspond to a fixed reduction of the action by 71g2 ϩ͓24TeϪTϩO͑eϪT͔͒ϩ ϩO͑g4͒ , 0.2S0 . ͫ 6 ͬ minimum of the potential to the other and back. An (49) example of such a path is the ‘‘sum ansatz’’ (Zinn-Justin, where the first term is the classical streamline interac- 1983), tion up to next-to-leading order, the second term is the quantum correction (one-loop) to the leading-order in- 1 ␶Ϫ␶I ␶Ϫ␶A x ͑␶͒ϭ 2Ϫtanh ϩtanh . (47) teraction, and the last term is the two-loop correction to sum 2g 2 2 ͫ ͩ ͪ ͩ ͪͬ the single-instanton interaction. 2 ϪT This path has the action SIA(T)ϭ1/g ͓1/3Ϫ2e If one tries to use the instanton result (46) in order to Ϫ2T ϩO(e )͔, where Tϭ͉␶IϪ␶A͉. It is qualitatively clear calculate corrections to Ectr , one encounters two prob- that if the two instantons are separated by a large time lems. First, the integral diverges at large T. This is sim- interval Tӷ1, the action SIA(T) is close to 2S0 . In the ply related to the fact that IA pairs with large separation opposite limit T 0, the instanton and the anti-instanton should not be counted as pairs, but as independent in- → annihilate, and the action SIA(T) should tend to zero. In stantons. This problem is easily solved; all one has to do that limit, however, the IA pair is at best an approxi- is subtract the square of the single-instanton contribu- mate solution of the classical equations of motion, and it tion. Second, once this subtraction has been performed, is not clear how the path should be chosen. the integral is dominated by the region of small T, The best way to deal with this problem is the ‘‘stream- where the action is not reliably calculable. This problem line’’ or ‘‘valley’’ method (Balitsky and Yung, 1986). In is indeed a serious one, stemming from the fact that Ectr this approach one starts with a well-separated IA pair is not directly related to tunneling, but is dominated by and lets the system evolve using the method of steepest perturbative contributions. In general, we expect Ectr to descent. This means that we have to solve have an expansion dx ␶͒ ␦S ␭͑ 2k ͑0͒ Ϫ2/͑6g2͒ 2k ͑2͒ f͑␭͒ ϭ , (48) Ectrϭ͚ g Ectr,kϩe ͚ g Ectr,kϩ¯ , (50) d␭ ␦x␭͑␶͒ k k where ␭ labels the path as we proceed from the initial where the first term is the perturbative contribution, the configuration x␭ϭ0(␶)ϭxsum(␶) down the valley to the second corresponds to one IA pair, and so on. However, vacuum configuration, and f(␭) is an arbitrary function perturbation theory in g is divergent (not even Borel that reflects the reparametrization invariance of the summable), so the calculation of the IA contribution re- streamline solution. A sequence of paths obtained by quires a suitable definition of perturbation theory. solving the streamline equation (48) numerically is One way to deal with this problem (going back to shown in Fig. 5 (Shuryak, 1988b). An analytical solution Dyson’s classical work on QED) is analytic continuation 2 to first order in 1/S0 can be found in Balitsky and Yung in g. For g imaginary (g negative), the IA contribution (1986). The action density sϭx˙ 2/2ϩV(x) corresponding is well defined [the integral over T is dominated by T 2 to the paths in Fig. 5 is shown in Fig. 6. We can see ϳϪlog(Ϫg )]. The IA contribution to Ectr is (Bogo- clearly how the two localized solutions merge and even- molny, 1980; Zinn-Justin, 1983) tually disappear as the configuration progresses down Ϫ1/͑3g2͒ the valley. Using the streamline solution, we find the e 2 E͑2͒ϭ log Ϫ ϩ␥ϩO g2 log͑g2͒ , instanton/anti-instanton action for large T (Faleev and ctr ␲g2 ͫ ͩ g2 ͪ „ …ͬ Silvestrov, 1995) (51)

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 338 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

where ␥ϭ0.577... is Euler’s constant. When we now TABLE I. Exact center-of-the-band energies Ectrϭ(E0 2 2 continue back to positive g , we get both real and imagi- ϩE1)/2 for different values of g [expressed in terms of N ϭ 2 nary contributions to Ectr . Since the sum of all contri- 1/(3g )] compared to the semiclassical estimate discussed in the text. butions to Ectr is certainly real, the imaginary part has to Ϫ1/(3g2) cancel against a small O(e ) imaginary part in the Nϵ1/(3g2) 4681012 perturbative expansion. This allows us to determine the (0) ex imaginary part Im Ectr of the analytically continued per- Ectr 0.4439 0.43797 0.44832 0.459178 0.467156 13 th turbative sum. Ectr 0.4367 0.44367 0.44933 0.459307 0.467173 From the knowledge of the imaginary part of pertur- bation theory, one can determine the large-order behav- (0) 2k (0) ior of the perturbation series Ectr ϭ͚kg Ectr,k (Brezin, In summary, Ectr is related to configurations with no Parisi, and Zinn-Justin, 1977; Lipatov, 1977). The coef- net topology, and in this case the calculation of instan- ficients are given by the dispersion integrals ton effects requires a suitable definition of the perturba- 1 ϱ dg2 tion series. This can be accomplished using analytic con- E͑0͒ ϭ Im͓E͑0͒ ͑g2͔͒ . (52) tinuation in the coupling constant. After analytic ctr,k ␲ ͵ ctr,k g2kϩ2 0 continuation, we can perform a reliable interacting in- Since the semiclassical result (51) is reliable for small g, stanton calculation, but the result has an imaginary part. we can calculate the large-order coefficients. Including This shows that the instanton contribution by itself is not the corrections calculated by Faleev and Silvestrov well determined—it depends on the definition of the (1995), we have perturbation sum. However, the sum of perturbative and 3kϩ1k 53 nonperturbative contributions is well defined (and real) E͑0͒ ϭ 1Ϫ ϩ . (53) and agrees very accurately with the numerical value of ctr,k ␲ ͩ 18k ¯ͪ Ectr . This result can be compared with the exact coefficients In gauge theories the situation is indeed very similar: (Brezin et al., 1977). For small k the result is completely there are both perturbative and nonperturbative contri- wrong, but already for kϭ5,6,7,8 the ratio of the butions to the vacuum energy, and the two contributions asymptotic result to the exact coefficients is 1.04, 1.11, are not clearly separated. However, in the case of gauge 1.12, 1.11. We conclude that instantons determine the theories, we do not know how to define perturbation large-order behavior of the perturbative expansion. This theory, so we are not yet able to perform a reliable cal- is in fact a generic result: the asymptotic behavior of culation of the vacuum energy similar to Eq. (54). perturbation theory is governed by semiclassical con- figurations (although not necessarily involving instan- tons). B. Fermions coupled to the double-well potential In order to check the instanton/anti-instanton result In this section we consider one fermionic degree of (51) against the numerical value of Ectr for different val- ues of g, we have to subtract the perturbative contribu- freedom ␺␣ (␣ϭ1,2) coupled to the double-well poten- tial. This model provides additional insight into the tion to Ectr . This can be done using analytic continua- tion and the Borel transform (Zinn-Justin, 1982), and vacuum structure, not only of quantum mechanics, but the result is in very good agreement with the instanton also of gauge theories. We shall see that fermions are calculation. A simpler way to check the instanton result intimately related to tunneling and that the fermion- was proposed by Faleev and Silvestrov (1995). These induced interaction between instantons leads to strong authors simply truncated the perturbative series at the instanton/anti-instanton correlations. Another motiva- Nth term. In this case, the best accuracy occurs when tion for studying fermions coupled to the double-well ͉NϪ1/3g2͉ϳ1 and the estimate for E is given by potential is that for a particular choice of the coupling ctr constant, the theory is supersymmetric. This means that N 3NeϪN perturbative corrections to the vacuum energy cancel, 2n ͑0͒ Ectrϭ ͚ g Ectr,nϩ log͑6N͒ϩ␥ and the instanton contribution is more easily defined. nϭ0 ␲ ͫ The model is defined by the action 1 2␲ 53 ϩ 1Ϫ , (54) 1 ͱ N N Sϭ dt͑x˙2ϩWЈ2ϩ␺␺˙ ϩcWЉ␺␴ ␺͒, (55) 3 ͬͩ 18 ͪ 2 ͵ 2 which is compared to the exact values in Table I. We where ␺ (␣ϭ1,2) is a two component spinor, dots de- observe that the result (54) is indeed very accurate, and ␣ note time and primes spatial derivatives, and WЈϭx(1 ϪN Ϫ1/3g2 that the error is on the order of e ϭe . Ϫgx). We shall see that the vacuum structure depends crucially on the Yukawa coupling c. For cϭ0 fermions decouple, and we recover the double-well potential 13 How can the perturbative result develop an imaginary part? studied in the previous sections, while for cϭ1 the clas- After analytic continuation, the perturbative sum is Borel sum- sical action is supersymmetric. The supersymmetry mable because the coefficients alternate in sign. If we define (0) transformation is given by Ectr by analytic continuation of the Borel sum, it will have an 2 imaginary part for positive g . ␦xϭ␨␴2␺, ␦␺ϭ␴2␨x˙ϪWЈ␨, (56)

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 339 where ␨ is a Grassmann variable. For this reason, W is instanton pairs. Between the two tunneling events, the usually referred to as the superpotential. The action (55) system has an excited fermionic state, which causes a can be rewritten in terms of two bosonic partner poten- new interaction between the instantons. For cϭ1, super- tials (Salomonson and Holton, 1981; Cooper, Khare, and symmetry implies that all perturbative contributions (in- Sukhatme, 1995). Nevertheless, it is instructive to keep cluding the zero-point oscillation) to the vacuum energy cancel. Using supersymmetry, one can calculate the the fermionic degree freedom, because the model has 15 many interesting properties that also apply to QCD, vacuum energy from the tunneling rate [Eq. (60)] (Salomonson and Holton, 1981). The result is where the action cannot be bosonized. Ϫ1/(3g2) 1 2 O(e ) and positive, which implies that supersym- As before, the potential Vϭ 2 WЈ has degenerate 16 minima connected by the instanton solution. The tunnel- metry is broken. While the dependence on g is what ing amplitude is given by we would expect for a gas of instanton/anti-instanton molecules, understanding the sign in the context of an 1 instanton calculation is more subtle (see below). Ϫ␤H ϪScl Tr͑e ͒ϭ d␶J ͱdet Fe , (57) It is an instructive exercise to calculate the vacuum ͵ ͱdet Ј O OB energy numerically (which is quite straightforward; we are still dealing with a simple quantum-mechanical toy where Scl is the classical action, B is the bosonic opera- O model). In general, the vacuum energy is nonzero, but tor (22), and F is the Dirac operator, O for c 1, the vacuum energy is zero up to exponentially d small→ corrections. Varying the coupling constant, one ϭ ϩc␴ WЉ͑x ͒. (58) OF dt 2 cl can verify that the vacuum energy is smaller than any power in g, showing that supersymmetry breaking is a As explained in Sec. II.A.3, has a zero mode, related OB nonperturbative effect. to translational invariance. This mode has to be treated For cÞ1, the instanton/anti-instanton contribution to separately, which leads to a Jacobian J and an integral the vacuum energy has to be calculated directly. Even over the corresponding collective coordinate ␶. The fer- 14 for cϭ1, where the result can be determined indirectly, mion determinant also has a zero mode, given by this is a very instructive calculation. For an instanton/ t 1 1 anti-instanton path, there is no fermionic zero mode. ␹I,AϭN exp ϯ dtЈcWЉ͑x ͒ . (59) Writing the fermion determinant in the basis spanned by ͵ cl ϯi ͩ Ϫϱ ͪ & ͩ ͪ the original zero modes of the individual pseudopar- Since the fermion determinant appears in the numerator ticles, we have of the tunneling probability, the presence of a zero 0 TIA mode implies that the tunneling rate is zero. det͑ F͒ZMZϭ , (61) The reason for this is simple: the two vacua have dif- O ͩTAI 0 ͪ ferent fermion number, so they cannot be connected by where TIA is the overlap matrix element, a bosonic operator. The tunneling amplitude is nonzero ϱ only if a fermion is created during the process (tϩc␴2WЉ„xIA͑t͒…͒␹I . (62ץTIAϭ ͵ dt␹A͑ ͗0,ϩ͉␺ϩ͉0,Ϫ͘, where ␺Ϯϭ(1/&)(␺1Ϯi␺2) and ͉0,Ϯ͘ Ϫϱ denote the corresponding eigenstates. Formally, we get Clearly, mixing between the two zero modes shifts the a finite result because the fermion creation operator ab- eigenvalues away from zero, and the determinant is non- sorbs the zero mode in the fermion determinant. As we zero. As before, we have to choose the correct shall see later, this mechanism is completely analogous instanton/anti-instanton path xIA(t) in order to evaluate to the axial U(1)A anomaly in QCD and baryon- TIA . Using the valley method introduced in the last sec- number violation in electroweak theory. For cϭ1, the tion, we find the ground-state energy (Balitsky and tunneling rate is given by (Salomonson and Holton, Yung, 1986) 1981) 1 1 g2 cϪ1 ϱ 2 Ϫ1/3g2 c 1 2 Eϭ ͓1ϪcϩO͑g ͔͒Ϫ e 2c d␶ Ϫ1/6g 2 2␲ ͩ 2 ͪ ͵0 ͗0,ϩ͉␺ϩ͉0,Ϫ͘ϭ e . (60) ͱ␲g2 2 This result can be checked by performing a direct calcu- ϫexp Ϫ2c͑␶Ϫ␶ ͒ϩ eϪ2␶ , (63) 0 g2 lation using the Schro¨ dinger equation. ͩ ͪ Let us now return to the calculation of the ground- state energy. For cϭ0, the vacuum energy is the sum of perturbative contributions and a negative nonperturba- 2 15The reason is that, for supersymmetry theories, the Hamil- tive shift O(eϪ1/(6g )) due to individual instantons. For tonian is the square of the supersymmetry generators Q␣ , H cÞ0, the tunneling amplitude [Eq. (60)] will only enter 1 ϭ ͕Q ,Q ͖. Since the tunneling amplitude ͗0,ϩ͉␺ ͉0,Ϫ͘ is squared, so one needs to consider instanton/anti- 2 ϩ Ϫ ϩ proportional to the matrix element of Qϩ between the two different vacua, the ground-state energy is determined by the square of the tunneling amplitude. 14In the supersymmetric case, the fermion zero mode is the 16This was indeed the first known example of nonperturbative superpartner of the translational zero mode. supersymmetry breaking (Witten, 1981).

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 340 T. Scha¨fer and E. V. Shuryak: Instantons in QCD where ␶ is the instanton/anti-instanton separation and sical vacuum of the theory. In the Hamiltonian formula- 2 exp(Ϫ2␶0)ϭcg /2. The two terms in the exponent inside tion, it is convenient to use the temporal gauge A0ϭ0 a a the integral correspond to the fermionic and bosonic in- (here we use matrix notation AiϭAi ␭ /2, where the teractions between instantons. One can see that fermi- SU(N) generators satisfy ͓␭a,␭b͔ϭ2ifabc␭c and are ons cut off the integral at large ␶. There is an attractive normalized according to Tr(␭a␭b)ϭ2␦ab). In this case, interaction, which grows with distance and forces instan- the conjugate momentum to the field variables Ai(x)is tons and anti-instantons to be correlated. Therefore, for 0Ai . The Hamiltonian isץjust the electric field Eiϭ cÞ0, the vacuum is no longer an ensemble of random given by tunneling events, but consists of correlated instanton/ anti-instanton molecules. 1 Hϭ d3x͑E2ϩB2͒, (65) The fact that both the bosonic and the fermionic in- 2g2 ͵ i i teraction are attractive means that the integral (63), just 2 2 like (46), is dominated by small ␶ where the integrand is where Ei is the kinetic and Bi the potential-energy not reliable. This problem can be solved as outlined in term. The classical vacuum corresponds to configura- the last section, by analytic continuation in the coupling tions with zero field strength. For non-Abelian gauge constant. As an alternative, Balitsky and Yung sug- fields this does not imply that the potential has to be gested shifting the integration contour in the complex ␶ constant, but limits the gauge fields to be ‘‘pure gauge,’’ plane, ␶ ␶ϩi␲/2. On this path, the sign of the bosonic (U͑xជ͒†. (66 ץA ϭiU͑xជ͒ interaction→ is reversed, and the fermionic interaction i i picks up a phase factor exp(ic␲). This means that there In order to enumerate the classical vacua, we have to is a stable saddle point, but the instanton contribution to classify all possible gauge transformations U(xជ ). This the ground-state energy is in general complex. The means that we have to study equivalence classes of maps imaginary part cancels against the imaginary part of the from 3-space R3 into the gauge group SU(N). In prac- perturbation series, and only the sum of the two contri- tice, we can restrict ourselves to matrices satisfying butions is well defined. U(xជ ) 1asx ϱ(Callan et al., 1978a). Such mappings A special case is the supersymmetric point cϭ1. In can be→ classified→ using an integer called the winding (or this case, perturbation theory vanishes and the contribu- Pontryagin) number, which counts how many times the tion from instanton/anti-instanton molecules is real, group manifold is covered, 1 Ϫ1/3g2 2 1 U͒ ץ†U͒͑U ץ†Eϭ e ͓1ϩO͑g ͔͒. (64) n ϭ d3x⑀ijk Tr͓͑U 2␲ W 24␲2 ͵ i j This implies that at the supersymmetry point cϭ1 there (U͔͒. (67 ץ†ϫ͑U is a well-defined instanton/anti-instanton contribution. k The result agrees with what one finds from the H In terms of the corresponding gauge fields, this number 1 ϭ 2 ͕Qϩ ,QϪ͖ relation or directly from the Schro¨ dinger is the Chern-Simons characteristic equation. 1 1 In summary, in the presence of light fermions, tunnel- 3 ijk a a abc a b c (jA ϩ f A A A . (68ץ nCSϭ d x⑀ A ing is possible only if the fermion number changes dur- 16␲2 ͵ ͩ i k 3 i j kͪ ing the transition. Fermions create a long-range attrac- Because of its topological meaning, continuous deforma- tive interaction between instantons and anti-instantons, tions of the gauge fields do not change n . In the case and the vacuum is dominated by instanton/anti- CS of SU(2), an example of a mapping with winding num- instanton ‘‘molecules.’’ It is nontrivial to calculate the ber n can be found from the ‘‘hedgehog’’ ansatz, contribution of these configurations to the ground-state energy because topologically trivial paths can mix with U͑xជ ͒ϭexp͓if͑r͒␶axˆa͔, (69) perturbative corrections. The contribution of molecules where rϭ x and xˆ aϭxa/r. For this mapping, we find is most easily defined if one allows the collective coordi- ͉ជ͉ nate (time separation) to be complex. In this case, there 2 df 1 sin͓2 f͑r͔͒ ϱ n ϭ dr sin2͑f ͒ ϭ f͑r͒Ϫ . (70) exists a saddle point where the repulsive bosonic inter- W ␲ ͵ dr ␲ͫ 2 ͬ action balances the attractive fermionic interaction and 0 molecules are stable. These objects give a nonperturba- In order for U(xជ ) to be uniquely defined, f(r) has to be tive contribution to the ground-state energy, which is in a multiple of ␲ at both zero and infinity, so that nW is general complex, except in the supersymmetric case indeed an integer. Any smooth function with f(r ϱ) where it is real and positive. ϭ0 and f(0)ϭn␲ provides an example of a function→ with winding number n. C. Tunneling in Yang-Mills theory We conclude that there is an infinite set of classical vacua enumerated by an integer n. Since they are topo- logically different, one cannot go from one vacuum to 1. Topology and classical vacua another by means of a continuous gauge transformation. Before we study tunneling phenomena in Yang-Mills Therefore there is no path from one vacuum to another theory, we have to become more familiar with the clas- such that the energy remains zero all the way.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 341

2. Tunneling and the Belavin-Polyakov-Schwartz-Tyupkin For finite-action configurations, the gauge potential has U†. In analogy ץinstanton to be pure gauge at infinity, A iU ␮→ ␮ Two important questions concerning the classical with the arguments given in the last section, all maps from the 3-sphere S (corresponding to ͉x͉ ϱ) into the vacua immediately come to mind. First, is there some 3 → physical observable that distinguishes between them? gauge group can be classified by a winding number n. † ␮U into Eq. (74), one finds that QץSecond, is there any way to go from one vacuum to an- Inserting A␮ϭiU other? The answer to the first question is positive but ϭn. most easily demonstrated in the presence of light fermi- Furthermore, if the gauge potential falls off rapidly at ons, so we shall come to it later. Let us now concentrate spatial infinity, then on the second question. We are going to look for a tunneling path in gauge d Qϭ dt d3xK ϭn ͑tϭϱ͒Ϫn ͑tϭϪϱ͒, theory which connects topologically different classical ͵ dt ͵ 0 CS CS vacua. From the quantum-mechanical example, we (76) know that we have to look for classical solutions of the Euclidean equations of motion. The best tunneling path which shows that field configurations with QÞ0 connect is the solution with minimal Euclidean action connecting different topological vacua. In order to find an explicit vacua with different Chern-Simons numbers. To find solution with Qϭ1, it is useful to start from the simplest these solutions, it is convenient to exploit the identity winding number nϭ1 configuration. As in Eq. (69), we ,U† with Uϭixˆ ␶ϩ , where ␶Ϯϭ(␶ ץcan take A ϭiU 1 4 a a ␮ ␮ ␮ ␮ ␮ ជ Sϭ 2 d xG␮␯G␮␯ a 2 4g ͵ ϯi). Then A␮ϭ2␩a␮␯x␯ /x , where we have introduced the ’t Hooft symbol ␩a␮␯ , 1 1 4 a ˜a a ˜a 2 ϭ 2 d x ϮG␮␯G␮␯ϩ ͑G␮␯ϯG␮␯͒ , (71) 4g ͵ ͫ 2 ͬ ⑀a␮␯ , ␮,␯ϭ1,2,3, ˜ ␦ , ␯ϭ4, where G␮␯ϭ1/2⑀␮␯␳␴G␳␴ is the dual field strength ten- ␩a␮␯ϭ a␮ (77) sor (the field tensor in which the roles of electric and ͭ Ϫ␦a , ␮ϭ4. ␯ magnetic fields are interchanged). Since the first term is

a topological invariant (see below) and the last term is We also define ␩¯ a␮␯ by changing the sign of the last two always positive, it is clear that the action is minimal if equations. Further properties of ␩a␮␯ are summarized in the field is (anti) self-dual, the Appendix, Sec. A.3. We can now look for a solution a a ˜a of the self-duality equation (72) using the ansatz A␮ G␮␯ϭϮG␮␯ . (72) 2 2 ϭ2␩a␮␯x␯f(x )/x , where f has to satisfy the boundary One can also show directly that the self-duality condi- condition f 1asx2 ϱ. Inserting the ansatz in Eq. 17 → → tion implies the equations of motion, D␮G␮␯ϭ0. This (72), we get is a useful observation because, in contrast to the equa- tion of motion, the self-duality equation (72) is a first- f͑1Ϫf ͒Ϫx2fЈϭ0. (78) order differential equation. In addition to that, one can show that the energy-momentum tensor vanishes for This equation is solved by fϭx2/(x2ϩ␳2), which gives self-dual fields. In particular, self-dual fields have zero the Belavin-Polyakov-Schwartz-Tyupkin instanton solu- (Minkowski) energy density. tion (Belavin et al., 1975), The action of a self-dual field configuration is deter- mined by the topological charge (or four-dimensional 2␩a␮␯x␯ Aa ͑x͒ϭ . (79) Pontryagin index) ␮ x2ϩ␳2 1 Qϭ d4xGa G˜a . (73) Here ␳ is an arbitrary parameter characterizing the size 32␲2 ͵ ␮␯ ␮␯ of the instanton. A solution with topological charge Q 2 2 From Eq. (71), we have Sϭ(8␲ ͉Q͉)/g for self-dual ϭϪ1 can be obtained by replacing ␩a␮␯ with ␩¯ a␮␯ . The fields. For finite-action field configurations, Q has to be corresponding field strength is an integer. This can be seen from the fact that the inte- grand is a total derivative, 192␳4 ͑Ga ͒2ϭ . (80) ␮␯ ͑x2ϩ␳2͒4 4 (␮K␮ϭ ͵ d␴␮K␮ , (74ץQϭ ͵ d x In our conventions, the coupling constant appears only 1 1 as a factor in front of the action. This convention is very (Aaϩ fabcAaAbAc . (75 ץK ϭ ⑀ Aa ␮ 16␲2 ␮␣␤␥ͩ ␣ ␤ ␥ 3 ␣ ␤ ␥ͪ convenient in dealing with classical solutions. For per- turbative calculations, it is more common to rescale the fields as A gA . In this case, there is a factor 1/g in ␮→ ␮ 17The reverse is not true, but one can show that non-self-dual the instanton gauge potential, which shows that the field solutions of the equations of motion are saddle points, not of the instanton is much stronger than an ordinary, per- local minima of the action. turbative field.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 342 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

a 4 Note that G␮␯ is well localized (it falls off as 1/x ) states but has to be a superposition of all vacua. This despite the fact that the gauge potential is long range, problem is similar to the motion of an electron in the A␮ϳ1/x. The invariance of the Yang-Mills equations periodic potential of a crystal. It is well known that the under coordinate inversion (Jackiw and Rebbi, 1976b) solutions form a band ␺␪ characterized by a phase ␪ implies that the singularity of the potential can be ෈͓0,2␲͔ (sometimes referred to as quasimomentum). shifted from infinity to the origin by means of a (singu- The wave functions are Bloch waves satisfying the peri- lar) gauge transformation Uϭixˆ ␶ϩ. The gauge poten- i␪n ␮ odicity condition ␺␪(xϩn)ϭe ␺␪(x). tial in singular gauge is given by Let us see how this band arises from tunneling events. ¯ 2 If instantons are sufficiently dilute, then the amplitude a x␯ ␩a␮␯␳ A ͑x͒ϭ2 . (81) i j ␮ x2 x2ϩ␳2 to go from one topological vacuum ͉ ͘ to another ͉ ͘ is given by This singularity at the origin is not physical; the field ␦ strength and topological charge density are smooth. NϩϪNϪϪjϩi ϪS NϩϩNϪ However, in order to calculate the topological charge ͗j͉exp͑ϪH␶͉͒i͘ϭ͚ ͚ ͑K␶e ͒ , Nϩ NϪ Nϩ!NϪ! from a surface integral over K␮ , we need to surround (83) the origin by a small sphere. The topology of this con- figuration is therefore located at the origin, not at infin- where K is the pre-exponential factor in the tunneling ity. In order to study instanton/anti-instanton configura- amplitude, and NϮ are the numbers of instantons and tions, we shall work mainly with such singular ant-instantons. Using the identity configurations. 1 2␲ The classical instanton solution has a number of de- ␦ ϭ d␪ei␪͑aϪb͒, (84) ab 2␲ ͵ grees of freedom known as collective coordinates. In the 0 case of SU(2), the solution is characterized by the in- we can write the sum over instantons and anti-instantons stanton size ␳, the instanton position z␮ , and three pa- as rameters which determine the color orientation of the 1 2␲ instanton. The group orientation can be specified in j ϪH i ϭ d ei␪͑iϪj͒ † ͗ ͉exp͑ ␶͉͒ ͘ ͵ ␪ terms of the SU(2) matrix U, A␮ UA␮U , or the cor- 2␲ 0 ab →1 a † b responding rotation matrix R ϭ 2 tr(U␶ U ␶ ), such that Aa RabAb . Due to the symmetries of the instan- ϫexp͓2K␶ cos͑␪͒exp͑ϪS͔͒. ␮→ ␮ ton configuration, ordinary rotations do not generate (85) new solutions. This result shows that the true eigenstates are the theta SU(3) instantons can be constructed by embedding vacua ␪ ϭ͚ ein␪ n . Their energy is the SU(2) solution. For ͉Q͉ϭ1, there are no genuine ͉ ͘ n ͉ ͘ SU(3) solutions. The number of parameters characteriz- E͑␪͒ϭϪ2K cos͑␪͒exp͑ϪS͒. (86) ing the color orientation is seven, not eight, because one of the SU(3) generators leaves the instanton invariant. The width of the zone is on the order of the tunneling For SU(N), the number of collective coordinates (in- rate. The lowest state corresponds to ␪ϭ0 and has nega- cluding position and size) is 4N. There exist exact tive energy. This is as it should be; tunneling lowers the n-instanton solutions with 4nN parameters, but they are ground-state energy. difficult to construct in general (Atiyah et al., 1977). A Does this result imply that in QCD there is a con- simple solution in which the relative color orientations tinuum of states without a mass gap? Not at all. Al- are fixed was given by ’t Hooft (unpublished), see Wit- though one can construct stationary states for any value ten (1977), Jackiw, Nohl, and Rebbi (1977), and the Ap- of ␪, they are not excitations of the ␪ϭ0 vacuum be- pendix Sec. A.1. cause in QCD the value of ␪ cannot be changed. As far We have explicitly constructed the tunneling path that as the strong interaction is concerned, different values of connects different topological vacua. The instanton ac- ␪ correspond to different worlds. Indeed, we can fix the tion is given by Sϭ(8␲2͉Q͉)/g2, implying that the tun- value of ␪ by adding an additional term, neling probability is i␪ a ˜ a 2 2 ϭ 2 G␮␯G␮␯ (87) Ptunnelingϳexp͑Ϫ8␲ /g ͒. (82) L 32␲ As in the quantum-mechanical example, the coefficient to the QCD Lagrangian. in front of the exponent is determined by a one-loop Does physics depend on the value of ␪? Naively, the calculation. interaction (87) violates both T and CP invariance. On the other hand, Eq. (87) is a surface term, and one might suspect that confinement somehow screens the effects of 3. The theta vacua the ␪ term. A similar phenomenon is known to oc- We have seen that non-Abelian gauge theory has a cur in three-dimensional compact electrodynamics periodic potential and that instantons connect the differ- (Polyakov, 1977). In QCD, however, one can show that ent vacua. This means that the ground state of QCD if the U(1)A problem is solved (there is no massless ␩Ј cannot be described by any of the topological vacuum state in the chiral limit) and none of the quarks is mass-

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 343 less, a nonzero value of ␪ implies that CP is broken between quarks would change sign. Clearly, it is impor- (Shifman, Vainshtein, and Zakharov, 1980a). tant to understand the properties of QCD with a non- Consider the expectation value of the CP-violating zero ␪ angle in more detail. observable ͗GG˜͘. Expanding the partition function in ˜ 2 powers of ␪, we have ͗GG͘ϭ␪(32␲ )␹top . Further- more, in Sec. V.E we shall prove an important low- 4. The tunneling amplitude energy theorem that determines the topological suscep- The next natural step is the one-loop calculation of tibility for small quark masses. Using these results, we the pre-exponent in the tunneling amplitude. In gauge have theory, this is a rather tedious calculation, which was done in the classic paper by ’t Hooft (1976b). Basically, m m ˜ 2 2 2 u d the procedure is completely analogous to what we did in ͗GG͘ϭϪ␪͑32␲ ͒f␲m␲ 2 (88) ͑muϩmd͒ the context of quantum mechanics. The field is ex- cl for two light flavors to leading order in ␪ and the quark panded around the classical solution, A␮ϭA␮ϩ␦A␮ .In masses. Similar estimates can be obtained for QCD, we have to make a gauge choice. In this case, it is CP-violating observables that are directly accessible to most convenient to work in a background field gauge cl experiment. The most severe limits on CP violation in D␮(A␯ )␦A␮ϭ0. the strong interaction come from the electric dipole of We have to calculate the one-loop determinants for the neutron. Current experiments imply that (Baluni, gauge fields, ghosts, and possible matter fields (which we 1979; Crewther, et al., 1979) shall deal with later). The determinants are divergent both in the ultraviolet, like any other one-loop graph, ␪Ͻ10Ϫ9. (89) and in the infrared, due to the presence of zero modes. The question of why ␪ is so small is known as the strong As we shall see below, the two are actually related. In fact, the QCD beta function is partly determined by zero CP problem. The status of this problem is unclear. As modes (while in certain supersymmetric theories, the long as we do not understand the source of CP violation beta function is completely determined by zero modes; in nature, it is not clear whether the strong CP problem see Sec. VIII.C.1). is expected to have a solution within the standard We already know how to deal with the 4Nc zero model, or whether there is some mechanism outside the modes of the system. The integral over the zero mode is standard model that adjusts ␪ to be small. traded for an integral over the corresponding collective One possibility is provided by the fact that the state variable. For each zero mode, we get one factor of the with ␪ϭ0 has the lowest energy. This means that if ␪ Jacobian ͱS0. The integration over the color orienta- becomes a dynamic variable, the vacuum can relax to tions is compact, so it just gives a factor, but the integral the ␪ϭ0 state (just as electrons can drop to the bottom over size and position, we have to keep. As a result, we of the conduction band by emitting phonons). This is the get a differential tunneling rate, basis of the axion mechanism (Peccei and Quinn, 1977). 8␲2 2Nc 8␲2 The axion is a hypothetical pseudoscalar particle that Ϫ5 dnIϳ 2 exp Ϫ 2 ␳ d␳dz, (90) couples to GG˜. The equations of motion for the axion ͩ g ͪ ͩ g ͪ field automatically remove the effective ␪ term, which is where the power of ␳ can be determined from dimen- now a combination of ␪QCD and the axion expectation sional considerations. value. Experimental limits on the axion coupling are The ultraviolet divergence is regulated using the very severe, but an ‘‘invisible axion’’ might still exist Pauli-Vilars scheme, which is the most convenient (Kim, 1979; Shifman, Vainshtein, and Zakharov, 1980b; method when dealing with fluctuations around non- Zhitnitsky, 1980; Dine and Fischler, 1983; Preskill, Wise, trivial classical field configurations (the final result can and Wilczek, 1983). be converted into any other scheme). This means that The simplest way to resolve the strong CP problem is the determinant det O of the differential operator O is 2 to assume that the mass of the u quark vanishes (pre- divided by det(OϩM ), where M is the regulator mass. sumably because of a symmetry not manifest in the stan- Since we have to extract 4Nc zero modes from det O, 4N dard model). Unfortunately, this possibility appears to this gives a factor M c in the numerator of the tunnel- be ruled out phenomenologically, but there is no way to ing probability. know for sure before this scenario is explored in more In addition to that, there will be a logarithmic depen- detail on the lattice. More recently, it was suggested that dence on M coming from the ultraviolet divergence. To QCD might undergo a phase transition near ␪ϭ0,␲.In one-loop order, it is just the logarithmic part of the po- cl the former case, there is some support for this idea from larization operator. For any classical field A␮ the result lattice simulations (Schierholz, 1994), but the instanton can be written as a contribution to the effective action model and lattice measurements of the topological sus- (Brown and Creamer, 1978; Vainshtein et al., 1982), ceptibility do not suggest any singularity around ␪ϭ0. 2 g2 The latter limit ␪ϭ␲ also conserves CP and has a num- ␦S ϭ log͑M␳͒S͑Acl͒. (91) NZM 3 8␲2 ber of interesting properties (Snyderman and Gupta, 1981): in this world the instanton-induced interaction In the background field of an instanton, the classical ac-

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 344 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

2 2 tion cancels the prefactor g /8␲ , and exp(Ϫ␦SNZM) became apparent that loop diagrams involving external ϳ(M␳)Ϫ2/3. Now we can collect all terms in the exponent vector and axial-vector currents could not be regulated of the tunneling rate, in such a way that all the currents remained conserved. From the triangle diagram involving two gauge fields 8␲2 dn ϳexp Ϫ ϩ4N log͑M␳͒ and the flavor singlet axial current, one finds I ͩ g2 c Nf (j5 ϭ Ga G˜ a , (95 ץ Nc ␮ ␮ 2 ␮␯ ␮␯ Ϫ log͑M␳͒ ␳Ϫ5d␳dz 16␲ 3 ͪ ␮ 5 where j␮ϭq¯␥␮␥5q with qϭ(u,d,s,...). This result is 8␲2 not modified at higher orders in the perturbative expan- ϵexp Ϫ ␳Ϫ5d␳dz , (92) ͩ g2͑␳͒ͪ ␮ sion. At this point, the gauge field on the right-hand side is some arbitrary background field. The fact that the fla- where we have recovered the running coupling constant vor singlet current has an anomalous divergence was (8␲2)/g2(␳)ϭ(8␲2)/g2Ϫ(11N /3)log(M␳). Thus the in- c quite welcome in QCD because it seemed to explain the frared and ultraviolet divergent terms combine to give the coefficient of the one-loop beta function, b absence of a ninth Goldstone boson, the so-called U(1)A puzzle. ϭ11Nc/3, and the bare charge and the regulator mass M can be combined into a running coupling constant. At Nevertheless, there are two apparent problems with two-loop order, the renormalization group requires the Eq. (95) if we want to understand the U(1)A puzzle. miracle to happen once again, and the nonzero mode The first is that the right-hand side is proportional to the -␮K␮ ,soitapץ ,determinant can be combined with the bare charge to divergence of the topological current give the two-loop beta function in the exponent and the pears that we can still define a conserved axial current. one-loop running coupling in the pre-exponent. The other is that, since the right-hand side of the The remaining constant was calculated by ’t Hooft anomaly equation is just a surface term, it seems that the (1976b) and Bernard et al. (1977). The result is anomaly does not have any physical effects.

2 2Nc The answer to the first problem is that, while the to- 0.466 exp͑Ϫ1.679Nc͒ 8␲ dnIϭ 2 pological charge is gauge invariant, the topological cur- ͑NcϪ1͒!͑NcϪ2͒! ͩ g ͪ rent is not. The appearance of massless poles in correla- 2 4 8␲ d zd␳ tion functions of K␮ does not necessarily correspond to ϫexp Ϫ . (93) massless particles. The answer to the second question is ͩ g2͑␳͒ͪ ␳5 of course related to instantons. Because QCD has topo- The tunneling rate dnA for anti-instantons is of course logically distinct vacua, surface terms are relevant. identical. Using the one-loop beta function, one can In order to see how instantons can lead to the non- write the result as conservation of axial charge, let us calculate the change

dnI d␳ in axial charge, ϳ ͑␳⌳͒b, (94) d4z ␳5 Q ϭQ tϭϩ ϪQ tϭϪ ϭ d4x j5 (␮ ␮ . (96ץ ͵ ϱ͒ 5͑ ϱ͒ 5͑ 5 ⌬ and because of the large coefficient bϭ(11Nc/3)ϭ11, the exponent overcomes the Jacobian and small-size in- In terms of the fermion propagator, ⌬Q5 is given by stantons are strongly suppressed. On the other hand, there appears to be a divergence at large ␳. This is re- (tr S͑x,x͒␥ ␥ . (97 ץ Q ϭ d4xN⌬ lated to the fact that the perturbative beta function is 5 ͵ f ␮ „ ␮ 5… not applicable in this regime. We shall come back to this The fermion propagator is the inverse of the Dirac op- question in Sec. III. Ϫ1 erator, S(x,y)ϭ͗x͉(iD” ) ͉y͘. For any given gauge field, we can determine the propagator in terms of the D. Instantons and light quarks eigenfunctions iD” ␺␭ϭ␭␺␭ of the Dirac operator, † 1. Tunneling and the U(1) anomaly ␺␭͑x͒␺␭͑y͒ A S͑x,y͒ϭ . (98) ͚ ␭ When we considered the topology of gauge fields and ␭ the appearance of topological vacua, we posed the ques- Using the eigenvalue equation, we can now evaluate tion whether the different vacua could be physically dis- ⌬Q5 , tinguished. In the presence of light fermions, there is a ␺ ͑x͒␺†͑x͒ simple physical observable that distinguishes between 4 ␭ ␭ ⌬Q5ϭNf d x tr ͚ 2␭␥5 . (99) the topological vacua, the axial charge. This observation ͵ ͩ ␭ ␭ ͪ helped to clarify the mechanism of chiral anomalies and For every nonzero ␭, ␥ ␺ is an eigenvector with eigen- showed how perturbative and nonperturbative effects 5 ␭ value Ϫ␭. But this means that ␺ and ␥ ␺ are orthogo- are related in the breaking of classical symmetries. ␭ 5 ␭ nal, so only zero modes can contribute to Eq. (99): Anomalies first appeared in the context of perturba- tion theory (Adler, 1969; Bell and Jackiw, 1969), when it ⌬Q5ϭ2Nf͑nLϪnR͒, (100)

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 345

where NL,R is the number of left and right-handed zero stantons) explains the axial anomaly. Integrating the modes, and we have used the fact that the eigenstates anomaly equation (95), we find that ⌬Q5 is related to are normalized. the topological charge Q. On the other hand, from Eq. The crucial property of instantons, originally discov- (100) we know that ⌬Q5 counts the number of left- ered by ’t Hooft, is that the Dirac operator has a zero handed zero modes minus the number of right-handed zero modes. But this is exactly what instantons do: every mode iD” ␺0(x)ϭ0 in the instanton field. For an instan- ton in the singular gauge, the zero-mode wave function instanton contributes one unit to the topological charge is and has a left-handed zero mode, while anti-instantons have QϭϪ1 and a right-handed zero mode. There is another way to look at this process, known as x ϩ ␳ 1 ␥• 1 ␥5 the ‘‘infinite hotel story’’ (Gribov, 1981; see also Shif- ␺0͑x͒ϭ 2 2 3/2 ␾, (101) ␲ ͑x ϩ␳ ͒ ͱx2 2 man, 1989). Let us consider the gauge potential of an instanton as a fixed background field and diagonalize the where ␾␣mϭ⑀␣m/& is a constant spinor in which the ជ SU(2) color index ␣ is coupled to the spin index m time-dependent Dirac Hamiltonian i␣ជ •D (again, it is ϭ1,2. Let us briefly digress in order to show that Eq. most convenient to work in the temporal gauge). The (101) is indeed a solution of the Dirac equation. First, presence of a four-dimensional, normalizable zero mode observe that18 implies that there is one left-handed state that crosses from positive to negative energy during the tunneling 1 event, while one right-handed state crosses the other ͑iD͒2ϭ ϪD2ϩ ␴ G . (102) way. This can be seen as follows: In the adiabatic ap- ” 2 ␮␯ ␮␯ ͩ ͪ proximation, solutions of the Dirac equation are given (Ϯ) (Ϯ) We can now use the fact that ␴␮␯G␮␯ ϭϯ␥5␴␮␯G␮␯ by (Ϯ) for (anti) self-dual fields G␮␯ . In the case of a self-dual t gauge potential, the Dirac equation iD” ␺ϭ0 then im- ␺i͑xជ ,t͒ϭ␺i͑xជ ,tϭϪϱ͒exp Ϫ dtЈ⑀i͑tЈ͒ . (105) plies (␺ϭ␹Lϩ␹R), ͩ ͵Ϫϱ ͪ

1 The only way we can have a four-dimensional, normal- 2 ͑ϩ͒ 2 t ϪD ϩ ␴␮␯G␮␯ ␹Lϭ0, ϪD ␹Rϭ0, (103) izable wave function is if ⑀i is positive for ϱ and ͩ 2 ͪ negative for t Ϫϱ. This explains how axial charge→ can → and vice versa (ϩ Ϫ,L R) for anti-self-dual fields. be violated during tunneling. No fermion ever changes 2 ↔ ↔ Since ϪD is a positive operator, ␹R has to vanish, and its chirality; all states simply move one level up or down. the zero mode in the background field of an instanton The axial charge comes, so to say, from the ‘‘bottom of has to be left handed, while it is right handed in the case the Dirac sea.’’ of an anti-instanton.19 A general analysis of the solutions In QCD, the most important consequence of the of Eq. (103) was given by ’t Hooft (1976b) and Jackiw anomaly is the fact that the would-be ninth Goldstone and Rebbi (1977). In practice, the zero mode is most boson, the ␩Ј, is massive even in the chiral limit. The m way the ␩ acquires its mass is also intimately related easily found by expanding the spinor ␹ as ␹␣ Ј (ϩ) ␣m with instantons, and we shall come back to this topic a ϭM␮(␶␮ ) . For (multi) instanton gauge potentials of a a number of times during this review. Historically, the first ␯log ⌸(x) (see Appendix A1), it isץthe form A␮ϭ␩¯ ␮␯ convenient to make the ansatz (Grossman, 1977) attempt to understand the origin of the ␩Ј mass from the anomaly was based on anomalous Ward identities ⌽ x͒ (Veneziano, 1979); see Sec. V.E. Saturating these Ward m ͑ ͑ϩ͒ ␣m -␮ ͑␶␮ ͒ . (104) identities with hadronic resonances and using certain adץ␹␣ ϭͱ⌸͑x͒ ͩ ⌸͑x͒ͪ ditional assumptions, one can derive the Witten- Substituting this ansatz in Eq. (103) shows that ⌽(x) has Veneziano relation (Veneziano, 1979; Witten, 1979b) to satisfy the Laplace equation ᮀ⌽(x)ϭ0. A solution f 2 that leads to a normalizable zero mode is given by 4 ␲ 2 2 2 2 2 ␹ ϭ d x͗Q͑x͒Q͑0͒͘ϭ ͑m ϩm Ϫ2m ͒. ⌽(x)ϭ␳ /x , from which we finally obtain Eq. (101). top ͵ ␩ ␩Ј K 2Nf Again, we can obtain an SU(3) solution by embedding (106) the SU(2) result. In this relation, we have introduced an important new We can now see how tunneling between topologically different configurations (described semiclassically by in- quantity, the topological susceptibility ␹top , which mea- sures fluctuations of the topological charge in the QCD vacuum. The combination of meson masses on the right- hand side corresponds to the part of the ␩Ј mass that is 18We use Euclidean Dirac matrices that satisfy ͕␥ ,␥ ͖ ␮ ␯ not due to the strange-quark mass. ϭ2␦␮␯ . We also have ␴␮␯ϭi/2͓␥␮ ,␥␯͔ and ␥5ϭ␥1␥2␥3␥4 . 19This result is not an accident. Indeed, there is a mathemati- There are several subtleties in connection with the cal theorem (the Atiyah-Singer index theorem) that requires Witten-Veneziano relation. In Sec. V.E, we shall show that QϭnLϪnR for every species of chiral fermion. In the case that in QCD with massless flavors the topological charge of instantons, this relation was proven by Schwarz (1977); see is screened and ␹topϭ0. This means that the quantity on also the discussion by Coleman (1977). the left-hand side of the Witten-Veneziano relation is

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 346 T. Scha¨fer and E. V. Shuryak: Instantons in QCD the topological susceptibility in pure gauge theory20 the quantum-mechanical toy model of Sec. II.B. Never- (without quarks). But in pure gauge theory, the ␩Ј cor- theless, there are important differences. In particular, relation function is pathological (see Sec. VI.C.4), so the we shall see that, in QCD, chiral symmetry breaking ␩Ј mass on the right-hand-side of the Witten-Veneziano implies that isolated instantons can have a nonzero am- relation has to be determined in full QCD. This means plitude (Sec. IV.D). that the left-hand side and the right-hand side of the Consider the fermion propagator in the instanton field Witten-Veneziano relation are actually defined in differ- 21 ϩ ϩ ent theories. Nevertheless, the Witten-Veneziano rela- ␺0͑x͒␺0 ͑y͒ ␺␭͑x͒␺␭ ͑y͒ S͑x,y͒ϭ ϩ , (109) tion provides a reasonable estimate of the quenched to- im ␭Þ͚0 ␭ϩim pological susceptibility (Sec. III.C.1), and effective Lagrangians that incorporate the Witten-Veneziano re- where we have written the zero-mode contribution sepa- lation provide a good description of the pseudoscalar rately. Suppose there are Nf light-quark flavors, so that N meson spectrum. We shall also show that the Witten- the instanton amplitude is proportional to m f (or, more Veneziano relation can be derived within the instanton generally, to ⌸fmf). Instead of the tunneling amplitude, liquid model using the mean-field approximation (Sec. let us calculate a 2Nf-quark Green’s function ¯ IV.G). An analog of the Witten-Veneziano relation in ͗⌸f␺f(xf)⌫␺f(yf)͘, containing one quark and antiquark full QCD is described in Sec. V.E. A more detailed of each flavor. Contracting all the quark fields, we obtain study of the ␩Ј correlation function in different instan- the Green’s function by multiplying the tunneling ampli- ton ensembles will be given in Sec. VI.C.4. tude by Nf fermion propagators. Every propagator has a zero-mode contribution with one power of the fermion 2. Tunneling amplitude in the presence of light fermions mass in the denominator. As a result, the zero-mode contribution to the Green’s function is finite in the chiral The previous subsection was a small detour from the limit.22 calculation of the tunneling amplitude. Now we return The result can be written in terms of an effective La- to our original problem and study the effect of light grangian (’t Hooft, 1976b; see Sec. IV.F, where we give a quarks on the tunneling rate. Quarks modify the weight more detailed derivation). It is a nonlocal 2Nf-fermion in the Euclidean partition function by the fermionic de- interaction, where the quarks are emitted or absorbed in terminant zero-mode wave functions. In general, it has a fairly complicated structure, but under certain assumptions, it det͓D” ͑A␮͒ϩmf͔, (107) can be significantly simplified. First, if we limit ourselves ͟f to low momenta, the interaction is effectively local. Sec- which depends on the gauge fields through the covariant ond, if instantons are uncorrelated, we can average over derivative. Using the fact that nonzero eigenvalues come their orientation in color space. For SU(3) color and in pairs, we can write the determinant as Nfϭ1, the result is (Shifman, Vainshtein, and Zakharov, 1980c) ␯ 2 2 det͓D” ϩm͔ϭm ͟ ͑␭ ϩm ͒, (108) ␭Ͼ0 4 2 3 N ϭ1ϭ d␳n0͑␳͒ m␳Ϫ ␲ ␳ q¯ RqL , (110) where ␯ is the number of zero modes. Note that the L f ͵ ͩ 3 ͪ integration over fermions gives a determinant in the nu- where n (␳) is the tunneling rate without fermions. m 0 merator. This means that (as 0) the fermion zero Note that the zero-mode contribution acts like a mass mode makes the tunneling amplitude→ vanish and indi- term. This is quite natural because, for N ϭ1, there is vidual instantons cannot exist. f only one chiral U(1) symmetry, which is anomalous. In We have already seen what the reason for this phe- A contrast to the case N Ͼ0, the anomaly can therefore nomenon is: During the tunneling event, the axial f charge of the vacuum changes by two units, so instan- generate a fermion mass term. tons have to be accompanied by fermions. Indeed, it was For Nfϭ2, the result is pointed out by ’t Hooft that the tunneling amplitude is 4 ϭ d␳n ͑␳͒ m␳Ϫ ␲2␳3q¯ q nonzero in the presence of external quark sources, be- Nfϭ2 0 ͟ f,R f,L L ͵ f ͩ 3 ͪ cause zero modes in the denominator of the quark ͫ propagator may cancel against zero modes in the deter- 3 4 2 ϩ ␲2␳3 u¯ ␭au d¯ ␭ad minant. This is completely analogous to the situation in 32 ͩ 3 ͪ ͩ R L R L

3 a a 20 Ϫ u¯ ␴ ␭ u d¯ ␴ ␭ d , (111) It is usually argued that the Witten-Veneziano relation is 4 R ␮␯ L R ␮␯ L ͪͬ derived in the large-Nc approximation to QCD and that ␹top ϭO(1) in this limit. That does not really solve the problem, 22 however. In order to obtain a finite topological susceptibility, Note that Green’s functions involving more than 2Nf legs one has to set Nfϭ0, even if Nc ϱ. are not singular as m 0. The Pauli principle always ensures 21 → → This means that a priori it is not even defined how the two that no more than 2Nf quarks can propagate in zero-mode numbers should be compared. states.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 347

a 23 a 2 where the ␭ are color Gell-Mann matrices. One can an estimate for the gluon condensate, ͗0͉(G␮␯) ͉0͘. easily check that the interaction is SU(2)ϫSU(2) invari- From an analysis of the charmonium spectrum, Shifman ant, but U(1)A is broken. This means that the ’t Hooft et al. (1979) obtained Lagrangian provides another derivation of the U(1)A anomaly. Furthermore, in Secs. III and IV we shall ar- a 2 4 ͗0͉͑G␮␯͒ ͉0͘Ӎ0.5 GeV . (113) gue that the importance of this interaction goes far be- yond the anomaly and that it explains the physics of It is difficult to assess the accuracy of this number. The chiral symmetry breaking and the spectra of light had- analysis has been repeated many times, including many rons. more channels. Reinders, Rubinstein, and Yazaki (1985) Finally, we need to include the effects of nonzero agree with the value of Eq. (113) and quote an error of modes on the tunneling probability. One effect is that 25%. On the other hand, a recent analysis gives the coefficient in the beta function is changed to b a 2 4 ͗0͉(G␮␯) ͉0͘ϭ(1.02Ϯ0.1) GeV , about twice the origi- ϭ11Nc/3Ϫ2Nf/3. In addition to that, there is an overall nal Shifman, Vainshtein, and Zakharov value (Narison, constant that was calculated by ’t Hooft (1976b) and 1996). Carlitz and Creamer (1979a), The tunneling rate can now be estimated using the Nf 2 following simple idea. If the nonperturbative fields con- n͑␳͒ϳ1.34͑m␳͒ ͓1ϩNf͑m␳͒ log͑m␳͒ϩ¯͔, (112) tributing to the gluon condensate are dominated by where we have also specified the next-order correction (weakly interacting) instantons, the condensate is simply in the quark mass. Note that at two-loop order we not proportional to the instantons density, because every only get the two-loop beta function in the running cou- single instanton contributes a finite amount pling but also one-loop anomalous dimensions in the 4 a 2 2 ͐d x(G␮␯) ϭ32␲ . Therefore the value of the gluon quark masses. condensate provides an upper limit for the instanton We should emphasize that, in this section, we have density, studied the effect of fermions on the tunneling rate only for widely separated, individual instantons. But light fer- dN 1 IϩA a 2 Ϫ4 mions induce strong correlations between instantons, nϭ 4 р 2 ͗͑G␮␯͒ ͘Ӎ1fm . (114) and the problem becomes very complicated. Many state- d x 32␲ ments that are correct in the case of pure gauge theory, e.g., the fact that tunneling lowers the ground-state en- Another estimate of the instanton density can be ob- ergy, are no longer obvious in the theory with quarks. tained from the topological susceptibility. This quantity But before we try to tackle these problems, we should measures fluctuations of the topological charge in a like to review what is known phenomenologically about four-volume V, instantons in QCD. ͗Q2͘ ␹topϭ lim . (115) V ϱ V III. PHENOMENOLOGY OF INSTANTONS →

A. How often does tunneling occur in the QCD vacuum? On average, the topological charge vanishes, ͗Q͘ϭ0, but generally in a given configuration QÞ0 (see Fig. 7). In order to assess the importance of instantons in the Topological fluctuations provide an important character- QCD vacuum, we have to determine the total tunneling istic of the vacuum in pure gauge QCD. However, in the rate in QCD. Unfortunately, the standard semiclassical presence of massless quarks, the topological charge is theory discussed in the last section is not able to answer screened, and ␹topϭ0 (see Sec. V.E). The value of ␹top this question. The problem is related to the presence of in quenched QCD can be estimated using the Witten- large-size instantons for which the action is of order one. Veneziano relation (106) to give The naive use of the one-loop running coupling leads to f 2 an infrared divergence in the semiclassical rate, which is ␲ 2 2 2 4 ␹topϭ ͑m␩ϩm␩ Ϫ2mK͒ϭ͑180 MeV͒ . (116) a simple consequence of the Landau pole. Before we 2Nf Ј discuss any attempts to improve on the theoretical esti- mate, we should like to study phenomenological esti- If one assumes that instantons and anti-instantons are mates of the instanton density in QCD. uncorrelated, the topological susceptibility can be esti- The first attempt along these lines (Shifman, mated as follows. The topological charge in some vol- Vanshtein, and Zakharov, 1978) was based on informa- ume V is QϭNIϪNA . For a system with Poissonian tion on properties of the QCD vacuum obtained from statistics, the fluctuations in the particle numbers are QCD sum rules. We cannot go into details of the sum- ⌬NI,AӍͱNI,A. This means that for a random system of rule method, which is based on using dispersion theory to match experimental information with the operator product expansion (OPE) prediction for hadronic corre- 23Again, we use conventions appropriate for dealing with lation functions. See the reviews of Reinders, Rubin- classical fields. In standard perturbative notations, the fields stein, and Yazaki (1985), Narison (1989), Shifman are rescaled by a factor of g, and the condensate is given by a 2 (1992). The essential point is that the method provides ͗0͉(gG␮␯) ͉0͘.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 348 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

critical size we have to integrate the rate in order to get the phenomenological instanton density24

␳max d␳n0͑␳͒ϭnphen . (117) ͵0 Ϫ4 Using nphenϭ1fm , Shifman et al. concluded that ␳max Ӎ1 fm. This is a very pessimistic result because it implies that the instanton action is not large, so the semiclassical approximation is useless. Also, if instantons are that large, they overlap strongly, and it makes no sense to speak of individual instantons. There are two possible ways in which the semiclassical approximation can break down as the typical size be- comes large. One possibility is that (higher-loop) pertur- bative fluctuations start to grow. We have discussed these effects in the double-well potential (see Sec. FIG. 7. Distribution of topological charges for an ensemble of II.A.4), but in gauge theory, the two-loop ͓O(1/S0)͔ 5000 thermalized configurations in pure gauge SU(3) at ␤ϭ6.1, corrections to the semiclassical result are not yet known. from Alles et al., 1997. Another possibility is that nonperturbative (multi- instanton etc.) effects become important. These effects can be estimated from the gluon condensate. The inter- instantons, we expect ␹topϭN/V. Using the phenomeno- N V action of an instanton with an arbitrary, weak external logical estimate from above, we again get ( / ) a ext Ϫ4 Ӎ1fm . field G␮␯ is given by (see Sec. IV.A.1) How reliable are these estimates? Both methods suf- 2␲2␳2 fer from uncertainties that are hard to assess. For the S ϭ ␩¯ a UabGb ext , (118) int g2 ␮␯ ␮␯ gluon condensate, there is no systematic method of separating perturbative and nonperturbative contribu- where U is the matrix that describes the instanton ori- tions. The sum-rule method effectively determines con- entation in color space. This is a dipole interaction, so to tributions to the gluon condensate with momenta below first order it does not contribute to the average action. a ext a certain separation scale. This corresponds to instanton To second order in G␮␯ , one has (Shifman et al., fluctuations above a given size, which in itself is not a 1980b) problem, since the rate of small instantons can be deter- ␲4␳4 mined perturbatively, but the value of the separation a 2 n͑␳͒ϭn0͑␳͒ 1ϩ 4 ͗͑G␮␯͒ ͘ϩ¯ . (119) scale (␮ϳ1 GeV) is not very well determined. In any ͫ 2g ͬ case, the connection of the gluon condensate with in- From our knowledge of the gluon condensate, we can stantons is indirect. Other fluctuations might very well now estimate for what size nonperturbative effects be- play a role. The estimate of the instanton density from come important. Using the Shifman, Vainshtein, and the (quenched) topological susceptibility relies on the Zakharov value, we see that for ␳Ͼ0.2 fm the interac- assumption that instantons are distributed randomly. In tion with vacuum fields (of whatever origin) is not neg- the presence of light quarks, this assumption is certainly ligible. Unfortunately, for ␳Ͻ0.2 fm the total density of incorrect (that is why an extrapolation from quenched to instantons is too small as compared to the phenomeno- real QCD is necessary). logical estimate. However, it is important to note the sign of the cor- rection. The nonperturbative contribution leads to a B. The typical instanton size and the instanton tunneling rate that grows even faster than the semiclas- liquid mode sical rate. Accounting for higher-order effects by expo- nentiating the second-order contribution, Shuryak Next to the tunneling rate, the typical instanton size is (1982a) suggested that the critical size be estimated from the most important parameter characterizing the instan- the modified condition ton ensemble. If instantons are too large, it does not 4 4 ␳max ␲ ␳ make any sense to speak of individual tunneling events, 2 d␳n0͑␳͒exp 4 ͗͑Ga␮␯͒ ͘ ϭnphen . (120) and semiclassical theory is inapplicable. If instantons are ͵0 ͫ 2g ͬ too small, then semiclassical theory is good, but the tun- neling rate is strongly suppressed. The first estimate of the typical instanton size was made by Shifman et al. 24This procedure cannot be entirely consistent, since simply (1978), based on the estimate of the tunneling rate given cutting off the size integration violates many exact relations above. such as the trace anomaly (see Sec. IV.C). Nevertheless, given If the total tunneling rate can be calculated from the all the other uncertainties, this method provides a reasonable semiclassical ’t Hooft formula, we can ask up to what first estimate.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 349

Since the rate grows faster, the critical size is shifted to a on the lattice. Before we come to direct instantons smaller value, searches, we should like to discuss the determination of the topological susceptibility, which requires measure- ␳maxϳ1/3 fm. (121) ments of the total topological charge inside a given vol- If the typical instanton is indeed small, we obtain a com- ume. This has become a very technical subject, and we pletely different perspective on the QCD vacuum: will not be able to go into much detail [see the nice, (1) Since the instanton size is significantly smaller than albeit somewhat dated, review by Kronfeld (1988)]. The the typical separation R between instantons, ␳/R standard techniques used to evaluate the topological ϳ1/3, the vacuum is fairly dilute. The fraction of charge are the (i) (naive) field-theoretical, (ii) geomet- spacetime occupied by strong fields is only a few ric, and (iii) fermionic methods. Today, these methods percent. are usually used in conjunction with various improve- (2) The fields inside the instanton are very strong, G␮␯ ments, like cooling, blocking, or improved actions. 2 ӷ⌳ . This means that the semiclassical approxima- The field-theoretic method is based on the naive lat- tion is valid, and the typical action is large: tice discretization of the topological charge density ˜ 2 2 GGϳ⑀␮␯␳␴ tr͓U␮␯U␳␴͔, where U␮␯ is the elementary S0ϭ8␲ /g ͑␳͒ϳ10Ϫ15ӷ1. (122) plaquette in the ␮␯ plane. The method is simple to implement, but it has no topological meaning, and the Higher-order corrections are proportional to 1/S0 naive topological charge Q is not even an integer. In and presumably small. addition to that, the topological susceptibility suffers (3) Instantons retain their individuality and are not de- from large renormalization effects and mixes with other stroyed by interactions. From the dipole formula, operators, in particular, the unit operator and the gluon one can estimate condensate (Campostrini, DiGiacomo, and Panagopo- ulos, 1988). S – ӶS ͉␦ int͉ϳ͑2 3͒ 0 . (123) There are a number of ‘‘geometric’’ algorithms that ensure that Q has topological significance (Luescher, (1) Nevertheless, interactions are important for the 1982; Woit, 1983; Phillips and Stone, 1986). This means structure of the instanton ensemble, since that Q is always an integer and that the topological charge can be expressed as a surface integral. All these methods are based on fixing the gauge and using some exp͉␦Sint͉ϳ20ӷ1. (124) interpolation procedure to reconstruct a smooth gauge This implies that interactions have a significant ef- potential from the discrete lattice data. For a finite lat- fect on correlations among instantons; the instanton tice with the topology of a four-dimensional torus, the ensemble in QCD is not a dilute gas, but an inter- topology of the gauge fields resides in the transition acting liquid. functions that connect the gauge potential on the bound- aries. The geometric method provides a well-defined to- Improved estimates of the instanton size can be ob- pological charge for almost all gauge configurations. In tained from phenomenological applications of instan- the continuum, different topological sectors are sepa- tons. The average instanton size determines the struc- rated by configurations with infinite action. On the lat- ture of chiral symmetry breaking, in particular the values of the quark condensate, the pion mass, its decay tice, however, different sectors are separated by excep- constant, and its form factor. We shall discuss these ob- tional finite-action configurations called dislocations. servables in greater detail in the next sections. Although expected to be unimportant for sufficiently In particular, the consequences of the vacuum struc- smooth fields, dislocations may spoil the continuum limit ture advocated here were studied in the context of the on the lattice (Pugh and Teper, 1989). ‘‘random-instanton liquid model.’’ The idea is to fix Fermionic methods for calculating the topological N/Vϭ1fmϪ4 and ␳ϭ1/3 fm and add the assumption charge rely on the connection between instantons and that the distribution of instanton positions, as well as fermionic zero modes. Exceptionally small eigenvalues color orientations, is completely random. This is not of the Dirac operator on the lattice have been identified necessarily in contradiction with the observation (124) (Smit and Vink, 1987, 1988; Laursen, Smit, and Vink, that interactions are important, as long as they do not 1990). Furthermore, it was demonstrated that the corre- induce strong correlations among instantons. The ran- sponding Dirac eigenvectors have the correct chirality dom model is sufficiently simple that one can study a and are spatially correlated with instantons. Fermionic large number of hadronic observables. The agreement methods are not sensitive to dislocations, but they suffer with experimental results is quite impressive, thus pro- from problems connected with the difficulty of defining viding support for the underlying parameters. chiral fermions on the lattice. In particular, the (almost) zero modes connected with instantons are not exactly C. Instantons on the lattice chiral, and the topological charge defined through a fer- mionic expectation value does suffer from renormaliza- 1. The topological charge and susceptibility tion (for both Wilson and staggered fermions). For this The most direct way to determine the parameters of reason, fermionic methods have never been pursued the instanton liquid is provided by numerical simulations very vigorously [see Vink (1988) for a rare exception].

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 350 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

Recently, some progress in constructing chiral fermions on the lattice has been made. See, for example, Kaplan (1992) and Narayanan and Neuberger (1995). These methods may provide improved measurements of the topological susceptibility (Narayanan and Vranas, 1997). Since most of the difficulties with the field-theoretical and geometrical algorithms are related to fluctuations on very short scales, it is natural to supplement these algo- rithms with some sort of smoothing procedure. The best known method of this type is cooling (Hoek, 1986; Hoek, Teper, and Waterhouse, 1987). In the cooling method, one locally minimizes the action. This operation quickly eliminates short-range quantum fluctuations and eventually leads to a smooth configuration, correspond- ing to the classical content of the original field. It has been verified that these configurations are indeed domi- nated by instantons25 (Hoek et al., 1987; Chu et al., FIG. 8. Scaling test of the topological susceptibility in pure 1994). It has also been checked that in the cooled con- gauge SU(3), from Alles et al., 1997. The improvement from (0) (2) QL to QL is clearly visible. Inside errors, the results are figurations the field-theoretic definition of the topologi- 2 cal charge agrees with the more sophisticated, geometri- independent of the bare charge ␤ϭ6/g . cal methods (Alles, DiGiacomo, and Gianetti, 1990; Wiese, 1990). Unfortunately, the cooling method also suffers from systematic uncertainties. If the simplest field-theoretic method is used, ␹top mixes with the unit Wilson action is used, instantons gradually shrink and operator and the gluon condensate. Early studies of the finally fall through the lattice. Improved lattice actions scaling behavior of ␹top can be found in Woit (1983), can make instantons stable (de Forcrand, Perez, and Ishikawa et al. (1983), and Fox et al. (1985). A more de- Statamescu, 1995), but instanton anti-instanton pairs still tailed study of scaling and the role of mixing effects [in annihilate during the cooling process. pure gauge SU(2)] was recently carried out by Alles, The study of topological objects on the lattice is part Campostrini, and DiGiacomo (1993). These authors use of a larger effort to find improved or even ‘‘perfect’’ ‘‘heating’’ to investigate the effect of fluctuations on lattice actions and operators.26 An example of such a classical instanton configurations. As quantum fluctua- method is the ‘‘inverse blocking’’ procedure considered tions are turned on, one can study the renormalization by Hasenfratz, DeGrand, and Zhu (1996). From the of the topological charge. For comparison, if a configu- field configuration on the original coarse lattice, one ration with no topology is heated, it is mainly sensitive constructs a smoother configuration on a finer lattice by to mixing. Alles et al. conclude that in their window, ␤ an approximate inverse renormalization-group transfor- ϭ(2/g2)ϭ2.45Ϫ2.525, everything scales correctly and mation. The method has the advantage that it gives a 5 4 4 ␹topϭ3.5(4)ϫ10 ⌳ Ӎ(195 MeV) , where we have used larger action for dislocations than the standard Wilson L ⌳LATϭ8 MeV. For comparison, the result from cooling action does, thus suppressing undesirable contributions is about 30% larger. This discrepancy gives a rough es- to the topological charge. First tests of improved topo- timate of the uncertainty in the calculation. logical charges are very promising, correctly recovering An example of the use of an improved topological instantons with sizes as small as 0.8a. Significantly im- charge operator is shown in Figs. 7 and 8. Figure 7 shows proved values for the topological susceptibility will hopefully be forthcoming. the distribution of topological charges in pure gauge Whatever method is used to define Q and measure SU(3). As expected, the distribution is a Gaussian with the topological susceptibility on the lattice, one has to zero mean. A scaling test of the topological susceptibil- ity is shown in Fig. 8. The result clearly shows the reduc- test the behavior of ␹top as the lattice spacing is taken to zero, a 0. If the topological susceptibility is a physical tion in the statistical error achieved using the improved → operator and the quality of the scaling behavior. The quantity, it has to exhibit scaling towards the continuum 4 limit. In the geometrical method, Q itself is not renor- topological susceptibility is ␹topϭ͓175(5) MeV͔ .On malized, and ␹ is expected to show logarithmic scaling the other hand, the geometric method (and preliminary top results from inverse blocking) gives larger values, for (if it were not for the contribution of dislocations). If the 4 example ␹topϭ(260 MeV) in pure-gauge SU(3) simula- tions (Grandy and Gupta, 1994). We conclude that lat- tice determinations are consistent with the phenomeno- 25This ‘‘experimental’’ result also shows that all stable classi- cal solutions of the Yang-Mills equations are of multi- logical value of ␹top , but that the uncertainty is still instanton type. rather large. Also, the scaling behavior of the topologi- 26Classically improved (perfect) actions have no discretiza- cal susceptibility extracted from improved (perfect) op- tion errors of order an for some (all) n, where a is the lattice erators or fermionic definitions still needs to be estab- spacing. lished in greater detail.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 351

FIG. 9. A typical slice through a lattice configuration: (a), (b), before cooling; (c), (d), after 25 cooling sweeps, from Chu et al., 1994. Figs. (a) and (c) show the fields strength G2, while Figs. (b) and (d) show the topologi- cal charge density GG˜. The lat- tice units are aϭ0.16 fm before and aϭ0.14 fm after 25 cooling sweeps.

A by-product of the measurements of Alles et al. is a Nevertheless, cooling has the advantage of providing new determination of the gluon condensate in pure smooth gauge configurations that are easily interpreted gauge SU(2), in terms of continuum fields. 2 2 8 4 4 A typical field configuration after the quantum noise ͗G ͘ϭ͑4␲ ͒ϫ0.38͑6͒ϫ10 ⌳ Ӎ1.5 GeV . (125) L has disappeared is shown in Fig. 9 (Chu et al., 1994). The This number is significantly larger than the Shifman, left panel shows the field strength on a slice through the Vainshtein, and Zakharov value given in the last section, lattice. One can clearly identify individual classical ob- but it is consistent with other lattice data, for example jects. The right panel shows the distribution of the topo- those of Campostrini, DiGiacomo, and Gunduc (1989). logical charge, demonstrating that the classical configu- However, when comparing the lattice data to the rations are indeed instantons and anti-instantons. Fixing Shifman-Vainshtein-Zakharov estimate one should keep physical units from the (quenched) rho-meson mass, in mind that the two results are based on very different Chu et al. conclude that the instanton density in pure physical observables. While the Shifman-Vainshtein- gauge QCD is (1.3Ϫ1.6) fmϪ4. This number is indeed Zakharov value is based on the OPE of a hadronic cor- close to the estimate presented above. relator, and the value of the separation scale is fairly The next question concerns the average size of an in- well determined, lattice results are typically based on the stanton. A qualitative confirmation of the instanton liq- value of the average plaquette, and the separation scale uid value ␳ϳ1/3 fm was first obtained by Polikarpov and is not very well defined. Veselov (1988). More quantitative measurements (Chu et al., 1994) are based on fitting of the topological charge 2. The instanton liquid on the lattice correlation function after cooling. The fit is very good The topological susceptibility provides only a global and gives an average size ␳ϭ0.35 fm. characterization of the instanton ensemble. In addition, More detail is provided by measurements of the in- we should like to identify individual instantons and stanton size distribution. Lattice studies of this type study their properties. This is true in particular for theo- were recently performed for pure gauge SU(2) by ries with light quarks, where the total charge is sup- Michael and Spencer (1995) and de Forcrand, Perez, pressed because of screening effects (see Sec. V.E). and Stamatescu (1997). The results obtained by Michael Most studies of instantons on the lattice are based on and Spencer on two different size lattices are shown in the cooling method. As already mentioned, cooling is Fig. 10(a). The agreement between the two measure- not an ideal method for this purpose because instantons ments is best for large instantons, while it is not so good and anti-instantons annihilate during cooling. While this for small ␳. That is of course exactly as expected; instan- process does not affect the total charge, it does affect the tons of size ␳ϳa fall through the lattice during cooling. density of instantons and anti-instantons. Ultimately, The most important result is the existence of a relatively improved operators are certainly the method of choice. sharp maximum in the size distribution together with a

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 352 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

FIG. 11. The distribution of the separation of like and unlike instanton pairs in pure gauge SU(2), from Michael and Spen- cer, 1995. The distance to the closest neighbor (after cooling) is given in lattice units aӍ0.08 fm.

3. The fate of large-size instantons and the beta function We have seen that both phenomenology and the available lattice data suggest that instantons larger than ␳Ӎ1/3 fm are strongly suppressed in QCD. In Sec. II.C.4 we saw that this result could not be understood from the leading-order semiclassical formula. This leaves essen- tially three possibilities: The instanton distribution is regulated by higher-order quantum effects, by classical instanton interactions, or by the interaction of instan- FIG. 10. Instanton size distribution in pure gauge SU(2), from tons with other classical objects (e.g., monopoles or Michael and Spencer (1995) and Shuryak (1995). The size ␳ is strings). given in fm, where lattice units have been fixed from the glue- 4 2 4 2 The possible role of repulsive interactions between in- ball mass m0ϩϩϭ1.7 GeV: ᭿,16,4/gϭ2.4; ᭹,24,4/g ϭ2.5. The dotted and dashed lines simply serve to guide the stantons will be discussed in Sec. IV (this is also what eye. The open circles and squares come from an interacting the open dots in Fig. 10 are based on). It is hard to instanton calculation, while the solid curve corresponds to the speculate on the role of other classical fields, although parametrization discussed in the text. we shall try to summarize some work in this direction in the next section. Here, we should like to discuss the pos- strong suppression of large-size instantons. The physical sibility that the size distribution is regulated by quantum mechanism for this effect is still not adequately under- fluctuations (Shuryak, 1995). If this is the case, gross fea- stood. In Fig. 10 we compare the lattice data with an tures of the size distribution can be studied by consider- instanton liquid calculation, in which large instantons ing a single instanton rather than an interacting en- are suppressed by a repulsive core in the instanton in- semble. teraction (Shuryak, 1995). The Gell-Mann–Low beta function is defined as a de- In addition to the size distribution, Michael and Spen- rivative of the coupling constant g over the logarithm of cer (1995) also studied correlations among instantons. the normalization scale ␮ at which g is determined, g g3 g5ץ They found that, on average, unlike pairs were closer than like pairs, R(IA) 0.7 R(II) . This clearly sug- ␤͑g͒ϭ ϭϪb ϪbЈ ϩ . (126) ¯ log ␮ 16␲2 ͑16␲2͒2 ץ ͘Ӎ ͗ ͘ ͗ gests an attractive interaction between instantons and anti-instantons. The distribution of pseudoparticle sepa- In QCD with Nc colors and Nf light flavors, we have b 2 rations is shown in Fig. 11. There are very few IA-pairs ϭ11Nc/3Ϫ2Nf/3 and bЈϭ34Nc /3Ϫ13NcNf/3ϩNf /Nc . with very small separation, because these pairs easily Remember that the tunneling amplitude is n(␳) annihilate during cooling. Like pairs show an enhance- ϳ␳Ϫ5 exp͓Ϫ(8␲2)/g2(␳)͔. In the weak-coupling domain, ment at small R, a feature that is not understood at one can use the one-loop running coupling and n(␳) present. ϳ␳bϪ5⌳b. This means that the strong growth of the size After identifying instantons on the lattice, the next distribution in QCD is related to the large value of b step is to study the importance of instantons for physical Ӎ9. observables. We shall discuss an example of this line of For pedagogical reasons, we should like to start our research in Sec. VI.E, where we present hadronic corre- discussion in the domain of the phase diagram where b lation functions in cooled configurations. Another ex- is small and this dangerous phenomenon is absent. For ample can be found in Thurner, Feurstein, and Markum Ncϭ3 colors, b is zero if Nfϭ33/2Ӎ16. When b is small (1997), who show that there is a strong correlation be- and positive, it turns out that the next coefficient bЈ is tween the quark condensate and the location of instan- negative (Belavin and Migdal, 1974; Banks and Zaks, tons after cooling. 1982), and therefore the beta function has a zero at

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 353

g2 b * ϭϪ . (127) 16␲2 bЈ Since b is small, so is g , and the perturbative calcula- * tion is reliable. If the beta function has a zero, the cou- pling constant will first grow as one goes to larger dis- tances but then stop running (‘‘freeze’’) as the critical value g is approached. In this case, the large-distance * behavior of all correlation functions is governed by the infrared fixed point, and the correlators decay as frac- tional powers of the distance. In this domain the instan- ton contribution to the partition function is of the order exp(Ϫ16␲2/g2 )ϳexp(Ϫ͉bЈ/b͉), so it is exponentially * small if b is small.27 What happens if Nf is reduced, so that we move away from the bϭ0 line? The infrared fixed point survives in some domain (the conformal region), but the value of the critical coupling grows. Eventually, non-perturbative FIG. 12. Nonperturbative beta function from SU(3) lattice effects (instantons in particular) grow exponentially, and gauge theory with Wilson action: ᮀ, Gupta, 1992; ᭹, Blum the perturbative result (127) is unreliable. As we shall et al., 1995. The solid line shows the fit discussed in the text. discuss in Sec. IX.D, existing lattice data suggest that the boundary of the conformal domain is around Nfϭ7 for Ncϭ3 (Iwasaki et al., 1996). be obtained with28 ⌳ϭ0.66 fmϪ1, pϭ3.5, and Cϭ4.8, There is no unique way to define the beta function in shown by the solid line in Fig. 10. In real QCD, the the nonperturbative regime. In our context, a preferred coupling cannot freeze because the theory certainly has definition is provided by the value of instanton action no infrared fixed point. Nevertheless, in order to make S ϭ8␲2/g2(␳) as a function of the instanton size. This inst the instanton density convergent, one does not need the quantity can be studied directly on the lattice by heating beta function to vanish. It is sufficient that the coupling (i.e., adding quantum fluctuations to) a smooth instan- Ͻ ton of given size. This program is not yet implemented, constant runs more slowly, with an effective b 5. but one can get some input from the measured instanton There are some indications from lattice simulations that this is indeed the case, and that there is a consistent size distribution. If Sinstӷ1, the semiclassical expression Ϫ5 trend from N ϭ0toNϭ16. These results are based on n(␳)ϳ␳ exp(ϪSinst) should be valid, and we can ex- f f tract the effective charge from the measured size distri- a definition of the nonperturbative beta function based bution. on pre-asymptotic scaling of hadronic observables. Ide- We have already discussed lattice measurements of ally, the lattice scale is determined by performing simu- n(␳) in Sec. III.B. These results are well reproduced lations at different couplings g, fixing the scale a from using the semiclassical size distribution with a modified the asymptotic (perturbative) relation g(a). Scaling be- running coupling constant (Shuryak, 1995), havior is established by studying many different had- ronic observables. Although asymptotic scaling is often 8␲2 bЈ violated, a weaker form of scaling might still be present. ϭbLϩ log L, (128) g2͑␳͒ b In this case, the lattice scale a at a given coupling g is determined by measuring a hadronic observable in units where (for Nfϭ0) the coefficients are bϭ11Nc/3, bЈ of a and then fixing a to give the correct experimental 2 ϭ17Nc /3 as usual, but the logarithm is regularized ac- value. This procedure makes sense as long as the result cording to is universal (independent of the observable). Performing simulations at different g, one can determine the func- 1 1 p tion g(a) and the beta function. Lϭ log ϩCp . (129) p ͫͩ ␳⌳ͪ ͬ Lattice results for both pure gauge (Gupta, 1992; open points) and Nfϭ2 (Blum et al., 1995) SU(3) are shown For small ␳, Eq. (128) reduces to the perturbative run- in Fig. 12. In order to compare different theories, it is ning coupling, but for large ␳ the coupling stops running convenient to normalize the beta function to its asymptotic (g 0) value. The ratio in a manner controlled by the two parameters C and p. → A good description of the measured size distribution can

28The agreement is even more spectacular in the O(3) 27 model. In this case the instanton size distribution is measured Since Nf is large, the vacuum consists of a dilute gas of 3 instanton/anti-instanton molecules, so the action is twice that over a wider range in ␳ and shows a very nice n(␳)ϳ1/␳ for a single instanton. behavior, which is what one would expect if the coupling stops running.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 354 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

d͑1/g2͒ d͑1/g2͒ Ϫ1 ting the dipole field of an instanton into two halves. This R ͑g͒ϭ (130) ␤ d log͑a͒ ͩ d log͑a͒ ͯ ͪ means that merons have long-range fields. g 0 → Another way to introduce fractional charge is by con- tends to 1 as g 0(6/g2 ϱ). The two-loop correction sidering twisted boundary conditions (’t Hooft, 1981a). → → is positive, so R 1 from above. However, in the non- In this case, one finds fracton solutions (also known as ’t ␤→ perturbative region the results show the opposite trend, Hooft fluxes) with topological charges quantized in units R g2 with ␤ dropping significantly below 1 around (6/ ) 1/Nc . Gonzales-Arroyo and Montero (1996) suggested Ӎ6. This means that the coupling constant runs slower that confinement is produced by an ensemble of these than the one-loop formula suggests. At somewhat objects glued together. In order to study this hypothesis, 2 smaller 6/g , R␤ displays a rapid upward turn, which is they measured the string tension in cooled and uncooled known as the transition to the strong-coupling regime. configurations with twisted boundary conditions. They This part of the data is clearly dominated by lattice ar- found that fractionally charged objects could indeed be tifacts. The significant reduction of the beta function (by identified, and that their number roughly scaled with the about 50%) observed for intermediate values of g is pre- string tension. Clearly, there are many problems that cisely what is is needed to explain the suppression of need to be understood, in particular, what the role of the large-size instantons. Furthermore, the reduction of R␤ boundary condition is and how regions with twisted is even larger for Nfϭ2. This might very well be a pre- boundary conditions can be glued together. cursor of the infrared fixed point. At some critical Nf we An important attempt at understanding confinement expect R␤ to touch zero. As Nf is further increased, the is based on the role of magnetic monopoles (see below). zero should move to weaker coupling (to the right) and One can make monopolelike configuration (more pre- reach infinity at Nfϭ33/2. cisely, dyonlike, since the fields are self-dual) from in- stantons by lining up identically oriented instantons.29 D. Instantons and confinement Another possibility is a chain of strongly correlated in- stanton anti-instanton pairs, which might create an infi- After the discovery of instantons, it was hoped that nitely long monopole loop. Under normal circum- instantons might help us to understand confinement in stances, however, these objects have very small entropy, QCD. This hope was mainly inspired by Polyakov’s and they are not found in the simulations discussed in proof that instantons lead to confinement in three- Sec. V. dimensional compact QED (Polyakov, 1987). However, The possible role of very large instantons was dis- there are important differences between three- and four- cussed by Diakonov and Petrov (1996). These authors dimensional theories. In three dimensions, the field of propose that instantons can cause confinement if the size an instanton looks like a magnetic monopole (with B distribution behaves as n(␳)ϳ1/␳3. This can be under- ϳ1/r2), while in four dimensions it is a dipole field that stood as follows. In Sec. IV.H, we shall show that the falls off as 1/r4. mass renormalization of a heavy quark due to instantons 3 For a random ensemble of instantons, one can calcu- is ⌬MQϳ(N/V)␳ . For typical instanton radii this con- late the instanton contribution to the heavy-quark po- tribution is not very important, but if the size distribu- 3 tential. In the dilute gas approximation, the result is de- tion has a 1/␳ tail, then ⌬MQ is linearly divergent, a termined by the Wilson loop in the field of an individual possible signature of confinement. This is very intrigu- instanton (Callan, Dashen, and Gross, 1978b). The cor- ing, but again, there are a number of open questions. responding potential is Vϳx2 for small x, but tends to a For one, the 1/␳3 distribution implies that the total vol- constant at large distances. This result was confirmed by ume occupied by instantons is infinite. Very large instan- numerical simulations in the random ensemble tons would also introduce long-range correlations, which (Shuryak, 1989), as well as the mean-field approximation are inconsistent with the expected exponential decay of (Diakonov, Petrov, and Pobylitsa, 1989). The main in- gluonic correlation functions. stanton effect is a renormalization of the heavy-quark Whatever the mechanism of confinement may turn mass ␦MQϭ50– 70 MeV. The force ͉dV/dx͉ peaks at x out to be, it is clear that instantons should be affected by Ӎ0.5 fm, but even at this point it is almost an order of confinement in some way. One example of this line of magnitude smaller than the string tension (Shuryak, reasoning is the idea that confinement might be the rea- 1989). son for the suppression of large-size instantons. A re- Later attempts to explain confinement in terms of in- lated, more phenomenological suggestion is that instan- stantons (or similar objects) fall roughly into three dif- tons provide a dynamic mechanism for bag formation ferent categories: objects with fractional topological (see Shuryak, 1978b; Callan, Dashen, and Gross, 1979). charge, strongly correlated instantons, and the effects of A lattice measurement of the suppression of instantons very large instantons. in the field of a static quark was recently performed by Classical objects with fractional topological charge the Vienna group (see Faber et al., 1995, and references were first seriously considered by Callan et al. (1978a), therein). The measured distribution of the topological who proposed a liquid consisting of instantons and merons. Merons are singular configurations with topo- logical charge Qϭ1/2 (Alfaro, Fubini, and Furlan, 1976). 29An example for this is the finite-temperature caloron solu- Basically, one can interpret merons as the result of split- tion; see Sec. VII.A.1.-

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 355

Lattice simulations that combine Abelian projection with the cooling technique or improved topological charges have indeed observed a strong positive correla- tion between monopoles and instantons (Feurstein, Markum, and Thurner, 1997). In order to understand this phenomenon in greater detail, a number of authors have studied monopole trajectories associated with indi- vidual instantons or instanton pairs (Chernodub and Gubarev, 1995; Hart and Teper, 1995; Suganuma et al., 1995). It was observed that each instanton was associ- ated with a monopole loop. For instanton anti-instanton pairs, these loops might either form two separate loops FIG. 13. Squared topological charge density around a static, or one common loop, depending on the relative color heavy quark-antiquark pair in SU(3) with Nfϭ3, from Faber orientation of the two instantons. Since then, it has been 3 et al., 1995. The results were obtained on an 8 ϫ4 lattice at shown that the small loops associated with individual 2 6/g ϭ5.6. The topological charge was measured after five cool- instantons do not correspond to global minima of the ing steps. gauge-fixing functional and should be discarded (Brower, Orginos, and Tan, 1996). On the other hand, charge is shown in Fig. 13. The instanton density around monopole trajectories connecting two or more instan- the static charge is indeed significantly suppressed, by tons appear to be physical. The main physical question is about a factor of two. Since instantons give a vacuum then whether these monopole loops can percolate to energy density on the order of Ϫ1 GeV/fm3, this effect form the long monopole loops responsible for confine- alone generates a significant difference in the nonpertur- ment. bative energy density (bag constant) inside and outside of a hadron. IV. TOWARDS A THEORY A very interesting picture of confinement in QCD is OF THE INSTANTON ENSEMBLE based on the idea that the QCD vacuum behaves like a dual superconductor, formed by the condensation of A. The instanton interaction magnetic monopoles (Mandelstam, 1976). Although the details of this mechanism lead to many serious questions 1. The gauge interaction (DelDebbio et al., 1997), there is some evidence in favor of this scenario from lattice simulations. There are no In the last section we argued that the density of in- semiclassical monopole solutions in QCD, but mono- stantons in the QCD vacuum is quite significant, imply- poles can arise as singularities after imposing a gauge ing that interactions among them are essential to an un- condition (’t Hooft, 1981b). The number and the loca- derstanding of the instanton ensemble. In general, it is tion of monopole trajectories will then depend on the clear that field configurations containing both instantons gauge choice. In practice, the so-called maximal Abelian and anti-instantons are not exact solutions of the equa- gauge has turned out to be particularly useful (Kronfeld, tions of motion and that the action of an instanton/anti- Schierholz, and Wiese, 1987). The maximal Abelian instanton pair is not equal to twice the single instanton gauge is specified by the condition that the norms of the action. The interaction is defined by off-diagonal components of the gauge fields be minimal, IA 1 2 2 2 SintϭS͑A␮ ͒Ϫ2S0 , (131) e.g., tr͓(A␮ ) ϩ(A ) ͔ϭmin in SU(2). This leaves one ␮ IA U(1) degree of freedom unfixed, so in this preferred where A␮ is the gauge potential of the instanton/anti- IA subgroup one can identify magnetic charges, study their instanton pair. Since A␮ is not an exact solution, there trajectories, and evaluate their contribution to the (Abe- is some freedom in choosing an ansatz for the gauge lian) string tension. The key observations are (a) that field. This freedom corresponds to finding a convenient the Abelian string tension (in maximal Abelian gauge) is parametrization of the field configurations in the vicinity numerically close to the full non-Abelian string tension, of an approximate saddle point. In general, we have to and (b) that it is dominated by the longest monopole integrate over all field configurations anyway, but the loops (Smit and van der Sijs, 1989; Suzuki, 1993). ansatz determines the way we split coordinates into ap- We can not go into the details of these calculations, proximate zero modes and nonzero modes. which would require a review in themselves, but only For well separated IA pairs, the fields are not strongly mention some ideas on how instantons and monopoles distorted and the interaction is well defined. For very might be correlated. If monopoles are responsible for close pairs, on the other hand, the fields are strongly color confinement, then their paths should wind around modified and the interaction is not well determined. In the color flux tubes, just as electric-current lines wind addition to that, if the instanton and anti-instanton begin around magnetic-flux tubes in ordinary superconductors. to annihilate, the gauge fields become perturbative and We have already mentioned that color flux tubes expel should not be included in semiclassical approximations. topological charges (see Fig. 13). This suggests that in- We comment on the present understanding of these stantons and monopoles should be (anti?) correlated. questions below.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 356 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

The interaction between instantons at large distances In terms of this angle, the dipole interaction is given by was derived by Callan, Dashen, and Gross (1978a; see 2 2 2 ¨ 32␲ ␳1␳2 also Forster, 1977). They began by studying the interac- S ϭϪ ͉u͉2͑1 – 4 cos2 ␪͒. (137) tion of an instanton with a weak, slowly varying external int g2 R4 Ga field ( ␮␯)ext . In order to ensure that the gauge field is The orientational factor dϭ1 – 4 cos2 ␪ varies between 1 localized, the instanton is put in the singular gauge. One and Ϫ3. We also observe that the dipole interaction van- then finds ishes when we average over all angles ␪. 2␲2 The dipole interaction is the leading part of the inter- S ϭϪ ␳2␩¯a ͑Ga ͒ . (132) int g2 ␮␯ ␮␯ ext action at large distances. In order to determine the in- teraction at intermediate separation, we have to evalu- This result can be interpreted as the external field inter- ate the total action for an IA pair. This is most easily acting with the color magnetic ‘‘dipole moment’’ 2 2 2 a done for the so-called sum ansatz (2␲ /g )␳ ␩¯␮␯ of the instanton. Note that the interac- IA I A tion vanishes if the external field is self-dual. If the ex- A␮ ϭA␮ϩA␮ , (138) ternal field is taken to be an anti-instanton at distance where again both the instanton and the anti-instanton R, the interaction is are in the singular gauge. The action Sϭ1/4͐d4xG2 can 2 ˆ ˆ 32␲ R␮R␯ easily be split into the free part 2S0 and the interaction. S ϭϩ ␳2␳2 ␩¯ a ␩b Rab , (133) 2 int g2 I A ␮␳ ␯␳ R4 Systematically expanding in 1/R , one finds (Diakonov and Petrov, 1984) Rab Rab where is the relative color orientation matrix 2 1 a † b ˆ 8␲ ϭ 2 tr(U␶ U ␶ ) and R is the unit vector connecting the IA 2 ˆ 2 SintϭϪ 2 ͉͑u͉ Ϫ4͉u•R͉ ͒ centers of the instanton and anti-instanton. The dipole- g ͭ dipole form of the interaction is quite general; the inter- 4␳2␳2 15␳2␳2͑␳2ϩ␳2͒ action of topological objects in other theories, such as 1 2 1 2 1 2 ϫ 4 Ϫ 6 skyrmions or O(3) instantons, is of similar type. ͫ R 2R ͬ Instead of considering the dipole interaction as a clas- 2 2 2 2 9␳ ␳ ͑␳ ϩ␳ ͒ sical phenomenon, we can also look at the interaction as 2 1 2 1 2 Ϫ8 ϩ͉u͉ 6 ϩO͑R ͒ (139) a quantum effect, due to gluon exchanges between the 2R ͮ instantons (Zakharov, 1992). The linearized interaction for the IA interaction and (132) corresponds to an instanton vertex that describes 2 2 2 2 2 the emission of a gluon from the classical instanton field. 8␲ ␳1␳2͑␳1ϩ␳2͒ SII ϭ ͑9ϩ3͉u͉2Ϫ4͉uជ ͉2͒ ϩO͑RϪ8͒ The amplitude for the exchange of one gluon between int g2 2R6 an instanton and an anti-instanton is given by (140) 4␲4 for the instanton-instanton (II) interaction. Here, uជ de- ␳2␳2RabRcd␩b ␩d ͗Ga ͑x͒Gc ͑0͒͘, (134) g2 1 2 I A ␮␯ ␣␤ ␮␯ ␣␤ notes the spatial components of the four-vector u␮ .To first order, we find the dipole interaction in the IA chan- a c where ͗G␮␯(x)G␣␤(0)͘ is the free gauge-field propaga- nel and no interaction between two instantons. To next tor, order, there is a repulsive interaction for both IA and II. ac Clearly, it is important to understand to what extent a c 2␦ ͗G␮␯͑x͒G␣␤͑0͒͘ϭ 2 6 ͑g␯␣x␮x␤ϩg␮␤x␯x␣ this interaction is unique or if it depends on details of ␲ x the underlying ansatz for the gauge potential. It was also realized that the simple sum ansatz leads to certain arti- Ϫg␯␤x␮x␣Ϫg␮␣x␯x␤͒. (135) facts, for example, that the field strength at the centers Inserting Eq. (135) into (134) gives the dipole interac- of the two instantons becomes infinite and that there is tion to first order. Summing all n gluon exchanges allows an interaction between pseudoparticles of the same one to exponentiate the result and reproduce the full charge. One of us (Shuryak, 1988a) therefore proposed dipole interaction (133). the ‘‘ratio’’ ansatz In order to study this interaction in greater detail, it is 2 xϪz 2 xϪz useful to introduce some additional notation. We can ab b ␳I͑ I͒␯ ab b ␳A͑ A͒␯ 2RI ␩¯␮␯ 4 ϩ2RA ␩␮␯ 4 characterize the relative color orientation in terms of the ͑xϪzI͒ ͑xϪzA͒ ϩ ϩ Aa ϭ , four-vector u␮ϭ(1/2i)tr(U␶␮ ), where ␶␮ ϭ(␶ជ,Ϫi). ␮ 2 2 ␳I ␳A Note that for the gauge group SU(2), u is a real vector 1ϩ ϩ ␮ xϪz 2 xϪz 2 with u2ϭ1, whereas for SU(NϾ2), u is complex and ͑ I͒ ͑ A͒ ␮ (141) ͉u͉2р1. Also note that, for SU(NϾ2), the interaction is completely determined by the upper 2ϫ2 block of the whose form was inspired by ’t Hooft’s exact multi- SU(N) matrix U. We can define a relative color orien- instanton solution. This ansatz ensures that the field tation angle ␪ by strength is regular everywhere and that (at least if they have the same orientation) there is no interaction be- ˆ 2 ͉u•R͉ cos2 ␪ϭ . (136) tween pseudoparticles of the same charge. Deriving an ͉u͉2 analytic expression for the interaction in the ratio ansatz

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 357

evolution of the pair and f(␭) is a function that depends on the parametrization of the streamline. The initial conditions are ␾(ϱ)ϭ␾c and ␾Ј(ϱ)ϭ␾0 , where ␾c is the classical solution corresponding to a well-separated instanton/anti-instanton pair and ␾0 is the translational zero mode. In QCD, the streamline equations were solved by Verbaarschot (1991) using the conformal invariance of the classical equations of motion. Conformal invariance implies that an instanton/anti-instanton pair in the sin- gular gauge at some distance R is equivalent to a regular-gauge instanton and a singular-gauge anti- instanton with the same center but different sizes. We do not want to go into details of the construction, but refer the reader to the original work. It turns out that the resulting gauge configurations are very close to the ansatz originally proposed by Yung (1988), 1 ␳2 1 Aa ϭ2␩a x ϩ2Rab␩b x , ␮ ␮␯ ␯ x2ϩ␳2␭ ␮␯ ␯ ␭ x2͑x2ϩ␳2/␭͒ (143)

where ␳ϭͱ␳1␳2 is the geometric mean of the two in- stanton radii and ␭ is the conformal parameter 2 2 2 2 2 2 2 1/2 R ϩ␳1ϩ␳2 ͑R ϩ␳1ϩ␳2͒ ␭ϭ ϩ 2 2 Ϫ1 . (144) 2␳1␳2 ͩ 4␳ ␳ ͪ 1 2 FIG. 14. Gauge field and fermion interaction for an instanton/ Note that it is large not only if the distance R signifi- anti-instanton pair: (a) the gluonic interaction in the stream- cantly exceeds the mean (geometric) size, but also if R is line (dash-dotted curve) and ratio ansatz (short-dashed curve). small and one instanton is much larger than another. The interaction is given in units of the single instanton action (This latter situation is important when we consider sup- S0 for the most attractive (cos ␪ϭ1) and most repulsive pression of large-size instantons.) (cos ␪ϭ0) orientations. The solid line shows the original The interaction for this ansatz is given by (Verbaar- streamline interaction supplemented by the core discussed in schot, 1991). Sec. V.C. (b) Fermionic overlap matrix element in the stream- 2 line (solid curve) and ratio ansatz (dotted curve). The matrix 8␲ 4 S ϭ ͕Ϫ4 1Ϫ␭4ϩ4␭2log͑␭͒ elements are given in units of the geometric mean of the in- IA g2 ͑␭2Ϫ1͒3 „ … stanton radii. 2 ˆ 2 2 2 ϫ͓͉u͉ Ϫ4͉u•R͉ ͔ϩ2„1Ϫ␭ ϩ͑1ϩ␭ ͒log͑␭͒… is somewhat tedious. Instead, we have calculated the in- ϫ͓͉͑u͉2Ϫ4͉u Rˆ ͉2͒2ϩ͉u͉4ϩ2͑u͒2͑u*͒2͔͖, teraction numerically and parametrized the result; see • Shuryak and Verbaarschot (1991) and Fig. 14. We ob- (145) serve that both the sum and the ratio ansatz lead to the which is also shown in Fig. 14. For the repulsive orien- dipole interaction at large distances, but the amount of tation, the interaction is similar to the ratio ansatz, and repulsion at short distance is significantly reduced in the the average interaction is repulsive. For the most attrac- ratio ansatz. tive orientation, however, the interaction approaches Given these ambiguities, the question is whether one Ϫ2S0 at short distance, showing that the instanton and can find the best possible ansatz for the gauge field of an anti-instanton annihilate each other and the total action IA pair. This question was answered with the introduc- of the pair approaches zero. tion of the streamline or valley method (Balitsky and It is interesting to note that the Yung ansatz, at least Yung, 1986). The basic idea is to start from a well sepa- to order 1/R6, leads to the same interaction as the per- rated instanton/anti-instanton pair, which is an approxi- turbative method (134) if carried out to higher order mate solution of the equations of motion, and then let (Arnold and Mattis, 1991; Balitsky and Scha¨fer, 1993; the system evolve under the force given by the variation Diakonov and Petrov, 1994). This problem is related to of the action (Shuryak, 1988b). For a scalar field theory, the calculation of instanton-induced processes at high the streamline equation is given by energy (like multigluon production in QCD, or the d␾ ␦S baryon-number-violating cross section in electroweak f͑␭͒ ϭ , (142) theory). To leading order in E/M , where E is the d␭ ␦␾ sph bombarding energy and Msph the sphaleron barrier (see where ␭ is the collective variable corresponding to the Sec. VIII.B), the cross section is given by

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 358 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

minant of the matrix (147) can be interpreted as the sum 4 ip•R ␴totϭdisc͵ d Re ͵͵d⍀1d⍀2n͑␳1͒n͑␳2͒ of all closed diagrams consisting of (zero-mode) quark exchanges between pseudoparticles. In other words, the ϫeϪSint͑R,⍀12͒, (146) logarithm of the Dirac operator (147) is the contribu- tions of the ’t Hooft effective interaction to the vacuum where ⍀iϭ(zi ,␳i ,Ui) are the collective coordinates as- sociated with the instanton and anti-instanton, and disc energy. f(s)ϭ(1/2i)͓f(sϩi⑀)Ϫf(sϪi⑀)͔ denotes the disconti- Due to the chirality of the zero modes, the matrix nuity of the amplitude after continuation to Minkowski elements TII and TAA between instantons of the same space, sϭp2Ͼ0. Given the agreement of the streamline charge vanish. In the sum ansatz, we can use the equa- method and the perturbative calculation to leading and tions of motion to replace the covariant derivative in next-to-leading order, it has been suggested that the be- (148) by an ordinary one. The dependence on the rela- tive orientation is then given by T ϭ(u Rˆ )f(R). This havior of ␴tot at high energy be used to define the inter- IA • means that, like the gluonic dipole interaction, the fer- action Sint (Diakonov and Polyakov, 1993). The behav- ior of this quantity is still debated, but it is generally mion overlap is maximal if cos ␪ϭ1. The matrix element believed that the baryon-number-violating cross section can be parametrized by does not reach the unitarity bound. In that case, the 1 4.0 interaction would have to have some repulsion at short T ϭi͑u R͒ . (149) IA • ␳ ␳ ͓2.0ϩR2/͑␳ ␳ ͔͒2 distance, unlike the streamline solution. I A I A Another possible way to make sense of the short- A parametrization of the overlap matrix element for the distance part of the IA interaction in the streamline streamline gauge configuration can be found in Shuryak method is to use analytic continuation in the coupling and Verbaarschot (1992). The result is compared with constant, as discussed in Sec. II.A. Allowing g2 Ϫg2 the sum ansatz in Fig. 14 (for cos ␪ϭ1). We observe that gives a new saddle point in the complex g plane at→ which the matrix elements are very similar at large distance but the IA interaction is repulsive and the semiclassical differ at short distance. method is under control. Using these results, we may write the contribution of an instanton/anti-instanton pair to the partition function 2. The fermionic interaction as

In the presence of light fermions, instantons interact 4 2 with each other not only through their gauge fields, but ZIAϭV4 ͵ d zdU exp͓Nf log͉TIA͑U,z͉͒ also through fermion exchanges. The basic idea in deal- ing with the fermionic interaction is that the Dirac spec- ϪSint͑U,z͔͒. (150) trum can be split into quasizero modes, linear combina- Here, z is the distance between the centers, and U is the tions of the zero modes of the individual instantons, and relative orientation of the pair. The fermionic part is nonzero modes. Here we shall focus on the interaction attractive, while the bosonic part is either attractive or due to approximate zero modes. The interactions due to repulsive, depending on the orientation. If the interac- nonzero modes and the corrections due to interference tion is repulsive, there is a real saddle point for the z between zero and nonzero modes were studied by integral, whereas for the attractive orientation there is Brown and Creamer (1978) and Lee and Bardeen only a saddle point in the complex z plane (as in Sec. (1979). II.B). In the basis spanned by the zero modes, we can write The calculation of the partition function (150) in the the Dirac operator as saddle-point approximation was recently attempted by Shuryak and Velkovsky (1997). They find that for a 0 TIA iDϭ , (147) large number of flavors, NfϾ5, the ground-state energy ” ͩT 0 ͪ AI oscillates as a function of Nf . The period of the oscilla- where we have introduced the overlap matrix elements tion is 4, and the real part of the energy shift vanishes TIA , for even Nfϭ6,8,... . The reason for these oscillations is exactly the same as in the case of supersymmetry quan- 4 † tum mechanics: the saddle point gives a complex contri- TIAϭ d x␺0,I͑xϪzI͒iD” ␺0,A͑xϪzA͒. (148) ͵ bution with a phase that is proportional to the number of flavors. Here, ␺0,I is the fermionic zero mode (101). The matrix elements have the meaning of a hopping amplitude for a quark from one pseudoparticle to another. Indeed, the B. Instanton ensembles amplitude for an instanton to emit a quark is given by the amputated zero-mode wave function iD” ␺0,I . This In Sec. II.C.4 we studied the semiclassical theory of shows that the matrix element (148) can be written as instantons, treating them as very rare (and therefore in- two quark-instanton vertices connected by a propagator, dependent) tunneling events. However, as emphasized † Ϫ1 ␺0,IiD” (iD” ) iD” ␺0,A . At large distance, the overlap in Sec. III.A, in QCD instantons are not rare, so one 3 matrix element decreases as TIAϳ1/R , which corre- cannot just exponentiate the results obtained for a single sponds to the exchange of a massless quark. The deter- instanton. Before we study the instanton ensemble in

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 359

QCD, we should like to discuss a simple physical anal- ferent bands, so the crystal is an insulator. In QCD, this ogy. In this picture, we think of instantons as atoms and means that chiral symmetry is not broken. light quarks as valence electrons. Another analogy with liquids concerns the question of A system like this can exist in many different phases, density stabilization. In order for the system to saturate, e.g., as a gas, liquid, or solid. In theories with massless we need a short-range repulsive force in addition to the quarks, instantons have ‘‘unsaturated bonds’’ and can- attractive, long-range dipole interaction. We have al- not appear individually. Isolated instantons in the form ready mentioned that the nature of the short-range in- of an atomic gas can only exist if there is a nonzero teraction and the (possibly related) question of the fate quark mass. The simplest ‘‘neutral’’ object is an of large instantons in QCD are not well understood. instanton/anti-instanton (IA) molecule. Therefore, if the Lattice simulations indicate that large instantons are strongly suppressed, but for the moment we have to in- instanton density is low (for whatever reason—high tem- clude this result in a phenomenological manner. perature, a large Higgs expectation value, etc.) the sys- Let us summarize this qualitative discussion. Depend- tem should be in a phase consisting of IA molecules. ing on the density, the instanton ensemble is expected to Below, we shall also argue that this is the case if the be in a gas, liquid, or solid phase. The phase boundaries density is not particularly small, but the interactions are will in general depend on the number of colors and fla- strong and favor the formation of molecules, for ex- vors. Large Nc favors a crystalline phase, while large Nf ample, if the number of light quarks Nf exceeds some favors a molecular gas. Neither case is phenomenologi- critical value. If the instanton ensemble is in the molecu- cally acceptable as a description of real QCD, with two lar phase, then quarks are bound to molecules and can- light and one intermediate mass flavor. We therefore not propagate over large distances. This means that expect (and will show below) that the instanton en- there are no eigenmodes with almost zero virtuality, the semble in QCD is in the liquid phase. ‘‘conductivity’’ is zero, or chiral symmetry remains un- broken. The liquid phase differs from the gas phase in many C. The mean-field approximation important respects. The density is determined by the in- teractions and cannot be arbitrarily small. A liquid has a In order to study the structure of the instanton en- surface tension, etc. As we shall see below, the instanton semble in a more quantitative fashion, we consider the ensemble in QCD has all these properties. In QCD we partition function for a system of instantons in pure also expect that chiral symmetry is spontaneously bro- gauge theory, ken. This means that in the ground state there is a pre- NϩϩNϪ ferred direction in flavor space, characterized by the 1 Zϭ ͟ ͓d⍀i n͑␳i͔͒exp͑ϪSint͒. (151) quark condensate ͗q¯q͘. This preferred orientation can Nϩ!NϪ! i ͵ only be established if quarks form an infinite cluster. In Here, N are the numbers of instantons and anti- terms of our analogy, this means that electrons are de- Ϯ instantons, ⍀ ϭ(z ,␳ ,U ) are the collective coordinates localized and the conductivity is nonzero. In the instan- i i i i of the instanton i, n(␳) is the semiclassical instanton ton liquid phase, quarks are delocalized because the in- distribution function (93), and S is the bosonic instan- stanton zero modes become collective. int ton interaction. In general, the dynamics of a system of At very high density, interactions are strong and the pseudoparticles governed by Eq. (151) is still quite com- ensemble is expected to form a four-dimensional crystal. plicated, so we have to rely on approximation schemes. In this case, both Lorentz and gauge invariance would There are a number of techniques well known from sta- be broken spontaneously; clearly, this phase is very dif- tistical mechanics that can be applied to the problem, for ferent from the QCD vacuum. Fortunately, however, ex- example, the mean-field approximation or the varia- plicit calculations (Diakonov and Petrov, 1984; Shuryak tional method. These methods are expected to be reli- and Verbaarschot, 1990) show that the instanton liquid able as long correlations between individual instantons crystallizes only if the density is pushed to about two are weak. orders of magnitude larger than the phenomenological The first such attempt was made by Callan, Dashen, value. At physical densities, the crystalline phase is not and Gross (1978a). These authors included only the di- favored: although the interaction is attractive, the en- pole interaction, which, as we noted above, vanishes on tropy is too small as compared to the random liquid.30 average and produces an attractive interaction to next The electronic structure of a crystal consists of several order in the density. In this case, there is nothing to bands. In our case, the Fermi surface lies between dif- balance the growth n(␳)ϳ␳bϪ5 of the semiclassical in- stanton distribution. In order to deal with this problem, Ilgenfritz and Mu¨ ller-Preußker (1981) introduced an ad 30Diakonov and Petrov (1984) suggested that the instanton hoc ‘‘hard core’’ in the instanton interaction (see also liquid crystallizes in the limit of a large number of colors, N c ¨ ϱ, because the interaction is proportional to the charge Munster, 1982). The hard core automatically excludes → 2 2 large instantons and leads to a well-behaved partition 8␲ /g , which is of order Nc . As long as instantons do not disappear altogether (the action is of order 1), the interaction function. It is important to note that one cannot simply becomes increasingly important. However, little is known cut the size integration at some ␳c , but has to introduce about the structure of large-Nc QCD. the cutoff in a way that does not violate the conformal

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 360 T. Scha¨fer and E. V. Shuryak: Instantons in QCD invariance of the classical equations of motion. This ␯V 1/2 bϪ4 guarantees that the results do not spoil the renormaliza- ␳2ϭ , ␯ϭ , (157) ␤␥2N 2 tion properties of QCD, most notably the trace anomaly ͩ ͪ (see below). In practice, Ilgenfritz and Mu¨ ller-Preußker which determines the dimensionless diluteness of the en- chose semble, ␳4(N/V)ϭ␯/(␤␥2). Using the pure gauge beta 2 2 2 1/4 function bϭ11, ␥ Ӎ25 from above and ␤Ӎ15, we get ϱ ͉zIϪzA͉Ͻ͑a␳I ␳A͒ ␳4(N/V)ϭ0.01, even more dilute than phenomenology Sintϭ 2 2 1/4, (152) ͭ 0 ͉zIϪzA͉Ͼ͑a␳I ␳A͒ requires. The instanton density can be fixed from the second self-consistency requirement, (N/V)ϭ2␮0 (the which leads to an excluded volume in the partition func- factor 2 comes from instantons and anti-instantons). We tion, controlled by the dimensionless parameter a. get We do not go into the details of their results, but present the next development (Diakonov and Petrov, N ϭ⌳4 ͓C ␤2Nc⌫͑␯͒͑␤␯␥2͒Ϫ␯/2͔2/͑2ϩ␯͒, (158) 1984). In this work, the arbitrary core is replaced by the V PV Nc interaction in the sum ansatz [see Eqs. (139) and Eq. where C is the prefactor in Eq. (93). The formula (140)]. The partition function is evaluated using a trial Nc distribution function. If we assume that correlations be- shows that ⌳PV is the only dimensionful parameter. The final results are tween instantons are not very important, then a good trial function is given by the product of single-instanton N 1 ␹ Ӎ ϭ͑0.65⌳ ͒4, ͑␳2͒1/2ϭ0.47⌳Ϫ1Ӎ R, distributions ␮(␳), top V PV PV 3 NϩϩNϪ 1 ␤ϭS0Ӎ15, (159) Z1ϭ ͟ ͵ d⍀i␮͑␳i͒ Nϩ!NϪ! i consistent with the phenomenological values for ⌳PV 300 MeV. It is instructive to calculate the free energy 1 Ӎ NϩϩNϪ as a function of the instanton density. Using Fϭ ϭ ͑V␮0͒ , (153) Nϩ!NϪ! Ϫ(1/V)log Z, we have where ␮0ϭ͐d␳␮(␳). The distribution ␮(␳) is deter- N N ␯ mined from the variational principle ␦ log Z /␦␮ϭ0. In Fϭ log Ϫ 1ϩ . (160) 1 V ͩ 2V␮ ͪ ͩ 2ͪ quantum mechanics a variational wave function always ͭ 0 ͮ provides an upper bound on the true ground-state en- The instanton density is determined by minimizing the -N/V)͔ϭ0. The vacuum energy den)ץ͓/Fץ ,ergy. The analogous statement in statistical mechanics is free energy known as Feynman’s variational principle. Using con- sity is given by the value of the free energy at the mini- vexity mum, ⑀ϭF0 . We find N/Vϭ2␮0 as above and ZϭZ exp͓Ϫ͑SϪS ͔͒ уZ exp͑Ϫ SϪS ͒, (154) b N 1͗ 1 ͘ 1 ͗ 1͘ ⑀ϭϪ . (161) 4 ͩVͪ where S1 is the variational action, one can see that the variational vacuum energy is always higher than the true Estimating the value of the gluon condensate in a dilute one. instanton gas, ͗G2͘ϭ32␲2(N/V), we see that Eq. (161) In our case, the single-instanton action is given by S1 is consistent with the trace anomaly. Note that, for non- ϭlog͓␮(␳)͔ while ͗S͘ is the average action calculated interacting instantons (with the size integration regular- from the variational distribution (153). Since the varia- ized in some fashion), one would expect ⑀ӍϪ(N/V), tional ansatz does not include any correlations, only the which is inconsistent with the trace anomaly and shows average interaction enters, the importance of respecting classical scale invariance. The second derivative of the free energy with respect 27 N 2 2 2 2 c 2 to the instanton density, the compressibility of the in- ͗Sint͘ϭ␥ ␳1␳2 , ␥ ϭ 2 ␲ (155) 4 Nc Ϫ1 stanton liquid, is given by 2F 4 Nץ for both IA and II pairs. Clearly, Eq. (155) is of the same form as the hard core (152) discussed above, only ϭ , (162) ͪ N/V͒2 ͯ b ͩ V͑ץ 2 the dimensionless parameter ␥ is determined from the n0 interaction in the sum ansatz. Applying the variational where n0 is the equilibrium density. This observable is principle, one finds (Diakonov and Petrov, 1984) also determined by a low-energy theorem based on bro- ␤␥2 ken scale invariance (Novikov et al., 1981), ␮͑␳͒ϭn͑␳͒exp Ϫ N␳2␳2 , (156) ͫ V ͬ 4 d4x͗G2͑0͒G2͑x͒͘ϭ͑32␲2͒ ͗G2͘. (163) where ␤ϭ␤(␳¯) is the average instanton action and ␳2 is ͵ b the average size. We observe that the single-instanton Here, the left-hand side is given by an integral over the distribution is cut off at large sizes by the average instan- field-strength correlator, suitably regularized and with ton repulsion. The average size ␳2 is determined by the the constant disconnected term ͗G2͘2 subtracted. For a 2 Ϫ1 2 self-consistency condition ␳ ϭ␮0 ͐d␳␮(␳)␳ . The re- dilute system of instantons, the low-energy theorem sult is gives

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 361

4 vacuum, however, chiral symmetry is spontaneously bro- ͗N2͘Ϫ͗N͘2ϭ ͗N͘. (164) ¯ b ken, and the quark condensate ͗qq͘ is nonzero. The quark condensate is the amplitude for a quark to flip its Here, ͗N͘ is the average number of instantons in a vol- chirality, so we expect that the instanton density is con- ume V. The result (164) shows that density fluctuations trolled not by the current masses, but by the quark con- in the instanton liquid are not Poissonian. Using the densate, which does not vanish as m 0. general relation between fluctuations and the compress- Given the importance of chiral symmetry→ breaking, ibility gives the result (162). This means that the form of we shall discuss this phenomenon on a number of differ- the free energy near the minimum is determined by the ent levels. In this section, we should like to give a simple renormalization properties of the theory. Therefore the qualitative derivation following (Shuryak, 1982b). Ear- functional form (160) is more general than the mean- lier works on chiral symmetry breaking by instantons field approximation used to derive it. How reliable are the numerical results derived from are those of Caldi (1977) and Carlitz and Creamer the mean-field approximation? The accuracy of the (1979a, 1979b); see also the review of Diakonov (1995). mean-field approximation can be checked by doing sta- The simplest case is QCD with just one light flavor, tistical simulations of the full partition function. Nfϭ1. In this theory, the only chiral symmetry is the (We shall come to this approach in Sec. V.) Another axial U(1)A symmetry, which is broken by the anomaly. question concerns the accuracy of the sum ansatz. This This means that there is no spontaneous symmetry can be checked explicitly by calculating the induced cur- breaking, and the quark condensate appears at the level cl rent j␮ϭD␮G␮␯ in the classical gauge configurations of a single instanton. The condensate is given by (Shuryak, 1985). This current measures the failure of the gauge potential to be a true saddle point. In the sum ͗q¯q͘ϭi d4x tr͓S͑x,x͔͒. (165) ansatz, the induced current gives a sizable contribution ͵ to the action, which means that this ansatz is not a good In the chiral limit, nonzero modes do not contribute to starting point for a self-consistent solution. the quark condensate. Using the zero-mode propagator In principle, this problem is solved in the streamline † S(x,y)ϭϪ␺0(x)␺0(y)/(im), we find that the contribu- method because, by construction, j␮ is orthogonal to tion of a single instanton to the quark condensate is 31 quantum fluctuations. However, applying the varia- given by Ϫ1/m. Multiplying this result by the density of tional method to the streamline configurations (Ver- instantons, we have ͗q¯q͘ϭϪ(N/V)/m. Since the in- baarschot, 1991) is not satisfactory either, because the stanton density is proportional to the quark mass m, the ensemble contains too many close pairs and too many quark condensate is finite in the chiral limit. large instantons. The situation is different for N Ͼ1. The theory still In summary, phenomenology and the lattice seem to f has an anomalous U(1) symmetry, which is broken by favor a fairly dilute instanton ensemble. This is well re- A instantons. The corresponding order parameter produced by the mean-field approximation based on the det (q¯ q ) (where f is the flavor index) appears al- sum ansatz, but the results are not really self-consistent. f f,L f,R ready at the one-instanton level. But in addition to that, How to generate a fully consistent ensemble in which there is a chiral SU(N ) ϫSU(N ) symmetry that is large instantons are automatically suppressed remains f L f R spontaneously broken to SU(N ) . This effect cannot an open problem. Nevertheless, as long as large instan- f V be understood on the level of a single instanton: the tons are excluded in a way that does not violate the contribution to q¯q is still (N/V)/m, but the density of symmetries of QCD, the results are likely to be indepen- ͗ ͘ instantons is proportional to (N/V)ϳmNf. dent of the precise mechanism that leads to the suppres- Spontaneous symmetry breaking has to be a collective sion of large instantons. effect involving infinitely many instantons. This effect is most easily understood in the context of the mean-field D. The quark condensate in the mean-field approximation method. For simplicity, we consider small-size instan- tons. Then the tunneling rate is controlled by the Proceeding from pure glue theory to QCD with light vacuum expectation value of the 2Nf-fermion operator quarks, one has to deal with the much more complicated (111) in the ’t Hooft effective Lagrangian. This vacuum problem of quark-induced interactions. Not only does expectation value can be estimated using the ‘‘vacuum the fermion determinant induce a very nonlocal interac- dominance’’ (or factorization) approximation, tion, but the very presence of instantons cannot be un- ¯ ¯ 1 derstood in the single-instanton approximation. Indeed, ͗␺⌫1␺␺⌫2␺͘ϭ 2 ͑Tr͓⌫1͔Tr͓⌫2͔ϪTr͓⌫1⌫2͔͒ as discussed in Sec. II.D, the semiclassical instanton den- N sity is proportional to the product of fermion masses and ϫ͗q¯q͘2, (166) therefore vanishes in the chiral limit m 0. In the QCD → where ⌫1,2 is a spin, isospin, color matrix and N ϭ4NfNc is the corresponding degeneracy factor. Using 31Except for the soft mode leading to the trivial gauge con- this approximation, we find that the factor ⌸fmf in the figuration, but the integration over this mode can be done ex- instanton density should be replaced by ⌸fmf* , where plicitly. the effective quark mass is given by

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 362 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

2 1 2 2 4 mf*ϭmfϪ ␲ ␳ ͗q¯ fqf͘. (167) i d x tr͓S͑x,x͔͒ϭϪi . (172) 3 ͵ ͚␭ ␭ϩim Thus, if chiral symmetry is broken, the instanton density We can immediately make a few important observations N is O„(m*) f…, finite in the chiral limit. concerning the spectrum of the Dirac operator: The next question is whether it is possible to make (1) Since the Dirac operator is Hermitian, the eigenval- this estimate self-consistent and calculate the quark con- ues ␭ are real. The inverse propagator (iD” densate from instantons. If we replace the current mass Ϫ1 32 ϩim) , on the other hand, consists of a Hermitian by the effective mass in the quark propagator, the con- and an anti-Hermitian piece. tribution of a single instanton to the quark condensate is (2) For every nonzero eigenvalue ␭ with eigenvector given by 1/m* and, for a finite density of instantons, we ␺␭ , there is another eigenvalue Ϫ␭ with eigenvec- expect tor ␥5␺␭ . ͑N/V͒ (3) This implies that the fermion determinant is posi- ͗q¯q͘ϭϪ . (168) tive: combining the nonzero eigenvalues in pairs, we m* get This equation, taken together with Eq. (167), gives a ͑i␭Ϫm͒ϭ ͑i␭Ϫm͒͑Ϫi␭Ϫm͒ϭ ͑␭2ϩm2͒. self-consistent value for the quark condensate, ␭Þ͟0 ␭у͟0 ␭у͟0 1 (173) ͗q¯q͘ϭϪ ͱ3N/2V. (169) (4) Only zero modes can be unpaired. Since ␥ ␺ ϭ ␲␳ 5 0 Ϯ␺0 , zero-mode wave functions have to have a Using the phenomenological values (N/V)ϭ1fmϪ4 and definite chirality. We have already seen this in the ␳ϭ0.33 fm, we get ͗q¯q͘ӍϪ(215 MeV)3, quite consis- case of instantons, where the Dirac operator has a tent with the experimental value ͗q¯q͘ӍϪ(230 MeV)3. left-handed zero mode. The effective quark mass is given by m* Using the fact that nonzero eigenvalues are paired, we ϭ␲␳(2/3)1/2(N/V)1/2 170 MeV. The self-consistent Ӎ can write the trace of the quark propagator as pair of equations (167), (168) has the general form of a 2m gap equation. We shall provide a more formal derivation 4 i d x tr͓S͑x,x͔͒ϭϪ 2 2 . (174) of the gap equation in Sec. IV.F. ͵ ␭у͚0 ␭ ϩm We have excluded zero modes since they do not contrib- ute to the quark condensate in the limit m 0 (for N → f E. Dirac eigenvalues and chiral symmetry breaking Ͼ1). In order to determine the average quark conden- sate, we introduce the spectral density ␳(␯)ϭ͓͚␭␦(␯ In this section we shall get a different and more mi- Ϫ␭)͔. We then have croscopic perspective on the formation of the quark con- ϱ 2m densate. The main idea is to study the motion of quarks ͗q¯q͘ϭϪ d␭␳͑␭͒ 2 2 . (175) in a fixed gauge field and then average over all gauge- ͵0 ␭ ϩm field configurations. This approach is quite natural from This result shows that the order in which we take the the point of view of the path integral (and lattice gauge chiral and thermodynamic limits is crucial. In a finite theory). Since the integral over the quark fields can al- system, the integral is well behaved in the infrared, and ways be performed exactly, quark observables are deter- the quark condensate vanishes in the chiral limit. This is mined by the exact quark propagator in a given gauge consistent with the observation that there is no sponta- configuration, averaged over all gauge fields. neous symmetry breaking in a finite system. A finite spin In a given gauge-field configuration, we can determine system, for example, cannot be magnetized if there is no ␮ external field. If the energy barrier between states withץthe spectrum of the Dirac operator iD” ϭ͓i ϩA␮(x)͔␥␮ , different magnetization is finite and there is no external iD” ␺␭ϭ␭␺␭ , (170) field that selects a preferred magnetization, the system will tunnel between these states and the average magne- where ␺␭ is an eigenstate with ‘‘virtuality’’ ␭. In terms of tization is zero. Only in the thermodynamic limit can the the spectrum, the quark propagator S(x,y)ϭ Ϫ1 system develop a spontaneous magnetization. Ϫ͗x͉iD” ͉y͘ is given by However, if we take the thermodynamic limit first, we † ␺␭͑x͒␺␭͑y͒ can have a finite density of eigenvalues arbitrarily close S͑x,y͒ϭϪ . (171) ͚ ␭ϩim to zero. In this case, the ␭ integration is infrared diver- ␭ gent as m 0, and we get a finite quark condensate, Using the fact that the eigenfunctions are normalized, → q¯q ϭϪ␲␳͑␭ϭ0͒, (176) we obtain the quark condensate, ͗ ͘ a result known as the Banks-Casher relation (Banks and Casher, 1980). This expression shows that quark conden- 32We shall give a more detailed explanation for this approxi- sation is connected with quark states of arbitrarily small mation in Sec. VI.C.2. virtuality.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 363

Studying chiral symmetry breaking requires an under- which has the same parametric dependence on (N/V) standing of quasizero modes, the spectrum of the Dirac and ␳ as the result in the previous section, only the co- operator near ϭ0. If there is only one instanton, the ␭ efficient is slightly different. In addition to that, we can spectrum consists of a single zero mode plus a continu- identify the effective mass m* introduced in the previ- ous spectrum of nonzero modes. If there is a finite den- ous section with a weighted average of the eigenvalues, sity of instantons, the spectrum is complicated, even if (m*)Ϫ1ϭNϪ1͚␭Ϫ1. the ensemble is very dilute. In the chiral limit, fluctua- It is very interesting to study the spectral density of tions of the topological charge are suppressed, so one the Dirac operator at small virtualities. Similar to the can think of the system as containing as many instantons density of states near the Fermi surface in condensed- as anti-instantons. The zero modes are expected to mix, matter problems, this quantity controls the low-energy so that the eigenvalues spread over some range of virtu- excitations of the system. If chiral symmetry is broken, alities ⌬␭. If chiral symmetry is broken, the natural value the spectral density at ␭ϭ0 is finite. In addition, chiral of the quark condensate is of order (N/V)/⌬␭. perturbation theory predicts the slope of the spectrum There is a useful analogy from solid-state physics. In (Smilga and Stern, 1993), condensed matter, atomic bound states may become de- localized and form a band. The material is a conductor if 1 ͗q¯q͘2 N2Ϫ4 ¯ f the Fermi surface lies inside a band. Such zones also ␳͑␭͒ϭϪ ͗qq͘ϩ 2 4 ͉␭͉ϩ¯ , (181) ␲ 32␲ f␲ Nf exist in disordered systems like liquids, but in this case they do not have a sharp boundary. which is valid for Nfу2. The second term is connected In the basis spanned by the zero modes of the indi- with the fact that for NfϾ2 there is a Goldstone boson cut in the scalar-isovector ( -meson) correlator, while vidual instantons the Dirac operator reduces to the ma- ␦ the decay ␦ ␲␲ is not allowed for two flavors. The trix → result implies that the spectrum is flat for Nfϭ2 but has 0 TIA a cusp for NfϾ2. iDϭ , (177) ” ͩT 0 ͪ AI already introduced in Sec. IV.A.2. The width of the zero-mode zone in the instanton liquid is governed by F. Effective interaction between quarks the off-diagonal matrix elements TIA of the Dirac opera- tor. The matrix elements depend on the relative color and the mean-field approximation orientation of the pseudoparticles. If the interaction be- In this section we should like to discuss chiral symme- tween instantons is weak, the matrix elements are dis- try breaking in terms of an effective, purely fermionic tributed randomly with zero average, but their variance theory that describes the effective interaction between is nonzero. Averaging T T* over the positions and IA IA quarks generated by instantons (Diakonov and Petrov, orientations of a pair of pseudoparticles, one gets 1986). For this purpose, we shall have to reverse the 2␲2 N␳2 strategy used in the last section and integrate over the 2 ͉͗TIA͉ ͘ϭ . (178) gauge field first. This will leave us with an effective 3Nc V theory of quarks that can be treated with standard If the matrix elements are distributed according to a many-body techniques. Using these methods allows us 33 Gaussian unitary ensemble, the spectral density is a to study not only chiral symmetry breaking, but also the semicircle, formation of quark-antiquark bound states in the instan- N ␭2 1/2 ton liquid. ␳͑␭͒ϭ 1Ϫ . (179) For this purpose we rewrite the partition function of ␲␴V 4␴2 ͩ ͪ the instanton liquid From the Casher-Banks formula, we then get the follow- N ϩN ing result for the quark condensate: 1 ϩ Ϫ Zϭ ͟ ͵ ͓d⍀in͑␳i͔͒ 1/2 Nϩ!NϪ! i 1 3Nc N ͗q¯q͘ϭϪ ӍϪ͑240 MeV͒3, (180) ␲␳ ͩ 2 Vͪ ϫ ϪS Dϩm Nf exp͑ int͒det͑ ” ͒ (182)

in terms of a fermionic effective action 33In the original work (Diakonov and Petrov, 1985) from ϩim͒␺ץwhich these arguments are taken, it was assumed that the spec- Zϭ d␺d␺† exp d4x␺†͑i trum had a Gaussian shape, with the width given above. How- ͵ ͩ ͵ ” ͪ ever, for a random matrix the correct result is a semicircle. In reality, the spectrum of the Dirac operator for a system of ϫ ͟ ͑⌰IϪimf͒͟ ͑⌰AϪimf͒ , (183) randomly distributed instantons is neither a semicircle nor a ͳ I,f A,f ʹ Gaussian; it has a nonanalytic peak at ␭ϭ0 (Shuryak and Ver- baarschot, 1990). This does not qualitatively change estimate ͒ ␾ ͑xϪzץϭ d4x ␺†͑x͒i ⌰ for the quark condensate. I,A ͵ „ ” I,A I,A …

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 364 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

YϩϩYϪ , which acts like a mass term. This can be seen ϫ d4y † yϪz i y ␺͑ ͒…, (184) explicitly by first performing the Grassmann integration”ץ ␾I,A͑ I,A͒„ ͵ which describes quarks interacting via the ’t Hooft ver- d␤Ϯ NϮ Zϭ exp NϮ log Ϫ1 tices ⌰I,A . The expectation value ͗•͘ corresponds to an ͵ 2␲ ͫ ͩ i␤ϮV ͪ average over the distribution of instanton collective co- d4k k2␸Ј2͑k͒ ordinates. Formally, Eq. (183) can be derived by ‘‘fermi- ϩN V tr log kϩ␥ ␤ c ͵ ͑2␲͒4 ͩ ” Ϯ Ϯ N ͪͬ onizing’’ the original action (see Nowak, 1991). In prac- c tice, it is easier to check the result by performing the (188) integration over the quark fields and verifying that one recovers the fermion determinant in the zero-mode ba- and then doing the saddle-point integral. Varying with sis. respect to ␤ϭ␤Ϯ gives the gap equation (Diakonov and Here, however, we want to use a different strategy Petrov, 1986) 4 2 and exponentiate the ’t Hooft vertices ⌰I,A in order to d k M ͑k͒ N derive the effective quark interaction. For this purpose ϭ , (189) ͵ ͑2␲͒4 k2ϩM2͑k͒ 4N V we calculate the average in Eq. (183) with respect to the c 2 2 variational single-instanton distribution (156). There are where M(k)ϭ␤k ␸Ј (k)/Nc is the momentum- no correlations, so only the average interaction induced dependent effective quark mass. The gap equation de- by a single instanton enters. For simplicity, we only av- termines the effective constituent mass M(0) in terms of erage over the position and color orientation and keep the instanton density N/V. For the parameters (159), the average size ␳ϭ␳¯ fixed, the effective mass is MӍ350 MeV. We can expand the gap equation in the instanton density (Pobylitsa, 1989). 4 For small N/V, one finds M(0)ϳ␳(N/2VN )1/2, which YϮϭ d z dU͟ ⌰I,A . (185) c ͵ ͵ f parametrically behaves like the effective mass m* intro- In order to exponentiate Y , we insert factors of unity duced above. Note that a dynamic mass is generated for Ϯ arbitrarily small values of the instanton density. This is ͐d⌫Ϯ͐d␤Ϯ /(2␲)exp͓i␤Ϯ(YϮϪ⌫Ϯ)͔ and integrate over ⌫ using the saddle-point method to obtain expected for Nfϭ1, since there is no spontaneous sym- Ϯ metry breaking and the effective mass is generated by d␤Ϯ the anomaly at the level of an individual instanton. ␺ץZϭ d␺d␺† exp d4x␺†i ͵ ͩ ͵ ” ͪ ͵ 2␲

Nϩ ϫexp͑i␤ Y ͒exp N log Ϫ1 2. The effective interaction for two or more flavors Ϯ Ϯ ϩͩ ͩ i␤ Vͪ ͪ ͫ ϩ In the context of QCD, the more interesting case is the one of two or more flavors. For N ϭ2, the effective ϩ͑ϩ Ϫ͒ , f ↔ ͬ ’t Hooft vertex is a four-fermion interaction 4 where we have neglected the current quark mass. In this d ki partition function, the saddle-point parameters ␤ play YϮϭ ͟ 4 ki␸Ј͑ki͒ Ϯ ͫ iϭ1,4 ͵ ͑2␲͒ ͬ the role of an activity for instantons and anti-instantons. 1 4 4 1. The gap equation for Nfϭ1 ϫ͑2␲͒ ␦ ͚ ki 2 ͩ i ͪ 4͑Nc Ϫ1͒ The form of the saddle-point equations for ␤Ϯ de- 2NcϪ1 pends on the number of flavors. The simplest case is ϫ ͑␺†␥ ␶Ϫ␺͒2 N ϭ ͩ 2N Ϯ a f 1, where the Grassmann integration is quadratic. c The average over the ’t Hooft vertex is most easily per- 1 formed in momentum space: ϩ ͑␺†␥ ␴ ␶Ϫ␺͒2 , (190) 8N Ϯ ␮␯ a ͪ c d4k Ϫ YϮϭ dU where ␶ ϭ(␶ជ,i) is an isospin matrix and we have sup- ͵ ͑2␲͒4 ͵ a pressed the momentum labels on the quark fields. In the † † k ϫ␺ ͑k͓͒k”␾I,A͑k͒␾I,A͑k͒k”͔␺͑k͒, (186) long-wavelength limit 0, the ’t Hooft vertex (190) corresponds to a local four-quark→ interaction34 where ␾(k) is the Fourier transform of the zero-mode profile (see Appendix A.2). Performing the average over 1 2NcϪ1 ϭ␤͑2␲␳͒4 ͓͑␺†␶Ϫ␺͒2 the color orientation, we get L 4͑N2Ϫ1͒ ͩ 2N a c c d4k 1 2 2 † YϮϭ ͵ 4 k ␸Ј ͑k͒␺ ͑k͒␥Ϯ␺͑k͒, (187) ͑2␲͒ Nc 34The structure of this interaction is identical to that one where ␥Ϯϭ(1Ϯ␥5)/2 and ␸Ј(k) is defined in the Ap- given in Eq. (111), as one can check using Fierz identities. The pendix. Clearly, the saddle-point equations are symmet- only new ingredient is that the overall constant ␤ is determined ric in ␤Ϯ , so that the average interaction is given by self-consistently from a gap equation.-

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 365

1 Varying with respect to ␤␴ gives the same gap equation ϩ͑␺†␥ ␶Ϫ␺͒2͔ϩ ͑␺†␴ ␶Ϫ␺͒2 . (191) 2 2 5 a 4N ␮␯ a ͪ as in the Nfϭ1 case, now with M(k)ϭ␤␴k ␸Ј (k). We c 2 2 also find (N/V)ϭ2 f␴ ␤ where fϭ2Nc(Nc Ϫ1)/(2Nc This Lagrangian is of the type first studied by Nambu Ϫ1). Expanding everything in (N/V), one can show and Jona-Lasinio (1961). It is widely used as a model for that M(0)ϳ(N/V)1/2, ␴ϳ(N/V)1/2, and ␤ϳconst. chiral symmetry breaking and as an effective description The fact that the gap equation is independent of Nf is for low-energy chiral dynamics (see the reviews of Vogl a consequence of the mean-field approximation. It im- and Weise, 1991; Klevansky, 1992; Hatsuda and Kuni- plies that even for Nfϭ2 chiral symmetry is spontane- hiro, 1994). ously broken for arbitrarily small values of the instanton Unlike the Nambu–Jona-Lasinio model, however, the density. As we shall see in the next section, this is not interaction has a natural cutoff parameter ⌳ϳ␳Ϫ1, and correct if the full fermion determinant is included. If the the coupling constants in Eq. (191) are determined in instanton density is too small, the instanton ensemble terms of a physical parameter, the instanton density forms a molecular gas, and chiral symmetry is unbroken. (N/V). The interaction is attractive for quark-antiquark However, as we shall show in Sec. V, the mean-field pairs with the quantum numbers of the ␲ and ␴ meson. approximation is quite useful for physically interesting If the interaction is sufficiently strong, the vacuum is values of the instanton density. rearranged, quarks condense, and a light (Goldstone) The quark condensate is given by pion is formed. The interaction is repulsive in the d4k M͑k͒ pseudoscalar-isoscalar [the SU(2) singlet ␩ ] and scalar- ͗q¯q͘ϭϪ4N . (195) Ј c ͵ ͑2␲͒4 M2͑k͒ϩk2 isovector ␦ channel, showing the effect of the U(1)A anomaly. Note that, to first order in the instanton den- Solving the gap equation numerically, we get ͗q¯q͘Ӎ 3 sity, there is no interaction in the vector ␳,␻,a ,f chan- Ϫ(255 MeV) . It is easy to check that ͗q¯q͘ 1 1 1/2 Ϫ1 nels. We shall explore the consequences of this interac- ϳ(N/V) ␳ , in agreement with the results obtained tion in much greater detail below. in Secs. IV.D and IV.E. The relation (195) was first de- In the case of two (or more) flavors the Grassmann rived by Diakonov and Petrov using somewhat different integration cannot be done exactly, since the effective techniques see (Sec. VI.B.3). The procedure for three flavors is very similar, so we action is more than quadratic in the fermion fields. In- do not need to go into detail here. Let us simply quote stead, we perform the integration over the quark fields the effective Lagrangian for N ϭ3 (Nowak, Verbaar- in mean-field approximation. This procedure is consis- f schot, and Zahed, 1989a), tent with the approximations used to derive the effective interaction (185). The mean-field approximation is most 1 ϭ␤͑2␲␳͒6 ⑀ ⑀ 2 f1f2f3 g1g2g3 easily derived by decomposing fermion bilinears into a L 6Nc͑NcϪ1͒ constant and a fluctuating part. The integral over the fluctuations is quadratic and can be done exactly. Tech- 2Ncϩ1 † † † ϫ ͑␺ ␥ϩ␺g ͒͑␺ ␥ϩ␺g ͒͑␺ ␥ϩ␺g ͒ nically, this can be achieved by introducing auxiliary sca- ͩ2N ϩ4 f1 1 f2 2 f3 3 c lar fields L ,R into the path integral35 and then shifting a a 3 the integration variables in order to linearize the inter- † † ϩ ͑␺f ␥ϩ␺g ͒͑␺f ␥ϩ␴␮␯␺g ͒ action. Using this method, the four-fermion interaction 8͑Ncϩ2͒ 1 1 2 2 becomes † † Ϫ 2 † Ϫ ϫ͑␺ ␥ ␴ ␺ ͒ϩ͑ϩ Ϫ͒ , (196) ͑␺ ␶ ␥ ␺͒ 2͑␺ ␶ ␥ ␺͒L ϪL L , (192) f ϩ ␮␯ g3 a Ϫ → a Ϫ a a a 3 ↔ ͪ † Ϫ 2 † Ϫ ͑␺ ␶a ␥ϩ␺͒ 2͑␺ ␶a ␥ϩ␺͒RaϪRaRa . (193) which was first derived in slightly different form by Shif- → man et al. (1980c). So far, we have neglected the current In the mean-field approximation, the L ,R integration a a quark mass dependence and considered the SU(N ) can be done using the saddle-point method. Since f symmetric limit. Real QCD is intermediate between the isospin and parity are not broken, only ␴ϭL ϭR can 4 4 N ϭ2 and N ϭ3 cases. Flavor mixing in the instanton have a nonzero value. At the saddle point, the free en- f f liquid with realistic values of quark masses was studied ergy FϭϪ(1/V)log Z is given by by Nowak et al. (1989a) to which we refer the reader for d4k more details. Fϭ4N log͓k2ϩ␤␴k2␸Ј2͑k͔͒ c ͵ ͑2␲͒4 Before we discuss the spectrum of hadronic excita- tions let us briefly summarize the last three subsections. 2 2Nc͑Nc Ϫ1͒ N ␤V A random system of instantons leads to spontaneous Ϫ2 ␤␴2Ϫ log . (194) 2N Ϫ1 V ͩ N ͪ chiral symmetry breaking. If the system is not only ran- c dom, but also sufficiently dilute, this phenomenon is most easily studied using the mean-field approximation. 35In the mean-field approximation, we do not need to intro- We have presented the mean-field approximation in three slightly different versions: one using a schematic duce auxiliary fields T␮␯ in order to linearize the tensor part of the interaction, since T␮␯ cannot have a vacuum expectation model in Sec. IV.D, one using random matrix arguments value. in Sec. IV.E, and one using an effective quark model in

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 366 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

2 2 this section. The results are consistent and illustrate the To this order in the quark masses, fKϭf␲ . The mass phenomenon of chiral symmetry breaking in comple- matrix in the flavor singlet and octet sector is more com- mentary ways. Of course, the underlying assumptions of plicated. One finds randomness and diluteness have to be checked. We shall 1 4 1 1 2 1 come back to this problem in Sec. V. Vϭ m2 Ϫ m2 ␩2ϩ m2 ϩ m2 ␩2 2 ͩ 3 K 3 ␲ ͪ 8 2 ͩ 3 K 3 ␲ ͪ 0

1 4& Nf N ϩ ͑m2 Ϫm2 ͒␩ ␩ ϩ ␩2 . (201) G. Bosonization and the spectrum 2 3 ␲ K 0 8 f2 ͩ V ͪ 0 of pseudoscalar mesons ␲ The last term gives the anomalous contribution to the ␩Ј We have seen that the ’t Hooft interaction leads to the mass. It agrees with the effective Lagrangian originally spontaneous breakdown of chiral symmetry and gener- proposed by Veneziano (1979) and leads to the Witten- ates a strong interaction in the pseudoscalar meson Veneziano relation [with ␹topӍ(N/V)] channels. We shall discuss mesonic correlation functions f2 m2 ϩm2Ϫ2m2 ͒ϭ2N N/V͒. (202) in great detail in Sec. VI and now consider only the ␲͑ ␩Ј ␩ K f͑ consequences for the spectrum of pseudoscalar mesons. Diagonalizing the mass matrix for mϭ5 MeV and ms The pseudoscalar spectrum is most easily obtained by ϭ120 MeV, we find m␩ϭ527 MeV, m␩Јϭ1172 MeV, bosonizing the effective action. In order to describe the and a mixing angle ␪ϭϪ11.5°. The ␩Ј mass is somewhat U(1)A anomaly and the scalar ␴ channel correctly, one too heavy, but given the crude assumptions, the result is has to allow for fluctuations of the number of instantons. certainly not bad. One should also note that the result The fluctuation properties of the instanton ensemble can corresponds to an ensemble of uncorrelated instantons. be described by the following ‘‘coarse-grained’’ partition In full QCD, however, the topological susceptibility is function (Nowak, Verbaarschot, and Zahed, 1989b): zero and correlations between instantons have to be b nϩ͑z͒ϩnϪ͑z͒ present (see Sec. V.E). S ϭ d4z͓nϩ͑z͒ϩnϪ͑z͔͒ log Ϫ1 eff 4 ͵ ͫ ͩ n ͪ ͬ 0 H. Spin-dependent interactions induced by instantons 1 ϩ ͵ d4z͓nϩ͑z͒ϪnϪ͑z͔͒2 2n0 The instanton-induced effective interaction between light quarks produces large spin-dependent effects. In 4 † ϩ ¯ this section, we wish to compare these effects with other ϩim͒␺Ϫn ͑z͒⌰I͑z͒”ץϩ d z͓␺ ͑i ͵ spin-dependent interactions in QCD and study the effect ϪnϪ͑z͒⌰¯ ͑z͔͒, (197) of instantons on spin-dependent forces in heavy-quark A systems. In QCD, the simplest source of spin-dependent where nϮ(z) is the local density of instantons/anti- effects is the hyperfine interaction generated by the one- ¯ instantons and ⌰I,A(z) is the ’t Hooft interaction (184) gluon exchange potential averaged over the instanton orientation. This partition ␣s ␲ function reproduces the low-energy theorem (164) as VOGEϭϪ ͑␭a␭a͒͑␴ជ ␴ជ ͒␦3͑rជ͒. (203) ij m m 6 i j i j well as the relation ␹topϭ(N/V) expected for a dilute i j system in the quenched approximation. In addition to This interaction has at least two phenomenologically im- that, the divergence of the flavor-singlet axial current is portant features: (a) The (␴ជ ␴ជ )(␭a␭a) term is twice as 5 ϩ Ϫ ␮j␮ϭ2Nf͓n (z)Ϫn (z)͔, consistent with the large in mesons as it is in baryons, and (b) the ratio ofץ given by axial U(1)A anomaly. spin-dependent forces in strange and nonstrange system Again, the partition function can be bosonized by in- is controlled by the inverse (constituent) quark mass. troducing auxiliary fields, linearizing the fermion inter- For comparison, the nonrelativistic limit of the action, and performing the integration over the quarks. ’t Hooft effective interaction is In addition to that, we expand the fermion determinant ␲2␳2 ͑m*͒2 3 in derivatives of the meson fields in order to recover VinstϭϪ 1ϩ ͑1ϩ3␴ជ ␴ជ ͒␭a␭a ij 6 m*m* ͩ 32 i j i j ͪ kinetic terms for the meson fields. This procedure gives i j the standard nonlinear ␴-model Lagrangian. To leading a a 1Ϫ␶i ␶j order in the quark masses, the pion and kaon masses ϫ ␦3͑rជ͒. (204) satisfy the Gell-Mann, Oakes, Renner relations ͩ 2 ͪ 2 2 The spin-dependent part of Vinst clearly shares the at- f␲m␲ϭϪ2m͗q¯q͘, (198) tractive features mentioned above. The dependence on 2 2 fKmKϭϪ͑mϩms͒͗q¯q͘, (199) the effective mass comes from having to close two of the with the pion-decay constant external legs in the three-flavor interaction (196). How- ever, there are important differences in the flavor de- 4 2 d k M ͑k͒ pendence of the one-gluon exchange and instanton in- f2 ϭ4N Ӎ͑100 MeV͒2. ␲ c ͵ ͑2␲͒4 ͓k2ϩM2͑k͔͒2 teractions. In particular, there is no ’t Hooft interaction (200) between quarks of the same flavor (uu, dd,orss). Nev-

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 367 ertheless, as shown by Shuryak and Rosner (1989), the reflection of the fact that dilute instantons do not con- potential provides a description of spin splittings in the fine. Also, the magnitude of the potential is on the order octet and decuplet that is as good as the one-gluon ex- of 100 MeV, too small to be of much importance in char- change. The instanton-induced potential has two addi- monium or bottonium systems. The spin-dependent part tional advantages over the one-gluon-exchange potential of the heavy-quark potential is (Dorokhov et al., 1992). First, we do not have to use an ss 1 (uncomfortably large value of the strong-coupling con- VQQ͑rជi ,rជj͒ϭϪ ͑␴ជ iٌជ i͒͑␴ជ jٌជ j͒V͑rជiϪrជj͒, (209 4MiMj stant ␣s , and the instanton potential does not have a (phenomenologically unwanted) large spin-orbit part. and it, too, is too small to be important phenomenologi- In addition to that, instantons provide genuine three- cally. More important is the instanton-induced interac- body forces. These forces act only in uds singlet states, tion in heavy-light systems. This problem was studied in like the flavor singlet ⌳ [usually identified with the some detail by Chernyshev et al. (1996). The effective ⌳(1405)] or the hypothetical dilambda (H-dibaryon).36 potential between the heavy and the light quark is given Another interesting question concerns instanton- by induced forces between heavy quarks (Callan et al., * a a 1978b). For heavy quarks, the dominant part of the in- ⌬MQmq ␭Q␭q V ͑rជ͒ϭ 1ϩ teraction is due to nonzero modes, which we have com- qQ 2͑N/V͒N ͫͩ 4 ͪ c pletely neglected in the discussion above. These effects spin a a ⌬MQ ␭Q␭q can be studied using the propagator of an infinitely Ϫ ␴ជ ␴ជ ␦3͑rជ͒, (210) ⌬M q Q 4 ͬ heavy quark, Q

1ϩ␥4 where ⌬MQ is the mass renormalization of the heavy S͑x͒ϭ ␦3͑rជ͒⌰͑␶͒P exp i A dx , (205) 2 ͩ ͵ ␮ ␮ͪ quark and 16␲ N ␳2 in the field of an instanton. Here, P denotes a path- ⌬Mspinϭ ϫ0.193 (211) Q N ͩ V ͪ M ordered integral, and we have eliminated the mass of the c Q heavy quark using a Foldy-Wouthuysen transformation. controls the hyperfine interaction. This interaction gives The phase accumulated by a heavy quark in the field of very reasonable results for spin splittings in heavy-light a single instanton (in singular gauge) is mesons. In addition to that, instantons generate many- ϱ body forces which might be important in heavy-light U͑rជ͒ϭP exp i A4dx4 baryons. We conclude that instantons can account for ͵Ϫϱ ͩ ͪ spin-dependent forces required in light-quark spectros- ϭcos͓F͑r͔͒ϩirជ•␶ជ sin͓F͑r͔͒, with copy without the need for large hyperfine interactions. Instanton-induced interactions are not very important in r heavy-quark systems, but may play a role in heavy-light F͑r͒ϭ␲ 1Ϫ , (206) ͩ ͱr2ϩ␳2 ͪ systems. where rជ is the spatial distance between the instanton and the heavy quark, and rϭ͉rជ͉. From this result, we can V. THE INTERACTING INSTANTON LIQUID determine the mass renormalization of the heavy quark due to the interaction with instantons in the dilute-gas A. Numerical simulations approximation (Callan et al., 1978b; Diakonov et al., 1989; Chernyshev, Nowak, and Zahed, 1996): In the last section we discussed an analytic approach to the statistical mechanics of the instanton ensemble 16␲ N 3 based on the mean-field approximation. This approach ⌬MQϭ ␳ ϫ0.552Ӎ70 MeV. (207) Nc ͩ V ͪ provides important insights into the structure of the in- In a similar fashion, one can determine the spin- stanton ensemble and the qualitative dependence on the independent part of the heavy-quark potential (for color interaction. However, the method ignores correlations singlet q¯q systems), among instantons, which are important for a number of phenomena, such as topological charge screening (Sec. 1 V.E), chiral symmetry restoration (Sec. VII.B), and had- V ͑rជ ,rជ ͒ϭ d␳n͑␳͒ d3r Tr͓U͑rជ Ϫrជ͒ QQ i j ͵ ͵ 3 i ronic correlation functions (Sec. VI). † In order to go beyond the mean-field approximation ϫU ͑rជjϪrជ͒Ϫ1͔. (208) and study the instanton liquid with the ’t Hooft interac- The potential (208) rises quadratically at short distance tion included to all orders, we have performed numeri- but levels off at a value of 2⌬MQ for rϾ5␳. This is a cal simulations of the interacting instanton liquid (Shuryak, 1988a; Shuryak and Verbaarschot, 1990; Nowak et al., 1989a; Scha¨fer and Shuryak, 1996a). In 36Takeuchi and Oka (1991) found that instanton-induced these simulations, we make use of the fact that the forces made the H unbound, a quite welcome conclusion, since quantum-field-theory problem is analogous to the statis- so far it has eluded all searches. tical, mechanics of a system of pseudoparticles in four

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 368 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

dimensions. The distribution of the 4NcN collective co- where the first factor, the high-momentum part, is the ordinates associated with a system of N pseudoparticles product of contributions from individual instantons cal- can be studied using standard Monte Carlo techniques culated in the Gaussian approximation, whereas the (e.g., the Metropolis algorithm), originally developed for low-momentum part associated with the fermionic zero simulations of statistical systems containing atoms or modes of individual instantons is calculated exactly. TIA molecules. is the NϩϫNϪ matrix of overlap matrix elements intro- These simulations have a number of similarities to lat- duced in Sec. IV.A.2. As emphasized before, this deter- tice simulations of QCD [see the textbooks of Creutz minant contains the ’t Hooft interaction to all orders. (1983), Rothe (1992), and Montvay and Mu¨ nster In practice, it is simpler to study the instanton en- (1995)]. As in lattice gauge theory, we consider systems semble for a fixed particle number NϭNϩϩNϪ . This in a finite four-dimensional volume, subject to periodic means that instead of the grand-canonical partition func- tion (212), we consider the canonical ensemble boundary conditions. This means that both approaches N ϩN share finite-size problems, especially the difficulty of 1 ϩ Ϫ working with realistic quark masses. Both methods are ZNϭ ͟ ͓d⍀in͑␳i͔͒ Nϩ!NϪ! i also formulated in Euclidean space, which means that it is difficult to extract real-time (in particular nonequilib- Nf rium) information. However, in contrast to the lattice, ϫexp͑ϪSint͒ det͑D” ϩmf͒ (214) ͟f spacetime is continuous, so we have no problems with chiral fermions. Furthermore, the number of degrees is for different densities and determine the ground state by drastically reduced, and meaningful (unquenched) simu- minimizing the free energy FϭϪ1/V log ZN of the sys- lations of the instanton ensemble can be performed in a tem. Furthermore, we shall consider only ensembles few minutes on an average workstation. Finally, using with no net topology. The two constraints N/Vϭconst the analogy with an interacting liquid, it is easier to de- and QϭNϩϪNϪϭ0 do not affect the results in the velop an intuitive understanding of the numerical simu- thermodynamic limit. The only exceptions are of course lations. fluctuations of Q and N, the topological susceptibility The instanton ensemble is defined by the partition and the compressibility of the instanton liquid, respec- tively. In order to study these quantities, one has to con- function sider appropriately chosen subsystems (see Sec. V.E). N ϩN 1 ϩ Ϫ In order to simulate the partition function (214), we Zϭ ͓d⍀ n͑␳ ͔͒ ͚ N !N ! ͵ ͟ i i generate a sequence ͕⍀i͖j (iϭ1,...,N; jϭ1,...,Nconf)of Nϩ ,NϪ ϩ Ϫ i configurations according to the weight function N f p(͕⍀i͖)ϳexp(ϪS), where

ϫexp͑ϪSint͒ det͑D” ϩmf͒, (212) NϩϩNϪ ͟f SϭϪ ͚ log͓n͑␳i͔͒ϩSintϩtr log͑D” ϩmf͒ (215) describing a system of pseudoparticles interacting via iϭ1 the bosonic action and the fermionic determinant. Here, is the total action of the configuration. This is most eas- 4 d⍀iϭdUid zid␳i is the measure in the space of collec- ily accomplished using the Metropolis algorithm: a new tive coordinates (color orientation, position, and size) configuration is generated using some microreversible associated with a single instanton, and n(␳) is the single- procedure ͕⍀i͖j ͕⍀i͖jϩ1 . The configuration is always instanton density [Eq. (93)]. accepted if the new→ action is smaller than the old one, The gauge interaction between instantons is approxi- and it is accepted with the probability exp(Ϫ⌬S)ifthe mated by a sum of pure two-body interactions Sint new action is larger. Alternatively, one can generate the 1 ϭ 2 ͚IÞJSint(⍀IJ). Genuine three-body effects in the in- ensemble using other techniques, e.g., the Langevin stanton interaction are not important as long as the en- (Nowak et al., 1989a), heat bath, or microcanonical semble is reasonably dilute. This part of the interaction methods. is fairly easy to deal with. The computational effort is similar to that for a statistical system with long-range B. The free energy of the instanton ensemble forces. The fermion determinant, however, introduces nonlo- Using the sequence of configurations generated by the cal interactions among many instantons. Changing the Metropolis algorithm, it is straightforward to determine coordinates of a single instanton requires the calculation expectation values by averaging measurements in many of the full N-instanton determinant, not just N two-body configurations, interactions. Evaluating the determinant exactly is a N quite formidable problem. In practice we proceed as in 1 Sec. IV.F and factorize the determinant into a low- and ͗ ͘ϭ lim ͚ ͕͑⍀i͖͒. (216) O N ϱ N jϭ1 O a high-momentum part, →

NϩϩNϪ This is how the quark and gluon condensates, as well as the hadronic correlation functions discussed in this and det͑D” ϩmf͒ϭ ͟ 1.34␳i det͑TIAϩmf͒, (213) ͩ i ͪ the following section, have been determined. However,

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 369

more work is required to determine the partition func- where NϭNϩϩNϪ . The free-energy density is finally tion, which gives the overall normalization of the instan- given by FϭϪ(1/V)log Z, where V is the four-volume ton distribution. The knowledge of the partition func- of the system. The pressure and the energy density are tion is necessary in order to calculate the free energy related to F by pϭϪF and ⑀ϭTdp/dTϪp. and the thermodynamics of the system. In practice, the partition function is most easily evaluated using the thermodynamic integration method (Kirkwood, 1931). For this purpose we write the total action as C. The instanton ensemble If correlations among instantons are important, the S͑␣͒ϭS0ϩ␣⌬S, (217) variational (or mean-field) partition function Z0 is not expected to provide an accurate estimate for the parti- which interpolates between a solvable action S0 and the tion function. This is certainly the case in the presence of full action S(␣ϭ1)ϭS0ϩ⌬S. If the partition function light fermions, in particular at finite temperature. In this for the system governed by the action S0 is known, the full partition function can be determined from section we want to present numerical results obtained from simulations of the instanton liquid at zero tempera- 1 ture. log Z͑␣ϭ1͒ϭlog Z͑␣ϭ0͒Ϫ ͵ d␣Ј͗0͉⌬S͉0͘␣Ј , As discussed in Sec. IV.A.1, a general problem in the 0 interacting instanton model is the treatment of very (218) close instanton anti-instanton pairs. In practice, we have where the expectation value ͗0͉•͉0͘␣ depends on the decided to deal with this difficulty by introducing a phe- coupling constant ␣. The obvious choice for decompos- nomenological short-range repulsive core, ing the action of the instanton liquid would be to use the 8␲2 A single-instanton action, S0ϭ͚i log͓n(␳i)͔, but this does S ϭ ͉u͉2, core g2 ␭4 not work since the ␳ integration in the free partition (222) function is not convergent. We therefore consider the 2 2 2 2 1/2 R2ϩ␳ ϩ␳ ͑R2ϩ␳ ϩ␳ ͒2 following decomposition: I A I A ␭ϭ ϩ 2 2 Ϫ1 , 2␳I␳A ͩ 4␳ ␳ ͪ I A NϩϩNϪ 2 ␳i into the streamline interaction. Here, ␭ is the conformal S͑␣͒ϭ ͚ Ϫlog͓n͑␳i͔͒ϩ͑1Ϫ␣͒␯ parameter (144) and A controls the strength of the core. iϭ1 ͩ ␳¯2 ͪ This parameter essentially governs the dimensionless di- luteness fϭ␳4(N/V) of the ensemble. The second pa- ϩ␣„Sintϩtr log͑D” ϩmf͒…, (219) rameter of the instanton liquid is the scale ⌳QCD in the where ␯ϭ(bϪ4)/2 and ␳2 is the average size squared of instanton size distribution, which fixes the absolute 2 the instantons with the full interaction included. The ␳i units. term serves to regularize the size integration for ␣ϭ0. It We have defined the scale parameter by fixing the does not affect the final result for ␣ϭ1. The specific instanton density to be N/Vϭ1fmϪ4. This means that, form of this term is irrelevant; our choice here is moti- in our units, the average distance between instantons is 1 vated by the fact that S(␣ϭ0) gives a single-instanton fm by definition. Alternatively, one can proceed as in distribution with the correct average size ␳2. The ␣ϭ0 lattice gauge simulations and use an observable such as partition function corresponds to the variational single- the ␳ meson mass to set the scale. Using N/V is very instanton distribution convenient and, as we shall see in the next section, using the ␳ or nucleon mass would not make much of a differ- 1 ence. We use the same scale-setting procedure for all NϩϩNϪ Z0ϭ ͑V␮0͒ , QCD-like theories, independent of N and N . This pro- Nϩ!NϪ! c f vides a simple prescription for comparing dimensional quantities in theories with different matter content. ϱ ␳2 The remaining free parameter is the (dimensionless) ␮0ϭ ͵ d␳n͑␳͒exp Ϫ␯ , (220) 0 ͩ ␳¯2 ͪ strength of the core A, which determines the (dimen- sionless) diluteness of the ensemble. In Scha¨fer and where ␮ is the normalization of the one-body distribu- 0 Shuryak (1996a), we chose Aϭ128, which gives f tion. The full partition function obtained from integrat- ϭ␳¯4(N/V)ϭ0.12 and ␳¯ϭ0.43 fm. As a result, the en- ing over the coupling ␣ is semble is not quite as dilute as phenomenology seems to 1 demand [(N/V)ϭ1fmϪ4 and ␳¯ϭ0.33 fm] but compa- log Zϭlog͑Z0͒ϩN d␣Ј N V ϭ Ϫ4 ¯ ͵0 rable to the lattice result ( / ) (1.4– 1.6) fm and ␳ ϭ0.35 fm (Chu et al., 1994). The average instanton ac- 2 ␳ 1 tion is SӍ6.4, while the average interaction is Sint /N ϫ͗0͉␯ Ϫ ͓Sintϩtr log͑Dϩmf͔͉͒0͘␣ , ¯2 N ” Ј Ӎ1.0, showing that the system is still semiclassical and ␳ that interactions among instantons are important, but (221) not dominant.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 370 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

At the minimum of the free energy we can also deter- mine the quark condensate [see Fig. 15(c)]. In quenched QCD, we have ͗q¯q͘ϭϪ(251 MeV)3, while a similar simulation in full QCD gives ͗q¯q͘ϭϪ(216 MeV)3,in good agreement with the phenomenological value.

D. Dirac spectra

We have already emphasized that the distribution of eigenvalues of the Dirac operator iD” ␺␭ϭ␭␺␭ is of great interest for many phenomena in QCD. In this section, we wish to study the spectral density of iD” in the instan- ton liquid for different numbers of flavors. Before we present the results, let us review a few general argu- ments. First, since the weight function contains the fer- Nf mion determinant detf(iD” )ϭ(⌸i␭i) , it is clear that small eigenvalues will be suppressed if the number of flavors is increased. This can also be seen from the Smilga-Stern result [Eq. (181)]. For Nfϭ2 the spectral density at ␭ ϭ0 is flat, while for NfϾ2 the slope of spectrum is posi- tive. In general, one might therefore expect that there is a critical number of flavors (smaller than Nfϭ17, where asymptotic freedom is lost) for which chiral symmetry is restored. There are a number of arguments that this in- deed happens in non-Abelian gauge theories (see Sec. IX.D). Let us only mention the simplest one here. The number of quark degrees of freedom is Nqϭ4NcNf , while, if chiral symmetry is broken, the number of low- 2 FIG. 15. Free energy, average instanton action, and quark con- energy degrees of freedom (‘‘pions’’) is N␲ϭNf Ϫ1. If densate as a function of the instanton density in the pure gauge chiral symmetry is still broken for NfϾ12, this leads to theory, from Scha¨fer and Shuryak, 1996a. All dimensionful the unusual situation that the effective number of de- quantities are given in units of the scale parameter ⌳QCD . grees of freedom at low energy is larger than the num- ber of elementary degrees of freedom at high energy. In Detailed simulations of the instanton ensemble in this case it is hard to see how one could ever have a QCD are discussed by Scha¨fer and Shuryak (1996a). As transition to a phase of weakly interacting quarks and an example, we show the free energy versus the instan- gluons, since the pressure of the low-temperature phase ton density in pure gauge theory (without fermions) in is always bigger. Fig. 15. At small density the free energy is roughly pro- In Fig. 16 we show the spectrum of the Dirac operator portional to the density, but at larger densities repulsive in the instanton liquid for Nfϭ0,1,2,3 light flavors (Ver- interactions become important, leading to a well-defined baarschot, 1994a). Clearly, the results are qualitatively 37 minimum. We also show the average action per instan- consistent with the Smilga-Stern theorem for Nfу2. In ton as a function of density. The average action controls addition to that, the trend continues for NfϽ2, where the probability exp(ϪS) to find an instanton, but has no the result is not applicable. We also note that for Nf minimum in the range of densities studied. This shows ϭ3 (massless) flavors, a gap starts to open up in the that the minimum in the free energy is a compromise spectrum. In order to check whether this gap indicates between maximum entropy and minimum action. chiral symmetry restoration in the infinite-volume limit, Fixing the units such that N/Vϭ1fmϪ4, we have ⌳ one has to investigate finite-size scaling. The problem ϭ270 MeV, and the vacuum energy density generated was studied in more detail by Scha¨fer and Shuryak by instantons is ⑀ϭϪ526 MeV/fm3. We have already (1996a), who concluded that chiral symmetry was re- stressed that the vacuum energy is related to the gluon stored in the instanton liquid between Nfϭ4 and Nf condensate by the trace anomaly. Estimating the gluon ϭ5. Another interesting problem is the dependence on condensate from the instanton density, we have ⑀ϭ the dynamic quark mass in the chirally restored phase 3 crit Ϫb/4(N/V)ϭϪ565 MeV/fm , which is in good agree- NfϾNf . If the quark mass is increased, the influence ment with the direct determination of the energy den- of the fermion determinant is reduced, and eventually sity. Not only the depth of the free energy, but also its curvature (the instanton compressibility) is fixed from 37 the low-energy theorem (162). The compressibility de- Numerically, the slope in the Nfϭ3 spectrum appears to be termined from Fig. 15 is 3.2(N/V)Ϫ1, which can be com- too large, but it is not clear how small ␭ has to be for the pared with 2.75(N/V)Ϫ1 from the low-energy theorem. theorem to be applicable.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 371

i␪␯ ϪS Zϭ͚ e dA␮e detf͑D” ϩmf͒ ␯ ͵␯

i␪ ␯ ϪS Ј 2 2 ϭ͚ ͑e det ͒ ͵ dA␮e ͟ ͑␭nϩmf ͒. (224) ␯ M ␯ n,f

Here, ␯ is the winding number of the configuration, is the mass matrix, and ⌸Ј denotes the product of allM ei- genvalues with the zero modes excluded. The result shows that the partition function depends on ␪ only through the combination ei␪ det , so the vacuum en- ergy is independent of ␪ if one ofM the quark masses van- ishes. The fact that ␹top vanishes implies that fluctuations in the topological charge are suppressed, so instantons, and anti-instantons have to be correlated. Every instanton is surrounded by a cloud of anti-instantons which com- pletely screens its topological charge, analogous to De- FIG. 16. Spectrum of the Dirac operator for different values of bye screening in ordinary plasmas. This fact can be seen the number of flavors N , from Verbaarschot, 1994b. The ei- f most easily by using the bosonized effective Lagrangian genvalue is given in units of the scale parameter ⌳QCD and the distribution function is normalized to one. (201). For simplicity we consider instantons to be point- like, but, in contrast to the procedure in Sec. IV.G, we do allow the positions of the pseudoparticles to be cor- ‘‘spontaneous’’ symmetry breaking is recovered. As a related. The partition function is given by Nowak et al., consequence, QCD has an interesting phase structure as 1989b; Kikuchi and Wudka, 1992; Dowrick and McDou- a function of the number of flavors and their masses, gall, 1993; and Shuryak and Verbaarschot, 1995 as even at zero temperature. N ϩN ␮ ϩ Ϫ NϩϩNϪ 0 4 Zϭ d zi exp͑ϪSeff͒, (225) N ͚,N N !N ! ͟i E. Screening of the topological charge ϩ Ϫ ϩ Ϫ

where ␮0 is the single-instanton normalization (153) and Another interesting phenomenon associated with dy- the effective action is given by namical quarks is topological charge screening. This ef- fect is connected with properties of the ␩Ј meson, strong ͱ2Nf CP violation, and the structure of QCD at finite ␪ angle. S ϭi d4x ␩ Qϩ d4x ͑␩ ,␩ ͒. (226) eff ͵ f 0 ͵ 0 8 Topological charge screening can be studied in a num- ␲ L ber of complementary ways. Historically, it was first dis- Here, the topological charge density is Q(x)ϭ͚Qi␦(x cussed on the basis of Ward identities associated with Ϫzi), and (␩0 ,␩8) is the nonanomalous part of the the anomalous U(1)A symmetry (Veneziano, 1979). Let pseudoscalarL meson Lagrangian with the mass terms us consider the flavor singlet correlation function ⌸␮␯ given in Eq. (201). We can perform the sum in Eq. (225) ϭ͗q¯␥␮␥5q(x)q¯␥␯␥5q(0)͘. Taking two derivatives and and, keeping only the quadratic terms, integrate out the using the anomaly relation (95), we can derive the low- meson fields and determine the topological charge cor- energy theorem relator in this model. The result is m q¯q 4 ͗ ͘ ␹ ϭ d x͗Q͑x͒Q͑0͒͘ϭϪ N 2Nf N top ͵ 2N ͗Q͑x͒Q͑0͒͘ϭ ␦4͑x͒Ϫ ͓cos2͑␾͒D͑m ,x͒ f ͩ V ͪ f2 V ␩Ј ͭ ␲ m2 ϩ d4x͗q¯␥ q͑x͒q¯␥ q͑0͒͘. (223) N2 ͵ 5 5 2 4 f ϩsin ͑␾͒D͑m␩ ,x͔͒ͮ , (227) Since the correlation function on the right-hand side 2 does not have any massless poles in the chiral limit, the where D(m,x)ϭm/(4␲ x)K1(mx) is the (Euclidean) topological susceptibility ␹ ϳm as m 0. More gener- propagator of a scalar particle and ␾ is the ␩Ϫ␩Ј mixing top → ally, ␹top vanishes if there is at least one massless quark angle. The correlator (227) has an obvious physical in- flavor. terpretation. The local terms is the contribution from a Alternatively, we can use the fact that the topological single instanton located at the center, while the second susceptibility is the second derivative of the vacuum en- term is the contribution from the screening cloud. One ergy with respect to the ␪ angle. Writing the QCD par- can easily check that the integral of the correlator is of 2 tition function as a sum over all topological sectors and order m␲ ,so␹topϳm in the chiral limit. We also ob- extracting the zero modes from the fermion determi- serve that the screening length is given by the mass of nant, we have the ␩Ј.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 372 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

Detailed numerical studies of topological charge screening in the interacting instanton model were per- formed by Shuryak and Verbaarschot (1995). The au- thors verified that complete screening took place if one of the quark masses went to zero and that the screening length was consistent with the ␩Ј mass. They also ad- dressed the question of how the ␩Ј mass could be ex- tracted from topological charge fluctuations. The main idea was not to study the limiting value of ͗Q2͘/V for large volumes but to determine its dependence on V for small volumes VϽ1fm4. In this case, one has to worry about possible surface effects. It is therefore best to con- sider the topological charge in a segment H(l4) 3 4 ϭl4ϫL of the torus L (a hypercube with periodic boundary conditions). This construction ensures that the surface area of H(l4) is independent of its volume. Us- ing the effective meson action introduced above, we ex- pect (in the chiral limit) N 1 2 3 Ϫm␩ l4 KP͑l4͒ϵ͗Q͑l4͒ ͘ϭL ͑1Ϫe Ј ͒. (228) ͩ V ͪ m␩Ј

Numerical results for KP(l4) are shown in Fig. 17. The solid line shows the result for a random system of instan- tons with a finite topological susceptibility ␹top FIG. 17. Pseudoscalar and scalar correlators KP,S(l4)asa Ӎ(N/V), and the dashed curve is a fit using the param- 3 function of the length l4 of the subvolume l4ϫL , from etrization (228). Again, we clearly observe topological Shuryak and Verbaarschot, 1995. Screening implies that the charge screening. Furthermore, the ␩Ј mass extracted correlator depends only on the surface, not on the volume of from the fit is m␩Јϭ756 MeV (for Nfϭ2), quite consis- the torus. This means that in the presence of screening, the tent with what one would expect. The figure also shows correlator goes to a constant. The results were obtained for the behavior of the scalar correlation function, related to Ncϭ3 and muϭmdϭ10 MeV and msϭ150 MeV. The upper the compressibility of the instanton liquid. The instanton solid line corresponds to a random system of instantons, while number NϭNϩϩNϪ is of course not screened, but the the other solid line shows the parametrization discussed in the fluctuations in N are reduced by a factor 4/b due to the text (the dashed line in the upper panel shows a slightly more interactions [see Eq. (162)]. For a more detailed analysis sophisticated parametrization). Note the qualitative difference of the correlation function, see Shuryak and Verbaar- between the data for topological (upper panel) and number schot (1995). fluctuations (lower panel). The topological charge correlator is We conclude that the topological charge in the instan- flat, corresponding to charge screening, while the number fluc- ton liquid is completely screened in the chiral limit. The tuations are only reduced in size as compared to the random ␩Ј mass is not determined by the topological suscepti- ensemble. bility, but by fluctuations of the charge in small subvol- umes. Here, jh(x) is a local operator with the quantum num- bers of a hadronic state h. We shall concentrate on me- sonic and baryonic currents of the type

ab¯a b jmes͑x͒ϭ␦ ␺ ͑x͒⌫␺ ͑x͒, (230) VI. HADRONIC CORRELATION FUNCTIONS abc aT b c jbar͑x͒ϭ⑀ ͓␺ ͑x͒C⌫␺ ͑x͔͒⌫Ј␺ ͑x͒. (231) A. Definitions and generalities Here, a,b,c are color indices and ⌫,⌫Ј are isospin and In a relativistic field theory, current correlation func- Dirac matrices. At zero temperature, we shall focus ex- tions determine the spectrum of hadronic resonances. In clusively on correlators for spacelike (or Euclidean) 2 addition to that, hadronic correlation functions provide separation ␶ϭͱϪx . The reason is that spacelike corr- a bridge between hadronic phenomenology, the under- elators are exponentially suppressed rather than oscilla- lying quark-gluon structure, and the structure of the tory at large distance. In addition, most theoretical ap- QCD vacuum. The available theoretical and phenom- proaches, like the operator product expansion (OPE), enological information was recently reviewed by lattice calculations, or the instanton model deal with Eu- Shuryak (1993), so we give only a brief overview here. clidean correlators. In the following, we consider hadronic point-to-point Hadronic correlation functions are completely deter- correlation functions, mined by the spectrum (and the coupling constants) of the physical excitations with the quantum numbers of ⌸h͑x͒ϭ͗0͉jh͑x͒jh͑0͉͒0͘. (229) the current jh . For a scalar correlation function, we have

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 373 the standard dispersion relation loop, gluon exchanges as well as nonperturbative effects. All of these effects are hidden in the vacuum average. 2 n ͑ϪQ ͒ Im ⌸͑s͒ Correlators of isosinglet meson currents jIϭ0 ⌸͑Q2͒ϭ ds ␲ ͵ sn͑sϩQ2͒ ϭ(1/&)(u¯⌫uϩd¯⌫d) receive an additional two-loop, or disconnected, contribution 2 ϩa0ϩa1Q ϩ¯ , (232) ab ba ⌸Iϭ0͑x͒ϭ͗Tr͓S ͑0,x͒⌫S ͑x,0͒⌫͔͘ where Q2ϭϪq2 is the Euclidean momentum transfer aa bb and we have indicated possible subtraction constants ai . Ϫ2͗Tr͓S ͑0,0͒⌫͔Tr͓S ͑x,x͒⌫͔͘. (237) The spectral function ␳(s)ϭ(1/␲)Im ⌸(s) can be ex- In an analogous fashion, baryon correlators can be ex- pressed in terms of physical states, pressed as vacuum averages of three-quark propagators. At short distance, asymptotic freedom implies that the 2 3 4 correlation functions are determined by free-quark ␳͑sϭϪq ͒ϭ͑2␲͒ ␦ ͑qϪqn͒͗0͉jh͑0͉͒n͘ ͚n propagation. The free-quark propagator is given by

ϫ n j†͑0͒ 0 , (233) i ␥•x ͗ ͉ h ͉ ͘ S ͑x͒ϭ . (238) 0 2␲2 x4 where ͉n͘ is a complete set of hadronic states. Correla- tion functions with nonzero spin can be decomposed This means that mesonic and baryonic correlation func- 6 into Lorentz-covariant tensors and scalar functions. tions at short distance behave as ⌸mesϳ1/x and ⌸bar Fourier-transforming Eq. (232) gives a spectral repre- ϳ1/x9, respectively. Deviations from asymptotic free- sentation of the coordinate-space correlation function dom at intermediate distances can be studied using the operator product expansion. The basic idea (Wilson, 1969) is to expand the product of currents in Eq. (229) ⌸͑␶͒ϭ ds ␳͑s͒D͑ͱs,␶͒. (234) ͵ into a series of coefficient functions cn(x) multiplied by local operators (0) On Here, D(m,␶) is the Euclidean propagator of a scalar particle with mass m, ⌸͑x͒ϭ cn͑x͒͗ n͑0͒͘. (239) ͚n O m D͑m,␶͒ϭ K ͑m␶͒. (235) 4␲2␶ 1 From dimensional considerations it is clear that the most singular contributions correspond to operators of the Note that, except for possible contact terms, subtraction lowest possible dimension. Ordinary perturbative contri- constants do not affect the coordinate-space correlator. butions are contained in the coefficient of the unit op- For large arguments, the correlation function decays ex- erator. The leading nonperturbative corrections are con- ponentially, ⌸(␶)ϳexp(Ϫm␶), where the decay is gov- trolled by the quark and gluon condensates of erned by the lowest pole in the spectral function. This is dimension three and four. the basis of hadronic spectroscopy on the lattice. In In practice, the OPE for a given correlation function practice, however, lattice simulations often rely on more is most easily determined using the short-distance ex- complicated sources in order to improve the suppression pansion of the propagator in external, slowly varying 38 of excited states. While useful in spectroscopy, these quark and gluon fields (Novikov, et al., 1985b), correlation functions mix short- and long-distance ef- ab ab fects and are not as interesting theoretically. i␦ ␥•x ␦ mf Sab͑x͒ϭ Ϫ ϩqa͑0͒q¯b͑0͒ Correlation functions of currents built from quark f 2␲2 x4 4␲2 x2 fields only (like the meson and baryon currents intro- i x ϩ x duced above) can be expressed in terms of the full quark k ab k ␥• ␴␣␤ ␴␣␤␥• Ϫ 2 ͑␭ ͒ G␣␤ 2 ϩ¯ . propagator. For an isovector meson current jIϭ1ϭu¯⌫d 32␲ x (where ⌫ is only a Dirac matrix), the correlator is given by the ‘‘one-loop’’ term (240) The corrections to the free propagator have an obvious ab ba ⌸Iϭ1͑x͒ϭ͗Tr͓S ͑0,x͒⌫S ͑x,0͒⌫͔͘. (236) interpretation in terms of the interaction of the quark with the external quark and gluon fields. There is an The averaging is performed over all gauge configura- enormous literature about QCD sum rules based on the tions, with the weight function det(D” ϩm)exp(ϪS). Note OPE. See the reviews of Reinders et al. (1985), Narison that the quark propagator is not translation invariant before the vacuum average is performed, so the propa- gator depends on both arguments. Also note that while 38This expression was derived in the Fock-Schwinger gauge a Eq. (236) has the appearance of a one-loop (perturba- x␮A␮ϭ0. The resulting hadronic correlation functions are of tive) graph, it includes arbitrarily complicated, multi- course gauge invariant.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 374 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

(1989), and Shifman (1992). The general idea is easily tions. However, it is also interesting to study the vacuum explained. If there is a window in which the OPE [Eq. expectation value of the propagator39 (239)] has reasonable accuracy and the spectral repre- ͗S͑x͒͘ϭS ͑x͒ϩ␥ xS ͑x͒. (242) sentation [Eq. (234)] is dominated by the ground state, S • V one can match the two expressions in order to extract From the definition of the quark condensate, we have ground-state properties. In general, the two require- ͗q¯q͘ϭSS(0), which means that the scalar component of ments are in conflict with each other, so the existence of the quark propagator provides an order parameter for a sum-rule window has to be established in each indi- chiral symmetry breaking. To obtain more information, vidual case. we can define a gauge-invariant propagator by adding a The main sources of phenomenological information Wilson line, about the correlation functions are summarized as fol- x ¯ lows (Shuryak, 1993): Sinv͑x͒ϭ ␺͑x͒P exp A␮͑xЈ͒dx␮Ј ␺͑0͒ . (243) ͳ ͩ ͵0 ͪ ʹ (1) Ideally, the spectral function is determined from an experimentally measured cross section using the op- This object has a direct physical interpretation because tical theorem. This is the case, for example, in the it describes the propagation of a light quark coupled to vector-isovector (rho-meson) channel, where the an infinitely heavy, static source (Shuryak, 1982a; necessary input is provided by the ratio Shuryak and Verbaarschot, 1993b; Chernyshev et al., 1996). It therefore determines the spectrum of heavy- ϩ Ϫ light mesons (with the mass of the heavy quark sub- ␴„e e ͑Iϭ1 hadrons͒… R͑s͒ϭ → , (241) tracted) in the limit where the mass of the heavy quark ␴͑eϩeϪ ␮ϩ␮Ϫ͒ → goes to infinity. where s is the invariant mass of the lepton pair. 1. The propagator in the field of a single instanton Similarly, in the axial-vector (a1 meson channel) the spectral function below the ␶ mass can be deter- The propagators of massless scalar bosons, gauge mined from the hadronic decay width of the ␶ lepton fields, and fermions in the background field of a single 40 ⌫(␶ ␯␶ϩhadrons). instanton can be determined analytically (Brown et al., → (2) In some cases, the coupling constants of a few reso- 1978; Levine and Yaffe, 1979). We do not go into details nances can be extracted indirectly, for example us- of the construction, which is quite technical, but only ing low-energy theorems. In this way, the approxi- provide the main results. mate shape of the pseudoscalar ␲,K,␩,␩Ј and some We have already seen that the quark propagator in glueball correlators can be determined. the field of an instanton is ill behaved because of the (3) Ultimately, the best source of information about presence of a zero mode. Of course, the zero mode does hadronic correlation functions is the lattice. At not cause any harm, since it is compensated by a zero in present most lattice calculations use complicated the tunneling probability. The remaining non-zero-mode nonlocal sources. Exceptions can be found part of the propagator satisfies the equation in Chu et al. (1993a, 1993b) and Leinweber (1995a, nz † 1995b). So far, all results have been obtained in the iD” S ͑x,y͒ϭ␦͑xϪy͒Ϫ␺0͑x͒␺0͑y͒, (244) quenched approximation. which ensures that all modes in Snz are orthogonal to In general, given the fundamental nature of hadronic the zero mode. Formally, this equation is solved by correlators, all models of hadronic structure or the QCD 1ϩ␥5 1Ϫ␥5 vacuum should be tested against the available informa- Snz͑x,y͒ϭDជ ⌬͑x,y͒ ϩ⌬͑x,y͒Dឈ , ” x ͩ 2 ͪ ” yͩ 2 ͪ tion on the correlators. We shall discuss some of these models as we go along. (245) where ⌬(x,y) is the propagator of a scalar quark in the fundamental representation, ϪD2⌬(x,y)ϭ␦(x,y).

B. The quark propagator in the instanton liquid 39The quark propagator is of course not a gauge-invariant As we have emphasized above, the complete informa- object. Here, we imply that a gauge has been chosen or the tion about mesonic and baryonic correlation functions is propagator is multiplied by a gauge string. Also note that, be- encoded in the quark propagator in a given gauge-field fore averaging, the quark propagator has a more general Dirac configuration. Interactions among quarks are repre- structure, S(x)ϭEϩP␥5ϩV␮␥␮ϩA␮␥␮␥5ϩT␮␯␴␮␯ . This de- sented by the failure of expectation values to factorize, composition, together with positivity, is the basis of a number of exact results about correlation functions (Weingarten, 1983; e.g., S(␶)2 Þ S(␶) 2. In the following, we shall con- ͗ ͘ ͗ ͘ Vafa and Witten, 1984). struct the quark propagator in the instanton ensemble, 40The result is easily generalized to ’t Hooft’s exact multi- starting from the propagator in the background field of a instanton solution, but much more effort is required to con- single instanton. struct the quark propagator in the most general (Atiyah, From the quark propagator, we calculate the Hitchin, Drinfeld, and Manin, 1977) instanton background ensemble-averaged meson and baryon correlation func- (Corrigan, Goddard, and Templeton, 1979).

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 375

Equation (245) is easily checked using the methods we † 1 ␺0͑x͒␺0͑y͒ employed in order to construct the zero-mode solution; S͑x,y͒ϭ ϭ ϩSnz͑x,y͒ see Eq. (102). The scalar propagator does not have any iD” ϩim m zero modes, so it can be constructed using standard techniques. The result (in singular gauge) is (Brown ϩm⌬͑x,y͒ϩ¯ . (249) et al., 1978) 2. The propagator in the instanton ensemble 1 1 1 In this section we extend the results of the last section ⌬͑x,y͒ϭ 2 2 4␲ ͑xϪy͒ ͱ1ϩ␳2/x2 ͱ1ϩ␳2/y2 to the more general case of an ensemble consisting of many pseudoparticles. The quark propagator in an arbi- 2 Ϫ ϩ ␳ ␶ •x␶ •y trary gauge field can always be expanded as ϫ 1ϩ . (246) ͩ x2y2 ͪ SϭS ϩS AS ϩS AS AS ϩ , (250) For an instanton located at z, one has to make the ob- 0 0 ” 0 0 ” 0 ” 0 ¯ vious replacements x (xϪz) and y (yϪz). The propagator in the field→ of an anti-instanton→ is obtained where the individual terms have an obvious interpreta- by interchanging ␶ϩ and ␶Ϫ. If the instanton is rotated tion as arising from multiple gluon exchanges with the by the color matrix Rab, then ␶Ϯ has to be replaced by background field. If the gauge field is a sum of instanton A ϭ A (Rab␶b,ϯi). contributions, ␮ ͚I I␮ , then Eq. (250) becomes Using the result for the scalar quark propagator and the representation (245) of the spinor propagator intro- SϭS0ϩ S0A” IS0ϩ S0A” IS0A” JS0ϩ¯ (251) duced above, the non-zero-mode propagator is given by ͚I ͚I,J

1 1 Snz͑x,y͒ϭ S ͑x,y͒ ϭS ϩ S ϪS ϩ S ϪS SϪ1 S ϪS 2 2 2 2 0 0 ͚ ͑ I 0͒ ͚ ͑ I 0͒ 0 ͑ J 0͒ ͱ1ϩ␳ /x ͱ1ϩ␳ /y ͫ I IÞJ

2 Ϫ ϩ 2 ␳ ␶ •x␶ •y ␳ Ϫ1 Ϫ1 ϫ 1ϩ ϪD ͑x,y͒ ϩ ͚ ͑SIϪS0͒S0 ͑SJϪS0͒S0 ͑SKϪS0͒ ͩ x2y2 ͪ 0 x2y2 IÞJ,JÞK Ϫ ϩ Ϫ ϩ ϩ . (252) ␶ •x␶ •␥␶ •͑xϪy͒␶ •y ¯ ϫ ␥ ͩ ␳2ϩx2 ϩ Here, I,J,K,... refer to both instantons and anti- Ϫ ϩ Ϫ ϩ ␶ •x␶ •͑xϪy͒␶ •␥␶ •y instantons. In the second line, we have resummed the ϩ ␥ , (247) ␳2ϩx2 Ϫͪͬ contributions corresponding to an individual instanton. SI refers to the sum of zero and non-zero-mode compo- where ␥Ϯϭ(1Ϯ␥5)/2. The propagator can be general- nents. At large distances from the center of the instan- ized to arbitrary instanton positions and color orienta- ton, SI approaches the free propagator S0 . Thus Eq. tions in the same way as the scalar quark propagator (252) has a nice physical interpretation: Quarks propa- discussed above. gate by jumping from one instanton to the other. If At short distances, as well as far away from the instan- ͉xϪzI͉Ӷ␳I , and ͉yϪzI͉Ӷ␳I for all I, the free propaga- ton, the propagator reduces to the free one. At interme- tor dominates. At large distances, terms involving more diate distances, the propagator is modified due to gluon and more instantons become important. exchanges with the instanton field, In the QCD ground state, chiral symmetry is broken. The presence of a condensate implies that quarks can ␥•͑xϪy͒ 1 propagate over large distances. Therefore we cannot ex- Snz͑x,y͒ϭϪ Ϫ pect that truncating the series (252) will provide a useful 2␲2͑xϪy͒4 16␲2͑xϪy͒2 approximation to the propagator at low momenta. Fur- ˜ thermore, we know that spontaneous symmetry break- ϫ͑xϪy͒␮␥␯␥5G␮␯ϩ¯ . (248) ing is related to small eigenvalues of the Dirac operator. This result is consistent with the OPE of the quark A good approximation to the propagator is obtained by propagator in a general background field [see Eq. (240)]. assuming that (SIϪS0) is dominated by fermion zero It is interesting to note that all the remaining terms are modes, regular as (xϪy)2 0. This has important consequences for the OPE of hadronic→ correlators in a general, self- † ␺I͑x͒␺I ͑y͒ dual background field (Dubovikov and Smilga, 1981). ͑S ϪS ͒͑x,y͒Ӎ . (253) I 0 im Finally, we need the quark propagator in the instan- ton field for small but nonvanishing quark mass. Ex- panding the quark propagator for small m, we get In this case, the expansion (252) becomes

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 376 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

S͑x,y͒ӍS0͑x,y͒ † ␺I͑x͒␺I ͑y͒ ␺I͑x͒ ϩ ϩ ͚I im I͚ÞJ im

† ␺J͑y͒ , Ϫim͒␺ ͑r͒ ϩץϫ d4r␺†͑r͒͑Ϫi ͩ͵ I ” J ͪ im ¯ (254) which contains the overlap integrals TIJ defined in Eq. (148). This expansion can easily be summed to give

S͑x,y͒ӍS0͑x,y͒

1 † ϩ͚ ␺I͑x͒ ␺J͑y͒. (255) I,J TIJϩimDIJϪim␦IJ 4 † Here, DIJϭ͐d r␺I(r)␺J(r)Ϫ␦IJ arises from the restric- tion IÞJ in the expansion (252). The quantity mDIJ is small in both the chiral expansion and the packing frac- tion of the instanton liquid and will be neglected in what follows. Comparing the resummed propagator (255) with the single-instanton propagator (253) shows the im- FIG. 18. Scalar and vector components of the quark propaga- portance of chiral symmetry breaking. While Eq. (253) tor, normalized to the free vector propagator. The dashed lines is proportional to 1/m, the diagonal part of the full Ϫ1 show the result of the mean-field approximation, while the propagator is proportional to (T )IIϭ1/m*. data points were obtained in different instanton ensembles. The result (255) can also be derived by inverting the Dirac operator in the basis spanned by the zero modes of the individual instantons: the propagator in the mean-field approximation. The mean-field propagator can be obtained in several ways. 1 Most easily, we can read off the propagator directly S͑x,y͒ӍS0͑x,y͒ϩ͚ ͉I͗͘I͉ ͉J͗͘J͉. (256) I,J iD” ϩim from the effective partition function (188). We find The equivalence of Eqs. (255) and (256) can be easily p” ϩiM͑p͒ S͑p͒ϭ (258) seen using the fact that, in the sum ansatz, the derivative p2ϩM2͑p͒ in the overlap matrix element TIJ can be replaced by a covariant derivative. with the momentum-dependent effective quark mass The propagator (255) can be calculated either numeri- ␤ N cally or using the mean-field approximation introduced M͑p͒ϭ p4␸Ј2͑p͒. (259) 2N V in Sec. IV.F. We shall discuss the mean-field propagator c in the following section. For our numerical calculations, Here, ␤ is the solution of the gap equation (189). Origi- we have improved the zero-mode propagator by adding nally, the result (258) was obtained by Diakonov and the contributions from nonzero modes to first order in Petrov from the Dyson-Schwinger equation for the the expansion (252). The result is quark propagator in the large-Nc limit (Diakonov and Petrov, 1986). At small momenta, chiral sym- S͑x,y͒ϭS ͑x,y͒ϩSZMZ͑x,y͒ 0 metry breaking generates an effective mass M(0) 2 NZM ϭ(␤/2Nc)(N/V)(2␲␳) . The quark condensate was al- ϩ ͓SI ͑x,y͒ϪS0͑x,y͔͒. (257) ͚I ready given in Eq. (195). At large momenta, we have M(p)ϳ1/p6 and constituent quarks become free current How accurate is this propagator? We have seen that the quarks. propagator agrees with the general OPE result at short For comparison with our numerical results, it is useful distance. We also know that it accounts for chiral sym- to determine the mean-field propagator in coordinate metry breaking and spontaneous mass generation at space. Fourier-transforming the result (258) gives large distances. In addition to that, we have performed a 1 p4 number of checks on the correlation functions that are S ͑x͒ϭ dp J ͑px͒, (260) sensitive to the degree to which (257) satisfies the equa- V 4␲2x ͵ p2ϩM2͑p͒ 2 tions of motion, for example, by testing whether the vec- 1 p3M͑p͒ tor correlator is transverse (the vector current is con- SS͑x͒ϭ 2 dp 2 2 J1͑px͒. (261) served). 4␲ x ͵ p ϩM ͑p͒ The result is shown in Fig. 18. The scalar and vector 3. The propagator in the mean-field approximation components of the propagator are normalized to the In order to understand the propagation of quarks in free propagator. The vector component of the propaga- the ‘‘zero-mode zone’’ it is very instructive to construct tor is exponentially suppressed at large distance, show-

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 377

FIG. 19. Pion correlation function in various approximations FIG. 20. Eta-prime meson correlation functions. The various and instanton ensembles. Upper panel: solid curve, the phe- curves and data sets are labeled as in Fig. 19. nomenological expectation; short-dashed curve, the operator product expansion; dotted curve, the single instanton; long- a a ␶ dashed curve, the mean-field approximation; ᮀ, data in the j␲ϭq¯␶ ␥5q, j␳ϭq¯ ␥␮q, j␩ ϭq¯␥5q, (262) random instanton ensemble. Lower panel: comparison of dif- 2 ns ferent instanton ensembles; ᮀ, random; ᭹, quenched; ᭿, un- quenched; ᭡, ratio ansatz; star, lattice; crown, interacting. where qϭ(u,d). Here, we consider only the nonstrange ␩Ј and refer the reader to Scha¨fer (1996), and Shuryak and Verbaarschot (1993a) for a discussion of SU(3) fla- ing the formation of a constituent mass. The scalar com- vor symmetry breaking and ␩Ϫ␩Ј mixing. The three ponent again shows the breaking of chiral symmetry. At channels discussed here are instructive because the short distances, the propagator is consistent with the instanton-induced interaction is attractive for the pion 2 2 ¯ coordinate-dependent quark mass mϭ␲ /3•x ͗␺␺͘ in- channel, repulsive for the ␩Ј, and (to first order in the ferred from the OPE equation (240). At large distances instanton density) does not affect the ␳ meson. The ␲, ␳, the exponential decay is governed by the constituent and ␩Ј are therefore representative of a much larger set mass M(0). of correlation functions. In addition, these three mesons For comparison, we also show the quark propagator have special physical significance. The pion is the light- in different instanton ensembles. The result is very simi- est hadron, connected with the spontaneous breakdown lar to the mean-field approximation; the differences are of SU(Nf)LϫSU(Nf)R chiral symmetry. The ␩Ј is sur- mainly due to different values of the quark condensate. prisingly heavy, a fact related to the anomalous U(1)A The quark propagator is not very sensitive to correla- symmetry. The ␳ meson, finally, is the lightest non- tions in the instanton liquid. In Shuryak and Verbaar- Goldstone particle and the first ␲␲ resonance. schot (1993b), and Chernyshev et al. (1996), the quark Phenomenological predictions for the correlation propagator was also used to study heavy-light mesons in functions are shown in Figs. 19–21 (Shuryak, 1993). All the 1/MQ expansion. The results are very encouraging, correlators are normalized to the free ones, R(␶) and we refer the reader to the original literature for 0 0 ϭ⌸⌫(␶)/⌸⌫(␶), where ⌸⌫(␶)ϭTr͓⌫S0(␶)⌫S0(Ϫ␶)͔. details. At short distances, asymptotic freedom implies that this ratio approaches one. At large distances, the correlators C. Mesonic correlators are exponential and R is small. At intermediate dis- tances, R depends on the quark-quark interaction in that channel. If RϾ1, we shall refer to the correlator as 1. General results and the operator product expansion attractive, while RϽ1 implies repulsive interactions. A large number of mesonic correlation functions have The normalized pion correlation function R␲ is signifi- been studied in the instanton model, and clearly this is cantly larger than one, showing a strongly attractive in- not the place to list all of them. Instead, we have decided teraction and a light bound state. The rho-meson cor- to discuss three examples that are illustrative of the relator is close to one out to fairly large distances x techniques and the most important effects. We shall con- Ӎ1.5 fm, a phenomenon referred to as ‘‘superduality’’ sider the ␲, ␳, and ␩Ј channels, related to the currents by Shuryak (1993). The ␩Ј channel is strongly repulsive,

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 378 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

power corrections in the ␲ channel are both attractive, so the OPE prediction has the correct tendency, but un- derpredicts the rise for xϾ0.25 fm. In the ␳-meson chan- nel, the leading corrections have a tendency to cancel each other, in agreement with the superduality phenom- enon mentioned above.

2. The single-instanton approximation After this prelude we come to the instanton model. We start by considering instanton contributions to the short-distance behavior of correlation functions in the single-instanton approximation41 (Shuryak, 1983). The main idea is that if xϪy is small compared to the typical instanton separation R, we expect that the contribution from the instanton IϭI closest to the points x and y * will dominate over all others. For the propagator in the zero-mode zone, this implies

1 † S͑x,y͒ϭ͚ ␺I͑x͒ ␺J͑y͒ IJ ͩTϩimͪ IJ † 1 ␺I ͑x͒␺I ͑y͒ † * * FIG. 21. Rho-meson correlation functions. The various curves Ӎ␺I ͑x͒ ␺I ͑y͒Ӎ , * ͩTϩimͪ m* and data sets are labeled as in Fig. 19. The dashed squares I I * * * (265) show the noninteracting part of the rho meson correlator in the interacting ensemble. where we have approximated the diagonal matrix ele- Ϫ1 Ϫ1 Ϫ1 ment by its average, (Tϩim)I I ӍN ͚I(Tϩim)II , showing an enhancement only at intermediate distances * * due to mixing with the ␩. and introduced the effective mass m* defined in Sec. Theoretical information about the short-distance be- IV.E, (m*)Ϫ1ϭNϪ1͚␭Ϫ1. In the following we shall use havior of the correlation functions comes from the op- the mean-field estimate m*ϭ␲␳(2n/3)1/2. As a result, erator product expansion (Shifman et al., 1979). For the the propagator in the single-instanton approximation ␲ and ␳ channels, we have looks like the zero-mode propagator of a single instan- 2 ton, but for a particle with an effective mass m*. 11 ␣s͑x͒ ␲ ⌸OPE͑x͒ϭ⌸0 ͑x͒ 1ϩ Ϫ m͗q¯q͘x4 The ␲ and ␩Ј correlators receive zero-mode contribu- ␲ ␲ ͩ 3 ␲ 3 tions. In the single-instanton approximation, we find (Shuryak, 1983) 1 11␲3 2 4 ϩ ͗G ͘x ϩ ␣s͑x͒ 4 2 ץ SIA 6␳ 1 81 384 ⌸ ͑x͒ϭϮ d␳n͑␳͒ x2͒2͑ץ ␲,␩Ј ͵ ␲2 ͑m*͒2 ¯ 2 2 6 ϫ͗␺␺͘ log͑x ͒x ϩ••• , (263) 4␰2 ␰2 ␰ 1ϩ␰ ͪ ϫ ϩ log , (266) ͭ x4 ͩ1Ϫ␰2 2 1Ϫ␰ͪͮ 2 ␣s͑x͒ ␲ ⌸OPE͑x͒ϭ⌸0͑x͒ 1ϩ Ϫ m͗q¯q͘x4 2 2 2 2 ␳ ␳ ͩ ␲ 4 where ␰ ϭx /(x ϩ4␳ ). There is also a non-zero-mode contribution to these correlation functions. It was calcu- 1 7␲3 lated by Shuryak (1989), but numerically it is not very Ϫ ͗G2͘x4ϩ ␣ ͑x͒ 384 81 s important. We show the result [Eq. (266)] in Fig. 19. For simplic- ¯ 2 2 6 ity, we have chosen n(␳)ϭn0␦(␳Ϫ␳0) with the standard ϫ͗␺␺͘ log͑x ͒x ϩ¯ , (264) Ϫ4 ͪ parameters n0ϭ1fm and ␳0ϭ0.33 fm. The pion cor- relator is similar to the OPE prediction at short dis- where we have restricted ourselves to operators of di- tances, xՇ0.25 fm, but follows the phenomenological mension up to six and the leading-order perturbative corrections. We have also used the factorization hypoth- esis for the expectation value of four-quark operators. 41We should like to distinguish this method from the dilute- Note that, to this order, the nonstrange eta-prime, and gas approximation. In the dilute-gas approximation, we sys- pion correlation functions are identical. This result tematically expand the correlation functions in terms of the shows that QCD sum rules cannot account for the one- (two-, three-, etc.) instanton contribution. In the presence U(1) anomaly. A of light fermions (for NfϾ1), however, this method is useless The OPE predictions [Eqs. (263) and (264)] are also because there is no zero-mode contribution to chirality- shown in Figs. 19–21. The leading quark and gluon violating operators from any finite number of instantons.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 379 result out to larger distances, xӍ0.4 fm. In particular, Note that the value of m͗q¯q͘ is ‘‘anomalously’’ large in we find that even at very short distances, xӍ0.3 fm, the the dilute-gas limit. This means that the contribution regular contributions to the correlator coming from in- from dimension-four operators is attractive, in contra- stanton zero modes are larger than the singular contri- diction with the OPE prediction based on the canonical butions included in the OPE. The fact that nonperturba- values of the condensates. tive corrections not accounted for in the OPE are An interesting observation is the fact that Eq. (269) is particularly large in spin-zero channels (␲, ␩Ј, ␴, and the only singular term in the dilute-gas approximation scalar glueballs) was first emphasized by Novikov et al. correlation function. In fact, the OPE of any mesonic (1981). For the ␩Ј, the instanton contribution is strongly correlator in any self-dual field contains only dimension- repulsive. This means that, in contrast to the OPE, the four operators (Dubovikov and Smilga, 1981). This single-instanton approximation at least qualitatively ac- means that for all higher-order operators, either the Wil- counts for the U(1)A anomaly. The single-instanton ap- son coefficient vanishes [as it does, for example, for the proximation was extended to the full pseudoscalar nonet abc a b c triple gluon condensate ͗f G␮␯G␯␳G␳␮͘] or the matrix by Shuryak (1983). It was shown that the substitution elements of various operators of the same dimension 43 m* m*ϩms gives a good description of SU(3) flavor cancel each other. This is a very remarkable result be- → symmetry breaking and the ␲,K,␩ correlation functions. cause it helps to explain the success of QCD sum rules In the ␳-meson correlator, zero modes cannot contrib- based on the OPE in many channels. In the instanton ute since the chiralities do not match. Nonvanishing con- model, the gluon fields are very inhomogeneous, so one tributions come from the non-zero-mode propagator would expect the OPE to fail for xϾ␳. The Dubovikov- (247) and from interference between the zero-mode part Smilga result shows that quarks can propagate through and the leading mass correction in Eq. (249): very strong gauge fields (as long as they are self-dual) SIA nz nz without suffering strong interactions. ⌸␳ ͑x,y͒ϭTr͓␥␮S ͑x,y͒␥␮S ͑y,x͔͒

† ϩ2Tr͓␥␮␺0͑x͒␺0͑y͒␥␮⌬͑y,x͔͒. (267) 3. The random-phase approximation The latter term survives even in the chiral limit because The single-instanton approximation clearly improves m the factor in the mass correction is cancelled by the on the short-distance behavior of the ␲,␩Ј correlation 1/m from the zero mode. Also note that the result cor- functions as compared to the OPE. However, in order to responds to the standard dilute-gas approximation, so describe a true pion bound state one has to resum the true multi-instanton effects are not included. After aver- 42 attractive interaction generated by the ’t Hooft vertex. aging over the instanton coordinates, we find (Andrei This is most easily accomplished using the random- and Gross, 1978) phase approximation (RPA), which corresponds to iter- ating the average ’t Hooft vertex in the s channel (see ץ ␳4 12 ⌸SIA͑x͒ϭ⌸0ϩ d␳n͑␳͒ x2͒ Fig. 2). The solution of the Bethe-Salpeter equation can͑ץ ␳ ␳ ͵ ␲2 x2 be written as (Diakonov and Petrov, 1986; Hutter, 1995; ␰ 1ϩ␰ Kacir, Prakash, and Zahed, 1996) ϫ log . (268) ͭ x2 1Ϫ␰ ͮ RPA MFA int ⌸␲ ͑x͒ϭ⌸␲ ͑x͒ϩ⌸␲ , (271) The result is also shown in Fig. 21. As in the OPE, the ⌸RPA x ϭ⌸MFA x . correlator is attractive at intermediate distances. The ␳ ͑ ͒ ␳ ͑ ͒ MFA correlation function does not go down at larger dis- Here, ⌸⌫ denotes the mean-field (noninteracting) tances, since the dilute-gas approximation does not ac- part of the correlation functions count for a dynamically generated mass. It is very in- MFA ¯ ¯ structive to compare the result to the OPE in greater ⌸⌫ ͑x͒ϭTr͓S͑x͒⌫S͑Ϫx͒⌫͔, (272) detail. Expanding Eq. (268), we get where S¯(x) is the mean-field propagator discussed in ␲2x4 Sec. VI.B.3. In the ␳-meson channel, the ’t Hooft vertex ⌸SIA͑x͒ϭ⌸0͑x͒ 1ϩ d␳n͑␳͒ . (269) ␳ ␳ ͩ 6 ͵ ͪ vanishes and the correlator is given by the mean-field contribution only. The interacting part of the ␲,␩Ј cor- This agrees exactly with the OPE, provided we use the relation functions is given by average values of the operators in the dilute-gas ap- Ϯ1 proximation, ⌸int ͑x͒ϭ d4qeiq•x⌫ ͑q͒ ⌫ ͑q͒, (273) ␲,␩Ј ͵ 5 ϯ 5 1 C5͑q͒ ͗G2͘ϭ32␲2 d␳n͑␳͒, and ͵ where the elementary loop function C5 and the vertex function ⌫5 are given by m͗q¯q͘ϭϪ͵ d␳n͑␳͒. (270) 43 ¯ 2 Here, we do not consider radiative corrections like ␣s͗␺␺͘ because we evaluate the OPE in a fixed (classical) background 42There is a mistake by an overall factor of 3/2 in the original field without taking into account radiative corrections to the work. background field.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 380 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

4 First of all, ␳4(N/V) is a useful expansion parameter V d p M1M2͑M1M2Ϫp1•p2͒ C ͑q͒ϭ4N , 5 cͩ Nͪ ͵ ͑2␲͒4 ͑M2ϩp2͒͑M2ϩp2͒ only if the instanton liquid is random and the role of 1 1 2 2 (274) correlations is small. As discussed in Sec. IV.B, if the density is very small, the instanton liquid is in a molecu- 4 d p ͱM1M2͑M1M2Ϫp1•p2͒ lar phase, while it is in a crystalline phase if the density is ⌫ ͑q͒ϭ4 . (275) 5 ͵ 4 2 2 2 2 large. Clearly, the RPA is expected to fail in both of ͑2␲͒ ͑m1ϩp1͒͑M2ϩp2͒ these limits. In general, the RPA corresponds to using Here, p ϭpϩq/2, p ϭpϪq/2, and M ϭM(p ) are 1 2 1,2 1,2 linearized equations for the fluctuations around a mean- the momentum-dependent effective quark masses. We field solution. In our case, low-lying meson states are have already stressed that in the long-wavelength limit, collective fluctuations of the chiral order parameter. For the effective interaction between quarks is of the Nambu–Jona-Lasinio type. Indeed, the pion correlator isovector scalar mesons, the RPA is expected to be good in the random-phase approximation to the Nambu– because the scalar mean field (the condensate) is large Jona-Lasinio model is given by and the masses are light. For isosinglet mesons, the fluc- tuations are much larger, and the RPA is likely to be NJL J5͑q͒ less useful. ⌸ ͑x͒ϭ d4qeiq•x , (276) ␲ ͵ Ϫ 1 GJ5͑q͒ In the following we shall therefore discuss results where J (q) is the pseudoscalar loop function from numerical calculations of hadronic correlators in 5 the instanton liquid. These calculations go beyond the ⌳ 4 2 d p ͑M Ϫp1•p2͒ RPA in two ways: (i) the propagator includes genuine J q ϭ4N , (277) 5͑ ͒ c ͵ 4 2 2 2 2 ͑2␲͒ ͑M ϩp1͒͑M ϩp2͒ many-instanton effects and non-zero-mode contribu- tions; and (ii) the ensemble is determined using the full G is the four-fermion coupling constant and the (fermionic and bosonic) weight function, so it includes (momentum-independent) constituent mass M is the so- lution of the Nambu–Jona-Lasinio gap equation. The correlations among instantons. In addition, we shall also loop function (277) is divergent and is regularized using consider baryonic correlators and three-point functions a cutoff ⌳. that are difficult to handle in the RPA. Clearly, the RPA correlation functions (273) and We discuss correlation function in three different en- (276) are very similar; the only difference is that in the sembles, the random, the quenched, and the fully inter- instanton liquid both the quark mass and the vertex are acting instanton ensembles. In the random model, the momentum dependent. The momentum dependence of underlying ensemble is the same as in the mean-field the vertex ensures that no regulator is required. The approximation, only the propagator is more sophisti- loop integral is automatically cut off at ⌳ϳ␳Ϫ1. We also cated. In the quenched approximation, the ensemble in- note that the momentum dependence of the effective cludes correlations due to the bosonic action, while the interaction at the mean-field level is very simple; the fully interacting ensemble also includes correlations in- vertex function is completely separable. duced by the fermion determinant. In order to check the The mean-field correlation functions are shown in dependence of the results on the instanton interaction, Figs. 19–21. The coordinate-space correlators are easily we study correlation functions in two different, un- determined by Fourier-transforming the result (273). quenched ensembles, one based on the streamline inter- We also show the Nambu–Jona-Lasinio result for the action (with a short-range core) and one based on the pion correlation function. In this case, the correlation ratio ansatz interaction. The bulk parameters of these function at short distances is not very meaningful, since ensembles are compared in Table II. We note that the the cutoff ⌳ eliminates all contributions to the spectral ratio ansatz ensemble is denser than the streamline en- function above that scale. semble. The pion correlator nicely reproduces the phenom- We are interested not only in the behavior of the cor- enological result. The ␩Ј correlation function has the relation functions, but also in numerical results for the correct tendency at short distances but becomes un- ground-state masses and coupling constants. For this physical for xϾ0.5 fm. The result in the ␳-meson chan- purpose we have fitted the correlators using a simple nel is also close to phenomenology, despite the fact that ‘‘pole-plus-continuum’’ model for the spectral functions. it corresponds to two unbound constituent quarks. In the case of the pion, this leads to the following pa- rametrization of the correlation function: 3 ϱ ⌸ x͒ϭ␭2 D m ,x͒ϩ dssD ͱs,x͒, (278) ␲͑ ␲ ͑ ␲ 2 ͵ ͑ 4. The interacting instanton liquid 8␲ s0

What is the quality and the range of validity of the where ␭␲ is the pion coupling constant defined in Table random-phase approximation? The RPA is usually mo- III and s0 is the continuum threshold. Physically, s0 tivated by the large- Nc approximation, and the dimen- roughly represents the position of the first excited state. sionless parameter that controls the expansion is Resolving higher resonances requires high-quality data 4 ␳ (N/V)/(4Nc). This parameter is indeed very small, and more sophisticated techniques. The model spectral but in practice there are additional parameters that de- function used here is quite popular in connection with termine the size of corrections to the RPA. QCD sum rules. It provides a surprisingly good descrip-

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 381

TABLE II. Bulk parameters density nϭN/V, average size ␳¯, diluteness ␳¯4(N/V), and quark con- densate ͗q¯q͘ in the different instanton ensembles. RILMϭrandom instanton liquid model. We also give the value of the Pauli-Vilars scale parameter ⌳4 that corresponds to our choice of units, n ϵ1fmϪ4.

Unquenched Quenched RILM Ratio ansatz (unquenched)

N/V 0.174⌳4 0.303⌳4 1.0 fm4 0.659⌳4 ␳¯ 0.64⌳Ϫ1 0.58⌳Ϫ1 0.33 fm 0.66⌳Ϫ1 (0.42 fm) (0.43 fm) (0.59 fm) ␳¯4(N/V) 0.029 0.034 0.012 0.125 ͗q¯q͘ 0.359⌳3 0.825⌳3 (264 MeV)3 0.882⌳3 (219 MeV)3 (253 MeV)3 (213 MeV)3 ⌳ 306 MeV 270 MeV - 222 MeV tion of all measured correlation functions, not only in and all fall somewhat short of the phenomenological re- the instanton model but also on the lattice (Chu et al., sult at intermediate distances xӍ1 fm. We have deter- 1993b; Leinweber, 1995a). mined the ␳-meson mass and coupling constant from a Correlation functions in the different instanton en- fit similar to Eq. (278). The results are given in Table IV. sembles were calculated by Shuryak and Verbaarschot The ␳-meson mass is somewhat too heavy in the random (1993a), Scha¨fer et al. (1994), and Scha¨fer and Shuryak and quenched ensembles but in good agreement with (1996a), to whose articles we refer the reader for more the experimental value m␳ϭ770 MeV in the un- details. The results are shown in Figs. 19–21 and sum- quenched ensemble. marized in Table IV. The pion correlation functions in Since there are no interactions in the ␳-meson channel the different ensembles are qualitatively very similar. to first order in the instanton density, it is important to The differences are mostly due to different values of the study whether the instanton liquid provides any signifi- quark condensate (and the physical quark mass) in the cant binding. In the instanton model, there is no confine- different ensembles. Using the Gell-Mann, Oaks, Ren- ment, and m␳ is close to the two- (constituent) quark ner relation, one can extrapolate the pion mass to the threshold. In QCD, the ␳ meson is also not a true bound physical value of the quark masses (see Table IV). The state but a resonance in the 2␲ continuum. To deter- results are consistent with the experimental value in the mine whether the continuum contribution in the instan- streamline ensemble (both quenched and unquenched) ton liquid is predominantly from two-pion or two-quark but clearly too small in the ratio ansatz ensemble. This is states would require the determination of the corre- a reflection of the fact that the ratio ansatz ensemble is sponding three-point functions, which has not yet been not sufficiently dilute. done. Instead, we have compared the full correlation In Fig. 21 we also show the results in the ␳ channel. function with the noninteracting (mean-field) correlator The ␳-meson correlator is not affected by instanton zero (272), where we use the average (constituent-quark) modes to first order in the instanton density. The results propagator determined in the same ensemble (see Fig. in the different ensembles are fairly similar to each other 21). This comparison provides a measure of the strength

TABLE III. Definition of various currents and hadronic matrix elements referred to in this work.

Channel Current Matrix element Experimental value

a a a b ab 3 ␲ j␲ϭq¯␥5␶ q ͗0͉j␲͉␲ ͘ϭ␦ ␭␲ ␭␲Ӎ(480 MeV) a a ␶ a b ab j ϭq¯␥ ␥ q ͗0͉j ͉␲ ͘ϭ␦ q␮f␲ f␲ϭ93 MeV ␮5 ␮ 5 2 ␮5 a a a b ab ␦ j␦ϭq¯␶ q ͗0͉j␦͉␦ ͘ϭ␦ ␭␦ ␴ j␴ϭq¯q ͗0͉j␴͉␴͘ϭ␭␴ ␩ j ϭq¯␥ q 0 j ␩ ϭ␭ ns ␩ns 5 ͗ ͉ ␩ns͉ ns͘ ␩ns ␶a m2 a ¯ a b ab ␳ ␳ j␮ϭq␥␮ q ͗0͉j␮͉␳ ͘ϭ␦ ⑀␮ g␳ϭ5.3 2 g␳ a 2 a ␶ ma j ϭq¯␥ ␥ q a b ab 1 a1 ␮5 ␮ 5 ͗0͉j ͉a ͘ϭ␦ ⑀ ga ϭ9.1 2 ␮5 1 ␮ g 1 a1 abc a b c N N ␩1ϭ⑀ (u C␥␮u )␥5␥␮d ͗0͉␩1͉N(p,s)͘ϭ␭1 u(p,s) abc a b c N N ␩2ϭ⑀ (u C␴␮␯u )␥5␴␮␯d ͗0͉␩2͉N(p,s)͘ϭ␭2 u(p,s) abc a b c ⌬ ⌬ ␩␮ϭ⑀ (u C␥␮u )u ͗0͉␩␮͉N(p,s)͘ϭ␭ u␮(p,s)

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 382 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

TABLE IV. Meson parameters in the different instanton en- global properties of the instanton ensemble, in particu- sembles. RILMϭrandom instanton liquid model. All quanti- lar its diluteness. Good phenomenology demands ␳¯4n ties are given in units of GeV. The current quark mass is mu Ӎ0.03, as originally suggested by Shuryak (1982c). The ϭmdϭ0.1⌳. Except for the pion mass, no attempt has been properties of the ␳ meson are essentially independent of made to extrapolate the parameters to physical values of the the diluteness but show sensitivity to IA correlations. quark mass. These correlations become crucial in the ␩Ј channel. Ratio ansatz Finally, we compare the correlation functions in the Unquenched Quenched RILM (unquenched) instanton liquid to lattice measurements reported by Chu et al. (1993a, 1993b; see Sec. VI.E). These correla- m␲ 0.265 0.268 0.284 0.128 tion functions were measured in quenched QCD, so they m␲ (extr.) 0.117 0.126 0.155 0.067 should be compared to the random or quenched instan- ␭␲ 0.214 0.268 0.369 0.156 ton ensembles. The results agree very well in the pion

f␲ 0.071 0.091 0.091 0.183 channel, while the lattice correlation function in the rho- meson channel is somewhat more attractive than the m␳ 0.795 0.951 1.000 0.654 correlator in the instanton liquid. g␳ 6.491 6.006 6.130 5.827 m 1.265 1.479 1.353 1.624 a1 5. Other mesonic correlation functions g 7.582 6.908 7.816 6.668 a1 m␴ 0.579 0.631 0.865 0.450 After discussing the ␲,␳,␩Ј in some detail we com- ment only briefly on other correlation functions. The re- m␦ 2.049 3.353 4.032 1.110 m maining scalar states are the isoscalar ␴ and the isovec- ␩ns 1.570 3.195 3.683 0.520 tor ␦ (the f0 and a0 according to the notation of the Particle Data Group). The sigma correlator has a dis- connected contribution, which is proportional to ͗q¯q͘2 of the interaction. We observe that there is an attractive at large distances. In order to determine the lowest reso- interaction generated in the interacting liquid due to nance in this channel, the constant contribution has to correlated instanton/anti-instanton pairs [see the discus- be subtracted, which makes it difficult to obtain reliable sion in Sec. VII.B.2, in particular, Eq. (332)]. This is results. Nevertheless, we find that the instanton liquid consistent with the fact that the interaction is consider- favors a (presumably broad) resonance around 500–600 ably smaller in the random ensemble. In the random MeV. The isovector channel is in many ways similar to model, the strength of the interaction grows as the en- the ␩Ј. In the random ensemble, the interaction is too semble becomes more dense. However, the interaction repulsive, and the correlator becomes unphysical. This in the full ensemble is significantly larger than in the problem is solved in the interacting ensemble, but the ␦ random model at the same diluteness. Therefore most of is still very heavy, m␦Ͼ1 GeV. the interaction is due to dynamically generated pairs. The remaining nonstrange vectors are the a1 , ␻, and The situation is drastically different in the ␩Ј channel. f1 . The a1 mixes with the pion, which allows a determi- Among the ϳ40 correlation functions calculated in the nation of the pion decay constant f␲ (as does a direct random ensemble, only the ␩Ј and the isovector-scalar ␦ measurement of the ␲Ϫa1 mixing correlator). In the in- discussed in the next section, are completely unaccept- stanton liquid, disconnected contributions in the vector able. The correlation function decreases very rapidly channels are small. This is consistent with the fact that and becomes negative at xϳ0.4 fm. This behavior is in- the ␳ and the ␻, as well as the a1 and the f1 , are almost compatible with the positivity of the spectral function. degenerate. The interaction in the random ensemble is too repulsive, Finally, we can also include strange quarks. SU(3) fla- and the model ‘‘overexplains’’ the U(1)A anomaly. vor breaking in the ’t Hooft interaction nicely accounts The results in the unquenched ensembles (closed and for the masses of the K and the ␩. More difficult is a open points) significantly improve the situation. This is correct description of ␩-␩Ј mixing, which can only be related to dynamic correlations between instantons and achieved in the full ensemble. The random ensemble anti-instantons (topological charge screening). The also has a problem with the mass splittings among the single-instanton contribution is repulsive, but the contri- vectors ␳, K*, and ␾ (Shuryak and Verbaarschot, bution from pairs is attractive (Scha¨fer et al., 1995). Only 1993a). This is related to the fact that flavor symmetry if correlations among instantons and anti-instantons are breaking in the random ensemble is so strong that the sufficiently strong will the correlators be prevented from strange and nonstrange constituent-quark masses are al- becoming negative. Quantitatively, the ␦ and ␩ns masses most degenerate. This problem is improved (but not in the streamline ensemble are still heavier than their fully solved) in the interacting ensemble. experimental values. In the ratio ansatz, on the other hand, the correlation functions even show an enhance- D. Baryonic correlation functions ment at distances on the order of 1 fm, and the fitted masses are too light. This shows that the ␩Ј channel is After discussing quark-antiquark systems in the last very sensitive to the strength of correlations among in- section, we now proceed to three-quark (baryon) chan- stantons. nels. As emphasized by Shuryak and Rosner (1989), the In summary, pion properties are mostly sensitive to existence of a strongly attractive interaction in the pseu-

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 383

TABLE V. Definition of nucleon and delta correlation functions.

Correlator Definition Correlator Definition N ¯ ⌬ ¯ ⌸1 (x) ͗tr„␩1(x)␩1(0)…͘ ⌸1 (x) ͗tr„␩␮(x)␩␮(0)…͘ N ˆ ¯ ⌬ ˆ ¯ ⌸2 (x) ͗tr„␥•x␩1(x)␩1(0)…͘ ⌸2 (x) ͗tr„␥•x␩␮(x)␩␮(0)…͘ N ¯ ⌬ ˆ ˆ ¯ ⌸3 (x) ͗tr„␩2(x)␩2(0)…͘ ⌸3 (x) ͗tr„x␮x␯␩␮(x)␩␯(0)…͘ N ˆ ¯ ⌬ ˆ ˆ ˆ ¯ ⌸4 (x) ͗tr„␥•x␩2(x)␩2(0)…͘ ⌸4 (x) ͗tr„␥•xx␮x␯␩␮(x)␩␯(0)…͘ N ¯ ⌸5 (x) ͗tr„␩1(x)␩2(0)…͘ N ˆ ¯ ⌸6 (x) ͗tr„␥•x␩1(x)␩2(0)…͘ doscalar quark-antiquark (pion) channel also implies an dominant at short distances. The scalar components of attractive interaction in the scalar quark-quark (di- the diagonal correlators, as well as the off-diagonal cor- quark) channel. This interaction is phenomenologically relation functions, are sensitive to chiral symmetry very desirable because it explains not only why the spin- breaking, and the OPE starts at order ͗q¯q͘ or higher. 1/2 nucleon is lighter than the spin-3/2 delta but also why Single-instanton corrections to the correlation functions lambda is lighter than sigma. were calculated by Dorokhov and Kochelev (1990), and Forkel and Banerjee (1993).44 Instantons introduce ad- 1. Nucleon correlation functions ditional, regular contributions in the scalar channel and violate the factorization assumption for the four-quark The proton current can be constructed by coupling a condensates. As in the pion case, both of these effects d quark to a uu diquark. The diquark has the structure increase the attraction already seen in the OPE. ⑀ u C⌫u , which requires that the matrix C⌫ be sym- N abc b c The correlation function ⌸2 in the interacting en- metric. This condition is satisfied for the V and T semble is shown in Fig. 22. There is a significant en- gamma matrix structures. The two possible currents hancement over the perturbative contribution which (with no derivatives and the minimum number of quark corresponds to a tightly bound nucleon state with a large fields) with positive parity and spin 1/2 are given by 45 coupling constant. Numerically, we find mNϭ1.019 (Ioffe, 1981) GeV (see Table VI). In the random ensemble, we have measured the nucleon mass at smaller quark masses and ␩ ϭ⑀ uaC␥ ub͒␥ ␥ dc, 1 abc͑ ␮ 5 ␮ found mNϭ0.960Ϯ0.30 GeV. The nucleon mass is fairly insensitive to the instanton ensemble. However, the a b c ␩2ϭ⑀abc͑u C␴␮␯u ͒␥5␴␮␯d . (279) strength of the correlation function depends on the in- stanton ensemble. This is reflected by the value of the It is sometimes useful to rewrite these currents in terms nucleon coupling constant, which is smaller in the inter- of scalar and pseudoscalar diquarks, acting model. Figure 22 also shows the nucleon correlation function a b c a b c ␩1,2ϭ͑2,4͕͒⑀abc͑u Cd ͒␥5u ϯ⑀abc͑u C␥5d ͒u ͖. measured in a quenched lattice simulation (Chu et al., (280) 1993b). The agreement with the instanton liquid results is quite impressive, especially given the fact that, before N Nucleon correlation functions are defined by ⌸␣␤(x) the lattice calculations were performed, there was no ϭ͗␩␣(0)␩¯␤(x)͘, where ␣,␤ are the Dirac indices of the phenomenological information on the value of the nucleon currents. In total, there are six different nucleon nucleon coupling constant and the behavior of the cor- correlators: the diagonal ␩1␩¯ 1 , diagonal ␩2␩¯ 2 , and off- relation function at intermediate and large distances. ¯ diagonal ␩1␩2 correlators, each contracted with either The fitted position of the threshold is E0Ӎ1.8 GeV, the identity or ␥•x (see Table V). Let us focus on the larger than the mass of the first nucleon resonance, the first two of these correlation functions. For more detail, Roper N*(1440), and above the ␲⌬ threshold E0 we refer the reader to Scha¨fer et al. (1994) and refer- ϭ1.37 GeV. This might indicate that the coupling of the ences therein. The OPE predicts (Belyaev and Ioffe, nucleon current to the Roper resonance is small. In the 1982) case of the ␲⌬ continuum, this can be checked directly using the phenomenologically known coupling con- N ⌸ ␲2 stants. The large value of the threshold energy also im- 1 ¯ 3 N0 ϭ ͉͗qq͉͘␶ , (281) plies that there is little strength in the (unphysical) ⌸2 12 three-quark continuum. The fact that the nucleon is ⌸N 1 ␲4 2 2 4 2 6 N0 ϭ1ϩ ͗G ͘␶ ϩ ͉͗q¯q͉͘ ␶ . (282) ⌸2 768 72 44The latter paper corrects a few mistakes in the original work by Dorokhov and Kochelev. The vector components of the diagonal correlators re- 45Note that this value corresponds to a relatively large cur- ceive perturbative quark-loop contributions, which are rent quark mass, mϭ30 MeV.-

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 384 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

TABLE VI. Nucleon and delta parameters in the different instanton ensembles. RILMϭrandom instanton liquid model. All quantities are given in units of GeV. The current quark mass is muϭmdϭ0.1⌳. Ratio ansatz Unquenched Quenched RILM (unquenched)

mN 1.019 1.013 1.040 0.983 1 ␭N 0.026 0.029 0.037 0.021 2 ␭N 0.061 0.074 0.093 0.048 m⌬ 1.428 1.628 1.584 1.372 ␭⌬ 0.027 0.040 0.036 0.026

⌬ a b c ␩␮ϭ⑀abc͑u C␥␮u ͒u . (283) However, the spin structure of the correlator ⌬ ⌬ ⌬ ⌸␮␯;␣␤(x)ϭ͗␩␮␣(0)␩¯␯␤(x)͘ is much richer. In general, there are ten independent tensor structures, but the ␮ ⌬ Rarita-Schwinger constraint, ␥ ␩␮ϭ0, reduces this number to four; see Table V. The OPE predicts ⌬ FIG. 22. Nucleon and delta correlation functions ⌸N and ⌸⌬ . ⌸ ␲2 2 2 1 ¯ 3 Curves labeled as in Fig. 19. ⌬0 ϭ4 ͉͗qq͉͘␶ , (284) ⌸2 12 deeply bound can also be demonstrated by comparing ⌸⌬ 25 1 ␲4 2 2 4 ¯ 2 6 the full nucleon correlation function with that of three ⌬0 ϭ1Ϫ ͗G ͘␶ ϩ6 ͉͗qq͉͘ ␶ , (285) ⌸2 18 768 72 noninteracting quarks (see Fig. 22). The full correlator is significantly larger than the noninteracting (mean-field) which implies that power corrections, in particular those result, indicating the presence of a strong, attractive in- due to the quark condensate, are much larger for the teraction. delta than for the nucleon. On the other hand, nonper- turbative effects due to instantons are much smaller. Some of this attraction is due to the scalar diquark The reason is that, while there are large, direct instanton content of the nucleon current. This raises the question contributions in the nucleon, there are none in the delta. whether the nucleon (in our model) is a strongly bound The delta correlation function in the instanton liquid diquark very loosely coupled to a third quark. In order is shown in Fig. 22. The result is qualitatively different to check this, we have decomposed the nucleon correla- from that for the nucleon channel: the correlator at in- tion function into quark and diquark components. Using termediate distance, xӍ1 fm, is significantly smaller and the mean-field approximation, that means treating the close to perturbation theory. This is in agreement with nucleon as a noninteracting quark-diquark system, for the results of the lattice calculation (Chu et al., 1993b). which we get the correlation function labeled (dq)(q) Note that, again, this is a quenched result, which should in Fig. 22. We observe that the quark-diquark model be compared to the predictions of the random instanton N explains some of the attraction seen in ⌸2 but falls model. short of the numerical results. This means that, while The mass of the delta resonance is too large in the diquarks may play some role in making the nucleon random model, but closer to experiment in the un- bound, there are substantial interactions in the quark- quenched ensemble. Note that just as for the nucleon, diquark system. Another hint of the qualitative role of part of this discrepancy is due to the value of the current diquarks is provided by the values of the nucleon cou- mass. Nevertheless, the delta-nucleon mass splitting in 1,2 the unquenched ensemble is m ϪmNϭ409 MeV, still pling constants ␭N . Using Eq. (280), we can translate ⌬ s,p too large as compared to the experimental value 297 these results into the coupling constants ␭N of nucleon currents built from scalar or pseudoscalar diquarks. We MeV. Just as for the ␳ meson, there is no interaction in the delta channel to first order in the instanton density. find that the coupling to the scalar diquark current ␩s ϭ⑀ (uaC␥ db)uc is an order of magnitude larger than However, if we compare the correlation function with abc 5 the mean-field approximation based on the full propaga- the coupling to the pseudoscalar current ␩p a b c tor (see Fig. 22), we find evidence for substantial attrac- ϭ⑀abc(u Cd )␥5u . This is in agreement with the idea that the scalar diquark channel is very attractive and tion between the quarks. Again, more detailed checks, that these configurations play an important role in the for example, concerning the coupling to the ␲N con- nucleon wave function. tinuum, are necessary.

2. Delta correlation functions E. Correlation functions on the lattice In the case of the delta resonance, there exists only The study of hadronic (point-to-point) correlation one independent current, given by (for the ⌬ϩϩ) functions on the lattice was pioneered by the MIT group

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 385

(Chu et al., 1993a, 1993b), who measured correlation functions of the ␲, ␦, ␳, a1 , N, and ⌬ in quenched QCD. The correlation functions were calculated on a 163ϫ24 lattice at 6/g2ϭ5.7, corresponding to a lattice spacing of aӍ0.17 fm. A more detailed investigation of baryonic correlation functions on the lattice was subsequently carried out by Leinweber (1995a, 1995b). We have al- ready shown some of the results of the MIT group in Figs. 19–22. The correlators were measured for dis- tances up to ϳ1.5 fm. Using the parametrization intro- duced above, they extracted ground-state masses and coupling constants and found good agreement with phe- nomenological results. What is even more important, they found the full correlation functions to agree with the predictions of the instanton liquid, even in channels (like the nucleon and delta) where no phenomenological information was available. In order to check this result in greater detail, they also studied the behavior of the correlation functions under FIG. 23. Behavior of pion (P) and nucleon (N) correlation cooling (Chu et al., 1994). The cooling procedure was functions under cooling, from Chu et al., 1994: Left panels, monitored by studying a number of gluonic observables, results in the original ensemble; center panels, after 25 cooling like the total action, the topological charge, and the Wil- sweeps; right panels, after 50 cooling sweeps. The solid lines son loop. From these observables, the authors concluded show fits to the data based on a pole-plus-continuum model for that the configurations were dominated by interacting the spectral function. The dotted and dashed lines show the instantons after ϳ25 cooling sweeps. Instanton/anti- individual contributions from the pole and the continuum instanton pairs were continually lost during cooling, and parts. after ϳ50 sweeps the topological charge fluctuations were consistent with a dilute gas. The characteristics of cantly heavier, m2ϩϩ /m0ϩϩӍ1.4, and the pseudoscalar the instanton liquid have already been discussed in Sec. is heavier still, m0Ϫϩ /m0ϩϩϭ1.5– 1.8 (Bali et al., 1993). III.C.2. After 50 sweeps the action was reduced by a (iii) The scalar glueball is much smaller than other glue- factor ϳ300, while the string tension (measured from balls. The size of the scalar is r0ϩϩӍ0.2 fm, while r2ϩϩ 7ϫ4 Wilson loops) had dropped by a factor of 6. Ӎ0.8 fm (de Forcrand and Liu, 1992). For comparison, a The behavior of the pion and nucleon correlation similar measurement for the ␲ and ␳ mesons gives 0.32 functions under cooling is shown in Fig. 23. The behav- fm and 0.45 fm, indicating that spin-dependent forces ior of the ␳ and ⌬ correlators is quite similar. During the are stronger between gluons than between quarks. cooling process, the scale was readjusted by keeping the Gluonic currents with the quantum numbers of the lowest glueball states are the field strength squared (S nucleon mass fixed. This introduced only a small uncer- ϭ0ϩϩ), the topological charge density (Pϭ0Ϫϩ), and tainty; the change in scale was ϳ16%. We observe that the energy-momentum tensors (Tϭ2ϩϩ): the correlation functions are stable under cooling. They agree almost within error bars. This can also be seen 1 j ϭ͑Ga ͒2, j ϭ ⑀ Ga Ga , from the extracted masses and coupling constants. While S ␮␯ P 2 ␮␯␳␴ ␮␯ ␳␴ mN and m␲ are stable by definition, m␳ and g␳ change 1 a a a by less than 2%, ␭␲ by 7%, and ␭N by 1%. Only the 2 jTϭ ͑G␮␯͒ ϪG0␣G0␣ . (286) delta mass is too small after cooling; it changes by 27%. 4 The short-distance behavior of the corresponding corre- F. Gluonic correlation functions lation functions is determined by the OPE (Novikov et al., 1980): to give One of the most interesting problems in hadronic ␲2 ⌸ ͑x͒ϭ⌸0 1Ϯ ͗fabcGa Gb Gc ͘x6ϩ spectroscopy is whether one can identify glueballs, S,P S,Pͩ 192g2 ␮␯ ␯␤ ␤␮ ¯ͪ bound states of pure glue, among the spectrum of ob- served hadrons. This question has two aspects. In pure (287) 2 glue theory, stable glueball states exist, and they have 25␲ ⌸ ͑x͒ϭ⌸0 1ϩ ͗2 Ϫ ͘log͑x2͒x8ϩ , been studied for a number of years in lattice simulations. T Tͩ 9216g2 O1 O2 ¯ͪ In full QCD, glueballs mix with quark states, making it (288) difficult to identify glueball candidates unambiguously. where we have defined the operators 1 Even in pure gauge theory, lattice simulations still re- abc b c 2 abc b c 2 O ϭ(f G␮␣G␯␣) , and 2ϭ(f G␮␯G␣␤) , and the quire large numerical efforts. Nevertheless, a few results free correlation functionsO are given by appear to be firmly established (Weingarten, 1994): (i) 4 4 The lightest glueball is the scalar 0ϩϩ, with a mass in the 384g 24g ⌸ ͑x͒ϭ͑Ϯ͒ , ⌸ ͑x͒ϭ . (289) 1.5–1.8 GeV range. (ii) The tensor glueball is signifi- S,P ␲4x8 T ␲4x8

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 386 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

Power corrections in the glueball channels are remark- ably small. The leading-order power correction 2 4 46 O(͗G␮␯͘/x ) vanishes, while radiative corrections of 2 2 4 the form ␣s log(x )͗G␮␯͘/x [not included in Eq. (287)], or higher-order power corrections like abc a b c 2 ͗f G␮␯G␯␳G␳␮͘/x , are very small. On the other hand, there is an important low-energy theorem that controls the large-distance behavior of the scalar correlation function (Novikov et al., 1979), 128␲2 d4x⌸ ͑x͒ϭ ͗G2͘, (290) ͵ S b where b denotes the first coefficient of the beta function. In order to make the integral well defined, we have to subtract the constant term ϳ͗G2͘2 as well as singular (perturbative) contributions to the correlation function. Analogously, the integral over the pseudoscalar correla- tion functions is given by the topological suscepti- 4 bility ͐d x⌸P(x)ϭ␹top . In pure gauge theory ␹top 2 2 Ӎ(32␲ )͗G ͘, while in unquenched QCD ␹topϭO(m) (see Sec. V.E). These low-energy theorems indicate the presence of rather large nonperturbative corrections in FIG. 24. Scalar and pseudoscalar glueball correlation func- the scalar glueball channels. This can be seen as follows: tions. Curves labeled as in Fig. 19. We can incorporate the low-energy theorem into the sum rules by using a subtracted dispersion relation sor in the self-dual field of an instanton is zero. Note ⌸͑Q2͒Ϫ⌸͑0͒ 1 Im ⌸͑s͒ that the perturbative contributions in the scalar and ϭ ds . (291) Q2 ␲ ͵ s͑sϩQ2͒ pseudoscalar channels have opposite sign, while the clas- sical contribution has the same sign. We therefore find, In this case, the subtraction constant acts like a power to first order in the instanton density, the three scenarios correction. In practice, however, the subtraction con- discussed in Sec. VI.C: attraction in the scalar channel, stant totally dominates ordinary power corrections. For repulsion in the pseudoscalar, and no effect in the tensor example, using pole dominance, the scalar glueball cou- channel. The single-instanton prediction is compared ϩϩ pling ␭Sϭ͗0͉jS͉0 ͘ is completely determined by the with the OPE in Fig. 24. We can clearly see that classical 2 2 2 2 subtraction, ␭S/mSӍ(128␲ /b)͗G ͘. fields are much more important than the power correc- For this reason, we expect instantons to give a large tions. contribution to scalar glueball correlation functions. Ex- Quantum corrections to this result can be calculated panding the gluon operators around the classical fields, from the second term in Eq. (292), using the gluon we have propagator in the instanton field (L. S. Brown et al., a 1978; Levine and Yaffe, 1979). The singular contribu- ⌸ ͑x,y͒ϭ͗0͉G2cl͑x͒G2cl͑y͉͒0͘ϩ͗0͉G ,cl͑x͒ S ␮␯ tions correspond to the OPE in the instanton field. x y ab b,cl There is an analog of the Dubovikov-Smilga result for ϫ͓D␮D␣D␯␤͑x,y͔͒ G␣␤ ͑y͉͒0͘ϩ¯ , glueball correlators: In a general self-dual background (292) field, there are no power corrections to the tensor cor- ab where D␮␯(x,y) is the gluon propagator in the classical relator (Novikov et al., 1980). This is consistent with the background field. If we insert the classical field of an result (288), since the combination ͗2 1Ϫ 2͘ vanishes instanton, we find (Novikov et al., 1979; Shuryak, 1982c; in a self-dual field. Moreover, the sumO of theO scalar and Scha¨fer and Shuryak, 1995a) pseudoscalar glueball correlators does not receive any power corrections, while the difference does, starting at 3ץ 8192␲2 SIA O(G3). ⌸S,P ͑x͒ϭ d␳n͑␳͒ 4 2 3 -x ͒ Numerical calculations of glueball correlators in dif͑ץ ␳ ͵ ¨ ␰6 10Ϫ6␰2 3 1ϩ␰ ferent instanton ensembles were performed by Schafer ϫ 6 2 2 ϩ log , (293) and Shuryak (1995a). At short distances, the results ͭ x ͩ ͑1Ϫ␰ ͒ ␰ 1Ϫ␰ͪͮ were consistent with the single-instanton approximation. where ␰ is defined as in Eq. (266). There is no classical At larger distances, the scalar correlator was modified contribution in the tensor channel, since the stress ten- due to the presence of the gluon condensate. This means that (like the ␴ meson), the correlator has to be sub- tracted, and the determination of the mass is difficult. In 2 46There is a ͗G ͘␦4(x) contact term in the scalar glueball the pure gauge theory we find m0ϩϩӍ1.5 GeV and ␮␯ 3 correlators which, depending on the choice of sum rule, may ␭0ϩϩϭ16Ϯ2 GeV . While the mass is consistent with enter momentum space correlation functions. QCD sum-rule predictions, the coupling is much larger

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 387 than would be expected from calculations that do not enforce the low-energy theorem (Narison, 1984; Bagan and Steele, 1990). In the pseudoscalar channel the correlator is very re- pulsive, and there is no clear indication of a glueball state. In the full theory (with quarks) the correlator is modified due to topological charge screening. The non- perturbative correction changes sign and a light (on the glueball mass scale) state, the ␩Ј, appears. Nonpertur- bative corrections in the tensor channel are very small. Isolated instantons and anti-instantons have a vanishing energy-momentum tensor, so the result is entirely due to interactions. In Scha¨fer and Shuryak (1995a) we also measured glueball wave functions. The most important result is that the scalar glueball is indeed small, r0ϩϩϭ0.2 fm, while the tensor is much bigger, r2ϩϩϭ0.6 fm. The size of the scalar is determined by the size of an instanton, whereas in the case of the tensor the scale is set by the average distance between instantons. This number is comparable to the confinement scale, so the tensor wave function is probably not very reliable. On the other hand, the scalar is much smaller than the confinement scale, so the wave function of the 0ϩϩ glueball may pro- vide an indication of the importance of instantons in FIG. 25. Hadronic wave functions of the pion, rho meson, pro- pure gauge theory. ton, and delta resonances in the random instanton ensemble.

G. Hadronic structure and n-point correlators This can be seen from Fig. 25, where we show results obtained in the random instanton ensemble (Scha¨fer So far, we have focused on two-point correlation in and Shuryak, 1995a). In particular, we observe that the the instanton liquid. However, in order to study had- pion and the proton, as well as the ␳ and the ⌬, have ronic properties like decay widths, form, structure func- essentially the same size. In the case of the pion and the tions, etc., we have to calculate n-point correlators. proton, this is in agreement with lattice data reported by There is no systematic study of these objects in the lit- Chu et al. (1991). These authors also find that the ␳ me- erature. In the following, we discuss two exploratory at- son is significantly larger (they did not study the ⌬). tempts and point out a number of phenomenologically A more detailed comparison with the lattice data re- interesting questions. veals some of the limitations of the instanton model. Both of our examples are related to the question of Lattice wave functions are linear at the origin, not qua- hadronic sizes. Hadronic wave functions (or Bethe- dratic as in the instanton liquid. Presumably, this is due Salpeter amplitudes) are defined by three-point correla- to the lack of a perturbative Coulomb interaction be- tors of the type tween the quarks in the instanton model. Also, lattice xϩy wave functions decay faster at large distances, which ¯ i͐ A͑xЈ͒dxЈ ⌸␲͑x,y͒ϭ͗0͉T͑d͑x͒Pe x ␥5u͑xϩy͒ might be related to the absence of a string potential in ¯ the instanton liquid. ϫd͑0͒␥5u͑0͒)͉0͘ Let us now focus on another three-point function that x ϱ is more directly related to experiment. The pion electro- → ␺͑y͒eϪm␲x. (294) → magnetic form factor is determined by the correlation These wave functions are not directly accessible to ex- function periment, but they have been studied in a number of ϩ 3 Ϫ ⌫␮͑x,y͒ϭ͗j ͑Ϫx/2͒j ͑y͒j ͑x/2͒͘, (295) lattice gauge simulations, both at zero (Velikson and 5 ␮ 5 Ϯ Weingarten, 1985; Chu, Lissia, and Negele, 1991; Hecht where j5 are the pseudoscalar currents of the charged 3 and DeGrand, 1992; Gupta, Daniel, and Grandy, 1993) pion and j␮ is the third component of the isovector- and at finite temperature (Bernard et al., 1992; Schramm vector current. In this case, the correlator is completely and Chu, 1993). In the single-instanton approximation, determined by the triangle diagram. In the context of we find that light states (like the pion or the nucleon) QCD sum rules, the pion form factor is usually analyzed receive direct instanton contributions, so their size is using a three-point function built from axial-vector cur- controlled by the typical instanton size. Particles like the rents (because the pseudoscalar sum rule is known to be ␳ or the ⌬, on the other hand, are not sensitive to direct unreliable). In the single-instanton approximation, the instantons and are therefore less bound and larger in problem was recently studied by Forkel and Nielsen size. (1995), where it was shown that, when direct instantons

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 388 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

0 0 pectedly small, gAӍ0.35. The quark model suggests gA Ӎ1, a discrepancy which is known as the ‘‘proton spin crisis.’’ 0 Since gA is defined by a matrix element of the flavor singlet current, one might suspect that the problem is somehow related to the anomaly. In fact, if it were not for the anomaly, one could not transfer polarization 0 from quarks to gluons and gA could not evolve. All this suggests that instantons might be crucial in understand- ing the proton spin structure. While some attempts in this direction have been made (Forte and Shuryak, 1991; Dorokhov, Kochelev, and Zubov, 1993), a realistic cal- culation is still lacking. Another question concerns the flavor structure of the unpolarized quark sea in the nucleon. From measure- ments of the Gottfried sum rule, the quark sea is known to be flavor asymmetric; there are more d than u quarks in the proton sea. From a perturbative point of view this FIG. 26. The size of the pion, determined from the inverse is puzzling because gluons are flavor blind, so a radia- monopole mass in the pion electromagnetic form, as a function tively generated sea is flavor symmetric. However, as of the instanton size in the random instanton ensemble, from pointed out by Dorokhov and Kochelev (1993), quark Blotz and Shuryak, 1997. The horizontal lines show the experi- pairs produced by instantons are flavor asymmetric. For mental uncertainty. example, a valence u quark generates d¯d,¯ss sea, but no u¯u pairs. Since a proton has two valence u quarks, this gives the right sign of the observed asymmetry. are included (as in Sec. VI.C.2), the pion form factor can also be extracted from the pseudoscalar correlator (295). Using the standard instanton liquid parameters, these VII. INSTANTONS AT FINITE TEMPERATURE authors calculated the pion form factor for Q2 A. Introduction ϳ1 GeV, obtaining good agreement with experimental data. 1. Finite-temperature field theory and the caloron solution Recently, the three-point function (295) in the instan- ton liquid was also calculated numerically (Blotz and In the previous sections we have shown that the in- Shuryak, 1997). In this case, one can go to arbitrarily stanton liquid model provides a mechanism for chiral large distances and determine the pion form factor at symmetry breaking and describes a large number of small momentum. In this regime, the pion form factor hadronic correlation functions. Clearly, it is of interest 2 2 2 to generalize the model to finite temperature and/or has a simple monopole shape, F␲(Q )ϭM /(Q ϩM2), with a characteristic mass close to the rho-meson density. This will allow us to study the behavior of had- mass. However, at intermediate momenta, Q2ϳ1, the rons in matter, the mechanism of the chiral phase tran- pion form factor is less sensitive to the meson cloud and sition, and possible nonperturbative effects in the high- largely determined by the quark-antiquark interaction temperature/density phase. Extending the methods described in the last three sections to nonzero tempera- inside the pion. In the instanton model, the interaction is ture is fairly straightforward. In Euclidean space, finite dominated by single-instanton effects, and the range is temperature corresponds to periodic boundary condi- controlled by the instanton size. This can be seen from tions on the fields. Basically, all we have to do is replace Fig. 26, where we show the fitted monopole mass as a all the gauge potentials and fermionic wave functions by function of the instanton size. Clearly, the two are their TÞ0 periodic counterparts. Nevertheless, doing so strongly correlated, and the instanton model reproduces leads to a number of interesting and very nontrivial phe- the experimental value of the monopole mass if the nomena, which we describe in detail below. Extending mean instanton size is ␳Ӎ0.35 fm. the instanton model to finite density is more difficult. In Finally, let us mention a few experimental observables Euclidean space a finite chemical potential corresponds that might be interesting in connection with instanton to a complex weight in the functional integral, so apply- effects in hadronic structure. These are observables that ing standard methods from statistical mechanics is less are sensitive to the essential features of the ’t Hooft straightforward. While some work on the subject has interaction, the strong correlation between quark and been done (Carvalho, 1980; Chemtob, 1981; Shuryak, gluon polarization and flavor mixing. The first is the fla- 1982d; Abrikosov, 1983), many questions remain to be 0 vor singlet axial coupling constant of the nucleon gA , understood. which is a measure of the quark contribution to the Before we study the instanton liquid at TÞ0, we nucleon spin. This quantity can be determined in polar- should like to give a very brief introduction to finite- ized, deep-inelastic scattering and was found to be unex- temperature field theory in Euclidean space (Shuryak,

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 389

1980; Gross, Pisarski, and Yaffe, 1981; McLerran, 1986; Periodic instanton configurations can be constructed Kapusta, 1989). The basic object is the partition function by lining up zero-temperature instantons along the imaginary time direction with the period ␤. Using Zϭ eϪ␤H Tr͑ ͒, (296) ’t Hooft’s multi-instanton solution (A7), the explicit ex- where ␤ϭ1/T is the inverse temperature, H is the pression for the gauge field is given by Harrington and Hamiltonian, and the trace is performed over all physi- Shepard (1978) cal states. In Euclidean space, the partition function can a a Ϫ1 (␯⌸ ͑x͒, (303ץA ϭ␩¯ ⌸͑x͒ be written as a functional integral, ␮ ␮␯ where ␤ ¯ 3 2 Zϭ DA␮ D␺D␺ exp Ϫ d␶ d x , ␲␳ sinh͑2␲r/␤͒ ͵per ͵aper ͩ ͵0 ͵ Lͪ ⌸͑x͒ϭ1ϩ . (304) (297) ␤r cosh͑2␲r/␤͒Ϫcos͑2␲␶/␤͒ where the gauge fields and fermions are subject to Here, ␳ denotes the size of the instanton. The solution periodic/antiperiodic boundary conditions (303) is sometimes referred to as the caloron. It has to- pological charge Qϭ1 and action Sϭ8␲2/g2 indepen- A␮͑rជ,␤͒ϭA␮͑rជ,0͒, (298) dent of temperature.48 A caloron with QϭϪ1 can be ¯ a a ␺͑rជ,␤͒ϭϪ␺͑rជ,0͒, ␺¯͑rជ,␤͒ϭϪ␺¯͑rជ,0͒. (299) constructed by making the replacement ␩␮␯ ␩␮␯ .Of course, as T 0 the caloron field reduces to the→ field of The boundary conditions imply that the fields can be an instanton.→ In the high-temperature limit T␳ӷ1, how- i␻n␶ expanded in a Fourier series ␾(rជ,␶)ϭe ␾n(rជ), where ever, the field looks very different (Gross et al., 1981). In ␻nϭ2␲nT and ␻nϭ(2nϩ1)␲T are the Matsubara fre- this case, the caloron develops a core of size O(␤) quencies for boson and fermions. 2 where the fields are very strong, G␮␯ϳO(␤ ). In the As an example, it is instructive to consider the free intermediate regime, O(␤)ϽrϽO(␳2/␤), the caloron propagator of a massless fermion at finite temperature. looks like a (T-independent) dyon with unit electric and Summing over all Matsubara frequencies we get magnetic charges, i ͑Ϫ1͒n ˆ aˆ i a a r r (͚ 2 2 . (300) E ϭB Ӎ . (305ץ•␥ ST͑r,␶͒ϭ 2 4␲ n r ϩ͑␶Ϫn␤͒ i i r2 The sum can easily be performed, and in the spatial di- In the far region, rϾO(␳2/␤), the caloron resembles a a a 3 rection one finds three-dimensional dipole field, Ei ϭBi ϳO(1/r ). At finite temperature, tunneling between degenerate i␥ជ •rជ S ͑r,0͒ϭ z exp͑Ϫz͒ classical vacua is related to the anomaly in exactly the T 2r4 2␲ same way as it is at Tϭ0. This means that, during tun- ͑zϩ1͒ϩ͑zϪ1͒exp͑Ϫ2z͒ neling, the fermion vacuum is rearranged and the Dirac ϫ , (301) operator in the caloron field has a normalizable left- ͓1ϩexp͑Ϫ2z͔͒2 handed zero mode. This zero mode can be constructed where zϭ␲rT. This result shows that, at finite tempera- from the zero modes of the exact n-instanton solution ture, the propagation of massless fermions in the spatial (Grossman, 1977). The result is direction is exponentially suppressed. The screening mass mϭ␲T is the lowest Matsubara frequency for fer- 1 ⌽͑x͒ 1Ϫ␥5 (306) , ⑀ ␥ ץ␺aϭ ͱ⌸͑x͒ mions at finite temperature.47 The energy ␲T acts like a i ␮ͩ⌸͑x͒ͪͩ 2 ␮ͪ aj 2&␲␳ ij (chiral) mass term for spacelike propagation. For bosons, on the other hand, the wave functions are peri- where ⌽(x)ϭ(⌸(x)Ϫ1)cos(␲␶/␤)/cosh(␲r/␤). Note odic, the lowest Matsubara frequency is zero, and the that the zero-mode wave function also shows an expo- propagator is not screened. The propagator in the tem- nential decay exp(Ϫ␲rT) in the spatial direction, despite poral direction is given by the fact the eigenvalue of the Dirac operator is exactly zero. This will have important consequences for instan- 3 2 i␥4 y 1ϩcos ͑y͒ ton interactions at nonzero temperature. S ͑0,␶͒ϭ , (302) T 2␲2␶3 2 sin3͑y͒ with yϭ␲␶T. Clearly, there is no suppression factor for 2. Instantons at high temperature propagation in the temporal direction. At finite temperature, just as at Tϭ0, the instanton density is controlled by fluctuations around the classical 47Let us mention another explanation for this fact which does not use the Matsubara formalism. In the real-time formalism, 48 ST(r,0) is the spatial Fourier transform of ͓1/2Ϫnf(Ek)͔, The topological classification of smooth gauge fields at finite where nf is the Fermi-Dirac distribution. For small momenta, temperature is more complicated than at Tϭ0 (Gross et al., kϽT, the two terms cancel and there is no power-law decay. 1981). The situation simplifies in the absence of magnetic In other words, the 1/r3 behavior disappears because the con- charges. In this case topological charge is quantized just as it is tributions from real and virtual fermions cancel each other. at zero temperature.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 390 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

caloron solution. In the high-temperature plasma phase, can be split into two pieces, ␦ϭ␦1ϩ␦2 , where the gluoelectric fields in the caloron are Debye screened, 2 2 so we expect that instantons are strongly suppressed at ϪD „A͑␳͒… ϪD „A͑␳͒… ␦ ϭTr log ϪTr log , (312) ͪ 2ץϪ ͩ ͪ 2ץhigh temperature (Shuryak, 1978b). The perturbative 1 T ͩ Ϫ Debye mass in the quark-gluon plasma is (Shuryak, 1978a) 2 2 ϪD „A͑␳,T͒… ϪD „A͑␳͒… 2 2 2 ␦2ϭTrT log 2 ϪTrT log 2 . ͪ ץϪ ͩ ͪ ץmDϭ͑Nc/3ϩNf/6͒g T . (307) ͩ Ϫ Normal O(1) electric fields are screened at distances (313) 3 1/(gT), while the stronger O(1/g) nonperturbative Here TrT includes an integration over R ϫ͓0,␤͔, and fields of the instantons should be screened for sizes ␳ A(␳,T) and A(␳) are the gauge potentials of the cal- Ϫ1 ϾT . An explicit calculation of the quantum fluctua- oron and the instanton, respectively. The two terms ␦1 tions around the caloron was performed by Pisarski and and ␦2 correspond to the two terms in the exponent in Yaffe (1980). Their result is the semiclassical result (308). n͑␳,T͒ϭn͑␳,Tϭ0͒ It was shown by Shuryak (1982d) that the physical origin of the first term is scattering of particles in the 1 heat bath on the instanton field. The forward scattering ϫexp Ϫ ͑2N ϩN ͒͑␲␳T͒2ϪB͑␭͒ ͩ 3 c f ͪ amplitude T(p,p) of a scalar quark can be calculated using the standard Lehmann-Symanzik-Zimmermann with reduction formula Nc Nf B͑␭͒ϭ 1ϩ Ϫ ͩ 6 6 ͪ 2 2 4 4 ip•͑xϪy͒ ,y͒ץx⌬͑x,y͒ץTr„T͑p,p͒…ϭ ͵ d xd ye Tr͑ ␭2 0.15 ϫ Ϫlog 1ϩ ϩ (308) (314) 3 ͑1ϩ0.15␭Ϫ3/2͒8 ͫ ͩ ͪ ͬ where ⌬(x,y) is the scalar quark propagator introduced where ␭ϭ␲␳T. As expected, large instantons, ␳ӷ1/T, in Sec. VI.B.1. By subtracting the trace of the free are exponentially suppressed. This means that the in- propagator and going to the physical pole p2ϭ0, one stanton contribution to physical quantities like the en- gets a very simple answer, ergy density (or pressure, etc.) is of the order Tr T͑p,p͒ ϭϪ4␲2␳2. (315) 1/T d␳ „ … b 4 b ⑀͑T͒ϳ 5 ͑␳⌳͒ ϳT ͑⌳/T͒ . (309) Since the result is just a constant, there is no problem ͵0 ␳ with analytic continuation to Minkowski space. Integrat- At high temperature, this is small compared to the en- ing the result over a thermal distribution, we get 4 ergy density of an ideal gas, ⑀(T)SBϳT . It was emphasized by Shuryak and Velkovsky (1994) d3p 1 that, although the Pisarski-Yaffe result contains only ␦1ϭ Tr T͑p,p͒ ͵ ͑2␲͒3 2p͑exp͑p/T͒Ϫ1͒ „ … one dimensionless parameter ␭, its applicability is con- trolled by two separate conditions: 1 1 ϭϪ ͑␲␳T͒2ϭϪ ␭2. (316) ␳Ӷ1/⌳, Tӷ⌳. (310) 6 6 The first condition ensures the validity of the semiclas- The constants appearing in the result are easily inter- sical approximation, while the second justifies the per- preted: ␳2 comes from the scattering amplitude, while turbative treatment of the heat bath. In order to illus- the temperature dependence enters via the integral over trate this point, we should like to discuss the derivation the heat bath. Note also that the scattering amplitude of the semiclassical result [Eq. (308)] in somewhat has the same origin (and the same dependence on greater detail. Our first point is that the finite- Nc ,Nf) as the Debye mass. temperature correction to the instanton density can be Formally, the validity of this perturbative calculation split into two parts of different physical origin. For this requires that g(T)Ӷ1, but in QCD this criterion is sat- purpose, let us consider the determinant of a scalar field isfied only at extremely high temperatures. We would in the fundamental representation.49 The temperature- argue, however, that the accuracy of the calculation is dependent part of the one-loop effective action controlled by the same effects that determine the valid- ity of the perturbative result for the Debye mass. Avail- ϪD2 ϪD2 log det ϭlog det ϩ␦ (311) able lattice data (Irback et al., 1991) suggest that screen- ͯͪ 2ץϪ ͩ ͯͪ 2ץϪ ͩ T Tϭ0 ing sets in right after the phase transition, and that the perturbative prediction for the Debye mass works to about 10% accuracy for TϾ3TcӍ500 MeV. Near the 49The quark and gluon (non-zero-mode) determinants can be critical temperature one expects that the Debye mass reduced to the determinant of a scalar field in the fundamental becomes small; screening disappears together with the and adjoint representation (Gross et al., 1981). plasma. Below Tc the instanton density should be deter-

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 391

mined by the scattering of hadrons, not quarks and glu- 3. Instantons at low temperature ons, on the instanton. We shall return to this point in the The behavior of the instanton density at low tempera- next section. ture was studied by Shuryak and Velkovsky (1994). At The second term in the finite-temperature effective action [Eq. (313)] has a different physical origin. It is temperatures well below the phase transition the heat determined by quantum corrections to the colored cur- bath consists of weakly interacting pions. The interac- rent (Brown and Creamer, 1978), multiplied by the tion of the pions in the heat bath with instantons can be T-dependent variation of the instanton field, the differ- determined from the effective Lagrangian (111). For ence between the caloron and the instanton. For ␭ small, two flavors, this Lagrangian is a product of certain four- the correction is given by fermion operators and the semiclassical single-instanton density. In the last section, we argued that the semiclas- 1 2 3 sical instanton density is not modified at small tempera- ␦2ϭϪ ␭ ϩO͑␭ ͒. (317) 36 tures. The temperature dependence is then determined The result has the same sign and the same dependence by the T dependence of the vacuum expectation value of the four-fermion operators. For small T, the situation on the parameters as ␦1 , but a smaller coefficient. Thus finite-T effects not only lead to the appearance of the can be simplified even further because the wavelength of usual (perturbative) heat bath, they also modify the the pions in the heat bath is large, and the four-fermion strong O(1/g) classical gauge field of the instanton. In operators can be considered local. Euclidean space, we can think of this effect as arising The temperature dependence of the expectation value from the interaction of the instanton with its periodic of a general four-fermion operator ͗(q¯Aq)(q¯Bq)͘ can ‘‘mirror images’’ along the imaginary-time direction. be determined using soft-pion techniques (Gerber and 2 2 However, below Tc , gluon correlators are exponentially Leutwyler, 1989). To order T /f␲ the result is deter- suppressed since glueballs states are very heavy. There- mined by the matrix element of the operator between fore we expect that for TϽTc the instanton field is not one-pion states, integrated over a thermal pion distribu- modified by the boundary conditions. Up to effects of tion. The one-pion matrix element can be calculated us- the order O͓exp(ϪM/T)͔, where M is the mass of the ing the reduction formula and current-algebra commu- lowest glueball, there is no instanton suppression due to tators. These methods allow us to prove the general the change in the classical field below Tc . formula (Eletsky, 1993)

2 T a a ͗͑q¯Aq͒͑q¯Bq͒͘Tϭ͗͑q¯Aq͒͑q¯Bq͒͘0Ϫ 2 ͗͑q¯͕⌫5 ,͕⌫5 ,A͖͖q͒͑q¯Bq͒͘0 (318) 96f␲

2 2 T a a T a a Ϫ 2 ͗͑q¯Aq͒͑q¯͕⌫5 ,͕⌫5 ,B͖͖q͒͘0Ϫ 2 ͗͑q¯͕⌫5 ,A͖q͒͑q¯͕⌫5 ,B͖q͒͘0 , (319) 96f␲ 48f␲

where A,B are arbitrary flavor-spin-color matrices and 3 a a K ϭq¯ ␶iq q¯ ␶iq ϩ q¯ ␶i␭aq q¯ ␶i␭aq ⌫5ϭ␶ ␥5 . Using this result, we obtain the instanton 2 L L L L 32 L L L L density at low temperature

9 i a i a 2 2 Ϫ q¯ ␴ ␶ ␭ q q¯ ␴ ␶ ␭ q . (322) 4 1 T 128 L ␮␯ L L ␮␯ L n͑␳,T͒ϭn͑␳͒ ␲2␳3 ͗K ͘ 1Ϫ ͩ 3 ͪ ͫ 1 0 4 ͩ 6f2 ͪ ␲ 1 T2 Although the vacuum expectation values of these two Ϫ͗K ͘ 1ϩ , (320) operators are unknown, it is clear that (barring unex- 2 0 12 6f2 ͩ ␲ ͪͬ pected cancellations) the T dependence should be rather weak, most likely inside the range nϭn0„1 where we have defined the two operators (␶ are isospin 2 2 ϮT /(6f␲)…. Furthermore, if one estimates the expecta- matrices) tion values using the factorization assumption (166), the 2 2 T dependence cancels to order T /f␲ . 3 Summarizing the last two sections, we conclude that K ϭq¯ q q¯ q ϩ q¯ ␭aq q¯ ␭aq 1 L L L L 32 L L L L the instanton density is expected to be essentially con- stant below the phase transition, but exponentially sup- 9 pressed at large temperature. We shall explore the Ϫ q¯ ␴ ␭aq q¯ ␴ ␭aq , (321) 128 L ␮␯ L L ␮␯ L physical consequences of this result in the remainder of

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 392 T. Scha¨fer and E. V. Shuryak: Instantons in QCD this section. Numerical evidence for our conclusions that comes from lattice simulations will be presented in Sec. VII.D.

4. Instanton interaction at finite temperature In order to study an interacting system of instantons at finite temperature, we need to determine the bosonic and fermionic interaction between instantons at TÞ0. This subject was studied by Diakonov and Mirlin (1988) and later, in much greater detail, by Shuryak and Ver- baarschot (1991), whom we follow here. As for Tϭ0, the gluonic interactions between an instanton and an anti-instanton is defined as SintϭS͓AIA͔Ϫ2S0 , where we have to specify an ansatz AIA for the gauge potential of the IA configuration. The main difficulty at TÞ0is that Lorenz invariance is lost, and the interaction de- pends on more variables: the spatial and temporal sepa- ration r,␶ of the instantons, the sizes ␳I ,␳A , the relative orientation matrix U, and the temperature T. There are also more invariant structures that can be constructed from the relative orientation vector u␮ . In the case of color SU(2), we have

ˆ 2 ˆ 4 SIAϭs0ϩs1͑u•R͒ ϩs2͑u•R͒ 2 ˆ 2 2 ϩs3„u Ϫ͑u•R͒ Ϫu4… 2 ˆ 2 2 2 ϩs4„u Ϫ͑u•R͒ Ϫu4… ˆ 2 2 ˆ 2 2 ϩs5͑u•R͒ „u Ϫ͑u•R͒ Ϫu4…, (323) FIG. 27. Instanton/anti-instanton interaction at finite tempera- where siϭsi(r,␶,␳I ,␳A ,␤). Because of the reduced ture: (a), the pure gauge interaction in units of S0 as a function symmetry of the problem, it is difficult to implement the of the spatial separation in units of ␳. The temperature is given streamline method. Instead, we shall study the interac- in MeV for ␳ϭ0.33 fm. (b) the fermionic overlap matrix ele- tion in the ratio ansatz. The gauge-field potential is ment TIA for spatial separation r; (c) same element for tem- given by a straightforward generalization of Eq. (141). poral separation ␶. Except for certain limits, the interaction cannot be ob- tained analytically. We therefore give a parametrization of the numerical results obtained by Shuryak and Ver- change, but it becomes more short range as the tempera- baarschot (1991): ture increases. This is a consequence of the core in the instanton gauge field discussed above. At temperatures 8␲2 4.0 ␤2 2 TϽ1/(3␳) the interaction is essentially isotropic. As we SIAϭ ͉u͉ g2 ͭ ͑r2ϩ2.0͒2 ␤2ϩ5.21 shall see below, this is not the case for the fermionic matrix elements. 1.66 0.72 log͑r2͒ ␤2 Ϫ ϩ ͉u͉2 The fermion determinant is calculated from the over- ͫ͑1ϩ1.68r2͒3 ͑1ϩ0.42r2͒4ͬ ␤2ϩ0.75 lap matrix elements TIA with the finite-temperature, 16.0 2.73 zero-mode wave functions. Again, the orientational de- ϩ Ϫ ϩ pendence is more complicated at TÞ0. We have ͫ ͑r2ϩ2.0͒2 ͑1ϩ0.33r2͒3ͬ T ϭu f ϩuជ rˆf (325) ␤2 IA 4 1 • 2 ϫ ͉u Rˆ ͉2 where f ϭf (r,␶,␳ ,␳ ,␤). The asymptotic form of T ␤2ϩ0.24ϩ11.50r2/͑1ϩ1.14r2͒ • i i I A IA for large temperatures ␤ 0,R ϱ can be determined ␤ 1 1 analytically. The result→ is (Khoze→ and Yung, 1991; ϩ0.36 log 1ϩ ͩ r ͪ ͑1ϩ0.013r2͒4 ␤2ϩ1.73 Shuryak and Verbaarschot, 1991) 2 as ␲ ␲␶ ␲r 2 ˆ 2 2 f1 ϭi sin exp Ϫ , (326) ϫ͉͑u͉ Ϫ͉u•R͉ Ϫ͉u4͉ ͒ͮ . (324) ␤ ͩ ␤ ͪ ͩ ␤ ͪ 2 This parametrization is shown in Fig. 27. We observe as ␲ ␲␶ ␲r f ϭi cos exp Ϫ . (327) that the qualitative form of the interaction does not 2 ␤ ͩ ␤ ͪ ͩ ␤ ͪ

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 393

one includes the bosonic interaction and the pre- exponential factor from taking into account fluctuations around the saddle point, the result looks different. The partition function after integrating over 10 of the 11 col- lective coordinates is shown in Fig. 28(b) for tempera- tures in the range (0.6Ϫ1.0)Tc . We observe that there is a maximum in the partition function at the symmetric point ␶ϭ1/(2T) if the temperature is bigger than TmolecӍ0.2␳Ӎ120 MeV. The temperature dependence of the partition function integrated over all variables is shown in Figs. 28(c) and 28(d). This means that there is a qualitative difference be- tween the status of instanton/anti-instanton molecules at low and high (TϾTmolec) temperatures. At low tem- peratures, saddle points in the instanton/anti-instanton separation exist only if the collective coordinates are analytically continued into the complex plane50 (as in Sec. II.A). At high temperatures there is a real saddle point in the middle of the Matsubara box ͓␶ϭ1/(2T)͔, FIG. 28. Mean-field results at finite temperature, from Velk- which gives a real contribution to the free energy. It is ovsky and Shuryak, 1996. (a) Fermionic overlap matrix ele- important that this happen close to the chiral phase tran- ment TIA(␶,rϭ0, TϭTc) as a function of the distance ␶ be- tween the centers of the instanton and anti-instanton in the sition. In fact, we shall argue that the phase transition is time direction; (b) Same for the partially integrated partition caused by the formation of these molecules. 2N function ZЈ ϭ͐d10⍀T feϪSint (Z is not yet integrated over mol IA B. Chiral symmetry restoration Matsubara time) at different temperatures Tϭ(0.6– 1.0)Tc ,in steps of ⌬Tϭ0.04T . The dashed line shows the behavior of c 1. Introduction to QCD phase transitions TIA . (c) Fermionic overlap matrix element after integration over Matsubara time; (d) partition function after integration Before we come to a detailed discussion of the instan- over Matsubara time. ton liquid at finite temperature, we should like to give a brief summary of what is known about the phase struc- A parametrization of the full result is shown in Fig. 27 ture of QCD at finite temperature. It is generally be- (Scha¨fer and Shuryak, 1996b). The essential features of lieved that at high temperature (or density) QCD under- the result can be seen from the asymptotic form [Eqs. goes a phase transition from hadronic matter to the (326) and (327)]: At large temperature, TIA is very an- quark-gluon plasma. In the plasma phase, color charges isotropic. The matrix element is exponentially sup- are screened (Shuryak, 1978a) rather then confined, and pressed in the spatial direction and periodic along the chiral symmetry is restored. At sufficiently high tem- temporal axis. The exponential decay in the spatial di- perature, perturbation theory should be applicable, and rection is governed by the lowest Matsubara frequency the physical excitations are quarks and gluons. In this ␲T. The qualitative behavior of TIA has important con- case, the thermodynamics of the plasma are governed by sequences for the structure of the instanton liquid at T the Stefan-Boltzmann law, just like ordinary blackbody Þ0. In particular, the overlap matrix element favors the radiation. formation of instanton/anti-instanton molecules ori- This basic scenario has been confirmed by a large ented along the time direction. number of lattice simulations. As an example, we show These configurations were studied in greater detail by the equation of state for Nfϭ2 (Kogut-Susskind) QCD Velkovsky and Shuryak (1996). These authors calculate in Fig. 29 (Blum et al., 1995). The transition temperature the IA partition function (150) from the bosonic and is Tc(Nfϭ2)Ӎ150 MeV. The energy density and pres- fermionic interaction at finite temperature. The integra- sure are small below the phase transition, but the energy tion over the collective coordinates was performed as density rises quickly to its perturbative (Stefan- follows: The point rϭ0 (same position in space) and the Boltzmann) value. The pressure, on the other hand, lags most attractive relative orientation (Uϭ1, cos ␪ϭ1) are behind and remains small up to TӍ2Tc . maxima of the partition function, so one can directly use In Fig. 29 the energy density and pressure are mea- the saddle-point method for 10 integrals out of 11 (three sured with respect to the perturbative vacuum. How- over relative spatial distance between the centers and ever, we have repeatedly emphasized that in QCD there over seven relative orientation angles). The remaining is a nonperturbative vacuum energy density (the bag integral over the temporal distance ␶ is more compli- pressure) even at Tϭ0. In order to compare a noninter- cated and has to be done numerically. In Fig. 28(a) we show the dependence of the overlap 50 matrix element TIA on ␶ for TϭTc , rϭ0, and Uϭ1. The two maxima at ␶Ӎ␳ and ␶Ӎ3␳ in Fig. 28(a) are not Even at Tc , TIA does not have a maximum at the sym- really physical; they are related to the presence of a core in the metric point ␶ϭ1/(2T), but a minimum. However, when ratio ansatz interaction.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 394 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

transition is provided by universality arguments. For this purpose, we have to identify an order parameter that characterizes the transition. In QCD, there are two lim- its in which this can be achieved. For infinitely heavy quarks, the Polyakov line (Polyakov, 1978) provides an order parameter for the deconfinement transition, while for massless quarks the fermion condensate is an order parameter for spontaneous chiral symmetry breaking. The Polyakov line is defined by

␤ ͗L͑xជ ͒͘ϭ P exp i d␶A0͑xជ ,␶͒ , (330) ͳ ͩ ͵0 ͪʹ FIG. 29. Equation of state for Nfϭ2 QCD with Kogut- which can be interpreted as the propagator of a heavy Susskind fermions, from Blum et al., 1995. quark. The free energy of a single static quark (minus the free energy of the vacuum) is given by FqϪF0ϭ acting gas of quarks and gluons with the hadronic phase, ϪT log(͉͗L(xជ)͉͘). For ͗L͘ϭ0, the free energy of an iso- we have to shift the pressure in the high-T phase by this lated quark is infinite, and the theory is in the confining amount, p p ϪB. Since the pressure from ther- QGP→ QGP phase. For ͗L͘Þ0 the free energy is finite, and quarks mal hadrons below Tc is small, a rough estimate of the are screened rather than confined. transition temperature can be obtained from the re- The Polyakov line has a nontrivial symmetry. Under quirement that the (shifted) pressure in the plasma gauge transformations in the center of the gauge group, phase be positive, pQGPϾ0. From this inequality we ex- Z ʚSU(N ), local observables are invariant but the Nc c pect the plasma phase to be favored for TϾTc 2 1/4 Polyakov line picks up a phase L zL with z෈ZN . ϭ͓(90B)/(Nd␲ )͔ , where Nd is the effective number → c of degrees of freedom in the quark-gluon-plasma phase. The deconfinement transition was therefore related to spontaneous breakdown of the Z center symmetry. For Nfϭ2 we have Ndϭ37 and TcӍ180 MeV. Nc Lattice simulations indicate that in going from Following Landau, long wave excitations near the phase transition should be governed by an effective Z sym- quenched QCD to QCD with two light flavors the tran- Nc sition temperature drops by almost a factor of two, from metric Lagrangian for the Polyakov line (Svetitsky and Tc(Nfϭ0)Ӎ260 MeV to Tc(Nfϭ2)Ӎ150 MeV. The Yaffe, 1982). Since long-range properties are deter- number of degrees of freedom in the plasma phase, on mined by the lowest Matsubara modes, the effective ac- the other hand, increases only from 16 to 37 (and Tc tion is defined in three dimensions. For Ncϭ2, this Ϫ1/4 does not vary much with Nd , TcϳNd ). This implies means that the transition is in the same universality class that there are significant differences between the pure as the dϭ3 Ising model, which is known to have a gauge and unquenched phase transitions. second-order phase transition. For Ncу3, on the other In particular, the low transition temperature observed hand, we expect the transition to be first order.51 These for Nfϭ2,3 suggests that the energy scales are quite dif- expectations are supported by lattice results. ferent. We have already emphasized that the bag pres- For massless quarks, chiral symmetry is exact, and the sure is directly related to the gluon condensate quark condensate ͗q¯q͘ provides a natural order param- (Shuryak, 1978b). This relation was studied in greater eter. The symmetry of the order parameter is deter- detail by Deng (1989), Adami, Hatsuda, and Zahed mined by the transformation properties of the matrix (1991), and Koch and Brown (1993). From the canonical Uijϭ͗q¯iqj͘. For Nfϭ2 flavors, this symmetry is given by energy-momentum tensor and the trace anomaly, the SU(2)ϫSU(2)ϭO(4), so the transition is governed by gluonic contributions to the energy density and pressure the effective Lagrangian for a four-dimensional vector are related to the electric- and magnetic-field strengths, field in three dimension (Pisarski and Wilczek, 1984; Wilczek, 1992; Rajagopal and Wilczek, 1993). The O(4) 1 g2b 2 2 2 2 ⑀ϭ ͗B ϪE ͘Ϫ 2 ͗E ϩB ͘, (328) Heisenberg magnet is known to have a second-order 2 128␲ phase transition, and the critical indices have been de- 1 g2b termined from both numerical simulations and the ⑀ ex- pϭ ͗B2ϪE2͘ϩ ͗E2ϩB2͘. (329) pansion. For N у3, however, the phase transition is pre- 6 128␲2 f dicted to be first order. Using the available lattice data one finds that in pure From these arguments, one expects the schematic gauge theory the gluon condensate essentially disap- phase structure of QCD in the mudϪms (with mud pears in the high-temperature phase, while in full QCD ϵmuϭmd) plane to look as shown in Fig. 30 (Brown (Nfϭ2,3) about half of the condensate remains. G. et al., 1990; Iwasaki et al., 1996). The upper right-hand Brown has referred to this part of the gluon condensate corner corresponds to the first-order pure gauge phase as the ‘‘hard glue’’ or ‘‘epoxy.’’ It plays an important role in keeping the pressure positive despite the low 51 transition temperature. Under certain assumptions, the NcϾ3 phase transition can More general information on the nature of the phase be second order; see Pisarski and Tytgat (1997).

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 395

ies confirmed (within errors) the O(4) values of the critical indices for the Nfϭ2 chiral transition with both staggered (Karsch and Laermann, 1994) and Wilson fer- mions (Iwasaki et al., 1996), more recent work concludes that 1/␦ (defined as q¯q ϳm1/␦) is consistent with zero ͗ ͘Tc (Ukawa, 1997). If this result is correct, it would imply that there is no Nfϭ2 second-order phase transition, and the only second-order line in Fig. 30 corresponds to the boundary of the first-order region, with standard Ising indices. Are there any possible effects that could upset the expected O(4) scenario for the Nfϭ2 transition? One possibility discussed in the literature is related to the fate of the U(1)A symmetry at Tc . At zero temperature the axial U(1)A symmetry is explicitly broken by the chiral anomaly and instantons. If instantons are strongly suppressed (Pisarski and Wilczek, 1984) or rearranged

FIG. 30. Schematic phase diagram of Nfϭ3 QCD with dy- (Shuryak, 1994) at Tc , then the U(1)A symmetry might namic quark masses mudϵmuϭmd and ms , from Iwasaki effectively be restored at the phase transition. In this et al., 1996: ᭹, simulations (with Wilson fermions) that found a case, the masses of the U(1)A partners of the ␲,␴, the first-order transition; ș, simulations that found a smooth cross- ␦,␩Ј, could become sufficiently small that fluctuations of over. The (approximate) location of QCD with realistic masses these modes would affect the universality arguments is shown by a star. given above. In particular, if there were eight rather than four light modes at the phase transition, the transi- transition. Presumably, this first-order transition extends tion would be expected to be first order. Whether this is to lower quark masses before it ends in a line of second- the correct interpretation of the lattice data remains un- order phase transitions. The first-order Nfϭ3 chiral clear at the moment. We shall return to the question of phase transition is located in the lower left corner and U(1)A breaking at finite temperature in Sec. VII.C.3 continues in the mass plane before it ends in another below. Let us only note that the simulations are very line of second-order phase transitions. At the left edge difficult and that there are several possible artifacts. For there is a tricritical point where this line meets the line example, there could be problems with the extrapolation of Nfϭ2 second-order phase transitions extending down to small masses. Also, first-order transitions need not from the upper left corner. have an order parameter, and it is difficult to distinguish Simulations suggest that the gap between the first- very weak first-order transitions from second-order tran- order chiral and deconfinement transitions is very wide, sitions. extending from mϵmuϭmdϭmsӍ0.2 GeV to m Completing this brief review of general arguments Ӎ0.8 GeV. This is in line with the arguments given and lattice results on the chiral phase transition, let us above, indicating that there are important differences also comment on some of the theoretical approaches. between the phase transitions in pure gauge and full Just as perturbative QCD describes the thermodynamics QCD. Nevertheless, one should not take this distinction of the plasma phase at very high temperature, effective too literally. In the presence of light quarks, there is no chiral Lagrangians provide a very powerful tool at low deconfinement phase transition in a strict, mathematical temperature. In particular, chiral perturbation theory sense. From a practical point of view, however, decon- predicts the temperature dependence of the quark and finement plays an important role in the chiral transition. gluon condensates as well as of the masses and coupling In particular, the equation of state shows that the jump constants of light hadrons (Gerber and Leutwyler, in energy density is dominated by the release of 37 1989). These results are expressed as an expansion in 2 2 quark and gluon degrees of freedom. T /f␲ . It is difficult to determine what the range of va- There are many important questions related to the lidity of these predictions is, but the approach certainly phase diagram that still have to be resolved. First of all, has to fail as one approaches the phase transition. we have to determine the location of real QCD (with The phase transition has also been studied in various two light, one intermediate-mass, and several heavy fla- effective models, for example, in the linear sigma model, vors) on this phase diagram. While results using stag- the chiral quark model, or the Nambu-Jona-Lasinio gered fermions (Brown et al., 1990) seem to suggest that model (Hatsuda and Kunihiro, 1985; Bernard and QCD lies outside the range of first-order chiral phase Meissner, 1988; Klimt, Lutz, and Weise, 1990). In the transitions and shows only a smooth crossover, simula- Nambu–Jona-Lasinio model, the chiral transition con- tions using Wilson fermions (Iwasaki et al., 1991) with nects the low-temperature phase of massive constituent realistic masses find a first-order transition. quarks with the high-temperature phase of massless A more general problem is to verify the structure of quarks (but of course not gluons). The mechanism for the phase diagram and to check the values of the critical the transition is similar to that for a transition from a indices near second-order transitions. While earlier stud- superconductor to a normal metal: The energy gain due

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 396 T. Scha¨fer and E. V. Shuryak: Instantons in QCD to pairing disappears once a sufficient number of sin͑␲T␶͒ 2Nf positive-energy states is excited. We have seen that, at det͑D͒ϳ . (331) ” ͯ cosh͑␲Tr͒ͯ zero temperature, the instanton liquid leads to a picture of chiral symmetry breaking which closely resembles the This expression is maximal for rϭ0 and ␶ϭ1/(2T), Nambu–Jona-Lasinio model. Nevertheless, we shall ar- which is the most symmetric orientation of the gue that the mechanism for the phase transition is quite instanton/anti-instanton pair on the Matsubara torus. different. In the instanton liquid, it is the ensemble itself Since the fermion determinant controls the probability of the configuration, we expect polarized molecules to that is rearranged at Tc . This means not only that we have nonzero occupation numbers, but also that the ef- become important at finite temperature. The effect should be largest when the IA pairs exactly fit onto the fective interaction itself changes. torus, i.e., 4␳ ␤. Using the zero-temperature value ␳ Finally, a number of authors have extended random Ӎ Ӎ0.33 fm, we get TӍ150 MeV, close to the expected matrix models to finite temperature and density (Jack- transition temperature for two flavors. son and Verbaarschot, 1996; Novak, Papp, and Zahed, In general, the formation of molecules is controlled by 1996; Wettig, Scha¨fer, and Weidenmu¨ ller, 1996). It is im- the competition between minimum action, which favors portant to distinguish these models from random matrix correlations, and maximum entropy, which favors ran- methods at T,␮ϭ0. In this case, there is evidence that domness. Determining the exact composition of the in- certain observables,52 like scaled correlations between stanton liquid as well as the transition temperature re- eigenvalues of the Dirac operator, are universal and can quires the calculation of the full partition function, be described in terms of suitably chosen random matrix including the fermion-induced correlations. We shall do ensembles. The effects of nonzero temperature and den- this using a schematic model in Sec. VII.B.3 and using sity, on the other hand, are included in a rather sche- numerical simulations in Sec. VII.B.4. matic fashion, by putting terms like (␲Tϩi␮) into the Before we come to this, we should like to study what Dirac operator.53 This procedure is certainly not univer- physical effects are caused by the presence of molecules. sal. From the point of view of the instanton model, the Qualitatively, it is clear why the formation of molecules entries of the random matrix correspond to matrix ele- leads to chiral symmetry restoration. If instantons are ments of the Dirac operator in the zero-mode basis. If bound into pairs, then quarks mostly propagate from we include the effects of a nonzero ␮,T in the schematic one instanton to the corresponding anti-instanton and form mentioned above, we assume that at ␮,TÞ0 there back. In addition to that, quarks propagate mostly along are no zero modes in the spectrum. But this is not true. the imaginary-time direction, so all eigenstates are well We have explicitly constructed the zero mode for the localized in space and no quark condensate is formed. caloron configuration in Sec. VII.A.1, for ␮Þ0 [see Car- Another way to see this is by noting that the Dirac op- valho (1980), and Abrikosov (1983)]. As we shall see erator essentially decomposes into 2ϫ2 blocks corre- below, in the instanton model the phase transition is not sponding to the instanton/anti-instanton pairs. This caused by a constant contribution to the overlap matrix means that the eigenvalues will be concentrated around elements, but by specific correlations in the ensemble. some typical Ϯ␭ determined by the average size of the pair, so the density of eigenvalues near ␭ϭ0 vanishes. 2. The instanton liquid at finite temperature We have studied the eigenvalue distribution in a sche- and IA molecules matic model of random and correlated instantons in In Sec. VII.A.3, we argued that the instanton density Scha¨fer et al. (1995), and a random matrix model of the remains roughly constant below the phase transition. transition based on these ideas was discussed by Wettig This means that the chiral phase transition has to be et al. (1996). caused by a rearrangement of the instanton ensemble. The effect of molecules on the effective interaction Furthermore, we have shown that the gluonic interac- between quarks at high temperature was studied by tion between instantons remains qualitatively un- Scha¨fer et al. (1995), using methods similar to those in- changed even at fairly high temperatures. This suggests troduced in Sec. IV.F. In order to determine the inter- that fermionic interactions between instantons drive the action in a quark-antiquark (meson) channel with given phase transition (Ilgenfritz and Shuryak, 1994; Scha¨fer quantum numbers, it is convenient to calculate both the et al., 1995; Scha¨fer and Shuryak, 1996a). direct and the exchange terms and Fierz-rearrange the The mechanism for this transition is most easily un- exchange term into an effective direct interaction. The derstood by considering the fermion determinant for resulting Fierz symmetric Lagrangian reads (Scha¨fer one instanton/anti-instanton pair (Scha¨fer et al., 1995). et al., 1995) Using the asymptotic form of the overlap matrix ele- 2 ¯ a 2 ¯ a 2 ments specified above, we have mol symϭG 2 ͓͑␺␶ ␺͒ Ϫ͑␺␶ ␥5␺͒ ͔ L ͭ Nc

1 a a 52On the other hand, macroscopic observables, like the aver- Ϫ ͓͑␺¯␶ ␥ ␺͒2ϩ͑␺¯␶ ␥ ␥ ␺͒2͔ 2N2 ␮ ␮ 5 age level density (and the quark condensate) are not expected c to be universal. 2 53 ¯ 2 See Wettig et al. (1996) for an attempt to model the results ϩ 2 ͑␺␥␮␥5␺͒ ϩ 8 , (332) of the instanton liquid in terms of a random matrix ensemble. Nc ͮ L

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 397 with the coupling constant Since the interaction is unchanged, so is the form of the single-instanton distribution, 1 2 2 Gϭ n͑␳ ,␳ ͒d␳ d␳ ͑2␲␳ ͒ ͑2␲␳ ͒ . (333) 2 ͵ 1 2 1 2 8T2 1 2 Ϫ␤␥ NT IA ␮͑␳͒ϭn͑␳,T͒exp ␳2␳2 , (334) V Here, is the color-octet part of the interaction and ␶a ͫ 3 ͬ L8 is a four-vector with components (␶ជ,1). Also, n(␳1 ,␳2) where n(␳,T) is the semiclassical result (308) and the is the tunneling probability for the IA pair and TIA the four-dimensional volume is given by VϭV3 /T. The T corresponding overlap matrix element. The effective La- dependence of the distribution functions modifies the grangian (332) was determined by averaging over all self-consistency equations for ␳2 and N/V. Following possible molecule orientations, with the relative color Diakonov and Mirlin (1988), we can expand the coeffi- orientation cos(␪)ϭ1 fixed. Near the phase transition, cient B(␭) in (308) to order T2: n(␳,T) molecules are polarized in the temporal direction, Lor- 1 11 2 entz invariance is broken, and vector interactions are Ӎexp͓Ϫ 3( 6 NcϪ1)(␲␳T) ͔n(␳,Tϭ0). In this case, the ¯ 2 ¯ 2 self-consistency condition for the average size is given modified according to (␺␥␮⌫␺) 4(␺␥0⌫␺) . Like the zero-temperature effective→ Lagrangian (111), by the interaction (332) is SU(2)ϫSU(2) symmetric. Since 1 11 N␤␥2 Ϫ1 2 2 2 2 molecules are topologically neutral, the interaction is ␳ ϭ␯ NcϪ1 ␲ T ϩ T␳ . (335) ͫ3 ͩ 6 ͪ V ͬ 3 also U(1)A symmetric. This does not mean that U(1)A symmetry is restored in the molecular vacuum. Even a For Tϭ0, this gives ␳2ϭ͓(␯V)/(␤␥2N)͔1/2 as before, N very small O(m f) fraction of random instantons will while for large TϾ(␲␳)Ϫ1 we have ␳2ϳ1/T2. The in- still lead to U(1)A breaking effects of order O(1) (see stanton density follows from the self-consistency equa- Sec. VII.C.3). If chiral symmetry is restored, the effec- tion for ␮0 . For large T we have tive interaction (332) is attractive not only in the pion 2 Ϫ␯ channel, but also in the other scalar-pseudoscalar chan- 1 11 T N/VϭC ␤2Nc⌳4⌫͑␯͒ N Ϫ1 ␲2 , (336) nels ␴, ␦, and ␩Ј. Furthermore, unlike the ’t Hooft in- Nc ͫ3 ͩ 6 c ͪ ⌳2ͬ teraction, the effective interaction in the molecular so that N/Vϳ1/TbϪ4, which is what one would expect vacuum also includes an attractive interaction in the vec- tor and axial-vector channels. If molecules are unpolar- from simply cutting the size integration at ␳Ӎ1/T. ized, the corresponding coupling constant is a factor of 4 The situation is somewhat more interesting if one ex- smaller than the scalar coupling. If they are fully polar- tends the variational method to full QCD (Ilgenfritz and ized, only the longitudinal vector components are af- Shuryak, 1989; Nowak, Verbaarschot, and Zahed, fected. In fact, the coupling constant is equal to the sca- 1989c). In this case, an additional temperature depen- lar coupling. A more detailed study of the quark dence enters through the T dependence of the average interaction in the molecular vacuum was performed by fermionic overlap matrix elements. More importantly, Scha¨fer et al. (1995) and Scha¨fer and Shuryak (1995b), the average determinant depends on the instanton size, in which hadronic correlation functions in the spatial and temporal direction were calculated in the schematic det͑D” ͒ϭ͟ ͑␳mdet͒, model mentioned above. We shall discuss the results be- I low. 1 1/2 m ϭ␳3/2 I͑T͒ d␳n͑␳͒␳ , (337) det ͫ2 ͵ ͬ where I(T) is the angle and distance-averaged overlap 3. Mean-field description at finite temperature matrix element TIA . The additional ␳ dependence modi- In the following two sections we wish to study the fies the instanton distribution (334) and introduces an statistical mechanics of the instanton liquid at finite tem- additional nonlinearity into the self-consistency equa- perature. This is necessary not only in order to study tion. As a result, the instanton density at large T de- thermodynamic properties of the system, but also to de- pends crucially on the number of flavors. For Nfϭ0,1, termine the correct ensemble for calculations of had- the density drops smoothly with N/Vϳ1/T2a and a ronic correlation functions. We first extend the mean- ϭ(bϪ4ϩ2Nf)/(2ϪNf) for large T. For Nfϭ2, the in- field calculation of Sec. IV.F to finite temperature. In stanton density goes to zero continuously at the critical the next section we shall study the problem numerically, temperature Tc , whereas for NfϾ2 the density goes to using the methods introduced in Sec. V.B. zero discontinuously at Tc . This behavior can be under- For pure gauge theory, the variational method was stood from the form of the gap equation for the quark extended to finite temperature by Diakonov and Mirlin condensate. We have ͗q¯q͘ϳconst͗q¯q͘NfϪ1, which, for (1988) and Kanki (1988). The gluonic interaction be- NfϾ2, cannot have a solution for arbitrarily small ͗q¯q͘. tween instantons changes very little with temperature, In the mean-field approximation, the instanton en- so we shall ignore this effect. In this case, the only dif- semble remains random at all temperatures. This of ference as compared to zero temperature is the appear- course implies that instantons cannot exist above Tc . ance of the perturbative suppression factor (308) (for We have already argued that this is not correct, and that TϾTc , although Diakonov and Mirlin used it for all T). instantons can be present above Tc , provided they are

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 398 T. Scha¨fer and E. V. Shuryak: Instantons in QCD bound in molecules (or larger clusters). In order to in- clude this effect in a variational calculation, Ilgenfritz and Shuryak introduced a ‘‘cocktail’’ model (Ilgenfritz and Shuryak, 1989; 1994), in which the instanton en- semble consists of a random and a molecular compo- nent. The composition of the instanton liquid is deter- mined by minimizing the free energy with respect to the two concentrations. As above, the distribution functions for random in- stantons is given by

2 2 2 Nf ␮͑␳͒ϭn͑␳͒exp͓Ϫ␬␳ ͑␳anaϩ2␳mnm͔͒͑mdet␳͒ , (338) where na,m are the densities of the random and molecu- 2 lar contributions and ␳ a,m are the corresponding aver- age radii. The parameter ␬ϭ␤␥2 characterizes the aver- age repulsion between instantons. The distribution of instantons bound in molecules is given by

␮͑␳1 ,␳2͒ϭn͑␳1͒n͑␳2͒

2 2 2 2 ϫexp͓Ϫ␬͑␳1ϩ␳2͒͑␳anaϩ2␳mnm͔͒ ϫ ͑T T* ͒Nf , (339) ͗ IA IA ͘ FIG. 31. Chiral restoration phase transitions for two massless Nf flavors and two different core parameters R . The upper panel where Im(Nf ,T)ϵ͗(TIATIA* ) ͘ is the average determi- c nant for an instanton/anti-instanton pair, with the rela- shows the T dependence of the densities: solid curve, na ; dot- tive orientation cos ␪ϭ1 fixed. Summing the contribu- ted curve, nm . In the middle panel the T dependence is shown: tions from both the random and the molecular solid curve, the total pressure p; dotted curve, the contribu- components, we find that the self-consistency condition tions of the pion gas/quark-gluon plasma; dash-dotted curve, contribution of instantons. The lower panel shows the energy for the instanton size becomes density (solid curve), which is modified by the instanton con- 2 2 2 2 2 tribution (dash-dotted curve). ␳m ␣ 1 2͑␳a͒ na 4͑␳m͒ nm ϭ , ϭ ϩ , (340) 2 ␤ ␬ ␤ ␣ -T). In order to proץ/pץ)␳a the energy density ⑀ϭϪpϩT where ␣ϭb/2Ϫ1 and ␤ϭb/2ϩ3Nf/4Ϫ2. Using this re- vide a more realistic description of the thermodynamics sult, one can eliminate the radii and determine the nor- of the chiral phase transition, Ilgenfritz and Shuryak malizations: added the free energy of a noninteracting pion gas in the 2 2 broken phase and a quark-gluon plasma in the symmet- A Im͑Nf ,T͒C ⌫ ͑␣͒ ric phase. ␮0,mϭ ␣ , Aϭ ␣ , ͓naϩ͑2␣/␤͒nm͔ ͑4␬␤͒ Minimizing the free energy gives a set of rather cum- (341) bersome equations. It is clear that in general there will be two phases, a low-temperature phase containing a Nf/2 Bna mixture of molecules and random instantons, and a ␮ ϭ , 0,a ͓n ϩ͑2␣/␤͒n ͔␤/2ϩNf/8 chirally restored high-temperature phase consisting of a m molecules only. The density of random instantons is sup- pressed as the temperature increases because the aver- C⌫͑␤͒ I͑T͒ Nf/2 ␤ Nf/8Ϫ␤/2 age overlap matrix element decreases. Molecules, on the Bϭ ␤ . (342) ͑2␬͒ ͩ 2 ͪ ͩ ␬ ͪ other hand, are favored at high T, because the overlap for the most attractive orientation increases. Both of Finally, the free energy is given by these results are simple consequences of the anisotropy of the fermion wave functions. 1 Na e␮0,aV4 Numerical results for N ϭ2 are shown in Fig. 31. In FϭϪ log Zϭ log f V V ͩ N ͪ practice, the average molecular determinant I (N ,T) 4 4 a m f was calculated by introducing a core into the IA inter- Nm e␮0,mV4 action. In order to assess the uncertainty involved, we ϩ log , (343) V ͩ N ͪ show the results for two different cores, Rcϭ␳ and 2␳. 4 m The overall normalization was fixed such that N/Vϭ1.4 and the instanton density is determined by minimizing F fmϪ4 at Tϭ0. Figure 31 shows that the random compo- with respect to the densities na,m of random and corre- nent dominates the broken phase and that the density of lated instantons. The resulting free energy determines instantons is only weakly dependent on T below the the instanton contribution to the pressure pϭϪF and phase transition. The number of molecules is small for

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 399

TϽTc but jumps up at the transition. The total instan- ton density above Tc ,(N/V)ϭ2nm turns out to be comparable to that at Tϭ0. The importance of the molecular component above Tc can be seen from the temperature dependence of the pressure. For Tϭ(1Ϫ2)Tc , the contribution of mol- ecules (dash-dotted line) is crucial in order to keep the pressure positive. This is the same phenomenon we al- ready mentioned in our discussion of the lattice data. If the transition temperature is as low as Tcϭ150 MeV, then the contribution of quarks and gluons is not suffi- cient to match the Tϭ0 bag pressure. The lower panel in Fig. 31 shows the behavior of the energy density. The jump in the energy density at Tc is ⌬⑀Ӎ(0.5 Ϫ1.0) GeV/fm3, depending on the size of the core. Al- though most of the latent heat is due to the liberation of quarks and gluons, a significant part is generated by molecules.

4. Phase transitions in the interacting instanton model In this section, we go beyond this schematic model and study the phase transition using numerical simula- tions of the interacting instanton liquid (Scha¨fer and Shuryak, 1996a). This means that we do not have to make any assumptions about the nature of the impor- tant configurations (molecules, larger clusters,...), nor do we have to rely on variational estimates to determine their relative importance. Neither, are we limited to a FIG. 32. Instanton density, free energy, and quark condensate simple two-phase picture with a first-order transition. as a function of the temperature in the instanton liquid with In Fig. 32 we show the instanton density, free energy, two light and one intermediate mass flavor, from Scha¨fer and and quark condensate for the physically relevant case of Shuryak, 1996a. The instanton density is given in units of the Ϫ4 two light and one intermediate-mass flavor. In the ratio zero-temperature value n0ϭ1fm , while the free energy and ansatz the instanton density at zero temperature is given the quark condensate are given in units of the Pauli-Vilars by N/Vϭ0.69⌳4. Taking the density to be 1 fmϪ4 at T scale parameter, ⌳ϭ222 MeV. ϭ0 fixes the scale parameter ⌳ϭ222 MeV and deter- mines the absolute units. The temperature dependence of the instanton density54 is shown in Fig. 32(a). It shows means that the phase transition is indeed caused by a a slight increase at small temperatures, starts to drop transition within the instanton liquid, not by the disap- around 115 MeV, and becomes very small for T pearance of instantons. This point is illustrated in Fig. Ͼ175 MeV. The free energy closely follows the behav- 33, which shows projections of the instanton liquid at ior of the instanton density. This means that the Tϭ74 MeV and Tϭ158 MeV, below and above the instanton-induced pressure first increases slightly, but phase transition. The figures are projections of a cube Ϫ1 3 Ϫ1 then drops and eventually vanishes at high temperature. Vϭ(3.0⌳ ) ϫT into the 3-4 plane. The positions of This behavior is expected for a system of instantons, but instantons and anti-instantons are denoted by ϩ/Ϫ sym- if all fluctuations are included, the pressure should al- bols. The lines connecting them indicate the strength of ways increase as a function of the temperature. the fermionic overlap (‘‘hopping’’) matrix elements. Be- The temperature dependence of the quark condensate low the phase transition, there is no clear pattern. In- is shown in Fig. 32(c). At temperatures below T stantons are unpaired, part of molecules or larger clus- ϭ100 MeV it is practically temperature independent. ters. Above the phase transition, the ensemble is Above that, ͗q¯q͘ starts to drop and becomes very small dominated by polarized instanton/anti-instanton mol- above the critical temperature TӍ140 MeV. Note that ecules. at this point the instanton density is N/Vϭ0.6 fmϪ4, Figure 34 shows the spectrum of the Dirac operator. slightly more than half the zero-temperature value. This Below the phase transition it has the familiar flat shape near the origin and extrapolates to a nonzero density of eigenvalues at ␭ϭ0. Near the phase transition the eigen- 54 value density appears to extrapolate to 0 as ␭ 0, but The instanton density is of course sensitive to our assump- → tions concerning the role of the finite-temperature suppression there is a spike in the eigenvalue density at ␭ϭ0. This factor. In practice, we have chosen a functional form that in- spike contains the contribution from unpaired instan- terpolates between no suppression below Tc and full suppres- tons. In the high-temperature phase, we expect the den- Nf sion above Tc , with a width ⌬Tϭ0.3Tc . sity of random instantons to be O(m ), giving a con-

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 400 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

FIG. 33. Typical instanton ensembles for Tϭ75 and 158 MeV, from Scha¨fer and Shuryak, 1996a. The plots show projections of a four-dimensional (3.0⌳Ϫ1)3ϫTϪ1 box into the 3-4 plane. The positions of instantons and anti-instanton are indicated by ϩ and Ϫ symbols: Dashed lines, fermionic overlap matrix ele- ments TIA⌳Ͼ0.40; solid lines, matrix elements TIAϾ0.56; heavy solid lines, matrix elements TIAϾ0.64. FIG. 34. Spectrum of the Dirac operator for different tem- peratures Tϭ75, 130, and 158 MeV, from Scha¨fer and Shuryak, 1996a. Eigenvalues are given in units of the scale tribution of the form ␳(␭)ϳmNf␦(␭) to the Dirac parameter. The normalization of the spectra is arbitrary (but spectrum. These eigenvalues do not contribute to the identical). quark condensate in the chiral limit, but they are impor- tant for U(1)A-violating observables. 55 The nature of the phase transition for different num- The results of simulations with Nfϭ2,3,5 flavors bers of flavors and values of the quark masses was stud- with equal masses are summarized in the phase diagram ¨ ied by Scha¨fer and Shuryak (1996a). The order of the 35 (Schafer and Shuryak, 1996a). For Nfϭ2 there is a second-order phase transition which turns into a line of phase transition is most easily determined by studying first-order transitions in the m-T plane for N Ͼ2. If the order-parameter fluctuations near T . For a first-order f c system is in the chirally restored phase (TϾT )atm transition one expects two (meta) stable phases. The sys- c ϭ0, we find a discontinuity in the chiral order parameter tem tunnels back and forth between the two phases, if the mass is increased beyond some critical value. with tunneling events signaled by sudden jumps in the Qualitatively, the reason for this behavior is clear. While order parameter. Near a second-order phase transition, increasing the temperature increases the role of correla- on the other hand, the order parameter shows large fluc- tions caused by fermion determinant, increasing the tuations. These fluctuations can be studied in greater de- quark mass has the opposite effect. We also observe that tail by measuring the scaling behavior of the mean- increasing the number of flavors lowers the transition square fluctuations (the scalar susceptibility) with the temperature. Again, increasing the number of flavors current mass and temperature. Universality makes defi- means that the determinant is raised to a higher power, nite predictions for the corresponding critical exponents. so fermion-induced correlations become stronger. For The main conclusion in Scha¨fer and Shuryak (1996a) Nfϭ5 we find that the transition temperature drops to was that the transition in QCD is consistent with a zero and the instanton liquid has a chirally symmetric nearby (Nfϭ2) second-order phase transition with O(4) critical indices. For three flavors with physical masses, the transition is either very weakly first order or 55 The case Nfϭ4 is omitted because the contribution of just a rapid crossover. As the number of flavors is in- small-size (semiclassical) instantons to the quark condensate is creased, the transition becomes more strongly first or- very small and the precise location of the phase boundary hard der. to determine.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 401

(Bochkarev and Shaposhnikov, 1986; Eletskii and Ioffe, 1993; Hatsuda, Koike, and Lee, 1993). At finite tempera- tures, however, the predictive power of QCD sum rules is very limited, because additional assumptions about the temperature dependence of the condensates and the shape of the spectrum are needed. There is an extensive literature on spacelike screening masses on the lattice, but only very limited information on temporal, point-to- point correlation functions (Boyd et al., 1994). At finite temperature, the heat bath breaks Lorenz FIG. 35. Phase diagram of the instanton liquid in the TϪm invariance, and correlation functions in the spatial and plane for different numbers of flavors. The open and closed temporal direction are independent. In addition, me- squares show points on the phase diagram where we have per- sonic and baryonic correlation functions have to obey formed simulations. For Nfϭ2, the open squares mark points where we found large fluctuations of the order parameter, in- periodic or antiperiodic boundary conditions, respec- dicative of a nearby second-order phase transition (marked by tively, in the temporal direction. In the case of spacelike correlators, one can still go to large x and filter out the a star). In the cases Nfϭ3 and 5, open and closed squares mark points in the chirally broken and restored phases, respec- lowest exponents known as screening masses. While tively. The (approximate) location of the discontinuity is these states are of theoretical interest and have been shown by the dashed line. studied in a number of lattice calculations, they do not correspond to poles of the spectral function in energy. In order to look for real bound states, one has to study ground state, provided the dynamic quark mass is less temporal correlation functions. However, at finite tem- than some critical value. peratures the periodic boundary conditions restrict the Studying the instanton ensemble more closely shows that, in this case, all instantons are bound into mol- useful range of temporal correlators to the interval ␶ Ͻ1/(2T) (about 0.6 fm at TϭTc). This means that there ecules. The molecular vacuum at Tϭ0 and large Nf has a somewhat different theoretical status from that of the is no direct procedure for extracting information about the ground state. The underlying physical reason is molecular vacuum for small N and large T. In the high- f clear: at finite temperatures excitations are always temperature phase, large instantons are suppressed and present. Below we explore how much can be learned long-range interactions are screened. This is not the case from temporal correlation functions in the interacting at Tϭ0 and N large, where these effects may contrib- f instanton liquid. In the next section we also present the ute to chiral symmetry breaking (see the discussion in corresponding screening masses. Sec. IX.D). Unfortunately, little is known about QCD with differ- ent numbers of flavors from lattice simulations. There 2. Temporal correlation functions are some results on the phase structure of large-Nf QCD that we shall discuss in Sec. IX.D. In addition to that, Finite-temperature correlation functions in the tem- there are some recent data by the Columbia group poral direction are shown in Fig. 36. These correlators (Chen and Mawhinney, 1997) on the hadron spectrum were obtained from simulations of the interacting in- ¨ for Nfϭ4. The most important result is that chiral stanton liquid (Schafer and Shuryak, 1995b). Correlators symmetry-breaking effects were found to be drastically in the random-phase approximation were studied by smaller than those for Nfϭ0,2. In particular, the mass Velkovsky and Shuryak (1996). The results, shown in splittings between chiral partners such as ␲Ϫ␴, Fig. 36, are normalized to the corresponding noninter- ␳Ϫa , N( 1 ϩ)ϪN( 1 Ϫ), were found to be four to five acting correlators, calculated from the free TÞ0 propa- 1 2 2 gator (300). Figures 36(a) and 36(b) show the pion and times smaller. This agrees well with what was found in sigma correlators for different temperatures below the interacting instanton model, but more work in this (open symbols) and above (closed symbols) the chiral direction is certainly needed. phase transition. The normalized ␲ and ␴ correlators are larger than one at all temperatures, implying that there C. Hadronic correlation functions at finite temperature is an attractive interaction even above Tc . In particular, the value of the pion correlator at ␶Ӎ0.6 fm, which is 1. Introduction not directly affected by the periodic boundary condi- Studying the behavior of hadronic correlation func- tions, is essentially temperature independent. This sug- tions at finite temperature is of great interest in connec- gests that there is a strong (␲,␴)-like mode present even tion with possible modifications of hadronic properties above Tc . Scha¨fer and Shuryak (1996b) tried to deter- in hot hadronic matter. In addition to that, the structure mine the properties of this mode from a simple fit to the of correlation functions at intermediate distances di- measured correlation function, similar to the Tϭ0 pa- rectly reflects on changes in the interaction between rametrization (278). Above Tc , the mass of the ␲-like quarks and gluons. There is very little phenomenological mode is expected to grow, but the precise value is hard Ϫ2 information on this subject, but the problem has been to determine. The coupling constant is ␭␲Ӎ1fm at T Ϫ2 studied extensively in the context of QCD sum rules ϭ170 MeV, as compared to ␭␲Ӎ3fm at Tϭ0.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 402 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

FIG. 36. Temporal correlation functions at TÞ0, normalized to free thermal correlators: (a) the pseudoscalar (pion) correlator; (b) the isoscalar scalar ␴; (c) the isovector vector (␳); and (d) the axial vector (a1). Correlators in the chirally symmetric phase (TуTc) are shown as solid points, below Tc as open points. ᭝, Tϭ0.43Tc ; ᮀ, Tϭ0.60Tc ; ˝, T ϭ0.86Tc ; ᭡, Tϭ1.00Tc ; ᭿, Tϭ1.13Tc .

The T dependence of the ␴ correlator is more pro- Scha¨fer and Shuryak (1995b) also studied baryon cor- nounced because there is a disconnected contribution relation functions at finite temperatures. Because the which tends to the square of the quark condensate at different nucleon and delta correlation functions have large distance. Above Tc , chiral symmetry is restored different transformation properties under the chiral and the ␴ and ␲ correlation functions are equal up to SU(2)ϫSU(2) and U(1)A symmetries, one can distin- small corrections due to the current quark masses. guish different modes of symmetry restoration. The Vector and axial-vector correlation functions are main possibilities are (i) that all resonances simply dis- shown in Figs. 36(c) and 36(d). At low T the two are appear, (ii) that all states become massless as T Tc ,or very different, while above Tc they become indistin- → (iii) that above Tc all states occur in parity doublets. In guishable, again in accordance with chiral symmetry res- the nucleon channel, we find clear evidence for the sur- toration. In the vector channel, the changes in the cor- vival of a massive nucleon mode. There is also support relation function indicate the ‘‘melting’’ of the for the presence of a degenerate parity partner above resonance contribution. At the lowest temperature, Tc , so the data seem to favor the third possibility. there is a small enhancement in the correlation function In summary, we find that correlation functions in at ␶Ӎ1 fm, indicating the presence of a bound state strongly attractive, chiral even channels are remarkably separated from the two-quark (or two-pion) continuum. stable as a function of temperature, despite the fact that T However, this signal disappears at Ӎ100 MeV, imply- the quark condensate disappears. There is evidence for ing that the ␳ meson coupling to the local current be- 56 the survival of (␲,␴)-like modes even above Tc . These comes small. This is consistent with the idea that had- modes are bound by the effective quark interaction gen- rons ‘‘swell’’ in hot and dense matter. At low erated by polarized instanton molecules [see Eq. (332)]. temperature, the dominant effect is mixing between the In channels that do not receive large nonperturbative vector and axial-vector channels (Dey, Eletsky, and contributions at Tϭ0 (like the ␳, a1 , and ⌬), the reso- Ioffe, 1990). In particular, there is a pion contribution to nances disappear, and the correlators can be described the vector correlator at finite T, which is most easily V in terms of the free-quark continuum (possibly with a observed in the longitudinal vector channel ⌸44(␶) [in V residual ‘‘chiral’’ mass). Fig. 36 we show the trace ⌸␮␮(␶) of the vector cor- relator].

3. U(1)A breaking

56Note, however, that in the instanton model, there is no con- So far, we have not discussed correlation functions finement and the amount of binding in the ␳-meson channel is related to the U(1)A anomaly, in particular the ␩ and ␩Ј presumably small. In full QCD, the ␳ resonance might there- channels. A number of authors have considered the pos- fore be more stable as the temperature increases. sibility that the U(1)A symmetry is at least partially re-

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 403 stored near the chiral phase transition (Shuryak, 1994; Huang and Wang, 1996; Kapusta, Kharzeev, and McLer- ran, 1996). Given the fact that the ␩ЈϪ␲ mass differ- ence is larger than all other meson mass splittings, any tendency towards U(1)A restoration is expected to lead to rather dramatic observable effects. Possible signa- tures of a hot hadronic phase with partially restored FIG. 37. Leading contributions to flavor mixing in the ␩Ϫ␩Ј U(1) symmetry that might be produced in heavy-ion A system below and above the chiral phase transition. collisions are anomalous values of the ␩/␲ and ␩Ј/␲ ratios, as well as an enhancement in the number of low- mass dilepton pairs from the Dalitz decay of the ␩Ј. effective Lagrangian reproduces the Witten-Veneziano relation f2 m2 ϭ␹ . In a quenched ensemble, we can In general, we know that isolated instantons disappear ␲ ␩Ј top above the chiral phase transition. In the presence of ex- further identify ␹topӍ(N/V). The temperature depen- dence of c is usually estimated from the semiclassical ternal sources, however, isolated tunneling events can 2 still occur (see Sec. II.D.2). The density of random in- tunneling amplitude n(␳)ϳexp͓Ϫ(8/3)(␲␳T) ͔.Asa N result, the strength of the anomaly is reduced by a factor stantons above T is expected to be O(m f), but the c ϳ5 near T . If the anomaly becomes weaker, the eigen- contribution of isolated instantons to the expectation c ¯ states are determined by the mass matrix. In that case, value of the ’t Hooft operator detf(␺l␺R) (and other the mixing angle is close to the ideal value ␪ϭϪ54.7° U(1)A-violating operators) is of order O(1) (Evans, and the nonstrange ␩NS is almost degenerate with the Hsu, and Schwetz, 1996; Lee and Hatsuda, 1996). The pion. presence of isolated zero modes in the spectrum of the There are several points in this line of argument that Dirac operator above Tc can be seen explicitly in Fig. are not entirely correct. The strength of the ’t Hooft ¨ 34. The problem was studied in greater detail by Schafer term is not controlled by the topological susceptibility (1996), where it was concluded that the number of (al- (␹topϭ0 in massless QCD), ␹top is not proportional to most) zero modes above Tc scales correctly with the dy- the instanton density (for the same reason), and, at least namic quark mass and the volume. below Tc , the semiclassical estimate for the instanton A number of groups have measured U(1)A-violating density is not applicable. Only above Tc do we expect observables at finite temperature on the lattice. Most instantons to be suppressed. However, chiral symmetry work focuses on the the susceptibility ␹␲Ϫ␹␦ (Chan- restoration affects the structure of flavor mixing in the drasekharan, 1995; Boyd, 1996; Bernard et al., 1997). ␩Ϫ␩Ј system (see Fig. 37). The mixing between the Above Tc , when chiral symmetry is restored, this quan- strange and nonstrange eta is controlled by the light- tity is a measure of U(1)A violation. Most of the pub- quark condensate, so ␩NS and ␩S do not mix above Tc . lished results conclude that U(1)A remains broken, al- As a result, the mixing angle is not close to zero, as it is though recent results by the Columbia group have 57 at Tϭ0, but close to the ideal value. Furthermore, the questioned that conclusion (Christ, 1996). In any case, anomaly can only affect the nonstrange ␩, not the one should keep in mind that all results involve extrapo- strange one. Therefore, if the anomaly is sufficiently lations to mϭ0, and that both instanton and lattice strong, the ␩NS will be heavier than the ␩S . simulations suffer from certain artifacts in this limit. This phenomenon can also be understood from the Phenomenological aspects of the U(1)A anomaly at effective Lagrangian (344). The determinant is third or- finite temperature are usually discussed in terms of the der in the fields, so it only contributes to mass terms if effective Lagrangian (Pisarski and Wilczek, 1984), some of the scalar fields have a vacuum expectation 1 value. Above Tc , only the strange scalar has a vacuum † † ␮⌽ ͒ ϪTr ͑⌽ϩ⌽ ͒ expectation value, so only light flavors mix, and only theץ␮⌽͒͑ץ͑ ϭ Tr L 2 „ … „M … ␩NS receives a contribution to its mass from the ϩV͑⌽⌽†͒ϩc͑det ⌽ϩdet ⌽†͒, (344) anomaly. This effect was studied more quantitatively by Scha¨fer (1996). Singlet and octet, as well as strange and where ⌽ is a meson field in the (3,3) representation of nonstrange, eta correlation functions in the instanton U(3)ϫU(3), V(⌽⌽†)isaU(3)ϫU(3) symmetric po- liquid are shown in Fig. 38. Below T the singlet corre- tential (usually taken be quartic), is a mass matrix, c lation function is strongly repulsive, while the octet cor- and c controls the strength of the UM(1) breaking inter- A relator shows some attraction at larger distance. The off- action. If the coupling is chosen as cϭ␹ /(12f3 ), the top ␲ diagonal correlator is small and positive, corresponding to a negative mixing angle. The strange and nonstrange

57 eta correlation functions are very similar, which is a sign In addition, it is not clear whether lattice simulations ob- for strong flavor mixing. This is also seen directly from serve the peak in the spectrum at ␭ϭ0, which is due to isolated instantons. The Columbia group has measured the valence the off-diagonal correlator between ␩S and ␩NS . mass dependence of the quark condensate, which is a folded Above Tc , the picture changes. The off-diagonal version of the Dirac spectrum (Chandrasekharan, 1995). The singlet-octet correlator changes sign, and its value at in- result looks very smooth, with no hint of an enhancement at termediate distances ␶Ӎ0.5 fm is significantly larger. small virtuality. The strange and nonstrange eta correlators are very dif-

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 404 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

FIG. 38. Eta-meson correlation functions at TÞ0, normalized to free thermal correlators: (a) the flavor singlet; (b) the octet and off- diagonal singlet-octet; (c) the nonstrange eta correlation function; (d) the strange and off- diagonal strange-nonstrange eta correlation functions. The correlators are labeled as in Fig. 36.

ferent from each other. The nonstrange correlation correlators) the spectrum of the lowest states can be de- function is very repulsive, but no repulsion is seen in the termined with good accuracy. Although it is not directly strange channel. This clearly supports the scenario pre- related to the spectrum of physical excitations, the struc- sented above. Near Tc the eigenstates are essentially the ture of spacelike screening masses is of theoretical inter- strange and nonstrange components of the ␩, with the est and has been investigated in a number of lattice (Tar ␩S being the lighter of the two states. This picture is not and Kogut, 1987; Gocksch, 1991) and theoretical realized completely; ⌸S,NS does not vanish, and the sin- (Eletsky and Ioffe, 1988; Hansson and Zahed, 1992; glet eta is still somewhat more repulsive than the octet Koch et al., 1992; Hansson, Sporre, and Zahed, 1994) eta correlation function. This is due to the fact that the works. light-quark mass does not vanish. In particular, in this At finite temperatures, the antiperiodic boundary con- simulation, the ratio (muϩmd)/(2ms)ϭ1/7, which is ditions in the temporal direction require the lowest Mat- about three times larger than the physical mass ratio. subara frequency for fermions to be ␲T. This energy It is difficult to provide a quantitative analysis of tem- acts like a mass term for propagation in the spatial di- poral correlation functions in the vicinity of the phase rection, so quarks effectively become massive. At as- transition. At high temperatures the temporal direction ymptotically large temperatures, quarks propagate only in a Euclidean box becomes short, and there is no in the lowest Matsubara mode, and the theory under- unique way to separate out the contribution from ex- goes dimensional reduction (Appelquist and Pisarski, cited states. Nevertheless, under some simplifying as- 1981). The spectrum of spacelike screening states is then sumptions one can try to translate the correlation func- determined by a three-dimensional theory of quarks tions shown in Fig. 38 into definite predictions with chiral mass ␲T, interacting via the three- concerning the masses of the ␩ and ␩Ј. For definiteness, dimensional Coulomb law and the nonvanishing space- we use ideal mixing above T and fix the threshold for c like string tension (Borgs, 1985; Manousakis and Polo- the perturbative continuum at 1 GeV. In this case, the nyi, 1987). masses of the strange and nonstrange components of Dimensional reduction at large T predicts almost de- the ␩ at Tϭ126 MeV are given by m ϭ(0.420 ␩S generate multiplets of mesons and baryons with screen- Ϯ0.120) GeV and m ϭ(1.250Ϯ0.400) GeV. ␩S ing masses close to 2␲T and 3␲T. The splittings of me- sons and baryons with different spins can be understood in terms of the nonrelativistic spin-spin interaction. The 4. Screening masses resulting pattern of screening states is in qualitative Correlation functions in the spatial direction can be agreement with lattice results even at moderate tem- studied at arbitrarily large distances, even at finite tem- peratures TӍ1.5Tc . The most notable exception is the perature. This means that (in contrast to the temporal pion, whose screening mass is significantly below 2␲T.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 405

FIG. 39. Spectrum of spacelike screening masses in the instan- ton liquid as a function of the temperature, from Scha¨fer and Shuryak, 1996b. The masses are given in units of the lowest fermionic Matsubara frequency ␲T. The dotted lines corre- spond to the screening masses mϭ2␲T and 3␲T for mesons and baryons in the limit when quarks are noninteracting.

Screening masses in the instanton liquid are summa- FIG. 40. Lattice measurements of the temperature depen- rized in Fig. 39. First of all, the screening masses clearly dence of the topological susceptibility in pure gauge SU(3): (a) show the restoration of chiral symmetry as T T : chi- → c from Chu and Schramm, 1995; (b) from Alles et al., 1997. The ral partners like the ␲ and ␴ or the ␳ and a1 become dash-dotted line labeled P-Y in (a) corresponds to the semi- degenerate. Furthermore, the mesonic screening masses classical (Pisarski-Yaffe) result [Eq. (308)], while the dashed are close to 2␲T above Tc , while the baryonic ones are curve shows the rescaled formula discussed in the text. The close to 3␲T, as expected. Most of the screening masses solid line labeled PCAC is not very relevant for pure gauge are slightly higher than n␲T, consistent with a residual theory. In (b), the circles and squares correspond to two dif- chiral quark mass on the order of 120–140 MeV. The ferent topological charge operators. most striking observation is the strong deviation from this pattern seen in the scalar channels ␲ and ␴, with D. Instantons at finite temperature: Lattice studies screening masses significantly below 2␲T near the chiral Very few lattice simulations have focused on the role phase transition. The effect disappears around T of instantons at finite temperature, and all the results Ӎ1.5Tc . We also find that the nucleon-delta splitting that have been reported were obtained in the quenched does not vanish at the phase transition, but decreases approximation (Teper, 1986; Hoek et al., 1987; Chu and smoothly. Schramm, 1995; Ilgenfritz, Mu¨ ller-Preussker, and Meg- In summary, the pattern of screening masses seen in giolaro, 1995). In this section we shall concentrate on the instanton liquid is in agreement with the results of results obtained by the cooling method, in particular, the 58 lattice calculations. In particular, the attractive interac- work of Chu and Schramm (1995). In this work, the tem- tion provided by instanton molecules accounts for the perature was varied by changing the number of time fact that the ␲,␴ screening masses are much smaller than slices N␶ϭ2,4,...,16, while the spatial extent of the lat- 2␲T near Tc . tice and the coupling constant ␤ϭ6 were kept fixed. The topological susceptibility was calculated from to- pological charge fluctuations ͗Q2͘/V in the cooled con- 58There is one exception, which concerns the screening figurations. In the quenched theory, correlations be- masses in the longitudinal ⌸44 and transverse ⌸ii vector chan- tween instantons are not very important, and the nels. In agreement with dimensional reduction, we find m ␳i topological susceptibility provides a good estimate of the Ͼm , while lattice results reported by Tar and Kogut (1987) ␳4 instanton density. Figure 40(a) shows their results as a 4 find the opposite pattern. function of T.AtTϭ0, ␹topӍ(180 MeV) , in agree-

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 406 T. Scha¨fer and E. V. Shuryak: Instantons in QCD ment with the phenomenological value. The topological 1. The O(2) sigma model susceptibility is almost temperature independent below The O(2) model is also known as a the dϭ2 Heisen- the critical temperature (T Ӎ260 MeV in quenched c berg magnet or the XY model. It describes a two- QCD), but drops very fast above Tc . The temperature dimensional spin vector Sជ governed by the Hamiltonian dependence of ␹top above Tc is consistent with the De- bye screening suppression factor (308), but with a E 1 (Sជ ͒2, (345 ץshifted temperature, T2 (T2ϪT2). Clearly, these re- ϭ d2x͑ c T 2t ͵ ␮ sults support the arguments→ presented in Sec. VII.A.3. The conclusions of Chu and Schramm are consistent together with the constraint Sជ 2ϭ1. Here, tϭT/J is a di- with results reported by Ilgenfritz et al. (1995) and Alles, mensionless parameter and J the coupling constant. In D’Elia, and DiGiacomo (1996). We show the results of this section we follow the more traditional language of Alles et al. in Fig. 40(b). statistical mechanics rather than Euclidean quantum Chu and Schramm also extracted the average instan- field theory. This means that the coordinates are x,y and ton size from the correlation function of the topological the weight factor in the functional integral is charge density. Below T , they found ␳ϭ0.33 fm, inde- c exp(ϪE/T). Of course, one can always switch to field pendent of temperature, while at Tϭ334 MeV, they got theory language by replacing the energy by the action a smaller value, ␳ϭ0.26 fm. This result is in good agree- and the temperature by the coupling constant g2. ment with the Debye screening dependence discussed The statistical sum is Gaussian except for the con- above. straint. We can make this more explicit by parametrizing Finally, they considered instanton contributions to the ជ pressure and spacelike hadronic wave functions. They the two-dimensional spin vector S in the form S1 found that instantons contributed roughly 15% of the ϭcos ␪, S2ϭsin ␪. In this case, the energy is given by 2 2 -␮␪) , which would describe a nonץ)pressure at Tϭ334 MeV and 5% at Tϭ500 MeV. While E/Tϭ(1/2t)͐d x spacelike hadronic wave functions were dominated by interacting scalar field if it were not for the fact that ␪ is instantons at Tϭ0 (see Sec. VI.G), this was not true at a periodic variable. It is often useful to define the theory directly on the lattice. The partition function is given by TϾTc . This is consistent with the idea that spacelike wave functions above T are determined by the space- c d␪x et al. Zϭ exp ␤ ͓cos͑␪ Ϫ␪ ˆ ͒Ϫ1͔ , like string tension (Koch , 1992), which disappears ͟ ͚ x xϩe␮ ͵ ͩ x 2␲ ͪ x,eˆ during cooling. ͩ ␮ ͪ Clearly, studies with dynamic fermions are of great (346) interest. Some preliminary results have been obtained which is automatically periodic in ␪. The lattice spacing by Ilgenfritz et al. (private communication). Using a a provides an ultraviolet cutoff. We should also note ‘‘gentle cooling’’ algorithm with only a few cooling itera- that the O(2) model has a number of physical applica- tions in order to prevent instantons and anti-instantons tions. First of all, the model obviously describes a two- from annihilating each other, they found evidence for an dimensional magnet. In addition, fluctuations of the or- anticorrelation of topological charges in the time direc- der parameter in superconducting films of liquid 4He tion above Tc . This would be the first direct lattice evi- and the dynamics of dislocations in the melting of two- dence for the formation of polarized instanton mol- dimensional crystals are governed by effective O(2) ecules in the chirally symmetric phase. models. A two-dimensional theory with a continuous symme- try cannot have an ordered phase at nonzero tempera- tures. This means that, under ordinary circumstances, VIII. INSTANTONS IN RELATED THEORIES two-dimensional models cannot have a phase transition A. Two-dimensional theories at finite temperatures (Mermin and Wagner, 1966). The O(2) model is special because it has a phase transition Although two-dimensional theories should logically at TcӍ(␲J)/2Ͼ0 (although, in agreement with the be placed between the simplest quantum-mechanical Mermin-Wagner theorem, the transition is not charac- systems and Yang-Mills theories, we have postponed terized by a local order parameter). Both the low- and their discussion up to now in order not to disrupt the the high-temperature phase are disordered, but the main line of the review. Nevertheless, topological ob- functional form of the spin-correlation function K(x) jects play an important role in many low-dimensional ϭ͗Sជ (x)Sជ (0)͘ changes. In some sense, the whole region models. We do not want to give an exhaustive survey of TϽTc is critical because the correlation function exhib- these theories, but have selected two examples, the its a power-law decay (Beresinsky, 1971). For TϾTc the O(2) and O(3) models, which, in our opinion, provide spin correlator decays exponentially, and the theory has a few interesting lessons for QCD. As far as other theo- a mass gap. ries are concerned, we refer the reader to the extensive The mechanism of this phase transition was clarified literature, in particular on the Schwinger model (Smilga, in the seminal paper by Koesterlitz and Thouless (1973; 1994a; Steele, Subramanian, and Zahed, 1995) and two- see also the review of Kogut, 1979). Let us start with the dimensional QCD with fundamental or adjoint fermions low-temperature phase. In terms of the angle variable ␪, (Smilga, 1994b). the spin-correlation function is given by

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 407

rather than the power-law divergence observed in ordi- ␪͒2 ץϪ1 i␪͑x͒ Ϫi␪͑0͒ Ϫ1/͑2t͒͐d2x͑ K͑x͒ϭN ͵ d␪͑x͒e e e ␮ . nary phase transitions.59 (347) There is an obvious lesson we should like to draw from this: the two phases of the O(2) model resemble At low temperatures, the system is dominated by spin the two phases found in QCD (Sec. VII.B.4). In both waves, and we expect that fluctuations in ␪ are small. In cases the system is described by an (approximately) ran- this case we can ignore the periodic character of ␪ for dom ensemble of topological objects in one phase and the moment. Using the propagator for the ␪ field, by bound pairs of pseudoparticles with opposite charge G(r)ϭϪ1/(2␲)log(r/a), we have in the other. The only difference is that the high- and K͑r͒ϭexp͓tG͑r͔͒ϳrϪt/2␲. (348) low-temperature phases are interchanged. This is essen- tially due to the fact that temperature has a different The correlator shows a power law, with a temperature- meaning in both problems. In the O(2) model, instan- dependent exponent ␩ϭT/(2␲J). tons are excited at high temperature, while in QCD they In order to understand the phase transition in the are suppressed. O(2) model, we have to go beyond Gaussian fluctua- tions and include topological objects. These objects can be classified by a winding number 2. The O(3) sigma model 1 The O(2) model can be generalized to a dϭ2 spin (qϭ dxជ ٌជ ␪. (349 2␲ Ͷ • system with a three-dimensional (or, more generally, N-dimensional) spin vector Sជ of unit length. Unlike the Solutions with qϭϮ1 are called (anti) vortices. A solu- O(2) model, the O(3) model is not just a free field tion with qϭn is given by ␪ϭn␣ with ␣ϭarctan(y/x). theory for a periodic variable. In fact, in many ways the The energy of a vortex is model resembles QCD much more than the O(2) model -␪ 2 ␲n2 does. First of all, for NϾ2 the rotation group is nonץ E͑qϭ1͒ ␲ dr ϭ ϭ log͑R/a͒, (350) -t Abelian, and spin waves [with (NϪ1) polarizations] in ͪ␣ץͩ T t ͵ r teract. In order to treat the model perturbatively, it is where R is the IR cutoff. Since the energy is logarithmi- useful to decompose the vector field in the form Sជ cally divergent, one might think that vortex configura- ϭ(␴,␲ជ ), where ␲ជ is an (NϪ1)-dimensional vector field tions are irrelevant. In fact, they are crucial for the dy- and ␴ϭͱ 2. The perturbative analysis of the O(N) namics of the phase transition. 1Ϫ␲ជ model gives the beta function (Polyakov, 1975) The reason is that it is not the energy, but the free energy FϭEϪTS, which is relevant for the statistical NϪ2 sum. The entropy of an isolated vortex is essentially the ␤͑t͒ϭϪ t2ϩO͑t3͒, (352) logarithm of all possible vortex positions, given by S 2␲ 2 ϭlog͓(R/a) ͔. For temperatures TϾTcӍ(␲J)/2, entropy which shows that just like QCD, the O(N) model is dominates over energy, and vortices are important. The asymptotically free for NϾ2. In the language of statisti- presence of vortices implies that the system is even more cal mechanics, the beta function describes the evolution disordered, and the spin correlator decays exponentially. of the effective temperature as a function of the scale. A What happens to the vortex gas below Tc? Although negative beta function then implies that if the tempera- isolated vortices have infinite energy, vortex-antivortex ture is low at the atomic scale a, the effective tempera- pairs (molecules) have finite energy EӍ␲/ ture grows as one goes go to larger scales. Eventually, (2t)͓log(R/a)ϩconst͔, where R is the size of the mol- the reduced temperature is tϭO(1) (T is comparable to ecule. At low temperatures, molecules are strongly sup- J), fluctuations are large, and long-range order is de- pressed, but as the temperature increases they become stroyed. Unlike the Nϭ2 model, the O(3) model has more copious. Above the critical temperature, mol- only one critical point, tϭ0, and correlation functions ecules are ionized and a vortex plasma is formed. decay exponentially for all temperatures TϾ0. There is yet another way to look at this transition. We Furthermore, like non-Abelian gauge theory, the can decompose any field configuration into the contribu- O(3) model has classical instanton solutions. The topol- tion of vortices and a smooth field. Using this decompo- ogy is provided by the fact that field configurations are sition, one can see that the O(2) sigma model is equiva- maps from two-dimensional spacetime (compacted to a lent to a two-dimensional Coulomb gas. The Koesterlitz- Thouless transition describes the transition from a system of dipoles to an ionized plasma. The correlation 59The interested reader can find further details in review talks length of the spin system is nothing but the Debye on spin models given at the annual Lattice meetings. Using screening length in the Coulomb plasma. At the transi- cluster algorithms to fight critical slowdown, one can obtain tion point, the screening length has an essential singular- very accurate data. In Wolff (1989), the correlation length ity (Koesterlitz and Thouless, 1973) reaches about 70 lattice units, confirming Eq. (351). Neverthe- less, not all Koesterlitz-Thouless results are reproduced: e.g., const ␩ ␰ϳexp (351) the value of index ␩ (defined by K(r)ϳ1/r as T Tc)isnot ͩ ͉TϪT ͉1/2ͪ 1/4 but noticeably larger. → c

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 408 T. Scha¨fer and E. V. Shuryak: Instantons in QCD sphere) into another sphere which describes the orienta- theory. Because of the divergence in the instanton den- tion of Sជ . The winding number is defined by sity, the topological susceptibility in the O(3) has no continuum limit. This conclusion is supported by lattice 1 simulations (Michael and Spencer, 1994). The simula- Sជ ͒. (353) tions also indicate that, for large-size instantons, the size ץSជ ϫ ץ͑ qϭ d2x⑀ Sជ 8␲ ͵ ␮␯ • ␮ ␯ distribution is dNϳd␳/␳Ϫ3. This power differs from the Similar to QCD, solutions with integer winding number semiclassical result, but it agrees with the ␳ dependence and energy Eϭ4␲J͉q͉ can be found from the self- coming from the Jacobian alone, without the running duality equation coupling in the action. This result supports the idea of a frozen coupling constant discussed in Sec. III.C.3.

(␯Sជ. (354ץ␮Sជ ϭϮ⑀␮␯Sជϫץ A solution with qϭ1 can be constructed by means of the B. Instantons in electroweak theory stereographic projection In the context of this review, we cannot provide a 2 2 2␳x1 2␳x2 x Ϫ␳ detailed discussion of instantons and baryon-number Sជ ϭ , , . (355) ͩ x2ϩ␳2 x2ϩ␳2 x2ϩ␳2 ͪ violation in electroweak theory. Nevertheless, we briefly touch on this subject because electroweak theory pro- At large distances all spins point up, in the center the vides an interesting theoretical laboratory. The coupling spin points down, and at intermediate distances x all ϳ␳ constant is small, the instanton action is large, and the spins are horizontal, pointing away from the center. Like semiclassical approximation is under control. Unfortu- QCD, the theory is scale invariant on the classical level, nately, this means that under ordinary conditions tun- and the energy of an instanton is independent of the size neling events are too rare to be of physical importance. O ␳. Instantons in the (2) model are sometimes called Interesting questions arise when one tries to increase the Skyrmions, in analogy with the static (three- tunneling rate, e.g., by studying scattering processes with d dimensional) solitons introduced by Skyrme in the collision energies close to the barrier height, or pro- ϭ O 4 (4) model (Skyrme, 1961). cesses in the vicinity of the electroweak phase transition. The next logical step is the analog of the ’t Hooft Another interesting problem is what happens if we con- calculation of fluctuations around the classical instanton sider the Higgs expectation value to be a free parameter. solution. The result of the semiclassical calculation is When the Higgs vacuum expectation value is lowered, (Polyakov, 1975) electroweak theory becomes a strongly interacting 2 2 theory, and we encounter many of the problems we have d xd␳ d xd␳ to deal with in QCD. dNinstϳ 3 exp͑ϪE/T͒ϳ , (356) ␳ ␳ Electroweak theory is an SU(2)LϫU(1) chiral gauge which is divergent both for large and small radii.60 The theory coupled to a Higgs doublet. For simplicity we result for large ␳ is of course not reliable, since it is shall neglect the U(1) interactions in the following, i.e., based on the one-loop beta function. set the Weinberg angle to zero. The most important dif- Because the O(3) model shows so many similarities ference, as compared to QCD, is the fact that gauge with QCD, it is natural to ask whether one can learn invariance is spontaneously broken. In the ground state, anything of relevance for QCD. Indeed, the O(3) model the Higgs field acquires an expectation value, which has been widely used as a testing ground for new meth- gives masses to the W bosons as well as to the quarks ods in lattice gauge theory. The numerical results pro- and charged leptons. If the Higgs field has a nonzero vide strong support for the renormalization-group analy- vacuum expectation value the instanton is, strictly speaking, not a solution of the equations of motion. sis. For example, Caracciolo, Edwards, and Sokal (1995) Ϫ1 studied the O(3) model at correlation lengths as large Nevertheless, it is clear that if ␳Ӷv (where v is the as ␰/aϳ105. The results agree with state-of-the-art the- Higgs vacuum expectation value), the gauge fields are oretical predictions (based on the three-loop beta func- much stronger than the Higgs field, and there should be an approximate instanton solution.61 In the central re- tion and an overall constant determined from the Bethe Ϫ1 Ϫ1 ansatz) with a very impressive accuracy, on the order of gion xϽmW ,mH , the solution can be found by keep- a few percent. ing the instanton gauge field fixed and solving the equa- Unfortunately, studies of instantons in the O(3) tions of motion for the Higgs field. The result is model have not produced any significant insights. In par- 2 ticular, there are no indications that small-size (semiclas- x 0 ␾ϭ U , (357) sical) instantons play any role in the dynamics of the x2ϩ␳2 ͩ v/&ͪ

60This result marks the first nonperturbative ultraviolet diver- 61This notion can be made more precise using the constrained gence ever discovered. It is similar to the divergence of the instanton solution (Affleck, 1981). This technique is similar to density of instanton molecules for large Nf discussed in Sec. the construction that defines the streamline solution for an in- IX.D. stanton anti-instanton pair.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 409 where U is the color orientation matrix of the instanton. where F(⑀) is called the ‘‘holy grail’’ function. At low The result shows that the instanton makes a hole of size energy, F(0)ϭ1 and baryon-number violation is ␳ in the Higgs condensate. Scale invariance is lost, and strongly suppressed. The first correction is F(⑀)ϭ1 the instanton action depends on the Higgs vacuum ex- Ϫ 9 ⑀4/3ϩO(⑀2), indicating that the cross section rises pectation value (’t Hooft, 1976b) 8 with energy. Clearly, the problem is the behavior of F(⑀) near ⑀ϭ1. Most authors now seem to agree that 8␲2 2 2 2 Sϭ 2 ϩ2␲ ␳ v . (358) F(⑀) will not drop below FӍ1/2, implying that baryon- g number violation in pp collisions will remain unobserv- The tunneling rate is pϳexp(ϪS), and large instantons able (Maggiore and Shifman, 1992a, 1992b; Veneziano, with ␳ӷvϪ1 are strongly suppressed. 1992; Zakharov, 1992; Diakonov and Petrov, 1994). The The loss of scale invariance also implies that, in elec- question is of interest also for QCD because the holy troweak theory, the height of the barrier separating dif- grail function at ⑀ϭ1 is related to the instanton/anti- ferent topological vacua can be determined. There is a instanton interaction at short distance. In particular, tak- static solution with winding number 1/2, corresponding ing into account unitarity in the multi-W (or multigluon) to the top of the barrier, called the sphaleron production process corresponds to an effective (Klinkhammer and Manton, 1984). The sphaleron en- instanton/anti-instanton repulsion; see Sec. IV.A.1. ergy is EsphӍ4mW /␣WӍ10 TeV. The other question of interest for QCD is what hap- Electroweak instantons also have fermionic zero pens if the Higgs vacuum expectation value v is gradu- modes, and as usual the presence of these zero modes is ally reduced. As v becomes smaller the theory moves connected with the axial anomaly. Since only left- from the weak-coupling to the strong-coupling regime. handed fermions participate in weak interactions, nei- The vacuum structure changes from a very dilute system ther vector nor axial-vector currents are conserved. The of IA molecules to a more dense (and more interesting) ’t Hooft vertex contains all twelve weak doublets, nonperturbative vacuum. Depending on the number of light fermions, one should eventually reach a confining, ͑␯e ,e͒, ͑␯␮ ,␮͒, ͑␯␶ ,␶͒,3*͑u,d͒, QCD-like phase. In this phase, leptons are composite, but the low-energy effective action is probably similar to 3*͑c,s͒,3*͑t,b͒, (359) the one in the Higgs phase (Abbott and Farhi, 1981; Claudson, Farhi, and Jaffe, 1986). Unfortunately, the where the factors of three come from color. Each dou- importance of nonperturbative effects has never been blet provides one fermionic zero mode, the flavor de- studied. pending on the isospin orientation of the instanton. The Let us comment only on one element that is absent in ’t Hooft vertex violates both baryon and lepton number. QCD, the scalar-induced interaction between instan- These processes are quite spectacular because all fami- tons. Although the scalar interaction is of order O(1) lies have to be involved, for example, and therefore suppressed with respect to the gauge in- Ϫ2 ¯ ϩ ϩ ϩ teraction O(g ), it is long range if the Higgs mass is uϩd dϩ¯sϩ2¯cϩ3¯tϩe ϩ␮ ϩ␶ , (360) Ϫ1 → small. For mH ӷRӷ␳ the IA interaction is (Yung, uϩd u¯ϩ2¯sϩ¯cϩ¯tϩ2b¯ϩ␯ ϩ␯ ϩ␶ϩ. (361) 1988) → e ␮ Note that ⌬Bϭ⌬LϭϪ3, so BϩL is violated, but B ␳2 1 ϪL is conserved. Unfortunately, the probability of such S ϭ4␲2a2␳2 1ϩ ͓2͑u Rˆ ͒2Ϫ1͔ ϩO . Higgs ͫ R2 • ͬ ͩ R4 ͪ an event is tiny, proportional to the square of the tun- 2 2 Ϫ169 (363) neling amplitude Pϳexp(Ϫ16␲ /gw)ϳ10 (’t Hooft, 1976a), many orders of magnitude smaller than any Unlike the gluonic dipole interaction, it does not vanish known radioactive decay. ˆ 2 if averaged over all orientations, ͗(u•R) ͘ϭ1/4. This Many authors have discussed the possibility of in- means that the scalar interaction can provide coherent creasing the tunneling rate by studying processes near attraction for distances RmHϽ1, which is of the order 2 4 2 the electroweak phase transition (Kuzmin, Rubakov, v ␳ n/mH where n is the instanton density. This is large and Shaposhnikov, 1985) or scattering processes involv- if the Higgs mass is small. ing energies close to the sphaleron barrier EӍEsph . Another unusual feature of Yung interaction (363) is Since E Ӎ10 TeV, this energy would have been acces- ˆ sph that it is repulsive for u•Rϭ1 (which is the most attrac- sible at the SSC and will possibly be within reach at the tive orientation for the dipole interaction). This would LHC. The latter idea became attractive when it was re- suggest that, for a light Higgs mass, there is no small-R alized that associated multi-Higgs and W production in- problem. This question was studied by Velkovsky and creases the cross section (Espinosa, 1990; Ringwald, Shuryak (1993). For the complete Yung ansatz (which is 1990). On general grounds one expects a good approximation to the full streamline solution) the approximate result (363) is valid only for RϾ10␳, 4␲ E ˆ ␴ ϳexp Ϫ F , (362) while for smaller separation the dependence on u•R is ⌬͑BϩL͒ ͫ ␣ ͩ E ͪͬ W sph reversed.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 410 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

C. Supersymmetric QCD 1. The instanton measure and the perturbative beta function

Supersymmetry (SUSY) is a powerful theoretical con- The simplest (Nϭ1) supersymmetric non-Abelian cept, which is of great interest in constructing field theo- gauge theory is SU(2) SUSY gluodynamics, defined by ries beyond the standard model. In addition, supersym- 1 i metric field theories provide a very useful theoretical a a a b ϭϪ 2 F␮␯F␮␯ϩ 2 ␭ ͑D” ͒ab␭ , (364) laboratory and have been responsible for a number of L 4g 2g important advances in our understanding of the ground a state of strongly coupled field theories. We have already where the gluino field ␭ is a Majorana fermion in the seen one example of the usefulness of supersymmetry in adjoint representation of SU(2). More complicated isolating instanton effects in the context of SUSY quan- theories can be constructed by adding additional matter tum mechanics, in Sec. II.B. In that case, SUSY implies fields and scalars. N extended supersymmetry has N that perturbative contributions to the vacuum energy gluino fields as well as additional scalars. Clearly, super- vanish and allows a precise definition of the instanton/ symmetric gluodynamics has ␪ vacua and instanton solu- anti-instanton contribution. In the following, we shall tions. The only difference as compared to QCD is that see that supersymmetry can be used in very much the the fermions carry adjoint color, so there are twice as same way in the context of gauge-field theories. many fermion zero modes. If the model contains scalars In general, one should keep in mind that SUSY theo- fields that acquire a vacuum expectation value, instan- ries are just ordinary field theories with a very specific tons are approximate solutions, and the size integration matter content and certain relations between different is automatically cut off by the scalar vacuum expectation coupling constants. Eventually, we hope to understand value. These theories usually resemble electroweak non-Abelian gauge theories for all possible matter sec- theory more than they do QCD. tors. In particular, we want to know how the structure of At first glance, instanton amplitudes seem to violate the theory changes as one goes from QCD to supersym- supersymmetry: the number of zero modes for gauge metric generalizations, where many exact statements fields and fermions does not match, while scalars have about instantons and the vacuum structure are known. no zero modes at all. However, one can rewrite the tun- Deriving these results often requires special techniques neling amplitude in manifestly supersymmetric form that go beyond the scope of this review. For details of (Novikov et al., 1983). We shall not do this here, but the supersymmetric instanton calculus we refer the stick to the standard notation. The remarkable observa- reader to the extensive review by Amati et al. (1988). tion is that the determination of the tunneling amplitude Nevertheless, we have tried to include a number of in- in SUSY gauge theory is actually simpler than in QCD. teresting results and explain them in standard language. Furthermore, with some additional input, one can deter- As theoretical laboratories, SUSY theories have sev- mine the complete perturbative beta function from the eral advantages over ordinary field theories. We have already mentioned one of them: Nonrenormalization tunneling amplitude. theorems imply that many quantities do not receive per- The tunneling amplitude is given by turbative contributions, so instanton effects are more 2␲ 2␲ ng/2 easily identified. In addition to that, SUSY gauge theo- n͑␳͒ϳexp Ϫ MngϪnf/2 ͩ ␣ ͪ ͩ ␣ ͪ ries usually have many degenerate classical vacua. These degeneracies cannot be lifted to any order in perturba- d␳ 4 k 2 tion theory, and instantons often play an important role ϫd x 5 ␳ d ␰f , (365) ␳ ͟f in determining the ground state. In most cases, the clas- sical vacua are characterized by scalar field vacuum ex- where all factors can be understood from the ’t Hooft pectation values. If the scalar field vacuum expectation calculation discussed in Sec. II.C.4. There are ngϭ4Nc value is large, one can perform reliable semiclassical cal- bosonic zero modes that have to be removed from the culations. Decreasing the scalar vacuum expectation determinant and these give one power of the regulator value, one moves towards strong coupling, and the dy- mass M each. Similarly, each of the nf fermionic zero namics of the theory is nontrivial. Nevertheless, super- modes gives a factor M1/2. Introducing collective coordi- symmetry restricts the functional dependence of the ef- nates for the bosonic zero modes gives a Jacobian ͱS0 fective potential on the scalar vacuum expectation value for every zero mode. Finally, d2␰ is the integral over the (and other parameters, like masses or coupling con- fermionic collective coordinates, and ␳k is the power of stants), so that instanton calculations can often be con- ␳ needed to give the correct dimension. Supersymmetry tinued into the strong-coupling domain. now ensures that all non-zero-mode contributions ex- Ultimately, we should like to understand the behavior actly cancel. More precisely, the subset of SUSY trans- of SUSY QCD as we introduce soft supersymmetry formations that does not rotate the instanton field itself breaking terms and send the masses of the gluinos and mixes fermionic and bosonic non-zero modes but anni- squarks to infinity. Not much progress has been hilates zero modes. This is why all non-zero modes can- achieved in this direction, but at least for small breaking cel, but zero modes can be unmatched. Note that as a the calculations are feasible and some lessons have been result of this cancellation, the power of M in the tunnel- learned. ing amplitude is an integer.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 411

Renormalizability demands that the tunneling ampli- One immediately recognizes two interesting special tude be independent of the regulator mass. This means cases. For Nϭ4, the beta function vanishes, and the that the explicit M dependence of the tunneling ampli- theory is conformal. In the case Nϭ2, the denominator tude and the M dependence of the bare coupling have to vanishes, and the one-loop result for the beta function is cancel. As in QCD, this allows us to determine the one- exact. loop coefficient of the beta function bϭ(4ϪN)Nc ϪNf . Again note that b is an integer, a result that 2. Nϭ1 supersymmetric QCD would appear very mysterious if we did not know about instanton zero modes. In the previous section we used instantons only as a In supersymmetric theories one can even go one step tool to simplify a perturbative calculation. Naturally, the further and determine the beta function to all loops (No- next question is whether instantons cause important dy- vikov et al., 1983; Vainshtein et al., 1986). For that pur- namic effects in SUSY theories. Historically, there has pose let us write down the renormalized instanton mea- been a good deal of interest in SUSY breaking by in- sure stantons, first discovered in the quantum-mechanical model discussed in Sec. II.B. The simplest field theory in ng/2 2␲ n Ϫn 2␲ n /2 which SUSY is broken by instantons is the n͑␳͒ϳexp Ϫ M g f /2 Z g ͩ ␣ ͪ ͩ ␣ ͪ g SU(2)ϫSU(3) model (Affleck, Dine, and Seiberg, 1984a, R 1985; Vainshtein, Shifman, and Zakharov, 1985). How- d␳ Ϫ1/2 4 k 2 ever, nonperturbative SUSY breaking does not take ϫ ͟ Zf d x 5 ␳ ͟ d ␰f , (366) ͩ f ͪ ␳ f place in supersymmetric QCD, so we shall not discuss it where we have introduced the field renormalization fac- any further. An effect that is more interesting in our context is tors Zg,f for the bosonic/fermionic fields. Again, non- renormalization theorems ensure that the tunneling am- gluino condensation in SUSY gluodynamics [Eq. (364); plitude is not renormalized at higher orders (the Novikov et al., 1983]. For SU(2) color, the gluino con- densate is most easily determined from the correlator cancellation between the non-zero-mode determinants a a␣ b b␤ persists beyond one loop). For gluons the field renor- ͗␭␣␭ (x)␭␤␭ (0)͘. In SU(N), one has to consider a a␣ malization (by definition) is the same as the charge the N-point function of ␭␣␭ . In supersymmetric theo- ries, gluinos have to be massless, so the tunneling ampli- renormalization Zgϭ␣R /␣0 . Furthermore, supersym- metry implies that the field renormalization is the same tude is zero. However, in the N-point correlation func- for gluinos and gluons. This means that the only new tion all zero modes can be absorbed by external sources, quantity in Eq. (366) is the anomalous dimension of the similar to the axial anomaly in QCD. Therefore there is a nonvanishing one-instanton contribution. In agree- quark fields, ␥␺ϭd log Zf /d log M. Again, renormalizability demands that the amplitude ment with SUSY Ward identities, this contribution is x be independent of M. This condition gives the independent. Therefore one can use cluster decomposi- 3 Novikov-Shifman-Vainshtein-Zakharov beta function tion to extract the gluino condensate ͗␭␭͘ϭϮA⌳ [in (Novikov et al., 1983) which, in the case Nϭ1, reads SU(2)], where A is a constant that is fixed from the single-instanton calculation. 3 g 3NcϪNfϩNf␥␺͑g͒ There are a number of alternative methods for calcu- ␤͑g͒ϭϪ 2 2 2 . (367) 16␲ 1ϪNcg /8␲ lating the gluino condensate in SUSY gluodynamics. For example, it has been suggested that ͗␭␭͘ can be calcu- The anomalous dimension of the quarks has to be cal- lated directly using configurations with fractional charge culated perturbatively. To leading order, it is given by (Cohen and Gomez, 1984). In addition to that, one can include matter fields, make the theory Higgs-like, and g2 N2Ϫ1 c 4 then integrate out the matter fields (Novikov et al., ␥␺͑g͒ϭϪ 2 ϩO͑g ͒. (368) 8␲ Nc 1985a; Shifman and Vainshtein, 1988). This method gives a different coefficient for the gluino condensate, a The result (367) agrees with explicit calculations up to problem that was recently discussed by Kovner and Shif- three loops (Jack, Jones, and North, 1997). Note that the man (1997). beta function is scheme dependent beyond two loops, so The next interesting theory is Nϭ1 SUSY QCD, in order to make a comparison with high-order pertur- where we add Nf matter fields (quarks ␺ and squarks ␾) bative calculations, one has to translate from the Pauli- in the fundamental representation. Let us first look at Vilars scheme to a more standard perturbative scheme, the Novikov-Shifman-Vainshtein-Zakharov beta func- e.g., MS. tion. For Nϭ1, the beta function blows up at g2 2 * In theories without quarks, the Novikov-Shifman- ϭ8␲ /Nc , so the renormalization-group trajectory can- Vainshtein-Zakharov result determines the beta func- not be extended beyond this point. Recently, Kogan and tion completely. For N-extended supersymmetric gluo- Shifman have suggested that at this point the standard dynamics, we have phase meets the renormalization-group trajectory of a different (nonasymptotically free) phase of the theory 3 g Nc͑4ϪN͒ (Kogan and Shifman, 1995). The beta function vanishes ␤͑g͒ϭϪ . (369) 16␲2 1Ϫ 2ϪN͒N g2/ 8␲2͒ at g2 /(8␲2)ϭ͓N (3N ϪN )͔/͓N (N2Ϫ1)͔, where we ͑ c ͑ * c c f f c

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 412 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

finity, and Nfϭ1 SUSY QCD does not have a stable ground state. The vacuum can be stabilized by adding a mass term. For nonzero quark mass m the vacuum en- ergy is given by

16m⌳5 32⌳10 4⌳5 2 ⑀ ϭ2m2v2Ϫ ϩ ϭ2 mvϪ , vac g4v2 g8v6 ͩ g4v3 ͪ (371)

where the first term is the classical Higgs mass term, the second is the one-instanton contribution, and the third is due to molecules. In this case, the theory has a stable FIG. 41. Instanton contributions to the effective potential in ground state at v2ϭϮ2⌳5/2/(g2m1/2). Since SUSY is un- supersymmetric QCD. broken, the vacuum energy is exactly zero. Let us note that in the semiclassical regime mӶ⌳Ӷv everything is have used the one-loop anomalous dimension. This is under control: instantons are small ␳ϳvϪ1, individual reliable if g is small, which we can ensure by choosing instantons are rare, molecules form a dilute gas, nR4 * N ϱ and N in the conformal window 3N /2ϽN ϳ(v/⌳)10, and the instanton and anti-instanton inside a c→ f c f Ͻ3Nc . Recently, Seiberg has shown that the conformal molecule are well separated, RIA /␳ϳ1/ͱg. Supersym- point exists for all Nf in the conformal window (even if metry implies that the result remains correct even if we Nc is not large) and has clarified the structure of the leave the semiclassical regime. This means, in particular, theory at the conformal point (Seiberg, 1994). that all higher-order instanton corrections [O(⌳15) etc] Let us now examine the vacuum structure of SUSY have to cancel exactly. Checking this explicitly might QCD for Ncϭ2 and Nfϭ1. In this case we have one provide a very nontrivial check of the instanton calculus. Majorana fermion (the gluino), one Dirac fermion (the Recently, significant progress has been made in deter- quark), and one scalar squark (or Higgs) field. The mining the structure of the ground state of Nϭ1 super- quark and squark fields do not have to be massless. We symmetric QCD for arbitrary Nc and Nf (Intriligator shall denote their mass by m, while v is the vacuum and Seiberg, 1996). Seiberg showed that the situation expectation value of the scalar field. In the semiclassical discussed above is generic for NfϽNc , a stable ground 5 regime vӷ⌳, the tunneling amplitude (366) is O(⌳ ). state can only exist for nonzero quark mass. For Nf 4 2 The ’t Hooft effective Lagrangian is of the form ␭ ␺ , ϭNcϪ1, the mÞ0 contribution to the potential is an containing four quark and two gluino zero modes. As instanton effect. In the case NfϭNc , the theory has chi- usual, instantons give an expectation value to the ral symmetry breaking (in the chiral limit m 0) and 62 → ’t Hooft operator. However, due to the presence of confinement. For NfϭNcϩ1, there is confinement but Yukawa couplings and a Higgs vacuum expectation no chiral symmetry breaking. Unlike QCD, the ’t Hooft value, they can also provide expectation values for op- anomaly matching conditions can be satisfied with mass- erators with fewer than six fermion fields. In particular, less fermions. These fermions can be viewed as elemen- combining the quarks with a gluino and a squark tad- tary fields in the dual (or ‘‘magnetic’’) formulation of the pole, we can construct a two-gluino operator. Instantons theory. For Ncϩ1ϽNfϽ3Nc/2, the magnetic formula- therefore lead to gluino condensation (Affleck, Dine, tion of the theory is IR free, while for 3Nc/2ϽNf and Seiberg, 1984b). Furthermore, using two more Ͻ3Nc the theory has an infrared fixed point. Finally, for Yukawa couplings one can couple the gluinos to exter- NfϾ3Nc , asymptotic freedom is lost and the electric nal quark fields. Therefore, if the quark mass is nonzero, theory is IR free. we get a finite density of individual instantons O(m⌳5/v2); see Fig. 41. As in ordinary QCD, there are also instanton/anti- 3. Nϭ2 supersymmetric gauge theory instanton molecules. The contribution of molecules to The work of Seiberg has shown that supersymmetry the vacuum energy can be calculated either directly or sufficiently constrains the effective potential in Nϭ1 su- indirectly, using the single-instanton result and argu- persymmetric QCD that the possible vacuum states for ments based on supersymmetry. This provides a nice all N and N can be determined. For Nϭ2 extended check on the direct calculation because in this case we c f supersymmetric QCD, these constraints are even more have a rigorous definition of the contribution from mol- powerful. Witten and Seiberg have been able to deter- ecules. Calculating the graph shown in Fig. 41(b), one mine not just the effective potential, but the complete finds (Yung, 1988) 10 IA 32⌳ ⑀ ϭ , (370) 62 vac g8v6 There is no instanton contribution to the superpotential, but instantons provide a constraint on the allowed vacua in the which agrees with the result originally derived by Af- quantum theory. To our knowledge, the microscopic mecha- fleck et al. (1984b) using different methods. The result nism for chiral symmetry breaking in this theory has not been implies that the Higgs expectation value is driven to in- clarified.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 413 low-energy effective action (Seiberg and Witten, 1994). For kϭ1, this was first checked by Finnel and Pouliot Again, the techniques used in this work are outside the (1995) in the case of SU(2) and by Ito and Sasakura scope of this review. However, instantons play an impor- (1996) in the more general case of SU(Nc). The basic tant role in this theory, so we should like to discuss a few idea is to calculate the coefficient of the interesting results. ’t Hooft interaction ϳ␭2␺2. The gluino ␭ and the Nϭ2 supersymmetric gauge theory contains two Ma- Higgsino ␺ together have eight fermion zero modes. a a a jorana fermions ␭␣ , ␺␣ and a complex scalar ␾ , all in Pairing zero modes using Yukawa couplings of the type the adjoint representation of SU(Nc). In the case of N (␭␺)␾ and the nonvanishing Higgs vacuum expectation ˜ ϭ2 SUSY QCD, we add Nf multiplets (q␣ ,˜q␣ ,Q,Q)of value, we can see that instantons induce a four-fermion ˜ quarks q␣ ,˜q␣ and squarks Q,Q in the fundamental rep- operator. In an impressive tour de force, the calculation resentation. In general, gauge invariance is broken and of the coefficient of this operator was recently extended the Higgs field develops an expectation value ͗␾͘ to the two-instanton level63 (Dorey, Khoze, and Mattis, 3 ϭa␶ /2. The vacua of the theory can be labeled by a 1996a; Aoyama et al., 1996). For Nfϭ0, the result is (gauge-invariant) complex number uϭ(1/2)͗tr ␾2͘.If 15⌳4 9!⌳8 the Higgs vacuum expectation value a is large (aӷ⌳), 2 2 S4fϭ dx͑␺ ␭ ͒ 2 6 ϩ 12 2 10 ϩ¯ , (375) the semiclassical description is valid and uϭ(1/2)a2.In ͵ ͫ8␲ a 3ϫ2 ␲ a ͬ Ϫ1 this case, instantons are small ␳ϳa and the instanton which agrees with the Witten-Seiberg solution. This is ensemble is dilute n␳4ϳ⌳4/a4Ӷ1. also true for NfÞ0, except in the case Nfϭ4, where a In the semiclassical regime, the effective Lagrangian is discrepancy appears. This is the special case in which the given by coefficient of the perturbative beta function vanishes. 1 1 Seiberg and Witten assume that the nonperturbative sd 2 ␮ † ␾ ͒ ␶(a) is the same as the corresponding bare coupling in ץ␮␾͒͑ץeffϭ Im Ϫ Љ͑␾͒ ͑F ͒ ϩ͑ L 4␲ ͫ F ͩ 2 ␮␯ the Lagrangian. But the explicit two-instanton calcula- 1 tion shows that even in this theory the charge is renor- ,␭¯ ϩ ٞ͑␾͒␭␴␮␯␺Fsd malized by instantons (Dorey, Khoze, and Mattisץ␺¯ϩi␭ץϩi␺ ” ” F ␮␯ ͪ & 1996b). In principle, these calculations can be extended 1 order by order in the instanton density. The result pro- 2 2 ϩ ЉЉ͑ ␾ ͒␺ ␭ ϩ auxϩ¯ , (372) vides a very nontrivial check on the instanton calculus. 4 F ͬ L For example, in order to obtain the correct two- sd ˜ instanton contribution, one has to use the most general where F ϭF␮␯ϩiF␮␯ is the self dual part of the field ␮␯ (ADHM) two-instanton solution, not just a linear super- strength tensor, aux contains auxiliary fields, and ‘‘•••’’ denotes higher derivativeL terms. Note that the effective position of two instantons. low-energy Lagrangian contains only the light fields. In Instantons also give a contribution to the expectation 2 the semiclassical regime, this is the U(1) part of the value of ␾ . Pairing off the remaining zero modes, we gauge field (the ‘‘photon’’) and its superpartners. Using see that the semiclassical relation uϭa2/2 receives a cor- arguments based on electric-magnetic duality, Seiberg rection (Finnel and Pouliot, 1995) and Witten determined the exact prepotential (␾). 2 4 8 F a ⌳ ⌳ From the effective Lagrangian, we can immediately read uϭ ϩ 2 ϩO 6 . (376) off the effective charge at the scale a, 2 a ͩ a ͪ ␶͑a͒ 4␲i ␪ More interesting are instanton corrections to the effec- Љ͑a͒ϭ ϭ ϩ , (373) tive charge ␶. The solution of Seiberg and Witten can be F 2 g2͑a͒ 2␲ written in terms of an elliptic integral of the first kind, which combines the coupling constant g and the ␪ angle. 2 2 4 The Witten-Seiberg solution also determines the anoma- K͑ͱ1Ϫk ͒ uϪͱu Ϫ4⌳ ␶͑u͒ϭi , k2ϭ . (377) lous magnetic moment ٞ and the four-fermion vertex K͑k͒ uϩͱu2Ϫ4⌳4 ЉЉ. In general, the structureF of the prepotential is given Fby In the semiclassical domain, this result can be written as the one-loop perturbative contribution plus an infinite ϱ i 4ϪN ␾2 i ⌳ ͑4ϪNf͒k ͑ f͒ 2 2 series of k-instanton terms. Up to the two-instanton ͑␾͒ϭ ␾ log 2 Ϫ ͚ k␾ . F 8␲ ͩ ⌳ ͪ ␲ kϭ1 F ͩ ␾ ͪ level, we have [for a more detailed discussion of the (374) nonperturbative beta function, see Bonelli and Matone (1996)] The first term is just the perturbative result with the one-loop beta functions coefficient. As noted in Sec. 8␲ 2 2a2 3⌳4 3ϫ5ϫ7⌳8 ϭ log Ϫ Ϫ ϩ . (378) VIII.C.1, there are no corrections from higher loops. In- g2 ␲ ͫ ͩ ⌳2 ͪ a4 8a8 ¯ͬ stead, there is an infinite series of power corrections. The coefficient is proportional to ⌳(4ϪNf)k, which is Fk exactly what one would expect for a k-instanton contri- 63Supersymmetry implies that there is no instanton/anti- bution. instanton contribution to the prepotential.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 414 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

It is interesting to note that instanton corrections tend to well as numerical calculations that take the ’t Hooft in- accelerate the growth of the coupling constant g. This is teraction into account to all orders. consistent with what was found in QCD by considering The progress made in understanding the physics of how small-size instantons renormalize the charge of a instantons in lattice calculations has been of equal im- larger instanton (Callan et al., 1978a). However, the re- portance. We now have data concerning the total den- sult is opposite to the trend discussed in Sec. III.C.3 sity, the typical size, the size distribution, and correla- (based on the instanton size distribution and lattice beta tions between instantons. Furthermore, there are function), which suggests that in QCD the coupling runs detailed checks on the mechanism of U(1)A violation more slowly than suggested by perturbation theory. and on the behavior of many more correlation functions If the Higgs vacuum expectation value is reduced, the under cooling. Recent investigations have begun to fo- instanton corrections in Eq. (377) start to grow and com- cus on many interesting questions, like the effects of pensate for the perturbative logarithm. At this point the quenching, correlations of instantons with monopoles, expansion (378) becomes unreliable, but the exact solu- etc. tion of Seiberg and Witten is still applicable. In the semi- In the following we shall first summarize the main re- classical regime, the spectrum of the theory contains sults concerning the structure of the QCD vacuum and monopoles and dyons with masses proportional to ␶.As its hadronic excitations, then discuss the effects of finite the Higgs vacuum expectation value is reduced, these temperature, and finally try to place QCD in a broader particles can become massless. In this case, the expan- context, comparing the vacuum structure of QCD with sion of the effective Lagrangian in terms of the original other non-Abelian field theories. (electrically charged) fields breaks down, but the theory can be described in terms of their (magnetically charged) dual partners. B. Vacuum and hadronic structure

The instanton liquid model is based on the assump- IX. SUMMARY AND DISCUSSION tion that nonperturbative aspects of the gluonic vacuum, like the gluon condensate, the vacuum energy density, A. General remarks or the topological susceptibility, are dominated by small- size (␳Ӎ1/3 fm) instantons. The density of tunneling Finally, we should like to summarize the main points events is nӍ1fmϪ4. These numbers imply that the of this review, discuss some of the open problems, and gauge fields are very inhomogeneous, with strong fields Ϫ1 Ϫ2 provide an outlook. In general, semiclassical methods in (G␮␯ϳg ␳ ) concentrated in small regions of space- quantum mechanics and field theory are well developed. time. In addition, the gluon fields are strongly polarized, We can reliably calculate the contribution of small-size the field strength locally being either self-dual or anti- (large-action) instantons to arbitrary Green’s functions. self-dual. Problems arise when we leave this regime and attempt Quark fields, on the other hand, cannot be localized to calculate the contribution from large instantons or inside instantons. Isolated instantons have unpaired chi- close instanton/anti-instanton pairs. While these prob- ral zero modes, so the instanton amplitude vanishes if lems can be solved rigorously in some theories (as in quarks are massless. In order to get a nonzero probabil- quantum mechanics or in some SUSY field theories), in ity, quarks have to be exchanged between instantons QCD-like theories we still face a number of unresolved and anti-instantons. In the ground state, zero modes be- problems and therefore have to follow a somewhat more come completely delocalized, and chiral symmetry is phenomenological approach. Nevertheless, the main broken. As a consequence, quark-antiquark pairs with point of this review is that important progress has been the quantum numbers of the pion can travel infinitely made in this context. The phenomenological success of far, and we have a Goldstone pion. the instanton liquid model is impressive, and initial at- This difference in the distribution of vacuum fields tempts to check the underlying assumptions explicitly on leads to significant differences in gluonic and fermionic the lattice are very encouraging. correlation functions. Gluonic correlators are much With this review, we want not only to acquaint the more short range, and as a result the mass scale for glue- reader with the theory of instantons in QCD, but also to balls, m0ϩϩӍ1.5 GeV, is significantly larger than the draw attention to the large number of observables, in typical mass of non-Goldstone mesons, m␳ϭ0.77 GeV. particular hadronic correlation functions at zero and fi- The polarized gluon fields lead to large spin splittings for nite temperatures, that have already been calculated in both glueballs and ordinary mesons. In general, we can the instanton liquid model. While some of these predic- group all hadronic correlation functions into three tions have been compared with phenomenological infor- classes: (i) Those that receive direct instanton contribu- mation or lattice results, many others still await confron- tions that are attractive, ␲,K,0ϩϩ glueball, N,..., (ii) tation with experiment or the lattice. The instanton those with direct instantons effects that are repulsive, liquid calculations were made possible by a number of ␩Ј,␦,0ϩϪ glueball,..., and (iii) correlation functions ϩϩ technical advances. We now have a variety of ap- with no direct instanton contributions ␳,a1 ,2 glueball, proaches at our disposal, including the single instanton, ⌬,... . Aswe have repeatedly emphasized throughout the mean-field and random-phase approximations, as this review, already this simple classification based on

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 415

first-order instanton effects gives a nontrivial under- structure is still very far from being complete. Clearly, standing of the bulk features of hadronic correlation the most important question concerns the mechanism of functions. confinement and its role in hadronic structure. In the In addition, the instanton liquid allows us to go into meantime, experiments continue to provide interesting much greater detail. Explicit calculations of the full cor- new puzzles: the spin of the nucleon, the magnitude and relation functions in a large number of hadronic chan- polarization of the strange sea, the isospin asymmetry in nels have been performed. These calculations require the light u,d quark sea, etc. only two parameters to be fixed. One is the scale param- eter ⌳ and the other characterizes the scale at which the effective repulsion between close pairs sets in. With C. Finite temperature and chiral restoration these parameters fixed from global properties of the vacuum, we not only find a very satisfactory description Understanding the behavior of hadrons and hadronic of the masses and couplings of ground-state hadrons, but matter at high temperature is the ultimate goal of the we also reproduce the full correlation function whenever experimental heavy-ion program. These studies comple- they are available. Of course, the instanton model ment our knowledge of hadronic structure at zero tem- reaches its limits as soon as perturbative or confinement perature and density and provide an opportunity to ob- effects become dominant. This is the case, for example, serve rearrangements in the structure of the QCD when one attempts to study bound states of heavy vacuum directly. quarks or tries to resolve high-lying radial excitations of Generalizing the instanton liquid model to finite tem- light hadrons. peratures is straightforward in principle. Nevertheless, How does this picture compare with other approaches the role of instantons at finite temperatures has been to hadronic structure? As far as the methodology is con- reevaluated during the past few years. There is evidence cerned, the instanton approach is close to (and to some that instantons are not suppressed near Tc , but disap- extent a natural outgrowth of) the QCD sum-rule pear only at significantly higher temperatures. Only method. Moreover, the instanton effects explain why the after instantons disappear does the system become OPE works in some channels and fails in others (those a perturbative plasma.64 with direct instanton contributions). The instanton liq- In addition, we have argued that the chiral transition uid provides a complete picture of the ground state, so is due to the dynamics of the instanton liquid itself. The that no assumptions about higher-order condensates are phase transition is driven by a rearrangement of the in- required. It also allows the calculations to be extended stanton liquid, going from a (predominantly) random to large distances, so that no matching is needed. phase at small temperature to a correlated phase of In the quark sector, the instanton model provides a instanton/anti-instanton molecules at high temperature. picture similar to the Nambu and Jona-Lasinio model. Without having to introduce any additional parameters, There is an attractive quark-quark interaction that this picture provides the correct temperature scale for causes quarks to condense and binds them into light me- the transition and agrees with standard predictions con- sons and baryons. However, the instanton liquid pro- cerning the structure of the phase diagram. vides a more microscopic mechanism, with a more direct If instantons are bound into topologically neutral connection to QCD, and relates the different coupling pairs at TϾT , they no longer generate a quark conden- constants and cutoffs in the Nambu and Jona-Lasinio c sate. However, they still contribute to the gluon conden- model. sate, the effective interaction between quarks, and the Instead of going into comparisons with the plethora of equation of state. Therefore instanton effects are poten- hadronic models that have been proposed over the tially very important in understanding the plasma at years, let us emphasize two points that we feel are im- moderate temperatures Tϭ(1–3)T . We have begun portant. Hadrons are not cavities which are empty inside c to explore some of these consequences in greater detail, (devoid of nonperturbative fields) as the bag model sug- in particular the behavior of spatial and temporal corre- gests. Indeed, matrix elements of electric and magnetic lation functions across the transition region. While fields inside the nucleon can be determined from the spacelike screening masses essentially agree with the re- trace anomaly (Shuryak, 1978b; Ji, 1995), showing that sults of lattice calculations, interesting phenomena are the density of instantons and the magnitude of the gluon seen in temporal correlation functions. We find evidence condensate inside the nucleon is only reduced by a few that certain hadronic modes survive in the high- percent. Hadrons are excitations of a very dense me- temperature phase. Clearly, much work remains to be dium, and this medium should be understood first. Fur- thermore, spin splittings in glueballs (2ϩϩϪ0ϩϩ,...) and light hadrons (␳Ϫ␲,...) are not small, so it makes 64We should mention that even at asymptotically high tem- no sense to treat them perturbatively. This was directly perature there are nonperturbative effects in QCD, related to checked on the lattice: spin splittings are not removed the physics of magnetic (three-dimensional) QCD. However, by cooling, which quickly eliminates all perturbative these effects are associated with the scale g2T, which is small contributions. compared to the typical momenta of the order T. This means What is the prospect for future work on hadronic that the corrections to quantities like the equation of state are structure? Of course, our understanding of hadronic small.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 416 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

FIG. 42. Schematic phase diagram of QCD (a) and supersymmetric QCD (b) as a func- tion of the number of colors Nc and the num- ber of flavors Nf .

done in order to improve our understanding of the high- dom instantons are pushed to larger sizes. Since in this temperature phase. regime the semiclassical approximation becomes unreli- able, we do not know how to calculate the rate of ran- dom instantons at large N . D. The big picture f For strongly correlated pairs (molecules) the trend is Finally, we should like to place QCD in a broader exactly opposite. The density of pairs is essentially inde- perspective and discuss what is known about the phase pendent of the quark condensate and only determined structure of non-Abelian gauge theories (both ordinary by the interaction of the two instantons. From purely dimensional considerations one expects the density of and supersymmetric) based on the gauge group SU(Nc) molecules to be dn ϳd␳⌳2b␳2bϪ5, which means that with Nf quark flavors. For simplicity, we shall restrict m ourselves to zero temperature and massless fermions. the typical size becomes smaller as Nf is increased This means that the theory has no dimensional param- (Shuryak, 1987). If NfϾ11Nc/2, we have bϽ2, and the eters other than ⌳. The phase diagram of ordinary and density of pairs is ultraviolet divergent [see the dashed line in Fig. 42(a)]. This phenomenon is similar to the UV SUSY QCD in the Nc-Nf plane is shown in Fig. 42. For divergence in the O(3) nonlinear ␴ model. Both are simplicity, we have plotted Nc and Nf as if they were continuous variables.65 We should emphasize that, while examples of UV divergencies of a nonperturbative na- the location of the phase boundaries can be rigorously ture. Most likely they do not have significant effects on established in the case of SUSY QCD, the phase dia- the physics of the theory. Since the typical instanton size gram of ordinary QCD is just a guess, guided by some of is small, one would expect that the contribution of mol- the results mentioned below. ecules at large Nf could be reliably calculated. However, Naturally, we are most interested in the role of instan- since the binding inside the pair increases with Nf , the separation of perturbative and nonperturbative fluctua- tons in these theories. As Nf is increased above the value 2 or 3 (relevant to real QCD), the two basic com- tions becomes more and more difficult. ponents of the instanton ensemble, random (individual) Rather than speculate about these effects, let us go instantons and strongly correlated instanton/anti- back and consider very large Nf . The solid line labeled instanton pairs (molecules), are affected in very differ- bϭ0 in Fig. 42 corresponds to a vanishing first coeffi- ent ways. Isolated instantons can only exist if the quark cient of the beta function, bϭ(11/3)NcϪ(2/3)Nfϭ0in condensate is nonzero, and the instanton density con- QCD and bϭ3NcϪNfϭ0 in SUSY QCD. Above this tains the factor (͉͗q¯q͉͘␳3)Nf which comes from the fer- line, the coupling constant decreases at large distances, mion determinant. As a result, small-size instantons are and the theory is IR free. Below this line, the theory is expected to have an infrared fixed point (Belavin and strongly suppressed as Nf is increased. This suppression factor does not affect instantons with a size larger than Migdal, 1974; Banks and Zaks, 1982). As discussed in ␳ϳ q¯q Ϫ1/3. This means that as N is increased, ran- Sec. III.C.3, this is due to the fact that the sign of the the ͉͗ ͉͘ f 2 second coefficient of the beta function bЈϭ34Nc /3 Ϫ13NcNf/3ϩNf /Nc is negative while the first one is 65In a sense, at least the number of flavors is a continuous positive. As a result, the beta function has a zero at g2 /(16␲2)ϭϪb/bЈ. This number determines the limit- variable. One can gradually remove a massless fermion by in- * creasing its mass. ing value of the charge at large distances. Note that the

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 417

fixed point is not destroyed by higher-order perturbative changes or large-size instantons, the lattice has to in- effects, since we can always choose a scheme where clude the relevant scales,66 which is not the case in the higher-order coefficients vanish. The presence of an IR present lattice simulations. fixed point implies that the theory is conformal, which The present lattice results more closely resemble the means correlation functions show a power-law decay at results in the interacting instanton liquid discussed in large distances. There is no mass gap, and the long- Sec. VII.B.4. There we found a line of (rather robust distance behavior is characterized by the set of critical first-order) transitions that touches Tϭ0 near NfӍ5. A exponents. large drop in the quark condensate in going from Nf Where is the lower boundary of the conformal do- ϭ2,3 to Nfϭ4 observed by the Columbia group (Chen main? A recent perturbative study based on the 1/Nf and Mawhinney, 1997) may very well be the first indica- expansion (Gracey, 1996) suggested that the IR fixed tion of this phenomenon. point might persist all the way down to Nfϭ6 (for Nc Again, there is no inconsistency between the interact- ϭ3). However, the critical coupling would become ing instanton calculation and the results of Appelquist larger and nonperturbative phenomena might become et al. The interacting instanton liquid model calculation important. For example, Appelquist, Terning, and takes into account the effects of small instantons only. If Wijewardhana (1996) argued that if the coupling con- small instantons do not break chiral symmetry, then stant reached a critical value the quark-antiquark inter- long-range Coulomb forces or large instantons can still 67 action could be sufficiently strong to break chiral sym- be responsible for chiral symmetry breaking. This metry. In their calculation, this would happen for Nc mechanism was studied by a number of authors (Bar- f ducci et al., 1988; Aoki et al., 1990), and the correspond- Ӎ4Nc [see the dashed line in Fig. 42(a)]. It was then realized that instanton effects could also ing quark condensate is about an order of magnitude smaller68 than the one observed for N ϭ2. We would be important (Appelquist and Selipsky, 1997). If the f therefore argue that, in a practical sense, QCD has two critical coupling is small, even large instantons have a different phases with chiral symmetry breaking, one in large action, Sϭ8␲2/g2 ӷ1, and the semiclassical ap- * which the quark condensate is large and generated by proximation is valid. As usual, we expect random instan- small-size instantons and one in which the condensate is tons to contribute to chiral symmetry breaking. Accord- significantly smaller and due to Coulomb forces or large ing to estimates made by Appelquist and Selipsky instantons. The transition regime is indicated by a wavy (1997), the role of instantons is comparable to that of line in Fig. 42(a). Further studies of the mechanisms of perturbative effects in the vicinity of Nfϭ4Nc . Chiral chiral symmetry breaking for different Nf are needed symmetry breaking is dominated by large instantons before final conclusions can be drawn. with size ␳ϳ͉͗q¯q͉͘Ϫ1/3Ͼ⌳Ϫ1, while the perturbative re- Finally, although experiment tells us that confinement Ϫ1 gime ␳Ͻ⌳ contributes very little. For even larger in- and chiral symmetry breaking go together for Nfϭ2,3, stantons, ␳ӷ͉͗q¯q͉͘Ϫ1/3, fermions acquire a mass due to the two are independent phenomena, and it is conceiv- chiral symmetry breaking and effectively decouple from able that there are regions in the phase diagram where gluons. This means that for large distances the charge only one of them takes place. It is commonly believed evolves as in pure gauge theory, and the IR fixed point is that confinement implies chiral symmetry breaking, but only an approximate feature, useful for analyzing the not even that is entirely clear. In fact, SUSY QCD with 69 theory above the decoupling scale. NfϭNcϩ1 provides a counterexample. Little is known about the phase structure of multifla- For comparison we also show the phase diagram of vor QCD from lattice simulations. Lattice QCD with up (Nϭ1) supersymmetric QCD [Fig. 42(b)]. As discussed to 240 flavors was studied by Iwasaki et al. (1996), who in Sec. VIII.C, the phase structure of these theories was showed that, as expected, the theory is trivial for bϾ0. recently clarified by Seiberg and collaborators. In this The paper also confirms the existence of an infrared fixed point for N у7(Nϭ3). In Fig. 42(a) we have f c 66This cannot be done by simply tuning the bare coupling to marked these results by open squares. Other groups the critical value for chiral symmetry breaking ␤Ӎ1 because have studied QCD with Nfϭ8 (Brown et al., 1992), 12 lattice artifacts create a chirally asymmetric and confining (Kogut and Sinclair, 1988) and 16 (Damgaard et al., phase already at ␤Ӎ4 – 5. Therefore one has to start at weak 1997) flavors. All of these simulations find a chirally coupling and then go to sufficiently large physical volumes to asymmetric and confining theory at strong coupling and reach the chiral symmetry-breaking scale. 67 a bulk transition to a chirally symmetric phase (at ␤ The situation is different at large temperatures because in 2 that case both Coulomb forces and large instantons are Debye ϭ2Nc /g Ͼ4.73, 4.47, and 4.12, respectively). It may appear that these results are in contradiction screened. 68This can be seen from the fact that these authors need an with the results of Appelquist et al. mentioned above, unrealistically large value of ⌳ Ӎ500 MeV to reproduce according to which chiral symmetry should be broken QCD the experimental value of f␲ . for NfϽ12 (Ncϭ3), but this is not the case, since the 69It is usually argued that anomaly matching shows that con- condensate is expected to be exponentially small. In finement implies chiral symmetry breaking for NcϾ2, but other words, in order to reproduce the subtle mecha- again SUSY QCD provides examples in which anomaly- nism of chiral breaking by large-distance Coulomb ex- matching conditions work out in subtle and unexpected ways.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 418 T. Scha¨fer and E. V. Shuryak: Instantons in QCD case, the lower boundary of the conformal domain is at sults of studying instantons in the vicinity of the phase Nfϭ(3/2)Nc . Below this line the dual theory based on transition in full QCD were reported. As an example for the gauge group SU(NfϪNc) loses asymptotic freedom. the use of improved actions we mention results of Colo- In this case, the excitations are IR free ‘‘dual quarks’’ or rado group (DeGrand, Hasenfratz, and Kovacs, 1997) in composite light baryons in terms of the original theory. pure gauge SU(2). They find that chiral symmetry Remarkably, the t’ Hooft matching conditions between breaking and confinement are preserved by inverse the original (short-distance) theory and the dual theory blocking (unlike ‘‘cooling’’) preserves even close based on a completely different gauge group work out instanton–antiinstanton pairs. The instanton size distri- exactly. The theory becomes confining for NfϭNcϩ1 bution is peaked at ␳Ӎ0.2 fm, and large instantons are and NfϭNc . In the first case, chiral symmetry is pre- suppressed. served, and the low-energy excitations are massless Low-lying eigenstates of the Dirac operator were baryons. In the second case, instantons modify the ge- studied by the MIT group (Ivanenko and Negele, 1997). ometry of the space of possible vacua, the point where They find that the corresponding wave functions are the excitations are massless baryons is not allowed, and spatially correlated with the locations of instantons, pro- chiral symmetry has to be broken. Note that, in ordinary viding support for the picture of the quark condensate as QCD, the ’t Hooft matching conditions cannot be satis- a collective state built from instanton zero modes. In fied for a confining phase without chiral symmetry addition to that, they studied the importance of low- breaking (for NcÞ2). For an even smaller number of lying states in hadronic correlation functions. They dem- 6 flavors, 0ϽNfрNcϪ1, massless SUSY QCD does not onstrate that the lowest ϳ100 modes (out of ϳ10 ) are have a stable ground state. The reason is that instanton/ sufficient to quantitatively reproduce the hadronic anti-instanton molecules generate a positive vacuum en- ground state contribution to the ␳ and ␲ meson correla- ergy density, which decreases with the Higgs expectation tion functions. value, so that the ground state is pushed to infinitely In addition to that, first attempts were made to study large Higgs vacuum expectation value. instantons in the vicinity of the finite temperature phase We have not shown the phase structure of SUSY transition in full QCD (de Forcrand, Perez, Hetrick, and QCD with NϾ1 gluinos. In some cases (e.g., Nϭ4, Nf Stamatescu, 1998). Using the cooling technique to iden- ϭ0orNϭ2, Nfϭ4), the beta function vanishes and the tify instantons, they verified the T dependence of the theory is conformal, although instantons may still cause instanton density discussion in Sec. VII.A. They observe a finite charge renormalization. As already mentioned, polarized instanton-antiinstanton pairs above Tc, but the low-energy spectrum of the Nϭ2 theory was re- these objects do not seem to dominate the ensemble. cently determined by Seiberg and Witten. The theory This point definitely deserves further study, using im- does not have chiral symmetry breaking or confinement, proved methods and smaller quark masses. but it contains monopoles/dyons which become massless In the main text we stated that there is no confine- as the Higgs vacuum expectation value is decreased. ment in the instanton model. Recently, it was claimed This can be used to trigger confinement when the theory that instantons generate a linear potential with a slope is perturbed to Nϭ1. close to the experimental value 1 GeV/fm (Fukushima, To summarize, there are many open questions con- Suganuma, Tanaka, Toki, and Sasaki, 1997). This cerning the phase structure of QCD-like theories and prompted Chen, Negele, and Shuryak (1998) to reinves- many issues to be explored in future studies, especially tigate the issue, and perform high statistics numerical on the lattice. The location of the lower boundary of the calculations of the heavy-quark potential in the instan- conformal domain and the structure of the chirally ton liquid at distances up to 3 fm. The main conclusions asymmetric phase in the domain Nfϭ4 – 12 should cer- are (i) the potential is larger and significantly longer tainly be studied in more detail. Fascinating results have range than the dilute gas result Eq. (208); (ii) a random clarified the rich (and sometimes rather exotic) phase ensemble with a realistic size distribution leads to a po- structure of SUSY QCD. To what extent these results tential that is linear even for large R Ӎ 3 fm; (iii) the will help our understanding of nonsupersymmetric theo- slope of the potential is still too small, KӍ200 MeV/fm. ries remains to be seen. In any event, it is certainly clear This means that the bulk of the confining forces still has that instantons and anomalies play a very important, if to come from some (as of yet?) unidentified objects with not dominant, role in both cases. small action. These objects may turn out to be large in- Note added in proof. stantons, but that would still imply that the main contri- Since this manuscript was prepared, the subject of in- bution is not semiclassical. Nevertheless, the result that stantons in QCD has continued to see many interesting the heavy quark potential is larger than expected is good developments. We would like to briefly mention some of news for the instanton model. It implies that even these, related to instanton searches on the lattice, the weakly bound states and resonances can be addressed relation of instantons with confinement, instantons and within in the model. charm quarks, instantons at finite chemical potential, We also did not discuss the role of charm quarks in and instantons in supersymmetric theories. the QCD vacuum. However, the color field inside a 2 Significant progress was made studying topology on small-size instantons G␮␯Ӎ1 GeV is comparable to the 2 2 the lattice using improved operators, renormalization charm quark mass squared mc Ӎ2 GeV , so one might group techniques, and fermionic methods. Also, first re- expect observable effects due to the polarization of

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 419 charm quark inside ordinary hadrons. Recently, it was that (at least in the case Nϭ2) this is not the correct way suggested that CLEO observations of an unexpectedly to take the large Nc limit (if we want to keep the physics large branching ratio B ␩ЈK (as well as inclusive B unchanged). ␩Јϩ•••) provide a smoking→ gun for such effects (Hal- A comparison of the running of the effective charge in →perin and Zhitnitsky, 1998). The basic idea is that these Nϭ2, 1 SUSY QCD and QCD was performed by Ran- decays proceed via Cabbibo-unsuppressed b ¯ccs dall, Rattazzi, and Shuryak (1998). In Nϭ2 SUSY QCD, transition, followed by ¯cc ␩Ј. The charm content→ of the Seiberg-Witten solution shows that instantons accel- the ␩Ј in the instanton liquid→ was estimated by Shuryak erate the growth of the coupling. As a result, the cou- and Zhitnitsky (1997), and the result is consistent with pling blows up at a scale ⌳NPϾ⌳QCD where the pertur- what is needed to understand the CLEO data. Another bative coupling is still small. A similar phenomenon interesting possibility raised in Halperin and Zhitnitsky takes place in QCD if the instanton correction to the running coupling is estimated from the formula of Cal- (1997) is that polarized charm quarks give a substantial lan, Dashen, and Gross. This might help to explain why contribution to the spin ͚ of the nucleon. A recent in- in QCD the nonperturbative scale ⌳ Ӎ⌳ Ӎ1 GeV stanton calculation only finds a contribution in the range NP ␹ SB Ͼ⌳ Ӎ200 MeV. ⌬c/͚ϭϪ(0.08–0.20) (Blotz and Shuryak, 1997), but the QCD value of ⌬c remains an interesting question for future deep inelastic scattering experiments (e.g., Compass at ACKNOWLEDGMENTS CERN). In the context of (polarized) nucleon structure It is a pleasure to thank our friends and collaborators functions we should also mention interesting work on who have contributed to this review or to the work pre- leading and nonleading twist operators (see Balla, sented here. In particular, we should like to thank our Polyakov, and Weiss, 1997, and references therein). collaborators J. Verbaarschot, M. Velkovsky, and A. Finally, Bjorken (1997) discussed the possibility that Blotz. We should like to thank M. Shifman for many instantons contribute to the decay of mesons containing valuable comments on the original manuscript. We have real ¯cc pairs (Bjorken, 1997). A particularly interesting also benefitted from discussions with G. E. Brown, D. I. case is the ␩c, which has three unusual 3-meson decay Diakonov, J. Negele, M. Novak, A. V. Smilga, A. Vain- channels (␩Ј␲␲, ␩␲␲, and KK␲), which contribute shtein, and I. Zahed. roughly 5% each to the total width. This fits well with ¯ ¯ ¯ the typical instanton vertex uuddss. In general, all of APPENDIX: BASIC INSTANTON FORMULAS these observables offer the chance to detect nonpertur- bative effects deep inside the semi-classical domain. 1. Instanton gauge potential Initial efforts were made to understand the instanton liquid at finite chemical potential (Scha¨fer, 1997). The We use the following conventions for Euclidean gauge a a suggestion made in this work is that the role that mol- fields: The gauge potential is A␮ϭA␮(␭ /2), where the ecules play in the high temperature phase is now played SU(N) generators satisfy ͓␭a,␭b͔ϭ2ifabc␭c and are a b ab by more complicated ‘‘polymers’’ that are aligned in the normalized according to tr͓␭ ␭ ͔ϭ2␦ . The covariant -␮ϪiA␮ and the fieldץtime direction. More importantly, it was suggested that derivative is given by D␮ϭ instanton lead to the formation of diquark condensates strength tensor is in high density matter (Alford, Rajagopal, and Wilczek, (␯A␮Ϫi͓A␮ ,A␯͔. (A1ץ␮A␯ϪץRapp, Scha¨fer, Shuryak, and Velkovsky, 1997). In F␮␯ϭ ;1997 the high density phase chiral symmetry is restored, but SU(3) color is broken by a Higgs mechanism. In our conventions, the coupling constant is absorbed into the gauge fields. Standard perturbative notation Instanton effects in SUSY gauge theories continue to corresponds to the replacement A gA . The single- be a very active field. For N ϭ 2 (Seiberg-Witten) theo- ␮ ␮ instanton solution in regular gauge is→ given by ries the n-instanton contribution was calculated explic- itly (Dorey, Khoze, and Mattis, 1997; Dorey, Hol- 2␩a␮␯x␯ lowood, Khoze, and Mattis, 1997), and as a by-product Aa ϭ , (A2) ␮ x2ϩ 2 these authors also determine the (classical) n-instanton ␳ measure in the cases N ϭ 1 and N ϭ 0 (non-SUSY). and the corresponding field strength is Yung also determined the one-instanton contribution to higher derivative operators beyond the SW effective La- 2 a 4␩a␮␯␳ grangian (Yung, 1997). Another very interesting result is G␮␯ϭϪ 2 2 2 , (A3) the generalization of the Seiberg-Witten solution to an ͑x ϩ␳ ͒ arbitrary number of colors Nc (Douglas and Shenker, 192␳4 1995). This result sheds some light on the puzzling prob- a 2 ͑G␮␯͒ ϭ 2 2 4 . (A4) lem of instantons in the large Nc limit. The large Nc ͑x ϩ␳ ͒ 2 limit is usually performed with g Nc held fixed, and in that case instantons amplitudes are suppressed by The gauge potential and field strength in singular gauge exp(ϪNc). However Douglas and Shenker (1995) found are

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 420 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

2 2␩¯ a␮␯x␯␳ with Aa ϭ , (A5) ␮ x2͑x2ϩ␳2͒ 1 ϱ 2 2 M͑r͒ϭ dpp ͉␸͑p͉͒ J1͑pr͒. (A16) r ͵0 4␳2 Ga ϭϪ ␮␯ ͑x2ϩ␳2͒2 The Fourier transform of the zero-mode profile is given by x␣x␯ x␮x␣ ϫ ␩¯ Ϫ2␩¯ Ϫ2␩¯ . (A6) d a␮␯ a␮␣ x2 a␣␯ x2 ␸͑p͒ϭ␲␳2 I ͑x͒K ͑x͒ϪI ͑x͒K ͑x͒ . ͩ ͪ dx „ 0 0 1 1 …ͯ xϭp␳/2 Finally, an n-instanton solution in singular gauge is (A17) given by a 3. Properties of ␩ symbols (␯ ln ⌸͑x͒, (A7ץA␮ϭ␩¯ a␮␯ n 2 We define four-vector matrices ␳i ⌸͑x͒ϭ1ϩ . (A8) ͚ xϪz ͒2 Ϯ iϭ1 ͑ i ␶␮ ϭ͑␶ជ,ϯi͒, (A18) Note that all instantons have the same color orientation. where ␶a␶bϭ␦abϩi⑀abc␶c and For a construction that gives the most general ϩ Ϫ a n-instanton solution, see Atiyah et al. (1977). ␶␮ ␶␯ ϭ␦␮␯ϩi␩a␮␯␶ , (A19) Ϫ ϩ ¯ a 2. Fermion zero modes and overlap integrals ␶␮ ␶␯ ϭ␦␮␯ϩi␩a␮␯␶ , (A20) with the symbols given by In singular gauge, the zero-mode wave function ␩ iD” ␾0ϭ0 is given by ␩a␮␯ϭ⑀a␮␯ϩ␦a␮␦␯4Ϫ␦a␯␦␮4 , (A21) 1 ⌽ (U ⑀ , (A9) ␩¯ a␮␯ϭ⑀a␮␯Ϫ␦a␮␦␯4ϩ␦a␯␦␮4 . (A22 ץ ␾ ϭ ͱ⌸ a␯ ͫ”ͩ⌸ͪͬ ab ␮b 2&␲␳ ␯␮ The ␩ symbols are (anti) self-dual in the vector indices where ⌽ϭ⌸Ϫ1. For the single-instanton solution, we get 1 1 ␩a␮␯ϭ ⑀␮␯␣␤␩a␣␤ , ␩¯a␮␯ϭϪ ⑀␮␯␣␤␩¯a␣␤ ␳ 1 1Ϫ␥5 x” 2 2 ␾a␯͑x͒ϭ 2 2 3/2 Uab⑀␮b . ␲ ͑x ϩ␳ ͒ ͫͩ 2 ͪ ͱx2ͬ ␯␮ (A10) ␩a␮␯ϭϪ␩a␯␮ . (A23) The instanton-instanton zero-mode density matrices are We have the following useful relations for contractions involving ␩ symbols 1 1Ϫ␥5 ␾ ͑x͒ ␾†͑y͒ ϭ ␸ ͑x͒␸ ͑y͒ x␥ ␥ y I i␣ J j␤ 8 I J ͩ ” ␮ ␯ ” 2 ͪ ij ␩a␮␯␩b␮␯ϭ4␦ab , (A24) Ϫ ϩ † ͑UI␶␮ ␶␯ UJ͒␣␤ , (A11) ␩a␮␯␩a␮␳ϭ3␦␯␳ , (A25) i 1ϩ␥5 ␾ ͑x͒ ␾† ͑y͒ ϭϪ ␸ ͑x͒␸ ͑y͒ x␥ y I i␣ A j␤ 2 I A ͩ ” ␮” 2 ͪ ␩ ␩ ϭ12, (A26) ij a␮␯ a␮␯ Ϫ † ͑UI␶ U ͒␣␤ , (A12) ␮ A ␩a␮␯␩a␳␭ϭ␦␮␳␦␯␭Ϫ␦␮␭␦␯␳ϩ⑀␮␯␳␭ , (A27)

i 1Ϫ␥5 ␾ ͑x͒ ␾†͑y͒ ϭ ␸ ͑x͒␸ ͑y͒ x␥ y ␩ ␩ ϭ␦ ␦ ϩ⑀ ␩ , (A28) A i␣ I j␤ 2 A I ͩ ” ␮ ” 2 ͪ a␮␯ b␮␳ ab ␯␳ abc c␯␳ ij ϩ † ␩ ␩¯ ϭ0. (A29) ͑UA␶␮ UI ͒␣␤ , (A13) a␮␯ b␮␯ with The same relations hold for ␩¯ a␮␯ , except for ␳ 1 ␸͑x͒ϭ . (A14) ␩¯ a␮␯␩¯ a␳␭ϭ␦␮␳␦␯␭Ϫ␦␮␭␦␯␳Ϫ⑀␮␯␳␭ . (A30) ␲ ͱx2͑x2ϩ␳2͒3/2 The overlap matrix element is given by Some additional relations are

4 † ⑀abc␩b␮␯␩c␳␭ϭ␦␮␳␩a␯␭Ϫ␦␮␭␩a␯␳ TAIϭ ͵ d x␾A͑xϪzA͒iD” ␾I͑xϪzI͒ ϩ␦␯␭␩a␮␳Ϫ␦␯␳␩a␮␭ , (A31) Ϫ † 1 d ϭr␮ Tr͑UI␶␮ UA͒ 2 M͑r͒, (A15) 2␲ r dr ⑀␭␮␯␴␩a␳␴ϭ␦␳␭␩a␮␯ϩ␦␳␯␩a␭␮ϩ␦␳␮␩a␯␭ . (A32)

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 421

4. Group integration Appelquist, T., and S. B. Selipsky, 1997, Phys. Lett. B 400, 364. In order to perform averages over the color group, we Appelquist, T., J. Terning, and L. C. R. Wijewardhana, 1996, Phys. Rev. Lett. 77, 1214. need the following integrands over the invariant Arnold, P. B., and M. P. Mattis, 1991, Mod. Phys. Lett. A 6, SU(N ) measure: c 2059. Atiyah, M. F., N. J. Hitchin, V. G. Drinfeld, and Y. I. Manin, 1 † 1977, Phys. Lett. A 65, 185. ͵ dUUijUklϭ ␦jk␦li , (A33) Nc Atiyah, M. F., and N. S. Manton, 1989, Phys. Lett. B 222, 438. Bagan, E., and S. Steele, 1990, Phys. Lett. B 243, 413. Bali, G. S., K. Schilling, A. Hulsebos, A. C. Irving, C. Michael, † † ͵ dUUijUklUmnUop and P. W. Stephenson, 1993, Phys. Lett. B 309, 378. Balitskii, I. I., and V. M. Braun, 1993, Phys. Lett. B 314, 237. Balitskii, I., and V. Braun, 1995, Phys. Lett. B 346, 143. 1 1 a a ϭ 2 ␦jk␦li␦no␦mpϩ 2 ͑␭ ͒jk͑␭ ͒li Balitsky, I. I., and A. Scha¨fer, 1993, Nucl. Phys. B 404, 639. Nc 4͑NcϪ1͒ Balitsky, I. I., and A. V. Yung, 1986, Phys. Lett. B 168, 113. b b Balla, J., M. V. Polyakov, and C. Weiss, 1997, hep-ph/9707515. ϫ͑␭ ͒no͑␭ ͒mp . (A34) Baluni, V., 1979, Phys. Rev. D 19, 2227. Additional results can be found in Creutz (1983). These Banks, T., and A. Casher, 1980, Nucl. Phys. B 169, 103. results can be rearranged using the SU(N) Fierz trans- Banks, T., and A. Zaks, 1982, Nucl. Phys. B 82, 196. formation, Barducci, A., R. Casalbuoni, S. D. Curtis, D. Dominici, and R. Gatto, 1988, Phys. Rev. D 38, 238. Belavin, A. A., and A. A. Migdal, 1974, preprint (Landau a a 2 ͑␭ ͒ij͑␭ ͒klϭϪ ␦ij␦klϩ2␦jk␦il . (A35) Inst.) 74-0894. Nc Belavin, A. A., A. M. Polyakov, A. A. Schwartz, and Y. S. Tyupkin, 1975, Phys. Lett. B 59, 85. REFERENCES Bell, J. S., and R. Jackiw, 1969, Nuovo Cimento A 60, 47. Belyaev, V. M., and B. L. Ioffe, 1982, Sov. Phys. JETP 83, 976. Abbott, L. F., and E. Farhi, 1981, Phys. Lett. B 101, 69. Beresinsky, V., 1971, Zh. Eksp. Teor. Fiz. 61, 1144. Abrikosov, A. A., 1983, Sov. J. Nucl. Phys. 37, 459. Bernard, C., T. Blum, C. DeTar, S. Gottlieb, U. M. Heller, J. Actor, A., 1979, Rev. Mod. Phys. 51, 461. E. Hetrick, K. Rummukainen, R. Sugar, D. Toussaint, and M. Adami, C., T. Hatsuda, and I. Zahed, 1991, Phys. Rev. D 43, Wingate, 1997, Phys. Rev. Lett. 78, 598. 921. Bernard, C. W., N. H. Christ, A. H. Guth, and E. J. Weinberg, Adler, S. L., 1969, Phys. Rev. 177, 2426. 1977, Phys. Rev. D 16, 2967. Affleck, I., 1981, Nucl. Phys. B 191, 429. Bernard, C., M. C. Ogilvie, T. A. DeGrand, C. DeTar, S. V. Affleck, I., M. Dine, and N. Seiberg, 1984a, Phys. Lett. B 140, Gottlieb, A. Krasnitz, R. L. Sugar, and D. Toussaint, 1992, 59. Phys. Rev. Lett. 68, 2125. Affleck, I., M. Dine, and N. Seiberg, 1984b, Nucl. Phys. B 241, Bernard, V., and U. G. Meissner, 1988, Phys. Rev. D 38, 1551. 493. Bjorken, J. D., 1997, hep-ph/9706524 Affleck, I., M. Dine, and N. Seiberg, 1985, Nucl. Phys. B 256, Blask, W., U. Bohn, M. Huber, B. Metsch, and H. Petry, 1990, 557. Z. Phys. A 337, 327. Aleinikov, A., and E. Shuryak, 1987, Yad. Fiz. 46, 122. Blotz, A., and E. Shuryak, 1997a, Phys. Rev. D 55, 4055. Alfaro, V. D., S. Fubini, and G. Furlan, 1976, Phys. Lett. B 65, Blotz, A., and E. Shuryak, 1997b, hep-ph/9710544. 163. Blum, T., L. Ka¨rkka¨innen, D. Toussaint, and S. Gottlieb, 1995, Alford, M., K. Rajagopal, and F. Wilczek, 1997, Phys. Rev. D 51, 5133. hep-ph/9711395. Bochkarev, A. I., and M. E. Shaposhnikov, 1986, Nucl. Phys. B Alles, B., A. DiGiacomo, and M. Gianetti, 1990, Phys. Lett. B 268, 220. 249, 490. Bogomolny, E. B., 1980, Phys. Lett. B 91, 431. Alles, B., M. Campostrini, and A. DiGiacomo, 1993, Phys. Bonelli, G., and M. Matone, 1996, Phys. Rev. Lett. 76, 4107. Rev. D 48, 2284. Borgs, C., 1985, Nucl. Phys. B 261, 455. Alles, B., M. D’Elia, and A. DiGiacomo, 1996, Nucl. Phys. B Boyd, G., 1996, preprint, hep-lat/9607046. 494, 281. Boyd, G., S. Gupta, F. Karsch, and E. Laerman, 1994, Z. Phys. Amati, D., K. Konishi, Y. Meurice, G. C. Rossi, and G. Ven- C 64, 331. eziano, 1988, Phys. Rep. 162, 169. Brezin, E., G. Parisi, and J. Zinn-Justin, 1977, Phys. Rev. D 16, Ambjorn, J., and P. Olesen, 1977, Nucl. Phys. B 170, 60. 408. Andrei, N., and D. J. Gross, 1978, Phys. Rev. D 18, 468. Brower, R. C., K. N. Orginos, and C.-I. Tan, 1996, Phys. Rev. Aoki, K. I., M. Bando, T. Kugo, M. G. Mitchard, and H. Na- D 55, 6313. katani, 1990, Prog. Theor. Phys. 84, 683. Brown, F. R., F. P. Butler, H. Chen, N. H. Christ, Z.-H. Dong, Aoyama, H., T. Harano, H. Kikuchi, I. Okouchi, M. Sato, and W. Schaffer, L. I. Unger, and A. Vaccarino, 1990, Phys. Rev. S. Wada, 1997, Prog. Theor. Phys. Suppl. 127,1. Lett. 65, 2491. Aoyama, H., T. Harano, M. Sato, and S. Wada, 1996, Phys. Brown, F. R., H. Chen, N. H. Christ, Z. Dong, R. D. Mawhin- Lett. B 388, 331. ney, W. Schaffer, and A. Vaccarino, 1992, Phys. Rev. D 46, Appelquist, T., and R. D. Pisarski, 1981, Phys. Rev. D 23, 2305. 5655.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 422 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

Brown, G. E., and M. Rho, 1978, Comments Nucl. Part. Phys. Crewther, R. J., P. D. Vecchia, G. Veneziano, and E. Witten, 18,1. 1979, Phys. Lett. B 88, 123. Brown, L. S., R. D. Carlitz, D. B. Creamer, and C. Lee, 1978, Damgaard, P. H., U. M. Heller, A. Krasnitz, and P. Olesen, Phys. Rev. D 17, 1583. 1997, Phys. Lett. B 400, 169. Brown, L. S., and D. B. Creamer, 1978, Phys. Rev. D 18, 3695. de Forcrand, F., and K.-F. Liu, 1992, Phys. Rev. Lett. 69, 245. de Forcrand, P., M. G. Perez, J. E. Hetrick, and I.-O. Caldi, D., 1977, Phys. Rev. Lett. 39, 121. Stamatescu, 1997, hep-lat/9802017. Callan, C. G., R. Dashen, and D. J. Gross, 1976, Phys. Lett. B de Forcrand, P., M. G. Perez, and I.-O. Statamescu, 1995, Nucl. 63, 334. Phys. Proc. Suppl. 47, 777. Callan, C. G., R. Dashen, and D. J. Gross, 1978a, Phys. Rev. D de Forcrand, P., M. G. Perez, and I.-O. Stamatescu, 1997, Nucl. 17, 2717. Phys. B 499, 409. Callan, C. G., R. F. Dashen, and D. J. Gross, 1978b, Phys. Rev. DeGrand, T., A. Hasenfratz, and T. G. Kovacs, 1997, D 18, 4684. hep-lat/9710078. Callan, C. G., R. F. Dashen, and D. J. Gross, 1979, Phys. Rev. DeGrand, T., R. Jaffe, K. Johnson, and J. Kiskis, 1975, Phys. D 19, 1826. Rev. D 12, 2060. Campostrini, M., A. DiGiacomo, and Y. Gunduc, 1989, Phys. DelDebbio, L., M. Faber, J. Greensite, and S. Olejnik, 1997, Lett. B 225, 393. Nucl. Phys. Proc. Suppl. 53, 141. Campostrini, M., A. DiGiacomo, and H. Panagopoulos, 1988, Deng, Y., 1989, Nucl. Phys. B (Proc. Suppl.) 9, 334. Phys. Lett. B 212, 206. Dey, M., V. L. Eletsky, and B. L. Ioffe, 1990, Phys. Lett. B 252, 620. Caracciolo, S. R., A. P. Edwards, and A. D. Sokal, 1995, Phys. Diakonov, D. I., 1995, in International School of Physics, En- Rev. Lett. 75, 1891. rico Fermi, Course 80, Varenna, Italy. Carlitz, R. D., and D. B. Creamer, 1979a, Ann. Phys. (N.Y.) Diakonov, D., 1996, Prog. Part. Nucl. Phys. 36,1. 118 , 429. Diakonov, D. I., and A. D. Mirlin, 1988, Phys. Lett. B 203, 299. Carlitz, R. D., and D. B. Creamer, 1979b, Phys. Lett. B 84, 215. Diakonov, D. I., and V. Y. Petrov, 1984, Nucl. Phys. B 245, Carvalho, C. A. D., 1980, Nucl. Phys. B 183, 182. 259. Chandrasekharan, S., 1995, Nucl. Phys. B (Proc. Suppl.) 42, Diakonov, D. I., and V. Y. Petrov, 1985, Sov. Phys. JETP 62, 475. 204. Chemtob, M., 1981, Nucl. Phys. B 184, 497. Diakonov, D. I., and V. Y. Petrov, 1986, Nucl. Phys. B 272, Chen, D., and R. D. Mawhinney, 1997, Nucl. Phys. B (Proc. 457. Suppl.) 53, 216. Diakonov, D. I., and V. Y. Petrov, 1994, Phys. Rev. D 50, 266. Chen, D., J. Negele, and E. Shuryak, 1998, ‘‘Instanton-induced Diakonov, D. I., and V. Y. Petrov, 1996, in Continuous Ad- static potential in QCD, revisited,’’ MIT preprint. vances in QCD 1996 (World Scientific, Singapore). Chernodub, M. N., and F. V. Gubarev, 1995, JETP Lett. 62, Diakonov, D. I., V. Y. Petrov, and P. V. Pobylitsa, 1988, Nucl. 100. Phys. B 306, 809. Chernyshev, S., M. A. Nowak, and I. Zahed, 1996, Phys. Rev. Diakonov, D., V. Petrov, and P. Pobylitsa, 1989, Phys. Lett. B D 53, 5176. 226, 372. Christ, N. 1996, talk given at RHIC Summer Study 96, Diakonov, D. I., and M. Polyakov, 1993, Nucl. Phys. B 389, Brookhaven National Laboratory. 109. Christos, G. A., 1984, Phys. Rep. 116, 251. Dine, M., and W. Fischler, 1983, Phys. Lett. B 120, 137. Christov, C. V., A. Blotz, H. Kim, P. Pobylitsa, T. Watabe, T. Dorey, N., T. J. Hollowood, V. V. Khoze, and M. P. Mattis, Meissner, E. Ruiz-Ariola, and K. Goeke, 1996, Prog. Part. 1997, hep-th/9709072. Nucl. Phys. 37, 91. Dorey, N., V. Khoze, and M. Mattis, 1996a, Phys. Rev. D 54, Chu, M. C., J. M. Grandy, S. Huang, and J. W. Negele, 1993a, 2921. Phys. Rev. Lett. 70, 225. Dorey, N., V. Khoze, and M. Mattis, 1996b, Phys. Rev. D 54, Chu, M. C., J. M. Grandy, S. Huang, and J. W. Negele, 1993b, 7832. Phys. Rev. D 48, 3340. Dorey, N., V. V. Khoze, and M. P. Mattis, 1997, Chu, M. C., J. M. Grandy, S. Huang, and J. W. Negele, 1994, hep-th/9708036. Phys. Rev. D 49, 6039. Dorokhov, A. E., and N. I. Kochelev, 1990, Z. Phys. C 46, 281. Chu, M.-C., M. Lissia, and J. W. Negele, 1991, Nucl. Phys. B Dorokhov, A. E., and N. I. Kochelev, 1993, Phys. Lett. B 304, 360, 31. 167. Chu, M. C., and S. Schramm, 1995, Phys. Rev. D 51, 4580. Dorokhov, A. E., N. I. Kochelev, and Y. A. Zubov, 1993, Int. Claudson, M., E. Farhi, and R. L. Jaffe, 1986, Phys. Rev. D 34, J. Mod. Phys. A 8, 603. 873. Dorokhov, A. E., Y. A. Zubov, and N. I. Kochelev, 1992, Sov. Cohen, E., and C. Gomez, 1984, Phys. Rev. Lett. 52, 237. J. Part. Nucl. 23, 522. Coleman, S., 1977, ‘‘The uses of instantons,’’ Proceedings of Douglas, M. R., and S. H. Shenker, 1995, Nucl. Phys. B 447, the 1977 School of Subnuclear Physics, Erice (Italy), repro- 271. duced in Aspects of Symmetry (Cambridge University Press, Dowrick, N., and N. McDougall, 1993, Nucl. Phys. B 399, 426. Cambridge, England, 1985), p. 265. Dubovikov, M. S., and A. V. Smilga, 1981, Nucl. Phys. B 185, Cooper, F., A. Khare, and U. Sukhatme, 1995, Phys. Rep. 251, 109. 267. Eguchi, T., P. B. Gilkey, and A. J. Hanson, 1980, Phys. Rep. Corrigan, E., P. Goddard, and S. Templeton, 1979, Nucl. Phys. 66, 213. B 151, 93. Eletskii, V. L., and B. L. Ioffe, 1993, Phys. Rev. D 47, 3083. Creutz, M., 1983, Quarks, gluons and lattices (Cambridge Uni- Eletsky, V. L., 1993, Phys. Lett. B 299, 111. versity Press, Cambridge). Eletsky, V. L., and B. L. Ioffe, 1988, Sov. J. Nucl. Phys. 48, 384.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 423

Espinosa, O., 1990, Nucl. Phys. B 343, 310. Ilgenfritz, E.-M., and E. V. Shuryak, 1989, Nucl. Phys. B 319, Evans, N., S. D. H. Hsu, and M. Schwetz, 1996, Phys. Lett. B 511. 375, 262. Ilgenfritz, E.-M., and E. V. Shuryak, 1994, Phys. Lett. B 325, Faber, M., H. Markum, S. Olejnik, and W. Sakuler, 1995, Nucl. 263. Phys. B (Proc. Suppl.) 42, 487. Intriligator, K., and N. Seiberg, 1996, Nucl. Phys. B (Proc. Faleev, S. V., and P. G. Silvestrov, 1995, Phys. Lett. A 197, 372. Suppl.) 45,1. Feurstein, M., H. Markum, and S. Thurner, 1997, Phys. Lett. B Ioffe, B. L., 1981, Nucl. Phys. B 188, 317. 396, 203. Irback, A., P. LaCock, D. Miller, B. Peterson, and T. Reisz, Feynman, R. P., and A. R. Hibbs, 1965, Quantum Mechanics 1991, Nucl. Phys. B 363, 34. and Path Integrals (McGraw-Hill, New York). Ishikawa, K., G. Schierholz, H. Schneider, and M. Teper, 1983, Finnel, D., and P. Pouliot, 1995, Nucl. Phys. B 453, 225. Phys. Lett. B 128, 309. Forkel, H., and M. K. Banerjee, 1993, Phys. Rev. Lett. 71, 484. Ito, K., and N. Sasakura, 1996, Phys. Lett. B 382, 95. Forkel, H., and M. Nielsen, 1995, Phys. Lett. B 345, 55. Iwasaki, Y., K. Kanaya, S. Kaya, S. Sakai, and T. Yoshie, 1996, Fo¨ rster, D., 1977, Phys. Lett. B 66, 279. Z. Phys. C 71, 343. Forte, S., and E. V. Shuryak, 1991, Nucl. Phys. B 357, 153. Iwasaki, Y., K. Kanaya, S. Sakai, and T. Yoshie, 1991, Phys. Fox, I. A., J. P. Gilchrist, M. L. Laursen, and G. Schierholz, Rev. Lett. 67, 1491. 1985, Phys. Rev. Lett. 54, 749. Ivanenko, T. L., and J. W. Negele, 1997, hep-lat/9709130. Fried, D., and K. Uhlenbeck, 1984, Instantons and Four- Jack, I., D. R. T. Jones, and C. G. North, 1997, Nucl. Phys. B Manifolds (Springer, New York). 486, 479. Fukushima, M., H. Suganuma, A. Tanaka, H. Toki, and S. Jackiw, R., C. Nohl, and C. Rebbi, 1977, Phys. Rev. D 15, 1642. Sasaki, 1997, hep-lat/9709133. Jackiw, R., and C. Rebbi, 1976a, Phys. Rev. Lett. 37, 172. Jackiw, R., and C. Rebbi, 1976b, Phys. Rev. D 14, 517. Georgi, H., and A. Manohar, 1984, Nucl. Phys. B 234, 189. Jackiw, R., and C. Rebbi, 1977, Phys. Rev. D 16, 1052. Gerber, P., and H. Leutwyler, 1989, Nucl. Phys. B 321, 387. Jackson, A. D., and J. J. M. Verbaarschot, 1996, Phys. Rev. D Geshkenbein, B., and B. Ioffe, 1980, Nucl. Phys. B 166, 340. 53, 7223. Gocksch, A., 1991, Phys. Rev. Lett. 67, 1701. Ji, X., 1995, Phys. Rev. Lett. 74, 1071. Gonzalez-Arroyo, A., and A. Montero, 1996, Phys. Lett. B Kacir, M., M. Prakash, and I. Zahed, 1996, preprint, 387, 823. hep-ph/9602314. Gracey, J. A., 1996, Phys. Lett. B 373, 178. Kanki, T., 1988, in Proceedings of the International Workshop Grandy, J., and R. Gupta, 1994, Nucl. Phys. B (Proc. Suppl.) on Variational Calculations in Quantum Field Theory, edited 34, 164. by L. Polley and D. Pottinger (World Scientific, Singapore), Gribov, V. N., 1981, preprint, KFKI-1981-66, Budapest. p. 215. Gross, D. J., R. D. Pisarski, and L. G. Yaffe, 1981, Rev. Mod. Kaplan, D. B., 1992, Phys. Lett. B 288, 342. Phys. 53, 43. Kapusta, J., 1989, Finite Temperature Field Theory (Cambridge Grossman, B., 1977, Phys. Lett. A 61, 86. University Press, Cambridge, England). Gupta, R., 1992, ‘‘Scaling, the renormalization group and im- Kapusta, J., D. Kharzeev, and L. McLerran, 1996, Phys. Rev. proved lattice actions,’’ preprint LA-UR-92-917, Los Alamos. D 53, 5028. Gupta, R., D. Daniel, and J. Grandy, 1993, Phys. Rev. D 48, Karsch, F., and E. Laermann, 1994, Phys. Rev. D 50, 6954. 3330. Khoze, V., and A. Yung, 1991, Z. Phys. C 50, 155. Halperin, I., and A. Zhitnitsky, 1997, hep-ph/9706251. Kikuchi, H., and J. Wudka, 1992, Phys. Lett. B 284, 111. Halperin, I., and A. Zhitnitsky, 1998, Phys. Rev. Lett. 80, 438. Kim, J. E., 1979, Phys. Rev. Lett. 43, 103. Hansson, T. H., and I. Zahed, 1992, Nucl. Phys. B 374, 277. Kirkwood, J. G., 1931, J. Chem. Phys. 22, 1420. Hansson, T. H., M. Sporre, and I. Zahed, 1994, Nucl. Phys. B Klevansky, S. P., 1992, Rev. Mod. Phys. 64, 649. 427, 545. Klimt, S., M. Lutz, and W. Weise, 1990, Phys. Lett. B 249, 386. Harrington, B. J., and H. K. Shepard, 1978, Phys. Rev. D 17, Klinkhammer, F. R., and N. S. Manton, 1984, Phys. Rev. D 30, 2122. 2212. Hart, A., and M. Teper, 1995, Phys. Lett. B 371, 261. Koch, V., and G. E. Brown, 1993, Nucl. Phys. A 560, 345. Hasenfratz, A., T. DeGrand, and D. Zhu, 1996, Nucl. Phys. B Koch, V., E. V. Shuryak, G. E. Brown, and A. D. Jackson, 478, 349. 1992, Phys. Rev. D 46, 3169. Hatsuda, T., Y. Koike, and S. H. Lee, 1993, Nucl. Phys. B 394, Koesterlitz, J. M., and D. J. Thouless, 1973, J. Phys. C 6, 118. 221. Kogan, I. I., and M. Shifman, 1995, Phys. Rev. Lett. 75, 2085. Hatsuda, T., and T. Kunihiro, 1985, Prog. Theor. Phys. 74, 765. Kogut, J. B., 1979, Rev. Mod. Phys. 51, 659. Hatsuda, T., and T. Kunihiro, 1994, Phys. Rep. 247, 221. Kogut, J. B., and D. K. Sinclair, 1988, Nucl. Phys. B 295, 465. Hecht, M., and T. DeGrand, 1992, Phys. Rev. D 46, 2115. Kovner, A., and M. Shifman, 1997, Phys. Rev. D 56, 2396. Hoek, J., 1986, Phys. Lett. B 166, 199. Kronfeld, A. S., 1988, Nucl. Phys. B (Proc. Suppl.) 4, 329. Hoek, J., M. Teper, and J. Waterhouse, 1987, Nucl. Phys. B Kronfeld, A., G. Schierholz, and U. J. Wiese, 1987, Nucl. Phys. 288, 589. B 293, 461. Huang, Z., and X.-N. Wang, 1996, Phys. Rev. D 53, 5034. Kuzmin, V. A., V. A. Rubakov, and M. E. Shaposhnikov, 1985, Hutter, M., 1995, Z. Phys. C 74, 131. Phys. Lett. B 155, 36. Ilgenfritz, E., and M. Mu¨ ller-Preussker, 1981, Nucl. Phys. B Landau, L. D., and E. M. Lifshitz, 1959, Quantum Mechanics 184, 443. (Pergamon, Oxford). Ilgenfritz, E.-M., M. Mu¨ ller-Preußker, and E. Meggiolaro, Laursen, M., J. Smit, and J. C. Vink, 1990, Nucl. Phys. B 343, 1995, Nucl. Phys. B (Proc. Suppl.) 42, 496. 522.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 424 T. Scha¨fer and E. V. Shuryak: Instantons in QCD

Lee, C., and W. A. Bardeen, 1979, Nucl. Phys. B 153, 210. Polyakov, A., 1987, Gauge Fields and Strings (Harwood, Chur, Lee, S. H., and T. Hatsuda, 1996, Phys. Rev. D 54, 1871. Switzerland). Leinweber, D. B., 1995a, Phys. Rev. D 51, 6369. Preskill, J., M. B. Wise, and F. Wilczek, 1983, Phys. Lett. B Leinweber, D. B., 1995b, Phys. Rev. D 51, 6383. 120, 133. Levine, H., and L. G. Yaffe, 1979, Phys. Rev. D 19, 1225. Pugh, D. J. R., and M. Teper, 1989, Phys. Lett. B 224, 159. Lipatov, L. N., 1977, JETP Lett. 22, 104. Rajagopal, K., and F. Wilczek, 1993, Nucl. Phys. B 399, 395. Luescher, M., 1982, Commun. Math. Phys. 85, 39. Rajaraman, R., 1982, Solitons and Instantons (North-Holland, Maggiore, M., and M. Shifman, 1992a, Phys. Rev. D 46, 3550. Amsterdam). Maggiore, M., and M. Shifman, 1992b, Nucl. Phys. B 371, 177. Randall, L., R. Rattazzi, and E. Shuryak, 1998, Mandelstam, S., 1976, Phys. Rep. 23, 245. hep-ph/9803258. Manousakis, E., and J. Polonyi, 1987, Phys. Rev. Lett. 58, 847. Rapp, R., T. Scha¨fer, E. V. Shuryak, and M. Velkovsky, 1997, McLerran, L., 1986, Rev. Mod. Phys. 58, 1021. hep-ph/9711396. Mermin, N., and H. Wagner, 1966, Phys. Rev. Lett. 16, 1133. Reinders, L. J., H. Rubinstein, and S. Yazaki, 1985, Phys. Rep. Michael, C., and P. S. Spencer, 1994, Phys. Rev. D 50, 7570. 127,1. Michael, C., and P. S. Spencer, 1995, Phys. Rev. D 52, 4691. Ringwald, A., 1990, Nucl. Phys. B 330,1. Montvay, I., and G. Mu¨ nster, 1995, Quantum Fields on a Lat- Ringwald, A., and F. Schrempp, 1994, in Proceedings of the tice (Oxford University Press, Oxford). International Seminar Quarks 94, Vladimir, Russia, preprint, Mu¨ nster, G., 1982, Z. Phys. C 12, 43. hep-ph/9411217. Nambu, Y., and G. Jona-Lasinio, 1961, Phys. Rev. 122, 345. Rothe, H. J., 1992, Lattice Gauge Theories, An Introduction Narayanan, R., and H. Neuberger, 1995, Nucl. Phys. B 443, (World Scientific, Singapore). 305. Salomonson, P., and J. W. V. Holton, 1981, Nucl. Phys. B 196, 509. Narayanan, R., and P. Vranas, 1997, Nucl. Phys. B 506, 373. Savvidy, G. K., 1977, Phys. Lett. B 71, 133. Narison, S., 1984, Z. Phys. C 26, 209. Scha¨fer, T., 1996, Phys. Lett. B 389, 445. Narison, S., 1989, QCD Spectral Sum Rules (World Scientific, Scha¨fer, T., 1997, hep-ph/9708256. Singapore). Scha¨fer, T., and E. V. Shuryak, 1995a, Phys. Rev. Lett. 75, Narison, S., 1996, Phys. Lett. B 387, 162. 1707. Novak, M., G. Papp, and I. Zahed, 1996, Phys. Lett. B 389, 341. Scha¨fer, T., and E. V. Shuryak, 1995b, Phys. Lett. B 356, 147. Novikov, V. A., 1987, Sov. J. Nucl. Phys. 46, 366. Scha¨fer, T., and E. V. Shuryak, 1996a, Phys. Rev. D 53, 6522. Novikov, V. A., M. A. Shifman, A. I. Vainshtein, and V. I. Scha¨fer, T., and E. V. Shuryak, 1996b, Phys. Rev. D 54, 1099. Zakharov, 1979, Nucl. Phys. B 165, 67. Scha¨fer, T., E. V. Shuryak, and J. J. M. Verbaarschot, 1994, Novikov, V. A., M. A. Shifman, A. I. Vainshtein, and V. I. Nucl. Phys. B 412, 143. Zakharov, 1980, Nucl. Phys. B 165, 55. Scha¨fer, T., E. V. Shuryak, and J. J. M. Verbaarschot, 1995, Novikov, V. A., M. A. Shifman, A. I. Vainshtein, and V. I. Phys. Rev. D 51, 1267. Zakharov, 1981, Nucl. Phys. B 191, 301. Schierholz, G., 1994, Nucl. Phys. Proc. Suppl. 37A, 203. Novikov, V. A., M. A. Shifman, A. I. Vainshtein, and V. I. Schramm, S., and M.-C. Chu, 1993, Phys. Rev. D 48, 2279. Zakharov, 1983, Nucl. Phys. B 229, 381. Schwarz, A. S., 1977, Phys. Lett. B 67, 172. Novikov, V. A., M. A. Shifman, A. I. Vainshtein, and V. I. Seiberg, N., 1994, Phys. Rev. D 49, 6857. Zakharov, 1985a, Nucl. Phys. B 260, 157. Seiberg, N., and E. Witten, 1994, Nucl. Phys. B 426, 19. Novikov, V. A., M. A. Shifman, A. I. Vainshtein, and V. I. Shifman, M. A., 1989, Sov. Phys. Usp. 32, 289. Zakharov, 1985b, Fortschr. Phys. 32, 585. Shifman, M. A., 1992, Vacuum Structure and QCD Sum Rules Novikov, V. A., M. A. Shifman, V. B. Voloshin, and V. I. (North-Holland, Amsterdam). Zakharov, 1983, Nucl. Phys. B 229, 394. Shifman, M., 1994, Instantons in Gauge Theories (World Sci- Nowak, M., 1991, Acta Phys. Pol. B 22, 697. entific, Singapore). Nowak, M. A., J. J. M. Verbaarschot, and I. Zahed, 1989a, Shifman, M. A., and A. I. Vainshtein, 1988, Nucl. Phys. B 296, Nucl. Phys. B 324,1. 445. Nowak, M. A., J. J. M. Verbaarschot, and I. Zahed, 1989b, Shifman, M. A., A. I. Vainshtein, and V. I. Zakharov, 1978, Phys. Lett. B 228, 115. Phys. Lett. B 76, 971. Nowak, M. A., J. J. M. Verbaarschot, and I. Zahed, 1989c, Shifman, M. A., A. I. Vainshtein, and V. I. Zakharov, 1979, Nucl. Phys. B 325, 581. Nucl. Phys. B 147, 385, 448. Olejnik, S., 1989, Phys. Lett. B 221, 372. Shifman, M. A., A. I. Vainshtein, and V. I. Zakharov, 1980a, Peccei, R., and H. R. Quinn, 1977, Phys. Rev. D 16, 1791. Nucl. Phys. B 166, 493. Phillips, A., and D. Stone, 1986, Commun. Math. Phys. 103, Shifman, M., A. Vainshtein, and V. Zakharov, 1980b, Nucl. 599. Phys. B 165, 45. Pisarski, R. D., and M. Tytgat, 1997, preprint, hep-ph/9702340. Shifman, M., A. Vainshtein, and V. Zakharov, 1980c, Nucl. Pisarski, R. D., and F. Wilczek, 1984, Phys. Rev. D 29, 338. Phys. B 163, 46. Pisarski, R. D., and L. G. Yaffe, 1980, Phys. Lett. B 97, 110. Shuryak, E., 1978a, Zh. E´ ksp. Teor. Fiz. [Sov. Phys. JETP 74, Pobylitsa, P., 1989, Phys. Lett. B 226, 387. 408]. Polikarpov, M. I., and A. I. Veselov, 1988, Nucl. Phys. B 297, Shuryak, E. V., 1978b, Phys. Lett. B 79, 135. 34. Shuryak, E., 1980, Phys. Rep. 61, 71. Polyakov, A. M., 1975, Phys. Lett. B 59, 79. Shuryak, E. V., 1981, Phys. Lett. B 107, 103. Polyakov, A., 1977, Nucl. Phys. B 120, 429. Shuryak, E. V., 1982a, Nucl. Phys. B 198, 83. Polyakov, A. M., 1978, Phys. Lett. B 72, 477. Shuryak, E. V., 1982b, Nucl. Phys. B 203, 93.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998 T. Scha¨fer and E. V. Shuryak: Instantons in QCD 425

Shuryak, E. V., 1982c, Nucl. Phys. B 203, 116. Takeuchi, S., and M. Oka, 1991, Phys. Rev. Lett. 66, 1271. Shuryak, E. V., 1982d, Nucl. Phys. B 203, 140. Tar, C. D., and J. Kogut, 1987, Phys. Rev. D 36, 2828. Shuryak, E. V., 1983, Nucl. Phys. B 214, 237. Teper, M., 1986, Phys. Lett. B 171, 86. Shuryak, E. V., 1985, Phys. Lett. B 153, 162. Thomas, A. W., S. Theberge, and G. A. Miller, 1981, Phys. Shuryak, E. V., 1987, Phys. Lett. B 196, 373. Rev. D 24, 216. Shuryak, E. V., 1988a, Nucl. Phys. B 302, 559, 574, 599. Thurner, S., M. Feurstein, and H. Markum, 1997, Phys. Rev. D Shuryak, E. V., 1988b, Nucl. Phys. B 302, 621. 56, 4039. Shuryak, E. V., 1988c, The QCD Vacuum, Hadrons and the Ukawa, A., 1997, Nucl. Phys. B (Proc. Suppl.) 53, 106. Superdense Matter (World Scientific, Singapore). Vafa, C., and E. Witten, 1984, Nucl. Phys. B 234, 173. Shuryak, E. V., 1989, Nucl. Phys. B 319, 521, 541. Vainshtein, A. I., M. A. Shifman, and V. I. Zakharov, 1985, Shuryak, E. V., 1993, Rev. Mod. Phys. 65,1. Sov. Phys. Usp. 28, 709. Shuryak, E. V., 1994, Comments Nucl. Part. Phys. 21, 235. Vainshtein, A. I., V. I. Zakharov, V. A. Novikov, and M. A. Shuryak, E. V., 1995, Phys. Rev. D 52, 5370. Shifman, 1982, Sov. Phys. Usp. 25, 195. Shuryak, E. V., and J. L. Rosner, 1989, Phys. Lett. B 218, 72. Vainshtein, A. I., V. I. Zakharov, V. A. Novikov, and M. A. Shuryak, E. V., and M. Velkovsky, 1994, Phys. Rev. D 50, Shifman, 1986, Sov. J. Nucl. Phys. 43, 294. 3323. Velikson, B., and D. Weingarten, 1985, Nucl. Phys. B 249, 433. Shuryak, E. V., and M. Velkovsky, 1997, preprint Velkovsky, M., and E. V. Shuryak, 1993, Phys. Lett. B 304, hep-ph/9703345. 281. Shuryak, E. V., and J. J. M. Verbaarschot, 1990, Nucl. Phys. B Velkovsky, M., and E. Shuryak, 1996, Phys. Rev. D 56, 2766. 341,1. Veneziano, G., 1979, Nucl. Phys. B 159, 213. Shuryak, E. V., and J. J. M. Verbaarschot, 1991, Nucl. Phys. B Veneziano, G., 1992, Mod. Phys. Lett. A 7, 1666. 364, 255. Verbaarschot, J. J. M., 1991, Nucl. Phys. B 362, 33. Shuryak, E. V., and J. J. M. Verbaarschot, 1992, Phys. Rev. Verbaarschot, J. J. M., 1994a, Act. Phys. Pol. 25, 133. Lett. 68, 2576. Verbaarschot, J. J. M., 1994b, Nucl. Phys. B 427, 534. Shuryak, E. V., and J. J. M. Verbaarschot, 1993a, Nucl. Phys. B Vink, J. C., 1988, Phys. Lett. B 212, 483. 410, 55. Vogl, U., and W. Weise, 1991, Prog. Part. Nucl. Phys. 27, 195. Shuryak, E. V., and J. J. M. Verbaarschot, 1993b, Nucl. Phys. B Weingarten, D., 1983, Phys. Rev. Lett. 51, 1830. 410, 37. Weingarten, D., 1994, Nucl. Phys. B (Proc. Suppl.) 34, 29. Shuryak, E. V., and J. J. M. Verbaarschot, 1995, Phys. Rev. D Wettig, T., A. Scha¨fer, and H. Weidenmu¨ ller, 1996, Phys. Lett. 52, 295. B 367, 28. Shuryak, E. V., and A. R. Zhitnitsky, 1997, hep-ph/9706316. Wiese, U. J., 1990, Nucl. Phys. B (Proc. Suppl.) 17, 639. Skyrme, T., 1961, Proc. R. Soc. London, Ser. A 262, 207. Wilczek, F., 1992, Int. J. Mod. Phys. A 7, 3911. Smilga, A., 1994a, Phys. Rev. D 49, 5480. Wilson, K. G., 1969, Phys. Rev. 179, 1499. Smilga, A., 1994b, Phys. Rev. D 49, 6836. Witten, E., 1977, Phys. Rev. Lett. 38, 121. Smilga, A. V., 1996, Phys. Rep. 291,1. Witten, E., 1979a, Nucl. Phys. B 149, 285. Smilga, A., and J. Stern, 1993, Phys. Lett. B 318, 531. Witten, E., 1979b, Nucl. Phys. B 156, 269. Smit, J., and A. J. van der Sijs, 1989, Nucl. Phys. B 355, 603. Witten, E., 1981, Nucl. Phys. B 188, 513. Smit, J., and J. C. Vink, 1987, Nucl. Phys. B 284, 234. Witten, E., 1983, Nucl. Phys. B 223, 433. Smit, J., and J. C. Vink, 1988, Nucl. Phys. B 298, 557. Wohler, C. F., and E. V. Shuryak, 1994, Phys. Lett. B 333, 467. Snyderman, N., and S. Gupta, 1981, Phys. Rev. D 24, 542. Woit, P., 1983, Phys. Rev. Lett. 51, 638. Steele, J., A. Subramanian, and I. Zahed, 1995, Nucl. Phys. B Wolff, U., 1989, Nucl. Phys. B 322, 759. 452, 545. Yung, A., 1997, hep-th/9705181. Suganuma, H., K. Itakura, H. Toki, and O. Miyamura, 1995, Yung, A. V., 1988, Nucl. Phys. B 297, 47. Nucl. Phys. Proc. Suppl. 47, 302. Zakharov, V. I., 1992, Nucl. Phys. B 371, 637. Suzuki, T., 1993, Nucl. Phys. B (Proc. Suppl.) 30, 176. Zhitnitsky, A. R., 1980, Sov. J. Nucl. Phys. 31, 260. Svetitsky, B., and L. G. Yaffe, 1982, Nucl. Phys. B 210, 423. Zinn-Justin, J., 1981, J. Math. Phys. 22, 511. ’t Hooft, G., 1976a, Phys. Rev. Lett. 37,8. Zinn-Justin, J., 1982, ‘‘The principles of instanton calculus: A ’t Hooft, G., 1976b, Phys. Rev. D 14, 3432. few applications,’’ in Recent Advances in Field Theory, Les ’t Hooft, G., 1981a, Commun. Math. Phys. 81, 267. Houches, Session XXXIX, edited by J.-B. Zuber and R. Stora ’t Hooft, G., 1981b, Nucl. Phys. B 190, 455. (North Holland, Amsterdam), p. 39. ’t Hooft, G., 1986, Phys. Rep. 142, 357. Zinn-Justin, J., 1983, Nucl. Phys. B 218, 333.

Rev. Mod. Phys., Vol. 70, No. 2, April 1998