New Methods in QFT and QCD: From Large-N Orbifold Equivalence to Bions and Resurgence
We present a broad conceptual introduction to some new ideas in non-perturbative QFT. The large- orbifold-orientifold equivalence connects a natural large- limit of QCD to QCD with adjoint fermions. QCD(adj) with periodic boundary conditions and double-trace deformation of Yang-Mills theory satisfy large- volume independence, a type of orbifold equivalence. Certain QFTs that satisfy volume independence at exhibit adiabatic continuity at finite-, and also become semi-classically calculable on small . We discuss the role of monopole-instantons, and magnetic and neutral bion saddles in connection to mass gap, and center and chiral symmetry realizations. Neutral bions also provide a weak coupling semiclassical realization of infrared-renormalons. These considerations help motivate the necessity of complexification of path integrals (Picard-Lefschetz theory) in semi-classical analysis, and highlights the importance of hidden topological angles. Finally, we briefly review the resurgence program, which potentially provides a novel non-perturbative continuum definition of QFT. All these ideas are continuously connected.
Keywords:large- orbifold and orientifold equivalence, volume independence, double-trace deformations, adiabatic continuity, semi-classical calculability, magnetic and neutral bions, Picard-Lefschetz theory of path integration, and resurgence
This review is a basic introduction to some new methods and ideas in quantum gauge theories in four dimensions, and sigma models in two dimensions, with an underlying emphasis on making progress in the understanding of quantum chromodynamics (QCD). The review has six short sections, each of which describes the most important concepts underlying these methods. The ideas are illustrated with simple examples. In fact, all six stories are continuously connected, and each borrows a new idea/perspective from the previous one. Our intention is to provide a framework in quantum field theory (QFT) in which all reasonable methods, lattice field theory, supersymmetric field theory, and semi-classical continuum methods in (non-supersymmetric) QFT, among others, are treated on a similar footing, and can be used in conjunction to explain some of the mysteries of strongly interacting QFT.
2 Large- Orbifold and Orientifold Equivalences
Large- orbifold/orientifold equivalence is an exact equivalence between certain sub-sectors of seemingly unrelated quantum field theories, in the limit of large-, where is the number of internal degrees of freedom. can be either the rank of the color or global symmetry group. These equivalences provide new and important insights into the dynamics of strongly coupled theories. Interesting examples of equivalences are: (i) those relating a theory on an infinite lattice to a theory on a finite-size lattice, or even on a single-site lattice, (unitary matrix model) Eguchi:1982nm (); GonzalezArroyo:1982hz (); Kovtun:2007py (); or (ii) in the continuum, relating a theory on to a theory compactified on ; or (iii) equivalences relating a supersymmetric theory to a non-supersymmetric one Kachru:1998ys (); Bershadsky:1998cb (); Armoni:2003gp (); Armoni:2003fb (); Armoni:2004ub (); Kovtun:2003hr (); Kovtun:2004bz (); Unsal:2006pj (); Kovtun:2005kh (). See Fig.1.
The main idea is the following. One starts with some “parent” theory, and constructs “daughter” theories, by using certain orbifold/orientifold projections, which amount to retaining only those fields which are invariant under a particular discrete symmetry group of the parent theory. For discrete abelian projections, the planar graphs of the daughter theories and the parent theory coincide (up to a simple rescaling of the coupling constant), which implies the coincidence of their planar perturbative expansions at Kachru:1998ys (); Bershadsky:1998cb (). Clearly, such a perturbative equivalence is an invitation to study possible non-perturbative realizations of this idea Armoni:2003gp (); Armoni:2003fb (). Refs. Kovtun:2003hr (); Kovtun:2004bz () proved that the validity of the non-perturbative equivalence relies on certain symmetry realizations, as described below.
To appreciate the implications of such equivalences, consider a simple example. Start with Super-Yang-Mills (SYM) with gauge group . The field content of the theory consists of a gauge boson, , and a Weyl fermion, , in the adjoint representation. Two non-trivial projections of the parent theory lead to two distinct gauge theories, see Fig 1 Unsal:2006pj (). Let denote the symplectic form which is real and anti-symmetric. There are two natural projections, by and where is fermion number modulo two, which may be realized by the following constraints on the fields:
QCD(AS) is a non-supersymmetric gauge theory with a single Dirac fermion in the two-index antisymmetric tensor representation . Clearly, the two daughter theories are different QFTs with different matter content, and different global symmetries. The parent and -daughter theories are supersymmetric, while the -daughter theory is non-supersymmetric. In addition to the presence vs. absence of supersymmetry in the two daughter theories, the global discrete chiral symmetry and the center symmetries of these two daughter theories are also different.
Before stating the main result, we note that the daughter theories also possess a non-trivial symmetry, charge conjugation , which is the image of the global part of the gauge symmetry in the parent QFT. (Think of embedding into , and note that the -action interchanges .) This symmetry will be used to define a neutral sector in the daughter theory, which constitutes the image of operators in the parent theory. Clearly, -odd operators are not in the image. In general orbifold projections, the discrete symmetry defining the neutral sector is the cyclic permutation symmetry. The operators transforming nontrivially under , or , are also called ‘twisted’ operators Kachru:1998ys (); Bershadsky:1998cb ().
Necessary and sufficient conditions Kovtun:2003hr (); Kovtun:2004bz (): There exists an exact equivalence between the neutral sectors of parent-daughter pairs as well as daughter-daughter pairs provided:
The symmetry in the parent theory used in the projection is not spontaneously broken.
The symmetry defining the neutral sector in the daughter theory is not spontaneously broken.
In the present example, provided the charge-conjugation is unbroken in QCD(AS), and the fermion number modulo two, , is unbroken in , the -even subsector of QCD(AS) is exactly equivalent to the bosonic sub-sector of SYM Unsal:2006pj (). The -realization is studied on the lattice DeGrand:2006qb (), demonstrating the existence of both -broken and unbroken phases.
Why is this useful? QCD(AS) is a natural generalization of QCD to a large- limit Armoni:2003gp () , due to the simple fact that for SU(3), the fundamental (F) and antisymmetric (AS) representations coincide: . So, one can take a large- limit of QCD in either of two ways. With two index irreps, fermions are unsuppressed in the large- limit: they have degree of freedom, just like gauge bosons, and unlike fermions in the fundamental representation. The exact equivalence in the limit allows one to convert knowledge about supersymmetric theories into improved understanding of QCD or QCD-like theories. For example, one immediate prediction is that the chiral condensate in SYM and in QCD(AS) must agree up to corrections. The condensate is calculable in SYM Davies:2000nw () because it is an element of the chiral ring, so we learn something non-trivial about the non-supersymmetric QCD(AS) theory: .
Another interesting aspect of the equivalence is that it sometimes has counterintuitive consequences, which are conceptually important in QFT. A natural setting where this happens is the following. Assume equivalence is valid. Then, if either the parent or daughter theory has a global symmetry which protects an observable, then in the other theory (which does not possess the same global symmetry) the image observable behaves as if it is protected by the higher symmetry of the companion Armoni:2007kd (); Unsal:2007fb (). Why is this so?
Consider the following example of this kind of argument. In SYM, because of the trace anomaly relation, the gluon condensate determines the ground state energy, and as such, it is an order parameter for dynamical supersymmetry breaking: . This theory has non-zero Witten index, hence, supersymmetry is unbroken Witten:1982df (). This implies that the gluon condensate must vanish. As a result of equivalence, in non-supersymmetric QCD(AS) the gluon condensate must also be zero, at leading order in the expansion, scaling as Unsal:2007fb ():
This is a very surprising result! In confining gauge theories, the natural scaling of (properly normalized) non-extensive observables is . For example, the gluon condenstate in pure Yang-Mills theory is parametrically , and one would expect to see this “natural” scaling in QCD(AS) as well. Furthermore, QCD(AS) is non-supersymmetric, and hence one cannot use the tools of supersymmetry. This means that, even in a supersymmetric theory, there must be a non-perturbative mechanism for the vanishing of the condensate, which does not rely on supersymmetry. Recall that both QCD(AS) and SYM are vector-like theories and hence the fermion determinant is positive semi-definite. Also is positive definite, in the Euclidean setting. This implies that the gluon condensate is the average of a positive observable with respect to a positive measure Shifman:1978bx (); Callan:1977gz (); Schafer:1996wv ().
A puzzle: The natural QCD intuition is that the gluon condensate is positive-definite in any confining vector-like theory. From the perspective of supersymmetric QFT, the gluon condensate is zero in theories which do not break supersymmetry. But SYM is both supersymmetric and vector-like, and it does not break supersymmetry. This raises an interesting question: is it possible to explain the vanishing of the gluon condensate without using the supersymmetry (SUSY) algebra, for example by using semi-classical arguments?
This is the type of puzzling implication of large- orbifold/orientifold equivalence that is the most challenging. Taking such challenges seriously may lead to a deeper understanding of QFT and path integrals, as we argue below. The resolution of this puzzle, which we discuss in §.6.1 below, challenges our present understanding of the semi-classical representation of the path integral. Our resolution introduces new concepts such as (contributing) complex saddles and hidden topological angles.
3 Large- Volume Independence
Birth, Death and re-birth of large- reduced models: (The subtitle is borrowed from a recent talk by Gonzalez-Arroyo.) In 1982, Eguchi and Kawai (EK) came up with one of the most remarkable ideas in large- gauge theory Eguchi:1982nm (). They proved that certain properties of Yang-Mill theory, formulated on a periodic lattice with -sites, are independent of the lattice size in the limit, and can be reduced to a matrix model on -site lattice, provided center-symmetry in the matrix model is not spontaneously broken, and translation symmetry in the lattice field theory is unbroken Yaffe:1981vf (). From the modern point of view, this is an orbifold equivalence. For example, starting with a unitary matrix model with gauge group, and orbifolding by a proper , one ends up with a four-dimensional quiver (or lattice) with gauge theory on sites, Kovtun:2007py (). The orbifold equivalence gives a mapping of a matrix model to Yang-Mills theory, and if valid, can be useful both numerically and analytically.
There was important progress in the early 1980s developing the EK idea Gross:1982at (); Parisi:1982gp (); Das:1982ux (). In the end, the original EK proposal fails due to broken (center) symmetry at weak coupling of the lattice theory (which is continuously connected to the continuum). Two important modifications were introduced: (i) quenched (QEK) Bhanot:1982sh (), and twisted (TEK) GonzalezArroyo:1982ub (); GonzalezArroyo:1982hz (). Eventually it was understood that even these modifications were not enough to realize the full reduction, but a newer version of TEK works, see below. See Bringoltz:2008av () for QEK, and Azeyanagi:2007su (); Teper:2006sp () for TEK; and Lucini:2012gg () for a review.
However, this subject is now not only reborn, it is also flourishing, with both analytic and numerical progress, and useful spin-offs. QCD(adj) and deformed-Yang-Mills Kovtun:2007py (); Unsal:2008ch () and a new version of TEK GonzalezArroyo:2010ss () work. One important aspect of TEK that had already been known for some time was the relation to non-commutative field theory Ambjorn:2000cs (). Newer and related ideas connected to large-N reduction include the idea of adiabatic continuity, rigorous semi-classics, and resurgence theory. Each will be explored in subsequent sections.
Why does the original EK idea fail? In an interesting way, the physical reason why the original EK reduction does not work predates the proposal itself. And the reason why modern variants of the EK proposal must work also predates the proposal. These reasons can be seen without elaborate calculations, and can be explained almost without formulas. The main results are actually conceptual realizations rather than calculations.
Consider compactifying Yang-Mills theory on to , instead of to a four-torus . is a thermal circle, with circumference equal to the inverse temperature. It is well-known that if we increase the temperature in pure Yang-Mills theory, the theory moves from the confined to the deconfined phase, which is experimentally observed. The associated symmetry is center-symmetry, , under which the order parameter Wilson line holonomy of the gauge field in the compact direction, , transforms non-trivially under . At small circle radius, where the analysis is reliable due to weak coupling, Gross, Pisarski and Yaffe (GPY) calculated the one loop effective potential Gross:1980br ():
Clearly, the minus sign in front of the one-loop effective potential indicates the center-instability of the deconfined phase, because the potential is minimized for . This means that on , volume independence is achieved above a critical radius, , but must fail below a critical size Yaffe:1981vf (); Kiskis:2003rd (). (This is sometimes called partial reduction.) This also means that even if the large- theory is compactified on , where , the equivalence must fail regardless of the radius of . Nothing changes as one moves to a QCD-like theory. The message is:
In all QCD-like theories, with all representations of fermions, and for all infinite Lie groups, Eguchi-Kawai reduction is destined to fail in a thermal set-up at sufficiently small thermal circle. Thermal compactification with is in direct conflict with large- reduction.
3.1 Boundary Conditions as an Idea, and Center Stability in QCD(adj)
Why it must work? Despite the fact that thermal fluctuations are in conflict with the large- reduction idea, quantum fluctuations may lead to a different behavior. We can motivate and illustrate the idea using supersymmetry, but in fact the eventual argument does not require SUSY. Consider a gauge boson and a fermion in the adjoint representation of some gauge group . Then, imposing periodic boundary conditions on fermions, boundary conditions which respect supersymmetry, one is guaranteed that a potential for the Wilson line is not induced to any loop order in perturbation theory. This vanishing hides a simple secret. In fact, the one-loop potential is
The really important point is not the cancellation itself, but the existence of the part (which has nothing to do with supersymmetry). It is just the consequence of periodic boundary conditions, , for fermions. This tells us that the quantum fluctuations (as opposed to thermal fluctuations) of an adjoint fermion undoes the center-destabilizing effect of gauge fluctuations. This is the main idea underneath periodic boundary conditions. Taking one more step in this direction, i.e., with fermions, one ends up with center-stability which is the crucial necessary and sufficient condition for the validity of large- volume independence.
This simple idea provides a solution of the Eguchi-Kawai proposal in QCD(adj). Thanks to the orientifold equivalence between QCD(AS) and QCD(adj), this is also the resolution of the problem for QCD Kovtun:2007py (). In fact, even for (or SYM), center-symmetry remains intact due to non-perturbatively induced neutral bion effects, as discussed in §5.
Lattice simulations with one-site or few-site large- matrix models confirm this proposal Bringoltz:2009kb (); Azeyanagi:2010ne (); Hietanen:2009ex (), as does an earlier simulation with massive adjoint fermions Cossu:2009sq (), emulating at . In the end, the resolution is a simple realization. We also note that the potential Eq.6 was derived earlier in the context of gauge-Higgs unification in Hosotani:1988bm () in extra-dimensional model building, although this work did not discuss the implication of this potential for large- reduction or non-perturbative QCD(adj) dynamics.
3.2 Perturbative Intuition Behind Volume Independence
Instead of providing a rigorous background for volume independence, we would like to provide an intuitive perturbative explanation for why it works the way it does Unsal:2010qh (). The interesting point is the relation between the Kaluza-Klein spectrum of perturbative modes and its dependence on the realization of center symmetry.
If center-symmetry is broken, then the eigenvalues of the Wilson line clump together. Fig. 2a shows the standard form of Kaluza-Klein towers, in which discrete momenta are quantized in units of , the usual text-book picture.
If center symmetry is unbroken, the eigenvalues of the Wilson line are evenly distributed around the unit circle. This results in a much finer “KK-spectrum”, due to “momentum-color intertwining”. The discrete momenta are quantized in units of . See Fig. 2b.
At fixed , and as , nothing special takes place in the center-broken background or with the standard KK spectrum. But for the center-symmetric background, the spacing in the Kaluza-Klein spectrum approaches zero, and the spectrum becomes indistinguishable from the perturbative spectrum of the theory on ! In other words, the scale disappears from perturbation theory as one takes the limit. One can interpret as the effective size of the compact dimension, and as one takes the limit, observables in the center-singlet sector (all local operators), and the spectrum of the theory, behave as if the theory is living on , despite the fact that the theory still lives on a space with size . A similar structure appears in all toroidal compactifications on , provided center symmetry is unbroken. With multiple dimensions compactified, TEK realizes an even more efficient version of this color-momentum intertwining, due to non-commutativity of the background Wilson lines in different directions.
Scales, two regimes, adiabatic continuity and universality: For center-symmetric confining theories on , there are two different regimes:
Here, “incalculable” means inaccessible with known analytical techniques. Since nature and numerical lattice simulations know how to “calculate” in this regime, this may just mean that the proper mathematical framework and physical ideas are yet to be developed. The first regime Eq.7 is the “real thing”, which we would like to understand. The second regime Eq.8 is the “cartoon” of the real thing, but it is calculable. These two regimes are very often adiabatically connected, with no phase transition in between. In cases where adiabatic continuity is believed to hold, all non-perturbative observables agree qualitatively, thus, the two regimes are in the same universality class. But we cannot yet prove this in generality.
3.3 A Puzzle and Emergent (Fermionic) Symmetry
A working large- volume independence must have a dramatic spectral implication. In QCD(adj), recall that periodic boundary conditions for fermions imply that, in the operator formalism, we are calculating a twisted (or graded) partition function, instead of the thermal partition function, namely,
This object is familiar from SUSY QFT, where it is supersymmetric Witten index Witten:1982df (). In non-supersymmetric theories, its role has been understood only recently Unsal:2007fb (); Unsal:2007vu (); Unsal:2007jx (); Basar:2013sza (). It can be used to find the phases of a theory as a function of the spatial compactification radius (which now has no thermal interpretation). Unlike thermal compactification, where the theory must move to a deconfined phase at small circle-radius, in spatial compactification the quantum fluctuations measured as a function of radius may lead to very different small- behavior. In particular, in QCD(adj), at infinite-, there is no phase transition as a function of radius, i.e, .
Volume independence seems to be puzzling in conjunction with the Hagedorn spectrum of hadronic states. If a theory has Hagedorn growth Hagedorn:1965st (), with density of states where ellipsis are sub-leading power-law and exponential factors, then the partition function must diverge above a critical temperature, , parametrically related to the deconfinement temperature. is the higher limit of metastability, , see Fig.4, at which the meta-stable confined phase ceases to exist. The actual deconfinement temperature is slightly lower. Regardless, the Hagedorn growth almost always demands the existence of a phase transition, while volume independence demands the opposite, the absence of any phase transition Basar:2013sza (). How can these two contradictory demands be reconciled?
The most reasonable resolution of this puzzle is an approximate spectral degeneracy between the states in the bosonic and fermionic sectors, and , similar to the SYM case. Note that not only the leading Hagedorn growth, but all power law corrections and all sub-leading exponentials must cancel between the two factors. The resolution of such a high level of conspiracy in the case seems to be feasible with the emergence of a fermionic symmetry at large , where there is no supersymmetry: namely, , up to a possible difference Basar:2013sza (); Basar:2014jua (), . The potential conflict with the Coleman-Mandula theorem is resolved because at the -matrix is trivial, and hadrons are free.
4 Double-Trace Deformations: Volume Independence vs. Adiabatic Continuity
There is another solution to the volume independence proposal Unsal:2008ch (). So far, this approach does not seem to be as practically useful, but conceptually it adds a new element at large-. Furthermore, at it leads to semi-classical calculability in Yang-Mills and QCD-like theories with one fermion flavor Shifman:2008ja (), and a regime continuously connected to large- and .
Main idea Unsal:2008ch (): Consider deforming, by the addition of a double-trace operator, the Yang-Mills theory compactified on small :
The double-trace deformation is : it is as important as the action itself at large-, and hence is not a small perturbation of the action.
In fact, it cannot be a small perturbation because it changes the phase structure of the theory: it turns the deconfined phase into the confined phase. As such, one may at first guess that the deformed theory has nothing to do with the original theory. This would be a natural point of objection for any sensible quantum field theorist.
But something striking happens. The effect of the deformation on observables such as the mass gap, spectra, string tension, topological susceptibility (neutral sector, center-singlet observables) is suppressed by . In other words, somewhat surprisingly, at large- its effect on observables is a small perturbation of order . For a proof of this statement using loop equations, see Unsal:2008ch ().
In the volume dependent regime, the deformed theory is analytically calculable, and this regime is adiabatically connected to the regime, with no gauge-invariant order parameter that can distinguish the two regimes. This is the realization of adiabatic continuity.
In this sense, the double-trace deformation Eq.12 defies standard intuition. It is a special structure. Veneziano once coined for us the special status of these deformations: “It is like a good samaritan, does the good deed, and disappears from the scene.” But why is this so?
The reason is , is a positive twisted (center non-singlet) square operator. The + sign forces stability of the center-symmetry, for which is an order parameter. In the Schwinger-Dyson (or loop) equations, one of the operators enters in some connected correlator, but the other part is always a spectator. Factorization, which is guaranteed thanks to center-stability, is then invoked, and results in: . But is already zero because of stabilization of the corresponding symmetry and the correction term vanishes at . This is the magic of center-stabilizing double-trace deformation. This construction also generalizes to other symmetry stabilizing double-trace operators Cherman:2010jj (). Double-trace deformations have also been used in exploring partial center-symmetry breaking phases Ogilvie:2007tj (); Myers:2007vc (); Myers:2009df (); Ogilvie:2012is ().
4.1 Deformed Yang-Mills and adiabatic continuity
The dYM theory in the small- regime provides an example of semi-classsically calculable confinement, which is adiabatically connected to pure Yang-Mills on , and strong coupling non-abelian confinement Unsal:2008ch (); Shifman:2008ja (). In this sense, it is one of the most useful arenas to study non-perturbative dynamics. To illustrate the basic idea, here we discuss some non-perturbative aspects of gauge theories.
At small , due to weak coupling and the deformation potential, the theory is Higgsed down to by a center-symmetric Wilson line
where denotes fourth component of gauge field along Cartan subalgebra. Since the center symmetry is the only global symmetry that can distinguish the small and large circle regimes, and it is unbroken, the deformed theory exhibits adiabatic continuity. The Wilson line behaves as a compact adjoint Higgs field, in contrast with the Polyakov model in which the adjoint Higgs field is non-compact and algebra valued Polyakov:1976fu (); Deligne:1999qp (). This subtle difference in the topology of field space has two significant consequences: first, instead of having just one type of monopole-instanton, there are two different types, and . Second, there exists a topological angle. One type of monopole-instanton is the regular 3d one, and the other is a twisted (affine) one Lee:1997vp (); Kraan:1998sn ().These defects carry two types of topological quantum numbers, magnetic and topological charge, , give by and . The monopole-instanton operators are given by
Here denotes the dual photon field, defined through the abelian duality relation, , and the form of the monopole-instanton amplitudes account for long-range magnetic Coulomb interactions. Since fluctuations around background holonomy field is gapped by deformation, we set it to zero in the discussion of long-distance physics. Note that the action of the monopole-instanton is of the 4d-instanton action, , with in the present example, and is the 4d-instanton amplitude.
At , the theory on realizes confinement and a mass gap due to the monopole-instanton mechanism Unsal:2008ch (), as in the Polyakov mechanism on i.e., Debye screening in a magnetically charged plasma Polyakov:1976fu (). The long-distance effective theory is
The mass gap (inverse Debye length) is given by .
Turning on the angle introduces a sign problem in the Euclidean description, and in the dual description in Eq.16, alters . For , the monopole-instanton induced gap dies off due to destructive topological interference, between and , and at leading order in semi-classics, the theory is gapless Unsal:2012zj (). To all orders in semi-classics the theory must either be gapless, or gapped and two-fold degenerate. The latter choice is realized due to the magnetic bion mechanism, inducing a term , originating from the correlated 2-event Unsal:2012zj (). The theory has two isolated vacua, and exhibits spontaneous CP-breaking. This is indeed compatible with the theory on , where the theory has a CP-symmetry at , which is believed to be spontaneously broken by the CP-odd condensate . This is again a manifestation of the adiabatic continuity idea.
The fact that confinement is realized by a magnetic bion mechanism has a dramatic impact. The theory has two kinds of domain walls (lines) L, L for which crossing the line, while for an external unit charge, the monodromy of the dual photon is . It is shown in Anber:2015kea () that the string is a composite, domain line, due to the term. Consider a segment of , and a well-separated pair parallel to L. Clearly, can fuse into , and reduce the energy of the system considerably. This forms line, where the joining points are the quark and anti-quark. Since the tension of and are exactly degenerate, separating quarks on the wall does not cost any energy, i.e, quarks become liberated on the wall. Such a scenario was proposed by S.J. Rey and Witten in the mid 1990s Witten:1997ep () using the vacuum structure and ideas about confinement in supersymmetric theories. The dYM theory and QCD(adj) (discussed in the next section) are the first concrete realization of this idea in a set-up where the confining dynamics is analytically understood.
The small circle limit of deformed YM provides the first realization of the Polyakov mechanism Polyakov:1976fu () and generalization thereof in a locally 4d setting, thirty years after the original idea Unsal:2008ch ()! As emphasized, dYM has intrinsically 4d aspects not present in the 3d Polyakov model. There are a number of interesting recent works studying dYM, for example, Thomas:2011ee (); Thomas:2012ib (); Thomas:2012tu (); Zhitnitsky:2013hs (); Li:2014lza (), and Anber:2013xfa (); Anber:2015wha (); Anber:2015bba (), and see Vairinhos:2010ha (); Vairinhos:2011gv () for initial lattice studies.
5 Magnetic and Neutral Bions in QCD(adj): Confinement and Center-Stability
QCD(adj) compactified on does not break its center-symmetry when fermions are endowed with periodic boundary conditions, for any value of . In the small-circle regime, the theory has monopole-instantons, similar to dYM. To keep the analogy with the dYM example, we continue to work with theory. In contrast with the bosonic case, each monopole-instanton operator has fermion zero modes dictated by Nye-Singer index theorem on Nye:2000eg (); Poppitz:2008hr () (also see Poppitz:2009tw (); Poppitz:2009uq (); Bruckmann:2003ag (); Misumi:2014raa (); GarciaPerez:2009mg ()). Thus, the monopole-instanton operators become:
This implies that the monopole-instantons cannot induce a mass gap and confinement in QCD(adj) Unsal:2007jx (). Note that the 4d instanton has fermion zero modes.
At a superficial level, QCD(adj) on resembles the Polyakov model coupled to massless Dirac fermions on . But the former confines, and the latter does not. To understand the difference, first, recall the important result Affleck:1982as ().
Affleck, Harvey, Witten : The Polyakov model coupled to massless Dirac fermions on remains gapless non-perturbatively, and does not confine. The work of AHW was viewed as the death of the Polyakov model in theories with fermions. Yet, in QCD(adj) on , it has been shown that the theory confines, and induces a mass gap for gauge fluctuations Unsal:2007jx (). What is the major difference between the two? This is most easily explained in terms of global symmetries and the presence/absence of anomalies, and monopole-operators.
On 4-manifold , the theory has an anomalous chiral symmetry, reduced down to due to instantons. The action of the on the multi-fermion part of the monopole-operator Eq.18 is negation, but since is anomaly free, the invariance of the monopole operator demands one to shift , i.e. there is a topological shift symmetry associated with monopole-operators which forbids , but permits , . Thus, the generation of a mass gap for gauge fluctuations is permitted Unsal:2007vu (). In contrast, on a 3-manifold is non-anomalous, and the action of on the multi-fermion part of the monopole operator is continuous, say by . The invariance of the monopole operator demands that , and this continuous shift symmetry protects the gaplessness of the dual photon Affleck:1982as (). Relatedly, does not exist on Unsal:2007vu ().
On , at second order in the semi-classical expansion, there are two types of correlated two-events (or bion events):
Magnetic bions Unsal:2007jx (); Anber:2011de () and neutral bions Poppitz:2011wy (); Argyres:2012ka (); Argyres:2012vv () are correlated-two-events with magnetic and topological charges and . The neutral bion is indistinguishable from the perturbative vacuum in the usual topological sense. But below, we discuss the meaning of the factor, which arises as a hidden topological angle, which can distinguish the two saddles.
The magnetic and neutral bion saddles are non-self-dual. If the monopole-size is , the bion saddles have a characteristic size , parametrically larger than the monopole size, but parametrically smaller than the inter-monopole separation. The long distance effective theory is described by the proliferation of one- and two-events, and is given by (for theories, the holonomy field decouples from the IR-dynamics)
Magnetic bions lead to a mass gap for gauge fluctuations and a finite string tension, i.e. confinement. To our knowledge, this is the first analytic demonstration of confinement in a locally 4d (non-supersymmetric) QCD-like gauge theory, and it is a relatively unanticipated mechanism. Neither monopole-instantons, as in Polyakov model Polyakov:1976fu (), nor monopole or dyon particles, as in Seiberg-Witten theory Seiberg:1994rs (), is the origin of confinement. Instead, the mechanism is induced by non-selfdual bion saddles. Furthermore, the bion-induced potential has two minima within the fundamental domain of , with associated with discrete chiral symmetry breaking and two vacua.
Neutral bions generate a non-perturbative potential for the Wilson line, and generate a potential between the eigenvalues of Wilson lines, i.e, preferring center-stability. In the theory, since the perturbative contribution to the Wilson line potential is absent, center stability is realized via the neutral bion mechanism.
For , where is the lower boundary of the conformal window, the theory is expected to exhibit non-abelian chiral symmetry breaking on . At weak coupling, this symmetry is not spontaneously broken, and the small- regime realizes confinement with discrete chiral symmetry breaking, but without non-abelian chiral symmetry breaking Unsal:2007vu (); Unsal:2007jx (). As one increases , (i.e., at stronger coupling) the monopole-induced term can lead to continuous chiral symmetry breaking, as in the Nambu-Jona-Lasinio model, but now realized as a zero temperature quantum phase transition. This happens at the boundary of the semi-classically justified long-distance effective theory. Unfortunately, there is no known microscopically reliable method to explore this regime. Nonetheless, it is encouraging that the reliable effective theory based on monopole-instantons and bions generates confinement, discrete chiral symmetry breaking and has the seed of non-abelian chiral symmetry breaking on QCD-like theories on . Models emulating this physical set-up also exhibit non-abelian chiral symmetry breaking Nishimura:2009me (); Shuryak:2012aa (); Liao:2012tw (); Larsen:2015tso (); Misumi:2014raa (). For other roles of the bion saddles in diverse theories, see Misumi:2014bsa (); Misumi:2014jua (); Nitta:2015tua (); Nitta:2014vpa ().
6 Toward Picard-Lefschetz Theory of Path Integration
Some aspects of the bion saddles in SYM and QCD(adj) raises interesting puzzles. For example, how can bions (which are presumably constructed as approximate solutions) be responsible for exact results such as the vanishing of vacuum energy or a condensate in SYM. Further, consider the following facts, which at first sight seem to suggest a possible incompatibility between SUSY and semi-classical analysis:
It is a widely known fact that in supersymmetric theories, the ground state energy is positive-semidefinite non-perturbatively, and zero to all orders in perturbation theory.
The standard textbook treatment of path integrals [see the beautiful book by Coleman Coleman (), or reviews on QCD Schafer:1996wv ()], presents a formalism in which only real saddles are accounted for. Does this mean that semi-classical analysis and supersymmetry are incompatible with each other? The answer is no, but the answer requires a significant revision of conventional semi-classics, broadening the perspective to explicitly include complex saddles.
Euclidean path integral in quantum mechanics: Assume one is given a Euclidean path integral over real paths/fields. Refs.Behtash:2015zha (); Behtash:2015loa () propose that the semi-classical analysis of generic Euclidean path integrals necessarily requires complexification of the path space, ,
Here denotes the complexified path space, and one should consider path integration over the middle-dimensional cycles for which . This means that in order to find the possible set of saddles that may contribute to the path integral, instead of solving the real classical Euclidean equations of motion, one should solve the holomorphic classical Euclidean equation of motion. For example, in quantum mechanics of a particle in a potential , to find saddles, one should solve the holomorphic Newton equations in the inverted potential:
where is holomorphic. The second one of these, the textbook recipe, misses some of the most important contributing saddles. Note that the first equation in Eq.24 is a coupled set of equations, which generically do not decouple, for the real and imaginary parts of the path. This simple fact has interesting consequences Behtash:2015zha (); Behtash:2015loa ().
The relation between holomorphization and Picard-Lefschetz theory is the following. If we (boldly) translate standard knowledge of steepest descent cycles in finite dimensions AGZV2 (); Pham () to infinite dimensions, we find a complex version of gradient flow Witten:2010cx (). For the real space path integral, this amounts to solving the complex gradient flow equations:
The holomorphic equations of motion are just the vanishing condition on the right hand side of the complex gradient flow (or Picard-Lefshetz) equations. The resulting solutions form a more complete basis for saddles which may contribute to path integral.
The practical implications of these formal ideas have not yet been fully realized in QFT. However, in the path integral formulation of quantum mechanical systems with bosonic and Grassmann fields (emulating the flavor degrees of freedom of QCD), it has been demonstrated that the counterpart of the magnetic bion is an exact real saddle non-selfdual solution, and the counterpart of the neutral bion is an exact complex non-selfdual solution to the holomorphic Newton equations in the inverted potential Behtash:2015zha (); Behtash:2015loa ().
6.1 Hidden Topological Angles and Complex Saddles
A hidden topological angle (HTA) is an invariant angle associated with saddle points of the complexified path integral and their descent manifolds (Lefschetz thimbles)Behtash:2015kva (); Behtash:2015kna (). The HTA is distinct from theta-parameter in the lagrangian. But in a way similar to the usual -parameter, which turns the instanton fugacity into a complex fugacity, , the HTA also plays a similar role in the path integral saddle space. For real saddles, the HTA is zero. The HTA plays a crucial role in the dynamics, e.g, gluon condensate, vacuum energy, and center stability, as discussed below. It is a conceptually new ingredient in the more rigorous understanding of semi-classics.
Now, we can explain the resolution of the problem that we set in §2: the vanishing and anomalously small gluon condensate in, respectively, SYM and non-supersymmetric QCD(AS). This also provides the resolution of the puzzle stated at the beginning of §6, concerning an apparent incompatibility of SUSY with semi-classical analysis. To highlight the surprising aspect of this analysis, we recall the folklore that the gluon condensate can only receive positive semi-definite contributions in a semi-classical expansion (when the topological theta angle is set to zero) in a vector-like theory, see Callan:1977gz (); Schafer:1996wv (); Shifman:1978bx ().
Consider SYM on small where it is semi-classically calculable. At leading order in semi-classics, the gluon condensate is zero because in the monopole-instanton background, the measure has two unpaired fermion zero modes, and the Grassmann integral over them gives zero. At second order, there are non-vanishing contributions. Magnetic bions contribute positively, but there is an extra phase associated with the neutral bion. As such, at second order in semi-classics, one has
By the trace anomaly, this translates to a result for the vacuum energy density:
The HTA, the angle in the factor accompanying the neutral bion contribution, provides the resolution of both the puzzles mentioned in §2 and §6, which actually turn out to be the same problem due to the trace anomaly relation. The contribution of the real saddle to the vacuum energy is negative, but there also exists a complex saddle, whose action is complex due to the HTA, and its contribution to the vacuum energy is positive. Moreover, the real part of the actions of the real and complex saddles are the same, so the two contributions cancel. This is the mechanism by which the semi-classical expansion is consistent with the SUSY algebra. Both the complex saddle and the HTA are crucial for the argument. This example also demonstrates that in using Lefschetz thimbles, for example, either in Euclidean semi-classics or real time semi-classics (with sign problems) Tanizaki:2014xba (); Cherman:2014sba () or in lattice simulations Cristoforetti:2012su (); Cristoforetti:2013wha (); Fujii:2013sra (); Aarts:2014nxa (), all thimbles whose Stokes multipliers are non-zero must be summed over. Numerical evidence for the correctness of this perspective is also given in Kanazawa:2014qma (); Tanizaki:2015rda (); Alexandru:2015xva (); Hayata:2015lzj (); Fujii:2015bua ().
In QM models with Grassmann fields, it is possible to construct exactly the non-selfdual bion solutions. They are solutions to the second-order (non-BPS) classical equations, with finite action, and the imaginary part of their action is always a multiple of . Furthermore, it has been shown that they are the exact form of the (approximate) correlated two-instanton events, and can be identified with Lefschetz thimbles Behtash:2015kva (); Behtash:2015kna (). Interestingly, if one restricts to real solutions it is not possible to find such exact saddle solutions, and the standard practice in the past was to construct approximate instanton/anti-instanton quasi-solutions. The situation is much improved now.
The relation between the instanton type BPS solutions and our exact non-selfdual solutions is also of importance and gives more insight into the full story. The position of a BPS instanton solution is a moduli, but with two such events, their separation is a quasi-zero mode, a parametrically small non-Gaussian mode, that needs to be integrated exactly to get the correct correlated two-event amplitude. In the background of multi-instantons, we consider decomposing the full space of fields into
where is the full thimble in field space, and are the Gaussian fluctuations of the fluctuation operator, and are exact and quasi-zero mode directions. As shown in Behtash:2015kna (), the relative angle between the magnetic and neutral bion thimbles is
which is a -worth of hidden topological angle structure. This is the origin of the vanishing of the gluon condensate and the vanishing vacuum energy in the semi-classical domain of SYM. The HTA also explains the parametric smallness of the gluon condensate in non-supersymmetric QCD(AS)Behtash:2015kna (). Next, we discuss the role of the neutral bion and the HTA in center-symmetry realization.
6.2 Calculability and the Deconfinement Phase Transition
Studying deconfinement or the center-symmetry-changing phase transition in continuum QCD-like theories by reliable analytical methods proved to be impossible until recently. Perhaps the state of the art perspective on the subject is still the one expressed by Gross-Pisarski-Yaffe (GPY) from 1981 Gross:1980br (): “It is hardly surprising that we cannot explore the transition as the temperature is lowered, from the unconfined to the confined phase using solely weak coupling techniques.” One possible way around this obstacle, by considering the theory on small , is found in Aharony:2003sx (). In this context, since the theory is at finite volume, a genuine phase transition is only possible at , where the thermodynamic limit is achieved Aharony:2003sx (). In Poppitz:2012sw (); Poppitz:2012nz (), a new idea resolving the GPY-problem is presented on the thermodynamic setting, so that it works for arbitrary gauge group . For related works, see Poppitz:2013zqa (); Anber:2014lba (); Anber:2013doa (); Anber:2012ig (); Anber:2011gn (); Shuryak:2013tka ().
The Main Idea Poppitz:2012sw (); Poppitz:2012nz (): The goal is to find a calculable phase transition and gain insight into the transition in pure Yang-Mills theory. Consider SYM, with a gauge boson and a fermion, and continuously connect this theory to pure Yang-Mills by turning on a mass term for the fermions. At , using the twisted partition function Eq.9 on (i.e., non-thermal periodic boundary conditions for fermions), this theory exhibits analyticity as a function of ; hence, no phase transitions Witten:1982df (); Davies:2000nw (). In fact, Eq.9 is the Witten index, and is a constant, independent of . At , the theory reduces to pure YM. Consider finite and small . If one takes , fixed, the theory lands on a confining phase, and the center is unbroken due to the non-perturbative center-stabilizing potential induced by the neutral bion. If one takes fixed, , the theory lands on a center-broken (deconfinement) regime, due to the perturbative one-loop center-destabilizing GPY-potential. This means that the two limits do not commute, and there must exist a calculable phase transition in between. Indeed there is, as shown in Fig. 4(left), for any simple gauge group. More precisely, the deconfinement phase transition in pure Yang-Mills theory is continuously connected to a quantum (non-thermal) phase transition in mass-deformed SYM theory. This transition can be studied in a controlled way.
The surprising outcome of this idea is that the mechanism governing the phase transition is universal, and valid for all simple groups. As an example, here we express the induced potential in dimensionless variablles, , just for the theory:
Neutral bions favor center stability and generate repulsion among the eigenvalues of the Wilson line (forcing in Eq.13.) Here, the crucial point is the HTA associated with the neutral bion Eq.20. Were it not for the HTA, the neutral bion would lead to a center-destabilizing effect! At finite , both the perturbative one-loop potential, and the monopole-instantons prefer center breaking, leading to an attraction among the eigenvalues of Wilson line. This contradicts some folklore that monopole-instantons favor confining Wilson line holonomy.
The semi-classical phase transition is driven by the competition between these three effects. Fig.4(right) shows contour plots for the potential for the trace of the gauge theory Wilson line in three regimes: (i) confined phase; (ii) phase between the limits of meta-stability (where confined or deconfined holonomy minima coexist, and one of these is the the global minimum) and (iii) deconfined phase. All three regimes have been observed in lattice simulations Bringoltz:2005xx (). We remind the reader that this figure is not a model. There are no tuned parameters; it is the result of a controlled semi-classical computations.
This framework can also be used to investigate the effect of the topological -angle dependence on the phase transition and critical temperatures, see Poppitz:2012nz (); Anber:2013sga (). The most interesting results is that , the critical temperature, is a multi-branched function, which has a minimum at . These results inspired various lattice studies, which exhibit remarkable matching to the theoretical predictions D'Elia:2012vv (); Bonati:2013tt (); D'Elia:2013eua (); Bonati:2015sqt (). The semi-classically calculable realization of center-symmetry also inspired related works in the strong coupling regime, see Liu:2015ufa (); Liu:2015jsa (); Larsen:2015vaa (), and lattice simulations Bergner:2014dua ().
7 Resurgence, Renormalons and Neutral Bions
In this final section, we present a brief overview of the resurgence program in non-trivial QFTs such as bosonic non-linear sigma models in two dimensions (, Grassmannian, , principal chiral model (PCM)), and Yang-Mills and one-flavor QCD-like theories in four dimensions Dunne:2015eaa (). The link to the previous sections is the fact that the neutral bion can be identified as the semi-classical realization of the infrared (IR) renormalons in these asymptotically free QFTs Argyres:2012ka (); Argyres:2012vv (); Dunne:2012ae (); Dunne:2012zk (), and that the Lefschetz thimble construction naturally encodes resurgence. These asymptotically free theories are incalculable on , (), for reasons explained below, and are all believed to be gapped. We describe here the analysis for the two dimensional sigma model, and we comment later on the similarities and differences for the other theories. In fact, the structure that emerges for all of these theories on exhibits a surprising level of universality.
The deep puzzles concerning these theories, emanating from profound work in the late 1970s and early 1980s 'tHooft:1977am (); Beneke:1998ui (); Callan:1977gz (); Schafer:1996wv (), are all very similar. Recently, progress has been made in the non-perturbative understanding of these theories, by combining the idea of adiabatic continuity described in §4 above, and the mathematical formalism of resurgence. Resurgence is a general mathematical formalism that unifies perturbative and non-perturbative expansions into a single unified framework in the form of a trans-series, such that the entire trans-series is internally self-consistent with respect to Borel summation and analytic continuation of the couplings, and naturally incorporates Stokes phenomena Ecalle:1981 (); bh90 (); Costin:2009 (); Aniceto:2013fka (); Dunne:2015eaa (). These ideas have also had a profound impact on the study of matrix models and string theory Marino:2008ya (); Pasquetti:2009jg (); Aniceto:2011nu (); Aniceto:2014hoa (); Garoufalidis:2010ya ().
It is well known that perturbation theory in almost all interesting QFTs is a divergent asymptotic expansion, even after regularization and renormalization. A powerful method to extract physical meaning from such a divergent expansion is the Borel transform, and the Borel sum. The Borel transform maps the (typically factorially divergent) series to a new convergent series: , and the Borel sum is the Laplace integral: . Formally, assigns a value to , but in almost all interesting QM examples, and for the above QFTs, has singularities at on the positive Borel axis, which means that as defined by the integral is ambiguous, depending on how one deforms the contour around the singularity. In certain cases, such as the QM cubic oscillator or the Stark effect problem, there is such a singularity but it has a clear physical meaning in terms of the instability of a metastable state, so the ambiguity is easily resolved by the physical condition of causality. But in other theories, such as the symmetric double-well, or periodic potential, or in asymptotically free QFTs, the systems are stable, and yet the ambiguous, non-perturbative imaginary contribution remains. In these cases, it implies that Borel resummed perturbation theory is incomplete on its own. Note that the pathological ambiguity has the non-perturbative form .
In QM with instantons there is another, lesser known, pathology, coming not from perturbation theory but from the non-perturbative ”instanton-gas” expansion itself. The one-instanton sector is well-defined and unambiguous Coleman (), but in the two-instanton sector, when one calculates the instanton-antiinstanton amplitude, via a procedure that was referred to as the BZJ-prescription in Argyres:2012ka (); Dunne:2012zk (), due to important works by Bogomolny Bogomolny:1980ur () and Zinn-Justin ZinnJustin:1981dx (), one finds that the amplitude is also multi-fold ambiguous. This prescription was partly a black-box until recently, and did not always produced a sensible result. It is now better understood using Lefschetz thimble techniques and complexification of path integral Behtash:2015loa (); Behtash:2015kva (), and again the net result is that the two imaginary ambiguous non-perturbative terms cancel one another, restoring consistency of the full ”trans-series” representation of the (real) physical quantity being computed:
up to terms of higher order. [This connection between different saddles is also related to an argument of Lipatov Lipatov:1976ny (), see also Balitsky:1985in ().] The ambiguities at leading order cancel, and the sum of perturbative and non-perturbative sectors is meaningful, ambiguity-free and accurate. In principle, this is just the ”tip-of-the-iceberg”: e.g., it has now been shown in a class of QM models that these resurgent cancellations occur to all orders of the trans-series expansion Dunne:2013ada (); Dunne:2014bca (). An obvious but deep question is: can this idea work in QFT, in particular to cure the IR-renormalon problem of asymptotically free QFT?
’t Hooft’s IR renormalon puzzle : In the asymptotically free QFTs mentioned above, there are Borel singularities that are parametrically closer to the origin of the Borel -plane, on , than the singularity 'tHooft:1977am (); Beneke:1998ui (). The singularity is related to the factorial combinatorial growth of the number of Feynman diagrams, while the (more dominant) IR renormalon singularity is related to divergences coming from phase space integration, at small internal momenta, smaller than . ’t Hooft called these singularities “IR renormalons” in the hope that they would be associated with a semiclassical saddle, such as an instanton. For example, the singularities are located at , but the IR-renormalon singularities are located at , where is the leading coefficient of the renormalization group beta function. Due to this mis-matched factor of , it appears that the cancellation mechanism at work in QM cannot possibly work in QFT on . However, there is another well-known problem in , Yang-Mills etc., coming from the instanton-gas picture itself. There are instantons, but the so called “dilute instanton gas approximation” must be regulated, because the instanton size is a moduli parameter, whose integration leads to another IR divergence.
The Main Idea Argyres:2012ka (); Argyres:2012vv (); Dunne:2012ae (); Dunne:2012zk (): The key step is to combine the concept of adiabatic continuity Unsal:2007vu (); Unsal:2007jx () discussed above, with resurgence theory. Asymptotically free QFTs can be brought into a calculable semi-classical domain on , and with appropriate boundary conditions or deformation this can be continuously connected to the strongly coupled domain of interest. In the calculable semi-classical domain, all non-perturbative aspects of field theory become tractable, and should not differ dramatically from the behavior on . Once a small circle compactification respecting continuity is found, which is counter-part of center-symmetric background in gauge theory, (note that high temperature limit of thermal compactification never achieves this), something magical occurs. E.g., the 2d instanton in the model, with action , splits up into kink-instantons with action , which is now the minimal action semi-classical saddle Bruckmann:2007zh (); Brendel:2009mp (); Dunne:2012ae (). At second order in semi-classical analysis, similar to Eq.20, there are two types of bion: we refer to them as charged and neutral bions. The neutral bions possess zero topological charge and zero “magnetic” charge, and can thus mix with perturbation theory. In the zero flavor theory, the neutral bion amplitude is two-fold ambiguous, and Borel-resummed perturbation theory (which transmutes to the combinatorics in reduced twisted compactification Dunne:2012ae (); Anber:2014sda () is also two fold ambiguous. Remarkably, these two ambiguities cancel each other exactly Dunne:2012ae (); Dunne:2012zk (). Correspondingly, in the Borel plane the leading singularity is located at , which, since for , is the same as the location of ’t Hooft’s elusive renormalon ambiguity :
Here is the strong scale of the theory. Both the ambiguity and the jump can be interpreted as Stokes phenomenon in field space.