Physical Review D

Finding interacting systems which can capture some features of gravity, but which remain simple enough to solve, has been of great interest in the context of gauge/gravity duality. A system of this type was proposed by Sachdev, Ye and Kitaev (SYK), consisting of a quantum mechanical model of N Majorana fermions interacting with random interactions. An interesting feature of this theory in the large N limit is that at low energies it develops an approximate conformal symmetry, which is similar to certain extremal black holes that develop a nearly AdS₂ region. This scenario can then be understood as a nearly AdS₂/CFT₁ duality.

In this paper, the authors expand the SYK model to an arbitrary number of interacting fermions at a time, and compute in detail two and four point correlation functions of the fundamental fermions, thus finding the spectrum of physical excitations. They explicitly show how the emergent conformal symmetry develops, and why the system displays chaotic behavior. Since it was proposed several years ago, the SYK model, and its extensions, have been an important tool in analyzing various holographic aspects of lower dimensional gravitational theories.

Gravitational waves as a signature prediction of the general theory of relativity, date back to the work of Albert Einstein in the year following the theory’s initial formulation. Almost exactly 100 years later, gravitational waves were directly observed by the LIGO and Virgo Collaborations in 2015. The discovery, named GW150914, was many decades in the making as the two detection instruments, located hundreds of miles apart in the United States, were honed to a precision that still defies one’s ordinary sensibility. The present two papers explain the necessary and painstaking work done to interpret the raw signal from the finely tuned detectors as arising from the merger of binary black holes of definite mass. These papers established for the first time the existence of gravitational waves and the occurrence of astrophysical black-hole binary mergers.

Gravitational waves as a signature prediction of the general theory of relativity, date back to the work of Albert Einstein in the year following the theory’s initial formulation. Almost exactly 100 years later, gravitational waves were directly observed by the LIGO and Virgo Collaborations in 2015. The discovery, named GW150914, was many decades in the making as the two detection instruments, located hundreds of miles apart in the United States, were honed to a precision that still defies one’s ordinary sensibility. The present two papers explain the necessary and painstaking work done to interpret the raw signal from the finely tuned detectors as arising from the merger of binary black holes of definite mass. These papers established for the first time the existence of gravitational waves and the occurrence of astrophysical black-hole binary mergers.

The discovery of the Higgs boson in 2012 was heralded as confirmation of the last piece of the Standard Model. Proposed by Peter Higgs and others in the early 1960s, the search for this hypothetical field involved a huge experimental and theoretical effort, culminating in the discovery of a new particle with mass around 125 GeV at the Large Hadron Collider. However, it remains an open question whether the observed particle is exactly as predicted in the Standard Model. The ATLAS and CMS collaborations have produced a large number of papers studying the new resonance, and there is an ongoing effort to test its properties in ever more detail.

Here, we have chosen two key papers published by the collaborations in 2014 for our anniversary selection. By studying the decay of the Higgs boson to two Z bosons, and subsequently four leptons, the experiments made not only a precise mass measurement, but also excluded with high confidence the possibility that the particle was a pseudoscalar or a higher-spin boson, thus confirming its spin-0, scalar nature in accordance with theory.

The discovery of the Higgs boson in 2012 was heralded as confirmation of the last piece of the Standard Model. Proposed by Peter Higgs and others in the early 1960s, the search for this hypothetical field involved a huge experimental and theoretical effort, culminating in the discovery of a new particle with mass around 125 GeV at the Large Hadron Collider. However, it remains an open question whether the observed particle is exactly as predicted in the Standard Model. The ATLAS and CMS collaborations have produced a large number of papers studying the new resonance, and there is an ongoing effort to test its properties in ever more detail.

Here, we have chosen two key papers published by the collaborations in 2014 for our anniversary selection. By studying the decay of the Higgs boson to two Z bosons, and subsequently four leptons, the experiments made not only a precise mass measurement, but also excluded with high confidence the possibility that the particle was a pseudoscalar or a higher-spin boson, thus confirming its spin-0, scalar nature in accordance with theory.

In this paper, Banks and Seiberg put in context several better- and lesser-known conjectures about (symmetry) properties of consistent quantum theories of gravity: (i) models of quantum gravity have no global symmetries, (ii) all continuous gauge groups are compact, and (iii) all allowed electric and magnetic charges are part of the spectrum(the “completeness hypothesis”). The authors provide strong arguments for the validity of these conjectures, mostly based on very general black hole physics, and show how they are interrelated. This not only set the conjectures on a strong footing but also allowed them to discover new observables in quantum field theories and properties of supergravities.

Thanks to this fundamental analysis these conjectures have become hallmark properties for the study and classifications of quantum field theories with a consistent quantum gravity completion, such as the very active swampland program.

The Muon (g-2) Collaboration reports the final results of the precision measurements of the muon anomalous magnetic moment at Brookhaven National Laboratory. In a multi-year effort comprising five runs the Collaboration achieved a precision of 0.54 ppm, making the muon anomalous magnetic moment one of the most precisely known observables in physics. It is also one of the most accurately computed quantities on the theory side, with all three fundamental interactions: electromagnetic, weak, and strong contributing; and with a theory error comparable to the experimental one. Intriguingly, there is about a 2.5 sigma discrepancy between theory and experiment, hinting that as-yet-unknown physics beyond the standard model might contribute. In order to resolve or strengthen the discrepancy a new measurement is under way at Fermi National Laboratory with the aim of decreasing the error by a factor of four. This, in turn, spurred an initiative to decrease the theory error, which is dominated by hadronic, i.e., strong interaction, uncertainties, by a similar amount.

Is our theoretical understanding of cosmology, as embodied in the standard Lambda-CDM model, consistent with precision cosmological data? In this paper by Tegmark et al, the standard model of cosmology is confronted with precision data. In particular, parameters of the standard model are extracted from CMB data from WMAP and power spectrum data from SDSS. The authors show that the standard model of cosmology is indeed compatible with the data. Further, they also show that generic expectations from inflationary models are also borne out by the data. Limits are established on the contribution of massive neutrinos to dark matter and a host of constraints on dark matter and dark energy are presented.

Ever since the Nobel Prize-winning discovery of the accelerated expansion of the Universe, there has been great interest in finding de Sitter (dS) vacua with a small (positive) cosmological constant in string theory that would fit the cosmological data. However, it is notoriously difficult to construct compactifications with a four-dimensional dS component in string theory, so much so, that a no-go theorem states that no such solutions exist.

Based on previous work on warped compactifications with additional sources (fluxes) that successfully circumvent the no-go theorem, Kachru, Kallosh, Linde, and Trivedi (KKLT) describe the construction of a compactification with its moduli, like the volume of the compact space, dynamically stabilized. This provides the first stable 4D anti-de Sitter (AdS) solution (which has a negative cosmological constant) with a four-dimensional AdS scale that is much larger than the compactification scale such that 4D effective field theory is consistent. In a second step they show how to uplift the cosmological constant to a positive value and thus realize a four-dimensional dS solution in string theory with a controlled breaking of supersymmetry. The resulting dS space is metastable, but the authors show that the solution exists long enough to agree with the lifetime of our Universe.

This construction triggered tremendous activity that continues to the present day, leading to concepts like the string landscape and the possibility of incorporating inflation in string theory. It is the subject of renewed scrutiny in the context of the so-called swampland program, which is one of the most exciting current debates in string theory and in which the KKLT result is still a centerpiece almost 20 years after its initial publication.

Despite much effort and the apparent potential of string theory, its connection to realistic phenomenology remains a challenge. One theoretical puzzle is the so-called hierarchy problem, namely how large differences between the various mass scales, such as the weak interaction and Planck scales, are generated in a unified fundamental theory. A promising mechanism to naturally obtain large ratios of scales employs warped spacetimes - where a warped higher dimensional spacetime generates different energy scales in four dimensions. While this mechanism and its implications had been studied as an effective model, a possible underlying microscopic origin had not previously been proposed in string theory. In this very influential paper, Giddings, Kachru and Polchinski demonstrated for the first time that hierarchies can be generated from warp factors in superstring theory. They showed how this occurs in both orientifold compactifications of type-IIB string theory and F-theory compactifications on Calabi-Yau fourfolds. In each case, the hierarchy of scales is fixed by a choice of certain fluxes in the compact manifold, which is a foundation of extensive studies of flux compactifications in string theory. This paper provided an important step in the endeavor to obtain the Standard Model from string theory.

High-energy physics involving quarks is complicated by the presence of multiple energy scales which enter into the calculations. Both low-energy “soft” and collinear degrees of freedom lead to large logarithms that depend on the ratios of disparate energy scales, and reliable perturbative calculations require that these logarithmic terms be resummed via renormalization group running. In the early 2000’s, collaborators Bauer, Fleming, Pirjol and Stewart developed an effective field theory approach below the heavy quark mass that implements these steps in a consistent manner, using an expansion in powers of small transverse momentum over the heavy quark mass that can account for the multiple scales.

Their original work investigated the decay of a heavy B meson to a light meson. Now known as SCET (Soft Collinear Effective Theory), the approach has since been applied to baryon decays, heavy quark production, jet physics, Higgs boson production, and other processes. This technique remains a subject of ongoing investigations and a tool used in many analyses today.

Coalescences of binary compact astrophysical objects, such as black holes, are primary sources of gravitational waves that have recently become accessible to detection by sensitive experiments. In general relativity, the binary problem, in its elemental form, is the two-body problem, which has resisted exact solution. In the present two papers, Alessandra Buonanno and Thibault Damour map the two-body problem to a one-body problem of a particle moving in an effective metric. The mapping allows for the separation of the dynamics into separate phases - conservative motion and radiation-reaction. In the latter phase, the authors show that there is a transition from an adiabatic quasicircular inspiral to a sudden “plunge” ending in final coalescence. The formalism of the effective one-body problem, while nonperturbative, is amenable to approximation schemes such as the post-Newtonian expansion, to which it is adapted.

These papers initiated a program that continues to be vigorously pursued to the present day and remains one of the few analytical approaches to the increasingly relevant problem of experimental detection of gravitational radiation from astrophysical binary coalescences. These results have already contributed to the experimental identification of binary coalescences via their gravitational wave signatures by LIGO.

Coalescences of binary compact astrophysical objects, such as black holes, are primary sources of gravitational waves that have recently become accessible to detection by sensitive experiments. In general relativity, the binary problem, in its elemental form, is the two-body problem, which has resisted exact solution. In the present two papers, Alessandra Buonanno and Thibault Damour map the two-body problem to a one-body problem of a particle moving in an effective metric. The mapping allows for the separation of the dynamics into separate phases - conservative motion and radiation-reaction. In the latter phase, the authors show that there is a transition from an adiabatic quasicircular inspiral to a sudden “plunge” ending in final coalescence. The formalism of the effective one-body problem, while nonperturbative, is amenable to approximation schemes such as the post-Newtonian expansion, to which it is adapted.

These papers initiated a program that continues to be vigorously pursued to the present day and remains one of the few analytical approaches to the increasingly relevant problem of experimental detection of gravitational radiation from astrophysical binary coalescences. These results have already contributed to the experimental identification of binary coalescences via their gravitational wave signatures by LIGO.

In contemporary models the decay of the inflation field (and other scalar fields) at the end of inflation is the source of all matter and radiation in the present Universe. This process was historically called “reheating” and is a cornerstone of the explanation of the Universe as we observe it. However, despite three decades of research, a theory of reheating is still far from complete. The inflaton field has to decay completely to avoid ending up with a cold and empty universe.

In this groundbreaking paper the authors provide an expansive and detailed fundamental theory of reheating, which includes several different phases dubbed preheating. The theory includes, in particular, nonperturbative effects of parametric resonances caused by the oscillating inflaton field, along with exponential growth of particle production and Bose condensation. They also take into account the expanding Universe and back reactions. Bose condensation happens during a narrow parametric resonance, although the most effective particle production takes place during a broad parametric resonance. These effects are difficult to study analytically, even in the simplest models, but with this paper the authors provided the basis for every model of reheating to this day. This includes the possibility of GUT model baryogenesis as well as the creation of topological defects during this phase.

Although superstring theory has been investigated intensively as a unified theory including quantum gravity, its reliable and manageable formulation remains for the most part only perturbative. However, it appears that to be able to describe the real world — the observed spectrum of elementary particles, with a small cosmological constant and a plausible mechanism for cosmic inflation — a nonperturbative formulation of string theory is needed. An important paper by Banks, Fischler, Shenker and Susskind proposed a precise equivalence between an infinite dimensional supersymmetric matrix quantum mechanics model and the uncompactified eleven-dimensional M theory in the infinite momentum limit, which in turn is equivalent to the strongly coupled type IIA string theory. This conjecture was the first attempt to obtain a nonperturbative formulation of a quantum theory which includes gravity. The proposal provides an example of gauge-theory/gravity duality where the gauge theory is a model of ordinary quantum mechanics with no spatial dimensions. By studying the dual matrix model, the conjecture can provide clues about the degrees of freedom of quantum gravity theories and the emergent geometries. This paper has been very influential in initiating this research direction, which continues to date.

In this and a companion paper, Phys. Rev. D 49, 3352 (1994), Larry McLerran and Raju Venugopalan proposed a classical model for the scattering of large nuclei at high energies. They argued that the color charges of the valence quarks can be treated as static sources which propagate along the light cone. A high density of soft gluons is generated, and can be described as a classical gauge field. This provides an energy scale much larger than the usual QCD scale, and so a small QCD coupling. Fluctuations in the color charge are treated as Gaussian random variables, and give the quark and gluon distribution functions to lowest order in the coupling. The McLerran-Venugopalan model is the starting point of an effective theory known as the color glass condensate (CGC), which describes the initial state of heavy ion collisions at high energies.

Initial explorations of black hole entropy understood it to be a concept dependent on the particularities of the defining equations of Einstein’s theory of general relativity. In this paper, Robert Wald showed that a notion of entropy can be defined in any diffeomorphism-invariant theory of gravity independent of spacetime dimension provided they support stationary black hole solutions with horizons satisfying some mild conditions. The entropy defined by Wald is the Noether charge derived from the horizon Killing field and coincides with the usual notion of entropy when restricted to Einstein gravity. When an initial stationary black hole is perturbed and evolves into a different stationary black hole final state, the entropy change is given by the Noether flux. Thus the analog of the second law of black hole thermodynamics is a statement about the positivity of the flux.

A key element of modern physics is the breaking of the electroweak gauge group SU(2) X U(1) to the electromagnetic group U(1)_em. In the standard model this is accomplished by a single Higgs doublet, but other scenarios are possible, involving additional Higgs bosons or strong dynamics. In 1991 Michael Peskin and Tatsu Takeuchi published a detailed analysis of the so-called oblique parameters S, T, and U, which measure possible deviations from the standard model due to quantum corrections that would arise from new electroweak physics. This work, based in a formalism developed by Kennedy and Lynn, collected and extended early brief results from Peskin, Takeuchi and other authors. Their parameters are still widely used today to constrain a variety of models, particularly in view of the Higgs particle discovered at the LHC.

In a paper published in PRD 30, 2212, Klinkhamer and Manton find a static yet unstable approximate solution of the equations of the Weinberg-Salam model, which they dub the sphaleron. The solution is localized in space and has a finite energy. The vacuum structure of the Weinberg-Salam model is complicated, comprising a multiplicity of degenerate vacua separated by energy barriers. The sphaleron is a saddle point sitting at the top of these barriers. Transitions between adjacent vacua entail a change in the baryon number B and the lepton number L, in the amount of ΔB = ΔL = n_f, the number of flavors. Thus, transitions driven by the sphaleron conserve B-L while neither B nor L are conserved separately. This mechanism provides a possible explanation for the observed baryon and lepton number asymmetries in the Universe.

Understanding the evolution of the Universe remains an outstanding problem. Regardless of the inflation model, quantum effects are likely to be very important in the early universe. In the absence of a consistent quantum theory of gravity, a potential approach to analyze the initial conditions for inflation is through the wave function of the Universe, which was proposed for the first time by Hartle and Hawking in “Wave function of the Universe”, published in PRD. The authors of the influential paper proposed that the quantum state of a spatially closed universe can be described by a wave function which is given by a path integral over all compact four-dimensional geometries which have a given compact three-dimensional geometry as a boundary, and interpret this wave function as the amplitude for that three-dimensional geometry to arise from a single point. In other words, this is the amplitude for the Universe to appear from nothing. While not complete, this approach was the first attempt to apply quantum mechanics to the Universe, and it has been studied ever since.

Before this paper, spontaneous symmetry breaking was conceived as arising from the properties of the classical theory - namely, from the existence of a degenerate set of ground states of the classical potential that were connected to each other by the action of a certain (broken) subgroup of the theory’s symmetry group. In this paper, Sidney Coleman and Erick Weinberg construct a theory where spontaneous symmetry breaking arises from purely quantum effects. In their theory, a self-interacting charged scalar field has a unique classical ground state invariant under electromagnetic gauge transformations. Nevertheless, they find that quantum effects induce a potential that breaks the gauge symmetry spontaneously and gives rise to a massive photon and a scalar particle in analogy with the classical Higgs potential but now due to purely quantum effects. Moreover, unlike the Higgs potential, there are no dimensionful parameters in their classical theory and the masses arise from quantum effects alone.

While working with H. Tye on a possible solution of the “magnetic monopole problem” (why don’t we see any?) through supercooling, when a scalar field is trapped in a false vacuum with finite energy density and negative pressure, A. Guth realized that this would lead to an exponential expansion of the Universe which he dubbed inflation.

He also realized that this would solve two of the biggest puzzles of hot big bang cosmology: 1. The horizon problem (why is the universe so uniform despite many regions are causally disconnected), and 2. the flatness problem (why is the universe nearly flat when small deviations would have grown significantly with time). This could only be achieved by enormously fine-tuned and unstable ad hoc initial conditions. However, in this groundbreaking paper, A. Guth showed how an inflationary universe arises and solves both problems naturally by growing from a tiny causally connected patch and exponentially stretching space into flatness, setting off a vigorous theoretical and observational research activity and providing the framework for our best understanding of the early Universe until today.

Hard exclusive processes in hadron scattering, those where the final state particles are completely specified, provide a key window to the structure of quantum chromodynamics. Following the development of QCD, Brodsky and Lepage put forth a firm theoretical basis for analyzing such processes in this paper. Their work confirmed earlier power-counting rules and established a framework for performing perturbative calculations related to a wide range of topics, including elastic scattering, form factors, particle decays, and hadron structure. Along the way it helped confirm the accuracy and validity of QCD as a theory of the strong force.

The formulation of non-Abelian gauge theories on a spacetime lattice by Kenneth G. Wilson, Phys. Rev. D 10, 2445 (1974), provided a nonperturbative regularization of these theories. In this paper, Michael Creutz pioneered the use of Monte Carlo techniques to solve the SU(2) gauge theory numerically. He showed that the string tension exhibits a crossover at intermediate couplings from strong coupling behavior to approximate scaling following from the weak coupling renormalization group. This work initiated the field of numerical simulations of lattice QCD, to this day the only way to obtain nonperturbative results in QCD from first principles.

Known neutrinos come in three varieties - electron, muon, and tau - but these identities change or oscillate, as neutrinos propagate, even in vacuum. In this paper, Lincoln Wolfenstein showed that neutrino flavor conversion is modified when neutrinos propagate through matter and these effects become noticeable and relevant to experiments when the propagation medium is dense or the distance is large. Matter effects, first uncovered here, are crucial to analyzing phenomena involving astrophysical and solar neutrinos as well as neutrinos propagating through the earth.

Just as a black hole evaporates quantum mechanically through particle production, evincing an effective temperature inversely proportional to the gravitational acceleration at its surface, William Unruh, in a groundbreaking paper, showed that a detector, accelerating uniformly in the vacuum state of conventional quantum field theories in flat spacetime, will detect particles, in contrast with inertial detectors, which, as expected, do not detect particles in the vacuum state. The uniformly accelerated detector responds as if it were in a thermal bath with an effective temperature proportional to its constant proper acceleration, analogous to what happens for the black hole.

This work lent considerable support to the notion of quantum instability of black holes, or more generally systems with event horizons, against particle creation and is one of the initial forays into the study of quantum aspects of strongly gravitating objects.

Pseudoparticles, or instantons as they came to be known, are exact solutions of the Euclidean field equations of Yang-Mills theories. They were found just the year before the publication of this seminal paper by Gerard ‘t Hooft. Instantons became important objects in mathematics and especially for nonperturbative aspects of the quantum dynamics of gauge theories, such as dynamical symmetry breaking, nontrivial vacuum structure and charge-parity violation, baryon number violation, and axion physics that provides an attractive scenario for dark matter. The paper opens an analytical window into nonperturbative gauge theories and the field of instantons, leading to many applications that continue to be topics of vigorous research today.

Over the years, black holes - massive objects abundant in the Universe - have been researched extensively theoretically and, more recently, also experimentally. In this seminal paper, Jacob D. Beckenstein pioneered a very important thermodynamic approach to their understanding, whereby the proportionality of the entropy to the surface area of the black hole was established. So many years later, this result still provides a gateway into the depths of the physics of black holes.