Search

We show that, in the Nelson-Barr solution to the strong CP-problem, a naturally light scalar can arise. It gives rise to a completely new phenomenology beyond that of the celebrated QCD axion, if this field constitutes dark matter, as the CKM elements vary periodically in time. We also discuss how the model can be tested using quantum sensors, in particular using nuclear clocks, which leads to an interesting synergy between different frontiers of physics., Comment: 8 pages

Theories in which the Peccei-Quinn phase transition occurs after inflation tend to suffer from problematic domain walls. One possible solution involves a small, explicit breaking ot the symmetry. But this raises other potential issues. We review some aspects of axion domain walls, focussing especially on this proposed solution. We argue, in disagreement with some recent literature, that there is little axion radiation from the system until the domains actually collapse. The same applies to gravitational waves and electromagnetic radiation. The final stages of the collapse yields small numbers of extremely energetic axions, which interact only rarely with ordinary matter, and are thus relatively harmless. We then note that, if one accepts a remarkable coincidence, this solution can be acceptable. We consider a possible explanation of the required coincidence, Comment: 12 pages

We review arguments that supersymmetric QCD with $N$ colors and $N_f \ge 0$ massive quarks confines. We explain that this remains the case for small soft breakings and small supersymmetric masses for any quarks. These results continue to hold at large $N$. We then consider these theories with large soft breakings -- real QCD -- describing the problem in terms of the direct computation of the linear potential between a heavy quark and antiquark. We provide naive arguments that confinement persists, and then give a more general treatment. The features required for the naive arguments are generally believed to hold at large $N$. Extending them to finite $N$ requires only mild assumptions., Comment: 13 pages. Loopholes for confinement argument elaborated. Errors in discussion of two dimensional model corrected. Other minor corrections

Any would-be Peccei-Quinn (PQ) symmetry is vulnerable to various types of explicit breaking. It has long been recognized that these can disrupt the axion solution to the strong CP problem. There have also been suggestions that, under certain circumstances, these can lead to a surprisingly large mass for the axion. Two types of corrections to this computation have been widely considered: higher dimension symmetry breaking operators, in theories where there is a complex field responsible for the spontaneous breaking of the PQ symmetry, and small instantons. Motivated by situations where small instantons dominate the $\theta$ potential of non-abelian gauge theories, we formulate the question of axion quality in terms of constraints on a Wilsonian effective action at scales somewhat above the scale of QCD. In this language, the standard axion mass computation assumes a nearly exact PQ symmetry in this action. The higher dimension operators and/or small instantons represent symmetry breaking terms in the Wilsonian action. This picture permits a uniform treatment of the problem of {\it axion quality}. If one assumes order one CP violation at high energies, then solving the strong CP problem constrains the axion potential terms in this Wilsonian action; in particular, the axion mass must be extremely close to its "standard" value., Comment: 9 pages

Recently there have been proposals to understand features of QCD such as confinement and chiral symmetry breaking by considering supersymmetric versions of the theory with various patterns of soft breaking. In this note we recall that with small soft breakings, SUSY QCD is suggestive of observed features of real QCD. But we outline some of the challenges to establishing these features of the theory with large soft breakings. It appears difficult to argue for confinement or chiral symmetry breaking; at best, one can say that {\it if} the non-supersymmetric theory does not confine and/or break chiral symmetry, phase transitions would appear inevitable as one increases the soft breakings. We also discuss, at large N, where confinement would imply chiral symmetry breaking, the challenges to establishing confinement., Comment: 21 pages Footnote and references added

Lattice gauge theory simulations are our principal probe of the masses of the light quarks. Results from such computations are the primary evidence against the $m_u=0$ solution to the strong CP problem. The large-$N$ approximation offers an independent approach to light quarks. We extend existing literature, noting that one can determine the parameters of the non-linear sigma model through second order in quark mass, rule out the $m_u=0$ hypothesis, and make predictions for outputs of lattice calculations and phenomenological fits. A crucial feature of this analysis is a Wilsonian effective action at scales above the $\eta^\prime$ mass. One can self-consistently test the validity of aspects of this framework, and it may well be good to the part-in-three level. We also note consistency with some phenomenological fits and existing lattice results., Comment: 21 pages, no figures. Contribution to a Festschrift in honor of Peter Suranyi

We consider aspects of the low energy description of global cosmic strings. In the non-relativistic limit, we study the extent to which these are described by the combination of the Nambu-Gotto and Kalb-Ramond actions. While in a formal sense this is the case for infiniately long strings, for more realistic situations, the collective coordinates of the system do not provide a complete description for more than brief intervals; as time evolves it becomes necessary to include low momentum goldstone excitations as well., Comment: 11 pages Reference corrected, comments added to clarify that the focus is on the classical effective field theory; matching to simulations would be of interest but is not the subject here

We consider some aspects of the Transplanckian Censorship Conjecture, which states that for theories of quantum gravity which yield a quintessence universe, there is a limit on the lifetime of the state not too different than the current Hubble horizon. We consider some aspects of tunneling in the presence of time-varying fields, and note that lifetimes can be parameterically quite large. We consider what it might mean to find a quintessence potential in string theory, and argue that such states are likely to have very long lifetimes., Comment: 11 pages. References added and repaired

Axions have for some time been considered a plausible candidate for dark matter. They can be produced through misalignment, but it has been argued that when inflation occurs before a Peccei-Quinn transition, appreciable production can result from cosmic strings. This has been the subject of extensive simulations. But there are reasons to be skeptical about the possible role of axion strings. We review and elaborate on these questions, and argue that parametrically strings are already accounted for by the assumption of random misalignment angles. We review and elaborate on these questions, and provide several qualitative arguments that parametrically strings are already accounted for by the assumption of random misalignment angles. The arguments are base on considerations of the collective modes of the string solutions, on computations of axion radiation in particular models, and reviews of simulations., Comment: 31 pages, 3 figures; published version

There have been many attempts to construct de Sitter space-times in string theory. While arguably there have been some successes, this has proven challenging, leading to the de Sitter swampland conjecture: quantum theories of gravity do not admit stable or metastable de Sitter space. Here we explain that, within controlled approximations, one lacks the tools to construct de Sitter space in string theory. Such approximations would require the existence of a set of (arbitrarily) small parameters, subject to severe constraints. But beyond this one also needs an understanding of big-bang and big-crunch singularities that is not currently accessible to standard approximations in string theory. The existence or non-existence of metastable de Sitter space in string theory remains a matter of conjecture., Comment: 20 pages. v2: references added; v3, v4: typos fixed

It has been suggested that scattering cross sections at very high energies for producing large numbers of Higgs particles may exhibit factorial growth, and that curing this growth might be relevant to other questions in the Standard Model. We point out, first, that the question is inherently non-perturbative; low orders in the formal perturbative expansion do not give a good approximation to the scattering amplitude for sufficiently large N for any fixed, small value of the coupling. Focusing on $\lambda \phi^{4}$ theory, we argue that there may be a systematic approximation scheme for processes where N particles near threshold scatter to produce N particles, and discuss the leading contributions to the scattering amplitude and cross sections in this limit. Scattering amplitudes do not grow as rapidly as in perturbation theory. Additionally, partial and total cross sections do not show factorial growth. In the case of cross sections for $2 \to N$ particles, there is no systematic large N approximation available. That said, we provide evidence that non-perturbatively, there is no factorial growth in partial or total cross sections., Comment: 22 pages, 3 figures

We study the Debye mass, $m_D$, and the topological susceptibility, $\chi$, at high temperatures in non-abelian gauge theory. Both exhibit, at some order in the perturbation expansion, infrared sensitivity. As a result, a perturbative analysis can at best provide an estimate of these quantities, subject to some uncertainty. The size of these uncertainties, particularly in the case of $\chi$, has been the subject of some debate. For the perturbative free energy, reframing an analysis of Braaten and Pisarski, the estimate and the associated error, can be understood in terms of a {\it Wilsonian} effective action for the low energy effective three dimensional theory. This action can be obtained completely from a perturbative calculation, which terminates at a finite order. This action provides the desired estimate. The size of the error follows from dimensional analysis in the low energy theory. The Debye length computation and its error can be obtained from a similar study of a non-relativistic effective theory for an adjoint scalar in three dimensions. $\chi$ requires a four dimensional analysis involving finite temperature instantons, but again the dominant sources of uncertainty are three dimensional, and we provide a procedure to estimate and an associated error. This uncertainty is of order $g^4 $ relative to the leading semiclassical result, and in situations of interest is small., Comment: 19 pages, 6 figures

The peculiar value of $\theta$ is a challenge to the notion of an anthropic landscape. We briefly review the possibility that a suitable axion might arise from an anthropic requirement of dark matter. We then consider an alternative suggestion of Kaloper and Terning that $\theta$ might be correlated with the cosmological constant. We note that in a landscape one expects that $\theta$ is determined by the expectation value of one or more axions. We discuss how a discretuum of values of $\theta$ might arise with an energy distribution dominated by QCD, and find the requirements to be quite stringent. Given such a discretuum, we find limited circumstances where small $\theta$ might be selected by anthropic requirements on the cosmological constant., Comment: 18 pages, 2 figures

If the QCD axion is a significant component of dark matter, and if the universe was once hotter than a few hundred MeV, the axion relic abundance depends on the function $\chi(T)$, the temperature-dependent topological susceptibility. Uncertainties in this quantity induce uncertainties in the axion mass as a function of the relic density, or vice versa. At high temperatures, theoretical uncertainties enter through the dilute instanton gas computation, while in the intermediate and strong coupling regime, only lattice QCD can determine $\chi(T)$ precisely. We reassess the uncertainty on the instanton contribution, arguing that it amounts to less than $20\%$ in the effective action, or a factor of 20 in $\chi$ at $T=1.5$ GeV. We then combine the instanton uncertainty with a range of models for $\chi(T)$ at intermediate temperatures and determine the impact on the axion relic density. We find that for a given relic density and initial misalignment angle, the combined uncertainty amounts to a factor of 2-3 in the zero-temperature axion mass., Comment: 15 pages, published version

We study the large $N$ $\theta$ dependence and the $\eta^\prime$ potential in supersymmetric QCD with small soft SUSY-breaking terms. Known exact results in SUSY QCD are found to reflect a variety of expectations from large $N$ perturbation theory, including the presence of branches and the behavior of theories with matter (both with $N_f \ll N$ and $N_f \sim N$). However, there are also striking departures from ordinary QCD and the conventional large $N$ description: instanton effects, when under control, are not exponentially suppressed at large $N$, and branched structure in supersymmetric QCD is always associated with approximate discrete symmetries. We suggest that these differences motivate further study of large $N$ QCD on the lattice., Comment: 20 pages; v2: references updated; v3: expanded introduction, accepted for publication in JHEP

This talk focuses on the role of light scalars in cosmology, both Nambu Goldstone bosons and pseudo moduli. The former include QCD axions, which might constitute the dark matter, and more general axions, which, under certain conditions, might play the role of inflatons, implementing {\it natural inflation}. The latter are the actors in (generalized) hybrid inflation. They rather naturally yield large field inflation, even mimicking chaotic inflation for suitable ranges of parameters., Comment: Talk at Nambu Memorial Symposium, 2016; 17 pages

We study some aspects of perturbation theory in $N=1$ supersymmetric abelian gauge theories with massive charged matter. In general gauges, infrared (IR) divergences and nonlocal behavior arise in 1PI diagrams, associated with a $1/k^4$ term in the propagator for the vector superfield. We examine this structure in supersymmetric QED. The IR divergences are gauge-dependent and must cancel in physical quantities like the electron pole mass. We demonstrate that cancellation takes place in a nontrivial way, amounting to a reorganization of the perturbative series from powers of $e^2$ to powers of $e$. We also show how these complications are avoided in cases where a Wilsonian effective action can be defined., Comment: 28 pages, 4 figures

Landscape analyses often assume the existence of large numbers of fields, $N$, with all of the many couplings among these fields (subject to constraints such as local supersymmetry) selected independently and randomly from simple (say Gaussian) distributions. We point out that unitarity and perturbativity place significant constraints on behavior of couplings with $N$, eliminating otherwise puzzling results. In would-be flux compactifications of string theory, we point out that in order that there be large numbers of light fields, the compactification radii must scale as a positive power of $N$; scaling of the radii and couplings with $N$ may also be necessary for perturbativity. We show that in some simple string theory settings with large numbers of fields, for fixed $R$ and string coupling, one can bound certain sums of squares of couplings by order one numbers. This may argue for strong correlations, possibly calling into question the assumption of random distributions. We consider implications of these considerations for classical and quantum stability of states without supersymmetry, with low energy supersymmetry arising from tuning of parameters, and with dynamical breaking of supersymmetry., Comment: 18 pages. Further modifications added in the interest of clarity and precision

Solutions to the strong CP problem typically introduce new scales associated with the spontaneous breaking of symmetries. Absent any anthropic argument for small $\bar\theta$, these scales require stabilization against ultraviolet corrections. Supersymmetry offers a tempting stabilization mechanism, since it can solve the "big" electroweak hierarchy problem at the same time. One family of solutions to strong CP, including generalized parity models, heavy axion models, and heavy $\eta^\prime$ models, introduces $\mathbb{Z}_2$ copies of (part of) the Standard Model and an associated scale of $\mathbb{Z}_2$-breaking. We review why, without additional structure such as supersymmetry, the $\mathbb{Z}_2$-breaking scale is unacceptably tuned. We then study "SUZ$_2$" models, supersymmetric theories with $\mathbb{Z}_2$ copies of the MSSM. We find that the addition of SUSY typically destroys the $\mathbb{Z}_2$ protection of $\bar\theta=0$, even at tree level, once SUSY and $\mathbb{Z}_2$ are broken. In theories like supersymmetric completions of the twin Higgs, where $\mathbb{Z}_2$ addresses the little hierarchy problem but not strong CP, two axions can be used to relax $\bar\theta$., Comment: 18 pp, v2: refs added

The possibility of a landscape of metastable vacua raises the question of what fraction of vacua are truly long lived. Naively any would-be vacuum state has many nearby decay paths, and all possible decays must be suppressed. An interesting model of this phenomena consists of $N$ scalars with a random potential of fourth order. Here we show that the scaling of the typical minimal bounce action with $N$ is readily understood, and differs from statements in the literature. We discuss the extension to more realistic landscape models., Comment: 15 pages References added, some typos fixed

Solutions to strong CP based on chiral structure have been subject to the most careful scrutiny and critique. Basic theoretical issues include hierarchy and fine-tuning problems, quality and genericity of symmetries, and compatibility with solutions to the electroweak hierarchy problem. We study the similar set of challenges for solutions to strong CP based on spontaneous CP violation and the Nelson-Barr mechanism. Some of our observations have appeared in the literature previously, and others are new; our purpose is to collect and analyze the issues as a whole and provide an assessment of the most plausible settings for the Nelson-Barr solution., Comment: 19 pp, 2 figures. v2: expanded discussion of strong dynamics, added references. Version published in JHEP

Naturalness has for many years been a guiding principle in the search for physics beyond the Standard Model, particularly for understanding the physics of electroweak symmetry breaking. However, the discovery of the Higgs particle at 125 GeV, accompanied by exclusion of many types of new physics expected in natural models has called the principle into question. In addition, apart from the scale of weak interactions, there are other quantities in nature which appear unnaturally small and for which we have no proposal for a natural explanation.We first review the principle, and then discuss some of the conjectures which it has spawned. We then turn to some of the challenges to the naturalness idea and consider alternatives., Comment: Prepared for Annual Review of Nuclear and Particle Science. 34 pages. Some comments added on effective field theories, alternative hierarch solutions

Recently it was shown that in QCD-like theories with $N_f > N$, where $N_f$ is the number of light flavors and $N$ is the number of colors, there are correlation functions that vanish in perturbation theory and at short distances receive dominant, calculable contributions from small instantons. Here we extend the set of such objects to theories with $N_f = N$, which includes real QCD, and discuss their application as a calibration of lattice computations at small quark mass. We revisit the related issue of the $u$ quark mass and its additive renormalization by small instantons, and discuss an alternative test of $m_u=0$ on the lattice., Comment: 22 pages, 2 figures. v3: reordered discussion of finite correlators and low energy constants, version published in PRD. v2: amended discussion of calculable vs incalculable effects for $N_f\leq N$

The BICEP2 observations, if confirmed, point to a high scale of inflation and large field excursions during the inflationary era. Non-compact string moduli spaces are a suggestive setting for these phenomena. While unlikely to be described by weak coupling models, one can write down plausible effective field theories compatible with known features of cosmology. These models can be viewed as generalizations to a large field regime of hybrid inflation. We note close parallels to small and large field axion models. This paper outlines the requirements for successful modular inflation, and gives examples of effective field theories which satisfy them. The required tunings are readily characterized. These models can also be thought of as models of chaotic inflation, in a way we describe. In the modular framework, one would expect that any would-be Peccei-Quinn symmetry would likely be badly broken during inflation, and the axion would have Hubble scale mass; in this situation, isocurvature fluctuations would be suppressed and the initial misalignment angle would be fixed, rather than being a random variable., Comment: 21 pages, latex, citations added; role of BICEP2 in discussion modified in light of joint Planck/BICEP2 analysis. Some clarification as to nature of the models considered, supersymmetric effective action during inflation, additional references

The discovery of a large tensor-to-scalar ratio by the BICEP2 experiment points to large field excursions during inflation. One framework that predicts large $r$ is monodromy inflation. While discussed mainly in the context of string theory, the phenomenon can be illustrated and studied in the well-understood framework of SUSY QCD with a large number of colors. We discuss the requirements for viable inflation as well as various difficulties for model building, including tunneling, tuning, and the species problem., Comment: 15 pages, 1 figure

We study some consequences of coupling supersymmetric theories to (super)gravity. To linear order, the couplings are determined by the energy-momentum supermultiplet. At higher orders, the couplings are determined by contact terms in correlation functions of the energy-momentum supermultiplet. We focus on the couplings of one particular field in the supergravity multiplet, the auxiliary field $M$. We discuss its linear and quadratic (seagull) couplings in various supersymmetric theories. In analogy to the local renormalization group formalism, we provide a prescription for how to fix the quadratic couplings. They generally arise at two-loops in perturbation theory. We check our prescription by explicitly computing these couplings in several examples such as mass-deformed ${\cal N}$ = 4 and in the Coulomb phase of some theories. These couplings affect the Lagrangians of rigid supersymmetric theories in curved space. In addition, our analysis leads to a transparent derivation of the phenomenon known as Anomaly Mediation. In contrast to previous approaches, we obtain both the gaugino and scalar masses of Anomaly Mediation by relying just on classical, minimal supergravity and a manifestly local and supersymmetric Wilsonian point of view. Our discussion naturally incorporates the connection between Anomaly Mediation and supersymmetric $AdS_4$ Lagrangians. This note can be read without prior familiarity with Anomaly Mediated Supersymmetry Breaking (AMSB)., Comment: 35 pages, 3 figures

It is sometimes stated that $n_s = 0.98$ in hybrid inflation; sometimes that it predicts $n_s >1$. A number of authors have consider aspects of Planck scale corrections and argued that they affect these predictions. Here we consider these systematically, describing the situations which can yield $n_s =0.96$, and the extent to which this result requires additional tuning., Comment: 12 pages, 1 figure

The phenomenon known as ``anomaly mediation" can be understood in a variety of ways. Rather than an anomaly, certain gaugino bilinear terms are required by local supersymmetry and gauge invariance (the derivation of these terms is in some cases related to anomalies in scale invariance or $R$ symmetries). We explain why the gaugino bilinear is required in supersymmetric gauge theories with varying number of colors and flavors. By working in the Higgs phase, gauging a flavor group, or working below the scale of gaugino condensation, each of these theories has a local effective description in which we can identify the bilinear term, establishing its necessity in the microscopic theory. For example, in theories that exhibit gaugino condensation, the potential in the very low energy theory is supersymmetric precisely due to the relation between the nonperturbative superpotential and the gaugino bilinear terms. Similarly, the gravitino mass appears from its coupling to the gaugino bilinear., Comment: 13 pg. v2: minor typos corrected; v3: references added, accepted for publication in JHEP

It is sometimes argued that a virtue of pushing the supersymmetry breaking scale above 1 PeV is that no particular flavor structure is required in the soft sector in order to evade bounds on flavor-changing neutral currents. However, without flavor structure, suppressing generic Planck-suppressed contributions to proton decay requires even higher SUSY scales, of order $10^{11}$ ($10^9$) GeV for degenerate (mini-split) gauginos and scalars. With flavor structure, the question of whether proton decay or flavor symmetries are more constraining is model-dependent, but it straightforward to find simple models where both constraints are satisfied for much lower SUSY scales., Comment: 14 pages, 1 figure. v2: expanded discussion of alignment models in sec. 3. accepted for publication in JHEP

Supersymmetry and string theory suggest the existence of light moduli. Their presence, or absence, controls the realization of supersymmetry at low energies. If there are no such fields, or if all such fields are fixed in a supersymmetric fashion, the conventional thermal production of LSP dark matter is possible, as is an anomaly-mediated ("mini-split") spectrum. On the other hand, the axion solution to the strong CP problem is not operative, and slow roll inflation appears difficult to implement. If there are light moduli, a mini-split spectrum is less generic, WIMP dark matter appears atypical, and the supersymmetry scale is likely tens of TeV or higher., Comment: v2: references added, correction to fig 1, conclusions unchanged. version accepted for publication in PRD

Dynamical, metastable supersymmetry breaking appears to be a generic phenomena in supersymmetric field theories. It's simplest implementation is within the so-called "retrofitted O'Raifeartaigh Models". While seemingly flexible, model building in these theories is significantly constrained. In gauge-mediated versions, if the approximate $R$ symmetry of the theory is spontaneously broken, the messenger scale is fixed; if explicitly broken by retrofitted couplings, a very small dimensionless number is required; if supergravity corrections are responsible for the symmetry breaking, at least two moderately small couplings are required, and there is a large range of possible messenger scales. In gravity mediated versions, achieving $m_{3/2} \approx M_Z$ is a problem of discrete tuning. With plausible assumptions, one can't achieve this to better than a factor of 100, perhaps accounting for a {\it little hierarchy} and the surprisingly large value of the Higgs mass., Comment: 15 pages

We comment on aspects of discrete anomaly conditions focussing particularly on $R$ symmetries. We review the Green-Schwarz cancellation of discrete anomalies, providing a heuristic explanation why, in the heterotic string, only the "model-independent dilaton" transforms non-linearly under discrete symmetries; this argument suggests that, in other theories, multiple fields might play a role in anomaly cancellations, further weakening any anomaly constraints at low energies. We provide examples in open string theories of non-universal discrete anomalies at low energies. We then consider the fact that $R$ symmetries are necessarily broken at low energies. We exhibit dynamical models, in which fields charged under the Standard Model gauge group (for example, a doublet and a triplet) gain roughly equal masses, but where the doublet and the triplet possess different discrete charges and the low-energy anomaly conditions fail., Comment: 13 pages; matches version published in JHEP; references added, expanded discussion about anomaly universality and gauge coupling unification

Most discussion of metastable dynamical supersymmetry breaking (MDSB) has focussed on low energy breaking, as in gauge mediation. It is of interest to consider possible implications for intermediate scale breaking ("gravity mediation"), especially as the early LHC results suggest the possibility that supersymmetry, if broken at relatively low energies, might be tuned. A somewhat high scale for susy breaking could ameliorate the usual flavor problems of gravity mediation, resolve the question of cosmological moduli, and account for a Higgs with mass well above $M_Z$. We study MDSB in gravity mediation, especially retrofitted models in which discrete $R$ symmetries play an important role, considering questions including implications of symmetries for $B$ and $A$ terms, and the genericity of split supersymmetry., Comment: 16 pages. Notational errors fixed, references added

Recently, questions have been raised about the role of Lorentz invariance in false vacuum decay. It has been argued that infinities may arise in an integration over Lorentz-boosted final states. This suggestion motivates a Minkowski-space analysis of the decay rate. We attempt to illuminate features of the amplitude computation, and argue that the total rate including excitations is both finite and Lorentz invariant., Comment: 23 pages

We explore some issues in slow roll inflation in situations where field excursions are small compared to $M_p$. We argue that for small field inflation, minimizing fine tuning requires low energy supersymmetry and a tightly constrained structure. Hybrid inflation is {\it almost} an inevitable outcome. The resulting theory can be described in terms of a supersymmetric low energy effective action and inflation completely characterized in terms of a small number of parameters. Demanding slow roll inflation significantly constrains these parameters. In this context, the generic level of fine tuning can be described as a function of the number of light fields, there is an upper bound on the scale of inflation, and an (almost) universal prediction for the spectral index. Models of this type need not suffer from a cosmological moduli problem., Comment: 16 pages. issues with single field inflation identified and clarified; role of discrete R symmetries clarified; error in discussion of a non-hybrid model corrected

We revisit two longstanding puzzles in supersymmetric gauge theories. The first concerns the question of the holomorphy of the coupling, and related to this the possible definition of an exact (NSVZ) beta function. The second concerns instantons in pure gluodynamics, which appear to give sensible, exact results for certain correlation functions, which nonetheless differ from those obtained using systematic weak coupling expansions. For the first question, we extend an earlier proposal of Arkani-Hamed and Murayama, showing that if their regulated action is written suitably, the holomorphy of the couplings is manifest, and it is easy to determine the renormalization scheme for which the NSVZ formula holds. This scheme, however, is seen to be one of an infinite class of schemes, each leading to an exact beta function; the NSVZ scheme, while simple, is not selected by any compelling physical consideration. For the second question, we explain why the instanton computation in the pure supersymmetric gauge theory is not reliable, even at short distances. The semiclassical expansion about the instanton is purely formal; if infrared divergences appear, they spoil arguments based on holomorphy. We demonstrate that infrared divergences do not occur in the perturbation expansion about the instanton, but explain that there is no reason to think this captures all contributions from the sector with unit topological charge. That one expects additional contributions is illustrated by dilute gas corrections. These are infrared divergent, and so difficult to define, but if non-zero give order one, holomorphic, corrections to the leading result. Exploiting an earlier analysis of Davies et al, we demonstrate that in the theory compactified on a circle of radius beta, due to infrared effects, finite contributions indeed arise which are not visible in the formal limit that beta goes to infinity., Comment: 28 pages, two references added, one typo corrected

If supersymmetry turns out to be a symmetry of nature at low energies, the first order of business to measure the soft breaking parameters. But one will also want to understand the symmetry, and its breaking, more microscopically. Two aspects of this problem constitute the focus of these lectures. First, what sorts of dynamics might account for supersymmetry breaking, and its manifestation at low energies. Second, how might these features fit into string theory (or whatever might be the underlying theory in the ultraviolet). The last few years have seen a much improved understanding of the first set of questions, and at least a possible pathway to address the second., Comment: Lectures for the 2010 TASI summer school

This article reviews the subject of supersymmetry and its breaking. The emphasis is on recent developments in metastable, dynamical supersymmetry breaking, which permit the construction of promising models of particle physics., Comment: Review article prepared for Reports on Progress in Physics, 53 pages

While axions seem ubiquitous in critical string theories, whether they might survive in any string theoretic description of nature is a difficult question. With some mild assumptions, one can frame the issues in the case that there is an approximate supersymmetry below the underlying string scale. The problem of axions is then closely tied to the question of how moduli are fixed. We consider, from this viewpoint, the possibility that supersymmetry is broken at an intermediate scale, as in "gravity mediation," at a low scale, as in gauge mediation, and at a very high scale, to model the possibility that there is no low energy supersymmetry. Putative mechanisms for moduli fixing can then be systematically classified, and at least for intermediate and high scale breaking, light axions appear plausible. In the course of this work, we are lead to consider aspects of moduli fixing and supersymmetry breaking, and we revisit the possibility of very large extra dimensions., Comment: 18 pages; v2, updated and added references

Discrete R symmetries are interesting from a variety of points of view. They raise the specter, however, of domain walls, which may be cosmologically problematic. In this note, we describe some of the issues. In many schemes for supersymmetry breaking, as we explain, satisfying familiar constraints such as suppression of gravitino production, insures that the domain walls are readily inflated away. However, in others, they form after inflation. In these cases, it is necessary that they annihilate. We discuss possible breaking mechanisms for the discrete symmetries, and the constraints they must satisfy so that the walls annihilate effectively., Comment: 13 pages

We revisit the question of whether or not one can perform reliable semiclassical QCD computations at zero temperature. We study correlation functions with no perturbative contributions, and organize the problem by means of the operator product expansion, establishing a precise criterion for the validity of a semiclassical calculation. For $N_f>N$, a systematic computation is possible; for $N_f

We prove a general bound on the superpotential in theories with broken supersymmetry and broken R-symmetry, 2|W|< f_a F, where f_a and F are the R-axion and Goldstino decay constants, respectively. The bound holds for weakly coupled as well as strongly coupled theories, thereby providing an exact result in theories with broken supersymmetry. We briefly discuss several possible applications., Comment: 20 pages

If nature exhibits low energy supersymmetry, discrete (non-$Z_2$) R symmetries may well play an important role. In this paper, we explore such symmetries. We generalize gaugino condensation, constructing large classes of models which are classically scale invariant, and which spontaneously break discrete R symmetries (but not supersymmetry). The order parameters for the breaking include chiral singlets. These simplify construction of models with metastable dynamical supersymmetry breaking. We explain that in gauge mediation, the problem of the cosmological constant makes "retrofitting" particularly natural -- almost imperative. We describe new classes of models, with interesting scales for supersymmetry breaking, and which allow simple solutions of the $\mu$ problem. We argue that models exhibiting such R symmetries can readily solve not only the problem of dimension four operators and proton decay, but also dimension five operators. On the other hand, in theories of "gravity mediation", the breaking of R symmetry is typically of order $M_p$, R parity is required to suppress dimension four $B$ and $L$ violating operators, and dimension five operators remain problematic., Comment: 17 pages, latex; references to earlier work added; typos fixed, minor corrections and revisions, journal version

In supersymmetric theories, the presence of axions usually implies the existence of a non-compact, (pseudo)moduli space. In gauge mediated models, the axion would seem a particularly promising dark matter candidate. The cosmology of the moduli then constrains the gravitino mass and the axion decay constant; the former can't be much below 10 MeV; the latter can't be much larger than 10^{13} GeV. Axinos, when identifiable, are typically heavy and do not play an important role in cosmology., Comment: final version published in Phys Rev D, 22 pages, 3 figures

Invoking the Peccei-Quinn (PQ) solution to the strong CP problem substitutes the puzzle of why $\theta_{qcd}$ is so small with the puzzle of why the PQ symmetry is of such high quality. Cosmological and astrophysical considerations raise further puzzles. This paper explores this issues in several contexts: string theory and field theory, and theories without and with low energy supersymmetry. Among the questions studied are whether requiring axion dark matter can account for the quality of the PQ symmetry, to which the answer is sometimes yes. In non-supersymmetric theories, we find $f_a = 10^{12}$ GeV is quite plausible. In gauge mediation, cosmological constraints on pseudomoduli place $f_a$ in this range, and require that the gravitino mass be of order an MeV., Comment: 17 pages, 1 figure

These lectures, given at the Cargese Summer school in 2008, provide an introduction to dynamical supersymmetry breaking and gauge mediation, with emphasis on the recent appreciation of the possible role of metastable supersymmetry breaking, and the evolving understanding of General Gauge Mediation. The underlying focus is on how supersymmetry might be realized at the Large Hadron Collider., Comment: 32 pages; lectures at Cargese school, 2008

Supersymmetric vacua are stable. It is interesting to ask: how long-lived are vacua which are nearly supersymmetric? This question is relevant if our universe is approximately supersymmetric. It is also of importance for a number of issues of the physics of the landscape and eternal inflation. In this note, we distinguish a variety of cases. In all of them the decay is slow. For a flat space theory decaying to a deep AdS vacuum, the leading behavior of the decay amplitude, if a thin wall approximation is valid, is ${\cal A} = \gamma e^{-2 \pi^2/({\rm Re} m_{3/2})^2}$ (where the phase of $m_{3/2}$ is defined in the text) for ${\rm Re} m_{3/2}>0$, and zero otherwise. Metastable supersymmetry breaking generally yields parametrically more rapid decays. For nearly supersymmetric decays, we will see that it is necessary to compute subleading terms in the exponential to extraordinarily high accuracy before one can meaningfully discuss the prefactor., Comment: 19 pages