Search

The Standard Model can be defined quantitatively by running parameters in a mass-independent renormalization scheme at a fixed reference scale. We provide a set of simple interpolation formulas that give the fundamental Lagrangian parameters in the $\bar{\rm MS}$ scheme at a renormalization scale of 200 GeV, safely above the top-quark mass and suitable for matching to candidate new physics models at very high mass scales using renormalization group equations. These interpolation formulas take as inputs the on-shell experimental quantities, and use the best available calculations in the pure $\bar{\rm MS}$ scheme. They also serve as an accounting of the parametric uncertainties for the short-distance Standard Model Lagrangian. We also include an interpolating formula for the W boson mass., 18 pages, python code included as ancillary file

Loop diagram calculations typically rely on reduction to a finite set of master integrals in $4 - 2ε$ dimensions. It has been shown that for any problem, the masters can be chosen so that their coefficients are finite as $ε\rightarrow 0$. I propose a definition of renormalized $ε$-finite master integrals, which incorporate ultraviolet divergence subtractions in a specific way. A key advantage of this choice is that in expressions for physical observables, expansions to positive powers in $ε$ are never needed. As an example, I provide the subtractions for general three-loop self-energy integrals. The differential equations method is used to compute numerically the renormalized $ε$-finite master integrals for arbitrary external momentum invariant, in special cases with internal masses equal to a single scale or zero. These include the ones necessary for the three-loop QCD corrections to the self-energies of the W, Z, and Higgs bosons. In principle, the same method should provide for numerical computation of general three-loop self energies with any masses., 33 pages. v2 includes one additional even threshold topology

Solutions to the $��$ problem in supersymmetry based on the Kim-Nilles mechanism naturally feature a Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) axion with decay constant of order the geometric mean of the Planck and TeV scales, consistent with astrophysical limits. We investigate minimal models of this type with two gauge-singlet fields that break a Peccei-Quinn symmetry, and extensions with extra vectorlike quark and lepton supermultiplets consistent with gauge coupling unification. We show that there are many anomaly-free discrete symmetries, depending on the vectorlike matter content, that protect the Peccei-Quinn symmetry to sufficiently high order to solve the strong CP problem. We study the axion couplings in this class of models. Models of this type that are automatically free of the domain wall problem require at least one pair of strongly interacting vectorlike multiplets with mass at the intermediate scale, and predict axion couplings that are greatly enhanced compared to the minimal supersymmetric DFSZ models, putting them within reach of proposed axion searches., 50 pages. v2: references added. v3: accepted by PRD, minor changes in Section V, references added

I study supersymmetric models in which the QCD gauge group is the remnant diagonal subgroup from the spontaneous breaking of an $SU(3) \times SU(3)$ gauge group at a multi-TeV scale. In renormalizable models with soft supersymmetry breaking, the scalar potential is shown to have global minima with the required gauge symmetry breaking pattern. In addition to a massive color octet vector boson, this framework predicts 3 color octet spin-0 sgluons, and 4 color octet gluinos with both Dirac and Majorana mass terms. One of the gluino mass eigenstates typically has a coupling to quark-squark pairs that is at least as large as the prediction of minimal supersymmetry, but it need not be the lightest one., 30 pages

We present an implementation and numerical study of the Standard Model couplings, masses, and vacuum expectation value (VEV), using the pure $\overline{\rm{MS}}$ renormalization scheme based on dimensional regularization. Here, the $\overline{\rm{MS}}$ Lagrangian parameters are treated as the fundamental inputs, and the VEV is defined as the minimum of the Landau gauge effective potential, so that tadpole diagrams vanish, resulting in improved convergence of perturbation theory. State-of-the-art calculations relating the $\overline{\rm{MS}}$ inputs to on-shell observables are implemented in a consistent way within a public computer code library, SMDR (Standard Model in Dimensional Regularization), which can be run interactively or called by other programs. Included here for the first time are the full 2-loop contributions to the Fermi constant within this scheme and studies of the minimization condition for the VEV at 3-loop order with 4-loop QCD effects. We also implement, and study the scale dependence of, all known multi-loop contributions to the physical masses of the Higgs boson, the W and Z bosons, and the top quark, the fine structure constant and weak mixing angle, and the renormalization group equations and threshold matching relations for the gauge couplings, fermion masses, and Yukawa couplings., Comment: 26 pages. The SMDR code can be obtained from https://davidgrobertson.github.io/SMDR/ v2: references added. v3: code updated with 3-loop QCD corrections to Higgs, W, Z masses. v4: The parameterization of the complex pole masses of W,Z has been changed in the code

The complex pole mass of the top quark is presented at full two-loop order in the Standard Model, augmenting the known four-loop QCD contributions. The input parameters are the MS-bar Yukawa and gauge couplings, the Higgs self-coupling, and the Higgs vacuum expectation value (VEV). Here, the VEV is defined as the minimum of the full effective potential in Landau gauge, so that tadpoles vanish. This is an alternative to earlier results that instead minimize the tree-level potential, resulting in a VEV that is gauge-fixing independent but accompanied by negative powers of the Higgs self-coupling in perturbative expansions. The effects of non-zero Goldstone boson mass are eliminated by resummation. I also study the renormalization scale dependence of the calculated pole mass., 24 pages. v2: references added, no other changes

We discuss the systematic evaluation of 3-loop vacuum integrals with arbitrary masses. Using integration by parts, the general integral of this type can be reduced algebraically to a few basis integrals. We define a set of modified finite basis integrals that are particularly convenient for expressing renormalized quantities. The basis integrals can be computed numerically by solving coupled first-order differential equations, using as boundary conditions the analytically known special cases that depend on only one mass scale. We provide the results necessary to carry this out, and introduce an implementation in the form of a public software package called 3VIL (3-loop Vacuum Integral Library), which efficiently computes the numerical values of the basis integrals for any specified masses. 3VIL is written in C, and can be linked from C, C++, or FORTRAN code., Comment: 36 pages. v3: Exact analytic form of eq. (5.39) is now included. v2: A new version of 3VIL, v2.0, contains significant restructuring and improvements compared to v1.0. Source code and documentation for 3VIL v2.0 are available at http://www.niu.edu/spmartin/3VIL/ or http://faculty.otterbein.edu/DRobertson/3VIL/

I obtain the complex pole squared mass of the Z boson at full two-loop order in the Standard Model in the pure MS-bar renormalization scheme. The input parameters are the running gauge couplings, the top-quark Yukawa coupling, the Higgs self-coupling, and the vacuum expectation value that minimizes the Landau gauge effective potential. The effects of non-zero Goldstone boson mass are resummed. Within a reasonable range of renormalization scale choices, the scale dependence of the computed pole mass is found to be comparable to the current experimental uncertainty, but the true theoretical error is likely somewhat larger., 15 pages. arXiv admin note: text overlap with arXiv:1503.03782

I provide a calculation at full two-loop order of the complex pole squared mass of the W boson in the Standard Model in the pure MS-bar renormalization scheme, with Goldstone boson mass effects resummed. This approach is an alternative to earlier ones that use on-shell or hybrid renormalization schemes. The renormalization scale dependence of the real and imaginary parts of the resulting pole mass are studied. Both deviate by about $\pm 4$ MeV from their median values as the renormalization scale is varied from 50 GeV to 200 GeV, but the theory error is likely larger. A surprising feature of this scheme is that the 2-loop QCD correction has a larger scale-dependence, but a smaller magnitude, than the 2-loop non-QCD correction, unless the renormalization scale is chosen very far from the top-quark mass., 17 pages. v2: benchmark input parameters changed, reference added

We investigate the implications of models that achieve a Standard Model-like Higgs boson of mass near 125 GeV by introducing additional TeV-scale supermultiplets in the vector-like 10+\bar{10} representation of SU(5), within the context of gauge-mediated supersymmetry breaking. We study the resulting mass spectrum of superpartners, comparing and contrasting to the usual gauge-mediated and CMSSM scenarios, and discuss implications for LHC supersymmetry searches. This approach implies that exotic vector-like fermions t'_{1,2}, b',and \tau' should be within the reach of the LHC. We discuss the masses, the couplings to electroweak bosons, and the decay branching ratios of the exotic fermions, with and without various unification assumptions for the mass and mixing parameters. We comment on LHC prospects for discovery of the exotic fermion states, both for decays that are prompt and non-prompt on detector-crossing time scales., Comment: 32 pages. v2: references added, figure caption 5.3 corrected

These notes, based on work with Herbi Dreiner and Howie Haber, discuss how to do practical calculations of cross sections and decay rates using two-component fermion notation, as appropriate for supersymmetry and other beyond-the-Standard-Model theories. Included are a list of two-component fermion Feynman rules for the Minimal Supersymmetric Standard Model, and some example calculations., Comment: 50 pages. This paper uses the -+++ metric. Errata, and an otherwise identical version using the +--- metric, are available at https://www.niu.edu/spmartin/TASI11/

We study the reach of the Large Hadron Collider with 1/fb of data at sqrt{s} = 7 TeV for several classes of supersymmetric models with compressed mass spectra, using jets and missing transverse energy cuts like those employed by ATLAS for Summer 2011 data. In the limit of extreme compression, the best limits come from signal regions that do not require more than 2 or 3 jets and that remove backgrounds by requiring more missing energy rather than higher effective mass., Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1105.4304

The sign of the supersymmetric Higgs mass ��is usually taken as an independent input parameter in analyses of the supersymmetric standard model. I study the role of theories of supersymmetry breaking in determining the sign of ��as an output. Models with vanishing soft scalar couplings at the apparent gauge coupling unification scale are known to predict positive ��. I investigate more general results for the sign of ��as a function of the holomorphic soft scalar couplings, and compare to predictions of models with gaugino mass dominance at higher scales. In a significant region of the B_0/m_{1/2} versus A_0/m_{1/2} plane including A_0 = B_0 = 0, ��must be positive. In another region, ��is definitely negative. Only in a smaller intermediate region does knowledge of the supersymmetry breaking mechanism not permit a definite prediction of the sign of ��. The last region will shrink considerably as the top quark mass becomes more accurately known., 18 pages

The prospects for discovering and studying signals of low-scale supersymmetry breaking models at the Tevatron Run II and beyond are explored. These models include gauge-mediated supersymmetry breaking as the most compelling and concrete realization, but more generally are distinguished by the presence of a nearly massless Goldstino as the lightest supersymmetric particle. The next-lightest supersymmetric particle(s) (NLSP) decays to its partner and the Goldstino. Depending on the supersymmetry breaking scale, these decays can occur promptly or on a scale comparable to or larger than the size of a detector. A systematic analysis based on a classification in terms of the identity of the NLSP and its decay length is presented. The various scenarios are discussed in terms of signatures and possible event selection criteria. The Run II and beyond discovery and exclusion reaches, including the effects of background, are detailed for the most compelling cases. In addition to standard event selection criteria based on missing energy and photons, leptons, jets, taus, tagged b-jets, or reconstructed Z-bosons, more exotic signals of metastable NLSPs such as displaced photons, large negative impact parameter tracks, kink tracks, both opposite and same-sign highly ionizing tracks, time of flight measurements, charge-changing tracks, charge-exchange tracks, and same-sign di-top events are investigated. The interesting possibility of observing a Higgs boson signal in events that are efficiently "tagged" by the unique signatures of low-scale supersymmetry breaking is also considered., Comment: 98 pages. Preprint numbers added, style macros updated (resulting in 4 more pages due to typesetting), no other changes

There are many theoretical reasons why heavy quasi-stable charged particles might exist. Pair production of such particles at the Tevatron can produce highly ionizing tracks (HITs) or fake muons. In gauge-mediated supersymmetry breaking, sparticle production can lead to events with a pair of quasi-stable sleptons, a significant fraction of which will have the same electric charge. Depending on the production mechanism and the decay chain, they may also be accompanied by additional energetic leptons. We study the relative importance of the resulting signals for the Tevatron Run II. The relative fraction of same-sign tracks to other background-free signals is an important diagnostic tool in gauge-mediated supersymmetry breaking that may provide information about mass splittings, tan(beta) and the number of messengers communicating supersymmetry breaking., 15 pages, latex, 5 figures

I provide a pedagogical introduction to supersymmetry. The level of discussion is aimed at readers who are familiar with the Standard Model and quantum field theory, but who have had little or no prior exposure to supersymmetry. Topics covered include: motivations for supersymmetry, the construction of supersymmetric Lagrangians, superspace and superfields, soft supersymmetry-breaking interactions, the Minimal Supersymmetric Standard Model (MSSM), R-parity and its consequences, the origins of supersymmetry breaking, the mass spectrum of the MSSM, decays of supersymmetric particles, experimental signals for supersymmetry, and some extensions of the minimal framework., 160 pages. Version 7 (January 2016) contains many updates and improvements. Errata, source files, and a version with larger type (12 pt, 179 pages) can be found at http://www.niu.edu/spmartin/primer/

We present a refined and expanded analysis of the CDF $ee����+ \Et$ event as superpartner production, assuming the lightest neutralino is the lightest supersymmetric particle. A general low-energy Lagrangian is constrained by a minimum cross section times branching ratio into two electrons and two photons, kinematics consistent with the event, and LEP1-LEP130 data. We examine how the supersymmetric parameters depend on the kinematics, branching ratios and experimental predictions with a selectron interpretation of the event, and discuss to what extent these are modified by other interpretations. Predictions for imminent CERN LEP upgrades and the present and future Fermilab Tevatron are presented. Finally, we briefly discuss the possible connection to other phenomena including a light stop, the neutralino relic density, the shift in $R_b$ and the associated shift in $��_s$, and implications for the form of the theory., 57 pages, LaTeX, uses epsf.sty, 19 figures. Version accepted for publication in Phys. Rev. D, with minor changes and a few clarifications

It has been shown recently that the introduction of an unphysical $��$-scalar mass $\tilde{m}$ is necessary for the proper renormalization of softly broken supersymmetric theories by dimensional reduction ($\drbar$). In these theories, both the two-loop $��$-functions of the scalar masses and their one-loop finite corrections depend on $\tilde{m}^2$. We find, however, that the dependence on $\tilde{m}^2$ can be completely removed by slightly modifying the \drbar renormalization scheme. We also show that previous \drbar calculations of one-loop corrections in supersymmetry which ignored the $\tilde{m}^2$ contribution correspond to using this modified scheme., 7 pages, LTH-336, NUB-3094-94TH, KEK-TH-404

In the minimal supersymmetric standard model, the existence of R-parity is not required for the internal consistency of the theory and might therefore be regarded as ad hoc. I catalog some simple conditions which are sufficient to guarantee that R-parity survives as an unbroken gauged discrete subgroup of the continuous gauge symmetry in certain extensions of the minimal supersymmetric standard model. If these criteria are met, R-parity is automatic. [Based on a talk given at the International Workshop on Supersymmetry and Unification of Fundamental Interactions (SUSY93) at Northeastern University, Boston, March 29-April 1 1993.], 9 pages, NUB-3063-93TH

This is the Snowmass2021 Energy Frontier (EF) Beyond the Standard Model (BSM) report. It combines the EF topical group reports of EF08 (Model-specific explorations), EF09 (More general explorations), and EF10 (Dark Matter at Colliders). The report includes a general introduction to BSM motivations and the comparative prospects for proposed future experiments for a broad range of potential BSM models and signatures, including compositeness, SUSY, leptoquarks, more general new bosons and fermions, long-lived particles, dark matter, charged-lepton flavor violation, and anomaly detection., Comment: 108 pages + 38 pages references and appendix, 37 figures, Report of the Topical Group on Beyond the Standard Model Physics at Energy Frontier for Snowmass 2021. The first nine authors are the Conveners, with Contributions from the other authors