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Robert Shrock

Publications and source records attributed to Robert Shrock.

At least 19 recordsLinked to original sources
arXiv2025

Chromatic Zeros on the Limit $G^{(p,\ell)}_\infty$ of the Family $G^{(p,\ell)}_m$ of Hierarchical Graphs

We calculate the continuous accumulation set ${\cal B}_q(p,\ell)$ of zeros of the chromatic polynomial $P(G^{(p,\ell)}_m,q)$ in the limit $m \to \infty$, on a family of graphs $G^{(p,\ell)}_m$ defined such that $G^{(p,\ell)}_m$ is obtained from $G^{(p,\ell)}_{m-1}$ by replacing each edge (i.e., bond) on $G^{(p,\ell)}_m$ by $p$ paths each of length $\ell$ edges, starting with the tree graph $T_2$. Our method uses the property that the chromatic polynomial $P(G,q)$ of a graph $G$ is equal to the $v=-1$ evaluation of the partition function of the $q$-state Potts model, together with (i) the property that $Z(G^{(p,\ell)}_m,q,v)$ can be expressed via an exact closed-form real-space renormalization (RG) group transformation in terms of $Z(G^{(p,\ell)}_{m-1},q,v')$, where $v'=F_{(p,\ell),q}(v)$ is a rational function of $v$ and $q$ and (ii) ${\cal B}_q(p,\ell)(v)$ is the locus in the complex $q$-plane that separates regions of different asymptotic behavior of the $m$-fold iterated RG transformation $F_{(p,\ell),q}(v)$ in the $m \to \infty$ limit. Thus, our results involve calculations of region diagrams in the complex $q$-plane showing the type of behavior that occurs in the $m \to \infty$ limit of the $m$-fold iterated RG transformation mapping $F_{(p,\ell),q}(v)$ starting with the initial value $v=v_0=-1$. Calculations are presented of the maximal point $q_c(G^{(p,\ell)}_\infty)$ at which the locus ${\cal B}_q$ crosses the real-$q$ axis, as well as several other points at which, depending on $p$ and $\ell$, the locus ${\cal B}_q$ crosses this axis. We give explicit results for a variety of $(p,\ell)$ cases and observe a number of interesting features. Calculations of the ground-state degeneracy of the Potts antiferromagnet at $q_c(G^{(p,\ell)}_\infty)$ are presented. This work extends a previous study with R. Roeder of the $(p,\ell)=(2,2)$ case to higher $p$ and $\ell$ values.

cond-mat.stat-mech↗
arXiv2025

Jones Polynomials and their Zeros for a Family of Knots and Links

We calculate Jones polynomials $V(H_r,t)$ for a family of alternating knots and links $H_r$ with arbitrarily many crossings $r$, by computing the Tutte polynomials $T(G_+(H_r),x,y)$ for the associated graphs $G_+(H_r)$ and evaluating these with $x=-t$ and $y=-1/t$. Our method enables us to circumvent the generic feature that the computational complexity of $V(L_r,t)$ for a knot or link $L_r$ for generic $t$ grows exponentially rapidly with $r$. We also study the accumulation set of the zeros of these polynomials in the limit of infinitely many crossings, $r \to \infty$.

math-ph↗
arXiv2025

Baryon Number Violation: From Nuclear Matrix Elements to BSM Physics

Processes that violate baryon number, most notably proton decay and $n\bar n$ transitions, are promising probes of physics beyond the Standard Model (BSM) needed to understand the lack of antimatter in the Universe. To interpret current and forthcoming experimental limits, theory input from nuclear matrix elements to UV complete models enters. Thus, an interplay of experiment, effective field theory, lattice QCD, and BSM model building is required to develop strategies to accurately extract information from current and future data and maximize the impact and sensitivity of next-generation experiments. Here, we briefly summarize the main results and discussions from the workshop "INT-25-91W: Baryon Number Violation: From Nuclear Matrix Elements to BSM Physics," held at the Institute for Nuclear Theory, University of Washington, Seattle, WA, January 13-17, 2025.

hep-ph↗
arXiv2025

Pion Decay

The decays of pions, the lightest particles composed of quarks, provide important insight into fundamental questions in particle physics including basic symmetries, the universality of fermionic weak interactions, and aspects of the strong interaction described by Quantum Chromodynamics (QCD). An introduction to the theoretical and experimental study of charged and neutral pion decays is presented.

hep-ph↗
arXiv2024

Exact Potts/Tutte Polynomials for Hammock Chain Graphs

We present exact calculations of the $q$-state Potts model partition functions and the equivalent Tutte polynomials for chain graphs comprised of $m$ repeated hammock subgraphs $H_{e_1,...,e_r}$ connected with line graphs of length $e_g$ edges, such that the chains have open or cyclic boundary conditions (BC). Here, $H_{e_1,...,e_r}$ is a hammock (series-parallel) subgraph with $r$ separate paths along ``ropes'' with respective lengths $e_1, ..., e_r$ edges, connecting the two end vertices. We denote the resultant chain graph as $G_{\{e_1,...,e_r\},e_g,m;BC}$. We discuss special cases, including chromatic, flow, and reliability polynomials. In the case of cyclic boundary conditions, the zeros of the Potts partition function in the complex $q$ function accumulate, in the limit $m \to \infty$, onto curves forming a locus ${\cal B}$, and we study this locus.

cond-mat.stat-mech↗
arXiv2024

Potts Partition Function Zeros and Ground State Entropy on Hanoi Graphs

We study properties of the Potts model partition function $Z(H_m,q,v)$ on $m$'th iterates of Hanoi graphs, $H_m$, and use the results to draw inferences about the $m \to \infty$ limit that yields a self-similar Hanoi fractal, $H_\infty$. We also calculate the chromatic polynomials $P(H_m,q)=Z(H_m,q,-1)$. From calculations of the configurational degeneracy, per vertex, of the zero-temperature Potts antiferromagnet on $H_m$, denoted $W(H_m,q)$, estimates of $W(H_\infty,q)$, are given for $q=3$ and $q=4$ and compared with known values on other lattices. We compute the zeros of $Z(H_m,q,v)$ in the complex $q$ plane for various values of the temperature-dependent variable $v=y-1$ and in the complex $y$ plane for various values of $q$. These are consistent with accumulating to form loci denoted ${\cal B}_q(v)$ and ${\cal B}_v(q)$, or equivalently, ${\cal B}_y(q)$, in the $m \to \infty$ limit. Our results motivate the inference that the maximal point at which ${\cal B}_q(-1)$ crosses the real $q$ axis, denoted $q_c$, has the value $q_c=(1/2)(3+\sqrt{5} \, )$ and correspondingly, if $q=q_c$, then ${\cal B}_y(q_c)$ crosses the real $y$ axis at $y=0$, i.e., the Potts antiferromagnet on $H_\infty$ with $q=(1/2)(3+\sqrt{5} \, )$ has a $T=0$ critical point. Finally, we analyze the partition function zeros in the $y$ plane for $q \gg 1$ and show that these accumulate approximately along parts of the sides of an equilateral triangular with apex points that scale like $y \sim q^{2/3}$ and $y \sim q^{2/3} e^{\pm 2\pi i/3}$. Some comparisons are presented of these findings for Hanoi graphs with corresponding results on $m$'th iterates of Sierpinski gasket graphs and the $m \to \infty$ limit yielding the Sierpinski gasket fractal.

cond-mat.stat-mech↗
arXiv2024

Effects of Neutron-Antineutron Transitions in Neutron Stars

We analyze effects of neutron-antineutron transitions in neutron stars, specifically on (i) cooling, (ii) rotation rate, and (iii) for binary pulsars, the increase in the orbital period. We show that these effects are negligibly small.

hep-ph↗
arXiv2023

A Comparative Study of Criticality Conditions for Anomalous Dimensions using Exact Results in an ${\cal N}=1$ Supersymmetric Gauge Theory

Two of the conditions that have been suggested to determine the lower boundary of the conformal window in asymptotically free gauge theories are the linear condition, $\gamma_{\bar\psi\psi,IR}=1$, and the quadratic condition, $\gamma_{\bar\psi\psi,IR}(2-\gamma_{\bar\psi\psi,IR})=1$, where $\gamma_{\bar\psi\psi,IR}$ is the anomalous dimension of the operator $\bar\psi\psi$ at an infrared fixed point in a theory. We compare these conditions as applied to an ${\cal N}=1$ supersymmetric gauge theory with gauge group $G$ and $N_f$ pairs of massless chiral superfields $\Phi$ and $\tilde \Phi$ transforming according to the respective representations ${\cal R}$ and $\bar {\cal R}$ of $G$. We use the fact that $\gamma_{\bar\psi\psi,IR}$ and the value $N_f = N_{f,cr}$ at the lower boundary of the conformal window are both known exactly for this theory. In contrast to the case with a non-supersymmetric gauge theory, here we find that in higher-order calculations, the linear condition provides a more accurate determination of $N_{f,cr}$ than the quadratic condition when both are calculated to the same finite order of truncation in a scheme-independent expansion.

hep-th↗
arXiv2023

Anomalous Dimensions at an Infrared Fixed Point in an SU($N_c$) Gauge Theory with Fermions in the Fundamental and Antisymmetric Tensor Representations

We present scheme-independent calculations of the anomalous dimensions $γ_{\barψψ,IR}$ and $γ_{\barχχ,IR}$ of fermion bilinear operators $\barψψ$ and $\barχχ$ at an infrared fixed point in an asymptotically free SU($N_c$) gauge theory with massless Dirac fermion content consisting of $N_F$ fermions $ψ^a_i$ in the fundamental representation and $N_{A_2}$ fermions $χ^{ab}_j$ in the antisymmetric rank-2 tensor representation, where $i,j$ are flavor indices. For the case $N_c=4$, $N_F=4$, and $N_{A_2}=4$, we compare our results with values of these anomalous dimensions measured in a recent lattice simulation and find agreement.

hep-lat↗
arXiv2023

Fitting a Self-Interacting Dark Matter Model to Data Ranging From Satellite Galaxies to Galaxy Clusters

We present a fit to observational data in an asymmetric self-interacting dark matter model using our recently calculated cross sections that incorporate both $t$-channel and $u$-channel exchanges in the scattering of identical particles. We find good fits to the data ranging from dwarf galaxies to galaxy clusters, and equivalent relative velocities from $\sim 20$ km/sec to $\gtrsim 10^3$ km/s. We compare our results with previous fits that used only $t$-channel exchange contributions to the scattering.

hep-ph↗
arXiv2023

Study of the Ultraviolet Behavior of an O($N$) $|\vec ϕ|^6$ Theory in $d=3$ Dimensions

We study the ultraviolet (UV) behavior of an O($N$) $|\vec ϕ|^6$ theory in $d=3$ spacetime dimensions, focusing on the question of the range in $N$ over which the perturbative beta function exhibits robust evidence of a UV zero in the $|\vec ϕ|^6$ coupling, $g$. The four-loop $(4\ell)$ beta function is known to have a (scheme-independent) UV zero at $g=g_{_{UV,4\ell}}$, which is reliably calculable for large $N$. For our analysis we use the six-loop beta function calculated in the minimal subtraction scheme. We find that this six-loop beta function has a UV zero, $g_{_{UV,6\ell}}$, if $N > N_c$, where $N_c \simeq 796$, and we calculate $g_{_{UV,6\ell}}$. To investigate the reliability of the result in the region of $N \gtrsim N_c$, we apply three methods: (i) calculation of the fractional difference between $g_{_{UV,4\ell}}$ and $g_{_{UV,6\ell}}$, (ii) a Padé approximant, and (iii) an assessment of scheme dependence. Our results provide quantitative measures of the range of $N$ over which the six-loop beta function has a UV zero and of the $1/N$ corrections to the value of $g$ at the UV zero for large but finite $N$. If one imposes a benchmark requirement that the fractional difference between $g_{_{UV,4\ell}}$ and $g_{_{UV,6\ell}}$ must be less than 15 \%, then our results show that this requirement is satisfied for $N \gtrsim 2 \times 10^3$. The possible role of nonperturbative effects is also noted.

hep-th↗
arXiv2023

Measures of Spin Ordering in the Potts Model with a Generalized External Magnetic Field

We formulate measures of spin ordering in the $q$-state ferromagnetic Potts model in a generalized external magnetic field that favors or disfavors spin values in a subset $I_s = \{1,...,s\}$ of the total set of $q$ values. The results are contrasted with the corresponding measures of spin ordering in the case of a conventional external magnetic field that favors or disfavors a single spin value out of total set of $q$ values. Some illustrative calculations are included.

cond-mat.stat-mech↗
arXiv2023

Search for an Ultraviolet Zero in the Seven-Loop Beta Function of the $λϕ^4_4$ Theory

We investigate whether the seven-loop beta function of the $λϕ^4_4$ theory exhibits evidence for an ultraviolet zero. In addition to a direct analysis of the beta function, we calculate and study Padé approximants and discuss effects of scheme transformations on the results. Confirming and extending our earlier studies of the five-loop and six-loop beta functions, we find that in the range of $λ$ where the perturbative calculation of the seven-loop beta function is reliable, the theory does not exhibit evidence for an ultraviolet zero.

hep-ph↗
arXiv2022

On General-n Coefficients in Series Expansions for Row Spin-Spin Correlation Functions in the Two-Dimensional Ising Model

We consider spin-spin correlation functions for spins along a row, $R_n = \langle σ_{0,0}σ_{n,0}\rangle$, in the two-dimensional Ising model. We discuss a method for calculating general-$n$ expressions for coefficients in high-temperature and low-temperature series expansions of $R_n$ and apply it to obtain such expressions for several higher-order coefficients. In addition to their intrinsic interest, these results could be useful in the continuing quest for an ordinary differential equation whose solution would determine $R_n$, analogous to the known ordinary differential equation whose solution determines the diagonal correlation function $\langle σ_{0,0}σ_{n,n}\rangle$ in this model.

math-ph↗
arXiv2022

Cross Section Calculations in Theories of Self-Interacting Dark Matter

We study an asymmetric dark matter model with self-interacting dark matter consisting of a Dirac fermion $χ$ coupled to a scalar or vector mediator, such that the reaction $χ+ χ\to χ+ χ$ is well described by perturbation theory. We compute the scattering cross section $σ$, the transfer cross section $σ_T$, and the viscosity cross section $σ_V$ for this reaction. As one part of our study, we give analytic and numerical comparisons of results obtained with the inclusion of both $t$-channel and $u$-channel exchanges and results obtained in an approximation that has often been used in the literature that includes only the $t$-channel contribution. The velocity dependences of these cross sections are studied in detail and shown to be in accord with observational data.

hep-ph↗
arXiv2022

Theories and Experiments for Testable Baryogenesis Mechanisms: A Snowmass White Paper

The baryon asymmetry of the Universe is one of the central motivations to expect physics beyond the Standard Model. In this Snowmass white paper, we review the challenges and opportunities in testing some of the central paradigms that predict physics at scales low enough to expect new experimental data in the next decade. Focusing on theoretical ideas and some of their experimental implications, in particular, we discuss neutron-antineutron transformations, flavor observables, next generation colliders, future neutron facilities, gravitational waves, searches for permanent electric dipole moments, $0νββ$ decay and some future large underground experiments as methods to test post-sphaleron baryogenesis, electroweak baryogenesis, mesogenesis and low scale leptogenesis. Finally, we comment on the cases where high scale physics can be probed through some of these same mechanisms.

hep-ph↗
arXiv2022

The Present and Future Status of Heavy Neutral Leptons

The existence of non-zero neutrino masses points to the likely existence of multiple SM neutral fermions. When such states are heavy enough that they cannot be produced in oscillations, they are referred to as Heavy Neutral Leptons (HNLs). In this white paper we discuss the present experimental status of HNLs including colliders, beta decay, accelerators, as well as astrophysical and cosmological impacts. We discuss the importance of continuing to search for HNLs, and its potential impact on our understanding on key fundamental questions, and additionally we outline the future prospects for next-generation future experiments or upcoming accelerator run scenarios.

hep-ph↗