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Robert Shrock

Publications and source records attributed to Robert Shrock.

At least 37 records · Page 2Linked to original sources
arXiv2021

Upper limits on branching ratios of the lepton-flavor-violating decays $τ\to \ell γγ$ and $τ\to \ell X$

From analysis of data produced by the BABAR experiment, the first upper bounds (90% C.L.) were obtained on the branching ratios $Br(τ\to e γγ)<2.5 \times 10^{-4}$ and $Br(τ\to μγγ)<5.8 \times 10^{-4}$. In addition, improved upper bounds (95% C.L.) were found on branching ratios $Br(τ\to e X)<1.4 \times 10^{-3}$ and $Br(τ\to μX) <2.0 \times 10^{-3}$, where $X$ is an undetected weakly interacting boson with mass $m_X<1.6$ GeV/$c^2$.

hep-ph↗
arXiv2021

Exact Results for Average Cluster Numbers in Bond Percolation on Infinite-Length Lattice Strips

We calculate exact analytic expressions for the average cluster numbers $\langle k \rangle_{Λ_s}$ on infinite-length strips $Λ_s$, with various widths, of several different lattices, as functions of the bond occupation probability, $p$. It is proved that these expressions are rational functions of $p$. As special cases of our results, we obtain exact values of $\langle k \rangle_{Λ_s}$ and derivatives of $\langle k \rangle_{Λ_s}$ with respect to $p$, evaluated at the critical percolation probabilities $p_{c,Λ}$ for the corresponding infinite two-dimensional lattices $Λ$. We compare these exact results with an analytic finite-size correction formula and find excellent agreement. We also analyze how unphysical poles in $\langle k \rangle_{Λ_s}$ determine the radii of convergence of series expansions for small $p$ and for $p$ near to unity. Our calculations are performed for infinite-length strips of the square, triangular, and honeycomb lattices with several types of transverse boundary conditions.

cond-mat.stat-mech↗
arXiv2020

Exponential Growth Constants for Spanning Forests on Archimedean Lattices: Values and Comparisons of Upper Bounds

We compare our upper bounds on the exponential growth constant $ϕ(Λ)$ characterizing the asymptotic behavior of spanning forests on Archimedean lattices $Λ$ with recently derived upper bounds. Our upper bounds on $ϕ(Λ)$, which are very close to the respective values of $ϕ(Λ)$ that we have calculated, are shown to be significantly better for these lattices than the new upper bounds.

math.CO↗
arXiv2020

$|Δ\mathcal{B}| =2$: A State of the Field, and Looking Forward--A brief status report of theoretical and experimental physics opportunities

The origin of the matter-antimatter asymmetry apparently obligates the laws of physics to include some mechanism of baryon number ($\mathcal{B}$) violation. Searches for interactions violating $\mathcal{B}$ and baryon-minus-lepton number $\mathcal{(B-L)}$ represent a rich and underutilized opportunity. These are complementary to the existing, broad program of searches for $\mathcal{L}$-violating modes such as neutrinoless double $β$-decay which could provide deeper understandings of the plausibility of leptogenesis, or $\mathcal{B}$-violating, $\mathcal{(B-L)}$-conserving processes such as proton decay. In particular, a low-scale, post-sphaleron violation mechanism of $\mathcal{(B-L)}$ could provide a \textit{testable} form of baryogenesis. Though theoretically compelling, searches for such $\mathcal{(B-L)}$-violating processes like $Δ\mathcal{B}=2$ dinucleon decay and $n\rightarrow\bar{n}$ remain relatively underexplored experimentally compared to other rare processes. By taking advantage of upcoming facilities such as the Deep Underground Neutrino Experiment and the European Spallation Source, this gap can be addressed with new intranuclear and free searches for neutron transformations with very high sensitivity, perhaps greater than three orders of magnitude higher than previous experimental searches. This proceedings reports on recent theoretical and experimental advances and sensitivities of next-generation searches for neutron transformations were detailed as part of the Amherst Center for Fundamental Interactions Workshop, "Theoretical Innovations for Future Experiments Regarding Baryon Number Violation," directly coordinated with the Rare Processes and Precision Measurements Frontier.

hep-ph↗
arXiv2020

Effect of Scheme Transformations on a Beta Function with Vanishing One-Loop Term

It is commonly stated that because terms in the beta function of a theory at the level of $\ell \ge 3$ loops and higher are scheme-dependent, it is possible to define scheme transformations that can be used to remove these terms, at least in the vicinity of zero coupling. We prove that this is not, in general, possible in the situation where a beta function is not identically zero but has a vanishing one-loop term.

hep-th↗
arXiv2020

Renormalization-Group Behavior of $ϕ^3$ Theories in $d=6$ Dimensions

We investigate possible renormalization-group fixed points at nonzero coupling in $ϕ^3$ theories in six spacetime dimensions, using beta functions calculated to the four-loop level. We analyze three theories of this type, with (a) a one-component scalar, (b) a scalar transforming as the fundamental representation of a global ${\rm SU}(N)$ symmetry group, and (c) a scalar transforming as a bi-adjoint representation of a global ${\rm SU}(N) \otimes {\rm SU}(N)$ symmetry. We do not find robust evidence for such fixed points in theories (a) or (b). Theory (c) has the special feature that the one-loop term in the beta function is zero; implications of this are discussed.

hep-th↗
arXiv2020

Nucleon Decay and $n$-$\bar n$ Oscillations in a Left-Right Symmetric Model with Large Extra Dimensions

We study baryon-number-violating processes, including proton and bound neutron decays and $n-\bar n$ oscillations, in a left-right-symmetric (LRS) model in which quarks and leptons have localized wavefunctions in extra dimensions. In this model we show that, while one can easily suppress baryon-number-violating nucleon decays well below experimental bounds, this does not suppress $n-\bar n$ transitions, which may occur at levels comparable to current limits. This is qualitatively similar to what was found in an extra-dimensional model with a Standard-Model low-energy effective field theory (SMEFT). We show that experimental data imply a lower limit on the mass scale $M_{n \bar n}$ characterizing the physics responsible for $n-\bar n$ oscillations in the LRS model that is significantly higher than in the extra-dimensional model using a SMEFT and explain the reason for this. Our results provide further motivation for new experiments to search for $n - \bar n$ oscillations.

hep-ph↗
arXiv2020

Scheme-Independent Series for Anomalous Dimensions of Higher-Spin Operators at an Infrared Fixed Point in a Gauge Theory

We consider an asymptotically free vectorial gauge theory, with gauge group $G$ and $N_f$ fermions in a representation $R$ of $G$, having an infrared fixed point of the renormalization group. We calculate scheme-independent series expansions for the anomalous dimensions of higher-spin bilinear fermion operators at this infrared fixed point up to $O(Δ_f^3)$, where $Δ_f$ is an $N_f$-dependent expansion variable. Our general results are evaluated for several special cases, including the case $G={\rm SU}(N_c)$ with $R$ equal to the fundamental and adjoint representations.

hep-ph↗
arXiv2020

Asymptotic Behavior of Spanning Forests and Connected Spanning Subgraphs on Two-Dimensional Lattices

We calculate exponential growth constants $ϕ$ and $σ$ describing the asymptotic behavior of spanning forests and connected spanning subgraphs on strip graphs, with arbitrarily great length, of several two-dimensional lattices, including square, triangular, honeycomb, and certain heteropolygonal Archimedean lattices. By studying the limiting values as the strip widths get large, we infer lower and upper bounds on these exponential growth constants for the respective infinite lattices. Since our lower and upper bounds are quite close to each other, we can infer very accurate approximate values for these exponential growth constants, with fractional uncertainties ranging from $O(10^{-4})$ to $O(10^{-2})$. We show that $ϕ$ and $σ$, are monotonically increasing functions of vertex degree for these lattices.

cond-mat.stat-mech↗
arXiv2019

Baryon-Number-Violating Nucleon and Dinucleon Decays in a Model with Large Extra Dimensions

It is known that limits on baryon-violating nucleon decays do not, in general, imply corresponding suppression of $n - \bar n$ transitions. In the context of a model with fermions propagating in higher dimensions, we investigate a related question, namely the implications of limits on $ΔL=-1$ proton and bound neutron decays mediated by four-fermion operators for rates of nucleon decays mediated by $k$-fermion operators with $k =6$ and $k=8$. These include a variety of nucleon and dinucleon decays to dilepton and trilepton final states with $ΔL=-3, \ -2, \ 1$, and $2$. We carry out a low-energy effective field theory analysis of relevant operators for these decays and show that, in this extra-dimensional model, the rates for these decays are strongly suppressed and hence are in accord with experimental limits.

hep-ph↗
arXiv2019

$q$-Plane Zeros of the Potts Partition Function on Diamond Hierarchical Graphs

We report exact results concerning the zeros of the partition function of the Potts model in the complex $q$ plane, as a function of a temperature-like Boltzmann variable $v$, for the $m$'th iterate graphs $D_m$ of the Diamond Hierarchical Lattice (DHL), including the limit $m \to \infty$. In this limit we denote the continuous accumulation locus of zeros in the $q$ planes at fixed $v = v_0$ as ${\mathcal B}_q(v_0)$. We apply theorems from complex dynamics to establish properties of ${\mathcal B}_q(v_0)$. For $v=-1$ (the zero-temperature Potts antiferromagnet, or equivalently, chromatic polynomial), we prove that ${\mathcal B}_q(-1)$ crosses the real-$q$ axis at (i) a minimal point $q=0$, (ii) a maximal point $q=3$ (iii) $q=32/27$, (iv) a cubic root that we give, with the value $q = q_1 = 1.6388969..$, and (v) an infinite number of points smaller than $q_1$, converging to $32/27$ from above. Similar results hold for ${\mathcal B}_q(v_0)$ for any $-1 < v < 0$ (Potts antiferromagnet at nonzero temperature). The locus ${\mathcal B}_q(v_0)$ crosses the real-$q$ axis at only two points for any $v > 0$ (Potts ferromagnet). We also provide computer-generated plots of ${\mathcal B}_q(v_0)$ at various values of $v_0$ in both the antiferromagnetic and ferromagnetic regimes and compare them to numerically computed zeros of $Z(D_4,q,v_0)$.

math-ph↗
arXiv2019

Improved Upper Limits on Baryon-Number Violating Dinucleon Decays to Dileptons

We consider effects of $n-\bar n$ oscillations and resultant matter instability due to dinucleons decays. We point out that existing upper bounds on the rates for the dinucleon decays $nn \to 2π^0$, $nn \to π^+π^-$, and $np \to π^+π^0$ imply upper bounds on the rates for dinucleon decays to dileptons $nn \to e^+ e^-$, $nn \to μ^+μ^-$, $nn \to ν_\ell \barν_\ell$, and $np \to \ell^+ ν_\ell$, where $\ell=e, \ μ, \ τ$. We present estimates for these upper bounds. Our bounds are substantially stronger than corresponding limits from direct searches.

hep-ph↗
arXiv2019

Improved Lower Bounds on Partial Lifetimes for Nucleon Decay Modes

In the framework of a baryon-number-violating effective Lagrangian, we calculate improved lower bounds on partial lifetimes for proton and bound neutron decays, including $p \to \ell^+ \ell'^+ \ell'^-$, $n \to \barν\ell^+ \ell'^-$, $p \to \ell^+ ν\barν$, and $n \to \barν\barνν$, where $\ell$ and $\ell'$ denote $e$ or $μ$, with both $\ell = \ell'$ and $\ell \ne \ell'$ cases. Our lower bounds are substantially stronger than the corresponding lower bounds from direct experimental searches. We also present lower bounds on $(τ/B)_{p \to \ell^+γ}$, $(τ/B)_{n \to \barνγ}$, $(τ/B)_{p \to \ell^+ γγ}$, and $(τ/B)_{n \to \barνγγ}$. Our method relies on relating the rates for these decay modes to the rates for decay modes of the form $p \to \ell^+ M$ and $n \to \barνM$, where $M$ is a pseudoscalar or vector meson, and then using the experimental lower bounds on the partial lifetimes for these latter decays.

hep-ph↗
arXiv2019

Ultraviolet to Infrared Evolution and Nonperturbative Behavior of ${\rm SU}(N) \otimes {\rm SU}(N-4) \otimes {\rm U}(1)$ Chiral Gauge Theories

We analyze the ultraviolet to infrared evolution and nonperturbative properties of asymptotically free ${\rm SU}(N) \otimes {\rm SU}(N-4) \otimes {\rm U}(1)$ chiral gauge theories with $N_f$ copies of chiral fermions transforming according to $([2]_N,1)_{N-4} + ([\bar 1]_N,[\bar 1]_{N-4})_{-(N-2)} + (1,(2)_{N-4})_N$, where $[k]_N$ and $(k)_N$ denote the antisymmetric and symmetric rank-$k$ tensor representations of SU($N$) and the rightmost subscript is the U(1) charge. We give a detailed discussion for the lowest nondegenerate case, $N=6$. These theories can exhibit both self-breaking of a strongly coupled gauge symmetry and induced dynamical breaking of a weakly coupled gauge interaction symmetry due to fermion condensates produced by a strongly coupled gauge interaction. A connection with the dynamical breaking of ${\rm SU}(2)_L \otimes {\rm U}(1)_Y$ electroweak gauge symmetry by the quark condensates $\langle \bar q q\rangle$ due to color SU(3)$_c$ interactions is discussed. We also remark on direct-product chiral gauge theories with fermions in higher-rank tensor representations.

hep-th↗
arXiv2019

Large-$N_c$ and Large-$N_F$ Limits of SU($N_c$) Gauge Theories with Fermions in Different Representations

We present calculations of certain limits of scheme-independent series expansions for the anomalous dimensions of gauge-invariant fermion bilinear operators and for the derivative of the beta function at an infrared fixed point in SU($N_c$) gauge theories with fermions transforming according to two different representations. We first study a theory with $N_f$ fermions in the fundamental representation and $N_{f'}$ fermions in the adjoint or symmetric or antisymmetric rank-2 tensor representation, in the limit $N_c \to \infty$, $N_f \to \infty$ with $N_f/N_c$ fixed and finite. We then study the $N_c \to \infty$ limit of a theory with fermions in the adjoint and rank-2 symmetric or antisymmetric tensor representations.

hep-th↗
arXiv2018

Study of Exponential Growth Constants of Directed Heteropolygonal Archimedean Lattices

We infer upper and lower bounds on the exponential growth constants $α(Λ)$, $α_0(Λ)$, and $β(Λ)$ describing the large-$n$ behavior of, respectively, the number of acyclic orientations, acyclic orientations with a unique source vertex, and totally cyclic orientations of arrows on bonds of several $n$-vertex heteropolygonal Archimedean lattices $Λ$. These are, to our knowledge, the best bounds on these growth constants. The inferred upper and lower bounds on the growth constants are quite close to each other, which enables us to derive rather accurate values for the actual exponential growth constants. Combining our new results for heteropolygonal Archimedean lattices with our recent results for homopolygonal Archimedean lattices, we show that the exponential growth constants $α(Λ)$, $α_0(Λ)$, and $β(Λ)$ on these lattices are monotonically increasing functions of the lattice coordination number. Comparisons are made with the corresponding growth constants for spanning trees on these lattices. Our findings provide further support for the Merino-Welsh and Conde-Merino conjectures.

cond-mat.stat-mech↗
arXiv2018

Asymptotic Behavior of Acyclic and Cyclic Orientations of Directed Lattice Graphs

We calculate exponential growth constants describing the asymptotic behavior of several quantities enumerating classes of orientations of arrow variables on the bonds of several types of directed lattice strip graphs $G$ of finite width and arbitrarily great length, in the infinite-length limit, denoted {G}. Specifically, we calculate the exponential growth constants for (i) acyclic orientations, $α(\{G\})$, (ii) acyclic orientations with a single source vertex, $α_0(\{G\})$, and (iii) totally cyclic orientations, $β(\{G\})$. We consider several lattices, including square (sq), triangular (tri), and honeycomb (hc). From our calculations, we infer lower and upper bounds on these exponential growth constants for the respective infinite lattices. To our knowledge, these are the best current bounds on these quantities. Since our lower and upper bounds are quite close to each other, we can infer very accurate approximate values for the exponential growth constants, with fractional uncertainties ranging from $O(10^{-4})$ to $O(10^{-2})$. Further, we present exact values of $α(tri)$, $α_0(tri)$, and $β(hc)$ and use them to show that our lower and upper bounds on these quantities are very close to these exact values, even for modest strip widths. Results are also given for a nonplanar lattice denoted $sq_d$. We show that $α(\{G\})$, $α_0(\{G\})$, and $β(\{G\})$ are monotonically increasing functions of vertex degree for these lattices. We also study the asymptotic behavior of the ratios of the quantities (i)-(iii) divided by the total number of edge orientations as the number of vertices goes to infinity. A comparison is given of these exponential growth constants with the corresponding exponential growth constant $τ(\{G\})$ for spanning trees. Our results are in agreement with inequalities following from the Merino-Welsh and Conde-Merino conjectures.

cond-mat.stat-mech↗