Search

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., Comment: 57 pages, latex, 26 figures

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., Comment: 43 pages, 17 figures

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., Comment: 6 pages, latex

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., Comment: 9 pages, latex

We present scheme-independent calculations of the anomalous dimensions $\gamma_{\bar\psi\psi,IR}$ and $\gamma_{\bar\chi\chi,IR}$ of fermion bilinear operators $\bar\psi\psi$ and $\bar\chi\chi$ at an infrared fixed point in an asymptotically free SU($N_c$) gauge theory with massless Dirac fermion content consisting of $N_F$ fermions $\psi^a_i$ in the fundamental representation and $N_{A_2}$ fermions $\chi^{ab}_j$ 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., Comment: 15 pages latex, 5 figures

We study the ultraviolet (UV) behavior of an O($N$) $|\vec \phi |^6$ 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 $|\vec \phi |^6$ 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 Pad\'e 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., Comment: 10 pages, latex

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., Comment: 22 pages, 6 figures, accepted for publication in Physica A

We investigate whether the seven-loop beta function of the $\lambda \phi^4_4$ theory exhibits evidence for an ultraviolet zero. In addition to a direct analysis of the beta function, we calculate and study Pad\'e 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 $\lambda$ where the perturbative calculation of the seven-loop beta function is reliable, the theory does not exhibit evidence for an ultraviolet zero., Comment: 6 pages, latex, 2 figures. arXiv admin note: substantial text overlap with arXiv:1610.03733

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., Comment: 9 pages, 8 figures, matches the publication in Phys. Rev. D

We consider spin-spin correlation functions for spins along a row, $R_n = \langle \sigma_{0,0}\sigma_{n,0}\rangle$, 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 $\langle \sigma_{0,0}\sigma_{n,n}\rangle$ in this model., Comment: No change in original results; by request of a referee, the paper was expanded to include transcriptions of results in terms of equivalent elliptic modulus variables

We study an asymmetric dark matter model with self-interacting dark matter consisting of a Dirac fermion $\chi$ coupled to a scalar or vector mediator, such that the reaction $\chi + \chi \to \chi + \chi$ is well described by perturbation theory. We compute the scattering cross section $\sigma$, the transfer cross section $\sigma_T$, and the viscosity cross section $\sigma_V$ 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., Comment: 15 pages, 10 figures

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., Comment: 82 pages, 34 figures. Contribution to Snowmass 2021

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, $0\nu \beta \beta $ 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., Comment: White paper to be submitted to the Snowmass Process' Rare Processes and Precision Measurements Frontier, Baryon and Lepton Number Violation Topical Group (This v2 version corrects a missing author and an author's name which was misspelled. Apologies to these authors for this oversight on the part of the submitter!)

We discuss some higher-loop studies of renormalization-group flows and fixed points in various quantum field theories., Comment: 10 pages, Latex, written version of a talk given via zoom at the XXXIII Int. Workshop on High Energy Physics, Kurchatov IHEP, to appear in the workshop proceedings

From analysis of data produced by the BABAR experiment, the first upper bounds (90% C.L.) were obtained on the branching ratios $Br(\tau \to e \gamma\gamma)<2.5 \times 10^{-4}$ and $Br(\tau \to \mu \gamma\gamma)<5.8 \times 10^{-4}$. In addition, improved upper bounds (95% C.L.) were found on branching ratios $Br(\tau \to e X)<1.4 \times 10^{-3}$ and $Br(\tau \to \mu X) <2.0 \times 10^{-3}$, where $X$ is an undetected weakly interacting boson with mass $m_X<1.6$ GeV/$c^2$., Comment: 8 pages, 2 figures

We calculate exact analytic expressions for the average cluster numbers $\langle k \rangle_{\Lambda_s}$ on infinite-length strips $\Lambda_s$, 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 $\langle k \rangle_{\Lambda_s}$ and derivatives of $\langle k \rangle_{\Lambda_s}$ with respect to $p$, evaluated at the critical percolation probabilities $p_{c,\Lambda}$ for the corresponding infinite two-dimensional lattices $\Lambda$. We compare these exact results with an analytic finite-size correction formula and find excellent agreement. We also analyze how unphysical poles in $\langle k \rangle_{\Lambda_s}$ 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., Comment: 29 pages, latex

We compare our upper bounds on the exponential growth constant $\phi(\Lambda)$ characterizing the asymptotic behavior of spanning forests on Archimedean lattices $\Lambda$ with recently derived upper bounds. Our upper bounds on $\phi(\Lambda)$, which are very close to the respective values of $\phi(\Lambda)$ that we have calculated, are shown to be significantly better for these lattices than the new upper bounds., Comment: 5 pages, latex. arXiv admin note: text overlap with arXiv:2002.07150, arXiv:1810.07357

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 $\beta$-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 $\Delta\mathcal{B}=2$ 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., Comment: Amherst Center for Fundamental Interactions workshop proceedings with short and long abstracts

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., Comment: 6 pages, latex

We investigate possible renormalization-group fixed points at nonzero coupling in $\phi^3$ 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., Comment: 9 pages

We use $\bar p p$ and $e^+e^-$ annihilation data to further strengthen lower bounds on the partial lifetimes for the baryon-number-violating dinucleon decays $nn \to e^+ e^-$ and $nn \to \mu^+\mu^-$., Comment: 5 pages, latex

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., Comment: 10 pages, latex. arXiv admin note: text overlap with arXiv:1911.05102

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 $O(\Delta_f^3)$, where $\Delta_f$ 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., Comment: 22 pages, latex, 7 figures

We calculate exponential growth constants $\phi$ and $\sigma$ 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 $\phi$ and $\sigma$, are monotonically increasing functions of vertex degree for these lattices., Comment: 33 pages, 30 tables. arXiv admin note: text overlap with arXiv:1810.07357

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 $\Delta L=-1$ 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 $\Delta L=-3, \ -2, \ 1$, 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., Comment: 29 pages, latex

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 $Z(D_4,q,v_0)$., Comment: 41 pages, 16 figures. Comments welcome!

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 $nn \to 2\pi^0$, $nn \to \pi^+\pi^-$, and $np \to \pi^+\pi^0$ imply upper bounds on the rates for dinucleon decays to dileptons $nn \to e^+ e^-$, $nn \to \mu^+\mu^-$, $nn \to \nu_\ell \bar\nu_\ell$, and $np \to \ell^+ \nu_\ell$, where $\ell=e, \ \mu, \ \tau$. We present estimates for these upper bounds. Our bounds are substantially stronger than corresponding limits from direct searches., Comment: 6 pages, 3 figures

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 $p \to \ell^+ \ell'^+ \ell'^-$, $n \to \bar\nu \ell^+ \ell'^-$, $p \to \ell^+ \nu\bar\nu$, and $n \to \bar\nu \bar\nu \nu$, where $\ell$ and $\ell'$ denote $e$ or $\mu$, 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 $(\tau/B)_{p \to \ell^+\gamma}$, $(\tau/B)_{n \to \bar\nu \gamma}$, $(\tau/B)_{p \to \ell^+ \gamma\gamma}$, and $(\tau/B)_{n \to \bar\nu \gamma\gamma}$. 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 $n \to \bar\nu M$, where $M$ is a pseudoscalar or vector meson, and then using the experimental lower bounds on the partial lifetimes for these latter decays., Comment: 11 pages, 8 figures

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., Comment: 22 pages, latex

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., Comment: 18 pages, RevTex

We infer upper and lower bounds on the exponential growth constants $\alpha(\Lambda)$, $\alpha_0(\Lambda)$, and $\beta(\Lambda)$ 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 $\Lambda$. 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 $\alpha(\Lambda)$, $\alpha_0(\Lambda)$, and $\beta(\Lambda)$ 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., Comment: 45 pages, latex. arXiv admin note: text overlap with arXiv:1810.07357

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, $\alpha(\{G\})$, (ii) acyclic orientations with a single source vertex, $\alpha_0(\{G\})$, and (iii) totally cyclic orientations, $\beta(\{G\})$. 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 $\alpha(tri)$, $\alpha_0(tri)$, and $\beta(hc)$ 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 $\alpha(\{G\})$, $\alpha_0(\{G\})$, and $\beta(\{G\})$ 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 $\tau(\{G\})$ for spanning trees. Our results are in agreement with inequalities following from the Merino-Welsh and Conde-Merino conjectures., Comment: 74 pages, 24 tables

In previous work we have presented scheme-independent calculations of physical properties of operators at a conformally invariant infrared fixed point in an asymptotically free gauge theory with gauge group $G$ and $N_f$ fermions in a representation $R$ of $G$. Here we generalize this analysis to the case of fermions in multiple representations, focusing on the case of two different representations. Our results include the calculation of the anomalous dimensions of gauge-invariant fermion bilinear operators, and the derivative of the beta function, evaluated at the infrared fixed point. We illustrate our results in an SU($N_c$) gauge theory with $N_F$ fermions in the fundamental representation and $N_{Adj}$ fermions in the adjoint representation.

We present the first analytic scheme-independent series calculations of anomalous dimensions of several types of baryon operators at an infrared fixed point (IRFP) in an asymptotically free SU(3) gauge theory with $N_f$ fermions. Separately, for an asymptotically free gauge theory with a gauge group $G$ and $N_f$ fermions in a representation $R$ of $G$, we consider physical quantities at an IRFP, including the anomalous dimension of gauge-invariant fermion bilinears and the derivative of the beta function. These quantities have been calculated in series expansions whose coefficients have been proved to be scheme-independent at each order. We illustrate the scheme independence using a variety of schemes, including the RI$^\prime$ scheme and several types of momentum subtraction (MOM) schemes., Comment: 16 pages, 4 figures

We study duality in $\mathcal{N}=1$ supersymmetric QCD in the non-Abelian Coulomb phase, order-by-order in scheme-independent series expansions. Using exact results, we show how the dimensions of various fundamental and composite chiral superfields, and the quantities $a$, $a/c$, and $b$ at superconformal fixed points of the renormalization group emerge in scheme-independent series expansions in the electric and magnetic theories. We further demonstrate that truncations of these series expansions to modest order yield very accurate approximations to these quantities., Comment: 4 figures

We consider a vectorial, asymptotically free SU($N_c$) gauge theory with $N_f$ fermions in a representation $R$ having an infrared (IR) fixed point. We calculate and analyze Pad\'e approximants to scheme-independent series expansions for physical quantities at this IR fixed point, including the anomalous dimension, $\gamma_{\bar\psi\psi,IR}$, to $O(\Delta_f^4)$, and the derivative of the beta function, $\beta'_{IR}$, to $O(\Delta_f^5)$, where $\Delta_f$ is an $N_f$-dependent expansion variable. We consider the fundamental, adjoint, and rank-2 symmetric tensor representations. The results are applied to obtain further estimates of $\gamma_{\bar\psi\psi,IR}$ and $\beta'_{IR}$ for several SU($N_c$) groups and representations $R$, and comparisons are made with lattice measurements. We apply our results to obtain new estimates of the extent of the respective non-Abelian Coulomb phases in several theories. For $R=F$, the limit $N_c \to \infty$ and $N_f \to \infty$ with $N_f/N_c$ fixed is considered. We assess the accuracy of the scheme-independent series expansion of $\gamma_{\bar\psi\psi,IR}$ in comparison with the exactly known expression in an ${\cal N}=1$ supersymmetric gauge theory. It is shown that an expansion of $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^4)$ is quite accurate throughout the entire non-Abelian Coulomb phase of this supersymmetric theory., Comment: 28 pages, 3 figures

We present scheme-independent calculations of the derivative of the beta function, denoted $\beta'_{IR}$, at a conformally invariant infrared (IR) fixed point, in several asymptotically free chiral gauge theories, namely SO($4k+2$) with $2 \le k \le 4$ with respective numbers $N_f$ of fermions in the spinor representation, and E$_6$ with fermions in the fundamental representation., Comment: 6 pages latex, 4 figures

We study properties of asymptotically free vectorial gauge theories with gauge groups $G={\rm SO}(N_c)$ and $G={\rm Sp}(N_c)$ and $N_f$ fermions in a representation $R$ of $G$, at an infrared (IR) zero of the beta function, $\alpha_{IR}$, in the non-Abelian Coulomb phase. The fundamental, adjoint, and rank-2 symmetric and antisymmetric tensor fermion representations are considered. We present scheme-independent calculations of the anomalous dimensions of (gauge-invariant) fermion bilinear operators $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^4)$ and of the derivative of the beta function at $\alpha_{IR}$, denoted $\beta'_{IR}$, to $O(\Delta_f^5)$, where $\Delta_f$ is an $N_f$-dependent expansion variable. It is shown that all coefficients in the expansion of $\gamma_{\bar\psi\psi,IR}$ that we calculate are positive for all representations considered, so that to $O(\Delta_f^4)$, $\gamma_{\bar\psi\psi,IR}$ increases monotonically with decreasing $N_f$ in the non-Abelian Coulomb phase. Using this property, we give a new estimate of the lower end of this phase for some specific realizations of these theories.

We study the possibility of an ultraviolet (UV) zero in the six-loop beta function of an O($N$) $\lambda |\vec \phi|^4$ field theory in $d=4$ spacetime dimensions. For general $N$, in the range of values of $\lambda$ where a perturbative calculation is reliable, we find evidence against such a UV zero in this six-loop beta function., Comment: 15 pages, 1 figure

We consider an asymptotically free, vectorial, ${\cal N}=1$ supersymmetric gauge theory with gauge group $G$ and $N_f$ pairs of chiral superfields in the respective representations ${\cal R}$ and $\bar {\cal R}$ of $G$, having an infrared fixed point (IRFP) of the renormalization group at $\alpha_{IR}$. We present exact results for the anomalous dimensions of various (gauge-invariant) composite chiral superfields $\gamma_{{\Phi}_{\rm prod}}$ at the IRFP and prove that these increase monotonically with decreasing $N_f$ in the non-Abelian Coulomb phase of the theory and that scheme-independent expansions for these anomalous dimensions as powers of an $N_f$-dependent variable, $\Delta_f$, exhibit monotonic and rapid convergence to the exact $\gamma_{{\Phi}_{\rm prod}}$ throughout this phase. We also present a scheme-independent calculation of the derivative of the beta function, $d\beta/d\alpha |_{\alpha=\alpha_{IR}}$, denoted $\beta'_{IR}$, up to $O(\Delta_f^3)$ for general $G$ and ${\cal R}$, and, for the case $G={\rm SU}(N_c)$, ${\cal R}=F$, we give an analysis of the properties of $\beta'_{IR}$ calculated to $O(\Delta_f^4)$., Comment: 29 pages, 13 figures

We study a vectorial asymptotically free gauge theory, with gauge group $G$ and $N_f$ massless fermions in a representation $R$ of this group, that exhibits an infrared (IR) zero in its beta function, $\beta$, at the coupling $\alpha=\alpha_{IR}$ in the non-Abelian Coulomb phase. For general $G$ and $R$, we calculate the scheme-independent series expansions of (i) the anomalous dimension of the fermion bilinear, $\gamma_{\bar\psi\psi,IR}$, to $O(\Delta_f^4)$ and (ii) the derivative $\beta' = d\beta/d\alpha$, to $O(\Delta_f^5)$, both evaluated at $\alpha_{IR}$, where $\Delta_f$ is an $N_f$-dependent expansion variable. These are the highest orders to which these expansions have been calculated. We apply these general results to theories with $G={\rm SU}(N_c)$ and $R$ equal to the fundamental, adjoint, and symmetric and antisymmetric rank-2 tensor representations. It is shown that for all of these representations, $\gamma_{\bar\psi\psi,IR}$, calculated to the order $\Delta_f^p$, with $1 \le p \le 4$, increases monotonically with decreasing $N_f$ and, for fixed $N_f$, is a monotonically increasing function of $p$. Comparisons of our scheme-independent calculations of $\gamma_{\bar\psi\psi,IR}$ and $\beta'_{IR}$ are made with our earlier higher $n$-loop values of these quantities, and with lattice measurements. For $R=F$, we present results for the limit $N_c \to \infty$ and $N_f \to \infty$ with $N_f/N_c$ fixed. We also present expansions for $\alpha_{IR}$ calculated to $O(\Delta_f^4)$., Comment: 40 pages, latex, 12 figures

We consider an asymptotically free vectorial SU($N_c$) gauge theory with $N_f$ massless fermions in a representation $R$, having an infrared fixed point (IRFP) of the renormalization group at $\alpha_{IR}$ in the conformal non-Abelian Coulomb phase. The cases with $R$ equal to the fundamental, adjoint, and symmetric rank-2 tensor representation are considered. We present scheme-independent calculations of the anomalous dimension $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^4)$ and $\beta'_{IR}$ to $O(\Delta_f^5)$ at this IRFP, where $\Delta_f$ is an $N_f$-dependent expansion parameter. Comparisons are made with conventional $n$-loop calculations and lattice measurements. As a test of the accuracy of the $\Delta_f$ expansion, we calculate $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^3)$ in ${\cal N}=1$ SU($N_c$) supersymmetric quantum chromodynamics and find complete agreement, to this order, with the exactly known expression. The $\Delta_f$ expansion also avoids a problem in which an IRFP may not be manifest as an IR zero of a higher $n$-loop beta function., Comment: 6 pages, latex, 3 figures

We investigate whether the beta function of the finite-$N$ Gross-Neveu model, as calculated up to the four-loop level, exhibits evidence for an infrared zero. As part of our analysis, we calculate and analyze Pad\'e approximants to this beta function and evaluate effects of scheme dependence. From our study, we find that in the range of coupling where the perturbative calculation of the four-loop beta function is reliable, it does not exhibit robust evidence for an infrared zero., Comment: 10 pages

We investigate whether the six-loop beta function of the $\lambda \phi^4_4$ theory exhibits evidence for an ultraviolet zero. As part of our analysis, we calculate and analyze Pad\'e approximants to this beta function. Extending our earlier results at the five-loop level, we find that in the range of $\lambda$ where the perturbative calculation of the six-loop beta function is reliable, the theory does not exhibit robust evidence for an ultraviolet zero., Comment: 7 pages latex, 2 figures

We present a scheme-independent calculation of the infrared value of the anomalous dimension of the fermion bilinear, $\gamma_{\bar\psi\psi,IR}$ in an SU(3) gauge theory as a function of the number of fermions, $N_f$, via a series expansion in powers of $\Delta_f$, where $\Delta_f=(16.5-N_f)$, to order $\Delta_f^4$. We perform an extrapolation to obtain the first determination of the exact $\gamma_{\bar\psi\psi,IR}$ from continuum field theory. The results are compared with calculations of the $n$-loop values of this anomalous dimension from series in powers of the coupling and from lattice measurements., Comment: 5 page latex, 1 figure

We calculate the value of the coupling at the infrared zero of the beta function of an asymptotically free SU(3) gauge theory at the five-loop level as a function of the number of fermions. Both a direct analysis of the beta function and analyses of Pad\'e approximants are used for this purpose. We then calculate the value of the five-loop anomalous dimension, $\gamma_m$, of the fermion bilinear at this IR zero of the beta function., Comment: 5 pages, latex

We present an integral formalism for constructing scheme transformations in a quantum field theory. We apply this to generate several new useful scheme transformations. A comparative analysis is given of these scheme transformations in terms of their series expansion coefficients and their resultant effect on the interaction coupling, in particular at a zero of the beta function away from the origin in coupling-constant space., Comment: 12 pages, latex

We analyze patterns of dynamical symmetry breaking in strongly coupled chiral gauge theories with direct-product gauge groups $G$. If the gauge coupling for a factor group $G_i \subset G$ becomes sufficiently strong, it can produce bilinear fermion condensates that break the $G_i$ symmetry itself and/or break other gauge symmetries $G_j \subset G$. Our comparative study of a number of strongly coupled direct-product chiral gauge theories elucidates how the patterns of symmetry breaking depend on the structure of $G$ and on the relative sizes of the gauge couplings corresponding to factor groups in the direct product., Comment: 25 pages, latex

We study asymptotically free SU($N_c$) gauge theories with ${\cal N}=1$ supersymmetry, including the purely gluonic theory and theories with $N_f$ copies of a pair of massless chiral superfields in the respective representations $R$ and $\bar R$ of SU($N_c$). The cases in which $R$ is the fundamental representation and the symmetric and antisymmetric rank-2 tensor representation are considered. We calculate Pad\'e approximants to the beta functions for these theories in the $\overline{\rm DR}$ scheme up to four-loop order for the gluonic theory and up to three-loop order for the theories with matter superfields and compare results for IR zeros and poles with results from the NSVZ beta function. Our calculations provide a quantitative measure, for these theories, of how well finite-order perturbative results calculated in one scheme reproduce properties of a known beta function calculated in a different scheme., Comment: 17 pages latex