Searcharxiv⌕ Search

arXiv subjects

Robert Shrock

Publications and source records attributed to Robert Shrock.

At least 91 records · Page 5Linked to original sources
arXiv2012

Some Exact Results on Bond Percolation

We present some exact results on bond percolation. We derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice $Λ$ by $\ell$ bonds connecting the same adjacent vertices, thereby yielding the lattice $Λ_\ell$. This relation is used to calculate the bond percolation threshold on $Λ_\ell$. We show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality $d \ge 2$ but changes it on a one-dimensional lattice and quasi-one-dimensional infinite-length strips. We also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the $N \to \infty$ limits of several families of $N$-vertex graphs. Finally, we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as $N \to \infty$.

cond-mat.stat-mech↗
arXiv2012

An Analysis of Scheme Transformations in the Vicinity of an Infrared Fixed Point

We give a detailed analysis of the effects of scheme transformations in the vicinity of an exact or approximate infrared fixed point in an asymptotically free gauge theory with fermions. We list necessary conditions that such transformations must obey and show that, although these can easily be satisfied in the vicinity of an ultraviolet fixed point, they constitute significant restrictions on scheme transformations at an infrared fixed point. We construct acceptable scheme transformations and use these to study the scheme-dependence of an infrared fixed point, making comparison with our previous three-loop and four-loop calculations of the location of this point in the $\bar{MS}$ scheme. We also use an illustrative hypothetical exact $β$ function to investigate how accurately analyses of finite-order series expansions probe an infrared fixed point and the effect of a scheme transformation on these. Some implications of our work are discussed.

hep-th↗
arXiv2012

Scheme Transformations in the Vicinity of an Infrared Fixed Point

We analyze the effect of scheme transformations in the vicinity of an exact or approximate infrared fixed point in an asymptotically free gauge theory with fermions. We show that there is far less freedom in carrying out such scheme transformations in this case than at an ultraviolet fixed point. We construct a transformation from the $\bar{MS}$ scheme to a scheme with a vanishing three-loop term in the $β$ function and use this to assess the scheme dependence of an infrared fixed point in SU($N$) theories with fermions. Implications for the anomalous dimension of the fermion bilinear operator are also discussed.

hep-ph↗
arXiv2012

Confinement contains condensates

Dynamical chiral symmetry breaking and its connection with the generation of hadron masses has historically been viewed as a vacuum phenomenon. We argue that confinement makes such a position untenable. If quark-hadron duality is a reality in QCD, then condensates, those quantities that were commonly viewed as constant empirical mass-scales that fill all spacetime, are instead wholly contained within hadrons; viz., they are a property of hadrons themselves and expressed, e.g., in their Bethe-Salpeter or light-front wave functions. We explain that this paradigm is consistent with empirical evidence, and incidentally expose misconceptions in a recent Comment.

nucl-th↗
arXiv2012

Comparison of Some Exact and Perturbative Results for a Supersymmetric SU($N_c$) Gauge Theory

We consider vectorial, asymptotically free ${\cal N}=1$ supersymmetric SU($N_c$) gauge theories with $N_f$ copies of massless chiral super fields in various representations and study how perturbative predictions for the lower boundary of the infrared conformal phase, as a function of $N_f$, compare with exact results. We make use of two-loop and three-loop calculations of the beta function and anomalous dimension of the quadratic chiral super field operator product for this purpose. The specific chiral superfield contents that we consider are $N_f$ copies of (i) $F+\bar F$, (ii) $Adj$, (iii) $S_2+\bar S_2$, and (iv) $A_2 + \bar A_2$, where $F$, $Adj$, $S_2$, and $A_2$ denote, respectively, the fundamental, adjoint, and symmetric and antisymmetric rank-2 tensor representations. We find that perturbative results slightly overestimate the value of $N_{f,cr}$ relative to the respective exact results for these representations, i.e., slightly underestimate the interval in $N_f$ for which the theory has infrared conformal behavior. Our results provide a measure of how closely perturbative calculations reproduce exact results for these theories.

hep-ph↗
arXiv2012

The Structure of Chromatic Polynomials of Planar Triangulation Graphs and Implications for Chromatic Zeros and Asymptotic Limiting Quantities

We present an analysis of the structure and properties of chromatic polynomials $P(G_{pt,\vec m},q)$ of one-parameter and multi-parameter families of planar triangulation graphs $G_{pt,\vec m}$, where ${\vec m} = (m_1,...,m_p)$ is a vector of integer parameters. We use these to study the ratio of $|P(G_{pt,\vec m},τ+1)|$ to the Tutte upper bound $(τ-1)^{n-5}$, where $τ=(1+\sqrt{5} \ )/2$ and $n$ is the number of vertices in $G_{pt,\vec m}$. In particular, we calculate limiting values of this ratio as $n \to \infty$ for various families of planar triangulations. We also use our calculations to study zeros of these chromatic polynomials. We study a large class of families $G_{pt,\vec m}$ with $p=1$ and $p=2$ and show that these have a structure of the form $P(G_{pt,m},q) = c_{_{G_{pt}},1}λ_1^m + c_{_{G_{pt}},2}λ_2^m + c_{_{G_{pt}},3}λ_3^m$ for $p=1$, where $λ_1=q-2$, $λ_2=q-3$, and $λ_3=-1$, and $P(G_{pt,\vec m},q) = \sum_{i_1=1}^3 \sum_{i_2=1}^3 c_{_{G_{pt}},i_1 i_2} λ_{i_1}^{m_1}λ_{i_2}^{m_2}$ for $p=2$. We derive properties of the coefficients $c_{_{G_{pt}},\vec i}$ and show that $P(G_{pt,\vec m},q)$ has a real chromatic zero that approaches $(1/2)(3+\sqrt{5} \ )$ as one or more of the $m_i \to \infty$. The generalization to $p \ge 3$ is given. Further, we present a one-parameter family of planar triangulations with real zeros that approach 3 from below as $m \to \infty$. Implications for the ground-state entropy of the Potts antiferromagnet are discussed.

math-ph↗
arXiv2011

Chromatic Polynomials of Planar Triangulations, the Tutte Upper Bound, and Chromatic Zeros

Tutte proved that if $G_{pt}$ is a planar triangulation and $P(G_{pt},q)$ is its chromatic polynomial, then $|P(G_{pt},τ+1)| \le (τ-1)^{n-5}$, where $τ=(1+\sqrt{5} \,)/2$ and $n$ is the number of vertices in $G_{pt}$. Here we study the ratio $r(G_{pt})=|P(G_{pt},τ+1)|/(τ-1)^{n-5}$ for a variety of planar triangulations. We construct infinite recursive families of planar triangulations $G_{pt,m}$ depending on a parameter $m$ linearly related to $n$ and show that if $P(G_{pt,m},q)$ only involves a single power of a polynomial, then $r(G_{pt,m})$ approaches zero exponentially fast as $n \to \infty$. We also construct infinite recursive families for which $P(G_{pt,m},q)$ is a sum of powers of certain functions and show that for these, $r(G_{pt,m})$ may approach a finite nonzero constant as $n \to \infty$. The connection between the Tutte upper bound and the observed chromatic zero(s) near to $τ+1$ is investigated. We report the first known graph for which the zero(s) closest to $τ+1$ is not real, but instead is a complex-conjugate pair. Finally, we discuss connections with nonzero ground-state entropy of the Potts antiferromagnet on these families of graphs.

math-ph↗
arXiv2011

Some Results on Vector and Tensor Meson Mixing in a Generalized QCD-like Theory

We present results for the mixing of quark flavor eigenstates to form low-lying vector and tensor meson mass eigenstates in a generalized QCD-like theory with $\ell$ massless or light, degenerate quarks and one quark of substantial mass, $m_Q$. We show that as $m_Q/Λ$ gets large, where $Λ$ is the confinement scale, the angle governing this mixing approaches $\arctan(1/\sqrt{\ell})$. We give a generalization of the Gell-Mann Okubo mass relation for this theory. A comparison of the $\ell=2$ results with data on $ω$-$ϕ$ and $f_2(1275)$-$f_2'(1525)$ mixing is included.

hep-ph↗
arXiv2011

Technicolor Models with Color-Singlet Technifermions and their Ultraviolet Extensions

We study technicolor models in which all of the technifermions are color-singlets, focusing on the case in these fermions transform according to the fundamental representation of the technicolor gauge group. Our analysis includes a derivation of restrictions on the weak hypercharge assignments for the technifermions and additional color-singlet, technisinglet fermions arising from the necessity of avoiding stable bound states with exotic electric charges. Precision electroweak constraints on these models are also discussed. We determine some general properties of extended technicolor theories containing these technicolor sectors.

hep-ph↗
arXiv2011

Higher-Loop Corrections to the Infrared Evolution of a Gauge Theory with Fermions

We consider a vectorial, asymptotically free gauge theory and analyze the effect of higher-loop corrections to the beta function on the evolution of the theory from the ultraviolet to the infrared. We study the case in which the theory contains $N_f$ copies of a fermion transforming according to the fundamental representation and several higher-dimensional representations of the gauge group. We also calculate higher-loop values of the anomalous dimension of the mass, $γ_m$ of $\barψψ$ at the infrared zero of the beta function. We find that for a given theory, the values of $γ_m$ calculated to three- and four-loop order, and evaluated at the infrared zero computed to the same order, tend to be somewhat smaller than the value calculated to two-loop order. The results are compared with recent lattice simulations.

hep-ph↗
arXiv2011

Exact Results on Potts/Tutte Polynomials for Families of Networks with Edge and Vertex Inflations

We derive exact relations between the Potts model partition function, or equivalently the Tutte polynomial, for a network (graph) $G$ and a network obtained from $G$ by (i) by replacing each edge (i.e., bond) of $G$ by two or more edges joining the same vertices, and (ii) by inserting one or more degree-2 vertices on edges of $G$. These processes are called edge and vertex inflation, respectively. The physical effects of these edge and vertex inflations are discussed. We also present exact calculations of these polynomials for families of networks obtained via the operation (ii) on a subset of the bonds of the network. Applications of these results include calculations of some network reliability polynomials. In addition, we evaluate our results to calculate various quantities of structural interest such as numbers of spanning trees, etc., and to determine their asymptotic behavior for large networks.

cond-mat.stat-mech↗
arXiv2011

Exact Potts/Tutte Polynomials for Polygon Chain Graphs

We present exact calculations of Potts model partition functions and the equivalent Tutte polynomials for polygon chain graphs with open and cyclic boundary conditions. Special cases of the results that yield flow and reliability polynomials are discussed. We also analyze special cases of the Tutte polynomials that determine various quantities of graph-theoretic interest.

cond-mat.stat-mech↗
arXiv2010

Patterns of Dynamical Gauge Symmetry Breaking

We construct and analyze theories with a gauge symmetry in the ultraviolet of the form $G \otimes G_b$, in which the vectorial, asymptotically free $G_b$ gauge interaction becomes strongly coupled at a scale where the $G$ interaction is weakly coupled and produces bilinear fermion condensates that dynamically break the $G$ symmetry. Comparisons are given between Higgs and dynamical symmetry breaking mechanisms for various models.

hep-ph↗
arXiv2010

Exact Results on Potts Model Partition Functions in a Generalized External Field and Weighted-Set Graph Colorings

We present exact results on the partition function of the $q$-state Potts model on various families of graphs $G$ in a generalized external magnetic field that favors or disfavors spin values in a subset $I_s = \{1,...,s\}$ of the total set of possible spin values, $Z(G,q,s,v,w)$, where $v$ and $w$ are temperature- and field-dependent Boltzmann variables. We remark on differences in thermodynamic behavior between our model with a generalized external magnetic field and the Potts model with a conventional magnetic field that favors or disfavors a single spin value. Exact results are also given for the interesting special case of the zero-temperature Potts antiferromagnet, corresponding to a set-weighted chromatic polynomial $Ph(G,q,s,w)$ that counts the number of colorings of the vertices of $G$ subject to the condition that colors of adjacent vertices are different, with a weighting $w$ that favors or disfavors colors in the interval $I_s$. We derive powerful new upper and lower bounds on $Z(G,q,s,v,w)$ for the ferromagnetic case in terms of zero-field Potts partition functions with certain transformed arguments. We also prove general inequalities for $Z(G,q,s,v,w)$ on different families of tree graphs. As part of our analysis, we elucidate how the field-dependent Potts partition function and weighted-set chromatic polynomial distinguish, respectively, between Tutte-equivalent and chromatically equivalent pairs of graphs.

cond-mat.stat-mech↗
arXiv2010

Infrared Evolution and Phase Structure of a Gauge Theory Containing Different Fermion Representations

We study the evolution of an asymptotically free vectorial SU($N$) gauge theory from the ultraviolet to the infrared and the resultant phase structure in the general case in which the theory contains fermions transforming according to several different representations of the gauge group. We discuss the sequential fermion condensation and dynamical mass generation that occur, and comment on the effect of bare fermion mass terms.

hep-ph↗
arXiv2010

Condensates in Quantum Chromodynamics and the Cosmological Constant

Casher and Susskind have noted that in the light-front description, spontaneous chiral symmetry breaking in quantum chromodynamics (QCD) is a property of hadronic wavefunctions and not of the vacuum. Here we show from several physical perspectives that, because of color confinement, quark and gluon QCD condensates are associated with the internal dynamics of hadrons. We discuss condensates using condensed matter analogues, the AdS/CFT correspondence, and the Bethe-Salpeter/Dyson-Schwinger approach for bound states. Our analysis is in agreement with the Casher and Susskind model and the explicit demonstration of "in-hadron" condensates by Roberts et al., using the Bethe-Salpeter/Dyson-Schwinger formalism for QCD bound states. These results imply that QCD condensates give {\it zero} contribution to the cosmological constant, since all of the gravitational effects of the in-hadron condensates are already included in the normal contribution from hadron masses.

hep-th↗