arXiv subjects
Publications and source records attributed to Robert Shrock.
We analyze $n - \bar n$ oscillations in generic models with large extra dimensions in which standard-model fields propagate and fermion wavefunctions have strong localization. We find that in these models $n - \bar n$ oscillations might occur at levels not too far below the current limit.
An approximation that is often used in fits to reactor and atmospheric neutrino data and in some studies of future neutrino oscillation experiments is to assume one dominant scale, $Δm^2$, of neutrino mass squared differences, in particular, $Δm^2_{atm} \sim 3 \times 10^{-3}$ eV$^2$. Here we investigate the corrections to this approximation arising from the quantity $Δm^2_{sol}$ relevant for solar neutrino oscillations, assuming the large mixing angle solution. We show that for values of $\sin^2(2θ_{13}) \sim 10^{-2}$ (in the range of interest for long-baseline neutrino oscillation experiments with either intense conventional neutrino beams such as JHF-SuperK or a possible future neutrino factory) and for $Δm^2_{sol} \sim 10^{-4}$ eV$^2$, the contributions to $ν_μ\toν_e$ oscillations from both CP-conserving and CP-violating terms involving $\sin^2(Δm^2_{sol}L/(4E))$ can be comparable to the terms involving $\sin^2(Δm^2_{atm}L/(4E))$ retained in the one-$Δm^2$ approximation. Accordingly, we emphasize the importance of performing a full three-flavor, two-$Δm^2$ analysis of the data on $ν_μ\to ν_e$ oscillations in a conventional-beam experiment and $ν_e \to ν_μ$, $\barν_e \to \barν_μ$ oscillations at a neutrino factory. We also discuss a generalized analysis method for the KamLAND reactor experiment, and note how the information from this experiment can be used to facilitate the analysis of the subsequent data on $ν_μ\to ν_e$ oscillations. Finally, we consider the analysis of atmospheric neutrino data and present calculations of matter effects in a three-flavor, two-$Δm^2$ framework relevant to this data and to neutrino factory measurements.
We present exact calculations of the Potts model partition function $Z(G,q,v)$ for arbitrary $q$ and temperature-like variable $v$ on self-dual strip graphs $G$ of the square lattice with fixed width $L_y$ and arbitrarily great length $L_x$ with two types of boundary conditions. Letting $L_x \to \infty$, we compute the resultant free energy and complex-temperature phase diagram, including the locus ${\cal B}$ where the free energy is nonanalytic. Results are analyzed for widths $L_y=1,2,3$. We use these results to study the approach to the large-q limit of ${\cal B}$.
We consider the $q$-state Potts model on families of self-dual strip graphs $G_D$ of the square lattice of width $L_y$ and arbitrarily great length $L_x$, with periodic longitudinal boundary conditions. The general partition function $Z$ and the T=0 antiferromagnetic special case $P$ (chromatic polynomial) have the respective forms $\sum_{j=1}^{N_{F,L_y,λ}} c_{F,L_y,j} (λ_{F,L_y,j})^{L_x}$, with $F=Z,P$. For arbitrary $L_y$, we determine (i) the general coefficient $c_{F,L_y,j}$ in terms of Chebyshev polynomials, (ii) the number $n_F(L_y,d)$ of terms with each type of coefficient, and (iii) the total number of terms $N_{F,L_y,λ}$. We point out interesting connections between the $n_Z(L_y,d)$ and Temperley-Lieb algebras, and between the $N_{F,L_y,λ}$ and enumerations of directed lattice animals. Exact calculations of $P$ are presented for $2 \le L_y \le 4$. In the limit of infinite length, we calculate the ground state degeneracy per site (exponent of the ground state entropy), $W(q)$. Generalizing $q$ from ${\mathbb Z}_+$ to ${\mathbb C}$, we determine the continuous locus ${\cal B}$ in the complex $q$ plane where $W(q)$ is singular. We find the interesting result that for all $L_y$ values considered, the maximal point at which ${\cal B}$ crosses the real $q$ axis, denoted $q_c$ is the same, and is equal to the value for the infinite square lattice, $q_c=3$. This is the first family of strip graphs of which we are aware that exhibits this type of universality of $q_c$.
We calculate Jones polynomials $V_L(t)$ for several families of alternating knots and links by computing the Tutte polynomials $T(G,x,y)$ for the associated graphs $G$ and then obtaining $V_L(t)$ as a special case of the Tutte polynomial. For each of these families we determine the zeros of the Jones polynomial, including the accumulation set in the limit of infinitely many crossings. A discussion is also given of the calculation of Jones polynomials for non-alternating links.
We discuss the high-energy behavior of cross sections in theories with large extra dimensions and low-scale quantum gravity, addressing two particular issues: (i) the tension of the D-branes, and (ii) bounds on the cross section and their relation to approximations in the mode sum over Kaluza-Klein-graviton exchanges.
We present exact solutions for the zero-temperature partition function of the $q$-state Potts antiferromagnet (equivalently, the chromatic polynomial $P$) on tube sections of the simple cubic lattice of fixed transverse size $L_x \times L_y$ and arbitrarily great length $L_z$, for sizes $L_x \times L_y = 2 \times 3$ and $2 \times 4$ and boundary conditions (a) $(FBC_x,FBC_y,FBC_z)$ and (b) $(PBC_x,FBC_y,FBC_z)$, where $FBC$ ($PBC$) denote free (periodic) boundary conditions. In the limit of infinite-length, $L_z \to \infty$, we calculate the resultant ground state degeneracy per site $W$ (= exponent of the ground-state entropy). Generalizing $q$ from ${\mathbb Z}_+$ to ${\mathbb C}$, we determine the analytic structure of $W$ and the related singular locus ${\cal B}$ which is the continuous accumulation set of zeros of the chromatic polynomial. For the $L_z \to \infty$ limit of a given family of lattice sections, $W$ is analytic for real $q$ down to a value $q_c$. We determine the values of $q_c$ for the lattice sections considered and address the question of the value of $q_c$ for a $d$-dimensional Cartesian lattice. Analogous results are presented for a tube of arbitrarily great length whose transverse cross section is formed from the complete bipartite graph $K_{m,m}$.
We present exact calculations of the partition function of the q-state Potts model for general q and temperature on strips of the square lattice of width L_y=3 vertices and arbitrary length L_x with periodic longitudinal boundary conditions, of the following types: (i) (FBC_y,PBC_x)= cyclic, (ii) (FBC_y,TPBC_x)= Möbius, (iii) (PBC_y,PBC_x)= toroidal, and (iv) (PBC_y,TPBC_x)= Klein bottle, where FBC and (T)PBC refer to free and (twisted) periodic boundary conditions. Results for the L_y=2 torus and Klein bottle strips are also included. In the infinite-length limit the thermodynamic properties are discussed and some general results are given for low-temperature behavior on strips of arbitrarily great width. We determine the submanifold in the {\mathbb C}^2 space of q and temperature where the free energy is singular for these strips. Our calculations are also used to compute certain quantities of graph-theoretic interest.
We present exact calculations of the partition function of the $q$-state Potts model on (i) open, (ii) cyclic, and (iii) Möbius strips of the honeycomb (brick) lattice of width $L_y=2$ and arbitrarily great length. In the infinite-length limit the thermodynamic properties are discussed. The continuous locus of singularities of the free energy is determined in the $q$ plane for fixed temperature and in the complex temperature plane for fixed $q$ values. We also give exact calculations of the zero-temperature partition function (chromatic polynomial) and $W(q)$, the exponent of the ground-state entropy, for the Potts antiferromagnet for honeycomb strips of type (iv) $L_y=3$, cyclic, (v) $L_y=3$, Möbius, (vi) $L_y=4$, cylindrical, and (vii) $L_y=4$, open. In the infinite-length limit we calculate $W(q)$ and determine the continuous locus of points where it is nonanalytic. We show that our exact calculation of the entropy for the $L_y=4$ strip with cylindrical boundary conditions provides an extremely accurate approximation, to a few parts in $10^5$ for moderate $q$ values, to the entropy for the full 2D honeycomb lattice (where the latter is determined by Monte Carlo measurements since no exact analytic form is known).
We discuss implications of improved upper bounds on the $|ΔL|=2$ processes (i) $K^+ \to π^- μ^+ μ^+$, from an experiment at BNL, and (ii) $μ^- \to e^+$ conversion, from an experiment at PSI. In particular, we address the issue of constraints on neutrino masses and mixing, and on supersymmetric models with R-parity violation.
We present exact calculations of partition function $Z$ of the $q$-state Potts model with next-nearest-neighbor spin-spin couplings, both for the ferromagnetic and antiferromagnetic case, for arbitrary temperature, on $n$-vertex strip graphs of width $L_y=2$ of the square lattice with free, cyclic, and Möbius longitudinal boundary conditions. The free energy is calculated exactly for the infinite-length limit of these strip graphs and the thermodynamics is discussed. Considering the full generalization to arbitrary complex $q$ and temperature, we determine the singular locus ${\cal B}$ in the corresponding ${\mathbb C}^2$ space, arising as the accumulation set of partition function zeros as $n \to \infty$.
We calculate matter effects on neutrino oscillations relevant for long baseline experiments. In particular, we compare the results obtained with constant density along the neutrino path versus results obtained by incorporating the actual density profiles in the Earth. We study the dependence of the oscillation signal on both $E/Δm^2_{atm}$ and on the angles in the leptonic mixing matrix. We also comment on the influence of $Δm^2_{sol}$ on the oscillations. The results show quantitatively how, as a function of these input parameters, matter effects can cause significant (25 %) changes in the oscillation probabilities. An important conclusion is that matter effects can be useful in amplifying certain neutrino oscillation signals and helping one to obtain measurements of mixing parameters and the magnitude and sign of $Δm^2_{atm}$.
partial abstract: The $q$-state Potts model partition function (equivalent to the Tutte polynomial) for a lattice strip of fixed width $L_y$ and arbitrary length $L_x$ has the form $Z(G,q,v)=\sum_{j=1}^{N_{Z,G,λ}}c_{Z,G,j}(λ_{Z,G,j})^{L_x}$, where $v$ is a temperature-dependent variable. The special case of the zero-temperature antiferromagnet ($v=-1$) is the chromatic polynomial $P(G,q)$. Using coloring and transfer matrix methods, we give general formulas for $C_{X,G}=\sum_{j=1}^{N_{X,G,λ}}c_{X,G,j}$ for $X=Z,P$ on cyclic and Möbius strip graphs of the square and triangular lattice. Combining these with a general expression for the (unique) coefficient $c_{Z,G,j}$ of degree $d$ in $q$: $c^{(d)}=U_{2d}(\frac{\sqrt{q}}{2})$, where $U_n(x)$ is the Chebyshev polynomial of the second kind, we determine the number of $λ_{Z,G,j}$'s with coefficient $c^{(d)}$ in $Z(G,q,v)$ for these cyclic strips of width $L_y$ to be $n_Z(L_y,d)=(2d+1)(L_y+d+1)^{-1} {2L_y \choose L_y-d}$ for $0 \le d \le L_y$ and zero otherwise. For both cyclic and Möbius strips of these lattices, the total number of distinct eigenvalues $λ_{Z,G,j}$ is calculated to be $N_{Z,L_y,λ}={2L_y \choose L_y}$. We point out that $N_{Z,L_y,λ}=2N_{DA,tri,L_y}$ and $N_{P,L_y,λ}=2N_{DA,sq,L_y}$, where $N_{DA,Λ,n}$ denotes the number of directed lattice animals on the lattice $Λ$.
We present exact calculations of the zero-temperature partition function (chromatic polynomial) and $W(q)$, the exponent of the ground-state entropy, for the $q$-state Potts antiferromagnet with next-nearest-neighbor spin-spin couplings on square lattice strips, of width $L_y=3$ and $L_y=4$ vertices and arbitrarily great length $L_x$ vertices, with both free and periodic boundary conditions. The resultant values of $W$ for a range of physical $q$ values are compared with each other and with the values for the full 2D lattice. These results give insight into the effect of such non-nearest neighbor couplings on the ground state entropy. We show that the $q=2$ (Ising) and $q=4$ Potts antiferromagnets have zero-temperature critical points on the $L_x \to \infty$ limits of the strips that we study. With the generalization of $q$ from ${\mathbb Z}_+$ to ${\mathbb C}$, we determine the analytic structure of $W(q)$ in the $q$ plane for the various cases.
We present exact solutions for the zero-temperature partition function (chromatic polynomial $P$) and the ground state degeneracy per site $W$ (= exponent of the ground-state entropy) for the $q$-state Potts antiferromagnet on strips of the square lattice of width $L_y$ vertices and arbitrarily great length $L_x$ vertices. The specific solutions are for (a) $L_y=4$, $(FBC_y,PBC_x)$ (cyclic); (b) $L_y=4$, $(FBC_y,TPBC_x)$ (Möbius); (c) $L_y=5,6$, $(PBC_y,FBC_x)$ (cylindrical); and (d) $L_y=5$, $(FBC_y,FBC_x)$ (open), where $FBC$, $PBC$, and $TPBC$ denote free, periodic, and twisted periodic boundary conditions, respectively. In the $L_x \to \infty$ limit of each strip we discuss the analytic structure of $W$ in the complex $q$ plane. The respective $W$ functions are evaluated numerically for various values of $q$. Several inferences are presented for the chromatic polynomials and analytic structure of $W$ for lattice strips with arbitrarily great $L_y$. The absence of a nonpathological $L_x \to \infty$ limit for real nonintegral $q$ in the interval $0 < q < 3$ ($0 < q < 4$) for strips of the square (triangular) lattice is discussed.
We present exact calculations of the zero-temperature partition function (chromatic polynomial) $P$ for the $q$-state Potts antiferromagnet on triangular lattice strips of arbitrarily great length $L_x$ vertices and of width $L_y=3$ vertices and, in the $L_x \to \infty$ limit, the exponent of the ground-state entropy, $W=e^{S_0/k_B}$. The strips considered, with their boundary conditions ($BC$) are (a) $(FBC_y,PBC_x)=$ cyclic, (b) $(FBC_y,TPBC_x)=$ Möbius, (c) $(PBC_y,PBC_x)=$ toroidal, and (d) $(PBC_y,TPBC_x)=$ Klein bottle, where $F$, $P$, and $TP$ denote free, periodic, and twisted periodic. Exact calculations of $P$ and $W$ are also given for wider strips, including (e) cyclic, $L_y=4$, and (f) $(PBC_y,FBC_x)=$ cylindrical, $L_y=5,6$. Several interesting features are found, including the presence of terms in $P$ proportional to $\cos(2πL_x/3)$ for case (c). The continuous locus of points ${\cal B}$ where $W$ is nonanalytic in the $q$ plane is discussed for each case and a comparative discussion is given of the respective loci ${\cal B}$ for families with different boundary conditions. Numerical values of $W$ are given for infinite-length strips of various widths and are shown to approach values for the 2D lattice rapidly. A remark is also made concerning a zero-free region for chromatic zeros.
In this paper we present exact calculations of the partition function $Z$ of the $q$-state Potts model and its generalization to real $q$, for arbitrary temperature on $n$-vertex strip graphs, of width $L_y=2$ and arbitrary length, of the triangular lattice with free, cyclic, and Möbius longitudinal boundary conditions. These partition functions are equivalent to Tutte/Whitney polynomials for these graphs. The free energy is calculated exactly for the infinite-length limit of the graphs, and the thermodynamics is discussed. Considering the full generalization to arbitrary complex $q$ and temperature, we determine the singular locus ${\cal B}$ in the corresponding ${\mathbb C}^2$ space, arising as the accumulation set of partition function zeros as $n \to \infty$. In particular, we study the connection with the T=0 limit of the Potts antiferromagnet where ${\cal B}$ reduces to the accumulation set of chromatic zeros. Comparisons are made with our previous exact calculation of Potts model partition functions for the corresponding strips of the square lattice. Our present calculations yield, as special cases, several quantities of graph-theoretic interest.
Let $P(G,q)$ be the chromatic polynomial for coloring the $n$-vertex graph $G$ with $q$ colors, and define $W=\lim_{n \to \infty}P(G,q)^{1/n}$. Besides their mathematical interest, these functions are important in statistical physics. We give a comparative discussion of exact calculations of $P$ and $W$ for a variety of recursive families of graphs, including strips of regular lattices with various boundary conditions and homeomorphic expansions thereof. Generalizing to $q \in {\mathbb C}$, we determine the accumulation sets of the chromatic zeros constituting the continuous loci of points on which $W$ is nonanalytic. Various families of graphs with the property that the chromatic zeros and/or their accumulation sets (i) include support for $Re(q) < 0$; (ii) bound regions and pass through $q=0$; and (iii) are noncompact are discussed, and the role of boundary conditions is analyzed. Some corresponding results are presented for Potts model partition functions for nonzero temperature, equivalent to the full Tutte polynomials for various families of graphs.