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Exact Solution for Three-Dimensional Ising Model

Zhang, Degang分类： 物理学 >> 普通物理:统计和量子力学,量子信息等

Three-dimensional Ising model in zero external field is exactly solved by operator algebras, similar to the Onsager's approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along both $(0 1 0)$ and $(0 0 1)$ directions and the screw boundary condition along the $(1 0 0)$ direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. A order-disorder transition in the infinite crystal occurs at a temperature $T=T_c$ determined by the condition: $\sinh\frac{2J}{k_BT_c}\sinh\frac{2(J_1+J_2)}{k_BT_c}=1$, where $(J_1 J_2 J)$ are the interaction energies in three directions, respectively. The analytical expressions for the internal energy and the specific heat are also given. It is also shown that the thermodynamic properties of 3D Ising model with $J_1=J_2$ are connected to those in 2D Ising model with the interaction energies $(J_1 J_{2D})$ by the relation $(\frac{J_{2D}}{k_BT})^*=(\frac{J}{k_BT})^*-\frac{J_1}{k_BT}$, where $x^*=\frac{1}{2}{\rm ln coth} x={\rm tanh}^{-1}(e^{-2x})$. |

Mayer-Cluster Expansion of Instanton Partition Functions and Thermodynamic Bethe Ansatz

Carlo Meneghelli; Gang Yang分类： 物理学 >> 基本粒子与场物理学

In [18] Nekrasov and Shatashvili pointed out that the N = 2 instanton partition function in a special limit of the Ω-deformation parameters is characterized by certain thermodynamic Bethe ansatz (TBA) like equations. In this work we present an explicit derivation of this fact as well as generalizations to quiver gauge theories. To do so we combine various techniques like the iterated Mayer expansion, the method of expansion by regions, and the path integral tricks for non-perturbative summation. The TBA equations derived entirely within gauge theory have been proposed to encode the spectrum of a large class of quantum integrable systems. We hope that the derivation presented in this paper elucidates further this completely new point of view on the origin, as well as on the structure, of TBA equations in integrable models. |

Cutting through form factors and cross sections of non-protected operators in N = 4 SYM

Dhritiman Nandan; Christoph Sieg; Matthias Wilhelm; Gang Yang分类： 物理学 >> 基本粒子与场物理学

: We study the form factors of the Konishi operator, the prime example of nonprotected operators in N = 4 SYM theory, via the on-shell unitarity method. Since the Konishi operator is not protected by supersymmetry, its form factors share many features with amplitudes in QCD, such as the occurrence of rational terms and of UV divergences that require renormalization. A subtle point is that this operator depends on the spacetime dimension. This requires a modification when calculating its form factors via the on-shell unitarity method. We derive a rigorous prescription that implements this modification to all loop orders and obtain the two-point form factor up to two-loop order and the three-point form factor to one-loop order. From these form factors, we construct an IR-finite crosssection-type quantity, namely the inclusive decay rate of the (off-shell) Konishi operator to any final (on-shell) state. Via the optical theorem, it is connected to the imaginary part of the two-point correlation function. We extract the Konishi anomalous dimension up to two-loop order from it. |

On-Shell Methods for the Two-Loop Dilatation Operator and Finite Remainders

Florian Loebbert; Dhritiman Nandan; Christoph Sieg; Matthias Wilhelm; Gang Yang分类： 物理学 >> 基本粒子与场物理学

We compute the two-loop minimal form factors of all operators in the SU(2) sector of planar N = 4 SYM theory via on-shell unitarity methods. From the UV divergence of this result, we obtain the two-loop dilatation operator in this sector. Furthermore, we calculate the corresponding finite remainder functions. Since the operators break the supersymmetry, the remainder functions do not have the property of uniform transcendentality. However, the leading transcendentality part turns out to be universal and is identical to the corresponding BPS expression. The remainder functions are shown to satisfy linear relations which can be explained by Ward identities of form factors following from R-symmetry |

Master integrals for the four-loop Sudakov form factor

Rutger H. Boels; Bernd A. Kniehl; , Gang Yang分类： 物理学 >> 基本粒子与场物理学

The light-like cusp anomalous dimension is a universal function in the analysis of infrared divergences. In maximally (N = 4) supersymmetric Yang– Mills theory (SYM) in the planar limit, it is known, in principle, to all loop orders. The non-planar corrections are not known in any theory, with the first appearing at the four-loop order. The simplest quantity which contains this correction is the four-loop two-point form factor of the stress tensor multiplet. This form factor was largely obtained in integrand form in a previous work for N = 4 SYM, up to a free parameter. In this work, a reduction of the appearing integrals obtained by solving integration-by-parts (IBP) identities using a modified version of Reduze is reported. The form factor is shown to be independent of the remaining parameter at integrand level due to an intricate pattern of cancellations after IBP reduction. Moreover, two of the integral topologies vanish after reduction. The appearing master integrals are cross-checked using independent algebraic-geometry techniques explored in the Mint package. The latter results provide the basis of master integrals applicable to generic form factors, including those in Quantum Chromodynamics. Discrepancies between explicitly solving the IBP relations and the MINT approach are highlighted. Remaining bottlenecks to completing the computation of the four-loop non-planar cusp anomalous dimension in N = 4 SYM and beyond are identified. |

Color-Kinematics Duality and Sudakov Form Factor at Five Loops

Gang Yang分类： 物理学 >> 基本粒子与场物理学

Using color-kinematics duality, we construct for the first time the full integrand of the five-loop Sudakov form factor in N = 4 super-Yang-Mills theory, including non-planar contributions. This result also provides a first manifestation of the color-kinematics duality at five loops. The integrand is explicitly ultraviolet finite when D < 26/5, coincident with the known finiteness bound for amplitudes. If the double-copy method could be applied to the form factor, this would indicate an interesting ultraviolet finiteness bound for N = 8 supergravity at five loops. The result is also expected to provide an essential input for computing the five-loop non-planar cusp anomalous dimension. |

Two-Loop SL(2) Form Factors and Maximal Transcendentality

Florian Loebbert; Christoph Sieg; Matthias Wilhelm; Gang Yang分类： 物理学 >> 基本粒子与场物理学

Form factors of composite operators in the SL(2) sector of N = 4 SYM theory are studied up to two loops via the on-shell unitarity method. The non-compactness of this subsector implies the novel feature and technical challenge of an unlimited number of loop momenta in the integrand’s numerator. At one loop, we derive the full minimal form factor to all orders in the dimensional regularisation parameter. At two loops, we construct the complete integrand for composite operators with an arbitrary number of covariant derivatives, and we obtain the remainder functions as well as the dilatation operator for composite operators with up to three covariant derivatives. The remainder functions reveal curious patterns suggesting a hidden maximal uniform transcendentality for the full form factor. Finally, we speculate about an extension of these patterns to QCD. |

The four-loop non-planar cusp anomalous dimension in N = 4 SYM

Rutger H. Boels; Tobias Huber; Gang Yang分类： 物理学 >> 基本粒子与场物理学

The light-like cusp anomalous dimension is a universal function that controls infrared divergences in quite general quantum field theories. In the maximally supersymmetric Yang-Mills theory this function is fixed fully by integrability to the three-loop order. At four loops a non-planar correction appears which we obtain for the first time from a numerical computation of the Sudakov form factor. Key ingredients are widely applicable methods to control the number-theoretic aspects of the appearing integrals. Our result shows explicitly that quadratic Casimir scaling breaks down at four loops. |

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