In this note we compute the non-planar one loop anomalous dimension of restricted Schur polynomials that belong to the sl(2) sector of N=4 super Yang-Mills theory and have a bare dimension of order N. Although the details are rather different, ultimately the problem of diagonalizing the dilatation operator in the sl(2) sector can be reduced to the su(2) sector problem. In this way we establish the expected dynamical emergence of the Gauss Law for giant gravitons and further show that the dilatation operator reduces to a set of decoupled harmonic oscillators.
We present the result of a full direct component calculation for the non-planar contribution to the four-loop anomalous dimension of the Konishi operator in N =4 Supersymmetric Yang-Mills theory. The result contains only zeta(5) term and is proportional to zeta(5) contribution in the planar case, which comes purely from wrapping corrections. We have extended also our previous calculations for the leading transcendental contribution arXiv:0811.0607 on non-planar case and have found the same results up to a common factor. It allows us to suggest that the non-planar contribution to the four-loop …
This review is devoted to collecting some results on the high spin expansion of (minimal) anomalous dimension. Thanks to the recent rationale on integrability, planar ${\cal N}=4$ super Yang-Mills theory (or its $\text{AdS}_5\times\text{S}^5$ string counterpart) represents a very practicable field. Here the attention will be restricted to its sector of twist operators, although the analysis tools are quite general (in integrable theories). Some structures and ideas turn out to be general also for other sectors or gauge theories.
We consider twist-3 operators in the sl(2) sector of N=4 SYM built out of three scalar fields with derivatives. We extract from the Bethe Ansatz equations of this sector the exact lowest anomalous dimension gamma(s) of scaling fields for several values of the operator spin s. We propose compact closed expressions for the spin dependence of gamma(s) up to the four loop level and show that they obey a simple new twist-3 transcendentality principle. As a check, we reproduce the four loop universal cusp anomalous dimension governing the logarithmic large spin limit of gamma(s).
It is shown how the BRST quantization can be applied to a gauge invariant sector of theories with anomalously broken symmetries. This result is used to show that shifting the anomalies to a classically trivial sector of fields (Wess-Zumino mechanism) makes it possible to quantize the physical sector using a standard BRST procedure, as for a non anomalous theory. The trivial sector plays the role of a topological sector if the system is quantized without shifting the anomalies.
We consider non-planar one-loop anomalous dimensions in maximally supersymmetric Yang-Mills theory and its marginally deformed analogues. Using the basis of Bethe states, we compute matrix elements of the dilatation operator and find compact expressions in terms of off-shell scalar products and hexagon-like functions. We then use non-degenerate quantum-mechanical perturbation theory to compute the leading $1/N^2$ corrections to operator dimensions and as an example compute the large $R$-charge limit for two-excitation states through subleading order in the $R$-charge. Finally, we numerically s…
We classify the 3-point functions of local gauge-invariant single-trace operators in the scalar sector of planar N=4 supersymmetric Yang-Mills involving at least one su(3) operator. In the case of two su(3) and one su(2) operators, the tree-level 3-point function can be expressed in terms of scalar products of su(3) Bethe vectors. Moreover, if the second level Bethe roots of one of the su(3) operators is trivial (set to infinity), this 3-point function can be written in a determinant form. Using the determinant representation, we evaluate the structure constant in the semi-classical limit, whe…
We compute the non-planar contribution to the anomalous dimension of the eight moment of the twist-2 operators in N=4 supersymmetric Yang-Mills theory at four loops. This result was obtained from the calculations of some elements of the anomalous dimension matrix for twist-2 operators with the help of a relations between its elements and eigenvalues. The properties of the obtained result and a possible relations with a planar results are discussed.
We have studied a (1+2)-dimensional Lorentz-violating model which is obtained from the dimensional reduction of the nonbirefringent sector of the CPT-even electrodynamics of the standard model extension (SME). The planar theory contains a gauge sector and a scalar sector which are linearly coupled by means of a Lorentz-invariance violating (LIV) vector, $S^μ$, while the kinetic terms of both sectors are affected by the components of a Lorentz-violating symmetric tensor, $κ^{μν}$. The energy-momentum tensor reveals that both sectors present energy stability for sufficiently small values of the …
The hidden sector of heterotic $M$-theory vacua whose gauge bundle contains an anomalous $U(1)$ factor is discussed in detail. The mathematical formalism for computing the inhomogeneous transformation of the dilaton and Kähler moduli axions under an anomalous $U(1)$ transformation is presented. Along with matter multiplets, which transform homogeneously under $U(1)$, the relevant part of the $U(1)$ invariant low energy hidden sector Lagrangian is presented and analyzed. A detailed mathematical formalism is given for rotating these field to a new basis of chiral superfields with normalized kine…
We compute the general form of the six-loop anomalous dimension of twist-two operators with arbitrary spin in planar N=4 SYM theory. First we find the contribution from the asymptotic Bethe ansatz. Then we reconstruct the wrapping terms from the first 35 even spin values of the full six-loop anomalous dimension computed using the quantum spectral curve approach. The obtained anomalous dimension satisfies all known constraints coming from the BFKL equation, the generalised double-logarithmic equation, and the small spin expansion.
This paper discusses the dilaton, Kähler moduli and hidden sector matter chiral superfields of heterotic $M$-theory vacua in which the hidden sector gauge bundle is chosen to be a line bundle with an anomalous U(1) structure group. For simplicity of notation, the theory is compactified on a Calabi-Yau threefold with $h^{1,1}=1$, although all methods and results apply to more general heterotic compactifications. After introducing a non-perturbative $F$-term potential and coupling to supergravity, the canonically normalized scalar and fermion mass eigenstates, evaluated around a fixed supersymme…
We compute the fermionic contribution to the photon-quark form factor to four-loop order in QCD in the planar limit in analytic form. From the divergent part of the latter the cusp and collinear anomalous dimensions are extracted. Results are also presented for the finite contribution. We briefly describe our method to compute all planar master integrals at four-loop order.
We compute the non-planar contribution to the universal anomalous dimension of twist-two operators in N=4 supersymmetric Yang-Mills theory at four loops through Lorentz spin eighteen. Exploiting the results of this and our previous calculations along with recent analytic results for the cusp anomalous dimension and some expected analytic properties, we reconstruct a general expression valid for arbitrary Lorentz spin. We study various properties of this general result, such as its large-spin limit, its small-x limit, and others. In particular, we present a prediction for the non-planar contrib…
We present the result of a full direct component calculation for the first three even moments of the non-planar contribution into the four-loop anomalous dimension of twist-2 operators in maximally extended N=4 supersymmetric Yang-Mills theory. Obtained result complete our previous calculations in arXiv:0902.4646 and gives the usual result for the higher moments on the contrary to degenerate one in the case of Konishi. We have proposed a general form of zeta(5) and zeta(3) parts of the full non-planar anomalous dimension of twist-2 operators. As by product, we have obtained the first moment of…
We study the integrability properties of planar N=2 superconformal field theories in four dimensions. We show that the spin chain associated to the planar dilation operator of N=2 superconformal QCD fails to be integrable at two loops. In our analysis we focus on a closed SU(2|1) sector, whose two-loop spin chain we fix by symmetry arguments (up to a few undetermined coefficients). It turns out that the Yang-Baxter equation for magnon scattering is not satisfied in this sector. On the other hand, we suggest that the closed SU(2,1|2) sector, which exists in any N=2 superconformal gauge theory, …
The anomaly implies an obstruction to a fully chiral covariant calculation of the effective action in the abnormal parity sector of chiral theories. The standard approach then is to reconstruct the anomalous effective action from its covariant current. In this work we use a recently introduced formulation which allows to directly construct the non trivial chiral invariant part of the effective action within a fully covariant formalism. To this end we develop an appropriate version of Chan's approach to carry out the calculation within the derivative expansion. The result to four derivatives, i…
Following a model recently investigated by Veneziano and Wosiek we briefly introduce Planar Quantum Mechanics (PQM). Then, we present high precision numerical results in the sectors with two and three fermions. We confirm, that the transition point in the 't Hooft's coupling constant lambda in these sectors occurs at lambda = 1, as was expected.
We compute the nonplanar contribution to the universal anomalous dimension of the SU(4)-singlet twist-two operators in N=4 supersymmetric Yang-Mills theory at four loops through Lorentz spin 18. From this, we numerically evaluate the nonplanar contribution to the four-loop lightlike cusp anomalous dimension and derive the transcendental $ζ_3$ and $ζ_5$ parts of the universal anomalous dimension for arbitrary Lorentz spin in analytic form. As for the lightlike cusp anomalous dimension and the $ζ_5$ part of the universal anomalous dimension, we confirm previous results.
We study the maximally-helicity-violating (MHV) six-gluon scattering amplitude in planar $\mathcal{N} = 4$ super-Yang-Mills theory at finite coupling when all three cross ratios are small. It exhibits a double logarithmic scaling in the cross ratios, controlled by a handful of "anomalous dimensions" that are functions of the coupling constant alone. Inspired by known seven-loop results at weak coupling and the integrability-based pentagon OPE, we present conjectures for the all-order resummation of these anomalous dimensions. At strong coupling, our predictions agree perfectly with the string …
In this article we calculate the five-loop anomalous dimension of twist-two operators in the planar N=4 SYM theory. Firstly, using reciprocity, we derive the contribution of the asymptotic Bethe ansatz. Subsequently, we employ the first finite-size correction for the AdS5xS5 sigma model to determine the wrapping correction. The anomalous dimension found in this way passes all known tests provided by the NLO BFKL equation and double-logarithmic constraints. This result thus furnishes an infinite number of experimental data for testing the veracity of the recently proposed spectral equations for…
In this work analyze the physical consistency of a nonbirefringent Lorentz-violating planar model via the analysis of the pole structure of its Feynman propagators. The nonbirefringent planar model, obtained from the dimensional reduction of the CPT-even gauge sector of the standard model extension, is composed of a gauge and a scalar fields, being affected by Lorentz-violating (LIV) coefficients encoded in the symmetric tensor $κ_{μν}$. The propagator of the gauge field is explicitly evaluated and expressed in terms of linear independent symmetric tensors, presenting only one physical mode. T…
We study noncommutative field theories at finite temperature to learn more about the degrees of freedom in the non-planar sector of these systems. We find evidence for winding states. At temperatures for which the thermal wavelength is smaller than the noncommutativity scale, there is a drastic reduction of the degrees of freedom in the non-planar sector. In this regime, the non-planar sector has thermodynamics resembling that of a 1+1 dimensional field theory.
We show that supersymmetry Ward identities contain an anomalous term which takes the form of a surface term in Hilbert space. In the one-instanton sector the anomalous term is the integral of a total rho-derivative where rho is the instanton's size. There are cases where the anomalous term is non-zero, and cannot be modified by subtractions. This constitutes a supersymmetry anomaly. The derivation is based on Feynman rules suitable for any non-perturbative sector of a weakly-coupled, renormalizable gauge theory.
We add the Wess-Zumino-Witten term to the N=3 massive nonlinear sigma model and study the leading logarithms in the anomalous sector. We obtain the leading logarithms to six loops for π^0 --> γ^*γ^* and to five loops for γ^*πππ. In addition we extend the earlier work on the mass and decay constant to six loops and the vector form factor to five loops. We present numerical result for the anomalous processes and the vector form factor. In all cases the series are found to converge rapidly.
The CPT-even abelian gauge sector of the Standard Model Extension is represented by the Maxwell term supplemented by $(K_{F})_{μνρσ}F^{μν}F^{ρσ}$, where the Lorentz-violating background tensor, $(K_{F})_{μνρσ}$, possesses the symmetries of the Riemann tensor. In the present work, we examine the planar version of this theory, obtained by means of a typical dimensional reduction procedure to $(1+2)$ dimensions. The resulting planar electrodynamics is composed of a gauge sector containing six Lorentz-violating coefficients, a scalar field endowed with a noncanonical kinetic term, and a coupling t…
In the β-deformed N=4 supersymmetric SU(N) Yang-Mills theory we study the class of operators O_J = Tr(Φ_i^J Φ_k), i\neq k and compute their exact anomalous dimensions for N,J\to\infty. This leads to a prediction for the masses of the corresponding states in the dual string theory sector. We test the exact formula perturbatively up to two loops. The consistency of the perturbative calculation with the exact result indicates that in the planar limit the one--loop condition g^2=h\bar{h} for superconformal invariance is indeed sufficient to insure the {\em exact} superconformal invariance of the t…
Bethe ansatz equations have been proposed for the asymptotic spectral problem of AdS_4/CFT_3. This proposal assumes integrability, but the previous verification of weak-coupling integrability covered only the su(4) sector of the ABJM gauge theory. Here we derive the complete planar two-loop dilatation generator of N=6 superconformal Chern-Simons theory from osp(6|4) superconformal symmetry. For the osp(4|2) sector, we prove integrability through a Yangian construction. We argue that integrability extends to the full planar two-loop dilatation generator, confirming the applicability of the Beth…
We derive the O(p6) Chiral Perturbation Theory Lagrangian in the massless quark limit for a class of gravity dual models of Quantum Chromodynamics with the chiral symmetry broken through boundary conditions. The odd O(p6) couplings are related to the O(p4) low-energy constants (LEC's) in the even-parity sector. Some combinations of even O(p6) couplings are found to be universal and independent of the peculiarities of the model. These relations turn out to be the manifestation at low energies of a broader relation between anomalous and even-parity amplitudes.
The result for the six-loop anomalous dimension of twist-three operators in the planar N=4 SYM theory is presented. The calculations were performed along the paper arXiv:0912.1624. This result provides a new data for testing the proposed spectral equations for planar AdS/CFT correspondence.
We study spin models underlying the non-planar dynamics of ${\cal N}=4$ SYM gauge theory. In particular, we derive the non-local spin chain Hamiltonian generating dilatations in the gauge theory at leading order in $g_{\rm YM}^2 N$ but exact in ${1\over N}$. States in the spin chain are characterized by a spin-configuration and a linking variable describing how sites in the chain are connected. Joining and splitting of string/traces are mimicked by a twist operator acting on the linking variable. The results are applied to a systematic study of non-planar anomalous dimensions and operator mixi…
We review the role of integrability in certain aspects of N=4 SYM which go beyond the planar spectrum. In particular, we discuss integrability in relation to non-planar anomalous dimensions, multi-point functions and Maldacena-Wilson loops.
We present results for the planar contribution to the four-loop anomalous dimension for a general N=2 supersymmetric Chern-Simons theory in three dimensions. These results should facilitate higher-order superconformality checks for theories relevant for the AdS/CFT correspondence.
We show how to generalize the $SU(2)$ WZW models to allow for open and unoriented sectors. The construction exhibits some novel patterns of Chan-Paton charge assignments and projected spectra that reflect the underlying current algebra.
We consider a BMN operator with one scalar, phi, and one vector, D_{m}Z, impurity field and compute the anomalous dimension both at planar and torus levels. This "mixed" operator corresponds to a string state with two creation operators which belong to different SO(4) sectors of the background. The anomalous dimension at both levels is found to be the same as the scalar impurity BMN operator. At planar level this constitutes a consistency check of BMN conjecture. Agreement at the torus level can be explained by an argument using supersymmetry and supression in the BMN limit. The same argument …
The typical transverse momentum $Q_s(t)$ (or "saturation" momentum) acquired by a hard particle propagating through a $\mathcal{N}=4$ SYM plasma increases over time like $t^γ$, with an anomalous exponent $γ>1/2$ characteristic of super-diffusion. This anomalous exponent is a function of the 't Hooft coupling $λ=g^2N_c$. Recently, a method has been proposed to systematically compute the perturbative series of $γ(λ)$ at weak coupling. This method relies on the traveling wave interpretation of the time evolution of $Q_s(t)$ and on the dominance of soft-collinear radiative corrections at large tim…
Using path integral method (Fujikawa's method) we calculate anomalies in noncommutative gauge theories with fermions in the bi-fundamental and adjoint representations. We find that axial and chiral gauge anomalies coming from non-planar contributions are derived in the low noncommutative momentum limit $\widetilde{p}^μ(\equiv θ^{μν}p_ν) \to 0$. The adjoint chiral fermion carries no anomaly in the non-planar sector in $D=4k (k=1,2,...,)$ dimensions. It is naturally shown from the path integral method that anomalies in non-planar sector originate in UV/IR mixing.
We start from a Lorentz non-invariant Abelian-Higgs model in 1+3 dimensions, and carry out its dimensional reduction to $D=1+2$. The planar model resulting thereof is composed by a Maxwell-Chern-Simons-Proca gauge sector, a massive scalar sector, and a mixing term (involving the fixed background, $v^μ$) that realizes Lorentz violation for the reduced model. Vortex-type solutions of the planar model are investigated,revealing charged vortex configurations that recover the usual Nielsen-Olesen configuration in the asymptotic regime. The Aharonov-Casher Effect in layered superconductors, that sho…
Within the context of $N=1$ supersymmetric heterotic $M$-theory, we present a "freeze-in" mechanism for producing dark matter via a "moduli portal" between the observable and hidden sectors. It is assumed that the observable sector consists of the MSSM or some physically acceptable extension of it, while the hidden sector is chosen so as to satisfy all physical and mathematical constraints. Dark matter production processes are examined for two fundamental types of hidden sectors; those whose gauge bundle structure group contains an anomalous $U(1)$ and those whose structure group is non-Abelia…
We show that renormalized non-commutative scalar field theories do not reduce to their planar sector in the limit of large non-commutativity. This follows from the fact that the RG equation of the Wilson-Polchinski type which describes the genus zero sector of non-commutative field theories couples generic planar amplitudes with non-planar amplitudes at exceptional values of the external momenta. We prove that the renormalization problem can be consistently restricted to this set of amplitudes. In the resulting renormalized theory non-planar divergences are treated as UV divergences requiring …
We present the full analytic result for the three-loop angle-dependent cusp anomalous dimension in QCD. With this result, infrared divergences of planar scattering processes with massive particles can be predicted to that order. Moreover, we define a closely related quantity in terms of an effective coupling defined by the light-like cusp anomalous dimension. We find evidence that this quantity is universal for any gauge theory, and use this observation to predict the non-planar $n_{f}$-dependent terms of the four-loop cusp anomalous dimension.
Anomalous Dimension of a General Effective Gauge Theory II: Fermionic Sector
PSI Center for Neutron and Muon Sciences（瑞士保罗谢勒研究所中子与缪子科学中心）；Dipartimento di Fisica e Astronomia “G. Galilei”, Università degli Studi di Padova（帕多瓦大学伽利略物理与天文学系）；Istituto Nazionale di Fisica Nucleare, Sezione di Padova（意大利国家核物理研究所帕多瓦分部）
We consider four-dimensional N=2 superconformal field theories based on ADE quiver diagrams. We use the procedure of hep-th/0206079 and compute the exact anomalous dimensions of operators with large U(1)_R charge to all orders in perturbation in the planar limit. The results are in agreement with the string computation in the dual pp-wave backgrounds.
We study structure constants of gauge invariant operators in planar N=4 Yang-Mills at one loop with the motivation of determining features of the string dual of weak coupling Yang-Mills. We derive a simple renormalization group invariant formula characterizing the corrections to structure constants of any primary operator in the planar limit. Applying this to the scalar SO(6) sector we find that the one loop corrections to structure constants of gauge invariant operators is determined by the one loop anomalous dimension Hamiltonian in this sector. We then evaluate the one loop corrections to s…
We study a gravitational model whose vacuum sector is invariant under conformal transformations. In this model we investigate the anomalous gravitational coupling of the large-scale matter. In this kind of coupling the large-sale matter is taken to couple to a metric which is different but conformally related to the metric appearing explicitly in the vacuum sector. The effect of the conformal symmetry breaking of the large-scale matter would lead in general to a variable strength of the anomalous gravitational coupling. This feature is used to derive a relativistic particle concept which share…
We use the principle of maximal transcendentality and the universal nature of subleading infrared poles to extract the analytic value of the four-loop collinear anomalous dimension in planar ${\cal N}=4$ super-Yang-Mills theory from recent QCD results, obtaining $\hat{\cal G}_{0}^{(4)} = - 300 ζ_7 - 256 ζ_2 ζ_5 - 384 ζ_3 ζ_4$. This value agrees with a previous numerical result to within 0.2 percent. It also provides the Regge trajectory, threshold soft anomalous dimension and rapidity anomalous dimension through four loops.
Positivity properties of observables in planar maximally supersymmetric Yang-Mills theory
Max-Planck-Institut für Physik（马克斯·普朗克物理研究所）；Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München（慕尼黑大学阿诺德·索末菲理论物理中心）
Anomalous Dimension of a General Effective Gauge Theory I: Bosonic Sector
CERN（欧洲核子研究中心）；Dipartimento di Fisica e Astronomia “G. Galilei”, Università degli Studi di Padova（帕多瓦大学物理与天文学系）；Istituto Nazionale di Fisica Nucleare, Sezione di Padova（意大利国家核物理研究所帕多瓦分部）
The familiar trace identity associated with the scale transformation xxxx on the Lagrangian density for a noninteracting massive real scalar field in 2 + 1 dimensions is shown to be violated on a single plate on which the Dirichlet boundary condition xxxx is imposed.It is however respected in : i. 1 + 1 dimensions in both free space and on a single plate on which the Dirichlet boundary condition xxxx holds; and, ii. in 2 + 1 dimensions in free space, i.e. the unconstrained configuration.On the plate where xxxx, the modified trace identity is shown to be anomalous with a numerical coefficient f…
In this note we compute the non-planar one loop anomalous dimension of restricted Schur polynomials that have a bare dimension of O(N). This is achieved by mapping the restricted Schur polynomials into states of a specific U(N) irreducible representation. In this way the dilatation operator is mapped into a u(n) valued operator and, as a result, can easily be diagonalized. The resulting spectrum is reproduced by a classical model of springs between masses.
Letting the mass depend on the spin-field coupling as $M^2=m^2-(eg/2c^2)F_{αβ}S^{αβ}$, we propose a new set of relativistic planar equations of motion for spinning anyons. Our model can accommodate any gyromagnetic ratio $g$ and provides us with a novel version of the Bargmann-Michel-Telegdi equations in 2+1 dimensions. The system becomes singular when the field takes a critical value, and, for $g\neq2$, the only allowed motions are those which satisfy the Hall law. For each $g\neq2,0$ a secondary Hall effect arises also for another critical value of the field. The non-relativistic limit of ou…
We compute the anomalous dimension of a length-five operator at five-loop order in the SU(2) sector of N=4 SYM theory in the planar limit. This is critical wrapping order at five loops. The result is obtained perturbatively by means of N=1 superspace techniques. Our result from perturbation theory confirms explicitly the formula conjectured in arXiv:0901.4864 for the five-loop anomalous dimension of twist-three operators. We also explicitly obtain the same result by employing the recently proposed Y-system.
In this article we study the action of the one loop dilatation operator on operators with a classical dimension of order N. These operators belong to the su(2) sector and are constructed using two complex fields Y and Z. For these operators non-planar diagrams contribute already at the leading order in N and the planar and large N limits are distinct. The action of the one loop and the two loop dilatation operator reduces to a set of decoupled oscillators and factorizes into an action on the Z fields and an action on the Y fields. Direct computation has shown that the action on the Y fields is…
We report on a systematic study of scalar field three-point functions in planar SU(N) N=4 super Yang-Mills theory. The motivation for this work is to provide sufficient data for future conjectures on the higher-loop form of the structure constants possibly involving integrability. For this we have computed a sample of 70 structure constants at one-loop order involving primary operators of up to and including length five built entirely from scalar fields. We observe in all 17 cases occurring in our sample that the one-loop structure constant of two protected chiral primary operators and one unp…
We construct a new representation for two- and three-point correlators of operators from sl(2) sector of planar N = 4 SYM. The spin and twist of operators are arbitrary. We start with the correlation function of light-ray operators and carry out a projection to particular local operators using the method of Separated Variables. With the same calculation we obtain polynomials which are dual to wave functions of sl(2,R) spin-chain.
We show that dual conformal symmetry, mainly studied in planar $\mathcal N = 4$ super-Yang-Mills theory, has interesting consequences for Feynman integrals in nonsupersymmetric theories such as QCD, including the nonplanar sector. A simple observation is that dual conformal transformations preserve unitarity cut conditions for any planar integrals, including those without dual conformal symmetry. Such transformations generate differential equations without raised propagator powers, often with the right hand side of the system proportional to the dimensional regularization parameter $ε$. A nont…
