arXiv subjects

Daniel Z. Freedman

Publications and source records attributed to Daniel Z. Freedman.

68 records · Page 4Linked to original sources

arXiv1998

Gauge Boson Exchange in $AdS_{d+1}$

We study the amplitude for exchange of massless gauge bosons between pairs of massive scalar fields in Anti-de Sitter space. In the AdS/CFT correspondence this amplitude describes the contribution of conserved flavor symmetry currents to 4-point functions of scalar operators in the boundary conformal theory. A concise, covariant (Y2K-compatible) derivation of the gauge boson propagator in $\AdS_{d+1}$ is given. Techniques are developed to calculate the two bulk integrals over AdS space leading to explicit expressions or convenient, simple integral representations for the amplitude. The amplitude contains leading power and sub-leading logarithmic singularities in the gauge boson channel and leading logarithms in the crossed channel. The new methods of this paper are expected to have other applications in the study of the Maldacena conjecture.

hep-th↗

arXiv1998

General Scalar Exchange in AdS_d+1

The scalar field exchange diagram for the correlation function of four scalar operators is evaluated in anti-de Sitter space, $AdS_{d+1}$. The conformal dimensions $Δ_i$, $i=1,...,4$ of the scalar operators and the dimension $Δ$ of the exchanged field are arbitrary, constrained only to obey the unitarity bound. Techniques similar to those developed earlier for gauge boson exchange are used, but results are generally more complicated. However, for integer $Δ_i, Δ$, the amplitude can be presented as a multiple derivative of a simple universal function. Results simplify if further conditions hold, such as the inequalities, $Δ< Δ_1+Δ_3$ or $Δ<Δ_2+Δ_4$. These conditions are satisfied, with $<$ replaced by $\le$, in Type IIB supergravity on $AdS_5\times S_5$ because of selection rules from SO(6) symmetry. A new form of interaction is suggested for the marginal case of the inequalities. The short distance asymptotics of the amplitudes are studied. In the direct channel the leading singular term agrees with the double operator product expansion. Logarithmic singularities occur at sub-leading order in the direct channel but at leading order in the crossed channel. When the inequalities above are violated, there are also $(\log)^2$ singularities in the direct channel.

hep-th↗

arXiv1998

Comments on 4-point functions in the CFT/AdS correspondence

We study the four--point function of chiral primaries corresponding to the dilaton--axion sector in supergravity in the $AdS_5$/CFT$_4$ correspondence. We find relations between some of the supergravity graphs and compute their leading singularities. We discuss the issue of logarithmic singularities and their significance for the OPE structure of the CFT.

hep-th↗

arXiv1998

Correlation functions in the CFT(d)/AdS(d+1) correpondence

Conformal techniques are applied to the calculation of integrals on AdS(d+1) space which define correlators of composite operators in the superconformal field theory on the d-dimensional boundary. The 3-point amplitudes for scalar fields of arbitrary mass and gauge fields in the AdS supergravity are calculated explicitly. For 3 gauge fields we compare in detail with the known conformal structure of the SU(4) flavor current correlator of the N=4, d=4 SU(N) SYM theory. Results agree with the free field approximation as would be expected from superconformal non-renormalization theorems. In studying the Ward identity relating to for (non-marginal) scalar composite operators O^I, we find that there is a subtlety in obtaining the normalization of from the supergravity action integral.

hep-th↗

arXiv1998

Field Theory Tests for Correlators in the AdS/CFT Correspondence

The order g^2 radiative corrections to all 2- and 3-point correlators of the composite primary operators Tr X^k are computed in {\cal N} = 4 supersymmetric Yang-Mills theory with gauge group SU(N). Corrections are found to vanish for all N. For k=2 this is a consequence of known superconformal nonrenormalization theorems, and for general k the result confirms an N-to-infinity, fixed large g^2N supergravity calculation and further conjectures in hep-th/9806074. A 3-point correlator involving {\cal N} = 4 descendents of Tr X^2 is calculated, and its order g^2 contribution also vanishes, giving evidence for the absence of radiative corrections in correlators of descendent operators.

hep-th↗

arXiv1996

Conformal Symmetry and the Chiral Anomaly

Two-loop contributions to the anomalous correlation function of three chiral currents are calculated by a method based on the conformal properties of massless field theories. The method was previously applied to virtual photon diagrams in quantum electrodynamics, and it is extended here to diagrams with scalars and chiral spinors in the abelian Higgs model and in the SU(3)xSU(2)xU(1) standard model. In each case there are nonvanishing contributions to the gauge current correlator from self-energy insertions, vertex insertions and nonplanar diagrams, but their sum exactly vanishes. The two-loop contribution to the anomaly therefore also vanishes, in agreement with the Adler-Bardeen theorem. An application of the method to the correlator of the R and Konishi axial currents in supersymmetric gauge theories which was reported in hep-th/9608125 is discussed here. The net two-loop contribution to this correlator also vanishes.

hep-th↗

arXiv1995

Consequences of Supergravity with Gauged $\rm U(1)_R$ Symmetry

The structure of gauged R supergravity Lagrangians is reviewed, and we consider models with a hidden sector plus light fields of the MSSM. A simple potential for the hidden sector is presented which has a global minimum with zero cosmological constant and spontaneously broken SUSY and R-symmetry. The $\rm U(1)_R$ vector multiplet acquires a Planck scale mass through the Higgs mechanism, and it decouples at low energy. Due to very interesting cancellations, the $\rm U(1)_R$ D-terms also drop out at low energy. Thus no direct effects of the gauging of R-symmetry remain in the low energy effective Lagrangian, and this result is model independent, requiring only that R-symmetry be broken at the Planck scale and $ = 0$, where $D$ is the auxiliary field of the $\rm U(1)_R$ vector multiplet. The low energy theory is fairly conventional with soft SUSY breaking terms for the MSSM fields. As a remnant of the gauging of R-symmetry, it also contains light fields, some required to cancel R-anomalies and others from the hidden sector.

hep-ph↗

arXiv1995

Spatial Geometry of Non-Abelian Gauge Theory in $2 + 1$ Dimensions

The Hamiltonian dynamics of $2 + 1$ dimensional Yang-Mills theory with gauge group SU(2) is reformulated in gauge invariant, geometric variables, as in earlier work on the $3 + 1$ dimensional case. Physical states in electric field representation have the product form $Ψ_{\mathrm{phys}} [E^{a i}] = \exp ( i Ω[ E ] / g ) F [G_{ij}]$, where the phase factor is a simple local functional required to satisfy the Gauss law constraint, and $G_{ij}$ is a dynamical metric tensor which is bilinear in $E^{a k}$. The Hamiltonian acting on $F [ G_{ij} ]$ is local, but the energy density is infinite for degenerate configurations where $\det G (x)$ vanishes at points in space, so wave functionals must be specially constrained to avoid infinite total energy. Study of this situation leads to the further factorization $F [G_{ij} ] = F_c [ G_{ij} ] \mathcal R [ G_{ij} ]$, and the product $Ψ_c [E] \equiv \exp (i Ω[ E ] / g ) F_c [G_{ij}]$ is shown to be the wave functional of a topological field theory. Further information from topological field theory may illuminate the question of the behavior of physical gauge theory wave functionals for degenerate fields.

hep-th↗

arXiv1994

Spatial Geometry and the Wu-Yang Ambiguity

We display continuous families of SU(2) vector potentials $A_i^a(x)$ in 3 space dimensions which generate the same magnetic field $B^{ai}(x)$ (with det $B\neq 0$). These Wu-Yang families are obtained from the Einstein equation $R_{ij}=-2G_{ij}$ derived recently via a local map of the gauge field system into a spatial geometry with $2$-tensor $G_{ij}=B^a{}_i B^a{}_j\det B$ and connection $Γ_{jk}^i$ with torsion defined from gauge covariant derivatives of $B$.

hep-th↗

arXiv1994

Some beautiful equations of mathematical physics

The basic ideas and the important role of gauge principles in modern elementary particle physics are outlined. There are three theoretically consistent gauge principles in quantum field theory: the spin-1 gauge principle of electromagnetism and the standard model, the spin-2 gauge principle of general relativity, and the spin-3/2 gauge principle of supergravity. (Dirac Prize lecture, November 1993. Two figures are unavailable, but should not be essential.)

hep-th↗

arXiv1994

Spatial Geometry of Hamiltonian Gauge Theories

The Hamiltonians of $SU(2)$ and $SU(3)$ gauge theories in 3+1 dimensions can be expressed in terms of gauge invariant spatial geometric variables, i.e., metrics, connections and curvature tensors which are simple local functions of the non-Abelian electric field. The transformed Hamiltonians are local. New results from the same procedure applied to the $SU(2)$ gauge theory in 2+1 dimensions are also given.

hep-th↗

arXiv1994

The Wu-Yang Ambiguity Revisited

Several examples are given of continuous families of SU(2) vector potentials $A_i^a(x)$ in 3 space dimensions which generate the same magnetic field $B^{ai}(x)$ (with det $B\neq 0$). These Wu-Yang families are obtained from the Einstein equation $R_{ij}=-2G_{ij}$ derived recently via a local map of the gauge field system into a spatial geometry with $2$-tensor $G_{ij}=B^a{}_i B^a{}_j\det B$ and connection $Γ_{jk}^i$ with torsion defined from gauge covariant derivatives of $B$.

hep-th↗

arXiv1994

Physical States in Canonically Quantized Supergravity

We discuss the canonical quantization of $N=1$ supergravity in the functional Schrodinger representation. Although the form of the supersymmetry constraints suggests that there are solutions of definite order $n$ in the fermion fields, we show that there are no such states for any finite $n$. For $n=0$, a simple scaling argument definitively excludes the purely bosonic states discussed by D'Eath. For $n>0$, the argument is based on a mode expansion of the gravitino field on the quantization 3-surface. It is thus suggested that physical states in supergravity have infinite Grassmann number. This is confirmed for the free spin-3/2 field, for which we find that states satisfying the gauge constraints contain an infinite product of fermion mode operators.

hep-th↗

arXiv1992

Conformal Symmetry and Differential Regularization of the Three-Gluon Vertex

The conformal symmetry of the QCD Lagrangian for massless quarks is broken both by renormalization effects and the gauge fixing procedure. Renormalized primitive divergent amplitudes have the property that their form away from the overall coincident point singularity is fully determined by the bare Lagrangian, and scale dependence is restricted to $δ$-functions at the singularity. If gauge fixing could be ignored, one would expect these amplitudes to be conformal invariant for non-coincident points. We find that the one-loop three-gluon vertex function $Γ_{μνρ}(x,y,z)$ is conformal invariant in this sense, if calculated in the background field formalism using the Feynman gauge for internal gluons. It is not yet clear why the expected breaking due to gauge fixing is absent. The conformal property implies that the gluon, ghost and quark loop contributions to $Γ_{μνρ}$ are each purely numerical combinations of two universal conformal tensors $D_{μνρ}(x,y,z)$ and $C_{μνρ}(x,y,z)$ whose explicit form is given in the text. Only $D_{μνρ}$ has an ultraviolet divergence, although $C_{μνρ}$ requires a careful definition to resolve the expected ambiguity of a formally linearly divergent quantity. Regularization is straightforward and leads to a renormalized vertex function which satisfies the required Ward identity, and from which the beta-function is easily obtained. Exact conformal invariance is broken in higher-loop orders, but we outline a speculative scenario in which the perturbative structure of the vertex function is determined from a conformal invariant primitive core by interplay of the renormalization group equation and Ward identities.

hep-th↗