Beta-function Coefficient Research Articles

Properties of the \u0454-expansion, Lagrange inversion and associahedra and the O (1) model

We discuss properties of the є-expansion in d = 4 − є dimensions. Using Lagrange inversion we write down an exact expression for the value of the Wilson-Fisher fixed point coupling order by order in є in terms of the beta function coefficients. The є-expansion is combinatoric in the sense that the Wilson-Fisher fixed point coupling at each order depends on the beta function coefficients via Bell polynomials. Using certain properties of Lagrange inversion we then argue that the є-expansion of the Wilson-Fisher fixed point coupling equally well can be viewed as a geometric expansion which is controlled by the facial structure of associahedra. We then write down an exact expression for the value of anomalous dimensions at the Wilson-Fisher fixed point order by order in є in terms of the coefficients of the beta function and anomalous dimensions. We finally use our general results to compute the values for the Wilson-fisher fixed point coupling and critical exponents for the scalar O (1) symmetric model to O(є7).

Conformal phase diagram of complete asymptotically free theories

We investigate the ultraviolet and infrared fixed point structure of gauge-Yukawa theories featuring a single gauge coupling, Yukawa coupling and scalar self-coupling. Our investigations are performed using the two loop gauge beta function, one loop Yukawa beta function, and one loop scalar beta function. We provide the general conditions that the beta function coefficients must abide for the theory to be complete asymptotically free while simultaneously possessing an infrared stable fixed point. We also uncover special trajectories in coupling space along which some couplings are both asymptotically safe and infrared conformal.

Chiral sine-Gordon model

We investigate the chiral sine-Gordon model using the renormalization group method. The chiral sine-Gordon model is a model for G-valued fields and describes a new class of phase transitions, where G is a compact Lie group. We show that the model is renormalizable by means of a perturbation expansion and we derive beta functions of the renormalization group theory. The coefficients of beta functions are represented by the Casimir invariants. The model contains both asymptotically free and ultraviolet strong-coupling regions. The beta functions have a zero which is a bifurcation point that divides the parameter space into two regions; they are the weak-coupling region and the strong-coupling region. A large-N model is also considered. This model is reduced to the conventional sine-Gordon model that describes the Kosterlitz-Thouless transition near the fixed point. In the strong-coupling limit, the model is reduced to a U(N) matrix model.

Resurgence and dynamics of O(N) and Grassmannian sigma models

We study the non-perturbative dynamics of the two dimensional O(N ) and Grassmannian sigma models by using compactification with twisted boundary conditions on $$ \mathbb{R}\times {S}^1 $$ , semi-classical techniques and resurgence. While the O(N) model has no instantons for N > 3, it has (non-instanton) saddles on $$ {\mathbb{R}}^2 $$ , which we call 2d-saddles. On $$ \mathbb{R}\times {S}^1 $$ , the resurgent relation between perturbation theory and non-perturbative physics is encoded in new saddles, which are associated with the affine root system of the o(N ) algebra. These events may be viewed as fractionalizations of the 2d-saddles. The first beta function coefficient, given by the dual Coxeter number, can then be intepreted as the sum of the multiplicities (dual Kac labels) of these fractionalized objects. Surprisingly, the new saddles in O(N ) models in compactified space are in one-to-one correspondence with monopole-instanton saddles in SO(N ) gauge theory on $$ {\mathbb{R}}^3\times {S}^1 $$ . The Grassmannian sigma models Gr(N, M ) have 2d instantons, which fractionalize into N kink-instantons. The small circle dynamics of both sigma models can be described as a dilute gas of the one-events and two-events, bions. One-events are the leading source of a variety of non-perturbative effects, and produce the strong scale of the 2d theory in the compactified theory. We show that in both types of sigma models the neutral bion emulates the role of IR-renormalons. We also study the topological theta angle dependence in both the O(3) model and Gr(N, M ), and describe the multi-branched structure of the observables in terms of the theta-angle dependence of the saddle amplitudes, providing a microscopic argument for Haldane’s conjecture.

One-loop renormalization and theSmatrix

In four-dimensional theories with massless particles, the one-loop amplitudes can be expressed in terms of a basis of scalar integrals and rational terms. Since the scalar bubble integrals are the only UV divergent integrals, the sum of the bubble coefficients captures the 1-loop UV behavior. In particular, in a renormalizable theory the sum of the bubble coefficients equals the tree-level amplitude times a proportionality constant that is related to the one-loop beta function coefficient beta_0. In this paper, we study how this proportionality is achieved from the viewpoint of on-shell amplitudes. For n-point MHV amplitude in (super) Yang-Mills theory, we demonstrate the existence of a hidden structure in each individual bubble coefficient which directly leads to systematic cancellations within the sum of them. The origin of this structure can be attributed to the collinear poles within a two-particle cut. Due to the cancellation, the one-loop beta function coefficient can be identified as a sum over the residues of unique collinear poles in particular two-particle cuts. Using CSW recursion relations, we verify the generality of this statement by reproducing the correct proportionality factor from such cuts for n-point split-helicity N^kMHV amplitudes.

The two faces of anomaly mediation

Anomaly mediation is a ubiquitous source of supersymmetry (SUSY) breaking which appears in almost every theory of supergravity. In this paper, we show that anomaly mediation really consists of two physically distinct phenomena, which we dub "gravitino mediation" and "Kahler mediation". Gravitino mediation arises from minimally uplifting SUSY anti-de Sitter (AdS) space to Minkowski space, generating soft masses proportional to the gravitino mass. Kahler mediation arises when visible sector fields have linear couplings to SUSY breaking in the Kahler potential, generating soft masses proportional to beta function coefficients. In the literature, these two phenomena are lumped together under the name "anomaly mediation", but here we demonstrate that they can be physically disentangled by measuring associated couplings to the goldstino. In particular, we use the example of gaugino soft masses to show that gravitino mediation generates soft masses without corresponding goldstino couplings. This result naively violates the goldstino equivalence theorem but is in fact necessary for supercurrent conservation in AdS space. Since gravitino mediation persists even when the visible sector is sequestered from SUSY breaking, we can use the absence of goldstino couplings as an unambiguous definition of sequestering.

On-shell superamplitudes in $ \mathcal{N} < 4 $ SYM

We present an on-shell formalism for superamplitudes of pure N<4 super Yang-Mills theory. Two superfields, Phi and Phi^+, are required to describe the two CPT conjugate supermultiplets. Simple truncation prescriptions allow us to derive explicit tree-level MHV and NMHV superamplitudes with N-fold SUSY. Any N=0,1,2 tree superamplitudes have large-z falloffs under super-BCFW shifts, except under [Phi,Phi^+>-shifts. We show that this `bad' shift is responsible for the bubble contributions to 1-loop amplitudes in N=0,1,2 SYM. We evaluate the MHV bubble coefficients in a manifestly supersymmetric form and demonstrate for the case of four external particles that the sum of bubble coefficients is equal to minus the tree superamplitude times the 1-loop beta-function coefficient. The connection to the beta-function is expected since only bubble integrals capture UV divergences; we discuss briefly how the minus sign arises from UV and IR divergences in dimensional regularization. Other applications of the on-shell formalism include a solution to the N^{K}MHV N=1 SUSY Ward identities and a clear description of the connection between 6d superamplitudes and the 4d ones for both N=4 and N=2 SYM. We outline extensions to N<8 supergravity.

RτAND HIGGS DECAY WIDTH IN CORGI APPROACH, BASED ON THE LEADING-b APPROXIMATION

We consider Adler D-function for the vector and scalar correlators with recent QCD analysis at N4LO approximation. The portion of perturbative coefficients of these observable containing the leading power of b, the first beta-function coefficient, is resummed to all-orders. The factorial behavior of coefficients allows us to use the Borel transformation to reveal the existent singularities as renormalons. CORGI approach is employed to avoid the renormalization dependence of the observable. In addition to the CORGI approach, the calculations of the standard perturbative QCD approach which uses the [Formula: see text] scheme with a physical choice of renormalization scale, are also presented. A comparison between the results of the standard and CORGI approaches for Rτand Higgs decay width is done. The comparison anticipates a better result for the CORGI approach. As an adjunct to these studies, the difference between the all order and fixed order result at N4LO approximation is used to estimate the uncertainty in [Formula: see text] extracted from Rτmeasurements. We find at N4LO approximation a smaller uncertainty with respect to the previous result which has been done at N4LO approximation.

UA(1)anomaly at high temperature: The scalar-pseudoscalar splitting in QCD

We estimate the splitting between the spatial correlation lengths in the scalar and pseudoscalar channels in QCD at high temperature. The splitting is due to the contribution of the instanton/anti-instanton chains in the thermal ensemble, even though instanton contributions to thermodynamic quantities are suppressed. The splitting vanishes at asymptotically high temperatures as $\Delta M/M\propto (\Lambda_{QCD}/T)^b$, where $b$ is the beta function coefficient.

DiscreteR-symmetry anomalies in heterotic orbifold models

Anomalies of discrete R symmetries appearing in heterotic orbifold models are studied. We find that the mixed anomalies for different gauge groups satisfy the universal Green-Schwarz condition, indicating that these anomalies are canceled by the Green-Schwarz mechanism. An exact relation between the anomaly coefficients of the discrete R symmetries and one-loop beta-function coefficients is obtained. We also find that the discrete R symmetries have a good chance to be unbroken down to the supersymmetry breaking scale. Even below this scale a Z{sub 2} subgroup is unbroken, and it may be an origin of the R parity of the minimal supersymmetric standard model. Relations between the R-symmetry anomalies and T-duality anomalies are also investigated.

NNLO QCD corrections to the [formula omitted] matrix elements using interpolation in [formula omitted

One of the most troublesome contributions to the NNLO QCD corrections to B ¯ → X s γ originates from three-loop matrix elements of four-quark operators. A part of this contribution that is proportional to the QCD beta-function coefficient β 0 was found in 2003 as an expansion in m c / m b . In the present paper, we evaluate the asymptotic behaviour of the complete contribution for m c ≫ m b / 2 . The asymptotic form of the β 0 -part matches the small- m c expansion very well at the threshold m c = m b / 2 . For the remaining part, we perform an interpolation down to the measured value of m c , assuming that the β 0 -part is a good approximation at m c = 0 . Combining our results with other contributions to the NNLO QCD corrections, we find B ( B ¯ → X s γ ) = ( 3.15 ± 0.23 ) × 10 −4 for E γ > 1.6 GeV in the B ¯ -meson rest frame. The indicated error has been obtained by adding in quadrature the following uncertainties: Non-perturbative (5%), parametric (3%), higher-order perturbative (3%), and the interpolation ambiguity (3%).

Scheme independence to all loops

The immense freedom in the construction of Exact Renormalization Groups means that the many non-universal details of the formalism need never be exactly specified, instead satisfying only general constraints. In the context of a manifestly gauge invariant Exact Renormalization Group for SU(N) Yang-Mills, we outline a proof that, to all orders in perturbation theory, all explicit dependence of beta function coefficients on both the seed action and details of the covariantization cancels out. Further, we speculate that, within the infinite number of renormalization schemes implicit within our approach, the perturbative beta function depends only on the universal details of the setup, to all orders.

On Nonperturbative Exactness of Konishi Anomaly and the Dijkgraaf-Vafa Conjecture

In this paper we study the nonperturbative corrections to the generalized Konishi anomaly that come from the strong coupling dynamics of the gauge theory. We consider U(N) gauge theory with adjoint and Sp(N) or SO(N) gauge theory with symmetric or antisymmetric tensor. We study the algebra of chiral rotations of the matter field and show that it does not receive nonperturbative corrections. The algebra implies Wess-Zumino consistency conditions for the generalized Konishi anomaly which are used to show that the anomaly does not receive nonperturbative corrections for superpotentials of degree less than 2l+1 where 2l = 3c(Adj)−c(R) is the one-loop beta function coefficient. The superpotentials of higher degree can be nonperturbatively renormalized because of the ambiguities in the UV completion of the gauge theory. We discuss the implications for the Dijkgraaf-Vafa conjecture.

All-orders infrared freezing of observables in perturbative QCD

We consider a Borel sum definition of all-orders perturbation theory for Minkowskian QCD observables such as the ${R}_{{e}^{+}{e}^{\ensuremath{-}}}$ ratio, and show that both this perturbative component and the additional nonperturbative all-orders operator product expansion (OPE) component can remain separately well-defined for all values of energy $\sqrt{s},$ with the perturbative component dominating as $\stackrel{\ensuremath{\rightarrow}}{s}\ensuremath{\infty},$ and with both components contributing as $\stackrel{\ensuremath{\rightarrow}}{s}0.$ In the infrared $\stackrel{\ensuremath{\rightarrow}}{s}0$ limit the perturbative correction to the parton model result for ${R}_{{e}^{+}{e}^{\ensuremath{-}}}$ has an all-orders perturbation theory component which smoothly freezes to the value $\mathcal{R}(0)=2/b,$ where $b=(33\ensuremath{-}{2N}_{f})/6$ is the first QCD beta-function coefficient, with ${N}_{f}$ flavors of massless quark. For freezing one requires ${N}_{f}&lt;9.$ The freezing behavior is manifested by the ``contour-improved'' or ``analytic perturbation theory'' (APT), in which an infinite subset of analytical continuation terms are resummed to all-orders. We show that for the Euclidean Adler-$D$ function, ${D(Q}^{2}),$ the perturbative component remains defined into the infrared if all the renormalon singularities are taken into account, but no analogue of the APT reorganization of perturbation theory is possible. We perform phenomenological comparisons of suitably smeared low-energy data for the ${R}_{{e}^{+}{e}^{\ensuremath{-}}}$ ratio, with the perturbative freezing predictions, and find good agreement.

All-orders infrared freezing of Re+e− in perturbative QCD

writing R e + e −( s) = 3 Σ f Q f 2(1 + R( s)), with Q f denoting the electric charge of quark flavour f, we show that for N f < 9 flavours of massless quark, the perturbative correction R( s) to the parton model result smoothly approaches from below the infra-red limit R(0) = 2 b , as s→0 . Here b = (33 - N f )/6 is the first QCD beta-function coefficient. This freezing holds to all-orders in perturbation theory. The s-dependence can be written in closed form in terms of the Lambert W function.

Zero modes, beta functions and IR/UV interplay in higher-loop QED

We analyze the relation between the short-distance behavior of quantum field theory and the strong-field limit of the background field formalism, for QED effective Lagrangians in self-dual backgrounds, at both one and two loop. The self-duality of the background leads to zero modes in the case of spinor QED, and these zero modes must be taken into account before comparing the perturbative beta function coefficients and the coefficients of the strong-field limit of the effective Lagrangian. At one-loop this is familiar from instanton physics, but we find that at two-loop the role of the zero modes, and the interplay between IR and UV effects in the renormalization, is quite different. Our analysis is motivated in part by the remarkable simplicity of the two-loop QED effective Lagrangians for a self-dual constant background, and we also present here a new independent derivation of these two-loop results.

Expanding running coupling effects in the hard Pomeron

We study QCD hard processes at scales of order k^2 > Lambda^2 in the limit in which the beta-function coefficient - b is taken to be small, but alphas(k) is kept fixed. The (nonperturbative) Pomeron is exponentially suppressed in this limit, making it possible to define purely perturbative high-energy Green's functions. The hard Pomeron exponent acquires diffusion and running coupling corrections which can be expanded in the b parameter and turn out to be dependent on the effective coupling b alphas^2 Y. We provide a general setup for this b-expansion and we calculate the first few terms both analytically and numerically.

All-orders infrared freezing of Re+e− in perturbative QCD

We consider the behaviour of the perturbative QCD corrections to the R e + e − ratio, in the limit that the c.m. energy s vanishes. Writing R e +e − (s)=3∑ fQ f 2(1+ R(s)) , with Q f denoting the electric charge of quark flavour f, we find that for N f <9 flavours of massless quarks, the perturbative correction R(s) to the parton model result smoothly approaches from below the infrared limit R(0)=2/b , as s→0. Here b=(33−2 N f )/6 is the first QCD beta-function coefficient. This freezing holds to all-orders in perturbation theory. The s-dependence can be written analytically in closed form in terms of the Lambert W function.

Renormalon-inspired resummations for vector and scalar correlators — estimating the uncertainty in αs( M2τ) and α( M2Z)

We perform an all-orders resummation of the QCD Adler D-function for the vector correlator, in which the portion of perturbative coefficients containing the leading power of b, the first beta-function coefficient, is resummed. To avoid a renormalization scale dependence when we match the resummation to the exactly known next-to-leading order (NLO), and next-NLO (NNLO) results, we employ the Complete Renormalization Group Improvement (CORGI) approach in which all RG-predictable ultraviolet logarithms are resummed to all-orders, removing all dependence on the renormalization scale. We can also obtain fixed-order CORGI results. Including suitable weight-functions we can numerically integrate these results for the D-function in the complex energy plane to obtain so-called “contour-improved” results for the ratio R and its tau decay analogue R τ . We use the difference between the all-orders and fixed-order (NNLO) results to estimate the uncertainty in α s ( M 2 Z ) extracted from R τ measurements, and find α s ( M 2 Z )=0.120±0.002. We also estimate the corresponding uncertainty in α( M 2 Z ) arising from hadronic corrections by considering the uncertainty in R( s), in the low-energy region, and compare with other estimates. Analogous resummations are also given for the scalar correlator. As an adjunct to these studies we show how fixed-order contour-improved results can be obtained analytically in closed form at the two-loop level in terms of the Lambert W-function and hypergeometric functions.

RG-flows and open/closed string duality

We discuss the interpaly between IR and UV divergences in theories with open and unoriented strings in view of the AdS/CFT correspondence. We start by deriving general formulas for the computation of threshold corrections to gauge couplings in generic configurations with open and unoriented strings. These allow us to discuss the IR/UV correspondence between beta-function coefficients and ``dilaton'' tadpoles for several brane configurations probed by D3-branes. Finally we comment on the AdS supergravity descriptions of gauge theories that are (super)conformal in the large N limit.