Field Anomalous Dimension Research Articles

Field redefinitions and infinite field anomalous dimensions

Field redefinitions are commonly used to reduce the number of operators in the Lagrangian by removing redundant operators and transforming to a minimal operator basis. We give a general argument that such field redefinitions, while leaving the S-matrix invariant and consequently finite, lead not only to infinite Green’s functions, but also to infinite field anomalous dimensions γϕ. These divergences cannot be removed by counterterms without reintroducing redundant operators.

Anomalous Ward identities for on-shell amplitudes at the conformal fixed point

Conformal symmetry underlies many massless quantum field theories, but little is known about the consequences of this powerful symmetry for on-shell scattering amplitudes. Working in a dimensionally-regularised ϕ3 model at the conformal fixed point, we show that the on-shell renormalised amplitudes satisfy anomalous conformal Ward identities. Each external on-shell state contributes two terms to the anomaly. The first term is proportional to the elementary field anomalous dimension, and thus involves only lower-loop information. We show that the second term can be given as the convolution of a universal collinear function and lower-order amplitudes. The computation of the conformal anomaly is therefore simpler than that of the amplitude at the same perturbative order, which gives our anomalous conformal Ward identities a strong predictive power in perturbation theory. Finally, we show that our result is also of practical importance for dimensionally-regularised amplitudes away from the conformal fixed point.

QCD cusp anomalous dimension: Current status

Calculation results for the HQET field anomalous dimension and the QCD cusp anomalous dimension, as well as their properties, are reviewed. The HQET field anomalous dimension [Formula: see text] is known up to four loops. The cusp anomalous dimension [Formula: see text] is known up to three loops, and its small-angle and large-angle asymptotics up to four loops. Some (but not all) color structures at four loops are known with the full [Formula: see text]-dependence. Some simple contributions are known at higher loops. For the [Formula: see text] asymptotics of [Formula: see text] (the light-like cusp anomalous dimension) and the [Formula: see text]-term of the small-[Formula: see text] expansion (the Bremsstrahlung function), the [Formula: see text] SYM results are equal to the highest-weight parts of the QCD results. There is an interesting conjecture about the structure of [Formula: see text] which holds up to three loops; at four loops it holds for some color structures and breaks down for other ones. In the cases when it holds, it related highly nontrivial functions of [Formula: see text], and it cannot be accidental; however, the reasons of this conjecture and its failures are not understood. The cusp anomalous dimension at the Euclidean angle [Formula: see text] is related to the static quark–antiquark potential due to conformal symmetry; in QCD, this relation is broken by an anomalous term proportional to the [Formula: see text]-function. Some new results are also presented. Using the recent four-loop result for [Formula: see text], here we obtain analytical expressions for some terms in the four-loop on-shell renormalization constant of the massive quark field [Formula: see text] which were previously known only numerically. We also present two new contributions to [Formula: see text], [Formula: see text] at five loops and to the quark–antiquark potential at four loops.

Four-loop QCD cusp anomalous dimension at small angle

We calculate the small angle expansion of the four-loop QCD cusp anomalous dimension. As a byproduct of our calculation, we also obtain the four-loop anomalous dimension of the heavy-quark field in HQET. The validity of the calculational setup is cross-checked by the independent calculation of the four-loop QCD beta-function from heavy-quark-gluon vertex renormalization in HQET. We check the obtained results for the cusp anomalous dimension and heavy-quark field anomalous dimension against available analytical and numerical results. Finally, we find that the maximal transcendentality contribution to the QCD Bremsstrahlung function coincides, up to a factor 3/2, with the Bremsstrahlung function in mathcal{N} = 4 supersymmetric Yang-Mills theory, at least, through 4 loops.

Three-loop order approach to flat polymerized membranes.

We derive the three-loop order renormalization group equationsthat describe the flat phase of polymerized membranes within the modified minimal subtraction scheme, following the pioneering one-loop order computation of Aronovitz and Lubensky [Phys. Rev. Lett. 60, 2634 (1988)10.1103/PhysRevLett.60.2634] and the recent two-loop order one of Coquand, Mouhanna, and Teber [Phys. Rev. E 101, 062104 (2020)2470-004510.1103/PhysRevE.101.062104]. We analyze the fixed points of these equationsand compute the associated field anomalous dimension η at three-loop order. Our results display a marked proximity with those obtained using nonperturbative techniques and reexpanded in powers of ε=4-D. Moreover, the three-loop order value that we get for η at the stable fixed point, η=0.8872, in D=2, is compatible with known theoretical results and within the range of accepted numerical values.

Towards general scalar-Yukawa renormalisation group equations at three-loop order

For arbitrary four-dimensional quantum field theories with scalars and fermions, renormalisation group equations in the overline{mathrm{MS}} scheme are investigated at three-loop order in perturbation theory. Collecting literature results, general expressions are obtained for field anomalous dimensions, Yukawa interactions, as well as fermion masses. The renormalisation group evolution of scalar quartic, cubic and mass terms is determined up to a few unknown coefficients. The combined results are applied to compute the renormalisation group evolution of the gaugeless Litim-Sannino model.

General scalar renormalisation group equations at three-loop order

For arbitrary scalar QFTs in four dimensions, renormalisation group equations of quartic and cubic interactions, mass terms, as well as field anomalous dimensions are computed at three-loop order in the overline{mathrm{MS}} scheme. Utilising pre-existing literature expressions for a specific model, loop integrals are avoided and templates for general theories are obtained. We reiterate known four-loop expressions, and from those derive β functions for scalar masses and cubic interactions. As an example, the results are applied to compute all renormalisation group equations in U(n) × U(n) scalar theories.

Critical models with N≤4 scalars in d=4−ε

We adopt a combination of analytical and numerical methods to study the renormalization group flow of the most general field theory with quartic interaction in $d=4-\epsilon$ with $N=3$ and $N=4$ scalars. For $N=3$, we find that it admits only three nondecomposable critical points: the Wilson-Fisher with $O(3)$ symmetry, the cubic with $H_3=(\mathbb{Z}_2)^3\rtimes S_3$ symmetry, and the biconical with $O(2)\times \mathbb{Z}_2$. For $N=4$, our analysis reveals the existence of new nontrivial solutions with discrete symmetries and with up to three distinct field anomalous dimensions.

Six-dimensional ultraviolet completion of the ℂℙ(N) σ model at two loops

We extend the recent one-loop analysis of the ultraviolet completion of the [Formula: see text] nonlinear [Formula: see text] model in six dimensions to two-loop order in the [Formula: see text] scheme for an arbitrary covariant gauge. In particular we compute the anomalous dimensions of the fields and [Formula: see text]-functions of the four coupling constants. We note that like Quantum Electrodynamics (QED) in four dimensions the matter field anomalous dimension only depends on the gauge parameter at one loop. As a nontrivial check we verify that the critical exponents derived from these renormalization group functions at the Wilson–Fisher fixed point are consistent with the [Formula: see text] expansion of the respective large [Formula: see text] exponents of the underlying universal theory. Using the Ward–Takahashi identity we deduce the three-loop [Formula: see text] renormalization group functions for the six-dimensional ultraviolet completeness of scalar QED.

Four-loop cusp anomalous dimension in QED

The 4-loop CF3TFnl and 5-loop CF4TFnl terms in the HQET field anomalous dimension γh are calculated analytically (the 4-loop one agrees with the recent numerical result [1]). The 4-loop CF3TFnl and 5-loop CF4TFnl terms in the cusp anomalous dimension Γ(φ) are calculated analytically, exactly in φ (the φ → ∞ asymptotics of the 4-loop one agrees with the recent numerical result [2]). Combining these results with the recent 4-loop dF Fnl contributions to γh and to the small-φ expansion of Γ(φ) up to φ4 [3], we now have the complete analytical 4-loop result for the Bloch-Nordsieck field anomalous dimension in QED, and the small-φ expansion of the 4-loop QED cusp anomalous dimension up to φ4.

Complete renormalization of QCD at five loops

We present new analytical five-loop Feynman-gauge results for the anomalous dimensions of ghost field and -vertex, generalizing the known values for SU(3) to a general gauge group. Together with previously published results on the quark mass and -field anomalous dimensions and the Beta function, this completes the 5-loop renormalization program of gauge theories in that gauge.

The Five-loop Beta Function for a General Gauge Group

We present results for the gluon field anomalous dimension in perturbative QCD and derive the corresponding Beta function at five-loop order. All given results are valid for a general gauge group.

Five-loop quark mass and field anomalous dimensions for a general gauge group

We present analytical five-loop results for the quark mass and quark field anomalous dimensions, for a general gauge group and in the overline{mathrm{MS}} scheme. We confirm the values known for the gauge group SU(3) from an independent calculation, and find full agreement with results available from large-Nf studies.

Six loop analytical calculation of the field anomalous dimension and the critical exponent η in O(n)-symmetric φ4 model

We report on a completely analytical calculation of the field anomalous dimension γφ and the critical exponent η for the O(n)-symmetric φ4 model at the record six loop level. We successfully compare our result for γφ with n=1 with the predictions based on the method of the Borel resummation combined with a conformal mapping (Kazakov et al., 1979 [40]). Predictions for seven loop contribution to the field anomalous dimensions are given.

The a-function for gauge theories

The a-function is a proposed quantity defined for quantum field theories which has a monotonic behaviour along renormalisation group flows, being related to the beta-functions via a gradient flow equation involving a positive definite metric. We construct the a-function at four loop order for a general gauge theory with fermions and scalars, using only one and two loop beta-functions; we are then able to provide a stringent consistency check on the general three-loop gauge beta-function. In the case of an N=1 supersymmetric gauge theory, we present a general condition on the chiral field anomalous dimension which guarantees an exact all-orders expression for the a-function; and we verify this up to fifth order (corresponding to the three-loop anomalous dimension).

Quark mass and field anomalous dimensions to O α s 5 $$ \mathcal{O}\left({\alpha}_s^5\right) $$

We present the results of the first complete analytic calculation of the quark mass and field anomalous dimensions to ${\cal O}(\alpha_s^5)$ in QCD.

Three-loop field renormalization for scalar field theory with Lorentz violation

Applying the counterterm method in minimal subtraction scheme we calculate the three-loop quantum correction to field anomalous dimension in a Lorentz-violating O(N) self-interacting scalar field theory. We compute the Feynman diagrams using dimensional regularization and ϵ-expansion techniques. As this approximation corresponds to a three-loop term, to our knowledge this is the first time in literature in which such a loop level is attained for a LV theory.

Higgs self-coupling beta-function in the Standard Model at three loops

We present the results for three-loop beta-function for the Higgs self-coupling calculated within the unbroken phase of the Standard Model. We also provide the results for three-loop beta-function for Higgs mass parameter, which was easily extracted from our calculation. Our results coincide with that of recent paper of K. Chetyrkin and M. Zoller (2013) [1]. In addition, the expression for the Higgs field anomalous dimension is given.

Anomalous dimensions of gauge fields and gauge coupling beta-functions in the Standard Model at three loops

We present the results for three-loop gauge field anomalous dimensions in the SM calculated in the background field gauge within the unbroken phase of the model. The results are valid for the general background field gauge parameterized by three independent parameters. Both quantum and background fields are considered. The former are used to find three-loop anomalous dimensions for the gauge-fixing parameters, and the latter allow one to obtain the three-loop SM gauge beta-functions. Independence of beta-functions of gauge-fixing parameters serves as a validity check of our final results.

Epsilon expansion for multicritical fixed points and exact renormalisation group equations

The Polchinski version of the exact renormalisation group equations is applied to multicritical fixed points, which are present for dimensions between two and four, for scalar theories using both the local potential approximation and its extension, the derivative expansion. The results are compared with the epsilon expansion by showing that the nonlinear differential equations may be linearised at each multicritical point and the epsilon expansion treated as a perturbative expansion. The results for critical exponents are compared with corresponding epsilon expansion results from standard perturbation theory. The results provide a test for the validity of the local potential approximation and also the derivative expansion. An alternative truncation of the exact RG equation leads to equations which are similar to those found in the derivative expansion but which gives correct results for critical exponents to order ε and also for the field anomalous dimension to order ε2. An exact marginal operator for the full RG equations is also constructed.