Momentum Subtraction Research Articles

NSVZ scheme with the higher derivative regularization for [formula omitted] SQED

The exact NSVZ relation between a β-function of N=1 SQED and an anomalous dimension of the matter superfields is studied within the Slavnov higher derivative regularization approach. It is shown that if the renormalization group functions are defined in terms of the bare coupling constant, this relation is always valid. In the renormalized theory the NSVZ relation is obtained in the momentum subtraction scheme supplemented by a special finite renormalization. Unlike the dimensional reduction, the higher derivative regularization allows to fix this finite renormalization. This is made by imposing the conditions Z3(α,μ=Λ)=1 and Z(α,μ=Λ)=1 on the renormalization constants of N=1 SQED, where Λ is a parameter in the higher derivative term. The results are verified by the explicit three-loop calculation. In this approximation we relate the DR¯ scheme and the NSVZ scheme defined within the higher derivative approach by the finite renormalization.

Exponential renormalisation. II. Bogoliubov's R-operation and momentum subtraction schemes

This article aims at advancing the recently introduced exponential method for renormalisation in perturbative quantum field theory. It is shown that this new procedure provides a meaningful recursive scheme in the context of the algebraic and group theoretical approach to renormalisation. In particular, we describe in detail a Hopf algebraic formulation of Bogoliubov's classical R-operation and counterterm recursion in the context of momentum subtraction schemes. This approach allows us to propose an algebraic classification of different subtraction schemes. Our results shed light on the peculiar algebraic role played by the degrees of Taylor jet expansions, especially the notion of minimal subtraction and oversubtractions.

Two loop QCD vertices at the symmetric point

We compute the triple gluon, quark-gluon and ghost-gluon vertices of QCD at the symmetric subtraction point at two loops in the MSbar scheme. In addition we renormalize each of the three vertices in their respective momentum subtraction schemes, MOMggg, MOMq and MOMh. The conversion functions of all the wave functions, coupling constant and gauge parameter renormalization constants of each of the schemes relative to MSbar are determined analytically. These are then used to derive the three loop anomalous dimensions of the gluon, quark, Faddeev-Popov ghost and gauge parameter as well as the beta-function in an arbitrary linear covariant gauge for each MOM scheme. There is good agreement of the latter with earlier Landau gauge numerical estimates of Chetyrkin and Seidensticker.

UV behavior of scalar fields

Exceptional ultraviolet properties of scalar fields are discussed. Novel technique for computing them within the momentum subtraction renormalization scheme is presented. Anomalous dimension of scalar mass in ϕ 4 theory is computed in this scheme. It is demonstrated that this anomalous dimension is singular in the limit of zero mass. The singularity is related to the naturalness problem.

One-loop corrections to the photon propagator in curved-space QED

We calculate and discuss the one-loop corrections to the photon sector of QED interacting to a background gravitational field. At high energies the fermion field can be taken as massless and the quantum terms can be obtained by integrating conformal anomaly. We present a covariant local expression for the corresponding effective action, similar to the one obtained earlier for the gravitational sector. At the moderate energies the quantum terms can be obtained through the heat kernel method. In this way we derive the exact one-loop beta-function for the electric charge in the momentum subtraction scheme and explore both massless and large mass limits. The relation between the two approaches is shown and the difference discussed in view of the possible applications to cosmology and astrophysics.

The strong coupling and its running to four loops in a minimal MOM scheme

We introduce the minimal momentum subtraction (MiniMOM) scheme for QCD. Its definition allows the strong coupling to be fixed solely through a determination of the gluon and ghost propagators. In Landau gauge this scheme has been implicit in the early studies of these propagators, especially in relation to their non-perturbative behaviour in the infrared and the associated infrared fixed-point. Here we concentrate on its perturbative use. We give the explicit perturbative definition of the scheme and the relation of its β-function and running coupling to the MS¯ scheme up to 4-loop order in general covariant gauges. We also demonstrate, by considering a selection of Nf=3 examples, that the apparent convergence of the relevant perturbative series can in some (though not all) cases be significantly improved by re-expanding the MS¯ coupling version of this series in terms of the MiniMOM coupling, making the MiniMOM coupling also of potential interest in certain phenomenological applications.

Mass-dependentβ-function

Threshold effects related to fermion masses are considered for an all-order β-function based on a background field momentum subtraction scheme. Far away from all thresholds, the suggested β-function reduces to the conjectured all-order form inspired by the Novikov-Shifman-Vainshtein-Zakharov β-function of N = 1 supersymmetric gauge theories with a fixed integer number of fermion flavors. At (formally) infinite masses the corresponding pure Yang-Mills β-function is recovered. We discuss applications to the phase diagram of non-Abelian field theories.

Strong-coupling constant at three loops in momentum subtraction scheme

In this paper we compute the three-loop corrections to the β function in a momentum subtraction (MOM) scheme with a massive quark. The calculation is performed in the background field formalism applying asymptotic expansions for small and large momenta. Special emphasis is devoted to the relation between the coupling constant in the MOM and MS ¯ schemes as well as their ability to describe the phenomenon of decoupling. It is demonstrated by an explicit comparison that the MS ¯ scheme can be consistently used to relate the values of the MOM-scheme strong-coupling constant in the energy regions higher and lower than the massive-quark production threshold. This procedure obviates the necessity to know the full mass dependence of the MOM β function and clearly demonstrates the equivalence of both schemes for the description of physics outside the threshold region.

Minkowski solution of Dyson-Schwinger equations in momentum subtraction scheme

Using the Green function integral representation the Dyson-Schwinger equations are solved directly in Minkowski space. Essential ideas of the spectral techniques are discussed and applied on two renormalizable models: the Yukawa theory with massive pseudoscalar meson and conventional spinor QED. Within the momentum subtraction procedure, the appropriate renormalization is performed analytically which leads to the usual dispersion formulation. The electron propagator obtained in this frame is compared with the solution of Euclidean Dyson-Schwinger equation and with the perturbation theory results as well. The proposed method has the advantage of obtaining solutions in both the space- and {\bf{time}}-like regimes of momenta. In addition,when the coupling constant increases we find some un-expected discrepancy between the Euclidean and integral representation solutions. Especially, when the solutions of the original momentum DSE's exhibit the signal for confinement then the spectral approach gives no solution for Lehmann function of confined particle.

Quark-gluon vertex from lattice QCD

The quark-gluon vertex in Landau gauge is studied in the quenched approximation using the Sheikholeslami-Wohlert (SW) fermion action with mean-field improvement coefficients in the action and for the quark fields. We see that the form factor that includes the running coupling is substantially enhanced in the infrared, over and above the enhancement arising from the infrared suppression of the quark propagator alone. We define two different momentum subtraction renormalisation schemes -- \MOMT (asymmetric) and \MOMB\ (symmetric) -- and determine the running coupling in both schemes. We find Lambda_msbar(N_f=0)=300(+150/-180)(55)(30) MeV from the asymmetric scheme. This is somewhat higher than other determinations of this quantity, but the uncertainties -- both statistical and systematic -- are large. In the symmetric scheme, statistical noise prevents us from obtaining a meaningful estimate for Lambda_msbar.

Quark-gluon vertex in a momentum subtraction scheme

We compute the quark-gluon vertex in quenched QCD, in the Landau gauge using an off-shell mean-field O ( a)-improved fermion action. The running coupling is calculated in an ‘asymmetric’ momentum subtraction scheme ( MOM ). We obtain a crude estimate for Λ MS = 170 ± 65 MeV, which is considerably lower than other determinations of this quantity. However, substantial systematic errors remain.

Remarks on the determination of the Landau gauge OPE for the asymmetric three gluon vertex

We compute a compact operator product expansion (OPE) formula describing power corrections to the perturbative expression for the asymmetric momentum subtraction--$(\mathrm{MOM}\ifmmode \tilde{}\else \~{}\fi{}\ensuremath{-})$ renormalized running coupling constant up to the leading logarithm. By the use of a phenomenological hypothesis leading to the factorization of the condensates through a perturbative vacuum insertion, the only relevant condensate in the game is $〈{A}^{2}〉.$ The validity of the OPE formula is tested by searching for a good-quality coherent description of previous lattice evaluations of the $\mathrm{MOM}\ifmmode \tilde{}\else \~{}\fi{}$-renormalized gluon propagator and running coupling.

The algebraic method

Combining the effect of an intermediate renormalization prescription (zero momentum subtraction) and the background field method (BFM), we show that the algebraic renormalization procedure needed for the computation of radiative corrections within non-invariant regularization schemes is drastically simplified. The present technique is suitable for gauge models and, here, is applied to the Standard Model. The use of the BFM allows a powerful organization of the counterterms and avoids complicated Slavnov–Taylor identities. Furthermore, the Becchi–Rouet–Stora–Tyutin (BRST) variation of background fields plays a special role in disentangling Ward–Takahashi identities (WTI) and Slavnov–Taylor identities (STI). Finally, the strategy to be applied to physical processes is exemplified for the process b→ sγ.

Non-perturbative renormalization in lattice field theory

A review the strategies which have been developped in recent years to solve the non-perturbative renormalization problem in lattice field theories. Although the techniques are general, the focus will be on applications to lattice QCD. I discuss the momentum subtraction and finite volume schemes, and their application to scale dependent renormalizations. The problem of finite renormalizations is illustrated with the example of explicit chiral symmetry breaking, and I give a short status report concerning Symanzik's improvement programme to O( a).

Two-loop QCD vertices and three-loop MOM β functions

We present numerical results for the two-loop QCD corrections to the ghost gluon vertex, the quark gluon vertex, and the triple gluon vertex in the massless limit at the symmetric point. We restrict ourselves to the tensor structures existing in the tree level. The corrections are used to examine different coupling constant definitions in momentum subtraction schemes and the corresponding three-loop β functions.

Exact mass-dependent two-loop [formula omitted] in the background MOM renormalization scheme

A two-loop calculation of the renormalization group β-function in a momentum subtraction scheme with massive quarks is presented using the background field formalism. The results have been obtained by using a set of new generalized recurrence relations proposed recently by one of the authors (O.V.T.). The behavior of the mass-dependent effective coupling constant is investigated in detail. Compact analytic results are presented.

A momentum subtraction scheme for two-nucleon effective field theory

We introduce a momentum subtraction scheme which obeys the power counting of Kaplan, Savage, and Wise (KSW), developed for systems with large scattering lengths, a. Unlike the power divergence subtraction scheme, coupling constants in this scheme obey the KSW scaling for all μ R >1/ a. We comment on the low-energy theorems derived by Cohen and Hansen. We conclude that there is no obstruction to using perturbative pions for momenta p> m π .

Three-loop beta function and non-perturbative αs in asymmetric momentum scheme

We determine the three-loop coefficient of the beta function in the asymmetric momentum subtraction scheme in Landau gauge. This scheme is convenient for lattice studies of \alpha_s, the running coupling constant of QCD. We present high statistics lattice results for \alpha_s in the SU(3) Yang-Mills theory without quark, compare with the three-loop running and extract the value of the corresponding \Lambda_\msbar parameter. We estimate the systematic error coming from four-loop terms. We obtain the result: \Lambda_\msbar = 295 (5) (15) {a^{-1}(\beta=6.0) \over 1.97 GeV} MeV.

Hard-soft renormalization of the massless Wess-Zumino model

We show that in a Wilsonian renormalization scheme with zero-momentum subtraction point the massless Wess-Zumino model satisfies the non-renormalization theorem; the finite renormalization of the superpotential appearing in the usual non-zero momentum subtraction schemes is thus avoided. We give an exact expression of the beta and gamma functions in terms of the Wilsonian effective action; we prove the expected relation β=3 gγ. We compute the beta function at the first two loops, finding agreement with previous results.

Lattice calculation of αs in momentum scheme

We compute the flavorless running coupling constant of QCD from the three gluon vertex in the (regularisation independent) momentum subtraction renormalisation scheme. This is performed on the lattice with high statistics. The expected color dependence of the Green functions is verified. There are significant $O(a^2\mu^2)$ effects which can be consistently controlled. Scaling is demonstrated when the renormalisation scale is varied between 2.1 GeV and 3.85 GeV. Scaling when the lattice spacing is varied is very well satisfied. The resulting flavorless conventional two loop $\Lambda^{(c)}_{\bar{MS}}$ is estimated to be, respectively for the MOM and $\widetilde{MOM}$ scheme, 361(6) MeV and 345(6) MeV, while the three loop results are, depending on $\beta_2: \Lambda^{(3)}_{\bar{MS}} = (412 - 59 \frac{\beta_2} {\beta_{2, \bar{MS}}} \pm 6 MeV) $ and $\Lambda^{(3)}_{\bar{MS}} = (382- 46\frac{\beta_2} {\beta_{2, \bar{MS}}}\pm 5 MeV) $. A preliminary computation of $\beta_2$ in the $\widetilde{MOM}$ scheme leads to $\Lambda^{(3)}_{\bar{MS}} = 303(5)\frac {a^{-1}(\beta=6.0)} {1.97 GeV}$ MeV.