Field Renormalization Constant Research Articles

Coefficients at powers of logarithms in the higher-derivatives and minimal-subtractions-of-logarithms renormalization scheme

For renormalizable theories with a single coupling constant regularized by higher derivatives we investigate the coefficients at powers of logarithms present in the renormalization constants assuming that divergences are removed by minimal subtractions of logarithms. According to this higher-derivatives and minimal-subtractions-of-logarithms ($\mathrm{HD}+\mathrm{MSL}$) renormalization prescription, the renormalization constants include only powers of $\mathrm{ln}\mathrm{\ensuremath{\Lambda}}/\ensuremath{\mu}$, where $\mathrm{\ensuremath{\Lambda}}$ and $\ensuremath{\mu}$ are the dimensionful regularization parameter and the renormalization point, respectively. We construct general explicit expressions for arbitrary coefficients at powers of this logarithm present in the coupling constant renormalization and in the field renormalization constant which relate them to the $\ensuremath{\beta}$-function and (in the latter case) to the corresponding anomalous dimension. To check the correctness, we compare the results with the explicit four-loop calculation made earlier for $\mathcal{N}=1$ supersymmetric quantum electrodynamics and (for the supersymmetric case) rederive a relation between the renormalization constants following from the Novikov, Shifman, Vainshtein, and Zakharov equation.

Fighting off field dependence in MSSM Higgs-mass corrections of order alpha _t,alpha _s and alpha _t^2

The connection between gauge and Higgs sectors makes supersymmetric extensions of the Standard Model predictive frameworks for the derivation of Higgs masses. In this paper, we study the contamination of such predictions by field-renormalization constants, in the MSSM with two-loop gaugeless corrections of mathcal {O}big (alpha _{t,b},alpha _s,,alpha _{t,b}^2big ) and full momentum dependence, and demonstrate how strict perturbative expansions allow to systematically neutralize the dependence on such unphysical objects. On the other hand, the popular procedure consisting in an iterative pole search remains explicitly dependent on field counterterms. We then analyze the magnitude of the intrinsic uncertainty that this feature implies for the iterative method, both in non-degenerate and near-degenerate regimes, and conclude that this strategy does not improve on the predictions of the more straightforward expansion. We also discuss several features related to the inclusion of the orders alpha _{t,b},alpha _s and alpha _{t,b}^2 in the so-called ‘fixed-order’ approach, such as the resummation of UV-logarithms for heavy supersymmetric spectra.

Matching the heavy-quark fields in QCD and HQET at four loops

The QCD/HQET matching coefficient for the heavy-quark field is calculated up to four loops. It must be finite; this requirement produces analytical results for some terms in the four-loop on-shell heavy-quark field renormalization constant which were previously only known numerically. The effect of a non-zero lighter-flavor mass is calculated up to three loops. A class of on-shell integrals with two masses is analyzed in detail. By specifying our result to QED, we obtain the relation between the electron field and the Bloch--Nordsieck field with four-loop accuracy.

Hadron mass scaling near thes-wave threshold

The influence of a two-hadron threshold is studied for the hadron mass scaling with respect to some quantum chromodynamics parameters. A quantum mechanical model is introduced to describe the system with a one-body bare state coupled with a single elastic two-body scattering. The general behavior of the energy of the bound and resonance state near the two-body threshold for a local potential is derived from the expansion of the Jost function around the threshold. It is shown that the same scaling holds for the nonlocal potential induced by the coupling to a bare state. In $p$ or higher partial waves, the scaling law of the stable bound state continues across the threshold describing the real part of the resonance energy. In contrast, the leading contribution of the scaling is forbidden by the nonperturbative dynamics near the $s$-wave threshold. As a consequence, the bound state energy is not continuously connected to the real part of the resonance energy. This universal behavior originates in the vanishing of the field renormalization constant of the zero-energy resonance in the $s$ wave. A proof is given for the vanishing of the field renormalization constant, together with a detailed discussion.

Structure of Near-Thresholds-Wave Resonances

We study the structure of two-body s-wave bound states as well as resonances in the threshold energy region. We focus on the single-channel scattering where the scattering length and the effective range are given by real numbers. It is shown that, in the energy region where the effective range expansion is valid, the properties of resonances are constrained only by the position of the pole. We find that the compositeness defined through the analytic continuation of the field renormalization constant is purely imaginary and normalized for resonances. We discuss the interpretation of this quantity by examining the structure of the hadron resonance Λc(2595) in the πΣc scattering. We show that the Λc(2595) resonance requires an unnaturally large effective range and hence it is not likely a πΣc molecule.

Compositeness of dynamically generated states in a chiral unitary approach

The structure of dynamically generated states in the chiral unitary approach is studied from a viewpoint of their compositeness. We analyze the properties of bound states, virtual states, and resonances in a single-channel chiral unitary approach, paying attention to the energy dependence of the chiral interaction. We define the compositeness of a bound state using the field renormalization constant which is given by the overlap of the bare state and the physical state in the nonrelativistic quantum mechanics, or by the residue of the bound state propagator in the relativistic field theory. The field renormalization constant enables one to define a normalized quantitative measure of compositeness of the bound state. Applying this scheme to the chiral unitary approach, we find that the bound state generated by the energy-independent interaction is always a purely composite particle, while the energy-dependent chiral interaction introduces the elementary component, depending on the value of the cutoff parameter. This feature agrees with the analysis of the effective interaction by changing the cutoff parameter. A purely composite bound state can be realized by the chiral interaction only when the bound state lies at the threshold or when the strength of the two-body attractive interaction is infinitely large. The natural renormalization scheme, introduced by the property of the loop function and the matching with the chiral low-energy theorem, is shown to generate a bound state which is dominated by the composite structure when the binding energy is small.

Renormalization of quark propagators from twisted-mass lattice QCD atNf=2

We present results concerning the nonperturbative evaluation of the renormalization constant for the quark field, ${Z}_{q}$, from lattice simulations with twisted-mass quarks and three values of the lattice spacing. We use the regularization-invariant momentum-subtraction (RI'-MOM) scheme. ${Z}_{q}$ has very large lattice spacing artefacts; it is considered here as a test bed to elaborate accurate methods which will be used for other renormalization constants. We recall and develop the nonperturbative correction methods and propose tools to test the quality of the correction. These tests are also applied to the perturbative correction method. We check that the lattice-spacing artefacts indeed scale as ${a}^{2}{p}^{2}$. We then study the running of ${Z}_{q}$ with particular attention to the nonperturbative effects, presumably dominated by the dimension-two gluon condensate $⟨{A}^{2}⟩$ in Landau gauge. We show indeed that this effect is present, and not small. We check its scaling in physical units, confirming that it is a continuum effect. It gives a $\ensuremath{\sim}4%$ contribution at 2 GeV. Different variants are used in order to test the reliability of our result and estimate the systematic uncertainties. Finally, combining all our results and using the known Wilson coefficient of $⟨{A}^{2}⟩$, we find ${g}^{2}({\ensuremath{\mu}}^{2})⟨{A}^{2}{⟩}_{{\ensuremath{\mu}}^{2}\mathrm{CM}}=2.01(11){(}_{\ensuremath{-}0.73}^{+0.61}){\mathrm{GeV}}^{2}$ at $\ensuremath{\mu}=10\text{ }\text{ }\mathrm{GeV}$, the local operator ${A}^{2}$ being renormalized in the $\overline{\mathrm{MS}}$ scheme. This last result is in fair agreement within uncertainties with the value independently extracted from the strong coupling constant. We convert the nonperturbative part of ${Z}_{q}$ from the regularization-invariant momentum-subtraction (RI'-MOM) scheme to $\overline{\mathrm{MS}}$. Our result for the quark field renormalization constant in the $\overline{\mathrm{MS}}$ scheme is ${Z}_{q}^{\overline{\text{ }\mathrm{MS}}\text{ }\mathrm{pert}}((2\text{ }\text{ }\mathrm{GeV}{)}^{2},{g}_{\mathrm{bare}}^{2})=0.750(3)(7)\ensuremath{-}0.313(20)({g}_{\mathrm{bare}}^{2}\ensuremath{-}1.5)$ for the perturbative contribution and ${Z}_{q}^{\overline{\mathrm{MS}}\text{ }\mathrm{nonperturbative}}((2\text{ }\text{ }\mathrm{GeV}{)}^{2},{g}_{\mathrm{bare}}^{2})=0.781(6)(21)\ensuremath{-}0.313(20)({g}_{\mathrm{bare}}^{2}\ensuremath{-}1.5)$ when the nonperturbative contribution is included.

Pole mass, width, and propagators of unstable fermions

The concepts of pole mass and width are extended to unstable fermions in the general framework of parity-nonconserving gauge theories, such as the standard model. In contrast with the conventional on-shell definitions, these concepts are gauge independent and avoid severe unphysical singularities, properties of great importance since most fundamental fermions in nature are unstable particles. General expressions for the unrenormalized and renormalized dressed propagators of unstable fermions and their field-renormalization constants are presented.

Massive overlap fermions on anisotropic lattices

We formulate the massive overlap fermions on anisotropic lattices. We find that the dispersion relation for the overlap fermion resembles the continuum form in the low-momentum region once the bare parameters are properly tuned. The quark self-energy and the quark field renormalization constants are calculated to one-loop in bare lattice perturbation theory. We argue that massive domain wall quarks might be helpful in lattice QCD studies on heavy-light hadron spectroscopy.

Massive domain wall fermions on four-dimensional anisotropic lattices

We formulate the massive domain wall fermions on anisotropic lattices. For the massive domain wall fermion, we find that the dispersion relation assumes the usual form in the low momentum region when the bare parameters are properly tuned. The quark self-energy and the quark field renormalization constants are calculated to one-loop in bare lattice perturbation theory. For light domain wall fermions, we verified that the chiral mode is stable against quantum fluctuations on anisotropic lattices. This calculation serves as a guidance for the tuning of the parameters in the quark action in future numerical simulations.

Infinite reduction of couplings in non-renormalizable quantum field theory

I study the problem of renormalizing a non-renormalizable theory with a reduced, eventually finite, set of independent couplings. The idea is to look for special relations that express the coefficients of the irrelevant terms as unique functions of a reduced set of independent couplings lambda, such that the divergences are removed by means of field redefinitions plus renormalization constants for the lambda's. I consider non-renormalizable theories whose renormalizable subsector R is interacting and does not contain relevant parameters. The "infinite" reduction is determined by i) perturbative meromorphy around the free-field limit of R, or ii) analyticity around the interacting fixed point of R. In general, prescriptions i) and ii) mutually exclude each other. When the reduction is formulated using i), the number of independent couplings remains finite or slowly grows together with the order of the expansion. The growth is slow in the sense that a reasonably small set of parameters is sufficient to make predictions up to very high orders. Instead, in case ii) the number of couplings generically remains finite. The infinite reduction is a tool to classify the irrelevant interactions and address the problem of their physical selection.

Physical renormalization condition for the quark-mixing matrix

We investigate the renormalization of the quark-mixing matrix in the Electroweak Standard Model. The corresponding counterterms are gauge independent as can be shown using an extended BRS symmetry. Using rigid SU(2)L symmetry, we prove that the ultraviolet-divergent parts of the invariant counterterms are related to the field renormalization constants of the quark fields. We point out that for a general class of renormalization schemes rigid SU(2)L symmetry cannot be preserved in its classical form, but is renormalized by finite counterterms. Finally, we discuss a genuine physical renormalization condition for the quark-mixing matrix that is gauge independent and does not destroy the symmetry between quark generations.

Topics in electroweak physics

We briefly discuss five topics in Precision Electroweak Physics: i) the recently proposed Effective Scheme of Renormalization, ii) evidence for electroweak bosonic corrections derived from the radiative correction Δr eff , iii) an approach to estimate the scale of new physics in a hypothetical Higgs-less scenario, iv) simple and accurate formulae for s eff 2, M W , φ l , and their physical applications, v) a recent proposal concerning the field renormalization constant for unstable particles.

Field renormalization constant for unstable particles

A recently proposed normalization condition for the imaginary part of the self-energy of an unstable particle is shown to lead to a closed expression for the field renormalization constant Z. In turn, the exact expression for Z is necessary, in some important cases, in order to avoid power-like infrared divergences in high orders of perturbation theory. In the same examples, the width plays the r\^ole of an infrared cutoff and, consequently, Z is not an analytic function of the coupling constant.

Effective lagrangian for the two-dimensional [formula omitted]P 2 model at low temperatures

Recent Monte Carlo simulations for the two-dimensional R P N−1 models have shown that the matrix non-linear sigma models (NL σM) have a long-distance behavior which is different from that for the corresponding vector models (S N−1 ). This discrepancy cannot be explained by the renormalization group (RG) applied to the naive continuum lagrangians (O( N)-(vector NL σ) model). Motivated by this fact, we study the low-temperature long-distance effective lagrangian for the two-dimensional R P 2 model by means of the saddle-point (SP) plus renormalization methods. Introducing a matrix order parameter Q via the Hubbard-Stratonovich transformation we analyze the system of a macroscopic effective continuum theory for the Q-field. After taking the O(1)×O(2)-invariant SP for the potential we renormalize the massive fluctuations suppressed by the cut-off. This procedure allows us to find, in addition to the O(3)-model, a new term for the massless modes. Thanks to the new term, it is expected that the infrared RG behavior for the R P 2 model could be distinguished from the S 2 model. The effective lagrangian itself turns out to be perturbatively non-renormalizable. The RG analysis is formulated in the space of general O(2)-invariant non-linear lagrangians with an infinite number of coupling constants. The first-order RG analysis using the same field-renormalization constant as of the O(3)-model suggests that at long distances the effective lagrangian in the vicinity of zero temperature deviates from the O(3)-model.

Spontaneous symmetry breaking on the lattice generated by Yukawa interaction

We study by numerical simulation a lattice Yukawa model with naive fermions at intermediate values of the Yukawa coupling constant y when the nearest-neighbour coupling κ of the scalar field Φ is very weakly ferromagnetic ( κ ≈ 0) or even antiferromagnetic ( κ < 0) and the non-vanishing value of 〈Φ〉 is generated by the Yukawa interaction. The renormalized Yukawa coupling y R achieves here its maximal value and this y-region is thus of particular importance for lattice investigations of strong Yukawa interaction. However, here the scalar field propagators have a very complex structure caused by fermion loop corrections and by the proximity of phases with antiferromagnetic properties. We develop methods for analyzing these propagators and for extracting the physical observables. We find that going into the negative κ region, the scalar field renormalization constant becomes small and y R does not see, to exceed the unitarity bound, making the existence of a non-trivial fixed point in the investigated Yukawa model quite unlikely.

Gauge invariance and renormalization

The renormalization of Abelian and non-Abelian local gauge theories is discussed. It is recalled that whereas Abelian gauge theories are invariant to local c-number gauge transformations δA μ ( x) = ∂ μ ,…, with □ Λ = 0, and to the operator gauge transformation δA μ ( x) = ∂ μ φ( x), …, δφ( x) = α −1 ∂· A( x), with □ φ = 0, non-Abelian gauge theories are invariant only to the operator gauge transformations δA μ(x) ∼ ▪ μC(x) , …, introduced by Becchi, Rouet and Stora, where ▪ μ is the covariant derivative matrix and C is the vector of ghost fields. The renormalization of these gauge transformation is discussed in a formal way, assuming that a gauge-invariant regularization is present. The naive renormalized local non-Abelian c-number gauge transformation δA μ(x) = (Z 1/Z 3)gA μ(x) × Λ(x)+∂ μ Λ(x) , …, is never a symmetry transformation and is never finite in perturbation theory. Only for Λ(x) = (Z 3/Z 1)L with L finite constants or for Λ(x) = Ω z ̃ 3C(x) with Ω a finite constant does it become a finite symmetry transformation, where z ̃ 3 is the ghost field renormalization constant. The renormalized non-Abelian Ward-Takahashi (Slavnov-Taylor) identities are consequences of the invariance of the renormalized gauge theory to this formation. It is also shown how the symmetry generators are renormalized, how photons appear as Goldstone bosons, how the (non-multiplicatively renormalizable) composite operator A μ × C is renormalized, and how an Abelian c-number gauge symmetry may be reinstated in the exact solution of many asymptotically fr ee non-Abelian gauge theories.

Particle Mixing andJP=2+Meson Decays

Particle mixing for meson fields of arbitrary spin is considered in a field-theoretic context. Expressions for the field renormalization constants and mixing parameters are given in terms of the physical masses, assuming Okubo-type octet breaking. Broken-$\mathrm{SU}(3)$ coupling constants are evaluated for the ${2}^{+}$ meson decays and the predicted partial widths compared with experiment. Satisfactory agreement is found.

Renormalisation d’un doublet de bosons en interaction avec un doublet de fermions

A study is made of the remormalization of the system in which two bosons of the same tensorial type interact with two different pairs of unlike fermions. The renormalization constants of the vertices are first explicitly defined. Then to renormalize the two boson fields and boson masses, 3 + 3 renormalization constants are required. These are determined explicity as a function of the unrenormalized vacuum polarization operators. A prescription for the renormalization of these operators is given. The high-energy limits of the boson propagators are related to the field renormalization constants. The resulting formulae generalize, in matrix form, the usual formulae for the single boson theory. Some care must be taken for the generalization to the case of unstable bosons. The case of two bosons interacting with a single pair of fermions is also briefly discussed.