A Simple Introduction to Soft Resummation

We clarify the notion of Wilsonian renormalization group (RG) invariance in supersymmetric gauge theories, which states that the low-energy physics can be kept fixed when one changes the ultraviolet cutoff, provided appropriate changes are made to the bare coupling constants in the Lagrangian. We first pose a puzzle on how a quantum modified constraint (such as Pf(Q^i Q^j) = \Lambda^{2(N+1)} in SP(N) theories with N+1 flavors) can be RG invariant, since the bare fields Q^i receive wave function renormalization when one changes the ultraviolet cutoff, while we naively regard the scale \Lambda as RG invariant. The resolution is that \Lambda is not RG invariant if one sticks to canonical normalization for the bare fields as is conventionally done in field theory. We derive a formula for how \Lambda must be changed when one changes the ultraviolet cutoff. We then compare our formula to known exact results and show that their consistency requires the change in \Lambda we have found. Finally, we apply our result to models of supersymmetry breaking due to quantum modified constraints. The RG invariance helps us to determine the effective potential along the classical flat directions found in these theories. In particular, the inverted hierarchy mechanism does not occur in the original version of these models.

In conventional treatments, predictions from fixed-order perturbative QCD calculations cannot be fixed with certainty due to ambiguities in the choice of the renormalization scale as well as the renormalization scheme. In this paper we present a general discussion of the constraints of the renormalization group (RG) invariance on the choice of the renormalization scale. We adopt the RG based equations, which incorporate the scheme parameters, for a general exposition of RG invariance, since they simultaneously express the invariance of physical observables under both the variation of the renormalization scale and the renormalization scheme parameters. We then discuss the self-consistency requirements of the RG, such as reflexivity, symmetry, and transitivity, which must be satisfied by the scale-setting method. The Principle of Minimal Sensitivity (PMS) requires the slope of the approximant of an observable to vanish at the renormalization point. This criterion provides a scheme-independent estimation, but it violates the symmetry and transitivity properties of the RG and does not reproduce the Gell-Mann-Low scale for QED observables. The Principle of Maximum Conformality (PMC) satisfies all of the deductions of the RG invariance - reflectivity, symmetry, and transitivity. Using the PMC, all non-conformal $\{\beta^{\cal R}_i\}$-terms (${\cal R}$ stands for an arbitrary renormalization scheme) in the perturbative expansion series are summed into the running coupling, and one obtains a unique, scale-fixed, scheme-independent prediction at any finite order. The PMC scales and the resulting finite-order PMC predictions are both to high accuracy independent of the choice of initial renormalization scale, consistent with RG invariance. [...More in the text...]

We present a combined resummation for the transverse momentum distribution of a colorless final state in perturbative QCD, expressed as a function of transverse momentum pT and the scaling variable x. Its expression satisfies three requirements: it reduces to standard transverse momentum resummation to any desired logarithmic order in the limit pT → 0 for fixed x, up to power suppressed corrections in pT; it reduces to threshold resummation to any desired logarithmic order in the limit x → 1 for fixed pT, up to power suppressed correction in 1 − x; upon integration over transverse momentum it reproduces the resummation of the total cross cross at any given logarithmic order in the threshold x → 1 limit, up to power suppressed correction in 1 − x. Its main ingredient, and our main new result, is a modified form of transverse momentum resummation, which leads to threshold resummation upon integration over pT , and for which we provide a simple closed-form analytic expression in Fourier-Mellin (b, N ) space. We give explicit coefficients up to NNLL order for the specific case of Higgs production in gluon fusion in the effective field theory limit. Our result allows for a systematic improvement of the transverse momentum distribution through threshold resummation which holds for all pT, and elucidates the relation between transverse momentum resummation and threshold resummation at the inclusive level, specifically by providing within perturbative QCD a simple derivation of the main consequence of the so-called collinear anomaly of SCET.

The renormalization scale-setting problem in QCD

We derive one-loop renormalization group (RG) invariant observables and analyze their phenomenological implications in the MSSM and its \mu problem solving extensions, U(1)' model and NMSSM. We show that there exist several RG invariants in the gauge, Yukawa and soft-breaking sectors of each model. In general, RG invariants are highly useful for projecting experimental data to messenger scale, for revealing correlations among the model parameters, and for probing the mechanism that breaks supersymmetry. The Yukawa couplings and trilinear soft terms in U(1)' model and NMSSM do not form RG invariants though there exist approximate invariants in low tan(beta). In the NMSSM, there are no invariants that contain the Higgs mass-squareds. We provide a comparative analysis of RG invariants in all three models and analyze their model-building and phenomenological implications by a number of case studies.

A valid prediction for a physical observable from quantum field theory should be independent of the choice of renormalization scheme—this is the primary requirement of renormalization group invariance (RGI). Satisfying scheme invariance is a challenging problem for perturbative QCD (pQCD), since a truncated perturbation series does not automatically satisfy the requirements of the renormalization group. In a previous review, we provided a general introduction to the various scale setting approaches suggested in the literature. As a step forward, in the present review, we present a discussion in depth of two well-established scale-setting methods based on RGI. One is the ‘principle of maximum conformality’ (PMC) in which the terms associated with the β-function are absorbed into the scale of the running coupling at each perturbative order; its predictions are scheme and scale independent at every finite order. The other approach is the ‘principle of minimum sensitivity’ (PMS), which is based on local RGI; the PMS approach determines the optimal renormalization scale by requiring the slope of the approximant of an observable to vanish. In this paper, we present a detailed comparison of the PMC and PMS procedures by analyzing two physical observables Re+e− and up to four-loop order in pQCD. At the four-loop level, the PMC and PMS predictions for both observables agree within small errors with those of conventional scale setting assuming a physically-motivated scale, and each prediction shows small scale dependences. However, the convergence of the pQCD series at high orders, behaves quite differently: the PMC displays the best pQCD convergence since it eliminates divergent renormalon terms; in contrast, the convergence of the PMS prediction is questionable, often even worse than the conventional prediction based on an arbitrary guess for the renormalization scale. PMC predictions also have the property that any residual dependence on the choice of initial scale is highly suppressed even for low-order predictions. Thus the PMC, based on the standard RGI, has a rigorous foundation; it eliminates an unnecessary systematic error for high precision pQCD predictions and can be widely applied to virtually all high-energy hadronic processes, including multi-scale problems.

Our numerical experiments aim to study the applicability of the renormalization group (RG) idea to multichain lattice systems. The systems are composed of multiple chains on a simple cubic lattice. The nearest-neighbor interaction is assumed between nonbonded chain elements. The RG parameter space is spanned by the inverse of the chain length 1/n, the solvent condition η, and the chain number concentration ρ. We first tentatively determine parameter flows by using two RG invariants (weak conditions). Then we examine whether the flows satisfy the RG requirements for the correlations (strong conditions). Simultaneous satisfaction will guarantee us the applicability of the RG idea. We found that the RG idea is applicable to the systems in the very good solvent region but becomes inapplicable to those in the poorer solvent regions and to the shorter spatial distances. In the range where the linearized RG defined by a fixed point in the good solvent region works successfully, we arrive at scaling forms in several fashions. One of them corresponds to Freed’s crossover form. The RG appears to work even in the nonlinear range, i.e., concentrated solutions outside the semidilute region. The RG failure in the poor solvent region is briefly discussed.

We calculate nonperturbative renormalization factors at hadronic scale for $\ensuremath{\Delta}S=2$ four-quark operators in quenched domain-wall QCD using the Schr\"odinger functional method. Combining them with the nonperturbative renormalization group running by the Alpha Collaboration, our result yields the fully nonperturbative renormalization factor, which converts the lattice bare ${B}_{K}$ to the renormalization group invariant (RGI) ${\stackrel{^}{B}}_{K}$. Applying this to the bare ${B}_{K}$ previously obtained by the CP-PACS Collaboration at ${a}^{\ensuremath{-}1}\ensuremath{\simeq}2,3,4\text{ }\text{ }\mathrm{GeV}$, we obtain ${\stackrel{^}{B}}_{K}=0.782(5)(7)$ [equivalent to ${B}_{K}^{\overline{\mathrm{MS}}}(\mathrm{NDR},2\text{ }\text{ }\mathrm{GeV})=0.565(4)(5)$ by two-loop running] in the continuum limit, where the first error is statistical and the second is systematic due to the continuum extrapolation. Except the quenching error, the total error we have achieved is less than 2%, which is much smaller than the previous ones. Taking the same procedure, we obtain ${m}_{u,d}^{\mathrm{RGI}}=5.613(66)\text{ }\text{ }\mathrm{MeV}$ and ${m}_{s}^{\mathrm{RGI}}=147.1(17)\text{ }\text{ }\mathrm{MeV}$ [equivalent to ${m}_{u,d}^{\overline{\mathrm{MS}}}(2\text{ }\text{ }\mathrm{GeV})=4.026(48)\text{ }\text{ }\mathrm{MeV}$ and ${m}_{s}^{\overline{\mathrm{MS}}}(2\text{ }\text{ }\mathrm{GeV})=105.6(12)\text{ }\text{ }\mathrm{MeV}$ by four-loop running] in the continuum limit.

We study inflation as a ‘cosmic’ renormalization-group (RG) flow. The flow, which encodes the dependence on the background metric, is described by a running coupling α, which parametrizes the slow roll, a de Sitter free, analytic beta function and perturbation spectra that are RG invariant in the superhorizon limit. Using RG invariance as a guiding principle, we classify the main types of flows according to the properties of their spectra, without referring to their origins from specific actions or models. Novel features include spectra with essential singularities in α and violations of the relation r + 8n t = 0 to the leading order. Various classes of potentials studied in the literature can be described by means of the RG approach, even when the action includes a Weyl-squared term, while others are left out. In known cases, the classification helps identify the models that are ruled out by data. The RG approach is also able to generate spectra that cannot be derived from standard Lagrangian formulations.

We establish a relation between bijective functions and renormalization group transformations and find their renormalization group invariants. For these functions, taking into account that they are globally one-to-one, we propose several improved approximations (compared with the power series expansion) based on this relation. We propose using the obtained approximations to improve the subsequent approximations of physical quantities obtained, in particular, by one of the main calculation techniques in theoretical physics, i.e., by perturbation theory. We illustrate the effectiveness of the renormalization group approximation with several examples: renormalization group approximations of several analytic functions and calculation of the nonharmonic oscillator ground-state energy. We also generalize our approach to the case of set maps, both continuous and discrete.

We study constraints among coupling constants of the standard model obtained in the noncommutative geometry (NCG) method. First, we analyze the evolution of the Higgs boson mass under the renormalization group by adopting the idea of \'Alvarez et al. For this analysis we derive two certain constraints by modifying Connes's way of constructing the standard model. Next, we find renormalization group invariant (RGI) constraints in the NCG method. We also consider the relation between the condition that a constraint among coupling constants of a model becomes RGI and the condition that the model becomes multiplicative renormalizable by using a simple example.

We prove the renormalization group (RG) invariance of the pole mass with respect to the RG functions of the minimal subtraction (MS) scheme and illustrate this in the case of the neutral scalar field theory both in the symmetric phase and in the broken symmetry phase. In the case of the broken symmetry phase, the correct RG invariance of the pole mass is shown to appear after the substitution of the vacuum expectation value (VEV) explicitly as a function of the Lagrangian parameters.

The QCD renormalization group equation and the elimination of fixed-order scheme-and-scale ambiguities using the principle of maximum conformality

We propose a novel method for renormalization group improvement of thermally resummed effective potential. In our method, β-functions are temperature dependent as a consequence of the divergence structure in resummed perturbation theory. In contrast to the ordinary MS¯ scheme, the renormalization group invariance of the resummed finite-temperature effective potential holds order by order, which significantly mitigates a notorious renormalization scale dependence of phase transition quantities such as a critical temperature even at the one-loop order. We also devise a tractable method that enables one to incorporate temperature-dependent higher-order corrections by fully exploiting the renormalization group invariance. Published by the American Physical Society 2024

We determine the strong coupling constant αs from the τ hadronic width using a renormalization group summed (RGS) expansion of the QCD Adler function. The main theoretical uncertainty in the extraction of αs is due to the manner in which renormalization group invariance is implemented, and the as yet uncalculated higher order terms in the QCD perturbative series. We show that new expansion exhibits good renormalization group improvement and the behavior of the series is similar to that of the standard CIPT expansion. The value of the strong coupling in [Formula: see text] scheme obtained with the RGS expansion is [Formula: see text]. The convergence properties of the new expansion can be improved by Borel transformation and analytic continuation in the Borel plane. This is discussed elsewhere in these issues.