Dimensional Regularization Research Articles

Non-Wilsonian ultraviolet completion via transseries

We study some of the implications for the perturbative renormalization program when augmented with the Borel–Ecalle resummation. We show the emergence of a new kind of nonperturbative fixed point for the scalar [Formula: see text] model, representing an ultraviolet self-completion by transseries. We argue that this completion is purely non-Wilsonian and it depends on one arbitrary constant stemming from the transseries solution of the renormalization group equation. On the other hand, if no fixed points are demanded through the adjustment of this arbitrary constant, we end up with an effective theory in which the scalar mass is quadratically-sensitive to the cutoff, even working in dimensional regularization. Complete decoupling of the scalar mass to this energy scale can be used to determine a physical prescription for the Borel–Laplace resummation of the renormalons in nonasymptotically free models. We also comment on possible orthogonal scenarios available in the literature that might play a role when no fixed points exist.

Stability of asymptotically de Sitter and anti-de Sitter black holes in 4D regularized Einstein–Gauss–Bonnet theory

The regularized four-dimensional Einstein-Gauss-Bonnet model has been recently proposed in [D. Glavan and C. Lin, Phys. Rev. Lett. \textbf{124}, 081301 (2020)] whose formulation is different of the Einstein theory, allowing us to bypass the Lovelock theorem. The action is formulated in higher dimensions ($D>4$) by adding the Gauss-Bonnet correction to the conventional Einstein-Hilbert action with a cosmological constant. The four-dimensional spacetime is constructed through dimensional regularization by taking the limit $D\rightarrow 4$. We find explicitly the parametric regions of stability of black holes for the asymptotically flat and (anti-)de Sitter spacetimes by analyzing the time-domain profiles for gravitational perturbations in both vector and scalar channels. In addition to the known eikonal instability we find the instability due to the positive cosmological constant. On the contrary, asymptotically anti-de Sitter black holes have no other instability than the eikonal one.

\textsf {SCHOONSCHIP}, the Largest Time Equation and the Continuous Dimensional Regularisation

\textsf {SCHOONSCHIP}, the Largest Time Equation and the Continuous Dimensional Regularisation

Modern methods of calculations of bremsstrahlung in the interaction of elementary particles

. The problem of real bremsstrahlung calculation is considered using the modern methods of regularization of divergencies. In particular, we calculate soft photon bremsstrahlung in the most general form using the method of dimensional regularization of infrared divergences. The general calculation algorithm of hard photon bremsstrahlung is described. It is shown that the contribution of hard bremsstrahlung can be separated into the finite and divergent parts. The divergent part can be factorized with the contribution of the initial process in the Born approximation. It is shown that a good choice of kinematic variables makes an analytic covariant calculation of the divergent part of the hard bremsstrahlung possible. In a particular case, an algorithm for determining the kinematic constraints on the invariants is described. A numerical analysis of the radiative corrections for gauge bosons production processes in the case of electron-photon collisions is performed. It is discovered that the contribution of the finite part of bremsstrahlung at high collision energies reaches 20 per cent and must be taken into account in calculations of radiative corrections. The results obtained can be used in various calculations, including covariant ones, performed in the context of confirmation of the Standard Model theoretical predictions or searching for manifestations of alternative gauge models.

Loop corrections to celestial amplitudes

We study the effect of loop corrections to conformal correlators on the celestial sphere at null infinity. We first analyze finite one-loop celestial amplitudes in pure Yang-Mills theory and Einstein gravity. We then turn to our main focus: infrared divergent loop amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory. We compute the celestial one-loop amplitude in dimensional regularization and show that it can be recast as an operator acting on the celestial tree-level amplitude. This extends to any loop order and the re-summation of all planar loops enables us to write down an expression for the all-loop celestial amplitude. Finally, we show that the exponentiated all-loop expression given by the BDS formula gets promoted on the celestial sphere to an operator acting on the tree-level conformal correlation function, thus yielding, the celestial BDS formula.

Spectral functions in the ϕ4 -theory from the spectral Dyson-Schwinger equations

We develop a non-perturbative functional framework for computing real-time correlation functions in strongly correlated systems. The framework is based on the spectral representation of correlation functions and dimensional regularisation. Therefore, the non-perturbative spectral renormalisation setup here respects all symmetries of the theories at hand. In particular this includes space-time symmetries as well as internal symmetries such as chiral symmetry, and gauge symmetries. Spectral renormalisation can be applied within general functional approaches such as the functional renormalisation group, Dyson-Schwinger equations, and two- or $n$-particle irreducible approaches. As an application we compute the full, non-perturbative, spectral function of the scalar field in the $\phi^4$-theory in $2+1$ dimensions from spectral Dyson-Schwinger equations. We also compute the $s$-channel spectral function of the full $\phi^4$-vertex in this theory.

Diffeomorphism invariance demands conformal anomalies

We study a series of the Wess-Zumino actions obtained by repeatedly integrating conformal anomalies with respect to the conformal-factor field that appear at higher loops. We show that they arise as physical quantities required to make nonlocal loop correction terms diffeomorphism invariant. Specifically, in a conformally flat spacetime $ds^2=e^{2\phi}(-d\eta^2 + d{\bf x}^2)$, we find that effective actions are described in terms of momentum squared expressed as a physical $Q^2 = q^2/e^{2\phi}$ for $q^2$ measured by the flat metric, which recalls the relationship between physical momentum and comoving momentum in cosmology. It is confirmed by calculating the effective action of QED in such a curved spacetime at the 3-loop level using dimensional regularization. The same applies to the case of QCD, in which we show that the effective action can be summarized in the form of the reciprocal of a running coupling constant squared described by the physical momentum. We also see that the same holds for renormalizable quantum conformal gravity and that conformal anomalies are indispensable for formulating the theory.

Critical Fermi surfaces in generic dimensions arising from transverse gauge field interactions

We study critical Fermi surfaces in generic dimensions arising from coupling finite-density fermions with transverse gauge fields, by applying the dimensional regularization scheme developed previously [Phys. Rev. B 92, 035141 (2015)]. We consider the cases of $U(1)$ and $U(1)\times U(1)$ transverse gauge couplings, and extract the nature of the renormalization group (RG) flow fixed points as well as the critical scalings. Our analysis allows us to treat a critical Fermi surface of a generic dimension $m$ perturbatively in an expansion parameter $\epsilon =\left (2-m \right ) /\left (m+1 \right).$ One of our key results is that although the two-loop corrections do not alter the existence of an RG flow fixed line for certain $U(1)\times U(1)$ theories, which was identified earlier for $m=1$ at one-loop order, the third-order diagrams do. However, this fixed line feature is also obtained for $m>1$, where the answer is one-loop exact due to UV/IR mixing.

Renormalization group equations and the recurrence pole relations in pure quantum gravity

In the framework of dimensional regularization, we propose a generalization of the renormalization group equations in the case of the perturbative quantum gravity that involves renormalization of the metric and of the higher order Riemann curvature couplings. The case of zero cosmological constant is considered. Solving the renormalization group (RG) equations we compute the respective beta functions and derive the recurrence relations, valid at any order in the Newton constant, that relate the higher pole terms 1/(d−4)n to a single pole 1/(d−4) in the quantum effective action. Using the recurrence relations we find the exact form for the higher pole counter-terms that appear in 2, 3 and 4 loops and we make certain statements about the general structure of the higher pole counter-terms in any loop. We show that the complete set of the UV divergent terms can be consistently (at any order in the Newton constant) hidden in the bare gravitational action, that includes the terms of higher order in the Riemann tensor, provided the metric and the higher curvature couplings are renormalized according to the RG equations.

A novel algorithm for nested summation and hypergeometric expansions

We consider a class of sums over products of Z-sums whose arguments differ by a symbolic integer. Such sums appear, for instance, in the expansion of Gauss hypergeometric functions around integer indices that depend on a symbolic parameter. We present a telescopic algorithm for efficiently converting these sums into generalized polylogarithms, Z-sums, and cyclotomic harmonic sums for generic values of this parameter. This algorithm is illustrated by computing the double pentaladder integrals through ten loops, and a family of massive self-energy diagrams through mathcal{O}left({epsilon}^6right) in dimensional regularization. We also outline the general telescopic strategy of this algorithm, which we anticipate can be applied to other classes of sums.

One-Loop order effects from one extra universal dimension on λϕ 4 theory

The self-interacting λ ϕ 4 scalar field theory is the prototype in the study of basic renormalisation issues in quantum field theories. In this paper we explore the one-loop order divergent impact to this theory from one universal extra dimension, , to the self-energy, four point vertex and the beta function. Such effects come as an infinite number of UV divergences corresponding to an infinite superposition of excited KK particles around the loop. We show that dimensional regularisation is adequate enough to control all of them in terms of the product of the one dimensional inhomogeneous Epstein zeta function times the gamma function. From the analytical properties of these functions, the UV divergences are extracted and the counterterms defined; the latter turn out to be of canonical dimension four at the Lagrangian level. We use both, the MS-scheme and a mass-dependent subtraction scheme to remove divergences. Only the latter manifestly satisfy the decoupling theorem.

Logarithmic enhancements in conformal perturbation theory and their real time interpretation

We study various corrections of correlation functions to leading order in conformal perturbation theory, both on the cylinder and on the plane. Many problems on the cylinder are mathematically equivalent to those in the plane if we give the perturbations a position dependent scaling profile. The integrals to be done are then similar to those in the study of correlation functions with one additional insertion at the center of the profile. We will be primarily interested in the divergence structure of these corrections when computed in dimensional regularization. In particular, we show that the logarithmic divergences (enhancements) that show up in the plane under these circumstances can be understood in terms of resonant behavior in time dependent perturbation theory, for a transition between states that is induced by an oscillatory perturbation on the cylinder.

An exponential regulator for rapidity divergences

Finding an efficient and compelling regularization of soft and collinear degrees of freedom at the same invariant mass scale, but separated in rapidity is a persistent problem in high-energy factorization. In the course of a calculation, one encounters divergences unregulated by dimensional regularization, often called rapidity divergences. Once regulated, a general framework exists for their renormalization, the rapidity renormalization group (RRG), leading to fully resummed calculations of transverse momentum (to the jet axis) sensitive quantities. We examine how this regularization can be implemented via a multi-differential factorization of the soft-collinear phase-space, leading to an (in principle) alternative non-perturbative regularization of rapidity divergences. As an example, we examine the fully-differential factorization of a color singlet's momentum spectrum in a hadron-hadron collision at threshold. We show how this factorization acts as a mother theory to both traditional threshold and transverse momentum resummation, recovering the classical results for both resummations. Examining the refactorization of the transverse momentum beam functions in the threshold region, we show that one can directly calculate the rapidity renormalized function, while shedding light on the structure of joint resummation. Finally, we show how using modern bootstrap techniques, the transverse momentum spectrum is determined by an expansion about the threshold factorization, leading to a viable higher loop scheme for calculating the relevant anomalous dimensions for the transverse momentum spectrum.

The mass quadrupole moment of compact binary systems at the fourth post-Newtonian order

The mass-type quadrupole moment of inspiralling compact binaries (without spins) is computed at the fourth post-Newtonian (4PN) approximation of general relativity. The multipole moments are defined by matching between the field in the exterior zone of the matter system and the PN field in the near zone, following the multipolar-post-Minkowskian (MPM)-PN formalism. The matching implies a specific regularization for handling infra-red (IR) divergences of the multipole moments at infinity, based on the Hadamard finite part procedure. On the other hand, the calculation entails ultra-violet (UV) divergences due to the modelling of compact objects by delta-functions, that are treated with dimensional regularization (DR). In future work we intend to systematically study the IR divergences by means of dimensional regularization as well. Our result constitutes an important step in the goal of obtaining the gravitational wave templates of inspiralling compact binary systems with 4PN/4.5PN accuracy.

D→4 Einstein-Gauss-Bonnet gravity and beyond

A 'novel' pure theory of Einstein-Gauss-Bonnet gravity in four-spacetime dimensions can be constructed by rescaling the Gauss-Bonnet coupling constant, seemingly eluding Lovelock's theorem. Recently, however, the well-posedness of this model has been called into question. Here we apply a `dimensional regularization' technique, first used by Mann and Ross to write down a D→2 limit of general relativity, to the case of pure Einstein-Gauss-Bonnet gravity. The resulting four-dimensional action is a particular Horndeski theory of gravity matching the result found via a Kaluza-Klein reduction over a flat internal space. Some cosmological solutions of this four-dimensional theory are examined. We further adapt the technique to higher curvature Lovelock theories of gravity, as well as a low-energy effective string action with an α′ correction. With respect to the D→4 limit of the α′-corrected string action, we find we must also rescale the dilaton to have a non-singular action in four dimensions. Interestingly, when the conformal rescaling Φ is interpreted as another dilaton, the regularized string action appears to be a special case of a covariant multi-Galileon theory of gravity.

Large-x resummation of off-diagonal deep-inelastic parton scattering from d-dimensional refactorization

The off-diagonal parton-scattering channels g + γ* and q + ϕ* in deep-inelastic scattering are power-suppressed near threshold x → 1. We address the next-to-leading power (NLP) resummation of large double logarithms of 1 − x to all orders in the strong coupling, which are present even in the off-diagonal DGLAP splitting kernels. The appearance of divergent convolutions prevents the application of factorization methods known from leading power resummation. Employing d-dimensional consistency relations from requiring 1/ϵ pole cancellations in dimensional regularization between momentum regions, we show that the resummation of the off-diagonal parton-scattering channels at the leading logarithmic order can be bootstrapped from the recently conjectured exponentiation of NLP soft-quark Sudakov logarithms. In particular, we derive a result for the DGLAP kernel in terms of the series of Bernoulli numbers found previously by Vogt directly from algebraic all-order expressions. We identify the off-diagonal DGLAP splitting functions and soft-quark Sudakov logarithms as inherent two-scale quantities in the large-x limit. We use a refactorization of these scales and renormalization group methods inspired by soft-collinear effective theory to derive the conjectured soft-quark Sudakov exponentiation formula.

Tree-level splitting amplitudes for a gluon into four collinear partons

We compute in conventional dimensional regularisation the tree-level splitting amplitudes for a gluon parent which splits into four collinear partons. This is part of the universal infrared behaviour of the QCD scattering amplitudes at next-to-next-to-next-to-leading order (N3LO) in the strong coupling constant. Combined with our earlier results for a quark parent, this completes the set of tree-level splitting amplitudes required at this order. We also study iterated collinear limits where a subset of the four collinear partons become themselves collinear.

One-loop corrections to the primordial tensor spectrum from massless isocurvature fields

We study one-loop corrections to the two-point correlation function of tensor perturbations in primordial cosmology induced by massless spectator matter fields. Using the Schwinger-Keldysh formalism in cosmological perturbation theory, we employ dimensional regularization and cutoff regularization to study the finite quantum corrections at one-loop arising from isocurvature fields of the massless scalar, fermion and abelian gauge field which are freely propagating on the FRW spacetime. For all cases, we find a logarithmic running of the form frac{C}{q^3}frac{H^4}{M_p^4} log left(frac{H}{mu}right) where C is a negative constant related to the beta function, H is the Hubble parameter at horizon exit and μ is the renormalization scale.

Power expansion for heavy quarkonium production at next-to-leading order in e+e\u2212 annihilation

We study heavy quarkonium production associated with gluons in e+e− annihilation as an illustration of the perturbative QCD (pQCD) factorization approach, which incorporates the first nonleading power in the energy of the produced heavy quark pair. We show how the renormalization of the four-quark operators that define the heavy quark pair fragmentation functions using dimensional regularization induces “evanescent” operators that are absent in four dimensions. We derive closed forms for short-distance coefficients for quark pair production to next-to-leading order ( {alpha}_s^2 ) in the relevant color singlet and octet channels. Using non-relativistic QCD (NRQCD) to calculate the heavy quark pair fragmentation functions up to v4 in the velocity expansion, we derive analytical results for the differential energy fraction distribution of the heavy quarkonium. Calculations for {}^3{S}_1^{left[1right]} and {}^1{S}_0^{left[8right]} channels agree with analogous NRQCD analytical results available in the literature, while several color-octet calculations of energy fraction distributions are new. We show that the remaining corrections due to the heavy quark mass fall off rapidly in the energy of the produced state. To explore the importance of evolution at energies much larger than the mass of the heavy quark, we solve the renormalization group equation perturbatively to two-loop order for the {}^1{S}_0^{left[8right]} case.

Running gravitational couplings, decoupling, and curved spacetime renormalization

We propose to slightly generalize the DeWitt-Schwinger adiabatic renormalization subtractions in curved space to include an arbitrary renormalization mass scale $\mu$. The new predicted running for the gravitational couplings are fully consistent with decoupling of heavy massive fields. This is a somewhat improvement with respect to the more standard treatment of minimal (DeWitt-Schwinger) subtractions via dimensional regularization. We also show how the vacuum metamorphosis model emerges from the running couplings.