Power Counting Arguments Research Articles

An Effective Field Theory Study of Medium Heavy Quark Evolution

The evolution of hard probes in a medium is a complex multiscale problem that significantly benefits from the use of Effective Field Theories (EFTs). Within the EFT framework, we aim to define a series of EFTs in a way that addresses each energy scale individually in separate steps. However, studying hard probes in a medium presents challenges. This is because an EFT is typically constructed by formulating the most general Lagrangian compatible with the problem’s symmetries. Nevertheless, medium effects may not always be encoded adequately in an effective action. In this paper, we construct an EFT that is valid for studying the evolution of a heavy quark in a QCD plasma containing few other heavy quarks, where degrees of freedom with an energy of the order of the temperature scale are integrated out. Through this example, we explicitly demonstrate how to handle the doubling of degrees that arise in non-equilibrium field theory. As a result, we derive a Fokker–Planck equation using only symmetry and power counting arguments. The methods introduced in this paper will pave the way for future developments in the study of quarkonium suppression.

Power counting energy flow polynomials

Power counting is a systematic strategy for organizing collider observables and their associated theoretical calculations. In this paper, we use power counting to characterize a class of jet substructure observables called energy flow polynomials (EFPs). EFPs provide an overcomplete linear basis for infrared-and-collinear safe jet observables, but it is known that in practice, a small subset of EFPs is often sufficient for specific jet analysis tasks. By applying power counting arguments, we obtain linear relationships between EFPs that hold for quark and gluon jets to a specific order in the power counting. We test these relations in the parton shower generator Pythia, finding excellent agreement. Power counting allows us to truncate the basis of EFPs without affecting performance, which we corroborate through a study of quark-gluon tagging and regression.

CFTs blueshift tensor fluctuations universally

The strong constraints of conformal symmetry cause any nearly-conformal sector to blueshift tensor fluctuations in cosmology. Hidden sectors with approximate conformal symmetry, which may be quite large, are a well-motivated extension of physics beyond the Standard Models of particle physics and cosmology. They can therefore lead to a detectable shift in the tensor tilt for next-generation CMB and gravitational wave experiments. We compute the leading-order contribution to the in-in graviton two-point function from virtual loops in such sectors to demonstrate this universal effect. In units where a single conformally-coupled scalar is 1, limits from Stage-IV CMB experiments could bound the size of this extra sector to be smaller than ∼1015, under a plausible calculational assumption backed by a simple power counting argument. This would be sufficient to rule out N-Naturalness as a complete resolution of the hierarchy problem.

Patterns of C- and CP-violation in hadronic η and η′ three-body decays

We construct hadronic amplitudes for the three-body decays η(′) → π+π−π0 and η′ → ηπ+π− in a non-perturbative fashion, allowing for C- and CP-violating asymmetries in the π+π− distributions. These amplitudes are consistent with the constraints of analyticity and unitarity. We find that the currently most accurate Dalitz-plot distributions taken by the KLOE-2 and BESIII collaborations confine the patterns of these asymmetries to a relative per mille and per cent level, respectively. Our dispersive representation allows us to extract the individual coupling strengths of the C- and CP-violating contributions arising from effective isoscalar and isotensor operators in η(′) → π+π−π0 and an effective isovector operator in η′ → ηπ+π−, while the strongly different sensitivities to these operators can be understood from chiral power counting arguments.

The P_{cs}(4459) pentaquark from a combined effective field theory and phenomenological perspective

The observation of the P_{cs}(4459) by the LHCb collaboration adds a new member to the set of known hidden-charm pentaquarks, which includes the P_c(4312), P_c(4440) and P_c(4457). The P_{cs}(4459) is expected to have the light-quark content of a Lambda baryon (I=0, S=-1), but its spin is unknown. Its closeness to the {bar{D}}^* Xi _c threshold – 4478,{mathrm{MeV}} in the isospin-symmetric limit – suggests the molecular hypothesis as a plausible explanation for the P_{cs}(4459). While in the absence of coupled-channel dynamics heavy-quark spin symmetry predicts the two spin-states of the {bar{D}}^* Xi _c to be degenerate, power counting arguments indicate that the coupling with the nearby {bar{D}} Xi _c' and {bar{D}} Xi _c^* channels might be a leading order effect. This generates a hyperfine splitting in which the J=tfrac{3}{2}{bar{D}}^* Xi _c pentaquark will be lighter than the J=tfrac{1}{2} configuration, which we estimate to be of the order of 5-15,{mathrm{MeV}}. We also point out an accidental symmetry between the P_{cs}(4459) and P_c(4440/4457) potentials. Finally, we argue that the spectroscopy and the J/psi Lambda decays of the P_{cs}(4459) might suggest a marginal preference for J = tfrac{3}{2} over J = tfrac{1}{2}.

Quantum stability of generalized Proca theories

We establish radiative stability of generalized Proca effective field theories. While standard power-counting arguments would conclude otherwise, we find non-trivial cancellations of leading order corrections by explicit computation of divergent one-loop diagrams up to four-point. These results are crosschecked against an effective action based generalized Schwinger–DeWitt method. Further, the cancellations are understood as coming from the specific structure of the theory through a decoupling limit analysis which at the same time allows for an extension of the results to higher orders.

Note on the absence of R2 corrections to Newton’s potential

We consider Einstein gravity with the addition of $R^2$ and $R^{\mu \nu} R_{\mu \nu}$ interactions in the context of effective field theory, and the corresponding scattering amplitudes of gravitons and minimally-coupled heavy scalars. First, we recover the known fact that graviton amplitudes are the same as in Einstein gravity. Then we show that all amplitudes with two heavy scalars and an arbitrary number of gravitons are also not affected by these interactions. We prove this by direct computations, using field redefinitions known from earlier applications in string theory, and with a combination of factorisation and power-counting arguments. Combined with unitarity, these results imply that, in an effective field theory approach, the Newtonian potential receives neither classical nor quantum corrections from terms quadratic in the curvature.

High-precision numerical solution of the Fermi polaron problem and large-order behavior of its diagrammatic series

We introduce a simple determinant diagrammatic Monte Carlo algorithm to compute the ground-state properties of a particle interacting with a Fermi sea through a zero-range interaction. The fermionic sign does not cause any fundamental problem when going to high diagram orders, and we reach order $N=30$. The data reveal that the diagrammatic series diverges exponentially as $(-1/R)^{N}$ with a radius of convergence $R<1$. Furthermore, on the polaron side of the polaron-dimeron transition, the value of $R$ is determined by a special class of three-body diagrams, corresponding to repeated scattering of the impurity between two particles of the Fermi sea. A power-counting argument explains why finite $R$ is possible for zero-range interactions in three dimensions. Resumming the divergent series through a conformal mapping yields the polaron energy with record accuracy.

Two-loop massive quark jet functions in SCET

We calculate the O( {alpha}_s^2 ) corrections to the primary massive quark jet functions in Soft-Collinear Effective Theory (SCET). They are an important ingredient in factorized predictions for inclusive jet mass cross sections initiated by massive quarks emerging from a hard interaction with smooth quark mass dependence. Due to the effects coming from the secondary production of massive quark-antiquark pairs there are two options to define the SCET jet function, which we call universal and mass mode jet functions. They are related to whether or not a soft mass mode (zero) bin subtraction is applied for the secondary massive quark contributions and differ in particular concerning the infrared behavior for vanishing quark mass. We advocate that a useful alternative to the common zero-bin subtraction concept is to define the SCET jet functions through subtractions related to collinear-soft matrix elements. This avoids the need to impose additional power counting arguments as required for zero-bin subtractions. We demonstrate how the two SCET jet function definitions may be used in the context of two recently developed factorization approaches to treat secondary massive quark effects. We clarify the relation between these approaches and in which way they are equivalent. Our two-loop calculation involves interesting technical subtleties related to spurious rapidity divergences and infrared regularization in the presence of massive quarks.

Splittings of low-lying charmonium masses at the physical point

We present high-precision results from lattice QCD for the mass splittings of the low-lying charmonium states. For the valence charm quark, the calculation uses Wilson-clover quarks in the Fermilab interpretation. The gauge-field ensembles are generated in the presence of up, down, and strange sea quarks, based on the improved staggered (asqtad) action, and gluon fields, based on the one-loop, tadpole-improved gauge action. We use five lattice spacings and two values of the light sea quark mass to extrapolate the results to the physical point. An enlarged set of interpolating operators is used for a variational analysis to improve the determination of the energies of the ground states in each channel. We present and implement a continuum extrapolation within the Fermilab interpretation, based on power-counting arguments, and thoroughly discuss all sources of systematic uncertainty. We compare our results for various mass splittings with their experimental values, namely, the 1S hyperfine splitting, the 1P-1S splitting and the P-wave spin-orbit and tensor splittings. Given the uncertainty related to the width of the resonances, we find excellent agreement.

Systematics of azimuthal anisotropy harmonics in proton–nucleus collisions at the LHC from the Color Glass Condensate

Simple power counting arguments in the dilute-dense framework of the Color Glass Condensate (CGC) Effective Field Theory predict that even and odd azimuthal anisotropy harmonics of two-particle correlations in proton–nucleus collisions at the LHC will respectively satisfy v2n2{2}∝Nch0 and v2n+12{2}∝Nch, where Nch denotes the number of charged particles. We show that these expectations are borne out qualitatively, and even quantitatively, within systematic uncertainties, for v2 and v4 in comparisons with data from the ATLAS collaboration. We also observe that ATLAS data for the v3 azimuthal harmonic are in excellent agreement with our qualitative expectation; quantitative comparisons are numerically challenging at present. The lessons from this study fully complement those gained by the recent comparison of the CGC dilute-dense framework [1] to data from the PHENIX collaboration on small system collisions at RHIC.

Kinetic theory of pattern formation in mixtures of microtubules and molecular motors.

In this study we formulate a theoretical approach, based on a Boltzmann-like kinetic equation, to describe pattern formation in two-dimensional mixtures of microtubular filaments and molecular motors. Following the previous work by Aranson and Tsimring [Phys. Rev. E 74, 031915 (2006)PLEEE81539-375510.1103/PhysRevE.74.031915] we model the motor-induced reorientation of microtubules as collision rules, and devise a semianalytical method to calculate the corresponding interaction integrals. This procedure yields an infinite hierarchy of kinetic equations that we terminate by employing a well-established closure strategy, developed in the pattern-formation community and based on a power-counting argument. We thus arrive at a closed set of coupled equations for slowly varying local density and orientation of the microtubules, and study its behavior by performing a linear stability analysis and direct numerical simulations. By comparing our method with the work of Aranson and Tsimring, we assess the validity of the assumptions required to derive their and our theories. We demonstrate that our approximation-free evaluation of the interaction integrals and our choice of a systematic closure strategy result in a rather different dynamical behavior than was previously reported. Based on our theory, we discuss the ensuing phase diagram and the patterns observed.

Power countings versus physical scalings in disordered elastic systems—case study of the one-dimensional interface

We study the scaling properties of a one-dimensional interface at equilibrium, at finite temperature and in a disordered environment with a finite disorder correlation length. We focus our approach on the scalings of its geometrical fluctuations as a function of its length. At large lengthscales, the roughness of the interface, defined as the variance of its endpoint fluctuations, follows a power-law behaviour whose exponent characterises its superdiffusive behaviour. In 1+1 dimensions, the roughness exponent is known to be the characteristic 2/3 exponent of the Kardar–Parisi–Zhang (KPZ) universality class. An important feature of the model description is that its Flory exponent, obtained by a power counting argument on its Hamiltonian, is equal to 3/5 and thus does not yield the correct KPZ roughness exponent. In this work, we review the available power-counting options, and relate the physical validity of the exponent values that they predict, to the existence (or not) of well-defined optimal trajectories in a large-size or low-temperature asymptotics. We identify the crucial role of the ‘cut-off’ lengths of the problem (the disorder correlation length and the system size), which one has to carefully follow throughout the scaling analysis. To complement the latter, we device a novel Gaussian variational method (GVM) scheme to compute the roughness, taking into account the effect of a large but finite interface length. Interestingly, such a procedure yields the correct KPZ roughness exponent, instead of the Flory exponent usually obtained through the GVM approach for an infinite interface. We explain the physical origin of this improvement of the GVM procedure and discuss possible extensions of this work to other disordered systems.

The gradient flow in λϕ 4 theory

A gradient flow equation for $\lambda\phi^{4}$ theory in $D=4$ is formulated. In this scheme the gradient flow equation is written in terms of the renormalized probe variable $\Phi(t,x)$ and renormalized parameters $m^{2}$ and $\lambda$ in a manner analogous to the higher derivative regularization. No extra divergence is induced in the interaction of the probe variable $\Phi(t,x)$ and the 4-dimensional dynamical variable $\phi(x)$ which is defined in renormalized perturbation theory. The finiteness to all orders in perturbation theory is established by power counting argument in the context of $D+1$ dimensional field theory. This illustrates that one can formulate the gradient flow for the simple but important $\lambda\phi^{4}$ theory in addition to the well-known Yang-Mills flow, and it shows the generality of the gradient flow for a wider class of field theory.

Uncertainty Analysis and Order-by-Order Optimization of Chiral Nuclear Interactions

Chiral effective field theory (chiEFT) provides a systematic approach to describe low-energy nuclear forces. Moreover, chiEFT is able to provide well-founded estimates of statistical and systematic uncertainties -- although this unique advantage has not yet been fully exploited. We fill this gap by performing an optimization and statistical analysis of all the low-energy constants (LECs) up to next-to-next-to-leading order. Our optimization protocol corresponds to a simultaneous fit to scattering and bound-state observables in the pion-nucleon, nucleon-nucleon, and few-nucleon sectors, thereby utilizing the full model capabilities of chiEFT. We study the effect on other observables by demonstrating error-propagation methods that can easily be adopted by future works. We employ mathematical optimization and implement automatic differentiation to attain efficient and machine-precise first- and second-order derivatives of the objective function with respect to the LECs. We use power-counting arguments to estimate the systematic uncertainty that is inherent to chiEFT and we construct chiral interactions at different orders with quantified uncertainties. Statistical error propagation is compared with Monte Carlo sampling showing that statistical errors are in general small compared to systematic ones. In conclusion, we find that a simultaneous fit to different sets of data is critical to (i) identify the optimal set of LECs, (ii) capture all relevant correlations, (iii) reduce the statistical uncertainty, and (iv) attain order-by-order convergence in chiEFT. Furthermore, certain systematic uncertainties in the few-nucleon sector are shown to get substantially magnified in the many-body sector; in particlar when varying the cutoff in the chiral potentials. The methodology and results presented in this Paper open a new frontier for uncertainty quantification in ab initio nuclear theory.

Discrete renormalization group for SU(2) tensorial group field theory

This article provides a Wilsonian description of the perturbatively renormalizable tensorial group field theory introduced in Comm. Math. Phys. 330 (2014), 581–637 (arXiv:1303.6772 [hep-th]). It is a rank-3 model based on the gauge group SU(2), and as such is expected to be related to Euclidean quantum gravity in three dimensions. By means of a power-counting argument, we introduce a notion of dimensionality of the free parameters dening the action. General flow equations for the dimensionless bare coupling constants can then be derived, in terms of a discretely varying cut-off, and in which all the so-called melonic Feynman diagrams contribute. Linearizing around the Gaussian fixed point allows to recover the splitting between relevant, irrelevant, and marginal coupling constants. Pushing the perturbative expansion to second order for the marginal parameters, we are able to determine their behaviour in the vicinity of the Gaussian fixed point. Along the way, several technical tools are reviewed, including a discussion of combinatorial factors and of the Laplace approximation, which reduces the evaluation of the amplitudes in the UV limit to that of Gaussian integrals.

Hořava gravity with mixed derivative terms

Horava gravity has been constructed so as to exhibit anisotropic scaling in the ultraviolet, as this renders the theory power-counting renormalizable. However, when coupled to matter, the theory has been shown to suffer from quadratic divergences. A way to cure these divergences is to add terms with both time and space derivatives. We consider this extended version of the theory in detail. We perform a perturbative analysis that includes all modes, determine the propagators and discuss how including mixed-derivative terms affects them. We also consider the Lifshitz scalar with mixed-derivative terms as a toy model for power counting arguments and discuss the influence of such terms on renormalizability.

Multiple scattering of heavy quarks in dense matter and the parametric prominence of drag

The case of heavy quark propagation in dense extended matter is studied in the multiple scattering formalism of the higher twist energy loss scheme. We consider the case of deep inelastic scattering off a large nucleus. The hard lepton scatters off a heavy quark fluctuation within one of the nucleons. This heavy quark then propagates through the dense medium, multiply scattering off the gluon field of the remaining nucleons in its path. We consider the fictitious process where a heavy quark propagates through the nucleus without radiation. Invoking Soft-Collinear Effective Theory power counting arguments, we consider the case of a "semi-hard" heavy-quark where the mass is of the order of the out-going momentum and larger than the transverse momentum imparted per unit length due to scattering. In this limit, it is found that longitudinal momentum exchanges (quantified by the transport coefficient $\hat{e}$) have a comparable effect on the off-shellness of the propagating quark, as the transverse momentum exchanges (quantified by $\hat{q}$) which constitute the leading cause of off-shellness for propagating light quarks or gluons. Consequences of this new hierarchy for the propagation of the heavy quark in dense matter are discussed.

Coordinate-space singularities of massless gauge theories

The structure of singularities in perturbative massless gauge theories is investigated in coordinate space. The pinch singularities in coordinate-space integrals occur at configurations of vertices which have a direct interpretation in terms of physical scattering of particles in real space-time in the same way as for the loop momenta in the case of momentum-space singularities. In the analysis of vertex functions in coordinate space, the well-known factorization into hard, soft, and jet functions is found. By power-counting arguments, it is found that coordinate-space integrals of vertex functions have logarithmic divergences at worst.

An effective field theory for non-relativistic Majorana neutrinos

Heavy Majorana neutrinos enter in many scenarios of physics beyond the Standard Model: in the original seesaw mechanism they provide a natural explanation for the small masses of the Standard Model neutrinos and in the simplest leptogenesis framework they are at the origin of the baryonic matter of the universe. In this paper, we develop an effective field theory for non-relativistic Majorana particles, which is analogous to the heavy-quark effective theory. Then, we apply it to the case of a heavy Majorana neutrino decaying in a hot and dense plasma of Standard Model particles, whose temperature is much smaller than the mass of the Majorana neutrino but still much larger than the electroweak scale. The neutrino width gets zero-temperature contributions that can be computed from in-vacuum matrix elements, and thermal corrections. Only the latter will be addressed. Symmetry and power counting arguments made manifest by the effective field theory restrict the form of the thermal corrections and simplify their calculation. The final result agrees with recent determinations obtained with different methods. The effective field theory presented here is suitable to be used for a variety of different models involving non-relativistic Majorana fermions.