Naive Dimensional Analysis Research Articles

Tritium β decay and proton-proton fusion in pionless effective field theory

The Gamow-Teller and Fermi matrix elements, 〈GT〉 and 〈F〉, respectively, for tritium β decay are calculated to next-to-leading order (NLO) in pionless effective field theory in the absence of Coulomb interactions and isospin violation giving the leading order predictions 〈GT〉0=0.9807 and 〈F〉0=1. Using an experimentally determined value for the tritium β decay GT matrix element, the two-body axial current low energy constant is fixed at NLO yielding L1,A=6.01±2.08fm3 at the renormalization scale of the physical pion mass, which agrees with predictions based on naive dimensional analysis. The impact of L1,A on proton-proton fusion is also discussed. Finally, the consequences of Wigner-SU(4) spin-isospin symmetry are considered for the Gamow-Teller matrix element. Published by the American Physical Society 2024

On the strong coupling problem in cosmologies with “strong gravity in the past”

In this paper, we examine the potential strong coupling problem at early times in a bouncing cosmological model with “strong gravity in the past” (Jordan frame), which is conformally related to inflation (Einstein frame). From naive-dimensional analysis in the Jordan frame, one would conclude that the quantum strong coupling energy scale can be lower than the classical energy scale. However, from the Einstein frame prospective, this should not be the case. We illustrate this point by calculation in the Jordan frame which shows cancellations of the dangerous contributions in the tree level amplitude.

Unitarization of infinite-range forces: graviton-graviton scattering

A method to unitarize the scattering amplitude produced by infinite-range forces is developed and applied to Born terms. In order to apply S-matrix techniques, based on unitarity and analyticity, we first derive an S-matrix free of infrared divergences. This is achieved by removing a divergent phase factor due to the interactions mediated by the massless particles in the crossed channels, a procedure that is related to previous formalisms to treat infrared divergences. We apply this method in detail by unitarizing the Born terms for graviton-graviton scattering in pure gravity and we find a scalar graviton-graviton resonance with vacuum quantum numbers (JPC = 0++) that we call the graviball. Remarkably, this resonance is located below the Planck mass but deep in the complex s-plane (with s the usual Mandelstam variable), so that its effects along the physical real s axis peak for values significantly lower than this scale. This implies that the corrections to the leading-order amplitude in the gravitational effective field theory are larger than expected from naive dimensional analysis for s around and above the peak position. We argue that the position and width of the graviball are reduced when including extra light fields in the theory. This could lead to phenomenological consequences in scenarios of quantum gravity with a large number of such fields or, in general, with a low-energy ultraviolet completion. We also apply this formalism to two non-relativistic potentials with exact known solutions for the scattering amplitudes: Coulomb scattering and an energy-dependent potential obtained from the Coulomb one with a zero at threshold. This latter case shares the same J = 0 partial-wave projected Born term as the graviton-graviton case, except for a global factor. We find that the relevant resonance structure of these examples is reproduced by our methods, which represents a strong indication of their robustness.

Perturbative calculations of deuteron form factors

We calculate the deuteron charge, electric quadrupole, and magnetic form factors up to next-to-next-to-leading order in chiral effective field theory, treating subleading corrections, especially that of chiral nuclear forces, in perturbation theory. We examine the power counting based on naive dimensional analysis by investigating the ultraviolet cutoff variation of these form factors. We find that the N${}^2$LO magnetic form factor shows significant cutoff dependence, suggesting the contact current operator responsible be enhanced. After promotion to N${}^2$LO, it indeed renormalizes the magnetic form factor. This is in agreement with a previous work based on renormalization-group analysis. For the charge and quadrupole moments, perturbative calculation allows us to study how they scale with multiple low-energy parameters such as the pion mass, deuteron binding momentum, and momentum transfer.

Renormalization of one-pion exchange in chiral effective field theory for antinucleon-nucleon scattering

The renormalization of iterated one-pion exchange (OPE) is studied in chiral effective field theory $(\ensuremath{\chi}\mathrm{EFT})$ for the antinucleon-nucleon $(\overline{N}N)$ system. The OPE potential is cut off at a certain distance and contact interactions are represented by a complex spherical well with the same radius. We investigate the dependence on the cutoff radius of the phase shifts, inelasticities, and mixing angles for the low partial waves in $\overline{N}N$ scattering. We show that renormalization requires additional contact interactions compared to the expectation based on naive dimensional analysis. Results after renormalization are compared with the state-of-the-art energy-dependent partial-wave analysis of $\overline{N}N$ data. We compare our conclusions with applications of $\ensuremath{\chi}\mathrm{EFT}$ to the nucleon-nucleon system.

Renormalization of CP -violating nuclear forces

Electric dipole moments of nuclei, diamagnetic atoms, and certain molecules are induced by CP-violating nuclear forces. Naive dimensional analysis predicts these forces to be dominated by long-range one-pion-exchange processes, with short-range forces entering only at next-to-next-to-leading order in the chiral expansion. Based on renormalization arguments we argue that a consistent picture of CP-violating nuclear forces requires a leading-order short-distance operator contributing to ${}^1S_0$-${}^3P_0$ transitions, due to the attractive and singular nature of the strong tensor force in the ${}^3P_0$ channel. The short-distance operator leads to $\mathcal O(1)$ corrections to static and oscillating, relevant for axion searches, electric dipole moments. We discuss strategies how the finite part of the associated low-energy constant can be determined in the case of CP violation from the QCD theta term by the connection to charge-symmetry violation in nuclear systems.

Covariant nucleon-nucleon contact Lagrangian up to order O(q4)

We adopt a covariant version of the naive dimensional analysis and construct the chiral two-nucleon contact Lagrangian constrained by Lorentz, parity, charge conjugation, hermitian conjugation, and chiral symmetries. We show that at $\mathcal{O}(q^0)$, $\mathcal{O}(q^2)$, $\mathcal{O}(q^4)$, where $q$ denotes a generic small quantity, there are 4, 13, and 23 terms, respectively. We find that by performing $1/m_N$ expansions, the covariant Lagrangian reduces to the conventional non-relativistic one, which includes 2, 7, and 15 terms at each corresponding order.

Analysis of general power counting rules in effective field theory

We derive the general counting rules for a quantum effective field theory (EFT) in $\mathsf{d}$ dimensions. The rules are valid for strongly and weakly coupled theories, and predict that all kinetic energy terms are canonically normalized. They determine the energy dependence of scattering cross sections in the range of validity of the EFT expansion. We show that the size of cross sections is controlled by the $\Lambda$ power counting of EFT, not by chiral counting, even for chiral perturbation theory ($\chi$PT). The relation between $\Lambda$ and $f$ is generalized to $\mathsf{d}$ dimensions. We show that the naive dimensional analysis $4\pi$ counting is related to $\hbar$ counting. The EFT counting rules are applied to $\chi$PT, low-energy weak interactions, Standard Model EFT and the non-trivial case of Higgs EFT.

Large BR(h → τ μ) in the MSSM?

We study how large the rate of the lepton-flavor violating Higgs decay h->tau mu can be in the (R-parity conserving) MSSM. We make no assumptions, such as universality or alignment, about the flavor structure of the MSSM. We only assume that all couplings and, in particular, the trilinear scalar ones, are perturbative. We take into account lower bounds on the bino and slepton masses from tau->mu gamma and h->gamma gamma as well as upper bounds on the trilinear scalar couplings from the requirement that the global minimum is not charge breaking. We find that in highly fine-tuned regions of parameter space, the ratio BR(h->tau mu)/BR(h->tau tau) can be enhanced by about three orders of magnitude above the estimate from naive dimensional analysis, but still about two orders of magnitude below the present bound. Thus, if h->tau mu is experimentally established to be close to present bounds, the MSSM will be excluded.

Binding energy of theX(3872)at unphysical pion masses

Chiral extrapolation of the $X(3872)$ binding energy is investigated using the modified Weinberg formulation of chiral effective field theory for the $D \bar{D}^*$ scattering. Given its explicit renormalisability, this approach is particularly useful to explore the interplay of the long- and short-range $D \bar{D}^*$ forces in the $X(3872)$ from studying the light-quark (pion) mass dependence of its binding energy. In particular, the parameter-free leading-order calculation shows that the $X$-pole disappears for unphysical large pion masses. On the other hand, without contradicting the naive dimensional analysis, the higher-order pion-mass-dependent contact interaction can change the slope of the binding energy at the physical point yielding the opposite scenario of a stronger bound $X$ at pion masses larger than its physical value. An important role of the pion dynamics and of the 3-body $D\bar{D}\pi$ effects for chiral extrapolations of the $X$-pole is emphasised. The results of the present study should be of practical value for the lattice simulations since they provide a non-trivial connection between lattice points at unphysical pion masses and the physical world.

TeV Scale Lepton Number Violation and Baryogenesis

Contrary to the common lore based on naive dimensional analysis, the seesaw scale for neutrino masses can be naturally in the TeV range, with small parameters coming from radiative corrections. We present one such class of type-I seesaw models, based on the left-right gauge group SU(2)L × SU(2)R × U(1)B-L realized at the TeV scale, which fits the observed neutrino oscillation parameters as well as other low energy constraints. We discuss how the small parameters of this scenario can arise naturally from one loop effects. The neutrino fits in this model use quasi-degenerate heavy Majorana neutrinos, as also required to explain the matter-antimatter asymmetry in our Universe via resonant leptogenesis mechanism. We discuss the constraints implied by the dynamics of this mechanism on the mass of the right-handed gauge boson in this class of models with enhanced neutrino Yukawa couplings compared to the canonical seesaw model and find a lower bound of mWR ≥ 9.9 TeV for successful leptogenesis assuming maximal CP asymmetry for each flavor. We also present a model with explicit neutrino mass fit, where the lower bound goes up to 13.1 TeV due to less than maximal primordial CP asymmetry predicted by the model.

On the power counting in effective field theories

We discuss the systematics of power counting in general effective field theories, focusing on those that are nonrenormalizable at leading order. As an illuminating example we consider chiral perturbation theory gauged under the electromagnetic U(1) symmetry. This theory describes the low-energy interactions of the octet of pseudo-Goldstone bosons in QCD with photons and has been discussed extensively in the literature. Peculiarities of the standard approach are pointed out and it is shown how these are resolved within our scheme. The presentation follows closely our recent discussion of power counting for the electroweak chiral Lagrangian. The systematics of the latter is reviewed and shown to be consistent with the concept of chiral dimensions. The results imply that naive dimensional analysis (NDA) is incomplete in general effective field theories, while still reproducing the correct counting in special cases.

Naive dimensional analysis counting of gauge theory amplitudes and anomalous dimensions

We show that naive dimensional analysis (NDA) is equivalent to the result that L-loop scattering amplitudes have perturbative order N=L+Δ, with a shift Δ that depends on the NDA-weight of operator insertions. The NDA weight of an operator is defined in this Letter, and the general NDA formula for perturbative order N is derived. The formula is used to explain why the one-loop anomalous dimension matrix for dimension-six operators in the Standard Model effective field theory has entries with perturbative order ranging from 0 to 4. The results in this Letter are valid for an arbitrary effective field theory, and they constrain the coupling constant dependence of anomalous dimensions and scattering amplitudes in a general effective field theory.

Naive dimensional analysis in holography

Naive dimensional analysis (NDA) is a widely used ansatz to estimate coupling constants among composite states emerging from dynamics of a strongly coupled gauge theory. However, the validity of NDA is still unclear because of the difficulty in calculating these quantities in strongly coupled theories. We examine the NDA ansatz using gauge/string duality, by estimating glueball coupling constants from gravitational description. The NDA scaling rule for coupling constants of some types of glueballs is verified and extended by both generic estimation and numerical calculations. The scaling rule verified in this article can be applied to some class of quiver gauge theories as well, not just to gauge theories with a single gauge group $SU(N_c)$.

Higgs production and decay processes via loop diagrams in various 6D Universal Extra Dimension models at LHC

We calculate loop-induced Higgs production and decay processes which are relevant for the LHC in various six-dimensional Universal Extra Dimension models. More concretely, we focus on Higgs production through gluon fusion and Higgs decay into two photons induced by loop diagrams. They are one-loop leading processes and the contribution of Kaluza-Klein particles is considered to be significant. These processes are divergent in six dimensions. Therefore, we employ a momentum cutoff, whose size is fixed from the validity of perturbative calculation through naive dimensional analysis. In these six-dimensional Universal Extra Dimension models, the Higgs production cross section through gluon fusion strongly increases considerably and the Higgs decay width into two photons decrease somehow. In particular in the case of the compactification on Projective Sphere, these effects are remarkable. The deviation of the $h^{(0)} \rightarrow 2 \gamma$ signal from the prediction of the Standard model is much larger than that in the case of the five-dimensional minimal UED model. We also consider threshold corrections in the two processes and these effect are noteworthy when we choose low cutoff scale in 6D UED. Comparing our calculation to the latest LHC results which were published at the Lepton-Photon 2011 is performed briefly.

Renormalizing chiral nuclear forces: Triplet channels

We discuss the subleading contact interactions, or counterterms, of the triplet channels of nucleon-nucleon scattering in the framework of chiral effective field theory, with $S$ and $P$ waves as the examples. The triplet channels are special in that they allow the singular attraction of one-pion exchange to modify Weinberg's original power-counting (WPC) scheme. With renormalization group invariance as the constraint, our power counting for the triplet channels can be summarized as a modified version of naive dimensional analysis in which, when compared with WPC, all of the counterterms in a given partial wave (leading or subleading) are enhanced by the same amount. More specifically, this means that WPC needs no modification in ${}^{3}\phantom{\rule{-0.16em}{0ex}}{S}_{1}\ensuremath{-}{}^{3}\phantom{\rule{-0.16em}{0ex}}{D}_{1}$ and ${}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{1}$, whereas a two-order enhancement is necessary in both ${}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{0}$ and ${}^{3}\phantom{\rule{-0.16em}{0ex}}{P}_{2}\ensuremath{-}{}^{3}\phantom{\rule{-0.16em}{0ex}}{F}_{2}$.

Renormalizing chiral nuclear forces: A case study of3P0

We discuss in this paper the subleading contact interactions, or counterterms, in the 3P0 channel of nucleon-nucleon scattering up to O(Q^3), where, already at leading order, Weinberg's original power counting (WPC) scheme fails to fulfill renormalization group invariance due to the singular attraction of one-pion exchange. Treating the subleading interactions as perturbations and using renormalization group invariance as the criterion, we investigate whether WPC, although missing the leading order, could prescribe correct subleading counterterms. We find that the answer is negative and, instead, that the structure of counterterms agrees with a modified version of naive dimensional analysis. Using 3P0 as an example, we also study the cutoffs where the subleading potential can be iterated together with the leading one.

If no Higgs then what?

In the absence of a Higgs boson, the perturbative description of the Standard Model ceases to make sense above a TeV. Heavy spin-1 fields coupled to W and Z bosons can extend the validity of the theory up to higher scales. We carefully identify regions of parameter space where a minimal addition — a single spin-1 SU(2)custodial-triplet resonance — allows one to retain perturbative control in all channels. Elastic scattering of longitudinal W and Z bosons alone seems to permit a very large cut-off beyond the Naive Dimensional Analysis expectation. We find however that including scattering of the spin-1 resonances then leads to an earlier onset of strong coupling. Most importantly for LHC searches, we define a self-consistent set-up with a well-defined range of validity without recourse to unitarization schemes whose physical meaning is obscure. We discuss the LHC phenomenology and the discovery reach for these electroweak resonances and mention the possibility of a nightmare scenario with no Higgs nor resonance within the LHC reach. Finally, we discuss the effects of parity breaking in the heavy resonance sector which reduces the contributions to the S parameter.

Corrections to Bekenstein-Hawking Entropy— Quantum or not-so Quantum?

Hawking radiation and Bekenstein-Hawking entropy are the two robust predictions of a yet unknown quantum theory of gravity. Any theory which fails to reproduce these predictions is certainly incorrect. While several approaches lead to Bekenstein-Hawking entropy, they all lead to different sub-leading corrections. In this article, we ask a question that is relevant for any approach: Using simple techniques, can we know whether an approach contains quantum or semi-classical degrees of freedom? Using naive dimensional analysis, we show that the semi-classical black-hole entropy has the same dimensional dependence as the gravity action. Among others, this provides a plausible explanation for the connection between Einstein’s equations and thermodynamic equation of state, and that the quantum corrections should have a different scaling behavior.

Natural units for nuclear energy density functional theory

Naive dimensional analysis based on chiral effective theory, when adapted to nuclear energy density functionals, prescribes natural units and a hierarchy of contributions that could be used to constrain fits of generalized functionals. By applying these units, a large sample of Skyrme parametrizations is examined for naturalness, which is signaled by dimensionless coupling constants of order one. The bulk of the parameters are found to be natural, with an underlying scale consistent with other determinations. Significant deviations from unity are associated with deficiencies in the corresponding terms of particular functionals or with an incomplete optimization procedure.