Order In Chiral Perturbation Theory Research Articles

The three-pion K-matrix at NLO in ChPT

The three-particle K-matrix, K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{K} $$\\end{document}df,3, is a scheme-dependent quantity that parametrizes short-range three-particle interactions in the relativistic-field-theory three-particle finite-volume formalism. In this work, we compute its value for systems of three pions in all isospin channels through next-to-leading order in Chiral Perturbation Theory, generalizing previous work done at maximum isospin. We obtain analytic expressions through quadratic order (or cubic order, in the case of zero isospin) in the expansion about the three-pion threshold.

Phase diagram of QCD matter with magnetic field: domain-wall Skyrmion chain in chiral soliton lattice

QCD matter in strong magnetic field exhibits a rich phase structure. In the presence of an external magnetic field, the chiral Lagrangian for two flavors is accompanied by the Wess-Zumino-Witten (WZW) term containing an anomalous coupling of the neutral pion π0 to the magnetic field via the chiral anomaly. Due to this term, the ground state is inhomogeneous in the form of either chiral soliton lattice (CSL), an array of solitons in the direction of magnetic field, or domain-wall Skyrmion (DWSk) phase in which Skyrmions supported by π3[SU(2)] ≃ ℤ appear inside the solitons as topological lumps supported by π2(S2) ≃ ℤ in the effective worldvolume theory of the soliton. In this paper, we determine the phase boundary between the CSL and DWSk phases beyond the single-soliton approximation, within the leading order of chiral perturbation theory. To this end, we explore a domain-wall Skyrmion chain in multiple soliton configurations. First, we construct the effective theory of the CSL by the moduli approximation, and obtain the ℂP1 model or O(3) model, gauged by a background electromagnetic gauge field, with two kinds of topological terms coming from the WZW term: one is the topological lump charge in 2+1 dimensional worldvolume and the other is a topological term counting the soliton number. Topological lumps in the 2+1 dimensional worldvolume theory are superconducting rings and their sizes are constrained by the flux quantization condition. The negative energy condition of the lumps yields the phase boundary between the CSL and DWSk phases. We find that a large region inside the CSL is occupied by the DWSk phase, and that the CSL remains metastable in the DWSk phase in the vicinity of the phase boundary.

A → πππ decay at next-to-leading order in chiral perturbation theory

We discuss the construction of the two-flavour axion-pion effective Lagrangian at the next-to-leading order (NLO) in chiral perturbation theory and present, as a phenomenological application, the calculation of the decay rate of a GeV-scale axion-like particle via the channel a → πππ. Through the NLO calculation, we assess the range of validity of the effective field theory and show that the chiral expansion breaks down just above the kinematic threshold. Alternative non-perturbative approaches are called for in order to extend the chiral description of axion-pion interactions.

Thermal axions with multi-eV masses are possible in low-reheating scenarios

We revise cosmological mass bounds on hadronic axions in low-reheating cosmological scenarios, with a reheating temperature T RH ≤ 100 MeV, in light of the latest cosmological observations. In this situation, the neutrino decoupling would be unaffected, while the thermal axion relic abundance is suppressed. Moreover, axions are colder in low-reheating temperature scenarios, so that bounds on their abundance are possibly loosened. As a consequence of these two facts, cosmological mass limits on axions are relaxed.Using state-of-the-art cosmological data and characterizing axion-pion interactions at the leading order in chiral perturbation theory, we find in the standard case an axion mass bound m a < 0.26 eV. However, axions with masses m a ≃ 1 eV, or heavier, would be allowed for reheating temperatures T RH ≲ 80 MeV. Multi-eV axions would be outside the mass sensitivity of current and planned solar axion helioscopes and would demand new experimental approaches to be detected.

Breakdown of Chiral Perturbation Theory for the Axion Hot Dark Matter Bound.

We show that the commonly adopted hot dark matter bound on the axion mass m_{a}≲1 eV is not reliable, since it is obtained by extrapolating the chiral expansion in a region where the effective field theory breaks down. This is explicitly shown via the calculation of the axion-pion thermalization rate at the next-to-leading order in chiral perturbation theory. We finally advocate a strategy for a sound extraction of the axion hot dark matter bound via lattice QCD techniques.

Scattering of two and three physical pions at maximal isospin from lattice QCD

We present the first direct N_f=2 lattice QCD computation of two- and three-pi ^+ scattering quantities that includes an ensemble at the physical point. We study the quark mass dependence of the two-pion phase shift, and the three-particle interaction parameters. We also compare to phenomenology and chiral perturbation theory (ChPT). In the two-particle sector, we observe good agreement to the phenomenological fits in s- and d-wave, and obtain M_pi a_0 = -0.0481(86) at the physical point from a direct computation. In the three-particle sector, we observe reasonable agreement at threshold to the leading order chiral expansion, i.e. a mildly attractive three-particle contact term. In contrast, we observe that the energy-dependent part of the three-particle quasilocal scattering quantity is not well described by leading order ChPT.

Application of effective field theory to finite-volume effects in aμHVP

One of the more important systematic effects affecting lattice computations of the hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon, $a_\mu^{\rm HVP}$, is the distortion due to a finite spatial volume. In order to reach sub-percent precision, these effects need to be reliably estimated and corrected for, and one of the methods that has been employed for doing this is finite-volume chiral perturbation theory. In this paper, we argue that finite-volume corrections to $a_\mu^{\rm HVP}$ can, in principle, be calculated at any given order in chiral perturbation theory. More precisely, once all low-energy constants needed to define the Effective Field Theory representation of $a_\mu^{\rm HVP}$ in infinite volume are known to a given order, also the finite-volume corrections can be predicted to that order in the chiral expansion.

Light clusters in dilute heavy-baryon admixed nuclear matter

We study the composition of nuclear matter at sub-saturation densities, non-zero temperatures, and isospin asymmetry, under the conditions characteristic of binary neutron star mergers, stellar collapse, and low-energy heavy-ion collisions. The composition includes light clusters with mass number Ale 4, a heavy nucleus (^{56}{Fe}), the varDelta -resonances, the isotriplet of pions, as well as the Lambda hyperon. The nucleonic mean-fields are computed from a zero-range density functional, whereas the pion-nucleon interactions are treated to leading order in chiral perturbation theory. We show that with increasing temperature and/or density the composition of matter shifts from light-cluster to heavy baryon dominated one, the transition taking place nearly independent of the magnitude of the isospin. Our findings highlight the importance of simultaneous treatment of light clusters and heavy baryons in the astrophysical and heavy-ion physics contexts.

Nπ -state contamination in lattice calculations of the nucleon pseudoscalar form factor

The nucleon-pion-state contribution in the QCD 3-point function of the pseudoscalar density is calculated to leading order in chiral perturbation theory. It predicts a nucleon-pion-state contamination in lattice estimates for the pseudoscalar form factor $G_{\rm P}(Q^2)$ determined with the plateau method. Depending on the momentum transfer $Q^2$ the contamination varies between -20% and +50% for a source-sink separation of 2 fm. The nucleon-pion-state contamination also causes violations in the generalized Goldberger-Treiman relation among the pseudoscalar and the axial nucleon form factors, the dominant source being the nucleon-pion-state contamination in the induced pseudoscalar form factor $\tilde{G}_{\rm P}(Q^2)$. Comparing the chiral perturbation theory predictions with lattice results of the PACS collaboration we find reasonable agreement even for source-sink separations as small as 1.3 fm.

|Vus| from Kℓ3 decay and four-flavor lattice QCD

Using HISQ $N_f=2+1+1$ MILC ensembles with five different values of the lattice spacing, including four ensembles with physical quark masses, we have performed the most precise computation to date of the $K\to\pi\ell\nu$ vector form factor at zero momentum transfer, $f_+^{K^0\pi^-}(0)=0.9696(15)_\text{stat}(12)_\text{syst}$. This is the first calculation that includes the dominant finite-volume effects, as calculated in chiral perturbation theory at next-to-leading order. Our result for the form factor provides a direct determination of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{us}|=0.22333(44)_{f_+(0)}(42)_\text{exp}$, with a theory error that is, for the first time, at the same level as the experimental error. The uncertainty of the semileptonic determination is now similar to that from leptonic decays and the ratio $f_{K^+}/f_{\pi^+}$, which uses $|V_{ud}|$ as input. Our value of $|V_{us}|$ is in tension at the 2--$2.6\sigma$ level both with the determinations from leptonic decays and with the unitarity of the CKM matrix. In the test of CKM unitarity in the first row, the current limiting factor is the error in $|V_{ud}|$, although a recent determination of the nucleus-independent radiative corrections to superallowed nuclear $\beta$ decays could reduce the $|V_{ud}|^2$ uncertainty nearly to that of $|V_{us}|^2$. Alternative unitarity tests using only kaon decays, for which improvements in the theory and experimental inputs are likely in the next few years, reveal similar tensions. As part of our analysis, we calculated the correction to $f_+^{K\pi}(0)$ due to nonequilibrated topological charge at leading order in chiral perturbation theory, for both the full-QCD and the partially-quenched cases. We also obtain the combination of low-energy constants in the chiral effective Lagrangian $[C_{12}^r+C_{34}^r-(L_5^r)^2](M_\rho)=(2.92\pm0.31)\cdot10^{-6}$.

Nπ -state contamination in lattice calculations of the nucleon axial form factors

The nucleon-pion-state contribution to QCD two-point and three-point functions used in lattice calculations of the nucleon axial form factors are studied in chiral perturbation theory. For small quark masses this contribution is expected to be the dominant excited-state contamination at large time separations. To leading order in chiral perturbation theory the results depend on only two experimentally known low-energy constants and the nucleon-pion-state contribution to the form factors can be estimated. The nucleon-pion-state contribution to the axial form factor $G_{\rm A}(Q^2)$ is at the 5 percent level for a source-sink separation of 2 fm and shows almost no dependence on the momentum transfer $Q^2$. In contrast, for the induced pseudoscalar form factor $\tilde{G}_{\rm P}(Q^2)$ the nucleon-pion-state contribution shows a rather strong dependence on $Q^2$ and leads to a 10 to 40 percent underestimation of $\tilde{G}_{\rm P}(Q^2)$ at small momentum transfers. The ChPT results can be used to analytically remove the nucleon-pion-state contribution from lattice data. Performing this removal for lattice data generated by the PACS collaboration we find agreement with experimental data and the predictions of the pion-pole dominance model. The removal works surprisingly well even for source-sink separations as small as 1.3 fm.

Light-Neutrino Exchange and Long-Distance Contributions to 0ν2β Decays: An Exploratory Study on ππ→ee.

We present an exploratory lattice QCD calculation of the neutrinoless double beta decay ππ→ee. Under the mechanism of light-neutrino exchange, the decay amplitude involves significant long-distance contributions. The calculation reported here, with pion masses m_{π}=420 and 140MeV, demonstrates that the decay amplitude can be computed from first principles using lattice methods. At unphysical and physical pion masses, we obtain that amplitudes are 24% and 9% smaller than the predication from leading order chiral perturbation theory. Our findings provide the lattice QCD inputs and constraints for effective field theory. A follow-on calculation with fully controlled systematic errors will be possible with adequate computational resources.

New spectrum of negative-parity doubly charmed baryons: Possibility of two quasistable states

The discovery of $\Xi_{cc}^{++}$ by the LHCb Collaboration triggers predictions of more doubly charmed baryons. By taking into account both the $P$-wave excitations between the two charm quarks and the scattering of light pseudoscalar mesons off the ground state doubly charmed baryons, a set of negative-parity spin-1/2 doubly charmed baryons are predicted already from a unitarized version of leading order chiral perturbation theory. Moreover, employing heavy antiquark-diquark symmetry the relevant low-energy constants in the next-to-leading order are connected with those describing light pseudoscalar mesons scattering off charmed mesons, which have been well determined from lattice calculations and experimental data. Our calculations result in a spectrum richer than that of heavy mesons. We find two very narrow $J^P=1/2^-$ $\Omega_{cc}^P$, which very likely decay into $\Omega_{cc}\pi^0$ breaking isospin symmetry. In the isospin-1/2 $\Xi_{cc}^P$ sector, three states are predicted to exist below 4.2~GeV with the lowest one being narrow and the other two rather broad. We suggest to search for the $\Xi_{cc}^{P}$ states in the $\Xi_{cc}^{++}\pi^-$ mode. Searching for them and their analogues are helpful to establish the hadron spectrum.

Realistic shell-model calculations for p -shell nuclei including contributions of a chiral three-body force

In this paper we present an evolution of our derivation of the shell-model effective Hamiltonian, namely introducing effects of three-body contributions. More precisely, we consider a three-body potential at next-to-next-to-leading order in chiral perturbation theory, and the induced three-body forces that arise from many-body correlations among valence nucleons. The first one is included, in the derivation of the effective Hamiltonian for one- and two-valence nucleon-systems, at first order in the many-body perturbation theory. Namely, we include only the three-body interaction between one or two valence nucleons and those belonging to the core. For nuclei with more than two valence particles, both induced - turned on by the two-body potential - and genuine three-body forces come into play. Since it is difficult to perform shell-model calculations with three-body forces, these contributions are estimated for the ground-state energy only. In order to establish the reliability of our approximations, we focus attention on nuclei belonging to the p shell, aiming to benchmark our calculations against those performed with the ab initio no-core shell-model. The obtained results are satisfactory, and pave the way to the application of our approach to nuclear systems with heavier masses.

Optimized chiral N2LO interactions in nuclear matter

We employ modern two- and three-nucleon interactions derived in chiral perturbation theory (ChPT) at next to-next-to leading order (N2LO), to calculate the energy per particle of symmetric nuclear matter and pure neutron matter in the framework of the microscopic Brueckner-Hartree-Fock approach. In particular, we present results concerning two optimized versions at N2LO of chiral potentials ( $ \mathrm{N2LO}_{opt}$ ), fitted to properties of light nuclei. We also employ the recently developed $\mathrm{N2LO}_{sat}$ interaction which has been calculated at the same order of ChPT but fitted in a very different way compared with the $\mathrm{N2LO}_{opt}$ interactions. We find that using these potentials, in general, it is not possible to reproduce all the saturation properties of nuclear matter. In particular the behaviour of the symmetry energy ( $E_{sym}$ predicted by the interactions considered is quite soft. This is shown comparing our results with the empirical constraints on $E_{sym}$ obtained from the data analysis of the excitation energies of isobaric analog states in nuclei and from experimental data of the neutron skin thickness of heavy nuclei. We finally confront our results with similar calculations performed by other research groups using nuclear chiral interactions at various order of ChPT and employing different many-body methods.

Three-particle Nππ state contribution to the nucleon two-point function in lattice QCD

The three-particle $N\ensuremath{\pi}\ensuremath{\pi}$ state contribution to the QCD two-point function of standard nucleon interpolating fields is computed to leading order in chiral perturbation theory. Using the experimental values for two low-energy coefficients, the impact of this contribution on lattice QCD calculations of the nucleon mass is estimated. The impact is found to be at the per mille level at most and negligible in practice.

Casimir squared correction to the standard rotator Hamiltonian for the O(n) sigma-model in the delta-regime

In a previous paper we found that the isospin susceptibility of the O(n) sigma-model calculated in the standard rotator approximation differs from the next-to-next-to leading order chiral perturbation theory result in terms vanishing like 1/ℓ, for ℓ = Lt/L → ∞ and further showed that this deviation could be described by a correction to the rotator spectrum proportional to the square of the quadratic Casimir invariant. Here we confront this expectation with analytic nonperturbative results on the spectrum in 2 dimensions, by Balog and Hegedüs for n = 3, 4 and by Gromov, Kazakov and Vieira for n = 4, and find good agreement in both cases. We also consider the case of 3 dimensions.

Can a Linear Sigma Model Describe Walking Gauge Theories at Low Energies?

In recent years, many investigations of confining Yang Mills gauge theories near the edge of the conformal window have been carried out using lattice techniques. These studies have revealed that the spectrum of hadrons in nearly conformal ("walking") gauge theories differs significantly from the QCD spectrum. In particular, a light singlet scalar appears in the spectrum which is nearly degenerate with the PNGBs at the lightest currently accessible quark masses. This state is a viable candidate for a composite Higgs boson. Presently, an acceptable effective field theory (EFT) description of the light states in walking theories has not been established. Such an EFT would be useful for performing chiral extrapolations of lattice data and for serving as a bridge between lattice calculations and phenomenology. It has been shown that the chiral Lagrangian fails to describe the IR dynamics of a theory near the edge of the conformal window. Here we assess a linear sigma model as an alternate EFT description by performing explicit chiral fits to lattice data. In a combined fit to the Goldstone (pion) mass and decay constant, a tree level linear sigma model has a Χ2/d.o.f. = 0.5 compared to Χ2/d.o.f. = 29.6 from fitting nextto-leading order chiral perturbation theory. When the 0++ (σ) mass is included in the fit, Χ2/d.o.f. = 4.9. We remark on future directions for providing better fits to the σ mass.

Elastic and inelastic pion-nucleon scattering to fourth order in chiral perturbation theory

We extend our previous study of elastic pion-nucleon scattering in the framework of chiral perturbation theory by performing a combined analysis of the reactions $\pi N \to \pi N$ and $\pi N \to \pi \pi N$. The calculation is carried out to fourth order in the chiral expansion using the heavy baryon approach and the covariant formulation supplemented with a modified version of the extended on-mass-shell renormalization scheme. We demonstrate that a combined fit to experimental data in both channels leads to a reduced amount of correlations between the low-energy constants. A satisfactory description of the experimental data in both channels is obtained, which is further improved upon including tree-level contributions of the $\Delta$(1232) resonance. We also explore a possibility of using the empirical information about $\pi N$ subthreshold parameters obtained recently by means of the Roy-Steiner equations to stabilize the fits.

The N/D method with non-perturbative left-hand-cut discontinuity and the [formula omitted]NN partial wave

In this letter we introduce an integral equation that allows to calculate the exact left-hand-cut discontinuity for an uncoupled S-wave partial-wave amplitude in potential scattering for a given finite-range potential. In particular this is applied here to the S01 nucleon–nucleon (NN) partial wave. The calculation of Δ(A) is completely fixed by the potential because short-range physics (corresponding to integrated out degrees of freedom within the low-energy Effective Field Theory) does not contribute to Δ(A). The results obtained from the N/D method for a partial-wave amplitude are rigorous, since now the discontinuities along the left-hand cut and right-hand cut are exactly known. This solves in this case the open question with respect to the N/D method and the effect on the final result of the non-perturbative iterative diagrams in the evaluation of Δ(A). The solution of this problem also implies the equivalence of the N/D method and the Lippmann–Schwinger (LS) equation for the nonsingular one-pion exchange S01NN potential (Yukawa potential). The equivalence between the N/D method with one extra subtraction and the LS equation renormalized with one counterterm or with subtractive renormalization also holds for the singular attractive S01NN potentials calculated by including higher orders in Chiral Perturbation Theory (ChPT). However, the N/D method is more flexible and, rather straightforwardly, it allows to evaluate partial-wave amplitudes with a higher number of extra subtractions, that we fix in terms of shape parameters within the effective range expansion. We give results up to three extra subtractions in the N/D method, which provide a rather accurate reproduction of the S01NN phase shifts when the NNLO ChPT potential is employed. Our new method then provides a general theory to renormalize non-perturbatively singular and regular potentials in scattering that can be extended to higher partial waves as well as to coupled channel scattering.