arXiv subjects
Publications and source records attributed to Andrzej Czarnecki.
We prove that the cyclic inequality $\sum\limits_{i=1}^{i=n}\left(\frac{x_i}{x_{i+1}}\right)^k\geq\sum\limits_{i=1}^{i=n}\frac{x_i}{x_{σ(i)}}$ holds for $k$ in a specific range dependant on the permutation $σ$. We also show that the same is not true for the Sahpiro-type generalizations.
Decay rates of the positronium molecule $\text{Ps}_{2}$ into two photons and into an electron-positron pair are determined. Previous studies find that these rates are very different, \[ Γ\left(\text{Ps}_{2}\to e^{+}e^{-}\right)/Γ\left(\text{Ps}_{2}\toγγ\right)\simeq250\text{ (previous studies)}. \] This is puzzling since both processes have two body final states and are of the same order in the fine structure constant. We propose a simple calculational method and test it with the well-established decay of the positronium ion into an electron and a photon. We then employ it to correct predictions for both these $\text{Ps}_{2}$ decays. We find that previous studies overestimated the $e^{+}e^{-}$ and underestimated the $γγ$ channel by factors of about 5.44 and 3.93 respectively. Our results give $Γ\left(\text{Ps}_2\to e^+e^-\right)/Γ\left(\text{Ps}_2 \to γγ\right) \simeq 11.7$.
Recently, $O(α^3)$ corrections to the muon decay rate and $O(α_s^3)$ to heavy quark decays have been determined by Fael, Schönwald, and Steinhauser. This is the first such perturbative improvement of these important quantities in more than two decades. We reveal and explain a symmetry pattern in these new corrections, and confirm the most technically difficult parts of their evaluation.
Contrary to what was previously believed, two-loop radiative corrections to the $g$-factor of an electron bound in a hydrogen-like ion at $\mathcal{O}\left(α^2 (Zα)^5\right)$ exhibit logarithmic enhancement. This previously unknown contribution is due to a long-distance light-by-light scattering amplitude. Taking an effective field theory approach, and using the Euler-Heisenberg Lagrangian, we find $Δg = -\left( \fracαπ \right)^2\left(Zα\right)^5 \frac{56 π}{135}\ln Zα$.
Pion beta decay, $π^+\to π^0 e^+ν(γ)$, provides a theoretically clean $\pm 0.3\%$ determination of the CKM matrix element $V_{\text{ud}}$. That value falls short of super-allowed nuclear beta decays where an order of magnitude better precision already exists. We advocate a new strategy for utilizing pion beta decay, based on its role in determining ${V_{\text{us}} / V_{\text{ud}}}$ via the ratio $R_V=Γ\left(K\to πl ν(γ)\right)/Γ\left(π^+\to π^0 e^+ν(γ)\right)$. $R_V$ measures $ { f_+^K(0) |V_{\text{us}}| / f_+^π(0) |V_{\text{ud}}| }$ and is insensitive to the Fermi constant and some radiative corrections. Its dependence on the ratio of form factors may also prove useful for lattice gauge theory calculations. Employing a lattice based value ${ f_+^K(0) / f_+^π(0) } = 0.970(2)$, we find $V_{\text{us}}/V_{\text{ud}}=0.22918(88)$ compared to $V_{\text{us}}/V_{\text{ud}}=0.23131(45)$ obtained from $R_A=Γ(K\to μν(γ))/Γ(π\to μν(γ))$. Those vector and axial-vector based $V_{\text{us}}/V_{\text{ud}}$ values exhibit a 2.2$σ$ discrepancy. That tension may be relieved by a shift in the lattice $ { f_+^K(0) / f_+^π(0) } $ towards the consistency range 0.961(4), changes in experimental input or new physics. Other implications of $R_V$ and $R_A$ are also discussed.
When a muon bound in an atom decays, there is a small probability that the daughter electron remains bound. That probability is evaluated. Surprisingly, a significant part of the rate is contributed by the negative energy component of the wave function, neglected in a previous study. A simple integral representation of the rate is presented. In the limit of close muon and electron masses, an analytic formula is derived.
The universal radiative corrections common to neutron and super-allowed nuclear beta decays (also known as ``inner'' corrections) are revisited in light of a recent dispersion relation study that found $+2.467(22)\%$, i.e.~about $2.4σ$ larger than the previous evaluation. For comparison, we consider several alternative computational methods. All employ an updated perturbative QCD four-loop Bjorken sum rule (BjSR) defined QCD coupling supplemented with a nucleon form factor based Born amplitude to estimate axial-vector induced hadronic contributions. In addition, we now include hadronic contributions from low $Q^2$ loop effects based on duality considerations and vector meson resonance interpolators. Our primary result, $2.426(32)\%$ corresponds to an average of a Light Front Holomorphic QCD approach and a three resonance interpolator fit. It reduces the dispersion relation discrepancy to approximately $1.1σ$ and thereby provides a consistency check. Consequences of our new radiative correction estimate, along with that of the dispersion relation result, for CKM unitarity are discussed. The neutron lifetime-$g_A$ connection is updated and shown to suggest a shorter neutron lifetime $< 879$ s. We also find an improved bound on exotic, non-Standard Model, neutron decays or oscillations of the type conjectured as solutions to the neutron lifetime problem, $\text{BR}(n\to \text{exotics}) < 0.16 \%$.
Experimental studies of neutron decay, $n\to pe\barν$, exhibit two anomalies. The first is a 8.6(2.1)s, roughly $4σ$ difference between the average beam measured neutron lifetime, $τ_n^\text{beam}=888.0(2.0)$s, and the more precise average trapped ultra cold neutron determination, $τ_n^\text{trap}=879.4(6)$s. The second is a $5σ$ difference between the pre2002 average axial coupling, $g_A$, as measured in neutron decay asymmetries $g_A^\text{pre2002}=1.2637(21)$, and the more recent, post2002, average $g_A^\text{post2002}=1.2755(11)$, where, following the UCNA collaboration division, experiments are classified by the date of their most recent result. In this study, we correlate those $τ_n$ and $g_A$ values using a (slightly) updated relation $τ_n(1+3g_A^2)=5172.0(1.1)$s. Consistency with that relation and better precision suggest $τ_n^\text{favored}=879.4(6)$s and $g_A^\text{favored}=1.2755(11)$ as preferred values for those parameters. Comparisons of $g_A^\text{favored}$ with recent lattice QCD and muonic hydrogen capture results are made. A general constraint on exotic neutron decay branching ratios, $<0.27\%$, is discussed and applied to a recently proposed solution to the neutron lifetime puzzle.
We present a construction (and classification) of certain invariant 2-forms on the real symplectic group. They are used to define a symplectic form on the quotient by a maximal torus and to "lift" a symplectic structure from a symplectic manifold to the bundle of frames. This is a by-product of a failed attempt to prove certain finiteness theorems for basic symplectic cohomologies. In the last part of the paper we include a valid proof.
We classify all integrable complex structures on 6-dimensional Lie algebras of the form $\mathfrak{g}\times\mathfrak{g}$.
The current status of electroweak radiative corrections to the muon anomalous magnetic moment is discussed. Asymptotic expansions for some important electroweak two loop top quark triangle diagrams are illustrated and extended to higher order. Results are compared with the more general integral representation solution for generic fermion triangle loops coupled to pseudoscalar and scalar bosons of arbitrary mass. Excellent agreement is found for a broader than expected range of mass parameters.
We compute corrections to the gyromagnetic factor of an electron bound in a hydrogen-like ion at order alpha^2(Z alpha)^5. This result removes a major uncertainty in predictions for silicon and carbon ions, used to determine the atomic mass of the electron.
We calculate the decay rate and spectrum for the decay of parapositronium into three photons. This process violates charge conjugation symmetry. It is forbidden within quantum electrodynamics but proceeds through the weak interaction. The branching ratio we find is fifty orders of magnitude smaller than the previous estimate.
We consider the interactions in a mesonic system, referred here to as `tetron', consisting of two heavy quarks and two lighter antiquarks (which may still be heavy in the scale of QCD), i.e. generally $Q_a Q_b \bar q_c \bar q_d$, and study the existence of bound states below the threshold for decay into heavy meson pairs. At a small ratio of the lighter to heavier quark masses an expansion parameter arises for treatment of the binding in such systems. We find that in the limit where all the quarks and antiquarks are so heavy that a Coulomb-like approximation can be applied to the gluon exchange between all of them, such bound states arise when this parameter is below a certain critical value. We find the parametric dependence of the critical mass ratio on the number of colors $N_c$, and confirm this dependence by numerical calculations. In particular there are no stable tetrons when all constituents have the same mass. We discuss an application of a similar expansion in the large $N_c$ limit to realistic systems where the antiquarks are light and their interactions are nonperturbative. In this case our findings are in agreement with the recent claims from a phenomenological analysis that a stable $b b \bar u \bar d$ tetron is likely to exist, unlike those where one or both bottom quarks are replaced by the charmed quark.
We compute the decay rate of dimuonium into a neutral pion and a photon. We find that approximately one in 10^5 ortho-dimuonia decays into this channel. We also determine the contribution of the virtual photon-pion loop to the hyperfine splitting in dimuonium and reproduce its leading effect in the anomalous magnetic moment of the muon.
We classify the 6-dimensional Lie algebras of the form $g\times g$ that admit integrable complex structure. We also endow a Lie algebra of the kind $o(n)\oplus o(n)$ with such a complex structure. The motivation comes from geometric structures á la Sasaki on $g$-manifolds.
We present examples of foliations with infinite dimensional basic symplectic and com- plex cohomologies, along with a general sufficient condition for such phenomena. This puts re- strictions on possible generalizations of several finiteness results from Riemannian foliations to any broader class. The examples are also noteworthy for the unusual behaviour of their basic de Rham cohomology.
We compute an $\mathcal{O}\left(α^2(Zα)^6\right)$ contribution to the hydrogen-atom Lamb shift arising from the light-by-light scattering. Analogous diagrams, with one atomic electric field insertion replaced by an external magnetic field, contribute to the gyromagnetic factor of the bound electron at $\mathcal{O}\left(α^2(Zα)^4\right)$. We also calculate the contribution to the gyromagnetic factor from the muon magnetic loop.