Spin-label Order Parameter Calibrations for Slow Motion

Calibrations are given to extract orientation order parameters from pseudo-powder electron paramagnetic resonance line shapes of ¹⁴N-nitroxide spin labels undergoing slow rotational diffusion. The nitroxide z-axis is assumed parallel to the long molecular axis. Stochastic-Liouville simulations of slow-motion 9.4-GHz spectra for molecular ordering with a Maier–Saupe orientation potential reveal a linear dependence of the splittings, \(2A_{\hbox{max} }\) and \(2A_{\hbox{min} }\), of the outer and inner peaks on order parameter \(S_{zz}\) that depends on the diffusion coefficient \(D_{{{\text{R}} \bot }}\) which characterizes fluctuations of the long molecular axis. This results in empirical expressions for order parameter and isotropic hyperfine coupling: \(S_{zz} = s_{1} \times \left( {A_{\hbox{max} } - A_{\hbox{min} } } \right) - s_{o}\) and \(a_{o}^{{}} = \tfrac{1}{3}\left( {f_{\hbox{max} } A_{\hbox{max} } + f_{\hbox{min} } A_{\hbox{min} } } \right) + \delta a_{o}\), respectively. Values of the calibration constants \(s_{1}\), \(s_{\text{o}}\), \(f_{\hbox{max} }\), \(f_{\hbox{min} }\) and \(\delta a_{o}\) are given for different values of \(D_{{{\text{R}} \bot }}\) in fast and slow motional regimes. The calibrations are relatively insensitive to anisotropy of rotational diffusion \((D_{{{\text{R}}//}} \ge D_{{{\text{R}} \bot }} )\), and corrections are less significant for the isotropic hyperfine coupling than for the order parameter.     

Higher Spin Dirac Operators Between Spaces of Simplicial Monogenics in Two Vector Variables

The higher spin Dirac operator \(\mathcal{Q}_{k,l}\) acting on functions taking values in an irreducible representation space for \(\mathfrak{so}(m)\) with highest weight \((k+\frac{1}{2},l+\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2})\), with k, l?∈?\(\mathbb{N}\) and \(k\geqslant l\), is constructed. The structure of the kernel space containing homogeneous polynomial solutions is then also studied.     

Leptoquark mechanism of neutrino masses within the grand unification framework

We demonstrate the viability of the one-loop neutrino mass mechanism within the framework of grand unification when the loop particles comprise scalar leptoquarks (LQs) and quarks of the matching electric charge. This mechanism can be implemented in both supersymmetric and non-supersymmetric models and requires the presence of at least one LQ pair. The appropriate pairs for the neutrino mass generation via the up-type and down-type quark loops are \(S_3\)–\(R_2\) and \(S_{1,\,3}\)–\(\tilde{R}_2\), respectively. We consider two distinct regimes for the LQ masses in our analysis. The first regime calls for very heavy LQs in the loop. It can be naturally realized with the \(S_{1,\,3}\)–\(\tilde{R}_2\) scenarios when the LQ masses are roughly between \(10^{12}\) and \(5 \times 10^{13}\) GeV. These lower and upper bounds originate from experimental limits on partial proton decay lifetimes and perturbativity constraints, respectively. Second regime corresponds to the collider accessible LQs in the neutrino mass loop. That option is viable for the \(S_3\)–\(\tilde{R}_2\) scenario in the models of unification that we discuss. If one furthermore assumes the presence of the type II see-saw mechanism there is an additional contribution from the \(S_3\)–\(R_2\) scenario that needs to be taken into account beside the type II see-saw contribution itself. We provide a complete list of renormalizable operators that yield necessary mixing of all aforementioned LQ pairs using the language of SU(5). We furthermore discuss several possible embeddings of this mechanism in SU(5) and SO(10) gauge groups.     

The envelope travelling wave solutions to the Gerdjikov–Ivanov model

Valence universal multireference coupled cluster (VUMRCC) method via eigenvalue independent partitioning has been applied to estimate the effect of three-body transformed Hamiltonian (\(\widetilde{{H}}_3\)) on ionisation potentials through full connected triple excitations \(S_{3}^{(1,0) }\). \(\widetilde{H}_3 \) is constructed using CCSDT1-A model of Bartlett et al for the ground-state calculation. Contribution of transformed Hamiltonian through full connected triples \(\overline{\widetilde{H}_3 S_3^{\left( {1,0} \right) }}\) involves huge amount of computational operations that is time-consuming. Investigation on \(\hbox {Cl}_{2}\) and \(\hbox {F}_{2}\) molecules using cc-pVDZ and cc-pVTZ basis sets shows that the above effect varies from 0.001 eV to around 0.5 eV, suggesting that inclusion of \(\overline{\widetilde{H} _3 S_3^{\left( {1,0} \right) } }\) is essential for highly accurate calculations.     

CP violations in predictive neutrino mass structures

We study the CP-violation effects from two types of neutrino mass matrices with (i) \((M_\nu )_{ee}=0\), and (ii) \((M_\nu )_{ee}=(M_\nu )_{e\mu }=0\), which can be realized by the high-dimensional lepton number violating operators \(\bar{\ell }_R^c\gamma ^\mu L_L (D_\mu \Phi )\Phi ^2\) and \(\bar{\ell }_R^c l_R (D_\mu {\Phi })^2\Phi ^2\), respectively. In (i), the neutrino mass spectrum is in the normal ordering with the lightest neutrino mass within the range \(0.002\,\mathrm{eV}\lesssim m_0\lesssim 0.007\,\mathrm{eV}\). Furthermore, for a given value of \(m_0\), there are two solutions for the two Majorana phases \(\alpha _{21}\) and \(\alpha _{31}\), whereas the Dirac phase \(\delta \) is arbitrary. For (ii), the parameters of \(m_0\), \(\delta \), \(\alpha _{21}\), and \(\alpha _{31}\) can be completely determined. We calculate the CP-violating asymmetries in neutrino–antineutrino oscillations for both mass textures of (i) and (ii), which are closely related to the CP-violating Majorana phases.     

Gravitational pressure,apparent horizon and thermodynamics of FLRW universe in the teleparallel gravity

In the context of the teleparallel equivalent of general relativity the concept of gravitational pressure and gravitational energy-momentum arisen in a natural way. In the case of a Friedmann–Lemaitre–Robertson–Walker space FLRW we obtain the total energy contained inside the apparent horizon and the radial pressure over the apparent horizon area. We use these definitions to written a thermodynamics relation \(T_{A}dS_{A} = dE_{A}+P_{A}dV_{A}\) at the apparent horizon, where \(E_{A}\) is the total energy inside the apparent horizon, \(V_{A}\) is the areal volume of the apparent horizon, \(P_{A}\) is the radial pressure over the apparent horizon area, \(S_{A}\) is the entropy which can be assumed as one quarter of the apparent horizon area only for a non stationary apparent horizon. We identify \(T_{A}\) as the temperature at the surface of the apparent horizon. We shown that for all expanding accelerated FLRW model of universe the radial pressure is positive.     

Rapid communication: $$K_{S}^{0}$$ Production from beryllium target using 120 $$\hbox {GeV}/\hbox {c}$$GeV/c protons beam interactions at the MIPP experiment

We have measured the cross-section for the \(K_{S}^{0}\) production from beryllium target using 120 \(\hbox {GeV}/\hbox {c}\) protons beam interactions at the main injector particle production (MIPP) experiment at Fermilab. The data were collected with target having a thickness of 0.94% of the nuclear interaction length. The \(K_{S}^{0}\) inclusive differential cross-section in bins of momenta is presented covering momentum range from \(0.4\,\hbox {GeV}/\hbox {c}\) to \(30\,\hbox {GeV}/\hbox {c}\). The measured inclusive \(K_{S}^{0}\) production cross-section amounts to \(39.54\pm 1.46\delta _{\mathrm {stat}}\pm 6.97\delta _{\mathrm {syst}}\) mb and the value is compared with the prediction of FLUKA hadron production model.     

Entanglement Rates and the Stability of the Area Law for the Entanglement Entropy

We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed \({\mathfrak{gl}(1|1)}\) spin-chain and its continuum limit—the \({c=-2}\) symplectic fermions theory—and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245–288, 2013) and Gainutdinov et al. (Nucl Phys B 871:289–329, 2013). Our main result is that the algebra of local Hamiltonians, the Jones–Temperley–Lieb algebra JTL _N , goes over in the continuum limit to a bigger algebra than \({\boldsymbol{\mathcal{V}}}\), the product of the left and right Virasoro algebras. This algebra, \({\mathcal{S}}\)—which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field \({S(z,\bar{z})\equiv S_{\alpha\beta} \psi^\alpha(z)\bar{\psi}^\beta(\bar{z})}\), with a symmetric form \({S_{\alpha\beta}}\) and conformal weights (1,1). We discuss in detail how the space of states of the LCFT (technically, a Krein space) decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the continuum limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL _N in the \({\mathfrak{gl}(1|1)}\) spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of \({\mathfrak{sp}_{N-2}}\). The semi-simple part of JTL _N is represented by \({U \mathfrak{sp}_{N-2}}\), providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL _N image represented in the spin-chain. On the continuum side, simple modules over \({\mathcal{S}}\) are identified with “fundamental” representations of \({\mathfrak{sp}_\infty}\).     

Likelihood analysis of the pMSSM11 in light of LHC 13-TeV data

We use MasterCode to perform a frequentist analysis of the constraints on a phenomenological MSSM model with 11 parameters, the pMSSM11, including constraints from \(\sim 36\)/fb of LHC data at 13 TeV and PICO, XENON1T and PandaX-II searches for dark matter scattering, as well as previous accelerator and astrophysical measurements, presenting fits both with and without the \((g-2)_\mu \) constraint. The pMSSM11 is specified by the following parameters: 3 gaugino masses \(M_{1,2,3}\), a common mass for the first-and second-generation squarks \(m_{\tilde{q}}\) and a distinct third-generation squark mass \(m_{\tilde{q}_3}\), a common mass for the first-and second-generation sleptons \(m_{\tilde{\ell }}\) and a distinct third-generation slepton mass \(m_{\tilde{\tau }}\), a common trilinear mixing parameter A, the Higgs mixing parameter \(\mu \), the pseudoscalar Higgs mass \(M_A\) and \(\tan \beta \). In the fit including \((g-2)_\mu \), a Bino-like \(\tilde{\chi }^0_{1}\) is preferred, whereas a Higgsino-like \(\tilde{\chi }^0_{1}\) is mildly favoured when the \((g-2)_\mu \) constraint is dropped. We identify the mechanisms that operate in different regions of the pMSSM11 parameter space to bring the relic density of the lightest neutralino, \(\tilde{\chi }^0_{1}\), into the range indicated by cosmological data. In the fit including \((g-2)_\mu \), coannihilations with \(\tilde{\chi }^0_{2}\) and the Wino-like \(\tilde{\chi }^\pm _{1}\) or with nearly-degenerate first- and second-generation sleptons are active, whereas coannihilations with the \(\tilde{\chi }^0_{2}\) and the Higgsino-like \(\tilde{\chi }^\pm _{1}\) or with first- and second-generation squarks may be important when the \((g-2)_\mu \) constraint is dropped. In the two cases, we present \(\chi ^2\) functions in two-dimensional mass planes as well as their one-dimensional profile projections and best-fit spectra. Prospects remain for discovering strongly-interacting sparticles at the LHC, in both the scenarios with and without the \((g-2)_\mu \) constraint, as well as for discovering electroweakly-interacting sparticles at a future linear \(e^+ e^-\) collider such as the ILC or CLIC.     

Light deflection and Gauss–Bonnet theorem: definition of total deflection angle and its applications

In this paper, we re-examine the light deflection in the Schwarzschild and the Schwarzschild–de Sitter spacetime. First, supposing a static and spherically symmetric spacetime, we propose the definition of the total deflection angle \(\alpha \) of the light ray by constructing a quadrilateral \(\varSigma ^4\) on the optical reference geometry \({\mathscr {M}}^\mathrm{opt}\) determined by the optical metric \(\bar{g}_{ij}\). On the basis of the definition of the total deflection angle \(\alpha \) and the Gauss–Bonnet theorem, we derive two formulas to calculate the total deflection angle \(\alpha \); (1) the angular formula that uses four angles determined on the optical reference geometry \({\mathscr {M}}^\mathrm{opt}\) or the curved \((r, \phi )\) subspace \({\mathscr {M}}^\mathrm{sub}\) being a slice of constant time t and (2) the integral formula on the optical reference geometry \({\mathscr {M}}^\mathrm{opt}\) which is the areal integral of the Gaussian curvature K in the area of a quadrilateral \(\varSigma ^4\) and the line integral of the geodesic curvature \(\kappa _g\) along the curve \(C_{\varGamma }\). As the curve \(C_{\varGamma }\), we introduce the unperturbed reference line that is the null geodesic \(\varGamma \) on the background spacetime such as the Minkowski or the de Sitter spacetime, and is obtained by projecting \(\varGamma \) vertically onto the curved \((r, \phi )\) subspace \({\mathscr {M}}^\mathrm{sub}\). We demonstrate that the two formulas give the same total deflection angle \(\alpha \) for the Schwarzschild and the Schwarzschild–de Sitter spacetime. In particular, in the Schwarzschild case, the result coincides with Epstein–Shapiro’s formula when the source S and the receiver R of the light ray are located at infinity. In addition, in the Schwarzschild–de Sitter case, there appear order \({\mathscr {O}}(\varLambda m)\) terms in addition to the Schwarzschild-like part, while order \({\mathscr {O}}(\varLambda )\) terms disappear.     

On a Pre-Lie Algebra Defined by Insertion of Rooted Trees

Let K be a field of characteristic zero. For \({n \in \mathbb{N}^{*}}\) , let \({\mathcal{T}^{\,\prime}_{n}}\) be the vector space of non-planar rooted trees with n vertices (Foissy in Bull Sci Math 126, no. 3, 193–239; no. 4, 249–288, 2002). Let \({\vartriangleright}\) be the left pre-Lie product of insertion of a tree inside another defined on infinitesimal characters of the graded Hopf algebra \({\mathcal{H}}\) introduced by Calaque, Ebrahimi-Fard and Manchon. Let \({\mathcal{T}^{\,\prime}=\oplus_{n\geq 2}\mathcal{T}^{\,\prime}_{n}}\) . In this work, we first prove that \({(\mathcal{T}^{\,\prime}, \vartriangleright)}\) a pre-Lie algebra generated by the two ladders E ₁ and E ₂ where E ₁ is the ladder with one edge and E ₂ is the ladder with two edges. Second, we prove that \({(\mathcal{T}^{\,\prime}, \vartriangleright)}\) is not a free pre-Lie algebra, and we exhibit a family of relations.     

Theoretical analysis of somatostatin receptor 5 with antagonists and agonists for the treatment of neuroendocrine tumors

We report on SSTR5 receptor modeling and its interaction with reported antagonist and agonist molecules. Modeling of the SSTR5 receptor was carried out using multiple templates with the aim of improving the precision of the generated models. The selective SSTR5 antagonists, agonists and native somatostatin SRIF-14 were employed to propose the binding site of SSTR5 and to identify the critical residues involved in the interaction of the receptor with other molecules. Residues Q2.63, D3.32, Q3.36, C186, Y7.34 and Y7.42 were found to be highly significant for their strong interaction with the receptor. SSTR5 antagonists were utilized to perform a 3D quantitative structure–activity relationship study. A comparative molecular field analysis (CoMFA) was conducted using two different alignment schemes, namely the ligand-based and receptor-based alignment methods. The best statistical results were obtained for ligand-based (\({q}^{2} = 0.454\), \({r}^{2}\) = 0.988, noc = 4) and receptor-guided methods (docked mode 1:\({q}^{2} = 0.530\), \({r}^{2} = 0.916\), noc = 5), (docked mode 2:\({q}^{2}\) = 0.555, \({r}^{2 }= 0.957\), noc = 5). Based on CoMFA contour maps, an electropositive substitution at \(\hbox {R}^{1}\), \(\hbox {R}^{2}\) and \(\hbox {R}^{4}\) position and bulky group at \(\hbox {R}^{4}\) position are important in enhancing molecular activity.     

Saturation spectra of low lying states of Nitrogen: reconciling experiment with theory

The hyperfine constants of the levels 2p ² \((^{3}\)P)3s ⁴P _J , 2p ² \((^3\)P)3p ⁴P\(^o_J\) and 2p ² \((^3\)P)3p ⁴D\(^o_J\), deduced by Jennerich et al. [Eur. Phys. J. D 40, 81 (2006)] from the observed hyperfine structures of the transitions 2p ² \((^3\)P)3s ⁴P _J \(\rightarrow\) 2p ² \((^3\)P)3p ⁴P\(^o_{J'}\) and 2p ² \((^3\)P)3s ⁴P _J \(\rightarrow\) 2p ² \((^3\)P)3p ⁴D\(^o_{J'}\) recorded by saturation spectroscopy in the near-infrared,strongly disagree with the ab initio values of Jönsson et al. [J. Phys. B: At. Mol.Opt. Phys. 43, 115006 (2010)].We propose a new interpretation of the recorded weak spectral lines. If the latter are indeed reinterpreted as crossover signals, a new set of experimental hyperfine constants is deduced, in very good agreement with the ab initio predictions.     

Spectral Deformation for Two-Body Dispersive Systems with e.g. the Yukawa Potential

We find an explicit closed formula for the k’th iterated commutator \({\text{ad}_{A}^{k}}(H_{V}(\xi ))\) of arbitrary order k ? 1 between a Hamiltonian \(H_{V}(\xi )=M_{\omega _{\xi }}+S_{\check V}\) and a conjugate operator \(A=\frac{\mathfrak{i}}{2}(v_{\xi}\cdot\nabla+\nabla\cdot v_{\xi})\), where \(M_{\omega _{\xi }}\) is the operator of multiplication with the real analytic function ω _ξ which depends real analytically on the parameter ξ, and the operator \(S_{\check V}\) is the operator of convolution with the (sufficiently nice) function \(\check V\), and v _ξ is some vector field determined by ω _ξ . Under certain assumptions, which are satisfied for the Yukawa potential, we then prove estimates of the form \(\| {{\text{ad}_{A}^{k}}(H_{V}(\xi ))(H_{0}(\xi )+\mathfrak{i} )}\|\leqslant C_{\xi }^{k}k!\) where C _ξ is some constant which depends continuously on ξ. The Hamiltonian is the fixed total momentum fiber Hamiltonian of an abstract two-body dispersive system and the work is inspired by a recent result [3] which, under conditions including estimates of the mentioned type, opens up for spectral deformation and analytic perturbation theory of embedded eigenvalues of finite multiplicity.     

Thermodynamic implications of the gravitationally induced particle creation scenario

A rigorous thermodynamic analysis has been done as regards the apparent horizon of a spatially flat Friedmann–Lemaitre–Robertson–Walker universe for the gravitationally induced particle creation scenario with constant specific entropy and an arbitrary particle creation rate \(\Gamma \). Assuming a perfect fluid equation of state \(p=(\gamma -1)\rho \) with \(\frac{2}{3} \le \gamma \le 2\), the first law, the generalized second law (GSL), and thermodynamic equilibrium have been studied, and an expression for the total entropy (i.e., horizon entropy plus fluid entropy) has been obtained which does not contain \(\Gamma \) explicitly. Moreover, a lower bound for the fluid temperature \(T_f\) has also been found which is given by \(T_f \ge 8\left( \frac{\frac{3\gamma }{2}-1}{\frac{2}{\gamma }-1}\right) H^2\). It has been shown that the GSL is satisfied for \(\frac{\Gamma }{3H} \le 1\). Further, when \(\Gamma \) is constant, thermodynamic equilibrium is always possible for \(\frac{1}{2}<\frac{\Gamma }{3H} < 1\), while for \(\frac{\Gamma }{3H} \le \text {min}\left\{ \frac{1}{2},\frac{2\gamma -2}{3\gamma -2} \right\} \) and \(\frac{\Gamma }{3H} \ge 1\), equilibrium can never be attained. Thermodynamic arguments also lead us to believe that during the radiation phase, \(\Gamma \le H\). When \(\Gamma \) is not a constant, thermodynamic equilibrium holds if \(\ddot{H} \ge \frac{27}{4}\gamma ^2 H^3 \left( 1-\frac{\Gamma }{3H}\right) ^2\), however, such a condition is by no means necessary for the attainment of equilibrium.     

Regularity of the Speed of Biased Random Walk in a One-Dimensional Percolation Model

We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelson-Fisk and Häggström established for this model a phase transition for the asymptotic linear speed \(\overline{\hbox {v}}\) of the walk. Namely, there exists some critical value \(\lambda _{\hbox {c}}>0\) such that \(\overline{\hbox {v}}>0\) if \(\lambda \in (0,\lambda _{\hbox {c}})\) and \(\overline{\hbox {v}}=0\) if \(\lambda \ge \lambda _{\hbox {c}}\). We show that the speed \(\overline{\hbox {v}}\) is continuous in \(\lambda \) on \((0,\infty )\) and differentiable on \((0,\lambda _{\hbox {c}}/2)\). Moreover, we characterize the derivative as a covariance. For the proof of the differentiability of \(\overline{\hbox {v}}\) on \((0,\lambda _{\hbox {c}}/2)\), we require and prove a central limit theorem for the biased random walk. Additionally, we prove that the central limit theorem fails to hold for \(\lambda \ge \lambda _{\hbox {c}}/2\).     

Modularity of logarithmic parafermion vertex algebras

The parafermionic cosets \(\mathsf {C}_{k} = {\text {Com}} ( \mathsf {H} , \mathsf {L}_{k}(\mathfrak {sl}_{2}) )\) are studied for negative admissible levels k, as are certain infinite-order simple current extensions \(\mathsf {B}_{k}\) of \(\mathsf {C}_{k}\). Under the assumption that the tensor theory considerations of Huang, Lepowsky and Zhang apply to \(\mathsf {C}_{k}\), irreducible \(\mathsf {C}_{k}\)- and \(\mathsf {B}_{k}\)-modules are obtained from those of \(\mathsf {L}_{k}(\mathfrak {sl}_{2})\). Assuming the validity of a certain Verlinde-type formula likewise gives the Grothendieck fusion rules of these irreducible modules. Notably, there are only finitely many irreducible \(\mathsf {B}_{k}\)-modules. The irreducible \(\mathsf {C}_{k}\)- and \(\mathsf {B}_{k}\)-characters are computed and the latter are shown, when supplemented by pseudotraces, to carry a finite-dimensional representation of the modular group. The natural conjecture then is that the \(\mathsf {B}_{k}\) are \(C_2\)-cofinite vertex operator algebras.     

The Structure of the Kac–Wang–Yan Algebra

The Lie algebra \({\mathcal{D}}\) of regular differential operators on the circle has a universal central extension \({\hat{\mathcal{D}}}\). The invariant subalgebra \({\hat{\mathcal{D}}^+}\) under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum \({\hat{\mathcal{D}}^+}\)-module with central charge \({c \in \mathbb{C}}\), and its irreducible quotient \({\mathcal{V}_c}\), possess vertex algebra structures, and \({\mathcal{V}_c}\) has a nontrivial structure if and only if \({c \in \frac{1}{2}\mathbb{Z}}\). We show that for each integer \({n > 0}\), \({\mathcal{V}_{n/2}}\) and \({\mathcal{V}_{-n}}\) are \({\mathcal{W}}\)-algebras of types \({\mathcal{W}(2, 4,\dots,2n)}\) and \({\mathcal{W}(2, 4,\dots, 2n^2 + 4n)}\), respectively. These results are formal consequences of Weyl’s first and second fundamental theorems of invariant theory for the orthogonal group \({{\rm O}(n)}\) and the symplectic group \({{\rm Sp}(2n)}\), respectively. Based on Sergeev’s theorems on the invariant theory of \({{\rm Osp}(1, 2n)}\) we conjecture that \({\mathcal{V}_{-n+1/2}}\) is of type \({\mathcal{W}(2, 4,\dots, 4n^2 + 8n + 2)}\), and we prove this for \({n = 1}\). As an application, we show that invariant subalgebras of \({\beta\gamma}\)-systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.