A survey is presented summarizing the empirical evidence for and interpretations of a first-order liquid-gas phase transition in nuclear matter. Earlier developments and the present state of knowledge about the extraction of the critical point for such a transition, primarily from the systematics of multifragmention data, are outlined. By analogy with a Van der Waals equation of state, the empirically deduced critical temperature and pressure permit to draw a schematic picture of the underlying nuclear potential. More detailed approaches to the liquid-gas transition using self-consistent nuclear Hartree-Fock and variational calculations are described. Critical exponents are reported. Then chiral effective field theory, as the low-energy realization of QCD, is discussed in the context of nuclear thermodynamics. Its implications for the liquid-gas transition in symmetric nuclear matter as well as in neutron-rich matter are reviewed.

Motivated by the recent paper arXiv:2602.11815 [nucl-th], we calculate in these notes the spectral functions (i.e. imaginary parts) of the NN-potentials as they arise from 2π-exchange with single and double Roper resonance excitation. In contrast to the full one-loop calculation, the spectral functions of the isoscalar and isovector central and tensor potentials can be given in simple analytical form. The pertinent momentum-space potentials V_C(q), W_C(q), V_T(q), W_T(q) are obtained via subtracted dispersion relations. This representation allows also to include a regulator function, that tames high-momentum components of the chiral 2π-exchange. The calculation is extended to 2π-exchange with combined Δ(1232)-isobar and Roper resonance excitation.

In this paper, the Born series for the s-wave scattering a0 is calculated for a class of central potentials V(r) up to sixth order in a dimensionless coupling strength g. Examples of exponentially decaying potentials as well as truncated potentials involving a single length-scale a are considered. In certain favorable cases, the exact result for the g-dependent s-wave scattering length a0=A0(g)a can be given in terms of special functions. The poles of A0(g) at increasing positive values of g correspond to the thresholds, where s-wave bound-states occur successively. Further scattering problems, where A0(g) is solvable in terms of elementary functions, are also presented.

Evidences are updated and strengthened for the two-scales picture of low-energy nucleon structure as a compact `hard' valence quark core surrounded by a `soft' cloud of quark-antiquark pairs (the meson cloud). These considerations are quantified by a spectral analysis of the mean-squared radii associated with the isoscalar and isovector electric form factors of the nucleon. Further supporting arguments come from corresponding studies of the axial and mass form factors and their inferred radii. Separating low-mass (mesonic) and high-mass (short-range) contributions in the spectral representations of each of these form factors, we conclude that a central core with an r.m.s. radius of about 1/2 fm results consistently as the common feature in all cases. Implications are discussed for baryonic matter at densities beyond that of equilibrium nuclear matter.

In this work the matrix exponential function is solved analytically for the special orthogonal groups SO(n) up to n=9. The number of occurring k-th matrix powers gets limited to 0≤k≤n−1 by exploiting the Cayley–Hamilton relation. The corresponding expansion coefficients can be expressed as cosine and sine functions of a vector-norm V and the roots of a polynomial equation that depends on a few specific invariants. Besides the well-known case of SO(3), a quadratic equation needs to be solved for n=4,5, a cubic equation for n=6,7, and a quartic equation for n=8,9. As an interesting subgroup of SO(7), the exceptional Lie group G2 of dimension 14 is constructed via the matrix exponential function through a remarkably simple constraint on an invariant, ξ=1. The traces of the SO(n)-matrices arising from the exponential function are sums of cosines of several angles. This feature confirms that the employed method is equivalent to exponentiation after diagonalization, but avoids complex eigenvalues and eigenvectors and operates only with real-valued quantities.

We examine the behavior of charged pions in neutron-rich matter using heavy-baryon chiral perturbation theory. This study is motivated by the prospect that pions, or pion-like excitations, may be relevant in neutron-rich matter encountered in core-collapse supernovae and neutron star mergers. We find, as previously expected, that the $\pi^-$ mass increases with density and precludes s-wave condensation at $n_B\lesssim n_{\text{sat}}$, where $n_{\text{sat}} \simeq 0.16 \text{fm}^{-3}$ is the nuclear saturation density, and the mass of the $\pi^+$ mode decreases with density. The uncertainty in these predictions increases rapidly for $n_B \gtrsim n_{\text{sat}} $ because low energy constants associated with the two-pion-two-nucleon operators in chiral perturbation theory are poorly constrained. We find that these uncertainties are especially large in symmetric nuclear matter and should be included in the analysis of pion-nucleus interactions at low energy and pionic atoms. In neutron-rich matter, accounting for the self-energy difference between neutrons and protons related to the nuclear symmetry energy has several effects. It alters the power counting of certain higher-order contributions to the pion self-energy. Previously unimportant but attractive diagrams are enhanced and result in a modest reduction of the pion masses. Furthermore, in the low-wavelength limit, a collective mode with the quantum numbers of the $\pi^+$ appears.

Information on the phase structure of strongly interacting matter at high baryon densities can be gained from observations of neutron stars and their detailed analysis. In the present work Bayesian inference methods are used to set constraints on the speed of sound in the interior of neutron stars, based on recent multi-messenger data in combination with limiting conditions from nuclear physics at low densities. Two general parametric representations are introduced for the sound speed $c_s$ in order to examine the independence with respect to choices for the parametrisation of Priors. Credible regions for neutron star properties are analysed, in particular with reference to the quest for possible phase transitions in cold dense matter. The evaluation of Bayes factors implies extreme evidence for a violation of the conformal bound, $c_s^2 \leq 1/3$, inside neutron stars. Given the presently existing data base, it can be concluded that the occurrence of a first-order phase transition in the core of even a two-solar-mass neutron star is unlikely, while a continuous crossover cannot be ruled out. At the same time it is pointed out that the discovery of a superheavy neutron star with a mass $ M \sim 2.3 - 2.4\, M_\odot$ would strengthen evidence for a phase change in the deep interior of the star.

With the aim of exploring the evidence for or against phase transitions in cold and dense baryonic matter, the inference of the sound speed and equation-of-state for dense matter in neutron stars is extended in view of recent new observational data. The impact of the heavy (2.35 $M_\odot$) black widow pulsar PSR J0952-0607 and of the unusually light supernova remnant HESS J1731-347 is inspected. In addition a detailed re-analysis is performed of the low-density constraint based on chiral effective field theory and of the perturbative QCD constraint at asymptotically high densities, in order to clarify the influence of these constraints on the inference procedure. The trace anomaly measure, $\Delta = 1/3 - P/\varepsilon$, is also computed and discussed. A systematic Bayes factor assessment quantifies the evidence (or non-evidence) of low averaged sound speeds $(c_s^2 \leq 0.1)$, a prerequisite for a phase transition, within the range of densities realized in the core of neutron stars. One of the consequences of including PSR J0952-0607 in the data base is a further stiffening of the equation-of-state, resulting for a 2.1 solar-mass neutron star in a reduced central density of less than five times the equilibrium density of normal nuclear matter at the 68\% level. The evidence against small sound speeds in neutron star cores is further strengthened. Within the inferred 68\% posterior credible bands, only a weak first-order phase transition with a coexistence density interval $\Delta n/n \lesssim 0.2$ would be compatible with the observed data.

In this work the matrix exponential function is solved analytically for the special orthogonal groups $SO(n)$ up to $n=9$. The number of occurring $k$-th matrix powers gets limited to $0\leq k \leq n-1$ by exploiting the Cayley-Hamilton relation. The corresponding expansion coefficients can be expressed as cosine and sine functions of a vector-norm $V$ and the roots of a polynomial equation that depends on a few specific invariants. Besides the well known case of $SO(3)$, a quadratic equation needs to be solved for $n=4,5$, a cubic equation for $n=6,7$, and a quartic equation for $n=8,9$. As an interesting subgroup of $SO(7)$, the exceptional Lie group $G_2$ of dimension $14$ is constructed via the matrix exponential function through a remarkably simple constraint on an invariant, $\xi=1$. The calculation of the trace of the $SO(n)$-matrices arising from the exponential function, results in a sum of cosines of several angles, which specify the associated conjugation class as a point on a maximal torus.

We introduce optimal observables to measure the Pauli form factor F_2 of the τ -lepton in the pair-production process e^- e^+ →τ ^-τ ^+ from the intensity distribution of the decay products. The spin-density matrix for the production process is calculated in QED up to order α ^3 including virtual photon-loops and soft bremsstrahlung, as well the γ Z^0 interference. We find that the decay channel (ρ ^-ν _τ )× (ρ ^+ ν̅_τ ) yields the best resolution for Re F_2(s) and Im F_2(s) due to its high branching fraction. We also study the bias that is introduced in the determination of F_2 , if the production spin-density matrix is taken in tree-level (one-photon exchange) approximation.

The well known analytical formula for SU(2) matrices $$U = \exp (i \vec \tau \!\cdot \! \vec \varphi \,) = \cos |\vec \varphi \,|\mathbf{1} + i\vec \tau \!\cdot \! {\hat{\varphi }} \, \sin |\vec \varphi \,|$$ is extended to the SU(3) group with eight real parameters. The resulting analytical formula involves the sum over three real roots of a cubic equation, corresponding to the so-called irreducible case, where one has to employ for solution the trisection of an angle. When going to the special unitary group SU(4) with 15 real parameters, the analytical formula involves the sum over four real roots of a quartic equation. The associated cubic resolvent equation with three positive roots belongs again to the irreducible case. Furthermore, by imposing the pertinent condition on SU(4) matrices one can also treat the symplectic group Sp(2) with ten real parameters. Since there the roots occur as two pairs of opposite sign, this simplifies the analytical formula for Sp(2) matrices considerably. An outlook to the situation with quasi-analytical formulas for SU(5), SU(6) and Sp(3) is also given.

We systematically calculate the radiative corrections of order α/π to elastic muon-proton scattering at low momentum transfers. These include vacuum polarization, photon-loop form factors of the muon and the proton, two-photon exchange corrections and soft photon radiation. In particular, we discuss these corrections for the kinematics of the upcomimg AMBER experiment with a 100 GeV muon beam energy. It is found that for the ratio to the Born cross section, only the minor terms from the photon-loop form factors of the proton and two-photon exchange depend on the proton structure predetermined by the strong interactions. Since a prominent role among the radiative corrections is played by soft photon radiation, the calculation of the bremsstrahlung process μp → μpγ should be extended beyond the soft photon approximation and tailored to the specific experimental conditions.

The phase structure of baryonic matter is investigated with focus on the role of fluctuations beyond the mean-field approximation. The prototype test case studied is the chiral nucleon-meson model, with added comments on the chiral quark-meson model. Applications to nuclear matter include the liquid-gas phase transition. Extensions to high baryon densities are performed for both nuclear and neutron matter. The role of vacuum fluctuations is systematically explored. It is pointed out that such fluctuations tend to stabilize the hadronic phase characterized by spontaneously broken chiral symmetry, shifting the chiral restoration transition to very high densities. This stabilization effect is shown to be further enhanced by additional dynamical fluctuations treated with functional renormalisation group methods.

From the subleading contributions to the chiral three-nucleon (3N) interaction [published in Phys. Rev. C 77, 064004 (2008) and Phys. Rev. C 84, 054001 (2011)], their first-order contributions to the energy per particle of isospin-symmetric nuclear matter and pure neutron matter are derived in an analytical way. For the variety of short-range and long-range terms that constitute the subleading chiral 3N force the pertinent closed 3-ring, 2-ring, and 1-ring diagrams are evaluated. While 3-ring diagrams vanish by a spin-trace and the results for 2-ring diagrams can be given in terms of elementary functions of the ratio Fermi momentum over pion mass, one ends up in most cases for the closed 1-ring diagrams with one-parameter integrals. The same treatment is applied to the subsubleading chiral three-nucleon interactions as far as these have been constructed up to now.

The g-factor and static quadrupole moment of the nuclides $$^{135}$$ Pr, $$^{105}$$ Pd, and $$^{187}$$ Au in the wobbling motion are investigated in the particle-rotor model as functions of the total spin I. The g-factor of $$^{105}\mathrm {Pd}$$ increases with increasing I, due to the negative gyromagnetic factor of a valence-neutron. This behavior is in contrast to the decreasing g-factor of the other two nuclides, $$^{135}$$ Pr and $$^{187}$$ Au, which feature a valence-proton. The static quadrupole moment Q depends on all three expectation values of the total angular momentum. It is smaller in the yrast band than in the wobbling band for the transverse wobblers $$^{135}$$ Pr and $$^{105}$$ Pd, while larger for the longitudinal wobbler $$^{187}$$ Au.

The long-range terms of the subleading chiral three-nucleon force [published in Phys. Rev. C77, 064004 (2008)] are specified to the case of three neutrons. From these 3n-interactions an effective density-dependent neutron-neutron potential V_med in pure neutron matter is derived. Following the division of the pertinent 3n-diagrams into two-pion exchange, two-pion-one-pion exchange and ring topology, all self-closings and concatenations of two neutron-lines to an in-medium loop are evaluated. The momentum and k_n-dependent potentials associated with the spin-operators 1, σ⃗_1·σ⃗_2, σ⃗_1·q⃗ σ⃗_2·q⃗, i( σ⃗_1+σ⃗_2)· (q⃗×p⃗ ), (σ⃗_1·p⃗ σ⃗_2·p⃗+σ⃗_1·p⃗ ' σ⃗_2·p⃗ ') and σ⃗_1· (q⃗×p⃗ )σ⃗_2· (q⃗×p⃗ ) are expressed in terms of functions, which are either given in closed analytical form or require at most one numerical integration. The subsubleading chiral 3N-force is treated in the same way. The obtained results for V_med are helpful to implement the long-range chiral three-body forces into advanced neutron matter calculations.

The interaction between hyperons and nucleons has a wide range of applications in strangeness nuclear physics and is a topic of continuing great interest. These interactions are not only important for hyperon-nucleon scattering but also essential as basic input to studies of hyperon-nuclear few- and many-body systems including hypernuclei and neutron star matter. We review the systematic derivation and construction of such baryonic forces from the symmetries of quantum chromodynamics within non-relativistic SU(3) chiral effective field theory. Several applications of the resulting potentials are presented for topics of current interest in strangeness nuclear physics.

Three-body forces acting on a $$\varLambda $$ hyperon in a nuclear medium are investigated, with special focus on the so-called hyperon puzzle in neutron stars. The hyperon–nucleon two-body interaction deduced from SU(3) chiral effective field theory is employed at next-to-leading order. Hyperon–nucleon three-body forces are approximated using saturation by decuplet baryons and are transcribed to density-dependent effective two-body interactions. These together are taken as input in a Brueckner–Bethe–Goldstone equation with explicit treatment of the $$\varLambda N\leftrightarrow \varSigma N$$ and $$\varLambda NN\leftrightarrow \varSigma NN$$ coupled channels. Single-particle potentials of a $$\varLambda $$ hyperon in symmetric nuclear matter and neutron matter are calculated. With parameters of the $$\varLambda NN$$ three-body force constrained by hypernuclear phenomenology, extrapolations to high baryon density are performed. By comparison of the $$\varLambda $$ and neutron chemical potentials at densities characteristic of the core of neutron stars it is found that the combined repulsive effects of two- and three-body correlations can make the appearance of $$\varLambda $$ hyperons in neutron stars energetically unfavourable, thus potentially offering a possible answer to a longstanding query.

The static quadrupole moments (SQMs) of nuclear chiral doublet bands are investigated for the first time taking the particle-hole configuration π(1h11/2)⊗ν(1h11/2)−1 with triaxial deformation parameters in the range 260∘≤γ≤270∘ as examples. The behavior of the SQM as a function of spin I is illustrated and analyzed. It is found that in the region of chiral vibrations the SQMs of doublet bands are strongly varying with I, whereas in the region of static chirality the SQMs of doublet bands are almost constant. Hence, the measurement of SQMs provides a new criterion for distinguishing the modes of nuclear chirality. Moreover, in the high-spin region the SQMs can be approximated by an analytic formula with a proportionality to cos⁡γ for both doublet bands. This provides a way to extract experimentally the triaxial deformation parameter γ for chiral bands from the measured SQMs.

In this work the elastic scattering of two nucleons is calculated in chiral effective field theory at next-to-leading order taking into account the coupled N$\Delta$-, $\Delta$N- and $\Delta\Delta$-channels. To solve the coupled channel scattering equation one needs as input the potentials for all combinations of these initial and final states. Up to next-to leading order these (transition) potentials arise from one-pion exchange, two-pion exchange and contact interactions. For the two-pion exchange we give analytic expressions for the spectral functions derived from all contributing one-loop diagrams. The forms of the contact potentials at leading and next-to-leading order are determined. We perform a fit of the low energy constants, that belong to the $NN \rightarrow NN$ contact potential and contribute up to next-to-leading order to $S$- and $P$-waves of NN scattering only. The influence of the $\Delta$-isobar dynamics entering through the coupled channels is studied in detail.