Holographic dense QCD in the Veneziano limit

The V-QCD model as discussed above already includes dense quark matter, but not nuclear matter. Baryons in the holographic approach are obtained through solitonic “instanton” configurations of the non-Abelian gauge fields living on the D-branes. While it is possible to construct such solitons in V-QCD (see Sec. 3.4), building a dense matter out of solitons is an extremely challenging problem. Therefore we resort to a simple approach with a homogeneous non-Abelian gauge field Rozali:2007rx ; Li:2015uea with two flavors, $A^{i}=h(r)\,\sigma^{i}\,,$ where $i=1,2,3$ are Lorentz indices, $r$ is the holographic coordinate, and $\sigma^{i}$ are the Pauli matrices (in flavor space). Applying this method in V-QCD Ishii:2019gta leads to the phase diagram of Fig. 2 (left). Interestingly the speed of sound in Fig. 2 (right) is high in the nuclear matter phase: the EOS is stiff. This will help building viable models that can support highly massive neutron stars.

The V-QCD model however cannot produce a fully feasible EOS for use in neutron star physics, because the homogeneous nuclear matter approach fails in the crust of the stars where densities are relatively low. Indeed, for densities below the nuclear saturation density $n_{s}\approx 0.16\,\mathrm{fm}^{-3}$ nucleons are clearly separated, and cannot be reliably modeled by a homogeneous distribution. Our approach is simply to replace holography in this regime by “traditional” approaches based on effective theory. It is actually difficult to derive a feasible EOS at low densities from holography (see however Kovensky:2021ddl ). We choose to use the popular APR Akmal:1998cf and HS(DD2) Hempel:2009mc ; Typel:2009sy EOSs at low densities (i.e., $n\lesssim 1.5n_{s}$) together with V-QCD at higher densities, and use them to construct families of “hybrid” EOSs at $T=0$ Ecker:2019xrw ; Jokela:2020piw ; Demircik:2021zll .

After completing the EOSs the simplest application is to analyze the structure of neutron stars. Results can be confronted with observations of neutron star masses and radii, as well as the deformation of neutron stars during the merger event GW170817 Abbott:2018exr . The mass measurements, in particular the result $M=2.08\pm 0.07M_{\odot}$ Fonseca:2021wxt for the pulsar J0740+6620 indicates that the maximum mass of neutron stars must be around $2M_{\odot}$ or higher. The effect of these constraints on the hybrid EOSs are demonstrated in Fig. 3 (left) as the pink band. We also added the constraints from NICER measurements of J0740+6620 (blue and black bands) Miller:2021qha ; Riley:2021pdl , and the EOSs for three examples of hybrid EOSs as the red curves. The right plot shows the same for mass-radius curves of nonrotating neutron stars.

Rapidly rotating neutron stars were analyzed using the V-QCD EOSs in Demircik:2020jkc . This study was motivated by the gravitational wave observation GW190425 Abbott:2020khf : a merger of a heavy ($\sim 23M_{\odot}$) black hole with a lighter ($\sim 2.6M_{\odot}$) companion. The lighter object falls in the “mass gap”, it is not clear whether it is a black hole or a neutron star. We found that our EOSs are compatible with the interpretation of this object as a neutron star, but it had to be rapidly rotating: the rotation frequency would need to be clearly higher than that of the fastest known pulsar. For the ratio of maximal masses of rotating and static neutron stars we found

a number slightly higher than in models without phase transition Breu:2016ufb .

While the EOS of QCD has large uncertainties at the densities of neutron star cores, much less is know about transport properties (see the review Schmitt:2017efp ). Transport is important for neutron star cooling, for damping of oscillations of the stars, and is also estimated to be relevant in neutron star mergers in the most violent period, right after the merger.

We computed estimates for transport properties using the D3-D7 model and V-QCD in Hoyos:2020hmq ; Hoyos:2021njg , and compared to perturbation theory. The results from holography were seen to deviate significantly from those obtained from perturbation theory. This suggests that one should be cautious when trying to estimate neutron star transport based on any perturbative analysis. Our result for the shear (left) and bulk (right) viscosities are shown in Fig. 4.

As commented above, the precise way of introducing nucleons in holographic models is via soliton configurations of non-Abelian gauge/fields in the bulk, which arise from the DBI actions Witten:1998xy ; Gross:1998gk . Such constructions have been carried out, for example, in the Witten-Sakai-Sugimoto model Kim:2006gp ; Hata:2007mb ; Bolognesi:2013nja and in the hard wall bottom-up model Pomarol:2007kr ; Pomarol:2008aa . The coupling of the soliton to the tachyon field, which characterizes chiral symmetry breaking, was studied in Gorsky:2013dda ; Gorsky:2015pra . The setup for the baryon solution in V-QCD was established in Jarvinen:2022mys . This required, among other things, a thorough study of the possible Chern-Simons action in the bottom-up framework, in the presence of nontrivial tachyon field. Numerical results for the soliton will appear soon Jarvinen:2022xxx . After fitting the model parameters to thermodynamics, meson masses, and the pion decay constant, the classical mass of the soliton was found to be near $1$ GeV, in good agreement with the observed masses of the nucleons.

State-of-the-art neutron star merger simulations require EOSs which depend, apart from density, the temperature and the charge fraction. The temperature dependence is important because shocks created during the merger heat up the matter significantly, and the charge fraction is important because the rapid merger process drives the matter outside $\beta$-equilibrium.

Consequently, the cold hybrid V-QCD EOS of Ecker:2019xrw ; Jokela:2020piw was extended to finite $T$ and outside of $\beta$-equilibrium in Demircik:2021zll . A key ingredient in this construction was a van-der-Waals model (see, e.g., Vovchenko:2017zpj ) which was adjusted such that it agrees exactly with the V-QCD EOS at $T=0$. It therefore provides an extrapolation of the V-QCD dense nuclear matter EOS to finite $T$. At low densities, and for the dependence on the charge fraction, we used the HS(DD2) model Hempel:2009mc ; Typel:2009sy , another key ingredient in this construction. Since both the quark matter and nuclear matter phases are described by V-QCD (apart from mild temperature dependence in the nuclear matter phase coming from the van-der-Waals extrapolation), the phase transition and the nuclear-quark matter mixed phase are determined by a single model, V-QCD.

The EOS was constructed in three variants, which roughly represent the leftover freedom in the parameters of V-QCD after fitting the model to lattice data.¹¹1The EOS files can be downloaded from the CompOSE database Typel:2013rza ; Typel:2022lcx for the three variants: DEJ(DD2-VQCD) soft, DEJ(DD2-VQCD) intermediate, DEJ(DD2-VQCD) stiff. In each of these models, the nuclear to quark matter transition was seen to end in a critical point with

which are close to estimates obtained in simpler holographic models DeWolfe:2010he ; Knaute:2017opk ; Critelli:2017oub ; Cai:2022omk .

The finite temperature V-QCD EOSs were used in state-of-the-art neutron star merger simulations in Tootle:2022pvd . That is, we evolved four dimensional general relativity and hydrodynamics by using the holographic EOS as an input. The chirp mass for the mergers was chosen to match with the GW170817 event, and we ran simulations with different mass ratios. The simulations used the Frankfurt University/Kadath (FUKA) Papenfort:2021hod for initial data and the Frankfurt/Illinois (FIL) code Most:2019kfe for evolution.

We focused on the production of quark matter in the merger. With the V-QCD EOSs, isolated neutron stars are fully composed of nuclear matter, but quark matter can be generated during the merger process, in particular right before a potential collapse into a black hole. We classified quark matter production into three stages: hot, warm and cold quark matter. In the first stage, hot quark matter is produces in the hottest (but not densest) regions of the hypermassive neutron star right after the merger. Warm quark matter is produced later due to complex merger dynamics in regions that are neither hottest nor densest. Finally, cold quark matter is produced in the dense and cold core right before the collapse to a black hole.