Conformal Window Research Articles

Defect induced heavy meson dynamics in the QCD conformal window

Defect induced heavy meson dynamics in the QCD conformal window

Exploring the conformal transition from above and below

We consider conformal transitions arising from the merging of IR and UV fixed points, expected to occur in QCD with a large enough number of flavors. We study the smoothness of physical quantities across this transition, being mostly determined by the logarithmic breaking of conformal invariance. We investigate this explicitly using holography where approaching the conformal transition either from outside or inside the conformal window (perturbed by a mass term) is characterized by the same dynamics. The mass of spin-1 mesons and Fπ are shown to be continuous across the transition, as well as the dilaton mass. This implies that the lightness of the dilaton cannot be a consequence of the spontaneous breaking of scale invariance when leaving the conformal window. Our analysis suggests that the light scalar observed in QCD lattice simulations is a qq¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ q\\overline{q} $$\\end{document} meson that becomes light since the qq¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ q\\overline{q} $$\\end{document}-operator dimension reaches its minimal value.

Low-energy theorem revisited and OPE in massless QCD

We revisit a low-energy theorem (LET) of NSVZ type in SU(N) QCD with Nf massless quarks derived in [1] by implementing it in dimensional regularization. The LET relates n-point correlators in the l.h.s. to n + 1-point correlators with the extra insertion of TrF2 at zero momentum in the r.h.s. We demonstrate that, for 2-point correlators of an operator O in the l.h.s., the LET implies that, in general, the integrated 3-point correlator in the r.h.s. needs in perturbation theory an infinite additive renormalization in addition to the multiplicative one. We relate the above counterterm to a corresponding divergent contact term in a certain coefficient of the OPE of TrF2 with O in the momentum representation, thus extending to any operator O an independent argument that first appeared for O = TrF2 in [2]. Finally, we demonstrate that in the asymptotically free phase of QCD the aforementioned counterterm in the LET is actually finite nonperturbatively after resummation to all perturbative orders. We also briefly recall the implications of the LET in the gauge-invariant framework of dimensional regularization for the perturbative and nonperturbative renormalization in large-N QCD. The implications of the LET inside and above the conformal window of SU(N) QCD with Nf massless quarks will appear in a forthcoming paper.

Reclaiming the Lost Conformality in a Non-Hermitian Quantum 5-State Potts Model.

Conformal symmetry, emerging at critical points, can be lost when renormalization group fixed points collide. Recently, it was proposed that after collisions, real fixed points transition into the complex plane, becoming complex fixed points described by complex conformal field theories (CFTs). Although this idea is compelling, directly demonstrating such complex conformal fixed points in microscopic models remains highly demanding. Furthermore, these concrete models are instrumental in unraveling the mysteries of complex CFTs and illuminating a variety of intriguing physical problems, including weakly first-order transitions in statistical mechanics and the conformal window of gauge theories. In this work, we have successfully addressed this complex challenge for the (1+1)-dimensional quantum 5-state Potts model, whose phase transition has long been known to be weakly first order. By adding an additional non-Hermitian interaction, we successfully identify two conjugate critical points located in the complex parameter space, where the lost conformality is restored in a complex manner. Specifically, we unambiguously demonstrate the radial quantization of the complex CFTs and compute the operator spectrum, as well as new operator product expansion coefficients that were previously unknown.

Schlieren measurements of shock train flow fields in a supersonic cylindrical isolator at Mach 2

In a supersonic cylindrical isolator at Mach 2, the structures and frequency characteristics of shock train flow fields were experimentally studied by the schlieren measurement method. According to the design principle of parallel light through schlieren windows in a cylindrical duct, a high-precision conformal optical window pair was designed and integratively processed before. Based on a self-built pipeline structure with conformal windows in a direct-connect wind tunnel under adjustable back-pressure conditions, the shock surfaces in a cylindrical isolator at Mach 2 were first captured by the schlieren method. Then, the schlieren photographs were corrected by a nonlinear image transformation algorithm for the restoration of real shock train structures, and the experimental results were compared with numerical simulation results quantitatively. Finally, the shock train positions were calculated by an image recognition algorithm to analyze the self-excited oscillation frequency characteristics of shock train structures. The methods and experiments in this study enriched optical observation methods of supersonic flows through non-rectangular cross-section isolators in scramjet.Graphical abstract

Scaling results for charged sectors of near conformal QCD

We provide the leading near conformal corrections on a cylinder to the scaling dimension of the lowest-lying fixed isospin charge Q operators defined at the lower boundary of the quantum chromodynamics conformal window, Δ˜Q=Δ˜Q*+(mσ4πν)2QΔ3B1+(mπ(θ)4πν)4Q23(1−γ)B2+O(mσ4,mπ8,mσ2mπ4). Here, Δ˜Q/r is the classical ground state energy of the theory on R×Sr3 at fixed isospin charge while Δ˜Q* is the scaling dimension at the leading order in the large charge expansion. In the conformal limit mσ=mπ=0, the state-operator correspondence implies Δ˜Q=Δ˜Q*. The near-conformal corrections are expressed in powers of the dilaton and pion masses in units of the chiral symmetry breaking scale 4πν with the θ-angle dependence encoded directly in the pion mass. The characteristic Q-scaling is dictated by the quark mass operator anomalous dimension γ and the one characterizing the dilaton potential Δ. The coefficients Bi with i=1,2 depend on the geometry of the cylinder and properties of the nearby conformal field theory. Published by the American Physical Society 2024

Effective potential and vacuum stability in the Litim-Sannino model

We revisit the scalar potential in the Litim-Sannino model. We compute for the first time the full quantum corrections to the classical potential and show that they significantly ameliorate the stability analysis at the UV fixed point. The quantum effective potential is computed at two-loop order and the numerical precision is further improved using resummations and parameter optimisations. As a result, we find a consistent widening of the UV conformal window across various approximations.

Dependence of the Landau gauge ghost-gluon-vertex on the number of flavors

The gauge-boson, ghost and fermion propagators as well as the gauge-boson-ghost vertex function are studied for SU(N), Sp(2N), and SO(N) gauge groups. We solve a set of coupled Dyson-Schwinger equations in Landau gauge for a variable, fractional number Nf of massless fermions in the fundamental representation. For large Nf we find a phase transition from a chirally broken into a chirally symmetric phase that is consistent with the behavior expected inside the conformal window. Even in the presence of fermions the gauge-boson-ghost-vertex dressing remains small. In the conformal window this vertex shows the expected power law behavior. It does not assume its tree-level value in the far infrared, but the respective dressing function is a constant greater than 1. Published by the American Physical Society 2024

Hydrodynamization in hybrid Bjorken flow attractors

Hybrid fluid models, consisting of two sectors with more weakly and more strongly self-interacting degrees of freedom coupled consistently as in the semi-holographic framework, have been shown to exhibit an attractor surface for Bjorken flow. Retaining only the simple viscid fluid descriptions of both sectors, we find that, on the attractor surface, the hydrodynamization times of both subsectors decrease with increasing total energy density at the respective point of hydrodynamization following a conformal scaling, reach their minimum values, and subsequently rise rapidly. The minimum values are obtained when the respective energy densities are of the order of the inverse of the dimensionful inter-system coupling. Restricting to attractor curves which can be matched to glasma models at a time set by the saturation scale for both p-p and Pb-Pb collisions, we find that the more weakly coupled sector hydrodynamizes much later, and the strongly coupled sector hydrodynamizes earlier in p-p collisions, since the total energy densities at the respective hydrodynamization times of these sectors fall inside and outside of the conformal window. This holds true also for phenomenologically relevant solutions that are significantly away from the attractor surface at the time we match to glasma models.

QCD with an infrared fixed point: The pion sector

The possibility that gauge theories with chiral symmetry breaking below the conformal window exhibit an infrared fixed point is explored. With this assumption three aspects of pion physics are reproduced if the quark mass anomalous dimension at the infrared fixed point is γ*=1. First, by matching the long-distance scalar adjoint correlation function. Second, by perturbing the fixed point by a small quark mass, the mq-dependence of the pion mass is reproduced by renormalization group arguments. Third, consistency of the trace anomaly and the Feynman-Hellmann theorem, for small mq, imply the same result once more. This suggests the following picture for the conformal window; close to its upper boundary γ* is zero and grows as the number of fermions is reduced until its lower boundary γ*=1 is reached, where chiral symmetry breaking sets in. Below, the strongly coupled gauge theory with γ*=1 is infrared dual to the free theory of pions. A possible dilaton sector of the scenario will be addressed in a companion paper. Published by the American Physical Society 2024

Dynamics of superconformal axion: Quality and scalegenesis

We explore a dynamical mechanism to realize the emergence of a global U(1)PQ symmetry and its spontaneous breaking at an intermediate scale for an axion solution to the strong CP problem. Such a dynamics is provided by a new supersymmetric QCD near the middle of conformal window that couples to fields spontaneously breaking the U(1)PQ symmetry. A large anomalous dimension of the U(1)PQ breaking fields leads to the suppression of explicit U(1)PQ-violating higher dimensional operators. The U(1)PQ breaking vacuum is generated at a scale hierarchically smaller than the Planck scale by a non-perturbative effect. The U(1)PQ breaking drives the conformal breaking, and all the new quarks become massive. The axion potential is generated by the ordinary color SU(3)C effect as the U(1)PQ symmetry is only anomalous under the SU(3)C. The saxion direction is stabilized by supersymmetry breaking and cosmologically harmless.

Flow evolution of mixed layer on convex curvature wall under hypersonic conditions

With the continuous upgrading of hypersonic vehicles, a new requirement for designing imaging window i.e. conformal window for improving aerodynamic characteristics, is put forward, in which the supersonic cooling film and optical window are required to maintain the same curvature shape as the aircraft body. In this work, the mixed-layer flow evolution on a convex wall (CV) is investigated. A nanoparticle-based planar laser scattering technique is used to design the flow field structure of the mixed layer in <i>Ma</i> = 6 hypersonic static wind tunnel, and the location of the mixed-layer instability is studied by combing fractal dimension. The results of pressure, and impulse of compression (<i>I</i><sub>p</sub>) evolution along the flow direction are obtained by numerical simulation, showing that the total incoming pressure (<i>P</i><sub>0</sub>) has a significant effect on the flow evolution of the mixed layer: as <i>P</i><sub>0</sub> increases, the ratio of static pressure (RSP) decreases, that the position of the mixed-layer instability is delayed, and that the flow velocity of the typical vortex structure increases. The favorable gradient existing at the CV wallleads the pressure to drop along the flow direction, and the pressure is enhanced when the supersonic air film along the tangential direction of the wall is under the operating condition. However, as <i>P</i><sub>0</sub> increases, the RSP decreases, and the lifting effect of the pressure on the CV decreases. The flow field is affected by the expansion effect of the CV, and <i>I</i><sub>p</sub> decreases along the flow direction. The supersonic air film can weaken the expansion effect on the CV and thus suppressing the decrease of <i>I</i><sub>p</sub>. The change rate of <i>I</i><sub>p</sub> (Δ<i>I</i><sub>p</sub>) is significantly affected by <i>P</i><sub>0</sub>, in a range of bending impulse |<i>I</i><sub><i>Φ</i></sub>| = 0.191–3.62, Δ<i>I</i><sub>p</sub> decreases from 178.67% to 12.02% when <i>P</i><sub>0</sub> = 0.5 MPa, and Δ<i>I</i><sub>p</sub> decreases from 40.38% to 5.64% when <i>P</i><sub>0</sub> = 1.0 MPa. Δ<i>I</i><sub>p</sub> decreases as |<i>I</i><sub><i>Φ</i></sub>| increases, but the decrease becomes less as <i>P</i><sub>0</sub> increases. The results reveal the flow evolution law of hypersonic mixed layer under the influence of convex curvature, and provide a certain reference for designing the shape of hypersonic vehicle to achieve aerodynamic drag reduction and thermal protection characteristics.

On the Anomalous Dimension in QCD

The anomalous dimension γm=1 in the infrared region near the conformal edge in the broken phase of the large Nf QCD has been shown by the ladder Schwinger–Dyson equation and also by the lattice simulation for Nf=8 and for Nc=3. Recently, Zwicky made another independent argument (without referring to explicit dynamics) for the same result, γm=1, by comparing the pion matrix element of the trace of the energy-momentum tensor π(p2)|(1+γm)·∑i=1Nfmfψ¯iψi|π(p1)=π(p2)|θμμ|π(p1)=2Mπ2 (up to trace anomaly) with the estimate of π(p2)|2·∑i=1Nfmfψ¯iψi|π(p1)=2Mπ2 through the Feynman–Hellmann theorem combined with an assumption Mπ2∼mf characteristic of the broken phase. We show that this is not justified by the explicit evaluation of each matrix element based on the dilaton chiral perturbation theory (dChPT): π(p2)|2·∑i=1Nfmfψ¯iψi|π(p1)=2Mπ2+[(1−γm)Mπ2·2/(1+γm)]=2Mπ2·2/(1+γm)≠2Mπ2 in contradiction with his estimate, which is compared with π(p2)|(1+γm)·∑i=1Nfmfψ¯iψi|π(p1)=(1+γm)Mπ2+[(1−γm)Mπ2]=2Mπ2 (both up to trace anomaly), where the terms in [] are from the σ (pseudo-dilaton) pole contribution. Thus, there is no constraint on γm when the σ pole contribution is treated consistently for both. We further show that the Feynman–Hellmann theorem is applied to the inside of the conformal window where dChPT is invalid and the σ pole contribution is absent, and with Mπ2∼mf2/(1+γm) instead of Mπ2∼mf, we have the same result as ours in the broken phase. A further comment related to dChPT is made on the decay width of f0(500) to ππ for Nf=2. It is shown to be consistent with the reality, when f0(500) is regarded as a pseudo-NG boson with the non-perturbative trace anomaly dominance.

Dynamic responses and damage evolution of conformal radome composites under blast loading

AbstractIn this paper, the dynamic responses and damage evolution of two types of conformal radome composites (CRCs), namely single‐window conformal radome (SWCR) and multi‐window conformal radome (MWCR), under blast loading, were investigated using numerical methods. The results show that the local dynamic response of the conformal window is dominating. The window deformation of SWCR is larger than that of MWCR under the same scaled distance, and the MWCR exhibits better blast resistance. The increase in window size causes the reduction of the bending stiffness of CRCs. The obvious stress concentration appears in the notch of carved foam cores, caused by the window effect. The conformal windows are weak parts for the CRCs under the blast loading. It was revealed that the damages of CRCs are mainly concentrated at the edges, center, and diagonal areas of the conformal windows.Highlights Designed two types of conformal radome composites with hierarchical configuration. Established the finite element (FE) models of two types of conformal radome composites Revealed the dynamic responses and damage evolution of conformal radome composites under blast loading.

Supersymmetric Cardy formula and the Weak Gravity Conjecture in AdS/CFT

The Weak Gravity Conjecture (WGC) in anti-de Sitter spacetime (AdS) asserts the existence of an operator in the boundary conformal field theory (CFT) whose scaling dimension-to-charge ratio satisfies a certain upper bound. This bound is specified by the ratio of the conformal central charge c and the flavor central charge kF. We propose a modified bound in AdS5/CFT4, determined by a combination of two central charges 3c − 2a instead of c. This combination arises in the Cardy-like limit of the 4d superconformal index, which captures the Bekenstein-Hawking entropy of large BPS black holes in AdS5. Using the new bound, we find that certain superconformal field theories (SCFTs) that are previously thought to violate the AdS WGC, including SQCDs in the conformal window, do satisfy the WGC. We check this version of the WGC against all possible superconformal gauge theories with SU(N) gauge group admitting a large N limit when the superpotential is absent. We conjecture the modified version of the WGC is a generic property of any 4d SCFT, regardless of the existence of a weakly coupled gravity dual or a large N limit.

Symplectic lattice gauge theories in the grid framework: Approaching the conformal window

Symplectic lattice gauge theories in the grid framework: Approaching the conformal window

Hierarchical and multi-material conformal composite structure for microwave transmission and blast-resistance

Two types of conformal composite panels, namely single-window conformal structure (SWCS) and multi-window conformal structure (MWCS), were designed and manufactured to achieve microwave transmission and blast-resistance simultaneously. These conformal panels have characteristics of multiple-materials, co-cured hierarchical structures and smooth external surface. Adopting woven lattice sandwich structure in conformal window lets that microwave transmission exceeding 80% covers bands from 4.5 GHz to 9.7 GHz and 16.5 GHz to 18 GHz. In field explosion experiments indicate that hierarchical stiffened sandwich structure with multi-windows exhibits better blast resistance. The MWCS remains structural integrity after repeated explosions with scaled distance varying from 2.565 m/kg1/3 to 0.646 m/kg1/3. Typical damage patterns of face blackening, face hole, matrix cracking, and breakage of conformal windows were revealed.

Mass-induced confinement near the sill of the conformal window

We revisit standard arguments for hyperscaling of the spectrum when a non-zero fermion mass is introduced to a gauge-fermion theory which is conformal in the infrared limit. With some general assumptions, we argue that the induced confinement scale will be significantly enhanced near the edge of the conformal to confining transition. This enhancement can allow for the fermion mass to be arbitrarily small compared to the confinement scale. This scale separation may allow for apparent spontaneous breaking of chiral symmetry within the conformal window, which may be of interest for construction of dilaton effective field theories in this regime.

Design and simulation of a conformal optical window pair by the quasi-quantitative Schlieren measurement in a cylindrical isolator

To realize the quasi-quantitative Schlieren measurement of the shock train structure across the cylindrical isolator in the supersonic flow field, based on the theory of Schlieren measurement and light transmission, we propose the design scheme of conformal windows and discuss the machining feasibility with the window’s parameters. The relationship between the Schlieren imaging effect, the deflection of parallel light in the Schlieren apparatus, and the conformal optical window aberrations was researched, putting forward the optical simulation model of the conformal optical windows and the quasi-quantitative Schlieren measurement. Moreover, considering the actual wind tunnel test conditions, under the speed of Mach 2 at the isolator’s inlet, the distortion brought by conformal optical windows in the Schlieren observation region was simulated by the computational fluid dynamics method. Finally, the image correction algorithm for the distorted picture caused by the conformal windows was presented, and the quasi-quantitative Schlieren measurement through the conformal optical window pair was conducted for the experimental verification. The results reveal the quasi-quantitative Schlieren measurement effect across the high-precision conformal optical window pair of the cylindrical isolator, providing guidance on maintaining the balance between the manufacturing feasibility and the observation performance of conformal windows.

Guide to anomaly-mediated supersymmetry-breaking QCD

We present a careful study of the chiral symmetry breaking minima and the baryonic directions in supersymmetric QCD ($SU({N}_{c})$ with ${N}_{f}$ flavors) perturbed by anomaly mediated supersymmetry breaking (AMSB). For the s-confining case of ${N}_{f}={N}_{c}+1$ and most of the free-magnetic phase (${N}_{f}\ensuremath{\le}1.43{N}_{c}$) we find that naive tree level baryonic runaways are stabilized by loop effects. Runaways are present, however, for the upper end of the free magnetic phase (${N}_{f}\ensuremath{\gtrsim}1.43{N}_{c}$) and into conformal window, signaling the existence of incalculable minima at large field values of $\mathcal{O}(\mathrm{\ensuremath{\Lambda}})$. Nevertheless, the chiral symmetry breaking points are locally stable, and are expected to continuously connect to the vacua of QCD for large SUSY breaking. The case of ${N}_{f}={N}_{c}$ requires particular care due to the inherently strongly coupled nature of the quantum modified moduli space. Due to the incalculability of critical K\"ahler potential terms, the stability of the chiral symmetry breaking point along baryonic directions cannot be determined for ${N}_{f}={N}_{c}$. With the exception of this case, all theories to which AMSB can be applied (${N}_{f}<3{N}_{c}$) possess stable chiral symmetry breaking minima, and all theories with ${N}_{f}\ensuremath{\lesssim}1.43{N}_{c}$ (aside from ${N}_{f}={N}_{c}$) are protected from runaways to incalculable minima.