Phase transitions between confinement and higgs phases in ${\cal N}=1\,\, SU(N_c)$ SQCD with $1\leq N_F\leq N_c$ quark flavors
Victor L. Chernyak
备注：
8 pages. Talk given at the International Conference on Particle Physics and Cosmology (professor V.A. Rubakov memorial conference), October 02-07, 2023, Yerevan, Armenia.\\
Considered is the standard 4-dimensional ${\cal N}=1\,\,SU(N_c)$ SQCD with $1\leq N_F\leq N_c$ quark flavors with masses $m_{Q,i}$. {\bf The gauge invariant order parameter $ρ$ is introduced in \cite{ch1} distinguishing confinement (with $ρ=0$) and higgs (with $ρ\neq 0$) phases}. Using a number of independent arguments for different variants of transition between the confinement and higgs regimes in these theories, it is shown that {\bf transitions between these regimes are not crossovers but the phase transitions}.
In \cite{FS} the very special $SU(N_c)$ QCD theory with $N_F=N_c$ defective scalar "quarks" was considered by E.Fradkin and S.H.Shenker. The conclusion of \cite{FS} was that {\bf the transition between the confinement and higgs regimes is the analytic crossover}. And although the theory considered in \cite{FS} was very specific, the experience shows that up to now there is a widely spread opinion that this conclusion has general applicability.
This model \cite{FS} is criticized in \cite{ch1},\cite{ch2} as incompatible with and very different from the standard non-SUSY $SU(N_c)\,\,N_F=N_c$ QCD theory with standard scalar quarks with all their physical degrees of freedom. It is emphasized that this model \cite{FS} is really {\bf the Stuckelberg $SU(N_c)$ YM-theory with no dynamical electric quarks and massive all $N_c^2-1$ electric gluons with fixed by hands nonzero masses}. There is no genuine confinement in this theory, it stays permanently in the completely higgsed (i.e. condensed) by hands phase only. And this is a reason for a crossover in this theory. While in the theory with standard scalar quarks there is the phase transition between the confinement (at $0 < |v|\ll Λ_{QCD}$) and higgs (at $|v|\ggΛ_{QCD}$) regimes.
We study 3d $\mathcal{N}=2$ SQCD with symplectic and orthogonal gauge groups and adjoint matter. For $USp(2n)$ with two fundamentals and $SO(N)$ with one vector these models have been recently shown to s-confine. Here we corroborate the validity of this proposal by relating it to the confinement of $USp(2n)$ with four fundamentals and an antisymmetric tensor, using exact mathematical results coming from the analysis of the partition function on the squashed three-sphere. Our analysis allows us to conjecture new s-confining theories for a higher number of fundamentals and vectors, in presence o…
We revisit the study of confining vacua in the softly broken N=2 supersymmetric QCD, in the light of some recent developments in our understanding of the dynamics of N=2 gauge theories. These vacua are characterized by an effective magnetic $SU(r)$ gauge group ($0\leq r\leq N_f/2$) and are referred to sometimes as the $r$ vacua. We further clarify the meaning of $r \leftrightarrow N_{f}-r$ duality arising from the matching of semi-classical and quantum vacua. A particular attention is paid to certain singular SCFT's of N=2 SQCD, driven into confinement phase by the adjoint mass deformation $μ\…
We study nonabelian vortices (flux tubes) in SU(N) gauge theories, which are responsible for the confinement of (nonabelian) magnetic monopoles. In particular a detailed analysis is given of ${\cal N}=2$ SQCD with gauge group SU(3) deformed by a small adjoint chiral multiplet mass. Tuning the bare quark masses (which we take to be large) to a common value $m$, we consider a particular vacuum of this theory in which an SU(2) subgroup of the gauge group remains unbroken. We consider $5 \ge N_f \ge 4$ flavors so that the SU(2) sub-sector remains non asymptotically free: the vortices carrying nona…
We consider the holographic dual of SQCD in the conformal phase. It is based on a higher derivative gravity theory, which ensures the correct field theory anomalies. This is then related to a six dimensional gravity theory via S^1 compactification. Some speculations are then made about the correspondence, Seiberg duality, and the nature of confinement from a holographic perspective.
Mass Spectra in ${\cal N}=1$ SQCD, in ${\cal N}=1$ SQCD-type theory and in softly broken ${\cal N}=2\rightarrow {\cal N}=1$ SQCD. And problems with the ${\cal N}=1$ Seiberg duality
We consider N=1 supersymmetric quantum chromodynamics (SQCD) with the gauge group U(N_c) and N_c+N quark flavors. N_c flavors are massless; the corresponding squark fields develop (small) vacuum expectation values (VEVs) on the Higgs branch. Extra N flavors are endowed with small (and equal) mass terms. We study this theory through its Seiberg's dual: U(N) gauge theory with N_c +N flavors of "dual quark" fields plus a gauge-singlet mesonic field M. The original theory is referred to as "quark theory" while the dual one is termed "monopole theory." The suggested mild deformation of Seiberg's pr…
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We consider N=2 SQCD with the U(N) gauge group and N_f flavors (N_f>N) perturbed by an N=2 breaking deformation - a small mass term μfor the adjoint matter. We study r-vacua, with the constraint $2N_f/3 < r \le N. At large values of the parameter ξ\simμm (m is a typical value of the quark masses) r quark flavors condense, by construction. The effective low-energy theory with the gauge group U(r)\times U(1)^{N-r} is at weak coupling. Upon reducing ξthe original theory undergoes a crossover transition from weak to strong coupling. As the original theory becomes strongly coupled, at low energies …
Recently we discovered and discussed non-Abelian duality in the quark vacua of N=2 super-Yang-Mills theory with the U(N) gauge group and N_f flavors (N_f>N). Both theories from the dual pair support non-Abelian strings which confine monopoles. Now we introduce an N=2-breaking deformation, a mass term μ{\mathcal A}^2 for the adjoint fields. Starting from a small deformation we eventually make it large which enforces complete decoupling of the adjoint fields. We show that the above non-Abelian duality fully survives in the limit of N=1 SQCD, albeit some technicalities change. For instance, non-A…
We calculate the three-loop Adler $D$-function of ${\cal N}=1$ SQCD regularized by higher covariant derivatives and find the subtraction scheme in which the exact NSVZ-like relation for this function that was recently proposed is valid.
We study ${\cal N}=2$ SU(2) supersymmetric QCD with massive hypermultiplets deformed in the Nekrasov-Shatashvili limit of the Omega-background. The prepotential of the low-energy effective theory is determined by the WKB solution of the quantum Seiberg-Witten curve. We calculate the deformed Seiberg-Witten periods around the massless monoplole point explicitly up to the fourth order in the deformation parameter.
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A new solution in the string background dual to N=1 SQCD-like theories is presented. The gauge coupling in this solution has walking property. The Wilson loop calculations show that quark anti-quark potential makes phase transitions. Additionally the effect of flavours on other solutions in this background is investigated by considering some unflavoured solutions and perturbing them with small parameter $x=\frac{N_{f}}{N_{c}}$.
This paper continues our studies in arXiV:1608.06452 [hep-th] of ${\cal N}=1$ gauge theories in the strongly coupled regimes. We also consider here the ${\cal N}=1$ SQCD-like theories with $SU(N_c)$ colors (and their Seiberg's dual), with $N_F$ flavors of light quarks and $N_F^2$ additional colorless flavored scalars $Φ^j_i$, but now with $N_F$ in the range $N_F>3N_c$. The mass spectra of these direct and dual theories in various vacua are calculated within the dynamical scenario introduced by the author in [8]. It assumes that quarks in such ${\cal N}=1$ SQCD-like theories without elementary …
Exact field theory dualities can be implemented by duality domain walls such that passing any operator through the interface maps it to the dual operator. This paper describes the S-duality wall of four-dimensional ${\cal N}=2$ SU(N) SQCD with 2N hypermultiplets in terms of fields on the defect, namely three-dimensional ${\cal N}=2$ SQCD with gauge group U(N-1) and 2N flavours, with a monopole superpotential. The theory is self-dual under a duality found by Benini, Benvenuti and Pasquetti, in the same way that T[SU(N)] (the S-duality wall of ${\cal N}=4$ super Yang-Mills) is self-mirror. The d…
We consider the lagrangian description of Argyres-Douglas theories of type $A_{2N-1}$, which is a $SU(N)$ gauge theory with an adjoint and one fundamental flavor. An appropriate reformulation allows us to map the moduli space of vacua across the duality, and to dimensionally reduce. Going down to three dimensions, we find that the adjoint SQCD "abelianizes": in the infrared it is equivalent to a $\mathcal{N}=4$ linear quiver theory. Moreover, we study the mirror dual: using a monopole duality to "sequentially confine" quivers tails with balanced nodes, we show that the mirror RG flow lands on …
New types of confining phase emerge when some singular SCFT's appearing as infrared fixed points of N=2 supersymmetric QCD (SQCD) are deformed by an N=1 adjoint mass term. We make further checks on the Gaiotto-Seiberg-Tachikawa (GST) description of these vacua against the symmetry and vacuum counting argument, and show that the GST variables correctly describe these systems, brought into confinement phase by the N=1 perturbation. Several examples of such vacua, USp(2N) and SU(N) theories with four flavors and SO(N) theories with one or two flavors, are discussed.
We derive an exact unified formula for all condensates (quark and monopole) in the hybrid r vacua in N=2 supersymmetric QCD slightly deformed by a μA^2 term. The gauge group is assumed to be U(N) and the number of the quark flavors N_f subject to the condition N< N_f < 2N. In the r vacua r quarks and N-r-1 monopoles from non-overlapping subgroups of U(N) develop vacuum expectation values (r<N). We then briefly review possible dynamical regimes (confinement, screening, and "instead of confinement") in the hybrid r vacua in μ-deformed N=2 SQCD (the small-μlimit).
New types of confinement phase emerge as singular SCFT's appearing as infrared-fixed-points of N=2 supersymmetric QCD (SQCD) are perturbed by an N=1 adjoint mass term. Based on a recent remarkable work on infrared-fixed-point SCFT of highest criticalities by Gaiotto, Seiberg and Tachikawa, we discuss physics of certain confining systems in SU(N), USp(2N) or SO(N) gauge theories. These show features different from a straightforward dual superconductivity picture of confinement a' la 't Hooft and Mandelstam, which might suggest a new venue in exploring the quark confinement mechanism in the real…
We obtain gravity duals to supersymmetric gauge theories in two and three spacetime dimensions with unquenched flavor. The supergravity solutions are generated by a set of color branes wrapping a compact cycle in a Calabi-Yau threefold, together with another set of flavor branes extended along the directions orthogonal to the cycle wrapped by the color branes. We construct supergravity backgrounds which include the backreaction induced by a smeared set of flavor branes, which act as delocalized dynamical sources of the different supergravity fields.
We construct holographic duals to N=1 SQCD with a quartic superpotential and unquenched massive flavors. Our backgrounds are generated by D5-branes wrapping two-dimensional submanifolds of an internal space. The flavor degrees of freedom are introduced by means of D5-branes extended along two-dimensional calibrated surfaces, and act as sources of the different supergravity fields. The backgrounds we get include the backreaction of the flavor branes and generalize the geometries obtained so far to the case in which the fundamental matter is massive. The supergravity solutions we find are regula…
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Considered are ${\cal N}=2\, SU(N_c)$ or $U(N_c)$ SQCD with $N_F<2N_c-1$ quark flavors with the quark mass term $m{\rm Tr}\,({\bar Q} Q)$ in the superpotential. ${\cal N}=2$ supersymmetry is softly broken down to ${\cal N}=1$ by the mass term $μ_{\rm x}{\rm Tr}\,(X^2)$ of colored adjoint scalar partners of gluons, $μ_{\rm x}\llΛ_2$ ($Λ_2$ is the scale factor of the $SU(N_c)$ gauge coupling). There is a large number of different types of vacua in these theories with both unbroken and spontaneously broken flavor symmetry, $U(N_F)\rightarrow U({\rm n}_1)\times U({\rm n}_2)$. We consider in this p…
The three-loop Adler $D$-function for ${\cal N}=1$ SQCD in the $\overline{\mbox{DR}}$ scheme is calculated. It appears that the result does not satisfy NSVZ-like equation which relates the $D$-function to the anomalous dimension of the matter superfields. However this NSVZ-like equation can be restored by a special tuning of the renormalization scheme. Also we demonstrate that the $D$-function defined in terms of the bare coupling does not satisfy the NSVZ-like equation in the case of using the regularization by dimensional reduction. The scheme-dependence of the $D$-function written in the fo…
We calculate the Adler $D$-function for ${\cal N}=1$ SQCD in the three-loop approximation using the higher covariant derivative regularization and the NSVZ-like subtraction scheme. The recently formulated all-order relation between the Adler function and the anomalous dimension of the matter superfields defined in terms of the bare coupling constant is first considered and generalized to the case of an arbitrary representation for the chiral matter superfields. The correctness of this all-order relation is explicitly verified at the three-loop level. The special renormalization scheme in which…
The three-loop Adler $D$-function for ${\cal N}=1$ SQCD in the $\overline{\mbox{DR}}$ scheme is calculated starting from the three-loop result recently obtained with the higher covariant derivative regularization. For this purpose, for the theory regularized by higher derivatives we find a subtraction scheme in which the Green functions coincide with the ones obtained with the dimensional reduction and the modified minimal subtraction prescription for the renormalization of the SQCD coupling constant and of the matter superfields. Also we calculate the $D$-function in the $\overline{\mbox{DR}}…
We propose a supersymmetric gradient flow in ${\cal N}=1$ SQCD in four dimensions. The flow equation is derived in the superfield formalism and is also given for component fields of the Wess-Zumino gauge in a gauge covariant manner. We find that the flow for the component fields is supersymmetric in a sense that the flow time derivative and any supersymmetry transformation commute with each other up to a gauge transformation.
It is widely considered that the classical Higgs branch of 4d $\mathcal{N}=2$ SQCD is a well understood object. However there is no satisfactory understanding of its structure. There are two complications: (1) the Higgs branch chiral ring contains nilpotent elements, as can easily be checked in the case of $\mathrm{SU}(N)$ with 1 flavour. (2) the Higgs branch as a geometric space can in general be decomposed into two cones with nontrivial intersection, the baryonic and mesonic branches. To study the second point in detail we use the recently developed tool of magnetic quivers for five-brane we…
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Considered are ${\cal N}=2\, SU(N_c)$ or $U(N_c)$ SQCD with $N_F<2N_c-1$ equal mass quark flavors. ${\cal N}=2$ supersymmetry is softly broken down to ${\cal N}=1$ by the mass term $μ_{\rm x}{\rm Tr}\,(X^2)$ of colored adjoint scalar partners of gluons, $μ_{\rm x}\llΛ_2$ ($Λ_2$ is the scale factor of the $SU(N_c)$ gauge coupling). There is a large number of different types of vacua in these theories with both unbroken and spontaneously broken global flavor symmetry, $U(N_F)\rightarrow U({\rm n}_1)\times U({\rm n}_2)$. We consider in this paper the large subset of these vacua with the unbroken …
Supersymmetric gauge theories have had a significant impact on our understanding of QCD and of field theory in general. The phases of N=1 supersymmetric QCD (SQCD) are discussed, and the possibility of similar phases in non-supersymmetric QCD is emphasized. It is described how duality in SQCD links many previously known duality transformations that were thought to be distinct, including Olive-Montonen duality of N=4 supersymmetric gauge theory and quark-hadron duality in (S)QCD. A connection between Olive-Montonen duality and the confining strings of Yang-Mills theory is explained, in which a …
We study the dynamics of 2+1 dimensional theories with ${\cal N}=1$ supersymmetry. In these theories the supersymmetric ground states behave discontinuously at co-dimension one walls in the space of couplings, with new vacua coming in from infinity in field space. We show that the dynamics near these walls is calculable: the two-loop effective potential yields exact results about the ground states near the walls. Far away from the walls the ground states can be inferred by decoupling arguments. In this way, we are able to follow the ground states of ${\cal N}=1$ theories in 2+1 dimensions and …
This paper continues our previous study of similar theories in \cite{ch5}. We also consider here the ${\cal N}=1$ SQCD-like theories with $SU(N_c)$ colors (and their Seiberg's dual with $SU(N_F-N_c)$ dual colors) and $N_F$ flavors of light quarks, and with $N_F^2$ additional colorless flavored fields $Φ^j_i$, but now with $N_F$ in the range $2N_c<N_F<3N_c$. The multiplicities of various vacua and quark and gluino condensates in these vacua are found. The mass spectra of the direct and Seiberg's dual theories in various vacua are calculated within the dynamical scenario which assumes that quark…
This article continues 1205.0410 [hep-th]. Considered is the N=1 SQCD-like theory with SU(N_c) colors and 3N_c/2 <N_F <2N_c flavors of light quarks Q_i,\bar Q_j, and with the additional N^2_F colorless flavored fields Phi_{ij} with the large mass parameter μ_Phi>>Lambda. The mass spectra of this Phi theory (and its dual variant, the dPhi theory) are calculated at different values of μ_Phi/Lambda >>1 within the dynamical scenario which implies that quarks can be in two different phases only: either this is the HQ (heavy quark) phase where they are confined, or they are higgsed at appropriate va…
We study supersymmetric Yang-Mills theories on the three-sphere, with massive matter and Fayet-Iliopoulos parameter, showing second order phase transitions for the non-Abelian theory, extending a previous result for the Abelian theory. We study both partition functions and Wilson loops and also discuss the case of different $R$-charges. Two interpretations of the partition function as eigenfunctions of the $A_{1} $ and free $A_{N-1}$ hyperbolic Calogero-Moser integrable model are given as well.
The Cachazo-Svrcek-Witten (CSW) rule for efficiently calculating gauge theory amplitudes is extended to N=1 supersymmetric QCD (SQCD), incorporating massless quarks, in a way preserving the manifest supersymmetry. Using this extended CSW rule, we obtain compact expressions of all the one-loop MHV amplitudes in SQCD including one or two external quark-antiquark (chiral-antichiral multiplets) pairs. The collinear sigularities of the five point amplitudes are investigated to confirm the consistency of the results.
Considered is the ${\cal N}=1$ SQCD-like theory with $SU(N_c)$ colors and $0< N_F<2N_c$ flavors of equal mass $0< m_Q\llΛ_Q$ quarks. Besides, it includes $N^2_F$ additional colorless but flavored fields $Φ_{i}^{j}$, with the large mass parameter $μ_Φ\ggΛ_Q$. The mass spectra of this $Φ$-theory are first directly calculated at $0<N_F<N_c$ where the quarks are weakly coupled, in all different vacua with the unbroken or spontaneously broken flavor symmetry $U(N_F)\rightarrow U(n_1)\times U(n_2)$. Further, the mass spectra of this direct $Φ$-theory and its Seiberg's dual variant, the $dΦ$-theory, …
Recent discoveries in supersymmetric gauge theories have significant implications for our understanding for QCD and of field theory in general. The phases of N=1 supersymmetric QCD (SQCD) are discussed, and the possibility of similar phases in non-supersymmetric QCD is emphasized. It is described how duality in SQCD links many previously known duality transformations that were thought to be distinct, including Olive-Montonen duality of N=4 supersymmetric gauge theory and quark-hadron duality in (S)QCD. A link between Olive-Montonen duality and the confining strings of (S)QCD is explained, in w…
We suggest how to consistently calculate the anomalous dimension $γ_*$ of the $\barψψ$ operator in finite order perturbation theory at an infrared fixed point for asymptotically free theories. If the $n+1$ loop beta function and $n$ loop anomalous dimension are known then $γ_*$ can be calculated exactly and fully scheme independently through $O(Δ_f^n )$ where $Δ_f = \bar{N_f} - N_f$ and $N_f$ is the number of flavors and $\bar{N}_f$ is the number of flavors above which asymptotic freedom is lost. For a supersymmetric theory the calculation preserves supersymmetry order by order in $Δ_f$. We th…
We consider a model which in a certain limit reduces to the large N ${\cal N}=1$ supersymmetric SU(N) gauge theory without matter. The gaugino condensate in this model is controlled by the dynamics of an additional singlet superfield. Using this model we explicitly construct BPS domain walls arising due to the chiral symmetry breaking. In particular, in the large N limit we obtain the exact shapes of the domain walls corresponding to solitons, and also of the domain walls interpreted as D-branes on which the SQCD string can end, whose existence was previously argued by Witten in the context of…
Considered is the direct ${\cal N}=1$ SQCD-like $Φ$-theory with $SU(N_c)$ colors and $3N_c/2< N_F<2N_c$ flavors of light quarks ${\overline Q},\,Q$. Besides, it includes $N^2_F$ additional colorless but flavored fields $Φ_{i}^{j}$ with the large mass parameter $μ_Φ\ggΛ_Q$, interacting with quarks through the Yukawa coupling. In parallel, is considered its Seiberg's dual variant, i.e. the $dΦ$-theory with $(N_F-N_c)$ dual colors, $N_F$ flavors of dual quarks ${\overline q},\,{q}$ and $N_F^2$ elementary mion fields $M^i_j\rightarrow ({\overline Q}_j Q^i)$. In considered here vacua, the quarks of…
It has been shown that non-Abelian solitonic vortex string supported in four-dimensional (4D) N = 2 supersymmetric QCD (SQCD) with the U(2) gauge group and $N_f = 4$ quark flavors becomes a critical superstring. This string propagates in the ten-dimensional space formed by a product of the flat 4D space and an internal space given by a Calabi-Yau noncompact threefold, namely, the conifold. The spectrum of low lying closed string states in the associated type IIA string theory was found and interpreted as a spectrum of hadrons in 4D N = 2 SQCD. In particular, the lowest string state appears to …
We consider the interplay between brane constructions and type IIA, IIB or M-theory geometries on Calabi-Yau (CY) and G_2 holonomy manifolds. This is related to N=1 (and N=2) gauge theories in four dimensions. We first discuss simple geometric transitions corresponding to brane set ups involving orthogonal (or parallel) Neveu-Schwarz branes that approach each other. This is related to confinement and Seiberg duality in SQCD. In particular, we argue that in type IIA, a CP^1 of singularities and one unit of Ramond-Ramond (RR) flux is dual to a D6 brane wrapped on a Lens space, describing the UV …
Using a proposal of Maldacena one describes the large N limit of gauge theories in terms of supergravity solutions on anti-de Sitter space. From this point of view we discuss a possible scenario for quark confinement in gauge theory by describing hadrons as strongly curved universes. In particular an interpretation of black hole as a bag model in SQCD is discussed. One relates the mystery of curvature singularities in classical general relativity with the mystery of quark confinement. The AdS bag model is defined by computing the probe membrane action in supergravity background. It naturally i…
It has been shown that non-Abelian solitonic vortex strings supported in four-dimensional (4D) ${\mathcal N}=2\;$ supersymmetric QCD (SQCD) with the U($N=2$) gauge group and $N_f=4$ quark flavors behave as critical superstrings. In addition to four translational moduli non-Abelian strings under consideration carry six orientational and size moduli. Together they form a ten-dimensional space required for a superstring to be critical. The target space of the string sigma model is a product of the flat four-dimensional space $\mathbb{R}^4$ and a Calabi-Yau non-compact threefold $Y_6$, namely, the…
We study new supergravity solutions related to large-$N_c$ ${\cal N}=1$ supersymmetric gauge field theories with a large number $N_f$ of massive flavors. We use a recently proposed framework based on configurations with $N_c$ color D5 branes and a distribution of $N_f$ flavor D5 branes, governed by a function $N_f S(r)$. Although the system admits many solutions, under plausible physical assumptions the relevant solution is uniquely determined for each value of $x\equiv N_f/N_c$. In the IR region, the solution smoothly approaches the deformed Maldacena-Núñez solution. In the UV region it appro…
Color confinement is one of the central issues in QCD so that there are various interpretations of this feature. In this paper we have adopted the interpretation that colored particles are not subject to observation just because colored states are unphysical in the sense of Eq. (2.16). It is shown that there are two phases in QCD distinguished by different choices of the gauge parameter. In one phase, called the "confinement phase", color confinement is realized and gluons turn out to be massive. In the other phase, called the "deconfinement phase", color confinement is not realized, but the g…
By using the gauge-invariant but path-dependent, variables formalism, we consider a recently proposed topologically massive $U{\left( 1 \right)_{\cal W}} \times U{(1)_{\cal Y}}$ Chern-Simons-Higgs theory in $2+1$ dimensions. In particular, we inspect the impact of a Chern-Simons mixing term between two Abelian gauge fields on physical observables. We pursue our investigation by analysing the model in two different situations. In the first case, where we integrate out the massive excitation and consider an effective model for the massless field, we show that the interaction energy contains a li…
We calculate circular Wilson loop expectation value of pure ${\cal N}=1$ super Yang-Mills from the Klebanov-Strassker-Tseytlin solution of supergravity and the proposed gauge/gravity duality. The calculation is performed numerically via searching world-sheet minimal surface. It is shown that Wilson loop exhibits area law for large radius which implies that ${\cal N}=1$ super Yang-Mills is confined at large distance or low energy scale. Meanwhile, Wilson loop exhibits logarithmic behavior for small radius and it indicates asymptotical freedom of ${\cal N}=1$ super Yang-Mills at short distance o…
We determine the static potential for a heavy quark-antiquark pair from gluodynamics in curved space-time. Our calculation is done within the framework of the gauge-invariant, path-dependent, variables formalism. The potential energy is the sum of a Yukawa and a linear potential, leading to the confinement of static charges.
We elaborate on s-confinement phases in three-dimensional $\mathcal{N}=2$ supersymmetric gauge theory, especially focusing on the $SU(N)$ and $USp(2N)$ gauge theories with anti-symmetric tensors and (anti-)fundamental matters. This will elucidate a quantum structure of the Coulomb moduli space of vacua. We stress the importance of so-called dressed Coulomb branch operators for describing these s-confinement phases. The 3d s-confinement phases are highly richer than the 4d ones since there is no chiral anomaly constraint on the matter contents.
We consider type IIB supergravity backgrounds which describe marginal deformations of the Coulomb branch of N=4 super Yang-Mills theory with SO(4) x SO(2) global symmetry. Wilson loop calculations indicate that certain deformations enhance the Coulombic attraction between quarks and anti-quarks at the UV conformal fixed-point. In the IR region, these deformations can induce a transition to linear confinement.
The confinement mechanism proposed earlier by the author is applied to problem of arising the so-called scale $Λ_{QCD}$ within the framework of QCD. The natural physical assumption consists of that $1/Λ_{QCD}\,\sim\,<r>$ where $<r>$ is a characteristic size of hadron (radius of confinement). The above confinement mechanism allows us to calculate $<r>$ for mesons in terms of quark and gluonic degrees of freedom and this permits to conclude that $Λ_{QCD}$ should slightly change from hadron to hadron.
