Progress in vacuum susceptibilities and their applications to the chiral phase transition of QCD
Zhu-Fang Cui, Feng-Yao Hou, Yuan-Mei Shi, Yong-Long Wang, Hong-Shi Zong
aa r X i v : . [ h e p - ph ] M a y Progress in vacuum susceptibilities and their applications to the chiralphase transition of QCD
Zhu-Fang Cui a,f, ∗ , Feng-Yao Hou b,f , Yuan-Mei Shi a,c,f , Yong-Long Wang a,d , Hong-Shi Zong a,e,f, ∗ a Department of Physics, Nanjing University, Nanjing 210093, China b Institute of Theoretical Physics, CAS, Beijing 100190, China c Department of Physics and electronic engineering, Nanjing Xiaozhuang University, Nanjing 211171, China d Department of Physics, School of Science, Linyi University, Linyi 276005, China e Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093, China f State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, CAS, Beijing, 100190, China
Abstract
The QCD vacuum condensates and various vacuum susceptibilities are all important parameters which char-acterize the nonperturbative properties of the QCD vacuum. In the QCD sum rules external field formula,various QCD vacuum susceptibilities play important roles in determining the properties of hadrons. In thispaper, we review the recent progress in studies of vacuum susceptibilities together with their applications tothe chiral phase transition of QCD. The results of the tensor, the vector, the axial-vector, the scalar, and thepseudo-scalar vacuum susceptibilities are shown in detail in the framework of Dyson-Schwinger equations.
Keywords: vacuum susceptibility, Dyson-Schwinger equations, chiral phase transition ∗ Corresponding author
Email addresses: [email protected] (Zhu-Fang Cui), [email protected] (Hong-Shi Zong)
Preprint submitted to Annals of Physics August 27, 2018 ontents1 Introduction 32 Derivation of the vector vacuum susceptibility 63 Calculations of various vacuum susceptibilities 9 . Introduction Nowadays, the commonly accepted theory that describes the strong interaction is Quantum Chromody-namics (QCD). Due to the asymptotic freedom nature of QCD, in the high energy and high momentum trans-fer region reliable calculations can be done using the perturbative theory. Numerous comparisons betweenthe results from theoretical calculations and experimental measurements have shown that QCD is correct inthe high energy field. However, in the low energy and low momentum transfer region the coupling constant α s of QCD becomes large and running, consequently things have to be treated non-perturbatively; andthanks to the non-perturbative nature, the QCD vacuum is not trivial. People believe that the non-trivialproperties of QCD vacuum are closely linked to spontaneous chiral symmetry breaking and confinement,which are the two main features of low-energy hadron physics. In order to characterize the QCD vacuum,people often introduce various vacuum condensates (such as the two-quark condensate, the gluon conden-sate, the mixed quark gluon condensates, and four-quark condensate, etc.) and many vacuum susceptibilities(such as the tensor vacuum susceptibility, the vector and the axial-vector vacuum susceptibilities, the scalarand the pseudo-scalar vacuum susceptibilities) phenomenologically [1, 2, 3, 4, 5, 6]. In this paper, we reviewthe recent progress in studies of vacuum susceptibilities together with their applications to the chiral phasetransition of QCD. The results of various vacuum susceptibilities are shown in detail in the framework ofDyson-Schwinger equations (DSEs).Metaphorically speaking, we can regard the QCD system as a black box, one way to study the QCDvacuum is to cause a perturbation by adding an external field to the system, such that we can learn somethingabout the QCD vacuum indirectly by studying its responses to the external field. In the theory of QCD,the quark propagator is the simplest Green function (two-point Green function), and its linear responsesto the external fields reflect the properties of QCD vacuum directly. QCD vacuum susceptibilities arejust the parameters that are associated with the linear responses of the quark propagator to the externalfields, which can then characterize the non-perturbative properties of QCD vacuum. In the QCD sumrules external field formula, QCD vacuum susceptibilities play important roles in determining the hadronproperties [2, 3, 5, 6]. For example, the strong and parity-violating pion-nucleon coupling depends cruciallyupon the vacuum susceptibilities of pion [7, 8], and the tensor vacuum susceptibility is closely related to thetensor charge of nucleon [9, 10, 11, 12, 13, 14, 15, 16, 17]. The nonlinear susceptibilities are also found tobe correlated with the cumulant of baryon-number fluctuations in experiments [18, 19, 20, 21, 22]. Due tothe importance of QCD vacuum susceptibilities in low energy hadron physics field, many different methodsand models to study various QCD vacuum susceptibilities have been utilized over the years. Take the scalarvacuum susceptibility as an example, there are Lattice QCD calculations [23, 24, 25, 26], calculations usingthe DSEs method [27, 28, 29], the multi-flavor Schwinger model [30, 31], the linear sigma model [32, 33],the (Polyakov loop extended) Nambu–Jona-Lasinio ((P)NJL) model [34, 35, 36], the nonlocal chiral quark In the experimental side, the physicists often measure the linear responses that are proportional to the perturbations, suchas many susceptibilities, conductivity, etc. Therefore, in order to compare with experimental results the calculations of variouslinear responses are very important in the theoretical side. ) [41, 42, 43]. Based on this, people have also studied theaxial-vector and the tensor vacuum susceptibilities in the framework of DSEs and under the rainbow-ladderapproximation [44, 45], while the authors of Ref. [46] studied the scalar vacuum susceptibility beyond therainbow-ladder approximation, as well as the pseudo-scalar vacuum susceptibility [47]. Here we want to stressthat the authors of Ref. [47] derived a model independent result for the pseudo-scalar vacuum susceptibilityusing the isovector–pseudo-scalar vacuum polarization, whereas the authors of Ref. [48] obtained modelindependent results for the vector and axial-vector vacuum susceptibilities using the vector and axial-vectorWard-Takahashi identities (WTIs). In Ref. [49], the authors calculated the tensor vacuum susceptibilityemploying the Ball-Chiu (BC) vertex [50, 51], and also studied the dressing effect of the quark-gluon vertexon the tensor vacuum susceptibility. Their results show that the tensor vacuum susceptibility obtained inthe BC vertex approximation is reduced by about 10% compared to its rainbow-ladder approximation value,which means that the dressing effect of the quark-gluon vertex is not very large in the calculation of thetensor vacuum susceptibility within the DSEs framework. These works promoted our understandings of thefive kinds of vacuum susceptibilities, and up to now, only the scalar and the tenser vacuum susceptibilitiesare still model-dependent, to choose a more reliable model then becomes very important.It is commonly accepted that with increasing temperature and/or quark chemical potential, stronglyinteracting matter will undergo a phase transition from the hadronic phase to the quark-gluon plasma(QGP) phase, which is also expected to appear in the ultra-relativistic heavy ion collisions [52], and thechiral symmetry which has been broken dynamically is also supposed to restore partially [53]. This is a veryimportant as well as a very hot topic. One of the main goals for RHIC (Relativistic Heavy Ion Collider) andLHC (Large Hadron Collider) is just to create and then study such a new state of matter. These studiesalso play crucial roles in researches on the evolution of the early universe. The properties of the QCDvacuum will change when the temperature and/or quark chemical potential vary, as a result the vacuumcondensates and vacuum susceptibilities which reflect the nature of the QCD vacuum should also change withtemperature and/or quark chemical potential. Theoretically, the QCD phase transitions via the calculationsof temperature and quark chemical potential dependence of the two-quark condensate or QCD vacuumsusceptibilities can then be studied. In the standard definition, an order parameter is a quantity which is zeroat one side of a phase transition and nonzero at the other side, so that the two-quark condensate in the chirallimit is such a quantity for the chiral phase transition (therefore, in some cases it is also referred to as “chiralcondensate”) [54]. However, while away from the chiral limit the two-quark condensate is no longer an exactorder parameter (since there is no exact phase transition), although it can be stretched to encompass first4rder transitions where the quantity has jumps from one nonzero value to another. Various susceptibilities,which are related to the derivatives of the two-quark condensate, are usually used to locate the transition,and hence are also important quantities to characterize the properties of the system. For example, the scalarsusceptibility is often used to describe the QCD chiral symmetry restoration at finite temperature and finitechemical potential [23, 28, 35, 55, 56, 57, 58, 59, 60], while studies of the temperature and chemical potentialdependence of the quark number susceptibility can provide useful information to the vicinity of the criticalend point (CEP), where the first order phase transition meets with the crossover and a second order phasetransition takes place [61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 46, 79, 80, 81, 82].The cases with finite temperature and chemical potential represents a big branch in the field of studieson QCD vacuum susceptibilities. Hence it is difficult to give a comprehensive presentation on its theoriesand applications to the chiral phase transition of QCD. In this paper, we mainly discuss the calculationsof various QCD vacuum susceptibilities at zero temperature and zero chemical potential, and then presentsome results in studies of the chiral phase transition of QCD. We will review the QCD sum rules externalfield formula in a nutshell, then take the constant vector field as an example to give the general derivationof the vector vacuum susceptibility, and obtain a model independent expression at last. We will also discussthe tensor, the vector, the axial-vector, the scalar, and the pseudo-scalar vacuum susceptibilities in detailtogether with their applications. From the model independent form of the vacuum susceptibilities, wesee that their calculations are closely linked to the quark propagator (two-point Green function) and thecorresponding vertex function (three-point Green function). Lattice QCD, which is commonly acceptedas a first principle calculation of QCD itself, can give good results for the two-point Green functions, butfor three- and more points Green functions there is still a long way to go. Accordingly, in order to dothe calculations we have to, at present, turn to some effective models or approaches that are based onQCD itself. In this paper we mainly use the framework of DSEs approach, which is a very useful andsuccessful non-perturbative method to treat the non-perturbative strong interactions as well as the propertiesof hadrons [83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96]. It must be emphasized here that, DSEsis an infinite series of coupled non-perturbative integral equations that the Green functions of continuousspace-time should satisfy, of which a given n -point Green function is related with the m -point Green function,where m > n . In order to solve practical problems, people have to take “appropriate” approximations tothe DSEs. So far, the most commonly used approximation scheme is the rainbow-ladder approximation.The rainbow approximation means that in the DSE for quark propagator the full quark-gluon vertex isreplaced by the bare vertex, while the ladder approximation is that in the Bethe-Salpeter equation (BSE) ofthe quark-meson vertex the quark–anti-quark scattering kernel is replaced by its lowest order perturbativecontribution, details of which can be found in Refs. [83, 85, 88]. The rainbow-ladder approximation is selfconsistent [97]. Besides, rainbow-ladder approximation also promises the DSEs to satisfy the axis vector Moreover, the two-quark condensate is actually unmeasurable, while various susceptibilities are reachable in the experiments. A good approximation should holds the original symmetries of QCD itself as many as possible, and meanwhile, easy tocalculate. In the global color symmetry model (GCM)[86, 98], if adopting the saddle point approximation, it can be found that the
2. Derivation of the vector vacuum susceptibility using the QCD sum rules external fieldformula
In this section we derive a model independent expression of the vector vacuum susceptibility using theQCD sum rules external field formula. To be specific, we will work in Euclidean space throughout this paper.The vacuum susceptibilities are closely related to the linear responses of the full quark propagator to theexternal fields, so we add to the Lagrangian a term that the quark current coupled to an external field, J Γ ( y ) V Γ ( y ) ≡ ¯ q ( y )Γ q ( y ) V Γ ( y ) , (1)here J Γ ( y ) is the quark current, V Γ ( y ) is the external field, q ( y ) is the quark field, Γ is a matrix in thedirect product space of Dirac, flavor, and color. The linear response term of the full quark propagator tothe external field can be written as [40], G cc ′ Γ αβ,q ( x ) = h ˜0 | T [ q cα ( x )¯ q c ′ β (0)]˜0 i J Γ = G cc ′ Γ ,P Tαβ,q ( x ) + G cc ′ Γ ,NPαβ,q ( x ) , (2)where ( cc ′ ) and ( αβ ) are the color and spinor indices of the quark, | ˜0 i is the exact vacuum; G Γ ,NPq ( x ) and G Γ ,P Tq ( x ) denote the non-perturbative and perturbative effects of the full quark propagator G cc ′ Γ αβ,q ( x ) underan external field, respectively. In the paper we adopt such a concept that, G cc ′ Γ αβ,q ( x ) is calculated in theNambu vacuum, of which dynamical chiral symmetry breaking (DCSB) and quark confinement are two ofthe most important features; while G Γ ,P Tq ( x ) is calculated in the Wigner vacuum, where the system holdschiral symmetry (in the chiral limit case) and the dressed quarks are not confined (more discussions can befound in Sec. 3.1 of this paper, and Refs. [40, 83, 88]). As we will show in the following part, G Γ ,NPq ( x ) isrelated to the vacuum susceptibilities and hard to calculate directly, while G cc ′ Γ αβ,q ( x ) and G Γ ,P Tq ( x ) can becalculated, and then we can get G Γ ,NPq ( x ) as well as the vacuum susceptibilities indirectly.In the QCD sum rules theory, the vacuum susceptibility is defined as [2, 3, 5, 8], G cc ′ Γ ,NPαβ,q ( x ) ≡ h ˜0 | : q cα ( x )¯ q c ′ β (0) : | ˜0 i J Γ ≡ −
112 (Γ V Γ ) αβ δ cc ′ χ Γ H ( x ) h | : ¯ q (0) q (0) : | i , (3) equation which quark propagator satisfy is just the unrenormalized DSE under rainbow approximation, while the one whichquark-meson vertex satisfy is just the unrenormalized BSE under ladder approximation. i represents the perturbative vacuum, χ Γ is the vacuum susceptibility, while H ( x ) is a function thatcharacterize the non-local property of the non-local two-quark condensate, and H (0) = 1. From Eq. (3)we can see that, in order to calculate χ Γ we should know G cc ′ Γ ,NPαβ,q ( x ) first, which means we should know G Γ ,P Tq ( x ) and G Γ q ( x ) simultaneously. Therefore, how to solve G Γ ,P Tq ( x ) and G Γ q ( x ) self-consistently becomesthe key to get a model independent result. Next we will take a constant vector field V µ as an example toshow how to get a model independent expression for the vector vacuum susceptibility [40].When there is an external field, a term which is the coupling of the vector current and the vector externalfield should be added to the action of the system, namely, ∆ S ≡ R d x ¯ q ( x ) iγ µ q ( x ) V µ ( x ). In this case, thedressed quark propagator in the chiral limit is G [ V ]( x ) = Z D ¯ q D q D A q ( x )¯ q (0) exp (cid:26) − S [¯ q, q, A ] − Z d x ¯ q ( x ) γ µ q ( x ) V µ ( x ) (cid:27) , (4)where S [¯ q, q, A ] = Z d x (cid:26) ¯ q ( ∂ − ig λ a A a ) q + 14 F aµν F aµν (cid:27) , (5)and F aµν = ∂ µ A aν − ∂ ν A aµ + gf abc A bµ A cν . Here we do not show the gauge fixing term, as well as the ghost fieldand its integration measure. Considering the linear response of the quark propagator to the external field,we can obtain G [ V ]( x ) = Z D ¯ q D q D A [ q ( x )¯ q (0)] exp {− S [¯ q, q, A ] }− Z D ¯ q D q D A (cid:20) q ( x )¯ q (0) Z d y ¯ q ( y ) γ µ q ( y ) V µ ( x ) (cid:21) exp {− S [¯ q, q, A ] } + · · ·≡ h ˜0 | T τ [ q ( x )¯ q (0)] | ˜0 i − Z d y h ˜0 | T τ [ q ( x )¯ q (0)¯ q ( y ) γ µ q ( y )] | ˜0 iV µ ( y ) + · · · , (6) T τ represents the time-ordering operation on Euclidean time τ , G ( x ) ≡ h ˜0 | T τ [ q ( x )¯ q (0)] | ˜0 i = G [ V = 0]( x ) isthe full quark propagator without an external field ( V µ = 0), while G V ( x ) ≡ h ˜0 | T τ [ q ( x )¯ q (0)]˜0 i V ≡ − Z d y h ˜0 | T τ [ q ( x )¯ q (0)¯ q ( y ) γ µ q ( y )] | ˜0 iV µ ( y ) (7)is the linear response term of the quark propagator to the external field. Hence, now the main task is toknow R d y h ˜0 | T τ [ q ( x )¯ q (0)¯ q ( y ) γ µ q ( y )] | ˜0 i , we can try to do Taylor expansion of the inverse of dressed quarkpropagator to V µ [40, 44, 45], G [ V ] − = G [ V ] − (cid:12)(cid:12) V µ =0 + δ G − [ V ] δ V µ (cid:12)(cid:12)(cid:12)(cid:12) V µ =0 V µ + · · · ≡ G − + Γ µ V µ + · · · , (8)and then only keep the first order term of the external field, G [ V ] = G − G Γ µ V µ G + · · · . (9)Here the vector vertex Γ µ is defined asΓ µ ( y , y ; z ) ≡ (cid:20) δ G [ V ]( y , y ) − δ V µ ( z ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) V µ =0 . (10)7he derivation of Eq. (9) is general, so is applicable to both full and perturbative quark propagators. Inconfiguration space it can be written explicitly as G [ V ]( x ) = G ( x ) − Z d u Z d u G ( x, u )Γ µ ( u , u ) G ( u , V µ + · · · = G ( x ) − Z d u Z d u G ( x, u ) (cid:20)Z d p (2 π ) e ip · ( u − u ) Γ µ ( p, (cid:21) G ( u , V µ + · · · = G ( x ) − Z d p (2 π ) e ip · x G ( p )Γ µ ( p, G ( p ) V µ + · · · . (11)Then the linear response term of the full quark propagator G Z ( x ) to an external field is, G V ( x ) ≡ h ˜0 | T τ [ q ( x )¯ q (0)] | ˜0 i V = − Z d p (2 π ) e ip · x G ( p )Γ µ ( p, G ( p ) V µ . (12)and similarly for the perturbative quark propagator G Z,P T ( x ), G V ,P T ( x ) ≡ h | T τ [ q ( x )¯ q (0)] | i Z = − Z d u Z d u G P T ( x, u ) (cid:20)Z d p (2 π ) e ip · ( u − u ) Γ P Tµ ( p, (cid:21) V µ G P T ( u , − Z d p (2 π ) e ip · x G P T ( p )Γ P Tµ ( p, V µ G P T ( p ) , (13)where G P T ( x ) ≡ h | T τ [ q ( x )¯ q (0)] | i , and Γ P Tµ ( p ; 0) is the corresponding perturbative vertex.Now let’s turn to the calculation of vacuum susceptibilities, the definition of the vector vacuum suscep-tibility χ V is, h ˜0 | : q aα (0)¯ q bβ (0) : | ˜0 i NP V · V ≡ − ( − i )12 ( γ · V ) αβ δ ab χ V h ˜0 | : ¯ q (0) q (0) : | ˜0 i , (14)where G ab V ,NPαβ,q (0) ≡ h ˜0 | : q aα (0)¯ q bβ (0) : | ˜0 i NP V is the local two-quark condensate. It can be seen that thetwo-quark condensate is an explicitly gauge-invariant quantity, and is indeed computed in a gauge-invariantway for example in Lattice QCD simulations. That it depends on the renormalization scheme is on the otherhand obvious. If we multiply δ ab γ βαν on both sides of Eq. (14), then h ˜0 | : ¯ q (0) γ ν q (0) : | ˜0 i NP V · V = ( − i ) χ V V ν h ˜0 | : ¯ q (0) q (0) : | ˜0 i = (cid:2) h ˜0 | T [¯ q (0) γ ν q (0)] | ˜0 i V − h | T [¯ q (0) γ ν q (0)] | i P T V (cid:3) · V . (15)From Eqs. (12) and (13) we can find that (cid:2) h ˜0 | T [¯ q (0) γ ν q (0)] | ˜0 i V (cid:3) · V = tr DC [ γ ν G Γ · V G ] , (16) (cid:2) h | T [¯ q (0) γ ν q (0)] | i P T V (cid:3) · V = tr DC (cid:2) γ ν G P T Γ P T · V G P T (cid:3) . (17) Without truncation, DSEs is equivalent to QCD; However, it is impossible to solve infinite coupled equations, to choose abetter truncation scheme is then not only necessary but also challenging. Concerning the choice of gauge, we will use Landaugauge throughout this paper, which has many advantages [83, 88, 95, 96, 99, 100, 101, 102, 103], for example, it is a fixedpoint of the renormalization group; that gauge for which sensitivity to model-dependent differences between Ansatze for thefermion-gauge boson vertex are least noticeable; and a covariant gauge, which is readily implemented in simulations of Latticeregularized QCD (see, e.g., Refs. [104, 105, 106] and citations therein and thereto). Importantly, capitalisation on the gaugecovariance of Schwinger functions obviates any question about the gauge dependence of gauge invariant quantities. χ V h ˜0 | : ¯ q (0) q (0) : | ˜0 i = i (cid:2) tr DC [ γ ν G Γ µ G ] − tr DC [ γ ν G P T Γ P Tµ G P T ] (cid:3) V µ V ν V . (18)Here we note that, from the viewpoint of Feynman diagram, tr DC [ γ ν G Γ µ G ] and tr DC [ γ ν G P T Γ P Tµ G P T ] arerespectively the values of the full and the perturbative vector vacuum polarizations at zero momentum, sothe calculation of the vector vacuum susceptibility comes down to the calculation of the vector vacuumpolarization. Here we want to stress that, as pointed out in Ref. [40], the derivation in this section is generaland model independent for a constant external field, and so it can also be used to derive other vacuumsusceptibilities similarly, such as the tensor vacuum susceptibility. However, for the axial-vector vacuumsusceptibility, we need to take a variable external field, details can be found in Ref. [44]. The significanceof the works in Refs. [40, 44] is that they give theoretically model independent results of various vacuumsusceptibilities, which provides a good starting point for further model calculations. In the next section wewill discuss the tensor, the vector, the axial-vector, the scalar, and the pseudo-scalar vacuum susceptibilitiesrespectively.
3. Calculations of various vacuum susceptibilities
After the model independent expressions of QCD vacuum susceptibilities are obtained, the next stepis to use these expressions to calculate the values of these vacuum susceptibilities. From Eq. (18) weknow that, in order to calculate the vector vacuum susceptibility, we must first know the quark propagator G ( P ) (two-point Green function) and the vertex function Γ µ ( P, Using the functional path integral techniques, we can get the DSEs for Green functions from the La-grangian (details can be found in the textbooks of Quantum Field Theory). Since the derivation of functionalpath integral does not rely on the perturbative theory, DSEs is strict (in the weak coupling limit, DSEs givethe standard Feynman diagram expansion of the Green functions). The DSE of the renormalized quark9ropagator is G ( p ) − = Z ( iγ · p + m bm ) + Z Z Λ q g D µν ( p − q ) γ µ λ a G ( q )Γ aν ( q, p ) , (19)here g s D abµν ( p ) is the renormalized gluon propagator, λ a is the Gell-Mann matrices with a the color index,Γ aν ( q, p ) is the renormalized quark-gluon vertex, m bm is the bare mass of current quark and is related toΛ, R Λ q ≡ R Λ d q/ (2 π ) represents some regularization scheme that keeps translation invariance, where Λ isthe mass scale, we should take the limit Λ → ∞ at last. Z ( µ , Λ ) and Z ( µ , Λ ) are the renormalizationconstants of the quark-gluon vertex and quark wave function, which depend on both the renormalizationpoint µ and the renormalization mass scale Λ. According to the Lorentz structure analysis, the solution ofthe quark DSE Eq. (19) has the following form G ( p ) − ≡ iγ · pA ( p , µ ) + B ( p , µ ) . (20)Eq.(19) must be solved under certain renormalization condition. Since the asymptotic freedom nature ofQCD, people usually demand that at a big space-like momentum square there should be G ( p ) − (cid:12)(cid:12) p = µ = iγ · p + m ( µ ) , (21)where m ( µ ) is the renormalized current quark mass at µ . When chiral symmetry is explicitly broken, therenormalized mass and the bare mass satisfy this relation, m ( µ ) = m bm (Λ) /Z m ( µ , Λ ), here Z m ( µ , Λ )is the renormalization constant of mass; while there is no explicit chiral symmetry breaking, Z m bm = 0,which is just the case of chiral limit. Most of the following discussions on the vacuum susceptibilities are forthe chiral limit case.It can be seen clearly from Eq. (19) that, the quark propagator, the gluon propagator, and the quark-gluon vertex are related to each other. The gluon propagator and the quark-gluon vertex also satisfy theirown DSEs, which are coupled to the Green functions with more points. Therefore, in order to obtain aclosed equation of the quark propagator, we must make some appropriate approximations to the gluonpropagator and the quark-gluon vertex. On the gluon propagator side, the researches of its DSE have madegreat progress, and the gluon propagator in Landau gauge is for all phenomenological purposes sufficientlywell-known [83, 88, 95, 96, 101, 99, 102, 100, 107, 108, 109]. The lack of knowledge refers much more tothe quark-gluon vertex. It is generally believed that, in studies of the color confinement, the DCSB, andthe properties of bound states, the non-perturbative quark-gluon vertex plays an important role [110, 111,112, 113, 114, 115, 116]. One of the most complete investigations of the quark-gluon vertex can be foundin Ref. [100], and further guides as well as some important recent works and studies beyond the rainbow-ladder truncation can be found in Refs. [117, 118, 119] (and references therein). Therefore, when studying thedynamical chiral symmetry breaking problems of the quark propagator, people usually choose an appropriategluon propagator model as input. A gluon propagator model often contains some parameters, which can befixed by fitting the low-energy hadron physics experiments (such as the mass and decay constant of pion).For the quark-gluon vertex, the case is more complicated. The most simple approximation is just the rainbowapproximation, namely, use the bare quark-gluon vertex γ µ to replace the full one Γ µ ( q, p ). This is a big10pproximation, since the momentum dependence of the quark-gluon vertex is completely neglected, and ofcourse, the non-perturbative information is also lost completely. In studies of low-energy hadron physics,such as the calculation of hadron masses, decay constants, and other physical quantities, it is necessary tosolve the BSE that the quark-meson vertex satisfy. As mentioned above, the rainbow approximation of thequark DSE and the ladder approximation of the quark-meson vertex BSE are consistent with each other,and over many years of studies it has been demonstrated that although the rainbow-ladder approximationis the most simple one, it can give successful descriptions to some meson properties. The reason why it cannot describe the scalar meson well has also become known. There have been many attempts that try tostudy the scalar meson beyond the rainbow-ladder approximation. However, this is rather difficult work,since if we do some approximations to the quark-gluon vertex of the quark DSE, we also need to finda corresponding quark-meson BSE that is consistent with these approximations, which is really a difficultwork. As we mentioned earlier, for studies of scalar vacuum susceptibility people should go beyond therainbow-ladder approximation. The authors of Ref. [46] noted that in the calculations of the scalar vacuumsusceptibility, only the scalar vertex with zero momentum exchange Γ( p,
0) is used, rather than the full scalarvertex Γ( p, q ). Using this feature and the WTI, they give a method that can describe the scalar vacuumsusceptibility beyond the rainbow-ladder approximation. In the following we will introduce the equations ofthe quark propagator and the quark-meson vertex under rainbow-ladder approximation first, then present amethod to go beyond it.Under the rainbow approximation and taking the chiral limit ( m bm = 0), and substituting Eq. (20) intoEq. (19), it is easy to know the equations that A ( p , µ ) and B ( p , µ ) satisfy[ A ( p , µ ) − Z ] p = 43 Z Λ q g s D ( p − q ) A ( q , µ ) q A ( q , µ ) + B ( q , µ ) (cid:20) p · q + 2 p · ( p − q ) q · ( p − q )( p − q ) (cid:21) ,B ( p , µ ) = 4 Z Λ q g s D ( p − q ) B ( q , µ ) q A ( q , µ ) + B ( q , µ ) , (22)where we take the Landau gauge for the gluon propagator D abµν ( p ) = δ ab (cid:0) δ µν − p µ p ν /p (cid:1) D ( p ). Afterinputting the model gluon propagator, Eq. (22) can be solved by numerical iteration. From Eq. (22) wecan see clearly that it has two distinct solutions: the Nambu-Goldstone solution (or Nambu solution) that B ( p , µ ) = 0 and the Wigner-Weyl solution (or Wigner solution) that B ( p , µ ) ≡
0. In the Nambu-Goldstone phase: 1) chiral symmetry breaking occurs dynamically, the originally massless current quarksobtain masses by DCSB; 2) quarks are confined, since the quark propagator does not have the Lehmannrepresentation. While in the Wigner phase, there is no DCSB, and the quarks are not confined [83, 86, 88].In the Wigner phase, Eq. (22) becomes,[ A ′ ( p , µ ) − Z ′ ] p = 43 Z Λ q g s D ( p − q ) q A ′ ( q , µ ) (cid:20) p · q + 2 p · ( p − q ) q · ( p − q )( p − q ) (cid:21) , (23) For example, by the constraint of the longitudinal WTI and the requirement that the vertex has no kinematic singularity,people have developed the Ball-Chiu (BC) and Curtis-Pennington (CP) approximations for fermion-boson vertex in the gaugetheory [50, 51, 120]. In recent years, people also try to use the constraint of the transverse WTI [121, 122]. A ′ ( p , µ ) represents the vector part of the self-energy function. Therefore, the chiral limit case of thequark propagator in the Wigner phase is G ( W ) ( p ) = − iγ · pA ′ ( p , µ ) p . (24)In the ladder approximation, quark-meson vertex (three-point Green function) also satisfies its own BSE.For the tensor vertex, its BSE isΓ mµν ( p,
0) = Z T σ µν − Z Λ q g s D ηζ ( p − q ) γ η G ( q )Γ mµν ( q, G ( q ) γ ζ , (25)where the superscript m denote “meson”, and then Γ means quark-gluon vertex while Γ m represents quark-meson vertex. Similar to the quark propagator, we choose such a the renormalization conditionΓ mµν ( p, (cid:12)(cid:12) p = µ = σ µν . (26)According to Lorentz structure analysis, the general form of Γ mµν can be written as the followingΓ mµν ( p,
0) = σ µν Λ ( p , µ ) + σ µγ p γ p ν Λ ( p , µ ) + γ µ p ν Λ ( p , µ )+ γ · pp µ p ν Λ ( p , µ ) + ip µ p ν Λ ( p , µ ) . (27)Now substituting Eq. (27) into Eq. (25) and after some algebra, we get five coupled integral equations of thescalar function Λ i ( p , µ )( i = 1 , · · · , , , ( p , µ ) ≡
0. So that [45]Γ mµν ( p,
0) = σ µν Λ ( p , µ ) + σ µγ p γ p ν Λ ( p , µ ) , (28)The BSE that the vector vertex satisfy isΓ mµ ( p,
0) = Z γ µ − Z Λ q g s D ηζ ( p − q ) γ η G ( q )Γ mµ ( q, G ( q ) γ ζ . (29)Here one way to solve this is to write down the general Lorentz structure of Γ mµ ( p,
0) and then do the numericaliteration, another more direct way is to use the Ward identity (which is obtained via the Ward-Takahashiidentity by taking the infrared limit of the photon)Γ mµ ( p,
0) = − ∂G ( p ) − ∂p µ , (30)by calculating the partial derivative of the quark propagator to the momentum p , we can know the vectorvertex directly. Similar treatment can also be used to the axial-vector vertex, for the details please seeRef. [44].For the scalar vertex, the rainbow-ladder approximation is not a good one and we have to go beyond it.Let us see its BSE first, Γ m ( p,
0) = 1 + Z d q (2 π ) K ( q, p )[ G ( q )Γ m ( q, G ( q )] , (31)here K ( q, p ) is the quark–anti-quark scattering kernel, which is a four-point Green function that containsmany non-perturbative information. As mentioned above, when beyond the rainbow-ladder approximation itis very difficult to find the corresponding BSE of the vertex. But if take a closer look we can find that only thescalar vertex (it is known from the functional path integral theory that taking the derivative of the two-point12reen function with respect to the scalar external field will give the scalar vertex) Γ m ( p,
0) = ∂G ( p ) − ∂m | m =0 is needed to calculate the scalar vacuum susceptibility. The authors of Ref. [46] skillfully used this point.By the DSE of quark propagator, Eq. (19), it is easy to know the equation of scalar vertexΓ m ( p,
0) = 1 + Σ ( p ) + Σ ( p ) , here Σ ( p ) = − Z Λ q g D µν ( p − q ) γ µ G ( q )Γ m ( q, G ( q )Γ ν ( q, p ) , Σ ( p ) = 43 Z Λ q g D µν ( p − q ) γ µ G ( q )Λ ν ( q, p ) , (32)where the summation over color is carried out to be 4 / ν ( q, p ) = ∂ Γ ν ( q,p ) ∂m .Using the vector Ward-Takahashi identity, ik ν Γ ν ( q, p ) = G ( q ) − − G ( p ) − , k = q − p (33)we can know another identity that Λ ν ( q, p ) and Γ m ( p ) satisfy, ik ν Λ ν ( q, p ) = Γ m ( q, − Γ m ( p, . (34)This is an important step towards the consistent solutions of quark propagator and the scalar vertex.By the Lorentz structure analysis, the inverse of the quark propagator G ( p ) − and the scalar vertex Γhas such general forms, G ( p ) − = iγ · pA ( p ) + B ( p ) , (35)Γ m ( p,
0) = iγ · pC ( p ) + D ( p ) . (36)And according to the works of Ball and Chiu [50, 51], we take the following forms of Γ ν ( q, p ) and Λ ν ( q, p )Γ µ ( q, p ) = Σ A γ µ + ( q + p ) µ [ 12 γ · ( q + p )∆ A − i ∆ B ]Λ µ ( q, p ) = Σ C γ µ + ( q + p ) µ [ 12 γ · ( q + p )∆ C − i ∆ D ] , where Σ F = 12 [ F ( q ) + F ( p )] , ∆ F = F ( q ) − F ( p ) q − p , (37)and F = A, B, C, D . Here it should be pointed out that, there are also many other forms of vertex ansatzthat can also satisfy the vector WTI. The main reasons that the authors of Ref. [46] chose the BC vertex isthat: firstly, there is no kinematic singularity; secondly, the momenta p and q are symmetrical. The mainpurpose of their work is to discuss the differences of the results between the bare vertex and a corrected one. Actually, the QCD vertex does not satisfy this identity, but rather the more complicated Slavnov-Taylor identity (STI),which can be derived from the Becchi-Rouet-Stora invariance of QCD and corresponds to the WTI of QED. The difficultchallenge is to find an Ansatz for the unknown renormalised propagators, vertices, etc., of QCD which satisfy the DSEs andrespect the STIs of the theory. Progress can be made by satisfying a subset of the DSEs and STIs, and then supplementingthese with information gleaned from Lattice QCD, phenomenology, etc. However, without ghosts Eq. (33) is identical to thecorresponding WTI of QED [83]. ν ( q, p ) into the DSE of the quark propagator Eq. (19), we can find the numericalsolutions of the vector function A and the scalar function B of the quark propagator. Then by substitutingΓ ν ( q, p ), Λ ν ( q, p ), A , and B into Eq. (32) we can solve the scalar vertex. This method can be generalized tothe calculations of vacuum susceptibilities at finite temperature and finite quark chemical potential. In thefollowing part we will show the details of the gluon propagators and the numerical results. In this part we discuss the calculations and results of various vacuum susceptibilities in detail, namely,the tensor, the vector, the axial-vector, the scalar, and the pseudo-scalar vacuum susceptibilities.
Studies of tensor vacuum susceptibility are closely related to the tensor charge of nucleon [10]. Thereare some calculations related to the tensor vacuum susceptibility, however, the results have shown that thetheoretical treatments of this quantity is subtle and different approaches may lead to different outputs, evenhave different sign. For example, within the framework of QCD sum rules, H. X. He and X. D.Ji (1995, 1996)got about 0.002 GeV [11, 12], V. M. Belyaev and A. Oganesian (1997) got -0.008 GeV [13], L. S. Kisslinger(1999) got 0.0072 ∼ [15], A. P. Bakuleva and S. V. Mikhailov (2000) got − . ± . for the non-local condensates sum rule and − . ± . for standard sum rule [16]; while W.Broniowski et al (1998) got -(0.0083 ∼ using the chiral constituent model [14], and H. T. Yang et al (2003) got -(0.0014 ∼ within global color symmetry model [17]. Clearly, in order to get areliable theoretical prediction of the tensor charge, we should determine the tensor vacuum susceptibility asprecise as possible. The work of Ref. [45] has provided a general treatment of the issue, and in Ref. [49] thecalculations have been performed by employing the Ball-Chiu (BC) vertex [50, 51]. The results of Ref. [49]show that the tensor vacuum susceptibility obtained in the BC vertex approximation is reduced by about 10%compared to its rainbow-ladder approximation value, which means that the dressing effect of the quark-gluonvertex is not very large in the calculation of the tensor vacuum susceptibility within the DSEs framework.Through the previous general discussions of the vacuum susceptibilities (Sec. 2 of this paper) we knowthat, if we introduce a term that the quark tensor current coupled to a constant tensor external field¯ q ( x ) σ µν q ( x ) Z µν ( Z µν is the tensor external field), using the QCD sum rules approach we can obtain thegeneral expression of the tensor vacuum susceptibility (here is the chiral limit case, but it is easy to generalizeto the case with nonzero current quark mass) χ Z = (cid:8) T r [ σ ηζ G Γ · ZG ] − T r [ σ ηζ G P T Γ P T · ZG P T ] (cid:9) Z ηζ h ˜0 | : ¯ q (0) q (0) : | ˜0 i . (38)The expression in the earlier literatures is [14, 15, 16, 17] χ ′ Z = (cid:8) T r [ σ ηζ Gσ · ZG ] − T r [ σ ηζ G P T σ · ZG P T ] (cid:9) Z ηζ h ˜0 | : ¯ q (0) q (0) : | ˜0 i , (39)Comparing these two equations it is easy to see that, the difference between χ Z and χ ′ Z comes from thefact that in the previous calculations instead of using the exact dressed vertex Γ µν and the perturbative one14 P Tµν , they used the bare vertex σ µν , in the course of which the non-perturbative and perturbative dressingeffects on the vertex were ignored. Here we want to stress that, both the full and the perturbative tensorvertexes have complicated momentum dependence. Especially for the full tensor vertex, there may existmass singularities of various meson bound states, so compared with the bare vertex it contains quite a lotof sophisticated non-perturbative effects of QCD.The discussions in Ref. [45] are under the rainbow-ladder approximation, and the renormalization pro-cedure is necessary. The renormalized version of the tensor vacuum susceptibility is (for the details pleasesee Ref. [45]) χ ZR = (cid:8) T r [ σ ηζ G R Γ R · ZG R ] − T r [ σ ηζ G P TR Γ P TR · ZG P TR ] (cid:9) Z T Z ηζ h ˜0 | : ¯ q R (0) q R (0) : | ˜0 i , (40)where G R , Γ R , G P TR and Γ
P TR represent the renormalized versions of the full quark propagator, the full tensorvertex, the perturbative quark propagator and the perturbative tensor vertex, respectively. h ˜0 | : ¯ q R (0) q R (0) : | ˜0 i is the renormalized two-quark condensate.To show the numerical difference between χ Z and χ ′ Z , we need to specify a model gluon propagator. Theone Ref. [45] selected is the popular as well as famous Maris-Tandy model [87], which reads g s D ( k ) = 4 π ω Dk e − k /ω + 4 π γ m π ln (cid:20) τ + (cid:16) k / Λ QCD (cid:17) (cid:21) F ( k ) , (41)here F = (cid:2) − exp( − k / [4 m t ]) (cid:3) /k , τ = e − γ m = 12 / (33 − N f ) is the anomalous dimension of mass.For N f = 4, Λ N f =4 QCD = 0 .
234 GeV, and the renormalization point is chosen as µ = 19 GeV. As Ref. [87]pointed out, the parameters ω and m t are not independent, to chose m t = 0 . ω = 0 . ≤ k there is g s D ( k ) ≃ πα s ( k ), where α s ( k ) is the “running” couplingconstant of QCD. In Ref. [87] they chose D = 1 .
25 GeV , and there studies show that this set of parameterscan give very good results for the properties of pion and kaon.Once the model gluon propagator is chosen, we can then calculate the quark propagator and the vertexthrough numerical iteration, as shown in Fig. 1 and Fig. 2 respectively. The final numerical results of thetensor vacuum susceptibilities are χ Z a = − . , χ ′ Z a = − . , (42)here a ≡ − h ˜0 | : ¯ q R (0) q R (0) : | ˜0 i .We see that, χ Z and χ ′ Z are indeed different, but the difference is not very large. Let us now analyze thereasons for this. As stated above, the difference between them comes from the choices of vertexes. We knowthat in the perturbative region the bare vertex σ µν can be regarded as the zeroth approximation of the dressedvertex Γ µν , and therefore it is physically interesting to analyze the effects caused by their differences in thenon-perturbative region. It can be seen clearly from Fig. 2 that, for large enough momentum square s = p ,Λ ( s, µ ) tends to 1 while Λ ( s, µ ) tends to 0, which means that in the large momentum region Γ µν tendsto σ µν ; however, for small s , Λ ( s, µ ) differs from 1 and Λ ( s, µ ) differs from 0, although the differencesare not significant. Therefore, the dressed vertex Γ µν ( p,
0) obtained in the rainbow-ladder approximation ofthe DSEs approach is similar to that of the bare vertex in the small momentum region, and it is just this15 -3 -2 -1 -0.20.00.20.40.60.81.01.21.41.61.82.02.2 A(s, ) B(s, ) A ( s , ) B ( s , ) s(GeV ) Figure 1: The scalar functions A ( s, µ ) and B ( s, µ ) [GeV] of the dressed quark propagator, taken from Ref. [45]. Here s = p . -3 -2 -1 ( s , ) s(GeV ) -3 -2 -1 -0.050-0.045-0.040-0.035-0.030-0.025-0.020-0.015-0.010-0.0050.0000.005 ( s , ) s(GeV ) Figure 2: The tensor vertex functions Λ ( s, µ ) and Λ ( s, µ )[GeV − ], taken from Ref. [45]. Here s = p . small difference that accounts for the small numerical difference between the numerical values of χ Z and χ ′ Z .The small magnitude of the numerical difference between χ Z and χ ′ Z suggests that the non-perturbativedressing effects on the dressed tensor vertex are not important, and therefore in this case to use σ µν asa replacement of Γ µν ( p,
0) is a rather good approximation. This is quite different from the case of vectorvacuum susceptibility [40].
For the vector vacuum susceptibility χ V and the axial-vector vacuum susceptibility χ A , in order todistinguish them from the later vertex wave function χ ν , here we use different symbols κ V and κ A instead.According to the previous derivations, κ V can also be written as κ V h ¯ qq i = Z d z Z d P (2 π ) e iP · z V µ ( z ) V ν ( z ) V ( z ) (cid:2) Π Vµν ( P ) − Π V,P Tµν ( P ) (cid:3) , (43)16here Π Vµν ( P ) is the vector vacuum polarizationΠ Vµν ( P ) = − Z Z Λ q tr [ γ µ χ ν ( q ; P )] , (44)while for κ A , κ A h ¯ qq i = Z d z Z d P (2 π ) e iP · z V µ ( z ) V ν ( z ) V ( z ) (cid:2) Π Aµν ( P ) − Π A,P Tµν ( P ) (cid:3) , (45)with Π Aµν ( P ) the axial-vector vacuum polarizationΠ Aµν ( P ) = − Z Z Λ q tr [ γ γ µ χ ν ( q ; P )] , (46)in which the trace is to be taken in color and Dirac space. V µ ( z ) and V µ ( z ) represent the variable vector andaxial-vector external fields, χ ν , χ ν are the exact vector and axial-vector vertex wave functions, which canbe written as G Γ G using the no external field versions of the full quark propagator G and the correspondingvertex functions Γ. Z is the renormalization constant of the quark wave function. According to the WTI,the vertex renormalization constant is also equal to Z . h ¯ qq i is he chiral quark condensate.As can be seen from Eqs. (43) and (45), in order to obtain the non-perturbative vector and axial-vectorvacuum susceptibilities, we should subtract Π V,P Tµν ( P ) and Π A,P Tµν ( P ) respectively, which arises from theperturbative effects. In other words, when calculating the vacuum susceptibilities, the mean values of theperturbative vacuum should be subtracted. This is very important and is deserving of additional attention.It is well known that the separation of the perturbative and the non-perturbative contributions from themean values of the vacuum is somewhat arbitrary. Usually, this arbitrariness is avoided by introducing somenormalization point [123]. In such a formula, the condensates will depend on the choice of the normalizationpoint. There are also other methods besides this one, for example, in studies of the mixed quark-gluoncondensate, the authors of Ref. [124] identified the perturbative vacuum with the Wigner vacuum, since bothof them are trivial in the sense that there are no chiral symmetry breaking and confinement, in contrast toNambu-Goldstone vacuum (the non-trivial vacuum) which corresponds to DCSB (more details can be foundin Ref. [124]). In Refs. [40, 44, 45], the authors adopted the viewpoint of Ref. [124] to calculate the vector,the axial-vector, and the tensor vacuum susceptibilities in the framework of rainbow-ladder approximation ofthe DSEs approach. For example, in the calculation of the vector vacuum susceptibility, Π Vµν ( P ) in Eq. (43)is calculated in the Nambu-Goldstone vacuum configuration, while Π V,P Tµν ( P ) is calculated in the Wignervacuum configuration. It is obvious that this calculation depends on the rainbow-ladder approximation ofthe DSEs approach. In the literature, there are few theoretical studies related to the vector and axial-vectorvacuum susceptibilities, among them, L. S. Kisslinger determined them using a three-point formalism withinthe method of QCD sum rules [15], M. Harada et al discussed the effective degrees of freedom at chiralrestoration and the vector manifestation in hidden local symmetry theory [125], and K. Jo et al calculatedvector susceptibility and QCD phase transition in anti-de Sitter (AdS)/QCD models [126]. In the following,we will show how the authors of Ref. [48] obtain model independent results for the vector and axial-vectorvacuum susceptibilities using the vector and axial-vector WTIs.Let us discuss the vector vacuum susceptibility κ V first. The conservation of the vector current ensures17hat the vector vacuum polarization is purely transverse, namely,Π Vµν ( P ) = (cid:18) δ µν − P µ P ν P (cid:19) Π VT ( P ) . (47)If we take the constant external field limit V µ ( z ) ≡ V µ , and use the following integration equation Z d l δ ( l ) p · lk · ll f ( k, p, l ) = 14 Z d l δ ( l ) p · kf ( k, p, l ) , (48)we can get κ V h ¯ qq i = 34 h Π VT ( P = 0) − Π V,P TT ( P = 0) i . (49)It can be seen that the vector vacuum susceptibility is closely related to the vector vacuum polarization atzero total momentum. Now contracting both sides of Eq. (44) with δ µν , we then haveΠ VT ( P = 0) = − Z Z Λ q tr [ γ µ χ µ ( q ; P = 0)] . (50)The case for Π V,P TT ( P = 0) is similar.In the chiral limit case, the wave function of the vector vertex satisfy the WTI iP µ χ µ ( q ; P ) = G ( q − ) − G ( q + ) , (51)where q ± = q ± P/
2. Expanding the right side of Eq. (51) to P µ and taking the limit P µ → χ µ ( q ; P = 0) = i ∂G ( q ) ∂q µ . (52)Substituting Eq. (52) into the right side of Eq. (50), and adopting the following parametrization of the quarkpropagator G ( q ) G ( q ) = 1 iγ · kA ( q ) + B ( q ) = − iγ · kσ V ( q ) + σ S ( q ) , (53)we obtain Π VT ( P = 0) = − Z Z Λ q tr (cid:20) iγ µ ∂G ( q ) ∂q µ (cid:21) = − Z Z Λ q [2 σ V ( q ) + q dσ V ( q ) dq ] . (54)Since for large q , σ V ∼ q − , this integral is quadratically divergent. However, this divergence is notgenuine. Note that the integrand is a total divergence, so the above integral vanishes if a translationinvariant regularization is adopted, and henceΠ VT ( P = 0) = 0 . (55)For similar reasons we can get Π V,P TT ( P = 0) = 0 too. Thus we may draw the conclusion that the vectorvacuum susceptibility is zero so long as the Ward identity is satisfied. In other words, the vanishing of the Actually, in numerical calculations of the vacuum polarization using DSEs people usually employ a cutoff to regularize theultraviolet divergence in Eq. (44), for the details please see Ref. [48] and references therein. − ) is very close to this model independent one.The case for the axial-vector vacuum polarization is a little bit more complicated, since the axial-vectorvertex contains a massless bound state pole. Nevertheless, the analysis is still straightforward, and thegeneral form of the axial-vector vacuum polarization can be written asΠ Aµν ( P ) = (cid:18) δ µν − P µ P ν P (cid:19) Π AT ( P ) − P µ P ν P Π AL ( P ) , (56)which contains both a transverse part and a longitudinal part. Following the same procedure of deducingthe vector vacuum susceptibility, we can get κ A h ¯ qq i = 34 h Π AT ( P = 0) − Π A,P TT ( P = 0) i − h Π AL ( P = 0) − Π A,P TL ( P = 0) i . (57)From Eqs. (56) and (46), we know that (cid:18) δ µν − P µ P ν P (cid:19) Π AT ( P ) − P µ P ν P Π AL ( P ) = − Z Z Λ q tr [ γ γ µ χ ν ( q ; P )] . (58)Now we will determine the axial-vector vacuum polarization at P = 0. In the chiral limit, the WTI forthe axial-vector vertex can be expressed as − iP µ χ µ ( q ; P ) = G ( q + ) γ + γ G ( q − ) . (59)where we also express the identity in terms of the dressed axial-vector vertex wave function χ µ instead ofthe dressed vertex Γ µ . Since χ µ possesses a longitudinal massless bound state pole, its generally expressionis then [54] χ µ ( q ; P ) = γ χ Rµ ( q ; P ) + ˜ χ µ ( q ; P ) + f π P µ P χ π ( q ; P ) . (60)where χ Rµ ( q ; P ) = γ µ F R + γ · qq µ G R − σ µν q ν H R , functions F R , G R , H R are regular at P → P µ ˜ χ µ ∼ O ( P ), f π is the pion decay constant in the chiral limit, and χ π ( q ; P ) is the canonically normalized Bethe-Salpeterwave functions of the massless bound state which take a general form χ π ( q ; P ) = 2 γ [ iE π ( q ; P ) + γ · P F π ( q ; P ) + γ · qq · P G π ( q ; P ) + σ µν q µ P ν H π ( q ; P )] . (61)In the chiral limit, f π is defined as f π P µ = Z Z Λ q tr [ γ γ µ χ π ( q ; P )] . (62)It is apparent that as P →
0, we can expand the right side of Eq. (62) to O ( P µ ), and then f π = − Z Z Λ q (cid:18) F π ( q ; 0) + 14 q G π ( q ; 0) (cid:19) . (63)Contracting both sides of Eq. (58) with the projector P µν = δ µν − P µ P ν P , we can get3Π AT ( P ) + 3Π AL ( P ) = − Z Z Λ q tr (cid:20) P µν γ γ µ ( γ χ Rν ( q ; P ) + ˜ χ ν ( q ; P ) + f π P µ P χ π ( q ; P )) (cid:21) . (64)19ot let us focus on the limit of Eq. (64) when P →
0. After some algebra we find that, the first term in theright side of Eq. (64) vanishes, and the second term also vanishes thanks to P µ ˜ χ µ ∼ O ( P ), while the thirdterm is found to be − f π N c Z R Λ q ( F π ( q ; 0) + q G π ( q ; 0)), which equals 3 f π after making use of Eq. (63).Therefore we obtain Π AT ( P = 0) + Π AL ( P = 0) = f π . (65)Similarly, contracting both sides of Eq. (58) with the projector P µν = δ µν − ζ P µ P ν P ( ζ = 4) gives3Π AT ( P ) + ( ζ − AL ( P ) = − Z Z Λ q tr (cid:20) P µν γ γ µ ( γ χ Rν ( q ; P ) + ˜ χ ν ( q ; P ) + f π P µ P χ π ( q ; P )) (cid:21) . (66)In the limit P →
0, the first term of Eq. (66) is (4 − ζ )4 N c Z R Λ q ( F R ( q ; 0) + q G R ( q ; 0)), the second termdoes not contribute, while the third term is ( ζ − f π . Now using Eq. (65) we then getΠ AT ( P = 0) = 4 N c Z Z Λ q [ F R ( q ; 0) + 14 q G R ( q ; 0)] . (67)Note that the result is independent of ζ .Now substituting the general form of χ µ in Eq. (60) into Eq. (59), and expand both sides to O (1) and O ( P µ ) respectively, we can get the corresponding Goldberger-Treiman relation. Functions F R ( q ; 0) , G R ( q ; 0)can be expressed using the vector part of the quark propagator and the BS wave function of pion (detailscan be found in Ref. [54]) F R ( q ; 0) = − σ V ( q ) − f π F π ( q ; 0) ,G R ( q ; 0) = − dσ V ( q ) dq − f π G π ( q ; 0) . (68)Substituting Eq. (68) into Eq. (67), and using Eq. (63), we getΠ AT ( P = 0) = − N c Z Z Λ q [2 σ V ( q ) + q dσ V ( q ) dq ] + f π = f π . (69)To obtain this result we have used the previous discussion that when using a translation invariant regular-ization, the integral to q in Eq. (69) is 0.Combining Eqs. (65) and (69), we haveΠ AT ( P = 0) = f π , Π AL ( P = 0) = 0 . (70)The subtraction term Π A,P TT ( P = 0) and Π A,P TL ( P = 0) can be obtained using similar methods. Note thatthe perturbative axial-vector vertex function has no pion pole term. We obtainΠ A,P TT ( P = 0) = 0 , Π A,P TL ( P = 0) = 0 . (71)The axial-vector vacuum susceptibility is then κ A h ¯ qq i = 34 f π . (72)From Eqs. (55) and (70), we found that in the chiral limit caseΠ VT ( P = 0) − Π AT ( P = 0) = − f π . (73)20hich is just the result from Weinberg sum rules [127].In summary, in this part we have derived model independent results of the vector and the axial-vectorvacuum susceptibilities, which will play an important role in the related calculations of the QCD sum rulesexternal field method. Using the vector and the axial-vector WTIs, Ref. [48] has demonstrated that in thechiral limit the vector vacuum susceptibility is 0, while the axial-vector vacuum susceptibility equals 3 / Studies on the QCD phase transitions play important roles in the studies of the evolution of the earlyuniverse and the high energy heavy ion collisions. The results of the Lattice QCD show that the QCD “phasetransition” in the early universe is not a real phase transition, but a smooth crossover [58]. In Ref. [58] theyadopted the scalar vacuum susceptibility χ as an indicator, and studied the temperature dependence of χ .The scalar vacuum susceptibility is very important in studies of QCD phase transitions, however, it is knownthat there is a square divergence in the χ ( T ) calculation. In order to eliminate it, a usual treatment is tosubtract χ ( T = 0) from χ ( T ), since the divergence only exists in χ ( T = 0), and the temperature dependentpart is not divergent [57, 58]. Obviously, such method is not applicable in the zero temperature case, butthe scalar vacuum susceptibility is meaningful at zero temperature. We will give a new treatment to thesquare divergence in χ ( T ) later.In QCD, the two-quark condensate associated with f flavor of quarks is defined as [128] h ¯ qq i f ( m f ; µ, Λ) = Z ( µ, Λ) N c tr D Z Λ q G f ( q ; m f ; µ ) , (74)where m f ( µ ) is the renormalized current quark mass, with µ the renormalization scale; Z is the renor-malization constant of the mass term in the Lagrangian, which depends implicitly on the gauge parameter.In the chiral limit case the two-quark condensate can be used as an order parameter to describe the chiralphase transition, the scalar vacuum susceptibility (or chiral susceptibility) just characterize its response tothe change of quark mass [31, 129] χ ( µ ) := ∂∂m ( µ ) h ¯ qq i ( m ; µ, Λ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ m =0 = Z ( µ, Λ) N c tr D Z Λ q ∂∂m G ( q ; m ; µ ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ m =0 , (75)here ˆ m is the auxiliary current quark mass that can be regarded as renormalization invariant. The order ofintegration and differentiation can be interchanged because the theory is properly regularized. Now use theWard identity ∂∂m G ( q ; m ; µ ) = − G ( q ; m ; µ )Γ( q, µ ) G ( q ; m ; µ ) (76)then χ ( µ ) = − Z ( µ, Λ) N c tr D Z Λ q G ( q ; 0; µ )Γ( q, µ ) G ( q ; 0; µ ) , (77)where Γ is the renormalized fully-dressed scalar vertex, which satisfies the inhomogeneous BSEΓ( k, P ; µ ) = Z I D + Z Λ q [ S ( q + )Γ ( q, P ) S ( q − )] sr K rstu ( q, k ; P ) . (78)21ith k the relative and P the total momenta of the quark–anti-quark pair; q ± = q ± P/ r, s, t, u representcolor and Dirac indices; and K is the fully-amputated quark–anti-quark scattering matrix.The quantity m Π ( P ) is renormalization point invariant in QCD, where Π ( P ) is the scalar vacuumpolarization Π ( P ; µ ) = Z N c tr D Z Λ q G ( q + )Γ( q, P ) G ( q − ) . (79)To compare with Eq. (77), we can get the general result χ ( µ ) = − Π ( P = 0; ˆ m = 0 , µ ) . (80)Hitherto we have not specified a regularization procedure for the scalar vacuum susceptibility. In thisconnection it is noteworthy that if a hard cutoff is used, then in the chiral limit of a non-interacting theory − Π ( P = 0; ˆ m = 0) = N c π Λ . (81)Ref. [128] pointed out that, this result can be traced to the dependence on current quark mass in Eq. (74).On the other hand, the Pauli-Villars regularization would yield zero, which is then the procedure that theauthors of Ref. [46] recommend and employ in models that preserve the one-loop renormalization-groupbehavior of QCD, and in this case the scalar vacuum susceptibility is defined self-consistent.For the calculation of χ ( µ ) we only need the scalar vertex at P = 0, at which total momentum it has thegeneral form Γ ( k, m ) = iγ · kC ( k ; m ) + D ( k ; m ) . (82)Owing to the Ward identity, Eq. (76), it is not necessary to solve the inhomogeneous BSE since C ( k ; m ) = ∂∂m A ( k ; m ); D ( k ; m ) = ∂∂m B ( k ; m ) . (83)Hence a solution of the gap equation suffices completely to fix Γ ( k, g D µν ( q ) = 4 π t µν ( q ) D q ω exp( − q ω ) , (84)with t µν ( q ) the transverse momentum projection operator. Eq. (84) delivers an ultraviolet finite model gapequation. Hence, the regularization mass scale Λ can be moved to infinity and the renormalization constantsequal to one.The active parameters in Eq. (84) are D and ω , but they are not independent. In reconsidering arenormalization group improved rainbow-ladder fit to a selection of ground state observables [87], Ref. [93]noted that a change in D is compensated by an alteration of ω . This feature has further been elucidated andexploited in Refs. [130, 131]. For the range [0 . , .
5] GeV, the fitted low-energy observables are approximatelyconstant along the trajectory ωD = (0 . =: m g (85)Herein, we employ ω = 0 . D = m g /ω = 1 . corresponds to what might be calledthe real-world reference value for the bare vertex. Here we should note that, if we choose Eq. (84) as the22odel gluon propagator, but for the quark-gluon vertex using the BC ansatz Eq. (37), there is no reasonto suppose that ω and D still satisfy the relation in Eq. (85), since the BC vertex contains considerableunknown non-perturbative information of interaction. Therefore, we need to refit the parameter D . Aftersome tries, it has been found that if one assumes D = 0 . , we can get the value of the two-quarkcondensate shown in the fourth column of Table 1. Hence we postulate that D = 0 . is the real worldreference value under the BC vertex. Table 1: Results obtained for selected quantities with ω = 0 . D parameter value indicated: A (0), M (0) are p = 0 in-vacuum values of the scalar functions A ( p ) and M ( p ) = B ( p ) /A ( p ); the quark condensate is definedwith m = 0 in Eq. (74); and χ is obtained from Eq. (90). A(0) is dimensionless, but all other entries are quoted in GeV. Thecalculations reported herein were performed in the chiral limit, and all are taken from Ref. [46] Vertex √ D A (0) M (0) − ( h ¯ qq i ) / √ χ Rainbow-Ladder 1 1.3 0.40 0.25 0.39Ball-Chiu 1 / √ Vµν ( P ) = N c tr D Z Λ q iγ µ G ( q + ) i Γ ν ( q, P ) G ( q − ) , (86)here Γ ν is the vector quark–anti-quark vertex that satisfies an inhomogeneous BSE similar to Eq. (78). Thispolarization couples to the photon and hence must be transverse while contain no mass term. Considertherefore [48] 14 Π Vµµ ( P = 0) = N c D Z Λ q iγ µ G ( q ) i Γ µ ( q, G ( q )= − N c D Z Λ q iγ µ ∂∂q µ G ( q )= − N c Z Λ q q ddq (cid:2) ( q ) σ V ( q ) (cid:3) , (87)where the vector Ward identity was used in the second line and we have written the dressed quark propagatorin the following form G ( p ) = − iγ · p σ V ( p ) + σ S ( p ) . (88)If we consider a free field and use a hard cutoff, then Eq. (87) becomes −
14 Π V µµ ( P = 0) = N c π Λ ; (89)23amely, the same sort of divergence is encountered as in the scalar vacuum polarization, Eq. (81). Naturally,any regularization scheme that preserves the vector Ward-Takahashi would yield zero as the result [132]. So,considering Eq. (87), we can define the regularized scalar vacuum susceptibility as χ = − Π (0; m = 0 , Λ) + 12 Π
Vµµ (0; m = 0 , Λ) (90)= − N c π Z ∞ ds s (cid:26) D ( s )[ σ s ( s ) − sσ s ( s ) ]+2 s C ( s ) σ V ( s ) σ S ( s ) + 2 σ V ( s ) + sσ ′ V ( s ) (cid:27) . Here, as long as each term in Eq. (90) is regularized independently in a valid fashion, then this is a “nugatory”transformation: since the photon is massless and hence in any valid scheme there must be Π
Vµµ (0; 0 , Λ) = 0.In other words, with this definition we can move on to calculate the susceptibility without specifying aparticular regularization scheme. The last two contributions in the integrand act as the Pauli-Villars terms,and Z = 1 in the ultraviolet-finite model.Now we will discuss the numerical results. Owing to the Gaussian form of the gluon propagator, Eq. (84),all the relevant integrations converge rapidly. Fig. 3 shows the functions obtained by solving the gap equation,and Fig. 4 gives the results which describe the P = 0 scalar vertex. All the results were obtained with theappropriate real world values of the interaction strength: D = 1 GeV for the bare vertex, and D = 0 . for the BC vertex, as listed in Table 1. -4 -3 -2 -1 A ( p ) p ( GeV ) -4 -3 -2 -1 B ( p ) p ( GeV ) Figure 3: The dressed quark propagator, taken from Ref. [46]. Left panel: A ( p ) = 1 /Z ( p ), where Z ( p ) is the wave functionrenormalization function; Right panel: B ( p ) [GeV], the scalar piece of the dressed-quark self-energy. In both panels, thedashed curve is calculated in the rainbow-ladder truncation, with I = D/ω = 4; while the solid curve is calculated with theBC vertex ansatz, and I = 2. It can be seen clearly from Fig. 3 that, the BC vertex ansatz has a quantitative impact on the magnitudeand pointwise evolution of the gap equations solution. This can be anticipated from Ref. [133]. Moreover,the pattern of behavior can be understood from Ref. [112]: the feedback arising through the ∆ B term inthe BC vertex always acts to alter the domain upon which A ( p ) and B ( p ) differ significantly in magnitudefrom their respective free-particle values, especially in the intermediate momentum region. The behaviors of24 -4 -3 -2 -1 C ( p ) p ( GeV ) -4 -3 -2 -1 D ( p ) p ( GeV ) Figure 4: P = 0 scalar vertex, taken from Ref. [46]. Left panel: C ( p ); Right panel: D ( p ). In both panels, the dashed curve iscalculated in the rainbow-ladder truncation, with I = D/ω = 4; while the solid curve is calculated with the BC vertex ansatz,and I = 2. C ( p ) and D ( p ) shown in Fig. 4 can be understood easily from those of A ( p ) and B ( p ), so that there isno need to discuss it further.The integrand in Eq. (90) is depicted in Fig. 5 for each vertex ansatz at the associated real worldinteraction strength. The resulting scalar vacuum susceptibilities are presented in the fifth column of Table 1.It can be found from this figure that the regularization has served to eliminate the far-ultraviolet tail of theintegrand, thereby ensuring convergence of the integral. We have varied the detailed form of the regularizingsubtraction, namely, using free-field propagators and vertices instead of gap and BSE solutions, and thebehavior of the integrand changes a little. Moreover, at real world values of the interaction strength forboth ansatzs the integrands have negative support in the infrared and positive support for p > (0 . GeV ) .These results tell us that when studying the scalar vacuum susceptibility, to go beyond the rainbow-laddertruncation is important and necessary, which then deserves further investigation.In Fig. 6 we depict the evolution of the scalar vacuum susceptibility with increasing interaction strength, I = D/ω . For I = 0 there is no interaction and then the “vacuum” is unperturbed by a small change inthe current quark mass, hence the susceptibility keeps zero. With increasing I the susceptibility will growsince the interaction in the ¯ qq channel is attractive and therefore magnifies the associated pairing. This isequivalent to stating that the scalar vertex is enhanced above its free field value. Then the growth continuesand accelerates until at some critical value I c the susceptibility becomes infinite, namely, a divergence appear.The critical values are I c = 1 .
93 for the rainbow-ladder approximation, and I c = 1 .
41 for the BC vertexansatz.This divergence can be understood in this way: the models we have defined contain a dimensionlessparameter I , which characterizes the interaction strength, and a current quark mass, which is an explicitsource of chiral symmetry breaking. In the general theory of phase transitions, the latter is analogous to anexternal magnetic field while 1 / I is kindred to a temperature. Consider the free energy for such theories f ( t, m ), where t = [1 − I c / I ]. If such a theory possesses a second-order phase transition, then the free energy25 -4 -3 -2 -1 -1.0-0.50.00.51.0 p ( GeV ) Figure 5: Integrand in Eq. (90), taken from Ref. [46]. Dashed curve: calculated in rainbow-ladder truncation, with I = D/ω =4; solid curve: calculated with BC vertex ansatz, and I = 2. is a homogeneous function of its arguments in the neighborhood of t = 0 = m . From this it follows that thetheory’s magnetization exhibits the following behavior (e.g., see the Appendix of Ref. [27] for the details): M ( t,
0) = t β , t → + , (91)and the associated magnetic susceptibility evolves in this way M (0 , m ) ∝ m − (1 − /δ ) . (92)For a mean-field theory, β = 1 /
2, and δ = 3.Now the nature of the critical interaction strength is easy to understand. In the class of theories weare considering, the quark condensate is analogous to the magnetization, and it is attended by the scalarvacuum susceptibility. For I < I c , the interaction is insufficient to generate a nonzero scalar term in thedressed-quark self-energy in the absence of a current quark mass; namely, DCSB is impossible and the modelrealizes chiral symmetry in the Wigner mode. But the situation changes at I c , and for I > I c a B = 0solution is always possible. Moreover, the behaviors of the susceptibility show that each model undergoesa second-order phase transition and realizes chiral symmetry in the Nambu-Goldstone mode for interactionstrengths above their respective values of I c . These observations emphasize the usefulness of the scalarvacuum susceptibility. With the bare vertex people can construct the pressure explicitly, and thereby show The pressure is defined as the negative of the effective action, and then the ground state of a system is that configuration forwhich the pressure is a global maximum. For the rainbow-ladder approximation, the corresponding pressure is just constructedfrom the famous CJT effective action [134]. .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.00.51.01.52.02.53.0 Figure 6: Dependence of the scalar vacuum susceptibility on the interaction strength in Eq. (84), taken from Ref. [46]; viz., I := D/ω : dashed curve is the RL vertex, while solid curve is BC vertex that for I > I c the DCSB solution is dynamically favored because it corresponds to the configuration of themaximum pressure. Nevertheless, the diagrammatic content of the BC vertex ansatz is not knowable andhence an expression for the pressure cannot be derived. In this case we may rely on the behavior of somesusceptibilities to conclude that DCSB is favored, as illustrated within many model studies. Once past thecritical point, the susceptibility decreases as I increases. This is because the magnitude of the condensateorder parameter grows in tandem with I and, therefore, the influence of any perturbation associated with acurrent quark mass must steadily diminish. We will do some related discussions in Sec. 4 of this paper. Forthe theoretical studies of the scalar and the following pseudo-scalar vacuum susceptibilities using the QCDsum rules, please refer to Ref. [15]. The color-singlet current-current correlators (or equivalently, the associated vacuum polarizations) aredirectly related to physical observables, and hence play important roles in QCD. For example, the vector vac-uum polarization is coupled to real and virtual photons, so that it is basic to the analysis and understandingof the process e + e − → hadrons [135, 136]. The standard Lattice QCD methodology related to the correlatorscan be found in Ref. [137]; on the other hand, correlators are also amenable to analysis via the operatorproduct expansion and are therefore fundamental in the application of QCD sum rules. In the latter connec-tion, the pseudo-scalar vacuum susceptibility (or the pion susceptibility) plays a role in the QCD sum rulesestimate of numerous meson-hadron couplings, for example, the strong and parity-violating pion-nucleoncouplings, g πNN and f πNN , respectively [7, 138, 139]. Furthermore, the pseudo-scalar vacuum susceptibilityis also a probe of QCD vacuum structure as the scalar susceptibility which is discussed in Sec. 3.2.3 of thispaper [46], while its analysis is more subtle, with conflicts and misconceptions being common in different27odel calculations [7, 8, 33, 139, 140].In this part we derive a model independent result for the pseudo-scalar vacuum susceptibility using theisovector–pseudo-scalar vacuum polarization [47], which can be written asΠ jk ( P ; ζ ) = Z N c tr Z Λ q i γ τ j G ( q + ) i Γ k ( q ; P ) G ( q − ) . (93)where ζ represents the renormalization scale, the trace is over flavor and spinor indices, τ j is the Pauli matri-ces of the isospin, P is the total momentum of the quark–anti-quark pair; G ( q ± ) represents the dressed-quarkpropagator, and Γ is the fully dressed pseudo-scalar vertex, both of which depend on the renormalizationpoint. Physical quantities obtained from Eq. (93) are then gauge invariant. The propagator can be obtainedfrom the gap equation, Eq. (19).The pseudo-scalar vertex is solved from an inhomogeneous BSE, namely[Γ j ( k ; P )] tu = Z [ 12 γ τ j ] tu + Z Λ q [ χ j ( q ; P )] sr K rstu ( q, k ; P ) , (94)here k is the relative momentum, r, s, t, and u represent color, flavor, and spinor indices, and χ µ ( k ; P ) = G ( k + )Γ µ ( k ; P ) G ( k − ) , (95)with k ± = k ± P/
2, and K ( q, k ; P ) is the fully amputated two-particle irreducible quark–anti-quark scatteringkernel. Now, let us consider the case with a space-time independent pseudo-scalar source, ~s = 0, associatedwith the term Z d x ¯ q ( x ) i γ ~τ~s q ( x ) (96)in the action, we can define a pseudo-scalar vacuum condensate, the gauge-invariant as well as properlyrenormalized form of which in QCD is h ¯ qq i µ ( ~s , m ; ζ, Λ) = Z N c tr Z Λ q i γ τ µ G ( q ; ~s , m ; ζ ) . (97)We can see that it is analogous to the vacuum quark condensate, Eq. (74). When ˆ m = 0, the DCSB can beexpressed via the Higgs mechanism as − h ¯ qq i ζ = lim m → h ¯ qq i ( m ; ζ, Λ) = 0 . (98)Eq. (98) then defines what we mean by an isoscalar-scalar configuration: isovector–pseudo-scalar correlationsare by convention measured with respect to this configuration. These observations highlight the importanceof the pseudo-scalar vacuum susceptibility χ µν ( ζ ) = ∂∂s µ h ¯ qq i ν ( ~s , m ; ζ, Λ) (cid:12)(cid:12)(cid:12)(cid:12) ~s =0 . (99)Then following the previous discussions, we can know that χ µν ( ζ ) = − µν ( P = 0; ~s = 0 , ˆ m, ζ ) . (100)Hitherto we have not specified a regularization procedure for the susceptibility, actually it can rigorously bedefined via a Pauli-Villars procedure, as discussed in Ref. [46].28ow let us discuss the value of χ µν ( ζ ) in the neighborhood of the chiral limit, therein we may write [54] i Γ µ ( k ; 0) = 12 iγ τ µ E R ( k ; 0) + r π m π Γ π µ ( k ; 0) . (101)where E R ( k ; P ) is a part of the inhomogeneous pseudo-scalar vertex that is regular as P + m π → π µ ( k ; P ) is the pion bound state’s canonically normalized Bethe-Salpeter amplitude, and r π ( ζ ) determinedby iδ µν r π ( ζ ) = h | ¯ q γ τ ν q | π µ i = Z N c tr Z Λ q γ τ ν χ µπ ( q ; P ) (102)is the residue of this bound state in the inhomogeneous pseudo-scalar vertex. Using this, the pion’s leptonicdecay constant can be expressed as δ µν f π P σ = h | ¯ q γ γ σ τ ν q | π µ i = Z N c tr Z Λ q γ γ σ τ ν χ µπ ( q ; P ) . (103) E R ( k ; P ) can be determine via the axial-vector WTI P ν Γ µ ν ( k ; P ) + 2 m ( ζ ) i Γ µ ( k ; P ) = G ( k + ) − iγ τ µ + 12 iγ τ µ G ( k − ) − , (104)with Γ µ ν the inhomogeneous axial-vector vertex. At P = 0 with ˆ m = 0 there is no pole contribution on theleft side and hence m ( ζ ) E R ( k ; P = 0) = B ( k ; m ; ζ ) . (105)in other words, this regular piece of the pseudo-scalar vertex is completely determined by the scalar partof the ˆ m = 0 quark self-energy. Using the systematic, non-perturbative, and symmetry-preserving DSEtruncation scheme introduced in Refs. [84, 141], we can verify this equation order-by-order via the gap andBethe-Salpeter equations.Now substituting Eq. (101) into Eq. (100), we then get χ µν m ∼ = δ µν χ ( ζ ) (106) χ ( ζ ) = χ π ( ζ ) + χ R ( ζ ) + O ( ˆ m ) , (107)so that, in the neighbourhood of ˆ m = 0, the pseudo-scalar susceptibility splits into a sum of two terms: thefirst one expresses the contribution of the pion pole, O ( ˆ m − ), and can be expressed in the closed form χ π ( ζ ) = 2 r π ( ζ ) m π ˆ m =0 = − h ¯ qq i ζ m ( ζ ) , (108)of which the last equality is proved in Ref. [54]; the second term O ( ˆ m ) can be determined via m ( ζ ) χ R ( ζ ) δ µν ˆ m ∼ = Z N c tr Z Λ q iγ τ ν G ( q ) i γ τ µ B ( q , m ) G ( q ) = δ µν h ¯ qq i ( m ; ζ, Λ) , (109)where h ¯ qq i ( m ; ζ, Λ) is the vacuum quark condensate defined in Eq. (74), and this entails χ R ( ζ ; m ) = χ ( ζ ) + O ( ˆ m ) , (110)where χ ( ζ ) is the scalar vacuum susceptibility defined in Eq. (75). Hence we arrive at a model independentconsequence of chiral symmetry and the pattern by which it is broken in QCD, namely, χ ( ζ ) ˆ m ∼ = − h ¯ qq i ζ m ( ζ ) + χ ( ζ ) + O ( ˆ m ) . (111)29 able 2: Pseudo-scalar vacuum susceptibility and related quantities computed using the two kernels of the BSE described inconnection with Eqs. (37) and (84), also similar to those of Table 1. Dimensioned quantities are listed in GeV, and all are takenfrom Ref. [46] Vertex √ D ω − ( h ¯ qq i ζ ) / f π m p χ π p χ R Rainbow-Ladder 1 0.5 0.25 0.091 0.0050 1.77 0.39Ball-Chiu 1 / √ L (2) N SU R (2) symmetry and, moreover, that ascribing scalar-isoscalar quantum numbersto the QCD vacuum is a convention contingent upon the form of the current-quark mass term. It followsthat, the massless action cannot distinguish between the continuum of sources specified byconstant × Z d x ¯ q ( x ) e iγ ~τ~θ q ( x ) , | θ | ∈ [0 , π ) . (112)Therefore, the regular part of the vacuum susceptibility must be identical when measured as the responseto any one of these sources, so that there is χ R = χ for all choices of ~θ . This is the content of the so-called“Mexican hat” potential, which is used in building effective models for QCD. The magnitude of χ dependson whether the chiral symmetry is dynamically broken and the strength of the interaction as measured withrespect to the critical value required for DCSB, as discussed in Sec. 3.2.3 of this paper [46]. When thesymmetry is dynamically broken, then the Goldstone modes appear, by convention, in the pseudo-scalar–isovector channel, and thus the pole contributions appear in χ but not in the chiral susceptibility. It isvalid to draw an analogy with the Weinberg sum rules [48, 127]. Eq. (111) also provides a novel, modelindependent perspective on a mismatch between the evaluation of the pseudo-scalar vacuum susceptibilityusing either a two-point or a three-point sum rules. The two-point study in Ref. [140] produces the pion polecontribution, χ π , which is also the piece emphasized in Ref. [33], whereas a three-point method in Ref. [139]isolates the regular piece, χ R , since a vacuum saturation ansatz is implemented in the derivation. Thus theiranalyses are not essentially in conflict, but emphasize different and independent pieces of the susceptibility,which can be distinguished. However, only the regular piece should be retained in a sum rules estimateof the pion-nucleon coupling constants [138]. We note in closing that other vacuum susceptibilities can beanalyzed similarly.
4. Susceptibilities and the chiral phase transition of QCD
The chiral phase transition of QCD and a possible CEP on the QCD phase diagram have been drawingmuch attention on both experimental and theoretical sides; for example, beam energy scan (BES) programat the RHIC [142, 143, 144, 145], and some theoretical studies [23, 57, 58, 60, 146, 147, 148, 149, 150,151, 152, 153, 154, 155, 156, 157, 158, 159]. To study the chiral phase transition of QCD, in the chirallimit the two-quark condensate can serve as an order parameter. Nevertheless, the current quark mass is30mall but not zero in the real world, so when studying the partial restoration of chiral symmetry at finitetemperature and quark chemical potential, people also need to resort to some alternative order parameters.When there is a coexistence of different phases, one of the candidates is the pressure difference (the bagconstant) [160, 161, 162, 163]. As discussed above, various susceptibilities are the parameters that relatedto the linear responses of the system to the external fields, which can then characterize the properties of thesystem, as well as to serve as indicators of corresponding phase transitions. In this section, we focus on theapplications of some QCD susceptibilities to the chiral phase transition of QCD.In quantum field theory, the dynamic properties of a system are fully characterized by the generatingfunctional, which corresponds to the partition function in thermodynamics. When the system is in a certainphase, the generating functional is usually analytic for some choice of parameters, such as the current massesof the fermions, the chemical potential, and the temperature. The generating functional often exhibits a non-analytic character while the phase transitions occur. So the location and characteristics of the chiral phasetransition in the system can be determined by the behaviors of this quantity with respect to the correspondingparameters. In this case, a phase transition, in which the first order derivative of the generating functionalwith some of the parameters is discontinuous, is referred to as first order or discontinuous phase transition.Second order or continuous phase transition exhibits continuity in first order derivative and a discontinuityor infinity in second order derivative. Nevertheless, thanks to the non-perturbative properties of QCD inthe low energy region, such discussions are difficult and complicated. Accordingly, researchers often resortto some effective models, for example, see Refs. [43, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172]. Inthe following, we take the NJL model, which is successful as well as popular in chiral phase transitionstudies [173, 174, 175, 176, 177, 178, 179, 161, 180, 181, 182], as an example, and give some discussionson the applications of some susceptibilities [36], similar discussions with in the framework of QED can befound in Ref. [183]. For the details of the NJL model, please see Refs. [173, 184, 185, 186].The Lagrangian of the NJL model is (here we still work in the Euclidean space, and take the number offlavors N f = 2 while the number of colors N c = 3) L NJL = L + g L int = ¯ q ( i ∂ − m ) q + g [(¯ qq ) + (¯ qiγ τ q ) ] , (113)where g is a coupling constant with the dimension of mass − , and the flavor and color indices are suppressed.Now we introduce the definitions of four kinds of susceptibilities: the chiral susceptibility χ s , the quarknumber susceptibility χ q , the vector-scalar susceptibility χ vs , and another auxiliary susceptibility χ m . Formathematical convenience, we first introduce these susceptibilities in the free quark gas case, where theinteraction term in the Lagrangian is zero, L int = 0. Denoting them with the superscript (0) , their definitions31nd expressions are then χ (0) s ≡ − ∂ h ¯ qq i f ∂m = N c N f π Z Λ0 (cid:20) m p βE g ( µ ) + p E f ( µ ) (cid:21) d p, (114) χ (0) q ≡ ∂ h q † q i f ∂µ = N c N f π Z Λ0 p βg ( µ )d p, (115) χ (0) vs ≡ ∂ h ¯ qq i f ∂µ = N c N f π Z Λ0 mp βE h ( µ )d p, (116) χ (0) m ≡ − ∂ h q † q i f ∂m = χ (0) vs , (117)where Λ is a three-momentum cutoff which is introduced to avoid the ultraviolet divergence in the NJLmodel, β = 1 /T , E = p m + p , g ( µ ) + h ( µ ) = 2 n ( µ )(1 − n ( µ )), g ( µ ) − h ( µ ) = 2 m ( µ )(1 − m ( µ )), f ( µ ) = 1 − n ( µ ) − m ( µ ), and n ( m, p, µ ) = 11 + exp [ β ( E − µ )] , (118) m ( m, p, µ ) = 11 + exp [ β ( E + µ )] , (119)the subscript f represents the free quark gas systems. It should be noted that χ (0) m and χ (0) vs have the sameanalytical expression, which is reasonable from the viewpoint of statistical mechanics, χ (0) m = χ (0) vs = TV ∂ ∂m∂µ ln Z f , (120)with Z f the QCD partition function in the free quark gas case.In the interacting case, these susceptibilities are coupled with each other, χ s ≡ − ∂ h ¯ qq i ∂m = χ (0) s ( µ )(1 + 2 gχ s ) − gN c χ (0) vs ( µ ) χ m , (121) χ q ≡ ∂ h q † q i ∂µ = 2 gχ (0) vs ( µ ) χ vs + χ (0) q ( µ )(1 − gN c χ q ) , (122) χ vs ≡ ∂ h ¯ qq i ∂µ = 2 gχ (0) s ( µ ) χ vs + χ (0) vs ( µ )(1 − gN c χ q ) , (123) χ m ≡ − ∂ h q † q i ∂m = χ (0) m ( µ )(1 + 2 gχ s ) − gN c χ (0) q ( µ ) χ m . (124)Using the iterative method, we can obtain the numerical results of these susceptibilities. For example,the vector-scalar susceptibility is the response of the effective quark mass ( M = m − g h ¯ qq i ) to the chemicalpotential µ , and its results are shown in Fig. 7. The parameters used in this section are taken from Ref. [185],namely, m = 5 . g = 5 .
074 GeV − , and Λ = 631 MeV.It can be seen from Fig. 7 that, when T is smaller than a critical value T c = 35 MeV, there always existsa convergent discontinuity of χ vs , corresponding to a first order phase transition; when T = T c , χ vs displaysa sharp and narrow divergent peak, which implies a second-order phase transition, or in other words, hereis a CEP on the phase diagram; when T > T c , the discontinuity disappears and a rather broad peak offinite height is shown, corresponding to the crossover region. Comparing this with the results of the chiralsusceptibility χ s and the quark number susceptibility χ q , shown in Fig. 8, we can conclude that at lowtemperature, the first order phase transition occurs at almost the same chemical potential; while in the32
40 260 280 300 320 340 360 380 40000.511.522.5 x 10 µ [MeV] χ vs [ M e V ] T=15MeVT=35MeVT=55MeVT=80MeVT=110MeV
Figure 7: χ vs at different µ and T , taken from Ref. [36]. crossover region, if we pick the peak of these susceptibilities as the artificial critical points, we see that theytend to occur at different chemical potentials as the temperature increases.The calculated results of χ vs , χ s , and χ q in the crossover region are shown in Figs. 9 and 10, respectively.We can find that they exhibit different behaviors: the chiral susceptibility χ s exhibits an obvious band, so itis convincing to define the peak of χ s as the artificial critical points; in the high T and/or low µ region, thevector-scalar susceptibility χ vs tends to vanish; while the global shape of the quark number susceptibility χ q is just similar to the ones of χ s and χ vs , but it is non-vanishing in the high T and/or high µ region whosebehavior is closely linked to the quark number density. Therefore, χ q can not describe the crossover propertywell in the high T and/or high µ region.It is very interesting and meaningful to compare the results of these susceptibilities with that of thethermal susceptibility χ T = ∂ h ¯ qq i /∂T . For mathematical convenience, we also define χ n = ∂ h q † q i /∂T . Bythe same process employed above, we obtain a set of coupled equations for χ T and χ n as χ T ≡ ∂ h ¯ qq i ∂T = ∂ h ¯ qq i ∂M ∂M∂T + ∂ h ¯ qq i ∂µ ∂µ∂T + (cid:0) ∂ h ¯ qq i ∂T (cid:1) M,µ =2 gχ (0) s ( µ ) χ T − gN c χ (0) vs ( µ ) χ n + M βχ (0) q ( µ ) − µβχ (0) m ( µ ) , (125) χ n ≡ ∂ h q † q i ∂T = ∂ h q † q i ∂M ∂M∂T + ∂ h q † q i ∂µ ∂µ∂T + (cid:0) ∂ h q † q i ∂T (cid:1) M,µ =2 gχ (0) m ( µ ) χ T − gN c χ (0) q ( µ ) χ n − µβχ (0) q ( µ ) + N c N f π Z Λ0 p Eβ h ( µ )d p, (126)with E = p M + p . The behavior of χ T in the crossover region is shown in Fig. 10, which is very similar tothat of χ s .Now we compare the results of different susceptibilities in a chiral phase diagram of QCD, as presented in33
40 260 280 300 320 340 360 380 400012345678 x 10 µ [MeV] χ s [ M e V ] T=15MeVT=35MeVT=55MeVT=80MeVT=110MeV
240 260 280 300 320 340 360 380 40001234567 x 10 µ [MeV] χ q [ M e V ] T=15MeVT=35MeVT=55MeVT=80MeVT=110MeV
Figure 8: χ s and χ q at different µ and T , taken from Ref. [36]. T [MeV] µ [MeV] χ vs [ M e V ] T [MeV] µ [MeV] χ s [ M e V ] Figure 9: χ vs and χ s in the crossover region, taken from Ref. [36]. Fig. 11. The corresponding lines in the crossover region are determined by the peaks χ vs , χ T , and χ s . Hereit should be stressed that, actually there is a very large number of studies investigating the phase diagramtogether with the CEP of QCD, and in various models [52, 161, 162, 171, 187, 188, 189, 190, 191, 192, 193] aswell as functional and Lattice QCD studies [23, 194, 195, 196, 197, 198, 199, 200] the results vary quite widely,most of them find a higher value for T c and a lower value for µ c of the CEP we plotted here, as summarised inFig. 12 (here µ B = 3 µ ), which is taken from Fig. 4 of Ref. [52]. The points in Fig. 12 represent effective modeland Lattice QCD predictions, the two dashed lines are parabolas with slopes corresponding to Lattice QCDcalculations, and the red circles are locations of the freeze-out points for heavy ion collisions at correspondingcenter of mass energies per nucleon (indicated by labels in GeV). It is shown clearly that, different modelsor parameter sets give quite different predictions (please refer to Ref. [52] for more details of these pointsand lines). The parameters we adopted here (NJL89a in Fig. 12) are just for the convenience of showing thequalitative results of different susceptibilities more explicitly. For a recent parameter-independent attemptof such studies, please see Ref. [201].We see from Fig. 11 that these susceptibilities split when the system tends to lower quark chemical34
50 100 150 200 250 300 100 150 200 25000.511.522.5 x 10 T [MeV] µ [MeV] χ q [ M e V ] T [MeV] µ [MeV] χ T [ M e V ] Figure 10: χ q and χ T in the crossover region, taken from Ref. [36]. potential and/or higher temperature. As we discussed above, up to now there is no exact order parameterfor studies of QCD chiral phase transition when beyond the chiral limit, and different susceptibilities wouldcharacterize different physical properties of the system. Hence, related topics deserve further discussions.
5. Summary
In the Dyson-Schwinger equations (DSEs) framework of QCD, we summarize studies of various vacuumsusceptibilities together with some of their applications to the chiral phase transition of QCD. In Section 1,we give a brief introduction of the vacuum susceptibilities together with their related applications. Then ageneral derivation of the vacuum susceptibility using the QCD sum rules external field formula is explicatedin the following Section 2, as well as a model independent expression, which is expressed with the quarkpropagator (two-point Green function) and the corresponding vertex function (three-point Green function).In the next Section 3, we review the calculations of the quark propagator and the vertex function underthe framework of DSEs-BSEs first, and then use these theoretical methods and results to discuss the calcu-lations of the tensor, the vector, the axis-vector, the scalar, and the pseudo-scalar vacuum susceptibilitiesfurther. For the tensor vacuum susceptibility, we introduce the calculation results of the Maris-Tandy gluonpropagator model using the rainbow-ladder approximation of DSEs. For the vector and axial-vector vac-uum susceptibilities, we present the model independent results obtained with the help of the vector andaxial-vector Ward-Takahashi identities: the vector vacuum susceptibility is strictly 0, while the axial-vectorvacuum susceptibility equals 3 /
50 100 150 200 250 300 350 400050100150200250 µ [MeV] T [ M e V ] First−order transitionCEPM=1/2M Peak of χ vs Peak of χ s Peak of χ T Figure 11: Phase diagram obtained according to different susceptibilities, taken from Ref. [36]. the help of an effective model of QCD then, in Section 4 we show the applications of some susceptibilities tothe QCD chiral phase transition as well as some discussions of the results.Here it also needs to be pointed out that, in the QCD sum rules external field approach, all kinds of vac-uum condensates and vacuum susceptibilities are introduced as independent phenomenological parameters,and their values are determined by fitting the theoretical predictions from QCD sum rules to the experi-mental results. The authors of Ref. [202] proposed for the first time that the four-quark condensate mayrelate to the corresponding vacuum susceptibility. If this conclusion is correct, the QCD sum rules externalfield formula can then introduce less independent parameters, which will certainly increase its predictiveness.Last but not least, it should be noted that up to now people still do not know how to rigorously define someof the vacuum condensates (including but not limited to the four-quark condensate) from the first principleof QCD. There is no doubt that, the issues related to the four-quark condensate and the relations betweenthe four-quark condensate and the corresponding vacuum susceptibility are worth further studying. And ofcourse, so are many discussions in this paper.
Acknowledgments
This work is supported in part by the National Natural Science Foundation of China (under Grant11275097, 11475085, and 11247219), the Jiangsu Planned Projects for Postdoctoral Research Funds (underGrant No. 1402006C), and the National Natural Science Foundation of Jiangsu Province of China (underGrant BK20130078). 36 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)
CO94 NJL01 NJL89bCJT02 NJL89aLR04 RM98LSM01HB02 LTE03LTE04 3NJL05INJL98 LR01 PNJL06
130 9 5 217 T ,MeV µ B , MeV Figure 12: Comparison of model and Lattice QCD predictions for the location of the possible CEP on the QCD phase diagram,taken from Ref. [52] (please refer to this paper for more details).
ReferencesReferences [1] M. A. Shifman, A. Vainshtein, V. I. Zakharov, Qcd and resonance physics. theoretical foundations,Nucl. Phys. B 147 (1979) 385–447.[2] I. Balitsky, A. Yung, Proton and neutron magnetic moments from qcd sum rules, Phys. Lett. B 129(1983) 328–334.[3] B. Ioffe, A. V. Smilga, Nucleon magnetic moments and magnetic properties of the vacuum in qcd,Nucl. Phys. B 232 (1984) 109–142.[4] L. J. Reinders, H. Rubinstein, S. Yazaki, Hadron properties from qcd sum rules, Phys. Rep. 127 (1985)1–97.[5] A. Bakulev, A. Radyushkin, Nonlocal condensates and qcd sum rules for the pion form factor, Phys.Lett. B 271 (1991) 223–230.[6] S. Mikhailov, A. Radyushkin, The pion wave function and qcd sum rules with nonlocal condensates,Phys. Rev. D 45 (1992) 1754.[7] E. Henley, W. Hwang, L. Kisslinger, The weak parity-violating pion-nucleon coupling, Phys. Lett. B367 (1996) 21–27.[8] H.-s. Zong, S. Qi, W. Chen, W.-m. Sun, E.-g. Zhao, Pion susceptibility of the qcd vacuum from aneffective quark–quark interaction, Phys. Lett. B 576 (2003) 289–296.379] R. L. Jaffe, X.-d. Ji, Chiral-odd parton distributions and polarized drell-yan process, Phys. Rev. Lett.67 (1991) 552.[10] R. L. Jaffe, X.-d. Ji, Chiral-odd parton distributions and drell-yan processes, Nucl. Phys. B 375 (1992)527–560.[11] H. He, X. Ji, Tensor charge of the nucleon, Phys. Rev. D 52 (1995) 2960–2963.[12] H.-x. He, X.-d. Ji, Qcd sum rule calculation for the tensor charge of the nucleon, Phys. Rev. D 54(1996) 6897.[13] V. Belyaev, A. Oganesian, A note on the qcd vacuum tensor susceptibility, Phys. Lett. B 395 (1997)307–310.[14] W. Broniowski, M. Polyakov, H.-C. Kim, K. Goeke, Tensor susceptibilities of the vacuum from con-stituent quarks, Phys. Lett. B 438 (1998) 242–247.[15] L. S. Kisslinger, Vector, axial, tensor, and pseudoscalar vacuum susceptibilities, Phys. Rev. C 59(1999) 3377.[16] A. P. Bakulev, S. V. Mikhailov, Qcd vacuum tensor susceptibility and properties of transverselypolarized mesons, Eur. Phys. J. C 17 (2000) 129–135.[17] H.-t. Yang, H.-s. Zong, J.-l. Ping, F. Wang, Tensor susceptibility of the qcd vacuum from an effectivequark–quark interaction, Phys. Lett. B 557 (2003) 33–37.[18] R. V. Gavai, S. Gupta, Pressure and nonlinear susceptibilities in qcd at finite chemical potentials,Phys. Rev. D 68 (2003) 034506.[19] R. V. Gavai, S. Gupta, On the critical end point of qcd, Phys. Rev. D 71 (2005) 114014.[20] R. V. Gavai, S. Gupta, Simple patterns for nonlinear susceptibilities near T c , Phys. Rev. D 72 (2005)054006.[21] S. Gupta, X. Luo, B. Mohanty, H. G. Ritter, N. Xu, Scale for the phase diagram of quantum chromo-dynamics, Science 332 (2011) 1525–1528.[22] A.-M. Zhao, Z.-F. Cui, Y. Jiang, H.-S. Zong, Nonlinear susceptibilities under the framework of dyson-schwinger equations, Phys. Rev. D 90 (2014) 114031.[23] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, C. Ratti, K. K. Szab´o, Is there still any t cmystery in lattice qcd? results with physical masses in the continuum limit iii, J. High Energy Phys.09 (2010) 073.[24] S. Plumari, W. M. Alberico, V. Greco, C. Ratti, Recent thermodynamic results from lattice qcdanalyzed within a quasiparticle model, Phys. Rev. D 84 (2011) 094004.3825] S. Bors´anyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti, K. Szab´o, Fluctuations of conserved charges atfinite temperature from lattice qcd, J. High Energy Phys. 01 (2012) 138.[26] M. D’Elia, M. Mariti, F. Negro, Susceptibility of the qcd vacuum to cp -odd electromagnetic backgroundfields, Phys. Rev. Lett. 110 (2013) 082002.[27] D. Blaschke, A. H¨oll, C. D. Roberts, S. Schmidt, Analysis of chiral and thermal susceptibilities, Phys.Rev. C 58 (1998) 1758.[28] M. He, Y. Jiang, W.-m. Sun, H.-s. Zong, Chiral susceptibility in an effective interaction model, Phys.Rev. D 77 (2008) 076008.[29] M. He, F. Hu, W.-m. Sun, H.-s. Zong, Crossover from a continuum study of chiral susceptibility, Phys.Lett. B 675 (2009) 32–37.[30] A. V. Smilga, J. Stern, et al., On the spectral density of euclidean dirac operator in qcd, Phys. Lett.B 318 (1993) 531–536.[31] A. V. Smilga, J. Verbaarschot, Scalar susceptibility in qcd and the multiflavor schwinger model, Phys.Rev. D 54 (1996) 1087.[32] G. Chanfray, M. Ericson, P. Guichon, Scalar susceptibility and chiral symmetry restoration in nuclei,Phys. Rev. C 68 (2003) 035209.[33] G. Chanfray, M. Ericson, Fluctuations of the quark densities in nuclei, Eur. Phys. J. A 16 (2003)291–297.[34] C. Sasaki, B. Friman, K. Redlich, Susceptibilities and the phase structure of a chiral model withpolyakov loops, Phys. Rev. D 75 (2007) 074013.[35] Y. Zhao, L. Chang, W. Yuan, Y.-x. Liu, Chiral susceptibility and chiral phase transition in nambu–jona-lasinio model, Eur. Phys. J. C 56 (2008) 483–492.[36] Y.-l. Du, Z.-f. Cui, Y.-h. Xia, H.-s. Zong, Discussions on the crossover property within the nambu–jona-lasinio model, Phys. Rev. D 88 (2013) 114019.[37] S.-i. Nam, Scalar susceptibility from the instanton vacuum with meson-loop corrections, Phys. Rev.D 79 (2009) 014008.[38] R. T. Andr´es, A. G. Nicola, Scalar susceptibilities and electromagnetic thermal mass differences inchiral perturbation theory, Prog. Part. Nucl. Phys. 67 (2012) 337–342.[39] A. G. Nicola, J. Pel´aez, J. R. de Elvira, Scalar susceptibilities and four-quark condensates in the mesongas within chiral perturbation theory, Phys. Rev. D 87 (2013) 016001.[40] H.-s. Zong, F.-y. Hou, W.-m. Sun, J.-l. Ping, E.-g. Zhao, Modified approach for calculating vacuumsusceptibility, Phys. Rev. C 72 (2005) 035202. 3941] W. Rantner, X.-g. Wen, Spin correlations in the algebraic spin liquid: implications for high-t csuperconductors, Phys. Rev. B 66 (2002) 144501.[42] J.-F. Li, H.-T. Feng, Y. Jiang, W.-M. Sun, H.-S. Zong, Calculation of the staggered spin correlationin the framework of the dyson-schwinger approach, Phys. Rev. D 87 (2013) 116008.[43] J.-F. Li, F.-Y. Hou, Z.-F. Cui, H.-T. Feng, Y. Jiang, H.-S. Zong, Influence of gauge boson mass on thestaggered spin susceptibility, Phys. Rev. D 90 (2014) 073013.[44] H.-s. Zong, Y.-m. Shi, W.-m. Sun, J.-l. Ping, Modified approach for calculating axial vector vacuumsusceptibility, Phys. Rev. C 73 (2006) 035206.[45] Y.-m. Shi, K.-p. Wu, W.-m. Sun, H.-s. Zong, J.-l. Ping, Modified approach for calculating tensorvacuum susceptibility, Phys. Lett. B 639 (2006) 248–257.[46] L. Chang, Y.-x. Liu, C. D. Roberts, Y.-m. Shi, W.-m. Sun, H.-s. Zong, Chiral susceptibility and thescalar ward identity, Phys. Rev. C 79 (2009) 035209.[47] L. Chang, Y.-x. Liu, C. D. Roberts, Y.-m. Shi, W.-m. Sun, H.-s. Zong, Vacuum pseudoscalar suscep-tibility, Phys. Rev. C 81 (2010) 032201.[48] L. Chang, Y.-x. Liu, W.-m. Sun, H.-s. Zong, Revisiting the vector and axial-vector vacuum suscepti-bilities, Phys. Lett. B 669 (2008) 327–330.[49] Y.-m. Shi, H.-x. Zhu, W.-m. Sun, H.-s. Zong, Calculation of tensor susceptibility beyond rainbow-ladder approximation, Few-Body Systems 48 (2010) 31–39.[50] J. S. Ball, T.-W. Chiu, Analytic properties of the vertex function in gauge theories. i, Phys. Rev. D22 (1980) 2542.[51] J. S. Ball, T.-W. Chiu, Analytic properties of the vertex function in gauge theories. ii, Phys. Rev. D22 (1980) 2550.[52] M. Stephanov, QCD phase diagram: an overview, PoS LAT2006:024,2006, arXiv: hep-lat/0701002(2006).[53] U. W. Heinz, Concepts of heavy-ion physics, arXiv: hep-ph/0407360 (2004).[54] P. Maris, C. D. Roberts, P. C. Tandy, Pion mass and decay constant, Phys. Lett. B 420 (1998)267–273.[55] C. Bernard, T. Burch, C. DeTar, J. Osborn, S. Gottlieb, E. Gregory, D. Toussaint, U. Heller, R. Sugar,Qcd thermodynamics with three flavors of improved staggered quarks, Phys. Rev. D 71 (2005) 034504.[56] M. Cheng, N. Christ, S. Datta, J. Van der Heide, C. Jung, F. Karsch, O. Kaczmarek, E. Laermann,R. Mawhinney, C. Miao, et al., Transition temperature in qcd, Phys. Rev. D 74 (2006) 054507.4057] Y. Aoki, Z. Fodor, S. Katz, K. Szabo, The qcd transition temperature: Results with physical massesin the continuum limit, Phys. Lett. B 643 (2006) 46–54.[58] Y. Aoki, G. Endr˝odi, Z. Fodor, S. Katz, K. Szabo, The order of the quantum chromodynamicstransition predicted by the standard model of particle physics, Nature 443 (2006) 675–678.[59] M. Cheng, N. Christ, M. Clark, J. Van der Heide, C. Jung, F. Karsch, O. Kaczmarek, E. Laermann,R. Mawhinney, C. Miao, et al., Study of the finite temperature transition in 3-flavor qcd, Phys. Rev.D 75 (2007) 034506.[60] Y. Aoki, S. Bors´anyi, S. D¨urr, Z. Fodor, S. D. Katz, S. Krieg, K. Szabo, The qcd transition temperature:results with physical masses in the continuum limit ii, J. High Energy Phys. 06 (2009) 088.[61] S. Gottlieb, W. Liu, D. Toussaint, R. L. Renken, R. L. Sugar, Quark-number susceptibility of high-temperature qcd, Phys. Rev. Lett. 59 (1987) 2247–2250.[62] S. Gottlieb, W. Liu, R. Renken, R. Sugar, D. Toussaint, Fermion-number susceptibility in lattice gaugetheory, Phys. Rev. D 38 (1988) 2888.[63] R. Gavai, J. Potvin, S. Sanielevici, Quark-number susceptibility in quenched quantum chromodynam-ics, Phys. Rev. D 40 (1989) 2743.[64] T. Kunihiro, Quark-number susceptibility and fluctuations in the vector channel at high temperatures,Physics Letters B 271 (1991) 395–402.[65] C. Bernard, T. Blum, C. DeTar, S. Gottlieb, K. Rummukainen, U. M. Heller, J. Hetrick, D. Toussaint,R. L. Sugar, Two-flavor staggered fermion thermodynamics at n t = 12, Phys. Rev. D 54 (1996) 4585.[66] S. Gottlieb, U. M. Heller, A. Kennedy, S. Kim, J. Kogut, C. Liu, R. Renken, D. Sinclair, R. Sugar,D. Toussaint, et al., Thermodynamics of lattice qcd with two light quarks on a 16 × π -and k-meson bethe-salpeter amplitudes, Phys. Rev. C 56 (1997) 3369.[86] P. C. Tandy, Hadron physics from the global color model of qcd, Prog. Part. Nucl. Phys. 39 (1997)117–199.[87] P. Maris, P. C. Tandy, Bethe-salpeter study of vector meson masses and decay constants, Phys. Rev.C 60 (1999) 055214. 4288] C. D. Roberts, S. M. Schmidt, Dyson-Schwinger equations: Density, temperature and continuumstrong QCD, Prog. Part. Nucl. Phys. 45 (2000) S1–S103.[89] R. Alkofer, L. Von Smekal, The infrared behaviour of qcd green’s functions: Confinement, dynamicalsymmetry breaking, and hadrons as relativistic bound states, Phys. Rep. 353 (2001) 281–465.[90] R. Alkofer, P. Watson, H. Weigel, Mesons in a poincare covariant bethe-salpeter approach, Phys. Rev.D 65 (2002) 094026.[91] P. Maris, C. D. Roberts, Dyson–Schwinger equations: a tool for hadron physics, Int. J. Mod. Phys. E12 (2003) 297–365.[92] R. Alkofer, W. Detmold, C. Fischer, P. Maris, Analytic properties of the landau gauge gluon andquark propagators, Phys. Rev. D 70 (2004) 014014.[93] P. Maris, A. Raya, C. Roberts, S. Schmidt, Facets of confinement and dynamical chiral symmetrybreaking, Eur. Phys. J. A 18 (2004) 231–235.[94] R. Alkofer, M. Kloker, A. Krassnigg, R. F. Wagenbrunn, Aspects of the confinement mechanism incoulomb-gauge qcd, Phys. Rev. Lett. 96 (2006) 022001.[95] C. Roberts, Hadron properties and Dyson–Schwinger equations, Prog. Part. Nucl. Phys. 61 (2008)50–65.[96] I. C. Clo¨et, C. D. Roberts, Explanation and prediction of observables using continuum strong { QCD } ,Prog. Part. Nucl. Phys. 77 (2014) 1 – 69.[97] H.-s. Zong, W.-m. Sun, A note on failure of the ladder approximation to qcd, Phys. Lett. B 640 (2006)196–200.[98] R. Cahill, C. D. Roberts, Soliton bag models of hadrons from qcd, Phys. Rev. D 32 (1985) 2419.[99] L. von Smekal, A. Hauck, R. Alkofer, A solution to coupled dyson–schwinger equations for gluons andghosts in landau gauge, Annals of Physics 267 (1998) 1–60.[100] R. Alkofer, C. S. Fischer, F. J. Llanes-Estrada, K. Schwenzer, The quark–gluon vertex in landau gaugeqcd: its role in dynamical chiral symmetry breaking and quark confinement, Annals of Physics 324(2009) 106–172.[101] A. Bashir, L. Chang, I. C. Clo¨et, B. El-Bennich, Y.-X. Liu, et al., Collective perspective on advancesin Dyson-Schwinger Equation QCD, Commun.Theor.Phys. 58 (2012) 79–134.[102] S. Strauss, C. S. Fischer, C. Kellermann, Analytic structure of the landau-gauge gluon propagator,Phys. Rev. Lett. 109 (2012) 252001.[103] A. Bashir, R. Bermudez, L. Chang, C. D. Roberts, Dynamical chiral symmetry breaking and thefermion gauge-boson vertex, Phys. Rev. C 85 (2012) 045205.43104] P. O. Bowman, U. M. Heller, D. B. Leinweber, M. B. Parappilly, A. G. Williams, J. Zhang, Unquenchedquark propagator in landau gauge, Phys. Rev. D 71 (2005) 054507.[105] I. Bogolubsky, E.-M. Ilgenfritz, M. M¨uller-Preussker, A. Sternbeck, Lattice gluodynamics computationof landau-gauge green’s functions in the deep infrared, Phys. Lett. B 676 (2009) 69–73.[106] A. Cucchieri, T. Mendes, G. M. Nakamura, E. M. S. Santos, Feynman gauge on the lattice: Newresults and perspectives, AIP Conference Proceedings 1354 (2011) 45–50.[107] S.-x. Qin, L. Chang, Y.-x. Liu, C. D. Roberts, D. J. Wilson, Interaction model for the gap equation,Phys. Rev. C 84 (2011) 042202.[108] A. C. Aguilar, D. Binosi, J. Papavassiliou, Unquenching the gluon propagator with schwinger-dysonequations, Phys. Rev. D 86 (2012) 014032.[109] D. Binosi, L. Chang, J. Papavassiliou, C. D. Roberts, Bridging a gap between continuum-qcd and abinitio predictions of hadron observables, Physics Letters B 742 (2015) 183 – 188.[110] J. Skullerud, A. Kizilers¨u, Quark-gluon vertex from lattice qcd, J. High Energy Phys. 09 (2002) 013.[111] J. I. Skullerud, P. O. Bowman, A. Kızılers¨u, D. B. Leinweber, A. G. Williams, Nonperturbativestructure of the quark-gluon vertex, J. High Energy Phys. 04 (2003) 047.[112] M. Bhagwat, A. H¨oll, A. Krassnigg, C. Roberts, P. Tandy, Aspects and consequences of a dressed-quark-gluon vertex, Phys. Rev. C 70 (2004) 035205.[113] M. Bhagwat, P. Tandy, Quark-gluon vertex model and lattice-qcd data, Phys. Rev. D 70 (2004)094039.[114] J.-I. Skullerud, P. O. Bowman, A. Kızılers¨u, D. B. Leinweber, A. G. Williams, Quark–gluon vertex inarbitrary kinematics, Nucl. Phys. B-Proceedings Supplements 141 (2005) 244–249.[115] H.-w. Lin, Quark-gluon vertex with an off-shell o (a)-improved chiral fermion action, Phys. Rev. D 73(2006) 094511.[116] A. Kızılers¨u, D. B. Leinweber, J.-I. Skullerud, A. G. Williams, Quark–gluon vertex in general kine-matics, Eur. Phys. J. C 50 (2007) 871–875.[117] E. Rojas, J. de Melo, B. El-Bennich, O. Oliveira, T. Frederico, On the quark-gluon vertex and quark-ghost kernel: combining lattice simulations with dyson-schwinger equations, J. High Energy Phys. 10(2013) 193.[118] A. C. Aguilar, D. Binosi, D. Ibanez, J. Papavassiliou, New method for determining the quark-gluonvertex, Phys. Rev. D 90 (2014) 065027.[119] R. Williams, The quark-gluon vertex in landau gauge bound-state studies arXiv: 1404.2545 (2014).44120] D. Curtis, M. Pennington, Truncating the schwinger-dyson equations: How multiplicative renormaliz-ability and the ward identity restrict the three-point vertex in qed, Phys. Rev. D 42 (1990) 4165.[121] H.-x. He, F. C. Khanna, Y. Takahashi, Transverse ward–takahashi identity for the fermion-bosonvertex in gauge theories, Phys. Lett. B 480 (2000) 222–228.[122] H.-x. He, Identical relations among transverse parts of variant green functions and the full vertices ingauge theories, Phys. Rev. C 63 (2001) 025207.[123] V. Novikov, M. A. Shifman, A. Vainshtein, V. I. Zakharov, Wilson’s operator expansion: can it fail?,Nucl. Phys. B 249 (1985) 445–471.[124] H.-s. Zong, J.-l. Ping, H.-t. Yang, X.-f. L¨u, F. Wang, Calculation of vacuum properties from the globalcolor symmetry model, Phys. Rev. D 67 (2003) 074004.[125] M. Harada, Y. Kim, M. Rho, C. Sasaki, Effective degrees of freedom at chiral restoration and thevector manifestation in hls theory, Nuclear Physics A 727 (2003) 437–463.[126] K. Jo, Y. Kim, H. K. Lee, S.-J. Sin, Vector susceptibility and qcd phase transition in ads/qcd models,Journal of High Energy Physics 2008 (2008) 040.[127] S. Weinberg, Precise relations between the spectra of vector and axial-vector mesons, Phys. Rev. Lett.18 (1967) 507.[128] K. Langfeld, H. Markum, R. Pullirsch, C. Roberts, S. Schmidt, Concerning the quark condensate,Phys. Rev. C 67 (2003) 065206.[129] P. Chakraborty, M. G. Mustafa, M. H. Thoma, Chiral susceptibility in the hard thermal loop approx-imation, Phys. Rev. D 67 (2003) 114004.[130] G. Eichmann, R. Alkofer, I. Clo¨et, A. Krassnigg, C. Roberts, Perspective on rainbow-ladder truncation,Phys. Rev. C 77 (2008) 042202.[131] G. Eichmann, I. Clo¨et, R. Alkofer, A. Krassnigg, C. Roberts, Toward unifying the description of mesonand baryon properties, Phys. Rev. C 79 (2009) 012202.[132] C. J. Burden, J. Praschifka, C. D. Roberts, Photon polarization tensor and gauge dependence inthree-dimensional quantum electrodynamics, Phys. Rev. D 46 (1992) 2695.[133] C. J. Burden, C. D. Roberts, A. G. Williams, Singularity structure of a model quark propagator, Phys.Lett. B 285 (1992) 347–353.[134] J. M. Cornwall, R. Jackiw, E. Tomboulis, Effective action for composite operators, Phys. Rev. D 10(1974) 2428.[135] C. Roberts, A. Williams, G. Krein, On the implications of confinement, Int. J. Mod. Phys. A 7 (1992)5607–5624. 45136] R. J. Gonsalves, Perturbative qcd at high energy colliders, Int. J. Mod. Phys. E 17 (2008) 870–890.[137] M. Bhagwat, A. Hoell, A. Krassnigg, C. Roberts, S. Wright, Schwinger functions and light-quarkbound states, Few-Body Systems 40 (2007) 209–235.[138] L. Reinders, H. Rubinstein, S. Yazaki, Hadron couplings to goldstone bosons in qcd, Nucl. Phys. B213 (1983) 109–121.[139] M. B. Johnson, L. S. Kisslinger, Hadronic couplings via qcd sum rules using three-point functions:Vacuum susceptibilities, Phys. Rev. D 57 (1998) 2847.[140] V. Belyaev, Y. I. Kogan, Axial and vector constants of the nucleon octet in qcd, Phys. Lett. B 136(1984) 273–278.[141] H. Munczek, Dynamical chiral symmetry breaking, goldstone’s theorem, and the consistency of theschwinger-dyson and bethe-salpeter equations, Phys. Rev. D 52 (1995) 4736.[142] B. Mohanty, S. Collaboration, et al., Star experiment results from the beam energy scan program atthe rhic, J. Phys. G: Nucl. Part. Phys. 38 (2011) 124023.[143] L. Kumar, Results from the star beam energy scan program, Nucl. Phys. A 862 (2011) 125–131.[144] L. Kumar, Star results from the rhic beam energy scan-i, Nucl. Phys. A 904 (2013) 256c–263c.[145] J. T. Mitchell, The rhic beam energy scan program: results from the phenix experiment, Nucl. Phys.A 904 (2013) 903c–906c.[146] R. D. Pisarski, Phenomenology of the chiral phase transition, Phys. Lett. B 110 (1982) 155–158.[147] R. D. Pisarski, F. Wilczek, Remarks on the chiral phase transition in chromodynamics, Phys. Rev. D29 (1984) 338.[148] A. Barducci, R. Casalbuoni, S. De Curtis, R. Gatto, G. Pettini, Chiral phase transitions in qcd forfinite temperature and density, Phys. Rev. D 41 (1990) 1610.[149] J. Berges, D.-U. Jungnickel, C. Wetterich, Two flavor chiral phase transition from nonperturbativeflow equations, Phys. Rev. D 59 (1999) 034010.[150] B.-J. Schaefer, J. Wambach, Susceptibilities near the qcd (tri) critical point, Phys. Rev. D 75 (2007)085015.[151] R. A. Lacey, N. N. Ajitanand, J. M. Alexander, P. Chung, W. G. Holzmann, M. Issah, A. Tara-nenko, P. Danielewicz, H. St¨ocker, Has the qcd critical point been signaled by observations at the bnlrelativistic heavy ion collider?, Phys. Rev. Lett. 98 (2007) 092301.[152] B.-J. Schaefer, J. Wambach, Renormalization group approach towards the qcd phase diagram, Physicsof Particles and Nuclei 39 (2008) 1025–1032. 46153] B.-J. Schaefer, M. Wagner, Three-flavor chiral phase structure in hot and dense qcd matter, Phys.Rev. D 79 (2009) 014018.[154] C. S. Fischer, Deconfinement phase transition and the quark condensate, Phys. Rev. Lett. 103 (2009)052003.[155] C. S. Fischer, J. A. Mueller, Chiral and deconfinement transition from dyson-schwinger equations,Phys. Rev. D 80 (2009) 074029.[156] C. S. Fischer, J. Luecker, J. A. Mueller, Chiral and deconfinement phase transitions of two-flavour qcdat finite temperature and chemical potential, Phys.Lett.B 702:438-441,2011 (2011).[157] C. S. Fischer, J. Luecker, Propagators and phase structure of nf=2 and nf=2+1 qcd, Phys.Lett. B718 (2012) 1036–1043.[158] J. Luecker, C. S. Fischer, Two-flavor qcd at finite temperature and chemical potential in a functionalapproach, Prog. Part. Nucl. Phys. 67 (2012) 200–205.[159] C. S. Fischer, J. Luecker, C. A. Welzbacher, Phase structure of three and four flavor qcd, Phys. Rev.D 90, (2014) 034022.[160] Y. Jiang, H. Gong, W.-m. Sun, H.-s. Zong, Wigner solution of the quark gap equation at nonzerocurrent quark mass and partial restoration of chiral symmetry at finite chemical potential, Phys. Rev.D 85 (2012) 034031.[161] Z.-f. Cui, C. Shi, Y.-h. Xia, Y. Jiang, H.-s. Zong, The Wigner solution of quark gap equation andchiral phase transition of QCD at finite temperature and nonzero chemical potential, Eur. Phys. J. C73 (2013) 2612.[162] C. Shi, Y.-l. Wang, Y. Jiang, Z.-f. Cui, H.-s. Zong, Locate qcd critical end point in a continuum modelstudy, J. High Energy Phys. 7 (2014) 014.[163] P.-l. Yin, Z.-f. Cui, H.-t. Feng, H.-s. Zong, The chiral phase transition of qed around the criticalnumber of fermion flavors, Annals of Physics 348 (2014) 306.[164] O. Scavenius, A. Mocsy, I. Mishustin, D. Rischke, Chiral phase transition within effective models withconstituent quarks, Phys. Rev. C 64 (2001) 045202.[165] C. N. Leung, S.-Y. Wang, Gauge independence and chiral symmetry breaking in a strong magneticfield, Annals of Physics 322 (2007) 701–708.[166] S. Roessner, C. Ratti, W. Weise, Polyakov loop, diquarks, and the two-flavor phase diagram, Phys.Rev. D 75 (2007) 034007.[167] B.-J. Schaefer, J. M. Pawlowski, J. Wambach, Phase structure of the polyakov-quark-meson model,Phys. Rev. D 76 (2007) 074023. 47168] K. Kashiwa, H. Kouno, M. Matsuzaki, M. Yahiro, Critical endpoint in the polyakov-loop extended njlmodel, Phys. Lett. B 662 (2008) 26–32.[169] T. K. Herbst, J. M. Pawlowski, B.-J. Schaefer, The phase structure of the polyakov–quark–mesonmodel beyond mean field, Phys. Lett. B 696 (2011) 58–67.[170] H.-t. Feng, B. Wang, W.-m. Sun, H.-s. Zong, Characteristic of chiral phase transition in qed 3 at zerodensity, Phys. Rev. D 86 (2012) 105042.[171] B. Wang, Z.-F. Cui, W.-M. Sun, H.-S. Zong, A model study of the chiral phase diagram of qcd,Few-Body Systems 55 (2014) 47–56.[172] Z.-f. Cui, C. Shi, W.-m. Sun, Y.-l. Wang, H.-s. Zong, The Wigner Solution and QCD Phase Transitionsin a Modified PNJL Model, Eur. Phys. J. C 74 (2014) 2782.[173] A. Masayuki, Y. Koichi, Chiral restoration at finite density and temperature, Nucl. Phys. A 504 (1989)668–684.[174] S. Klimt, M. Lutz, W. Weise, Chiral phase transition in the su (3) nambu and jona-lasinio model,Phys. Lett. B 249 (1990) 386–390.[175] D. Gomez Dumm, N. Scoccola, Chiral phase transition in a covariant nonlocal njl model, Phys. Lett.B 506 (2001) 267–274.[176] P. Costa, M. C. Ruivo, C. A. de Sousa, Thermodynamics and critical behavior in the nambu˘jona-lasinio model of qcd, Phys. Rev. D 77 (2008) 096001.[177] J. K. Boomsma, D. Boer, Influence of strong magnetic fields and instantons on the phase structure ofthe two-flavor nambu˘jona-lasinio model, Phys. Rev. D 81 (2010) 074005.[178] B. Hiller, J. Moreira, A. A. Osipov, A. H. Blin, Phase diagram for the nambu˘jona-lasinio model with’t hooft and eight-quark interactions, Phys. Rev. D 81 (2010) 116005.[179] M. N. Chernodub, Spontaneous electromagnetic superconductivity of vacuum in a strong magneticfield: Evidence from the nambu˘jona-lasinio model, Phys. Rev. Lett. 106 (2011) 142003.[180] P. D. Powell, G. Baym, Axial anomaly and the three-flavor nambu˘jona-lasinio model with confine-ment: Constructing the qcd phase diagram, Phys. Rev. D 85 (2012) 074003.[181] M. Zubkov, Schwinger–dyson equation and njl approximation in massive gauge theory with fermions,Annals of Physics 354 (2015) 72–88.[182] S. Shi, Y.-C. Yang, Y.-H. Xia, Z.-F. Cui, X.-J. Liu, H.-S. Zong, Dynamical chiral symmetry breakingin the njl model with a constant external magnetic field, Phys. Rev. D 91 (2015) 036006.[183] P.-l. Yin, Y.-m. Shi, Z.-f. Cui, H.-t. Feng, H.-s. Zong, Continuum study of various susceptibilitieswithin thermal qed3