Regge spectra of excited mesons, harmonic confinement and QCD vacuum structure

Regge spectra of excited mesons, harmonic confinement and QCD vacuum structure

Sergei N. Nedelko111nedelko@theor.jinr.ru, Vladimir E. Voronin222voronin@theor.jinr.ru Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia
Abstract

An approach to QCD vacuum as a medium describable in terms of statistical ensemble of almost everywhere homogeneous Abelian (anti-)self-dual gluon fields is briefly reviewed. These fields play the role of the confining medium for color charged fields as well as underline the mechanism of realization of chiral and symmetries. Hadronization formalism based on this ensemble leads to manifestly defined quantum effective meson action. Strong, electromagnetic and weak interactions of mesons are represented in the action in terms of nonlocal -point interaction vertices given by the quark-gluon loops averaged over the background ensemble. New systematic results for the mass spectrum and decay constants of radially excited light, heavy-light mesons and heavy quarkonia are presented. Interrelation between the present approach, models based on ideas of soft wall AdS/QCD, light front holographic QCD, and the picture of harmonic confinement is outlined.

pacs:
12.38.Aw, 12.38.Lg, 12.38.Mh, 11.15.Tk

I Introduction

Almost forty five years ago Feynman, Kislinger, and Ravndal noticed Feynman () that the Regge spectrum of meson and baryon masses could be universally described by assuming the four-dimensional harmonic oscillator potential acting between quarks and antiquarks. During subsequent years the idea of four-dimensional harmonic oscillator re-entered the discussion about quark confinement several times in various ways. Leutwyler and Stern developed the formalism devoted to the covariant description of bilocal meson-like fields combined with the idea of harmonic confinement Leutwyler:1977vz (); Leutwyler:1977vy (); Leutwyler:1977pv (); Leutwyler:1977cs (); Leutwyler:1978uk (). Considerations of paper Feynman () and Leutwyler-Stern formalism Leutwyler:1977vz (); Leutwyler:1977vy (); Leutwyler:1977pv (); Leutwyler:1977cs (); Leutwyler:1978uk () can be seen as the forerunners to at present time very popular soft wall AdS/QCD models Karch:2006pv () and the light front holographic QCD deTeramond:2005su (); Brodsky:2006uqa (); deTeramond:2008ht (). In recent years, the approaches to confinement based on the ideas of soft wall AdS/QCD model and light front holography demonstrated an impressive phenomenological success Karch:2006pv (); deTeramond:2005su (); Brodsky:2006uqa (); deTeramond:2008ht (); Swarnkar (); Gutsche:2015xva (). The crucial for phenomenology features of these approaches are the particular dilaton profile and the harmonic oscillator form of the confining potential as the function of fifth coordinate . All these approaches begin with different motivation but finally come to the Schrödinger type differential equation with the harmonic potential in defining the wave functions and mass spectrum of mesons and baryons.

The physical origin of the above-mentioned particular form of dilaton profile in AdS/QCD and light front holography as well as the harmonic potential in the Stern-Leutwyler studies and, hence, the Laguerre polynomial form of the meson wave functions, could not be identified within these approaches themselves. The preferable form of the dilaton profile and/or the potential are determined by the phenomenological requirement of Regge character of the excited meson mass spectrum Karch:2006pv (); Feynman (); Leutwyler:1977vz (); Leutwyler:1977vy (); Leutwyler:1977pv (); Leutwyler:1977cs (); Leutwyler:1978uk ().

The approach presented in this paper has been developed in essence twenty years ago EN1 (). It clearly incorporates the idea of harmonic confinement both in terms of elementary color charged fields and the composite colorless hadron field. The distinctive feature of the present approach is that it basically links the concept of harmonic confinement and Regge character of hadron mass spectrum to the specific class of nonperturbative gluon configurations – almost everywhere homogeneous Abelian (anti-)self-dual gluon fields. A posteriori a close interrelation of the Abelian (anti-)self-dual fields and the hadronization based on harmonic confinement can be read off the papers Leutwyler1 (); Leutwyler2 (); Minkowski (); Leutwyler:1977vz (); Leutwyler:1977vy (); Leutwyler:1977pv (); Leutwyler:1977cs (); Leutwyler:1978uk (). In brief, the line of arguments is as follows (for more detailed exposition see NV ()).

An important benchmark has been the observation of Pagels and Tomboulis Pagels () that Abelian self-dual fields describe a medium infinitely stiff to small gauge field fluctuations, i.e. the wave solutions for the effective quantum equations of motion are absent. This feature was interpreted as suggestive of confinement of color. Strong argumentation in favour of the Abelian (anti-)self-dual homogeneous field as a candidate for the global nontrivial minimum of the effective action originates from the papers Minkowski (); Leutwyler2 (); NG2011 (); Pawlowski (); George:2012sb (). In particular, Leutwyler has shown that the constant gauge field is stable (tachyon free) against small quantum fluctuations only if it is Abelian (anti-)self-dual covariantly constant field Minkowski (); Leutwyler2 (). Nonperturbative calculation of the effective potential within the functional renormalization group Pawlowski () supported the earlier one-loop results on existence of the nontrivial minimum of the effective action for the Abelian (anti-)self-dual field.

The eigenvalues of the Dirac and Klein-Gordon operators in the presence of Abelian self-dual field are purely discrete, and the corresponding eigenfunctions of quarks and gluons are of the bound state type. This is a consequence of the fact that these operators contain the four-dimensional harmonic oscillator, acting as a confining harmonic potential. Eigenmodes of the color charged fields have no (quasi-)particle interpretation but describe field fluctuations decaying in space and time. The consequence of this property is that the momentum representation of the translation invariant part of the propagator of the color charged field in the background of (anti-)self-dual Abelian gauge field is entire analytical function. The absence of pole in the propagator was treated as the absence of the particle interpretation of the charged field Leutwyler1 (). However just the absence of a single quark or anti-quark in the spectrum can not be considered as sufficient condition for confinement. One has to explain the most peculiar feature of QCD – the Regge character of the physical spectrum of colorless hadrons. Usually Regge spectrum is related to the string picture of confinement, justified in two complementary ways and limits: classical relativistic rotating string connecting massless quark and antiquark, and the linear potential between nonrelativistic heavy quark and antiquark with the area law for the temporal Wilson loop as a relevant criterion for static quark confinement. Neither the homogeneous Abelian (anti-)self-dual field itself nor the form of gluon propagator in the presence of this background had the clue to linear quark-antiquark potential. Nevertheless, the analytic structure of the gluon and quark propagators and assumption about the randomness of the background field ensemble led both to the area law for static quarks and the Regge spectrum for light hadrons.

Randomness of the ensemble of almost everywhere homogeneous Abelian (anti-)self-dual gluon fields has been taken into account implicitly in the model of hadronization developed in EN1 (); EN () via averaging of the quark loops over the parameters of the random fields. The nonlocal quark-meson vertices with the complete set of meson quantum numbers were determined in this model by the form of the color charged gluon propagator. The spectrum of mesons displayed the Regge character both with respect to total angular momentum and radial quantum number of the meson. The reason for confinement of a single quark and Regge spectrum of mesons turned out to be the same – the analytic properties of quark and gluon propagators.

This result has almost completed the quark confinement picture based on the random almost everywhere homogeneous Abelian (anti-)self-dual fields. Self-duality of the fields plays the crucial role in this picture. This random field ensemble represents a medium where the color charged elementary excitations exist as quickly decaying in space and time field fluctuations but the collective colorless excitations (mesons) can propagate as plain waves (particles). It should be stressed that in this formalism any meson looks much more like a complicated collective excitation of a medium (QCD vacuum) involving quark, antiquark and gluon fields than a nonrelativistic quantum mechanical bound state of charged particles (quark and anti-quark) due to some potential interaction between them. Within this relativistic quantum field description the Regge spectrum of color neutral collective modes appeared as a ”medium effect” as well as the suppression (confinement) of a color charged elementary modes.

However, besides this dynamical color charge confinement, a correct complete picture must include the limit of static quark-antiquark pair with the area law for the temporal Wilson loop. In order to explore this aspect an explicit construction of the random domain ensemble was suggested in paper NK1 (), and the area law for the Wilson loop was demonstrated by the explicit calculation. Randomness of the ensemble (in line with Olesen7 ()) and (anti-)self-duality of the fields are crucial for this result.

In this paper we briefly review the approach to confinement, chiral symmetry realization and bosonization based on the representation of QCD vacuum in terms of the statistical ensemble of almost everywhere homogeneous Abelian (anti-)self-dual gluon fields, systematically calculate the spectrum of radial meson excitations and their decay constants and outline the possible relation between the formalism of soft wall AdS/QCD and light-front holography, and this, at first sight, different approach.

The paper is organized as follows. Section II is devoted to motivation of the approach. Derivation of the effective meson action is considered in section III. Results for the masses, transition and decay constants of various mesons are presented in section IV. In the section V we outline possible relation between the present hadronization approach and the formalism of the soft wall AdS/QCD model, light front holographic QCD, compare the quark and gluon propagators of the present approach with the results of functional renormalization group (FRG) and Dyson-Schwinger equations (DSE). Important technical details are given in the appendices.

Ii Domain wall networks as QCD vacuum

The primary phenomenological basis of the present approach is the existence of nonzero condensates in QCD, first of all – the scalar gluon condensate . In order to incorporate this condensate into the functional integral approach to quantization of QCD one has to choose appropriate conditions for the functional space of gluon fields to be integrated over (see, e.g., Ref.faddeev ()). Besides the formal mathematical content, these conditions play the role of substantial physical input which, together with the classical action of QCD, complements the statement of the quantization problem. In other words, starting with the very basic representation of the Euclidean functional integral for QCD,

 Z=N∫FBDA∫ΨDψD¯ψexp{−S[A,ψ,¯ψ]}, (1)

one has to specify integration spaces for gluon and for quark fields. Bearing in mind a nontrivial QCD vacuum structure encoded in various condensates, one have to define permitting gluon fields with nonzero classical action density,

 FB={A:limV→∞1V∫Vd4xg2Faμν(x)Faμν(x)=B2}.

It is assumed that the constant may have a nonzero value. The gauge fields that satisfy this condition have a potential to provide the vacuum with the whole variety of condensates.

An analytical approach to definition and calculation of the functional integral can be based on separation of modes responsible for nonzero condensates from the small perturbations . This separation must be supplemented with gauge fixing. Background gauge fixing condition is the most natural choice. To perform separation, one inserts identity

 1=∫BDBΦ[A,B]∫QDQ∫ΩDωδ[Aω−Qω−Bω]δ[D(Bω)Qω]

in the functional integral and arrives at

 Z = N′∫BDB∫ΨDψD¯ψ∫QDQdet[D(B)D(B+Q)]δ[D(B)Q]e−SQCD[B+Q,ψ,¯ψ] = ∫BDBexp{−Seff[B]}.

Thus defined quantum effective action has a physical meaning of the free energy of the quantum field system in the presence of the background gluon field . In the limit global minima of determine the class of gauge field configurations representing the equilibrium state (vacuum) of the system.

Quite reliable argumentation in favour of (almost everywhere) homogeneous Abelian (anti-)self-dual fields as dominating vacuum configurations was put forward by many authors Pagels (); Leutwyler2 (). As it has already been mentioned in Introduction, nonperturbative calculation of QCD quantum effective action within the functional renormalization group approach Pawlowski () supported the one-loop result Pagels (); Minkowski (); Leutwyler2 () and indicated the existence of a minimum of the effective potential for nonzero value of Abelian (anti-)self-dual homogeneous gluon field.

Ginzburg-Landau (GL) approach to the quantum effective action indicated a possibility of the domain wall network formation in QCD vacuum resulting in the dominating vacuum gluon configuration seen as an ensemble of densely packed lumps of covariantly constant Abelian (anti-)self-dual field NK1 (); NG2011 (); NV (); George:2012sb (). Nonzero scalar gluon condensate postulated by the effective potential

 Ueff = Λ412Tr(C1˘f2+43C2˘f4−169C3˘f6), (2)

with being a scale of QCD and , leads to the existence of twelve discrete degenerate global minima of the effective action (see Fig.1),

 ˘Aμ∈{˘B(kl)μ| k=0,1,…,5; l=0,1},  ˘B(kl)μ=−12˘nkB(l)μνxν, ~B(l)μν=12εμναβB(l)αβ=(−1)lB(l)μν, ˘nk=T3 cos(ξk)+T8 sin(ξk),  ξk=2k+16π, (3)

where and correspond to the self-dual and anti-self-dual field respectively, matrix belongs to Cartan subalgebra of with six values of the angle corresponding to the boundaries of the Weyl chambers in the root space of .

The minima are connected by the parity and Weyl group reflections. Their existence indicates that the system is prone to the domain wall formation. To demonstrate the simplest example of domain wall interpolating between the self-dual and anti-self-dual Abelian configurations, one allows the angle between chromomagnetic and chromoelectric fields to vary from point to point in and restricts other degrees of freedom of gluon field to their vacuum values. In this case Ginsburg-Landau Lagrangian leads to the sine-Gordon equation for with the standard kink solution (for details see Ref. NG2011 (); NV ())

 ω(xν)=2 arctan(exp(μxν)).

Away from the kink location vacuum field is almost self-dual () or anti-self-dual (). Exactly at the wall it becomes purely chromomagnetic (). Domain wall network is constructed by means of the kink superposition. In general kink can be parametrized as

 ζ(μi,ηiνxν−qi)=2πarctanexp(μi(ηiνxν−qi)),

where is inverse width of the kink, is a normal to the wall and are coordinates of the wall. A single lump in two, three and four dimensions is given by

 ω(x)=πk∏i=1ζ(μi,ηiνxν−qi).

for , respectively. The general kink network is then given by the additive superposition of lumps

 ω=π∞∑j=1k∏i=1ζ(μij,ηijνxν−qij).

Topological charge density distribution for a network of domain walls with different width is illustrated in Fig.2.

Based on this construction, the measure of integration over the background field can be constructively represented as the infinite dimensional (in the infinite volume) integral over the parameters of domain walls in the network: their positions, orientations and widths, with the weight determined by the effective action. It should be noted that chronologically the explicit construction of the domain wall network is the most recent development of the formalism that have been studied in the series of papers EN (); EN1 (); NK1 (); NK4 (); NK6 (), in which the domain wall defects in the homogeneous Abelian (anti-)self-dual field were taken into account either implicitly or in an explicit but simplified form with the spherical domains. The practical calculations in the next sections will be done within combined implementation of domain model given in paper NK4 (): propagators in the quark loops are taken in the approximation of the homogeneous background field and the quark loops are averaged over the background field, the correlators of the background field are calculated in the spherical domain approximation.

Iii Hadronization within the domain model of QCD vacuum

The haronization formalism based on domain model of QCD vacuum was elaborated in the series of papers EN1 (); EN (); NK1 (); NK4 (). We refer to these papers for most of the technical details omitted in this brief presentation. It has been shown that the model embraces static (area law) and dynamical quark confinement (propagators in momentum representation are entire analytical functions) as well as spontaneous breaking of chiral symmetry by the background domain structured field itself. problem was resolved without introducing the strong CP-violation NK6 (). Estimation of masses of light, heavy-light mesons and heavy quarkonia along with their orbital excitations EN1 (); EN (); NK4 () demonstrated promising phenomenological performance. However, calculations in Refs. EN (); NK4 () have been done neglecting a mixing between radially excited meson fields. Below we present results of calculation refined in this respect.

In the spherical domain approximation, the background gluon fields are represented by the ensemble of domain-structured fields with the strength tensor NK1 (); NK4 ()

 Faμν(x)=N∑k=1n(k)aB(k)μνθ(1−(x−zk)2/R2),B(k)μνB(k)μρ=B2δνρ,B=2√3Λ2, ~B(k)μν=±B(k)μν,^n(k)=t3cosξk+t8sinξk,ξk∈{π6(2k+1), k=0,…,5},

where is the space-time coordinate of the -th domain center, scale and mean domain radius are parameters of the model related to the scalar gluon condensate and topological susceptibility of pure Yang-Mills vacuum, respectively NK1 ().

The measure of integration over ensemble of background fields is defined as NK1 (); NK4 ()

 ∫BdB… = ∏k124π2limV→∞1V∫Vd4zk2π∫0dφk∫π0dθksinθk × ∫2π0dξk3,4,5∑l=0,1,2δ(ξk−(2l+1)π6)∫π0dωk∑n=0,1δ(ωk−πn)…

Once the measure is specified, one can return to the functional integral (1) and integrate out fluctuation part of the gluon fields :

 Z = ∫dB∫ΨDψD¯ψ∫QDQδ[D(B)Q]ΔFP[B,Q]e−SQCD[Q+B,ψ,¯ψ] = ∫dB∫ΨDψD¯ψexp{∫dx¯ψ(i⧸∂+g⧸B−m)ψ}W[j],

where is the local quark current. Recalling the definition of Green functions,

 Ga1…anμ1…μn(x1,…,xn|B)=1gnδnlnW[j]δja1μ1(x1)…δjanμn(xn)∣∣ ∣∣j=0,

we arrive at the representation

 W[j|B] = exp{∑ngnn!∫d4x1…∫d4xnja1μ1(x1)…janμn(xn)Ga1…anμ1…μn(x1,…,xn|B)},

where by construction the gauge coupling constant and the exact renormalized -point gluon Green functions of pure gauge theory in the presence of the background field appear to be renormalized within appropriate renormalization scheme. It is needless to say that the functional form of these Green functions, gluon propagator in particular, has been a subject of many investigations carried out over decades. Quite reliable information about two- and three-point Euclidean Green functions was obtained within the functional renormalization group, Lattice QCD as well as calculations based on Dyson-Schwinger equations.

At this step one has to set up the approximation scheme. We truncate the exponent in up to the four-fermion interaction term. Interaction between standard local color charged quark currents is described by the product of the renormalized coupling constant squared and exact gluon propagator which will be approximated by the gluon propagator in the presence of homogeneous Abelian (anti-)self-dual field. Radiative corrections due to the gluon and ghost field fluctuations are neglected (for more details see Refs. EN1 (); EN (); NK4 ()). It should be noted that omitted radiative corrections can be represented in terms of the standard for pure gluodynamics set of Feynman graphs for gluon polarization function but the internal lines in the graphs correspond to the gluon and ghost propagators in the background field . In other words, the approximation in use corresponds to the lowest (tree level) order with respect to perturbative fluctuations , but the background field (vacuum field ) itself is taken into account exactly.

The randomness of domain ensemble is taken into account implicitly by means of averaging the nonlocal meson-meson interaction vertices over all possible configurations of the homogeneous background field at the final stage of derivation of the effective meson action EN (); NK4 ().

Relevant truncated part of QCD functional integral reads

 Z=∫BdB∫ΨDψD¯ψexp{∫d4x¯ψ(i⧸∂+g⧸B−m)ψ+L}, (4) L=g22∫d4x∫d4y Gabμν(x,y|B)jaμ(x)jbν(y),

where is a diagonal quark mass matrix. By means of the standard Fierz transformation of color, Dirac and flavour matrices the four-quark interaction can be rewritten as

 L=g22∑J,cCJ∫d4x∫d4yG(x−y)JJc(x,y|B)JJc(y,x|B),

where numerical coefficients are different for different spin-parity . Here bilocal color neutral quark currents,

 JJc(x,y|B)=¯ψ(x)λcΓJexp{i2xμ^Bμνyν}ψ(y),

are singlets with respect to the local background gauge transformations. In the center of quark mass coordinate system bilocal currents take the form

 JJc(x,y|B)→JJc(x,z|B)=¯ψf(x)λcΓJexp(izμ\lx@stackrel↔Dμff′(x))ψf′(x), (5) \lx@stackrel↔Dff′μ=ξf\lx@stackrel←Dμ−ξf′\lx@stackrel→Dμ,  \lx@stackrel←Dμ(x)=\lx@stackrel←∂μ+i^Bμ(x),  \lx@stackrel→Dμ(x)=\lx@stackrel→∂μ−i^Bμ(x), ξf=mf′mf+mf′, ξf′=mfmf+mf′,

and their interaction is described by the action EN1 ()

 S=g22∑J,cCJ∫d4x∫d4zG(z)J†Jc(x,z|B)JJc(x,z|B), (6) G(z)=14π2z2exp{−14Λ2z2}, (7)

where - center of quark mass coordinates, and - relative coordinates of quark and antiquark. It has to be noted here that quark fields are seen as pure fluctuations describable in terms of four-dimentional harmonic oscillator eigenmodes of the bound state type NV (); NK1 (); NK4 () in . Interpretation of the quark field in terms of point-like particle is simply does not exist in the confining background under consideration. Function originates from the gluon propagator in the presence of the homogeneous Abelian (anti-)self-dual gluon field EN1 (). It differs from the free massless scalar propagator by the Gaussian exponent, which completely changes the IR properties of the propagator but leaves its UV asymptotic behaviour unchanged. In momentum representation it takes the form

 ~G(p)=1p2(1−e−p2/Λ2). (8)

It is important that nonzero gluon condensates and represented by the Abelian (anti-)self-dual vacuum field remove the pole from the propagator which can be treated as dynamical confinement of the color charged fields Leutwyler1 ().

The quark propagator in the homogeneous as well as domain structured NK1 () Abelian (anti-)self-dual gluon field also demonstrates confinement. Momentum representation of the translation invariant part of the quark propagator in the presence of the homogeneous field,

 S(x,y)=exp(−i2xμBμνyν)H(x−y),

is an entire analytical function of momentum:

 ~Hf(p)=12vΛ2∫10dse(−p2/2vΛ2)s(1−s1+s)m2f/4vΛ2[s1−s2pαγα±isγ5γαfαβpβ (9) fαβ=^n2vΛ2Bαβ, v=diag(16,16,13),  ^Bρμ^Bρν=4v2Λ4δμν.

The propagator has a rich Dirac structure including not only the vector and scalar parts but also the pseudoscalar, axial vector and tensor terms (flavour index is omitted for the sake of brevity)

 ~H(p)=m2vΛ2HS(p2)∓γ5m2vΛ2HP(p2)+γαpα2vΛ2HV(p2)±iγ5γαfαβpβ2vΛ2HA(p2)+σαβmfαβ4vΛ2HT(p2). (10)

Here ”” corresponds to self-dual and anti-self-dual background field configurations. One can easily reconstruct explicit form of functions from Eq.(III). More detailed description of different form factors, particularly the scalar one, and their role in the chiral symmetry realization will be given in section V. This structure of the quark propagator plays important role for successful description of the meson spectrum, especially for the ground state light mesons.

There are two equivalent ways to derive effective meson action based on the functional integral (4) with the interaction term taken in the form (6). The first one is bosonization of the functional integral in terms of bilocal meson-like fields (see for example Ref. roberts ()). We shall return to this option in the discussion section. Another, more elucidative way is to decompose the bilocal currents (5) over complete set of functions orthogonal with the weight determined by function originating from the gluon propagator (7) in Eq.(6)

 JaJ(x,z)=∞∑n,l=0(z2)l/2fnlμ1…μl(z)JaJlnμ1…μl(x).

Here is the radial quantum number and is the orbital momentum. Coefficient quark currents have to describe intrinsic structure of the collective meson-like excitations with complete set of quantum numbers. The form of interaction (6) and natural requirement of diagonality (with respect to and ) of the four-quark interaction, expressed in terms of the currents , indicate the choice of

 fnlμ1…μl=Lnl(z2)T(l)μ1…μl(nz),nz=z/√z2. (11)

Here are irreducible tensors of four-dimensional rotational group, and generalized Laguerre polynomials obey relation

 ∫∞0duρl(u)Lnl(u)Ln′l(u)=δnn′,ρl(u)=ule−u.

The weight comes from the gluon propagator (7). Nonlocal quark currents with complete set of meson quantum numbers can be explicitly calculated and depend only on the center of mass coordinate  EN (); NK4 (),

 JaJlnμ1…μl(x)=¯q(x)VaJlnμ1…μl(\lx@stackrel↔D(x)Λ)q(x), VaJlnμ1…μl(x)=CJlnMaΓJFnl(\lx@stackrel↔D2(x)Λ2)T(l)μ1…μl(1i\lx@stackrel↔D(x)Λ), (12) Fnl(s)=sn∫10dttn+lexp(st), C2Jln=CJl+12ln!(n+l)!,CS/P=2CV/A=19,

where and are flavour and Dirac matrices respectively. The four-fermion interaction takes the form of an infinite sum of the current-current interactions diagonal with respect to all quantum numbers

 L=g22∑aJln∫d4xJ†aJln(x)JaJln(x).

It has to be stressed that the nonlocal quark currents are invariant with respect to the local gauge transformations of the background gauge field as the vertices (12) depend on the covariant derivatives.

The truncated QCD functional integral can be rewritten in terms of the composite colorless meson fields by means of the standard bosonization procedure: introduce the auxiliary meson fields, integrate out the quark fields, perform the orthogonal transformation of the auxiliary fields that diagonalizes the quadratic part of the action and, finally, rescale the meson fields to provide the correct residue of the meson propagator at the pole corresponding to its physical mass (if any). More details can be found in Ref. EN (); EN1 (); NK4 (). The result can be written in the following compact form

 Z=N∫DϕQexp{−Λ22h2Qg2∫d4xϕ2Q(x)−∞∑k=21kWk[ϕ]}, (13) Wk[ϕ]=∑Q1…QkhQ1…hQk∫d4x1…∫d4xkΦQ1(x1)…ΦQk(xk)Γ(k)Q1…Qk(x1,…,xk), (14) ΦQ(x)=∫d4p(2π)4eipxOQQ′(p)~ϕQ′(p), (15)

where condensed index denotes all relevant meson quantum numbers and indices. Integration variables in the functional integral (13) correspond to the physical meson fields that diagonalize the quadratic part of the effective meson action (14) in momentum representation, which is achieved by means of orthogonal transformation .

Interactions between physical meson fields are described by -point nonlocal vertices ,

 Γ(2)Q1Q2=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(2)Q1Q2(x1,x2)−Ξ2(x1−x2)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(1)Q1G(1)Q2, Γ(3)Q1Q2Q3=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(3)Q1Q2Q3(x1,x2,x3)−32Ξ2(x1−x3)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(2)Q1Q2(x1,x2)G(1)Q3(x3) +12Ξ3(x1,x2,x3)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(1)Q1(x1)G(1)Q2(x2)G(1)Q3(x3), Γ(4)Q1Q2Q3Q4=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(4)Q1Q2Q3Q4(x1,x2,x3,x4)−43Ξ2(x1−x2)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(1)Q1(x1)G(3)Q2Q3Q4(x2,x3,x4) −12Ξ2(x1−x3)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(2)Q1Q2(x1,x2)G(2)Q3Q4(x3,x4) +Ξ3(x1,x2,x3)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(1)Q1(x1)G(1)Q2(x2)G(2)Q3Q4(x3,x4) −16Ξ4(x1,x2,x3,x4)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(1)Q1(x1)G(1)Q2(x2)G(1)Q3(x3)G(1)Q4(x4),

subsequently tuned to the physical meson representation by means of corresponding orthogonal transformations . Vertices are expressed via 1-loop diagrams which include nonlocal quark-meson vertices (12) and quark propagators (III) :

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G(k)Q1…Qk(x1,…,xk)=∫dBTrVQ1(x1|B)S(x1,x2)…VQk(xk|B)S(xk,x1),

Bar denotes integration over all configurations of the background fields. As it is illustrated in Fig.3, vertex functions include, in general, several one-loop diagrams correlated via the background field. In the simplified model of spherical domains, the -point correlator is given by a volume of overlap of four-dimensional hyperspheres NK1 (); NK4 ().

It has to be noted that though all Dirac structures, besides the vector and scalar ones, are nullified by the integration of the propagator (10) over the background field, all of them give highly nontrivial contribution to the quark loops (products of several propagators). For example, the two-point correlators responsible for the mass spectrum contain not only the one gluon (fluctuation !) exchange interaction hidden in the vertex but also additional , , , and interactions effectively generated by the background gluon field .

It has to be stressed that the terms linear in meson fields are absent in (13). The linear terms naturally vanish for all mesons besides the scalar ones, and their elimination for the scalar fields requires solution of an infinite system of equations

 Λ2Φ(0)Q1=∞∑k=1gkk∑Q1…QkΦ(0)Q2…Φ(0)QkΓ(k)Q1…Qk, (16)

where and can be treated as an infinite set of scalar quark condensates labelled by the radial quantum number . As we shall discuss in section V, solution of this system of equations leads to the interesting details of the chiral symmetry realization in the presence of the background field under consideration. Actual calculations further below will be done with constant mass which from now on will be treated as the infrared limit of the running nonperturbative quark masses considered as parameters of the model.

The mass spectrum of mesons and quark-meson coupling constants are determined by the quadratic part of the effective meson action via equations

 1=g2Λ2~ΠQ(−M2Q|B), (17) h−2Q=ddp2~ΠQ(p2)|p2=−M2Q, (18)

where is the diagonalized two-point correlator put on mass shell:

 ~ϕ†Q(−p)[OT(p)~Γ(2)(p)O(p)]QQ′~ϕQ′(p)|p2=−M2Q=~ΠQ(−M2Q)~ϕ†Q(−p)~ϕQ(p)|p2=−M2Q.

Explicit construction of will be discussed in the next section. Solution of Eq. (17) identifies the position of the pole in the propagator of the meson with quantum numbers . Definition (18) of the meson-quark coupling constant provides correct residue at the pole.

The free parameters of the model are the IR limits of the running renormalized strong coupling constant , quark masses , , , , and the scales and . By construction, the coupling constant and the quark masses correspond to the background Feynman gauge condition and momentum subtraction (MOM) renormalization scheme at subtraction point . The scale and mean domain size are related to the scalar gluon condensate and topological susceptibility of pure gluodynamics respectively,

 ⟨αsF2⟩=23Λ4π,χYM=172Λ8R4128π2.

It should be noted that decomposition (11) and (12) attributes the same radial form factor to the mesons with different spin-parity . Moreover, the form of appears to be the same for all quarkonium-like collective excitations with different quark content and spin-parity such as and mesons. On the contrary, the physical meson states correspond to the momentum dependent transformed basis and respectively transformed quark current

 faJlnp(z)=∞∑n′=0Onn′aJl(p)fln′(z),  OaJlOTaJl=I, ~JaJ(p,z)=∞∑nl(z2)l/2faJnlp(z)Onn′aJl~JaJln′(p),

where is an orthogonal transformation of the initial basis taking into account two-point function . All this means that though ab initio the basic property of quark-meson interaction form factor is set up by gluon propagator, it is the quark loop that defines its final physical form which is different for different mesons.

Iv Masses and decay constants of mesons

iv.1 Mass spectrum of radial excitations of light, heavy-light mesons and heavy quarkonia

Meson masses are defined by the algebraic equation (17). This equation emerges as follows. In the momentum representation, the quadratic part of the effective action pseudoscalar and vector meson fields with zero orbital momentum has the form

 S2=−12∫d4p(2π)4~ΦaV0nμ(−p)[Λ2δaa′δμμ′δnn′−g2~Γ(2)μμ′aV0n,a′V0n′(p)]Φa′V0n′μ′(p) −12∫d4p(2π)4~ΦaP0n(−p)[Λ2δaa′δnn′−g2~Γ(2)aP0n,a′P0n′(p2)]~Φa′P0n′(p),

where vector two point correlator has the structure

 ~Γ(2)μμ′aV0n,a′V0n′(p)=~Γ(2)aV0n,a′V0n′(p2)δμμ′+~LaV0n,a′V0n′(p2)pμpμ′. (19)

Vector fields (see Eq.(15)) are subject to the on-shell condition

 pμϕaV0nμ=0, p2=−M2aV0n,

while the mass () is determined by (17) with

 ~ΠQ(p)