Glueball is one of the most crucial predictions from quantum chromodynamics (QCD), whose non-Abelian feature makes it possible to form bound states of gauge bosons, i.e., glueballs made of two/three gluons (gg, ggg, etc.), [1]. The gauge field plays a more important dynamical role in glueballs than that in the standard hadrons, therefore studying particles like glueballs offers a good opportunity of understanding nonperturbative aspects of QCD. The glueball spectra has attracted much attention for four decades [1], and it has been widely investigated by using various nonperturbative methods. For example, glueballs have been studied by using lattice QCD [2–11], by using effective models like flux tube model [12] and MIT bag model [13–17], by using QCD sum rules [18–32] as well as by using the relativistic many-body approach [33–35]. There are also some other analyses of glueballs in Refs. [36–44]. For more information, please refer to review papers [45–47].

On the other hand, the spin and mass of the glueball can be constrained from high energy scattering data. Regge trajectories α(t)=α0+α′t of the glueball have been used to fit high energy pp and pp¯ scattering cross section. The C-parity even glueball, Pomeron exchange gives the lightest J=2++ glueball mass M=t=1.92  GeV. Analogy with the “Pomeron,” C-parity odd “odderon” contributing to large odd amplitude was proposed in the 1970s in describing the high energy pp and pp¯ scattering [48,49]. The odderon was regarded as a three-gluon state: Oabcμνσ(k1,k2,k3)=dabcGaμ(k1)Gbν(k2)Gcν(k3)(1)where the lower indices refer to color and the upper ones refer to the Lorentz structure, and dabc is the fundamental symmetric tensor in SU(3). The evidence for the identification of the odderon has been debated for a longtime. Recently, the D0 collaboration and TOTEM Collaboration announced the evidence of a t-channel exchanged C-parity odd odderons in pp and pp¯ scattering [50]. Especially the odderon’s contribution at the dip-bump region is very essential. The mass of 3-- odderon M3--=3.001  GeV and dacay width Γ3--=2.984  GeV are extracted by using the dipole (DP) Regge model to fit the scattering data [51–54].

In Ref. [32], the oddball spectra has been calculated by using the QCD sum rule. In this work, we are going to investigate the glueball spectra in the framework of holographic QCD, which is based on the gravity/gauge duality, or anti-de Sitter/conformal field theory (AdS/CFT) correspondence [55–57]. AdS/CFT correspondence offers a new possibility to tackle the difficulty of strongly coupled gauge theories [58–61]. Many efforts from both top-down and bottom-up approaches have been paid on examining the nonperturbative properties of QCD [62], e.g., QCD equation of state [63–65], phase transitions [66–72], fluid properties of quark-gluon plasma, meson spectra [72–84], baryon spectra [85–87], as well as the glueball sector [88–111]. In Refs. [112,113], by linearizing the fluctuations around a classical σ-model coupled to gravity in d+1 dimensions, a gauge invariant (diffeomorphism invariant) formalism for calculating the spectra of scalar glueballs and tensor glueballs was developed, which was initially proposed in Refs. [114–116]. This algorithmic formalism was tested and some nontrivial applications were given in Refs. [117–126]. The glueball mass spectra and decay rate in the Sakai-Sugimoto model have been investigated in Refs. [127–129]. Glueballs and oddballs spectra have also been widely studied by using the bottom-up approach, where some studies are based on hard-wall [73] and soft-wall holographic QCD models [74] with the conformal AdS5 background metric.

A realistic nonconformal holographic QCD model should reveal both the spontaneous chiral symmetry breaking and color charge confinement or linear confinement, which are two main features of QCD in the low energy regime. In the top-down approach, the Sakai-Sugimoto (SS) model or D4–D8 brane system [75,76] is one of the most successful nonconformal holographic QCD models. In the bottom-up approach, the dynamical holographic QCD (DhQCD) model constructed in Refs. [67,102,130] can simultaneously describe both chiral symmetry breaking and linear confinement, where the gluon dynamics background is solved by the coupling between the gravity and the dilaton field Φ(z), which is responsible for the gluon condensate and confinement, and the scalar field X(z) is introduced to mimic chiral dynamics. Evolution of the dilaton field and scalar field in 5-dimensional space-time resemble the renormalization group from ultraviolet (UV) to infrared (IR). This dynamical holographic QCD model describes the scalar glueball spectra and the light meson spectral quite well [67,102,130]. Except the dynamical holographic QCD model, there are several other nonconformal holographic QCD models in the same gravity-dilaton system which can well describe nonperturbative QCD properties, e.g., the Gubser model [131–133] and the improved holographic QCD model [134–136] with inputting of a dilaton potential, and the refined model [137] and Dudal model [138] with inputting of a deformed metric.

In the gravity-dilaton system, the metric, the dilaton field, and the dilaton potential are self-consistently solved from each other through the Einstein field equations and the equation of motion of the dilaton field. In principle, the three types of models, (A) inputting the form of the dilaton field, (B) inputting the deformed metric, and (C) inputting the dilaton potential, should be equivalent to describe the background at zero temperature and zero density. We will compare the spectra of glueballs/oddballs including scalar, vector, as well as tensor states and their excitations with those from the lattice QCD, and compare thermodynamic properties with the lattice QCD results for pure gluon system and/or 2+1 flavors system in these three types of models. Though these models have been separately investigated on thermodynamics and scalar/tensor glueballs, however, it is still worthy to check the consistency of the models and whether they can simultaneously describe the glueball spectra and the equation of state. Only those models which can simultaneously describe the glueball spectra and pure gluon system’s equation of state are candidates of dual theory of gluodynamics. It is still difficult to find the dual theory of full QCD, the existence of dual theory of pure gluodynamics would be quite encouraging.

The paper is organized as follows: we introduce the general Einstein-Maxwell-dilaton framework in Sec. II. Then in Sec. III we introduce five different models in the gravity-dilaton system. In Sec. IV we introduce the glueball and oddball operator and calculate the mass spectra in these models and we compare the results of mass spectra with lattice results, results from the QCD sum rule, and results extracted from high energy scattering data. In Sec. V we compare thermodynamic properties of these models with lattice results. Finally, a summary is given in Sec. VI.

To keep the self-consistency of investigating the glueball spectra as well as further studies on QCD matter at finite temperature and finite chemical potential, we first introduce the general framework of the Einstein-Maxwell-dilaton (EMD) system, which comes back to the gravity-dilaton coupling system at zero chemical potential. The total action of the 5-dimensional holographic QCD model including glueball/oddball excitations is Stotals=Sbs+Sgs,(2)where Sbs is the action for the background in the string frame, and Sgs is the action describing glueballs in the string frame.

The Einstein-Maxwell-dilaton action Sbs for the background in the string frame is Sbs=12κ52∫d5x-gse-2Φ[Rs+4gsMN∂MΦ∂NΦ-Vs(Φ)-h(Φ)4e4Φ3gsMM˜gsNN˜FMNFM˜N˜],(3)where s denotes the string frame, κ52=8πG5, the G5 is the 5-dimensional Newton constant. The gs is the determinant of the metric in the string frame: gs=det(gMN), and the metric tensor in the string frame is extracted from ds2=L2e2As(z)z2(-f(z)dt2+dz2f(z)+dy12+dy32+dy32),(4)where L is the curvature radius of the asymptotic AdS5 space-time. For simplicity, we set L=1 in the following calculations. The Rs is the Ricci curvature scalar in the string frame. The scalar field Φ(z) is the dilaton field which depends only on the coordinate z, FMN is the field strength of the U(1) gauge field AM: FMN=∂MAN-∂NAM.(5)The 5-dimensional field AM is dual to baryon number current. h(Φ) describes the coupling strength of AM in the theory, Vs(Φ) represents the potential of the dilaton field in the string frame. h(Φ) and Vs(Φ) are the functions that depends only on the value of Φ.

In the gravity-dilaton system, the metric, the dilaton field, and the dilaton potential can be self-consistently solved from each other through the Einstein field equations and the equation of motion of the dilaton field. At zero temperature and zero chemical potential, the function f(z)=1 and At(z)=0, then Eq. (14) to Eq. (18) can be simplified: AE′′-6zAE′+4z2+3AE′2+L2e2AEVϕ3z2=0,(19)AE′′-AE′(-2z+AE′)+ϕ′26=0,(20)ϕ′′+ϕ′(-3z+3AE′)-L2e2AEz2dVϕ(ϕ)dϕ=0,(21)where Eq. (21) is the constraint. Under the condition that we have proper boundary conditions, if we input (A) the form of the dilaton field ϕ(z), or (B) the function AE(z), or (C) the dilaton potential Vϕ(ϕ), we can solve the other two. In principle, these three types of models of EMD system are totally equivalent to describe the background in the vacuum. However, at finite temperature and finite chemical potential, the situation will become different. If we input Vϕ(ϕ), the form of Vϕ(ϕ) is independent of the temperature/chemical potential, from Eq. (14) and Eq. (18), we can solve different functions AE(z) and ϕ(z) at different temperature/chemical potential, which can be denoted by AET,μ(T,μ,z) and ϕT,μ(T,μ,z). On the other hand, if we input AE(z) [or ϕ(z)], whose form is independent of temperature/chemical potential, we can derive Vϕ(ϕ) with temperature/chemical potential dependence, which can be denoted by VϕT,μ(T,μ,ϕ). The two descriptions, that are equivalent at vacuum, now become distinct from each other at finite temperature/chemical potential. From now on, we call fixing Vϕ(ϕ) “description A,” fixing AE(z) or ϕ(z) is denoted by “description B.”

It is more convenient to solve the system in the Einstein frame from Eqs. (19)–(21). In the following we list two sets of vacuum solutions of Vϕ(ϕ), AE(z), and ϕ(z) that satisfy the EOMs.

In this section, we discuss the spectra of glueballs and oddballs. There are two different methods to investigate the glueball spectra in holographic QCD. The first method can be called the “glueball fluctuations method” as in Refs. [63,94,97,109,122,135], where the scalar and tensor fluctuations are treated as the 5-dimensional fields dual to scalar and tensor glueballs respectively. Another method, which is used in this work, can be called the “glueball excitations method,” where the action of the related glueball fields are introduced and the glueballs are treated as excitations from the background as used in Refs. [99,100,103,104,144], where the glueball spectra are treated in the same way as the meson spectra in the bottom-up framework. In practice, the glueball fluctuations method is more widely used in the top-down framework, and the glueball excitations method is more widely used in the bottom-up framework. We adopt the glueball excitations method in this work in order to keep the way of treating the meson spectra and glueball spectra on an equal footing in the deformed holographic QCD models.

Although we could consider more generic Lagrangians [145], we still use the simple action describing scalar, vector, and tensor glueballs/oddballs here. In the string frame, the action is Sgs=-cg∫d5x-gse-pΦ×{[12gsMN∂MS∂NS+12e-cr.m.ΦMS,52S2]+[14gsMM˜gsNN˜(∂MVN-∂NVM)(∂M˜VN˜-∂N˜VM˜)+12e-cr.m.ΦMV,52V2]+[12∇LTMN∇LTMN-∇LTLM∇NTNM+∇MTMN∇NT-12∇MT∇MT+12e-cr.m.ΦMT,52(TMNhMN-T2)]+terms for high spin fields (spin S ⩾ 3)},(65)where s denotes the string frame, cg describes the coupling strength of glueballs part in the whole theory. The fields S, VM, and TMN are 5-dimensional fields that are dual to scalar glueball, vector glueball, and spin-2 glueball operators respectively. T=gsMNTMN, and TMN satisfies the following constraints ∇MTMN=0,T=0,Tμν=1z2e2AsT˜μν,TMz=0.(66)As in Ref. [146], the parameter p is introduced to make a distinction between glueballs (oddballs) with different P-parity: {p=1,for even parity,p=-1,for odd parity.(67)Also we introduce a z dependent modified 5-dimensional mass: M52(z)=e-cr.m.ΦM52,(68)where cr.m. is a constant. The M52 is listed in Table I given by the AdS5/CFT4 correspondence dictionary. The AdS5/CFT4 duality gives one-to-one correspondence between 4-dimensional operators in the N=4 super Yang-Mills theory and the spectrum of the type IIB string theory on AdS5×S5. Based on the AdS/CFT dictionary, the conformal dimension of a q-form operator at the ultraviolet (UV) boundary is related to the 5-dimensional mass square M52 of its dual field in the bulk as follows [55–57]: M52=(Δ-q)(Δ+q-4).(69)