Empirical and lattice-QCD studies indicate that, at nonzero temperature and baryon chemical potential, QCD undergoes two intertwined but conceptually distinct phase transitions: deconfinement, whereby color degrees of freedom become manifest over hadronic scales, and chiral-symmetry restoration characterized by the melting of the quark condensate. Clarifying the interplay between these two phenomena is essential for interpreting experimental observables in heavy-ion collisions, elucidating the equation of state of dense matter in compact stars and constraining scenarios for the evolution of the early Universe [1–4].

For physical quark masses, lattice-QCD studies establish that QCD matter undergoes a smooth analytic crossover at vanishing baryon chemical potential [5–8]. In the two-flavor chiral limit, by contrast, the transition becomes a true phase transition whose order—second or first—depends on the relative hierarchy between chiral-symmetry restoration and the effective restoration of the axial U(1) anomaly [5]. Besides its theoretical significance, mapping the QCD phase diagram in the quark-mass plane is essential for locating the critical end point (CEP), a primary objective of present heavy-ion-collision experiments; in particular, the chiral-limit transition temperature provides an upper bound for possible CEP temperatures at nonzero baryon chemical potential [9]. Since lattice simulations cannot be carried out directly in the exact chiral limit, analyses that exploit universal scaling functions are indispensable for extrapolating physical results to that limit.

Recent lattice-QCD results have revealed the anticipated O(N) scaling behavior [10–12] of the chiral crossover. The connection based on the universal scaling equation is restricted to a finite domain referred to as the scaling window. The width of this window is still under debate. According to lattice-QCD results [13,14], the scaling region is located at relatively large values of the pion mass. Reference [15], for example, explores the conformal window across different flavor numbers and suggests that, within the 3D O(4) universality class, the physical pion mass may lie close to the critical point, with scaling behavior potentially extending into the high-temperature regime up to 300 MeV. In contrast, another lattice-QCD study [16] indicates that the scaling window may also extend below the physical quark mass and, based on the scaling function, provides an estimate of the transition temperature in the chiral limit. Consistently, alternative approaches such as Dyson-Schwinger equations (DSEs) [9,17,18] and the functional renormalization group (FRG) [19,20] also demonstrate the presence of scaling behavior for quark masses below the physical point.

In the low-energy regime where QCD phase transitions take place, the strong coupling strength renders conventional perturbative techniques inadequate. Thus, it is essential to develop nonperturbative methods to explore the underlying physics. Among such methods, lattice QCD—formulated from first principles—has been recognized as a reliable tool. However, at large baryon densities, lattice QCD suffers from the notorious sign problem [21], which has yet to be satisfactorily resolved. This limitation has motivated the search for alternative nonperturbative approaches.

In the 1990s, ’t Hooft proposed a preliminary form of the holographic principle [22], which was subsequently integrated with string theory by Susskind [23]. The AdS/CFT correspondence, first conjectured by Maldacena [24], has shed new light on solving strongly coupled problems in gauge field theories. In the study of QCD matter, a celebrated achievement of holographic approaches is the derivation of the universal lower bound η/s=1/(4π) for the shear viscosity to entropy density ratio, providing key insight into the nearly perfect-fluid behavior of the quark-gluon plasma from a theoretical perspective [25–27]. Employing a bottom-up strategy that prioritizes phenomenology over top-down string-theoretic origins, researchers have constructed a variety of holographic models—including light-front holographic QCD [28], holographic model of QCD in the Veneziano limit [29], and Einstein–Maxwell–Dilaton (EMD) setups [30–36]—to capture hadron spectra, thermodynamics, transport coefficients, and the QCD phase transition with remarkable success. From those phenomenological studies, it appears that the EMD model provides an adequate description of gluodynamics.

It has been shown that spontaneous chiral-symmetry breaking, as another key low-energy feature of QCD, can be effectively described within the hard-wall and soft-wall AdS/QCD models [37,38]. A finite chiral condensate is dynamically generated, lifting the degeneracy between chiral partners in the hadron spectrum [39,40]. As the temperature increases, this condensate is progressively suppressed; once it vanishes, chiral symmetry is restored and the chiral phase transition takes place [41,42].

The principal advantage of this model is that by elevating the four-dimensional (4D) global symmetry to a five-dimensional (5D) gauge symmetry, it naturally incorporates the conserved currents and the q¯q bilinear operator. This makes it straightforward to explore scenarios with arbitrary quark masses and finite charge densities. Studies have extended from simple two-flavor systems to Nf=2+1 [43–45], and more recently to Nf=2+1+1 [46]. The phase diagram in the quark-mass plane (mu/d-ms) can be summarized in a holographic version [47] of the so-called “Columbia plot,” which qualitatively agrees well with that obtained by combining the lattice simulations and other effective studies [8,48]. By introducing the baryon and isospin chemical potentials μB and μI through the corresponding conserved currents, one can probe regimes of high baryon and isospin densities where recent studies predict the appearance of additional phases, e.g., the quarkyonic and pion-condensed phases [49–52]. A magnetic field can be introduced by constructing the electromagnetic current from the baryon and isospin currents; inverse magnetic catalysis then emerges naturally, without fine-tuning of the model settings [53,54]. Furthermore, as derived in [55], the model can be reduced to a four-dimensional chiral perturbation theory. The matching of the near Tc behavior of its Goldstone modes [56–58] with those of the four-dimensional chiral perturbation theory [59] provides explicit evidence for this equivalence. Overall, the soft-wall model offers an excellent starting point for exploring chiral dynamics in QCD, serving in effect as a five-dimensional analog of chiral perturbation theory.

Although the above analyses have thoroughly validated the soft-wall model’s consistency and effectiveness in describing low-energy QCD, its universality has yet to be fully tested. While the critical exponents have been proved to be of mean-field level [47], with β=1/2, δ=3 in the two-flavor limit, the holographic investigation of mass scaling behavior of the pseudocritical temperature, as well as of the scaling function itself, remains largely undeveloped. Given that universal scaling functions enable chiral extrapolation in lattice QCD and are important for constraining holographic model building (while fitting at the physical point is valuable, it alone cannot fully validate a holographic model’s robustness), in this work, we aim to investigate the scaling behavior of the chiral phase transition temperature Tc with respect to different quark and pion masses. We will calculate the scaling function and test the scaling window in the holographic model, which may provide further constraints for building a realistic holographic QCD model. In this work, we will focus on the two-flavor limit for simplicity.

The organization of this paper is as follows: In Sec. II, we briefly review the soft-wall model and chiral phase transition in this model. In Sec. III, we numerically extract the scaling functions and test the universal relations of those functions as a check of the self-consistency of the soft-wall AdS/QCD models. In Sec. IV, we give analyses of the scaling behavior of the pseudocritical temperature. An additional constraint is given for constructing a more realistic holographic QCD model for chiral dynamics. Finally, a brief summary will be given in Sec. V.

In this section, we provide a brief overview of the soft-wall model and introduce several modifications to explore its universal properties. As discussed above, the action of the soft-wall model is constructed by promoting the 4D global chiral symmetry SU(2)L⊗SU(2)R to 5D gauge symmetry. Since we focus on the two-flavor limit, i.e., considering only the u and d quarks, the action of the soft-wall model is given as follows: S5D=∫d5xge-ΦTr{|DMX|2-VX(|X|)-14g52(FL2+FR2)}.(1)

In the above action, M represents spacetime indices, g is the determinant of the metric gMN, X is an Nf×Nf matrix-valued scalar field, and Φ is the dilaton field. As in previous studies, we set the dilaton field as Φ=μg2r2,(2)with r being the holographic dimension. The 5D coupling constant g5 can be determined as g52=12π2Nc by matching the large-momentum expansion of the vector current Jμa=q¯γμtaq correlation in AdS/QCD with that from the perturbative calculation [37]. The covariant derivative DM and the field strength tensors FMNL,R are defined as follows: DMX=∂MX-iLMX+iXRM,FMNL=∂MLN-∂NLM-i[LM,LN],FMNR=∂MRN-∂NRM-i[RM,RN],(3)with LM, RM being the 5D left- and right-handed gauge potentials, which are dual to the left- and right-handed chiral currents. VX(|X|) represents the scalar potential, which may generally include a coupling to the dilaton field. Since we consider the degenerate two-flavor case (mu=md) only, the expectation of X can be decomposed as X=12χI2, with I2 the 2×2 identity matrix. The factor 1/2 is introduced to ensure that the kinetic term in χ is canonical. Then we denote the scalar potential in terms of χ as V(χ)=Tr[VX(X)]. Since we do not consider the case with finite densities or consider the vector perturbations, the relevant part of the action reduces to S5D=∫d5xge-Φ[12χ′2-V(χ)].(4)

To study chiral dynamics in this model, a specific form of the background metric is necessary. In consideration of the symmetries of the four-dimensional theory at finite temperature, the metric ansatz is adopted as follows: ds2=e2A(r)(f(r)dt2-dx2-1f(r)dr2).(5)A simple consideration is to take the AdS-Schwarzschild black hole solution with A(r)=-ln(r),f(r)=1-r4rh4.(6)Here, rh denotes the black hole horizon, at which the blackening factor satisfies f(rh)=0. This can be mapped to the temperature through the Hawking temperature defined as T=|f′(rh)4π|=1πrh.(7)

To solve this model, one has to derive the equation of motion for χ from the action Eq. (4), which reads χ′′+(3A′+f′f-Φ′)χ′-e2AfV,χ(χ)=0,(8)with V,χ(χ)≡∂χV(χ). Here we have assumed that the system is in an equilibrium state, so χ depends on r only. The overprime  ′ in the above equation denotes differentiation with respect to r.

It is not difficult to derive from the above equation the near boundary (r=0) and near horizon (r=rh) expansion of χ as (9)χ(r→0)=mqζr+⋯+r3σζ+O(r4),(9a)χ(r→rh)=c0-V′(c0)4rh(r-rh)+O(r-rh)2,(9b)where mq and σ≡⟨q¯q⟩ are two independent integration constants in the UV region, dual to the quark mass and the chiral condensate, respectively. c0 is an integration constant in the IR region and c0≡χ(rh). In Ref. [60], ζ is identified as a normalization constant with the value ζ=Nc2π, which is obtained by mapping the correlation of the scalar operator to that from the 4D perturbative calculation. Here, we will take Nc=3. One of the integration constants in the IR region is discarded due to its association with a divergent solution at rh. Thus, c0 is kept as the only coefficient corresponding to a regular solution at the horizon.

This study focuses on the scaling behavior of the chiral phase transition temperature. By utilizing the asymptotic expansions of the solutions near the UV boundary r=0 and the IR horizon r=rh as provided in Eqs. (9a) and (9b), the boundary-valued problem can be reformulated as an iterative problem following the procedure established in our previous work [42,57]. This enables the application of the “shooting method” [61] to extract the chiral condensate σ. Before one can work it out, the scalar potential should be specified as well.

Following our previous study [42], we choose the scalar potential to be V(χ)=12m52(r)χ2+λ8χ4.(10)Here, one can assume a general coupling between the scalar field and the dilaton field, such as g2(ϕ)χ2 and g4(ϕ)χ4 with g2, g4 two coupling functions. In this case, effective r-dependent m52(r) and λ(r) can be introduced. That is why we treat m52 and λ as r-dependent functions in the above potential. Following Ref. [45], we will take Model I:  m52(r)=-3-μc2r2(11)and a constant λ, which we will call “model I.” Here, the leading term -3 comes from the AdS/CFT prescription M52=(d-p)(d+p-4) with the dimension d=3, and p=0 for the scalar q¯q operator. Following Ref. [62], we will take Model II:  m52(r)=-3[1+γtanh(κΦ)](12)and a constant λ, which we will call “model II.” The light meson spectra have been carefully studied in these two models, and the model parameters are fitted as shown in Table I for model I and in Table II for model II, which are taken from Refs. [45,62], respectively. The quark mass enters the two models through the boundary expansion of χ, and the physical values mphy in Tables I and II give pion masses around the experimental value (we take 139.6 MeV). Here, the pion masses can be obtained by considering the perturbation of the Goldstone modes (for details, please refer to Refs. [37,38]).

Inserting the parameters in Tables I and II and into the two models, one can obtain the chiral condensate as a function of the temperature. Furthermore, since the quark mass appears in the boundary condition, it is also possible to vary its value, and correspondingly, one gets the function σ(mq,T). We plot the results in Fig. 1. In the upper, middle panels of the figure, the results of models I and II are presented, respectively. For each model, we show two different cases, with a physical quark mass and in the chiral limit. For the physical case, we tune the quark mass to get a pion mass of around 139.6 MeV, while in the chiral limit, we take mq=0 and correspondingly mπ=0 for its Goldstone nature. The vacuum values of the condensate σs and the second-order transition temperature are σs=0.01475  GeV3, Tc0=163.3  MeV for model I, and σs=0.02749  GeV3, Tc0=160.9  MeV for model II. For the physical quark masses, the pseudocritical temperatures extracted from the maxima of |dσ/dT| are Tc=164.0  MeV for model I and Tc=161.2  MeV for model II, while those from the maxima of the chiral susceptibility are Tc=167.4  MeV for model I and Tc=163.9  MeV for model II (red dots). To eliminate the influence of other differences between the models, we plot the chiral condensate σ/σs (where σs is the zero-temperature condensate) as a function of T/Tc across all the models, alongside lattice-QCD data (the filled black triangles) from Ref. [6], as shown in Fig. 2. The results reveal a systematic trend: Models I and II (dashed and dotted-dashed lines) track the lattice data reasonably well near the pseudocritical temperature (T/Tc<1), but deviate significantly in the high-temperature crossover region (T/Tc>1). We explored parameter variations to improve the agreement, but the effect was marginal. The main reason is that the crossover transition at the physical quark mass appears too sharp in these two models, as illustrated in the figures, exhibiting an overly rapid decrease of the chiral condensate at high temperatures.

To broaden the transition region, we note that the chiral condensate in the soft-wall model is sensitive to the quartic coupling. As demonstrated in Ref. [47], a stronger quartic coupling term suppresses the chiral condensate more significantly at low temperatures. Specifically, if one introduces a quartic coefficient that depends on the coordinate r (induced by its coupling to the dilaton) and allows it to grow with increasing r, the low-temperature condensate is suppressed more strongly than its high-temperature counterpart. Furthermore, softening the potential at large χ (e.g., by including a logarithmic term) enables the condensate to approach its T=0 limit more gradually. The interplay between these two mechanisms is expected to effectively broaden the temperature profile of the condensate. Motivated by this insight, we propose a refined model formulation, defined as follows: Model III:  V(χ)=12m52χ2+12λ(r)χ2ln(1+bχ2),m52(r)=-3-μc2r2,λ(r)=a0−a1tanh(a2r2).(13)We will call this model “model III.” As can be seen from the setup of these models, in addition to μg, which accounts for linear confinement [38], an additional scale is required to describe spontaneous chiral-symmetry breaking [42].

Upon selecting parameter set 1 of model III from Table III and solving the equations of motion, we obtain the chiral condensate in this model. As illustrated in the bottom panel of Fig. 1 and the green solid line in Fig. 2, model III (set 1) yields a broader crossover region, as expected. Compared to the other two models, model III with parameter set 1 (green solid line) achieves substantially better agreement with lattice-QCD results across the entire temperature range. It confirms that the functional form of the scalar potential itself—specifically, the logarithmic correction and r dependence in model III—is essential for capturing the quantitative thermodynamics of the chiral crossover. For later convenience, we also obtain the vacuum condensate σs and the second-order transition temperature Tc0 for model III set 1 as σs=0.002918  GeV3, Tc0=145.5  MeV. Correspondingly, for physical quark masses, the pseudocritical temperature extracted from the maxima of |dσ/dT| is Tc=147.0  MeV for model III set 1 (black dots), while that from the maxima of the chiral susceptibility is Tc=151.5  MeV for model III set 1 (red dots).

The broadening of the crossover region, which helps describe the lattice data at the physical quark mass in model III set 1, is also important for investigating scaling behaviors, particularly the Tc–m scaling. Since the primary objective of this work is to rigorously investigate these scaling behaviors, we introduce two additional parameter sets denoted as sets 2 and 3 (listed in Table III), which produce even broader crossover regions and provide a systematic assessment of how the crossover region affects the Tc–m scaling, as discussed in the following sections. As shown in the purple solid line in Fig. 2, model III set 2 produces a mild broadening of the phase transition region. Qualitatively, its deviation from the lattice-QCD data remains approximately within 1σ for T<1.1Tc (with only a few data points slightly exceeding this bound). It should be emphasized that the “1σ” bound referred to here is a qualitative description rather than a rigorously defined statistical quantity. As shown in the blue solid line in Fig. 2, model III set 3 exhibits a larger deviation, remaining approximately within 5σ for T<1.1Tc. We qualitatively regard this as an upper bound on the effect of broadening the phase transition region. For later convenience, we also obtain σs=0.001688  GeV3, Tc0=163.1  MeV for set 2, and σs=0.002175  GeV3, Tc0=144.4  MeV for set 3.

In summary, we have shown that the soft-wall model, with appropriate modifications, can accurately describe the chiral phase transition at the physical quark mass. Based on this, we will proceed to study the scaling properties in the subsequent sections.

As discussed in the last section, with arbitrarily small quark mass, the chiral phase transition turns into a crossover. When the transition point lies sufficiently close to the critical point, it can be strongly influenced by the critical point. A scaling window and universal scaling behavior appear in this region, which might be relevant to the search for the CEP of the QCD phase diagram if the CEP belongs to the same universality class. In this context, it has been verified that the chiral condensate near the critical point obeys the following scaling laws: σ(T,mq=0)∼(Tc0-T)β,σ(Tc0,mq)∼mq1/δ,(14)with β=1/2 and δ=3 in the two-flavor limit [47]. However, these scaling laws capture the critical behavior only along certain directions, namely, holding mq fixed for β and T fixed for δ. To get more information about how the critical point governs the crossover region, one has to extract the scaling functions. This section is therefore devoted to their determination.

As already mentioned in the above sections, by taking a tiny but finite quark mass, the chiral phase transition turns from a second-order transition into a continuous crossover. Although the continuous crossover is not a real phase transition, it is a smooth and rapid change between two different phases, and one can define a pseudocritical temperature in different ways. Generally, the pseudocritical temperature also obeys a certain kind of scaling law. In this section, we will investigate the scaling behavior of the transition temperature.

The critical point of the two-flavor soft-wall AdS/QCD model is systematically studied in Ref. [47], and it is proved to exhibit mean-field scaling laws with exponents β=1/2, δ=3, when tuning either the temperature or the quark mass in its vicinity. However, the universal scaling functions that arise when the temperature and quark mass are varied simultaneously, together with the scaling behavior of the pseudocritical temperature itself, remain largely unexplored in this model. To verify the effectiveness and internal consistency of this holographic model, we present a careful study of the near-critical scaling behavior of various chiral order parameters.

Under different model settings and physical conditions, we find that the rescaled magnetization M/h1/δ in the chiral limit collapses onto a single, universal curve in the chiral limit mq→0, defining the scaling function fG(z). Quantitatively, this scaling function obtained numerically from holography is in excellent agreement with its four-dimensional mean-field counterpart. Furthermore, we derive a perturbative equation to evaluate the chiral susceptibility, from which we obtain another universal scaling function fχ(z). It is found that the universal relationship fχ(z)=1δfG(z)-zβδfG′(z) holds in all three models, confirming the theoretical consistency of this holographic framework.

Furthermore, we extract the pseudocritical temperature from the peak of the susceptibility and from the inflection point of the chiral condensate. In contrast to the claim in Ref. [68], the three models yield markedly different pseudocritical temperatures when extracted from the two order parameters, although the results coincide in the chiral limit. In this context, our results are consistent with the view in Ref. [69], which suggests that different order parameters can yield significantly different pseudocritical temperatures. Adopting the susceptibility-peak definition of the transition temperature, we numerically verify that in all three holographic models the pseudocritical temperature obeys the universal scaling law Tc-Tc0∼mq1/βδ or Tc-Tc0∼mπ2/βδ. The scaling region in these models can extend below the physical quark mass. These findings support the consistency of soft-wall models with mean-field universality. Moreover, we find that the percentage temperature defined by the 79% [the value is set by the ratio fχ(0)/fχ,max], converges to its chiral-limit value much faster than the pseudocritical temperature, offering a practical observable for determining Tc0.

While the three holographic models share identical critical exponents and scaling functions in the chiral limit, their scaling coefficients span orders of magnitude. Compared with Tc scaling from DSE [9], FRG [70], and lattice QCD [16], models I and II yield a slope, i.e., the scaling coefficient, that is roughly one third as large, a discrepancy that appears to afflict earlier soft-wall constructions as well. It is interesting to see that by softening the scalar potential with certain logarithmic terms and r-dependent coefficients, model III reproduces a Tc scaling in quantitative agreement with the DSE results [9]. This improvement can be attributed to the logarithmic modification in the scalar potential. While models I and III are equivalent at the critical point (sharing the same subleading expansion), the logarithmic term in model III effectively incorporates higher-order interactions that become relevant for finite quark masses, driving the model toward better agreement with QCD phenomenology. These findings might offer valuable guidance for refining future model constructions in the soft-wall AdS/QCD. It is worth noting that due to the complexity of the numerical computations and the high dimensionality of the parameter space, this work does not present a single parameter set that simultaneously reproduces both the chiral condensate and the Tc scaling with high precision. Nevertheless, we anticipate that future optimization strategies, such as those employing machine learning algorithms, will enable more refined fittings. Finally, in this study, we have restricted our attention to the two-flavor chiral critical point. Extending the analysis to the entire critical lines in the quark-mass phase diagram, and examining the resulting consequences for the critical end point within the same holographic framework, constitutes an interesting direction for future work.