Since the discovery of a new scalar particle with a mass of about 125 GeV by the ATLAS and CMS collaborations at the Large Hadron Collider (LHC) in 2012 [1−3], its observed properties have been consistent, within current theoretical and experimental uncertainties, with expectations for the Standard Model (SM) Higgs boson. Nevertheless, such a state can also be accommodated in a wide range of theories beyond the Standard Model (BSM). Although no direct evidence of BSM physics has yet emerged at the LHC, the precision of Higgs coupling measurements and the constraints from searches for new resonances still leave substantial parameter space for new physics (NP) interpretations. Many BSM scenarios predict an extended Higgs sector, making it an essential task for LHC Run 3 and future experiments to determine whether the discovered scalar boson is part of a richer Higgs structure. Notably, these extended frameworks may feature not only heavier Higgs resonances but also lighter scalar states. The search for such light additional Higgs bosons is therefore of great importance for uncovering the underlying mechanism of electroweak symmetry breaking and for revealing possible signs of NP.

The current mass measurement of the 125 GeV Higgs boson is GeV [4]. To investigate its properties in detail and to assess whether it is the only fundamental scalar, the LHC has performed precision measurements across multiple decay channels. In particular, the ATLAS and CMS collaborations have observed the Higgs boson in various bosonic and fermionic modes [3−6], establishing its spin-parity quantum numbers and measuring its production cross sections. In the SM, the Higgs couples directly to W and Z bosons and indirectly to photons via loop effects, making the , , and channels particularly sensitive for experimental studies. The measured signal strengths in these channels are [3, 5−8]

1

Owing to the Yukawa interactions, the Higgs boson couples most strongly to third-generation fermions, leading to dominant decays into and . The corresponding measured signal strengths are reported in [3, 6, 7, 9−11].

2

In addition to the established Higgs boson at 125 GeV, several experimental studies have pointed to the possibility of a lighter scalar resonance near 95 GeV, renewing interest in extended Higgs sectors. Searches for additional low-mass Higgs states have been carried out at LEP [12−14], the Tevatron [15], and the LHC [16−22]. In such searches, the experimental signal strength μ is defined relative to that of a hypothetical SM Higgs boson of the same mass, even if no SM Higgs exists at that mass. Specifically, the experimental signal strengths at are:

3

where ϕ denotes a hypothetical BSM scalar, and is the cross section for a SM-like Higgs of the same mass [23, 24].

Analyses by the CMS Collaboration in the diphoton channel at and 13 TeV, using datasets of 19.7 fb and 35.9 fb , revealed a local excess near 95.3 GeV with a significance of approximately 2.8σ [16, 17]. Using the full Run 2 dataset and improved event classification, a more recent CMS study reports a local excess at 95.4 GeV with a significance of 2.9σ [22], which is broadly compatible with ATLAS results based on 80 fb of data [19]. The combined ATLAS+CMS signal strength can be expressed as

4

Furthermore, LEP reported a 2.3σ local excess in the channel, consistent with a scalar particle with a mass of approximately 95 GeV [13]. The corresponding signal strength is

5

From a theoretical perspective, various BSM frameworks have been proposed to account for the observed excesses near 95 GeV [25−29]. Previous studies [30, 31] have shown that the diphoton rate can be several times larger than the SM prediction for a scalar of the same mass in the Next-to-Minimal Supersymmetric Standard Model (NMSSM). Similarly, the Two-Higgs-Doublet Model extended with an additional real singlet (N2HDM) has been extensively explored as a possible explanation for the CMS and LEP excesses [32, 33]. Other scenarios, such as the SSM with CP conservation or violation [34−36], singlet-extended or radiative Higgs models [37], and gauge-extended frameworks like the model [38], have also been investigated as viable possibilities.

Despite the success of the SM in describing particle interactions, it still leaves several fundamental questions unanswered, such as the nature of dark matter, the origin of neutrino masses, and the hierarchy problem. These open issues have long motivated the exploration of BSM physics. Supersymmetry (SUSY), once regarded as one of the most elegant and theoretically appealing extensions, offers a natural framework to address these problems through its fermion–boson symmetry. However, the persistent null results in supersymmetric particle searches over the past few decades have substantially reduced its experimental viability. The reported 750 GeV excess in 2015 initially generated considerable excitement as a potential signal of new physics (NP), but the 2016 data did not confirm this excess [39, 40], further decreasing overall interest in supersymmetric frameworks. Meanwhile, attention shifted toward dark matter, another compelling aspect of NP. Although significant progress has been made in dark matter searches, stringent direct-detection limits have excluded most of the viable parameter space [41, 42], leaving only narrow regions open for exploration. Consequently, interest has increasingly turned to non-supersymmetric extensions, which can offer simpler and more economical explanations. Specifically, the observation of a scalar lighter than 125 GeV would not necessarily point to a supersymmetric origin but could instead arise naturally in certain non-supersymmetric gauge extensions.

In the VLFM, the right-handed and left-handed fields of vector-like fermions have the same transformation properties under the gauge group of the SM, allowing them to acquire gauge-invariant mass terms [43]. Consequently, they can evade constraints from Higgs production cross sections and from direct searches at the LHC. The mixing between vector-like fermions and SM fermions induces corrections to the couplings of fermions to the W, Z, and Higgs bosons [44]. As a result, the Glashow–Iliopoulos–Maiani (GIM) mechanism is generically violated, leading to flavor-changing neutral currents (FCNCs) at tree level. Moreover, vector-like fermions can introduce new CP-violating sources, contributing to the electric dipole moments (EDMs) of leptons, quarks, and the neutron [45]. Finally, the introduced vector-like quarks can help alleviate the gauge hierarchy problem in the SM. The VLFM accordingly offers rich phenomenology.

The remainder of this paper is organized as follows. In Sec. II.A, we introduce the main components of the VLFM, while Sec. II.B presents the relevant formulae and mass matrices required for the one-loop corrections to the CP-even Higgs mass-squared matrix. In Sec. III, we derive the decay widths and corresponding signal strengths of the SM-like Higgs boson with a mass around 125 GeV, including the channels , , and . Sec. IV is devoted to the analysis of a lighter Higgs-like scalar near 95 GeV, where we present its dominant decay modes and together with the corresponding signal strengths. In Sec. V, we present a numerical analysis over the relevant parameter space, and Sec. VI summarizes our main results and conclusions. For completeness, some lengthy analytical expressions are collected in Appendix A.

At the LHC, the dominant production mode of the Higgs boson is gluon fusion. In the SM, this process proceeds at leading order (LO) via a one-loop diagram involving virtual top quarks, while subleading contributions from bottom quarks are strongly suppressed. The inclusive cross section has been computed up to next-to-next-to-leading order (NNLO) in QCD [53], which increases the LO prediction by about 80-100%. Because the gluon-fusion rate is loop-induced, it is highly sensitive to new physics (NP): any heavy particle that couples significantly to the Higgs can modify the amplitude and hence the production rate. In the VLFM, the LO decay width for the process is given by [54−59]

42

where . The explicit expressions for the Yukawa couplings and are given by

43

with

44

Here, label the SM up- and down-type quark generations, and , respectively, while labels the vector-like quarks introduced in the VLFM extension. denotes the Higgs mixing matrix. The Yukawa couplings and describe the Higgs interactions with the vector-like fermions, which induce mixing with the SM quarks.

The form factor in Eq. (42) is defined as

45

with

46

The Higgs diphoton decay is mediated by loop diagrams. In the SM, the leading-order (LO) contributions arise from loops of charged gauge bosons, , and top quarks. Within the -VLFM framework, additional effects are induced by the vector-like fermions , , and that mix with the third-generation fermions. The decay width can be written as

47

where the sum runs over the charged fermions . Here, denotes the color factor ( for quarks, for leptons), and is the corresponding electric charge.

The Higgs-fermion couplings, , are obtained by diagonalizing the extended fermion mass matrices. In the leptonic sector, the coupling to the mass eigenstate is

48

with

49

The coupling of the Higgs boson to W bosons is determined by the Higgs mixing matrix.

50

The loop function for spin-1 particles is given by:

51

For the neutral CP-even Higgs boson with a mass of approximately , the decay modes and are kinematically allowed. After summing over all kinematically accessible final states of the off-shell vector bosons, the corresponding partial decay widths are given by [60−63].

52

Throughout, we adopt the abbreviations and , with denoting the Weinberg angle. Furthermore, e denotes the electromagnetic coupling constant.

The coupling is:

53

and the form factor is given as

54

The partial decay width of the neutral CP-even Higgs boson of mass into a fermion pair is given at LO by [64, 65]

55

Both ATLAS and CMS have observed a mild excess in Higgs-boson production with decay to the diphoton channel, relative to the SM expectation. The signal strength for a given production mode and decay channel, normalized to the corresponding SM prediction, is defined as [66]:

56

where ggF and VBF denote gluon-gluon fusion and vector-boson fusion, respectively. When normalized to their SM values, the NP-to-SM ratios of the Higgs production cross sections are given by the corresponding ratios of the relevant partial and total decay widths.

57

Here, the total decay width of the 125 GeV Higgs boson within the NP framework is given by

58

where contributions from rare or invisible decay channels are neglected, and denotes the total decay width of the SM Higgs boson. Using Eqs. (56) and (57), the signal strengths for the Higgs decay channels in the VLFM can be expressed as

59

with and .

The signal strengths of the 95-GeV scalar excess in the VLFM are defined as the ratios of the corresponding production cross sections and branching ratios to their SM expectations:

60

with

61

In the narrow-width approximation, the signal strengths, normalized to the SM predictions, are given by Eq. (60). "NP" refers to the predictions in our BSM model ( VLFM), while "SM" refers to the hypothetical SM Higgs with the same mass [30]. denotes the total decay width of the 95 GeV scalar in the VLFM, while corresponds to the total width of an SM Higgs boson with the same mass. The partial decay widths are evaluated analogously to Eqs. (42), (47) and (55), with replaced by .

In our numerical analysis, we assume that the lightest CP-even Higgs boson has a mass around , while the next-to-lightest CP-even Higgs boson is identified with the SM-like state at . The masses and associated signal strengths of these states are then discussed within the VLFM framework, subject to the following phenomenological constraints:

1. The second-lightest CP-even Higgs boson is taken to be the observed Higgs boson, with mass as reported by the PDG [4].

2. The third-generation fermion masses are required to match their SM values after mixing with the vector-like fermions [4].

3. For the vector-like fermions in the VLFM, we take them to be at the TeV scale [67−69].

4. Motivated by previous analyses of extensions with vector-like fermions, we allow the mass to be at the TeV scale in our numerical study, as such values have been shown to be phenomenologically viable in Refs. [70, 71].

These constraints are imposed in the parameter scan to ensure consistency with current experimental data. For the subsequent numerical analysis, we adopt the following representative benchmark scenario:

62

Here , , and denote the Yukawa couplings of the top quark, bottom quark, and tau lepton, respectively, defined as , , and .

The benchmark points in this work are not chosen arbitrarily but are determined by the model structure and experimental constraints. In the VLFM model considered in this paper, the CP-even neutral scalar sector is described by a mass matrix, and it must simultaneously accommodate a scalar consistent with the SM Higgs and a light scalar around 95 GeV [4]. These two mass conditions impose stringent constraints on the parameter space: the parameters directly related to the scalar masses are confined to relatively narrow intervals, thereby significantly reducing their freedom.

On this basis, we further combine the Higgs signal strength measurements at 125 GeV and the excess around 95 GeV and perform a joint fit across multiple decay channels. This procedure imposes additional experimental constraints on the parameter space, further restricting the region compatible with all observations.

In contrast, parameters that are not directly related to the masses of these two scalars but mainly affect their decay properties have relatively broader allowed ranges and retain some degree of variability. Although the benchmark point we present corresponds to the optimal fit obtained from the analysis, these parameters can still vary within certain ranges while maintaining a good description of the experimental data.

The benchmark point given in this work corresponds to the best-fit point obtained from the analysis. Therefore, the apparent concentration of these parameter values mainly originates from the compression of the parameter space required to simultaneously satisfy the 95 GeV and 125 GeV experimental constraints, rather than from any ad hoc choice. In addition, the constrained parameter ranges are consistent with the typical physical scales expected in the model.

Using this benchmark, the VLFM predictions are tested against Higgs data via a analysis, with points having within the range of the best-fit value considered favored. The is defined as

63

Here, denotes the experimental value of the corresponding observable, while represents its theoretical prediction. The term stands for the total uncertainty on the observable, including statistical and systematic experimental errors as well as theoretical uncertainties. The fit includes the Higgs boson mass, its signal strengths in the , , , , and channels, and the potential excess at in the and final states. By accounting for multiple experimental constraints from diverse physical processes, the parameter space obtained through the analysis is more tightly constrained and reliable.

The parameters and represent the Yukawa couplings that link the SM-like and vector-like down-type quarks. Specifically, couples the SM right-handed down quarks ( ) to the vector-like left-handed components ( ), while couples the left- and right-handed components of the vector-like quarks ( and ). After the singlet scalars S and acquire vacuum expectation values (VEVs) and , these couplings contribute to the mass-mixing structure of the extended quark sector, affecting both the quark masses and their interactions with the Higgs and bosons. We therefore treat them as scan parameters in our analysis. Figure 1 shows the results of the parameter scan with the benchmark set , , , , , , , and . The scan ranges are and . The best-fit point ( ) is indicated by a black dot, and the , , and confidence regions are represented by ■, ♦, and ▲, respectively.

As shown in Fig. 1(a), the allowed region in the plane forms a narrow arc-shaped band, indicating a strong correlation between the two parameters. The viable points cluster in the ranges and , with the best-fit point at approximately (0.391, 0.310). The (■) region lies along the outer side of the arc and is somewhat shorter, while the (♦) and (▲) regions lie inward along the same trajectory and cover progressively larger portions. Within this correlated region, smaller values are more frequently found at relatively larger , indicating a mild preference for larger .

In Fig. 1(b), the points form a narrow, descending band, revealing a clear anticorrelation between and . The lowest- points are shown as ■, transitioning to ♦ and then ▲ toward the band edges. This pattern implies that when one signal strength increases, the other must decrease accordingly. As a result, they cannot vary independently and are strongly constrained by precise Higgs measurements.

In Fig. 1(c), a diagonal band of allowed points is observed, indicating a clear positive correlation between the two Higgs masses. As increases from about 94.1 to 94.9 GeV, rises in parallel from roughly 125.1 to 125.6 GeV. The band is relatively broad, suggesting that both masses can shift together within a finite range while remaining consistent with the constraints. The best-fit point, indicated by the black dot, is located at approximately and , a result that is particularly significant, as it simultaneously aligns with the 95 GeV scalar excess and the observed 125 GeV Higgs boson.

The parameter governs the mixing between the vector-like lepton and the tau lepton, while primarily determines the mass of the vector-like lepton. Figure 2 presents the distribution obtained from a scan over and within and , using the benchmark parameters , , , , , , , and . The best-fit point ( , •) is indicated, and the (■), - (♦), and - (▲) regions highlight the parameter space favored by the fit.

In Fig. 2(a), the allowed parameter space forms a tilted, wedge-like strip extending from the upper left to the lower right of the ( , ) plane, indicating a clear inverse correlation between the two couplings. The best-fit point lies within the compact 1σ region, while the 2σ and 3σ regions expand outward along the same orientation. This stratified pattern suggests that only correlated variations in and are compatible with the Higgs signal strength measurements, thereby imposing stringent constraints on the Yukawa structure in the lepton sector.

In Fig. 2(b), we map the viable points in the plane spanned by the signal strength of and the mass of the 125 GeV Higgs boson. The results exhibit a series of nested, semicircular layers centered on , reflecting the strong experimental preference for a Higgs-boson mass consistent with that of the SM-like Higgs boson. The region (■) is tightly concentrated near this central value, while the (♦) and (▲) regions form successive shells that extend outward, indicating a gradual deterioration of the fit quality as the mass departs from the preferred point. The best-fit point, marked by the black dot at , lies extremely close to the PDG world average of , demonstrating excellent agreement between the model prediction and current Higgs measurements. This pattern indicates that, with the Higgs mass fixed near its measured value, the signal strength can vary only within a very narrow range because of strong experimental constraints.

In Fig. 2(c), we show the correlation between the diphoton signal strength of the 95 GeV scalar, , and its mass . The distribution of points forms an inclined band, indicating a positive correlation: increases from approximately 94.1 GeV to 95.0 GeV as increases. The global best-fit point is located at the lower boundary of this region. The symbols and their colors represent the CL, with ■ denoting the most favored region, followed by ♦ and then ▲. This correlation reflects intrinsic model constraints that link the diphoton rate and the mass of the 95 GeV scalar.

Similarly, in the up-type quark sector, the parameters and characterize the Yukawa interactions responsible for the coupling between the SM top quark and a vector-like fermion. They determine the strength of this interaction, thereby influencing the structure of the up-type quark mass matrix and the mixing pattern in the extended quark sector. Both parameters are included as scan variables in the numerical analysis. In Fig. 3, with , , , , , , , and fixed, we perform a parameter scan over and across and . The best-fit point is found at , while the surrounding regions correspond approximately to the ( ), - ( ), and - ( ) confidence levels.

Figure 3(a) displays the fitted parameter distribution in the plane. The points align along a well-defined descending strip, reflecting a compensating relation between the two couplings. The global best-fit point, marked by the black dot at , lies within the compact region, which is successively enclosed by the broader and confidence regions. This nested structure highlights the statistical hierarchy of the fit and demonstrates that Higgs observables can be satisfied only through a tightly correlated adjustment of and .

Figure 3(b) shows the correlation between and . Compared with Fig. 3(a), the distribution here is noticeably broader, with the outer regions extending into a long descending tail. The best-fit point lies near the center of the innermost domain, and any deviation from this point quickly worsens the fit. This pattern indicates that, although some flexibility exists, the signal strength remains tightly correlated with .

Figure 3(c) presents the distribution of points in the plane of the diphoton signal strength versus its mass ( , ). The points trace a tilted, semicircular arc: is constrained within a narrow range of 124.8-125.4 GeV, while varies close to the SM expectation, spanning 0.95-1.10. It reveals a compact ■ core, successively surrounded by ♦ and ▲ layers that form nested semicircular envelopes, with the outer layer spanning the widest arc. The global best-fit point at ( lies well inside the core. A mild anticorrelation emerges along the arc, with decreasing slightly as increases, indicating constrained joint variations rather than independent scatter.

In Fig. 3(d), the allowed points cluster into a diagonal band extending from the upper-left to the lower-right of the ( , ) plane. The strip exhibits a compact ■ core is enclosed by ♦ and then ▲ regions, forming a slender corridor of viable points. A clear anticorrelation is observed: larger values of correspond to smaller , indicating that the two observables are tightly constrained by the fit. The best-fit point, ( , lies near the upper boundary of the ■ core, highlighting a preference for a slightly heavier mass accompanied by a reduced signal strength.

In the VLFM, and denote the VEVs of the singlet scalars S and , respectively. These VEVs break the extra gauge symmetry and generate masses for the new gauge boson as well as the vector-like fermions through their Yukawa interactions. They also play an important role in determining the scalar mass spectrum. Therefore, we take and as key parameters in our numerical analysis. In Fig. 4, we perform an extended scan over the scalar VEVs, and , while keeping , , , , , , , and fixed. The resulting distribution is shown, with the best-fit point at (•) and the , - , and - confidence regions indicated by ■, ♦, and ▲, respectively.

Figure 4(a) displays the viable region in the plane. The points align along a narrow diagonal band, revealing a strong positive correlation between the two VEVs. The distribution is coded by using both color and shape, with a compact ■ core, surrounded by ♦, and ▲ outer layers, corresponding to increasing values. The best-fit point, , lies near the center of the innermost region, representing the most statistically favored vacuum configuration.

Figure 4(b) depicts the correlation between the Higgs signal strength in the channel and the mass of the lightest scalar, . The points trace a curved band with a negative slope, in which larger values correspond to smaller . The best-fit point, , lies near the densest region. The point distribution reflects the fit quality: points near the center of the band agree better with the data, whereas those farther away correspond to poorer fits along the same trend. Even small deviations of from unity induce noticeable shifts in , indicating that the light-scalar sector is highly sensitive to precision Higgs measurements.

Figure 4(c) shows the distribution of points in the plane. The values of span , whereas the SM-like Higgs mass is tightly constrained to a narrow window around . The points exhibit a layered structure rather than a uniform spread: a compact ■ core is surrounded by a ♦ band and further enclosed by a ▲ envelope, delineating progressively less-favored regions of parameter space. The global best-fit point is located at , lying within the innermost region. This pattern highlights that the Higgs mass is tightly constrained, in contrast to the greater flexibility of the diphoton rate.

In the VLFM, denotes the gauge coupling of the additional symmetry, while the parameter quantifies the gauge mixing between hypercharge and . In Fig. 5, rather than fixing and as in the previous analyses, we allow them to vary over the broader ranges and , keeping all other parameters fixed: , , , , , , , and . The resulting point distribution is shown in terms of the values, with the best-fit point at (indicated by •). The confidence regions are represented by ■ ( , ), ♦ ( - , ), and ▲ ( - , ).

Figure 5(a) presents the correlation between and the Higgs signal strength . The scanned points form a positively correlated band, indicating that larger generally enhances . Fit quality is encoded by color and shape: the most-favored points (■) cluster in the lower-left region of the band, while less-favored points (♦ and ▲) lie farther along the same trend.

Figure 5(b) plots versus . The points lie within a horizontally elongated band with a slight negative slope: those in the region (■) are concentrated at the lower edge of the band, while the (♦) and (▲) regions extend upward. This pattern highlights the sensitivity of to gauge mixing effects.

Figure 5(c) shows the relationship between the masses of the two CP-even Higgs bosons, and . The points are confined to a narrow diagonal strip, consistent with a scenario in which one scalar has a mass near 95 GeV, aligning with the potential excess, while the other corresponds to the observed Higgs boson at 125 GeV. The fit quality is indicated by a color gradient along the diagonal: the blue area encompasses the global best-fit point at ( , ) (94.17,125.20) GeV, suboptimal points appear in green, and poorer fits in red. This blue-to-red gradient clearly illustrates the variation in and underscores the excellent agreement of the model predictions with the Higgs data.

In the parameter ranges corresponding to Fig. 4, GeV and GeV; in Fig. 5, and . Since , the correction is dominated by the suppression factor . For the representative parameter choices used in Figs. 4 and 5 ( ), the correction is at the level.

In this work, we perform a systematic study of the VLFM. The gauge symmetry of this framework is extended to . Compared with the SM, the particle content is enlarged by three generations of right-handed neutrinos ( ) and two singlet Higgs fields (ϕ and S). In addition, the model introduces one generation of vector-like quarks, a vector-like lepton, and a vector-like neutrino. With significantly fewer free parameters than supersymmetric extensions, the VLFM provides a theoretically consistent and economical framework for exploring new physics (NP) beyond the SM.

In the scalar sector, the CP-even components of one Higgs doublet (H) and two Higgs singlets (ϕ, S) mix to form a mass-squared matrix. The lightest eigenstate can account for the 95 GeV excess reported at the LHC, while the second-lightest state requires one-loop corrections to reproduce the observed 125 GeV Higgs mass. Using the Higgs signal strengths measured in the , , , , and channels by ATLAS and CMS, we perform a numerical fit to the model. Our results show that the predictions of the VLFM framework are in better agreement with the experimental data than those of the SM. Viable parameter points exist within the confidence level of a fit, simultaneously accommodating both the 95 GeV excess and the 125 GeV Higgs boson. The gauge couplings , , the singlet VEVs , , and the new Yukawa couplings are found to be tightly constrained and highly sensitive to the fit results.

Furthermore, within the VLFM framework, an inert right-handed neutrino can satisfy constraints from dark-matter direct-detection experiments, and the model may additionally provide a new avenue to address the observed lepton-flavor universality violation at tree level. In future work, we will further investigate the VLFM to determine its viable parameter space.

Here, we use the down-type quark as an example and present selected analytic expressions for illustration.

A1

with the auxiliary functions defined as

A2