Resonant top pair searches at the LHC: a window to electroweak phase transition

The dynamics of electroweak phase transition could have profound consequences for particle physics and cosmology. We study the prospects for the HL-LHC to probe the strong first-order electroweak phase transition (SFOEWPT) regime in the type-I 2HDM. We focus on the Higgstrahlung channel $pp\to ZH/A$ with a resonant top-quark pair final state $H/A\to t\bar t$. We find that the top-quark pair final state renders the largest sensitivity to the SFOEWPT regime, in comparison to the other Higgstrahlung searches already performed by ATLAS and CMS, that focus on the $H/A\to bb$ and $H\to WW$ final states. We also derive the complementarity of the Higgstrahlung searches with other relevant classes of searches at the HL-LHC and compare them with the gravitational wave sensitivity at LISA.


I. INTRODUCTION
The electroweak symmetry appears exact in the early Universe at high temperatures.
However, at temperatures around 100 GeV, the Higgs field develops a vacuum expectation value, spontaneously breaking this symmetry. At this time, the Universe goes through a transition from a symmetric to a broken phase. While the Standard Model (SM) predicts a continuous transition [1], new physics can generate a first-order phase transition. As yet, it is unknown how the electroweak phase transition (EWPT) occurred -whether it was violent or calm; first-order phase transition or smooth crossover.
First-order phase transition is a violent phenomenon that can display relevant consequences for the evolution of the Universe. In particular, this transition could provide the required out-of-equilibrium conditions to generate the baryon asymmetry of the Universe via electroweak baryogenesis [2][3][4][5][6]. If the first-order phase transition was strong enough, it could have generated relic gravitational waves (GW) that might be probed in future GW detectors, such as in the space-based LISA mission [7,8]. This transfiguration in the EWPT profile usually requires novel degrees of freedom close to the electroweak scale, displaying sizable interactions with the Higgs boson [9,10]. Thus, precision measurements for the Higgs sector and the search for new heavy scalars at colliders are also important avenues for probing models which trigger first-order electroweak phase transition. The Higgs pair production pp → hh plays a central role in these studies, as it provides a direct probe to the Higgs potential through resonant and non-resonant searches [11][12][13][14][15]. Thus, this subject provides an exciting research arena on the interface between particle physics and cosmology.
The strength of the phase transition is correlated with the potential upliftment of the true vacuum compared to the symmetric one at zero temperature [16][17][18]. This gauge-invariant property sheds light on the phase transition pattern analytically. In particular, it is possible to derive that the large order parameter, ξ c ≡ v c /T c ≳ 1, favors light scalar masses [10].
This grants extra motivation for resonant searches at the Large Hadron Collider (LHC).
Besides scalar masses under the TeV scale, the analytical structure of the beyond the SM effects on the vacuum upliftment, leading to strong first-order electroweak phase transition (SFOEWPT), ξ c ≳ 1, results in a distinct hierarchy of masses among the new scalar states.
Due to the preference for large mass hierarchy among the scalar modes, it is likely that at least one of the scalar states is above the top-quark pair threshold. As recently shown, the gluon fusion production channel gg → A/H → tt plays a leading role in these phenomenological studies, granted by its large event rate [10]. Another relevant channel is the resonant Higgstrahlung production A → ZH [34], which is enhanced in the preferred scalar mass regime for SFOEWPT, m H < m H ± ≈ m A . However, the current experimental analyses explore the A → ZH searches only through the decays H → bb and H → W W with Z → ℓℓ [35]. The flipped channel H → ZA is also analyzed with A → bb and Z → ℓℓ [36].
For A → ZH or H → ZA channels, the corresponding heavy scalar decay to top-quark pair H/A → tt can also be an important signature for the SFOEWPT parameter space.
Compared to the already explored scalar decays to bb and W + W − , the tt final state covers a different mass spectrum, which can significantly improve the sensitivity of A → ZH and H → ZA channels. In the present work, we scrutinize the sensitivity of the scalar decays to top-quark pair for probing SFOEWPT in the 2HDM. Special attention will be devoted to the Higgstrahlung mode pp → ZH/A at the high luminosity LHC (HL-LHC).
This paper is organized as follows. In Section II, we introduce the two Higgs doublet model. In Section III, we discuss the leading contributions for the 2HDM signature pp → ZH/A with top-quark pair final state H/A → tt. The corresponding HL-LHC sensitivity is derived in Section IV. In Section V, we study the complementarity between collider and gravitational wave experiments to probe the electroweak phase transition profile in the 2HDM. In particular, we scrutinize the relevance of the Higgstrahlung channel with top pair final states H/A → tt with respect to other relevant classes of searches at the HL-LHC. We also contrast the collider with the gravitational wave sensitivity. We summarize in Section VI. Further details on the complementary channel gg → H/A → tt are presented in Section A.

II. TWO HIGGS DOUBLET MODEL
In this work, we consider the CP-conserving 2HDM with a softly broken Z 2 symmetry, where the scalar potential can be written as [37] Expanding around the VEVs, the two SU (2) L doublets can be expressed by where the vacuum expectation values v i are connected to the SM VEV by v 2 1 + v 2 2 = v 2 ≈ (246 GeV) 2 . After electroweak symmetry breaking, the model provides five physical mass eigenstates from the scalar sector: two CP-even neutral scalars h and H, where h is identified as the SM Higgs boson, a CP-odd neutral scalar A, and a pair of charged scalars H ± . The mass and gauge eigenstates are related by the rotation angle β for the charged and CP-odd sectors, tan β ≡ v 2 /v 1 , and by the angle α for the CP-even sector The rotation matrix is given by where s x ≡ sin x and c x ≡ cos x. The charged and neutral massless Goldstone bosons are denoted by G ± and G 0 , respectively.
The physical parameters of 2HDM can be chosen as The parameters t β ≡ tan β and c β−α ≡ cos(β − α) control the coupling strength of the scalar particles to fermions and gauge bosons, displaying critical phenomenological relevance.
Whereas the Higgs-gauge couplings scale as g hV V ∝ s β−α and g HV V ∝ c β−α , the fermion interactions hinge on both t β and c β−α . The fermionic couplings depend on the Z 2 charge assignment in the Yukawa sector. In this study, we focus on the Type-I scenario where all fermions couple solely to Φ 2 . When confronted with current experimental constraints, the alignment limit c β−α → 0 is preferred [38,39]. Furthermore, electroweak precision measurements also put strong constraints on the 2HDM parameter space, which pushes  Between these two terms, there are three leading effects that guarantee the dominance of the 1 Note that we grouped two diagrams with two different types of the propagators in Fig. 1 (c), as they share the same scaling behavior as we will discuss in the following. Similar for Fig. 1 (e). We decompose the signal in its leading contributions: i) loop-induced gluon fusion production gg → ZH (red); ii) b-quark initiated production bb → ZH (blue); and iii) Drell-Yan like contribution qq → Z * → ZH (green). For illustration, we consider the type-I 2HDM with maximally allowed mixing angles, giving the current LHC data: c β−α ≈ 0.3. In addition, we assume m H = 600 GeV, m A = 750 GeV, and t β = 1. gluon fusion channel: i) the gluon fusion channel is driven by the large parton distribution function of initial state gluons; ii) it has larger initial state color factor; and iii) the triangle and box diagrams are enhanced by the sizable top Yukawa coupling, shown respectively in Fig. 1 (a) and Fig. 1 (b). The last term, the Drell-Yan-like mode, generally produces subleading corrections. Its cross-section is hampered by the absence of a resonant scalar mode A → ZH and the dependence with the mixing angle σ ZH DY ∝ c 2 β−α . The suppressed rate becomes apparent even when adopting maximally allowed mixing angles. We depict this scenario in Fig. 2, considering the Type-I 2HDM with c β−α ≈ 0.3. Remarkably, the type-II scenario leads to further depleted Drell-Yan production, as the experimental constraints tend to confine the model parameters further towards the alignment limit, c β−α → 0.
Although resonant production gg → A/H → ZH/A typically results in high rates, the other gluon fusion contributions present relevant effects that need to be included for a robust simulation. In particular, the triangle and box diagrams depicted in Fig. 1 (a) and Fig. 1 (b) display sizable interference effects, which significantly depend on the scalar and pseudoscalar particle widths. To illustrate it, we present in Fig. 3 the parton level distributions 650 700 750 800 850 900 950 1000 ZH results in leading effects, followed by its interference with the box contributions. In general, there is also interference between the signal with the ttZ background. However, as checked numerically, such interference generates only subleading rates for the allowed 2HDM parameter space.

IV. ANALYSIS
In this section, we derive the sensitivity to the Higgstrahlung signals pp → Z(ℓℓ)H(tt) and For any other values of masses and widths obtained in our parameter scan, interpolation will be used to extract the corresponding results. To easily cover all other possible choices of parameters, the generation of resonant and interference terms for the signal is performed separately for contributions with distinct dependence on t β and c β−α . Hence, for any other case, we can scale the result according to the corresponding dependence on t β and c β−α . The dependence of each diagram on t β and c β−α is listed in Tab. I. 3 The procedure described in this paragraph will allow for a more effective scan of the model parameters. In particular, this will be relevant for the broad parameter space scan of the type-I 2HDM performed in Contribution Fig. 1(a) Fig. 1(b) Fig. 1(c) Fig. 1(d) Fig. 1(e) Fig. 1(f) Fig. 1 Fig. 1, where Section V.
In our event analysis, we require three isolated leptons with p T ℓ > 20 GeV and |η ℓ | < 2.5.
We require one charged lepton pair of the same flavor and opposite sign, whose invariant for, which constitute different partitions. We chose the combination that minimizes where  In Fig. 5, we present the resulting upper limit on the signal cross-section for Z(ℓℓ)H(tt) and Z(ℓℓ)A(tt) production at 95% confidence level (CL). The results are shown for several hypotheses of scalar and pseudoscalar widths. 4 We assume the HL-LHC with integrated luminosity L = 3 ab −1 . The binned likelihood analysis is sensitive to the width dependence, leading to weaker results in the augmented Γ H,A /m H,A regime. This is mostly a result of the enhanced negative contribution from the triangle-box interference for larger widths. The large interference effects and broader resonance from large width lead to suppressed signal events, which can be observed in Fig. 3 and Fig. 4. While we present the results in Fig. 5 following a model-independent approach similar to the CMS study in Ref. [46], in Sec. V C we perform a uniform parameter space scan considering the 2HDM in the type-I scenario.

V. ELECTROWEAK PHASE TRANSITION IN THE 2HDM
In this section, we study the complementarity between collider and gravitational wave experiments to probe the phase transition profile in the 2HDM. We first define the one-loop effective potential at finite temperature in Section V A, then discuss important ingredients for calculating the EWPT and gravitational wave sensitivity at space-based experiments in Section V B. Finally, we analyze in Section V C the relevance of the Higgstrahlung channel with top pair final states H/A → tt with respect to other relevant classes of searches at the HL-LHC and also contrast the collider with the gravitational wave sensitivity.

A. Finite Temperature Effective Potential
In order to study the thermal history of the 2HDM, one needs to study the loop-corrected effective potential at finite temperature. It is defined by the addition of the tree-level potential V 0 , the zero-temperature one-loop corrections from the Coleman-Weinberg potential V CW with the respective counter terms V CT , and the one-loop thermal corrections V T . Thus, the final potential can be written as The zero-temperature one-loop correction in the MS have the form [48] where In principle, the tree-level scalar masses and mixing angles are shifted by the addition of the Coleman-Weinberg potential. To make our parameter space scan more efficient, we follow a renormalization prescription that requires these parameters to match their tree-level values [20,49]. We achieve this by adding the counter-terms imposing the following renormalization conditions at zero-temperature where ω tree generically stands for the minimum of the tree-level potential for the ϕ i fields.
Eq. (10) requires that the zero-temperature minimum matches the tree-level value. Similarly, Eq. (11) demands that T = 0 masses and mixing angles do not change with respect to the tree-level numbers.
The one-loop thermal corrections read [50] where f , V T , and V L indicate respectively the sum over fermions, transverse gauge bosons (W T , Z T ), and longitudinal modes of gauge bosons and scalars (W L , Z L , γ L , Φ 0 , Φ ± ). The second line of Eq. (12) corresponds to the daisy contributions, following the Arnold-Espinosa scheme [20,50]. Finally, the thermal functions J + and J − are given by

B. Thermal History and Gravitational Waves
The rich structure of the 2HDM potential grants distinct phase transition processes. For successful baryogenesis, it is critical to prevent the baryon number generated during the phase transition from being significantly washed out. This imposes the electroweak phase transition to be strong first-order [51,52] where T c is the critical temperature and v c is the critical Higgs VEV obtained when the would-be true vacuum and false vacuum are degenerate.
The occurrence of phase transition depends on the false to true vacuum tunneling rate, which is given by [53,54] Γ(T ) ≈ T 4 S 3 2πT where S 3 denotes the three-dimensional Euclidean action that can be expressed as To obtain the nucleation rate, we need to calculate the scalar field ϕ bubble profile by solving the bounce equation with boundary conditions lim r→∞ ϕ(r) = 0 and lim r→0 dϕ(r) dr = 0.
The nucleation onsets at temperature T n , where one bubble nucleates per horizon volume.
This condition can be approximated for EWPT as S 3 (T n )/T n ≈ 140 [56].
We can now define the quantities β and α that describe the dynamical properties of the phase transition, which can be used to model the strength of stochastic gravitational wave signals [7,57]. The inverse time duration of phase transition β/H n is defined as where H n is the Hubble constant at T n . The other important parameter α characterizes the ratio between the latent heat released during the phase transition (ϵ) with respect to the radiation energy density (ρ rad ), α ≡ ϵ/ρ rad . These parameters are given by with g ⋆ being the number of relativistic degrees of freedom in the thermal plasma. Here, ∆ denotes the difference between true and false vacua.
We are able to estimate the sensitivity of gravitational wave experiments, adopting the signal-to-noise ratio (SNR) measure [57] where T is the duration of the mission and Ω Sens is the sensitivity profile of the particular GW experiment [8]. In our analysis, we consider the space-based LISA experiment, assuming T = 5 years and SNR > 10 to characterize signal detection [57].

C. Probing EWPT in the 2HDM
In this section, we examine the sensitivity of the resonant top pair searches to the pa- The theoretical and experimental limits to the model are implemented with ScannerS v2.0.0 [58,59]. With this package, we require constraints from perturbative unitarity [60][61][62], boundness from below [63], vacuum stability [64,65] [35,36]. In Fig. 6, we present the projection for these analyses to the HL-LHC luminosity L = 3 ab −1 . The projected upper bounds on the cross-section for corresponding processes are compared with the crosssection times the branching fraction of each parameter point obtained from ScannerS [59].
Both the bottom and W -boson final states will provide relevant sensitivity to SFOEWPT.
Interestingly, the sensitivity of these channels is mostly restricted to the m H,A < 350 GeV  pp → Z(ℓℓ)A(tt) analyzes, as described in Section III. We observe that the Higgstrahlung searches with tt final state largely extend the sensitivity to SFOEWPT.
In the left two panels of Fig. 7, we present a detailed comparison among the bb, W W , and tt final states for the pp → ZH/A channel. The number in each region indicates the fraction of points in our uniformly random scan that can be excluded by the corresponding searches. They are the fraction of points currently allowed, under theoretical and experimental constraints, that will be probed at the HL-LHC. The first panel focuses on the SFOEWPT regime with ξ c > 1 and the second one represents the points that can display gravitational wave signals at LISA with signal-to-noise ratio SNR > 10. We observe from these Venn diagrams that the Higgstrahlung channels with top pair final states pp → ZH(tt) and ZA(tt) cover a large portion of the allowed parameter space. Therefore, the inclusion of this channel in forthcoming experimental analyses is strongly motivated.
In the right two panels of Fig. 7 searches. The results from these other studies were obtained from Ref. [10]. We added further details for the gg → H/A → tt searches in the Section A. The HL-LHC will be able to cover ≈ 80% of the available ξ c > 1 parameter space in the Type-I 2HDM scenario.
While the considered channels present relevant complementarities, the Higgstrahlung mode and gluon fusion scalar production with subsequent decay to top-quark pair (or charged Higgs production with fermionic decay H ± → tb) result in leading sensitivities.

VI. SUMMARY
The thermal history of electroweak symmetry breaking in the Universe could have pro-   shape from the signal-only sample changes into a bump-dip shape when accounting for the interference with the QCD top-quark pair production. Thus, it is necessary to include these effects to generate robust phenomenological modeling for the signal sample.
Current ATLAS and CMS analyses for resonant top-quark pair production account for the interferences with the SM tt production [46,85]. In particular, the more recent CMS