Probing Quartic Higgs Self-Interaction

The Higgs self-interactions play a crucial role for exploring the underlying mechanism of electroweak symmetry breaking and the nature of the phase transition involved. In this article, we propose to probe the quartic Higgs self-interaction at lepton and hadron colliders, via the di-Higgs productions. We analyze the contributions of the quartic Higgs coupling, including the renormalization of the cubic Higgs coupling and the modification of the $VVhh$ form factor, to the vector-boson-fusion and the vector-boson associated di-Higgs productions at one-loop level. Such an effect is independent of the choice of gauge-fixing, if the quartic Higgs coupling is decoupled from the other ones in the contexts considered. Notably, a combination of these two di-Higgs productions is important for optimizing the collider sensitivities to probe the quartic Higgs coupling. With this guideline, we explore the ILC and CLIC sensitivities, and find that the ILC has a potential to measure the quartic Higgs coupling, normalized by its SM value, with a precision of $\sim \pm 25$ (500 GeV, 4 ab$^{-1}$ + 1 TeV, 2.5 ab$^{-1}$) and $\sim \pm20$ (500 GeV, 4 ab$^{-1}$ + 1 TeV, 8 ab$^{-1}$), at $1\sigma$ C.L., after marginalizing the cubic Higgs coupling in the $\chi^2$ analysis. The dependence on the renormalization scheme of the cubic Higgs coupling is discussed.


I. INTRODUCTION
The Higgs self-interaction is one of the most important targets for experimentalists to measure at colliders. In the standard model (SM), the Higgs potential V SM ¼ −μ 2 H † H þ λðH † HÞ 2 is fully determined by the electroweak scale v ¼ 246 GeV and the Higgs mass m h ¼ 125 GeV, with λ ¼ m 2 h =2v 2 and μ 2 ¼ m 2 h =2. The cubic and quartic Higgs couplings are then completely fixed, For many reasons, new physics may enter the Higgs potential, driving the electroweak phase transition (EWPT) and yielding a deviation of the Higgs self-couplings from the SM prediction. In a general context, such a deviation can be parametrized as with κ 3 and κ 4 being free parameters. Pinning down the Higgs self-couplings with precision therefore is vital for probing the underlying physics and the nature of EWPT. The measurements of the cubic Higgs coupling via various di-Higgs productions have been extensively studied so far. At hadron colliders, the main channels include gluon-fusion production, vector boson-fusion (VBF) production, top pair-associated production, and vector bosonassociated (VBA) production. At lepton colliders, the dominant channels are the Z boson-associated production and the VBF production. The LHC has no sensitivity to the SM cubic Higgs coupling yet. But, the high-luminosity LHC, say, L ¼ 3 ab −1 @14 TeV, is expected to be able to probe it with a precision of ∼Oð1Þ in the gluon-fusion production [1,2], with an improvement of earlier analyses (see, e.g., Refs. [3,4]). At a future 100 TeV hadron collider (for discussions at 27 TeV, see, e.g., Ref. [5]), the cubic Higgs coupling could be measured with a higher precision. For example, in the gluon-fusion channel, the cubic Higgs coupling could be measured with a precision of percent level [6,7]. The VBF production is found to be not quite sensitive [8,9]. The analysis for the top pair-associated production [10,11] and VBA productions [12] at a future hadron collider are still absent. As for the lepton colliders, the International Linear Collider (ILC) is able to measure the cubic Higgs coupling with a precision of 27% in the Zhh production at 500 GeV with L ¼ 4 ab −1 and a precision of 14% in the ννhh production at 1 TeV with L ¼ 2.5 ab −1 [13]. The Compact Linear Collider (CLIC) is able to measure the cubic Higgs coupling with a precision of 54% in the ννhh channel, with L ¼ 1.5 ab −1 data at 1.4 TeV, and 29%, with L ¼ 2 ab −1 data at 3 TeV [14].
To fully pin down the Higgs potential, we also need to measure the quartic Higgs coupling. The traditional wisdom for this is to measure the tri-Higgs productions. However, such measurements are known to be difficult, even at a future 100 TeV hadron collider [15], due to the tiny cross section of tri-Higgs production and its weak dependence on the quartic Higgs coupling. The recent studies on the tri-Higgs productions in the most promising decay channel bbbbγγ showed that the sensitivity to probe κ 4 in the high-luminosity phase of the future hadron collider, say, 30 ab −1 @100 TeV, is ∼Oð10Þ [16,17] (for studies on the tri-Higgs searches in different decay channels, see Refs. [18,19]). This motivates the proposal in this article, say, to probe the quartic Higgs coupling via its loop corrections to the di-Higgs productions. We expect a combination of the di-Higgs and tri-Higgs measurements in the future to improve the precision of measuring the quartic Higgs coupling.
For the di-Higgs productions at colliders, there are two types of one-loop effects involving the quartic Higgs coupling. 1 Both of them are independent of the choice of gauge fixing. The first type is the renormalization of the cubic Higgs coupling λ 3 2 , which is universal for different di-Higgs processes. The rest of the diagrams belong to the second type. They are irreducible and finite, yielding nontrivial corrections to the form factor of the relevant vertices such as VVhh. The two types of diagrams are reminiscent of the self-energy and the vertex corrections induced by the cubic Higgs coupling in the single Higgs production [20,21]. But, there exists a generic difference. The one-loop correction of the quartic Higgs coupling to the cubic Higgs coupling is logarithmically divergent. Its renormalization necessarily introduces a renormalizationscheme dependence on the interpretation of the experimental constraints for the cubic Higgs coupling.
A full treatment of these di-Higgs productions at oneloop level needs to embed the κ scheme, essentially a parametrization of new physics corrections to the Higgs self-couplings, into an effective field theory (EFT) for the Higgs boson (for a review, see, e.g., Ref. [22]), and then take into account the loop effects from all relevant particles. Here, the EFT could be either the SM EFT, in which new particles get decoupled at a high-energy scale, or the Higgs Effective Field Theory (HEFT), which is known to describe the IR limit of some composite Higgs models (for a review, see, e.g., Ref. [23]), dilaton constructions [24,25], the SM extension with a nondecoupling heavy singlet scalar [26], etc. In these contexts, the quartic Higgs coupling is generally decoupled from other couplings relevant to the di-Higgs productions. In the HEFT, with its potential given by VðhÞ ¼ P n a n ðh=vÞ n , this feature is generic. In the SM Effective Field Theory (SMEFT), the quartic Higgs coupling becomes decoupled as long as more than one higherdimensional operator is turned on. 3 Interestingly, we observe that the one-loop diagrams with no quartic Higgs coupling involved (the summation of which is expected to be independent of gauge fixing and to involve the SM couplings and κ 3 only), though interfering with the tree-level κ 3 diagrams and the one-loop κ 4 diagrams, yield a next-to-leading-order (NLO) impact only for both the κ 3 and κ 4 sensitivity analysis at lepton colliders after a proper renormalization for λ 3 . So, we will ignore such diagrams below. 4 The QCD loop diagrams may yield nontrivial effects for the analysis at hadron colliders. In this paper, for a given di-Higgs process, we assume a universal QCD K factor, which is independent of the corrections of the Higgs self-couplings.
The rest of the paper is organized as follows. In Sec. II, we will calculate the one-loop effects of the quartic Higgs coupling in renormalizing the cubic Higgs coupling and in correcting the VVhh form factor. We will also discuss how to extract the κ 4 sensitivity in a way that is less dependent on the λ 3 renormalization scheme. The numerical calculations of the VBF and VBA di-Higgs productions at both lepton and hadron colliders are presented in Sec. III. We will analyze the sensitivities of the di-Higgs probe to the quartic Higgs coupling at the ILC and CLIC in Sec. IV. We will conclude in Sec. V. 1 Unlike other di-Higgs productions, the gluon-fusion one does not involve the quartic coupling until the two-loop level. But we will not specify this subtlety below, upon the understanding. 2 The quartic Higgs coupling also renormalizes the Higgs mass. But it can be fully resolved by the physical Higgs mass. 3 For discussions on the SMEFT phenomenology with O 6 turned on, see, e.g., [27]. 4 Though a quinary Higgs coupling may appear in the BSM physics often, it has no contributions to the di-Higgs production at one-loop level, except renormalizing the cubic Higgs coupling. In that case, the effects of the quinary Higgs coupling can be fully absorbed by the counter-term.

II. ONE-LOOP EFFECTS OF THE QUARTIC HIGGS COUPLING
The one-loop effects of the quartic Higgs coupling include (i) renormalizing the cubic Higgs coupling and (ii) modifying the form factor of the relevant vertices. The relevant Feynman diagrams are shown in Figs. 1-3, respectively. As discussed above, in general, we can assume that the quartic Higgs coupling is decoupled from other couplings involved in the di-Higgs productions, including the cubic Higgs coupling and the quartic couplings involving both Higgs and Goldstone bosons (for justification regarding this in the SMEFT, please see Appendix A). Then, its quantum corrections to the di-Higgs productions are automatically guaranteed to be independent of the choice of gauge fixing. For the diagrams renormalizing the cubic Higgs coupling, no Goldstone bosons nor gauge bosons are involved where the gauge fixing is applied. These diagrams will contribute to the di-Higgs productions in a universal way. For the diagrams modifying the VVhh form factor, though both the gauge bosons and Goldstone bosons are involved, their summation yields a cancellation of the gauge dependence. These diagrams are finite and will contribute to the VBA and VBF di-Higgs productions. For the diagrams modifying the tthh, gtthh (or Ztthh), and ggtthh form factors, again, no Goldstone bosons nor gauge bosons are involved. These diagrams are finite and will contribute to the gluon fusion and top quark-associated di-Higgs productions.
Computing the diagrams in Fig. 1 with the dimensional regularization, we obtain the tri-Higgs vertex where λ 3 is the renormalized cubic Higgs coupling and γ ¼ 0.577… is the Euler constant. We use δ 3 to denote the counterterm schematically. This counterterm can arise from the higher-dimensional operators in the SMEFT (e.g., the dimension-6 operator O 6 ) or the h 3 term in the HEFT. Their coefficients then match onto the couplings between the Higgs field and the new fields in a UV-complete model which have been integrated out to define the EFT. The renormalized cubic Higgs coupling λ 3 can be defined by properly choosing the p 2 j values for Γðp 2 1 ; p 2 2 ; p 2 3 Þ. Since the three Higgs legs cannot be on shell at the same time, we will consider two schemes: (i) Scheme 1.-Set p 2 j ¼ 0, and define λ 3 ≡ Γð0; 0; 0Þ. Equation (2.1) then becomes (ii) Scheme 2.-Set p 2 1;2 ¼ m 2 h , p 2 3 ¼ 4m 2 h , and define λ 3 ≡ Γðm 2 h ; m 2 h ; 4m 2 h Þ. In any di-Higgs productions, the cubic Higgs coupling always has two on-shell Higgs legs, and the third one is characterized by the di-Higgs invariant mass p 2 :

ð2:3Þ
This choice is effectively equivalent to the MS renormalization scheme with μ ¼ 0.67m h . The one-loop corrections of the quartic Higgs coupling to the VVhh form factor is a summation of three terms in the R ξ gauge ð2:4Þ Here, F i denotes the contribution of the ith diagram in Fig. 2 with the momentum of the incoming gauge bosons denoted as k 1 and k 2 : ð2:5Þ After a contraction with external massless fermion current or massive gauge bosons that are on shell, only the q μ q ν term is left in F 2 . Then, the summation of F 1 and F 2 leads to a cancellation of ξ dependence, as we expected. One can also check that F½HHVV is UV finite, similar to the case of F½HVV discussed in Ref. [28].
The calculation of the one-loop corrections of the quartic Higgs coupling to the tthh, gtthh, and ggtthh form factors is straightforward, based on the diagrams in Fig. 3. We do not show the results here, since below we will focus on the VBF and VBA di-Higgs productions.
With the renormalized cubic Higgs coupling λ 3 and the modified VVhh form factor, we can parametrize the deviation of the cross section σ from the SM prediction σ 0 in the relevant di-Higgs productions as The first two terms denote the contributions from the cubic Higgs coupling only, at the leading order that arises from the tree level. The rest arises from the interference between the κ 4 one-loop corrections and the tree-level amplitudes. We neglect the quadratic term in κ 4 , given that it results from the interference between one-loop amplitudes. Then, the cubic and quartic Higgs couplings can be probed by measuring the di-Higgs production cross sections at colliders.
The interpretation of the collider sensitivities for probing κ 3 depends on the λ 3 renormalization scheme. But such a scheme dependence can be largely suppressed for κ 4 , by marginalizing κ 3 in the χ 2 analysis. This can be understood in the following way. Consider N ≥ 2 observables fO i g, which depend on two parameters x and y linearly: The two parameters can be fit using the χ 2 analysis, with Here, σ i is the measurement uncertainty of O i . Then, the marginalized constraint for one of the two parameters, say, y, can be obtained by integrating x out, given by ð2:9Þ Here, M is the inverse of the covariance matrix for x and y. At 1σ C:L:, we have Δχ 2 ¼ 1, which yields To match with the discussions on the Higgs selfcouplings, we can make replacements: ðx; yÞ → ðκ 3 ; κ 4 Þ and Since the left side of this equation is independent of the λ 3 renormalization scheme at the leading order, ΔC ij should be nearly scheme independent, given that κ 4 by definition is a parameter independent of κ 3 or λ 3 . Then, we are able to obtain Δκ 4 by applying Eq. (2.10), with the scheme dependence suppressed, if all pairs of fC ðiÞ 41 ; C ðjÞ 41 g are calculated with proper precisions. Note that the "if" condition is important for suppressing the linear-level scheme dependence. For example, if one were to combine the di-Higgs productions discussed above with the single Higgs productions in the analysis, the two-loop contributions of the quartic Higgs coupling to the latter channels would need to be incorporated. The nonlinear terms in Eq. (2.6), if turned on, may weaken this argument. But the scheme dependence introduced is of next-to-next-toleading order (NNLO) and could be further suppressed if the NNLO nonlinear terms, such as the ones proportional to κ 2 4 , are properly calculated. If there are two observables only, the formula for Δκ 4 is reduced to Here, jσ i =C i 31 j and jσ j =C j 31 j represent the precision of measuring κ 3 via O i and O j , respectively, with κ 4 being turned off. An interesting observation is that a larger jΔC ij j tends to yield a higher precision for the κ 4 measurement. This can be the case when the two observables O i and O j constrain the κ 3 − κ 4 plane in two clearly separated directions. Below, we will show how to optimize the measurement precision for κ 4 using this guideline.

III. ANALYSES AT LEPTON AND HADRON COLLIDERS
In this section, we calculate the one-loop contributions of the quartic Higgs coupling in the VBF and VBA di-Higgs productions at both lepton and hadron colliders. We use FEYNRULE [29] to generate the model file. The cross sections are then calculated with FEYNARTS3.8 and FORMCALC9.5 [30] using a factorization scale of m h ¼ 125 GeV, where the LOOPTOOLS [31] is linked to calculate the loop integral. The electroweak input parameters in the analysis are chosen as G F ¼ 1.1663787 × 10 −5 GeV −2 , m Z ¼ 91.1876 GeV, and m W ¼ 80.385 GeV [32]. For consistency checks, we compare the tree-level cross sections with those given by MADGRAPH@AMC2.3.3 [33] and CALCHEP3.6.27 [34]. Also, we have checked the values of the squared one-loop amplitudes at some given points in the phase space by comparing with the results calculated by hand.

A. Lepton colliders
At lepton colliders, the main di-Higgs production processes include the Z-associated production e þ e − → Zhh and the VBF production e þ e − → ννhh. Though they could be kinematically turned on, the VBF production e þ e − → e þ e − hh and the top pair-associated production e þ e − → tthh suffer a suppression of cross section. So, we will focus on the former two channels. Figure 4 shows their leadingorder cross sections in the SM, as functions of the center-ofmass energy ffiffi ffi s p , with an unpolarized initial state. The cross section for the Zhh process reaches the peak at ffiffi ffi s p ∼ 500 GeV and then slowly decreases due to an s-channel suppression. As for the VBF production of ννhh, due to the t-channel contributions mediated by the W boson, its cross section keeps growing up to a few TeV. In Table I, we show the leading-order SM cross sections and the coefficients defined in Eq. (2.6) for these two processes, in different collider configurations. The cubic Higgs coupling is renormalized in scheme 1. Note that the beam polarization does not modify the values of C 3a and C 4b but changes the total cross section only. As we demonstrated in Sec. II, the ΔC ij defined in Eq. (2.11) is independent of the λ 3 renormalization scheme at the linear level. Particularly, a larger jΔC ij j tends to yield a higher precision for the κ 4 measurement, after κ 3 is marginalized. For optimizing the collider sensitivities and potentially its configuration design, therefore, it is helpful to have the information on jΔC ij j for various observable pairs available. In Fig. 5, we show ΔC ij for the observables available in the Zhh and ννhh channels. The dashed and solid lines denote the cases in which the two observables are from the same and different channels, respectively. The red and blue colors represent different choices for the reference observable O j . Then, we show the ffiffi ffi s p dependence of ΔC ij by varying ffiffi ffi s p from 500 GeV to 3 TeV for O i . Interestingly, the two observables, if arising from the Zhh and ννhh channels separately, result in a jΔC ij j of Oð10 −2 Þ. This is several times or even one order larger than that obtained in the complementary cases and is not sensitive to the value of ffiffi ffi s p . Indeed, such a pair of observables has clearly separated degenerate directions at the κ 3 − κ 4 plane. A combination of them will be very important for optimizing the sensitivities to probe κ 4 .

B. Hadron colliders
The main di-Higgs production processes at hadron colliders include the gluon-fusion production (gg → hh), the top pair-associated production (pp →tthh), the VBF production (pp → hhjj), and the VBA production (pp → Vhh, V ¼ Z, W). For all of these processes, the cross sections increase as ffiffi ffi s p increases from 14 to 100 TeV. At 100 TeV, the gluon-fusion cross section is around 1 pb;   [35,36]. For illustration purposes, we will focus on the VBA and VBF productions. Table II shows the leading-order cross sections in the SM and the coefficients defined in Eq. (2.6) for the VBA and VBF productions, at 14 and 100 TeV. Here, we find the contribution from the VBF productions at hadron colliders by imposing a set of universal VBF selection cuts as [9] p T;j > 25 GeV; ΔR jj > 4; M jj > 600 GeV ð3:1Þ except a rapidity cut jη j j < 4.5 at 14 TeV and jη j j < 10 at 100 TeV. The cubic Higgs coupling is renormalized in scheme 1. Similar to the analyses at lepton colliders, the knowledge on jΔC ij j is helpful for optimizing the sensitivities at the hadron collider to probe the quartic Higgs coupling. In Fig. 6, we show ΔC ij for the observable pairs, which are available in the Zhh and jjhh channels. 5 We use the red and blue colors to denote the Zhh and the jjhh as O i , respectively. The lines of different styles (solid, dashed, and dotted-dashed) represent different reference observables O j for a given O i . Then, we show the ffiffi ffi s p dependence of ΔC ij by varying ffiffi ffi s p from 14 to 100 TeV for O i . The two observables, if arising from the Zhh=Whh and the jjhh at the hadron collider separately, result in a jΔC ij j of Oð10 −2 Þ. This magnitude is several times or even one order larger than that obtained in the cases in which both observables are from the Zhh=Whh channels or both are from the jjhh channel and is not very sensitive to the value of ffiffi ffi s p . These observations are similar to what we had at lepton colliders. So, a combination of such a pair of observables is very important for optimizing the sensitivities to probe κ 4 at hadron colliders. This conclusion can be generalized to the combination of two observables that are defined at lepton colliders and hadron colliders, separately. As is shown in Fig. 6, the jjhh and the Zhh at hadron colliders can result in a jΔC ij j of Oð10 −2 Þ as well, by pairing with the Zhh and the ννhh at lepton colliders, respectively.

IV. COLLIDER SENSITIVITIES TO THE HIGGS SELF-COUPLINGS
In this section, we reinterpret the projected precisions of the di-Higgs measurements as the sensitivities to probe both the cubic and quartic Higgs couplings. At hadron colliders, the sensitivity of measuring the VBA production at the LHC is poor [12], and the study on a future collider is absent. The VBF production, on the other hand, is not quite sensitive to the cubic Higgs coupling [8,9], and new analysis strategies have yet to be developed. So, we will focus on the analysis at lepton colliders. For simplification, we assume that the Higgs self-couplings only yield negligible modifications for the signal efficiency of the SM contributions. Then, the projected precisions as summarized in Table III can be directly applied to our analysis below, using the parametrization in Eq. (2.6). For the convenience of discussions, we define two ILC scenarios: (i) ILC1 ¼ ILC (500 GeV, 4 ab −1 þ1 TeV, 2.5 ab −1 [13]); (ii) ILC2 ¼ ILC (500 GeV, 4 ab −1 þ 1 TeV, 8 ab −1 [37]). Figure 7 shows the sensitivity contours of measuring κ 3 and κ 4 at 1σ C:L:, at the ILC and CLIC. Here, the cubic Higgs coupling is renormalized in scheme 1. In this figure, the yellow region is defined by the perturbative unitarity bound of the hh → hh scattering (the derivation is presented in Appendix B). This unitarity requirement sets a range between ∼ AE65 for κ 4 , within which κ 3 is allowed to vary from ∼ −9 to ∼7. The brown and blue circles represent the sensitivities of the ILC1 and the ILC2, respectively. In both scenarios, the ILC yields an exclusion limit for κ 3 and κ 4 well within the perturbative regime. 6 This can be understood, since the ILC sensitivities benefit a lot from: (i) the combination of the Zhh and ννhh observables, which are characterized by relatively large jΔC ij j values of Oð10 −2 Þ (ii) the good precisions for measuring the Zhh at 500 GeV (almost maximized cross section, high luminosity) and the ννhh at 1 TeV (large cross section, high luminosity). The purple circle represents the CLIC sensitivities by combining the measurements of Zhh at 1.4 GeV and ννhh at 1.4 and 3 TeV. As a comparison, it is difficult for the CLIC to reach an exclusion limit for κ 4 within the perturbative regime. Its sensitivities suffer from both the suppressed Zhh cross section at a higher beam energy scale and the relatively low luminosity.
Given its potential in probing the Higgs self-interactions, let us look into the ILC analysis and sensitivities in more detail. In Fig. 8, we present the ILC sensitivities in both scheme 1 and scheme 2 of the λ 3 renormalization, with and without the nonlinear terms in Eq. (2.6). At the linear level, the exclusion contours at the κ 3 − κ 4 plane are an ellipse with the major axis being close to the κ 4 direction. As is indicated in the left panel, the change from scheme 1 to scheme 2 yields a counterclockwise rotation for the exclusion contours. The nonlinear terms deform these ellipses. Compared to scheme 2, the ellipse orientation in scheme 1 restricts κ 3 to be smaller and makes the nonlinear effects less important. In the right panel, the sensitivities to probe κ 3 and κ 4 are shown by marginalizing κ 4 and κ 3 , respectively, in the χ 2 fit. The κ 3 sensitivity depends strongly on the λ 3 renormalization scheme by definition, while the κ 4 sensitivity is nearly scheme independent at the linear level, as we advertised in Sec. II. The scheme dependence is mainly introduced via the nonlinear terms in Eq. (2.6) in this context. The allowed ranges for κ 4 can then vary by a few percent between scheme 1 and scheme 2.
To illustrate the scheme dependence more clearly, we present in Fig. 9 the κ 4 sensitivity in the ILC2 scenario for a wide range of the renormalization scale μ, say, from 50 to 500 GeV. The difference between the yellow and the light blue regions shows the scheme dependence introduced via nonlinear terms in Eq. (2.6). Scheme 1 turns out to yield the almost minimal discrepancy between the linear and nonlinear results. For a smaller or larger renormalization scale, the constrained region shifts downward, yielding a less positive upper limit and a more negative lower limit. In a scenario with lower sensitivities, say ILC1, the nonlinear effects would be more significant.
In all, the ILC has a potential to probe jκ 4 j as small as ∼25 in the ILC1 scenario and ∼20 in the ILC2 scenario, respectively, at 1σ C:L:, in a λ 3 renormalization scheme in which nonlinear effects are minimized. Such a sensitivity is comparable to the one that could be achieved by measuring the tri-Higgs production at a high-luminosity future hadron collider, say, 30 ab −1 @100 TeV [16,17]. FIG. 7. The sensitivity contours of measuring κ 3 and κ 4 at 1σ C:L:, at the ILC and CLIC. The yellow region is perturbatively unitarity safe. As a benchmark, we indicate the region that is favored by first-order EWPT in the SMEFT with the O 6 and O 8 operators turned on (the discussions are presented in Appendix C) in orange.

V. CONCLUSION
The Higgs self-interactions play a crucial role for exploring the underlying mechanisms of electroweak symmetry breaking and the nature of the phase transition involved. Motivated by this, we proposed to probe the quartic Higgs self-interaction at lepton and hadron colliders, via the di-Higgs productions. We analyzed the corrections of the quartic Higgs coupling to the VBF and VBA di-Higgs productions at the one-loop level. Such an effect is independent of the gauge fixing, if the quartic Higgs coupling is decoupled from other couplings in the given context. In the calculations, we ignored the one-loop diagrams with no quartic Higgs coupling involved. These diagrams yield a NLO impact only for the sensitivity analysis of κ 3 and κ 4 at lepton colliders, after a proper renormalization for λ 3 . One notable observation in the analysis is that the observables from the VBF and VBA di-Higgs productions probe the κ 3 − κ 4 plane in two clearly separated directions, at both lepton and hadron colliders. A combination of these two channels therefore is important for optimizing the collider sensitivities. With this guideline, we analyzed the ILC and CLIC sensitivities. We are able to extract the sensitivity on κ 4 , which is nearly independent of the λ 3 renormalization scheme at the linear level, by marginalizing the cubic Higgs coupling in the χ 2 analysis. Then, in a λ 3 renormalization scheme in which the nonlinear effects are almost minimized, we found that the ILC has the potential to measure the quartic Higgs coupling, normalized by its SM value, with a marginalized precision of ∼ AE25 in the ILC1 scenario and ∼ AE20 in the ILC2 scenario at 1σ C:L.
The collider sensitivities could be further improved by utilizing the di-Higgs invariant mass distribution of the di-Higgs events. In the analysis pursued, we have assumed that new physics does not significantly modify the kinematics of the SM di-Higgs events. To look into this further, we show the SM cross sections and the values of C 3a and C 4b in the di-Higgs invariant mass bins of e − e þ → Zhh and e − e þ → ννhh at ILC, in Tables IV and V of Appendix D, respectively. It is easy to see, though the C 41 C 31 defined in the Zhh channel is not very sensitive to the m hh values, a relatively small m hh value yields a more negative C 41 C 31 in the ννhh channel and hence a larger jΔC ij j between the two channels. Additionally, both channels become more sensitive to κ 3 in the low m hh region, with a larger jC 31 j value. According to Eq. (2.11), therefore, the collider sensitivities could be further improved by requiring relatively small m hh for the di-Higgs events. Furthermore, if Eq. (2.12) is applied to the pair of observables Zhh at 500 GeV and ννhh at 1 TeV, we can check Thus, by improving the measurement precision for the Zhh at 500 GeV, if sizably, the sensitivities for probing κ 4 could be significantly improved. We need to keep in mind that the di-Higgs productions could be contaminated by some other new physics, via, e.g., the wave function renormalization of gauge bosons or the Higgs boson, the definition shift of the electroweak parameters, or the introduction of new vertices. Here, we have turned off all of these effects and simply assumed that they can be constrained sufficiently well for our purpose, by the electroweak and Higgs precision measurements at future colliders (for recent studies, see, e.g., Refs. [40][41][42][43]).
Given the significance of measuring the Higgs selfinteractions in particle physics, it is worthwhile to pursue a more systematic and complete analysis on its collider sensitivities. We can extend the analysis from lepton colliders to hadron colliders, particularly to the next-generation hadron colliders. More di-Higgs production channels can be taken into account, such as the gluon-fusion and top quark-associated processes, in that case. The leading-order effects of the quartic Higgs coupling appear at two-and oneloop level, respectively. We may also incorporate the tri-Higgs productions at both lepton colliders and hadron colliders in the analysis. The observables arising from these channels could be characterized by a jΔC ij j of Oð10 −2 Þ as well and further improve the marginalized precision of κ 4 . Additionally, the quartic Higgs coupling contributes to the single Higgs productions (e.g., Zh and ννh) at two-loop level, which in turn may facilitate the probe for the quartic Higgs coupling. To end the discussion, we would stress again that to probe κ 4 by combining the di-Higgs productions and other Higgs channels the C 41 =C 31 for both need to be calculated with proper precisions, to suppress the scheme dependence of the λ 3 renormalization at least at the linear level. We leave a full study on these to future work. Note added.-Recently, Ref. [44] appeared, and it partially overlaps with this one in analyzing the one-loop corrections of the quartic Higgs coupling to the Zhh and ννhh productions at lepton colliders. But our work is different from Ref. [44] in the following aspects: (i) We developed a general guideline for optimizing the collider sensitivities of probing the quartic Higgs coupling, based on Eq. (2.11).
(ii) We analyzed the one-loop corrections of the quartic Higgs coupling to the Zhh=Whh and jjhh productions at hadron colliders as well. (III) We presented the ILC sensitivities for probing κ 4 by marginalizing κ 3 in the χ 2 analysis and discuss the scheme dependence of the λ 3 renormalization in detail.

APPENDIX A: GENERAL HIGGS POTENTIAL AND COUPLINGS IN SMEFT
In this Appendix, we find the relation between Higgs self-couplings and Higgs-Goldstone couplings in SMEFT. A general Higgs potential here can be parametrized as a general function, Given the electroweak scale v and the Higgs mass m h , the first two derivatives of the general function are determined by where the subscript v denotes the quantity evaluated at h ¼ v, π i ¼ 0. After substituting (A2), the cubic couplings of Higgs and Goldstone bosons turn out to depend on the general function only up to the second derivative, So, for an arbitrary Higgs potential in (A1), they remain the same as in the SM. The cubic Higgs coupling and the quartic Higgs-Goldstone-boson quartic couplings further depend on the third derivative of the general function, where κ 3 ≡ v 2 F 000 v =3F 00 v . If the Higgs potential only includes one higher-dimensional operator, say ðH † HÞ 3 , κ 3 is then proportional to its coefficient. In the general case with more than one higher-dimensional operator, κ 3 determines one of their linear combinations. For the later case, the quartic Higgs coupling receives new independent contribution from the fourth derivative of the potential, where As in (A4), the same combination κ 4 enters into the coupling for hhhπ 0ðþÞ π 0ð−Þ . Such a relation can be generalized further when the potential has enough independent terms, namely, the self-coupling of n Higgs is correlated with the coupling of a pair of Goldstone bosons and (n − 1) Higgs.
In summary, for the κ 4 dependence of the di-Higgs production cross section, we only need to consider the one-loop diagram from quartic Higgs coupling, while quartic couplings involving both Higgs and Goldstone are irrelevant.

APPENDIX B: PERTURBATIVE UNITARITY BOUND
To have the perturbative calculation still be reliable, Higgs self-couplings need to satisfy the perturbative unitarity bound. The scattering amplitude for hh → hh at tree level is where the Mandelstam variables s ¼ E 2 , t ¼ −ðE 2 − 4m 2 h Þ sin 2 θ=2, and u ¼ −ðE 2 − 4m 2 h Þ cos 2 θ=2 in the center-of-mass frame. The partial wave amplitudes are then computed as [45,46] a l ðEÞ ¼ 1 2 where the additional factor of 1=2 comes from normalization of the symmetric initial, final states, and kinematic factor βðx; y; zÞ ¼ ðx 2 þ y 2 þ z 2 − 2xy − 2yz − 2xzÞ 1=2 . For the s wave, l ¼ 0, we find The s-wave unitarity condition requires jRea 0 ðEÞj < 1=2. In the high-energy limit E 2 ≫ m 2 h , λ 4 contributes at the leading order. Thus, we can obtain jλ 4 j < 16π, namely, j1 þ κ 4 j < 16πv 2 3m 2 h ¼ 65: ðB4Þ λ 3 starts to dominate at the low energy, and the amplitude reaches a peak at some scale. Assuming the peak amplitude satisfies the s-wave unitarity condition, we can find the range of κ 3 for a given κ 4 satisfying Eq. (B4).

APPENDIX C: FIRST-ORDER ELECTROWEAK PHASE TRANSITION: A BENCHMARK
The nature of EWPT could have a strong correlation with the Higgs potential at zero temperature. For illustrating the collider capability in probing the EWPT nature, we analyze a simplified model in the SMEFT [47] (we will tolerate the potential uncertainties caused by such a simplified treatment [48]; for discussions in more general contexts, see, e.g., Ref. [49]), as the benchmark. Here, the temperature-dependent term results from an expansion of thermal mass for the SM particles. a 0 ∼ 3 is defined by the SM physics. The firstorder EWPT requires the coexistence of two degenerate vacua, characterized by v T ¼ 0 and v T ¼ v c ≠ 0 at the critical temperature T c , with v c satisfying the following condition: Here, m 2 h ¼ 2λv 2 þ 3c 6 v 4 Λ 2 þ 3c 8 v 6 Λ 4 is the squared Higgs mass at zero temperature. We then scan over fc 6 ; c 8 g, to extract out the fκ 3 ; κ 4 g region where a first-order EWPT is favored, using the relation The favored region is marked in orange in Fig. 7. In the case with c 8 ¼ 0, the orange region is reduced to the bottom boundary, which is consistent with the results obtained in Ref. [47], in which only the O 6 operator is turned on.