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Following the discovery of the Higgs boson at the Large Hadron Collider (LHC) [1, 2], precise measurements of its properties, including mass, spin, and couplings to gauge bosons and fermions, became critically important [3−9]. To date, these measurements have been consistent with the expectations of the Standard Model (SM) [10, 11]. However, the trilinear and quartic Higgs self-couplings, denoted as
λ3H andλ4H , respectively, which represent a fundamental aspect of the SM that connects the Higgs mechanism and the stability of our Universe [12], are still subject to large uncertainties.Significant efforts have been dedicated to improving the measurement of the Higgs self-coupling. The most direct approach involves measuring the cross section of Higgs boson pair production, predominantly through gluon-gluon fusion (ggF). At the leading order (LO), this process occurs via a top-quark loop. In the large top-quark mass (
mt ) limit, the cross section of ggF Higgs boson pair production is known up to next-to-next-to-next-to-leading-order (N3 LO) QCD corrections [13−17], with the calculation of the soft gluon resummation effect also being studied [18−20]. When considering the fullmt dependence, only next-to-leading-order (NLO) QCD corrections are available [21−24], while estimates of the finitemt effects at next-to-next-to-leading-order (NNLO) have been conducted [25−29]. A comprehensive simulation of events necessitates a fully differential calculation of the Higgs boson pair production and decay to thebˉbγγ final state at QCD NLO [30]. Furthermore, NLO electroweak corrections have been explored [31−37]. The subdominant channel—vector boson fusion (VBF)—has also been computed up to N3LO in QCD [38−42].The current constraints on the trilinear Higgs self-coupling extracted from the Run 2 dataset of Higgs boson pair production at the LHC by the ATLAS and CMS collaborations are
−0.6<κλ3H<6.6 [43] and−1.24<κλ3H< 6.49 [11], respectively, within theκ framework [44], whereκλ3H=λ3H/λSM3H , withλSM3H being the SM value of the trilinear Higgs self-coupling. Meanwhile, the process of single Higgs boson production and decay depends on the Higgs self-coupling only via higher-order electro-weak (EW) corrections and can also impose certain constraints [45−50]. Measurement based on the differential fiducial cross section in bins of the Higgs boson transverse momentum sets a constraint of−5.4<κλ3H< 14.9 [51]. Combined analyses of single- and double-Higgs production result in the constraint−0.4<κλ3H< 6.3 at the 95% confidence level, assuming that new physics changes only the Higgs self-coupling [43].These constraints stem from considering the cross section of Higgs boson pair production as a function of the Higgs self-coupling. Indeed, even with higher-order QCD corrections, the cross section is a quadratic function of the trilinear Higgs self-coupling,
λ3H . Nevertheless, higher-order EW corrections introduce contributions from Feynman diagrams containing one or more triple Higgs or quadruple Higgs vertices, leading to a distinct functional dependence on the Higgs self-coupling. Specifically, new quartic and cubic power dependencies on the trilinear Higgs self-coupling emerge, which has a significant impact on the constraints, given that the current upper limit is notably large.In practical calculations, maintaining an explicit dependence on the Higgs self-coupling can be challenging given that it is typically treated as a derived parameter in the conventional calculation of EW corrections, particularly during the renormalization process [35, 52]. Even if one can perform the renormalization by taking the Higgs self-coupling as a primary parameter, it is not clear how the relation
m2H=2λv2 can be implemented and the Higgs self-coupling can be rescaled. Different choices lead to different expressions for the cross sections. Therefore, the calculation of the EW corrections in the SM cannot be extended to the case with generalλ3H andλ4H , and the cross section with higher power (beyond quadratic) dependence on the Higgs self-couplings in the generalκ framework is still lacking.To address this challenge, we propose a renormalization procedure that explicitly retains the Higgs self-couplings at each step and introduce renormalization of the coupling modifier. By combining this procedure with analytical and numerical calculations of the complex one-loop and two-loop amplitudes, respectively, we derive the cross sections of the Higgs boson pair production in both the ggF and VBF channels as functions of the Higgs self-couplings. Our findings indicate that incorporating higher power dependencies of the cross section on the Higgs self-couplings can reduce the upper limit of
κλ3H by approximately 20%. -
The LO contribution to the ggF Higgs boson pair production
g(p1)g(p2)→H(p3)H(p4) arises from the top-quark induced triangle and box Feynman diagrams, which are of orderλ3H andλ03H , respectively. Therefore, the LO cross section at the 13 TeV LHC can be expressed asσκλggF,LO=(4.72κ2λ3H−23.0κλ3H+35.0)fb.
(1) Here, the subscript
λ3H inκλ3H signifies the deviation of the trilinear Higgs self-coupling from its SM value. Below, we also introduceκλ4H to denote the modification of the quartic Higgs self-coupling. The SM cross section is recovered whenκλ3H=1 andκλ4H=1 . This quadratic functional form persists even with the inclusion of higher-order QCD corrections [53], e.g.,σκλggF,NNLO−FT=(10.8κ2λ3H−49.6κλ3H+70.0)fb.
(2) In this expression, the full one-loop real contributions are merged with other NNLO QCD corrections in the large
mt limit. It is worth noting that the effects of QCD corrections are substantial, with each term inκλ3H more than doubling in value compared to the LO expression.The EW corrections change the above functional form in two aspects. First, the coefficients of the quadratic, linear, and constant terms are altered by the corrections induced by virtual gauge bosons. However, the overall impact is relatively minor, typically amounting to only a few percent, as reported in Ref. [35]. Given this negligible influence on the constraints related to the Higgs self-coupling, these corrections are deemed insignificant for the purposes of our study and are consequently omitted. Second, higher power or new dependence on the Higgs self-couplings arises from the corrections induced by virtual Higgs bosons. Fig. 1 shows some typical two-loop Feynman diagrams, which give contributions of order
λ33H ,λ23H ,λ4Hλ3H ,λ4H to the amplitudes. As a result, the cross section contains new quartic and cubic powers ofλ3H and starts to be sensitive to the quartic Higgs self-couplingλ4H . Our objective is to assess these corrections, denoted byδσκλEW , and their impact on the constraints on the Higgs self-couplings.Figure 1. Typical two-loop Feynman diagrams of order
λ33H (a),λ23H (b),λ4Hλ3H (c) , andλ4H (d) , respectively.Our calculations for two-loop diagrams were realized as follows. We used FeynArts [54] to generate the Feynman diagrams and corresponding amplitudes. The amplitudes were expressed as a linear combination of two tensor structures with the coefficients called form factors [55]. After performing the Dirac algebra with FeynCalc [56−58], we obtained scalar integrals for each form factor. Rather than reducing all scalar integrals to master integrals and establishing differential equations for these master integrals, we directly computed the scalar integrals for specific phase space points employing the numerical package AMFlow [59, 60]. This decision was motivated by the intricate nature of constructing differential equations, particularly with general kinematic dependencies, which can be exceedingly time-consuming. Even if the differential equation is derived, obtaining an analytical solution appears infeasible with current technologies owing to the presence of massive propagators. Numerical solutions of these differential equations often suffer from accuracy loss. By contrast, direct numerical calculation at each phase space point ensures accuracy. The primary challenge lies in covering the entire phase space efficiently. Fortunately, the process of
gg→HH is dominated by S-wave scattering, making the amplitude insensitive to the scattering angle. Moreover, its dependence on the scattering energy is also weak, except in very high-energy regions, as shown below. These characteristics enable the generation of a grid with a limited data set. This grid can be used to accurately calculate the amplitude at any point in the phase space.Note that the sum of all one-particle irreducible two-loop diagrams is finite. However, the sum of one-particle reducible two-loop diagrams contains ultraviolet divergences. They will cancel after considering the contributions from the counter-terms in renormalization.
In the SM, the Lagrangian for the Higgs sector can be expressed as
LH=(Dμϕ0)†(Dμϕ0)+μ20(ϕ†0ϕ0)−λ0(ϕ†0ϕ0)2,
(3) where
ϕ0 denotes the bare Higgs doublet andDμ is the covariant derivative. The relations between the bare quantities and their renormalized counterparts are given byϕ0=Z1/2ϕϕ,μ20=Zμ2μ2, andλ0=Zλλ. The EW gauge symmetry is spontaneously broken once the Higgs field develops a non-vanishing vacuum expectation value
v . Taking the unitary gauge, we can express the Higgs field asϕ=1√2(0H+Zvv),
(4) where
Zv is the renormalization constant for the vacuum expectation value. The renormalized Lagrangian in theκ framework after EW gauge symmetry breaking is given byLκH=12Zϕ(∂μH)2−(−12Zμ2ZϕZ2vμ2v2+14ZλZ2ϕZ4vλv4)−(ZλZ2ϕZ3vλv3−Zμ2ZϕZvμ2v)H−(32ZλZ2ϕZ2vλv2−12Zμ2Zϕμ2)H2−Zκ3HZλZ2ϕZvλ3HvH3−14Zκ4HZλZ2ϕλ4HH4+⋯,
(5) where the ellipsis represents the terms involving EW gauge bosons. Note that the letter
λ is solely used for the Higgs self-coupling in the SM butλ3H≡κλ3Hλ andλ4H≡κλ4Hλ denote the Higgs self-couplings that could be modified by new physics. We have added the renormalization constantsZκ3H andZκ4H for the coupling modifiersκλ3H andκλ4H to account for potential new physics effect in renormalization. Following the general principle of theκ framework, we have assumed that the new physics does not affect the vacuum expectation value and Higgs mass when we rescale the Higgs self-couplings. The second term in the first line of Eq. (5) contains no field and thus can be safely dropped. WritingZ=1+δZ , the third term can be expanded as(μ2v−λv3)H+[(δZμ2+δZϕ+δZv)μ2v−(δZλ+2δZϕ+3δZv)λv3]H+⋯
(6) where we have neglected higher-order corrections that are products of two
δZ 's. The renormalization condition is set such that there is no tadpole contribution. This condition requiresμ2=λv2 at the tree level and(δZμ2−δZλ−δZϕ−2δZv)μ2v+T=0 at the one-loop level, withT being the contribution from the one-loop tadpole diagrams. The vacuum expectation value appears always in the form ofZ1/2ϕZvv and is closely related to the massive gauge boson mass. Therefore,δZv would be determined only after considering the renormalization of the EW gauge sector. Given that we focus on the corrections induced by the Higgs self-couplings, we can simply takeδZv+δZϕ/2=0 . We adopt dimensional regularization, i.e., the space-time dimension is set asd=4−2ϵ to regulate the ultraviolet divergence, andμR is selected as the renormalization scale. The tadpole diagram is evaluated to beT=3λ3Hv16π2m2H(1ϵ+lnμ2Rm2H+1).
(7) The mass of the Higgs boson,
mH , can be determined from the quadratic term in Eq. (5) as follows:12(∂μH)2−μ2H2+12δZϕ(∂μH)2−(32δZλ+52δZϕ−12δZμ2+3δZv)μ2H2≡12(∂μH)2−12m2HH2+12δZϕ(∂μH)2−12(δZm2H+δZϕ)m2HH2.
(8) On the right-hand side, we have introduced the classical mass terms. By comparing both sides, it is straightforward to obtain that
m2H=2μ2 andδZm2H≡32δZλ+32δZϕ−12δZμ2+3δZv . Applying the on-shell renormalization condition for the Higgs field, we obtainδZm2H=3λ4H16π2(1ϵ+lnμ2Rm2H+1)+9λ23Hv2m2H18π2(1ϵ+lnμ2Rm2H+2−π√3),δZϕ=9λ23Hv28π2√3−2π/3√3m2H.
(9) Combining the above equations, we derive results for the other renormalization constants, namely
δZμ2 andδZλ . Then, the counter-term for the triple Higgs interaction in Eq. (5) is given byδλ3H≡δZλ+2δZϕ+δZv+δZκ3H=−3λ3H16π2(1ϵ+lnμ2Rm2H+1)+3λ4H16π2(1ϵ+lnμ2Rm2H+1)+3λ23Hv216π2m2H(6ϵ+6lnμ2Rm2H+21−4√3π)+δZκ3H
(10) with
δZκ3H≡Zκ3H−1 .Including the contribution of counter-terms, we obtain the following result with explicit Higgs self-coupling dependence for the one-particle reducible diagrams:
MLOgg→H∗→HH{316π21ϵ(−2λ4H−λ3H+6λ23Hv2m2H)+δZκ3H+316π2lnμ2Rm2H[−2λ4H−λ3H+6λ23Hv2m2H]−9λ23H8π2v2s−m2H[β(ln(1−β1+β)+iπ)+sm2H(1−2π3√3)+5π3√3−1]+3λ23H16π2v2m2H(21−4√3π)−9λ23Hv24π2C0[m2H,m2H,s,m2H,m2H,m2H]−3λ4H16π2[β(ln(1−β1+β)+iπ)+5−2π√3]−3λ3H16π2},
(11) where
s=(p1+p2)2,β=√1−4m2H/s , andC0[m2H,m2H,s,m2H,m2H,m2H] is a scalar integral that can be calculated using Package-X [61].MLOgg→H∗→HH is the LO amplitude which contains the Higgs self-coupling. In the SM, the divergences in the first line vanish andδZκ3H is not needed. In the generalκ framework, it is essential to includeδZκ3H in renormalization. The presence ofδZκ3H for general Higgs self-couplings demonstrates that the cross section in theκ framework cannot be derived from the result in the SM, as mentioned in the introduction. We adopt the¯MS scheme to subtract the divergences. As a result, the coupling modifier is scale-dependent. If it is expressed in terms of the value at the scalemH , then an additional contribution from its perturbative expansion exactly cancels the aboveln(μ2R/m2H) term.The
κ framework was initially established based on the signal strength obtained from experimental results. Here, we present a field definition of theκ framework in the Higgs sector that enables higher-order calculations. A more systematical approach is to use the Higgs effective field theory (HEFT) with the electroweak chiral Lagrangian [62], which can be considered as an upgrade of theκ framework to a quantum field theory that provides a general EFT description of the electroweak interactions with the presently known elementary particles under a cutoff scale of approximately a few TeV [63]. The physical Higgs fieldH is introduced as a singlet underSU(2)L×U(1)Y and chiral symmetry. Our strategy, which includes the choice of unitary gauge and the implementation of theκ parameter in the symmetry-broken phase, is equivalent to the application of HEFT in Higgs boson pair production. This equivalence can be easily verified by comparing our Lagrangian in Eq. (5) and the one in Eq. (2.5) of Ref. [64].At this point, we can compute the finite part of the squared amplitudes. We set the two-dimensional grid as a function of the Higgs velocity
β andcosθ withθ the scattering angle. The value ofβ ranges from 0 to 1;cosθ also ranges from 0 to 1 given that the squared amplitudes are symmetric underθ→π−θ . The grids for the LO squared amplitudes andλ dependent EW corrections are shown in Fig. 2. Note that the squared amplitudes are stable against the change within the regionβ<0.6 . For larger values ofβ , the LO squared amplitudes rise dramatically and then start to drop whenβ>0.9 . By contrast, theλ dependent correction first decreases and then increases whenβ is larger than 0.85. These variations are mainly due to the large logarithmslni(1−β) . Therefore, we constructed the grid as a function oflni(1−β) forβ≥0.96 . We tested the grid by comparing the generated values1 and those obtained by direct high-precision computation at some phase space points that were not on the grid lattice. We found good agreement at the per-mille level. We used the grid to calculate the LO total cross section by performing the convolution with the parton distribution function (PDF) and phase space integrations. Comparing this to the result obtained using analytical expressions or the OpenLoops package [65−67], we found a relative difference lower thanO(10−3) .The LO cross section of the VBF channel also exhibits a quadratic dependence on the trilinear Higgs coupling. The higher power dependence can be obtained by calculating the one-loop diagrams with an additional Higgs propagator. The calculation procedure is standard except for the renormalization, as explained above. We implemented the analytical results using the proVBFHH program [41, 68]. We employed the QCDLoop package [69] to evaluate the scalar one-loop integrals.
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In our numerical calculations, we set
v=(√2GF)−1/2 with the Fermi constantGF=1.16637×10−5GeV−2 , the Higgs boson massmH=125 GeV, and the top quark massmt=173 GeV. For the VBF channel, we set the EW gauge boson masses asMW=80.379 GeV andMZ=91.1876 GeV. We used thePDF4LHC15_nlo_100_pdfas PDF set [70] and the associating strong couplingαs . The default renormalization scale inαs and factorization scale in the PDF were chosen to beμR,F=mHH/2 in the ggF channel andμR,F=√−q2i in the VBF channel, withmHH being the Higgs pair invariant mass andqi being the transferred momenta from quark lines.The EW corrections that contain higher power dependence on the Higgs self-coupling are given by
δσκλggF,EW=(0.075κ4λ3H−0.158κ3λ3H−0.006κ2λ3Hκλ4H−0.058κ2λ3H+0.070κλ3Hκλ4H−0.149κλ4H)fb
(12) for the ggF channel and
δσκλVBF,EW=(0.0215κ4λ3H−0.0324κ3λ3H−0.0019κ2λ3Hκλ4H−0.0043κ2λ3H+0.0151κλ3Hκλ4H−0.0211κλ4H)fb
(13) for the VBF channel. We computed all the
O(λi3H),i≥2 contributions in the amplitude. The aboveκ2λ3H terms arise because we aimed to keep the cancellation relation between theO(λ3H) andO(1) amplitudes at LO. The cubicκ3λ3H and quarticκ4λ3H terms appear for the first time up to this perturbative order. Although their coefficients are small, they provide notable corrections to the cross section ifκλ3H is chosen to be much larger than 1. As shown in Table 1, theλ dependent corrections in the ggF (VBF) channel reach 91% (82%) of the LO cross section forκλ3H=6 .κλ3H κλ4H ggF VBF σκλLO σκλNNLO−FT δσκλEW σκλLO σκλNNNLO δσκλEW 1 1 16.7 31.2 −0.225 1.71 1.69 −2.30×10−2 3 1 8.59 18.4 1.28 3.59 3.53 8.35×10−1 6 1 67.3 161 60.6 25.1 24.6 20.7 1 3 16.7 31.2 −0.393 1.71 1.69 −3.89×10−2 1 6 16.7 31.2 −0.646 1.71 1.69 −6.27×10−2 3 3 8.59 18.4 1.30 3.59 3.53 8.50×10−1 6 6 67.3 161 61.0 25.1 24.6 20.7 Table 1. Cross sections (in fb) of ggF and VBF Higgs boson pair production for different values of
κλ3H andκλ4H at the 13 TeV LHC.In addition, there is a new dependence on the quartic Higgs self-coupling
λ4H . Because this dependence is only linear and the corresponding coefficients are small, their contributions are negligible. According to Table 1, the cross section varies by 0.6% whenκλ4H changes from 1 to 6 while keepingκλ3H=6 . As a consequence, we do not expect that a meaningful constraint on the quartic Higgs self-coupling can be extracted from Higgs boson pair production at the LHC.Next, we compare our results with those reported in Refs. [31, 71]. The authors of these papers obtained similar expressions for the cross sections in the ggF channel. However, they assumed that the triple and quartic Higgs self-couplings are modified by one dimension-six and one dimension-eight operators
2 . They performed calculations, especially renormalization, in terms of the coefficients of higher-dimensional operators and then transformed the results onto the basis ofκλ3H andκλ4H . These results cannot be directly compared with the experimental analysis in theκ framework.Fig. 3 shows different perturbative predictions for the cross sections of both ggF and VBF Higgs boson pair productions at the 13 TeV LHC as a function of
κλ3H=κλ4H=κλ . It is evident that higher-order perturbative corrections dramatically change the functional form. The current experimental upper limit set by the ATLAS (CMS) collaboration onκλ is6.6 (6.49) based on the theoretical predictions at QCD NNLO in the ggF channel and NNNLO in the VBF channel; see Table 1. TakingδσκλEW corrections into account and assuming that the QCD and EW corrections are factorizable, the upper limit is narrowed down to 5.4 (5.37). These limits are almost the same when keepingκλ4H=1 . If the scale uncertainties are considered [74], the upper limit would span in the range(6.5, 6.8) for the ATLAS results, which would decrease to(5.4, 5.6) after including higher power dependence. Concerning the CMS results, the upper limit changes from (6.40, 6.67) to (5.31, 5.48). The lower limits are only slightly modified.Figure 3. (color online) Cross sections of Higgs boson pair production at the 13 TeV LHC including both ggF and VBF processes as a function of
κλ3H=κλ4H=κλ . The black line represents the LO results, whereas the green line denotes the results with (N)NNLO QCD corrections in the ggF (VBF) channel. The red line indicates the results including higher power dependence on the Higgs boson self-coupling. The current and improved upper limits are labeled by points A and B, respectively.Lastly, the Higgs boson pair invariant mass
mHH distributions are shown in Fig. 4. The peak position moves from 400 GeV to 260 GeV whenκλ varies from 1 to 6, which indicates that the Higgs bosons tend to be produced with very low velocity in the case of largeκλ values. This feature could help set optimal cuts in the experimental analysis to enhance sensitivity. From this figure, we also observe that theλ dependent corrections have a great impact on the distributions, especially in the smallmHH region. Therefore, they should be included in future studies. -
The precise shape of the Higgs potential constitutes a fundamental enigma in particle physics. The current limits on the Higgs self-coupling are predominantly derived under the assumption that the cross section of the Higgs boson pair production is a quadratic function of the self-coupling. We found that the functional form should be generalized to include quartic and cubic power dependencies on the Higgs self-coupling that arise from higher-order quantum corrections induced by virtual Higgs bosons.
We propose a proper renormalization procedure to explicitly retain the Higgs self-couplings at each calculation step and introduce renormalization of the coupling modifiers to ensure the cancellation of all ultraviolet divergences. We present numerical results of the cross sections of both the ggF and VBF channels at the LHC including higher power dependencies on the Higgs self-coupling. With these improved functional forms, we demonstrate that the upper limit set by the ATLAS (CMS) collaboration on the trilinear Higgs self-coupling normalized to its SM value can be reduced from 6.6 (6.49) to 5.4 (5.37). This more precise constraint is achieved without analyzing additional data, underscoring the critical importance of incorporating higher power dependencies on the Higgs self-coupling in the cross section.
Furthermore, we found it difficult to derive any useful constraint on the quartic Higgs self-coupling solely from Higgs boson pair production. To probe the quartic self-coupling, alternative channels such as triple Higgs boson production may be explored, necessitating collider facilities with energies higher than those of the LHC to provide insights into this aspect of the Higgs potential.
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We express our gratitude to Huan-Yu Bi, Yan-Qing Ma, and Huai-Min Yu for comparing the numerical results of the two-loop amplitudes. We also thank Shan Jin, Yefan Wang, and Lei Zhang for helpful discussions.
Improved constraints on Higgs boson self-couplings with quartic and cubic power dependencies of the cross section
- Received Date: 2024-11-12
- Available Online: 2025-02-15
Abstract: Precise determination of the Higgs boson self-couplings is essential for understanding the mechanism underlying electroweak symmetry breaking. However, owing to the limited number of Higgs boson pair events at the LHC, only loose constraints have been established to date. Current constraints are based on the assumption that the cross section is a quadratic function of the trilinear Higgs self-coupling within the