Top-Yukawa-induced Corrections to Higgs Pair Production

Higgs-boson pair production at hadron colliders is dominantly mediated by the loop-induced gluon-fusion process $gg\to HH$ that is generated by heavy top loops within the Standard Model with a minor per-cent level contamination of bottom-loop contributions. The QCD corrections turn out to be large for this process. In this note, we derive the top-Yukawa-induced part of the electroweak corrections to this process and discuss their relation to an effective trilinear Higgs coupling with integrated out top-quark contributions.

are the Higgs mass and self-interactions of the Higgs field. The production of Higgs-boson pairs is the first class of processes that offers the direct access to the trilinear self-coupling of the Higgs boson as a first step towards the reconstruction of the full Higgs potential. At the Large Hadron Collider (LHC), the dominant Higgs-boson pair production mechanism is provided by the gluon-fusion process gg → HH, while the other production modes as vector-boson fusion (VBF) qq → qqHH, double Higgs-strahlung qq → W/Z + HH and double Higgs bremsstrahlung off top quarks qq, gg → ttHH are suppressed by at least one order of magnitude [6]. The individual production cross sections roughly follow the pattern of single-Higgs boson production but are in general smaller by about three orders of magnitude. Since the trilinear Higgs coupling contributes only to a subset of diagrams of each production process the sensitivity to the trilinear Higgs coupling is reduced due to the dominance of the continuum diagrams. The slope of the gluon-fusion cross section as a function of the trilinear Higgs coupling λ follows the rough behaviour ∆σ/σ ∼ −∆λ/λ around the SM prediction [6][7][8]. This implies that the uncertainties of the production cross section are immediately translated to the uncertainties of the extracted trilinear self-coupling so that the reduction of the theoretical uncertainties of the Higgs pair production cross section is crucial for an accurate extraction of the trilinear self-interaction from the experimental measurements. This feature translates to a similar situation for the distributions as well. The trilinear coupling develops a significant contribution for Higgs-pair production closer to the production threshold, while it dies out for large invariant Higgs-pair masses. In the last range, however, statistics will be small in experiment so that the bulk of reconstructed events will emerge from the region closer to the threshold.
The gluon-fusion mechanism gg → HH is mediated by top-and to a much lesser extent bottom-quark loops, see Fig. 1. The full next-to-leading-order (NLO) QCD corrections have been calculated by a time-consuming numerical integration of the corresponding two-loop integrals, since there are no systematic analytical methods to calculate the corresponding two-loop integrals [9,10]. Similar to the single-Higgs case they enhance the cross section by about 100%. Because the invariant mass of the final-state Higgs-boson pair is significantly larger than in the single-Higgs case, the heavy top-quark limit (HTL) works less reliably for Higgs-boson pairs. The full NLO QCD corrections result in a decrease of the total cross section by about 15%, due to finite NLO top mass effects beyond the heavy-top limit, at the LHC for a c.m. energy of 14 TeV. This shows that the heavy-top limit for the relative QCD corrections [7] works still quite well for the total cross section also in the Higgs-pair case. For the exclusive cross section at large invariant Higgs-pair masses, however, the finite mass effects at NLO can reach a level of −30%. The next-to-NLO (NNLO) QCD corrections to the total cross section have been obtained in the heavy top-quark limit. They imply an additional moderate rise of the total cross section by about 20% [11]. Recently, the next-to-NNLO (N 3 LO) QCD corrections to the total cross section became available and turned out to be small, affecting the total cross section at the few per-cent level only [12]. NNLO top mass effects have been estimated to about 5% by means of a heavy top-quark expansion of the 2-loop virtual corrections [13]. Beyond NNLO, the next-to-next-to-leading-logarithmic (NNLL) soft and collinear gluon resummation contributes 5-10% to the total cross section [14]. The factorization and renormalization scale dependence has been reduced to about 5%. In order to obtain an estimate of the residual theoretical uncertainties, however, the uncertainties due to the scheme and scale choice of the virtual top mass have to be taken into account as well. These latter effects increase the theoretical uncertainties to a level of 20-25% [10]. The electroweak corrections to this process are unknown. They are expected in the 10%-range for the total cross section, but larger in the tails of the distributions.
In this work we investigate the electroweak corrections induced by the top-Yukawa coupling as a uniquely defined contribution to the full electroweak corrections. In Section 2, we will define our notation and the corresponding leading-order (LO) result for gg → HH. In Section 3, we describe the effective Higgs (pair) couplings to gluons in the HTL and the effective trilinear Higgs coupling within the effective-potential approach, where the top contributions are integrated out. Section 4 describes the NLO calculation and Section 5 our results with a discussion of our findings. In Section 6, we conclude. • λ Figure 1: Diagrams contributing to Higgs-boson pair production via gluon fusion. The contribution of the trilinear Higgs coupling is marked in red.
The LO Higgs pair production via gluon fusion is mediated by heavy top-loop contributions and a marginal contribution of bottom loops, see Fig. 1. In this work we neglect the bottom-loop contributions and take into account the top loops only. The Higgs-boson pair production cross section at LO is given by where L gg denotes the gluonic parton luminosity given in terms of the gluon densities at the factorization scale µ F , and the integration boundary is given by τ 0 = 4M 2 H /s, where s denotes the hadronic center-of-mass (c.m.) energy squared and M H the Higgs mass. The scale Q 2 = M 2 HH is defined in terms of the invariant mass M HH of the Higgs pair at LO.

The LO partonic cross section can be cast into the form
and the symmetry factor 1/2 for the identical Higgs bosons in the final state is included.
involves the trilinear Higgs coupling that is related to the Higgs mass and the vacuum expectation value (vev) v at LO, where the vev is related to the Fermi constant G F = 1/( √ 2v 2 ). The factor α s (µ R ) denotes the strong coupling at the renormalization scale µ R . The form factors F △ of the LO triangle diagrams and F , G of the LO box diagrams can be found in Refs. [15]. In the HTL they approach simple expressions, F △ → 2/3, F → −2/3 and G → 0.

Effective Lagrangians
In this section we address the effective gluonic single-and double-Higgs couplings as well as the effective Higgs self-couplings after integrating out the heavy-top contributions, i.e. the effective couplings valid in the HTL at the leading order of an inverse large top-mass expansion.

Gluonic Higgs couplings
In the HTL, the top-Yukawa-induced electroweak corrections to the effective Hgg and HHgg couplings can be obtained as where G a µν denotes the gluonic field-strength tensor and H the SM Higgs field. The radiatively corrected coefficients are given by where C 1 describes the genuine corrections to the Hgg and HHgg vertices [16] (see Fig. 2a) and C 2 the universal top-Yukawa-induced correction related to the Higgs wave-function and vacuum expectation value [17] (see Fig. 2b). This yields the explicit effective Hgg and HHgg couplings, where This effective Lagrangian describes the electroweak corrections induced by x t to the Hgg and HHgg vertices in the HTL and will be used in this limit in the following. We would like to point out explicitly that the square root of the wave-function counterterm of the external Higgs boson(s) is already taken into account in this effective Lagrangian.

Higgs self-couplings
The starting point of effective Higgs self-couplings is the effective one-loop corrected Higgs potential involving virtual top-quark effects of the SM [18], with the bare Higgs self-coupling λ 0 , the SM Higgs doublet in unitary gauge, the loop coefficient and the field-dependent top-mass parameter The expression above for the effective Higgs potential involves the 't Hooft scaleμ. After minimization of the effective Higgs potential by the tadpole equation, and the renormalization of the Higgs mass, the effective Higgs trilinear (quartic) self-coupling can be obtained by the third (fourth) derivative of this effective Higgs potential with respect to the physical Higgs field H, with These effective NLO couplings are the relevant Higgs self-interactions in the HTL and will be compared with the full triple-vertex corrections within this work.

Top-Yukawa-induced electroweak corrections to Higgs pair production
The top-Yukawa-induced electroweak corrections arise from NLO diagrams involving topquark loops as shown in Fig. 3, where the tadpole diagrams are displayed explicitly. For simplicity, we will use the relative corrections of Eq.
while the LO form factor G does not receive top-Yukawa-induced electroweak corrections in our approach, since G vanishes in the HTL. The top-Yukawa-induced radiative corrections in Eq. (18) read as where the vertex, self-energy and counterterm corrections are given by We are adopting the scalar integrals in n = 4 − 2ǫ dimensions, . (21) In the expression of Eq. (20), the self-energy Σ H (Q 2 ) and its derivative Σ ′ H (Q 2 ), the tadpole term T 1 /v, the trilinear Higgs-coupling counterterm δλ HHH and the Higgs-mass counterterm δM 2 H are given by where the self-energies Σ H , Σ W and the Higgs-mass counterterm include tadpole contributions as well, and we only kept terms of O(m 4 t ) and O(m 2 t ) for the counterterms to be consistent. For the calculation, we have used the alternative tadpole-scheme of Ref. [19] 1 and implemented the electroweak parameters in the G F scheme, i.e. choosing G F , M Z , M W as input parameters for the electroweak gauge sector, while the Weinberg angle θ W and the QED coupling α are derived quantities. In addition, we have taken into account that the effective Lagrangian of Eq. (8) contains the wave-function renormalization of the external Higgs fields that has to be compensated in the corrections ∆ HHH to avoid double counting. Within our electroweak renormalization, the trilinear coupling is given by its LO expression in terms of the renormalized Higgs mass and vacuum expectation value of Eq. (5). We will compare the explicit NLO result of Eq. (18) to the corresponding one using the effective trilinear coupling λ ef f HHH , i.e. adding the corresponding matching term with ∆λ HHH of Eq. (17) to avoid double counting and using the effective coupling λ HHH → λ ef f HHH of Eq. (16) for the triangle coefficient C △ in Eq. (3) in both the LO and NLO expressions.
The relative electroweak corrections to the Higgs-pair production cross section are defined by expanding the expression of Eq. (3) up to NLO by using the corrected form factors of Eq. (18) at the parton level, such that the hadronic cross section is corrected as Within this expression we will either use the LO expression of the triple Higgs coupling λ HHH of Eq. (5) or the radiatively-corrected effective coupling λ ef f HHH of Eq. (16) with the according form of the radiative corrections as shown in Eq. (23).   Fig. 4 as a function of the invariant Higgs-pair mass M HH . The full lines denote the real parts and the dashed line the imaginary part. The blue curves exhibit the real and imaginary parts of ∆ HHH in terms of the LO trilinear Higgs coupling, while the red curve shows the correction factor after introducing the effective coupling λ ef f HHH . The size of the correction factor shows that the effective trilinear coupling does not capture the dominant part of the electroweak corrections so that its use is not supported by our results.

Results
The relative electroweak corrections originating from the top-Yukawa-induced contributions are shown in Fig. 5 for the differential cross section as a function of the invariant Higgs-pair mass M HH . The radiative corrections close to the production threshold turn out to be large. This is due to the vanishing of the matrix element in the leading term in the large inverse top-mass expansion of the LO expression of Eq. (3) so that the LO matrix element is highly suppressed at threshold. This suppression, however, is lifted by the radiative corrections to the effective trilinear Higgs coupling λ ef f HHH or, equivalently, the mismatch of electroweak corrections to the triangle and box diagrams. However, Fig. 5 Figure 5: The relative top-Yukawa-induced electroweak corrections to the differential Higgspair production cross section as a function of the invariant Higgs-pair mass M HH . The blue curve shows the electroweak correction factor using the LO trilinear Higgs coupling of Eq. (5) and the red curve the corrections factor involving the effective coupling λ ef f HHH of Eq. (16). The black dotted line at the value 1 is inserted to guide the eye. The electroweak corrections factor is independent of the hadronic c.m. energy and scale choices in the QCD part of the differential cross section dσ/dM HH so that it is valid for any hadronic energy as a pure rescaling factor. support the use of the effective trilinear Higgs coupling λ ef f HHH to improve the perturbative result. Thus, the naive argument that the effective trilinear Higgs coupling induces a SM contribution to κ λ , is not supported by our results, but the inclusion of the complete electroweak corrections is mandatory instead. We observe that the electroweak corrections appear with opposite sign close to the threshold between the options of using the LO and the effective trilinear coupling.
The effect of the top-Yukawa-induced electroweak corrections on the total integrated hadronic cross section amounts to so that the corrections induce an effect of about 0.2% on the total cross section, if the LO-like trilinear Higgs coupling λ HHH is adopted. The bulk of these corrections cannot be absorbed in the effective triple Higgs coupling, but the latter option leads to an artificial increase of the relative electroweak corrections.

Conclusions
In this note we have investigated the electroweak corrections to Higgs-pair production via gluon fusion induced by top-quark contributions. While keeping the full top-mass dependence in the triple-Higgs vertex and self-energy corrections, we have worked in the HTL for the radiative corrections to the effective ggH(H) vertices for the relative corrections. The top-Yukawa-induced NLO electroweak corrections to the total gluon-fusion cross section amount to about 0.2%. After integrating out the top-quark contributions an effective trilinear Higgs coupling can be defined in terms of the effective Higgs potential that is dressed with contributions scaling with the fourth power of the top mass. This is known already starting from the Coleman-Weinberg potential [18]. This effective trilinear Higgs coupling can be introduced in the full calculation of electroweak corrections as well and leads to a modification of the counterterms in order to remove potential double counting of corrections. However, introducing this effective coupling the remaining electroweak corrections turn out to be larger than in the case of the LO-like triple Higgs coupling.