Radiative Corrections to Aid the Direct Detection of the Higgsino-like Neutralino Dark Matter: Spin-Independent Interactions

The lightest neutralino ($\tilde{\chi}_1^0$) is a good Dark Matter (DM) candidate in the R-parity conserving Minimal Supersymmetric Standard Model (MSSM). In this work, we consider the light higgsino-like neutralino as the Lightest Stable Particle (LSP), thanks to rather small higgsino mass parameter $\mu$. We then estimate the prominent radiative corrections to the neutralino-neutralino-Higgs boson vertices. We show that for higgsino-like $\tilde{\chi}_1^0$, these corrections can significantly influence the spin-independent direct detection cross-section, even contributing close to 100\% in certain regions of the parameter space. These corrections, therefore, play an important role in deducing constraints on the mass of the higgsino-like lightest neutralino DM, and thus the $\mu$ parameter.

At the Large Hadron Collider (LHC), the discovery of the Higgs boson has established the Standard Model (SM) physics.However, in spite of strong motivation for physics beyond the standard model, no hints for the same have been observed so far.The constraints on the supersymmetric spectrum [11][12][13][14][15][16][17], while generally dependent on the nature of the low-lying states, have been raising concerns about the "naturalness" requirements [6][7][8][9][10].In this regard, within the minimal supersymmetric standard model (MSSM) paradigm, the constraint on the µ parameter is of particular interest.This has been widely studied in literature in the light of LHC. 1 In the minimal construct, a small higgsino mass parameter µ (of O(100) GeV) is of relevance. 2Assuming that the gaugino mass parameters M 1 and M 2 are in the ballpark of multi-TeV, such a scenario leads to a compressed higgsino spectrum.In the R-parity conserving scenario, where the lightest supersymmetric particle (LSP) is stable.This leads to the higgsino-like lightest neutralino being a Dark Matter (DM) candidate.Such a scenario attracts rather weak constraints from the electroweakino searches at the LHC, as the decay of the next two heavier (higgsino-like) states lead to soft SM particles in the final states at the collider.
In the present work, we revisit the implications of the spin-independent direct detection constraints on the higgsino-like ( χ0 1 ).As the coupling of CP-even neutral Higgs bosons with a pair of χ0 1 is vanishingly small at the tree-level, the contribution to the spin-independent neutralino-nucleus interaction process is suppressed. 3Consequently, the radiative contributions to the scattering process need to be considered in order to accurately estimate the relevant cross-sections.While such a scenario has been previously considered in the literature, in the context of pure-higgsinos, the importance of radiative corrections to the direct detection process has received some attention [47][48][49][50]. 4  We improve the estimation of the radiative corrections to the spin-independent direct detection crosssection, by incorporating the contributions from the gauge bosons, the Higgs bosons, the respective superpartners and the third generation (s)quarks to the relevant vertices involving neutralino and Higgs bosons.
Further, we renormalize the chargino-neutralino sector using the use the on-shell renormalization scheme and estimate the relevant vertex counterterms, thus paving the way towards a full one-loop treatment to the neutralino-Higgs boson vertices.
In order to satisfy the thermal relic abundance of Ω DM h 2 ≃ 0.12, as required by cosmological considerations [59,60], the higgsino-like neutralino LSP must be around 1 TeV and can be lowered further in the 1 Within the high scale supersymmetric models, a moderate value of the µ parameter may be realized in the focus point region [3,[18][19][20][21][22]. 2 Note that, the fine-tuning measure, estimated following the "electroweak" naturalness criteria, is stated to be about O(10 − presence of co-annihilation [33].Below this mass scale, it is generally under abundant.However, there are viable non-thermal production scenarios, where adequate production of such DM may be possible in the early Universe [61,62].Further, the presence of additional DM components, e.g., axions, may contribute to the DM abundance [44,63,64].In this work, we will not concern ourselves with satisfying the thermal relic abundance in the early Universe.We will only focus on the impact of certain radiative corrections on such a DM candidate in the light of direct DM searches.Note that if the LSP constitutes only a fraction of the required DM relic abundance (and, therefore, the local DM density), the constraint on the DM-nucleon scattering cross-section from direct searches will be relaxed in the same proportion.This article is organized as follows: in II, the chargino-neutralino spectrum of interest has been described, and the tree-level interactions between χ0 1 and the CP-even Higgs bosons are described.Following this, in III, the generalities of (spin-independent direct detection of DM and the implications in the context of a higgsino-like DM candidate have been discussed.Subsequently, in IV, we present the important electroweak radiative corrections to the vertices involving neutralino and the Higgs bosons and study its impact on the spin-independent DM-nucleon cross-sections.We also reflect on the parameter region where such corrections are significant and comment on the implications on the viable region for the higgsino mass parameter µ.Finally, in V, we summarize the results and conclude.

II. THE FRAMEWORK
In this section, we briefly discuss the chargino-neutralino sector in the MSSM; in particular, we focus on the parameter region with rather small higgsino mass parameter µ and light higgsino-like states.
The mass matrix M c can be expressed as, Here M 2 and µ stand for the supersymmetry breaking SU (2) wino mass parameter and the supersymmetric higgsino mass parameter, respectively.M W is the mass of the W boson, and tan β is the ratio of the vevs of the up-type and the down-type CP-even neutral Higgs bosons.The matrix M c can be diagonalized with a bi-unitary transformation using the unitary matrices U and V to obtain, The eigenstates are ordered such that m χ+ The left-and right-handed components of these mass eigenstates, the charginos ( χ+ i with i ∈ {1, 2}), are where P L and P R are the usual projectors, ψ − j = ψ − † j , and summation over j is implied.For the neutralino states, in the gauge eigenbasis [consisting of the bino ( B 0 ), neutral wino ( W 3 ), and down-type and up-type neutral higgsinos ( h0 1 and h0 2 respectively)], , the mass term takes the following form [65]: The neutralino mass matrix M n is given by, In the above equation s W , s β , c W and c β stand for sin θ W , sin β, cos θ W and cos β respectively while θ W is the weak mixing angle.M Z is the mass of the Z boson, and M 1 is the supersymmetry breaking U (1) Y gaugino (bino) mass parameter.M n can be diagonalized by a unitary matrix N to obtain the masses of the neutralinos as follows, The eigenstates ( χ0 i ) are ordered according to the respective mass eigenvalues as follows, These eigenstates satisfy χ0c i = χ0 i , where the superscript c stands for charge conjugation.The left-handed components of these mass eigenstates, the Majorana neutralinos, χ0 i (i ∈ {1, 2, 3, 4}), may be obtained as, where summation over j is again implied.
The analytical expressions corresponding to the chargino and the neutralino mass eigenvalues have been obtained in the literature [66,67].However, a numerical estimation of the eigenvalues is straightforward and convenient, especially in the case of the neutralinos.
In the region of interest in this article, the higgsino mass parameter is rather small in comparison with the gaugino and the wino mass parameters, i.e., |µ| ≪ |M 1 |, M 2 ; the masses of the light higgsino-like particles may be approximately given by [47,68] 6 m χ± In the above expression, subscripts a(s) refer to anti-symmetric (symmetric) combinations of up-type ( h0 2 ) and down-type ( h0 1 ) higgsinos constituting the respective mass eigenstates.Here, the symmetric and anti-symmetric states refer to the higgsino-like states with compositions without and with a relative sign between N i3 and N i4 respectively.It has been pointed out in the literature that, due to the mixing effects, the mass differences ∆m 1 = m χ± 1 − m χ0 1 may become very small in certain regions of the parameter space [27,29,71,72].In the present context, we will consider the mass differences ∆m 1 , ∆m 2 ≫ O(1 MeV).
Thus, as we will discuss in the next section, in the direct detection experiments, only the elastic scattering of χ0 1 with the nucleon will be relevant.

B. Neutralino-Higgs boson(s) Interaction : Tree-level and at One-loop
As we will elaborate in III, for the spin-independent direct detection of χ0 1 , the relevant vertices involve the lightest neutralino and the CP-even Higgs bosons.The gauge symmetry of the MSSM, particularly the electroweak gauge group, prohibits any superpotential term with two higgsino states and a Higgs boson in the gauge eigenbasis.Therefore, the tree-level interaction term, in the gauge eigenbasis, involves one higgsino, one gaugino, and a Higgs boson.Consequently, in the mass eigenbasis, the tree-level vertex takes the following form [65], where In the above expressions, h 1 and h 2 denote the 125 GeV Higgs boson and the heavy Higgs boson mass eigenstates, respectively, g 2 denotes the SU (2) gauge coupling, and α denotes the mixing angle in the CPeven Higgs sector.Note that in the above equations, N 11 and N 12 denote the bino and wino composition of the lightest neutralino mass eigenstate, while N 2 13 and N 2 14 denote the respective down-type and up-type Higgsino fractions in the χ0 1 respectively.For higgsino-like χ0 1 , the gaugino fraction is very small compared to the higgsino fraction, i.e., Thus, the tree-level vertex involving the CP-even Higgs bosons is suppressed by a rather small gaugino component of the mixing matrix (i.e., N 11 and N 12 ).As pointed out, the radiative corrections to these vertices play an important role in the spin-independent direct detection process; this will be illustrated in IV.
To treat the interaction Lagrangian at one-loop level, the counterterm Lagrangian L CT should be added to the tree-level Lagrangian L tree .The lagrangian, thus takes the following form : where L Born is written using the renormalized fields, and L CT involves the contributions from the relevant counterterms.The "bare" and the renormalized neutralino mass eigenstates are related as follows where the index j has been summed over j ∈ {1, 2, 3, 4}.The wave-function renormalization counterterms δZ ij are determined using the on-shell renormalization schemes [73], a comparison among different variants can be found in ref. [74].Similarly, for the CP-even neutral Higgs bosons the 'bare" and the renormalized mass eigenstates are related as follows, where the index j has been summed over j ∈ {1, 2}.The on-shell renormalization prescription is used to determine the wave-function renormalization counterterms δZ H ij .With the above relations, the counterterm Lagrangian L CT relevant for the present discussion can be obtained as follows where the renormalized fields are used in the counterterm Lagrangian; we have dropped the respective superscript.At the one-loop level, the neutralinos, charginos, gauge bosons, and the Higgs bosons contribute to the χ0 1 , χ0 1 , h i vertices.Further, there can be sizable contributions from the third-generation (s)quarks, thanks to the sizable Yukawa couplings, as will be discussed in IV.Note that for all our benchmark scenarios, as described in IV, the lightest eigenvalue of the neutralino mass matrix M n is positive.Consequently, for the benchmarks presented in IV (I and II),

C. Constraints on the Parameter Space
A small µ ≪ |M 1 |, M 2 , as discussed above, leads to a compressed spectrum with three closely spaced states χ0 1 , χ0 2 , χ± 1 .The LHC sets stringent limits on chargino and neutralino masses, from pair productions of charginos and neutralinos and the subsequent decay of those to χ0 1 and SM particles.These limits are sensitive to the mass difference of the heavier chargino and neutralino states and the LSP ( χ0 1 ).As for small mass splittings, the relevant bounds on the compressed spectrum can be found in ref. [15,[75][76][77].
For 300 (600) GeV higgsino-like neutralinos, ∆m 1 0.3 (0.2) GeV is disfavored [77], and the constraints weaken for heavier mass.Searches targetting mass splittings around the electroweak scale may be found in ref. [11][12][13][14][15][16][17], where decays of the heavier neutralinos into on-shell gauge bosons or Higgs bosons and the LSP, as well as their three-body decays have been considered.However, as |M 1 |, M 2 ≫ |µ| in our context, these constraints are not very relevant to the present discussion.We have considered the following constraints on the spectrum for our benchmark scenarios presented in IV.
• We have constrained the lightest CP-even Higgs mass m h within the range : 122 ≤ m h (GeV) ≤ 128 [78][79][80].Note that the experimental uncertainty is about 0.25 GeV and the uncertainty in the theoretical estimation of the Higgs mass is about ±3 GeV, see e.g.[81] and references there.
• The squarks and the sleptons masses have been assumed to be above 1.5 TeV, and the gluino mass is kept above 2.2 TeV, respecting the constraints from the LHC.
• However, we have relaxed the constraints on the relic density of DM (i.e., Ω DM ≃ 0.12).As χ0 1 may not constitute all of the DM, the constraint from indirect searches on the higgsino-like DM [36,41,42] has also been relaxed.

III. DIRECT DETECTION OF DARK MATTER: IMPLICATIONS FOR A HIGGSINO-LIKE LSP A. Generalities of Direct Detection
In this section, we describe the generalities of spin-independent direct detection and sketch the implications for the higgsino-like χ0 1 − nucleon scattering.In the context of direct detection, the differential event-rate per unit time at a detector, as a function of the nuclear recoil energy E R , is given by, In the above equation, ρ χ0 1 is the local density (≃ 0.3 GeV cm −3 ), n T is the number of target nuclei in the detector and σ(v, E R ) denotes the scattering cross-section with the nucleus.Further, f E ( v) denotes the velocity distribution in the Earth's rest frame and f E ( v) = f ( v + v E ), where f is the distribution function in the galactic rest frame and v E is the velocity of Earth with respect to the galactic rest frame and v esc is the escape velocity of our galaxy.Further, is the minimum speed of the DM particle required to impart a recoil energy E R , where m T is the mass of the target nucleus, and M r is the reduced mass of the DM-nucleus system.The cross-section with a nucleus (atomic number A and charge Z) is given by, where m T is the mass of the target nucleus, q 2 = 2m T E R is the square of the momentum transfer, and F stands for the form factor, which will be taken as the Woods Saxon form factor [91].Further, σ is the (spin-independent) DM-nucleus scattering cross-section.In the present context only spin-independent cross-section is relevant, which, at zero momentum transfer is given by σ 0 .

B. Dark Matter-nucleon spin-independent elastic scattering
In the following, we briefly discuss the relevant parton level effective Lagrangian leading to the the spin-independent interaction [92], In the above equation, the first term in the right-hand-side (RHS) receives contributions largely from the scattering processes mediated by the Higgs bosons.In particular, in the limit of no mixing in the squark sector, the contribution from the squark sector to this operator vanishes [92,93].The next term captures the effect of squark-mediated s−channel scattering processes.Further, L g eff denotes the relevant effective interactions with gluons which contribute to the spin-independent neutralino nucleon scattering process [92,93].
We now focus on the implications for a higgsino-like χ0 Consequently, the gaugino fraction in the lightest neutralino is typically O(10 −2 ).Thus, the tree-level Higgs boson exchange contributions, while small, can be significant and generally non-negligible.The tree-level contributions from the (s−channel) squark-mediated processes are suppressed by an additional factor of a rather small gaugino fraction in χ0 1 and/or an additional factor of Yukawa coupling for the first two generation (s)quarks as compared to the tree-level Higgs boson exchange processes.Further, we have considered the first two generations of squarks to be very heavy (≫ O(2)) TeV for all our benchmark scenarios, as will be described in IV.Therefore, contributions from the respective squark-mediated processes (and their contributions to the neutralino-gluon effective operators) remain sub-dominant in the present context.In the following, we first describe the Higgs exchange contribution, as the focus of the present study is on the radiative corrections to the neutralino-Higgs boson vertices.
The effective parton-level interactions, as mentioned in (20) leads to the following effective interaction Lagrangian with the nucleon N ∈ {n, p}, where n and p stand for neutron and proton, respectively : where f N denotes the effective coupling and ψ N denotes the field describing the nucleon N .The important contributions from the two CP-even neutral Higgs boson mediated processes in the spin-independent cross-section σ SI comes from its contribution to the coefficient λ q .The contribution from the two CP-even neutral Higgs bosons λ H q is given by, In this expression, C i = C L i as mentioned in 14 and C iq denotes the coupling of the same Higgs boson and quark(q) [65].The respective contribution to the spin-independent elastic scattering cross-section may be expressed in terms of their contribution to the effective interaction strength f N [92,94,95], where [92,96], In the above equation, f N Tq denotes the contribution of the (light) quarks q ∈ {u, d, s} to the mass m N of the nucleon N . 7urther , g S denotes gauge coupling for the strong interaction; m qj and C i qj denote the mass of the j-th squark and its coupling with the i-th (CP-even) neutral Higgs boson, respectively.The heavy quarks ({c, b, t}) contribute to f N through the loop-induced interactions with gluons.In 23, the first term includes a contribution from the effective neutralino-quark interactions; the second and the third term includes the contributions from the effective interaction with the gluon fraction.In particular, the second term (proportional to f N T G λ H q mq ) and the third term (proportional to f N T G m N T q) include the relevant contributions from the heavy quarks and all the squarks to the Higgs bosons-gluon effective vertices respectively.
A brief discussion on various other important contributions to the DM-nucleon scattering is in order.In addition to the contribution from the Higgs boson exchange processes, there are tree-level contributions to f N from squark exchange processes.As already mentioned, in the present discussion we assume the (first two generations of) squarks to be very heavy.In such a scenario, the dominant contribution to the spinindependent neutralino-nucleon interaction is mediated by the Higgs bosons.Further, the contribution to f (H) N from the term proportional to T q, which incorporates the squark contributions to the effective vertices involving Higgs bosons and gluons, is suppressed for heavy squark masses.In the present context, the only the third generation squarks are relatively light, around 1.5 TeV.Regarding other important radiative corrections, the supersymmetric-QCD corrections to the Higgs and the down-type quark vertices [100]; the one-loop corrections to the neutralino-gluon interactions originating from the triangle vertex corrections involving (s)quarks at the Higgs-gluon-gluon vertex, and also the box-diagrams involving (s)quarks can be sizable [92].These contributions have been implemented in the numerical package micrOMEGAS [91] following ref.[92]. 8Further, contributions from the box diagrams to the DM-quark scattering involving electroweak gauge bosons have been considered in the literature [48][49][50].In ref. [50], it has been shown that the tree-level Higgs boson exchange contribution to the (higgsino-like) neutralino-nucleon scattering dominates over these contributions when the gaugino mass parameters are less than O(5 − 10) TeV, as is relevant in the present context.
As is evident from the discussion above, the interaction rate is proportional to f 2 N , which involves the square of the LSP-Higgs bosons vertices C i .As discussed above, several dominant one-loop contributions to the scattering process have been estimated and incorporated in the publicly available packages, e.g., micrOMEGAs [91,101,102].However, a detailed estimation of the one-loop corrections from the modification of the neutralino-Higgs boson vertices C i have not received adequate attention. 9In particular, for an almost pure higgsino-like LSP, which is often relevant for a natural supersymmetric spectrum, the small (but generally non-vanishing) gaugino fractions imply that the tree-level value of C i to be small.Therefore, radiative corrections to the same vertices can play a crucial role in the estimation of the cross-section.In 1, some of the important diagrams contributing to the vertex correction have been depicted.We consider all the triangle diagrams involving charginos, neutralinos, gauge bosons and Higgs bosons which contribute to the vertex corrections to the χ1 − χ1 − h i vertices.Further, as the Yukawa couplings for the third generation (s)quarks are large, contributions from the third generation (s)quarks have also been considerd.
As the loop diagrams with two fermions and one boson are generally Ultra-Violet (UV) divergent, we have included the vertex counterterms, and ensured the UV finiteness of the overall contributions.Note that the wave function renormalization counterterms also include the effect of mixing of the tree-level fields (due to radiative corrections from the two-point functions) appearing in the external lines.The complete set of radiative contributions considered in this work have been described in Appendix B, and the counterterms have been mentioned in Appendix C.

IV. RESULTS
In this section, we present the results highlighting the importance of the radiative corrections to the vertices involving neutralino and Higgs bosons, as discussed in III, to the (spin-independent) direct detection process in the context of a higgsino-like χ0 1 .

A. Implementation
We begin by describing the procedure to compute the radiative corrections.The steps have been sketched in the flowchart shown in 2.
The parameters are read from the output file and the relevant radiative corrections are numerically evaluated using those parameters.The input parameters M 1 , M 2 and µ in the chargino-neutralino sector are varied to obtain different benchmark scenarios, as presented in I and II.The details of the benchmark scenarios will be discussed in the next subsection.
• To evaluate the radiative corrections to the χ0 1 − χ0 1 − h i vertices, we have used the publicly available packages FeynArts (version 3.11) [104,105], FormCalc (version 9.10) [106], and LoopTools (version 2.15) [106].In particular, the Feynman diagrams are evaluated using FeynArts, and the vertex corrections are calculated using FormCalc.Further, the radiative contributions are expressed in terms of the Passarino-Veltman integrals (briefly discussed in Appendix A) and numerically evaluated for the benchmark scenarios using FormCalc and LoopTools.Further, the UV finiteness of the radiatively corrected vertex factors (including the counter-term contributions) have been numerically checked using the packages mentioned above.Finally, the numerical results are stored in data files.
To determine the relevant counterterms, we have used the on-shell renormalization scheme.We have used the relevant counterterms implemented in FormCalc, as described in ref. [107].
• To evaluate the direct detection cross-sections micrOMEGAs [91,101,108,109] (version 5.2.1) [102] has been used.We generated the model files for micrOMEGAs using SARAH (version 4.14.5)package [110] on the Mathematica platform.We modify the relevant vertices in the code to include the radiatively corrected vertices.For each benchmark, then, the corrected vertices are read from the output file, as described above, using a subroutine.Thus, the radiatively corrected vertices are used to evaluate the spin-independent direct detection cross-section.

B. Benchmark Scenarios
In this subsection, the benchmark scenarios have been discussed.The benchmark points have been described in I and II.
As we focus on the higgsino-like χ0 1 DM, |µ| ≪ |M 1 |, M 2 have been set for all the benchmark scenarios.The tree-level vertices C L/R i , as described in 14, are proportional to the product of the gaugino and higgsino components of χ0 1 .Thus, the tree-level spin-independent cross-section (σ SI ) is sensitive to the variation in the gaugino-higgsino mixing.With |M 1 |, M 2 , and |µ| fixed, the gaugino-higgsino mixing is sensitive to the signs of M 1 and µ.Consequently, the tree-level spin-independent cross-sections and the relative contributions from the radiative corrections to the χ0 1 − χ0 1 − h i vertex factors can be very different even for very similar chargino-neutralino masses.In the benchmark scenarios, with |µ| ≪ |M 1 |, M 2 ; we have varied the sign of µ and M 1 to illustrate this variation.Further, the order of M 1 and M 2 have been altered to study the effect of the variation in the gaugino components.
The benchmark points BP-1a to BP-6a, as shown in I reflect scenarios with |µ| = 300 GeV.Setting 1 , χ0 2 , χ± 1 are closely spaced and are higgsino-like states.For the benchmark scenarios BP-1b to BP-6b, as shown in II, a heavier |µ| = 600 GeV has been considered.As discussed above, to illustrate the variation in the gaugino components of χ0 1 for very similar particle spectra, the sign of µ and the sign of M 1 have been varied.For BP-1a and 2a, with µ = 300 GeV and µ = −300 GeV respectively, M 1 is set to −5 TeV.For BP-3a and 4a, with µ = 300 GeV and µ = −300 GeV respectively, M 1 is set to 5 TeV.For all these benchmark scenarios, we fix M 2 = 4 TeV.For BP-5a and 6a, with µ = 300 GeV and M 2 = 5 TeV, while M 1 assumes −4 TeV and 4 TeV respectively.BP-1b to BP-6b resembles BP-1a to BP-6a respectively, only with |µ| = 600 GeV.Note that for BP-1 to BP-4 (a and b), while for BP-5 and BP-6 (a and b) |M 1 | < M 2 .For all these benchmark scenarios tan β = 10, the masses of the Higgs bosons and the third-generation squarks, which are also relevant for the present study, have been kept fixed.Further, constraints from LHC on such compressed spectra have been taken into account.
For all the benchmark scenarios, χ0 1 corresponds to a mass eigenstate with positive eigenvalue.In the benchmark scenarios with a negative µ parameter, i.e.BP-2a, BP-2b, BP-4a and BP-4b, χ0 1 is the symmetric higgsino-like state.For all other benchmarks (with positive µ parameter) χ0 1 is the antisymmetric higgsino-like state.Note that, irrespective of the sign of M 1 , the gaugino components in χ0 1 are reduced  substantially for negative µ (where χ0 1 is the symmetric state) as compared to positive µ (where χ0 1 is the symmetric state).This is evident from comparing the wino and the bino components (N 12 , N 11 respectively) of χ0 1 in BP-1a(b) and BP-2a(b) respectively.In particular, the wino component is reduced by approximately 50% and 25% for benchmarks BP-1a(b) and BP-2a(b), respectively.The bino content, which contributes subdominantly, follows a similar trend, although by a smaller margin.As the tree-level χ0 1 − χ0 1 −h i vertices are directly proportional to the gaugino fraction, the change in sign of the higgsino mass  parameter µ leads to a significant change in the tree-level spin-independent direct detection cross-section.

BP
. σ SI denotes spinindependent cross-section (with proton) including the radiative corrections and ∆σ SI (%) denotes the percentage contribution to the same from the radiative corrections under consideration.In the third and the fourth column title, "Total" refers to total percentage correction to C L/R i , "CT" refers to the percentage contribution from the counter-term vertex, "Loop" denotes the percentage contribution from the one-loop diagrams, and "SQ" denotes the percentage contribution from the third-generation quarks and squarks running in the loops.
In this section, we discuss the numerical results.The radiative corrections to the χ0 1 − χ0 1 − h i vertices for the benchmark scenarios, as described in I and II, have been computed and have been presented in III.In III, the one-loop corrected χ0 ) for the respective benchmark scenarios (as mentioned in the first column) have been presented in the second column.In the third and the fourth column the percentage contribution from the radiative corrections to the χ0 1 -proton spin-independent scattering cross-sections ∆C × 100% have been described for i = 1 and i = 2 respectively.Note that in the present scenario, the results are similar for χ0 1 -neutron spin-independent scattering cross-sections.For estimating the radiative corrections, contributions from the loops involving all the neutralinos and charginos, gauge bosons, Higgs bosons, and third-generation (s)quarks have been considered.
Individual contributions from all the loops, counterterms, and also the third generation (s)quarks to the respective vertices have been mentioned.Finally, the radiatively corrected χ0 1 − nucleon cross-section and the percentage contribution to the same ∆σ SI = σ SI − σ SI tree σ SI tree × 100% have been presented in the fifth column.In the above discussion, the subscript "tree" denotes the respective quantities without including the radiative corrections considered in this article.As discussed in the previous section, we have used FeynArts, FormCalc, and LoopTools for the numerical evaluation of the radiative contributions and the relevant counterterms.
As described in I and II, for all the benchmark scenarios χ0 1 is dominantly higgsino-like.The higgsino fraction (HF = |N 13 | 2 + |N 14 | 2 ) is above 99%.The radiative corrections to the χ0 contributes dominantly to the spin-independent cross-section σ SI .The contribution to the spinindependent cross-section σ SI from the heavy Higgs boson h 2 is only about 3% for all the benchmark scenarios.This is because m h 2 ≫ m h 1 (about ten times) in the present context.Therefore, its contribution to the χ0 ) is suppressed, as can be inferred from 22. Thus, for all the benchmark scenarios, the percentage corrections to the cross-section σ SI are approximately twice that of the percentage corrections to the χ0 The radiative corrections to χ0 1 − χ0 1 − h 1 /h 2 vertices are significant for all the benchmark scenarios and vary between approximately 9%-40% for the light Higgs boson vertex and between approximately 5%-21% for the vertex involving the heavy Higgs boson.Comparing the first eight benchmarks (BP-1a to BP-4b), the percentage change in the χ0 1 − χ0 1 − h 1 vertices are significant for the benchmarks with negative µ (BP-2a, BP-2b and BP-4a, BP-4b), as compared to their counterparts with positive µ (BP-1a, BP-1b and BP-3a, BP-3b).Let us consider BP-1a(b) and BP-2a(b).While BP-1a(b) and BP-2a(b) only differ by the sign of µ, thus, the percentage contribution to C L/R 1 from the radiative correction for BP-2a(b) is significantly higher as compared to BP-1a(b).This is largely because of a substantial reduction in the tree-level vertex factor for BP-2a and BP-2b, while the radiative corrections are also marginally higher.Note that, for positive (negative) µ, χ0 1 is the symmetric (anti-symmetric) higgsino-like state.A similar argument explains the larger percentage corrections in the context of BP-4a(b), as compared to BP-3a(b).It follows from 14, that for the symmetric states, the respective tree-level vertex suffers from cancellation between two terms proportional to N 13 and N 14 respectively.In all the benchmark scenarios, the dominant loop contributions to C L/R 1 come from the triangle loops involving two vector bosons and one neutralino/chargino.Further, the third generation (s)quarks contribute significantly thanks to the large Yukawa couplings.The contributions from the loops involving, in particular, two quarks and one squark tend to negate the contributions from the loops involving the vector bosons and one neutralino/ chargino.In BP-5a(b) and BP-6a(b), the difference in the percentage contribution to C L/R 1 is largely attributed to the cancellation from the (s)quark loop.Further, contributions from the vertex counterterms are substantial.In particular, we find sizable contributions from the terms proportional to the diagonal and off-diagonal wave-function renormalization counterterms.
The vertex counterterms are evaluated following the implementation in FormCalc [107].The details have been discussed in Appendix C. On-shell renormalization schemes have been adopted for the neutralinochargino sector [73].In particular, for BP1a to BP-4b, two chargino masses and the heaviest neutralino mass (CCN [4]) have been used as on-shell input masses.For BP-5a, BP-5b, BP-6a, and BP-6b, two chargino masses and the third neutralino mass (CCN [3]) have been used as on-shell input masses.This ensures that there is always a bino-like neutralino among the input masses [74,111].The respective contributions from the counterterms have been shown in III.Note that we have used tree-level masses for all the neutralinos and charginos, including χ0 1 for the eastimation of the spin-independent scattering crosssection.This ensures that the percentage corrections to the cross-section reflects only the contributions from the vertex corrections, which we intend to illustrate.

As the spin-independent cross-section of χ0
1 with the nucleons (protons and neutrons) receive dominant contributions from the light Higgs boson-mediated processes, the percentage corrections to the crosssections are about twice the respective percentage corrections to the χ0 1 − χ0 1 − h 1 vertex.These crosssections can be enhanced by up to about 100% for the benchmark scenarios.This highlights the importance of these corrections in the present context.Note that, as mentioned in the I and elaborated further in III, certain important loop corrections to the Higgs bosons-nucleon interactions, which contribute to the effective neutralino-nucleon effective operators (see 21 and 20), have been already included in micrOMEGAs.Thus, the cross-sections computed using the one-loop corrections to χ0 1 − χ0 1 − h 1 /h 2 vertices also effectively include certain two-loop contributions.
These corrections are also included in the cross-sections with which we have compared the final results after including the vertex corrections.Thus, the percentage corrections to the cross-sections, as mentioned in the III, solely come from the corrections to the χ0 1 − χ0 1 − h 1 /h 2 vertices.Here, M 2 is taken as 4 TeV and the other parameters are assumed to be the same as mentioned in Table I.

Assuming that χ0
1 constitutes the entirety of DM, we have further considered the implications of these large corrections for the viability of sub-TeV higgsino-like DM in light of stringent limits from the direct detection experiments.We consider the DM-nucleon (proton) cross-section limits from the LUX-ZEPLIN (LZ) experiment [37] and compare the status of the benchmark scenarios after including the radiative corrections as shown in Fig. 3(a).We find that, thanks to the radiative corrections, benchmark point BP-1a is pushed above the lower limit of the 1σ sensitivity band (dotted line), and BP-1b is pushed close to the 1σ band.Benchmark points BP-2a and BP-2b are pushed close to the 1σ band while lying below it.The benchmark BP-3a falls on the exclusion line (solid line) and is close to being ruled out after the corrections are added, and BP-3b is also close to the exclusion limit.As for benchmark BP4a, it is pushed above the 1σ lower band, and BP-4b is pushed close to it.BP-5a and BP-5b are also pushed closer to the 1σ band of the exclusion region; Finally, BP6a and BP6b benchmarks are pushed above the 1σ band and close to the exclusion limit when the corrections are added.Although, to estimate the overall impact on the scattering cross-section all the radiative corrections need to be considered together, the above discussion aims to demonstrate the relative importance of the vertex corrections, in comparison with the same cross-section evaluated using the tree-level vertices C i . 10To demonstrate the significance of the radiative corrections on constraining the Higgsino mass parameter µ in the present context, we further vary the µ parameter keeping all the other relevant parameters the same as BP-3a(or b).The cross-sections with the radiatively corrected χ0 1 -χ0 ) and the respective tree-level vertices (C L/R i tree ) have been used to obtain the dashed red line and the dot-dashed orange line respectively in Fig. 3(a).As demonstrated in the figure, the dashed red line intersects the 90% confidence limit from the LZ experiment [37] for a heavier m χ0 1 , as compared to the dot-dashed orange line.As in the present context m χ0 1 ≃ |µ| (as |µ| ≪ |M 1 |, M 2 ), therefore, the constraint on the µ parameter is improved.This is further illustrated in Fig. 3

Light higgsino-like χ0
1 fits well within the framework of natural supersymmetry.In this article, we have considered higgsino-like χ0 1 DM within R-parity conserving MSSM and have studied the importance of a class of radiative corrections to the χ0 1 − χ0 1 − h 1 /h 2 vertices in the context of spin-independent 10 Note that by changing the stop-stop-Higgs boson soft-supersymmetry-breaking trilinear term Tt to -4 TeV, the light Higgs mass m h 1 , as computed by SPheno, becomes about 125 GeV.We have checked that using m h 1 ≃ 125 GeV, with the above modifications to the stop sector parameters, does not affect the vertex corrections and the percentage corrections to the direct detection cross-section appreciably.For most of the benchmark scenarios, which assume m h 1 ≃ 123 GeV, using the parameters as mentioned above lead to variations in the percentage correction to the neutralino-proton cross-section (∆σSI) by less than ∼3%.Further, note that while using m h 1 ≃ 125 GeV, keeping all the other parameters as the benchmark scenarios, does not change in the vertex corrections appreciably, and thus, the percentage change in the spin-independent cross-sections (∆σSI ) are also well below a percent. 11The cases for other combinations of signs of M1 and µ are not shown as their cross-sections lie below the LZ bounds in the parameter space of our interest.

direct detection. The tree-level couplings between χ0
1 and the CP-even neutral Higgs bosons (h 1 , h 2 ), in such a scenario, are suppressed by small gaugino-higgsino mixing.However, as demonstrated in this article, the radiative contributions to these vertices (including the respective counterterms) from the loops involving the charginos, neutralinos, gauge bosons, and Higgs bosons can have significant implications for direct detection.Further, third-generation (s)quark contributions are significant and tend to cancel the former to some extent in the parameter region considered in this article.For the benchmark scenarios presented, the radiatively corrected vertices can be enhanced by about 40% compared to the respective treelevel vertices.The spin-independent cross-section of χ0 1 with the nucleons (protons and neutrons), which receives a significant contribution from the CP-even neutral Higgs boson mediated processes through the respective effective operators, thus, can be enhanced by about 100% in certain benchmark scenarios.We further illustrate that the corrections are sensitive to the sign of µ and the choice of the gaugino mass parameters M 1 and M 2 , even though |µ| ≪ |M 1 |, M 2 .Note that, the "tree-level" cross-section in such scenarios is quite sensitive to the small gaugino admixture in the χ0 1 .Thus, generally, the constraint on the mass of sub-TeV higgsino-like χ0 1 , after including these corrections, is sensitive to the sign of µ and the choice of the gaugino mass parameters M 1 and M 2 .As mentioned in the Introduction, in the sub-TeV mass region, the thermal relic abundance of a higgsino-like χ0 1 LSP is inadequate to fulfill the required relic abundance of DM (Ω DM h 2 = 0.12 [60]).Thus, assuming only thermal production of χ0 1 will lead to a dilution of the direct detection constraints on χ0 1 , in proportion to the relative abundance of χ0 1 .However, considering the possibility of non-thermal production of χ0 1 in the early Universe, there is a possibility that χ0 1 constitutes the entire DM.In any scenario, the result demonstrates the significance of the complete vertex corrections to the χ0 1 − χ0 1 − h 1 /h 2 vertices in the spin-independent scattering cross-section of a higgsino-like χ0 1 DM.In this appendix, we summarize the Passarino-Veltman functions [112], which appear in the radiative corrections, as described in Appendix B. We follow the convention of refs.[113,114].The Passarino-Veltman C functions have the following form: We have used the following abbreviation: where μ denotes a parameter with dimension of mass.Further, Contraction with p µ 1 , then, gives where the momenta and masses are as shown in 4. Further, where , for i ∈ {1, 2}.In the expressions above, B 0 is given by where ∆ := 2 4−d − γ E + log 4π, γ E is the Euler-Mascheroni constant and d stands for space-time dimension.

APPENDIX B
In this Appendix, we discuss the radiative corrections to the χ0 1 − χ0 1 − h i vertices originating from the triangle diagrams.In particular, generic expressions for contributions from scalar bosons, vector bosons, and fermions running in the loops have been provided.In the following discussion, F and F ′ denote fermions, S and S ′ are used for scalar bosons, and V denotes vector bosons.Further, q 2 denotes the square of the momentum transferred from the incident χ0 1 to the quarks in the nucleons, and d stands for space-time dimension.Here, G and G ± refer to the neutral and charged Goldstone Bosons respectively.
We have evaluated the expressions using Package-X (version 2.1.1)[113], and have also checked some of these expressions by explicit calculations.Feynman gauge has been used for the calculation.The vertices may be found in ref. [65].Topology-(1a): The respective Feynman diagram is shown in Fig. 5(a). where where where where ( where where where, where where where λ h i qt qs = C h i , qt , qs are defined as where (B.28) (1) where where (3) For third-generation quarks and squarks: i, j = 3 and s ∈ {1, 2}.Topology-(2a): The respective Feynman diagram is shown in Fig. 6(a). where ( where Topology-(2b): The respective Feynman diagram is shown in Fig. 6(b). where where where Topology-(3a): The respective Feynman diagram is shown in Fig. 7(a).
where  where (2)  where  where where

APPENDIX C
The counterterm Lagrangian for the χ0 1 − χ0 1 − h i interaction (L CT ) is given as follows: where δC L i = δC R * i for i ∈ {1, 2}.
In the above expression, δC R 1 is given by, We use the following abbreviations in this section: Re takes the real part of loop integrals but does not affect the complex couplings.For the notations, we have closely followed [107] and [73].The relevant counter-terms have been listed below.
For the Higgs sector, the following counterterms are relevant [107]: In the gauge boson sector, the relevant counter-terms are as follows [107]: The counterterms to the gaugino and higgsino mass parameters are determined from the charginoneutralino sector.In the In the CCN(n) scheme, these are given by [73,74,107,111]:

FIG. 2 .
FIG. 2. The flow-chart for implementation of the relevant corrections to the neutralino-Higgs boson(s) vertices.

1
have been tabulated for |µ| = 300 GeV.HF stands for higgsino fraction.The fixed input parameters are: the mass of the pseudoscalar Higgs boson m A = 1.414TeV, and tan β = 10.The gluino mass parameter M 3 = 3 TeV.The trilinear coupling for two stops with the Higgs boson is set as T t = −3 TeV.The soft-supersymmetry-breaking mass parameters for the left-type and the right-type stop and sbottom squarks are as follows: m QL = 2.69 TeV, m tR = 2.06 TeV and m bR = 2.50 TeV.As for the physical masses, the charged Higgs boson mass M H ± = 1.416TeV, the CP-even Higgs mixing angle α = sin −1 (−0.1).For all the benchmarks, the third generation squark mass and mixing parameters are taken as: the lightest stop mass m t1 = 2.05 TeV, the heaviest stop mass m t2 = 2.71 TeV, the lightest sbottom mass m b1 = 2.50 TeV, the heaviest sbottom mass m b2 = 2.69 TeV.

1
have been tabulated for |µ| = 600 GeV.HF stands for higgsino fraction.The fixed input parameters are: the mass of the pseudoscalar Higgs boson m A = 1.414TeV, and tan β = 10.The gluino mass parameter M 3 = 3 TeV.The trilinear coupling for two stops with the Higgs boson is set as T t = −3 TeV.The soft-supersymmetry-breaking mass parameters for the left-type and the right-type stop and sbottom squarks are as follows: m QL = 2.69 TeV, m tR = 2.06 TeV and m bR = 2.50 TeV.As for the physical masses, the charged Higgs boson mass M H ± = 1.416TeV, the CP-even Higgs mixing angle α = sin −1 (−0.1).For all the benchmarks, the third generation squark mass and mixing parameters are taken as: the lightest stop mass m t1 = 2.05 TeV, the heaviest stop mass m t2 = 2.71 TeV, the lightest sbottom mass m b1 = 2.50 TeV, the heaviest sbottom mass m b2 = 2.69 TeV.

FIG. 3 .
FIG. 3. Panel (a) shows the comparison of the shift of various benchmark points (Table III) before and after adding the vertex corrections (C L/R i ) with the direct detection bound of LUX-ZEPLIN (LZ) experiment [37].The circled points depict the corrected cross-sections (σ SI ) and the uncircled ones are without the corrections (σ SI tree ).Panel (b) shows the shift in the µ parameter for different values of M 1 as constrained by LZ (2022) [37] after adding the vertex corrections (C L/R i ).The change in the constraint on the µ parameter (for µ ≪ M 1 ,M 2 ) corresponding to the cross-section after adding the vertex corrections C L/R i is shown by the solid line, the dashed line represents the case (b) with positive M 1 and µ (µ ≪ M 1 , M 2 ).In this figure, the constraint on µ parameter is shown to vary with respect to M 1 .We have assumed, as in the benchmark scenarios, tan β = 10, M 2 = 4 TeV and M A = 1.414TeV; the other parameters are also kept the same as mentioned in Table I and II.As shown in Fig.3(b), for M 1 = 2 TeV, µ 493 GeV (as shown by the dashed line) is excluded by the direct detection experiment LZ, when tree-level χ0 1 -χ0 1 -h 1 /h 2 vertices are used to estimate the respective cross-sections.While estimating the cross-section using the radiatively corrected vertices (C L/R i ), the constraint shifts to µ 593 GeV (as shown by the solid line), a shift of 100 GeV.Likewise, the bound on µ shifts from 230 GeV to 291 GeV for M 1 = 5 TeV.Thus, the constraint on the µ parameter space (with µ ≪ M 1 ,M 2 ) becomes more stringent by about 60-100 GeV, as illustrated in this figure.11

FIG. 4 .
FIG. 4. The above figure shows the mass and momentum convention for the Passarino-Veltman functions.
1 , as we consider in the present context.For such states, the higgsino fraction is much greater than the gaugino fraction (i.e., |N 13 |, |N 14 | ≫ |N 11 |, |N 12 |)., the tree-level coupling involving two neutralinos and the Higgs bosons (see 14) are small, as these are suppressed by a factor of the gaugino component of the higgsino-like neutralino state.Note that, in the present context we consider |M 1 |, M 2 5 TeV and the higgsino mass parameter |µ| 1 TeV.