Cancellation in Dark Matter-Nucleon Interactions: the Role of Non-Standard-Model-like Yukawa Couplings

Extensive searches to probe the particle nature of dark matter (DM) have been going on for some decades now but, so far, no conclusive evidence has been found. Among various options, the Weakly Interacting Massive Particles (WIMP) remains one of the prime possibilities as candidates for DM near the TeV scale. Taking a phenomenological view, such null results may be explained for a generic WIMP in a Higgs-portal scenario if we allow the light-quark Yukawa couplings to assume non-Standard Model (non-SM)-like values. This follows from a cancellation among different terms in the DM-nucleon scattering which can, in turn, lead to a vanishingly small direct-detection cross section. It might also lead to isospin violation in the DM-nucleon scattering. Such non-SM values of light-quark Yukawa couplings may be probed in the high luminosity run of the LHC.


I. INTRODUCTION
In the search for new physics (NP), a large class of current and future experiments are dedicated to test the particle nature of dark matter (DM), especially by probing the ones known as the Weakly Interacting Massive Particles (WIMPs) (see e.g., [1,2]). Among them, the direct detection (DD) experiments rely upon the scattering of a DM particle off a detector nucleus, thereby looking for the recoiled nucleus. The list of current experiments in this direction includes Xenon1T [3], LUX [4], PandaX-II [5], Super-CDMS [6,7] (also see [8]). The WIMP-nucleus scattering can be divided into spin-independent (SI) and spin-dependent parts. Because of its coherent nature, the SI part is enhanced by a factor of A 2 , where A is the mass number of the nucleus, making the SI scattering more relevant to the experiments (see, e.g., [9,10]).
There have been some attempts to identify ways that would allow the WIMP scenarios to explain the non-observation of any DM signal above the neutrino floor. The Higgs mediated models have drawn significant attention in this regard [19][20][21][22][23][24][25][26][27][28][29][43][44][45][46][47][48][49][50]. They are relevant in many favoured beyond-the-SM (BSM) scenarios. The SM-like Higgs scalar can lead to large contributions at the microscopic level through its coupling with the DM and quarks. One may find parts of the parameter spaces where the couplings of the DM to Z or the Higgs boson may be highly suppressed or even zero identically. Similarly, in the models with an extended Higgs sector, destructive interference between light and heavy CP-even the DD experiments. Finally, we conclude in Sec. V.

II. SPIN INDEPENDENT DIRECT DETECTION CROSS SECTION: HIGGS EXCHANGES
Neglecting the nuclear structure effects, the DM-nucleus SI elastic scattering cross section at zero momentum transfer for a real scalar dark matter φ can be written as, where, A and Z are the mass number and the atomic number of the nucleus, respectively, M φ is mass of the DM and m r is the reduced mass. At the microscopic level, the effective DM-quark scattering can be read from the following interaction term: where q denotes a quark and f q , the corresponding scattering coefficient. In a simple model of H-portal DM, we can set f q = λ φ y q /m 2 H where y q is the Yukawa coupling of the quark q and λ φ is the effective DM coupling to the SM Higgs boson. Similarly, the DM-nucleon interaction can be expressed as, where q /m q ) with m N being the nucleon mass. The effective interaction of the DM particle with a nucleon (proton or neutron) can be obtained from the expectation value of the operator in Eq. (2) with respect to the initial and final nucleon states (N ≡ p or n) [78]. Here, one may use the fact that nucleon mass is determined from the trace of the energy-momentum tensor. Generically, for q = u, d, s the factor f (N) q can be expressed as For the heavier quarks, f can be evaluated through one-loop contributions due to scattering off gluons [78,79]: In fact, to the leading order, the effective HG µν G µν vertex at small momentum transfer can also be used for the above computation [80]. The dominant QCD corrections should also be taken into account in that case. Altogether, one can cast the effective DM-nucleon scattering coefficient as [29,81], The parameters, f (N) q (q ∈ u, d, s) can be determined from lattice QCD calculations [82]. For the heavier quarks, the leading order QCD correction becomes C q = 1 + 11α s (m q )/4π. We use the following values of f (N) q [82,83], which lead to f (N) G ∼ 0.92 (ignoring the differences between nucleons N = p, n). 1 It should be noted that the above numerical values are subject to some uncertainties as they are evaluated using the hadronic data.
As indicated in the Introduction, we allow the light-quark Yukawa couplings to deviate from their respective SM values, and more importantly, allow them to attain negative values which are not in violation with any experimental observation so far. We use this freedom to delineate the tentative range for y s or y c in terms of the other SM-like Yukawa couplings so that λ N = 0 can be achieved. Since C q is very close to 1, here, we may simply assume C q = 1 to get a qualitative picture. Thus, by substituting the values for f (N) q in Eq. (6), one would get a typical regime for the Yukawa couplings where λ N would be vanishingly small, i.e., In this illustrative example, we have allowed only y s and y c to take non-SM values for simplicity and further assumed that y s cancels the light quark contribution and y c cancel the heavier quark contribution separately. Obviously, this assumption is only a choice that we make for the illustration and not a requirement. One may tune any one or more light-quark Yukawa couplings to attain λ N = 0. The above equation simplifies to for N = p. Notably, for this set of parameters, λ p = λ n , implying that the DM-neutron scattering cross section will not vanish identically, and hence, a degree of isospin violation would be observed. We will further discuss about this possibility in Sec. IV. A similar cancellation condition can be achieved for the neutron as well by using the respective form factors where one finds y s = −0.857 y SM s . If one only allows either y c or y s to take values such that the DM-nucleon effective coupling in Eq. (6) vanishes, then a relatively larger negative values would be required, which may need some fine-tuning. Similarly, y u and/or y d can also be considered to be negative to achieve the same result.

III. NON-SM LIGHT-QUARK YUKAWA COUPLINGS: EXAMPLES AND EXPERIMENTAL TESTS
In this section, we illustrate how the non-SM-like Yukawa couplings can be generated through dimension-6 operators at the tree level in an effective theory framework. The current LHC measurements prevent large variations in y t, b [87] but leave space for deviations in the Yukawa couplings of the first two generations of quarks (for large changes to the top Yukawa couplings, see [74]). A discussion on the collider tests of non-SM values of y q follows in the next section.

A. Non-SM-like Light-quark Yukawa Couplings and Higher Dimensional Operators
In this example, we include a particular type of effective dimension-6 operators at some NP scale Λ in the quark Yukawa interaction Lagrangian, where Here, Y u, d H are the Wilson coefficients determined by the details of the NP model. In general, Y q and Y q H are assumed to be 3 × 3 matrices in the generation space. The SM Higgs doublet is denoted by H (H ≡ iτ 2 H * ), q L is the left-handed SU(2) quark doublet and u R , d R are the right-handed up-and downtype quarks, respectively. After the electroweak symmetry breaking (EWSB), using H = 0 h + v with v 174 GeV, one obtains the quark mass matrix M q and the corresponding Yukawa coupling matrix in the mass basis by considering unitary rotations of both left-and right-handed quark fields. For simplicity, we ignore flavour mixings and assume diagonal NP Yukawa couplings, i.e., Y q H = I 3×3 in the same basis. Though this is not true in general, it suffices for the present purpose. Now, we see that for a quark (q), its physical mass (m q ) and Yukawa coupling (y q ) to the physical Higgs boson (h) become non-aligned, i.e., , a few comments are in order.
• It is clear that, when a higher-order operator [like the one shown in Eq. (11)] is added to the SM Lagrangian, the fermion mass and Yukawa coupling become two independent quantities in the physical basis. In other words, it gives us the freedom to modify the quark-Higgs Yukawa couplings without perturbing the quark masses.
• The sign of the Yukawa couplings y q depends on the sign of the Wilson coefficients Y q H . In particular, for the first two generations of quarks (u, d, s, c), one may find that m q /v εY q H , thus the respective Yukawa couplings may naturally become negative.
• To achieve the correct size and sign of m q with a negative y q [see e.g., Eq. (8)], one may use Eqs. (12) and (13) to obtain, This sets an upper bound on the NP scale Λ -for a perturbative choice of Y q H ∼ O (1), the NP scale Λ should not be much larger than a few TeV. For example, Eq. (14) implies that Λ should be lower than about 2.9 TeV if we take the charm quark (i.e., set m q = m SM c and Y q H = 1). On the contrary, y q > 0 can only set a lower bound on Λ. Thus the choice of negative values for the Yukawa couplings is more natural and predictive compared to the positive values.
• Usually, if one considers with full generality, Y q and Y q H cannot be diagonalized simultaneously in the mass basis. As a result, Higgs mediated flavour changing neutral couplings among the SM quarks, which are otherwise extremely constrained by experiments, may appear. However, such flavour changing couplings may be suppressed if a definite flavour symmetry or some flavour selection rules are applied [75]. Since it hardly has any impact in the present case, we will not discuss it anymore.
In general, the effective Lagrangian can be considered to have three parts: where L SM is the usual SM part, L NP includes the new physics terms (except the DM part) responsible mainly for the nonalignment of the Yukawa couplings of the physical Higgs boson to SM quarks once the heavy fields are integrated out, and L DM refers to the terms involving DM interactions. Usually, these three parts may couple with each other in specific scenarios. For example, here we will see the NP-SM couplings producing the necessary deviations in Yukawa couplings at the tree level. As mentioned in Sec. II, we consider a real singlet scalar φ as the DM particle while the SM Higgs acts as the portal. An additional Z 2 symmetry ensures the stability of φ . Apart from the usual kinetic term in L SM , the potential term for the model can be written as, In principle, there should be a self-interacting φ 4 term also. However, we can safely ignore it, since it will have no effect in determining either the relic density or the DD cross section. After EWSB, φ doesn't get any VEV. Hence, the φ -mass term can be defined as, The only interaction term connecting the dark sector with the SM is (2λ Hφ v)hφ 2 . The phenomenology of this model has been widely studied. But as discussed earlier, it is quite difficult to satisfy the current direct detection bounds with this simple extension and this motivates us to propose L NP .
In L NP , we introduce the VL fermions and argue that the non-SM-like Yukawa couplings for the light quarks can be generated at the tree level once the heavy VL fermion degrees are integrated out. Based on our previous discussion it is clear that we want non-SM-like Yukawa couplings, specifically for the 2 nd generation quarks to the SM Higgs scalar. Thus, we consider a SM-like set up to specify the underlying NP theory [74,75] that includes only one generation of VL quarks: an SU(2) VL quark doublet Q = (C, S)(3, 2, 1/6) and the corresponding up-type and down-type SU(2) singlets C(3, 1, 2/3) and S(3, 1, −1/3), carrying the same quantum numbers as the SM quark doublets and singlets. Further, we assume the VL quarks are in their mass basis, with M Q,C, S 2 TeV, putting them well above the current LHC bounds. In general, we may write down the NP Lagrangian for the interactions among the VL quarks and the SM states, as follows.
Here q L refers to the second-generation SM quarks. This Lagrangian can lead to the desired dimension-6 operators in Eq. (11) after integrating out heavy the VL quarks. The dimension-6 couplings Y c H ,Y s H , as defined in Eq. (11), can be obtained from the diagrams shown in Fig. 1 and are given below: 2 Thus, if all the VL quarks have masses M C ∼ M S ∼ M Q ∼ 2 TeV and the new physics couplings λ NP ∼ O(1), the Yukawa couplings of the second-generation quarks can be considered for modification, as shown in Eq. (13).
C. Tests of the Non-standard Light-quark Yukawa Couplings at the LHC and Beyond The SM Higgs couplings to the massive vector bosons, the third generation quarks and the τ-lepton and its effective couplings to two gluons or two photons are known with some accuracy (roughly 10% − 20%) [72]. Recently the ATLAS and CMS collaborations announced the first observation of a Higgs decaying to two µ's [106,107]. However, how the Higgs couples to the other light fermions has never been probed experimentally -the LHC is yet to observe a direct Higgs decay to a pair of electrons or any of the first two generation of quarks. Because of the Higgs mechanism, the Yukawa coupling of a fermion is proportional to its mass in the SM. This makes the Yukawa couplings of the light fermions harder to probe. However, with the high luminosity run of the LHC (HL-LHC), there could be some chance of probing them. In recent times, several possibilities have been explored for testing these couplings at the LHC [76,77,[108][109][110][111][112][113][114][115][116][117][118][119][120][121][122], especially for c and also for s-quarks, in parallel to the ongoing experimental efforts [123][124][125][126][127][128]. Some of these proposals consider looking at rare Higgs decays to light flavoured mesons (like J/ψ or φ , etc.) [108,110,113,119]. Even though these can offer clean signals, such strategies suffer from low signal rates due the small decay rates involved. There are channels with relatively larger signal cross sections [like pp → W /Zh → W /Z(cc) [109,111,112], pp → hc [114], pp → hh → (cc)(γγ) [121] etc.] that require light-quark jet tagging. Ref. [118] uses a refined triggering strategy and some machine learning techniques to probe pp → h → ccγ and estimates the HL-LHC reach as y c /y SM c < 8. The process pp → hγ is used in Ref. [122]. Ref. [116] points out the possibility of using the charge asymmetry in pp → W ± h to constrain the light-quark Yukawa couplings. Ref. [120] employs a combination of the above ideas. Ref. [117] looks at the Higgs p T distribution in pp → h + j( j b ) → γγ + j( j b ). In the gluongluon fusion process, the Higgs is mostly produced centrally, i.e., about y h ≈ 0. For a non-negligible y u , the Higgs would also be produced via uū fusion. However, since only u is a valance quark of proton, not u, production from the uū fusion would peak in forward region, i.e., around higher |y h |. Hence, Ref. [115] considers the idea of using both p T and rapidity distributions of the Higgs to obtain bounds on the light Yukawa couplings. Here, we quote the projected reach in the y q values (q = u, d, s, c) at the LHC with 3000 fb −1 of integrated luminosity from Ref [73]: These proposals are somewhat competitive in nature as the projected limits obtained for either the 300 fb −1 LHC or HL-LHC are similar. Notice that the limits are put on the absolute values of the Yukawa couplings as the processes involved are insensitive to the sign of the light-quark Yukawa couplings. There are, however, some processes that have some sensitivity on the signs. In the process where a Higgs is produced with a jet (pp → h j), the shape of the p T distribution of the Higgs depends on the production mode. Ref. [76] utilizes this to estimate that the HL-LHC can restrict y c /y SM c to be within [−0.6, 3]. Here, the interference between cand tquark loops gives rise to a term linear in y c in the cross section of the gg → h j subprocess, making it somewhat sensitive to the sign of y c . Ref. [77] also considers the p T distribution of Higgs. They consider the process pp → h → 4 at the next-to-leading order (NLO) where the quark fusion and gluon fusion subprocesses can interfere. For the HL-LHC, they find −1550 < y u /y SM u < 700 and −800 < y d /y SM d < 300. There are other ways to probe the light-quark Yukawa couplings than at the LHC or HL-LHC. For example, Ref. [129] indicates that FCNC transitions in the Kaon sector could restrict the down-quark Yukawa coupling to be within the rage 0.4 < |y d /y SM d | < 1.7. It is possible to probe the light-quark Yukawa couplings by measuring isotope shifts, that are affected by Higgs exchange, in atomic clock transitions [130]. Ref. [68] points out that a discovery of H-portal dark matter could let us put bounds on these couplings. The future colliders, especially the leptonic ones, could also offer us a better handle in measuring the Higgs couplings in general [73]. This is mainly because the lepton colliders are clean and allow us to reconstruct the processes far more accurately than their hadron counterparts. For example, Ref. [131] considers probing via hadronic event shapes at lepton colliders and shows that light-quark Yukawa couplings greater than |0.09 × y SM b | might be excluded in an e + e − collider of centre-of-mass energy 250 GeV with an integrated luminosity of 5 ab −1 . In the subsequent analysis, we compute σ SI with non-SM values of y c and y s , especially with negative values of y c and y s , so that the SI scattering cross section may become vanishingly small. Hence, the future generation experiments like XENONnT or LZ could only assert our proposal through their blindness to find any signal in the σ SI -M DM plane. But, on the other hand, parts of the parameter space with non-SM values for y c (this includes the negative values as well) can be tested at the HL-LHC as claimed in Refs. [73,76]. This complementarity between the DM and LHC searches might help in testing our proposal.

IV. RELIC DENSITY AND DIRECT DETECTION OF DM
For numerical analysis, we use the code micrOMEGAs [81,132] to evaluate the relic density and DD cross section. The dominant QCD corrections in the SI DM-nucleon scattering are already included in the code. In our model, the main two free parameters are M φ and λ Hφ . Additionally, we consider y c and y s also as free parameters. In our computation, we set λ Hφ = 0.02. The valid parameter space should comply with the observed relic abundance data [133,134], For the singlet scalar DM φ , one can easily solve the Boltzmann equation to get the corresponding relic abundance. The Boltzmann equation is given by, where H is the Hubble constant and σ e f f v is the thermal averaged cross section of the DM annihilation to the SM particles. In this scenario, only the s-channel process shown in Fig. 2 keeps the DM in thermal equilibrium. The variation of Ω φ h 2 as a function of the DM mass is depicted in Fig. 3. The dependence of the relic density on the variations of charm and strange quark Yukawa couplings is negligible. The blue dotted line in the figure represents the central value of the DM relic density obtained from the PLANCK data. We see that, in addition to the resonance dip, this model is also able to satisfy the astrophysical data nicely for M φ ∼ 100 ± 10 GeV. Even though the present DD bounds exclude a real singlet scalar DM in this region, the non-SM Yukawa couplings allow us to retain it, as we see from Fig. 4. Fig. 4 validates our qualitative conclusions from Sec. II. Here, we show the variation of SI DD cross section as a function of M φ . We consider two situations: (i) both y c and y s can assume non-SM values while all other Yukawa couplings are fixed to their SM values and (ii) only y c can assume non-SM values. One may simply use Eq. (6) to find y q such that the DM-nucleon SI scattering cross-section becomes small enough to evade limits from the future-generation experiments. In the first scenario, absolute values of y c and y s can be much closer to their SM values while in the second case, we find Here the numerical value for y c is slightly away from that in Eq. (9), due to the inclusion of QCD corrections. Similar results are obtained in Fig. 4(b) as well. Fig. 4(c) shows the variation of σ φ −n SI with respect to M φ , for the same set of y c and y s where λ p → 0, i.e., with y c = −1.875y SM c and y s = −0.770y SM s , keeping all other quark Yukawa couplings fixed at their SM values. The blue line in Fig. 4(c) shows that even though σ φ −n SI can't be vanishingly small for this set of parameter points, the DM-neutron scattering cross section can be below the proposed direct detection bounds for M φ ≥ 50 GeV. This, clearly, reflects isospin violation to a large extent, following from the constraint λ p → 0.
However, the IVDM scenario can be realized in a more general way and a large violation can be observed for moderate to large non-SM-like first-generation Yukawa couplings. Since f (N) u, d values are different for neutron and proton, one may easily get a parameter space where f p = f n . A particular useful scenario appears when f n / f p takes a negative value, as it then offers some significant cancellations in the DM-nucleus scattering cross section. Though a larger value of y d or y u can easily lead to f n / f p > 0, but negative values can only be achieved in a narrow domain which can be computed using Eq. (6). For example, considering only y d to assume non-SM values, the following range for the same coupling can be observed. where, Note that X p > X n for f (N) q as shown in Eq. (7). For numerical estimation, we can consider a benchmark ratio, e.g., f n / f p ≈ −0.7. This particular value has some importance to relax tensions between different results for low DM mass. One may easily check that, y d /y SM d ≈ −11.6 can lead to the above ratio.

V. CONCLUSION
Direct searches of WIMPs as a form of dark matter have been underway for a long time. Due to the present and projected experimental sensitivities towards the spin-independent direct dark matter detection cross section, the available parameter spaces in simple H-portal dark matter models are significantly reduced or threatened to be ruled out. However, there exist a few small regions where the DM interactions with nucleons can be very tiny, thus escaping the ever impinging bounds from the DM searches. One such example is the so-called "Blind spots" where either the DM couplings with Higgs scalar vanish, or there is some destructive interference among diagrams involving different neutral scalars. In this paper, we have realized another route in the same direction, where the Higgs boson couplings with the nucleons become vanishingly small. Apparently, such a requirement can be realized easily, if the lightquark Yukawa couplings are allowed to assume non-SM values in presence of some new physics; in particular, negative values -a possibility allowed by the current experiments. We consider some higher dimensional operators or, in particular, dimension-6 operators involving the SM fields that let the lightquark Yukawa couplings to be negative without disturbing the respective quark masses. Importantly, the new physics scale is bounded and for perturbative values of new couplings, can only be of the order of a few TeV. We also consider a specific realization of this with vector-like quarks with masses about 2 TeV. In this set-up, we consider a real SM-singlet scalar as the DM candidate. In the absence of any discovery, generally, such a simple DM set up would be excluded completely for M DM ≤ 1 TeV from the projected sensitivity of the proposed LZ or XENONnT experiments. Here we observe that resultant SI DM-nucleon scattering cross section can be made vanishingly small for all values of DM mass. Needless to say, our observation would be unchanged for any other DM candidate in the Higgs-portal models or in the models where the Higgs produces the dominant contribution to the direct detection process, since our argument does not depend on the DM-DM-Higgs couplings and, hence, it can take O(1) values as well. Isospinviolation in the DM-nucleon scattering can also be realized for negative values of the first-generation light-quark Yukawa couplings. Usually, the exclusion limits are obtained assuming isospin-conserving dark matter and hence can be much relaxed when the DM couples differently to protons and neutrons. Probing such non-standard values of the Yukawa couplings of the first two generations of quarks is hard even for the HL-LHC, though there exist studies that aim to narrow down the allowed values of the lightquark Yukawa couplings, even the negative values. For example, it has been argued that, at the LHC, it might be possible to pin the charm-quark Yukawa coupling within [−0.6, 3]y SM c with 3000 fb −1 of integrated luminosity in a largely model independent manner. Hence, even though the future generation of dark matter search experiments based on the dark matter-nucleon scattering is blind to our proposal, it might be tested at the HL-LHC.