Improved limits for Higgs-portal dark matter from LHC searches
Searches for invisible Higgs decays at the Large Hadron Collider constrain dark matter Higgs-portal models, where dark matter interacts with the Standard Model fields via the Higgs boson. While these searches complement dark matter direct-detection experiments, a comparison of the two limits depends on the coupling of the Higgs boson to the nucleons forming the direct-detection nuclear target, typically parameterized in a single quantity . We evaluate using recent phenomenological and lattice-QCD calculations, and include for the first time the coupling of the Higgs boson to two nucleons via pion-exchange currents. We observe a partial cancellation for Higgs-portal models that makes the two-nucleon contribution anomalously small. Our results, summarized as , show that the uncertainty of the Higgs–nucleon coupling has been vastly overestimated in the past. The improved limits highlight that state-of-the-art nuclear physics input is key to fully exploiting experimental searches.
pacs:Dark matter, WIMPs, chiral Lagrangians
After the initial discovery of a new particle by the ATLAS and CMS collaborations at the Large Hadron Collider (LHC) in 2012 Aad:2012tfa (); Chatrchyan:2012xdj (), its mass has been further constrained to sub-GeV accuracy, Aad:2015zhl (), and so far neither studies of its spin and parity Khachatryan:2014kca (); Aad:2015mxa () nor its branching fractions Khachatryan:2016vau () have revealed significant deviations from the Standard Model (SM) prediction for the Higgs boson . An interesting consequence is that events with large missing transverse momentum would be expected if the Higgs boson decayed to long-lived non-SM states that do not leave a signature in the detector, so-called invisible Higgs decays. The absence of such observations sets limits on the invisible decay channels Aad:2014iia (); Chatrchyan:2014tja (); Aad:2014wza (); Aad:2015uga (); Aad:2015txa (); Aad:2015pla (); Khachatryan:2016whc (), which provide stringent constraints on beyond-SM physics.
One particular example concerns Higgs-portal models for dark matter, see, e.g., Kanemura:2010sh (); Djouadi:2011aa (); Djouadi:2012zc (); Beniwal:2015sdl (), in which the dark-matter candidate , a weakly interacting massive particle (WIMP), interacts with the SM fields via exchange of the Higgs boson. With the nature of dark matter still a major puzzle, the quest to understand its composition is being vigorously pursued, besides collider signatures, in direct-detection experiments looking for the WIMP scattering off atomic nuclei, as well as indirect searches Baudis:2016qwx (). As long as the WIMP mass fulfills , limits on the invisible decay width of the Higgs boson are in a one-to-one correspondence with direct-detection limits on the WIMP–nucleon cross section , provided the nuclear physics input is sufficiently under control.
Such comparison relies on the coupling of the Higgs boson to a single nucleon, which proceeds via the nucleon matrix elements for the scalar current
for the light quarks (with quark masses and nucleon mass ), as well as through heavy-quark loops coupling to gluon fields, see Fig. 1. Integrating out the heavy quarks at leading order in the strong coupling constant leads to an effective coupling Shifman:1978zn ()
to describe the Higgs–nucleon interaction. Both recent LHC analyses constraining Higgs-portal dark matter from invisible Higgs decays Aad:2015pla (); Khachatryan:2016whc () use a central value Young:2009zb () with variations covering (motivated by Toussaint:2009pz (), see Djouadi:2011aa (); see also Sage:2015wfa ()). Given recent progress in lattice-QCD calculations Durr:2015dna (); Yang:2015uis (); Abdel-Rehim:2016won (); Bali:2016lvx () and pion–nucleon () phenomenology Alarcon:2011zs (); Hoferichter:2015dsa (); Hoferichter:2016ocj (); RuizdeElvira:2017stg (), this large range in no longer reflects the current knowledge of the scalar couplings of the nucleon. Consequently, the limits for derived from the searches for invisible Higgs decays can be significantly improved. In the first part of this paper, we provide a more detailed assessment of the current situation, including corrections to Eq. (2) regarding isospin violation and higher orders in for the heavy-quark loops.
In addition, the standard decomposition (2) does not reflect that atomic nuclei are strongly interacting many-nucleon systems, so that not only the one-body (single-nucleon) matrix elements enter. As first pointed out in Prezeau:2003sv (), corrections to this picture from two-body currents, where the Higgs couples to a pion exchanged between two nucleons, can become important. Such corrections are conveniently addressed within chiral effective field theory (EFT) Epelbaum:2008ga (); Machleidt:2011zz (), an expansion around the chiral limit of QCD in terms of momenta and quark masses, as successfully demonstrated in previous applications to WIMP–nucleus scattering Cirigliano:2012pq (); Menendez:2012tm (); Klos:2013rwa (); Baudis:2013bba (); Cirigliano:2013zta (); Hoferichter:2015ipa (); Hoferichter:2016nvd (); Korber:2017ery () (related work using chiral EFT for WIMP–nucleon interactions Hoferichter:2015ipa () is restricted to the coupling to a single nucleon Bishara:2016hek (); Bishara:2017pfq () or focuses on the calculation of the nuclear states based on chiral EFT forces Gazda:2016mrp ()). In this work, we extend previous calculations limited to xenon Hoferichter:2016nvd () to a broad range of target nuclei and show that the corrections from the coupling to two nucleons are largely coherent and can thus be absorbed into a redefinition of the single-nucleon . The combination of the one- and two-body contributions to then allows us to compare the limits on Higgs-portal models from collider and direct-detection experiments using state-of-the-art nuclear physics input.
Ii Scalar couplings of the nucleon
The evaluation of the quark-level operators from Fig. 1 between nucleon states requires knowledge of the scalar couplings defined in Eq. (1). To arrive at Eq. (2), the couplings of the heavy quarks are eliminated in favor of the light quarks’ according to the following procedure Shifman:1978zn (): at , the QCD trace anomaly for active degrees of freedom reads
with gluon field-strength tensor . Integrating out the heavy quarks then yields for each flavor
and Eq. (2) follows.
Modifications of the leading result (2) arise from two sources: isospin-breaking and perturbative corrections. For - and -quarks the dominant isospin-breaking effects cancel in the sum , with remaining effects of the size Crivellin:2013ipa ()
where denotes the portion of the proton–neutron mass difference proportional to . In a nucleus, this would lead to a relative correction
where / refer to proton/neutron number, , and we have inserted values for a typical Xe isotope. Thus, isospin-violating effects in are small Crivellin:2015bva () and will be neglected in the following. The perturbative corrections have been worked out in detail in Hill:2014yxa (). Here, we use the full result for the quark
as well as the correction for .
In this way, we are left with the determination of the scalar couplings of the light quarks in the isospin limit. In general, these couplings cannot be measured in experiment due to the absence of a scalar source, leaving lattice QCD as the only viable option. However, an important exception concerns the scalar couplings of - and -quarks, which the Cheng–Dashen low-energy theorem Cheng:1970mx (); Brown:1971pn () relates to the scattering amplitude, offering a unique opportunity to test the results of lattice-QCD calculations with experiment. More precisely, the low-energy theorem allows one to extract the -term from the isoscalar amplitude evaluated at a particular kinematic point in the subthreshold region. The required analytic continuation can be performed in a stable manner using constraints from analyticity and unitarity in the form of Roy–Steiner equations Hoferichter:2015dsa (); Ditsche:2012fv (); Hoferichter:2012wf (); Hoferichter:2015tha (); Hoferichter:2015hva (); Hoferichter:2016duk (); Siemens:2016jwj (), with the result that ultimately the value of is fully determined by the amplitude at threshold, i.e. the scattering lengths. These, in turn, are known to high accuracy from measurements in pionic atoms Gotta:2008zza (); Strauch:2010vu (); Hennebach:2014lsa (); Baru:2010xn (); Baru:2011bw (). The result Hoferichter:2015dsa () agrees with calculations based on chiral perturbation theory, once the low-energy constants are extracted from a phase-shift solution consistent with the pionic-atom scattering lengths Alarcon:2011zs ().
Unfortunately, these results from phenomenology are in tension with the most advanced lattice calculations Durr:2015dna (); Yang:2015uis (); Abdel-Rehim:2016won (); Bali:2016lvx (), see Table 1. These results correspond to (nearly) physical quark masses, thereby eliminating a major source of systematic uncertainty in earlier calculations, but finite-volume corrections, discretization effects, and excited-state contamination may not be under sufficient control yet. It has also been suggested that the tension could be due to deficient input for the scattering lengths in the phenomenological analysis, which could again be checked on the lattice Hoferichter:2016ocj (), but the fact that low-energy scattering data corroborate the pionic-atom values makes this explanation appear unlikely RuizdeElvira:2017stg ().
For a similar cross check of lattice results with phenomenology would need to rely on relations, but the slow convergence of the expansion prohibits a meaningful, quantitative test. Table 1 also lists the heavy-quark couplings as well as the resulting values for with and without perturbative corrections, which prove to be only of moderate size. In particular, the direct determination of from Abdel-Rehim:2016won () is consistent with the perturbative calculation.
To combine the results of Table 1 into a single value for , we perform a naive average of the four lattice values, take as the mean between lattice and phenomenology with an error sufficiently large to cover both, and increase the error in to be able to accommodate a similar potential bias as for . We include the perturbative corrections, but add an additional error , about half the shift observed in Table 1 when adding the higher-order terms. Altogether, this leads to the one-body (1b) contribution
Iii Two-body currents
The spin-independent limits on derived from direct-detection experiments, see, e.g., Angloher:2015ewa (); Agnese:2015nto (); Agnes:2015ftt (); Amole:2016pye (); Armengaud:2016cvl (); Tan:2016zwf (); Akerib:2016vxi (); Aprile:2017iyp (), assume a simple relation with the differential WIMP–nucleus cross section
where is the momentum transfer, the relative velocity, and a nuclear form factor normalized to . This decomposition ignores subleading corrections, such as isospin violation or two-body effects, which in general all come with an independent nuclear form factor Hoferichter:2016nvd (). For the special case of Higgs-mediated WIMP–nucleus scattering, isospin-breaking corrections are small, see Eq. (7). In contrast, the leading effect of two-body currents is expected to be coherent (scaling with ), so that it can be included as an effective shift in . This contribution is crucial to ensure consistency: direct-detection limits on convert limits on the WIMP–nucleus rate to an effective WIMP–nucleon cross section based on Eq. (10), effectively subsuming the coupling to two nucleons. The cross section derived from limits on invisible Higgs decays needs to be consistent with this convention, and thus has to include the two-nucleon effects as well. Note that if the nuclear structure factors used in the interpretation of the direct-detection experiments accounted for two-body contributions (as is the case for the recent spin-dependent limits Aprile:2013doa (); Uchida:2014cnn (); Fu:2016ega (); Akerib:2017kat ()), the resulting WIMP–nucleon cross section could be interpreted as a true one-body quantity, and then the transition from collider limits would not require adding a two-body correction.
The leading two-body diagrams in chiral EFT are shown in Fig. 2. In the derivation, the couplings to the scalar current and the trace anomaly lead to two different responses Hoferichter:2016nvd (). For the scalar case, diagram is suppressed by two orders because there is no scalar source in the leading chiral Lagrangian, while the subleading Lagrangian does not produce a single-pion vertex. The validity of the underlying chiral EFT counting could be checked by comparing to nuclear -terms from lattice QCD, see Beane:2013kca () for such a calculation at heavy pion masses. Similarly, diagram requires an insertion of the quark-mass matrix, which also suppresses this contribution by two orders in the chiral counting. This leaves only the coupling to the pion in flight in diagram . For , diagram also enters, diagram is again only next-to-next-to-leading order, but there is a leading-order contribution to diagram , which was not included in Hoferichter:2016nvd ().
The terms in diagram are directly related to the contact operators that enter in the potential. Indeed, one can show that combining them with parts of diagram , they yield the leading potential Hoferichter:2017 (). Taken together with momentum-dependent one-body corrections at the same order, which can be identified with the kinetic-energy operator , one finds a combined contribution for the response proportional to
where is the binding energy of the nucleus represented by , obtained from interactions only. This adds to the remaining parts from diagram (both the scalar and responses) to an effective two-body shift in Hoferichter:2016nvd (); Hoferichter:2017 ()
We have evaluated the two-body contribution in harmonic-oscillator states using the occupation numbers that result from a large-scale shell-model diagonalization with state-of-the-art nuclear interactions (see Hoferichter:2016nvd (); Hoferichter:2017 () for details) for a wide range of nuclei that includes all stable xenon, argon, germanium, and silicon isotopes. We observe a robust coherence proportional to of the two-nucleon structure factors, leading to
where the two terms correspond to those in Eq. (12), and in each term the first uncertainty is from the variation over the different isotopes. For the binding-energy term, we take the experimental energy, corrected for the Coulomb contribution following Duflo:1995ep (). This effectively includes higher-order terms, e.g., due to interactions. For the first term in Eq. (12), the second uncertainty is due to the truncation in the chiral expansion and from the use of the harmonic-oscillator model to evaluate . We have taken this uncertainty significantly larger than that naively estimated by the chiral expansion to account for possible cancellations of different contributions, as would be manifest between kinetic and potential energies. We have also estimated the uncertainty in the many-body calculation by using the occupation numbers corresponding to two nuclear interactions, and the effect is less than .
Numerically, we observe a partial cancellation between the two terms, to the effect that the size of the two-body corrections is reduced from the expected few-percent level to below . Combined with the Higgs coupling to a single nucleon, we find
which is our final result.
Iv Impact on LHC exclusion limits
The relation between limits for the branching fraction of invisible Higgs decays and depends on the nature of the Higgs portal. For instance, for a scalar, vector, or fermion portal , , , respectively, with Higgs doublet and scalar, vector, or fermion fields , , , one finds
The consequences of the improved coupling are illustrated in Fig. 3, which is inspired by analogous figures in the most recent ATLAS Aad:2015pla () and CMS Khachatryan:2016whc () studies. For definiteness, we adopt the CMS limit (at confidence level). Even on a logarithmic scale the impact due to the improved nuclear physics input used is striking. Only a narrow band remains in the previously considered parameter space, tightening considerably the limits that result from experimental searches. Our results highlight that a careful determination of the nuclear physics related to the dark matter interactions is key for a correct determination of the limits that searches for invisible Higgs decays impose on Higgs-portal models for dark matter.
In summary, we have evaluated the Higgs–nucleon coupling, , using state-of-the-art phenomenological and lattice-QCD calculations, and including the coupling of the Higgs to two nucleons as predicted by chiral EFT. Our result, , reduces dramatically the uncertainty in the excluded region derived from experiment. This highlights that the nuclear physics input is key to fully exploiting the consequences of experimental searches, and that it needs to be treated consistently in limits from collider experiments and direct detection.
Acknowledgements.We thank Hans-Werner Hammer, David B. Kaplan, and Jordy de Vries for valuable discussions. Financial support by the DOE (Grant No. DE-FG02-00ER41132), the DFG (Grant SFB 1245), the ERC (Grant No. 307986 STRONGINT), MEXT (Priority Issue on Post-K computer), and JICFuS is gratefully acknowledged. AS thanks the INT for its hospitality during the completion of this work.
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