Spin-Dependent Light Dark Matter Constraints from Mediators
SSpin-Dependent Light Dark Matter Constraints from Mediators
Harikrishnan Ramani
1, 2 and Graham Woolley Berkeley Center for Theoretical Physics, University of California, Berkeley, CA 94720 Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Department of Physics, University of California, Berkeley, CA 94720
A bevy of light dark matter direct detection experiments have been proposed, targeting spin-independent dark matter scattering. In order to be exhaustive, non-standard signatures that havebeen investigated in the WIMP window including spin-dependent dark matter scattering also need tobe looked into in the light dark matter parameter space. In this work, we promote this endeavor byderiving indirect limits on sub-GeV spin-dependent dark matter through terrestrial and astrophysicallimits on the forces that mediate this scattering.
I. INTRODUCTION
Repeated null results in the highly motivatedWIMP window have inspired vistas into lighter darkmatter below the GeV range, e.g [1] and referencestherein. New direct detection (DD) strategies [2–7]with lower energy thresholds to target this mass range,as well as experiments to probe the mediator itself [8–10] have been proposed recently.Spin-dependent (SD) dark matter models are wellmotivated in literature [11–13] and couple only to thenuclear spin. Since most common spin-independent(SI) DM detector targets have an even number of (andhence paired) protons and neutrons, they are not assensitive to spin-dependent DM. SD interactions havebeen investigated around the GeV range and above in[14–21] with odd-p or odd-n isotopes. However, all DDproposals for light DM hitherto have explored only theSI case. Intriguingly, since light dark matter detectionnecessarily involves lighter nuclear targets, the loss incoherence will not be as limiting as in WIMP DM de-tection. Also, self-interaction constraints, which limitthe DM - mediator coupling could also be relaxed ifthe DM self-interaction involves suppression factors.Thus, the viability of models of DM that domi-nantly interact with the SM through spin-dependentcouplings below the GeV scale becomes an interestingquestion. Purely SD interactions are hard to model-build even around the GeV scale[13, 22]. Furthermore,as pointed out in [3, 23], stringent limits on a light mediator already set severe limits on MeV scale DMalbeit it was explored only in the SI context. In thisletter, we use a similar strategy to set limits on SDDD cross-section and explore the viability of param-eter space to determine if future SD DD experimentsare well-motivated.
II. MODELS
Purely spin-dependent interactions between DMand SM target are mediated by a pseudoscalar , anaxial-vector or a dipole interaction mediated by a darkvector. This mediator can further couple in a plethoraof ways with DM. An exhaustive list of operators wasstudied in detail in [13, 22] and the operators thatproduce solely spin-dependent interactions are enu-merated in Table. I. Added to this are 3 operators thatrepresent DM charge - SM dipole scattering which aremomentum dependent.The operators are broadly organized in terms of therelevant mediator - pseudoscalar, axial-vector or a vec-tor coupled to nucleons through dipole interactions.Also tabulated are the DD cross-section as well as thedark matter self-interaction induced in both the mass-less and massive limits. The "operator" term whichis typically used only for the massive mediator limits,is used both in the massless and massive limits forbrevity.In the same vein as in [3] the strategy is to minimizeself-interaction so as to evade bullet-cluster limits[24] a r X i v : . [ h e p - ph ] M a y DM coupling SM coupling mediator Operator Direct Detection Self-InteractionMassless Massive Massless Massive φ a ¯ Nγ Na pseudoscalar O s q q q χχa O f3 B µ B µ a O v ¯ χγ χa O f4 q q φ † ∂ µ φA µ ¯ Nγ µ γ NA µ axial-vector O s v q v q χγ µ χA µ O f B † ν ∂ µ B ν A µ O v B † ν ∂ ν B µ A µ O v6 q q q ¯ χγ µ γ χA µ O f8 q q (cid:15) σνρµ B σ ∂ ν B ρ A µ O v ¯ χσ µν χF µν ¯ Nσ µν NF µν vector O f q q φ † ∂ µ φA µ O s q q q ¯ χγ µ χA µ O f B † ν ∂ µ B ν A µ O v TABLE I. Operators that produce solely spin-dependent interactions are enumerated. They are broadly organized interms of the mediator - pseudoscalar, axial-vector or vector through dipole coupling. Also tabulated are the DD cross-section as well as the dark matter self-interaction induced in both the massless and massive limits. and maximize the DD cross-section so as to set themost conservative limits arising from mediators.Among the 4 unique options relevant to pseu-doscalar mediators, O s , O f , O v have the same cross-section behavior and differ only in the spin of DM.Thus we shall consider only O f and the rest differonly by O (1) numbers. O f will be considered in itsown right and hence these two operators will be con-sidered.Among the axial-vector mediated operators, O f , O v have identical cross-section behavior and hence only O f will be considered. O s , O f , O v have the sameself-interaction cross-section as O f but have velocitysuppressed DD cross-section and hence will always bemore strongly constraining. O v is relevant only forvector dark matter and will be considered in its ownright. In the SM, Z exchange can generate some ofthese operators, however for light DM, Z decay posessevere constraints on the coupling and hence the cross-sections. Hence we we will consider only a new axial-vector mediator to generate these interactions.Finally, the vector dipole coupling to the SMproduces O f and three other operators. Whilethe SM photon is the ideal mediator, it generi- cally produces dipole-dipole as well as dipole-chargeand charge-charge interactions between the "milli-charge" DM and the SM, the latter of which is spin-independent[25]. The relative ratio of SD and SI in-teractions in the m χ (cid:28) m N limit is, σ SD σ SI ∼ m χ Z m N (1)where Z eff is the effective momentum-exchange-dependent charge DM sees and is given by, Z eff = a q a q Z (2)This is Thomas-Fermi screening, with a the Bohr ra-dius. Apart from the lack of coherent-enhancementthere is also a momentum suppression. However formuch smaller DM masses, i.e. m χ (cid:28) a v ∼ MeV,there is large amount of screening and Z eff → a q Z and at some mass, DM will only scatter with the SMdipole. However, for SD to overcome SI one has to goall the way down to m χ ∼ keV or below. There arestrong constraints on milli-charge particles at thesemasses from stellar constraints. The same statementsare true for the vanilla dark photon with kinetic mix-ing.In order to see SD signal first, we need to introducea new vector that couples only through dipole interac-tions. The limits on such a force, that couples purelythrough the dipole with nuclei has not been studiedin literature, but in general they should be compa-rable with the pseudoscalar. Furthermore, the DDcross-section as well as self-interaction cross-sectionare comparable with other operators considered, andhence we postpone analyzing this to future work. III. PSEUDOSCALAR MEDIATOR
In this section we will concern ourselves with pseu-doscalar mediator denoted a . The nuclear couplingcommon to O f and O f is g N a ¯ N γ N . These can begenerated from quark couplings, L ⊃ g q i a ¯ q i γ q i . (3)Limits on g q i a ¯ q i γ q i from flavor physics are summa-rized in [26]. For limit setting purposes, we considerthe quark-Yukawa coupling scenario. These can beconverted into limits on nucleons through [27, 28].Finally constraints from supernovae cooling are de-rived for pseudoscalars in [29].These constraints are summarized in Fig. 1. Weshow constraints from meson decays both in the visi-ble and invisible decay regimes for both neutron andproton couplings. Also shown are limits from super-nova cooling. Interestingly, there is a pseudoscalartrapping window, between the supernova limits andthe meson limits. Projected constraints from the pro-posed GANDHI experiment [10] from nuclear decaysthrough an invisible pseudoscalar could close this win-dow upto m a = 4 MeV.We will next summarize the DD and self-interactioncross-sections for pseudoscalar mediated operators. A. O f Starting with the Lagrangian,
L ⊃ a ( g χ ¯ χχ + g N ¯ N γ N ) (4) - - - - m a [ GeV ] g N � � ��� � � ��� � � � � ������������� FIG. 1. Terrestrial and astrophysical limits on the pseu-doscalar Yukawa coupling to nucleons are plotted as afunction of pseudoscalar mass m a . Terrestrial limits areadapted from [26] and organized into proton and neutroncouplings as well as into visible and invisible decay chan-nels for the pseudoscalar. SN1987A constraints are from[29]. Also shown are projected limits from [10]. The transfer cross-section is given by [3] σ born T ∼ g χ πm χ v { log(1 + R ) − R / (1 + R ) } (5)where R = m χ vm φ , and the per-nucleon reference DDcross-section by, σ DD = g N g χ π m χ v m N ( m φ + m χ v ) (6) B. O f L ⊃ a ( g χ ¯ χγ χ + g N ¯ N γ N ) (7)The transfer cross-section is given by, σ born T = g χ (cid:0) (cid:0) R + 1 (cid:1) log (cid:0) R + 1 (cid:1) + (cid:0) R − R − (cid:1) R (cid:1) m χ R ( R + 1) (8) σ DD = g χ g N π m χ v m N ( m φ + m χ v ) (9) IV. AXIAL-VECTOR MEDIATOR
As explained before, limits from Z decays force usto consider a BSM axial-vector. Starting with theLagrangian, L ⊃ g N ¯ N γ µ D µ γ N (10)Strict limits were derived in [30] from the UV com-pletion required to cancel anomalies on axial-vectorsand will not be summarized here. There, explicit con-straints were derived for currents coupled to right-handed quarks, but differ by an overall factor of 2with the axial-vector case. Next, we consider opera-tors generated by the axial-vector, O f and O v . A. O f The DM interaction in this case is,
L ⊃ g χ ¯ χγ µ D µ γ χ (11)The transfer cross-section is given by, [3] σ born T = 6 g χ πm χ v { log(1 + R ) − R / (1 + R ) } (12)and finally, the DD cross-section is given by[22], σ DD = 12 g N g χ π m χ ( m A + m χ v ) (13) B. O v The transfer cross-section is given by σ born T =3 g χ (cid:0) (cid:0) R + 1 (cid:1) log (cid:0) R + 1 (cid:1) + (cid:0) R − R − (cid:1) R (cid:1) m χ R ( R + 1) (14)and finally, the DD cross-section is given by, σ DD = 3 g N g χ π m χ v ( m A + m χ v ) (15) V. RESULTS
Using the DD and self-interaction cross-sections inthe previous sections, we set limits on the per-nucleonDD cross-section using the following procedure. Ifthe DM species under consideration makes up all ofDM, i.e. f DM = 1 , then the coupling of DM to themediator, g χ is constrained through the relevant self-interaction constraint from the Bullet Cluster system.We assume this cross-section to be σ born T m χ ≤ cm gram (16)Furthermore, the typical relative velocity in this sys-tem is assumed to be v ∼ . . If the DM species issubcomponent, f DM ∼ . , then the self-interactionconstraints no longer hold. g χ = 1 is instead im-posed for perturbativity. Constraints on g N , the SM-mediator coupling are obtained from the terrestrialand supernovae limits summarized in the previous sec-tions. For the pseudoscalar coupling, we take the min-imum of the visible and invisible channels for limit-setting purposes.With upper limits on both g χ and g N , one can ob-tain the maximum DD cross-section using the rel-evant formulae in the previous sections as a func-tion of m χ , the DM mass and m a / m A , the medi-ator mass. Finally, we scan all relevant mediatormass to find the largest allowed σ DD for a given m χ . This cross-section is plotted in Fig. 2 for scat-tering with protons, (top-left), with neutrons, (top-right) and for sub-component DM, (bottom-left) and(bottom-right). Different colors correspond to the 4operators considered, O f , O f , O f , O v . Projected lim-its from the GANDHI experiment that can probe thepseudoscalar trapping window are also shown. Fur-ther, current DD constraints on SD proton scatteringcompiled from [14–18] and for neutron scattering from[16, 19–21] are displayed.For pseudoscalar mediated models, unsurprisingly O f relaxes constraints compared to O f . Furthermore,constraints on σ n are stricter than σ p owing to thedisparate factors involved in converting pseudoscalarquark couplings to pseudoscalar proton and neutron - - - - - m χ [ GeV ] σ p [ c m ] � � � � � � � � ������ ������� ����������� � �� = � - - - - - m χ [ GeV ] σ n [ c m ] � � � � � � � � ������ ������� ����������� � �� = � - - - - - m χ [ GeV ] σ p [ c m ] � � � � � � � � ������ ������� ����������� � �� = ���� - - - - - m χ [ GeV ] σ n [ c m ] � � � � � � � � ������������� ����������� � �� = ���� FIG. 2. Limits on per-nucleon reference DD cross-section for scattering with proton-spin (
Top-Left ) and neutron-spin(
Top-Right ) are shown as a function of DM mass m χ for various operators considered in the text. Also shown areprojected improvements (dotted lines) on constraints on the mediator from the GANDHI experiment [10]. Currentconstraints from [14–18] for σ p and [16, 19–21] for σ n are also displayed. Bottom Panel : Similar analysis but forsub-component DM where DM self-interactions are not constraining. couplings. For axial-vector mediators, O f sets bet-ter constraints compared to O v . Comparing to spin-independent limits derived in [3], even the most pes-simistic limits are tighter, owing to the tight con-straints on the mediator in the case of axial-vectorsand the momentum suppression in the DD cross-section in the case of pseudoscalar mediator. Also,unlike spin-independent cross-sections, limits do notrelax around the GeV mass. Furthermore, relaxingthe self-interaction constraints by looking at subcom-ponent DM, relaxes the DD cross-section limit differ-ently for O f and O f , with the latter having a muchlower limit. This is because the pseudoscalar media-tor also suppresses the self-interaction cross-section bypowers of momentum exchange whereas for the axial- vector mediator no such suppression exists.The neutrino floor for light dark matter experi-ments is target dependent and expected to be in the σ N ∼ − − − cm range[23, 31]. Thus, thissets up a region of viable parameter space that canshow signals solely in experiments sensitive to spin-dependent scattering, or to mediator searches likeGANDHI. ACKNOWLEDGEMENTS
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