Direct detection of vector dark matter through electromagnetic multipoles
OOctober 7, 2020 IPMU20-0077, TUM-HEP 1269/20
Direct detection of vector dark matter throughelectromagnetic multipoles
Junji Hisano a,b,c
Alejandro Ibarra d Ryo Nagai e,f a Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University,Furo-cho Chikusa-ku, Nagoya, 464-8602 Japan b Department of Physics, Nagoya University, Furo-cho Chikusa-ku, Nagoya, 464-8602 Japan c Kavli IPMU (WPI), UTIAS, University of Tokyo, Kashiwa, 277-8584, Japan d Physik-Department, Technische Universität München, James-Franck-Straße, 85748 Garching, Ger-many e Dipartimento di Fisica e Astronomia, Universita’ degli Studi di Padova, Via Marzolo 8, 35131Padova, Italy f Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Padova, Via Marzolo 8, 35131 Padova,Italy
E-mail: [email protected], [email protected],[email protected]
Abstract:
Dark matter particles, even if they are electrically neutral, could interact withthe Standard Model particles via their electromagnetic multipole moments. In this paper,we focus on the electromagnetic properties of the complex vector dark matter candidate,which can be described by means of seven form factors. We calculate the differentialscattering cross-section with nuclei due to the interactions of the dark matter and nuclearmultipole moments, and we derive upper limits on the former from the non-observation ofdark matter signals in direct detection experiments. We also present a model where thedark matter particle is a gauge boson of a dark SU (2) symmetry, and which contains heavynew fermions, charged both under the dark SU (2) symmetry and under the electromagnetic U (1) symmetry. The new fermions induce at the one loop level electromagnetic multipolemoments, which could lead to detectable signals in direct detection experiments. a r X i v : . [ h e p - ph ] O c t ontents B.1 XENON1T/nT 15B.2 SuperCDMS 17B.3 CRESST-III 17B.4 PICO-60 18
C Analytic expressions for the electromagnetic form factors 18
There is mounting evidence for the existence of dark matter in galaxies, clusters of galaxiesand the Universe at large scale (see e.g. [1–3]). All the current body of evidence fordark matter arises from its gravitational interactions with ordinary matter. However, it isgenerically expected from particle physics models that the dark matter particle could haveadditional interactions with our visible sector apart from gravity.A simple possibility is that the dark matter interacts electromagnetically. Obviously,the dark matter must be dark. However, electromagnetic interactions are not precluded,as long as they are sufficiently weak to be compatible with current cosmological and as-trophysical observations, as well as with current direct, indirect and collider searches. Infact, in many models the dark matter particle has electromagnetic interactions, providedthere exist “portal” particles, which interact both with the photon and with the dark matterparticle. A renown example is millicharged dark matter [4] (for an overview, see e.g. [5]).In this case, the “portal” particle is a hidden-photon, which interacts with the dark matterparticle, as well as with the Standard Model photon (via kinetic mixing).– 1 –ark matter particles can also interact electromagnetically even if the electric chargeis exactly zero, via higher electromagnetic multipole moments [6–11]. For instance, a Diracfermion dark matter candidate acquires via loops a magnetic dipole moment, a chargeradius, and an anapole moment when it has a Yukawa coupling with an electromagneticallycharged scalar, which acts in this case as “portal” particle [12–19]. For Majorana fermiondark matter, only the anapole moment can be generated, since it is the only multipolethat violates the charge-conjugation symmetry [20, 21]. The electromagnetic interactions,despite the loop suppression, can lead to detectable detection rates in a direct detectionexperiment, and can be crucial for assessing the detection prospects of some dark matterframeworks, notably those where the dark matter interacts only with leptons or with heavyquarks.In this paper we focus on the complex vector as dark matter candidate (some explicitmodels can be found e.g. in [22–35]). In these frameworks, the dark matter particleinteracts at tree level with the Standard Model through the exchange of heavy fermionsor heavy scalars (possibly mixing with the Standard Model Higgs). However, spin-1 darkmatter particles could also interact electromagnetically with the Standard Model throughtheir multipole moments. In this paper we will investigate in a model independent way theelectromagnetic properties of vector dark matter and their implications for direct detectionexperiments.The paper is organized as follows. In Section 2 we discuss the general form of theelectromagnetic interactions of a vector dark matter particle with a nucleus. In Section 3we calculate the implications for direct detection experiments of the vector electromagneticmultipole moments, and we derive upper limits on the various form factors from exper-iments. In Section 4 we present a concrete model of vector dark matter with non-zeroelectromagnetic interactions. Finally, in Section 5 we present our conclusions.
We consider a massive complex vector field V µ with mass m V as dark matter candidate.The effective interaction Lagrangian of an on-shell vector field V µ with the electromagneticvector field A µ was systematically analyzed in [36] (for earlier works, see [37, 38]). Keepingterms up to dimension six it reads: L /e = ig A m V (cid:104) ( V † µν V µ − V † µ V µν ) ∂ λ F λν − V † µ V ν (cid:3) F µν (cid:105) + g A m V V † µ V ν ( ∂ µ ∂ ρ F ρν + ∂ ν ∂ ρ F ρµ )+ g A m V (cid:15) µνρσ ( V † µ ←→ ∂ ρ V ν ) ∂ λ F λσ + iκ A V † µ V ν F µν + iλ A m V V † λµ V µν F νλ + i ˜ κ A V † µ V ν ˜ F µν + i ˜ λ A m V V † λµ V µν ˜ F νλ , (2.1)– 2 – VV (cid:104)(cid:28)(cid:96)(cid:59)(cid:50)(cid:105) (cid:77)(cid:109)(cid:43)(cid:72)(cid:50)(cid:109)(cid:98) (cid:104)(cid:28)(cid:96)(cid:59)(cid:50)(cid:105) (cid:77)(cid:109)(cid:43)(cid:72)(cid:50)(cid:109)(cid:98)
047 (2 . ≤ pR D ≤ . . (2.8)where j ( x ) is a spherical Bessel function of the first kind, R = (cid:113) c + π a − s (with c = (1 . A / − . fm, a = 0 . fm and s = 0 . fm) and R D (cid:39) . A / fm. Further, µ N = e/ m p denotes the nuclear magneton, and ¯ µ T is the weighted dipole moment for thetarget nuclei, defined as: ¯ µ T = (cid:32)(cid:88) i f i µ i S i + 1 S i (cid:33) / , (2.9)where f i , µ i , and S i are the elemental abundance, nuclear magnetic moment, and spin,respectively, of the isotope i [41].We note that the terms proportional to µ V , d V are enhanced by a factor /E R , andthe terms proportional to µ V , d V , Q V , g A by a factor /v . The term proportional to d V isdoubly enhanced by / ( E R v ) . These enhancements have important implications for directdetection experiments, as we will discuss in the next section. We assume that the dark matter in our Galaxy is in the form of N vectors, V i , i = 1 ...N ,with mass m V i and number density in the Solar System n i , such that ρ loc = N (cid:88) i n i m V i . (3.1)– 4 – able 1 . C , P , and CP properties of the various vector dark matter electromagnetic multipolemoments. Form factors in Eq. (2.2)
C P CP µ V , Q V , g A + + + d V , ˜ Q V + − − g A − − + We will keep the discussion general and we will not specify how many of these componentsare real and how many are complex, or whether the complex components are symmetric orasymmetric. In our numerical analysis we will adopt ρ loc = 0 . − .The differential event rate at a direct detection experiment reads: dRdE R = 1 m T (cid:90) d v vf Lab ( (cid:126)v ) N (cid:88) i =1 n i dσ i dE R , (3.2)where dσ i /dE R is the dark matter-nucleus differential cross section, discussed in Section 2,and f Lab ( (cid:126)v ) denotes the DM velocity distribution in the laboratory frame. For the latter,we will adopt a Maxwell-Boltzmann distribution in the galactic frame, truncated at theescape velocity from the Galaxy, v esc : f Lab ( (cid:126)v ) = f ( (cid:126)v + (cid:126)v E ) , (3.3)with (cid:126)v E the velocity of the Earth in the galactic frame and f ( (cid:126)v ) = (cid:40) N e − v /v ( | (cid:126)v | < v esc )0 ( | (cid:126)v | > v esc ) , (3.4)with N = π / v (cid:20) erf (cid:18) v esc v (cid:19) − v esc √ πv e − v esc v (cid:21) . (3.5)Hereafter we take v esc = 544 km s − , v = 220 km s − and v E = 232 km s − . Finally, wecalculate the number of events at a given direct detection experiment integrating dR/dE R over the recoil energy, taking into account the corresponding detection efficiency.We show in Figure 2 the 90% C.L. upper limits on the various vector dark matterelectromagnetic multipole moments from the non-observation of a dark matter signal atthe XENON1T [42], SuperCDMS [43], PICO-60 [44] and CRESST-III [45] experiments,alongside with the expected sensitivity of the XENONnT experiment [46]. Details of thecalculation are given in Appendix B. For the plots we have assumed that the dark matteris constituted by one complex vector ( V and V † ), that interacts with the nucleus via one ofthe form factors in Eq. (2.2) only. For the real vector, only the form factors g A and g A arenon-vanishing. We find that XENON1T sets the most stringent limits for m V (cid:38) GeV,– 5 –hile CRESST-III for m V ∼ − GeV. We also find that the scattering rate is for mostexperiments dominated by the dark matter interaction with the nuclear charge, as a resultof the enhancement by Z ; the dark matter interaction with the nuclear magnetic dipolemoment only plays a role for the PICO experiment.Let us note that other search strategies could set more stringent limits on the formfactors, notably collider search experiments, and could cover the low mass region untestedby direct detection experiments. A detailed analysis will be presented elsewhere [47]. We extend the Standard Model (SM) gauge group with a non-abelian SU (2) D gauge sym-metry and a U (1) X global symmetry. We assume that the symmetry is spontaneouslybroken by the vacuum expectation value of a spin-0 field Φ D , doublet under SU (2) D andwith charge / under U (1) X . The vacuum possesses a remnant global U (1) D symmetry,corresponding to the generator T D + X , with T D and X being respectively generators ofthe SU (2) D and the U (1) X symmetries. This remnant symmetry ensures the stability ofthe lightest among all particles charged under the U (1) D symmetry. In this case, theseare the three massive W D bosons, which are absolutely stable.In this simple model the W D bosons only interact with the Standard Model through theHiggs portal interaction ( H † H )(Φ † D Φ D ) . To couple the W D bosons to the electromagneticfield, we augment the model with extra fermions, Ψ l and Ψ e , charged under U (1) Y andunder the dark sector symmetries SU (2) D and U (1) X . The particle content of the modeland the charges under the different symmetry groups are summarized in Table 2. The Lagrangian of the model reads: L = L SM + L kin + L mass − V , (4.1)where L kin = ¯Ψ l iγ µ (cid:32) ∂ µ + ig D (cid:88) a =1 W aDµ T a − ig (cid:48) B µ (cid:33) Ψ l + ¯Ψ e iγ µ (cid:0) ∂ µ − ig (cid:48) B µ (cid:1) Ψ e , (4.2) −L mass = m Ψ l ¯Ψ l P L Ψ l + m Ψ e ¯Ψ e P L Ψ e + ¯Ψ l Φ D ( λ L P L + λ R P R ) Ψ e + y ¯Ψ l ( iτ Φ ∗ D ) l R + h . c . , (4.3) V = µ D (Φ † D Φ D ) + λ D † D Φ D ) + λ DH † D Φ D )( H † H ) , (4.4)where T a = τ a / , with τ a ( a = 1 , , being the SU (2) D Pauli matrices, and P L,R =(1 ∓ γ ) / are the projection operators. g (cid:48) and B µ denote U (1) Y coupling and gauge boson,respectively. l R denotes the right-handed SM lepton. Here g D is real, while m Ψ l , m Ψ e , λ L , The U (1) D symmetry is analogous to the electromagnetic U (1) symmetry, which arises after the spon-taneous breaking of the SU (2) L × U (1) Y symmetry by the Higgs field, doublet under SU (2) L and withhypercharge 1/2. The only difference is that we assume the U (1) X symmetry to be global instead of local,as the U (1) Y symmetry. A similar setup was discussed in Ref. [32] in the context of multicomponent dark matter scenarios. – 6 – - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Figure 2 . Current upper limits in the various vector dark matter electromagnetic multipolemoments from the XENON1T, SuperCDMS, PICO-60 and CRESST-III data. We also show theprojected sensitivity of the XENONnT experiment. – 7 – R , and y are in general complex quantities. We denote the components of the SU (2) D doublets as Ψ l = (cid:32) Ψ N Ψ E (cid:33) , Φ D = (cid:32) ϕ + iϕ ϕ + iϕ (cid:33) . (4.5)The Yukawa interaction with coupling y induces the mixing between Ψ N and the SM leptonafter SU (2) D symmetry breaking. In our numerical analysis, we take y = 0 for simplicity.For µ D < , the SU (2) D symmetry breaks spontaneously. We work in the gauge where (cid:104) ϕ (cid:105) = v D / √ , (cid:104) ϕ , , (cid:105) = 0 . Then, the W D bosons acquire a common mass m W D = g D v D / and the fermion mass terms become: −L mass = m Ψ l ( ¯Ψ N ) R (Ψ N ) L + (cid:16) ( ¯Ψ E ) R ( ¯Ψ e ) R (cid:17) M E (cid:32) (Ψ E ) L (Ψ e ) L (cid:33) + h . c . , (4.6)where we have defined left- and right-handed fields in the usual manner, (Ψ) L,R = P L,R Ψ ,and M E = (cid:32) m Ψ l λ L √ v Dλ R √ v D m Ψ e (cid:33) . (4.7)The mass matrix M E can be diagonalized by the field transformation: (cid:32) (Ψ E ) L (Ψ E ) L (cid:33) = V † L (cid:32) (Ψ E ) L (Ψ e ) L (cid:33) , (cid:32) (Ψ E ) R (Ψ E ) R (cid:33) = V † R (cid:32) (Ψ E ) R (Ψ e ) R (cid:33) , (4.8)where V L,R are unitary matrices satisfying V † R M E V L = diag ( m E , m E ) , (4.9)with m E , being positive and real, and ordered such that m E ≤ m E . The Dirac massterm for Ψ N is taken to be real and non-negative via an appropriate phase rotation. Finally,one finds the following interaction Lagrangian of the mass eigenstate fields with the photonand the SU (2) D gauge vectors: L int = − g D √ (cid:18) ¯Ψ iE [( V L ) i P L + ( V R ) i P R ] γ µ Ψ N W − Dµ + h.c. (cid:19) − e Ψ N γ µ Ψ N A µ − e ¯Ψ iE γ µ Ψ iE A µ . (4.10)Generalizations to a larger number of fields are straightforward.The quantum numbers for the physical particles in this setup are summarized in table 2.The states W ± D = ( W D ∓ iW D ) / √ , Ψ E and Ψ E transform non-trivially under the remnant U (1) D and the lightest among them will be absolutely stable. In this work we will assume m E , m E > m W D , such that W ± D are dark matter candidates ( Ψ E decays into W D andSM particles via the Yukawa coupling Eq. (4.3), with rate proportional to y ). Notice that W D does not carry a U (1) D charge and can decay. Concretely, the Lagrangian Eq. (4.10)induces a kinetic mixing term between W D and A µ of the form L ⊃ − (cid:15) W Dµν F µν , (4.11)– 8 – able 2 . Particle content and charge assignments for the model present in Section 4. Gauge eigenstates Spin SU (2) D U (1) X SU (3) C SU (2) L U (1) Y W D Φ D / Ψ l − / -1 Ψ e − -1Mass eigenstates Spin U (1) D SU (3) C U (1) EM W ± D = √ ( W D ± iW D ) ± W D = W D h D Ψ N − E − − E − − with (cid:15) given by: (cid:15) = eg D π (cid:20) log m N m E − (cid:16) | ( V R ) | + | ( V L ) | (cid:17) log m E m E (cid:21) , (4.12)Redefining the vector fields in the usual manner to bring the kinetic terms into their canon-ical form, one finds the coupling in the Lagrangian ∼ e(cid:15)J µ em W D , that induces the decay of W D into Standard Model fermions.Due to the assignments of gauge charges of the fields Ψ l and Ψ e , one generically expectsthese particles to be in thermal equilibrium with the plasma of Standard Model particles.Accordingly, the complex vector dark matter candidates W ± D are also expected to be inthermal equilibrium with the SM. Therefore, for appropriate model parameters W ± D couldaccount for the whole dark matter of the Universe via the mechanism of thermal freeze-out.The dark matter candidates W ± D in our galaxy interact with the Standard Model parti-cles at tree level via the Higgs portal, or at the one loop-level via the electroweak interactionsof the fermions Ψ l and Ψ e . Direct detection of vector dark matter through the Higgs portalinteractions was discussed e.g. in [26]. Here we assume that the Higgs portal interactionsare negligibly small, and we focus on the implications for direct detection experiments ofthe electroweak interactions induced at the quantum level by the fermions Ψ , E . In whatfollows we will consider only the electromagnetic interactions, discussed in section 3, sincedark matter interactions with nuclei induced by weak multipoles are expected to be sub-dominant.The electromagnetic multipole moments can be readily computed from the diagramsin Fig. 3 and read: µ V = − eg D π m W ± D (cid:88) i =1 (cid:16) r N − r E i (cid:17)(cid:20)(cid:18) | ( V L ) i | + | ( V R ) i | (cid:19) G (1) µ ( r N , r E i ) – 9 – iE N iE VV †
The C and P conserving electromagnetic multipoles ( µ V , d V , and g A ) vanish when themasses of the particles in the loop are degenerate, m N = m E i . This fact was emphasized byRef. [48]. iv) For fixed r N , r E i , the dipole moments ( µ V , d V ) scale as m − W ± D , the quadrupolemoments ( Q V , ˜ Q V ) as m − W ± D and g A and g A are independent of m W ± D .– 11 – - - -
10 100 1000 10 - - - Figure 5 . Differential event rate for a xenon target at recoil energy E R = 10 keV as a functionof the dark matter mass, assuming m Ψ l = 2 m Ψ e = 10 m W D g D = 1 , λ R = − and λ L = 1 . ( CP conserving point, left panel) or λ L = 1 . e i π ( CP violating point, right panel). We show in Figure 4 a scan plot with the predicted values of the form factors as afunction of the dark matter mass m W D , taking for concreteness g D = 1 , and the remainingparameters in the ranges: m Ψ e m W D = [1 , ,m Ψ l m W D = [1 , , | λ L,R | = [0 , , Arg [ λ L,R ] = [0 , π ] . (4.19)We also show in the Figure the upper limits on the form factors from various experimentsfrom Figure 2 (determined assuming that only one form factor contributes to the scatter-ing). Notably, there are portions of the parameter space which can be probed by currentexperiments, even in the absence of Higgs portal interactions, due especially to the inter-actions induced by the CP violating moment d V and by the CP conserving moments µ V , Q V and g A . This is a consequence of the enhancement of the scattering rate induced bythese electromagnetic multipoles at low relative velocities, cf. Eq. (2.2), especially by theelectric dipole moment, which is doubly enhanced by /E R and by /v .To investigate the relative effect of the different multipoles in the differential event rate,we show in Figure 5 the contributions of the different terms in Eq. (2.2) for a xenon targetat recoil energy E R = 10 keV, for some exemplary parameters conserving CP (left panel) orviolating CP (right panel). Concretely, we take m Ψ l /m W D = 10 , m Ψ e /m W D = 5 , g D = 1 , λ R = − , as well as λ L = 1 . for the CP conserving case and λ L = 1 . e i π for the CP violating case. For the CP violating case, the d V contribution dominates over the wholerange of masses analyzed; for the CP conserving case, the µ V contribution dominates for m W D (cid:38) GeV, while g A dominates for smaller masses; this is due to the contribution tothe interaction vertex from the kinetic mixing.– 12 – X E N O N T G e V T e V T e V T e V T e V | μ V / μ N | = - - - - - X E N O N n T - C R E SS T - III X E N O N T G e V T e V T e V T e V T e V | d V / e | = - [ f m ] - - - - X E N O N n T - C R E SS T - III
Figure 6 . Impact of direct detection experiments for the model described in Section 4 assuming m Ψ l = 2 m Ψ e , g D = 1 , λ R = − , as well as λ L = 1 . for the CP conserving case (left panel) and λ L = 1 . e i π for the CP violating case (right panel). The black lines represent isocontours of themagnetic dipole moment (left panel) and the electric dipole moment (right panel), while the bluelines represent isocontours of min { m N , m E } . The green and red regions correspond to the
90 % exclusion limits from the XENON1T and CRESST-III experiments, respectively; the green linecorresponds to the future prospect by XENONnT experiment.
Finally, in Figure 6 we investigate the impact of direct detection experiments in probingthe parameter space of the model. As before, we fix for concreteness m Ψ l = 2 m Ψ e , g D = 1 , λ R = − , as well as λ L = 1 . for the CP conserving case (left panel) and λ L = 1 . e i π for the CP violating case (right panel), and we show as black lines the isocontours of themagnetic dipole moment ( | µ V /µ N | , left panel) and the electric dipole moment ( | d V /e | [ fm ] ,right panel). The blue dashed lines are contours of the smallest mass between m N and m E .On the other hand, the green and red regions correspond to the
90 % exclusion limits fromthe XENON1T and CRESST-III experiments, respectively; the green line corresponds tothe future prospect by XENONnT experiment. As shown in the Figure, current experimentsprobe a significant part of the parameter space, especially for the CP violating case, wherefermions as heavy as 100 TeV in the loop can induce electric dipole moments at the reachof current experiments. As mentioned in Section 2 this is due to the double enhancementof the scattering rate mediated by the electric dipole moment by /E R and by /v . We have presented a comprehensive study of electromagnetic multipole moments of thecomplex vector dark matter candidate, and we have studied their implications for directdetection experiments. We have parametrized the electromagnetic interactions of the vec-tor dark matter by means of seven form factors and we have calculated the differentialscattering cross-section of the vector dark matter with the nucleus via the interactions of– 13 –heir multipole moments.We have set upper limits on the vector dark matter electromagnetic multipole momentsfrom the non-observation of an excess of nuclear recoils in direct detection experiments. Fordark matter masses above ∼ GeV the strongest constraints are set by the XENON1Texperiment, and below that mass by the CRESST-III experiment. The strongest limits arisefor a dark matter mass (cid:39) GeV and read | µ V /µ N | < × − , | Q V /e | < × − fm , | d V /e | < × − fm , | ˜ Q V /e | < × − fm , g A < × − and g A < × − .Lastly, we have constructed a concrete model of vector DM where the interactions withthe Standard Model are dominated by the electromagnetic multipole moments. The modelis based on three ingredients: i) a “dark” non-Abelian gauge symmetry, which is sponta-neously broken, ii) a new U (1) global symmetry, and iii) new matter particles, chargedboth under the electromagnetic U (1) symmetry and the “dark” non-Abelian symmetry. Wefind that after the spontaneous breaking of the “dark” non-Abelian symmetry, there is aremnant symmetry that stabilizes the vector dark matter against decay. Also, the newmatter particles generate via quantum effects the electromagnetic multipole moments forthe vector dark matter. We have found that, despite the loop suppression of the multi-pole moments, the vector dark matter could be detected in current experiments throughtheir electromagnetic interactions with the nuclei, even when the fermion mass lie in themulti-TeV range. Acknowledgments
We thank Tomohiro Abe for comments on the manuscript. The work of J.H. and R.N. wassupported by JSPS KAKENHI (Grant Number 20H01895 (J.H.) and 19K14701 (R.N.)) andby Grant-in-Aid for Scientific research from the Ministry of Education, Science, Sports,and Culture (MEXT), Japan (Grant Numbers 16H06492 (J.H.) and 18H05542 (R.N.)).The work of J.H. was also supported by JSPS Core-to-Core Program (Grant NumbersJPJSCCA20200002), and World Premier International Research Center Initiative (WPIInitiative), MEXT, Japan. The work of R.N. was also supported by the University ofPadua through the “New Theoretical Tools to Look at the Invisible Universe” project andby Istituto Nazionale di Fisica Nucleare (INFN) through the “Theoretical AstroparticlePhysics” (TAsP) project. A.I. would like to thank the KMI for hospitality during theinitial stages of this work. The work of A.I. has been partially supported by the DeutscheForschungsgemeinschaft (DFG, German Research Foundation) under Germany’s ExcellenceStrategy – EXC-2094 – 390783311.
A Effective interactions of an on-shell complex vector field with the elec-tromagnetic field . In this appendix we review the general structure of the
V V † γ vertex derived in [36],where V is a complex vector field, and we present the effective interaction Lagrangian of anon-shell vector field V µ with the electromagnetic field A µ , keeping terms up to dimension– 14 –ix. The general interaction vertex reads: Γ αβµV V † γ ( q, ¯ q, p ) /e = f A ( p ) Q µ g αβ − f A ( p ) m V Q µ p α p β + f A ( p )( p α g µβ − p β g µα )+ if A ( p )( p α g µβ + p β g µα ) + if A ( p ) (cid:15) µαβρ Q ρ − f A ( p ) (cid:15) µαβρ p ρ − f A ( p ) m V Q µ (cid:15) αβρσ p ρ Q σ , (A.1)where q µ , ¯ q µ and p µ are, respectively, the 4-momenta of the fields V , V † and A , and Q µ = q µ − ¯ q µ . Here, we have imposed the conditions ∂ µ V µ = 0 , ∂ µ A µ = 0 . The first condition is justified for on-shell V µ , since it is derived from the equation ofmotion, which for (free) V µ is reads ( (cid:3) + m V ) V ν = ∂ ν ( ∂ µ V µ ) . The second condition isjustified because the scalar component does not contribute to the scattering amplitude.The form factors are regular at p = 0 . Let us note that f A , (0) = 0 due to the U (1) EM gauge invariance. Further, f A (0) corresponds to the electromagnetic charge of V , and it iszero in our case. f Ai ( p ) ( i = 1 , , · · · , are expanded around p (cid:39) as f A ( p ) = p m V ( g A + λ A ) + O ( p ) ,f A ( p ) = λ A + O ( p ) ,f A ( p ) = κ A + λ A + O ( p ) ,f A ( p ) = p m V g A + O ( p ) ,f A ( p ) = p m V g A + O ( p ) ,f A ( p ) = ˜ κ A − ˜ λ A + O ( p ) ,f A ( p ) = − ˜ λ A O ( p ) . (A.2)The coefficients g A , g A , g A , λ A , ˜ κ A , ˜ λ A , and κ A parametrize the strength of the effec-tive interactions of an on-shell vector V µ with the electromagnetic field A µ , as defined inEq. (2.1). B Direct detection event rate
We summarize in this Appendix how we calculated the expected number of signal eventsin the experiments employed in our analysis.
B.1 XENON1T/nT
The XENON collaboration uses a liquid xenon detector with a dual-phase time projectionchamber (TPC). The signal from nucleon recoils can be efficiently discriminated from the– 15 –ackground signals from the ratio between the primary ( S ) and secondary ( S ) scintil-lation light signals. The scintillation light is converted into photoelectrons (PE) by thephotomultiplier tubes (PMT).The signal rate in number of photoelectrons n can be calculated from [49]; dRdn = (cid:90) E min R E max R dE R (cid:15) ( E R ) Poiss ( n | ν ( E R )) dRdE R , (B.1)where dR/dE R denotes the differential event rate for the dark matter scattering off a Xenucleus, while E min R = 4 . keV and E max R = 40 . keV. Further, (cid:15) ( E R ) is the detectionefficiency, which we take from Fig. 1 of Ref. [42]. Finally, ν ( E R ) is the expected numberof PEs for a given recoil energy E R , which we obtain from the S1 yield given in the lowerleft panel of Fig. 13 of Ref. [46]. In our analysis, we focus on the central detector regionwith a mass of 0.65 t and consider only events between the median of the nuclear recoilband and the lower σ quantile. This approach reduces the background level, as argued inRefs. [50, 51]. We therefore multiply the detection efficiency by an additional factor 0.475to take into account our reference region. Next, we determine the differential event rate asthe function of the primary scintillation light yield. The differential event rate reads: dRdS = ∞ (cid:88) n =1 Gauss ( S | n, √ nσ PMT ) dRdn . (B.2)Here, σ PMT is the average single-PE resolution of the photomultipliers, for which we con-servatively take σ PMT = 0 . [52, 53]. Finally the expected number of events in the energybin [ S min1 , S max1 ] is obtained from N th [ S min1 , S max1 ] = w exp (cid:90) S max1 S min1 dRdS . (B.3)where w exp is the exposure of the experiment.Following Refs. [50, 51], we divide the signal region into two parts, which correspondto S ∈ [3 , PE and S ∈ [35 , PE, respectively. For each energy bin, we calculate theTest Statistic (TS) function, defined as (see e.g. [54]):TS = − (cid:20) L ( N th ) L bkg (cid:21) , (B.4)with L ( N th ) = 1 N obs ! ( N th + N bkg ) N obs exp (cid:8) − ( N th + N bkg ) (cid:9) , (B.5)and L bkg ≡ L (0) . N obs and N bkg are the numbers of the observed and background events,respectively.The XENON1T collaboration has reported in [42] the latest results of their search,using an exposure w exp = 278 . days × . ton. Following Refs. [50, 51], we adopt ( N obs , N bkg ) = (0 , . for the first energy bin and ( N obs , N bkg ) = (2 , . for the secondenergy bin. Finally, we derive the 90% CL upper limit on the number of signal events byrequiring TS > . in each energy bin. – 16 –o estimate the future prospect of XENONnT with an exposure w exp = 20 t · yrs [46],we follow Ref. [55] and we apply the maximum gap method [56] under the assumptionof zero observed events. Namely, we require − exp( − N th ) ≥ . , which corresponds to N th (cid:46) . . B.2 SuperCDMS
The SuperCDMS detector consists of 15 Ge target crystals, each instrumented with ion-ization and phonon detectors. The measured ionization and phonon energies are used toderive the recoil energy and the ionization yield. The information from the ionization yieldcan be used to distinguish signal from background.Following Ref. [54], we estimate the DM event rate as N th = w exp (cid:90) E max E min dE R , (cid:15) ( E R ) dRdE R , (B.6)where dR/dE R denotes the differential scattering rate of dark matter particles off a Genucleus, E min = 1 . keV, E max = 10 keV, w exp is the exposure, and (cid:15) ( E R ) is the efficiency,which we take from Fig. 1 of [43].The SuperCDMs collaboration presented in [43] the results of their first search, based onan exposure w exp = 577 kg · days , reporting N obs = 11 . On the other hand, the backgroundsin their experiment are not fuly understood, therefore we conservatively take N bkg = 0 inthe derivation of the upper limit of signal events. The C.L. limit corresponds to N th < . [57]. B.3 CRESST-III
The CRESST-III experiment employs a CaWO crystal target as cryogenic calorimeters.The discrimination of the dark matter signal from the background is performed by measur-ing simultaneously the phonon/heat and the scintillation light signals. The design aims toachieve a low threshold for the recoil energy, smaller than eV.The expected event rate in the energy bin [ E min , E max ] , can be calculated from N th = w exp (cid:90) E max E min (cid:88) i = { Ca , O , W } f i (cid:15) i ( E R ) dR i dE R . (B.7)Here dR i /dE R denotes the differential cross section for the DM scattering off the nucleus i =Ca, O, W, and w exp is the exposure. Further, (cid:15) i ( E R ) is the detector efficiency for thenucleus i , which we read from the data implemented in DDCalc-2.0.0 [58, 59]. Finally, f i denotes the mass fraction for element i : f Ca = 0 . , f O = 0 . , and f W = 0 . ,respectively.In the first run (from / − / ), five detectors reached/exceeded the designgoal [60]. Among the five detectors, the detector called “detector A” achieved the lowestenergy threshold (cid:39) eV [44]. The results from the detector A give the largest sensitivityto low mass dark matter candidates. The total exposure was w exp = 5 . kg × days [61].– 17 –o derive the 90% C.L. limit, we consider 10 energy bins of uniform size in log-scalebetween eV and keV, and simply assume that the number of signal events follows aPoisson distribution with N obs = { , , , , , , , , , } . To determine the
90 %
C.L. limit on the event rate we require that the Poisson likelihoods L ( N th ) satisfy χ ( N th ) = − L ( N th ) + 2 ln L ( N obs ) < . (B.8)in each of the bins, where the Poisson likelihood is calculated from − L ( N th ) = 2 N th − N obs + N obs ln N obs N th . (B.9) B.4 PICO-60
The PICO-60 collaboration employs a C F superheated liquid detector. The expectedevent number can be calculated from N th = w exp (cid:88) i = { C , F } (cid:90) ∞ dE R f i P i ( E R ) dR i dE R , (B.10)Here dR i /dE R denotes the differential cross section for the DM scattering off the nucleus i =C, F, and w exp is the exposure. Further, P ( E R ) is the bubble nucleon efficiency for givena recoil energy E R . We read the efficiency for F and C from Figure 3 of Ref. [45]. Finally, f i denotes the mass fraction for element i : f C = 0 . and f F = 0 . , respectively.The results of the dark matter search were reported in Ref. [45] for an exposure w exp =48 . kg × . days. The collaboration reported the observation of 3 candidate events (basedon the single bubble selection) while the number of background events is determined to be . ± . . We estimate the 90% C.L. bound on N th by inserting N obs = 3 and N bkg = 1 into Eq. (B.4) and by requiring that TS > . . We obtain N th (cid:46) . . as 90% C.L. boundon the event rate. C Analytic expressions for the electromagnetic form factors
The loop function G are given by: G (1) µ ( x, y ) = − − λ (cid:16) ( x − y ) − ( x + y ) (cid:17) + ( x − y ) log (cid:18) x y (cid:19) , (C.1) G (2) µ ( x, y ) = 4 xyλ ( x, y ) − xyx − y log (cid:18) x y (cid:19) , (C.2) G (1) Q ( x, y ) = − (cid:16) λ ( x, y ) (cid:16) x − x (2 y + 3) + y − y + 2 (cid:17) + 1 (cid:17) + 2 (cid:16) x − x ( y + 1) + y ( y − (cid:17) x − y ) log (cid:18) x y (cid:19) , (C.3) G (2) Q ( x, y ) = − xyλ ( x, y ) + 4 xyx − y log (cid:18) x y (cid:19) , (C.4) We used
Package-X [62] to evaluate the one-loop diagrams. – 18 – (1)1 ( x, y ) = 43 λ ( x, y ) κ ( x, y ) (cid:16) x − y ) + ( x − y ) (8 − x + y )) − x + y ) + 9( x + y ) (cid:17) + 43 (cid:16) x − y ) − x + y ) + 3 (cid:17) κ ( x, y ) − (cid:16) x − y ) − ( x + y ) + 2 x − y (cid:17) log (cid:16) x y (cid:17) , (C.5) G (2)1 ( x, y ) = − xyκ ( x, y ) − λ ( x, y ) κ ( x, y ) xy (cid:16) ( x − y ) − x + y ) + 2 (cid:17) + 43 (cid:16) xyx − y (cid:17) log (cid:18) x y (cid:19) , (C.6) G (1) d ( x, y ) = 8 x y λ ( x, y ) , (C.7) G ˜ Q ( x, y ) = − x y λ ( x, y ) , (C.8) G ( x, y ) = λ ( x, y ) (cid:18) − x + x (cid:0) y + 4 (cid:1) − y + 4 y − (cid:19) − (cid:0) x − y (cid:1) log (cid:18) y x (cid:19) − , (C.9)where λ and κ are defined as κ ( x, y ) = (1 − x − y ) − x y , (C.10) λ ( x, y ) = 1 κ / ( x, y ) log (cid:32) κ / ( x, y ) − (1 − x − y )2 xy (cid:33) . (C.11) References [1] G. Bertone, D. Hooper and J. Silk,
Particle dark matter: Evidence, candidates andconstraints , Phys. Rept. (2005) 279 [ hep-ph/0404175 ].[2] L. Bergström,
Nonbaryonic dark matter: Observational evidence and detection methods , Rept. Prog. Phys. (2000) 793 [ hep-ph/0002126 ].[3] J. Silk et al., Particle Dark Matter: Observations, Models and Searches . Cambridge Univ.Press, Cambridge, 2010, 10.1017/CBO9780511770739.[4] B. Holdom,
Two U(1)’s and Epsilon Charge Shifts , Phys. Lett. B (1986) 196.[5] S. Davidson, S. Hannestad and G. Raffelt,
Updated bounds on millicharged particles , JHEP (2000) 003 [ hep-ph/0001179 ].[6] J. Bagnasco, M. Dine and S. D. Thomas, Detecting technibaryon dark matter , Phys. Lett. B (1994) 99 [ hep-ph/9310290 ].[7] M. Pospelov and T. ter Veldhuis,
Direct and indirect limits on the electromagneticform-factors of WIMPs , Phys. Lett. B (2000) 181 [ hep-ph/0003010 ].[8] K. Sigurdson, M. Doran, A. Kurylov, R. R. Caldwell and M. Kamionkowski,
Dark-matterelectric and magnetic dipole moments , Phys. Rev. D (2004) 083501 [ astro-ph/0406355 ].[9] E. Masso, S. Mohanty and S. Rao, Dipolar Dark Matter , Phys. Rev. D (2009) 036009[ ].[10] V. Barger, W.-Y. Keung and D. Marfatia, Electromagnetic properties of dark matter: Dipolemoments and charge form factor , Phys. Lett. B (2011) 74 [ ]. – 19 –
11] T. Banks, J.-F. Fortin and S. Thomas,
Direct Detection of Dark Matter ElectromagneticDipole Moments , .[12] N. Weiner and I. Yavin, UV completions of magnetic inelastic and Rayleigh dark matter forthe Fermi Line(s) , Phys. Rev. D (2013) 023523 [ ].[13] K. Fukushima and J. Kumar, Dipole Moment Bounds on Dark Matter Annihilation , Phys.Rev. D (2013) 056017 [ ].[14] J. Kopp, L. Michaels and J. Smirnov, Loopy Constraints on Leptophilic Dark Matter andInternal Bremsstrahlung , JCAP (2014) 022 [ ].[15] A. Ibarra and S. Wild, Dirac dark matter with a charged mediator: a comprehensive one-loopanalysis of the direct detection phenomenology , JCAP (2015) 047 [ ].[16] R. Primulando, E. Salvioni and Y. Tsai,
The Dark Penguin Shines Light at Colliders , JHEP (2015) 031 [ ].[17] P. Sandick, K. Sinha and F. Teng, Simplified Dark Matter Models with Charged Mediators:Prospects for Direct Detection , JHEP (2016) 018 [ ].[18] J. Herrero-Garcia, E. Molinaro and M. A. Schmidt, Dark matter direct detection of afermionic singlet at one loop , Eur. Phys. J. C (2018) 471 [ ].[19] J. Hisano, R. Nagai and N. Nagata, Singlet Dirac Fermion Dark Matter with Mediators atLoop , JHEP (2018) 059 [ ].[20] B. Kayser and A. S. Goldhaber, CPT and CP Properties of Majorana Particles, and theConsequences , Phys. Rev. D (1983) 2341.[21] E. Radescu, Comments on the Electromagnetic Properties of Majorana Fermions , Phys. Rev.D (1985) 1266.[22] G. Servant and T. M. Tait, Is the lightest Kaluza-Klein particle a viable dark mattercandidate? , Nucl. Phys. B (2003) 391 [ hep-ph/0206071 ].[23] H.-C. Cheng, J. L. Feng and K. T. Matchev,
Kaluza-Klein dark matter , Phys. Rev. Lett. (2002) 211301 [ hep-ph/0207125 ].[24] J. Hubisz and P. Meade, Phenomenology of the littlest Higgs with T-parity , Phys. Rev. D (2005) 035016 [ hep-ph/0411264 ].[25] A. Birkedal, A. Noble, M. Perelstein and A. Spray, Little Higgs dark matter , Phys. Rev. D (2006) 035002 [ hep-ph/0603077 ].[26] T. Hambye, Hidden vector dark matter , JHEP (2009) 028 [ ].[27] J. Hisano, K. Ishiwata, N. Nagata and M. Yamanaka, Direct Detection of Vector DarkMatter , Prog. Theor. Phys. (2011) 435 [ ].[28] H. Davoudiasl and I. M. Lewis,
Dark Matter from Hidden Forces , Phys. Rev. D (2014)055026 [ ].[29] C. Gross, O. Lebedev and Y. Mambrini, Non-Abelian gauge fields as dark matter , JHEP (2015) 158 [ ].[30] A. Karam and K. Tamvakis, Dark matter and neutrino masses from a scale-invariantmulti-Higgs portal , Phys. Rev. D (2015) 075010 [ ].[31] S.-M. Choi, H. M. Lee, Y. Mambrini and M. Pierre, Vector SIMP dark matter withapproximate custodial symmetry , JHEP (2019) 049 [ ]. – 20 –
32] F. Elahi and S. Khatibi,
Multi-Component Dark Matter in a Non-Abelian Dark Sector , Phys.Rev. D (2019) 015019 [ ].[33] T. Abe, M. Fujiwara, J. Hisano and K. Matsushita,
A model of electroweakly interactingnon-abelian vector dark matter , .[34] E. Nugaev and A. Shkerin, Unveiling complex vector dark matter by magnetic field , .[35] F. Elahi and M. Mohammadi Najafabadi, Neutron Decay to a Non-Abelian Dark Sector , .[36] K. Hagiwara, R. D. Peccei, D. Zeppenfeld and K. Hikasa, Probing the Weak Boson Sector in e + e − → W + W − , Nucl. Phys.
B282 (1987) 253.[37] K. Gaemers and G. Gounaris,
Polarization Amplitudes for e + e − → W + W − and e + e − → ZZ , Z. Phys. C (1979) 259.[38] G. Gounaris et al., Triple gauge boson couplings , in
AGS / RHIC Users Annual Meeting ,pp. 525–576, 1, 1996, hep-ph/9601233 .[39] R. H. Helm,
Inelastic and Elastic Scattering of 187-Mev Electrons from Selected Even-EvenNuclei , Phys. Rev. (1956) 1466.[40] J. D. Lewin and P. F. Smith,
Review of mathematics, numerical factors, and corrections fordark matter experiments based on elastic nuclear recoil , Astropart. Phys. (1996) 87.[41] S. Chang, N. Weiner and I. Yavin, Magnetic Inelastic Dark Matter , Phys. Rev.
D82 (2010)125011 [ ].[42]
XENON collaboration,
Dark Matter Search Results from a One Ton-Year Exposure ofXENON1T , Phys. Rev. Lett. (2018) 111302 [ ].[43]
SuperCDMS collaboration,
Search for Low-Mass Weakly Interacting Massive Particles withSuperCDMS , Phys. Rev. Lett. (2014) 241302 [ ].[44]
CRESST collaboration,
First results from the CRESST-III low-mass dark matter program , Phys. Rev. D (2019) 102002 [ ].[45]
PICO collaboration,
Dark Matter Search Results from the Complete Exposure of thePICO-60 C F Bubble Chamber , Phys. Rev.
D100 (2019) 022001 [ ].[46]
XENON collaboration,
Physics reach of the XENON1T dark matter experiment , JCAP (2016) 027 [ ].[47] In preparation.[48] A. B. Lahanas and V. C. Spanos,
Static quantities of the W boson in the MSSM , Phys. Lett.
B334 (1994) 378 [ hep-ph/9405298 ].[49]
XENON100 collaboration,
Likelihood Approach to the First Dark Matter Results fromXENON100 , Phys. Rev.
D84 (2011) 052003 [ ].[50]
GAMBIT collaboration,
Global analyses of Higgs portal singlet dark matter models usingGAMBIT , Eur. Phys. J.
C79 (2019) 38 [ ].[51] P. Athron, J. M. Cornell, F. Kahlhoefer, J. Mckay, P. Scott and S. Wild,
Impact of vacuumstability, perturbativity and XENON1T on global fits of Z and Z scalar singlet dark matter , Eur. Phys. J.
C78 (2018) 830 [ ]. – 21 – XENON collaboration,
Lowering the radioactivity of the photomultiplier tubes for theXENON1T dark matter experiment , Eur. Phys. J.
C75 (2015) 546 [ ].[53] P. Barrow et al.,
Qualification Tests of the R11410-21 Photomultiplier Tubes for theXENON1T Detector , JINST (2017) P01024 [ ].[54] M. Cirelli, E. Del Nobile and P. Panci, Tools for model-independent bounds in direct darkmatter searches , JCAP (2013) 019 [ ].[55] S. J. Witte and G. B. Gelmini,
Updated Constraints on the Dark Matter Interpretation ofCDMS-II-Si Data , JCAP (2017) 026 [ ].[56] S. Yellin,
Finding an upper limit in the presence of unknown background , Phys. Rev.
D66 (2002) 032005 [ physics/0203002 ].[57] F. Ferrer, A. Ibarra and S. Wild,
A novel approach to derive halo-independent limits on darkmatter properties , JCAP (2015) 052 [ ].[58] http://ddcalc.hepforge.org/ .[59]
The GAMBIT Dark Matter Workgroup collaboration,
DarkBit: A GAMBIT modulefor computing dark matter observables and likelihoods , Eur. Phys. J.
C77 (2017) 831[ ].[60] M. Mancuso et al.,
A Low Nuclear Recoil Energy Threshold for Dark Matter Search withCRESST-III Detectors , J. Low. Temp. Phys. (2018) 441.[61]
CRESST collaboration,
Description of CRESST-III Data , .[62] H. H. Patel, Package-X: A Mathematica package for the analytic calculation of one-loopintegrals , Comput. Phys. Commun. (2015) 276 [ ].].